Finite N=1 SUSY field theories and dimensional regularization

Volume ! 79, number 4

PHYSICS LETTERS B

30 October 1986

FINITE N- - 1 SUSY FIELD THEORIES AND DIMENSIONAL REGULARIZATION D.I. K A Z A K O V dspen Center/'or Phystcs, Aspen, CO 81611, US4 and Joint lnstttute for Nuclear Research. Dubna, 101000 Moscow, USSR Received 14 July 1986

A s~mplealgorithm Is proposed to construct fimte N= 1 SUSY field theories w~thlndlmensmnal regularlzatmn.This is achieved by choosingthe Yukawa couphngs to be y,=g( o~ +al~e+e+ot~e-~+ ), where g is the gauge couphng and the coefficientso~I),o~, etc, are calculatedorder by order ofpenurbatmn theory It ~sshownthat the theory can be made fimte m all orders of perturbation theory

I In our previous papers [ 1,2] we have shown how one can construct finite N = 1 SUSY field theories by fine-tuning of Yukawa couplings. The procedure is the following. We consider an arbitrary N = 1 SUSY field theory formulated m terms of N = 1 superfields. The action contains both the gauge and Yukawa type interactions with the couphngs g and y,, respectively To reduce the n u m b e r of independent divergences we use the background field gauge. In this gauge the only uncorrelated divergences are those of the gauge and the chiral superfield propagators. If these propagators are finite then the anomalous dxmensmns vamsh and so do the fl-functions. That means that the theory is finite. It has been shown [ 3,4] that this theory is one-loop finite if: (1) The set of chlral superfields obeys the sum rule

T ( R ) =3Cc~ ,

(1)

three and higher loop finiteness [5,6] In fact the only divergence that one should take care of is that of the choral f e l d propagators. The gauge propagator is finite whenever this is true for the chiral ones. Due to supersymmetry the following relaUon holds [ 7 ] :

flg=g2 [ZRT(R)-3C~]+~y,

(hereafter g stands for g2/16zr- and y, for y;/167t in a usual fashion). Eq. (2) exhibits the fact that if all the anomalous dimensions y , = 0 in L loops then the gauge fl-function fig = 0 in L + 1 loops [ 8 ]. This has been checked m two- and three-loop order by exphclt calculation [ 5 ]. The anomalous dimensions of independently renormahzed chlral superfields are 7,({Y:}, g) = Z B]:y:+B',og /

R

where T(R) for a given representation R a of the gauge group G is defined as Tr R a R t ' = d a h T ( R ) , and Cc, IS the value of the group Caslmir operator. ( n ) The Yukawa couphngs y,=o~,g, where o~, are some constants. In th~s case the theory is also two-loop fimte [3,4] but the above c o n d m o n s are not enough to achieve ' Permanent addres 352,

(2)

1-g

+ ~ B'2,hy/y~+ ~ B'2/Y:g+B%g 2 + ..., 11,

(3)

!

where y/ and g are the renormahzed couphngs, z= 1,2,...,N,j,k= 1,2,...,M. We can achieve fimteness, i.e. vanishing of all 7, by choosing y: to be yj = % g + d,g2 q_pjg3 q_....

(4)

In one-loop order we have 0370-2693/86/$ 03.50 © Elsevier Scaence Publishers B.V. (North-Holland Physics Publishing Division)

Volume 179, number 4

PHYSICS LETTERS B

B'~,a/+ B'I 0 = 0 .

(5)

30 October 1986

fit'=0,

Ze=I,

g~aRE=g.

I

This system of linear equations has a solution if the rank of the matrix B~ equals the n u m b e r o f independently renormallzed fields ( N ) . In two loops we have

~, B'~,c~,+ ~ B~,~ec,a~+ ~ B'2,a/+B'~o = 0 . /

/J,

(6)

1

The existence o f the solution to eq. (6) is guaranteed by that o f eq. (5) due to the a p p e a r a n c e o f the same matrix BI in the homogeneous part. Evidently this procedure will go on in all orders o f perturbation theory. The calculations up to three loops in a n u m b e r o f models have been p e r f o r m e d [2 ]. The same procedure has been also discussed in a recent paper [ 9 ]. 2 In the present note we would like to propose a simpler algorithm to obtain a finite theory which explores the advantages of d i m e n s i o n a l regularlzatIon. Again for the same reason as above, we take care o f c h l r a l field propagators only. Let the p e r t u r b a t i v e expansion be D,({y,}, =1+

g, p2, E) ~ 1 ,,=1 ~

TO prove that eq. (9) actually solves the problem, we consider eqs. ( 7 ) order-by-order in the loop expansion.

1 loop fl~l~ = 0 =~ gB~,RE = g , l/e:

C]~(y,g) l . . . . . ~ = 0 .

Due to eq. ( 8 ) this is nothing else but eq. (5) and has a solution if rank B~ = N .

2 loops Due to eq. (2) fl~g2)=0, that means that again gBARZ= g

1/e2: The coefficient C'22(y, g) should vanish when Y/= c¢~g because otherwise we will obtain a nonlocal divergence o f the type (C~2/E) In p2 which is forbidden due to the r e n o r m a h z a b I l i t y o f the theory. This divergence cannot be e h m l n a t e d in the usual fashion because all one-loop counterterms vanish. So C;z(y, g) should have the form C~2(y, g) =C] ~(y, g).F~(y, g).

(10)

l/e: C ~ (y, g) l . . . . .

0c52(y, g)

(C',,,(y,g) +C ',.... ,(y,g)+... \

~-ff

E" J

+C~'l~Y'g) +c',,o(y,g)).

(7)

where the d i m e n s i o n o f the s p a c e - t i m e is taken to be 4 - 2E. The coefficients are not arbitrary but are governed by C',,t due to r e n o r m a h z a b l l i t y o f the theory. At one-loop o r d e r (see eq. ( 3 ) )

C ' ~ ( y , g ) = - ( ~ B'lyl+B',og).

It is possible if the matrix OC'~2/OyI has the rank = N. But due to eqs. ( 8 ) , (10) we have

OC'22(y,g)/Oyll . . . . ~= -B'~.F~(y,g) l,.. . . . So if F~(y, g) l . . . . . ~ is not zero, we have the

same requirement we already had at one loop, i.e. rank

(8)

Finiteness now means vanishing o f all C',,~, k = 1, , n. This can be achieved by choosing Yukawa couplings to be y B A R E = g B A R E ( Odb + OU,1 E q'- O[ 5 E 2 -~- .. ) .

We see that even if C'l(y,g)l,_,,,e¢O we can achieve finiteness properly choosing the p a r a m e t e r s ce~.

(9)

Eq. ( 9 ) means that nonlinear g d e p e n d e n c e o f eq. (4) is now transformed into that o f e It happens also due to eq. (2) that

Bt = N . This algorithm will obviously work in all loops. All the coefficients C',,~, 2<~k<~n vanish due to the r e n o r m a h z a b l h t y properties and absence o f lower o r d e r counterterms. To achieve the absence o f a simple pole C'~ we have in our disposal a set o f p a r a m eters a J,,. Actually as in eq. (10 )

C',,,,(y, g) = C'll (Y, g).F',,(y, g),

(1 1 )

and in n loops the 1/E term is 353

Volume 179, number 4

PHYSICS LETTERS B

unified models. S o m e SU5 and SU6 e x a m p l e s do already exist [ l 1 - 1 3 ] . T h e f i m t e n e s s r e q u i r e m e n t h a p p e n s to be a strong c o n s t r a i n t a n d leaves us only four realistic posslbihties: SUs, SU6, SO1o and E6. T h e n u m b e r o f g e n e r a t i o n s is also restricted by eq. (1)

OC',,,, ~,, ...... ,,++

..... ,,

D u e to eq. (1 1 ) it is r e d u c e d to

C ' , , l ( y , g ) l . . . . . ~,+

-

~

'

''

'

•

H e n c e in any o r d e r n we h a v e the s a m e m a t r i x B~ and the only thing to be s h o w n is that all F',(y, g) I, ,~,,~,¢ 0. To p r o v e this one can use the fact that C',,,, equals to (-)"C',,(,,~, w h e r e C',,~,,~ is the o r d e r n t e r m in the e x p a n s i o n Z,=I+

c~(y, g)

~

~,

,

V=I

C',(y, g) = ~ C',¢~){y, g}~ . ,I,

30 October 1986

v

T h e coefficients C~ obey the pole e q u a t i o n s . U s i n g t h e m one can p r o v e that

[141. It was suggested [ 11 ] that the low energy l i m i t o f a f n i t e superstrlng theory w o u l d itself be a finite fourd i m e n s i o n a l theory. If so t h e n it w o u l d be v e r y m t e r esting to find out w h e t h e r the p r o p o s e d f i m t e S U S Y m o d e l s m a y e m e r g e f r o m the superstring or they give us a l t e r n a t i v e ways o f finite low energy u m f i c a t i o n . It is a pleasure to t h a n k the A s p e n C e n t e r for Physics w h e r e this w o r k was c o m p l e t e d for k i n d hospitality. I a m grateful to M a r k G r l s a r u for helpful discussions and r e a d i n g the m a n u s c r i p t . U s e f u l conv e r s a t i o n s with D.V. Shlrkov, A A. V l a d i m i r o v and O.V. T a r a s o v are also a c k n o w l e d g e d

References F',, 1, ,~,,,,~F',,

~ 1,_~,,~ . . . . .

F'~ 1,,_,~,,~= 1 .

T h i s c o m p l e t e s o u r proof. 3 We have d e m o n s t r a t e d that a theory can be m a d e f i m t e by i m p o s i n g constraints on the Y u k a w a couphngs N e c e s s a r y a n d sufficient c o n d i t i o n s are d e f i n e d already in o n e - l o o p order: C o n d i t i o n (l) 5 2 ~ ( R ) = 3 C G . C o n & t l o n ( 11 ). rank BL = N C o n d i t i o n ( n i ) . c~6 >/0. W o r k i n g in superfields as has b e e n already m e n t t o n e d o n e - l o o p f i m t e n e s s leads to a t w o - l o o p one. T h a t m e a n s that all c~] = 0. N o n v a m s h i n g o f c~,, k~> 2 then m a y well be a reflectton o f S U S Y b r e a k i n g by d i m e n s i o n a l r e g u l a r i z a t i o n in higher loops [ 2,10 ]. It is w o r t h m e n t I o m n g that the p r o p o s e d algor i t h m based on eq. (9) is s i m p l e r t h a n that based on eq. (4) [ 1 ]. W h a t one actually has to do is to c h o o s e the bare Y u k a w a couplings in a way as to cancel l/e t e r m s in chiral field p r o p a g a t o r s m e v e r y o r d e r o f perturbation theory T h e resulting theory wtll be f i m t e m any l o o p o r d e r a n d to any o r d e r o f g. T h e coefficients c~~, d j, p ' e t c , o f e q . (4) are in one to o n e corr e s p o n d e n c e with c~¢, o f e q . (9) T h e p r o p o s e d finite S U S Y field t h e o r i e s can be used for c o n s t r u c t i n g finite s u p e r s y m m e t r i c g r a n d

354

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