<p>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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ξ</mi></math> and space-time dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi></math>, 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.</p>

Pakistan

Anam Ashraf

M. Jamil Aslam

Faisal Akram

Spain

Mexico

Adnan Bashir

Landau-Khalatnikov-Fradkin transformations in quantum electrodynamics: For perturbation theory and dynamical mass generation

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 <math><msup><mi>γ</mi><mo>*</mo></msup><msup><mi>π</mi><mn>0</mn></msup><mi>γ</mi></math> transition form factor <math><msup><mi>G</mi><mrow><msup><mi>γ</mi><mo>*</mo></msup><msup><mi>π</mi><mn>0</mn></msup><mi>γ</mi></mrow></msup><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> for a wide range of photon momentum transfer squared <math><msup><mi>Q</mi><mn>2</mn></msup></math>. 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 from becoming stable asymptotic states. The Bethe-Salpeter amplitude is expressed without a spectral density function, which encapsulates its low- and large-momentum behavior. 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 <math><msup><mi>G</mi><mrow><msup><mi>γ</mi><mo>*</mo></msup><msup><mi>π</mi><mn>0</mn></msup><mi>γ</mi></mrow></msup><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> and find it consistent with direct QCD-based studies, as well as most available experimental data. It slightly exceeds the conformal limit for large <math><msup><mi>Q</mi><mn>2</mn></msup></math>, 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.

China

M. Atif Sultan

Jiayin Kang

Lei Chang

Neutral pion to two-photons transition form factor revisited

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 <math><mrow><mi>π</mi></mrow></math>, <math><msub><mi>η</mi><mi>c</mi></msub></math>, <math><msub><mi>η</mi><mi>b</mi></msub></math> and the hypothetical strangeonium <math><msub><mi>η</mi><mi>s</mi></msub><mo>(</mo><mi>s</mi><mover><mi>s</mi><mo>¯</mo></mover><mo>)</mo></math>. 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 <math><mi>D</mi></math> term are also computed. We show that the <math><mi>D</mi></math> term for the pseudoscalar mesons is bounded within the <math><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>)</mo></math> range; these bounds correspond to the massless (chiral limit) and infinitely massive cases, respectively. Considering the current interest in the GFFs, understanding the <math><mrow><mi>D</mi></mrow></math> term of pseudoscalar mesons and their GFFs can provide an important first step for future endeavors in the field.

Zanbin Xing

Khépani Raya

Gravitational form factors of pseudoscalar mesons in a contact interaction

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 <math><msub><mi>Z</mi><mn>1</mn></msub></math>, <math><msub><mi>Z</mi><mn>2</mn></msub></math> and <math><msub><mi>Z</mi><mn>3</mn></msub></math>. 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 <math><mn>8</mn><mo>/</mo><mn>3</mn></math> times that for the standard QED in four dimensions, implying a higher Landé <math><mi>g</mi></math>-factor. We expect our perturbative calculation of the fermion-photon vertex to serve as a guide for any nonperturbative construction of this Green function, invariably required in the Schwinger-Dyson equation studies of the subject. We also comment on the Landau-Khalatnikov-Fradkin transformations of the massive fermion propagator and provide a comparison with its one-loop calculation.

Victor Miguel Banda Guzmán

United States

Luis Albino

Dania Rodríguez-Tzintzun

One loop reduced QED for massive fermions within an innovative formalism

We compute a 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 a 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 12 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.

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

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 <math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>ξ</mi></mrow></msub></mrow></math> 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 <math><mn>0</mn><mo>≤</mo><mi>ξ</mi><mo>≤</mo><mn>1</mn></math> considered here. The chiral quark condensate reveals a mild gauge dependence in the region of <math><mi>ξ</mi></math> investigated. We comment on how to build and improve upon this exploratory study in future in conjunction with generalized gauge covariance relations for QCD.

Brazil

José Roberto Lessa

Colombia

Fernando E. Serna

Bruno El-Bennich

Portugal

Orlando Oliveira

Gauge dependence of the quark gap equation: An exploratory study

We compute the electromagnetic charged kaon form factor in the timelike 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 timelike 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 timelike probing photon, the electromagnetic form factor is saturated by the <math><mi>ρ</mi><mo>(</mo><mn>770</mn><mo>)</mo></math> and <math><mi>ϕ</mi><mo>(</mo><mn>1020</mn><mo>)</mo></math> resonances. We assume <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></math> 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.

A. S Miramontes

Timelike electromagnetic kaon form factor

We present an evaluation of the <math><msup><mi>π</mi><mo>±</mo></msup></math> and <math><msup><mi>K</mi><mo>±</mo></msup></math> box contributions to the hadronic light-by-light piece of the muon’s anomalous magnetic moment, <math><msub><mi>a</mi><mi>μ</mi></msub></math>. 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 <math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow></msub></mrow></math> the obtained EFFs are practically equivalent. Our analysis yields the combined estimates of <math><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><msup><mrow><mi>π</mi></mrow><mrow><mo>±</mo></mrow></msup><mo>−</mo><mi>box</mi></mrow></msubsup><mo>=</mo><mo>−</mo><mo>(</mo><mn>15.6</mn><mo>±</mo><mn>0.2</mn><mo>)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup></mrow></math> and <math><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><msup><mrow><mi>K</mi></mrow><mrow><mo>±</mo></mrow></msup><mo>−</mo><mi>box</mi></mrow></msubsup><mo>=</mo><mo>−</mo><mo>(</mo><mn>0.48</mn><mo>±</mo><mn>0.02</mn><mo>)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup></mrow></math>, in full agreement with results previously obtained within the DSE formalism and other contemporary estimates.

Ángel Miramontes

Pablo Roig

Pion and Kaon box contribution to <math><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>HLbL</mi></mrow></msubsup></mrow></math>

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.

Eduardo Rojas

Fernando Serna

Roberto Silveira

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

A continuum approach to quantum chromodynamics (QCD), based upon Schwinger-Dyson (SD) and Bethe-Salpeter (BS) equations, is employed to provide a tightly constrained prediction for the <math><mrow><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mi>γ</mi></mrow><mrow><mo>*</mo></mrow></msup><mo>→</mo><mo>{</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>,</mo><mi>η</mi><mo>,</mo><msup><mrow><mi>η</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>}</mo></mrow></math> transition form factors (TFFs) and their corresponding pole contribution to the hadronic light-by-light (HLbL) piece of the anomalous magnetic moment of the muon (<math><msub><mi>a</mi><mi>μ</mi></msub></math>). This work relies on a practical and well-tested quark-photon vertex Ansatz approach to evaluate the TFFs for arbitrary spacelike photon virtualities, in the impulse approximation. The numerical results are parametrized meticulously, ensuring a reliable evaluation of the HLbL contributions to <math><msub><mi>a</mi><mi>μ</mi></msub></math>. We obtain <math><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>−</mo><mtext>pole</mtext></mrow></msubsup><mo>=</mo><mspace></mspace><mo>(</mo><mn>6.14</mn><mo>±</mo><mn>0.21</mn><mo>)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup></mrow></math>, <math><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>η</mi><mo>−</mo><mtext>pole</mtext></mrow></msubsup><mo>=</mo><mo>(</mo><mn>1.47</mn><mo>±</mo><mn>0.19</mn><mo>)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup></mrow></math>, <math><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><msup><mrow><mi>η</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>−</mo><mtext>pole</mtext></mrow></msubsup><mo>=</mo><mo>(</mo><mn>1.36</mn><mo>±</mo><mn>0.08</mn><mo>)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup></mrow></math>, yielding a total value of <math><msubsup><mi>a</mi><mi>μ</mi><mrow><msup><mi>π</mi><mn>0</mn></msup><mo>+</mo><mi>η</mi><mo>+</mo><msup><mi>η</mi><mo>′</mo></msup><mo>−</mo><mtext>pole</mtext></mrow></msubsup><mo>=</mo><mo>(</mo><mn>8.97</mn><mo>±</mo><mn>0.48</mn><mo>)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>10</mn></mrow></msup></math>, compatible with contemporary determinations. Notably, we find that <math><msubsup><mi>a</mi><mi>μ</mi><mrow><msub><mi>η</mi><mi>c</mi></msub><mo>+</mo><msub><mi>η</mi><mi>b</mi></msub><mo>−</mo><mtext>pole</mtext></mrow></msubsup><mo>≈</mo><msubsup><mi>a</mi><mi>μ</mi><mrow><msub><mi>η</mi><mi>c</mi></msub><mo>−</mo><mtext>pole</mtext></mrow></msubsup><mo>=</mo><mo>(</mo><mn>0.09</mn><mo>±</mo><mn>0.01</mn><mo>)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>10</mn></mrow></msup></math>, which might not be negligible once the percent precision in the computation of the light pseudoscalars is reached.

Contribution of neutral pseudoscalar mesons to <math><msubsup><mi>a</mi><mi>μ</mi><mtext>HLbL</mtext></msubsup></math> within a Schwinger-Dyson equations approach to QCD