Heavy quark form factors in the large $\beta_0$ limit

Heavy quark form factors are calculated at $\beta_0 \alpha_s \sim 1$ to all orders in $\alpha_s$ at the first order in $1/\beta_0$. The $n_f^2 \alpha_s^3$ terms in the recent results [arXiv:1611.07535] for the vector form factors are confirmed, and $n_f^{L-1} \alpha_s^L$ terms for higher $L$ are predicted.


Introduction
Quark form factors are building blocks for various production cross sections and decay widths in QCD. Recently massive-quark vector form factors have been calculated to to 3 loops [1].
We'll 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 [2,3]). 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 be easily obtained from n L−1 f terms ( Fig. 1). We shall consider only the first 1/β 0 order 3 . Figure 1: Diagrams producing the highest degree of n f in each order of perturbation theory. 1 A.G.Grozin@inp.nsk.su 2 Also Novosibirsk State University, Novosibirsk, Russia 3 In some cases it is possible to obtain results for 1/β 2 0 corrections, see, e. g., [4,5].
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) with k 1,2 m, these currents can be expanded in HQET ones [6,7]: where the leading HQET currents arẽ andÕ i are local and bilocal dimension-4 HQET operators with appropriate quantum numbers. The HQET current renormalization constantZ does not depend on the Dirac structure and is a function of the Minkowski angle ϑ: v 1 · v 2 = cosh ϑ = w. The coefficients in (5) can be obtained by matching the on-shell matrix elements (k 1,2 = 0) in QCD and HQET: (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 H 0 i = F i /F i = F i . The renormalized matching coefficients are UV divergences cancel in the ratio F i /Z as well as in the ratioF i /Z. Both F i andF i contain IR divergences which cancel in the ratio F i /F i because HQET is constructed to reproduce the IR behaviour of QCD (F i have no loop corrections because their UV and IR divergences cancel each other). The dependence of H i (µ, µ ) on µ and µ is determined by the RG equations. Their solution can be written as where for any anomalous dimension γ(α s ) = γ 0 α s /(4π) + γ 1 (α s /(4π)) 2 + · · · we define 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). Figure 2: On-shell massive self-energy integrals and off-shell HQET ones.

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 m 2 − (k i + mv) 2 − i0 ( Table 1). The corresponding HQET diagram (Fig. 2b) has HQET denominators −2k i · v − 2ω − i0 instead of massive ones. First we perform 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 [8]: 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 − ni (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 (14). 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− nj 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 [8] were used, e. g., in [9, 10]).

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 [12,4] It is convenient to write the functions f i (ε, u) in the form usual for on-shell massive QCD problems (see [3]) We calculate the vertex function (Fig. 1) and multiply it by Z os Q with the 1/β 0 accuracy (see [3]). Reducing on-shell massive QCD integrals to off-shell HQET ones by the inversion relation (17) and then to the master integrals by IBP [11], we obtain where (the same function appears also in the 1-loop self-energy integral with arbitrary masses m 1,2 and arbitrary p 2 , where both indices are equal to 1 [13]). At ϑ = 0 this result agrees with the result of [14] at m 1 = m 2 , see also [3] 4 . Re-expressing the bare form factors (26) 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) : 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 (29) at u = 0 is ϑ/ sinh ϑ) we obtain the anomalous dimension corresponding to Z/Z [12,4]: 4 Note a typo: the unnumbered formula below (8.93) should read These anomalous dimensions at the 1/β 0 order are [15,16] 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 coefficient of ε 0 all f nm except f n0 and f 0m cancel. The coefficients f n0 form K γn−γ (α s (µ)), see (9); we have [4] where the Borel images of the perturbative series forĤ i are The integral (36) is not well defined because of poles at the integration contour. The leading renormalon ambiguities are given by the residues at u = 1/2 [17] (see also [3]). It is easy to calculate these residues because F (29) at u = 1/2 is just 2/(w + 1): where As demonstrated in [17], these IR renormalon ambiguities are compensated by the UV renormalon ambiguities in the matrix elements of the HQET operatorsÕ i in (5). The hypergeometric function F (29) has been expanded in u to all orders [13], the coefficients are expressed via Nielsen polylogarithms S nm (x). The result [13] is written for the case of an Euclidean angle 5 ; 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 [19], but then the symmetry ϑ → −ϑ will not be explicit. Acknowledgements. I am grateful to M. Steinhauser for useful comments and hospitality in Karlsruhe, where the major part of this work was done; to J. M. Henn for useful discussions and hospitality in Mainz; and to M. Yu. Kalmykov for bringing ref. [13] to my attention and discussions related to it.
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 [20]: where τ is a flavour matrix with Tr τ = 0. The currents with γ AC 5 Γ n have anomalous dimensions γ n , because they can be obtained from the case of massless quarks; γ HV 5 Γ n is just Γ 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 2 loops. In particular, Z 0 (α s ) = 1. In HQET currents with γ AC 5 and with γ HV 5 have the same anomalous dimensionγ, and the finite renormalization factor similar to (41) is 1. In the large β 0 limit (see (34)) At the leading 1/β 0 order we may use these formulae for flavour singlet currents, too. The matrix γ AC 5 Γ n has the same property (4) but with η = −(−1) n . From our results (26)-(28) we see that, indeed,Ĥ 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 <Q(mv 2 )|J µ |Q(mv 1 )> = F A 1ū2 γ AC 5 γ µ u 1 + F A 2ū2 γ AC 5 u 1 (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 (26), (27) with N m (ε, u) = −2(3 − 2ε)(1 − u), see, e. g., [3]. And indeed, from (28), (44-45) we obtain

B Expansion of the hypergeometric function F
We can also find several terms of this expansion using the Mathematica package HypExp [21] (which uses HPL [22]). This results in