On the role of heavy flavor parton distributions at high energy colliders

We compare `fixed flavor number scheme' (FFNS) and `variable flavor number scheme' (VFNS) parton model predictions at high energy colliders. Based on our recent LO- and NLO-FFNS dynamical parton distributions, we generate radiatively two sets of VFNS parton distributions where also the heavy quark flavors h=c,b,t are considered as massless partons within the nucleon. By studying the role of these distributions in the production of heavy particles (h\bar{h}, t\bar{b}, hW^{+-}, Higgs--bosons, etc.) at high energy ep, p\bar{p} and pp colliders, we show that the VFNS predictions are compatible with the FFNS ones (to within about 10-20% at LHC, depending on the process) when the invariant mass of the produced system far exceeds the mass of the participating heavy quark flavor.

In a recent publication [1] we updated the dynamical leading order (LO) and next-toleading order (NLO) parton distributions of [2]. These analyses were undertaken within the framework of the so called 'fixed flavor number scheme' (FFNS) where, besides the gluon, only the light quarks q = u, d, s are considered as genuine, i.e. massless partons within the nucleon. This factorization scheme is fully predictive in the heavy quark h = c, b, t sector where the heavy quark flavors are produced entirely perturbatively from the initial light quarks and gluons -as required experimentally, in particular in the threshold region.
However, even for very large values of Q 2 , Q 2 m 2 c,b , these FFNS predictions are in remarkable agreement with DIS data [1,2] and, moreover, are perturbatively stable, despite the common belief that 'non-collinear' logarithms ln(Q 2 /m 2 h ) have to be resummed.
In many situations, however, calculations within this factorization scheme become unduly complicated. For example, the single top production process at hadron colliders via W -gluon fusion requires the calculation of the subprocess ug → d tb at LO and of ug → d tb g etc. at NLO. It thus becomes expedient to consider for such calculations the so called 'variable flavor number scheme' (VFNS) where also the heavy quarks h = c, b, t are taken to be massless partons within the nucleon. In this scheme, the above FFNS calculations simplify considerably, i.e. one needs merely ub → dt at LO and ub → dtg etc. at the NLO of perturbation theory [3]. The VFNS is characterized by increasing the number n f of massless partons by one unit at Q 2 = m 2 h starting from n f = 3 at Q 2 = m 2 c , i.e. c(x, m 2 c ) =c(x, m 2 c ) = 0. The matching conditions at LO and NLO are fixed by continuity relations [4] at the respective thresholds Q 2 = m 2 h . Thus the 'heavy' n f > 3 quark distributions are perturbatively uniquely generated from the n f −1 ones via the massless renormalization group Q 2 -evolutions. The running strong coupling can be approximated by the common NLO 'asymptotic' solution with β 0 = 11 − 2n f /3 and β 1 = 102 − 38n f /3, which turns out to be sufficiently accurate [1] for our relevant Q 2 -region, Q 2 > ∼ 2 GeV 2 . Since β 0,1 are not continuous for different flavor numbers n f , the continuity of α s (Q 2 ) requires to choose different values for the integration constant Λ for different n f , Λ (n f ) , which are fixed by the common α s (Q 2 ) matchings at the should become insensitive to this, somewhat arbitrary, input selection [5] whose virtues are the fulfillment of the standard sum-rule constraints together with reasonable shapes and sizes of the various input distributions. As we shall see, this expectation is based on the fact that at Q 2 m 2 h the VFNS distributions are dominated by their radiative evolution rather than by the specific input at Q 2 = m 2 h . In other words, because of the long evolution distance, input differences get 'evolved away' at Q 2 m 2 h where the universal perturbative QCD splittings dominate.
As a first test of the VFNS 'heavy' quark distributions we consider charm and bottom electroproduction processes, since deep inelastic structure functions play an instrumental role in determining parton distributions. In Figs. 1 and 2 we compare the VFNS with the FFNS predictions for F c 2 (x, Q 2 ) and F b 2 (x, Q 2 ), respectively, using 1 µ 2 = Q 2 + 4m 2 c,b for the FFNS although our results are not very sensitive to this specific choice of the factorization and renormalization scales. As usual, µ 2 = Q 2 in the VFNS. Notice furthermore that the NLO-VFNS predictions for xh (short-dashed curves) are very similar to the ones for the O(α s ) quark and gluon contributions almost cancel. As expected [6] the discrepancies between the predictions for xh(x, Q 2 ) in the VFNS and for (2e 2 h ) −1 F h 2 (x, Q 2 ) in the FFNS in the relevant kinematic region (small x, large Q 2 ) never disappear and can amount to as much as about 30% at very small-x, even at W 2 ≡ Q 2 ( 1 x − 1) far above threshold, i.e. W 2 W 2 th = (2m h ) 2 . This is due to the fact that here the ratio of the threshold energy W th ≡ √ŝ th of the massive subprocess (γ * g → hh, etc.) and the mass of the produced heavy quark √ŝ th /m h = 2 is not sufficiently high to exclude significant contributions from the threshold region. Even for the lightest heavy quark, h = c, such non-relativistic GeV 2 due to significant β c < 0.9 contributions, and the situation becomes worse for h = b (cf. Fig. 4 of [6]). This is in contrast to processes where one of the produced particles is much heavier than the other one, like the weak CC contribution [7,8]  in the former case of hh production. Thus the single top production rate in W + g → tb is dominated by the (beyond-threshold) relativistic region where βb > 0.9 and therefore is expected to be well approximated by W + b → t where b is an effective massless parton within the nucleon. In Fig. 3 we compare the LO FFNS [7,8] predictions for 1 2 F CC 2,tb (x, Q 2 ) with the corresponding VFNS ones for ξb(ξ, Q 2 +m 2 t ) where the latter refers to the W + b → t transition using the slow rescaling variable [9] (Notice that the fully massive NLO FFNS QCD corrections to W + g → tb are unfortunately not available in the literature.) As expected the differences between the two schemes are here less pronounced than in the case of cc and bb electroproduction in Figs. 1 and 2. These results indicate that one may resort to the simpler VFNS with its massless h(x, µ 2 ) distributions to estimate rather reliably the production rates of heavy quarks, gauge bosons, Higgs scalars, etc. at Tevatron and LHC energies.
As a next test of these VFNS distributions we therefore turn to hadronic W ± production and present in Fig. 4 their NLO predictions for σ(pp → W ± X) as compared to the data [10,11,12,13,14] and to predictions based on the NLO CTEQ6.5 distributions [15]. Also shown in this figure is a comparison of our LO FFNS and VFNS predictions. Although quantitatively slightly different, the dominant light quark contributions in the FFNS (ud → W + , us → W + , etc.) are due to the same subprocesses as in the VFNS, but the relevant heavy quark contributions have been calculated via gs(d) →cW + , gu → bW + , etc. as compared to cs(d) → W + ,bu → W + etc. in the VFNS. Here we again expect the VFNS with its effective massless 'heavy' quark distributions h(x, µ 2 ) to be adequate, since nonrelativistic contributions from the threshold region in the FFNS are suppressed due to The LO gluon induced heavy quark contributions to W ± production in the FFNS are obtained from a straightforward calculation of the differential cross section [16] dσ(ŝ)/dt which yieldŝ LO )] and the relevant CKM matrix element(s) V qq are taken from [17]. The corresponding total W ± hadronic production cross section relevant for Fig. 4 is then given by have been presented in [18], but questioned in [19]. Here we just mention that we fully confirm the LO results for W t production obtained in [19] at Tevatron and LHC energies.
Taking into account that the K ≡ NLO/LO factor is expected [19] to be in the range of 1.2 -1.3, our LO-FFNS predictions in Fig. 4 imply equally agreeable NLO predictions as the (massless quark) NLO-VFNS ones [20] shown by the solid and dashed-dotted curves in Fig. 4.
In Table 1 Table 1 for obtaining the total NLO-FFNS predictions without committing any significant error. The resulting total rate for W + +W − production at LHC of 192.7 ± 4.7 nb is comparable to our NLO-VFNS prediction in Table 1  uncertainty estimates of our NLO predictions at LHC: and, for completeness, at LO where the subscript pdf refers to the 1σ uncertainties of our parton distribution functions [1]. For comparison, the NLO-VFNS prediction of CTEQ6.5 [15] is 202 nb with an uncertainty of 8%, taking into account a pdf uncertainty of slightly more than 2σ. Similarly, MRST [21] predict about 194 nb. From these results we conclude that, for the time being, the total W ± production rate at LHC can be safely predicted within an uncertainty of about 10% irrespective of the factorization scheme.
It is also interesting to study the dependence of the FFNS predictions for the contribu- other VFNS distributions, e.g., those of [5]. This is illustrated more quantitatively in Fig. 7 where we compare our c-and b-distributions, together with the important gluondistribution, with the ones of CTEQ6 [26] and CTEQ6.5 [15] in the sea-and gluon-relevant The ratios for the light u-and d-distributions are even closer to 1 than the ones shown in Fig. 7, typically between 0.95 and 1.05 which holds in particular for the CTEQ6 distributions when compared to our ones. Incidentally the VFNS under consideration and commonly used [5,26] is also referred to as the zero-mass VFNS. Sometimes one uses an improvement on this, now known as the general-mass VFNS [15,21,27,28,29,30], where mass-dependent corrections are maintained in the hard cross sections. This latter factorization scheme interpolates between the strict zero-mass VFNS, used in our evolution to Q 2 m 2 h , and the (experimentally required) FFNS used for our input at Q 2 = m 2 h . As expected and shown in Fig. 7, scheme (input) differences at lower Q 2 = O(m 2 h ) only marginally affect the asymptotic results at Q 2 = M 2 W m 2 c,b where the CTEQ6.5 parametrizations [15] (corresponding to a general-mass VFNS) become very similar to the ones of CTEQ6 [26] and our GJR-VFNS (corresponding to the zero-mass VFNS). As stated repeatedly before, this is essentially due to the dominance of the large evolution effects over the minor differences involved at the lower scales, e.g. at   (4) and (5). The total NLO-FFNS rates have been obtained by adopting an expected [19] K-factor of 1.2 for the subleading gluon initiated LO rates involving the heavy c and b quarks.   : Predictions for the total W + + W − production rates at pp colliders with the data taken from [10,11,12,13,14]. The LO and NLO GJR parton distributions in the VFNS have been generated from the FFNS ones [1] as described in the text. The NLO-VFNS CTEQ6.5 distributions are taken from [15]. The adopted momentum scale is µ R = µ F ≡ µ = M W . The scale uncertainty of our NLO GJR predictions, due to varying µ according to 1 2 M W ≤ µ ≤ 2M W , amounts to less than 2% at √ s = 1.96 TeV, for example. The shaded region around our central GJR predictions is due to the ±1σ uncertainty implied by our dynamical NLO parton distributions [1].