arXiv subjects

Konstantin Stepanyantz

Publications and source records attributed to Konstantin Stepanyantz.

At least 19 recordsLinked to original sources

arXiv2026

Leading Yukawa terms in the three-loop anomalous dimension and four-loop $\beta$-function of ${\cal N}=1$ supersymmetric theories for various renormalization prescriptions

For an arbitrary renormalizable ${\cal N}=1$ supersymmetric theory with a single gauge coupling regularized by higher covariant derivatives we calculate the three-loop contribution to the anomalous dimension of the matter superfields proportional to the sixth powers of Yukawa couplings for a wide class of renormalization prescriptions. Next, taking into account that in the case of using the higher covariant derivatives supplemented by minimal subtractions of logarithms the NSVZ relation is valid in all orders, we calculate the corresponding four-loop contribution to the gauge $\beta$-function and investigate its scheme dependence. In particular, we construct the so-called ``minimal'' renormalization prescription for which the renormalization group functions have the simplest possible form provided that the exact equations for the gauge and Yukawa $\beta$-functions following from supersymmetry remain valid.

hep-th↗

arXiv2026

Some approximate renormalization group invariants for supersymmetric extensions of the Standard Model and the Yukawa unification

For supersymmetric extensions of the Standard Model we construct some expressions that include Yukawa couplings for the third and second generations and receive relatively small quantum corrections. This implies that they slightly depend on scale and are therefore approximate renormalization group invariants. Using these invariants we try to analyse possible relations between the Yukawa couplings at the unification scale $M_X$ as well as the predictions for values of $\mbox{tg}\,\beta$ and $\alpha(M_X)$. In particular, we suggest two variants of such relations and investigate whether they agree with the experimental values of elementary particle masses. It is demonstrated that the Yukawa unification for the third and second generations consistent with them can be achieved by adding exotic superfields forming 3 representations $5+\bar{5}$ of the group $SU(5)$ to the MSSM field content. We argue that this may indicate the possible underlying $E_6$ gauge symmetry.

hep-ph↗

arXiv2025

The renormalization group invariants and exact results for various supersymmetric theories

Some recent all-loop results on the renormalization of supersymmetric theories are summarized and reviewed. In particular, we discuss how it is possible to construct expressions which do not receive quantum corrections in all orders for certain ${\cal N}=1$ supersymmetric theories. For instance, in ${\cal N}=1$ SQED+SQCD there is a renormalization group invariant combination of two gauge couplings. For the Minimal Supersymmetric Standard Model there are two such independent combinations of the gauge and Yukawa couplings. We investigate the scheme-dependence of these results and verify them by explicit three-loop calculations. We also argue that the all-loop exact $\beta$-function and the corresponding renormalization group invariant can exist in the $6D$, ${\cal N}=(1,0)$ supersymmetric higher derivative gauge theory interacting with a hypermultiplet in the adjoint representation.

hep-th↗

arXiv2025

Relation between leading divergences in nonrenormalizable $4D$ supersymmetric theories

We consider an ${\cal N}=1$ nonrenormalizable supersymmetric gauge theory with the superpotential quartic in the chiral matter superfields. With the help of the Slavnov's higher covariant derivative regularization it is demonstrated that (in the lowest nontrivial order) the leading power divergent quantum correction to the gauge coupling constant is given by an integral of double total derivatives with respect to the loop momenta. The result obtained after calculating this integral turned out to be proportional to the corresponding quantum correction to the kinetic term of the matter superfields. More exactly, in the considered approximation the quadratically divergent contributions to the gauge coupling and to the kinetic term of the chiral matter superfields are related by an equation analogous to the exact NSVZ $\beta$-function for the renormalizable case.

hep-th↗

arXiv2025

Structure of renormalization constants for theories with multiple couplings in the MS-like subtraction schemes

For theories with multiple couplings we construct simple expressions for the four-dimensional (or, in general, integer-dimensional) renormalization constants assuming that all divergences are logarithmical. These expressions allow relating all coefficients at $\varepsilon$-poles, logarithms, and (if exist) mixed terms to the coefficients of the renormalization group functions in any order of the perturbation theory for MS-like renormalization prescriptions. The result admits such a formulation in that $\varepsilon$-poles and $\ln\Lambda/\mu$ enter on the same footing. For theories with two and three couplings we present explicit expressions for the pole/logarithm structure of renormalization constants in the lowest orders of the perturbation theory. They are verified by comparisons with the two-loop explicit calculation for ${\cal N}=1$ SQCD+SQED and also with the previously known three-loop calculations for the $\varphi^4$-theory with two couplings.

hep-th↗

arXiv2025

Exact expressions for the renormalization constants in the MS-like schemes

We briefly review how it is possible to derive some exact expressions for the renormalization constants for the MS-like renormalization prescriptions using the arguments based on the renormalization group. These expressions are obtained for a version of the dimensional technique in which the dimensionful parameter $\Lambda$ differs from the renormalization scale $\mu$. They encode the equations relating the coefficients at higher $\varepsilon$-poles, powers of $\ln \Lambda/\mu$, and mixed terms of the structure $\varepsilon^{-q} \ln^p \Lambda/\mu$ to the coefficients of the renormalization group functions (i.e. of the $\beta$-function and the anomalous dimension). The general results are verified by some multiloop calculations.

hep-th↗

arXiv2025

Three-loop verification of the equations relating running of the gauge couplings in ${\cal N}=1$ SQCD+SQED

We verify a recently derived equations relating the renormalization group running of two gauge couplings in ${\cal N}=1$ SQCD+SQED by the explicit three-loop calculation. It is demonstrated that these equations are really valid in the HD+MSL scheme. In other words, if a theory is regularized by higher covariant derivatives and the renormalization is made by minimal subtractions of logarithms, the analogs of the strong and electromagnetic gauge couplings do not run independently. However, in the $\overline{\mbox{DR}}$ scheme the considered equations do not hold starting from the three-loop order, where the scheme dependence becomes essential. Therefore, they are valid only for a certain set of the renormalization prescriptions. We prove that all of them can be obtained from the HD+MSL scheme by finite renormalizations which satisfy a special constraint and illustrate how this works in the three-loop approximation.

hep-th↗

arXiv2024

All-loop renormalization group invariants for MSSM

For MSSM from the gauge couplings, Yukawa couplings, and the coefficient $\mu$ in the part of superpotential quadratic in the Higgs superfields we construct combinations which (for certain renormalization prescriptions) do not depend on the renormalization point in all loops. In other words, these combinations are the renormalization group invariants. Similar invariants are also constructed for NMSSM. The derivation is based on the nonrenormalization of the superpotential and the NSVZ equations. We argue that the scale invariance of the considered combinations takes place in the class of the HD+MSL schemes. This fact has been verified in the lowest orders, up to and including the one in which the dependence on the renormalization prescription becomes essential. It is also demonstrated that in the $\overline{\mbox{DR}}$ scheme the renormalization group invariance does not take place starting from the approximation, where the scheme dependence manifests itself.

hep-ph↗

arXiv2024

Higher $\varepsilon$-poles and logarithms in the MS-like schemes from the algebraic structure of the renormalization group

We investigate the structure of renormalization constants within the MS-like renormalization prescriptions for a version of dimensional regularization in which the dimensionful regularization parameter $\Lambda$ differs from the renormalization point $\mu$. Namely, we rewrite the all-loop equations relating coefficients at higher $\varepsilon$-poles and higher powers of $\ln\Lambda/\mu$ to the coefficients of the renormalization group functions in a simple unified form. It is argued that this form follows from the algebraic structure of the renormalization group.

hep-th↗

arXiv2024

Algebraic structure of the renormalization group in the renormalizable QFT theories

We consider the group formed by finite renormalizations as an infinite-dimensional Lie group. It is demonstrated that for the finite renormalization of the gauge coupling constant its generators $\hat L_n$ with $n\ge 1$ satisfy the commutation relations of the Witt algebra and, therefore, form its subalgebra. The commutation relations are also written for the more general case when finite renormalizations are made for both the coupling constant and matter fields. We also construct the generator of the Abelian subgroup corresponding to the changes of the renormalization scale. The explicit expressions for the renormalization group generators are written in the case when they act on the $\beta$-function and the anomalous dimension. It is explained how the finite changes of these functions under the finite renormalizations can be obtained with the help of the exponential map.

hep-th↗

arXiv2024

The gauge coupling unification in Grand Unified Theories based on the group $E_8$

We consider a theory with the gauge group $E_8$ assuming that the gauge symmetry breaking pattern is $E_8 \to E_7 \times U_1 \to E_6 \times U_1 \to SO_{10} \times U_1 \to SU_5 \times U_1 \to SU_3 \times SU_2 \times U_1$ and vacuum expectation values are acquired only by components of the representations 248. It is demonstrated that in this case there are several options for the relations between the gauge couplings of the resulting theory, but only one of them gives $\alpha_3 = \alpha_2$ and $\sin^2\theta_W = 3/8$. Also, it is the only option for which the resulting theory can include all MSSM superfields.

hep-ph↗

arXiv2023

Do we understand the internal spaces of second quantized fermion and boson fields, with gravity included? Relation with strings theories

The article proposes the description of internal spaces of fermion (quarks and leptons and antiquarks and antileptons) and boson (photons, weak bosons, gluons, gravitons and scalars) second quantized fields in a unique way if they all are massless. The internal spaces are described by ``basis vectors'', which are the superposition of odd (for fermions) and even (for bosons) products of the operators $\gamma^ {a}$. For an arbitrary symmetry $SO(d-1,1)$ of the internal spaces, it is the number of fermion fields (they appear in families and have their Hermitian conjugated partners in a separate group) equal to the number of boson fields (they appear in two orthogonal groups), manifesting a kind of supersymmetry, which differ of the string supersymmetry. On the assumption that fermions and bosons are active (they have momenta different from zero) only in $d=(3+1)$ ordinary space-time, bosons present vectors if they carry the space index $\mu=(0,1,2,3)$, and present scalars if they carry the index $\sigma \ge 5$. The author discusses this theory's latest achievements, with a trial to understand whether the extension to strings or to odd-dimensional spaces can lead to a new kind of supersymmetry. This model, named {\it spin-charge-family} theory, manifests in a long series of papers on the phenomenological success of the theory in elementary particle physics and cosmology.

physics.gen-ph↗

arXiv2023

Higher logarithms and $\varepsilon$-poles for the MS-like renormalization prescriptions

We consider a version of dimensional regularization (reduction) in which the dimensionful regularization parameter $\Lambda$ is in general different from the renormalization scale $\mu$. Then in the scheme analogous to the minimal subtraction the renormalization constants contain $\varepsilon$-poles, powers of $\ln\Lambda/\mu$, and mixed terms of the structure $\varepsilon^{-q}\ln^{p}\Lambda/\mu$. For the MS-like schemes we present explicit expressions for the coefficients at all these structures which relate them to the coefficients in the renormalization group functions, namely in the $\beta$-function and in the anomalous dimension. In particular, for the pure $\varepsilon$-poles we present explicit solutions of the 't~Hooft pole equations. Also we construct simple all-loop expressions for the renormalization constants (also written in terms of the renormalization group functions) which produce all $\varepsilon$-poles and logarithms and establish a number of relations between various coefficients at $\varepsilon$-poles and logarithms. The results are illustrated by some examples.

hep-th↗

arXiv2023

A condition for the reduction of couplings in the $P = \frac{1}{3}Q$ supersymmetric theories

We demonstrate that in the $P=\frac{1}{3}Q$ supersymmetric theories the renormalization group invariance of the ratio $\lambda^{ijk}/e$ (of the Yukawa couplings to the gauge coupling) is equivalent to a simple relation between the anomalous dimensions of the quantum gauge superfield, of the Faddeev--Popov ghosts, and of the matter superfields, which should be valid in each order of the perturbation theory. In the one- and two-loop approximations it is verified explicitly. Presumably, in higher orders this relation can be satisfied for the planar supergraphs under a certain renormalization prescription. Assuming that it is valid we rewrite the exact equation for the (corresponding contribution to the) anomalous dimension of the matter superfields in the theories under consideration in a different (but equivalent) form.

hep-th↗

arXiv2023

The gauge coupling unification in the flipped $E_8$ GUT

The gauge coupling unification is investigated at the classical level under the assumptions that the gauge symmetry breaking chain is $E_8\to E_7\times U_1 \to E_6\times U_1 \to SO_{10}\times U_1 \to SU_5 \times U_1 \to SU_3 \times SU_2 \times U_1$ and only components of the representations 248 of $E_8$ can acquire vacuum expectation values. We demonstrate that there are several options for the relations between the gauge couplings of the resulting theory, but the only symmetry breaking pattern corresponds to $\alpha_3=\alpha_2$ and $\sin^2\theta_W=3/8$. Moreover, only for this option the particle content of the resulting theory includes all MSSM superfields. It is also noted that this symmetry breaking pattern corresponds to the case when all representation which acquire vacuum expectation values have the minimal absolute values of the relevant $U_1$ charges.

hep-ph↗

arXiv2023

Quantum properties of gauge theories with extended supersymmetry formulated in ${\cal N}=1$ superspace

We analyse quantum properties of ${\cal N}=2$ and ${\cal N}=4$ supersymmetric gauge theories formulated in terms of ${\cal N}=1$ superfields and investigate the conditions imposed on a renormalization prescription under which the non-renormalization theorems are valid. For this purpose in these models we calculate the two-loop contributions to the anomalous dimensions of all chiral matter superfields and the three-loop contributions to the $\beta$-functions for an arbitrary ${\cal N}=1$ supersymmetric subtraction scheme supplementing the higher covariant derivative regularization. We demonstrate that, in general, the results do not vanish due to the scheme dependence, which becomes essential in the considered approximations. However, the two-loop anomalous dimensions vanish if a subtraction scheme is compatible with the structure of quantum corrections and does not break the relation between the Yukawa and gauge couplings which follows from ${\cal N}=2$ supersymmetry. Nevertheless, even under these conditions the three-loop contribution to the $\beta$-function does not in general vanishes for ${\cal N}=2$ supersymmetric theories. To obtain the purely one-loop $\beta$-function, one should also chose an NSVZ renormalization prescription. The similar statements for the higher loop contributions are proved in all orders.

hep-th↗

arXiv2022

Coefficients at powers of logarithms in the HD+MSL renormalization scheme

For renormalizable theories with a single coupling constant regularized by higher derivatives we investigate the coefficients at powers of logarithms present in the renormalization constants assuming that divergences are removed by minimal subtractions of logarithms. According to this (HD+MSL) renormalization prescription the renormalization constants include only powers of $\ln\Lambda/\mu$, where $\Lambda$ and $\mu$ are the dimensionful regularization parameter and the renormalization point, respectively. We construct general explicit expressions for arbitrary coefficients at powers of this logarithm present in the coupling constant renormalization and in the field renormalization constant which relate them to the $\beta$-function and (in the latter case) to the corresponding anomalous dimension. To check the correctness, we compare the results with the explicit four-loop calculation made earlier for ${\cal N}=1$ SQED and (for the supersymmetric case) rederive a relation between the renormalization constants following from the NSVZ equation.

hep-th↗

arXiv2022

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

Three-loop $\beta$-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 $\beta$-functions defined in terms of the bare couplings. After that, we construct the three-loop $\beta$-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 $\overline{\mbox{DR}}$ scheme. Also this can be considered as an indepedent confirmation of the $\overline{\mbox{DR}}$ results.

hep-ph↗