Large-x structure of physical evolution kernels in Deep Inelastic Scattering

The modified evolution equation for parton distributions of Dokshitzer, Marchesini and Salam is extended to non-singlet Deep Inelastic Scattering coefficient functions and the physical evolution kernels which govern their scaling violation. Considering the x->1 limit, it is found that the leading next-to-eikonal logarithmic contributions to the physical kernels at any loop order can be expressed in term of the one-loop cusp anomalous dimension, a result which can presumably be extended to all orders in (1-x), and has eluded so far threshold resummation. Similar results are shown to hold for fragmentation functions in semi-inclusive e+ e- annihilation. Gribov-Lipatov relation is found to be satisfied by the leading logarithmic part of the modified physical evolution kernels.

In [15], the result for the leading contribution to this quantity in the x → 1 limit was derived, which resums all logarithms at the leading eikonal level, and nicely summarizes analytically in momentum space the standard results [16,17] of threshold resummation: where r = 1−x x (with rQ 2 ≡ W 2 the final state "jet" mass), B DIS δ (a s ) is related to the the quark form factor, and J (Q 2 ), the "physical Sudakov anomalous dimension" (a renormalization scheme invariant quantity), is given by: is the universal "cusp" anomalous dimension [18] (see also [19]), with a s ≡ αs 4π the M S coupling, is the beta function (with β 0 = 11 3 C A − 2 3 n f ) and is the usual final state "jet function" anomalous dimension. It should be noted that j 1 = A 1 (the one loop cusp anomalous dimension), and also that both A(a s ) and B(a s ) (in contrast to J (Q 2 )) are renormalization scheme-dependent quantities. The renormalization group invariance of J (Q 2 ) yields the standard relation: where L x ≡ ln(1−x) and a s = a s (Q 2 ), from which the structure of all the eikonal logarithms in K(x, a s (Q 2 )) can be derived. A term like L p x 1−x arising from J (rQ 2 ) r in eq.(1.2) must be interpreted as usual as a standard +-distribution. All the eikonal logarithms are thus absorbed into the single scale (1 − x)Q 2 (see also [20][21][22] and section VI-E in [11]). However, no analogous result holds [6] at the next-to-eikonal level (except [2] at largeβ 0 ). In this note, I show that the leading next-to-eikonal logarithmic contributions to the physical evolution kernel at a given order in a s can actually be determined in term of lower order leading eikonal coefficients, representing the first step towards threshold resummation at the next-to-eikonal level. This result is obtained by extending the approach of [23,24] (which deals with parton distributions) to the DIS coefficient functions themselves.

The modified physical kernel
I consider the class of modified physical evolution equations: where for book-keeping purposes I introduced the parameter λ, which shall eventually be set to its physically meaningful value λ = 1, in straightforward analogy to the modified evolution equation for parton distributions of [24]. I note that K(x, a s , λ = 0) ≡ K(x, a s ), the 'standard' physical evolution kernel. Eq.(2.1) allows to determine K(x, a s , λ) given K(x, a s ) (or vice-versa). Indeed, expanding C 2 (y, Q 2 /z λ , µ 2 F ) around z = 1, keeping the other two variables fixed, and reporting into eq.(2.1), one easily derives the following relation between K(x, a s , λ) and K(x, a s ): where only terms with a single overall factor of λ need actually to be kept up to next-toeikonal order, since one can check terms with more factors of λ, which are associated to more factors of ln x, are not relevant to determine the next-to-eikonal logarithms in the physical kernel. In the rest of the paper (except section 4) I shall therefore simply use: Eq.(2.3) can be solved perturbatively. Setting: (and similarly for K(x, a s )), one gets: The K i (x)'s are determined in term of splitting functions and coefficient functions as follows [14]: . Consider now the x → 1 limit. The one-loop splitting function is given by [25]: with A 1 = 4C F , and 1 : Moreover, at the next-to-eikonal level we have, dropping from now on δ function contributions: with [26]: Also: From eq.(2.6) one can derive [6,7] the following expansions for x → 1:

Two loop kernel
From eq.(2.6), (2.7), (2.9) and (2.11) one deduces: Then eq.(2.5) yields for x → 1: Thus, setting λ = 1, one finds that the leading next-to-eikonal logarithm in K 1 (x, λ = 1) vanishes, yielding the relation: which is correct [6,7]. This finding is not surprising: up to two loop, the leading next-toeikonal logarithm is contributed only by the splitting function, since b 11 = 0 (e.g. h 21 = C 2 ), and one effectively recovers the result (eq.(2.10)) holding [24] for the two loop splitting function. The situation however changes drastically at three loop, where the leading nextto-eikonal logarithm is contributed by the coefficient function rather then the splitting function, and the crucial question is whether the leading next-to-eikonal logarithm still vanishes for λ = 1.

Three loop kernel
Eq.(2.5) yields for x → 1: Requiring h 32 (λ), the coefficient of the O(L 2 x ) term, to vanish for λ = 1 predicts: which is indeed the correct [6,7] value. I stress that this result is not a consequence of the relation [24,27,28] C 3 = 2A 1 A 2 for P 2 (x). Indeed it is well-known [29] that the P i (x)'s, and in particular P 2 (x), have only a single next-to-eikonal logarithm: and thus P 2 (x) cannot contribute to the double logarithm in K 2 (x). Rather, h 32 is contributed by the coefficient functions in eq.(2.6), and eq.(3.5) yields a prediction for the O(L 2 x ) term in c 2 (x).

Five loop kernel
One can similarly predict the leading next-to-eikonal logarithm in the five loop physical kernel (which depends on the four loop coefficient function). Using eq.(2.3), the coefficient of the O(L 4 x ) term in K 4 (x, λ) is found to be given by: where k 43 = −A 1 β 3 0 (again consistent with eq. (1.7)). Requiring this coefficient to vanish for λ = 1 predicts 2 : (3.10)
One can further show [30] that the moment space functional relation which accounts for leading logarithms at all orders in (1 − x) is: (3.14) Eq.(3.12) results from expanding the right hand side of eq.(3.14) to first order in ∆N ≡ λK(N, a s ). It is interesting that eq.(3.14) is identical to the functional relation 3 obtained [28,32] for the splitting functions in the conformal limit (where the splitting functions coincide with the K i 's).

Leading next-to-next-to-eikonal logarithms
It can be checked [30] that similar methods allow to predict using eq.(2.2) the leading logarithmic contributions at the next-to-next-to-eikonal level, i.e. the coefficient of the . The crucial new point, however, is that the leading term in the eikonal expansion has to be defined in term of the one-loop splitting function prefactor p qq (x) (eq.(2.8)), instead of 1/r as in eq.(2.12). Namely, keeping only leading logarithms at each eikonal order, the predicted f c ji coefficients (j = i + 1, i ≥ 0) are defined 4 by: which are seen to be correct using eq.(3.26) in [7]. The latter equation also makes it likely that similar leading logarithmic predictions can be obtained to any order in (1 − x), using the same prefactor p qq (x) as in eq.(4.1) to define the leading term in the eikonal expansion. Indeed, one derives for instance [30] the O((1 − x)) 2 coefficients in eq.(4.1) (with g 10 = 0): which are correct [7]. I note that f c 21 and g 21 coincide (like h 21 ) with the splitting functions contributions.

Fragmentation functions in e + e − annihilation
Similar results hold for physical evolution kernels associated to fragmentation functions in semi-inclusive e + e − annihilation (SIA), provided one sets λ = −1 in the analogue of eq.(2.1): where C T denotes a generic non-singlet SIA coefficient function. I first note that threshold resummation in this case [33] leads at the leading eikonal level to an equation similar to eq.(1.2): where x should now be identified to Feynman-x rather then Bjorken-x, and I used the results of [34] which imply that the "physical Sudakov anomalous dimension" J (Q 2 ) is the same for structure and fragmentation functions. The statement above eq.(5.1) then follows from the following two observations: i) The predictions in eq.(3.3), (3.5), (3.8) and (3.10) depend only upon coefficients of leading eikonal logarithms in the physical evolution kernels. ii) Eq.(3.26) in [7] shows that the latter coefficients are identical for deep-inelastic structure functions and for e + e − fragmentation functions (consistently with the remark below eq.(5.2)), but that the coefficients of the leading next-to-eikonal logarithms are equal only up to a sign change (in an expansion in 1/r) between deep-inelastic structure functions and fragmentation functions.

Conclusion
A modified 5 evolution equation for DIS non-singlet structure functions, analoguous to the one used in [24] for parton distributions, but which deals with the physical scaling violation and coefficient functions, has been proposed. It allows to relate the leading nextto-eikonal logarithmic contributions in the momentum space physical evolution kernel to coefficients of leading eikonal logarithms at lower loop order (depending only upon the one-loop cusp anomalous dimension A 1 ), which represents the first step towards threshold resummation at the next-to-eikonal level. This result also explains the observed [6,7] universality of the leading next-to-eikonal logarithmic contributions to the physical kernels of the various non-singlet structure functions, linking them to the known [35] universality of the eikonal contributions. Similar results hold at the next-to-next-to-eikonal level with a proper definition of the leading eikonal piece, and can presumably be extended to leading logarithmic contributions at all orders in (1 − x). Analogous results are obtained for fragmentation functions in semi-inclusive e + e − annihilation.
One may ask to what extent the success of the present approach may be attributed, as suggested in [24,32] for the splitting functions case, to the classical nature [36] of soft radiation. In fact, the main result of this paper for the (modified) DIS physical evolution kernel can be summarized (barring the δ-function contribution) by the following equation: where the second term (the "subleading logarithms") is contributed by all powers in (1− x) except the leading eikonal one. The first term in eq.(6.1) accounts for the leading logarithmic contributions to the modified kernel (together with some subleading logarithms) to all powers in (1 − x) at any given loop order, and implies leading logarithmic contributions are actually absent beyond O(1 − x) power. This term has the remarkable effective oneloop splitting function form 4C F a phys (1 − x)Q 2 ) p qq (x), with the "physical coupling" a phys (Q 2 ) ≡ 1 4C F J (Q 2 ). As pointed out in [32], the x 1−x part of the one-loop prefactor (eq.(2.8)) should be interpreted as corresponding to universal classical radiation, a QCD manifestation of the Low-Burnett-Kroll theorem [36], while the 1 − x part represents a genuine quantum contribution. Now, it is clear that at the next-to-eikonal level, the 1 − x part of the prefactor is irrelevant: only the "classical" 1/r part is required to separate those leading logarithms in the standard (λ = 0) physical evolution kernel which are correctly predicted in the present approach (the h ji in eq.(4.1)), hence "inherited" in the sense of [32], from the "primordial" ones (those which at each loop order carry the same color factors as the leading O(1/(1−x)) eikonal logarithms, and can thus be absorbed into the definition of the leading term). However, it appears from the results of section 4 that, at next-to-next-to-eikonal level, the full one-loop prefactor has to be used into the definition of the leading term to properly isolate the "inherited" next-to-next-to-eikonal logarithms (the f c ji in eq.(4.1)). Moreover, although the "inherited" f c ji are purely "classical" (like the h ji ), the "inherited" g ji at the O((1 − x) 2 ) level are a mixture of "quantum" and "classical". Indeed, setting f q 10 = 1 2 k 10 and f q 21 = 1 2 k 21 (the "quantum parts" of the O(1 − x) coefficients), one finds g 21 = g q 21 + g c 21 , with g q 21 = k 10 f q 10 = 1 2 k 2 10 and g c 21 = − 1 6 k 2 10 ; and g 32 = g q 32 + g c 32 , with g q 32 = 1 2 k 21 f q 10 + k 10 f q 21 = 3 4 k 21 k 10 and g c 32 = − 1 4 k 21 k 10 . In both cases g c ji = − 1 3 g q ji , which shows the "inherited" g ji coefficients are actually dominantly "quantum".
It can be further checked [30] that the very same first term in eq.(6.1) also accounts for the leading logarithmic contributions to the λ = −1 modified SIA physical evolution kernel to all powers in (1− x), which implies that the leading logarithmic parts of the modified DIS and SIA physical evolution kernels satisfy Gribov-Lipatov relation [37], namely we have: where J (1 − x)Q 2 LL = A a s ((1 − x)Q 2 ) LL = A 1 as(Q 2 ) 1+as(Q 2 )β 0 Lx is the leading logarithmic contribution to eq.(1.7). Indeed, once transformed back to the standard (λ = 0) physical kernels, eq.(6.2) is consistent with eq.(3.26) in [7] at least to next-to-next-to-eikonal order, and is probably correct to all orders in (1 − x) (with identically vanishing contributions beyond O(1 − x) order). On the other hand, contrary to the splitting functions case where it has been checked up to three loops [32,38], a full Gribov-Lipatov relation K(x, a s , λ = 1) = K T (x, a s , λ = −1) does not seem to hold for subleading logarithms beyond the leading eikonal level.
The resummation of the subleading logarithmic contributions at next-to-eikonal order in eq.(6.1), not adressed here, remains an open issue: the present method does not work for them, except in the conformal limit, where one recovers the results of [24].