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Beyond the Standard Model: Supersymmetry and String Theory
Contents
1 The Standard Model and Its Discontents 11.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Standard Model as an Effective Field Theory . . . . . . . .. . . . 2
1.2.1 Integrating out theW andZ Bosons . . . . . . . . . . . . . . . 31.2.2 What Might the Standard Model Come From? . . . . . . . . . . 4
1.3 Experimental Status of Lepton and Baryon Conservation . . .. . . . . 51.3.1 Dimension five: Lepton Number Violation and Neutrino Mass . 51.3.2 Other Dimension Five Operators . . . . . . . . . . . . . . . . . 61.3.3 Dimension Six Operators: Proton Decay . . . . . . . . . . . . .6
1.4 A puzzle at the renormalizable level . . . . . . . . . . . . . . . . .. . 71.5 The Hierarchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Dark Matter and Dark Energy . . . . . . . . . . . . . . . . . . . . . . 81.7 Summary: Limitations of the Standard Model . . . . . . . . . . .. . . 9
2 Of Anomalies, Strong CP and Axions 92.1 The Chiral Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Return to QCD . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 A Two Dimensional Detour . . . . . . . . . . . . . . . . . . . 132.1.3 The Anomaly In Two Dimensions . . . . . . . . . . . . . . . . 132.1.4 The CPN Model: An Asymptotically Free Theory . . . . . . . . 142.1.5 The LargeN Limit . . . . . . . . . . . . . . . . . . . . . . . . 152.1.6 The Role of Instantons . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Real QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.1 The Theory and its Symmetries . . . . . . . . . . . . . . . . . 192.2.2 Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.3 Real QCD and the U(1) Problem . . . . . . . . . . . . . . . . . 24
2.3 The Strong CP Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 θ-dependence of the Vacuum Energy . . . . . . . . . . . . . . . 252.3.2 The Neutron Electric Dipole Moment . . . . . . . . . . . . . . 26
2.4 Possible Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.1 The Axion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 An Introduction to Supersymmetry 29
4 An Overview of Supersymmetry 304.1 The Supersymmetry Algebra and its Representations . . . . .. . . . . 304.2 N=1 Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 N=2 Theories: Exact Moduli Spaces . . . . . . . . . . . . . . . . . . .374.4 A Still Simpler Theory: N=4 Yang Mills . . . . . . . . . . . . . . . .. 394.5 Aside: Choosing a Vacuum . . . . . . . . . . . . . . . . . . . . . . . . 414.6 Supersymmetry and the Hierarchy Problem . . . . . . . . . . . . .. . 424.7 Supersymmetry in Superspace . . . . . . . . . . . . . . . . . . . . . . 434.8 Component Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . 434.9 Soft Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . . 434.10 Cancellation of Quadratic Divergences . . . . . . . . . . . . . .. . . . 434.11 More on the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.12 LSP as the Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . 474.13 Local Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5 Spontaneous Supersymmetry Breaking 515.1 Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.1.1 F and D Type Breaking . . . . . . . . . . . . . . . . . . . . . . 51
6 Non-Renormalization Theorems 51
7 Beyond Perturbation Theory 517.1 Supersymmetric QCD and Its Symmetries . . . . . . . . . . . . . . . .517.2 Nf < Nc: The Superpotential . . . . . . . . . . . . . . . . . . . . . . . 517.3 Nf = Nc − 1: Instanton computation of the Superpotential . . . . . . . 517.4 Nf < Nc: Gaugino Condensation . . . . . . . . . . . . . . . . . . . . 517.5 Supersymmetry Breaking: The(3, 2) model . . . . . . . . . . . . . . . 51
8 Seiberg Duality 51
9 Introduction 53
10 The Bosonic String 54
11 The Superstring 54
12 Type II Strings 54
13 Heterotic String 54
14 Compactification of String Theory 5414.1 Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5414.2 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5414.3 Calabi-Yau compactifications . . . . . . . . . . . . . . . . . . . . . .. 54
15 Dynamics of String Theory at Weak Coupling 5415.1 Non-Renormalization Theorems . . . . . . . . . . . . . . . . . . . . .5415.2 Gaugino Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.3 Obstacles to A String Phenomenology . . . . . . . . . . . . . . . .. . 54
16 Strings at Strong Coupling: Duality 5416.1 Perturbative Dualities . . . . . . . . . . . . . . . . . . . . . . . . . .. 5416.2 D-Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5416.3 Strong-Weak Coupling Dualities: Some Evidence . . . . . . .. . . . . 54
16.3.1 IIB Self Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 5416.3.2 Catalog of other string Dualities . . . . . . . . . . . . . . . . .54
16.4 IIA → 11 Dimensional Supergravity (M Theory) . . . . . . . . . . . . 5416.5 Strongly Coupled Heterotic String . . . . . . . . . . . . . . . . . .. . 5416.6 AdS-CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
17 Coda: Where are We headed 5517.1 Some alternatives to Supersymmetry . . . . . . . . . . . . . . . .. . . 55
17.1.1 Technicolor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.1.2 Large Extra Dimensions . . . . . . . . . . . . . . . . . . . . . 5517.1.3 Warped Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 5517.1.4 String Phenomenology: The Landscape? . . . . . . . . . . . .5517.1.5 The Tevatron, The LHC and the ILC . . . . . . . . . . . . . . . 55
18 Appendices 5518.1 Left Handed Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 5518.2 Gauge Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5518.3 Path Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1 The Standard Model and Its Discontents
1.1 The Standard Model
The Standard Model is extremely successful. At any given time, you will find a fewpeople excited about some slight discrepancy between theory and experiment, but thisjust serves to highlight the extraordinary level of agreement.
It is amazing that at a microscopic level, almost everythingwe know about natureis described by such a simple structure. To write down the model, one first specifiesthe gauge group:SU(3) × SU(2) × U(1). Correspondingly, there are gauge bosons,Aa
µ, W iµ, andBµ. The description of the matter content is particularly compact if one
describes the fermions in terms of left-handed fields (as explained in Wess and Bagger,and also in Peskin, in four dimensions, a general spin-1/2 field, whether “Majorana” or“Dirac” can be written in terms of left-moving fields). Below,we list the fields, using anotation where the first number in the parenthesis refers to theSU(3) representation, thesecond theSU(2) representation, and the subscript is theU(1) charge (“hypercharge”):Quark doublets:Qf = (3, 2)1/3
Antiquark singlets:uf = (3, 1)−4/3 df = (3, 1)2/3
Lepton doublets:Lf = (1, 2)−1
Lepton singlets:ef = (1, 1)2
In addition to the fermions, (and the bosons which are required by the gauge symme-try), the simplest version of the Standard Model contains a single Higgs field, a scalar,φ:Higgs scalar:φ = (1, 2)1
Specifying the interactions is not much more complicated. First, there are gauge-invariant kinetic terms for all of the fields. For the quark doublets, for example, theseare:
iQσµDµQ∗ (1)
where
DµQ = (∂µ − iAµ − iWµ − i1
3Bµ)Q (2)
andAµ is a matrix-valued field,Aµ = AaµT
a, as isWµ = W iµT
i. In terms of these fields,the field strengths can be written elegantly as:
Fµν = ∂µAν − ∂νAµ + i[Aµ, Aν ]. (3)
and similarly for theSU(2) andU(1) fields. The kinetic terms for the gauge fields, interms of these matrix-valued fields, take the form:
LA = − 1
2g2TrF 2
µν . (4)
1
Before including the Higgs fields, at the level of dimension four terms, i.e. of renor-malizable terms, this is all we can write. Including the Higgs field, however, there areadditional possibilities. Apart from the kinetic term for the Higgs field, we can write apotential:
V (φ) = −µ2|φ|2 + λ|φ|4. (5)
Again, this is completely gauge invariant.There are also Yukawa couplings of the fermions to the Higgs field:
Lyuk = yUf,f ′Qf uf ′φ+ yD
f,f ′Qf df ′σ2φ∗ + yL
f,f ′Lf ef ′φ. (6)
By redefining fields, we can simplify the matricesyU , yD andyL; it is customary to takeyU andyL diagonal, and there is a conventional form foryD (the Kobiyashi-Maskawamatrix) which we will describe in section ***.
If we require renormalizability, i.e. we require that all ofthe terms in the lagrangianbe of dimension four or less, this is all we can write. It is remarkable that this simplestructure incorporates over a century of investigation of elementary particles. Workingthrough the consequences of this theory is very challenging, and is the subject of manyexcellent books and reviews. In the following sections, we will focus on a few featureswhich might be relevant to what comes beyond.
Exercise: Compute the mass of the Higgs field as a function ofµ andλ. Discuss pro-duction of Higgs particles (you do not need to do detailed calculations, but indicate therelevant Feynman graphs and make at least crude estimates ofcross sections) ine+e−,µ+µ− andpp annihilation. Keep in mind that because some of the Yukawa couplingsare extremely small, there may be processes generated by loop effects which are biggerthan processes which arise at tree level.
1.2 The Standard Model as an Effective Field Theory
The standard model has some remarkable properties. The renormalizable terms respecta variety of symmetries, all of which are observed to hold to ahigh degree in nature:
• Baryon number:
Q→ ei α3Q u→ e−i α
3 u d→ e−i α3 d (7)
• Three separate lepton numbers:
Lf → e−iαfLf ef → eiαf ef . (8)
We did notimpose these symmetries. They are simply consequences of gauge invari-ance, and the fact that there are only so many renormalizableterms we can write. These
2
symmetries are said to be “accidental,” since they don’t seem to result from any deepunderlying principle.
This is already a triumph. As we will see when we consider possible extensions ofthe Standard Model, this did not have to be. But this success raises the question: whyshould we impose the requirement of renormalizability?
In the early days of quantum field theory, renormalizabilitywas sometimes presentedas a sacred principle. There was a view that field theories were fundamental, and shouldmake sense in and of themselves. Much effort was devoted to understanding whetherthe theories existed in the limit that the cutoff was taken toinfinity.
But there was an alternative model for understanding field theories, provided byFermi’s original theory of the weak interactions. In this theory, the weak interactions aredescribed by a lagrangian of the form:
Lweak =Gf√
2JµJµ (9)
Here the currents,Jµ, are bilinears in the fermions; they include terms likeQσµT aQ∗.This theory, like the Standard Model, was very successful. It took some time to actuallydetermine the form of the currents, but for forty years, all experiments in weak interac-tions could be summarized in a lagrangian of this form. Only with the first indicationsof theZ boson ine+e− annihilation were deviations observed.
The four fermi theory is non-renormalizable. Taken seriously as a fundamental the-ory, it predicts violations of unitarity at TeV energy scales. But from the beginning,the theory was viewed as aneffective field theory, valid only at low energies. WhenFermi first proposed the theory, he assumed that the weak forces were due to exchangeof particles – what we now know as theW andZ bosons.
1.2.1 Integrating out theW andZ Bosons
Within the Standard Model, we can derive the Fermi theory, and we can also understandthe deviations. A traditional approach is to examine the Feynman diagram:
This can be understood as a contribution to a scattering amplitude, but it is best un-derstood here as a contribution to the effective action of the quarks and leptons. Thecurrents of the Fermi theory are just the gauge currents which describe the coupling at
3
each vertex. The propagator, in the limit of very small momentum transfer, is just a con-stant. In coordinate space, this corresponds to a space-time δ-function. The interactionis local. The effect is just to give the four Fermi lagrangian. One can consider effectsof small finite momentum by expanding the propagator in powers of q2. This will givefour fermi operators with derivatives. These are suppressed by powers ofMW ; thereeffects are very tiny at low energies. Still, in principle, they are there, and in fact themeasurement of such terms provided the first hints of the existence of theZ boson.
This can also be derived in the path integral approachHere insert handout from218.
The lesson is that, prior to 198?, one could view QED + QCD + the Fermi theory asa perfectly acceptable theory of the interactions. The theory would have to be viewed,however, as an effective theory, valid only up to an energy scale of order100 GeV orso. Sufficiently precise experiments would require inclusion of operators of dimensionhigher than four. The natural scales for these operators would be the weak scale.
1.2.2 What Might the Standard Model Come From?
As successful as the Standard Model is, it is likely that, like the four-Fermi theory, itis the low energy limit of some underlying, more “fundamental” theory. In the secondhalf of this book, our model for this theory will be String Theory. Consistent theories ofstrings, for reasons which are somewhat mysterious, are theories which describe generalrelativity and gauge interactions. Unlike field theory, String Theory is a finite theory. Itdoes not require a cutoff for its definition. In principle, all physical questions have well-defined answers within the theory. If this is the correct picture for the origin of nature atextremely short distances, then the Standard Model is just its low energy limit. When westudy string theory, we will understand in some detail how such a structure can emerge.For now, the main lesson we should take concerns the requirement of renormalizability:the Standard Model should be viewed as an effective theory, valid up to some energyscale,Λ. Renormalizability is not a constraint we impose upon the theory; rather, weshould include operators of dimension five or higher with coefficients scaled by inversepowers ofΛ. The question of the value ofΛ is an experimental one. From the success ofthe Standard Model, as we will see, we know that the cutoff is large. From string theory,we might imagine thatΛ ≈ Mp = 1.2 × 1018 GeV. But, as we will now describe, wehave experimental evidence that there is new physics which we must include at scaleswell belowMp. We will also see that there are theoretical reasons to believe that thereshould be new physics at TeV energy scales.
4
1.3 Experimental Status of Lepton and Baryon Conservation
1.3.1 Dimension five: Lepton Number Violation and Neutrino Mass
To proceed systematically, we should write operators of dimension five, six, and so on.At the level of dimension five, we can write several terms which violate lepton number:
L =1
Λγf,f ′φφLfL
′f + c.c.. (10)
With non-zeroφ, these terms give rise to neutrino masses. In the past few years, per-suasive evidence has emerged that the neutrinos do have non-zero masses and mixings.This comes from the study of neutrinos coming from the sun (the “solar neutrinos”)and neutrinos produced in the upper atmosphere by cosmic rays (which produce pionswhich subsequently decay to muons andνµ’s, whose decays in turn produce electrons,νµ’s, andνe’s. Accelerator and reactor experiments are now firming up this picture. Thebest current values for the neutrino masses and mixings are:
mν ... (11)
It is conceivable that these masses are not described by the lagrangian of eqn. 10.Instead, the masses might be “Dirac,” by which one means thatthere might be additionaldegrees of freedom, which by analogy to thee fields we could labelν, with very tinyYukawa couplings to the normal neutrinos. This would truly represent a breakdown ofthe Standard Model: even at low energies, we would have been missing basic degrees offreedom. But this does not seem likely. If there are singlet neutrinos,N , nothing wouldprevent them from gaining a “Majorana” mass:
Lmaj = MNN. (12)
As for the leptons and quarks, there would also be a coupling of ν to N . There wouldnow be amass matrix for the neutrinos, involving bothN andν. For simplicity, considerthe case of just one generation. Then this matrix would have the form:
Mν =(M yvyv 0
). (13)
Such a matrix has one large eigenvalue, of orderM , and one small one, of ordery2v2
M.
This provides a natural way to understand the smallness of the neutrino mass; it is re-ferred to as the “seesaw mechanism.” Alternatively, we can think of integrating out theright-handed neutrino, and generating the operator of eqn.[10].
It seems more plausible that the observed neutrino mass is Majorana than Dirac, butthis is a question that hopefully will be settled in time by experiment. If it is Majorana,
5
this suggests that there is another scale in physics, well below the Planck scale. For evenif the new Yukawa couplings are of order one, the neutrino mass is of order
mν = 10−5 eV(Mp/Λ) (14)
If the Yukawa’s are small, as are many of the quark Yukawa couplings, the scale can bemuch smaller.
1.3.2 Other Dimension Five Operators
There is another class of dimension five operators which can appear in the effectivelagrangian. These are magnetic and electric dipole operators. For example, for themuon magnetic dipole moment, one can have a term:
Lg−2 =e
ΛFµνµσ
µνµ+ c.c., (15)
whereFµν is the electromagnetic field (in terms of the fundamentalSU(2) andU(1)fields, one can write similar gauge-invariant combinationswhich reduce to this at lowenergies). The muon magnetic moment is measured to extremely high precision, and itsStandard Model contribution is calculated with comparableprecision. As of this writing,there is a 2.6σ discrepancy between the two. Whether this reflects new physics or not isuncertain.
Similar operators can also generate an electric dipole moment for the leptons or thequarks:
Ld =e
ΛFµνµσ
µνµ+ c.c. (16)
where
Fµν =1
2εµνρσF
ρσ (17)
These operators violate CP. There are strong experimental limits on the electric dipolemoment of the neutron and of the electron which constrain thecoefficient.
Operators where one of the muons is replaced by an electron can generate rare pro-cesses such asµ→ e+ γ, on which there are quite stringent experimental bounds.
1.3.3 Dimension Six Operators: Proton Decay
Proceeding to dimension six, we can write numerous terms which violate baryon num-ber, as well as additional lepton number violating interactions:
Lbv =1
Λ2Qσµu∗Lσµd
∗ + . . . (18)
6
This can lead to processes such asp→ πe. Experiments deep underground set limits oforder1033 years on this process. Correspondingly, the scaleΛ must be larger than1015
GeV.So viewing the Standard Model as an effective field theory, wesee that there are
many possible non-renormalizable operators which might appear, but most have scaleswhich are tightly constrained by experiment. One might hope– or despair – that theStandard Model will provide a complete description of nature up to scales many ordersof magnitude larger than we can hope to probe in experiment.
We now turn to some challenges for the Standard Model.
1.4 A puzzle at the renormalizable level
There is another puzzle, which we will touch upon now but return to and discuss ingreater detail later. There are a set of operators ofdimension four, which we don’t usualdiscuss when we introduce non-abelian gauge theories:
Lθ = θF F (19)
where
Fµν =1
2εµνρσF
ρσ. (20)
We usually ignore these because classically they are inconsequential; they are totalderivatives and do not modify the equations of motion. But as we will see they havereal effects at the quantum level. Because they are CP violating, they turn out to behighly constrained. From the limits on the neutron electricdipole moment, one canshow thatθ < 10−9. This is the first real puzzle we have encountered. Why such asmall dimensionless number? Answering this question, as wewill see, may point tonew physics.
1.5 The Hierarchy Problem
The second very puzzling feature is the Higgs field. As of thiswriting, the Higgs fieldis the one piece of the Standard Model which has not been seen.Indeed, the structurewe have postulated, a single Higgs doublet with a particularpotential, might be viewedas somewhat artificial. We could have included several doublets, or perhaps tried toexplain the breaking of the gauge symmetry through some morecomplicated dynamics.But there is a more serious question associated with fundamental scalar fields, raisedlong ago by Ken Wilson. This problem is often referred to as the “hierarchy problem.”
Consider, first, the one loop corrections to the electron massin QED. These arelogarithmically divergent. In other words,
δm = amoα
4πln(Λ). (21)
7
We can understand this result in simple terms. In the limitmo → 0, the theory has anadditional symmetry, a chiral symmetry, under whiche ande transform by independentphases. This symmetry forbids a mass term, so the result mustbe linear in the (bare)mass. So on dimensional grounds, any divergence is at most logarithmic. This actuallyresolves a puzzle of classical physics, where the self-energy of the electron is linearlydivergent.
But for scalars, there is no such symmetry, and corrections tomasses are quadrati-cally divergent. One can see this quickly for the Higgs self-coupling, which gives riseto a mass correction of the form:
δm2 = λ2∫ d4k
(2π)4(k2 −m2)(22)
If we view the standard model as an effective field theory, this integral should be cut offat a scale where new physics enters. We have argued that this might occur at, say,1014
GeV. But in this case the correction to the Higgs mass is gigantic compared to the Higgsmass itself.
1.6 Dark Matter and Dark Energy
In recent years, astronomers and astrophysicists have presented persuasive evidence thatthe energy density of the universe is largely in some unfamiliar form; about30% somenon-baryonic pressureless matter (the dark matter) and about 60% in some form withnegative pressure (the dark energy). The latter might be a cosmological constant (ofwhich more later). The former could well be some new type of weakly interacting parti-cle. Dark matter would indicate the existence of additional, possibly quite light degreesof freedom in nature. The dark energy is totally mysterious.If it represents a cosmologi-cal constant, it is even more puzzling than the hierarchy problem we described before. Acosmological constant represents, in essence, the energy density of the vacuum. In fieldtheory this isquartically divergent; it is the first divergence one encounters in quantumfield theory. At one loop, it is given by an expression of the form:
Λ =∑
i
(−1)F∫ d3k
2π3
1
2
√k2 +m2
i (23)
where the sum is over all particle species (including spins). If one cuts this off, again at1014 GeV, one gets a result of order
Λ = 1054 Gev4 (24)
The measured value of the dark energy density, by contrast, is
Λ = 10−47 GeV4. (25)
This wide discrepancy is probably one of the most troubling problems facing funda-mental physics today.
8
1.7 Summary: Limitations of the Standard Model
2 Of Anomalies, Strong CP and Axions
In the previous chapter, we mentioned that the Standard Model has an additional param-eter, theθ parameter of QCD. This parameter is dimensionless. It is alsoCP violating.The θ term in the QCD lagrangian, on the other hand, is a total divergence, and it istempting to ignore it. The reasons why this term is dynamically important are inter-esting and subtle. They are connected with a class of phenomena known as anomalies.Usually, the term anomaly is used to refer to the violation atthe quantum level of asymmetry which is valid classically. Anomalies will be a recurring theme of all beyondthe Standard Model physics, and particularly in string theory. In this chapter, we explainhow anomalies arise in four dimensional field theories in some detail, and mention howanomalies arise and the role they play and space-times of different dimensionalities.
2.1 The Chiral Anomaly
Before considering real QCD, consider a simpler theory, with only a single flavor ofquark. Before making any field redefinitions, the lagrangian takes the form:
L = − 1
4g2F 2
µν + q 6q∗ + q 6q∗mqq +m∗q∗q∗. (26)
Here, we have written the lagrangian in terms of two-component fermions, and notedthat a priori, the mass need not be real,
m = |m|eiθ. (27)
In this chapter, it will sometimes be convenient to work withfour component fermions,and it is valuable to make contact with this language in any case. In terms of these:
L = Re m qq + Im m qqγ5q. (28)
In order to bring the mass term to the conventional form, withnoγ5’s, one could tryto redefine the fermions:
q → e−iθ/2q q → e−iθ/2q. (29)
But in field theory, transformations of this kind are potentially fraught with difficultiesbecause of the infinite number of degrees of freedom.
A simple calculation shows that there is a difficulty, one of the simplest manifes-tations of an anomaly. Suppose, first, thatM is very large. In that case we want tointegrate out the quarks and obtain a low energy effective theory. To do this, we studythe path integral:
Z =∫
[dAµ]∫
[dq][dq]eiS (30)
9
Again supposem = eiθM , whereθ is small andM is real. In order to makem real, wecan again make the transformations:q → qe−iθ/2; q → qe−iθ/2 (in four component lan-guage, this isq →−iθ/2γ5 q).) The result of integrating out the quark, i.e. of performingthe path integral overq andq can be written in the form:
Z =∫
[dAµ]∫eiSeff (31)
HereSeff is the effective action which describes the interactions ofgluons at scales wellbelowM .
file=tasi001.eps,angle=-90,width=7cm
Fig. 1. The triangle diagram associated with the four dimensional anomaly.
Because the field redefinition which eliminatesθ is just a change of variables in thepath integral, one might expect that there can be noθ-dependence in the effective action.But this is not the case. To see this, suppose thatθ is small, and instead of makingthe transformation, treat theθ term as a small perturbation, and expand the exponential.Now consider a term in the effective action with two externalgauge bosons. This isgiven by the Feynman diagram in fig. 1. The corresponding termin the action is givenby
δLeff = −iθ2g2MTr(T aT b)
∫ d4k
(2π)4Trγ5
1
6k+ 6q1 −M6ε1
1
6k −M6ε2
1
6k− 6q2 −M(32)
Here, as in the figure, theqi’s are the momenta of the two photons, while theε’s are theirpolarizations anda andb are the color indices of the gluons. To perform the integral,itis convenient to introduce Feynman parameters and shift thek integral, giving:
δLeff = −iθg2MTr(T aT b)∫dα1dα2
∫ d4k
(2π)4Trγ5( 6k − α1 6q1 + α2 6q2+ 6q1 +M) 6ε1
(33)
× ( 6k − α1 6q1 + α2 6q2 +M) 6ε2( 6k − α1 6q1 + α2 6q2− 6q2 +M)
(k2 −M2 +O(q2i ))
3
For smallq, we can neglect theq-dependence of the denominator. The trace in thenumerator is easy to evaluate, since we can drop terms linearin k. This gives, afterperforming the integrals over theα’s,
δLeff = g2M2θTr(T aT b)εµνρσqµ1 q
ν2ε
ρ1ε
σ2
∫ d4k
(2π)4
1
(k2 −M2)3. (34)
This corresponds to a term in the effective action, after doing the integral overk andincluding a combinatoric factor of two from the different ways to contract the gaugebosons:
δLeff =1
32π2θTr(FF ). (35)
10
Now why does this happen? At the level of the path integral, the transformationwould seem to be a simple change of variables, and it is hard tosee why this shouldhave any effect. On the other hand, if one examines the diagram of fig. 1, one sees thatit contains terms which are linearly divergent, and thus it should be regulated. A simpleway to regulate the diagram is to introduce a Pauli-Villars regulator, which means thatone subtracts off a corresponding amplitude with some very large massΛ. However, wehave just seen that the result is independent ofΛ! This sort of behavior is characteristicof an anomaly.
Consider now the case thatm � ΛQCD. In this case, we shouldn’t integrate out thequarks, but we still need to take into account the regulator diagrams. For smallm, theclassical theory has an approximate symmetry under which
q → eiαq q → eiαq (36)
(in four component language,q → eiαγ5q). In particular, we can define a current:
jµ5 = qγ5γµq, (37)
and classically,∂µj
µ5 = mqγ5q. (38)
Under a transformation by an infinitesmal angleα one would expect
δL = α∂µjµ5 = mαqγ5q. (39)
But what we have just discovered is that the divergence of the current contains another,m-independent, term:
∂µjµ5 = mqγ5q +
1
32π2FF . (40)
This anomaly can be derived in a number of other ways. One can define, for example,the current by “point splitting,”
jµ5 = q(x+ iε)ei
∫ x+ε
xdxµAµq(x) (41)
Because operators in quantum field theory are singular at short distances, the Wilsonline makes a finite contribution. Expanding the exponentialcarefully, one recovers thesame expression for the current. A beautiful derivation, closely related to that we haveperformed above, is due to Fujikawa, described in [58]. Hereone considers the anomalyas arising from a lack of invariance of the path integral measure. One carefully evalu-ates the Jacobian associated with the change of variablesq → q(1 + iγ5α), and showsthat it yields the same result[58]. We will do a calculation along these lines in a twodimensional model shortly.
The anomaly has a number of important consequences:
11
• πo decay: the divergence of the axial isospin current,
(j35)
µ = uγ5γµu− dγ5γ
µd (42)
has an anomaly due to electromagnetism. This gives rise to a coupling of theπo
to two photons, and the correct computation of the lifetime was one of the earlytriumphs of the theory of quarks with color.
• Anomalies in gauge currents signal an inconsistency in a theory. They mean thatthe gauge invariance, which is crucial to the whole structure of gauge theories(e.g. to the fact that they are simultaneously unitary and lorentz invariant) is lost.The absence of gauge anomalies is one of the striking ingredients of the standardmodel, and it is also crucial in extensions such as string theory.
2.1.1 Return to QCD
What we have just learned is that, if in our simple model above,we require that thequark masses are real, we must allow for the possible appearance in the lagrangian ofthe standard model, of theθ-terms of eqn.??. 1 This term, however, can be removedby aB + L transformation. What are the consequences of these terms? Wewill focuson the strong interactions, for which these terms are most important. At first sight, onemight guess that these terms are in fact of no importance. Consider, first, the case ofQED. Then ∫
d4xFF (43)
is a the integral of a total divergence,
FF = ~E · ~B =1
2∂µε
µνρσA
νF ρσ. (44)
As a result, this term does not contribute to the classical equations of motion. Onemight expect that it does not contribute quantum mechanically either. If we think of theEuclidean path integral, configurations of finite action have field strengths,Fµν whichfall off faster than1/r2 (wherer is the Euclidean distance), andA which falls off fasterthan1/r, so one can neglect surface terms inLθ.
(A parenthetical remark: This is almost correct. However, if there are magneticmonopoles there is a subtlety, first pointed out by Witten[59]. Monopoles can carryelectric charge. In the presence of theθ term, there is an extra source for the electric field
1In principle we must allow a similar term, for the weak interactions. However,B + L is a classicalsymmetry of the renormalizable interactions of the standard model. This symmetry is anomalous, and canbe used to remove the weakθ term. In the presence of higher dimensionB + L-violating terms, this is nolonger true, but any effects ofθ will be extremely small, suppressed bye−2π/αw as well as by powers ofsome large mass scale.
12
at long distances, proportional toθ and the monopole charge. So the electric charges aregiven by:
Q = nee−eθnm
2π(45)
wherenm is the monopole charge in units of the Dirac quantum.)In the case of non-Abelian gauge theories, the situation is more subtle. It is again
true thatFF can be written as a total divergence:
FF = ∂µKµ Kµ = εµνρσ(AaνF
aρσ − 2
3fabcAa
νAbρA
cσ). (46)
But now the statement thatF falls faster than1/r2 does not permit an equally strongstatement aboutA. We will see shortly that there are finite action configurations – finiteaction classical solutions – whereF ∼ 1
r4 , butA → 1r, so that the surface term cannot
be neglected. These terms are called instantons. It is because of this thatθ can have realphysical effects.
2.1.2 A Two Dimensional Detour
Before considering four dimensions with all of its complications, it is helpful to considertwo dimensions. Two dimensions are often a poor analog for four, but for some of theissues we are facing here, the parallels are extremely close. In these two dimensionalexamples, the physics is more manageable, but still rich.
2.1.3 The Anomaly In Two Dimensions
Consider, first, electrodynamics of a massless fermion in twodimensions. Let’s investi-gate the anomaly. The point-splitting method is particularly convenient here. Just as infour dimensions, we write:
jµ5 = ψ(x+ iε)ei
∫ x+ε
xAρdxρ
γµψ(x) (47)
For very smallε, we can pick up the leading singularity in the product ofψ(x + ε)ψ byusing the operator product expansion, and noting that (using naive dimensional analysis)the leading operator is the unit operator, with coefficient proportional to1/ε. We canread off this term by taking the vacuum expectation value, i.e. by simply evaluating thepropagator. String theorists are particularly familiar with this Green’s function:
〈ψ(x+ ε)ψ(x)〉 =1
2π
6εε2
(48)
Expanding the factor in the exponential to orderε gives
∂µjµ5 = naive piece +
i
2π∂µερA
ρtr6εε2γµγ5. (49)
13
Taking the trace givesεµνεν ; averagingε over angles(< εµεν >= 1
2ηµνε
2), yields
∂µjµ5 =
1
4πεµνF
µν . (50)
Exercise: Fill in the details of this computation, being careful about signs and factors of2.
This is quite parallel to the situation in four dimensions. The divergence of thecurrent is itself a total derivative:
∂µjµ5 =
1
2πεµν∂
µAν . (51)
So it appears possible to define a new current,
Jµ = jµ5 − 1
2πεµν∂
µAν (52)
However, just as in the four dimensional case, this current is not gauge invariant. Thereis a familiar field configuration for whichA does not fall off at infinity: the field ofa point charge. Indeed, if one has charges,±θ at infinity, they give rise to a constantelectric field,Foi = eθ. Soθ has a very simple interpretation in this theory.
It is easy to see that physics is periodic inθ. Forθ > q, it is energetically favorableto produce a pair of charges from the vacuum which shield the charge at∞.
2.1.4 The CPN Model: An Asymptotically Free Theory
The model we have considered so far is not quite like QCD in at least two ways. First,there are no instantons; second, the couplinge is dimensionful. We can obtain a theorycloser to QCD by considering theCPN model (our treatment here will follow closelythe treatment in Peskin and Schroeder’s problem 13.3[58]).This model starts with a setof fields,zi, i = 1, . . . N + 1. These fields live in the spaceCPN . This space is definedby the constraint: ∑
i
|zi|2 = 1; (53)
in addition, the pointzi is equivalent toeiαzi. To implement the first of these constraints,we can add to the action a lagrange multiplier field,λ(x). For the second, we observethat the identification of points in the “target space,”CPN , must hold at every point inordinary space-time, so this is aU(1) gauge symmetry. So introducing a gauge field,Aµ, and the corresponding covariant derivative, we want to study the lagrangian:
L =1
g2[|Dµzi|2 − λ(x)(|zi|2 − 1)] (54)
14
Note that there is no kinetic term forAµ, so we can simply eliminate it from the actionusing its equations of motion. This yields
L =1
g2[|∂µzj|2 + |z∗j∂µzj|2] (55)
It is easier to proceed, however, keepingAµ in the action. In this case, the action isquadratic inz, and we can integrate out thez fields:
Z =∫
[dA][dλ][dzj]exp[−L] =∫
[dA][dλ]e∫
d2xΓeff [A,λ] (56)
=∫
[dA][dλ]exp[−Ntr log(−D2 − λ) − 1
g2
∫d2xλ]
2.1.5 The LargeN Limit
By itself, the result of eqn. 56 is still rather complicated. The fieldsAµ andλ havecomplicated, non-local interactions. Things become much simpler if one takes the “largeN limit”, a limit where one takesN → ∞ with g2N fixed. In this case, the interactionsof λ andAµ are suppressed by powers ofN . For largeN , the path integral is dominatedby a single field configuration, which solves
δΓeff
δλ= 0 (57)
or, setting the gauge field to zero,
N∫ d2k
(2π)2
1
k2 + λ=
1
g2, (58)
giving
λ = m2 = M exp[− 2π
g2N]. (59)
Here,M is a cutoff required because the integral in eqn. 58 is divergent. This result isremarkable. One has exhibited dimensional transmutation:a theory which is classicallyscale invariant contains non-trivial masses, related in a renormalization-group invariantfashion to the cutoff. This is the phenomenon which in QCD explains the masses of theproton, neutron, and other dimensionful quantities. So thetheory is quite analogous toQCD. We can read off the leading term in theβ-function from the familiar formula:
m = Me−∫
dgβ(g) (60)
so, with
β(g) = − 1
2πg3bo (61)
15
we havebo = 1.But most important for our purposes, it is interesting to explore the question ofθ-
dependence. Again, in this theory, we could have introduceda θ term:
Lθ =θ
2π
∫d2εµνF
µν , (62)
whereFµν can be expressed in terms of the fundamental fieldszj. As usual, this isthe integral of a total divergence. But precisely as in the case of 1 + 1 dimensionalelectrodynamics we discussed above, this term is physically important. In perturbationtheory in the model, this is not entirely obvious. But using our reorganization of thetheory at largeN , it is. The lowest order action forAµ is trivial, but at one loop (order1/N ), one generates a kinetic term forA through the usual vacuum polarization loop:
Lkin =N
2πm2F 2
µν . (63)
At this order, the effective theory consists of the gauge field, then, with couplinge2 =2πm2
N, and some coupling to a dynamical, massive fieldλ. As we have already argued,
θ corresponds to a non-zero background electric field due to charges at infinity, and thetheory clearly can have non-trivialθ-dependence.
There is, in addition, the possibility of including other light fields, for example mass-less fermions. In this case, one can again have an anomalousU(1) symmetry. There isthen noθ-dependence, since it is possible to shield any charge at infinity. But there isnon-trivial breaking of the symmetry. At low energies, one has now a theory with afermion coupled to a dynamicalU(1) gauge field. The breaking of the associatedU(1)in such a theory is a well-studied phenomenon[60].Exercise: Complete Peskin and Schroeder, Problem 13.3.
2.1.6 The Role of Instantons
There is another way to think about the breaking of theU(1) symmetry andθ-dependencein this theory. If one considers the Euclidean functional integral, it is natural to look forstationary points of the integration, i.e. for classical solutions of the Euclidean equationsof motion. In order that they be potentially important, it isnecessary that these solutionshave finite action, which means that they must be localized inEuclidean space and time.For this reason, such solutions were dubbed “instantons” by’t Hooft. Such solutionsare not difficult to find in theCPN model; we will describe them briefly below. Thesesolutions carry non-zero values of the topological charge,
1
2π
∫d2xεµνFµν = n (64)
16
and have an action2πn. As a result, they contribute to theθ-dependence; they give acontribution to the functional integral:
Zinst = e−2πn
g2 einθ∫
[dδzj]e−δzi
δ2Sδziδzj
δzj+ . . . (65)
It follows that:
• Instantons generateθ-dependence
• In the largeN limit, instanton effects are, formally, highly suppressed, muchsmaller that the effects we found in the largeN limit
• Somewhat distressingly, the functional integral above cannot be systematicallyevaluated. The problem is that the classical theory is scaleinvariant, as a resultof which, instantons come in a variety of sizes.
∫[dδz] includes an integration
over all instanton sizes, which diverges for large size (i.e. in the infrared). Thisprevents a systematic evaluation of the effects of instantons in this case. At hightemperatures[61], it is possible to do the evaluation, and instanton effects are,indeed, systematically small.
It is easy to construct the instanton solution in the case ofCP 1. Rather than write thetheory in terms of a gauge field, as we have done above, it is convenient to parameterizethe theory in terms of a single complex field,φ. One can, for example, defineφ = z1/z2.Then, with a bit of algebra, one can show that the action forφ takes the form:
L = (∂µφ∂µφ∗)
1
1 + φ∗φ− φ∗φ
(1 + φ∗φ)2.(66)
One can think of the fieldφ as living on the space with metric given by the term inparenthesis,gφφ∗ . One can show that this is the metric one obtains if one stereographi-cally maps the sphere onto the complex plane. This mapping, which you may have seenin your math methods courses, is just:
z =x1 + ix2
1 − x3
; (67)
The inverse is
x1 =z + z∗
1 + |z|2 x2 =z − z∗
i(1 + |z|2) (68)
x3 =|z|2 − 1
|z|2 + 1.
It is straightforward to write down the equations of motion:
∂2φgφφ + ∂µφ(∂µφ∂g
∂φ+ ∂µφ
∂g
∂φ) = 0 (69)
17
Now calling the space time coordinatesz = x1 + ix2, z∗ = x1 − ix2, you can see that ifφ is analytic, the equations of motion are satisfied! So a simple solution, which you cancheck has finite action, is
φ(z) = ρz. (70)
In addition to evaluating the action, you can evaluate the topological charge,
1
2π
∫d2xεµνF
µν = 1 (71)
for this solution. More generally, the topological charge measures the number of timesthatφ maps the complex plane into the complex plane; forφ = zn, for example, one haschargen.Exercise: Verify that the action of eqn. 66 is equal to
L = gφ,φ∗∂µφ∂µφ∗ (72)
whereg is the metric of the sphere in complex coordinates, i.e. it isthe line elementdx2
1 + dx22 + dx2
3 expressed asgz,zdz dz + gz,z∗dz dz∗ + gz∗zdz
∗ dz + gdz∗dz∗dz∗dz∗. A
model with an action of this form is called a “Non-linear Sigma Model;” the idea is thatthe fields live on some “target” space, with metricg. Verify eqns. 66,68.
More generally,φ = az+bcz+d
is a solution with action2π. The parametersa, . . . d arecalled collective coordinates. They correspond to the symmetries of translations, dila-tions, and rotations, and special conformal transformations (forming the groupSL(2, C)).In other words, any given finite action solution breaks the symmetries. In the path inte-gral, the symmetry of Green’s functions is recovered when one integrates over the col-lective coordinates. For translations, this is particularly simple. If one studies a Green’sfunction,
< φ(x)φ(y) >≈∫d2xoφcl(x− xo)φcl(y − yo)e
−So (73)
The precise measure is obtained by the Fadeev-Popov method.Similarly, the integrationover the parameterρ yields a factor
∫dρρ−1e
− 2πg2(ρ) . . . (74)
Here the first factor follows on dimensional grounds. The second follows from renormalization-group considerations. It can be found by explicit evaluation of the functional determi-nant[62]. Note that, because of asymptotic freedom, this means that typical Green’sfunctions will be divergent in the infrared.
There are many other features of this instanton one can consider. For example,one can consider adding massless fermions to the model, by simply coupling them ina gauge-invariant way toAµ. The resulting theory has a chiralU(1) symmetry, whichis anomalous. In the presence of an instanton, one can easilyconstruct normalizable
18
fermion zero modes (the Dirac equation just becomes the statement thatψ is analytic).As a result, Green’s functions computed in the instanton background do not respect theaxialU(1) symmetry. But rather than get too carried away with this model(I urge youto get a little carried away and play with it a bit), let’s proceed to four dimensions, wherewe will see very similar phenomena.
2.2 Real QCD
The model of the previous section mimics many features of real QCD. Indeed, we willsee that much of our discussion can be carried over, almost word for word, to the ob-served strong interactions. This analogy is helpful, giventhat in QCD we have no ap-proximation which gives us control over the theory comparable to that which we foundin the largeN limit of theCPN model. As in that theory:
• There is aθ parameter, which appears as an integral over the divergenceof a non-gauge invariant current.
• There are instantons, which indicate that there should be real θ-dependence. How-ever, instanton effects cannot be considered in a controlled approximation, andthere is no clear sense in whichθ-dependence can be understood as arising frominstantons.
• There is another approach to the theory, which shows that theθ-dependence is real,and allows computation of these effects. In QCD, this is related to the breaking ofchiral symmetries.
2.2.1 The Theory and its Symmetries
While it is not in the spirit of much of this school, which is devoted to the physics ofheavy quarks, it is sufficient, to understand the effects ofθ, to focus on only the lightquark sector of QCD. For simplicity in writing some of the formulas, we will considertwo light quarks; it is not difficult to generalize the resulting analysis to the case of three.It is believed that the masses of theu andd quarks are of order5 MeV and10 MeV,respectively, much lighter than the scale of QCD. So we first consider an idealization ofthe theory in which these masses are set to zero. In this limit, the theory has a symmetrySU(2)L × SU(2)R. This symmetry is spontaneously broken to a vectorSU(2). Thethree resulting Goldstone bosons are theπ mesons. Calling
q =(ud
)q =
(ud
), (75)
the twoSU(2) symmetries act separately onq andq (thought of as left handed fermions).The order parameter for the symmetry breaking is believed tobe the condensate:
Mo = 〈qq〉. (76)
19
This indeed breaks the symmetry down to the vector sum. The associated Goldstonebosons are theπ mesons. One can think of the Goldstone bosons as being associatedwith a slow variation of the expectation value in space, so wecan introduce a compositeoperator
M = qq = Moei
πa(x)τafπ
(1 00 1
)(77)
The quark mass term in the lagrangian is then (for simplicitywriting mu = md = mq;more generally one should introduce a matrix)
mqM. (78)
ExpandingM in powers ofπ/fπ, it is clear that the minimum of the potential occurs forπa = 0. Expanding to second order, one has
m2πf
2π = mqMo. (79)
But we have been a bit cavalier about the symmetries. The theory also has twoU(1)’s;
q → eiαq q → eiαqq → eiαq q → e−iαq (80)
The first of these is baryon number and it is not chiral (and is not broken by the conden-sate). The second is the axialU(1)5; It is also broken by the condensate. So, in additionto the pions, there should be another approximate Goldstoneboson. The best candidateis theη, but, as we will see below (and as you will see further in Thomas’s lectures), theη is too heavy to be interpreted in this way. The absence of thisfourth (or in the case ofthree light quarks, ninth) Goldstone boson is called theU(1) problem.
TheU(1)5 symmetry suffers from an anomaly, however, and we might hopethat thishas something to do with the absence of a corresponding Goldstone boson. The anomalyis given by
∂µjµ5 =
2
32π2FF (81)
Again, we can write the right hand side as a total divergence,
FF = ∂µKµ (82)
where
Kµ = εµνρσ(AaνF
aρσ − 2
3fabcAa
νAbρA
cσ). (83)
So if it is true that this term accounts for the absence of the Goldstone boson, we needto show that there are important configurations in the functional integral for which therhs does not vanish rapidly at infinity.
20
2.2.2 Instantons
It is easiest to study the Euclidean version of the theory. This is useful if we are interestedin very low energy processes, which can be described by an effective action expandedabout zero momentum. In the functional integral,
Z =∫
[dA][dq][dq]e−S (84)
it is natural to look for stationary points of the effective action, i.e. finite action, classicalsolutions of the theory in imaginary time. The Yang-Mills equations are complicated,non-linear equations, but it turns out that, much as in theCPN model, the instantonsolutions can be found rather easily[63]. The following tricks simplify the construction,and turn out to yield the general solution. First, note that the Yang-Mills action satisfiesan inequality:
∫(F ± F )2 =
∫(F 2 + F 2 ± 2FF ) =
∫(2F 2 + 2FF ) ≥ 0. (85)
So the action is bounded∫FF , with the bound being saturated when
F = ±F (86)
i.e. if the gauge field is (anti) self dual.2 This equation is a first order equation, and itis easy to solve if one first restricts to anSU(2) subgroup of the full gauge group. Onemakes the ansatz that the solution should be invariant undera combination of ordinaryrotations and globalSU(2) gauge transformations:
Aµ = f(r2) + h(r2)~x · ~τ (87)
where we are using a matrix notation for the gauge fields. One can actually make abetter guess: define the gauge transformation:
g(x) =x4 + i~x · ~τ
r(88)
and takeAµ = f(r2)g∂µg
−1 (89)
Then plugging in the Yang-Mills equations yields:
f =r2
r2 + ρ2(90)
2This is not an accident, nor was the analyticity condition inthe CPN case. In both cases, we canadd fermions so that the model is supersymmetric. Then one can show that if some of the supersymmetrygenerators,Qα annihilate a field configuration, then the configuration is a solution. This is a first ordercondition; in the Yang-Mills case, it implies self-duality, and in theCPN case it requires analyticity.
21
whereρ is an arbitrary quantity with dimensions of length. The choice of origin hereis arbitrary; this can be remedied by simply replacingx → x − xo everywhere in theseexpressions, wherexo represents the location of the instanton.ExerciseCheck that eqns. 89,90 solve 86.
¿From this solution, it is clear why∫∂µK
µ does not vanish for the solution: whileA is a pure gauge at infinity, it falls only as1/r. Indeed, sinceF = F , for this solution,
∫F 2 =
∫F 2 = 32π2 (91)
This result can also be understood topologically.g defines a mapping from the “sphereat infinity” into the gauge group. It is straightforward to show that
1
32π2
∫d4xFF (92)
counts the number of timesg maps the sphere at infinity into the group (one for thisspecific example;n more generally). We do not have time to explore all of this indetail; I urge you to look at Sidney Coleman’s lecture, “The Uses of Instantons”[64].To actually do calculations, ’t Hooft developed some notations which are often moreefficient than those described above[62].
So we have exhibited potentially important contributions to the path integral whichviolate theU(1) symmetry. How does this violation of the symmetry show up? Let’sthink about the path integral in a bit more detail. Having found a classical solution, wewant to integrate about small fluctuations about it:
Z = e− 8π2
g2 eiθ∫
[dδA][dq][dq]eiδ2S (93)
NowS contains an explicit factor of1/g2. As a result, the fluctuations are formally sup-pressed byg2 relative to the leading contribution. The one loop functional integral yieldsa product of determinants for the fermions, and of inverse square root determinants forthe bosons. Consider, first, the integral over the fermions. It is straightforward, if chal-lenging, to evaluate the determinants[62]. But if the quark masses are zero, the fermionfunctional integrals are zero, because there is a zero mode for each of the fermions, i.e.for bothq andq there is a normalizable solution of the equation:
6Du = 0 6Du = 0 (94)
and similarly ford andd. It is straightforward to construct these solutions:
u =ρ
(ρ2 + (x− xo)2)3/2ζ (95)
whereζ is a constant spinor, and similarly foru, etc.
22
This means that in order for the path integral to be non-vanishing, we need to includeinsertions of enoughq’s andq’s to soak up all of the zero modes. In other words, non-vanishing Green’s functions have the form
〈uudd〉 (96)
and violate the symmetry. Note that the symmetry violation is just as predicted from theanomaly equation:
∆Q5 = 41
π2
∫d4xFF = 4 (97)
However, the calculation we have described here is not self consistent. The diffi-culty is that among the variations of the fields we need to integrate over are changes inthe location of the instanton (translations), rotations ofthe instanton, and scale transfor-mations. The translations are easy to deal with; one has simply to integrate overxo (onemust also include a suitable Jacobian factor[64]). Similarly, one must integrate overρ.There is a power ofρ arising from the Jacobian, which can be determined on dimen-sional grounds. For our Green’s function above, for example, which has dimension6,we have (if all of the fields are evaluated at the same point),
∫dρρ−7. (98)
However, there is additionalρ-dependence because the quantum theory violates the scalesymmetry. This can be understood by replacingg2 → g2(ρ) in the functional integral,and using
e−8π2g2(ρ) ≈ (ρM)bo (99)
for smallρ. For3 flavor QCD, for example,bo = 9, and theρ integral diverges for largeρ. This is just the statement that the integral is dominated bythe infrared, where theQCD coupling becomes strong.
So we have provided some evidence that theU(1) problem is solved in QCD, but noreliable calculation. What aboutθ-dependence? Let us ask first aboutθ-dependence ofthe vacuum energy. In order to get a non-zero result, we need to allow that the quarksare massive. Treating the mass as a perturbation, we obtain
E(θ) = CΛ9QCDmumd cos(θ)
∫dρρ−3ρ9. (100)
So again, we have evidence forθ-dependence, but cannot do a reliable calculation. Thatwe cannot do a calculation should not be a surprise. There is no small parameter in QCDto use as an expansion parameter. Fortunately, we can use other facts which we knowabout the strong interactions to get a better handle on both theU(1) problem and thequestion ofθ-dependence.
Before continuing, however, let us consider the weak interactions. Here there isa small parameter, and there are no infrared difficulties, sowe might expect instanton
23
effects to be small. The analog of theU(1)5 symmetry in this case is baryon number.Baryon number has an anomaly in the standard model, since all of the quark doubletshave the same sign of the baryon number. ’t Hooft realized that one could actually useinstantons, in this case, to compute the violation of baryonnumber. Technically, thereare no finite action Euclidean solutions in this theory; thisfollows, as we will see ina moment, from a simple scaling argument. However, ’t Hooft realized that one canconstruct important configurations of non-zero topological charge by starting with theinstantons of the pure gauge theory and perturbing them. If one simply takes such aninstanton, and plugs it into the action, one necessarily finds a correction to the action ofthe form
δS =1
g2v2ρ2. (101)
This damps theρ integral at largeρ, and leads to a convergent result. Affleck showedhow to develop this into a systematic computation[66]. Notethat from this, one can seethat baryon number violation occurs in the standard model, and that the rate is incrediblysmall, proportional toe−2 π
αW .
2.2.3 Real QCD and the U(1) Problem
In real QCD, it is difficult to do a reliable calculation which shows that there is not anextra Goldstone boson, but the instanton analysis we have described makes clear thatthere is no reason to expect one. Actually, while perturbative and semiclassical (in-stanton) techniques have no reason to give reliable results, there are two approximationmethods techniques which are available. The first is largeN , where one now allows theN of SU(N) to be large, withg2N fixed. In contrast to the case ofCPN , this doesnot permit enough simplification to do explicit computations, but it does allow one tomake qualitative statements about the theory. available inQCD. Witten has pointed outa way in which one can at relate the mass of theη (or η′ if one is thinking in terms ofSU(3)×SU(3) current algebra) to quantities in a theory without quarks. The point is tonote that the anomaly is an effect suppressed by a power ofN , in the largeN limit. Thisis because the loop diagram contains a factor ofg2 but not ofN . So, in largeN , it canbe treated as a perturbation, and the theη is massless.∂µj
µ5 is like a creation operator
for theη, so (just like∂µjµ 35 is a creation operator for theπ meson), so one can compute
the mass if one knows the correlation function, at zero momentum, of
〈∂µjµ5 (x)∂µj
µ5 (y)〉 ∝ 1
N2〈F (x)F (x)F (y)F (y)〉 (102)
To leading order in the1/N expansion, this correlation function can be computed in thetheory without quarks. Witten argued that while this vanishes order by order in perturba-tion theory, there is no reason that this correlation function need vanish in the full theory.Attempts have been made to compute this quantity both in lattice gauge theory and using
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the AdS-CFT correspondence recently discovered in string theory. Both methods givepromising results.
So the U(1) problem should be viewed as solved, in the sense that absent any argu-ment to the contrary, there is no reason to think that there should be an extra Goldstoneboson in QCD.
The second approximation scheme which gives some control ofQCD is known aschiral perturbation theory. The masses of theu, d ands quarks are small compared tothe QCD scale, and the mass terms for these quarks in the lagrangian can be treated asperturbations. This will figure in our discussion in the nextsection.
2.3 The Strong CP Problem
2.3.1 θ-dependence of the Vacuum Energy
The fact that the anomaly resolves theU(1) problem in QCD, however, raises anotherissue. Given that
∫d4xFF has physical effects, the theta term in the action has physical
effects as well. Since this term is CP odd, this means that there is the potential for strongCP violating effects. These effects should vanish in the limit of zero quark masses, sincein this case, by a field redefinition, we can removeθ from the lagrangian. In the presenceof quark masses, theθ-dependence of many quantities can be computed. Consider, forexample, the vacuum energy. In QCD, the quark mass term in the lagrangian has theform:
Lm = muuu+mddd+ h.c. (103)
Were it not for the anomaly, we could, by redefining the quark fields, takemu andmd to be real. Instead, we can define these fields so that there is no θF F term in theaction, but there is a phase inmu andmd. Clearly, we have some freedom in makingthis choice. In the case thatmu andmd are equal, it is natural to choose these phases tobe the same. We will explain shortly how one proceeds when themasses are different(as they are in nature). We can, by convention, takeθ to be the phase of the overalllagrangian:
Lm = (muuu+mddd) cos(θ/2) + h.c. (104)
Now we want to treat this term as a perturbation. At first order, it makes a contri-bution to the ground state energy proportional to its expectation value. We have alreadyargued that the quark bilinears have non-zero vacuum expectation values, so
E(θ) = (mu +md)eiθ〈qq〉. (105)
While, without a difficult non-perturbative calculation, wecan’t calculate the sepa-rate quantities on the right hand side of this expression, wecan, using current algebra,relate them to measured quantities. A simple way to do this isto use the effective la-grangian method (which will be described in more detail in Thomas’s lectures). The
25
basic idea is that at low energies, the only degrees of freedom which can readily beexcited in QCD are the pions. So parameterizeqq as
qq = Σ =< qq > eiπa(x)σa
2fπ (106)
We can then write the quark mass term as
Lm = eiθTrMqΣ. (107)
Ignoring theθ term at first, we can see, plugging in the explicit form forΣ, that
m2πf
2π = (mu +md) < qq > . (108)
So the vacuum energy, as a function ofθ, is:
E(θ) = m2πf
2π cos(θ). (109)
This expression can readily be generalized to the case of three light quarks by similarmethods. In any case, we now see that there is real physics inθ, even if we don’tunderstand how to do an instanton calculation. In the next section, we will calculate amore physically interesting quantity: the neutron electric dipole moment as a functionof θ.
2.3.2 The Neutron Electric Dipole Moment
As Scott Thomas will explain in much greater detail in his lectures, the most interestingphysical quantities to study in connection with CP violationare electric dipole moments,particularly that of the neutron,dn. It has been possible to set strong experimental limitson this quantity. Using current algebra, the leading contribution to the neutron electricdipole moment due toθ can be calculated, and one obtains a limitθ < 10−9[65]. Theoriginal paper on the subject is quite readable. Here we outline the main steps in thecalculation; I urge you to work out the details following thereference. We will simplifythe analysis by working in an exactSU(2)-symmetric limit, i.e. by takingmu = md =m. We again treat the lagrangian of [104] as a perturbation. Wecan also understandhow this term depends on theπ fields by making an axialSU(2) transformation on thequark fields. In other words, a backgroundπ field can be thought of as a small chiraltransformation from the vacuum. Then, e.g., for theτ3 direction,q → (1 + iπ3τ3)q (theπ field parameterizes the transformation), so the action becomes:
m
fπ
π3(qγ5q + θqq) (110)
In other words, we have calculated a CP violating coupling of the mesons to the pions.
26
file=tasi002.eps,angle=-90,width=7cm
Fig. 2. Diagram in which CP-violating coupling of the pion contributes todn.
This coupling is difficult to measure directly, but it was observed in [65] that thiscoupling gives rise, in a calculable fashion, to a neutron electric dipole moment. Con-sider the graph of fig. 2. This graph generates a neutron electric dipole moment, if wetake one coupling to be the standard pion-nucleon coupling,and the second the couplingwe have computed above. The resulting Feynman graph is infrared divergent; we cutthis off atmπ, while cutting off the integral in the ultraviolet at the QCD scale. Becauseof this infrared sensitivity, the low energy calculation isreliable. The exact result is:
dn = gπNN−θmumd
fπ(mu +md)〈Nf |qτaq|Ni〉 ln(MN/mπ)
1
4π2MN . (111)
The matrix element can be estimated using ordinarySU(3), yielding dn = 5.2 ×10−16θ cm. The experimental bound givesθ < 10−9−10. Understanding why CP vi-olation is so small in the strong interactions is the “strongCP problem.”
2.4 Possible Solutions
What should our attitude towards this problem be? We might argue that, after all, someYukawa couplings are as small as10−5, so why is10−9 so bad. On the other hand, wesuspect that the smallness of the Yukawa couplings is related to approximate symme-tries, and that these Yukawa couplings are telling us something. Perhaps there is someexplanation of the smallness ofθ, and perhaps this is a clue to new physics. In thissection we review some of the solutions which have been proposed to understand thesmallness ofθ.
2.4.1 The Axion
Perhaps the most popular explanation of the smallness ofθ involves a hypothetical par-ticle called the axion. We present here a slightly updated version of the original idea ofPeccei and Quinn[73].
Consider the vacuum energy as a function ofθ, eqn. [105]. This energy has aminimum atθ = 0, i.e. at the CP conserving point. As Weinberg noted long ago, thisis almost automatic: points of higher symmetry are necessarily stationary points. As itstands, this observation is not particularly useful, sinceθ is a parameter, not a dynamicalvariable. But suppose that one has a particle,a, with coupling to QCD:
Laxion = (∂µa)2 +
(a/fa + θ)
32π2FF (112)
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fa is known as the axion decay constant. Suppose that the rest ofthe theory possesses asymmetry, called the Peccei-Quinn symmetry,
a→ a+ α (113)
for constantα. Then by a shift ina, one can eliminateθ. E(θ is now V (a/fa), thepotential energy of the axion. It has a minimum atθ = 0. The strong CP problem issolved.
One can estimate the axion mass by simply examiningE(θ).
m2a ≈ m2
πfπ2
f 2a
. (114)
If fa ∼ TeV , this yields a mass of order KeV. Iffa ∼ 1016 GeV, this gives a mass oforder10−9 eV. As for theθ-dependence of the vacuum energy, it is not difficult to getthe factors straight using current algebra methods. A collection of formulae, with greatcare about factors of2, appears in [74]
Actually, there are several questions one can raise about this proposal:
• Should the axion already have been observed? In fact, as Scott Thomas will ex-plain in greater detail in his lectures, the couplings of theaxion to matter can beworked out in a straightforward way, using the methods of current algebra (inparticular of non-linear lagrangians). All of the couplings of the axion are sup-pressed by powers offa. So if fa is large enough, the axion is difficult to see. Thestrongest limit turns out to come from red giant stars. The production of axions is“semiweak,” i.e. it only is suppressed by one power offa, rather than two powersof mW ; as a result, axion emission is competitive with neutrino emission untilfa > 1010 GeV or so.
• As we will describe in a bit more detail below, the axion can becopiously pro-duced in the early universe. As a result, there is anupper bound on the axiondecay constant. In this case, as we will explain below, the axion could constitutethe dark matter.
• Can one search for the axion experimentally[75]? Typically,the axion couples notonly to theFF of QCD, but also to the same object in QED. This means that ina strong magnetic field, an axion can convert to a photon. Precisely this effect isbeing searched for by a group at Livermore (the collaboration contains membersfrom MIT, University of Florida) and Kyoto. The basic idea isto suppose that thedark matter in the halo consists principally of axions. Using a (superconducting)resonant cavity with a high Q value in a large magnetic field, one searches for theconversion of these axions into excitations of the cavity. The experiments havealready reached a level where they set interesting limits; the next generation ofexperiments will cut a significant swath in the presently allowed parameter space.
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• The coupling of the axion toFF violates the shift symmetry; this is why theaxion can develop a potential. But this seems rather paradoxical: one postulatesa symmetry, preserved to some high degree of approximation,but which is nota symmetry; it is at least broken by tiny QCD effects. Is this reasonable? Tounderstand the nature of the problem, consider one of the ways an axion can arise.In some approximation, we can suppose we have a global symmetry under which ascalar field,φ, transforms asφ→ eiαφ. Suppose, further, thatφ has an expectationvalue, with an associated (pseudo)-Goldstone boson,a. We can parameterizeφ as:
φ = faeia/fa |〈φ〉| = fa (115)
If this field couples to fermions, so that they gain mass from its expectation value,then at one loop we generate a couplingaF F from integrating out the fermions.This calculation is identical to the corresponding calculation for pions we dis-cussed earlier. But we usually assume that global symmetriesin nature are ac-cidents. For example, baryon number is conserved in the standard model simplybecause there are no gauge-invariant, renormalizable operators which violate thesymmetry. We believe it is violated by higher dimension terms. The global sym-metry we postulate here is presumably an accident of the samesort. But for theaxion, the symmetry must beextremely good. For example, suppose one has asymmetry breaking operator
φn+4
Mnp
(116)
Such a term gives a linear contribution to the axion potential of order fn+3a
Mnp
. If fa ∼1011, this swamps the would-be QCD contribution (m2
πf2π
fa) unlessn > 12[76]!
This last objection finds an answer in string theory. In this theory, there are axions,with just the right properties, i.e. there are symmetries inthe theory which areexactin perturbation theory, but which are broken by exponentially small non-perturbativeeffects. The most natural value forfa would appear to be of orderMGUT −Mp. Whetherthis can be made compatible with cosmology, or whether one can obtain a lower scale,is an open question.
3 An Introduction to Supersymmetry
In a standard advanced field theory course, one learns about anumber of symmetries:Poincare invariance, global continuous symmetries, discrete symmetries, gauge symme-tries, approximate and exact symmetries. The latter symmetries all have the propertythat they commute with Lorentz transformations, and in particular they commute withrotations. So the multiplets of the symmetries always contain particles of the same spin;in particular, they always consist of either bosons or fermions.
29
For a long time, it was believed that these were the only allowed symmetries; thisstatement was even embodied in a theorem, known as the Coleman-Mandula theorem.However, physicists studying theories based on strings stumbled on a symmetry whichrelated fields of different spin. Others quickly worked out simple field theories with thisnew symmetry: supersymmetry.
A good introduction to the formal theory of supersymmetry isprovided by[?,?].Here we will mention just a few important features. Supersymmetric field theories canbe formulated in dimensions up to eleven. These higher dimensional theories will beimportant when we consider string theory. For now, we consider theories in four dimen-sions. The supersymmetry charges, because they change spin, must themselves carryspin – they are spin-1/2. They transform as doublets under the lorentz group, like thespinorsχ andχ∗. There can be 1,2,4 or 8 such spinors; correspondingly, the symmetryis said to beN = 1, 2, 4 or 8 supersymmetry. Like generators of an ordinary group, thesupersymmetry generators obey an algebra; unlike an ordinary bosonic group, however,the algebra involves anticommutators as well as commutators (it is said to be “graded”).
4 An Overview of Supersymmetry
4.1 The Supersymmetry Algebra and its Representations
In this lecture, we will collect a few facts that will be useful in the subsequent discus-sion. We won’t attempt a thorough introduction to the subject. This is provided, forexample, by Lykken’s lectures[13,14], Wess and Bagger’s text[15], and Appendix B ofPolchinski’s text[16].
Supersymmetry, even at the global level, is remarkable, in that the basic algebrainvolves the translation generators:
{QAα , Q
∗Bβ} = 2σµ
αβδABPµ (117)
{QAα , Q
∗βB} = εαβX
AB. (118)
TheXAB ’s are Lorentz scalars, antisymmetric inA,B, known as central charges.If nature is supersymmetric, it is likely that the low energysymmetry isN = 1,
corresponding to only one possible value for the indexA above. OnlyN = 1 supersym-metry has chiral representations. In addition,N > 1 supersymmetry, as we will see, isessentially impossible to break; this is not the case forN = 1. ForN = 1, the basicrepresentations of the supersymmetry algebra, on masslessfields, are
• Chiral superfields fields:(φ, ψα), a complex fermion and a chiral scalar
• Vector superfields:(λ,Aµ), a chiral fermion and a vector meson, both, in general,in the adjoint representation of the gauge group
30
• The gravity supermultiplet:(ψµ,α, gµν), a spin-3/2 particle, the gravitino, and thegraviton.
N=1 supersymmetric field theories are conveniently described using superspace. Thespace consists of bosonic coordinates,xµ, and Grassman coordinates,θα, θ
∗α. In the case
of global supersymmetry, the description is particularly simple. The supersymmetrygenerators, classically, can be thought of as operators on functions ofxµ, θ, θ∗:
Qα = ∂α − iσµααθ
∗β∂µ; Qα = −∂α + iθ∗ασµαα∂µ. (119)
A general superfield,Φ(x, θ, θ) contains many terms, but can be decomposed intotwo irreducible representations of the algebra, corresponding to the chiral and vectorsuperfields described above. To understand these, we need tointroduce one more set ofobjects, the covariant derivatives,Dα andDα. These are objects which anti-commutewith the supersymmetry generators, and thus are useful for writing down invariant ex-pressions. They are given by
Dα = ∂αiσµααθ
∗α∂µ; Dα = −∂α − iθασµαα∂µ. (120)
With this definition, chiral fields are defined by the covariant condition:
DαΦ = 0. (121)
Chiral fields are annihilated by the covariant derivative operators. In general, thesecovariant derivatives anticommute with the supersymmetryoperators,Qα, so the condi-tion 121 is a covariant condition. This is solved by writing
Φ = Φ(y) = φ(y) +√
2θψ(y) + θ2F (y). (122)
wherey = xµ + iθσµθ. (123)
Vector superfields form another irreducible representation of the algebra; they satisfythe condition
V = V † (124)
Again, it is easy to check that this condition is preserved bysupersymmetry transforma-tions.V can be expanded in a power series inθ’s:
V = iχ− iχ† − θσµθ∗Aµ + iθ2θλ− iθ2θλ+1
2θ2θ2D. (125)
In the case of aU(1) theory, gauge transformations act by
V → V + iΛ − iΛ† (126)
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whereΛ is a chiral field. So, by a gauge transformation, one can eliminateχ. This gaugechoice is known as the Wess-Zumino gauge. This gauge choice breaks supersymmetry,much as choice of Coulomb gauge in electrodynamics breaks Lorentz invariance.
In the case of aU(1) theory, one can define a gauge-invariant field strength,
Wα = −1
4D2DαV. (127)
In Wess-Zumino gauge, this takes the form
Wα = −iλα + θαD − σµνβαFµνθβ + θ2σµ
αβ∂µλ
∗β. (128)
This construction has a straightforward non-Abelian generalization in superspace, whichis described in the references. When we write the lagrangian in terms of componentfields below, the non-abelian generalization will be obvious.
4.2 N=1 Lagrangians
One can construct invariant lagrangians by noting that integrals over superspace areinvariant up to total derivatives:
δ∫d4x
∫d4θ h(Φ,Φ†, V ) =
∫d4xd4θ (εαQ
α + εαQα)h(Φ,Φ†, V ) = 0. (129)
For chiral fields, integrals overhalf of superspace are invariant:
δ∫d4xd2θf(Φ) = (εαQ
α + εαQα)f(Φ). (130)
The integrals over theQα terms vanish when integrated overx andd2θ. TheQ∗ termsalso give zero. To see this, note thatf(Φ) is itself chiral (check), so
Q∗αf ∝ θασµ
αα∂µf. (131)
We can then write down the general renormalizable, supersymmetric lagrangian:
L =1
g(i)2
∫d2θW (i)2
α +∫d4θΦ†
ieqiV Φi
∫d2θW (Φi) + c.c. (132)
The first term on the right hand side is summed over all of the gauge groups, abelian andnon-abelian. The second term is summed over all of the chiralfields; again, we havewritten this for aU(1) theory, where the gauge group acts on theΦi’s by
Φi → e−qiΛΦi (133)
but this has a simple non-abelian generalization.W (Φ) is a holomorphic function of theΦi’s (it is a function ofΦi, notΦ†
i ).
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In terms of component fields, his lagrangian takes the form, in the Wess-Zuminogauge:
L = −1
4g−2
a F a2µν − iλaσµDµλ
a∗ ++|Dµφi|2− iψiσµDµψ
∗i +
1
2g2(Da)2 +Da
∑
i
φ∗iT
aφi
(134)
+ F ∗i Fi − Fi
∂W
∂φi
+ cc +∑
ij
1
2
∂2W
∂φi∂φj
ψiψj + i√
2∑
λaψiTaφ∗
i .
The scalar potential is found by solving for the auxiliaryD andF fields:
V = |Fi|2 +1
2g2a
(Da)2 (135)
with
Fi =∂W
∂φ∗i
Da =∑
i
(gaφ∗iT
aφi). (136)
In this equation,V ≥ 0. This fact can be traced back to the supersymmetry algebra.Starting with the equation
{Qα, Qβ} = 2Pµσµ
αβ, (137)
Multiplying by σo and take the trace:
QαQα +QαQα = E. (138)
If supersymmetry is unbroken,Qα|0〉 = 0, so the ground state energy vanishesif andonly if supersymmetry is unbroken. Alternatively, consider the supersymmetry transfor-mation laws forλ andψ. One has, under a supersymmetry transformation with parame-ter ε,
δψ =√
2εF + . . . δλ = iεD + . . . (139)
So if eitherF orD has an expectation value, supersymmetry is broken.We should stress that these statements apply to global supersymmetry. We will dis-
cuss the case of local supersymmetry later, but, as we will see, many of the lessons fromthe global case extend in a simple way to the case in which the symmetry is a gaugesymmetry.
We can now very easily construct a supersymmetric version ofthe standard model.For each of the gauge fields of the usual standard model, we introduce a vector super-field. For each of the fermions (quarks and leptons) we introduce a chiral superfield withthe same gauge quantum numbers. Finally, we need at least twoHiggs doublet chiralfields; if we introduce only one, as in the simplest version ofthe standard model, the re-sulting theory possesses gauge anomalies and is inconsistent. In other words, the theoryis specified by giving the gauge group (SU(3) × SU(2) × U(1)) and enumerating thechiral fields:
Qf , uf , df Lf , ef HU , HD. (140)
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The gauge invariant kinetic terms, auxiliaryD terms, and gaugino-matter Yukawacouplings are completely specified by the gauge symmetries.The superpotential can betaken to be:
W = HU(ΓU)f,f ′Qf Uf ′ +HD(ΓD)f,f ′QfDf ′HD(ΓE)f,f ′Lf ef ′ . (141)
As we will discuss shortly, this is not the most general lagrangian consistent with thegauge symmetries. It does yield the desired quark and leptonmass matrices, withoutother disastrous consequences.Exercise: Consider the case of one generation. Show that if
〈HU〉 = 〈HD〉 =(
0v
), (142)
(all others vanishing), then〈Da〉 = 0; 〈F i〉 = 0. (143)
Study the spectrum of the model. Show that the superpartnersof theW andZ are de-generate with the corresponding gauge bosons. (Note that for the massive gauge bosons,the multiplet includes an additional scalar). Show that thequarks and leptons gain mass,and are degenerate with their scalar partners.
The fact that the states fall into degenerate multiplets reflects that for this set ofground states (parameterized byv), supersymmetry is unbroken. That supersymmetryis unbroken follows from the fact that the energy is zero, by our earlier argument. It canalso be understood by examining the transformation laws forthe fields. For example,
δφi = ζα[Qα, φi] =√
2iζψ (144)
but the right hand side has no expectation value. Similarly,
δψi =√
2ζFi +√
2iσµζ∂µφ. (145)
The last term vanishes by virtue of the homogeneity of the ground state; the first vanishesbecauseFi = 0. Similar statements hold for the other possible transformations.
This is our first example of a moduli space. Classically, at least, the energy is zerofor any value ofv. So we have a one parameter family of ground states. These statesare physically inequivalent, since, for example, the mass of the gauge bosons dependson v. We will shortly explain why, in field theory, it is necessaryto choose a particularv, and why there are not transitions between states of different v (in any approximationin which degeneracy holds). As we will see later in these lectures, generically classicalmoduli spaces are not moduli spaces at the quantum level.
One can also read off from the lagrangian the couplings, not only of ordinary fields,but of their superpartners. For example, there is a Yukawa coupling of the gauginos tofermions and scalars, whose strength is governed by the corresponding gauge couplings.
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file=tasi99a.eps,angle=-90,width=7cm
Fig. 3. Some of the vertices in a supersymmetric theory. Dashed lines denote scalars, solid linesfermions.
There are also quartic couplings of the scalars, with gauge strength. These are indicatedin fig. 3.
Before turning to the phenomenology of this “Minimal Supersymmetric StandardModel,” (MSSM), it is useful to get some more experience withthe properties of su-persymmetric theories. With the limited things we know, we can already derive somedramatic results. First, we can write down the most general globally supersymmetric la-grangian, with terms with at most two derivatives, but not restricted by renormalizability(in the rest of this section, the lower caseφ refers both to the chiral field and its scalarcomponent):
L =∫d4θK(φ†
i , φi) +∫d2θW (φi) + cc +
∫d2θf(φi)W
2α + cc. (146)
HereK is a general function known as the Kahler potential.W andf are necessarilyholomorphic functions of the chiral fields. One can consider terms involving the covari-ant derivatives,Dα, but these correspond to terms with more than two derivatives, whenwritten in terms of component fields.
We will often be interested in effective lagrangians of thissort, for example, in study-ing the low energy limit of string theory. From the holomorphy of W andf , as well asfrom the symmetries of the models, one can often derive remarkable results. Consider,for example, the “Wess-Zumino” model, a model with a single chiral field with super-potential
W =1
2mφ2 +
1
3λφ3. (147)
For generalm andλ, this model has no continuous global symmetries. Ifm = 0, is hasan “R” symmetry, a symmetry which does not commute with supersymmetry:
φ→ e2i3
αφ θ → eiαθ dθ → e−iαθ. (148)
Under this transformation,W → e2iαW (149)
so∫d2θW is invariant. This transformation does not commute with supersymmetry;
recalling the form ofQα in terms ofθ’s, one sees that
Qα ∼ ∂
∂θα
+ . . .→ eiαQα. (150)
Correspondingly, the fermions and scalars in the multiplet transform differently: thescalar has the sameR charge as the superfield,2/3, whileψ has R charge one unit less
35
than that of the scalar, i.e.
φ→ e2iα/3φ ψ → e−iα/3ψ. (151)
It is easy to check that this a symmetry of the lagrangian, written in terms of the compo-nent fields. Correspondingly, in the quantum theory,
Qα ≈∫d3x(σµ
αα∂µφψ∗α + ψαF ) → eiαQα. (152)
Symmetries of this type will play an important role in much ofwhat follows. Ingeneral, in a theory with several chiral fields, one has
φi → eiαRiφi W (φi) → e2iαW (φi). (153)
If there are vector multiplets in the model, the gauge bosonsare neutral under the sym-metry, while the gauginos have charge+1. We will also be interested in discrete versionsof these symmetries (in which, essentially, the parameterα takes on only some discretevalues). In the case of the Wess-Zumino model, for non-zerom, a discrete subgroupsurvives for whichα = 3nπ, i.e.φ→ φ, ψ → −ψ. More elaborate discrete symmetrieswill play an important role in our discussions.
Even in the Wess-Zumino model with non-zerom, we can exploit the power of con-tinuous symmetries, by thinking of the couplings as if they were themselves backgroundvalues for some chiral fields. If we assignR charge+1 to φ, we can make the theoryR-invariant if we assignR charge−1 to λ. In practice, we might be interested in theorieswhereλ is the scalar component of a dynamical field (this will often be the case in stringtheory) or we may simply view this as a trick[17]. In either case, we can immediatelysee that there are no corrections to theφ3 term in the superpotential in powers ofλ. Thereason is that the superpotential must be a holomorphic function ofφ and λ (so it cannotinvolve, say,λλ†), and it must respect theR symmetry. Becauseλ is the small parameterof the theory, we have proven a powerful non-renormalization theorem: the superpoten-tial cannot be corrected to any order in the coupling constant. This non-renormalizationtheorem was originally derived by detailed consideration of the properties of Feynmangraphs. What is crucial to this argument is thatW is a holomorphic function ofφ andthe parameters of the lagrangian.
This is not to say that nothing in the effective action of the theory is corrected fromits lowest order value; non-holomorphic quantities are renormalized. For example:
∫d4θφ†φf(λ†λ) (154)
is allowed. In the Wess-Zumino model, this means that all of the renormalizations aredetermined by wave function renormalization. Finally, we should note that ifm = 0 attree level, no masses are generated for fermions or scalars in loops.
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4.3 N=2 Theories: Exact Moduli Spaces
We have already encountered an extensive vacuum degeneracyin the case of the MSSM.Actually, the degeneracy is much larger; there is a multiparameter family of such flatdirections involving the squark, slepton and Higgs fields. For the particular example,we saw that classically the possible ground states of the theory are labeled by a quantityv. States with differentv are physically distinct; the masses of particles, for example,depend onv. In non-supersymmetric theories, one doesn’t usually contemplate suchdegeneracies, and even if one had such a degeneracy, say, at the classical level, onewould expect it to be eliminated by quantum effects. We will see that in supersymmetrictheories, these flat directions almost always remain flat in perturbation theory; non-perturbatively, they are sometimes lifted, sometimes not.Moreover, such directionsare ubiquitous The space of degenerate ground states of a theory is called the “modulispace.” The fields whose expectation values label these states are called the moduli. Insupersymmetric theories, such degeneracies are common, and are often not spoiled byquantum corrections.
In theories withN = 1 supersymmetry, detailed analysis is usually required to de-termine whether the moduli acquire potentials at the quantum level. For theories withmore supersymmetries (N > 1 in four dimensions;N ≥ 1 in five or more dimensions),one can usually show rather easily that the moduli space is exact. Here we consider thecase ofN = 2 supersymmetry in four dimensions. These theories can also be describedby a superspace, in this case built from two Grassman spinors, θ and θ. There are twobasic types of superfields[13], called vector and hyper multiplets. The vectors are chiralwith respect to bothDα andDα, and have an expansion, in the case of aU(1) field:
ψ = φ+ θαWα + θ2D2φ†, (155)
whereφ is anN = 1 chiral multiplet andWα is anN = 1 vector multiplet. The fact thatφ† appears as the coefficient of theθ2 term is related to an additional constraint satisfiedbyψ[13]. This expression can be generalized to non-abelian symmetries; the expressionfor the highest component ofψ is then somewhat more complicated[13]; we won’t needthis here.
The theory possesses anSU(2) R symmetry under whichθ and θ form a doublet.Under this symmetry, the scalar component ofφ, and the gauge field, are singlets, whileψ andλ form a doublet.
I won’t describe the hypermultiplets in detail, except to note that from the perspec-tive ofN = 1, they consist of two chiral multiplets. The two chiral multiplets transformas a doublet of theSU(2). The superspace description of these multiplets is more com-plicated[13,14].
In the case of a non-abelian theory, the vector field,ψa, is in the adjoint representa-tion of the gauge group. For these fields, the lagrangian has avery simple expression in
37
superspace:
L =∫d2θd2θ ψaψa, (156)
or, in terms ofN = 1 components,
L =∫d2θ W 2
α +∫d4θφ†eV φ. (157)
The theory with vector fields alone has a classical moduli space, given by the values ofthe fields for which the scalar potential vanishes. Here thisjust means that theD fieldsvanish. Written as a matrix,
D = [φ, φ†], (158)
which vanishes for diagonalφ, i.e. for
φ =a
2
(1 00 −1
). (159)
In quantum field theory, one must choose a value ofa. This is different than in thecase of quantum mechanical systems with a finite number of degrees of freedom; thisdifference will be explained below. As in the case of the MSSM, the spectrum dependson a. For a given value ofa, the massless states consist of aU(1) gauge boson, twofermions, and a complex scalar (essentiallya), i.e. there is one light vector multiplet.The masses of the states in the massive multiplets depend ona.
For many physically interesting questions, one can focus onthe effective theory forthe light fields. In the present case, the light field is the vector multiplet,ψ. Roughly,
ψ ≈ ψaψa = a2 + aδψ3 + . . . (160)
What kind of effective action can we write forψ? At the level of terms with up to fourderivatives, the most general effective lagrangian has theform:3
L =∫d2θd2θf(ψ) +
∫d8θH(ψ, ψ†). (161)
Terms with covariant derivatives correspond to terms with more than four derivatives,when written in terms of ordinary component fields.
The first striking result we can read off from this lagrangian, with no knowledge ofH andf , is that there is no potential forφ, i.e. the moduli space is exact. This statementis true perturbatively and non-perturbatively!
One can next ask about the functionf . This function determines the effective cou-pling in the low energy theory, and is the object studied by Seiberg and Witten[18]. Wewon’t review this whole story here, but indicate how symmetries and the holomorphy of
3This, and essentially all of the effective actions we will discuss, should be thought of as Wilsonianeffective actions, obtained by integrating out heavy fieldsand high momentum modes.
38
f provide significant constraints (Michael Peskin’s TASI 96 lectures provide a conciseintroduction to this topic[19]). It is helpful, first, to introduce a background field,τ ,which we will refer to as the “dilaton,” with coupling
L =∫d2θd2θτψaψa (162)
where
τ = θ +i
g2+ . . . . (163)
τ is a chiral field. For our purposes,τ need not be subject to the same constraint as thevector superfield. Classically, the theory has an R-symmetry under whichψa rotates bya phase,ψa → eiαψa. But this symmetry is anomalous. Similarly, shifts in the real partof τ (θ) are symmetries of perturbation theory. This insures that there is only a one-loopcorrection tof . This follows, first, from the fact that any perturbative corrections tofmust beτ -independent. A term of the form
c ln(a)ψ2 (164)
respects the symmetry, since the shift of the logarithm justgenerates a contribution pro-portional toFF , which vanishes in perturbation theory. Beyond perturbation theory,however, we expect corrections proportional toae−τ , since this is invariant under thenon-anomalous symmetry. It is these corrections which wereworked out by Seiberg andWitten.
4.4 A Still Simpler Theory: N=4 Yang Mills
N = 4 Yang Mills theory is an interesting theory in its own right: it is finite and con-formally invariant. It also plays an important role in Matrix theory, and is central to ourunderstanding of the AdS/CFT correspondence.N = 4 Yang Mills has sixteen super-charges, and is even more tightly constrained than theN = 2 theories. There does notexist a convenient superspace formulation for this theory,so we will find it necessary toresort to various tricks. First, we should describe the theory. In the language ofN = 2supersymmetry, it consists of one vector and one hyper multiplet. In terms ofN = 1 su-perfields, it contains three chiral superfields,φi, and a vector multiplet. The lagrangianis
L =∫d2θW 2
α +∫d4θ
∑φ†
ieV φi +
∫d2θφa
i φbjφ
ckεijkε
abc. (165)
In the above description, there is a manifestSU(3) × U(1) R symmetry. Under thissymmetry, theφi’s haveU(1)R charge2/3, and form a triplet of theSU(3). But thereal symmetry is larger – it isSU(4). Under this symmetry, the four Weyl fermionsform a 4, while the six scalars transform in the6. Thinking of these theories as thelow energy limits of toroidal compactifications of the heterotic string will later give us a
39
heuristic understanding of thisSU(4): it reflects theO(6) symmetry of the compactifiedsix dimensions. In string theory, this symmetry is broken bythe compactification lattice;this reflects itself in higher derivative, symmetry breaking operators.
In theN = 4 theory, there is, again, no modification of the moduli space,pertur-batively or non-perturbatively. This can be understood in avariety of ways. We canuse theN = 2 description of the theory, defining the vector multiplet to contain theN = 1 vector and one (arbitrarily chosen) chiral multiplet. Thenan identical argumentto that given above insures that there is no superpotential for the chiral multiplet alone.TheSU(3) symmetry then insures that there is no superpotential for any of the chiralmultiplets. Indeed, we can make an argument directly in the language ofN = 1 super-symmetry. If we try to construct a superpotential for the lowenergy theory in the flatdirections, it must be anSU(3)-invariant, holomorphic function of theφi’s. But there isno such object.
Similarly, it is easy to see that there no corrections to the gauge couplings. Forexample, in theN = 2 language, we want to ask what sort of function,f , is allowed in
L =∫d2θd2θf(ψ). (166)
But the theory has aU(1) R invariance under which
ψ → e2/3iαψ θ → eiαθ θ → e−iαθ (167)
Already, then ∫d2θd2θψψ (168)
is the unique structure which respects these symmetries. Now we can introduce a back-ground dilaton field,τ . Classically, the theory is invariant under shifts in the real partof τ , τ → τ + β. This insures that there are no perturbative corrections tothe gaugecouplings. More work is required to show that there are no non-perturbative correctionseither.
One can also show that the quantityH in eqn. [161] is unique in this theory, againusing the symmetries. The expression[20,21]:
H = c ln(ψ) ln(ψ†), (169)
respects all of the symmetries. At first sight, it might appear to violate scale invariance;given thatψ is dimensionful, one would expect a scale,Λ, sitting in the logarithm.However, it is easy to see that one integrates over the full superspace, anyΛ-dependencedisappears, sinceψ is chiral. Similarly, if one considers theU(1) R-transformation,the shift in the lagrangian vanishes after the integration over superspace. To see thatthis expression is not renormalized, one merely needs to note that any non-trivialτ -dependence spoils these two properties. As a result, in the case ofSU(2), the fourderivative terms in the lagrangian are not renormalized. Note that this argument is non-perturbative.
40
4.5 Aside: Choosing a Vacuum
It is natural to ask: why in field theory, in the presence of moduli, does one have tochoose a vacuum? In other words, why aren’t their transitionbetween states with differ-ent expectation values for the moduli?
file=tasi99b.eps,angle=-90,width=7cm
Fig. 4. In a ferromagnetic, the spins are aligned, but the direction is arbitrary.
This issue is most easily understood by considering a different problem: rotationalinvariance in a magnet. Consider fig. 4, where we have considered a ferromagnet withspins aligned in two different directions, one oriented at an angleθ relative to the other.We can ask: what is the overlap of the two states, i.e. what is〈θ|0〉? For a single site,the overlap between the state|+〉 and the rotated state is:
〈+|eiτ1θ/2|+〉 = cos(θ/2). (170)
If there areN such sites, the overlap behaves as
〈θ|0〉 ∼ (cos(θ/2))N (171)
i.e. it vanishes exponentially rapidly with the volume.For a continuum field theory, states with differing values ofthe order parameter,v,
also have no overlap in the infinite volume limit. This is illustrated by the theory of ascalar field with lagrangian:
L =1
2(∂µφ)2. (172)
For this system, the expectation valueφ = v is not fixed. The lagrangian has a symmetry,φ→ φ+ δ, for which the charge is just
Q =∫d3xΠ(~x) (173)
whereΠ is the canonical momentum. So we want to study
〈v|0〉 = 〈0|eiQ|0〉. (174)
We must be careful how we take the infinite volume limit. We will insist that this bedone in a smooth fashion, so we will define:
Q =∫d3x∂oφe
−~x2/V 2/3
(175)
= − i∫ d3k
(2π)3
√ωk
2
(V 1/3
√π
)3
e−~k2V 2/3/4[a(~k) − a†(~k)].
41
Now, one can evaluate the matrix element, using
eA+B = eAeBe12[A,B]
(provided that the commutator is a c-number), giving
〈0|eiQ|0〉 = e−cv2V 2/3
, (176)
wherec is a numerical constant. So the overlap vanishes with the volume. You canconvince yourself that the same holds for matrix elements oflocal operators. This resultdoes not hold in0 + 1 and1 + 1 dimensions, because of the severe infrared behavior oftheories in low dimensions. This is known to particle physicists as Coleman’s theorem,and to condensed matter theorists as the Mermin-Wagner theorem.
{QAα , Q
∗Bβ} = 2σµ
αβδABPµ (177)
{QAα , Q
∗βB} = εαβX
AB. (178)
TheXAB ’s are Lorentz scalars, antisymmetric inA,B, known as central charges.We will focus, principally, onN = 1 supersymmetry; this means that the indexA
above takes only one value.
4.6 Supersymmetry and the Hierarchy Problem
In our discussion of the Standard Model, we argued that corrections to scalar massesare quadratically divergent because, unlike fermions, there is no symmetry in the limitin which there mass becomes small. But supersymmetry is a symmetry which relatesbosons to fermions, so in a world in which the laws of nature were supersymmetric,scalar masses might be stable against large radiative corrections. This feature of super-symmetry has aroused great interest.
But there is another reason to think that supersymmetry mightbe important to thehierarchy problem. In fact, stability is not all we want. It is all well and good to saythat if a theory has two dimensional parameters, one orders of magnitude smaller thananother, that quantum effects will maintain this hierarchy. But supersymmetry is proneto the appearance of such hierarchies.
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4.7 Supersymmetry in Superspace
4.8 Component Lagrangians
4.9 Soft Supersymmetry Breaking
4.10 Cancellation of Quadratic Divergences
To get some practice with supersymmetric theories, let’s check that in a simple model,the quadratic divergences do indeed cancel. Take aU(1) theory, with (massless) chiralfieldsφ+ andφ−. Before doing any computation, it is easy to see that providedwe workin a way which preserves supersymmetry, there can be no quadratic divergence. In thelimit that the mass term vanishes, the theory has a chiral symmetry under whichφ+ andφ− rotate by the same phase,
φ± → eiαφ±. (179)
This symmetry forbids a mass term in the superpotential,Λφ+φ−, the only way a super-symmetric mass term could appear. The actual diagrams we need to compute are shownin fig. ??. Since we are only interested in the mass, we can take the external momentumto be zero. It is convenient to choose Landau gauge for the gauge boson. In this gaugethe gauge boson propagator is
Dµν = −i(gµν −qµqνq2
)1
q2(180)
so the first diagram vanishes. The second, third and fourth are straightforward to workout from the basic lagrangian. One finds:
Ib = g2(i)(−i) 3
(2π)4
∫ d4k
k2(181)
Ic = g2(i)(−i)(√
2)2
(2π)4
∫ d4k
k4tr(kµσ
µkν σν) (182)
= − 4g2
(2π)4
∫ d4k
k2(183)
Ic = g2(i)(−i) 1
(2π)4
∫ d4k
k2. (184)
It is easy to see that the sum,Ia + Ib + Ic + Id = 0.Clearly if nature is supersymmetric, supersymmetry is broken. We want, then, to ask,
what is the likely scale of supersymmetry breaking? We can modify our computation
43
above so as to address this question. Suppose that supersymmetry breaking induces amass for the scalars,m2, but the fermions remain massless. Then onlyId changes;
Id → g2
(2π)4
∫ d4k
k2 − m2(185)
= −i g2
(2π)4
∫ d4kE
k2E + m2
= m2 independent +ig2
16π2m2 ln(Λ2/m2).
We have worked here in Minkowski space, and I have indicated factors ofi to assistthe reader in obtaining the correct signs for the diagrams. In the second line, we haveperformed a Wick rotation. In the third, we have separated off a mass-independent part,since we know that this is cancelled by the other diagrams.
Exercise: Verify the expressions forIa − Id.
Summarizing, the one loop mass shift is
δm2 = − g2
16π2m2 ln(Λ2/m2). (186)
Note that the mass shift is proportional tom2, the supersymmetry breaking mass, as wewould expect since supersymmetry is restored asm2 → 0. In the context of the standardmodel, we see that the scale of supersymmetry breaking cannot be much larger than thethe Higgs mass scale itself. Roughly speaking, it can’t be much larger than this scale byfactors of order1/
√αW , i.e., factors of order6.
A second reason to suspect that low energy supersymmetry might have somethingto do with nature comes from string theory. From the lecturesat this school, you havesurely gained the sense that supersymmetry is intrinsic to string theory. It is natural tosuspect that any consistent fundamental theory must be supersymmetric.
By itself, this is not enough to argue that supersymmetry should survive to low ener-gies. But it has been known for some time that there are a vast array of supersymmetricsolutions of string theory. A general feature of these solutions is that if supersymmetryis unbroken in some lowest order approximation, the theory remains supersymmetricto all orders. Moreover, non-supersymmetric solutions areproblematic: typically thevacuum is unstable already at one loop. While one cannot claimto have understoodhow string dynamics might break supersymmetry and choose some particular groundstate, it is almost impossible to imagine how a sensible ground state could emerge ina non-supersymmetric vacuum. So if string theory describesnature, low energy super-symmetry is almost certainly a prediction.
Finally, there are some small experimental hints that supersymmetry might be true.The most dramatic of these is the unification of couplings [?]. To understand this, we first
44
introduce a supersymmetric extension of the standard model, known as the “MinimalSupersymmetric Standard Model,” or MSSM. In this model, thegauge symmetry is stilltaken to beSU(3) × SU(2) × U(1). Each gauge generator is now associated with avector multiplet, so there is a gaugino for each gauge boson.Similarly, all of the knownquarks and leptons are promoted to chiral multiplets. In other words, each left-handedfermion of the standard model now has a complex scalar partner with the same quantumnumbers. Finally, instead of one Higgs doublet, there must be two, each containing aboson and a fermion. Otherwise the model suffers from anomalies. We will denote thesebyHU andHD, with hypercharge±1, respectively.
Knowing the representation content of the theory, we can work out theβ-functionsof the different groups. The general expression for the one-loopβ-functions is
bo =11
3CA − 2
3
nf∑
i=1
ci2 −1
3
nφ∑
j=1
cj2. (187)
Here the sum overi runs over all of the left-handed fermions of the theory, while thatoverj runs over the scalars. For the gauginos,c2 = CA, so for a supersymmetric theorywith nf chiral fields in the fundamental representation we obtain
bo = 3CA − 1
2nf . (188)
For SU(3), with three generations, this givesbo = 3. Similarly, for SU(2), oneobtainsbo = −1 (remember to keep track of the two Higgs doublets). For theU(1),some care is required with the normalization of the charge. Let us assume that the threegauge groups are unified inSU(5). In that case, all of the generators must be normalizedin the same way. In a singlet generation, one has a5 and10. The5 contains thed quarkand the lepton doublet. For this representation, theSU(3) andSU(2) generators satisfytrT 2 = 1/2. The correspondingU(1) generator is then
Y =
√3
20diag(2/3, 2/3, 2/3,−1,−1). (189)
In other words,Y is related to the conventional hypercharge generator by:
Y =√
3/20Y g′ = g5
√3/5 (190)
(remember that the hypercharge boson is taken to couple toY/2). From this it followsthatsin2(θw) = 3/8.
With this information, one can now run the gauge couplings tohigh energy. Thesimplest thing to do is to assume that all of the new particlespredicted by supersymmetryhave the same mass (say a few hundred GeV up to a TeV). At one loop, the couplingssatify
α−1i (MZ) = α−1
i (MGUT ) +b(i)o
4πln(MGUT/MZ). (191)
45
Including also two loop corrections, one obtains quite goodagreement. The couplingsfail to unify in nonsupersymmetric theories. This might be acoincidence, but it is quitesuggestive.4
4.11 More on the MSSM
Above, we have introduced the fields of the MSSM, and explained their gauge quantumnumbers. Let us develop this model further. The lagrangian includes, first, the gaugeinvariant kinetic terms for all of the fields. In a supersymmetric theory, this includesYukawa couplings of gauginos to matter fields and quartic scalar couplings from theD2
terms. The usual Yukawa couplings of fermions arise from terms in the superpotential.These can be written in the form
W = HUQyuu+HDQyDd+HDLyLe. (192)
In this expression, the quark and lepton superfields are understood to carry a flavor index,and they’s are3× 3 matrices. IfHU andHD have non-zero masses, this gives mass forquarks and squarks, leptons and sleptons.
Exercise: Check that their are a set of supersymmetric ground states with equal, non-zero expectation values forHu andHD i.e., that the energy vanishes ifHU = HD =(
0v
). Check that one obtains equal masses for bosons and fermions,first for the quarks
and leptons, then for the gauge bosons and gauginos.
There are other terms which can also be present in the superpotential. These includethe “µ-term,” µHUHD. This is a supersymmetric mass term for the Higgs fields. Wewill see later that we needµ ∼MZ to have a viable phenomenology.
A set of dimension four terms which are permitted by the gaugesymmetries raisemuch more serious issues. For example, one can have terms
uf dgdhΓfgh +QfLgdHλ
fgh. (193)
These couplings violateB andL! This is our first serious setback. In the standard model,there is no such problem. The leading operators permitted bygauge invariance are fourfermi operators of dimension6, and it is easy to imagine that they are suppressed bysome very large mass scale.
If we are not going to simply give up, we need to suppressB andL violation at thelevel of dimension four terms. This presumably requires additional symmetries. Thereare various possibilities one can imagine.
4Since the weak and electromagnetic couplings are far betterknown thanαs, one usually computesαs, making some assumptions about thresholds both at the GUT scale and at the supersymmetric scale.One actually finds that, with the simplest assumptions for these, that one predicts a value ofαs slightlytoo large. This problem, and possible solutions, have been discussed in [?].
46
1. Global continuous symmetries: It is hard to see how such symmetries could bepreserved in any quantum theory of gravity, and indeed in string theory, there is atheorem which asserts that there are no global continuous symmetries [?].
2. Discrete symmetries: Discrete symmetries can be gauge symmetries, and indeedsuch symmetries are common in string theory. These symmetries are often “Rsymmetries,” symmetries which do not commute with supersymmetry.
A simple (though not unique), solution to the problem of baryon and lepton numberviolation by dimension four operators is known asR-parity or “matter parity.” Under thissymmetry, all ordinary particles are even, while their superparters are odd. Imposing thissymmetry immediately eliminates all of the dangerous operators. For example,
∫d2θudd ∼ ψuψd
˜d (194)
(we have changed notation again: the tilde here indicates the superpartner of the ordinaryfield, i.e., the squark). This operator is clearly odd under the symmetry.
More formally, we can define this symmetry as the transformation on superfields:
θα → −θα (195)
(Qf , uf df , Lf ef ) → −(Qf , uf df , Lf ef ) (196)
(HU , HD) → (HUHD). (197)
While imposing this symmetry solves the immediate problem, it also has a strikingconsequence: the lightest of the new particles predicted bysupersymmetry (the LSP) isstable. In order to avoid various cosmological disasters, this particle must be electricallyneutral. It is then, inevitably, very weakly interacting. This in turn means:
• The generic signature of R-parity conserving supersymmetric theories is eventswith missing energy.
• Supersymmetry is likely to produce an interesting dark matter candidate.
In most of what follows, we will assume a conservedR-parity.
4.12 LSP as the Dark Matter
A stable particle is not necessarily a good dark matter candidate. But we can make acrude calculation which indicates that the LSP density is ina suitable range to be thedark matter. Consider particles,X, with mass of order100 GeV interacting with weakinteraction strength. Their annihilation cross sections go asG2
FE2. So, in the early
universe, the corresponding interaction rate is of order
Γ ≈ ρXG2FE
2 ≈ ρXG2FT
2. (198)
47
These interactions drop out of equilibrium when
Γ ∼ H ∼ T 2/Mp, (199)
i.e., when
ρX ∼ G2F
Mp
∼ 10−9. (200)
T here is of order 1 GeV, soρX
ργ
∼ 10−9. (201)
This means that theX particles have a number density today similar to that of baryons.So if their masses are of order100 GeV, their density can be of order the closure density.This estimate is quite crude, but more careful studies indicate that the LSP can be thedark matter for a broad range of parameters.
So while it is disturbing that we need to impose additional symmetries in order toavoid proton decay, it is also exciting that this leads to a possible solution of one of themost critical problems of cosmology: the identity of the dark matter.
4.13 Local Supersymmetry
It would take many lectures (and a more expert lecturer) to give a proper expositionof N = 1 supergravity. Fortunately, there are only a few facts we will need to know.First, the terms in the effective action with at most two derivatives or four fermions arecompletely specified by three functions:
1. The Kahler potential,K(φ, φ†), a function of the chiral fields
2. The superpotential,W (φ), a holomorphic function of the chiral fields.
3. The gauge coupling functions,fa(φ), which are also holomorphic functions of thechiral fields.
The lagrangian which follows from these can be found, for example, in [?,?]. Let usfocus, first, on the scalar potential. This is given by
V = eK
[(∂W
∂φi
+∂K
∂φi
W
)gij
(∂W ∗
∂φ∗j
+∂K
∂φ∗j
W
)− 3|W |2
], (202)
where
gij =∂2K
∂φi∂φj
(203)
is the Kahler metric associated with the Kahler potential. In this equation, we haveadopted units in which
GN =1
8πM2. (204)
48
whereM ≈ 2 × 1018 GeV is the reduced Planck mass we encountered earlier.There is a standard strategy for building supergravity models. One introduces two
sets of fields, the “hidden sector fields,” which will be denoted byZi, and the “visiblesector fields,” denotedya. TheZi’s are assumed to be connected with supersymmetrybreaking, and to have only very small couplings to the ordinary fields, ya. In otherwords, one assumes that the superpotential,W , has the form
W = Wz(Z) +Wy(y), (205)
at least up to terms suppressed by1/M .One also usually assumes that the Kahler potential has a “minimal” form,
K =∑
z†i zi +∑
y†aya. (206)
One chooses (tunes) the parameters ofWZ so that
〈FZ〉 ≈MwM (207)
and〈V 〉 = 0. (208)
Note that this means that〈W 〉 ≈MWM
2p . (209)
To understand the structure of the low energy theory in such amodel, suppose firstthatW (y) = 0, and thatK is of the “minimal” form, eq. (206). Then
V (y) = eK(z)|〈W 〉|2∑
|ya|2. (210)
In other words, the scalars all have a common mass,
m2o = e〈K〉 |〈W 〉|2
M4≈ m2
3/2. (211)
Herem3/2 is the mass of the gravitino. Note that withF ∼MMZ ,m3/2 ∼MZ .If we now allow for a non-trivialWy, we find alsoA andBµ terms. For example,
the terms∂W
∂ya
yaWy = 3Wy (212)
if W is homogeneous of degree three. Additional contributions arise from⟨∂W
∂zi
⟩y∗jW
∗ + c.c. (213)
Exercise: The simplest model of the hidden sector is known as the “Polonyi model.” Inthis model,
W = m2(z + β) (214)
49
β = (2 +√
3M) . (215)
Verify that the minium of the potential forZ lies at
Z = (√
3 − 1)M (216)
and that
m3/2 = (m2/m)e(√
3−1)2/2 m2o = 2
√3m2
3/2 A = (3 −√
3)m3/2. (217)
So far, we have not addressed the question of gaugino mass. This can arise from anon-trivial gauge coupling function,
fa = cZ
M(218)
which gives a gluino mass, just as it would in the global case:
mλ =cFz
M. (219)
So these models have just the correct structure. They have soft breakings of thecorrect order of magnitude, and they exhibit, with our assumption of minimal kineticterm, the properties of universality and proportionality.Indeed, this is a highly predictiveframework, with only (assuming MSSM particle content)5 parameters. A large amountof work has been done on these models, including investigations of:5
1. Renormalization group evolution: one finds that there is a substantial region of theparameter space in whichSU(2) × U(1) is broken in the correct fashion.
2. Unification: The predictions for unification are far better than in the minimal stan-dard model. However, the prediction forαs tends to be somewhat larger thanobserved. This can be “fixed” by adding new thresholds at the high scale [?].
3. b-τ unification: One can also consider the possibility that theb andτ Yukawa’sunify. This is certainly not a general requirement of unification; it depends, ingrand unified models, on the Higgs content, and in string theory on the details ofcompactification. Still, this idea seems viable (and interesting).
4. Proton decay: with detailed assumptions about the susy spectrum and the structureof grand unification, it is possible to compute the proton lifetime and compare withexperimental limits.
5More detail on all of these points is provided in many excellent review articles. See, for example, JonBagger’s 1995 TASI lectures, and references therein [?].
50
5. Dark matter: Again, in a detailed model, one can identify the LSP and computeits couplings to matter. This permits a precise calculationof the abundance.
6. b → sγ: As we have already remarked, one can compute the rate for this processin particular models. One often finds that other constraintsare stronger.
5 Spontaneous Supersymmetry Breaking
5.1 Supersymmetry Breaking
5.1.1 F and D Type Breaking
6 Non-Renormalization Theorems
Statement of the theorems. Seiberg’s proof. Various applications.
7 Beyond Perturbation Theory
7.1 Supersymmetric QCD and Its Symmetries
7.2 Nf < Nc: The Superpotential
7.3 Nf = Nc − 1: Instanton computation of the Superpotential
7.4 Nf < Nc: Gaugino Condensation
7.5 Supersymmetry Breaking: The(3, 2) model
8 Seiberg Duality
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Part II: String/M Theory
52
9 Introduction
String theory and quantum gravity. A bit of history of stringtheory.
53
10 The Bosonic String
11 The Superstring
12 Type II Strings
13 Heterotic String
14 Compactification of String Theory
14.1 Tori
14.2 Orbifolds
14.3 Calabi-Yau compactifications
15 Dynamics of String Theory at Weak Coupling
15.1 Non-Renormalization Theorems
15.2 Gaugino Condensation
15.3 Obstacles to A String Phenomenology
16 Strings at Strong Coupling: Duality
16.1 Perturbative Dualities
16.2 D-Branes
16.3 Strong-Weak Coupling Dualities: Some Evidence
16.3.1 IIB Self Duality
16.3.2 Catalog of other string Dualities
16.4 IIA → 11 Dimensional Supergravity (M Theory)
16.5 Strongly Coupled Heterotic String
16.6 AdS-CFT
Very brief. Mainly AdS5.
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17 Coda: Where are We headed
17.1 Some alternatives to Supersymmetry
17.1.1 Technicolor
17.1.2 Large Extra Dimensions
17.1.3 Warped Spaces
17.1.4 String Phenomenology: The Landscape?
17.1.5 The Tevatron, The LHC and the ILC
18 Appendices
18.1 Left Handed Fermions
18.2 Gauge Symmetry
18.3 Path Integration
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59
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