It is well known that random matrix models for QCD atzeroimaginary chemical potential (or temper-ature)uhave the correctθ-dependence. In this chapter we have shown that this is not automatically the case foru 6= 0. We obtain the correctθ-dependence only after introducingν-dependent normal-ization factorsNνin the sum over topologies.
To explain this we have introduced the topological domain of the Dirac spectrum, which is defined as the part of the Dirac spectrum that is sensitive to the topological charge. We have shown that foru = 0 the topological domain coincides with the microscopic domain. This is also the case at u 6= 0for models for which noν-dependent normalization factors are needed to obtain the correct θ-dependence. However, for the model we analyzed that requires nontrivial normalization factors, the complete Dirac spectrum is inside the topological domain. This results in a partition function that gives universal behavior for small Dirac eigenvalues, but has bulk spectral correlations that depend both onu and on the topological charge. In the thermodynamic limit this leads to an additional u-dependent factor in the partition function at fixed topological charge which results in an incorrect θ-dependence of the partition function. To obtain a partition function with the usual behavior in
the chiral limit, one has to introduce additional ν-dependent normalization factors in the sum over topologies.
Our observations are of potential importance for lattice QCD at nonzero imaginary chemical po-tential or temperature. Depending on, e.g., the fermion formulation or the algorithm used, it could be that nontrivial normalization factors are needed in the sum over topological sectors, and these could even persist in the continuum limit. To find out whether such normalization factors might be nec-essary, it would be interesting to determine the topological domain as a function of the deformation parameters. This is feasible with current lattice technology. To be consistent with the general proper-ties of QCD, the topological domain should not extend beyond the microscopic domain. Future work will tell us if this interesting picture prevails.
Epilogue
Conclusions and outlook
The low-energy limit of QCD at sufficiently small quark masses in a finite volume can be viewed as a finite-volume expansion about a zero-dimensional theory that is uniquely determined by its universality class. Since QCD is in the same universality class as the chiral unitary ensemble of RMT, we can use predictions of RMT to describe the properties of QCD in this limit. The dimensionless quantities of RMT can be mapped to dimensionful quantities of QCD using the two leading-order low-energy constantsΣandF.
These constants can be determined from fits to Dirac eigenvalue correlation functions obtained from lattice QCD simulations in the εregime. The low-energy constant Σ can be extracted from the position of the lowest Dirac eigenvalue, and the low-energy constantF can be extracted from the shift of Dirac eigenvalues due to the presence of a nonzero imaginary chemical potential. The relevant eigenvalue correlation functions can be calculated efficiently in lattice QCD since the Dirac operator remains anti-Hermitian for nonzero imaginary chemical potential. Furthermore, it is sufficient to introduce nonzero imaginary chemical potential only for valence quarks, and therefore also existing lattice QCD configurations that were generated at zero chemical potential can be used to extract the low-energy constantFin this way.
In order to correct for the effect of the finite simulation volume we have calculated the partially quenched low-energy effective theory with imaginary chemical potential at NNLO in theε expan-sion. While at NLO the predictions of RMT still hold withΣandFreplaced by effective low-energy constantsΣeffandFeff, at NNLO also non-universal terms arise. We have discussed how to minimize these non-universal terms and the finite-volume corrections toΣandF by a suitable choice of lattice geometry. It was shown that at simulation volumes of approximately(2fm)4and atF = 90MeV an optimal result can be obtained by using an asymmetric lattice with one spatial dimension that is twice the size of the other dimensions. Since many lattice configurations are generated on a similar geom-etry, where the large spatial dimension is exchanged with the temporal dimension, one can minimize the effects of the finite simulation volume by a suitable rotation of the lattice. We performed such a rotation on dynamical two-flavor lattice configurations of the JLQCD collaboration and extractedΣ andF with good precision.
In future work we will use the exact form of the non-universal terms derived in chapter 5 to calcu-late analytic expressions for Dirac eigenvalue correlation functions at NNLO in theεexpansion. In this way we can reduce the systematic error of the fits and construct eigenvalue correlation functions that allow for an extraction of further low-energy constants.
In the second part of this thesis we discussed the role of topology in schematic random matrix mod-els. Such models can be used to obtain a schematic description of the chiral phase transition of QCD.
We have classified different schematic random matrix models according to the topological domain of their respective Dirac spectra. The topological domain was defined as the part of the eigenvalue spectrum that is sensitive to the topological charge. We have shown that additional normalization factors need to be included in the sum over topological sectors to remedy an unphysical suppression of topological fluctuations if the topological domain of the Dirac eigenvalues extends beyond the
microscopic domain
Recently, there has been progress in the formulation of schematic random matrix models at nonzero temperature and topology. In Ref. [94] a model was proposed that properly incorporates the depen-dence of the chiral phase transition on the number of quark flavors and allows for finite topological fluctuations. The model of Ref. [94] at fixed topological charge is equivalent to the model of Ref. [80].
Therefore the topological domain of its Dirac eigenvalues does not extend beyond the microscopic domain. This is in accordance with our results.
One-loop propagators at finite volume
In this chapter we discuss one-loop propagators at finite volume with periodic boundary conditions in dimensional regularization. These propagators were calculated originally in Refs. [30, 57]. The propagator of a scalar field with massmat finite volumeV is given by
G(x) = 1 V
X
k
eikx
k2+m2, (A.1)
where the sum is over all finite-volume momenta k. A related quantity of interest is the massless propagator
G(x) =¯ 1 V
X
k6=0
eikx
k2 , (A.2)
where the sum is over all nonzero finite-volume momentak. Since the constant mode is subtracted, G¯has no infrared singularity. All one-loop combinations ofGandG¯can be related to
Gr = Γ(r) V
X
k
1
(k2+m2)r (A.3)
and
G¯r = Γ(r) V
X
k6=0
1
(k2)r. (A.4)
In the following we first calculateGrat spacetime dimensiond= 1and extract the spectrum of the harmonic oscillator fromG0as an exercise. Then we calculateGrat arbitrary spacetime dimension d, and finally we relateG¯r toGr and give explicit formulas and numbers for common spacetime geometries.
A.1 Poisson’s sum over momenta
In this section we use Poisson’s summation formula
∞
X
n=−∞
e2πinϕ =
∞
X
n=−∞
δ(ϕ−n) (A.5)
to obtain a convenient expression for the sum over finite-volume momentak. The d-dimensional generalization of Eq. (A.5) is given by
X
~n∈Zd
e2πi~n·~ϕ= X
~n∈Zd
δ(d)(ϕ~−~n), (A.6)
and therefore the integral overϕwith a test functionf(ϕ)~ results in
where~nandϕ~ ared-dimensional vectors. The sum over finite-volume momenta~kwith components
~ki= 2πni