The th Smallest Dirac Operator Eigenvalue
and the Pion Decay Constant
We derive an analytical expression for the distribution of the th smallest Dirac eigenvalue in QCD with imaginary isospin chemical potential in the Dirac operator for arbitrary gauge field topology . Because of its dependence on the pion decay constant through the chemical potential in the epsilon-regime of chiral perturbation theory this can be used for lattice determinations of that low-energy constant. On the technical side we use a chiral Random-Two Matrix Theory, where we express the th eigenvalue distribution through the joint probability of the ordered smallest eigenvalues. The latter can be computed exactly for finite and infinite , for which we derive generalisations of Dyson’s integration Theorem and Sonine’s identity.
It is by now well known how the spontaneous breaking of chiral symmetry in QCD leads to remarkably strong predictions for the spectral properties of the Dirac operator in that theory. Based first exclusively on the relation to the effective field theory for the associated Nambu-Goldstone bosons at fixed gauge field topology , an intriguing relation to universal Random Matrix Theory (RMT) was also pointed out . It has subsequently become clear how these two alternative formulations are related, and all -point spectral correlation functions have been shown to be identical in these two formulations to leading order in a -expansion (where gives the extent of the space-time volume ) [3, 4]. This holds then also for individual distributions of Dirac operator eigenvalues .
If one seeks sensitivity to the pion decay constant it turns out to be useful to consider the Dirac operator of quark doublets with isospin chemical potential . Based on the chiral Lagrangian formulation , it has been suggested to use a spectral 2-point function of the two associated Dirac operators with imaginary isospin chemical potential. The advantage of imaginary chemical potential lies in the fact that the corresponding Dirac operator retains its anti-hermiticity. To leading order, all results can be expressed in terms of the simple finite-volume scaling variable . In this way, can be extracted from fits that vary and/or . There is also sensitivity to in other observables that couple to chemical potential [7, 8, 9, 10, 11].
The leading-order chiral Lagrangian computations of ref.  have been given a reformulation in terms of a chiral Random Two-Matrix Theory in ref. . In this way, all spectral correlation functions associated with the two Dirac operators and , with respective chemical potentials and , have been computed analytically in  for both the quenched and the full theory with light flavours. It also includes all spectral correlation functions where the imaginary isospin chemical potential only enters in the Dirac operator whose eigenvalues are being computed, while the gauge field configurations are obtained in the usual way at vanishing chemical potential. In analogy with what is being done when varying quark masses away from the value used for generating the gauge field configurations we call this “partial quenching”.
In an earlier paper  it was shown how all probability distributions of individual Dirac operator eigenvalues can be computed by means of a series expansion in higher -point spectral correlation functions. In reference  an explicit analytical formula was also given for the lowest non-zero eigenvalue distribution for arbitrary combinations of flavours and of the Dirac operators and , respectively, at gauge field topology , in an approach quite close to that of ref. . Two obvious questions remained open that we will answer in this paper: how to extend this to non-zero topology , and how to compute the distribution of the second, third or general th eigenvalue as these are known for zero chemical potential . However, the path of ref.  is not very suitable in particular for the derivation of the distributions of higher eigenvalues in a compact analytical manner. Our setup will follow closely ref.  at vanishing chemical potential. We shall present here a new formalism that immediately allows for the analytical determination of the distributions of these higher eigenvalues. The extension of the first approach  to for the first eigenvalue is presented in appendix C, as an alternative formulation and analytical check to part of our new approach. The benefit of our new results should be two-fold: while expressions for higher topology allow for an independent determination of and from different lattice configurations, the expressions for higher eigenvalues should allow for a better determination using the same configurations as for the first eigenvalue.
How much of the program proposed in  to determine on the lattice has been realised in the meantime? Based on a preliminary account  of ref. , the expansion of the first eigenvalue was first used in simulations in . However, the question remained how large the finite-volume corrections to the leading order (LO) -expansion are, in which the chiral Lagrangian-RMT correspondence holds. In a series of papers this question has been addressed and answered: in  the next to LO corrections (NLO) and in  next to next to LO (NNLO) corrections in the epsilon-expansion were computed. As a result of these computations at NLO all RMT expressions for arbitrary -point density correlations functions (and thus for all individual eigenvalues too) remain valid. The infinite volume expressions simply get renormalised by finite-volume corrections, one only has to replace and by and in the corresponding RMT expressions. Here the subscript “” for effective encodes the corrections that match those computed earlier in  and [21, 11], respectively. Only at NNLO non-universal, non-RMT corrections appear. It was further noticed in [18, 19], that the size of the corrections at each order depends considerably on the lattice geometry, in particular when using asymmetric geometries. In order to keep the NNLO corrections small, in  the authors used a specific optimised geometry where they could apply RMT predictions and effective couplings at NLO only, and they obtained realistic values for and from partially quenched lattice data for small chemical potential. Technically speaking was determined there from the first eigenvalue distribution at vanishing chemical potential, and from the shift of the eigenvalues compared to zero chemical potential. We refer to  for a more detailed discussion of these fits.
Motivated by these findings we have completed the computation for the th Dirac eigenvalues for all in the RMT setting, in order to have a more complete mathematical toolbox at hand.
Our paper is organised as follows. In the next section we briefly define the notation and remind the reader of the definition of chiral Random Two-Matrix Theory. We introduce a certain joint probability density and describe how it can be used to derive individual eigenvalue distributions. We give the explicit finite- solution here in terms of new polynomials and a new sequence of matrix model kernels. We take the scaling limit relevant for QCD in section 4, and write out explicitly and discuss the physically most important examples such as partially quenched results. Section 5 contains our conclusions and a suggestion for a quite non-trivial but important extension of these results. Because some of the relevant technical details have been described in ref. , we have relegated many of the mathematical details in this paper to appendices. In addition, in Appendix A we describe an explicit construction of the polynomials needed to compute the first eigenvalue distribution in sectors of non-trivial gauge field topology if one alternatively uses the method of ref. .
2 Chiral Random Two-Matrix Theory
Before turning to the relevant Random Two-Matrix Theory, we first briefly outline the set-up in the language of the gauge field theory. We are considering QCD at finite four-volume , and we assume that chiral symmetry is spontaneously broken at infinite volume. We consider two Dirac operators with different imaginary baryon (quark) chemical potential ,
When this is simply imaginary isospin chemical potential, but we can stay with the more general case. We thus consider light quarks coupled to quark chemical potential , and light quarks coupled to quark chemical potential . Let us first consider the conceptually simplest case where , and later comment on the changes needed to deal with partial quenching.
In the chiral Lagrangian framework the terms that depend on are easily written down on the basis of the usual correspondence with external vector sources. Going to the -regime of chiral perturbation theory in sectors of fixed gauge field topology , the leading term in the effective partition function including imaginary reads [6, 23] (see also  for QCD-like theories)
In (2.3) the matrix
is made out of the chemical potentials, and the quark mass matrix is
The partition function (2.3) is a simple zero-dimensional group integral. The leading contribution to the effective low-energy field theory at finite volume in the -regime is thus well known.
We consider now the limit in which while and are kept fixed. In this limit, to LO in the -expansion, the effective partition function of this theory and all the spectral correlation functions of its Dirac operator eigenvalues are completely equivalent to the chiral Random Two-Matrix Theory with imaginary chemical potential that was introduced in ref. . The equivalence for the two-point function follows from , for all higher density correlations it was proven in . Therefore, since we have proven  that the probability distribution of the th smallest eigenvalue can be computed in terms of this infinite sequence of spectral correlation functions, we are free to use the chiral Random Two-Matrix Theory when performing the actual analytical computation.
As already mentioned it has been shown in  that also to NLO in the -expansion the random matrix expressions  for density correlation functions remain valid, when replacing and by the renormalised constants and that encode the finite-volume corrections. Only to NNLO non-universal corrections to the random matrix setting appear.
The partition function of chiral Random Two-Matrix Theory is, up to an irrelevant normalisation factor, defined as
where are given by
The operator remains anti-Hermitian because the chemical potentials are imaginary, as shown explicitly. Both and are complex rectangular matrices of size , where both and are integers. The index corresponds to gauge field topology in the usual way. The aforementioned correspondence to chiral perturbation theory holds in the following microscopic large- limit:
In the framework of chiral Random Two-Matrix Theory it is particularly simple to consider the situation corresponding to what we call partial quenching. Here one simply considers eigenvalues of one of the matrices, say , that then does not enter into the actual integration measure of (2.6) by setting . In the language of the chiral Lagrangian, this needs to be done in terms of graded groups or by means of the replica method.
Referring to ref.  for details, we immediately write down the corresponding representation in terms of eigenvalues and of and , respectively,
up to an irrelevant (mass dependent) normalization factor. The integrand is the joint probability distribution function (jpdf), which is central for what follows:
Because the integration in eq. (2.6) was over and separately, the matrices now become coupled in the exponent. The corresponding unitary group integral leads to the determinant of modified -Bessel functions, and removes one of the initially two Vandermonde determinants, which is defined as . The precise connection between the constants and is given by
where the latter is defined for later convenience. We need the joint probability distribution to be normalised to unity, which is done trivially by dividing by (cf. eq. 2.9)).
3 The th Eigenvalue at Finite- for Arbitrary
We now follow the derivation of ref.  rather closely. We are able to do that because we focus here on the distributions of individual -eigenvalues only - which are those we may partially quench. For that purpose it is convenient to first consider the joint probability distribution of the smallest -eigenvalues, ordered such that :
This quantity is then used to generate the th -eigenvalue distribution through the following integration111Compared to  we are already working with squared variables here. Translating to that picture the integration bounds in eqs. (3.1) and (3.2) remain the same as in .
Note that for no integration is needed, and .
The computation of mixed or conditional individual eigenvalue distributions, e.g. to find the joint distribution of the first - and first -eigenvalue, remains an open problem.
We next proceed as in ref. , and integrate out all -eigenvalues exactly. Because of this we note that in eq. (3.1) we can replace the determinant over the Bessel functions by times its diagonal part, after having made use of the antisymmetry property of . After inserting a representation of the Bessel function in terms of a factorised infinite sum over Laguerre polynomials (see eq. (B.7) in ), we get
where the Laguerre polynomials now appear with their corresponding weight function due to the identity used. Next we include the set of masses, , into to form a larger Vandermonde determinant of size , and then replace it by a determinant of in general arbitrary Laguerre polynomials normalised to be monic
Here the index of the Laguerre polynomials is arbitrary. The monic Laguerre polynomials relate to ordinary Laguerre polynomials in a -independent manner:
In eq. (3.4) the inverse powers can be taken out of the determinant. Inserting this back into eq. (LABEL:intP) for we can use the orthogonality of the Laguerre polynomials in the integrated variables , killing the infinite sums from the expanded Bessel functions. The Laguerre polynomials in thus replace those in inside the determinant, times the norm from the integration. We obtain
where we have defined
where we have split the Vandermonde determinant into integrated and unintegrated variables, and defined the following constant
We can now change variables for , and then perform the shift to obtain integrations in eq. (3.13):
We thus obtain an integral with extra mass terms of flavour-type “1”, in addition to the shifted masses. The weight
is now of Laguerre-type corresponding to a fixed topological charge of , irrespective of the actual topological charge of the given gauge field sector we started with. We will therefore call spurious topology. Compared with the corresponding derivation in case of vanishing chemical potential , this can be seen to differ by one unit, compared to spurious topology at vanishing in . The reason for this difference is easily traced to the different integration measure for the -eigenvalues, which has one power less in the Vandermonde determinant compared to the case of vanishing chemical potential222Of course, the additional pieces due to the -integrations are what ensures equivalence to those corresponding one-matrix model results in the limit .. It is an interesting and quite non-trivial check on our present calculation that we recover the results of reference  in the limit of vanishing chemical potential. In particular, the shift from spurious topology to spurious topological charge in the integration measure will now arise due to recurrence relations of Laguerre polynomials. Some details of this will be given below.
When replacing the Vandermonde determinant in the variables as well as by a determinant containing Laguerre polynomials we thus choose polynomials in order to be able to exploit the orthogonality properties with respect to the measure eq. (3.19).
For the new masses times this is an easy task. We can include them into a bigger determinant of size , following the identity eq. (3.4). Here we replace the variables by variables , and the set of masses by the following set of masses:
and likewise we define
For computational simplicity we first set the degenerate masses to be different by adding small pairwise different constants, , and then set at the end of the computation. Also we may chose spurious topology in eq. (3.4). The prefactors in front of the Laguerre polynomials inside the determinant can be taken out.
To express the determinant of the shifted arguments in eq. (3.18) in terms of Laguerre polynomials requires a bit more algebra:
where for convenience we have defined , as well as
In the first step in eq. (3.27) we have used the invariance of the determinant to undo the shift in of the variables. This leads to a shift in variables and to linear combinations of the Laguerre polynomials in the masses. In the second step we have added columns from the left to the right to replace monic powers in and by polynomials . Because the determinant is not an invariant Vandermonde this leads to a further sum in the first rows, invoking the following new polynomials
The form given here is derived in Appendix A using identities for Laguerre polynomials. For later purpose, we note already that in the limit of zero chemical potential, in the limit , we obtain Laguerre polynomials of shifted mass from the :
In this way we recover, after the use of a few identities for Laguerre polynomials, the results of ref.  in the limit of vanishing chemical potential.
We now proceed with the integration over the variables in eq. (3.18). Using the rewriting discussed above, we have: