SFB/CPP-07-47 TTP07-21

Light quark mass effects in the on-shell renormalization constants

S. Bekavac^{(a)}, A. Grozin^{(b)}, D. Seidel^{(a)} and M. Steinhauser^{(a)}

(a) Institut für Theoretische Teilchenphysik, Universität Karlsruhe (TH), 76128 Karlsruhe, Germany (b) Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia

Abstract

We compute the three-loop relation between the pole and the minimally subtracted quark mass allowing for virtual effects from a second massive quark. We also consider the analogue effects for the on-shell wave function renormalization constant. PACS numbers: 12.38.Bx 12.38.-t 14.65.Dw 14.65.Fy

1 Introduction

Quark masses are fundamental parameters of the Standard Model (SM) and thus it is desirable to determine their numerical values with the highest possible precision. In order to do so it is necessary to fix a renormalization scheme which defines the quark mass. A renormalization scheme which is of particular importance in those situations where the quark mass is small as compared to the typical energy scale is the \overline{MS} scheme. Although it has no immediate physical interpretation, in general a good convergence in perturbation theory is observed. On the other hand, the most intuitive renormalization scheme is the on-shell scheme where the renormalized quark mass is defined as the pole of the propagator. It is well known that such a definition is plagued by numerically large long-distance effects which are connected to the renormalon corrections. As a consequence one observes a bad behaviour of the perturbative expansion if the pole mass is used as a parameter. Still, it plays a crucial role in all computations of threshold phenomena like energy levels or the decay of a quark-anti-quark bound state. In order to make contact between high-energy phenomena on one side and threshold processes on the other side it is necessary to have a precise relation between the pole and the \overline{MS} quark mass at hand. Important three-loop contributions are provided in this paper. As a by-product we also compute the same kind of corrections to the wave function renormalization constant.

The one-loop corrections to the relation between the \overline{MS} and on-shell mass have been computed almost 30 years ago in Ref. [?]. About ten years later, the two-loop corrections have been calculated in Ref. [?]. Shortly afterwards also the two-loop result for the onshell wave function renormalization constant Z_{2}^{OS} has been obtained [?]. Contributions with a single mass scale were expressed via master integrals exactly in d = 4 − 2ε; the ε-expansion of the only non-trivial master integral is known [?, ?] far enough to obtain O(ε^{4}) terms in Z_{m}^{OS} and Z_{2}^{OS}. Contributions with two mass scales were obtained up to O(1) only. Later, a reduction algorithm for two-loop two-scale on-shell integrals has been constructed, and expressions via master integrals exact in d have been obtained [?].^{1} The three-loop relation between the \overline{MS} and on-shell mass has been computed at the end of the nineties in a semi-numerical way in Refs. [?,?] where the off-shell fermion propagator has been considered for small and large external momenta. The on-shell quantities have been obtained with the help of a conformal mapping and Padé approximation. Later the results have been confirmed in Ref. [?] by an analytical on-shell calculation. The three-loop result for Z_{2}^{OS} has been obtained in Ref. [?]. Both analytical calculations have recently been rederived in Ref. [?]. In this paper we complete the three-loop results for the mass counterterm Z_{m}^{OS} and Z_{2}^{OS} by computing the contributions where a second mass scale is present through a closed quark loop. We would like to mention that the approximation linear in the mass ratio has been derived in Ref. [?] using the corresponding corrections to the static potential [?] and their cancellation in the relation between the \overline{MS} and 1S quark mass.

1The ε-expansion of the two non-trivial master integrals was only obtained up to O(1). One of these integrals was later expanded to O(ε^{5}) [?] (though only the O(ε) term is published). Both integrals were calculated [?] up to O(ε), as series in the mass ratio up to the sixth order.

We concentrate on the situation where the second quark mass is smaller than the external one, although our formulae can also be applied to the reversed situation. Then, however, often it is advantageous to perform a decoupling of the heavy mass leading to an effective theory where the latter is integrated out.

As far as the quark masses in the SM are concerned the corrections considered in this paper are of practical relevance for the bottom quark where the second mass scale is given by the charm quark. In this case we have for the mass ratio m_{c}/m_{b} ≈ 0.3. Thus the massless approximation does not provide a good result. In all other cases the effect from a lighter quark mass is negligible. Nevertheless, below generic results are presented.

In principle at three-loop order there is also a contribution involving two additional masses which are present in two closed fermion loops. However, for all practical applications the lightest quark mass can safely be neglected.

The remainder of the paper is organized as follows: In Section 2 we discuss the framework which is used for the calculation. Afterwards the results for the relation between the onshell and \overline{MS} quark mass and the wave function renormalization constant are discussed in Sections ?? and ??, respectively. Since the analytical expressions are quite involved we present in both cases handy approximation formulae of our results. Finally, Section ?? contains a simple application and our conclusions. Appendix ?? lists all master integrals in graphical from.

2 On-shell renormalization of quark mass and wave function

The formulae relevant for the computation of the renormalization constants for the mass and wave function have been derived in Refs. [?, ?]. For completeness we repeat the resulting expressions which read

Z_{m}^{OS} = 1 + Σ_{1}(M_{q}^{2}, M_{q}) ,  (1)
(Z_{2}^{OS})^{−1} = 1 + 2M_{q}^{2} \frac{∂}{∂q^{2}} Σ_{1}(q^{2}, M_{q})|_{q^{2}=M_{q}^{2}} + Σ_{2}(M_{q}^{2}, M_{q}) ,  (2)

where Z_{m}^{OS} and Z_{2}^{OS} are defined through

m_{q,0} = Z_{m}^{OS} M_{q} ,  (3)
ψ_{0} = \sqrt{Z_{2}^{OS}} ψ .  (4)

ψ is the quark field renormalized in the on-shell scheme with mass m_{q}, M_{q} is the on-shell mass and bare quantities are denoted by a subscript 0. Σ denotes the quark self-energy