Heavy-quark form factors in the large β0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _0$$\end{document} limit

Heavy-quark form factors are calculated at β0αs∼1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _0 \alpha _s \sim 1$$\end{document} to all orders in αs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _s$$\end{document} at the first order in 1/β0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\beta _0$$\end{document}. Using the inversion relation generalized to vertex functions, we reduce the massive on-shell Feynman integral to the HQET one. This HQET vertex integral can be expressed via a 2F1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_2F_1$$\end{document} function; the nth term of its ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} expansion is explicitly known. We confirm existing results for nlL-1αsL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_l^{L-1} \alpha _s^L$$\end{document} terms in the form factors (up to L=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L=3$$\end{document}), and we present results for higher L.


Introduction
Quark form factors are building blocks for various production cross sections and decay widths in QCD. Massive quark form factors are known up to two loops [1]; recently they have been calculated at three loops in the large N c limit [2].
We shall consider heavy-quark form factors in the large β 0 limit, where β 0 α s ∼ 1, and 1/β 0 is an expansion parameter (see the reviews [3][4][5]). A bare form factor can be written as (1) Keeping terms with the highest degree of β 0 in each order of perturbation theory, we get The leading coefficients a L ,L−1 can easily be obtained from n L−1 f terms ( Fig. 1). We shall consider only the first 1/β 0 order. 1 1 In some cases it is possible to obtain results for 1/β 2 0 corrections; see, e.g. [6][7][8]. a e-mail: A.G.Grozin@inp.nsk.su

Heavy-quark bilinear currents
We consider the QCD currents where Q 0 is a bare heavy-quark field. The antisymmetrized product of n γ matrices has the property All results for form factors of this current will explicitly depend on n and η.
In situations when the initial heavy-quark momentum p 1 and the final one p 2 can be written as p 1,2 = mv 1,2 + k 1,2 (m is the on-shell mass, v 2 1,2 = 1) with small residual momenta k 1,2 m, these currents can be expanded in HQET ones [9,10]: where the leading HQET currents arẽ and theÕ i are local and bilocal dimension-4 HQET operators with appropriate quantum numbers. Here h v 1,2 0 are two (unrelated) bare fields describing HQET quarks with the velocities v 1,2 having small (variable) residual momenta; the HQET Lagrangian explicitly contains v 1,2 . These reference velocities can be changed by arbitrary small vectors of order k i /m (reparametrization invariance). The HQET For our purpose it is convenient to choose v 1,2 = p 1,2 /m, i. e., both residual momenta k 1,2 = 0. Then the matrix elements ofÕ i vanish: non-zero expressions for these matrix elements (having dimensionality of energy) cannot be constructed, because we have no non-zero dimensionful parameters. The coefficients H i in (5) can be obtained by matching the on-shell matrix elements (k 1,2 = 0) in QCD and HQET: where u 1,2 are the Dirac spinors of the initial quark and the final one (all loop corrections toF i vanish because they contain no scale). Therefore the bare matching coefficients (in the relation similar to (5) but for the bare currents) are The renormalized matching coefficients are

Fig. 2
On-shell massive self-energy integrals and off-shell HQET ones where for any anomalous dimension γ (α s ) Matrix elements of the currents with n = 0, 1 can be written via smaller numbers of form factors: where F i with n = 0, η = 1 are used, and where F i with n = 1, η = −1 are used.

Inversion relations
On-shell massive self-energy integrals with one massive line and any number of massless ones in some cases can be expressed via similar off-shell HQET integrals. Suppose all massless lines can be drawn at one side of the massive one and the resulting graph is planar (e.g., the diagram in Fig. 2a). Lines of such a diagram subdivide the plane into a number of polygonal cells (plus the exterior); with each cell we can associate a loop momentum (flowing counterclockwise). Then outer massless edges of the diagram correspond to the denominators −k 2 i − i0; inner massless edges to −(k i − k j ) 2 − i0; and massive edges to Table 1). The corresponding HQET diagram (Fig. 2b) has HQET denominators −2k i · v − 2ω − i0 instead of massive ones. First we perform a Wick rotation of all loop momenta k i0 → ik i0 (in the v rest frame). Then, in Euclidean momentum space, we invert each loop momentum [11]:

Fig. 3 Examples of on-shell massive diagrams which cannot be transformed to off-shell HQET ones by inversion relations
Inversion transforms massive denominators to HQET ones (and vice versa) if we identify see Table 1. As a result, a massive on-shell diagram (Fig. 2a) becomes m − n i (the sum runs over all massive line segments, n i are their indices, i. e. the powers of the denominators) times the off-shell HQET diagram ( Fig. 2b) with ω = −(2m) −1 (15). The indices of all inner massless edges, as well as of all massive edges (which become HQET ones), remain intact (see Table 1). From the same table it is clear that the index of an outer massless edge becomes d − n i , where the sum runs over all edges of the cell to which this outer edge belongs (they can be all massless, or one of them can be massive). If there is a cell k i bounded only by inner massless edges, and maybe one massive one, then the denominator (k 2 i ) d− n j will appear (Fig. 3). This denominator does not correspond to any line, and hence the resulting integral is not a Feynman integral at all; in this case, the discussed relation becomes rather useless (though formally correct). The inversion relations [11] were used, e.g., in [12][13][14]).
The inversion relations can be generalized to similar vertex integrals; the masses of the initial particle and the final one may differ. At one loop (Fig. 4), the integrals The integrals I (17) have been investigated in [15]. Here we need only the integrals M (16) with m 1 = m 2 ; they reduce to the integrals I (17) with ω 1 = ω 2 , which are especially simple [15]: where is the one-loop HQET self-energy integral. We only need integer n 1,2 ; in this case all I reduce by IBP to 2 master integrals [15]: I (1, 0, n) (trivial) and I (1, 1, n) (given by (19)). Inversion relations can be generalized to diagrams with more external legs. For example, the one-loop massive box diagram with two on-shell legs and the corresponding offshell HQET one (Fig. 5) are related by M(n 1 , n 2 , n 3 , n 4 ; ϑ; m 1 ,

Large-β 0 limit
We need only terms with the highest degree of n f ; therefore, there is no need to distinguish between n f and n l = n f − 1, or any n f + const. The gluon propagator can be written as where the gluon self-energy is At this leading large β 0 order, the coupling constant renormalization is simple: The bare QCD matrix elements can be written in the form [6,16] It is convenient to write the functions f i (ε, u) in the form usual for on-shell massive QCD problems (see [5]) We calculate the vertex function ( Fig. 1) and multiply it by Z os Q with the 1/β 0 accuracy (see [5]). Reducing on-shell massive QCD integrals to off-shell HQET ones by the inversion relation (18) and then to the master integrals by IBP [15], we obtain where (the same function appears also in the one-loop self-energy integral with arbitrary masses m 1,2 and arbitrary p 2 , where both indices are equal to 1 [17]). At ϑ = 0 this result agrees with the result of [18] at m 1 = m 2 ; see also [5]. 2 Re-expressing the bare form factors (27) via the renormalized coupling we obtain We should have (see (8)) log F 0 = log(Z (α s (μ))/Z (α s (μ))) + log H (μ, μ) : (32) negative degrees of ε go to log(Z /Z ), non-negative ones to log H . The function is regular at the origin; expanding (b/(ε+b)) L in b, we obtain a quadruple sum. In the coefficient of ε −1 all f nm except f n0 cancel; differentiating this coefficient in log b (and using the fact that F (30) at u = 0 is ϑ/ sinh ϑ) we obtain the anomalous dimension corresponding to Z /Z [6,16]: These anomalous dimensions at the 1/β 0 order are [19,20] Our results satisfy this requirement ( f 1,2 (−b, 0) = 0 because the QCD current J does not mix with currents with other Dirac structures).
In the second formula in (8.95), the coefficient of R 0 should contain an extra factor 3. In both formulae in (8.96), their right-hand sides should be 1 + α s correction.
In the coefficient of ε 0 all f nm except f n0 and f 0m cancel. The coefficients f n0 form K γ n −γ (α s (μ)), see (9); we have [6] where the Borel images of the perturbative series forĤ i are The integral (37) is not well defined because of poles at the integration contour. The leading renormalon ambiguities are given by the residues at u = 1/2 [21] (see also [5]). It is easy to calculate these residues because F (30) at u = 1/2 is just 2/(w + 1): , where As demonstrated in [21], matrix elements of the QCD currents between ground-state mesons (pseudoscalar or vector) are unambiguous: the IR renormalon ambiguities of the leading matching coefficients H i are compensated by the UV renormalon ambiguities in the matrix elements of the 1/m suppressed HQET operatorsÕ i in (5) (see also [5]). The hypergeometric function F (30) has been expanded in u to all orders [17], the coefficients are expressed via Nielsen polylogarithms S nm (x). The result [17] is written for the case of an Euclidean angle 3 ; its analytical continuation to Minkowski angles is It is possible to re-express this expansion in terms of Nielsen polylogarithms of just one argument, see [23], but then the symmetry ϑ → −ϑ will not be explicit.

Appendix A: Anticommuting γ 5 and 't Hooft-Veltman γ 5
For flavour-nonsinglet currents one may use the anticommuting γ 5 without encountering contradictions; they are related to the currents with the 't Hooft-Veltman γ 5 by a finite renormalization [24][25][26]: where τ is a flavour matrix with Tr τ = 0. The currents with γ AC 5 n have anomalous dimensions γ n , because they can be obtained from the case of massless quarks; γ HV 5 n is just 4−n with reshuffled components. Equating the derivatives in d log μ we obtain where the anomalous dimensions γ n and γ 4−n differ starting from two loops. In particular, Z 0 (α s ) = 1. In HQET currents with γ AC 5 and with γ HV 5 have the same anomalous dimensioñ γ , and the finite renormalization factor similar to (A.2) is 1. In the large β 0 limit (see (35)) At the leading 1/β 0 order we may use these formulae for flavour singlet currents, too. The matrix γ AC 5 n has the same property (4) but with η = −(−1) n . From our results (27)- (29) we see that, indeed, Matrix elements of the currents with γ AC 5 and n = 0, 1 can be written via smaller numbers of form factors: where F i with n = 0, η = −1 are used, and where F i with n = 1, η = 1 are used. The divergence of the axial current is where the bare mass m 0 = Z os m m. Taking the matrix element of this equation we obtain The on-shell mass renormalization constant Z os m at the first 1/β 0 order is given by the formula similar to (27), (28) with N m (ε, u) = −2(3 − 2ε)(1 − u); see, e.g., [5]. And indeed, from (29) and H ··· (τ ) are harmonic polylogarithms (see [29][30][31]). Only one new polylogarithm appears at each order. In order to compare the expansion coefficients in (41) and in (B.10), we need to transform them to harmonic polylogarithms of the same argument, which we choose as x = e −ϑ . In (41), we first rewrite S nm (−x −1 ) via S nm (−x) using the formula from [23]; then we rewrite S nm (−x) via H ··· (−x) and then via H ··· (x); we rewrite log cosh(ϑ/2) (B.11) via H ··· (x); and finally we re-express products of harmonic polylogarithms via their linear combinations. In (B.10) we rewrite harmonic polylogarithms with ± indices [30] via normal ones with indices 0, ±1; substitute τ = (1 − x)/(1 + x) and re-express via H ··· (x); and finally convert products of harmonic polylogarithms to sums. All these steps are done in Mathematica using HPL [29,30]. We have checked that all the coefficients presented in (B.10) agree with (41).