We prove that three-dimensional N=1 supersymmetric Yang-Mills-Chern-Simons
theory is finite to all loops. This leaves open the possibility that different
regularization methods give different finite effective actions. We show that
for this model dimensional regularization and regularization by dimensional
reduction yield the same effective action.Comment: 6 pages, 1 figure, latex, espcrc2. Contribution to the Proceedings of
  the 30th Ahrenshoop Symposium on the Theory of Elementary Particles, edited
  by D. Lust, H.-J. Otto and G. Weigt, to appear in Nuclear Physics B,
  Proceedings Supplemen

't Hooft

Breitenlohner

Capper

Epstein

Ermushev

F. Ruiz Ruiz

Gates

Giavarini

Hepp

Kazakov

Lucchesi

P. van Nieuwenhuizen

Siegel

Speer

English

arXiv

Crossref

Supersymmetric Yang-Mills-Chern-Simons theory

© 1997 Elsevier Science B.V. All rights reserved. 
International Symposium Ahrenshoop on the Theory of Elementary Particles (30.1996.Buckow, Alemania)We prove that three-dimensional N=1 supersymmetric Yang Mills-Chern-Simons theory is finite to all loops. This leaves open the possibility that different regularization methods give different finite effective actions. We show that for this model dimensional regularization and regularization by dimensional reduction yield the same effective action.Depto. de Física TeóricaFac. de Ciencias FísicasTRUEpu

Ruíz Ruíz, Fernando

van Nieuwenhuizen, P.

Docta Complutense

ELSEVIER Nuclear Physics B (Proc. Suppl.) 56B (1997) 269-274 PROCEEBlNGS SUPPLEIVIENTS Supersymmetric Yang-Mills-Chern-Simons theory F. Ruiz Ruiza* and P. van Nieuwenhuizenbt aInstitut &r Theoretische Physik, Universitlt Heidelberg Philosophenweg 16, 69120 Heidelberg, Germany “Institute for Theoretical Physics, State University of New York at Stony Brook Stony Brook, NY 11794-3840, USA We prove that three-dimensional N= 1 supersymmetric Yang-Mills-Chern-Simons theory is finite to all loops. This leaves open the possibility that different regularization methods give different fmite effective actions. We show that for this model dimensional regularization and regularization by dimensional reduction yield the same effective action. 1. Introduction and conclusions 2. Superfields and components The supersymmetric regularization of gauge theories remains a major unsolved problem in su- persymmetry. In this contribution we consider supersymmetric Yang-Mills-Chern-Simons theory in three dimensions and prove that ordinary di- mensional regularization (or DReG) [l] and reg- ularization by dimensional reduction (or DReD) [2] preserve both supersymmetry and BRS invari- ance. We further show that they give the same Green functions. Our strategy is to first prove that the theory is finite to all loop orders, so that the regularized effective actions rDReG and rDReD are also renormalized effective actions and the difference AI’ = rDReG - rDReD is the differ- ence of two renormalized effective actions. Next we show that this difference vanishes. This, to- gether with the observations that DReG preserves at all stages the BRS identities of local gauge in- variance and that DReD preserves supersymme- try, implies the thesis. We will also see that nei- ther DReG nor DReD have problems in dealing with the &-‘“P that appears in the classical action of the model. The work reported here is based on ref. [3], to which we refer for more details. *Alexander von Humboldt Fellow. tResearch supported by NSF grant Phy 9309888. In three-dimensional N = 1 supersymmetry, su- perspace is parameterized by three real spacetime coordinates xp and two real anticommuting Ma- jorana spinor coordinates ea. Any vector zt@ can be represented as a symmetric rank-two vector z@ with indices a, ,8 = 1,2 through the relation $ = (+/J$“? where rp are the Dirac matrices. The spinor superderivative D, is defined by D, = 8, + i @&3,. Supersymmetry transformations are generated by the supercharge Qa = 0, - i Ofl&, and have the form 6xafl = a’@- 2i da@), &P = co, where a@ is a real commuting constant vec- tor, ea is an anticommuting constant Majorana spinor and E(??) = 3 (E’-V@ + ~fl0~). A super- field @(x, 8) transforms linearly under the action of the supercharge: 6* = @Q,S. The gauge field A& of a real, compact, semi- simple Lie algebra with completely antisymmetric structure constants fabc is part of a supermulti- plet described by a Majorana spinor gauge poten- tial l?z [4], The superfield rz defines a real vector gauge potential r;, and an imaginary spinor field strength Wz through the equations Besides A:,, the supermultiplet contains a real 0920-5632/97/%17.00 0 1997 Elsevier Science B.V. All rights reserved. PII: SO920-5632(97)00335-6 210 E Ruiz Ruiz, I! van NieuwenhuizenINuclear Physics B (Proc. Suppl.) 56B (1997) 269-274 scalar field HQ and two anticommuting Majorana spinors x”, and AZ, given by the projections onto P=O A& = I?$ X~=+v~~. We will work in the supersymmetric Landau gauge, characterized by the condition D‘T: = 0. In this gauge, the classical N = 1 Yang-Mills- Chern-Simons action has the form re = ; SYM+f'CS+SGF +&S , where S 1 - YM = -3~~2 s ffxd% waawz + $fabcraoyb@(D&) _ !/b~f~deraarbprz r;] SGF = f J d%d%t@'aDar;) and SEs=; d3xd% s ( f KFaa sI’; - KS SC’ > . Here m is a parameter with dimensions of mass, g is a dimensionless coupling constant, s is the BRS operator, Ca and Co are real anticommuting antighost and ghost superfields, and KiY and K$ are commuting external supersources coupled to the nonlinear BRS transforms srg and SC”. The BRS transformations that leave SYM, Scs and SGF invariant are given by s r; =i(v,c)" sB”=O s&=Ba sCa =-i fnbcCbCc , with Vgb = babD, +if acb I?: the spinor covariant superderivative, Ba a real Lagrange multiplier su- perfield imposing the gauge condition Dar: = 0, and s satisfying as usual s2 = 0. The components of Ba and Ca are defined by the projections b” = B” <; = iD,B” h” = -lD”Ba 2 those of Ca by replacing hatted antighosts with unhatted ghosts, and those of KiY and KZ, by ua = K” a ar ta=K$ I G” = -; DaKz, Kga = i D(, K;, r fyz = - ; DPD,K;, 7,” = i D,Kc p=-!.DSKa 2 c By construction, l?s is invariant under supersym- metry transformations. To formulate DReG, we work with component fields. The terms Scs, SYM, SG.F and SES in I’o are given in terms of components by 1 SYM = -$ + i fabc A;A;A;) - ; xaXa] SGF = s ( d3x s - r? i3,Vap + i GaAa - GoHa) , where Fiu is the field strength, Dib = cSab& + facbAE is the covariant derivative, and V; = A; + ; fabc xbypxc (1) Aa = A” + #9x” + ; f abc pbx” _ f fabc Hb$ - & fabcfcdyxb (xd-&) . (2) The action of s on components is obtained from the definition of the latter as projections and the F Ruiz Ruiz, I? van Nieuwenhuizen/Nuclear Physics B (Proc. Suppl.) 568 (1997) 269-274 271 action of s on superfields. It is given by r; : sy = ilpa - fdbc-&c sA; = (D,c)~ sH” = ‘J= + fabcHbcc _ ; fflbcj&,c sx” = _fnbqbcc Ba : sb” = 0 sp = 0 sh” = 0 *a c : $” = b” qp = i( sLja = ha CA : sca = - f pCbCc @ = fnbc@& SWa = _pcWbCc _ f pbc pbvc . The supersymmetry transformation laws for the components are obtained in the same way. Here we only present those for the components of the gauge multiplet: 6xa=Jfae--Ha~ 6A:, = Ey,XG + E (l&x)” bH” = -CA” 6Xa = - ; +‘7”F,:, E + fabcXb(XcE) . Power counting shows that there is only a fi- nite number of superficially divergent diagrams, thus proving that the theory is superrenormaliz- able. At one loop there are quadratic, linear and logarithmic divergences; at two loops there are linear and logarithmic divergences; and at three loops only logarithmic divergences survive. Fur- thermore, quadratically divergent one-loop dia- grams do not have internal gauge lines and the only primitively divergent two and three-loop 1PI diagrams are those in Table 1, where 0 denotes the superficial UV degree of divergence of the di- agram. ;I Table 1: Power counting for component fields The BRS identity for the full renormalized ef- fective action I takes the form +bdr ,+i<E+hg @ =o. where the sum is extended over @ = x” , Vi, HO, A”, CQ, cp”, wa. In what follows, we will write this equation as (I’, I’) = 0 and use the notation 0 for the Slavnov-Taylor operator: 0 = (I’e, ). An important property of 0 is that it commutes with the supersymmetry generator 6: [O, 61 = 0. We remark that I’ generates 1PI Green func- tions for the fields Vz and A* and not for the ele- mentary fields AZ and Aa. This is due to the fact that &s introduces external sources for the BRS variations of VE and A”, and not for those of A; and X”. To compute I, we use the Feynman rules for AZ and X” and treat Vi and A as composite fields defined by eqs. (1) and (2). It is not difficult to see that, given a 1PI diagram with superficial degree of divergence ti, all the diagrams that re- sult from replacing one or more of the external Af and/or Xa-lines with any of the composite fields on the right-hand side in eqs. (1) and (2) have superficial degree of divergence strictly less than 13. Regarding then Vi and An as composite fields does not worsen power counting. 3. Dimensional reduction and dimensional regularization In DReD, all the fields and matrices are kept three-dimensional and the momenta are contin- ued in the sense of ordinary DReG to d < 3. Be- cause the Dirac algebra is performed in three di- mensions, the Fierz identities remain valid and DReD manifestly preserves supersymmetry. The regularized action computed with DReD satisfies then bPDReD = 0. The BRS transformation for the gauge field in DReD, however, is not the same as in the unregularized theory. Indeed, whereas the first d < 3 components of the gauge field have the same BRS transformation law as the gauge field in the unregularized theory, the last 3 -d components transform as sAE = fnbcA;cc. Due to this fact, DReD does not manifestly preserve 212 E Ruiz Ruiz, I? van Nieuwenhuizen/Nuclear Physics B (Proc. SuppI,) 5.58 (1997) 269-274 BRS invariance. It may happen that at the end of all calculations, once the limit d + 3 has been taken, all effects due to the splitting of the gauge field into d and 3 -d components go away, but this is not what is meant by manifest BRS in- variance. Concerning the well known algebraic inconsistency [5] that occurs in products of three or more epsilons in DReD, we mention that it disappears in the limit d + 3, since contributions with three or more epsilons are finite by power counting at d = 3. To define DReG, we follow ref. [6] and treat &‘“P as purely three-dimensional [l]. This gives for the propagator of the gauge field in d 2 3 di- mensions A,&) = D&p) + &(p) , where (3) D,,(p) = _ gJrn m %Pv fl + i p’$b” - i P,PV p2 (p2 +m2 -io) Here gpv and $‘ are d-dimensional, objects with a tilde are three-dimensional and objects with a caret are (d-3)-dimensional. Since the propa- gator is the inverse of the kinetic term in the d- dimensional classical action and the BRS trans- formation for the gauge field is the same as in the regularized theory, DReG preserves BRS invariance [6] [7]. Hence, the DReG regular- ized effective action satisfies the BRS identity (rDReG , rDReG ) = 0. The complicated propaga- tor for the gauge field is the price for having a consistent treatment of E fi”P while manifestly pre- serving BRS invariance. As regards supersymme- try, it is well known that DReG does not mani- festly preserve it. 4. Perturbative finiteness To prove perturbative finiteness at one loop, we consider a one-loop 1PI diagram and denote by 2>(d) its value in DReG. According to eq. (3), if the diagram has an internal gauge line, ‘D(d) is the sum of two contributions: E)(d) = ao(d)+DR(d). The contribution Do(d) contains the SO(d) co- variant part D,, of all the gauge propagators. The contribution 2)~(d) contains at least one R,, and can be easily seen to be both UV and IR finite at d = 3 by power counting. Being finite at d = 3 and being at least linear in &, Z)R(d) vanishes as d + 3. We are thus left with only the SO(d) covariant ‘DD(~). If the diagram has no internal gauge line, D(d) is already SO(d) covariant. Us- ing that SO(d) covariant one-loop integrals have no poles when d is continued to a positive odd integer [S] completes the proof at one loop. This also proves that in the limit d + 3, 1PI Green functions at one loop are identical in DReG and in DReD. At two loops we proceed differently. Let us assume that the two-loop correction I’FReG to the effective action consists in the limit d + 3 of a divergent part I’g$$G and a finite part I$‘giG. Since rFReG satisfies the BRS identity @rFneG + (r?neG, r?neG) = 0 (4 and I’yReG is finite, the divergent part l?gfgG sat- isfies Ol?~$~G = 0. Because 1PI Feynman dia- grams with external sources as external lines are finite by power counting and there are no one- loop subdivergences, P~_$~G does not depend on the external sources and @I’~~~” = 0 reduces to sI’!$~~G = 0. Using the power counting in Table 1 and that contributions to two-loop 1PI diagrams from R,, are finite, we have that the most general form of r!$$G is l$?f;vG = &PC2 , where El, = m s [ dsx (~1 mxaxa + a2 X”#x” + a3 pAa + a4 Aa A" + a5 HaHa + a6 fnbcxapbf (5) + (117 fabcfcde(Xa7~xb) (n$x”)] and al,.. . , or are numerical coefficients. The equation SP2+ DReG = 0 is an equation in the co- efficients ai whose only solution is oi = 0. This completes the proof at two loops. The proof at three loops is analogous. Now F Ruiz Ruiz, P van Nieuwenhuizen/Nuclear Physics B (Proc. Suppl.) 56B (1997) 269-274 273 the only three-loop Lorentz invariant divergence is rFfjieuG = Ap- d_3 w3, with but Pa3 is not BRS invariant. At higher loops, finiteness follows from power counting and from absence of subdivergences. 5. A BRS invariant and supersymmetric effective action Since the theory is finite, every regularization method defines a renormalization scheme. We consider two renormalization schemes: scheme one uses DReG as regulator and performs no sub- tractions, scheme two uses DReD and performs no subtractions. We want to prove that the dif- ference Al? = rDReG - rDReD between the cor- responding renormalized effective actions is zero. We have seen in section 4 that this is the case at one loop. So let us consider the two-loop case. There is a general theorem in quantum field theory [9] that states that if two different renor- malization (not regularization) schemes yield the same Green functions up to k-l loops, then at k loops they give Green functions that can differ at most by a local finite polynomial in the external momenta of degree equal to the superficial overall UV degree of divergence Wk at k loops. This, and the power counting in Table 1, implies that the difference Al?, at two loops can at most be of the form rDReG 2 - rFReD = ps, , (6) with Pa, as in eq. (5). We observe that, since DReD preserves supersymmetry, IJyReD satisfies OReD = 0 . 2 (7) Acting with 6 on eq. (4), using eqs. (6) and (7), and recalling that [O,S] = 0 and that Al’, = 0, we obtain that @6Pa, = 0. Since Pap does not, de- pend on the external sources, 6P,, is independent of the external sources and WP,, = 0 reduces to s6P,, = 0, which is an equation in the coeffi- cients pi in PO,. Because c5P3, depends polyno- mially on the components of the gauge multiplet and their derivatives and has an overall factor of m, any nontrivial 6Pg, satisfying sSP,, = 0 must be m times a local BRS invariant of mass dimen- sion two. However, there are no such invariants. Hence, 6P,, = 0. The only supersymmetry in- variant that can be formed from Pa, is pzyy = (ym s [ d3x f x”&y” + x;“X” + A”A” - HnHa - $ fnbcfc%FYpxb) (X%&7] . At this point we have exhausted all the informa- tion given by BRS symmetry and supersymme- try. We determine the value of the coefficient (Y in Ply by means of an explicit calculation (see below) and find (Y = 0. At three loops, the difference is AI’s = aPa,. Since AI’s is not BRS invariant,, nor supersym- metric, the same arguments as used at the two- loop level are now powerful enough to conclude that cr = 0 without the need of any explicit com- putation. At higher loops, the difference AI’ van- ishes since at one, two and three loops it, vanishes and there are no overall divergences by power counting. We now compute CY in F$y. To do this, we evaluate the difference between the contri- butions from DReG and DReD to the selfen- ergy of the field Ha. The vertices with an H are H(x, H+v, H&c and H$xc. Using them, one can construct two-loop 1PI diagrams with the six topologies in Fig. 1. In fact,, since $ only propa- gates in cp and c into 2, and there is no four-point vertex containing the fields H, cp and ~3, no graphs with the topology of Fig. la can be constructed. The topologies in Figs. lb and lc, being prod- ucts of one-loop topologies, give the same contri- butions in DReG as in DReD, hence they do not, contribute to a. We are thus left with the topolo- gies in Figs. Id, le and If. Because one-loop sub- diagrams give the same contributions in DReG as in DReD, only the overall divergent part of the corresponding two-loop diagrams contribute to a. Since the two-loop diagrams are logarithmi- cally divergent, the contributions to cy come from setting in the numerators the external momen- tum fl and the mass m equal to zero, except, of course, for the overall factor m. The overall 214 E Ruiz Ruiz, P van Nieuwenhuizen/Nuclear Physics B (Pmt. Suppl.) S6B (1997) 269-274 -e- -03 -8- -&- -& -ii- (4 (4 6) Figure 1: Two-loop topologies for (HH)ipr divergent part of every diagram then reads s ddk ddq m N(k, q) -- @Id (2~)~ D(k,q,p,m) ’ It is very easy to see that the numerator N(k, q) always contains a trace over a fermion loop. This, and the observation that diagrams with internal gauge lines only occur in topology le and that their contributions separately cancel, implies that the overall divergence in DReG and DReD are the same except for the trace over the fermions. The trace of a sum of products of b and 8 can always be written as d-dimensional scalar prod- ucts k”, kq and q” times an overall trace of the unit matrix. After summing over diagrams, (Y can then be written as [.Y:=(trDRe~a-trDR~Dn) ’ s ddk ddq j(k2, kq,q”) f?@ (2n)d D(k,q,p,m) ’ where j(k2, kq,q”) is a polynomial of its argu- ments. Because the theory is finite, the integral is finite and therefore the difference due to the trace vanishes in the limit d + 3. Hence a! = 0. The equality of IDReG and FDReD is not ex- plained by local quantum field theory. One possi- ble explanation might be that there exists a third, as yet unknown, symmetry of the model. 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Supersymmetric Yang-Mills-Chern-Simons theory

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