We present the first lattice result of the near threshold $ \Lambda_c\Lambda_c $ scattering with $ I(J^P) = 0(0^+) $ . The calculation is performed on two $ N_f = 2+1 $ Wilson-Clover ensembles with pion mass $ m_\pi \sim 303 $ MeV and lattice spacing $ a = 0.07746 $ fm. Lüscher's finite volume method is utilized to extract the scattering parameters from the finite-volume spectrum. The coupled channel $ \Xi_{cc}N $ is ignored in the scattering analysis based on the observation that the energy levels computed from the $ \Lambda_c\Lambda_c $ and $ \Xi_{cc}N $ operators do not mix. The $ \Sigma_c\Sigma_c $ channel is not included either since the energy range explored in this study is well below its threshold. Our results indicate that the interaction in the $ \Lambda_c\Lambda_c $ single channel is repulsive, and the scattering length is determined to be $ a_0 = -0.21(4)(8) $ fm, where the first error is statistical and the second is systematic.

China

Hanyang Xing

Yiqi Geng

Chuan Liu

Liuming Liu

Peng Sun

Jiajun Wu

Zhicheng Yan

Ruilin Zhu

Lattice QCD study of Λ $\textit{c}$ Λ $\textit{c}$ scattering * Supported by NSFC (12293060, 12293061, 12293063, 12175279, 12435002, 12322503, 12175239). L. Liu also acknowledges the Strategic Priority Research Program of the Chinese Academy of Sciences with (XDB34030301) and Guangdong Major Project of Basic and Applied Basic Research (2020B0301030008). We also acknowledge the Science and Education Integration Young Faculty Project of University of Chinese Academy of Sciences (YSBR-101)

Parton distribution functions (PDFs) at large are challenging to extract from experimental data, yet they are essential for understanding hadron structure and searching for new physics beyond the Standard Model. Within the framework of the large momentum expansion of lattice quasi-PDFs, we investigate large PDFs, where the matching coefficient is factorized into the hard kernel, related to the active quark momentum , and the threshold soft function, associated with the spectator momentum (1 − ) . The renormalization group equation of the soft function enables the resummation of the threshold double logarithms ln (1 − ), which is crucial for a reliable and controllable calculation of large PDFs. Our analysis with pion valence PDFs indicates that perturbative matching breaks down when the spectator momentum (1 − ) approaches Λ , but remains valid when both and (1 − ) are much larger than Λ . Additionally, we incorporate leading renormalon resummation within the threshold framework, demonstrating good perturbative convergence in the region where both spectator and active quark momenta are perturbative scales.

United States

Xiangdong Ji

Poland

Yizhuang Liu

Yushan Su

Rui Zhang

Effects of threshold resummation for large-

We calculate the gravitational tensor-monopole moment of the momentum-current density <math><msup><mi>T</mi><mrow><mi>i</mi><mi>j</mi></mrow></msup></math> in the ground state of the hydrogen atom to order <math><mi>O</mi><mo>(</mo><mi>α</mi><mo>)</mo></math> in quantum electrodynamics (QED). The result is <math><msub><mi>τ</mi><mi>H</mi></msub><mo>/</mo><msub><mi>τ</mi><mn>0</mn></msub><mo>−</mo><mn>1</mn><mo>=</mo><mfrac><mrow><mn>4</mn><mi>α</mi></mrow><mrow><mn>3</mn><mi>π</mi></mrow></mfrac><mrow><mo>(</mo><mi>ln</mi><msup><mi>α</mi><mn>2</mn></msup><mo>−</mo><mn>0.028</mn><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>3.06</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>2</mn></mrow></msup></math> where <math><msub><mi>τ</mi><mn>0</mn></msub><mo>=</mo><msup><mi>ℏ</mi><mn>2</mn></msup><mo>/</mo><mn>4</mn><msub><mi>m</mi><mi>e</mi></msub></math> is the leading-order moment. The physics of the next-to-leading-order correction is similar to that of the famous Lamb shift for energy levels.

Jinghong Yang

Gravitational tensor-monopole moment of the hydrogen atom to order <math><mi>O</mi><mo>(</mo><mi>α</mi><mo>)</mo></math>

In this work, we investigate a coordinate space structure function   <math> <mi>E</mi> </math>  $$ \mathcal{E} $$ (z 2 m 2, λ) in the 2D U(N) Gross-Neveu model to the next-to-leading order in the large-N expansion. We analytically perform the twist expansion in the Bjorken limit through double Mellin representations. Hard and non-perturbative scaling functions are naturally generated in their Borel representations with detailed enumerations and explicit expressions provided to all powers. The renormalon cancellation at t = n between the hard functions at powers p and the non-perturbative functions at powers p + n are explicitly verified, and the issue of “scale-dependency” of the perturbative and non-perturbative functions is explained naturally. Simple expressions for the leading power non-perturbative functions are also provided both in the coordinate space and the momentum-fraction space (0 &lt; α &lt; 1) with “zero-mode-type” subtractions at α = 0 discussed in detail. In addition to the Bjorken limit, we also perform the threshold expansion of the structure function up to the next-to-next-to-leading threshold power exactly and investigate the resurgence relation between threshold and “Regge” asymptotics. We also prove that the twist expansion is absolutely convergent for any 0 &lt; z 2 &lt; ∞ and any λ ∈ iR  ≥0.

Bjorken and threshold asymptotics of a space-like structure function in the 2D U(N) Gross-Neveu model

We show that as a result of causality-constrained coordinate space analyticity, the Drell-Yan-shape transverse-momentum dependent soft factor in the exponential regulator allows Euclidean-type parametric representations without cuts, to all orders in perturbation theory. Moreover, it is identical to another soft factor defined with a single time-ordering that has a natural interpretation as a space-like form factor. Furthermore, this relation continues to hold for a larger class of TMD soft factors that interpolate between three different rapidity regulators: the off-light-cone regulator, the finite light-front length regulator, and the exponential regulator.

Revisiting the time-ordering issue of TMD soft factors: causality, coordinate space analyticity and new equalities

In this work, we investigate quantitative properties of correlation functions on the boundaries between two 2D Ising-like models with dual parameters β and <math><msup><mrow><mi>β</mi></mrow><mrow><mo>⋆</mo></mrow></msup></math>. Spin-spin correlators in such constructions without reflection symmetry with respect to translation-invariant directions are usually represented as <math><mn>2</mn><mo>×</mo><mn>2</mn></math> block Toeplitz determinants which are usually significantly harder than the scalar (<math><mn>1</mn><mo>×</mo><mn>1</mn></math> block) versions. Nevertheless, we show that for the specific <math><mi>β</mi><mo>/</mo><msup><mrow><mi>β</mi></mrow><mrow><mo>⋆</mo></mrow></msup></math> boundaries considered in this work, the symbol matrices allow explicit commutative Wiener-Hopf factorizations. As a result, the constants <math><mi>E</mi><mo>(</mo><mi>a</mi><mo>)</mo></math> and <math><mi>E</mi><mo>(</mo><mover><mrow><mi>a</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>)</mo></math> for the large n asymptotics still allow explicit representations that generalize the strong Szegö's theorem for scalar symbols. However, the Wiener-Hopf factors at different z do not commute. We will show that due to this non-commutativity, “logarithmic divergences” in the Wiener-Hopf factors generate certain “anomalous terms” in the exponential form factor expansions of the re-scaled correlators. Since our boundaries in the naive scaling limits can be formulated as certain integrable boundaries/defects in 2D massive QFTs, the results of this work facilitate detailed comparisons with bootstrap approaches.

Spin-spin correlators on the / boundaries in 2D Ising-like models: Exact analysis through theory of block Toeplitz determinants

In this paper, we show that the common hard kernel of double-log-type or threshold-type factorization for certain space-like parton correlators that arise in the context of lattice parton distributions, the heavy-light Sudakov hard kernel, has linear infrared (IR) renormalon. We explicitly demonstrate how this IR renormalon correlates with ultraviolet (UV) renormalons of next-to-leading power operators in two explicit examples: threshold asymptotics of space-like quark-bilinear coefficient functions and transverse momentum dependent (TMD) factorization of quasi wave function amplitude. Theoretically, the pattern of renormalon cancellation complies with general expectations to marginal asymptotics in the UV limit. Practically, this linear renormalon explains the slow convergence of imaginary parts observed in lattice extraction of the Collins-Soper kernel and signals the relevance of next-to-leading power contributions. Fully factorized, fully controlled threshold asymptotic expansion for space-like quark-bilinear coefficient functions in coordinate and moment space has also been proposed.

Renormalon cancellation and linear power correction to threshold-like asymptotics of space-like parton correlators

We revisit some aspects of the stringy approach to dipole-dipole interactions, scattering and entanglement in QCD, using the Nambu-Goto (NG) string, without recourse to holography. We first show that the potential between two static dipoles exchanging closed NG strings is attractive at all separations. Underlining the exchange there is an emergent entropy, that is dominated by the tachyon at large separation, and vanishes at short separation, a measure of the confinement-deconfinement transition. The same tachyon is dominant in the scattering amplitude, as a correlator of two Wilson loops for two fixed dipolelike hadrons separated at large rapidity gap, where the contribution of the world sheet fermions is included. While the tachyon causes the mean string bit density to grow exponentially with the rapidity, the total scattering cross section still satisfies the Froissart bound by quantum shadowing. The stringy scattering exchange also carries an entanglement entropy, that saturates when the bound is reached. For hadrons with varying dipole sizes, the tachyon exchange takes place in hyperbolic space in the conformal limit. The result for the full <math><mi>S</mi></math>-matrix is reminiscent of the one from Mueller’s evolved dipole wave function, for the total dipole-dipole cross section in perturbative QCD.

Maciej A. Nowak

Ismail Zahed

Nambu-Goto string in QCD: Dipole interactions, scattering, and entanglement

We analyze Mueller’s QCD dipole wave function evolution in the double logarithm approximation (DLA). Using complex analytical methods, we show that the distribution of the dipole in the wave function (gluon multiplicity distribution) asymptotically satisfies the Koba-Nielsen-Olesen (KNO) scaling, with a nontrivial scaling function <math><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo></math> with <math><mi>z</mi><mo>=</mo><mfrac><mi>n</mi><mover><mi>n</mi><mo>¯</mo></mover></mfrac></math>. The scaling function decays exponentially as <math><mn>2</mn><mo>(</mo><mn>2.55</mn><msup><mo>)</mo><mn>2</mn></msup><mi>z</mi><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mi>z</mi><mn>0.3917</mn></mfrac></mrow></msup></math> at large <math><mi>z</mi></math>, while its growth is log-normal as <math><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>ln</mi><mn>2</mn></msup><mi>z</mi></mrow></msup></math> for small <math><mi>z</mi></math>. A detailed analysis of the Fourier-Laplace transform of <math><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo></math> allows for performing the inverse Fourier transform and accessing the nonasymptotic bulk region around the peak. The bulk and asymptotic results are shown to be in good agreement with the measured hadronic multiplicities in DIS, as reported by the H1 Collaboration at HERA in the region of large <math><msup><mi>Q</mi><mn>2</mn></msup></math>. A numerical tabulation of <math><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo></math> is included. Remarkably, the same scaling function is found to emerge in the resummation of double logarithms in the evolution of jets. Using the generating function approach, we show why this is the case. The absence of KNO scaling in noncritical and super-renormalizable theories is briefly discussed. We also discuss the universal character of the entanglement entropy in the KNO scaling limit and its measurement using the emitted multiplicities in DIS and <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> annihilation.

Mueller’s dipole wave function in QCD: Emergent Koba-Nielsen-Olesen scaling in the double logarithm limit

Parton distribution functions (PDFs) at large x are poorly constrained by high-energy experimental data, but extremely important for probing physics beyond standard model at colliders. We study the calculation of PDFs at large-x through large-momentum P  z  expansion of the lattice quasi PDFs. Similar to deep-inelastic scattering, there are two distinct perturbative scales in the threshold limit where the matching coefficient can be factorized into a space-like jet function at scale P  z |1 − y| and a pair of heavy-light Sudakov form factors at scale P  z . The matching formula allows us to derive a full renormalization group resummation of large threshold logarithms, and the result is consistent with the known calculation to the next-to-next to leading order (NNLO). This paves the way for direct large-x PDFs calculations in lattice QCD. As by-products, we find that the space-like jet function is related to a time-like version calculated previously through analytic continuation, and the heavy-light Sudakov form factor, calculated here to NNLO, is a universal object appearing as well in the large momentum expansion of quasi transverse-momentum-dependent PDFs and quasi wave-function amplitudes.