Bridging light-cone and NRQCD approaches: asymptotic behavior of $B_c$ electromagnetic form factor
Yu Jia, Jian-Xiong Wang, Deshan Yang
备注：
v3, 39 pages, 5 figures. A minor error in equation (37) corrected, Fig. 4 redrawn. A new paragraph added to discuss the numerical significance of double logarithm term in O(alpha_s) short-distane coefficient for gamma*->J/psi+eta_c process, and one reference added
This work aims at illustrating that, for a class of leading-twist hard exclusive reactions involving two heavy quarkonia, the light-cone approach, when equipped with the strategy of refactorization of the light-cone distribution amplitude of quarkonium, can be employed to elegantly reproduce the corresponding predictions made in the nonrelativistic QCD (NRQCD) factorization approach, order by order in perturbative expansion. Taking the electromagnetic form factor of the $B_c$ meson at large momentum transfer, $Q^2$, as a concrete example, we compare the results obtained from both NRQCD-based and light-cone-based calculations through the next-to-leading order (NLO) in $α_s$, while at the leading order (LO) in both velocity and $1/Q^2$ expansion, and explicitly confirm their mutual agreement. As a byproduct, we apply our NLO result to explore certain features about the asymptotic behavior of the heavy-light meson form factor. We also address the major theoretical obstacles that prevent us from establishing an analogous equivalence between these two approaches for the double charmonium production process of phenomenological interest, $e^+e^-\to J/ψ+η_c$.
We show that the twist-2 light-cone wave-functions of eta_c and J/ψcan be factorized with nonrelativistic QCD(NRQCD) at one-loop level, where the nonperturbative effects are represented by NRQCD matrix elements. The factorization is achieved by expanding the small velocity v, which the c- or \bar c- quark moves inside a rest quarkonium with. At leading order of v the twist-2 light-cone wave-functions of eta_c and J/ψcan be factorized as the product of a perturbative function and a NRQCD matrix element. The perturbative function is calculated at one-loop level and free from any soft divergence.…
The presented paper is devoted to theoretical analysis of charmonium mesons $η_c$, $J/ψ$, $χ_{c0,1,2}$, and $h_c$ production in exclusive radiative $Z$ boson decays in the frameworks of Nonrelativistic Chromodynamic and amplitude expansion on the light-cone. In the framework of light-cone expansion amplitudes and widths of all processes are described with one universal relation. It is interesting to note that from this relation it follows automatically which component of $Z$ boson (either vector or axial) gives contribution to the width of the particular decay. It is shown that the effect of i…
The rare process of $W$ radiative decay into the $B_c$ meson can serve as an interesting play ground for testing two influential perturbative QCD approaches: non-relativistic quantum chromodynamics (NRQCD) and ligh-cone (LC) factorization. Since the $B_c$ meson is a genuine heavy quarkonium which is composed of two different species of heavy quarks, it is natural to employ NRQCD factorization to tackle this exclusive quarkonium production process; on the other hand, since the mass of the $W$ boson is much greater than that of the $B_c$, the collinear factorization is also a viable approach. To…
The light-cone distribution amplitudes (LCDAs) serve as important non-perturbative inputs for the study of hard exclusive processes. In this paper, we calculate ten LCDAs at twist-2 for the S-wave and P-wave $B_c$ mesons up to the next-to-leading order (NLO) of the strong coupling $α_s$ and leading order of the velocity expansion. Each one of these ten LCDAs is expressed as a product of a perturbatively calculable distribution and a universal NRQCD matrix-element. By use of the spin symmetry, only two NRQCD matrix-elements will be involved. The reduction of the number of non-perturbative input…
Predictions for $e^+e^-\to J/ψη_c$ from previous studies are made by taking charmonia as a nonrelativistic bound state and by using nonrelativistic QCD(NRQCD) approach. The predicted cross-section is smaller by an order of magnitude than the experimentally observed. We study the process by taking charm quark as a light quark and use light-cone wave-functions to parameterize nonperturbative effects related to charmonia. The total cross section of $e^+e^-\to J/ψη_c$ can be predicted, if these wave-functions are known. Motivated by studies of light-cone wave-functions of light hadrons, we make a …
It has been suggested in Ref. [A. E. Bondar and V. L. Chernyak, Phys. Lett. B 612, 215 (2005)] that the disagreement between theoretical calculations and experimental observations for the rate for the process e+ e- -> J/psi + eta_c at the B factories might be resolved by using the light-cone method to take into account the relative momentum of the heavy-quark and antiquark in the quarkonia. The light-cone result for the production cross section in Ref. [A. E. Bondar and V. L. Chernyak, Phys. Lett. B 612, 215 (2005)] is almost an order of magnitude larger than existing NRQCD factorization resul…
In a typical exclusive quarkonium production process, when the center-of-mass energy, $\sqrt{s}$, is much greater than the heavy quark mass $m$, large kinematic logarithms of $s/m^2$ will unavoidably arise at each order of perturbative expansion in the short-distance coefficients of the nonrelativistic QCD (NRQCD) factorization formalism, which may potentially harm the perturbative expansion. This symptom reflects that the hard regime in NRQCD factorization is too coarse and should be further factorized. We suggest that this regime can be further separated into "hard" and "collinear" degrees o…
In this paper, we calculate twist-2 light-cone distribution amplitudes (LCDAs) of the S-wave and P-wave quarkonia (namely $^1S_0$ state $η_{Q}$, $^3S_1$ state $J/ψ(Υ)$, $^1P_1$ state $h_{Q}$ and $^3P_J$ states $χ_{QJ}$ with $J=0,1,2$ and $Q=c,b$) to the next-leading order of the strong coupling $α_s$ and leading order of the velocity expansion $v$ in the non-relativistic QCD (NRQCD). We apply these LCDAs to some single quarkonium exclusive productions at large center-of-mass energy, such as $γ^*\to η_Qγ, χ_{QJ}γ~(J=0,1,2)$, $Z\to η_Qγ, χ_{QJ}γ~(J=0,1,2), J/ψ(Υ)γ, h_Qγ$ and $h\to J/ψγ$, by adop…
In this paper we have analyzed the production of pair charmonium mesons in the reactions $e^+ e^- \to J/ Ψχ_{c0}, e^+ e^-\to Ψ(2S) χ_{c0}$ at energy $\sqrt s = 10.6$GeV in the framework of the light cone formalism. In comparison with NRQCD the numerical results for the cross sections are in better agreement with experiment.
The typical binding energy of heavy hadron spectroscopy makes the system accessible to perturbative calculations in terms of non-relativistic QCD. Within NRQCD the predictions of heavy quarkonium energy levels rely on the accurate description of the static QCD potential $V_{\rm QCD}(r)$. Historically, heavy quarkonium spectroscopy was studied using phenomenological approaches such as the Cornell model $V_{\rm Cornell}=-κ/r+σ\, r$, which assumes a short-distance dominant Coulomb potential plus a liner rising potential that emerges at long distances. Such model works reasonably well in describin…
We test the nonrelativistic QCD factorization conjecture for inclusive quarkonium production at two loops by carrying out a covariant calculation of the nonrelativistic quantum chromodynamics (NRQCD) long-distance matrix element (LDME) for a heavy-quark pair in an $S$-wave, color-octet state to fragment into a heavy-quark pair in a color-singlet state of arbitrary orbital angular momentum. The NRQCD factorization conjecture for the universality of the LDME requires that infrared divergences that it contains be independent of the direction of the Wilson line that appears in its definition. We f…
In this paper the production of excited vector and pseudoscalar charmonium mesons in $\epem$ annihilation is analyzed in the framework of light cone. In particular the cross sections $\epem \to ψ(2S)η_c(1S), ψ(1S)η_c(2S),ψ(2S)η_c(2S)$ have been calculated. It is shown, that contrary to NRQCD the cross sections calculated in the framework of light cone agree with experimental data.
Within the nonrelativistic quantum chromodynamics (NRQCD) factorization formalism, we compute the relativistic corrections to the exclusive decays of bottomonia with even charge conjugation parity into $S$-wave charmonium pairs at leading order in the strong coupling constant. Relativistic corrections are resummed for a class of color-singlet contributions to all orders in the charm-quark velocity $v_c$ in the charmonium rest frame. Almost every process that we consider in this work has negative relativistic corrections ranging from -20 to -35,%. Among the various processes, the relativistic c…
This paper is devoted to the study of the processes e^+e^- \to J/Ψη_c, J/Ψη_c', ψ' η_c, ψ' η_c' within light cone formalism. It is shown that if one disregards the contribution of higher fock states, the twist-3 distribution amplitudes needed in the calculation can be unambiguously determined from the twist-2 distribution amplitudes and equations of motion. Using models of the twist-2 distribution amplitudes the cross sections of the processes under study have been calculated. The results of the calculation are in agreement with Belle and BaBar experiments. It is also shown that relativistic a…
In this work, we study the radiative decay of $Z$ boson to a $P$-wave quarkonium $H$ in association with a photon, where $H$ can be $χ_{QJ}$, $h_Q$ with $Q=c,b$ and $J=0,1,2$. The helicity amplitudes and the unpolarized decay widths are evaluated up to QCD next-to-next-to-leading order (NNLO) within the framework of nonrelativistic QCD (NRQCD). For the first time, we check the NRQCD factorization for $h_Q$ exclusive production at two-loop order. The leading logarithms (LL) of $m_Z^2/m_Q^2$ in the leading-twist short-distance coefficients, which may potentially ruin the perturbative convergence…
The recently-proposed quasi distributions point out a promising direction for lattice QCD to investigate the light-cone correlators, such as parton distribution functions (PDF) and distribution amplitudes (DA), directly in the $x$-space. Owing to its excessive simplicity, the heavy quarkonium can serve as an ideal theoretical laboratory to ascertain certain features of quasi-DA. In the framework of non-relativistic QCD (NRQCD) factorization, we compute the order-$α_s$ correction to both light-cone distribution amplitudes (LCDA) and quasi-DA associated with the lowest-lying quarkonia, with the …
We calculate the NLO corrections for the gluon fragmentation functions to a heavy quark-antiquark pair in ${^{1}\hspace{-0.6mm}S_{0}^{[1]}}$ or ${^{1}\hspace{-0.6mm}S_{0}^{[8]}}$ state within NRQCD factorization. We use integration-by-parts reduction to reduce the original expression to simpler master integrals (MIs), and then set up differential equations for these MIs. After calculating the boundary conditions, MIs can be obtained by solving the differential equations numerically. Our results are expressed in terms of asymptotic expansions at singular points of $z$ (light-cone momentum fract…
Analogous to NRQCD factorization for heavy quarkonium exclusive production, in this work we propose to employ the heavy-quark-effective-theory (HQET) factorization, which has been predominantly applied to account for exclusive $B$ decays, to study the exclusive production of the heavy-flavored mesons. We take $W\to B(D_s)+γ$ as a prototype process. The validity of the HQET factorization rests upon the presumed scale hierarchy: $m_W\sim m_b\gg Λ_{\rm QCD}$. Through an explicit analysis at next-to-leading order in $α_s$ yet at leading order in $1/m_b$, we verify that the decay form factors can i…
