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Two-loop renormalization of the effective field theory of a static quark

We give a recurrence relation for two-loop integrals encountered in the effective field theory of an infinitely heavy quark, Q, interacting with gluons and Nl massless quarks, q, from which we obtain exact two-loop results, in any dimension and covariant gauge, for the propagator of Q and the vertex function of the heavy-light current J = Q Gamma q, at zero q momentum. The anomalous dimension of the Q field agrees with the recent result of Broadhurst, Gray and Schilcher. The anomalous dimension of the current is gamma_J = d log Z_J / d log mu = - alpha_s/pi (1 + (127 + 56 zeta(2) - 10 Nl)/72) alpha_s/pi + O(alpha_s^2)) which gives the new two-loop correction to the result of Voloshin and Shifman.


Introduction
Recently an interesting new approach to QCD problems involving a heavy quark was proposed, namely the effective field theory (EFT) of a static quark [1][2][3][4][5]. (See also the review [6].) To leading order in the 1/m expansion, the EFT lagrangian is [1] where q 0 and A 0µ are the bare light-quark and gluon fields of conventional QCD and Q 0 is the bare static-quark field, which is a two-component spinor. (When used as a 4-component spinor, its lower components are assumed to vanish.) The free-field propagator of Q 0 is 1/ω (equivalently, (1 + γ 0 )/2ω, in 4-component form) where ω ≡ p 0 − m is the energy by which it is off-shell. Its coupling to gluons is given by the vertex ig 0 v µ t a , where v µ = (1, 0) is its velocity. The lagrangian (1) possess an SU(2) static-quark spin symmetry [4].
In this paper we consider the two-loop renormalization of EFT, using dimensional regularization in D ≡ 4−2ǫ dimensions. In the MS scheme the bare quantities in (1) are related to the corresponding renormalized quantities by 3 and Z α are the same as in QCD [7], with N L light-quark flavours, andμ 2 = µ 2 e γ /4π. Here we calculate, to two loops, the static-quark wave-function renormalization constant Z 2 and the renormalization constant Z J of the heavy-light current J 0 = Q 0 Γq 0 = Z J J, where Γ is some (irrelevant) gamma matrix. For Z 2 we obtain the same result as Broadhurst, Gray and Schilcher [8], who have independently extracted it from the gauge-invariant singularities of on-shell wavefunction renormalization of a finite-mass quark [9]. ¿From Z J we obtain the twoloop correction to the leading-order result [10,11] for the anomalous dimension γ J ≡ d log Z J /d log µ. This is needed for the renormalization-group analysis of the extrapolations of lattice computations of heavy-meson decay constants [12].
Renormalization of EFT corresponds closely to renormalization of the Wilson line [13][14][15][16][17], which coincides with the static-quark propagator in coordinate space. In particular the lagrangian (1) is essentially the same as that proposed in [14], for investigating Wilson lines, and used in the two-loop calculations of [16,17]. In [16] Aoyama calculated Z 2 in the Feynman gauge, at N L = 0; we disagree with his result. In [17] Korchemsky and Radyushkin calculated the two-loop anomalous dimension of a heavy-heavy current, which depends on the relative velocity of initial and final quarks; the corresponding one-loop result [13] has recently been rediscovered in EFT [5].

One-and two-loop integrals
The static-quark bare self-energy term −iΣ(ω) is given, to two loops, by the diagrams of Fig 1, where the blob in Fig 1a includes the light-quark, gluon and ghost loops in the gluon propagator. We take the light quarks as massless and easily evaluate the master one-loop EFT integral where we adopt the convention that massless lines have greek indices and static lines have roman ones. We find that which enables us to evaluate diagrams of the form of Fig 1a. The master two-loop EFT integral, for diagrams of the form of Fig 1b, is In degenerate cases one uses (2) and the corresponding result [18] for the master one-loop diagram of massless QCD to obtain I(α, 0, γ, p, q) = I(α, 2γ + p + q − D)I(γ, q) (4) This is an enormous simplification in comparison with on-shell massive QCD calculations [8,9], where this last diagram was the most difficult.
In general we use a recurrence relation to reduce the non-degenerate case of the master integral to a sum of known degenerate cases. Using the method of integration by parts [9,18], we have found that where, for example, A + raises α and Q − lowers q. Again this is simpler than the triangle relation [18] of massless QCD, because no more than three terms appear and no static-line index is increased. Now it is a matter of programming to reduce every one-loop integral to a rational function of D ≡ 4 − 2ǫ times and every two-loop integral from Fig 1 to a linear combination of Γ 2 1 and with coefficients which are rational functions of D. We used REDUCE [19] on a Vax and checked results with Mathematica [20] on a Macintosh. The programming effort and run times were far less than in the related QCD calculations [8,9].

Anomalous dimension of the static-quark field
Using these methods, it is straightforward to compute the two-loop corrections to the bare propagator S 0 (ω). We find that with coefficients and colour factors C A = N , C F = (N 2 − 1)/2N , T F = 1 2 , for an SU(N ) gauge group, or C A = 0, C F = T F = 1 for a U(1) gauge group.
We now renormalize the bare static-quark field Q 0 =μ −ǫ Z 1/2 2 Q using the minimal wave-function renormalization constant Z 2 which makes the renormalized propagator S(ω) = S 0 (ω)/ Z 2 finite. (Tildes are used here to distinguish EFT renormalization constants from those of QCD.) We work in the MS scheme, in which the bare coupling and gauge parameter are renormalized by [7] where α s ≡ g 2 /4π. We posit a minimal form Z 2 = 1+ (C 11 (a)/ǫ)α s /π + (C 22 (a)/ǫ 2 + C 21 (a)/ǫ)α 2 s /π 2 + O(α 3 s ) and determine its coefficients from (10). The renormalizability of the theory requires that this constant make S 0 (ω)/ Z 2 finite for all ω, which provides a highly non-trivial check on our result. Then, using the µ dependence of α s and a, implied by (11,12), we obtain the anomalous dimension γ F ≡ d log Z 2 /d log µ of the static Q field. As a check on the whole procedure we mirror it, step by step, by that for a massless quark, to obtain the corresponding QCD quantity γ F for the massless q field from the recurrence relations of [18]. Our result is where the latter form was independently obtained in [8] from the gauge-invariant singularities of on-shell wave-function renormalization of a finite-mass quark. Note that (13) vanishes in QED in the renormalized Yennie gauge [21] a = 3, for which there is no 'infrared catastrophe' [22]. From (13) we find that γ F agrees with the QCD result of Egorian and Tarasov [23] (misquoted in [7], which contains sign errors).
Setting a = 1 and N L = 0 in (13), we discover an error in [16], where the renormalization of the Wilson line was calculated (neglecting light-quark loops) in the Feynman gauge (where Fig 1b gives no contribution). The α 2 s /ǫ 2 terms in [16] are inconsistent with the renormalization group and the α 2 s /ǫ terms include the contributions of Figs 1c,d with the wrong sign. This incorrect result was used in [24], where the authors also omitted a factor of 2 from the two-loop term in the anomalous dimension. In view of these approximations and errors the two-loop analysis of [24] needs reconsideration.

Anomalous dimension of the heavy-light current
The simplest method for finding the anomalous dimension γ J of the current J 0 = Q 0 Γq 0 is to consider the bare vertex function with zero light-quark momentum. The diagrams of Fig 2 then differ from those of Fig 1 only in their numerator structure and the indices of their denominators. The effect is to multiply the gamma matrix Γ by a factor Γ 0 (ω), which is gauge-dependent and more complicated than (10). But it becomes much simpler when one multiplies by the external static-quark bare propagator S 0 (ω). We find that is gauge-invariant. The coefficients are independently of the spin structure of the current J.
The renormalization constant Z J = Z Γ Z 1/2 2 Z 1/2 2 may be calculated from the requirement that the renormalized vertex function Γ(ω) = Γ 0 (ω)/ Z Γ be finite. Equivalently, but more conveniently, we have This gives two highly non-trivial checks on our results. First, the α 2 s /ǫ 2 terms in (14) must be such that division by a constant makes V (ω) finite for all ω. Second, the anomalous dimension γ V = d log Z V /d log µ must be gauge-invariant, to ensure that γ J = γ V − 1 2 γ F + 1 2 γ F is also gauge-invariant. As a third check, we mirrored this EFT calculation, step by step, by the QCD calculation of γ qq , obtaining agreement with Tarrach [25]. These checks give us considerable confidence in our final result whose second form we have also verified from the quark-condensate contributions to the correlator of J [26].

Summary and conclusions
We have given a method for calculating one-scale one-and two-loop EFT integrals. The master two-loop integral can be reduced by the recurrence relation (6) to the degenerate cases (3,4,5), which yield the structures (7,8). We have used this method to calculate the static-quark propagator and a particular case of the heavy-light vertex function, obtaining (10,14) in an arbitrary spacetime dimension and in any covariant gauge. From these follow the anomalous dimensions of the static-quark propagator (13) and the heavy-light current (15). The first agrees with [8] and contradicts an old result for the Wilson line [16]. The second is also confirmed in [26]. It gives the radiative correction to the one-loop result of [10,11] and is necessary for matching QCD and EFT currents, with account of the radiative corrections of [1]. The lack of it in [12] limited the accuracy of extrapolation of lattice computation.
Novel features of EFT calculations include: the trivial simplification of diagrams such as Figs 1d and 2i, which are the most difficult in on-shell massive QCD calculations [8,9]; an algorithm more efficient than the triangle relation [18] of massless QCD; the appearance of ζ(2) = π 2 /6 in counterterms, via the relation (9) between the basic integral structures, in contrast to the situation in QCD, where ζ(3) is the first term to appear in the expansion.