Dimensional regularization of supergraphs

Volume 122B, number 3,4

PHYSICS LETTERS

10 March 1983

DIMENSIONAL REGULARIZATION OF SUPERGRAPHS L.V. AVDEEV and A.Yu. KAMENSHCHIK

Jomt Institute for NuclearResearch, Dubna, USSR Received 22 October 1982

The applicability region of supersymmetnc dimensional regularlzatmn Is determined in the Wess-Zummo model To achmve it, a consistent version of the regularlzatlon for supergraphs :s formulated.

The whole set of relations which defines (supersymmetric dimensional) regularization by dimensional reduction (RDR) [1] is known to be inconsistent [2]. In the component-field fonnahsm one succeeded in f'mdmg the " r e d u n d a n t " relations (Flerz 1dentines), giving them up makes the regularizatlon consistent [3], still allowing one to compute diagrams. We carry out the same programme for a superfield formulation o f the W e s s - Z u m i n o m o d e l where m the componentfield approach the RDR is equivalent [4] to conventional &menslonal regularization. Up to a def'mlte order the superfield RDR calculations using the contradictory set o f relations give the same results, as our consistent formulation does. Hence, up to that order they are reliable. In terms o f chlral superfields ~ ( x , 0) the action for the Wess Zumino model has the form

S=fdxlffd20620~(x,O)exp(-210~O)cb(x,O) +m tafd2OO?2(x'O)+ f d 2 ~ 2 ( x ' ~ ) )

+ 3-S

d20 ~3(x, O)

The notation for the two-component ant:commuting spinors is

1 Moscow State Umverslty, Moscow, USSR

0 031-9163/83/0000 0000/$ 03 00 © 1983 North-Holland

O~-'-OAeAB.~B, 02=OAeABO B, O~ = 0A e A ~ h,

b = o.p.,

Ocr~z~ = 0A eAB(O~)BAeA~ ~ , ('d.)[?A = eAB(% )BA ~Ah ,

etc.

The basic relations have the following form

eABeCD = 5ACSBD ~AD6BC ,

(2)

1 f d2OOAl..OAn = ~neeAlA2 ,

(3)

ou3 v + ov~u = 2guy 1, ~uov + ~6vou = 2guu 1,

(4)

trl =6AA=2,

(5)

eAB=--eBA , eABeBC=--fAC, 1

2

(6)

OAOB = :eABO ,

(7)

OAOBOC = 0 ,

(8)

and the same for eA/~ and 0A" Notme that (8) follows from ( 5 ) - ( 7 ) , which in turn can be derived from (2). Below we shall see that (2) has to be given up, and then ( 3 ) - ( 7 ) will be the axioms in place o f ( 2 ) - ( 4 ) . After the reduction to a nonmtegral dimension we get a projection operatorg~v with the property

guu = d = 4 - 2 e .

(9)

Since there are no vector fields in the model, we shall only need the d-d:menslonal o-matrices (they appear together with momenta m the form/5) obeying (41, just as in the conventional dimensional regulanzation. Eqs. (4), (9) and the cycliclty o f the traces lead to the 247

Volume 122B, number 3,4

PHYSICS LETTERS

following formulas for T-~(/al .../~2rn ) = t r (Ol2101a2 "" OU2m -1 -dU2m ) +- tr (°~1 °~s2 ""~dU2m-1 °tS2m )" 2m T+(/21 -'//2m) = ~ ( - ) n g u l u n T+(/22 "AZ/n• U2m), n=2 (10) T-(/R1 "#2m) = 0 .

(Ii)

Formula (11) is specified for d dimensions. It is analogous to the nullification o f any tr (')'5")'~ "-3'u2 m ) with anticommuting ~'5, (75,3'u}+ = 0, in the nonintegral-dimensional space. As we shall see below, noncorrespondence o f (11) to the four-dimensional hmit is not seen in the super-mvariance region. The set of equations ( 2 ) - ( 1 1 ) is contradictory. Multiply (2) by M A C N B D , where M = aul ..01.,2 m , N = OVl..3V2n Taking into account a consequence of the notation and (6), eAB ecDNBD = (°V2n" ~dvl )CA =- N R A " we obtain a relation, which would be an identity for the o-matrices of the four-dimensional space" A --- tr [ M ( N + N R ) ] - tr (M) tr (N) = 0 . But the calculation of the traces in the left-hand side with the use o f (10), (11) and (5), when M and N each contain four o-matrices, gives a nonzero answer" gl.llVl

It Implies that shifts of the 0-integration variables are not generally permissible, and therefore, supersymmetry o f the regularized action ( 1 } can be broken. The use o f ( 3 ) (11) only and the 0-shift ban m the supergraph techniques of ref. [5] lead to a consitent regularlzation scheme. It can be shown to be equivalent to component-field calculations [6] with anticommutmg 75- However, we cannot introduce differentiation with respect to simple 0-variables, since it would allow one to derive (2) from (7), the above contradiction following immediately. This fact does not permit us to construct a consistent version of the supergraph method with covariant derivatives [7] to treat vector superfields. Now we are to study the scope of theosupersymmetric dimensional regularization. Compare the expressions for supergraphs before momentum integration in the consistent RDR version with those obtained in four dimensions and thus possessing supersymmetry. We only consider the eo~, dp3(~3) and 4~2~ 2 dlagrmns because other graphs are convergent due to power-counting rules with no reference to supersymmetry. The only formula of those used which fails in four dimensions is (11 ). It is applied to compute the traces arising from closed cycles of lines in the diagrams. The corresponding four-dimensional formulas, for instance,

" glalV 4

T - ( # I "/a4) = 4Ieul u4 ' A=4

~- 4 det (/a 1 /24, v 1 .v4)v~0. gu4vl

"

gu4u4

To avoid this discrepancy, we have to exclude eq. (2), which allows only two values for spinor Indices. However, "simple" 0-variables entering (1) should be left two-component in the sense of (7), (8), It ensures the existence of an explicit 6-function for the 0-integraUon (3), That function is necessary for a correct construction of the propagator for the action (1) Therefore, we postulate ( 3 ) - ( 7 ) and (9), to make possible the use of (3)-(11). In the quasi-two-dimensional space [3], where (5) is true but (2) is not, a sum of two simple 0-variables is not a simple 0-variable (0 + ~)3 4= 0,because (3) and (6) give 4 fd2,

~oA eCEeDFf d

20 (0 + ~o)e(0 + ~0)e(0 + ~P)F

= eABeCD -- 6AC6BD + 6AD6BS ~ 0 . 248

10 March 1983

(12)

Involve the totally antisymmetric tensor. A nonzero contribution of that type after the integration over the internal momentum of the cycle can only be obtained if the latter has at least four independent external momenta The minimal even cycle with such a property includes six hnes and may contribute to the six-loop ~ , five-loop 4~3 and four-loop ~2~>2 graphs (or to the divergent parts o f the five-loop 4>2~'2 ones). Consequently, up to this lImit the difference between (11) and (12) does not tell on the results, and our unambiguous expressions have the form which is supersymmetrlc in the four-dimensional space. However, to prove their supersymmetry properties, distinctive features of that space can turn out to be necessary. Namely, exactly four values for gorentz mdices and two for splnor ones (2). In terms of the g u y tensors the former Implies that antisymmetrlzat~on over five Indices is Impossible in four dimensions.

Volume 122B, number 3,4

PHYSICS LETTERS

10 March 1983

Table 1 The RDR m the Wess-Zumlno model The minimum number of loops, when the corresponding properties of the RDR versmns can be broken. Versions

Consistent

Graphs

q,~,

Inconsistent qb3

dp2dp2

d~vergent parts 1 2 3 4

unamblguny supersymmetry four-dlmensmnal hmlt external justification

det(/21

gl5,p 1 . v 5 ) = O .

. 5 6

.

. . 5 5 5 5 IS unnecessary

(13)

In turn, eq. (2) allows additional simplifications in explesslons with spmor-mdex quantities. Thus, the supersymmetry-breaklng parts of the d-dimensional results may include evanescent momentum-combinations, to prove the nullification o f which in four dimensions one would need formula (13), and 0-structures, which would be zeros if formula (2) were true A detailed analysis shows that the evanescent 0structures cannot break supersymmetry properties o f the propagators and triple vertices and also the finiteness of the quartlc vertices. Therefore, nothing but the evanescent momentum-combinations can break supersymmetry. Such a combination has at least tenth power in momenta. On the other hand, the maximal power o f the numerator o f the L-loop diagram is 2L (for 4 ~ and 4~3) or 2L + 2 (for ~ 2 ~ 2 ) Hence, supersymmetry can be broken an the consistent RDR version from five (in the q~?band q93 graphs) or four loops on (In the 4~2~ 2 ones). Since the d02c~2 diagrams have only loganthrmc divergencies, only the evanescent combinations whlcll revolve no external momenta can contribute to their divergent parts. Due to antisymmetry o f ( 1 3 ) , for such a combination at least five independent internal momenta are reqmred. Thus, the four-loop quartic vertices remain finite. We now consider computations with the use of the whole set ( 2 ) - ( 9 ) and 0-shifts or by the supergraph method o f ref. [7], i.e. m the inconsistent RDR versions. The following order ofmampulations is natural First the four-dimensional 0- (or D- [7]) and o-algebra (2)--(8), (10), (12) are carried out, and then the d-

qb~

qb3

qb2~c)2

6 ~ oo 5

6 oo oo 5

5

fimte parts 4 4

4

dimensional m o m e n t u m Integration is. Here the evanescent 0-structures and momentum-combinations become a source o f alnblgulties rather than o f supersymmetry breaking. For the corresponding estimates one should use supergraph power-counting rules [5,

7]. Calculations in contradictory versions are surely reliable while they have an external justification, 1.e while they agree with a consistent scheme which preserves supersymmetry up to a definite order. It IS such a correspondence that we established above when studying super-lnvariance of our unambiguous formulation Therefore, in its lnvariance region one can also perform 0-shifts, considerably simplifying the colnputations, or employ techniques o f ref. [7]. These arguments justify the use o f the RDR in actual three- and four-loop calculations [8]. The results of the above analysis are summarized m table 1. The consistent (superfield or component-field) RDR formulation and the lnconststent one (with 0shifts or covarlant derivatives) are considered. Their following properties are under study 1 unamblgmty, 2. supersymmetry, 3. correspondence to the fourdimensional formulas such as (12), 4 accordmlce with a consistent scheme. For each property the minimal number of loops is pointed out, when at can be broken for the first time Our main conclusion is that for divergent parts o f the diagrams the RDR can only prove Incorrect from five loops on. Maybe the search for air internal justification cnterlon in the contradictory RDR versions would extend their scope and give a satisfactory regularIzation for vector superfields. We thank A A. Vladmfirov for useful discussions 249

Volume 122B, number 3,4

PHYSICS LETTERS

References [1] W. Siegel, Phys. Lett. 84B (1979) 193. [2] W. Siegel, Phys Lett. 94B (1980) 37. [3] L V. Avdeev, G.A Choehla and A A Vladlmlrov, Phys. Lett. 105B (1981) 272. [4] D.M. Capper, D.R.T. Jones and P van Nxeuwenhmzen, Nucl Phys. B167 (1980) 479 [5] K FuBkawa and W Lang, Nucl Phys. B88 (1975) 61, L Mezmcescu and V. Ogxevetsky, JINR preprmt E2-8277 (Sept 1974),Usp Fzl Nauk117 (1975) 637

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[6] T. Curtrlght and G Ghandour, Ann. Phys (NY) 106 (1977) 209, P K. Townsend and P. van Nleuwenhmzen, Phys Rev D20 (1979) 1832, E Sezgm, Nucl. Phys B162 (1980) 1 [7] M. Gnsarta, W. Siegel and M. Ro~ek, Nucl Phys B159 (1979) 429 [8] L F Abbott and M T Grlsaru, Nucl. Phys B169 (1980) 415, A Sen and M.K Sundaresan, Phys. Lett 101B (1981) 61, L V Avdeev, S G. Gonshny, A Yu. Kamenshchlk and S.A Latin, J1NR preprmt E2-82-342 (May 1982)