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Unitary matrix integral for QCD with real quarks and the GOE-GUE crossover

A unitary matrix integral that appears in the low-energy limit of QCD-like theories with quarks in real representations of the gauge group at finite chemical potential is analytically evaluated and expressed as a pfaffian. Its application to the GOE-GUE crossover in random matrix theory is discussed. An analogous unitary integral for QCD-like theories with quarks in pseudoreal representations of the gauge group is also evaluated.

In QCD, due to spontaneous breaking of chiral symmetry, the low-energy physics may be described by a nonlinear sigma model. In the so-called ε-regime [8,14], exact zero modes of the Nambu-Goldstone modes dominate the partition function and the infinite-dimensional path integral reduces to a finite-dimensional integral over a coset space. It is known since olden times that there are three patterns of chiral symmetry breaking in QCD, depending on the representation of quarks [15]. In QCDlike theories with quarks in real representations of the gauge group, the pattern is SU(2N f ) → SO(2N f ) where N f is the number of Dirac fermions [16,17]. The basic degrees of freedom at low energy are expressed through U T U where U is a matrix field that lives on U(2N f ). When a quark chemical potential µ is added, the lowenergy sigma model attains additional terms [17]. In this paper, we are interesting in evaluating the ε-regime partition function of QCD with real quarks when the chemical potential is different for each flavor, i.e., (µ 1 , · · · , µ N f ) are all distinct. This case was not covered in [17]. We will show that the partition function has a pfaffian form. We also argue that the integral formula has an application to the symmetry crossover between the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Unitary Ensemble (GUE) in RMT. Moreover, since U T U is an element of the Circular Orthogonal Ensemble in RMT [18], our result may have a potential application in this direction as well.
This paper is organized as follows. In section II the main analytical results of this paper are summarized. In section III some applications are illustrated. In section IV we give a derivation of the integral formulae presented in section II. In section V we give a formula (with-out proof) for a related unitary matrix integral that has applications to QCD-like theories with quarks in pseudoreal representations of the gauge group. Finally in section VI we conclude.

II. MAIN RESULTS
For an arbitrary Hermitian N × N matrix H with mutually distinct eigenvalues {e k } and an arbitrary nonzero γ ∈ C, we have for even N and for odd N , where dU denotes the normalized Haar measure, ∆ N (e) ≡ i<j (e i − e j ) is the Vandermonde determinant, Pf denotes a pfaffian, and erf(x) is the error function.
We performed an intensive numerical check of these formulae for N up to 10 by estimating the left hand sides of the formulae by Monte Carlo methods and verified their correctness. We used a Python library pfapack [19] for an efficient calculation of a pfaffian.
As a side remark we note that a similar pfaffian formula has been obtained for a unitary matrix integral considered in [20]. It follows from a slight generalization of [17] that the static part of the partition function for the low-energy limit of massless QCD with N f flavors of quarks in real representations with chemical potential where V 4 is the Euclidean spacetime volume, F is the pion decay constant, and is the chemical potential matrix. The same sigma model is expected to arise in the large-N limit of the chiral symplectic Ginibre ensemble [21] which has exactly the same symmetry as QCD with real quarks. Let us define the dimensionless variables Then a straightforward application of (1) yields which completely fixes the chemical potential dependence of the effective theory in the ε-regime.

B. GOE-GUE crossover
In the classical papers [22,23], Mehta and Pandey solved the random matrix ensemble intermediate between GOE and GUE. Their ingenious approach was to consider the random matrix where S is a Gaussian real symmetric matrix and T is a Gaussian Hermitian matrix. As α grows from zero, the level statistics evolve from GOE to GUE. The transition occurs at the scale α 2 ∼ 1/N where N is the matrix size. Mehta and Pandey successfully derived the joint probability distribution function of eigenvalues of H by using the Harish-Chandra-Itzykson-Zuber integral [9,10].
An alternative approach to the GOE-GUE transition would be to consider the random matrix where A is a Gaussian real anti-symmetric matrix. [Actually the ensemble (7) is essentially equivalent to (8), because the sum of two Gaussian real symmetric matrices is again a Gaussian real symmetric matrix with a modified variance.] Then the Gaussian weight for S and A reads as Upon diagonalization H = U EU † we end up with the unitary integral which exactly has the form (1) and (2). Carrying out the integrals, we immediately arrive at the joint eigenvalue density derived by Mehta and Pandey [22,23]. Our integral thus provides a way to solve the transitive ensemble without recourse to the Harish-Chandra-Itzykson-Zuber integral.

IV. DERIVATION OF THE FORMULAE
The derivation proceeds in three steps. A.
Step 1: the heat equation We adopt the method of heat equation [10]. Let us assume t > 0 and consider a function where we wrote P = U T U for brevity. Then The Laplacian over Hermitian matrices is defined by Then we have (assuming that repeated indices are summed) Comparison of (11) and (15) indicates that, if we set α = N (N − 1)/4, then holds. Then it is obvious that also satisfies the same differential equation as z N (t, H, U ). Using the invariance of the Haar measure it is easy to verify that Z N (t, H) depends on H only through its eigenvalues {e 1 , · · · , e N }. An important property of Z N (t, H) is its translational invariance. Namely, Z N (t, H) = Z N (t, H + a1 N ) for an arbitrary a ∈ R, which can be easily verified from the definition (18). This means that Z N (t, H) depends on {e k } only through the differences {e k − e }. For us it is beneficial to transform the Laplacian into the "polar coordinate" [24] where X denote the angular variables. Then To obtain the basic building block of Z N (t, H), we shall explicitly work out the N = 2 case in the next subsection.

B. Step 2: the N = 2 case
Let us look at N = 2, for which where E = diag(e 1 , e 2 ) and we dropped the U(1) phase of U because it decouples. Using the parametrization hence Next we employ the Hopf coordinate of S 3 : (x 0 , x 1 , x 2 , x 3 ) = (cos ξ 1 sin η, cos ξ 2 cos η, sin ξ 1 sin η, sin ξ 2 cos η) (24) which yields where we have defined From (20), we know that (29) fulfills the equation It is then easy to find a solution for N = 4 by treating Z 2 as a building block, e.g., [(e i − e j )Z 2 (t, e i , e j )] (33) for even N and for odd N , where C N is the not yet determined constant. By construction, C 2 = 1. We note that this Pfaffian form is consistent with the translational invariance of Z N (t, H). To fix the normalization uniquely, we need to examine the boundary condition at t = +0.

C. Step 3: saddle point analysis
To determine the normalization of the Pfaffian formula, let us perform the saddle point approximation of for t → +0, with E = diag(e 1 , e 2 , · · · , e N ). Apparently U = 1 N is the dominant saddle point. A closer inspection of the exponent of (35) shows that, in fact, any U of the form OV with O ∈ O(N ) and V = diag(e iθ1 , · · · , e iθ N ) ∈ U(1) N is also a saddle. Hence the saddle point manifold appears to be given by U(1) N × O(N ). However there is a subtlety: the 2 N elements diag(±1, · · · , ±1) belong to both U(1) N and O(N ). To avoid duplication, the correct saddle point manifold must be considered as When performing the saddle point integral, the volume of this manifold should be factored out, because the Gaussian integral is performed only for massive modes. Now the part of U(N ) over which the Gaussian integration is done may be parametrized as U = exp(i i<j θ ij S ij ), where S ij is a symmetric matrix such that (S ij ) pq = δ ip δ jq + δ iq δ jp . Then Therefore Recalling α = N (N −1)/4 and using the volume formulae [25][26][27] Vol(U(N )) = 2 N π N (N +1)/2 N k=1 Γ(k) , as well as the duplication formula Γ (2z) = π −1/2 2 2z−1 Γ (z) Γ z + 1 2 [28], we finally obtain Note that this formula is valid for both even N and odd N . Next we turn to (33) Matching (41) and (43) we obtain for even N . Similarly, for odd N , (34) yields which, combined with (41), leads to for odd N . Finally, introducing γ ≡ 2/t and using the formula we find that (33) and (34) are equivalent to (1) and (2), respectively. Thus we have proved (1) and (2) for γ > 0.
Since both sides of the equations (1) and (2) are analytic in γ, it is concluded that these equations are valid also for complex γ. This completes the proof. The equality (47) does not seem to be known in the literature, but can be shown by noting that both sides of (47) solve the heat equation with the same initial condition.

V. RELATED INTEGRAL
The methodology employed to prove (1) and (2) can be put to work to prove another integral formula with no fundamental difficulty. Let us state the main result. The proof has been published in [29].
Let N be an even positive integer and define an N × N antisymmetric matrix Then, for an arbitrary Hermitian N × N matrix H with mutually distinct eigenvalues {e k } and an arbitrary nonzero β ∈ C, we have

VI. CONCLUSIONS AND OUTLOOK
In this paper we have derived a pfaffian formula for a unitary matrix integral with the heat equation method, and used it to evaluate the low-energy partition function of QCD-like theories with reak quarks. We also showed that the classical result by Mehta and Pandey for the transitive ensemble between GOE and GUE can be reproduced with our formula. There are miscellaneous future directions of research, which we list below: Our integral is a special case of a more general integral U(N ) dU exp Tr(U T SU HU † SU * H * ) where H is a Hermitian matrix and S is a real symmetric matrix. Putting S = 1 N reproduces our integral. It is easily seen that this integral is a function of the eigenvalues of S and H only. It would be interesting to seek for an analytic formula for this integral.
In section III A we considered the partition function in the massless limit. Whether our integral formulae can be extended to the massive case is an open problem.
An extension to supergroups along the lines of [30][31][32] would be interesting.
Our formula is valid only for Hermitian H. Can we relax this constraint and generalize the formula to an arbitrary complex matrix? Methods such as those in [12,33] may prove helpful in this regard.