<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]

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 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

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 discuss the different Kimber-Martin-Ryskin (KMR) prescriptions for unintegrated parton distribution functions (uPDFs). We show that the strong-ordering (SO) and the angular-ordering (AO) cutoffs lead to strong discrepancies between the obtained cross sections. While the result obtained with the AO cutoff overestimates the heavy-flavor cross section by about a factor of 3, the SO cutoff gives the correct answer. We also solve the issue of the KMR uPDF definitions mentioned by Golec-Biernat and Staśto [Phys. Lett. B 781, 633 (2018)] and show that, in the case of the AO cutoff, the KMR uPDFs are ill defined.

Chile

Benjamin Guiot

Pathologies of the Kimber-Martin-Ryskin prescriptions for unintegrated PDFs: Which prescription should be preferred?

We previously investigated the compatibility of the $${\textit{Kimber}}$$ <math><mi>Kimber</mi></math> –$${\textit{Martin}}$$ <math><mi>Martin</mi></math> –$${\textit{Ryskin}}$$ <math><mi>Ryskin</mi></math>  ($${\textit{KMR}}$$ <math><mi>KMR</mi></math> ) $${\textit{unintegrated}}$$ <math><mi>unintegrated</mi></math>  parton distribution functions ($${\textit{UPDF}}$$ <math><mi>UPDF</mi></math> ) with the experimental data on the proton (longitudinal) structure functions ($${\textit{PSF}}$$ <math><mi>PSF</mi></math>  ($${\textit{PLSF}}$$ <math><mi>PLSF</mi></math> )). Recently $${\textit{Golec-Biernat}}$$ <math><mrow><mi>Golec</mi><mo>-</mo><mi>Biernat</mi></mrow></math>  and $${\textit{Stasto}}$$ <math><mi>Stasto</mi></math>  ($${\textit{GBS}}$$ <math><mi>GBS</mi></math> ) demonstrated that the differential version of $${\textit{KMR}}$$ <math><mi>KMR</mi></math>  prescription and the implementations of angular (strong) ordering ($${\textit{AOC}}$$ <math><mi>AOC</mi></math>  ($${\textit{SOC}}$$ <math><mi>SOC</mi></math> )) constraints, cause the negative-discontinuous $${\textit{UPDF}}$$ <math><mi>UPDF</mi></math>  with the ordinary parton distribution functions ($${\textit{PDF}}$$ <math><mi>PDF</mi></math> ) as the input, which leads to a sizable effect on the calculation of $${\textit{PSF}}$$ <math><mi>PSF</mi></math> . In the present work, we use the new $${\textit{MMHT2014}}$$ <math><mrow><mi>MMHT</mi><mn>2014</mn></mrow></math> -$${\textit{LO}}$$ <math><mi>LO</mi></math> -$${\textit{PDF}}$$ <math><mi>PDF</mi></math>  as the input and focus on the $${\textit{UPDF}}$$ <math><mi>UPDF</mi></math>  behaviors as was raised by $${\textit{GBS}}$$ <math><mi>GBS</mi></math> . The resulting $${\textit{PSF}}$$ <math><mi>PSF</mi></math>  and $${\textit{PLSF}}$$ <math><mi>PLSF</mi></math>  are compared with the $${\textit{MSTW2008}}$$ <math><mrow><mi>MSTW</mi><mn>2008</mn></mrow></math> -$${\textit{LO}}$$ <math><mi>LO</mi></math> -$${\textit{PDF}}$$ <math><mi>PDF</mi></math>  and $${\textit{MRST99}}$$ <math><mrow><mi>MRST</mi><mn>99</mn></mrow></math> -$${\textit{PDF}}$$ <math><mi>PDF</mi></math>  and the 2014 data given by the $${\textit{ZEUS}}$$ <math><mi>ZEUS</mi></math>  and $${\textit{H1}}$$ <math><mrow><mi>H</mi><mn>1</mn></mrow></math>  collaborations. The calculated $${\textit{PSF}}$$ <math><mi>PSF</mi></math>  and $${\textit{PLSF}}$$ <math><mi>PLSF</mi></math>  based on the integral prescription of the $${\textit{KMR}}$$ <math><mi>KMR</mi></math> -$${\textit{UPDF}}$$ <math><mi>UPDF</mi></math>  with the $${\textit{AOC}}$$ <math><mi>AOC</mi></math>  and the ordinary $${\textit{PDF}}$$ <math><mi>PDF</mi></math>  as the input are reasonably consistent with the experimental data. Therefore, they are approximately independent to the $${\textit{PDF}}$$ <math><mi>PDF</mi></math>  (no need to impose cutoff on the $${\textit{PDF}}$$ <math><mi>PDF</mi></math> ). At very small x regions because of the excess of gluons in the $${\textit{MMHT2014}}$$ <math><mrow><mi>MMHT</mi><mn>2014</mn></mrow></math> -$${\textit{LO}}$$ <math><mi>LO</mi></math> -$${\textit{PDF}}$$ <math><mi>PDF</mi></math>  and $${\textit{MSTW2008}}$$ <math><mrow><mi>MSTW</mi><mn>2008</mn></mrow></math> -$${\textit{LO}}$$ <math><mi>LO</mi></math> -$${\textit{PDF}}$$ <math><mi>PDF</mi></math> , an increase in $${\textit{PSF}}$$ <math><mi>PSF</mi></math>  and $${\textit{PLSF}}$$ <math><mi>PLSF</mi></math>  is achieved. Finally, according to the $${\textit{GBS}}$$ <math><mi>GBS</mi></math>  report the differential version by using the cutoff independent $${\textit{PDF}}$$ <math><mi>PDF</mi></math>  produces results far from the experimental data.

A phenomenological investigation of the integral and the differential versions of the $${\textit{Kimber}}$$ <math><mi>Kimber</mi></math> –$${\textit{Martin}}$$ <math><mi>Martin</mi></math> –$${\textit{Ryskin}}$$ <math><mi>Ryskin</mi></math>  $${\textit{unintegrated}}$$ <math><mi>unintegrated</mi></math>  parton distribution functions using two different constraints and the $${\textit{MMHT}}2014$$ <math><mrow><mi>MMHT</mi><mn>2014</mn></mrow></math>  $${\textit{PDF}}$$ <math><mi>PDF</mi></math>

A natural star product for 4-d <math><mi>κ</mi></math>-Minkowski space is used to investigate various classes of <math><mi>κ</mi></math>-Poincaré invariant scalar field theories with quartic interactions whose commutative limit coincides with the usual <math><msup><mi>ϕ</mi><mn>4</mn></msup></math> theory. <math><mrow><mi>κ</mi></mrow></math>-Poincaré invariance forces the integral involved in the actions to be a twisted trace, thus defining a Kubo-Martin-Schwinger (KMS) weight for the noncommutative (<math><mrow><msup><mrow><mi>C</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math>-)algebra modeling the <math><mi>κ</mi></math>-Minkowski space. In all the field theories, the twist generates different planar one-loop contributions to the 2-point function which are at most UV linearly diverging. Some of these theories are free of UV/IR mixing. In the others, UV/IR mixing shows up in non-planar contributions to the 2-point function at exceptional zero external momenta while staying finite at nonzero external momenta. These results are discussed together with the possibility for the KMS weight relative to the quantum space algebra to trigger the appearance of KMS state on the algebra of observables.

Timothé Poulain

Jean-Christophe Wallet

<math><mi>κ</mi></math>-Poincaré invariant quantum field theories with Kubo-Martin-Schwinger weight

For the present study, we have used the Martin-like potential for the quark confinement. Our predicted states in the S-wave, 23S1 (2605.86 MeV) and 21S0 (2521.72 MeV), are in very good agreement with experimental results of 2608±2.4±2.5 MeV and 2539.4±4.5±6.8 MeV, respectively, reported by the BABAR Collaboration. The calculated P-wave D meson states, 13P2 (2462.50 MeV), 13P1 (2407.56 MeV), 13P0 (2373.82 MeV) and 11P1 (2423.28 MeV), are in close agreement with experimental average (Particle Data Group) values of 2462.6±0.7 MeV, 2427±26±25 MeV, 2318±29 MeV and 2421.3±0.6 MeV, respectively. The pseudoscalar decay constant ( fP = 202.57 MeV) of the D meson is in very good agreement with the experiment as well as with the lattice predictions. The Cabibbo favoured nonleptonic decay branching ratios, BR (D0→K-π+) of 4.071% and BR (D0→K+π-) of 1.135×10-4 , are also in very good agreement with the respective experimental values of 3.91±0.08% and (1.48±0.07)×10-4 reported by CLEO Collaboration. The mixing parameters of the D0 – D¯0 oscillation, xq (5.14 ×10-3 ), yq (6.02 ×10-3 ) and RM (3.13 ×10-5 ), are in very good agreement with BaBar and Belle Collaboration results.

India

Manan Shah

Bhavin Patel

P.C. Vinodkumar

D meson spectroscopy and their decay properties using Martin potential in a relativistic Dirac formalism

We study photon-ALP conversion by resonance effects in the magnetized plasma of galaxy clusters and compare the predicted distortion of the cosmic microwave background spectrum in the direction of such objects to measurements of the thermal Sunyaev-Zeldovich effect. Using galaxy cluster models based on current knowledge, we obtain upper limits on the photon-ALP coupling constant g of â²Â ðª(10â11 GeVâ1). The constraints apply to the mass range of 2Â Â·Â 10â14 eV â²Â mALPÂ â²Â 3Â Â·Â 10â12 eV in which resonant photon-ALP conversions can occur. These limits are slightly stronger than current limits, and furthermore provide an independent constraint. We find that a next generation PRISM-like experiment would allow limits down to gÂ âÂ ðªÂ (10â14 GeVâ1), two orders of magnitude stronger than the currently strongest limits in this mass range.