Gluon distribution at very small x from C-even quarkonia production at the LHC

Abstract

C-parity-even quarkonia [eta] ^sub b,c^ and [chi] ^sub b,c^ with spin 0 and 2 are produced via two-gluon fusion. The expected cross section of the inclusive production of the quarkonia at the LHC, times the branching ratios of convenient decays, is up to tens of nanobarn per unit rapidity in the case of charmonia and around one nanobarn for the bottomonia. Measuring the quarkonia production as function of rapidity will allow to determine the gluon distribution function in nucleons in a very broad range of the Bjorken x from x~10^sup -2^ where it is already known, down to x~10^sup -6^ where it is totally unknown. The scale of the gluon distribution found from such measurements turns out to be rather low, Q ^sup 2^2.5-3 GeV^sup 2^, for charmonia and rather large, Q ^sup 2^20 GeV^sup 2^, for bottomonia. We evaluate the scale by studying the next-to-leading-order production cross sections.

Published for SISSA by Springer

Received: November 20, 2012

Revised: January 4, 2013 Accepted: January 23, 2013

Published: February 12, 2013

Gluon distribution at very small x from C-even quarkonia production at the LHC

Dmitri Diakonov,a,b M.G. Ryskina and A.G. Shuvaeva,b

aPetersburg Nuclear Physics Institute, Kurchatov National Research Centre, Gatchina, St. Petersburg 188300, Russia

bSt. Petersburg Academic University,

St. Petersburg 194021, Russia

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: C-parity-even quarkonia b,c and ~b,c with spin 0 and 2 are produced via

two-gluon fusion. The expected cross section of the inclusive production of the quarkonia at the LHC, times the branching ratios of convenient decays, is up to tens of nanobarn per unit rapidity in the case of charmonia and around one nanobarn for the bottomonia. Measuring the quarkonia production as function of rapidity will allow to determine the gluon distribution function in nucleons in a very broad range of the Bjorken x from x

102 where it is already known, down to x 106 where it is totally unknown. The

scale of the gluon distribution found from such measurements turns out to be rather low, Q2 2.5 3 GeV2, for charmonia and rather large, Q2 20 GeV2, for bottomonia. We

evaluate the scale by studying the next-to-leading-order production cross sections.

Keywords: Hadronic Colliders, NLO Computations

ArXiv ePrint: 1211.1578v2

c

JHEP02(2013)069

SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP02(2013)069

Web End =10.1007/JHEP02(2013)069

Contents

1 Introduction 1

2 Elementary cross sections of the c and ~c0,2 production 4

3 The scale parameter for the gluon distribution 6

4 Extracting gluon distribution from the C-even charmonia production 9

5 Gluon distribution from the C-even bottomonia production 12

6 Conclusions 13

A Cross sections of mesons production 13

1 Introduction

The small-x gluon distribution function in nucleons at a relatively low momentum scale is a fundamental quantity in high energy physics, determining the bulk of the collision processes. Apart from being of practical importance for evaluating the rate of many processes at high energies and of the background for new physics, the gluon distribution in nucleons has its own fundamental value as it collects many ne and subtle features of Quantum Chromodynamics. At a relatively low momentum scale and small x one expects the transition from the hard DGLAP regime to the soft nonperturbative pomeron [1] but their interplay is not fully understood. Theoretical models predict various gluon distribution functions g(x, Q2), therefore, knowing it one can discriminate between the models. However, the experimental knowledge of this fundamental quantity is so far limited.

At present, the low-x global parton analysis is based mainly on the deep inelastic scattering HERA data where quark (and antiquark) but not gluon distributions are measured directly. The gluon parton distribution functions (PDFs) are extracted from the derivative dF2(x, Q2)/d ln Q2 using the DGLAP evolution equation. For this reason the accuracy in the determination of the gluon densities is not too good. Moreover, in the range of very small x < 103 and at low momenta scale Q2 2 3 GeV2 the present-day gluon distri

butions are actually given by ad hoc extrapolations from the larger x data since this range has not been accessible by the previous data.

In gure 1 we plot the low-x extrapolations from the CT10 [2] and NNPDF [3, 4] gluon distributions. The drop of the gluon ux x g(x, 2.5 GeV2) at very small x is counterintuitive: on the contrary, one expects that it should be roughly a constant, which would correspond to a constant cross section for minijet production, or even rise as a small power of 1/x, see gure 1, left. The unexpected behaviour of g(x) may be a result of

1

JHEP02(2013)069

x g[LParen2]x, 20 GeV2[RParen2]

x g[LParen2]x, 2.5 GeV2[RParen2]

20

8

15

6

10

4

2

5

-4 -3 -2 -1 log[LParen1]x[RParen1]

-4 -3 -2 -1 0 log[LParen1]x[RParen1]

Figure 1. Gluon distribution function (times x) x g(x, Q2) for the scales Q2 = 2.5 GeV2 appropriate for charmonia production (left), and Q2 = 20 GeV2 appropriate for bottomonia production (right). The shadowed areas are spanning the extrapolations of the CT10 (upper side) and of the NNPDF (lower side) NLO parton distributions. The curves show our extrapolations to the small-x range assuming x g(x) const. (dot-dashed lines), x g(x) 1/x0.1 (solid lines) and x g(x) 1/x0.2

(dashed lines). On the right, the dashed line shows the extrapolation x g(x) 1/x0.24. The plots give the idea of the vast uncertainty in the present-day knowledge of the gluon distribution at very small x.

neglecting power and absorptive corrections that are probably non-negligible at relatively low Q2 2.5 GeV2. It should be noted that the MSTW low-x NLO gluon distribution [5]

becomes even negative at this low scale, which gives the idea of the uncertainty in the present-day knowledge.

The much higher energy of the LHC and a relatively low mass of the c and ~c mesons allows to probe the gluon distribution directly down to a few units of 106. Indeed, the c with spin 0 and ~c mesons with spin J = 0, 2 having positive C-parity are produced in the leading order (LO) via the simple gluon-gluon fusion gg c, ~c0, ~c2, and similarly for the bottomonia. The two-gluon fusion into spin-1 mesons such as J/ and ~c1 is

forbidden by the Landau-Yang selection rule [6], therefore the c(0+, 2980), ~c0(0++, 3415) and ~c2(2++, 3556) mesons are, in this sense, privileged. A survey of quarkonia production in high energy collisions can be found in ref. [7].

In the LO the inclusive production cross section of C-even quarkonia, integrated over the transverse momentum of a meson is given by a simple factorized equation [8]1

d(pp quarkonium)

dY = x1 g(x1, F ) x2 g(x2, F )

(gg quarkonium) , (1.1)

The last factor being the fusion cross section is given in section 2, and the values of x1,2

are found from the kinematics as

x1,2 = M

JHEP02(2013)069

quarkonium

s eY 4 104 eY . (1.2)

It means that for the pp collision energy s = 8 TeV, an LHCb experiment carried out in the rapidity range Y = 2 5 is in a position to measure gluon distribution with x as

small as the record 2.5 106, if the lightest c meson is used.

1If one does not sum over transverse momenta of the produced quarkonia there is, strictly speaking, no factorization even in the LO, and eq. (1.1) is replaced by a more complicated expression [8].

2

The C-even quarkonia production is not the only way to probe low x partons. One can measure the PDFs at low x at the LHC by observing di erent low-mass systems, such as the Drell-Yan lepton pairs, or open heavy-quark Q Q states. The advantage of the quarkonia is their direct coupling to gluons already in the LO. In the case of charmonia with their low mass 3 GeV one achieves almost the lowest possible scale where one can justify the

notion of the gluon distribution itself, and the use of perturbative QCD. In fact, it is not altogether clear beforehand if the gluon distribution at a relatively low scale corresponding to the charmonia production as measured in the pp collisions is not a ected by power corrections such as the absorptive e ects and/or the multiple gluon rescattering, and is not di erent from that measured, say, in the ep collisions at the same value of Bjorken x. This important question has to be answered experimentally. The bottomonia production corresponds to a higher scale, and there is most probably no such problem there. Therefore, comparing the gluon distributions obtained from charmonia and bottomonia production one would be able to judge about the possible nonlinear e ects of the gluon self-interactions at a relatively low scale.

There is also a theoretical problem with the simple LO eq. (1.1). Supposing the inclusive b- or c-quarkonium production cross section is measured to what precisely factorization scale F does the gluon distribution correspond when extracted from eq. (1.1)?

This is an important question since one expects that the PDFs depend strongly on the choice of F at low x because of the strong gluon bremsstrahlung there.

In general, after summing up all orders of the perturbation theory, the nal result should not depend on the choice of F that is used to separate the incoming PDFs from the hard matrix element

. Contributions with low virtuality, Q2 < 2F , of the incoming partons are included into the PDFs, while those with Q2 > 2F are assigned to the matrix element. However, at low x the probability to emit a new gluon in an interval F is

enhanced by the large value of the longitudinal phase space, that is by the large value of ln(1/x). In fact, the mean number of gluons in the interval ln F is

hni

sNc

ln

JHEP02(2013)069

1 x

[parenrightbigg]

ln 2F (1.3)

leading to the value of hni up to about 8, for the case ln(1/x) 8 and the commonly

practiced F scale variation from /2 to 2.2 Meanwhile, the next-to-leading (NLO), coe cient function (the hard matrix element), allows, by denition, the emission of only one additional parton. Therefore one cannot expect here a compensation between the contributions coming from the PDF and from the coe cient function. To that end one would need in this case to calculate the hard matrix element to the eighth order, which is not practical. [At large x the compensation is much better and provides reasonable stability of the predictions with respect to the variations of the scale F .]

To circumvent this di culty and to x the factorization scale in eq. (1.1), we use an approximate method following the recent ref. [9]. The method is recalled in section 3 where we also nd the best choice of the factorization scale F = 0 for the processes at hand: actually it determines at what scale parameter the gluon distributions are evaluated when

2For illustration we have use the values of x 103 104 and s(2 = 2.5 GeV2) 0.3.

3

the quarkonia production is measured. In the forthcoming section 2 we evaluate the cross sections for the elementary hard two-gluon fusion processes into C-even charmonia. It becomes possible after we go into some details of the inverse processes i.e. the decays of charmonia. In section 4 we discuss the resulting inclusive production cross section of C-even charmonia, and the ways to experimentally detect them. We stress that the absolute normalization of the gluon distribution obtained from the measurements we suggest, can be found even in the case when the experimental and/or theoretical normalization of the cross sections is poorly known. In section 5 we discuss briey the production of the bottomonium ~b2. The cross section is less than in the case of the charmonia production, however it can give an important independent information on the gluon distribution at a larger normalization scale. We summarize in section 6.

2 Elementary cross sections of the c and ~c0,2 production

In the literature, one can nd the LO two-gluon fusion cross sections gg M as well as

the NLO di erential cross sections gg M + g and gq M + q, expressed through the

charmonia radial wave function at the origin R0 (for the s-wave charmonium c) or the derivative at the origin R1 (for p-wave charmonia ~c0,2). In particular, the LO two-gluon fusion elementary cross sections are [1012]

LO(gg c) =

2 2s 3

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R20 M5 c

, (2.1)

LO(gg ~c0) = 122 2s

R21

M7~c0

, (2.2)

. (2.3)

The numerical values of the quantities R0 and R1 have been evaluated in the past by many authors in the nonrelativistic quark models. Depending on the details of the model used these quantities lie in the ranges R20 (0.5 1.0) GeV3 and R21 (0.07

0.14) GeV5. One can try to avoid model-dependent estimates and reduce the uncertainties in the couplings of two gluons to the charmonia by using the experimentally-known partial widths of the charmonia decays. We write the C and the C gg widths (including

the 1st order QCD radiative corrections) from ref. [13]:

( c ) = 4 Q4c 2em

f2 c M c

LO(gg ~c2) = 162 2s

R21

M7~c2

1 +

203 23[parenrightbigg] s

, (2.4)

( c gg) =

29 4 2s

f2 c M c

1 + 4.8 s

[parenrightBig]

, (2.5)

(~c0 ) = 4 Q4c 2em

f2~c0 M~c0

1 +

23 28 9

s

, (2.6)

(~c0 gg) =

29 4 2s

f2~c0 M~c0

1 + 8.77 s

[parenrightBig]

, (2.7)

4

t( ), keV exper( ), keV t(gg), MeV exper(hadrons), MeV

c 5.3 5.3 0.5 29.7 29.7 1.0

~0 2.3 2.3 0.23 10.3 10.3 0.6

~2 0.55 0.51 0.043 1.48 1.59 0.11Table 1. A simultaneous t to the radiative and to the hadronic widths of the C-even charmonia, eqs. (2.4)(2.9).

(~c2 ) =

415 4 Q4c 2em

f2~c2 M~c2

1 163 s

, (2.8)

1 4.827 s

[parenrightBig], (2.9)

where f c, f~c0 f~c0 are the relativistic matrix elements of the local heavy-quark currents creating (or annihilating) the appropriate mesons from the vacuum. In the nonrelativistic limit they are related to the wave functions at the origin [13]:

f2 c =

3

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(~c2 gg) =

4 15

29 4 2s

f2~c2 M~c2

R20M c , f2~c0 =

108

R21

M3~c0

, (2.10)

with f~c2 = f~c0. The system, however, is not fully nonrelativistic, and we relax this condition.

We now t the experimentally-known widths by eqs. (2.4)(2.9). We identify the gg widths with the total hadronic widths, ( c gg) ( c)tot, (~c0 gg) (~c0)tot

(1 0.0117), (~c2 gg) (~c0)tot (1 0.195) where in the parentheses be subtract the

branching ratios of the radiative decays, Br(~c0 J/ ) = 0.0117 and Br(~c2 J/ ) =

0.195. Here and below the experimental numbers are from the latest PDG listings [14]. We treat f c, f~c0, f~c2 and s as free tting parameters. The results of the t are presented in table 1, and are impressively good.

We nd the best-t values f c = 432 MeV, f~c0 = 240 MeV, f~c2 = 361 MeV and s = 0.335. Using these values in eq. (2.10) and eqs. (2.1)(2.3) we obtain the elementary gluon-fusion cross sections

(gg c) 344 nb, (2.11)

(gg ~c0) 62 nb, (2.12)

(gg ~c2) 140 nb. (2.13)

It should be kept in mind, though, that the QCD radiative corrections and the relativistic corrections to the charmonia decays appear to be rather large, therefore, the above cross sections extracted from the t to the charmonia widths carry theoretical uncertainties. An estimate of the corrections shows that eqs. (2.11)(2.13) may be correct up to a factor of two in either direction.

Indeed, the LO cross sections of the hard gluon fusion to charmonia can be derived alternatively from simple arguments. The decay of a spin-zero meson into two on-mass-shell gluons is described by only one helicity amplitude, call it A. In terms of this amplitude

5

the width of a meson with mass M is (M gg) =

A2

2 M whereas its production cross

section is LO = A2

16M2 . We account here for the fact that the standard gluon PDF already includes the sum over the 8 gluon colours and over 2 transverse polarizations. A similar relation between the two-gluon fusion cross section and the two-gluon decay width exists for the spin-2 ~c2 meson; the only di erence is the spin factor (2J + 1). Therefore, the cross sections of the hard subprocesses can be written as

(gg c)

2 ( c gg)

8M3 c

539 nb, (2.14)

(gg ~c0)

2 (~c0 gg)

8M3~c0

124 nb, (2.15)

JHEP02(2013)069

(gg ~c2)

52 (~c2 gg)

8M3~c2

85 nb , (2.16)

where for the numerical evaluation we have replaced the two-gluon widths by the phenomenological hadronic widths as above.

Comparing the estimates (2.11)(2.13) with the estimates (2.14)(2.16) one gets the idea of the theoretical uncertainty in evaluating the elementary cross sections. The rst derivation takes into account the radiative corrections to the charmonia decays but ignores them in the cross sections. The second derivation is based on the fact that the e ective Cgg vertex (C = ~c, c) is the same in the decay into two on-mass-shell gluons as in the fusion of two gluons (that should be considered as being on-mass-shell in the LO) into a charmonium. In both cases the radiative corrections seem to be the same. Therefore, we are inclined to trust more the second estimate (2.14)(2.16), given the experience of the rst one: it shows that the two-gluon decays can be well replaced by the total hadronic widths.

3 The scale parameter for the gluon distribution

To sketch the idea how to choose the appropriate scale, we start with the LO expression for the cross section. In the collinear approach, the cross section has the form

(F ) = PDF(F ) CLO PDF(F ), (3.1)

where CLO denotes the LO hard matrix element squared. The e ect of varying the scale from m to F in both PDFs can be expressed, to the rst order in s, as

(F ) = PDF(m)

CLO + s 2 ln

(PleftCLO + CLOPright)

[parenrightbigg]

PDF(m), (3.2)

where the splitting functions Pleft and Pright act on the left and on the right PDFs, respec

tively. Let us recall that in calculating the s correction in eq. (3.2), the integral over the transverse momentum (virtuality) of the parton in the LO DGLAP evolution is approximated by the pure logarithmic dk2/k2 form. That is to say, in the collinear approach, the Leading Log Approximation (LLA) is used.

6

2F m2

Let us now study the cross section at the NLO. First, we note that the original Feynman diagrams corresponding to the NLO matrix element CNLO formally do not depend on F .

However, we shall see below that in fact scale dependence appears. In the NLO we can write

(F ) = PDF(F ) (CLO + sCNLOcorr) PDF(F ), (3.3)

where we include the NLO correction to the coe cient function. In terms of Feynman diagrams it means that the gg c(~c) subprocess plus the 2 2 subprocesses,

gg c(~c) + g and qg c(~c) + q, are now calculated with better than the LLA

accuracy. However part of this contribution is already included, to the LLA accuracy, into the second term in eq. (3.2). Therefore this part should be now subtracted from CNLO. Moreover, this LLA part depends on the scale F . As a result, changing F redistributes the order s correction between the LO part (PDF CLO PDF) and the NLO

part (PDF sCNLOrem PDF).

We see that the part of the NLO correction that remains after the subtraction, CNLOrem(F ), depends now on the scale F as due to the F dependence of the LO LLA term that has been subtracted out. The trick is to choose an appropriate scale F = 0

such as to minimize the remaining NLO contribution CNLOrem(F ). To be more precise, we choose the value F = 0 such that as much as possible of the real NLO contribution (which has a ladder-like form and which is strongly enhanced by the large value of ln(1/x)) is included into the LO part where all the logarithmically enhanced s ln(1/x) terms are naturally collected by the incoming parton distributions.3

As shown in ref. [9], after the scale F = 0 is xed for the LO contribution the variation of the scale in the remaining NLO part does not change noticeably the predicted cross section. Moreover, it was shown that in the case of the Drell-Yan lepton pair production the NLO prediction with F = 0 is very close to the NNLO result.

We now determine the best value of the scale F = 0 for which the factorization eq. (1.1) is maximally correct. It will be in fact the scale parameter for the gluon distribution measured from the c, ~c inclusive production rates, if one uses eq. (1.1) to determine the gluon distribution.

As explained above, in order to nd the value of the appropriate scale F = 0 of the

LO contribution we have to know the cross section of hard subprocesses calculated at the NLO level. The di erential cross sections of the gg M + g and gq M + q subprocesses

as functions of the Mandelstam variables s and t are presented in refs. [12, 17, 18] and are collected in the appendix. We integrate them there over the available t interval, subtract the contributions generated by the last step of the LO DGLAP evolution up to the factorization scale F , convoluted with the LO cross sections. Finally, we choose the value of the factorization scale F = 0 such that it nullies the remaining NLO gg M + g and

gq M + q contributions.

3Actually our approach is rather close in spirit to the kt-factorization method. Using the known NLO result we account for the exact kt integration in the last cell adjacent to the LO hard matrix element (describing the gg ! c(~c) boson fusion), while the unintegrated parton distribution is generated by the last step of the DGLAP evolution, similarly to the prescription proposed in refs. [15, 16].

7

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subprocess = 0 ( ) = 0.1 = 0.2 = 0.3

gq c + q 3.3 3.1 2.9 2.75

gq ~(0) + q 2.4 2.3 2.2 2.1

gq ~(2) + q 2.9 2.8 2.6 2.5

gg c + g 3.3 3.0 2.75 2.5

gg ~(0) + g 2.4 2.1 1.9 1.7

gg ~(2) + g 2.9 2.5 2.2 1.9

Table 2. The best scale 20 (in GeV2) calculated from various subprocesses, depending on the power in the gluon ux assumed.

The value of 0 found by this method may be in fact di erent for various subprocesses. It depends also on the subprocess energy. Therefore we have to average the gg M + g

and gq M + q cross sections with the incoming parton ux F () driven by the PDF low

x behaviour. For low-x parton distributions, we assume a power behaviour, F () with the power 0 < < 0.3. Depending on the choice of , we present in table 2 the scale 2F = 20 that nullies the remaining NLO contribution of the gq c + q, gq ~c(J) + q

and of the gg c + g, gg ~c(J) + g subprocesses.In the case of an asymptotically high subenergy , when the ladder-type diagrams

dominate, the values of 0 are the same for both gq M +q and gg M +g subrpocesses.

However even the LHC energy is not su cient to reach the asymptotics. Actually the rapidity interval available at the LHC, Y 10, corresponds approximately to 0.1.

This value looks also as realistic for the gluon distribution at low x and relatively low scale

2.5 GeV2. For > 0 the value of 0 needed to nullify the remaining NLO contribution of

gq M + q subprocess is larger than that for the gg M + g case. Let us note, however,

that in the last case by changing the value of F we try to mimic by the LO-generated contribution also the terms that have the structure rather di erent from that generated by the LO evolution. This is not altogether consistent. Therefore we believe that the value of 0 calculated from the gq M + q subprocess whose Feynman diagram has the same

form as that generated by the DGLAP evolution, is more reliable.

It is interesting that for the pseudoscalar c production we get a larger value of 0 despite that its mass is less than that of ~c0,2. Owing to the unnatural parity of c, the production vertex contains an additional transverse momentum that enhances large-|t|

contributions. To compensate it, one has to take a larger 0.

We see from table 2 that we still have some 10 20% uncertainty in the value of

the appropriate scale 0 but this is much less than the usually used ad hoc interval from M/2 up to 2M. Moreover, when and if it comes to tting the data it will be possible to simultaneously specify/determine the value of and to x the appropriate scale F = 0

more precisely. At the moment we think that the power = 0.1 is the most realistic for a relatively low scale 2F = 2 3 GeV2.

8

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ddY * Br * Br, nb

30

25

20

15

10

5

0 1 2 3 4 5 Y

Figure 2. Cross section of the inclusive ~c2 production per unit rapidity Y , times the branching ratio of its decay into J/ , times the branching ratio of the J/ + decay, in nanobarns.

The shaded area corresponds to the gluon ux from the shaded area in gure 1, whereas the dot-dashed, solid and dashed curves correspond to the extrapolation using x g(x) const. ( = 0),

x g(x) 1/x0.1 ( = 0.1), and x g(x) 1/x0.2 ( = 0.2), respectively, shown in gure 1, left. The

scale parameter 20 = 2.5 GeV2 is assumed for the gluon distribution.

4 Extracting gluon distribution from the C-even charmonia production

Choosing the appropriate scale F = 0 from table 2 we strongly suppress the remaining higher s order contributions to the LO factorization eq. (1.1). Thus, the inclusive cross section of the c and ~c0,2 production

ddY = x1g(x1, 0) x2g(x2, 0)

(gg Ceven charmonium) (4.1)

will measure directly the product of gluon densities at the normalization point 0. The

values of x1,2 are found from eq. (1.2) while the values of

are given by eqs. (2.14)(2.16). For example, at s = 8 TeV and Y = 5 we have for the c production x1 = 0.055, x2 =

2.5 106, and for the ~c2 production x1 = 0.066, x2 = 3.0 106. At the upper side (x1)

the gluon distribution is rather accurately established, therefore from measuring the rate of the charmonia production one can nd the gluon densities at unprecedented low x2, see

in more detail below.

Let us briey discuss how to register the production of the c and ~c mesons. Of the three C-even charmonia considered the most favourable observational conditions seem to be for the ~c2 meson via an anomalously large radiative decay Br(~c2 J/ ) = 0.1950.008.

Actually the ~c2 inclusive production has been already observed at the LHCb via this particular decay channel [19]. The expected ~c2 production rate, times this branching ratio, times the branching ratio Br(J/ +) = (5.93 0.06) 102 is plotted in

gure 2. It appears to be quite large in the range of tens of nanobarns.

The ~c0 meson has a comparable production rate but a much smaller radiative decay branching ratio Br(~c0 J/ ) = 0.0117 0.008. The production times branching curves

for the ~c0 meson are similar to those shown in gure 2 but the overall scale is an order of

9

JHEP02(2013)069

magnitude less. Therefore, unless a good hadronic decay channel is found, the ~c0 meson cannot compete with its ~c2 cousin.

Finally, the c meson decays mainly into , K mesons and it is not easy to nd a meson decay mode that would not be plagued by the huge multi-meson combinatorial background at the LHC energies. Probably the best channel would be the decay into pp with the branching ratio Br( c pp) = (1.41 0.17) 103 or into

with the branching ratio

) = (0.94 0.32) 103. The c cross section times the pp branching is similar

and close in magnitude to what is presented in gure 2 for the ~c2 meson. It should be noted that the c production measures, paradoxically, the gluon distribution at a larger scale than the ~c2 meson (see section 3) and therefore the two measurements can be of independent value.

It is interesting that when in the whole x region probed by an experiment the gluon distribution has the power-law form g(x) 1/x1+ the production cross section is inde

pendent on the rapidity Y of the charmonia produced. However the hight of the plateau is extremely sensitive to the power at small x.

Remarkably, even if one does not know the absolute normalization of the experimental cross section for the charmonia production and/or of the theoretical cross section (4.1), one can still extract the absolute normalization of the gluon distribution at low x, by matching the measurements with the distribution in the medium-x range where it is already known.

To give the idea, let us consider the setup of the LHCb where particles with rapidity up to Y = 5 can be registered. Supposing it is found that the number of counts C(Y ) of the ~c2 mesons produced in a rapidity interval Y Y is roughly independent of Y ,

corresponding to the plateau in gure 2. It means that the gluon distribution at very small x has a power-law behaviour, x g(x) = a x . We want to nd the power and the absolute normalization a. We write

x1 g(x1) x2 g(x2) = N C(Y ), x1,2 =

Br( c

JHEP02(2013)069

M~c2

! = N C(Y ), (4.3)

where N is an unknown normalization factor.4 We see from eq. (4.3) that the power law

for g(x) is the only one leading to the number of counts C(Y ) independent of the rapidity.

However, at Y > 4 the number of counts will deviate from the plateau, see gure 2. This is where one of the two fusing gluons has a known ux x1 g(x1), see the solid part of the curve at the right-hand side of gure 3. At the end point of the solid curve, that is at x = xc 0.028 the gluon distribution is still known to an accuracy of a few percent.5

Therefore, one can determine the other gluons x2 g(x2) from eq. (4.2):

N

1 x2 g(x2) = C(Y )x1 g(x1) =

10

s eY , (4.2)

a2 M2~c2 s

a

N

x 2. (4.4)

4In fact N 1 is the elementary fusion cross section (2.13), times the integrated luminosity, times the registration e ciency.

5The accuracy can be estimated by comparing two known gluon distributions [24] at xc.

x g[LParen1]x[RParen1]

8

Y[Equal]4.25

Y[Equal]2

Y[Equal]0

Y[Equal]3

Y[Equal]4.75

-6 -5 -4 -3 -2 -1 0 log[LParen1]x[RParen1]

Figure 3. Finding the gluon distribution x g(x) from eq. (4.2). The solid part of the curve shows the known distribution at the normalization point 20 = 2.5 GeV2. The dashed part is the supposed power law at very small x, x g(x) = a x . The rectangular gates indicate the values of x1 (right point) and x2 (left point) in the product of the gluon distributions, corresponding to a given rapidity Y of the ~c2 meson produced. s = 8 TeV is assumed.

To nd numerically and a

N , one has to make two or more measurements at Y > 4, for example as shown in gure 3, and make a two-parameter t to eq. (4.4). Then, assuming the power behaviour of the gluon distribution all the way up to xc we equate

a

N

In this equation, the combination a

N is presumably known from the t above, and all the rest quantities are also known, except the normalization factor N . Therefore, eq. (4.5)

enables one to nd N and hence the absolute normalization of the gluon distribution a.

Alternatively, one can nd , a and N separately by solving the system of equa

tion (4.3) and two equations (4.4) evaluated at two di erent rapidities Y > 4.

If the actual behaviour of the gluon distribution at very low x is substantially di erent from power-like, this will be seen from the deviation from the at plateau in the production rate as function of Y . The data should be then analyzed accordingly, however in any case the absolute normalization of the gluon distribution will be possible to deduce even without knowing the absolute values of the charmonia production cross section by matching the data with the gluon distribution at x xc 0.028 where it is already known with a

reasonable accuracy.

We would like to remark that a good complement would be measuring charmonia production in a xed-target experiment with the LHC beams (AFTER@LHC) as it allows to observe charmonia with low pT and to extract the gluon distribution at x from a few units of 103 to x 1 [20, 21].6

6We thank J.-P. Lansberg for bringing our attention to this work.

11

6

4

x_2

2

x_1

JHEP02(2013)069

x c = 1

N

xc g(xc) . (4.5)

ddY * Br * Br, nb

2.0

1.5

1.0

0.5

0 1 2 3 4 5 Y

Figure 4. The expected cross section of the inclusive bottomonium ~b2(1P ) production per unit rapidity Y , times the branching ratio of its decay into (1S), times the branching ratio of the (1S) + decay, in nanobarns. The gluon distribution shown in gure 1, right, is assumed.

5 Gluon distribution from the C-even bottomonia production

The same theoretical considerations can be applied to measuring gluon distributions from the production of the C-even bb mesons, such as ~b2(1P )(2++, 9912). Like ~c2, the bottomonium ~b2 has a large branching ratio for the radiative decay, Br(~b2(1P ) (1S)) =

0.191 0.012 while the leptonic branching ratio for the is Br( (1S) +) = (2.48 0.05) 102. This decay cascade makes the observation of the bottomonium

~b2(1P ) possible.

At large heavy-quark masses, the gluon-fusion cross sections of the quarkonia production scale as (gg ~) 2s(M~)/M2~ and therefore the ~b cross section is expected to

be 20 times less than that of ~c. From the evaluation of the ~c2 production cross sec

tion (2.16) we estimate (gg ~b2(1P )) 4 nb. The ~b2(2P ) production cross section

must be 3 4 times smaller, according to the nonrelativistic estimate of R1(0).

However, the smaller elementary ~b2 production cross section is multiplied in eq. (4.1) by a larger gluon ux x1 g(x1, 20) x2 g(x2, 20) expected at the scale 20 appropriate for the bottomonia as contrasted to the charmonia. According to the derivation in section 3, the scale 20 is proportional to the mass squared of the quarkonium in question. It is known that at higher resolution scale the gluon distribution increases towards small x e ectively as a higher power x . Taking 0.25 and using table 2 we nd that for the ~b2 production

the gluon distribution scale is 20 20 GeV2; it is plotted in gure 1, right.

The expected cross section of the inclusive bottomonium ~b2(1P ) production, times the branching ratio of its decay into (1S), times the branching ratio of the (1S) +

decay, is plotted in gure 4. It is not a plateau anymore, even if one assumes a power-law behaviour of the gluon distribution at low x. We see that it is a few times less than the inclusive ~c2 production, times its branching ratios, see gure 2, but probably within reach.

The bottomonium ~b2(2P )(2++, 10269) can be observed via the radiative decay into the two s with the branching ratios Br(~b2(2P ) (1S)) = 0.0710.01 and Br(~b2(2P )

(2S)) = 0.162 0.024. The consequent leptonic branching ratios for the decays are

12

JHEP02(2013)069

Br( (1S) +) = (2.48 0.05) 102 and Br( (2S) +) = (1.93 0.17) 102,

respectively. Combining these decay cascades, the total registration rate of the ~b2(2P ) bottomonium is expected to be very similar to that shown in gure 4, being however 3 4

times less.

Although the production cross section (times the branching ratios) for ~b2(1P ) is several times less than that for ~c2(1P ), it would be easier to match the measured gluon distribution at very low x to that already known at larger values of x, see the end of section5. The smallest value of x accessible from the bottomonium production is xmin = 8.3106. It should be noted that the ~b2 must have a broad distribution in the transverse momenta as due to the typical double-logarithmic QCD form factor [8].

6 Conclusions

The inclusive production cross sections of C-even charmonia ~c2(2++, 3556) and c(0+, 2998) at the LHC, times the branching ratios of their convenient decay modes, Br(~c2 J/ ) Br(J/ +) and Br( c pp), respectively, are estimated to lie in

the range 5 30 nb, depending on what is the actual behaviour of the gluon distribution

at very low Bjorken x. Measuring the production of those charmonia, integrated over their transverse momenta will enable one to determine the fundamental quantity the gluon distribution in nucleons g(x, Q2) at an unprecedented low x 2.5 106 and relatively

low normalization scale Q2 = 2.5 3 Gev2. The absolute normalization of the gluon

distribution can be found by matching the measured charmonia yield with the gluon distribution at higher x where it is already known, even if the normalization of the experimental and/or theoretical cross sections are not well established.

Similarly, measuring the inclusive production of the bottomonium ~b2(2++, 9912) with the cross section times the branching ratios around 1 nb will allow to extract the gluon distribution at x 8.3 106 but a larger scale Q2 20 GeV2.

Combining the measurements of the two quarkonia production will give a rather full knowledge of the fundamental quantity the gluon distribution in a broad range of x and Q2. In particular, one will be in a position to judge if at Q2 = 2.5 3 Gev2

the nonlinearity (the gluon self-interaction) becomes important or not, and to discriminate between various theoretical models of the high-energy processes.

Acknowledgments

We thank Alexey A. Vorobyev and Mark Strikman for helpful discussions. This work has been supported in part by the grants RSGSS-4801.2012.2 and RFBR 11-02-00120-a. D.D. acknowledges partial support by the Japan Society for the Promotion of Sciences and thanks the RCNP at Osaka University where this work has been nalized, for hospitality.

A Cross sections of mesons production

We list below the LO and NLO cross sections of c and ~c mesons production [12, 17, 18]. We have checked the equations and present them in the form which is further on used for integrating the di erential cross sections over t.

13

JHEP02(2013)069

The LO contributions to the hard gg ~c(J =0, 2) and gg c(J = 0) subprocesses,

i.e. the LO two-gluon fusion cross sections are given by eqs. (2.1)(2.3).

The NLO 2 2 di erential cross sections are expressed through the quantity denot

ing the ~ + g, ~ + q or + g, + q energy squared, r =/M2. The variable z is dened by the equation t = (1 1/r + z)/2, where t is the momentum transfer squared from the

initial to the nal gluon or quark.

The NLO cross section is obtained by integrating the di erential cross section over t, that is translated in our notations into the integration over z. The divergencies of the integrands at z = 1 1/r or at z = 1 + 1/r reect the logarithmic divergencies of the

di erential cross sections at t 0 or at = M2 t 0. They are the collinear

singularities that are responsible for the evolution of the PDF,

d

NLO

dt[vextendsingle][vextendsingle]

1 LO K(1/r)r,

where K(x) is LO DGLAP splitting function for the gg or the qg channels. To avoid the double counting we subtract from the NLO cross sections the logarithmic part at z > 1 1/r 22F / (and at z < 1 + 1/r + 22F / if there is a singularity in) as being

attributed to the PDF, thereby removing the infrared divergency at t 0.

Since the NLO qg qM cross section is described by the same diagram as that

responsible for the LO DGLAP evolution we choose the scale F = 0 such that being integrated up to 0 the LO-generated contribution nullies the remaining NLO qg qM

cross section. By doing that we shift the major part of the corrections (enhanced by the large value of ln(1/x)) to the low x parton distributions. Below we list the NLO cross sections used in this derivation.

The gg g + ~c(0) di erential cross section is

d

(gg g + ~c(0))

dt =

where

N~gg(0) = 32

D~gg(0) = (rz + r + 1)4(rz + r 1)(rz r + 1)(rz r 1)4(r 1)4r .

14

JHEP02(2013)069

t0 =

3SR21 322M5~c(0)

F ~gg(0), F ~gg(0) = N~gg(0)/D~gg(0) (A.1)

3(154 r + 27) + (z2 + 3)4(z2 1)2r14 12(76z2 + 159)r3 (270z2 187)r2 + 2(87z4 + 848z2 + 649)r5 + (279z4 + 1004z2 663)r4 2(3z6 + 25z4 + 85z2 + 47)(z2 + 3)2(z2 1)r13 (36z6 + 893z4 + 3418z2 4875)r6 + 8(42z6 + 517z4 + 1268z2 507)r7

+2(69z6 531z4 6217z2 5193)(z2 1)r9 (81z8 + 1760z6 + 5858z4 + 16312z2 + 7669)r8+(18z10 + 1249z8 + 10424z6 + 34958z4 + 21726z2 + 2025)r10

4(48z10 + 661z8 + 3172z6 + 7958z4 + 5604z2 + 1757)r11+(9z12 + 212z10 + 1809z8 + 5952z6 + 11019z4 + 5516z2 + 1083)r12

[bracketrightbig]

,

We integrate it over t and obtain

NLO(gg g + ~c(0)) =

3SR21 64M5~c(0)

T ~gg(0) (A.2)

where

T ~gg(0) =

64[parenleftbigg][parenleftbigg]108 ln M2~2F(r2 r + 1)2(r + 1)4(r 1)2

(172r10 56r9 617r8 + 188r7 + 1104r6 508r5 + 302r4

+52r3 + 92r2 + 132r + 99) (r2 1)

12(9r11 31r9 + 14r8 + 40r7 10r6 176r5 + 42r4 + 7r3

+10r2 41r 24) ln(r)r[parenrightbigg][bracketrightbigg]

JHEP02(2013)069

/ h3(r + 1)5(r 1)4r2[bracketrightBig].

The gg g + ~c(2) di erential cross section is

d

(gg g + ~c(2))

dt =

3 3SR21 322M5~c(2)

F ~gg(2), F ~gg(2) = N~gg(2)/D~gg(2) (A.3)

where

N~gg(2) = 64

6(28r + 9) + (z2 + 3)4(z2 1)2r14+6(5z2 34)r3 5(36z2 + 103)r2 + 2(93z4 + 1405z2 + 2022)r4 2(519z4+1567z2+2762)r52(3z632z4+199z210)(z2+3)2(z21)r13 (24z6 + 3017z4 + 10102z2 + 3705)r6 + 4(315z6 + 2264z4 + 5635z2 + 2154)r7

2(27z8 + 178z6 + 760z4 + 7742z2 + 605)r8

4(93z8 + 1197z6 + 4055z4 281z2 + 216)r9

+(12z10 + 1099z8 + 8732z6 + 29186z4 + 6336z2 309)r10 2(21z10 + 482z8 + 4190z6 + 13432z4 + 837z2 + 1006)r11 +2(3z12 11z10 + 186z8 + 2634z6 + 5991z4 2927z2 + 780)r12

[bracketrightbig]

,

D~gg(2) = 3(rz + r + 1)4(rz + r 1)(rz r + 1)(rz r 1)4(r 1)4r .

We integrate it over t and obtain

NLO(gg g + ~c(2)) =

3 3SR21 64M5~c(2)

T ~gg(2) (A.4)

15

where

T ~gg(2) =

128[parenleftbigg][parenleftbigg]72 ln M2~2F(r2 r + 1)2(r + 1)4(r 1)2

(106r10 32r9 101r8 + 239r7 + 651r6 793r5 + 395r4

527r3 + 35r2 + 201r + 66) (r2 1)

12(6r11 22r9 + 8r8 74r7 31r6

11r5 + 204r4 86r3 17r2 5r 12) ln(r)r[parenrightbigg][bracketrightbigg] /

h9(r + 1)5(r 1)4r2[bracketrightBig] .

The gg g + c di erential cross section is d

(gg g + )

dt =

3SR2042M3 F gg, F gg = N gg/D gg (A.5)

where

N gg = (r4z4 + 6r4z2 + 9r4 12r3z2

4r3 + 6r2z2 + 6r2 4r + 9)(r2z2 + 3r2 2r 1)2 ,

D gg = (rz + r + 1)2(rz + r 1)(rz r + 1)(rz r 1)2(r 1)2r .

We integrate it over t and obtain

NLO(gg g + ) =

3SR20

8M3 T gg (A.6)

where

JHEP02(2013)069

T gg =

2[parenleftbigg][parenleftbigg]12 ln M2 2F(r2 r + 1)2(r + 1)2

(12r6 + 23r4 + 24r3 + 2r2 + 11)

(r2 1)

12(r7 5r5 2r4 r3 3r 2) ln(r)r[parenrightbigg][bracketrightbigg] /

3(r + 1)3(r 1)2r2[bracketrightbig].

The qg q + ~c(0) di erential cross section is d

(qg q + ~c(0))

dt =

32 sR21 92M5~c(0)

F ~qg(0), F ~qg(0) = N~qg(0)/D~qg(0) (A.7)

where

N~qg(0) = (r2(z2 2z + 5) + 2rz 2r + 1)(rz + r + 5)2 ,

D~qg(0) = 2(rz + r + 1)4(rz + r 1) .

16

We integrate it over t and obtain

NLO(qg g + ~c(0)) =

16 3SR21 9M5~c(0)

T ~qg(0) (A.8)

where

27 ln M2~2F(2r2 2r + 1)r

2(43r2 14r + 4)(r 1) 6(9r2 9r + 4) ln(r)r[bracketrightbigg] /(6r2) .

The qg q + ~c(2) di erential cross section is d

(qg q + ~c(2))

dt =

32 sR21 32M5~c(0)

T ~qg(0) =

JHEP02(2013)069

F ~qg(2), F ~qg(2) = N~qg(2)/D~qg(2) (A.9)

where

N~qg(2) =

r4(z4 + 2z2 + 8z + 5) r3(44z2 + 8z 36) + r2(22z2 48z + 34) +48rz 4r + 25

[bracketrightbig]

,

D~qg(2) = (3(rz + r + 1)4(rz + r 1)) . We integrate it over t and obtain

NLO(qg q + ~c(2)) =

16 sR21 3M5~c(2)

T ~qg(2) (A.10)

where

18 ln M2~2F(2r2 2r + 1)r

(53r2 16r + 20)(r 1) 3(12r2 12r + 5) ln(r)r[bracketrightbigg] /(9r2) .

The qg q + c di erential cross section is d

(qg q + )

dt =

4 sR20

92M3 c

T ~qg(2) =

F qg, F qg = N qg/D qg (A.11)

where

N qg = 2(z 1)r + 1 + (z2 2z + 5)r2 ,

D qg = (rz + r + 1)2(rz + r 1) .

We integrate it over t and obtain

NLO(qg g + ) =

2 3SR20

9M3 T qg (A.12)

17

where

r . (A.14)

The resulting scales 20 depending on the subprocess and on the power are presented in table 2.

In the case of qg qM subprocess there is no virtual loop corrections at this (NLO)

s order. However there are such a loop corrections for the gg M vertex. Moreover

for the real gg gM fusion we have the divergency at r 1 coming from the normal

soft gluon emission. This divergency is canceled by the virtual loop contribution. It is the well known Bloch-Nordsieck cancelation which in collinear approach (DGLAP evolution) is accounted for by the plus prescription in the evolution kernel, that is by the subtraction of the 1/(r 1) pole (and the ln(r 1)/(r 1) singularity) in our terms. In other words

the singular integral

[integraltext]

10 f(z)/(1 z)+ is treated as the

10 (f(z) f(1))/(1 z) where the

second term, f(1), accounts for the virtual loop contribution.7

In particular, in Feynman gauge the loop correction to gg M vertex contains the well

known Sudakov double logarithm, which reects a small probability to observe an exclusive gg M process without any additional gluon emission. However, here we consider the

inclusive cross section where such an additional emission is allowed. Therefore Sudakov logarithm is exactly cancelled between the real soft gluon emission and the virtual loop correction. In the planar gauge which provides the collinear factorization of inclusive pp M + . . . cross section [8] these double logarithms are hidden in the gluon self-

energy diagrams, that is in the incoming parton distributions also calculated under the plus prescription.8

Besides the self-energy loops there are the loop corrections to the gg M vertex

function. However using the phenomenological normalization to the two gluon decay width we already accounted for these non-singular correction in the value of corresponding width.

Recall also that actually we calculate the scale 0 based on the qg qM subprocess

which has no infrared singularity and no NLO loop correction.

7Another possibility is to subtract from the di erential cross section d

/dt the contribution generated by the LO DGLAP evolution. This completely eliminates the infrared divergency leading to the same nal result.

8We do not consider here the non-singular O( s) part of the gluon self-energy since by the convention this part is completely included into the incoming PDFs; instead of Z1/2 factor for each external leg the whole Z factor is absorbed in PDFs while the matrix element is calculated assuming the incoming partons to be on-mass-shell, that is with Z = 1.

18

ln M2 c2F(2r2 2r + 1) 2(ln(r) + 1)(r 1)r[bracketrightbigg]/r . (A.13)

After subtracting the logarithmical divergent parts of the NLO cross sections (attributed to the PDF) we have to average the remaining cross sections over the incoming subenergy, that is to integrate over r with the weight driven by the parton ux F (). Assuming the power behaviour x g(x) x of the low-x gluon distribution we obtain

the ux F r . Therefore, our goal is to choose such a scale 2F = 20 that nullies

the integral

[integraldisplay]

T qg =

JHEP02(2013)069

NLOqgqM(r, F )r dr

[integraltext]

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