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Introduction to SupersymmetryJoseph Conlon
Hilary Term 2010
1 Introduction and References
These notes accompany the supersymmetry lecures given in Hilary and Trinityterm 2009. There is nothing here that cannot be improved upon so here aresome suggestions for further reading:
• Weak Scale Supersymmetry, CUP (2006), by Howard Baer and XerxesTata.
The focus of this book is really supersymmetry at the LHC, written bytwo leading SUSY phenomenologists. If your interest in supersymmetryis primarily in the MSSM and its phenomenology, this is the book to buy.
• The Soft Supersymmetry Breaking Lagrangian: Theory and Applications,hep-ph/0312378, Chung et al.
This is a good review of the MSSM and supersymmetry breaking. It’sphenomenologically minded and is reasonably easy to read and dip in andout of. Not deep, but rather a broad first reference for TeV supersymmetryand its phenomenology.
• Part III Supersymmetry Course Notes, Fernando Quevedo and OliverSchlotterer, http://www.damtp.cam.ac.uk/user/fq201/susynotes.pdf.
These are the lecture notes for the 24 hour Part III supersymmetry courseat Cambridge given by Fernando Quevedo, which this course is in partbased upon. This should be at a similar level to the material presentedhere.
• Supersymmetry and Supergravity, Princeton University Press (1992), JuliusWess and Jonathan Bagger.
Commonly known just as Wess and Bagger, this book is a genuine classicbut is not dip-in reading. Dry but correct. Best used as a reference book.Very good on supergravity and has the 4d N=1 supergravity action inexplicit complete gory detail.
• Quantum Theory of Fields, Volume III, CUP(2000), Steven Weinberg.
Everything you ever wanted to know about supersymmetry. The advan-tage and disadvantage of these books is that Weinberg is one of the greatmasters of quantum field theory. You learn a lot, but you pay in bloodfor it.
• Modern Supersymmetry, OUP?(1995?), John Terning.
A good general textbook on supersymmetry. A book with ‘Modern’ in itstitle dates itself by its topics. As a consequence this book is particularlystrong on the aspects of non-perturbative supersymmetric field theory, butmuch weaker on supergravity.
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2 Motivations for Supersymmetry
Why is supersymmetry even worth considering as a symmetry of nature? Wepresent three reasons here, starting from the more formal and ending in themore phenomenological.
2.1 Susy is Special
We know nature likes symmetries. Symmetry principles underlie most of thedynamics of the Standard Model.
We are familiar with two basic types of symmetry. The particles of theStandard Model are defined as representations of the SO(3, 1) symmetry ofMinkowski space. The Minwoski space metric
ds2 = −dt2 + dx2 + dy2 + dz2 (1)
is invariant under SO(3, 1) Lorentz transformations of the coordinates. Thesetransformations are the Lorentz transformations and are the foundation ofspecial relativity. For example, under a Lorentz transformation given by anSO(3, 1) matrix Mµν , a vector field Aµ transforms as
Anewµ = MµνAoldν . (2)
Other representations are the spinor representation that describes fermions suchas the quarks and leptons and the (trivial) scalar representation which describesthe Higgs.
A second type of symmetry are internal symmetries. These may be eitherglobal symmetries (such as the approximate SU(2) isospin symmetry of thestrong interactions) or local (gauge) symmetries. An example of these is theSU(3) gauge symmetry of the strong interactions which transforms quark statesof different colour into one another. Internal symmetries do not change theLorentz indices of a particle: an SU(3)c rotation changes the colour of a quark,but not its spin.
It seems like a cute idea to try and combine these two types of symmetriesinto one big symmetry group, which would contain spacetime SO(3, 1) in onepart and internal symmetries such as SU(3) in another. In fact this was an activearea of research in prehistoric times (the 1960s), when there were attempts tocombine all known symmetries into a single group such as SU(6). However thisprogram was killed by a series of no-go results, culminating in the Coleman-Mandula theorem (1967):
Coleman-Mandula Theorem: The most general bosonic symmetry ofscattering amplitudes (i.e. of the S-matrix) is a direct product of the Poincareand internal symmetries.
G = GPoincare ×Ginternal
In other words, program over. There is no non-trivial way to combine space-time and internal symmetries.
2
However, like all good no-go theorems, the Coleman-Mandula theorem has aloophole. Not knowing better, Coleman and Mandula only considered bosonicsymmetry generators. In the early 1970s supersymmetry began to appear onthe scene, and in 1975 Haag, Lopuskanski and Sohnius extended the Coleman-Mandula theorem to include the case of fermionic symmetry generators, whichrelate particles of different spins.
The result?
Haag, Lopuskanski and Sohnius: The most general symmetry of theS-matrix is a direct product of super-Poincare and internal symmetries.
G = Gsuper−Poincare ×Ginternal
The super-Poincare algebra is the extension of the Poincare group to includetransformations that turn bosons into fermions and fermions into bosons: su-persymmetry transformations. So Coleman-Mandula were almost right, but notquite: there is one non-trivial way to extend the spacetime symmetries, andthat is to incorporate supersymmetry.
The upshot of this is that supersymmetry appears in a special way: it is theunique extension of the Lorentz group as a symmetry of scattering amplitudes.This represents one reason to take supersymmetry seriously as a possible newsymmetry of nature.
2.2 Strings and Quantum Gravity
Einstein’s theory of general relativity is described, just like other theories, by aLagrangian
LGR = 16π2M2P
∫d4x
√gR.
Unlike the Standard Model, general relativity is a non-renormalisable theory: allinteractions are suppressed by a scale MP = 2.4× 1018GeV = 1.7× 1014ELHC .This is somewhat analogous to the Fermi theory of weak interactions, whichis also non-renormalisable, and where interactions are suppressed by 1/M2
W ∼1/(100GeV)2).
At energy scales below MP , general relativity works just fine. However atenergy scales above MP , just as for Fermi theory above MW , general relativityautomatically loses predictive power. There are an infinite number of countert-erms to be included in loop diagrams (R2,R3, . . .) and no way to determine theircoefficients. As all such higher-derivative operators become equally importantabove MP , all predictivity is lost. This is the problem of quantum gravity.
Theorists rush where experimenters fear to tread, and despite (or in samecases because of) the absence of data lots of people have spent lots of timethinking about quantum gravity and how to formulate a consistent theory ofit. The upshot is that the most attractive approach is that of string theory.String theory succeeds, often in surprising ways, in taming the divergences ofquantum gravity and giving finite answers. It turns out supersymmetry plays
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a central and essential role in this: in any string theory, nature always lookssupersymmetric at sufficiently high energy scales.
If string theory is telling us something about nature, nature is supersym-metric at some energy scale, giving us good reason to regard supersymmetryas a genuine symmetry of nature. The only problem is that it gives us noreason why this scale should be any smaller than the quantum gravity scale,MP = 2.4× 1018GeV.
2.3 The Hierarchy Problem
The hierarchy problem is the one really good reason why there is a fair chancesupersymmetry will show up at the TeV energy scale.
The hierarchy problem is the problem of why the weak scale is so muchsmaller than the Planck scale. It is also the most serious theoretical problem ofthe Standard Model. To recall, in the Standard Model the electroweak symme-try is broken by a vacuum expectation value for the Higgs boson, 〈v〉 = 246GeV.The size of this vev determines the masses of the Z and W bosons, mW = gw√
2v.
The vev of the Higgs is determined by the parameters µ and λ in the Higgspotential,
VH = −µ2|φ|+ λ|φ|4. (3)
As in any quantum theory, the parameters in this Lagrangian are subject toquantum corrections. The actual value of the Higgs vev is determined not bythe classical values of µ and λ, but by those after quantum corrections. If theStandard Model is valid up to a scale Λ (i.e. no new particles appear until wereach the scale Λ) then we can estimate the size of quantum corrections to theterm µ2. The three largest contributions come from the Higgs self-loop, the Wloop and the top quark loop. Let’s focus on the top quark loop, shown in figure2.3.
Figure 1: Quantum Corrections to the Higgs Mass.
This diagram gives a correction to the mass term in (3). This amplitude
4
behaves as
M ∼ −i∫
d4k
(2π)4−iyt/k −mt
−iyt/k + /p−mt
→|k|→∞ y2t
∫d4k
(2π)41k2
∼ y2t
16π2
∫ Λ
mt
|k|dk ∼ y2t
16π2Λ2. (4)
If the Standard Model is valid up to the GUT scale (Λ ∼ 1016GeV), the quantumcorrection to the Higgs mass term is of size δµ2 ∼ y2
t
16π2 Λ2 m2W . Quantum
corrections to the Higgs potential are then enormous: we would expect thequantum-corrected Higgs vev to be around the GUT scale, not the weak scale.However, we know the masses of the W and Z bosons are O(102GeV) and notO(1016)GeV.
This is the hierarchy problem: why is the weak scale so much less than thePlanck scale? There are two options. Either there is an enormous cancellationgoing on, or the Standard Model is not valid up to the GUT scale and newphysics comes in at the TeV scale, cancelling the divergent terms in the Higgspotential.
TeV supersymmetry has the attractive feature of automatically taming theHiggs divergence. This finally makes it reasonable that supersymmetry is notjust a symmetry of nature, but a symmetry that could appear at the next majorcollider, the LHC.
3 SL(2, C) and dotted and undotted indices
Before we discuss the supersymmetry algebra it is helpful to establish notationand conventions. What we are doing here is setting up the appropriate languageto write and manipulate 2-component spinors. The SO(3, 1) Lorentz algebra is
[Pµ, P ν ] = 0.[Mµν , Pα] = i(Pµηνα − P νηµα) (5)
[Mµν ,Mρσ] = i(Mνσηνρ +Mνρηµσ −Mµρηνσ −Mνσηµρ). (6)
Let’s define Ji = 12εijkMjk where i, j, k = 1, 2, 3 as the generator of spatial
rotations and Ki = M0i as the generator of Lorentz boosts. We can use theseto rewrite the Lorentz algebra as
[Ji, Jj ] = εijkJk (7)[Ki,Kj ] = −iεijkJk (8)[Ji,Kj ] = iεijkKk. (9)
If we defineAi =
12(Ji + iKi),
Bi =12(Ji − iKi),
5
we can rewrite the Lorentz algebra as
[Ai, Aj ] = iεijkAk (10)[Bi, Bj ] = iεijkBk (11)[Ai, Bj ] = 0. (12)
This gives two copies of the SU(2) algebra, from which we infer that, at leastlocally,
SO(3, 1) ≡ SU(2)× SU(2). (13)
This tells us that we can label representations of SO(3, 1) by two SU(2) ’charges’:as SU(2) reps are labelled by a positive integer 2S + 1, where S is the spin.Examples are
Scalar ≡ (0, 0)
Left-handed spinor ≡ (12, 0)
Right-handed spinor ≡ (0,12)
vector ≡ (12,12). (14)
Note that we should really replace SO(3, 1) by the universal covering groupSpin(3, 1) to account for fermionic antiperiodicity.
For representing spinors it is helpful to use the map between Spin(3, 1) andSL(2,C). SL(2,C) is the group of special (unit determinant), linear (acts asa matrix) complex transformations on 2-component vectors. Briefly, left andright-handed spinors will be the fundamental and antifundamental representa-tions of SL(2,C).
Let’s first describe the map between Spin(3, 1) and SL(2,C). Suppose wehave a 4d vector,
X = (x0, x1, x2, x3). (15)
We can map this to a 2× 2 hermitian matrix as
X ≡ Xµσµ =
(x0 + x3 x1 − ix2
x1 + ix2 x0 − x3
), (16)
where σµ =(
1 00 1
),
(0 11 0
),
(0 i−i 0
),
(1 00 −1
)are the stan-
dard Pauli matrices. Note that this map is invertible. Note that det X =XµXµ = x2
0 − x21 − x2
2 − x23.
There are natural actions on both 4-component vectors and 2× 2 matrices.On the vector the natural action is
Xµ → ΛµνXν , (17)
where Λ is an SO(3, 1) matrix. On the matrix the natural action is
X → NXN†. (18)
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As the transformations are special they have unit determinant, and so det X′=
det X. This implies that using the vector/matrix correspondence the transfor-mations preserve the SO(3, 1) norm and thus correspond to Lorentz transfor-mations. There is therefore a map between SL(2,C) and SO(3, 1). You cancheck that the map is a homomorphism and preserves the group structure.
However, the map is not an isomorphism as
N = ±(
1 00 1
)both map to 1 ∈ SO(3, 1).
This reflects that the fact that SL(2,C) is equivalent to Spin(3, 1) which is infact the double cover of SO(3, 1).
• Exercise: Show that the explicit map from SL(2,C) → SO(3, 1) is givenby
Λµν (N) =12Tr[σµNσνN
†] . (19)
The relationship between SL(2,C) ≡ Spin(3, 1) and SO(3, 1) is very similar tothe relationship between SU(2) and SO(3): one is the double cover of the other.
3.1 Spinor Representations
The basic rule is that undotted indices correspond to left-handed ( 12 , 0) spinors
and dotted indices correspond to right-handed (0, 12 ) spinors.
The fundamental representation of SL(2,C) corresponds to the left-handedspinor and the anti-fundamental representation to the right-handed spinor:
ψ′
α = Nβαψβ (20)
χ′
α = (N∗)βαχβ . (21)
Generally in susy formulae there are two kinds of indices: spinor (α, β, α, β) andspacetime (µ, ν, ρ, σ). Spacetime indices are raised and lowered by the metrictensor gµν and transform via SO(3, 1) matrices Λµν .
Spinor indices are raised and lowered by the SU(2) invariant tensor εαβ ,given by
εαβ = εαβ =(
0 −11 0
)(22)
εαβ = εαβ =(
0 1−1 0
)(23)
εαβεβγ = δαγ . (24)
Raised indices transform as
(ψ′)α = ψβN−1 α
β (25)
χα = χβ(N∗)−1 α
β(26)
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An example of a ‘mixed’ tensor is σµαα, which transforms as
(σρ)αα = (Λ−1)ρν︸ ︷︷ ︸SO(3,1)
Nβα (σν)ββ(N
∗)βα︸ ︷︷ ︸SL(2,C)
(27)
We can also define (σµ)αα ≡ εαβεαβσµββ
. This has the standard form σµ =
(1,−σi). We can now define
(σµν)βα =i
4(σµσν − σν σµ)βα (28)
(σµν)αβ =i
4(σµσν − σνσµ)βα = (σµν)† (29)
These matrices satisfy the Lorentz algebra
[σµν , σρσ] = i (σµσηνρ + σνρηµσ − σµρηνσ − σνσηµρ) (30)
and we can use this to write general Lorentz transformations as
Nβα =
(e−
i2ωµνσ
µν)βα. (31)
Some useful identities are
σµσν + σν σµ = 2ηµν(
1 00 1
)(32)
Tr (σµσν) = 2ηµν (33)
σµαασββµ = 2δβαδ
βα (34)
σµν =12iεµνρσσρσ (35)
σµν =−12iεµνρσσρσ. (36)
Spinor products and Fierz identities are
χψ ≡ χαψα = −χαψα = ψαχα = ψχ. (37)
χψ ≡ χαψα = −χαψα = ψχ. (38)
ψψ = ψαψα = εαβψβψα = ψ2ψ1 − ψ1ψ2. (39)
For Grassmanian ψ, we can also write ψαψβ = 12εαβψψ. We can also enumerate
a useful set of Fierz identities:
(θψ)(θψ) = −12(ψψ)(θθ) = −1
2(θθ)(ψψ). (40)
ψσµνχ = −χσµνψ. (41)
ψαχβ =12εαβ (ψχ) +
12(σµνεT
)αβ
(ψσµνχ) . (42)
ψαχα =12
(ψσµχ)σµαα. (43)
(ψχ)† = (ψαχα)† = χα ∗ ψ∗,α = χαψα = χψ. (44)
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4 Supersymmetry Algebras
4.1 Form of the Supersymmetry Algebra
The supersymmetry algebra involves extending the ordinary Lorentz symme-try group of 4d quantum field theory by additional fermionic generators. Thefermionic generators are Weyl spinors.
We include N such generators QAα ,A = 1 . . . N into the algebra, as well astheir Hermitian conjugates Q ˙alpha. Here α and α are both spinor indices, trans-forming as the fundamental(anti-fundamental) of SU(2,C). Undotted indicescorrespond to left-handed fermions, and dotted indices correspond to right-handed fermions. A Weyl spinor has 2D/2 complex components, and so in 4dimensions has 4 real components.
A supersymmetry algebra with 1 spinor generator Qα is called N = 1 super-symmetric, one with two spinor generators Q1
α, Q2α called N = 2 supersymmet-
ric, etc. As a Weyl spinor in 4d has 4 real components, a 4d theory with N = 1supersymmetry has a total of 4 real supercharges. Note that this is dimension-dependent. For example, in 10 dimensions the smallest spinor representation isthe Majorana-Weyl representation, which has 8 complex components. N = 1supersymmetry in 10d therefore has sixteen supercharges, and N = 2 super-symmetry thirty-two supercharges.
The N = 1 supersymmetry algebra isQα, Qβ
= 2σm
αβPm, (45)
Qα, Qβ =Qα, Qβ
= 0, (46)
[Pm, Qα] =[Pm, Qα
]= 0, (47)
[Pm, Pn] = 0. (48)
As the susy generators are spinors, they naturally anticommute rather thancommute. There are also transformations involving the Lorentz generators Λαβ ,which are not shown here. These transformations can be summarised by thestatement that Qα transforms as a spinor under Lorentz transformations, so
[Qα,Mµν ] = (σµν)βαQβ . (49)
In general susy transformations commute with all internal symmetries (forexample flavour symmetries). The exceptions to this are known as R-symmetries(R-parity in the MSSM is an example of an R-symmetry). Such symmetries arisefrom automorphisms of the susy algebra.
Qα −→ eiγQα, (50)Qα −→ e−iγQα. (51)
If this is a symmetry of the Lagrangian (in general it isn’t), then it representsa U(1) R-symmetry. The generator R has commutation relations
[Qα, R] = Qα, (52)[Qα, R
]= −Qα. (53)
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Note that R-symmetries act differently on bosons and fermions in the samesupermultiplet (cf R parity in the MSSM, which has value 1 for Standard Modelparticles and −1 for their MSSM superpartners).
Suppose R|b〉 = r|b〉, and |f〉 = Q|b〉. Then
RQ|b〉 −QR|b〉 = Q|b〉, (54)
and so R(Q|b〉) = (r + 1)(Q|b〉), giving
R|f〉 = (r + 1)|f〉, for R|b〉 = r|b〉. (55)
It is relatively straightforward to extend the supersymmetry algebra to moregenerators. If we have N generators, the algebra becomes (Qα → QAα , A =1 . . . N)
QAα , QβB
= 2
(σµαβPµ
)δAB .
QAα , QBβ
= ZAB . (56)
Z is called the central charge of the algebra and it commutes with all othergenerators.[
ZAB , Pµ]
=[ZAB ,Mµν
]=[ZAB , QAα
]=[ZAB , ZCD
]= 0.
The antisymmetry of Z means it can only be present in algebras with extended(N ≥ 2) supersymmetry. Z can be diagonalised. As it is antisymmetric, Her-mitian transformations can be used to bring it to the canonical forms
N even Z =
0 q1 0 0 . . . . . . 0 0−q1 0 0 0 . . . . . . 0 00 0 0 q2 . . . . . . 0 00 0 q2 0 . . . . . . 0 0. . . . . . . . . . . . . . . . . . 0 00 0 0 0 . . . . . . 0 qN/20 0 0 0 . . . . . . −qN/2 0
(57)
N odd Z =
0 q1 0 0 . . . 0 0 0−q1 0 0 0 . . . 0 0 00 0 0 q2 . . . 0 0 00 0 q2 0 . . . 0 0 0. . . . . . . . . . . . . . . 0 0 00 0 0 0 . . . 0 qN/2 00 0 0 0 . . . −qN/2 0 00 0 0 0 0 0 0 0
(58)
4.2 Representations of the Supersymmetry Algebra
Given a symmetry group, the first important thing to do is to study the rep-resentations. For the Lorentz group, the study of the representations leads tothe classification of particles as either scalar, spinor or vector. For the gaugesymmetry groups SU(3) and SU(2) , the study of the representations underlies
10
the structure and interactions of the Standard Model. In a similar way thestructure and interactions of supersymmetric theories are determined to a largeextent by the representations of the susy algebra.
We start by proving an important if unsurprising result: the number ofbosons and fermions in any massive representation is identical.
Consider the operator (−1)NF , which acts on 1-particle states as
(−1)NF |b〉 = |b〉, (59)(−1)NF |f〉 = −|f〉. (60)
The generalisation to n-particle states is obvious. As Qα converts fermions intobosons and vice-versa,
(−1)NFQα = −Qα(−1)NF . (61)
Given a finite-dimensional representation, we therefore have
Tr[(−1)NF
QAα , QβB
]= Tr
[(−1)NF
(QAα QβB + QβBQ
Aα
)](62)
= Tr[−QAα (−1)NF QβB +QAα (−1)NF QβB
](63)
= 0. (64)
HoweverQAα , QβB
= 2
(σµαβPµ
)δAB , and so the left-hand side becomes
TR[(−1)NF 2
(σµαβPµ
)δAB
].
QAα commutes with Pµ, and so all states in the representation have the samemomentum. We can therefore pull the 4-vector 〈Pµ〉 out, to get
0 = 2〈Pµ〉σµαβδABTr
[(−1)NF
]. (65)
For this to hold for all α, β,A,B we need
Tr[(−1)NF
]= 0. (66)
That is, the representation contains identical numbers of bosons and fermions.
Note that this result required 〈Pµ〉 6= 0. This is not a mere technicality.For example, the vacuum of a quantum field theory has 〈Pµ〉 = 0, and for thevacuum Qα|vac〉 = 0: there are only bosonic states in the representation. In thespecific case of the vacuum Tr
[(−1)NF
]is known as the Witten index. Although
currently this seems a bit trivial, we shall see later that we can use this to giveimportant information about quantum field theories at strong coupling.
4.3 One-particle representations
Let us now consider one particle representations of the supersymmetry algebra.We will start with massless particles, and then move on to massive representa-tions. Recall that the algebra is
QAα , QβB
= 2
(σµαβPµ
)δAB .
11
For a massless particle we can boost to a light-like reference frame where the4-momentum is given by Pµ = (E, 0, 0, E). In this case
σµαβPµ = Eσ0 + Eσ3 (67)
= E
((1 00 1
)+(
1 00 −1
))(68)
= 2E(
1 00 0
)αβ
. (69)
The supersymmetry algebra is therefore given byQAα , QβB
= 4E
(1 00 0
)αβ
δAB . (70)
Specialising to N = 1 supersymmetry we obtainQ1, Q1
= 4E, (71)
Q2, Q2
= 0. (72)
Consider a state |P, λ〉 of momentum P and helicity λ. If we write a = Q1
2√E
and a† = Q1
2√E
, then we obtain the algebraa, a†
= 1, a, a = a†, a† = 0.
This is precisely the algebra of fermionic creation and annihilation operators,suggesting that we can form representations by starting at a vacuum state |Ω〉and acting with creation operators. Furthermore, as [Qα, J3] = 1
2σ3αβQβ (as Qα
transforms as a Weyl spinor), we have [Q1, J3] = 1
2Q1, and so [a, J3] = 12a. We
then obtain
J3a|P, λ〉 = (−[a, J3] + aJ3) |P, λ〉 (73)
=(−a
2+ aλ
)|P, λ〉 (74)
= a(λ− 12)|P, λ〉. (75)
Therefore a|P, λ〉 has helicity λ− 12 and adagger|P, λ〉 has helicity (λ+ 1
2 ). So cre-ation operators increase the helicity of a state by 1
2 and annihiliation operatorsdecrease the helicity by 1
2 .
Also note that Q2|P, λ〉 = 0 for any state. Why?
Q2, Q2 = 0 → 〈P, λ|Q2Q2 + Q2Q2|P, λ〉 = 0 (76)→ 〈P, λ|Q2Q2|P, λ〉 = −〈P, λ|Q2Q2|P, λ〉 (77)= ||Q2|P, λ〉||2 = −||Q2|P, λ〉||2. (78)
However positivity of the norm therefore requires Q2|P, λ〉 = 0 for all states.
We can now examine the representations of the susy algebra. We suppose wehave a state |Ω〉 in the representation annihilated by all annihilation operators.Then if |Omega〉 = |P, λ〉, a†|Omega〉 = |P, λ+ 1
2 〉 and a†a†|Ω〉 = 0. If a theory
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is also invariant under CPT (this is not an algebraic consequence, but does holdfor all relativistic quantum field theories) then each state must have a CPTconjugate. Normally the expression ‘susy representation’ will be used for thefull set of states,
|Ω〉+ a†|Omega〉+ |Omega′〉+ a|Ω′〉 >
where |Omega′ > is the CPT conjugate of |Omega >.
We can now enumerate the classic representations of N = 1 supersymmetry.
Chiral Multiplet
This has λ = 0 and so |Ω〉 = |P, 0〉. The states are
|Omega〉 = |P, 0〉, a†|Ω〉 = |P, 12〉, |Ω′〉 = |P, 0 >′, a|Ω′ >= |P,−1
2> .
The physical content of the chiral multiplet is one Weyl spinor and one com-plex scalar. The chiral multiplet is so called because it allows matter in chiralrepresentations of the gauge group. Examples of chiral multiplets in the MSSMare (λ = 0, λ = 1
2 ) being (squark, quark), (higgs, higgsino), (slepton, lepton).
Vector Multiplet
This has λ = 12 and so |Ω〉 = |P, 1
2 〉. The states are
|Omega〉 = |P, 12〉, a†|Ω〉 = |P, 1〉, |Ω
′〉 = |P,−1
2>
′, a|Ω
′>= |P,−1 > .
The physical content of the vector multiplet is one 2-component spinor (gaugino)and one vector boson (a gauge boson). The vector multiplet generalises gaugeinteractions to supersymmetric field theories. Examples of vector multiplets inthe MSSM are (λ = 1
2 , λ = 1) being (gluino, gluon), (photino, photon), (Wino,W).
Gravity Multiplet
This has λ = 3/2 and so |Ω〉 = |P, 3/2〉. The states are
|Omega〉 = |P, 3/2〉, a†|Ω〉 = |P, 2〉, |Ω′〉 = |P,−3/2 >
′, a|Ω
′>= |P,−2 > .
The physical content of the gravity multiplet is one spin 3/2 vector-spinor (grav-itino) and one spin 2 multiplet (the graviton). The gravity multiplet does notplay a role in globally supersymmetric quantum field theories, but is essentialto the formulation of locally supersymmetric theories (supergravity).
4.4 Massless Representations of N ≥ 2 supersymmetry
If Pµ = (E, 0, 0, E) the susy algebra wasQAα , QβB
= 4E
(1 00 0
)αβ
δAB .
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So for N > 1 susy we just haveQA1 , Q1B
= 4EδAB ,
QA2 , Q2B
= 0.
This simply generalises the N = 1 algebra so that we now have N copies offermionic creation and annihilation operators instead of just a single copy. As[QA1 , J3] = 1
2QA1 , we find
J3aA|P, λ >= aA(λ− 1
2)|P, λ >
for all A, and so each lowering operator reduces helicity by 12 and each rais-
ing operator increases helicity by 12 . The general state in a representation is
thereforeεAB...Ka†Aa
†B . . . a
†K |Ω >
where ε is antisymmetric and 0 ≤ K ≤ N . There are(NK
)states of helicity
λ +K/2. The ground state |Ω〉 has helicity λ and the top state a†1a†2 . . . a
†N |Ω〉
helicity λ+N/2. What are the classic representations of N ≥ 2 supersymmetry?
N=2 Hypermultiplet
This has |Ω〉 = |P,− 12 〉. It has the following states and helicities
a†1a†2|Ω〉 −1
2(79)
a†1|Ω〉, a†2|Ω〉 0, 0 (80)
|Ω〉 −12
(81)
This is the hypermultiplet. Important things to note are
1. As susy generators commute with gauge generators, the states of helicity± 1
2 have the same gauge charges. This representation (and in fact allN = 2 representations) is therefore non-chiral.
2. If the gauge representation is real, the N = 2 hypermultiplet is CPTself-conjugate.
3. The particle fermion is one spin-(1/2) fermion and one complex scalar.Including a CPT conjugate, we have two spin-(1/2) fermions and twocomplex scalars.
14
N=2 Vector multiplet
This has |Ω〉 = |P, 0〉. It has the following states and helicities
a†1a†2|Ω〉 +1 (82)
a†1|Ω〉, a†2|Ω〉 +1/2,+1/2 (83)
|Ω〉, |Ω′〉 0, 0 (84)a1|Ω′〉, a2|Ω′〉 −1/2,−1/2 (85)
a1a2|Ω′〉 −1. (86)
This is the vector multiplet, which contains a spin-1 gauge boson. The mattercontent of this multipet is one vector boson, one complex scalar and two spin-1/2fermions.
N=4 Vector Multiplet
There is only one important representation forN = 4 supersymmetry, the vectormultiplet. Here |Ω〉 = |P,−1〉. The states, helicities and multiplicities are
a†1a†2a
†3a
†4|Ω〉 +1 1 (87)
a†1a†2a
†3|Ω〉, +1/2 4 (88)
a†1a†2|Ω〉 0 6 (89)
a†1|Ω〉 −1/2 4 (90)|Ω〉 −1 1. (91)
This is the N = 4 vector multiplet. It consists of one vector boson, four spin-1/2fermions and three complex scalars. This is the multiplet present in the N = 4super Yang-Mills theory that has played such a prominent role in the AdS/CFTcorrespondence. N = 4 supersymmetry is maximal in global supersymmetry,as once we go above N = 4 we must necessarily include the gravitino in thespectrum, and this then also requires a graviton for a consistent theory.
N=8 Gravity Multiplet
This is essentially the unique representation of N = 8 supersymmetry. N = 8supersymmetry is often called maximal supersymmetry, as there is no consistentknown way of writing interacting theories for fundamental particles with spins
15
greater than 2. The states are
(a†)8|Ω〉 +2 1 (92)(a†)7|Ω〉, +3/2 8 (93)(a†)6|Ω〉 +1 28 (94)(a†)5|Ω〉 +1/2 56 (95)(a†)4|Ω〉 0 70 (96)(a†)3|Ω〉, −1/2 56 (97)(a†)2|Ω〉 −1 28 (98)(a†)1|Ω〉 −3/2 8 (99)
|Ω〉 −2 1. (100)
This is the gravity multiplet of N = 8 supersymmetry. The states consist of onegraviton, eight gravitini, 28 gauge bosons, 56 spin-1/2 fermions and 35 complexscalars.
With the N=8 gravity multiplet we conclude our study of massless represen-tations of the supersymmetry algebra. We now move to massive representations.
4.5 Massive One-Particle Representations
For massive particles we can boost to a frame where the 4-momentum is Pµ =(M, 0, 0, 0). In this case the susy algebra looks like (N=1)
Qα, Qβ
= 2
(σµαβPµ
)= 2E
(1 00 1
).
Writing Q1√2E, Q2√
2Eas a1, a2 and Q1√
2E, Q2√
2Eas a†1, a
†2, we have
a1, a†1 = 1, (101)
a2, a†2 = 1. (102)
In contrast to the massless case we now have two sets of fermionic creation andannihilation operators. In general, for the case of N susy generators (withoutcentral charges) we have 2N operators
aA1 , a†,B1 = δAB , (103)
aA2 , a†,B2 = δAB . (104)
Representations can be constructed as before by starting with a ground state|Ω〉 and acting with raising operators. However, there are some differencescompared to the massless case. First, as [Q, J3] = 1
2σ3Q, we have
[a1, J3] =12a1
[a2, J3] = −12a2
16
This means that a†1 increases the z spin by 1/2, while a†2 decreases it by 1/2.Second, as Pµ = (M, 0, 0, 0) the little group is SO(3) and particles in automat-ically in SO(3) ≈ SU(2) representations. So the total set of states in a massiveN=1 susy representation is given by
a†1a†2|Ω, J〉+ SO(3) rotations , (C)
a†1|Ω, J〉, a†2|Ω, J〉+ SO(3) rotations (B)
|Ω, J〉 + SO(3) rotations (A)
As we have already seen(a†1a
†2
)transforms as a 2 of SU(2). Therefore the
states (B) transform under SU(2) as 2× J = (J + 12 )⊕ (J− 1
2 ). Furthermore,a†1a
†2 is antisymmetric under interchange of a†1 and a†2 and so corresponds to the
εαβuαuβ 1 representation of SU(2). Therefore the SU(2) content of the states
isa†1a
†2|Ω, J〉 −→ J (2J + 1)
a†1|Ω, J〉, a†2|Ω, J〉 −→ (J +
12)⊕ (J − 1
2) (2J + 2) + 2J
|Ω, J〉 −→ J (2J + 1)
We therefore see that the massive susy reps have equal numbers of bosonic andfermionic states, as expected.
Let us consider some examples of massive representations of the supersym-metry algebra.
Massive Chiral Multiplet
We first suppose |Ω >= |P, 0〉. As states we have
Fermions: a†1|Ω〉, a†2|Ω〉
Bosons: |Ω〉, a†1a†2|Ω〉
These give one massive scalar and one spin-(1/2) fermion. In a supersymmetricfield theory, such a multiplet can be generated by giving a mass term to a chiralsuperfield, provided this is allowed by the gauge symmetries of the theory. Notethat all states in this representation are in the same gauge representation. Ifthis representation is real, then the multiplet can be CPT invariant, and in thiscase the particles have Majorana masses.
If the gauge representation is complex then CPT necessarily generates newparticles, and so the complete massive multiplet involves four fermionic and fourbosonic degrees of freedom, corresponding to a Dirac mass.
Vector Multiplet
In this case we start with |Ω〉 = |P, 12 〉. From our general analysis the matter
content of the theory is now found to be
1× Massive spin-1 vector 31×Real scalar 12×Spin 1/2 fermion 4
17
As expected there are equal numbers (4) of bosonic and fermionic states.
Note that there are the same total number of states as for 1 massless vectormultiplet and one massless chiral multiplet (why?).
4.6 Central Charges and BPS States
An interesting and important set of states occurs in the case that the super-symmetry algebra has central charges,
QLα, QMβ = εαβZLM . (105)
Here Z is antisymmetric, ZLM = −ZML. The antisymmetry of Z implies thatthis can only occur for N ≥ 2 supersymmetry. By unitary rotations we can putZ in a standard form (N even)
Z =
0 q1 0 0 . . . . . . 0 0−q1 0 0 0 . . . . . . 0 00 0 0 q2 . . . . . . 0 00 0 q2 0 . . . . . . 0 0. . . . . . . . . . . . . . . . . . 0 00 0 0 0 . . . . . . 0 qN/20 0 0 0 . . . . . . −qN/2 0
. (106)
We will study this algebra for the top 2× 2 block; the extension to the generalcase is then easy. We also will use Q for q1. The supersymmetry algebra givesus
Q11, Q
22 = Z, Q2
1, Q12 = −Z, Q1
1, Q2
2 = Z, Q2
1, Q1
2 = −Z.
QL1 , QM1 = QL2 , QM2 = 2MδLM
We can rewrite this algebra by defining
a1α =
1√2
(Q1α + εαβQ
2β
).
That is:a1 =
1√2
(Q1
1 + Q22
)a2 =
1√2
(Q1
2 − Q21
)Likewise we can define
b1 =1√2
(Q1
1 − Q22
)b2 =
1√2
(Q1
2 + Q21
)and so
a†1 =1√2
(Q1
1+Q2
2
), a†2 =
1√2
(Q1
2−Q2
1
).
18
b†1 =1√2
(Q11 −Q2
2
), b†2 =
1√2
(Q21 +Q2
1
).
These ladder operators satisfy the relations
a1, a1 = a2, a2 = b1, b1 = b2, b2 = a1, a2 = b1, b2 = 0.
However
a1, a†1 =
12Q1 + Q2
2
=12
(2M + 2M + Z + Z) = 2M + Z. (107)
and we also finda2, a
†2 = 2M + Z. (108)
Similar calculations giveb1, b†1 = −Z + 2M. (109)
and in general we have
aα, a†β = δαβ(2M + Z),
bα, b†β = δαβ(2M − Z). (110)
This tells us that 2M ≥ Z, as otherwise we could not have a positive definitenorm. To see this, suppose that 2M < Z. Then for a state |φ〉,
〈φ|b1b†1 + b†1b1|φ〉 < 0
and so||b†
1|φ〉||2 + ||b1|φ〉||2 < 0,
which implies that there is at least one state with negative norm.
Furthermore, as there can be no physical zero norm states, then if 2M = Zthen b and b† must annihilate the state, and are represented by the zero operator.
A state having 2M = Z is called BPS, and for such a state only the (a, a†)operators are active. In this case there are only four states in the representationin total (|Ω〉, a†1|Ω〉, a
†2|Ω〉, a
†1a
†2|Ω〉).
For general values of N, the algebra (110) becomes
aMα , a†,Lβ = δαβδ
ML (2M + ZL) (no summation over L),
bMα , b†,Lβ = δαβδ
ML (2M − ZL) (no summation over L). (111)
Here M,L = 1 . . . N/2, where N is the total number of supersymmetry genera-tors (e.g. N = 8 for maximal N = 8 supersymmetry). We can then reprise theabove arguments to see that
1. 2M ≥ Zi for all i.
2. A general state with 2M > Zi for all i has 22N states in the representationcoming from use of the 2N pairs of creation/annihilation operators. Sucha state is called a long multiplet.
19
3. A state with 2M = Zi is called BPS. If there are r central charges withvalue Zi, then there are 2N − r pairs of creation/annihilation operatorsand there are 22N−r states in the representation. Such a state is called ashort multiplet.
4. A state for which 2M = Zi for all i (which clearly requires Zi = Zj for alli, j) is called an ultrashort multiplet. Such a multiplet has a total of 2N
states, and is the smallest possible susy multiplet.
BPS multiplets have two very useful properties. They are smaller than reg-ular representations of the susy algebra and have mass/charge ratios that arefixed by the supersymmetry algebra. These properties are particularly usefulfor studying the strong coupling limit of supersymmetric theories, as these prop-erties are protected under a continuous variation in the gauge coupling. Thisallows results to be deduced at strong coupling, where we might not otherwisebe able to calculate.
5 Supersymmetric Field Theories
We now want to develop the formalism to construct and write supersymmetricfield theories. To do so it is convenient (although not necessary) to develop theidea of superspace.
We start by introducing fermionic Grassmann coordinates θα. These will beviewed as fermionic coordinates on superspace. The bosonic coordinates are theregular spatial coordinates xµ. We can write the supersymmetry algebra as[
ξαQα, ξαQα]
= 2(ξασ
mααξα
)Pm. (112)
i.e. [ξQ, ξQ
]= 2(ξσmξ)Pm. (113)
Note that these expressions involve commutators rather than anti-commutators.We can use this algebra to represent the supersymmetry algebra as translationson superspace.
Superspace is parametrised by
(Xm, θα, θα).
Let us define a group element (‘translations in superspace’) by
G(x, θ, θ) = exp
i−xmPm︸ ︷︷ ︸
spatial
+ θQ+ θQ︸ ︷︷ ︸fermionic
.
This represents superspace translations in both spatial and Grassmann direc-tions.
What happens if we have two such transformations in superspace? Consider
G(0, ξξ)G(xM , θ, θ) = ei[ξQ+ξQ]ei[−xmPm+θQ+θQ]. (114)
20
The Hausdorff formula is
eAeB = eA+B+ 12 [A,B]+higher commutators (115)
As [Q,P ] = 0 and [Q, Q] = P , all higher commutators vanish, and so it isstraightforward to multiply the two elements, obtaining
G(0, ξ, ξ)G(xM , θ, θ) = G(xM + iθσmξ − iξσmθ, θ + ξ, θ + ξ). (116)
This therefore induces a motion in parameter space
g(ξ, ξ)(xm, θ, θ) → (xm + iθσmξ − iξσmθ, θ + ξ, θ + ξ).
We can write this motion as
g(ξ, ξ) = ξQ+ ξQ = ξα(
∂
∂θα− iσmααθ
α∂m
)+ξα
(∂
∂θα− iθασαβmε
βα∂m
)(117)
These provide a representation of the supersymmetry algebra acting on super-space:
Qα =∂
∂θα− iσmααθ
α∂m, (118)
Qα =∂
∂θα− iθασm
αβεβα∂m, (119)
with Qα, Qα = 2iσαα (recall Pm = i∂m).
We can now define superfields. Superfields are essentially functions on su-perspace, understood as power series expansions in θ and θ.
Φ(x, θ, θ) = φ(x) + θλ(x) + θχ(x) + θθA(x) + θθB(x)+θσmθcm(x) + θθθψ(x) + θθθρ(x) + θθθθD(x). (120)
Superfields transform under supersymmetry transformations in the natural way:
δξΦ(x, θ, θ) ≡(ξQ+ ξQ
)F.
Different types of superfields are defined by imposing different restrictions onthe general superfield. To define these restrictions, it is very helpful to introducethe superspace derivatives Dα and Dα,
Dα =∂
∂θα+ iσmααθ
α∂m (121)
Dα = − ∂
∂θα− iθασmαα∂m. (122)
As an exercise, we can check that Dα and D satisfy the anticommutation rela-tions,
Dα, Dα = −2iσmαα∂m
21
Dα, Dβ = Dα, Dβ = 0.
Furthermore, D and Q anticommute:
D,Q = D, Q = D,Q = D, Q = 0.
We can now define the single most important form of superfield, the chiralsuperfield.
A chiral superfield is defined as a superfield whose components are restrictedby
DΦ = 0. (123)
Chiral superfields are the generalisation to field theory of the chiral multiplet.
The second most important kind of superfield is the vector superfield. Thisis defined by
V = V †.
Note that as D commutes with Q, the notion of a chiral superfield is preservedunder supersymmetry transformations. As (ξQ+ ξQ)† = (ξQ+ ξQ), the sameis also true of vector superfields: the nature of a superfield is preserved undersupersymmetry transformations.
Just as fields (scalars, spinors, vectors...) are the basic ingredients of or-dinary quantum field theory, so superfields are the basic ingredient of super-symmetric quantum field theory. Superfields contain within themselves severaldifferent fields, all related by supersymmetry transformations.
5.1 Chiral Superfields
Recall chiral superfields are defined by
DαΦ = 0.
As D is a linear differential operator, we immediately see that if Φ1 and Φ2 arechiral superfields, then
1. Φ1 + Φ2 is a chirla superfield.
2. Φ1Φ2 are chiral superfields.
Let us now look explicitly at the form of Φ(x). To solve DαΦ = 0, it is convenientto write ym = xm + iθσmθ. This allows us to re-express (exercise)
Dα =∂
∂θα+ iσmααθ
α∂m → ∂
∂θα+ 2iσmααθ
α ∂
∂ym,
Dα = − ∂
∂θα− iθασmαα∂m → − ∂
∂θα. (124)
These expressions seem a bit confusing (in fact contradictory) at first, and so weshould say a little about what they mean. They should be understood first asacting on a superfield Φ(xm, θ, θ), and then transformed as acting on a superfieldwritten as Φ(ym, θ, θ). For example, the transformed version of Dα should be
22
interpreted as the statement that Dαθ = Dαy = 0. The significance of this isthat it implies that the general chiral superfield can be expanded as a powerseries in y and θ,
Φ(y, θ, θ) = φ(y) +√
2θψ(y) + θθF (y). (125)
If we further expand ym = xm + iθσmθ, we then have
Φ(x, θ, θ) = φ(x) + iθσmθ∂mφ(x) +14θθθθ∂µ∂
µφ(x)
+√
2θψ(x)− i√2θθ∂mψ(x)σmθ + θθF (x). (126)
As an exercise, check this expansion.
Let us count degrees of freedom, writing Φ in the form
Φ = φ(y)︸︷︷︸complex scalar
+√
2θψ(y)︸ ︷︷ ︸fermions
+ θθF (y)︸ ︷︷ ︸auxiliary field
(127)
Off-shell, the scalar φ has 2 real degrees of freedom, the fermion ψ 4 real degreesof freedom, and the auxiliary scalar field F 2 real degrees of freedom. On-shell,these will reduce to 2 propagating bosonic degrees of freedom and 2 propagatingfermionic degrees of freedom. The equations of motion eliminate two of the fourfermionic degrees of freedom, and it turns out that the auxiliary field F is non-propagating can be integrated out. The vev of F is however of great interest asit is a measure of the magnitude of supersymmetry breaking that is present.
5.2 Constructing supersymmetric Lagrangians
We now want to construct supersymmetric Lagrangians. To do so, let us lookat how superfields transform under supersymmetry transformations. Let’s firstconsider a general superfield,
S(x, θ, θ) = φ(x) + θψ(x) + θχ(x) + θθM(x) + θθN(x)+θσµθVµ(x) + θθθλ(x) + θθθρ(x) + θθθθD(x). (128)
Under a susy transformation δS = (εQ+ εQ)S, giving
δφ = εψ + εχ,
δψ = 2εM + (σµε) (i∂µφ+ Vµ) ,δχ = 2εN − (εσµ)(i∂µφ− Vµ),
δM = ελ− i
2∂µψσ
µε,
δN = ερ+i
2εσµ∂µχ,
δλ = 2εD +i
2(σνσµε) ∂µVν + i(σµε)∂µM,
δVµ = εσµλ+ ρσµε+i
2(∂νψσµσνε− εσνσµ∂
ν χ) .
δρ = 2εD − i
2(σν σµε) ∂µVν + i (σµε) ∂µN,
δD =i
2∂µ(εσµλ− ρσµε
). (129)
23
These should be verified by the student. Although the above expressions area mild chore to derive, once done we can use these expressions to read off thetransformations for any of the constrained superfields, such as chiral superfields,vector superfields, etc.
One very important feature of the above transformations is that
δD = ∂µ ( ) . (130)
That is, the auxiliary component D always transforms as a total derivative. Thesignificance of this is that it implies that for any superfield S(x, θ, θ),∫
d4xd2θd2θS(x, θ, θ)
is invariant under supersymmetry transformations.
We now have a procedure to construct (part of ) supersymmetric Lagrangians.
1. Take any superfield
2. Take its D-term, and integrate over spacetime.
The resulting theory will automatically be invariant under supersymmetry trans-formations.
Now let’s extend this a bit further by restricting the general susy transfor-mations just to chiral superfields. In this case the susy transformations are
δφ =√
2εψ,δψ = i
√2σµε∂µφ+
√2εF,
δF = i√
2εσµ∂µψ. (131)
Note that now for chiral superfields the F-term transforms as a total derivative,and also recall that products of chiral superfields are also chiral superfields.Consequently any holomorphic function W (Φ) of chiral superfields is also achiral superfield, and ∫
d4xd2θW (Φ)
is invariant under supersymmetry transformations. Such a term is called anF-term, whereas terms of the form∫
d4xd2θd2θ K(Φ, Φ)
are called D-terms.
Suppose we have a set of n chiral superfields Φ1, . . . Phin. We can constructsupersymmetric Lagrangians using two functions
1. K(Φi,Φ†j) - a real, non-holomorphic function of superfields and their con-
jugates
2. W (Φi) - a complex holomorphic function of chiral superfields.
24
Then a supersymmetric Lagrangian is given by
L =∫d4xd2θd2θ K(Φi,Φ
†j)︸ ︷︷ ︸
D−term
+∫d4xd2θW (Φ)︸ ︷︷ ︸F−term
. (132)
This is in fact the most general supersymmetric Lagrangian for a theory of chiralsuperfields. What are the mass dimensions of fields?
Φ(y, θ, θ) = φ(y) +√
2θψ(y) + θθF (y). (133)
Here φ has mass dimension 1 as a scalar, and ψ as a fermion has mass dimension3/2. Consequently θ has mass dimension −1/2, while F has mass dimension 2.We also see that K has mass dimension 2 and W mass dimension 3.
A renormalisable theory therefore has
K =∑
cijΦiΦj,†, (134)
W =∑i
λiΦi +∑
mijΦiΦj︸ ︷︷ ︸mass terms
+∑
YijkΦiΦjΦk︸ ︷︷ ︸Yukawa couplings
. (135)
We will shortly discuss some simple supersymmetric field theories. Before doingso, we first state one truly remarkable result that explains why supersymmetrictheories are so attractive.
The superpotential is not renormalised at any order in perturbation theory.
It is tree-level exact up to non-perturbative corrections.
The Kahler potential by contrast gets renormalised at all orders in perturbationtheory.
This point is so important it is worth dwelling on. First, what is the physicsof this? The underlying reason for non-renormalisation is holomorphy. Pertur-bation theory is an expansion in a coupling constant, which is real, while thesuperpotential is holomorphic and complex. So a real expansion cannot be madeconsistent with holomorphy, and so the holomorphic quantity (the superpoten-tial) therefore cannot be renormalised. In contrast, the Kahler potential (whichis real) does indeed get renormalised at every order in perturbation theory.
Let us now make a slight diversion in order to understand better how thisoccurs. We will deal with the expansion in powers of gauge coupling.
We start by promoting gauge couplings to superfields. In string theory, thisalways happens - all coupling constants originate as the vevs of very weakly(gravitationally) coupled fields. In the context of globally supersymmetric fieldtheory, such ultra-weakly coupled fields are often called spurion fields.
Now consider the action of a non-Abelian gauge theory,
1g2
∫d4xF aµνF
a,µν + iθYM
∫d4xF aµν(∗F )a,µν . (136)
25
Scalar superfields are naturally chiral superfields involving complex components.The gauge coupling by itself is real, and is naturally complexified by includingthe θ-angle. So a gauge coupling corresponds to a chiral superfield
Φ = (1g2
+ iθYM ) + . . .
Now, we know from standard field theory arguments that the topological term∫d4xF aµν(∗F )a,µν is locally a total derivative and vanished for field configura-
tions that are continuously connected to the vacuum. So when we perform thepath integral ∫
DAe−1
g2 FµνFµν+iθFµν(∗F )µν
(137)
perturbative fluctuations about Aµ = 0 do not depend on the value of θ. (Notethat non-perturbative instantonic effects do depend on θ, and contribute tothe action terms that are suppressed as e−
8πg2 +iθ) As a result, the perturbative
expansion depends only on g−2 = Φ + Φ and not at all on θ.
Now comes the magic part. The superpotential is holomorphic, and so mustbe a holomorphic function of Φ. However, we have just established that pertur-bation theory is an expansion in Φ+ Φ, and so can never modify a holomorphicquantity.
We therefore conclude: the superpotential is not renormalised at any orderin perturbation theory.
This is an AMAZING result!
5.3 The Wess-Zumino Model
We now want to be a little more prosaic and examine in some detail the simplestexample of a supersymmetric field theory. We consider a theory of a single chiralsuperfield with
K = Φ†Φ,W = mΦ2 + gΦ3. (138)
What is the explicit form of this? Recall that we can expand Φ as
Φ = φ(y) +√
2θψ(y) + θθF (y)
= φ(x) +√
2θψ(x) + (θθ)F (x) + i(θσµθ)∂µφ
−14(θθ)(θθ)∂µ∂µφ−
i√2(θθ)∂µψ(x)σµθ. (139)
We can now look for the (θθ)(θθ) component of Φ†Φ (exercise!), and find
Φ†Φθθθθ =12(∂µφ)(∂µφ∗)−1
4φ∗ (∂µ∂µφ)−1
4φ (∂µ∂µφ∗)+|F |2+
i
2(ψσµ∂µψ + ∂µψσ
µψ).
(140)Integrating by parts, we obtain
Φ†Φθθθθ =[(∂µφ)(∂µφ†) + |F |2 − i(ψσµ∂µψ)
](141)
26
and so ∫d4xd2θd2θΦ†Φ =
∫d4x∂µφ∂
µφ† + |F |2 − iψσµ∂µψ. (142)
We see that this gives rise to standard kinetic terms for the scalar and fermionicfields. These we know and love. There is also an unusual term that we have notseen before, involving |F |2 but no kinetic terms for the F field.
Let us pause on understanding this, and instead consider the F-terms in-volving the superpotential. These take the basic form∫
d4x d2θW (Φ) (143)
We can again recall (again and again) the basic expansion
Φ = φ(y) +√
2θψ(y) + θθF (y). (144)
This time however, as y = x+ θσµθ, in terms of the expansion it is sufficient toreplace y by x, as none of thebarθ terms can survive the superspace integral. We can then expand in (Φ−φ),and write Φ = φ(x) + (Φ− φ)(x).
W (Φ) = W (φ) + (Φ− φ)︸ ︷︷ ︸√
2θψ+θθF
∂W
∂φ+
12(Φ− φ)2︸ ︷︷ ︸
− 12 (θθ)(ψψ)
∂2W
∂φ2. (145)
Then∫d4xd2θW (Φ) + c.c =
(∂W
∂φ|Φ=φF + c.c
)− 1
2
(∂2W
∂φ2ψψ + c.c
)(146)
The F-term Lagrangian density is then
L = ∂µφ∂µφ∗− i
(ψσµ∂µψ
)+ |F |2 +
(∂W
∂φF +
∂W ∗
∂φ∗F ∗)− 1
2
(∂2W
∂φ2ψψ + c.c
)(147)
We can combine (147) with (142) to see that the auxiliary field F appears inthe Lagrangian as
LF = FF ∗ + F∂W
∂φ+ F ∗
(∂W
∂φ
)∗. (148)
As there are no derivative terms for F , we see that F is not a propagating degreeof freedom. At least classically, we can simply solve for the equations of motionof F and integrate it out.1
The equations of motion give
0 =∂L∂F
= F ∗ +∂W
∂φ, (149)
1Quantum mechnically, we will have to be a little more careful when performing the pathintegral. In particular, it is necessary to account correctly for the measure when integratingover DF .
27
and soF ∗ = −∂W
∂φ. (150)
The Lagrangian LF therefore becomes
LF = −(∂W
∂φ
)(∂W ∗
∂φ∗
). (151)
We then obtain overall
L = ∂µφ∂µφ∗ − iψσµ∂µψ −
(∂W
∂φ
)(∂W ∗
∂φ∗
)− 1
2
(∂2W
∂φ2ψψ + c.c
). (152)
What does this tell us? There are kinetic terms for both scalar and fermionfields, and interaction terms between scalars and fermions. There is also howevera potential, and the potential is given by
VF = |F |2 =∣∣∣∣∂W∂φ
∣∣∣∣2 . (153)
There are some important features about this potential that will generalise toall supersymmetric models which do not include gravity.2
1. The scalar potential is positive semi-definite, and cannot be negative.
2. The scalar potential is given by the (sum over) squares of the susy breakingparameters, here the F-term.
3. The scalar potential vanishes if and only if supersymmetry is unbroken.
Let us now be a little more concrete. We take the specific example of
W =mΦ2
2+gΦ3
3. (154)
This is the superpotential appropriate to the Wess-Zumino model. Then thescalar potential is given by
V =∣∣∣∣∂W∂φ
∣∣∣∣2 = m2ΦΦ∗ + gm(ΦΦ∗,2 + Φ2Φ∗)+ g2Φ2Φ∗,2. (155)
Note that this scalar potential has
1. A mass term m2ΦΦ∗.
2. A quartic scalar interaction g2Φ2Φ∗,2.
Now consider the fermionic terms from the Lagrangian (152). We have
∂2W
∂φ2= m+ 2gΦ. (156)
2A sceptical reader may observe here that this is like proving results that hold for all fluidswith zero viscosity. The scepticism may have a point.
28
This then gives a fermionic interaction Lagrangian of
Lint = −12(mψψ +mψψ
)− g
(Φψψ + Ψψψ
). (157)
This gives a
1. mass term − 12
(mψψ +mψψ
)and so the fermion mass is m.
2. Yukawa coupling g(Φψψ + Φψψ).
In particular, notice - and absorb! - the fact that the coefficient of the trilinearYukawa coupling is essentially the square-root of the coefficient of the quarticscalar interaction.
XX FIT IN FEYNMAN DIAGRAMS XX
This is at the heart of the famous supersymmetric cancellations in the ra-diative corrections to the Higgs mass.
Note that supersymmetry does not mean that any interaction can be mod-ified simply by swapping bosons and fermions - it is instead far more sophisti-cated than that. Instead it relates the quartic scalar interaction to the trilinearYukawa interaction.
Let us now examine a little more closely the general structure of supersym-metric theories. Recall that for a chiral multiplet
Φ = φ(y) +√
2θψ(y) + θθF (y), (158)
then the supersymmetry transformations act as
δφ =√
2εψ,δψ =
√2iσµε∂µφ+
√2εF,
δF = i√
2εσµ∂µψ. (159)
Also recall that in general for a symmetry transformation Q to be unbroken , itmust preserve the vacuum and so
Q|0〉vac = 0. (160)
For a general field background, we see from (159) that we always require 〈ψ〉 = 0in order to preserve supersymmetry. This is no great hardship, as we require〈ψ〉 = 0 to preserve Lorentz symmetry. In a similar vein, the Lorentz condition∂µφ = 0 also eliminates a further term. For Lorentz-preserving field configura-tions we are then left with
δφ = 0,δψ =
√2εF,
δF = 0. (161)
From this it is straightforward to see that 〈F 〉 is an order parameter of super-symmetry breaking.
29
For a canonical Kahler potential it is straightforward to generalise the aboveanalysis for a single superfield to the case of n superfields. In this case theauxiliary field Lagrangian is
LF =∑i
FiF†i +
∑i
(Fi∂W
∂φi+ F †
i
∂W
∂φ†i
). (162)
From these we obtain that
F †i = −∂W
∂φ, Fi = −∂W
∂†i,
and so
V =∑i
∣∣∣∣∂W∂φi∣∣∣∣2 . (163)
For general forms of the Kahler potential, this instead becomes
V =∑
Kij
(∂W
∂φi
)(∂W
∂φ∗j
). (164)
Here the Kahler metric is defined by
Kij =∂2K
∂φi∂φj. (165)
General properties of (globally) supersymmetric theories are
1. Susy breaking is parametrised by the F-terms Fi = −∂W∂φi
. Supersymmetryis unbroken if and only if 〈Fi〉 = 0 for all i.
2. The potential is positive semi-definite,
V =∑
Kij
(∂W
∂φi
)(∂W
∂φ∗j
). (166)
3. The potential vanishes if and only if we have a supersymmetric solution.
One very natural question to ask about supersymmetric theories is whethersupersymmetric solutions always exist. This is not a question to deliberateon, as the answer is straightforward: No. There is a simple and well-knowncounterexample, the O’Raifertaigh model. This is defined by
K =3∑i=1
Φ†iΦi, (167)
W = gΦ1
(Φ2
3 −m2)
+MΦ2Φ3. (168)
The F-terms are then given by
−F ∗1 = ∂W
∂Φ1|Φ=φ = g(φ2
3 −m2), (169)
−F ∗2 = ∂W
∂Φ2|Φ=φ = Mφ3, (170)
−F ∗3 = ∂W
∂Φ3|Φ=φ = 2gφ1φ3 +Mφ2. (171)
30
It is easy to see that this theory cannot preserve supersymmetry as we cannotsolve 〈Fi〉 = 0 simultaneously.
As an exercise, study the potential and spectrum of this model. For sim-plicity take m2 < M2
2g2 . The minimum is found to be at 〈phi2〉 = 〈φ3〉 = 0, with〈φ1〉 unfixed.
The fermionic spectrum is given by
(0,M,M),
while the bosonic spectrum is given by
(0,√M2 + 2gm2,
√M2 − 2gm2).
The magnitude of the scalar potential is
Vmin = |F1|2 = g2m4. (172)
The spectrum is non-degenerate, with the lack of degeneracy measured bygm2 = |F1|. Note that the form the non-degeneracy takes is to split the scalarmasses by equal amounts above and below those of the fermions.
This is an example of a more general result, which is a supersymmetric sumrule that holds for such renormalisable, globally supersymmetric field theories.
STr(M2) =∑b
m2 −∑f
m2 = 0. (173)
That is, the sum over squared masses with a relative sign between bosonic andfermionic degrees of freedom always vanishes.
A second, and more trivial, example of supersymmetry breaking is the Fayetmodel. This is given by
K = Φ†Φ, (174)W = λΦ. (175)
This has an F-term Fφ = λ with V = |F |2 = λ62 = V0. This has a trivial (flat)potential, with the vacuum energy simply shifted up. It has a massless scalarand a massless fermion, and no non-trivial interactions.
5.4 General Properties of Supersymmetric Theories
We are familiar with the way that Goldstone bosons arise from the spontaneousbreaking of symmetries. Goldstone’s theorem tells us that when a global sym-metry is spontaneously broken, there remains a massless boson (a Goldstoneboson) and that the masslessness of this boson is a direct consequence of thesymmetry breaking.
Mutatis mutandis, the same is true of supersymmetry. The chief difference(pretty much the only difference really) is that supersymmetry is a fermionicsymmetry rather than a bosonic symmetry. The parameter of supersymme-try transformations, ε, transforms as a spinor under the Lorentz group. The
31
consequence of this is that any supersymmetric field theory with broken super-symmetry always has a massless fermion in its spectrum, a goldstino.
If you are awake reading this, this should sound bad. Massless particlescan be probed both cosmologically and in collider experiments, and if such amassless fermion existed then we would expect to know about it. There ishowever a catch. In any supersymmetric theory with gravity - in other words,any supersymmetric theory with a chance to describe the real world - there isa gravitino. The gravitino is the supersymmetric partner of the graviton. Themassless gravitino multiplet only has two degrees of freedom, but in theorieswith broken supersymmetry the gravitino becomes massive which required fourdegrees of freedom. It obtains the extra two degrees of freedom by eating thegoldstino, in a supersymmetric analogue of the Higgs mechanism imaginativelynamed the super-Higgs mechanism.
We don’t have time here for a full discussion of supergravity, but let us brieflymention how ideas from global supersymmetry extend to local supersymmetry.We will give results rather than derivations. First, chiral superfields remainchiral superfields.
Φ −→ Φ
The Kahler and superpotentials are in general non-renormalisable, so we shouldwrite
K = K(Φi,Φ†j) →M2
PK
(ΦiMP
,Φ†j
MP
), (176)
W = W (Φ) →M3PW
(ΦMP
). (177)
The F-terms again are the order parameters for supersymmetry breaking. How-ever these are now given by
FΦ =(∂ΦW +
(∂ΦK)M2P
W
)eK/2M
2P . (178)
It is customary to write this as a covariant derivative, and also to note thatF-terms can be raised and lowered using the Kahler metric,
F i = eK/2KijDjW ≡ eK/2M2PKij
(∂jW +
(∂jK
)W). (179)
As before supersymmetry is broken if FΦ 6= 0. However, one big (and phe-nomenologically important) difference is that the vacuum energy is no longerpositive semi-definite. Instead, it can be written as
V =∑i,j
KijFiF j︸ ︷︷ ︸∑
|F |2
− 3M2P
eK/M2P |W |2︸ ︷︷ ︸
supergravity term
(180)
The last term is specific to supergravity. There are two crucial consequencesof the negativity of the last term. First, supersymmetric solutions (F I = 0for all I) can have negative vacuum energy in supergravity. Secondly, solutionswith broken supersymmetry can have vanishing vacuum energy. Given that our
32
universe has vanishing vacuum energy, this implies that supersymmetry is notinconsistent with the world as we observe it.
The gravitino mass in supergravity is given by
m3/2 = eK/2M2P|W |M2P
. (181)
If the vacuum energy vanishes, then the gravitino mass is the order parameterof supersymmetry breaking. However also note that in supersymmetric theorieswith negative cosmological constant, the gravitino mass can be non-zero evenwhile the graviton remains massless. At first this seems a little paradoxical,but it is in fact a consequence of the properties of AdS space (i.e. that witha negative cosmological constant). In AdS space supersymmetry is consistentwith - and in fact requires - mass splittings between superpartners.
In theories with vanishing vacuum energy and broken supersymmetry, thegravitino mass is the order parameter of supersymmetry breaking and measuresthe magnitude of supersymmetry breaking. This is easy to see from the potential(180): if the vacuum energy vanishes, then
3m23/2M
2P = |F |2, (182)
and so the gravitino mass is a direct measure of the magnitude of the F-terms.As mentioned previously, the gravitino obtains its mass by eating the masslessgoldstino in the super-Higgs effect. The goldstino provides the two extra degreesof freedom necessary to convert a massless gravitino (two propagating degreesof freedom) into a massive gravitino (four propagating degrees of freedom).
Let us now describe one of the simplest non-trivial but important supergrav-ity theories. This is the simplest possible example of a no-scale model. It isspecified by (we put MP = 1)
K = −3 ln(T + T ), (183)W = W0. (184)
Here W0 is a constant. There is one superfield T , and note that the theory onlydepends on the real part of T : there is a symmetry T → T + iλ in the theory.Then
K,T =−3
T + T=⇒ K,T T =
3(T + T )2
. (185)
As a consequence we evaluate the F-term to get
FT = eK/2 (∂TW + ∂TKW ) (186)
=1
(T + T )3/2× −3W0
T + T= − 3W0
(T + T )5/2. (187)
By raising indices using the Kahler metric, we have FT = − W0(T+T )1/2 . It then
follows that
KT TFTF T =
3W 20
(T + T )3, (188)
33
and as a consequence
V = KT TFTF T − 3eKW 2 (189)
=3W 2
0
(T + T )3− 3W 2
0
(T + T )3= 0. (190)
The potential therefore vanishes for all values of the field T . The gravitino massis given by
m3/2 = eK/2W =W0
(T + T )3/2, (191)
and is non-zero for all values of T .
This model is a no-scale model. There are several key features that define itas a no-scale model. The three most important features of a no-scale model are
1. Vanishing cosmological constant - as we see above, the solution has zerovacuum energy.
2. Broken supersymmetry - FT 6= 0 and so the vacuum does not preservesupersymmetry. Equivalently, (given the vanishing vacuum energy) thegravitino mass is non-zero.
3. An unfixed flat direction. No scale models always have a residual unfixeddirection which, at least at tree level, remains massless. In this case thisis given by the field Re(T ). Note that Im(T ) is also massless, but this isguaranteed by the symmetry T → T + iλ.
Originally there was a hope that no-scale models could provide a possible solu-tion to the cosmological constant problem, as they combine a vanishing vacuumenergy with broken supersymmetry. However it was soon realised that this isonly a property of the tree-level theory, and at loop level the cosmological con-stant problem will return. At loop level the unfixed modulus T will also obtaina potential from corrections to the Kahler potential, thereby lifting the flatnessof the potential.
Another reason for their importantce is that no scale models frequently occurin string theory, in the context of dimensional reduction of ten dimensional stringtheory solutions to 4 dimensions. So while the properties of no-scale solutionsare non-generic if we consider the list of all possible supergravity theories, inthe context of string theories they in fact occur regularly. In this context theshift symmetry T → T + ıλ is perturbatively exact, and is equivalent to thesymmetry in the QCD theta angle.
5.5 Vector Superfields
So far we have constructed chiral superfields and used these to build up La-grangains describing supersymmetric theories of bosons and fermions. We nowwant to generealise this to theories involving gauge interactions and chargesparticles. To do so we need the concept of a vector multiplet.
The vector multiplet is defined by constraining the general multiplet by
V = V †.
34
This constrains the fields present in a general multiplet to be
V (x, θ, θ) = C(x) + i(θχ(x)− θχ(x))
+i
2[θθ(M(x) + iN(x))− θθ(M(x)− iN(x))
]−(θσmθ
)Vm(x) + iθθθ
[λ(x) +
i
2σm∂mχ(x)
](192)
−iθθθ[λ(x) +
i
2σm∂m ¯chi(x)
]+
12θθθθ(D(x) +
12C(x)).
Note the particular choice of fields that are used when writing the vector multi-plet - for example, we have written D(x)+ 1
2C(x) instead of D(x). This choiceis motivated in advance by the fact that the fields C(x), χ(x),M(x), N(x) areall unphysical and can be gauged away.
Let us make this explicit. It is manifest that given a chiral superfield Φ(x),Φ(x) + Φ†(x) is a vector superfield. In fact,
Φ + Φ† = (A+A†) +√
2(θψ + θψ
)+ θθF + θθF ∗
+iθσmθ∂m(A−A∗) +i√2θθθσm∂mψ
+i√2θθθσm∂mψ +
14θθθθ
(A+A†) . (193)
The thing to note in particular here is that the θσµθ part of Φ + Φ† has thecomponent i∂m(A − A†). This looks precisely like a conventional gauge trans-formation, with A(x) the scalar field that plays the role of the gauge parameter.
This motivates the supersymmetric generalisation of the notion of a gaugetransformation:
Conventional: Aµ → Aµ + ∂µφ for some φ,Susy: V (x) → V (x) + Φ(x) + Φ†(x) for some Φ.
Under this transformation, the elements of a vector superfield transform as
C(x) → C(x) + (A(x) +A∗(x)),
χ(x) → χ(x)− i√
2ψ(x),M + iN → (M + iN)− 2iF,
vm → vm − i∂m(A−A∗),λ → λ,
D → D. (194)
From this we see that this generalises the traditional notion of a gauge trans-formation to supersymmetric theories. We also see that we can drasticallysimplify the form of a vector superfield by going to a particular gauge (whereC = χ = M = N = 0). This gauge is traditionally calles Wess-Zumino gauge.
35
In it,
V = −θσmθvm(x) + iθθθλ(x)
−iθθθλ(x) +12(θθ)(θθ)D(x), (195)
V 2 = −12(θθ)(θθ)vµvµ, (196)
V 3 = 0. (197)
Note that the choice of gauge is not preserved under susy transformations. Ifyou act on a vector field in Wess-Zumino gauge with a supersymmetry transfor-mation parametrised by ε, the resulting field will no longer be in Wess-Zuminogauge.
We should view V as the superfield equivalent of the gauge potential, and(Φ + Φ∗) as the supersymmetric analogue of a gauge transformation.
As with electrodynamics, we want to construct Lagrangians that are invari-ant under gauge transformations. For Abelian theories, the answer is reasonablyeasy to guess. Suppose a chiral multiplet A transforms under a gauge transfor-mation as
A→ eiqΛ(x)A.
This is the natural transformation law, that generalises the scalar transformationφ(x) → eiqλ(x)φ(x). Recall that the kinetic term entering the Kahler potentialis A†A. We then have the transformation
A†A(x) → A†A(x)eiq(Λ−Λ†). (198)
We want to construct a gauge invariant kinetic term. Given that V transformsas V → V + i(Λ− Λ†), then under gauge transfomations
A†e−qVA→ A†e−qVA. (199)
This term then gives the gauge interactions between charged matter fields (A)and Abelian vector bosons V . Written out in components, we have
(Φ†e−qV Φ)θθθθ = FF ∗ + φ∂µ∂µφ∗ + i∂µψσ
µψ
−qvµ(
12ψσµψ +
i
2φ∗∂µφ−
i
2(∂µφ∗)φ
)(200)
+i√2q(φλψ − φ∗λψ
)+
12
(qD − 1
2q2vµv
µ
)φ∗φ.
These are the appropriate interactions to couple a complex scalar and chiralfermion to gauge bosons and gauginos.
We also need the kinetic terms for the gauge field itself: that is, we need tofind a supersymmetric multiplet that incorporates the FµνFµν term represent-ing the gauge kinetic term. The easiest way to do this is to find a superfieldthat includes the gauge field strength as one of its components. Consider thesuperfield
Wα = −14DDDαV. (201)
36
This is constructed from vector superfields and has the following properties.First, Wα is a chiral superfield. To see this, note that
DβWα = 0, (202)
as Dβ , Dβ = 0, and Wα by definition includes both D derivatives. Secondly,Wα is gauge invariant under the transformation
V → V + Φ + Φ†.
To see this, note that Wα transforms as
Wα → −14DDDα
(V + Φ + Φ†) .
Now DαΦ† = 0 as Φ† is an antichiral superfield. As a result we have
Wα → −14DDDαV −
14DDDαΦ,
= Wα −14D(DDα +DαD
)Φ, ( as DΦ = 0)
= Wα −14Dβ(DβDα +DαDβ
)Φ
= Wα −14DβDα, DβΦ
= Wα −14∂µ
(DβΦ
)σµαβ
= Wα. ( as DβΦ = 0). (203)
Therefore we establish that Wα is a chiral, gauge-invariant superfield.
Proving thatWα is gauge-invariant makes the evaluation ofWα a lot simpler.All we need to do is to go to Wess-Zumino gauge, in which the form of the vectorsuperfield simplifies dramatically. Having done this, we can then evaluate Wα
explicitly by taking the super-derivatives. If we do this, we find
Wα = −iλα(y) +D(y)θα + θθσµαα∂µλα
− i2
(σµσν) βα (∂µvν(y)− ∂νvµ(y))︸ ︷︷ ︸
Field strength Fµν
θβ . (204)
Here y = x + iθσθ. We see that Wα contains the gauge field strength Fµν .For this reason Wα is called the field strength superfield and plays a centralrole in construction supersymmetric gauge Lagrangians. As Wα involves thefield strength, W 2 ≡ WαW
α will contain the kinetic terms. As Wα is a chiralsuperfield, it is clear we need an F-term Lagrangian. We therefore take
12
∫d2θWαWα = WαWα|θθ = −iλσµ∂µλ−
14FµνFµν
+D2
2+i
8FαβF γδεαβγδ. (205)
37
To ensure the action is real we take∫d4x
14
(∫d2θWαWα +
∫d2θ WαW
α
)=∫d4x
(12D2 − 1
4FµνFµν − iλσµ∂µλ
).
(206)These are the supersymmetric kinetic terms for an Abelian gauge field (andgaugino).
There is an easy (and important) generalisation of this. Suppose we have achiral superfield
Φ(y) = (fR(y) + ifI(y)) +√
2θψ + (θθ)F. (207)
Suppose also that fR(x) 6= 0 and fI(x) 6= 0, while F = ψ = 0. We call Φ thegauge coupling superfield, and it does not have to be dynamical.3 We can nowwrite down a Lagrangian∫
d4x14
(∫d2θΦWαWα +
∫d2θΦ†WαW
α
)(208)
=∫d4x
(12fRD
2 − fR4FµνFµν − ifRλσ
µ∂µλ+fI8εαβγδF
αβF γδ).
Let us now write fR = 1g2 and fI = θYM , to then give∫
d4x
(1
2g2D2 − 1
4g2FµνFµν −
i
g2λσµ∂µλ+
θYM8
εαβγδFαβF γδ
). (209)
This Lagrangian now has both a gauge coupling 1g2 and also a topological θYM
term εαβγδFαβF γδ. The values of both of these couplings are controlled by the
vacuum expectatation value of the chiral multiplet Φ. This is an importantstatement and is the origin of several deep and powerful results. For an Abeliangauge theory, this however all seems slightly trivial - the topological term isalways trivial, and without matter there is no intrinisic meaning to a U(1)gauge coupling. The real power of this statement will come when we want toanalyse the structure of non-Abelian supersymmetric gauge theories.
In general, if we have a term∫d2θ f(Φ) (WαW
α + c.c) , (210)
then f(Φ) is called the gauge kinetic function. Re(f) gives the physical gaugecoupling and Im(f) gives the theta angle.
Let us draw an important general conclusion from this: any theory whichhas dynamical gauge couplings, in which f(Φ) depends on a dynamical superfieldΦ, also has a dynamical theta angle. What is the significance of a dynamicatheta angle? The theta angle controls the phase of QCD instanton effects, anda dynamical theta angle is precisely equivalent to an axion.
3This means we do not have to include a kinetic term for it. If we feel squiffy about this,we can include a kinetic term for φ and also include an arbitrary large mass term in thesuperpotential mΦΦ so that the field is non-dynamical.
38
Put another way, any theory (for example string theory) which has both su-persymmetry and dynamical gauge couplings automatically has axions in. Thismakes it clear that axions are an important generic expectation of the low energyphysics arising from string theory.
Let us also note here another point that we will develop more fully later.Consider the action ∫
d2θφWαWα + c.c (211)
Recall
Wα = −iλα(y) + . . . ,
Φ = φ(y) +√
2θψ(y) + (θθ)F (y). (212)
As a result, if FΦ 6= 0, then we have a term∫d4x
(FΦλλ+ FΦ† λλ
), (213)
which gives a Majorana mass for the gauginos. The significance of this, andwhat we learn from it, is that F-terms for the gauge kinetic function give gauginomasses. There is always a chiral gauge coupling superfield, and we can evaluatesupersymmetry breaking gaugino masses by evaluating the F-term of this field.
We have so far discussed how to construct supersymmetric Lagrangians forAbelian gauge groups in supersymmetric theories. For Abelian theories thereis another important term that can be added to the Lagrangian, the Fayet-Iliopoulos term,
LFI =∫d2θd2θξV. (214)
As V = . . .+ 12 (θθ)
(θθ)D(x), this gives a contribution to the Lagrangian of
LFI =∫d4x
12ξD(x). (215)
This is gauge-invariant as under V → V + Φ + Φ†, we have∫d4x
12ξD →
∫d4x
(12ξD +
14∂µ∂
µφ+14∂µ∂
µφ∗)
=∫d4x
12ξD(x), (216)
once we integrate by parts.
For Abelian theories, we have now enumerated the three basic kinds of termsinvolving vector fields:
1. Gauge kinetic terms
2. Gauge interactions of matter fields
3. Fayet-Iliopolous terms
39
We now combine these to see the full structure of the resulting Lagrangian andpotential.
We first want to make a comment that we will return to later. As in regularfield theory, we have a choice as to whether we put the gauge couplings in thekinetic terms or the interactions. This is the difference between
1g2FµνF
µν + φ∗Aµ∂µφ
andFµνF
µν + gφ∗Aµ∂µφ.
Classically, these Lagrangians are related by the field redefinition Aµ → gAµ.Quantum mechanically, this field rescaling is anomalous and the two Lagrangiansare different. We will discuss the quantum anomalies later.
We choose to put the gauge coupling in the kinetic term (this is more con-sistent with notions of holomorphy). So we have
L =1
4g2
(∫d2θWαWα + c.c
)+∫d4θ
(ξV +
∑i
Φ†ie−qV Φi
)(217)
(Here q is the charge but contains no factors of the gauge coupling).
We can write this out in total, to get∫d4xL =
∫d4x
(− 1
4g2FµνFµν −
i
g2λσµ∂µλ+
12g2
D2
)(218)
+(FF ∗ − ∂µφi∂
µφ∗i + i∂µψiσµψi)−Aµ
(12qiψiσ
µψi + iqiφ∗i ∂µφi
)+
i√2
(qiφiλψi − qiφ
∗i λψi
)+
12
(qiD −
12q2i vµv
µ
)φ∗iφi +
12ξD.
Let’s focus on the parts involving the non-dynamical D-term:∫d4x
12g2
D2 +D
2(qiφ∗iφi + ξ) . (219)
We solve to obtain D = − g2
2 (∑i qiφ
∗iφi + ξ), to give
VD =∫d4x
g2
2
(∑i
qi|φi|2 + ξ
)2
. (220)
This is called the D-term potential. Together with the F-term potential, it isone of the two major sources of the potential in supersymmetric field theories.The D-term potential is
1. Associated to gauge fields, and proportional to the (gauge coupling)2
2. Up to the FI terms, entirely determined by the gauge representations thatfields are in.
40
3. Positive semi-definite: if VD > 0 then supersymmetry is broken, as δλ ∼εD for a vector superfield.
D-terms can also lead to supersymmetry breaking. Let us consider some simplemodels. The first case is the absolute simplest, corresponding to a U(1) gaugetheory with no charged matter and a single FI term. In this case
L =∫d4x d2θWαW
α +∫d4x d2θd2θξV, (221)
and so
VD =g2ξ2
2.
This represents a constant flat potential with non-zero vacuum energy.
The second case is a slight generalisation to consider a U(1) theory, withan FI term and one chiral superfield of charge q). (However note that whileconsistent as a classical theory, quantum mechanically this theory is actuallyanomalous). The D-term potential is
VD =g2
2(q|φ|2 + ξ
)2. (222)
There are then two cases. First, if q and ξ have opposite signs then a susysolution can be found and is located at |φ| =
√−ξq . In this case the gauge group
is Higgsed due to the vev for φ. However, if q and ξ have identical signs thenno susy solution exists. In this case the minimum has broken supersymmetry,with 〈φ〉 = 0 and a mass term for φ coming from its coupling to the FI term.
5.6 Non-Abelian Gauge Theories
Finally we wish to construct non-Abelian supersymmetric gauge theories. Asbefore, the starting point is the vector multiplet V a. The difference is that nowthis has to have components V a for each generator of the gauge group. Chiralsuperfields now transform as
Φ′
= e−iΛΦ,
Φ′,† = Φ†eiΛ
†, (223)
with Λ = T aijΛa and T aij the generator in the Φ representation. So if we naivelygeneralise the Abelian coupling Φ†eV Φ, then it transforms as
Φ†eV Φ → Φ†eiΛ†eV
′
e−iΛΦ. (224)
We see that we can ensure a gauge invariant interaction by taking
eV′
= e−iΛ†eV eiΛ. (225)
Here V = VaTaij and so is matrix valued. An immediate worry here is that the
definition (225) is in terms of a specific representation of the gauge group, and inthis respect does not appear to be well defined. However recall that one case use
41
the Hausdorff formula to simplify the products of exponentials. This involvesmultiple commutators of the exponents of the form [A,B], [A, [A,B], . . .. Thesecan all be simplified by use of the group commutation relations, [T a, T b] =fabcT c, until eventually we end up with a single generator and a relationship ofthe form
V′= V
′
aTaij , (226)
where V′
a is now a direct function of the transformation parameters Λ and theoriginal components V . The fact that these simplifications only require use ofthe group commutation relations means that this relationship is well definedand is independent of the representation originally used in the definition (225).As a result we can directly generalise the matter-gauge interaction to the caseof non-Abelian theories, to give ∫
d4θΦ†eV Φ, (227)
with gauge transformations behaving as
Φ → eiΛΦ,
Φ† → Φ†e−iΛ†,
eV′
→ e−iΛ†eV eiΛ. (228)
In a similar vein we can also define the field strength superfield for non-Abeliantheories,
Wα = −14DDe−VDαe
V . (229)
The fact that e−V eV = 1 tells us that as
eV′
= e−iΛ†eV eiΛ, (230)
thene−V
′
= e−iΛe−V eiΛ†. (231)
As a result Wα transforms as
Wα → −14DD
(e−iΛe−V eiΛ
†)Dα
(e−iΛ
†eV eiΛ
)= e−iΛWαe
iΛ, (232)
where we have used DαΛ† = 0, DΛ = 0. It therefore follows that Tr(WαWα) is
gauge-invariant, and so ∫d2θ (f(Φ)WαWα + c.c) (233)
is a gauge invariant Lagrangian, which generalise the Abelian theory to a non-Abelian one.
The Lagrangian for a non-Abelian theory takes basically the same form asan Abelian one, with the obvious generalisations of Fµν = ∂µAν − ∂νAµ →∂µAν − ∂νAµ+ [Aµ, Aν ], and the presence of non-Abelian covariant derivatives.
42
One important area is the D-term. First, the coupling Φ†eV Φ now givesD(a)Φ†
iTaΦi on taking the (θθ)(θθ) component of V . As a result the D-term
action looks like(Da)2
2g2+Daֆ
iTaΦi, (234)
and so
VDa =g2
2
(∑i
Φ†iTaΦi
)2
. (235)
There is one contribution to the D-term potential for each generator of the gaugegroup, and so the total potential is given by
VD =g2
2
∑a
(∑i
φ†iTaΦi
)2
. (236)
If there are multiple non-Abelian gauge groups, then we additionally need tosum over each group.
A further important feature of non-Abelian groups is that there are no Fayet-Iliopoulos terms for non-Abelian gauge groups, as the FI term∫
d4x d2θ d2θ ξV (237)
is not gauge invariant for non-Abelian theories.
It is a general result for non-Abelian theories that the F-terms give thestructure of the vacuum. If we can solve the F-term equations, then it is alwayspossible to solve the D-term equations.
To summarise then:
1. Chiral superfields give matter fields, and have Lagrangians described bya superpotential W and Kahler potential K,
∫d2θW (Φ) and
∫d2θd2θK.
These generate an F-term potential VF =∑i
(∂W∂Φ
)2.
2. Vector superfields are used for gauge interactions, which come from anaction ∫
d2θ1g2WαWα +
∫d4θΦ†eV Φ. (238)
There is a D-term potential
VD =g2
2
∑a
(∑i
Φ†iTaΦi + ξ
)2
. (239)
The FI term is only non-zero for Abelian gauge theories.
6 The MSSM
Supersymmetry may be a wonderful symmetry, but is it relevant to nature?This question has existed for a long time - the Supersymmetric Standard Model
43
Table 1: Field content of the SSM/MSSM
Field SU(3) SU(2)W U(1)YQL,i=1,2,3 3 2 -1/6UR,i=1,2,3 3 1 2/3DR,i=1,2,3 3 1 -1/3Li=1,2,3 1 2 1/2ER,i=1,2,3 1 1 -1
Hu =(h+u
h0u
)1 2 -1/2
Hd =(h−dh0d
)1 2 1/2
was formulated by Fayet in 1977 and the first experimental bounds on super-symmetry appeared in 1978. If supersymmetry is present as a fundamentaltheory of nature, we have been waiting a long time for it.
Fortunately we have now developed enough formalism and technology tobe able to construct the MSSM (Minimal Supersymmetric Standard Model).The MSSM is one of the principal search targets for the LHC experiments andrepresents the supersymmetrisation of the Standard Model with minimal extraingredients.
The prime reason for the attractiveness of the MSSM is that it addresses thetechnical hierarchy problem of the Standard Model: what stabilises the Higgsmass against radiative corrections? [INSERT PICTURE] AS there is no massterm for fermions in the MSSM, there is also no mass term for bosons. Thereforeat any scale where supersymmetry is a good symmetry, bosons must remainmassless. The Higgs mass therefore cannot be parametrically larger than thesupersymmnetry breaking scale. An alternative way of putting this is that massterms are superpotential terms, and so in a supersymmetric theory are eitherpresent at tree level or not present at all. As a consequence, supersymmetricmass terms cannot be acquired at 1-loop and therefore masses in supersymmetryare radiatively stable.
What is the particle content of the MSSM? There are two basic types ofmultiplet present
1. Vector multiplets - there is one vector multiplet for each gauge generatorof the SU(3)× SU(2)× U(1) gauge group.
2. Chiral multiplets - there are chiral multiplets for each charged matter fieldpresent in the Standard Model.
Let us enumerate the chiral multiplets, together with their gauge charges. Theyare The immediate obvious point here is that the MSSM has two Higgs doubletsin contrast to the Standard Model which only has one. Every other field is asimple susy-ification of the Standard Model matter content. There is an easyway to see why two Higgs doublets are necessary in supersymmetry. We recallthat cancellation of gauge anomalies depends on the fermion sector of the theory.
44
In the Standard Model, the Higgs does not contribute to any anomalies becauseit is purely a scalar particle. However, once we promote the Higgs multiplet toa chiral multiplet, it contains a chiral fermion. This does contribute to gaugeanomalies, and as for a theory with one Higgs doublet this is the only extracontribution to gauge anomalies (since the susy partners of the quarks andleptons are bosons), the resulting theory is anomalous and sick. However withtwo Higgs doublets the contibrution from each cancels, and the resulting theoryis anomaly-free.
The MSSM is a renormalisable field theory. It has
• Constant (i.e. non field-dependent) gauge couplings, fa = τa = constant.
• Canonical Kahler potential K = Φ†eV Φ.
• Vanishing FI term, ξU(1)Y= 0 (otherwise we would break electromag-
netism in vacuo and give the photon a mass).
The gauge couplings of the MSSM have one important feature: extrapolatedfrom the weak scale, they converge at MGUT ∼ 2×1016GeV. This famous resultsuggests that there is a higher unified structure present at the extremely smalllength scales that correspond to the GUT scale. However, this is not necessarilymore than a suggestion - there is no direct evidence for GUTs and in particularproton decay, one of the characteristic signals of Grand Unified Theories, hasnot been observed.
Question: why is this plot meaningful? The significance of the plot is thatthree lines meet at a point, whereas two lines always meet at a point. However,there is no intrinsic notion of a gauge coupling for U(1) gauge fields - the ab-sence of a triple gauge boson vertex means that we can always rescale the gaugepotential and thereby modify what we mean by the gauge coupling. So given thatU(1)Y has no intrinsic coupling, why does the plot mean anything? (Somewhatelliptic hint: what is plotted is not 4π
g2Ybut 5
3 ×4πg2Y
.)
Answer: the embedding of U(1)Y into SU(5) provides a canonical normali-sation of the U(1).
What is the MSSM superpotential?
W = yu,ijHuQiLU
jR + yd,ijHdQ
iLD
jR + yL,ijHdL
jEjR + µHuHd. (240)
This has
• Yukawa couplings for up-type quarks (induced by Hu)
• Yukawa couplings for down-type quarks (induced by Hd)
• Yukawa couplings for leptons (induced by Hd)
• a supersymmetric Higgs mass term (the µ term)
The form of the superpotential is significantly restricted by holomorphy: in theStandard Model we could couple h or h∗ to get Yukawa couplings, but thisis forbidden by holomorphy (so no H∗
dQiLU
jR coupling for example). This is a
45
second reason why there have to be two Higgs doublets in the supersymmetricStandard Model, as with only one Higgs doublet there would be no way ofobtaining Yukawa couplings.
However, there are also extra terms which could in principle be added to thesuperpotential,
W /BL = λi LLER︸ ︷︷ ︸∆L=+1
+λ2 LQDR︸ ︷︷ ︸∆L=+1
+λ3 URDRDR︸ ︷︷ ︸∆B=−1
+λ4 LH2︸︷︷︸∆L=+1
(241)
These terms have the crucial, and perhaps embarrassing feature, of violatingeither lepton or baryon number. This matters of course because both leptonand baryon number are good global symmetries of the Standard Model and arefor all practical purposes exact. 4 These terms have to be highly restricted inorder to avoid proton decay. If all four such terms λ1, λ2, λ3, λ4 were non-zerothen the proton can decay via the following process
XXXXXXX
PROTON DECAY DIAGRAM
XXXXXX
The decay is mediated by a virtual squark d, and so the amplitude is sup-pressed by the squark propagator 1
m2d
and the lifetime suppressed by 1m4
d
. One
can do a full calculation, but it is simplest to observe that the only other di-mensionful scale is the proton mass, and so the lifetime is
τp ∼
(m4d
m5p
)∼(
1TeV1GeV
)4
10−23s( md
1TeV
)4
∼( md
1TeV
)4
10−10s. (242)
Such a lifetime is literally fatal, and we can see that this is entirely incompatiblewith the observed stability of the proton τp & 1034years. To avoid this we needto forbid some or all of the operators in W /BL. The simplest option is to forbidall operators via R-parity. R-parity is defined by
R = (−1)3(B−L)+2S =
+1 SM particles−1 SUSY particles (243)
However this may be overkill (to forbid proton decay we only need to forbid oneside of the diagram rather than both).
An implication of R parity is that the lightest supersymmetric particle (LSP)is stable, as its decays must involve another R-charged particle - which is notpossible.
Let us also make a note on nomenclature: supersymmetric particles arenamed by adding ‘o’ (fermions) or ‘s’ (scalars):
Quark → SquarkLepton → SleptonGluino → GluinoW/Z → Wino/Zino
4B+L can be violated by electroweak sphaleron processes which are suppressed by e− 8π2
g2Y M .
These may be important in the early universe when the temperature is comparable to theelectroweak scale, but at the current time give a proton lifetime of τ ∼ 1080years.
46
6.1 Yukawa Couplings
In the Standard Model we have the Yukawa couplings
ytφqL,ttR → mt = yt < v >, (244)mb = yb < v > . (245)
In the Standard Model there is then a single Higgs vev, and the different massesof the top and bottom quarks come from the different Yukawa couplings ofthe top and bottom quarks. The experimental fact that mt ∼ 170GeV andmb ∼ 4GeV then implies that yb/yt ∼ 1/40, and so the bottom Yukawa couplingis intrinsically small.
What about the MSSM? In the MSSM there are two Higgs doublets, whichcan have separate vevs 〈Hu〉 and 〈Hd〉. The mass of the up-type quarks comefrom the vev of the up-type Higgs, and the mass of the down-type quarks comefrom the vev of the down-type Higgs. So
mt = yt〈Hu〉, (246)mb = yb〈Hd〉, (247)me = ye〈Hd〉. (248)
As a result the up-type and down-type quarks obtain masses from differentvevs. The importance of this is that the physical Higgs scalar - the one thatgives mass to the W and Z bosons - is a combination of Hu and Hd. In theMSSM there are five Higgs scalars (8 from two complex doublets, of which 3 areeaten by the SU(2) gauge bosons). The lightest CP even combination of thesehas very similar couplings to the Standard Model Higgs.
An important (and unknown) property of the MSSM is tanβ, which is theratio of the up-type and down-type vevs,
tanβ =〈Hu〉〈Hd〉
. (249)
The overall magnitude of the vev, v =√〈Hu〉2 + 〈Hd〉2 is set by the mass of
the W and Z bosons. However the extent to which this vev consists of 〈Hu〉and 〈Hd〉 is undetermined, and this is what tanβ specifies. The light StandardModel-like Higgs is then given by
h ∼ sinβHu + cosβHd. (250)
tanβ is extremely important for the phenomenology of the MSSM, and gives oneof the main differences between the couplings of a standard Model Higgs and anMSSM Higgs. The reason is that for large values of tanβ, 〈Hd〉 is actually rathersmall. In particular, for tanβ ∼ 40, then 〈Hu〉 ∼ 170GeV and 〈Hd〉 ∼ 4GeV.In this case the bottom Yukawa coupling yb is close to unity, and as a resultthe bottom quark has an O(1)) coupling to Hd. This is important in certainloop processes, as it means that compared to the Standard Model certain loopprocesses are tanβ enhanced.
47
6.2 Supersymmetry Breaking
The ‘supersymmetry’ in MSSM is a misnomer: no superpartners have beenobserved and supersymmetry is necessarily broken. ‘MSSM’ refers to the su-persymmetric Standard Model supplemented by soft terms. Soft terms areterms in the Lagrangian that explicitly break supersymmetry, are added tothe Lagrangian of the supersymmetric Standard Model, but do not reintroducequadratic divergences. In supergravity models, the soft terms arise from spon-taneous breaking of supersymmetry and keeping the terms that survive in theMP →∞ limit.
The possible form of soft terms was classified by Girardello and Grisaru in1982. They are given by:
Scalar mass terms m2QQQ
∗ (251)Gaugino Masses Mλλλ (252)
Trilinear scalar A-terms Aφiφjφk( where ΦiΦjΦk ∈W ) (253)Bilinear scalar B-terms Bφiφj( where ΦiΦj ∈W ) (254)
Note that all the soft terms are dimensionful - all dimension 4 (marginal) susybreaking operators are hard. This is as one would expect, as marginal operatorsare relevant on all distance scales, whereas relevant operators are important atlow energies. Quadratic divergences are a feature of the high energy Lagrangian.Therefore marginal operators, which are important for high energy physics andmodify the physics at all energy scales, are able to reintroduce quadratic di-vergences. In contrast, relevant operators - and all dimensionful operators arerelevant - only affect long-distance physics and so cannot affect short-distancedivergences.
The phenomenological theory called the MSSM is then defined by a La-grangian
L = LSSM︸ ︷︷ ︸Supersymmetric Standard Model
+ Lsoft︸ ︷︷ ︸soft breaking terms
. (255)
The soft breaking terms are expected to have masses around the weak scale ifthey are to be useful for addressing the hierarchy problem. The soft Lagrangianis an expression not of triumph but of failure: it is not a fundamental theoryof supersymmetry breaking but instead a parametrisation of ignorance. Modelsof supersymmetry breaking aim at computing the parameters of the MSSM interms of a deeper underlying theory.
Supersymmetric phenomenology is primarily about understanding how togenerate the soft terms and the consequences of any given choice of soft terms.The rest of the Langrangian is entirely fixed by supersymmetry; the soft termsare what is left unknown. Supersymmetry is divine, whereas supersymmetrybreaking infernal. All the many possible different models of supersymmetry arefundamentally different soft Lagrangians.
The big picture of supersymmetric model building is as follows:
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48
INSERT PICTURE
XXXXXXXXX
The starting point of a supersymmetric model is an explicitly supersymmet-ric theory. This theory may have one or many (meta)stable vacuum solutions,and one or many of these solutions may break supersymmetry. This means thatthe supercharges do not annihilate the vacuum. As the original theory was ex-plicitly supersymmetric, the supersymmetry breaking is spontaneous. The fulltheory has two sectors:
• A hidden sector, in which the fields have non-zero F-terms. This is thesector of the theory in which supersymmetry is actually broken.
• The visible sector, which contains the MSSM fields. If the MSSM is ex-tended to contain other fields with Standard Model charges and weakscale masses, these would normally also count as part of the visible sector.Supersymmetry breaking is communicated from the hidden sector to thevisible sector by the mediator.
One natural question is: why not just have a single sector? Rather thanhave separate hidden and visible sectors, why not just break supersymmetry inthe MSSM itself and be done with it? The problem with this is the supertraceformula of global supersymmetry,
STr(M2) =∑i
m2b,i −m2
f,i = 0. (256)
Given the lightness of the MSSM fermions (mu,d ∼ O(5MeV)), this impliesthat some of the squarks and sleptons must also be extremely light, and in factsufficiently light to have already been observed.
Broadly speaking, there are two main ways of communicating supersymme-try breaking to the visible sector:
• Gauge mediation: supersymmetry breaking is communicated via renor-malisable gauge interactions.
• Gravity mediation: This is a complete misnomer, and would better becalled moduli mediation. Here supersymmetry breaking is communicatedby intrinsically non-renormalisable operators that couple the visible sectorand the hidden sector. Such operators are directly generated at the stringscale and require an understanding of the string theory in order to studythem properly.
It is useful to package both methods in a unified way based on how couplingsdepend on fields that break supersymmetry. The easiest case to consider is thatof gaugino masses and A-terms. These turn out to be closely related to gaugeand Yukawa couplings respectively. The appropriate terms in the superspaceLagrangians is ∫
d2θfa(Φ)WαWα +
∫d2θYαβγ(Φ)CαCβCγ (257)
49
Now
Wα = λα + . . . (258)Cα = φαi + θψαi + . . . (259)
If we expand the superfield fa(Φ), we have
fa(Φ) = fa(φ) + . . .+ θ2Ffa(Φ).
Whichever superfield determines the gauge coupling, its F-term gives the gaug-ino mass. The same is true for the Yukawa couplings and A-terms: the lowestcomponent of Yαβγ(Φ) gives the Yukawa couplings and the (θθ) component ofYαβγ(Φ) gives the A-term. What this tells us is that in a spontaneously brokensupersymmetric theory, we can extract gaugino masses and A-terms by
1. Looking at how gauge couplings and Yukawa couplings depend on thefields that break supersymmetry.
2. Extracting the F-terms of these expressions.
To make this more explicit, let’s consider gaugino masses. We have saidthese arise from the expression for gauge couplings. So what enters the value ofgauge couplings? What determines that, say, αSU(3) ∼ 0.11 when evaluated atMZ?
We said earlier that gauge couplings appear to be unified at a high scale,and then run down from that high scale to lowe energy scales. What determinesthe couplings at this high scale? In string models, the gauge couplings at thestring scale are typically set by a geometric feature of the compactification (forexample, the size of a cycle or the value of the string coupling constant). Ina low energy 4-dimensional theory, these parameters are themselves controlledby fields. For example, there is a dilaton field S such that Re(S) = 1
gsand a
volume modulus T such that Re(T ) is a measure of the volume of the compactspace. These fields control the high-scale geometry and thereby also control thehigh-scale gauge couplings.
In this case the chiral superfields that control the size of these cycles enterthe gauge couplings, for example as
fa =T
2π. (260)
(The linear dependence on T follows from the axionic nature of Im(T )) TakingT to be a dimensionful field, we then must write
fa =T
2πMX, (261)
where MX is a dimensionful scale that could naturally be the Planck scale MP
or the string scale MS . The classical gaugino Lagrangian then looks like
Re(T )2πMX
λ∂µλ+FT
2πMXλλ, (262)
50
and so the physical gaugino mass is given by
Mλ,physical =FT
MXRe(fa). (263)
The suppression by the scale MX appropriate to a non-renormalisable operatorindicates that this is a gravity-mediated contribution. A general worry aboutnon-renormalisable operators is that they cannot be computed at low energy -to compute them, you need to know the fundamental theory. This is true, andis an important motivation for understanding the fundamental theory better.
What else do gauge couplings depend on? We know that gauge couplingsrun at 1-loop, and that the beta function is determined by the number of lightspecies.
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INSERT PICTURE
XXXXXXXXXXX
Using a Wilsonian approach,
Re(fa(µ)) = Re(fa(Λ))︸ ︷︷ ︸string scale couplings
+ β ln(
Λ2
µ2
)︸ ︷︷ ︸
running from scale Λ to scale µ
. (264)
As a superfield equation, the 1-loop gauge kinetic function is then
fa = fa,0 + βa ln(
Λ2
µ2
). (265)
Now suppose there is a mass threshold present at a scale M . We can accommo-date this threshold using a superpotential
W = MQQ, (266)
which ’integrates out’ a quark species at the scale M . In this case the gaugekinetic function would look like
fa = fa,0 + β1 ln(
Λ2
µ2
)+ β2 ln
(M2
µ2
). (267)
We can rewrite this as
fa(µ) = fa,0 + (β1 − β2) ln(
Λ2
M2
)+ β2 ln
(Λ2
µ2
). (268)
When we introduced the mass threshold M , it was a fixed scale. Now let usinstead regard it as the vev of a field X,
W = XQQ, with 〈X〉 = M. (269)
Allowing the vev of X to be arbitrary, we can extend (268) into a superspaceequation,
fa(µ) = fa,0 + (β1 − β2) ln(
Λ2
M2
)+ β2 ln
(Λ2
µ2
). (270)
51
This encodes the dependence of the gauge couplings on X. However, as thegauge couplings depend on X, this also means that if FX 6= 0 then gauginomasses are generated. Therefore: if we have
W = FXX +XQQ, (so FX =∂W
∂X6= 0), (271)
then
FX∂Xfa = (β1 − β2)2FX
〈X〉, (272)
giving the unnormalised gaugino masses. In this case, the gaugino mass was in-duced at 1-loop (as it comes from the difference in β-functions) and is parametrisedby FX
〈X〉 .
The scenario we have just described is that of gauge mediation: the gener-ation of gaugino masses comes purely from renormalisable interactions and isa 1-loop effect. Let us think about the full expression for (Wilsonian) gaugecouplings:
fa(µ) = fa,0(Φ)︸ ︷︷ ︸Tree level gauge couplings
+ (∆β)i ln(
Λ2
X2
)︸ ︷︷ ︸Mass threshols
+ βlow ln(
Λ2UV
µ2
)︸ ︷︷ ︸
Running gauge couplings(273)
The entire expression should be understood as a superfield expression. We havelabelled the different parts of this expression as to what conventional naes areapplied to susy breaking originating in each of the different terms. This is reallythe absolute simplest way to understand the computation of gaugino masses,
1. What superfields do the gauge couplings depend on?
2. What are the F-terms of these fields?
There is a further subtlety to do with anomalies and the difference betweenWilsonian and 1PI couplings, which we will return to later.
Everything we have just said for gauge couplings also applies to A-terms.We now replace the gauge couplings fa(Φ)WαW
α with the Yukawa couplingsYijk(Φ)CiCjCk. The F-term of Yijk(Φ) gives trilinear A-terms, and so there isa contibution to the A-terms given by
Aijk → F I∂IYijk(Φ). (274)
In general, there are additional terms in the expression for A-terms associatedto the fact that the Yukawa couplings are not yet canonically normalised. Thekinetic terms for the matter fields are given by
∫d4θKij(Φ, Φ)CiC j and the
physical Yukawa couplings depend on Kij .
We can play a similar game for the matter fields. Consider a general Kahlerpotential ∫
d2θd2θQ†eVQK(Φ, Φ). (275)
Recall: the Kahler potential is not protected by holomorphy and is renormalisedat all orders in perturbation theory.
52
If we view this as a superfield expansion, and consider the simplest possibility∫d2θd2θQ†eVQ
Φ†ΦΛ2
, (276)
then we see that for non-zero FΦ that scalar masses arise from the (θθ)(θθ)component of Φ†Φ:
Φ†Φ =(FΦ,†FΦ)
Λ2(θθ)
(θθ)
=⇒ m2Q =
|F |2
Λ2. (277)
In fact, in general we see that the (θθ)(θθ) component of K(Φ, Φ) will give riseto scalar masses.
We can view the kinetic terms in a similar way. The kinetic terms (effectivelywavefunction normalisation) are modified at all orders in perturbation theory.We can write the Lagrangian at a given energy scale (i.e. in terms of theWilsonian renormalisation group) as
L =∫d2θ [fa(Φ)]1−loopWαW
α+∫d2θd2θ
(Q†eVQ
) [K(Φ,Φ†)
]all loop+
∫d2θYαβγ(Φ)CαCβCγ .
(278)Just as for the gauge couplings we can write the matter kinetic terms as
K(Φ,Φ†) = K0(Φ,Φ†)︸ ︷︷ ︸tree level high scale
+ K1−loop(Φ,Φ†)︸ ︷︷ ︸running due to thresholds
. (279)
Gravity mediation comes from the tree-level high scale kinetic terms (i.e. whatis the normalisation at the string scale?). Gauge mediated soft terms come fromrunning due to thresholds: squared masses arise at two loops.
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Let us make a couple of general comments about computing soft terms.
• Gravity mediated contributions need to be computed via the formalism ofsupergravity. There really is no option to this if you want to be doing ameaningful computation. For formulae, see Brignole, Ibanez and Munoz.
• Gauge mediated contributions are best computed not by working in atheory with broken supersymmetry and computing higher loop diagrams.Instead, it is better to work with supersymmetric theories and use tech-niques of analyticity in superspace.
Let us now discuss some general properties of gauge and gravity mediation.First of all we need the general expression
V =∑i,j
KijFiF j − 3m2
3/2M2P (280)
= |F |2 − 3m23/2M
2P . (281)
53
First, note that this implies that any consistent theory of supersymmetry break-ing necessarily includes supergravity: we cannot just use global supersymmetry,even though it it useful for thinking about some of the issues.
For gravity mediation, we have
f ∼ X
MP, K ∼ X†X
M2P
Q†Q.
=⇒ Mλ ∼FX
MP, MQ2 ∼ |FX |2
M2P
.
=⇒ FX ∼ (1011GeV)2 =⇒ |FX |2
MP∼ 1TeV, and m3/2 ∼ 1TeV.
For gauge mediation, we have
f ∼ f0 +β
16π2ln(M2P
X2
)
Mλ ∼β
16π2
FX
〈X〉, MQ2 ∼
(β
16π2
)2
|FX
X|2.
We need FX
〈X〉 ∼ 105GeV. 〈X〉 is arbitrary. Limiting cases are
• 〈X〉 ∼ 104GeV → 109GeV2 ∼ (104GeV)2 =⇒ m3/2 ∼ 0.1eV.
• 〈X〉 ∼ 1016GeV → FX ∼ 1021GeV2 ∼ (1010GeV)2 =⇒ m3/2 ∼(10GeV).
An important consequence is that in gauge mediation the gravitino is the lightestsupersymmetric particle.
Question: what is the strength of the coupling between the gravitino andmatter? Why not just MP ?
Answer : Not so fast! Recall the super Higgs effect. The gravitino becomesmassive (four degrees of freedom) by eating the goldstino. The longitudinalmodes of the gravitino are those of the goldstino, and have couplings set by thegoldstino.
So for gauge mediation, gravitino couplings are set by 〈X〉 rather than MP :low-scale gauge mediation has LSP-gravitino couplings suppressed b 1
(10 TeV)
rather than 1MP
.
6.3 MSSM Phenomenology
MSSM phenomenology is determined by the pattern of soft masses at the weakscale. This is set by
1. Soft masses determined at the high (string) scale
2. RGE equations that evolve these terms down to the weak scale.
54
By now this evolution is highly automated and is carried out by various programssuch as SoftSUSY. General features of the spectrum are
1. Coloured particles run up and tend to be heavy at low energies.
2. SU(2) charged particles tend to run down.
3. In the presence of large (as observed) top mass the Hu mass parameterruns negative and triggers radiative EWSB (Ibanez + Ross).
For practice gaining intuition as to the susy spectrum, go to kraml.web.cern.ch/kraml/comparison.
6.4 Flavour
One very important (possibly the most important) set of constraints on su-persymmetry come from flavour physics. In fact these constraints are ratherembarrassing, because flavour constraints show no evidence of any new physicsat the TeV scale. For this to hold, the form of any new physics must takea highly constrained form. Often this form is governed by the assumption ofMinimal Flavour Violation. The absence of any signs of new physics (or super-symmetry) through low energy flavour observables is known as the susy flavourand susy CP problems.
We will not give an exhaustive survey of all possible processes, but let usmention two. The first example is that of K0 − K0 mixing, namely mixingbetween the neutral K0 meson and its antipartner. K0 − K0 mixing can occurin the Standard Model, but only as a loop level process that is also suppressedby CKM factors.
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It can also occur in the MSSM, where it is also a loop level process. In thecase that flavour mixing is large, then the SUSY diagram can give a dominantcontribution to KK mixing. However this is directly excluded because theexperimental measurements of KK mixing are all consistent with no exoticcontribution to the mixing.
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In the diagram, x refers to mass insertions that interconvert between s andd squarks. The dimensionless measure of the mixing is
m2s −m2
d
msmd
.
For large mixings - i.e. a susy mass spectrum that does not commute with theYukawa flavour structure - this diagram gives a contribution to KK mixing that
55
is far larger than the experimental limits. The absence of such contributionsis known as the susy flavour problem. This problem is not restricted to the Kmeson system. For example, it also exists in systems of both D mesons (thoseinvolving charm quarks) and B mesons (involving b quarks), where oscillationsof neutral mesons do not shown signs of large new contributions from Beyond-the-Standar-Model physics.
Another type of process leading to flavour constraints are those involvingrare decays, for example the process µ→ eγ. This is forbidden in the StandardModel as lepton flavour is a conserved quantity (we neglect the tiny effectsof non-zero neutrino masses). However lepton flavour is only an accidentalsymmetry of the Standard Model and so can be badly violated in the MSSMwith general flavour violating terms. An example of an MSSM diagram thatleads to µ→ eγ is
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In the Standard Model the muon decays via weak processes in the decaymode µ→ eνµνe. As the weak interaction is highly suppressed at low energies,this gives a chance for susy contributions to have a measurable branching ra-tio. The measurement of the branching ratio µ → eγ is one of the targets ofthe MEG experiment (for example), aiming to probe parameter space down toBR(µtoeγ) . 10−12.
The general danger in both of these diagrams, which can lead to large unob-served susy contributions, is that the flavour basis does not align with the massbasis. This imposes significant constraints on the soft terms of the MSSM. Thebasic solutions to these flavour problems are
1. Degeneracy : universality of soft scalar masses. If the soft masses areuniversal, then flavour rotations do not alter the structure of the softmass matrix. In this case it is always possible to align the flavour andmass matrices, and so the dangerous flavour-violating processes are absent.This is the most commonly adopted solution of the supersymmetric flavourproblem.
2. Alignment: in this case the structure of soft masses is not degenerate, butis aligned with respect to the Yukawas. Specifically, the soft scalar massesare diagonalised by the same rotations that diagonalise the Yukawas. Inthis case it is again possible to go to a basis where the flavour structureof the soft masses is identical to that of the Yukawas, and so there are nodangerous flavour violating processes.
3. Decoupling: the dangerous processes come from loops involving susy par-ticles, and so involve factors of 1
m2q. If susy particles have masses much
larger than 1TeV, then such loops are small. For a flavour-anarchic spec-trum, this requires that the masses are m2
Q∼ (40TeV)2. However, this is
not a particularly good solution as this removes the motivation for super-symmetry in the first place.
56
An important general comment to make on the supersymmetric flavour con-straints is that the constraints on the first two generations are much more severethan the constraints on the third generation. This should not be at all surpris-ing - the first two generations are the lightest and involve cosmologically stableparticles, and it is much easier to produce K mesons than it is to produce topquarks!
6.5 CP violation
A further source of constraints on the susy spectrum comes from the requirementthat the susy spectrum does not introduce large new CP violating phases. CPviolation is associated to complex phases in the Lagrangian and these can appearin several places, most notably in gaugino masses, A-terms and the µ-term.
The simplest solution to the CP problem is to require that all phases areuniversal: that is, there are no relative phases between the different gauginomasses and the A-terms are all proportional to the Yukawa couplings and withthe same overall phases as the gaugino masses.
6.6 MSUGRA/CMSSM
One commonly used set of soft terms goes by the name of mSUGRA or CMSSM.Often these are taken to be interchangeable. However there are those who thinkthat it is vital to distinguish correctly between the two. There are also those whothink it is vital to distinguish between Marxist-Leninist-Trotskyist an Leninist-Marxist-Trotskyism. We shall neglect these important distinctions here. Thesesoft terms are defined by universal high-scale boundary conditions at the GUTscale (MGUT ∼ 2 × 1016GeV). These involve the simplest choice of soft termsthat satisfy the flavour and CP constraints.
1. Universal scalar masses m2
2. Universal gaugino masses M with M real.
3. Universal A-terms Aαβγ = AYαβγ with A real.
4. Higgs sector terms µ and Bµ are parametrised in a phenomenologicalfashion by tanβ and sign(µ).
We shall discuss the last point in more detail later. Let us just say here thatthe soft parameters µ and Bµ, after minimising the Higgs potential, becomeequivalent to the Z mass mZ , the ratio of the Higgs vevs 〈vu〉/〈vd〉 and the signof the µ parameter sign(µ). This spectrum is then evolved down to the weakscale to give the physical mass spectrum.
The universality of the scalar masses mean that the soft mass parametersfor up-type squarks, down-type squarks, leptons and Higgs fields is taken to bethe same at the GUT scale. The different low-scale value of the soft parameterscomes purely from renormalisation group evolution of these parameters fromshort to long distances. The evolution starts at the GUT scale and is terminated
57
at the weak scale. The reality of the gaugino masses and A terms is necessaryto ensure the absence of large new sources of CP violation.
In total mSUGRA has 5 parameters: m2, Mi, A, tanβ and sign(µ). By nowboth the renormalisation group evolution of the soft terms (through programssuch as softSUSY) and a scan over parameter space is highly automated.
The cosmological properties of supersymmetric spectra have also becomehighly automated with the development of programs such as microMEGAS.The R-parity conserving LSP is a WIMP dark matter candidate, and its thermalrelic abundance can be computed automatically by a numerical solution of theBoltzmann equations. Such a particle can be searched for either via directdetection experiments:
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These search for wimp-nucleon scattering from WIMPs in the galactic darkmatter halo. Another detection method is via indirect detection of annihilationsignals from WIMPs that annihilate in the galactice centre. As the annihilationrate goes as the square of the density, the ideal region to search for such WIMPsare area with a large dark matter density and small baryonic densities. For thesereason dwarf spheroidal galaxies are often used, as they have large mass-to-lightratios (which is a measure of the amount of dark matter present).
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Typical annihilation diagrams are shown above.
6.7 The Higgs Sector
The MSSM Higgs sector ((1, 2)−1/2 + (1, 2)1/2) has a potential coming from
• F-terms µHuHd
• D-terms g2(∑
i Φ†i (T
aΦ)i)2
• soft terms m2HH
∗uHu and BµHuHd
Putting these together, the tree level Higgs potential is
VHiggs =(m2Hu
+ µ2) (|h0u|2 + |h+
u |2)
+(m2Hd
+ µ2) (|h0d|2 + |h−d |
2)−Bµ
(h+u h
−d + h0
uh0d + h.c
)+g2
8
(|h+u |2 − |h0
u|2 + |h0d|2 − |h−d |
2)2
+ 4|h+u |2|h0
u|2 + 4|h0d|2|h−d |
2
−4(h+,∗u h−,∗d h0
uh0d + h0,∗
u h0,∗d h+
u h−d
)+g′2
8[|h+u |2 + |h0
u|2 − |h0d|2 − |h−d |
2]. (282)
58
The minimum of this potential has
〈h0u〉 =
√2vu, (283)
〈h0d〉 =
√2vd, (284)
where vu and vd are both real without loss of generality. As electric charge ispreserved in the vacuum, there can be no mass mixing between charged andneutral Higgs fields. To obtain the charged Higgs mass matrix, we can thereforecompute (the relevant fields are h+
u , h+,∗u , h−d , h
−,∗d )
M2h± =
∂2V
∂hu,±∂hd,±, with hu → vu, hd → vd, 〈h±u 〉 = 0, (285)
to obtain the charged Higgs mass matrix.
However we first study the neutral Higgs sector, putting hu,± = hd,± = 0.To do so we solve ∂V
∂h0u
= ∂V∂h0
d= 0 (and conjugates). This gives the following
equations:(m2Hu
+ µ2)h0u −Bµh0,∗
d +14
(g2 + g
′,2)h0u
(|h0u|2 − |h0
d|2)
= 0.(286)(m2Hd
+ µ2)h0d −Bµh0,∗
u − 14
(g2 + g
′,2)h0d
(|h0u|2 − |h0
d|2)
= 0. (287)
These equations can be solved to give
Bµ =
(m2Hu
+m2Hd
+ 2µ2)sin 2β
2,
µ2 =m2Hd−m2
Hutan2 β(
tan2 β − 1) − M2
Z
2. (288)
Here MZ is the Z mass, and we have used
M2Z =
(g2 + g
′,2) v2
u + v2d
2. (289)
This shows explicitly how Bµ and µ can be related to MZ and tanβ. Since MZ
is a known quantity, in practice we work backwards, and specify tanβ, usingthis to infer the value of µ and Bµ at the high scale.
We can solve for the charged Higgs mass matrix to get
M2h± =
(Bµ cotβ + g2
2 v2d −Bµ− g2
2 vuvd
−Bµ− g2
2 vuvd Bµ tanβ + g2
2 v2u
). (290)
The eigenvalues of this matrix are
m2G± = 0,
m2H± = Bµ (cotβ + tanβ) +M2
W . (291)
The zero eigenvalues correspond to the two Goldstone bosons of the brokenSU(2) symmetry. These are eaten by the SU(2) gauge bosons and become thelongitudinal components of the W± gauge bosons.
59
Now consider the neutral sector. Again a decoupling occurs - as the Higgssector is CP invariant, the real (CP even) and imaginary (CP odd) parts of theHiggs sector do not mix. For the imaginary parts we obtain
M2hI =
∂2V
∂hIu∂hId
=(Bµ cotβ BµBµ Bµ tanβ
), (292)
with
m2G0
= 0,
m2A = Bµ (cotβ + tanβ) . (293)
From this we see that (at tree level) m2H± = m2
A +m2W . The A particle is the
pseudoscalar neutral Higgs boson, as it is odd under parity.
We can do the same trick for the neutral CP even Higgs scalar. In this casethe mass matrix is
M2h0
R=(
m2A cos2 β +M2
Z sin2 β −(m2A +M2
Z
)sinβ cosβ
−(m2A +m2
Z) sinβ cosβ m2A sin2 β +M2
Z cos2 β
). (294)
There are two neutral CP even Higgs scalars, denoted by h,H. From the abovewe find that
m2h,H =
12
[(m2
A +m2Z)±
√(m2
A +M2Z)2 − 4m2
AM2Z cos2 2β
]. (295)
From this we obtain that
mh ≤ mA| cos 2β| ≤ mH , (296)mh ≤ MZ | cos 2β| ≤ mH . (297)
The lightest MSSM Higgs suffers from the very important tree-level constraint
mh < MZ . (298)
This constraint is important, as searches at LEP-II have ruled out the possibilityof a Standard Model Higgs lighter than the Z mass!
Note that the physical propagating mass eigenstates
(G±, G0)︸ ︷︷ ︸Goldstones
, H±︸︷︷︸charged Higgs
, A︸︷︷︸CP odd
, h,H︸︷︷︸CP even
are all related by mixing matrices to the original states. In fact,(G+
H+
)=
(cosβ sinβ− sinβ cosβ
)(h−,∗d
h+u
), (299)(
G0
A
)=
(sinβ − cosβcosβ sinβ
)(h0u,I
h0d,I
), (300)
(hH
)=
cosα sinalpha− sinα cosα
( h0u,R
h0d,R
). (301)
Here α is a messy (although explicit) expression and we do not write it out infull.
60
General Comments about the Higgs Sector
• The Higgs sector preserves CP so long as both µ and Bµ are real. Thisensures that the minimum of the Higgs potential occurs for 〈hu〉 and 〈hd〉both real and aligned, with no relative phase between them.
• Why is the tree-level Higgs mass sharply bounded by the Z mass? In theStandard Model, the Higgs mass can reach up to O(1000)GeV. In theStandard Model, we had
VH = −m2|φ|2 + λ|φ|4, (302)
and m2 and λ were both arbitrary. 〈v〉 =√
m2
2λ then determined theZ mass, and so there was one undetermined parameter left in the Higgspotential.
In the MSSM, the mass squared terms are still arbitrary, as they comefrom the soft masses which are only determined by the high energy the-ory. However, the quartic term is no longer free: this is instead fixed bythe D-term gauge interactions, which depend only on the (known) gaugecouplings. So once we know the vev (i.e. MZ) the mass of the Higgsis no longer arbitrary and is already determined. This is why there is amaximum mass of the MSSM Higgs: there is no leeway in the quarticterm.
• Note however that if we add an additional gauge singlet scalar field to thetheory, with a superpotential interaction
W = λSHuHd, (303)
then this generates an F-term quartic coming from
VF = |∂W∂S
|2 = λ2|Hu|2|Hd|2. (304)
This model is called the NMSSM and allows the Higgs mass constraintsof the MSSM to be relaxed, due to the additional quartic contributioncoming from FS .
• Radiative corrections, however, matter. It is a very imporant feature ofthe MSSM that there are large radiative corrections to the tree-level Higgspotential. These originate from the large top mass and its O(1) Yukawacoupling. There are loops involving both tops and stops that renormaliseboth the quadratic and quartic Higgs couplings. Examples of these areshown in the diagrams below
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In the decoupling limit (mH ,mA,mH± mh), we have as a result
m2h,1−loop = m2
h,tree +3g2m4
t
16π2M2W sin2 β
ln
[(1 +
m2tL
m2t
)(1 +
m2tR
m2t
)].
(305)
61
Here g is the SU(2) gauge coupling. Let us estimate this:
g2
4π|SU(2) ∼
130,
m4t
M2W
∼ 16M2Z , ln
[m2tLm2tR
m4t
]∼ 7(mt ∼ 800GeV).
Putting this together gives (for these numbers) approximately
m2h,1−loop = m2
h,tree +M2Z , (306)
and somh,1−loop ∼ 130GeV.
While the detailed value of the 1-loop Higgs mass will depend on thevalue of the left and right-handed stops, as well as the measured value ofthe top mass, the clear lesson from this is that the 1-loop corrections arecomparable to the tree-level terms and can raise the Higgs mass abovethe LEP exclusion limits - but not by much. One of the unambiguouspredictions of the MSSM is a light Higgs with a mass not much largerthan 130 GeV.
• In the large tanβ regime, the Yukawa coupling of the bottom quark alsobecomes O(1), as does the tau Yukawa coupling. This is because down-type Yukawa couplings are given by mb
mttanβ, and so approach unity in
the large tanβ regime. In this regime certain loop corrections can becomelarge, as they rely on loops involving the bottom quark Yukawa coupling,and so are strongly enhanced in the large tanβ limit.
Certain processes that are tanβ enhanced are gµ−2, where ∆g−2,MSSM ∼tanβ and the rare decay Bs → µ+µ−, where BR(Bs → µ+µ−) ∼ (tanβ)6.
7 Anomalies in Supersymmetric Theories
An anomaly is a classical symmetry of a theory that is not preserved quantummechanically. The classic example of an anomaly is the axial anomaly of QCD,associated to independent rotations of the left and right-handed quarks.
QL → eiθQL, (307)QR → eiφQR. (308)
The axial anomaly allows the decay π0 → γγ to proceed at a far more rapidrate than would otherwise be allowed (not suppressed by the scale of chiralsymmetry breaking).
Key general features of anomalies are
• They depend on the fermionic sector of the theory and in particular onthe number of massless fermions.
• They are 1-loop exact (this represents one of the few examples of a non-susy non-renormalisation theorem).
62
We are going to focus here specifically on gauge coupling anomalies in super-symmetric gauge theories.
Let us first of all state the problem, and then over the next few lectures goon to develop the solution. If we are interested in scattering amplitudes, it isnatural to ask what the gauge coupling is at a scale µ. This can be probed bya scattering diagram such as
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The amplitude for this diagram should be given by
A ∼ g2(t)t
. (309)
The coupling depends on the scale of the process, and runs with energy scale. Inparticular, we are interested in the 1PI coupling, which represents the ‘physical’gauge coupling (the coupling inferred from 2 → 2 scattering). This differs fromthe holomorphic coupling set by the gauge kinetic function, which is 1-loopexact, and is protected by the non-renormalisation theorems. Our aim here isto relate the two.
We want to relate the holomorphic action∫d2θ
1g2h
(Wα(Vh)Wα(Vh) + h.c) +∫d2θd2θZΦ†
heV Φh
to the physical action∫d4xd2θ
1g2c
(Wα (gcVc)Wα(gcVc) + h.c)+∫d4xd2θd2θ
((√ZΦh)†eV (
√ZΦh)
).
Classically, these are the same actions: we just define gcVc = Vh, gc = gh,Φc =
√ZΦh and then the two Lagrangians look the same. Classical physics is
not affected by any field redefinitions: we can rescale fields and recover identi-cal physics. This should ring alarm bells, as we know that many theories canhave classical symmetries which do not hold quantum mechanically (for exam-ple QCD is classically scale-invariant but quantum-mechanically has runningcouplings). So as quantum physicists we should squint suspiciously at this fieldredefinition and suspect it will not hold quantum mechanically.
This is indeed correct. The technical issue is one that also occurs with thechiral anomaly. A quantum theory is defined not just by an action, but by apath integral:
A =∫DΦe−S(Φ).
Even if S(Φ) = S(Φ′), it does not follow that DΦ = DΦ
′. This invariance
of the measure under transformations then leads to anomalous contributions tothe gauge couplings. This is very similar to the Fujikawa derivation of the chiralanomaly as due to the (non)-invariance of the measure under chiral transforma-
63
tions.
D (gcVc) 6= gcDVc (310)
D(√
ZΦ)
6=√ZDΦ. (311)
Before we deal with the more complicated case of supersymmetric gauge theo-ries, it is helpful first to review the Fujikawa derivation of the chiral anomaly.
Consider the path integral for fermions in the presence of a gauge field,
Z =∫DAµDψDψ exp
[− 1
4g2FµνF
µν + i
∫d4xψ(i /D)ψ
](312)
We use 4-dimensional Dirac spinors. The axial transformation is
ψ(x) → &exp[iα(x)γ5
]ψ(x),
ψ(x) → ψ(x) exp[−iα(x)γ5
]. (313)
For an infinitesimal transformation, the Lagrangian integrand behaves as
L → L− ∂µα(x)ψγµγ5ψ, (314)
and so is invariant for constant α. It also naively appears that
DψDψ → DψDψ, (315)
under axial rotations, and so naively it appears that the entire functional integralis invariant. However this last relation is only a naive one and in fact is seen tobe false under a careful analysis.
What’s the problem? First, we should be careful about what we meanby Dψ. ψ(x) is a field, and the actual modes that we can quantise are theeigenmodes o ψ. So we should first of all expand ψ(x) and ψ(x) in terms ofeigenmodes of the Dirac equation,
ψ(x) =∑n
anφn(x),
ψ(x) =∑n
φ†n(x)bn. (316)
Orthogonality here means that∫φ†n(x)φm(x)d4x = δn,m. The eigenmodes
φn(x) are defined by
(i /D)φn(x) = λnφn(x).φ†n(x)(i /D) = λnφ
†n(x). (317)
Here D is the covariant derivative, Dµ = ∂µ + ieAµ (for an Abelian theory).In general we do not need to know the explicit form of the eignefunctions,although for zero gauge field (Aµ = 0) these are just the standard plane waveeigenfunctions. The degrees of freedom are the coefficients of the eigenmodes,and so having carried out the decomposition we can write the functional integralas
DψDψ =∏m
dbmdam (318)
64
This is then a sum over all possible coefficients of the eigenfuntions, and makesthe general form of the functional integral Dψ explicit.
An infinitesimal chiral transformation takes ψ(x) → ψ′(x) = (1+iα(x)γ5)ψ(x).
Obviously this also gives an infinitesimal transformation on the expansion coef-ficients an, bm.
a′
m =∑n
∫d4xφ†m(x)
[(1 + iα(x)γ5)anφn(x)
]=
∑n
(δmn + Cmn) an. (319)
whereCmn =
∫d4xφ†miα(x)γ5φn(x).
This can be viewed as a coordinate change on the ‘coordinates’ am, bm. As theseare Grassmann coordinates, the measure therefore transforms as
DψDψ → DψDψ [det(1 + C)]−2. (320)
(as there is one transformation for Dψ and one for Dψ). So we need to evaluate
J = det(1 + C) = exp [tr log(1 + C)]
= exp
[∑n
Cnn + . . .+O(C2)
]. (321)
For infinitesimal α we only need the first term in this expression, and so have
lnJ = tr(C) = i
∫d4xα(x)
∑n
φ†n(x)γ5φn(x). (322)
Note that
• This is in some sense undefined, as it sums over an infinite tower of modesand therefore needs to be regulated.
• The sum basically counts ‘number of left-moving solutions of the Diracequation’ minus ‘number of right-moving solutions of the Dirac equation’.We can see this by the presence of γ5, which separates out left- and right-handed solutions with a relative minus sign.
• If we are smart, we now understand the answer. Massive modes alwayspair up into an equal number of chiral and antichiral states, and so cannotcontribute. Massless modes can have a net chirality which is given by anindex theorem and is determined by the Pontryagin number of the gaugebackground.
Let’s see how this arises explicitly. We insert a UV regulator that cuts off themassive modes. The natural choice for a UV regulator is to de-weight high
65
energy modes exponentaially with their mass. We can write∑n
φ†n(x)γ5φn(x) = lim
M→∞
∑n
φ†n(x)γ5φn(x)e−λ
2n/M
2
= limM→∞
Tr〈φn|γ5e(i /D)2/M2
|φn〉
= limM→∞
Tr〈x|γ5e(i /D)2/M2
|x〉. (323)
Now (i /D)2 = −D2+ e2σ
µνFµν (for a background field). As we have a (regulated)trace over all possible states, we can evaluate this in any basis we choose. Let’schoose the simplest possible basis of states, that of plane-waves.
1000
,
0100
,
0010
,
0001
eik·x.
So we have
limM→∞
∫d4k
(2π)4e−ik·xTr
(γ5e(−DµD
µ+ e2σ
µνFµν)2/M2)eik·x
We can move eik·x through and replace Dµ by kµ, to get
limM→∞
∫d4k
(2π)4Tr(γ5 exp
[(−kµkµ + e
2σµνFµν)
M2
]).
To get a non-zero trace over all four polarisation vectors, we need to bring downfour gamma matrices, and so we have
limM→∞
Tr (γ5[γµ, γν ]Fµν)2 1
(2M2)212!
∫d4k
(2π)4e−k
2/M2.
The∫d4k integral now cancels the residual factor of M−4, leaving an overall
finite answer of ∫d4x
−18π2
Fµν(∗F )µν (324)
This then implies that the integrand measure transforms under chiral transfor-mations as∫DAµDψDψ →
∫DAµD
(ψe−iαγ5
)D(eiαγ5ψ) = DAµDψDψ
(1 +
∫d4x
−iα8π2
Fµν(∗F )µν).
The integrand measure is therefore not invariant under chiral transformations.This actually makes perfect sense: the path integral counts (number of leftmovers) minus (number of right movers). This difference is precisely accountedfor by the instanton number. In non-zero instanton backgrounds, there really aremore left-movers than right-movers and axial symmetry is not a symmetry. Thisaccounts for the topolgical form of the chiral anomaly. Instanton backgroundsdo have different numbers of left and right-movers, and so have an intrinsicnotion of chirality - it is therefore natural that the measure should fail to beinvariant in such backgrounds.
66
7.1 The Konishi Anomaly
Let us now generalise this to the case of supersymmetric field theories. Our firstgeneralisation is to the Konishi anomaly, which is the supersymmetrisation ofthe chiral anomaly.
The path integral of a chiral multiplet in a gauge field background is∫DφDψDFDφDψDF exp
[−∫d4x|Dφ|2 + ψ /Dψ − FF
].
We want to perform a general rescaling Φ = eαΦ′. Naively,
DΦ = DφDψDF = Dφ′(det eα)Dψ
′(det eα)−2DF (det eα) ≡ DΦ
′J.
Here the negative powers of the determinant are due to the Grassmanian natureof the fermions. We now want to define and regularise the Jacobian. We proceedin the same way as before.
First, we expand the scalar field φ in a basis of eigenstates.
φ(x) =∑n
λnφn(x), so thatDφ =∏n
dλn.
The redefined field is then expanded in a modified basis of eigenstates,
eαφ(x) =∑
λ′
nφn(x), with λ′
n =∫φ†n(x)e
αφ(x) ≡ Cnmλm.
HereCnm =
∫φ†n(x)e
α(x)φm(x)dx.
As a result, ∏dλ
′= (detC)
∏dλ
and
(det c) = exp[tr log
∫φ†n(x)e
α(x)φm(x)]
(325)
= exp[tr∫φ†n(x)α(x)φm(x)
](326)
= exp
[∑n
∫α(x)φ†n(x)φn(x)
]. (327)
As a consequence we have
lnJ =∫d4xα(x)
∑n
φ†n(x)φn(x). (328)
We regulate this as before by cutting off the high momentum modes, which ismost easily carried out by using an exponential cutoff.∑
n
φ†n(x)φn(x) →∑n
e−λ2ntφ†n(x)φn(x) (329)
= Trφet(Dµ)2 . (330)
67
We have a similar effect for both the fermionic and F terms, so we have
lnJ = α(Trφet(Dµ)2 − Trψet(
/D)2 + TrF et(Dµ)2). (331)
We are now going to evaluate this explicitly. We work in Abelian gauge theoryand impose a constant electromagnetic field background (this does not signifi-cantly affect anything).
Fµν =
0 E 0 0−E 0 0 00 0 0 B0 0 −B 0
. (332)
Then Dµ = ∂µ + iAµ = i (−i∂µ +Aµ). We now make a gauge choice
A1 = −Ex2
2, A2 =
Ex1
2, A3 = −Bx4
2, A4 =
Bx3
2.
We also write pµ = −i∂µ. Then
Tret(Dµ)2 = Tr exp
[−t
((p1 −
Ex2
2
)2
+(p2 +
Ex1
2
)2
+(p3 −
Bx4
2
)2
+(p4 +
Bx3
2
)2)]
.
(333)(Note that pµ here is an operator). We now define
H(E) =(p1 −
Ex2
2
)2
+(p2 +
Ex1
2
)2
= p21 + p2
2 +(E
2
)2 (x2
1 + x22
)− E(p1x2 − p2x1), (334)
and likewise
H(B) =(p3 −
Bx4
2
)2
+(p4 +
Bx3
2
)2
. (335)
Now, as (x1, x2, p1, p2) are simulataneously diagonalisable, we can therefore split
Tret(Dµ)2 = Tre−tH(E)H(B)
= Tr(x1,x2)e−tH(E)Tr(x3,x4)e
−tH(B). (336)
To evaluate the traces we take
pµ =
√E
2
(aµ − a†µ
i√
2
), xµ =
√2E
(aµ + a†µ√
2
).
We can then write
H(E) =E
2×−1
2
[(a1 − a†1)
2 + (a2 − a†2)2]
+E
2× 1
2
[(a1 + a†1)
2 + (a2 + a†2)2]− E
2i
((a1 − a†1)(a2 + a†2)− (a2 − a†2)(a1 + a†1)
)=
E
4
[2(a1a
†1 + a†1a1) + 2(a†2a2 + a†2a2) + 2i(2a1a
†2 − 2a†1a2)
]=
E
2
[(a†1a1 + a1a
†1) + (a†2a2 + a†2a2) + 2i(a1a
†2 − a†1a2)
]=
E
2
[2a†1a1 + 2a†2a2 + [a1, a
†1] + [a2, a
†2] + 2i(a1a
†2 − a†1a2)
]. (337)
68
Now define aL = a1−ia2√2
, aR = a1+ia2√2
. This then gives
a†LaL = (a†1 + ia†2)(a1 − ia2)
=12
(a†1a1 + a†2a2 − i(a†1a2 − a†2a1)
).(338)
So then
H(E) = E
(2a†LaL +
[a1, a†1] + [a2, a
†2]
2
),
and soH(E) = E(2a†LaL + 1).
aL still has the canonical harmonic oscillator commutation relations, and so
Tre−tH(E) =∑nL,nR
e−tE(2nL+1).
The sum over both nL and nR reflects the fact that we need to sum over bothaL and aR excitations. The fact that this sum diverges reflects a divergencedue to infinite space-time volume. Let’s make this more precise. We go to finitevolume (radius L) and only want to sum over modes with
〈nL, nR|x2 + y2|nL, nR〉 < L2
Now
x2 + y2 =2
2E
((a1 + a†1
)2
+(a2 + a†2
)2)
=1
2E
(2(aLa
†L + a†LaL
)+ 2
(aRa
†R + a†RaR
)).
=1E
(2nL + 2nR + 2) . (339)
As a result x2 + y2 = 2E (nL + nR + 1), and so the condition that x2 + y2 < L2
translates to the condition that 2E (nL + nR + 1) < L2. As a a result we obtain
the bound
nR <EL2
2− nL − 1. (340)
Our regularised sum is now
Tre−tH(E) =∑nL,nR
e−tE(2nL+1)
=∑nL
(EL2
2− nL − 1
)e−tE(2nL+1). (341)
To leading order in L2 (i.e. at large volume), we have
EL2
2e−tE
(1 + e−2tE + (e−2tE)2 + . . .
)=
EL2
2e−tE
1− e−2tE+ . . .
=(πL2)E
4π1
sinh(tE). (342)
69
Going to infinite volume, we therefore have
Tre−tH(E) =14π
∫dx0dx1
E
sinh tE. (343)
Performing the same calculation for H(B), we then obtain
Tret(Dµ)2 =1
16π2
∫d4x
EB
sinh tE sinh tB(344)
This diverges as we take the regulator t → 0 (as we would expect). As this isjust a scalar field, we also get the same result for the F-term. So at this pointwe now have two components of the anomalous Jacobian,
lnJ = α
Trφet(Dµ)2︸ ︷︷ ︸yes
−Trψet/D2
+ TrF et(Dµ)2︸ ︷︷ ︸yes
. (345)
We next need the regulated Dirac determinant. The Dirac operator is
( /D)2 = D2µ −
i
4[γµ, γν ]Fµν
= (Dµ)2 +(
(E +B)σ3 00 (−E +B)σ3
). (346)
Any fermionic state can be written as a linear combination of the basic wave-functions
ψ(x)⊗
1000
, ψ(x)⊗
0100
.
We can therefore also factorise
TrL,Ret(/D)2 = TrL,Ret(Dµ)2e
(E +B)σ3 00 (−E +B)σ3
= Tret(Dµ)2)Tret(±E+B), (347)
where + applies for left-handed modes and − applies for right-handed modes.Now,
Tret(±E+B)σ3= et(±E+B) + e−t(±E+B)
= 2 cosh t(E ±B). (348)
So the fermionic heat kernel is then
TrL,Ret/D2
= Tret(Dµ)2Tret(±E+B)σ3
=1
16π2
∫d4x
EB × 2 cosh t(E ±B)sinhEt sinhBt
. (349)
We can now sum (left-handed fermion)
lnJ = α(Trφet(Dµ)2 − TrψL
et /D2+ TrF et(Dµ)2
)(350)
=α
16π2
∫d4xEB
2− 2 cosh t(E +B)sinhEt sinhBt
. (351)
70
and likewise (right-handed fermion)
ln J = α(Trφet(Dµ)2 − TrψR
et /D2+ TrF et(Dµ)2
)(352)
=α∗
16π2
∫d4xEB
2− 2 cosh t(E −B)sinhEt sinhBt
. (353)
Let’s look at evaluating this for small t.
cosh ε =eε + e−ε
2= 1 +
12
(ε2
2+ε2
2
)= 1 +
ε2
2.
and so
2− 2 cosh t(E −B) = 2− 2(
1 +t2(E −B)2
2
)= −t2(E −B)2. (354)
Therefore
(2− 2 cosh[t(E ±B)])EBsinhEt sinhBt
∼ − t2(E −B)2EB
EBt2
∼ −(E ±B)2. (355)
So, including the cutoff and regulating we emerge with
lnJ =α
16π2
∫d4x− (E +B)2.
This is now a finite and well-defined answer. Now,
FµνFµν = −2(E2 +B2),
Fµν(∗F )µν = −4EB.
So(E +B)2 = −1
2[FµνFµν + 2Fµν(∗F )µν ] .
This now precisely matches the expansion of the gauge kinetic term∫d2θWαW
α
that we encountered when discussing vector superfields.
Note that this expression should also have overall factors of the charge (es-sentially as Dµ → ∂µ − ieAµ etc), and the charge is the Casimir of the repre-sentation. So what we emerge with overall is
lnJ = − 116
∫d4xd2θ
2t2(Φ)8π2
ln (eα)WαWα. (356)
plus the contribution from ln J . Let us consider two special cases.
First, take the case where α is purely imaginary (α = iθ). Then
lnJ + ln J = iθ
∫d4x
16π2EB
−2[cosh t(E +B)− cosh t(E −B)]sinh tE sinh tB
= iθ
∫d4x
16π2− 2EB × t2(E +B)2 − t2(E −B)2
2t2EB
= iθ
∫d4x
16π2− 4EB. (357)
71
From this we see that we can write
lnJ + ln J = iθ
(1
16π2
∫d4x− 4EB
)= iθ
∫d4x
16π2Fµν(∗F )µν , (358)
as −4EB = Fµν(∗F )µν .
This result should not be a surprise at all, as what we have done here issimply to reproduce the chiral anomaly, which corresponds to a rephasing ofchiral fermions (and thus a rephasing of the chiral superfields).
Now let us take the case where α is purely real (α = θ). In this case we havenot a rephasing but a rescaling of the chiral superfields. We now have
lnJ + ln J = θ
∫d4x
16π2EB
2[2− cosh t(E +B)− cosh t(E −B)]sinh tE sinh tB
= θ
∫d4x
16π22EB
[−t2(E +B)2 − t2(E −B)2]2t2EB
= −θ∫
d4x
16π22(E2 +B2) = −θ
∫d4x
16π2FµνF
µν . (359)
What we now see is a correction to the physical gauge couplings. This correctionis known as the Konishi anomaly. What this says is that if we have a theorywith matter fields that are not canonically normalised, then the physical (1PI)gauge couplings receive an anomalous 1-loop contribution
1g2phys
(µ) =1
g2holo
(µ)− Ta(r)8π2
lnZr, (360)
where the Kahler potential looks like
K = . . .+ ZΦ†Φ + . . .
So we start with
L =∫d2θfa(Φ)WαW
α +∫d4θ
∑i
ZrΦ†reqV Φr, (361)
and this turns into
1g2phys
= Refa(Φ) +(∑r nrTa(r)− 3Ta(G))
16π2ln(
Λ2
µ2
)−∑r
Ta(r)8π2
ln detZr︸ ︷︷ ︸Konishi anomaly
.
(362)We now extend this to look at anomalous contributions to gauge couplingscoming from non canonically-normalised gauge fields.
7.2 Vector Multiplets and the NSVZ β Function
We now look at resclings associated to a vector multiplet. We first need to setup the action. For vector fields we need to gauge-fix, which we do via Faddeev-
72
Popov:
1 =∫Dgδ(DµV gµ − a) det(DµDµ)
=∫DgDcDcδ(DµV gµ − a)e−
∫d4xc(DµDµ)c. (363)
Now we can also insert
1 =∫Dae−
∫d4x ξ
2g2 a2∫
Dae−∫d4x ξ
2g2 a2,
giving overall
1 =
∫Dae−
∫d4x ξ
2g2 a2 ∫DgDcDcδ(DµV gµ − a)e−
∫d4xc(DµDµ)c∫
Dae−∫d4x ξ
2g2 a2
. (364)
Integrating over a, we get
1 =∫DgDcDce−
∫d4x
ξ(DµVgµ )2
2g2 +c(DµDµ)c∫Dae−
∫d4x ξ
2g2 a2
. (365)
As standard, we then have the path integral
Z =∫DVDgDcDce−
∫d4x
ξ(DµVgµ )2
2g2 +c(DµDµ)c−FµνFµν∫
Dae−∫d4x ξ
2g2 a2
. (366)
As the kinetic term is gauge-invariant, we can write (using DV g = DV )
Z =∫DV gDgDcDce−
∫d4x
ξ(DµVgµ )2
2g2 +c(DµDµ)c−FµνFµν∫
Dae−∫d4x ξ
2g2 a2
,
=∫Dg
∫ DV gDcDce− ∫d4x
ξ(DµVgµ )2
2g2 +c(DµDµ)c−FµνFµν∫
Dae−∫d4x ξ
2g2 a2
(367)
This is the standard Faddeev-Popov trick. Now note an important feature. Ifwe rescale the vector multiplet, then we also need to rescale the gauge-fixingfunction (as DµV gµ = a is the gauge-fixing function).
Now, the quadratic gauge field operator is, in Feynman gauge (ξ = 1),
Aa,µ[(Dµ)2abδµν − i(F cρσ)(M
ρσ)µν(tcG)ab]Aν,b.
This is the quadratic operator that acts on vector fields in a background gaugefield (i.e. generalising D2
µ and /D2 for scalar and spinor fields). It is this operator
that we have to use to regularise the path integral when we rescale the vectorfield measure.
So for the vector superfield we then have
lnJ = α(Trvet[(Dµ)2δµν−iFρσ(Mρσ)µν ] − Traet(Dµ)2 − Trλet(
/D)2 − Trλet /D2
+ TrDet(DµDµ)
).
(368)
73
Tra and TrD cancel, to give
lnJ = α(Trvet[(Dµ)2δµν−iFρσ(Mρsigma)µν ] − Trλet
/D2− Trλe
t /D2). (369)
Let’s evaluate this for the simplest case, an SU(2) gauge field, with a backgroundfor the T 3 generator. The T 3 generator just leads to simple charges for T 1 ±iT 2, which behave effectively as U(1) fields. We can focus on these particulargenerators to see what the modified coupling is.
We do something very similar to what we did previously for the fermionicfields. We write
F 3µν =
0 E 0 0−E 0 0 00 0 0 B0 0 −B 0
. (370)
We can decompose the vector field into four components, writing
A(x) =
A1
A2
A3
A4
. (371)
Just as for the spinor fields, we can decompose
Aµ(x) = A(x)⊗
1000
, (372)
and we have
−i(Fρσ)(Mρσ)µν =(
2iEσ2 00 2iBσ2
). (373)
Then the vector heat kernel becomes
Tret(Dµ)2 × Tr exp[
2itEσ2 00 2itBσ2
],
with the traces factorising. This gives
116π2
∫d4xEB
2(cosh 2tE + cosh 2tB)sinh tE sinh tB
.
We now combine these with the fermionic traces to give
lnJ =α
16π2
∫d4x2EB
[cosh 2tE + cosh 2tB − cosh t(E +B)− cosh t(E −B)]sinh tE sinh tB
.
(374)and so
lnJ =α
16π2
∫d4x2(E2 +B2) −→ α
16π2
∫d4xFµνF
µν . (375)
The ‘charge’ factor is T2(G) which again comes from the fact that the vectoris in the adjoint of the gauge group and so the appropriate Casimir is thatapplying to the adjoint. So the overall result is
D(gcVc) = D(Vc) exp[∫
d4y
∫d2θ
2T2(G)8π2
(ln gc)W a(gcVc)W a(gcVc)]. (376)
74
This gives an anomalous contribution to the gauge coupling for rescaling thevector superfield and making it canonically normalised.
For the physical coupling, we then have
1g2(µ)
= Re(fa(Φ))+βa
16π2ln(
Λ2
µ2
)+T (G)8π2
ln g−2a (Φ, Φ, µ)−
∑ Ta(r)8π2
ln detZr(Φ, Φ, µ).
(377)Eq (377) relates the physical coupling - the one that actually matters in de-termining the strength of scattering amplitudes - to the holomorphic couplingwhich is protected by non-renormalisation theorems. Anomalies are 1-loop ex-act, and this relationship can be integrated to give the (all-orders) NSVZ β-function,
β(g) = − 116π2
3T2(G)−∑r T2(r)(1− γr)
1− T2(G)g2
8π2
. (378)
Here γr is the anomalous dimension of the chiral superfields Φr. These anoma-lous dimensions get corrections at all orders in the loop expansion, as these areassociated to renormalisation of the kinetic term for Φr, which comes from theKahler potential and so is not protected against renormalisation. Note that thegauge field does not have an equivalent term: the holomorphic gauge kineticfunction is 1-loop exact, as is the anomalous rescaling, and so the correctionsterminate.
This β function is famous both as an ‘exact’ result in supersymmetric gaugetheories and as a guide to understanding the general properties of supersym-metric gauge theories. However one should use the ‘exactness’ with care asβ-functions are not scheme independent quantities. Fixed points are scheme-independent, and so the NSVZ β-function can (for example) be used to studythe phase diagram of a supersymmetric theory.
Finally, in supergravity there is an additional anomaly (the Kahler-Weylanomaly) that comes from the rescaling of g → λg, where g is the space-timemetric. This arises as the ‘natural’ metric kinetic term, in a basis where holo-morphy is manifest, is given not by the Einstein-Hilbert term but instead by akinetic term ∫
√geK/3R. (379)
Converting this to canonical Einstein gravity requires a rescaling of the metric, inthe same way that converting the holomorphic gauge kinetic term
∫fa(Φ)WαW
α
to a canonically normalised kinetic term∫WαW
α required a rescaling of thevector superfields.
Taking this metric rescaling into account, we obtain the supergravity gener-alisation of the NSVZ β function, given by the Kaplunovsky-Louis formula
1g2(µ)
= Re(fa(Φ))+βa
16π2ln(M2P
µ2
)+T (G)8π2
ln g−2(Φ, Φ, µ)+(∑nrTa(r)− T (G))
16π2K(Φ, Φ)−
∑ Ta(r)8π2
ln detZr.
(380)
75
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