Prompt $\eta _c$ meson production at the LHC in the NRQCD with $k_T$-factorization

Abstract

In the framework of the $k_T$-factorization approach, the prompt production of $\eta _c$ mesons at the LHC conditions is studied. Our consideration is based on the off-shell amplitudes for hard partonic subprocesses and on the nonrelativistic QCD (NRQCD) formalism for the formation of bound states. We try two latest parametrizations for noncollinear, or transverse momentum dependent (TMD) gluon densities derived from the Catani–Ciafaloni–Fiorani–Marchesini (CCFM) equation. We use the values of the nonperturbative matrix elements obtained from a combined fit of the $\eta _c$ and $J/\psi$ differential cross sections. Finally, we show an universal set of parameters that provides a reasonable simultaneous description for all of the available data on the prompt $J/\psi$ and $\eta _c$ production at the LHC.

1 Introduction

Since long ago, the production of quarkonium states in high energy hadronic collisions remains an area of intense attention from both theoretical and experimental sides. These processes are sensitive to the interaction dynamics both at small and large distances: the production of heavy quarks with high transverse momentum is followed by the formation of bound states with low relative quark momentum. Accordingly, the description of these processes involves both perturbative and non-perturbative methods. Our present work continues the line started in the previous publications [1,2,3]. We have already considered there the prompt production of $\psi '$, $\chi _c$, and $J/\psi$ mesons and now come to $\eta _c$ mesons.

The production of $\eta _c$ mesons turned out to be rather puzzling for conventional NRQCD calculations at the leading and next-to-leading orders (NLO) [4, 5]; see also the discussions in [8, 8, 8]. This time, the theory was very unlucky to have too few degrees of freedom. The essential parameters, the so called long distance matrix elements (LDMEs) or nonperturbative matrix elements (NMEs), have been already fixed from the data on the production and polarization of $J/\psi$ mesons. The correspondence between the $\eta _c$ and $J/\psi$ production parameters follows from the heavy quark spin symmetry rules and is explained in the next section. So, the theory lost its flexibility and made a prediction for $\eta _c$ by a huge factor off the measured cross section. The overall situation was even called ‘challenging’ [4]. Large efforts invested in extending the calculations to higher orders made no sensible effect. The aim of the present note is to show that the approach consistently used in [1,2,3] meets no troubles with the $\eta _c$ data.

Our solution implies a certain modification of the conventional NRQCD rules. We find the conventional rules rather unsatisfactory. Indeed, in all practical calculations with NRQCD, the final state gluons changing the color and other quantum numbers of the $Q\bar{Q}$ pair and bringing it to the color-singlet state are regarded as carrying no energy–momentum. This is in obvious contradiction with confinement which prohibits the emission of infinitely soft colored quanta. In reality, the heavy quark system must undergo a kind of final state interaction where the energy–momentum exchange must be larger than at least the typical confinement scale (say, $\Lambda _{QCD}$). This is not the matter of only kinematic corrections; without some finite energy–momentum transfer we cannot organise transition amplitudes with correct spin properties.

In the present work, we understand the long-distance transition amplitudes as consecutive color-electric dipole transitions in the spirit of multipole radiation theory. This has dramatic consequences for the polarization of the final state mesons. As a result of the accepted rules (see Sect. 2), the final state $J/\psi$ mesons come out nearly unpolarized, either because of the cancellation between the $^3P_1^{[8]}$ and $^3P_2^{[8]}$ contributions, or as a result of two successive color-electric (E1) dipole transitions in the chain $^3S_1^{[8]}\rightarrow {^3P}_{J}^{[8]}\rightarrow J/\psi$ with $J = 0, 1, 2$. This contrasts with the conventional NRQCD calculations which show that $J/\psi$ mesons produced from high-$p_T$ gluons as $^3P_1^{[8]}$ states carry strong transverse polarization [9]. With our completely different view on the $J/\psi$ depolarization mechanism, we finally arrive at a completely different set of the fitted NMEs.

As usual, to preserve the consistency with our previous studies we work in the $k_T$-factorization approach [10,11,12,13]. The kinematics of our processes is characterized by the double inequality $s \gg \mu ^2 \simeq \hat{s} \gg \Lambda ^2$ which shows that the typical parton interaction scale $\mu$ is much higher than the QCD parameter $\Lambda$, but is much lower than the total c.m.s. energy $\sqrt{s}$. In such a case, the perturbative QCD expansions in $\alpha _s$ may contain large coefficients ${{\mathcal {O}}}[\ln (s/\mu ^2)]={{\mathcal {O}}}[\ln (1/x)]$ which compensate the smallness of the coupling constant $\alpha _s(\mu ^2/\Lambda ^2)$. Resummation of the terms $[\alpha _s\ln (\mu ^2/\Lambda ^2)]^n$, $[\alpha _s\ln (1/x)]^n$, and $[\alpha _s \ln (\mu ^2/\Lambda ^2)\,\ln (1/x)]^n$ up to infinitely large n results in parton distributions ${{\mathcal {F}}}_i(x,{\mathbf k}_T^2,\mu ^2)$ that generalize the factorization of the hadronic matrix elements beyond the collinear approximation.

The parton evolution is described in our work with CCFM (Catani–Ciafaloni–Fiorani–Marchesini) equation [14,15,16,17]. The latter converges to BFKL (Balitsky–Fadin–Kuraev–Lipatov) equation [18,19,20] in the region of small x and to DGLAP equation at large x and provides a continuous interpolation between these two regimes. The typical values of x probed in the considered processes are of the order $x\sim m_T/\sqrt{s}$  with $m_T=(m^2+p_T^2)^{1/2}$. The arguments justifying the use of CCFM equation at large x (resp., large $p_T$) are therefore the same as for the collinear factorization approach based on DGLAP equation. At the same time, the soft gluon resummation implemented in our parton evolution algorithm [21] regularises infrared divergences and makes our approach usable even at small $p_T$. For the different aspects of using the $k_T$-factorization approach  the reader may consult the reviews [22,23,24,25].

2 Theoretical framework

As it was done previously for $\psi '$, $\chi _c$ and $J/\psi$ production [1,2,3], the present calculations are based on perturbative QCD and nonrelativistic bound state formalism (NRQCD). The production of $\eta _c$ mesons is dominated by the color singlet (CS) contribution that refers to the partonic subprocess

$$\begin{aligned} g^*(k_1)+g^*(k_2)\rightarrow \eta _c(p) \end{aligned}$$

(1)

with the respective cross section

$$\begin{aligned}&\sigma (pp\rightarrow \eta _c+X)\nonumber \\&\quad =\int \frac{2\pi }{x_1 x_2 s\,F}\, {{\mathcal {F}}}_g(x_1,{\mathbf{k}}_{1T}^2,\mu ^2)\,{{\mathcal {F}}}_g(x_2,{\mathbf{k}}_{2T}^2,\mu ^2)\nonumber \\&\qquad \times \; \bigl |\mathcal{M}(g^*g^*\rightarrow \eta _c)\bigr |^2\, d{\mathbf{k}}_{1T}^2\,d{\mathbf{k}}_{2T}^2\,dy_\eta \,\frac{d\phi _1}{2\pi } \,\frac{d\phi _2}{2\pi }, \end{aligned}$$

(2)

where ${k}_1$ and ${k}_2$ denote the initial gluon 4-momenta, $\phi _1$ and $\phi _2$ are the respective azimuthal angles, $y_\eta$ is the rapidity of $\eta _c$ meson, $x_1$ and $x_2$ are the gluon longitudinal momentum fractions, $\mathcal{M}(g^*g^*\rightarrow \eta _c)$ is the hard scattering amplitude, and ${{\mathcal {F}}}_g(x_i,{\mathbf{k}}_{iT}^2,\mu ^2)$ is the transverse momentum dependent (TMD, or unintegrated) gluon density in a proton at the scale $\mu ^2$. In accordance with the general definition [26], the off-shell gluon flux factor in (2) is taken as $F=2\lambda ^{1/2}(\hat{s},k_1^2,k_2^2)$, where $\hat{s}=(k_1+k_2)^2$.

In addition to the above, we have considered a number of color octet (CO) contributions and a contribution from the feed-down $h_c\rightarrow \eta _c\,X$ process. In all of the considered cases the initial gluons are taken off-shell. That means, they have nonzero transverse momentum and an admixture of longitudinal component in the polarization vector. Explicitly, the gluon spin density matrix is taken in the form [10, 11]: $\overline{\epsilon _g^\mu \epsilon _g^{*\nu }}=k_T^\mu k_T\nu /|k_T|^2$, where $k_T$ is the component of the gluon momentum perpendicular to the beam axis. In the collinear limit, when $k_T\rightarrow 0$, this expression converges to the ordinary $\overline{\epsilon _g^\mu \epsilon _g^{*\nu }}=-g^{\mu \nu }/2$. In all other respects, we follow standard QCD Feynman rules.

The CO terms refer to the perturbative production of a color-octet $c\bar{c}^{[8]}$ pair followed by nonperturbative gluon radiation bringing the intermediate $c\bar{c}^{[8]}$ state to a real (colorless) meson:

$$\begin{aligned} g^*(k_1)+g^*(k_2)\rightarrow c\bar{c}^{[8]}\rightarrow \eta _c(p) + \text{ soft }~\text{ gluons }. \end{aligned}$$

(3)

The intermediate color octet $c\bar{c}^{[8]}$ state can be either of $^1S_0$, $^3S_1$, $^3P_0$, $^3P_1$, $^3P_2$, or $^1P_1$, where we use standard spectroscopic notation. As usual, the hard production amplitudes contain spin and color projection operators that guarantee the proper quantum numbers of the state under consideration. The probabilities of the subsequent nonperturbative soft transitions are not calculable within the theory and are usually accepted as free model parameters. There are, however, certain restrictions coming from some general principles. Whenever calculable or not, the nonperturbative amplitudes must be identical for transitions in both directions (i.e., from vectors to scalars and vice versa), as it is motivated by the heavy quark spin symmetry (HQSS). The amplitudes can only differ by an overall normalizing factor representing the averaging over spin degrees of freedom. Thus, we strictly have from this property [27, 28]:

$$\begin{aligned} \left\langle {\mathcal {O}}^{\eta _c}\bigl [^1S_0^{[1]}\bigr ]\right\rangle= & {} \frac{1}{3} \left\langle {\mathcal {O}}^{J/\psi }\bigl [^3S_1^{[1]}\bigr ]\right\rangle \nonumber \\ \left\langle {\mathcal {O}}^{\eta _c}\bigl [^1S_0^{[8]}\bigr ]\right\rangle= & {} \frac{1}{3} \left\langle {\mathcal {O}}^{J/\psi }\bigl [^3S_1^{[8]}\bigr ]\right\rangle \nonumber \\ \left\langle {\mathcal {O}}^{\eta _c}\bigl [^3S_1^{[8]}\bigr ]\right\rangle= & {} \left\langle {\mathcal {O}}^{J/\psi }\bigl [^1S_0^{[8]}\bigr ]\right\rangle \nonumber \\ \left\langle {\mathcal {O}}^{\eta _c}\bigl [^1P_1^{[8]}\bigr ]\right\rangle= & {} 3 \left\langle {\mathcal {O}}^{J/\psi }\bigl [^3P_0^{[8]}\bigr ]\right\rangle \nonumber \\ \left\langle {\mathcal {O}}^{h_c} \bigl [^1P_1^{[1]}\bigr ]\right\rangle= & {} 3 \left\langle {\mathcal {O}}^{\chi _{c0}}\bigl [^3P_0^{[1]}\bigr ]\right\rangle \nonumber \\ \left\langle {\mathcal {O}}^{h_c} \bigl [^1S_0^{[8]}\bigr ]\right\rangle= & {} 3 \left\langle {\mathcal {O}}^{\chi _{c0}}\bigl [^3S_1^{[8]}\bigr ]\right\rangle . \end{aligned}$$

(4)

The above relations require a simultaneous fit for the $\eta _c$ and $J/\psi$ production data. This fit turned out to be impossible in the traditional NRQCD scheme. The calculated cross sections were either found to be at odds with the measurements [4] or at odds with theoretical principles [5].

The crucial point in the above papers is the presence of a large unwanted contribution to the $\eta _c$ production cross section from the intermediate $^3S_1^{[8]}$ state (unwanted, as the $\eta _c$ production cross section is saturated by the color singlet channel alone; a fact, already pointed out in [29]). The corresponding nonperturbative matrix element is an HQSS counterpart of the $^1S_0^{[8]}$ matrix element engaged in the production of $J/\psi$ mesons, where it is needed to make the outgoing $J/\psi$ meson unpolarised: this spinless state is employed to dilute strong $J/\psi$ polarization in other channels. Note by the way that the size of $\langle {\mathcal {O}}^{J/\psi }[^1S_0^{[8]}]\rangle$ matrix element used in [4] is in conflict with the NRQCD quark relative velocity counting rules.

Table 1 Sets of NME’s for $J/\psi$ production as determined from the different fits

Table 2 Sets of NME’s for $h_c$ production as determined from the different fits

Table 3 Sets of NME’s for $\eta _c$ production as determined from the different fits

In the present work, we exploit rather different understanding [30] of the $J/\psi$ depolarization mechanism. This mechanism is not connected to the choice of factorization scheme ($k_T$ or collinear), but represents a completely independent issue. We follow the interpretation of the nonperturbative matrix elements (NMEs) as consecutive color-electric dipole transitions in the spirit of multipole radiation theory.

Only a single E1 transition is needed to transform a P-wave state into an S-wave state and the structure of the respective ${^3P_J}\rightarrow {^3S_1}+g$ amplitudes is taken the same as for radiative decays of $\chi _{cJ}$ mesons [31, 32]:

$$\begin{aligned} {\mathcal {A}}(\chi _{c0}(p)\rightarrow & {} J/\psi (p{-}k)+ \gamma (k) \nonumber \\&\propto \,k_{\mu }\,p^{\mu }\,\varepsilon _{\nu }^{(J/\psi )}\, \varepsilon ^{\nu }_{(\gamma )}, \end{aligned}$$

(5)

$$\begin{aligned} {\mathcal {A}}(\chi _{c1}(p)\rightarrow & {} J/\psi (p{-}k)+ \gamma (k) \nonumber \\&\propto \epsilon ^{\mu \nu \alpha \beta }\,k_{\mu }\, \varepsilon _{\nu }^{(\chi _{c1})}\, \varepsilon _{\alpha }^{(J/\psi )}\, \varepsilon _{\beta }^{(\gamma )}, \end{aligned}$$

(6)

$$\begin{aligned} {\mathcal {A}}(\chi _{c2}(p)\rightarrow & {} J/\psi (p{-}k)+ \gamma (k) \nonumber \\&\propto p^{\mu }\,\varepsilon _{(\chi _{c2})}^{\alpha \beta }\, \varepsilon _{\alpha }^{(J/\psi )} \,[k_{\mu }\,\varepsilon _{\beta }^{(\gamma )} -k_{\beta }\,\varepsilon _{\mu }^{(\gamma )}]. \end{aligned}$$

(7)

The only difference is in the overall normalizing factor, as the electric charge is replaced with the color charge. The transformation of an S-wave state into another S-wave state (such as $J/\psi$ meson) requires two successive E1 transitions $\;{^3S_1}^{[8]}\rightarrow ~{^3P_J}^{[8]}+g$, $\;~{^3P_J}^{[8]}\rightarrow J/\psi +g$ proceeding via either of the three intermediate states: ${^3P_0}^{[8]}$, ${^3P_1}^{[8]}$, or ${^3P_2}^{[8]}$. Here we exploit the same effective coupling vertices as above (5)–(7) and only change the sign of the radiated momentum k for the ’backward’ case, $^3S_1\rightarrow ~^3P_J$.

The accepted rules (5)–(7) lead to the fact that the final state $J/\psi$ mesons come out unpolarized [30], either because of the cancellation between the $^3P_1^{[8]}$ and $^3P_2^{[8]}$ contributions, or as a result of two successive color-electric (E1) dipole transitions in the chain $^3S_1^{[8]}\rightarrow {^3P}_{J}^{[8]}\rightarrow J/\psi$ with $J = 0, 1, 2$. Note that this property remains true irrespective of the numerical values of NMEs and only follows from the spin algebra. Thus, we no longer need the diluting $^1S_0^{[8]}$ contribution to $J/\psi$  and can significantly reduce or even kill it. This means that we also get rid of the $^3S_1^{[8]}$ contribution to $\eta _c$ production.

As we have mentioned already, our parton distribution functions are derived from CCFM equation which converges to BFKL equation at small x and to DGLAP equation at large x. We benefit from having a continuous interpolation between the two different regimes and from the ease of including radiation corrections in the form of TMD distributions. The soft gluon resummation implemented in our parton evolution algorithm regularises infrared divergences and makes our approach usable even in the low $p_T$ region.

In the numerical analysis shown below, we tried two latest sets of TMD gluon densities in a proton, referred to as JH’2013 set 1 and JH’2013 set 2 [33]. These gluon densities were obtained from CCFM evolution equation [14,15,16,17] where the input parametrization (used as boundary conditions) was fitted to the proton structure function $F_2(x,Q^2)$. The resummation of large logarithmic terms proportional to $\ln 1/x$ (important at high energies $\sqrt{s}$, or, equivalently, at small $x \sim \mu /\sqrt{s}$) and $\ln 1/(1 - x)$ is included in our approach as part of the CCFM evolution of gluon densities (see [14,15,16,17] for more information).

We take from [34] the charmonia masses $m(\eta _c) =2.9839\ \hbox {GeV}$, $m(h_c) = 3.52538\ \hbox {GeV}$, $m(J/\psi ) = 3.0969\ \hbox {GeV}$ and the branching fractions $B(J/\psi \rightarrow \mu ^+ \mu ^-) =0.05961$ and $B(h_c \rightarrow \eta _c \gamma ) = 0.51$. The renormalization and factorization scales are set to $\mu _R^2=m^2+{\mathbf{p}}_T^2$ and $\mu _F^2=\hat{s}+{\mathbf{Q}}_T^2$, where m and ${\mathbf{p}}_T$ are the mass and transverse momentum of the produced charmonium, and ${\mathbf{Q}}_T$ is the transverse momentum of the initial off-shell gluon pair. The choice of $\mu _R$ is rather standard for charmonia production, while the unusual choice of $\mu _F$ is connected with the CCFM evolution (see [33] for details). The analytic expressions for the hard scattering amplitudes in (1) and (3) were obtained using the algebraic manipulation system form [35]. The multidimensional phase space integration has been performed by means of the Monte-Carlo technique using the routine vegas [36].

3 Numerical results

All the $J/\psi$ and $\eta _c$ production mechanisms have different shapes of transverse momentum distributions, that enabled us to extract the corresponding NMEs from the experimental data. Here, to determine the NMEs of $J/\psi$ mesons (as well as their $\eta _c$ counterparts) we performed a combined fit of $J/\psi$ and $\eta _c$ transverse momentum distributions using the latest CMS [37, 38], ATLAS [39] and LHCb data [40] collected at 7, 8 and 13 TeV. Here, the factorization principle seems to be on solid theoretical grounds because of not too low $p_T$ values for both $J/\psi$ and $\eta _c$ mesons. We do not impose any kinematic restrictions but the experimental acceptance. The fitting procedure was separately done (using fitting algorithm as implemented in gnuplot package [41]) in each of the rapidity subdivisions under the requirement that the NMEs be strictly positive, and then the mean-square average of the fitted values was taken. The corresponding uncertainties are estimated in a standard way using Student’s t-distribution at confidence level $P = 95$%. Note that we used the results of a global fit for the entire charmonium family (including, in particular, $\chi _{cJ}$ and $\psi '$ states) [44] to properly calculate the feed-down contributions from $h_c$, $\chi _{cJ}$, and $\psi '$ decays.

Fig. 1

Transverse momentum distribution of prompt $J/\psi$ mesons produced in pp collisions at $\sqrt{s} = 7\ \hbox {TeV}$ (upper plots) and $\sqrt{s} = 13\ \hbox {TeV}$ (lower plots). The shaded bands on the left panels represent the total uncertainties of our calculations (i.e. scale uncertainties and the uncertainties coming from NMEs fit, summed in quadrature), as estimated for JH’2013 set 2 gluon density. The relative contributions from the different production mechanisms are shown on the right panels. The experimental data are from CMS [37, 38]

Fig. 2

Transverse momentum distribution of prompt $\eta _c$ mesons produced in pp collisions at $\sqrt{s} = 7\ \hbox {TeV}$ (upper plots) and $\sqrt{s} =8\ \hbox {TeV}$ (lower plots). Shaded bands on the left panels represent the total uncertainties of our calculations (i.e. scale uncertainties and the uncertainties coming from NMEs fit, summed in quadrature), as estimated for JH’2013 set 2 gluon density. The relative contributions from the different production mechanisms are shown on the right panels. The experimental data are from LHCb [40]

For some (yet unrecognized) reasons, our $^1P_1^{[1]}$ production amplitude (needed to calculate the feed-down $h_c\rightarrow \eta _c\,X$) disagrees with the one found in the literature. Our calculation is off-shell, but has continuous on-shell limit that can be promptly compared with [45, 46]. The contribution is anyway small and unimportant numerically; but the discrepancy is still of interest from the academic point of view. For the lack of details presented in [45, 46], we cannot repeat their calculation. The details of our calculation are explained in the Appendix.

The numerical values of our NMEs for $J/\psi$ and $h_c$ mesons are written out in Tables 1 and 2. For comparison, we also present here several sets of NMEs [42, 43, 47, 48], obtained in the NLO NRQCD by other authors. The NMEs shown for $h_c$ mesons are translated from $\chi _c$ NMEs using HQSS formulas. The fits differ from one another by somehow differently selected data sets. The corresponding values of NMEs for $\eta _c$ meson are collected in Table 3. They can be easily obtained from Table 1 using the HQSS relations (4).

A comparison of our predictions with the experimental results is displayed in Figs. 1 and 2. The theoretical uncertainty bands include both scale uncertainties and the uncertainties coming from the NMEs fitting procedure. First of them were obtained by varying the $\mu _R$ scale around its default value by a factor of 2. This was accompanied with using the JH’2013 set $2+$ and JH’2013 set $2-$ in place of the JH’2013 set 2, in accordance with [33]. One can see that we have achieved a reasonably good agreement between our calculations and LHCb measurements (with both of the considered TMD gluons), simultaneously for the prompt $\eta _c$ and $J/\psi$ production data collected at different energies and in the whole $p_T$ range. The presented results can give a significant impact on the understanding of charmonia production within NRQCD.

4 Conclusions

We have considered the production of charmonium states at the LHC and found a consistent simultaneous description for the $J/\psi$ and $\eta _c$ data. Our fitted nonperturbative matrix elements are universal and strictly obey the heavy quark spin symmetry rules.

The fundamental difference with the ’traditional’ NRQCD scheme (which was unable to accommodate the whole data set) is in a different treatment of the nonperturbative color-octet transitions. The latter are described in our approach in terms of multipole radiation theory. Then the $J/\psi$ mesons are produced nearly unpolarized in all channels, thus making no special need in the diluting $^1S_0^{[8]}$ contribution to $J/\psi$ production and, accordingly, requiring no $^3S_1^{[8]}$ contribution to $\eta _c$ production. In the forthcoming paper [44] we are going to present a global fit for the entire charmonium family, including $J/\psi$, $\chi _{cJ}$, $\psi (2S)$ and $\eta _c$ mesons.

Appendix: Off-shell production amplitude for \(^1P_1^{[1]}\) state

In this section, we consider the gluon–gluon fusion subprocess

$$\begin{aligned} g(k_1,\epsilon _1,a) + g(k_2,\epsilon _2,b)\rightarrow g(k_3,\epsilon _3,c) + c\bar{c} (p,\epsilon ^\alpha ), \end{aligned}$$

(8)

where the symbols in the parentheses indicate the momentum, the polarization, and the color of the interacting quanta. The calculation of this subprocess at ${{\mathcal {O}}}(\alpha _s^3)$ relates to six Feynman diagrams:

$$\begin{aligned} {{\mathcal {M}}}_1= & {} \text{ tr }\left\{ \not \!\epsilon _1(\not \!p_c-\not \!k_1+m_c)\not \!\epsilon _2(-\not \!p_{\bar{c}}-\not \!k_3+m_c) \not \!\epsilon _3\;{{\mathcal {P}}}_S\right\} \nonumber \\&\times \left[ k_1^2-2(p_{c}k_1)\right] ^{-1}\left[ k_3^2 +2(p_{\bar{c}}k_3)\right] ^{-1}, \end{aligned}$$

(9)

$$\begin{aligned} {{\mathcal {M}}}_2= & {} \text{ tr }\left\{ \not \!\epsilon _1(\not \!p_c-\not \!k_1+m_c)\not \!\epsilon _3(-\not \!p_{\bar{c}}+\not \!k_2+m_c) \not \!\epsilon _2\;{{\mathcal {P}}}_S\right\} \nonumber \\&\times \left[ k_1^2-2(p_{c}k_1)\right] ^{-1}\left[ k_2^2 -2(p_{\bar{c}}k_2)\right] ^{-1}, \end{aligned}$$

(10)

$$\begin{aligned} {{\mathcal {M}}}_3= & {} \text{ tr }\left\{ \not \!\epsilon _3(\not \!p_c+\not \!k_3+m_c)\not \!\epsilon _1(-\not \!p_{\bar{c}}+\not \!k_2+m_c) \not \!\epsilon _2\;{{\mathcal {P}}}_S\right\} \nonumber \\&\times \left[ k_3^2+2(p_{c}k_3)\right] ^{-1}\left[ k_2^2 -2(p_{\bar{c}}k_2)\right] ^{-1}, \end{aligned}$$

(11)

$$\begin{aligned} {{\mathcal {M}}}_4= & {} \text{ tr }\left\{ \not \!\epsilon _2(\not \!p_c-\not \!k_2+m_c)\not \!\epsilon _1(-\not \!p_{\bar{c}}-\not \!k_3+m_c) \not \!\epsilon _3\;{{\mathcal {P}}}_S\right\} \nonumber \\&\times \left[ k_2^2-2(p_{c}k_2)\right] ^{-1} \left[ k_3^2+2(p_{\bar{c}}k_3)\right] ^{-1}, \end{aligned}$$

(12)

$$\begin{aligned} {{\mathcal {M}}}_5= & {} \text{ tr }\left\{ \not \!\epsilon _2(\not \!p_c-\not \!k_2+m_c)\not \!\epsilon _3(-\not \!p_{\bar{c}}+\not \!k_1+m_c) \not \!\epsilon _1\;{{\mathcal {P}}}_S\right\} \nonumber \\&\times \left[ k_2^2-2(p_{c}k_2)\right] ^{-1}\left[ k_1^2 -2(p_{\bar{c}}k_1)\right] ^{-1}, \end{aligned}$$

(13)

$$\begin{aligned} {{\mathcal {M}}}_6= & {} \text{ tr }\left\{ \not \!\epsilon _3(\not \!p_c+\not \!k_3+m_c)\not \!\epsilon _2(-\not \!p_{\bar{c}}+\not \!k_1+m_c) \not \!\epsilon _1\;{{\mathcal {P}}}_S\right\} \nonumber \\&\times \left[ k_3^2+2(p_{c}k_3)\right] ^{-1} \left[ k_1^2-2(p_{\bar{c}}k_1)\right] ^{-1}, \end{aligned}$$

(14)

$$\begin{aligned} {{\mathcal {M}}}= & {} {{\mathcal {M}}}_1+{{\mathcal {M}}}_2+{{\mathcal {M}}}_3+{{\mathcal {M}}}_4+{{\mathcal {M}}}_5+{{\mathcal {M}}}_6, \end{aligned}$$

(15)

with the property ${{\mathcal {M}}}_1={{\mathcal {M}}}_6$, ${{\mathcal {M}}}_2={{\mathcal {M}}}_5$, ${{\mathcal {M}}}_3={{\mathcal {M}}}_4$. The color factor is universal and is equal to $d^{abc}/4\sqrt{3}$. This set of diagrams is complete; no other diagrams can contribute at the order ${{\mathcal {O}}}(\alpha _s^3)$ to the production of a meson with the given quantum numbers $J^{PC}=1^{+-}$.

The amplitudes ${{\mathcal {M}}}_i$ contain spin projection operators which discriminate the spin-singlet and spin-triplet $c\bar{c}$ states:

$$\begin{aligned} {{\mathcal {P}}}_{S{=}0}= & {} (\not \!p_{\bar{c}}-m_c)\gamma _5(\not \!p_c+m_c)\cdot (2m_c)^{-3/2}, \end{aligned}$$

(16)

$$\begin{aligned} {{\mathcal {P}}}_{S{=}1}= & {} (\not \!p_{\bar{c}}-m_c)\not \!\epsilon _{\psi }(\not \!p_c+m_c)\cdot (2m_c)^{-3/2}, \end{aligned}$$

(17)

where $m_c$ is the charmed quark mass. These projectors are orthogonal to each other, as they should be: $tr\{{{\mathcal {P}}}_0 \overline{{{\mathcal {P}}}_1}\}=0$. For the $^1P_1^{[1]}$ state we evidently have to use the projector ${{\mathcal {P}}}_0$.

The orbital angular momentum L is associated with the relative momentum q of the quarks in a bound state. The relative momentum q is defined as

$$\begin{aligned} p_c=\frac{1}{2}p+q,\quad p_{\bar{c}}=\frac{1}{2}p-q. \end{aligned}$$

(18)

According to a general formalism developed in [49, 50], the terms showing no dependence on q are identified with the contributions to the $L=0$ state; the terms linear in $q^\alpha$ are related to the $L=1$ state with the proper polarization vector $\epsilon ^\alpha$ (see below); the quadratic terms $q^\alpha q^\beta$ refer to the $L=2$ state with the polarization tensor $\epsilon ^{\alpha \beta }$; and so on. The decomposition of ${{\mathcal {M}}}$ in powers of q is carried out by expanding the subprocess amplitude as

$$\begin{aligned} {{\mathcal {M}}}(q)={{\mathcal {M}}}|_{q{=}0} + q^\alpha (\partial {{\mathcal {M}}}/\partial q^\alpha )|_{q{=}0} + \cdots , \end{aligned}$$

(19)

where q is assumed to be a small quantity. The amplitude ${{\mathcal {M}}}(q)$ has to be multiplied by the bound state wave function $\Psi (q)$ and integrated over q. A term-by-term integration of Eq.(19) is performed using the relations

$$\begin{aligned}&\int \frac{d^3q}{(2\pi )^3}\Psi (q)=\frac{1}{\sqrt{4\pi }}{{\mathcal {R}}}(x{=}0), \end{aligned}$$

(20)

$$\begin{aligned}&\int \frac{d^3q}{(2\pi )^3}q^\alpha \Psi (q)= -i\epsilon ^\alpha \frac{\sqrt{3}}{\sqrt{4\pi }}{{\mathcal {R}}}'(x{=}0), \end{aligned}$$

(21)

etc., where ${{\mathcal {R}}}(x)$ is the radial wave function in the coordinate representation (the Fourier transform of $\Psi (q)$). This formula completes our derivation of the production matrix element. The resulting expression has been explicitly tested for gauge invariance by substituting the gluon momentum $k_i$ for the polarization vector $\epsilon _i$. We have observed gauge invariance even with off-shell initial gluons.