PRL 114, 142002 (2015)

PHYSICAL

REVIEW

week ending 10 APRIL 2015

LETTERS

Quark Mass Relations to Four-Loop Order in Perturbative QCD Peter Marquard,1 Alexander V. Smirnov,2 Vladimir A. Smirnov,3 and Matthias Steinhauser4 'Deutsches Elektronen Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany "Scientific Research Computing Center, Moscow State University, 119991 Moscow, Russia 3Skobeltsyn Institute o f Nuclear Physics, Moscow State University, 119991 Moscow, Russia 4Institut fu r Theoretische Teilchenphysik, Karlsruhe Institute o f Technology (KIT), 76128 Karlsruhe, Germany (Received 7 February 2015; published 7 April 2015) We present results for the relation between a heavy quark mass defined in the on-shell and minimal subtraction (MS) scheme to four-loop order. The method to compute the four-loop on-shell integral is briefly described and the new results are used to establish relations between various short-distance masses and the MS quark mass to next-to-next-to-next-to-leading order accuracy. These relations play an important role in the accurate determination of the MS heavy quark masses. DOI: 10.1103/PhysRevLett. 114.142002

PACS numbers: 12.38.Bx, 12.38.Cy, 14.65.Fy, 14.65.Ha

The precise knowledge of quark masses plays an impor­ tant role in many phenomenological applications. This is in particular true for the heavy top, bottom, and charm quarks. For example, the top quark mass enters as a crucial parameter the combined electroweak fits which have been used to obtain indirect information about the Higgs boson mass, and nowadays serve as consistency checks for the standard model, see, e.g., Refs. [1,2]. The uncertainty in the top quark mass is also dominant in the analyses of the stability of the electroweak vacuum [3-5], A prominent example where a precise bottom quark mass is required are fi-meson decays which are often proportional to the fifth power of mb. Precise charm and bottom quark masses are important to obtain accurate predictions for the Higgs boson decays into the respective quark flavors. Also in the context of top and bottom Yukawa coupling unification, precise mass values are indispensable since they serve as boundary conditions at low energies. Last but not least, quark masses enter the Lagrange density of the standard model as fundamental parameters. Thus, it is mandatory to obtain precise numerical values by comparing high-order theoreti­ cal predictions with precise experimental data. At lowest order in perturbation theory there is no need to fix the renormalization scheme for the quark masses. However, after including quantum corrections it is necessary to apply renormalization conditions which fix the renorm­ alization scheme. A natural scheme for heavy quark masses, i.e., the charm, bottom, and top quark masses, is the on-shell (OS) scheme where one requires that the inverse heavy quark propagator with momentum q has a zero at the position of the on-shell mass, M, i.e., for q2 — M 2. It is well known that perturbation theory has a bad convergence behavior in case the on-shell quark mass is used as a parameter. Another widely used renormalization scheme is based on minimal subtraction. This means that the mass parameter entering the quark propagator is defined in such a way that just divergent terms (and no finite contributions) are absorbed 0031-9007/15/114(14)/142002(6)

such that the quark propagator is finite (after wave function renormalization). In this Letter we consider four-loop corrections to the relation between the on-shell and the minimal subtraction (MS) definition of a heavy quark mass that allows for a precise conversion from one renormaliza­ tion scheme into the other. For the various heavy quarks, different methods relying on different quark mass definitions are used to extract the mass values. For example, in Ref. [6] low-moment sum rules have been used to extract directly the MS charm and bottom quark masses without any reference to the on-shell mass. On the other hand, physical observables inherently connected to the threshold, like T sum rules or top quark pair production close to threshold, rely on properly defined quark masses, like the potential subtracted (PS) [7], IS [8-10], or renormalon subtracted (RS) [11] definition. When comparing with experimental data, in a first step the corresponding mass values are extracted. Afterwards, they are converted to the MS definition. Note that the relation between the MS and the OS mass is an important ingredient to obtain the relation between the PS, IS, or RS masses and the MS mass. In this Letter we use the four-loop MS-OS relation to establish relations between the PS, IS, RS and the MS quark mass, which are necessary to obtain the latter with next-to-next-to-next-to-leading order (N3LO) accuracy. Note that there is a further definition of a threshold mass, the so-called kinetic mass [12] which has been used for quite a number of applications in B physics, (see, e.g., Ref. [13]). However, the relation to the on-shell mass is only known to two loops (NNLO). For this reason it is not considered in the following. In the following we first discuss the relation between the MS and OS quark mass. Afterwards we elaborate on the relation between the threshold (PS, 1S, and RS) and the MS mass. The latter is obtained by using as starting point the definition of the PS, IS, or RS masses which establishes a

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PHYSICAL

PRL 114, 142002 (2015)

REVIEW

relation to the pole mass. Afterwards the pole mass is replaced by the MS mass which leads to the desired relation between the short-distance masses. To derive a formula relating the MS and OS quark mass it is advantageous to start with relations between theses masses and the bare mass, m°, which are given by (we refrain from adding a superscript “OS” to the on-shell mass but use a capital letter. Similarly, a lowercase m without further superscript stands for the MS quark mass. For all other mass definitions we use a lowercase “m” and a superscript indicating the renormalization scheme.) m °= Z ™ sm,

m° = Z ° sM.

(1)

Z ^ s is known to four loops and can be found in Refs. [14-16]. By construction, the ratio of the two equations in (1) is finite which leads to

ZmW

m (fi) M ’

(2)

where zm depends on as(ji) and log(n/M) and has the following perturbative expansion

with Zm = 1. For a derivation of convenient formulas relating zm(jj) to the on-shell quark self energy we refer to Refs. [17-19] where it is shown that Z °s is obtained from the sum of scalar and vector contribution evaluated on-shell, i.e., Z ° s = l + X v (q2 = M 2) + X s (q2 = M 2).

(4)

One-, two-, and three-loop QCD results to Z °s have been computed in Refs. [17,20] and [18,19,21,22], respectively, and electroweak effects have been considered in Refs. [23-27]. The main task of this Letter is the compu­ tation of the four-loop QCD corrections to Z°s and, consequently, to zm- For convenience we introduce also the inverse relation to Eq. (2) as follows: M = m(n)cm{n).

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where V(q) is the perturbative contribution to the static heavy quark potential. 5m(ji A can be computed in pertur­ bative QCD and has the form 8m(jif ) = n f

CfCxs

,1 +

as

-7— a \

4K

+ A) ( 2 + l°g^ 2 ^ (8)

+ ■

where /?0 = 11 - 2n//3 is the one-loop coefficient of the QCD fi function and a, = 3 1 / 3 - 10«//9 the one-loop coefficient of the static potential. «/ is the number of massless quarks. is the factorization scale which is of the order of the soft scale. In this Letter we use ftf = 2 GeV for bottom and Hj = 20 GeV for top quarks. 8m(fij) is known to N3LO [28], which involves the three-loop corrections to the static potential, a-s [29-31]. The N3LO relation between mps and m is obtained by inserting Eq. (5) into Eq. (6). All ingredients are already expanded and the coefficients of (as)" have simply to be combined. In particular, the term in Eq. (6) involving a 3 is combined with the four-loop term in the m-M relation. One obtains an explicit formula to compute the PS mass in case the MS is given. For a given PS mass we solve this equation iteratively to obtain the MS mass. The 1S mass is defined as half the perturbative mass of a fictitious 13S| state, where it is assumed that the quark is stable. Thus, we have the following relation between the IS and on-shell mass [8-10] mi s = M + l £ rt|M E „.1;

(9)

where £ pt is the perturbative ground state energy which is available to third order [28,32,33], The last missing ingredient was the three-loop static potential which has been evaluated in Refs. [29-31]. The replacement a" -» a"e"~l implements the so-called e expansion which guar­ antees that the appropriate orders in the expansions of E f and M (in terms of m) are combined. The perturbative expansion of E f has the following form Cl M a2 I 1 Ic f t - K j f i e m1 - 1 = - 6 -

(5)

( 10) n> 0

The PS quark mass has been introduced in Ref. [7]. Its relation to the pole mass is given by mps = M —

(6)

with 8m(/if ) — - ^

2 J\-q\

Quark mass relations to four-loop order in perturbative QCD.

We present results for the relation between a heavy quark mass defined in the on-shell and minimal subtraction (MS[over ¯]) scheme to four-loop order...
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