For a general ${\cal N}=1$ supersymmetric gauge theory regularized by higher covariant derivatives we prove in all orders that the $β$-function defined in terms of the bare couplings is given by integrals of double total derivatives with respect to loop momenta. With the help of the technique used for this proof it is possible to construct a method for obtaining these loop integrals, which essentially simplifies the calculations. As an illustration of this method, we find the expression for the three-loop contribution to the $β$-function containing the Yukawa couplings and compare it with the result of the standard calculations made earlier. Also we briefly discuss, how the structure of the loop integrals for the $β$-function considered in this paper can be used for the all-loop perturbative derivation of the NSVZ relation in the non-Abelian case.
Two-loop Gell-Mann-Low function is calculated for N=1 supersymmetric Yang-Mills theory, regularized by higher covariant derivatives. The integrals, which define it, are shown to be reduced to total derivatives and can be easily calculated analytically.
Contribution of matter fields to the Gell-Mann-Low function for N=1 supersymmetric Yang-Mills theory, regularized by higher covariant derivatives, is obtained using Schwinger-Dyson equations and Slavnov-Tailor identities. A possible deviation of the result from the corresponding contribution in the exact Novikov, Shifman, Vainshtein and Zakharov $β$-function is discussed.
The integrals defining the two-loop beta-function for the general renormalizable N=1 supersymmetric Yang--Mills theory, regularized by higher covariant derivatives, are investigated. It is shown that they are given by integrals of double total derivatives. These integrals are not equal to zero due to appearing of delta-functions. These delta-functions allow to reduce the two-loop integrals to one-loop integrals, which can be easily calculated. The result agrees with the exact NSVZ beta-function and calculations made by different methods.
General expression for three-loop $β$-functions of $\cal N =$ 1 supersymmetric theories with multiple gauge couplings regularized by higher covariant derivatives
The contributions of the matter superfields and of the Faddeev--Popov ghosts to the $β$-function of ${\cal N}=1$ 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 …
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 ba…
For the general renormalizable N=1 supersymmetric Yang--Mills theory, regularized by higher covariant derivatives, a two-loop beta-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.
We calculate the Adler $D$-function for ${\cal N}=1$ 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…
The regularization scheme is proposed for the constrained Hamiltonian formulation of the gauge fields coupled to the chiral or axial fermions. The Schwinger terms in the regularized operator first-class constraint algebra are shown to be consistent with the covariant divergence anomaly of the corresponding current. Regularized quantum master equations are studied, and the Schwinger terms are found out to break down both nilpotency of the BRST-charge and its conservation law. Wess-Zumino consistency conditions are studied for the BRST anomaly and they are shown to contradict to the covariant Sc…
Structure of quantum corrections in ${\cal N}=1$ supersymmetric gauge theories is investigated in the case of using the regularization by higher covariant derivatives. It is demonstrated that this regularization allows revealing some interesting features which lead to the exact relations between the renormalization group functions. In particular, the NSVZ equation, which relates the $β$-function to the anomalous dimension of the matter superfields, naturally appears in this case. We briefly review the all-loop derivation of this equation and the construction of a simple renormalization prescri…
We consider a general non-Abelian renormalizable ${\cal N}=1$ supersymmetric gauge theory, regularized by higher covariant derivatives without breaking the BRST invariance, and calculate one-loop divergences for a general form of higher derivative regulator and of the gauge fixing term. It is demonstrated that the momentum integrals giving the one-loop $β$-function are integrals of double total derivatives independently of a particular choice of the higher derivative term. Evaluating them we reproduce the well-known result for the one-loop $β$-function. Also we find that the three-point ghost …
For the general renormalizable ${\cal N}=1$ 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 examin…
We discuss a covariant functional integral approach to the quantization of the bosonic string. In contrast to approaches relying on non-covariant operator regularizations, interesting operators here are true tensor objects with classical transformation laws, even on target spaces where the theory has a Weyl anomaly. Since no implicit non-covariant gauge choices are involved in the definition of the operators, the anomaly is clearly separated from the issue of operator renormalization and can be understood in isolation, instead of infecting the latter as in other approaches. Our method is of wi…
The two-loop anomalous dimension of the chiral matter superfields is calculated for a general ${\cal N}=1$ 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 o…
We consider the general ${\cal N}=1$ supersymmetric gauge theory with matter, regularized by higher covariant derivatives without breaking the BRST invariance, in the massless limit. In the $ξ$-gauge we obtain the (unrenormalized) expression for the two-point Green function of the quantum gauge superfield in the one-loop approximation as a sum of integrals over the loop momentum. The result is presented as a sum of three parts: the first one corresponds to the pure supersymmetric Yang--Mills theory in the Feynman gauge, the second one contains all gauge dependent terms, and the third one is th…
The determinant and higher loop terms, usually treated with the Pauli-Villars and higher covariant derivatives methods, in the background field method in 4 dimensions can hardly be regularized simultaneously. At the same time we observe that introduction of a scalar multiplier at the quadratic form, which is equivalent to a change of the measure in the functional integral, influences only the determinant part of the effective action. This allows one to choose the integration measure and the function in the regularized propagator in such a way as to make all terms in the expansion finite.
For a general renormalizable ${\cal N}=1$ supersymmetric gauge theory with a simple gauge group we verify the ultraviolet (UV) finiteness of the two-loop matter contribution to the triple gauge-ghost vertices. These vertices have one leg of the quantum gauge superfield and two legs corresponding to the Faddeev--Popov ghost and antighost. By an explicit calculation made with the help of the higher covariant derivative regularization we demonstrate that the sum of the corresponding two-loop supergraphs containing a matter loop is not UV divergent in the case of using a general $ξ$-gauge. In the …
We demonstrate that in non-Abelian ${\cal N}=1$ 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 inte…
The propagators of the Faddeev-Popov (FP) ghosts for Yang-Mills theories and perturbative quantum gravity in the covariant gauge are infrared (IR) divergent in de Sitter spacetime. We point out, however, that the modes responsible for these divergences will not contribute to loop diagrams in computations of time-ordered products in either Yang-Mills theories or perturbative quantum gravity. Therefore we propose that the IR divergent FP-ghost propagator should be regularized by a small mass term that is sent to zero in the end of any perturbative calculations. This proposal is equivalent to usi…
Regularization of Gauss-Bonnet Gravity in Riemann-Cartan Geometry
School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（基础物理与数学科学学院，杭州高等研究院，UCAS）；University of Chinese Academy of Sciences（中国科学院大学）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University（湖南师范大学物理与量子效应及应用协同创新中心）
In this article, various approaches to calculate covariant expressions for the bilinears of Dirac spinors are presented. For this purpose, algebraic equations defining Dirac spinors are discussed. Following that, a covariant approach for spacetime parameterization is presented and the equations defining Dirac spinors are written fully in terms of Lorentz scalars. After presenting how the tensorial bilinears can be reduced to combinations of scalar bilinears with appropriate Lorentz structures, a covariant recipe for the calculation of scalar bilinears is provided.
We calculate the three-loop Adler $D$-function of ${\cal N}=1$ SQCD regularized by higher covariant derivatives and find the subtraction scheme in which the exact NSVZ-like relation for this function that was recently proposed is valid.
Motivated by string theory connection, a covariant procedure for perturbative calculation of the partition function of the two-dimensional generalized $σ$-model is considered. The importance of a consistent regularization of the measure in the path integral is emphasized. The partition function is computed for a number of specific 2-manifolds: sphere, disk and torus.
Some of the basic issues related to the regularization and anomalies in gauge theory are reviewed, with particular emphasis on the recent development in lattice gauge theory. The generalized Pauli-Villars regularization is discussed from a view point of the covariant regularization of currents, and the construction of a regularized effective action in terms of covariant currents is compared with the lattice formulation of chiral Abelian theory.
Based on the covariant background field method, we calculate the ultraviolet counter\-terms up to two-loop order and discuss the renormalizability of the three-dimensional non-linear sigma models with arbitrary Riemannian manifolds as target spaces. We investigate the bosonic model and its supersymmetric extension. We show that at the one-loop level these models are renormalizable and even finite when the manifolds are Ricci-flat. However, at the two-loop order, we find non-renormalizable counterterms in all cases considered, so the renormalizability and finiteness of such models are completel…
A simplified version of Higher Covariant Derivative regularization for Yang-Mills theory is constructed. This may make Higher Covariant Derivative method more attractive for practical calculations.
It is widely accepted that the Feynman integral is one of the most promising methodologies for defining a generally covariant formulation of nonperturbative interacting quantum field theories (QFTs) without a fixed prearranged causal background. Recent literature suggests that if the spacetime metric is not fixed, e.g. because it is to be quantized along with the other fields, one may not be able to avoid considering the Feynman integral in the original Lorentz signature, without Wick rotation. Several mathematical phenomena are known, however, which are at some point showstoppers to a mathema…
A novel technique based on Schwinger's proper time method is applied to the Casimir problem of the M.I.T. bag model. Calculations of the regularized vacuum energies of massless scalar and Dirac spinor fields confined to a static and spherical cavity are presented in a consistent manner. While our results agree partly with previous calculations based on asymptotic methods, the main advantage of our technique is that the numerical errors are under control. Interpreting the bag constant as a vacuum expectation value, we investigate potential cancellations of boundary divergences between the canon…
The inverse time ordered Green's function is defined in a covariant three-dimensional formalism using the diagram approach.The equation of motion of a two-quark composite system in an external electromagnetic field up to the second order in the field strength is obtained. The polarizability of the $π$-meson is calculated using this formalism. The mean square radius of the two-quark system is calculated from the wave functions of the system. In addition, radiative corrections are calculated for both the polarizability and the mean square radius.
For N=1 supersymmetric quantum electrodynamics, regularized by higher derivatives, a method for summation of all Feynman diagrams defining the beta-function is presented. Using this method we prove that the beta-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 beta-function. The results are compared with the explicit three-loop calculation.
Canonical forms are given for the nilpotent BRS operator $\d$ and the covariant `loop space' derivative ${\cal D}_{\m}$ for the p-brane fields for all odd p. The defining characteristic of ${\cal D}_{\m}$ is that it is a functional derivative operator which generalizes the ordinary functional derivative and also commutes with $\d$. Methods of construction for the canonical forms are discussed.
The perturbative all-loop derivation of the NSVZ $β$-function for ${\cal N}=1$ 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 $β$-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 case of using the higher derivative regularization we construct the subtraction scheme which gives the NSVZ-like relation for the anomalous dimension of the photino mass in softly broken ${\cal N}=1$ SQED with $N_f$ flavors in all loops. The corresponding renormalization prescription is determined by simple boundary conditions imposed on the renormalization constants. It allows fixing an arbitrariness of choosing finite counterterms in every order of the perturbation theory in such a way that the renormalization group functions defined in terms of the renormalized coupling constant sati…
We consider the softly broken ${\cal N}=1$ supersymmetric electrodynamics, regularized by higher derivatives. For this theory we demonstrate that the renormalization of the photino mass is determined by integrals of double total derivatives in the momentum space in all orders. Consequently, it is possible to derive the NSVZ-like exact relation between the photino mass anomalous dimension and the anomalous dimension of the matter superfields in the rigid theory by direct summation of supergraphs. It is important that both these renormalization group functions are defined in terms of the bare co…
Three-loop quantum corrections to the effective action are calculated for N=1 supersymmetric electrodynamics, regularized by higher derivatives. Using the obtained results we investigate the anomaly puzzle in the considered model.
We present a formulation of the coupling of vector multiplets to N=2 supergravity which is symplectic covariant (and thus is not based on a prepotential) and uses superconformal tensor calculus. We do not start from an action, but from the combination of the generalised Bianchi identities of the vector multiplets in superspace, a symplectic definition of special Kahler geometry, and the supersymmetric partners of the corresponding constraints. These involve the breaking to super-Poincare symmetry, and lead to on-shell vector multiplets. This symplectic approach gives the framework to formulate…
We demonstrate the effectivity of the covariant background field method by means of an explicit calculation of the 3-loop beta-function for a pure Yang-Mills theory. To maintain manifest background invariance throughout our calculation, we stay in coordinate space and treat the background field non-perturbatively. In this way the presence of a background field does not increase the number of vertices and leads to a relatively small number of vacuum graphs in the effective action. Restricting to a covariantly constant background field in Fock-Schwinger gauge permits explicit expansion of all qu…
We verify the identity which relates the two-point Green functions of ${\cal N}=1$ 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 function of the gauge superfield is given by integrals of double total derivatives in the momentum space. It also allows to derive the NSVZ $β$-function exactly in all loops if the renormalization group functions are defined in terms of the bare coupling constant. In order to verify the considered ident…
We present the covariant BRST quantization of the super-D-string. The non-vanishing supersymmetric U(1) field strength ${\cal F}$ is essential for the covariant quantization of the super-D-string as well as for its static picture. A SO(2) parameter parametrizes a family of local supersymmetric (kappa symmetric) systems including the super-D-string with ${\cal F}\ne 0$ and the Green-Schwarz superstring with ${\cal F}= 0$. We suggest that $E^1$ (canonical conjugate of U(1) gauge field) plays a role of the order parameter in the Green-Schwarz formalism: the super-D-string exists for $E^1 \ne 0$ w…
In a recent paper, the BRST formalism for the gauge-fixed N=2 twistor-string was used to calculate Green-Schwarz supersring scattering amplitudes with an arbitrary number of loops and external massless states. Although the gauge-fixing procedure preserved the worldsheet N=2 superconformal invariance of the twistor-string, it broke the target-space SO(9,1) super-Poincaré invariance down to an SU(4)xU(1) subgroup. In this paper, generators for the SO(9,1) super-Poincaré transformations, as well as manifestly covariant vertex operators, are explicitly constructed out of the gauge-fixed matter fie…
Some calculations in supersymmetric theories, made with the higher derivative regularization, show that the beta-function is given by integrals of total derivatives. This is qualitatively explained for the N=1 supersymmetric electrodynamics in all orders.
The three-loop Adler $D$-function for ${\cal N}=1$ SQCD in the $\overline{\mbox{DR}}$ 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{\mbox{DR}}…
An extension of the non-local regularization scheme is formulated in the Sp(2) symmetric Lagrangian BRST quantization framework. It generates a systematic treatment of the anomalous quantum master equations and allows to substract the divergences as well as to calculate genuine higher loop BRST and anti-BRST anomalies.
An exact integral expression is found for the amplitude of a Bosonic string with ends separated by a fixed distance $R$ evolving over a time $T$ between arbitrary initial and final configurations. It is impossible to make a covariant subtraction of a covariant quantity which would render the amplitude non-zero. It is suggested that this fact (and not the tachyon) is responsible for the lack of a continuum limit of regularized random-surface models with target-space dimension greater than one. It appears consistent, however, to remove this quantity by hand. The static potential of Alvarez and A…
Some recent research of quantum corrections in ${\cal N}=1$ supersymmetric theories is briefly reviewed. The most attention is paid to the theories regularized by higher covariant derivatives. In particular, we discuss, how the NSVZ and NSVZ-like relations appear with this regularization and how one can construct the NSVZ scheme in all orders.
We continue the development of the effective covariant methods for calculating the heat kernel and the one-loop effective action in quantum field theory and quantum gravity. The status of the low-energy approximation in quantum gauge theories and quantum gravity is discussed in detail on the basis of analyzing the local Schwinger - De Witt expansion. It is argued that the low-energy limit, when defined in a covariant way, should be related to background fields with covariantly constant curvature, gauge field strength and potential. Some new approaches for calculating the low-energy heat kernel…
A covariant calculus for the construction of effective string theories is developed. Effective string theory, describing quantum string-like excitations in arbitrary dimension, has in the past been constructed using the principles of conformal field theory, but not in a systematic way. Using the freedom of choice of field definition, a particular field definition is made in a systematic way to allow an explicit construction of effective string theories with manifest exact conformal symmetry. The impossibility of a manifestly invariant description of the Polchinski-Strominger Lagrangian is demo…
This paper is an overview on our recent results in the calculation of the heat kernel in quantum field theory and quantum gravity. We introduce a deformation of the background fields (including the metric of a curved spacetime manifold) and study various asymptotic expansions of the heat kernel diagonal associated with this deformation. Especial attention is payed to the low-energy approximation corresponding to the strong slowly varying background fields. We develop a new covariant purely algebraic approach for calculating the heat kernel diagonal in low-energy approximation by taking into ac…
We describe an analytic continuation of the Euclidean Grosse-Wulkenhaar and LSZ models which defines a one-parameter family of duality covariant noncommutative field theories interpolating between Euclidean and Minkowski space versions of these models, and provides an alternative regularization to the usual Feynman prescription. This regularization allows for a matrix model representation of the field theories in terms of a complex generalization of the usual basis of Landau wavefunctions. The corresponding propagators are calculated and identified with the Feynman propagators of the field the…
Using our recently proposed covariant algebraic approach the heat kernel for a Laplace-like differential operator in low-energy approximation is studied. Neglecting all the covariant derivatives of the gauge field strength (Yang-Mills curvature) and the covariant derivatives of the potential term of third order and higher a closed formula for the heat kernel as well as its diagonal is obtained. Explicit formulas for the coefficients of the asymptotic expansion of the heat kernel diagonal in terms of the Yang-Mills curvature, the potential term and its first two covariant derivatives are obtain…
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 th…
We present a diagrammatic formulation of recently-revived covariant functional approaches to one-loop matching from an ultraviolet (UV) theory to a low-energy effective field theory. Various terms following from a covariant derivative expansion (CDE) are represented by diagrams which, unlike conventional Feynman diagrams, involve gauge-covariant quantities and are thus dubbed "covariant diagrams." The use of covariant diagrams helps organize and simplify one-loop matching calculations, which we illustrate with examples. Of particular interest is the derivation of UV model-independent universal…
We use our recently proposed algebraic approach for calculating the heat kernel associated with the Laplace operator to calculate the one-loop effective action in the non-Abelian gauge theory. We consider the most general case of arbitrary space-time dimension, arbitrary compact simple gauge group and arbitrary matter and assume a covariantly constant gauge field strength of the most general form, having many independent color and space-time invariants (Savvidy type chromomagnetic vacuum) and covariantly constant scalar fields as a background. The explicit formulas for all the needed heat kern…
We developed a gauge-covariant formulation of the non-equilibrium Green function method for the dynamical and/or non-uniform electromagnetic field by means of the deformational quantization method. Such a formulation is realized by replacing the Moyal product in the so-called Wigner space by the star product, and facilitates the order-by-order calculation of a gauge-invariant observable in terms of the electromagnetic field. An application of this formalism to the linear response theory is discussed.
The main results are: 1. A manifestly covariant technique for the calculation of De Witt coefficients is elaborated; 2. The coefficients $a_3$ and $a_4$ are calculated; 3. Covariant methods for the study of the nonlocal structure of the effective action are developed. The terms of first and second order in background fields in De Witt coefficients are calculated. The summation of these terms is carried out and nonlocal covariant expression for the Green function, the heat kernel and the effective action are obtained. It is shown that in the conform-invariant case the Green function is finite. …
We calculate the covariant one-loop quantum gravitational effective action for a scalar field model inspired by the recently proposed nonminimal natural inflation model. Our calculation is perturbative, in the sense that the effective action is evaluated in orders of background field, around a Minkowski background. The effective potential has been evaluated taking into account the finite corrections. An order-of-magnitude estimate of the one-loop corrections reveals that gravitational and non-gravitational corrections have same or comparable magnitudes.
At the three-loop level we analyze, how the NSVZ relation appears for ${\cal N}=1$ SQED regularized by the dimensional reduction. This is done by the method analogous to the one which was earlier used for the theories regularized by higher derivatives. Within the dimensional technique, the loop integrals cannot be written as integrals of double total derivatives. However, similar structures can be written in the considered approximation and are taken as a starting point. Then we demonstrate that, unlike the higher derivative regularization, the NSVZ relation is not valid for the renormalizatio…
We construct a de Sitter invariant photon propagator in general covariant gauges. Our result is a natural generalization of the Allen-Jacobson photon propagator in Feynman gauge. Our propagator reproduces the correct response to a point static charge and the one-loop electromagnetic stress-energy tensor, strongly suggesting that it is suitable for perturbative calculations on de Sitter.
