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SUPERSPACE or One thousand and one lessons in supersymmetry S James Gates, Jr Massachusetts Institute of Technology, Cambridge, Massachusetts (Present address: University of Maryland, College Park, Maryland) gatess@wam.umd.edu Marcus T Grisaru Brandeis University, Waltham, Massachusetts (Present address: McGill University, Montreal, Quebec) grisaru@physics.mcgill.ca Martin Roˇek c State University of New York, Stony Brook, New York rocek@insti.physics.sunysb.edu Warren Siegel University of California, Berkeley, California (Present address: State University of New York) warren@wcgall.physics.sunysb.edu Library of Congress Cataloging in Publication Data Main entry under title: Superspace : one thousand and one lessons in supersymmetry (Frontiers in physics ; v 58) Includes index Supersymmetry Quantum gravity Supergravity I Gates, S J II Series QC174.17.S9S97 1983 530.1’2 83-5986 ISBN 0-8053-3160-3 ISBN 0-8053-3160-1 (pbk.) Superspace is the greatest invention since the wheel [1] Preface Said Ψ to Φ, Ξ, and Υ: ‘‘Let’s write a review paper.’’ Said Φ and Ξ: ‘‘Great idea!’’ Said Υ: ‘‘Naaa.’’ But a few days later Υ had produced a table of contents with 1001 items Ξ, Φ, Ψ, and Υ wrote Then didn’t write Then wrote again The review grew; and grew; and grew It became an outline for a book; it became a first draft; it became a second draft It became a burden It became agony Tempers were lost; and hairs; and a few pounds (alas, quickly regained) They argued about ‘‘;’’ vs ‘‘.’’, about ‘‘which’’ vs ‘‘that’’, ‘‘˜’’ vs ‘‘ˆ’’, ‘‘γ’’ vs ‘‘Γ’’, ‘‘+’’ vs ‘‘-’’ Made bad puns, drew pictures on the blackboard, were rude to their colleagues, neglected their duties Bemoaned the paucity of letters in the Greek and Roman alphabets, of hours in the day, days in the week, weeks in the month Ξ, Φ, Ψ and Υ wrote and wrote * * * This must stop; we want to get back to research, to our families, friends and students We want to look at the sky again, go for walks, sleep at night Write a second volume? Never! Well, in a couple of years? We beg our readers’ indulgence We have tried to present a subject that we like, that we think is important We have tried to present our insights, our tools and our knowledge Along the way, some errors and misconceptions have without doubt slipped in There must be wrong statements, misprints, mistakes, awkward phrases, islands of incomprehensibility (but they started out as continents!) We could, probably we should, improve and improve But we can no longer wait Like climbers within sight of the summit we are rushing, casting aside caution, reaching towards the moment when we can shout ‘‘it’s behind us’’ This is not a polished work Without doubt some topics are treated better elsewhere Without doubt we have left out topics that should have been included Without doubt we have treated the subject from a personal point of view, emphasizing aspects that we are familiar with, and neglecting some that would have required studying others’ work Nevertheless, we hope this book will be useful, both to those new to the subject and to those who helped develop it We have presented many topics that are not available elsewhere, and many topics of interest also outside supersymmetry We have [1] A Oop, A supersymmetric version of the leg, Gondwanaland predraw (January 10,000,000 B.C.), to be discovered included topics whose treatment is incomplete, and presented conclusions that are really only conjectures In some cases, this reflects the state of the subject Filling in the holes and proving the conjectures may be good research projects Supersymmetry is the creation of many talented physicists We would like to thank all our friends in the field, we have many, for their contributions to the subject, and beg their pardon for not presenting a list of references to their papers Most of the work on this book was done while the four of us were at the California Institute of Technology, during the 1982-83 academic year We would like to thank the Institute and the Physics Department for their hospitality and the use of their computer facilities, the NSF, DOE, the Fleischmann Foundation and the Fairchild Visiting Scholars Program for their support Some of the work was done while M.T.G and M.R were visiting the Institute for Theoretical Physics at Santa Barbara Finally, we would like to thank Richard Grisaru for the many hours he devoted to typing the equations in this book, Hyun Jean Kim for drawing the diagrams, and Anders Karlhede for carefully reading large parts of the manuscript and for his useful suggestions; and all the others who helped us S.J.G., M.T.G., M.R., W.D.S Pasadena, January 1983 August 2001: Free version released on web; corrections and bookmarks added Contents Preface Introduction A toy superspace 2.1 Notation and conventions 2.2 Supersymmetry and superfields 2.3 Scalar multiplet 15 2.4 Vector multiplet 18 2.5 Other global gauge multiplets 28 2.6 Supergravity 34 2.7 Quantum superspace 46 Representations of supersymmetry 3.1 Notation 54 3.2 The supersymmetry groups 62 3.3 Representations of supersymmetry 69 3.4 Covariant derivatives 83 3.5 Constrained superfields 89 3.6 Component expansions 92 3.7 Superintegration 97 3.8 Superfunctional differentiation and integration 101 3.9 Physical, auxiliary, and gauge components 108 3.10 Compensators 112 3.11 Projection operators 120 3.12 On-shell representations and superfields 138 3.13 Off-shell field strengths and prepotentials 147 Classical, global, simple (N = 1) superfields 4.1 The scalar multiplet 149 4.2 Yang-Mills gauge theories 159 4.3 Gauge-invariant models 178 4.4 Superforms 181 4.5 Other gauge multiplets 198 4.6 N -extended multiplets 216 Classical N = supergravity 5.1 Review of gravity 232 5.2 Prepotentials 244 5.3 Covariant approach 267 5.4 Solution to Bianchi identities 292 5.5 Actions 299 5.6 From superspace to components 315 5.7 DeSitter supersymmetry 335 Quantum global superfields 6.1 Introduction to supergraphs 337 6.2 Gauge fixing and ghosts 340 6.3 Supergraph rules 348 6.4 Examples 364 6.5 The background field method 373 6.6 Regularization 393 6.7 Anomalies in Yang-Mills currents 401 Quantum N = supergravity 7.1 Introduction 408 7.2 Background-quantum splitting 410 7.3 Ghosts 420 7.4 Quantization 431 7.5 Supergravity supergraphs 438 7.6 Covariant Feynman rules 446 7.7 General properties of the effective action 452 7.8 Examples 460 7.9 Locally supersymmetric dimensional regularization 469 7.10 Anomalies 473 Breakdown 8.1 Introduction 496 8.2 Explicit breaking of global supersymmetry 500 8.3 Spontaneous breaking of global supersymmetry 506 8.4 Trace formulae from superspace 518 8.5 Nonlinear realizations 522 8.6 SuperHiggs mechanism 527 8.7 Supergravity and symmetry breaking 529 Index 542 INTRODUCTION There is a fifth dimension beyond that which is known to man It is a dimension as vast as space and as timeless as infinity It is the middle ground between light and shadow, between science and superstition; and it lies between the pit of man’s fears and the summit of his knowledge This is the dimension of imagination It is an area which we call, ‘‘the Twilight Zone.’’ Rod Serling 1001: A superspace odyssey Symmetry principles, both global and local, are a fundamental feature of modern particle physics At the classical and phenomenological level, global symmetries account for many of the (approximate) regularities we observe in nature, while local (gauge) symmetries ‘‘explain’’ and unify the interactions of the basic constituents of matter At the quantum level symmetries (via Ward identities) facilitate the study of the ultraviolet behavior of field theory models and their renormalization In particular, the construction of models with local (internal) Yang-Mills symmetry that are asymptotically free has increased enormously our understanding of the quantum behavior of matter at short distances If this understanding could be extended to the quantum behavior of gravitational interactions (quantum gravity) we would be close to a satisfactory description of micronature in terms of basic fermionic constituents forming multiplets of some unification group, and bosonic gauge particles responsible for their interactions Even more satisfactory would be the existence in nature of a symmetry which unifies the bosons and the fermions, the constituents and the forces, into a single entity Supersymmetry is the supreme symmetry: It unifies spacetime symmetries with internal symmetries, fermions with bosons, and (local supersymmetry) gravity with matter Under quite general assumptions it is the largest possible symmetry of the Smatrix At the quantum level, renormalizable globally supersymmetric models exhibit improved ultraviolet behavior: Because of cancellations between fermionic and bosonic contributions quadratic divergences are absent; some supersymmetric models, in particular maximally extended super-Yang-Mills theory, are the only known examples of fourdimensional field theories that are finite to all orders of perturbation theory Locally INTRODUCTION supersymmetric gravity (supergravity) may be the only way in which nature can reconcile Einstein gravity and quantum theory Although we not know at present if it is a finite theory, quantum supergravity does exhibit less divergent short distance behavior than ordinary quantum gravity Outside the realm of standard quantum field theory, it is believed that the only reasonable string theories (i.e., those with fermions and without quantum inconsistencies) are supersymmetric; these include models that may be finite (the maximally supersymmetric theories) At the present time there is no direct experimental evidence that supersymmetry is a fundamental symmetry of nature, but the current level of activity in the field indicates that many physicists share our belief that such evidence will eventually emerge On the theoretical side, the symmetry makes it possible to build models with (super)natural hierarchies On esthetic grounds, the idea of a superunified theory is very appealing Even if supersymmetry and supergravity are not the ultimate theory, their study has increased our understanding of classical and quantum field theory, and they may be an important step in the understanding of some yet unknown, correct theory of nature We mean by (Poincar´) supersymmetry an extension of ordinary spacetime syme metries obtained by adjoining N spinorial generators Q whose anticommutator yields a translation generator: {Q ,Q } = P This symmetry can be realized on ordinary fields (functions of spacetime) by transformations that mix bosons and fermions Such realizations suffice to study supersymmetry (one can write invariant actions, etc.) but are as cumbersome and inconvenient as doing vector calculus component by component A compact alternative to this ‘‘component field’’ approach is given by the superspace superfield approach Superspace is an extension of ordinary spacetime to include extra anticommuting coordinates in the form of N two-component Weyl spinors θ Superfields Ψ(x , θ) are functions defined over this space They can be expanded in a Taylor series with respect to the anticommuting coordinates θ; because the square of an anticommuting quantity vanishes, this series has only a finite number of terms The coefficients obtained in this way are the ordinary component fields mentioned above In superspace, supersymmetry is manifest: The supersymmetry algebra is represented by translations and rotations involving both the spacetime and the anticommuting coordinates The transformations of the component fields follow from the Taylor expansion of the translated and rotated superfields In particular, the transformations mixing bosons INTRODUCTION and fermions are constant translations of the θ coordinates, and related rotations of θ into the spacetime coordinate x A further advantage of superfields is that they automatically include, in addition to the dynamical degrees of freedom, certain unphysical fields: (1) auxiliary fields (fields with nonderivative kinetic terms), needed classically for the off-shell closure of the supersymmetry algebra, and (2) compensating fields (fields that consist entirely of gauge degrees of freedom), which are used to enlarge the usual gauge transformations to an entire multiplet of transformations forming a representation of supersymmetry; together with the auxiliary fields, they allow the algebra to be field independent The compensators are particularly important for quantization, since they permit the use of supersymmetric gauges, ghosts, Feynman graphs, and supersymmetric power-counting Unfortunately, our present knowledge of off-shell extended (N > 1) supersymmetry is so limited that for most extended theories these unphysical fields, and thus also the corresponding superfields, are unknown One could hope to find the unphysical components directly from superspace; the essential difficulty is that, in general, a superfield is a highly reducible representation of the supersymmetry algebra, and the problem becomes one of finding which representations permit the construction of consistent local actions Therefore, except when discussing the features which are common to general superspace, we restrict ourselves in this volume to a discussion of simple (N = 1) superfield supersymmetry We hope to treat extended superspace and other topics that need further development in a second (and hopefully last) volume We introduce superfields in chapter for the simpler world of three spacetime dimensions, where superfields are very similar to ordinary fields We skip the discussion of nonsuperspace topics (background fields, gravity, etc.) which are covered in following chapters, and concentrate on a pedagogical treatment of superspace We return to four dimensions in chapter 3, where we describe how supersymmetry is represented on superfields, and discuss all general properties of free superfields (and their relation to ordinary fields) In chapter we discuss simple (N = 1) superfields in classical global supersymmetry We include such topics as gauge-covariant derivatives, supersymmetric models, extended supersymmetry with unextended superfields, and superforms In chapter we extend the discussion to local supersymmetry (supergravity), relying heavily on the compensator approach We discuss prepotentials and covariant derivatives, the construction INTRODUCTION of actions, and show how to go from superspace to component results The quantum aspects of global theories is the topic of chapter 6, which includes a discussion of the background field formalism, supersymmetric regularization, anomalies, and many examples of supergraph calculations In chapter we make the corresponding analysis of quantum supergravity, including many of the novel features of the quantization procedure (various types of ghosts) Chapter describes supersymmetry breaking, explicit and spontaneous, including the superHiggs mechanism and the use of nonlinear realizations We have not discussed component supersymmetry and supergravity, realistic superGUT models with or without supergravity, and some of the geometrical aspects of classical supergravity For the first topic the reader may consult many of the excellent reviews and lecture notes The second is one of the current areas of active research It is our belief that superspace methods eventually will provide a framework for streamlining the phenomenology, once we have better control of our tools The third topic is attracting increased attention, but there are still many issues to be settled; there again, superspace methods should prove useful We assume the reader has a knowledge of standard quantum field theory (sufficient to Feynman graph calculations in QCD) We have tried to make this book as pedagogical and encyclopedic as possible, but have omitted some straightforward algebraic details which are left to the reader as (necessary!) exercises 534 BREAKDOWN f i + mG i = 6mκ−2 s − (Q + Q)AB d A d B = 3i κ−2 k A i G i + (Q + Q)AB d B = mG ij f j + i k A i d A + mG i (3s − m) + κ2Q AB ,i d A d B = (8.7.12) a.3 Wave equations We now find the linearized wave equations (8.3.36) that follow from (8.7.7) As in sec 8.3, we expand the fields in small fluctuations about their vacuum values For the remainder of this subsection, all quantities G and Q AB are evaluated at Φi = a i and ˜ i = Φ We find it useful to introduce shifted variables A ≡ A − G i Ai , ψ ≡ ψ − G i ψi , S ≡ S − G i F i (8.7.13) From (8.7.7a) we have S − A (s − m) − 2mA − 2mG i˜ i − (G ij f j + sG i )Ai = A • i ∂ α αψ • α − k A i G i λA α − 2mψ α − 2mG i ψ αi = (8.7.14a) (8.7.14b) From (8.7.7b), we find ˜ F i + m[(2A − A )G i + (3G i G j + G ij )A j + Ai ] = • ψ• i ∂ α α˜ αi + 2mG i ψ α (8.7.14c) + m(3G i G j + G ij )ψ α j + k A i λA α − i κ2Q AB ,i d A λB α = (8.7.14d) From (8.7.7c), we find A (Q + Q)AB D B + Q AB ,i d B Ai + Q AB ,i d B ˜ i + 3i κ−2 [k A i Ai − k A i˜ i + k A i G i (A + A )] = A (8.7.14e) 8.7 Supergravity and symmetry breaking • 535 • (Q + Q)AB i ∂ α α λ B α + Q AB ,i ( f i λB α − i d B ψ αi ) + 3κ−2 k A i (ψ αi + G i ψ α ) = (8.7.14f) We are left with the equations of the physical boson fields These simplify greatly if we use (8.7.14a,c,e); we find A =0 (8.7.14g) ˜ i + {3κ−2 (Q + Q)−1 A AB 3 (k A i − i κ2Q AC ,i d C )(k B j + i κ2Q BE ,j d E ) + ik A i ,j d A − m (G ik G kj + 3G i G k G kj + 3G ik G k G j − G ik jl G k G l ) A + m[(3s − m)δi j + 3(3s − 2m)G i G j ]}˜ j 3 + {3κ−2 (Q + Q)−1 AB (k A i − i κ2Q AC ,i d C )(k B j + i κ2Q BE ,j d E ) + κ2Q AB ,ij d A d B − m (3G ik G k G j + 3G i G jk G k + G ijk G k ) + m(3s − 2m)(G ij + 3G i G j )}Aj = (8.7.14h) • • (Q + Q)AB ∇αα f B αβ − 3κ−2 (k A i k B i + k B i k A i )AB β α • + 3κ−2 k A i G i X β α = (8.7.14i) • • • • • • where X αα = ∂ αα ImA − ImG j ∇αα Aj = ∂ αα ImA + (G j k B j − G j k B j )AB αα Here ∇αα is the Yang-Mills covariant derivative a.4 Bose masses We now discuss these results From (8.7.14g) we see that the complex scalar A is massless For the real part, this is no surprise: ReA is the trace of the graviton, which is massless because the cosmological term was assumed to vanish However, the imaginary part requires some care The pseudoscalar Im A ≡ ρ is not recognizable as one of the fields of the supergravity multiplet; it stands for −1 • ∂ · A where Aαα is the 536 BREAKDOWN divergence Im A = of the Im A − axial vector auxiliary field Thus the equation i Im G i A = should be replaced by • • Aαα − Im G i ∇αα Ai = (8.7.15) For many purposes, it makes little difference whether we use ImA or replace it with −1 • ∂ · A By dimensional analysis and Lorentz invariance, Aαα can enter the wave equation of the scalar fields ˜ i only through its divergence: A • ˜ i + cG i ∂ αα Aαα + = A • (8.7.16) Substituting in (8.7.15), we find ˜ (Ai + cG i ImG j Aj ) + = (8.7.17) • When we have A instead of Aαα , we get the same result, since instead of (8.7.16) we have ˜ i + cG i A ImA + = , (8.7.18) and using the A wave equation, we reobtain (8.7.17) However, if gauge invariance is broken the gauge field wave equation can get a spurious contribution from ImA that is not present when −1 ∂ · A is used instead Indeed, • • • substituting (8.7.15) into the first form of X αα (with ∂ αα ImA replaced by Aαα ) gives a zero contribution to the spin mass When gauge invariance is unbroken, we get no • contribution from the form with A as well: ∂ αα ImA does not affect the spin mass, and the vacuum expectation value of k B j is zero However, if gauge invariance is broken, • • the expectation value of k B j is not zero (equivalently, ∂ αα (G i Ai ) = G i ∇αα Ai ) and X gives a spurious contribution that must be removed by hand a.5 Fermi masses The component ψ corresponds to the γ-trace of the gravitino; we define the Goldstino as that combination of matter fields that couples to ψ (it makes no essential difference whether we use ψ or ψ , since we are always free to add terms to the gravitino) Thus we define η α ≡ G i ψ αi + 2m G i k A i λA α (8.7.19) 8.7 Supergravity and symmetry breaking 537 We also define ‘‘transverse’’ fields that are orthogonal to η: ψiT ≡ ψi − G i η i m λAT ≡ λA + (These satisfy G i ψiT + 2m dAη (8.7.20) G i k A i λAT = 0.) In terms of these, the spinor wave equations become: • i ∂ ααψ • α − 2m(ψ • • i ∂ α α η α + 2m(ψ + ηα) = (8.7.21a) + ηα) = (8.7.21b) α α • ψ• i ∇α α˜ αiT + m(G ij + G i G j )ψ α jT + (k A i − k A j G j G i − • i • i ∇α α λ B αT + [6κ−2 (Q + Q)−1 BA (k A j − κ2Q AB ,i d B )λA αT = i − [m(Q + Q)−1 BC Q CA ,i G i + (8.7.21c) κ2Q AC ,j d C ) − 2i d B ]ψ α jT i m d B k A l G l ]λA αT = (8.7.21d) Care must be taken to ensure that the mass operator on ψT , λT is restricted to the ‘‘transverse’’ subspace, i.e., preserves the orthogonality to η Observe that since the trace of the gravitino is a negative norm state, i.e., a ghost, its kinetic term has a minus sign relative to physical spinors (the same is true for the trace of the graviton; the whole φ multiplet has negative norm, as can be seen from the action (8.7.3)) Consequently, though the mass matrix in the ψ-η system (which is decoupled from the other spinors) does not vanish, both eigenvalues are zero (the mass matrix is not hermitian) Actually, we did not have to explicitly find the wave equation to arrive at this result: The condition that the Goldstino can be gauged away (that we can go to a U-gauge) implies that both the Goldstino η and the γ-trace of the gravitino ψ must have zero mass in the gauge that we use 538 BREAKDOWN a.6 Supertrace Having found the wave equations (8.7.14g-i,a8), and understood their significance, we can evaluate the supertrace The spin contribution is (recall that we are still in normal coordinates): 3 − 2[ik A i ,i d A − 3κ−2 (Q + Q)−1 AB (k A i − i κ2Q AC ,i d C )(k B i + i κ2Q BE ,i d E ) − m (G ij G ij + 3G i G j G ij + 3G ij G i G j − G ik ij G k G j ) + m(3s − m)N + 3(3s − 2m)(m − s)] (8.7.22a) where N ≡ δi i is the number of chiral multiplets The combined contribution of the spin and spin fields is: − 2[9κ−2 (Q + Q)−1 AB k A i k B i + i (Q + Q)−1 AB d C (k A i Q BC ,i − k A i Q BC ,i ) + ik A i ,i d A + G ij ik f k f j − (N + 1)m + (N − 1)3ms + tr ( 1 Qi Qj )f j f i] Q +Q Q +Q (8.7.22b) • The spin contribution, omitting the X αα term is given by the expression · 3κ−2 (Q + Q)−1AB (k A i k B i + k B i k A i ) and cancels the first term of (8.7.22b) (The normalization comes from the states of a spin particle and from the form (8.3.37) of the spin wave equation.) Finally, the spin contribution is just −4m Thus we get (using the gauge-invariance relations (8.7.9) to simplify some expressions) str M = − 2[ik A i ,i d A + G ij ik f k f j − (N − 1)(m − − i tr ( κ2 (Q + Q)AB d A d B ) 1 Q i )k A i d A + tr ( Qi Qj )f j f i] , Q +Q Q +Q Q +Q (8.7.23) in normal coordinates, or, in general, using coordinate invariance, we have str M = − 2[ik A i ;i d A + Ri j f j f i − (N − 1)(m − − i tr ( κ2 (Q + Q)AB d A d B ) 1 Q i )k A i d A + tr ( Qi Qj )f j f i] Q +Q Q +Q Q +Q (8.7.24) 8.7 Supergravity and symmetry breaking 539 We remind the reader that here 3/2 str M ≡ (−1)2J (2J + 1)M J (8.7.25) J =0 b Superfield computation of the supertrace If our only interest is the supertrace formula (8.7.71), we can obtain it with far less work using the technique developed in sec 8.4.b (Of course, in general we are interested in the mass matrices themselves, and not just the supertrace) We start with ln det(IK i j ) = − (N + 1)G + ln det(G i j ) + N ln(−φe −νtrV φ) (8.7.26) where differentiation is with respect to ˜ = φe −νtrV and not φ φ Before adding contributions from the gravitino mass and correcting for the axial vector auxiliary field (see below), the supertrace read from (8.4.9) is str M = − 2[ik A i ;i d A − (N + 1)ν tr d + R k l f l f k − (N + 1)(G i j f j f i + i G i k A i d A ) + tr (Q k 1 Ql )f l f k − i tr (Q i )k A i ] Q +Q Q +Q Q +Q (8.7.27) where we use (4.1.29,30): Γl = [ln det(G i j )]l R k l = [ln det(G i j )]k l , (8.7.28) The expression (8.7.27) has not made use of the vacuum conditions (8.7.8) or (8.7.12), and does not include either the spin contribution or the axial vector auxiliary field correction to the spin mass matrix discussed in subsec 8.7.a.4 As we saw in the previous section, the spin contribution must be included separately, since the γ-trace of the gravitino cannot contribute directly: the condition for the superHiggs mechanism to occur and for the gravitino to absorb the Goldstino in U-gauge requires the Goldstino-gravitino γ-trace system to be massless The spin correction, though somewhat subtle, can also be found without extensive computation As described in sec 8.7.a.4, we simply subtract κ−2 (Q + Q)−1AB k A i G i (k B j G j − k B j G j ) = − 2κ2 (Q + Q)AB d A d B (see 540 BREAKDOWN discussion following (8.7.22b) for an explanation of the factors) One further point deserves comment: When we rescaled φ to remove the potential g (see the beginning of sec 8.7.a), we lost sight of the contribution of the Fayet-Iliopoulos term When we make the shift G → G + ln(gge 3νtrV ), the ν tr d term in equation (8.7.27) is absorbed into the iG i k A i d A term as a consequence of R-invariance of g; it is most straightforward to work in the coordinate system where the Killing vectors take the form of usual gauge transformations: ν g tr (T A ) − g i (T A )i j a j = (8.7.29) and hence νtr (T A ) − [ln(gge 3νV )]i (T A )i j a j = (8.7.30) Using the vacuum equations, we can substitute into the supertrace (8.7.27) Including the gravitino and the spin correction term, we recover (8.7.24) c Examples We can use the supertrace formulae to study many cases of interest In particular, in extended supergravity theories we encounter ‘‘nonminimal’’ G and Q terms For example, in N = supergravity, which contains one physical chiral multiplet, three vec3 tor multiplets, three ( , 1) multiplets and the supergravity multiplet, G ∼ − ln(1 − ΦΦ) , Q∼ 1−Φ 1+Φ (8.7.31) We cannot treat the actual N = theory since a description of the interacting ( , 1) multiplet is not available, but (8.7.31) suggests looking at a system with one scalar multiplet and n vector multiplets V A , coupled to N = supergravity, with G as above and Q AB = 1−Φ δ + Φ AB (8.7.32) We find, with G = ∂2 G= (1 − aa)2 ∂Φ∂Φ (8.7.33) 8.7 Supergravity and symmetry breaking 541 the supertrace 3/2 (−1)2J (2J + 1)M J = − 2(n + 2)G f f = − 2(n + 2)m (8.7.34) J =0 Note that the Q and R terms in (8.7.17) combine because (Q + Q)2 = − 4(G )−1 [(1 + Φ)(1 + Φ)]−2 Unless a scalar potential g(Φ) is introduced, no supersymmetry breaking will occur However, it is possible to add such a term in N = supergravity, and there exist mechanisms to generate terms that act like a potential even in N = supergravity For N > the analogs of G and Q are expressed in terms of an overcomplete set of fields We may expect however that Q and G are related such that det(G i ) ∼ det(Q + Q)h(Φ )h(Φi ) where h(Φ ) is a holomorphic function In that case i j i we may also expect a simple result for the supertrace We can also construct models with a Fayet-Iliopoulos term and vanishing cosmological constant For example, consider G = ΦeV Φ + α2 ln[ΦeV Φ] + χχ + ln[(β + χ)(β + χ)] 3 (8.7.35) where Φ and χ are chiral fields, Φ transforming under the gauge transformation while χ is inert, and β is chosen so as to make the cosmological constant vanish (the potential and the Fayet-Iliopoulos term are included in G as the α-term) We find a solution to (8.7.8) with d = for some finite range of α (as can be verified by a perturbation expansion about α = 0) 542 INDEX INDEX Action, component scalar multiplet superconformal supergravity vector multiplet Actions, in gravity in supergravity Adler-Bardeen theorem Adler-Rosenberg method Algebra, superconformal super-deSitter super-Lie super-Poincar´ e supersymmetry Anholonomy coefficients Anomalies, in Yang-Mills currents local supersymmetry (super)conformal Anomaly cancellation Anomaly, chiral trace Antisymmetric tensor Auxiliary field Axial (n = 0) supergravity Axial-vector auxiliary field 15, 150, 331 15, 150, 302 245, 303, 312 255, 259, 309 23, 26, 162, 168, 306 238 299 407, 495 402, 478, 486 65 67 63 63 236, 249 401 489 474 494 407 473, 476, 479 186 16, 151, 162, 252, 326 257, 274, 288 246 Background field method Background-quantum splitting Background transformations Beta-function, vanishing of Bianchi identities Bianchi identities, solution of Bisection Breaking and auxiliary fields Breaking, radiative soft spontaneous BRST transformations 373 373, 377, 379, 382, 410 414 379, 412 369 22, 25, 29, 39, 140, 174, 181, 204, 292 25, 40, 176, 184, 294, 296 120, 123, 126 508 509 500 496, 506 342, 345 INDEX Casimir operator Catalyst ghost Central charge ‘‘Check’’ objects Chiral spinor superfield Chiral superfield Clifford vacuum Commutator algebra Commutator, graded Compensator, conformal density tensor Compensators Compensators, gravitino multiplet Components, auxiliary by expansion by projection covariant gauge of scalar multiplet of supergravity multiplet of vector multiplet physical Conformal invariance Conjugation, hermitian rest-frame Connection, central charge gauge isospin Lorentz Constraints, conformal breaking conformal supergravity conventional Poincar´ supergravity e representation-preserving solution of Contortion Converter Coset space, and σ-models and superspace 543 72, 87 426 64, 72 39, 251, 277 95, 123, 159, 188 89 69 320 56 240, 480 242,250,255,259,267,286 242, 274 112, 267 208 13, 108 10, 92 11, 94 24, 178 108 94 38, 245, 261, 322 160 108 65, 80, 240 57 123 86 18, 165, 170 86 36, 86, 235, 252 265, 274, 470 270 21, 35, 171, 237, 270, 276, 410, 470 274 172, 270, 278, 470 172, 276, 279, 470 41, 115, 273, 289, 298 163 117 74 544 INDEX Cosmological constant Cosmological term Covariant Feynman rules Covariant functional derivative Covariantization, of actions Covariantly chiral CP(n) models CPT Curvature 528 44, 312, 333 382, 446 384, 447 43, 300 172 113, 179 77 38, 236, 264 Degauged U (1) Degree of divergence Delta-function Density compensator Ψ Derivative, Dspinor superfunctional Derivatives, covariant DeSitter supersymmetry Determinant, vierbein Dilatation generator Divergences D-manipulation Doubling trick Duality, for the gravitino multiplet 289, 298 393 8, 97 250, 267, 269 9, 83 8, 56 101, 168 18, 24, 35, 165, 170, 235, 249, 269 67, 335 238 65, 81, 275 358, 452 48, 50, 360 386, 449 211 of minimal and n = − supergravity of nonminimal and chiral multiplets of tensor and chiral multiplets transformation 310 200 190 190, 204 Effective action Energy, positivity Energy-momentum tensor Euler number 47, 357, 373, 452 64, 497 473, 481 476 Faddeev-Popov ghost Fayet-Iliopoulos term Fermi-Feynman gauge Feynman rules Field equations Field strength 52, 340, 344, 381, 420, 432 178, 218, 308, 389, 514 342, 345 46,53,348,438 153,169,313 25,40,122,167 INDEX 545 Field strength, conformal gravitino multiplet supergravity Yang-Mills Field strengths, off-shell 124 206 244, 266 156, 167, 176 147 γ-trace Gauge, normal supersymmetric Gauge averaging Gauge fixing Gauge-restoring transformation Gauge transformations Gauge WZ model General coordinate transformations Ghost counting Goldstino Gravitino mass Gravitino multiplet 474, 481 156 37, 338, 415, 440 52, 341, 344 52, 341, 343, 428 115, 161, 164, 173 159 198 233 420 498, 509, 513, 522, 525, 527 333, 533 206 Hat objects Hidden ghost HyperKăhler manifold a Hypermultiplet 250, 282, 411 424, 432 158, 222 218 Improved tensor multiplet Index conventions Indices, flat isospin world Integral, Berezin superfunctional 191 7, 54, 542 35, 234, 252 55 35, 234, 252 8, 97 103 Jacobi identities 22 K gauge group Kăhler, manifold a potential Killing vectors 34, 170, 172, 270 155, 511 155, 511, 531 157, 514 Lagrange multiplier 203 546 INDEX Λ gauge group Legendre transform Lie derivative Light-cone, basis formalism Linear superfield Local scale transformations Locality in θ Lorentz generators Lorentz transformations, local orbital 159, 162, 173, 247, 279 191 232 55, 108 108, 142 91 240 48, 357 35, 76, 235, 249 35, 234 233 Mass, gauge invariant Mass matrices Measure, chiral general 26 532 301 300 Minimal (n = − ) supergravity 256, 287 Multiplet, gravitino N = scalar N = tensor N = vector N = Yang-Mills nonminimal scalar scalar tensor 3-form variant tensor variant vector vector 206 218 223 216 228, 369 199 15, 70, 149 186 193 203 201 18, 159, 185 Nielsen-Kallosh ghost Nonlinear realizations Nonlinear σ models 53, 376, 381, 434 117, 522 117, 154, 219 Nonminimal (n = 0, − ) supergravity 256, 287 No-renormalization theorem Normal coordinates 358 157, 533 O’Raiferteaigh model 507 Power-counting 358, 393, 454, 455 INDEX 547 Prepotential Prepotential, gravitino multiplet supergravity Yang-Mills Projection operators 147, 173 206 244 159, 173 120 Quantum transformations 378, 413, 431 Rarita-Schwinger field Recursion relations Reduction, product of D’s Regularization Regularization, by dimensional reduction inconsistencies in local dimensional Pauli-Villars point-splitting Representation, chiral irreducible off-shell on-shell superconformal super-deSitter super-Poincar´ e vector Ricci tensor R-transformations R-weight 246 547 85 393 394 397, 472 469 398, 404 399, 405 79, 165, 174, 284 120 13, 108, 143 13, 69, 138, 143 80 82 75 79 237 96, 153 96, 153, 169 Scalar potential Scale invariance Self-energy S-matrix Soft breaking terms Spurion S -supersymmetry Stueckelberg formalism Superanomaly Supercoordinate transformations Supercovariantization Supercurrent 153 240 49, 390, 443, 460 391, 463 502 500 66, 246 112 484 34 324 473, 480 548 INDEX Superdeterminant Superfield Superfield strength Superform Superhelicity SuperHiggs effect Superpotential Superscale transformations Supertrace Supertrace multiplet Supervector Symmetrization 99, 254 9, 75 140 28, 181 13, 73 498, 527 507 250, 271, 275 100, 513, 518, 538 473, 481, 486 34 7, 56 Tangent space Tangent-space basis Tensor calculus Time being, the Torsion Torsion, flat superspace Transformation superfield Transverse gauge 35, 86 183 326 250, 357, 384, 410, 433, 485 38, 236, 264 36, 87 96 440 U (1) covariant derivatives U-gauge 269 527 Variant representation Variation, covariant Vielbein determinant Vielbein, flat supergravity Vierbein Volkov-Akulov model 31, 201 168 42, 254, 255 28, 86 34 232, 246 522 Wess-Zumino gauge, supergravity Yang-Mills Wess-Zumino model Weyl tensor 38, 246, 261, 317 20, 161, 163 150 237 ...Library of Congress Cataloging in Publication Data Main entry under title: Superspace : one thousand and one lessons in supersymmetry (Frontiers in physics ; v 58) Includes index Supersymmetry Quantum... is represented on superfields, and discuss all general properties of free superfields (and their relation to ordinary fields) In chapter we discuss simple (N = 1) superfields in classical global supersymmetry. .. existence in nature of a symmetry which unifies the bosons and the fermions, the constituents and the forces, into a single entity Supersymmetry is the supreme symmetry: It unifies spacetime symmetries