<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

Renormalization group flows are studied between 5d SCFTs engineered by (p, q) 5-brane webs with large numbers of external 5-branes. A general expression for the free energy on S 5 in terms of single-valued trilogarithm functions is derived from their supergravity duals, which are characterized by the 5-brane charges and additional geometric parameters. The additional geometric parameters are fixed by regularity conditions, and we show that the solutions to the regularity conditions extremize a trial free energy. These results are used to survey a large sample of   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (105) renormalization group flows between different 5d SCFTs, including Higgs branch flows and flows that preserve the SU(2) R- symmetry. In all cases the free energy changes monotonically towards the infrared, in line with a 5d F -theorem.

United States

Martin Fluder

Christoph Uhlemann

Evidence for a 5d F-theorem

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?

The infrared formula relates the Schur index of a 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 theory to its wall-crossing invariant, a.k.a. BPS monodromy. A further extension of this formula, proposed by Córdova, Gaiotto and Shao, includes contributions by various types of line and surface defects. We study BPS monodromies in the presence of vortex surface defects of arbitrary vorticity for general class   <math> <mi>S</mi> </math>  $$ \mathcal{S} $$  theories of type A 1 engineered by UV curves with at least one regular puncture. The trace of these defect BPS monodromies is shown to coincide with the action of certain q-difference operators acting on the trace of the (pure) 4d BPS monodromy. We use these operators to develop a “bootstrap” (of traces) of BPS monodromies, relying only on their infrared properties, thereby reproducing the very general ultraviolet characterization of the Schur index.

Japan

Switzerland

Pietro Longhi

An infrared bootstrap of the Schur index with surface defects

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>

We use recently established AdS6/CFT5 dualities to count the microstates of magnetically charged AdS6 × S 2 × Σ black holes in Type IIB. The near-horizon limit is described by solutions with AdS2 ×   <math> <msub> <mi>Σ</mi> <msub> <mi>g</mi> <mn>1</mn> </msub> </msub> </math>  $$ {\varSigma}_{{\mathfrak{g}}_1} $$  ×   <math> <msub> <mi>Σ</mi> <msub> <mi>g</mi> <mn>2</mn> </msub> </msub> </math>  $$ {\varSigma}_{{\mathfrak{g}}_2} $$  × S 2 × Σ geometry, where   <math> <msub> <mi>Σ</mi> <msub> <mi>g</mi> <mi>i</mi> </msub> </msub> </math>  $$ {\varSigma}_{{\mathfrak{g}}_i} $$  are Riemann surfaces of constant curvature and Σ is a further Riemann surface over which the geometry is warped. Our results show that the topologically twisted indices of the proposed dual superconformal field theories precisely reproduce the Bekenstein-Hawking entropy of this class of black holes. This provides further support for a prescription to compute fivedimensional topologically twisted indices put forth recently, and for the proposed dualities. We confirm the N 4 scaling found in the sphere partition functions and extend previous matches of sphere partition functions to AdS6 solutions with monodromy.