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Heavy-quark contributions to the ratio F


DESY 08-002 January 2008

ISSN 0418-9833

Heavy-quark contributions to the ratio FL/F2 at low x
arXiv:0801.1502v1 [hep-ph] 9 Jan 2008
Alexey Yu. Illarionov?
Scuola Internazionale Superiore di Studi Avanzati, Via Beirut, 2–4, 34014 Trieste, Italy

Bernd A. Kniehl? , Anatoly V. Kotikov?
II. Institut f¨ ur Theoretische Physik, Universit¨ at Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany

Abstract
i (x, Q2 ) We study the heavy-quark contribution to the proton structure functions F2 i 2 and FL (x, Q ), with i = c, b, for small values of Bjorken’s x variable at next-to-lading i /F i that are useful to order and provide compact formulas for their ratios Ri = FL 2 2 i extract F2 (x, Q ) from measurements of the doubly di?erential cross section of inclusive deep-inelastic scattering at DESY HERA. Our approach naturally explains why Ri is approximately independent of x and the details of the parton distributions in the small-x regime.

PACS: 12.38.-t, 12.38.Bx, 13.66.Bc, 13.85.Lg

Electronic address: illario@sissa.it. Electronic address: kniehl@desy.de. ? Electronic address: kotikov@theor.jinr.ru; on leave of absence from the Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia.
?

?

1

1

Introduction

The totally inclusive cross section of deep-inelastic lepton-proton scattering (DIS) depends on the square s of the centre-of-mass energy, Bjorken’s variable x = Q2 /(2pq ), and the inelasticity variable y = Q2 /(xs), where p and q are the four-momenta of the proton and the virtual photon, respectively, and Q2 = ?q 2 > 0. The doubly di?erential cross section is parameterized in terms of the structure function F2 and the longitudinal structure function FL , as 2πα2 d2 σ = {[1 + (1 ? y )2 ]F2 (x, Q2 ) ? y 2FL (x, Q2 )}, 4 dx dy xQ (1)

where α is Sommerfeld’s ?ne-structure constant. At small values of x, FL becomes nonnegligible and its contribution should be properly taken into account when the F2 is i extracted from the measured cross section. The same is true also for the contributions F2 i and FL of F2 and FL due to the heavy quarks i = c, b. Recently, the H1 [1, 2, 3] and ZEUS [4, 5, 6] Collaborations at HERA presented new c b c data on F2 and F2 . At small x values, of order 10?4 , F2 was found to be around 25% of F2 , which is considerably larger than what was observed by the European Muon Collaboration (EMC) at CERN [7] at larger x values, where it was only around 1% of F2 . Extensive c theoretical analyses in recent years have generally served to establish that the F2 data can be described through the perturbative generation of charm within QCD (see, for example, the review in Ref. [8] and references cited therein). In the framework of Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) dynamics [9], there are two basic methods to study heavy-?avour physics. One of them [10] is based on the massless evolution of parton distributions and the other one on the photon-gluon fusion (PGF) process [12]. There are also some interpolating schemes (see Ref. [13] and c references cited therein). The present HERA data on F2 [1, 2, 3, 4, 5, 6] are in good agreement with the modern theoretical predictions. c b In earlier HERA analyses [1, 4], FL and FL were taken to be zero for simplicity. Four c years ago, the situation changed: in the ZEUS paper [5], the FL contribution at next-toleading order (NLO) was subtracted from the data; in Refs. [2, 3], the H1 Collaboration introduced the reduced cross sections σ ? ii = xQ4 y2 d2 σ ii i 2 = F ( x, Q ) ? F i (x, Q2 ) 2 2πα2 [1 + (1 ? y )2] dx dy 1 + (1 ? y )2 L (2)

i for i = c, b and thus extracted F2 at NLO by ?tting their data. Very recently, a similar analysis, but for the doubly di?erential cross section d2 σ ii /(dx dy ) itself, has been performed by the ZEUS Collaboration [6]. i i In this letter, we present a compact formula for the ratio Ri = FL /F2 , which greatly i 2 ii simpli?es the extraction of F2 from measurements of d σ /(dx dy ).

2

2

Master formula

We now derive our master formula for Ri (x, Q2 ) appropriate for small values of x, which has the advantage of being independent of the parton distribution functions (PDFs) fa (x, Q2 ), with parton label a = g, q, q, where q generically denotes the light-quark ?avours. In the small-x range, where only the gluon and quark-singlet contributions matter, while the non-singlet contributions are negligibly small, we have1
i Fk (x, Q2 ) = a=g,q,q l=+,? l l Ck,a (x, Q2 ) ? xfa (x, Q2 ),

(3)

where l = ± labels the usual + and ? linear combinations of the gluon and quarkl singlet contributions, Ck,a (x, Q2 ) are the DIS coe?cient functions, which can be calculated perturbatively in the parton model of QCD, ? is the renormalization scale appearing in the strong-coupling constant αs (?), and the symbol ? denotes convolution according to 1 the usual prescription, f (x) ? g (x) = x (dy/y )f (y )g (x/y ). Massive kinematics requires l 2 that Ck,a = 0 for x > bi = 1/(1+4ai), where ai = m2 i /Q . We take mi to be the solution of mi (mi ) = mi , where mi (?) is de?ned in the modi?ed minimal-subtraction (MS) scheme. l Exploiting the small-x asymptotic behaviour of fa (x, Q2 ) [12],
l fa (x, Q2 ) → x→ 0

1 x1+δl

,

(4)

Eq. (3) can be rewritten as
i Fk (x, Q2 ) ≈ l l Mk,a (1 + δl , Q2 )xfa (x, Q2 ), a=g,q,q l=+,?

(5)

where
l Mk,a (n, Q2 ) = 0

bi l dx xn?2 Ck,a (x, Q2 )

(6)

is the Mellin transform, which is to be analytically continued from integer values n to real values 1 + δl . As demonstrated in Ref. [14], HERA data support the modi?ed Bessel-like behavior of PDFs at low x values predicted in the framework of the so-called generalized + ? double-asymptotic scaling regime. In this approach, one has Mk,a (1, Q2 ) = Mk,a (1, Q2 ) l if Mk,a (n, Q2 ) are devoid of singularities in the limit δl → 0, as in our case. De?ning ± l Mk,a (1, Q2 ) = Mk,a (1, Q2 ) and using fa (x, Q2 ) = l=± fa (x, Q2 ), Eq. (5) may be simpli?ed to become i (x, Q2 ) ≈ Fk Mk,a (1, Q2 )xfa (x, Q2 ). (7)
a=g,q,q

A further simpli?cation is obtained by neglecting the contributions due to incoming light quarks and antiquarks in Eq. (7), which is justi?ed because they vanish at LO and are
Here and in the following, we suppress the variables ? and mi in the argument lists of the structure and coe?cient functions for the ease of notation.
1

3

numerically suppressed at NLO for small values of x. One is thus left with the contribution due to PGF [12], i Fk (x, Q2 ) ≈ Mk,g (1, Q2 )xfg (x, Q2 ). (8) In fact, the non-perturbative input fg (x, Q2 ) does cancels in the ratio Ri (x, Q2 ) ≈ ML,g (1, Q2 ) , M2,g (1, Q2 )

(9)

which is very useful for practical applications. Through NLO, Mk,g (1, Q2 ) exhibits the structure Mk,g (1, Q2 ) = e2 i a(?) Mk,g (1, ai ) + a(?) Mk,g (1, ai ) + Mk,g (1, ai ) ?2 × ln 2 mi + O(a3 ), (10)
(0) (1) (2)

where ei is the fractional electric charge of heavy quark i and a(?) = αs (?)/(4π ) is the couplant. Inserting Eq. (10) into Eq. (9), we arrive at our master formula Ri (x, Q2 ) ≈ ML,g (1, ai ) + a(?) ML,g (1, ai ) + ML,g (1, ai ) ln(?2 /m2 i) M2,g (1, ai ) + a(?) M2,g (1, ai ) + M2,g (1, ai ) ln(?2 /m2 i) + O(a2 ). (11)
(0) (1) (2) (0) (1) (2)

We observe that the right-hand side of Eq. (11) is independent of x, a remarkable feature that is automatically exposed by our procedure. In the next two sections, we present (j ) compact analytic results for the LO (j = 0) and NLO (j = 1, 2) coe?cients Mk,g (1, ai), respectively.

3

LO results

The LO coe?cient functions of PGF can be obtained from the QED case [15] by adjusting coupling constants and colour factors, and they read [16, 17] C2,g (x, a) = ?2x{[1 ? 4x(2 ? a)(1 ? x)]β ? [1 ? 2x(1 ? 2a) CL,g (x, a) = 8x2 [(1 ? x)β ? 2axL(β )], where β= 1? 4ax , 1?x L(β ) = ln 1+β . 1?β (13)
(0) (0)

+ 2x2 (1 ? 6a ? 4a2 )]L(β )},

(12)

4

√ 1? b √ , t= 1+ b we perform the Mellin transformation in Eq. (6) to ?nd √ J (a) = ? b ln t,

where

Using the auxiliary formulas ? if m = 0 ? 1 ? 2aJ (a), b b m [1 ? 2 a ? 4 a (1 + 3 a ) J ( a )] , if m = 1 , x β= 2 ? b2 2 0 [(1 + 3a)(1 + 10a) ? 6a(1 + 6a + 10a )J (a)], if m = 2 ?3 if m = 0 ? J (a), b b m [1 ? (1 + 2 a ) J ( a )] , if m = 1 , ? x L(β ) = ? 2 b2 2 0 ? 3 [3(1 + 2a) ? 2(1 + 4a + 6a )J (a)], if m = 2

(14)

(15)

(16)

2 (0) M2,g (1, a) = [1 + 2(1 ? a)J (a)], 3 4 (0) ML,g (1, a) = b[1 + 6a ? 4a(1 + 3a)J (a)]. 3 At LO, the small-x approximation formula thus reads Ri ≈ 2bi 1 + 6ai ? 4ai (1 + 3ai )J (ai ) . 1 + 2(1 ? ai )J (ai )

(17)

(18)

4

NLO results

The NLO coe?cient functions of PGF are rather lengthy and not published in print; they are only available as computer codes [18]. For the purpose of this letter, it is su?cient to work in the high-energy regime, de?ned by ai ? 1, where they assume the compact form [19] (j ) (j ) Ck,g (x, a) = βRk,g (1, a), (19) with 8 (1) R2,g (1, a) = CA [5 + (13 ? 10a)J (a) + 6(1 ? a)I (a)], 9 16 (1) RL,g (1, a) = ? CA b{1 ? 12a ? [3 + 4a(1 ? 6a)]J (a) + 12a(1 + 3a)I (a)}, 9 (2) (0) Rk,g (1, a) = ?4CA Mk,g (1, a),

(20)

where CA = N for the colour gauge group SU(N), J (a) is de?ned by Eq. (16), and √ 1 (21) I (a) = ? b ζ (2) + ln2 t ? ln(ab) ln t + 2 Li2 (?t) . 2 Here, ζ (2) = π 2 /6 and Li2 (x) = ? 0 (dy/y ) ln(1 ? xy ) is the dilogarithmic function. Using Eq. (14) for m = 0, we ?nd the Mellin transform (6) of Eq. (19) to be Mk,g (1, a) = [1 ? 2aJ (a)]Rk,g (1, a). 5
(j ) (j ) 1

(22)

c Table 1: Values of F2 (x, Q2 ) extracted from the H1 measurements of σ ? cc at low [3] and 2 2 ?3 high [2] values of Q (in GeV ) at various values of x (in units of 10 ) using our approach at NLO for ?2 = ξQ2 with ξ = 1, 100. The LO results agree with the NLO results for ξ = 1 within the accuracy of this table. For comparison, also the results determined in Refs. [2, 3] are quoted. Q2 x H1 ?2 = Q2 ?2 = 100 Q2 12 0.197 0.435 ± 0.078 0.433 0.432 12 0.800 0.186 ± 0.024 0.185 0.185 25 0.500 0.331 ± 0.043 0.329 0.329 25 2.000 0.212 ± 0.021 0.212 0.212 60 2.000 0.369 ± 0.040 0.368 0.368 60 5.000 0.201 ± 0.024 0.200 0.200 200 0.500 0.202 ± 0.046 0.201 0.201 200 1.300 0.131 ± 0.032 0.130 0.130 650 1.300 0.213 ± 0.057 0.212 0.213 650 3.200 0.092 ± 0.028 0.091 0.091

5

Results

We are now in a position to explore the phenomenological implications of our results. As for our input parameters, we choose mc = 1.25 GeV and mb = 4.2 GeV. While the LO result for Ri in Eq. (18) is independent of the unphysical mass scale ?, the NLO formula (11) does depend on it, due to an incomplete compensation of the ? dependence of a(?) by the terms proportional to ln(?2 /Q2 ), the residual ? dependence being formally beyond NLO. In order to estimate the theoretical uncertainty resulting from this, we put ?2 = ξQ2 and vary ξ . Besides our default choice ξ = 1, we also consider the extreme choice ξ = 100, which is motivated by the observation that NLO corrections are usually large and negative at small x values [20]. A large ξ value is also advocated in Ref. [21], where the choice ξ = 1/xa , with 0.5 < a < 1, is proposed. i We now extract F2 (x, Q2 ) (i = c, b) from the H1 measurements of the reduced cross sections in Eq. (2) at low (12 < Q2 < 60 GeV2 ) [3] and high (Q2 > 150 GeV2 ) [2] values of Q2 using the LO and NLO results for Ri derived in Sections 3 and 4, respectively. Our NLO results for ?2 = ξQ2 with ξ = 1, 100 are presented for i = c, b in Tables 1 and 2, respectively, where they are compared with the values determined by H1. We refrain 2 2 from showing our results for other popular choices, such as ?2 = 4m2 i , Q + 4mi because they are very similar. We observe that the theoretical uncertainty related to the freedom in the choice of ? is negligibly small and ?nd good agreement with the results obtained by the H1 Collaboration using a more accurate, but rather cumbersome procedure [2, 3]. The experimental data from the ZEUS Collaboration [6] do not allow for such an analysis because they do not come in the form of Eq. (2). In order to assess the signi?cance of and the theoretical uncertainty in the NLO corrections to Ri , we show in Fig. 1 the Q2 dependences of Rc , Rb , and Rt evaluated at LO 6

b Table 2: Values of F2 (x, Q2 ) extracted from the H1 measurements of σ ? bb at low [3] and high [2] values of Q2 (in GeV2 ) at various values of x (in units of 10?3 ) using our approach at NLO for ?2 = ξQ2 with ξ = 1, 100. The LO results agree with the NLO results for ξ = 1 within the accuracy of this table. For comparison, also the results determined in Refs. [2, 3] are quoted. Q2 x H1 ?2 = Q2 ?2 = 100 Q2 12 0.197 0.0045 ± 0.0027 0.0047 0.0046 12 0.800 0.0048 ± 0.0022 0.0048 0.0048 25 0.500 0.0123 ± 0.0038 0.0124 0.0124 25 2.000 0.0061 ± 0.0024 0.0061 0.0061 60 2.000 0.0190 ± 0.0055 0.0190 0.0190 60 5.000 0.0130 ± 0.0047 0.0130 0.0130 200 0.500 0.0413 ± 0.0128 0.0400 0.0400 200 1.300 0.0214 ± 0.0079 0.0212 0.0212 650 1.300 0.0243 ± 0.0124 0.0238 0.0238 650 3.200 0.0125 ± 0.0055 0.0125 0.0125 2 2 from Eq. (18) and at NLO from Eq. (11) with ?2 = 4m2 i , Q + 4mi . We observe from Fig. 1 that the NLO predictions are rather stable under scale variations and practically coincide with the LO ones in the lower Q2 regime. On the other hand, for Q2 ? 4m2 i , the NLO predictions overshoot the LO ones and exhibit an appreciable scale dependence. We encounter the notion that the ?xed-?avour-number scheme used here for convenience is bound to break down in the large-Q2 regime due to unresummed large logarithms of the form ln(Q2 /m2 i ). In our case, such logarithms do appear linearly at LO and quadratically at NLO. In the standard massless factorization, such terms are responsible for the Q2 evolution of the PDFs and do not contribute to the coe?cient functions. In fact, in the variable-?avour-number scheme, they are MS-subtracted from the coe?cient functions and absorbed into the Q2 evolution of the PDFs. Thereafter, the asymptotic large-Q2 dependences of Ri at NLO should be proportional to αs (Q2 ) and thus decreasing. This is familiar from the Callan-Gross ratio R = FL /(F2 ? FL ), as may be seen from its (x, Q2 ) parameterizations in Ref. [22]. Fortunately, this large-Q2 problem does not a?ect our results in Tables 1 and 2 because the bulk of the H1 data is located in the range of moderate Q2 values. Furthermore, Ri enters Eq. (2) with the suppression factor y 2 /[1 + (1 ? y )2]. The ratio Rc was previously studied in the framework of the kt -factorization approach [17] and found to weakly depend on the choice of unintegrated gluon PDF and to be approximately x independent in the small-x regime (see Fig. 8 in Ref. [17]). Both features are inherent in our approach, as may be seen at one glance from Eq. (11). The prediction for Rc from Ref. [17], which is included in Fig. 1 for comparison, agrees well with our results in the lower Q2 range, but it continues to rise with Q2 , while our results reach maxima, beyond which they fall. In fact, the kt -factorization approach is likely to overestimate Rc 2 2 for Q2 ? 4m2 i , due to the unresummed large logarithms of the form ln(Q /mi ) discussed above.

7

Figure 1: Rc , Rb , and Rt evaluated as functions of Q2 at LO from Eq. (18) (dot-dashed 2 2 2 lines) and at NLO from Eq. (11) with ?2 = 4m2 i (dashed lines) and ? = Q + 4mi (solid lines). For comparison, the prediction for Rc in the kt -factorization approach (dot-dotdashed line) [17] is also shown.

8

6

Conclusions

i i In this letter, we derived a compact formula for the ratio Ri = FL /F2 of the heavy?avour contributions to the proton structure functions F2 and FL valid through NLO at small values of Bjorken’s x variable. We demonstrated the usefulness of this formula by c b extracting F2 and F2 from the doubly di?erential cross section of DIS recently measured by the H1 Collaboration [2, 3] at HERA. Our results agree with those extracted in Refs. [2, 3] well within errors. In the Q2 range probed by the H1 data, our NLO predictions agree very well with the LO ones and are rather stable under scale variations. Since we worked in the ?xed-?avour-number scheme, our results are bound to break down for Q2 ? 4m2 i, which manifests itself by appreciable QCD correction factors and scale dependences. As is well known, this problem is conveniently solved by adopting the variable-?avour-number scheme, which we leave for future work. Our approach also simply explains the feeble dependence of Ri on x and the details of the PDFs in the small-x regime.

Acknowledgements
We are grateful to Sergei Chekanov, Vladimir Chekelian, Achim Geiser, Leonid Gladilin, and Zakaria Merebashvili for useful discussions. A.Yu.I. is grateful to the Scuola Internazionale Superiore di Studi Avanzati (SISSA), where most of his work has been done. A.V.K. was supported in part by the Alexander von Humboldt Foundation and the Heisenberg-Landau Programme. This work was supported in part by BMBF Grant No. 05 HT4GUA/4, HGF Grant No. NG–VH–008, DFG Grant No. KN 365/7–1, and RFBR Grant No. 07-02-01046-a.

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