Quark Mass and Field Anomalous Dimensions to ${\cal O}(\alpha_s^5)$

We present the results of the first complete analytic calculation of the quark mass and field anomalous dimensions to ${\cal O}(\alpha_s^5)$ in QCD.


Introduction
The quark masses depend on a renormalization scale.The dependence is usually referred to as "running" and is governed by the quark mass anomalous dimension, γ m , defined as: where a s = α s /π = g 2 /(4π 2 ), g is the renormalized strong coupling constant and µ is the normalization scale in the customarily used MS renormalization scheme.Up to and including four loop level the anomalous dimension is known since long [1][2][3][4][5].In this paper we will describe the results of calculation of γ m and a related quantity -the quark field anomalous dimension -in the five-loop order.The evaluation of the quark mass anomalous dimension with five-loop accuracy has important implications.The Higgs boson decay rate into charm and bottom quarks is proportional to the square of the respective quark mass at the scale of m H and the uncertainty from the presently unknown 5-loop terms in the running of the quark mass is of order 10 −3 .This is comparable to the precision advocated for experiments e.g. at TLEP [6].Similarly, the issue of Yukawa unification is affected by precise predictions for the anomalous quark mass dimension.
The paper is organized as follows.The next section deals with the overall set-up of the calculations.Then we present our results (Section 3), and a brief discussion (Section 4) as well as a couple of selected applications (Section 5).Our short conclusions are given Section 6.

Technical preliminaries
To calculate γ m one needs to find the so-called quark mass renormalization constant, Z m , which is defined as the ratio of the bare and renormalized quark masses, viz.
Within the MS scheme [7,8] the coefficients (Z m ) ij are just numbers [9]; ǫ ≡ 2−D/2 and D stands for the space-time dimension.Combining eqs.(1.1,2.1) and using the RG-invariance of of m 0 , one arrives at the following formula for γ m : To find Z m one should compute the vector and scalar parts of the quark self-energy Σ V (p2 ) and Σ S (p 2 ).In our convention, the bare quark propagator is proportional to Requiring the finiteness of the renormalized quark propagator and keeping only massless and terms linear in m q , one arrives at the following recursive equations to find Z m where K ǫ {f (ǫ)} stands for the singular part of the Laurent expansion of f (ǫ) in ǫ near ǫ = 0 and Z 2 is the quark wave function renormalization constant.Eqs.(2.3) express Z m through massless propagator-type (that is dependent on one external momentum only) Feynman integrals (FI), denoted as p-integrals below.
Eqs. (2.3) require the calculation of a large number1 the five-loop p-integrals to find Z m and Z 2 to O(α 5 s ).At present there exists no direct way to analytically evaluate five-loop p-integrals.However, according to (2.1) for a given five-loop p-integral we need to know only its pole part in ǫ in the limit of ǫ → 0. A proper use of this fact can significantly simplify our task.The corresponding method-so-called Infrared Rearrangement (IRR)-first suggested in [11] and elaborated further in [12][13][14] allows to effectively decrease number of loops to be computed by one 2 .In its initial version IRR was not really universal; it was not applicable in some (though rather rare) cases of complicated FI's.The problem was solved by elaborating a special technique of subtraction of IR divergences -the R * -operation [15,16].This technique succeeds in expressing the UV counterterm of every L-loop Feynman integral in terms of divergent and finite parts of some (L-1)-loop massless propagators.
In our case L = 5 and, using IRR, one arrives at at around 10 5 four-loop p-integrals.These can, subsequently, be reduced to 28 four-loop masterp-integrals, which are known analytically, including their finite parts, from [17,18] as well as numerically from [19].
We need, thus, to compute around 10 5 p-integrals.Their singular parts, in turn, can be algebraically reduced to only 28 master 4-loop p-integrals.The reduction is based on evaluating sufficiently many terms of the 1/D expansion [20] of the corresponding coefficient functions [21].
All our calculations have been performed on a SGI ALTIX 24-node IB-interconnected cluster of eight-cores Xeon computers using parallel MPI-based [22] as well as thread-based [23] versions of FORM [24].

Results
Our result for the anomalous dimension reproduce well-known results obtained in [4,5].The boxed terms in (3.4) are in full agreement with the results derived previously on the basis of the 1/n f method in [25][26][27].
For completeness we present below the result for the quark field anomalous dimension The above result is presented for the Feynman gauge; the coefficients (γ 2 ) i with i ≤ 3 can be found in [28] (for the case of a general covariant gauge and SU(N) gauge group).

Discussion
In numerical form γ m reads Note that significant cancellations between n 0 f and n 1 f terms for the values of n f around 3 or so persist also at five-loop order.As a result we observe a moderate growth of the series in a s appearing in the quark mass anomalous dimension at various values of active quark flavours (recall that even for scales as small as 2 GeV a s ≡ αs π ≈ 0.1).with a 15 years old prediction based on the "Asymptotic Páde Approximants" (APAP) method [29] (the boxed term below was used as the input) Unfortunately, this impressively good agreement does not survive for fixed values of n f due to severe cancellations between different powers of n f as one can see from the Table 1.

RGI mass
Eq. (4.7) naturally leads to an important concept: the RGI mass which is often used in the context of lattice calculations.The mass is µ and scheme independent; in any (mass-independent) scheme The function c s (x) is used, e.g, by the ALPHA lattice collaboration to find the MS mass of the strange quark at a lower scale, say, m s (2 GeV) from the m RGI s mass determined from lattice simulations (see, e.g.[40]).For example, setting a s (µ = 2 GeV) = αs(µ) π = 0.1, we arrive at (h counts loops): 1 + 0.0895 h 2 + 0.0137 h 3 + 0.00195 h 4 +(0.00157− 0.000011 β 4 ) h 5  (5.2) In order to have an idea of effects due the five-loop term in (5.2) one should make a guess about β4 .By inspecting lower orders in a s + 1.777 a 2 s + 4.4711 a 3 s + 20.990 a4 s + β4 a 5 s one can assume a natural estimate of β 4 as laying in the interval 50 − 100.With this choice we conclude that the (apparent) convergence of the above series is quite good even at a rather small energy scale of 2 GeV.On the other hand, the authors of [30] estimate β4 in the n f = 3 QCD as large as -850!With such a huge and negative value of β4 the five loop term in (5.2) would amount to 0.01092 and, thus, would significantly exceed the four-loop contribution (0.00195).

Higgs decay into quarks
The decay width of the Higgs boson into a pair of quarks can be written in the form where µ is the normalization scale and R S is the spectral density of the scalar correlator, known to α 4 s from [41] where we set a s = α s /π = 0.0360 (for the Higgs mass value M H = 125 GeV and α s (M Z ) = 0.118).Expression (5.3) depends on two phenomenological parameters, namely, α s (M H ) and the quark running mass m q .In what follows we consider, for definiteness, the dominant decay mode H → bb.To avoid the appearance of large logarithms of the type ln µ 2 /M 2 H the parameter µ is customarily chosen to be around M H .However, the starting value of m b is usually determined at a much smaller scale (typically around 5-10 GeV [42]).The evolution of m b (µ) from a lower scale to µ = M h is described by a corresponding RG equation which is completely fixed by the quark mass anomalous dimension γ(α s ) and the QCD beta function β(α s ) (for QCD with n f = 5).In order to match the O(α s 4 ) accuracy of ( 5 is around -2 (for β4 = 100).This should be contrasted to the parametric uncertainties coming from the input parameters α s (M Z ) = 0.1185(6) [44] and m b (m b ) = 4.169(8) GeV [45] which correspond to ± 1 and ± 4 respectively.We conclude, that the O(α 4 s ) terms in (5.4), (5.5)) are of no phenomenological relevancy at present.But, the situation could be different if the project of TLEP [6] is implemented.For instance, the uncertainty in α s (M Z ) could be reduced to ±2 and Higgs boson branching ratios with precisions in the permille range are advertised.

Table 1 :
The exact results for (γ m ) 4 together with the predictions made with the help of the original APAP method and its two somewhat modified versions.