Precise quark masses from sum rules

Matthias Steinhauser

Institut für Theoretische Teilchenphysik, Universität Karlsruhe (TH) 76128 Karlsruhe, Germany

In this contribution an improved analysis is described to extract precise charm and bottom quark masses from experimental and theoretical moments of the photon polarization function. The obtained \overline{MS} mass values read m_{c}(3 GeV) = 0.986(13) GeV and m_{b}(10 GeV) = 3.609(25) GeV.

1 Introduction

The theory of strong interaction has the strong coupling constant and the quark masses as fundamental input parameters. The latter constitute an essential input for the evaluation of weak decay rates of heavy mesons and for quarkonium spectroscopy. Furthermore, decay rates and branching ratios of a light Higgs boson — as suggested by electroweak precision measurements — depend critically on the masses of the charm and bottom quarks. Last not least, confronting the predictions for these masses with experiment is an important task for all variants of Grand Unified Theories. To deduce the values in a consistent way from different experimental investigations and with utmost precision is thus a must for current phenomenology.

The method described in this contribution goes back to 1977 [?] and was applied to next-to-next-to-leading order (NNLO) in Ref. [?]. The NNNLO analysis, including updated experimental input, was presented in Ref. [?].

2 Moments

The basic object which enters our analysis is the photon polarization function defined through

(−q^{2}g_{μν} + q_{μ}q_{ν}) Π(q^{2}) = i ∫ dx e^{iqx}⟨0|T j_{μ}(x)j_{ν}^{†}(0)|0⟩ ,  (1)

with j_{μ} being the electromagnetic current. The normalized total cross section for hadron production in e^{+}e^{−} annihilation is then given by

R(s) = \frac{σ(e^{+}e^{−} → hadrons)}{σ_{pt}} = 12π Im [Π(q^{2} = s + iϵ)] ,  (2)

where σ_{pt} = 4πα^{2}/(3s). In the following we add a subscript Q to indicate the contribution from the heavy quark Q.

The idea for extracting a quark mass value m_{Q} is based on moments constructed from Π_{Q}. On one hand one can compute the Taylor expansion of Π_{Q}(q^{2}) around q^{2} = 0 and obtain the so-called “theory-moments” from

M_{n} = \frac{12π^{2}}{n!} (\frac{d}{dq^{2}} )^{n} Π_{Q}(q^{2})|_{q^{2}=0} .  (3)

The three-loop contribution to Π_{Q}(q^{2}) up to n = 8 within QCD has been computed in Refs. [?, ?] and the four-loop calculation for n = 0 and n = 1 has been performed in Refs. [?, ?]. In the analysis of Ref. [?] also two-loop QED corrections and non-perturbative contributions have been considered. The latter shows a visible effect only in the case of the charm quark.

From dimensional considerations we have m_{Q} ∼ (M_{n})^{\frac{1}{2n}} which implies a stronger dependence of m_{Q} on variations of M_{n} for smaller values of n. Furthermore, higher values of n require a careful theoretical treatment of the threshold region and the construction of an effective theory. The analysis performed in Ref. [?] is restricted to n = 1, 2, 3 and 4. Note that precise mass values can only be obtained for the three lowest moments since the non-perturbative contributions become too big already for n = 4.

One of the major advantages of the method discussed in this paper is that we can adopt the \overline{MS} scheme for the quark mass entering Eq. (3) and thus directly extract the corresponding value for the mass.

In order to extract experimental moments one exploits the analyticity of Π_{Q} and arrives at

M_{n} = ∫ \frac{ds}{s^{n+1}} R_{Q}(s) ,  (4)

where R_{Q} naturally divides into three parts: At lower energies one has the narrow resonances which are the J/Ψ and Ψ′ for charm the Υ(nS) (n = 1, …, 4) in the case of the bottom quark. The corresponding contributions to M_{n} are obtained with the help of the narrow width approximation for R(s)

R^{res}(s) = \frac{9πM_{R}Γ_{ee}}{α^{2}} (\frac{α}{α(s)})^{2} δ(s − M_{R}^{2}) ,  (5)

where the electronic widths Γ_{ee} are known at the 1-2% level.

The second part is called threshold region and extends in the case of the charm quark from 3.73 GeV to about 5 GeV. In this region the cross section shows a rapid variation and can not be described by perturbation theory. Measurements from the BES collaboration from 2001 [?] and 2006 [?] provide excellent data for R(s) with an uncertainty of about 4%. In order to obtain R_{c} one has to subtract the contribution from the light quarks which is explained in detail in Ref. [?].

The treatment of the bottom threshold region is quite similar. Measurements of R from threshold up to 11.24 GeV have been performed by the CLEO Collaboration more than 20 years ago [?], with a systematic error of 6%. No radiative corrections were applied. The average value derived from the four data points below threshold amounts to R̄ = 4.559±0.034(stat.) which is 28% larger than the prediction from perturbative QCD (pQCD). However, a later result of CLEO [?] at practically the same energy, R(10.52 GeV) = 3.56 ± 0.01 ± 0.07, is significantly more precise and in perfect agreement with theory. Applying a rescaling factor of 1/1.28 to the old CLEO data not only enforces agreement between old and new CLEO data and pQCD in the region below the Υ(4S), it leads, in addition, also to excellent agreement between theory and experiment above threshold around 11.2 GeV where pQCD should be applicable also to bottom production. Further support to our approach is provided by the CLEO measurement of the cross section for bottom quark production at \sqrt{s} = 10.865 GeV which is given by σ_{b}(\sqrt{s} = 10.865 GeV) = 0.301±0.002±0.039 nb [?]. The