m_u+m_d From Isovector Pseudoscalar Sum Rules

We revisit the isovector pseudoscalar sum rule determination of m_u+m_d, using families of finite energy sum rules known to be very accurately satisfied in the isovector vector channel. The sum rule constraints are sufficiently strong to allow a determination of both m_u+m_d and the excited resonance decay constants. The corresponding Borel transformed sum rules are also very well satisfied, providing a non-trivial consistency check on the treatment of direct instanton contributions. We obtain [m_u+m_d](2 GeV)=7.8\pm 1.1 MeV (in the MS bar scheme), only marginally compatible with the most recent sum rule determinations, but in good agreement with recent unquenched lattice extractions.


I. INTRODUCTION
Because of the Ward identity ∂ µ A µ ud = (m u + m d )ūiγ 5 d, a study of the correlator allows one, in principle, to determine m u + m d [1]. A number of such studies have been performed [1][2][3][4], the most recent (Refs. [3] (BPR) and [4] (P98)) employing, respectively, 3- Recent unquenched lattice simulations, in contrast, yield [5,6] [m u + m d ](2 GeV) = 6.88 +.28 where the errors do not reflect the uncertainty involved in using perturbative versions of the renormalization constants. Because the consistency of the lattice and sum rule determinations is not particularly good, we revisit the sum rule treatment of Π ud .
In this paper, we study Π ud using Borel transformed and finite energy sum rules (BSR's and FESR's). The BSR's have the form [7] where M is the Borel mass, s 0 the "continuum threshold", and B [Π ′′ ud ] (M 2 ) the Borel where s 0 is arbitrary and w(s) is any function analytic in the region of the contour.
The π contribution to ρ ud , [ρ ud (s)] π = 2f 2 π m 4 π δ (s − m 2 π ), with f π = 92.4 MeV, is very accurately known. The decay constants of the π(1300) and π(1800), needed to describe the remaining contributions to ρ ud below s ∼ 4 GeV 2 , are not known, and need to be determined as part of the sum rule analysis.
The LHS of Eq. (4) can be evaluated using the OPE, provided that M is sufficiently large compared to the QCD scale. The condition that s 0 be similarly large, though necessary, is not sufficient for the OPE to be employed reliably on the LHS of Eq. (6) since, except for extremely large s 0 , the OPE is expected to break down near the timelike real axis [8]. For the isovector vector (IVV) channel, this breakdown can be seen explicitly using the very precise spectral data available from hadronic τ decay [9]: FESR's involving w(s) = s k with k = 0, 1, 2, 3 (which fail to suppress contributions from the region of the circle |s| = s 0 near the timelike real axis) are rather poorly satisfied at scales 2 GeV 2 < s 0 < m 2 τ [10]. The breakdown of the OPE, however, turns out to be very closely localized to the vicinity of the timelike axis: FESR's based on weights having even a single zero at s = s 0 are very accurately satisfied over this whole range [10]. Thus at the scales 2 GeV 2 < s 0 < 4 GeV 2 of interest to us, the supplementary constraint w(s 0 ) = 0 must be imposed in order to obtain reliable FESR's. We call such FESR's "pinch-weighted", or pFESR's.
The OPE representation of Π ud (Q 2 ) is known up to dimension D = 6, with the dominant D = 0 perturbative contribution known to 4-loop order [11,12]. Working with Π ′′ (Q 2 ), which allows logarithms to be summed via the scale choice µ 2 = Q 2 , one has [11,12] whereā ≡ a(Q 2 ) = α s (Q 2 )/π,m k ≡ m k (Q 2 ), with α s (Q 2 ) and m(Q 2 ) the running coupling and running mass at scale µ 2 = Q 2 in the MS scheme, Ω 4 and Ω ss 3 are the RG invariant modifications of aG 2 and m ss s defined in Ref. [11], and ρ V SA in Eq. (9) describes the deviation of the four-quark condensates from their vacuum saturation values. We have dropped D = 2 contributions, which are suppressed by two additional powers of m u,d , and The Borel transforms of the above expressions are well known, and may be found in Refs. [11,12].
In scalar and pseudoscalar channels, direct instanton contributions are potentially important, but are not incorporated in the OPE representation of Π ud [13,14]. We estimate their size using the instanton liquid model [15]. ILM contributions to the theoretical side of the Π ud BSR are given by where ρ I ≃ (1/0.6 GeV) is the average instanton size and K i are the MacDonald functions. ILM contributions play only a small (few percent) role in the BSR analysis at scales We employ the following values for OPE/ILM input: ρ I = 1/(0.6 GeV) [14,15]; [11,12]; gqσF q = 0.8 ± 0.2 GeV 2 qq [18]; and ρ V SA = 0 → 10. The D = 0 and 4 OPE contributions are evaluated via the contour-improvement prescription [19], using the analytic solutions for α s (Q 2 ) and m(Q 2 ) obtained from the 4loop-truncated versions of the β [20] and γ [21] functions.
The quality of this match is shown in Fig. 2 for the w N family and in Fig. 3 for the w D family.
The result of Eq. (12) is compatible with the P98 results corresponding to s 0 ≃ 4 GeV 2 but significantly smaller than that corresponding to s 0 = 2 GeV 2 . Since, in spite of optimization, the match for w N is best where that for w D is worst, and vice versa, it appears that some modification of the P98 spectral ansatz is also required.
In this work, we aim to determine simultaneously the excited resonance decay constants, f π(1300) ≡ f 1 and f π(1800) ≡ f 2 , which characterize the modifications of the spectral ansatz, and m u + m d . To this end, we perform a combined w N and w D pFESR analysis 3 . Our spectral ansatz is where B 1,2 (s) are standard Breit-Wigner forms for the π(1300) and π(1800). We employ The errors labelled "Γ" result from varying the input resonance parameters within the PDG2000 errors, and are due essentially entirely to the (large) uncertainty on the π(1300) width. Those labelled "theory" reflect uncertainties in the OPE input and our estimate of the error associated with truncating the D = 0 series at O(a 3 ). Those labelled "method" are obtained by studying the impact of employing different analysis windows in s 0 and A, and performing separate w N and w D analyses. Further details of the analysis, and a breakdown of the separate error contributions will be given elsewhere [24]. The OPE+ILM/spectral integral match corresponding to these results, shown in Fig. 4 for the w N family and Fig. 5 for the w D family, is obviously excellent.
As noted above, the ILM contributions play a non-negligible role in the pFESR analysis.
In fact, if one removes ILM contributions, an equally good OPE/spectral integral match is to be compared to the central value given in Eq. (14). The agreement is excellent. The stability of the BSR analysis, shown in Fig. 6 24.4 ± 1.5 [25] with recent determinations of m s using hadronic τ data [26]. A recent summary [27] gives 83 MeV < m s (2 GeV) < 130 MeV, which corresponds to 6.8 < [m u +  14), (15) and (16). The labelling of the hadronic integrals, OPE integrals and the A = 0, 2 and 4 cases, is as for Fig.1 above.
FIG. 5. The optimized OPE+ILM/spectral integral match, as in Fig. 4, except for the w D rather than w N weight family.