arXiv subjects

Taushif Ahmed

Publications and source records attributed to Taushif Ahmed.

At least 19 recordsLinked to original sources

arXiv2026

Phase-space integrals through Mellin-Barnes representation

We compute angular phase-space integrals with three and four denominators analytically, working within dimensional regularisation via the Mellin-Barnes (MB) representation. The approach converts multifold MB integrals into real parametric integrals and expresses all results in terms of Goncharov polylogarithms (GPLs). For three denominators, all-massless results are obtained to $\mathcal{O}(ε^2)$ and the single-massive case to $\mathcal{O}(ε)$; for four denominators, both the massless and single-massive cases are solved to $\mathcal{O}(ε^0)$. Integrals with multiple massive momenta follow from a partial fraction decomposition reducing them to the single-massive case. Recursion relations relating integrals with higher denominator powers to master integrals are derived. These are essential ingredients to solving full phase-space integrals.

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arXiv2025

Angular phase-space integrals with four denominators through Mellin--Barnes

We compute four-denominator angular phase-space integrals using the Mellin--Barnes (MB) technique in dimensional regularisation. Independent of the scattering process, an angular integral can be categorised based on the nature of the momenta appearing in the denominators. We address all scenarios involving fully massless and massive momenta. We present a partial fraction decomposition that relates angular integrals with multiple massive momenta to those with a single massive momentum. By solving six- and seven-fold MB integrals, we express the final results up to the finite order in the dimensional regulator in terms of Goncharov polylogarithms.

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arXiv2025

NNLO phase-space integrals for semi-inclusive deep-inelastic scattering

We evaluate the phase-space integrals that arise in double real emission diagrams for semi-inclusive deep-inelastic scattering at next-to-next-to-leading order (NNLO) in QCD. Utilizing the reverse unitarity technique, we convert these integrals into loop integrals, allowing us to employ integration-by-parts identities and reduce them to a set of master integrals. The master integrals are then solved using the method of differential equations and expressed in terms of Goncharov polylogarithms. By examining the series expansion in the dimensional regulator, we discover additional relations among some of the master integrals. As an alternative approach, we solve the master integrals by decomposing them into angular and radial components. The angular parts are evaluated using Mellin-Barnes representation, while special attention is given to the singular structures of the radial integrals to handle them accurately. Here the results are provided in terms of one-fold integrals over classical polylogarithms. This approach provides a clearer understanding of the origin of soft and collinear singularities.

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arXiv2025

Two-loop helicity amplitudes for diphoton production with massive quark loop

We compute two-loop helicity amplitudes in QCD for diphoton production through quark- and gluon-initiated channels, accounting for a massive internal quark loop by keeping its full mass dependence. Using physical projectors, we directly decompose the amplitude into its helicity components. By renormalising the heavy quark mass in on-shell, and other quantities in $\overline{\rm MS}$ schemes, we obtain finite remainders. This work paves the way for calculating the cross-section for diphoton production at higher orders in QCD with a massive quark loop, employing different subtraction schemes. The effect of a heavy quark is expected to play a crucial role in high-luminosity LHC.

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arXiv2024

Phase-space integrals through Mellin-Barnes representation

This letter introduces a novel analytical approach to calculating phase-space integrals, crucial for precision in particle physics. We develop a method to compute angular components using multifold Mellin-Barnes integrals, yielding results in terms of Goncharov polylogarithms for integrals involving three denominators. Our results include expressions for massless momenta up to $\cal{O}(ε^2)$ and for one massive momentum up to $\cal{O}(ε)$. Additionally, we derive recursion relations that reduce integrals with higher powers of denominators to simpler ones. We detail how to combine the angular part with the radial one which requires a careful handling of singularities.

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arXiv2024

Two-loop non-planar four-point topology with massive internal loop

We study a set of two-loop non-planar master integrals needed for the NNLO QCD corrections to diphoton and dijet production at hadron colliders. The top-sector topology contains an internal massive fermion loop and is known to contain elliptic curves. Leveraging the method of differential equations, we provide a comprehensive discussion for deriving an $ε$-factorized differential equation related to the most intricate sector within the Feynman integral family. Despite the dependence on multiple scales and the presence of two elliptic sectors, we demonstrate how to leverage the properties of their maximal cuts and the factorization of the Picard-Fuchs operator to deal with the complexity of the analytic computation. In particular, we construct a transformation matrix that brings the differential equations into a format enabling the convenient expression of analytic results in terms of Chen's iterated integrals.

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arXiv2024

Two-loop Vertices with Vacuum Polarization Insertion

We present the analytic evaluation of the second-order corrections to the massive form factors, due to two-loop vertex diagrams with a vacuum polarization insertion, with exact dependence on the external and internal fermion masses, and on the squared momentum transfer. We consider vector, axial-vector, scalar and pseudoscalar interactions between the external fermion and the external field. After renormalization, the finite expressions of the form factors are expressed in terms of polylogarithms up to weight three.

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arXiv2023

A note on quark and gluon energy-momentum tensors

We discuss the constraints on quark and gluon energy-momentum tensors in QCD that follow from the requirement of Renormalisation-Group invariance of the traces of these operators. Our study covers the most general form of the latter traces, while the energy-momentum tensors themselves are only subjected to very mild constraints. We derive Renormalisation-Group equations for the two finite independent functions of the strong coupling constant and renormalisation scale of minimal subtraction which completely define the energy-momentum tensors. We demonstrate that previously proposed definitions of the renormalized quark and gluon energy-momentum tensors are special cases of our results assuming no explicit dependence on the renormalisation scale. Finally, we present $\overline{\mathrm{MS}}$-renormalised quark and gluon energy-momentum tensors at four-loop order.

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arXiv2021

On the $ZH$ amplitudes at two loop in QCD

Through this article, we show that the amplitudes for the $b\bar{b} \rightarrow ZH$ in the presence of bottom-Higgs Yukawa coupling can be constructed, up to a few anomalous diagrams, solely from a set of vector form factors of properly grouped classes of diagrams. Thereby, we can completely bypass the subtle issues involving the chiral quantity $γ_5$. In the second part, we demonstrate that while computing the $q\bar{q} \rightarrow ZH$ amplitudes in Higgs effective field theory employing a non-anticommuting $γ_5$ in dimensional regularization, we need to introduce a four-point effective composite operator to the renormalized Lagrangian in order to retain the physical properties of the amplitude.

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arXiv2021

Rapidity distribution at soft-virtual and beyond for n-colorless particles to N$^4$LO in QCD

We present a systematic framework to study the threshold contributions of the differential rapidity distribution for the production of any number of colorless particles in the hadronic colliders. This has been achieved based on the universality structure of the soft enhancements associated with the real emissions, along with the factorization property of the differential cross section and the renormalization group invariance. In this formalism, we present a universal soft-collinear operator to compute the soft virtual differential cross section for a generic $2\rightarrow n$ scattering process up to next-to-next-to-next-to-next-to-leading order (N$^4$LO) in perturbative QCD. We also provide a universal operator to perform the threshold resummation to next-to-next-to-next-to-leading logarithmic (N$^3$LL) accuracy. We explicitly present the approximate analytical results of the rapidity distributions at N$^4$LO and N$^3$LL for the Higgs boson production through gluon fusion and bottom quark annihilation, and also for the Drell-Yan production at the hadronic collider. \textcolor{black}{We extend our framework to include the next to threshold contributions for the diagonal partonic channels.

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arXiv2021

Two-loop amplitudes for di-Higgs and di-pseudo-Higgs productions through quark annihilation in QCD

Through this article, we present the two-loop massless QCD corrections to the production of di-Higgs and di-pseudo-Higgs boson through quark annihilation in the large top quark mass limit. Within dimensional regularisation, we employ the non-anticommuting $γ_5$ and treat it under the Larin prescription. We discover the absence of any additional renormalisation, so-called contact renormalisation, that could arise from the short distance behaviour of two local operators. This finding is in corroboration with the operator product expansion. By examining the results, we discover the lack of similarity in the highest transcendentality weight terms between these finite remainders and that of a pair of half-BPS primary operators in maximally supersymmetric Yang-Mills theory. We need these newly computed finite remainders to calculate observables involving di-Higgs or di-pseudo-Higgs at the next-to-next-to-leading order. We implement the results to a numerical code for further phenomenological studies.

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arXiv2021

A pitfall in applying non-anticommuting $γ_5$ in $q\overline{q} \rightarrow ZH$ amplitudes

In computing the two-loop QCD corrections to a class of Feynman diagrams for the process $q\overline{q} \rightarrow ZH$ in Higgs effective field theory, we discover a striking phenomenon. We find the need for an additional local composite operator in the renormalised Lagrangian while employing a non-anticommuting $γ_5$ in dimensional regularisation. The computation using anticommuting $γ_5$, however, does not require any such amendment.

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arXiv2021

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

The renormalization constant $Z_J$ of the flavor-singlet axial-vector current with a non-anticommuting $γ_5$ in dimensional regularization is determined to order $α_s^3$ in QCD with massless quarks. The result is obtained by computing the matrix elements of the operators appearing in the axial-anomaly equation $\big[\partial_μ J^μ_{5} \big]_{R} = \frac{α_s}{4 π}\, n_f\, {\displaystyle \mathrm{T}_{F}} \, \big[F \tilde{F} \big]_{R}$ between the vacuum and a state of two (off-shell) gluons to 4-loop order. Furthermore, through this computation, the conjectured equality between the $\overline{\mathrm{MS}}$ renormalization constant $Z_{F\tilde{F}}$ associated with the operator $\big[F \tilde{F} \big]_{R}$ and that of $α_s$ is verified explicitly to hold true at 4-loop order. This equality automatically ensures a relation between the respective anomalous dimensions, $γ_{\scriptscriptstyle J} = \frac{α_s}{4 π}\, n_f\, {\displaystyle \mathrm{T}_{F}} \, γ_{\scriptscriptstyle FJ} $, at order $α_s^4$ given the validity of the axial-anomaly equation which was used to determine the non-$\overline{\mathrm{MS}}$ piece of $Z_J$ for the particular $γ_5$ prescription in use.

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arXiv2020

Two-loop Doubly Massive Four-Point Amplitude Involving a half-BPS and Konishi Operator

The two-loop four-point amplitude of two massless SU(N) colored scalars and two color singlet operators with different virtuality described by a half-BPS and Konishi operators is calculated analytically in maximally supersymmetric Yang-Mills theory. We verify the ultraviolet behaviour of the unprotected composite operator and exponentiation of the infrared divergences with correct universal values of the anomalous dimensions in the modified dimensional reduction scheme. The amplitude is found to contain lower transcendental weight terms in addition to the highest ones and the latter has no similarity with similar amplitudes in QCD.

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arXiv2020

Form factors with two operator insertions and the principle of maximal transcendentality

We present the first calculations of two-point two-loop form factors (FFs) with a two identical operators insertion in maximally supersymmetric Yang-Mills theory. In this article, we consider the supersymmetry protected half-BPS primary and unprotected Konishi operators. Unlike the FFs of a single operator insertion of the half-BPS primary, the FFs involving two half-BPS operators are found to contain lower transcendentality weight terms in addition to the highest ones. Moreover, in contrast to Sudakov FFs, the highest weight terms of the FFs of a double half-BPS no longer match with that of a double Konishi. We also find that the principle of maximal transcendentality, which dictates the presence of identical highest weight terms in the scalar FFs of half-BPS and quark/gluon FFs in QCD, does not hold true anymore for insertions of two identical operators. We discover the absence of any additional ultraviolet counterterm that could arise from the contact interaction between two composite operators.

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arXiv2020

Soft-virtual correction and threshold resummation for $n$-colorless particles to fourth order in QCD: Part I

We present the general form of a universal soft-collinear distribution operator to compute the soft-virtual cross-section to next-to-next-to-next-to-next-to-leading order (N$^4$LO) in QCD for a process with any number of final state colorless particles in hadron colliders. By acting this universal operator on the pure virtual corrections, which need to be computed explicitly for a process, the soft-virtual cross-section can be obtained. The operator is constructed by exploiting the factorization and renormalization group evolution of amplitudes in QCD, and the universality of soft gluon contributions. We also provide the hard coefficient to perform the threshold resummation to next-to-next-to-next-to-leading logarithmic (N$^3$LL) accuracy. Furthermore, we present the approximate analytical results of the soft-virtual cross-sections at N$^4$LO and N$^3$LL for the Higgs boson production through gluon fusion and bottom quark annihilation, and also for the Drell-Yan production at the hadronic collider.

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arXiv2020

Polarized $q \bar{q} \rightarrow Z +$Higgs amplitudes at two loops in QCD: the interplay between vector and axial vector form factors and a pitfall in applying a non-anticommuting $γ_5$

We consider QCD corrections to two loops for the polarized amplitudes of $q{\bar q}\to Z +$ Higgs boson. First we show how the polarized amplitudes of $b \bar{b} \rightarrow Z h$ associated with a non-vanishing $b$-quark Yukawa coupling and a scalar or pseudoscalar Higgs boson $h$ can be built up solely from vector form factors (FF) of properly grouped classes of diagrams, bypassing completely the need of explicitly manipulating $γ_5$ in dimensional regularization (up to a few "anomalous", i.e., triangle diagrams). We determine the contributions of the triangle diagrams in the heavy top limit. We present the analytic results of the vector FF and the triangle-diagram contributions to the axial vector FF, which are sufficient for deriving the two-loop QCD amplitudes for $b \bar{b} \rightarrow Z h$ with a CP-even and CP-odd Higgs boson $h$. We derive the respective Ward identity for these amplitudes, which are subsequently verified to two-loop order in QCD using these FF. In addition, the FF of a class of corrections to $q \bar{q} \rightarrow ZH$ proportional to the top-Yukawa coupling are obtained analytically to two-loop order in QCD in the heavy-top limit using the Higgs-gluon effective Lagrangian where the top quark is integrated out. We address a pitfall that occurs when applying the non-anticommutating $γ_5$ prescription to this class of contributions that has been overlooked so far in the literature. We attribute this issue to the fact that the absence of certain heavy-mass expanded diagrams in the infinite-mass limit of a scattering amplitude with an axial vector current depends on the particular $γ_5$ prescription in use.

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arXiv2020

Four-particle scattering amplitudes in QCD at NNLO to higher orders in the dimensional regulator

We compute all helicity amplitudes for four-particle scattering in massless QCD with $n_f$ fermion flavours to next-to-next-to-leading order (NNLO) in perturbation theory. In particular, we consider all possible configurations of external quarks and gluons. We evaluate the amplitudes in terms of a Laurent series in the dimensional regulator to the order required for future next-to-next-to-next-to-leading order (N$^3$LO) calculations. The coefficients of the Laurent series are given in terms of harmonic polylogarithms that can readily be evaluated numerically. We present our findings in the conventional dimensional regularisation and in the t'Hooft-Veltman schemes.

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