arXiv subjects

Shashank Bhatnagar

Publications and source records attributed to Shashank Bhatnagar.

At least 19 recordsLinked to original sources

arXiv2026

Exclusive production of P-wave charmonia through two virtual photons in electron-positron annihilation

We present a relativistic study of the exclusive double-charmonium processes $e^-e^+ \to \gamma^*\gamma^* \to \chi_{c1}+\eta_c$ and $e^-e^+ \to \gamma^*\gamma^* \to h_c+h_c$ at $\sqrt{s}=10.6$ GeV in the framework of the $4\times4$ Bethe-Salpeter equation. Since these channels are forbidden in single-photon annihilation, they proceed purely through two-photon quark-rearrangement mechanisms and thus provide a clean probe of relativistic quarkonium dynamics. Using parameters fixed from mass spectroscopy in our earlier work, we obtain leading-order predictions for the total cross sections and their energy dependence. The predicted cross sections are of order $10^{-3}$ fb and exhibit a hierarchy that originates from the radial structure of the meson wave functions, which determines the overlap integrals contributing to the production amplitude. While the absolute magnitudes of the cross sections differ from NRQCD estimates, the qualitative ordering of the channels is consistent with NRQCD expectations. We find a smooth power-law falloff of the cross section for both $\chi_{c1}+\eta_c$ and $h_c+h_c$. These results provide benchmark predictions for future searches at high-luminosity $e^-e^+$ colliders.

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arXiv2025

$\gamma, J/\psi$ production through ISR process in electron-positron annihilation at B-factories

We study the Initial state radiation (ISR) process, $e^- e^+ \rightarrow^*\gamma\rightarrow \gamma +J/\psi$ observed at high luminosity $e^- e^+$ colliders such as B-factories, with the ISR photon being emitted through the initial electron or positron, prior to their annihilation. The emitted $J/\Psi$ is produced through the photon fragmentation process, with the produced $J/\psi$ in turn decaying into a pair of leptons, $\mu^-+\mu^+$ through the leptonic decay process, $J/\Psi\rightarrow \gamma^*\rightarrow \gamma \mu^- \mu^+$. The cross section for the $J/\Psi$ production, and the branching ratio for the leptonic decay of $J/\Psi\rightarrow \mu^- \mu^+$, both calculated in the framework of Bethe-Salpeter equation are used to work out the cross section for the complete process, $e^- e^+ \rightarrow \gamma +J/\psi\rightarrow \gamma+\mu^- +\mu^+$. Our results are compared with the BaBar data, and other models. This study is then extended to production and decays of $\Upsilon(1S)$ through the ISR process.

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arXiv2024

$(J/\psi, J/\psi)$, and $(\eta_c, \eta_c)$ production through two intermediate photons in electron-positron annihilation at B-factories

We study the processes, $e^- e^+ \rightarrow \gamma^* \gamma^* \rightarrow J/\psi +J/\psi$, and $e^- e^+ \rightarrow \gamma^* \gamma^* \rightarrow \eta_c+ \eta_c$ at $\sqrt{s}=10.6$ GeV in the framework of $4\times 4$ Bethe-Salpeter equation. For $J/\psi+J/\psi$ production, the dominant contribution is through fragmentation process, while for $\eta_c+\eta_c$ production, the quark rearrangement diagrams contribute. Our results of cross section for $J/\psi+J/\psi$ and $\psi(2S)+\psi(2S)$ are compatible with the experimental upper limits set by Belle Collaboration, while in the absence of experimental data for $\eta_c(1S)+\eta_c(1S)$, and $\eta_c(2S)+\eta_c(2S)$ production, we have given theoretical prediction of their cross sections, and compared with NRQCD prediction.

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arXiv2024

Radiative $E1$ transitions between $^3P_1$ and $^3S_1$ quarkonium states

In this work we study the E1 decay processes, $^3P_1$ $\rightarrow$ $^3S_1\gamma$, and $^3S_1$ $\rightarrow$ $^3P_1\gamma$ in the framework of Bethe-Salpeter equation and calculate their decay widths. We have used algebraic forms of Salpeter wave functions obtained through analytic solutions of mass spectral equations for ground and excited states of $^3S_1$, and $^3P_1$ equal mass quarkonia in approximate harmonic oscillator basis to do analytic calculations of their decay widths. These decay widths have been compared with data and other models.

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arXiv2022

($\gamma$, $\chi_{cJ}(J=0,1)$) and ($\gamma$, $\eta_c$) production in electron-positron annihilation at $\sqrt{s}=10.6$ GeV and $4.6$ GeV in the framework of Bethe-Salpeter equation

In present work we study the production of ground and excited charmonium states in $e^- e^+ \rightarrow \gamma\chi_{cJ}(nP)(J=0,1)$, and $e^- e^+ \rightarrow \gamma \eta_c(nS)$, through leading order (LO) diagrams, which proceed through exchange of a virtual photon that couples to $\gamma$ and $\eta_c/\chi_{cJ}$ through the triangular quark loop diagram, in the framework of $4\times 4$ Bethe-Salpeter equation (BSE), at center of mass energies $\sqrt{s}=10.6$ GeV(Belle energy), and 4.6 GeV (BESIII energy). The amplitude simplifies to a general form required by Lorentz-covariance, in terms of its form factors. The cross sections for these processes with leading order (tree level) diagrams alone at $\sqrt{s}$=10.6 GeV provide a sizable contribution, which might be mainly due to the BSE being a fully relativistic approach that incorporates the relativistic effect of quark spins and can also describe internal motion of constituent quarks within the hadron in a relativistically covariant manner. Our results are compared with recent Belle data at 10.6 GeV, and BESIII data at 4.6 GeV, as well as other models. Plots of cross sections versus the center-of mass energy, $\sqrt{s}$ reveal mild fluctuations in their behaviour for all the three processes in the low energy ($\sqrt{s}$ = 4 - 6 GeV) region, and are analyzed in terms of the form factors.

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arXiv2022

$J/\Psi$ and $\chi_{cJ}(J=0,1)$ production in electron-positron annihilation at $\sqrt{s}=10.6$ GeV in the framework of Bethe-Salpeter equation

In present work we study the production of ground and excited charmonium pairs in $e^- e^+ \rightarrow \Psi(nS)+ \chi_{cJ}(nP)$ for $J=0,1$ and $n=1,2$, through leading order (LO) tree-level diagrams $\sim O(\alpha_{em} \alpha_s)$, which proceed through exchange of a virtual photon and an internal gluon line connecting two quark lines (in the triangle quark loop part of the diagram), in the framework of $4\times 4$ Bethe-Salpeter equation, at center of mass energy, $\sqrt{s}=10.6 GeV.$ We have tried to show that the cross sections for this process calculated with use of the four leading order $\sim O(\alpha_{em} \alpha_s)$ diagrams using BSE approach come close to experimental data, which might be mainly due to the consistent treatment of motion of quarks inside the hadrons in the BSE framework.

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arXiv2021

$E1$ and M1 radiative transitions involving heavy-light axial, pseudoscalar and vector quarkonia in the framework of Bethe-Salpeter equation

This work is an extension of our previous work in \cite{bhatnagar20} to calculate M1 transitions, $0^{-+}\rightarrow 1^{--} \gamma$, and E1 transitions involving axial vector mesons such as, $1^{+-} \rightarrow 0^{-+}\gamma$, and $0^{-+}\rightarrow 1^{+-} \gamma $ for which very little data is available as of now. We make use of the general structure of the transition amplitude, $M_{fi}$ derived in our previous work \cite{bhatnagar20} as a linear superposition of terms involving all possible combinations of $++$, and $--$ components of Salpeter wave functions of final and initial hadrons. In the present work, we make use of leading Dirac structures in the hadronic Bethe-Salpeter wave functions of the involved hadrons, which makes the formulation more rigorous. We evaluate the decay widths for both the above mentioned $M1$ and $E1$ transitions. We have used algebraic forms of Salpeter wave functions obtained through analytic solutions of mass spectral equations for ground and excited states of $1^{--}$,$0^{-+}$ and $1^{+-}$ heavy-light quarkonia in approximate harmonic oscillator basis to do analytic calculations of their decay widths. We have compared our results with experimental data, where ever available, and other models.

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arXiv2020

Mass spectra and decay constants of heavy-light axial quarkonia in the framework of Bethe-Salpeter equation

In this work we calculate the mass spectrum and decay constants of ground and excited states of heavy-light $P$-wave mesons such as $1^{++}$ and $1^{+-}$, with quark composition, $c\overline{u}, c\overline{s}, b\overline{u}, b\overline{s}$, and $b\overline{c}$ in the framework of a QCD motivated Bethe-Salpeter equation (BSE) by making use of the exact treatment of the spin structure $(\gamma_{\mu}\bigotimes\gamma_{\mu})$ in the interaction kernel, In this $4\times 4$ BSE framework, the coupled Salpeter equations for $Q\overline{q}$ are first solved for the confining part of interaction, and are shown to decouple under heavy-quark approximation. Then the one-gluon-exchange interaction is perturbatively incorporated, leading to their mass spectral equations. The analytic forms of wave functions obtained from these equations are then used for calculation of leptonic decay constants of ground and excited states of $1^{++}$, and $1^{+-}$ as a test of these wave functions and the over all framework.

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arXiv2020

Radiative decays of heavy-light quarkonia through $M1$, and $E1$ transitions in the framework of Bethe-Salpeter equation

In this work we study the radiative decays of heavy-light quarkonia through M1 and E1 transitions, that involve quark-triangle diagrams with two hadron vertices, and are difficult to evaluate in BSE-CIA. We have expressed the transition amplitude, $M_{fi}$ as a linear superposition of terms involving all possible combinations of $++$, and $--$ components of Salpeter wave functions of final and initial hadron, with coefficients being related to results of pole integrals over complex $σ$- plane. We evaluate the decay widths for $M1$ transitions ($^3S_1 \rightarrow ^1S_0 +γ$), and $E1$ transitions ($^3S_1 \rightarrow ^1P_0 +γ$ and $^1P_0 \rightarrow ^3S_1 +γ$). We have used algebraic forms of Salpeter wave functions obtained through analytic solutions of mass spectral equations for ground and excited states of $0^{++},1^{--}$, and $0^{-+}$ heavy-light quarkonia in approximate harmonic oscillator basis, to calculate their decay widths. The input parameters used by us were obtained by fitting to their mass spectra. We have compared our results with experimental data and other models, and found reasonable agreements.

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arXiv2019

Double charmonium productions in electron-positron annihilation using Bethe-Salpeter approach

We calculate the double charmonium production cross section within the framework of $4\times 4$ Bethe-Salpeter Equation in the electron-positron annihilation, at center of mass energy $\sqrt{s}= 10.6$GeV, that proceeds through the exchange of a single virtual photon. In this calculation, we make use of the full Dirac structure of 4D BS wave functions of these charmonia, with the incorporation of all the Dirac covariants (both leading and sub-leading). The calculated cross-sections for the double charmonium productions for final states, ($J/Ψ, η_c$); ($Ψ',η_c$) ; ($J/Ψ, η_c'$); and ($Ψ',η_c'$) are close to experimental data, and in broad agreement with results of other theoretical models.

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arXiv2019

Analytic approach to calculations of mass spectra and decay constants of heavy-light quarkonia in the framework of Bethe-Salpeter equation

This work is an extension of the work in \cite{bhatnagar18} to ground and excited states of $0^{++}, 0^{-+}$, and $1^{--}$ of heavy-light ($c\overline{u}, c\overline{s}, b\overline{u}, b\overline{s}$, and $b\overline{c}$) quarkonia in the framework of a QCD motivated Bethe-Salpeter equation (BSE) by making use of the exact treatment of the spin structure $(γ_μ\bigotimesγ_μ)$ in the interaction kernel, in contrast to the approximate treatment of the same in our previous works \cite{hluf16, bhatnagar18}), which is a substantial improvement over our previous works \cite{hluf16,bhatnagar18}. In this $4\times 4$ BSE framework, the coupled Salpeter equations for $Q\overline{q}$ (that are more involved than the equal mass ($Q\overline{Q}$) mesons) are first shown to decouple for the confining part of interaction, under heavy-quark approximation, and analyically solved, and later the one-gluon-exchange interaction is perturbatively incorporated, leading to their mass spectral equations. The analytic forms of wave functions obtained from these equations are then used for calculation of leptonic decay constants of ground and excited states of $0^{-+}$, and $1^{--}$ as a test of these wave functions and the over all framework.

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arXiv2017

Analytical calculations of ground and excited states of unequal mass heavy pseudoscalar and vector mesons mass spectra using Bethe-Salpeter formalism

Considering the fact that, a wide range of variation in masses of various states of heavy pseudoscalar and vector mesons can be seen in different models and the experimental data on masses of many of these states is not yet currently available, in this paper, we study the analytical calculation of heavy pseudoscalar and vector mesons mass spectra ( such as $η_{c}$, $B_{c}$, $η_{b}$, $J/ψ$, $B^{*}_{c}$ and $Υ$) and their corresponding wave functions in the framework of Bethe-Salpeter equation under the covariant instantaneous ansatz. The parameters of the framework were determined by a fit to the heavy pseudoscalar and vector mesons mass spectrum of ground states. These input parameters so fixed were found to give good agreements with data on mass spectra of ground and excited states of these mesons mass spectra.

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arXiv2017

An approach to calculation of mass spectra and two-photon decays of $c\overline{c}$ mesons in the framework of Bethe-Salpeter Equation

In this work we calculate the mass spectrum of charmonium for $1P,...,3P$ states of $0^{++}$ , $1^{++}$, as well as for $1S,...,4S$ states of $0^{-+}$, and $1S, ...,5D$ states of $1^{--}$ along with the two-photon decay widths of ground and first excited state of $0^{++}$ quarkonia for the process, $O^{++}\rightarrow γγ$ in the framework of a QCD motivated Bethe-Salpeter Equation. In this $4\times 4$ BSE framework, the coupled Salpeter equations are first shown to decouple for the confining part of interaction, under heavy-quark approximation, and analyically solved, and later the one-gluon-exchange interaction is perturbatively incorporated leading to mass spectral equations for various quarkonia. The analytic forms of wave functions obtained are used for calculation of two-photon decay widths of $χ_{c0}$. Our results are in reasonable agreement with data (where ever available) and other models.

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arXiv2017

Radiative decay widths of ground and excited states of vector charmonium and bottomonium

In this work we study the radiative decay widths of vector quarkonia for the process of $J/ψ(nS)\rightarrowη_{c}(nS)γ$ and $Υ(nS)\rightarrowη_{b}(nS)γ$ (for principal quantum numbers $n=1, 2, 3$) in the framework of Bethe-Salpeter equation under the covariant instantaneous ansatz using a $4\times 4$ form of BSE. The parameters of the framework were determined by a fit to the mass spectrum of ground states of pseudoscalar and vector quarkonia, such as; $η_{c}$, $η_{b}$, $J/ψ$ and $Υ$. These input parameters so fixed were found to give good agreements with data on mass spectra of ground and excited states of pseudoscalar and vector quarkonia, leptonic decay constants of pseudoscalar and vector quarkonia, two photon decays and two gluon decays of pseudoscalar quarkonia in our recent paper. With these input parameters so fixed, the radiative decay widths of ground (1S) and excited (2S, 3S) states of heavy vector quarkonia ($J/Ψ$ and $Υ$) are calculated and found to be in reasonable agreement with data.

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arXiv2016

Spectroscopy of ground and excited states of pseudoscalar and vector charmonium and bottomonium

In this work we calculate the mass spectrum, weak decay constants, two photon decay widths, and two gluon decay widths of ground (1S), and radially excited (2S,3S,...) states of pseudoscalar charmoniuum and bottomonium such as $η_{c}$ and $η_{b}$, as well as the mass spectrum and leptonic decay constants of ground state (1S), excited (2S,1D,3S,2D,4S,and 3D) states of vector charmonium and bottomonium such as $J/ψ$, and $Υ$, using the formulation of Bethe-Salpeter equation under covariant Instantaneous Ansatz (CIA). Our results are in good agreement with data (where ever available) and other models. In this framework, from the beginning, we employ a $4\times 4$ representation for two-body ($q\overline{q}$) BS amplitude for calculating both the mass spectra as well as the transition amplitudes. However, the price we have to pay is to solve a coupled set of equations for both pseudoscalar and vector quarkonia, which we have explicitly shown get decoupled in the heavy-quark approximation, leading to mass spectral equation with analytical solutions for both masses, as well as eigenfunctions for all the above states, in an approximate harmonic oscillator basis. Further, in the present framework of BSE, the hadron-quark vertex function (and the full 4D BS wave function) for pseudoscalar and vector quarkonia used for calculation of various processes, accommodates all the Dirac structures from their complete set in a natural manner.

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arXiv2015

Decay rates of charmonia within a quark-antiquark confining potentials

In this work, we investigate the spectroscopy and decay rates of charmonia within the framework of non-relativistic Schrödinger equation by employing an approximate inter quark-antiquark potential. The spin hyperfine, spin-orbit and tensor components of the one gluon exchange interaction are employed to compute the spectroscopy of the excited S states and few low-lying P and D waves. The resultant wave functions at zero inter quark separation as well as some finite separation are employed to predict the di-gamma, di-leptonic and di-gluon decay rates of charmonia states by using the conventional Van Royen-Weisskopf formula. The di-gamma and di-leptonic decay widths are also computed by incorporating the relativistic corrections of order $v^4$ within the NRQCD formalism. We have observed that the NRQCD predictions with their matrix elements computed at finite radial separation yielded results which are found to be in better agreement with experimental value for both di-gamma and di-leptonic decays. The same scenario is seen in the case when di-gamma and di-leptonic decay widths are computed with Van Royen-Weisskopf formula. It is also observed that the di-gluon decay width with the inclusion of binding energy effects are in better agreement with the experimental data available for 1S-2S and 1P. The di-gluon decay width of 3S and 2P waves waves are also predicted. Thus, the present study of decay rates clearly indicates the importance of binding energy effects.

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arXiv2013

Relevance of various Dirac covariants in hadronic Bethe-Salpeter wave functions in electromagnetic decays of ground state vector mesons

In this work we have employed Bethe-Salpeter equation (BSE) under covariant instantaneous ansatz (CIA) to study electromagnetic decays of ground state equal mass vector mesons: $ρ$, $ω$, $ϕ$, $ψ$ and $Y$ through the process $V\rightarrowγ*\rightarrow e^+ + e^-$. We employ the generalized structure of hadron-quark vertex function $Γ$ which incorporates various Dirac structures from their complete set order-by-order in powers of inverse of meson mass. The electromagnetic decay constants for the above mesons are calculated using the leading order (LO) and the next-to-leading order (NLO) Dirac structures. The relevance of various Dirac structures in this calculation is studied.

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arXiv2012

Cross section for double charmonium production in electron-positron annihilation at energy $\sqrt{s}$= 10.6 GeV

In this work we study the process $e^{+}+e^{-}\longrightarrow J/Ψ+η_{c} $ at energy $\sqrt{s}= 10.6 GeV$ observed recently at B-factories whose measurements were made by Babar and Belle groups. We calculate the cross section for this process in the Bethe-salpeter formalism under Covariant Instantaneous Anstaz (CIA). To simplify our calculation, the heavy quark approximation $(q<<M, P M)$ is employed in the quark and gluon propagators. In the exclusive process of $e^{+}e^{-}$ annihilation into two heavy quarkonia, the cross section calculated in this scenario is compatible with the experimental data of Babar and Belle.

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