Dimensional Regularization Research Articles

The maxcut of the sunrise with different masses in the continuous Minkoskean dimensional regularisation

We evaluate the maxcut of the two loops sunrise amplitude with three different masses by direct use of the loop momenta in the Minkoskean (as opposed to the usual Euclidean) continuous dimension regularisation, obtaining in that way six related but different functions expressed in the form of one dimensional finite integrals. We then consider the 4th order homogeneous equation valid for the maxcut, and show that for arbitrary dimension d the six functions do satisfy the equation separately. We further discuss the d=2,3,4 cases, verifying that only four of them are linearly independent. The equal mass limit is also shortly considered.

One-loop QCD amplitudes in the Feynman-diagram gauge

Scattering amplitudes for the massless QCD process, qq¯→q′q¯′, are calculated in the one-loop order in the Feynman-Diagram (FD) gauge, where gluons are quantized on the light cone with opposite direction of the three-momenta. We find non-decoupling of the Faddeev-Popov ghosts and nonconventional UV singularities in dimensional regularization. The known QCD amplitudes with asymptotic freedom are reproduced only after summing propagator and vertex corrections. By quantizing gluons in the Feynman gauge on the FD gauge background, we obtain the one-loop improved FD gauge amplitudes. Published by the American Physical Society 2024

Z boson emission by an electron and the decays of Z boson into fermions in a de Sitter universe

In this paper we develop a method to obtain the rates for the decay of a Z boson into fermions in de Sitter geometry. Our general results obtained in de Sitter space-time allow us to obtain the Minkowski limits for the transition rates when the expansion parameter vanishes. Another important result reported in the present study is related to the emission of Z bosons by electrons and positrons. All the processes were studied by implementing perturbative methods that allow us to define the transition amplitudes in the first order of perturbation theory. The variation of probabilities and rates in terms of expansion parameter and particles masses is also given, pointing out that the processes that generate particle production are possible only in the early universe. For computing the transition rates and probabilities we use the dimensional regularization method and the minimal substraction method.

Low-energy theorem revisited and OPE in massless QCD

We revisit a low-energy theorem (LET) of NSVZ type in SU(N) QCD with Nf massless quarks derived in [1] by implementing it in dimensional regularization. The LET relates n-point correlators in the l.h.s. to n + 1-point correlators with the extra insertion of TrF2 at zero momentum in the r.h.s. We demonstrate that, for 2-point correlators of an operator O in the l.h.s., the LET implies that, in general, the integrated 3-point correlator in the r.h.s. needs in perturbation theory an infinite additive renormalization in addition to the multiplicative one. We relate the above counterterm to a corresponding divergent contact term in a certain coefficient of the OPE of TrF2 with O in the momentum representation, thus extending to any operator O an independent argument that first appeared for O = TrF2 in [2]. Finally, we demonstrate that in the asymptotically free phase of QCD the aforementioned counterterm in the LET is actually finite nonperturbatively after resummation to all perturbative orders. We also briefly recall the implications of the LET in the gauge-invariant framework of dimensional regularization for the perturbative and nonperturbative renormalization in large-N QCD. The implications of the LET inside and above the conformal window of SU(N) QCD with Nf massless quarks will appear in a forthcoming paper.

On gauge amplitudes first appearing at two loops

We study scattering amplitudes in massless non-abelian gauge theory where all outgoing gluons have positive helicity. It has been argued recently by Costello that for a particular fermion representation (8 fundamentals plus one antisymmetric-tensor representation in SU(N)) the one-loop amplitudes vanish identically. We show that this vanishing leads to previously-observed identities among one-loop color-ordered partial amplitudes. We then turn to two loops, where Costello has computed the all-plus amplitudes for this theory, as rational functions of the kinematics for any number of gluons using the celestial chiral algebra (CCA) bootstrap. We show that in dimensional regularization, these two-loop amplitudes are not rational, and they are not even finite as ϵ → 0. However, the finite remainder for four gluons agrees with the formula by Costello. In addition, we provide a mass regulator for the infrared-divergent loop integrals; with this regulator, the CCA bootstrap formula is recovered exactly. Finally, we use the CCA bootstrap to compute the double-trace terms in the theory at two loops for an arbitrary number of gluons.

Loops in anti de Sitter space

We discuss general one and two-loop banana diagrams and one-loop diagrams with external lines with arbitrary masses on the anti de Sitter spacetime by using methods of AdS quantum field theory in the dimensional regularization approach. The banana diagrams explicitly computed in this paper are indeed the necessary ingredients for the evaluation of the two-loop effective potential of the Standard Model and can be used to extend the flat space results in presence of a negative cosmological constant. In the one-loop case we also compute the effective potential for an O(N) model in d = 4 dimension as an explicit function of the cosmological constant Λ, both exactly and perturbatively up to order Λ. In the two-loop case we show the explicit calculation is possible thanks to a remarkable discrete Källén-Lehmann formula which we found and proved sometimes ago and whose domain of applicability we extend in the present paper.

Echoes of Veltman criteria on the next-two-Higgs-doublet model

We investigate how the Next-Two-Higgs Doublet Model extension (N2HDM) should look if we are to address the naturalness problem using dimensional regularization. In such a model, new Higgs states are predicted, namely: three CP-even h1,2,3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_{1,2,3}$$\\end{document}, one CP-odd A, and a pair of charged Higgs boson H±\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^\\pm $$\\end{document}. Our calculations of the overall quadratic divergences have been performed with full consistency with the latest data from the Large Hadron Collider (LHC) concerning the observed 125 GeV Higgs boson, alongside precision electroweak data tests and lower mass limits on charged Higgs boson. It is shown that the quadratically divergent quantum corrections δi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta _i$$\\end{document} (i=1,2,3) for the three CP-even Higgs bosons are controllably small, though hidden fine-tuning might still be required. This reveals a significant impact on the model parameter space, Higgs spectrum mass and notably the singlet-doublet admixture.

Renormalization of nonlocal gluon operators on the lattice

Wen study the renormalization of a complete set of gauge-invariant gluon nonlocal operators in lattice perturbation theory. We determine the mixing pattern under renormalization of these operators using symmetry arguments, which extend beyond perturbation theory. Additionally, we derive the renormalization factors of the operators within the modified minimal subtraction (MS¯) scheme up to one loop. To enable a nonperturbative renormalization procedure, we investigate a suitable version of the modified regularization-invariant (RI′) scheme, and we calculate the conversion factors from that scheme to MS¯. The computations are performed by employing both dimensional and lattice regularizations, using the Wilson gluon action. This work is relevant to nonperturbative studies of the gluon parton distribution functions (PDFs) on the lattice. Published by the American Physical Society 2024

Next-to-next-to-leading order evolution of polarized parton densities in the Larin scheme

In many calculations involving polarized twist-2 parton densities to higher order in the strong coupling constant one uses the Larin scheme to describe chiral effects in dimensional regularization. Upon forming observables, the scheme dependence cancels. Still one needs a corresponding regularization scheme to compute the contributing building blocks, like massless and massive Wilson coefficients, as well as the massive three-loop operator matrix elements used in the variable flavor number scheme. These are matched to the evolved parton distribution functions in the Larin scheme. Starting with suitable input distributions we provide the solution of scale evolution of the different polarized parton distribution functions in Bjorken x space for a wide range of virtualities Q2 in the Larin scheme, at next-to-leading, at next-to-next-to-leading order for the first time. We also illustrate the deviation between the parton distributions in the Larin and MS¯ schemes numerically. Published by the American Physical Society 2024

Conformal anomalies and renormalized stress tensor correlators for nonconformal theories

We analyze the proposal of defining the Weyl anomaly for classically nonconformal theories as gmn⟨Tmn⟩−⟨gmnTmn⟩, originally put forward by Duff, in the case of a scalar field with quartic self-interaction in 4d. We work in the context of dimensional regularization in curved background to two loops (first order in the coupling). We review the original regularized but not renormalized prescription and its ambiguities; we argue that it cannot be extended to the interacting theory as it fails to provide a finite result. We then propose an alternative prescription via renormalized expectation values. At one loop, our candidate reproduces the local heat kernel result, while its extension to interacting theories contains nonlocal contributions. Published by the American Physical Society 2024

A computation of two-loop six-point Feynman integrals in dimensional regularization

We compute three families of two-loop six-point massless Feynman integrals in dimensional regularization, namely the double-box, the pentagon-triangle, and the hegaxon-bubble family. This constitutes the first analytic computation of two-loop master integrals with eight scales. We use the method of canonical differential equations. We describe the corresponding integral basis with uniform transcendentality, the relevant function alphabet, and analytic boundary values at a particular point in the Euclidean region up to the fourth order in the regularization parameter ϵ. The results are expressed as one-fold integrals over classical polylogarithms. We provide a set of supplementary files containing our results in machine-readable form, including a proof-of-concept implementation for numerical evaluations of the one-fold integrals valid within a subset of the Euclidean region.

One-loop five-parton amplitudes in the NMRK limit

We analyse the real part of one-loop five-parton amplitudes in the next-to-multi-Regge kinematic (NMRK) limit, to leading power, and to finite order in the dimensional regularisation parameter. To leading logarithmic (LL) accuracy, it is known that five-parton amplitudes in this limit are given to all-orders by a single factorised expression, in which the pair of partons which are not well-separated in rapidity are described by a two-parton emission vertex. In this study, we investigate the one-loop amplitudes at next-to-leading logarithmic (NLL) accuracy, and find that is has a more complex structure. In particular, it is found that the purely gluonic amplitudes are compatible with an analogous factorisation of individual colour structures. From the one-loop amplitudes we extract one-loop two-parton emission vertices, which are functions of a subset of the momenta of the amplitude. In the multi-Regge kinematic (MRK) limit, the vertices themselves factorise into the known one-loop single-parton emission vertices and Lipatov vertex, with rapidity dependence governed by the one-loop gluon Regge trajectory, as required by compatibility with the known MRK limit of amplitudes. The one-loop two-parton emission vertices are necessary ingredients for the construction of the next-to-next-to leading order (NNLO) jet impact factors in the BFKL framework.

Rational terms of UV origin to all loop orders

Numerical approaches to computations typically reconstruct the numerators of Feynman diagrams in four dimensions. In doing so, certain rational terms arising from the (D − 4)-dimensional part of the numerator multiplying ultraviolet (UV) poles in dimensional regularisation are not captured and need to be obtained by other means. At one-loop these rational terms of UV origin can be computed from a set of process-independent Feynman rules. Recently, it was shown that this approach can be extended to two loops. In this paper, we show that to all loop orders it is possible to compute rational terms of UV origin through process-independent vertices that are polynomial in masses and momenta.

Monte Carlo evaluation of divergent one-loop integrals without contour deformation

Reference (Pittau and Webber in Eur Phys J C 82(1):55, https://doi.org/10.1140/epjc/s10052-022-10008-6, 2022) introduces a method for computing numerically four-dimensional multi-loop integrals without performing an explicit analytic contour deformation around threshold singularities. In this paper, we extend such a technique to massless scalar one-loop integrals regularized in the framework of dimensional regularization. A two-loop example is also discussed.

Loops in de Sitter space

We discuss general one and two-loops banana diagrams with arbitrary masses on the de Sitter spacetime by using direct methods of dS quantum field theory in the dimensional regularization approach. In the one-loop case we also compute the effective potential for an O(N) model in d = 4 dimension as an explicit function of the cosmological constant Λ, both exactly and perturbatively up to order Λ. For the two-loop case we show that the calculation is made easy thanks to a remarkable Källén-Lehmann formula that has been in the literature for a while. We discuss the divergent cases at d = 3 using a contiguity formula for generalized hypergeometric functions and we extract the dominant term at d = 4 proving a general formula to deal with a divergent hypergeometric series.

An introduction to semi-automated matrix element computation in particle physics

This paper presents a semi-automated method to compute matrix elements at the Leading Order and the Next-to-Leading Order in the ABC model of particle physics. The ABC model consists of three scalar particles and they interact only when all three particles are present in the interaction. In the Next-to-Leading Order calculations, one often has to deal with ultraviolet divergences. Some of the techniques such as Wick’s rotation, Feynman parameterisation, and dimensional regularization which are useful in Next-to-Leading Order computations are presented by applying them on the ABC model calculations and also implemented in python based programs using Computer Algebra System.

The stationary Klein-Gordon equation with a delta-like source: A generalized function approach

This work aims to initiate a discussion on finding solutions to non-homogeneous differential equations in terms of generalized functions. For simplicity, we conduct the analysis within the specific context of the stationary Klein-Gordon equation with a point-like source, identifying a generalized function that solves such an equation and aligns with the solution obtained through the Fourier approach with dimensional regularization. In addition to being regular at the source singularity, a notable advantage of our solution is its presentation as a single expression, eliminating the need for piecewise definitions. The arguments presented here are applicable to a broader range of situations, offering a novel approach to addressing divergences in field theories using generalized functions. Moreover, we anticipate that the approach introduced in this work could provide a new method for handling Green functions regularized at coincident points, thereby simplifying the renormalization process in a wide range of theories.

Conservative Black Hole Scattering at Fifth Post-Minkowskian and First Self-Force Order

We compute the fifth post-Minkowskian (5PM) order contributions to the scattering angle and impulse of classical black hole scattering in the conservative sector at first self-force order using the worldline quantum field theory formalism. This challenging four-loop computation required the use of advanced integration-by-parts and differential equation technology implemented on high-performance computing systems. Use of partial fraction identities allowed us to render the complete integrand in a fully planar form. The resulting function space is simpler than expected: In the scattering angle, we see only multiple polylogarithms up to weight three and a total absence of the elliptic integrals that appeared at 4PM order. All checks on our result, both internal—cancellation of dimensional regularization poles and preservation of the on-shell condition—and external—matching the slow-velocity limit with the post-Newtonian (PN) literature up to 5PN order and matching the tail terms to the 4PM loss of energy—are passed. Published by the American Physical Society 2024

Toward multiloop local renormalization within causal loop-tree duality

Renormalization is a well-known technique to get rid of ultraviolet (UV) singularities. When relying on dimensional regularization, these become manifest as ε poles, allowing one to define counterterms with useful recursive properties. However, this procedure requires one to work at “integral level” and poses difficulties to achieve a smooth combination with seminumerical approaches. This article is devoted to the development of an integrand-level renormalization formalism, better suited for semi- or fully numerical calculations. Starting from the loop-tree duality, we keep the causal representations of the integrands of multiloop Feynman diagrams and explore their UV behavior. Then, we propose a strategy that allows one to build local counterterms, capable of rendering the expressions integrable in the high-energy limit and in four space-time dimensions. Our procedure was tested on diagrams up to three loops, and we found a remarkably smooth cancellation of divergences. The results of this work constitute a powerful step toward a fully local renormalization framework in quantum field theory. Published by the American Physical Society 2024

Coulomb Branch Amplitudes from a Deformed Amplituhedron Geometry

The amplituhedron provides, via geometric means, the all-loop integrand of scattering amplitudes in maximally supersymmetric Yang-Mills theory. Unfortunately, dimensional regularization, used conventionally for integration, breaks the beautiful geometric picture. This motivates us to propose a “deformed” amplituhedron. Focusing on the four-particle amplitude, we introduce two deformation parameters, which can be interpreted as particle masses. We provide evidence that the mass pattern corresponds to a specific choice of vacuum expectation values on the Coulomb branch. The deformed amplitude is infrared finite, making the answer well defined in four dimensions. Leveraging four-dimensional integration techniques based on differential equations, we compute the amplitude up to two loops. In the limit where the deformation parameters are taken to zero, we recover the known Bern-Dixon-Smirnov amplitude. In the limit where only one deformation parameter is taken to zero, we find a connection to the angle-dependent cusp anomalous dimension. Published by the American Physical Society 2024