Origin of Quantum Chromodynamics · More importantly, the diagram in the previous slide is for spin 3/2 baryons. Although baryons are fermion, the 3/2 spin baryons seem not to support

Origin of Quantum Chromodynamics

Thitipat Sainapha

November 15, 2018

Thitipat Sainapha Origin of QCD November 15, 2018 1 / 48

Overview

1 Strong InteractionYukawa effective theoryQuark modele−p+ Scattering

2 Construct the QCDRenormalization group flowAsymptotic freedom


Yukawa theory (I)

Before 1930s, there are only 2 fundamental forces in nature: Gravity andElectromagnetism.

Gravity are described by Newton’s law of gravitation, modified byEinstein’s general relativity

Electromagnetism are studied through Maxwell’s theory, also successfullyquantized becoming quantum electrodynamics (QED)


Yukawa theory (II)

At the beginning of 20th century, there are several discoveries aboutradioactivity and atomic structure

There should exist other kinds of forces: One binds nucleons inside nucleuswhile another explains how nucleus radiates high energy e−, known as thefamous nuclear β-decay.

β-decay can be effectively understood by 4-Fermi theory and its UVcomplete theory, electroweak theory.


Yukawa theory (III)

The last type was found to be very strong coupled with respects to allother forces, this is why it called the strong interaction

As 4-Fermi theory in weak interaction case, strong interaction is effectivelydescribed by Yukawa theory

LYukawa = LFermion(ψ)− 1

2π( + m2)π − gπψψ (1)

where ψ is some hadron such as proton and neutron, whereas π is scalarfield known as pion field


Yukawa theory (IV)

Steal technique from QED, the force between two hadrons (nucleons)determined by exchanging of pion, pictorially represented by t-channelFeynman diagram

πUsing Feynman rule to find the relevant truncated S-matrix (each vertexgives coupling constant g up to factor i, and internal line gives bosonicFeynman propagator)


Yukawa theory (VI)

We end up with something like

∼ −g2 i

k2 −m2 + iε(2)

To get ordinary potential in classical theory, we will calculate the Fouriertransformation of (2) in static limit (Born approximation)

V (r) = g2

∫d3k

(2π)3e ik·r

−(k2 + m2) + iε(3)


Yukawa theory (VII)

After working with complex analysis, the integral (3) can be done bywriting the coutour and using Cauchy’s residual theorem, we obtain

V (r) = − g2

4πre−imr (4)

The form in (4) is formally known as Yukawa potential which was usedto understand the binding energy inside nucleus. We see from (4) thatstrong force has short-distance behavior. Note that in m→ 0 limit, thisexpression reduces to long distance Coulomb-like potential.


Yukawa theory (VIII)

This model was very successful to describe strong interaction. This ledYukawa-san received the Nobel prize in 1949

This model is okay if we think that hadrons like proton and neutron areelementary particle. It’s not!


Quark model (I)

Gell-Mann and Zweig independently studied SU(3) group, they foundmathematically that both mesons and baryons lie on SU(3) Cartan diagram

Figure: Eightfold way retreived fromhttps://en.wikipedia.org/wiki/Eightfold Way (physics)


Quark model (II)

Baryons can be constructed by 3⊗ 3⊗ 3 representation while mesons canbe constructed 3⊗ 3 representation.

This gives us the hint for baryons and mesons’ substructure so-calledquarks

However, why should believe that SU(3) group theory really describeshadrons?


Quark model (III)

Figure: Baryons decuplet retreived fromhttps://en.wikipedia.org/wiki/Eightfold Way (physics)


Quark model (IV)

Historically, we didn’t discovered Ω− in the diagram in previous slide atthat time. The later discovering of these particle ensured the correctnessof eightfold way led Gell-Mann and Zweig recieved the Nobel prize in 1969.

More importantly, the diagram in the previous slide is for spin 3/2 baryons.Although baryons are fermion, the 3/2 spin baryons seem not to supportthis fact.

This implies that there should be hidden anti-symmetric degree of freedomcalled color proposed by Nambu.


Quark model (V)

The spectra of strong interacting particles, in fact, are much more strangethan we expected. If we plot graph between spin with mass square. Wewill find linear behavior known as Regge’s trajectories

Figure: Regge’s slope retreived fromhttp://insti.physics.sunysb.edu/∼siegel/reggepart/reggepart.html


Quark model (VI)

The effective model to understand the Regge’s trajectories is there existsstring-like structure gluing them together. This historically was the originof string theory as the theory of strong interaction. Presumably, this is thereason why many people , like Nambu and Gross, who established stronginteraction also contributed in string theories.

This assumption currently has changed a lot: The string-like structure isnon-Abelian electric flux tube understood in context of dualsuperconductor model with monopoles.


e−p+ scattering (I)

The another weird phenomenon of strong interaction comes fromelectron-proton scattering

In 1911, Rutherford and his team did very famous gold foil experiment,they used classical scattering theory to estimate the cross section formula.

His cross section need to be more precise by working with more accuratescattering process. Such a process need the smallest nuclei and thesmallest probe as possible. That are proton and electron.


e−p+ scattering (II)

First, we will pretend that proton is an elementary particle. By this naivetreatment, this scattering process can be studied by QED

γ

p+

e−

p+

e−

As we have seen from Yukawa theory, in massless internal propagator case(photon is massless), this graph will give Coulomb-like potential. Thus, wecall this graph the Coulomb scattering.


e−p+ scattering (III)

To get quantum correction, we will modify the Feynman of QED vertex tobe proportional to generalized QED vertex function Γµ defined as

Γµ(q) = F1(q2)γµ +iσµν

2mpqνF2(q2) (5)

where F1 and F2 are called form factors. Note that the reason why we candecompose general vertex function this way is suggested from Gordon’sidentity. At leading order, F1(0) = 1 and F2(0) = 0. On the other hands,at 1-loop order, F2(0) = αe

2π , this gives quantum correction to magneticmoment called anomalous magnetic moment. (Dirac’s equation predictedthe gyro-magnetic ratio is 2 while 1-loop QED predicted 2 + αe

π )


e−p+ scattering (IV)

By this parametrization, we can compute the tree-level elastic Coulombscattering cross section (in lab frame, since mp >> me , we can think thatonly electron travels but proton stays at rest)

dσ

dΩ=

α2e

4E 2sin4 θ2

E ′

E

[(F 21 −

q2

4m2p

F 22

)cos2

θ

2− q2

2m2p

(F1 + F2)2sin2θ

2

](6)

known as Rosenbluth formula.


e−p+ scattering (V)

The formula (6) may fit very well with experiments but not in all regime.Obviously, it is because proton is not an elementary particle but compositeparticle containing quarks.

γ

p+

e−

p+

e−


e−p+ scattering (VI)

As we have already mentioned that in low energy limit, QEDapproximation works well. However, in very high enough energy, electroncan collide with proton hard enough so that proton break apart into jet ofquarks. Then, traditional parametrization will no longer be able to use.This is known as Deep inelastic scattering (DIS).

To fix this, we need to understand what really happened at collision region.


e−p+ scattering (VII)

Feynman suggested that the collision of proton or other hadrons can berelated to the collision of things inside that hadron (valence quarks, seaquarks and gluons). He call it parton.

The partonic momentum is related to proton’s momentum as

pµi = xPµ. (7)

where pµi is 4-momentum of parton of species i, Pµ be proton’s4-momentum and Bjorken value x.


e−p+ scattering (VIII)

The amazing fact about parton model is that form factors , also crosssection, are approximately independent of energy scale we are doingexperiments. This behavior is also widely known as Bjorken scaling.

Actually, we will discuss about these phenomenon as the strong constraintfor constructing the quantum theory of strong interaction soon!


e−p+ scattering (IX)

Figure: Bjorken scaling retrieved fromhttps://www.science20.com/a quantum diaries survivor/wolf prize in physics awarded to james bjorken-152846Thitipat Sainapha Origin of QCD November 15, 2018 24 / 48

Overview

1 Strong InteractionYukawa effective theoryQuark modele−p+ Scattering

2 Construct the QCDRenormalization group flowAsymptotic freedom


Renormalization group flow (I)

Before we go through the problem mentioned in previous section, let’sstudy about the useful ingredients parallelly discovered first.

At the beginning of QED era, people got very confused about the problemof divergences appearing at all loop level QFT calculation. For example,the quantum correction of Coulomb potential turned out to be infinite, sothe shift of energy level between 2s orbital and 2p orbital is also infinite!Infinite Lamb shift!


Renormalization group flow (II)

Many people including Dirac himself threw QED into trash bin, waitinguntil Feynman, Schwinger and Tomonaga to rescue it.

The key idea is they argued that infinities are not observable, also allparameters such as mass, coupling constant, field. The observables mustbe finite. Thus, the parameters appearing in the Lagrangian all haveinfinity in them. To get observable, all of that parameters must berenormalized.

Suppose we have any operator unrenormalized O0 called bare operator, wecan renormalize this operator as

O = ZOO0 = (1 + δO)O0 (8)


Renormalization group flow (III)

That implies any observable parameter can be thought as its bareparameter in different scale.

This leads us to relate to value of parameter at two different scalesdetermined by first order differential equation known as renormalizationgroup equation, discovered firstly by Gell-Mann and Low.

As in Hamiltonian mechanics, first order differential equation can be usedto understand the flow of variable in phase space-like structure. (How thename renormalization group flow or RG flow coming from)


Renormalization group flow (IV)

Let’s consider the pure Yang-Mills theory with Lagrangian density

LYM = −1

2Tr [(∂µAν − ∂νAµ − ig [Aµ,Aν ])2] (9)

This Lagrangian is singular because of gauge redundant degree of freedom.Definition of Hamiltonian is then not unique. This non-uniqueness makesus impossible to quantize the theory. To fix this, we need to add extra termto fix gauge sometimes known as Fermi gauge fixing term −1

ξTr [(∂µAµ)2]

with non-dynamical field ξ so-called Nakanishi-Lautrup auxiliary field


Renormalization group flow (V)

Since YM is non-Abelian gauge theory, additional term makes Jacobian ofpath integral non-trivial. The cost of non-Abelian gauge fixing is we needto introduce more fields which are scalar field but satisfy Fermi-Diracstatistics, c and c says.

The existence of this field violates spin-statistics theorem and make theorybecoming non-unitary; hence, this fields are called Faddeev-Popov ghostand anti-ghost, respectively.

The existence of ghosts modified YM theory to be

LYM = −12Tr [(∂µAν − ∂νAµ − ig [Aµ,Aν ])2]− 1

ξTr [(∂µAµ)2]+

2Tr [cc − g(∂µc)[Aµ, c]](10)


Renormalization group flow (VI)

Note that Lagrangian (10) is no longer invariant under gauge symmetrybut two new symmetry birth: U(1) ghost symmetry and BRST symmetry.

BRST symmetry forms so-called BRST algebra which is so much similar tosuperalgebra. These two symmetry led Kugo and Ojima to postulate thestrong constraint on how quark are confined. We already discussed theevidences of confinement behavior in context of Regge’s trajectories.(Regge’s slope implies that potential energy somehow is linearproportionality and no quark can be escape from this potential!)


Renormalization group flow (VII)

Again all parameters in Lagrangian (10) are infinite. We need torenormalize all of that things. In pure YM, there are no infrared cutoff (IRdivergence are not regularized since generally fermion’s mass will act as IRregulator). We need to perform the renormalized at non-zero Euclideanenergy scale. Namely, p2 = −µ2

We define bare gluon propagator to be

D0(−µ2)abµν =i

µ2Z3

(gµν +

kµkνµ2

)δab (11)


Renormalization group flow (VIII)

and bare ghost propagator

G0(−µ2)abµν = − i

µ2Z3δab (12)

Also define charge renormalization constant Z1 in term of bare three-pointvertex function

Γabc0 (p, q, r)µνρ = Z−11 Γabc(p, q, r)µνρ (13)


Renormalization group flow (IX)

For ghost vertex, we have

Γabc0 (p, q, r)µνρ = Z−11 Γabc(p, q, r)µνρ (14)

According to Ward-Takahashi identity, these identity gives the relationamong renormalization constants (11)-(14)

Z3

Z1=

Z3

Z1

(15)


Renormalization group flow (X)

Ward-Takahashi identity implies that the way we renormalize the gluonpropagator is the same way we renormalize ghost. Thus ghost is harmfuland useful at the same times, one can compute the 1-loop gluon vacuumpolarization contributions, only gluon loop amplitudes alone can’t begauge invariant. After we add ghost loop contribution, the total amplitudebecomes manifestly gauge invariant!


Renormalization group flow (XI)

The last relevant renormalization factor comes from longitudinal part ofpropagator

(D−10 )abµν = ...+i

ξ0kµkν , (16)

This longitudinal mode is unphysical since this mode is proportional to4-momentum of gluon corresponding to pure gauge degree of freedom.Thus, after quantization procedure, we can gauge away this degree offreedom by choosing gauge fixing choice such that ξ = ξ0 = 0 calledLandau gauge.


Renormalization group flow (XII)

From all renormalization factors we defined previously, we can extract thefactors for renormalized wave functions and renormalized coupling constant

Aaµ = Z

−1/23 (A0)aµ

caµ = Z−1/23 (c0)aµ,

(17)

andg = Z

3/23 Z−11 g0 (18)


Renormalization group flow (XII)

Suppose our interests is any 1-particle irreducible contribution (1PI)containing any numbers and any kinds of fields and parameters. However,The bare 1PI Green’s function will independent of change of energy scalebecause bare variables are fixed at only some scale (scale of Lagrangiandensity). Mathematically,

µ∂

∂µΓ(n)0 (Λ, g0) = 0, (19)

Using chain’s rule, we finally obtain[µ∂

∂µ+ β(g)

∂

∂g− nγ(g)

]Γ(n)(g , µ) = 0 (20)

Equation (21) is the Gell-Mann-Low renormalization group equation wewere talking about.


Renormalization group flow (XIII)

Each new functions are defined by

β(g) ≡ µ∂g∂µ

(21)

γ(g) ≡ 1

2µ∂ln(Z3)

∂µ(22)

The function (23) are called anomalous dimension determined thederiation of conformal scaling, e.g. if we do classically conformaltransformation p → λp. In quantum level, 1PI Green’s function will changeby classical scaling with extra factor proportional to gamma function.


Renormalization group flow (XIV)

More important differentiable function is β function because it isdetermined the change of coupling constant as energy scale changing.Thus, the running of coupling encoded inside these powerful function.There are 3 important case depending on value of β.

β > 0 coupling constant increases as increasing of energy scale.

β = 0 theory is scaleless, e.g. conformal field theory.

β < 0 coupling constant decrease as increasing of energy scale.


Asymptotic freedom (I)

One might naturally ask that why case of running coupling of stronginteraction should be?

The answer can be obtained from the hints specifically Bjorken scalingbehavior. Bjorken scaling is occured at high energy deep inelasticscattering (experiment done by SLAC). Bjorken scaling implies the runningof cross section (determined by Gell-Mann-Low equation) must be verysmall. Alternatively, β(g∞) ≈ 0 with g∞ coupling at ultra high energylevel. Finally, we reach the fact the strong interaction must be UV stableor asymptotically free!


Asymptotic freedom (II)

Coleman and Gross proved that no theory without non-Abelian gauge fieldcan be asymptotically free. All we have to do is just calculate the 1-looprenormalization constant of Yang-Mills theory.

Z3 can be extracted from 1-loop gluon vacuum polarization (includingfermion loop graph) We obtain

Z3 = 1 +1

ε

g2

16π2

(13

3CA −

8

3Nf TF

)(23)

For ghost vacuum polarization, we get

Z3 = 1 +1

ε

g2

16π2

[3

2CA

](24)


Asymptotic freedom (III)

On the other hand, Z1 can be found through calculation the 3-pointfunction

Z1 = 1 +1

ε

g2

16π2

(−3

2CA

)(25)

Finally, Z1 is not necessary to compute because we can calculatealternatively by using Ward-Takahashi identity (16)


Asymptotic freedom (IV)

Solving for β function by

β(g) = −gµ ∂

∂µ

(Z 3/2

Z1

)(26)

After long calculation, we can solve perturbatively to get 1-loop betafunction

β(g) = −ε2g − g3

16π2

(11

3CA −

4

3Nf TF

)(27)

where ε is UV cut off setting to be zero after ending the full calculation.CA is Casimir operator of adjoint representation defined to beCA = C (Adj), d(R)C (R) = T (R)d(G ) and Tr [T a

RTbR ] = T (R)δab.


Asymptotic freedom (V)

For SU(N) gauge theory. d(G ) = N2 − 1, T (Fund) = 12 and T (Adj) = N.

We can easily get the full expression of beta function in SU(N) gaugetheory

β(g) = − g3

16π2

(11

3N − 2

3NF

). (28)

We found that for the case that N > 2Nf11 , the theory will be

asymptotically free.


Asymptotic freedom (VI)

Our current knowledge is number of flavors Nf = 6, then for SU(N) for Ngreater than 1, everything will be fine. This also implies Coleman andGross proof. Then what N should it be? Since we know that color degreeof freedom has been discovered. For baryon decuplet state to beanti-symmetric, it should be color singlet. According to group theory, theonly possible way that 3 baryons can have color singlet is color symmetryshould belong to SU(3)

The discovering of asymptotic freedom led to the origin of quantum theoryof strong interaction known as quantum chromodynamic (QCD). Thus,this discovering due to the work of David Gross and his advisee FrankWilzcek is worth the Nobel prize in 2004! 30 years after published theirpaper.


References

Gross, D.J. and Wilczek, F. Physical Review D. 8 (1973)

Schwartz, M.D. Quantum field thoery and the standard model; New York,NY:Cambridge University Press (2014)


The End


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Origin of Quantum Chromodynamics · More importantly, the diagram in the previous slide is for spin 3/2 baryons. Although baryons are fermion, the 3/2 spin baryons seem not to support