<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

The GLR-MQ equation is a nonlinear evolution equation that takes into account the shadowing effect, which tames the growth of the gluon at small-x. In this study, we analytically solve for the first time the nonlinear GLR-MQ equation using the homogeneous balance method. The definite solution of the GLR-MQ equation is obtained by fitting the MSTW2008LO gluon distribution data. We find that the geometric scaling is an inherent property of our analytical solution. Furthermore, the gluon distribution functions derived from our solution are capable of accurately reproducing the MSTW2008LO data across most regions. These results indicate that our analytical solution from the homogeneous balance method is valid to describe the gluon behavior at small-x and moderate <math><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup></math> regions. Moreover, the saturation scale <math><msub><mrow><mi>Q</mi></mrow><mrow><mi>s</mi></mrow></msub></math> has been extracted from our analytical solution, we find that the energy-dependent saturation scale obeys the exponential law <math><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mspace width="0.2em"></mspace><mo>∝</mo><mspace width="0.2em"></mspace><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup><msup><mrow><mi>e</mi></mrow><mrow><mi>λ</mi><mi>Y</mi></mrow></msup></math>. We also perform a numerical simulation using the analytical results at <math><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>5</mn><mspace width="0.25em"></mspace><msup><mrow><mtext>GeV</mtext></mrow><mrow><mn>2</mn></mrow></msup></math> as the initial condition. We find that our analytical results are consistent with the numerical results in small-x.

China

Yanbing Cai

Xiaopeng Wang

Xurong Chen

New analytical solution of the nonlinear Gribov-Levin-Ryskin-Mueller-Qiu equation

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 numerically study for the first time the nonlinear GLR-MQ evolution equations for nuclear parton distribution function (nPDFs) to next-to-leading order accuracy and quantify the impact of gluon recombination at small <math><mi>x</mi></math>. Using the nCTEQ15 nPDFs as input, we confirm the importance of the nonlinear corrections for small <math><mi>x</mi><mo>≲</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math>, whose magnitude increases with a decrease of <math><mi>x</mi></math> and an increase of the atomic number <math><mi>A</mi></math>. We find that at <math><mi>x</mi><mo>=</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>5</mn></mrow></msup></math> and for heavy nuclei, after the upward evolution from <math><msub><mi>Q</mi><mn>0</mn></msub><mo>=</mo><mn>2</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> to <math><mi>Q</mi><mo>=</mo><mn>10</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, the quark singlet <math><mi>Ω</mi><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> and the gluon <math><mi>G</mi><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> distributions become reduced by 9%–15%, respectively. The relative effect is much stronger for the downward evolution from <math><msub><mi>Q</mi><mn>0</mn></msub><mo>=</mo><mn>10</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> to <math><mi>Q</mi><mo>=</mo><mn>2</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, where we find that <math><mi>Ω</mi><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> is suppressed by 40%, while <math><mi>G</mi><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> is enhanced by 140%. These trends propagate into the <math><msubsup><mi>F</mi><mn>2</mn><mi>A</mi></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> nuclear structure function and the <math><msubsup><mi>F</mi><mi>L</mi><mi>A</mi></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> longitudinal structure function, which after the downward evolution become reduced by 45% and enhanced by 80%, respectively. Our analysis indicates that the nonlinear effects are most pronounced in <math><msubsup><mi>F</mi><mi>L</mi><mi>A</mi></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> and are already quite sizable at <math><mi>x</mi><mo>∼</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math> for heavy nuclei. We have checked that our conclusions very weakly depend on the choice of input nPDFs. In particular, using the EPPS21 nPDFs as input, we obtain quantitatively similar results.

Germany

J. Rausch

Finland

V. Guzey

Australia

M. Klasen

Numerical evaluation of the nonlinear Gribov-Levin-Ryskin-Mueller-Qiu evolution equations for nuclear parton distribution functions

We develop a novel Monte Carlo parton branching algorithm based on the Gribov-Levin-Ryskin (GLR) equation. The formulations of both forward evolution and backward evolution for the GLR equation are presented. The results from the Monte Carlo implementation of the GLR equation are in full agreement with its numerical solutions. Our work thus paves the way for developing an event generator that embodies the saturation effect.

Yu Shi

Shu-Yi Wei

Jian Zhou

Parton shower generator based on the Gribov-Levin-Ryskin equation

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 causes which can limit the accuracy of the predictions based on the conventional PDFs when including in global parton analyses the data at moderate scales $\mu $. The first is the existence of power corrections ${\cal O} (Q_0^2/\mu ^2)$ due to the double counting of contributions arising from the region below the input scale $Q_0$. The second concerns the possible inclusion of the BFKL re-summation of the $(\alpha _{\mathrm {s}}\ln (1/x))^n$ terms. The third is the treatment of the heavy-quark thresholds. We show how to include the heavy-quark masses ($m_h$ with $h=c,b,t$) in DGLAP evolution which provides the correct <span class="it">smooth</span> behaviour through the threshold regions and how to subtract the low parton virtuality $|k^2|\lt Q^2_0$ contributions from the DIS and Drell–Yan NLO coefficient functions in order to avoid the double counting.

United Kingdom

A.D. Martin

M.G. Ryskin

Comments on Global Parton Analyses

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>

We argue that if the elastic proton–proton cross section increases with energy, the Froissart-like high energy behaviour of the elastic amplitude (which corresponds to a ‘black disk’ of radius <math><mi>R</mi><mo>(</mo><mi>s</mi><mo>)</mo><mo>=</mo><mi>c</mi><mi>ln</mi><mo>⁡</mo><mi>s</mi><mo>−</mo><mi>β</mi><mi>ln</mi><mo>⁡</mo><mo>(</mo><mi>ln</mi><mo>⁡</mo><mi>s</mi><mo>)</mo></math>) is the only possibility to satisfy the unitarity equation at each value of the impact parameter, b. Otherwise the cross section of events with Large Rapidity Gaps grows faster than the total cross section at the same b. That is, these ‘gap’ events require maximal growth of the high-energy (asymptotic) cross section and of the interaction radius <math><mi>R</mi><mo>(</mo><mi>s</mi><mo>)</mo></math> in order to be consistent with unitarity.