Introduction to Quantum Field Theory

and

Matter under Extreme Conditions

Prof. Dr. David Blaschke

Institute for Theoretical Physics, University of Wroclaw, Poland

Bogoliubov Laboratory for Theoretical Physics,JINR Dubna, Russia

Summer Semester 2007

Abstract

The series of lectures gives an introduction to the modern formulation of quan-tum field theories using Feynman path integrals. The formalism is developedfirst for the vacuum case and is then generalized to the conditions of finite tem-peratures, densities and strong fields with special emphasis on phase transitions,processes of particle creation in Quantum Field Theories of strong, electro-weakand gravitational interactions. Applications of the formalism are considered inthe physics of condensed matter, plasma physics, ultrarelativistic heavy-ion colli-sions, high-intensity optical and X-ray lasers and the physics of compact objectssuch as neutron stars and black holes.

Contents

1 Quantum Fields at Zero Temperature 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Minkowski Space Conventions . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Four Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Dirac Matrices . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Free Particle Solutions . . . . . . . . . . . . . . . . . . . . 6

1.4 Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.1 Free-Fermion Propagator . . . . . . . . . . . . . . . . . . . 101.4.2 Green Function for the Interacting Theory . . . . . . . . . 131.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Path Integral in Quantum Mechanics . . . . . . . . . . . . . . . . 141.6 Functional Integral in Quantum Field Theory . . . . . . . . . . . 16

1.6.1 Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . 171.6.2 Lagrangian Formulation of Quantum Field Theory . . . . 191.6.3 Quantum Field Theory for a Free Scalar Field . . . . . . . 201.6.4 Scalar Field with Self-Interactions . . . . . . . . . . . . . . 211.6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.7 Functional Integral for Fermions . . . . . . . . . . . . . . . . . . . 231.7.1 Finitely Many Degrees of Freedom . . . . . . . . . . . . . 231.7.2 Fermionic Quantum Field . . . . . . . . . . . . . . . . . . 261.7.3 Generating Functional for Free Dirac Fields . . . . . . . . 281.7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.8 Functional Integral for Gauge Field Theories . . . . . . . . . . . . 311.8.1 Faddeev-Popov Determinant and Ghosts . . . . . . . . . . 341.8.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.9 Dyson-Schwinger Equations . . . . . . . . . . . . . . . . . . . . . 421.9.1 Photon Vacuum Polarization . . . . . . . . . . . . . . . . . 421.9.2 Fermion Self Energy . . . . . . . . . . . . . . . . . . . . . 471.9.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.10 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . 501.10.1 Quark Self Energy . . . . . . . . . . . . . . . . . . . . . . 50

1

1.10.2 Dimensional Regularization . . . . . . . . . . . . . . . . . 531.10.3 Regularized Quark Self Energy . . . . . . . . . . . . . . . 571.10.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

1.11 Renormalized Quark Self Energy . . . . . . . . . . . . . . . . . . 601.11.1 Renormalized Lagrangian . . . . . . . . . . . . . . . . . . 601.11.2 Renormalization Schemes . . . . . . . . . . . . . . . . . . 631.11.3 Renormalized Gap Equation . . . . . . . . . . . . . . . . . 661.11.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

1.12 Dynamical Chiral Symmetry Breaking . . . . . . . . . . . . . . . 681.12.1 Euclidean Metric . . . . . . . . . . . . . . . . . . . . . . . 681.12.2 Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . . . 731.12.3 Mass Where There Was None . . . . . . . . . . . . . . . . 75References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2 Quantum Fields at Finite Temperature and Density 842.1 Ensembles and Partition Function . . . . . . . . . . . . . . . . . . 84

2.1.1 Partition function in Quantum Statistics and QuantumField Theory . . . . . . . . . . . . . . . . . . . . . . . . . 85

2.1.2 Equivalence of Path Integral and Statistical Operator rep-resentation for the Partition function . . . . . . . . . . . . 86

2.1.3 Bosonic Fields . . . . . . . . . . . . . . . . . . . . . . . . . 902.1.4 Fermionic Fields . . . . . . . . . . . . . . . . . . . . . . . 932.1.5 Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . 95

2.2 Interacting Fermion Systems:Hubbard-Stratonovich Trick . . . . . . . . . . . . . . . . . . . . . 982.2.1 Walecka Model . . . . . . . . . . . . . . . . . . . . . . . . 982.2.2 Nambu–Jona-Lasinio (NJL) Model . . . . . . . . . . . . . 982.2.3 Mesonic correlations at finite temperature . . . . . . . . . 1062.2.4 Matsubara frequency sums . . . . . . . . . . . . . . . . . . 107

3 Particle Production by Strong Fields 1123.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.2 Dynamics of pair creation . . . . . . . . . . . . . . . . . . . . . . 113

3.2.1 Creation of fermion pairs . . . . . . . . . . . . . . . . . . . 1133.2.2 Creation of boson pairs . . . . . . . . . . . . . . . . . . . . 118

3.3 Discussion of the source term . . . . . . . . . . . . . . . . . . . . 1193.3.1 Properties of the source term . . . . . . . . . . . . . . . . 1193.3.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 121

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4 Problem sets - statistical QFT 1274.1 Partition function - Introduction . . . . . . . . . . . . . . . . . . . 1274.2 Partition function - Fermi systems . . . . . . . . . . . . . . . . . . 128

4.3 Partition function - Quark matter . . . . . . . . . . . . . . . . . . 129

5 Projects 1305.1 Symmetry breaking. Goldstone-Theorem. Higgs-Effect . . . . . . 130

5.1.1 Spontaneous symmetry breaking: Complex scalar field . . 1305.1.2 Electroweak symmetry breaking: Higgs mechanism . . . . 131

Chapter 1

Quantum Fields at ZeroTemperature

1.1 Introduction

This is the first part of a series of lectures whose aim is to provide the tools forthe completion of a realistic calculation in quantum field theory (QFT) as it isrelevant to Hadron Physics.

Hadron Physics lies at the interface between nuclear and particle (high en-ergy) physics. Its focus is an elucidation of the role played by quarks and gluonsin the structure of, and interactions between, hadrons. This was once parti-cle physics but that has since moved to higher energy in search of a plausiblegrand unified theory and extensions of the so-called Standard Model. The onlyhigh-energy physicists still focusing on hadron physics are those performing thenumerical experiments necessary in the application of lattice gauge theory, andthose pushing at the boundaries of applicability of perturbative QCD or tryingto find new kinematic regimes of validity.

There are two types of hadron: baryons and mesons: the proton and neu-tron are baryons; and the pion and kaon are mesons. Historically the namesdistinguished the particle classes by their mass but it is now known that thereare structural differences: hadrons are bound states, and mesons and baryonsare composed differently. Hadron physics is charged with the responsibility ofproviding a detailed understanding of the differences.

To appreciate the difficulties inherent in this task it is only necessary to re-member that even the study of two-electron atoms is a computational challenge.This is in spite of the fact that one can employ the Schrodinger equation for thisproblem and, since it is not really necessary to quantize the electromagnetic field,the underlying theory has few complications.

The theory underlying hadron physics is quantum chromodynamics (QCD),and its properties are such that a simple understanding and simple calculations

1

are possible only for a very small class of problems. Even on the domain for whicha perturbative application of the theory is appropriate, the final (observable)states in any experiment are always hadrons, and not quarks or gluons, so thatcomplications arise in the comparison of theory with experiment.

The premier experimental facility for exploring the physics of hadrons is theThomas Jefferson National Accelerator Facility (TJNAF), in Newport News, Vir-ginia. Important experiments are also performed at the Fermilab National Ac-celerator Facility (FermiLab), in Batavia, Illinois, and at the Deutsches Elek-tronensynchrotron (DESY) in Hamburg. These facilities use high-energy probesand/or large momentum-transfer processes to explore the transition from thenonperturbative to the perturbative domain in QCD.

On the basis of the introduction to nonperturbative methods in QFT in thevacuum, we will develop in the second part of the lecture series the tools for ageneralization to the situation many-particle systems of hadrons at finite tem-peratures and densities in thermodynamical equilibrium within the Matsubaraformalism.

The third part in the series of lectures is devoted to applications for theQCD phase transition from hadronic matter to a quark-gluon plasma (QGP)in relativistic heavy-ion collisions and in the interior of compact stars. Dataare provided from a completed program of experiments at the CERN-SPS andpresently running programs at RHIC Brookhaven. The future of this directionwill soon open a new domain of energy densities (temperatures) at CERN-LHC(2007) and at the future GSI facility FAIR, where construction shall be completedin 2015. The CBM experiment will then allow insights into the QCD phasetransition at relatively low temperatures and high baryon densities, a situationwhich bears already similarities with the interior of compact stars, formed insupernova explosions and observed as pulsars in isolation or in binary systems.Modern astrophysical data have an unprecendented level of accuracy allowing fornew stringent constraints on the behavior of the hadronic equation of state athigh densities.

The final part of lectures enters the domain of nonequilibrium QFT and willfocus on a particular problem which, however, plays a central role: particle pro-duction in strong time-dependent external fields. The Schwinger mechanism forpair production as a strict result of quantum electrodynamics (QED) is still notexperimentally verified. We are in the fortunate situation that developments ofmodern laser facilities in the X-ray energy domain with intensities soon reachingthe Schwinger limit for electron- positron pair creation from vacuum will allownew insights and experimental tests of approaches to nonperturbative QFT inthe strong field situation. These insights will allow generalizations for the otherfield theories such as the Standard Model and QCD, where still the puzzles ofinitial conditions in heavy-ion collisions and the origin of matter in the Universeremain to be solved.

1.2 Minkowski Space Conventions

In the first part of this lecture series I will use the Minkowski metrics. Later Iwill employ a Euclidean metric because that is most useful and appropriate fornonperturbative calculations.

1.2.1 Four Vectors

Normal spacetime coordinates are denoted by a contravariant four-vector:

xµ := (x0, x1, x2, x3) ≡ (t, x, y, z). (1.1)

Throughout: c = 1 = h, and the conversion between length and energy is just:

1fm = 1/(0.197327 GeV) = 5.06773 GeV−1 . (1.2)

The covariant four-vector is obtained by changing the sign of the spatial compo-nents of the contravariant vector:

xµ := (x0, x1, x2, x3) ≡ (t,−x,−y,−z) = gµνxν , (1.3)

where the metric tensor is

gµν =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

. (1.4)

The contracted product of two four-vectors is (a, b) := gµνaµbν = aνb

ν: i.e., acontracted product of a covariant and a contravariant four-vector. The Poincare-invariant length of any vector is x2 := (x, x) = t2 − ~x2.

Momentum vectors are similarly defined:

pµ = (E, px, py, pz) = (E, ~p). (1.5)

and(p, k) = pνk

ν = EpEk − ~p~k . (1.6)

Likewise,(x, p) = tE − ~x~p . (1.7)

The momentum operator

pµ := i∂

∂xµ= (i

∂

∂t,1

i~∇) =: i∇µ (1.8)

transforms as a contravariant four-vector, and I denote the four-vector analogueof the Laplacian as

∂2 := −pµpµ =∂

∂xµ

∂

∂xµ. (1.9)

The contravariant four-vector associated with the electromagnetic field is

Aµ(x) = (φ(x), ~A(x)) (1.10)

with the electric and magnetic field strengths obtained from

F µν = ∂νAµ − ∂µAν . (1.11)

For example,

Ei = F 0i; ~E = −~∇φ− ∂

∂t~A. (1.12)

Similarly, Bi = εijkF jk; j, k = 1, 2, 3. Analogous definitions hold in QCD for thechromomagnetic field strengths.

1.2.2 Dirac Matrices

The Dirac matrices are indispensable in a manifestly Poincare covariant descrip-tion of particles with spin; i.e., intrinsic angular momentum, such as fermionswith spin 1

2. The Dirac matrices are defined by the Clifford Algebra

γµ, γν = 2gµν (1.13)

and one common 4 × 4 representation is [each entry represents a 2 × 2 matrix]

γ0 =

[

1 00 −1

]

, ~γ =

[

0 ~σ−~σ 0

]

, (1.14)

where ~σ are the usual Pauli Matrices:

σ1 =

[

0 11 0

]

, σ2 =

[

0 −ii 0

]

, σ3 =

[

1 00 −1

]

, (1.15)

and 1 = diag[1, 1]. Clearly γ†0 = γ0; and ~γ† = −~γ. NB. These properties are notspecific to this representation; e.g., γ1γ1 = −1, an analogue of the properties ofa purely imaginary number.

In discussing spin, two combinations of Dirac matrices frequently appear:

σµν =i

2[γµ, γν] , γ5 = iγ0γ1γ2γ3 = γ5 , (1.16)

and I note that

γ5σµν =i

2εµνρσσρσ , (1.17)

with εµνρσ the completely antisymmetric Levi-Civita tensor: ε01223 = +1, εµνρσ =−εµνρσ . In the representation introduced above,

γ5 =

[

0 11 0

]

. (1.18)

Furthermore,γ5, γµ = 0 , γ†5 = γ5 . (1.19)

γ5 plays a special role in the discussion of parity and chiral symmetry, two keyaspects of the Standard Model.

The “slash” notation is a frequently used shorthand:

γµAµ =: A/ = γ0A0 − ~γ ~A , (1.20)

γµpµ =: p/ = γ0E − ~γ~p , (1.21)

γµpµ =: i∇/ ≡ i∂/ = iγ0 ∂

∂t+ i~γ ~∇ = iγµ

∂

∂xµ. (1.22)

(1.23)

The following identities are important in evaluating the cross sections fordecay and scattering processes:

trγ5 = 0 , (1.24)

tr1 = 4 , (1.25)

tra/b/ = 4(a, b) , (1.26)

tra/1a/2a/3a/4 = 4[(a1, a2)(a3, a4) − (a1, a3)(a2, a4) + (a1, a4)(a2, a3)] ,(1.27)

tra/1 . . . a/n = 0 , for n odd , (1.28)

trγ5a/b/ = 0 , (1.29)

trγ5a/1a/2a/3a/4 = 4iεµνρσaµ1a

ν2a

ρ3a

σ4 , (1.30)

γµa/γµ = −2a/ , (1.31)

γµa/b/γµ = 4(a, b) , (1.32)

γµa/b/c/γµ = −2c/b/a/ . (1.33)

(1.34)

They can all be derived using the fact that Dirac matrices satisfy the Cliffordalgebra. For example (remember trAB = trBA):

tra/b/ = aµbνtrγµγν (1.35)

= aµbν1

2tr[γµγν + γνγµ] (1.36)

= aµbν1

2tr[2gµν1] (1.37)

= aµbν1

22gµν4 = 4(a, b) . (1.38)

(1.39)

1.3 Dirac Equation

The unification of special relativity (Poincare covariance) and quantum mechan-ics took some time. Even today many questions remain as to a practical imple-mentation of a Hamiltonian formulation of the relativistic quantum mechanics ofinteracting systems. The Poincare group has ten generators: the six associatedwith the Lorentz transformations – rotations and boosts – and the four associatedwith translations. Quantum mechanics describes the time evolution of a systemwith interactions, and that evolution is generated by the Hamiltonian. However,if the theory is formulated with an interacting Hamiltonian then boosts will al-most always fail to commute with the Hamiltonian and thus the state vectorcalculated in one momentum frame will not be kinematically related to the statein another frame. That makes a new calculation necessary in every frame andhence the discussion of scattering, which takes a state of momentum p to anotherstate with momentum p′, is problematic. (See, e.g., Ref. [2]).

The Dirac equation provides the starting point for a Lagrangian formulationof the quantum field theory for fermions interacting via gauge boson exchange.For a free fermion

[i∂/ −m]ψ = 0 , (1.40)

where ψ(x) is the fermion’s “spinor” – a four component column vector, whilein the presence of an external electromagnetic field the fermion’s wave functionobeys

[i∂/ − eA/ −m]ψ = 0 , (1.41)

which is obtained, as usual, via “minimal substitution:” pµ → pµ − eAµ inEq. (1.40). These equations have a manifestly covariant appearance but provingtheir covariance requires the development of a representation of Lorentz trans-formations on spinors and Dirac matrices:

ψ′(x′) = S(Λ)ψ(x) , (1.42)

Λνµγ

µ = S−1(Λ)γνS(Λ) , (1.43)

S(Λ) = exp[− i

2σµν ω

µν] , (1.44)

where ωµν are the parameters characterising the particular Lorentz transforma-tion. (Details can be found in the early chapters of Refs. [1, 3].)

1.3.1 Free Particle Solutions

As usual, to obtain an explicit form for the free-particle solutions one substitutesa plane wave and finds a constraint on the wave number. In this case thereare two qualitatively different types of solution, corresponding to positive andnegative energy. (An appreciation of the physical reality of the negative energy

solutions led to the prediction of the existence of antiparticles.) One inserts

ψ(+) = e−i(k,x) u(k) , ψ(−) = e+i(k,x) v(k) , (1.45)

into Eq. (1.40) and obtains

(k/ −m) u(k) = 0 , (k/ +m) v(k) = 0 . (1.46)

Assuming that the particle’s mass in nonzero then working in the rest frameyields

(γ0 − 1) u(m,~0) = 0 , (γ0 + 1) v(m,~0) = 0 . (1.47)

There are clearly (remember the form of γ0) two linearly-independent solutionsof each equation:

u(1)(m,~0) =

1000

, u(2)(m,~0) =

0100

, v(1)(m,~0) =

0010

, v(2)(m,~0) =

0001

.

(1.48)The solution in an arbitrary frame can be obtained simply via a Lorentz boostbut it is even simpler to observe that

(k/ −m) (k/ +m) = k2 −m2 = 0 , (1.49)

with the last equality valid when the particles are on shell, so that the solutionsfor arbitrary kµ are: for positive energy (E > 0),

u(α)(k) =k/ +m

√

2m(m+ E)u(α)(m,~0) =

(

E +m

2m

)1/2

φα(m,~0)

σ · k√

2m(m+ E)φα(m,~0)

,(1.50)

with the two-component spinors, obviously to be identified with the fermion’sspin in the rest frame (the only frame in which spin has its naive meaning)

φ(1) =

(

10

)

, φ(2) =

(

01

)

; (1.51)

and, for negative energy,

v(α)(k) =−k/ +m

√

2m(m + E)v(α)(m,~0) =

σ · k√

2m(m + E)χα(m,~0)

(

E +m

2m

)1/2

χα(m,~0)

,(1.52)

with χ(α) obvious analogues of φ(α) in Eq. (1.51). (This approach works becauseit is clear that there are two, and only two, linearly-independent solutions ofthe momentum space free-fermion Dirac equations, Eqs. (1.46), and, for the ho-mogeneous equations, any two covariant solutions with the correct limit in therest-frame must give the correct boosted form.)

In quantum field theory, as in quantum mechanics, one needs a conjugatestate to define an inner product. For fermions in Minkowski space that conjugateis ψ(x) = ψ†(x)γ0, which satisfies

ψ(i←

∂/ +m) = 0 . (1.53)

This equation yields the following free particle spinors in momentum space

u(α)(k) = u(α)(m,~0)k/ +m

√

2m(m + E)(1.54)

v(α)(k) = v(α)(m,~0)−k/ +m

√

2m(m + E), (1.55)

where I have used γ0(γµ)†γ0 = γµ, which relation is easily derived and is particu-larly important in the discussion of intrinsic parity; i.e., the transformation prop-erties of particles and antiparticles under space reflections (an improper Lorentztransformation).

The momentum space free-particle spinors are orthonormalised

u(α)(k) u(β)(k) = δαβ u(α)(k) v(β)(k) = 0v(α)(k) v(β)(k) = −δαβ v(α)(k) u(β)(k) = 0

. (1.56)

It is now possible to construct positive and negative energy projection oper-ators. Consider

Λ+(k) :=∑

α=1,2

u(α)(k) ⊗ u(α)(k) . (1.57)

It is plain from the orthonormality relations, Eqs. (1.56), that Λ+(k) projectsonto positive energy spinors in momentum space; i.e.,

Λ+(k) u(α)(k) = u(α)(k) , Λ+(k) v(α)(k) = 0 . (1.58)

Now, since

∑

α=1,2

u(α)(m,~0) ⊗ u(α)(m,~0) =

(

1 00 0

)

=1 + γ0

2, (1.59)

then

Λ+(k) =1

2m(m + E)(k/ +m)

1 + γ0

2(k/ +m) . (1.60)

Noting that for k2 = m2; i.e., on shell,

(k/ +m) γ0 (k/ +m) = 2E (k/ +m) , (1.61)

(k/ +m) (k/ +m) = 2m (k/ +m) , (1.62)

one finally arrives at the simple closed form:

Λ+(k) =k/ +m

2m. (1.63)

The negative energy projection operator is

Λ−(k) := −∑

α=1,2

v(α)(k) ⊗ v(α)(k) =−k/ +m

2m. (1.64)

The projection operators have the following important properties:

Λ2±(k) = Λ±(k) , (1.65)

tr Λ±(k) = 2 , (1.66)

Λ+(k) + Λ−(k) = 1 . (1.67)

Such properties are a characteristic of projection operators.

1.4 Green Functions

The Dirac equation is a partial differential equation. A general method for solvingsuch equations is to use a Green function, which is the inverse of the differentialoperator that appears in the equation. The analogy with matrix equations isobvious and can be exploited heuristically.

Equation (1.41):[i∂/x − eA/(x) −m]ψ(x) = 0 , (1.68)

yields the wave function for a fermion in an external electromagnetic field. Con-sider the operator obtained as a solution of the following equation

[i∂/x′ − eA/(x′) −m]S(x′, x) = 1 δ4(x′ − x) . (1.69)

It is immediately apparent that if, at a given spacetime point x, ψ(x) is a solutionof Eq. (1.68), then

ψ(x′) :=∫

d4xS(x′, x)ψ(x) (1.70)

is a solution of[i∂/x′ − eA/(x′) −m]ψ(x′) = 0 ; (1.71)

i.e., S(x′, x) has propagated the solution at x to the point x′.

This effect is equivalent to the application of Huygen’s principle in wave me-chanics: if the wave function at x, ψ(x), is known, then the wave function atx′ is obtained by considering ψ(x) as a source of spherical waves that propagateoutward from x. The amplitude of the wave at x′ is proportional to the amplitudeof the original wave, ψ(x), and the constant of proportionality is the propagator(Green function), S(x′, x). The total amplitude of the wave at x′ is the sum overall the points on the wavefront; i.e., Eq. (1.70).

This approach is practical because all physically reasonable external fieldscan only be nonzero on a compact subdomain of spacetime. Therefore the solu-tion of the complete equation is transformed into solving for the Green function,which can then be used to propagate the free-particle solution, already found,to arbitrary spacetime points. However, obtaining the exact form of S(x′, x) isimpossible for all but the simplest cases.

1.4.1 Free-Fermion Propagator

In the absence of an external field Eq. (1.69) becomes

[i∂/x′ −m]S(x′, x) = 1 δ4(x′ − x) . (1.72)

Assume a solution of the form:

S0(x′, x) = S0(x′ − x) =

∫

d4p

(2π)4e−i(p,x

′−x) S0(p) , (1.73)

so that substituting yields

(p/ −m)S0(p) = 1 ; i.e., S0(p) =p/ +m

p2 −m2. (1.74)

To obtain the result in configuration space one must adopt a prescription forhandling the on-shell singularities in S(p), and that convention is tied to theboundary conditions applied to Eq. (1.72). An obvious and physically sensi-ble definition of the Green function is that it should propagate positive-energy-fermions and -antifermions forward in time but not backwards in time, and viceversa for negative energy states.

As we have already seen, the wave function for a positive energy free-fermionis

ψ(+)(x) = u(p) e−i(p,x) . (1.75)

The wave function for a positive energy antifermion is the charge-conjugate ofthe negative-energy fermion solution:

ψ(+)c (x) = C γ0

(

v(p) ei(p,x))∗

= C v(p)T e−i(p,x) , (1.76)

where C = iγ2γ0 and (·)T denotes matrix transpose. (This defines the operationof charge conjugation.) It is thus evident that our physically sensible S0(x

′ − x)can only contain positive-frequency components for x′0 − x0 > 0.

One can ensure this via a small modification of the denominator of Eq. (1.74):

S0(p) =p/ +m

p2 −m2→ p/ +m

p2 −m2 + iη, (1.77)

with η → 0+ at the end of all calculations. Inserting this form in Eq. (1.73) isequivalent to evaluating the p0 integral by employing a contour in the complex-p0

that is below the real-p0 axis for p0 < 0, and above it for p0 > 0. This prescriptiondefines the Feynman propagator.

To be explicit:

S0(x′ − x) =

∫

d3p

(2π)3ei~p·(~x

′−~x) 1

2ω(~p)

×∫ ∞

−∞

dp0

2π

[

e−ip0(x′

0−x0)

p/ +m

p0 − ω(~p) + iη− e−ip

0(x′0−x0)

p/ +m

p0 + ω(~p) − iη

]

,

(1.78)

where the energy ω(~p) =√~p2 +m2. The integrals are easily evaluated using

standard techniques of complex analysis, in particular, Cauchy’s Theorem.Focusing on the first term of the sum inside the square brackets, it is apparent

that the integrand has a pole in the fourth quadrant of the complex p0-plane. Forx′0 − x0 > 0 we can evaluate the p0 integral by considering a contour closed bya semicircle of radius R → ∞ in the lower half of the complex p0-plane: theintegrand vanishes exponentially along that arc, where p0 = −iy, y > 0, because(−i) (−iy) (x′0 −x0) = −y (x′0 −x0) < 0. The closed contour is oriented clockwiseso that

∫ ∞

−∞

dp0

2πe−ip

0(x′0−x0)

p/ +m

p0 − ω(~p) + iη+= (−) i e−ip

0(x′0−x0)(p/ +m)

∣

∣

∣

p0=ω(~p)−iη+

= −i e−iω(~p)(x′0−x0) (γ0ω(~p) − γ · ~p+m)

= −i e−iω(~p)(x′0−x0) 2mΛ+(p) . (1.79)

For x′0 − x0 < 0 the contour must be closed in the upper half plane but thereinthe integrand is analytic and hence the result is zero. Thus

∫ ∞

−∞

dp0

2πe−ip

0(x′0−x0)

p/ +m

p0 − ω(~p) + iη+= −i θ(x′0 − x0) e−iω(~p)(x′

0−x) 2mΛ+(~p) .

Observe that the projection operator is only a function of ~p because p0 = ω(~p).Consider now the second term in the brackets. The integrand has a pole in

the second quadrant. For x′0−x0 > 0 the contour must be closed in the lower half

plane, the pole is avoided and hence the integral is zero. However, for x′0−x0 < 0the contour should be closed in the upper half plane, where the integrand isobviously zero on the arc at infinity. The contour is oriented anticlockwise sothat

∫ ∞

−∞

dp0

2πe−ip

0(x′0−x0)

p/ +m

p0 + ω(~p) − iη+= i θ(x0 − x′0) e−ip

0(x′0−x)(p/ +m)

∣

∣

∣

p0=−ω(~p)+iη+

= i θ(x0 − x′0) e+iω(~p)(x′0−x) (−γ0ω(~p) − γ · ~p+m)

= i θ(x0 − x′0) e+iω(~p)(x′0−x) 2mΛ−(−~p) . (1.80)

Putting these results together, changing variables ~p → −~p in Eq. (1.80), wehave

S0(x′ − x) = −i

∫

d3p

(2π)3

m

ω(~p)

[

θ(x′0 − x0) e−i(p,x′−x) Λ+(~p)

+ θ(x0 − x′0) ei(p,x′−x)Λ−(~p)

]

, (1.81)

[(pµ) = (ω(~p), ~p)] which, using Eqs. (1.58) and its obvious analogue for Λ−(~p),manifestly propagates positive-energy solutions forward in time and negative en-ergy solutions backward in time, and hence satisfies the physical requirementstipulated above.

Another useful representation is obtained merely by observing that

∫ ∞

−∞

dp0

2πe−ip

0(x′0−x) p/ +m

p0 − ω(~p) + iη+=

∫ ∞

−∞

dp0

2πe−ip

0(x′0−x) γ

0ω(~p) − ~γ · ~p +m

p0 − ω(~p) + iη+

=∫ ∞

−∞

dp0

2πe−ip

0(x′0−x) 2mΛ+(~p)

1

p0 − ω(~p) + iη+,

(1.82)

and similarly

∫ ∞

−∞

dp0

2πe−ip

0(x′0−x) p/ +m

p0 + ω(~p) − iη+=

∫ ∞

−∞

dp0

2πe−ip

0(x′0−x) −γ0ω(~p) − ~γ · ~p+m

p0 + ω(~p) − iη+

=∫ ∞

−∞

dp0

2πe−ip

0(x′0−x) 2mΛ−(−~p) 1

p0 + ω(~p) − iη+,

(1.83)

so that Eq. (1.78) can be rewritten

S0(x′ − x) =

∫

d4p

(2π)4e−i(p,x

′−x) m

ω(~p)

[

Λ+(~p)1

p0 − ω(~p) + iη− Λ−(−~p) 1

p0 + ω(~p) − iη

]

.

(1.84)

The utility of this representation is that it provides a single Fourier amplitudefor the free-fermion Green function; i.e., an alternative to Eq. (1.77):

S0(p) =m

ω(~p)

[

Λ+(~p)1

p0 − ω(~p) + iη− Λ−(−~p) 1

p0 + ω(~p) − iη

]

, (1.85)

which is indispensable in making a connection between covariant perturbationtheory and time-ordered perturbation theory: the second term generates the Z-diagrams in loop integrals.

1.4.2 Green Function for the Interacting Theory

We now return to Eq. (1.69):

[i∂/x′ − eA/(x′) −m]S(x′, x) = 1 δ4(x′ − x) , (1.86)

which defines the Green function for a fermion in an external electromagneticfield. As mentioned, a closed form solution of this equation is impossible in allbut the simplest field configurations. Is there, nevertheless, a way to construct asystematically improvable approximate solution?

To achieve that one rewrites the equation:

[i∂/x′ −m]S(x′, x) = 1 δ4(x′ − x) + eA/(x′)S(x′, x) , (1.87)

which, it is easily seen by substitution, is solved by

S(x′, x) = S0(x′ − x) + e∫

d4y S0(x′ − y)A/(y)S(y, x) (1.88)

= S0(x′ − x) + e∫

d4y S0(x′ − y)A/(y)S0(y − x)

+e2∫

d4y1

∫

d4y2 S0(x′ − y1)A/(y1)S0(y1 − y2)A/(y2)S0(y2 − x) + . . .

(1.89)

This perturbative expansion of the full propagator in terms of the free propagatorprovides an archetype for perturbation theory in quantum field theory. Oneobvious application is the scattering of an electron/positron by a Coulomb field,which is a worked example in Sec. 2.5.3 of Ref. [3]. Equation (1.89) is a firstexample of a Dyson-Schwinger equation [4].

This Green function has the following interpretation:

1. It creates a positive energy fermion (antifermion) at spacetime point x;

2. Propagates the fermion to spacetime point x′; i.e., forward in time;

3. Annihilates this fermion at x′.

The process can equally well be viewed as

1. The creation of a negative energy antifermion (fermion) at spacetime pointx′;

2. Propagation of the antifermion to the spacetime point x; i.e., backward intime;

3. Annihilation of this antifermion at x.

Other propagators have similar interpretations.

1.4.3 Exercises

1. Prove these relations for on-shell fermions:

(k/ +m) γ0 (k/ +m) = 2E (k/ +m) ,

(k/ +m) (k/ +m) = 2m (k/ +m) .

2. Obtain the Feynman propagator for the free-field Klein Gordon equation:

(∂2x +m2)φ(x) = 0 , (1.90)

in forms analogous to Eqs. (1.81), (1.84).

1.5 Path Integral in Quantum Mechanics

Local gauge theories are the keystone of contemporary hadron and high-energyphysics. Such theories are difficult to quantise because the gauge dependence isan extra non-dynamical degree of freedom that must be dealt with. The modernapproach is to quantise the theories using the method of functional integrals,attributed to Feynman and Kac. References [3, 5] provide useful overviews of thetechnique, which replaces canonical second-quantisation.

It is customary to motivate the functional integral formulation by reviewingthe path integral representation of the transition amplitude in quantum mechan-ics. Beginning with a state (q1, q2, . . . , qN) at time t, the probability that onewill obtain the state (q′1, q

′2, . . . , q

′N) at time t′ is given by (remember, the time

evolution operator in quantum mechanics is exp[−iHt], where H is the system’sHamiltonian):

〈q′1, q′2, . . . , q′N ; t′|q1, q2, . . . , qN ; t〉 = limn→∞

N∏

α=1

∫ n∏

i=1

dqα(ti)∫ n+1∏

i=1

dpα(ti)

2π

× exp

in+1∑

j=1

pα(tj)[qα(tj) − qα(tj−1)] − εH(p(tj),qtj

+ qtj−1

2)

,(1.91)

where tj = t + jε, ε = (t′ − t)/(n + 1), t0 = t, tn+1 = t′. A compact notation iscommonly introduced to represent this expression:

〈q′t′|qt〉J =∫

[dq]∫

[dp] exp

i∫ t′

tdτ [p(τ)q(τ) −H(τ) + J(τ)q(τ)]

, (1.92)

where J(t) is a classical external “source.” NB. The J = 0 exponent is nothingbut the Lagrangian.

Recall that in Heisenberg’s formulation of quantum mechanics it is the oper-ators that evolve in time and not the state vectors, whose values are fixed at agiven initial time. Using the previous formulae it is a simple matter to prove thatthe time ordered product of n Heisenberg position operators can be expressed as

〈q′t′|TQ(t1) . . . Q(tn)|qt〉 =∫

[dq]∫

[dp] q(t1) q(t2) . . . q(tn) exp

i∫ t′

tdτ [p(τ)q(τ) −H(τ)]

. (1.93)

NB. The time ordered product ensures that the operators appear in chronologicalorder, right to left.

Consider a source that “switches on” at ti and “switches off” at tf , t < ti <tf < t′, then

〈q′t′|qt〉J =∫

dqidqf 〈q′t′|qf tf 〉 〈qf tf |qiti〉J 〈qiti|qt〉 . (1.94)

Alternatively one can introduce a complete set of energy eigenstates to resolvethe Hamiltonian and write

〈q′t′|qt〉 =∑

n

〈q′|φn〉 e−iEn(t′−t) 〈φn|q〉 (1.95)

t′ →t →

−i∞+i∞

= 〈q′|φ0〉 e−iE0(t′−t) 〈φ0|q〉 ; (1.96)

i.e., in either of these limits the transition amplitude is dominated by the groundstate.

It now follows from Eqs. (1.94), (1.96) that

limt′ →t →

−i∞+i∞

〈q′t′|qt〉e−iE0(t′−t) 〈q′|φ0〉 〈φ0|q〉

=∫

dqidqf 〈φ0|qf tf〉 〈qf tf |qiti〉J 〈qiti|φ0〉 =: W [J ] ;

(1.97)i.e., the ground-state to ground-state transition amplitude (survival probability)in the presence of the external source J . From this it will readily be apparentthat, with tf > t1 > t2 > . . . > tm > ti,

δmW [J ]

δJ(t1) . . . δJ(t1)

∣

∣

∣

∣

∣

J=0

= im∫

dqidqf 〈φ0|qf tf 〉 〈qf tf |TQ(t1) . . . Q(tn)|qiti〉 〈qiti|φ0〉,(1.98)

which is the ground state expectation value of a time ordered product of Heisen-berg position operators. The analogues of these expectation values in quantumfield theory are the Green functions.

The functional derivative introduce here: δδJ(t)

, is defined analogously to the

derivative of a function. It means to write J(t) → J(t) + ε(t); expand thefunctional in ε(t); and identify the leading order coefficient in the expansion.Thus for

Hn[J ] =∫

dt′ J(t′)n (1.99)

δHn[J ] = δ∫

dt′ J(t′)n =∫

dt′ [J(t′) + ε(t′)]n −∫

dt′ J(t′)n (1.100)

=∫

dt′ n J(t′)n−1 ε(t′) + [. . .]. (1.101)

So that taking the limit δJ(t′) = ε(t′) → δ(t− t′) one obtains

δHn[J ]

δJ(t)= n J(t)n−1 . (1.102)

The limit procedure just described corresponds to defining

δJ(t)

δJ(t′)= δ(t− t′) . (1.103)

This example, which is in fact the “product rule,” makes plain the very closeanalogy between functional differentiation and the differentiation of functions sothat, with a little care, the functional differentiation of complicated functions isstraightforward.

Another note is in order. The fact that the limiting values in Eqs. (1.96),(1.97) are imaginary numbers is a signal that mathematical rigour may be foundmore easily in Euclidean space where t→ −itE . Alternatively, at least in princi-ple, the argument can be repeated and made rigorous by making the replacementEn → En − iη, with η → 0+. However, that does not help in practical calcula-tions, which proceed via Monte-Carlo methods; i.e., probability sampling, whichis only effective when the integrand is positive definite.

1.6 Functional Integral in Quantum Field The-

ory

The Euclidean functional integral is particularly well suited to a direct numericalevaluation via Monte-Carlo methods because it defines a probability measure.However, to make direct contact with the perturbation theory of canonical second-quantised quantum field theory, I begin with a discussion of the Minkowski spaceformulation.

1.6.1 Scalar Field

I will consider a scalar field φ(t, x), which is customary because it reduces thenumber of indices that must be carried through the calculation. Suppose thata large but compact domain of space is divided into N cubes of volume ε3 andlabel each cube by an integer α. Define the coordinate and momentum via

qα(t) := φα(t) =1

ε3

∫

Vα

d3x φ(t, x) , qα(t) := φα(t) =1

ε3

∫

Vα

d3x∂φ(t, x)

∂t; (1.104)

i.e., as the spatial averages over the cube denoted by α.The classical dynamics of the field φ is described by a Lagrangian:

L(t) =∫

d3xL(t, x) →N∑

α=1

ε3 Lα(φα(t), φα(t), φα±s(t)) , (1.105)

where the dependence on φα±s(t); i.e., the coordinates in the neighbouring cells,is necessary to express spatial derivatives in the Lagrangian density, L(x).

With the canonical conjugate momentum defined as in a classical field theory

pα(t) :=∂L

∂φα(t)= ε3

∂Lα

∂φα(t)=: ε3πα(t) , (1.106)

the Hamiltonian is obtained as

H =∑

α

pα(t) qα(t) − L(t) =:∑

α

ε3Hα , (1.107)

Hα(πα(t), φα(t), φα±s(t)) = πα(t) φα(t) − Lα . (1.108)

The field theoretical equivalent of the quantum mechanical transition ampli-tude, Eq. (1.91), can now be written

limn→∞,ε→0+

N∏

α=1

∫ n∏

i=1

dφα(ti)∫ n∏

i=1

ε3dπα(ti)

2πexp

in+1∑

j=1

ε∑

α

ε3

πα(tj)φα(tj) − φα(tj−1)

ε

− Hα

(

πα(tj),φα(tj) + φα(tj−1)

2,φα±s(tj) + φα±s(tj−1)

2

)

=:∫

[Dφ]∫

[Dπ] exp

i∫ t′

tdτ∫

d3x

[

π(τ, ~x)∂φ(τ, ~x)

∂τ−H(τ, ~x)

]

, (1.109)

where, as classically, π(t, ~x) = ∂L(t, ~x)/∂φ(t, ~x) and its average over a spacetimecube is just πα(t). Equation (1.109) is the transition amplitude from an initialfield configuration φα(t0) := φα(t) to a final configuration φα(tn+1) := φα(t′).

Generating Functional

In quantum field theory all physical quantities can be obtained from Green func-tions, which themselves are determined by vacuum-to-vacuum transition ampli-tudes calculated in the presence of classical external sources. The physical orinteracting vacuum is the analogue of the true ground state in quantum mechan-ics. And, as in quantum mechanics, the fundamental quantity is the generatingfunctional:

W [J ] :=1

N∫

[Dφ]∫

[Dπ] exp

i∫

d4x [π(x)φ(x) −H(x) +1

2iηφ2(x) + J(x)φ(x)]

,

(1.110)where N is chosen so that W [0] = 1, and a real time-limit is implemented andmade meaningful through the addition of the η → 0+ term. (NB. This subtractsa small imaginary piece from the mass.)

It is immediately apparent that

1

inδnW [J ]

δJ(x1)δJ(x2) . . . δJ(xn)

∣

∣

∣

∣

∣

J=0

=〈0|Tφ(x1)φ(x2) . . . φ(xn)|0〉

〈0|0〉 =: G(x1, x2, . . . , xn) ,

(1.111)where |0〉 is the physical vacuum. G(x1, x2, . . . , xn) is the complete n-point Greenfunction for the scalar quantum field theory; i.e., the vacuum expectation valueof a time-ordered product of n field operators. The appearance of the word“complete” means that this Green function includes contributions from productsof lower-order Green functions; i.e., disconnected diagrams.

The fact that the Green functions in a quantum field theory may be defined viaEq. (1.111) was first observed by Schwinger [6] and does not rely on the functionalformula for W [J ], Eq. (1.110), for its proof. However, the functional formulismprovides the simplest proof and, in addition, a concrete means of calculating thegenerating functional: numerical simulations.

Connected Green Functions

It is useful to have a systematic procedure for the a priori elimination of dis-connected parts from a n-point Green function because performing unnecessarywork, as in the recalculation of m < n-point Green functions, is inefficient. Aconnected n-point Green function is given by

Gc(x1, x2, . . . , xn) = (−i)n−1 δnZ[J ]

δJ(x1)δJ(x2) . . . δJ(xn)

∣

∣

∣

∣

∣

J=0

, (1.112)

where Z[J ], defined viaW [J ] =: expiZ[J ] , (1.113)

is the generating functional for connected Green functions.

It is instructive to illustrate this for a simple case (recall that I have alreadyproven the product rule for functional differentiation):

Gc(x1, x2) = (−i) δ2Z[J ]

δJ(x1) δJ(x2)

∣

∣

∣

∣

∣

J=0

= − δ2 lnW [J ]

δJ(x1) δJ(x2)

∣

∣

∣

∣

∣

J=0

= − δ

δJ(x1)

[

1

W [J ]

δW [J ]

δJ(x2)

]∣

∣

∣

∣

∣

J=0

= +1

W 2[J ]

δW [J ]

δJ(x1)

δW [J ]

δJ(x2)

∣

∣

∣

∣

∣

J=0

− 1

W [J ]

δ2W [J ]

δJ(x1) δJ(x2)

∣

∣

∣

∣

∣

J=0

= i〈0|φ(x1)|0〉

〈0|0〉 i〈0|φ(x2)|0〉

〈0|0〉 − i2〈0|Tφ(x1)φ(x2)|0〉

〈0|0〉= −G(x1)G(x2) +G(x1, x2) . (1.114)

1.6.2 Lagrangian Formulation of Quantum Field Theory

The double functional integral employed above is cumbersome, especially sinceit involves the field variable’s canonical conjugate momentum. Consider then aHamiltonian density of the form

H(x) =1

2π2(x) + f [φ(x), ~∇φ(x)] . (1.115)

In this case Eq. (1.110) involves∫

[Dπ] exp

i∫

d4x [−1

2π2(x) + π(x)φ(x)]

= ei∫

d4x [φ(x)]2∫

[Dπ] exp

− i

2

∫

d4x [π(x) − φ(x)]2

= ei∫

d4x [φ(x)]2 × N , (1.116)

where “N” is simply a constant. (This is an example of the only functionalintegral than can be evaluated exactly: the Gaussian functional integral.) Hence,with a Hamiltonian of the form in Eq. (1.115), the generating functional can bewritten

W [J ] =N

N∫

[Dφ] exp

i∫

d4x [L(x) +1

2iηφ2(x) + J(x)φ(x)]

.(1.117)

Recall now that the classical Lagrangian density for a scalar field is

L(x) = L0(x) + LI(x) , (1.118)

L0(x) =1

2[∂µφ(x) ∂µφ(x) −m2φ2(x)] , (1.119)

with LI(x) some functional of φ(x) that usually does not depend on any deriva-tives of the field. Hence the Hamiltonian for such a theory has the form inEq. (1.115) and so Eq. (1.117) can be used to define the quantum field theory.

1.6.3 Quantum Field Theory for a Free Scalar Field

The interaction Lagrangian vanishes for a free scalar field so that the generatingfunctional is, formally,

W0[J ] =1

N∫

[Dφ] exp

i∫

d4x(

1

2[∂µφ(x) ∂µφ(x) −m2φ2(x) + iηφ2(x)] + J(x)φ(x)

)

.

(1.120)The explicit meaning of Eq. (1.120) is

W0[J ] = limε→0+

1

N∫

∏

α

dφα exp

i∑

α

ε4∑

β

ε41

2φαKαβ φβ +

∑

α

ε4 Jαφα

,

(1.121)where α, β label spacetime hypercubes of volume ε4 and Kαβ is any matrix thatsatisfies

limε→0+

Kαβ = [−∂2 −m2 + iη]δ4(x− y) , (1.122)

where αε→0+→ x, β

ε→0+→ y and∑

α ε4 ε→0+→ ∫

d4x; i.e., Kαβ is any matrix whosecontinuum limit is the inverse of the Feynman propagator for a free scalar field.(NB. The fact that there are infinitely many such matrices provides the scopefor “improving” lattice actions since one may choose Kαβ wisely rather than forsimplicity.)

Recall now that for matrices whose real part is positive definite

∫

Rn

n∏

i=1

dxi exp−1

2

n∑

i,j=1

xiAij xj +n∑

i=1

bi xi

=(2π)n/2√

detAexp1

2

n∑

i,j=1

bi (A−1)ij bj =

(2π)n/2√detA

exp

1

2btA−1b

.(1.123)

Hence Eq. (1.121) yields

W0[J ] = limε→0+

1

N ′1√

detAexp1

2

∑

α

ε4∑

β

ε4 Jα1

iε8(K−1)αβ Jβ , (1.124)

where, obviously, the matrix inverse is defined via∑

γ

Kαγ (K−1)γβ = δαβ . (1.125)

Almost as obviously, consistency of limits requires

limε→0+

1

ε4δαβ = δ4(x− y) , lim

ε→0+

∑

α

ε4 =∫

d4x , (1.126)

so that, with

O(x, y) := limε→0+

1

ε8(K−1)αβ , (1.127)

the continuum limit of Eq. (1.125) can be understood as follows:

limε→0+

∑

γ

ε4 Kαγ1

ε8(K−1)γβ = lim

ε→0+

1

ε4δαβ

⇒∫

d4w [−∂2x −m2 + iη]δ4(x− w)O(w, y) = δ4(x− y)

.·· [−∂2x −m2 + iη]O(x, y) = δ4(x− y) . (1.128)

Hence O(x, y) = ∆0(x− y); i.e., the Feynman propagator for a free scalar field:

∆0(x− y) =∫ d4p

(2π)4e−i(q,x−y)

1

q2 −m2 + iη. (1.129)

(NB. This makes plain the fundamental role of the “iη+” prescription in Eq. (1.77):it ensures convergence of the expression defining the functional integral.)

Putting this all together, the continuum limit of Eq. (1.124) is

W [J ] =1

Nexp

− i

2

∫

d4x∫

d4y J(x) ∆0(x− y) J(y)

. (1.130)

1.6.4 Scalar Field with Self-Interactions

A nonzero interaction Lagrangian, LI [φ(x)], provides for a self-interacting scalarfield theory (from here on I will usually omit the constant, nondynamical nor-malisation factor in writing the generating functional):

W [J ] =∫

[Dφ] exp

i∫

d4x [L0(x) + LI(x) + J(x)φ(x)]

= exp

[

i∫

d4xLI

(

1

i

δ

δJ(x)

)]

∫

[Dφ] exp

i∫

d4x [L0(x) + J(x)φ(x)]

(1.131)

= exp

[

i∫

d4xLI

(

1

i

δ

δJ(x)

)]

exp

− i

2

∫

d4x∫

d4y J(x) ∆0(x− y) J(y)

,

(1.132)

where

exp

[

i∫

d4xLI

(

1

i

δ

δJ(x)

)]

:=∞∑

n=0

in

n!

[

LI

(

1

i

δ

δJ(x)

)]n

. (1.133)

Equation (1.132) is the basis for a perturbative evaluation of all possibleGreen functions for the theory. As an example I will work through a first-ordercalculation of the complete 2-point Green function in the theory defined by

LI(x) = − λ

4!φ4(x) . (1.134)

The generating functional yields

W [0] =

1 − iλ

4!

∫

d4x

(

δ

δJ(x)

)4

exp

− i

2

∫

d4u∫

d4v J(u) ∆0(u− v) J(v)

∣

∣

∣

∣

∣

∣

J=0

= 1 − iλ

4!

∫

d4x 3[i∆0(0)]2 . (1.135)

The 2-point function is

δ2W [J ]

δJ(x1) δJ(x2)= −i∆0(x1 − x2) + i

λ

8

∫

d4x [i∆0(0)]2[i∆0(x1 − x2)]

+iλ

2

∫

d4x [i∆0(0)][i∆0(x1 − x)][i∆0(x− x2)] . (1.136)

Using the definition, Eq. (1.111), and restoring the normalisation,

G(x1, x2) =1

i21

W [0]

δ2W [J ]

δJ(x1) δJ(x2)(1.137)

= i∆0(x1 − x2) − iλ

2

∫

d4x [i∆0(0)][i∆0(x1 − x)][i∆0(x− x2)] ,(1.138)

where I have useda + λb

1 + λc= a + λ(b− ac) + O(λ2) . (1.139)

This complete Green function does not contain any disconnected parts becausethe vacuum is trivial in perturbation theory; i.e.,

〈0|φ(x)|0〉〈0|0〉 := G(x)|J=0 =

1

i

δW [J ]

δJ(x)

∣

∣

∣

∣

∣

J=0

= 0 (1.140)

so that the field does not have a nonzero vacuum expectation value. This isthe simplest demonstration of the fact that dynamical symmetry breaking is aphenomenon inaccessible in perturbation theory.

1.6.5 Exercises

1. Repeat the derivation of Eq. (1.114) for Gc(x1, x2, x3).

2. Prove Eq. (1.131).

3. Derive Eq. (1.136).

4. Prove Eq. (1.140).

1.7 Functional Integral for Fermions

1.7.1 Finitely Many Degrees of Freedom

Fermionic fields do not have a classical analogue: classical physics does not con-tain anticommuting fields. In order to treat fermions using functional integralsone must employ Grassmann variables. Reference [7] is the standard source for arigorous discussion of Grassmann algebras. Here I will only review some necessaryideas.

The Grassmann algebra GN is generated by the set of N elements, θ1 , . . . , θNwhich satisfy the anticommutation relations

θi, θj = 0 , i, j = 1, 2, . . . , N . (1.141)

It is clear from Eq. (1.141) that θ2i = 0 for i = 1, . . . , N . In addition, the elements

θi provide the source for the basis vectors of a 2n-dimensional space, spannedby the monomials:

1, θ1, . . . , θN , θ1θ2, . . . θN−1θN , . . . , θ1θ2 . . . θN ; (1.142)

i.e., GN is a 2N -dimensional vector space. (NB. One can always choose the p-degree monomial in Eq. (1.142): θi1θi2 . . . θIN , such that i1 < i2 < . . . < iN .)Obviously, any element f(θ) ∈ GN can be written

f(θ) = f0+∑

i1

f1(i1) θi1+∑

i1,i2

f2(i1, i2) θi1θi2+. . .+∑

i1,i2,...,iN

fN(i1, i2, . . . , iN ) θ1θ2 . . . θN ,

(1.143)where the coefficients fp(i1, i2, . . . , ip) are unique if they are chosen to be fullyantisymmetric for p ≥ 2.

Both “left” and “right” derivatives can be defined on GN . As usual, theyare linear operators and hence it suffices to specify their operation on the basiselements:

∂

∂θsθi1θi2θip = δsi1θi2 . . . θip − δsi2θi1 . . . θip + . . .+ (−)p−1δsipθi1θi2 . . . θip−1

,(1.144)

θi1θi2θip

←

∂

∂θs= δsipθi1 . . . θip−1

− δsip−1θi1 . . . θip−2

θip + . . .+ (−)p−1δsi1θi2θip .(1.145)

The operation on a general element, f(θ) ∈ GN , is easily obtained. It is alsoobvious that

∂

∂θ1

∂

∂θ2f(θ) = − ∂

∂θ2

∂

∂θ1f(θ) . (1.146)

A definition of integration requires the introduction of Grassmannian lineelements: dθi, i = 1, . . . , N . These elements also satisfy Grassmann algebras:

dθi, dθj = 0 = θi, dθj , i, j = 1, 2, . . .N . (1.147)

The integral calculus is completely defined by the following two identities:∫

dθi = 0 ,∫

dθi θi = 1 , i = 1, 2, . . . , N . (1.148)

For example, it is straightforward to prove, using Eq. (1.143),∫

dθN . . . dθ1 f(θ) = N ! fN(1, 2, . . . , N) . (1.149)

In standard integral calculus a change of integration variables is often used tosimplify an integral. That operation can also be defined in the present context.Consider a nonsingular matrix (Kij), i, j = 1, . . . , N , and define new Grassmannvariables ξ1, . . . , ξN via

θi =N∑

i=j

Kij ξj . (1.150)

With the definition

dθi =N∑

j=1

(K−1)ji dξj (1.151)

one guarantees∫

dθi θj = δij =∫

dξi ξj . (1.152)

It follows immediately that

θ1θ2 . . . θN = (detK) ξ1ξ2 . . . ξN (1.153)

dθNdθN−1 . . . dθ1 = (detK−1) dξNdξN−1 . . . dξ1 , (1.154)

and hence∫

dθN . . . dθ1 f(θ) = (detK−1)∫

dξN . . . dξ1 f(θ(ξ)) . (1.155)

In analogy with scalar field theory, for fermions one expects to encounterintegrals of the type

I :=∫

dθN . . . dθ1 exp

N∑

i,j=1

θiAijθj

, (1.156)

where (Aij) is an antisymmetric matrix. NB. Any symmetric part of the matrix,A,cannot contribute since:

∑

i,j

θiAijθjrelable

=∑

j,i

θjAjiθi (1.157)

use sym.=

∑

i,j

θjAijθi (1.158)

anticom.= −

∑

i,j

θiAijθj . (1.159)

·..∑

i,j

θiAijθj = 0 for A = At . (1.160)

Assume for the moment that A is a real matrix. Then there is an orthogonalmatrix S (SSt = I) for which

StAS =

0 λ1 0 0 . . .−λ1 0 0 0 . . .

0 0 0 λ2 . . .0 0 −λ2 0 . . .. . . . . . . . . . . . . . .

=: A . (1.161)

Consequently, applying the linear transformation θi =∑Ni=1 Sij ξj and using

Eq. (1.155), we obtain

I =∫

dξN . . . dξ1 exp

N∑

i,j=1

ξiAijξj

. (1.162)

Hence

I =

∫

dξN . . . dξ1 exp

2 [λ1ξ1ξ2 + λ2ξ3ξ4 + . . .+ λN/2ξN−1ξN ]

= 2N/2λ1λ2 . . . λN , N even∫

dξN . . . dξ1 exp

2 [λ1ξ1ξ2 + λ2ξ3ξ4 + . . .+ λ(N−1)/2ξN−2ξN−1]

= 0 , N odd

(1.163)

i.e., since detA = det A,I =

√det 2A , . (1.164)

Equation (1.164) is valid for any real matrix, A. Hence, by the analyticfunction theorem, it is also valid for any complex matrix A.

The Lagrangian density associated with the Dirac equation involves a field ψ,which plays the role of a conjugate to ψ. If we are express ψ as a vector in GN

then we will need a conjugate space in which ψ is defined. Hence it is necessaryto define θ1, θ2, . . . , θN such that the operation θi ↔ θi is an involution of thealgebra onto itself with the following properties:

i) (θi) = θiii) (θiθj) = θj θiiii) λ θi = λ∗ θi , λ ∈ C .

(1.165)

The elements of the Grassmann algebra with involution are θ1, θ2, . . . , θN , θ1, θ2 . . . , θN ,each anticommuting with every other. Defining integration via obvious analogywith Eq. (1.148) it follows that

∫

dθNdθN . . . dθ1dθ1 exp

−N∑

i,j=1

θiBijθj

= detB , (1.166)

for any matrix B. (NB. This is the origin of the fermion determinant in thequantum field theory of fermions.) This may be compared with the analogousresult for commuting real numbers, Eq. (1.123):

∫

RN

N∏

i=1

dxi exp−πN∑

i,j=1

xiAij xj =1√

detA. (1.167)

1.7.2 Fermionic Quantum Field

To describe a fermionic quantum field the preceding analysis must be generalisedto the case of infinitely many generators. A rigorous discussion can be found inRef. [7] but here I will simply motivate the extension via plausible but formalmanipulations.

Suppose the functions un(x) , n = 0, . . . ,∞ are a complete, orthonormalset that span a given Hilbert space and consider the Grassmann function

θ(x) :=∞∑

n=0

un(x) θn , (1.168)

where θn are Grassmann variables. Clearly

θ(x), θ(y) = 0 . (1.169)

The elements θ(x) are considered to be the generators of the “Grassman algebra”G and, in complete analogy with Eq. (1.143), any element of G can be uniquelywritten as

f =∞∑

n=0

∫

dx1dx2 . . . dxN θ(x1)θ(x2) . . . θ(xN ) fn(x1, x2, . . . , xN) , (1.170)

where, for N ≥ 2, the fn(x1, x2, . . . , xN ) are fully antisymmetric functions of theirarguments.

In another analogy, the left- and right-functional-derivatives are defined viatheir action on the basis vectors:

δ

δθ(x)θ(x1)θ(x2) . . . θ(xn) =

δ(x− x1) θ(x2) . . . θ(xn) − . . .+ (−)n−1δ(x− xn) θ(x1) . . . θ(xn−1) ,(1.171)

θ(x1)θ(x2) . . . θ(xn)

←

δ

δθ(x)=

δ(x− xn) θ(x1) . . . θ(xn−1) − . . .+ (−)n−1δ(x− x1) θ(x2) . . . θ(xn) ,(1.172)

cf. Eqs. (1.144), (1.145).

Finally, we can extend the definition of integration. Denoting

[Dθ(x)] := limN→∞

dθN . . . dθ2dθ1 , (1.173)

consider the “standard” Gaussian integral

I :=∫

[Dθ(x)] exp∫

dxdy θ(x)A(x, y)θ(y)

(1.174)

where, clearly, only the antisymmetric part of A(x, y) can contribute to the result.Define

Aij :=∫

dxdy ui(x)A(x, y)uj(y) , (1.175)

thenI = lim

N→∞

∫

dθN . . . dθ2dθ1 e∑N

i=1θiAijθj (1.176)

so that, using Eq. (1.164),

I = limN→∞

√

det 2AN , (1.177)

where, obviously, AN is the N × N matrix in Eq. (1.175). This provides adefinition for the formal result:

I =∫

[Dθ(x)] exp∫

dxdy θ(x)A(x, y)θ(y)

=√

Det2A , (1.178)

where I will subsequently identify functional equivalents of matrix operationsas proper nouns. The result is independent of the the basis vectors since allsuch vectors are unitarily equivalent and the determinant is cylic. (This meansthat a new basis is always related to another basis via u′ = Uu, with UU † =I. Transforming to a new basis therefore introduces a modified exponent, nowinvolving the matrix UAU †, but the result is unchanged because detUAU † =detA.)

In quantum field theory one employs a Grassmann algebra with an involution.In this case, defining the functional integral via

[Dθ(x)][Dθ(x)] := limN→∞

dθNdθN . . . dθ2dθ2 dθ1dθ1 , (1.179)

one arrives immediately at a generalisation of Eq. (1.166)

∫

[Dθ(x)][Dθ(x)] exp

−∫

dxdy θ(x)B(x, y) θ(x)

= DetB , (1.180)

which is also a definition.The relation

ln detB = tr lnB , (1.181)

valid for any nonsingular, finite dimensional matrix, has a generalisation thatis often used in analysing quantum field theories with fermions. Its utility isto make possible a representation of the fermionic determinant as part of thequantum field theory’s action via

DetB = exp Tr LnB . (1.182)

I note that for an integral operator O(x, y)

TrO(x, y) :=∫

d4x trO(x, x) , (1.183)

which is an obvious analogy to the definition for finite-dimensional matrices.Furthermore a functional of an operator, whenever it is well-defined, is obtainedvia the function’s power series; i.e., if

f(x) = f0 + f1 x + f2 x2 + [. . .] , (1.184)

then

f [O(x, y)] = f0 δ4(x− y) + f1O(x, y) + f2

∫

d4wO(x, w)O(w, y) + [. . .] . (1.185)

1.7.3 Generating Functional for Free Dirac Fields

The Lagragian density for the free Dirac field is

Lψ0 (x) =∫

d4x ψ(x) (i∂/ −m)ψ(x) . (1.186)

Consider therefore the functional integral

W [ξ, ξ] =∫

[Dψ(x)][Dψ(x)] exp

i∫

d4x[

ψ(x)(

i∂/ −m + iη+)

ψ(x) + ψ(x)ξ(x) + ξ(x)ψ(x)]

.

(1.187)

Here ψ(x), ψ(x) are identified with the generators of G, with the minor additionalcomplication that the spinor degree-of-freedom is implicit; i.e., to be explicit,one should write

∏4r=1[Dψr(x)]

∏4s=1[Dψs(x)]. This only adds a finite matrix

degree-of-freedom to the problem, so that “DetA” will mean both a functionaland a matrix determinant. This effect will be encountered again; e.g., with theappearance of fermion colour and flavour. In Eq. (1.187) I have also introducedanticommuting sources: ξ(x), ξ(x), which are also elements in the Grassmannalgebra, G.

The free-field generating functional involves a Gaussian integral. To evaluatethat integral I write

O(x, y) = (i∂/ −m+ iη+)δ4(x− y) (1.188)

and observe that the solution of∫

d4wO(x, w)P (w, y) = I δ4(x− y) (1.189)

i.e., the inverse of the operator O(x, y) is (see Eq. 1.72) precisely the free-fermionpropagator:

P (x, y) = S0(x− y) . (1.190)

Hence I can rewrite Eq. (1.187) in the form

W [ξ, ξ] =∫

[Dψ(x)][Dψ(x)] exp

i∫

d4xd4y[

ψ′(x)O(x, y)ψ′(y) − ξ(x)S0(x− y)ξ(y)]

(1.191)where

ψ′(x) := ψ(x) +∫

d4w ξ(w)S0(w − x) , ψ(x) := ψ(x) +∫

d4wS0(x− w) ξ(w) .

(1.192)Clearly, ψ′(x) and ψ′(x) are still in G and hence related to the original variablesby a unitary transformation. Thus changing to the “primed” variables introducesa unit Jacobian and so

W [ξ, ξ] = exp

−i∫

d4xd4y ξ(x)S0(x− y) ξ(y)

×∫

[Dψ′(x)][Dψ′(x)] exp

i∫

d4xd4y ψ′(x)O(x, y)ψ′(y)

(1.193)

= det[−iS−10 (x− y)] exp

−i∫

d4xd4y ξ(x)S0(x− y) ξ(y)

(1.194)

=1

N ψ′

exp

−i∫

d4xd4y ξ(x)S0(x− y) ξ(y)

, (1.195)

where N ψ0 := det[iS0(x− y)]. Clearly.

N ψ0 W [ξ, ξ]

∣

∣

∣

ξ=0=ξ= 1 . (1.196)

The 2 point Green function for the free-fermion quantum field theory is noweasily obtained:

N ψ0

δ2W [ξ, ξ]

iδξ(x) (−i)δξ(y)

∣

∣

∣

∣

∣

ξ=0=ξ

=〈0|Tψ(x)ˆψ(y)|0〉

〈0|0〉

=∫

[Dψ(x)][Dψ(x)]ψ(x)ψ(y) exp

i∫

d4x ψ(x)(

i∂/ −m+ iη+)

ψ(x)

= i S0(x− y) ; (1.197)

i.e., the inverse of the Dirac operator, with exactly the Feynman boundary con-ditions.

As in the example of a scalar quantum field theory, the generating functionalfor connected n-point Green functions is Z[ξ, ξ], defined via:

W [ξ, ξ] =: exp

iZ[ξ, ξ]

. (1.198)

Hitherto I have not illustrated what is meant by “DetO,” where O is anintegral operator. I will now provide a formal example. (Rigour requires a carefulconsideration of regularisation and limits.) Consider a translationally invariantoperator

O(x, y) = O(x− y) =∫ d4p

(2π)4O(p) e−i(p,x−y). (1.199)

Then, for f as in Eq. (1.185),

f [O(x− y)] =∫

d4p

(2π)4

f0 + f1O(p) + f2O2(p) + [. . .]

e−i(p,x−y)

=∫

d4p

(2π)4f(O(p)) e−i(p,x−y) . (1.200)

I now apply this to N ψ0 := Det[iS0(x−y)] and observe that Eq. (1.182) means

one can begin by considering TrLn iS0(x− y). Writing

S0(p) = m∆0(p2)

[

1 +p/

m

]

, ∆0(p2) =1

p2 −m2 + iη+, (1.201)

the free fermion propagator can be re-expressed as a product of integral operators:

S0(x− y) =∫

d4wm∆0(x− w)F(w− y) , (1.202)

with ∆0(x− y) given in Eq. (1.129) and

F(x− y) =∫ d4p

(2π)4

[

1 +p/

m

]

e−i(p,x−y) . (1.203)

It follows that

TrLn iS0(x− y) = Tr

Ln im∆0(x− y) + Ln[

δ4(x− y) + F(x− y)]

. (1.204)

Using Eqs. (1.183), (1.200), one can express the second term as

TrLn[

δ4(x− y) + F(x− y)]

=∫

d4x∫

d4p

(2π)4tr ln [1 + F(p)]

=∫

d4x∫

d4p

(2π)42 ln

[

1 − p2

m2

]

(1.205)

and, applying the same equations, the first term is

TrLn im∆0(x− y) =∫

d4x∫

d4p

(2π)42 ln

[

im∆0(p2)]2, (1.206)

where in both cases∫

d4x measures the (infinite) spacetime volume. Combiningthese results one obtains

LnN ψ0 = TrLn iS0(x− y) =

∫

d4x∫

d4p

(2π)42 ln ∆0(p2) , (1.207)

where the factor of 2 reflects the spin-degeneracy of the free-fermion’s eigenvalues.(Including a “colour” degree-of-freedom, this would become “2Nc,” where Nc isthe number of colours.)

1.7.4 Exercises

1. Verify Eq. (1.149).

2. Verify Eqs. (1.153), (1.154).

3. Verify Eqs. (1.163), (1.164).

4. Verify Eqs. (1.166).

5. Verify Eq. (1.181).

6. Verify Eq. (1.197).

7. Verify Eq. (1.205).

1.8 Functional Integral for Gauge Field Theo-

ries

To begin I will consider a non-Abelian gauge field theory in the absence of cou-plings to matter field, which is described by a Lagrangian density:

L(x) = −1

4F aµν(x)F µν

a (x), (1.208)

F aµν(x) = ∂µB

aν − ∂νB

aµ + gfabcB

bµB

cν , (1.209)

where g is a coupling constant and fabc are the structure constants of SU(Nc):i.e., with T a : a = 1, . . . , N2

c − 1 denoting the generators of the group

[Ta, Tb] = ifabcTc . (1.210)

In the fundamental representation Ta ≡ λa

2, where λa are the generalization

of the eight Gell-Mann matrices, while in the adjoint representation, relevant tothe realization of transformations on the gauge fields,

T abc = −ifabc . (1.211)

An obvious guess for the form of the generating functional is

W [J ] =∫

[DBaµ] exp

i∫

d4x[L(x) + Jµa (x)Baµ]

, (1.212)

where, as usual, Jµa (x) is a (classical) external source for the gauge field. Itwill immediately be observed that this is a Lagrangian-based expression for thegenerating functional, even though the Hamiltonian derived from Eq. (1.208)is not of the form in Eq. (1.115). It is nevertheless (almost) correct (I willmotivate the modifications that need to be made to make it completely correct)and provides the foundation for a manifestly Poincare covariant quantisation ofthe field theory. Alternatively, one could work with Coulomb gauge, build theHamiltonian and construct W [J ] in the canonical fashion, as described in Sec.??, but then covariance is lost. The Coulomb gauge procedure gives the same S-matrix elements (Green functions) as the (corrected-) Lagrangian formalism andhence they are completely equivalent. However, manifest covariance is extremelyuseful as it often simplifies the allowed form of Green functions and certainlysimplifies the calculation of cross sections. Thus the Lagrangian formulation ismost often used.

The primary fault with Eq. (1.212) is that the free-field part of the Lagrangiandensity is singular: i.e., the determinant encountered in evaluating the free-gauge-boson generating functional vanishes, and hence the operator cannot be inverted.

This is easily demonstrated. Observe that∫

d4xL0(x) = −1

4

∫

d4x[∂µBaν (x) − ∂νB

aµ(x)][∂µBν

a(x) − ∂νBµa (x)](1.213)

= −1

2

∫

d4xd4yBaµ(x)

[−gµν∂2 + ∂µ∂ν ]δ4(x− y)

Baν(y)(1.214)

=:1

2

∫

d4xd4yBaµ(x)Kµν(x, y)Ba

ν(y) . (1.215)

The operator Kµν(x, y) thus determined can be expressed

Kµν(x− y) =∫ d4q

(2π)4

(

−gµνq2 + qµqν)

e−i(p,x−y) (1.216)

from which it is apparent that the Fourier amplitude is a projection operator; i.e.,a key feature of Kµν(x, y) is that it projects onto the space of transverse gaugefield configurations:

qµ(gµνq2 − qµqν) = 0 = (gµνq2 − qµqν)qν , (1.217)

where qµ is the four-momentum associated with the gauge field. It follows thatthe W0[J ] obtained from Eq. (1.212) has no damping associated with longitudinalgauge fields and is therefore meaningless. A simple analogy is

∫ ∞

−∞dx∫ ∞

−∞dye−(x−y)2 , (1.218)

in which the integrand does not damp along trajectories in the (x, y)-plane relatedvia a spatial translation: (x, y) → (x + a, y + a). Hence there is an overalldivergence associated with the translation of the centre of momentum: X =(x + y)/2. Y = (x− y): X → X + (a, a), Y → Y ,

∫ ∞

−∞dx∫ ∞

−∞dye−(x−y)2 =

∫ ∞

−∞dX

∫ ∞

−∞dY e−Y

2

(1.219)

=(∫ ∞

−∞dX

)√π = ∞ . (1.220)

The underlying problem, which is signalled by the behavior just identified, is thegauge invariance of the action:

∫ L(x); i.e., the action is invariant under localfield transformation

Bµ(x) := igBaµ(x)T a → G(x)Bµ(x)G−1(x) + [∂µG(x)]G−1(x) , (1.221)

G(x) = exp−igT aΘa(x) . (1.222)

This means that, given a reference field configuration Bµ(x), the integrand inthe generating functional, Eq. (1.212), is constant along the path through thegauge field manifold traversed by the applying gauge transformations to Bµ(x).Since the parameters characterizing the gauge transformations, Θa, are contin-uous functions, each such gauge orbit contains an uncountable infinity of gaugefield configurations. It is therefore immediately apparent that the generatingfunctional, as written, is is undefined: it contains a multiplicative factor pro-portional to the length (or volume) of the gauge orbit. (NB. While there is, inaddition, an uncountable infinity of distinct reference configurations, the actionchanges upon any shift orthogonal to a gauge orbit.)

Returning to the example in Eq. (1.219), the analogy is that the “Lagrangiandensity” l(x, y) = (x − y)2 is invariant under translations; i.e., the integrand isinvariant under the operation

gs(x, y) = exp

s∂

∂x+ s

∂

∂y

(1.223)

(x, y) → (x′, y′) = gs(x, y)(x, y) = (x + s, y + s) . (1.224)

Hence, given a reference point P = (x0, y0) = (1, 0), the integrand is constantalong the path (x, y) = P0 + (s, s) through the (x, y)-plane. (This is a translationof the centre-of-mass: X0 = (x0 +y0)/2 = 1/2 → X0 +s.) Since s∈(−∞,∞), this

path contains an uncountable infinity of points, and at each one the integrandhas precisely the same value. The integral thus contains a multiplicative factorproportional to the length of the translation path, which clearly produces aninfinite (meaningless) result for the integral. (NB. The value of the integrandchanges upon a translation orthogonal to that just identified.)

1.8.1 Faddeev-Popov Determinant and Ghosts

This problem with the functional integral over gauge fields was identified byFaddeev and Popov. They proposed to solve the problem by identifying andextracting the gauge orbit volume factor.

A Simple Model

Before describing the procedure in quantum field theory it can be illustratedusing the simple integral model. One begins by defining a functional of our “fieldvariable”, (x, y), which intersects the centre-of-mass translation path once, andonly once:

f(x, y) = (x+ y)/2 − a = 0 . (1.225)

Now define a functional ∆f such that

∆f [x, y]∫ ∞

−∞daδ((x + y)/2 − a) = 1 . (1.226)

It is clear that ∆f [x, y] is independent of a:

∆f [x + a′, y + a′]−1 =∫ ∞

−∞daδ((x + a′ + y + a′)/2 − a) (1.227)

a=a−a′=

∫ ∞

−∞daδ((x + y)/2 − a) (1.228)

= ∆f [x, y]−1 . (1.229)

Using ∆f the model generating functional can be rewritten

∫ ∞

−∞dx∫ ∞

−∞dye−l(x,y) =

∫ ∞

−∞da∫ ∞

−∞dx∫ ∞

−∞dye−l(x,y)∆f [x, y]δ((x+ y)/2 − a)

(1.230)and now one performs a centre-of-mass translation: x → x′ = x + a, y → y′ =y + a, under which the action is invariant so that the integral becomes

∫ ∞

−∞da∫ ∞

−∞dx∫ ∞

−∞dye−l(x,y)∆f [x, y]δ((x+ y)/2 − a) (1.231)

=∫ ∞

−∞da∫ ∞

−∞dx∫ ∞

−∞dye−l(x,y)∆f [x + a, y + a]δ((x + y)/2) . (1.232)

Now making use of the a−independence of ∆f [x, y], Eq. (1.227),

=[∫ ∞

−∞da∫ ] ∫ ∞

−∞dx∫ ∞

−∞dye−l(x,y)∆f [x, y]δ((x+ y)/2) . (1.233)

In this last line the “volume” or “path length” has been explicitely factored out atthe cost of introducing a δ-function, which fixes the centre-of-mass; i.e., a singlepoint on the path of translationally equivalent configurations, and a functional∆f , which, we will see, is the analogue in this simple model of the Fadeev-Popovdeterminant. Hence the “correct” definition of the generating functional for thismodel is

w(~j) =∫ ∞

−∞dx∫ ∞

−∞dye−l(x,y)+jxx+jyy ∆f [x, y]δ((x+ y)/2) . (1.234)

Gauge Fixing Conditions

To implement this idea for the real case of non-Abelian gauge fields one envisagesan hypersurface, lying in the manifold of all gauge fields, which intersects eachgauge orbit once, and only once. This means that if

fa[Baµ(x); x] = 0 , a = 1, 2, . . . , N 2

c − 1 , (1.235)

is the equation describing the hypersurface, then there is a unique element in eachgauge orbit that satisfies one Eq. (1.235), and the set of these unique elements,none of which cannot be obtained from another by a gauge transformation, formsa representative class that alone truly characterises the physical configuration ofgauge fields. The gauge-equivalent, and therefore redundant, elements are absent.

Equations (1.235) can also be viewed as defining a set of non-linear equationsfor G(x), Eq (??). This in the sense that for a given field configuration, Bµ(x),it is always possible to find a unique gauge transformation, G1(x), that yields agauge transformed field BG1

µ (x), from Bµ(x) via Eq. (??), which is the one andonly solution of fa[B

a,G1

µ (x); x] = 0. Equation (1.235) therefore defines a gaugefixing condition.

In order for Eqs. (1.235) to be useful it must be possible that, when given aconfiguration Bµ(x), for which fa[Bµ(x); x] 6= 0, the equation

fa[BGµ (x); x] = 0 (1.236)

can be solved for the gauge transformation G(x). To see a consequence of thisrequirement consider a gauge configuration Bb

µ(x) that almost, but not quite,satisfies Eqs. (1.235). Applying an infinitesimal gauge transformation to thisBbµ(x) the requirement entails that it must be possible to solve

fa[Bbµ(x) → Bb

µ(x) − g fbdcBdµ(x) δΘc(x) − ∂µδΘ

b(x); x] = 0 (1.237)

for the infinitesimal gauge transformation parameters δΘa(x). Equation (1.237)can be written (using the chain rule)

fa[Bbµ(x)] −

∫

d4yδfa[B

bµ(x); x]

δBcν(y)

[

δcd∂ν + gfcedBeν(y)

]

δΘd(y) = 0 . (1.238)

This looks like the matrix equation ~f = O ~θ, which has a solution for ~θ if, andonly if, detO 6= 0, and a similar constraint follows from Eq. (1.238): the gaugefixing conditions can be solved if, and only if,

DetMf := Det

−δfa[Bbµ(x); x]

δBcν(y)

[

δcd∂ν − gfcdeBeν(y)

]

=: Det

−δfa[Bbµ(x); x]

δBcν(y)

[Dν(y)]cd

6= 0 . (1.239)

Equation (1.239) is the so-called admissibility condition for gauge fixing condi-tions.

A simple illustration is provided by the lightlike (Hamilton) gauges, which arespecified by

nµBaµ(x) = 0 , n2 > 1 , a = 1, 2, . . . , (N 2 − 1) . (1.240)

Choosing (nµ) = (1, 0, 0, 0), the equation for G(x) is, using Eq. (??)

∂

∂tG(t, ~x) = −G(t, ~x)B0(t, ~x) , (1.241)

and this nonperturbative equation has the unique solution

G(t, ~x) = T exp

−∫ t

−∞dsB0(s, ~x)

, (1.242)

where T is the time ordering operator. One may compare this with Eq. (1.238),which only provides a perturbative [in g] solution. While that may be an advan-tage, Eq. (1.242) is not a Poincare covariant constraint and that makes it difficultto employ in explicit calculations.

A number of other commonly used gauge fixing conditions are

∂µBaµ(x) = 0 , Lorentz gauge

∂µBaµ(x) = Aa(x) , Generalised Lorentz gauge

nµBaµ(x) = 0 , n2 < 0 , Axial gauge

nµBaµ(x) = 0 , n2 = 0 , Light-like gauge

~∇ · ~Ba(x) = 0 , Coulomb gauge

(1.243)

and the generalised axial, light-like and Coulomb gauges, where an arbitraryfunction, Aa(x), features on the r.h.s.

All of these choices satisfy the admissability condition, Eq. (1.239), for smallgauge field variations but in some cases, such as Lorentz gauge, the uniquenesscondition fails for large variations; i.e., those that are outside the domain of per-turbation theory. This means that there are at least two solutions: G1, G2, ofEq. (1.236), and perhaps uncountably many more. Since no nonperturbative so-lution of any gauge field theory in four spacetime dimensions exists, the actualnumber of solutions is unknown. If it is infinite then the Fadde’ev-Popov defini-tion of the generating functional fails in that gauge. These additional solutionsare called Gribov Copies and their existence raises questions about the correctway to furnish a nonperturbative definition of the generating functional [9], whichare currently unanswered.

Isolating and Eliminating the Gauge Orbit Volume

To proceed one needs a little information about the representation of non-Abeliangroups. Suppose u is an element of the group SU(N). Every such element can be

characterised by (N 2 − 1) real parameters: Θa, a = 1, . . . , N2 − 1. Let G(u) bethe representation of u under which the gauge fields transform; i.e., the adjointrepresentation, Eqs. (??), (??). For infinitesimal tranformations

G(u) = I − igT aΘa(x) + O(Θ2) (1.244)

where T a are the adjoint representation of the Lie algebra of SU(N), Eq. (??).Clearly, if u, u′ ∈ SU(N) then uu′ ∈ SU(n) and G(u)G(u′) = G(uu′). (Theseare basic properties of groups.)

To define the integral over gauge fields we must properly define the gauge-field “line element”. This is the Hurwitz measure on the group space, which isinvariant in the sense that du′ = d(u′u). In the neighbourhood of the identityone may always choose

du =∏

a

dΘa (1.245)

and the invariance means that since the integration represents a sum over allpossible values of the parameters Θa, relabelling them as Θa cannot matter. Itis now possible to quantise the gauge field; i.e., properly define Eq. (??).

Consider ∆f [Baµ] defined via, cf. Eq. (??),

∆f [Bbµ]∫

∏

x

du(x)∏

x,a

δ[faBb,uµ (x); x] = 1 , (1.246)

where Bb,uµ (x) is given by Eq. (??) with G(x) → u(x). ∆f [B

aµ] is gauge invariant:

∆−1f [Bb,u

µ ] =∫

∏

x

du′(x)∏

x,a

δ[faBb,u′uµ (x); x]

=∫

∏

x

d(u′(x)u(x))∏

x,a

δ[faBb,u′uµ (x); x]

=∫

∏

x

du′′(x)∏

x,a

δ[faBb,u′′

µ (x); x] = ∆−1f [Bb

µ] . (1.247)

Returning to Eq. (??), one can write

W [0] =∫

∏

x

du(x)∫

[DBaµ] ∆f [B

aµ]∏

x,b

δ[fbBa,uµ (x); x] expi

∫

d4xL(x) .

(1.248)Now execute a gauge transformation: Ba

µ(x) → Ba,u−1

µ (x), so that, using theinvariance of the measure and the action, one has

W [0] = [∫

∏

x

du(x)]∫

[DBaµ] ∆f [B

aµ]∏

x,b

δ[fbBaµ(x); x] expi

∫

d4xL(x) ,

(1.249)

where now the integrand of the group measure no longer depends on the groupelement, u(x) (cf. Eq. (??)). The gauge orbit volume has thus been identifiedand can be eliminated so that one may define

W [Jaµ ] =∫

[DBaµ] ∆f [Ba

µ]∏

x,b

δ[fbBaµ(x); x] expi

∫

d4x [L(x) + Jµa (x)Baµ(x)] .

(1.250)Neglecting for now the possible existence of Gribov copies, Eq. (1.250) is thefoundation we sought for a manifestly Poincare covariant quantisation of thegauge field. However, a little more work is needed to mould a practical tool.

Ghost Fields

A first step is an explicit calculation of ∆f [Baµ]. Since it always appears multi-

plied by a δ-function it is sufficient to evaluate it for those field configurationsthat satisfy Eq. (1.235). Recalling Eqs. (1.238), (1.239), for infinitesimal gaugetransformations

fa[Bb,uµ (x); x] = fa[B

bµ(x); x] +

∫

d4y (Mf)acΘc(y)

= 0 +∫

d4y (Mf)acΘc(y) . (1.251)

Hence, using the definition, Eq. (1.246),

∆−1f [Bb

µ] =∫

∏

x,a

dΘa(x) δ[∫

d4y (Mf)acΘc(y)]

(1.252)

and changing variables: Θa → Θa = (Mf)acΘc(y), this gives

∆−1f [Bb

µ] = DetMf

∫

∏

x,a

dΘa(x) δ[

Θa(y)]

, (1.253)

where the determinant is the Jacobian of the transformation, so that

∆f [Bbµ] = DetMf . (1.254)

Now recall Eq. (1.180). This means Eq. (1.254) can be expressed as a func-tional integral over Grassmann fields: φa, φb, a, b = 1, . . . , (N2 − 1),

∆f [Bbµ] =

∫

[Dφb][Dφa] exp

−∫

d4xd4y φb(x) (Mf)bc(x, y)φa(y)

, . (1.255)

Consequently, absorbing non-dynamical constants into the normalisation,

W [Jaµ ] =∫

[DBaµ] [Dφb][Dφa]

∏

x,b

δ[fbBaµ(x); x]

× exp

i∫

d4x [L(x) + Jµa (x)Baµ(x)] + i

∫

d4xd4y φb(x) (Mf)bc(x, y)φa(y)

.

(1.256)

The Grassmann fields φa, φb are the Fadde’ev-Popov Ghosts. They are anessential consequence of gauge fixing.

As one concrete example, consider the Lorentz gauge, Eq. (1.243), for which

(ML)ab = δ4(x− y)[

δab ∂2 − gfabc ∂µBc

µ(x)]

(1.257)

and therefore

W [Jaµ] =∫

[DBaµ] [Dφb][Dφa]

∏

x,a

δ[∂µBaµ(x)] exp

i∫

d4x[

−1

4F µνa (x)F a

µν(x)

− ∂µφa(x) ∂νφa(x) + gfabc[∂µφa(x)]φb(x)Bc

µ(x) + Jµa (x)Baµ(x)

]

.(1.258)

This expression makes clear that a general consequence of the Fadde’ev-Popovprocedure is to introduce a coupling between the gauge field and the ghosts. Thusthe ghosts, and hence gauge fixing, can have a direct impact on the behaviour ofgauge field Green functions.

As another, consider the axial gauge, for which

(MA)ab = δ4(x− y)[

δab nµ ∂µ]

, . (1.259)

Expressing the related determinant via ghost fields it is immediately apparentthat with this choice there is no coupling between the ghosts and the gauge fieldquanta. Hence the ghosts decouple from the theory and may be discarded asthey play no dynamical role. However, there is a cost, as always: in this gaugethe effect of the delta-function,

∏

x,a δ[nµBa

µ(x)], in the functional integral is tocomplicate the Green functions by making them depend explicitly on (nµ). Eventhe free-field 2-point function exhibits such a dependence.

It is important to note now that this decoupling of the ghost fields is tied toan “accidental” elimination of the fabc term in Dµ(y), Eq. (1.239). That term isalways absent in Abelian gauge theories, for which quantum electrodynamics isthe archetype, because all generators commute and analogues of fabc must vanish.Hence in Abelian gauge theories ghosts decouple in every gauge.

To see how δ[fbBaµ(x); x] influences the form of Green functions, consider

the generalised Lorentz gauge:

∂ν Baµ(x) = Aa(x) , (1.260)

where Aa(x) are arbitrary functions. The Fadde’ev-Popov determinant is thesame in generalised Lorentz gauges as it is in Lorentz gauge and hence

∆GL[Baµ] = ∆L[Ba

µ] , (1.261)

where the r.h.s. is given in Eq. (1.257). The generating functional in this gaugeis therefore

W [Jaµ ] =∫

[DBaµ] [Dφb][Dφa]

∏

x,a

δ[∂µBaµ(x) − Aa(x)] exp

i∫

d4x[

−1

4F µνa (x)F a

µν(x)

− ∂µφa(x) ∂νφa(x) + gfabc[∂µφa(x)]φb(x)Bc

µ(x) + Jµa (x)Baµ(x)

]

.

Gauge invariance of the generating functional, Eq. (1.246), means that one canintegrate over the Aa(x), with a weight function to ensure convergence:

∫

[DAa] exp

− i

2λ

∫

d4xAa(x)Aa(x)

, (1.262)

to arrive finally at the generating functional in a covariant Lorentz gauge, speci-fied by the parameter λ:

W [Jaµ , ξag , ξ

ag ] =

∫

[DBaµ] [Dφb][Dφa]

exp

i∫

d4x[

−1

4F µνa (x)F a

µν(x) − 1

2λ[∂µBa

µ(x)] [∂νBaν (x)]

− ∂µφa(x) ∂νφa(x) + gfabc[∂µφa(x)]φb(x)Bc

µ(x)

+Jµa (x)Baµ(x) + ξag (x)φa(x) + φa(x)ξag (x)

]

, (1.263)

where ξag , ξag are anticommuting external sources for the ghost fields. (NB. Tocomplete the definition one should add convergence terms, iη+, for every field or,preferably, work in Euclidean space.)

Observe now that the free gauge boson piece of the action in Eq. (1.263) is

1

2

∫

d4x d4y Baµ(x)Kµν(x− y;λ)Ba

ν(y)

:=1

2

∫

d4x d4y Baµ(x)

[gµν∂2 − ∂µ∂ν(1 − 1

λ) − igµν η+]δ4(x− y)

Baν (y)(1.264)

The operator Kµν(x− y;λ) thus defined can be expressed

Kµν(x− y) =∫

d4q

(2π)4

(

−gµν(q2 + iη+) + qµqν[1 − 1

λ])

e−i(q,x−y), (1.265)

cf. Eq. (??), and now

qµKµν = − 1

λqν (1.266)

so that in this case the action does damp variations in the longitudinal compo-nents of the gauge field. Kµν(x − y;λ) is the inverse of the free gauge bosonpropagator; i.e., the free gauge boson 2-point Green function, Dµν(x − y), isobtained via ∫

d4wKµρ (x− w)Dρν(w − y) = gµν δ4(x− y) , (1.267)

and hence

Dµν(x− y) =∫ d4q

(2π)4

(

−gµν + (1 − λ)qµqν

q2 + iη+

)

1

q2 + iη+e−i(q,x−y). (1.268)

The obvious λ dependence is a result of the presence of δ[fbBaµ(x); x] in the

generating functional, and this is one example of the δ-function’s direct effect onthe form of Green functions: they are, in general, gauge parameter dependent.

1.8.2 Exercises

1. Verify Eq. (1.257).

2. Verify Eq. (1.258).

3. Verify Eq. (1.259).

4. Verify Eq. (1.268).

1.9 Dyson-Schwinger Equations

It has long been known that, from the field equations of quantum field theory, onecan derive a system of coupled integral equations interrelating all of a theory’sGreen functions [6, 10]. This tower of a countable infinity of equations is calledthe complex of Dyson-Schwinger equations (DSEs). This intrinsically nonpertur-bative complex is vitally important in proving the renormalisability of quantumfield theories and at its simplest level provides a generating tool for perturba-tion theory. In the context of quantum electrodynamics (QED) I will illustrate anonperturbative derivation of two equations in this complex. The derivation ofothers follows the same pattern.

1.9.1 Photon Vacuum Polarization

Generating Functional for QED

The action for QED with Nf flavours of electomagnetically active fermion, is

S[Aµ, ψ, ψ] =∫

d4x

Nf∑

f=1

ψf(

i∂/−mf0 + ef0A/

)

ψf − 1

4FµνF

µν − 1

2λ0

∂µAµ(x) ∂νAν(x)

,

(1.269)where: ψf(x), ψf(x) are the elements of the Grassmann algebra that describethe fermion degrees of freedom, mf

0 are the fermions’ bare masses and ef0 theircharges; and Aµ(x) describes the gauge boson [photon] field, with

Fµν = ∂µAν − ∂νAµ , (1.270)

and λ0 the bare covariant-Lorentz-gauge fixing parameter. (NB. To describe anelectron the physical charge ef < 0.)

The derivation of the generating functional in Eq. (1.263) can be employedwith little change here. In fact, in this context it is actually simpler becauseghost fields decouple. Combining the procedure for fermions and gauge fields,

described in Secs. 1.7, ?? respectively, one arrives at

W [Jµ, ξ, ξ] =∫

[DAµ] [Dψ][Dψ]

exp

i∫

d4x[

−1

4F µν(x)Fµν(x) − 1

2λ0∂µAµ(x) ∂νAν(x)

+Nf∑

f=1

ψf(

i∂/−mf0 + ef0A/

)

ψf +Jµ(x)Aµ(x) + ξf(x)ψf(x) + ψf(x)ξf(x)]

,

where Jµ is an external source for the electromagnetic field, and ξf , ξf are ex-ternal sources for the fermion field that, of course, are elements in the Grass-mann algebra. (NB. In Abelian gauge theories there are no Gribov copies in thecovariant-Lorentz-gauges.)

Functional Field Equations

As described in Sec. 1.6.1, it is advantageous to work with the generating func-tional of connected Green functions; i.e., Z[Jµ, ξ, ξ] defined via

W [Jµ, ξ, ξ] =: exp

iZ[Jµ, ξ, ξ]

. (1.271)

The derivation of a DSE now follows simply from the observation that the integralof a total derivative vanishes, given appropriate boundary conditions. Hence, forexample,

0 =∫

[DAµ] [Dψ][Dψ]δ

δAµ(x)exp

i(

S[Aµ, ψ, ψ] +∫

d4x[

ψfξf + ξfψf + AµJ

µ]

)

=∫

[DAµ] [Dψ][Dψ]

δS

δAµ(x)+ Jµ(x)

exp

i(

S[Aµ, ψ, ψ] +∫

d4x[

ψfξf + ξfψf + AµJ

µ]

)

=

δS

δAµ(x)

[

δ

iδJ,δ

iδξ,− δ

iδξ

]

+ Jµ(x)

W [Jµ, ξ, ξ] , (1.272)

where the last line has meaning as a functional differential operator on the gen-erating functional.

Differentiating Eq. (1.269) gives

δS

δAµ(x)=[

∂ρ∂ρgµν −

(

1 − 1

λ0

)

∂µ∂ν

]

Aν(x) +∑

f

ef0 ψf(x)γµψ

f(x) (1.273)

so that the explicit meaning of Eq. (1.272) is

−Jµ(x) =[

∂ρ∂ρgµν −

(

1 − 1

λ0

)

∂µ∂ν

]

δZ

δJν(x)+∑

f

ef0

(

− δZ

δξf(x)γµ

δZ

δξf(x)+

δ

δξf(x)

[

γµδ iZ

δξf(x)

])

,

(1.274)

where I have divided through by W [Jµ, ξ, ξ]. Equation (1.274) represents a com-pact form of the nonperturbative equivalent of Maxwell’s equations.

One-Particle-Irreducible Green Functions

The next step is to introduce the generating functional for one-particle-irreducible(1PI) Green functions: Γ[Aµ, ψ, ψ], which is obtained from Z[Jµ, ξ, ξ] via a Leg-endre transformation

Z[Jµ, ξ, ξ] = Γ[Aµ, ψ, ψ] +∫

d4x[

ψfξf + ξfψf + AµJ

µ]

. (1.275)

A one-particle-irreducible n-point function or “proper vertex” contains no con-tributions that become disconnected when a single connected m-point Greenfunction is removed; e.g., via functional differentiation. This is equivalent to thestatement that no diagram representing or contributing to a given proper vertexseparates into two disconnected diagrams if only one connected propagator is cut.(A detailed explanation is provided in Ref. [3], pp. 289-294.)

A simple generalisation of the analysis in Sec. 1.6.1 yields

δZ

δJµ(x)= Aµ(x) ,

δZ

δξ(x)= ψ(x) ,

δZ

δξ(x)= −ψ(x) , (1.276)

where here the external sources are nonzero. Hence Γ in Eq. (1.275) must satisfy

δΓ

δAµ(x)= −Jµ(x) ,

δΓ

δψf(x)= −ξf (x) ,

δΓ

δψf(x)= ξf(x) . (1.277)

(NB. Since the sources are not zero then, e.g.,

Aµ(x) = Aµ(x; [Jµ, ξ, ξ]) ⇒ δA

δJµ(x)6= 0 , (1.278)

with analogous statements for the Grassmannian functional derivatives.) It iseasy to see that setting ψ = 0 = ψ after differentiating Γ gives zero unless thereare equal numbers of ψ and ψ derivatives. (This is analogous to the result forscalar fields in Eq. (1.140).)

Now consider the product (with spinor labels r, s, t)

−∫

d4zδ2Z

δξfr (x)ξht (z)

δ2Γ

δψht (z)ψg

s(y)

∣

∣

∣

∣

∣ ξ = ξ = 0ψ = ψ = 0

. (1.279)

Using Eqs. (1.276), (1.277), this simplifies as follows:

=∫

d4zδψht (z)

δξfr (x)

δξgs(y)

δψht (z)

∣

∣

∣

∣

∣ ξ = ξ = 0ψ = ψ = 0

=δξgs (y)

δξfr (x)

∣

∣

∣

∣

∣

ψ = ψ = 0= δrsδ

fgδ4(x− y) .

(1.280)

Now returning to Eq. (1.274) and setting ξ = 0 = ξ one obtains

δΓ

δAµ(x)

∣

∣

∣

∣

∣

ψ=ψ=0

=[

∂ρ∂ρgµν −

(

1 − 1

λ0

)

∂µ∂ν

]

Aν(x)− i∑

f

ef0tr[

γµSf(x, x; [Aµ])

]

,

(1.281)after making the identification

Sf(x, y; [Aµ]) = − δ2Z

δξf(y)ξf(x)=

δ2Z

δξf(x)ξf(y)(no summation on f) , (1.282)

which is the connected Green function that describes the propagation of a fermionwith flavour f in an external electromagnetic field Aµ (cf. the free fermion Greenfunction in Eq. (1.197).) I observe that it is a direct consequence of Eq. (1.279)that

Sf(x, y; [A])−1 =δ2Γ

δψf(x)δψf (y)

∣

∣

∣

∣

∣

ψ=ψ=0

, (1.283)

and it is a general property that such functional derivatives of the generatingfunctional for 1PI Green functions are related to the associated propagator’sinverse. Clearly the vacuum fermion propagator or connected fermion 2-pointfunction is

Sf(x, y) := Sf (x, y; [Aµ = 0]) . (1.284)

Such vacuum Green functions are of primary interest in quantum field theory.To continue, one differentiates Eq. (1.281) with respect to Aν(y) and sets

Jµ(x) = 0, which yields

δ2Γ

δAµ(x)δAν(y)

∣

∣

∣

∣

∣ Aµ = 0ψ = ψ = 0

=[

∂ρ∂ρgµν −

(

1 − 1

λ0

)

∂µ∂ν

]

δ4(x− y)

−i∑

f

ef0tr

γµδ

δAν(y)

δ2Γ

δψf(x)δψf(x)

∣

∣

∣

∣

∣

ψ=ψ=0

−1

.

(1.285)

The l.h.s. is easily understood. Just as Eqs. (1.283), (1.284) define the inverseof the fermion propagator, so here is

(D−1)µν(x, y) :=δ2Γ

δAµ(x)δAν(y)

∣

∣

∣

∣

∣ Aµ = 0ψ = ψ = 0

. (1.286)

The r.h.s., however, must be simplified and interpreted. First observe that

δ

δAν(y)

δ2Γ

δψf(x)δψf (x)

∣

∣

∣

∣

∣

ψ=ψ=0

−1

=

−∫

d4ud4w

δ2Γ

δψf(x)δψf (w)

∣

∣

∣

∣

∣

ψ=ψ=0

−1δ

δAν(y)

δ2Γ

δψf (u)δψf(w)

δ2Γ

δψf(w)δψf(x)

∣

∣

∣

∣

∣

ψ=ψ=0

−1

,

(1.287)

which is an analogue of the result for finite dimensional matrices:

d

dx

[

A(x)A−1(x) = I]

= 0 =dA(x)

dxA−1(x) + A(x)

dA−1(x)

dx

⇒ dA−1(x)

dx= −A−1(x)

dA(x)

dxA−1(x) .(1.288)

Equation (1.287) involves the 1PI 3-point function

ef0Γfµ(x, y; z) :=δ

δAν(z)

δ2Γ

δψf(x)δψf (y). (1.289)

This is the proper fermion-gauge-boson vertex. At leading order in perturbationtheory

Γfν(x, y; z) = γν δ4(x− z) δ4(y − z) , (1.290)

a result which can be obtained via the explicit calculation of the functional deriva-tives in Eq. (1.289).

Now, defining the gauge-boson vacuum polarisation:

Πµν(x, y) = i∑

f

(ef0)2∫

d4z1 d4z2 tr

[

γµSf(x, z1)Γfν(z1, z2; y)Sf(z2, x)

]

, (1.291)

it is immediately apparent that Eq. (1.285) may be expressed as

(D−1)µν(x, y) =[

∂ρ∂ρgµν −

(

1 − 1

λ0

)

∂µ∂ν

]

δ4(x− y) + Πµν(x, y) . (1.292)

In general, the gauge-boson vacuum polarisation, or “self-energy,” describes themodification of the gauge-boson’s propagation characteristics due to the presenceof virtual fermion-antifermion pairs in quantum field theory. In particular, thephoton vacuum polarisation is an important element in the description of processsuch as ρ0 → e+e−.

The propagator for a free gauge boson was given in Eq. (1.268). In thepresence of interactions; i.e., for Πµν 6= 0 in Eq. (1.292), this becomes

Dµν(q) =−gµν + (qµqν/[q2 + iη])

q2 + iη

1

1 + Π(q2)− λ0

qµqν

(q2 + iη)2, (1.293)

where I have used the “Ward-Takahashi identity:”

qµ Πµν(q) = 0 = Πµν(q) qν , (1.294)

which means that one can write

Πµν(q) =(

−gµνq2 + qµqν)

Π(q2) . (1.295)

Π(q2) is the polarisation scalar and, in QED, it is independent of the gauge param-eter, λ0. (NB. λ0 = 1 is called “Feynman gauge” and it is useful in perturbativecalculations because it obviously simplifies the Π(q2) = 0 gauge boson propagatorenormously. In nonperturbative applications, however, λ0 = 0, “Landau gauge,”is most useful because it ensures the gauge boson propagator itself is transverse.)

Ward-Takahashi identities (WTIs) are relations satisfied by n-point Greenfunctions, relations which are an essential consequence of a theory’s local gaugeinvariance; i.e., local current conservation. They can be proved directly from thegenerating functional and have physical implications. For example, Eq. (1.295)ensures that the photon remains massless in the presence of charged fermions. (Adiscussion of WTIs can be found in Ref. [1], pp. 299-303, and Ref. [3], pp. 407-411;and their generalisation to non-Abelian theories as “Slavnov-Taylor” identities isdescribed in Ref. [5], Chap. 2.)

In the absence of external sources for fermions and gauge bosons, Eq. (1.291)can easily be represented in momentum space, for then the 2- and 3-point func-tions that appear explicitly must be translationally invariant and hence can besimply expressed in terms of Fourier amplitudes. This yields

iΠµν(q) = −∑

f

(ef0)2∫

d4`

(2π)dtr[(iγµ)(iSf (`))(iΓf(`, `+ q))(iS(`+ q))] . (1.296)

It is the reduction to a single integral that makes momentum space representa-tions most widely used in continuum calculations.

In QED the vacuum polarisation is directly related to the running couplingconstant. This connection makes its importance obvious. In QCD the connectionis not so direct but, nevertheless, the polarisation scalar is a key component inthe evaluation of the strong running coupling.

In the above analysis we saw that second derivatives of the generating func-tional, Γ[Aµ, ψ, ψ], give the inverse-fermion and -photon propagators and that thethird derivative gave the proper photon-fermion vertex. In general, all derivativesof this generating functional, higher than two, produce the corresponding properGreen’s functions, where the number and type of derivatives give the number andtype of proper Green functions that it can serve to connect.

1.9.2 Fermion Self Energy

Equation (1.274) is a nonperturbative generalisation of Maxwell’s equation inquantum field theory. Its derivation provides the model by which one can obtain

an equivalent generalisation of Dirac’s equation. To this end consider that

0 =∫

[DAµ] [Dψ][Dψ]δ

δψf (x)exp

i(

S[Aµ, ψ, ψ] +∫

d4x[

ψgξg + ξgψg + AµJ

µ]

)

=∫

[DAµ] [Dψ][Dψ]

δS

δψf(x)+ ξf(x)

exp

i(

S[Aµ, ψ, ψ] +∫

d4x[

ψgξg + ξgψg + AµJ

µ]

)

=

δS

δψf(x)

[

δ

iδJ,δ

iδξ,− δ

iδξ

]

+ ηf(x)

W [Jµ, ξ, ξ] (1.297)

=

[

ξf(x) +

(

i∂/−mf0 + ef0γ

µ δ

iδJµ(x)

)

δ

iδξf(x)

]

W [Jµ, ξ, ξ] . (1.298)

This is a nonperturbative, funcational equivalent of Dirac’s equation.One can proceed further. A functional derivative with respect to ξf : δ/δξf(y),

yields

δ4(x− y)W [Jµ] −(

i∂/−mf0 + ef0γ

µ δ

iδJµ(x)

)

W [Jµ]Sf(x, y; [Aµ]) = 0 ,(1.299)

after setting ξf = 0 = ξf , where W [Jµ] := W [Jµ, 0, 0] and S(x, y; [Aµ]) is definedin Eq. (1.282). Now, using Eqs. (1.271), (1.277), this can be rewritten

δ4(x− y) −(

i∂/−mf0 + ef0A/(x; [J ]) + ef0γ

µ δ

iδJµ(x)

)

Sf(x, y; [Aµ]) = 0 ,(1.300)

which defines the nonperturbative connected 2-point fermion Green function(This is clearly the functional equivalent of Eq. (1.86).)

The electromagentic four-potential vanishes in the absence of an externalsource; i.e., Aµ(x; [J = 0]) = 0, so it remains only to exhibit the content of theremaining functional differentiation in Eq. (1.300), which can be accomplishedusing Eq. (1.287):

δ

iδJµ(x)Sf(x, y; [Aµ]) =

∫

d4zδAν(z)

iδJµ(x)

δ

δAν(z)

δ2Γ

δψf(x)δψf(y)

∣

∣

∣

∣

∣

ψ=ψ=0

−1

= −ef0∫

d4z d4u d4wδAν(z)

iδJµ(x)Sf (x, u) Γν(u, w; z)S(w, y)

= −ef0∫

d4z d4u d4w iDµν(x− z)Sf(x, u) Γν(u, w; z)S(w, y) ,

(1.301)

where, in the last line, I have set J = 0 and used Eq. (1.286). Hence, in theabsence of external sources, Eq. (1.300) is equivalent to

δ4(x− y) =(

i∂/−mf0

)

Sf(x, y)

− i (ef0)2∫

d4z d4u d4wDµν(x, z) γµ S(x, u) Γν(u, w; z)S(w, y) = δ4(x− y) .(1.302)

Just as the photon vacuum polarisation was introduced to simplify, or re-express, the DSE for the gauge boson propagator, Eq. (1.291), one can define afermion self-energy:

Σf(x, z) = i(ef0)2∫

d4u d4wDµν(x, z) γµ S(x, u) Γν(u, w; z) , (1.303)

so that Eq. (1.302) assumes the form

∫

d4z[(

i∂/x −mf0

)

δ4(x− z) − Σf(x, z)]

S(z, y) = δ4(x− y) . (1.304)

Again using the property that Green functions are translationally invariantin the absence of external sources, the equation for the self-energy can be writtenin momentum space:

Σf (p) = i (ef0)2∫ d4`

(2π)4Dµν(p− `) [iγµ] [iSf(`)] [iΓfν(`, p)] . (1.305)

In terms of the self-energy, it follows from Eq. (1.304) that the connected fermion2-point function can be written in momentum space as

Sf(p) =1

p/−mf0 − Σf (p) + iη+

. (1.306)

Equation (1.305) is the exact Gap Equation. It describes the manner in whichthe propagation characteristics of a fermion moving through the ground stateof QED (the QED vacuum) is altered by the repeated emission and reabsorp-tion of virtual photons. The equation can also describe the real process ofBremsstrahlung. Furthermore, the solution of the analogous equation in QCDits solution provides information about dynamical chiral symmetry breaking andalso quark confinement.

1.9.3 Exercises

1. Verify Eq. (1.277).

2. Verify Eq. (1.287).

3. Verify Eq. (1.296).

4. Verify Eq. (1.300).

1.10 Perturbation Theory

1.10.1 Quark Self Energy

A key feature of strong interaction physics is dynamical chiral symmetry breaking(DCSB). In order to understand it one must first come to terms with explicitchiral symmetry breaking. Consider then the DSE for the quark self-energy inQCD:

−iΣ(p) = −g20

∫ d4`

(2π)4Dµν(p− `)

i

2λaγµ S(`) Γaν(`, p) , (1.307)

where I have suppressed the flavour label. The form is precisely the same as thatin QED, Eq. (1.305), with the only difference being the introduction of the colour(Gell-Mann) matrices: λa; a = 1, . . . , 8 at the fermion-gauge-boson vertex. Theinterpretation of the symbols is also analogous: Dµν(`) is the connected gluon2-point function and Γaν(`, `

′) is the proper quark-gluon vertex.The one-loop contribution to the quark’s self-energy is obtained by evaluating

the r.h.s. of Eq. (1.307) using the free quark and gluon propagators, and thequark-gluon vertex:

Γa (0)ν (`, `′) =

i

2λaγν , (1.308)

which appears to be a straightforward task.To be explicit, the goal is to calculate

−iΣ(2)(p) = −g20

∫

d4k

(2π)4

(

−gµν + (1 − λ0)kµkν

k2 + iη+

)

1

k2 + iη+

× i

2λaγµ

1

6k + p/−m0 + iη+

i

2λaγµ (1.309)

and one may proceed as follows. First observe that Eq. (1.309) can be re-expressedas

−iΣ(2)(p) = −g20 C2(R)

∫ d4k

(2π)4

1

(k + p)2 −m20 + iη+

1

k2 + iη+

×

γµ (6k + p/+m0) γµ − (1 − λ0) (6k − p/+m0) − 2 (1 − λ0)(k, p)6kk2 + iη+

(1.310)

where I have used

1

2λa

1

2λa = C2(R) Ic ; C2(R) =

N2c − 1

2Nc, (1.311)

with Nc the number of colours (Nc = 3 in QCD) and Ic, the identity matrix incolour space. Now note that

2 (k, p) = [(k + p)2 −m20] − [k2] − [p2 −m2

0] (1.312)

and hence

−iΣ(2)(p) = −g20 C2(R)

∫ d4k

(2π)4

1

(k + p)2 −m20 + iη+

1

k2 + iη+

γµ (6k + p/+m0) γµ + (1 − λ0) (p/−m0)

+ (1 − λ0) (p2 −m20)

6kk2 + iη+

− (1 − λ0) [(k + p)2 −m20]

6kk2 + iη+

.

(1.313)

Focusing on the last term:∫

d4k

(2π)4

1

(k + p)2 −m20 + iη+

1

k2 + iη+[(k + p)2 −m2

0]6k

k2 + iη+

=∫

d4k

(2π)4

1

k2 + iη+

6kk2 + iη+

= 0 (1.314)

because the integrand is odd under k → −k, and so this term in Eq. (1.313)vanishes, leaving

−iΣ(2)(p) = −g20 C2(R)

∫

d4k

(2π)4

1

(k + p)2 −m20 + iη+

1

k2 + iη+

γµ (6k + p/+m0) γµ + (1 − λ0) (p/−m0)

+ (1 − λ0) (p2 −m20)

6kk2 + iη+

.

(1.315)

Consider now the second term:

(1 − λ0) (p/−m0)∫

d4k

(2π)4

1

(k + p)2 −m20 + iη+

1

k2 + iη+.

In particular, focus on the behaviour of the integrand at large k2:

1

(k + p)2 −m20 + iη+

1

k2 + iη+

k2→±∞∼ 1

(k2 −m20 + iη+) (k2 + iη+)

. (1.316)

The integrand has poles in the second and fourth quadrant of the k0-plane butvanishes on any circle of radius R → ∞. That means one may rotate the contouranticlockwise to find

∫ ∞

0dk0 1

(k2 −m20 + iη+) (k2 + iη+)

=∫ i∞

0dk0 1

([k0]2 − ~k2 −m20 + iη+)([k0]2 − ~k2 + iη+)

k0→ik4= i∫ ∞

0dk4

1

(−k24 − ~k2 −m2

0) (−k24 − ~k2)

. (1.317)

Performing a similar analysis of the∫ 0−∞ part one obtains the complete result:

∫

d4k

(2π)4

1

(k2 −m20 + iη+) (k2 + iη+)

= i∫

d3k

(2π)3

∫ ∞

−∞

dk4

2π

1

(−~k2 − k24 −m2

0) (−~k2 − k24).

(1.318)

These two steps constitute what is called a “Wick rotation.”The integral on the r.h.s. is defined in a four-dimensional Euclidean space;

i.e., k2 := k21 + k2

2 + k23 + k2

4 ≥ 0 . . . k2 is nonnegative. A general vector in thisspace can be written in the form:

(k) = |k| (cosφ sin θ sin β, sinφ sin θ sin β, cos θ sin β, cos β) , (1.319)

and clearly k2 = |k|2. In this space the four-vector measure factor is∫

d4Ek f(k1, . . . , k4) =

1

2

∫ ∞

0dk2k2

∫ π

0dβ sin2β

∫ π

0dθ sin θ

∫ 2π

0dφ f(k, β, θ, φ)

(1.320)Returning now to Eq. (1.316) the large k2 behaviour of the integral can be

determined via∫

d4k

(2π)4

1

(k + p)2 −m20 + iη+

1

k2 + iη+≈ i

16π2

∫ ∞

0dk2 1

(k2 +m20)

=i

16π2lim

Λ→∞

∫ Λ2

0dx

1

x +m20

=i

16π2lim

Λ→∞ln(

1 + Λ2/m20

)

→ ∞ ;

(1.321)

i.e., after all this work, the result is meaningless: the one-loop contribution tothe quark’s self-energy is divergent!

Such “ultraviolet” divergences, and others which are more complicated, ap-pear whenever loops appear in perturbation theory. (The others include “in-frared” divergences associated with the gluons’ masslessness; e.g., consider whatwould happen in Eq. (1.321) with m0 → 0.) In a renormalisable quantum fieldtheory there exists a well-defined set of rules that can be used to render per-turbation theory sensible. First, however, one must regularise the theory; i.e.,introduce a cutoff, or some other means, to make finite every integral that ap-pears. Then every step in the calculation of an observable is rigorously sensible.Renormalisation follows; i.e, the absorption of divergences, and the redefinitionof couplings and masses, so that finally one arrives at S-matrix amplitudes thatare finite and physically meaningful.

The regularisation procedure must preserve the Ward-Takahashi identities(the Slavnov-Taylor identities in QCD) because they are crucial in proving thata theory can be sensibly renormalised. A theory is called renormalisable if, andonly if, there are a finite number of different types of divergent integral so thatonly a finite number of masses and coupling constants need to be renormalised.

1.10.2 Dimensional Regularization

The Pauli-Villars prescription is favoured in QED and that is described, for ex-ample, in Ref. [3]. In perturbative QCD, however, “dimensional regularisation”is the most commonly used procedure and I will introduce that herein.

The key to the method is to give meaning to the divergent integrals by chang-ing the dimension of spacetime. Returning to the exemplar, Eq. (1.316), thismeans we consider

T =∫

dDk

(2π)D1

(k + p)2 −m20 + iη+

1

k2 + iη+(1.322)

where D is the dimension of spacetime and is not necessarily four.Observe now that

1

aα bβ=

Γ(α + β)

Γ(α) Γ(β)

∫ 1

0dx

xα−1 (1 − x)β−1

[a x + b (1 − x)]α+β, (1.323)

where Γ(x) is the gamma-function: Γ(n+ 1) = n!. This is an example of what iscommonly called “Feynman’s parametrisation,” and one can now write

T =∫ 1

0dx

∫ dDk

(2π)D1

[(k − xp)2 −m20(1 − x) + p2x(1 − x) + iη+]2

.(1.324)

(1.325)

The momentum integral is well-defined for D = 1, 2, 3 but, as we have seen, notfor D = 4. One proceeds under the assumption that D is such that the integralis convergent then a shift of variables is permitted:

Tk→k−xp

=∫ 1

0dx

∫ dDk

(2π)D1

[k2 − a2 + iη+]2, (1.326)

where a2 = m20(1 − x) − p2x(1 − x).

I will consider a generalisation of the momentum integral:

In =∫

dDk

(2π)D1

[k2 − a2 + iη+]n, (1.327)

and perform a Wick rotation to obtain

In =i

(2π)D(−1)n

∫

dDk1

[k2 + a2]n. (1.328)

The integrand has an O(D) spherical symmetry and therefore the angular inte-grals can be performed:

SD :=∫

dΩD =2πD/2

Γ(D/2). (1.329)

Clearly, S4 = 2π2, as we saw in Eq. (1.321). Hence

In = i(−1)n

2D−1πD/21

Γ(D/2)

∫ ∞

0dk

kD−1

(k2 + a2)n. (1.330)

Writing D = 4 + 2ε one arrives at

In =i

(4π)2(−a2)2−n

(

a2

4π

)εΓ(n− 2 − ε)

Γ(n). (1.331)

(NB. Every step is rigorously justified as long as 2n > D.) The important pointfor continuing with this procedure is that the analytic continuation of Γ(x) isunique and that means one may use Eq. (1.331) as the definition of In wheneverthe integral is ill-defined. I observe that D = 4 is recovered via the limit ε→ 0−

and the divergence of the integral for n = 2 in this case is encoded in

Γ(n− 2 − ε) = Γ(−ε) =1

−ε − γE + O(ε) ; (1.332)

i.e., in the pole in the gamma-function. (γE is the Euler constant.)Substituting Eq. (1.331) in Eq. (1.326) and setting n = 2 yields

T = (gνε)2 i

(4π)2

Γ(−ε)(4π)ε

∫ 1

0dx

[

m20

ν2(1 − x) − p2

ν2x(1 − x)

]ε

, (1.333)

wherein I have employed a nugatory transformation to introduce the mass-scaleν. It is the limit ε → 0− that is of interest, in which case it follows that (xε =exp ε ln x ≈ 1 + ε ln x)

T = (gνε)2 i

(4π)2

−1

ε− γE + ln 4π −

∫ 1

0dx ln

[

m20

ν2(1 − x) − p2

ν2x(1 − x)

]

= (gνε)2 i

(4π)2

−1

ε− γE + ln 4π + 2 − ln

m20

ν2−(

1 − m20

p2

)

ln

[

1 − p2

m20

]

.

(1.334)

It is important to understand the physical content of Eq. (1.334). While it isonly one part of the gluon’s contribution to the quark’s self-energy, many of itsproperties hold generally.

1. Observe that T (p2) is well-defined for p2 < m20; i.e., for all spacelike mo-

menta and for a small domain of timelike momenta. However, at p2 = m20,

T (p2) exhibits a ln-branch-point and hence T (p2) acquires an imaginarypart for p2 > m2

0. This imaginary part describes the real, physical processby which a quark emits a massless gluon; i.e., gluon Bremsstrahlung. InQCD this is one element in the collection of processes referred to as “quarkfragmentation.”

2. The mass-scale, ν, introduced in Eq. (1.333), which is a theoretical artifice,does not affect the position of the branch-point, which is very good becausethat branch-point is associated with observable phenomena. While it mayappear that ν affects the magnitude of physical cross sections because itmodifies the coupling, that is not really so: in going to D = 2n − ε di-mensions the coupling constant, which was dimensionless for D = 4, hasacquired a mass dimension and so the physical, dimensionless coupling con-stant is α := (gνε)2/(4π). It is this dependence on ν that opens the doorto the generation of a “running coupling constant” and “running masses”that are a hallmark of quantum field theory.

3. A number of constants have appeared in T (p2). These are irrelevant becausethey are eliminated in the renormalisation procedure. (NB. So far we haveonly regularised the expression. Renormalisation is another step.)

4. It is apparent that dimensional regularisation gives meaning to divergentintegrals without introducing new couplings or new fields. That is a ben-efit. The cost is that while γ5 = iγ0γ1γ2γ3 is well-defined with particularproperties for D = 4, a generalisation to D 6= 4 is difficult and hence so isthe study of chiral symmetry.

D-dimensional Dirac Algebra

When one employs dimensional regularisation all the algebraic manipulationsmust be performed before the integrals are evaluated, and that includes the Diracalgebra. The Clifford algebra is unchanged in D-dimensions

γµ , γν = 2 gµν ; µ, ν = 1, . . . , D − 1 , (1.335)

but nowgµν g

µν = D (1.336)

and hence

γµ γν = D 1D , (1.337)

γµ γν γµ = (2 −D) γν , (1.338)

γµ γν γλ γµ = 4 gνλ1D + (D − 4) γν γλ , (1.339)

γµ γν γλ γρ γµ = −2 γρ γλ γν + (4 −D) γν γλ γρ , (1.340)

where 1D is the D ×D-dimensional unit matrix.It is also necessary to evaluate traces of products of Dirac matrices. For a

D-dimensional space, with D even, the only irreducible representation of theClifford algebra, Eq. (1.335), has dimension f(D) = 2D/2. In any calculation itis the (anti-)commutation of Dirac matrices that leads to physically important

factors associated with the dimension of spacetime while f(D) always appears asa common multiplicative factor. Hence one can just set

f(D) ≡ 4 (1.341)

in all calculations. Any other prescription merely leads to constant terms of thetype encountered above; e.g., γE, which are eliminated in renormalisation.

The D-dimensional generalisation of γ5 is a more intricate problem. However,I will not use it herein and hence will omit that discussion.

Observations on the Appearance of Divergences

Consider a general Lagrangian density:

L(x) = L0(x) +∑

i

giLi(x) , (1.342)

where L0 is the sum of the free-particle Lagrangian densities and Li(x) representsthe interaction terms with the coupling constants, gi, written explicitly. Assumethat Li(x) has fi fermion fields (fi must be even since fermion fields always appearin the pairs ψ, ψ), bi boson fields and n∂i derivatives. The action

S =∫

d4xL(x) (1.343)

must be a dimensionless scalar and therefore L(x) must have mass-dimensionM4. Clearly a derivative operator has dimension M . Hence, looking at the free-particle Lagrangian densities, it is plain that each fermion field has dimensionM3/2 and each boson field, dimension M 1. It follows that a coupling constantmultiplying the interaction Lagrangian density Li(x) must have mass-dimension

[gi] = M4−di , di =3

2fi + bi + n∂i . (1.344)

It is a fundamentally important fact in quantum field theory that if there iseven one coupling constant for which

[gi] < 0 (1.345)

then the theory possesses infinitely many different types of divergences and hencecannot be rendered finite through a finite number of renormalisations. Thisdefines the term nonrenormalisable.

In the past nonrenormalisable theories were rejected as having no predictivepower: if one has to remove infinitely many infinite constants before one candefine a result then that result cannot be meaningful. However, the modern viewis different. Chiral perturbation theory is a nonrenormalisable “effective theory.”However, at any finite order in perturbation theory there is only a finite number

of undetermined constants: 2 at leading order; 6 more, making 8 in total, whenone-loop effects are considered; and more than 140 new terms when two-loopeffects are admitted. Nevertheless, as long as there is a domain in some physicalexternal-parameter space on which the one-loop corrected Lagrangian densitycan be assumed to be a good approximation, and there is more data that canbe described than there are undetermined constants, then the “effective theory”can be useful as a tool for correlating observables and elucidating the symmetriesthat underly the general pattern of hadronic behaviour.

1.10.3 Regularized Quark Self Energy

I can now return and re-express Eq. (1.315):

−iΣ(2)(p) = −(g0νε)2 C2(R)

∫ dDk

(2π)4

1

ν2ε

1

(k + p)2 −m20 + iη+

1

k2 + iη+

γµ (6k + p/+m0) γµ + (1 − λ0) (p/−m0)

+ (1 − λ0) (p2 −m20)

6kk2 + iη+

.

(1.346)

It can be separated into a sum of two terms, each proportional to a differentDirac structure:

Σ(p/) = ΣV (p2) p/+ ΣS(p2) 1D , (1.347)

that can be obtained via trace projection:

ΣV (p2) =1

f(D)

1

p2trD [p/Σ(p/)] , ΣS(p2) =

1

f(D)trD [Σ(p/)] . (1.348)

These are the vector and scalar parts of the dressed-quark self-energy, and theyare easily found to be given by

p2 ΣV (p2) = −i (g0νε)2 C2(R)

∫ dDk

(2π)4

1

ν2ε

1

[(k + p)2 −m20 + iη+] [k2 + iη+]

×[

(2 −D)(p2 + pµkµ) + (1 − λ0)p2 + (1 − λ0)(p

2 −m20)

pµkµ

k2 + iη+

]

.

(1.349)

ΣS(p2) = −i (g0νε)2 C2(R)

∫

dDk

(2π)4

1

ν2ε

m0(D − 1 + λ0)

[(k + p)2 −m20 + iη+] [k2 + iη+]

. (1.350)

(NB. As promised, the factor of f(D) has cancelled.)

These equations involve integrals of the general form

I(α, β; p2, m2) =∫

dDk

(2π)4

1

ν2ε

1

[(k + p)2 −m2 + iη+]α [k2 + iη+]β,(1.351)

Jµ(α, β; p2, m2) =∫

dDk

(2π)4

1

ν2ε

kµ

[(k + p)2 −m2 + iη+]α [k2 + iη+]β,(1.352)

the first of which we have already encountered in Sec. 1.10.2. The general resultsare (D = 4 + 2ε)

I(α, β; p2, m2) =i

(4π)2

(

1

p2

)α+β−2Γ(α + β − 2 − ε) Γ(2 + ε− β)

Γ(α) Γ(2 + ε)

×(

− p2

4πν2

)ε (

1 − m2

p2

)2+ε−α−β

2F1(α + β − 2 − ε, 2 + ε− β, 2 + ε;1

1 − (m2/p2)) ,

(1.353)

Jµ(α, β; p2, m2) = pµi

(4π)2

(

1

p2

)α+β−2Γ(α + β − 2 − ε) Γ(3 + ε− β)

Γ(α) Γ(3 + ε)

×(

− p2

4πν2

)ε (

1 − m2

p2

)2+ε−α−β

2F1(α + β − 2 − ε, 3 + ε− β, 3 + ε;1

1 − (m2/p2))

=: pµ J(α, β; p2, m2) , (1.354)

where 2F1(a, b, c; z) is the hypergeometric function.Returning again to Eqs. (1.349), (1.350) it is plain that

ΣV (p2) = −i (g0νε)2 C2(R)

[

2(1 + ε) J(1, 1) − (1 + 2ε) I(1, 1) − (p2 −m20) J(1, 2)

]

+λ0

[

(p2 −m20) J(1, 2) − I(1, 1)

]

, (1.355)

ΣS(p2) = −i (g0νε)2 C2(R)m0 (3 + λ0 + 2ε) I(1, 1; p2, m2) , (1.356)

where I have omitted the (p2, m20) component in the arguments of I, J .

The integrals explicitly required are

I(1, 1) =i

(4π)2

−1

ε+ ln 4π − γE − ln

(

−p2

ν2

)

+ 2

−m2

p2ln

(

−m2

p2

)

−(

1 − m2

p2

)

ln

(

1 − m2

p2

)

, (1.357)

J(1, 1) =i

(4π)2

1

2

−1

ε+ ln 4π − γE − ln

(

−p2

ν2

)

+ 2

−m2

p2

(

2 − m2

p2

)

ln

(

−m2

p2

)

−

1 − 2m2

p2+

[

m2

p2

]2

ln

(

1 − m2

p2

)

− m2

p2

,

(1.358)

J(1, 2) =i

(4π)2

1

p2

−m2

p2ln

(

−m2

p2

)

+m2

p2ln

(

1 − m2

p2

)

+ 1

. (1.359)

Using these expressions it is straightforward to show that the λ0-independentterm in Eq. (1.355) is identically zero in D = 4 dimensions; i.e., for ε− > 0−, andhence

ΣV (p2) = λ0(gνε)2

(4π)2C2(R)

1

ε− ln 4π + γE + ln

m20

ν2

−1 − m20

p2+

(

1 − m40

p4

)

ln

(

1 − p2

m2

)

. (1.360)

It is obvious that in Landau gauge: λ0 = 0, in four spacetime dimensions:ΣV (p2) ≡ 0, at this order. The scalar piece of the quark’s self-energy is alsoeasily found:

ΣS(p2) = m0(gνε)2

(4π)2C2(R)

−(3 + λ0)

[

1

ε− ln 4π + γE + ln

m20

ν2

]

+2(2 + λ0) − (3 + λ0)

(

1 − m20

p2

)

ln

(

1 − p2

m20

)

. (1.361)

Note that in Yennie gauge: λ0 = −3, in four dimensions, the scalar piece of theself-energy is momentum-independent, at this order. We now have the completeregularised dressed-quark self-energy at one-loop order in perturbation theoryand its structure is precisely as I described in Sec. 1.10.2. Renormalisation mustfollow.

One final observation: the scalar piece of the self-energy is proportional to thebare current-quark mass, m0. That is true at every order in perturbation theory.Clearly then

limm0→0

ΣS(p2, m20) = 0 (1.362)

and hence dynamical chiral symmetry breaking is impossible in perturbationtheory.

1.10.4 Exercises

1. Verify Eq. (1.321).

2. Verify Eq. (1.328).

3. Verify Eq. (1.334).

4. Verify Eqs. (1.349) and (1.350).

5. Verify Eqs. (1.360) and (1.361).

1.11 Renormalized Quark Self Energy

Hitherto I have illustrated the manner in which dimensional regularisation is em-ployed to give sense to the divergent integrals that appear in the perturbativecalculation of matrix elements in quantum field theory. It is now necessary torenormalise the theory; i.e., to provide a well-defined prescription for the elimi-nation of all those parts in the calculated matric element that express the diver-gences and thereby obtain finite results for Green functions in the limit ε → 0−

(or with the removal of whatever other parameter has been used to regularise thedivergences).

1.11.1 Renormalized Lagrangian

The bare QCD Lagrangian density is

L(x) = −1

2∂µB

aν (x)[∂µBνa(x) − ∂νBµa(x)] − 1

2λ∂µB

aν (x)∂µB

aν (x)

−1

2g fabc [∂µBνa(x) − ∂νBµa(x)]Bb

µ(x)Bcν(x)

−1

4g2fabcfadeB

bµ(x)Bc

ν(x)Bµd(x)Bνe(x)

−∂µφa(x)∂µφa(x) + g fabc ∂µφa(x)φb(x)Bµc(x)

+qf(x)i∂/qf(x) −mf qf(x)qf (x) +1

2g qf(x)λa/Ba(x)qf (x) ,(1.363)

where: Baµ(x) are the gluon fields, with the colour label a = 1, . . . , 8; φa(x), φa(x)

are the (Grassmannian) ghost fields; qf (x), qf(x) are the (Grassmannian) quarkfields, with the flavour label f = u, d, s, c, b, t; and g, mf , λ are, respectively, thecoupling, mass and gauge fixing parameter. (NB. The QED Lagrangian density isimmediately obtained by setting fabc ≡ 0. It is clearly the non-Abelian nature ofthe gauge group, SU(Nc), that generates the gluon self-couplings, the triple-gluonand four-gluon vertices, and the ghost-gluon interaction.)

The elimination of the divergent parts in the expression for a Green functioncan be achieved by adding “counterterms” to the bare QCD Lagrangian density,one for each different type of divergence in the theory; i.e., one considers therenormalised Lagrangian density

LR(x) := L(x) + Lc(x) , (1.364)

with

Lc(x) = C3YM1

2∂µB

aν (x)[∂µBνa(x) − ∂νBµa(x)] + C6

1

2λ∂µB

aν (x)∂µB

aν (x)

+C1YM1

2g fabc [∂µBνa(x) − ∂νBµa(x)]Bb

µ(x)Bcν(x)

+C51

4g2fabcfadeB

bµ(x)Bc

ν(x)Bµd(x)Bνe(x)

+C3 ∂µφa(x)∂µφa(x) − C1g fabc ∂µφ

a(x)φb(x)Bµc(x)

−C2F qf(x)i∂/qf (x) + C4m

f qf(x)qf(x) − C1F1

2g qf(x)λa/Ba(x)qf(x) .

(1.365)

To prove the renormalisability of QCD one must establish that the coefficients, Ci,each understood as a power series in g2, are the only additional terms necessaryto remove all the ultraviolet divergences in the theory at every order in theperturbative expansion.

In the example of Sec. 1.10.2 I illustrated that the divergent terms in the reg-ularised self-energy are proportional to g2. This is a general property and henceall of the Ci begin with a g2 term. The Ci-dependent terms can be treated just asthe terms in the original Lagrangian density and yield corrections to the expres-sions we have already derived that begin with an order-g2 term. Returning to theexample of the dressed-quark self-energy this means that we have an additionalcontribution:

∆Σ(2)(p/) = C2F p/− C4m , (1.366)

and one can choose C2F , C4 such that the total self-energy is finite.The renormalisation constants are introduced as follows:

Zi := 1 − Ci (1.367)

so that Eq. (1.365) becomes

LR(x) = −Z3Y M

2∂µB

aν (x)[∂µBνa(x) − ∂νBµa(x)] − Z6

2λ∂µB

aν (x)∂µB

aν (x)

−Z1Y M

2g fabc [∂µBνa(x) − ∂νBµa(x)]Bb

µ(x)Bcν(x)

−Z5

4g2fabcfadeB

bµ(x)Bc

ν(x)Bµd(x)Bνe(x)

−Z3 ∂µφa(x)∂µφa(x) + Z1 g fabc ∂µφ

a(x)φb(x)Bµc(x)

+Z2F qf(x)i∂/qf (x) − Z4m

f qf(x)qf (x) +Z1Y M

2g qf (x)λa/Ba(x)qf(x) ,

(1.368)

NB. I have implicitly assumed that the renormalisation counterterms, and hencethe renormalisation constants, are flavour independent. It is always possible tochoose prescriptions such that this is so.

I will now introduce the bare fields, coupling constants, masses and gauge

fixing parameter:

Bµa0 (x) := Z

1/23YM Bµa(x) , qf0 (x) := Z

1/22F qf (x) ,

φa0(x) := Z1/23 φa(x) , φa0(x) := Z

1/23 φa(x) ,

g0YM := Z1YM Z−3/23YM g , g0 := Z1 Z

−13 Z

−1/23YM ,

g0F := Z1F Z−1/23YM Z−1

2F g , g05 := Z1/25 Z−1

3YM g

mf0 := Z4 Z

−12F m

f , λ0 := Z−16 Z3YM λ .

(1.369)

The fields and couplings on the r.h.s. of these definitions are called renormalised,and the couplings are finite and the fields produce Green functions that are finiteeven in D = 4 dimensions. ( NB. All these quantities are defined in D = 4 + 2ε-dimensional space. Hence one has for the Lagrangian density: [L(x)] = MD, andthe field and coupling dimensions are

[q(x)] = [q(x)] = M 3/2+ε , [Bµ(x)] = M1+ε ,

[φ(x)] = [φ(x)] = M1+ε , [g] = M−ε ,

[λ] = M0 , [m] = M1 .)

(1.370)

The renormalised Lagrangian density can be rewritten in terms of the barequantities:

LR(x) = −1

2∂µB

a0ν(x)[∂µBνa

0 (x) − ∂νBµa0 (x)] − 1

2λ0

∂µBa0ν(x)∂µB

a0ν(x)

−1

2g0YM fabc [∂µBνa

0 (x) − ∂νBµa0 (x)]Ba

0µ(x)Ba0ν(x)

−1

4g205fabcfadeB

b0µ(x)Bc

0ν(x)Bµd0 (x)Bνe

0 (x)

−∂µφa0(x)∂µφa0(x) + g0 fabc ∂µφa0(x)φb0(x)Bµc

0 (x)

+qf0 (x)i∂/qf0 (x) −mf0 q

f0 (x)qf0 (x) +

1

2g0F q

f0 (x)λa/Ba

0(x)qf0 (x) .

(1.371)

It is apparent that now the couplings are different and hence LR(x) is not invariantunder local gauge transformations (more properly BRST transformations) unless

g0YM = g0 = g0F = g05 = g0 . (1.372)

Therefore, if the renormalisation procedure is to preserve the character of thegauge theory, the renormalisation constants cannot be completely arbitrary butmust satisfy the following “Slavnov-Taylor” identities:

g0YM = g0 ⇒ Z3YM

Z1YM=Z3

Z1

,

g0YM = g0F ⇒ Z3YM

Z1YM=Z2F

Z1F,

g0YM = g05 ⇒ Z5 =Z2

1YM

Z3YM.

(1.373)

In QED the second of these equations becomes the Ward-Takahashi identity:Z1F = Z2F .

1.11.2 Renormalization Schemes

At this point we can immediately write an expression for the renormalised dressed-quark self-energy:

Σ(2)R (p/) =

(

ΣV (p2) + C2F

)

p/+ ΣS(p2) − C4m . (1.374)

The subtraction constants are not yet determined and there are many ways onemay choose them in order to eliminate the divergent parts of bare Green functions.

Minimal Subtraction

In the minimal subtraction (MS) scheme one defines a dimensionless coupling

α :=(gνε)2

4π(1.375)

and considers each counterterm as a power series in α with the form

Ci =∞∑

j=1

j∑

k=1

C(2j)i,k

1

εk

(

α

π

)j

, (1.376)

where the coefficients in the expansion may, at most, depend on the gauge pa-rameter, λ.

Using Eqs. (1.360), (1.361) and (1.374) we have

Σ(2)R (p/) = p/

α

πλ

1

4C2(R)

[

1

ε− ln 4π + γE + ln

m2

ν2(1.377)

−1 − m2

p2+

(

1 − m4

p4

)

ln

(

1 − p2

m2

)]

+ C2F

(1.378)

+mα

π

1

4C2(R)

−(3 + λ)

[

1

ε− ln 4π + γE + ln

m2

ν2

]

(1.379)

+2(2 + λ) − (3 + λ)

(

1 − m2

p2

)

ln

(

1 − p2

m2

)

− C4

. (1.380)

Now one chooses C2F , C4 such that they cancel the 1/ε terms in this equationand therefore, at one-loop level,

Z2F = 1 − C2F = 1 +α

πλ

1

4C2(R)

1

ε, (1.381)

Z4 = 1 − C4 = 1 +α

π(3 + λ)

1

4C2(R)

1

ε, (1.382)

and hence Eq. (1.380) becomes

Σ(2)R (p/) = p/

α

πλ

1

4C2(R)

[(

− ln 4π + γE + lnm2

ν2

)

(1.383)

−1 − m2

p2+

(

1 − m4

p4

)

ln

(

1 − p2

m2

)]

+ C2F

(1.384)

+mα

π

1

4C2(R)

−(3 + λ)

(

− ln 4π + γE + lnm2

ν2

)

(1.385)

+2(2 + λ) − (3 + λ)

(

1 − m2

p2

)

ln

(

1 − p2

m2

)

− C4

, (1.386)

which is the desired, finite result for the dressed-quark self-energy.It is not common to work explicitly with the counterterms. More often one

uses Eq. (1.371) and the definition of the connected 2-point quark Green function(an obvious analogue of Eq. (1.282)):

iSfR(x, y;m, λ, α) = −δ2ZR[Jaµ , ξ, ξ]

δξf(y)δξf(x)= 〈0|qf(x) qf(y)|0〉 = Z−1

2F 〈0|qf0 (x) qf0 (y)|0〉(1.387)

to writeSR(p/;m, λ, α) = lim

ε→0

Z−12F S0(p/;m0, λ0, α0; ε)

, (1.388)

where in the r.h.s. m0, λ0, α0 have to be substituted by their expressions in termsof the renormalised quantities and the limit taken order by order in α.

To illustrate this using our concrete example, Eq. (1.388) yields

1 − ΣV R(p2;m, λ, α)

p/−m

1 + ΣSR(p2;m, λ, α)/m

= limε→0−

(

Z2F

[

1 − ΣV (p2;m0, λ0, α0)]

p/−m0

[

1 + ΣS(p2;m0, λ0, α0)/m0

])

(1.389)

and taking into account that m = Z−14 Z2F m0 then

1 − ΣV R(p2;m, λ, α) = limε→0−

Z2F

1 − ΣV (p2;m0, λ0, α0)

, (1.390)

1 + ΣSR(p2;m, λ, α)/m = limε→ε−

Z4

1 + ΣS(p2;m0, λ0, α0)/m0

.(1.391)

Now the renormalisation constants, Z2F , Z4, are chosen so as to exactly cancelthe 1/ε poles in the r.h.s. of Eqs. (1.390), (1.391). Using Eqs. (1.360), (1.361),Eqs. (1.381), (1.382) are immediately reproduced.

Modified Minimal Subtraction

The modified minimal substraction scheme MS is also often used in QCD. It takesadvantage of the fact that the 1/ε pole obtained using dimensional regularisation

always appears in the combination

1

ε− ln 4π + γE (1.392)

so that the renormalisation constants are defined so as to eliminate this combi-nation, in its entirety, from the Green functions. At one-loop order the renor-malisation constants in the MS scheme are trivially related to those in the MSscheme. At higher orders there are different ways of defining the scheme and therelation between the renormalisation constants is not so simple.

Momentum Subtraction

In the momentum subtraction scheme (µ-scheme) a given renormalised Greenfunction, GR, is obtained from its regularised counterpart, G, by subtractingfrom G its value at some arbitrarily chosen momentum scale. In QCD that scaleis always chosen to be a Euclidean momentum: p2 = −µ2. Returning to ourexample of the dressed-quark self-energy, in this scheme

ΣAR(p2;µ2) := ΣA(p2; ε) − ΣA(p2; ε) ;A = V, S , (1.393)

and so

Σ(2)V R(p2;µ2) =

α(µ)

πλ(µ)

1

4C2(R)

−m2(µ)

(

1

p2+

1

µ2

)

+

(

1 − m4(µ)

p4

)

ln

(

1 − p2

m(µ)2

)

−(

1 − m4(µ)

µ4

)

ln

(

1 +µ2

m2(µ)

)

,

(1.394)

Σ(2)SR(p2;µ2) = m(µ)

α(µ)

π1

4C2(R) −[3 + λ(µ)]

×[(

1 − m2(µ)

p2

)

ln

(

1 − p2

m2(µ)

)

−(

1 +m2(µ)

µ2

)

ln

(

1 +µ2

m2(µ)

)]

(1.395)

where the renormalised quantities depend on the point at which the renormalisa-tion has been conducted. Clearly, from Eqs. (1.390), (1.391), the renormalisationconstants in this scheme are

Z(2)2F = 1 + Σ

(2)V (p2 = −µ2; ε) (1.396)

Z(2)4 = 1 − Σ

(2)S (p2 = −µ2; ε)/m(µ) . (1.397)

It is apparent that in this scheme there is at least one point, the renormali-sation mass-scale, µ, at which there are no higher order corrections to any of theGreen functions: the corrections are all absorbed into the redefinitions of the cou-pling constant, masses and gauge parameter. This is valuable if the coefficients of

the higher order corrections, calculated with the parameters defined through themomentum space subtraction, are small so that the procedure converges rapidlyon a large momentum domain. In this sense the µ-scheme is easier to understandand more intuitive than the MS or MS schemes. Another advantage is the mani-fest applicability of the “decoupling theorem,” Refs. [11], which states that quarkflavours whose masses are larger than the scale chosen for µ are irrelevant.

This last feature, however, also emphasises that the renormalisation constantsare flavour dependent and that can be a nuisance. Nevertheless, the µ-scheme isextremely useful in nonperturbative analyses of DSEs, especially since the flavourdependence of the renormalisation constants is minimal for light quarks when theEuclidean substraction point, µ, is chosen to be very large; i.e., much larger thantheir current-masses.

1.11.3 Renormalized Gap Equation

Equation (1.307) is the unrenormalised QCD gap equation, which can be rewrit-ten as

−iS−10 (p) = −i(p/−m0) + g2

0

∫

d4`

(2π)4Dµν

0 (p− `)i

2λaγµ S0(`) Γa0ν(`, p) . (1.398)

The renormalised equation can be derived directly from the generating functionaldefined using the renormalised Lagrangian density, Eq. (1.368), simply by repeat-ing the steps described in Sec. 1.9.2. Alternatively, one can use Eqs. (1.371) toderive an array of relations similar to Eq. (1.387):

Dµν0 (k) = Z3YM Dµν

R (k) , Γa0ν(k, p) = Z−11F ΓaRν(k, p) , (1.399)

and others that I will not use here. (NB. It is a general feature that propagatorsare multiplied by the renormalisation constant and proper vertices by the inverseof renormalisation constants.) Now one can replace the unrenormalised couplings,masses and Green functions by their renormalised forms:

−iZ−12 S−1

R (p) = −ip/ + iZ−12F Z4mR

+Z21F Z

−13YM Z−2

2F g2R

∫

d4`

(2π)4Z3YM Dµν

R (p− `)i

2λaγµ Z2F SR(`)Z−1

1F ΓaRν(`, p) ,

(1.400)

which simplifies to

−iΣR(p) = i(Z2F − 1) p/− i(Z4 − 1)mR

−Z1F g2R

∫

d4`

(2π)4DµνR (p− `)

i

2λaγµ SR(`) ΓaRν(`, p)

=: i(Z2F − 1) p/− i(Z4 − 1)mR − iΣ′(p) , (1.401)

where Σ′(p) is the regularised self-energy.In the simplest application of the µ-scheme one would choose a large Euclidean

mass-scale, µ2, and define the renormalisation constants such that

ΣR(/p+ µ = 0) = 0 (1.402)

which entails

Z2F = 1 + Σ′V (p/+ µ = 0) , Z4 = 1 − Σ′S(p/+ µ = 0)/mR(µ) , (1.403)

where I have used Σ′(p) = Σ′(p) p/ + Σ′S(p). (cf. Eqs. (1.396), (1.397).) This issimple to implement, even nonperturbatively, and is always appropriate in QCDbecause confinement ensures that dressed-quarks do not have a mass-shell.

On-shell Renormalization

If one is treating fermions that do have a mass-shell; e.g., electrons, then anon-shell renormalisation scheme may be more appropriate. One fixes the renor-malisation constants such that

S−1R (p/)

∣

∣

∣

/p=mR

= p/−mR , (1.404)

which is interpreted as a constraint on the pole position and the residue at thepole; i.e., since

S−1(p/) = [/p−mR − ΣR(p/)]|/p=mR− [/p−mR]

[

d

d/pΣR(p/)

]∣

∣

∣

∣

∣

/p=mR

+ . . . (1.405)

then Eq. (1.404) entails

ΣR(p/)|/p=mR= 0 ,

d

d/pΣR(p/)

∣

∣

∣

∣

∣

/p=mR

= 0 . (1.406)

The second of these equations requires

Z2F = 1 + Σ′V (m2R) + 2m2

R

d

dp2Σ′V (p2)

∣

∣

∣

p2=m2R

+ 2mRd

dp2Σ′S(p2)

∣

∣

∣

∣

∣

p2=m2R

(1.407)

and the first:Z4 = Z2F − Σ′V (m2

R) − Σ′S(m2R) . (1.408)

1.11.4 Exercises

1. Using Eqs. (1.360), (1.361) derive Eqs. (1.381), (1.382).

2. Verify Eqs. (1.396), (1.397).

3. Verify Eq. (1.401).

4. Verify Eq. (1.403).

5. Beginning with Eq. (1.404), derive Eqs. (1.407), (1.408). NB. p/p/ = p2 andhence d

dp/f(p2) = d

dp/f(p/p/) = 2 p/ d

dp2f(p2).

1.12 Dynamical Chiral Symmetry Breaking

This phenomenon profoundly affects the character of the hadron spectrum. How-ever, it is intrinsically nonperturbative and therefore its understanding is bestsought via a Euclidean formulation of quantum field theory, which I will nowsummarise.

1.12.1 Euclidean Metric

I will formally describe the construction of the Euclidean counterpart to a givenMinkowski space field theory using the free fermion as an example. The La-grangian density for free Dirac fields is given in Eq. (1.186):

Lψ0 (x) =∫ ∞

−∞dt∫

d3x ψ(x)(

i∂/ −m+ iη+)

ψ(x) . (1.409)

Recall now the observations at the end of Sec. 1.5 regarding the role of the iη+

in this equation: it was introduced to provide a damping factor in the generatingfunctional. The alternative proposed was to change variables and introduce aEuclidean time: t→ −itE . As usual

∫ ∞

−∞dt f(t) =

∫ ∞

−∞d(−itE) f(−itE) = −i

∫ ∞

−∞dtE f(−itE) (1.410)

if f(t) vanishes on the curve at infinity in the second and fourth quadrants ofthe complex-t plane and is analytic therein. To complete this Wick rotation,however, we must also determine its affect on i∂/, since that is part of the “f(t);”i.e., the integrand:

i∂/ = iγ0 ∂

∂t+ iγi

∂

∂xiM→E→ iγ0 ∂

∂(−itE)+ iγi

∂

∂xi

= −γ0 ∂

∂tE− (−iγi) ∂

∂xi(1.411)

=: −γEµ∂

∂xEµ, (1.412)

where (xEµ ) = (~x, x4 := −it) and the Euclidean Dirac matrices are

γE4 = γ0 γEi = −iγi . (1.413)

These matrices are Hermitian and satisfy the algebra

γEµ , γEν = 2δµν ;µ, ν = 1, . . . , 4, (1.414)

where δµν is the four-dimensional Kroncker delta. Henceforth I will adopt thenotation

aE · bE =4∑

µ=1

aEµ bEν . (1.415)

Now, assuming that the integrand is analytic where necessary, one arrivesformally at

∫ ∞

−∞dt∫

d3x ψ(x)(

i∂/ −m+ iη+)

ψ(x)

= −i∫ ∞

−∞dtE

∫

d3x ψ(xE)(

−γE · ∂E −m+ iη+)

ψ(xE) . (1.416)

Intepreted naively, however, the action on the r.h.s. poses problems: it is not realif one inteprets ψ(xE) = ψ†(xE) γ4.

Let us study its involution

AE =∫

d4xE ψ(xE) (−γE · ∂E −m+ iη+)ψ(xE) (1.417)

where ψr(xE) and ψs(x

E) are intepreted as members of a Grassmann algebra withinvolution, as described in Sec. 1.7.1:

AE =∫

d4xE ψr(xE) (−[γE · ∂E→]rs −mδrs + iη+ δrs)ψs(xE)

=∫

d4xE ψs(xE)

(

−[γE · ∂E←]rs −m∗ δrs − iη+δrs

)

ψr(xE) . (1.418)

We know that the mass is real: m∗ = m, and that as an operator the gradient isanti-Hermitian; i.e.,

∂E← = −∂E→ (1.419)

but that leaves us with

[γEµ ]rs = [(γEµ )T]sr . (1.420)

If we define

(γEµ )T := −CE γEµ C†E , (1.421)

where CE = γ2γ4 is the Euclidean charge conjugation matrix then

AE = AE (1.422)

when ψ and ψ are members of a Grassmann algebra with involution and we takethe limit η+ = 0, which we will soon see is permissable.

Return now to the generating functional of Eq. (1.187):

W [ξ, ξ] =∫

[Dψ(x)][Dψ(x)] exp

i∫

d4x[

ψ(x)(

i∂/ −m + iη+)

ψ(x) + ψ(x)ξ(x) + ξ(x)ψ(x)]

.

(1.423)

A Wick rotation of the action transforms the source-independent part of theintegrand, called the measure, into

exp

−∫

d4xE[

ψ(xE)(

γ · ∂ +m− iη+)

ψ(xE)]

(1.424)

so that the Euclidean generating functional

WE[ξ, ξ] :=∫

[Dψ(xE)][Dψ(xE)]

× exp

−∫

d4x[

ψ(xE) (γ · ∂ +m)ψ(xE) + ψ(xE)ξ(xE) + ξ(xE)ψ(xE)]

(1.425)

involves a positive-definite measure because the free-fermion action is real, as Ihave just shown. Hence the “+iη+” convergence factor is unnecessary here.

Euclidean Formulation as Definitive

Working in Euclidean space is more than simply a pragmatic artifice: it is possibleto view the Euclidean formulation of a field theory as definitive (see, for example,Refs. [12, 13, 14, 15]). In addition, the discrete lattice formulation in Euclideanspace has allowed some progress to be made in attempting to answer existencequestions for interacting gauge field theories [15].

The moments of the Euclidean measure defined by an interacting quantumfield theory are the Schwinger functions:

Sn(xE,1, . . . , xE,n) (1.426)

which can be obtained as usual via functional differentiation of the analogue ofEq. (1.425) and subsequently setting the sources to zero. The Schwinger functionsare sometimes called Euclidean space Green functions.

Given a measure and given that it satisfies certain conditions [i.e., the Wight-man and Haag-Kastler axioms], then it can be shown that the Wightman func-tions, Wn(x1, . . . , xn), can be obtained from the Schwinger functions by analyticcontinuation in each of the time coordinates:

Wn(x1, . . . , xn) = limxi4→0

Sn([~x1, x14 + ix0

1], . . . , [~xn, xn4 + ix0n]) (1.427)

with x01 < x0

2 < . . . < x0n. These Wightman functions are simply the vacuum ex-

pectation values of products of field operators from which the Green functions[i.e., the Minkowski space propagators] are obtained through the inclusion of step-[θ-] functions in order to obtain the appropriate time ordering. (This is describedin some detail in Refs. [13, 14, 15].) Thus the Schwinger functions contain all ofthe information necessary to calculate physical observables.

This notion is used directly in obtaining masses and charge radii in latticesimulations of QCD, and it can also be employed in the DSE approach sincethe Euclidean space DSEs can all be derived from the appropriate Euclideangenerating functional using the methods of Sec. 1.9 and the solutions of theseequations are the Schwinger functions.

All this provides a good reason to employ a Euclidean formulation. Another isa desire to maintain contact with perturbation theory where the renormalisationgroup equations for QCD and their solutions are best understood [16].

Collected Formulae for Minkowski ↔ Euclidean Transcription

To make clear my conventions (henceforth I will omit the supescript E used todenote Euclidean four-vectors): for 4-vectors a, b:

a · b := aµ bν δµν :=4∑

i=1

ai bi , (1.428)

so that a spacelike vector, Qµ, has Q2 > 0; the Dirac matrices are Hermitian anddefined by the algebra

γµ, γν = 2 δµν ; (1.429)

and I useγ5 := − γ1γ2γ3γ4 (1.430)

so thattr [γ5γµγνγργσ] = −4 εµνρσ , ε1234 = 1 . (1.431)

The Dirac-like representation of these matrices is:

~γ =

(

0 −i~τi~τ 0

)

, γ4 =

(

τ 0 00 −τ 0

)

, (1.432)

where the 2 × 2 Pauli matrices are:

τ 0 =

(

1 00 1

)

, τ 1 =

(

0 11 0

)

, τ 2 =

(

0 −ii 0

)

, τ 3 =

(

1 00 −1

)

. (1.433)

Using these conventions the [unrenormalised] Euclidean QCD action is

S[B, q, q] =∫

d4x

1

4F aµνF

aµν +

1

2λ∂ ·Ba ∂ ·Ba +

Nf∑

f=1

qf

(

γ · ∂ +mf + ig1

2λa γ ·Ba

)

qf

,

(1.434)where F a

µν = ∂µBaν − ∂νB

aµ − gf abcBb

µBcν. The generating functional follows:

W [J, ξ, ξ] =∫

dµ(q, q, B, ω, ω) exp∫

d4x[

q ξ + ξ q + Jaµ Aaµ

]

, (1.435)

with sources: η, η, J , and a functional integral measure

dµ(q, q, B, ω, ω) := (1.436)∏

x

∏

φ

Dqφ(x)Dqφ(x)∏

a

Dωa(x)Dωa(x)∏

µ

DBaµ(x) exp(−S[B, q, q] − Sg[B, ω, ω]) ,

where φ represents both the flavour and colour index of the quark field, and ωand ω are the scalar, Grassmann [ghost] fields. The normalisation

W [η = 0, η = 0, J = 0] = 1 (1.437)

is implicit in the measure. As we saw in Sec. 1.8.1, the ghosts only couple directlyto the gauge field:

Sg[B, ω, ω] =∫

d4x[

−∂µωa ∂µωa − gf abc ∂µωa ωbBc

µ

]

, (1.438)

and restore unitarity in the subspace of transverse [physical] gauge fields. I notethat, practically, the normalisation means that ghost fields are unnecessary in thecalculation of gauge invariant observables using lattice-regularised QCD becausethe gauge-orbit volume-divergence in the generating functional, associated withthe uncountable infinity of gauge-equivalent gluon field configurations in the con-tinuum, is rendered finite by the simple expedient of only summing over a finitenumber of configurations.

It is possible to derive every equation introduced above, assuming certainanalytic properties of the integrands. However, the derivations can be sidesteppedusing the following transcription rules:

Configuration Space

1.∫ M

d4xM → −i∫ E

d4xE

2. /∂ → iγE · ∂E

3. /A → −iγE · AE

4. AµBµ → −AE ·BE

5. xµ∂µ → xE · ∂E

Momentum Space

1.∫ M

d4kM → i∫ E

d4kE

2. /k → −iγE · kE

3. /A → −iγE · AE

4. kµqµ → −kE · qE

5. kµxµ → −kE · xE

These rules are valid in perturbation theory; i.e., the correct Minkowski spaceintegral for a given diagram will be obtained by applying these rules to theEuclidean integral: they take account of the change of variables and rotationof the contour. However, for the diagrams that represent DSEs, which involvedressed n-point functions whose analytic structure is not known as priori, theMinkowski space equation obtained using this prescription will have the rightappearance but it’s solutions may bear no relation to the analytic continuationof the solution of the Euclidean equation. The differences will be nonperturbativein origin.

1.12.2 Chiral Symmetry

Gauge theories with massless fermions have a chiral symmetry. Its effect can bevisualised by considering the helicity: λ ∝ J · p, the projection of the fermion’sspin onto its direction of motion. λ is a Poincare invariant spin observable thattakes a value of ±1. The chirality operator can be realised as a 4× 4-matrix, γ5,and a chiral transformation is then represented as a rotation of the 4 × 1-matrixquark spinor field

q(x) → eiγ5θ q(x) . (1.439)

A chiral rotation through θ = π/2 has no effect on a λ = +1 quark, qλ= + → qλ= +,but changes the sign of a λ = −1 quark field, qλ=− → − qλ=−. In compositehadrons this is manifest as a flip in their parity: JP=+ ↔ JP=−; i.e., a θ = π/2chiral rotation is equivalent to a parity transformation. Exact chiral symmetrytherefore entails that degenerate parity multiplets must be present in the spec-trum of the theory.

For many reasons, the masses of the u- and d-quarks are expected to be verysmall; i.e., mu ∼ md ΛQCD. Therefore chiral symmetry should only be weaklybroken, with the strong interaction spectrum exhibiting nearly degenerate paritypartners. The experimental comparison is presented in Eq. (1.440):

N(12

+, 938) π(0−, 140) ρ(1−, 770)

N(12

−, 1535) a0(0+, 980) a1(1

+, 1260). (1.440)

Clearly the expectation is very badly violated, with the splitting much too largeto be described by the small current-quark masses. What is wrong?

Chiral symmetry can be related to properties of the quark propagator, S(p).For a free quark (remember, I am now using Euclidean conventions)

S0(p) =m− iγ · pm2 + p2

(1.441)

and as a matrix

S0(p) → eiγ5θS0(p)eiγ5θ =−iγ · pp2 +m2

+ e2iγ5θm

p2+m2(1.442)

under a chiral transformation. As anticipated, for m = 0, S0(p) → S0(p); i.e., thesymmetry breaking term is proportional to the current-quark mass and it can bemeasured by the “quark condensate”

−〈qq〉 :=∫

d4p

(2π)4tr [S(p)] ∝

∫

d4p

(2π)4

m

p2 +m2, (1.443)

which is the “Cooper-pair” density in QCD. For a free quark the condensatevanishes if m = 0 but what is the effect of interactions?

As we have seen, interactions dress the quark propagator so that it takes theform

S(p) :=1

iγ · p+ Σ(p)=

−iγ · pA(p2) +B(p2)

p2A2(p2) +B2(p2), (1.444)

where Σ(p) is the self energy, expressed in terms of the scalar functions: A andB, which are p2-dependent because the interaction is momentum-dependent. Onthe valid domain; i.e., for weak-coupling, they can be calculated in perturbationtheory and at one-loop order, Eq. (1.395) with p2 → −(pE)2 µ2, m2(µ),

B(p2) = m

(

1 − αSπ

ln

[

p2

m2

])

, (1.445)

which is ∝ m. This result persists: at every order in perturbation theory everymass-like correction to S(p) is ∝ m so that m is apparently the only source ofchiral symmetry breaking and 〈qq〉 ∝ m → 0 as m → 0. The current-quarkmasses are the only explicit chiral symmetry breaking terms in QCD.

However, symmetries can be “dynamically” broken. Consider a point-particlein a rotationally invariant potential V (σ, π) = (σ2 +π2−1)2, where (σ, π) are theparticle’s coordinates. In the state depicted in Fig. 1.1, the particle is stationaryat an extremum of the action that is rotationally invariant but unstable. In theground state of the system, the particle is stationary at any point (σ, π) in thetrough of the potential, for which σ2 +π2 = 1. There are an uncountable infinityof such vacua, |θ〉, which are related one to another by rotations in the (σ, π)-plane. The vacua are degenerate but not rotationally invariant and hence, ingeneral, 〈θ|σ|θ〉 6= 0. In this case the rotational invariance of the Hamiltonian isnot exhibited in any single ground state: the symmetry is dynamically brokenwith interactions being responsible for 〈θ|σ|θ〉 6= 0.

-1

-0.5

0

0.5

1-1

-0.5

0

0.5

1

0

0.5

1

1.5

-1

-0.5

0

0.5

1

•

Figure 1.1: A rotationally invariant but unstable extremum of the Hamiltonianobtained with the potential V (σ, π) = (σ2 + π2 − 1)2.

1.12.3 Mass Where There Was None

The analogue in QCD is 〈qq〉 6= 0 when m = 0. At any finite order in perturbationtheory that is impossible. However, using the Dyson-Schwinger equation [DSE]for the quark self energy [the QCD “gap equation”]:

iγ · pA(p2) +B(p2) = Z2 iγ · p + Z4m

+Z1

∫ Λ d4`

(2π)4g2Dµν(p− `) γµ

λa

2

1

iγ · `A(`2) +B(`2)Γaν(`, p) , (1.446)

depicted in Fig. 1.2, it is possible to sum infinitely many contributions. Thatallows one to expose effects in QCD which are inaccessible in perturbation the-ory. [NB. In Eq. (1.446), m is the Λ-dependent current-quark bare mass and

∫ Λ

represents mnemonically a translationally-invariant regularisation of the integral,with Λ the regularisation mass-scale. The final stage of any calculation is to re-move the regularisation by taking the limit Λ → ∞. The quark-gluon-vertex andquark wave function renormalisation constants, Z1(ζ

2,Λ2) and Z2(ζ2,Λ2), dependon the renormalisation point, ζ, and the regularisation mass-scale, as does themass renormalisation constant Zm(ζ2,Λ2) := Z2(ζ

2,Λ2)−1Z4(ζ2,Λ2).]

The quark DSE is a nonlinear integral equation for A and B, and it is thenonlinearity that makes possible a generation of nonperturbative effects. The

Σ=

D

γΓS

Figure 1.2: DSE for the dressed-quark self-energy. The kernel of this equationis constructed from the dressed-gluon propagator (D - spring) and the dressed-quark-gluon vertex (Γ - open circle). One of the vertices is bare (labelled by γ)as required to avoid over-counting.

kernel of the equation is composed of the dressed-gluon propagator:

g2Dµν(k) =

(

δµν −kµkνk2

)

G(k2)

k2, G(k2) :=

g2

[1 + Π(k2)], (1.447)

where Π(k2) is the vacuum polarisation, which contains all the dynamical infor-mation about gluon propagation, and the dressed-quark-gluon vertex: Γaµ(k, p).The bare (undressed) vertex is

Γaµ(k, p)bare = γµλa

2. (1.448)

Once Dµν and Γaµ are known, Eq. (1.446) is straightforward to solve by iteration.One chooses an initial seed for the solution functions: 0A and 0B, and evaluatesthe integral on the right-hand-side (r.h.s.). The bare propagator values: 0A = 1and 0B = m are often adequate. This first iteration yields new functions: 1A and

1B, which are reintroduced on the r.h.s. to yield 2A and 2B, etc. The procedureis repeated until nA = n+1A and nB = n+1B to the desired accuracy.

It is now easy to illustrate DCSB, and I will use three simple examples.

Nambu–Jona-Lasinio Model

The Nambu–Jona-Lasinio model [17] has been popularised as a model of low-energy QCD. The commonly used gap equation is obtained from Eq. (1.446) viathe substitution

g2Dµν(p− `) → δµν1

m2G

θ(Λ2 − `2) (1.449)

in combination with Eq. (1.448). The step-function in Eq. (1.449) provides amomentum-space cutoff. That is necessary to define the model, which is not

renormalisable and hence can be regularised but not renormalised. Λ thereforepersists as a model-parameter, an external mass-scale that cannot be eliminated.

The gap equation is

iγ · pA(p2) +B(p2)

= iγ · p+m +4

3

1

m2G

∫

d4`

(2π)4θ(Λ2 − `2) γµ

−iγ · `A(`2) +B(`2)

`2A2(`2) +B2(`2)γµ ,(1.450)

where I have set the renormalisation constants equal to one in order to completethe defintion of the model. Multiplying Eq. (1.450) by (−iγ · p) and tracing overDirac indices one obtains

p2A(p2) = p2 +8

3

1

m2G

∫

d4`

(2π)4θ(Λ2 − `2) p · ` A(`2)

`2A2(`2) +B2(`2), (1.451)

from which it is immediately apparent that

A(p2) ≡ 1 . (1.452)

This property owes itself to the the fact that the NJL model is defined by a four-fermion contact interaction in configuration space, which entails the momentum-independence of the interaction in momentum space.

Simply tracing over Dirac indices and using Eq. (1.452) one obtains

B(p2) = m+16

3

1

m2G

∫ d4`

(2π)4θ(Λ2 − `2)

B(`2)

`2 +B2(`2), (1.453)

from which it is plain that B(p2) = constant = M is the only solution. This, too,is a result of the momentum-independence of the model’s interaction. Evaluatingthe angular integrals, Eq. (1.453) becomes

M = m+1

3π2

1

m2G

∫ Λ2

0dx x

M

x +M2= m+M

1

3π2

1

m2G

C(M2,Λ2) ,(1.454)

C(M2,Λ2) = Λ2 −M2 ln[

1 + Λ2/M2]

(1.455)

Λ defines the model’s mass-scale and I will henceforth set it equal to one sothat all other dimensioned quantities are given in units of this scale, in whichcase the gap equation can be written

M = m+M1

3π2

1

m2G

C(M2, 1) . (1.456)

Irrespective of the value of mG, this equation always admits a solution M 6= 0when the current-quark mass m 6= 0.

Consider now the chiral-limit, m = 0, wherein the gap equation is

M = M1

3π2

1

m2G

C(M2, 1) . (1.457)

This equation admits a solution M ≡ 0, which corresponds to the perturbativecase considered above: when the bare mass of the fermion is zero in the beginningthen no mass is generated via interactions. It follows from Eq. (1.443) that thecondensate is also zero. This situation can be described as that of a theorywithout a mass gap: the negative energy Dirac sea is populated all the way upto E = 0.

Suppose however that M 6= 0 in Eq. (1.457), then the equation becomes

1 =1

3π2

1

m2G

C(M2, 1) . (1.458)

It is easy to see that C(M 2, 1) is a monotonically decreasing function of M witha maximum value at M = 0: C(0, 1) = 1. Consequently Eq. (1.458) has a M 6= 0solution if, and only if,

1

3π2

1

m2G

> 1 ; (1.459)

i.e., if and only if

m2G <

Λ2

3π2' (0.2GeV )2 (1.460)

for a typical value of Λ ∼ 1 GeV. Thus, even when the bare mass is zero, the NJLmodel admits a dynamically generated mass for the fermion when the couplingexceeds a given minimum value, which is called the critical coupling: the chiralsymmetry is dynamically broken! (In this presentation the critical coupling isexpressed via a maximum value of the dynamical gluon mass, mG.) At thesestrong couplings the theory exhibits a nonperturbatively generated gap: the ini-tially massless fermions and antifermions become massive via interaction withtheir own “gluon” field. Now the negative energy Dirac sea is only filled up toE = −M , with the positive energy states beginning at E = +M ; i.e., the theoryhas a dynamically-generated, nonperturbative mass-gap ∆ = 2M . In additionthe quark condensate, which was zero when evaluated perturbatively becausem = 0, is now nonzero.

Importantly, the nature of the solution of Eq. (1.456) also changes qualita-tively when mG is allowed to fall below it’s critical value. It is in this way thatdynamical chiral symmetry breaking (DCSB) continues to affect the hadronicspectrum even when the quarks have a small but nonzero current-mass.

Munczek-Nemirovsky Model

The gap equation for a model proposed more recently [18], which is able torepresent a greater variety of the features of QCD while retaining much of thesimplicity of the NJL model, is obtained from Eq. (1.446) by using

G(k2)

k2= (2π)4Gδ4(k) (1.461)

in Eq. (1.447) with the bare vertex, Eq. (1.448). Here G defines the model’smass-scale.

The gap equation is

iγ · pA(p2) +B(p2) = iγ · p+m +Gγµ−iγ · pA(p2) +B(p2)

p2A2(p2) +B2(p2)γµ ,(1.462)

where again the renormalisation constants have been set equal to one but in thiscase because the model is ultraviolet finite; i.e., there are no infinities that mustbe regularised and subtracted. The gap equation yields the following two coupledequations:

A(p2) = 1 + 2A(p2)

p2A2(p2) +B2(p2)(1.463)

B(p2) = m + 4B(p2)

p2A2(p2) + B2(p2), (1.464)

where I have set the mass-scale G = 1.Consider the chiral limit equation for B(p2):

B(p2) = 4B(p2)

p2A2(p2) +B2(p2). (1.465)

The existence of a B 6≡ 0 solution; i.e., a solution that dynamically breaks chiralsymmetry, requires

p2A2(p2) +B2(p2) = 4 , (1.466)

measured in units of G. Substituting this identity into equation Eq. (1.463) onefinds

A(p2) − 1 =1

2A(p2) ⇒ A(p2) ≡ 2 , (1.467)

which in turn entailsB(p2) = 2

√

1 − p2 . (1.468)

The physical requirement that the quark self energy be real in the spacelikeregion means that this solution is only acceptable for p2 ≤ 1. For p2 > 1 onemust choose the B ≡ 0 solution of Eq. (1.465), and on this domain we then findfrom Eq. (1.463) that

A(p2) = 1 +2

p2A(p2)⇒ A(p2) =

1

2

(

1 +√

1 + 8/p2

)

. (1.469)

Putting this all together, the Munczek-Nemirovsky model exhibits the dynamicalchiral symmetry breaking solution:

A(p2) =

2 ; p2 ≤ 11

2

(

1 +√

1 + 8/p2)

; p2 > 1(1.470)

B(p2) =

√1 − p2 ; p2 ≤ 1

0 ; p2 > 1 .(1.471)

which yields a nonzero quark condensate. Note that the dressed-quark self-energyis momentum dependent, as is the case in QCD.

It is important to observe that this solution is continuous and defined for allp2, even p2 < 0 which corresponds to timelike momenta, and furthermore

p2A2(p2) +B2(p2) > 0 , ∀ p2 . (1.472)

This last fact means that the quark described by this model is confined: the prop-agator does not exhibit a mass pole! I also note that there is no critical couplingin this model; i.e., the nontrivial solution for B always exists. This exemplifies acontemporary conjecture that theories with confinement always exhibit DCSB.

In the chirally asymmetric case the gap equation yields

A(p2) =2B(p2)

m+B(p2), (1.473)

B(p2) = m+4 [m+B(p2)]2

B(p2)([m +B(p2)]2 + 4p2). (1.474)

The second is a quartic equation for B(p2) that can be solved algebraically withfour solutions, available in a closed form, of which only one has the correct p2 →∞ limit: B(p2) → m. Note that the equations and their solutions always have asmooth m→ 0 limit, a result owing to the persistence of the DCSB solution.

Renormalization-Group-Improved Model

Finally I have used the bare vertex, Eq. (1.448) and G(Q) depicted in Fig. 1.3, insolving the quark DSE in the chiral limit. If G(Q = 0) < 1 then B(p2) ≡ 0 is theonly solution. However, when G(Q = 0) ≥ 1 the equation admits an energeticallyfavoured B(p2) 6≡ 0 solution; i.e., if the coupling is large enough then even in theabsence of a current-quark mass, contrary to Eq. (1.445), the quark acquires amass dynamically and hence

〈qq〉 ∝∫

d4p

(2π)4

B(p2)

p2A(p2)2 +B(p2)26= 0 for m = 0 . (1.475)

These examples identify a mechanism for DCSB in quantum field theory. Thenonzero condensate provides a new, dynamically generated mass-scale and if itsmagnitude is large enough [−〈qq〉1/3 need only be one order-of-magnitude largerthan mu ∼ md] it can explain the mass splitting between parity partners, andmany other surprising phenomena in QCD. The models illustrate that DCSB islinked to the long-range behaviour of the fermion-fermion interaction. The sameis true of confinement.

The question is then: How does the quark-quark interaction behave at largedistances in QCD? It remains unanswered.

0 1 10 100Q (GeV)

0.0

0.2

0.4

0.6

0.8

1.0

G(Q)

Figure 1.3: Illustrative forms of G(Q): the behaviour of each agrees with per-turbation theory for Q > 1 GeV. Three possibilities are canvassed in Sec.1.12.3:G(Q = 0) < 1; G(Q = 0) = 1; and G(Q = 0) > 1.

Bibliography

[1] J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill,New York, 1964).

[2] B.D. Keister and W.N. Polyzou, “Relativistic Hamiltonian dynamics in nu-clear and particle physics,” Adv. Nucl. Phys. 20, 225 (1991).

[3] C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, NewYork, 1980).

[4] C.D. Roberts and A.G. Williams, “Dyson-Schwinger Equations and theirApplication to Hadronic Physics,” Prog. Part. Nucl. Phys. 33, 477 (1994).

[5] P. Pascual and R. Tarrach, Lecture Notes in Physics, Vol. 194, QCD: Renor-

malization for the Practitioner (Springer-Verlag, Berlin, 1984).

[6] J.S. Schwinger, “On The Green’s Functions Of Quantized Fields: 1 and 2,”Proc. Nat. Acad. Sci. 37 (1951) 452; ibid 455.

[7] F.A. Berezin, The Method of Second Quantization (Academic Press, NewYork, 1966).

[8] L.D. Faddeev and V.N. Popov, “Feynman Diagrams For The Yang-MillsField,” Phys. Lett. B 25 (1967) 29.

[9] V.N. Gribov, “Quantization Of Nonabelian Gauge Theories,” Nucl. Phys. B139 (1978) 1.

[10] F.J. Dyson, “The S Matrix In Quantum Electrodynamics,” Phys. Rev. 75(1949) 1736.

[11] K. Symanzik, “Infrared Singularities And Small Distance Behavior Analy-sis,” Commun. Math. Phys. 34 (1973) 7; T. Appelquist and J. Carazzone,“Infrared Singularities And Massive Fields,” Phys. Rev. D 11 (1975) 2856.

[12] K. Symanzik in Local Quantum Theory (Academic, New York, 1969) editedby R. Jost.

82

[13] R.F. Streater and A.S. Wightman, A.S., PCT, Spin and Statistics, and AllThat, 3rd edition (Addison-Wesley, Reading, Mass, 1980).

[14] J. Glimm and A. Jaffee, Quantum Physics. A Functional Point of View(Springer-Verlag, New York, 1981).

[15] E. Seiler, Gauge Theories as a Problem of Constructive Quantum Theoryand Statistical Mechanics (Springer-Verlag, New York, 1982).

[16] D.J. Gross, “Applications Of The Renormalization Group To High-EnergyPhysics,” in Proc. of Les Houches 1975, Methods In Field Theory (NorthHolland, Amsterdam, 1976) pp. 141-250.

[17] Y. Nambu and G. Jona-Lasinio, “Dynamical Model Of Elementary ParticlesBased On An Analogy With Superconductivity. I,II” Phys. Rev. 122 (1961)345, 246

[18] H.J. Munczek and A.M. Nemirovsky, Phys. Rev. D 28 (1983) 181.

Chapter 2

Quantum Fields at FiniteTemperature and Density

2.1 Ensembles and Partition Function

In equilibrium statistical mechanice, one normally encounters three types of en-semble. The microcanonical ensemble is used to describe an isolated systemwhich has a fixed energy E, a fixed particle number N , and a fixed volume V .The canonical ensemble is used to describe a system in contact with a heat reser-voir at temperature T . The system can freely exchange energy with the reservoir.Thus T , N , and V are fixed. In the grand canonical ensemble, the system canexchange particles as well as energy with a reservoir. In this ensemble, T , V , andthe chemical potential µ are fixed variables.

In the latter two ensembles T−1 = β may be thought of as a Lagrange mul-tiplier which determines the mean energy of the system. Similarly, µ may bethought of as a Lagrange multiplier which determines the mean number of par-ticles in the system. In a relativistic quantum system where particles can becreated and destroyed, it is most straightforward to compute observables in thegrand canonical ensemble. So we therefore use this ensemble. This is withoutloss of generality since one can pass over to either of the other ensembles byperforming a Laplace transform on the variable µ and/or the variable β.

Consider a system described by a Hamiltonian H and a set of conservednumber operators Ni. In relativistic QED, for example, the number of electronsminus the number of positrons is a conserved quantity, not the electron number orthe positron number separately. These number operators must be Hermitian andmust commute with H as well as with each other. Also, the number operatorsmust be extensive in order that the usual macroscopic thermodynamic limit canbe taken. The statistical density matrix is

ρ = exp[

−β(H − µiNi)]

, (2.1)

84

where a summation over i is implied. The ensemble average of an operator A is

A =TrρA

Trρ. (2.2)

The grand canonical partition function is

Z = Trρ . (2.3)

The function Z = Z(T, V, µ1, µ2, . . .) is the single most important function inthermodynamics. From it all other standard thermodynamic properties may bedetermined. For example the equations of state for, e.g., the pressure P , theparticle number N , the entropy S, and the energy E are, in the infinite volumelimit,

P = T∂ lnZ

∂V, (2.4)

Ni = T∂ lnZ

∂µi, (2.5)

P =∂(T lnZ)

∂T, (2.6)

E = −PV + TS + µiNi . (2.7)

Note that the notion of ensembles as introduced here for the situation in thermo-dynamical equilibrium can be extended to the nonequilibrium situation, where ageneralized Gibbs ensemble can be introduced for systems which are in a nonequi-librium situation that can be characterized by further observables such as currentsor reaction variables. These additional observables shall be accounted for by anenlarged set of Lagrange multipliers thus arriving at a statistical operator of thenonequilibrium state, also called relevant statistical operator within the Zubarevformalism.

2.1.1 Partition function in Quantum Statistics and Quan-

tum Field Theory

In order to calculate the partition using methods of quantum field theory, werecall that in quantum statistics

Z = Tr e−β(H−µiNi) =∫

dφa〈φa|e−β(H−µiNi)|φa〉 , (2.8)

where the sum runs over all (eigen-)states. This has an appearance very similarto the transition amplitude (time evolution operator) in Quantum Field Theorywhne one switches to an imaginary time variable τ = i t and limit the integrationover τ to the region between 0 and β. The trace operation means that we have to

integrate over all fields φa. Finally, if the system admits some conserved charge,then we must make the replacement

H(π, φ) → K(pi, φ) = H(π, φ) − µN (π, φ) , (2.9)

where N (π, φ) is the conserved charge density.In fact, we can express the partition function Z as a functional integral over

fields and their conjugate momenta. This fundamental formula reads

Z =∫

Dπ∫

periodicDφ exp

∫ β

0

∫

d3x

(

i π∂φ

∂τ−H(π, φ) + µN (π, φ)

)

. (2.10)

The term “periodic” means that the integration over the field is constrained sothat φ(~x, 0) = φ(~x, β). This is a consequence of the trace operation, settingφa(~x) = φ(~x, 0) = φ(~x, β). There is no restriction on the π integration. Thegeneralization of (2.10) to an arbitrary number of fields and conserved charges isobvious.

In the following subsection we show the equivalence of the expressions (2.10) and (2.8) for the partition function. The key lesson is that the quantization inthe Path integral representation is provided by the integration over all alternativeclassical field configurations (under given constraints) whereas in the statisticaloperator representation the notion of field operators has to be introduced.

2.1.2 Equivalence of Path Integral and Statistical Opera-

tor representation for the Partition function

Be φ(~x, 0) a field operator in the Schrodinger picture at time t = 0 and π(~x, 0) thecorresponding canonically conjugated field momentum operator. For eigenstates| φ〉 of the field holds the eigenvalue equation

φ(~x, 0) | φ〉 = φ(~x) | φ〉 , (2.11)

where φ(~x) is the “eigenvalue” corresponding to the field operator. For the eigen-states of the fields completeness and orthonormality shall hold

∫

dφ(~x) | φ〉〈φ | = 1 (2.12)

〈φa | φb〉 = δ [φa(~x) − φb(~x)] . (2.13)

For the field momentum operator and its eigenstates | π〉 holds analogously

π(~x, 0) | π〉 = π(~x) | π〉 (2.14)∫

dπ(~x)

2π| π〉〈π | = 1 (2.15)

〈πa | πb〉 = δ [πa(~x) − πb(~x)] . (2.16)

The transition amplitude between coordinate and momentum eigenstates in quan-tum mechanics is (h = 1)

〈x | p〉 = eipx . (2.17)

The generalization to the quantum field theory case follows by going over to aninfinite number of degrees of freedom

∑

pixi →∫

d3x π(~x)φ(~x) thus arriving at

〈φ | π〉 = exp[

i∫

d3x π(~x)φ(~x)]

. (2.18)

For a dynamical description of the system we require the Hamiltonian operator

H =∫

d3xH(

∧π, ∧φ)

. (2.19)

Consider the state | φa〉 at t = 0. At a later time tf this state has evolved to

e−Htf | φa〉. The transition amplitude of the state | φa〉 to the state | φb〉 at time

tf is therefore given by 〈φb | e−Htf | φa〉.In order to express the quantum statistical partition function, we are inter-

ested in the case that the system returns at t = tf to the initial state at t = 0.The sign of the state is not an observable and is left undetermined at this stage.

e−iHtf | φa〉 → ± | φa〉 (2.20)

In order to evaluate the transition amplitude the time interval (0, tf) is decom-posed into equidistant parts of the length 4t = tf/N . At each time step weintroduce a complete set of field and field-momentum states

〈φa | e−iHtf | φa〉 = limN→∞

∫

(

N∏

i=1

dπidφi2π

)

×〈φa | πN〉〈πN | e−iH4t | φN〉〈φN | πN−1〉×〈πN−1 | e−iH4t | φN−1〉 × . . .

×〈φ2 | π1〉〈π1 | e−iH4t | φ1〉〈φ1 | φa〉 (2.21)

We make use of the following expressions

〈φ1 | φa〉 = δ (φ1 − φa) (2.22)

〈φi+1 | πi〉 = exp[

i∫

d3x πi(~x)φi+1(~x)]

. (2.23)

For 4t→ 0 the exponential function can be expanded

〈πi | e−iH4t | φi〉 ' 〈πi | (1 − H4t) | φi〉= 〈πi | φi〉(1 −Hi4t)= (1 −Hi4t) exp

[

i∫

d3x πi(~x)φi(~x)]

, (2.24)

whereHi =

∫

d3xH(

π〉(~§), φ〉(~§))

. (2.25)

Taken all expressions together yields

〈φa | e−iHtf | φa〉 = limN→∞

∫

(

N∏

i=1

dπidφi2π

)

δ (φ1 − φa)

× exp−i4tN∑

j=1

∫

d3x[H(π|, φ|) −

−πjφj+1 − φj

4t ] (2.26)

Here holds φN+1 = φa = φ1. In the continuum limit we obtain

〈φa | e−iHtf | φa〉 =∫

Dπφ(~x,tf )=±φa

∫

φ(~x,0)=φa

Dφ exp

i

tf∫

0

dt∫

d3~x

(

π∂φ

∂t−H(φ, π)

)

=∫

Dπφ(~x,tf )=±φa

∫

φ(~x,0)=φa

Dφ exp

i

tf∫

0

dt∫

d3~xL(φ, π)

(2.27)

The notations Dπ and Dφ stand for the Functional Integration over fields andtheir conjugate momenta.

The partition function of a quantum statistical system is defined as

Z = Tre−β(H−µiNi)=

∫

dφa〈φa | e−β(H−µiNi) | φa〉 (2.28)

The expression under the integral shows formal equivalence to the time evolutionoperator (2.27) except for the factor i in the exponent. The formal equivalence canbe made still closer by introducing the imaginary time τ = it. Ni is an operatorcorresponding to conserved charges in the system, such as baryon number. Theseconservation laws can be incorporated into the formalism as constraints by themethod of Lagrangian multipliers. This is done by the replacement

H(π, φ) → K(π, φ) = H(π, φ) − µ〉N〉(π, φ) (2.29)

The result for the partition function reads

Z =∫

[ dπ]∫

±

[dφ] exp

β∫

0

dτ∫

d3x

(

iπ∂φ

∂τ−H(π, φ) + µ〉N〉(π, φ)

)

(2.30)

The index ± stands for the symmetry of the fields at the borders of the imaginarytime interval which is determined up to a phase factor φ(~x, 0) = ±φ(~x, β), wherethe upper sign stands for Bosons, while the lower is for Fermions.

The next step is the transition to the Fourier representation

φ(x, τ) =

(

β

V

) 1

2 ∞∑

n=−∞

∑

p

ei(p·x+ωnτ)φn(p) (2.31)

which allows to get rid of differential operators in the action functional andtransform it to an algebraic expression of momenta and frequencies. Due tothe (anti)periodicity on the imaginary time interval the conjugate frequenciesbecome discrete. For bosonic fields φ(~x, 0) = φ(~x, β) is fulfilled for

ωn = 2nπT (2.32)

The ωn are denoted as Matsubara-Frequencies. For fermionic fields a similarargument leads to the fermionic Matsubara frequencies ωn = (2n+ 1)πT .

2.1.3 Bosonic Fields

Neutral Scalar Field

The most general renormalizable Lagrangian for a neutral scalar field is

L =1

2∂µφ∂

µφ− 1

2m2φ2 − U(φ) , (2.33)

where the potential isU(φ) = gφ3 + λφ4 , (2.34)

and λ ≥ 0 for stability of the vacuum. The momentum conjugate to the field is

π =∂L

∂(∂0φ)=∂φ

∂t, (2.35)

and the Hamiltonian is

H = π∂φ

∂t− L =

1

2π2 +

1

2(∇φ)2 +

1

2m2φ2 + U(φ) . (2.36)

There is no conserved charge.The first step in evaluating the partition function is to return to the discretized

version

Z = limN→∞

(

ΠNi=1

∫ ∞

−∞

dπi2π

∫

periodicdφi

)

(2.37)

exp

N∑

j=1

∫

d3x[

iπj(φj+1 − φj) − ∆τ(

1

2π2j +

1

2(∇φj)2 +

1

2m2φ2

j + U(φj))]

.

The momentum integrations can be done immediately since they are just productsof Gaussian integrals. Divide position space into M 3 little cubes with V = L3,L = aM , a→ 0, M → ∞, M an integer.

For convenience, and to ensure that Z remains explicitely dimensionless ateach step in the calculation, we write πj = Aj/(a

3 ∆τ)1/2 and integrate Aj from−∞ to +∞. For each cube we obtain

∫ ∞

−∞

dAj2π

exp

−1

2A2j + i

(

a3

∆τ

)1/2

(φj+1 − φj)Aj

= (2π)−1/2 exp

[

−a3(φj+1 − φj)2

2∆τ

]

. (2.38)

Thus far we have

Z = limM,N→∞

(2π)−M3N/2

∫

[

ΠNi=1dφi

]

exp

∆τN∑

j=1

∫

d3x

−1

2

(

(φj+1 − φj)

∆τ

)2

− 1

2(∇φj)2 − 1

2m2φ2

j − U(φj)

.(2.39)

Returning to the continuum limit, we obtain

Z = N ′∫

periodicDφ exp

(

∫ β

0dτ∫

d3xL)

. (2.40)

The Lagrangian is expressed as a functional of φ and its first derivatives. Theformula (2.39) expresses Z as a functional integral over φ of the exponential ofthe action in imaginary time. The normalization constant is irrelevant, sincemultiplication of Z by any constant does not change the thermodynamics.

Next, we turn to the case of noninteracting fields with U(φ) = 0. Interactionswill be discussed separately. Define

S =∫ β

0dτ∫

d3xL = −1

2

∫ β

0dτ∫

d3x

(

∂φ

∂τ

)2

+ (∇φ)2 +m2φ2

. (2.41)

Integrating by parts and taking note of the periodicity of φ, we obtain

S = −1

2

∫ β

0dτ∫

d3xφ

(

− ∂2

∂τ 2−∇2 +m2

)

φ . (2.42)

The field can be decomposed into a Fourier series according to

φ(~x, τ) =

(

β

V

)1/2 ∞∑

n=−∞

∑

~p

ei(~p~x+ωnτ)φn(~p) , (2.43)

where ωn = 2πnT , due to the constraint of periodicity that φ(~x, β) = φ(~x, 0) forall ~x. The normalization of (2.43) is chosen conveniently so that each Fourieramplitude is dimensionless. Substituting (2.43) into (2.42), and noting thatφ−n(−~p) = φ∗n(~p) as required by the reality of φn(~p), we find

S = −1

2β2∑

n

∑

~p

(ω2n + ω2)φn(~p)φ∗n(~p) (2.44)

with ω =√~p2 +m2. The integrand depends only on the amplitude of φ and not

its phase. The phases can be integrated out to get

Z = N ′ΠnΠ~p

[∫ ∞

−∞dAn(~p) exp

[

−1

2β2(ω2

n + ω2)A2n(~p)

]]

= N ′ΠnΠ~p

[

2π/(β2(ω2n + ω2))

]1/2. (2.45)

Ignoring an overall multiplicative factor independent of β and V , which does notaffect the thermodynamics, we arrive at

Z = ΠnΠ~p

[

β2(ω2n + ω2)

]−1/2. (2.46)

More formally one can arrive at this result by using the general rules for Gaussianfunctional integrals over commuting (bosonic) variables, derived before in theQFT chapter, since (2.40) and (2.42) can be expressed as

Z = N ′∫

Dφ exp[

−1

2(φ,Dφ)

]

= N ′constant(det D)−1/2 , (2.47)

where D = β2(ω2n + ω2) in (~p, ωn) space and (φ,Dφ) denotes the inner product

on the function space.Thus far we have

lnZ = −1

2

∑

n

∑

~p

ln[

β2(ω2n + ω2)

]

. (2.48)

Note that the sum over n is divergent. This unfortunate feature stems fromour careless handling of the integration measure Dφ. A more rigorous treatmentusing the proper definition of Dφ gives a finite result. In order to handle (2.48),we make use of

ln[

(2πn)2 + β2ω2)]

=∫ β2ω2

1

dΘ2

Θ2 + (2πn)2+ ln

[

1 + (2πn)2]

, (2.49)

where the last term is β− independent and thus can be ignored. Furthermore,

∞∑

−∞

1

n2 + (Θ/2π)2=

2π2

Θ

(

1 +2

eΘ − 1

)

, (2.50)

hence

lnZ = −∑

~p

∫ βω

1dΘ

(

1

2+

1

eΘ − 1

)

. (2.51)

Carrying out the Θ integral, and throwing away a β− independent piece, wefinally arrive at

lnZ = V∫

d3p

(2π)3

[

−1

2βω − ln(1 − e−βω)

]

, (2.52)

from which we obtain immediately the well-known expression for the ideal Bosegas (µ = 0), once we subtract the divergent expressions for the zero-point energy

E0 = − ∂

∂βlnZ0 = V

∫

d3p

(2π)3

ω

2, (2.53)

and for the zero-point pressure

P0 = T∂

∂VlnZ0 = −E0

V, (2.54)

which are typical for the quantum field-theoretical treatment. With this subtrac-tion the vacuum is defined as the state with zero energy and pressure.

2.1.4 Fermionic Fields

Dirac fermions are descibed by a four-spinor field ψ with a Lagrangian density

L = ψ(i∂/−m)ψ

= ψ†γ0

(

iγ0 ∂

∂t+ i~γ · ~∇−m

)

ψ . (2.55)

The momentum conjugate to this field is

Π =∂L

∂(∂ψ/∂t)= iψ† , (2.56)

because γ0γ0 = 1. Thus, somewhat paradoxically, ψ and ψ† must be treated asindependent entities in the Hamiltonian formulation. The Hamiltonian densityis found by standard procedures,

H = Π∂ψ

∂t− L = ψ†

(

i∂

∂t

)

ψ − L = ψ(−i~γ · ~∇ +m)ψ , (2.57)

and the partition function is

Z = Tr e−β(H−µQ) , (2.58)

with the conserved charge Q =∫

d3xψ†ψ. The functional integral representationreads

Z =∫

Dψ†Dψ exp

[

∫ β

0dτ∫

d3xψ†(

−γ0 ∂

∂τ+ i~γ · ~∇−m+ µγ0

)

ψ

]

(2.59)

As with bosons, it is most convenient to work in (~p, ωn) space instead of (~x, τ)space, i.e.,

ψα(~x, τ) =

(

β

V

)1/2 ∞∑

n=−∞

∑

~p

ei(~p~x+ωnτ)ψα;n(~p) , (2.60)

where now ωn = (2n + 1)πT due to the antiperiodicity of the (Grassmannian)Fermion field at the borders of the fundamental strip 0 ≤ τ ≤ β in the imaginarytime, ψ(~x, 0) = −ψ(~x, β).

Now we are ready to evaluate the fermionic partition function (2.59),

Z =[

ΠnΠ~pΠα

∫

idψ†α;n(~p)dψα;n(~p)]

eS ,

S =∑

n

∑

~p

iψ†α;n(~p)Dαρψρ;n(~p) ,

D = −iβ[

(−iωn + µ) − γ0~γ · ~p−mγ0]

, (2.61)

using our knowledge about Grassmannian integration of Gaussian functional in-tegrals, resulting in

Z = det D . (2.62)

Employing the identityln detD = Tr lnD , (2.63)

and evaluating the determinant in Dirac space explicitly (Exercise !), one finds

lnZ = 2∑

n

∑

~p

ln

β2[

(ωn + iµ)2 + ω2]

. (2.64)

Since both positive and negative frequencies have to be summed over, the latterexpression can be put in a form analogous to the above expression in the bosoniccase,

lnZ =∑

n

∑

~p

ln[

β2(

ω2n + (ω − µ)2

)]

+ ln[

β2(

ω2n + (ω + µ)2

)]

. (2.65)

In the further evaluation we can go similar steps as in the bosonic case, with twoexceptions: (1) the presence of a chemical potential, splitting the contributions ofparticles and antiparticles; (2) the Matsubara frequencies are now odd multiplesof πT , so that the infinite sum to be exploited reads

∞∑

n=−∞

1

(2n + 1)2π2 + Θ2=

1

Θ

(

1

2− 1

eΘ + 1

)

. (2.66)

Integrating over the auxiliary variable Θ, and dropping terms independent of βand µ, we finally obtain

lnZ = 2V∫

d3p

(2π)3

[

βω + ln(1 + e−β(ω−µ)) + ln(1 + e−β(ω+µ))]

. (2.67)

Notice that the factor 2 corresponding to the spin- 12

nature of the fermions comesout automatically. Separate contributions from particles (µ) and antiparticles(-µ) are evident. Finally, the zero-point energy of the vacuum also appears inthis formula.

2.1.5 Gauge Fields

Quantizing the Electromagnetic Field

In this chapter we would like to understand how the Faddeev-Popov ghosts asintroduced to eliminate divergences of the gauge freedom will act in the puregauge theory to eliminate unphysical degrees of freedom and allow to derive theblackbody radiation law with the correct number of physical degrees of freedom.

We start with recalling the path integral formulation of QED for the photonfield Aµ(x) with the field strength tensor

Fµν = ∂µAν − ∂νAµ (2.68)

and the free action functional

S = −1

4

∫

d4xFµνFµν . (2.69)

This action is invariant under gauge transformations

Aµ(x) → Aµ(x) = A′µ(x) + ∂µω(x) , (2.70)

where ω(x) is a scalar function which parametrizes the gauge transformations.The momenta conjugate to the space components of Ai(x) are, up to a sign, thecomponents Ei(x) = Ei(x) of the electric field

πi = −Ei = −F0i , (2.71)

while the magnetic field B(x) is

Bi = εijk∂jAk . (2.72)

We work in an axial gauge, A3 = 0 to be specific. The momenta π1 and π2 areindependent variables; E3 is not an independnet variable, but it is a function of π1

and π2, which may be computed from Gauss’s law ∇·E = 0. There are thus twodynamical variables A1 and A2 with conjugate momenta π1 and π2. We defineπ3 = −E3(π1, π2), but π3 is not to be interpreted as a conjugate momentum. Thepartition function is written as a Hamiltonian path integral

Z =∫

D(π1, π2)∫

Ai(0)=Ai(β)D(A1, A2) exp

[

∫ β

0d4x(iπ1∂τA

1 + iπ2∂τA2 −H)

]

,(2.73)

where we have used the notation∫ β

0d4x =

∫ β

0dτ∫

d3x , (2.74)

while the Hamiltonian density H is

H =1

2(E2 + B2) =

1

2

(

π21 + π2

2 + E23(π1, π2) + B2

)

(2.75)

Equation (2.73) is then transformed by using

1 =∫

Dπ3δ(π3 + E3(π1, π2)) , (2.76)

and

δ(π3 + E3(π1, π2)) = δ(∇ · π) det

(

∂(∇ · π)

∂π3

)

= det[

∂3δ3(x − y)

]

δ(∇ · π) . (2.77)

In the following step, one inserts an integral representation of δ(∇ · π)

δ(∇ · π) =∫

DA4 exp

[

i∫ β

0d4xA4(∇ · π)

]

, (2.78)

where A4 = iA0, and we work in Euclidean space now: xµ = (x,x4) = (x, τ),A4 = (A,A4). Performing the π− integration, we are left with

Z =∫

D(A1, A2, A4) det[

∂3δ3(x − y)

]

exp

[

∫ β

0d4x

(

1

2(i∂τA− i∇A4)2 − 1

2B2

)

]

,

(2.79)where A = (A1,A2, 0). Note that the argument of the exponential is

1

2E2 − 1

2B2 = L . (2.80)

The A− integration is rendered more aesthetic by inserting

1 =∫

DA3δ(A3) , (2.81)

and the partition function assumes the form

Z =∫

DAµδ(A3) det[

∂3δ3(x − y)

]

exp

(

∫ β

0d4xL

)

. (2.82)

The axial gauge A3 = 0 is not a particularly convenient gauge to use for practicalcomputations. Furthermore, it is not immediately apparent that (2.82) is a gaugeinvariant expression for Z.

Take an arbitrary gauge specified by F = 0, where F is some function of Aµ

and its derivatives. For the gauge above, F = A3. For this gauge, (2.82) is givenby

Z =∫

DAµδ(F ) det

(

∂F

∂α

)

exp

(

∫ β

0d4xL

)

. (2.83)

Equation (2.83) is manifestly gauge invariant: L is invariant, the gauge fixingfactor times the Jacobian of the transformation δ(F ) det(∂F/∂α) is invariant,and the integration is over all four components of the vector potential. Equation(2.83) reduces to (2.82) in the case of the axial gauge A3 = 0. We know this iscorrect since it was derived from first principles in the Hamiltonian formulation

of the gauge theory, Z = Tr e−βH

Blackbody radiation

It is important to verify that (2.83) describes blackbody radiation with two po-larization degrees of freedom. We will do this here in the axial gauge A3 = 0, theFeynman gauge is left as an exercise.

In the axial gauge, we rewrite (2.79) as

Z =∫

D(A0, A1, A2) det(∂3)eS0

S0 =1

2

∫

dτ∫

d3x(A0, A1, A2)

×

∇2 −∂1∂∂τ

−∂2∂∂τ

−∂1∂∂τ

∂22 + ∂2

3 + ∂2

∂τ2 ∂1∂2

−∂2∂∂τ

−∂1∂2 ∂21 + ∂2

3 + ∂2

∂τ2

A0

A1

A2

. (2.84)

We can express the determinant of ∂3 as a functional integral over a complexghost field C: that is, a Grassmann field with spin-0,

det(∂3) =∫

DCDC exp

(

∫ β

0dτ∫

d3xC∂3C

)

. (2.85)

These ghost fields C and C are not physical fields since they do not appearin the Hamiltonian. Furthermore, since they are anticommuting scalar fieldsthey violate the connection between spin and statistics. It is simply a convenientfunctional integral representation of the determinant of an operator. The greatestapplicability of these ficticious ghost fields will be to non-Abelian gauge theories,see also Sect. 5.2.1. of Le Bellac [2].

In frequency-momentum space the partition function is expressed as

lnZ = ln det(βp3) −1

2ln det(D) ,

D = β2

p2 −ωnp1 −ωnp2

−ωnp1 ω2n + p2

2 + p23 −p1p2

−ωnp1 −p1p2 ω2n + p2

1 + p23

.

Carrying out the determinantal operation,

lnZ =1

2Tr ln(β2p2

3) −1

2Tr ln

[

β6p23(ω

2n + p2)2

]

= ln(

ΠnΠp

[

β2(ω2n + p2)

]−1)

= 2V∫ d3p

(2π)3

[

−1

2βω − ln(1 − e−βω)

]

. (2.86)

Here, ω = |p|. Comparison with the result for the scalar field case shows that(2.86) describes massless bosons with two spin degrees of freedom in thermalequilibrium; in other words, blackbody radiation.

2.2 Interacting Fermion Systems:

Hubbard-Stratonovich Trick

So far we have dealt with free quantum fields in the absence of interactions andhave obtained nice closed expressions for the thermodynamic potential, i.e., there-fore also for the generating functionals of the thermodynamic Green functions.However, once we switch on the interactions in our model field theories, there isonly a very limited class of soluble models, in general we have to apply approx-imations. The most common technique is based on perturbation theory, whichrequires a small parameter. For strong interactions at low momentum transfer(the infrared region), the coupling is nonperturbatively strong and alternative,nonperturbative methods have to be invoked. One of the strategies, which is espe-cially suitable for the treatment of quantum field theories within the path integralformulation is based on the introduction of collective variables (auxialiary fields)by an exact integral transformation due to Stratonovich and Hubbard which al-lows to eliminate (integrate out) the elementary degrees of freedom. Generally,the (dual) coupling of the auxialiary fields is weak so that perturbative expansionsof the nonlinear effective action make sense and provide useful results already atlow orders of this expansion.

A general class of interactions for which the Hubbard-Stratonovich (HS) trans-formation is immediately applicable, are four-fermion couplings of the current-current type

Lint = G(ψψ)2 . (2.87)

A Fermi gas with this typ of interaction serves as a model for electronic supercon-ductivity (Bardeen-Cooper-Schrieffer (BCS) model, 1957) or for chiral symmetrybreaking in quark matter (Nambu–Jona-Lasinio (NJL) model, 1961).

The HS-transformation for (2.87) reads

exp[

G(ψψ)2]

= N∫

Dσ exp

[

σ2

4G+ ψψσ

]

(2.88)

and allows to bring the functional integral over fermionic fields into a quadratic(Gaussian) form so that fermions can be integrated out. This is also calledBosonization procedure.

2.2.1 Walecka Model

This notes for this subsection are provided separately.

2.2.2 Nambu–Jona-Lasinio (NJL) Model

Here we will present an application of the HS technique to the NJL model forquark matter at finite densities and temperatures. This is possible since the

interaction of this model is of the current-current form and therefore the HS trickfor the bosonization of 4-fermion interactions applies. The Lagrangian density isgiven by

L = qiα(i∂/δijδαβ −M0ijδαβ + µij,αβγ

0)qjβ

+ GS

8∑

a=0

[

(qτ af q)2 + (qiγ5τ

af q)

2]

+ GD

∑

k,γ

[

(qiαεijkεαβγqCjβ)(qCi′α′εi′j′kεα′β′γqj′β′)

+ (qiαiγ5εijkεαβγqCjβ)(qCi′α′iγ5εi′j′kεα′β′γqj′β′)

]

, (2.89)

where from here on the quark spinor is qiα, with the flavor index i = u, d, s andα = r, g, b stands for the color degree of freedom. M 0

ij = diag(m0u, m

0d, m

0s) is

the current quark mass matrix in flavor space and µij,αβ is the chemical potentialmatrix in color and flavor space. The grand cononical thermodynamical potentialΩ = T lnZ is known once we manage to evaluate the partition function in a rea-sonable approximation. A closed solution, even for the simple model Lagrangion(2.89), is not possible. The bosonization of the partition function proceeds asfollows

Z =∫

DqDq exp∫

d4xL

=∫

DqDqΠi=u,d,sDφiΠkγ=ur,dg,sbD∆kγD∆†kγ exp∫

d4xLHS (2.90)

whehre the effective ’linearized’ Lagrangian is given by

LHS = qiα[

i∂/δijδαβ − (M0ij − φiδij)δαβ + µij,αβγ

0]

qjβ

−∑

i=u,d,s

φ2i

8GS

−∑

kγ=ur,dg,sb

|∆kγ|24GD

+ qiα∆kγ

2qCjβ + qCiα

∆†kγ2qjβ, (2.91)

The fermionic quark spinor fields can be grouped into bi-spinors, formed by theoriginal and the charge-conjugated, transposed anti-spinor. This way, the ac-tion functional can be given the quadratic form which allows to integrate outthe elementary quark fields using the Gaussian Functional Integration rule, thusleaving us after this exact Hubbard-Stratonovich transformation with an alter-native representation of the partition function in terms of collective fields φi and∆kγ.

The next step is the mean-field approximation, which consists of replacing thecollective fields by those values which make the action functional extremal andneglecting the fluctuations around them, i.e. dropping the functional integrationover these collective fields. This yields the thermodynamic potential

Ω(T, µ) =φ2u + φ2

d + φ2s

8GS+

|∆ud|2 + |∆us|2 + |∆ds|24GD

− T∑

n

∫ d3p

(2π)3

1

2Tr ln

(

1

TS−1(iωn, ~p)

)

+ Ωe − Ω0. (2.92)

Here S−1(p) is the inverse propagator of the quark fields at four momentump = (iωn, ~p),

S−1(iωn, ~p) =

[

p/−M + µγ0 ∆kγ

∆†kγ p/−M − µγ0

]

, (2.93)

and ωn = (2n + 1)πT are the Matsubara frequencies for fermions. The thermo-dynamic potential of ultrarelativistic electrons,

Ωe = − 1

12π2µ4Q − 1

6µ2QT

2 − 7

180π2T 4, (2.94)

has been added to the potential, and the vacuum contribution,

Ω0 = Ω(0, 0) =φ2

0u + φ20d + φ2

0s

8GS

−2Nc

∑

i

∫

d3p

(2π)3

√

M2i + p2, (2.95)

has been subtracted in order to get zero pressure in vacuum. Using the identityTr(ln(D)) = ln(det(D)) and evaluating the determinant (see Appendix A), weobtain

ln det(

1

TS−1(iωn, ~p)

)

= 218∑

a=1

ln

(

ω2n + λa(~p)

2

T 2

)

. (2.96)

The quasiparticle dispersion relations, λa(~p), are the eigenvalues of the Hermitianmatrix,

M =

[

−γ0~γ · ~p− γ0M + µ γ0∆kγC

γ0C∆†kγ −γ0~γT · ~p+ γ0M − µ

]

, (2.97)

in color, flavor, and Nambu-Gorkov space. This result is in agreement with [?, ?].Finally, the Matsubara sum can be evaluated on closed form [?],

T∑

n

ln

(

ω2n + λ2

a

T 2

)

= λa + 2T ln(1 + e−λa/T ), (2.98)

leading to an expression for the thermodynamic potential on the form

Ω(T, µ) =φ2u + φ2

d + φ2s

8GS+

|∆ud|2 + |∆us|2 + |∆ds|24GD

−∫

d3p

(2π)3

18∑

a=1

(

λa + 2T ln(

1 + e−λa/T))

+ Ωe − Ω0. (2.99)

It should be noted that (14) is an even function of λa, so the signs of the quasi-particle dispersion relations are arbitrary. In this paper, we assume that there areno trapped neutrinos. This approximation is valid for quark matter in neutronstars, after the short period of deleptonization is over.

Equations (2.94), (2.95), (2.97), and (2.99) form a consistent thermodynamicmodel of superconducting quark matter. The independent variables are µ andT . The gaps, φi, and ∆ij, are variational order parameters that should be de-termined by minimization of the grand canonical thermodynamical potential,Ω. Also, quark matter should be locally color and electric charge neutral, so atthe physical minima of the thermodynamic potential the corresponding numberdensities should be zero

nQ = − ∂Ω

∂µQ= 0, (2.100)

n8 = − ∂Ω

∂µ3= 0, (2.101)

n3 = − ∂Ω

∂µ8= 0. (2.102)

The pressure, P , is related to the thermodynamic potential by P = −Ω at theglobal minima of Ω. The quark density, entropy and energy density are thenobtained as derivatives of the thermodynamical potential with respect to µ, Tand 1/T , respectively.

The numerical solutions to be reported in this Section are obtained with thefollowing set of model parameters, taken from Table 5.2 of Ref. [?] for vanishing’t Hooft interaction,

m0u,d = 5.5 MeV , (2.103)

m0s = 112.0 MeV , (2.104)

GSΛ2 = 2.319 , (2.105)

Λ2 = 602.3 MeV . (2.106)

With these parameters, the following low-energy QCD observables can be repro-duced: mπ = 135 MeV, mK = 497.7 MeV, fπ = 92.4 MeV. The value of thediquark coupling strength GD = ηGS is considered as a free parameter of themodel. Here we present results for η = 0.75 (intermediate coupling) and η = 1.0(strong coupling).

Quark masses and pairing gaps at zero temperature

The dynamically generated quark masses and the diquark pairing gaps are deter-mined selfconsistently at the absolute minima of the thermodynamic potential,in the plane of temperature and quark chemical potential. This is done for both

300 350 400 450 500 550µ [MeV]

100

200

300

400

500

600

∆, M

[M

eV]

Ms

Mu

Md

Mu, M

d∆

ud∆

us, ∆

ds

300 350 400 450 500 550µ [MeV]

100

200

300

400

500

600

∆, M

[M

eV]

Ms

Mu

Md

Mu, M

d∆

ud∆

us, ∆

ds

Figure 2.1: Gaps and dynamical quark masses as a function of µ at T=0 forintermediate diquark coupling, η = 0.75 (left) and for strong diquark coupling,η = 1 (right).

the strong and the intermediate diquark coupling strength. In Figs. 1 and 2we show the dependence of masses and gaps on the quark chemical potential atT = 0 for η = 0.75 and η = 1.0, resp. A characteristic feature of this dynam-ical quark model is that the critical quark chemical potentials where light andstrange quark masses jump from their constituent mass values down to almosttheir current mass values do not coincide. With increasing chemical potential thesystem undergoes a sequence of two transitions: (1) vacuum → two-flavor quarkmatter, (2) two-flavor → three-flavor quark matter. The intermediate two-flavorquark matter phase occurs within an interval of chemical potentials typical forcompact star interiors. While at intermediate coupling the asymmetry betweenof up and down quark chemical potentials leads to a mixed NQ-2SC phase be-low temperatures of 20-30 MeV, at strong coupling the pure 2SC phase extendsdown to T=0. Simultaneously, the limiting chemical potentials of the two-flavorquark matter region are lowered by about 40 MeV. Three-flavor quark matteris always in the CFL phase where all quarks are paired. The robustness of the2SC condensate under compact star constraints, with respect to changes of thecoupling strength, as well as to a softening of the momentum cutoff by a form-factor, has been recently investigated within a different parametrization [?] withsimilar trend: for η = 0.75 and NJL formfactor the 2SC condensate does notoccur for moderate chemical potentials while for η = 1.0 it occurs simultaneouslywith chiral symmetry restoration. Fig. 3 shows the corresponding dependencesof the chemical potentials conjugate to electric (µQ) and color (µ8) charges. Allphases considered in this work have zero n3 color charge for µ3 = 0.

300 350 400 450 500 550 600µ [MeV]

-250

-200

-150

-100

-50

0

µ Q, µ

8 [M

eV] µ

Q, η=0.75µ

8, η=0.75µ

Q, η=1µ

8, η=1

0 200 400p [MeV]

0

100

200

300

400

500

E [

MeV

]

ug-drub-sr, db-sgur-dg-sb

0 200 400 600p [MeV]

ub-srdb-sg

Figure 2.2: Left: Chemical potentials µQ and µ8 at T=0 for both values of thediquark coupling η = 0.75 and η = 1. All phases considered in this work have zeron3 color charge for µ3 = 0, hence µ3 is omitted in the plot. Right: Quark-quarkquasiparticle dispersion relations. For η = 0.75, T = 0, and µ = 480 MeV (leftpanel) there is a forbidden energy band above the Fermi surface. All dispersionrelations are gapped at this point in the µ − T plane, see Fig. 5. There is noforbidden energy band for the ub − sr, db − sq, and ur − dg − sb quasiparticlesat η = 1, T = 84 MeV, and µ = 500 MeV (right panel). This point in the µ− Tplane constitute a part of the gapless CFL phase.

Dispersion relations and gapless phases

In Fig. 4 we show the quasiparticle dispersion relations of different excitationsat two points in the phase diagram: (I) the CFL phase (left panel), where thereis a finite energy gap for all dispersion relations. (II) the gCFL phase (rightpanel), where the energy spectrum is shifted due to the assymetry in the chemicalpotentials, such that the CFL gap is zero and (gapless) excitations with zeroenergy are possible. In the present model, this phenomenon occurs only at ratherhigh temperatures, where the condensates are diminished by thermal fluctuations.

Phase diagram

The thermodynamical state of the system is characterized by the values of theorder parameters and their dependence on T and µ. Here we illustrate thisdependency in a phase diagram. We identify the following phases:

1. NQ: ∆ud = ∆us = ∆ds = 0;

2. NQ-2SC: ∆ud 6= 0, ∆us = ∆ds = 0, 0¡χ2SC¡1;

3. 2SC: ∆ud 6= 0, ∆us = ∆ds = 0;

350 400 450 500 550µ [MeV]

0

10

20

30

40

50

60

70

80

T [

MeV

]

g2SC

175

NQ-2SC CFL

0.90.8

gCFL

guSC

2SC

χ2SC = 1.0

0.7

NQ

Ms=

200

MeV

300 350 400 450 500 550µ [MeV]

0

20

40

60

80

100

120

T [

MeV

]

NQ

2SC CFL

guSC

gCFL

175

Ms=

200

MeV

150

g2SC

Figure 2.3: Left: Phase diagram of neutral three-flavor quark matter for inter-mediate diquark coupling η = 0.75. First-order phase transition boundaries areindicated by bold solid lines, while thin solid lines correspond to second-orderphase boundaries. The dashed lines indicate gapless phase boundaries. The vol-ume fraction, χ2SC , of the 2SC component of the mixed NQ-2SC phase is denotedwith thin dotted lines, while the constituent strange quark mass is denoted withbold dotted lines. Right: Phase diagram of neutral three-flavor quark matter forstrong diquark coupling η = 1. Line styles as in previous Figures.

4. uSC: ∆ud 6= 0, ∆us 6= 0, ∆ds = 0;

5. CFL: ∆ud 6= 0, ∆ds 6= 0, ∆us 6= 0;

and their gapless versions.The resulting phase diagrams for intermediate and strong coupling are given

in Figs. 5 and 6, resp. and constitute the main result of this work, which issummarized in the following statements:

1. Gapless phases occur only at high temperatures, above 50 MeV (interme-diate coupling) or 60 MeV (strong coupling).

2. CFL phases occur only at rather high chemical potential, well above thechiral restoration transition, i.e. above 464 MeV (intermediate coupling) or426 MeV (strong coupling).

3. Two-flavor quark matter for intermediate coupling is at low temperatures(T¡20-30 MeV) in a mixed NQ-2SC phase, at high temperatures in the pure2SC phase.

4. Two-flavor quark matter for strong coupling is in the 2SC phase with ratherhigh critical temperatures of ∼ 100 MeV.

5. The critical endpoint of first order chiral phase transitions is at (T,µ)=(44MeV, 347 MeV) for intermediate coupling and at (92 MeV, 305 MeV) forstrong coupling.

2.2.3 Mesonic correlations at finite temperature

In the previous section we have seen how the concept of order parameters can beintroduced in quantum field theory in the mean-field approximation. By analysisof the gap equations describing the minima of the thermodynamical potential inthe space of order parameters we have we could investigate the phenomenon ofspontaneous symmetry breaking, indicated by a nonvanishing value of the orderparameter (gap). Important examples being: chiral symmetry breaking (massgap) and superconductivity (energy gap). At finite temperature and density thevalues of these gaps change and their vanishing indicates the restoration of asymmetry. As the symmetries prevailing under given thermodynamical condi-tions of temperature and chemical potential (density) characterize a phase of thesystem, we have thus acquainted ourselves with a powerful quantum field the-oretical method of analysing phase transitions. The results of such an analysisare summarized in phase diagrams. A prominent example is the phase diagramof QCD in the temperature- density plane, shown schematically in Fig. ??. Itexhibits two major domains: Hadronic matter (confined quarks and gluons) andthe Quark-gluon plasma (QGP), separated by the phenomenon of quark (andgluon) deconfinement under investigation in heavy-ion collisions, but also in thephysics of compact stars and in simulations of Lattice-gauge QCD on modern Ter-aflop computers. The hadronic phase is subdivided into a nuclear matter phaseat low temperatures with a gas-liquid transition (similar to the van-der-Waalstreatment of real gases) and a ’plasma phase’ of a hot hadron gas with a multi-tude of hadronic resonances. The QGP phase is subdivided into a quark matterphase at low temperatures where most likely gluons are still condensed (confined)and the strongly correlated Fermi-liquid of quarks shoild exhibit the phenomenonof superconductivity/ superfludity with a Bose condensate of Cooper pairs of .diquarks. At asymptotic temperatures and densities one expects a system offree quarks and gluons (due to asymptotic freedom of QCD). In any real situ-ation in terrestrial experiments or in astrophysics, one expects to be quite farfrom this ideal gas state. As the nature of the confinement-deconfinement tran-sition is not yet clarified, we must speculate what to expect in the vicinity of theconjectured deconfinement phase transition. In the lattice-QCD simulations onehas found evidence for strong correlations even above the critical temperatureTc ∼ 170 MeV, which is defined by an increase of the effective number of degreesof freedom ε(T )/T 4 by about one order of magnitude in a close vicinity of Tc.Indications for strong correlations in the QGP one has also found in recent RHICexperiments, where from particle production and flow one has found fast ther-malization and an extremely low viscosity (’perfect fluid’). One speaks about an

’sQGP phase’ at temperatures Tc <∼ T <∼ 2 Tc.In order to investigate the phase diagram experimentally, one has to correlate

the observables with the regions. As the detectors are situated in the ’vacuum’at zero temperature and chemical potential where confinement prevails, no direct

Quark Matter

Novae

AGS Brookhaven

SIS Darmstadt

CERN-SPS

Super-

Quark-Gluon-Plasma

CO

NFI

NE

ME

NT

1 3

[T

=14

0 M

eV]

H

[n =0.16 fm ]ο

1.5

0.1

-3

DECONFINEMENT

FAIR (Project)

RHIC, LHC (construction)

Hadron gas

Nuclear matter

QC

D -

Lat

tice

Gau

ge T

heor

y

Neutron / Quark Stars

Baryon Density

Tem

pera

ture

Big Bang

Heavy Io

n Collisions

COLOR SUPERCONDUCTIVITY

Figure 2.4: The conjectured phase diagram of QCD and places where to investi-gate it.

detection of quarks and gluons from the plasma phase is possible and one hasto conjecture about the properties of the sQGP phase(s) from the traces whichthey might leave in the hadronic and electromagnetic spectra emitted by thehadronizing fireball formed in a heavy-ion collision or by the neutrino-emittingcooling processes of compact stars.

As a prerequisite for discussing hadronic spectra and their modifications, wewill study the mesonic spectral function(s), and for their analysis within effectivemodels like the NJL or MN models, we need to evaluate loop diagrams at finitetemperature. A basic ingredient are Matsubara frequency sums.

2.2.4 Matsubara frequency sums

In computing Feynman diagrams with internal fermion lines we shall encounterfrequency sums, and we have to learn how to evaluate them. Let us denote thequantity (ω2

n + E2p)−1, which appears in the free Fermion propagator

SF (iωn,p) = −∫ ∞

−∞

dp0

2π

ρF(p0,p)

iωn − p0

=m − p/

ω2n − E2

p

(2.107)

by ∆(iωn, Ep), and its mixed representation by ∆(τ, Ep), suppressing the sub-script F for simplicity

∆(τ, Ep) = T∑

n

e−iωnτ∆(iωn, Ep) . (2.108)

It can easily be checked (Exercise !) that

∆(τ, Ep) =1

2Ep

[

(1 − n(Ep))e−Epτ − n(Ep)e

Epτ]

=∑

s=±1

s

2Ep(1 − f(sEp))e

−sEpτ , (2.109)

where the Fermi-Dirac distribution n(p0) is

n(p0) =1

eβ|p0| + 1. (2.110)

One should note the absolute value of p0 in (2.110), in contrast with the definitionof f(p0) = 1/[exp(βp0) + 1]. Note also that

f(E) = n(E) , f(−E) = 1 − n(E) ,

1 − f(E) − f(−E) = 0 . (2.111)

In frequency space, the formula corresponding to (2.109) is

∆(iωn, Ep) =1

ω2n − E2

p

=∑

s=±1

∆s(iωn, Ep

=∑

s=±1

− s

2Ep

1

iωn − sEp. (2.112)

The frequency sums are performed by following the methods of analytic continua-tion iωn → k0 and contour integration with a function having simple poles at thediscrete frequencies k0 = iωn with unit residuum and convergence for |k0| → ∞.Let us give two general examples for Matsubara sums of one-loop diagrams withtwo external (amputated) legs.

• Fermion-boson case, see Fig. 2.2.4:

T∑

n

∆s1(iωn, E1)∆s2(i(ω − ωn), E2) = − s1s2

4E1E2

1 + f(s1E1) − f(s2E2)

iω − s1E1 − s2E2

(2.113)

T∑

n

ωn∆s1(iωn, E1)∆s2(i(ω − ωn), E2) = − is2

4E2

1 + f(s1E1) − f(s2E2)

iω − s1E1 − s2E2

(2.114)

φ2

φ1

φ

q

k

q−k

Figure 2.5: One-loop diagram for the fermion-antifermion polarization function.

φ2

φ1

q−k

k

qφ

Figure 2.6: One-loop diagram for the fermion self-energy.

• Fermion-antifermion case, see Fig. 2.2.4:

T∑

n

∆s1(iωn, E1)∆s2(i(ω − ωn), E2) = − s1s2

4E1E2

1 − f(s1E1) − f(s2E2)

iω − s1E1 − s2E2

(2.115)

T∑

n

ωn∆s1(iωn, E1)∆s2(i(ω − ωn), E2) = − is2

4E2

1 − f(s1E1) − f(s2E2)

iω − s1E1 − s2E2

(2.116)

Note that you can obtain unify these formula using the rule: f(sE) → −f(sE)when replacing a bosonic by a fermionic line.

Exercise: Verify these expressions!

Bibliography

[1] J.I. Kapusta, Finite-temperature Field Theory, Cambridge University Press,1989

[2] M. Le Bellac, Thermal Field Theory, Cambridge University Press, 1996

[3] D. Blaschke, et al., Phys. Rev. D 72 (2005) 065020.

111

Chapter 3

Particle Production by StrongFields

3.1 Introduction

Particle production under the influence of strong, time-dependent external fieldsas in ultrarelativistic heavy-ion collisions and high-intensity lasers raises a numberof challenging problems. One of these interesting questions is how to formulatea kinetic theory that incorporates the mechanism of particle creation [1, 2, 3, 4,5, 6, 7, 8, 9]. Within the framework of a flux tube model[4, 10, 11, 12], a lot ofpromising research has been carried out during the last years. In the scenariowhere a chromo-electric field is generated by a nucleus-nucleus collision, the pro-duction of parton pairs can be described by the Schwinger mechanism[13, 14, 15].The produced charged particles generate a field which, in turn, modifies the ini-tial electric field and may cause plasma oscillations. The interesting questionof the back reaction of this field has been analyzed within a field theoreticalapproach[16, 17, 18]. The results of a simple phenomenological considerationbased on kinetic equations and the field-theoretical treatment[5, 19, 20, 21] agreewith each other. The source term which occurs in such a modified Boltzmannequation was derived phenomenologically in Ref.[22]. However, the systematicderivation of this source term in relativistic transport theory is not yet fully car-ried out. For example, recently it was pointed out that the source term may havenon-Markovian character[6, 23] even for the case of a constant electric field.

In the present lecture, a kinetic equation is derived in a consistent field theo-retical treatment for the time evolution of the pair creation in a time-dependentand spatially homogeneous electric field. This derivation is based on the Bogoli-ubov transformation for field operators between the asymptotic in-state and theinstantaneous state. In contrast with phenomenological approaches, but in agree-ment with[6, 23], the source term of the particle production is of non-Markoviancharacter. The kinetic equation derived reproduces the Schwinger result in the

112

low density approximation in the weak field limit.

3.2 Dynamics of pair creation

3.2.1 Creation of fermion pairs

In this section we demonstrate the derivation of a kinetic equation with a sourceterm for fermion-antifermion production. As an illustrative example we considerelectron-positron creation, however the generalization to quark-antiquark paircreation in a chromoelectric field in Abelian approximation is straightforward.For the description of e+e− production in an electric field we start from the QEDlagrangian

L = ψiγµ(∂µ + ieAµ)ψ −mψψ − 1

4FµνF

µν , (3.1)

where F µν is the field strength, the metric is taken as gµν = diag(1,−1,−1,−1)and for the γ– matrices we use the conventional definition [24]. In the followingwe consider the electromagnetic field as classical and quantize only the matterfield. Then the Dirac equation reads

(iγµ∂µ − eγµAµ −m)ψ(x) = 0 . (3.2)

We use a simple field-theoretical model to treat charged fermions in an externalelectric field charactarized by the vector potential Aµ = (0, 0, 0, A(t)) with A(t) =A3(t). The electric field

E(t) = E3(t) = −A(t) = −dA(t)/dt (3.3)

is assumed to be time-dependent but homogeneous in space (E1 = E2 = 0). Thisquasi-classical electric field interacts with a spinor field ψ of fermions. We lookfor solutions of the Dirac equation where eigenstates are represented in the form

ψ(±)pr (x) =

[

iγ0∂0 + γkpk − eγ3A(t) +m]

χ(±)(p, t) Rr eipx, (3.4)

where k = 1, 2, 3 and the superscript (±) denotes eigenstates with the positiveand negative frequencies. Herein Rr (r = 1, 2) is an eigenvector of the matrixγ0γ3

R1 =

010

−1

, R2 =

10

−10

, (3.5)

so that R+r Rs = 2δrs . The functions χ(±)(p, t) are related to the oscillator-type

equation

χ(±)(p, t) = −(

ω2(p, t) + ieA(t))

χ(±)(p, t) , (3.6)

where we define the total energy ω2(p, t) = ε2⊥ + P 2

‖ (t), the transverse energy

ε2⊥ = m2 + p2

⊥ and P‖(t) = p‖ − eA(t). The solutions χ(±)(p, t) of Eq. (3.6)for positive and negative frequencies are defined by their asymptotic behavior att0 = t→ −∞, where A(t0) = 0. We obtain

χ(±)(p, t) ∼ exp ( ± iω0(p) t) , (3.7)

where the total energy in asymptotic limit is given as ω0(p) = ω0(p, t0) =limt→−∞

ω(p, t). Note that the system of the spinor functions (3.4) is complete

and orthonormalized. The field operators ψ(x) and ψ(x) can be decomposed inthe spinor functions (3.4) as follows:

ψ(x) =∑

r,p

[

ψ(−)pr (x) bpr(t0) + ψ

(+)pr (x) d+

−pr(t0)]

. (3.8)

The operators bpr(t0), b+pr(t0) and dpr(t0), d

+pr(t0) describe the creation and anni-

hilation of electrons and positrons in the in-state |0in> at t = t0, and satisfy theanti-commutation relations [24]

bpr(t0), b+p′r′(t0) = dpr(t0), d+p′r′(t0) = δrr′ δpp′ . (3.9)

The evolution affects the vacuum state and mixes states with positive and nega-tive energies resulting in non-diagonal terms that are responsible for pair creation.The diagonalization of the hamiltonian corresponding to a Dirac-particle (Eq.(3.2)) in the homogeneous electric field (3.3) is achieved by a time-dependentBogoliubov transformation

bpr(t) = αp(t) bpr(t0) + βp(t) d+−pr(t0) ,

dpr(t) = α−p(t) dpr(t0) − β−p(t) b+−pr(t0)

(3.10)

with the condition|αp(t)|2 + |βp(t)|2 = 1 . (3.11)

Here, the operators bpr(t) and dpr(t) describe the creation and annihilation ofquasiparticles at the time t with the instantaneous vacuum |0t >. Clearly, theoperator system b(t0), b+(t0); d(t0), d

+(t0) is unitary non-equivalent to the systemb(t), b+(t); d(t), d+(t). The substitution of Eqs. (3.10) into Eq. (3.8) leads to thenew representation of the field operators

ψ(x) =∑

r,p

[

Ψ(−)pr (x) bpr(t) + Ψ

(+)pr (x) d+

−pr(t)]

. (3.12)

The link between the new Ψ(±)pr (x) and the former (3.4) basis functions is given

by a canonical transformation

ψ(−)pr (x) = αp(t) Ψ

(−)pr (x) − β∗p(t) Ψ

(+)pr (x) ,

ψ(+)pr (x) = α∗p(t) Ψ(+)(x)pr + βp(t) Ψ

(−)pr (x) .

(3.13)

Therefore it is justified to assume that the functions Ψ(±)pr have a spin structure

similar to that of ψ(+)pr in Eq. (3.4),

Ψ(±)pr (x) =

[

iγ0∂0 + γkpk − eγ3A(t) +m]

φ(±)p (x) Rr e±iΘ(t)eipx, (3.14)

where the dynamical phase is defined as

Θ(p, t) =∫ t

t0dt′ω(p, t′) . (3.15)

The functions φ(±)p are yet unknown. The substitution of Eq. (3.14) into Eqs.

(3.13) leads to the relations

χ(−)(p, t) = αp(t) φ(−)p (t) e−iΘ(p,t) − β∗p(t) φ

(+)p (t) eiΘ(p,t) ,

χ(+)(p, t) = α∗p(t) φ(+)p (t) eiΘ(p,t) + βp(t) φ

(−)p (t) e−iΘ(p,t) .

(3.16)

Now we are able to find explicit expressions for the coefficients αp(t) and βp(t).Taking into account that the functions χ(±)(p, t) are defined by Eq. (3.6), weintroduce additional conditions to Eqs. (3.16) according to the Lagrange method

χ(−)(p, t) = −iω(p, t)[

αp(t) φ(−)p (t) e−iΘ(p,t) + β∗p(t) φ

(+)p (t) eiΘ(p,t)

]

,

χ(+)(p, t) = iω(p, t)[

α∗p(t) φ(+)p (t) eiΘ(p,t) − βp(t) φ

(−)p (t) e−iΘ(p,t)

]

.

(3.17)

Differentiating these equations with respect to time, using Eqs. (3.6) and (3.16)and then choosing as an Ansatz

φ(±)p (t) =

√

√

√

√

ω(p, t) ± P‖(t)

ω(p, t), (3.18)

we obtain the following differential equations for the coefficients

αp(t) =eE(t)ε⊥2ω2(p, t)

β∗p(t) e2iΘ(p,t) ,

β∗p(t) = − eE(t)ε⊥2ω2(p, t)

αp(t) e−2iΘ(p,t) .

(3.19)

As the result of the Bogoliubov transformation we obtained the new coefficientsof the instantaneous state at the time t. The relations between them read

αp(t) =1

2√

ω(p, t) (ω(p, t) − P‖(t))

(

ω(p, t) χ(−)(p, t) + i χ(−)(p, t))

eiΘ(p,t) ,

β∗p(t) = − 1

2√

ω(p, t) (ω(p, t) − P‖(t))

(

ω(p, t) χ(−)(p, t) − i χ(−)(p, t))

e−iΘ(p,t) .

(3.20)

It is convenient to introduce new operators which absorb the dynamical phase

Bpr(t) = bpr(t) e−iΘ(p,t) , (3.21)

Dpr(t) = dpr(t) e−iΘ(p,t) (3.22)

satisfying the anti-commutation relations:

Bpr(t), B+p′r′(t) = Dpr(t), D

+p′r′(t) = δrr′ δpp′ . (3.23)

It is easy to show that these operators satisfy the Heisenberg-like equations ofmotion

dBpr(t)

dt= − eE(t)ε⊥

2ω2(p, t)D+−pr(t) + i [H(t), Bpr(t)] ,

dDpr(t)

dt=

eE(t)ε⊥2ω2(p, t)

B+−pr(t) + i [H(t), Dpr(t)] ,

(3.24)

where H(t) is the hamiltonian of the quasiparticle system

H(t) =∑

r,p

ω(p, t)(

B+pr(t) Bpr(t) −D−pr(t) D

+−pr(t)

)

. (3.25)

The first term on the r.h.s. of Eqs. (3.24) is caused by the unitary non-equivalenceof the in-representation and the quasiparticle one.

Now we explore the evolution of the distribution function of electrons withthe momentum p and spin r defined as

fr(p, t) =< 0in|b+pr(t) bpr(t)|0in >=< 0in|B+pr(t) Bpr(t)|0in > . (3.26)

According to the charge conservation the distribution functions for electrons andpositrons are equal fr(p, t) = fr(p, t), where

fr(p, t) =< 0in|d+−pr(t) d−pr(t)|0in >=< 0in|D+

−pr(t) D−pr(t)|0in > . (3.27)

The distribution functions (3.26) and (3.27) are normalized to the total numberof pairs N(t) of the system at a given time t

∑

r,p

fr(p, t) =∑

r,p

fr(p, t) = N(t) . (3.28)

Time differentiation of Eq. (3.26) leads to the following equation

dfr(p, t)

dt= −eE(t)ε⊥

ω2(p, t)ReΦr(p, t) . (3.29)

Herein, we have used the equation of motion (3.24) and evaluated the occurringcommutator. The function Φr(p, t) in Eq. (3.29) describes the creation and

annihilation of an electron-positron pair in the external electric field E(t) and isgiven as

Φr(p, t) =< 0in|D−pr(t) Bpr(t)|0in > . (3.30)

It is straightforward to evaluate the derivative of this function. Applying theequations of motion (3.24), we obtain

dΦr(p, t)

dt=eE(t)ε⊥2ω2(p, t)

[

2fr(p, t) − 1]

− 2iω(p, t) Φr(p, t) , (3.31)

where because of charge neutrality of the system, the relation fr(p, t) = fr(p, t)is used. The solution of Eq. (3.31) may be written in the following integral form

Φr(p, t) =ε⊥2

∫ t

t0dt′

eE(t′)

ω2(p, t′)

[

2fr(p, t′) − 1

]

e2i[Θ(p,t′)−Θ(p,t)] . (3.32)

The functions Θ(p, t) and Θ(p, t′) in Eq. (3.32) can be taken at t0 (see thedefinition (3.15)). Hence with A(t0) = 0, the function Φr(p, t) vanishes at t0.Inserting Eq. (3.32) into the r.h.s of Eq. (3.29) we obtain

dfr(p, t)

dt=eE(t)ε⊥2ω2(p, t)

∫ t

−∞dt′eE(t′)ε⊥ω2(p, t′)

[

1 − 2fr(p, t′)]

cos(

2[Θ(p, t) − Θ(p, t′)])

.

(3.33)Since the distribution function obviously does not depend on spin (3.33), we candefine: fr = f . With the substitution f(p, t) → F (P , t), where the 3-momentumis now defined as P (p⊥, P‖(t)), the kinetic equation (3.33) is reduced to its finalform:

dF (P , t)

dt=∂F (P , t)

∂t+ eE(t)

∂F (P , t)

∂P‖(t)= S(−)(P , t) , (3.34)

with the Schwinger source term

S(−)(P , t) =eE(t)ε⊥2ω2(p, t)

∫ t

−∞dt′eE(t′)ε⊥ω2(p, t′)

[

1 − 2F (P , t′)]

cos(

2[Θ(p, t) − Θ(p, t′)])

.

(3.35)Recently in Ref.[6], a kinetic equation similar to (3.34) has been derived within aprojection operator formalism for the case of a time-independent electric fieldwhere it was first noted that this source term has non-Markovian character.As well as in this method the multiple pair creation is not considered. Thepresence of the Pauli blocking factor [1 − 2F (P , t)] in the source term has beenobtained earlier in Ref.[18]. We would like to emphasize the closed form of thekinetic equation in the present work where the source term does not includethe anomalous distribution functions (3.30) for fermion-antifermion pair creation(annihilation).

3.2.2 Creation of boson pairs

In this subsection we derive the kinetic equation with source term for the case ofpairs of charged bosons in a strong electric field.

The Klein-Gordon equation reads

(

(∂µ + ieAµ)(∂µ + ieAµ) +m2)

φ(x) = 0 . (3.36)

The solution of the Klein-Gordon equation in the presence of the electric fielddefined by the vector potential Aµ = (0, 0, 0, A(t)) is taken in the form [24]

φ(±)p (x) = [2ω(p)]−1/2 eixpg(±)(p, t) , (3.37)

where the functions g(±)(p, t) satisfy the oscillator-type equation with a variablefrequency

g(±)(p, t) + ω2(p, t) g(±)(p, t) = 0 . (3.38)

Solutions of Eq. (3.38) for positive and negative frequencies are defined by theirasymptotic behavior at t0 = t→ −∞ similarly to Eq. (3.7).

The field operator in the in-state is defined as

φ(x) =∫

d3p [ φ(−)p (x) ap(t0) + φ

(+)p (x) b+vp(t0) ] . (3.39)

The diagonalization of the hamiltonian corresponding to the instantaneous stateisachieved by the transition to the quasiparticle representation. The Bogoliubovtransformation for creation and annihilation operators of quasiparticles has thefollowing form

ap(t) = αp(t) ap(t0) + βp(t) b+−p(t0) ,

b−p(t) = α−p(t) bp(t0) + β−p(t) a+−p(t0)

(3.40)

with the condition|αp(t)|2 − |βp(t)|2 = 1 . (3.41)

The derivation of the Bogoliubov coefficients α and β is similar to that given inthe previous subsection. We obtain the equations of motion for the coefficientsof the canonical transformation (3.40) as follows

αp(t) =ω(p, t)

2ω(p, t)β∗p(t) e

2iΘ(p,t) , (3.42)

βp(t) =ω(p, t)

2ω(p, t)α∗p(t) e

2iΘ(p,t) . (3.43)

Following the derivation for the case of fermion production, we arrive at thefinal result for the source term in the bosonic case

S(+)(P , t) =eE(t)ε⊥2ω2(p, t)

∫ t

−∞dt′eE(t′)ε⊥ω2(p, t′)

[

1 + 2F (P , t′)]

cos(

2[Θ(p, t) − Θ(p, t′)])

,

(3.44)which differs from the fermion case just by the sign in front of the distributionfunction due to the different statistics of the produced particles.

3.3 Discussion of the source term

3.3.1 Properties of the source term

We can combine the results for fermions (3.35) and for bosons (3.44) into a singlekinetic equation

dF(±)(P , t)

dt=∂F(±)(P , t)

∂t+ eE(t)

∂F(±)(P , t)

∂P3

= S(±)(P , t) . (3.45)

Here, the upper (lower) sign corresponds to the Bose-Einstein (Fermi-Dirac)statistics. Based on microscopic dynamics, these kinetic equations are exactwithin the approximation of a time-dependent homogeneous electric field andthe neglect of collisions. The Schwinger source terms (3.35) and (3.44) are char-acterized by the following features:

1. The kinetic equations (3.45) are of non-Markovian type due to the explicitdependence of the source terms on the whole pre-history via the statisticalfactor 1± 2F (P , t) for fermions or bosons, respectively. The memory effectis expected to lead to a modification of particle pair creation as comparedto the (Markovian) low-density limit, where the statistical factor is absent.

2. The difference of the dynamical phases, Θ(p, t)−Θ(p, t′), under the integrals(3.35) and (3.44) generates high frequency oscillations.

3. The appearance of such a source term leads to entropy production dueto pair creation (see also Rau[6]) and therefore the time reversal symmetryshould be violated, but it does not result in any monotonic entropy increase(in absence of collisions).

4. The source term and the distribution functions have a non-trivial momen-tum dependence resulting in the fact that particles are produced not onlyat rest as assumed in previous studies, e.g. Ref.[19].

5. In the low density limit and in the simple case of a constant electric fieldwe reproduce the pair production rate given by Schwinger’s formula

Scl = limt→+∞

(2π)−3g∫

d3P S(P , t) =e2E2

4π3exp

(

− πm2

|eE|)

. (3.46)

As noted above, Rau[6] found that the source term has a non-Markovianbehaviour by deriving the production rate within a projector method. In thelimit of a constant field our results agree with Ref.[6]. In our approach the electricfield is treated as a general time dependent field and hence there is no a priori

limitation to constant fields. However our result allows to explore the influenceof any time-dependent electric field on the pair creation process. It is importantto note that in general this time dependence should be given by a selfconsistentsolution of the coupled field equations, namely the Dirac (Klein-Gordon) equationand the Maxwell equation. This would incorporate back reactions as mentioned inthe introduction. However, the solution of such a system of equations is beyondthe scope of this work. Herein we will restrict ourselves to the study of somefeatures of the new source term.

Finally we remark that the source term is characterized by two time scales:the memory time

τmem ∼ ε⊥eE

(3.47)

and the production interval

τprod = 1/ < S(±) > , (3.48)

with < S(±) > denoting the time averaged production rate. As long as E <<m2/e < ε2

⊥/e, the particle creation process is Markovian: τmem << τprod. Thisresults for constant fields agree with those of Rau[6].

-3.0 -1.0 1.0 3.0p|| /ε

-2.0

-1.0

0.0

1.0

S/ε

-10.0

0.0

10.0

20.0

30.0

S/ε

Figure 3.1: The pair production rate as a function of parallel momentum fora constant, strong field (upper plot: E0 = 1.5) and a weak field (lower plot:E0 = 0.5) at t = 0.

3.3.2 Numerical results

In order to study the new source term, we consider two different cases, namely aconstant field and a time dependent electric field. For the numerical evaluationwe start with Eq. (3.35) assuming a dilute system, F = 0, and

S(p‖, t) =S(p‖, t)

ε⊥=

E(t)

2ω2(p‖, t)

∫ t

−∞dt′

E(t′)

ω2(p‖, t′)cos

(

2[Θ(p‖, t) − Θ(p‖, t′)])

,

(3.49)where we have introduced dimensionless variables

E(t) = eE(t)/ε2⊥ , t = tε⊥ , (3.50)

p‖ = p‖/ε⊥ , ω = ω/ε⊥ . (3.51)

This notation is convenient to distinguish the weak field (E < 1) and strong field(E > 1) limits. A particularly simple result is obtained if we assume a constantfield,

A(t) = A(t)/ε⊥ = tE0/e , (3.52)

where the electric field does not depend on time and the energy is given as

ω20(p‖, t) = 1 + (p‖ − E0t)

2 . (3.53)

For the source term we obtain

S(p‖, t) =E2

0

2ω20(p‖, t)

∫ t

−∞dt′

1

ω20(p‖, t′)

cos(

2∫ t

t′dt′′ω0(p‖, t

′′))

. (3.54)

In Fig. 3.1 we plot the particle production rate as function of the parallelmomentum p‖ for a weak constant field and a strong field, respectively. The

rates are normalized to be of the order of one for E0 = 0.5 at zero values of bothmomentum and time. The production rate is positive when particles are producedwith positive momenta. Pairs with negative momenta are moving against thefield and hence get annihilated, clearly to be seen in the negative productionrate. These results mainly agree with those of Ref.[6], since no time dependencewas considered. Note that using the prefactor and field strengths chosen by Rau,we reproduce exactly the results given in Ref.[6].

In considering the time dependence of the source term, we go beyond theanalysis of Ref.[6]. In Fig. 3.2, we display the time dependence of the productionrate at zero momentum. The maximum of the production rate is concentratedaround zero and shows an oscillating behaviour for large times. Indeed it ispossible to write Eq. (3.54) in terms of the Airy function because the constantfield provides an analytical solution for the dynamical phase difference using theenergy given in Eq. (3.53). The production of pairs for strong fields is larger

-3.0 2.0 7.0tε

-2.0

-1.0

0.0

1.0

S/ε

-10.0

0.0

10.0

20.0

30.0

S/ε

Figure 3.2: The pair production rate as a function of time for a constant, strongfield (upper plot: E0 = 1.5) and a weak field (lower plot: E0 = 0.5) at zeroparallel momentum.

Figure 3.3: The pair production rate, S(p‖, t), as a function of time and parallel

momentum for a time dependent weak electric field charactarized by E0 = 0.5,σ = 1 and τ = 0 . All plotted values are dimensionless as described in the text.

Figure 3.4: The pair production rate, S(p‖, t), as a function of time and parallel

momentum for a time dependent strong electric field charactarized by E0 = 1.5,σ = 1 and τ = 0 . All plotted values are dimensionless as described in the text.

than that of weak fields, and the typical time period of the oscillations is smaller.

The situation changes if we allow the electric field to be time dependent. Weassume a Gaussian field at the dimensionsless time τ with the width σ = σε⊥,

E(t) = E0e−(t−τ)2/σ2

(3.55)

and obtain

A(t) = −E0σ

√π

2

[

Erf[(t− τ)/σ] − Erf[(−τ − t0)/σ]]

. (3.56)

The occuring error function is defined as

Erf(z) =2√π

∫ z

0e−x

2

dx . (3.57)

Using this Ansatz for the field strength in Eq. (3.49) we obtain the numericalresults plotted in Figs. 3.3 and 3.4. Therein all occuring values are dimensions-less. The electric field is non zero within a certain width σ = 1 for τ = 0 aroundt = 0. Therefore the oscillations for times beyond the time where the electricalfield is finite are damped out. The pair production rate is peaked around smallmomenta for both weak and strong fields. For strong fields the distribution isshifted to positive momenta but remains still close to small parallel momenta. Itis important to point out that the production of particles happens not only atrest (p‖ = 0) what is assumed in many works also addressing the back reactionproblem, e.g.[19]. In contrary we find a non-trivial momentum dependence of thepair creation rate depending on the field strength and on time.

3.4 Summary

We have derived a quantum kinetic equation within a consistent field theoreticaltreatment which contains the creation of particle-antiparticle pairs in a time-dependent homogeneous electric field. For both fermions and bosons we obtain asource term providing a kinetic equation of non-Markovian character. The sourceterm is characterized by a pair production rate that contains a time integrationover the evolution of the distribution function and therefore involves memoryeffects.

For the simple case of a constant electric field in low density limit and Marko-vian approximation, we analytically and numerically reproduce the results ofearlier works[6]. Since our approach is not restricted to constant fields, we haveexplored the dependence of the source term on a time dependent (model) electricfield with a Gaussian shape. Within these two different Ansatze for the field,we have performed investigations of the time structure of the source term. Theproduction of pairs does not happen at rest only. We observe a non-trivial mo-mentum dependence of the source term depending on the field strength and ontime.

The particle production source is dominated by two time scales: the memorytime and the production time. The numerical results mainly show oscillationsdue to the dynamical phase and urge the need to include the Maxwell equationto determine the electric field by physical boundary conditions (back reactions),work in this direction is in progress. Furthermore, it would be of great interest toextend this approach to the QCD case to explore the impact of a non-Markoviansource term on the pre-equilibrium physics in ultrarelativistic heavy-ion collisions.

Bibliography

[1] N.B. Naroshni and A.I. Nikishov, Yad. Fiz. 11 (1970) 1072 (Sov. J. Nucl.Phys. 11 (1970) 596); V.S. Popov and M.S. Marinov, Yad. Fiz. 16 (1972)809 (Sov. J. Nucl. Phys. 16 (1974) 449); V.S. Popov, Zh. Eksp. Teor. Fiz.62 (1972) 1248 (Sov. Phys. JETP 35 (1972) 659); M.S. Marinov and V.S.Popov, Fort. d. Phys. 25 (1977) 373.

[2] A. Bia las and W. Czyz, Phys. Rev. D 30 (1984) 2371; 31 (1985) 198;Z. Phys. C 28 (1985) 255; Nucl. Phys. B 267 (1985) 242; Acta Phys. Pol. B17(1986) 635.

[3] K. Kajantie and T. Matsui, Phys. Lett. B 164 (1985) 373.

[4] G. Gatoff, A.K. Kerman, and T. Matsui, Phys. Rev. D 36 (1987) 114.

[5] F. Cooper, J.M. Eisenberg, Y. Kluger, E. Mottola, and B. Svetitsky,Phys. Rev. D 48 (1993) 190.

[6] J. Rau, Phys.Rev. D 50 (1994) 6911.

[7] R.S. Bhalerao and V. Ravishankar, Phys.Lett. B 409 (1997) 38.

[8] W. Greiner, B. Muller, and J. Rafelski, Quantum Electrodynamics of Strong

Fields (Springer-Verlag, Berlin, 1985).

[9] A.A. Grib, S.G. Mamaev, and V.M. Mostepanenko, Vacuum quantum effects

in strong external fields, (Atomizdat, Moscow, 1988).

[10] S. Nussinov, Phys. Rev. Lett. 34 (1975) 1286.

[11] B. Andersson, G. Gustafson, G. Ingelman, and T. Sjostrand, Phys. Rep. 97(1993) 31.

[12] N.K. Glendenning and T. Matsui, Phys. Rev. D 28 (1983) 2890.

[13] F. Sauter, Z. Phys. 69 (1931) 742.

[14] W. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714.

125

[15] J. Schwinger, Phys. Rev. 82 (1951) 664.

[16] F. Cooper and E. Mottola, Phys. Rev. D 40 (1989) 456.

[17] Y. Kluger, J.M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola,Phys. Rev. Lett. 67 (1991) 2427.

[18] Y. Kluger, J.M. Eisenberg, B. Svetitsky, F. Cooper, and E. Mottola,Phys. Rev. D 45 (1992) 4659.

[19] Y. Kluger, J.M. Eisenberg, and B. Svetitsky, Int. J. Mod. Phys. E 2 (1993)333 (here further references may be found).

[20] F. Cooper, S. Habib, Y. Kluger, E. Mottola, J.P. Paz, and P.R. Anderson,Phys. Rev. D 50 (1994) 2848.

[21] J.M. Eisenberg, Phys. Rev. D 51 (1995) 1938.

[22] Ch. Best and J.M. Eisenberg, Phys. Rev. D 47 (1993) 4639.

[23] J. Rau and B. Muller, Phys. Rep. 272 (1996) 1.

[24] J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill,New York, 1964).

Chapter 4

Problem sets - statistical QFT

4.1 Partition function - Introduction

1.1. Construct the statistical operator ρ for the the grand canonical ensem-ble from the principle of maximum information entropy (Jaynes) S =−Trρ ln ρ −→ Max.

1.2. The Hamiltonian form of the Path Integral representation of the partitionfunction for a scalar field theory is given by (τ = it)

Z(T, V, µ) =∫

Dπ∫

φ(0)=φ(β)Dφ exp

∫ β

0dτ∫

Vd3x

[

iπ∂φ

∂τ−H(π, φ) − µN

]

,

H(π, φ) =1

2π2 +

1

2m2φ2 + U(φ)

Find the Lagrangian formulation by evaluating the path integral over thefield momentum π in the case µ = 0.

1.3. The pressure for a noninteracting scalar field is

p(T, V, µ) =1

βVlnZ0(T, V, µ) =

1

2βV

∑

n

∑

p

ln[β2(ω2n + ω2(p))] .

Perform the summation over Matsubara frequencies ωn = 2πnT, β = 1/Texplicitely!

127

4.2 Partition function - Fermi systems

2.1 The partition function of a Fermi-gas is given by Z = detD, whereD = −iβγ0[γ0(−iωn + µ) − ~γ · ~p − m]. Use the representation of the 4x4Dirac gamma matrices and evaluate the determinant to show the result:

detDirac

D = β4[(ωn + iµ)2 + ω2(p)]2 ,

where ω2(p) = p2 +m2.

2.2 The pressure of a noninteracting Fermi-gas is given by (β = 1/T )

p(T, V, µ) =1

βVlnZ0(T, V, µ) =

2

βV

∑

n

∑

p

lnβ2[(ωn + iµ)2 + ω2(p)] .

perform the summation over the Matsubara- Frequencies ωn = (2n+ 1)πT ,and show the result

p(T, V, µ) = 2∫

d3p

(2π)3ω(p) + T ln[1 + e−β(ω(p)−µ)] + T ln[1 + e−β(ω(p)+µ)]

2.3 Evaluate the pressure of a massless ideal Fermi-gas (e.g. Neutrinos) byperforming the momentum integration with the result:

p = −Ω =µ4

12π2+µ2T 2

6+

7π2T 4

180.

Use the relation ε = Ts− p+µn with s = ∂Ω/∂T and n = ∂Ω/∂µ to proveε = 3p for this case.

2.4 The pressure of a cold, interacting neutron gas in the relativistic mean-field(Walecka) model is given by

P =1

8π2

[

2

3E∗Fp

3F −m∗n

2E∗FpF +m∗n4 ln (

E∗F + pFm∗n

)]

+1

2

(

g2ω

m2ω

)

n2 − 1

2

(

g2σ

m2σ

)

n2s

n =p3F

3π2, EF∗ =

√

p2F +m∗n

2 , m∗n = mn −(

g2σ

m2σ

)

ns

ns =m∗n2π2

[

E∗FpF −m∗n2 ln (

E∗F + pFm∗n

)]

.

Show that this system of equations can be given a closed form! Derive theexpression for the energy density!

4.3 Partition function - Quark matter

The grand canonical thermodynamical potential for quark matter within thenonlocal chiral quark model is given by

Ω(φ;µ, T ) =φ2 − φ2

0

4 G1− 6

π2

∫ ∞

0dq q2

Eφ(q) − Eφ0(q)

+T ln

[

1 + exp

(

−Eφ(q) − µ

T

)]

+ T ln

[

1 + exp

(

−Eφ(q) + µ

T

)]

.(4.1)

The dispersion relation is Eφ(q) =√

q2 + (m + φ g(q))2 where the mass gap

(order parameter, chiral gap) φ = φ(T, µ) depends on temperature T and chemicalpotential µ and φ0 = φ(0, 0) is the value of the order parameter in the vacuumat T = µ = 0. The coupling constant is G1 = 3.761/Λ2 where Λ = 1.025 GeVis the range of the separable interaction with the Gaussian formfactor g(q) =exp(−q2/Λ2) and m = 2.41 MeV is the current quark mass.

3.1 Find the T = 0 limit of Eq. (1) and derive the condition for the minimumof the thermodynamical potential with respect to a variation of the orderparameter φ (gap equation)! Solve this gap equation for µ = 0 in order tofind the value of φ0!

3.2 Solve the gap equation φ(µ) for 0 ≤ µ ≤ 500 MeV and find the criticalchemical potential µc where the mass gap jumps to a very low value (chiralsymmetry restoration).

3.3 Insert the solution of the gap equation φ(µ) in Eq. (1) and evaluate thepressure p(µ) = −Ω(φ;µ, T = 0) by momentum integration for 0 ≤ µ ≤ 500MeV.

3.4 Perform a fit of the quark matter pressure for µ > µc to a bag model

pBag(µ) =µ4

2π2− B (4.2)

and discuss whether a stable quark matter core inside a neutron star ispossible by comparing with solutions from the Diploma thesis by ThomasKlahn!

3.5 Perform the same steps for Lorentzian and cut-off (Nambu–Jona-Lasinio)form factor models! The parameters can be taken from the referencehttp://arXiv.org/abs/hep-ph/0602238.What is the influence of the form of the interaction on the structure of aneutron star with quark matter core?

Chapter 5

Projects

5.1 Symmetry breaking. Goldstone-Theorem.

Higgs-Effect

5.1.1 Spontaneous symmetry breaking: Complex scalarfield

Spontaneous symmetry breaking is a very general phenomenon in nature, bestexplained on the example of a simple model, the complex scalar field with negativemass-squared

L = ∂µΦ∗∂µΦ −m2Φ∗Φ − λ(Φ∗Φ)2 (5.1)

The potential energy density corresponding to this problem depends on the valuesof the parameters λ and m. Because of stability, λ has to be positive. Then theshape of the potential depends on the sign of the mass term. For positive m2 itis given in the upper graph of Fig. 5.1.1, while for m2 = −c2 < 0 it has the shapeof a Mexican hat (bottom of a wine-bottle), see lower graph in Fig. 5.1.1. TheLagrangian (5.1) has a global U(1) symmetry, i.e. the replacement Φ → Φeiα

with a real phase α, independent on the location x, leaves it invariant. But anyground state at the bottom of the potential given by

Φ0 = Φ1,0 + iΦ2,0 (5.2)

will break this rotational symmetry, see lower graph in Fig. 5.1.1.

1. Find the ground state of the system at T = 0 and the masses m1 and m2

of the elementary excitations in the mean field approximation!

2. Evaluate the thermodynamical potential (and the pressure) at finite tem-perature!

3. Show that the symmetry gets restored at finite temperature in a secondorder phase transition!

130

σ)

(x,σ)

π)(y,

π)(y,

(x,

(b)

(a)

Figure 5.1: Potential with spontaneous symmetry breaking.

4. Formulate and prove the Goldstone theorem!

5.1.2 Electroweak symmetry breaking: Higgs mechanism

Modern gauge theories attempt to unify forces of nature, such as weak and elec-tromagnetic forces are unified in the Weinberg-Salam model. In this model, ascalar field (Higgs field) breaks spontaneously the gauge symmetry and gives massto the vector bosons (observed as W± and Z0). Describe this phenomenon in thehigh-temperature mean-field approximation to the Lagrangian

L = (∂µ − ie Aµ)Φ∗(∂µ + ie Aµ)Φ + c2Φ∗Φ − λ(Φ∗Φ)2 − 1

4F µνFµν , (5.3)

where Fµν = ∂µ Aν − ∂ν Aµ.

Bibliography

[1] J.I. Kapusta, Finite-temperature field theory, Cambridge University Press(1989), chapter 7 and chapter 9.

132