Sypersymmetries and Quantum Symmetries
Dubna 2007
K.Stepanyantz
Moscow 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
S


1
1 4
m 4

4
2
ab
4
 2V
 2V t
2
t
Re
tr
d
x
d

W
C
W

d
x
d


e



e


d
x
d




h
.
c
.
a
b




2e2
4
2

where  and  are chiral scalar matter superfields, V is a real scalar
gauge superfield, and
Wa 
1
D(1   5 ) D[e 2V (1   5 ) Da e 2V ]
32
The action is invariant under the gauge transformations

e2V  ei e2V ei ;
  ei;
  e  i 
t
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.
Quantization

We use the background field method: e2V  e e2V e , where  is a background
  Vbg.
field. Using the background gauge invariance it is possible to set   Vbg ;
Backlground covariant derivatives are given by
Dbg  e 


1
(1   5 ) De ;
2
Dbg  e
1
(1   5 ) De ;
2
i
Dbg ,    (C  ) ab Dbg , a , Dbg ,b 
2
The gauge is fixed by adding the following term:
S gf  
1
tr  d 4 x d 4 VDbg2 Dbg2 V  VDbg2 Dbg2 V 
2
32e
The corresponding ghost Lagrangian is
Sc  i tr  d 4 x d 4
(c  c


)V c  c   cthV (c  c  ) 
Also it is necessary to add the Nielsen-Kallosh ghosts
1
4
4
  
tr
d
x
d

B
e e B
4e 2 
In order to calculate quantum corrections we also introduce additional sources 0
SB 
S0 

1 4
4
 2V
 2V t
d
x
d


e



  h.c.
0
0 e
4

Differentiation with respect to additional sources allows calculating vacuum
expectation value of some copmposite operators.
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
Dbg2 ,  

1
4
4
S  2 tr Re  d x d  V
2e
2n
n 1
V
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
Z   d
  det PV (V ,Vbg , M i )  exp  i(S  S  S gf  S gh  SB  S0 ) 
ci
i
where the coeffecients satisfy the condition  ci  1;
i
c M
i
2
i
0
i
The regularization breaks the BRST-invariance. Therefore, it is
necessary to use a special subtraction scheme, which restore the SlavnovTaylor 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.
Calculating matter contribution by Schwinger-Dyson
equations and Slavnov-Taylor identities
Schwinger-Dyson equation for the two-point Green function of the gauge
field can be graphically presented as
 2

 Vbg  Vbg
+contributions of the quantum
gauge field and ghosts
+
Feynman rules are (massless case for simplicity):
2 2
2
A propagator:     Dx Dx  8
x y
4 2G ( 2 )
xy
Vertexes (obtained by solving the Slavnov-Taylor identities):

 3

a

  Vbg , yz x

1
a 2
2 8
2 8
2
a 
2

2 8
2 8
  eT  1/ 2  Dy  xy Dy  yz  F (q )  eT q G '(q ) D  5 D  D y  xy Dy  yz 
32
 p 0
1
 eT a  Dy2 xy8 Dy2 yz8  G (q 2 )
8

 3
 a

  Vbg , y0 zx


1 a b
a 2
2 8 8
2
2
2 8
c 8
2
  2eT  1/ 2  Dy  xy yz  F (q )  eT D Cbc Dy  Dy  xy Dy yz  f (q )
8
 p 0
1
1
eT a q  G '(q 2 ) D   5 D  Dy2 xy8  yz8   eT a  Dy2 xy8  yz8  G (q 2 )
16
4
Exact beta-function and new identity
Substituting the solution of Slavnov-Taylor identities to the Schwinger-Dyson
equations (in the massless case) we find
1
d4 p
 )
4
2
1
   tr  d 
V
(

p
)


V
(
p
)
d
(

,
bg
1/
2
p
8
(2 )4
Gell-Mann-Low function is then given by


 d ( ,  p ) 

d ( ,  )
p
 ln p
We obtained that
d
d
d 4q 1  d
16 f 
1
2 2
d  8 C ( R)
ln(
q
G
)


  ( PV )  ( gauge)
d ln 
d ln   (2 ) 4 q 2  dq 2
G 
(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
 2 3C2  2C ( R)(1   ( )) 
 ( )  
C
2 1  2
2

in the substitution

16 p 2 f ( p 2 )
 ( )   ( )  lim
p 0
G( p 2 )
New identity in N=1 supersymmetric Yang-MIlls theory with
matter
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):
f (q 2 )
 d q q 2G ( q 2 )  0
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:
Da , z Dz2
 3
8
8
a
tr  d x d y Vbg , y D Vbg , x
z  x , p 0  0
2


4i  jz  Vbg , y0, x
where the sources are assumed to be expressed in terms of fields.
4
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:
The corresponding function f can be found by calculating diagrams of the type
nonchiral
chiral
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.
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
d
d 4q  2

q 2n
2
d 2  48  0
q
(1

)
2
n

 
d ln   (2 ) 4 

2n

  2(n  1) 1  k
 2n


1

2n
 2n 1  k

2n

2
1

d 1 ( ,  )  d 2 ln
p
p
where
d 4k 1 d 

(q  k ) 2 n
2
(
q

k
)
(1

)
2
n

 (2 )4 k 2 dk 2 
 
1



The integrand is again a total derivative in the four dimensional sperical
coordinates! Really,
d 4k 1 d
1
2
f
(
k
)

f (k 2  )  f (k 2  0) 
2 
 (2 )4 k 2 dk 2
16
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
3C2 2 3C22 3
 ( )  

 O( 4 )
2
2
4
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.)
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,
f (q 2 )
 d q q2G(q2 )  0 in the massless case
4
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.