On the heavy-quark content of the nucleon

Physics Letters B 291 (1992) 325-328 PHYSICS LETTERS B North-Holland

On the heavy-quark content of the nucleon

E. L a e n e n ~, S. R i e m e r s m a , J. Smi th

Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11794-3840, USA

and

W.L. v a n N e e r v e n

Instituut Lorentz, University of Leyden, P.O. Box 9506, NL-2300 RA Leyden, The Netherlands

Received 6 July 1992

Recently QCD corrections to the light (massless)-quark and heavy (massive)-quark structure functions of the proton have been computed up to O (as 2 ). Using these results we present the O(a 2) contributions to the ratio of the heavy- quark content of the nucleon with respect to the sum of the heavy-quark plus light-quark content for charm and bottom quarks. We find that, at small x, the heavy-quark fraction is surprisingly large and remarkably insensitive to changes in the mass factorization scale and/or the choice of parton densities.

One of the most important objectives of exper- iments at HERA, which have just started to take data, is an accurate determination of the gluon density g(x , Q2) [1 ]. Here Q denotes the momentum transfer in the process in which the gluon partici- pates, and x the fraction of the proton momentum carried by the gluon. The large center-of-mass energy (v/S = 314 GeV) for electron-proton collisions at HERA ensures that for the first time the small-x region (x < 10 -2) will be probed. This region is very important for many reasons, see ref. [2] for an overview, in particular for making predictions for experimental rates at the future LHC and SSC colliders. For an accurate interpretation of the col- lected data a better theoretical understanding of the gluon density is necessary. With this in view the perturbative QCD contributions to the structure functions F2 (x, Q2) and FL (x, Q2) which contain light (massless) partons, and the corresponding structure functions F2 (x, Q2, m 2), FL (x, Q2, m 2 ) which contain heavy (massive) partons (m is the heavy-quark

Address after 1 September 1992: Fermi National Ac- celerator Laboratory, P.O. Box 500, Batavia, IL 60610, USA.

mass) have been calculated to O ( a 2) [3,4]. The magnitudes of these quantities impact on the gluon density through the momentum sum rule. We should stress that the perturbative QCD contributions to the heavy-quark structure functions have been computed assuming only light (massless) partons in the initial state, i.e., we did not assume an "intrinsic" heavy- quark density #l. A comparison of both mechanisms has e.g. been made in ref. [7]. In most parton densities currently available, an intrinsic heavy-quark content of the nucleon has been considered and parametrized in a variety of ways with the conclusion that even at small x the heavy-quark content in the nucleon is small. In this letter we will consider the "perturbative" heavy-quark content using the structure functions mentioned earlier, both at O(as) and O(a 2 ). We find that this content is surprisingly large at small x in both cases. This result is stable under

#1 To consider both would require a precisely formulated factorization theorem for heavy-quark production, valid for both large m and large Q, which would separate the "perturbative" from the "non-perturbative" production mechanisms. It has been argued in this context that in fact the intrinsic heavy-quark production mechanisms are suppressed [5,6 ].

0370-2693/92/$ 05.00 (~) 1992-Elsevier Science Publishers B.V. All rights reserved 325

Volume 291, number 3 PHYSICS LETTERS B 24 September 1992

changes in the mass factorization scale, the sensitiv- ity decreasing when the O(a 2 ) terms are included. In addition, a different choice of parton densities does not significantly change the result.

We first review the definitions of the light- and heavy-quark structure functions. The light-quark ones are defined by

1

l ~min

×'k,i(~ Q2, M2) whereas for the heavy-quark ones we have

(1)

1

Fk(x'Q2'm2) = Z. f ..~_fd~ , ( ¢ , M 2)

l ~min

X.~k,i(~,Q2,M2) , ( 2 )

where ~mi, = x(4m 2 + Q2)/QZ and the index k = 2, L whereas i ranges over the partons (quarks, anti- quarks and gluons). In the above expressions M denotes the mass factorization scale, and f are the partonic densities. The light- and heavy-quark (Wilson) coefficient functions Ck,i and .¢'k,i, respectively, have been computed through O (a2) in ref. [3], and in ref. [4 ] respectively, using both the MS and DIS schemes. In this letter we will use these functions in the MS factorization scheme. The renormalization scale/t also enters via the running coupling constant in the perturbative expansions of the coefficient functions. For simplicity we have chosen/t = M.

It is convenient to define the ratios (as in ref. [4] )

K~k i) (x, Q2, m 2 )

~,(i) (x, Q2, m: ) = " ~ , ( 3 )

f~ i~ (x, Q2, m2) + f~ i~ (x, Q2)

for k = 2, L. These K's are a measure o f the per- centage of the (perturbative) heavy-quark content in the nucleon as compared with the total light-parton plus heavy-quark content. The index i indicates the order in as to which the C.k,i (x/~, Q2, M 2) and the .~k,i(x/~,Q2, M 2) in (1) and (2) are calculated.

We will present results for K2 and KL for charm (m = me = 1.5GeV/c 2) a n d b o t t o m (m = mb = 4.75 GeV/c 2 ) quarks and investigate the dependence of these quantities on M, x, Q2 and i. The results for the heavy-quark content influence the gluon density through the momentum sum rule. In principle, one should take a gluon density then compute the heavy- quark content and compare the result with experimental data. Depending on whether the result is too large or too small the gluon density should be changed and the heavy-quark component redetermined until a "fixed point" is reached. Due to the lack of any data we will simply examine the results from some currently available parametrizations, to be specified in the following.

We start with the charm quark taking the u, d and s-quarks as massless. In fig. l a we show the x- dependence of the O(as) contributions [i = 1 in (3) ] to K2 (x, Q2, m E ) for various choices of the mass factorization scale M, at Q2 = l0 (GeV/c) 2, and Q2 = 100 (GeV/c) 2. The corresponding results for KL(X, Q2,m 2) are shown in fig. lb. We have chosen the range o f factorization scale for c-quarks to be ½(Q2 + 4m2)1/2 < M < 2(Q 2 + 4m~) 1/2. For

these curves we have used the Morfin-Tung (MT) parametrizations [8] of the parton densities, where we have adopted Fit B1 in table /4 given for the MS scheme with A4 = 0.194 GeV/c. Further we use the two-loop running coupling constant a s ( M 2) in eq. (10) of ref. [9] with interpolation across the b- quark threshold so that A4 = 0.194 GeV/c should be replaced by A5 = 0.126 GeV/c) .

From these figures we note first that, at fixed Q2, the magnitude of KL is always smaller than that of the corresponding K2, and that both increase as the scale Q2 increases. From a physical point of view this can be interpreted by saying that higher Q2 probes smaller distance scales, so more virtual heavy-quark- antiquark pairs are produced. Another prominent feature is the large size of K2 (x, Q2, m 2) at small x for both values of Q2, although one should take the values in the region 10 -4 < X < 10 -3 with some cau- tion since the parton densities used in (1) and (2) are then extrapolated far outside the x-region where they have been fit to any data. All these figures show a clear, but certainly not too large, dependence on the factorization scale M.

Let us now consider the results for the case of

326


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Fig. 1. (a) The x-dependence of the ratio K~l)(x, Q2,m2 c) for Q2 = 10(GeV/c)2 (lower set of curves) and for Q2 = 100 (GeV/c) 2 (higher set of curves) and three mass factorization scales, i.e., M = ½ (Q2 + 4m2)1/2 (the long-dashed curve), M = (Q2 + 4m2)U2 (the solid curve), and M = 2(Q 2 + 4m2) 1/2 (the short-dashed curve). (b) The x-dependence of the ratio K~ l) (x, Q2, rnc 2) for Q2 = 10 (GeV/c)2 and Q2 = 100 (GeV/c)2. The notation is the same as in (a).

. 4 ~ i 1 1 1 l l I I I I l l I I I I I I B B I F I E l l 1 1 1

' , X .2 - - " , X

X " , X

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" . x ". ~k

o ~ f ~ H , ~ l , , , , , , , l " ~ ' - - L , , " ,~ , . 0 0 0 1 .001 .1 .01

X

Fig. 2. (a) The x-dependence of the ratio K(22)(x, Q2, m2c) for Q2 = 10(GeV/c)2 and Q2 = 100(GeV/c)2. (b) The

x-dependence of the ratio KL (2) (x, Q2, m 2) for Q2 = l0 (GeV/c) 2 and Q2 = 100 (GeV/c) 2. The notation is the same as in fig. la.

327


charm product ion when the O (a~) contributions are included. We have plot ted in fig. 2a the sum of the O(as ) and the O ( a 2) contributions [i = 2 in (3) ] to /(2 (x, Q2, m E) as a function o f x for various choices of the mass factorization scale M, at Q2 = 10 (GeV/c ) 2 and Q2 = 100 (GeV/c ) 2. The corresponding results for KL (x, Q2, m 2 ) are shown in fig. 2b. Regarding these figures the same features hold as for figs. la, lb. Comparing the plots with i = 1 and i = 2 we can make the following addit ional observations: there is a marked increase in the K ' s as Q2 is increased from 10 to 100 (GeV/c ) 2. However, the ratio K,!E)/K,! l) is larger for the Q2 = 10 (GeV/c ) 2 curves, indicating that the O (a 2) corrections are more important for the smaller Qz values. Another general feature is that the plateau behaviour o f the K~ 1) as x ~ 0 is d iminished when we analyze the K~ 2) , which is a consequence of the high-energy behaviour of the Wilson coefficient functions (,k.i and-¢k.i in O ( a 2 ) [3,4]. Note also that the mass factorization scale dependence is generally reduced when we include the O ( a 2) corrections. Since the ratio in (3) depends on a choice for the parton densities we have recomputed some of the results with the MS HMRS [ 10 ] parametrizat ions, and checked that Kz (x, Q2, m 2 ) hardly changes. Thus we conclude that there is remarkably little variat ion in the K-rat ios under changes in the mass factorization scale and choice of parton densities, and that the charm quark presence in the proton is quite large [e.g. roughly (15-20)% at x "~ 0.1, for Q2 between 10 and 100 (GeV/c)2] .

We now comment on the si tuation for the b-quark without showing any plots. Its mass we have taken to be m = mb ---- 4.75 GeV/c 2, where we consider the u, d, s and c-quarks to be massless. Varying the factorization scale over the range ½ (Q2 + m2)1/2 < M < 2 (Q2 + m 2 ) l/z, we found very little change in the b-quark K's. The latter are smaller than their c-quark counterparts in this part icular Q2 range, but grow very rapidly as Q2 increases. For example, at x = 10 -4 and Q2 = 100 (GeV/c ) 2 the values are K2 ~ 3% and KL ~-, 2%. The numerical values do not level out as x decreases to 10 -4 which indicates that the plateau is at even smaller x.

Thus we conclude that the currently available parametr izat ions for the l ight-parton densities yield a large perturbat ive QCD heavy-quark content in the proton, which is stable under changes in the factor-

ization scale and choice of the partonic densities. Whether the results presented here should be given any more "absolute" meaning requires further study; as stated in the above, an i terative procedure where the gluon density is refitted, with which the heavy- quark content is compared to data, etc., should be tried. The final result of which (the "fixed point") would lead to the experimental value for the heavy- quark content of the nucleon, although it is not clear precisely what the value should be.

The work in this paper was supported in part under the contract NSF 91-08054. Financial support was also provided by the Netherlands Organization for Scientific Research (NWO) and the Texas Nat ional Research Laboratory Commission.

References

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On the heavy-quark content of the nucleon