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Radiative Corrections to the Single Spin Asymmetry in Heavy Quark Photoproduction? N.Ya.Ivanov?

Yerevan Physics Institute, Alikhanian Br.2, 375036 Yerevan, Armenia We analyze in the framework of pQCD the properties of the single spin asymmetry in heavy ?avor production by linearly polarized photons. At leading order, the parallel- perpendicular asymmetry in azimuthal distributions of both charm and bottom quark is predicted to be about 20% in a wide region of initial energy. Using the soft gluon resummation formalism, we have calculated the nextto-leading order corrections to the asymmetry to next-to-leading logarithmic accuracy. It is shown that radiative corrections practically do not a?ect the Born predictions for the azimuthal asymmetry at energies of the ?xed target experiments. Both leading and next-to-leading order predictions for the asymmetry are insensitive to within few percent to theoretical uncertainties in the QCD input parameters: mQ , ?R , ?F , ΛQCD and in the gluon distribution function. Our analysis shows that nonperturbative corrections to a B -meson azimuthal asymmetry due to the gluon transverse motion in the target are negligible. We conclude that measurements of the single spin asymmetry would provide a good test of pQCD applicability to heavy ?avor production at ?xed target energies.

arXiv:hep-ph/0102328v1 27 Feb 2001

I. INTRODUCTION

Presently, the basic spin-averaged characteristics of heavy ?avor hadro-, photo- and electroproduction are known exactly up to the next-to-leading order (NLO) [1–9]. Two main results of the exact pQCD calculations can be formulated as follows. First, the NLO corrections are large; they increase the leading order (LO) predictions for both charm and bottom production cross sections approximately by a factor of 2. For this reason, one could expect that the higher order corrections as well as the nonperturbative contributions can be essential in these processes, especially for the c-quark case. Second, the ?xed order predictions are very sensitive to standard uncertainties in the input QCD parameters. In fact, the total uncertainties associated with the unknown values of the heavy quark mass, mQ , the factorization and renormalization scales, ?F and ?R , ΛQCD and the parton distribution functions are so large that one can only estimate the order of magnitude of the NLO predictions for total cross sections [7,8]. For this reason, it is very di?cult to compare directly, without additional assumptions, the ?xed order predictions for spin-averaged cross sections with experimental data and thereby to test the pQCD applicability to the heavy quark production. Since the spin-averaged characteristics of heavy ?avor production are not well de?ned quantitatively in pQCD it is of special interest to study those spin-dependent observables which are stable under variations of input parameters of the theory [10]. In this report we analyze the charm and bottom production by linearly polarized photons, namely the reactions γ ↑ +N → Q(Q) + X. (1.1)

We consider the single spin asymmetry parameter, A(s), which measures the parallel-perpendicular asymmetry in the quark azimuthal distribution: A(s) = 1 dσ (s, ? = 0) ? dσ (s, ? = π/2) . Pγ dσ (s, ? = 0) + dσ (s, ? = π/2) (1.2)

√ Here dσ (s, ?) ≡ dσ (s, ?) , Pγ is the degree of linear polarization of the incident photon beam, s is the centre of d? mass energy of the process (1.1) and ? is the angle between the beam polarization direction and the observed quark transverse momentum.

? ?

Contribution to XV International Seminar on High Energy Physics Problems, Dubna, Russia, Sept. 25-29, 2000 E-mail: nikiv@uniphi.yerphi.am

1

The properties of the single spin asymmetry at Born level as well as the contributions of nonperturbative e?ects (such as the gluon transverse motion in the target and the heavy quark fragmentation) have been considered in [10]. Using the methods of Refs. [11–14] for the threshold resummation of soft gluons, we have also calculated the nextto-leading order corrections to A(s) at next-to-leading logarithmic (NLL) level [15]. The main results of our analysis can be formulated as follows: ? At ?xed target energies, the LO predictions for azimuthal asymmetry (1.2) are not small and can be tested experimentally. For instance,

2 LO A(s = 400GeV2 ) |LO Charm ≈ A(s = 400GeV ) |Bottom ≈ 0.18.

(1.3)

? Radiative corrections practically do not a?ect the Born predictions for A(s) at ?xed target energies. ? At energies su?ciently above the production threshold, both leading and next-to-leading order predictions for A(s) are insensitive (to within few percent) to uncertainties in the QCD parameters: mQ , ?R , ?F , ΛQCD and in the gluon distribution function. This implies that theoretical uncertainties in the spin-dependent and spinaveraged cross sections (the numerator and denominator of the fraction (1.2), respectively) cancel each other with a good accuracy. ? Nonperturbative corrections to the b-quark azimuthal asymmetry A(s) due to the gluon transverse motion in the target are negligible. Because of the smallness of the c-quark mass, the kT -kick corrections to A(s) in the charm case are larger; they are of order of 20%. We conclude that the single spin asymmetry is an observable quantitatively well de?ned, rapidly convergent in pQCD and insensitive to nonperturbative contributions. Measurements of asymmetry parameters would provide a good test of the ?xed order QCD applicability to heavy ?avor production.

II. SINGLE SPIN ASYMMETRY AT LEADING ORDER

At leading order, O(αem αS ), the only partonic subprocess which is responsible for heavy quark photoproduction is the two-body photon-gluon fusion: γ (kγ ) + g (kg ) → Q(pQ ) + Q(pQ ). The cross section corresponding to the Born diagrams is [10]:

2 e2 ?2 2(1 ? β 2 )(β 2 ? x ?2 ) d2 σ ? Born ? Q αem αS (?R ) 1 + x + (1 + Pγ cos 2?) , (? s, t, ?, ?2 R) = C 2 2 2 d? xd? s ? 1?x ? (1 ? x ? )

(2.1)

(2.2)

where Pγ is the degree of the photon beam polarization; ? is the angle between the observed quark transverse 2 momentum, pQ,T , and the beam polarization direction. In (2.2) C is the color factor, C = TF =Tr(T aT a )/(Nc ? 1) = 1/2, and eQ is the quark charge in units of electromagnetic coupling constant. We use the following de?nition of partonic kinematical variables: s ? = (kγ + kg ) ; u ? = (kg ? pQ ) ; β= 1? 4 m2 ; s ?

2 2

? = (kγ ? pQ )2 ; t ? ? m2 t x ?=1+2 ; s ? s ? 2 2 β ?x ?2 ; pQ,T = 4

(2.3)

where m is the heavy quark mass. Unless otherwise stated, the CTEQ5 [16] parametrization of the gluon distribution function is used. The default values of the charm and bottom mass are mc = 1.5 GeV and mb = 4.75 GeV; Λ4 = 300 MeV and Λ5 = 200 MeV. The default values of the factorization scale ?F chosen for the A(s) asymmetry calculation are ?F |Charm = 2mc for 2

the case of charm production and ?F |Bottom = mb for the bottom case [9,17]. For the renormalization scale, ?R , we use ?R = ?F . Let us discuss the pQCD predictions for the asymmetry parameter de?ned by (1.2). Our calculations of A(s) at LO for the c- and b-quark are presented by solid lines in Fig.1. One can see that at energies su?ciently above the production threshold the single spin asymmetry A(Eγ ) depends weekly on Eγ , Eγ = (s ? m2 N )/2mN .

0.05 0.2 0.15 A(Eγ ) 0.1 0.05 Charm production Solid: QCD LO 2 Dashed: LO + kT kick, < k2 T >= 0.5 GeV 2 2 Dotted: LO + kT kick, < kT >= 1 GeV 0.05 0.1 0.2 Eγ , TeV 0.5 1 0.1 0.2 0.5 0.2 0.15 A(Eγ ) 0.1 0.05 Bottom production Solid: QCD LO 2 Dashed: LO + kT kick, < k2 T >= 0.5 GeV 2 2 Dotted: LO + kT kick, < kT >= 1 GeV 0.1 0.15 0.2 0.3 Eγ , TeV 0.5 0.7 1 0.1 0.15 0.2 0.3 0.5 0.7

FIG. 1. Single spin asymmetry, A(Eγ ), in c- and b- quark production as a function of beam energy Eγ = (s ? m2 N )/2mN ; the QCD LO predictions with and without the inclusion of kT -kick e?ect.

The most interesting feature of LO predictions for A(Eγ ) is that they are practically insensitive to uncertainties in QCD parameters. In particular, changes of the charm quark mass in the interval 1.2 < mc < 1.8 GeV a?ect the quantity A(Eγ ) by less than 6% at energies 40 < Eγ < 1000 GeV. Remember that analogous changes of mc lead to variations of total cross sections from a factor of 10 at Eγ = 40 GeV to a factor of 3 at Eγ = 1 TeV. The extreme choices mb = 4.5 and mb = 5 GeV lead to 3% variations of the parameter A(Eγ ) in the case of bottom production at energies 250 < Eγ < 1000 GeV. The total cross sections in this case vary from a factor of 3 at Eγ = 250 GeV to a factor of 1.5 at Eγ = 1 TeV. The changes of A(Eγ ) are less than 3% for choices of ?F in the range 1 2 mb < ?F < 2mb . For the total cross sections, such changes of ?F lead to a factor of 2.7 at Eγ = 250 GeV and of 1.7 at Eγ = 1 TeV. We have veri?ed also that all the CTEQ3-CTEQ5 versions of the gluon distribution function [16] as well as the CMKT parametrization [18] lead to asymmetry predictions which coincide with each other with accuracy better than 1.5%

III. NONPERTURBATIVE CONTRIBUTIONS

Let us discuss how the pQCD predictions for single spin asymmetry are a?ected by nonperturbative contributions due to the intrinsic transverse motion of the gluon in the target. In our analysis, we use the MNR model [17] parametrization of the gluon transverse momentum distribution, kg = z kN + kg,T , (3.1)

where kN is the target momentum in the γN centre of mass system. According to [17], the primordial transverse momentum, kg,T , has a random Gaussian distribution:

2 1 1 d2 N kT = exp ? 2 2 N d2 kT π kT kT

,

(3.2)

2 2 where kT ≡ kg,T . It is evident that the inclusion of this e?ect results in a dilution of azimuthal asymmetry. In [9,17], the parametrization (3.2) (so-called kT -kick) have been used to describe the single inclusive spectra and the 2 < 2 GeV2 . QQ correlations. It was found that in charm photoproduction 0.5 < kT

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Our calculations of the parameter A(s) at LO with the kT -kick contributions are presented in Fig.1 by dashed 2 2 ( kT = 0.5 GeV2 ) and dotted ( kT = 1 GeV2 ) curves. So, we can conclude that nonperturbative corrections to the b-quark asymmetry parameter (1.2) due to the kT -kick e?ect practically do not a?ect predictions of the underlying perturbative mechanism: photon-gluon fusion. Our calculations of the pT - and xF -distributions of the azimuthal asymmetry in heavy quark photoproduction are given in [10].

IV. NEXT-TO-LEADING ORDER CORRECTIONS

The perturbative expansion for the γg cross section, σ ? (ρ, ?2 ), is usually written in terms of scaling functions: σ ? (ρ, ?2 ) =

2

αem αS (?2 )e2 Q m2

∞

4παS (?2 )

k=0

k

k

c(k,l) (ρ) lnl

l=0

?2 , m2

(4.1)

where ρ = 4m2 /s ?, s ? = (kγ + kg ) ; c(0,0) (ρ) corresponds for the Born contribution (2.2), c(1,1) (ρ) and c(1,0) (ρ) describe 2 the order-αS corrections, factorization scale (?F = ?R = ?) dependent and independent, respectively. To calculate the NLO corrections to the single spin asymmetry A(s), we need to take into account the virtual O(αem α2 S ) corrections to the Born process (2.1) and the real gluon emission in the photon-gluon fusion: γ (kγ ) + g (kg ) → Q(pQ ) + Q(pQ ) + g (pg ).

2

(4.2)

? The scale dependent term c(1,1) (ρ) which is the coe?cient of ln m 2 can be expressed explicitly in terms of the Born cross section using the renormalization group arguments [3]. Contribution of the scale independent cross secton, c(1,0) (ρ), near the threshold can be obtained with help of the soft gluon resummation method [12–14]. To the next-to leading logarithmic accuracy, the soft gluon contribution to the photon-gluon fusion can be written in a factorized form as:

s ?2

d2 σ ? ?1 , u ?1 , u ?, t ?1 s ?, t ?1 ≈ B Born s ?1 d? dt u1

∞

?1 + u δ s ?+ t ?1 +

n=1

αS (?) π

n

?1 , u ?, t ?1 K (n) s

,

(4.3)

?1 = t ? ? m2 , u ?1 , u where t ?1 = u ? ? m2 and B Born s ?, t ?1 describes the Born level γg cross section: ?1 , u B Born s ?, t ? 1 = e2 Q αem αS ?1 4 m2 s ? t u ?1 m2 s ? + + 1? ? ? ? u ?1 t1 t1 u ?1 t1 u ?1 (4.4)

At NLO, O(αem α2 S ), the soft gluon corrections to NLL accuracy in the MS factorization scheme are (cf. Ref. [14]): ?1 , u K (1) s ?, t ?1 = 2CA ln s ?4 /m2 s ?4 ?u ?1 m2 +

+

1 s ?4 ?2 m2

CA ln

+

?1 t u ?1

+ ReLβ ? ln

?2 m2

? 2CF (ReLβ + 1) + (4.5)

δ (? s4 ) CA ln where CA = Nc = 3, CF = in (4.5) are de?ned as:

2 Nc ?1 2Nc

ln

,

4 ?1 + u ,s ?4 = s ?+ t ?1 and β = = 3

√ 1 ? ρ. The function Lβ and the plus-distribution + iπ , (4.6)

Lβ =

1 ? 2 m2 / s ? ln β

1?β 1+β

lnl s ?4 /m2 s ?4

= lim

+

?→0

lnl s ?4 /m2 1 θ (? s4 ? ?) + lnl+1 s ?4 l+1

? m2

δ (? s4 ) .

(4.7)

4

Eq.(4.5) describes very well the threshold behavior of the exact partonic cross sections c(1,0) (ρ) and c(1,1) (ρ) and gives, at NLO, practically whole contribution to the unpolarized bottom production in photon-hadron reactions at ?xed target energies, Eγ < ? 1 TeV. All terms in the r.h.s. of Eq.(4.5) originate from the so-called collinear pg,T → 0, and soft, pg → 0 , components of cross section. Since the azimuthal angle ? is the same for both γg and QQ centre of mass systems in the collinear and soft limits, Eq.(4.3) can be generalized to the spin-dependent case substituting the spin-averaged Born cross section ?1 , u ?1 , u ?, t ?1 , ? . by the ?-dependent one: B Born s ?, t ?1 ?→ B Born s The results of our calculations of the single spin asymmetry A(s) at NLO to NLL accuracy in c- and b-quark production are presented by dashed line in Fig.2. The details of calculations as well as the higher order predictions for A(s) will be given in [15].

20 0.19 0.18 A(Eγ ) 0.17 0.16 Charm production Solid: QCD LO Dashed: QCD NLO 20 30 50 70 Eγ , GeV 100 150 200 A(Eγ ) 30 50 70 100 150 200 0.19 0.18 0.17 0.16 Bottom production Solid: QCD LO Dashed: QCD NLO 0.15 0.2 0.3 0.5 0.7 1 0.15 0.2 0.3 0.5 0.7 1

Eγ , TeV

FIG. 2. Single spin asymmetry, A(Eγ ), in c- and b-quark production at LO (solid curve) and at NLO to NLL accuracy (dashed curve).

One can see from Fig.2 that the NLO NLL and Born predictions for A(s) coincide with each other with accuracy better than 2%. We have veri?ed that the azimuthal asymmetry is independent (to within few percent) of theoretical uncertainties in the QCD input parameters (mQ , ?R , ?F and ΛQCD ) at NLO too.

V. CONCLUSION

Our analysis shows that the NLO corrections practically do not a?ect the Born predictions for the single spin asymmetry in heavy ?avor production by linearly polarized photons at ?xed target energies. So, the quantity A(s) is an observable quantitatively well de?ned, rapidly convergent and insensitive to nonperturbative contributions. Measurements of the azimuthal asymmetry in bottom production would be a good test of the conventional parton model based on pQCD.

ACKNOWLEDGMENTS

The research described in this paper was carried out in collaboration with A. Capella and A.B. Kaidalov. Author would like to thank A.V. Efremov for useful discussion. I also wish to thank Organizing Committee of XV ISHEPP for invitation and ?nancial support and, once more, E.B. Plekhanov for warm hospitality. This work was supported in part by the grant NATO PST.CLG.977275.

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