<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

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]

A correction to this paper has been published: https://doi.org/10.1007/JHEP01(2024)191

United Kingdom

Jonathan Davies

Italy

Martin Jung

Stefan Schacht

Erratum to:

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 propose novel methods to determine the <math><mrow><mi>ϒ</mi><mo>(</mo><mn>4</mn><mi>S</mi><mo>)</mo><mo>→</mo><msup><mrow><mi>B</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>B</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></math> and <math><mrow><mi>ϒ</mi><mo>(</mo><mn>4</mn><mi>S</mi><mo>)</mo><mo>→</mo><msup><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mover><mrow><mi>B</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>0</mn></mrow></msup></mrow></math> decay rates. The precision to which they and their ratio are known yields at present a limiting uncertainty around 2% in measurements of absolute <math><mi>B</mi></math> decay rates, and thus in a variety of applications, such as precision determinations of elements of the Cabibbo-Kobayashi-Maskawa matrix and flavor-symmetry relations. The new methods we propose are based in one case on exploiting the <math><mi>ϒ</mi><mo>(</mo><mn>5</mn><mi>S</mi><mo>)</mo></math> datasets, in the other case on the different average number of charged tracks in <math><msup><mi>B</mi><mo>±</mo></msup></math> and <math><msup><mi>B</mi><mn>0</mn></msup></math> decays. We estimate future sensitivities using these methods and discuss possible measurements of <math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>/</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>u</mi></mrow></msub></mrow></math> at the (HL-)LHC.

Germany

Florian Bernlochner

Munira Khan

United States

Greg Landsberg

Zoltan Ligeti

Novel approaches to determine <math><mrow><msup><mrow><mi>B</mi></mrow><mrow><mo>±</mo></mrow></msup></mrow></math> and <math><msup><mi>B</mi><mn>0</mn></msup></math> meson production fractions

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 perform a detailed model-independent phenomenological analysis of $$ \overline{B}\to \overline{D}D $$ decays. Employing an SU(3) analysis including symmetry-breaking contributions together with a conservative power counting for various suppression effects, we obtain updated Standard Model predictions for all branching fractions and CP asymmetries from a fit to the available experimental data, testable at Belle II and the LHC experiments. These results include all relevant suppressed contributions, thereby providing upper limits on subleading Standard Model (SM) effects like “penguin pollution”, enabling searches for physics beyond the SM. Importantly, allowing in the same fit for the production fractions of charged and neutral $\textit{B}$ mesons to be different, we find $\textit{f}$ /$\textit{f}$ = 0$\textit{.}$86 ± 0$\textit{.}$05, which is 2$\textit{.}$5$\textit{σ}$ away from unity, which, if confirmed, would have important phenomenological consequences.

$$ \overline{B}\to \overline{D}D $$ decays and the extraction of $\textit{f}$

In this work, we study the structure of the leading order Martin–Ryskin–Watt (MRW) unintegrated parton distribution function (UPDF) and explain in detail why there exists discrepancy between the two different definitions of this UPDF model, i.e., the integral (I-MRW) and differential (D-MRW) MRW UPDFs. We perform this investigation with both angular and strong ordering cutoffs. The derivation footsteps of obtaining the I-MRW UPDF from the D-MRW ones are numerically performed, and the reason of such non-equivalency between the two forms is clearly explained. We show and find out that both methods suggested in the papers by Golec-Biernat and Staśto as well as that of Guiot have shortcomings, and only the combination of their prescriptions can give us the same UPDF structure from both of these two different versions of the MRW UPDF, namely I-MRW and the D-MRW UPDFs.

Ramin Kord Valeshabadi

M. Modarres

On the ambiguity between differential and integral forms of the Martin–Ryskin–Watt unintegrated parton distribution function model

We analyze in detail the angular distributions in  $${\bar{B}}\rightarrow D^*\ell {{\bar{\nu }}}$$  <math> <mrow> <mover> <mrow> <mi>B</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>→</mo> <msup> <mi>D</mi> <mo>∗</mo> </msup> <mi>ℓ</mi> <mover> <mrow> <mi>ν</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   decays, with a focus on lepton-flavour non-universality. We investigate the minimal number of angular observables that fully describes current and upcoming datasets, and explore their sensitivity to physics beyond the Standard Model (BSM) in the most general weak effective theory. We apply our findings to the current datasets, extract the non-redundant set of angular observables from the data, and compare to precise SM predictions that include lepton-flavour universality violating mass effects. Our analysis shows that the number of independent angular observables that can be inferred from current experimental data is limited to only four. These are insufficient to extract the full set of relevant BSM parameters. We uncover a  $$\sim 4\sigma $$  <math> <mrow> <mo>∼</mo> <mn>4</mn> <mi>σ</mi> </mrow> </math>   tension between data and predictions that is hidden in the redundant presentation of the Belle 2018 data on  $${\bar{B}}\rightarrow D^*\ell {{\bar{\nu }}}$$  <math> <mrow> <mover> <mrow> <mi>B</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>→</mo> <msup> <mi>D</mi> <mo>∗</mo> </msup> <mi>ℓ</mi> <mover> <mrow> <mi>ν</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   decays. This tension specifically involves observables that probe  $$e-\mu $$  <math> <mrow> <mi>e</mi> <mo>-</mo> <mi>μ</mi> </mrow> </math>   lepton-flavour universality. However, we find inconsistencies in these data, which renders results based on it suspicious. Nevertheless, we discuss which generic BSM scenarios could explain the tension, in the case that the inconsistencies do not affect the data materially. Our findings highlight that  $$e-\mu $$  <math> <mrow> <mi>e</mi> <mo>-</mo> <mi>μ</mi> </mrow> </math>   non-universality in the SM, introduced by the finite muon mass, is already significant in a subset of angular observables with respect to the experimental precision.

Christoph Bobeth

Marzia Bordone

Nico Gubernari

Danny Dyk

Lepton-flavour non-universality of  $${\bar{B}}\rightarrow D^*\ell {{\bar{\nu }}}$$  <math> <mrow> <mover> <mrow> <mi>B</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>→</mo> <msup> <mi>D</mi> <mo>∗</mo> </msup> <mi>ℓ</mi> <mover> <mrow> <mi>ν</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   angular distributions in and beyond the Standard Model

We provide updated predictions for the hadronic decays  $$\bar{B}_s^0\rightarrow D_s^{(*)+} \pi ^-$$  <math> <mrow> <msubsup> <mover> <mrow> <mi>B</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mi>s</mi> <mn>0</mn> </msubsup> <mo>→</mo> <msubsup> <mi>D</mi> <mi>s</mi> <mrow> <mo>(</mo> <mrow></mrow> <mo>∗</mo> <mo>)</mo> <mo>+</mo> </mrow> </msubsup> <msup> <mi>π</mi> <mo>-</mo> </msup> </mrow> </math>   and  $$\bar{B}^0\rightarrow D^{(*)+} K^-$$  <math> <mrow> <msup> <mrow> <mover> <mrow> <mi>B</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mn>0</mn> </msup> <mo>→</mo> <msup> <mi>D</mi> <mrow> <mo>(</mo> <mrow></mrow> <mo>∗</mo> <mo>)</mo> <mo>+</mo> </mrow> </msup> <msup> <mi>K</mi> <mo>-</mo> </msup> </mrow> </math>  . They are based on  $${\mathcal {O}}(\alpha _s^2)$$  <math> <mrow> <mi>O</mi> <mo>(</mo> <msubsup> <mi>α</mi> <mi>s</mi> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </math>   results for the QCD factorization amplitudes at leading power and on recent results for the  $$\bar{B}_{(s)} \rightarrow D_{(s)}^{(*)}$$  <math> <mrow> <msub> <mover> <mrow> <mi>B</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> <mo>→</mo> <msubsup> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow></mrow> <mo>∗</mo> <mo>)</mo> </mrow> </msubsup> </mrow> </math>   form factors up to order  $$\mathcal{O}(\Lambda _\mathrm{QCD}^2/m_c^2)$$  <math> <mrow> <mi>O</mi> <mo>(</mo> <msubsup> <mi>Λ</mi> <mrow> <mi>QCD</mi> </mrow> <mn>2</mn> </msubsup> <mo>/</mo> <msubsup> <mi>m</mi> <mi>c</mi> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </math>   in the heavy-quark expansion. We give quantitative estimates of the matrix elements entering the hadronic decay amplitudes at order  $$\mathcal{O}(\Lambda _\mathrm{QCD}/m_b)$$  <math> <mrow> <mi>O</mi> <mo>(</mo> <msub> <mi>Λ</mi> <mi>QCD</mi> </msub> <mo>/</mo> <msub> <mi>m</mi> <mi>b</mi> </msub> <mo>)</mo> </mrow> </math>   for the first time. Our results are very precise, and uncover a substantial discrepancy between the theory predictions and the experimental measurements. We explore two possibilities for this discrepancy: non-factorizable contributions larger than predicted by the QCD factorization power counting, and contributions beyond the Standard Model. We determine the  $$f_s/f_d$$  <math> <mrow> <msub> <mi>f</mi> <mi>s</mi> </msub> <mo>/</mo> <msub> <mi>f</mi> <mi>d</mi> </msub> </mrow> </math>   fragmentation fraction for the CDF, D0 and LHCb experiments for both scenarios.

Tobias Huber

A puzzle in  $$\varvec{\bar{B}_{(s)}^0 \rightarrow D_{(s)}^{(*)+} \lbrace \pi ^-, K^-\rbrace }$$  <math> <mrow> <msubsup> <mover> <mrow> <mi>B</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mn>0</mn> </msubsup> <mo>→</mo> <msubsup> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow></mrow> <mo>∗</mo> <mo>)</mo> <mo>+</mo> </mrow> </msubsup> <mrow> <mo>{</mo> <msup> <mi>π</mi> <mo>-</mo> </msup> <mo>,</mo> <msup> <mi>K</mi> <mo>-</mo> </msup> <mo>}</mo> </mrow> </mrow> </math>   decays and extraction of the  $$\varvec{f_s/f_d}$$  <math> <mrow> <msub> <mi>f</mi> <mi>s</mi> </msub> <mo>/</mo> <msub> <mi>f</mi> <mi>d</mi> </msub> </mrow> </math>   fragmentation fraction