Supersymmetric Theory and Models
aa r X i v : . [ h e p - ph ] N ov SCIPP 17/11
Supersymmetric Theory and Models
Howard E. Haber and Laurel Stephenson Haskins Santa Cruz Institute for Particle Physics,University of California, Santa Cruz, CA 95064, USA Racah Institute of Physics,Hebrew University, Jerusalem 91904, Israel
In these introductory lectures, we review the theoretical tools used inconstructing supersymmetric field theories and their application to phys-ical models. We first introduce the technology of two-component spinors,which is convenient for describing spin- fermions. After motivating whya theory of nature may be supersymmetric at the TeV energy scale, weshow how supersymmetry (SUSY) arises as an extension of the Poincar´ealgebra of spacetime symmetries. We then obtain the representationsof the SUSY algebra and discuss its simplest realization in the Wess-Zumino model. In order to have a systematic approach for obtainingsupersymmetric Lagrangians, we introduce the formalism of superspaceand superfields and recover the Wess-Zumino Lagrangian. These meth-ods are then extended to encompass supersymmetric abelian and non-abelian gauge theories coupled to supermatter. Since supersymmetryis not an exact symmetry of nature, it must ultimately be broken. Wediscuss several mechanisms of SUSY-breaking (both spontaneous and ex-plicit) and briefly survey various proposals for realizing SUSY-breakingin nature. Finally, we construct the the Minimal Supersymmetric exten-sion of the Standard Model (MSSM), and consider the implications forthe future of SUSY in particle physics. Contents
Supersymmetric Theory and Models
11. Introduction to the TASI-2016 Supersymmetry Lectures . . . . . . . . . . . . 32. Spin-1/2 fermions in quantum field theory . . . . . . . . . . . . . . . . . . . . 42.1. Two-component spinor technology . . . . . . . . . . . . . . . . . . . . . . 62.2. Correspondence between the two- and four-component spinor notations . 162.3. Feynman Rules for Dirac and Majorana fermions . . . . . . . . . . . . . 202.4. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 . Motivation for TeV-scale supersymmetry . . . . . . . . . . . . . . . . . . . . . 283.1. Why the TeV scale? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2. The modern principle of naturalness . . . . . . . . . . . . . . . . . . . . 303.3. Avoiding quadratic UV-sensitivity with elementary scalars . . . . . . . . 314. Supersymmetry: first steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.1. Review of the Poincar´e algebra . . . . . . . . . . . . . . . . . . . . . . . 324.2. The supersymmetry (SUSY) algebra . . . . . . . . . . . . . . . . . . . . 344.3. Representations of the N = 1 SUSY algebra . . . . . . . . . . . . . . . . 374.4. Consequences of super-Poincar´e invariance . . . . . . . . . . . . . . . . . 444.5. Supersymmetric theories of spin-0 and spin- particles . . . . . . . . . . 494.6. The SUSY algebra realized off-shell . . . . . . . . . . . . . . . . . . . . . 524.7. Counting bosonic and fermionic degrees of freedom . . . . . . . . . . . . 544.8. Lessons from the Wess-Zumino Model . . . . . . . . . . . . . . . . . . . 554.9. Appendix: Constructing the states of a supermultiplet . . . . . . . . . . 554.10. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605. Superspace and Superfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.1. Superspace coordinates and translations . . . . . . . . . . . . . . . . . . 615.2. Expansion of the superfield in powers of θ and θ † . . . . . . . . . . . . . 645.3. Spinor covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 655.4. Chiral superfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.5. Constructing the SUSY Lagrangian . . . . . . . . . . . . . . . . . . . . . 705.6. R -invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.7. Grassmann integration and the SUSY action . . . . . . . . . . . . . . . . 755.8. Improved ultraviolet behavior of supersymmetry . . . . . . . . . . . . . . 775.9. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796. Supersymmetric gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.1. Vector superfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2. Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.3. Gauge-invariant interactions . . . . . . . . . . . . . . . . . . . . . . . . . 866.4. Generalizing to more than one chiral superfield . . . . . . . . . . . . . . 876.5. SUSY Yang-Mills theory coupled to supermatter . . . . . . . . . . . . . . 896.6. The SUSY Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.7. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927. Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937.1. Spontaneous SUSY breaking . . . . . . . . . . . . . . . . . . . . . . . . . 937.2. Mass Sum rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.3. The origin of SUSY-breaking dynamics . . . . . . . . . . . . . . . . . . . 997.4. A phenomenological approach: soft SUSY-breaking . . . . . . . . . . . . 1027.5. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068. Supersymmetric extension of the Standard Model (MSSM) . . . . . . . . . . . 1078.1. Field content of the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.2. The superpotential of the MSSM . . . . . . . . . . . . . . . . . . . . . . 1118.3. Supersymmetry breaking in the MSSM . . . . . . . . . . . . . . . . . . . 1148.4. The MSSM parameter count . . . . . . . . . . . . . . . . . . . . . . . . . 1158.5. The MSSM particle spectrum . . . . . . . . . . . . . . . . . . . . . . . . 1178.6. The Higgs sector of the MSSM . . . . . . . . . . . . . . . . . . . . . . . 1208.7. Unification of gauge couplings . . . . . . . . . . . . . . . . . . . . . . . . 1248.8. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289. Supersymmetry Quo Vadis? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 . Introduction to the TASI-2016 Supersymmetry Lectures These lectures were first presented at the 2016 Theoretical Advanced StudyInstitute (TASI-2016) in Boulder, CO. Four ninety-minute lectures weregiven, with the aim of presenting the basic theoretical techniques of su-persymmetry needed for the construction of a supersymmetric extension ofthe Standard Model of particle physics. The lectures were pitched at anelementary level, assuming that the students were well versed in quantumfield theory, gauge theory and the Standard Model, but with no assumedprior knowledge of supersymmetry. Nevertheless, some aspects of theselectures may also be useful to the reader with some prior knowledge ofsupersymmetry.It is possible to introduce the technology of supersymmetry theory usingfour-component spinor notation that is familiar to all students of quantumfield theory. However, it is our view that employing two-component spinornotation greatly simplifies the presentation of the theoretical structure ofsupersymmetry in 3+1 spacetime dimensions. Thus, in Section 2, we in-troduce the two-component spinor notation in some detail and discuss howit is related to the better known four-component spinor notation. Thismaterial is based heavily on a comprehensive review of Dreiner, Haberand Martin that is presented in Ref. [1]. In this review, it is shown thatpractical calculations in quantum field theory can be carried out entirelywithin the framework of the two-component spinor notation, which includethe development of Feynman rules for two-component spinors. However,at the end of Section 1, we are slightly less ambitious and revert to four-component fermion notation for the purpose of computing scattering anddecay processes. In particular, we provide a translation between two andfour-component spinor notation, and develop four-component spinor Feyn-man rules that treat both Dirac and Majorana fermions on the same footing.In Section 3, we present the motivation for TeV-scale supersymmetry.Namely, why is it that we feel compelled to introduce a supersymmetricextension of the Standard Model, despite the great success of the StandardModel in describing collider data and the absence of significant evidence fornew physics beyond the Standard Model. With this motivation in mind,we are ready to explore the theoretical aspects of supersymmetry.Since this is not a review article, we do not feel compelled to present acomprehensive list of references. Nevertheless, it is instructive to assemblea list of books and lecture notes on supersymmetry, many of which we havefound quite useful in preparing these lectures. Thus, we draw your attention3o the following books listed in Refs. [2–29] and the following reviews andlecture notes listed in Ref. [30–49]. The reader is warned that conventionsvary widely among these references. Apart from the two possible choices forthe spacetime metric (either the mostly minus metric used in these lecturesor the mostly plus metric), there are many different choices in the definitionof a variety of quantities, often involving different choices of signs. Of thesemany conventions, we believe that the ones employed in these lecture notesare probably closest to those that appear in Ref. [33]. In Section 4, we show how the algebra of the Poincar´e group can be ex-tended to obtain the supersymmetry (SUSY) algebra. The representationsof the N = 1 SUSY algebra are elucidated, and the Wess-Zumino model ispresented as the simplest realization of a supersymmetric field theory. InSection 5, we take some of the mystery out of constructing a SUSY La-grangian by introducing the concepts of superspace and superfields. Thisformalism allows one to construct supersymmetric field theories without anyguesswork. In Section 6, the formalism of supersymmetric gauge theoriesis developed. In Section 7, we examine supersymmetry breaking, which isnecessary for accommodating the observation that the elementary particlesobserved today are not each accompanied by an equal-mass superpartner.Finally, in Section 8, we construct the Minimal Supersymmetric extensionof the Standard Model (MSSM). We end these lectures in Section 9 with abrief discussion of what lies ahead for supersymmetry.
2. Spin-1/2 fermions in quantum field theory
We begin these lectures with a treatment of spin- fermions in quantumfield theory. In most introductory courses in relativistic quantum fieldtheory, the student first encounters fermion fields in the treatment of a rel-ativistic theory of electrons and photons. The electron is represented by afour-component Dirac fermion field, and the free field electron Lagrangianyields the Dirac equation. The four components represent two degrees offreedom corresponding to the electron and two degrees of freedom corre-sponding to the positron. Feynman rules for quantum electrodynamics aredeveloped and the vector-like nature of the e + e − coupling to photons leadsto some important simplifications. We also note that although Ref. [38] employs the mostly plus metric, one can obtain aversion of Martin’s Supersymmetry Primer in the mostly minus metric by changing oneline in the LaTeX source code. This alternative version of the Primer closely matchesthe conventions employed in these lectures. EM and are thus represented by Dirac fermion fields.The neutrinos are massless, but only the left-handed neutrinos and right-handed antineutrinos are present in the theory. Thus, one can still use four-component fermion fields (by applying the appropriate chiral projectionoperators on the neutrino fields). Hence, the four-component techniques ofquantum electrodynamics are easily accommodated and Feynman rules forthe fermion fields are obtained in a straightforward manner.However, the observation of neutrino mixing phenomena implies thatneutrinos are massive, which requires new physics beyond the StandardModel of the electroweak interactions. Models of neutrino mass ofteninclude neutral self-conjugate fermion states with two degrees of free-dom, called Majorana fermions. Such states can be described using four-component fermion fields that are constrained by an appropriate conjuga-tion condition. However, the resulting field theory description of systems ofMajorana and Dirac fermions is somewhat awkward. Moreover, the Feyn-man rules for interacting Majorana fermions require some care.Returning to first principles, one can ask how spin- fermions arise inquantum field theory. In Section 2.1, we shall demonstrate that the fun-damental building blocks employed in constructing spin- quantum fieldsare two-component spinors corresponding to the two-dimensional represen-tations of the Lorentz group. A neutral Majorana fermion is then repre-sented by a two-component fermion field. Dirac fermions arise when oneconsiders theories of two mass-degenerate two-component fermions, whichcan be combined to make a charged four-component Dirac fermion. This iscompletely analogous to the case of spin-0 bosons, in which a neutral bo-son is represented by a real scalar field and a charged boson is representedby a complex scalar field (whose real and imaginary parts constitute twomass-degenerate real scalars).The development of two-component spinor technology has a number ofbenefits. First, it provides an elegant unified description of Majorana andDirac fermions. Second, it is very convenient to employ the two-componentspinor formalism in theories of chiral interactions. Finally, it will proveespecially useful in developing the formalism of supersymmetry, which is5he main focus of these lectures.Because most students see the four-component spinor formalism firstand are therefore more familiar with it, we shall devote Section 2.2 to thetranslation between the two- and four-component formalisms. Finally, inSection 2.3 we demonstrate how Feynman rules involving four-componentfermion fields can be extended to incorporate Majorana fermions.This section is based on a comprehensive review of Dreiner, Haber andMartin [1], where many references to the original literature can be found. Two-component spinor technology
Orthochronous Lorentz transformations
Quantum spin- fields transform under a two-dimensional irreducible rep-resentation of the Lorentz group. Thus, we first examine the propertiesthat define a Lorentz transformation [50]. Under an active Lorentz trans-formation, Λ µν , a four-vector p µ transforms as p ′ µ = Λ µν p ν . (2.1)The condition that g µν p µ p ν is invariant under Lorentz transformations im-plies that Λ µν g µρ Λ ρλ = g νλ . (2.2)That is, Λ ∈ O(3,1). Eq. (2.2) implies that Λ possesses the following twoproperties: (i) det Λ = ± | Λ | ≥
1. Thus, Lorentz trans-formations fall into four disconnected classes denoted by a pair of signs, (cid:0) sgn[det Λ] , sgn[Λ ] (cid:1) . The proper orthochronous Lorentz transformationscorrespond to (+ , +) and are continuously connected to the identity.The most general proper orthochronous Lorentz transformation, char-acterized by a rotation angle θ about an axis b n ( ~θ ≡ θ b n ) and a boostvector ~ζ ≡ ˆ v tanh − β (where ˆ v ≡ ~v / | ~v | is the unit velocity vector and β ≡ | ~v | /c ), is a 4 × (cid:0) − iθ αβ s αβ (cid:1) = exp (cid:16) − i ~θ · ~s − i ~ζ · ~k (cid:17) , (2.3)where θ αβ is antisymmetric, with θ i ≡ ǫ ijk θ jk , ζ i ≡ θ i = − θ i , and( s αβ ) µν = i ( g αµ g βν − g βµ g αν ) , (2.4)with s i ≡ ǫ ijk s jk and k i ≡ s i = − s i . We have employed a notationwhere the lower case Latin indices i, j, k = 1 , , ǫ = +1. In our conventions, the Minkowski metric tensor is g µν = diag(1 , − , − , − Henceforth, we shall work in particle physics units where ~ = c = 1. s µν are antisymmetric 4 × i.e. , s µν = − s νµ ,and satisfy the commutation relations,[ s αβ , s ρσ ] = i ( g βρ s ασ − g αρ s βσ − g βσ s αρ + g ασ s βρ ) . (2.5)It follows from eqs. (2.3) and (2.4) that an infinitesimal orthochronousLorentz transformation is given byΛ µν ≃ δ µν + θ µν ≃ ( × − i ~θ · ~s − i ~ζ · ~k ) µν , (2.6)where × is the 4 × θ µν = − θ νµ .2.1.2. Finite-dimensional Representations of the Lorentz Group
A generic spin- s field Φ transforms asΦ( x ) → Φ ′ ( x ′ ) = M R (Λ)Φ( x ) , (2.7)where M R ≡ exp (cid:0) − iθ µν S µν (cid:1) and the S µν constitute finite-dimensionalirreducible matrix representations of the Lie algebra of the Lorentz group.The S µν satisfy the same commutation relations as the s µν given ineq. (2.5). It is convenient to denote the six independent generators de-fined by the S µν as S i ≡ ǫ ijk S jk , K i ≡ S i , (2.8)where i, j, k = 1 , ,
3. The S i generate three-dimensional rotations in spaceand the K i generate the Lorentz boosts. It then follows that M R ≡ exp (cid:16) − i ~θ · ~S − i ~ζ · ~K (cid:17) . (2.9)The S i and K i satisfy the commutation relations,[ S i , S j ] = ǫ ijk S k , (2.10)[ S i , K j ] = ǫ ijk K k , (2.11)[ K i , K j ] = − ǫ ijk S k . (2.12)We define the following linear combinations of the generators, ~S + ≡ ( ~S + i ~K ) , ~S − ≡ ( ~S − i ~K ) , (2.13)which satisfy the commutation relations,[ S i + , S j + ] = iǫ ijk S k + , (2.14)[ S i − , S j − ] = iǫ ijk S k − , (2.15)[ S i ± , S j ∓ ] = 0 , (2.16)corresponding to two independent (complexified) SU(2) Lie algebras. Thus,the representations of the Lorentz algebra are characterized by ( s , s ),where the s i are half-integers. For example, (0 ,
0) corresponds to a scalarfield and ( , ) corresponds to a four-vector field.7.1.3. Two-component spinors
Spin-1/2 fermion fields transform under the spinor representations, ( , ~S + = ~σ and ~S − = 0, and (0 , ) corresponding to ~S + = 0and ~S − = ~σ . That is, the Lorentz transformation matrices acting onspinor fields may be written in terms of the Pauli spin matrices σ , σ , and σ as follows,( ,
0) : M = exp (cid:16) − i ~θ · ~σ − ~ζ · ~σ (cid:17) , (2.17)which via a similarity transformation is equivalent to the matrix represen-tation, ( M − ) T = iσ M ( iσ ) − , and(0 , ) : [ M − ] † = exp (cid:16) − i ~θ · ~σ + ~ζ · ~σ (cid:17) , (2.18)which via a similarity transformation is equivalent to the matrix represen-tation, M ∗ = iσ [ M − ] † ( iσ ) − .Thus, the Lorentz transformation law for two-component ( ,
0) fieldscan be written in two equivalent ways, ξ ′ α = M αβ ξ β , ξ ′ α = [( M − ) T ] αβ ξ β , (2.19)where α, β = 1 ,
2. Likewise, the Lorentz transformation law for two-component (0 , ) fields can be written in two equivalent ways, ξ ′ † ˙ α = [( M − ) † ] ˙ α ˙ β ξ † ˙ β , ξ ′ † ˙ α = [ M ∗ ] ˙ α ˙ β ξ † ˙ β . (2.20)The (0 , ) fields are related to the ( ,
0) fields by hermitian conjugation, ξ † ˙ α ≡ ( ξ α ) † , ξ † ˙ α ≡ ( ξ α ) † . (2.21)It is conventional to employ undotted indices for the spinor components of( ,
0) fields and dotted indices for the spinor components of (0 , ) fields.As noted below eqs. (2.17) and (2.18), respectively, each of the twoequivalent representation matrices, M and ( M − ) T in the case of ( , M − ) † and M ∗ in the case of (0 , ), are related by a similarity trans-formation involving the antisymmetric matrices, iσ = (cid:18) − (cid:19) = ǫ αβ = ǫ ˙ α ˙ β , (2.22)and ( iσ ) − = − iσ = ǫ αβ = ǫ ˙ α ˙ β , (2.23)which define the epsilon symbols with undotted and dotted indices. Notethat the epsilon symbols with raised and lowered indices differ by an overallsign. Moreover, they can be used to raise and lower the spinor indices,8 α = ǫ αβ ξ β , ξ α = ǫ αβ ξ β , ξ † ˙ α = ǫ ˙ α ˙ β ξ † ˙ β , ξ † ˙ α = ǫ ˙ α ˙ β ξ † ˙ β . (2.24)The products of two epsilon symbols with undotted and with dottedindices, respectively, satisfy, ǫ αβ ǫ γδ = − δ γα δ δβ + δ δα δ γβ , (2.25) ǫ ˙ α ˙ β ǫ ˙ γ ˙ δ = − δ ˙ γ ˙ α δ ˙ δ ˙ β + δ ˙ δ ˙ α δ ˙ γ ˙ β , (2.26)where δ ˙ β ˙ α = δ βα and the two-index symmetric Kronecker delta symbol withundotted indices is defined by δ = δ = 1 and δ = δ = 0. In particular, ǫ αγ ǫ γβ = δ βα , ǫ ˙ α ˙ γ ǫ ˙ γ ˙ β = δ ˙ β ˙ α . (2.27)Finally, we introduce the σ -matrices: σ µα ˙ β = ( × ; ~σ ) , σ µ ˙ αβ = ( × ; − ~σ ) , (2.28)where × is the 2 × M † ) ˙ α ˙ β σ µ ˙ βγ M γδ = Λ µν σ ν ˙ αδ , ( M − ) αβ σ µβ ˙ γ [( M − ) † ] ˙ γ ˙ δ = Λ µν σ να ˙ δ . (2.29)Note that the matrix M and its inverse have the same spinor index structure(and likewise for the matrix M † and its inverse).We will sometimes find it useful to relate the σ µ and σ µ matrices usingthe identities σ µα ˙ α = ǫ αβ ǫ ˙ α ˙ β σ µ ˙ ββ , σ µ ˙ αα = ǫ αβ ǫ ˙ α ˙ β σ µβ ˙ β . (2.30)The significance of σ µ is that Lorentz 4-vectors can be built from spinorbilinears. For example, χ α ( x ) σ µα ˙ β ξ ˙ β ( x ) transforms as a Lorentz 4-vector, χ ′ α ( x ′ ) σ µα ˙ β ξ ′ † ˙ β ( x ′ ) = χ α ( x )[ M − σ µ ( M − ) † ] α ˙ β ξ † ˙ β ( x ) (2.31)= Λ µν χ ( x ) α σ να ˙ β ξ † ˙ β ( x ) , (2.32)after making use of eq. (2.29). Spinor indices can be suppressed by adoptinga summation convention where we contract indices as follows: αα and ˙ α ˙ α . (2.33)For example, ξη ≡ ξ α η α , (2.34) ξ † η † ≡ ξ † ˙ α η † ˙ α , (2.35) ξ † σ µ η ≡ ξ † ˙ α σ µ ˙ αβ η β , (2.36) ξσ µ η † ≡ ξ α σ µα ˙ β η † ˙ β . (2.37)9n particular, for anticommuting spinors, ηξ ≡ η α ξ α = − ξ α η α = + ξ α η α = ξη . (2.38) η † ξ † ≡ η † ˙ α ξ † ˙ α = − ξ † ˙ α η † ˙ α = ξ † ˙ α η † ˙ α = ξ † η † . (2.39)The behavior of spinor products under hermitian conjugation is note-worthy,( ξ Σ η ) † = η † Σ r ξ † , ( ξ Σ η † ) † = η Σ r ξ † , ( ξ † Σ η ) † = η † Σ r ξ , (2.40)where in each case Σ stands for any sequence of alternating σ and σ matri-ces, and Σ r is obtained by reversing the order of the σ and σ matrices thatappear in Σ.From the sigma matrices, one can construct the antisymmetrized prod-ucts, ( σ µν ) αβ ≡ i (cid:0) σ µα ˙ γ σ ν ˙ γβ − σ να ˙ γ σ µ ˙ γβ (cid:1) , (2.41)( σ µν ) ˙ α ˙ β ≡ i (cid:16) σ µ ˙ αγ σ νγ ˙ β − σ ν ˙ αγ σ µγ ˙ β (cid:17) . (2.42)With this notation, we may write the ( ,
0) and (0 , ) transformationmatrices, respectively, as M = exp (cid:0) − iθ µν σ µν (cid:1) , (2.43)( M − ) † = exp (cid:0) − iθ µν σ µν (cid:1) , (2.44)where the θ µν are defined below eq. (2.3).Consider a pure boost of an on-shell two-component spinor from its restframe to the frame where p µ = ( E p , ~p ), with E p = ( | ~p | + m ) / . In thiscase, setting θ ij = 0 (corresponding to no rotation), we obtain, M = exp (cid:16) − ~ζ · ~σ (cid:17) = r p · σm = ( E p + m ) × − ~σ · ~p p m ( E p + m ) , (2.45)( M − ) † = exp (cid:16) + ~ζ · ~σ (cid:17) = r p · σm = ( E p + m ) × + ~σ · ~p p m ( E p + m ) . (2.46)The matrix square roots, √ p · σ and √ p · σ , appearing in eqs. (2.45) and(2.46) are defined to be the unique non-negative definite hermitian matriceswhose squares are equal to the non-negative definite hermitian matrices p · σ and p · σ , respectively. Note that p · σ and p · σ are non-negative matrices due to the implicit mass-shell conditionsatisfied by p µ . Useful identities
The following identities can be used to systematically simplify expressionsinvolving products of σ and σ matrices, σ µα ˙ α σ ˙ ββµ = 2 δ βα δ ˙ β ˙ α , (2.47) σ µα ˙ α σ µβ ˙ β = 2 ǫ αβ ǫ ˙ α ˙ β , (2.48) σ µ ˙ αα σ ˙ ββµ = 2 ǫ αβ ǫ ˙ α ˙ β , (2.49)[ σ µ σ ν + σ ν σ µ ] αβ = 2 g µν δ βα , (2.50)[ σ µ σ ν + σ ν σ µ ] ˙ α ˙ β = 2 g µν δ ˙ α ˙ β , (2.51) σ µ σ ν σ ρ = g µν σ ρ − g µρ σ ν + g νρ σ µ + iǫ µνρκ σ κ , (2.52) σ µ σ ν σ ρ = g µν σ ρ − g µρ σ ν + g νρ σ µ − iǫ µνρκ σ κ , (2.53)where ǫ = − ǫ = +1 in our conventions. The traces of alternatingproducts of σ and σ matrices are given by,Tr[ σ µ σ ν ] = Tr[ σ µ σ ν ] = 2 g µν , (2.54)Tr[ σ µ σ ν σ ρ σ κ ] = 2 ( g µν g ρκ − g µρ g νκ + g µκ g νρ + iǫ µνρκ ) , (2.55)Tr[ σ µ σ ν σ ρ σ κ ] = 2 ( g µν g ρκ − g µρ g νκ + g µκ g νρ − iǫ µνρκ ) . (2.56)Traces involving an odd number of σ and σ matrices cannot arise, sincethere is no way to connect the spinor indices consistently. Additional iden-tities involving σ µν and σ µν can be found in Ref. [1].Finally, we examine some useful identities involving bilinear spinorquantities. Although the two-component spinor fields appearing in theselectures are anticommuting, one also may encounter commuting two-component spinor wave functions. Thus, it is convenient to denote anarbitrary two-component spinor by z i , and a sign factor, ( − A = +1[ − z z = − ( − A z z (2.57) z † z † = − ( − A z † z † (2.58) z σ µ z † = ( − A z † σ µ z (2.59) z σ µ σ ν z = − ( − A z σ ν σ µ z (2.60) z † σ µ σ ν z † = − ( − A z † σ ν σ µ z † (2.61) z † σ µ σ ρ σ ν z = ( − A z σ ν σ ρ σ µ z † . (2.62)In many cases, it is convenient to rewrite a product of two bilinear spinorquantities in terms of products in which the individual spinors appear in a11ifferent order. Below, we provide five different Fierz identities, which arevalid for both commuting and anticommuting spinors,( z z )( z z ) = − ( z z )( z z ) − ( z z )( z z ) , (2.63)( z † z † )( z † z † ) = − ( z † z † )( z † z † ) − ( z † z † )( z † z † ) , (2.64)( z σ µ z † )( z † σ µ z ) = − z z )( z † z † ) , (2.65)( z † σ µ z )( z † σ µ z ) = 2( z † z † )( z z ) , (2.66)( z σ µ z † )( z σ µ z † ) = 2( z z )( z † z † ) . (2.67)An exhaustive list of Fierz identities can be found in Appendix B of Ref. [1].2.1.5. Free field theories of two-component fermions
The ( ,
0) spinor field ξ α ( x ) describes a neutral Majorana fermion. Thefree-field Lagrangian is: L = iξ † σ µ ∂ µ ξ − m ( ξξ + ξ † ξ † ) , (2.68)which is hermitian up to a total divergence since we can rewrite the aboveLagrangian as L = iξ † σ µ ↔ ∂ µ ξ − m ( ξξ + ξ † ξ † ) + total divergence , (2.69)where ξ † σ µ ↔ ∂ µ ξ ≡ ξ † σ µ ( ∂ µ ξ ) − ( ∂ µ ξ ) † σ µ ξ .Generalizing to a multiplet of two-component fermion fields, ˆ ξ αi ( x ),labeled by flavor index i , the free Lagrangian is L = i ˆ ξ † i σ µ ∂ µ ˆ ξ i − M ij ˆ ξ i ˆ ξ j − M ij ˆ ξ † i ˆ ξ † j , (2.70)where hermiticity implies that M ij ≡ ( M ij ) ∗ is a complex symmetric ma-trix. To identify the physical fermion fields, we express the so-called inter-action eigenstate fields , ˆ ξ αi ( x ), in terms of mass-eigenstate fields ξ ( x ) = Ω − ˆ ξ ( x ) , (2.71)where Ω is unitary and chosen such thatΩ T M Ω = m = diag( m , m , . . . ) , (2.72)where the m i are non-negative real numbers. In linear algebra, this is calledthe Takagi diagonalization of a complex symmetric matrix M [51, 52]. Tocompute the values of the diagonal elements of m , we note thatΩ T M M † Ω ∗ = m . (2.73)Since M M † is hermitian, it can be diagonalized by a unitary matrix. Thus,the m i of the Takagi diagonalization are the non-negative square-roots ofthe eigenvalues of M M † . In terms of the mass eigenstate fields, L = iξ † i σ µ ∂ µ ξ i − m i ( ξ i ξ i + ξ † i ξ † i ) . (2.74) Subsequently, it was recognized in Refs. [53, 54] that the Takagi diagonalization wasfirst established for nonsingular complex symmetric matrices by Autonne [55]. xample 1 (The Seesaw Mechanism [56–60]). The seesaw Lagrangian for the two-component fermions ψ and ψ is L = i (cid:0) ψ † σ µ ∂ µ ψ + ψ † σ µ ∂ µ ψ (cid:1) − M ij ψ i ψ j − M ij ψ † i ψ † j , (2.75)where M ij = (cid:18) m D m D M (cid:19) , (2.76)and (without loss of generality) m D and M are real and positive. TheTakagi diagonalization of this matrix isΩ T M Ω = M D , (2.77)where Ω = (cid:18) i cos θ sin θ − i sin θ cos θ (cid:19) , M D = (cid:18) m − m + (cid:19) , (2.78)with m ± = (cid:20)q M + 4 m D ± M (cid:21) (2.79)and sin 2 θ = 2 m D p M + 4 m D . (2.80)If M ≫ m D , then the corresponding fermion masses are m − ≃ m D /M and m + ≃ M , with sin θ ≃ m D /M . The mass eigenstates, χ i are given by ψ i = Ω ij χ j ; to leading order in m d /M , iχ ≃ ψ − m D M ψ , χ ≃ ψ + m D M ψ . (2.81)Indeed, one can check that: m D ( ψ ψ + ψ ψ ) + M ψ ψ + h . c . ≃ (cid:20) m D M χ χ + M χ χ + h . c . (cid:21) , (2.82)which corresponds to a theory of two Majorana fermions—one very lightand one very heavy ( the seesaw ).In any theory containing a multiplet of fields, one can check for theexistence of global symmetries. The simplest case is a theory of a pair oftwo-component ( ,
0) fermion fields χ and η , with the free-field Lagrangian, L = iχ † σ µ ∂ µ χ + iη † σ µ ∂ µ η − m ( χη + χ † η † ) . (2.83)13he Lagrangian given in eq. (2.83) possesses a U(1) global symmetry, χ → e iθ χ and η → e − iθ η . That is, χ and η are oppositely charged. Thecorresponding mass matrix is M = (cid:18) mm (cid:19) . (2.84)Performing the Takagi diagonalization yields two degenerate two-component fermions of mass m . However, the corresponding mass-eigenstates are not eigenstates of charge. Together, χ and η † constitute asingle (four-component) Dirac fermion .More generally, consider a collection of charged Dirac fermions repre-sented by pairs of two-component interaction eigenstate fields ˆ χ αi ( x ), ˆ η iα ( x ),with L = i ˆ χ † i σ µ ∂ µ ˆ χ i + i ˆ η † i σ µ ∂ µ ˆ η i − M ij ˆ χ i ˆ η j − M ij ˆ χ † i ˆ η † j , (2.85)where M is a complex matrix with matrix elements denoted by M ij (notethe placement of the flavor indices i and j ), and M ij ≡ ( M ij ) ∗ .We denote the mass eigenstate fields by χ i and η i and the unitarymatrices L and R , such that ˆ χ i = L ik χ k and ˆ η i = R ik η k , and L T M R = m = diag( m , m , . . . ) , (2.86)where the m i are non-negative real numbers. This is the singular valuedecomposition of a complex matrix (see, e.g., Refs. [53, 54]). Noting that R † ( M † M ) R = m , (2.87)the diagonal elements of m are the non-negative square roots of the corre-sponding eigenvalues of M † M . In terms of the mass eigenstate fields, L = iχ † i σ µ ∂ µ χ i + iη † i σ µ ∂ µ η i − m i ( χ i η i + χ † i η † i ) . (2.88)2.1.6. Fermion–scalar interactions
The most general set of interactions with the scalars of the theory ˆ φ I arethen given by: L int = − ˆ Y Ijk ˆ φ I ˆ ψ j ˆ ψ k − ˆ Y Ijk ˆ φ I ˆ ψ † j ˆ ψ † k , (2.89)where ˆ Y Ijk ≡ ( ˆ Y Ijk ) ∗ and ˆ φ I ≡ ( ˆ φ I ) ∗ . The flavor index I runs over acollection of real scalar fields ˆ ϕ i and pairs of complex scalar fields ˆΦ j and This is the analog of a free field theory of a complex scalar boson Φ with a mass term, L mass = − m | Φ | . Writing Φ = ( φ + iφ ) / √
2, we can write Lagrangian in terms of φ and φ with a diagonal mass term. But, φ and φ do not correspond to states ofdefinite charge. j ≡ ( ˆΦ j ) ∗ (where a complex field and its conjugate are counted sepa-rately). The Yukawa couplings ˆ Y Ijk are symmetric under interchange of j and k .The mass-eigenstate basis ψ is related to the interaction-eigenstate basisˆ ψ by a unitary transformation,ˆ ψ ≡ ˆ ξ ˆ χ ˆ η = U ψ ≡ Ω 0 00 L
00 0 R ξχη , (2.90)where Ω, L , and R are constructed as described previously. Likewise a uni-tary transformation yields the scalar mass-eigenstates via ˆ φ = V φ . Thus,in terms of mass-eigenstate fields: L int = − Y Ijk φ I ψ j ψ k − Y Ijk φ I ψ † j ψ † k , (2.91)where Y Ijk = V J I U mj U nk ˆ Y Jmn .2.1.7.
Fermion–gauge boson interactions
In the gauge-interaction basis for the two-component fermions the corre-sponding interaction Lagrangian is given by L int = − g a A µa ˆ ψ † i σ µ ( T a ) ij ˆ ψ j , (2.92)where the index a labels the (real or complex) vector bosons A µa and issummed over. If the gauge symmetry is unbroken, then the index a runsover the adjoint representation of the gauge group, and the ( T a ) ij are her-mitian representation matrices of the gauge group acting on the fermions.There is a separate coupling g a for each simple group or U(1) factor of thegauge group G.In the case of spontaneously broken gauge theories, one must diagonalizethe vector boson squared-mass matrix. The form of eq. (2.92) still applieswhere A aµ are gauge boson fields of definite mass, although in this casefor a fixed value of a , the product g a T a is some linear combination ofthe original g a T a of the unbroken theory. That is, the hermitian matrixgauge field ( A µ ) ij ≡ A aµ ( T a ) ij appearing in eq. (2.92) can always be re-expressed in terms of the physical mass eigenstate gauge boson fields. If an For a U (1) gauge group, the T a are replaced by real numbers corresponding to the U(1)charges of the ( ,
0) fermions. In terms of mass-eigenstate fermion fields, L int = − A µa ψ † i σ µ ( G a ) ij ψ j , (2.93)where G a = g a U † T a U (no sum over a ).The case of gauge interactions of charged Dirac fermions can be treatedas follows. Consider pairs of ( ,
0) interaction-eigenstate fermions ˆ χ i andˆ η i that transform as conjugate representations of the gauge group (hencethe difference in the flavor index heights). The Lagrangian for the gaugeinteractions of Dirac fermions can be written in the form: L int = − g a A µa ˆ χ † i σ µ ( T a ) ij ˆ χ j + g a A µa ˆ η † i σ µ ( T a ) ji ˆ η j , (2.94)where the A aµ are gauge boson mass-eigenstate fields. Here we have usedthe fact that if ( T a ) ij are the representation matrices for the ˆ χ i , then the ˆ η i transform in the complex conjugate representation with generator matrices − ( T a ) ∗ = − ( T a ) T . In terms of mass-eigenstate fermion fields, L int = − A µa h χ † i σ µ ( G aL ) ij χ j − η † i σ µ ( G aR ) ji η j i , (2.95)where G aL = g a L † T a L and G aR = g a R † T a R (no sum over a ). Correspondence between the two-component and four-component spinor notations
Most pedagogical treatments of calculations in particle physics employ four-component Dirac spinor notation, which combines distinct irreducible rep-resentations of the Lorentz symmetry algebra. Parity-conserving theoriessuch as QED and QCD and their Feynman rules are especially well-suited tofour-component spinor notation. In light of the widespread familiarity withfour-component spinor techniques, we provide in this section a translationbetween two-component and four-component spinor notation. In terms of the physical gauge boson fields, A aµ T a consists of a sum over real neutralgauge fields multiplied by hermitian generators, and complex charged gauge fields mul-tiplied by non-hermitian generators. For example, in the electroweak Standard Model,G = SU(2) × U(1) with gauge bosons and generators W aµ and T a = τ a for SU(2), and B µ and Y for U(1), where the τ a are the usual Pauli matrices. After diagonalizing thegauge boson squared-mass matrix, gW aµ T a + g ′ B µ Y = g √ W + µ T + + W − µ T − ) + g cos θ W (cid:0) T − Q sin θ W (cid:1) Z µ + e Q A µ , where Q = T + Y is the generator of the unbroken U(1) EM , T ± ≡ T ± i T , and e = g sin θ W = g ′ cos θ W . The massive gauge boson charge-eigenstate fields of thebroken theory consist of a charged massive gauge boson pair, W ± ≡ ( W ∓ iW ) / √ Z ≡ W cos θ W − B sin θ W , and the massless photon, A ≡ W sin θ W + B cos θ W . From two-component to four-component spinor notation
The correspondence between the two-component and four-componentspinor language is most easily exhibited in the basis in which γ is diagonal(this is called the chiral representation). Employing 2 × γ µ = σ µα ˙ β σ µ ˙ αβ ! , γ ≡ iγ γ γ γ = (cid:18) − δ αβ δ ˙ α ˙ β (cid:19) . (2.96)The chiral projections operators are P L ≡ (1 − γ ) , (2.97) P R ≡ (1 + γ ) . (2.98)In addition, we identify the generators of the Lorentz group in the reducible( , ⊕ (0 , ) representation Σ µν ≡ i γ µ , γ ν ] = (cid:18) σ µν αβ σ µν ˙ α ˙ β (cid:19) , (2.99)where Σ µν satisfies the duality relation, γ Σ µν = iǫ µνρτ Σ ρτ .A four-component Dirac spinor field, Ψ( x ), is made up of two mass-degenerate two-component spinor fields, χ α ( x ) and η α ( x ) as follows:Ψ( x ) ≡ χ α ( x ) η † ˙ α ( x ) ! . (2.100)Note that P L and P R project out the upper and lower components, respec-tively. The Dirac conjugate field Ψ and the charge conjugate field Ψ c aredefined by Ψ( x ) ≡ Ψ † A = (cid:0) η α ( x ) , χ † ˙ α ( x ) (cid:1) , (2.101)Ψ c ( x ) ≡ C Ψ T ( x ) = η α ( x ) χ † ˙ α ( x ) ! , (2.102)where the Dirac conjugation matrix A and the charge conjugation matrix C satisfy Aγ µ A − = γ µ † , C − γ µ C = − γ µ T . (2.103)It is conventional to impose two additional conditions:Ψ = A − Ψ † , (Ψ c ) c = Ψ . (2.104) In most textbooks, Σ µν is called σ µν . Here, we use the former symbol so that there isno confusion with the two-component definition of σ µν . A † = A , C T = − C , ( AC ) − = ( AC ) ∗ . (2.105)In the chiral representation, A and C are explicitly given by A = (cid:18) δ ˙ α ˙ β δ αβ (cid:19) , C = ǫ αβ ǫ ˙ α ˙ β ! . (2.106)Note the numerical equalities, A = γ and C = iγ γ , although theseidentifications do not respect the structure of the undotted and dottedindices specified above.Finally, we note the following results, which are easily derived: A Γ A − = η A Γ Γ † , η A Γ = ( +1 , for Γ = , γ µ , γ µ γ , Σ µν , − , for Γ = γ , Σ µν γ , (2.107) C − Γ C = η C Γ Γ T , η C Γ = ( +1 , for Γ = , γ , γ µ γ , − , for Γ = γ µ , Σ µν , Σ µν γ . (2.108)2.2.2. Four-component spinor bilinear covariants
The Dirac bilinear covariants are quantities that are quadratic in the Diracspinor fields and transform irreducibly as Lorentz tensors. These may beconstructed from the corresponding quantities that are quadratic in thetwo-component spinors. To construct a translation table between the two-component spinor and four-component spinor forms of the bilinear covari-ants, we first define two Dirac spinor fields,Ψ ( x ) ≡ χ ( x ) η † ( x ) ! , Ψ ( x ) ≡ χ ( x ) η † ( x ) ! , (2.109)where spinor indices have been suppressed. It follows that,Ψ Ψ = η χ + χ † η † , (2.110)Ψ γ Ψ = − η χ + χ † η † , (2.111)Ψ γ µ Ψ = χ † σ µ χ + η σ µ η † , (2.112)Ψ γ µ γ Ψ = − χ † σ µ χ + η σ µ η † , (2.113)Ψ Σ µν Ψ = 2( η σ µν χ + χ † σ µν η † ) , (2.114)Ψ Σ µν γ Ψ = − η σ µν χ − χ † σ µν η † ) . (2.115)The above results can be used to to obtain the translations given in Table 1.18 able 1.: Relating the Dirac bilinear covariants written in terms of four-component Dirac spinor fields to the corresponding quantities expressed in termsof two-component spinor fields using the notation of eq. (2.100). These resultsapply to both commuting and anticommuting spinors. In the latter case, one mayalternatively write Ψ γ µ P R Ψ = − η † σ µ η , etc. [cf. eq. (2.59)]. Ψ P L Ψ = η χ Ψ c P L Ψ c = χ η Ψ P R Ψ = χ † η † Ψ c P R Ψ c = η † χ † Ψ c P L Ψ = χ χ Ψ P L Ψ c = η η Ψ P R Ψ c = χ † χ † Ψ c P R Ψ = η † η † Ψ γ µ P L Ψ = χ † σ µ χ Ψ c γ µ P L Ψ c = η † σ µ η Ψ c γ µ P R Ψ c = χ σ µ χ † Ψ γ µ P R Ψ = η σ µ η † Ψ Σ µν P L Ψ = 2 η σ µν χ Ψ c Σ µν P L Ψ c = 2 χ σ µν η Ψ Σ µν P R Ψ = 2 χ † σ µν η † Ψ c Σ µν P R Ψ c = 2 η † σ µν χ † When Ψ = Ψ , the bilinear covariants listed in eqs. (2.110)–(2.115) areeither hermitian or anti-hermitian. Using eq. (2.107), it follows that ΨΓΨis hermitian for Γ = × , iγ , γ µ , γ µ γ , Σ µν , and i Σ µν γ .One can also define Majorana bilinear covariants. A four-componentMajorana fermion field is defined by the condition,Ψ M ( x ) = Ψ cM ( x ) = C Ψ TM ( x ) = ξ α ( x ) ξ ˙ α † ( x ) ! . (2.116)Eqs. (2.110)–(2.115) and the results of Table 1 may also be applied to four-component Majorana spinors, Ψ M and Ψ M , by setting ξ ≡ χ = η , and ξ ≡ χ = η , respectively. This implements the Majorana condition givenin eq. (2.116) and imposes additional restrictions on the Majorana bilinearcovariants. In particular, the anticommuting Majorana four-componentfermion fields satisfy the following additional identities,Ψ M Ψ M = Ψ M Ψ M , (2.117)Ψ M γ Ψ M = Ψ M γ Ψ M , (2.118)Ψ M γ µ Ψ M = − Ψ M γ µ Ψ M , (2.119)Ψ M γ µ γ Ψ M = Ψ M γ µ γ Ψ M , (2.120)Ψ M Σ µν Ψ M = − Ψ M Σ µν Ψ M , (2.121)Ψ M Σ µν γ Ψ M = − Ψ M Σ µν γ Ψ M . (2.122)19f Ψ M = Ψ M ≡ Ψ M , then eqs. (2.117)–(2.122) yieldΨ M γ µ Ψ M = Ψ M Σ µν Ψ M = Ψ M Σ µν γ Ψ M = 0 . (2.123)One additional useful result for Majorana fermion fields is:Ψ M γ µ P L Ψ M = − Ψ M γ µ P R Ψ M . (2.124) Feynman Rules for Dirac and Majorana fermions
The application of four-component fermion techniques in parity-violatingtheories is straightforward for processes involving Dirac fermions. However,the inclusion of Majorana fermions involves some subtleties that requireelucidation. In light of the widespread familiarity with four-componentspinor techniques, we shall develop four-component fermion Feynman rulesthat treat Dirac and Majorana fermions on equal footing [1, 61–63]. Consider first the Feynman rule for the four-component fermion propa-gator. Virtual Dirac fermion lines can either correspond to Ψ or Ψ c . Here,there is no ambiguity in the propagator Feynman rule, since for free Diracfermion fields, h | T [Ψ( x )Ψ( y )] | i = h | T [Ψ c ( x )Ψ c ( y )] | i , (2.125)so that the Feynman rules for the propagator of a Ψ and Ψ c line, exhib-ited below, are identical. The same rule also applies to a four-componentMajorana fermion Ψ M . p i (/ p + m ) p − m + iǫ Consider next a set of neutral Majorana fermions Ψ Mi and chargedDirac fermions Ψ i , Ψ Mi = ξ i ξ † i ! , Ψ i = χ i η † i ! , (2.126)interacting with a neutral scalar φ or vector boson A µ . The interactionLagrangian in terms of two-component fermions is L int = − ( λ ij ξ i ξ j + λ ij ξ † i ξ † j ) φ − ( κ ij χ i η j + κ ij χ † i η † j ) φ − G ij ξ † i σ µ ξ j A µ − [( G L ) ij χ † i σ µ χ j + ( G R ) ij η † i σ µ η j ] A µ , (2.127)where λ is a complex symmetric matrix with λ ij ≡ λ ∗ ij , κ is an arbitrarycomplex matrix with κ ij ≡ ( κ ij ) ∗ , and G , G L and G R are hermitian ma-trices. Converting to four-component spinor notation (see Problem 1), theresulting Feynman rules are shown below. For a comprehensive set of two-component fermion Feynman rules, see Ref. [1]. Ψ Mj Ψ Mi − i ( λ ij P L + λ ij P R ) A µ Ψ Mj Ψ Mi − iγ µ [ G ij P L − G j i P R ] φ Ψ j Ψ i or φ Ψ cj Ψ ci − i ( κ ij P L + κ ji P R ) A µ Ψ j Ψ i − iγ µ [( G L ) ij P L + ( G R ) ij P R ]or or A µ Ψ cj Ψ ci iγ µ [( G L ) ij P L + ( G R ) ij P R ]21he arrows on the Dirac fermion lines depict the flow of the conservedcharge. A Majorana fermion is self-conjugate, so its arrow simply reflectsthe structure of L int ; i.e. , Ψ M [Ψ M ] is represented by an arrow pointingout of [into] the vertex. The arrow directions determine the placement ofthe u and v spinors in an invariant amplitude.For vertices involving Dirac fermions, one has a choice of either using theDirac field or its charge conjugated field. The Feynman rules correspondingto these two choices are related, due to the following identity,Ψ ci ΓΨ cj = − Ψ Ti C − Γ C Ψ Tj = Ψ j C Γ T C − Ψ i = η C Γ Ψ j ΓΨ i , (2.128)where we have used eq. (2.108). Note that the extra minus sign that arises inthe penultimate step above is due to the anticommutativity of the fermionfields.Next, consider the interaction of fermions with charged bosons Φ and W (assumed to have charge equal to that of χ and η † ). The correspondinginteraction Lagrangian is given by: L int = − Φ[( κ ) ij ξ i η j + ( κ ) ij ξ † i χ † j ] − Φ † [( κ ) ij ξ i χ j + ( κ ) ij ξ † ii η † j ] − W µ [( G ) ji χ † j σ µ ξ i − ( G ) ij ξ † i σ µ η j ] − W † µ [( G ) j i ξ † i σ µ χ j − ( G ) ij η † j σ µ ξ i ] , (2.129)where κ , κ , G and G are complex matrices. Converting to four-component spinor notation, the corresponding Feynman rules are: Φ Ψ Mi Ψ j or Φ Ψ Mi Ψ cj − i ( κ ij P L + κ ij P R ) Φ Ψ Mi Ψ j or Φ Ψ Mi Ψ cj − i ( κ ij P L + κ ij P R )22 Ψ Mj Ψ i − iγ µ ( G ij P L − G ji P R ) W Ψ Mj Ψ ci iγ µ ( G ji P R − G ij P L )or or W Ψ Mj Ψ i − iγ µ ( G ij P L − G ji P R )or or W Ψ Mj Ψ ci iγ µ ( G ji P R − G ij P L )When the interaction Lagrangians given in eqs. (2.127) and (2.129) areconverted to four-component spinor notation (see Problems 1 and 2 at theend of this section), there is an equivalent form in which L int is writ-ten in terms of charge-conjugated Dirac four-component fields [after usingeq. (2.128)]. Thus, the Feynman rules involving Dirac fermions can taketwo possible forms, as shown above. As previously noted, the directionof an arrow on a Dirac fermion line indicates the direction of the fermion23harge flow (whereas the arrow on the Majorana fermion line is uncon-nected to charge flow). However, we are free to choose either a Ψ or Ψ c lineto represent a Dirac fermion at any place in a given Feynman graph. Forany decay or scattering process, a suitable choice of either the Ψ-rule orthe Ψ c -rule at each vertex (the choice can be different at different vertices)will guarantee that the arrow directions on fermion lines flow continuouslythrough the Feynman diagram. Then, to evaluate an invariant amplitude,one should traverse any continuous fermion line (either Ψ or Ψ c ) by movingantiparallel to the direction of the fermion arrows.For a given process, there may be a number of distinct choices forthe arrow directions on the Majorana fermion lines, which may dependon whether one represents a given Dirac fermion by Ψ or Ψ c . However,different choices do not lead to independent Feynman diagrams. Whencomputing an invariant amplitude, one first writes down the relevant Feyn-man diagrams with no arrows on any Majorana fermion line. The numberof distinct graphs contributing to the process is then determined. Finally,one makes some choice for how to distribute the arrows on the Majoranafermion lines and how to label Dirac fermion lines (either as the field Ψ orits charge conjugate Ψ c ) in a manner consistent with the Feynman rules forthe vertices previously given. The end result for the invariant amplitude(apart from an overall unobservable phase) does not depend on the choicesmade for the direction of the fermion arrows.Using the above procedure, the Feynman rules for the external fermionwave functions are the same for Dirac and Majorana fermions: • u ( ~p , s ): incoming Ψ [or Ψ c ] with momentum ~p parallel to the arrowdirection, • ¯ u ( ~p , s ): outgoing Ψ [or Ψ c ] with momentum ~p parallel to the arrowdirection, • v ( ~p , s ): outgoing Ψ [or Ψ c ] with momentum ~p anti-parallel to the arrowdirection, • ¯ v ( ~p , s ): incoming Ψ [or Ψ c ] with momentum ~p anti-parallel to the arrowdirection.We now consider the application of the Feynman rules presented aboveto some 2 → Since the charge of Ψ c is opposite in sign to the charge of Ψ, the corresponding arrowdirections of the Ψ and Ψ c lines must point in opposite directions. xample 2 (Ψ( p )Ψ( p ) → Φ( k )Φ( k ) via Ψ M -exchange). Here,Φ is a charged scalar. The contributing Feynman graphs are:Ψ M ΨΨ c Ψ M ΨΨ c Following the arrows on the fermion lines in reverse, the invariant amplitudeis given by, i M = ( − i ) ¯ v ( ~p , s )( κ P L + κ ∗ P R ) (cid:20) i ( / p − / k + m ) t − m + i ( / k − / p + m ) u − m (cid:21) × ( κ P L + κ ∗ P R ) u ( ~p , s ) , (2.130)where t ≡ ( p − k ) , u ≡ ( p − k ) and m is the Majorana fermionmass. The sign of each diagram is determined by the relative permutationof spinor wave functions appearing in the amplitude (the overall sign of theamplitude is unphysical). In the present example, in both terms appearingin eq. (2.130), the spinor wave functions appear in the same order (first ~p and then ~p ), implying a relative plus sign between the two terms.One can check that i M is antisymmetric under interchange of the twoinitial electrons. This is most easily verified by taking the transpose ofthe invariant amplitude (the latter is a complex number whose value isnot changed by transposition). It is convenient to adopt the convention inwhich the (commuting) u and v spinor wave functions are related via, v ( ~p , s ) = C ¯ u ( ~p , s ) T , u ( ~p , s ) = C ¯ v ( ~p , s ) T , (2.131)¯ v ( ~p , s ) = − u ( ~p , s ) T C − , ¯ u ( ~p , s ) = − v ( ~p , s ) T C − . (2.132)where C is the charge conjugation matrix. Using eqs. (2.131) and (2.132),the transposed amplitude can be simplified by employing the relation,¯ v ( ~p , s )Γ u ( ~p , s ) = − η C Γ ¯ v ( ~p , s )Γ u ( ~p , s ) , (2.133)which is a consequence of eq. (2.108). Example 3 (Ψ( p )Ψ c ( p ) → Ψ M ( p )Ψ M ( p ) via charged Φ-exchange). In addition to a possible s -channel annihilation graph, the contributingFeynman graphs can be represented by either diagram set (i) or diagramset (ii) shown below, where each set contains a t -channel and u -channelgraph, respectively. 25iagram set (i): Ψ M Ψ M ΨΨ c Ψ M Ψ M ΨΨ c Diagram set (ii): Ψ M Ψ M ΨΨ Ψ M Ψ M ΨΨThe amplitude is evaluated by following the arrows on the fermion linesin reverse. Either diagram set (i) or set (ii) may be chosen to evaluate theinvariant amplitude. We again employ eq. (2.108) to derive the relation,¯ v ( ~p , s )Γ v ( ~p , s ) = − η C Γ ¯ u ( ~p , s )Γ u ( ~p , s ) , (2.134)which can be used in comparing the invariant amplitude obtained by usingdiagram sets (i) and (ii). One can check that the invariant amplitudesresulting from diagram sets (i) and (ii) differ by an overall minus sign,which is unphysical. The overall minus sign arises due to the fact thatthe corresponding order of the spinor wave functions differs by an oddpermutation [e.g., for the t -channel graphs, compare 3142 and 3124 for (i)and (ii) respectively]. For the same reason, there is a relative minus signbetween the t -channel and u -channel graphs for either diagram set [e.g.,compare 3142 and 4132 in diagram set(i)].If s -channel annihilation contributes, its contribution to the invariantamplitude is easily obtained. Relative to the t -channel graph of diagramset (ii) above, the s -channel graph shown below comes with an extra minussign (since 2134 is odd with respect to 3124).26 M Ψ M ΨΨIn the computation of the unpolarized cross-section, non-standard spinprojection operators can arise in the evaluation of the interference terms(see Appendix D of Reference [32]), such as X s u ( ~p , s ) v T ( ~p , s ) = (/ p + m ) C T , X s ¯ u T ( ~p , s )¯ v ( ~p , s ) = C − (/ p − m ) , which requires additional manipulation of the charge conjugation matrix C .However, these non-standard spin projection operators can be avoided byjudicious use of spinor wave function product relations of the kind obtainedin eqs. (2.133) and (2.134). Problems
Problem 1.
Convert the interaction Lagrangian given by eq. (2.127) tofour-component spinor notation. Show that the end result is L int = − ( λ ij Ψ Mi P L Ψ Mj + λ ij Ψ iM P R Ψ jM ) φ − Ψ j ( κ ij P L + κ ij P R )Ψ i φ − Ψ Mi γ µ (cid:2) ( G a ) ij P L − ( G a ) ji P R (cid:3) Ψ Mj − (cid:2) ( G aL ) ij Ψ i γ µ P L Ψ j + ( G aR ) ij Ψ i γ µ P R Ψ j (cid:3) A aµ , (2.135) where the Ψ Mj are a set of (neutral) Majorana four-component fermionsand the Ψ j are a set of Dirac four-component fermions. Problem 2.
Convert the interaction Lagrangian given by eq. (2.129) tofour-component spinor notation. Show that the end result is L int = − (cid:2) ( κ ) ij Ψ j P L Ψ Mi + ( κ ) ij Ψ j P R Ψ iM (cid:3) Φ − (cid:2) ( G ) ji Ψ j γ µ P L Ψ Mi + ( G ) ij Ψ j γ µ P R Ψ iM (cid:3) W µ + h . c . (2.136) Problem 3.
Derive eq. (2.133). Then, verify that the invariant amplitudegiven by eq. (2.130) is antisymmetric under the interchange of the twoinitial electrons.
Problem 4.
Derive eq. (2.134). Then, verify that the invariant amplitudefor the scattering process considered in Example 3 obtained from diagramsets (i) and (ii), respectively, differ by an overall minus sign. . Motivation for TeV-scale supersymmetry The Standard Model (SM) of particle physics has been remarkably success-ful for describing the observed behavior of the fundamental particles andtheir interactions [64]. Indeed, there are no definitive departures from theStandard Model observed in experiments conducted at high energy colliderfacilities. Nevertheless, some fundamental microscopic phenomena mustnecessarily lie outside of the purview of the SM. These include: neutrinoswith non-zero mass [65]; dark matter [66]; the suppression of CP-violationin the strong interactions (the so-called strong CP problem [67]); gauge cou-pling unification [68]; the baryon asymmetry of the universe [69]; inflationin the early universe [70]; dark energy [71]; and the gravitational interac-tion. None of these phenomena can be explained within the framework ofthe SM alone.Consequently, the SM should be regarded at best as a low-energy effec-tive field theory [72], which is valid below some high energy scale. That is,new high energy scales must exist where more fundamental physics resides.In this section, we explain why one might expect to find this new physics atthe TeV scale. We discuss the principle of naturalness , and how supersym-metry provides a natural mechanism for avoiding the quadratic sensitivityof the squared-masses of elementary scalar particles to ultraviolet physics.
Why the TeV scale?
The classical gravitational interaction lies outside the SM. Using the fun-damental constants, ~ , c and Newton’s gravitational constant G N , one canconstruct a quantity with the units of energy called the Planck scale, M PL c ≡ (cid:18) ~ c G N (cid:19) / ≃ . × GeV . (3.1)The significance of the Planck scale can be seen as follows. At the Planckenergy scale, the quantum mechanical aspects of gravity can no longer beneglected. The gravitational energy of a particle of mass m , evaluated atits Compton wavelength, r c = ~ / ( mc ),Φ ∼ G N m r c = G N m c ~ < ∼ mc , (3.2)must be below 2 mc to avoid particle-antiparticle pair creation by the grav-itational field. Hence, up to O (1) constants, we conclude that m < ∼ M PL . Note that for m = M PL , the Schwarzschild radius r s ≡ G N m/c ≃ r c , which providesadditional evidence that the quantum mechanical nature of gravity cannot be neglectedat energy scales above the Planck scale. GeV. Henceforth, we shalldefine Λ to be the lowest energy scale at which the SM breaks down.The predictions made by the SM depend on a number of parametersthat must be taken as input to the theory. These parameters cannot be pre-dicted, since their values are sensitive to unknown ultraviolet (UV) physics.In the 1930s, it was already appreciated that a critical difference exists be-tween the behavior of boson and fermion masses [73]. Fermion masses arelogarithmically sensitive to UV physics [74] due to the chiral symmetry ofmassless fermions, which implies that the radiative correction to the tree-level fermion mass is of the form, δm F ∼ m F ln(Λ /m F ) , (3.3)which vanishes in the limit of m F →
0. In contrast, no such symmetry existsfor bosons (in the absence of supersymmetry), and consequently we expectquadratic sensitivity of the boson squared-mass to UV physics, δm B ∼ Λ . These observations have important consequences for the fundamentalphysics that describes the Higgs boson. In the SM, the Higgs boson squared-mass is given by m h = λv and the W boson squared-mass is m W = g v ,where h Φ i = v/ √ λ is the Higgs self-coupling [cf. eq. (3.5)], and g is theSU(2) gauge coupling. Together, these imply that m h m W = 4 λg , (3.4)which one would expect to be roughly of O (1). The Higgs boson with mass125 GeV satisfies this expectation.However, the existence of the Higgs boson is a consequence of a spon-taneously broken scalar potential, V (Φ) = − µ (Φ † Φ) + λ (Φ † Φ) , (3.5)where µ = λv at the minimum of the scalar potential. The parameter µ is quadratically sensitive to Λ. Hence, to obtain v = 246 GeV in a29heory where v ≪ Λ requires a significant fine-tuning of the ultravioletparameters of the fundamental theory. Indeed, the one-loop contributionsto the squared mass parameter µ are expected to be of order ( g / π )Λ .Setting this quantity to be of order of v (to avoid an unnatural cancellationbetween the tree-level parameter and the loop corrections) yieldsΛ ≃ πv/g ∼ O (1 TeV) . (3.6)Thus, a natural theory of electroweak symmetry breaking (EWSB) appearsto require new TeV scale physics beyond the SM associated with the EWSBdynamics. The modern principle of naturalness
This principle of naturalness was first introduced by Weisskopf in a paperpublished in 1939 [73]. In the abstract of this 1939 paper, Weisskopf wrote,“the self-energy of charged particles obeying Bose statistics is found to bequadratically divergent...,” and concluded that in theories of elementarybosons, new phenomena must enter at an energy scale of m/e (where e is the relevant coupling). In modern particle physics, naturalness is oftenassociated with the question, “how do we understand the magnitude of theEWSB scale?” In the absence of new physics beyond the SM, its naturalvalue would be the Planck scale (or perhaps the grand unification scale orthe seesaw scale that controls neutrino masses).There have been a number of theoretical proposals to explain the ori-gin of the EWSB energy scale: (1) naturalness is restored by a symmetryprinciple–supersymmetry (SUSY)–which ties the bosons to the more well-behaved fermions [75, 76]; (2) the Higgs boson is an approximate Gold-stone boson, the only other known mechanism for keeping an elementaryscalar light [77]; (3) the Higgs boson is a composite scalar, with an inverselength of order the TeV-scale [77]; (4) extra spatial dimensions beyond threeprovide new mechanisms for naturally large hierarchies of scales [78, 79];(5) classical scale invariance and its minimal violation via quantum anoma-lies [80–85] can generate a Higgs mass via dimensional transmutation [86];and (6) the EWSB scale arises due to some vacuum selection mechanism(either anthropic [87] or cosmological [88, 89]). Finally, maybe none of theseexplanations are relevant, and the EWSB energy scale is simply the resultof some initial condition whose origin will never be discernible.Of course, these are lectures on supersymmetry. Thus, we shall motivateSUSY at the TeV scale as a potential solution of the so-called hierarchyproblem: why is the scale of EWSB so much smaller than the Planck scale?30 .3. Avoiding quadratic UV-sensitivity with elementary scalars
First, consider a lesson from history. The electron self-energy in classi-cal electromagnetism goes like e /a , where a is the classical radius of theelectron. For a point-like electron, a →
0; hence the electron self-energy di-verges linearly. In the quantum theory, fluctuations of the electromagneticfields (in the “single electron theory”) generate a quadratic divergence. Ifthese divergences are not canceled, one would expect QED to break downat an energy of order m e /e , far below the Planck scale.The linear and quadratic divergences will cancel exactly if one makesa bold hypothesis: the existence of the positron (with a mass equal tothat of the electron but of opposite charge). Weisskopf was the first todemonstrate this cancellation in 1934 [74]. This is an historical examplein which a symmetry implies the existence of a partner particle that cancelsthe dangerously large UV contribution to the particle mass.The motivation for SUSY may be viewed analogously [90, 91], with theelectron playing the role of SM particles and the positron playing the roleof superpartners. SUSY associates a fermionic superpartner with every SMparticle and vice versa, thus doubling the SM spectrum. SUSY relates theself-energy of the elementary scalar boson to the self-energy of its fermionicpartner. Since the latter is only logarithmically sensitive to Λ, we concludethat the quadratic sensitivity of the scalar squared-mass to UV physicsmust exactly cancel. Naturalness is restored!However, since no superpartners degenerate in mass with the corre-sponding SM particles exist in nature, SUSY must be a broken symmetry.Although the fundamental origin of SUSY-breaking is yet to be understood,the effective scale of SUSY-breaking cannot be much larger than of order afew TeV, if SUSY is responsible for the origin of the EWSB scale.The absence of any evidence for SUSY at the LHC [92] is a cause forsome concern [48]. This has led to some discussion of the so-called littlehierarchy problem [93–95] which reflects the observation that the effectiveSUSY-breaking mass scale is somewhat separated from the scale of EWSB.Nevertheless, if evidence for supersymmetric phenomena in the TeV ormulti-TeV regime were to be eventually established at the LHC or at afuture collider facility (with an energy reach beyond the LHC [96]), it wouldbe viewed as a spectacularly successful explanation of the large hierarchybetween the (multi-)TeV scale and Planck scale. In this case, the remaininglittle hierarchy would perhaps be regarded as a less pressing issue. Actually the cancellation was not present in the initial publication, but thanks to aletter from Wendell Furry, the correct result was published in an erratum. . Supersymmetry: first steps The supersymmetry algebra is a generalization of the Lie algebra of thePoincar´e group of spacetime symmetries. In this section we begin by re-viewing the representations of the Poincar´e group. We then present the su-persymmetry algebra and examine its representations. The consequences ofsuper-Poincar´e invariance, in terms of the vacuum energy and the bosonicand fermionic degrees of freedom, are discussed. Finally, we exhibit howthese properties are manifested in the simplest supersymmetric field theoryof spin-0 and spin- particles (the so-called Wess-Zumino model [97]), anddemonstrate how the SUSY algebra is realized. Review of the Poincar´e algebra
The Poincar´e group consists of Lorentz transformations and spacetimetranslations [50]. That is, under a Poincar´e transformation, the spacetimecoordinates transform as x ′ µ = Λ µν x ν + a µ , where Λ is given by eq. (2.3)and a µ is a constant four-vector. Under a Lorentz transformation Λ and aspacetime translation a , the field ψ α of spin s transforms as, ψ ′ α ( x ) = exp (cid:0) − iθ µν S µν (cid:1) αβ ψ β (cid:0) Λ − ( x − a ) (cid:1) , (4.1)where we have used x = Λ − ( x ′ − a ) and redefined the dummy variable x ′ by removing the prime. The Poincar´e algebra is obtained by considering aninfinitesimal Poincar´e transformation. Expanding in a Taylor series aboutΛ = × and a = 0, we may rewrite eq. (4.1) as ψ ′ α ( x ) ≃ (cid:2) + ia µ P µ − i θ µν ( L µν + S µν ) (cid:3) αβ ψ β ( x ) , (4.2)where is the unit operator, P µ ≡ i∂ µ and L µν ≡ i ( x µ ∂ ν − x ν ∂ µ ) are thelinear and angular momentum operators, respectively, and S µν depends onthe representation; for spin- two-component fermions, S µν = ( σ µν for ( ,
0) fields; σ µν for (0 , ) fields . (4.3)The Poincar´e algebra consists of ten generators P µ and J µν ≡ L µν + S µν (where J µν = − J νµ ), which obey the following commutation relations:[ P µ , P ν ] = 0 , (4.4) (cid:2) J µν , P λ (cid:3) = i ( g νλ P µ − g µλ P ν ) , (4.5) (cid:2) J αβ , J ρσ (cid:3) = i ( g βρ J ασ − g αρ J βσ − g βσ J αρ + g ασ J βρ ) . (4.6) The operators , P µ and L µν include an implicit factor of δ αβ , whereas the spinoperator S µν depends non-trivially on α and β (except for the case of spin zero, when S = 0). P µ and J µν ), which are given by P ≡ P µ P µ and w ≡ w µ w µ , (4.7)where w µ is the Pauli-Lubanski vector, w µ ≡ − ǫ µνρλ J νρ P λ , (4.8)in a convention where ǫ = 1. Explicitly, w µ = ( ~J · ~P ; P ~J + ~K × ~P ) , (4.9)where J i ≡ ǫ ijk J jk and K i ≡ J i . Note that w µ P µ = 0 and [ w µ , P ν ] = 0 . (4.10)The unitary representations of the Poincar´e algebra can be labeled bythe eigenvalues of P and w when acting on the physical states with non-negative energy P . The eigenvalue of P is m , where m is the mass of thephysical state. To see the physical interpretation of w , we first consider thecase of m = 0. In this case, it is convenient to evaluate w in the particlerest frame. In this frame, w µ = (0 ; m ~S ), where S i is defined in eq. (2.8).Hence, w = − m ~S , with eigenvalues − m s ( s + 1), s = 0 , , , . . . . Weconclude that massive (positive energy) states can be labeled by ( m, s ),where m is the mass and s is the spin of the state.If m = 0, the previous analysis is not valid, since we cannot evaluate w in the rest frame. Nevertheless, if we take the m → w = 0, or the corresponding states haveinfinite spin. We reject the second possibility (which does not appear to berealized in nature), in which case w = lim m → ( − m ~S ) = 0. Thus, wemust solve the equations, w = P = w µ P µ = 0. It is simplest to choose aframe in which P = P (1; 0 , ,
1) where P >
0. In this frame, it is easy toshow that w = w (1; 0 , , w µ = hP µ , (4.11)where h is called the helicity operator. In particular,[ h , P µ ] = [ h , J µν ] = 0 , (4.12)which means that the eigenvalues of h can be used to label states of the ir-reducible massless representations of the Poincar´e algebra. From eq. (4.11),we derive h = w P = ~J · ~P P = ~S · ~P | ~P | = ~S · ˆ P , (4.13) We define the differential operator L i ≡ ǫ ijk L jk . Then, noting that ~L = ~x × ~P , itfollows that ~L · ~P = 0. Hence, ~J · ~P = ( ~L + ~S ) · ~P = ~S · ~P . P = | ~P | for massless states. Eigenvalues of h arecalled the helicity (and are denoted by λ ); its spectrum consists of non-negative half-integers, λ = 0 , ± , ± , . . . . Under a CPT transformation, λ → − λ . Thus, in any quantum field theory realization of massless parti-cles, both ±| λ | helicity states must appear in the theory. It is common torefer to a massless (positive energy) state of helicity λ as having spin | λ | . The supersymmetry (SUSY) algebra
In the 1960s, Coleman and Mandula proved a very powerful no-go theoremthat showed that in quantum field theories in 3 + 1 dimensional space-time with a mass gap, the only possible symmetry incorporating Poincar´etransformations and a global internal symmetry group of transformationsmust be a trivial tensor product of the two groups [98]. Subsequently,Haag, Lopusza´nski and Sohnius proved that the only possible extensionof the Poincar´e algebra involves the addition of new fermionic generatorsthat transform either as a ( ,
0) or (0 , ) under the Lorentz algebra, de-noted by Q iα and its hermitian conjugate Q † ˙ αi ≡ ( Q iα ) † , respectively, where i = 1 , , . . . N [9, 99]. In these lectures, we shall focus exclusively on thecase of N = 1, in which case the subscript i can be dropped.We therefore begin by examining the structure of the N = 1 SUSYalgebra, which is obtained by adding one ( ,
0) and one (0 , ) generatorto the Poincar´e algebra, denoted by Q α and Q † ˙ α , respectively. These two-component spinor generators have no explicit dependence on the spacetimecoordinate and are thus invariant under spacetime translations. That is,exp ( − ia µ P µ ) Q α exp ( ia µ P µ ) = Q α , (4.14)exp ( − ia µ P µ ) Q † ˙ α exp ( ia µ P µ ) = Q † ˙ α , (4.15)where the a µ are real parameters. Working to first order in a µ , it fol-lows that the spinor generators must commute with the translation gener-ator P µ , [ Q α , P µ ] = [ Q † ˙ α , P µ ] = 0 . (4.16)The commutation relations given in eq. (4.16) can also be deduced byemploying the following algebraic argument. Using the known transforma-tion properties of Q α , Q † ˙ α and P µ under the Poincar´e algebra, it followsthat [ Q α , P µ ] must consist of generators whose transformation propertiesare consistent with the tensor product,( , ⊗ ( , ) = (1 , ) ⊕ (0 , ) , (4.17)34nder the Poincar´e algebra. But according to the Haag- Lopuszanski-Sohnius theorem, there are no (1 , ) generators. This argument still leavesopen the possibility that [ Q α , P µ ] ∝ σ µα ˙ β Q † ˙ β . However, it can be shownusing the Jacobi identity that the proportionality constant must be zero.The transformation properties of Q α and Q † ˙ α under the Poincar´e algebrayield their commutation relations with the J µν ,[ Q α , J µν ] = ( σ µν ) αβ Q β , [ Q † ˙ α , J µν ] = − Q † ˙ β ( σ µν ) ˙ β ˙ α . (4.18)The Coleman-Mandula theorem implies that one cannot obtain a consis-tent algebraic structure by postulating commutation relations for the Q α and Q † ˙ α . However, by declaring Q α and Q † ˙ α to be fermionic generators,one can postulate anticommutation relations for Q α and Q † ˙ α such that thegenerators { P µ , J µν , Q α , Q † ˙ α } form a closed algebraic system. We there-fore consider the three possible anticommutation relations, along with theirtransformation properties with respect to the Poincar´e algebra, { Q α , Q β } ( , ⊗ ( ,
0) = (1 , ⊕ (0 , , (4.19) { Q † ˙ α , Q † ˙ β } (0 , ) ⊗ (0 , ) = (0 , ⊕ (0 , , (4.20) { Q α , Q † ˙ β } ( , ⊗ (0 , ) = ( , ) . (4.21)Eqs. (4.19) and (4.20) imply that { Q α , Q β } = s ( σ µν ) αβ J µν + kδ αβ , (4.22) { Q † ˙ α , Q † ˙ β } = s ∗ ( σ µν ) ˙ α ˙ β J µν + k ∗ δ ˙ α ˙ β , (4.23)where s and k are complex numbers and eq. (4.23) is the hermitian con-jugate of eq. (4.22). Note that we have raised and/or lowered some ofthe spinor indices for convenience. Since [ Q α , P λ ] = [ Q † ˙ α , P λ ] = 0 and[ J µν , P λ ] = 0, it follows that s = 0. If we now lower all spinor indices,eqs. (4.22) and (4.23) with s = 0 yield { Q α , Q β } = kǫ βα , { Q † ˙ α , Q † ˙ β } = k ∗ ǫ ˙ β ˙ α , (4.24)and we conclude that k = 0, since the left-hand sides of the above equationsare symmetric under the interchange of spinor indices, whereas the righthand sides are antisymmetric. Hence, { Q α , Q β } = { Q † ˙ α , Q † ˙ β } = 0 . (4.25)Eq. (4.21) implies that the remaining anticommutation relation mustbe of the form { Q α , Q † ˙ β } = tσ µα ˙ β P µ , (4.26)35here t is a complex number. Multiplying eq. (4.26) by σ ν ˙ βα and usingTr( σ µ σ ν ) = 2 g µν , it follows that σ ˙ βαµ { Q α , Q † ˙ β } = 2 tP µ . (4.27)In particular, for µ = 0, eq. (4.27) relates the energy P to the SUSYgenerators: 2 tP = Q Q † + Q † Q + Q Q † + Q † Q . (4.28)Since P ≥ m for physical states of mass m and the right-hand side ofeq. (4.28) is positive semi-definite, it follows that t must be real and posi-tive. One can rescale the definition of the fermionic generators Q and Q † such that t = 2. In this convention, { Q α , Q † ˙ β } = 2 σ µα ˙ β P µ . (4.29)To summarize, the N = 1 SUSY algebra is spanned by the generators { P µ , J µν , Q α , Q † ˙ α } , which satisfy eqs. (4.4)–(4.6) and[ Q α , P µ ] = [ Q † ˙ α , P µ ] = 0 , (4.30)[ Q α , J µν ] = ( σ µν ) αβ Q β , (4.31) h Q † ˙ α , J µν i = − Q † ˙ β ( σ µν ) ˙ β ˙ α , (4.32) { Q α , Q β } = { Q † ˙ α , Q † ˙ β } = 0 , (4.33) { Q α , Q † ˙ β } = 2 σ µα ˙ β P µ . (4.34)Note that eqs. (4.30)–(4.34) are unchanged under the U(1) phase trans-formation, Q α → e − iχ Q α , Q † ˙ α → e iχ Q † ˙ α , (4.35)whereas the generators P µ and J µν are not transformed. One can thereforeextend the N = 1 SUSY algebra by adding a bosonic generator R such that e iχR Q α e − iχR = e − iχ Q α , (4.36) e iχR Q † ˙ α e − iχR = e iχ Q † ˙ α . (4.37)Expanding out to first order in χ , one easily derives the commutation rela-tions, [ R , Q α ] = − Q α , (4.38) h R , Q † ˙ α i = Q † ˙ α . (4.39) We reject the possibility of t = 0, in which case Q = Q † = 0 and the SUSY algebrareduces to the Poincar´e algebra.
36e therefore say that the generator Q α has an R -charge of −
1. Since P µ and J µν are uncharged under the U(1) R transformation, it follows that[ R , P µ ] = [ R , J µν ] = 0 . (4.40)Thus, eqs. (4.4)–(4.6), (4.30)–(4.34) and (4.38)–(4.40) define the maximallyextended N = 1 SUSY algebra, which includes an additional continuousU(1) R symmetry. Representations of the N = 1 SUSY algebra
In Section 4.1, we identified the two Casimir operators of the Poincar´e alge-bra, P and w , and noted that the representations of the Poincar´e algebracan be labeled by the eigenvalues of the Casimir operators acting on thephysical states. We saw that the massive representations can be labeledby their mass and spin, ( m, s ). For a fixed value of m , the correspond-ing spin- s representations are (2 s + 1)-dimensional. For massless states,we defined the helicity operator h = ~S · ˆ P [cf. eq. (4.13)], with eigenval-ues λ = 0 , ± , ± . . . . We also noted that λ changes sign under a CPTtransformation. Hence, the massless positive energy representations of thePoincar´e algebra are specified by | λ | . For the case of λ = 0, the corre-sponding representation is one-dimensional. For any non-zero choice for λ ,the corresponding representation is two-dimensional and reducible, as both ±| λ | helicity states must appear.The unitary representations of the N = 1 SUSY algebra can be de-termined by using similar techniques [100, 101]. First, we identify theCasimir operators, which commute with all the SUSY algebra generators, { P µ , J µν , Q α , Q † ˙ α } . It is clear that P is a Casimir operator, since Q α and Q † ˙ α commute with P µ . However, w is not a Casimir operator ofthe SUSY algebra. To establish this result, it is straightforward to use the(anti-)commutation relations of the SUSY algebra to prove that:[ w µ , Q α ] = i ( σ µν ) αβ Q β P ν , h w µ , Q † ˙ α i = i ( σ µν ) ˙ β ˙ α Q † ˙ β P ν . (4.41)Using these results, it is straightforward to derive:[ w , Q α ] = 2 iσ µναβ Q β w µ P ν − P Q α , (4.42) h w , Q † ˙ α i = 2 iσ µν ˙ β ˙ α Q † ˙ β w µ P ν − P Q † ˙ α . (4.43)Thus, w does not commute with the fermionic generators of the SUSYalgebra. One consequence of this result is that the representations of the37USY algebra consist of supermultiplets that contain particles of equal massbut with different spins.In order to deduce the possible spins that make up an irreducible super-multiplet, we shall identify a second Casimir operator of the N = 1 SUSYalgebra. We begin by defining the operator B µ ≡ w µ + Q † σ µ Q . (4.44)Using eqs. (4.33), (4.34) and (4.41), one can derive[ B µ , Q α ] = − P µ Q α , [ B µ , Q † ˙ α ] = P µ Q † ˙ α . (4.45)The four-vector operator B µ possesses some of the properties of the Pauli-Lubanski vector w µ . In particular,[ B µ , B ν ] = iǫ µνρλ B ρ P λ ; (4.46)[ B µ , P ν ] = 0; (4.47)[ B µ , J νλ ] = i (cid:0) g µν B λ − g µλ B ν (cid:1) . (4.48)One may be tempted to conjecture that B ≡ B µ B µ is a Casimir operatorof the SUSY algebra. However, [ B , Q α ] = 0, so we must look further.The structure of eq. (4.45) suggests that we define C µν ≡ B µ P ν − B ν P µ . (4.49)It then follows that[ C µν , Q α ] = [ C µν , Q † ˙ α ] = [ C µν , P λ ] = 0 , (4.50)where the first two commutators vanish as a consequence of eq. (4.45) andthe last commutator vanishes as a consequence of eq. (4.47). Moreover,eqs. (4.5) and (4.48) imply that P µ and B µ are Lorentz four-vectors, inwhich case C µν is a second-rank Lorentz tensor. Hence C ≡ C µν C µν = 2[ B P − ( B · P ) ] , (4.51)satisfies [ C , P µ ] = [ C , J µν ] = [ C , Q α ] = [ C , Q † ˙ α ] = 0 . (4.52)We conclude that P and C are the two Casimir operators of the N = 1SUSY algebra. Representations of the N = 1 SUSY algebra can thereforebe labeled by the eigenvalues of P and C when acting on the physicalstates. The eigenvalue of P is m , where m is the mass. To under-stand the physical meaning of C , we will consider massive and masslesssupermultiplets separately. As in the case of the Poincar´e algebra, we restrict our considerations to states ofnon-negative energy P . able 2.: States of an N = 1 massive supermultiplet of superspin j . Aninterpretation is provided for j = s and j = s + where s is a non-negative integer.The bosonic and fermionic degrees of freedom (D.o.f.) of the supermultipletcoincide and is equal to 2(2 j + 1). Spin D.o.f. Interpretation ( j = s ) Interpretation ( j = s + ) j j + 1) complex spin- s boson “complex” spin-( s + ) fermion j + j + 2 spin-( s + ) fermion real spin-( s + 1) boson j − j spin-( s − ) fermion real spin- s boson4.3.1. Massive N = 1 supermultiplets To see the physical interpretation of C , we first consider the case of m = 0,so that we are free to evaluate the Lorentz scalar C in the particle restframe. In this frame, B µ = ( Q † σ Q ; mS i + Q † σ i Q ) , (4.53)where S i is defined in eq. (2.8). We then compute, C = 2 (cid:2) B P − ( B · P ) (cid:3) = 2 m (cid:2) B − B (cid:3) = − m B i B i , (4.54)where B i B i ≡ | ~B | . Moreover, if we define the rest-frame operator, J i ≡ m B i = S i + 14 m Q † σ i Q , (4.55)then it follows from eq. (4.46) that[ J i , J j ] = iǫ ijk J k . (4.56)The eigenvalues of J i J i are j ( j + 1) for j = 0 , , , . . . . Hence, theeigenvalues of C = − m J i J i (4.57)are − m j ( j + 1). We conclude that for positive energy, timelike P µ , theunitary irreducible representations of the N = 1 SUSY algebra are labeledby ( m, j ), where j is called the superspin of the supermultiplet. The statesof an irreducible N = 1 massive supermultiplet of superspin j are exhibitedin Table 2. The explicit construction of these states and a discussion oftheir properties is presented in Section 4.9. Example 4 (The massive chiral supermultiplet, j = 0). For j = 0,only j = 0 is possible, in which case the massive supermultiplet is made upof two states of spin 0 and two states of spin . The two spin-0 states can becombined into a single complex scalar state, and the two spin- states canbe identified as the two components of a two-component Majorana fermion.In this case the j − row of Table 2 is not relevant.39t can be shown (see Problem 5) that the massive supermultiplet ofsuperspin consists of a (real) spin-1 boson, a (real) spin-0 boson and twomass-degenerate Majorana fermions, which can be combined into a singleDirac fermion (called a complex fermion in Table 2). As expected, in boththe j = 0 and j = cases exhibited above, the number of bosonic degreesof freedom of the supermultiplet equals the number of fermionic degrees offreedom.4.3.2. Massless N = 1 supermultiplets We now examine the case of zero-mass positive energy states, where P = 0and P >
0. If one multiplies eq. (4.34) by P ρ P λ σ ˙ γαρ σ ˙ βτγ , one can easilyderive the anticommutation relation, { P ρ σ ˙ γαρ Q α , P λ Q † ˙ β σ ˙ βτ } = 2 P P µ σ ˙ γτµ . (4.58)Thus, for P = 0 we have, h Ψ | { P ρ σ ˙ γαρ Q α , P λ Q † ˙ β σ ˙ βτ } | Ψ i = 0 , (4.59)for any state | Ψ i . In the space of one-particle states, only positively-normedstates exist. Noting that ( P µ σ ˙ αβµ Q β ) † = P µ Q † ˙ β σ ˙ βαµ , eq. (4.59) implies thatas operators on the space of one-particle states, P ρ σ ˙ γαρ Q α = P λ Q † ˙ β σ ˙ βτλ = 0 , for P = 0 . (4.60)Using this result, one can evaluate the Casimir operator C , defined ineq. (4.51), in the case of P = 0. In particular, using w µ P µ = 0 andeq. (4.60), C = − B · P ) = − ( Q † ˙ α σ ˙ αβµ Q β P µ ) = 0 . (4.61)The same conclusion can be obtained by choosing the standard referenceframe, P µ = P (1 ; 0 , , { Q , Q † } = 0 , { Q , Q † } = 4 P , (4.62) { Q , Q } = { Q , Q } = { Q , Q } = 0 , (4.63) { Q † , Q † } = { Q † , Q † } = { Q † , Q † } = 0 . (4.64)Hence, C = − B · P ) = − P ( Q † Q ) = P Q † Q † Q Q = 0 . (4.65)40q. (4.60) implies a number of other operator identities when acting onthe space of one-particle states. Using eq. (4.34), one easily derives[ Q α Q α , Q † ˙ β ] = 4 P µ σ µα ˙ β Q α , [ Q † ˙ α Q † ˙ α , Q β ] = − P µ σ µα ˙ β Q † ˙ β . (4.66)Applying eq. (4.60) then yields[ Q α Q α , Q † ˙ β ] = [ Q † ˙ α Q † ˙ α , Q β ] = 0 , for P = 0 . (4.67)Then, for any one-particle state | Ψ i , eqs. (4.33), (4.34) and (4.67) yield P µ σ µα ˙ α Q β Q β | Ψ i = { Q α , Q † ˙ α } Q β Q β | Ψ i = Q α Q † ˙ α Q β Q β | Ψ i = Q α [ Q † ˙ α , Q β Q β ] | Ψ i = 0 . (4.68)A similar computation of P µ σ µα ˙ α Q † ˙ β Q † ˙ β allows us to conclude that P µ Q β Q β | Ψ i = P µ Q † ˙ β Q † ˙ β | Ψ i = 0 , for P = 0 , (4.69)after multiplying through by σ ˙ ααν and evaluating the resulting trace. As weare only interested in positive energy states, we conclude that as operatorson the space of one-particle states, Q β Q β = Q † ˙ β Q † ˙ β = 0 , for P = 0 and P > . (4.70)In order to identify the massless supermultiplets of one-particle states,it is convenient to define L µ ≡ ( w µ + B µ ) = w µ + Q † σ µ Q . (4.71)Note [ Q α , P µ ] = [ Q † ˙ α , P µ ] = 0 and [ w µ , P ν ] = 0 imply that[ P µ , L ν ] = 0 . (4.72)Using eqs. (4.33), (4.34) and (4.41), one can easily derive[ L µ , Q α ] = − ( σ µ σ ν ) αβ Q β P ν , [ L µ , Q † ˙ α ] = ( σ ν σ µ ) ˙ β ˙ α Q † ˙ β P ν . (4.73)A straightforward computation then gives:[ L µ , L ν ] = iǫ µνρλ ( L ρ + Q † σ ρ Q ) P λ . (4.74)When P = 0, we impose the results of eq. (4.60) to obtain P µ L µ = [ L µ , Q α ] = [ L µ , Q † ˙ α ] = 0 , for P = 0 . (4.75)Moreover, if we employ the identity ǫ µνρλ σ ρ = i ( σ ν σ µ σ λ − σ λ σ µ σ ν ) , (4.76)41which is a consequence of eq. (2.52)], it then follows from eq. (4.60) that ǫ µνρλ Q † σ ρ QP λ = 0 , for P = 0 . (4.77)Hence, in the massless case, eq. (4.74) simplifies to[ L µ , L ν ] = iǫ µνρλ L ρ P λ , for P = 0 . (4.78)Finally, we evaluate L µ L µ for the positive energy massless one-particlestates. As in the analysis of the Poincar´e algebra, we shall assume that w µ w µ = lim m → ( − m ~S ) = 0. Using eq. (4.77), it follows that w µ Q † σ µ Q = − ǫ µνρλ J νρ P λ Q † σ µ Q = 0 . (4.79)In light of eq. (2.49), we obtain( Q † σ µ Q )( Q † σ µ Q ) = 2 ǫ ˙ α ˙ γ ǫ βτ Q † ˙ α Q β Q † ˙ γ Q τ = 2 ǫ ˙ α ˙ γ ǫ βτ Q † ˙ α [2 P µ σ µβ ˙ γ − Q † ˙ γ Q β ] Q τ = 2( Q † ˙ α Q † ˙ α )( Q β Q β ) − P µ Q † σ µ Q = 0 , (4.80)after applying the operator identities given in eqs. (4.60) and (4.70). Hence, L µ L µ = 0 , for P = 0 and P > . (4.81)When P = 0 and P >
0, the properties of L µ [cf. eqs. (4.72), (4.75),(4.78) and (4.81)] match precisely the properties of the Pauli-Lubanski vec-tor. Thus, we must solve the equations L = P = L µ P µ = 0. In areference frame in which P µ = P (1 ; 0 , ,
1) and P >
0, it follows that L µ = L (1 ; 0 , , L µ = K P µ , (4.82)where K ≡ L /P is called the superhelicity operator. More explicitly, ina frame where P µ = P (1 ; 0 , , K = h + 18 P (cid:16) Q † Q + Q † Q (cid:17) , (4.83)where h ≡ w /P = ~S · ˆ P is the usual helicity operator acting on masslessone-particle states. By virtue of eqs. (4.30) and (4.75), it follows that[ K , P µ ] = [ K , Q α ] = [ K , Q † ˙ α ] = 0 . (4.84)Hence, the states of the massless supermultiplet are eigenstates of K ,with possible eigenvalues κ = 0 , ± , ± , ± , . . . . In contrast, h does notcommute with Q α and Q † ˙ α . Thus, the different states of the massless su-permultiplet will have different helicities. We conclude that for positiveenergy, timelike P µ , the irreducible representations of the N = 1 SUSY al-gebra are labeled by the eigenvalue κ of the superhelicity operator, which is42 able 3.: States of an N = 1 massless supermultiplet of superhelicity κ and thecorresponding CPT conjugates which comprise an N = 1 massless supermultipletof superhelicity − κ + . An interpretation is provided for κ = s and κ = s − ,where s is a positive integer. In the special case of κ = , the scalar boson of thesupermultiplet is complex, whereas for κ = 1 , , , . . . , the bosonic member of thesupermultiplet is real with nonzero spin. In all cases, the number of bosonic andfermionic degrees of freedom (D.o.f.) coincide and are equal to 2. Helicities D.o.f. Interpretation ( κ = s ) Interpretation ( κ = s − ) κ , − κ s boson spin-( s − ) fermion κ − , − κ + s − ) fermion spin-( s −
1) bosoncalled the superhelicity of the massless supermultiplet. Moreover, an N = 1massless supermultiplet with superhelicity κ consists of two massless stateswith helicity κ and κ − , respectively. Any quantum field theory realization of supersymmetry respects CPTsymmetry. Since the helicity changes sign under a CPT transformation, itfollows that any irreducible massless supermultiplet with superhelicity κ must be accompanied by the corresponding CPT-conjugate states thatmake up an irreducible massless supermultiplet with superhelicity − κ + .Hence, without loss of generality, we can restrict the possible values of thesuperhelicity to κ = , , , . . . . These results are summarized in Table 3.The explicit construction of the states of an irreducible massless supermul-tiplet and a discussion of their properties is presented in Section 4.9. Example 5 (A massless chiral supermultiplet, with κ = ). Including the CPT-conjugates, this supermultiplet contains two states ofhelicity 0, and two states of helicity ± , respectively, which yields a mass-less complex scalar and a massless Majorana fermion. We recognize this asthe massless limit of a massive j = 0 chiral supermultiplet. Example 6 (a massless gauge supermultiplet, with κ = 1). Including the CPT-conjugates, this supermultiplet contains two states ofhelicity ± and two states of helicity ±
1, which yields a massless Majoranafermion and a massless spin-1 particle. This is a gauge supermultiplet (e.gthe photino and the photon of supersymmetric QED). In the literature, it is more common to define L µ = ( K + ) P µ , in which case thehelicities of the massless N = 1 supermultiplet are κ + and κ (e.g., see refs. [4, 11]).In our opinion, the definition of the superhelicity operator given in eq. (4.82) is cleaner.
43n Problem 8, you will show that a massless supermultiplet with κ = 2and its CPT-conjugates contains a massless spin- and a massless spin 2particle, which is realized in supergravity by the gravitino and the graviton,respectively. Consequences of super-Poincar´e invariance
A Poincar´e invariant quantum field theory respects the Poincar´e algebragenerated by { P µ , J µν } , which satisfy commutation relations given byeqs. (4.4)–(4.6). One of the basic postulates of Poincar´e-invariant quan-tum field theory states that a translationally-invariant, Lorentz-invariantvacuum | i exists such that [102], P µ | i = 0 , J µν | i = 0 . (4.85)In particular, h | P µ | i = 0. Indeed if h | P µ | i 6 = 0, then the vacuumwould not be invariant under Lorentz transformations. This is easily provenby taking the vacuum expectation value ofexp (cid:0) iθ ρτ J ρτ (cid:1) P µ exp (cid:0) − iθ ρτ J ρτ (cid:1) = Λ µν P ν , (4.86)where the θ ρτ = − θ ρτ parameterize the 4 × µν [cf. eqs. (2.3) and (2.4)]. Using J µν | i = 0, it follows that h | P µ | i = Λ µν h | P ν | i , (4.87)which holds for all Lorentz transformations Λ. Thus, it follows that h | P µ | i = 0.A super-Poincar´e invariant quantum field theory respects the SUSYalgebra generated by { P µ , J µν , Q α , Q † ˙ α } . The SUSY algebra genera-tors satisfy the commutation relations of the Poincar´e algebra and the(anti)commutation relations given by eqs. (4.30)–(4.34). Two importantconsequences can be established:1. The vanishing of the vacuum energy is a necessary and sufficientcondition for the existence of a global supersymmetric vacuum. In a theory governed by a supersymmetric action, for a fixed non-zero P µ the number of bosonic and fermionic degrees of freedom coincide. We address these two results in the next two subsections.44.4.1.
The vacuum energy of a globally supersymmetric theory
In order to prove that the vanishing of the vacuum energy is a necessary andsufficient condition for the existence of a global supersymmetric vacuum,we consider the anticommutation relations of the fermionic generators ofthe SUSY algebra, { Q α , Q † ˙ β } = 2 σ µα ˙ β P µ . (4.88)Following the derivation of eq. (4.28), P = h Q Q † + Q † Q + Q Q † + Q † Q i . (4.89)Since the right-hand side of eq. (4.28) is positive semi-definite (and neither Q nor Q † is the zero operator), it follows that h | P | i = 0 ⇐⇒ Q α | i = 0 . (4.90)In particular, Q α | i = 0 implies that the vacuum is supersymmetric,in the same way that P µ | i = J µν | i = 0 imply that the vacuum istranslationally-invariant and Lorentz-invariant. However, this proof is troubling for two separate reasons. First, supposethat the action of the theory is invariant under supersymmetric transfor-mations, but the vacuum is not preserved by supersymmetry. In this case, Q α | i 6 = 0, and we say that supersymmetry is spontaneously broken. Then,eq. (4.89) implies that h | P | i >
0, which contradicts eq. (4.85). Thus,it appears that the spontaneous breaking of supersymmetry is not pos-sible without breaking Lorentz invariance. Perhaps a more fundamentalobjection is that the concept of the vacuum energy is usually consideredto be unphysical in non-gravitational theories, as it is commonly assertedthat only energy differences are physical. Thus, it seems to be a matter ofconvention to choose the vacuum energy such that h | P | i = 0.To overcome the objections raised above, we re-examine the concept ofthe vacuum energy in relativistic (non-gravitational) quantum field theory.Using the Noether procedure, the conserved canonical energy-momentumtensor, T ( c ) µν can be obtained, which satisfies ∂ µ T ( c ) µν = 0. One can then Equivalently, h | { Q α , Q † ˙ β } | i = 0, by covariance with respect to the SUSY algebra,since there are no spinor quantities with one undotted and one dotted index that canappear on the right hand side of this equation. Hence, Q α | i = 0, which then yields h | P | i = 0. The arguments given here do not depend on whether one employs the canonical energymomentum tensor or the improved symmetrized energy-momentum tensor. h | T ( c ) µν | i = E g µν , (4.91)where E is typically UV divergent. Since the Hamiltonian density is identi-fied as H = T , it follows that E is the vacuum energy density. However,one is always free to define a new subtracted energy-momentum tensor, T µν ≡ T ( c ) µν − E g µν , (4.92)which is a Lorentz-covariant expression. By construction, ∂ µ T µν = 0 and h | T µν | i = 0 . (4.93)The energy-momentum tensor T µν plays a distinguished role in relativisticquantum field theory, since it can be used to construct the generators ofspacetime translations, P µ = Z d x T µ , (4.94)that satisfy h | P µ | i = 0. Indeed, P µ defined by eq. (4.94) is a four-vectorwith respect to Lorentz transformations. Likewise, one can construct adistinguished angular momentum tensor M µνλ that can be used to constructthe generators of Lorentz transformations J µν = Z d xM µν , (4.95)which satisfy h | J µν | i = 0.However, in a supersymmetric theory, another choice of the energy-momentum tensor is natural. The fermionic generators Q α and Q † ˙ α ofthe SUSY algebra are time-independent (conserved) quantities that areobtained by integrating the zeroth component of the supercurrents, Q α = Z d xJ α , Q † ˙ α = Z d xJ † ˙ α . (4.96)In a theory governed by a supersymmetric Lagrangian, the supercurrents J µα and J † ˙ α µ are related by supersymmetry to an energy-momentum tensor,denoted by T (SUSY) µν . Then, the proper interpretation of eq. (4.88) is [103] { Q α , Q † ˙ β } = 2 σ µα ˙ β Z d x T (SUSY) µ . (4.97) For example, in the quantum theory of free fields, the vacuum energy is set to zero bydefining the Hamiltonian density to be normal ordered. { Q α , Q † ˙ β } = 2 σ µα ˙ β P µ + 2 E σ α ˙ β , (4.98)where P µ is defined by eq. (4.94) and E ≡ Z d x h | T (SUSY)00 | i . (4.99)If E = 0 (which corresponds to T (SUSY) µν = T µν ), then we recover thestandard SUSY algebra, and the vacuum is supersymmetric. If E = 0,then eq. (4.98) is consistent with h | P µ | i = 0 (which is required by theLorentz-invariant vacuum) and with Q α | i 6 = 0. In particular, E serves asan order parameter for broken supersymmetry.Note that E ≥ E = h | Q Q † + Q † Q + Q Q † + Q † Q | i ≥ . (4.100)In supersymmetric theories, it is common to call E the vacuum energy.Thus, if supersymmetry is spontaneously broken, then this definition of thevacuum energy is not compatible with usual conventions of quantum fieldtheory in which the vacuum energy is defined to be zero.Although the conclusions obtained above are correct, the derivation ofeq. (4.98) is still somewhat formal. Indeed if the vacuum breaks super-symmetry, then the integrals in eq. (4.96) do not converge when integratedover an infinite volume (this is an infrared divergence), so strictly speak-ing the fermionic generators Q α and Q † ˙ α are undefined. Nevertheless,the supercurrents are conserved, as expected in a supersymmetric theorywith no explicit supersymmetry breaking. In section 7.1.3, we will demon-strate that given a supersymmetric Lagrangian, if the vacuum breaks su-persymmetry then a massless Goldstone fermion exists in the spectrum.The long range forces mediated by this massless particle are responsiblefor the non-convergence of the integrals in eq. (4.96). Equivalently, in aspontaneously-broken globally supersymmetric theory, applying Q α to thevacuum creates a zero-momentum massless fermionic state, which is a stateof infinite norm [25].4.4.2. Equality of bosonic and fermionic degrees of freedom insupersymmetric theories
In a theory governed by a supersymmetric action, for a fixed non-zero P µ the number of bosonic and fermionic degrees of freedom coincide. To prove Moreover, given a non-zero value for h | T (SUSY)00 | i , which is a constant by trans-lational invariance, one sees that E defined in eq. (4.99) also diverges in the infinitevolume limit. Q α or Q † ˙ α to a physicalstate changes that state by adding half a unit of spin. An explicit exampleof this behavior can be seen in eqs. (4.164) and (4.165). We can summarizethis behavior in the following schematic equations, Q α | B i = | F i , Q α | F i = | B i , (4.101)and similarly for the application of Q † ˙ α , where | B i is a bosonic state and | F i is a fermionic state. It is convenient to introduce an operator, denotedby ( − F , with the following properties:( − F | B i = | B i , ( − F | F i = − | F i . (4.102)Note that Q α ( − F | F i = − Q α | F i = − | B i , (4.103)( − F Q α | F i = ( − F | B i = | B i , (4.104)and similarly for the application of Q † ˙ α . It follows that Q α [and Q † ˙ α ] anti-commute with ( − F , { Q α , ( − F } = { Q † ˙ α , ( − F } = 0 . (4.105)Using eq. (4.105), we can evaluate the following trace over physicalstates, Tr h ( − F { Q α , Q † ˙ β } i = Tr h ( − F ( Q α Q † ˙ β + Q † ˙ β Q α ) i = Tr h − Q α ( − F Q † ˙ β + ( − F Q † ˙ β Q α i = Tr h − Q † ˙ β Q α ( − F + Q † ˙ β Q α ( − F i = 0 , (4.106)after a cyclic permutation within the trace at the penultimate step. Em-ploying eq. (4.34), we conclude thatTr( − F = 0 , for any fixed non-zero P µ . (4.107)For a fixed non-zero eigenvalue p µ obtained by applying the momentumoperator P µ to a physical state,Tr( − F = X { r } h p µ , { r }| ( − F | p µ , { r }i = N B ( p µ ) − N F ( p µ ) = 0 , (4.108)where { r } indicates all other quantum numbers of the physical state. Thus,the number of bosonic ( N B ) and fermionic ( N F ) degrees of freedom coin-cide. 48e have already observed that eq. (4.105) is satisfied by all positive en-ergy representations of the SUSY algebra. The proof above demonstratesthat the equality of bosonic and fermionic degrees of freedom in super-symmetric theories is far more general. Indeed, the only case where thisequality can break down is when P µ = 0, corresponding to the vacuumstate of the supersymmetric theory. Supersymmetric theories of spin-0 and spin- particles The simplest supermultiplet contains a complex scalar and a two-component (Majorana) fermion, of common mass m . The case of m = 0corresponds to superspin j = 0 and the case of m = 0 corresponds to super-helicity and its CPT-conjugate.4.5.1. The Wess-Zumino Lagrangian
A Lagrangian that respects the SUSY algebra is given by L = ( ∂ µ A ) † ( ∂ µ A ) + iψ † σ µ ∂ µ ψ − (cid:12)(cid:12)(cid:12)(cid:12) dWdA (cid:12)(cid:12)(cid:12)(cid:12) − " d WdA ψψ + (cid:18) d WdA (cid:19) † ψ † ψ † , (4.109)where A is a complex scalar, ψ and ψ † are two-component spinors, and W = W ( A ) [called the superpotential ] is a holomorphic function of A ( i.e. ,a function of A and not A † ). If W ( A ) is (at most) a cubic polynomialin A , then the above Lagrangian yields a renormalizable quantum fieldtheory called the Wess-Zumino model . For example, a simple quadraticsuperpotential, W = mA , describes a free theory of a complex scalarand a Majorana fermion of common mass | m | . An interacting theory isobtained by including a cubic term in the superpotential, W = mA + gA . (4.110)Without loss of generality, we can assume that m and g are non-negative(by appropriate rephasing of A and ψ ). Then, inserting eq. (4.110) intoeq. (4.109) yields the Wess-Zumino Lagrangian, L = ( ∂ µ A ) † ( ∂ µ A ) + iψ † σ µ ∂ µ ψ − m ( ψψ + ψ † ψ † ) − m ( A † A ) − g ( Aψψ + A † ψ † ψ † ) − mg ( A † A )( A + A † ) − g ( A † A ) . (4.111) For example, Witten showed that in an SU( N ) supersymmetric Yang-Mills theory,Tr( − F = N for the supersymmetric ground state [104]. Employing A for a complex scalar field rather than φ follows the notation first intro-duced in Ref. [2]. It should not be confused with the notation for a vector field, whichwill henceforth be denoted by V .
49s expected, the boson and fermion are mass-degenerate. Moreover, SUSYimposes relations among the couplings. In this model, we see that thequartic scalar coupling is the square of the Yukawa (scalar-fermion-fermion)coupling.In order to employ four-component Feynman rules, it is convenient toconvert the Wess-Zumino Lagrangian into four-component fermion form.Writing A = ( S + iP ) / √
2, where S and P are hermitian fields, we obtain L = ( ∂ µ S ) + ( ∂ µ P ) − m ( S + P ) + Ψ M ( iγ µ ∂ µ − m )Ψ M − g √ S Ψ M ψ M − iP Ψ M γ Ψ M ] − mg √ S ( S + P ) − g ( S + P ) . (4.112)Note that this Lagrangian separately conserves C, P and T. We identify S as a scalar and P as a pseudoscalar.4.5.2. Invariance of the Wess-Zumino Lagrangian with respect toSUSY transformations
The Wess-Zumino Lagrangian given by eq. (4.111) is invariant with respectto global supersymmetry transformations. Explicitly, these transformationsdepend on an infinitesimal Grassmann (anticommuting) two-componentspinor parameter ξ that is independent of the spacetime position x , δ ξ A = √ ξψ , (4.113) δ ξ ψ α = − i √ σ µ ξ † ) α ∂ µ A − √ ξ α (cid:18) dWdA (cid:19) † . (4.114)By hermitian conjugation, one also obtains δ ξ A † = √ ξ † ψ † , (4.115) δ ξ ψ † ˙ α = i √ ξσ µ ) ˙ α ∂ µ A † − √ ξ † ˙ α (cid:18) dWdA (cid:19) . (4.116)Applying these transformation laws to eq. (4.111), one obtains a result ofthe form δ ξ L = ∂ µ K µ . (4.117)That is, the action of the Wess-Zumino Model, S = R d x L , is invariantunder global SUSY transformations; i.e. , δ ξ S = 0.But, how do we know that the transformation laws just introducedcorrespond to SUSY transformations? Recall that for ordinary spacetimetranslations, e ia · P Φ( x ) e − ia · P = Φ( x + a ) , (4.118)50hich in infinitesimal form is given by i (cid:2) P µ , Φ( x ) (cid:3) = ∂ µ Φ( x ) , (4.119)where Φ = A or ψ . Equivalently, for an infinitesimal translation, δ a Φ( x ) ≡ Φ( x + a ) − Φ( x ) ≃ a µ ∂ µ Φ( x ) = ia µ (cid:2) P µ , Φ( x ) (cid:3) . (4.120)Likewise, since Q and Q † are the generators of SUSY-translations, we ex-pect δ ξ Φ( x ) = i (cid:2) ξQ + ξ † Q † , Φ( x ) (cid:3) . (4.121)Consider the commutator of two SUSY-translations:( δ η δ ξ − δ ξ δ η )Φ( x ) = (cid:20) i ( ηQ + η † Q † ) , (cid:2) i ( ξQ + ξ † Q † ) , Φ( x ) (cid:3)(cid:21) − ( ξ ←→ η )= (cid:20)(cid:2) i ( ηQ + η † Q † ) , i ( ξQ + ξ † Q † ) (cid:3) , Φ( x ) (cid:21) , (4.122)after employing the Jacobi identity for the double commutators. Using theSUSY algebra, (cid:2) ηQ , ξ † Q † (cid:3) = 2( ησ µ ξ † ) P µ . Note that the anticommutator has been converted into a commutator due tothe fact that η and ξ are anticommuting two-component spinors. Likewise, (cid:2) ηQ , ξQ (cid:3) = (cid:2) η † Q † , ξ † Q † (cid:3) = 0 . Hence, we end up with (cid:2) δ η , δ ξ (cid:3) Φ( x ) = 2( ξσ µ η † − η † σ µ ξ † ) (cid:2) P µ , Φ( x ) (cid:3) = − i ( ξσ µ η † − η † σ µ ξ † ) ∂ µ Φ( x ) . (4.123)Likewise, a similar computation yields, (cid:2) δ η , δ ξ (cid:3) A ( x ) = − i ( ξσ µ η † − η † σ µ ξ † ) ∂ µ A ( x ) , (4.124) (cid:2) δ η , δ ξ (cid:3) ψ α ( x ) = − i ( ξσ µ η † − η † σ µ ξ † ) ∂ µ ψ α + R , (4.125)where the remainder R vanishes after imposing the classical field equationsfor ψ α ( x ), as you will verify in Problem 10. We conclude that the SUSYalgebra is realized on-shell , i.e. , after employing the classical field equations.It is instructive to employ Noether’s theorem, which states that an in-variance of the action under a continuous symmetry implies the existence51f a conserved current. Since we have explicitly identified the SUSY trans-formations, we can use Noether’s theorem to determine the correspond-ing conserved supercurrent. Using δ ξ L = ∂ µ K µ , the resulting conservedNoether supercurrents are ξ α J µα + ξ † ˙ α J † µ ˙ α = X Φ δ ξ Φ δ L δ ( ∂ µ Φ) − K µ , (4.126)where the sum is taken over Φ = A , ψ . Note that the supercurrent hasboth a Lorentz index and a spinor index. Noether’s theorem states thatthe supercurrent is conserved after imposing the classical field equations .That is, ∂ µ J µα = ∂ µ J † µ ˙ α = 0 . (4.127)The supercharges are defined in the usual way (as previously noted): Q α = Z d xJ α , Q † ˙ α = Z d xJ † ˙ α . (4.128)These are expressions that depend on the fields A and ψ . One can nowemploy the canonical commutation relations of the boson field A and thecanonical anticommutation relations of the fermion field ψ to verify that { Q α , Q β } = { Q † ˙ α , Q † ˙ β } = 0 , { Q α , Q † ˙ β } = 2 σ µα ˙ β P µ , (4.129)where P µ is the Noether charge of spacetime translations given in eq. (4.94). The SUSY algebra realized off-shell
The SUSY transformation laws of the Wess-Zumino Lagrangian exhibitedin eqs. (4.113) and (4.114) are not in an optimal form for two reasons. First,in the case of a cubic superpotential W ( A ), the transformation law for ψ α is non-linear in the fields. Second, the SUSY algebra is only realized on-shell. We can address both these issues by introducing an auxiliary complexscalar field F ( x ). Consider the alternative Lagrangian, L = ( ∂ µ A ) † ( ∂ µ A ) + iψ † σ µ ∂ µ ψ + F † F + F dWdA + F † (cid:18) dWdA (cid:19) † − " d WdA ψψ + (cid:18) d WdA (cid:19) † ψ † ψ † . (4.130)The field F ( x ) is auxiliary since L does not depend on ∂ µ F and ∂ µ F † .That is, F and F † are non-dynamical fields.52e can trivially solve for F and F † using the classical field equations, ∂ L ∂F = 0 = ⇒ F † = − dWdA , (4.131) ∂ L ∂F † = 0 = ⇒ F = − (cid:18) dWdA (cid:19) † . (4.132)Hence, eqs. (4.131) and (4.132) yield, F † F + F dWdA + F † (cid:18) dWdA (cid:19) † = − (cid:12)(cid:12)(cid:12)(cid:12) dWdA (cid:12)(cid:12)(cid:12)(cid:12) . (4.133)Plugging this result back into eq. (4.130), we recover the general form ofthe Wess-Zumino Lagrangian given by eq. (4.109).The Lagrangian including the auxiliary fields given by eq. (4.130) isalso invariant under SUSY translations. The appropriately modified SUSYtransformation laws are now given by δ ξ A = √ ξψ , (4.134) δ ξ ψ α = − i √ σ µ ξ † ) α ∂ µ A + √ ξ α F , (4.135) δ ξ F = − i √ ξ † σ µ ∂ µ ψ . (4.136)By hermitian conjugation, one also obtains δ ξ A † = √ ξ † ψ † , (4.137) δ ξ ψ † ˙ α = i √ ξσ µ ) ˙ α ∂ µ A † + √ ξ † ˙ α F † , (4.138) δ ξ F † = i √ ∂ µ ψ † ) σ µ ξ . (4.139)Applying these transformation laws to eq. (4.130), one obtains a result ofthe form δ ξ L = ∂ µ K ′ µ , (4.140)where the explicit form for K ′ µ is to be determined in Problem 13. More-over, as you will verify in Problem 14, (cid:2) δ η , δ ξ (cid:3) Φ( x ) = − i ( ξσ µ η † − η † σ µ ξ † ) ∂ µ Φ( x ) , (4.141)for Φ = A , ψ and F without the need to impose the classical field equations.Thus, the Wess-Zumino Lagrangian with auxiliary fields included as ineq. (4.130) is invariant under SUSY translations, and the SUSY algebra isrealized off-shell , i.e. , without requiring that the fields satisfy their classicalfield equations.The following two observations will be particularly useful as we moveforward. First, note that the mass dimensions of the fields are given by53 A ] = 1, [ ψ ] = and [ F ] = 2, which is consistent with the requirementthat [ L ] = 4 (since the action is dimensionless in units of ~ = 1). Then,eqs. (4.134)–(4.136) are dimensionally consistent if [ ξ ] = . Second, notethat δ ξ F given in eq. (4.136) is a total derivative. Indeed, δ ξ F is a totalderivative as a consequence of dimensional analysis and the linearity of theSUSY transformation laws. This implies that δ ξ F must involve ∂ µ , since[ ∂ µ ] = 1. An important consequence of this observation is that R d x F isinvariant under SUSY transformations. Counting bosonic and fermionic degrees of freedom
It is instructive to count both the on-shell and off-shell bosonic andfermionic degrees of freedom in the Wess-Zumino model, which is a the-ory of a complex scalar and a two-component fermion.A complex scalar possesses two real degrees of freedom. Note thatapplying the classical field equations (in this case the inhomogeneous Klein-Gordon equation) does not affect the number of scalar degrees of freedom,but only the spacetime dependence of the scalar field. The two-componentfermion ψ α possesses two complex degrees of freedom, which yields fourreal degrees of freedom. Applying the classical field equations, iσ µ ∂ µ ψ = (cid:18) d WdA (cid:19) † ψ † , (4.142)which relate ψ and ψ † , thereby eliminating two of the four degrees of free-dom. By taking the derivative of eq. (4.142), one can eliminate ψ † usingthe hermitian conjugate of eq. (4.142). The resulting equation for ψ is theinhomogeneous Klein-Gordon equation, which does not further affect thenumber of fermionic degrees of freedom. Thus, the Wess-Zumino modelpossesses two on-shell bosonic and two fermionic degrees of freedom.The counting of the off-shell degrees of freedom can be performed byexamining the Lagrangian [eq. (4.130)] expressed in terms of the propagat-ing and auxiliary fields. In this case, we count two real degrees of freedomfor the complex scalar, four real degrees of freedom for the two-componentfermion and two real degrees of freedom for the complex auxiliary field F .That is, the Wess-Zumino model possesses four bosonic and four fermionicoff-shell degrees of freedom.Thus, the number of bosonic and fermionic degrees of freedom matchin both on-shell and off-shell counting. Equivalently, we can count ψ and ψ † as four independent degrees of freedom. If d W/dA = 0, then iσ µ ∂ µ ψ = 0 yields a relation between ψ and ψ . .8. Lessons from the Wess-Zumino Model
In our study of the Wess-Zumino model, we provided a Lagrangian thatincorporated the fields of a known supermultiplet. However, it was rathermysterious how this Lagrangian was obtained. It was even more mysterioushow we came up with the correct SUSY transformation laws for the variousfields. Moreover, it was quite laborious to verify that the proposed SUSYtransformation laws satisfy the SUSY algebra and the action is invariantunder super-Poincar´e transformations.We also learned that in order for the SUSY transformation laws torespect the SUSY algebra off-shell, one must introduce additional auxiliaryfields. One additional benefit of doing so is that the corresponding SUSYtransformation laws are now linear in all the fields. For this reason, weintroduced the auxiliary field F , which can be used to write down theSUSY translation-invariant quantity R d x F ( x ). This observation actuallyprovides an important clue for how to construct a SUSY Lagrangian.As we shall demonstrate in Section 5, it is possible to develop a for-malism in which, starting with a known supermultiplet, one can triviallyconstruct a Lagrangian that is invariant under super-Poincar´e transforma-tions. Moreover, this formalism will provide explicit forms for the SUSYtransformation laws that automatically respect the SUSY algebra. Appendix: Constructing the states of a supermultiplet
In this subsection, we provide further details on the construction of thestates of the massive and massless supermultiplets, which yields the resultspresented in Tables 2 and 3.4.9.1.
States of a massive supermultiplet of superspin j To construct the states of the massive supermultiplet, we note that in therest frame, the anticommutators given in eqs. (4.33) and (4.34) simplify to { Q , Q † } = { Q , Q † } = 2 m , (4.143) { Q , Q } = { Q , Q } = { Q , Q } = 0 , (4.144) { Q † , Q † } = { Q † , Q † } = { Q † , Q † } = 0 . (4.145)All states in a supermultiplet with superspin j are simultaneous eigenstatesof P , J i J i and J with eigenvalues m , j ( j +1) and j , respectively, wherethe possible values of j are − j, − j + 1 , . . . , j − , j .55or a fixed value of the superspin j , there exists a distinguished state ofthe supermultiplet that is a simultaneous eigenstate of P , J i J i and J ,denoted by | Ω i , which satisfies Q β | Ω i = 0 , S + | Ω i = 0 , (4.146)where S ± ≡ S ± iS . To verify that a state | Ω i exists that is annihilatedby Q β , let us assume the contrary. Suppose that a simultaneous eigenstateof P , J i J i and J , denoted by | Ψ i , is not annihilated by Q β . In the restframe, eq. (4.45) yields [ J i , Q β ] = [ J i , Q † ˙ β ] = 0 , (4.147)so it follows that Q β | Ψ i is also a simultaneous eigenstate of P , J i J i and J . By assumption, Q β | Ψ i is not annihilated by Q α , so we conclude that Q α Q β | Ψ i is also a simultaneous eigenstate of P , J i J i and J . But wenow arrive at a contradiction, since eq. (4.144) yields Q γ ( Q α Q β | Ψ i ) = 0 . (4.148)Consequently, there must be at least one state of the supermultiplet thatsatisfies Q β | Ω i = 0. Using eqs. (4.55) and (4.146), it follows that J i | Ω i = S i | Ω i . (4.149)If S + | Ω i = 0, then it follows that | Ω i is also a simultaneous eigenstate of ~S and S with corresponding eigenvalues j ( j + 1) and j . Moreover, this statemust be unique under the assumption that the superspin j supermultipletis an irreducible representation of the N = 1 supersymmetry algebra.Note that eq. (4.41) when evaluated in the rest frame yields:[ S i , Q α ] = iσ i αβ Q β , [ S i , Q † ˙ α ] = iσ i β ˙ α Q † ˙ β . (4.150)Hence, one can define additional states of the supermultiplet, | Ω( j ) i ≡ ( S − ) j − j | Ω i , for j = − j, − j + 1 , . . . , j − , j , (4.151)all of which satisfy Q α | Ω( j ) i = 0 , (4.152) Recall that if | s, m s i are eigenstates of ~S and S with corresponding eigenvalues s ( s + 1) and m s respectively, then S ± | s, m s i = p ( s ∓ m s )( s ± m s + 1) | s, m s ± i .
56s a result of eq. (4.150). As before, J i | Ω( j ) i = S i | Ω( j ) i as a conse-quence of eqs. (4.55) and (4.152). It follows that | Ω( j ) i is also a simul-taneous eigenstate of ~S and S with corresponding eigenvalues j ( j + 1)and j . That is, | Ω( j ) i = | j, j i , (4.153)where the rest-frame spin and its projection along the z -axis are explicitlyindicated.Starting from | Ω( j ) i = | j, j i , one can now construct the remainingstates of the massive supermultiplet by considering the series of states foreach possible value of j , | Ω( j ) i , Q † ˙ α | Ω( j ) i , Q † ˙ α Q † ˙ β | Ω( j ) i , . . . . Thisseries of states terminates due to eq. (4.144) and only four independentstates survive (for a given fixed value of j ), | Ω( j ) i , Q † | Ω( j ) i , Q † | Ω( j ) i , Q † Q † | Ω( j ) i . (4.154)All the states of eq. (4.154) are mass-degenerate (with mass m = 0).The spins of these states can be determined by applying the operators ~S and S . By virtue of eq. (4.153), we already know that | Ω( j ) i is a spin- j state with S -eigenvalue j . Next, one can use eq. (4.150) to derive:[ S i , Q † ˙ α Q † ˙ β ] = iQ † ˙ γ h σ i γ ˙ α Q † ˙ β − σ i γ ˙ β Q † ˙ α i , (4.155) h ~S , Q † ˙ α Q † ˙ β i = 2 iQ † ˙ γ h σ i γ ˙ α Q † ˙ β − σ i γ ˙ β Q † ˙ α i S i . (4.156)It immediately follows that:[ S i , Q † Q † ] = iQ † Q † Tr σ i = 0 , (4.157) h ~S , Q † Q † i = 2 iQ † Q † S i Tr σ i = 0 . (4.158)Applying eqs. (4.157) and (4.158) to the state | Ω( j ) i , it follows that Q † Q † | Ω( j ) i is also a spin- j state with S -eigenvalue j . This result iseasily understood. Noting that we can write Q † Q † = ǫ ˙ α ˙ β Q † ˙ α Q † ˙ β , (4.159)it follows that Q † Q † is a scalar operator. This is consistent with the factthat the antisymmetric part of the tensor product of two SU(2) spinorrepresentations is an SU(2) singlet. Thus, Q † Q † | Ω( j ) i and | Ω( j ) i possessthe same eigenvalues with respect to ~S and S .To determine the properties of Q † | Ω( j ) i and Q † | Ω( j ) i , we first notethat Q α is a spinor operator that imparts spin- to any state it acts on.Moreover, eq. (4.150) yields: S Q † | Ω( j ) i = ( j + ) Q † | Ω( j ) i , S Q † | Ω( j ) i = ( j − ) Q † | Ω( j ) i . (4.160)57ence, one can employ the standard results from the theory of angular mo-mentum addition in quantum mechanics, which relates the tensor productbasis to the total angular momentum basis. In particular, | j , m i = X m ,m | j , m i ⊗ | j , m i h j j ; m m | j m i , (4.161)where h j j ; m m | j m i are the Clebsch-Gordon (C-G) coefficients. Weemploy the Condon-Shortly phase conventions in which the C-G coefficientsare real and symmetric. In the present application, we require the followingtwo C-G coefficients (taking the upper and lower signs, respectively), (cid:12)(cid:12) , ± (cid:11) ⊗ (cid:12)(cid:12) j , m ∓ (cid:11) = (cid:18) j + ± m j + 1 (cid:19) / (cid:12)(cid:12) j + , m (cid:11) ∓ (cid:18) j + ∓ m j + 1 (cid:19) / (cid:12)(cid:12) j − , m (cid:11) , (4.162)Eqs. (4.153), (4.160), (4.157) and (4.158) imply that | Ω( j ) i = | j , j i , (4.163) Q † | Ω( j ) i = (cid:18) j + j + 12 j + 1 (cid:19) / (cid:12)(cid:12) j + , j + (cid:11) − (cid:18) j − j j + 1 (cid:19) / (cid:12)(cid:12) j − , j + (cid:11) , (4.164) Q † | Ω( j ) i = (cid:18) j − j + 12 j + 1 (cid:19) / (cid:12)(cid:12) j + , j − (cid:11) + (cid:18) j + j j + 1 (cid:19) / (cid:12)(cid:12) j − , j − (cid:11) , (4.165) Q † Q † | Ω( j ) i = | j , j i . (4.166)In particular, if j = j then eqs. (4.164) and (4.165) imply that Q † | Ω( j ) i and Q † | Ω( j ) i are orthogonal linear combinations of spin-( j ± ) states(although these states are eigenstates of S as shown in eq. (4.160)). If j = ± j then Q † | Ω( j ) i and Q † | Ω( − j ) i are states of spin-( j + ), since boththese states are eigenstates of ~S and S with eigenvalues ( j + )( j + )and ± ( j + ), respectively.Note that since [ P , Q α ] = [ P , Q † ˙ α ] = 0, it follows that all the statesof the supermultiplet, | Ω( j ) i , Q † | Ω( j ) i , Q † | Ω( j ) i , Q † Q † | Ω( j ) i ,are mass-degenerate, with common mass m . The states of an N = 1 massivesupermultiplet of superspin j are exhibited in Table 2.In summary, there are 4(2 j + 1) mass-degenerate states in a massivesupermultiplet of superspin j , which are explicitly given by eqs. (4.163)–(4.166), for j = − j, − j +1 , . . . , j − , j . In general, a massive supermultiplet58f superspin j is made up of 2(2 j + 1) states of spin j , 2 j + 2 states of spin( j + ) and 2 j states of spin ( j − ). The extra two states for the case ofspin-(2 j + 1) arise when j = ± j , in which cases Q † | Ω( j ) i and Q † | Ω( − j ) i are pure states of spin ( j + ) as previously noted. Note that the numberof fermionic and bosonic degrees of freedom of the massive supermultipletcoincide and is equal to 2(2 j + 1). These results are summarized in Table 2.4.9.2. States of a massless supermultiplet of superhelicity κ To construct the states of an irreducible massless supermultiplet, we choosethe standard reference frame, P µ = P (1 ; 0 , , P and thesuperhelicity operator K , with eigenvalues m and κ , respectively, wherethe possible values of κ = 0 , ± , ± , ± , . . . .For a fixed value of the superhelicity κ , there exists a distinct state ofthe supermultiplet, denoted by | Ω i , that satisfies: Q β | Ω i = 0 , K | Ω i = κ | Ω i . (4.167)To verify that a state | Ω i exists that is annihilated by Q β , let us assumethe contrary. Suppose that a state of the massless supermultiplet, denotedby | Ψ i exists that is not annihilated by Q β . Due to eq. (4.84), it followthat Q β | Ψ i must also be a state of the massless supermultiplet. Arguingas we did below eq. (4.146), we again arrive at a contradiction. Conse-quently, there must be at least one state of the supermultiplet that satisfies Q β | Ω i = 0. Moreover, a state that satisfies eq. (4.167) must be uniqueunder the assumption that the massless supermultiplet with superhelicity κ is an irreducible representation of the N = 1 SUSY algebra.The states of the massless supermultiplet are obtained by consideringthe series, | Ω i , Q † ˙ α | Ω i , Q † ˙ β Q † ˙ α | Ω i . (4.168)However, Q † ˙ β Q † ˙ α | Ω i = 0 as a result of eq. (4.70), and P λ Q † ˙ β σ ˙ βτλ | Ω i = 0as a consequence of eq. (4.60). Thus, in contrast to the massive super-multiplet, the massless supermultiplet contains only two states. These twostates are eigenvalues of the helicity operator h . To determine the cor-responding helicities, we shall employ the standard reference frame where P µ = P (1 ; 0 , , Q = Q † = 0, it follows59hat the massless N = 1 supermultiplet consists of the two states, | Ω i and Q † | Ω i . Using eqs. (4.83) and (4.167), the helicities of these two states canbe determined, h | Ω i = (cid:20) K − P (cid:16) Q † Q + Q † Q (cid:17)(cid:21) | Ω i = κ | Ω i , (4.169) hQ † | Ω i = (cid:20) K − P (cid:16) Q † Q + Q † Q (cid:17)(cid:21) Q † | Ω i = (cid:20) κQ † − P Q † (cid:16) P µ σ µ − Q † Q (cid:17) − P Q † (cid:16) P µ σ µ − Q † Q (cid:17)(cid:21) | Ω i = (cid:2) κ − ( σ − σ ) (cid:3) Q † | Ω i = ( κ − ) Q † | Ω i . (4.170)Indeed, the superhelicity κ is the maximal helicity of the massless N = 1supermultiplet. Thus, an irreducible N = 1 massless supermultiplet withsuperhelicity κ consists of two massless states with helicity κ and κ − ,respectively. These results are summarized in Table 3. Problems
Problem 5.
Show that the massive j = supermultiplet corresponds to areal vector field, a real scalar field and a Dirac fermion field. Problem 6.
Derive the following three commutation relations: [ B µ , Q α ] = − P µ Q α , [ B µ , Q † ˙ α ] = P µ Q † ˙ α , (4.171)[ B µ , B ν ] = iǫ µνρλ B ρ P λ , (4.172) where B µ is defined in eq. (4.53). Problem 7.
Derive the following twp commutation relations, [ L µ , Q α ] = − ( σ µ σ ν ) αβ Q β P ν , [ L µ , Q † ˙ α ] = ( σ ν σ µ ) ˙ β ˙ α Q † ˙ β P ν , (4.173) where L µ is defined in eq. (4.71). Problem 8.
Show that a massless supermultiplet with κ = 2 and its CPT-conjugates corresponds to a massless spin- and a massless spin 2 particle,which is realized in supergravity by the gravitino and the graviton. Problem 9.
Obtain the explicit form for K µ in eq. (4.117). roblem 10. Obtain an explicit expression for R ( x ) in eq. (4.125), andshow that it vanishes after imposing the classical field equations for ψ α ( x ) .Note that this computation is non-trivial and requires a judicious applicationof Fierz identities for two-component fermions (which can be found, e.g.,in Appendix B of Ref. [1]). Problem 11.
Obtain an explicit expression for J µα in terms of the fields A and ψ in the Wess-Zumino model. Problem 12.
Verify, for the Wess-Zumino model, that the Noether super-charges defined by eq. (4.128) satisfy the SUSY algebra [cf. eq. (4.129)].
Problem 13.
Obtain the explicit form for K ′ µ in eq. (4.140). Problem 14.
Starting from eqs. (4.134)–(4.136), verify that (cid:2) δ η , δ ξ (cid:3) Φ( x ) = − i ( ξσ µ η † − η † σ µ ξ † ) ∂ µ Φ( x ) , for Φ = A , ψ and F without the need to impose the classical field equations.
5. Superspace and Superfields
In the section we introduce superspace coordinates θ and θ † . The conceptof a supersymmetry transformation is then realized as a translation in su-perspace. We construct superfields [105–107], which can be expanded inpowers of θ and θ † ; the corresponding expansion coefficients are the fieldsof a supermultiplet. By introducing the spinor covariant derivative, one isable to define the derivative of a superfield that is covariant with respectto SUSY transformations. This allows us to define an irreducible chiralsuperfield by imposing a derivative constraint.Employing this formalism, we demonstrate how to construct a SUSYLagrangian for chiral superfields, and and show that the supersymmetricaction can be expressed as an integral over superspace. Finally, we discussthe improved ultraviolet behavior of SUSY and introduce the celebratednon-renormalization theorem of N = 1 supersymmetry [108, 109]. Superspace coordinates and translations
In Section 4 we indicated that we expect a SUSY translation to be similarto a space-time translation, where the SUSY generators Q , Q † replace the P µ of ordinary space-time translations: δ ξ Φ( x ) = i (cid:2) ξQ + ξ † Q † , Φ( x ) (cid:3) , (5.1)for Φ = A , ψ or F . But what exactly is being translated?61n this subsection, we extend spacetime by introducing Grassmann co-ordinates, θ α and θ † ˙ α . The result is an 8-dimensional superspace with co-ordinates ( x µ , θ α , θ † ˙ α ). The Grassmann coordinates are anticommutingcoordinates; i.e., they satisfy anticommutation relations, { θ α , θ β } = { θ † ˙ α , θ † ˙ β } = { θ α , θ † ˙ β } = 0 . (5.2)One can also define derivatives with respect to θ and θ † . It is convenientto introduce the following notation, ∂ α ≡ ∂∂θ α , ∂ † ˙ α ≡ ∂∂θ † ˙ α . (5.3)The derivatives with respect to θ and θ † are defined in the obvious way, ∂ α θ β = δ βα , ∂ † ˙ α θ † ˙ β = δ ˙ β ˙ α . (5.4)It then follows that ∂ α θ β = ∂ α ( ǫ βγ θ γ ) = − ǫ αβ , ∂ † ˙ α θ † ˙ β = ∂ † ˙ α ( ǫ ˙ β ˙ γ θ † ˙ γ ) = − ǫ ˙ α ˙ β . (5.5)Derivatives with respect to θ and θ † satisfy a modified Leibniz rule, ∂ α ( f g ) = ( ∂ α f ) g + ( − ε ( f ) f ( ∂ α g ) , (5.6) ∂ † ˙ α ( f g ) = ( ∂ † ˙ α f ) g + ( − ε ( f ) f ( ∂ † ˙ α g ) , (5.7)where ε ( f ) = ( , if f is Grassmann even , , if f is Grassmann odd , (5.8)and f is Grassmann even [odd] if it is a product of an even [odd] numberof anticommuting quantities. For example, ∂ α ( θθ ) = ∂ α (cid:0) ǫ γβ θ γ θ β (cid:1) = ǫ γβ ( δ γα θ β − δ βα θ γ ) = 2 θ α , (5.9) ∂ ˙ α ( θ † θ † ) = ∂ ˙ α (cid:0) ǫ ˙ β ˙ γ θ † ˙ γ θ † ˙ β (cid:1) = ǫ ˙ β ˙ γ ( δ ˙ γ ˙ α θ † ˙ β − δ ˙ β ˙ α θ † ˙ γ ) = − θ † ˙ α . (5.10)Likewise, one conventionally defines, ∂ α ≡ ∂∂θ α , ∂ † ˙ α ≡ ∂∂θ † ˙ α . (5.11)However, one needs to be careful since this notation leads to an unexpectedminus sign when relating the derivatives of eqs. (5.3) and (5.11), ∂ α = − ǫ αβ ∂ β , ∂ † ˙ α = − ǫ ˙ α ˙ β ∂ † ˙ β . (5.12)This is the one case where the rule for raising a spinor index given ineq. (2.24) does not apply. 62n order to define translations in superspace, we shall generalize thetranslation operator exp( ix · P ) to the super-translation operator, G ( x, θ, θ † ) = exp( ix · P + θQ + θ † Q † ) . (5.13)We can now extend the field operator, Φ( x ) = exp( ix · P )Φ(0) exp( − ix · P )to a superfield operator,Φ( x, θ, θ † ) = G ( x, θ, θ † )Φ(0 , , G − ( x, θ, θ † ) . (5.14)In this way, we can realize a supersymmetry transformation as a translationin superspace.Using the Baker-Campbell-Hausdorff formula [110],exp( A ) exp( B ) = exp (cid:0) A + B + [ A , B ] + · · · (cid:1) , (5.15)one can prove (see Problem 15), G ( y, ξ, ξ † ) G ( x, θ, θ † ) = G (cid:0) x + y + i ( ξσθ † − θσξ † ) , ξ + θ, ξ † + θ † (cid:1) . (5.16)Note the appearance in eq. (5.16) of an extra non-trivial spacetime trans-lation, i ( ξσθ † − θσξ † ). Hence, it follows that G ( y, ξ, ξ † )Φ( x, θ, θ † ) G − ( y, ξ, ξ † )= Φ (cid:0) x + y + i ( ξσθ † − θσξ † ) , ξ + θ, ξ † + θ † (cid:1) . (5.17)For infinitesimal y , ξ and ξ † , we can approximate G ( y, ξ, ξ † ) ≃ + i ( y · P + ξQ + ξ † Q † ) , (5.18)which allows us to rewrite the left-hand side of eq. (5.17) as G (cid:0) y, ξ, ξ † (cid:1) Φ (cid:0) x, θ, θ † (cid:1) G − (cid:0) y, ξ, ξ † (cid:1) ≃ (cid:0) + i (cid:0) y · P + ξQ + ξ † Q † (cid:1)(cid:1) Φ (cid:0) x, θ, θ † (cid:1) (cid:0) − i (cid:0) y · P + ξQ + ξ † Q † (cid:1)(cid:1) ≃ Φ (cid:0) x, θ, θ † (cid:1) + iy µ [ P µ , Φ] + i [ ξQ, Φ] + i (cid:2) ξ † Q † , Φ (cid:3) . (5.19)One can also Taylor expand the right-hand side of eq. (5.17), which to firstorder yieldsΦ (cid:0) x + y + i ( ξσθ † − θσξ † ) , ξ + θ, ξ † + θ † (cid:1) = Φ( x, θ, θ † ) + (cid:2) y µ + i ( ξσ µ θ † − θσ µ ξ † ) (cid:3) ∂ µ Φ( x, θ, θ † )+ (cid:0) ξ α ∂ α + ξ † ∂ † ˙ α (cid:1) Φ( x, θ, θ † ) , (5.20)where we have employed the derivatives defined in eq. (5.3). Comparingthe first-order terms of eqns. (5.19) and (5.20), we end up with expressionsfor the following commutators, (cid:2) Φ , P µ (cid:3) = i ∂ µ Φ , (5.21) (cid:2) Φ , ξQ (cid:3) = i ξ α (cid:0) ∂ α + i ( σ µ θ † ) α ∂ µ (cid:1) Φ , (5.22) (cid:2) Φ , ξ † Q † (cid:3) = − i (cid:16) ∂ † ˙ α + i ( θσ µ ) ˙ α ∂ µ (cid:17) ξ † ˙ α Φ , (5.23)63he above results motivate the introduction of the following differentialoperators, b P µ = i∂ µ , (5.24) b Q α = i∂ α − ( σ µ θ † ) α ∂ µ , (5.25) b Q † ˙ α = − i∂ † ˙ α + ( θσ µ ) ˙ α ∂ µ , (5.26)which allow us to succinctly rewrite eqs. (5.21)–(5.23) as follows: (cid:2) Φ , P µ (cid:3) = b P µ Φ , (5.27) (cid:2) Φ , ξQ (cid:3) = ( ξ b Q )Φ , (5.28) (cid:2) Φ , ξ † Q † (cid:3) = ( ξ † b Q † )Φ . (5.29)In eq. (4.121), we noted that the action of an infinitesimal SUSY trans-formation on any field Φ ( x ) was given by δ ξ Φ( x ) = i (cid:2) ξQ + ξ † Q † , Φ( x ) (cid:3) . Inlight of eqs. (5.28) and (5.29), we conclude that the action of an infinitesimalSUSY transformation on a superfield Φ (cid:0) x, θ, θ † (cid:1) is given by δ ξ Φ( x, θ, θ † ) = − i ( ξ b Q + ξ † b Q † )Φ( x, θ, θ † ) . (5.30) Expansion of the superfield in powers of θ and θ † Consider the Taylor expansion of a superfield, Φ( x, θ, θ † ), in powers of θ and θ † . The coefficients of this expansion will be functions of x , whichcan be interpreted as ordinary fields. Since θ and θ † are anticommutingcoordinates, this Taylor series terminates after a finite number of terms. Inparticular, since θ and θ † are anticommuting two-component spinor quan-tities, it follows that ( θ ) = ( θ ) = ( θ † ˙1 ) = ( θ † ˙2 ) = 0, whereas productssuch as θ θ and θ † θ † do not vanish. Indeed, it is easy to check that θ α θ β = − ǫ αβ θθ , θ † ˙ α θ † ˙ β = ǫ ˙ α ˙ β θ † θ † ,θ α θ β = ǫ αβ θθ , θ † ˙ α θ † ˙ β = − ǫ ˙ α ˙ β θ † θ † , where θθ ≡ θ α θ α and θ † θ † ≡ θ † ˙ α θ † ˙ α following the convention of eq. (2.33).Products such as θ α θ β θ γ = 0, since the spinor indices can assume at mosttwo different values. Finally, the following three results are noteworthy (seeProblem 17), ( θσ µ θ † ) θ β = − θθ ( σ µ θ † ) β (5.31)( θσ µ θ † ) θ † ˙ β = − θ † θ † ( θσ µ ) ˙ β (5.32)( θσ µ θ † )( θσ ν θ † ) = g µν ( θθ )( θ † θ † ) . (5.33)64ometimes, we shall write θθθ † θ † ≡ ( θθ )( θ † θ † ). In such products, thereshould be no ambiguity in omitting the parentheses.The Taylor series expansion of a complex superfield Φ( x, θ, θ † ) is there-fore given by,Φ( x, θ, θ † ) = f ( x ) + θζ ( x ) + θ † χ † ( x ) + θθm ( x ) + θ † θ † n ( x ) + θσ µ θ † V µ ( x )+( θθ ) θ † λ † ( x ) + ( θ † θ † ) θλ ( x ) + θθθ † θ † d ( x ) , (5.34)where f , m , n , V µ , and d are complex commuting bosonic fields and ζ , χ , λ and ψ are anticommuting two-component fermionic fields. The SUSYtransformation laws of the component fields can now be easily obtained(see Problem 18) by comparing both sides of eq. (5.30).Hence, there are 16 bosonic and 16 fermionic real degrees of freedom.If we impose the constraint, Φ † = Φ, then f , d and V µ are real bosonicfields, n † = m , ζ = χ and λ = ψ . In this case, there are 8 bosonic and 8fermionic real degrees of freedom. In both cases, there are too many de-grees of freedom to describe the supermultiplet of the Wess-Zumino model.This is because an unconstrained complex superfield, Φ( x, θ, θ † ), describesa reducible representation of the SUSY algebra. One must impose super-symmetric constraints to project out an irreducible supermultiplet. The superfield defined in eq. (5.34) is an example of a bosonic super-field, where the Taylor series coefficients of terms even in the number ofGrassmann coordinates are commuting bosonic fields and the coefficientsof terms odd in the number of Grassmann coordinates are anticommutingfermionic fields. Similarly, one can define a fermionic superfield, where theTaylor series coefficients of terms even in the number of Grassmann coordi-nates are anticommuting fermionic fields and the coefficients of terms oddin the number of Grassmann coordinates are commuting bosonic fields.
Spinor covariant derivatives
For a superfield Φ, it is easy to check that neither ∂ α Φ nor ∂ ˙ α Φ is a super-field, since ∂ α ( δ ξ Φ) = δ ξ ( ∂ α Φ) , ∂ † ˙ α ( δ ξ Φ) = δ ξ ( ∂ † ˙ α Φ) . (5.35)Note that if Φ is a bosonic superfield, then the hermitian conjugate of ∂ α Φis given by, ( ∂ α Φ) † = − ∂ † ˙ α Φ † , (5.36)where the minus sign above is related to the minus sign in eq. (5.5). A real superfield Φ yields an off-shell irreducible representation with superspin j = .More on this in Section 6.
65e therefore introduce spinor covariant derivatives D α and D ˙ α suchthat D α Φ and D ˙ α Φ are superfields, which implies the following conditionsmust be satisfied, D α ( δ ξ Φ) = δ ξ ( D α Φ) , D ˙ α ( δ ξ Φ) = δ ξ ( D ˙ α Φ) . (5.37)Using eq. (5.30) to express δ ξ Φ in terms of the operators b Q and b Q † definedin eqs. (5.25) and (5.26), respectively, one easily derives { D α , b Q β } = { D α , b Q † ˙ β } = { D ˙ α , b Q β } = { D ˙ α , b Q † ˙ β } = 0 . (5.38)To fix the explicit forms for the spinor covariant derivatives, we choosethe normalization of D α so that it has the form D α = ∂ α + . . . , where theellipsis refers to correction terms needed to satisfy eqs. (5.37) and (5.38).In the case of D α , it is customary to impose the condition,( D α Φ) † = D ˙ α Φ † , (5.39)where Φ is a bosonic superfield, in which case D ˙ α = − ∂ † ˙ α + . . . [cf. eq. (5.36)].The explicit forms for the spinor covariant derivatives that satisfy theabove conditions are given by, D α = ∂ α − i ( σ µ θ † ) α ∂ µ , (5.40) D ˙ α = − ∂ † ˙ α + i ( θσ µ ) ˙ α ∂ µ . (5.41)In particular, D and D satisfy the same anticommutation relations as b Q and b Q † (see Problem 21), { D α , D β } = { D ˙ α , D ˙ β } = 0 and { D α , D ˙ β } = 2 iσ µα ˙ β ∂ µ . (5.42)One can also define spinor covariant derivatives with a raised spinorindex. In this case, it is conventional to define, D α ≡ ǫ αβ D β = − ∂ α + i ( θ † σ µ ) α ∂ µ , (5.43) D ˙ α ≡ ǫ ˙ α ˙ β D β = ∂ † ˙ α − i ( σ µ θ ) ˙ α ∂ µ , (5.44)where we have employed eq. (5.12). That is, the spinor indices of D α and D ˙ α are raised in the conventional way according to eq. (2.24). Thefollowing differential operators will be useful later in these lectures, D = D α D α = − ∂ α ∂ α + 2 i ( ∂ α σ µα ˙ β θ † ˙ β ) ∂ µ + θ † θ † (cid:3) , (5.45) D = D ˙ α D ˙ α = − ∂ † ˙ α ∂ † ˙ α + 2 i ( θ α σ µα ˙ β ∂ † ˙ β ) ∂ µ + θθ (cid:3) , (5.46) Note that if Φ is a bosonic superfield, then D α Φ and D ˙ α Φ are fermionic superfields. This is in contrast to the rule for raising the spinor indices of ∂ α and ∂ † ˙ α specified ineq. (5.12), where an extra minus sign appears. (cid:3) ≡ ∂ µ ∂ µ . One can then derive the following identity (see Prob-lem 22), [ D , D ] = 4 iσ µα ˙ β ∂ µ [ D α , D ˙ β ] . (5.47)We have employed different notation for the conjugation of the variousdifferential operators that appear in this subsection. The relation of b Q † to b Q is hermitian conjugation in the same sense that ˆ P µ = i∂ µ [defined ineq. (5.24)] is an hermitian operator in quantum field theory with respect tothe inner product defined by the integration of complex fields over space-time. That is, the dagger on the differential operator b Q † denotes Hermitianconjugation with respect to the inner product defined by the integration ofcomplex superfields over superspace. In contrast, the relation of D to D is complex conjugation in the samesense that ∂ ∗ µ is the complex conjugate of ∂ µ . In the latter case, the dif-ferential operator ∂ µ is a real operator. That is, if we define ∂ ∗ µ to be thederivative operator that acts on the field φ such that( ∂ µ φ ) † = ∂ ∗ µ φ † , (5.48)then since ( ∂ µ φ ) † = ∂ µ φ † , it follows that ∂ ∗ µ = ∂ µ . In light of eq. (5.39), wecan therefore regard D as the complex conjugate of D . Chiral superfields
A chiral superfield is obtained by imposing the constraint D ˙ α Φ = 0 on ageneral superfield Φ. Such a constraint is covariant with respect to SUSYtransformations, and the end result is an irreducible superfield that cor-responds to the superspin j = 0 irreducible representation of the SUSYalgebra. Using eq. (5.41), the constraint yields a differential equation, D ˙ α Φ = (cid:2) − ∂ † ˙ α + i ( θσ µ ) ˙ α ∂ µ (cid:3) Φ( x, θ, θ † ) = 0 , (5.49)whose solution is of the formΦ( x, θ, θ † ) = exp( − iθσ µ θ † ∂ µ )Φ( x, θ ) . (5.50)We can expand Φ( x, θ ) in a (truncated) Taylor series in θ ,Φ( x, θ ) = A ( x ) + √ θψ ( x ) + θθF ( x ) , (5.51)where the factor of √ − iθσ µ θ † ∂ µ ) = 1 − iθσ µ θ † ∂ µ − ( θθ )( θ † θ † ) (cid:3) , (5.52) For further details, see Refs. [33, 38]. Integration over superspace will be treated inSection 5.7.
67e find after some algebraic manipulation a chiral superfield with the form,Φ( x, θ, θ † ) = A ( x ) + √ θψ ( x ) + θθF ( x ) − iθσ µ θ † ∂ µ A ( x ) − i √ θθ ) θ † σ µ ∂ µ ψ ( x ) − ( θθ )( θ † θ † ) (cid:3) A ( x ) . (5.53)Note that the chiral superfield Φ has dimension [Φ] = 1, in which caseit follows that the dimensions of the component fields are [ A ] = 1 and[ ψ ] = , as expected, whereas [ F ] = 2 after making use of the dimensionsof the Grassmann coordinates, [ θ ] = [ θ † ] = − .Given a chiral superfield Φ, its hermitian conjugate, Φ † , is an antichiralsuperfield, which is defined by the SUSY-covariant constraint, D α Φ † = 0.Using eq. (5.40), the latter constraint yields a differential equation, D α Φ † = (cid:2) ∂ α − i ( σ µ θ † ) α ∂ µ (cid:3) Φ † ( x, θ, θ † ) = 0 , (5.54)whose solution is of the formΦ † ( x, θ, θ † ) = exp( iθσ µ θ † ∂ µ )Φ † ( x, θ † ) . (5.55)We can expand Φ † ( x, θ † ) in a (truncated) Taylor series in θ † ,Φ † ( x, θ † ) = A † ( x ) + √ θ † ψ † ( x ) + θ † θ † F † ( x ) . (5.56)Plugging this result into eq. (5.55) and following the same procedure asbefore, we end up with,Φ † ( x, θ, θ † ) = A † ( x ) + √ θ † ψ † ( x ) + θ † θ † F † ( x ) + iθσ µ θ † ∂ µ A † ( x ) − i √ θ † θ † ) θσ µ ∂ µ ψ † ( x ) − ( θθ )( θ † θ † ) (cid:3) A † ( x ) . (5.57)Since Φ † is the hermitian conjugate of Φ, we can identify A † , ψ † and F † asthe hermitian conjugates of A , ψ and F .In calculations, it is often simpler to employ the so-called chiral repre-sentation , in which all superfield operators O are modified according to O chiral = exp( iθσ µ θ † ∂ µ ) O exp( − iθσ µ θ † ∂ µ ) . (5.58)In the chiral representation, b Q α = i∂ α , b Q † ˙ α = − i∂ † ˙ α + 2( θσ µ ) ˙ α ∂ µ , (5.59) D ˙ α = − ∂ † ˙ α , D α = ∂ α − i ( σ µ θ † ) α ∂ µ . (5.60)Thus, in the chiral representation, the requirement D ˙ α Φ = − ∂ † ˙ α Φ = 0 issimply the requirement that Φ is independent of θ † . In the chiral represen-tation, the chiral superfield will be denoted byΦ ( x, θ ) = A ( x ) + √ θψ ( x ) + θθF ( x ) . (5.61)68t then follows that the general expression for a chiral superfield isΦ( x, θ, θ † ) = exp( − iθσ µ θ † ∂ µ )Φ ( x, θ ) = Φ ( x − iθσ µ θ † , θ ) . (5.62)It is convenient to define the shifted spacetime coordinate, y ≡ x − iθσ µ θ † , (5.63)so that the chiral superfield is given by,Φ (cid:0) x, θ, θ † (cid:1) = Φ ( y, θ ) . (5.64)The SUSY transformation laws for the fields that appear in the chiralsuperfield can now be determined simply by inserting the expression for Φin the chiral representation given by eq. (5.61) into eq. (5.30). In performingthe computation, one employs the chiral representation expressions for b Q and b Q † given in eq. (5.59). You may verify (see Problem 24) that theresult of this calculation coincides with the SUSY transformation laws givenpreviously in eqs. (4.134)–(4.136).Likewise, one can define an antichiral representation in which O antichiral = exp( − iθσ µ θ † ∂ µ ) O exp( iθσ µ θ † ∂ µ ) . (5.65)In the antichiral representation, b Q † ˙ α = − i∂ † ˙ α , b Q α = i∂ α − σ µ θ † ) α ∂ µ ,D α = ∂ α , D ˙ α = − ∂ † ˙ α + 2 i ( θσ µ ) ˙ α ∂ µ . (5.66)Thus, in the antichiral representation, the requirement D α Φ † = ∂ α Φ † = 0is simply the requirement that Φ † is independent of θ . In the antichiralrepresentation, the antichiral superfield will be denoted byΦ ( x, θ † ) = A † ( x ) + √ θ † ψ † ( x ) + θ † θ † F † ( x ) . (5.67)It then follows that the general expression for an antichiral superfield isΦ † ( x, θ, θ † ) = exp( iθσ µ θ † ∂ µ )Φ ( x, θ † ) = Φ ( x + iθσ µ θ † , θ † ) . (5.68)It is convenient to define the shifted spacetime coordinate, y † ≡ x + iθσ µ θ † , (5.69)so that the antichiral superfield is given by,Φ † (cid:0) x, θ, θ † (cid:1) = Φ (cid:0) y † , θ † (cid:1) . (5.70)69 .5. Constructing the SUSY Lagrangian F -terms Ultimately, our goal is to construct an action that is invariant under SUSY.It is therefore sufficient to construct a Lagrangian that transforms underSUSY as a total derivative. In the literature, it is common to use thenomenclature F -term to denote the coefficient of the θθ term of a superfield.This is sometimes explicitly indicated as follows,[Φ] θθ = [Φ] F = F. (5.71)Recall that in eq. (4.136), we demonstrated that the auxiliary field F ( x )transforms as a total derivative under the SUSY transformation laws. But,this field is simply the coefficient of the θθ term of a chiral superfield! In-deed, the F -term of any chiral superfield transforms under a SUSY trans-formation as a total derivative. This means that such terms (and theirhermitian conjugates) are candidates for terms in a Lagrangian, which thenyields an action that is invariant under SUSY.To discover the relevant F -terms for constructing a SUSY Lagrangian,we first prove an important theorem. Theorem 1.
For any positive integers n and m , if Φ is a chiral superfield,then so is Φ n , whereas Φ n (Φ † ) m is not a chiral superfield. Proof.
We first note that D ˙ α Φ n = n Φ n − D ˙ α Φ = 0 , (5.72)which shows Φ n satisfies the defining constraint of a chiral superfield. Asimilar computation shows that Φ n (Φ † ) m does not satisfy the required con-straint.An important consequence of the above theorem is that X n ≥ [ a n Φ n ] F + h . c . (5.73)is a Lorentz scalar that transforms as a total divergence, and thus is acandidate for terms in a Lagrangian whose action is invariant under SUSY.5.5.2. Kinetic terms
To construct the kinetic terms of the SUSY Lagrangian, we define theoperator T , T Φ = − D Φ † , (5.74)70here D ≡ D ˙ α D ˙ α . Note that D ˙ α ( T Φ) = 0 (due to the anticommutationrelations satisfied by D ), so that T Φ is a chiral superfield. In the chiralrepresentation, with Φ = A + √ θψ + θθF , T Φ = F † − i √ θσ µ ∂ µ ψ † − θθ (cid:3) A † . (5.75)Hence, the F -component of Φ T Φ is given by,[Φ T Φ] F = − A (cid:3) A † + F † F + iψσ µ ∂ µ ψ † = ( ∂ µ A )( ∂ µ A † ) + F † F + iψ † σ µ ∂ µ ψ + total derivative , (5.76)which we recognize as the kinetic energy term of the Wess-Zumino La-grangian [cf. eq. (4.130)].5.5.3. Mass terms
To construct the mass terms of the SUSY Lagrangian, the following theoremis useful.
Theorem 2.
For any chiral superfield Φ , [Φ] F = − D Φ (cid:12)(cid:12)(cid:12)(cid:12) θ = θ † =0 = ∂ α ∂ α Φ (cid:12)(cid:12)(cid:12)(cid:12) θ = θ † =0 . (5.77) Proof.
Eq. (5.77) follows immediately from eq. (5.45).We can compute the F term of any holomorphic function of a chiral super-field, W (Φ), as follows. After making judicious use of the chain rule,[ W (Φ)] F = ∂ α ∂ α W (cid:12)(cid:12)(cid:12)(cid:12) θ = θ † =0 = ∂ α dWd Φ ∂ α Φ (cid:12)(cid:12)(cid:12)(cid:12) θ = θ † =0 = 14 (cid:26)(cid:18) d Wd Φ ∂ α Φ ∂ α Φ (cid:19) + dWd Φ ∂ α ∂ α Φ (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) θ = θ † =0 . (5.78)Noting that ( ∂ α Φ ∂ α Φ) θ = θ † =0 = − ψψ , eq. (5.78) yields,[ W (Φ)] F = − (cid:18) d Wd Φ (cid:19) Φ= A ψψ + (cid:18) dWd Φ (cid:19) Φ= A F . (5.79)Introducing the notation, dW/dA ≡ ( dW/d Φ) Φ= A , it follows that[ W (Φ)] F = − d WdA ψψ + dWdA F . (5.80)In the jargon of SUSY, W (Φ) is called the superpotential . For renormaliz-able theories, W (Φ) is at most cubic in Φ.71.5.4. The Wess-Zumino SUSY Lagrangian using F -terms Collecting the results of eqs. (5.76) and (5.80), we end up with, L = [Φ T Φ] F + (cid:8) [ W (Φ)] F + h . c . (cid:9) = ( ∂ µ A ) † ( ∂ µ A ) + iψ † σ µ ∂ µ ψ + F dWdA + F † (cid:18) dWdA (cid:19) † + F † F − " d WdA ψψ + (cid:18) d WdA (cid:19) † ψ † ψ † , (5.81)after dropping total derivative terms. We have thus recovered the Wess-Zumino Lagrangian that was previously written down in eq. (4.109).The proof that the Wess-Zumino action is supersymmetric, or equiva-lently, δ ξ L = ∂ µ K ′ µ , is now trivial since L was constructed from F -terms,which transform as total derivatives under SUSY transformations.5.5.5. An alternate form for the kinetic terms: D -terms and theK¨ahler potential The approach of subsection 5.5.2 is not the only supersymmetric way toconstruct the kinetic energy terms. Consider an unconstrained superfield V ( x, θ, θ † ). Expanding V as a Taylor series in θ and θ † , the highest ordernonvanishing term is proportional to ( θθ )( θ † θ † ). If we write V ( x, θ, θ † ) = · · · + ( θθ )( θ † θ † ) D ( x ) , (5.82)then one can show that δ ξ D ( x ) is a total derivative using dimensional anal-ysis as we did for δ ξ F ( x ) at the end of Section 4.6. Hence, D -terms canalso provide suitable terms for a SUSY Lagrangian.We shall denote the D -term by,[ V ] θθθ † θ † = [ V ] D = D , (5.83)using a notation analogous to that of eq. (5.71). The relevant theoremanalogous to eq. (5.77) is given below.
Theorem 3.
For any superfield V , [ V ] D = D D V (cid:12)(cid:12)(cid:12)(cid:12) θ = θ † =0 = ( ∂ † ˙ α ∂ † ˙ α )( ∂ α ∂ α ) V (cid:12)(cid:12)(cid:12)(cid:12) θ = θ † =0 . (5.84) Proof.
Eq. (5.84) follows immediately from eqs. (5.45) and (5.46).72or example, if Φ is a chiral superfield, one can show that (see Problem 25),[Φ † Φ] D = ( ∂ µ A )( ∂ µ A † ) + F † F + iψ † σ µ ∂ µ ψ + total derivative , (5.85)which again reproduces the kinetic energy terms of the Wess-Zumino La-grangian.Indeed, one can obtain candidate terms for a SUSY Lagrangian by con-sidering the θθθ † θ † component of an arbitrary function of a chiral superfieldand its complex conjugate. This function, denoted by K (Φ , Φ † ), is calledthe K¨ahler potential. Applying the chain rule as in our computation of[ W (Φ)] F [cf. eqs. (5.78)–(5.80)], one can calculate (see Problem 26),[ K (Φ , Φ † )] D = ∂ K∂A∂A † (cid:20) ( ∂ µ A )( ∂ µ A † ) + F † F + iψ † σ µ ↔ ∂ µ ψ (cid:21) − ∂ K∂A∂A † (cid:20) F ψ † ψ † + iψ † σ µ ψ∂ µ A † (cid:21) − ∂ K∂A ∂A † (cid:20) F † ψψ − iψ † σ µ ψ∂ µ A (cid:21) + 14 ∂ K∂A ∂A † ( ψψ )( ψ † ψ † ) + total derivative . (5.86)We conclude that the most general SUSY Lagrangian involving a chiralsuperfield Φ is given by L = [ K (Φ , Φ † )] D + (cid:8) [ W (Φ)] F + h . c . (cid:9) . (5.87)The auxiliary field F can be determined via its classical field equation,which yields F = (cid:18) ∂ K∂A∂A † (cid:19) − " ∂ K∂A ∂A † ψψ − (cid:18) dWdA (cid:19) † . (5.88)The case of K (Φ , Φ † ) = Φ † Φ reduces to the result of eq. (5.85) andcorresponds to the kinetic energy term of the Wess-Zumino model as notedabove. In this case, eq. (5.88) yields, F = − (cid:18) dWdA (cid:19) † , (5.89)which reproduces the result previously obtained in eq. (4.132).More complicated K¨ahler potentials yield non-renormalizable La-grangians. These arise in low-energy effective field theories (that includeoperators of dimension greater than four), in supersymmetric σ -models, andin supergravity. Such applications lie beyond the scope of these lectures.73 .6. R -invariance Recall that the SUSY algebra can be extended by added adding a bosonicU(1) R generator R such that [cf. eqs. (4.36)–(4.39)],[ R , Q α ] = − Q α , h R , Q † ˙ α i = Q † ˙ α . (5.90)The action of U(1) R on a superfield Φ can be represented by a differentialoperator b R acting on superspace,[Φ , R ] = b R Φ , (5.91)where b R ≡ θ α ∂ α − θ † ˙ α ∂ † ˙ α − n , with n ∈ R . (5.92)We call n the weight (or R -charge) of the superfield Φ. (For a real superfield,only n = 0 is possible.) Under a U(1) R transformation, δ a Φ = ia [ R ,
Φ] = − ia b R Φ . (5.93)Acting on a superfield Φ( x, θ, θ † ), b R Φ( x, θ, θ † ) = e ina Φ( x, e − ia θ, e ia θ † ) , (5.94)The differential operator b R satisfies the identities, D α b R = ( b R + 1) D α , (5.95) D ˙ α b R = ( b R − D ˙ α . (5.96)Hence, it follows that if Φ is a chiral [antichiral] superfield, then b R Φ is achiral [antichiral] superfield.Given a chiral superfield, Φ = A + √ θψ + θθF , in the chiral represen-tation, the U(1) R transformations of the component fields are: A → e ina A , (5.97) ψ → e i ( n − a ψ , (5.98) F → e i ( n − a F , (5.99)after employing eq. (5.94).
Theorem 4.
The kinetic energy term [Φ † Φ] D is automatically R -invariant,whereas [ W (Φ)] F is R -invariant if and only if W has R -charge equal to 2. Proof. If n = 2, then F is invariant under a U(1) R transformation, in lightof eq. (5.99). This result applies to any F -term. Example 7 (Wess-Zumino model with W (Φ) = m Φ + g Φ ). If m = 0, then the Wess-Zumino model is R -invariant with n = . If g = 0, then the Wess-Zumino model is R -invariant with n = . If both m = 0 and g = 0, then the Wess-Zumino model is not R -invariant.74 .7. Grassmann integration and the SUSY action
A supersymmetric action can be written as an integral over superspace.First, we introduce integration over anticommuting Grassmann variables.The rules of integration are [111], Z dθ = Z dθ † = 0 , Z θ dθ = Z θ † dθ † = 1 . (5.100)That is, integration over Grassmann variables is in some sense equivalentto differentiation.It is conventional to define d θ ≡ − dθ α dθ α , (5.101) d θ † ≡ − dθ † ˙ α dθ † ˙ α , (5.102) d θ ≡ d θd θ † , (5.103)which yields the following non-zero integrals, Z d θ ( θθ ) = Z d θ † ( θ † θ † ) = Z d θ ( θθ )( θ † θ † ) = 1 . (5.104)It follows that for a chiral superfield, Z d θ Φ( x, θ, θ † ) = Z d θ Φ ( x, θ ) = [Φ] F = − D Φ (cid:12)(cid:12)(cid:12)(cid:12) θ = θ † =0 . (5.105)Likewise, for an arbitrary superfield V ( x, θ, θ † ), Z d θ V ( x, θ, θ † ) = [ V ] D = D D V (cid:12)(cid:12)(cid:12)(cid:12) θ = θ † =0 . (5.106)Thus, the most general SUSY action involving a chiral superfield Φ is S = Z d x d θK (Φ , Φ † ) + Z d x d θ W (Φ) + Z d x d θ † W (Φ † ) . (5.107)Generalizations to theories with multiple chiral superfields are straight-forward. In the more general case, W is a holomorphic multivariable func-tion of the chiral superfields, and K is a multivariable function of the chiralsuperfields and their hermitian conjugates. For a renormalizable theory, W is at most a cubic multinomial, W (Φ i ) = X i a i Φ i + X i,j b ij Φ i Φ j + X i,j,k c ijk Φ i Φ j Φ k , (5.108)and K (Φ i , Φ † i ) = X i Φ † i Φ i . (5.109)75n special cases, one can convert an integral over “half” of superspace(e.g. integrals over d x d θ ) into an integral over the full superspace. Thekey observation is that for an arbitrary superfield V , Z d x d θ V ( x, θ, θ † ) = Z d x (cid:0) − D V (cid:1) . (5.110)On the left-hand side of eq. (5.110), the integration over d θ projects outall terms proportional to θθ . On the right-hand side, D = − ∂ α ∂ α up tototal derivative terms that can be dropped because we are integrating over d x . Hence, ∂ α ∂ α has the effect of projecting out all terms proportionalto θθ . Likewise, Z d x d θ † V ( x, θ, θ † ) = Z d x (cid:0) − D V (cid:1) . (5.111)Hence, it follows that Z d x d θ (cid:0) − D V (cid:1) = Z d x d θ V ( x, θ, θ † ) . (5.112)Eqs. (5.105) and (5.106) identify integrals over half of superspace as F -terms and integrals over the full superspace as D -terms. However,eq. (5.112) appears to blur the distinction between D -terms and F -terms.For example, in the Wess-Zumino Lagrangian, the kinetic energy term maybe written as an F -term, [Φ T Φ] F [cf. eq. (5.81)], or as a D -term, [Φ † Φ] D ,as in eqs. (5.85) and (5.87). However, consider the case of a half superspaceintegral of the superpotential given in eq. (5.107). If we attempt to convertthis into a full superspace integral using eq. (5.112), the end result is Z d x d θ W (Φ) = − Z d x d θ D − W (Φ) . (5.113)Due to the inverse differential operator, the integrand on the right-handside of eq. (5.113) is a non-local functional of chiral superfields. This pro-vides the distinction between F -terms and D -terms. In particular, any halfsuperspace integral that can be converted into a full superspace integralover a local functional of superfields will be called a D -term.Having written the action in eq. (5.107) as an integral over superspace(for D -terms) and half of superspace (for F -terms), one can obtain expres-sions for the Green functions of quantum chiral (and antichiral) superfields.The corresponding two-point functions provide expressions for the super-space propagators. One can then formulate a set of superspace Feynmanrules and develop a diagrammatic representation of the perturbative expan-sion of the Green functions. This was first carried out by Grisaru, Ro˘cek,76nd Siegel [108], and was applied to the perturbative computation of theeffective action. Indeed, such techniques are quite useful since a single su-pergraph (in which individual lines correspond to superfields) is equivalentto a large number of Feynman diagrams involving the corresponding com-ponent fields. A comprehensive treatment of these methods are beyond thescope of these lectures. For a pedagogical development of supergraphs andsuperspace Feynman rules, see e.g. Refs. [3, 4, 11, 112]. Improved ultraviolet behavior of supersymmetry
An attractive feature of supersymmetric quantum field theories is that theirultraviolet divergences are better behaved, as compared to ordinary quan-tum field theories. Ref. [108] demonstrated that the loop corrections tothe effective action of a supersymmetric theory of chiral superfields can beexpressed as an integral over the full superspace, X n Z d x · · · d x n Z d θ g n ( x , . . . , x n ) F ( x , θ, θ † ) · · · F n ( x n , θ, θ † ) , (5.114)where the F i ( x i , θ, θ † ) are local functionals of chiral and antichiral super-fields and their covariant derivatives, and the g n are translationally invari-ant functions on Minkowski space.Eq. (5.114) implies that D -terms are renormalized but F -terms are notrenormalized. Moreover, if F -terms are absent at tree-level, then they arenot generated at the loop level. Hence, the tree-level K¨ahler potential isrenormalized by radiative corrections, whereas there are no loop correctionsto the tree-level superpotential. This is the famous non-renormalizationtheorem of N = 1 supersymmetry. The proof of the non-renormalizationtheorem in Ref. [108] relies on the analysis of supergraphs in perturbationtheory, and is beyond the scope of these lectures. Heuristically, this theoremis a consequence of an exact cancellation between fermion and boson loopcontributions to the effective action due to supersymmetry.Note that the non-renormalization of the tree-level superpotential issimply a consequence of the fact that the integral of a product of chiral The proof of the non-renormalization theorem implicitly assumes that the function g n in eq. (5.114) is local. However, the non-renormalization theorem can fail if the super-symmetric theory contains massless fields as shown in Refs. [113–115], due to infrareddivergences. For example, the inverse Laplacian operator (cid:3) − (from a massless prop-agator) can appear, resulting in a non-local function g n in eq. (5.114). One can showthat the non-renormalization theorem holds for the Wilsonian effective action [116, 117],where the infrared effects are cut off [109, 118]. all of superspace in eq. (5.114) is zero due to eq. (5.100)[see Problem 29]. Moreover, the assumption that the F i in eq. (5.114) are local functionals of chiral and antichiral superfields is essential. Otherwise,one could employ eq. (5.113) and erroneously claim the existence of loopcorrections to the tree-level superpotential.We now briefly explore the consequence of the non-renormalization ofthe superpotential. Consider the action of the Wess-Zumino model, S WZ = Z d x Z d θ Φ † Φ + (cid:20)Z d x Z d θ (cid:0) m Φ + λ Φ (cid:1) + h . c . (cid:21) . (5.115)The non-renormalization theorem implies that renormalized fields and pa-rameters are related to bare fields and parameters as follows [119],Φ R = Z − / Φ , m R = Zm , λ R = Z / λ . (5.116)where the subscript R indicates renormalized quantities and the bare quan-tities have no subscript. Eq. (5.116) is equivalent to the statement that thesuperpotential is unrenormalized, W R (Φ R ) = W (Φ). That is, m R Φ R + λ R Φ R = m Φ + λ Φ . (5.117)Wave function renormalization is a consequence of the renormalization ofthe K¨ahler potential (Φ † Φ in the case of the Wess-Zumino model).The non-renormalization theorem does not assert that the parameters ofthe superpotential are not renormalized. Indeed, eq. (5.116) states that therenormalization of the parameters m and λ are governed by the wave func-tion renormalization constant Z . Moreover, the wave function renormal-ization constants of the component fields of the chiral superfield are equal(i.e., A R = Z − / A and ψ R = Z − / ψ ), as a consequence of supersymmetry.In Ref. [109], Seiberg offered a more intuitive understanding of the non-renormalization theorem, which also forbids nonperturbative corrections tothe Wilsonian effective action [cf. footnote 32]. Seiberg’s argument drawson the symmetry and holomorphy of the superpotential. Consider againthe example of the Wess-Zumino superpotential, W (Φ) = m Φ + λ Φ .Following Ref. [109], one can think of m and λ as the vacuum expectationvalues of chiral superfields, so that W must be holomorphic in m and λ aswell as in Φ. In light of Theorem 4 in Section 5.6, the theory is invariantunder an enhanced U(1) × U(1) R symmetry, with the charge assignmentsshown in Table 4. The fact that the superpotential is a holomorphic function of chiral superfields playsa critical role in Seiberg’s argument. In contrast, the renormalization of the K¨ahlerpotential is possible because the latter is a function of chiral and antichiral superfieldsand hence is not holomorphic. able 4.: Charge assignments under the U(1) × U(1) R symmetry. Φ Φ † m λ U(1) 1 − − − R − × U (1) R symmetry and holomorphy, correctionsto the Wilsonian effective superpotential must therefore be of the form m Φ f (cid:18) λ Φ m (cid:19) , (5.118)where f is an arbitrary holomorphic function. Eq. (5.118) is valid for ar-bitrary λ . Thus, we can take | λ | ≪
1, in which case perturbation theoryshould be valid. Expanding in powers of the coupling constant λ , the per-turbative expansion should have the following form, W eff = ∞ X n =0 a n λ n m n − Φ n +2 . (5.119)The terms in W eff are represented diagrammatically by one particle irre-ducible (1PI) supergraphs constructed from propagators and three-pointvertices proportional to λ . However, one cannot construct a one-loop (orhigher) supergraph that behaves like λ n Φ n +2 . It is easy to show that tree-level diagrams with n + 2 external legs, n vertices and n − λ n Φ n +2 . But, the only 1PI tree-level graphs are thosewith either two or three external legs! Hence, we conclude that a = , a = and a n = 0 for n ≥ That is W eff (Φ) = W tree (Φ), which is thestatement that the superpotential is not renormalized. Problems
Problem 15.
Prove that G ( y, ξ, ξ † ) G ( x, θ, θ † ) = G (cid:0) x + y + i ( ξσθ † − θσξ † ) , ξ + θ, ξ † + θ † (cid:1) . HINT : use the Baker-Campbell-Hausdorff formula given in eq. (5.15).
Problem 16.
Verify that when acting on a superfield Φ( x, θ, θ † ) , { b Q α , b Q β } = { b Q † ˙ α , b Q † ˙ β } = 0 , { b Q α , b Q † ˙ β } = 2 σ µα ˙ β b P µ . One can also conclude that a n = 0 for n ≥ W eff must have a smooth limit as m → roblem 17. Prove eqs. (5.31)–(5.33). The last result is an example of aFierz identity (see, e.g., Appendix B of Ref. [1] or Appendix A of Ref. [10]).
Problem 18.
Using eq. (5.30), obtain the SUSY transformation laws forthe bosonic component fields, f , m , n , V µ , and d , and the fermionic com-ponent fields, ζ , χ , λ and ψ , which appear in the complex superfield definedin eq. (5.34). Problem 19.
Suppose that Φ is a bosonic superfield. Verify that eq. (5.36)holds. Then, show that eqs. (5.40) and (5.41) satisfy eq. (5.39). Problem 20.
Suppose that Φ is a fermionic superfield. Show thateqs. (5.36) and (5.39) are modified as follows: ( ∂ α Φ) † = ∂ † ˙ α Φ † and ( D α Φ) † = − D ˙ α Φ † . Problem 21.
Show that the spinor covariant derivatives, as defined ineq. (5.40) and eq. (5.41), satisfy the following anticommutation relations, { D α , D β } = { D ˙ α , D ˙ β } = 0 and { D α , D ˙ β } = 2 iσ µα ˙ β ∂ µ . Problem 22.
Derive eq. (5.47).
Problem 23.
Prove that exp( − iθσ µ θ † ∂ µ ) = 1 − iθσ µ θ † ∂ µ − ( θθ )( θ † θ † ) (cid:3) , where (cid:3) ≡ ∂ µ ∂ µ . Problem 24.
Using eq. (5.30), one can obtain the SUSY transformationlaws for the component fields A , ψ and F in eq. (5.53). Perform the calcula-tion by working in the chiral representation and show that the SUSY trans-formation laws for A , ψ and F coincide with the results obtained previouslyin eqs. (4.134)–(4.136) for the fields of a superspin j = 0 supermultiplet. Problem 25. If Φ is a chiral superfield, show that [Φ ∗ Φ] D = ( ∂ µ A )( ∂ µ A ∗ ) + F ∗ F + iψ † σ µ ∂ µ ψ + total derivative . Problem 26.
Derive eq. (5.86).
Problem 27.
A linear superfield [105, 106], L ( x, θ, ¯ θ ) , is defined as a con-strained real scalar superfield that satisfies, D L ( x, θ, ¯ θ ) = D L ( x, θ, ¯ θ ) = 0 .Identify the component fields that make up the linear superfield. Show that ∂ µ V µ = 0 , where V µ is the component vector field of L . Check that thenumber of fermion and boson degrees of freedom of the linear superfield areequal. [HINT: the identity given by eq. (5.47) should be helpful.] roblem 28. Employing the operator T defined in eq. (5.74), show that Z d x d θ Φ T Φ = Z d x d θ Φ † Φ , (5.120) by converting the integral over half of superspace into an integral over thefull superspace. Use the above result to conclude that [Φ T Φ] F = [Φ † Φ] D . Problem 29. If Φ is a chiral superfield and Φ † is an antichiral superfield,show that Z d x d θ Φ( x, θ, θ † ) = Z d x d θ Φ † ( x, θ, θ † ) = 0 . (5.121)
6. Supersymmetric gauge theories
In this section, we discuss the supersymmetric extension of gauge theories.We begin with the vector superfield V , which contains the gauge fields aswell as their supersymmetric partners, the gauginos. We discuss the behav-ior of V under a gauge transformation, and the gauge-invariant interactionterms that couple the vector superfield with one or more chiral superfields.Both abelian and non-abelian gauge groups are treated. Finally, we con-struct the SUSY Lagrangians corresponding to QED and a non-AbelianSUSY Yang-Mill theory coupled to supersymmetric matter. Vector superfields
Imposing a reality condition on a complex superfield (which is a covariantconstraint with respect to SUSY transformations), we obtain the so-calledreal vector superfield, V ( x, θ, θ † ) = V † ( x, θ, θ † ) , (6.1)which will be employed in constructing supersymmetric gauge theories. Ex-panding in θ and θ † , V = C + iθχ − iθ † χ † + iθθ ( M + iN ) − iθ † θ † ( M − iN ) + θσ µ θ † V µ + i ( θθ ) θ † (cid:0) λ † − i σ µ ∂ µ χ (cid:1) − i ( θ † θ † ) θ ( λ − iσ µ ∂ µ χ † (cid:1) + ( θθ )( θ † θ † ) (cid:0) D − (cid:3) C (cid:1) , (6.2)where C , M , N , D and V µ are real bosonic fields, and χ and λ are two-component fermion fields. The various factors of i and are conventional,and the particular linear combination of fields chosen as coefficients of( θθ ) θ † , ( θ † θ † ) θ and ( θθ )( θ † θ † ) are convenient for later purposes [cf. foot-note 38]. Note that the superfield V is dimensionless, in which case it follows81hat the dimensions of the component fields are [ V µ ] = 1 and [ λ ] = , asexpected, whereas [ C ] = [ D ] = 0, and [ χ ] = after making use of thedimensions of the Grassmann coordinates, [ θ ] = [ θ † ] = − .The real vector field V µ is a candidate for a gauge boson of an abelianU(1) gauge theory. The corresponding field strength tensor is given by F µν = ∂ µ V ν − ∂ ν V µ . (6.3)Indeed, this can be shown to be one of the components of the field strengthsuperfield , which is defined by W α = − D D α V . (6.4)Note that D ˙ β W α = 0, so that W α is a spinor chiral superfield. Evaluatingthe above expression, and expressing it in the chiral representation, W α ( y, θ, θ † ) = − iλ α + θ α D − i ( σ µ σ ν θ ) α F µν − θθ ( σ µ ∂ µ λ † ) α , (6.5)where y ≡ x − iθσ µ θ † . The fermionic partner of the gauge boson, called thegaugino, is represented by the two-component spinor field λ . Remarkably,the fields C , M , N and χ that are coefficients in the Taylor expansion ofthe vector superfield V do not appear in eq. (6.5). The reason for this willbecome apparent in Section 6.2.One can work out the SUSY transformation laws of the fields, λ , F µν and D , by matching component fields on both sides of the following equa-tion, δ ξ W α = − i ( ξ b Q + ξ † b Q † ) W α . (6.6)The end result is δ ξ λ α = iξ α D + ( σ µ σ ν ) αβ ξ β F µν , (6.7) δ ξ F µν = i∂ µ ( ξσ ν λ † − λσ ν ξ † ) − i∂ ν ( ξσ µ λ † − λσ µ ξ † ) , (6.8) δ ξ D = ∂ µ ( ξσ µ λ † + λσ µ ξ † ) . (6.9)Note that the mass dimension of the D -term is given by [ D ] = 2. Hence,dimensional analysis implies that δ ξ D must be a total derivative, which isconfirmed in eq. (6.9). From the above transformation laws, we concludethat ( λ , λ † , F µν , D ) forms an irreducible supermultiplet (correspondingto superhelicity 1).To obtain the Lagrangian for the SUSY U(1) gauge theory, note that [ W α W α ] F + h . c . = i ( λσ µ ∂ µ λ † + λ † σ µ ∂ µ λ ) + D − F µν F µν = iλ † σ µ ∂ µ λ + D − F µν F µν + total derivative . (6.10)82his is the kinetic energy term for a U(1) gauge field V µ and its gauginosuperpartner λ . Both the gauge boson and gaugino are massless. The realscalar field D is not dynamical; it is an auxiliary field.The action corresponding to the Lagrangian of eq. (6.10) can be writtenas an integral over half of superspace. In particular, eq. (5.105) yields, L = Z d θ W α W α + h . c . (6.11)One can show that [ W α W α ] F and its hermitian conjugate term differ onlyby a total derivative. Hence, both terms contribute equally to the action,which is given by S = Z d x d θ W α W α . (6.12)It is sometimes convenient to turn this integral into an integration over thefull superspace. Using a trick analogous to the one employed in eq. (5.112),we end up with, S = Z d x d θ (cid:0) − D (cid:1) ( D α V ) W α = Z d x d θ d θ † ( D α V ) W α , (6.13)after using eq. (6.4) to rewrite one factor of W α in terms of V .It is instructive to count the degrees of freedom in the irreducible super-multiplet, ( λ , λ † , F µν , D ). On-shell, there are two real fermionic degreesof freedom associated with the massless gaugino, after imposing the La-grange field equations, iσ µα ˙ β ∂ µ λ β = 0 . (6.14)This matches the two real bosonic degrees of freedom corresponding to thetwo transverse polarizations of the massless gauge boson.To count the off-shell bosonic degrees of freedom, one must take intoaccount the Bianchi identity, ǫ µνρσ ∂ ν F ρσ = 0 , (6.15) Starting with two complex (or equivalently four real) degrees of freedom for the two-component gaugino field λ , eq. (6.14) relates the spinor components λ and λ , therebyreducing the number of real degrees of freedom from four to two. Although it appears that the Bianchi identity yields four constraints, since the space-time index µ is a free index, in fact only three constraints are independent. This isbecause one of the four constraints is redundant due to the identity, ǫ µνρσ ∂ µ ∂ ν F ρσ = 0,which is automatically satisfied as a result of the antisymmetry of the Levi-Civita tensor.Physically, the Bianchi identity implies that the three components of the electric fieldvector determine the three components of the magnetic field vector. F µν from six to three. Adding in the one real degree of freedom associated with D , we end up with a total of four real bosonic degrees of freedom, whichmatches the four real off-shell fermionic degrees of freedom correspondingto λ and λ † . Gauge invariance
The vector superfield V contains the familiar gauge field V µ . But it alsoincludes other component fields C , χ , M and N , whose meaning is lessobvious. As we will see, these latter fields turn out to be gauge artifacts.Thus, we must examine how gauge transformations of the gauge field theoryget promoted to gauge transformations of the vector superfield V .Let Λ( x, θ, θ † ) be a chiral superfield ( i.e. , D ˙ α Λ = 0) and let Λ † ( x, θ, θ † )be the corresponding antichiral superfield. Consider the transformation, V → V + i (Λ − Λ † ) . (6.16)We assert that eq. (6.16) is a supersymmetric generalization of the gaugetransformation of an abelian gauge theory, henceforth called a super gaugetransformation.With the help of eq. (5.42), it is straightforward to show that the fieldstrength superfield, W α , is invariant under a super gauge transformation.Moreover, if the Taylor series of Λ( x, θ, θ † ) is written as Λ( x, θ, θ † ) = e A ( x ) + √ θ e ψ ( x ) + θθ e F ( x ) − iθσ µ θ † ∂ µ e A ( x ) − i √ θθ ) θ † σ µ ∂ µ e ψ ( x ) − ( θθ )( θ † θ † ) (cid:3) e A ( x ) , (6.17)then the impact of the super gauge transformation given by eq. (6.16) onthe component fields of V is, C → C + i ( e A − e A ∗ ) , (6.18) χ → χ + √ e ψ , (6.19) M + iN → M + iN + 2 e F , (6.20) V µ → V µ + ∂ µ ( e A + e A ∗ ) , (6.21) λ → λ , (6.22) D → D . (6.23) In contrast to the chiral superfield Φ in eq. (5.53) whose mass dimension is 1, the chiralsuperfield Λ is dimensionless, as required for consistency in light of eq. (6.16). The invariance of λ and D under super gauge transformations is a consequence of theparticular choices made for the coefficients of ( θθ ) θ † , ( θ † θ † ) θ and ( θθ )( θ † θ † ) in eq. (6.2). V µ transformsby an ordinary gauge transformation. Moreover the field strength tensor F µν = ∂ µ V ν − ∂ ν V µ , the gaugino field λ , and the auxiliary field D are gaugeinvariant as one would anticipate (consistent with the fact that the fieldstrength superfield W is gauge invariant).One particularly useful gauge choice is to choose e A , e ψ and e F such that C = χ = M = N = 0 . (6.24)This is called the Wess-Zumino (WZ) gauge [120]. The existence of sucha gauge implies that the fields C , χ , M , and N are gauge artifacts, aspreviously stated. The main drawback of the WZ gauge is that it is nota supersymmetric gauge choice. That is, starting from the WZ gauge andperforming a SUSY transformation on the component fields of the vectorsuperfield V will yield new component fields that do not satisfy the WZgauge condition.The main benefit of the WZ gauge is that it provides enormous simpli-fication in many practical computations. In particular, applying the WZgauge condition [eq. (6.24)] to the vector superfield given in eq. (6.2), V WZ = θσ µ θ † V µ + i ( θθ )( θ † ¯ λ ) − i ( θ † θ † )( θλ ) + ( θθ )( θ † θ † ) D . (6.25)Computing the square of V WZ with the help of eq. (5.33) yields, V ( x, θ, θ † ) = ( θθ )( θ † θ † ) V µ V µ . (6.26)and V n WZ ( x, θ, θ † ) = 0 for n = 3 , , , . . . . This implies that the Taylor seriesfor the exponential of V WZ is a finite series and contains only three terms,exp(2 gV WZ ) = 1 + 2 gV WZ + 2 g V . (6.27)This result will be especially important when we consider gauge-invariantinteractions in Section 6.3.Finally, we consider the implications of R -invariance. Since V is a realsuperfield, it follows from eq. (5.94) that, b RV ( x, θ, θ † ) = V ( x, e − ia θ, e ia θ † ) . (6.28)In the Wess-Zumino gauge, the R transformations of the component fieldsare given by V µ → V µ , (6.29) λ → e ia λ , (6.30) D → D . (6.31)The Lagrangian of eq. (6.10) for the SUSY gauge theory is invariant under R transformations. In the present context, the presence of R -invariance isassociated with the chiral symmetry of the massless gaugino.85 .3. Gauge-invariant interactions
Suppose that Φ is a chiral superfield that is charged under the U(1) gaugegroup. Then the gauge transformations of the chiral superfield and thecorresponding antichiral superfield are given by,Φ → e − ig Λ Φ , Φ † → e ig Λ † Φ † , (6.32)where Λ is the chiral superfield gauge transformation parameter introducedin eq. (6.16). In the presence of gauge interactions, the kinetic energy termfor the chiral superfield given by eq. (5.85), L KE = [Φ † Φ] D = Z d θ Φ † Φ , (6.33)is not gauge invariant. But this deficiency is easily repaired. A gauge-invariant kinetic energy term with respect to the gauge transformationsgiven in eqs. (6.16) and (6.32) is given by, L KE = [Φ † e gV Φ] D = Z d θ Φ † e gV Φ . (6.34)The proof is left as an exercise (see Problem 31).Normally, the exponential, exp(2 gV ), would yield an infinite series ofterms. But, the series terminates in the Wess-Zumino gauge, as indicatedin eq. (6.27), and we get L KE =( D µ A )( D µ A ) † + iψ † σ µ D µ ψ + F † F + ig √ A † λψ − Aλ † ψ † ) + gAA † D + total derivative , (6.35)where D µ ≡ ∂ µ + igV µ is the usual gauge-covariant derivative. The pres-ence of the Yukawa interaction of the scalar-fermion-gaugino is especiallynoteworthy, with a coupling proportional to the gauge coupling g . Thisis a consequence of supersymmetry, which relates the gauge and Yukawacouplings that otherwise would be independent.Another manifestation of SUSY is revealed when we consider the termsof the Lagrangian involving the auxiliary fields F and D . Consider theLagrangian of the interacting gauge theory that consists of contributionsfrom eqs. (6.10) and (6.35). We can isolate those terms that involve F and D explicitly, L = (cid:26) [ W α W α ] F + h . c . (cid:27) + [Φ † e gV Φ] D = . . . + F † F + D + gAA † D . (6.36)86olving the Lagrange field equations for F and D , ∂ L ∂F = 0 ⇐⇒ F = 0 , (6.37) ∂ L ∂D = 0 ⇐⇒ D = − gA † A . (6.38)Inserting these results back into eq. (6.36) [where the terms not explicitlygiven can be found in eqs. (6.10) and (6.35)] yields the Lagrangian in termsof its physical fields, L = − F µν F µν + iλ † σ µ ∂ µ λ + ( D µ A )( D µ A ) † + iψ † σ µ D µ ψ + i √ g ( A † λψ − Aλ † ψ † ) − g ( A † A ) . (6.39)Thus, a potential for the scalar field A has been generated, V scalar = g ( A † A ) . (6.40)There is one more possible term, called the Fayet-Iliopoulos term [121],that can appear in a renormalizable SUSY U(1) gauge theory Lagrangian, L FI = 2 ξ [ V ] D = ξD + total divergence . (6.41)This modifies the form of D obtained in eq. (6.38), D = − gA † A − ξ , (6.42)which in turn modifies the scalar potential, V scalar = (cid:2) gA † A + ξ (cid:3) . (6.43)The existence of a quartic scalar coupling proportional to the square of thegauge coupling (in the presence or absence of a Fayet-Iliopoulos term) isanother manifestation of SUSY. Generalizing to more than one chiral superfield
With only one chiral superfield, it was not possible to include a superpo-tential W (Φ) in our gauge theory, since W is a holomorphic function of acharged field and hence not gauge-invariant. But, a theory with more thanone charged chiral superfield can admit a gauge invariant superpotential.For example, consider a set of charged chiral superfields Φ i with U(1)charges q i , which transform under U(1) asΦ i → e − igq i Λ Φ i . (6.44)87uppose that a gauge-invariant superpotential can be constructed, W (Φ i ).When we solve for the auxiliary field F i , we will obtain F i = − (cid:18) dWdA i (cid:19) † , (6.45)as before [cf. eq. (4.132)], which provides the F -term contributions to thescalar potential, V scalar ∋ X i (cid:12)(cid:12)(cid:12)(cid:12) dWdA i (cid:12)(cid:12)(cid:12)(cid:12) . (6.46)When we solve for the auxiliary field D , we obtain a contribution fromeach scalar A i , D = − ξ − X i q i gA † i A i . (6.47)The corresponding D -term contributions to the scalar potential are V scalar ∋ " ξ + X i gq i A † A . (6.48)Including both the F -term and D -term contributions yields the followingscalar potential, V scalar = X i (cid:12)(cid:12)(cid:12)(cid:12) dWdA i (cid:12)(cid:12)(cid:12)(cid:12) + " ξ + X i gq i A † A , (6.49)which can also be conveniently written as V scalar = X i F † i F i + D , (6.50)where F and D are given by eqs. (6.45) and (6.48), respectively. Note thatthe form of the scalar potential [either eq. (6.49) or (6.50)] makes clear that V scalar ≥
0. This observation will play an important role in the theory ofsupersymmetry breaking, which is treated in Section 7.The above results can now be used to construct the supersymmetricextension of QED. The superfield content of SUSY-QED consists of a realvector superfield V , a chiral superfield Φ + with charge q = 1, and a chiralsuperfield Φ − with charge q = −
1. The unique renormalizable, gauge-invariant superpotential is W (Φ + , Φ − ) = m Φ + Φ − . (6.51)The R -charges of both Φ + and Φ − can be chosen to be +1, in which case thetheory is also R -invariant. The construction of the SUSY-QED Lagrangianis left as an exercise (see Problem 32).88 .5. SUSY Yang-Mills theory coupled to supermatter
The construction of the supersymmetric generalization of Yang-Mills the-ory, i.e. , a non-abelian gauge theory coupled to matter, is more complicatedthan the case of an abelian gauge theory treated in previous sections. Inthis subsection, we will summarize the main modifications. The reader canfill in the details with the help of Refs. [10, 33].Consider a non-abelian compact simple Lie group G, with generators T a that satisfy commutation relations, (cid:2) T a , T b (cid:3) = if abc T c . (6.52)It is convenient to normalize the generators of the defining (fundamental)representation of G such that,Tr( T a T b ) = δ ab . (6.53)The vector superfield, V a , possesses an adjoint index a , which runs overthe generators of G. Thus, we can define the matrix gauge superfield, V ≡ V a T a . (6.54)The gauge transformation law for V given in eq. (6.16) is significantly morecomplicated in the case of a non-abelian gauge theory, e gV −→ e − ig Λ † e gV e ig Λ † , (6.55)where Λ ≡ (Λ a T a ) ij is the matrix chiral superfield gauge transformationparameter.The chiral superfields are now multiplets corresponding to representa-tion R of the gauge group G, transforming as Φ i → (cid:0) e − ig Λ (cid:1) ij Φ j , (6.56)which provides the generalization of eq. (6.32) to a nonabelian gauge group.Note that Φ † i (cid:0) e gV (cid:1) ij Φ j is gauge-invariant, if the gauge transformation lawfor V is given by eq. (6.55).Likewise, we define a matrix version of the nonabelian field-strengthsuperfield, W α ≡ W aα T a , where W α = − g D e − gV D α e gV . (6.57)Unlike the abelian case, W α is not gauge-invariant. However it transformsas an adjoint field, W α → e − ig Λ W α e ig Λ , (6.58) When acting on the Φ i , one employs the generators T a in the representation R .
89o that Tr( W α W α ) is gauge-invariant. In the WZ gauge, when expandedin component fields, W aα depends only on the physical fields, λ a , F µνa andthe auxiliary field D a , W aα = − iλ aα + θ α D a − i ( σ µ σ ν θ ) α F aµν − σ µ ( D µab λ † b ) α θθ , (6.59)where D µab ≡ δ ab ∂ µ + gf abc V cµ , (6.60)is the gauge-covariant derivative in the adjoint representation, and F aµν = ∂ µ V aν − ∂ ν V aµ − gf abc V bµ V cν (6.61)is the nonabelian field strength tensor. The SUSY Lagrangian
The Lagrangian for SUSY Yang-Mills theory coupled to supermatter isgiven by L = (cid:20) Z d θ Tr( W α W α ) + h . c . (cid:21) + Z d θ Φ † e gV Φ + (cid:20)Z d θ W (Φ k ) + h . c . (cid:21) . (6.62)In contrast to the abelian gauge theory, no Fayet-Iliopoloulos term is al-lowed since [ D a ] D carries an adjoint index and thus is not gauge invariant.The superpotential W (Φ k ) is assumed to be a gauge-invariant holomorphicfunction of the chiral superfields. The chiral superfields Φ k taken togethertransform under a reducible d -dimensional representation R = ⊕ k R k ofthe gauge group G, where d = P k dim R k . In terms of component fields,eq. (6.62) yields L = − F aµν F µνa + iλ † a σ µ ( D µ λ ) a + D a D a + F † i F i + ( D µ A ) i ( D µ A ) † i + iψ † i σ µ ( D µ ψ ) i + gA † i T aij A j D a + ig √ A † i T aij ψ j λ a − λ † a ψ † i T aij A j )+ F i dWdA i + F † i (cid:18) dWdA i (cid:19) † − d WdA i dA j ψ i ψ j − (cid:18) d WdA i dA j (cid:19) † ψ † i ψ † j , (6.63)where there is an implicit sum over repeated indices, and the labels i and j run over 1 , , . . . , d . The corresponding covariant derivative, when acting on In contrast to the abelian case, the expansion of W aα in terms of its component fieldsin the nonabelian case will necessarily contain gauge artifacts. After imposing the WZgauge condition, the expansion of W aα in terms of its component fields resembles thecorresponding expression of SUSY abelian gauge theory [cf. eq. (6.5)]. A i and ψ i , is D µ = ∂ µ + igT a V aµ , where is the d × d identity matrix and the generators T a are in the reducible representation R of the group G.Note that the interactions of the matter fermions and the gauginos withthe gauge fields are dictated by gauge invariance (via the gauge covariantderivative) and do not depend on supersymmetry. In contrast, the Yukawainteraction of the gaugino with the matter fermion and its scalar partner(with a coupling proportional to the gauge coupling g ) is a consequence ofsupersymmetry, and relates the gauge and Yukawa couplings that otherwisewould be independent.We can now eliminate the auxiliary fields F i and D a by employing theLagrange field equations. We end up with F i = − (cid:18) dWdA i (cid:19) † , D a = − gA † i T aij A j . (6.64)Substituting back into eq. (6.63) yields the following scalar potential, V scalar = X i (cid:12)(cid:12)(cid:12)(cid:12) dWdA i (cid:12)(cid:12)(cid:12)(cid:12) + g ( A † i T aij A j ) . (6.65)Equivalently, we can write: V scalar = D a D a + X i F † i F i . (6.66)Eqs. (6.65) and (6.66) provide the nonabelian generalization of eqs. (6.49)and (6.50). As in the abelian case, V scalar ≥ L = Z d θ (cid:2) K ( e gV Φ , Φ † ) + K (Φ , Φ † e gV ) (cid:3) + (cid:20)Z d θ W (Φ i ) + h . c . (cid:21) + (cid:20) Z d θ f ab (Φ) W αa W bα + h . c . (cid:21) , (6.67)where K is the K¨ahler potential and f ab (Φ) is a holomorphic function ofthe chiral superfields called the gauge kinetic function . In renormalizableglobal supersymmetry, the minimal versions of the K¨ahler potential andgauge kinetic function are used: K ( e gV Φ , Φ † ) = K (Φ , Φ † e gV ) = Φ † e gV Φ , (6.68) f ab (Φ) = δ ab . (6.69)The generalization of the SUSY Lagrangian to a theory based on a gaugegroup that is a direct product of compact simple Lie groups and U(1) factorsis straightforward. There is a gauge field strength tensor and a separategauge coupling constant corresponding to each group in the direct product.Details are left for the reader. 91 .7. Problems
Problem 30.
Show that W α is invariant under the gauge transformationof eq. (6.16). Problem 31.
Show that the kinetic energy term given by eq. (6.34) isinvariant under the gauge transformations for Φ and Φ † given in eq. (6.32)and V → V + i (Λ − Λ † ) . Problem 32.
Construct the full SUSY QED Lagrangian in the Wess-Zumino gauge. Show that the physical states of the theory consist of a Diracfermion (the “electron”), two complex scalar “selectrons,” usually denotedby e e L and e e R , a massless photon, and a massless photino. Check that thenumber of bosonic and fermionic degrees of freedom are equal, both off-shelland on-shell. Problem 33.
Consider the SUSY QED theory examined in Problem 32.However, this time do not impose the Wess-Zumino gauge condition. In-stead, explore the consequences of adding the following supersymmetricgauge fixing term [122–124], L GF = − α (cid:2) ( D V )( D V ) (cid:3) D , (6.70) where α is the gauge fixing parameter. Problem 34.
Starting from the case where the gauge group G is non-abelian, show that the gauge transformation law for the gauge superfield V ,as deduced from eq. (6.55), reduces to V → V + i (Λ − Λ † ) in the abelian limit.Likewise, show that W α as given in eq. (6.57) reduces to W α = − D D α V in the abelian limit. Problem 35.
Evaluate the contribution of the K¨ahler potential terms tothe Lagrangian given in eq. (6.67) in terms of the component fields. Showthat your result reduces to eq. (5.86) in the limit of g → . Problem 36.
Evaluate the contribution of the gauge kinetic function termsto the Lagrangian given in eq. (6.67) in terms of the component fields. Howdoes your result simplify in the abelian limit?
Problem 37.
Starting from eq. (6.67), solve for the auxiliary fields F i and D a using the Lagrange field equations. Using these results, determinethe form of the scalar potential that generalizes the results of eqs. (6.65)and (6.66). . Supersymmetry Breaking If supersymmetry were an exact symmetry of nature, then particles andtheir superpartners, which differ in spin by half a unit, would be degeneratein mass. Since superpartners have not (yet) been observed, supersymmetrymust be a broken symmetry. In light of the non-observation of supersym-metric particles at the LHC, the energy scale of supersymmetry breakingmust lie above 1 TeV.The fundamental mechanism responsible for supersymmetry breaking ispresently unknown. In Section 7.1, we describe some general considerationsrelated to SUSY breaking, and we examine several possible frameworks forthe spontaneous breaking of SUSY. In Section 7.2, we examine constraintson mass splittings within supermultiplets in the presence of SUSY-breaking.The possible origins of SUSY-breaking dynamics is surveyed in Section 7.3.Finally, in Section 7.4, we examine a more agnostic approach, in which thesupersymmetry of the effective low energy theory at the TeV scale is softlybroken. In such an approach, we identify the possible soft-supersymmetrybreaking terms that can appear in the Lagrangian, without making assump-tions about their fundamental origin.
Spontaneous SUSY breaking
In Section 4.2, we derived eq. (4.28), which states that the energy operator P for a supersymmetric theory is given by P = 12 t (cid:16) Q Q † + Q † Q + Q Q † + Q † Q (cid:17) , (7.1)where t is real and positive (conventionally, t = 2). Since the right-handside of eq. (7.1) is positive semi-definite, it follows that the vacuum energyis zero if and only if the vacuum is supersymmetric: h | P | i = 0 ⇐⇒ Q α | i = 0 . (7.2)Moreover, assuming the absence of fermion condensation, the vacuumenergy can be identified as the vacuum expectation value of the scalarpotential. That is, in the case of a supersymmetric vacuum, h | P | i = 0 ⇐⇒ h | V scalar | i = 0 . (7.3) That is, we assume the absence of a fermion bilinear covariant, with the properties ofa Lorentz scalar, that acquires a nonzero vacuum expectation value.
93o appreciate the significance of h | V scalar | i = 0, recall eq. (6.66), whichwe repeat below for the convenience of the reader, V scalar = D a D a + X i F ∗ i F i . (7.4)It follows that if the vacuum is supersymmetric, then the vacuum expecta-tion values of the auxiliary fields must vanish, h | F i | i = h | D a | i = 0 . (7.5)One can reach the same conclusion by considering the transformationlaws of the field components of a superfield. For a chiral superfield, thecomponent fermion field transforms according to, δ ξ ψ αi = i (cid:2) ξQ + ξ † Q † , ψ αi (cid:3) = − i √ σ µ ξ † ) α ∂ µ A i + √ ξ α F i . (7.6)By Lorentz invariance, h | ∂ µ A i | i = 0. Hence, h | (cid:2) ξQ + ξ † Q † , ψ αi (cid:3) | i = √ ξ α h | F i | i . (7.7)Thus, if Q α | i = 0 and Q † ˙ α | i = 0, then h | F i | i = 0. Likewise, for a realvector superfield, the component gaugino field transforms according to, δ ξ λ aα = i (cid:2) ξQ + ξ † Q † , λ aα (cid:3) = iξ α D a + ( σ µ σ ν ) αβ ξ β F aµν . (7.8)Since h | F aµν | i = 0 (again, by Lorentz invariance), it follows that h | (cid:2) ξQ + ξ † Q † , λ aα (cid:3) | i = iξ α h | D a | i . (7.9)Thus, if Q α | i = 0 and Q † ˙ α | i = 0, then h | D a | i = 0.If at least one of the components of the auxiliary fields F i or D a hasa nonzero vacuum expectation value, then SUSY is spontaneously bro-ken. Mechanisms of spontaneous SUSY breaking fall into two possible cat-egories: F -type breaking, if h | F i | i 6 = 0 for some i , and D -type breakingif h | D a | i 6 = 0 for some a .7.1.1. The O’Raifeartaigh mechanism ( F -type breaking) One way to spontaneously break SUSY is to construct a model in whichit is impossible to simultaneously solve the Lagrange field equations forall the components of the auxiliary fields, F i . This is the O’Raifeartaighmechanism [125], where the SUSY breaking arises entirely from a nonzero F -term vacuum expectation value. A well-known supersymmetric joke: a graduate student returns to the University forthe fall semester after spending a month at TASI earlier in the summer. The professorsays to the student, “Welcome back! I see that one of the lecture courses you attendedat TASI was an introduction to supersymmetry. So, did you learn anything useful fromthese lectures?” The student replies, “I learned how to spell O’Raifeartaigh’s name.” Implicitly, we are assuming here that if the D -term is present, then h D a i = 0. F † i = − dWdA i = 0 . (7.10)A solution to these equations corresponds to the existence of a choice of thescalar fields, A i , such that all the equations, F † i = 0, are fulfilled. Supposethat a solution, A i = v i , solves these equations. In light of eq. (7.4), thissolution must correspond to a minimum of the scalar potential, which weidentify as the vacuum (ground) state of the theory. Since F † i = 0 impliesthat F i = 0, we can conclude that h | F i | i = 0 (for all i ). If no solutionto eq. (7.10) exists, then it must be true that h | F i | i 6 = 0 for some i . Inthis latter case, SUSY must be spontaneously broken.The simplest O’Raifeartaigh model that exhibits F -term SUSY breakingcontains three chiral superfields and is treated in Problem 38.7.1.2. D -type breaking via the Fayet-Iliopoulos term Consider SUSY-QED with a superpotential given by eq. (6.51) and a Fayet-Iliopoulos term. Using eqs. (6.45) and (6.47), the resulting scalar potential[eq. (6.50)] is given by V scalar = | F + | + | F − | + D , (7.11)where F ± = − mA ± , D = − g (cid:0) | A + | − | A − | (cid:1) − ξ. (7.12)Suppose that m > gξ . One can check that the minimum of the scalarpotential occurs for h A + i = h A − i = 0. Moreover, at the scalar potentialminimum, h F + i = h F − i = 0, whereas h D i = − ξ = 0. Thus, in this modelSUSY breaking arises entirely from a nonzero D -term vacuum expectationvalue. Additional aspects of this model are treated in Problems 40 and 41.7.1.3. The goldstino
From Goldstone’s theorem, we know that the spontaneous breaking of acontinuous symmetry (with bosonic generators) gives rise to a masslessboson called the Nambu-Goldstone boson. Analogously, the spontaneousbreaking of supersymmetry, whose algebra contains fermionic generators,gives rise to a massless fermion called the Goldstone fermion, which is morecommonly known as the goldstino [126].
Theorem 5.
If SUSY is spontaneously broken, then there exists a masslessspin-1/2 fermion in the spectrum called the goldstino. roof. Although this theorem can be proven rigorously, independently ofperturbation theory, it is instructive to exhibit a proof based on a tree-levelanalysis of a SUSY nonabelian gauge theory coupled to supermatter. Thescalar potential is given by eq. (7.4) where [cf. eq. (6.64)], F i = − (cid:18) dWdA i (cid:19) † , D a = − gA † i T aij A j . (7.13)At the scalar potential minimum, where ∂V /∂A j = 0, the scalar fields areequal to their vacuum expectation values, A j = h A j i . Then,0 = (cid:18) ∂V∂A j (cid:19) h A i = − gA † i T aij D a (cid:12)(cid:12)(cid:12)(cid:12) h A i − X i ∂ W∂A i ∂A j F i (cid:12)(cid:12)(cid:12)(cid:12) h A i . (7.14)Hence, X i (cid:28) ∂ W∂A i ∂A j (cid:29) h F i i = − g h A i i † T aij h D a i . (7.15)The superpotential W must be a gauge invariant function of the chiralsuperfields. That is, W (Φ) = W ( e − ig Λ Φ) . (7.16)where Λ ≡ Λ a T a is the matrix chiral superfield gauge transformation pa-rameter. Taking Λ a infinitesimal and expanding to first order yields dWd Φ i T aij Φ j = 0 . (7.17)Evaluating the hermitian conjugate of this expression, setting θ = θ † = 0,and taking the vacuum expectation value of the resulting equation, we endup with h F i i T aji h A j i † = 0 . (7.18)The fermion masses can be determined from the SUSY Lagrangian givenby eq. (6.63) after setting the scalar fields to their vacuum expectationvalues, − L mass = (cid:28) ∂ W∂A i ∂A j (cid:29) ψ i ψ j − i √ g h A i i † T aij ψ j λ a + h . c . (7.19)= (cid:0) ψ i − iλ b (cid:1) (cid:28) ∂ W∂A i ∂A j (cid:29) √ g h A j i † T aji √ g h A i i † T bij ψ j − iλ b . (7.20)96sing eqs. (7.15) and (7.18), one can verify that the fermion mass matrixgiven in eq. (7.20) possesses a zero eigenvalue, (cid:28) ∂ W∂A i ∂A j (cid:29) √ g h A j i † T aji √ g h A i i † T bij h F j i √ h D a i = 0 , (7.21)under the assumption that at least one of the auxiliary field vacuum expec-tation values is nonzero. The corresponding eigenvector, (cid:16) h F j i , √ h D a i (cid:17) ,can be identified with the massless goldstino, e G . That is, e G = h F j i ψ j − i √ h D a i λ a . (7.22)The existence of the goldstino in the fermion mass spectrum is a con-sequence of the assumption that the vacuum is not invariant under SUSYtransformations, in which case at least one of the auxiliary field vacuumexpectation values is nonzero, as assumed below eq. (7.21). In contrast,if the vacuum is supersymmetric, then h F j i = h D a i = 0, in which caseeqs. (7.15) and (7.18) are trivially satisfied. Hence in this case, one cannotconclude that a zero eigenvalue of the fermion mass matrix exists. Mass Sum rules
If SUSY is broken, then there is no expectation that particles in a would-besupermultiplet are degenerate in mass. If the SUSY breaking is sponta-neous, then there is still some memory of supersymmetry in the propertiesof the SUSY-broken theory. In particular, the mass spectrum of the sponta-neously broken SUSY theory satisfies certain sum rules that reflect the factthe spontaneous breaking of the supersymmetry is inherently soft [127].To exhibit such sum rules, we return to the Lagrangian of the SUSYnonabelian gauge theory coupled to supermatter given in eq. (6.63). We setthe scalar fields and the auxiliary fields to their vacuum expectation valuesand compute the resulting tree-level mass spectrum.The spin-1 masses arise from L mass = ( D µ A )( D µ A ) † , (7.23)where D µ = ∂ µ + igT a V aµ . It is convenient to write the gauge boson squared-mass matrix as follows,( M ) ab = 2 g h A † i i T aij T bjk h A k i = 2 * ∂D a ∂A † k ∂D b ∂A k + , (7.24)97here we have made use of D a = − gA † i T aij A j [cf. eq. (6.64)]. Likewise, wecan rewrite the spin-1/2 mass matrix [previously obtained in eq. (7.20)] as, M = * − ∂F † i ∂A j + −√ (cid:28) ∂D a ∂A i (cid:29) −√ (cid:28) ∂D b ∂A j (cid:29) . (7.25)The spin-0 masses arise from the scalar potential, V ≡ V scalar . Identi-fying the terms quadratic in the scalar field, − L mass = 12 (cid:0) A i A † j (cid:1) * ∂ V∂A i ∂A † k + (cid:28) ∂ V∂A i ∂A ℓ (cid:29)* ∂ V∂A † j ∂A † k + * ∂ V∂A † j ∂A ℓ + A † k A ℓ . (7.26)The scalar squared-mass matrix given above will be denoted by M .The elements of the scalar squared-mass matrix can be rewritten interms of derivatives of the auxiliary fields F i and D a . For example, notingthat eq. (7.13) implies that F is a function of A † (and likewise, F † is afunction of A ), then it follows from eq. (7.4) that ∂ V∂A i ∂A † k = ∂F † m ∂A i ∂F m ∂A † k + ∂D a ∂A † k ∂D a ∂A i + D a ∂ D a ∂A † k ∂A i . (7.27)One can now evaluate the trace of the various squared-mass matrices,Tr M = 2 * ∂D a ∂A † k ∂D a ∂A k + , (7.28)Tr M † M = * ∂F i ∂A † k ∂F † i ∂A † k + + 4 * ∂D a ∂A † k ∂D a ∂A k + , (7.29)Tr M = 2 * ∂F † i ∂A k ∂F i ∂A † k + + 2 * ∂D a ∂A † k ∂D a ∂A k + + 2 * D a ∂ D a ∂A † k ∂A k + , (7.30)where there are implicit sums over each pair of repeated indices. We cansimplify the last term of eq. (7.30) using D a = − gA † i T aij A j to obtain.Tr M = 2 * ∂F i ∂A † k ∂F † i ∂A k + + 2 * ∂D a ∂A † k ∂D a ∂A k + − g h D a i Tr T a . (7.31)98t then follows thatTr( M − M + 3 M ) = − g h D a i Tr T a . (7.32)We recognize the left-hand side of eq. (7.32) as a supertrace, which isdefined as the following weighted sum of traces,Str M ≡ X J ( − J (2 J + 1) Tr M J , (7.33)where M J is the squared-mass matrix of real spin- J fields. Note the( − J factor, so that bosons contribute positively and fermions negativelyto the sum over J . As applied to a SUSY nonabelian gauge theory coupledto supermatter, the sum is taken over J = 0, and 1. Hence, eq. (7.32)assumes the following simple form,Str M = − g h D a i Tr T a . (7.34)The mass sum rule can provide a useful check on the phenomenologicalviability of theories with tree-level spontaneous supersymmetry breaking.Let us now see how this applies in several cases. The origin of SUSY-breaking dynamics
Models of tree-level spontaneous SUSY breaking
In the case of F -type breaking ( i.e. , the O’Raifeartaigh model), in which h F i i 6 = 0 and h D a i = 0, eq. (7.34) yieldsStr M = 0 . (7.35)For example, consider the matter sector of SUSY-QED, which contains twochiral supermultiplets [cf. eq. (6.51)]. The corresponding spectrum containsa four-component Dirac electron and its two complex scalar superpartners,the selectrons (denoted by e e and e e ). If SUSY is spontaneously broken byan F -term vacuum expectation value, then eq. (7.35) yields m e + m e = 2 m e , (7.36)so that one selectron would be heavier than the electron and the otherselectron would be lighter than the electron. Clearly, this is very bad forphenomenology, since experiment demands that all superpartner massesmust be significantly heavier than their SM counterparts.Consider next D -type breaking with h F i i = 0 and h D a i 6 = 0 in a non-abelian gauge theory. In this case, Tr T a = 0 and we again conclude that Note that complex fields are equivalent to two mass-degenerate real fields. M = 0. However, it turns out that when the scalar potential is min-imized, it is always possible to find a vacuum in which h D a i = 0. Hence, D -term SUSY-breaking is not possible in this case (see Problem 43).Finally, consider D -type breaking in a gauge theory with a U(1) factor.The Standard Model provides an example of this case. But in the StandardModel, the hypercharge generator satisfies Tr Y = 0 when summed over onegeneration of matter. Hence we again find that Str M = 0. It is possi-ble to construct models of D -type SUSY breaking via the Fayet-Iliopoulosterm ξ . In such models, h D i is proportional to ξ , as shown below eq. (7.12).However, no realistic models of this type are known.Based on the above considerations, we conclude that the mass sumrule severely constrains tree-level SUSY-breaking models. Indeed, no phe-nomenologically realistic tree-level spontaneously broken SUSY model hasever been successfully constructed.7.3.2. Gauge-mediated SUSY breaking
One way to avoid the tyranny of the mass sum rule is to consider models inwhich the radiative corrections to the tree-level masses are significant. Ingeneral, there is no reason why the radiative corrections should respect thetree-level relations derived in Section 7.2. For example, one can constructmodels with two distinct sectors of supermatter, which are coupled by theexchange of gauge bosons. The particles of the Standard Model (SM) residein one of the supermatter sectors, whereas the source of SUSY-breaking(SSB) is located in the second supermatter sector, whose characteristicmass scale, M SSB , is assumed to be significantly above 1 TeV. Indeed, inthis second supermatter sector, the masses of particles and their superpart-ners are split due to SUSY-breaking, while respecting the tree-level masssum rule obtained in eq. (7.34). In this case, tree-level SUSY-breakingis phenomenologically viable in light of the large characteristic mass scale M SSB that governs the SSB sector.In such a setup, SUSY is unbroken in the SM sector at tree level, inwhich case Str M = 0 is trivially satisfied (see Problem 42). However, thereexist radiative corrections to the sum rule induced by loops involving thesupermatter of the SSB sector. These corrections are responsible for SUSY-breaking in the SM sector and the corresponding mass splitting betweenthe SM particles and their superpartners. Moreover, these mass splittingsare totally radiative in nature and not subject to the tree-level sum rule ofeq. (7.34). Models can easily be constructed in which the masses of the SM100uperpartners are all raised above 1 TeV, thereby avoiding conflict with thecurrent LHC searches. The end result is SUSY-breaking in the SM that isphenomenologically viable.In the scenario outlined above, SUSY-breaking is communicated to theSM-sector via a messenger mechanism, in which the messengers consistsof gauge bosons that couple both to the SM sector and the SSB sector.Models of this type provide examples of gauge-mediated SUSY breaking(GMSB). Details of GMSB model building lie beyond the scope of theselectures. For further details, you may consult Refs. [39, 44, 46].7.3.3. Local supersymmetry and the super-Higgs mechanism
Another way of evading the tyranny of the mass sum rule is to considermodels with local supersymmetry.In these lectures, we have focused on theories with global supersymme-try, where the anticommuting SUSY translation parameter ξ is independentof the position x . Suppose we attempt to generalize this to local super-symmetry, where ξ = ξ ( x ). Since the spinorial SUSY generators satisfy { Q α , Q ˙ β } = 2 σ µα ˙ β P µ , a theory of local supersymmetry must also be invari-ant under local spacetime translations, in which the translation depends onthe position. A theory that possesses a local spacetime translation sym-metry is a theory of gravity! Hence, a locally supersymmetric theory is atheory of gravity plus supersymmetry, i.e. , supergravity [24, 28].We have already encountered the massless supermultiplet that containsthe spin-3/2 gravitino and the spin-2 graviton. Suppose we couple thissupermultiplet to ordinary supermatter. In addition, suppose that the localsupersymmetry is broken, which will generate a mass splitting within thegraviton supermultiplet. We require that the graviton remain massless,while the gravitino acquires mass. This can be accomplished via the superHiggs mechanism [128, 129].We have seen in Section 7.1.3 that in models of spontaneously-brokenglobal supersymmetry, the spectrum includes a massless goldstino. In mod-els of spontaneously-broken supergravity, the goldstino is “absorbed” bythe gravitino via the super-Higgs mechanism. Initially, a massless gravitinopossesses only two helicity states, λ = ± . In the super-Higgs mechanism,the goldstino provides λ = ± helicity states for a massive gravitino. Thatis, the goldstino is removed from the physical spectrum and the gravitinoacquires a mass (denoted by m / ). The gravitino now possesses the fourhelicity states, λ = ± , ± , as expected for a massive spin- particle.101n spontaneously broken supergravity, the tree-level mass sum rule ob-tained in eq. (7.34) is modified. For example, if N chiral supermultipletsare minimally coupled to supergravity, then [130],Str M = ( N − m / − κ h D a D a i ) − g h D a i T a , (7.37)where κ = (8 πG N ) / = (8 π ) / M − . Typical models of interest have h D a i = 0, in which case [131] ,Str M = 2( N − m / . (7.38)If m / > ∼ O (1 TeV), then one expects the superpartner masses of SM par-ticles to lie in the TeV regime.7.3.4. Gravity-mediated SUSY-breaking
Consider again the framework of two distinct sectors of supermatter thatare initially uncoupled. We identify one of the sectors as the SM sectorwhere the SM particles and their superpartners reside. In the second so-called “hidden” sector, SUSY is spontaneously broken.Supergravity models provide a natural mechanism for transmitting theSUSY breaking of the hidden sector to the particle spectrum of the SM sec-tor. In models of gravity-mediated SUSY breaking, gravity is the messengerof supersymmetry breaking [31, 132]. More precisely, SUSY breaking in theSM sector is mediated by effects of gravitational strength (suppressed by in-verse powers of the Planck mass). The induced mass splittings between theSM particles and their superpartners are of O ( m / ), whereas the gravitinocouplings are roughly gravitational in strength.Under certain theoretical assumptions on the structure of the K¨ahlerpotential (the so-called sequestered form introduced in Ref. [133]), SUSYbreaking is due entirely to the super-conformal (super-Weyl) anomaly,which is common to all supergravity models. This approach is calledanomaly-mediated supersymmetry breaking (AMSB). Indeed, anomaly me-diation is more generic than originally conceived, and provides a ubiquitoussource of supersymmetry breaking [134, 135]. A phenomenological approach: soft SUSY-breaking
If SUSY-breaking arises due to gauge-mediated SUSY-breaking or gravity-mediated SUSY-breaking, then we can formally integrate out the SSB sectorphysics at the mass scale M SSB that characterizes the fundamental SUSY-breaking dynamics. For example, in the case of gravity-mediated SUSY102reaking, we identify M SSB = M PL . In GMSB models, M SSB can be muchsmaller than M PL but still significantly larger than the scale of electroweaksymmetry breaking.The end result is an effective broken supersymmetric theory whoseLagrangian consists of supersymmetric terms and explicit SUSY-breakingterms. The explicit SUSY-breaking terms that are present in the effectivelow-energy theory (which is valid at energy scales below M SSB ) are “soft.”The meaning of soft in this context will be explained shortly.The phenomenological approach to SUSY-breaking takes the point ofview that the fundamental dynamics of SUSY-breaking is unknown. There-fore, we should simply parameterize SUSY breaking in the low-energy ef-fective theory by including all possible soft-SUSY-breaking terms. The co-efficients of these terms will be taken to be arbitrary (to be determined byexperiment). Ultimately, these parameters will provide clues to the struc-ture of the fundamental dynamics that is responsible for SUSY-breaking.7.4.1.
A catalog of soft-SUSY-breaking terms
The most general set of soft-SUSY-breaking terms in a super-Yang Millstheory coupled to supermatter was first elucidated by Girardello and Gris-aru in Ref. [136], − L soft = m ij A † i A j + (cid:2) m ab λ a λ b + h . c . (cid:3) + (cid:2) w ( A ) + h . c . (cid:3) , (7.39)where there is an implicit sum over repeated indices. The scalar squared-mass matrix m ij is hermitian and the gaugino mass matrix m ab is complexsymmetric. The function w ( A ) is a holomorphic cubic multinomial of thescalar fields, w ( A ) = c i A i + b ij A i A j + a ijk A i A j A k . (7.40)Note that c i = 0 in the absence of any gauge singlet fields. In the literature,the b ij are called the B -terms and the a ijk are called the A -terms. Notethe corresponding mass dimensions, [ b ij ] = 2 and [ a ijk ] = 1.Dimension-4 terms are not included in eq. (7.39), since non-supersymmetric dimension-4 terms would constitute a hard breaking ofsupersymmetry [137]. One interesting feature of eq. (7.39) is the ab-sence of non-supersymmetric fermion mass terms, m ij ψ i ψ j + h . c . , and non-holomorphic cubic terms in the scalar fields (e.g., A i A j A † k , etc.). Althoughsuch terms are technically soft in models with no gauge singlets [138–142],theses terms rarely arise in actual models of fundamental SUSY-breaking,or if present are highly suppressed [137]. Henceforth, we shall neglect them.103n general, there is no relation between w ( A ) and the superpotential,which under the assumption of renormalizability has the following genericform, W (Φ) = κ i Φ i + µ ij Φ i Φ j + λ ijk Φ i Φ j Φ k . (7.41)But, some models of fundamental SUSY breaking yield the relations, c i = Cκ i , b ij = Bµ ij , a ijk = Aλ ijk , (7.42)which relate the coefficients of w ( A ) to the coefficients of W (Φ).7.4.2. Soft vs. hard SUSY breaking and the reappearance ofquadratic divergences
Consider the one-loop effective potential for a gauge theory coupled tomatter, V eff ( A ) = V scalar ( A ) + V (1) ( A ) . (7.43)If we regulate the divergence of the one-loop correction by a momentumcutoff Λ, then [143] V (1) ( A ) = Λ π Str M i ( A ) + 164 π Str (cid:26) M i ( A ) (cid:20) ln (cid:18) M i ( A )Λ (cid:19) − (cid:21)(cid:27) , (7.44)where M i ( A ) are the relevant squared-mass matrices for spin 0, and 1,in which the scalar vacuum expectation values are replaced by the corre-sponding scalar fields, A .Eq. (7.44) implies that both in supersymmetric theories and in the caseof spontaneously broken SUSY (assuming in the latter that all U(1) gen-erators are traceless), we have Str M = 0, in which case the quadraticdivergences [i.e., the terms proportional to Λ in eq. (7.44)] cancel exactly!In Ref. [136], Girardello and Grisaru showed that if explicit SUSY breakingterms are present, then there is a catalog of possible explicit SUSY-breakingterms for which Str M i ( A ) is a constant independent of the scalar fields, A . Such terms shift the vacuum energy, but in the context of quantumfield theory they have no observable effect. Terms with such propertiesare deemed “soft,” and are given in eq. (7.39). In contrast, hard SUSY-breaking terms will generate quadratically divergent terms in V (1) that arescalar-field-dependent. This is a signal that some of the parameters of thelow-energy effective theory are quadratically sensitive to UV physics. Non-holomorphic cubic terms and mass terms of fermions that reside in a chiral su-permultiplet can generate quadratically divergent terms in V (1) that are linear in thescalar fields, A . However, if no gauge singlet fields exist in the model, then terms thatare linear in A are absent due to gauge invariance. Soft SUSY-breaking: an effective theory perspective
Consider a set of light chiral superfields Φ and a set of heavy chiral super-fields Ω associated with a mass scale M ≡ M SSB . Furthermore, assume thatSUSY-breaking is generated by an F -term that resides in the SSB sector, h F Ω i = f = 0 . (7.45)One can integrate out the physics of the SSB sector, as shown in the fol-lowing examples [136, 144, 145]. Example 8.
Consider a holomorphic cubic multinomial of chiral super-fields Φ, which we denote by e w (Φ). A possible term in the effective La-grangian is 1 M Z d θ Ω e w (Φ) , (7.46)since Ω e w (Φ) is a term in the superpotential. The factor of M − appears onthe basis of dimensional analysis. In particular, note the mass dimensions,[ e w ] = 3, [Ω]=1 and [ R d θ ] = 1.Since the vacuum expectation value of F Ω , denoted by h F Ω i = f , isnonzero, it follows that h Ω i ∋ θθf . Inserting this into eq. (7.46) yields,1 M Z d θ θθf e w (Φ) = fM e w ( A ) , (7.47)which produces the term, w ( A ) = ( f /M ) e w ( A ), in our catalog of δ L soft given in eq. (7.39).In order to achieve soft-SUSY-breaking masses in the low-energy effec-tive theory of order 1 TeV, one must require that f /M ∼ O (1 TeV). Forexample, in gravity-mediated SUSY breaking, M ∼ M PL , in which case f ∼ (10 GeV) . Note that f / identifies the energy scale of the funda-mental SUSY breaking. Example 9.
Another possible term in the effective Lagrangian is1 M Z d θ Φ † i (cid:0) e gV (cid:1) ij Φ j Ω † Ω , (7.48)which would contribute to the K¨ahler potential. Setting h Ω i = θθf andevaluating the result in the Wess-Zumino gauge, f M Z d θ ( θθ )( θ † θ † )Φ † i (cid:0) e gV (cid:1) ij Φ j = f M A † A . (7.49)Thus, the low-energy effective theory contains a scalar squared-mass termof order f /M , which we again recognize as one of the soft-SUSY-breakingterms of eq. (7.39). 105 xample 10.
Finally, one additional possible term in the effective La-grangian is 1 M Z d θ Ω Tr( W α W α ) , (7.50)which would contribute to the gauge kinetic function. Setting h Ω i = θθf , fM Z d θ θθ Tr( W α W α ) = − fM Tr( λ α λ α ) , (7.51)which yields a gaugino mass term of order f /M .We have thus demonstrated how the possible soft-SUSY-breaking termsof eq. (7.39) can arise in the low-energy effective theory after integratingout the physics associated with the SSB sector. Problems
Problem 38.
An O’Raifeartaigh model that exhibits F -term SUSY break-ing must involve at least three chiral superfields [125]. One of the simplestmodels of this type has the following superpotential, W (Φ , Φ , Φ ) = λ Φ (Φ − m ) + µ Φ Φ , (7.52) where λ is dimensionless and µ and m are mass parameters. Evaluate thecorresponding F -terms, F , F and F and write out the scalar potential, V scalar . Show that no solution for the scalar fields A A and A exist suchthat F = F = F = 0 . Conclude that SUSY is spontaneously broken. Problem 39.
Find the minimum of V scalar obtained in Problem 38, andverify that h | V scalar | i > . Identify the goldstino of this model. Finally,compute the mass spectrum of the fermions and bosons and verify that themass sum rule, eq. (7.35), is satisfied. Problem 40.
Show that in the case of SUSY-QED with a Fayet-Iliopoulosterm and m > gξ [cf. eqs. (7.11) and (7.12)], SUSY is broken and thegoldstino can be identified as the photino (the supersymmetric partner ofthe photon). In the case of m < gξ , is SUSY broken? Is the U(1) gaugesymmetry broken? Problem 41.
Referring back to Problem 40, determine the masses of theelectron and its scalar partners and the masses of the photon and photinoin the two cases of m > gξ and m < gξ , respectively. Evaluate Str M in both cases, and compare with eq. (7.34). Problem 42.
Show that the sum rule of eq. (7.34) is valid in the limit ofexact SUSY, i.e., when the masses of bosons and fermions are equal. roblem 43.
Show that in a SUSY nonabelian gauge theory that is cou-pled to supermatter, only F -type SUSY breaking is allowed. To prove thisstatement, assume that a solution to h F i i = 0 exists and show that one canalways find a choice of scalar fields A i that provide a solution to eq. (7.10)such that h D a i = 0 for all a . HINT : If the A i provide a solution to eq. (7.10), then so do the corre-sponding gauge transformed scalar fields, ( e − ig Λ ) ij A j . The key observationis that the superpotential is a holomorphic function of the scalar fields A i .Hence, one can generate additional solutions to eq. (7.10) by taking g com-plex, which will modify h D a i . Conclude that there must then be a set of A i such that h F i i = h D a i = 0 . See Ref. [2] for further details.
8. Supersymmetric extension of the Standard Model (MSSM)
With the necessary SUSY technology now in hand, we are ready to studyits realization in extensions to the SM. In this section, we describe theminimal supersymmetric extension of the Standard Model (MSSM). Muchof the presentation of this section follows Ref. [48], where many of therelevant references to the original literature can be found.In Section 8.1, we begin by presenting the MSSM field content. We thenspecify the SU(3) × SU(2) × U(1) gauge-invariant superpotential for the chi-ral superfields in Section 8.2. Given the superfield formalism developed inSections 5 and 6, all the supersymmetric interactions of the theory are nowdetermined. At this stage, the supersymmetry is still an exact symmetry.We introduce SUSY breaking in the MSSM in Section 8.3. Since the fun-damental origin of SUSY-breaking is unknown, we parametrize the SUSY-breaking by adding all possible soft-SUSY-breaking terms consistent withthe SU(3) × SU(2) × U(1) gauge symmetry and a discrete B − L symmetry.In Section 8.4, we count the number of parameters that govern the MSSM.The resulting MSSM particle spectrum and Higgs boson spectrum are ex-hibited in Sections 8.5 and 8.6, respectively. Finally, in Section 8.7, wedemonstrate the unification of gauge couplings in the MSSM.As in the SM, the neutrinos of the MSSM are massless. To incorporatemassive neutrinos, one can introduce right-handed neutrinos and employthe seesaw mechanism. It is then a simple matter to extend the MSSMby adding a SM singlet superfield that contains a right-handed neutrinoand the corresponding sneutrino superpartner. We shall not present thisconstruction in these lectures; for further details, see e.g. Ref. [146].107 .1. Field content of the MSSM
MSSM superfields and their component fields
The minimal supersymmetric extension of the Standard Model (MSSM)contains the fields of the two-Higgs-doublet extension of the SM and theircorresponding superpartners. The gauge fields and their superpartners arecontained in real vector supermultiplets. These gauge supermultiplets con-sist of the SU(3) × SU(2) × U(1) gauge bosons and their gaugino fermionicsuperpartners. The matter fields and their superpartners reside in chi-ral supermultiplets. The three generations of quark and lepton supermul-tiplets consist of left-handed quarks and leptons and their scalar super-partners (squarks and sleptons), and the corresponding antiparticles. TheHiggs supermultiplets consist of two complex Higgs doublets, their higgsinofermionic superpartners, and the corresponding antiparticles. The MSSMfields and their gauge quantum numbers are shown in Table 5.
Table 5.:
The fields of the MSSM and their SU(3) × SU(2) × U(1) quantum num-bers are listed. The electric charge is given in terms of the third component ofthe weak isospin T and U(1) hypercharge Y by Q = T + Y . For simplicity,only one generation of quarks and leptons is exhibited. The left-handed charge-conjugated quark and lepton fields are denoted by a superscript c . In particular, f cL ≡ P L f c = P L C ¯ f T = C ¯ f T R , following the notation of Ref. [147], where f is afour-component fermion field. The L and R subscripts of the squark and sleptonfields indicate the chirality of the corresponding fermionic superpartners. Field content of the MSSMSuper- Super- Bosonic Fermionicmultiplets field fields partners SU(3) SU(2) U(1)gluon/gluino b V g e g b V W ± , W f W ± , f W b V ′ B e B b L ( e ν L , e e − L ) ( ν, e − ) L − b E c ˜ e + R e cL b Q ( e u L , e d L ) ( u, d ) L / b U c e u ∗ R u cL ¯3 1 − / b D c e d ∗ R d cL ¯3 1 2 / b H d ( H d , H − d ) ( e H d , e H − d ) 1 2 − b H u ( H + u , H u ) ( e H + u , e H u ) 1 2 1108able 5 shows that one Higgs doublet superfield has hypercharge − ± i.e. depend only on chiralsuperfields and not their hermitian conjugates, it is important to keep trackof the quantum numbers of the chiral superfields of the model.8.1.2. Anomaly cancellation and the second Higgs doublet
The enlarged Higgs sector of the MSSM constitutes the minimal structureneeded to guarantee the cancellation of gauge anomalies generated by thehiggsino superpartners that can appear as internal lines in one-loop trianglediagrams with three external electroweak gauge bosons.Potentially problematic anomalies arise from one-loop
V V A and
AAA triangle diagrams with three external gauge bosons, and fermions runningaround the loop [where V refers to a γ µ (vector) vertex and A refers toa γ µ γ (axial vector) vertex]. An anomalous theory violates unitarity andfails as a consistent quantum field theory. Thus, we need to make sure allgauge anomalies cancel when summed over all triangle diagrams with fixedexternal gauge fields [148].The anomalies will cancel if certain group theoretical constraints aresatisfied. In particular, the trace of the product of the relevant generatorsappearing at the external vertices must vanish, W i W j B triangle ⇐⇒ Tr( T Y ) = 0 ,BBB triangle ⇐⇒ Tr( Y ) = 0 . In the Standard Model, the fermion contributions to Tr( Y ) sum to zero:Tr( Y ) SM = 3 (cid:0) + − + (cid:1) − − . (8.1)In contrast, in the MSSM, if we only add the higgsinos ( e H + u , e H u ), theresulting anomaly factor is Tr( Y ) = Tr( Y ) SM + 2 , leading to a gaugeanomaly. To cancel this, we must add a second higgsino doublet withopposite hypercharge, ( e H d , e H − d ).There is an independent argument for requiring the second Higgs dou-blet in the MSSM. With only one Higgs doublet, one cannot generate massfor both “up”-type and “down”-type quarks (and charged leptons) in a waythat is consistent with a holomorphic superpotential.109.1.3. Suppressed baryon and lepton number violation
It is an experimental fact that baryon number B and lepton number L are,to a very good approximation, global symmetries of nature. If neutrinosare Majorana fermions, then L -violation is present but strongly suppressed,with neutrino masses of order v /M , where v is the scale of electroweak sym-metry breaking and M ≫ v . No B -violation has yet been experimentallyobserved. Moreover, the current bounds on the proton lifetime suggest thatthe mass scale associated with baryon number violation cannot be belowabout 10 GeV, which is a characteristic scale of grand unification.One of the remarkable features of the SM is that the suppression of B and L -violating processes is a natural feature of the model. That is, theSM Lagrangian possesses an accidental global B − L symmetry due to thefact that all renormalizable terms of the Lagrangian (with dimension fouror less) that can be composed of SM fields preserve the B and L globalsymmetries. Indeed, B and L -violating operators composed of SM fieldsmust have dimension d = 5 or larger [149–151].For example, consider the dimension-five L -violating operator, L = − f mn M ( ǫ ij L mi H j )( ǫ kℓ L nk H ℓ ) + h . c . , (8.2)where f is a coefficient that depends on the lepton generation (labeled by m and n ), H j is the complex Higgs doublet field and L ai ≡ ( ν aL , ℓ aL ) is the dou-blet of two-component lepton fields. After electroweak symmetry breaking,the neutral component of the doublet Higgs field acquires a vacuum expec-tation value, and a Majorana mass matrix for the neutrinos is generated.The dimension-five term given by eq. (8.2) is generated by new physicsbeyond the SM at the scale M . Likewise, one can construct dimension-six B -violating operators composed of SM fields that allow, e.g., for protondecay, which is suppressed by v /M . Such terms can be generated, e.g.,in grand unified theories with a characteristic mass scale M G . In general, B and L -violating effects are suppressed by ( v/M ) d − , where M is the char-acteristic mass scale of the physics that generates the corresponding higherdimensional operator (of dimension d ).Unfortunately, the suppression of B and L -violation is not guaranteed ina generic supersymmetric extension of the Standard Model. For example,it is possible to construct gauge invariant supersymmetric dimension-four B and L -violating operators made up of fields of SM particles and their su-perpartners. Such operators, if present in the theory, would yield a protondecay rate many orders of magnitude larger than the current experimental110ound. To avoid this catastrophic prediction, one can introduce an addi-tional symmetry in the supersymmetric theory that will eliminate the B and L -violating operators of dimension d ≤
4. Further details are providedin the next subsection. Nevertheless, one must admit that the SM providesa more satisfying explanation for approximate B and L conservation thandoes its supersymmetric extension. The superpotential of the MSSM
Given the chiral and gauge superfield content of the MSSM, we must nowspecify the superpotential. The most general SU(3) × SU(2) × U(1) gauge-invariant superpotential (omitting the right-handed neutrino superfield) is W = ( h u ) mn b Q m · b H u b U cn + ( h d ) mn b H d · b Q m b D cn + ( h e ) mn b H d · b L m b E cn + µ b H u · b H d + W RPV , (8.3)where m and n label the generations. That is, h u , h d and h e are 3 × b H u · b H d ≡ ǫ ij b H ui b H dj = b H + u b H − d − b H u b H d . (8.4)The so-called µ -term above is the supersymmetric analog of the Higgs bosonsquared-mass term of the SM.In addition to the supersymmetric generalization of the SM Yukawacouplings and the µ -term, the gauge symmetries of the superpotential alsoallow for a number of new terms that violate B − L conservation. Asdiscussed in Section 8.1.3, this is in contrast to the SM where there areno B or L -violating interactions at the renormalizable level. The B − L violating terms of the supersymmetric model arise due to the presence of W RPV in eq. (8.3) and are given by, W RPV = ( λ L ) pmn b L p b L m b E cn + ( λ ′ L ) pmn b L p b Q m b D cn + ( λ B ) pmn b U cp b D cm b D cn + ( µ L ) p b H u b L p . (8.5)Note that the term proportional to λ B violates B , while the other threeterms violate L . The L -violating term proportional to µ L is the generaliza-tion of the µ b H u b H d term, in which the Y = − b H d isreplaced by the lepton supermultiplet b L p . Indeed, if L violation is present,then there is no distinction between b L and b H d , since the gauge quantumnumbers of these two superfields are identical.111f all terms in W RPV were allowed, the resulting model would predict aproton decay rate many orders of magnitude larger than the current exper-imental bound. This can be avoided by imposing an appropriate discretesymmetry that would eliminate the undesirable terms in W .The standard choice in constructing the MSSM is to set W RP V = 0.There are a number of ways to accomplish this. First, one one could di-rectly impose a B − L symmetry. Alternatively, one can set W RP V = 0 byintroducing a matter parity, under which b Q , b U c , b D c , b L and b E c are odd, and b H u and b H d are even. Finally, a third option is to impose an R -invariant su-perpotential. As discussed in Section 5.6, W is R -invariant if the R chargesof the chiral superfields are chosen such that R ( W ) = 2. Thus, if we choose R charges of + for b Q , b U c , b D c , b L , b E c and R charges of +1 for b H u , b H d ,then the condition of R -invariance sets W RPV = 0.One has to make sure that whichever symmetry one chooses to set W RPV = 0 is also consistent with the soft-SUSY-breaking terms that aresubsequently added to the model. In particular, in the case of the R -invariance, recall that R ( λ ) = 1, which forbids the gaugino mass term, m λ ( λλ + λ † λ † ) . (8.6)But phenomenology requires massive gauginos. This motivates the use of R -parity, described in the following subsection, rather than R -invariance.8.2.1. R -parity The gaugino mass term in eq. (8.6) is an allowed soft-SUSY-breaking term.If this term is added to a theory with an R -invariant superpotential, thenthe continuous U(1) R symmetry is broken down to a discrete Z symmetry,called R -parity [152, 153]. One can check that the R -parity of a particlewith baryon number B , lepton number L and spin S is given by R = ( − B − L )+2 S . (8.7)It is sufficient to impose R -parity invariance in order to set W RPV = 0, which is equivalent to imposing the B − L discrete symmetry. For theremainder of these lectures, we shall assume that R -parity is conserved.One can use eq. (8.7) to deduce the R -parity quantum numbers of allSM particles and their supersymmetric partners, R = ( +1 , for all SM particle particles , − , for all superpartners . (8.8) The effects of imposing matter parity and R -parity in the MSSM are identical for allrenormalizable interactions. R -parity in scattering and decay processes has a crit-ical impact on supersymmetric phenomenology. For example, any initialstate in a scattering experiment will involve ordinary ( R -even) particles.Consequently, it follows that supersymmetric particles must be producedin pairs. In general, these particles are highly unstable and decay intolighter states. Moreover, R -parity invariance also implies that the lightestsupersymmetric particle (LSP) is absolutely stable, and must eventuallybe produced at the end of a decay chain initiated by the decay of a heavyunstable supersymmetric particle.In order to be consistent with cosmological constraints, a stable LSP isalmost certainly electrically and color neutral. Consequently, the LSP in an R -parity-conserving theory is weakly interacting with ordinary matter, i.e .,it behaves like a stable heavy neutrino and will escape collider detectorswithout being directly observed. Thus, the canonical signature for conven-tional R -parity-conserving supersymmetric theories is missing (transverse)energy, due to the escape of the LSP. Moreover, the stability of the LSPin R -parity-conserving supersymmetry makes it a promising candidate fordark matter.8.2.2. MSSM parameters of the SUSY-conserving sector
The parameters of the SUSY-conserving sector consist of: (i) gauge cou-plings, g s , g , and g ′ , corresponding to the Standard Model gauge groupSU(3) × SU(2) × U(1) respectively; (ii) a SUSY-conserving higgsino mass pa-rameter µ ; and (iii) Higgs-fermion Yukawa coupling constants, λ u , λ d , and λ e , corresponding to the couplings of one generation of left- and right-handed quarks and leptons and their superpartners to the Higgs bosonsand higgsinos. Because there is no right-handed neutrino (or its superpart-ner) in the MSSM as defined here, a Yukawa coupling λ ν is not included.The complex µ parameter and Yukawa couplings enter via the most generalrenormalizable R -parity-conserving superpotential given by eq. (8.3) with W RPV = 0.One can now obtain the scalar potential from eq. (6.66) as applied tothe MSSM, V scalar = (cid:2) D a D a + ( D ′ ) (cid:3) + F ∗ i F i , (8.9)where the index a runs over the SU(3) and SU(2) gauge indices and D ′ isthe U(1) Y D -term. Focusing on the terms that depend on the Higgs bosonfields, one obtains, 113 Higgs = | µ | (cid:2) | H d | + | H u | (cid:3) + ( g + g ′ ) (cid:2) | H d | − | H u | (cid:3) + g | H ∗ d H u | . (8.10)Clearly h V Higgs i ≡ h | V Higgs | i ≥
0, as expected. Moreover, H d = H u = 0minimizes the Higgs scalar potential, which yields h V Higgs i = 0, correspond-ing to a supersymmetric vacuum. Thus, there is no SU(2) × U(1) breaking atthis stage. But after introducing soft SUSY-breaking terms, some of whichinvolve the Higgs fields, it will then be possible to spontaneously breakthe SU(2) × U(1) symmetry. Consequently, SUSY breaking and electroweaksymmetry breaking are intimately related in the MSSM.
Supersymmetry breaking in the MSSM
Following the rules of Girardello and Grisaru [136] that were presented inSection 7.4.1, we add the soft-SUSY-breaking terms, consistent with theSU(3) × SU(2) × U(1) gauge symmetry and the assumed R -parity invariance(for a review, see Ref. [43]). For simplicity, we consider in this section thecase of one generation of quarks, leptons, and their scalar superpartners.The supersymmetry-breaking sector contains the following sets of pa-rameters: (i) three complex gaugino Majorana mass parameters, M , M ,and M , associated with the SU(3), SU(2), and U(1) subgroups of theStandard Model; (ii) five squark and slepton squared-mass parameters, M e Q , M e U , M e D , M e L , and M e E , corresponding to the superpartners of thefive electroweak multiplets of left-handed fermion fields and their charge-conjugates, ( u, d ) L , u cL , d cL , ( ν , e − ) L , and e cL [cf. Table 5]; and (iii) threeHiggs-squark-squark and Higgs-slepton-slepton trilinear interaction terms,with complex coefficients T U ≡ λ u A U , T D ≡ λ d A D , and T E ≡ λ e A E (whichdefine the A -parameters). Following Ref. [35], it is conventional to sepa-rate out the factors of the Yukawa couplings in defining the A -parameters,originally motivated by a simple class of gravity-mediated SUSY-breakingmodels [31, 38, 132]. With this definition, if the A -parameters are para-metrically of the same order (or smaller) relative to other supersymmetry-breaking mass parameters, then only the third generation A -parameterswill be phenomenologically relevant.Finally, we have (iv) two real squared-mass parameters ( m and m )and one complex squared-mass parameter, m ≡ µB (the latter definesthe B -parameter), which appear in the tree-level scalar Higgs potential, V = ( m + | µ | ) H † d H d + ( m + | µ | ) H † u H u + ( m H u H d + h . c . )+ ( g + g ′ )( H † d H d − H † u H u ) + | H † d H u | . (8.11)114ote that the quartic Higgs couplings in eq. (8.11) are related to the gaugecouplings g and g ′ as a consequence of supersymmetry. The breaking ofthe electroweak symmetry SU(2) × U(1) to U(1) EM is only possible afterintroducing the supersymmetry-breaking Higgs squared-mass parameters m , m (which can be negative) and m . After minimizing the Higgsscalar potential, these three squared-mass parameters can be re-expressedin terms of the two Higgs vacuum expectation values, h H d i ≡ v d / √ h H u i ≡ v u / √
2, and the CP-odd Higgs mass m A [cf. eqs. (8.13) and (8.14)below]. One is always free to rephase the Higgs doublet fields such that v d and v u are both real and positive.The quantity, v d + v u = 4 m W /g = (2 G F ) − / ≃ (246 GeV) , is fixedby the Fermi constant, G F , whereas the ratiotan β = v u v d (8.12)is a free parameter such that 0 ≤ β ≤ π/
2. The tree-level conditionsfor the scalar potential minimum relate the diagonal and off-diagonal Higgssquared-mass parameters in terms of m Z = ( g + g ′ )( v d + v u ), the angle β ,and the CP-odd Higgs mass m A :sin 2 β = 2 m m + m + 2 | µ | = 2 m m A , (8.13) m Z = −| µ | + m − m tan β tan β − . (8.14)At this stage, one can already see the tension with naturalness, if theSUSY parameters, | m | , | m | and | µ | , are significantly larger than the scaleof electroweak symmetry breaking. In this case, m Z will be the difference oftwo large numbers, requiring some fine-tuning of the SUSY parameters inorder to produce the correct Z boson mass. In the literature, this tensionis referred to as the little hierarchy problem [93–95], previous noted inSection 3.3. One must also guard against the existence of charge and/orcolor breaking global minima due to non-zero vacuum expectation valuesfor the squark and charged slepton fields. This possibility can be avoidedif the A -parameters are not unduly large [154–160]. Additional constraintsmust also be respected to avoid directions in scalar field space in which thefull tree-level scalar potential can become unbounded from below [160]. The MSSM parameter count
The total number of independent physical parameters that define theMSSM (in its most general form) is quite large, primarily due to the soft-115upersymmetry-breaking sector. In particular, in the case of three gen-erations of quarks, leptons, and their superpartners, M e Q , M e U , M e D , M e L ,and M e E are hermitian 3 × A U , A D , and A E are complex3 × M , M , M , B , and µ are in general complexparameters. Finally, as in the Standard Model, the Higgs-fermion Yukawacouplings, λ f ( f = u , d , and e ), are complex 3 × M f = λ f v f / √
2, where v e ≡ v d [with v u and v d as defined above eq. (8.12)].However, not all these parameters are physical. Some of the MSSM pa-rameters can be eliminated by expressing interaction eigenstates in terms ofthe mass eigenstates, with an appropriate redefinition of the MSSM fieldsto remove unphysical degrees of freedom. The analysis of Refs. [161, 162]shows that the MSSM possesses 124 independent parameters. Of these, 18correspond to SM parameters (including the QCD vacuum angle, θ QCD ),one corresponds to a Higgs sector parameter (the analogue of the SM Higgsmass), and 105 are genuinely new parameters of the model. The latterinclude: five real parameters and three CP-violating phases in the gaug-ino/higgsino sector, 21 squark and slepton masses, 36 real mixing angles todefine the squark and slepton mass eigenstates, and 40 CP-violating phasesthat can appear in the squark and slepton interactions.Unfortunately, without additional restrictions on the 124 parameters,the MSSM is not a phenomenologically viable theory. In particular, ageneric point of the MSSM parameter space typically exhibits: (i) no con-servation of the separate lepton numbers L e , L µ , and L τ ; (ii) unsuppressedflavor-changing neutral currents (FCNCs) [163, 164]; and (iii) new sourcesof CP violation [165] that are inconsistent with the experimental bounds.For example, the strong suppression of FCNCs observed in nature impliesthat the off-diagonal matrix elements of the soft-SUSY-breaking squark andslepton squared-mass matrices are highly constrained [43, 45].In practice, various simplifying assumptions are imposed on the SUSY-breaking sector to reduce the number of parameters to a more manageableform, such that the constraints imposed by lepton and quark flavor changingand CP-violating processes are satisfied. For example, specific models ofgravity-mediated and gauge-mediated supersymmetry breaking introducea small number of fundamental parameters that provide the source forSUSY-breaking for the MSSM, consistent with the constraints due to flavor One of the benefits of GMSB models is that the SUSY-breaking is transmitted to theMSSM sector via gauge boson exchange, which is automatically flavor-conserving. M , M and M , theHiggs sector parameters m A and tan β , the Higgsino mass parameter µ , fivesquark and slepton squared-mass parameters for the degenerate first andsecond generations ( M e Q , M e U , M e D , M e L and M e E ), the five correspondingsquark and slepton squared-mass parameters for the third generation, andthree third-generation A -parameters ( A t , A b and A τ ). The first and sec-ond generation A -parameters can be neglected as their phenomenologicalconsequences are negligible. Such an approach assumes that new sourcesof flavor violation and/or CP-violation are either absent or negligible. The MSSM particle spectrum
Spin-1/2 superpartners
The superpartners of the gauge and Higgs bosons are fermions, whose namesare obtained by appending “ino” to the end of the corresponding SM par-ticle name. The gluino is the color-octet Majorana fermion partner of thegluon with mass M e g = | M | . The superpartners of the electroweak gaugeand Higgs bosons (the gauginos and higgsinos) can mix due to SU(2) × U(1)breaking effects. As a result, the physical states of definite mass are model-dependent linear combinations of the charged or neutral gauginos and hig-gsinos, called charginos and neutralinos, respectively (sometimes collec-tively called electroweakinos). The charginos are Dirac fermions, and theneutralinos are Majorana fermions.The tree-level mixing of the charged gauginos ( f W ± ) and higgsinos ( e H + u and e H − d ) is governed by a 2 × M C ≡ (cid:18) M gv u / √ gv d / √ µ (cid:19) . (8.15) In Ref. [168], the number of pMSSM parameters is reduced to ten by assuming onecommon squark mass parameter for the first two generations, a second common squarkmass parameter for the third generation, a common slepton mass parameter, and acommon third generation A parameter. The pMSSM approach has been recently extended to include additional CP-violatingSUSY-breaking parameters in Ref. [169]. M C [cf. eq. (2.86)]: U ∗ M C V − = diag( M e χ +1 , M e χ +2 ) , (8.16)where U and V are unitary matrices. The physical chargino states are Diracfermions and are denoted by e χ ± and e χ ± . These are linear combinations ofthe charged gaugino and higgsino states determined by the matrix elementsof U and V . The chargino masses correspond to the singular values of M C , i.e. , the positive square roots of the eigenvalues of M † C M C , M e χ +1 , e χ +2 = (cid:26) | µ | + | M | + 2 m W ∓ q ( | µ | + | M | + 2 m W ) − | µM − m W sin 2 β | (cid:27) , (8.17)where the states are ordered such that M e χ +1 ≤ M e χ +2 . The relative phase of µ and M is physical and potentially observable.The tree-level mixing of the neutral gauginos ( e B and f W ) and higgsinos( e H d and e H u ) is governed by a 4 × M N ≡ M − g ′ v d g ′ v u M gv d − gv u − g ′ v d gv d − µ g ′ v u − gv u − µ . (8.18)To determine the physical neutralino states and their masses, one mustperform a Takagi-diagonalization of the complex symmetric matrix M N [cf. eq. (2.77)]: W T M N W = diag( M e χ , M e χ , M e χ , M e χ ) , (8.19)where W is a unitary matrix. The physical neutralino states are Majoranafermions, and are denoted by e χ i ( i = 1 , . . . M e χ ≤ M e χ ≤ M e χ ≤ M e χ . The e χ i are the linear combinations ofthe neutral gaugino and higgsino states determined by the matrix elementsof W . The neutralino masses correspond to the singular values of M N , i.e. ,the positive square roots of the eigenvalues of M † N M N .8.5.2. Spin-0 superpartners
The superpartners of the quarks and leptons are spin-zero bosons: thesquarks, charged sleptons, and sneutrinos, respectively. For a given Diracfermion f , there are two superpartners, e f L and e f R , where the L and R f L,R ≡ (1 ∓ γ ) f ,respectively. (There is no e ν R in the MSSM.) However, e f L – e f R mixing ispossible, in which case e f L and e f R are not mass eigenstates.We first consider the squarks and the sleptons. For three generationsof squarks, one must diagonalize 6 × e q iL , e q iR ), where i = 1 , , M = M e Q + m q + L q m q X ∗ q m q X q M e R + m q + R q ! , (8.20)where X q ≡ A q − µ ∗ (cot β ) T q , (8.21)and T q = ( , for q = t , − , for q = b. (8.22)The diagonal squared-masses are governed by soft-SUSY-breakingsquared-masses M e Q and M e R ≡ M e U [ M e D ] for q = t [ b ], the correspond-ing quark masses m t [ m b ], and electroweak correction terms: L q ≡ ( T q − e q sin θ W ) m Z cos 2 β , (8.23) R q ≡ e q sin θ W m Z cos 2 β , (8.24)where e q = [ − ] for q = t [ b ].The off-diagonal squark squared-masses are proportional to the cor-responding quark masses and depend on tan β , the soft-SUSY-breaking A -parameters and the higgsino mass parameter µ . Assuming that the A -parameters are parametrically of the same order (or smaller) relative toother SUSY-breaking mass parameters, it then follows that e q L – e q R mixingeffects are small, with the possible exception of the third generation, wheremixing can be enhanced by factors of m t and m b tan β .In the case of third generation e q L – e q R mixing, the mass eigenstates (de-noted by e q and e q , with m ˜ q < m ˜ q ) are determined by diagonalizing the2 × M . The corresponding squared-masses and mixing angle are: m q , = h Tr M ∓ p (Tr M ) − M i , (8.25)sin 2 θ ˜ q = 2 m q | X q | m q − m q . (8.26)119he results above also apply to the charged sleptons with the substitutions: q → ℓ with T ℓ = − and e ℓ = −
1, and the replacement of the SUSY-breaking parameters: M e Q → M e L , M e D → M e E , and A q → A τ . For theneutral sleptons, e ν R does not exist in the MSSM, so e ν L is a mass eigenstate.In the case of three generations, the supersymmetry-breaking scalar-squared masses [ M e Q , M e U , M e D , M e L , and M e E ] and the A -parameters [ A U , A D , and A E ] are now 3 × × e f iL – e f jR mixing (for i = j ). In practice, since the e f L – e f R mixing is appreciable only for the third generation, this additionalcomplication can often be neglected. The Higgs sector of the MSSM
Having completed our tour of the superpartners of the SM particles, wenow focus of the Higgs sector of the MSSM [170–172]. We first providedetails of the structure of the Higgs sector based on a tree-level analysis.We then discuss the importance of radiative corrections, in light of theobserved Higgs boson with a mass of 125 GeV.8.6.1.
The tree-level MSSM Higgs sector
The tree-level scalar Higgs potential, previously given in eq. (8.11), is CP-conserving. This follows from the fact that m , the only potentially com-plex parameter that appears in eq. (8.11), can be chosen real and positiveby an appropriate rephasing of the Higgs fields.After minimizing the Higgs potential, as indicated above eq. (8.12), onecan identify the physical Higgs states. The five physical Higgs particlesconsist of a charged Higgs pair H ± = H ± d sin β + H ± u cos β , (8.27)one CP-odd neutral scalar A = √ (cid:0) Im H d sin β + Im H u cos β (cid:1) , (8.28)and two CP-even neutral scalar mass eigenstates that are determined bydiagonalizing the neutral CP-even Higgs scalar squared-mass matrix, M = (cid:18) m A sin β + m Z cos β − ( m A + m Z ) sin β cos β − ( m A + m Z ) sin β cos β m A cos β + m Z sin β (cid:19) . (8.29)The eigenstates of M are identified as the neutral CP-even Higgs bosons, h = − ( √ H d − v d ) sin α + ( √ H u − v u ) cos α , (8.30) H = ( √ H d − v d ) cos α + ( √ H u − v u ) sin α , (8.31)120hich defines the CP-even Higgs mixing angle α .All Higgs masses and couplings can be expressed in terms of two pa-rameters, usually chosen to be m A and tan β . The charged Higgs mass isgiven by m H ± = m A + m W . (8.32)The squared-masses of the CP-even Higgs bosons h and H are eigenvaluesof M . The trace and determinant of M yield, m h + m H = m A + m Z , m h m H = m A m Z cos β , (8.33)where the CP-even Higgs masses are given by m H,h = (cid:18) m A + m Z ± q ( m A + m Z ) − m Z m A cos β (cid:19) . (8.34)In the convention where 0 ≤ β ≤ π , it is standard practice to choose α to lie in the range | α | ≤ π . However, because the off-diagonal element of M is negative semi-definite, one finds that − π ≤ α ≤
0. More explicitly,the mixing angle α can be determined as a function of m A and tan β fromthe following expression and from eq. (8.34), cos α = s m A sin β + m Z cos β − m h m H − m h , (8.35)and sin α = − (1 − cos α ) / .In the expression for the couplings of the Higgs bosons with the gaugebosons, only the combination β − α appears. For example, the couplingof h to V V (where
V V = W + W − or ZZ ) relative to the correspondingcoupling of the SM Higgs boson, h SM , is given by, g hV V g h SM V V = sin( β − α ) . (8.36)Given the range of the angles α and β , it follows that 0 ≤ β − α ≤ π . Inparticular, the following expressions can be obtained,cos( β − α ) = m Z sin 2 β cos 2 β p ( m H − m h )( m H − m Z cos β ) . (8.37)sin( β − α ) = s m H − m Z cos βm H − m h . (8.38) The corresponding expressions for a general CP-conserving two Higgs doublet modelcan be found in Ref. [173] . W RPV = 0] by employing the last two terms ofeq. (6.63). Focusing on the Higgs interactions with third generation quarks,one obtains the so-called Type-II Higgs-quark interaction [174], L Yuk = − ǫ ij (cid:2) h b b R H di Q Lj + h t t R Q Li H uj (cid:3) + h . c . , (8.39)where Q L ≡ ( t L , b L ) is the quark doublet and i and j are SU(2) indices. Ineq. (8.39), we employ four-component quark fields, where q R,L ≡ P R,L q and P R,L = (1 ± γ ). The quark masses are identified by replacing the Higgsfields in eq. (8.39) with their corresponding vacuum expectation values, m b = h b v cos β/ √ , m t = h t v sin β/ √ . (8.40)The tree-level Yukawa couplings of the lightest CP-even Higgs boson tothird generation quark pairs are given by g hb ¯ b = − m b v sin α cos β = m b v (cid:2) sin( β − α ) − cos( β − α ) tan β (cid:3) , (8.41) g ht ¯ t = m t v cos α sin β = m t v (cid:2) sin( β − α ) + cos( β − α ) cot β (cid:3) . (8.42)It is straightforward to work out the couplings of the other Higgs bosons ofthe model to the quarks (and leptons). A comprehensive set of Feynmanrules for Higgs bosons in the MSSM can be found in Refs. [170, 171].In the limit of m A ≫ m Z , the expressions for the Higgs masses andmixing angle are given by, m h ≃ m Z cos β − m Z sin β m A , (8.43) m H ≃ m A + m Z sin β , (8.44) m H ± = m A + m W , (8.45)cos( β − α ) ≃ m Z sin 4 β m A . (8.46)Two consequences are immediately apparent. First, m A ≃ m H ≃ m H ± , (8.47)up to corrections of O ( m Z /m A ). Second, cos( β − α ) ≃
0, up to correctionsof O ( m Z /m A ). This is the decoupling limit of the MSSM Higgs sector,since at energy scales below the approximately common mass of the heavyHiggs bosons H ± , H , and A , the effective Higgs theory is equivalent to theone-doublet Higgs sector of the SM [175, 176]. Indeed, one can check thatin the limit of cos( β − α ) →
0, all the h couplings to SM particles approachtheir SM limits, as in the case of the hV V coupling exhibited in eq. (8.36)and in the case of the hq ¯ q couplings exhibited in eqs. (8.41) and (8.42).122.6.2. Impact of radiative corrections on the MSSM Higgs sector
The tree-level result for m h given in eq. (8.34) yields a startling prediction, m h ≤ m Z | cos 2 β | ≤ m Z . (8.48)This is clearly in conflict with the observed Higgs mass of 125 GeV. How-ever, the above inequality receives quantum corrections. The Higgs masscan be shifted due to loops of particles and their superpartners exhibitedbelow (an incomplete cancellation, which would have been exact if super-symmetry were unbroken). h h h ht e t , The impact of these corrections can be significant [177–179]. In particular,the qualitative behavior of the one-loop radiative corrections can be mosteasily seen in the limit of large top-squark masses. In this limit, both theoff-diagonal entries and the splitting between the two diagonal entries of thetop-squark squared-mass matrix [eq. (8.20)] are small in comparison to thesquare of the geometric mean of the two top-squark masses, M ≡ M e t M e t .In this case (assuming m A > m Z ), the predicted upper bound for m h isapproximately given by [180] m h < ∼ m Z + 3 g m t π m W (cid:20) ln (cid:18) M S m t (cid:19) + X t M S (cid:18) − X t M S (cid:19)(cid:21) , (8.49)where X t ≡ A t − µ cot β governs stop mixing (taking A t and µ real forsimplicity). The Higgs mass upper limit is saturated when tan β is large[ i.e. , cos (2 β ) ∼
1] and X t = √ M S , which defines the so-called maximalmixing scenario.A more complete treatment of the radiative corrections [181] shows thateq. (8.49) somewhat overestimates the true upper bound of m h . Thesemore refined computations, which incorporate renormalization group im-provement, and the two-loop and leading three-loop contributions, yieldan upper bound of m h < ∼
135 GeV in the region of large tan β (with anaccuracy of a few GeV) for m t = 175 GeV and M S < ∼ .7. Unification of gauge couplings
Grand unification theory (GUT) predicts the unification of gauge couplingsat some very high energy scale [29, 68, 147, 183]. The running of the cou-plings is dictated by the particle content of the effective theory that residesbelow the GUT scale. However, attempts to embed the Standard Modelin an SU(5) or SO(10) unified theory do not quite succeed. In particular,the three running gauge couplings (the strong QCD coupling g s and theelectroweak gauge couplings g and g ′ ) do not meet at a point, as shownby the dashed lines in Fig. 1. In contrast, in the case of the MSSM withsuperpartner masses of order 1 TeV, the renormalization group evolution ismodified above the SUSY-breaking scale. In this case, unification of gaugecouplings can be (approximately) achieved as illustrated by the red andblue lines in Fig. 1. Fig. 1.:
Renormalization group evolution of the inverse gauge couplings α − a ( Q )in the Standard Model (dashed lines) and the MSSM (solid lines). In the MSSMcase, α ( m Z ) is varied between 0.121 and 0.117, and the supersymmetric particlemass thresholds are between 500 GeV and 1.5 TeV, for the lower and upper solidlines, respectively. Two-loop effects are included. Taken from Ref. [38]. A quantitative assessment of the success of gauge coupling unificationcan be performed as follows. Since the electroweak gauge couplings g and g ′ are very well measured, first focus on these two couplings. For a given low-energy effective theory (below the GUT scale), we use the renormalizationgroup equations (RGEs) to determine the couplings g and g ′ as a function124f the energy scale. We then define M GUT to be the scale at which thesetwo couplings meet.We now assume that the unification of the three gauge couplings, g s , g and g ′ occurs at M GUT . Using the RGEs for the gauge couplings, we cannow run g s down to the electroweak scale and compare with the experi-mentally measured value.8.7.1. Normalization of the U(1) Y coupling In electroweak theory, the overall normalization of the U(1) Y coupling isa matter of convention. But, if the GUT group is simple and nonabelian,then the relative normalization of the U(1) Y coupling to the SU(2) gaugecoupling is fixed. We denote the SU(3) × SU(2) × U(1) Y gauge couplingsusing the proper GUT normalization by g , g and g respectively. Ourtask is to relate g with g ′ . To do so, let us begin by considering thecovariant derivative, D µ = ∂ µ + i X a g a T a A aµ . (8.50)If the gauge group is a direct product group, then different sets of generators T a are associated with with the different group factors, and we must use theappropriate g a depending on which generator it multiplies. In particular,for SU(2) × U(1) Y (below the GUT scale), g a T a A qµ ∋ gT W µ + g ′ Y B µ . (8.51)Above the GUT scale, the corresponding terms of the covariant derivativeare g a T a A qµ ∋ g U ( T W µ + T B µ ) , (8.52)where g U is the gauge coupling of the unifying GUT group and T is theproperly normalized hypercharge generator. In particular, the generatorsof the GUT group satisfy Tr( T a T b ) = T ( R ) δ ab , (8.53)where T ( R ) is a constant that depends on the representation R . We nowset the two covariant derivatives above equal at the GUT scale, g U ( T W µ + T B µ ) = gT W µ + g ′ Y B µ . (8.54) Once T ( R ) is fixed for one representation, it is then determined for all other rep-resentations. It is standard practice to fix T ( R ) = for the defining (fundamental)representation, although the argument presented below is independent of this choice. able 6.: The T and Y quantum numbers of the two-component fermion fieldsthat make up one generation of SM fermions. In computing the correspondingtraces, one must not forget the color factor of 3 that arises when tracing over the(suppressed) color index. Two-component fields T Y Tr( T ) Tr Y ψ Q
12 13 ) 3( ) ψ Q −
12 13 ) 3( ) ψ U − ) ψ D ) ψ L − ψ L − − ψ E g U = g = g = g at the GUT scale, it follows that g = g and g T = g ′ ( Y / T ( R ) only depends on the representation R ,eq. (8.53) yields Tr( T ) = Tr( T ) . Thus, g = g ′ Tr Y T ) . (8.55)The relevant quantum numbers are provided in Table 6.The traces in eq. (8.55) are evaluated by summing over one generationof fermions, under the assumption that it is made up of complete irreduciblerepresentations of the GUT group. Using the results of Table 6, we simplyadd up the last two columns. Including the appropriate color factor of 3when tracing over the suppressed color index, we obtain Tr( T ) = 2 andTr Y = . Thus, eq. (8.55) yields g = g ′ . (8.56)8.7.2. Gauge coupling running
We now examine the running of the gauge couplings in the one-loop ap-proximation, where the gauge couplings g i obey the differential equation, dg i dt = b i g i π , for i = 1 , , , (8.57) In an SU(5) GUT, one generation of fermions make up a 10-dimensional and thecomplex conjugate of a 5-dimensional representation of SU(5). In an SO(10) GUT, onegeneration of fermions (including the right-handed neutrino) comprise a 16 dimensionalspinor representation of SO(10). t = ln Q and Q is the energy scale. The solution to eq. (8.57) is1 g i ( m Z ) = 1 g U − b i π ln (cid:18) m Z M (cid:19) , (8.58)where M GUT is the GUT scale at which the three gauge couplings unify.Using eq. (8.58), the following two equations are obtained:sin θ W ( m Z ) = g ′ ( m Z ) g ( m Z ) + g ′ ( m Z ) = g ( m Z ) g ( m Z ) + g ( m Z )= 38 − π α ( m Z )( b − b ) ln (cid:18) M m Z (cid:19) , (8.59)ln (cid:18) M m Z (cid:19) = 32 π b + 3 b − b (cid:18) α ( m Z ) − α s ( m Z ) (cid:19) , (8.60)where e = g sin θ W , α ≡ e / π and α s ≡ g s / π .It is convenient to introduce the parameter, x ≡ (cid:18) b − b b − b (cid:19) . (8.61)Then, eqs. (8.59) and (8.60) yield,sin θ W ( m Z ) = 11 + 8 x (cid:20) x + α ( m Z ) α s ( m Z ) (cid:21) . (8.62)Once we know the value of x , we can use the above equation to determine α s ( m Z ) given the values of sin θ W and α , evaluated at m Z , α s ( m Z ) = α ( m Z )(1 + 8 x ) sin θ W ( m Z ) − x . (8.63)The value of x is determined from the values of the b i , which are givenby the following formula, b i = T f ( R k ) Y j = k d f ( R j ) + T s ( R k ) Y j = k d s ( R j ) − C ( G i ) , (8.64)where f , s stand for fermions and scalars, respectively, d ( R ) is the dimen-sion of the representation R , and the generators in representation R satisfy,Tr( T a T b ) = T ( R ) δ ab , ( T a T a ) ij = C ( G ) δ ij . (8.65)Note that, T ( R ) = (cid:20)q
35 12 Y (cid:21) = Y , (8.66)127here we have employed the properly normalized hypercharge generator, p / Y / C (G) = N for G=SU( N ), and C (G) = 0 forG=U(1).One can now assess the success or failure of gauge coupling unificationin the SM and in the MSSM. For details, see Problems 47 and 48. Asadvertised in Fig. 1, the gauge couplings do not unify when the SM isextrapolated to the GUT scale. In contrast, in the MSSM, the modifiedrunning of the gauge couplings due to the supersymmetric partners of theSM particles results in approximate unification. This success has oftenbeen touted as one of the motivations for TeV-scale supersymmetry.
Problems
Problem 44.
Starting with the SUSY Lagrangian for SUSY Yang Millstheory coupled to matter given in eq. (6.63), eliminate the auxiliary fieldsand obtain the Lagrangian of the MSSM prior to SUSY-breaking. For sim-plicity, you may consider only one generation of quarks and leptons andtheir superpartners. Then add in the soft-SUSY-breaking terms to obtainthe complete MSSM Lagrangian. Using this result, verify the mass spectrumof the supersymmetric particles obtained in Section 8.5.
Problem 45.
Using the results of Problem 44, verify the results obtainedin Section 8.6 for the MSSM Higgs sector. Write out the Feynman rulesfor the interaction of the Higgs bosons with the gauge bosons and with thequarks and leptons.
Problem 46.
Using the results of Problem 44, one can obtain the completeset of Feynman rules for the MSSM with one generation of quarks andleptons and their superpartners. Work out as many of the rules as you canand check your results against Ref. [188].
Problem 47.
Assuming N g generations of the quarks and leptons and N h copies of the SM Higgs boson, use eq. (8.64) to obtain b = N g − ,b = N h + N g − ,b = N h + N g . For a more precise analysis, we should extend the calculations of this subsection toinclude two-loop running of the gauge couplings [184]. One must also properly treatthreshold corrections at the TeV scale [185, 186] (due to mass splittings among super-partners) and at the GUT scale [187]. The latter are quite model-dependent and allowssome wiggle room in achieving precise gauge coupling unification. or the SM, we have N g = 3 and N h = 1 . Check that b = − , b = − and b = . Consequently, x = 23218 = 0 . In particular, note that x is independent of N g . Problem 48.
Show that the SM results of Problem 47 are modified in theMSSM as follows: b =2 N g − ,b = N h + 2 N g − ,b = N h + 2 N g . For the MSSM, we have N g = 3 and N h = 2 . Verify that b = − , b = 1 and b = , and consequently, x = . Using the values for α ( m Z ) and sin θ W ( m Z ) given in Ref. [189], evaluate α s using eq. (8.63). Show thatfor x = (as predicted by the MSSM), one obtains a value for α s ( m Z ) thatis quite close to the current world average [189]. Using x = 0 . , checkthat the corresponding SM prediction for α s ( m Z ) is significantly lower thanthe observed value.
9. Supersymmetry Quo Vadis?
In these lectures, time constraints have limited the number of topics that wehave been able to cover. The reader can consult the many fine books [2–29]and the reviews and lecture notes [30–49] already cited in Section 1 topursue various topics in supersymmetry in greater depth.In Section 8, we introduced the basics of the MSSM. But this is notthe only possible supersymmetric extension of the SM. For example, inthe MSSM as defined in Section 8, the neutrino is massless. There area number of ways to extend the MSSM to allow for massive neutrinos.For example, by relaxing the assumption of R -parity conservation, one canintroduce lepton number violating terms in the MSSM Lagrangian thatcan be used to incorporate massive neutrinos that are consistent with theneutrino oscillation data. Alternatively, one can start with the seesaw-extended SM and consider its supersymmetric extension [146, 193–201].Extensions of the MSSM have also been proposed to solve a variety oftheoretical problems. One such problem involves the µ parameter of the There is a huge literature on this subject. See, e.g., Refs. [190–192] and the referencescontained therein. µ is a SUSY- preserving parameter, it must be of orderthe effective SUSY-breaking scale of the MSSM to yield a consistent su-persymmetric phenomenology [202]. Any natural solution to the so-called µ -problem must incorporate a symmetry that enforces µ = 0 and a smallsymmetry-breaking parameter that generates a value of µ that is not para-metrically larger than the effective SUSY-breaking scale [203].A number of proposed mechanisms in the literature provide concreteexamples of a natural solution to the µ -problem of the MSSM (see, e.g. ,Refs. [202–206]). For example, one can replace µ by the vacuum expectationvalue of a new SU(3) × SU(2) × U(1) singlet scalar field. This can be achievedby adding a singlet chiral superfield to the MSSM. The end result is thenext-to-minimal supersymmetric extension of the SM, otherwise known asthe NMSSM, which is reviewed in Refs. [207, 208].Ultimately, in order to determine how nature chooses to incorporatesupersymmetry, one must discover evidence for supersymmetric particlesin experiments. The phenomenology of the MSSM and its extensions is ahuge subject that requires a separate lecture course. Since we have no timeto present a detailed treatment of supersymmetric phenomenology here, wecan only refer the reader to some of the excellent books and review articleson this subject (see e.g., Refs. [16, 17, 32, 47, 92]).As discussed in Section 3, supersymmetry was proposed to avoidquadratic UV-sensitivity in a theory with elementary scalars. To avoida significant fine-tuning of the fundamental parameters, which is requiredto explain the observed Higgs and Z boson masses, the SUSY-breakingscale should be not much larger than 1 TeV. Consequently, experimentscurrently being carried out at the Large Hadron Collider (LHC) should beon the verge of discovering supersymmetric particles. However, so far noevidence for SUSY has emerged from the LHC data.Figs. 2 and 3 summarize the limits on supersymmetric particle massesas of the spring of 2017. Because the LHC is a proton-proton collider, thestrongest SUSY mass bounds of about 2 TeV are obtained for the coloredsuperpartners (squarks and gluinos). Bounds on the top squark mass (whichplay an important role in assessing the degree of fine-tuning required toaccommodate the observed Higgs and Z boson masses) are closer to 1 TeV.Clearly some tension exists between the theoretical expectations for themagnitude of the SUSY-breaking parameters and the non-observation ofsupersymmetric phenomena. Hence, the title of this section, which is alsothe title of Ref. [211], where the theoretical implications of the present LHCdata for TeV-scale supersymmetry is reconsidered.130 ig. 2.: Mass reach of a representative selection of ATLAS searches for SUSYas of May, 2017. Taken from Ref. [209].
Mass Scale [GeV] χ∼ χ∼ W Z → ± χ∼ χ∼→ pp χ∼ χ∼ W H → ± χ∼ χ∼→ pp χ∼ χ∼ W Z → ± χ∼ χ∼→ pp χ∼ χ∼ντττ→ ± χ∼ χ∼→ pp χ∼ χ∼ντ ll → ± χ∼ χ∼→ pp χ∼ χ∼ν lll → ± χ∼ χ∼→ pp χ∼ χ∼ν lll → ± χ∼ χ∼→ pp χ∼ q → q~,q~q~ → pp χ∼ q → q~,q~q~ → pp χ∼ b → b~,b~b~ → pp χ∼ b → b~,b~b~ → pp χ∼ b → b~,b~b~ → pp χ∼ b → b~,b~b~ → pp χ∼ ± b W → b ± χ∼→ t~,t~t~ → pp χ∼ ± b W → b ± χ∼→ t~,t~t~ → pp χ∼ ± b W → b ± χ∼→ t~,t~t~ → pp χ∼ ± b W → b ± χ∼→ t~,t~t~ → pp (4-body) χ∼ b f f → t~,t~t~ → pp (4-body) χ∼ b f f → t~,t~t~ → pp (4-body) χ∼ b f f → t~,t~t~ → pp χ∼ c → t~,t~t~ → pp χ∼ c → t~,t~t~ → pp χ∼ c → t~,t~t~ → pp χ∼ t → t~,t~t~ → pp χ∼ t → t~,t~t~ → pp χ∼ t → t~,t~t~ → pp χ∼ t → t~,t~t~ → pp χ∼ t → t~,t~t~ → pp χ∼ t → t~,t~t~ → pp χ∼ t → t~,t~t~ → pp χ∼ qq (W/Z) → ) χ∼ / ± χ∼ qq( → g~,g~g~ → pp χ∼ qq (W/Z) → ) χ∼ / ± χ∼ qq( → g~,g~g~ → pp χ∼ qq W → ± χ∼ qq → g~,g~g~ → pp χ∼ qq W → ± χ∼ qq → g~,g~g~ → pp χ∼ qq W → ± χ∼ qq → g~,g~g~ → pp ± χ∼ bt → g~,g~g~ → pp χ∼ t c → t~ t → g~,g~g~ → pp χ∼ tt → g~,g~g~ → pp χ∼ tt → g~,g~g~ → pp χ∼ tt → g~,g~g~ → pp χ∼ tt → g~,g~g~ → pp χ∼ tt → g~,g~g~ → pp χ∼ tt → g~,g~g~ → pp χ∼ tt → g~,g~g~ → pp χ∼ tt → g~,g~g~ → pp χ∼ bb → g~,g~g~ → pp χ∼ bb → g~,g~g~ → pp χ∼ bb → g~,g~g~ → pp χ∼ qq → g~,g~g~ → pp χ∼ qq → g~,g~g~ → pp E W K G a ug i no s < 40 GeV) LSP - M
Mother (Max exclusion for M 2l soft SUS-16-048SUS-16-025 MultileptonSUS-16-039 Multilepton SUS-16-039SUS-16-024 Multilepton (tau dominated)SUS-16-039 x=0.5 Multilepton (tau enriched)SUS-16-039 x=0.5 Multilepton + 2l same-sign (flavour democratic)SUS-16-039 x=0.95 Multilepton (flavour democratic) SUS-16-039SUS-16-024 x=0.5 S qu a r k )s~,c~,d~,u~( L q~+ R q~ 0l(MT2) SUS-16-036SUS-16-015 )s~,c~,d~,u~( L q~+ R q~ 0l(MHT) SUS-16-033SUS-16-014 0lSUS-16-032 ) T α SUS-16-016 0l( 0l(MT2) SUS-16-036SUS-16-015 0l(MHT) SUS-16-033SUS-16-014 2l opposite-signSUS-17-001 x=0.5 0l(MT2)SUS-16-036 x=0.5 0l SUS-16-049SUS-16-029 x=0.5 1l SUS-16-051SUS-16-028 x=0.5SUS-16-031 1l soft < 80 GeV)
LSP - M
Mother (Max exclusion for M 0l SUS-16-049SUS-16-029 < 80 GeV)
LSP - M
Mother (Max exclusion for M 2l soft SUS-16-048SUS-16-025 < 80 GeV)
LSP - M
Mother (Max exclusion for M 0lSUS-16-049 < 80 GeV)
LSP - M
Mother (Max exclusion for M 0l(MT2)SUS-16-036 < 80 GeV)
LSP - M
Mother (Max exclusion for M 0lSUS-16-032 < 80 GeV)
LSP - M
Mother (Max exclusion for MSUS-16-030 0l 0l SUS-16-049SUS-16-029 1l SUS-16-051SUS-16-028 2l opposite-sign SUS-17-001SUS-16-027 ) T α SUS-16-016 0l( 0l(MT2) SUS-16-036SUS-16-015 0l(MHT) SUS-16-033SUS-16-014 G l u i no Multilepton SUS-16-041SUS-16-022 x=0.5 0l(MHT) SUS-16-033SUS-16-014 x=0.5 2l same-sign SUS-16-035SUS-16-020 = 20 GeV)
LSP - M
Interm. (M 2l same-sign SUS-16-035SUS-16-020 x=0.5) φ∆ LSP - M ± χ∼ (MSUS-16-030 0l = 20 GeV) LSP - M
Mother (M 1l(MJ)SUS-16-037SUS-16-030 0l Multilepton SUS-16-041SUS-16-022 2l same-sign SUS-16-035SUS-16-020 ) φ∆ T α SUS-16-016 0l( 0l(MT2) SUS-16-036SUS-16-015 0l(MHT) SUS-16-033SUS-16-014 ) T α SUS-16-016 0l( 0l(MT2) SUS-16-036SUS-16-015 0l(MHT) SUS-16-033SUS-16-014 0l(MT2) SUS-16-036SUS-16-015 0l(MHT) SUS-16-033SUS-16-014
Selected CMS SUSY Results* - SMS Interpretation Moriond '17 - ICHEP '16 = 13TeVsCMS Preliminary -1 L = 12.9 fb -1 L = 35.9 fb
LSP m ⋅ +(1-x) Mother m ⋅ = x Intermediate mFor decays with intermediate mass,0 GeV unless stated otherwise ≈ LSP
Only a selection of available mass limits. Probe *up to* the quoted mass limit for m*Observed limits at 95% C.L. - theory uncertainties not included
Fig. 3.:
Summary of exclusion limits in Simplified Model Spectra (SMS) fromCMS searches for SUSY as of March, 2017. Taken from Ref. [210]. r a c t i on o f M ode l s E xc l uded g~ t~ t~ b~ b~ q~ χ∼ χ∼ χ∼ χ∼ τ∼ τ∼ l~ ± χ∼ ± χ∼ M a ss [ G e V ] ATLAS − = 8 TeV, 20.3 fbs Fig. 4.:
A summary of the sensitivity of ATLAS to different types of supersym-metric particles in the 19 parameter pMSSM. Each vertical bar is a 1D projectionof the supersymmetric particle mass, with the color coding representing the frac-tion of models excluded by the ATLAS searches in each bin. This figure takenfrom Ref. [212].
The absence of evidence for supersymmetry in the LHC data can also beinterpreted in the context of the pMSSM, which was briefly introduced atthe end of Section 8.4. In a scan of the 19 parameter pMSSM performed bythe ATLAS Collaboration, the mass of each supersymmetric particle wasconstrained with an upper limit of 4 TeV, motivated to ensure a high densityof models in reach of the LHC. Lower limits on the supersymmetric particlemasses were also applied to avoid constraints from the LEP experiments.A summary of the sensitivity of the ATLAS Collaboration experiment todifferent types of supersymmetric particles in the pMSSM is shown in Fig. 4.Of course, the LHC program is still in its infancy. Two more years ofRun-2 data from 2017–2018 must be analyzed. After a two year shutdown,Run-3 follows from 2021–2023 according to the current planning schedule.The high luminosity (HL) phase of the LHC [213] will commence in 2026,with an anticipation of reaching 3000 fb − of data by the year 2038. Thisis a nearly 100-fold increase of the present LHC data sample. There is stillample room for the discovery of SUSY at the LHC during its lifetime.Thus, the experimental future of supersymmetry is still very much alive.Beyond the HL-LHC, there are possibilities of energy upgrades at the LHCby roughly a factor of two, and considerations of the next generation ofhadron colliders with a center of mass energy of 100 TeV. If the SUSY-breaking scale is somewhat higher than 1 TeV (but less than say, 10 TeV),132hen opportunities for discovery will be available at these future hadroncollider facilities [96].The theoretical future of supersymmetry is also quite bright. Even ifthe SUSY-breaking scale lies significantly above the TeV-scale, there arestill many opportunities for incorporating supersymmetry into the funda-mental theory of particle physics. In this latter scenario, SUSY would notbe relevant for explaining the origin of the scale of electroweak symmetrybreaking. Another explanation would be required, perhaps one of the othersuggested theoretical approaches mentioned in Section 3.For example, it may still be possible that some remnant of the super-symmetric particle spectrum survives down to the TeV-scale or below. Thisis the idea of split-SUSY [214–219], in which the squarks and sleptons aresignificantly heavier (perhaps by many orders of magnitude) than 1 TeV,whereas the fermionic superpartners of the gauge and Higgs bosons maybe kinematically accessible at the LHC. Of course, the SUSY-breaking dy-namics responsible for such a split-SUSY spectrum would not be relatedto the origin of the scale of electroweak symmetry breaking. Nevertheless,models of split-SUSY can account for the dark matter (which is assumedto be the LSP gaugino or higgsino) and gauge coupling unification, therebypreserving two of the desirable features of TeV-scale supersymmetry.There are many theoretical aspects of supersymmetry theory that liebeyond the scope of these lectures but deserve further exploration. Amongthese, non-perturbative approaches to supersymmetric theories, such asholomorphy and Seiberg duality, have been particularly fruitful. The powerof holomorphy was briefly exhibited in Section 5.8, when we reviewedSeiberg’s proof of the non-renormalization of the superpotential [109].There are many other applications of holomorphy, such as the computa-tion of exact β functions in supersymmetric gauge theories [116, 117, 220].As an effective tool in non-perturbative regimes, Seiberg duality elucidatesstrongly coupled gauge theories by relating them to dual weakly coupledgauge theories. In Refs. [20, 23, 221–223] one can find numerous applica-tions to the study of non-perturbative dynamics in strongly-coupled super-symmetric theories and in fundamental models of SUSY-breaking.Another flourishing area of research is that of scattering amplitudes[224, 225], where novel methods are being developed to facilitate compu-tations that were previously intractable using the traditional Feynman-diagrammatic approach. Here supersymmetric theories can serve as test-ing grounds for techniques that may eventually be extended to non-supersymmetric quantum field theories. For example, in N = 4 super-133ymmetric Yang-Mills theory (one of the few known examples of a finitequantum field theory in four spacetime dimensions), amplitudes are wellunderstood, making it a relatively simple arena in which to study new com-putational methods [226]. Moreover, tree-level gluon scattering amplitudesin N = 4 super Yang-Mills are identical to those in any other gauge the-ory, so it is reasonable to expect that methods developed for SUSY gaugetheories could be adapted to the computation of QCD amplitudes.Supersymmetry is also a powerful tool for analyzing a variety of prob-lems in mathematical physics, and plays a critical role in the formulationof string theory [26, 227–234]. Evidently, even in the absence of evidencefor SUSY at the TeV scale, it is very likely that supersymmetry will leadto important new insights, both in experimental and theoretical directions.With this in mind, it is our hope that these lectures have provided a modestintroduction into the fascinating world of supersymmetry. Acknowledgments
We would like to thank Zackaria Chacko, Andrew Cohen, Michael Dine,Herbi Dreiner, Stephen Martin, Raman Sundrum, and John Terning formany enlightening discussions. H.E.H. is grateful to Rouven Essig andIan Low for their invitation to present these lectures at TASI 2016, andtheir patience in waiting for these lecture notes to be completed. Thiswork is supported in part by the U.S. Department of Energy grant numberde-sc0010107. L.S.H. is also supported by the Israel Science Foundationunder grant no. 1112/17.
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