Hunting for the <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>X</mi></mrow><mrow><mi>b</mi></mrow></msub></math> via radiative decays

In this paper, we study radiative decays of <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>X</mi></mrow><mrow><mi>b</mi></mrow></msub></math> , the counterpart of the famous <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math> in the bottomonium-sector as a candidate for meson–meson molecule, into the <math altimg="si3.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mi>ϒ</mi><mo stretchy="false">(</mo><mi>n</mi><mi>S</mi><mo stretchy="false">)</mo></math> ( <math altimg="si4.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></math> ). Since it is likely that the <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>X</mi></mrow><mrow><mi>b</mi></mrow></msub></math> is below the <math altimg="si5.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><msup><mrow><mover accent="true"><mrow><mi>B</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>⁎</mo></mrow></msup></math> threshold and the mass difference between the neutral and charged bottom meson is small compared to the binding energy of the <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>X</mi></mrow><mrow><mi>b</mi></mrow></msub></math> , the isospin violating decay mode <math altimg="si6.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>X</mi></mrow><mrow><mi>b</mi></mrow></msub><mo stretchy="false">→</mo><mi>ϒ</mi><mo stretchy="false">(</mo><mi>n</mi><mi>S</mi><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></math> would be greatly suppressed. This will promote the importance of the radiative decays. We use the effective Lagrangian based on the heavy quark symmetry to explore the rescattering mechanism and calculate the partial widths. Our results show that the partial widths into <math altimg="si7.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mi>ϒ</mi><mo stretchy="false">(</mo><mi>n</mi><mi>S</mi><mo stretchy="false">)</mo></math> are about 1 keV, and thus the branching fractions may be sizeable, considering the fact the total width may also be smaller than a few MeV like the <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math> . These radiative decay modes are of great importance in the experimental search for the <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>X</mi></mrow><mrow><mi>b</mi></mrow></msub></math> particularly at hadron collider. An observation of the <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>X</mi></mrow><mrow><mi>b</mi></mrow></msub></math> will provide a deeper insight into the exotic hadron spectroscopy and is helpful to unravel the nature of the states connected by the heavy quark symmetry

"Covariance Matrix of the Pion Form Factor" of "Measurement of the e+e−→π+π−\mathrm e^+\mathrm e^-\rightarrow\mathrm\pi^+\mathrm\pi^- Cross Section between 600 and 900 MeV Using Initial State Radiation"

Statistical covariance matrix of the pion form factor $|F_\pi|^2$

"Pion Form Factor" of "Measurement of the e+e−→π+π−\mathrm e^+\mathrm e^-\rightarrow\mathrm\pi^+\mathrm\pi^- Cross Section between 600 and 900 MeV Using Initial State Radiation"

Pion form factor $|F_\pi|^2$ measured using the initial state radiation method. The data is corrected concerning vacuum polarization effects

"Table 1" of "Measurement of the e+e−→π+π−\mathrm e^+\mathrm e^-\rightarrow\mathrm\pi^+\mathrm\pi^- Cross Section between 600 and 900 MeV Using Initial State Radiation"

Results for the bare cross section $\sigma^\text{bare}_{\pi^+\pi^-}$ and the pion form factor together with their statistical uncertainties. The systematical uncertainties are given by 0.9% (see arXiv:1507.08188)

"Bare Cross Section" of "Measurement of the e+e−→π+π−\mathrm e^+\mathrm e^-\rightarrow\mathrm\pi^+\mathrm\pi^- Cross Section between 600 and 900 MeV Using Initial State Radiation"

Bare cross section $\sigma^\mathrm{bare}(e^+e^-\to\pi^+\pi^-(\gamma_\mathrm{FSR}))$ of the process $e^+e^-\to\pi^+\pi^-$ measured using the initial state radiation method. The data is corrected concerning final state radiation and vacuum polarization effects. The final state radiation is added using the Schwinger term at born level

"Bare Cross Section" of "Measurement of the e+e−→π+π−\mathrm e^+\mathrm e^-\rightarrow\mathrm\pi^+\mathrm\pi^- Cross Section between 600 and 900 MeV Using Initial State Radiation"

Bare cross section $\sigma^\mathrm{bare}(e^+e^-\to\pi^+\pi^-(\gamma_\mathrm{FSR}))$ of the process $e^+e^-\to\pi^+\pi^-$ measured using the initial state radiation method. The data is corrected concerning final state radiation and vacuum polarization effects. The final state radiation is added using the Schwinger term at born level

"Covariance Matrix of the Pion Form Factor" of "Measurement of the e+e−→π+π−\mathrm e^+\mathrm e^-\rightarrow\mathrm\pi^+\mathrm\pi^- Cross Section between 600 and 900 MeV Using Initial State Radiation"

Statistical covariance matrix of the pion form factor $|F_\pi|^2$

"Table 1" of "Measurement of the e+e−→π+π−\mathrm e^+\mathrm e^-\rightarrow\mathrm\pi^+\mathrm\pi^- Cross Section between 600 and 900 MeV Using Initial State Radiation"

Results of the BESIII measurement of the cross section $\sigma^{\rm bare}_{\pi^+\pi^-(\gamma_{\rm FSR})} \equiv \sigma^{\rm bare}(e^+e^-\rightarrow\pi^+\pi^-(\gamma_{\rm FSR}))$ and the squared pion form factor $|F_\pi|^2$. The errors are statistical only. The value of $\sqrt{s'}$ represents the bin center. The 0.9$\%$ systematic uncertainty is fully correlated between any two bins

"Pion Form Factor" of "Measurement of the e+e−→π+π−\mathrm e^+\mathrm e^-\rightarrow\mathrm\pi^+\mathrm\pi^- Cross Section between 600 and 900 MeV Using Initial State Radiation"

Pion form factor $|F_\pi|^2$ measured using the initial state radiation method. The data is corrected concerning vacuum polarization effects

"Covariance Matrix of the Bare Cross Section" of "Measurement of the e+e−→π+π−\mathrm e^+\mathrm e^-\rightarrow\mathrm\pi^+\mathrm\pi^- Cross Section between 600 and 900 MeV Using Initial State Radiation"

Statistical covariance matrix of the bare cross section $\sigma^\mathrm{bare}(e^+e^-\to\pi^+\pi^-(\gamma_\mathrm{FSR}))$