NSVZ Β-function Research Articles

A class of the NSVZ renormalization schemes for [formula omitted] SQED

For the N=1 supersymmetric electrodynamics we investigate renormalization schemes in which the NSVZ equation relating the β-function to the anomalous dimension of the matter superfields is valid in all loops. We demonstrate that there is an infinite set of such schemes. They are related by finite renormalizations which form a group and are parameterized by one finite function and one arbitrary constant. This implies that the NSVZ β-function remains unbroken if the finite renormalization of the coupling constant is related to the finite renormalization of the matter superfields by a special equation derived in this paper. The arbitrary constant corresponds to the arbitrariness of choosing the renormalization point. The results are illustrated by explicit calculations in the three-loop approximation.

Quantum properties of supersymmetric theories regularized by higher covariant derivatives

We investigate quantum corrections in = 1 non-Abelian supersymmetric gauge theories, regularized by higher covariant derivatives. In particular, by the help of the Slavnov–Taylor identities we prove that the vertices with two ghost legs and one leg of the quantum gauge superfield are finite in all orders. This non-renormalization theorem is confirmed by an explicit one-loop calculation. By the help of this theorem we rewrite the exact NSVZ β-function in the form of the relation between the β-function and the anomalous dimensions of the matter superfields, of the quantum gauge superfield, and of the Faddeev–Popov ghosts. Such a relation has simple qualitative interpretation and allows suggesting a prescription producing the NSVZ scheme in all loops for the theories regularized by higher derivatives. This prescription is verified by the explicit three-loop calculation for the terms quartic in the Yukawa couplings.

Supersymmetry, quantum corrections, and the higher derivative regularization

We investigate the structure of quantum corrections in N = 1 supersymmetric theories using the higher covariant derivative method for regularization. In particular, we discuss the non-renormalization theorem for the triple gauge-ghost vertices and its connection with the exact NSVZ β-function. Namely, using the finiteness of the triple gauge-ghost vertices we rewrite the NSVZ equation in a form of a relation between the β-function and the anomalous dimensions of the quantum gauge superfield, of the Faddeev-Popov ghosts, and of the matter superfields. We argue that it is this form that follows from the perturbative calculations, and give a simple prescription how to construct the NSVZ scheme in the non-Abelian case. These statements are confirmed by an explicit calculation of the three-loop contributions to the β-function containing Yukawa couplings. Moreover, we calculate the two-loop anomalous dimension of the ghost superfields and demonstrate that for doing this calculation it is very important that the quantum gauge superfield is renormalized non-linearly.

New form of the NSVZ relation at the two-loop level

Recently the exact NSVZ β-function was rewritten in the form of a relation between the β-function and the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields. It was also suggested that this form of the NSVZ equation follows from an underlying equation relating two-point Green functions of the theory. Here we demonstrate that this relation is satisfied at the two-loop level for the non-Abelian N=1 supersymmetric gauge theories in the case of using the simplest (BRST non-invariant) version of the higher covariant derivative regularization. Consequently, the integrals giving the two-loop β-function can be reduced to the one-loop integrals giving the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields.

Non-renormalization of the [formula omitted]-vertices in [formula omitted] supersymmetric theories

Using the Slavnov–Taylor identities we prove that the three-point ghost vertices with a single line of the quantum gauge superfield are not renormalized in all loops in N=1 supersymmetric gauge theories. This statement is verified by the explicit one-loop calculation made by the help of the BRST invariant version of the higher covariant derivative regularization. Using the restrictions to the renormalization constants which are imposed by the non-renormalization of the considered vertices we express the exact NSVZ β-function in terms of the anomalous dimensions of the Faddeev–Popov ghosts and of the quantum gauge superfield. In the expression for the NSVZ β-function obtained in this way the contributions of the Faddeev–Popov ghosts and of the matter superfields have the same structure.

Relation between two-point Green’s functions of $$\mathcal{N} = 1$$ SQED with N f flavors, regularized by higher derivatives, in the three-loop approximation

We verify the identity which relates the two-point Green’s functions of \(\mathcal{N} = 1\) SQED with N f flavors, regularized by higher derivatives, by explicit calculations in the three-loop approximation. This identity explains why, in the limit of the vanishing external momentum, the two-point Green’s function of the gauge superfield is given by integrals of double total derivatives in the momentum space. It also makes it possible to exactly derive the NSVZ β-function in all loops if the renormalization group functions are defined in terms of the bare coupling constant. In order to verify the considered identity, we use it to construct integrals giving the three-loop β-function starting from the two-point Green’s functions of matter superfields in the two-loop approximation. Then we demonstrate that the results for these integrals coincide with the sums of the corresponding three-loop supergraphs.

The NSVZ β-function in supersymmetric theories with different regularizations and renormalization prescriptions

We briefly review the calculations of quantum corrections related with the exact NSVZ $\beta$-function in ${\cal N}=1$ supersymmetric theories, paying especial attention to the scheme dependence of the results. It is explained, how the NSVZ relation is obtained for the renormalization group functions defined in terms of the bare coupling constant if a theory is regularized by higher derivatives. Also we describe, how to construct a special renormalization prescription which gives the NSVZ relation for the renormalization group functions defined in terms of the renormalized coupling constant exactly in all orders for Abelian supersymmetric theories, regularized by higher derivatives. The scheme dependence of the NSVZ $\beta$-function (for the renormalization group functions defined in terms of the renormalized coupling constant) is discussed in the non-Abelian case. It is shown that in this case the NSVZ $\beta$-function leads to a certain scheme-independent equality.

The NSVZ β-function and the Schwinger-Dyson equations for N $$ \mathcal{N} $$ = 1 SQED with N f flavors, regularized by higher derivatives

The effective diagram technique based on the Schwinger-Dyson equations is constructed for N=1 SQED with N_f flavors, regularized by higher derivatives. Using these effective diagrams, it is possible to derive the exact NSVZ relation between the beta-function and the anomalous dimension of the matter superfields exactly in all loops, if the renormalization group functions are defined in terms of the bare coupling constant. In particular, we verify that all integrals which give the beta-function defined in terms of the bare coupling constant are integrals of double total derivatives and prove some identities relating Green functions.

The higher derivative regularization and quantum corrections in [formula omitted] supersymmetric theories

We construct a new version of the higher covariant derivative regularization for a general N=2 supersymmetric gauge theory formulated in terms of N=1 superfields. This regularization preserves both supersymmetries of the classical action, namely, the invariance under the manifest N=1 supersymmetry and under the second hidden on-shell supersymmetry. The regularizing N=2 supersymmetric higher derivative term is found in the explicit form in terms of N=1 superfields. Thus, N=2 supersymmetry is broken only by the gauge fixing procedure. Then we analyze the exact NSVZ β-function and prove that in the considered model its higher loop structure is determined by the anomalous dimension of the chiral superfield Φ in the adjoint representation which is the N=2 superpartner of the gauge superfield V. Using the background field method we find that this anomalous dimension is related with the anomalous dimension of the hypermultiplet and vanishes if the effective action is invariant under N=2 background supersymmetry. As a consequence, in this case the higher loop contributions to β-function also vanish. The one-loop renormalization structure in the considered regularization is also studied by the explicit calculations of the one-loop renormalization constants.

Scheme independent consequence of the NSVZ relation for [formula omitted] SQED with [formula omitted] flavors

The exact NSVZ β-function is obtained for N=1 SQED with Nf flavors in all orders of the perturbation theory, if the renormalization group functions are defined in terms of the bare coupling constant and the theory is regularized by higher derivatives. However, if the renormalization group functions are defined in terms of the renormalized coupling constant, the NSVZ relation between the β-function and the anomalous dimension of the matter superfields is valid only in a certain (NSVZ) scheme. We prove that for N=1 SQED with Nf flavors the NSVZ relation is valid for the terms proportional to (Nf)1 in an arbitrary subtraction scheme, while the terms proportional to (Nf)k with k⩾2 are scheme dependent. These results are verified by an explicit calculation of a three-loop β-function and a two-loop anomalous dimension made with the higher derivative regularization in the NSVZ and MOM subtraction schemes. In this approximation it is verified that in the MOM subtraction scheme the renormalization group functions obtained with the higher derivative regularization and with the dimensional reduction coincide.

Multiloop calculations in supersymmetric theories with the higher covariant derivative regularization

Most calculations of quantum corrections in supersymmetric theories are made with the dimensional reduction, which is a modification of the dimensional regularization. However, it is well known that the dimensional reduction is not self-consistent. A consistent regularization, which does not break the supersymmetry, is the higher covariant derivative regularization. However, the integrals obtained with this regularization can not be usually calculated analytically. We discuss application of this regularization to the calculations in supersymmetric theories. In particular, it is demonstrated that integrals defining the β-function are possibly integrals of total derivatives. This feature allows to explain the origin of the exact NSVZ β-function, relating the β-function with the anomalous dimensions of the matter superfields. However, integrals for the anomalous dimension should be calculated numerically.

Derivation of the exact NSVZ β-function in N = 1 SQED regularized by higher derivatives by summation of Feynman diagrams

We describe some calculations, which demonstrate that the β-function in N = 1 supersymmetric theories, regularized by higher covariant derivatives, is given by integrals of double total derivatives. This allows to calculate one of the loop integrals analytically and obtain the NSVZ β-function in the considered approximation. For the N = 1 supersymmetric electrodynamics the factorization of integrals into integrals of double total derivatives can be proved in all orders using a special technique. As a result, the exact NSVZ β-function is obtained by explicit summation of Feynman diagrams without a redefinition of the coupling constant.

Derivation of the exact NSVZ β-function in [formula omitted] SQED, regularized by higher derivatives, by direct summation of Feynman diagrams

For N = 1 supersymmetric quantum electrodynamics, regularized by higher derivatives, a method for summation of all Feynman diagrams defining the β-function is presented. Using this method we prove that the β-function is given by an integral of a total derivative, which can be easily calculated. It is shown that surviving terms give the exact NSVZ β-function. The results are compared with the explicit three-loop calculation.

A new relation restricting the Green functions of N = 1 supersymmetric electrodynamics

For the N = 1 supersymmetric electrodynamics it is shown that if the Gell-Mann-Low function coincides with the exact Novikov-Shifman-Vainstein-Zakharov β-function (NSVZ β-function), then the sum of some effective diagrams with vertices of three and four points is zero.

\U0001d4a9 = 1 super Yang–Mills theories and wrapped branes

I consider supergravity solutions of D5 branes wrapped on supersymmetric 2-cycles and use them to discuss relevant features of four-dimensional \U0001d4a9 = 1 super Yang–Mills theories with gauge group SU(N). In particular, using a gravitational dual of the gaugino condensate, it is shown that it is possible to obtain the complete NSVZ β-function. It is also described how different aspects of the gauge theory are nicely encoded in this supergravity solution.

Formula omitted] and [formula omitted] super-Yang–Mills theories from wrapped branes

We consider supergravity solutions of D5-branes wrapped on supersymmetric 2-cycles and use them to discuss relevant features of four-dimensional N=1 and N=2 super-Yang–Mills theories with gauge group SU( N). In particular in the N=1 case, using a gravitational dual of the gaugino condensate, we obtain the complete NSVZ β-function. We also find non-perturbative corrections associated to fractional instantons with charge 2/ N. These non-perturbative effects modify the running of the coupling constant which remains finite even at small scales in a way that resembles to the soft confinement scenario of QCD.

Probing the Infrared Structure of Gauge Theories: A Padé-Approximant Approach

Padé-approximant treatments of the known terms of the QCD β-function are seen to develop possible infrared fixed point structure only if the number of fermion flavours is sufficiently large. This flavour threshold is seen to be between six and nine flavours, depending upon both the specific choice of approximant as well as on the presently-unknown five-loop β-function contribution. Below this flavour threshold, Padé approximants based upon the QCD β-function manifest the same infrared attractor structure as that which characterizes the exact NSVZ β-function of supersymmetric gluodynamics. Such infrared attractor structure is also seen to characterize Padé-approximant treatments of vector SU(N) gauge theory in the large N limit, suggesting common infrared dynamics for the strong and weak phases of this theory.

Infrared consistency of NSVZ and DRED supersymmetricgluodynamics

Padé approximant methods are applied to the known terms of the dimensional reduction (DRED)β-function for N = 1 supersymmetric SU(3) Yang-Mills theory.Each of the [N|M] approximants with N + M⩽4, M≠0constructed from this series exhibits a positive pole which precedes anypositive zeros of the approximant, consistent with the sameinfrared-attractor pole behaviour known to characterize the exact NSVZβ-function. A similar Padé-approximant analysis of truncations ofthe NSVZ series is shown to reproduceconsistently the geometric-seriespole of the exact NSVZ β-function.

N = 1 supersymmetry and the three-loop gauge β-function

We calculate the three-loop gauge β-function for an abelian N = 1 supersymmetric gauge theory, using DRED. We construct a coupling constant redefinition that relates the result to the corresponding term in the NSVZ β-function, and by generalising this redefinition to the non-abelian case we derive the DRED three-loop gauge β-function for the non-abelian case.