Diffusion wake accompanying a Mach cone is a unique feature of the medium response to projectiles traveling at a speed faster than the velocity of sound. This is also the case for jet-medium interaction inside the quark-gluon plasma in high-energy heavy-ion collisions. It leads to a depletion of soft hadrons in the opposite direction of the propagating jet and, recently, has been observed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>Z</mi></mrow></math>-jet events of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>Pb</mi><mo>+</mo><mi>Pb</mi></mrow></math> collisions at LHC. In dijet events, however, the diffusion wake of one jet usually overlaps with the medium-induced hadron enhancement of the other jet without a clear signal except a reduction of the hadron enhancement, unless there is a large rapidity gap between the two jets. We propose to use the rapidity asymmetry of jet-hadron correlations in dijets with a finite rapidity gap relative to that without, as a robust and background-free signal of the diffusion wake. The asymmetry emerges because the diffusion wake of one jet is shifted to a finite rapidity relative to the other jet. Consequently, a depletion of soft hadrons appears in the shifted rapidity region of the diffusion wake and an enhancement in the rapidity region of the other jet whose soft hadron enhancement is no longer or less reduced by the diffusion wake as in dijets without a rapidity gap. We predict the rapidity asymmetry using both theoretical and mixed-event background subtraction for different values of the rapidity gap within the CoLBT-hydro model. Future measurements of this rapidity asymmetry with high statistics data on dijets should provide more precise insights into the jet-induced diffusion wake and properties of the quark-gluon plasma.

China

United States

Zhong Yang

Xin-Nian Wang

Rapidity Asymmetry of Jet-Hadron Correlation as a Robust Signal of Diffusion Wake Induced by Dijets in High-Energy Heavy-Ion Collisions

Motivated by two new states <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>h</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1911</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>2300</mn><mo stretchy="false">)</mo></math> observed by BESIII, we have investigated the mass spectrum and the strong decay properties of the strangeonium mesons within the modified Godfrey-Isgur model by considering the screening effects. We have determined the free parameters using the masses and widths of the well-established <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>s</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></math> states <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mo stretchy="false">(</mo><mn>1020</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mo stretchy="false">(</mo><mn>1680</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>h</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1415</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>f</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">(</mo><mn>1525</mn><mo stretchy="false">)</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>ϕ</mi><mn>3</mn></msub><mo stretchy="false">(</mo><mn>1850</mn><mo stretchy="false">)</mo></math>. According to our results, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>h</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1911</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>2300</mn><mo stretchy="false">)</mo></math> could be well explained as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>h</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msup><mn>2</mn><mn>1</mn></msup><msub><mi>P</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>h</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msup><mn>3</mn><mn>1</mn></msup><msub><mi>P</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>s</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></math> states, respectively. Meanwhile, the possible assignments of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>2000</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mn>1870</mn><mo stretchy="false">)</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mo stretchy="false">(</mo><mn>2170</mn><mo stretchy="false">)</mo></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>3</mn><mn>3</mn></msup><msub><mi>S</mi><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mn>1</mn></msup><msub><mi>D</mi><mn>2</mn></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>2</mn><mn>3</mn></msup><msub><mi>D</mi><mn>1</mn></msub></math> are also discussed. Furthermore, the masses and widths of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mi>S</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn><mi>S</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mi>P</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mi>P</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn><mi>P</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mi>D</mi></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mi>D</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>s</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></math> states are also given and compared with various theoretical predictions, which will be helpful for the observations and confirmations of these states in the future.

Wei Hao

Pakistan

M. Atif Sultan

Li-Juan Liu

En Wang

Strangeonium spectrum with screening effects and interpretation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>h</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1911</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>2300</mn><mo stretchy="false">)</mo></math> observed by BESIII

We determine the low-energy constants <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mn>0</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mn>4</mn></mrow><mrow><mi>r</mi></mrow></msubsup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>L</mi><mn>5</mn><mi>r</mi></msubsup></math> of SU(3) chiral perturbation theory from a lattice QCD calculation of the scalar form factors of the pion with fully controlled systematics. Lattice results are computed on a large set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> gauge ensembles covering four lattice spacings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0.049</mn><mo>,</mo><mn>0.086</mn><mo stretchy="false">]</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math>, pion masses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>∈</mo><mo stretchy="false">[</mo><mn>130</mn><mo>,</mo><mn>350</mn><mo stretchy="false">]</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, and various large physical volumes. By determining the notorious quark-disconnected contributions with unprecedented precision and using a large range of source-sink separations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>sep</mi></mrow></msub><mo>∈</mo><mo stretchy="false">[</mo><mn>1.0</mn><mo>,</mo><mn>3.25</mn><mo stretchy="false">]</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math>, we are able for the first time to obtain the scalar radii from a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>z</mi></math>-expansion parametrization of the form factors rather than a simple linear approximation at small momentum transfer. The low-energy constants are obtained from the physical extrapolation of the radii using next-to-leading-order SU(3) next-to-leading-order chiral perturbation theory to parametrize the quark mass dependence. Systematic uncertainties are estimated via model averages based on the Akaike information criterion. Our determination of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mn>4</mn></mrow><mrow><mi>r</mi></mrow></msubsup></mrow></math> is the first lattice determination to obtain a result not compatible with zero.

Germany

Georg von Hippel

Konstantin Ottnad

Low-Energy Constants of Chiral Perturbation Theory from Pion Scalar Form Factors in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>-Flavor Lattice QCD with Controlled Errors

Parton distribution functions (PDFs) form an essential part of particle physics calculations. Currently, the most precise predictions for these nonperturbative functions are generated through fits to global data. A problem that several PDF fitting groups encounter is the presence of tension in datasets that appear to pull the fits in different directions. In other words, the best fit depends on the choice of dataset. Several methods to capture the uncertainty in PDFs in presence of seemingly inconsistent fits have been proposed and are currently in use. These methods are important to ensure that uncertainty in PDFs are not underestimated. Here we propose a novel method for estimating the uncertainty by introducing a generalized statistical model based on Bayesian hierarchical models, which is implemented via the Gaussian Mixture Model (GMM). The methodology is inspired by unsupervised machine learning techniques and is closely related to the statistical methods of ensemble learning and Bayesian model averaging. Using a toy model of PDFs, we demonstrate how the GMM can be used to faithfully reconstruct the likelihood associated with PDF fits, which can in turn be used to accurately determine the uncertainty on PDFs, especially in the presence of tension in the fitted datasets. We further show how this statistical model reduces to the usual <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>χ</mi><mn>2</mn></msup></math> likelihood function for a consistent dataset and provide measures to optimize the number of Gaussians in the GMM.

Mengshi Yan

Tie-Jiun Hou

Zhao Li

Kirtimaan Mohan

C.P. Yuan

Generalized statistical model for fits to parton distributions

Axionlike particles (ALPs) are hypothetical particles that serve as promising candidates for cold dark matter. Portals like inelastic axion scattering and axion propagated force have been employed to search for the upper limit of the ALPs’ coupling with standard model particles. Other methods, like the axion-fermion interaction in the CASPEr experiment, integrate the dark matter motion into the measurement. We suggest a new method for detecting dark matter axions based on axion-electron elastic scattering. In the pseudoscalar axion model, this interaction can be seen as an effective magnetic field, so high-sensitivity atomic magnetometers can be utilized to measure this interaction. The scattering cross section of this process is significantly amplified by the high number density and occupation number of axion dark matter. The upper limit of the electron-axion coupling coefficient obtained from this process can reach two orders of magnitude higher than previous results at low axion mass, and will exceed the astrophysics limits by using a developing magnetometer. This scattering process also provides an efficient way to detect local structures of dark matter.

Dongyi Yang

Chenxi Sun

Jianwei Zhang

Constraints on axion dark matter by spin-dependent macroscopic force

The singlet-doublet model is an economical model of weakly interacting dark matter. We revisit it in light of improved dark matter direct detection limits. We characterize the now well-defined regions of remaining parameter space with suppressed direct detection cross sections and discuss features of the spectrum accessible at the Large Hadron Collider. We discuss when and how parameters in these special regions might be realized as the result of renormalization group evolution when starting with generic ultraviolet initial conditions.

Italy

Prudhvi N. Bhattiprolu

Evan Petrosky

Aaron Pierce

Singlet-doublet dark matter revisited

We report on a search for sub-GeV dark matter (DM) particles interacting with electrons using the DAMIC-M prototype detector at the Modane Underground Laboratory. The data feature a significantly lower detector single <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></math> rate (factor 50) compared to our previous search, while also accumulating a 10 times larger exposure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo>∼</mo><mn>1.3</mn><mtext> </mtext><mtext> </mtext><mi>kg</mi><mtext>−</mtext><mi>day</mi></mrow></math>. DM interactions in the skipper charge-coupled devices (CCDs) are searched for as groups of two or three adjacent pixels with a total charge between 2 and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>4</mn><mtext> </mtext><mtext> </mtext><msup><mi>e</mi><mo>−</mo></msup></math>. We find 144 candidates of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></math> and 1 candidate of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>4</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></math>, where 141.5 and 0.071, respectively, are expected from background. With no evidence of a DM signal, we place stringent constraints on DM particles with masses between 1 and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1000</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><mo>/</mo><msup><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> interacting with electrons through an ultralight or heavy mediator. For large ranges of DM masses below <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mtext> </mtext><mi>GeV</mi><mo>/</mo><msup><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>, we exclude theoretically motivated benchmark scenarios where hidden-sector particles are produced as a major component of DM in the Universe through the freeze-in or freeze-out mechanisms.

K. Aggarwal

I. Arnquist

Argentina

N. Avalos

France

X. Bertou

Spain

N. Castelló-Mor

A.E. Chavarria

J. Cuevas-Zepeda

A. Dastgheibi-Fard

C. De Dominicis

O. Deligny

J. Duarte-Campderros

E. Estrada

R. Gaïor

Switzerland

E.L. Gkougkousis

T. Hossbach

L. Iddir

B.J. Kavanagh

B. Kilminster

A. Lantero-Barreda

Canada

I. Lawson

A. Letessier-Selvon

H. Lin

P. Loaiza

A. Lopez-Virto

R. Lou

K.J. McGuire

S. Munagavalasa

J. Noonan

D. Norcini

S. Paul

P. Privitera

P. Robmann

B. Roach

M. Settimo

R. Smida

M. Traina

R. Vilar

R. Yajur

D. Venegas-Vargas

C. Zhu

Y. Zhu

Probing Benchmark Models of Hidden-Sector Dark Matter with DAMIC-M

The radiative seesaw model with gauged <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi mathvariant="normal">U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>B</mi><mo>−</mo><mi>L</mi></mrow></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></math> extension is a well-motivated scenario that gives consistent predictions of active neutrino masses and the abundance of dark matter. Majorana masses of right-handed neutrinos, the lightest of which can be identified as dark matter, are given by the spontaneous breaking of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi mathvariant="normal">U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>B</mi><mo>−</mo><mi>L</mi></mrow></msub></math> gauge symmetry. We revisit this model with the latest constraints from dark-matter searches, neutrino oscillations, flavor experiments, and collider experiments. We examine several benchmark scenarios and identify the valid parameter space of this model, finding that there are still allowed regions under the latest experimental constraints. We present new viable benchmark scenarios for this model, i.e., the right-handed neutrino dark-matter scenario and the scalar dark-matter scenario. We also mention the testability of these benchmark scenarios in future experiments.

Japan

Shinya Kanemura

Yushi Mura

Guohao Ying

Revisiting the model for radiative neutrino masses with dark matter in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mstyle mathvariant="normal"><mi>U</mi></mstyle><mo stretchy="false">(</mo><mn>1</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>B</mi><mo>−</mo><mi>L</mi></mrow></msub></mrow></math> gauge theory

In this work, we systematically study the interactions of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math>-wave <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup><msup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></math> systems within the framework of chiral effective field theory in heavy hadron formalism. We calculate the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup><msup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></math> effective potentials up to next-to-leading order, explore the bound state formations, and investigate the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup><msup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></math> scattering properties, such as scattering rate, scattering length, and effective range. Our results show that all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo>=</mo><mn>1</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup><msup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></math> potentials are repulsive, preventing the formation of bound states, while the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo>=</mo><mn>0</mn></math> potentials are generally attractive. Specifically, we get two important observations: first, the shallow bound state is more likely to exist in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><msup><mn>0</mn><mo>+</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> system than in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><msup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><msup><mn>1</mn><mo>+</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> system; second, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mo>*</mo></msup><msup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><msup><mn>0</mn><mo>+</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mo>*</mo></msup><msup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><msup><mn>1</mn><mo>+</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> systems possess relatively large binding energies and positive scattering lengths, which suggests strong bound state formations in these channels. So, the attractions in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mo>*</mo></msup><msup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup><mo stretchy="false">[</mo><mi>I</mi><mo>=</mo><mn>0</mn><mo stretchy="false">]</mo></math> systems are deeper than those in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><msup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mi>I</mi><mo>=</mo><mn>0</mn><mo stretchy="false">]</mo></math> systems, thus we strongly recommend the future experiment to search for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mo>*</mo></msup><msup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup><mo stretchy="false">[</mo><mi>I</mi><mo>=</mo><mn>0</mn><mo stretchy="false">]</mo></math> tetraquark systems. We also investigate the coupled-channel effects on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>J</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow></math> systems and conclude that the inclusion of the coupled channels introduces small but visible influences. In addition, we also investigate the dependencies of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><msup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></math> binding energies on the contact low-energy coupling constants.

Zhe Liu

Hao Xu

Xiang Liu

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup><msup><mrow><mover accent="true"><mrow><mi>B</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></mrow></math> dynamics in chiral effective field theory

We present a lattice QCD calculation of the pion scalar form factor and associated radii with fully controlled systematics. Lattice results are computed on a large set of 17 gauge ensembles with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> Wilson Clover-improved sea quarks. These ensembles cover four values of the lattice spacing between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mn>0.049</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mn>0.086</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>, a pion mass range of 130–350 MeV, and various physical volumes. A precise determination of the notorious quark-disconnected contributions facilitates an unprecedented momentum resolution for the form factor, particularly on large and fine ensembles in the vicinity of physical quark mass. A large range of source-sink separations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1.0</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi><mo>≲</mo><msub><mi>t</mi><mi>sep</mi></msub><mo>≲</mo><mn>3.25</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> is used to reliably extract the relevant ground state matrix elements at vanishing and nonvanishing momentum transfer. This allows us for the first time to obtain the scalar radii from a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>z</mi></math>-expansion parametrization of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Q</mi><mn>2</mn></msup></math> dependence of the resulting form factors rather than a simple, linear approximation at small momentum transfer. The physical extrapolation for the radii is carried out using three-flavor NLO chiral perturbation theory to parametrize the quark mass dependence in terms of three low-energy constants, including the first lattice determination of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>L</mi><mn>4</mn><mi>r</mi></msubsup></math>. Systematic uncertainties on the final results related to the ground state extraction, form factor parametrization as well as the physical extrapolation are accounted for via model averages based on the Akaike information criterion. Our physical results for the pion singlet and octet scalar radii are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">⟨</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mo stretchy="false">⟩</mo></mrow><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>=</mo><mn>0.550</mn><mo stretchy="false">(</mo><mn>32</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>39</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mo stretchy="false">[</mo><mn>50</mn><msub><mrow><mo stretchy="false">]</mo></mrow><mrow><mtext>total</mtext></mrow></msub><mtext> </mtext><mtext> </mtext><msup><mrow><mi>fm</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">⟨</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mo stretchy="false">⟩</mo></mrow><mrow><mi>π</mi></mrow><mrow><mn>8</mn></mrow></msubsup><mo>=</mo><mn>0.443</mn><mo stretchy="false">(</mo><mn>16</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>45</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mo stretchy="false">[</mo><mn>47</mn><msub><mrow><mo stretchy="false">]</mo></mrow><mrow><mtext>total</mtext></mrow></msub><mtext> </mtext><mtext> </mtext><msup><mrow><mi>fm</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>, respectively. The values of the relevant low energy constants are given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mspace linebreak="goodbreak"></mspace><mn>116.5</mn><mo stretchy="false">(</mo><mn>5.5</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>13.7</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mo stretchy="false">[</mo><mn>14.8</mn><msub><mrow><mo stretchy="false">]</mo></mrow><mrow><mtext>total</mtext></mrow></msub><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>L</mi><mn>4</mn><mi>r</mi></msubsup><mo stretchy="false">(</mo><mi>μ</mi><mo>=</mo><mn>770</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><mo stretchy="false">)</mo><mo>=</mo><mo>+</mo><mn>0.38</mn><mo stretchy="false">(</mo><mn>09</mn><msub><mo stretchy="false">)</mo><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>15</mn><msub><mo stretchy="false">)</mo><mi>sys</mi></msub><mo stretchy="false">[</mo><mn>18</mn><msub><mo stretchy="false">]</mo><mtext>total</mtext></msub><mo>×</mo><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"><msubsup><mi>L</mi><mn>5</mn><mi>r</mi></msubsup><mo stretchy="false">(</mo><mi>μ</mi><mo>=</mo><mn>770</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi><mo stretchy="false">)</mo><mo>=</mo><mo>+</mo><mn>0.58</mn><mo stretchy="false">(</mo><mn>0.38</mn><msub><mo stretchy="false">)</mo><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>1.08</mn><msub><mo stretchy="false">)</mo><mi>sys</mi></msub><mo stretchy="false">[</mo><mn>1.14</mn><msub><mo stretchy="false">]</mo><mtext>total</mtext></msub><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math>. For the light quark contribution to the pion scalar radius, we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">⟨</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mo stretchy="false">⟩</mo></mrow><mrow><mi>π</mi></mrow><mrow><mi>l</mi></mrow></msubsup><mo>=</mo><mn>0.515</mn><mo stretchy="false">(</mo><mn>28</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>32</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>sys</mi></mrow></msub><mo stretchy="false">[</mo><mn>43</mn><msub><mrow><mo stretchy="false">]</mo></mrow><mrow><mtext>total</mtext></mrow></msub><mtext> </mtext><mtext> </mtext><msup><mrow><mi>fm</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>, resulting in a value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mover accent="true"><mo>ℓ</mo><mo stretchy="true">¯</mo></mover><mn>4</mn></msub><mo>=</mo><mn>3.99</mn><mo stretchy="false">(</mo><mn>15</mn><msub><mo stretchy="false">)</mo><mrow><mi>stat</mi></mrow></msub><mo stretchy="false">(</mo><mn>17</mn><msub><mo stretchy="false">)</mo><mi>sys</mi></msub><mo stretchy="false">[</mo><mn>23</mn><msub><mo stretchy="false">]</mo><mtext>total</mtext></msub></math> for the pertinent low-energy constant in two-flavor chiral perturbation theory.

Scalar size of the pion from lattice QCD