CC BY
25
UDC 530.1; 539.1
NSVZ SCHEME AND THE REGULARIZATION BY HIGHER DERIVATIVES
K. V. Stepanyantz
Department of Theoretical Physics, Moscow State University, Leninskie Gory, Moscow 119991, Russia.
E-mail: stepan@phys.msu.ru
The NSVZ scheme is constructed in all orders for the renormalization group functions defined in terms of the renormalized coupling constant for AbelianN = 1 supersymmetric theories regularized by higher derivatives. For the other renormalization prescriptions the scheme-independent consequences of the NSVZ relation are investigated. It is explained, why for the renormalization group functions defined in terms of the bare coupling constant the NSVZ relation is valid for all renormalization prescriptions in the case of using the higher derivative regularization.
Keywords: supersvmmetry, renormalization, ¡3-function, anomalous dimension.
1 Introduction
The NSVZ ^-function [1-4] is a relation between the ,0-function of N =1 supersymmetric theories and the anomalous dimensions of the matter superfields:
P(o) = —
a2 3C2 - T(R) + C(R)ijYji(a)/r
2n(1 - C2a/2n)
(1)
Here we use the notation
tr (TATB ) = T (R) SAB ; (TA)ik (TA)kj = C (R)j ;
çACD rBCD _ n XAB.
rAC D rBC D _ <;AB . _ £
f f = C2 o ; r = oaa.
(2)
For the particular case of N =1 supersymmetric electrodynamics (SQED) with Nf flavors the NSVZ ^-function takes the form [5,6]
£(a)
x2 Nf
l - i(a)).
(3)
The NSVZ ^-function was constructed using various general arguments: structure of instanton contributions [1, 3, 7], anomalies [2, 4, 8], the non-renormalization theorem for the topological term [9].
The NSVZ expression can be compared with the results of explicit calculations which in supersymmetric theories are mostly made using the regularization by the dimensional reduction [10]. (It should be noted that this regularization is either mathematically-inconsistent [11], or is not manifestly supersymmetric [12] and can break supersymmetry in higher loops [13, 14].) Using the dimensional reduction supplemented by the DR-scheme the ,0-function for general N = l
four-loop approximation [15-18]. The NSVZ ^-function
agrees with these calculations only in the one- and two-loop approximations. In the higher loops it is obtained only after a specially tuned finite renormalization [16,19].
It appears that a very convenient tool for calculating quantum corrections in supersymmetric theories is the higher covariant derivative regularization [20, 21]. (It also includes the PauliVillars regularization for removing the one-loop divergences [22,23].) Unlike the dimensional reduction, it is consistent and (if it is used for supersymmetric theories) does not break supersymmetry [24,25]. This regularization can be also formulated for N = 2 supersymmetric theories [26,27].
The explicit calculations made with the higher
N = 1
theories reveal an interesting feature of quantum corrections: integrals giving the ^-function defined in terms of the bare coupling constant are integrals of (double) total derivatives [28, 29]. (Note that in these integrals the external momentum vanishes.) The NSVZ relation appears after calculating the momentum integral of a total derivative. For Abelian supersymmetric theories this was proved exactly in all orders [30,31].
However, the renormalization group (RG) functions defined by the standard way in terms of the renormalized coupling constant [32] are scheme dependent. They satisfy the NSVZ relation only with a certain subtraction scheme, which is called the NSVZ scheme.
At present there is no general prescription how to construct this scheme in all orders with the dimensional reduction. However this can be easily done using the higher derivative regularization [33]. In the present paper we describe how this can be made.
n
2 N = 1 SQED with Nf flavors, regularized by higher derivatives
In this paper we consider N = 1 SQED with Nf flavors which is described by the action
1 r Nf 1 r
S = ^ Re d4x d20 WaWa + d4x dA°
0 i=a
x(<j>*ae2V <f><* + rae-2V ¿J, (4)
in the massless limit. Here V is a real gauge superfield, and (pa with a = 1,..., Nf are chiral matter superfields. In the Abelian case Wa = D2DaV/4. In order to introduce the higher derivative regularization we add the higher derivative term Sa to the classical action:
r(2) =/(04 ( - iV^ ^1/2
1 N/ /
x V(p, 0)d-1(ao, A/p) + - ]T [4>*a(-P, 0)
a=1
x^a(p,0) + 4>*a(-P,0)4a(p,0)) G(ao, A/p)), (10)
where r(2) is a part of the effective action corresponding to the two-point Green functions and d2n1/2 denotes a supersymmetric transversal projection operator.
In order to construct the renormalized coupling constant a(a0, A/p) we require finiteness of d-1(a0(a, A/p), A/p) in the limit A ^ to. The renormalization constant Z3 is then defined by
Sreg — S + SA,
where
1
Sa = J d4x d20Wa (R(82/A2) - 1) Wa
(5)
(6)
Z [J, n]= ^det PV (V, Mi)
J i
x exP | iSreg + iSgt + ^^^rnces |.
Nf ci
(7)
We require that the degrees of the Pauli-Villars determinants cI satisfy the constrains
E
ci
E ci M\
(8)
Mi = ai A, ai = ai (eo).
Let us define the functions d-1(a0, A/p) G(a0, A/p) according to the following equation:
Z3(a, A/p) = —.
ao
(11)
R - 1
derivatives. A convenient choice of this function is R = 1 + d2n/A2n.
By introducing Sa one regularizes all divergences beyond the one-loop approximation. The remaining one-loop divergencies can be removed by inserting the Pauli-Villars determinants into the generating functional [231:
Z
constructed by requiring finiteness of the renormalized two-point Green function ZG in the limit A ^ to.
3 NSVZ relation for the RG functions defined in terms of the bare coupling constant
The RG functions can be defined in terms of the bare coupling constant according to the following prescription:
p{ao(a, A/p)) =
dao
d ln A
a=const
Yij (ao(a, A/p)) = -
d ln Zj
d ln A
a=const
(12) (13)
due to which the remaining one-loop divergences cancel. The masses of the Pauli-Villars fields are chosen
A
independent of the bare coupling constant:
(9) and
where the derivatives should be calculated at a fixed value of the renormalized coupling constant. It is possible to prove [33] that these RG functions are scheme independent for a fixed regularization, but depend on the regularization. Moreover, in all loops they satisfy the NSVZ relation for Abelian supersymmetric theories, regularized by higher derivatives [30,31].
The NSVZ relation appears, because with the higher covariant derivative regularization loop integrals giving the ^-function defined in terms of the bare coupling constant are integrals of total derivatives [28]
and even integrals of double total derivatives [29]. (In
p=
0.) As a consequence, one of the momentum integrals can be calculated analytically, producing the NSVZ relation for the RG functions defined in terms of the bare coupling constant:
0
ß(a0)
d
a
d ln A
d 1(ao, A/p) — a-1
p=0
d
Nf (l —
n V d ln A
Nf (1 — Y(ao)).
ln G(a0, A/q)
q=0
(14)
Similar features are also valid in the non-Abelian case, but the calculations have been done only in the two-loop approximation [34-36].
4 The NSVZ scheme with the higher derivatives
In the previous section we consider the RG function defined in terms of the bare coupling constant. However, by standard way the RG functions are defined in terms of the renormalized coupling constant [32]:
ß( a(ao, A/p)) =
da
d ln p
a0=const
7i3 (a(ao, A/p)) =
d ln Zj
d ln p
a0=const
Z$(a, xo) = 1; Zi3 (a,xo) = l
ß(a) = ß(a) 7(a) = Y(a).
RG functions approximation
and the anomalous dimension in the three- and two-loop approximations, respectively. Let us present the results for various definitions of the RG functions and in various subtraction schemes.
The RG functions defined in terms of the bare coupling constant coincide with the RG functions defined in terms of the renormalized coupling constant in the NSVZ scheme and are given by the following expressions:
7NSVz(a)
2 1 n a a2 (1 v^
(a) = Y(a)---1--2(t: + Nf ) ci ln a;
n n2 \ 2 z—'
1=1
+f + O(a3);
(nsvz(a) = ß(a)a2Nf (l + - — 4 (1 + Nf n V n n2 \ 2
n
x ^ c; ln a; + Nf) + O(a3)) .
I =1
(19)
(20)
(15)
(16)
(In order to obtain these functions it is necessary to express the RHS via a0 and calculate the derivatives at a fixed value of the bare coupling constant.) The RG functions (15) and (16) are scheme-dependent. According to [33, 37] they coincide with the RG functions defined in terms of the bare coupling constant, if the boundary conditions
We see that in this scheme the NSVZ relation is really-satisfied in the considered approximation.
Let us also present the results for the RG functions defined in terms of the renormalized coupling constants for other subtraction schemes.
In the MOM scheme the dimensional reduction and the higher derivative regularizations give the same result [371
7mom(-) = -— + ßmom(a)
a a2(1 + Nf)
n a2N
2n2
+ O(c
2
(17)
x (1 — Z(3)))+ O(a3)) .
^ (1+a—i+3Nf
3\
(21)
(22)
are imposed on the renormalization constants, where xo is an arbitrary fixed value of lnA/p:
In the DR-scheme the result was obtained in Ref. [16] and is written as
(18)
DR
» = — a + + o(-3)
n 4n2
(23)
_ Due to the scheme-dependence the RG functions (3(a) and 7(a) satisfy the NSVZ relation only in a certain subtraction scheme, called the NSVZ scheme. This scheme is evidently fixed in all loops by the boundary conditions (17) if the theory is regularized by higher derivatives, because the functions ^d 7 satisfy the NSVZ relation in the case of using this regularization.
ßDR (a) =
a2Nf
a a2 (2 + 3Nf)
1 + - —
n
4n2
+ O(a3) .
Comparing all above expressions one can see that in the considered approximations only terms proportional to (Nf )2a4 in the (-function and to Nf a2 in the anomalous dimension are scheme dependent. The other terms coincide in all schemes.
in the three-loop
6 The NSVZ relation renormalizations
and
finite
Using the higher derivative regularization with Rk = 1 + k2n /A2n one c^ calculate the (-function
Different renormalization prescriptions can be related by finite renormalizations
a ^ a'(a); Z'(a', A/p) = z(a)Z(a, A/p),
(24)
3
n
under which the ¡-function (15) and the anomalous dimension (16) are changed as follows:
~ dm' ~
3' (a') = —(1(a) da
l'(a') =
d ln z da
•3(a)+3(a). (25)
Using these equations one can see [37] that if ¡3(a) and 7(a) satisfy the NSVZ relation, then
a'( da' ((a) =
a2Nf
1 — Y'(a')
n 1 — a2Nf (d ln z/da)/n
a=a(a')
Taking into account that quantum corrections to the
Nf
obtain
a'(a) — a = O(Nf); z(a) = O ((Nf )0)
(26)
Therefore, all scheme-dependent terms in the
( Nf ) 2 ( Nf ) o
dimension are scheme-independent. This means that the NSVZ relation is valid for terms proportional to
( Nf ) 1
to (Nf )a with a > 2 are scheme-dependent.
Similarly making finite renormalizations in the non-Abelian case one can see [38] that in L loops terms proportional to tr (C(R)L) satisfy the NSVZ relation for an arbitrary renormalization prescription. This implies that the NSVZ relation non-trivially constrains the divergences in spite of its scheme-dependence.
7 Conclusion
For Abelian supersymmetric theories, regularized by higher derivatives, the NSVZ ¡-function relates the scheme-independent RG functions defined in terms of the bare coupling constant. The NSVZ relation follows from the fact that the integrals which determine the 0-function defined in terms of the bare coupling constant can be written as integrals of double total derivatives.
For the RG functions defined in terms of the renormalized coupling constant, the NSVZ relation is valid only in a special NSVZ scheme. In the Abelian case the higher derivative regularization enables to give a simple prescription for constructing this scheme in all orders by imposing the boundary conditions (17).
Although the NSVZ relation is scheme-dependent, it has some scheme-independent consequences which non-trivially restrict the divergences.
Acknowledgement
The author is very grateful for collaboration and useful discussions of the material of this talk to A. L. Kataev. The work was supported by the RFBR grant No. 14-01-00695.
References
[1] Novikov V. A., Shifman M. A., Vainshtein A. I. and Zakharov V. I., 1983 Nucí. Phys. B 229 381.
[2] Jones D. R. T„ 1983 Phys. Lett. B 123 45.
[3] Novikov V. A., Shifman M. A., Vainshtein A. I. and Zakharov V. I., 1986 Phys. Lett. B 166 329; Sov. J. Nuel. Phys. 43 294; [Yad. Fiz. 43 459].
[4] Shifman M. A. and Vainshtein A. I., 1986 Nuel. Phys. B 277 456; Sov. Phys. JETP 64 428; [Zh. Eksp. Tear. Fiz. 91 723].
[5] Vainshtein A. I., Zakharov V. I. and Shifman M. A., 1985 JETP Lett. 42 224 [Pisma Zh. Eksp. Tear. Fiz. 42 182].
[6] Shifman M. A., Vainshtein A. I. and Zakharov V. I., 1986 Phys. Lett. B 166 334.
[7] Shifman M. A. and Vainshtein A. I., 1999 In Shifman, M. A.; ITEP lectures on particle physics and field theory, v. 2, 485-647. [hep-th/9902018],
[8] Arkani-Hamed N. and Murayama H., 2000 JHEP 0006 030.
[9] Kraus E„ Rupp C. and Sibold K., 2003 Nuel. Phys. B 661 83.
[10] Siegel W., 1979 Phys. Lett. B 84 193.
[11] Siegel W., 1980 Phys. Lett. B 94 (1980) 37.
[12] Avdeev L. V., Chochia G. A. and Vladimirov A. A., 1981 Phys. Lett. B 105 272.
[13] Avdeev L. V., 1982 Phys. Lett. B 117 317.
[14] Avdeev L. V. and Vladimirov A. A., 1983 Nuel. Phys. B 219 262.
[15] Avdeev L. V. and Tarasov O. V., 1982 Phys. Lett. B 112 356.
TSPU Bulletin. 2014. 12 (153)
[16] Jack I., Jones D. R. T. and North C. G., 1996 Phys. Lett. В 386 138.
[17] Jack I., Jones D. R. T. and North C. G., 1997 Nucl. Phys. В 486 479.
[18] Harlander R. V., Jones D. R. Т., Kant P., Mihaila L. and Steinhauser M., 2006 JEEP 0612 024.
[19] Jack I., Jones D. R. T. and Pickering A., 1998 Phys. Lett. В 435 61.
[20] Slavnov A. A., 1971 Nucl. Phys. В 31 301.
[21] Slavnov A. A., 1972 Theor. Math. Phys. 13 1064 [Tear. Mat. Fiz. 13 174].
[22] Faddeev L. D. and Slavnov A. A., "Gauge Fields. Introduction To Quantum Theory," Nauka, Moscow, 1978 and Front. Phys. 50 (1980) 1 [Front. Phys. 83 (1990) 1].
[23] Slavnov A. A., 1977 Theor. Math. Phys. 33 977 [Tear. Mat. Fiz. 33 210].
[24] Krivoshchekov V. K., 1978 Theor. Math. Phys. 36 745 [Teor. Mat. Fiz. 36 291].
[25] West P. C., 1986 Nucl. Phys. В 268 113.
[26] Krivoshchekov V. K„ 1984 Phys. Lett. В 149 128.
[27] Buchbinder I. L. and Stepanyantz К. V., 2014 Nucl. Phys. В 883 20.
[28] Soloshenko A. A. and Stepanyantz К. V., 2004 Theor. Math. Phys. 140 1264 [Teor. Mat. Fiz. 140 430].
[29] Smilga A. V. and Vainshtein A., 2005 Nucl. Phys. В 704 445.
[30] Stepanyantz К. V., 2011 Nucl. Phys. В 852 71.
[31] Stepanyantz К. V., 2014 JEEP 1408 096.
[32] Bogolyubov N. N. and Shirkov D. V., "Introduction To The Theory Of Quantized Fields," Nauka, Moscow, 1984 [Intersci. Monogr. Phys. Astron. 3 (1959) 1].
[33] Kataev A. L. and Stepanyantz К. V., 2013 Nucl. Phys. В 875 459.
[34] Pimenov А. В., Shevtsova E. S. and Stepanyantz К. V., 2010 Phys. Lett. В 686 293.
[35] Stepanyantz К. V., 2011 Phys. Part. Nucl. Lett. 8 321.
[36] Stepanyantz К. V., 2011 arXiv:1108.1491 [hep-th],
[37] Kataev A. L. and Stepanyantz К. V., 2014 Phys. Lett. В 730 184.
[38] Kataev A. L. and Stepanyantz К. V., 2014 "The NSVZ beta-function in super symmetric theories with different regularizations and renormalization prescriptions," arXiv:1405.7598 [hep-th], Theor.Math.Phys., to appear.
Received Ц.11.20Ц
К. В. Степаньянц
NSVZ СХЕМА И РЕГУЛЯРИЗАЦИЯ ВЫСШИМИ ПРОИЗВОДНЫМИ
Для абелевых N = 1 суперсимметричных теорий, регуляризоваииых высшими производными, NSVZ схема построена во всех порядках для ренормгрупповых функций, определенных в терминах перенормированной константы связи. Для других перенормировочных предписаний исследуются схемно-независимые следствия NSVZ соотношения. Объясняется, почему для ренормгрупповых функций, определенных в терминах голой константы связи, NSVZ соотношение справедливо при произвольных перенормировочных предписаниях в случае использования регуляризации высшими производными.
Ключевые слова: суперсимметрия, перенормировка, в-функция, аномальная размерность.
Степаньянц К. В., кандидат физико-математических наук, доцент. Московский Государственный Университет им. М. В. Ломоносова.
Ленинские горы 1, стр. 2, 119991 Москва, Россия. E-mail: stepan@phys.msu.ru
