Production of $D$-wave states of $\bar b c$ quarkonium at the LHC

The hadronic production of $D$-wave states of $\bar b c$ is studied. The relative yield of such states is estimated for kinematic conditions of LHC experiments.


II. CALCULATION TECHNIQUE
To estimate the production amplitude of D-wave B c states we use the analogous technique as for S and P waves, namely, we perform calculations within the color singlet model neglecting the internal velocities of quarks inside quarkonium (see for details [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]): where T is the amplitude of four heavy quark gluonic production with momenta p i in the leading-order approximation, which is contributed by 36 Feynman diagrams; q is the quark three-momentum in the B c meson rest frame, and Ψ(q) is the B c meson wave function.
For D-wave states the first two terms in (1) are equal to zero, and therefore an amplitude is proportional to the second derivative of the wave function at origin R (0) and to the second derivatives of T over q. The amplitudes for the spin singlet A jz (J = 2, j z = l z ) and for the spin triplet A Jjz (J = 1, 2, 3; j z = s z + l z ) can be expressed as follows (see also [7], where the D-wave B c production was studied in the fragmentation approach): where Π J, αβρ (j z ) = lz,sz αβ (l z ) ρ (s z ) · C Jjz szlz , ρ and αβ are vector and polarization tensors and C Jjz szlz are Clebsch-Gordan coefficients. The states with a definite spin value are constructed by operators and The spinors in (5) and (6) are expressed as follows: where pb = m b m b +mc P Bc , p c = mc m b +mc P Bc and k(q) is a Lorentz boost of four-vector (0, q) to the system where the B c momentum is equal to P Bc .
Amplitudes and their derivatives have been calculated numerically. To simplify the calculations we square and summarize amplitudes, keeping only a spin value S = 0 or S = 1.
The amplitude squared for the spin-singlet state with S = 0 (1 1 D 2 ) is given by the following equation: The sum of amplitudes squared for the spin-triplet states with S = 1 ( A more rigorous consideration of this process within NRQCD [25] implies that the final meson is no longer abc pair rather a superposition of Fock states: where 1 and 8 refer to color singlet and color octet states of the quark pair. 1 The NRQCD ( 3 D j ) relevant for the fist terms in the expansions (11) and (12) are related to the quarkonium wave function Therefore production of the first components in the expansions (11) and (12) can be described by formulas obtained from (1).
By analogy with the fragmentation case [7], it can be shown that within NRQCD the contributions to the gluonic production of the second and third terms in (11) and (12) are of the same order on α s and v, as the contribution of the first terms, and therefore they should be also included. Having an experience in calculation of gluonic production of S-and P -wavebc color singlets, it is not difficult to calculate the hard parts of appropriate production amplitudes for S-and P -wavebc color octets. Unfortunately the soft part of such amplitudes can not be accurately estimated, because values of the relevant NRQCD operators are unknown.
However understanding the kinematic behavior of such contributions even without knowing the exact normalization could help in the experimental search of D-wave B c states. We estimate here three additional NRQCD contributions enumerated in Eqs. (11) and (12) using the fairly defined hard parts of the processes normalized by coefficients which are extracted using a naive velocity scaling, as explained below.
To model the contributions of P -wave color octet states |bc( 1 P 1 , 8)g and |bc( 3 P j , 8)g to the discussed cross sections we use our tools for calculation of P -wave B c color singlet states.
We replace the color singlet wave function to color octet one and multiply the wave function 1 As noted in [7], in the above Fock state expansion there are also other O(v 2 ) states, but their production will be further suppressed by powers of v. 2 There are two widely used normalizations for O 1 matrix elements. One normalization method (BBL) inherits from the study [25]. Another one (PCGMM) is based on the work [26]. These two normalization methods relate to each other as follows: Since we consider our study to be a continuation of work [7] that uses the BBL normalization, we also use it in Eqs. (13) and (14). The exact determination of the discussed operators one can find for example in [27] and [28]. derivative squared at origin |R P (0)| 2 by the coefficient eff is an effective squared velocity of quarks in the B c meson: To model the contributions of S-wave color octet states |bc( 1 S 0 , 8)gg and |bc( 3 S 1 , 8)gg we use our tools for calculation of S-wave B c color singlet states, where we replace the color singlet wave function to color octet one and multiply the coordinate wave function squared at origin Constructing the contributions of S-wave color singlet states |bc( 1 S 0 , 1)gg and |bc( 3 S 1 , 1)gg , as in the previous case (16) we just rescale the wave function squared at origin The very similar approach was applied to estimate the color octet contribution to the B c P -wave production in the work [29]. Also in [29] the properties of color matrix for the gluonic bc color octet production were studied in details.
It is worth to remind that for gluonic bbcc production the replacement of the color singlet wave function ofbc-pair to the color octet wave function cannot be reduced to a simple scaling of the matrix element, because it essentially changes the relative contributions of different Feynman diagrams to the total amplitude. 3 The v 2 eff value we estimate as: 3 For the first time the color matrix for the process gg → bbcc was investigated in [30], where it was shown that such color matrix has 13 nonzero eigenvalues. Three of them correspond to the cases, where thebc-pair is in a color singlet state: (bc) 1 ⊗ (bc) 1 , (bc) 1 ⊗ (bc) 8−symmetric and (bc) 1 ⊗ (bc) 8−antisymmetric . The remaining ten eigenvalues correspond to the cases, where thebc-pair is in a color octet. We refer to the studies [20,29] for details.
where E is the averaged kinematic energy of quark inside the B c -meson and µ = mcm b mc+m b . Using the value E ≈ 0.35 GeV estimated in [31], we obtain that v 2 eff ≈ 0.15. To estimate the additional NRQCD contributions numerically we choose the following central values for K coefficients in Eqs. (15) to (17): In order to estimate the uncertainties of the additional contributions to hadronic production we vary the K P 8 value from 0.1 to 0.2, and the K S8,1 value from 0.015 to 0.03.
It should be emphasized that the normalization values proposed in Eqs. (15) to (19) cannot be regarded as reliable, and may drastically differ from values, which be will measured experimentally or predicted within a more rigorous approach. Nevertheless, we think that it is useful to demonstrate in this study, how the color octet contribution could influence the kinematic behavior of the D-wave B c meson yield at LHC experiments.
The calculation results have been tested for Lorentz invariance and gauge invariance. As it was already noted, the calculations were conducted within the method very close to ones applied to the study of S-and P -wave states. As in our previous researches the integration over phase space was carried out within the RAMBO algorithm [32].
While results of the latest calculations have been verified many times by other research groups [11-13, 16-19, 21, 22, 24] we have tried to minimize the possibility of error in our new work.

III. ESTIMATIONS OF RELATIVE YIELD
For the numerical estimations of cross sections we involve the wave functions and masses listed in Table I (see also Table IV where the predictions for masses of D-wave excitations are presented). The parameters of radial wave functions for 1S and 1D states are taken from works [5], [33,34], and [35,36]. Following most of the previous research on this topic, we choose the mass values of quarks in such a way as the mass of finalbc quarkonium is correct. We are motivated by the fact, that relative yield of 2S excitations was described quite well within this choice.
Using the wave function parameters from [5] and choosing the quark masses as in Table I we predict, that the relative yield of B c (1D) with respect to the direct B c (1S) in the gluonic fusion is about 0.5 ÷ 1.3 %, as seen from Table II, where the cross section values for the gluonic production are presented at different gluonic energies. As shown in Figure 1 the distributions over transverse momentum for D-wave states are quite similar to ones for S-wave states. It is worth mentioning, that the predicted ratio of states with spin S = 1 ( states with spin S = 0 (1 1 D 2 ) is in approximate accordance with a simple spin counting rule (see ratios in Table III): This feature allows us to use the prediction of quasipotential model [33,34] where wave functions are essentially different for 1 3 D 1 , 1 3 D 2 , 1 3 D 3 and 1 1 D 2 states, as well as for 1 3 S 1 and 1 1 S 0 states, even if contributions of different D states are not estimated separately. For this case we can approximately estimate the cross section ratio averaging the wave function values according to spin counting rules (see Table I): Usage of (21) and (22) calculated within the approach [33,34] leads to a little bit more optimistic values: the discussed relative yield is approximately 1.6 times higher. Another model, based on the quasipotential approach [35,36], predicts essentially lower values for wave functions at origin (see Table I and  estimating the relative yield. Using the wave functions from [35,36] increases the final value by 1.5 times comparing to [5].  (15,16,17) are able to crucially change the production properties. While the p T distribution shapes are more or less the same, the energy dependencies for the color octet contributions and for the color singlet contributions essentially differ: the color octet contributions decrease faster with energy, than the color singlet ones. Moreover, seems, that the shape of energy dependence is mostly determined by the color state ofbc-pair and practically does not depend on its orbital momentum.
Concerning the numerical values of the additional NRQCD contributions one can conclude that each of such contributions is of order of the direct color singlet production, as expected.
This circumstance testifies to the self-consistency of our calculations. Since there are three such additional contributions, they crucially increase the total cross section.
To obtain the proton-proton cross sections the gluonic cross sections are convoluted with CT14 PDFs [37]: To decrease uncertainties due to the scale choice and QCD corrections we present a relative yield of 1D states with respect to 1S states. The calculations are performed for forward and central kinematic regions. The forward one is restricted by cuts 2 < η < 4.5, p T < 10 GeV and nearly corresponds to LHCb conditions, while the central one is restricted by cuts 2 < η < 4.5, p T < 10 GeV, |η| < 2.5, 10 GeV < p T < 50 GeV and approximately corresponds to CMS or ATLAS conditions. The proton-proton energy of collision is chosen equal to 13 TeV.
When following the collinear approximation, one should always keep in mind the problem of transverse momentum of the initial gluons. Indeed, in some cases accounting the initial gluon transverse momenta crucially changes the production features (see, for example [38] or [39]).
However, we believe that in our case the dependence on the initial gluonic transverse momenta is more or less eliminated in the ratio σ(B c (D))/σ(B c (S).
The systematic uncertainty of the calculations is estimated with variation of the scale in the range E T /2 < µ < 2E T . It is well seen in Figures 3 and 4 that the relative yields are hardly dependent on scale variation. As it is seen in Table III within the applied model (the color singlet production in the gluon fusion subprocess) the relative yield value depends on kinematics: for the central region it is approximately twice as large. The use of the wave functions set [33,34] or [35,36] increases the predicted yield of D-wave states to 1 ÷ 1.8 %.
We emphasize ones again, that the naive normalization used in this research cannot be regarded as reliable, and may be drastically far from values, which will be measured experimentally or predicted within a more rigorous approach.

IV. CONCLUSIONS
The B c (2S) excitations have been observed at LHC in the B c π + π − spectrum [1][2][3][4], and this result stimulated us to estimate possibilities to search for B c (D) excitations in the same spectrum. At very large statistics it would be possible to distinguish two peaks in the B c π + π −  We have to conclude that an observation of the discussed states at LHC is a quite challenging experimental task due to the small relative yield. Table V: B c meson wave functions within the quasipotential models [33,34] and [35,36].