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LTH 329 Equivalence of Dimensional Reduction and Dimensional Regularisation 4 9 9 1 n a J 8 I. Jack, D. R. T. Jones and K. L. Roberts 2 1 v DAMTP, University of Liverpool, Liverpool L69 3BX, U.K. 9 4 3 1 For some years there has been uncertainty over whether regularisation by dimensional 0 4 reduction (DRED) is viable for non-supersymmetric theories. We resolve this issue by 9 / h showing that DRED is entirely equivalent to standard dimensional regularisation(DREG), p - to all orders in perturbation theory and for a general renormalisable theory. The two p e regularisation schemes are related by an analytic redefinition of the couplings, under which h : the β-functions calculated using DRED transform into those computed in DREG. The S- v i X matrix calculated using DRED is numerically equal to the DREG version, ensuring that r both schemes give the same physics. a January 1994 1. Introduction Dimensional reduction (DRED) was introduced some time ago[1] as a means of reg- ulating supersymmetric gauge theories which maintains manifest supersymmetry whilst retaining the elegant features of dimensional regularisation (DREG). However, its use has often been attended by controversy. Firstly, there are potential ambiguities in DRED associated with the treatment of the Levi-Civita symbol, ǫµνρσ and, relatedly, with the correct definition of γ [2]. We shall not concern ourselves with these problems here. Secondly, 5 questions have been raised regarding the unitarity of DRED. It was argued by van Damme and ’t Hooft[3]that whilst DRED preserves unitarity in the supersymmetric case, unitarity is broken when DRED is applied to non-supersymmetric theories. However, it is our con- tention that when DRED is applied in the fashion envisaged in Ref. [4] (which differs from that employed by van Damme and ’t Hooft in its treatment of counterterms) then unitarity is preserved for any theory. In Ref. [5], we presented evidence for our claim by computing the two-loop β-functions for a toy model using both DRED and DREG, and showing that the DRED β-functions could be transformed into those for DREG by a coupling constant redefinition. We noted that a regularisation scheme mentioned in Ref. [3] (their “System 3”), which differs from the version of DRED which they use, led to the same β-functions as our version of DRED. However it was by no means clear a priori that this scheme was equivalent to the version of DRED which we advocate. In this paper we shall show that System 3 is indeed equivalent to our version of DRED, in perfect generality and to all orders. Since it will also be clear that System 3 is equivalent to DREG up to coupling constant redefinition, also to all orders and for a general theory, we shall have established the exact equivalence of DRED and DREG and hence we shall have shown that DRED preserves unitarity. Moreover, we shall also see that the S-matrix computed using DRED is equivalent to that computed using DREG. 2. Dimensional regularisation and minimal subtraction Let us start by describing the standard dimensional regularisation procedure for a general theory. Consider a general renormalisable gauge theory coupled to scalar fields (the extension to fermions is straightforward, but introduces notational complications) with bare Lagrangian L(λ ,g ,α ) = L (g ,α )+∂ φi ∂ φi +λijφi φj +λijkφi φj φk +λijklφi φj φk φl B B B G B B µ B µ B B B B B B B B B B B B B (2.1) 1 in 4 dimensions. In Eq. (2.1), φi represents the bare scalar fields, and L subsumes all B G the gauge-field dependent terms in the Lagrangian. The bare gauge coupling is denoted g (assuming a simple group so that there is only one gauge coupling), and α represents B B the bare gauge-fixing parameter. The S-matrix for the theory is constructed in standard fashion from the one-particle-irreducible (1PI) n-point functions. However, it will be suffi- cient for our purposes to focus our attention on the 1PI n-point functions for n 4, since ≤ for a renormalisable theory in 4 dimensions these are the only ones which contain potential primitive divergences. Again, since the theory is renormalisable, each potentially divergent 1PI n-point function with n 4 is associated with some n-point coupling. For definite- ≤ ness and notational convenience we shall restrict our discussion to the 1PI functions with external scalar lines only; however, the extension to external gauge fields is trivial. The 1PI 4-point function corresponding to a 4-point coupling λijkl will be denoted Γijkl, with B a similar notation for 3- and 2-point functions. The theory is regulated by continuation to d = 4 ǫ dimensions. We can write, for example, − Γijkl = λijkl +Dijkl(λ ,g ,α ) (2.2) B B B B where Dijkl(λ ,g ,α ) = Dijkl(λ ,g ,α ) (2.3) B B B L B B B LX=1 with Dijkl(λ ,g ,α ) representing the sum of L-loop diagrams contributing to Γijkl. (Of L B B B course Dijkl and Dijkl also depend on ǫ and on the external momenta, but we shall not L writethisdependence explicitly.) Intermsofλ theexpression inEq.(2.2)hasdivergences B represented by poles in ǫ. Within standard DREG, we write λ as a Laurent series B AI(λ) λI = µkIǫ λI + n , (2.4) B ǫn (cid:0) nX=1 (cid:1) (where λI represents any of λijkl, λijk, λij, g or α, and where k takes, for example, the I value 1 when I denotes ijkl, 1 when I denotes ijk and 0 when I denotes ij) so that the 2 Green function Gijkl, defined by Gijkl = Z ii′ Z jj′ Z kk′ Z ll′Γi′j′k′l′ , (2.5) φ φ φ φ p p p p where Z is the wave-function renormalisation for the fields φ, becomes finite as a function φ of the renormalised couplings λI. Any momentum-dependent terms in the 1PI 4-point and 2 3-point functions with purely scalar external lines must be finite by renormalisability, and we have suppressed them; however this is not so for the 2-point case, for which we require the Green function Gij = Z ii′ Z jj′Γi′j′, (2.6) φ φ p p to be finite, where Γij = p2 δij +Fij(λ ) + λij +Dij(λ ) , (2.7) B B B (cid:2) (cid:3) (cid:2) (cid:3) with p an external momentum. In fact, Eq. (2.2) yields a very succinct formulation of the process of constructing counterterm diagrams, which will be useful later on in comparing DRED with van Damme and ’t Hooft’s System 3. First of all, in view of Eq. (2.5) it is convenient to define “dressed” versions of quantities by λ¯ij = Z ii′ Z jj′λi′j′, (2.8) φ φ p p and so on. Since Gijkl as given by Eqs. (2.5) is finite as a function of the renormalised couplings λI, we have P.P.[λ¯ijkl] = λ¯ijkl λijkl = P.P. D¯ijkl (2.9) B B − − { } where “P.P.” stands for “Pole Part” and we assume that we are working with minimal subtraction. (Thepolepart istakenonly aftersubstituting theexpressions Eq.(2.4)forλI B interms of the renormalisedcoupling.) Next, forconvenience, we define λI AIn(λ), P ≡ n=1 ǫn so that λI = λI +λI . (Here we set µ = 1 for convenience; we know that λIPcontains no B P P explicit µ-dependence.) Expanding D (λ ) around λ, g and α, we obtain L B λ¯ijkl λijkl = B − P.P. Z ii′ Z jj′ Z kk′ Z ll′λI1 ...λIl d ... d Di′j′k′l′(λ) , − { φ φ φ φ P P dλ dλ L } LX=1Xl=1 p p p p I1 Il (2.10) where ∂ ∂ ∂ ∂ ∂ ∂ λI λijkl +λijk +λij +g +α . (2.11) ∂λI ≡ ∂λijkl ∂λijk ∂λij ∂g ∂α Diagrammatically,eachterminthesumontheright-handsideofEq.(2.10)represents aset of diagrams obtained by inserting a counterterm λI in turn at each vertex corresponding P to the coupling λI in each L-loop diagram, together with an insertion of Z on every φ external line. Hencethe truncationof Eq.(2.10)attheorder inλI corresponpding toa given 3 loop order L, corresponds precisely to the process of computing counterterms; the right- hand side of Eq. (2.10) automatically produces the set of L-loop diagrams together with the correct associated subtraction diagrams with the correct symmetry factors. These are fairly obvious remarks but we have been unable to find this formulation in the literature. In a similar fashion, we have from Eqs. (2.6) and (2.7) that Zij = δij P.P. F¯ij . (2.12) φ − { } Eqs.(2.10)(withsimilarexpressionsforthe3-and2-pointcouplingsinvolvingDijk andDij respectively) and (2.12)clearly determine λI asLaurent series interms oftherenormalised B couplings as in Eq. (2.4). They also determine Z as a Laurent series φ p zij(λ) Z ij = δij + n . (2.13) φ ǫn p nX=1 3. Dimensional reduction We now turn to discussing two modifications of DREG (which in fact will turn out to be equivalent); namely DRED, and also van Damme and ’t Hooft’s System 3. The crucial distinction between DRED and DREG is that in DRED the continuation from 4 to d = 4 ǫ dimensions is made by compactification. Therefore the number of field com- − ponents remains unchanged, though they only depend on d co-ordinates. Consequently ǫ of the components of a vector multiplet become scalars, termed “ǫ-scalars”, and which we shall denote φ˜ı. Moreover, terms in the Lagrangian involving ǫ-scalars cannot be expected to renormalise in the same way as terms involving the corresponding vector fields, since they are not related by d-dimensional Lorentz invariance. Hence we must introduce new couplings for the terms in the Lagrangian involving ǫ-scalars. We call these “evanescent couplings”. We shall term the original fields and couplings in the Lagrangian “real”. In a natural extension of the notation for purely real couplings in Eq. (2.4), we denote them as λ˜ı˜k˜˜l, λ˜ı˜kl, λ˜ıjk, etc and moreover write them generically as λI˜. It was demonstrated E E E E in Ref. [5] that it is vital to maintain the distinction between evanescent and real couplings in order to show the equivalence between the DRED and DREG β-functions. The difference between our implementation of DRED and that adopted by van Damme and ’t Hooft in Ref. [3] (their “System 4”) resides in the counterterms assigned to evanescent couplings. The renormalisation of evanescent couplings involves computing graphs with 4 external ǫ-scalars. Our philosophy is that we should calculate the counterterms required to make graphs with external ǫ-scalars finite and use those counterterms to construct subtraction diagrams. This is equivalent to the usual diagram-by-diagram subtraction procedure where we subtract from each diagram subtraction diagrams obtained by replacing divergent subdiagrams by counterterm insertions with the same pole structure. System 4 of van Damme and ’t Hooft, on the other hand, involves replacing the genuine counterterms for evanescent couplings by the counterterms for the corresponding real couplings, and this is their interpretation of DRED. Henceforth, when we use the term DRED we will refer to the former version of the scheme. (We should stress that the two versions of DRED are equivalent for supersymmetric theories.[6][3]) In either case, the calculations are done using minimal subtraction, so that the counterterms contain only poles in ǫ. We must take into account factors of ǫ from the multiplicity of the ǫ scalars so that, for instance, a one-loop subdiagram with a divergent momentum integral giving an ǫ−1 pole but with a factor of ǫ due to the presence of ǫ-scalars in the loop would be finite and would not require a subtraction. Now let us turn to System 3. We consider first the theory obtained by performing the reduction to 4 E dimensions; this will have a Lagrangian L = L(λ )+L (λ ,(λ ) ) B E B E B − where L represents the extra terms involving the additional scalars (which we shall call E ǫ-scalars, although at present their multiplicity is E; shortly we shall set E = ǫ). These terms also involve bare evanescent couplings denoted by (λ ) . We now write the bare E B couplings in terms of the renormalised quantities by continuing to d = 4 ǫ dimensions − and computing the counterterms required to make the theory finite as a function of the renormalised couplings. We obtain AI (λ,λ )Em λI = µkIǫ λI + nm E , B ǫn (cid:0) nX=1mX=0 (cid:1) (3.1) A˜I˜ (λ,λ )Em (λI˜) = µlI˜ǫ λI˜ + nm E , E B E ǫn (cid:0) nX=1mX=0 (cid:1) where we assume that the counterterms are defined by minimal subtraction, so that the AI and A˜I˜ are independent of ǫ. k is as defined following Eq. (2.4), and l in similar nm nm I I˜ fashion. We also have expressions analogous to Eq. (2.13) for the wave-function renormalisation matrix for the real scalar fields and for the ǫ-scalars: zij (λ,λ )Em Z ij = δij + nm E , φ ǫn p nX=1mX=0 (3.2) z˜˜ı˜ (λ,λ )Em Z ˜ı˜= δ˜ı˜+ nm E . ǫ ǫn p nX=1mX=0 5 (There is no wave-function renormalisation matrix mixing real scalars and ǫ-scalars.) We now set E = ǫ, so that we have AI (λ,λ )ǫm λI = µkIǫ λI + nm E , B ǫn (cid:0) nX=1mX=0 (cid:1) A˜I˜ (λ,λ )ǫm (λI˜) = µlI˜ǫ λI˜ + nm E , E B E ǫn (cid:0) nX=1mX=0 (cid:1) (3.3) zij (λ,λ )ǫm Z ij = δij + nm E , φ ǫn p nX=1mX=0 z˜˜ı˜ (λ,λ )ǫm Z ˜ı˜= δ˜ı˜+ nm E . ǫ ǫn p nX=1mX=0 Next we define new couplings λ′I, λ′I˜ by writing E BI(λ′,λ′ ) λI = µkIǫ λ′I + n E B ǫn (cid:0) nX=1 (cid:1) (3.4) B˜I˜(λ′,λ′ ) (λI˜) = µlI˜ǫ λ′I˜+ n E E B E ǫn (cid:0) nX=1 (cid:1) and requiring that there is no explicit ǫ-dependence in BI and B˜I˜. The couplings λ′I, λ′I˜ n n E ′ ′ are those of van Damme and ’t Hooft’s scheme System 3. The relation between λ , λ and E λ, λ rapidly becomes very complex as one looks beyond one loop; we shall shortly give E an explicit expression up to two loops. It is important to note however that the relation between λ, λ and λ′, λ′ is analytic. Next we define Z′ij(λ′,λ′ ,ǫ) and Z′′kj(λ′,λ′ ,ǫ) by E E φ E φ E Z′′ik(λ′,λ′ ,ǫ) Z′kj(λ′,λ′ ,ǫ) = Z ij(λ,λ ,ǫ) (3.5) φ E φ E φ E q q p where we require z′ij(λ′,λ′ ) Z ′ij(λ′,λ′ ,ǫ) = δij + n E (3.6) φ E ǫn p X with z′ij independent of ǫ, and where Z′′ is finite as ǫ 0. These requirements specify Z′ φ → φ ′′ and Z uniquely. Similarly, we also define φ z˜′˜ı˜(λ′,λ′ ) √Z′˜ı˜(λ′,λ′ ,ǫ) = δ˜ı˜+ n E , (3.7) ǫ E ǫn X together with Z′′˜ı˜(λ′,λ′ ,ǫ), by Z′′ Z′ = √Z , with Z′′ analytic in ǫ and z˜′˜ı˜indepen- ǫ E ǫ ǫ ǫ ǫ n p p dent of ǫ. We will later show that the Laurent expansions of the bare couplings in Eq. (3.4) 6 ′ ′ are exactly those obtained using DRED, and that correspondingly Z and Z are the real φ ǫ scalar and ǫ-scalar wave-function renormalisations obtained using DRED. ′ ′ The β-functions for λ, λ , λ , and λ are defined by E E d d βI(λ,λ ,ǫ) = µ λI, βI˜(λ,λ ,ǫ) = µ λI˜ E dµ E E dµ E (3.8) d d β′I(λ′,λ′ ,ǫ) = µ λ′I, β′I˜(λ′,λ′ ,ǫ) = µ λ′I˜ E dµ E E dµ E and are given explicitly by βI(λ,λ ,ǫ) = k ǫλI + AI ǫm, βI˜(λ,λ ,ǫ) = l ǫλI˜ + A˜I˜ ǫm, E − I DI 1m E E − I˜ E DI˜ 1m mX=0 mX=0 β′I(λ′,λ′ ,ǫ) = k ǫλ′I + ′BI, β′I˜(λ′,λ′ ,ǫ) = l ǫλ′I˜+ ′B˜I˜ E − I DI 1 E E − I˜ E DI˜ 1 (3.9) where ∂ ∂ = k λJ. + l λJ˜. k . (3.10) DI J ∂λJ J˜ E ∂λJ˜ − I XJ XJ˜ E ′ ′ , and are defined in similar fashion. (In fact the effect of the operators DI˜ DI DI˜ D is to multiply an L-loop contribution to A , A˜ , B or B˜ by L.) By definition, the 1m 1m 1 1 ′ ′ β-functions β and β for λ and λ are related by coupling constant redefinition, and we have from Eq. (3.8) ∂ ∂ β′I(λ′,λ′ ,ǫ) = β. +β . λ′I(λ,λ ,ǫ), (3.11) E ∂λ E ∂λ E E (cid:0) (cid:1) or equivalently ∂ ∂ βI(λ,λ ,ǫ) = β′. +β′ . λI(λ′,λ′ ,ǫ). (3.12) E ∂λ′ E ∂λ′ E (cid:0) E(cid:1) Now the standard DREG β-function for λ, in the absence of the ǫ-scalars, is clearly given by βI (λ) = limβI(λ,λ ,ǫ) = AI . (3.13) DREG ǫ→0 E DI 10 So if we take the limit as ǫ 0 of Eq. (3.12), we obtain, using Eqs. (3.13) that: → ∂ ∂ βI (λ) = β′(λ′,λ′ ,0). +β′ (λ′,λ′ ,0). λI(λ′,λ′ ,0). (3.14) DREG E ∂λ′ E E ∂λ′ E (cid:0) E(cid:1) Our next step is to show that our prescription for DRED is equivalent to System 3. In order to do this, we need to consider the way in which the bare couplings are generated 7 diagrammatically, since our version of DRED depends crucially on the way in which counterterms are constructed for divergent subdiagrams. For the theory reduced from 4 to 4 E dimensions, the 1PI function corresponding to a real (but not 2-point) coupling λI − is given by an expression analogous to Eq. (2.2), ΓI(λ ,(λ ) ) = λI + EMDI (λ ,(λ ) ) (3.15) B E B B M B E B MX=0 where DI represents the sum of all graphs which produce a multiplicity factor of EM. M The 1PI function corresponding to an evanescent (but not 2-point) coupling λI˜ is similarly E given by ΓI˜(λ ,(λ ) ) = (λI˜) + EMD˜I˜ (λ ,(λ ) ) (3.16) B E B E B M B E B MX=0 The 1PI 2-point function for real fields is given by an expression analogous to Eq. (2.7), Γij = p2[δij + EMFij(λ ,(λ ) )]+[λij + EMDij (λ ,(λ ) )], (3.17) M B E B B M B E B MX=0 MX=0 and there is a similar expression for the 1PI 2-point function for ǫ-scalars. The definitions of λI , (λI˜) in terms of λI , λI˜ in Eq. (3.1), together with the expressions for Zij and B E B E φ Z˜ı˜ in Eq. (3.3), precisely ensure that the Green functions given by the “dressed” versions ǫ of the 1PI functions, GI Γ¯I and GI˜ Γ¯I˜, are finite as a function of λI, λI˜. (Dressed ≡ ≡ E versions of Green’s functions with external ǫ-scalars are defined in a similar way to Eq. ˜˜ı ji (2.8), but with factors of √Z instead of Z where appropriate.) To take the example ǫ φ ′ ′ of thereal 2-point coupling, Z ii Z jjpΓi′j′(λ ,(λ ) ) isguaranteed tobe finite. This φ φ B E B p p implies the separate finiteness of Z ii′(λ,λ ,ǫ) Z jj′(λ,λ ,ǫ) δi′j′ φ E φ E p+ EMFi′j′(pλI + AIn(cid:2)m(λ,λE)Em,λI˜ + A˜In˜m(λ,λE)Em) (3.18) M ǫn E ǫn MX=0 nX=1mX=0 nX=1mX=0 (cid:3) and Z ii′(λ,λ ,ǫ) Z jj′(λ,λ ,ǫ) λi′j′ φ E φ E B p+ EMDi′j′pλI + AIn(cid:2)m(λ,λE)Em,λI˜ + A˜In˜m(λ,λE)Em) . (3.19) M ǫn E ǫn MX=0 (cid:0) nX=1mX=0 nX=1mX=0 (cid:3) A fortiori, when we set E = ǫ in all the Green functions, we still have finite expressions. We then rewrite λI = λI + AInm(λ,λE)ǫm, (λI˜) = B n=1 m=0 ǫn E B P P 8 λI˜ + A˜In˜m(λ,λE)Em, in terms of λ′I, λ′I˜ according to Eq. (3.4), and at E n=1 m=0 ǫn E P P ′′ ′ ′ ′ ′ ′ the same time we rewrite Z (λ,λ ,ǫ) and Z (λ,λ ,ǫ) as Z (λ ,λ ,ǫ)Z (λ ,λ ,ǫ) and φ E ǫ E φ E φ E ′′ ′ ′ ′ ′ ′ Z (λ ,λ ,ǫ)Z (λ ,λ ,ǫ) according to Eqs. (3.5)–(3.7), obtaining expressions which are ǫ E ǫ E ′ ′ finite as functions of λ , λ . For instance, in the 2-point example above, we now have that E Z′′ii′(λ′,λ′ ,ǫ) Z′i′i′′(λ′,λ′ ,ǫ) Z′′jj′(λ′,λ′ ,ǫ) Z′j′j′′(λ′,λ′ ,ǫ) φ E φ E φ E φ E × q q q q δi′′j′′ + ǫMFi′′j′′ λ′J + BnJ(λ′,λ′E),λ′J˜+ BñJ˜(λ′,λ′E) (3.20) M ǫn E ǫn (cid:2) MX=0 (cid:0) X X (cid:1)(cid:3) and Z′′ii′(λ′,λ′ ,ǫ) Z′i′i′′(λ′,λ′ ,ǫ) Z′′jj′(λ′,λ′ ,ǫ) Z′j′j′′(λ′,λ′ ,ǫ) φ E φ E φ E φ E × q q q q λ′i′′j′′ + ǫMDi′′j′′ λ′J + BnJ(λ′,λ′E),λ′J˜+ BñJ˜(λ′,λ′E) (3.21) M ǫn E ǫn (cid:2) MX=0 (cid:0) X X (cid:1)(cid:3) are finite as functions of λ′, λ′ . We can then remove the factors of Z′′ and Z′′ in E φ ǫ q p ′′ Eqs. (3.20), (3.21) and the other Green functions, still leaving finite expressions, since Z φ ′′ and Z are analytic in ǫ. These resulting finite quantities can be regarded as deriving from ǫ ′ ′ ′ ′ ′ ′ 1PI functions “dressed” by Z (λ ,λ ,ǫ) and Z (λ ,λ ,ǫ). Hence we now have that φ E ǫ E BÎ(λ′,λ′ ) BJ(λ′,λ′ ) B˜J˜(λ′,λ′ ) λˆ′I + n E + ǫMDÎ (λ′J + n E ,λ′J˜+ n E ) ǫn M ǫn E ǫn X MX=0 X X λˆ′I˜+ Bˆ˜In˜(λ′,λ′E) + ǫMDˆ˜I˜ (λ′J + BnJ(λ′,λ′E),λ′J˜+ BñJ˜(λ′,λ′E)), E ǫn M ǫn E ǫn X MX=0 X X (3.22) Z′ij(λ′,λ′ )+ ǫMFîj(λ′J + BnJ(λ′,λ′E),λ′J˜+ BñJ˜(λ′,λ′E)), φ E M ǫn E ǫn MX=0 X X Z′˜ı˜(λ′,λ′ )+ ǫMFˆ˜˜ı˜(λ′J + BnJ(λ′,λ′E),λ′J˜+ BñJ˜(λ′,λ′E)) ǫ E M ǫn E ǫn MX=0 X X ik jl (where for example λîj λkl Z′ Z′ ) are finite as a function of λ′I, λ′I˜. The ≡ φ φ E q q “hatted” quantities will later turn out to be identical to quantities evaluated using DRED 9