Order $\alpha_s^2$ perturbative QCD corrections to the Gottfried sum rule

The order $\alpha_s^2$ perturbative QCD correction to the Gottfried sum rule is obtained. The result is based on numerical calculation of the order $\alpha_s^2$ contribution to the coefficient function and on the new estimate of the three-loop anomalous dimension term. The correction found is negative and rather small. Therefore it does not affect the necessity to introduce flavour-asymmetry between $\bar{u}$ and $\bar{d}$ antiquarks for the description of NMC result for the Gottfried sum rule.


Introduction
One of the still actively discussed problems of deep-inelastic scattering (DIS) is related to the consideration of the Gottfried sum rule [1], namely This sum rule is the first non-single (NS) moment of the difference of F 2 structure functions (SFs) of charged lepton-nucleon DIS which in general has the following definition: An extensive discussion of the current studies of this sum rule was given in the review of Ref. [2]. However, for the sake of completeness, we will remind the existing experimental situation,which is stimulating the continuation of the research of various subjects, related to the Gottfried sum rule.
In fact if the sea is flavour symmetric, namely u=d, one should have However, the most detailed analysis of muon-nucleon DIS data of NMC collaboration gives the following result [3]: It clearly indicates the violation of theoretical expression of Eq.(3) and necessitates more detailed investigations of different effects, related to the Gottfried sum rule. In this Letter we reconsider the question of studying the contributions of α 2 s , corrections to this sum rule previously raised in Ref. [4].

Available perturbative corrections
The status of the O(α s ) perturbative QCD corrections to I GSR was summarised in Ref. [5]. Following this review, we will extend its presentation to the order α 2 s level. It should be stressed that the renormalisation group equation for I GSR contains the anomalous dimension term: where A s = α s /(4π) and The first two scheme-independent coefficients in Eq. (6) are well-known: where C F = 4/3, C A = 3 and n f is the number of active flavours.
The corresponding anomalous dimension function has the canonical expansion However, like in the case of the first moments of SFs of νN DIS, the first coefficient of the NS anomalous dimension function of the first moment γ n=1 0 is identically equal to zero. The difference is starting to manifest itself from the two-loop level, where in order to get the corresponding result in case of anomalous dimension for I GSR it is necessary to make analytical continuation and to use the so-called (+) prescription (see e.g. Ref. [6]). In the case of γ n=1 1 this was done in Ref. [7] and Ref. [8] and result in the following analytical expression γ n=1 where the numerical value of ζ(3)=1.2020569 was taken into account. The perturbative corrections to I GSR can be obtained from the solution of the renormalisation group equation of Eq. (5): where the anomalous dimension term is defined as Since the first coefficient of γ n=1 I GSR is identically zero ( namely γ n=1 0 =0), there is no singularity in AD(A s ) and we can put in Eq. (12) the lower bound of integration δ = 0. In this case we obtain the following expression for the expansion of AD(A s ) up to O(α 2 s )-corrections: The only unknown terms here is the third coefficient γ n=1 2 of the anomalous dimension function γ n=1 I GSR (A s ), which in general is scheme-dependent. In the cases of n f = 3 and n f = 4 the numerical versions of Eq. (13) read where the scheme-dependent expression for γ n=1 2 is still unknown. Its value will be fixed in the next section using the results of calculations in the MS-scheme.
Few words should be added here on the perturbative theory expansion of C(A s ). From the general grounds it should have the following form: As was found in Ref. [9] its first coefficient is zero, namely C n=1 1 = 0. However, as will be shown in the next section the non-zero perturbative theory contribution is appearing at the two-loop level.

Calculations and estimates of the α 2 s contributions
We will start from the calculations of perturbative contribution to the coefficient function C(A s ) at the α 2 s -level. It can be obtained after applying (+) prescription to the results of Ref. [10]. Indeed, the order α 2 s correction to the coefficient function of I GSR is defined by taking the first moment from the sum where the expressions for the functions C (x, 1) were calculated in Ref. [10] and confirmed with the help of another technique in Ref. [11]. Integrating Eq. (17) numerically with arbitrary Casimir operators C A and C F , we obtain the following n f -independent and scheme-independent result Combining now Eq. (14) and Eq. (15) with Eq. (18) we find the following expressions for I GSR : where α s = α s (Q 2 ) is the NLO expression for MS coupling constant.
In order to get the feeling what might be the contribution of the terms proportional to γ n=1 2 we will avoid extrapolation procedure of the values of γ n 2 used in Ref. [4], calculated analytically for even n=2,4,..,14 in the works of Ref. [12]. Indeed, performing extrapolation from the even values of n for the NLO terms γ n 1 of the corresponding anomalous dimension function, we are obtaining the following estimate γ n=1 1 = 28.23, which is 10 times larger that the real value given in Eq. (10). Therefore, the used in Ref. [4] extrapolation procedure is considerably overestimating the value of the coefficient γ n=1 1 . The similar situation can occur in the case of using extrapolation procedure for fixing the value of γ n=1 2 . Indeed, following the ideas of Ref. [4] we get from extrapolation of the known even values for γ n 2 the following estimates: γ n=1 2 ≈ 361 for n f = 3 and γ n=1 2 ≈ 283 for n f = 4, which to our point of view might be unrealistically large.
Keeping in mind that only direct calculation of γ n=1 2 can give the real numerical value of this term, we nevertheless are proposing the following way of fixing uncalculated contribution to γ n=1 I GSR function. We noticed the following numerical pattern of the behaviour of anomalous dimension function for n ≥ 2: γ n 1 /γ n 2 ∼ 0.12 for n f =4 (see Ref. [4] and Ref. [13] especially). We have checked that for n f =3 the similar relation is γ n 1 /γ n 2 ∼ 0.09 for n ≥ 2. Hoping that these relations are also valid in case of n ≥ 1 , we estimate the values for

Comments on violation of the Gottfried sum rule
In the previous section we found that order α 2 s perturbative QCD corrections to the Gottfried sum rule are really small and can not describe violation of the theoretical prediction from its NMC experimental value. This, in turn, lead to the necessity of introduction of the effect of flavour asymmetry of antiquark distributions in the nucleon [3], namely This phenomenological result is important for fixing the corresponding d/u ratio in different sets of parton distribution functions, which are relevant to the LHC physics (for a review see e.g. Ref. [15] ). On the other side the consideration of available E866 data for the Drell-Yan production in proton-proton and proton-deuteron scattering has confirmed the effects of flavour asymmetry. Indeed the analysis of Ref. [14] gave the following number It was also noted in Ref. [14] that it is unlikely to receive additional contribution to Eq. (26)from the region above x = 0.35, since the sea is rather small in this region. However, the contribution to this whole integral from the unmeasured region x ≤ 0.015 is missed. The attempt to fix it was made in Ref. [16] using the extrapolation to small x region. As the result the authors of Ref. [16] suggested the manifestation of substantial contribution of twist-4 1/Q 2 -effects in Eq. (1). Note, that final E866 result is is closer to NMC result, than obtained in Ref. [16] extrapolated value, namely Ref. [16] = 0.09 ± 0.02 .
Therefore, in order to understand the status of their conclusion on the possibility of existence of substantial contribution of the 1/Q 2 -corrections to the Gottfried sum rule it is necessary to be more careful in performing extrapolations to low x-region. It is highly desirable to estimate the effects of higher-twist contributions to the Gottfried sum rule using any concrete model. However, one should keep in mind that there are also some other explanations of the observed deviation from the canonical value 1/3 for the Gottfried sum rule (see Ref. [17] and Ref. [2] for the review of other works on the subject).

Conclusions
In summary: we found non-zero O(α 2 s ) perturbative QCD contributions to the coefficient function of the Gottfried sum rule. We also estimated the effect due to nonzero value of the three-loop contribution to NS anomalous dimension function for n = 1 moment, which as we think is rather small. More detailed result can be obtained after completing the analytical calculations of the three-loop corrections to the NS kernel of DGLAP equation. This work is now in progress (see e.g. Ref. [18]). In any case the value of the α 2 s -correction is dominated by the calculated in this Letter contribution to the coefficient function, which is negative, but also small. Therefore, the existing NMC observation of the flavour asymmetry between d and u antiquarks is surviving. We hope that the possible future new HERA data might be useful for more detailed measurement of the Gottfried sum rule and for further studies of the effect of flavour asymmetry in the d/u ratio.