In the following we discuss the finite-volume corrections toΣandF and the coefficients of the non-universal terms at NNLO in the ε expansion. We explicitly consider the case ofNf = 2 and an asymmetric box with geometry
(a) L00 = 2L , L01 =L02 =L03 =L ,
(b) L03 = 2L , L00 =L01 =L02 =L . (5.59) This case is relevant for the numerical discussion of chapter 6. The three-flavor coupling constants L1, . . . , L5can be related to the two-flavor coupling constantsl1,l2, andl4by
l1 = 4L1+ 2L3, l2 = 4L2, l4 = 8L4+ 4L5, (5.60) see, e.g., Eqs. (3.15) and (3.16) of Ref. [63]. Therefore
Σeff
and
Feff2
F2 = 1− 4P2 F2√
V − 4P3 F2√
V + 8P2P3+ 8P32+ 4P22
F4V +8P4+ 16P5+ 4P6 F4V + 1
F4V [12l1+ 4l2] + P1
F4V [8l1+ 16l2]. (5.62)
In Ref. [64] the scale dependence of the coupling constantsliwithi= 1, . . . ,7is separated as
li =lir+γiλ , (5.63)
where
γ1= 1
3, γ2= 2
3, γ4= 2. (5.64)
It is straightforward to check that the divergences in Eqs. (5.61) and (5.62) cancel with this set ofγ1, γ2, andγ4. The renormalized coupling constantslri can be related to scale-independent constants¯li by
lri = γi 2(4π)2
¯li+ log(m2π/µ2)
, (5.65)
wheremπis the mass of the pion and
¯l1 =−2.3±3.7, ¯l2 = 6.0±1.3, ¯l4= 4.3±0.9, (5.66) see Ref. [64]. We perform the calculation at scaleV−1/4 so that
lir= γi
2(4π)2
h¯li+ log(m2π
√ V)
i
≈ γi
2(4π)2
¯li (5.67)
sincem2π√
V ≈1in theεexpansion (see Sec. 2.4). Note that the finite-volume corrections toΣand F are independent of the scale.
We calculate the renormalized two-loop propagatorP6rat scaleV−1/4numerically and obtain (a) P6r= 0.016(2), (b) P6r=−0.028(2) (5.68) for geometries(a) and(b), see Figs. 5.2 and 5.3. The finite-volume corrections toΣfor F = 90 MeV andL= 1.71fm at NLO and NNLO are given by
ΣNLOeff
Σ = 1.1454, ΣNNLOeff
Σ = 1.20(2), (5.69)
where the error is due to the uncertainty in¯l4. Note thatΣeffis independent of the choice of geometry (a)or(b). The finite-volume corrections toF forF = 90MeV andL= 1.71fm at NLO and NNLO are given by
(a) FeffNLO
F = 1.3192, FeffNNLO
F = 1.26(2), (b) FeffNLO
F = 1.06816, FeffNNLO
F = 1.07(4), (5.70) for geometries(a)and(b), where the error is due to the uncertainty in¯l1,¯l2, andP6r. Therefore we confirm the picture obtained in Sec. 4.5 at NLO that the finite-volume corrections toFcan be largely
m2√ V P6r(m2)
0.5 1.5 2.5
0 0.01 0.02
Polynomial fit Numerical data
Figure 5.2: Extrapolation ofP6r = limm→0P6r(m2)for geometry(a)at scaleV−1/4. We fit a poly-nomial of order four. The dashed lines enclose the one-sigma error band.
m2√ V P6r(m2)
0.5 1.5 2.5
−0.01
−0.02
−0.03
Polynomial fit Numerical data
Figure 5.3: Extrapolation ofP6r = limm→0P6r(m2)for geometry(b)at scaleV−1/4. We fit a poly-nomial of order four. The dashed lines enclose the one-sigma error band.
reduced by an asymmetric geometry with one large spatial dimension instead of one large temporal dimension.
The coefficients Υ2, . . . ,Υ5, whose corresponding non-universal terms do not depend on the chemical potential, are independent of the choice of geometry (a) or (b). This is expected since only the chemical potential has a preferred direction and breaks the permutation symmetry of all four dimensions. The coefficientsΥ6, . . . ,Υ10are affected by the choice of geometry(a)or(b)through the following combinations of propagators,
P4+ 2P5, P4+ 4P5, P5. (5.71)
We give their values
(a) P4+ 2P5 =−0.023, P4+ 4P5 =−0.033, P5 =−0.005,
(b) P4+ 2P5 =−0.003, P4+ 4P5 = 0.007, P5 = 0.005 (5.72) at scaleV−1/4 for geometry(a) and(b). Note that the contribution of the propagators is reduced significantly for the combinationsP4+ 2P5 andP4 + 4P5 in geometry(b). In chapter 6 we show
numerically that the non-universal effects are indeed smaller for geometry(b). The coefficientΥ1 does not have any physical effect.
Results from lattice QCD
In this chapter we use numerical data for the spectrum of the Dirac operator in order to determine the low-energy constantsΣandF. We employ the dynamical two-flavor configurations of JLQCD [32, 65] on a163×32lattice with lattice spacinga = 0.107(3) fm and quark massam = 0.002.
The quark fields and gluon fields have periodic boundary conditions in all four dimensions. In order to access the low-energy constantF we allow for valence quarks with nonzero imaginary chemical potentialiµ. The simulations are performed in two geometries
(a) L0 = 32a , L1 =L2 =L3 = 16a ,
(b) L3 = 32a , L0 =L1 =L2 = 16a . (6.1) In Sec. 6.1 we determine Σ by a fit to the distribution of the lowest Dirac eigenvalue without chemical potential. In Sec. 6.2 we determine F by a fit to the eigenvalue shift due to a nonzero chemical potential. For a detailed explanation of both eigenvalue correlation functions we refer to Sec. 3.4 of this thesis.
In Ref. [66] a similar method to obtain the low-energy constants F andΣ was used in an ex-ploratory, qualitative study. In the following we obtain quantitative results forΣandF. A publication containing the results of this chapter is in preparation [67].
6.1 The low-energy constant Σ
In this section we fit the distribution of the lowest Dirac eigenvalue to the analytic formula of RMT, see Sec. 3.4. The result of the fit is given by
a3Σeff = 0.00208(2), (6.2)
or in physical units
Σeff= (235(6)(1)MeV)3, (6.3)
where the errors are due to the uncertainty ina(left) and due to statistics (right). Note that the error inadominates the error inΣeff. SinceΣeffis independent of the choice of geometry(a)or(b), see chapter 5, we only perform the fit in one geometry.
The best fit, displayed in Fig. 6.1, has a χ2/dof = 2.9which is uncomfortably large. The sys-tematic errors in the shape of the lowest eigenvalue distribution due to non-universal terms in the finite-volume effective Lagrangian are thus quite large. Nevertheless, Σeff is not sensitive to the shape of the distribution but merely to the overall scale. Therefore we can assume that the fit gives a reasonable result.
We include the NNLO finite-volume corrections of Sec. 5.4 and find Σ = Σeff
1.20(2) = (221(6)(1)(1)MeV)3, (6.4)
aλ P1(λ)
0.003 0.009 0.015 0.021 0.027 15
45 75
105 Fit to RMT
Lattice data
Figure 6.1: Fit to lowest eigenvalue distributionP1(λ)in geometry(b)withχ2/dof= 2.9. The best fit is given bya3Σeff= 0.00208(2).
dˆ Pd( ˆd)
−0.4 −0.2 0 0.2 0.4 1
2 3
4 Gaussian fit
Lattice data
Figure 6.2: Gaussian fit to distributionPd( ˆd)foraµ= 0.01in geometry(a)withχ2/dof= 4.2.
where the rightmost error is due to the uncertainty in the finite-volume corrections. Therefore
mΣV = 0.452(4)(7), (6.5)
where the errors are due to statistics (left) and due to the uncertainty in the finite-volume corrections (right). We conclude that theεexpansion is indeed applicable for our choice ofV andm, see Sec. 2.4.