arXiv subjects

Adnan Bashir

Publications and source records attributed to Adnan Bashir.

At least 19 recordsLinked to original sources

arXiv2026

Hadronic vacuum polarization contribution to $a_\mu$ from functional methods with strong and electromagnetic isospin breaking

We present a continuum-QCD determination of the leading-order hadronic vacuum-polarization contribution to the anomalous magnetic moment of the muon within the Dyson-Schwinger and Bethe-Salpeter equation framework. The calculation incorporates pion back-reaction, a fully dressed quark-photon vertex with a dynamically generated $\rho$-resonance structure in the timelike region, and both strong and electromagnetic isospin breaking treated self-consistently at the quark level. We obtain $a_\mu^{\mathrm{HVP,LO}}(u+d+s+c)|_{\mathrm{ISB}} = 709.7 \times 10^{-10}$, in good agreement with recent lattice-QCD determinations, and find an isospin-breaking shift of $\Delta a_\mu^{\mathrm{HVP,LO}} = 4.5 \times 10^{-10}$ ($0.6\%$), demonstrating that isospin-breaking effects, while quantitatively modest, are not negligible. Including the bottom-quark contribution and an indicative estimate of the systematic uncertainties, we obtain our final result, $a_\mu^{\mathrm{HVP,LO}}(u+d+s+c+b)|_{\mathrm{ISB}} = (710.0 \pm 14.5) \times 10^{-10}$.

hep-ph↗

arXiv2026

Landau-Khalatnikov-Fradkin Transformations in Reduced Quantum Electrodynamics: Perturbative and Nonperturbative Dynamics of the Fermion Propagator

We present a comprehensive analysis of the Landau-Khalatnikov-Fradkin transformations for the charged fermion propagator in reduced quantum electrodynamics (RQED). Starting from the propagator in a reference gauge, we perform a gauge transformation to obtain its analytical expression valid to all orders in an arbitrary covariant gauge and also applicable in a nonperturbative context. This work complements and extends previous studies of quantum electrodynamics in various spacetime dimensions, for both massless and massive fermions. At the perturbative level, we expand the resulting expressions up to two-loop order for both massless and massive cases, and compare our results with those available in the literature wherever possible. We argue that the most suitable choice of the reference covariant gauge in RQED is $\xi=1/3$, as in this case the leading logarithmic contribution to the massless wave-function renormalization vanishes at one-loop order. This choice provides a direct connection between perturbation theory and the constraints imposed by multiplicative renormalizability on the massless fermion propagator. We also investigate the implications of the Landau-Khalatnikov-Fradkin transformations for the dynamically generated mass function of the fermion propagator. Finally, through numerical computation, we demonstrate that both the chiral fermion condensate and the fermion pole mass are gauge-invariant quantities.

hep-th↗

arXiv2025

Imaging the charge distributions of flavor-symmetric and -asymmetric mesons

We investigate the internal structure of a comprehensive set of pseudoscalar and vector mesons, including both flavor-symmetric and flavor-asymmetric systems, by reconstructing their charge distributions from electromagnetic form factors. To achieve this, we employ a Maximum Entropy Method optimized for charge distributions, utilizing previously published form factor data obtained within the Dyson-Schwingers and Bethe-Salpeter equations framework. Furthermore, we calculate the average distance between the valence quark and antiquark that constitute the meson, interpreting it as an estimate for both the meson's spatial size and the typical range of quark motion. Our results reveal that this distance for the lightest quarkonia is approximately five times larger than that for the heaviest. Moreover, due to spin effects, vector mesons exhibit sizes that are 5-15\% larger than their pseudoscalar counterparts.

hep-ph↗

arXiv2025

One-loop off-shell quark-gluon vertex in arbitrary gauge and dimensions: a streamlined approach through the second-order formalism of QCD

The standard Feynman rules used for perturbative calculations in quantum chromodynamics (QCD) are derived from a Lagrangian that is first-order in derivatives. It includes a three-point quark-gluon vertex which obscures the precise disentangled manner in which spin and momentum are interchanged during these interactions. An unambiguous understanding of this interchange is insightful for efficiently extracting physically relevant information from various Green's functions. To separate the scalar and spin degrees of freedom and gain physical insight from the outset, we examine the quark-gluon vertex using the less commonly employed second-order formalism of QCD. We compute this off-shell vertex in arbitrary space-time dimensions and covariant gauges by using scalar integrals with shifted dimensions, which include higher powers of the propagators, within a combined first- and second-order formalism. This approach naturally identifies the transverse components of the quark-gluon vertex, even before evaluating the tensor Feynman integrals. We also compute the on-shell version of this vertex using exclusively the second-order formalism, facilitating a precise identification of spin and momentum interchange. Through analyzing the Pauli form factor at $k^2=0$ (where $k$ represents the momentum of the external gluon), we find that only a specific set of second-order Feynman diagrams are relevant for calculating the electromagnetic and chromomagnetic dipole moments. These diagrams represent quantum processes in which the spin of the incoming quark changes only once due to interactions with the virtual gluons that form the quark-gluon vertex. All other interactions involve only momentum interchange (scalar interactions). Our results are in complete agreement with those obtained from the first-order formalism.

hep-ph↗

arXiv2025

Landau-Khalatnikov-Fradkin Transformations in Quantum Electrodynamics: For Perturbation Theory and Dynamical Mass Generation

We carry out a comprehensive analysis of the Landau-Khalatnikov-Fradkin transformations for a charged fermion propagator at the two-loop level in quantum electrodynamics (QED). Starting with an arbitrary covariant gauge $\xi$ and space-time dimension $d$, we provide its explicit expressions in three and four-dimensional QED. We begin with the tree-level fermion propagator in the Landau gauge and gauge-transform it to obtain an analytical expression for an all order result in an arbitrary covariant gauge. We expand it out to two-loops both for the massless and massive propagators in three and four space-time dimensions. In addition to comparing with all earlier results in the literature wherever possible, we also study constraints of multiplicative renormalizabilty of our results in four-dimensional QED which are logarithmically divergent. Finally, we analyze representative solutions of the fermion propagator which correspond to dynamical chiral symmetry breaking and mass generation in QED. We study the gauge dependence of these emergent solutions, that of the Euclidean pole mass and the chiral fermion condensate.

hep-ph↗

arXiv2024

Neutral pion to two-photons transition form factor revisited

Based upon a combined formalism of Schwinger-Dyson and Bethe-Salpeter equations in quantum chromodynamics (QCD), we propose a QCD kindred algebraic model for the dressed quark propagator, for the Bethe-Salpeter amplitude of the pion and the electromagnetic quark-photon interaction vertex. We then compute the $\gamma^{*}\pi^0\gamma$ transition form factor $G^{\gamma^{*}\pi^0\gamma}(Q^2)$ for a wide range of photon momentum transfer squared $Q^2$. The quark propagator is expanded out in its perturbative functional form but with dynamically generated dressed quark mass. It has complex conjugate pole singularities in the complex-momentum plane which is motivated by the solution of the quark gap equation with rainbow-ladder truncation of the infinite set of Schwinger-Dyson equations. This complex pole singularity structure of the quark propagator can be associated with a signal of confinement which prevents quarks to become stable asymptotic states. The Bethe-Salpeter amplitude is expressed without a spectral density function, which encapsulate its low and large momentum behaviour. The QCD evolution of the distribution amplitude is also incorporated into our model through the direct implementation of Efremov-Radyushkin-Brodsky-Lepage evolution equations. We include the effects of the quark anomalous magnetic moment in the description of the quark-photon vertex whose infrared enhancement is known to dictate hadronic properties. Once the QCD kindred model is constructed, we calculate the form factor $G^{\gamma^{*}\pi^0\gamma}(Q^2)$ and find it consistent with direct QCD-based studies as well as most available experimental data. It slightly exceeds the conformal limit for large $Q^2$ which might be attributed to the scaling violations in QCD. The associated interaction radius and neutral pion decay width turn out to be compatible with experimental data.

hep-ph↗

arXiv2024

Gravitational form factors of pseudoscalar mesons in a contact interaction

Given the unique role played by the gravitational form factors (GFFs) in unraveling the internal mechanics of hadrons, we examine the GFFs of ground state pseudoscalar mesons $\pi$, $\eta_c$, $\eta_b$ and the hypothetical {\em strangeonium} $\eta_s(s\bar{s})$. We adopt the coupled framework of Dyson-Schwinger and Bethe-Salpeter equations within a contact interaction, and employ a novel approach to the dressed amputated meson-meson scattering amplitude which makes connection with the energy-momentum tensor and with the GFFs. The resulting GFFs fulfill the anticipated symmetry constraints. The corresponding charge and mass radii and the $D-$term are also computed. We show that the $D-$term for the pseudoscalar mesons is bounded within the $(-1, -1/3)$ range; these bounds correspond to the massless (chiral limit) and infinitely massive cases, respectively. Considering the current interest in the GFFs, understanding the \textit{D}-term of pseudoscalar mesons and their GFFs can provide an important first step for future endeavors in the field.

hep-ph↗

arXiv2023

One loop reduced QED for massive fermions within an innovative formalism

We carry out a detailed study of the three-point fermion-photon interaction vertex at one loop order for massive fermions in reduced quantum electrodynamics. This calculation is carried out in arbitrary covariant gauges and space-time dimensions within a recently proposed innovative approach based upon an efficient combination of the first and second order formalisms of quantum electrodynamics. This procedure provides a natural decomposition of the vertex into its components which are longitudinal and transverse to the photon momentum. It also separates the spin and scalar degrees of freedom of a fermion interacting electromagnetically, allowing us to readily establish the gauge-independence of the Pauli form factor and compute it in an expeditious manner. All incoming and outgoing momenta are taken off-shell at the outset. However, we present results for cases of particular kinematic interest whenever required. For the sake of completeness, we also provide expressions for the massive fermion self energy and photon vacuum polarization, verifying known expressions for massless reduced quantum electrodynamics and computing the renormalization constants ${\cal Z}_1$, ${\cal Z}_2$ and ${\cal Z}_3$. As we provide general expressions for the computed Green functions, we readily reproduce and confirm the results for standard quantum electrodynamics. Comparing the two cases, we infer that the Pauli form factor for reduced quantum electrodynamics is $8/3$ times that for the standard QED in four dimensions, implying a higher Landé $g$-factor. We expect our perturbative calculation of the fermion-photon vertex to serve as a guide for any non-perturbative construction of this Green function, invariably required in the Schwinger-Dyson equation studies of the subject.

hep-ph↗

arXiv2023

Effect of the Quark-Gluon Vertex on Dynamical Chiral Symmetry Breaking

In this work, we investigate how the details of the quark-gluon interaction vertex affect the quantitative description of chiral symmetry breaking through the gap equation for quarks. We start from two gluon propagator models widely used in literature and constructed in direct connection with our gradually improved understanding of infrared quantum chromodynamics coupled with its exact one-loop limit. The gap equation is then solved by employing a variety of vertex \emph{Ansätze}, which have been constructed in order to implement some of the key aspects of quantum chromodynamics, namely, multiplicative renormalizability of the quark propagator, gauge invariance, matching with perturbation theory in the weak coupling regime, independence from unphysical kinematic singularities as well as manifestly correct transformation properties under charge conjugation and parity operations. On general grounds, all truncation schemes exhibit the same qualitative and quantitative pattern of chiral symmetry breaking, ensuring the overall robustness of this approach and its potentially reliable description of the hadron spectrum and properties.

nucl-th↗

arXiv2023

One-loop fermion-photon vertex in arbitrary gauge and dimensions: a novel approach

We compute one-loop electron-photon vertex with fully off-shell external momenta in an arbitrary covariant gauge and space-time dimension. There exist numerous efforts in literature where one-loop off-shell vertex is calculated by employing the standard first order Feynman rules in different covariant gauges and space-time dimensions of interest. The tensor structure which decomposes this three-point vertex into the components transverse and longitudinal to the photon momentum gets intertwined in this first order formalism. The Ward-Takahashi identity is explicitly invoked to untangle the two pieces and the results are expressed in a preferred basis of twelve spin-amplitudes. We propose a novel approach based upon an efficient combination of the first and second order formalisms of quantum electrodynamics to compute this one-loop vertex. Among some conspicuous advantages is the fact that this less known second order formalism separates the spin and scalar degrees of freedom of an electron interacting electromagnetically. More noticeably, the longitudinal and transverse contributions naturally disentangle from the onset in our approach. Moreover, this decomposition leads to identities between one-loop scalar Feynman integrals with higher powers in the propagators and shifted space-time dimensions that can be used to prove the Ward-Takahashi identity at one-loop order without the need to evaluate any Feynman integral. Additionally, this natural decomposition allows us to establish the gauge-independence of the Pauli form factor through explicit cancellations of scalar Feynman integrals that depend on the gauge parameter. These cancellations naturally lead to a compact expression for the Pauli form factor in arbitrary dimensions. Wherever necessary and insightful, we make comparisons with earlier works.

hep-ph↗

arXiv2023

Gauge dependence of the quark gap equation: an exploratory study

We study the gauge dependence of the quark propagator in quantum chromodynamics by solving the gap equation with a nonperturbative quark-gluon vertex which is constrained by longitudinal and transverse Slavnov-Taylor identities, the discrete charge conjugation and parity symmetries and which is free of kinematic singularities in the limit of equal incoming and outgoing quark momenta. We employ gluon propagators in renormalizable $R_ξ$ gauges obtained in lattice QCD studies. We report the dependence of the nonperturbative quark propagator on the gauge parameter, in particular we observe an increase, proportional to the gauge-fixing parameter, of the mass function in the infrared domain, whereas the wave renormalization decreases within the range $0 \leq ξ\leq 1$ considered here. The chiral quark condensate reveals a mild gauge dependence in the region of $ξ$ investigated. We comment on how to build and improve upon this exploratory study in future in conjunction with generalized gauge covariance relations for QCD.

hep-ph↗

arXiv2022

The time-like electromagnetic kaon form factor

We compute the electromagnetic charged kaon form factor in the time-like region by employing a Poincaré covariant formalism of the Bethe-Salpeter equation to study quark-antiquark bound states in conjunction with the Schwinger-Dyson equation for the quark propagator. Following a recent kindred calculation of the time-like electromagnetic pion form factor, we include the most relevant intermediate composite particles permitted by their quantum numbers in the interaction kernel to allow for a decay mechanism for the resonances involved. This term augments the usual gluon mediated interaction between quarks. For a sufficiently low energy time-like probing photon, the electromagnetic form factor is saturated by the $ρ(770)$ and $ϕ(1020)$ resonances. We assume $SU(2)$ isospin symmetry throughout. Our results for the absolute value squared of the electromagnetic form factor agree qualitatively rather well and quantitatively moderately so with available experimental data.

hep-ph↗

arXiv2022

Pion and Kaon box contribution to $a_μ^{\text{HLbL}}$

We present an evaluation of the $π^\pm$ and $K^\pm$ box contributions to the hadronic light-by-light piece of the muon's anomalous magnetic moment, $a_μ$. The calculation of the corresponding electromagnetic form factors (EFFs) is performed within a Dyson-Schwinger equations (DSE) approach to quantum chromodynamics. These form factors are calculated in the so-called rainbow-ladder (RL) truncation, following two different evaluation methods and, subsequently, in a further improved approximation scheme which incorporates meson cloud effects. The results are mutually consistent, indicating that in the domain of relevance for $a_μ$ the obtained EFFs are practically equivalent. Our analysis yields the combined estimates of $a_μ^{π^\pm-box}=-(15.6\pm 0.2)\times 10^{-11}$ and $a_μ^{K^\pm-\text{box}}=-(0.48\pm 0.02)\times 10^{-11}$, in full agreement with results previously obtained within the DSE formalism and other contemporary estimates.

hep-ph↗

arXiv2022

Dyson-Schwinger equations and the muon g-2

We present a brief introduction to the Dyson-Schwinger equations (DSEs) approach to hadron and high-energy physics. In particular, how this formalism is applied to calculate the electromagnetic form factors $γ^* γ^* \to \textbf{P}^0$ and $γ^* \textbf{P}^\pm \to \textbf{P}^\pm$ (with $\textbf{P}^\pm$ and $\textbf{P}^0$ charged and neutral ground-state pseudoscalar mesons, respectively) is discussed. Subsequently, the corresponding contributions of those form factors to the muon anomalous magnetic moment ($g-2$) are estimated. We look forward to promoting the DSE approach to address theoretical aspects of the muon $g-2$, highlighting some calculations that could be carried out in the future.

hep-ph↗

arXiv2022

Dressed quark-gluon vertex form factors from gauge symmetry

We present preliminary results on the longitudinal $and$ transverse form factors of the quark-gluon vertex as functions of the incoming and outgoing quark momenta and an angle $θ=2π/3$ between them. The expressions for these form factors were previously derived from Slavnov-Taylor identities, gauge covariance and multiplicative renormalizability that firmly constrain the fermion-boson vertex.

hep-ph↗

arXiv2021

The impact of transverse Slavnov-Taylor identities on dynamical chiral symmetry breaking

We extend earlier studies of transverse Ward-Fradkin-Green-Takahashi identities in QED, their usefulness to constrain the transverse fermion-boson vertex and their importance for multiplicative renormalizability, to the equivalent gauge identities in QCD. To this end, we consider transverse Slavnov-Taylor identities that constrain the transverse quark-gluon vertex and derive its eight associated scalar form factors. The complete vertex can be expressed in terms of the quark's mass and wave-renormalization functions, the ghost-dressing function, the quark-ghost scattering amplitude and a set of eight form factors. The latter parametrize the hitherto unknown nonlocal tensor structure in the transverse Slavnov-Taylor identity which arises from the Fourier transform of a four-point function involving a Wilson line in coordinate space. We determine the functional form of these eight form factors with the constraints provided by the Bashir-Bermudez vertex and study the effects of this novel vertex on the quark in the Dyson-Schwinger equation using lattice QCD input for the gluon and ghost propagators. We observe significant dynamical chiral symmetry breaking and a mass gap that leads to a constituent mass of the order of 500 MeV for the light quarks. The flavor dependence of the mass and wave-renormalization functions as well as their analytic behavior on the complex momentum plane is studied and as an application we calculate the quark condensate and the pion's weak decay constant in the chiral limit. Both are in very good agreement with their reference values.

nucl-th↗

arXiv2021

Landau-Khalatnikov-Fradkin transformations for the two loop massless quark propagator

Landau-Khalatnikov-Fradkin transformations (LKFTs) yield the gauge dependence of correlation functions within the class of linear covariant gauges. We derive the LKFT for the quark propagator and explicitly evaluate it up to the two loop level in the chiral limit. Although the number of diagrams to be evaluated is significantly larger than with the conventional computational scheme, the diagrams are simpler in nature, thereby leading to a considerably faster evaluation of the gauge dependent part than naively expected. Finally, we also resum the LKFT generated terms and compare our results with earlier work in the literature.

hep-th↗

arXiv2021

Dynamical diquarks in the ${\boldsymbol{γ^{(\ast)} p\to N(1535)\tfrac{1}{2}^-}}$ transition

The $γ^{(\ast)}+p \to N(1535) \tfrac{1}{2}^-$ transition is studied using a symmetry-preserving regularisation of a vector$\,\otimes\,$vector contact interaction (SCI). The framework employs a Poincaré-covariant Faddeev equation to describe the initial and final state baryons as quark+di\-quark composites, wherein the diquark correlations are fully dynamical, interacting with the photon as allowed by their quantum numbers and continually engaging in breakup and recombination as required by the Faddeev kernel. The presence of such correlations owes largely to the mechanisms responsible for the emergence of hadron mass; and whereas the nucleon Faddeev amplitude is dominated by scalar and axial-vector diquark correlations, the amplitude of its parity partner, the $N(1535) \tfrac{1}{2}^-$, also contains sizeable pseudoscalar and vector diquark components. It is found that the $γ^{(\ast)}+p \to N(1535) \tfrac{1}{2}^-$ helicity amplitudes and related Dirac and Pauli form factors are keenly sensitive to the relative strengths of these diquark components in the baryon amplitudes, indicating that such resonance electrocouplings possess great sensitivity to baryon structural details. Whilst SCI analyses have their limitations, they also have the virtue of algebraic simplicity and a proven ability to reveal insights that can be used to inform more sophisticated studies in frameworks with closer ties to quantum chromodynamics.

hep-ph↗