Bjorken sum rule in QCD with analytic coupling

We present details of study of the Bjorken polarized sum rule carried out recently in [1] within the range of energies where the data were collected by JLAB collaboration, 0.05 GeV2<Q2<3 GeV2. Three approaches to QCD with analytic (holomorphic) coupling are considered: Analytic Perturbation Theory (APT), Two-delta analytic QCD, and Three-delta lattice-motivated analytic QCD in the three-loop and four-loop MiniMOM schemes. The new frameworks with respective couplings give results which agree well with the experimental data for 0.5 GeV2<Q2<3GeV2 already when only one higher-twist term is taken into account.

For the evaluation of the LT contribution we consider, in addition to pQCD, analytic frameworks of QCD (anQCD) which are a useful tool to evaluate physical quantities at lowmomentum transfer. In these anQCD frameworks the running coupling has no spurious (Landau) singularities, unlike pQCD in the usual MS-like scheme. PQCD coupling a(Q 2 ) pt ≡ α s (Q 2 )/π possesses Landau singularities at small momenta |Q 2 | 1 GeV 2 , while the general principles of quantum field theories (QFT) require that the spacelike QCD observables D(Q 2 ), such as current correlators and structure functions, be holomorphic (analytic) functions of Q 2 throughout an entire generalized spacelike region Q 2 ∈ C\(−∞, 0]. The usual pQCD couplings a pt (Q 2 ) do not reflect these properties, hence the LT part of D(Q 2 ) evaluated in terms of a pt (Q 2 ) (as a truncated perturbation series) does not have these properties dictated by QFT. However, in anQCD we have a pt (Q 2 ) → A(Q 2 ), where A(Q 2 ) is the anQCD coupling holomorphic in Q 2 ∈ C\(−∞, 0]. Therefore, the evaluation of the LT contribution D eval. (Q 2 ) → F(A(Q 2 )) has the correct holomorphic properties (the same is true for the HT contribution).
Other considered analytic frameworks are the Two-delta analytic QCD (2δanQCD [13]) and the (lattice-motivated) Three-delta analytic QCD (3δanQCD [14,15]). They are less closely (than APT) based on the underlying pQCD coupling a pt (Q 2 ): the equality Im A(−σ − i ) = ρ (pt) 1 (σ) is enforced only for sufficiently large σ ≥ M 2 0 (where M 0 ∼ 1 GeV is a "pQCDonset" scale). For 0 < σ < M 2 0 , the otherwise unknown discontinuity function ρ 1 (σ) ≡ Im A(Q 2 = −σ − i ) is parametrized by two or three delta functions, respectively. Such a parametrization is partly motivated by the Padé approximant approach to the coupling A(Q 2 ). The renormalization schemes of the underlying pQCD coupling in 2δanQCD are constrained by the requirement that acceptable values of M 0 ∼ 1 GeV and A(0) ∼ 1 are obtained [13,16,17]. In 3δanQCD, the lattice calculations provide two conditions for the coupling at very low Q 2 < 1 GeV 2 , which give two additional constraints on the delta functions. In both 2δanQCD and 3δanQCD the coupling A(Q 2 ) practically agrees with the underlying pQCD coupling a pt ( The construction of the analytic analogs A n (Q 2 ) of powers a pt (Q 2 ) n , for general anQCD, was formulated for integer n in [18], and for general (noninteger) n in [19].

Bjorken sum rule
is the nonsinglet combination given by the difference between the proton and neutron polarized structure functions and integrated over the entire Bjorken-x interval This is the first moment of the nonsinglet contribution to the polarized structure functions. BSR can be written in terms of a sum of two terms, one coming from pQCD as an expansion of the running coupling a pt (Q 2 ) = α s (Q 2 )/π and the other is the twist-4 contribution dictated by the OPE [2] In the limit Q 2 → ∞ we have Γ p−n 1 (∞) = g A /6, where g A is the nucleon axial charge, g A = 1.2723 ± 0.0023 [22]. Higher-twist contributions are neglected here.
If we use the OPE formalism, the elastic contribution (at x = 1) to BSR (1) should in principle be included. It is convenient to exclude the elastic contribution, since the Q 2 -dependence of the nonsinglet inelastic BSR at low Q 2 is constrained by the Gerasimov-Drell-Hearn (GDH) sum rule [9], as was highlighted in [23]. Further, at high Q 2 the elastic contribution is not noticable [23]. Hence we investigate the behavior of the pure inelastic contribution as a continuation to low-energy regime [24].
The twist-2 contribution E NS (Q 2 ) in (2) is known up to N 3 LO [7]: We will evaluate this in analytic QCD frameworks, where the perturbation series (3) must be expressed as a nonpower series via the transformation a(Q 2 ) n → A (j) n (Q 2 ), where index j indicates the analytic QCD framework (j =APT, 2δanQCD, and 3δanQCD). For the numerical evaluation of A (j) n (Q 2 )'s, various programs [13,14,15,17] written in Mathematica are used. 2 We will use N f = 3 and various renormalization scales (RScl) µ 2 = Q 2 in the evaluation of the quantity E NS,j (Q 2 ). In such a case, the dependence on the RScl parameter C ≡ ln(µ 2 /Q 2 ) enters the coefficients e NS j → e NS j (C) and the couplings A (j) n (Q 2 exp(C)). The last term in BSR (2) has known evolution [25] in pQCD and, consequently, in general analytic versions These contributions are important in the low-energy regime Q 2 ∼ 1 GeV 2 . At very low Q 2 < 1 GeV 2 , they grow quickly and the OPE series diverges. This is a general problem in OPE. However, we can address this problem by replacing the OPE expression (2) at very low Q 2 < Q 2 0 (≈ 0.5 GeV 2 ) with a χPT-motivated expression [4] 3 : Here, κ X is the anomalous moment of the nucleon X (κ p = 1.793, κ n = 1.916), A and B are fit parameters. The first term (∼ Q 2 ) originates from the Gerasimov-Drell-Hearn sum rule [9]. At higher Q 2 (Q 2 0.5 GeV 2 ), we use OPE (2).

Numerical Results
Here we first test only the OPE approach (2) with pQCD MS LT term. We search at each order for a minimum scale Q 2 = Q 2 min where χ 2 is minimal, while the other fit parameter is µ p−n 4 . In order to fix the MS QCD scale Λ, we perform the standard extraction, i.e., Λ (N f =3) value is obtained from a reference value, a(Q 2 = M 2 Z ) = 0.1185/π [27]. The RGE evolution of a pt (Q 2 ) down from is carried out with four-loop MS beta function and with the corresponding three-loop quark threshold conditions [28] (at thresholds κm q (m q ) with κ = 1).
In Table 1 we present the obtained values of µ p−n 4,pQCD (only the statistical errors were considered), and Q 2 min , to various orders in the perturbation expansion (3). As noted in previous works [23,29,30], a duality between HT contribution and the order of a perturbation series appears: when we use higher order in pQCD, the HT contribution becomes smaller in its absolute value. But this apparent property is unstable, because the µ p−n 4,pQCD coefficient is very sensitive to Λ (pQCD) parameter at N 3 LO [23]. The extracted values are consistent with those obtained at LO in [31], and at NLO in [32].
In Fig. 1 we show the pQCD fit of BSR function Γ p−n 1 at NLO, N 2 LO and N 3 LO. Increasing the perturbation order, the range of applicability of pQCD becomes smaller, covering fewer points of data in the low-Q 2 region. as a function of Q 2 , to various orders of perturbation series (2). The newer data [5] have small statistical errors and are in black, and the older data [3,4,6] are in light grey (orange online). Now we employ analytic (holomorphic) QCD approaches in the fits. In Table 2 we show, for five different evaluations of the LT contribution E NS (Q 2 ), the resulting values of the fit parameters: HT coefficient µ p−n 4 (Q 2 in ); RScl parameter C ≡ ln(µ 2 /Q 2 ) of the LT contribution; matching point scale Q 2 0 ; parameter A in the χPT-motivated expression (5). The values of the B parameter of the χPT-motivated expression were extracted by the matching condition at Q 2 = Q 2 0 . The last column gives the values of χ 2 for the obtained curves. The resulting values of A are approximately consistent with the value A = 0.74 obtained in χPT calculations in [34] but not with the value A = 2.4 obtained in [35].
The obtained curves are shown in Fig. 2. As mentioned, these curves consist of two curves "glued together" at a matching point Q 2 0 : the OPE curve (2) for Q 2 ≥ Q 2 0 and the χPT-motivated curve (5) for Q 2 ≤ Q 2 0 . These curves indicate that the pQCD MS approach and, to a lesser extent, the (F)APT approach, give a visible slope discontinuity (kink) at the matching point Q 2 = Q 2 0 between the OPE and the χPT-motivated expression, i.e., they cannot bridge well the gap between the high and low-Q 2 regimes. However, 2δanQCD and 3δanQCD are able to bridge this gap without a visible kink, cf. Fig. 2.

Conclusions
In [1] we investigated the Bjorken polarized sum rule (BSR) Γ p−n 1 (Q 2 ) (with the elastic contribution excluded) as a function of squared momentum transfer Q 2 , at Q 2 ≤ 3 GeV 2 in various QCD approaches, and compared it with the available experimental data. For Q 2 ≥ Q 2 0 (≈ 0.5 GeV 2 ) we used the theoretical expressions for the leading-twist (LT) contribution plus one higher-twist (HT) term µ p−n 4 /Q 2 , Eqs. (3) and (2). For Q 2 ≤ Q 2 0 , the χPT-motivated expression (5) was used. The fit parameters were the HT coefficient µ p−n 4 (Q 2 in ) (at Q 2 in = 1 GeV 2 ), the renormalization scale (RScl) parameter C ≡ ln(µ 2 /Q 2 ) in the LT contribution, the transition scale Q 2 0 , and the free parameter A in the χPT-motivated expression (5). For the evaluation of the LT contribution at Q 2 ≥ Q 2 0 we used the usual MS pQCD, and various QCD versions with holomorphic infrared-safe coupling A(Q 2 ): (F)APT [10]; 2δanQCD [13,16,17]; and a latticemotivated 3δanQCD coupling in the three-and four-loop lattice MiniMOM scheme: 3l3δanQCD [14] and 4l3δanQCD [15]. It turned out that the latter three holomorphic (analytic) QCD versions give the best fit results and the lowest values of Q 2 0 ≈ 0.4-0.5 GeV 2 and χ 2 ≈ 5.0-5.5. (F)APT requires a higher value Q 2 0 ≈ 0.63 GeV 2 and gives χ 2 ≈ 13.5. The MS pQCD gives the worst results, χ 2 ≈ 24.; the principal reason for this lies in the fact that the MS pQCD coupling a pt (Q 2 ) has Landau singularities at positive Q 2 ≤ 0.37 GeV 2 , and this makes the evaluation of low-Q 2 BSR unreliable.
The presented evaluation of low-Q 2 BSR shows that it is imperative to use QCD frameworks whose couplings have no Landau singularities in this region.