Sypersymmetries and Quantum Symmetries Dubna 2007

K.StepanyantzMoscow State University

Department of Theoretical Physics

New identity for Green function in N=1 supersymmetric theories

N=1 supersymmetric theories

N=1 supersymmetric Yang-Mills theory with matter is described by the action

where and are chiral scalar matter superfields, is a real scalar gauge superfield, and

The action is invariant under the gauge transformations

2 25 5

1(1 ) [ (1 ) ]

32V V

a aW D D e D e

2 2 ; ;tV i V i i ie e e e e e

V

We investigate quantum corrections to the two-point Green function of the gauge superfield and to some correlators of composite operators exactly to all orders of the perturbation theory.

4 2 4 4 2 2 4 22

1 1Re . .

2 4 2

tab V V ta b

mS tr d x d W C W d x d e e d x d h c

e

QuantizationWe use the background field method: , where is a background field. Using the background gauge invariance it is possible to set . Backlground covariant derivatives are given by

The gauge is fixed by adding the following term:

The corresponding ghost Lagrangian is

Also it is necessary to add the Nielsen-Kallosh ghosts

In order to calculate quantum corrections we also introduce additional sources

Differentiation with respect to additional sources allows calculating vacuum expectation value of some copmposite operators.

5 5 , , ,

1 1(1 ) ; (1 ) ; ( ) ,2 2 2

abbg bg bg bg a bg b

iD e De D e De D C D D

2 2V Ve e e e

;bg bgV V

4 4 2 2 2 22

1

32gf bg bg bg bgS tr d x d VD D V VD D Ve

4 4 ( ) ( )cS i tr d x d c c V c c cthV c c

4 42

1

4BS tr d x d B e e Be

0

4 4 2 20 0

1. .

4

tV VS d x d e e h c

0

Higher derivative regularization

To regularize the theory we use the higher covariant derivative regularization. (A.A.Slavnov, Theor.Math.Phys. 23, (1975), 3; P.West, Nucl.Phys. B268, (1986), 113.) We add to the action the term

Then divergences remain only in the one-loop approximation. In order to regularize them, it is necessary to introduce Pauli-Villars determinants into the generating functional

where the coeffecients satisfy the condition

The regularization breaks the BRST-invariance. Therefore, it is necessary to use a special subtraction scheme, which restore the Slavnov-Taylor identities: A.A.Slavnov, Phys.Lett. B518, (2001), 195; Theor.Math.Phys. 130, (2002), 1; A.A.Slavnov, K.Stepanyantz, Theor.Math.Phys. 135, (2003), 673; 139, (2004), 599.

12,4 4

2 2

1Re

2

n

bg

n

DS tr d x d V V

e

0

det ( , , ) exp ( )ic

bg i gf gh Bi

Z d PV V V M i S S S S S S 21; 0i i i

i i

c c M

Calculating matter contribution by Schwinger-Dyson equations and Slavnov-Taylor identities

2

bg bgV V

+

Schwinger-Dyson equation for the two-point Green function of the gauge field can be graphically presented as

Feynman rules are (massless case for simplicity): A propagator:

Vertexes (obtained by solving the Slavnov-Taylor identities):

2 228

2 24 ( )x x

xyx y

D D

G

32 2 8 8 2 2 2 8 8 2

1/ 2, 0 0

2 2 8 8 2 8 8 25

12 ( ) ( )

8

1 1'( ) ( )

16 4

a a b cy xy yz bc y y xy y yza

bg y z x p

a ay xy yz y xy yz

eT D F q eT D C D D D f qV

eT q G q D D D eT D G q

32 2 8 2 8 2 2 2 8 2 8

1/ 2 5, 0

2 8 2 8 2

1( ) '( )

32

1( )

8

a ay xy y yz y xy y yza

bg y z x p

ay xy y yz

eT D D F q eT q G q D D D DV

eT D D G q

+contributions of the quantum gauge field and ghosts

Exact beta-function and new identity

Substituting the solution of Slavnov-Taylor identities to the Schwinger-Dyson equations (in the massless case) we find

Gell-Mann-Low function is then given by

We obtained that

(We did not calculate contribution of the gauge field and did not write contribution of Pauli-Villars fields). Then the Gell-Mann-Low function differs from NSVZ beta-function

in the substitution

44 2 1

1/ 24

1( ) ( ) ( , )

8 (2 ) bg

d ptr d V p V p d p

41 2 2

4 2 2

1 168 ( ) ln( ) ( ) ( )

ln ln (2 )

d d d q d fd C R q G PV gauge

d d q dq G

2

2

2

3 2 ( )(1 ( ))( )

2 1 2

C C R

C

2 2

20

16 ( )( ) ( ) lim

( )p

p f p

G p

( , ) ( , )ln

d dp pp

New identity in N=1 supersymmetric Yang-MIlls theory with matter

24

2 2

( )0

( )

f qd q

q G q

2 3,8 8

, , , 02, 0,

04a z za

bg y bg x z x pz bg y x

D Dtr d x d y V D V

i j V

HOWEVER, explicit calculations with the higher derivative regularization in the three- and four-loop approximations (A.Soloshenko, K.Stepanyantz, Theor.Math.Phys. 134, (2003), 377; A.Pimenov, K.Stepanyantz, Theor.Math.Phys. 140, (2006), 687.) show that all in integrals, defining the two-point Green function of the gauge field in the limit p 0, has integrads which are total derivatives. This can be only if the following identity takes place (massless case for simplicity):

This identity is also valid in non-Abelian theory (K.Stepanyantz, Theor.Math.Phys. 150, (2007), 377). It is nontrivial only in the three-loops. In the massive case this identity is much more complicated, but can be wriiten in the simple functional form:

where the sources are assumed to be expressed in terms of fields.

Three-loop verification of new identity in non-Abelian theory New identity and factorization of the integrands to total derivatives can be verified for special groups of diagrams (A.Soloshenko, K.Stepanyantz, Theor.Math.Phys. 140, (2004), 1264.) A way of proving in the Abelian case was scketched in K.Stepanyantz, Theor.Math.Phys. 146, (2006), 321. In the non-Abelian case there is a new type of diagrams:

It is also necessary to attach a line of the background gauge field to the line of the matter superfield by all possible ways.

The result is again an integral of a total derivative! Therefore, the new identity is also valid in this case.

The corresponding function f can be found by calculating diagrams of the type

chiralnonchiral

Structure of quantum correction in N=1 supersymmetic Yang-Mills theory

Are integrads, defining the two-loop function of N=1 SYM, also reduced to total derivatives with the higher covariant derivative regularization? (A.Pimenov, K.Stepanyantz, hep-th/0707.4006)

Two-loop Gell-Mann-Low function of N=1 SYM (without matter) is defined by the diagrams

A wavy line corresponds to the quantum gauge field, dashs correspond to Faddeev-Popov ghosts, dots correspond to Nielsen-Kallosh ghosts, and a cross denots a counterterm insersion.

Usual diagrams are obtained by adding two external lines of the background field by all possible ways.

Structure of quantum correction in N=1 supersymmetic Yang-Mills theory

In the limit p 0 two-loop contribution is where

The integrand is again a total derivative in the four dimensional sperical coordinates! Really,

Therefore, this seems to be a general feacture of all supersymmetric theories, althouht the reaon is so far unclear.

The corresponding two-loop Gell-Mann-Low function coincides with the well known expression

With the higher derivative regularization there are divergences only in the one-loop approximation. (similar to M.Shifman, A.Vainstein, Nucl.Phys. B277, (1986), 456.)

2 2 342 2

2

3 3( ) ( )

2 4

C CO

4

2 2 24 2 2 2

1 1( ) ( ) ( 0)

(2 ) 16

d k df k f k f k

k dk

12( , ) lnd dp p

1 14 42 22 2 2

2 22 0 4 4 2 2

1 22 2

2 2

1 ( )48 (1 ) ( ) (1 )ln (2 ) (2 )

2( 1) 1 2 1

n n

n n

n n

n n

d d q d k dq q kd q q kd k dk

k kn n

Conclusion and open questions

• With the higher derivtive regularizations integrals, defining the two-point Green function of the gauge field in the limit p 0, can be easily taken, because the integrands are total derivatives. It is a general feature of N=1 supersymmetric theories. Why it is so?

• There is a new identity in both Abelian and non-Abelian N=1 supersymmetric theories the matter superfields, for example,

in the massless case

It is not a consequence of the supersymmetric or gauge Slavnov-Taylor identities. The corresponding terms in the effective action are invariant under rescaling. What symmetry leads to this identity?

• New identity is nontrivial starting from the three-loop approximation.

24

2 2

( )0

( )

f qd q

q G q