Matching the heavy-quark fields in QCD and HQET at four loops

The QCD/HQET matching coefficient for the heavy-quark field is calculated up to four loops. It must be finite; this requirement produces analytical results for some terms in the four-loop on-shell heavy-quark field renormalization constant which were previously only known numerically. The effect of a non-zero lighter-flavor mass is calculated up to three loops. A class of on-shell integrals with two masses is analyzed in detail. By specifying our result to QED, we obtain the relation between the electron field and the Bloch--Nordsieck field with four-loop accuracy.


I. INTRODUCTION
Some classes of QCD problems with a single heavy quark can be examined in a simpler effective theory, the so-called heavy quark effective theory (HQET, see, e. g., [1][2][3]). Let us consider QCD with a single heavy flavor Q and n l light flavors (n f = n l + n h , n h = 1). The heavy-quark momentum can be decomposed as p = M v + k, where M is the on-shell Q mass, and v is some reference 4-velocity (v 2 = 1). In the case of QED, it is called Bloch-Nordsieck effective theory [4].
In the effective theory, the heavy quark (respectively lepton) is represented by the field h v . The MS renormalized fields Q(µ) and h v (µ) are related by [5] (1) where D µ ⊥ = D µ − v µ v · D, and the matching coefficient is given by Here Z os Q and Z os h are the on-shell field renormalization constants (they depend on the corresponding bare couplings and bare gauge-fixing parameters), and Z Q and Z h are the MS renormalization constants. The covariant-gauge fixing parameter is defined in such a way that the bare gluon propagator is given by (g µν − ξ 0 p µ p ν /p 2 )/p 2 ; it is renormalized by the gluon-field renormalization constant: 1 − ξ 0 = Z A (α s (µ), ξ(µ))(1 − ξ(µ)). The 1/M correction in (1) is fixed by reparametrization invariance [6].
The MS renormalized matching coefficient is obviously finite at ε → 0, because it relates the off-shell renormalized propagators in the two theories, which are both finite. The ultraviolet divergences cancel in the ratios Z Q /Z os Q and Z h /Z os h , because they relate renormalized fields; the infrared divergences cancel in Z os Q /Z os h , because HQET is constructed to reproduce the infrared behavior of QCD; the MS renormalization constants Z Q and Z h (purely off-shell quantities) are infrared finite. If we assume that all light flavors are massless we have Z os h = 1: all loop corrections vanish because they contain no scale, ultraviolet and infrared divergences of Z os h mutually cancel. Taking light-quark masses m i into account produces corrections suppressed by powers of m i /M , see Sect. III.
The matching coefficient satisfies the renormalizationgroup equation where the anomalous dimensions are defined as γ i = d log Z i /d log µ (i = Q, h). It is sufficient to obtain the initial condition z(µ 0 ) for some scale µ 0 ∼ M ; z(µ) for other renormalization scales µ can be found by solving Eq. (3). We choose to present the result for µ 0 = M . The heavy-quark field matching coefficient z(µ) has been calculated up to three loops [5]. When the matching coefficient is used within a quantity containing 1/ε divergences, terms with positive powers of ε in z(µ) are needed; such terms were not given in [5]. We present the four-loop result in Sect. II. Power corrections due to lighter-flavor masses up to three loops are obtained in Sect. III. The QED result, i. e. the four-loop relation between the lepton field and the Bloch-Nordsieck field, is discussed in Sect. IV. In Appendix A we provide analytic results for the decoupling coefficients for the strong coupling constant and the gluon field up to three-loop order including linear ε terms. Appendix B contains a detailed analysis of a class of on-shell integrals with two masses. It al-lows us, in particular, to obtain exact results for the three-loop term in the MS-on-shell mass relation with a closed massless and a closed lighter-flavor massive fermion loop (previously this term was only known as a truncated series in this mass ratio).
The on-shell heavy-quark field renormalization constant Z os Q depends on the bare coupling g (n f ) 0 , the bare gauge parameter ξ (n f ) 0 and the on-shell mass M : The two-loop expression is known exactly in ε [7]; it contains a single non-trivial master integral, further terms of its ε expansion are presented in [8,9]. The three-loop term has been calculated in [10,11]. At four loops, the terms with n 3 l and n 2 l are known analytically [12], and the remaining ones numerically [13]. Recently the QED-like color structures C 4 F , 3 , d F F n h have been calculated analytically [14]. Here and below we use the notation where , and the round brackets mean symmetriza- ). This result contains the same master integrals as the electron g − 2 [15,16].
In [15] they have been calculated numerically to 1100 digits, and analytical expressions have been reconstructed using PSLQ. In the case of the light-by-light contribution d F F n h the results contain ε 0 terms of 6 master integrals (known numerically to 1100 digits); all the remaining constants are completely expressed via known transcendental numbers (Note that the definition of the constant t 63 is missing in the journal article [14]; it is included in the version v3 of the arXiv publication.).
At four loops, some color structures are known analytically: A are known numerically [13].
We need to express the three terms in (4) in terms of the same set of variables, for which we choose α and ξ (n f ) 0 via these variables is straightforward, since the three-loop renormalization constants in QCD are well known. Expressing α (n l ) s (µ) and ξ (n l ) (µ) via the n f -flavor quantities requires decoupling relations up to O(ε) at three loops. For convenience we present explicit results in Appendix A.
The resulting matching coefficient z(M ) must be finite at ε → 0. This requirement together with the known results for Z Q and Z h leads to analytical expressions for the four-loop coefficients Z 4,0 , Z 4,1 , and Z 4,2 in (5) as well as for Z 4,3 , except two color structures C F C 3 A and d F A where the corresponding terms in γ h are not known analytically. The analytic results are presented in the tables I and II. We refrain from showing results for the n 2 l and n 3 l terms, which are already known since a few years [12]. Furthermore, we have introduced a n = Li n (1/2) (in particular a 1 = log 2); ζ n denotes the Riemann zeta function and ξ 0 = ξ (n f ) 0 . Analytical results for the color structures [14]. They agree with the expressions given in tables I and II. Numerical results for these coefficients are given in the tables V, VI, and VII of Ref. [13]. Good agreement is found.
Using the matching coefficient z(µ) together with quantities which contains 1/ε divergences, terms with positive powers of ε are needed. In order to get the finite four-loop contribution, we need the α L s term in z(µ) expanded up to ε 4−L . Our result for µ = M is given by   CF (TF n h ) 3 where α s = α given in Eqs. (28)(29)(30)(31)(32) of [14]. Their numerical values are given in Eqs. (5-9) of that paper. The finite four-loop terms of Eq. (7) are equal to the corresponding finite four-loop terms in Z os Q plus products of lower-loop quantities which are all known analytically. For 14 out of 23 color structures these coefficients in Z os Q are only known numerically [13]. We use these numerical values, together with their uncertainty estimates, from the tables V, VI, and VII of that paper. Note that in Ref. [13] Z os Q has been computed in an expansion in ξ up to the second order; 9 out of these 19 color structures are obviously gauge invariant, and 7 more seem to be either gauge-invariant or have at most linear ξ terms (though we know no explicit proof). The remaining 3 structures ( A T F n h ) may contain terms with higher powers of ξ, which are not known. The same is true for the corresponding terms in z(µ) in Eq. (7).
If we re-express z(M ) in Eq. (7) via α (n l ) s (M ), the terms up to three loops agree with [5]. (Note that positive powers of ε are not presented [5].) The α 4 s n 3 l term also agrees with [5]. After specifying the color factors to QCD with N c = 3 we obtain for ε = 0 In Landau gauge (ξ (n f ) = 1) at n l = 4 this gives while the naive nonabelianization [22] (large β 0 limit) pre- The comparison to Eq. (9) shows that up to four loops these predictions are rather good. The coefficients are all negative and grow very fast, which can be explained by the infrared renormalon at u = 1/2 [5]. This is the closest possible position of a renormalon singularity in the Borel plane u to the origin, and it leads to the fastest possible growth of perturbative terms (L − 1)! (β 0 /2) L (α s /π) L . The coefficients of powers of ξ are much smaller than the ξ-independent terms.

III. EFFECT OF A LIGHTER-FLAVOR MASS
Now we suppose that n m light flavors have a non-zero mass m, while the remaining n 0 = n l − n m light flavors are massless. In practice, n m = 1, e. g. c in b-quark HQET. In this case the massless result (7) for the matching coefficient should be multiplied by the additional factor , 0) = 1. This factor does not depend on the renormalization scale µ. In the expression we re-express all terms via α If we express z via α (n f ) s (µ), ξ (n f ) (µ), the coefficients will depend on µ. This dependence is determined by the renormalization-group equation together with Ultraviolet divergences cancel in each fraction in (11). On the other hand, the on-shell wave-function renormalization factors have extra infrared divergences at m = 0. However, z in Eq. (11) has a smooth limit for x → 0. In the following we illustrate the cancellation for infrared divergences at two-loop order. Similar mechanisms are also at work at higher loop orders. For dimensional reasons the two-loop corrections in Fig. 1a lead to log Z os h (m) ∼ g 4 0 m −4ε . Furthermore, we have log Z os h (0) = 0. Thus, the limit x → 0 is discontinuous. In QCD (Fig. 1b) we have log Z os Q (0) ∼ g 4 0 M −4ε for dimensional reasons. For m M there are 3 regions (see [28,29]): • Hard (all momenta ∼ M ): a regular series in m 2 , log Z os Q (m) hard = log Z os • Soft-hard (momentum of one m-line is ∼ m, all the remaining momenta are ∼ M ). If we take the term m from the numerator / k + m of the soft propagator, there is another factor m in the numerator of the hard mass-m propagator, and the soft-loop integral is ∼ m 2−2ε ; if we take / k instead, we have to expand the hard subdiagram in k up to the linear term, and the soft loop is ∼ m 4−2ε . We obtain log Z os • Soft (all momenta ∼ m): the leading term is the HQET one, the Taylor series is in x (not in As a result, log Z os Q (m) hard − log Z os Q (0) is smooth at x → 0; log Z os Q (m) soft-hard is subleading and hence smooth; log Z os Q (m) soft has the same discontinuity as log Z os h ; hence log z (12) has a smooth limit 1 at x → 0. ) has been calculated up to ε 0 in [7]; the result exact in ε has been obtained in [30]. The three-loop term has been calculated up to ε 0 in [31]. Some master integrals are only known as truncated series in x or as numerical interpolations, see [32] for detailed discussion of these master integrals. Exact results in x for the coefficient of C F T 2 F n m n 0 α 3 s can be obtained using the formulas of Appendix B. The ) at two loops has been calculated in [22], and at three loops in [33] (one of the master integrals is discussed in [34]; note that there are some typos in formulas in the journal version of [33] fixed later in arXiv).
Altogether we are now in the position to obtain z up to three loops. The expansion of z in terms of α (n f ) s (M ) and its decomposition into color factors is given by where The expansion of this function in x reads Note that the only terms with odd powers of x are x 1 and x 3 . The expansion in x −1 is given by For illustration we show in Fig. 2 The O(ε) term at two loops reads where L = log x. At three-loop order the C F T 2 F n m n 0 α 3 s term is known exactly via harmonic polylogarithms of x: where after the second equality sign we show the expansion in x. In principle, it is straightforward to obtain exact results in x also the four-loop C F T 3 F n n n 2 0 α 4 s term. However, we refrain from presenting such results because the remaining four-loop color structures are not known.
The remaining three-loop terms can be obtained in a series expansion in x with the help of the result from [31]. Including terms up to order x 8 gives 1213332979 83349000 13159 225 Starting from three loops the individual terms in Eq. (12) are gauge parameter dependent. However, ξ cancels in the threeloop expression for z . It might be that z is gauge invariant to all orders, but we have no proof of this conjecture.

IV. THE QED AND BLOCH-NORDSIECK HEAVY-LEPTON FIELDS
In QED the matching coefficient z(µ) is gauge invariant to all orders in α [5]. The proof given in this paper is literally valid only for n f = 1 lepton flavor, but can be easily generalized for any n f , as we demonstrate in the following.
The QED on-shell renormalization constant Z os ψ is gauge invariant to all orders [10,35,36]. Gauge dependence of the MS Z ψ can be found using the so-called LKF transformation [37,38] for arbitrary n f . In the gauge where the free photon propagator is the full bare lepton propagator reads where S L (x) is the Landau-gauge propagator. In the covariant gauge ∆(k) = (1 − ξ 0 )/(k 2 ) 2 , and∆(0) = 0 in dimensional regularization. The lepton fields renormalization does not depend on their masses, so, let us assume that all n f flavors are massless. The propagator has a single Dirac structure where S 0 (x) is the d-dimensional free propagator. Then re-expressing this result via the renormalized quantities, we obtain In QED Z A Z α = 1 due to Ward identities, hence d log((1 − ξ(µ))α(µ)) d log µ = −2ε exactly, and the anomalous dimension contains ξ only in the one-loop term.
In the Bloch-Nordsieck EFT with n l light lepton flavors Z os h is gauge-invariant (even if some of these flavors have non-zero masses). Gauge dependence of the MS Z h can be found using exponentiation. The full bare propagator is where w i are webs [39,40]. In QED all webs have even numbers of photon legs; all webs with > 2 legs are gauge invariant; all 2-leg webs except the trivial one (the free photon propagator) are gauge invariant, too. Therefore, re-expressing this result via the renormalized quantities, we obtain (26) Finally, in the abelian case ζ α (µ) = ζ A (µ) −1 due to Ward identities, hence (1 − ξ (n f ) (µ))α (n f ) (µ)) = (1 − ξ (n l ) (µ))α (n l ) (µ)), and we arrive at the conclusion that z(µ) is gauge invariant (some light flavors may be massive, this does not matter).
Let us in the following specify z(M ) from Eq (7) to QED. Setting C F = T F = d F F = 1 and C A = d F A = 0 we see that our four-loop result is indeed gauge invariant and is given by of Eq. (26) in [14]. Its numerical value is given in Eq. (15) in this paper.
Numerically, in pure QED (n l = 0) at ε = 0 we have where α = α (1) (M ), the MS QED coupling with one active flavor at µ = M , the on-shell electron mass. In contrast to the QCD case (7) the coefficients are numerically smaller and have different signs.

V. CONCLUSION
We have calculated the (finite) matching coefficient between the QCD heavy-quark field Q and the corresponding HQET field h v up to four loops. Explicit results are presented for µ = M ; results for different values of µ can be obtained with the help of (known) renormalization group equations. The effect of a non-zero light-flavor mass (e. g., c in b-quark HQET) is calculated up to three loops. We also present results for the matching constant in QED.
As a possible application of our results we want to mention the possibility to obtain the QCD heavy-quark propagator (say, in Landau gauge) from lattice QCD results for the HQET propagator. A heavy-quark field can be put onto the lattice only if M a 1, where a is the lattice spacing. On the other hand, in HQET simulations there is no lattice h v field at all. The HQET propagator is just a straight Wilson line, i. e. a product of lattice gauge links. It is therefore much easier to obtain the HQET propagator from lattice simulations.
After taking the continuum limit, one can get the continuum coordinate-space HQET propagator. Then the QCD heavyquark propagator can be obtained with the help of the matching coefficient z(µ), provided that 1/M n corrections can be neglected. Note that this can be done for arbitrarily heavy QCD quark, including the case when the use of the dynamic heavy-quark field on the lattice is impossible. The decoupling coefficients satisfy the renormalization group equations It is sufficient to have initial conditions, say, at µ = M for solving these equations. For the computation of z(M ) we need the decoupling coefficients up to α 3 s ε. Up to the order α 2 s expression exact in ε can be found in [41]. The finite three-loop results have been obtained in [42] in term of N c and in [43] for an arbitrary color group. The α 3 s ε terms were derived in the course of four-loop calculations [43][44][45]. However, results for an arbitrary color group, including positive powers of ε, are not explicitly presented in these publications. Therefore, we present them here: where α s = α (n f ) s (M ).

Appendix B: On-shell diagrams with two masses
Light-quark mass effects in the heavy-quark on-shell propagator diagrams arise for the first time at two loops, see Fig. 1b. The corresponding integral family can be defined as with p 2 = M 2 . If there are insertions to gluon lines in Fig. 1b containing only massless lines, such diagrams are expressed via the integrals (B1) with n 2 = n + lε, where l is the total number of loops in these insertions and n is integer (n 1,3,4 are always integer). These integrals have been studied in [30]. The IBP algorithm obtained there reduces them to four master integrals I 0,lε,1,1 = C lε , I 1,lε,1,0 = C lε , I 1,lε,1,1 = C lε , We set M = 1 and m = x.
It is more convenient to use the column vector j = I 0,lε,2,2 , I 2,lε,2,0 , I 2,lε,2,1 , I 1,lε,2,2 as master integrals instead of (B2). Differentiating them in m and reducing the results back to j [46], we obtain the differential equations In many cases such equations can be reduced to an ε-form [47] This makes their iterative solution to any order in ε almost trivial. Differential equations for on-shell sunsets I n1,0,n3,n4 were considered in [33,48], but they were not in ε-form. Several terms of small-x expansions were obtained from differential equations in [49]; it is easier to obtain them by calculating the corresponding residues in the Mellin-Barnes representation [30].