Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://nuclphys.sinp.msu.ru/conf/lpp14/220809/Stepanyantz.pdf Äàòà èçìåíåíèÿ: Wed Sep 16 16:31:13 2009 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 00:45:57 2012 Êîäèðîâêà:

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Fourteenth Lomonosov Conference on Elementary Particle Physics Moscow, August 3 ­ 25, 2009.

K.V.Stepanyantz
Moscow State University Department of Theoretical Physics

Revealing structure of quantum corrections in N = 1 sup ersymmetric theories using the Schwinger­Dyson equations
& ~ 1

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' Quantum corrections in supersymmetric theories are investigated for a long time, for example in the papers
L.V.Avdeev, O.V.Tarasov, Phys.Lett. 112B, A.Parkes, P.West, Phys.Lett. 138B, (1983), I.Jack, D.R.T.Jones, C.G.North, Phys.Lett Nucl.Phys. B486 (1997), 479; I.Jack, D.R.T.Jones, A.Pickering, Phys.Lett. (1982),356; 99; B386, (1996), 138; B435, (1998), 61.

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Most calculations were made with the dimensional reduction
W.Siegel, Phys.Lett. 84B, (1979), 193; 94B, (1980), 37.

With the dimensional reduction the -function was calculated even in the four-loop approximation. It is in agreement with the exact NSVZ -function 2 3C2 - T (R) + C (R)i j j i ()/r () = - 2 (1 - C2 /2 ) .

V.Novikov, M.A.Shifman, A.Vainstein, V.Zakharov, Nucl.Phys. B 229, (1983), 381; Phys.Lett. 166B, (1985), 329; M.Shifman, A.Vainshtein, Nucl.Phys. B 277, (1986), 456.

up to the redefinition of the coupling constant. &

~ 2

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' Calculations with the higher covariant derivatives Interesting features can be revealed with the higher covariant derivative regularization.
A.A.Slavnov, Theor.Math.Phys. 23, (1975), 3; P.West, Nucl.Phys. B268, (1986), 113.

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Usually integrals, obtained with the higher derivative regularization are very complicated. However for supersymmetric theories integrals, defining the -function, can be easily calculated, because the integrands are total derivatives.
A.Soloshenko, K.S. hep-th/0304083; A.Pimenov, K.S., Theor.Math.Phys. 147, (2006), 687.

Example: N=1 supersymmetric Yang-Mills theory with matter in the massless case is described by the action 1 1 4 2 ab S = 2 Re tr d x d Wa C Wb + 2e 4 1 + d4 x d2 ij k i j k + h.c. . 6 & d4 x d4 ( )i (e
2V

)i j j +

~ 3

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' Two-lo op -function for a general renormalizable N = 1 sup ersymmetric Yang-Mills theory Two-loop calculation gives the following result: 32 3 2 32 ( ) = - C2 + T (R)I0 + C2 I1 + C (R )i j C (R )j i I2 + 2 r kl ikl j +3 T (R)C2 I3 + 2 C (R)i j I4 + . . . , 4 r where we do not write the integral for the one-loop ghost contribution and the integrals I0 ­I4 are given below, and the following notation is used: tr (T A T B ) T (R) f
AC D B C D AB

$

;

(T A )i k (T A )k j C (R)i j ; r AA .

f

C2

AB

;

Taking into account Pauli­Villars contributions, Ii = Ii (0) -
I

Ii (MI ),

i = 0, 2, 3 ~ 4

where Ii are given by &

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' Factorization of integrands into total derivatives
I0 (M ) = 8 d4 q d 1d (2 )4 d ln q 2 dq 1 ln q 2 (1 + q 2
2m

$

2

/2m )2 + M

2

+

mq 2m /2m q 2 (1 + q 2m /2m ) M2 + - ; 2(q 2 (1 + q 2m /2m )2 + M 2 ) q 2 (1 + q 2m /2m )2 + M 2 I1 = 96 â
2

d 1d d4 q d4 k (2 )4 (2 )4 d ln k2 dk

2

1 q 2 (q + k)2 (1 + q

2n

/2n ) ;

â

1 (1 + (q + k)2n /2n )
2

n n+1 - (1 + k2n /2n ) (1 + k2n /2n )2
2

I2 (M ) = -64

d4 q d4 k d 1d (2 )4 (2 )4 d ln q 2 dq

q2 â k2 (1 + k2n /2n )

(1 + (q + k)2m /2m ) q 2 (1 + q 2m /2m )3 + â ((q + k)2 (1 + (q + k)2m /2m ) + M 2 ) (q 2 (1 + q 2m /2m )2 + M 2 )2 mq 2m /2m 2mq 2m /2m M 2 +2 - 2m /2m )2 + M 2 q (1 + q (q 2 (1 + q 2m /2m )2 + M 2 )
2

;

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~ 5

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' Factorization of integrands into total derivatives
I3 (M ) = 16
2

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d d4 q d4 k (2 )4 (2 )4 d ln

k (1 + q 2m /2m ) â q (q 2 (1 + q 2m /2m )2 + M 2 )

1 â (k + q )2 (1 + (q + k)2n /2n )

(1 + k2m /2m )3 -2 + (k (1 + k2m /2m )2 + M 2 )2
2

mk2m /2m 2mk2m /2m M 2 +2 -2 2m /2m )2 + M 2 k (1 + k (k (1 + k2m /2m )2 + M 2 ) 1d -2 k dk â
2

-

(q 2 (1 + q 1
2n

2m

2(1 + q 2m /2m )(1 + (q + k)2m /2m ) â /2m )2 + M 2 ) ((q + k)2 (1 + (q + k)2m /2m )2 + M 2 )
2

(1 + k
2

nk2n /2n + /2n ) (1 + k2n /2n )

; 1 k2 (q + k)2 (1 + k
m

I4 = 64

d4 q d4 k d 1d (2 )4 (2 )4 d ln q 2 dq 1
2m

2

2m

/2m )
2

â

1 â (1 + (q + k)

/2m )

(1 + q

2m

/2

mq 2m /2m + ) (1 + q 2m /2m )

.

&

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Two-lo op -function for a general renormalizable N = 1 sup ersymmetric Yang-Mills theory The integrals can be calculated using the identity d4 k 1 d 1 f (k 2 ) = (2 )4 k 2 dk 2 16
2

f (k 2 = ) - f (k 2 = 0) .

The result for the two-loop Gell-Mann­Low function is given by 2 3 2 () = - 3C2 - T (R) + - 3C2 + T (R)C2 + 2 (2 )2 2 + C (R)i j C (R)j r
i

2 C (R )i j - 8 3 r

ikl j kl

+ ....

Comparing this with the one-loop anomalous dimension gives the NSVZ result. The result also agrees with the DRED calculations. & ~ 7

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' Three-lo op calculation for SQED The notation is
(2)

$

=

d4 p 4 d 4 (2 )

-

1 V(-p) 2 16

1 /2

V(p) d-1 (, µ/p) + (1)

1 + ( )i (-p, ) j (p, ) (Z G)i j (, µ/p) . 4 The main result: (It was obtained as the equality of some well defined integrals) d - d-1 (0 , /p) - 0 1 =- d ln d p=0 1 d 1 = 1- ln G(0 , /q ) = d ln q =0 1 - ln Z (, /µ) = 1 - 0 (, q =0

d - 0 1 (, µ/)= ln 1d + ln Z G(, µ/q ) - d ln /µ) .

Therefore, with higher derivatives it is not necessary to redefine the coupling constant in order to obtain the agreement with the NSVZ -function. The ~ reason is that the integrands are again total derivatives. 8 & %


' Possible explanation One can try to explain the factorization of integrands into total derivatives substituting solutions of the Slavnov­Taylor identities into the Schwinger­Dyson equations. Here we consider only a contribution of the matter superfields: (omitting the regularization in order to avoid large expressions) B Vy V
A x

$

e = ( )i (T A )i j (e x B 4 Vy


2V

x

x )j + h.c. + gauge contribution.

Then we introduce the notation (U0 )
n=1 i



ij k

D2 j 2 2

k

D2 +2 2 2

j

k ; (2) ~ 9

(U1 )i
n=1

ij k j k .

&

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' We also need auxiliary sources for these values: SS
our ce

$ 1 4

=

d4 x d2 ()j (U1 )j +

d8 x ( )i (e 0

2V

+ U )i + h.c. 0

There is a very important identity D2 D2 (e - =- 2 ( )i 8 0
2V

+U

0 )i

= , ( )i

which allows to relate usual Green functions, and the functions, containing derivatives w.r.t. auxiliary sources. Then using the identity S A V - ( e2V + i 2 i ¯2 A a 2V i¯ A A jD ¯ - +U0 ) D V - 2D ( e + U0 ) Da V (T )i 2j 16 2 ¯2 D2 i A j ¯2 A D a A Da D ¯ - 2 (U1 ) (T )i D V + 2D V j . (3) 2j 2 16 16 2 d8 x VA ( e
2V

)i (T A )i j j =

d8 x

8

the considered contribution to the two-point function can be written as a sum ~ of some effective diagrams. 10 & %

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

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Vertexes here contain derivatives w.r.t. auxiliary sources 0 and . They are restricted by the Slavnov­Taylor identities. Substituting solution of these identities we obtain (in the massless case for simplicity) d - d-1 (0 , 0 , /p) - 0 d ln 1d â 2 2 ln(q 2 G2 ) + 2K + q dq
1 p=0

d = -2 T (R) d ln ¯ Da D 2 V
B

d4 q â (2 )4

Da V &

A

~ 11

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' The function K is defined by 1 2 ¯2 - (K )j i Dy (jy )j (w )i 2
8 yw

$

.

(4)

For a theory with cubic superpotential this function is nontrivial in the two-loop approximation. Factorization of some other values into total derivatives The explanation, given above, is not complete, because some explicit calculations show that the diagram, which is contributed by the transversal part of the Green function, in the lowest loops is also an integral of a total derivative. Therefore, it is necessary to explain, why this contribution disappear. More exactly, it is necessary to explain why in the in lowest loops ¯ Da D 2 1 dK T (R) + =0 4 d ln q=0 Da V &
A

V

B

~ 12

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Toward the explanation The diagram, which was not calculated, contains derivatives with respect to the auxiliary sources 0 . We consider only a part, corresponding to the cubic superpotential. It can be rewritten as a sum of two-loop effective diagrams: ¯ ¯ ¯ Da D2 Da D2 Da D2 Da V
A

+ V Da V
B

A

+...

Da VA VB VB These diagrams are calculated by the same method as above. The result is the following: The "bad" term with the function K is completely canceled. However, again there is a nontrivial contribution of the transversal part (staring at least from the three-loop approximation). Possibly, it can be zero, but so far there are no calculations, confirming this. & ~ 13

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Conclusion and op en questions

With the higher derivative regularization integrals, defining the -function, at least in the lowest loops can be easily taken, because the integrands are total derivatives. The result is in agreement with the NSVZ -function. The explanation can be possibly made by substituting solutions of the Slavnov­Taylor identities into the Schwinger­Dyson equations. The main problem is whether the factorization of integrands into total derivatives takes place only in the lowest loops or it holds exactly to all loops. Higher loops in the Schwinger­Dyson equations are essential and can possibly lead to explanation of some so far mysterious cancelations. ~ 14

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