Heavy-quark form factors in the large $\beta _0$ limit

Abstract

Heavy-quark form factors are calculated at $\beta _0 \alpha _s \sim 1$ to all orders in $\alpha _s$ at the first order in $1/\beta _0$. 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 ${}_2F_1$ function; the nth term of its $\varepsilon$ expansion is explicitly known. We confirm existing results for $n_l^{L-1} \alpha _s^L$ terms in the form factors (up to $L=3$), and we present results for higher L.

1 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 $\beta _0$ limit, where $\beta _0 \alpha _s \sim 1$, and $1/\beta _0$ is an expansion parameter (see the reviews [3,4,5]). A bare form factor can be written as

$$\begin{aligned} F = 1 + \sum _{L=1}^\infty \sum _{n=0}^{L-1} a_{Ln} \beta _0^n \left( \frac{g_0^2}{(4\pi )^{d/2}}\right) ^L. \end{aligned}$$

(1)

Keeping terms with the highest degree of $\beta _0$ in each order of perturbation theory, we get

$$\begin{aligned} F = 1 + \frac{1}{\beta _0} f\left( \frac{\beta _0 g_0^2}{(4\pi )^{d/2}}\right) + \mathcal {O}\left( \frac{1}{\beta _0^2}\right) . \end{aligned}$$

(2)

The leading coefficients $a_{L,L-1}$ can easily be obtained from $n_f^{L-1}$ terms (Fig. 1). We shall consider only the first $1/\beta _0$ order.¹

2 Heavy-quark bilinear currents

We consider the QCD currents

$$\begin{aligned} J_0 = \bar{Q}_0 \Gamma Q_0 = Z(\alpha _s^{(n_f)}(\mu )) J(\mu ),\quad \Gamma = \gamma ^{[\mu _1} \cdots \gamma ^{\mu _n]}, \end{aligned}$$

(3)

where $Q_0$ is a bare heavy-quark field. The antisymmetrized product of n $\gamma$ matrices has the property

$$\begin{aligned} \gamma ^\mu \Gamma \gamma _\mu = \eta (d - 2 n) \Gamma ,\quad \eta = (-1)^n. \end{aligned}$$

(4)

All results for form factors of this current will explicitly depend on n and $\eta$.

In situations when the initial heavy-quark momentum $p_1$ and the final one $p_2$ can be written as $p_{1,2} = m v_{1,2} + k_{1,2}$ (m is the on-shell mass, $v_{1,2}^2=1$) with small residual momenta $k_{1,2} \ll m$, these currents can be expanded in HQET ones [9, 10]:

$$\begin{aligned} J(\mu )= & {} \sum _{i=0}^2 H_i(\mu ,\mu ') \tilde{J}_i(\mu ') \nonumber \\&+ \frac{1}{2 m} \sum _i G_i(\mu ,\mu ') \tilde{O}_i(\mu ') + \mathcal {O}\left( \frac{1}{m^2}\right) , \end{aligned}$$

(5)

where the leading HQET currents are

(6)

and the $\tilde{O}_i$ 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 current renormalization constant $\tilde{Z}$ does not depend on the Dirac structure and is a function of the Minkowski angle $\vartheta$: $v_1 \cdot v_2 = \cosh \vartheta = w$.

Fig. 1

Diagrams producing the highest degree of $n_f$ in each order of perturbation theory

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 of $\tilde{O}_i$ 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:

$$\begin{aligned} \langle Q(p_2=mv_2)|J_0|Q(p_1=mv_1) \rangle= & {} \sum _{i=0}^2 F_i\,\bar{u}_2 \Gamma _i u_1, \nonumber \\ \langle Q(k_2=0)|\tilde{J}_{i0}|Q(k_1=0)\rangle= & {} \tilde{F}_i\,\bar{u}_2 \Gamma _i u_1,\quad \tilde{F}_i =1, \nonumber \\ \end{aligned}$$

(7)

where $u_{1,2}$ are the Dirac spinors of the initial quark and the final one (all loop corrections to $\tilde{F}_i$ vanish because they contain no scale). Therefore the bare matching coefficients (in the relation similar to (5) but for the bare currents) are $H^0_i = F_i/\tilde{F}_i = F_i$. The renormalized matching coefficients are

$$\begin{aligned} H_i(\mu ,\mu ') = H^0_i \frac{\tilde{Z}(\alpha _s^{(n_l)}(\mu '))}{Z(\alpha _s^{(n_f)}(\mu ))} = \frac{F_i \tilde{Z}}{\tilde{F}_i Z}. \end{aligned}$$

(8)

UV divergences cancel in the ratio $F_i/Z$ as well as in the ratio $\tilde{F}_i/\tilde{Z}$. Both $F_i$ and $\tilde{F}_i$ contain IR divergences which cancel in the ratio $F_i/\tilde{F}_i$ because HQET is constructed to reproduce the IR behaviour of QCD ($\tilde{F}_i$ have no loop corrections because their UV and IR divergences cancel each other).

The dependence of $H_i(\mu ,\mu ')$ on $\mu$ and $\mu '$ is determined by the RG equations. Their solution can be written as

$$\begin{aligned} H_i(\mu ,\mu ')= & {} \hat{H}_i \left( \frac{\alpha _s^{(n_f)}(\mu )}{\alpha _s^{(n_f)}(\mu _0)}\right) ^{\gamma _{n0}/(2\beta _0^{(n_f)})} K_{\gamma _n}^{(n_f)}(\alpha _s^{(n_f)}(\mu )) \nonumber \\&{}\times \left( \frac{\alpha _s^{(n_l)}(\mu ')}{\alpha _s^{(n_l)}(\mu _0)}\right) ^{-\tilde{\gamma }_0/(2\beta _0^{(n_l)})} K_{-\tilde{\gamma }}^{(n_l)}(\alpha _s^{(n_l)}(\mu ')), \nonumber \\ \end{aligned}$$

(9)

where for any anomalous dimension $\gamma (\alpha _s) = \gamma _0 \alpha _s/(4\pi ) + \gamma _1 (\alpha _s/(4\pi ))^2 + \cdots$ we define

$$\begin{aligned} K_\gamma (\alpha _s)= & {} \exp \int _0^{\alpha _s} \frac{\mathrm{d}\alpha _s}{\alpha _s} \left( \frac{\gamma (\alpha _s)}{2\beta (\alpha _s)} - \frac{\gamma _0}{2\beta _0} \right) \nonumber \\= & {} 1 + \frac{\gamma _0}{2\beta _0} \left( \frac{\gamma _1}{\gamma _0} - \frac{\beta _1}{\beta _0} \right) \frac{\alpha _s}{4\pi } + \cdots \end{aligned}$$

(10)

Matrix elements of the currents with $n=0$, 1 can be written via smaller numbers of form factors:

$$\begin{aligned} \langle Q(mv_2)|J|Q(mv_1) \rangle= & {} F^S \bar{u}_2 u_1, \nonumber \\ F^S= & {} F_0 + 2 F_1 + (2w-1) F_2 , \end{aligned}$$

(11)

where $F_i$ with $n=0$, $\eta =1$ are used, and

$$\begin{aligned} \langle Q(mv_2)|J^\mu |Q(mv_1) \rangle= & {} (F^V_1 + F^V_2) \bar{u}_2 \gamma ^\mu u_1 \nonumber \\&- F^V_2 \bar{u}_2 u_1 \frac{(v_1 + v_2)^\mu }{2},\end{aligned}$$

(12)

$$\begin{aligned} F^V_1= & {} F_0 + 2 F_1 - (2w-3) F_2,\nonumber \\ F^V_2= & {} - 4 (F_1 + F_2), \end{aligned}$$

(13)

where $F_i$ with $n=1$, $\eta =-1$ are used.

3 Inversion relations

Fig. 2

On-shell massive self-energy integrals and off-shell HQET ones

Table 1 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 $- k_i^2 - i0$; 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 $- 2 k_i\cdot v - 2 \omega - i0$ instead of massive ones. First we perform a Wick rotation of all loop momenta $k_{i0} \rightarrow i k_{i0}$ (in the v rest frame). Then, in Euclidean momentum space, we invert each loop momentum [11]:

$$\begin{aligned} k_i \rightarrow \frac{k_i}{k_i^2}. \end{aligned}$$

(14)

Inversion transforms massive denominators to HQET ones (and vice versa) if we identify

$$\begin{aligned} - 2 \omega = m^{-1}, \end{aligned}$$

(15)

see Table 1. As a result, a massive on-shell diagram (Fig. 2a) becomes $m^{-\sum n_i}$ (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 $\omega = - (2m)^{-1}$ (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 $d - \sum n_i$, 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 $(k_i^2)^{d-\sum n_j}$ 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]).

Fig. 3

Examples of on-shell massive diagrams which cannot be transformed to off-shell HQET ones by inversion relations

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

$$\begin{aligned}&M(n_1,n_2,n;\vartheta ;m_1,m_2) = \int \frac{\mathrm{d}^d k}{i\pi ^{d/2}} \end{aligned}$$

(16)

$$\begin{aligned}&\quad \times \frac{1}{[- k^2 - 2 m_1 v_1 \cdot k - i0]^{n_1} [- k^2 - 2 m_2 v_2 \cdot k - i0]^{n_2} (-k^2-i0)^n}, \nonumber \\&I(n_1,n_2,n;\vartheta ;\omega _1,\omega _2) = \int \frac{\mathrm{d}^d k}{i\pi ^{d/2}} \nonumber \\&\quad \times \frac{1}{[- 2 k \cdot v_1 - 2 \omega _1 - i0]^{n_1} [- 2 k \cdot v_2 - 2 \omega _2 - i0]^{n_2} (-k^2-i0)^n} \nonumber \\ \end{aligned}$$

(17)

are related by

$$\begin{aligned}&M(n_1,n_2,n;\vartheta ;m_1,m_2) = m_1^{-n_1} m_2^{-n_2} \nonumber \\&\quad \times I(n_1,n_2,d{-}n_1{-}n_2-n;\vartheta ; -\, (2 m_1)^{-1},- (2 m_2)^{-1}).\nonumber \\ \end{aligned}$$

(18)

Fig. 4

One-loop vertex 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 $\omega _1=\omega _2$, which are especially simple [15]:

$$\begin{aligned}&I(n_1,n_2,n;\vartheta ;\omega ,\omega ) = (-2\omega )^{d-n_1-n_2-2n} I(n_1+n_2,n) \nonumber \\&\quad \times \,_{3}F_{2}\left( \left. \begin{array}{c}n_1,n_2,\frac{d}{2}-n\\ \frac{n_1+n_2}{2},\frac{n_1+n_2+1}{2}\end{array}\right| \frac{1-\cosh \vartheta }{2}\right) , \end{aligned}$$

(19)

where

$$\begin{aligned} I(n_1,n) = \frac{\Gamma (-d+n_1+2n) \Gamma (d/2-n)}{\Gamma (n_1) \Gamma (n)} \end{aligned}$$

(20)

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 off-shell HQET one (Fig. 5)

$$\begin{aligned}&M(n_1,n_2,n_3,n_4;\vartheta ;m_1,m_2;q^2,q\cdot v_1,q\cdot v_2) = \int \frac{\mathrm{d}^d k}{i\pi ^{d/2}}\nonumber \\&\quad \times \frac{1}{(- k^2 - 2 m_1 v_1\cdot k)^{n_1} (- k^2 - 2 m_2 v_2\cdot k)^{n_2} (- (k+q)^2)^{n_3} (- k^2)^{n_4}}, \end{aligned}$$

(21)

$$\begin{aligned}&I(n_1,n_2,n_3,n_4;\vartheta ;\omega _1,\omega _2;q^2,q\cdot v_1,q\cdot v_2) = \int \frac{\mathrm{d}^d k}{i\pi ^{d/2}} \nonumber \\&\quad \times \frac{1}{( - 2 k \cdot v_1 - 2 \omega _1 )^{n_1}( - 2 k \cdot v_2 - 2 \omega _2 )^{n_2} (- (k+q)^2)^{n_3} (- k^2)^{n_4}} \end{aligned}$$

(22)

are related by

$$\begin{aligned}&M(n_1,n_2,n_3,n_4;\vartheta ;m_1,m_2;q^2,q\cdot v_1,q\cdot v_2) \nonumber \\&= m_1^{-n_1} m_2^{-n_2} (-q^2)^{n_3} I(n_1,n_2,n_3,\nonumber \\&\qquad d-n_1-n_2-n_3-n_4;\vartheta ; \nonumber \\&\qquad -(2 m_1)^{-1},-(2 m_2)^{-1};1/q^2,\nonumber \\&\qquad q\cdot v_1/(-q^2),\; q\cdot v_2/(-q^2)). \end{aligned}$$

(23)

Fig. 5

Box diagrams

4 Large-$\beta _0$ 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 $n_f+\mathrm {const}$. The gluon propagator can be written as

$$\begin{aligned} D_{\mu \nu }(k) = \frac{1}{k^2 (1-\Pi (k^2))} \left( g_{\mu \nu } - \frac{k_\mu k_\nu }{k^2}\right) , \end{aligned}$$

(24)

where the gluon self-energy is

$$\begin{aligned} \Pi (k^2)= & {} \beta _0 \frac{g_0^2}{(4\pi )^{d/2}} e^{-\gamma \varepsilon } \frac{D(\varepsilon )}{\varepsilon } (-k^2)^{-\varepsilon }, \\ D(\varepsilon )= & {} e^{\gamma \varepsilon } \frac{(1-\varepsilon ) \Gamma (1+\varepsilon ) \Gamma ^2(1-\varepsilon )}{(1-2\varepsilon ) (1-\frac{2}{3}\varepsilon ) \Gamma (1-2\varepsilon )} = 1 + \frac{5}{3} \varepsilon + \cdots \nonumber \end{aligned}$$

(25)

At this leading large $\beta _0$ order, the coupling constant renormalization is simple:

$$\begin{aligned}&\beta _0 \frac{g_0^2}{(4\pi )^{d/2}} e^{-\gamma \varepsilon } = b Z_\alpha (b) \mu ^{2\varepsilon }, \nonumber \\&b = \beta _0 \frac{\alpha _s(\mu )}{4\pi },\quad Z_\alpha = \frac{1}{1+b/\varepsilon }. \end{aligned}$$

(26)

The bare QCD matrix elements can be written in the form [6, 16]

$$\begin{aligned} F_i = \delta _{i0} + \frac{1}{\beta _0} \sum _{L=1}^\infty \frac{f_i(\varepsilon ,L\varepsilon )}{L} \Pi (-m^2)^L + \mathcal {O}\left( \frac{1}{\beta _0^2}\right) . \end{aligned}$$

(27)

It is convenient to write the functions $f_i(\varepsilon ,u)$ in the form usual for on-shell massive QCD problems (see [5])

$$\begin{aligned} f_i(\varepsilon ,u) = C_F \frac{e^{\gamma \varepsilon }}{D(\varepsilon )} \frac{\Gamma (1-2u) \Gamma (1+u)}{\Gamma (3-u-\varepsilon )} N_i(\varepsilon ,u). \end{aligned}$$

(28)

We calculate the vertex function (Fig. 1) and multiply it by $Z_Q^{\scriptstyle \mathrm {os}}$ with the $1/\beta _0$ 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

$$\begin{aligned} N_0(\varepsilon ,u)= & {} \biggl [ - \eta u \frac{n-2+\varepsilon }{w-1} - 2 (w+1) u (n-2)^2 \nonumber \\&- u \bigl ( \eta u + 4 (w+1) \varepsilon \bigr ) (n-2)\nonumber \\&+ 2 (2-u) \bigl ( w + (w+1) u \bigr ) \nonumber \\&- ( 6 w + 2 u + \eta u^2 ) \varepsilon \nonumber \\&+ 2 \bigl ( w - (w+1) u \bigr ) \varepsilon ^2 \biggr ] F \nonumber \\&+ \eta u \frac{n-2+\varepsilon }{w-1} + 2 (n-2)^2 + 4 \varepsilon (n-2) \nonumber \\&- 6 (1-u^2) + 2 (1-u) (5+2u) \varepsilon \nonumber \\&- 2 (1-2u) \varepsilon ^2, \nonumber \\ N_1(\varepsilon ,u)= & {} u \biggl [ \eta w \frac{n-2+\varepsilon }{w-1} - \eta u (n-2) - 2 + u + \varepsilon \nonumber \\&- \eta u \varepsilon \biggr ] F - \eta u \frac{n-2+\varepsilon }{w-1}, \nonumber \\ N_2(\varepsilon ,u)= & {} \eta u \frac{n-2+\varepsilon }{w-1} \nonumber \\&\times [1 - (1 + (w-1) u) F], \end{aligned}$$

(29)

where

$$\begin{aligned} F = \,_{2}F_{1}\left( \left. \begin{array}{c}1,1+u\\ 3/2\end{array}\right| \frac{1-w}{2}\right) \end{aligned}$$

(30)

(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 $\vartheta =0$ 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

$$\begin{aligned} F_i = \delta _{i0} + \frac{1}{\beta _0} \sum _{L=1}^\infty \frac{f_i(\varepsilon ,L\varepsilon )}{L} \left[ D(\varepsilon ) \left( \frac{\mu ^2}{m^2}\right) ^\varepsilon \frac{b}{\varepsilon +b}\right] ^L. \end{aligned}$$

(31)

We should have (see (8))

$$\begin{aligned} \log F_0 = \log (Z(\alpha _s(\mu ))/\tilde{Z}(\alpha _s(\mu ))) + \log H(\mu ,\mu )\,: \end{aligned}$$

(32)

negative degrees of $\varepsilon$ go to $\log (Z/\tilde{Z})$, non-negative ones to $\log H$. The function

$$\begin{aligned} f_0(\varepsilon ,u) D(\varepsilon )^{u/\varepsilon } \left( \frac{\mu ^2}{m^2}\right) ^u = \sum _{n,m=0}^\infty f_{nm} \varepsilon ^n u^m \end{aligned}$$

(33)

is regular at the origin; expanding $(b/(\varepsilon +b))^L$ in b, we obtain a quadruple sum. In the coefficient of $\varepsilon ^{-1}$ all $f_{nm}$ except $f_{n0}$ cancel; differentiating this coefficient in $\log b$ (and using the fact that F (30) at $u=0$ is $\vartheta /\sinh \vartheta$) we obtain the anomalous dimension corresponding to $Z/\tilde{Z}$ [6, 16]:

$$\begin{aligned} \gamma _n - \tilde{\gamma } = - 2 \frac{b}{\beta _0} f_0(-b,0) + \mathcal {O}\left( \frac{1}{\beta _0^2}\right) . \end{aligned}$$

(34)

These anomalous dimensions at the $1/\beta _0$ order are [19, 20]

$$\begin{aligned} \gamma _n= & {} 4 C_F \frac{b}{\beta _0} \frac{(1 + \frac{2}{3} b) \Gamma (2+2b)}{(1+b)^2 (2+b) \Gamma ^3(1+b) \Gamma (1-b)} \nonumber \\&{}\times (n-1) (3-n+2b) + \mathcal {O}\left( \frac{1}{\beta _0^2}\right) , \end{aligned}$$

(35)

$$\begin{aligned} \tilde{\gamma }= & {} 4 C_F \frac{b}{\beta _0} \frac{(1 + \frac{2}{3} b) \Gamma (2+2b)}{(1+b) \Gamma ^3(1+b) \Gamma (1-b)} (\vartheta \coth \vartheta - 1) \nonumber \\&{} + \mathcal {O}\left( \frac{1}{\beta _0^2}\right) . \end{aligned}$$

(36)

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 $\varepsilon ^0$ all $f_{nm}$ except $f_{n0}$ and $f_{0m}$ cancel. The coefficients $f_{n0}$ form $K_{\gamma _n-\tilde{\gamma }}(\alpha _s(\mu ))$, see (9); we have [6]

$$\begin{aligned} \hat{H}_i = \delta _{i0} + \frac{1}{\beta _0} \int _0^\infty du\,e^{-u/b} S_i(u) + \mathcal {O}\left( \frac{1}{\beta _0^2}\right) , \end{aligned}$$

(37)

where the Borel images of the perturbative series for $\hat{H}_i$ are

$$\begin{aligned} S_i(u) = \frac{1}{u} \left[ \left( e^{5/3} \frac{\mu _0^2}{m^2}\right) ^u f_i(0,u) - f_i(0,0)\right] . \end{aligned}$$

(38)

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 $2/(w+1)$:

$$\begin{aligned}&\Delta H_0 = \left( \frac{4}{w+1} - 3\right) \frac{\Delta \bar{\Lambda }}{2m},\quad \Delta H_1 = \frac{1}{w+1} \frac{\Delta \bar{\Lambda }}{2m}, \nonumber \\&\Delta H_2 = 0, \end{aligned}$$

(39)

where

$$\begin{aligned} \Delta \bar{\Lambda } = - 2 \frac{C_F}{\beta _0} e^{5/6} \Lambda _{\overline{\scriptstyle \mathrm {MS}}}. \end{aligned}$$

(40)

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 operators $\tilde{O}_i$ 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 angle³; its analytical continuation to Minkowski angles is

$$\begin{aligned} F= & {} \frac{1}{\sinh \vartheta (2 \cosh (\vartheta /2))^{2 u}} \biggl [ \frac{\sinh (\vartheta u)}{u} \nonumber \\&- e^{-\vartheta u} \sum _{n=1}^\infty u^n \sum _{m=1}^n (-2)^{n-m} S_{m,n-m+1}(-e^{\vartheta }) \nonumber \\&+ e^{\vartheta u} \sum _{n=1}^\infty u^n \sum _{m=1}^n (-2)^{n-m} S_{m,n-m+1}(-e^{-\vartheta }) \biggr ]. \end{aligned}$$

(41)

It is possible to re-express this expansion in terms of Nielsen polylogarithms of just one argument, see [23], but then the symmetry $\vartheta \rightarrow -\vartheta$ will not be explicit.

Appendices

Appendix A: Anticommuting $\gamma _5$ and ’t Hooft–Veltman $\gamma _5$

For flavour-nonsinglet currents one may use the anticommuting $\gamma _5$ without encountering contradictions; they are related to the currents with the ’t Hooft–Veltman $\gamma _5$ by a finite renormalization [24,25,26]:

$$\begin{aligned} ( \bar{q} \gamma _5^{\scriptstyle \mathrm {AC}} \Gamma _n \tau q)_\mu = Z_{2-n}(\alpha ^{(n_f)}(\mu )) ( \bar{q} \gamma _5^{\scriptstyle \mathrm {HV}} \Gamma _n \tau q )_\mu , \end{aligned}$$

(A.1)

where $\tau$ is a flavour matrix with $\mathop {\mathrm {Tr}}\nolimits \tau =0$. The currents with $\gamma _5^{\scriptstyle \mathrm {AC}} \Gamma _n$ have anomalous dimensions $\gamma _n$, because they can be obtained from the case of massless quarks; $\gamma _5^{\scriptstyle \mathrm {HV}} \Gamma _n$ is just $\Gamma _{4-n}$ with reshuffled components. Equating the derivatives in $d\log \mu$ we obtain

$$\begin{aligned} Z_{2-n}(\alpha _s) = K_{\gamma _n-\gamma _{4-n}}^{(n_f)}(\alpha _s), \end{aligned}$$

(A.2)

where the anomalous dimensions $\gamma _n$ and $\gamma _{4-n}$ differ starting from two loops. In particular, $Z_0(\alpha _s)=1$. In HQET currents with $\gamma _5^{\scriptstyle \mathrm {AC}}$ and with $\gamma _5^{\scriptstyle \mathrm {HV}}$ have the same anomalous dimension $\tilde{\gamma }$, and the finite renormalization factor similar to (A.2) is 1. In the large $\beta _0$ limit (see (35))

$$\begin{aligned} Z_n(\alpha _s)= & {} \exp \biggl [- \frac{8n}{\beta _0} \nonumber \\&\times \int _0^b \mathrm{d}b \frac{(1 + \frac{2}{3} b) \Gamma (2+2b)}{(1+b)^2 (2+b) \Gamma ^3(1+b) \Gamma (1-b)} \nonumber \\&+ \mathcal {O}\left( \frac{1}{\beta _0^2}\right) \biggr ]. \end{aligned}$$

(A.3)

At the leading $1/\beta _0$ order we may use these formulae for flavour singlet currents, too. The matrix $\gamma _5^{\scriptstyle \mathrm {AC}} \Gamma _n$ has the same property (4) but with $\eta =-(-1)^n$. From our results (27)–(29) we see that, indeed,

$$\begin{aligned} \hat{H}_{\gamma _5^{\scriptstyle \mathrm {AC}} \Gamma _n} = \left. \hat{H}_{\Gamma _n}\right| _{\eta \rightarrow -\eta } = \hat{H}_{\gamma _5^{\scriptstyle \mathrm {HV}} \Gamma _n} = \hat{H}_{\Gamma _{4-n}}. \end{aligned}$$

(A.4)

Matrix elements of the currents with $\gamma _5^{\scriptstyle \mathrm {AC}}$ and $n=0$, 1 can be written via smaller numbers of form factors:

$$\begin{aligned} \langle Q(mv_2)|J|Q(mv_1) \rangle= & {} F^P \bar{u}_2 \gamma _5^{\scriptstyle \mathrm {AC}} u_1, \nonumber \\ F^P= & {} F_0 - 2 F_1 - (2w+1) F_2, \end{aligned}$$

(A.5)

where $F_i$ with $n=0$, $\eta =-1$ are used, and

$$\begin{aligned} \langle Q(mv_2)|J^\mu |Q(mv_1) \rangle= & {} F^A_1 \bar{u}_2 \gamma _5^{\scriptstyle \mathrm {AC}} \gamma ^\mu u_1 \nonumber \\&+ F^A_2 \bar{u}_2 \gamma _5^{\scriptstyle \mathrm {AC}} u_1 \frac{(v_2 - v_1)^\mu }{2}, \nonumber \\ F^A_1= & {} F_0 + 2 F_1 + (2w-1) F_2,\nonumber \\ F^A_2= & {} 4 (F_1 - F_2), \end{aligned}$$

(A.6)

where $F_i$ with $n=1$, $\eta =1$ are used.

The divergence of the axial current is

$$\begin{aligned} i \partial _\mu (\bar{Q}_0 \gamma _5^{\scriptstyle \mathrm {AC}} \gamma ^\mu Q_0) = 2 m_0 \bar{Q}_0 \gamma _5^{\scriptstyle \mathrm {AC}} Q_0, \end{aligned}$$

(A.7)

where the bare mass $m_0 = Z_m^{\scriptstyle \mathrm {os}} m$. Taking the matrix element of this equation we obtain

$$\begin{aligned} F^A_1 + \frac{w-1}{2} F^A_2 = Z_m^{\scriptstyle \mathrm {os}} F^P. \end{aligned}$$

(A.8)

The on-shell mass renormalization constant $Z_m^{\scriptstyle \mathrm {os}}$ at the first $1/\beta _0$ order is given by the formula similar to (27), (28) with $N_m(\varepsilon ,u) = - 2 (3-2\varepsilon ) (1-u)$; see, e.g., [5]. And indeed, from (29), (A.5)–(A.6) we obtain

$$\begin{aligned} N^A_1 + \frac{w-1}{2} N^A_2 = N^P + N_m. \end{aligned}$$

(A.9)

Appendix B: Expansion of the hypergeometric function F

We can also find several terms of this expansion using the Mathematica package HypExp [27, 28] (which uses HPL [29, 30]). This results in

$$\begin{aligned} F= & {} \frac{1}{\sinh \vartheta } \biggl [ \vartheta - H_{-+}(\tau ) u - ( H_{-+-}(\tau ) - 2 H_{-+}(\tau ) l ) \frac{u^2}{2} \nonumber \\&- ( H_{-+--}(\tau ) - 2 H_{-+-}(\tau ) l + 2 H_{-+}(\tau ) l^2 ) \frac{u^3}{3} \nonumber \\&- \biggl ( H_{-+---}(\tau ) - 2 H_{-+--}(\tau ) l + 2 H_{-+-}(\tau ) l^2 \nonumber \\&- \frac{4}{3} H_{-+}(\tau ) l^3 \biggr ) \frac{u^4}{4} \nonumber \\&- \biggl ( H_{-+----}(\tau ) - 2 H_{-+---}(\tau ) l + 2 H_{-+--}(\tau ) l^2 \nonumber \\&- \frac{4}{3} H_{-+-}(\tau ) l^3 + \frac{2}{3} H_{-+}(\tau ) l^4 \biggr ) \frac{u^5}{5} \nonumber \\&\biggl ( H_{-+-----}(\tau ) - 2 H_{-+----}(\tau ) l + 2 H_{-+---}(\tau ) l^2 \nonumber \\&- \frac{4}{3} H_{-+--}(\tau ) l^3 + \frac{2}{3} H_{-+-}(\tau ) l^4 - \frac{4}{15} H_{-+}(\tau ) l^5 \biggr ) \frac{u^6}{6} \nonumber \\&- \cdots \biggr ], \end{aligned}$$

(B.10)

where

$$\begin{aligned}&\tau = \tanh \frac{\vartheta }{2},\quad l = \frac{1}{2} H_-(\tau ) = \log \cosh \frac{\vartheta }{2}, \nonumber \\&H_+(\tau ) = \vartheta , \end{aligned}$$

(B.11)

and $H_{\cdots }(\tau )$ 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 $x = e^{-\vartheta }$. 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 $H_{\cdots }(-x)$ and then via $H_{\cdots }(x)$; we rewrite $\log \cosh (\vartheta /2)$ (B.11) via $H_{\cdots }(x)$; 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, $\pm 1$; substitute $\tau =(1-x)/(1+x)$ and re-express via $H_{\cdots }(x)$; 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).

Appendix C: Vector form factors

The vector form factors $F^V_{1,2}$ (13) can be written in the form (27), (28); from (29), (13) we obtain

$$\begin{aligned} N^V_1(\varepsilon ,u)= & {} 2 [ 2 w + u - 3 u^2 - 3 w \varepsilon + 2 w u \varepsilon - (w-3) u^2 \varepsilon \nonumber \\&+ w \varepsilon ^2 - (w+1) u \varepsilon ^2 ] F \nonumber \\&- 2 \bigl [ 2 + u - 3 u^2 - 3 \varepsilon + 2 u \varepsilon \nonumber \\&+ 2 u^2 \varepsilon + \varepsilon ^2 - 2 u \varepsilon ^2 \bigr ], \end{aligned}$$

(C.12)

$$\begin{aligned} N^V_2(\varepsilon ,u)= & {} 4 u (1 + u - 2 u \varepsilon ) F. \end{aligned}$$

(C.13)

All loop corrections to $F^V_1$ vanish at $\vartheta =0$, and hence $N^V_1=0$ at $w=1$.

The form factor $F^V_1 = H^V_1/\tilde{Z}$, where $\tilde{Z}$ at the $1/\beta _0$ order is determined by the anomalous dimension (36), and $H^V_1$ contains only non-negative powers of $\varepsilon$. We choose $\mu =\mu '=\mu _0=m$. $H^V_1$ at $\varepsilon =0$ is given by the coefficients $f_{n0}$ (which produce $K_{-\tilde{\gamma }}$ (10)) and $f_{0n}$ (which produce $\hat{H}^V_1$ (37)); $\varepsilon ^n$ terms ($n>0$) require all $f_{nm}$. Writing the expansion (B.10) as $F = f_0 - f_1 u - f_2 u^2/2 - f_3 u^3/3 - \cdots$ we obtain up to four loops

$$\begin{aligned}&H^V_1 = 1 + C_F \frac{b}{\beta _0} \Biggl \{ - 2 w f_1 + (3 w + 1) f_0 - 4 \nonumber \\&\quad - \left( w f_2 + (3 w + 1) f_1 - \left( \frac{\pi ^2}{6} + 8 \right) w f_0 + \frac{\pi ^2}{6} + 8 \right) \varepsilon \nonumber \\&\quad - \biggl ( \frac{2}{3} w f_3 + \frac{3 w + 1}{2} f_2 + \left( \frac{\pi ^2}{6} + 8 \right) w f_1 \nonumber \\&\quad + \left( \frac{2}{3} \zeta _3 w - \frac{\pi ^2}{4} w - \frac{\pi ^2}{12} {-}16 w \right) f_0 - \frac{2}{3} \zeta _3 {+} \frac{\pi ^2}{3} + 16 \biggr ) \varepsilon ^2 \nonumber \\&\quad - \Biggl ( \frac{w}{2} f_4 + \left( w + \frac{1}{3} \right) f_3 + \left( \frac{\pi ^2}{12} + 4 \right) w f_2 \nonumber \\&\quad - \left( \frac{2}{3} \zeta _3 w - \frac{\pi ^2}{4} w - \frac{\pi ^2}{12} - 16 w \right) f_1 \nonumber \\&\quad - \left( \frac{\pi ^4}{80} w - \zeta _3 w - \frac{1}{3} \zeta _3 + \frac{2}{3} \pi ^2 w + 32 w \right) f_0 \nonumber \\&\quad + \frac{\pi ^4}{80} - \frac{4}{3} \zeta _3 + \frac{2}{3} \pi ^2 + 32 \Biggr ) \varepsilon ^3 + \cdots \nonumber \\&\quad - b \Biggl [ w f_2 {+} \left( \frac{19}{3} w + 1 \right) f_1 {-} \frac{1}{3} \left( 2 \pi ^2 w + \frac{209}{6} w {+} \frac{1}{2} \right) f_0 \nonumber \\&\quad + \frac{2}{3} \left( \pi ^2 + \frac{53}{3} \right) \nonumber \\&\quad + \Biggl ( 2 w f_3 {+} \frac{1}{2} \left( \frac{47}{3} w {+} 3 \right) f_2 + \left( \frac{3}{2} \pi ^2 w + \frac{281}{9} w + \frac{1}{3} \right) f_1 \end{aligned}$$

$$\begin{aligned}&\quad - \left( 8 \zeta _3 w + \frac{131}{36} \pi ^2 w + \frac{3}{4} \pi ^2 + \frac{5813}{108} w - \frac{203}{36} \right) f_0 \nonumber \\&\quad + 8 \zeta _3 + \frac{79}{18} \pi ^2 + \frac{1301}{27} \Biggr ) \varepsilon \nonumber \\&\quad + \Biggl ( \frac{7}{2} w f_4 + \frac{1}{3} \left( \frac{103}{3} w + 7 \right) f_3 \nonumber \\&\quad + \frac{1}{3} \left( \frac{19}{4} \pi ^2 w + \frac{317}{3} w + 1 \right) f_2 \nonumber \\&\quad + \frac{1}{3} \left( 46 \zeta _3 w + \frac{271}{12} \pi ^2 w + \frac{19}{4} \pi ^2 + \frac{6677}{18} w - \frac{203}{6} \right) f_1 \nonumber \\&\quad - \frac{1}{3} \biggl ( \frac{199}{80} \pi ^4 w + \frac{317}{3} \zeta _3 w + 23 \zeta _3 + \frac{1693}{36} \pi ^2 w + \frac{5}{12} \pi ^2 \nonumber \\&\quad + \frac{129389}{216} w - \frac{6563}{72} \biggr ) f_0 \nonumber \\&\quad + \frac{1}{3} \left( \frac{199}{80} \pi ^4 + \frac{386}{3} \zeta _3 + \frac{427}{9} \pi ^2 + \frac{27425}{54} \right) \Biggr ) \varepsilon ^2 + \cdots \Biggr ] \nonumber \\&\quad - b^2 \Biggl [ \frac{4}{3} w f_3 + \left( \frac{19}{3} w + 1 \right) f_2 \nonumber \\&\quad + \frac{1}{3} \left( 4 \pi ^2 w + \frac{203}{3} w + 1 \right) f_1 \nonumber \\&\quad - \frac{1}{3} \left( 28 \zeta _3 w + \frac{38}{3} \pi ^2 w + 2 \pi ^2 + \frac{4919}{54} w - \frac{139}{6} \right) f_0 \nonumber \\&+ \frac{1}{3} \left( 28 \zeta _3 + \frac{44}{3} \pi ^2 + \frac{1834}{27} \right) \nonumber \\&\quad + \biggl ( 6 w f_4 + 4 \left( \frac{52}{9} w + 1 \right) f_3 \nonumber \\&\quad + \frac{1}{2} \left( 7 \pi ^2 w + \frac{1171}{9} w + \frac{5}{3} \right) f_2 \nonumber \\&\quad + \left( 44 \zeta _3 w + \frac{359}{18} \pi ^2 w + \frac{7}{2} \pi ^2 + \frac{5366}{27} w - \frac{310}{9} \right) f_1 \nonumber \\ \end{aligned}$$

$$\begin{aligned}&\quad - \biggl ( \frac{92}{45} \pi ^4 w + \frac{1114}{9} \zeta _3 w + 22 \zeta _3 + \frac{4075}{108} \pi ^2 w + \frac{17}{36} \pi ^2 \nonumber \\&\quad + \frac{258445}{972} w - \frac{9473}{108} \biggr ) f_0 \nonumber \\&\quad + \frac{1}{9} \left( \frac{92}{5} \pi ^4 + 1312 \zeta _3 + \frac{2063}{6} \pi ^2 + \frac{43297}{27} \right) \biggr ) \varepsilon + \cdots \Biggr ] \nonumber \\&\quad - b^3 \Biggl [ 3 w f_4 + 2 \left( \frac{19}{3} w + 1 \right) f_3 + \left( 2 \pi ^2 w {+} \frac{203}{6} w {+} \frac{1}{2} \right) f_2 \nonumber \\&\quad + \left( 24 \zeta _3 w + \frac{38}{3} \pi ^2 w + 2 \pi ^2 + \frac{4955}{54} w - \frac{139}{6} \right) f_1 \nonumber \\&\quad - \biggl ( \frac{71}{60} \pi ^4 w + \frac{233}{3} \zeta _3 w + 12 \zeta _3 + \frac{203}{9} \pi ^2 w + \frac{\pi ^2}{3} \nonumber \\&\quad + \frac{34937}{324} w - \frac{6007}{108} \biggr ) f_0 \nonumber \\&\quad + \frac{1}{3} \left( \frac{71}{20} \pi ^4 + 269 \zeta _3 + \frac{206}{3} \pi ^2 + \frac{4229}{27} \right) + \cdots \Biggr ] \nonumber \\&\quad + \cdots \Biggr \}. \end{aligned}$$

(C.14)

The form factor $F^V_2=H^V_2$ is finite at $\varepsilon =0$ (this requirement explains why $N^V_2$ (C.13) vanishes at $u=0$). We obtain

$$\begin{aligned}&F^V_2 = C_F \frac{b}{\beta _0} \Biggl \{ 2 f_0 - 2 (f_1 - 4 f_0) \varepsilon \nonumber \\&\quad - \left( f_2 + 8 f_1 - \left( \frac{\pi ^2}{6} + 16 \right) f_0 \right) \varepsilon ^2 \nonumber \\&\quad - \frac{2}{3} \left( f_3 + 6 f_2 + \left( \frac{\pi ^2}{4} + 24 \right) f_1 {+} \left( \zeta _3 {-} \pi ^2 {-} 48 \right) f_0 \right) \varepsilon ^3 \nonumber \\&\quad + \cdots \nonumber \\&\quad - b \Biggl [ 2 f_1 - \frac{25}{3} f_0 {+} \left( 3 f_2 {+} \frac{74}{3} f_1 {-} \frac{1}{2} \left( 3 \pi ^2 {+} \frac{961}{9} \right) f_0 \right) \varepsilon \nonumber \\&\quad + \frac{1}{3} \biggl ( 14 f_3 + 86 f_2 + \left( \frac{19}{2} \pi ^2 + \frac{1105}{3} \right) f_1 \nonumber \\&\quad - \left( 46 \zeta _3 + \frac{233}{6} \pi ^2 + \frac{23545}{36} \right) f_0 \biggr ) \varepsilon ^2 + \cdots \Biggr ] \nonumber \\&\quad - b^2 \Biggl [ 2 f_2 + \frac{50}{3} f_1 - \frac{1}{3} \left( 4 \pi ^2 + \frac{317}{3} \right) f_0 \nonumber \\&\quad + \biggl ( 8 f_3 + \frac{149}{3} f_2 + \left( 7 \pi ^2 + \frac{1912}{9} \right) f_1 \nonumber \\&\quad - \left( 44 \zeta _3 + \frac{521}{18} \pi ^2 + \frac{18451}{54} \right) f_0 \biggr ) \varepsilon + \cdots \Biggr ] \nonumber \\&\quad - b^3 \Biggl [ 4 f_3 + 25 f_2 + \left( 4 \pi ^2 + \frac{317}{3} \right) f_1 \nonumber \\&\quad - \left( 24 \zeta _3 + \frac{50}{3} \pi ^2 + \frac{8609}{54} \right) f_0 + \cdots \Biggr ] + \cdots \Biggr \}. \end{aligned}$$

(C.15)

Using HPL [29, 30] we have successfully reproduced all $n_l^{L-1} \alpha _s^L$ terms with $L=1$, 2, 3 in $F^V_{1,2}$ from [2].