- Email: [email protected]
Invariant regularization for N = 2 superfield perturbation theory
Volume 149B, number 1,2,3
PHYSICS LETTERS
13 Dcc.c,nbct 1984
INVARIANT REGULARIZATION FOR N = 2 SUPERFIELD PERTURBATION THEORY
V.K. KRIVOSHCHEKOV Steklov Mathematical Institute, Moscow, USSR Received 23 August 1984
An invariant intermediate regulaxization scheme is formulated for N = 2 superfield perturbation theory m the framework of the background field method. The regularization incorporates higher derivative regularlzation with a speci,d one-loop one in such a way that both the gauge symmetry and N = 2 SUSY axe manifestly preserved
The issue of invariant regularization is an important aspect in the treatment of extended SUSY theories in the framework of the N = 2 unconstraaned superfield perturbation theory. It was noted in ref. [1] that the possibility of incorporating higher derivative regularization with a special one-loop regularization f o r N = 2 gauge SUSY theories requires further investigation. It is just the aim of this paper to clarify the point and to formulate an explicxt regularization scheme for N -- 2 superfield perturbation theory that manifestly preserves b o t h N = 2 SUSY and the gauge symmetry. In fact this regularization is a generalization of the method proposed for gauge theory in refs. [2,3] to the N = 2 SYM case. For N = 1 SUSY gauge theories a similar approach has been described in ref. [4] (cf. ref. [5]). This scheme acquires a particularly simple form when combined with the background field method so it can be directly applied to the N = 2 superfield perturbation theory developed in ref. [1 ]. The present note can be viewed as a complementary one to the paper mentioned above so we mostly use the notations and conventions introduced in it. Since the N = 2 superfield perturbation theory developed m ref. [1] in the framework of the background field method is rather complicated we first consider its N = 1 counterpart. The total quantum background action for the N = 1 SYM theory can be written in the following form: I~_M + IyQ_M + i(Fl)_p + iF(2_)p +r(1) +.(2) ~N-K ~N-K , i(I) F - P , i(2) F - P and i(1) N - K , I(N2)K are actions for F a d d e e v 128
Popov and Nlelsen-Kallosh ghost fields of the first and second generations [1 ]. We recall that the background feld method for the N = 1 unconstramed superfield has the following features, only the physical Y - M fields and the first generaUon F - P ghosts couple to all orders, the remainder contribute only at the one-loop order by virtue of their couplings to the background Yang-Mills superfields. The first stage of regulanzaUon consists Jn background covariant higher derwative regulanzatlon. In order to regularize all but the one-loop supergraphs it is sufficient to add to the Y - M quantum action IyQ_M a term manifestly mvanant under both the background and the quantum gauge transformations:
1 fd4x d20 Tr[(~_DZc/)2Wc~)(c=D2 z)Zw~) - (® B2~ B2w~)(~B2~ B2w~)],
(1)
where -7~a = D~ + lg[A~ +AQ, - ] and cb~ = D e + lg[AB, -] are total gauge and background-gauge covanant derivatives, and Wa = Wa(Ag +A Q) and WB = Wa(AB) are Y - M superfield strengths. The gauge-fix--B 2 ing function/7(0) = c/) V Q remains unchanged, and 1) 2) so do the F - P ghost actions IV_p,/(l~-P. To regularize the gauge-fixing lagrangian for IyQ_M + AIQ_M we choose a suitable operator in such a way that after 't Hooft averaging it gives rise to the background covariant gauge-fixing term
Volume 149B, number 1,2,3
PHYSICS LETTERS
.~g.f. ~ Tr[(~B 2 vQ)((-/) B2 V Q)
P~;I~IP(AB, vQ) = - i In f d~b exp (Tr
+ A-4(¢3 B2(~)B2 (~B 2 vQ)(Q)B2@ B2EZ)B2 vQ)]. X The corresponding N-K ghost action/(N1)_K IS also modlfmd to acquire the folm Tr
fd8z {e-VB~eVlS~ + A -4 e- vB [D2(e VBD2(e- VS~e VB)e- VB)] X
eVB[D2(e-VBD2(eVB~e-VB)eVB)]},
13 December 1984
(2)
where ~, g are chiral anticommutmg N-K ghost fields, exp(V B) = exp(-lu B) exp(lfi B) and VB is the background prepotential needed to terminate the "ghost for ghost" sequence. Here we have used the relation U3 B2 [exp(i~B)~0 exp(--i~8)] = exp(lt~B)(D2~0) exp(--lt~B). One can replace ~ by D2X~I), where ×~1) is unconstrained and then proceed as m ref. [1 ]. So as before, the first generation N-K ghosts contribute only at the one-loop level. There is no need to alter other actions. However, we emphasize that (1) and (2) do regularize all but the one-loop IPI supergraphs, as follows from the explicit power counting given mref. [4]. On the other hand the regularized effective action Is manifestly mvariant under background gauge transformations since its effect can be compensated for by a redefimtion of the ghost superfield which has the form of a homogeneous ghost background gauge (pregauge) transformation [1 ]. In the background field approach the special oneloop regulanzatlon acquires an extremely simple form. Since this method is in fact a varmnt of the well-known Pauli-Vfllars loop regularization the key point is the possibility to insert into the quantum quadratic actions appropriate mass type regulators respecting background gauge invariance [4]. Technically it is achieved through replacing the one-loop contribution to the 1PI vertex function generating functional by a regularized one. The one-loop contribution arising from a partially regularized quantum Y-M action can be present in the form [41
f d%d%'
82(.~_M + A,Q~_ M + ./~g.f.) l~(z)~(z,)~
VQ(z) VQ(z') In det-l/2Xo(A B, vQ).
!
= -i i,l-loop Y-M is invanant under background gauge transfor. mations because one can simultaneously perform the replacement ~O-~ exp(iK) ~Oexp(-iK). In order to regularize lP~;l.°l~lp it is necessary to replace d e t - l / 2 x 0 by det- I/2Xol]) (det Xi)-eil2, where Jr) = X 0 + rn? and E! ei + 1 -- 0, ~i elm2 = 0. As follows from ref. [41, this procedure guarantees one-loop divergence cancellations in the effective action provided the other quadratic actions are replaced in a similar way. The original background gauge invariance is maintained throughout by virtue of the ghost superfield redef'mitions: X(11)-~ exp(iK) X~1) exp(-iK), X~1) ~ exp(iK) X~1) exp(-iK), X~1) "~ exp(iAB) X~1) exp(-iAB), etc. Thus we have succeeded in constructing the manifestly regularized effective action for N = 1 SYM in the framework of the background field method. We note that this result is not merely instructive in view of furtherN= 2 SUSY generalization but it can be used to investigate the structure of the renormalization procedure in the background feld method and possibly to resolve the problem of anomalies in N = 1 SYM. Before discussing the N = 2 invariant regularization scheme we recall some results obtained in ref. [1 ] for the N = 2 superfield perturbation theory in the frame. work of the background field method. The total N = 2 SYM action after quantum background splitting was presented in a form similar to that for the N = 1 SYM case. Namely, it comprised IyB_M+ I~_ M actions accompanied by the actions for two generations of F - P ghosts and two generations of N - K ghosts arising from the 't Hooft averaging for the background covariant gauge-fixing terms. Although the solution of the constraints on N = 2 Y-M field strengths was given in an iterative fashion in the Y-M coupling constant it is perfectly adequate for perturbation theory. We emphasize that the background gauge invariance of the effective action is maintained throughout, due to suitable ghost redefinitions. It is just the latter property that permits one-loop invariant regularization Ala Slavnov in the N = 2 SYM case. The higher covariant derivative regularization is 129
Volume 149B, number 1,2,3
PHYSICS LETTERS
is performed along the same lines as for the N = 1 SYM theory and brings no complications. So the one-loop effective action contribution (to any given order in the Y - M coupling constant) can be presented in the form (3) and the relevant replacement of det's can be safely performed. This procedure affects neither N = 2 SUSY, since it is formulated by usingN = 2 unconstrained superfields, nor background gauge invariance, since the suitable redefinitions are introduced for ghost superfields. The inclusion of a hypermultiplet action coupled to the Y - M gives rise only to new ghost actions and can be treated in the same way. So we conclude that putting aside algebraic details the invariant regularizatlon scheme for the N = 2 SYM theory remains essentially the same as for its N = 1 counterpart.
130
13 December 1984
The author is indebted to A.A. Slavnov for useful discussion and thanks K.S. Stelle for bringing the problem to his attention.
References [1] P.S. Howe, K.S. Stelle and P.K. Townsend, Miraculous ultraviolet cancellations m supersymmetry made manifest, preprint ICTP]82-83/20 (1983). [2] A.A. Slavnov, Nucl. Phys. B31 (1971) 301;Teor. Mat. Fiz. 13 (1972) 1064 [3] A.A. Slavnov, Teor. Mat Fiz. 33 (1977) 977 [4] V.K Knvoshchekov, Teor. Mat. Fiz 36 (1978) 745. [5] A.A Slavnov, in. Superspace and supergravaty, eds. S.W. Hawking and M. Ro~ek (Cambridge U.P., London, 1981) p. 177
PHYSICS LETTERS
13 Dcc.c,nbct 1984
INVARIANT REGULARIZATION FOR N = 2 SUPERFIELD PERTURBATION THEORY
V.K. KRIVOSHCHEKOV Steklov Mathematical Institute, Moscow, USSR Received 23 August 1984
An invariant intermediate regulaxization scheme is formulated for N = 2 superfield perturbation theory m the framework of the background field method. The regularization incorporates higher derivative regularlzation with a speci,d one-loop one in such a way that both the gauge symmetry and N = 2 SUSY axe manifestly preserved
The issue of invariant regularization is an important aspect in the treatment of extended SUSY theories in the framework of the N = 2 unconstraaned superfield perturbation theory. It was noted in ref. [1] that the possibility of incorporating higher derivative regularization with a special one-loop regularization f o r N = 2 gauge SUSY theories requires further investigation. It is just the aim of this paper to clarify the point and to formulate an explicxt regularization scheme for N -- 2 superfield perturbation theory that manifestly preserves b o t h N = 2 SUSY and the gauge symmetry. In fact this regularization is a generalization of the method proposed for gauge theory in refs. [2,3] to the N = 2 SYM case. For N = 1 SUSY gauge theories a similar approach has been described in ref. [4] (cf. ref. [5]). This scheme acquires a particularly simple form when combined with the background field method so it can be directly applied to the N = 2 superfield perturbation theory developed in ref. [1 ]. The present note can be viewed as a complementary one to the paper mentioned above so we mostly use the notations and conventions introduced in it. Since the N = 2 superfield perturbation theory developed m ref. [1] in the framework of the background field method is rather complicated we first consider its N = 1 counterpart. The total quantum background action for the N = 1 SYM theory can be written in the following form: I~_M + IyQ_M + i(Fl)_p + iF(2_)p +r(1) +.(2) ~N-K ~N-K , i(I) F - P , i(2) F - P and i(1) N - K , I(N2)K are actions for F a d d e e v 128
Popov and Nlelsen-Kallosh ghost fields of the first and second generations [1 ]. We recall that the background feld method for the N = 1 unconstramed superfield has the following features, only the physical Y - M fields and the first generaUon F - P ghosts couple to all orders, the remainder contribute only at the one-loop order by virtue of their couplings to the background Yang-Mills superfields. The first stage of regulanzaUon consists Jn background covariant higher derwative regulanzatlon. In order to regularize all but the one-loop supergraphs it is sufficient to add to the Y - M quantum action IyQ_M a term manifestly mvanant under both the background and the quantum gauge transformations:
1 fd4x d20 Tr[(~_DZc/)2Wc~)(c=D2 z)Zw~) - (® B2~ B2w~)(~B2~ B2w~)],
(1)
where -7~a = D~ + lg[A~ +AQ, - ] and cb~ = D e + lg[AB, -] are total gauge and background-gauge covanant derivatives, and Wa = Wa(Ag +A Q) and WB = Wa(AB) are Y - M superfield strengths. The gauge-fix--B 2 ing function/7(0) = c/) V Q remains unchanged, and 1) 2) so do the F - P ghost actions IV_p,/(l~-P. To regularize the gauge-fixing lagrangian for IyQ_M + AIQ_M we choose a suitable operator in such a way that after 't Hooft averaging it gives rise to the background covariant gauge-fixing term
Volume 149B, number 1,2,3
PHYSICS LETTERS
.~g.f. ~ Tr[(~B 2 vQ)((-/) B2 V Q)
P~;I~IP(AB, vQ) = - i In f d~b exp (Tr
+ A-4(¢3 B2(~)B2 (~B 2 vQ)(Q)B2@ B2EZ)B2 vQ)]. X The corresponding N-K ghost action/(N1)_K IS also modlfmd to acquire the folm Tr
fd8z {e-VB~eVlS~ + A -4 e- vB [D2(e VBD2(e- VS~e VB)e- VB)] X
eVB[D2(e-VBD2(eVB~e-VB)eVB)]},
13 December 1984
(2)
where ~, g are chiral anticommutmg N-K ghost fields, exp(V B) = exp(-lu B) exp(lfi B) and VB is the background prepotential needed to terminate the "ghost for ghost" sequence. Here we have used the relation U3 B2 [exp(i~B)~0 exp(--i~8)] = exp(lt~B)(D2~0) exp(--lt~B). One can replace ~ by D2X~I), where ×~1) is unconstrained and then proceed as m ref. [1 ]. So as before, the first generation N-K ghosts contribute only at the one-loop level. There is no need to alter other actions. However, we emphasize that (1) and (2) do regularize all but the one-loop IPI supergraphs, as follows from the explicit power counting given mref. [4]. On the other hand the regularized effective action Is manifestly mvariant under background gauge transformations since its effect can be compensated for by a redefimtion of the ghost superfield which has the form of a homogeneous ghost background gauge (pregauge) transformation [1 ]. In the background field approach the special oneloop regulanzatlon acquires an extremely simple form. Since this method is in fact a varmnt of the well-known Pauli-Vfllars loop regularization the key point is the possibility to insert into the quantum quadratic actions appropriate mass type regulators respecting background gauge invariance [4]. Technically it is achieved through replacing the one-loop contribution to the 1PI vertex function generating functional by a regularized one. The one-loop contribution arising from a partially regularized quantum Y-M action can be present in the form [41
f d%d%'
82(.~_M + A,Q~_ M + ./~g.f.) l~(z)~(z,)~
VQ(z) VQ(z') In det-l/2Xo(A B, vQ).
!
= -i i,l-loop Y-M is invanant under background gauge transfor. mations because one can simultaneously perform the replacement ~O-~ exp(iK) ~Oexp(-iK). In order to regularize lP~;l.°l~lp it is necessary to replace d e t - l / 2 x 0 by det- I/2Xol]) (det Xi)-eil2, where Jr) = X 0 + rn? and E! ei + 1 -- 0, ~i elm2 = 0. As follows from ref. [41, this procedure guarantees one-loop divergence cancellations in the effective action provided the other quadratic actions are replaced in a similar way. The original background gauge invariance is maintained throughout by virtue of the ghost superfield redef'mitions: X(11)-~ exp(iK) X~1) exp(-iK), X~1) ~ exp(iK) X~1) exp(-iK), X~1) "~ exp(iAB) X~1) exp(-iAB), etc. Thus we have succeeded in constructing the manifestly regularized effective action for N = 1 SYM in the framework of the background field method. We note that this result is not merely instructive in view of furtherN= 2 SUSY generalization but it can be used to investigate the structure of the renormalization procedure in the background feld method and possibly to resolve the problem of anomalies in N = 1 SYM. Before discussing the N = 2 invariant regularization scheme we recall some results obtained in ref. [1 ] for the N = 2 superfield perturbation theory in the frame. work of the background field method. The total N = 2 SYM action after quantum background splitting was presented in a form similar to that for the N = 1 SYM case. Namely, it comprised IyB_M+ I~_ M actions accompanied by the actions for two generations of F - P ghosts and two generations of N - K ghosts arising from the 't Hooft averaging for the background covariant gauge-fixing terms. Although the solution of the constraints on N = 2 Y-M field strengths was given in an iterative fashion in the Y-M coupling constant it is perfectly adequate for perturbation theory. We emphasize that the background gauge invariance of the effective action is maintained throughout, due to suitable ghost redefinitions. It is just the latter property that permits one-loop invariant regularization Ala Slavnov in the N = 2 SYM case. The higher covariant derivative regularization is 129
Volume 149B, number 1,2,3
PHYSICS LETTERS
is performed along the same lines as for the N = 1 SYM theory and brings no complications. So the one-loop effective action contribution (to any given order in the Y - M coupling constant) can be presented in the form (3) and the relevant replacement of det's can be safely performed. This procedure affects neither N = 2 SUSY, since it is formulated by usingN = 2 unconstrained superfields, nor background gauge invariance, since the suitable redefinitions are introduced for ghost superfields. The inclusion of a hypermultiplet action coupled to the Y - M gives rise only to new ghost actions and can be treated in the same way. So we conclude that putting aside algebraic details the invariant regularizatlon scheme for the N = 2 SYM theory remains essentially the same as for its N = 1 counterpart.
130
13 December 1984
The author is indebted to A.A. Slavnov for useful discussion and thanks K.S. Stelle for bringing the problem to his attention.
References [1] P.S. Howe, K.S. Stelle and P.K. Townsend, Miraculous ultraviolet cancellations m supersymmetry made manifest, preprint ICTP]82-83/20 (1983). [2] A.A. Slavnov, Nucl. Phys. B31 (1971) 301;Teor. Mat. Fiz. 13 (1972) 1064 [3] A.A. Slavnov, Teor. Mat Fiz. 33 (1977) 977 [4] V.K Knvoshchekov, Teor. Mat. Fiz 36 (1978) 745. [5] A.A Slavnov, in. Superspace and supergravaty, eds. S.W. Hawking and M. Ro~ek (Cambridge U.P., London, 1981) p. 177