Pis'ma v ZhETF, vol.88, iss. 11, pp.743-747
© 2008 December 10
Production and polarization of T mesons in the ^-factorization approach in more detail
S. P. Baranov1), N. P. Zotov* p.N. Lebedev Institute of Physics RAS, 119991 Moscow, Russia * D. V. Skobeltzyn Institute of Nuclear Physics, 119991 Moscow, Russia Submitted 20 October 2008
In the framework of the fct-factorization approach, the production and polarization of T mesons at the Fermilab Tevatron is considered, and a comparision of the calculated double differential distributions and spin alignment parameter a with the DO experimental data is shown. We argue that measuring the double differential cross section and the polarization of upsilonium states can serve as a crucial test discriminating two competing theoretical approaches to the parton dynamics in QCD.
PACS: 12.38.Bx, 13.85.Ni, 14.40.Gx
1. Introduction. Nowadays, the production of heavy quarkonium states at high energies is under intense theoretical and experimental study [1, 2]. The production mechanism involves the physics of both short and long distances, and so, appeals to both perturba-tive and nonperturbative methods of QCD. This feature gives rise to two competing theoretical approaches known in the literature as the color-singlet [3, 4] and color-octet [5] models. According to the color-singlet approach, the formation of a colorless final state takes place already at the level of the hard partonic subprocess (which includes the emission of hard gluons when necessary). In the color-octet model, also known as nonrela-tivistic QCD (NRQCD), the formation of a meson starts from a color-octet QQ pair and proceeds via the emission of soft nonperturbative gluons. The former model has a well defined applicability range and has already demonstrated its predictive power in describing the J/\j) production at HERA, both in the collinear [6] and the fct-factorization [7] approaches. As it was shown in the analysis of recent ZEUS [8] data, there is no need in the color-octet contribution, neither in the collinear nor in the fct-factorization approach. The numerical estimates of the color octet contributions extracted from the analysis of Tevatron data are at odds with the HERA data, especially as far as the inelasticity parameter z = ¡E1 is concerned [9]. In the fct-factorization approach, the values of the color-octet contributions obtained as fits of the Tevatron data appear to be substantially smaller than the ones in the collinear scheme, or even can be neglected at all [10-13].
Recently, the results of new theoretical caclulations of the next-to-leading (NLO) and next-to-next-to-leading
e-mail: baranov8sci.lebedev.ru; zotov0theory.sinp.msu.ru
(NNLO) order corrections to colour singlet (CS) quarkonium production have been obtained in the framework of standard pQCD [14]. In the region of moderate px (Pt > lOGeV), these corrections enhance the color singlet production rate by one order of magnitude and even larger. These new results are in much better agreement with the fct-factorization predictions than it was seen for leading order collinear calculations.
In the present note we follow the guideline of our previous publication [15] and show a more detailed analysis of the production and polarization of T mesons at the Tevatron conditions using the fct-factorization approach.
2. Theoretical framework. In the fct-factorization approach, the cross section of a physical process is calculated as a convolution of the off-shell partonic cross section <j and unintegrated parton distribustions /r), which depend on both the longitudinal momentum fraction x and transverse momentum kx~.
Ppp = J g(x 1, fcfr, /¿2) Tg(x2, k^Ti^2) x
X CTgg{^X\ , X2 , fcjx i T i " *) d>Xi d>X2 dk-^'j* dk2'j' • (1)
In accordance with the fct-factorization prescriptions [16-19], the off-shell gluon spin density matrix is taken in the form
iPr = |fcT|2 = eTkvT/\kT\2. (2)
In all other respects, our calculations follow the standard Feynman rules.
In order to estimate the degree of theoretical uncertainty connected with the choice of unintegrated gluon density, we use two different parametrizations, which are known to show the largest difference with each other, namely, the ones proposed in Refs. [16, 19] and [20].
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S. P. Baranov, N. P. Zotov
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Fig.l. Differential cross section and spin alignement parameter a as functions of the T(IS') transverse momentum pT, integrated over four different rapidity intervals. The panels from top to bottom: \y\ < 0.6; 0.6 < \y\ < 1.2; 1.2 < \y\ < 1.8; 1.8 < \y\. Dashed histograms, dGRV gluon density; dash-dotted histograms, JB gluon density. Thin lines, the direct contribution only; thick lines, with the feed-down from states added. Experimental points: • DO [26]; o CDF [27]; * DO (preliminary) [28]
In the first case [16], the unintegrated gluon density is derived from the ordinary (collinear) density G(x, /t2) by differentiating it with respect to /t2 and setting /t2 = k^. Here we use the leading order Gltick-Reya-Vogt (LO GRV) set [21] as the input colinear density. In the following, this will be referred to as dGRV parametrisation. The other unintegrated gluon density [20] is obtained as a solution of leading order Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [19] in the double-logarithm approximation. Technically, it is calculated as a convolution of the ordinary gluon density with some universal weight factor. In the following, this will be referred to as JB parametrisation.
The production of T(1S) mesons in pp collisions can proceed via either direct gluon-gluon fusion or the pro-
duction of P-wave states Xb followed by their radiative decays xj^T+7. The direct mechanism corresponds to the partonic subprocess g+g "£+g which includes the emission of an additional hard gluon in the final state. The production of P-wave mesons is given by g+g Xb, and there is no emission of any additional gluons. As we have already argued in our previous publication [15], we see no need in taking the color-octet contributions into consideration.
The polarization state of a vector meson is characterized by the spin alignment parameter a which is defined as a function of any kinematic variable as
a{T) = {da/dV - 3daL/ctP)/{da/dV + daL/ctP), (3)
Production and polarization of X mesons
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Fig.2. Differential cross section and spin alignement parameter a as functions of the T(IS') rapidity y, integrated over three different intervals of pr- The panels from top to bottom: pr < 3 GeV; 3 < pr < 8 GeV; 8 < pr GeV. Notation of the curves is as in Fig.l. Recoil system is assumed everywhere
where a is the reaction cross section and ax, is the part of cross section corresponding to mesons with longitudinal polarization (zero helicity state). The limiting values a = 1 and a = — 1 refer to the totally transverse and totally longitudinal polarizations. We will be interested in the behavior of a as a function of the T transverse momentum: V = |pt|- The experimental definition of a is based on measuring the angular distributions of the decay leptons
dr(T^ß+ß-)/d cos в
a cos
(4)
where 9 is the polar angle of the final state muon measured in the decaying meson rest frame.
The definition of helicity and, consequently, the definition of a is frame-dependent. There are four commonly used different definitions of the helicity frame: these are the recoil, the target, the Collins-Soper, and the Gottfried-Jackson systems. In our analysis, we will basically use the recoil system (which, at the Tevatron conditions, is the same as the laboratory or proton-proton center-of-mass system), unless a different choice is explicitly declared.
When considering the polarization properties of X(1S) mesons originating from radiative decays of P-wave states, we rely upon the dominance of electric di-
pole El transitions2^. The corresponding invariant amplitudes can be written as [22]
iA(Xi T7) ex ^k^e^e™, (5)
iA(X2 T7) (X P^2)4T) M7) -*/j47)] ' (6)
with p and k being the momenta of the decaying meson and the emitted photon; and the
respective polarization vectors; and the antisym-
metric Levita-Civita tensor. This leads to the following relations between the production cross sections for different helicity states (see Eq. (14) in [22]):
0T(ft=o) = -B(xi^T7)
7Xl(W = l)
+ B(x 2^T7)
о"Т(|л.|=1) = B(Xi^l) "1
ТХ2(Л=0)
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2'ln our previous paper [15], two somewhat different models
w
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