Dimensional Regularization Research Articles

Renormalization and mixing of the Gluino-Glue operator on the lattice

We study the mixing of the Gluino-Glue operator in mathcal{N}=1 Supersymmetric Yang–Mills theory (SYM), both in dimensional regularization and on the lattice. We calculate its renormalization, which is not merely multiplicative, due to the fact that this operator can mix with non-gauge invariant operators of equal or, on the lattice, lower dimension. These operators carry the same quantum numbers under Lorentz transformations and global gauge transformations, and they have the same ghost number. We compute the one-loop quantum correction for the relevant two-point and three-point Green’s functions of the Gluino-Glue operator. This allows us to determine renormalization factors of the operator in the {overline{mathrm{MS}}} scheme, as well as the mixing coefficients for the other operators. To this end our computations are performed using dimensional and lattice regularizations. We employ a standard discretization where gluinos are defined on lattice sites and gluons reside on the links of the lattice; the discretization is based on Wilson’s formulation of non-supersymmetric gauge theories with clover improvement. The number of colors, N_c, the gauge parameter, beta , and the clover coefficient, c_{mathrm{SW}}, are left as free parameters.

Renormalization of the flavor-singlet axial-vector current and its anomaly in dimensional regularization

The renormalization constant ZJ of the flavor-singlet axial-vector current with a non-anticommuting γ5 in dimensional regularization is determined to order {alpha}_s^3 in QCD with massless quarks. The result is obtained by computing the matrix elements of the operators appearing in the axial-anomaly equation {left[{partial}_{mu }{J}_5^{mu}right]}_R=frac{alpha_s}{4pi }{n}_f{mathrm{T}}_F{left[Ftilde{F}right]}_R between the vacuum and a state of two (off-shell) gluons to 4-loop order. Furthermore, through this computation, the equality between the overline{mathrm{MS}} renormalization constant {Z}_{Ftilde{F}} associated with the operator {left[Ftilde{F}right]}_R and that of αs is verified explicitly to hold true at 4-loop order. This equality automatically ensures a relation between the respective anomalous dimensions, {gamma}_J=frac{alpha_s}{4pi }{n}_f{mathrm{T}}_F{gamma}_{FJ} , at order {alpha}_s^4 given the validity of the axial-anomaly equation which was used to determine the non- overline{mathrm{MS}} piece of ZJ for the particular γ5 prescription in use.

Renormalization functions of the tricritical O(N)-symmetric Φ6 model beyond the next-to-leading order in 1/N

We investigate higher-order corrections to the effective potential of the tricritical O(N)-symmetric model in 3-2ε dimensions in its phase exhibiting spontaneous breaking of its scale symmetry. The renormalization group β-function and the anomalous dimension γ of this model are computed up to the next-to-next-to-leading order in the1/N expansion technique and using a dimensional regularization in a minimal subtraction scheme.

A prescription for projectors to compute helicity amplitudes in D dimensions

This article discusses a prescription to compute polarized dimensionally regularized amplitudes, providing a recipe for constructing simple and general polarized amplitude projectors in D dimensions that avoids conventional Lorentz tensor decomposition and avoids also dimensional splitting. Because of the latter, commutation between Lorentz index contraction and loop integration is preserved within this prescription, which entails certain technical advantages. The usage of these D-dimensional polarized amplitude projectors results in helicity amplitudes that can be expressed solely in terms of external momenta, but different from those defined in the existing dimensional regularization schemes. Furthermore, we argue that despite being different from the conventional dimensional regularization scheme (CDR), owing to the amplitude-level factorization of ultraviolet and infrared singularities, our prescription can be used, within an infrared subtraction framework, in a hybrid way without re-calculating the (process-independent) integrated subtraction coefficients, many of which are available in CDR. This hybrid CDR-compatible prescription is shown to be unitary. We include two examples to demonstrate this explicitly and also to illustrate its usage in practice.

Mixed EW-QCD two-loop amplitudes for qoverline{q}to {mathrm{ell}}^{+}{mathrm{ell}}^{-} and \u03b35 scheme independence of multi-loop corrections

We perform a dedicated study of the qoverline{q} -initiated two-loop electroweak-QCD Drell-Yan scattering amplitude in dimensional regularization schemes for vanishing light quark and lepton masses. For the relative order α and αs one-loop Standard Model corrections, details of our comparison to the original literature are given. The infrared pole terms of the mixed two-loop amplitude are governed by a known generalization of the dipole formula and we show explicitly that exactly the same two-loop polarized hard scattering functions are obtained in both the standard ’t Hooft-Veltman-Breitenlohner-Maison γ5 scheme and Kreimer’s anticommuting γ5 scheme.

Two-loop renormalisation of gauge theories in 4D implicit regularisation and connections to dimensional methods

We compute the two-loop beta -function of scalar and spinorial quantum electrodynamics as well as pure Yang–Mills and quantum chromodynamics using the background field method in a fully quadridimensional setup using implicit regularization (IREG). Moreover, a thorough comparison with dimensional approaches such as conventional dimensional regularization (CDR) and dimensional reduction (DRED) is presented. Subtleties related to Lorentz algebra contractions/symmetric integrations inside divergent integrals as well as renormalisation schemes are carefully discussed within IREG where the renormalisation constants are fully defined as basic divergent integrals to arbitrary loop order. Moreover, we confirm the hypothesis that momentum routing invariance in the loops of Feynman diagrams implemented via setting well-defined surface terms to zero deliver non-abelian gauge invariant amplitudes within IREG just as it has been proven for abelian theories.

HepLib: A C++ library for high energy physics

HepLib is a C++ Library for computations in High Energy Physics, it works on top of GiNaC, a well-established C++ library used to perform symbolic computations. HepLib combines serval well-known packages to get high efficiency, including qgraf to generate Feynman aptitudes, FORM to perform Dirac/Color matrix related computations, and FIRE or KIRA for integration-by-parts (IBP) reduction. Another core feature of HepLib lies in the numerical evaluation of master integrals using sector decomposition, which is a general method widely used in high-order numerical computation and has been implemented in many public packages in many different languages, and we present another implementation in the language of C++ with many new features. We use GiNaC to handle the symbolic operations, and export the corresponding integrand into an optimized C++ code, that will be compiled internally and linked dynamically, a customizable numerical integrator is selected to perform the numerical integration, while the integrand can be evaluated in different float precisions, including the arbitrary precision supported by MPFR. Program summaryProgram Title:HepLibCPC Library link to program files:https://doi.org/10.17632/jspjxy3z66.1Developer's repository link:https://heplib.github.io/Licensing provisions:GPLv3Programming language:C++External libraries:GiNaC, qgraf, MPFR, Fermat, QHull, MinUit2, CUBAExternal programs:FORM, FIRE, KIRANature of problem: There are many independent programs or packages in high energy physics, which are written or developed in different programming languages, here we are trying to provide an integrated interface in the C++ language, for generating Feynman diagrams/amplitudes, performing algebraic simplifications on Dirac/Color matrix objects, and reducing scalar integrals to master integrals. The evaluation of master integrals is in the core part during the high-order calculations, one needs to resolve ultraviolet/infrared divergence within dimensional regularization and the singularities inside the integration domain while performing multiple dimensional numerical integrations.Solution method:HepLib uses qgraf to generate the Feynman diagrams/aptitudes, FORM to perform Dirac/Color matrix related computations, FIRE or KIRA for the IBP reduction. A C++ implementation of sector decomposition method is used to extract both ultraviolet and infrared singularity of the integral, the contour deformation is adopted to avoid the singularity inside the integration domain, a parallelized numerical integration is chosen to achieve high efficiency or performance.Restrictions: Depending on the complexity of the problem itself.

Functional renormalization and the MS¯ scheme

Working with scalar field theories, we discuss choices of regulator that, inserted in the functional renormalization group equation, reproduce the results of dimensional regularization at one and two loops. The resulting flow equations can be seen as nonperturbative extensions of the $\overline{\text{MS}}\,$ scheme. We support this claim by recovering all the multicritical models in two dimensions. We discuss a possible generalization to any dimension. Finally, we show that the method also preserves nonlinearly realized symmetries, which is a definite advantage with respect to other regulators.

Axial anomaly in very special relativity

In this paper we study the axial anomaly in Very Special Relativity Electrodynamics using Pauli-Villars and dimensional regularization of ultraviolet divergences and Mandelstam-Leibbrandt regularization of infrared divergences. We compute the anomaly in 2 and 4 dimensional space-time. We find that this procedure preserves the vector Ward identity(charge conservation) and reproduce the standard axial anomaly in 2 and 4 dimensions without corrections from VSR. Finally, we show how to obtain the anomaly in the path integral approach.

Role of generic scale invariance in a Mott transition from a U(1) spin-liquid insulator to a Landau Fermi-liquid metal

We investigate the role of generic scale invariance in a Mott transition from a U(1) spin-liquid insulator to a Landau Fermi-liquid metal, where there exist massless degrees of freedom in addition to quantum critical fluctuations. Here, the Mott quantum criticality is described by critical charge fluctuations, and additional gapless excitations are U(1) gauge-field fluctuations coupled to a spinon Fermi surface in the spin-liquid state, which turn out to play a central role in the Mott transition. An interesting feature of this problem is that the scaling dimension of effective leading local interactions between critical charge fluctuations differs from that of the coupling constant between U(1) gauge fields and matter-field fluctuations in the presence of a Fermi surface. As a result, there appear dangerously irrelevant operators, which can cause conceptual difficulty in the implementation of renormalization group (RG) transformations. Indeed, we find that the curvature term along the angular direction of the spinon Fermi surface is dangerously irrelevant at this spin-liquid Mott quantum criticality, responsible for divergence of the self-energy correction term in U(1) gauge-field fluctuations. Performing the RG analysis in the one-loop level based on the dimensional regularization method, we reveal that such extremely overdamped dynamics of U(1) gauge-field fluctuations, which originates from the emergent one-dimensional dynamics of spinons, does not cause any renormalization effects to the effective dynamics of both critical charge fluctuations and spinon excitations. However, it turns out that the coupling between U(1) gauge-field fluctuations and both matter-field excitations still persists at this Mott transition, which results in novel mean-field dynamics to explain the nature of the spin-liquid Mott quantum criticality. We discuss physical implications of effective one-dimensional spin dynamics and extremely overdamped gauge dynamics at the Mott quantum criticality.

The dimension-shift conjecture for one-loop amplitudes

A conjecture made by Bern, Dixon, Dunbar, and Kosower asserts a simple dimension shifting relationship between the one-loop structure of mathcal{N} = 4 MHV amplitudes and all-plus helicity amplitudes in pure Yang-Mills theory. We prove this conjecture to all orders in dimensional regularisation using unitarity cuts, and evaluate the form of these simplest one-loop amplitudes using a generalised D-dimensional unitarity technique which captures the full amplitude to all multiplicities.

Finite Tsallis gravitational partition function for a system of galaxies

In this paper, we will study the large scale structure formation using the gravitational partition function. We will assertively argue that the system of gravitating galaxies can be analyzed using the Tsallis statistical mechanics. The divergences in the Tsallis gravitational partition function can be removed using the mathematical riches of the generalization of the dimensional regularization (GDR). The finite gravitational partition function thus obtained will be used to evaluate the thermodynamics of the system of galaxies and thus, to understand the clustering of galaxies in the universe. The correlation function which is beleived to contain the information of clustering of galaxies will also be discussed in this formalism.

Cluster Algebras for Feynman Integrals.

We initiate the study of cluster algebras in Feynman integrals in dimensional regularization. We provide evidence that four-point Feynman integrals with one off-shell leg are described by a C_{2} cluster algebra, and we find cluster adjacency relations that restrict the allowed function space. By embedding C_{2} inside the A_{3} cluster algebra, we identify these adjacencies with the extended Steinmann relations for six-particle massless scattering. The cluster algebra connection we find restricts the functions space for vector boson or Higgs plus jet amplitudes and for form factors recently considered in N=4 super Yang-Mills. We explain general procedures for studying relationships between alphabets of generalized polylogarithmic functions and cluster algebras and use them to provide various identifications of one-loop alphabets with cluster algebras. In particular, we show how one can obtain one-loop alphabets for five-particle scattering from a recently discussed dual conformal eight-particle alphabet related to the G(4,8) cluster algebra.

Trace anomaly for Weyl fermions using the Breitenlohner-Maison scheme for \u03b3*

We revisit the computation of the trace anomaly for Weyl fermions using dimensional regularization. For a consistent treatment of the chiral gamma matrix γ* in dimensional regularization, we work in n dimensions from the very beginning and use the Breitenlohner-Maison scheme to define γ*. We show that the parity-odd contribution to the trace anomaly vanishes (for which the use of dimension-dependent identities is crucial), and that the parity-even contribution is half the one of a Dirac fermion. To arrive at this result, we compute the full renormalized expectation value of the fermion stress tensor to second order in perturbations around Minkowski spacetime, and also show that it is conserved.

Renormalization group theory of the generalized multi-vertex sine-Gordon model

Abstract We investigate the renormalization group theory of the generalized multi-vertex sine-Gordon model by employing the dimensional regularization method and also the Wilson renormalization group method. The vertex interaction is given by $\cos(k_j\cdot \phi)$, where $k_j$ ($j=1,2,\ldots,M$) are momentum vectors and $\phi$ is an $N$-component scalar field. The beta functions are calculated for the sine-Gordon model with multiple cosine interactions. The second-order correction in the renormalization procedure is given by the two-point scattering amplitude for tachyon scattering. We show that new vertex interaction with the momentum vector $k_{\ell}$ is generated from two vertex interactions with vectors $k_i$ and $k_j$ when $k_i$ and $k_j$ meet the condition $k_{\ell}=k_i\pm k_j$, called the triangle condition. A further condition $k_i\cdot k_j=\pm 1/2$ is required within the dimensional regularization method. The renormalization group equations form a set of closed equations when $\{k_j\}$ form an equilateral triangle for $N=2$ or a regular tetrahedron for $N=3$. The Wilsonian renormalization group method gives qualitatively the same result for beta functions.

The fifth-order post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach: Potential contributions

We calculate the potential contributions of the motion of binary mass systems in gravity to the fifth post–Newtonian order ab initio using coupling and velocity expansions within an effective field theory approach based on Feynman amplitudes starting with harmonic coordinates and using dimensional regularization. Furthermore, the singular and logarithmic tail contributions are calculated. We also consider the non–local tail contributions. Further steps towards the complete calculation are discussed. We calculate all but the rational O(ν2) contributions to the bound state energy for circular motion and periastron advance K(Eˆ,j). Comparisons are given to results in the literature.

Supersymmetry anomalies and the Wess–Zumino model in a supergravity background

We briefly recall the procedure for computing the Ward identities in the presence of a regulator which violates the symmetry being considered. We compute the first nontrivial correction to the supersymmetry Ward identity of the Wess–Zumino model in the presence of background supergravity using dimensional regularization. We find that the result can be removed using a finite local counter term and so there is no supersymmetry anomaly.

Emergent gravity as the eraser of anomalous gauge boson masses, and QFT-GR concord

In the same base setup as Sakharov's induced gravity, we investigate emergence of gravity in effective quantum field theories (QFT), with particular emphasis on the gauge sector in which gauge bosons acquire anomalous masses in proportion to the ultraviolet cutoff $\Lambda_\wp$. Drawing on the fact that $\Lambda_\wp^2$ corrections explicitly break the gauge and Poincare symmetries, we find that it is possible to map $\Lambda_\wp^2$ to spacetime curvature as a covariance relation and we find also that this map erases the anomalous gauge boson masses. The resulting framework describes gravity by the general relativity (GR) and matter by the QFT itself with $\log\Lambda_\wp$ corrections (dimensional regularization). This QFT-GR concord predicts existence of new physics beyond the Standard Model such that the new physics can be a weakly-interacting or even a non-interacting sector comprising the dark matter, dark energy and possibly more. The concord has consequential implications for collider, astrophysical and cosmological phenomena.

Two-loop renormalization and mixing of gluon and quark energy-momentum tensor operators

In this paper, we present one- and two-loop results for the renormalization of the gluon and quark gauge-invariant operators which appear in the definition of the QCD energy-momentum tensor, in dimensional regularization. To this end, we consider a variety of Green's functions with different incoming momenta. We identify the set of twist-2 symmetric traceless and flavor singlet operators which mix among themselves and we calculate the corresponding mixing coefficients for the nondiagonal components. We also provide results for some appropriate regularization-independent (RI')-like schemes, which address this mixing, and we discuss their application to nonperturbative studies via lattice simulations. Finally, we extract the one- and two-loop expressions of the conversion factors between the proposed RI' and the MSbar schemes. From our results regarding the MSbar-renormalized Green's functions, one can easily derive conversion factors relating numerous variants of RI'-like schemes to MSbar. To make our results easily accessible, we also provide them as Supplemental Material, in the form of a Mathematica input file and, also, an equivalent text file.

On Bound States in Quantum Field Theory

In this paper, a new method to describe the energy spectrums of bound states in Quantum Field Theory is presented. We point out that the fundamental field and its dual soliton combine together to form bound states and the soliton corresponds to the ghost particle in our regularization scheme which takes advantage of dimensional regularization and Pauli-Villars regularization. Based on this point of view, we discuss the bound states of massive Thirring model, the positronium (e+e−) in QED and the vector meson in QCD. We also give a new way to obtain the mass of soliton (quantum soliton) from the stationary condition (gap equation). Our results agree with experimental data to high precision. We argue that the hypothetic X17 particle in decay of 8Be and 4He is a soliton. For vector meson, we find the squared masses of ρ resonances are \(m^{2}(n)\sim (an^{1/3}-b)^{2}\) (n ∈ N) which coincide well with experiments.