We present the first complete next-to-leading-order calculation of the impact factors for hadroproduction of the $$ {}^1{S}_0^{\left[1\right]} $$ , $$ {}^1{S}_0^{\left[8\right]} $$ , and $$ {}^3{S}_1^{\left[8\right]} $$ NRQCD states within the BFKL formalism. We complete the recent virtual-correction computation presented in JHEP 12 (2024) 129 by that of the real-emission contributions. We observe the cancellation of the soft divergences between these real- and virtual-emission contributions and we note that the surviving collinear singularities are compatible with factorisation up to one loop for a novel class of processes where BFKL resummation can be applied. Our work indeed represents the first complete NLO quarkonium impact factor in the BFKL framework and paves the way to first next-to-leading-logarithmic-precision studies for hadroproduction of forward-backward quarkonium associated production at hadron colliders.

Poland

Michael Fucilla

France

Jean-Philippe Lansberg

Israel

Maxim Nefedov

Lech Szymanowski

Samuel Wallon

Complete NLO BFKL impact factors for quarkonium hadroproduction in NRQCD: the case of $$ ^1{S}_0^{\left[1\right]} $$ , $$ ^1{S}_0^{\left[8\right]} $$ , and $$ ^3{S}_1^{\left[8\right]} $$ states

We present an extension of the MadGraph5_aMC@NLO framework that enables the automated calculation of leading-order cross sections for S-wave quarkonium and leptonium production within the non-relativistic QCD (NRQCD) and non-relativistic QED (NRQED) factorisation formalisms. The framework has been validated against a variety of benchmark processes, demonstrating robustness and flexibility for phenomenological studies. A key advantage of this implementation is its seamless integration with existing MadGraph5_aMC@NLO features, allowing computations not only within the Standard Model but also in a wide range of Beyond the Standard Model or Effective Field Theory scenarios via a modified Universal Feynman Output (UFO) interface. Furthermore, the framework maintains compatibility with standard Monte Carlo event generators for parton showering and hadronisation. Through numerous examples, we highlight that theoretical studies of quarkonium processes require careful consideration: the impact of subleading contributions is often difficult to predict using simple counting arguments based solely on the hierarchy of couplings and velocity-scaling rules.

Alice Serri

Belgium

Chris Flett

Olivier Mattelaer

Hua-Sheng Shao

Lukas Simon

Automated event generation for S-wave quarkonium and leptonium production in NRQCD and NRQED

We explore the possibility of using ultra-periphe- ral proton-lead collisions at the LHC to study inclusive vector-quarkonium photoproduction, that occurs when a quasi-real photon emitted by a fully stripped lead ion breaks a proton to produce a vector quarkonium. Owing to the extremely large energies of the colliding hadrons circulating in the LHC, the range of accessible photon-nucleon centre-of-mass energies, $$W_{\gamma p}$$ , largely exceeds what has been and will be studied at lepton-hadron colliders, HERA and the EIC. We perform a tune to HERA photoproduction data, use this tune to predict the yields of photoproduced $$J/\psi $$ , and estimate the corresponding transverse-momentum reach at LHC experiments. We also model the hadroproduction background and demonstrate that inclusive photoproduction can be isolated at the LHC from such background by imposing constraints on the hadronic activity in the lead-going direction at mid, forward, or far-forward rapidities depending on the capability of the detector under consideration. We find that the resulting cross sections are large enough to be measured by ALICE, ATLAS, CMS, and LHCb. We estimate the background-to-signal ratio after isolation to be of the order of 0.001 and 0.1 in the low and large transverse-momentum regions, respectively. In addition, we propose and assess the Jacquet–Blondel method to reconstruct the photon-nucleon centre-of-mass energy and the fractional energy of the quarkonium with respect to the photon.

Ireland

Kate Lynch

Spain

Charlotte Hulse

Ronan McNulty

Inclusive photoproduction of vector quarkonium in ultra-peripheral collisions at the LHC

We cure the perturbative instability of the total-inclusive-photoproduction cross sections of vector S-wave quarkonia observed at high photon-proton-collision energies ( $$\sqrt{s_{\gamma p}}$$  <math> <msqrt> <msub> <mi>s</mi> <mrow> <mi>γ</mi> <mi>p</mi> </mrow> </msub> </msqrt> </math>  ) in Next-to-Leading Order (NLO) Collinear-Factorisation (CF) computations. This is achieved using High-Energy Factorisation (HEF) in the Doubly-Logarithmic Approximation (DLA), which is a subset of the Leading-Logarithmic Approximation (LLA) of HEF which resums higher-order QCD corrections proportional to  $$\alpha _s^n \ln ^{n-1} ({\hat{s}}{/}M^2)$$  <math> <mrow> <msubsup> <mi>α</mi> <mi>s</mi> <mi>n</mi> </msubsup> <msup> <mo>ln</mo> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>s</mi> <mo>^</mo> </mover> <mo>/</mo> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </math>   in the Regge limit  $${\hat{s}}\gg M^2$$  <math> <mrow> <mover> <mi>s</mi> <mo>^</mo> </mover> <mo>≫</mo> <msup> <mi>M</mi> <mn>2</mn> </msup> </mrow> </math>   with  $$M^2$$  <math> <msup> <mi>M</mi> <mn>2</mn> </msup> </math>   being the quarkonium mass and  $${\hat{s}}$$  <math> <mover> <mi>s</mi> <mo>^</mo> </mover> </math>   is the squared partonic-center-of-mass energy. Such a DLA is strictly consistent with the NLO and NNLO DGLAP evolutions of the parton distribution functions. By improving the treatment of the large- $${\hat{s}}$$  <math> <mover> <mi>s</mi> <mo>^</mo> </mover> </math>   asymptotics of the CF coefficient function, the resummation cures the unphysical results of the NLO CF calculation. The matching is directly performed in  $${\hat{s}}$$  <math> <mover> <mi>s</mi> <mo>^</mo> </mover> </math>   space using the Inverse-Error Weighting matching procedure which avoids any possible double counting. The obtained cross sections are in good agreement with data. In addition, the scale-variation uncertainty of the matched result is significantly reduced compared to the LO results. Our calculations also yield closed-form analytic limits for  $${\hat{s}}\gg M^2$$  <math> <mrow> <mover> <mi>s</mi> <mo>^</mo> </mover> <mo>≫</mo> <msup> <mi>M</mi> <mn>2</mn> </msup> </mrow> </math>   of the NLO partonic CF and numerical limits for contributions to those at NNLO scaling like  $$\alpha _s^2 \ln ({\hat{s}}/M^2)$$  <math> <mrow> <msubsup> <mi>α</mi> <mi>s</mi> <mn>2</mn> </msubsup> <mo>ln</mo> <mrow> <mo>(</mo> <mover> <mi>s</mi> <mo>^</mo> </mover> <mo>/</mo> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </math>  .

Melih Ozcelik

Curing the high-energy perturbative instability of vector-quarkonium-photoproduction cross sections at order  $$\alpha \alpha _s^3$$  <math> <mrow> <mi>α</mi> <msubsup> <mi>α</mi> <mi>s</mi> <mn>3</mn> </msubsup> </mrow> </math>   with high-energy factorisation

The goal of this white paper is to give a comprehensive overview of the rich field of forward physics. We discuss the occurrences of BFKL re-summation effects in special final states, such as Mueller–Navelet jets, jet–gap–jets, and heavy quarkonium production. It further addresses TMD factorization at low $x$ and the manifestation of a semi-hard saturation scale in (generalized) TMD PDFs. More theoretical aspects of low-$x$ physics, probes of the quark–gluon plasma, as well as the possibility to use photon–hadron collisions at the LHC to constrain hadronic structure at low $x$, and the resulting complementarity between LHC and the EIC are also presented. We also briefly discuss diffraction at colliders as well as the possibility to explore further the electroweak theory in central exclusive events using the LHC as a photon–photon collider.

Mexico

Martin Hentschinski

United States

Christophe Royon

Marco Alcazar Peredo

Cristian Baldenegro

Italy

Andrea Bellora

Renaud Boussarie

Francesco Giovanni Celiberto

Türkiye

Salim Cerci

Portugal

Grigorios Chachamis

Czechia

J.G. Contreras

Brazil

Sylvain Fichet

Gero von Gersdorff

Germany

Pablo González

Andreas van Hameren

Jamal Jalilian-Marian

Mats Kampshoff

United Kingdom

Valery Khoze

Michael Klasen

Spencer Robert Klein

Georgios Krintiras

Piotr Kotko

Krzysztof Kutak

Emilie Li

Constanin Loizides

Mohammed M.A. Mohammed

Melih A. Ozcelik

Alessandro Papa

CERN

Michael Pitt

Agustin Sabio Vera

Jens Salomon

Sebastian Sapeta

Gustavo Gil da Silveira

Victor Paulo Gonçalves

Mark Strikman

Deniz Sunar Cerci

Daniel Tapia Takaki

Marek Taševský

White Paper on Forward Physics, BFKL, Saturation Physics and Diffraction

We revisit inclusive <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> and ϒ photoproduction at lepton-hadron colliders, namely in the limit when the exchanged photon is quasi real. Our computation includes the next-to-leading-order (NLO) <math><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub></math> corrections to the leading-order contributions in v. Similarly to the case of NLO charmonium-hadroproduction processes, the resulting cross sections obtained in the <math><mover><mrow><mtext>MS</mtext></mrow><mo>‾</mo></mover></math> factorisation scheme are sometimes found to be negative. We show that the scale-fixing criterion which we derived in a previous study of <math><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub></math> production successfully solves this problem from the EicC all the way up to the FCC-eh energies. We then elaborate on how to study a scale uncertainty akin to that derived by scale variations when one fixes a scale. In turn, we investigate where both <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> and ϒ photoproduction could be used to improve our knowledge of gluon content of the proton at scales as low as a couple of GeV.

Alice Colpani Serri

China

Yu Feng

Carlo Flore

Yelyzaveta Yedelkina

Revisiting NLO QCD corrections to total inclusive / And ϒ Photoproduction Cross Sections in Lepton-proton Collisions

The energy dependence of the total hadroproduction cross section of pseudoscalar quarkonia is computed via matching Next-to-Leading Order (NLO) Collinear-Factorisation (CF) results with resummed higher-order corrections, proportional to   <math> <msubsup> <mi>α</mi> <mi>s</mi> <mi>n</mi> </msubsup> <msup> <mo>ln</mo> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </math>  $$ {\alpha}_s^n{\ln}^{n-1} $$ (1/z), to the CF hard-scattering coefficient, where z = M 2 /   <math> <mover> <mi>s</mi> <mo>̂</mo> </mover> </math>  $$ \hat{s} $$  with M and   <math> <mover> <mi>s</mi> <mo>̂</mo> </mover> </math>  $$ \hat{s} $$  being the quarkonium mass and the partonic center-of-mass energy squared. The resummation is performed using High-Energy Factorisation (HEF) in the Doubly-Logarithmic (DL) approximation, which is a subset of the leading logarithmic ln(1/z) approximation. Doing so, one remains strictly consistent with the NLO and NNLO DGLAP evolution of the PDFs. By improving the treatment of the small-z asymptotics of the CF coefficient function, the resummation cures the unphysical results of the NLO CF calculation. The matching is directly performed in the z-space and, for the first time, by using the Inverse-Error Weighting (InEW) matching procedure. As a by-product of the calculation, the NNLO term of the CF hard-scattering coefficient proportional to   <math> <msubsup> <mi>α</mi> <mi>s</mi> <mn>2</mn> </msubsup> </math>  $$ {\alpha}_s^2 $$ ln(1/z) is predicted from HEF.

Matching next-to-leading-order and high-energy-resummed calculations of heavy-quarkonium-hadroproduction cross sections

We present the reweighting of two sets of nuclear parton distribution functions (PDFs), nCTEQ15 and EPPS16, using a selection of experimental data on heavy-flavor meson [<math><msup><mi>D</mi><mn>0</mn></msup></math>, <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math>, <math><mi>B</mi><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi></math> and <math><mi>ϒ</mi><mo>(</mo><mn>1</mn><mi>S</mi><mo>)</mo></math>] production in proton-lead collisions at the LHC which were not used in the original determination of these nuclear PDFs. The reweighted PDFs exhibit significantly smaller uncertainties thanks to these new heavy-flavor constraints. We present a comparison with another selection of data from the LHC and relativistic heavy ion collider (RHIC) which were not included in our reweighting procedure. The comparison is overall very good and serves as a validation of these reweighted nuclear PDF sets, which we dub <math><mrow><mi>nCTEQ</mi><msub><mrow><mn>15</mn></mrow><mrow><mi>rwHF</mi></mrow></msub></mrow></math> and <math><mrow><mi>EPPS</mi><msub><mrow><mn>16</mn></mrow><mrow><mi>rwHF</mi></mrow></msub></mrow></math>. This indicates that the LHC and forward RHIC heavy-flavor data can be described within the standard collinear factorization framework with the same (universal) small-<math><mi>x</mi></math> gluon distribution. We discuss how we believe such reweighted PDFs should be used as well as the limitations of our procedure.

Aleksander Kusina

Ingo Schienbein

Reweighted nuclear PDFs using heavy-flavor production data at the LHC

We address the unphysical energy dependence of quarkonium-hadroproduction cross sections at Next-to-Leading Order (NLO) in  $$\alpha _s$$  <math> <msub> <mi>α</mi> <mi>s</mi> </msub> </math>   which we attribute to an over-subtraction in the factorisation of the collinear singularities inside the PDFs in the  $$\overline{\text {MS}}$$  <math> <mover> <mtext>MS</mtext> <mo>¯</mo> </mover> </math>   scheme. Such over- or under-subtractions have a limited phenomenological relevance in most of the scattering processes in particle physics. On the contrary, it is particularly harmful for  $$P_T$$  <math> <msub> <mi>P</mi> <mi>T</mi> </msub> </math>  -integrated charmonium hadroproduction which renders a wide class of NLO results essentially unusable. Indeed, in such processes,  $$\alpha _s$$  <math> <msub> <mi>α</mi> <mi>s</mi> </msub> </math>   is not so small, the PDFs are not evolved much and can be rather flat for the corresponding momentum fractions and, finally, some process-dependent NLO pieces are either too small or too large. We propose a scale-fixing criterion which avoids such an over-subtraction. We demonstrate its efficiency for  $$\eta _{c,b}$$  <math> <msub> <mi>η</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> </math>   but also for a fictitious light elementary scalar boson. Having provided stable NLO predictions for  $$\eta _{c,b}$$  <math> <msub> <mi>η</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> </math>    $$P_T$$  <math> <msub> <mi>P</mi> <mi>T</mi> </msub> </math>  -integrated cross sections,  $$\sigma ^{\mathrm{NLO}}_{\eta _Q}$$  <math> <msubsup> <mi>σ</mi> <msub> <mi>η</mi> <mi>Q</mi> </msub> <mi>NLO</mi> </msubsup> </math>  , and discussed the options to study  $$\eta _{b}$$  <math> <msub> <mi>η</mi> <mi>b</mi> </msub> </math>   hadroproduction, we argue that their measurement at the LHC can help better determine the gluon PDF at low scales and tell whether the local minimum in conventional NLO gluon PDFs around  $$x=0.001$$  <math> <mrow> <mi>x</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </math>   at scales below 2 GeV is physical or not.

Curing the unphysical behaviour of NLO quarkonium production at the LHC and its relevance to constrain the gluon PDF at low scales

We study the inclusive <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> production at large transverse momenta at lepton-hadron colliders in the limit when the exchange photon is quasi real, also referred to as photoproduction. Our computation includes the leading-<math><msub><mrow><mi>P</mi></mrow><mrow><mi>T</mi></mrow></msub></math> leading-v next-to-leading <math><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub></math> corrections. In particular, we consider the contribution from <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> plus another charm quark, by employing for the first time in quarkonium photoproduction the variable-flavour-number scheme. We also include a QED-induced contribution via an off-shell photon which remained ignored in the literature and which we show to be the leading contribution at high <math><msub><mrow><mi>P</mi></mrow><mrow><mi>T</mi></mrow></msub></math> within the reach of the EIC. In turn, we use our computation of <math><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>+</mo><mtext>charm</mtext></math> to demonstrate its observability at the future EIC and the EIC sensitivity to probe the non-perturbative charm content of the proton at high x.

Ukraine

Large- Inclusive Photoproduction of / in electron-proton collisions at HERA and the EIC