ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 6, с. 821-829
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
DOUBLE P-WAVE CHARMONIUM PRODUCTION IN e+e- ANNIHILATION
©2014 A. P. Martynenko1),2)*, A. M. Trunin2)
Received April 3,2013
On the basis of perturbative QCD and relativistic quark model we calculate relativistic and bound state corrections in the production processes of a pair of P-wave charmonium states. Relativistic factors in the production amplitude are taken into account connected with the relative motion of heavy quarks and the transformation law of the bound-state wave function to the reference frame of the moving P-wave mesons. Relativistic corrections to the quark bound-state wave functions in the rest frame are considered by means of the Breit-like potential. It turns out that the examined effects change essentially nonrelativistic results of the cross section for the reaction e+ + —> hc + XcJ at the center-of-mass energy л/s = 10.6 GeV.
DOI: 10.7868/S0044002714060105
1. INTRODUCTION
The large value of exclusive double charmonium production cross section measured at the Belle and BABAR experiments [1,2] reveals definite problems in theoretical description of these processes [3—5]. Many theoretical efforts were made in order to improve the calculation of the production cross section e+ + e- ^ J+ . They included the analysis of other production mechanisms for the state J/^ + + [6, 7] and the calculation of different corrections which could change essentially initial nonrelativistic result [8—15]. Despite evident successes achieved on the basis of nonrelativistic quantum chromodynamics (NRQCD), the light cone method, quark potential models for correcting the discrepancy between the theory and experiment, the double charmonium production in e+e- annihilation remains an interesting task. On the one hand, there are other production processes of the P- and D-wave charmonium states which can be investigated in the same way as a production of S-wave states. Recently the Belle and BABAR Collaborations discovered new charmonium-like states in e+e- annihilation [16, 17]. The nature of these numerous resonances remains unclear to the present. Some of them are considered as P- and D-wave excitations in the system (cc). On the other hand, the variety of the used approaches and model parameters in this problem raises the question about a comparison of obtained results that will lead to a better understanding of the quark—gluon
!)Samara State University, Russia.
2)Samara State Aerospace University, Russia.
E-mail: mart@ssu.samara.ru
dynamics and different mechanisms of the charmonium production. Two sources of the changing of nonrelativistic cross section for a double charmonium production are revealed to the present: the radiative corrections of order O(as) and relative motion of c quarks forming the bound states. Actual physical processes of the charmonium production require formation of hadronic particles in final states (bound states of cc), for which perturbative quantum chromo-dynamics cannot provide high-precision description. Further investigation of charmonium production can improve our understanding of heavy quark production and the formation of quark bound states.
This work continues our study of exclusive double charmonium production in e+e- annihilation in the case of a pure P-wave (cc) quarkonium on the basis of a relativistic quark model (RQM) [12,18-21]. Note that the term RQM specifies the approach in which the systematic account of corrections connected with relative motion of heavy quarks can be performed. Relativistic quark model provides a solution in many tasks of heavy quark physics. It uses a number of perturbative and nonperturbative parameters entering in the quark interaction operator. All observables can be expressed in terms of these parameters. In this way we can check predictions of any quark model and draw a conclusion about its successfulness. At the same time the existence of a large number of different quark models which are sometimes very complicated for the practical use put a question about the elaboration of unified model containing generally accepted structural elements. Another approach to the heavy quark physics which does not contain the ambiguities of the quark models was formulated in [22]. As any other model of strong interactions
The production amplitude of a pair of P-wave charmonium states in e+e- annihilation. Phc denotes the P-wave meson hc and Txaj denotes the P-wave meson Xcj. The wavy line shows the virtual photon and the dashed line corresponds to the gluon. r is the production vertex function.
of quarks and gluons the approach of NRQCD introduces in the theory a large number of matrix elements parameterizing nonperturbative dynamics of quarks. To a certain extent the microscopic picture of the quark—gluon interaction resident in quark models is changed by the global picture operating with numerous nonperturbative matrix elements. An improved determination of color-singlet NRQCD matrix elements for S-wave charmonium is presented in [23]. Their study evidently shows that the account of relative order v2 corrections significantly increases the values of the matrix elements of leading order in v. The correspondence between parameters of quark models and NRQCD which can be established opens the way for better understanding of quark—gluon interactions at small distances. In this sense both approaches complement each other and could reveal new aspects of color dynamics of quarks and gluons. Thus, the aim of this study consists in the extension of relativistic approach to the quarkonium production from [12, 18, 19] on the processes e+ + e- — hc + + %cJ and a calculation of relativistic corrections of order v2 and determination of the interrelationship with predictions of NRQCD.
2. GENERAL FORMALISM
We investigate the quarkonium production in the lowest-order perturbative quantum chromodynam-ics. The usual color-singlet mechanism is considered as a basic one for the pair charmonium production. The diagrams that give contributions to the amplitude of these processes in leading order of the QCD coupling constant as are presented in the figure. Two other diagrams can be obtained by corresponding permutations. There are two stages of the production process. In the first stage, which is described by perturbative QCD, the virtual photon j* produces four heavy c quarks and c antiquarks with the following four-momenta:
PV2 = \P±P, (P-P) = 0; qi-2 = ^Q±q, (q-Q) = 0,
(1)
where P(Q) are the total four-momenta, p = = LP(0, p), q = LP(0, q) are the relative four-momenta obtained from the rest-frame four-momenta (0, p) and (0, q) by the Lorentz transformation to the system moving with the momenta P, Q. The momenta pi;2 of the heavy quark c and antiquark c are not on the mass shell: pi 2 = P2/4 — — p2 = M2/4 — p2 = m2. Relation (1) describes the symmetrical escape of the c quark and cc antiquark from the mass shell. In the second nonperturbative stage, quark—antiquark pairs form the final mesons.
Let us consider the production amplitude of the P-wave vector state hc and P-wave states xcJ (J = = 0,1,2), which can be presented in the form [12, 19,21]:
M(p-,p+,P,Q) = x v(p+)Y0u(p-) :
8ir2cws(4m2)Qc 3s
f dp idq
(2)
(2n)3 J (2n)3
x Sp<! * Pc (p, p )rf (p, q, P, Q)* Pcj (q, Q)Yv +
V
XcJ
+ *Pjj (q, Q)rf(p, q, P, Q)*£(p, P)Yv
where a superscript P indicates the P-wave meson, as(4m2) is the QCD coupling constant, a is the fine structure constant and Qc is the c-quark electric charge, r1 , 2 are the vertex functions defined below. The production processes e+ + e- — hc + xcJ contain the quark bound states. The transition of free quarks to the (cc) mesons is described by specific wave functions. The relativistic P-wave functions of the bound quarks accounting for the transformation from the rest frame to the moving one with four momenta P, Q are
*Pc (p,P) =
* hc (p)
V1 — 1
+ Vi
m 2m
P2
p
2m(e(p) + m) 2m
c
e
e
X
x
x
x
x
X 75(1 + Vi)
Vl + 1 2
+ v p + JL
2m(e(p) + m) 2m
m 2m
(4)
V2 - 1
+ V2-
q
+
q
2 ' " 2m(e(q)+m) 2m x e*v(Q, Sz)(1 + Û2) x
t>2 + 1 2
+ V2
q
Q
2m(e(q) + m) 2m
where the hat is a notation for the contraction of the four-vector with the Dirac matrices, v1 = P/Mhc, v2 = Q/MXcJ; ep(Q, Sz) is the polarization vector of the spin-triplet state XcJ, e(p) = \/p2 + m2 and m is the c-quark mass. The relativistic functions (3), (4) and vertex functions r^ do not contain the 5(p2 — — M2/4 + m2). More complicated factor including the bound-state wave function in the rest frame presented in (3) and (4) plays the role of the 5 function. This means that instead of the substitutions Mhc = = 2e(p) and MXjJ = 2e(q) in the production amplitude we carry out the integration over the quark relative momenta p and q. The amplitude (2) is projected onto a color singlet state by replacing vi(0)uk(0) with a projection operator of the form vi(0)uk(0) = = Sik/V3. The relativistic wave functions in (3), (4) are equal to the product of the wave functions in the rest frame and the spin projection operators that are accurate at all orders in |p|/m [12, 21]. The expression of the spin projector in a slightly different form has been derived primarily in [24]. In this work spin projectors are written in terms of heavy quark momenta p1>2 lying on the mass shell. Our derivation of relations (3), (4) accounts for the transformation law of the bound-state wave functions from the rest frame to the moving one with four momenta P and
Q [12].
We omit here the intermediate expressions giving rise to our final relations (2)—(4) [12, 18]. The presence of the 5(p ■ P) function allows to make the integration over relative energy p° if we write the initial production amplitude as a convolution of the truncated amplitude with two Bethe—Salpeter (BS) meson wave functions. The BS wave function satisfies a two-body bound-state equation which is very complicated and has no known solution. A way
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