<p>Understanding the internal structure of nucleons and nuclei has been a topic of enduring interest in high-energy physics. Gravitational form factors (GFFs) provide an important portal for us to probe the energy-momentum/mass distribution of nucleons and nuclei. This paper presents the study of the photon and gluon momentum GFFs, also known as the A-GFFs, of relativistic hadrons using the Weizsäcker-Williams method. To begin, we express the photon A-GFFs in terms of charge form factors and discuss the corresponding photon radius. Furthermore, an integral relation between the gluon A-GFF and the Laplacian of dipole scattering amplitude is derived in the small-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> framework, and it allows us to unravel the gluon energy momentum distribution inside hadrons through measurements at the upcoming Electron-Ion Collider. In addition, we generalize the analysis to study the A-GFF of nuclei and propose employing the nuclear gluon mean square radius, together with the charge distribution, to constrain the neutron distribution for large nuclei. This work provides an interesting perspective into the fundamental structure of high-energy hadrons.</p>

China

Yoshikazu Hagiwara

Finland

Xuan-Bo Tong

Bo-Wen Xiao

Understanding gravitational form factors with the Weizsäcker-Williams method

<p>The simplified connection between Standard Model (SM) particles and light dark matter (LDM) can be introduced via spin-0 and spin-1 lepton-specific mediators. Moreover, in a mediator mass range from sub-MeV to sub-GeV, fixed-target facilities such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>NA</mi><mrow><mn>64</mn><mi>e</mi></mrow></mrow></math>, LDMX, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>NA</mi><mrow><mn>64</mn><mi>μ</mi></mrow></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi mathvariant="normal">M</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></math> can potentially probe such particles of the hidden sector via missing energy signatures that are described by the bremsstrahlunglike process involving leptons. We compare the Weizsaker-Williams (WW) approximation and the exact tree-level (ETL) approach for the bremsstrahlunglike mediator production cross section by choosing various parameters of the fixed-target experiments. We show that the relative difference between the total cross sections calculated in the WW and ETL approximation varies from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>%</mo></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo><mo>%</mo></math> for a muon mode and from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mo>−</mo><mn>20</mn><mo stretchy="false">)</mo><mo>%</mo></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mn>80</mn><mo stretchy="false">)</mo><mo>%</mo></math> for an electron mode. We argue that the main difference between two approaches for electron beam mode arises from peak forward region of the cross section. We also discuss the impact of parametrization of nuclear and atomic elastic form factors on the total cross section. In particular, we show that the employing different form-factor parametrization can lead to a uncertainty in the cross section at the level of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>≲</mo><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo><mo>%</mo></math>. That study may be helpful for the lepton fixed-target experiments that examine the bremsstrahlunglike production of the DM mediators.</p>

Russia

I.V. Voronchikhin

D.V. Kirpichnikov

Implication of the Weizsacker-Williams approximation for the dark matter mediator production

In addition to vector (<math><mrow><mi>V</mi></mrow></math>) type new particles extensively discussed previously, both <math><mi>C</mi><mi>P</mi></math>-even (<math><mi>S</mi></math>) and <math><mi>C</mi><mi>P</mi></math>-odd (<math><mi>P</mi></math>) spin-0 dark matter (DM) mediators can couple to muons and be produced in the bremsstrahlung reaction <math><mrow><msup><mrow><mi>μ</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>+</mo><mi>N</mi><mo>→</mo><msup><mrow><mi>μ</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>+</mo><mi>N</mi><mo>+</mo><mi>S</mi><mo>(</mo><mi>P</mi><mo>)</mo></mrow></math>. Their possible subsequent invisible decay into a pair of Dirac DM particles, <math><mi>S</mi><mo>(</mo><mi>P</mi><mo>)</mo><mo>→</mo><mi>χ</mi><mover><mi>χ</mi><mo>¯</mo></mover></math>, can be detected in fixed target experiments through missing energy signature. In this paper, we focus on the case of experiments using high-energy muon beams. For this reason, we derive the differential cross sections involved using the phase space Weiszäcker-Williams approximation and compare them to the exact-tree-level calculations. The formalism derived can be applied in various experiments that could observe muon-spin-0 DM interactions. This can happen in present and future proton beam-dump experiments such as NA62, SHIP, HIKE, and SHADOWS; in muon fixed target experiments as <math><mrow><mi>NA</mi><mrow><mn>64</mn><mi>μ</mi></mrow></mrow></math>, MUonE and M3; in neutrino experiments using powerful proton beams such as DUNE. In particular, we focus on the <math><mrow><mi>NA</mi><mrow><mn>64</mn><mi>μ</mi></mrow></mrow></math> experiment case, which uses a 160 GeV muon beam at the CERN Super Proton Synchrotron accelerator. We compute the derived cross sections, the resulting signal yields and we discuss the experiment projected sensitivity to probe the relic DM parameter space and the <math><mo>(</mo><mi>g</mi><mo>−</mo><mn>2</mn><msub><mo>)</mo><mi>μ</mi></msub></math> anomaly favored region considering <math><msup><mn>10</mn><mn>11</mn></msup></math> and <math><msup><mn>10</mn><mn>13</mn></msup></math> muons on target.

Switzerland

H. Sieber

D. V. Kirpichnikov

I. V. Voronchikhin

P. Crivelli

S. N. Gninenko

M. M. Kirsanov

JINR

N. V. Krasnikov

Spain

L. Molina-Bueno

S. K. Sekatskii

Probing hidden sectors with a muon beam: Implication of spin-0 dark matter mediators for the muon (<math><mrow><mi>g</mi><mo>−</mo><mn>2</mn></mrow></math>) anomaly and the validity of the Weiszäcker-Williams approach

In [1], we performed the first complete computation of the back-to-back inclusive dijet cross-section in Deeply Inelastic Scattering (DIS) at small x Bj to next-to-leading order (NLO) in the Color Glass Condensate effective field theory (CGC EFT). We demonstrate here that for dijets with relative transverse momentum P ⊥ and transverse momentum imbalance q ⊥, to leading power in q ⊥/P ⊥, the cross-section for longitudinally polarized photons can be fully factorized into the product of a perturbative impact factor and the non-perturbative Weizsäcker-Williams (WW) transverse momentum dependent (TMD) gluon distribution to NLO accuracy. The impact factor can further be expressed as the product of a universal soft factor which resums Sudakov double and single logs in P ⊥/q ⊥ and a coefficient function given by a remarkably compact analytic expression. We show that in the CGC EFT the WW TMD satisfies a kinematically constrained JIMWLK renormalization group evolution in rapidity. This factorization formula is valid to all orders in Q  s /q ⊥ for q ⊥, Q  s  ≪ P ⊥, where Q  s  is the semi-hard saturation scale that grows with rapidity.

France

Paul Caucal

United States

Farid Salazar

Björn Schenke

Poland

Tomasz Stebel

Raju Venugopalan

Back-to-back inclusive dijets in DIS at small x: gluon Weizsäcker-Williams distribution at NLO

The <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> production originating from elastic and inelastic photoproduction processes in <math><mrow><mi>p</mi><mtext>−</mtext><mi>p</mi></mrow></math> collisions at LHC energies is investigated, where the fragmentation processes are involved. An exact treatment is performed, which adopts the Martin-Ryskin method to weight the contribution from different channels, and can return to Weizsäcker-Williams approximation (WWA) when <math><msup><mi>Q</mi><mn>2</mn></msup><mo>→</mo><mn>0</mn></math>. The relevant kinematical relations are also achieved. We present a comprehensive analysis for the feature of WWA by comparing with the exact treatment. The results are expressed in <math><msup><mi>Q</mi><mn>2</mn></msup></math>, <math><mi>y</mi></math>, <math><mi>z</mi></math>, <math><msub><mi>p</mi><mi>T</mi></msub></math>, and <math><msub><mi>y</mi><mi>r</mi></msub></math> (rapidity) distributions, and the total cross sections are also estimated. The numerical results indicate that the incoherent-photon emission can provide a meaningful contribution to elastic photoproduction, and starts to play a very important role in the inelastic processes. The photoproduction and fragmentation processes can improve the contribution of <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> production in <math><mrow><mi>p</mi><mtext>−</mtext><mi>p</mi></mrow></math> collisions at LHC energies. Moreover, the WWA is only effective in very restricted domains, and the exact treatment is needed to deal accurately with the <math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> photoproduction, which can naturally avoid double counting and WWA errors.

Zhi-Lei Ma

Zhun Lu

Jia-Qing Zhu

Li Zhang

Elastic and inelastic <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math> photoproduction in <math><mrow><mi>p</mi><mtext>−</mtext><mi>p</mi></mrow></math> collisions at LHC energies: Features of the Weizsäcker-Williams approximation

The production of a hard dijet with small transverse momentum imbalance in semi-inclusive DIS probes the conventional and linearly polarized Weizsäcker-Williams (WW) transverse momentum dependent (TMD) gluon distributions. The latter, in particular, gives rise to an azimuthal dependence of the dijet cross section. In this paper we analyze the feasibility of a measurement of these TMDs through dijet production in DIS on a nucleus at an electron-ion collider. We introduce the mcdijet Monte Carlo generator to sample quark-antiquark dijet configurations based on leading-order parton level cross sections with WW gluon distributions that solve the nonlinear small-<math><mi>x</mi></math> QCD evolution equations. These configurations are fragmented to hadrons using PYTHIA, and final-state jets are reconstructed. We report on background studies and on the effect of kinematic cuts introduced to remove beam jet remnants. We estimate that with an integrated luminosity of 20 <math><mrow><msup><mrow><mi>fb</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>/</mo><mi>nucleon</mi></mrow></math> one can determine the distribution of linearly polarized gluons with a statistical accuracy of approximately 5%.

Adrian Dumitru

Vladimir Skokov

Thomas Ullrich

Measuring the Weizsäcker-Williams distribution of linearly polarized gluons at an electron-ion collider through dijet azimuthal asymmetries

We show that the diffractive forward production of two quarkonia, especially the <math><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> and <math><msub><mi>η</mi><mi>c</mi></msub></math> states, in proton-proton or proton-ion collisions can access the Weizsäcker-Williams gluon Wigner distribution of the proton. We use the hybrid factorization approach where the collinear, double gluon PDF is applied for one of the protons and the <math><msub><mi>k</mi><mi>T</mi></msub></math>-dependent (Wigner or GTMD) distribution for the other. The production of quarkonia is treated in the NRQCD framework. A particularly concise formula is obtained for double <math><msub><mi>χ</mi><mrow><mi>J</mi><mo>=</mo><mn>1</mn></mrow></msub></math> production.

Renaud Boussarie

Japan

Yoshitaka Hatta

Feng Yuan

Probing the Weizsäcker-Williams gluon Wigner distribution in <math><mi>p</mi><mi>p</mi></math> collisions

We apply point-particle effective field theory (PPEFT) to compute the leading shifts due to finite-sized source effects in the Coulomb bound energy levels of a relativistic spinless charged particle. This is the analogue for spinless electrons of calculating the contribution of the charge-radius of the source to these levels, and our calculation disagrees with standard calculations in several ways. Most notably we find there are two effective interactions with the same dimension that contribute to leading order in the nuclear size, one of which captures the standard charge-radius contribution. The other effective operator is a contact interaction whose leading contribution to δE arises linearly (rather than quadratically) in the small length scale, ϵ, characterizing the finite-size effects, and is suppressed by ( Zα ) 5 . We argue that standard calculations miss the contributions of this second operator because they err in their choice of boundary conditions at the source for the wave-function of the orbiting particle. PPEFT predicts how this boundary condition depends on the source’s charge radius, as well as on the orbiting particle’s mass. Its contribution turns out to be crucial if the charge radius satisfies ϵ ≲ ( Zα ) 2 a B , where a B is the Bohr radius, because then relativistic effects become important for the boundary condition. We show how the problem is equivalent to solving the Schrödinger equation with competing Coulomb, inverse-square and delta-function potentials, which we solve explicitly. A similar enhancement is not predicted for the hyperfine structure, due to its spin-dependence. We show how the charge-radius effectively runs due to classical renormalization effects, and why the resulting RG flow is central to predicting the size of the energy shifts (and is responsible for its being linear in the source size). We discuss how this flow is relevant to systems having much larger-than-geometric cross sections, such as those with large scattering lengths and perhaps also catalysis of reactions through scattering with monopoles. Experimental observation of these effects would require more precise measurement of energy levels for mesonic atoms than are now possible.

Canada

C. Burgess

Peter Hayman

Markus Rummel

Belgium

Matt Williams

László Zalavári

Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition