We construct an exact analytic solution of the revised small-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> helicity evolution equations from Borden, Kovchegov, and Li [Helicity evolution at small x: Quark to gluon and gluon to quark transition operators, <a href="http://dx.doi.org/10.1007/JHEP09(2024)037">J. High Energy Phys. 09 (2024) 037</a>], where the contributions of the quark-to-gluon and gluon-to-quark transition operators were newly included. These evolution equations are written in the large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> and -<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub></math> limit and are double-logarithmic, resumming powers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>s</mi></msub><msup><mi>ln</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mi>x</mi><mo stretchy="false">)</mo></math>. Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub></math> are the numbers of quark colors and flavors, respectively, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>s</mi></msub></math> is the strong coupling constant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> is the Bjorken-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> variable. Using our solution, we obtain analytic, small-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>, large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> and -<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub></math> expressions for the flavor singlet quark and gluon helicity parton distribution functions (PDFs) and for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>g</mi><mn>1</mn></msub></math> structure function as double-inverse Laplace transforms. We also extract analytic expressions for the eigenvalues of the matrix of polarized Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) anomalous dimensions and, subsequently, analytic expressions for each of the four individual polarized anomalous dimensions themselves (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi><msub><mi>γ</mi><mrow><mi>q</mi><mi>q</mi></mrow></msub><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mi>γ</mi><mrow><mi>q</mi><mi>G</mi></mrow></msub><mo>,</mo><mi mathvariant="normal">Δ</mi><msub><mi>γ</mi><mrow><mi>G</mi><mi>q</mi></mrow></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi><msub><mi>γ</mi><mrow><mi>G</mi><mi>G</mi></mrow></msub></math>): these expressions resum powers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>s</mi></msub><mo>/</mo><msup><mi>ω</mi><mn>2</mn></msup></math> to all orders at large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub></math> (with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ω</mi></math> the Mellin moment variable). We extract the leading small-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> asymptotic power-law growth of the helicity distributions, given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi><mi mathvariant="normal">Σ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>∼</mo><mi mathvariant="normal">Δ</mi><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>∼</mo><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>∼</mo><msup><mrow><mo stretchy="false">(</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo stretchy="false">)</mo></mrow><msub><mi>α</mi><mi>h</mi></msub></msup></math>, where the intercept <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>h</mi></msub></math> satisfies an algebraic equation. Although the algebraic complexity of the equation prevented us from obtaining a general analytic expression for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>h</mi></msub></math>, we determine <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>h</mi></msub></math> numerically for various values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub></math> (and, for the special case of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><msub><mi>N</mi><mi>c</mi></msub></math>, we determine <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>h</mi></msub></math> analytically). We further obtain the explicit asymptotic expressions for the helicity distributions, which yield numerical values for the ratio of the gluon helicity PDF to the flavor singlet quark helicity PDF in the small-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> asymptotic limit (for different <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>/</mo><msub><mi>N</mi><mi>c</mi></msub></math>). Just as for the analytic solution to small-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> helicity evolution equations in the large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> limit constructed in Borden and Kovchegov [Analytic solution for the revised helicity evolution at small x and large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi mathvariant="normal">N</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math>: New resummed gluon-gluon polarized anomalous dimension and intercept, <a href="http://dx.doi.org/10.1103/PhysRevD.108.014001">Phys. Rev. D 108, 014001 (2023)</a>] and for the iterative solution to the large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> and -<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub></math> helicity evolution equations constructed in Borden, Kovchegov, and Li [Helicity evolution at small x: Quark to gluon and gluon to quark transition operators, <a href="http://dx.doi.org/10.1007/JHEP09(2024)037">J. High Energy Phys. 09 (2024) 037</a>], we again find that all our predictions for polarized DGLAP anomalous dimensions are fully consistent with the existing finite-order calculations. Similar to the large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> case [Borden and Kovchegov, Analytic solution for the revised helicity evolution at small x and large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi mathvariant="normal">N</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math>: New resummed gluon-gluon polarized anomalous dimension and intercept, <a href="http://dx.doi.org/10.1103/PhysRevD.108.014001">Phys. Rev. D 108, 014001 (2023)</a>], our intercept <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>h</mi></msub></math> (evaluated at various <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub></math> values) exhibits a very slight disagreement with the predictions made within the infrared evolution equations framework in Bartels et al. [Flavor singlet contribution to the structure function G(1) at small x, <a href="http://dx.doi.org/10.1007/s002880050285">Z. Phys. C 72, 627 (1996)</a>], Blümlein and Vogt [The singlet contribution to the structure function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> at small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>, <a href="http://dx.doi.org/10.1016/0370-2693(96)00958-6">Phys. Lett. B 386, 350 (1996)</a>], Boussarie et al. [Proton spin structure at small-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>, <a href="http://dx.doi.org/10.1016/j.physletb.2019.134817">Phys. Lett. B 797, 134817 (2019)</a>].

United States

Jeremy Borden

Yuri V. Kovchegov

Analytic solution for the helicity evolution equations at small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> and large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub></math>

In this paper, we investigate the small-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> behavior of the flavor-singlet, leading-twist quark transverse-momentum-dependent parton distribution functions (TMDs) using the light-cone operator treatment. This formalism allows us to express TMD operators at small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> in terms of polarized dipole amplitudes, enabling a systematic approach to their small-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> evolution. We derive the evolution equations for these TMDs and solve them within the large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> approximation under the linearized, double-logarithmic approximation, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> represents the number of quark colors. Expanding on previous work on unpolarized and helicity TMDs, we present the small-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> asymptotics for a comprehensive set of TMDs, including the Sivers function, helicity worm-gear, transversity, pretzelosity, Boer-Mulders, and transversity worm-gear distributions. Our results provide a complete picture of the small-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> asymptotic behavior for all leading-twist flavor-singlet quark TMDs. We also discuss the implications of our findings for phenomenological applications and outline potential avenues for further research, particularly in understanding nonlinear effects and extending beyond the double-logarithmic approximation and large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> approximations.

Daniel Adamiak

M. Gabriel Santiago

Thailand

Yossathorn Tawabutr

Small-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> asymptotics of the leading-twist flavor-singlet quark TMDs

Many contributions to the decay rate of the inclusive radiative $$\overline{B}\to {X }_{s}\gamma $$ transition have been calculated to NNLO in QCD during the past decades. However, there are still a few unknown contributions from multi-parton final states which are formally NLO. In the present work, we compute those four-body $$b\to sq\overline{q }\gamma $$ contributions at NLO in QCD which need to be supplemented by the five-body $$b\to sq\overline{q }g\gamma $$ bremsstrahlung. This calculation formally completes the purely perturbative contributions to $$\overline{B}\to {X }_{s}\gamma $$ at NLO. Our results are obtained by applying modern techniques of integral reduction and evaluation of master integrals. In particular, the analytic integration over the four and five-particle phase space in the presence of a cut on the photon energy turns out to be technically involved. We give our results completely analytically in terms of multiple polylogarithms, including the dependence on the collinear logarithms which arise from the mass-regularisation of collinear divergences. The numerical impact of multi-parton corrections on the $$\overline{B}\to {X }_{s}\gamma $$ decay rate turns out to be small, owing to a partial cancellation between LO and NLO contributions.

Germany

Kevin Brune

Tobias Huber

Lars-Thorben Moos

Multi-parton contributions to $$\overline{B}\to {X }_{s}\gamma $$ at NLO

Description of spin-dependent hadronic processes at high energies in terms of parton helicities is a both effective and technically convenient means. In the present paper, we obtain explicit expressions for the parton helicities when either collinear or $\textit{KT}$ forms of QCD factorization are used. Starting our studies with calculation of the helicities in the double-logarithmic approximation (DLA) in the region of small $\textit{x}$ and large $$Q^2$$ , we generalize the results in order to obtain formulae valid at arbitrary $\textit{x}$ and $$Q^2$$ . We argue against using collinear factorization, when the parton orbital angular momenta are accounted for, and prove that $\textit{KT}$ factorization should be used instead. We also consider in detail the small-$\textit{x}$ asymptotics of the parton helicities, compare them with the DGLAP-asymptotics in LO, NLO, etc. and prove that the DGLAP asymptotics are less singular at small $\textit{x}$ than the Regge asymptotics

Russia

B. Ermolaev

Parton helicities at arbitrary $\textit{x}$ and $$Q^2$$ in double-logarithmic approximation

The small-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> regime remains a key testing ground for perturbative QCD and future lepton-hadron collider physics. We demonstrate that the long-standing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∼</mo><mn>30</mn><mo>%</mo></math> spread among global next-to-next-to-leading-order parton distribution functions (PDFs) for heavy-flavor parton densities at small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> has now converged below <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>≈</mo><mn>17</mn><mo>%</mo></math>. Using the latest nnpdf4.0, ct18, and msht20 sets, we quantify a maximal relative deviation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Δ</mi><mi>max</mi></msub><mo>≃</mo><mn>16.8</mn><mo>%</mo></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi><mo>=</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><mo>=</mo><mn>100</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>. This convergence, further stabilized by consistent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>s</mi></msub></math> and scale variations, marks a new precision frontier for the heavy-flavor sector. We evaluate the phenomenological impact for heavy-quark electroproduction at future <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>e</mi><mi>p</mi></math> colliders (LHeC and FCC–eh), showing that PDF-driven cross section uncertainties for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>−</mo></msup><mi>p</mi><mo stretchy="false">→</mo><msup><mi>e</mi><mo>−</mo></msup><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mi>X</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>−</mo></msup><mi>p</mi><mo stretchy="false">→</mo><msup><mi>e</mi><mo>−</mo></msup><mi>b</mi><mover accent="true"><mi>b</mi><mo stretchy="false">¯</mo></mover><mi>X</mi></math> have been reduced by nearly a factor of 2 compared to 2013 estimates.

Türkiye

S.O. Kara

Convergence of heavy-flavor parton distribution functions at small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> and its impact on future <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>e</mi><mi>p</mi></math> colliders

In this work, we investigate the dipion transition processes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>π</mi><mi>π</mi><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>J</mi><mo>=</mo><mn>0</mn></mrow></math>, 1, 2) within the framework of heavy hadron chiral perturbation theory, treating <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math> as a molecular state composed of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup><mo>+</mo><mi mathvariant="normal">H</mi><mo>.</mo><mi mathvariant="normal">c</mi><mo>.</mo></math> components. By analyzing the box and triangle loop diagrams with the nonrelativistic effective field theory power-counting rule, we demonstrate that box diagrams dominate these dipion transition processes. Branching ratios are calculated as functions of the mixing angle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>θ</mi></math>, which parametrizes the neutral and charged meson compositions of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math>. Our results indicate that the branching fractions for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>π</mi><mi>π</mi><msub><mi>χ</mi><mrow><mi>c</mi><mn>0</mn></mrow></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>π</mi><mi>π</mi><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>π</mi><mi>π</mi><msub><mi>χ</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub></math> are of the orders of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>10</mn><mrow><mo>−</mo><mn>5</mn></mrow></msup></math>, respectively. We also predict the ratios <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">B</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>π</mi><mi>π</mi><msub><mi>χ</mi><mrow><mi>c</mi><mn>0</mn><mo>/</mo><mn>2</mn></mrow></msub><mo stretchy="false">]</mo></mrow><mo>/</mo><mrow><mi mathvariant="script">B</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>π</mi><mi>π</mi><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo stretchy="false">]</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">B</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub><mo stretchy="false">]</mo></mrow><mo>/</mo><mrow><mi mathvariant="script">B</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msup><mi>π</mi><mn>0</mn></msup><msup><mi>π</mi><mn>0</mn></msup><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub><mo stretchy="false">]</mo></mrow></math>. The latter deviates from isospin-symmetry expectations, revealing various degrees of isospin violation. By studying the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>+</mo></msup><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> invariant mass spectra, we find a double-bump structure in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> invariant mass distributions of the process <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>+</mo></msup><msub><mi>χ</mi><mrow><mi>c</mi><mn>0</mn></mrow></msub></math> invariant mass distribution of the process <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><msub><mi>χ</mi><mrow><mi>c</mi><mn>0</mn></mrow></msub></math>, which could be tested by future experimental measurements.

China

Qi Wu

Zhong-Quan Sun

Dian-Yong Chen

Shi-Dong Liu

Gang Li

Dipion transitions from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>J</mi><mo>=</mo><mn>0</mn></mrow></math>, 1, 2)

We study the properties of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>1880</mn><mo stretchy="false">)</mo></math>, the structure around the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>p</mi><mo stretchy="false">¯</mo></mover><mi>p</mi></math> threshold that appears in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>3</mn><mo stretchy="false">(</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><mo stretchy="false">)</mo></mrow></math> invariant mass spectrum in the decay process of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi><mo stretchy="false">→</mo><mi>γ</mi><mn>3</mn><mo stretchy="false">(</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><mo stretchy="false">)</mo></mrow></math>. Nucleon-antinucleon rescattering is taken into account in our analysis, and the decay amplitude of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi><mo stretchy="false">→</mo><mi>γ</mi><mn>3</mn><mo stretchy="false">(</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><mo stretchy="false">)</mo></mrow></math> can be obtained by the distorted wave Born approximation. With these amplitudes, we analyze the contributions to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>1880</mn><mo stretchy="false">)</mo></math> from different partial waves. Our analysis suggests that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>1880</mn><mo stretchy="false">)</mo></math> should be isoscalar <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>0</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup></math>, and it is generated by the threshold behavior.

Qin-He Yang

Ling-Yun Dai

Ulf-G. Meißner

Quantum numbers of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>1880</mn><mo stretchy="false">)</mo></math>

We study inelastic dark matter produced via freeze-in through a light mediator with a mass splitting below the electron-positron threshold. In this regime, the heavier dark matter state is naturally long-lived compared to the age of the Universe and decays to the lighter state in association with photons. Given a light mediator, the dark matter abundance is directly related to the decay rate of the heavier dark matter. We show that observations of photons from the Galactic Center can effectively probe inelastic dark matter freeze-in with mediators at the 100 MeV scale and dark matter at the GeV scale.

Gordan Krnjaic

Canada

David McKeen

Riku Mizuta

Gopolang Mohlabeng

David E. Morrissey

Douglas Tuckler

X-rays from inelastic dark matter freeze-in

We investigate the possibility of having a vector boson dark matter in a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mi>X</mi></msub></math> extended two-Higgs doublet model (2HDM) setup. The gauge boson gains mass when a Standard Model (SM) singlet complex scalar, which is charged under the dark <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mi>X</mi></msub></math> symmetry, acquires a vacuum expectation value. This scalar acts as the connection between the SM sector and DM via the Higgs portal. An additional exact charge conjugation symmetry inhibits the mixing of this gauge boson with the photon, thereby confirming the stability of DM. On the other hand, 2HDM with type-I <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>Z</mi><mn>2</mn></msub></math> restriction can offer a nonstandard Higgs in the lighter mass range. This freedom allows us to accommodate dark matter mass in the (40–60)-GeV regime where the direct detection constraints are strongest. We study the dark matter phenomenology of such a model while taking care of all possible theoretical and experimental constraints.

India

Nandini Das

Juhi Dutta

Dilip Kumar Ghosh

Santosh Kumar Rai

Vector dark matter in a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mi>X</mi></msub></math> extended 2HDM

We quantify the effect of high-energy JIMWLK evolution on the deformed structure of heavy (Uranium) and intermediate (Ruthenium) nuclei. The soft gluon emissions in the high-energy evolution are found to drive the initially deformed nuclei towards a more spherical shape, although the evolution is slow, especially for the longest distance-scale quadrupole deformation. We confirm a linear relationship between the squared eccentricity $$\varepsilon _n^2$$ and the deformation parameter $$\beta _n^2$$ in central collisions across the energy range covered by the RHIC and LHC measurements. The applied JIMWLK evolution is found to leave visible signatures in the eccentricity evolution that can be observed if the same nuclei can be collided at RHIC and at the LHC, or in rapidity-dependent flow measurements. Our results demonstrate the importance of including the Bjorken-$\textit{x}$ dependent nuclear geometry when comparing simulations of the Quark Gluon Plasma evolution with precise flow measurements at high collision energies.

Finland

Heikki Mäntysaari

Pragya Singh

Energy dependence of the deformed nuclear structure at small-$\textit{x}$