The double pion photoproduction off nucleons near threshold is analyzed in a covariant baryon chiral perturbation theory up to next to leading order, where the <math><mrow><mi>Δ</mi><mo>(</mo><mn>1232</mn><mo>)</mo></mrow></math>, <math><msup><mi>N</mi><mo>*</mo></msup><mo>(</mo><mn>1400</mn><mo>)</mo></math>, and <math><mi>ρ</mi><mo>(</mo><mn>770</mn><mo>)</mo></math> resonances are included as explicit degrees of freedom. For the process <math><mi>γ</mi><mi>p</mi><mo>→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mn>0</mn></msup><mi>n</mi></math>, the chiral results of total cross sections, invariant-mass distributions, and beam-helicity asymmetry are in good agreement with the experimental data within uncertainties. For the process <math><mi>γ</mi><mi>p</mi><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><msup><mi>π</mi><mn>0</mn></msup><mi>p</mi></math>, the prediction of total cross section deviates from the existing experimental data. Once the final-state interaction of <math><mi>π</mi><mi>π</mi></math> in the isoscalar S-wave channel is taken into account, a good description of the cross section is achieved. The effect of the Roper resonance always turns out be negligible, and hence can be thrown away in future study of this process.

China

Kai-Ge Kang

Xiong-Hui Cao

De-Liang Yao

Han-Qing Zheng

Double pion photoproduction off nucleons in covariant chiral perturbation theory

The effective potential of the <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> model at large <math><mi>N</mi></math> limit is reinvestigated with varying pion mass and temperature. For large pion masses and high temperatures, we find the phenomenologically favored vacuum, located on the upper branch of the double-branched effective potential for physical <math><msub><mi>m</mi><mi>π</mi></msub></math>, moves to the lower branch and becomes no longer a local minimum but a saddle point. The existence and running of the tachyon pole are also discussed. These phenomena indicate that the applicable energy range of the <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> model is more and more limited as <math><msub><mi>m</mi><mi>π</mi></msub></math> becomes larger and the temperature goes higher. With the effective coupling constant defined from the effective potential, the possible correspondence between the two branches of the effective potential and the two phases of the theory (distinguished by positive or negative coupling) is verified even with nonzero explicit symmetry breaking and at finite temperature. Also, we generalize the <math><mi>N</mi><mo>/</mo><mi>D</mi></math> modified <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> model to study the thermal trajectory of the <math><mi>σ</mi></math> pole with the cross-channel contributions considered and find the thermal <math><mi>σ</mi></math> pole trajectory resembles its counterpart with varying pion masses at zero temperature.

Yuan-Lin Lyu

Qu-Zhi Li

Zhiguang Xiao

Revisiting <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> <math><mi>σ</mi></math> model at unphysical pion masses and high temperatures. II. The vacuum structure and thermal <math><mi>σ</mi></math> pole trajectory with cross-channel improvements

Roy-equation analysis on lattice data of <math><mi>π</mi><mi>π</mi></math> scattering phase shifts at <math><msub><mi>m</mi><mi>π</mi></msub><mo>=</mo><mn>391</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> reveals that the lowest <math><msub><mi>f</mi><mn>0</mn></msub></math> meson becomes a bound state under this condition. In addition, there is a pair of complex poles below threshold generated by crossing symmetry [X.-H. Cao et al., Phys. Rev. D 108, 034009 (2023)]. We use the <math><mi>N</mi><mo>/</mo><mi>D</mi></math> method to partially recover crossing symmetry of the <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> <math><mi>σ</mi></math> model amplitude at leading order of <math><mn>1</mn><mo>/</mo><mi>N</mi></math> expansion, and qualitatively reproduce the pole structure and pole trajectories with varying pion masses as revealed by Roy-equation analyses. The <math><mi>σ</mi></math> pole trajectory with varying temperature is also discussed and found to be similar to its properties when varying <math><msub><mi>m</mi><mi>π</mi></msub></math>. As the temperature increases, the complex <math><mi>σ</mi></math> poles firstly move from the second Riemann sheet to the real axis becoming two virtual state poles, and then one virtual state pole moves to the first sheet turning into a bound state pole and finally tends to the pion pole position at high temperature which is as expected from the chiral symmetry restoration. Our results provide further evidence that the lowest <math><msub><mi>f</mi><mn>0</mn></msub></math> state extracted from experiments and lattice data plays the role of the <math><mi>σ</mi></math> meson in the spontaneous breaking of chiral symmetry. Finally, we also briefly discuss the problems of the effective potential in the situation when <math><msub><mi>m</mi><mi>π</mi></msub></math> and the temperature become large.

Revisiting <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> <math><mi>σ</mi></math> model at unphysical pion masses and high temperatures

Recently, the LHCb experimental group found an exotic state   $ T^+_{cc} $   from the   $ T^+_{cc} $   process. A key question is whether it is just a molecule or may have a confined tetraquark ingredient. To investigate this, different methods were used, including a two-channel (  $ T^+_{cc} $   and   $ T^+_{cc} $  ) K-matrix unitarization and a single-channel Flatté-like parametrization method analyzed utilizing the pole counting rule and spectral density function sum rule. These analyses demonstrated that   $ T^+_{cc} $   is a molecular state, although the possibility that there may exist an elementary ingredient cannot be excluded, according to an approximate analysis of its production rate.

Chang Chen

Ce Meng

Some remarks on compositeness of   

The attractive interaction between   $ J/\psi $   and   $ J/\psi $   has to be strong enough if X(6900) is of the molecule type. We argue that since   $ J/\psi $   decays predominantly into a   $ J/\psi $   pair, the interactions between   $ J/\psi $   and   $ J/\psi $   may be significantly enhanced owing to the three point   $ J/\psi $   loop diagram. The enhancement originates from the anomalous threshold located at   $ J/\psi $   GeV  $ J/\psi $  , whose effect propagates into the s-channel partial wave amplitude in the vicinity of   $ J/\psi $   GeV. This effect may be helpful in the formation of the   $ J/\psi $   peak.

Ye Lu

Guang-You Qin

A discussion on the anomalous threshold enhancement of – Couplings and Peak  

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.

Zhi-Hui Guo

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

A near-threshold enhancement in the  $$D^{+}_{s}D^{-}_{s}$$  <math> <mrow> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> <msubsup> <mi>D</mi> <mi>s</mi> <mo>-</mo> </msubsup> </mrow> </math>   system, dubbed as X(3960), is observed by the LHCb collaboration recently. A combined analysis on  $$\chi _{c0}(3930)~(\rightarrow D^+ D^-)$$  <math> <mrow> <msub> <mi>χ</mi> <mrow> <mi>c</mi> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mn>3930</mn> <mo>)</mo> </mrow> <mspace width="3.33333pt"></mspace> <mrow> <mo>(</mo> <mo>→</mo> <msup> <mi>D</mi> <mo>+</mo> </msup> <msup> <mi>D</mi> <mo>-</mo> </msup> <mo>)</mo> </mrow> </mrow> </math>  ,  $$X(3960)~(\rightarrow D^{+}_{s}D^{-}_{s})$$  <math> <mrow> <mi>X</mi> <mrow> <mo>(</mo> <mn>3960</mn> <mo>)</mo> </mrow> <mspace width="3.33333pt"></mspace> <mrow> <mo>(</mo> <mo>→</mo> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> <msubsup> <mi>D</mi> <mi>s</mi> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> </mrow> </math>  , and  $$X(3915)~(\rightarrow J/\psi \omega )$$  <math> <mrow> <mi>X</mi> <mo>(</mo> <mn>3915</mn> <mo>)</mo> <mspace width="3.33333pt"></mspace> <mo>(</mo> <mo>→</mo> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> <mi>ω</mi> <mo>)</mo> </mrow> </math>   is performed using both a K-matrix approach of  $$D_{(s)}\bar{D}_{(s)}$$  <math> <mrow> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> <msub> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </mrow> </math>   four-point contact interactions and a model of Flatté-like parameterizations. The use of the pole counting rule and spectral density function sum rule indicate, under current statistics, that this  $$D^{+}_{s}D^{-}_{s}$$  <math> <mrow> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> <msubsup> <mi>D</mi> <mi>s</mi> <mo>-</mo> </msubsup> </mrow> </math>   near-threshold state has probably the mixed nature of a  $$c\bar{c}$$  <math> <mrow> <mi>c</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   confining state and  $$D^{+}_{s}D^{-}_{s}$$  <math> <mrow> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> <msubsup> <mi>D</mi> <mi>s</mi> <mo>-</mo> </msubsup> </mrow> </math>   continuum.

Yu Chen

Hao Chen

Taiwan

Hong-Rong Qi

On the nature of X(3960)

Analyses of the LHCb data on <math><mi>X</mi><mo>(</mo><mn>6900</mn><mo>)</mo></math> found in di-<math><mi>J</mi><mo>/</mo><mi>ψ</mi></math> and <math><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>ψ</mi><mo>(</mo><mn>3686</mn><mo>)</mo></math> systems are performed using a momentum-dependent Flatté-like parametrization. The use of the pole-counting rule and spectral density function sum rule give consistent conclusions that <math><mi>X</mi><mo>(</mo><mn>6900</mn><mo>)</mo></math> may not be a molecule of <math><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>ψ</mi><mo>(</mo><mn>3686</mn><mo>)</mo></math>. Nevertheless, it is still possible that <math><mi>X</mi><mo>(</mo><mn>6900</mn><mo>)</mo></math> is a molecule of higher states, such as <math><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>ψ</mi><mo>(</mo><mn>3770</mn><mo>)</mo></math>, <math><msub><mi>χ</mi><mrow><mi>c</mi><mn>0</mn></mrow></msub><msub><mi>χ</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub></math>, etc.

<math><mi>X</mi><mo>(</mo><mn>6900</mn><mo>)</mo></math> peak could be a molecular state

Neural networks are trained to judge whether or not an exotic state is a hadronic molecule of a given channel according its line-shapes. This method performs well in both trainings and validation tests. As applications, it is applied to study X(3872), X(4260) and  $$Z_c(3900)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>3900</mn> <mo>)</mo> </mrow> </mrow> </math>  . The results show that  $$Z_c(3900)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>3900</mn> <mo>)</mo> </mrow> </mrow> </math>   should be regarded as a  $${\bar{D}}^* D$$  <math> <mrow> <msup> <mrow> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mo>∗</mo> </msup> <mi>D</mi> </mrow> </math>   molecular state but X(3872) not. As for X(4260), it can not be a molecular state of  $$\chi _{c0}\omega $$  <math> <mrow> <msub> <mi>χ</mi> <mrow> <mi>c</mi> <mn>0</mn> </mrow> </msub> <mi>ω</mi> </mrow> </math>  . Some discussions on  $$X_1(2900)$$  <math> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>2900</mn> <mo>)</mo> </mrow> </mrow> </math>   are also provided.

Wen-Qi Niu

Identifying hadronic molecular states with a neural network

The hyperbolic version of Roy-Steiner equation describing low energy πN scatterings, with larger analyticity domain in the complex s plane is solved. The numerical results on phase shifts of low partial waves are in agreement with that of Hoferichter et al. [Phys. Rept. 625 (2016) 1]. A subthreshold pole in S 11 channel is found located at   <math> <msqrt> <mi>s</mi> </msqrt> </math>  $$ \sqrt{s} $$  = (918 ± 3) − i(163 ± 9) MeV.