We calculate the nucleon mass in a manifestly relativistic baryon chiral perturbation theory up to the leading two-loop order. Through dimensional counting analysis, we perform the chiral expansion and verify the validity of the extended-on-mass-shell scheme at the two-loop level. As a result, we obtain the complete chiral representation of the nucleon mass up to $$ \mathcal{O} $$ ($\textit{p}$ ), which preserves the original analytic properties and satisfies the correct power counting. The obtained chiral result is well-suited for chiral extrapolation and provides an excellent description of lattice QCD data across a broad range of pion masses. We find that the $$ \mathcal{O} $$ ($\textit{p}$ ) contribution is small, approximately 10 MeV, and varies only mildly with increasing pion mass, demonstrating good convergence of the nucleon mass up to pion masses of about 350 MeV at two-loop order.

China

Ze-Rui Liang

Han-Xue Chen

Feng-Kun Guo

Zhi-Hui Guo

De-Liang Yao

Chiral representation of the nucleon mass at leading two-loop order

<p>Many exotic charmoniumlike mesons have already been discovered experimentally, of which the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>Z</mi><mi>c</mi></msub></math> mesons with isospin 1 are prominent examples. We investigate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>+</mo><mo>±</mo></mrow></msup></math> states with flavor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mi>c</mi><mover accent="true"><mi>q</mi><mo stretchy="false">¯</mo></mover><mi>q</mi></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>q</mi><mo>=</mo><mi>u</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>d</mi></mrow></math>) in isospin 1 using lattice QCD. This is the first study of these mesons employing more than one volume and involving frames with nonzero total momentum. We utilize two <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> CLS ensembles with <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><mn>280</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>. The simulations are performed with unphysical light quark masses at a single lattice spacing of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>≃</mo><mn>0.086</mn><mtext> </mtext><mtext> </mtext><mrow><mi>fm</mi></mrow></mrow></math> and omit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ψ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>S</mi><mo stretchy="false">)</mo><mi>π</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ψ</mi><mo stretchy="false">(</mo><mn>3770</mn><mo stretchy="false">)</mo><mi>π</mi></math> and three-particle decay channels, so our results provide only qualitative insights. Resulting eigenenergies are compatible or just slightly shifted down with respect to noninteracting energies, where the most significant shifts occur for certain <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></math> states. Both channels <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mrow><mo>+</mo><mo>±</mo></mrow></msup></math> have a virtual pole slightly below the threshold if <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></math> is assumed to be decoupled from other channels. In addition, we perform a coupled channel analysis of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>π</mi></math> and <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></math> scattering with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>+</mo><mo>−</mo></mrow></msup></math> within an effective field theory framework. The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>π</mi></math> and <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></math> invariant-mass distributions from BESIII and finite-volume energies from several lattice QCD simulations, including this work, are fitted simultaneously. All fits yield two poles relatively close to the <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></math> threshold and reasonably reproduce the experimental <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>Z</mi><mi>c</mi></msub></math> peaks. They also reproduce lattice energies up to slightly above the <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></math> threshold, while reproduction at even higher energies is better for fits that put more weight on the lattice data. Our findings suggest that the employed effective field theory can reasonably reconcile the peaks in the experimental line shapes and the lattice energies, although those lie close to noninteracting energies. We also study <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>π</mi></math> scattering in s wave and place upper bounds on the phase shift.</p>

Slovenia

Mitja Sadl

Germany

Sara Collins

India

M. Padmanath

Sasa Prelovsek

Lin-Wan Yan

Charmoniumlike channels <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mo>+</mo></msup></math> with isospin 1 from lattice and effective field theory

We pursue the calculation of the model-independent component of the axion-photon-photon coupling in the (3) chiral perturbation theory up to next-to-leading order, with the emphasis on the isospin breaking effect. The mixing of the $$\pi ^0$$ - $$\eta $$ - $$\eta '$$ -axion system is revised as well by working out the complete linear isospin-breaking terms. Our calculation shows that the isospin-breaking correction to the axion-photon-photon coupling amounts to more than 15%, comparing with the result in the isospin limit.

Rui Gao

Jin Hao

Chun-Gui Duan

Spain

J. Oller

Hai-Qing Zhou

Isospin-breaking contribution to the model-independent axion-photon-photon coupling in

We calculate the amplitudes of the axion photoproduction off the nucleon, i.e., <math><mi>γ</mi><mi>N</mi><mo>→</mo><mi>a</mi><mi>N</mi></math>, within the framework of chiral effective field theory. Several different types of contributions are simultaneously included in our calculation, namely, the nucleon exchanges up to next-to-leading order, the <math><mi>a</mi><mi>γ</mi><mi>γ</mi></math> vertex, and the vector-meson exchanges in the <math><mi>t</mi></math> channel. We utilize the existing hadronic inputs as much as possible to fix the unknown couplings. A comprehensive study of the phenomenological discussions is then provided in this work. Different mechanisms in the <math><mi>γ</mi><mi>N</mi><mo>→</mo><mi>a</mi><mi>N</mi></math> processes manifest distinct behaviors in the total and differential cross sections, which could provide useful quantities to distinguish different axion models.

Xiong-Hui Cao

Comprehensive study of axion photoproduction off the nucleon in chiral effective field theory

We study the mass spectra and radiative decays of <math><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>0</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>(</mo><mn>2317</mn><mo>)</mo></mrow></math> and <math><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>1</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mn>2460</mn><mo>)</mo></mrow></math> in an unquenched framework. In addition to coupled channel effects between the <math><mrow><mi>c</mi><mrow><mover><mrow><mi>s</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></mrow></math> cores and <math><msup><mi>D</mi><mrow><mo>(</mo><mo>*</mo><mo>)</mo></mrow></msup><mi>K</mi></math> channels, <math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo>(</mo><mo>*</mo><mo>)</mo></mrow></msup><mi>K</mi><mtext>−</mtext><msup><mrow><mi>D</mi></mrow><mrow><mo>(</mo><mo>*</mo><mo>)</mo></mrow></msup><mi>K</mi></mrow></math> self-interactions are also considered in this work and we succeed to reproduce their mass spectra. Furthermore, we study the radiative decays of the <math><msubsup><mi>D</mi><mrow><mi>s</mi><mn>0</mn></mrow><mo>*</mo></msubsup><mo>(</mo><mn>2317</mn><mo>)</mo></math> and <math><msubsup><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow><mo>′</mo></msubsup><mo>(</mo><mn>2460</mn><mo>)</mo></math> by simultaneously including the compound structures of conventional <math><mi>c</mi><mover><mi>s</mi><mo>¯</mo></mover></math> cores and <math><msup><mi>D</mi><mrow><mo>(</mo><mo>*</mo><mo>)</mo></mrow></msup><mi>K</mi></math> components. We also calculate their bottom analogs with heavy quark symmetry. Our study offers useful insights into the important unquenched effects in the formation of <math><msubsup><mi>D</mi><mrow><mi>s</mi><mn>0</mn></mrow><mo>*</mo></msubsup><mo>(</mo><mn>2317</mn><mo>)</mo></math>, <math><msubsup><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow><mo>′</mo></msubsup><mo>(</mo><mn>2460</mn><mo>)</mo></math>, and the bottom counterparts.

Zi-Le Zhang

Zhan-Wei Liu

Si-Qiang Luo

Ping Chen

Masses and radiative decay widths of <math><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>0</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>(</mo><mn>2317</mn><mo>)</mo></mrow></math> and <math><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>1</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mn>2460</mn><mo>)</mo></mrow></math> and their bottom analogs

We study the axion and axion-like particle production from the η → ππa decay within the SU(3) chiral perturbation theory up to the one-loop level. The conventional SU(3) chiral low energy constants are found to be able to reabsorb all the divergences from the chiral loops in the η → ππa decay amplitude, and hence render the amplitude independent of the renormalization scale. The unitarized η → ππa decay amplitudes are constructed to take into account the ππ final-state interactions and also properly reproduce the perturbative results from the chiral perturbation theory. Detailed analyses between the perturbative amplitudes and the unitarized ones are given in the phenomenological discussions. By taking the values of the chiral low energy constants in literature, we predict the Dalitz distributions, the spectra of the ππ and aπ systems, and also the branching ratios of the η → ππa process by varying m  a  from 0 to m  η  − 2m  π .

Jin-Bao Wang

Zhun Lu

Axion production in the η → ππa decay within SU(3) chiral perturbation theory

Axion-pion scattering amplitudes at finite temperatures are calculated within chiral perturbation theory up to the one loop level. Unitarization procedure is implemented to these amplitudes in order to extend the applicable range of energy and temperature. The influence of the thermal axion-pion scattering amplitudes on the <math><mi>a</mi><mi>π</mi><mo>→</mo><mi>π</mi><mi>π</mi></math> cross sections and the axion thermalization rate is investigated, with the emphasis on the comparison with the zero-temperature-amplitude case. A brief discussion on the cosmological implication of the axion thermalization rate, which is calculated by using the <math><mi>a</mi><mi>π</mi><mo>→</mo><mi>π</mi><mi>π</mi></math> amplitudes at finite temperatures, is also given. The thermal corrections to the axion-pion scattering amplitudes can cause around a 10% shift of the determination of the axion decay constant <math><msub><mi>f</mi><mi>a</mi></msub></math> and its mass <math><msub><mi>m</mi><mi>a</mi></msub></math>, comparing with the results by using the <math><mi>a</mi><mi>π</mi><mo>→</mo><mi>π</mi><mi>π</mi></math> amplitudes at zero temperature.

Axion-pion scattering at finite temperature in chiral perturbation theory and its influence in axion thermalization

We study the <math><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>π</mi></math> and <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> coupled-channel system within a covariant framework. The <math><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>π</mi></math> and <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> invariant-mass distributions measured at 4.23 and 4.26 GeV by BESIII and the finite-volume energy levels from recent lattice QCD simulations are simultaneously fitted. Phase shifts and inelasticities of the <math><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>π</mi></math> and <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> scattering are predicted using the resulting amplitudes. Poles corresponding to the <math><msub><mi>Z</mi><mi>c</mi></msub><mo>(</mo><mn>3900</mn><mo>)</mo></math> state are found in the complex energy plane and their couplings with <math><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>π</mi></math> and <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> are determined. Our results indicate that the current lattice data do not preclude the existence of the <math><msub><mi>Z</mi><mi>c</mi></msub><mo>(</mo><mn>3900</mn><mo>)</mo></math> state.

Zhi-Yong Zhou

Reconciling experimental and lattice data of <math><msub><mi>Z</mi><mi>c</mi></msub><mo>(</mo><mn>3900</mn><mo>)</mo></math> in a <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>π</mi><mtext>−</mtext><mi>D</mi><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> coupled-channel analysis

Motivated by the recent observation of <math><msub><mi>T</mi><mrow><mi>c</mi><mover><mi>s</mi><mo>¯</mo></mover><mn>0</mn></mrow></msub><mo>(</mo><mn>2900</mn><msup><mo>)</mo><mn>0</mn></msup></math> and <math><msub><mi>T</mi><mrow><mi>c</mi><mover><mi>s</mi><mo>¯</mo></mover><mn>0</mn></mrow></msub><mo>(</mo><mn>2900</mn><msup><mo>)</mo><mrow><mo>+</mo><mo>+</mo></mrow></msup></math> in the <math><msub><mi>D</mi><mi>s</mi></msub><mi>π</mi></math> invariant mass distributions, we investigate <math><msup><mi>D</mi><mo>*</mo></msup><msup><mi>K</mi><mo>*</mo></msup></math> interactions in a coupled-channel approach. Within the hidden local symmetry formalism, a sizable attraction interaction is found in the <math><mi>J</mi><mo>=</mo><mn>0</mn></math> isospin triplet sector that can form a bound or a virtual state, which is consistent with the experimentally observed <math><msub><mi>T</mi><mrow><mi>c</mi><mover><mi>s</mi><mo>¯</mo></mover><mn>0</mn></mrow></msub><mo>(</mo><mn>2900</mn><mo>)</mo></math>. By reproducing a <math><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mi>ρ</mi><mtext>−</mtext><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> bound/virtual state with the pole mass equal to that of the <math><msub><mi>T</mi><mrow><mi>c</mi><mover><mi>s</mi><mo>¯</mo></mover><mn>0</mn></mrow></msub><mo>(</mo><mn>2900</mn><mo>)</mo></math> measured by LHCb in the sector <math><mo>(</mo><mi>I</mi><mo>,</mo><mi>J</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>)</mo></math>, we determine the unknown parameter in the loop function, and then search for possible poles in the sectors of <math><mi>I</mi><mo>=</mo><mn>1</mn></math>, <math><mi>J</mi><mo>=</mo><mn>1</mn></math>, 2 and <math><mi>I</mi><mo>=</mo><mn>0</mn></math>, <math><mi>J</mi><mo>=</mo><mn>0</mn></math>, 1, 2. The predicted resonances provide a useful reference for the future experimental studies of the <math><mo>(</mo><mi>C</mi><mo>,</mo><mi>S</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math> systems and can be also helpful to unravel the nature of the <math><msub><mi>T</mi><mrow><mi>c</mi><mover><mi>s</mi><mo>¯</mo></mover><mn>0</mn></mrow></msub><mo>(</mo><mn>2900</mn><mo>)</mo></math>.

Man-Yu Duan

Meng-Lin Du

En Wang

Dian-Yong Chen

Coupled-channel <math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>*</mo></mrow></msup><mtext>−</mtext><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>*</mo></mrow></msubsup><mi>ρ</mi></mrow></math> interactions and the origin of <math><msub><mi>T</mi><mrow><mi>c</mi><mover><mi>s</mi><mo>¯</mo></mover><mn>0</mn></mrow></msub><mo>(</mo><mn>2900</mn><mo>)</mo></math>

An extended Roy equation including a bound state pole is used to study <math><mi>π</mi><mi>π</mi></math> scatterings at unphysical large pion masses when <math><mi>σ</mi></math> becomes a bound state in one situation and stays as a broad resonance in the other case. The coupled integral equations at large pion masses are solved by taking the lattice driving terms and the Regge amplitudes as inputs. Relying on the solutions of Roy equations that respect unitarity, analyticity and crossing symmetry, we give predictions to the phase shifts with <math><mi>I</mi><mi>J</mi><mo>=</mo><mn>00</mn></math>, 11, 20 in the elastic energy region. We then perform analytic continuation into the complex <math><mi>s</mi></math> plane to search for various poles, all of which are inside the validity domain of the Roy equation. This is the first time that lattice data at unphysical large pion masses are analyzed within the rigorous Roy equation method.

Qu-Zhi Li

Han-Qing Zheng

Roy equation analyses of <math><mi>π</mi><mi>π</mi></math> scatterings at unphysical pion masses