Higher Covariant Derivative Regularization Research Articles

The structure of quantum corrections and exact results in supersymmetric theories from the higher covariant derivative regularization

The structure of quantum corrections and exact results in supersymmetric theories from the higher covariant derivative regularization

Three-loop β-functions and two-loop anomalous dimensions for MSSM regularized by higher covariant derivatives in an arbitrary supersymmetric subtraction scheme

Three-loop β-functions of the Minimal Supersymmetric Standard Model regularized by higher covariant derivatives are obtained for an arbitrary supersymmetric subtraction scheme. For this purpose we first calculate two-loop anomalous dimensions for all MSSM chiral matter superfields defined in terms of the bare couplings. Then we use the NSVZ equations for the renormalization group functions defined in terms of the bare couplings, which are valid in all orders in the case of using the higher covariant derivative regularization. This gives the three-loop β-functions defined in terms of the bare couplings. After that, we construct the three-loop β-functions and the two-loop anomalous dimensions standardly defined in terms of the renormalized couplings for an arbitrary subtraction scheme. As a nontrivial correctness test, we verify that for a certain renormalization prescription the general results reproduce the ones obtained earlier in the overline{mathrm{DR}} scheme. Also this can be considered as an indepedent confirmation of the overline{mathrm{DR}} results.

NSVZ $$\boldsymbol{\beta}$$-Function and NSVZ Scheme with the Higher Covariant Derivative Regularization in the Non-Abelian Case

NSVZ $$\boldsymbol{\beta}$$-Function and NSVZ Scheme with the Higher Covariant Derivative Regularization in the Non-Abelian Case

The all-loop perturbative derivation of the NSVZ beta -function and the NSVZ scheme in the non-Abelian case by summing singular contributions

The perturbative all-loop derivation of the NSVZ beta -function for {{mathcal {N}}}=1 supersymmetric gauge theories regularized by higher covariant derivatives is finalized by calculating the sum of singularities produced by quantum superfields. These singularities originate from integrals of double total derivatives and determine all contributions to the beta -function starting from the two-loop approximation. Their sum is expressed in terms of the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields. This allows obtaining the NSVZ equation in the form of a relation between the beta -function and these anomalous dimensions for the renormalization group functions defined in terms of the bare couplings. It holds for an arbitrary renormalization prescription supplementing the higher covariant derivative regularization. For the renormalization group functions defined in terms of the renormalized couplings we prove that in all loops one of the NSVZ schemes is given by the HD + MSL prescription.

NSVZ Relation and NSVZ Scheme in $$\mathcal{N} = 1$$ Non-Abelian Supersymmetric Gauge Theories

We briefly describe the perturbative derivation of the exact NSVZ $$\beta $$ -function for $$\mathcal{N} = 1$$ supersymmetric non-Abelian gauge theories in the case of using the higher covariant derivative regularization. It is demonstrated that with this regularization the NSVZ equation written in the form of a relation between the $$\beta $$ -function and the anomalous dimensions of the quantum superfields is satisfied by the renormalization group functions defined in terms of the bare couplings independently of a renormalization prescription. For the renormalization group functions defined in terms of the renormalized couplings this implies that one of the NSVZ schemes in all loops is given by the HD+MSL prescription, when the theory is regularized by higher derivatives and the renormalization constants contain only powers of $$ln\Lambda {\text{/}}\mu $$ . All these statements are verified by comparing certain terms in the three-loop $$\beta $$ -function with the corresponding contributions to anomalous dimensions of the quantum superfields.

Two-loop renormalization of the matter superfields and finiteness of mathcal{N} = 1 supersymmetric gauge theories regularized by higher derivatives

The two-loop anomalous dimension of the chiral matter superfields is calculated for a general mathcal{N} = 1 supersymmetric gauge theory regularized by higher covariant derivatives. We obtain both the anomalous dimension defined in terms of the bare couplings, and the one defined in terms of the renormalized couplings for an arbitrary renormalization prescription. For the one-loop finite theories we find a simple relation between the higher derivative regulators under which the anomalous dimension defined in terms of the bare couplings vanishes in the considered approximation. In this case the one-loop finite theory is also two-loop finite in the HD+MSL scheme. Using the assumption that with the higher covariant derivative regularization the NSVZ equation is satisfied for RGFs defined in terms of the bare couplings, we construct the expression for the three-loop β-function. Again, the result is written both for the β-function defined in terms of the bare couplings and for the one defined in terms of the renormalized couplings for an arbitrary renormalization prescription.

The Higher Covariant Derivative Regularization as a Tool for Revealing the Structure of Quantum Corrections in Supersymmetric Gauge Theories

We discuss why the Slavnov higher covariant derivative regularization appeared to be an excellent instrument for investigating quantum corrections in supersymmetric gauge theories. For example, it allowed to demonstrate that the $\beta$-function in these theories is given by integrals of double total derivatives and to construct the NSVZ renormalization prescription in all loops. It was also used for deriving the non-renormalization theorem for the triple gauge-ghost vertices. With the help of this theorem the exact NSVZ $\beta$-function was rewritten in a new form, which revealed its perturbative origin. Moreover, in the case of using the higher covariant derivative regularization it is possible to construct a method for obtaining the $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories, which simplifies the calculations in a great extent. This method is illustrated by an explicit two-loop calculation made in the general $\xi$-gauge.

The NSVZ \u03b2-function for theories regularized by higher covariant derivatives: the all-loop sum of matter and ghost singularities

The contributions of the matter superfields and of the Faddeev-Popov ghosts to the β-function of mathcal{N} = 1 supersymmetric gauge theories defined in terms of the bare couplings are calculated in all orders in the case of using the higher covariant derivative regularization. For this purpose we use the recently proved statement that the β-function in these theories is given by integrals of double total derivatives with respect to the loop momenta. These integrals do not vanish due to singularities of the integrands. This implies that the β-function beyond the one-loop approximation is given by the sum of the singular contributions, which is calculated in all orders for singularities produced by the matter superfields and by the Faddeev-Popov ghosts. The result is expressed in terms of the anomalous dimensions of these superfields. It coincides with the corresponding part of the new form of the NSVZ equation, which can be reduced to the original one with the help of the non-renormalization theorem for the triple gauge-ghost vertices.

The three-loop Adler D-function for mathcal{N}=1 SQCD regularized by dimensional reduction

The three-loop Adler D-function for mathcal{N}=1 SQCD in the overline{mathrm{DR}} scheme is calculated starting from the three-loop result recently obtained with the higher covariant derivative regularization. For this purpose, for the theory regularized by higher derivatives we find a subtraction scheme in which the Green functions coincide with the ones obtained with the dimensional reduction and the modified minimal subtraction prescription for the renormalization of the SQCD coupling constant and of the matter superfields. Also we calculate the D-function in the overline{mathrm{DR}} scheme for all renormalization constants (including the one for the electromagnetic coupling constant which appears due to the SQCD corrections). It is shown that the results do not satisfy the NSVZ-like equation relating the D-function to the anomalous dimension of the matter superfields. However, the NSVZ-like scheme can be constructed with the help of a properly tuned finite renormalization. It is also demonstrated that the three-loop D-function defined in terms of the bare couplings with the dimensional reduction does not satisfy the NSVZ-like equation for an arbitrary renormalization prescription. We also investigate a possibility to present the results in the form of the β-expansion and the scheme dependence of this expansion.

Two-loop renormalization of the Faddeev-Popov ghosts in mathcal{N}=1 supersymmetric gauge theories regularized by higher derivatives

For the general renormalizable mathcal{N}=1 supersymmetric gauge theory we investigate renormalization of the Faddeev-Popov ghosts using the higher covariant derivative regularization. First, we find the two-loop anomalous dimension defined in terms of the bare coupling constant in the general ξ-gauge. It is demonstrated that for doing this calculation one should take into account that the quantum gauge superfield is renormalized in a nonlinear way. Next, we obtain the two-loop anomalous dimension of the Faddeev-Popov ghosts defined in terms of the renormalized coupling constant and examine its dependence on the subtraction scheme.

New form of the exact NSVZ \u03b2-function: the three-loop verification for terms containing Yukawa couplings

We investigate a recently proposed new form of the exact NSVZ β-function, which relates the β-function to the anomalous dimensions of the quantum gauge superfield, of the Faddeev-Popov ghosts, and of the chiral matter superfields. Namely, for the general renormalizable mathcal{N} = 1 supersymmetric gauge theory, regularized by higher covariant derivatives, the sum of all three-loop contributions to the β-function containing the Yukawa couplings is compared with the corresponding two-loop contributions to the anomalous dimensions of the quantum superfields. It is demonstrated that for the considered terms both new and original forms of the NSVZ relation are valid independently of the subtraction scheme if the renormalization group functions are defined in terms of the bare couplings. This result is obtained from the equality relating the loop integrals, which, in turn, follows from the factorization of the integrals for the β-function into integrals of double total derivatives. For the renormalization group functions defined in terms of the renormalized couplings we verify that the NSVZ scheme is obtained with the higher covariant derivative regularization supplemented by the subtraction scheme in which only powers of ln Λ/μ are included into the renormalization constants.

The Adler D-function for [formula omitted] SQCD regularized by higher covariant derivatives in the three-loop approximation

We calculate the Adler D-function for N=1 SQCD in the three-loop approximation using the higher covariant derivative regularization and the NSVZ-like subtraction scheme. The recently formulated all-order relation between the Adler function and the anomalous dimension of the matter superfields defined in terms of the bare coupling constant is first considered and generalized to the case of an arbitrary representation for the chiral matter superfields. The correctness of this all-order relation is explicitly verified at the three-loop level. The special renormalization scheme in which this all-order relation remains valid for the D-function and the anomalous dimension defined in terms of the renormalized coupling constant is constructed in the case of using the higher derivative regularization. The analytic expression for the Adler function for N=1 SQCD is found in this scheme to the order O(αs2). The problem of scheme-dependence of the D-function and the NSVZ-like equation is briefly discussed.

Three-loop NSVZ relation for terms quartic in the Yukawa couplings with the higher covariant derivative regularization

We demonstrate that in non-Abelian N=1 supersymmetric gauge theories the NSVZ relation is valid for terms quartic in the Yukawa couplings independently of the subtraction scheme if the renormalization group functions are defined in terms of the bare couplings and the theory is regularized by higher covariant derivatives. The terms quartic in the Yukawa couplings appear in the three-loop β-function and in the two-loop anomalous dimension of the matter superfields. We have obtained that the three-loop contribution to the β-function quartic in the Yukawa couplings is given by an integral of double total derivatives. Consequently, one of the loop integrals can be taken and the three-loop contribution to the β-function is reduced to the two-loop contribution to the anomalous dimension. The remaining loop integrals have been calculated for the simplest form of the higher derivative regularizing term. Then we construct the renormalization group functions defined in terms of the renormalized couplings. In the considered approximation they do not satisfy the NSVZ relation for a general renormalization prescription. However, we verify that the recently proposed boundary conditions defining the NSVZ scheme in the non-Abelian case really lead to the NSVZ relation between the terms of the considered structure.

Derivation of the exact expression for theDfunction inN=1SQCD

We discuss various details of our derivation of the exact expression for the Adler D function in N=1 supersymmetric QCD (SQCD). This exact formula relates the D function to anomalous dimensions of the matter superfields. Our perturbative derivation refers to the D function defined in terms of the bare coupling constant in the case of using the higher covariant derivative regularization. The exact expression for this function is obtained by direct summation of supergraphs to all orders in the non-Abelian coupling constant. As we argued previously, our formula should be valid beyond perturbation theory too. The perturbative result we present here coincides with the general formula order by order. We discuss consequences for N=1 SQCD in the conformal window. It is noted that our exact relation can allow one to determine the (infrared) critical anomalous dimension of the Seiberg M field present in the dual theory.

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.

Quantum corrections in N = 1 supersymmetric theories with cubic superpotential, regularized by higher covariant derivatives

Using the higher covariant derivative regularization we calculate a two-loop β-function for the N = 1 supersymmetric Yang-Mills theory with the matter superfields, containing the cubic superpotential. It is found that all integrals, defining this function, are integrals of total derivatives.

Calculation of two-loop β-function for general [formula omitted] supersymmetric Yang–Mills theory with the higher covariant derivative regularization

For the general renormalizable N=1 supersymmetric Yang–Mills theory, regularized by higher covariant derivatives, a two-loop β-function is calculated. It is shown that all integrals, needed for obtaining this function, can be easily calculated, because they are integrals of total derivatives.