<p>We present analytical expressions for the transverse momentum dependent (unintegrated) gluon and quark densities in a proton derived at leading order of QCD running coupling and valid at both small and large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>. The calculations are performed using the Kimber-Martin-Ryskin/Watt-Martin-Ryskin prescription (in both differential and integral formulations) with different treatment of kinematical constraint, which reflects the angular and strong ordering conditions for parton emissions. As an input, the analytical solution of QCD evolution equations for conventional (collinear) parton distributions is applied, where the valence and nonsinglet quark parts obey the Gross-Llewellyn-Smith and Gottfried sum rules, and momentum conservation for the singlet quark and gluon densities is taken into account. Several phenomenological parameters are extracted from the combined fit to precision data on the proton structure function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> collected by the BCDMS, H1, and ZEUS Collaborations, comprising a total of 933 points from 5 datasets. Comparison with the numerical results obtained by other groups is presented, and the phenomenological application to the inclusive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi></math>-jet production at the LHC is given.</p>

JINR

A.V. Kotikov

Russia

A.V. Lipatov

Updating TMD parton densities in a proton within the Kimber-Martin-Ryskin approach

We renormalize the soft function entering the factorization and resummation of the $\textit{qg}$ parton-scattering channel of the Drell-Yan process near the kinematic threshold $$ \hat{s} $$ → $\textit{Q}$ at next-to-leading power in the expansion around $\textit{z}$ ≡ $\textit{Q}$ / $$ \hat{s} $$ = 1, and solve its renormalization-group equation.

Germany

Martin Beneke

Hong Kong

Yao Ji

Erik Sünderhauf

Xing Wang

Renormalization of the next-to-leading-power soft function for the Drell-Yan off-diagonal channel

Romania

Victor E. Ambrus

France

Maxim N. Chernodub

Corrigendum to “Acceleration as a circular motion along an imaginary circle: Kubo-Martin-Schwinger condition for accelerating field theories in imaginary-time formalism” [Physics Letters B 855 (2024) 138757/PLB 138795]

We investigate the formation of bound states of non-relativistic dark matter particles subject to long-range interactions through radiative capture. The initial scattering and final bound states are described by Coulomb potentials with different strengths, as relevant for non-abelian gauge interactions or theories featuring charged scalars. For bound states with generic quantum numbers $\textit{n}$ and $\textit{ℓ}$, we provide closed-form expressions for the bound-state formation (BSF) cross sections of monopole, dipole and quadrupole transitions, and of arbitrary multipole order when $\textit{ℓ}$ = $\textit{n}$ – 1. This allows us to investigate in detail a strong enhancement of BSF that occurs for initial states in a repulsive potential. For $\textit{ℓ}$ = $\textit{n}$ – 1 ≫ 1, we show that the BSF cross section for each single bound state violates the perturbative unitarity bound in the vicinity of a certain critical initial velocity, and provide an interpretation in terms of a smooth matching of classical trajectories. When summing the BSF cross section over all possible bound states in the final state, this leads to a unitarity violation below a certain velocity, but within the validity range of the weakly coupled non-relativistic description. We identify an effectively strong interaction as the origin of this unitarity violation, which is caused by an “anomalously” large overlap of scattering and bound-state wave functions in Coulomb potentials of different strength.

Tobias Binder

Lorenzo Ros

Mathias Garny

Stefan Lederer

Perturbative unitarity violation in radiative capture transitions to dark matter bound states

The non-observation of baryon number violation suggests that the scale of baryon-number violating interactions at zero temperature is comparable to the GUT scale. However, the pertinent measurements involve hadrons made of the first-generation quarks, such as protons and neutrons. One may therefore entertain the idea that new flavour physics breaks baryon number at a much lower scale, but only in the coupling to a third generation quark, leading to observable baryon-number violating b-hadron decay rates. In this paper we show that indirect constraints on the new physics scale ΛBNV from the existing bounds on the proton lifetime do not allow for this possibility. For this purpose we consider the three dominant proton decay channels p →   <math> <msup> <mi>ℓ</mi> <mo>+</mo> </msup> <msub> <mi>ν</mi> <mi>ℓ</mi> </msub> <mover> <mi>ν</mi> <mo>¯</mo> </mover> </math>  $$ {\ell}^{+}{\nu}_{\ell}\overline{\nu} $$ , p →   <math> <msup> <mi>π</mi> <mo>+</mo> </msup> <mover> <mi>ν</mi> <mo>¯</mo> </mover> </math>  $$ {\pi}^{+}\overline{\nu} $$  and p → π 0 ℓ + mediated by a virtual bottom quark.

Gael Finauri

United States

Alexey Petrov

Indirect constraints on third generation baryon number violation

In this paper, we address the reduced beauty cross section [<math><msubsup><mi>σ</mi><mi>red</mi><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math>] and the beauty structure function [<math><msubsup><mi>F</mi><mn>2</mn><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math>], to study the beauty content of a proton. We calculate <math><msubsup><mi>σ</mi><mi>red</mi><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup></math> and <math><msubsup><mi>F</mi><mn>2</mn><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup></math> in the <math><msub><mi>k</mi><mi>t</mi></msub></math>-factorization formalism by using the integral form of the Kimber-Martin-Ryskin and Martin-Ryskin-Watt unintegrated parton distribution function (KMR and MRW-UPDF) with the angular ordering constraint (AOC) and the <math><mi>M</mi><mi>M</mi><mi>H</mi><mi>T</mi><mn>2014</mn></math> <math><mi>P</mi><mi>D</mi><mi>F</mi></math> set as the input. Recently Guiot and van Hameren demonstrated that the upper limit, <math><msub><mi>k</mi><mi>max</mi></msub></math>, of the transverse-momentum integration performed in the <math><msub><mi>k</mi><mi>t</mi></msub></math>-factorization formalism should be almost equal to <math><mi>Q</mi></math>, where <math><mi>Q</mi></math> is the hard scale, otherwise it leads to an overestimation of the proton structure function [<math><msub><mi>F</mi><mn>2</mn></msub><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math>]. In the present work, we show that <math><msub><mi>k</mi><mi>max</mi></msub></math> cannot be equal to <math><mi>Q</mi></math> at low and moderate energy region, and also by considering the gluon and quark contributions to the same perturbative order and a physical gauge for the gluon, i.e., <math><msup><mi>A</mi><mi>μ</mi></msup><msubsup><mi>q</mi><mi>μ</mi><mo>′</mo></msubsup><mo>=</mo><mn>0</mn></math> in the calculation of <math><msubsup><mi>F</mi><mn>2</mn><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup></math> in the <math><msub><mi>k</mi><mi>t</mi></msub></math>-factorization formalism, we do not encounter any overestimation of the theoretical predictions due to different choices of <math><msub><mi>k</mi><mi>max</mi></msub><mo>&gt;</mo><mi>Q</mi></math>. Finally, the resulted <math><msubsup><mi>σ</mi><mi>red</mi><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup></math> and <math><msubsup><mi>F</mi><mn>2</mn><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup></math> are compared to the experimental data and the theoretical predictions. In general, the extracted <math><msubsup><mi>σ</mi><mi>red</mi><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup></math> and <math><msubsup><mi>F</mi><mn>2</mn><mrow><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover></mrow></msubsup></math> based on the KMR and MRW approaches are in perfect agreement with the experimental data and theoretical predictions at high energies, but at low and moderate energies, the one developed from the KMR approach has better consistency than that of the MRW approach.

Iran

N. Olanj

Phenomenological investigation of the beauty content of a proton in the framework of <math><msub><mi>k</mi><mi>t</mi></msub></math>-factorization using Kimber-Martin-Ryskin and Martin-Ryskin-Watt unintegrated parton distributions

We scrutinize the Sommerfeld enhancement in dark matter pair annihilation for p-wave and higher-ℓ partial waves. For the Yukawa potential these feature a super-resonant Breit-Wigner peak in their velocity-dependence close to Sommerfeld resonances as well as a universal scaling with velocity for all ℓ ≥ 1 that differs from the s-wave case. We provide a quantum mechanical explanation for these phenomena in terms of quasi-bound states sustained by the centrifugal barrier of the partial-wave potential, and give approximate WKB expressions capturing the main effects. The impact of quasi-bound states is exemplified for wino dark matter and models with light mediators, with a focus on indirect detection signals. We note that quasi-bound states can also explain similar peaks in the bound-state formation and self-scattering cross sections.

Enhancement of p-wave dark matter annihilation by quasi-bound states

We discuss the imaginary-time formalism for field theories in thermal equilibrium in uniformly accelerating frames. We show that under a Wick rotation of Minkowski spacetime, the Rindler event horizon shrinks to a point in a two-dimensional subspace tangential to the acceleration direction and the imaginary time. We demonstrate that the accelerated version of the Kubo-Martin-Schwinger (KMS) condition implies an identification of all spacetime points related by integer-multiple rotations in the tangential subspace about this Euclidean Rindler event-horizon point, with the rotational quanta defined by the thermal acceleration, <math><mi>α</mi><mo>=</mo><mi>a</mi><mo>/</mo><mi>T</mi></math>. In the Wick-rotated Rindler hyperbolic coordinates, the KMS relations reduce to standard (anti-)periodic boundary conditions in terms of the imaginary proper time (rapidity) coordinate. Our findings pave the way to study, using first-principle lattice simulations, the Hawking-Unruh radiation in geometries with event horizons, phase transitions in accelerating Early Universe and early stages of quark-gluon plasma created in relativistic heavy-ion collisions.

Victor E. Ambruş

Acceleration as a circular motion along an imaginary circle: Kubo-Martin-Schwinger condition for accelerating field theories in imaginary-time formalism

We calculate directly in position space the one-loop renormalization kernels of the soft operators O  γ  and O  g  that appear in the soft-quark contributions to, respectively, the subleading-power γγ → h and gg → h form factors mediated by the b-quark. We present an IR/rapidity divergence-free definition for O  g  and demonstrate that with a correspondent definition of the collinear function, a consistent factorization theorem is recovered. Using conformal symmetry techniques, we establish a relation between the evolution kernels of the leading-twist heavy-light light-ray operator, whose matrix element defines the B-meson light-cone distribution amplitude (LCDA), and O  γ  to all orders in perturbation theory. Application of this relation allows us to bootstrap the kernel of O  γ  to the two-loop level. We construct an ansatz for the kernel of O  g  at higher orders. We test this ansatz against the consistency requirement at two-loop and find they differ only by a particular constant.

Renormalization of the next-to-leading-power γγ → h and gg → h soft quark functions

A correction to this paper has been published: https://doi.org/10.1007/JHEP03(2022)080

Patrick Hager

Robert Szafron

Erratum to: Soft-collinear gravity beyond the leading power