about the three-loop relation between the ms and the pole quark masses

ELSEVIER Nuclear Physics B (Proc. Suppl.) 89 (2000) 52-57

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About the three-loop relation between the MS and the pole quark masses

Kirill Melnikov a and Timo van Ritbergen b

aStanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA

bInstitut ffir Theoretische Teilchenphysik, Universit~it Karlsruhe, D-76128 Karlsruhe, Germany

We give a brief overview of the recent analytical calculation of the relation between the MS quark mass and the pole quark mass at the third order of Quantum Chromodynamics. Other new results for typical on-shell QCD quantities beyond two-loops are briefly discussed as well, such as a 3-loop result for the quark wave function renormalization factor Z2 in the on-shell scheme.

1. I N T R O D U C T I O N

Perturbative QCD calculations with on-shell quarks in the initial or final state have received significant attention in recent years, spurred by a growing importance of the field of heavy quark physics. In spite of the non-perturbative effects that are inherent to QCD, these perturbative calculations play an important role in establish- ing accurate relations between theory and exper- iment. In heavy quark physics, one separates soft and hard contributions through an expansion in the inverse quark mass; familiar examples being HQET (for a review see e.g.[l]) and NRQCD [2]. It often happens that the leading term in such an expansion is determined by a perturbative calculation in full QCD and for this reason perturbative calculations with heavy on-shell quarks in the initial and final state are an integral part of present-day phenomenology.

The need for increasingly precise perturbative QCD calculations for phenomenological applica- tions has also led to a better understanding of more fundamental issues, such as issues related to the proper definition of quark masses.

It is important to realize that quarks are not freely observable particles (unlike leptons) and the notion of a quark mass relies on a theoret- ical construction. Different definitions of quark masses exist referring to different schemes for the renormalization of the QCD Lagrangian of the strong interactions. Two prominent mass definitions are the MS mass ~ and the pole mass

M. The pole mass M is the renormalized quark mass in the on-shell (OS) renormalization scheme, while the MS mass is the renormalized quark mass in the modified minimal subtraction scheme [3,4] which is intimately related to the use of dimensional regularization [5].

For calculations with on-shell quarks in the initial or final states the on-shell renormalization scheme may seem like the most "natural" scheme but there are nevertheless compelling reasons for not expressing physical QCD quantities in terms of on-shell renormalized quark masses.

To begin with, one should realize that the pole mass of a quark is not a physical quantity in a truly non-perturbative sense since confinement of quarks in QCD implies that there is no pole in the quark propagator beyond the perturbation theory. Furthermore, it is known that the pole mass suffers from an unusually strong infrared sensitivity (even though the pole mass definition is infrared finite order-by-order) which re- veals itself through contributions in the perturbative series for the pole mass that grow facto- rially at higher orders corresponding to a singularity close to the origin in the Borel plane [6,7]. This (infrared) renormalon singularity is important for phenomenology since it implies that, the pole quark mass can not be determined with an accuracy better than 5M ~ AQCD [6,7]. Never- theless, the pole mass remains to be important for processes where the on-shell mass definition has technical advantages since the pole mass can afterwards be eliminated in favor of a mass that

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K. Melnikov, T van Ritbergen/Nuclear Physics B (Proc. Suppl.) 89 (2000) 52-57 53

is free from long distance ambiguities such as e.g. the M--S mass which is a very convenient reference mass for high energy processes i.e. for processes with energy scales far above quark produc- tion threshold.

It is believed that the large perturbative cor- rections that are explicitly found for many low scale quantities are related to the adoption of an improper quark mass definition, such as the pole mass, and that the perturbative series can be im- proved by choosing a less infrared sensitive mass definition. This is supported by the fact that the renormalon pole which appears for the pole mass, is absent in the corresponding physical quantities. Nevertheless, it should be kept in mind that, up to date, this physical picture has been checked (for lower orders) only against the first non-trivial order of perturbation theory.

For completeness we should also mention that it often happens that the MS mass is not the op- timal choice to parameterize the low-scale processes and recently other short-distance low-scale masses have been proposed [8-10] that can be accurately determined from the analysis of the low energy data [11-13]. The extracted values of those low-scale masses can then be confronted with an MS reference mass once their relation to the MS mass is calculated with a sufficiently high precision, in practice requiring high perturbative orders. For technical reasons, these mass relations go via the pole mass and one needs to calculate the relation between the MS and the pole quark mass to a very high perturbative order to reach the required precision in such mass relations.

In order to obtain the 3-loop relation between the MS and the pole quark masses, and as a first step towards the calculation of other on-shell quantities at the third order, we need to calculate the 3-loop renormalization constants (Z-factors) of QCD in the on-shell scheme. A brief overview of the corresponding calculations [14,15] is given in the next section.

2 . O N - S H E L L Z - F A C T O R S

Quark masses in both, the MS and the on-shell, renormalization schemes are renormalized multi-

plicatively and the connection between renormalized and bare quark masses is defined as

m0 = Z,~ m, (1) OS m0 = M, (2)

where mo is the bare (un-renormalized) quark

mass, and Z Ms and Z °s are the renormalization factors for the quark mass in, respectively, the MS and the on-shell schemes.

In a similar way the quark wave function renormalization constant is defined,

¢0 = x/G ¢ (3)

where ~b0 and ¢ stand for respectively the bare and the renormalized quark fields and the wave function renormalization constant Z2 depends on the chosen renormalization scheme. Since the bare quark mass is (trivially) renormalization scheme independent the relation between the pole quark mass and the MS mass is easily expressed as the ratio

Zm °°s - , , ( 4 )

M ZMS

One sees that in order to calculate the relation between the MS and the pole quark mass in the three-loop order one needs to calculate the mass renormalization factors in both the MS and the on-shell schemes in the three-loop order. For- tunately, due to the very simple form of renormalization factors in the MS scheme, the renormalization factor Zm Ms, (and also Z MS) is already known at the 4-loop order of perturbation theory and can therfore be taken from the literature.

In comparison to the MS-scheme, the calculation of renormalization factors in the on- shell scheme is much more involved and for this reason the on-shell renormalization factors were known only in the two-loop approximation. Both the renormalization constants Zm °s and Z °s can be derived by considering the one-particle irreducible quark self-energy operator ~(p, M). This object can be parameterized by two independent functions:

~(p, M) = MP.1 (p 2, M) + (i5 - M)E2(p 2, M), (5)

54 K. Melnikov, 7:. van Ritbergen/Nuclear Physics B (Proc. Suppl.) 89 (2000) 52-57

r

A B C D

E

I J

v

K

Figure 1. Examples of three-loop quark propagator diagrams corresponding to eleven integration topologies. The diagrams in boxes are of the abelian type (i.e. QED-like).

so that the complete fermion propagator reads:

= i 15 - mo + Z(P, M)" (6)

Identifying the position of the pole with the pole mass (which is the renormalized mass in the on- shell scheme) and the residue with the wave function renormalization constant, one obtains

Z m = 1 + ~ l ( p 2 , M ) [ p 2 = M 2 ,

1 = 1 + . ,

Z2

+ 2 M 2 + r l(p2,M)lp==M . (7)

The pole mass in the above equation is calculated iteratively by using mass counterterms where ap- propriate, i.e. by calculating lower order diagrams with mass counterterm insertions.

Hence, both Z,~ and Z2 can be extracted from the heavy quark self-energy operator and its first derivative evaluated on-shell. This task becomes increasingly difficult when one goes to higher orders. In the present case we have about sixty diagrams to be calculated and the question about an efficient way to do the calculation becomes of

tremendous importance. An important observation is that all of the integrals that one can face while computing the on-shell fermion propagator can be expressed through eleven basic three-loop integrals (topologies) shown in Fig.1. Any integral that belongs to a certain topology is consid- ered to be a function of the powers of denomi- nators (both positive and negative integer powers are allowed) and one irreducible numerator in each case.

For each of the topologies one writes down a system of recurrence relations based on integration-by-parts identities that are derived using the fact that in dimensional regularization any integral of the total derivative vanishes:

0 = f ~k~ (/~ x propagators ) . (8)

Here ki is one of the three loop momenta and lj is either one of the loop momenta or the external momenta. Hence the starting set of equations consists of twelve recurrence relations for each of the topologies. One may solve these equations to yield a recursive algorithm that can be imple- mented in a computer program, such that after a sufficient number of iterations, all integrals with

K. Melnikov, T. van Ritbergen/Nuclear Physics B (Proc. Suppl.) 89 (2000) 52-57 55

general powers of the propagators axe expressed through a finite set of integrals that have propagators only raised to a power one or zero. These simpler integrals are generally called "primitive" integrals, and only these integrals in the whole calculation have to be computed by direct integration.

In order to convince the reader that the 3-loop calculation of both Zm and Z2 is feasible, and can even be done analytically, we should note that of the eleven topologies shown in Fig.l, four are of the abelian type, and these abelian topologies have been encountered previously in the "clas- sical" 3-loop calculation of the electron (9 - 2) in QED [16]. This means that one expects that most of the necessary integrals that one encoun- ters in the calculation of Zm and Z2 have been calculated already for the electron (g - 2). Fur- thermore, the experience from other multi-loop calculations in QCD shows that the remaining non-abelian topologies of Fig.1 are generally re- ducible to the abelian topologies when one uses integration-by-paxts identities.

This expectation was found to be correct: after solving the integration-by-parts relations for each of the eleven topologies of Fig.l, and calculation of all diagrams that contribute to the one-paxticle irreducible quark self-energy operator E(p,M), we found that in addition to 17 primitive integrals already known from the electron (g - 2) calculation [16] we needed only one genuinely new non-abelian primitive integral. Furthermore, we also needed to evaluate two of the master integrals of Ref. [16] to a higher order in the regulaxiza- tion parameter E. The corresponding results for these 3 primitive integrals can be found in Ref. [15]. We should also mention that as a check of our solution of the recurrence relations we have re-evaluated the three-loop anomalous magnetic moment of the electron and confirmed the result of Ref.[16].

The result for the 3-1oop relation between the MS mass and the Pole quark mass that we obtained by calculating Zm °s and subsequently using Eq. (4) reads:

~ ( M ) _ l _ 4 ( _ _ ~ ) M 3

- 28--g- + g 6 - V log2 -

+ ( 2332-----8 + ~3 + 81

-27r21og 2 2 + 9 6 7 2 61 7r 4 81 6--~ 7r 1944 1 8 "~ 9478333

81 l°g4 2 - ~ a 4 ) 93312

1439. 2 61 + - - 12 65¢5

587 21 22 2 644201 .2 +l-~Tr og2+~-~Tr log22 38880 ~

695 4 1~21og42+220 } + + 7 a 4 (9)

o¢ where a4 = ~] 1/(2'~n 4) = Li4(1/2) ~ 0.517479

r~=l and NL stands for the number of light quark fla-

OL(NL+I) { ~/f~ vors. ae -= s ~.~j is the MS strong cou- pling constant renormalized at the scale of the pole mass # = M in the theory with (NL + 1) ac- tive flavors. Numerically, we obtain for the mass relation:

~ ( M ) _ 1 - 4

(1.0414 NL - 14.3323)

(--0.65269 N~ + 26.9239 NL

-198.7068). (10)

First of all, we should note that a more general result in terms of color group invariants can be found in Ref.[14]; a full result for the underlying on-shell mass renormalization factor Zm °s can be found in Ref.[15].

The two-loop contribution in Eq.(9) coincides naturally with the well-established second order result [17,18]. Furthermore, the three-loop co- efficients in Eq.(10) are in good agreement with the recent numerical study of Ref.[19]. It is interesting to note how close our exact result is to the so-called large-/3o approximation which

56 K. Melnikov, 7:. van Ritbergen/Nuclear Physics B (Proc. Suppl.) 89 (2000) 52-57

leads to a three-loop coefficient of approximately -180 + 22NL - 0.65N 2. Finally it is worth not- ing that our result is also remarkably close to the FAC/PMS estimates of Ref.[20]. However, in spite of these numerical agreements one should realize that an exact analytic confirmation of our result remains as an open task.

2.1. Wave funct ion renormal izat ion

We have also calculated the on-shell wave function renormalization constant Z ° s at the third order of perturbative QCD. Compared to Zm °s this is a slightly more sophisticated calculation since it involves taking the first derivative with respect to the external momentum of the self energy operator, see Eq.(7). An interesting open question concerning this renormalization factor is related to the observation of Ref. [17] that at two loops Z °s is gauge independent, which was fol- lowed by a conjecture about the possible all-order gauge-independence of Z °s .

To elaborate a bit more on this issue, we should note that no formal proof has been given for a gauge independence of Z ° s in QCD. However, in the abelian case, i.e. QED, such a proof does exist and takes the form of the Johnson-Zumino identity [21-23]:

dlogZ2 _ie2 ~ dDk d ~ = _ (2~r)Dk4 = 0 , (11)

where ~ is the gauge fixing parameter in a general covariant gauge. We note that the right hand side of the above equation is zero by virtue of the fact that scale-less integrMs in dimensional regularization are defined to be zero. The importance of the choice of the infrared regulator (here: dimensional regularization) comes from the fact that, in contrast to Z °s , the wave function renormalization constant Z ° s is not infrared finite.

Our explicit cMculation for Z ° s in QCD [see Ref. [15] for a complete list of the results] shows that the gauge dependence appears in the three- loop order. This gauge dependence is sparse however and appears only in two non-abelian color structures. Furthermore, a very strong check on these color structures is obtained by relating Z °s to the analogous quantity in heavy quark effec-

tive theory, on the basis of which we may con- clude that the Johnson-Zumino identity does not extend to Quantum Chromodynamics.

A C K N O W L E D G M E N T S

We are grateful to D.J. Broadhurst and A. G. Grozin for useful conversations. This re- search was supported in part by the United States Department of Energy, contract DE- AC03-76SF00515, by BMBF under grant number BMBF-057KA92P, by Graduiertenkolleg "Teil- chenphysik" at the University of Karlsruhe and by the DFG Forschergruppe "Quantenfeldtheorie, Computeralgebra und Monte-Carlo-Simulation'.

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