x=x0, y=y0
=−2NfStrY2
∂0G(x¯ −y)
∂0G(x¯ −y)
− ∂02G(x¯ −y)G(x¯ −y)
+. . . (4.28) Thus we find
(4.25)=−V
2NfStrY2 Z
d4x ∂0G(x)¯ 2
+. . . , (4.29) where we have used the fact that the propagator is periodic in time. Therefore the corrections to the effective Lagrangian are given by
−NfStrCU0−1CU0 Z
d4x ∂0G(x)¯ 2
. (4.30)
Combining (4.23) and (4.30), we find that the fluctuations correct the leading-order contribution to the Lagrangian,
−F2
2 Str CU0−1CU0, (4.31)
to
−F2
2 StrCU0−1CU0
1−2Nf F2
G(0)¯ − Z
d4x ∂0G(x)¯ 2
. (4.32)
Thus at next-to-leading order we find an effective low-energy constantFeffgiven by Feff
F = 1−Nf F2
G(0)¯ − Z
d4x ∂0G(x)¯ 2
. (4.33)
This again agrees with the result for the unquenched partition function [23, 56].
4.5 The optimal lattice geometry
In Fig. 4.1 we show the finite-volume corrections at NLO to the low-energy constantsΣandF as a function of the box sizeLin a symmetric box. Note that the effects of the finite volume increase with the number of sea quark flavors Nf and that, depending on Nf, a box size of 3−5 fm is necessary to reduce the effects of the finite volume at NLO to about 10%. The effects are calculated atF = 90 MeV. In Fig. 4.2 we show the effect of an asymmetric box with Nf = 2 and L = 2 fm. An important message of this figure is that the magnitude of the finite-volume corrections can be significantly reduced by choosing one large spatial dimension instead of a large temporal dimension.
The reason for this behavior is that the chemical potential only affects the temporal direction, see
L/fm Σeff/Σ
1 2 3 4 5 6
1.1 1.2 1.3 1.4
L/fm Feff/F
1 2 3 4 5 6
1.1 1.2 1.3 1.4
Nf = 2 Nf = 3 Nf = 3
Figure 4.1: Volume-dependence at NLO of the low-energy constants Σeff(left) and Feff (right) in a symmetric box with dimensionsL0 =L1 =L2 =L3 =LatF = 90MeV.
L0/L
1 2 3 4 5
0.8 1.0 1.2 1.4
L3/L
1 2 3 4 5
0.8 1.0 1.2
1.4 Σeff/Σ
Feff/F
Figure 4.2: Effect of an asymmetric box with parametersNf = 2,L= 2fm, andF = 90MeV. We compare a large temporal dimensionL0withL1=L2 =L3=L(left) to a large spatial dimensionL3withL0 =L1 =L2 =L(right).
Eq. (3.2), and therefore breaks the permutation symmetry of the four dimensions. This manifests itself in the propagator
Z
d4x ∂0G(x)¯ 2
(4.34) which, as shown in Eq. (A.70), contains a term proportional toL20/√
V, whereL0 is the size of the temporal dimension. This term leads to an enhancement of the corrections in case of a large temporal dimension. Choosing instead one large spatial dimension, the finite-volume corrections are reduced, unless the asymmetry is too large. For the parameters used in Fig. 4.2, the optimal value isL3/L≈2.
This is good news. Many lattice simulations (at zero chemical potential) are performed withL1= L2=L3=LandL0= 2L. To determineF, it suffices to introduce the imaginary chemical potential in the valence sector. Therefore, one can take a suitable set of existing dynamical configurations and redefine L0 ↔ L3 before adding the chemical potential. This will minimize the finite-volume corrections for both Σ andF, at least for the parameter values chosen in Fig. 4.2. Note that this procedure increases the temperature of the system by a factor of two. One needs to check that the system does not end up in the chirally restored phase, in which our results no longer apply.
Next-to-leading-order corrections
In this chapter we calculate the partition function at next-to-next-to-leading order (NNLO) in the εexpansion. We allow for nonzero imaginary chemical potential and consider its contribution to leading order. In this way we obtain next-to-leading-order finite-volume corrections to the low-energy constantsΣandF. At this order 8additional low-energy constantsL1, . . . , L8 need to be included.1 In general the low-energy constantsL1, . . . , L8 are scale-dependent. We renormalize the theory and confirm that the scale dependence of the coupling constantsL1, . . . , L8is the same as in the ordinarypexpansion [58].
At NNLO in theεexpansion there are new terms in the finite-volume effective Lagrangian that are not present in the universal limit. We give their coefficients in terms of finite-volume propagators.
In the last section of this chapter we discuss the finite-volume corrections to Σ andF and the coefficients of the non-universal terms in the special case ofNf = 2and an asymmetric box. This case is relevant for the numerical analysis of chapter 6.
A publication containing the results of this chapter is in preparation [59].
5.1 The partition function
In this section we express the partition function at NNLO in theεexpansion in terms of one-loop and two-loop propagators. For simplicity we restrict the discussion to the sea-quark sectorNv = 0. The terms at next-to-leading order in the Lagrangian with imaginary chemical potential, see Refs. [57]
and [58], are given by
L4=−L1(Tr[∇µU−1∇µU])2−L2Tr[∇µU−1∇νU] Tr[∇µU−1∇νU]
−L3Tr[∇µU−1∇µU∇νU−1∇νU] +
2Σ F2
L4Tr[∇µU−1∇µU] Tr[M U−1+M†U] +
2Σ F2
L5Tr[∇µU−1∇µU(M U−1+M†U)]
− 2Σ
F2 2
L6 Tr[M U−1+M†U]2
− 2Σ
F2 2
L7 Tr[M U−1−M†U]2
− 2Σ
F2 2
L8Tr[M U−1M U−1+M†U M†U]− 2Σ
F2 2
H2TrM†M , (5.1) whereH2 corresponds to a contact termthat is needed in the renormalization of one-loop graphs.
The field-strength tensors defined byLµ andRµare not included since they vanish in the case of
1In order to distinguish the low-energy constantsL1,L2,L3from the length of the corresponding spatial dimension, we rename the lengthLitoL0iwithi= 1,2,3in the remainder of this chapter.
imaginary chemical potential [58]. The invariant measure relevant to this order is given by
see Sec. 2.5. We perform the expansion in terms of fieldsξand average over the fields using computer algebra2. The resulting expression is given in terms of the massless finite-volume propagator in dimensional regularization defined in App. A,
G(x) =¯ 1
where the sum is over all nonzero momenta. We use the identity
∂ρ2G(x)¯
x=0 = 1
V (5.4)
and finally express the result in terms of the propagatorsP1, . . . , P6 defined below.
Propagators
The relevant propagators for the partition function at NNLO are defined as P1=V ∂20G(0)¯ , P2 =√
VG(0)¯ , P3=√
V[∂02G(x)] ¯¯ G(x), P4 = ¯G(x)2,
P5= [∂02G(x¯ +y)] ¯G(x) ¯G(y), P6 =V[∂02G(x)] ¯¯ G(x)2, (5.5) where the integral over open spacetime coordinatesx,y, andzis implied. If we impose conservation of momentum and use
(∂L0µ)L0µG¯r = 2Γ(r+ 1)
where ∂L0µ denotes the partial derivative w.r.t. L0µand no sum overµis implied, we can relate the one-loop propagatorsP1, . . . , P5 toG¯rwhich is defined in App. A. We find For convenience we state the result of App. A explicitly as
G¯r= lim
2We use a C++ library for tensor algebra developed by the author of this thesis.
withshape coefficientsβn. We expressG¯0,G¯1, andG¯2in terms of shape coefficients and find P1 =−1
2L00(∂L0
0)β0+1
4, P2=−β1,
P3 = 1 4β1+1
2L00(∂L0
0)β1, P4=−2λ+β2+log(√
V) (4π)2 , P5 =−1
4P4−1 4L00(∂L0
0)β2− 2
(16π)2 , (5.9)
where we borrow the definition ofλfrom Ref. [58], λ= 1
(4π)2 1
d−4 −1 2
1 + Γ0(1) + log(µ2) + log(4π)
(5.10) with number of spacetime dimensionsd. We explicitly include the dependence on the scaleµwhich we define with positive mass dimension in this thesis. The two-loop propagatorP6 is calculated in Sec. 5.3. The result is given by
P6 =P6r+1 3λ−10
3 λP1, (5.11)
whereP6ris finite and depends only on the shape of the spacetime box.
The averaged Lagrangian at NNLO in theεexpansion and to leading order in the imaginary chem-ical potentialC2can be written as
L¯=−Σeff 2 Tr
M†U0+U0−1M
− Feff2
2 Tr CU0−1CU0+F˜eff2 2 Tr C2
+ ¯Ln(U0, M, C), (5.12)
whereL¯ncontains new terms of orderε8that are not present in the universal limit, L¯n= Υ1F2Tr[C]2+ Υ2VΣ2(Tr[M†U0]2+ Tr[U0−1M]2)
+ Υ3VΣ2(Tr[(M†U0)2] + Tr[(U0−1M)2])
+ Υ4VΣ2Tr[U0−1M] Tr[M†U0] + Υ5VΣ2Tr[M†M] + Υ6VΣF2Tr[U0−1CU0C](Tr[M†U0] + Tr[U0−1M]) + Υ7VΣF2Tr[C(M†CU0+U0−1CM)
+U0−1CU0(M†U0C+CU0−1M)]
+ Υ8VΣF2Tr[C](Tr[U0{M†, C}] + Tr[U0−1{C, M}]) + Υ9VΣF2Tr[C2({U0, M†}+{M, U0−1})]
+ Υ10VΣF2Tr[C2](Tr[M†U0] + Tr[U0−1M]). (5.13) In the following we expressΣeff,Feff,F˜eff, andΥ1, . . . ,Υ10in terms of the propagatorsP1, . . . , P6. Note thatF˜effandΥ1 do not couple toU0and therefore have no physical effect. They are, however, needed in the renormalization of the coupling constants discussed in Sec. 5.2.
Finite-volume effective couplings
In the following we state the resulting expressions forΣeff,Feff,F˜eff, andΥ1, . . . ,Υ10. The effective chiral condensate is given by
Σeff which agrees with Eqs. (22) and (23) of Ref. [57]. The effective coupling constants
Feff2 contain the two-loop propagatorP6. The effective coupling constants of the non-universal terms are given by