Topology in the form of instantons and dyons is an important ingredient of the QCD vacuum [1–3]. The main reason is that the Dirac operator for field configurations with nonzero topological charge has a generic number of exact zero modes, which induce the chiral condensate for massless quarks. At nonzero quark mass, the total number of instantons and anti-instantons is even more important. This number scales with the four-dimensional spacetime volume V, unlike the topological charge, which scales as V. If instantons and anti-instantons are not strongly overlapping, they give rise to near-zero modes [4] which determine the value of the chiral condensate. Given the importance of topology for the QCD partition function, it is puzzling that the conjugate parameter, the so-called θ angle, is consistent with zero according to all available experimental evidence. Nevertheless, theories with nonzero θ angle have received a great deal of attention both as theories beyond the standard model as well as from a purely theoretical perspective [5–10].

In [11,12] we have resolved an interesting apparent puzzle for one-flavor QCD at zero θ angle: the chiral condensate should be independent of the sign of the quark mass m, but the condensate expressed in terms of the Dirac eigenvalues appears to be an odd function of m. The point is that this function still needs to be averaged over gauge fields, and this average depends on the quark mass through the fermion determinant in such a way that the chiral condensate eventually becomes an even function of m. This resolution is reviewed in Sec. II: at negative quark mass, the statistical weight in the average over gauge fields becomes negative, which leads to exponentially large oscillations that can move the discontinuity of the chiral condensate away from the support of the Dirac spectrum and could be shown to yield a mass-independent condensate. In the present paper we extend the work of [11,12] to arbitrary θ angle and to more than one flavor, as already sketched in [13]. Let us emphasize that for nonzero θ the statistical weight is not only negative but becomes complex.

To be able to obtain explicit results, most of our calculations are performed in the ϵ domain of QCD (also called the microscopic domain). In this domain the Compton wavelength of the pion is much larger than the size of the box so that the kinetic term of the chiral Lagrangian can be ignored and only the mass term remains. This chiral Lagrangian can also be obtained from a random matrix theory with the same global symmetries as QCD [14,15], which makes it possible to find analytical expressions for the spectral density of the Dirac operator using powerful random matrix techniques. Another benefit of working in the ϵ domain is that the chiral condensate can be obtained from the spectral density without any additional regularization. This is important because the spectral density of the Dirac operator is renormalization-group and gauge invariant, while the chiral condensate is only gauge invariant.

The structure of this paper is as follows. In Sec. II we review the mass dependence of the chiral condensate for one- and two-flavor QCD. The sign problem for QCD at nonzero θ angle is discussed in Sec. III. A decomposition of the spectral density and the chiral condensate is introduced in Sec. IV. In Sec. V we derive general analytical expressions for the contributions of the zero modes and of the quenched part of the Dirac spectrum to the chiral condensate and show that each of them increases exponentially with the volume at nonzero θ angle, but that their sum remains finite. The one-flavor case is worked out in detail in Sec. VI, where we also obtain an expression for the spectral density of the Dirac operator at fixed θ angle. The two-flavor case is discussed in Sec. VII, where we derive analytical expressions for the spectral density and the chiral condensate at fixed θ angle. Concluding remarks are made in Sec. VIII, and technical details are given in several Appendixes. In particular, in Appendix A we obtain identities for sums of products of three and four Bessel functions which, as far as we know, are new.

Some notes on notation: on the macroscopic scale, the Dirac eigenvalues and the quark masses are denoted by λ and m, while on the microscopic scale they are denoted by x and u, respectively. For functions of these variables, such as the partition function Z, the spectral density ρ or the chiral condensate Σ, we use the same symbol on the macroscopic and the microscopic scales to simplify the notation. The corresponding functions are of course different, but it should be clear from the arguments of the function what is meant in every case. Also, when we give results for partition functions, we drop irrelevant normalization factors.

We consider QCD with Nf quark flavors and quark mass matrix M=diag(m1,…,mNf), which we allow to be complex for the time being. The mass matrix appears in the QCD Lagrangian in the form ψ¯RMψL+ψ¯LM†ψR, where ψR/L denotes quark fields of definite chirality. For a given gauge-field configuration with topological charge ν, the fermion determinant is ∏λn>0det(λn2+MM†)×{detνM,ν≥0,det-νM†,ν<0,(1)where the λn are the eigenvalues of the Dirac operator, and the second factor is due to the presence of |ν| exact zero modes. The partition function for fixed topological charge ν is given by the average of the fermion determinant over gauge-field configurations with fixed ν. Defining Z^|ν|(|M|)=⟨∏λn>0det(λn2+MM†)⟩ν,(2)which only depends on |M|=diag(|m1|,…,|mNf|) and |ν|, the partition function reads Zν(M)=Z^|ν|(|M|)×{detνM,ν≥0,det-νM†,ν<0.(3)The partition function at fixed θ angle is then given by Z(M,θ)=∑ν=-∞∞eiνθZν(M)=∑ν=-∞∞eiν(θ+∑k=1Nfφk)det|M||ν|Z^|ν|(|M|),(4)where mk=|mk|eiφk defines the phase φk of mk. It is clear that Z(M,θ) is a periodic function of θ, and if ∑kφk is a multiple of π, it is also even in θ. It only depends on the sum of the phases of the quark masses, and this sum can be absorbed in a redefinition of the θ angle. The same statement is true for the spectral density, whose mass dependence comes only from the fermion determinant in the statistical measure.

Therefore, from a mathematical point of view, it suffices to derive results for real and non-negative quark masses. However, we will sometimes consider the “physical” situation where one of the quark masses is taken to be negative. This case can be obtained by shifting θ→θ+π in the mathematical result. Nevertheless, our results assume mk≥0 unless stated otherwise.

The chiral condensate of flavor k is defined as Σ(mk)=-⟨ψ¯kψk⟩=-⟨ψ¯kRψkL⟩-⟨ψ¯kLψkR⟩=1NdV(ddmk+ddmk*)logZ(M),(5)where we have suppressed the dependence of Σ(mk) on M and where Nd (with d for “degenerate”) is the number of quarks whose mass equals mk. Note that Eq. (5) is valid both at fixed ν and at fixed θ. Let us make two remarks here. First, for a real and negative mass, the derivatives in Eq. (5) simply lead to an extra sign (compared with the result for a positive mass). Second, for a genuinely complex mass mk=|mk|eiφk, one can rewrite the derivatives in terms of |mk| and φk. For the partition function (4) at fixed θ the derivative with respect to φk can be rewritten as a derivative with respect to θ.

Let us begin with the case of one flavor of mass m. Since the free energy is extensive in V, the QCD partition function, obtained by expanding the action to lowest order in m, is given by [5] Z(m,θ)=Z(m=0,θ)exp[mVΣcosθ+O(m2V)],(6)where Σ is the absolute value of the chiral condensate in the limit m=0 and θ=0. Since the mass-independent factor Z(m=0,θ) does not contribute to the chiral condensate we will ignore it below. Equation (6) is valid for both positive and negative quark masses. This has the consequence that the chiral condensate from Eq. (5), Σ(m)=Σcosθ,(7)does not change sign when m becomes negative. Assuming nonvanishing Σ, at first sight this appears to be in contradiction with the Banks-Casher formula [16], which expresses the chiral condensate in terms of the eigenvalues λn of the Dirac operator starting from the relation Σ(m)=⟨1V∑n1iλn+m⟩,(8)where the average is over gauge-field configurations. Equation (8) is valid both at fixed ν and at fixed θ. The eigenvalues either are zero, giving a term proportional to 1/m, or occur in pairs ±λn, which yield terms of the form 2m/(λn2+m2). Thus the function Σ(m) appears to be odd in m. The resolution of this puzzle is that the statistical weight in Eq. (8) contains the fermion determinant, which leads to an additional mass dependence. This resolution was fully worked out in Refs. [11,12] by an explicit computation of the spectral density and the chiral condensate in the ϵ domain of QCD. Employing identities for sums of products of Bessel functions, the expressions could be summed to give the spectral density at θ=0. For negative mass the resulting expression is increasing exponentially with V and oscillating with a period that scales as 1/V. As we know from QCD at nonzero chemical potential [17], exactly such behavior of the spectral density can eliminate a discontinuity of the chiral condensate. However, as has already been observed in [18], the contributions of both the zero modes and the nonzero modes diverge exponentially with the volume. It turns out [11,12] that these divergent contributions cancel identically, resulting in a chiral condensate that remains constant in the ϵ domain, i.e., for mΛQCDV≪1.

Let us turn to two-flavor QCD. In this case the full flavor symmetry is U(2)×U(2), with the axial U(1) group broken by the anomaly and the SU(2)×SU(2) subgroup broken spontaneously by the chiral condensate. In the ϵ domain of QCD, the partition function of the resulting Nambu-Goldstone modes, which interact according to a chiral Lagrangian, simplifies to [5] Z(M,θ)=∫SU(Nf)dUexp[VΣ Re(eiθ/NfTrMU)],(9)which actually holds for any Nf. The measure dU is the normalized Haar measure. In the thermodynamic limit, the U field aligns itself with the chiral condensate. For Nf=2, the simplest case is when the two masses are equal to a common mass m. For mcos(θ/2)>0 the saddle-point solution is U=1, but for mcos(θ/2)<0 it is given by U=-1. To leading order in the thermodynamic limit this results in logZ(m,m,θ)≈|m|VΣ≫12VΣ|mcos(θ/2)|.(10)Because of the absolute value, the chiral condensate as defined in Eq. (5) acquires a discontinuity at m=0, Σ(m)≈|m|VΣ≫1sign(m)Σ|cos(θ/2)|.(11)

Another simple case is that of unequal quark masses and θ=0, where we have in leading order [5] logZ(m1,m2,θ=0)≈|mk|VΣ≫1VΣ|m1+m2|.(12)The phase diagram of this case is shown in Fig. 1. The two phases are separated by the line m1+m2=0 on which the pion mass becomes zero, known as the Dashen phenomenon [19]. For three flavors, this line changes to a finite region where pions condense and CP symmetry is spontaneously broken [20]. It can also become a finite region when pion condensation occurs in the case of Wilson fermions [21].

The phase diagram at fixed θ has to be contrasted to the one at fixed topological charge ν, where we have in leading order [5] logZν(m1,m2)≈|mk|VΣ≫1VΣ(|m1|+|m2|).(13)The sign of the chiral condensate at fixed ν always changes sign when one of the masses changes its sign, unlike at fixed θ=0, where the sign change occurs at the Dashen line m1=-m2. One aim of the present work is to understand this difference. In Fig. 2 we highlight the different behavior of the two-flavor chiral condensate at fixed topological charge ν (left) and fixed θ angle (right).

Finally, let us mention one particular limit of the two-flavor case. When one of the quark masses becomes large, the two-flavor theory reduces to the one-flavor theory. In the chiral Lagrangian we then have U→(e-iφ00eiφ)(14)with φ fixed at φ=θ/2, resulting in the one-flavor partition function (6).

Generically, QCD at nonzero θ angle has a sign problem, which originates from the weight factor exp[iν(θ+∑kφk)] in Eq. (4). The sign problem is absent only for cos(θ+∑kφk)=1, and it is most severe for cos(θ+∑kφk)=-1, in which case the weight factor is (-1)ν. For real (but possibly negative) quark masses, these two conditions translate into cosθ=±sign detM.

While these two extreme cases are already apparent from Eq. (4), the severity of the sign problem in the general case can be measured by the ratio of the partition function with a phase and the phase-quenched partition function. Explicit analytical results for this ratio can be obtained in the ϵ domain.

For one flavor the ratio is given by exp[-ΔF(m,θ)]=Z(m,θ)Z(|m|,0)=eVΣ(mcosθ-|m|).(15)Therefore exponential cancellations take place at θ≠0. The function ΔF(m,θ)=VΣ|m|(1-sign(m)cosθ) is shown in Fig. 3 for m>0. In agreement with the general argument above, it assumes its maximum at cosθ=-sign(m), where the sign problem is most severe, while the sign problem is absent for cosθ=sign(m). The free energy is a smooth function of the quark mass and the θ angle, which reflects the fact that the one-flavor theory has no phase transition.

For two flavors the ratio is given by exp[-ΔF(m1,m2,θ)]=Z(m1,m2,θ)Z(|m1|,|m2|,0).(16)The ϵ-domain result for the two-flavor partition function is given by the integral over SU(2) in Eq. (9). In terms of the microscopic variables uk=mkVΣ it becomes [5] Z(u1,u2,θ)=2I1(u12+u22+2u1u2cosθ)u12+u22+2u1u2cosθ(17)with Ik the modified Bessel function of the first kind. Again the difference ΔF assumes its maximum at cosθ=-sign(m1m2); see the plots in Fig. 4. Therefore the sign problem is most severe either at θ=0 when both masses have opposite signs or at θ=π when both masses have the same sign. This can also be seen in the exponentially increasing oscillations of the level density when increasing the θ angle; see Fig. 9 below. The sign problem is absent only for cosθ=sign(m1m2).

We illustrate the free-energy difference in Fig. 4. The partition function (17) is monotonically decreasing (increasing) in θ∈[0,π] for quark masses of equal (opposite) signs. This carries over to an increase (decrease) of the free energy. The free-energy difference is also strictly increasing (decreasing) with respect to the moduli of the masses for equal (opposite) signs. For equal masses it simplifies to ΔF(u,u,θ)=log[I1(2u)cos(θ/2)I1(2ucos(θ/2))],(18)where the monotonicity can be checked easily.

The free-energy difference has the thermodynamic limit lim|u1|,|u2|≫1ΔF(u1,u2,θ)=|u1|+|u2|-u12+u22+2u1u2cosθ,(19)where only the Bessel functions had to be approximated. (There are subleading logarithmic corrections to this result.) For equal masses the thermodynamic limit becomes lim|u|≫1ΔF(u,u,θ)=2|u|(1-|cos(θ/2)|),(20)which has a kink at the Dashen point θ=π.

In the limit of zero quark masses, the chiral condensate at fixed topological charge has a discontinuity on the support of the Dirac spectrum, which is dense in the thermodynamic limit. We have to understand how this discontinuity can be moved away from the support of the spectrum at nonzero θ angle. When the spectral density is positive, this is certainly not possible. However, the averaging procedure to obtain the spectral density includes the fermion determinant. This determinant is not positive definite at nonzero θ angle, and thus the spectral density is generically not positive definite. Moreover, the spectral density is normalized with respect to the partition function at θ≠0, which is exponentially smaller than the phase-quenched partition function, and therefore may result in a spectral density that increases exponentially with the volume. Indeed, we know from QCD at nonzero chemical potential [17] and from QCD-like theories with indefinite measure [18] that the discontinuity of the chiral condensate can be shifted when the spectral density oscillates with an amplitude that is exponentially large in V and with a period that scales as 1/V. As a first step toward understanding this behavior, we decompose the spectral density and the chiral condensate into various contributions.

We denote the spectral density of the Dirac operator at fixed topological charge ν by ρν(λ,M). The spectral density at fixed θ is defined by ρ(λ,M,θ)=∑νPν(M,θ)ρν(λ,M),(21)where Pν is the statistical weight to find a gauge-field configuration with topological charge ν, Pν(M,θ)=Zν(M)eiνθ∑νZν(M)eiνθ=Zν(M)eiνθZ(M,θ).(22)To obtain a more detailed picture we split the spectral density into a zero-mode part and a nonzero-mode part, ρ(λ,M)=ρzm(λ,M)+ρnzm(λ,M).(23)This splitting is valid both at fixed ν and at fixed θ. Equation (21) holds separately for the zero-mode and nonzero-mode parts.

The zero-mode part of the density at fixed ν is ρνzm(λ)=|ν|δ(λ).(24)We will see in the next section that ρzm gives a contribution to the chiral condensate that diverges exponentially if a sign problem is present. This contribution must be canceled by a similar contribution of the nonzero-mode part ρnzm to obtain a finite condensate. The question is what part of ρnzm is responsible for this cancellation. Obviously there is no unique answer to this question, but we know that this cancellation also has to take place in the quenched approximation. Therefore we decompose ρnzm at fixed ν into a quenched part (obtained by setting Nf=0) and a dynamical part (the remainder), ρνnzm(λ,M)=ρνq(λ)+ρνd(λ,M).(25)The ϵ-domain result for the quenched part in terms of the microscopic variable x=λVΣ reads [22] ρνq(x)=|x|2[Jν2(x)-Jν+1(x)Jν-1(x)](26)with Jk the Bessel function of the first kind. The dynamical part depends, in addition to x, also on the quark masses, which on the microscopic scale we collect in u→=(u1,…,uNf). Explicit expressions can be found in [23,24].

An alternative to Eq. (25) would be to split ρnzm at fixed θ into a phase-quenched part [12], obtained by letting M→|M| and θ→0, and an oscillating remainder. The cancellation is then achieved by the phase-quenched part, but this part is more complicated than the quenched part, and therefore we do not consider it in this paper.

The chiral condensate is obtained from the spectral density by the relation Σ(m,M)=1V∫-∞∞dλρ(λ,M)iλ+m,(27)where m is a valence quark mass. It is convenient to distinguish m from the sea quark masses in M, but at the end of the calculation m will usually be set equal to one of the sea quark masses. Again, Eq. (27) holds both at fixed ν and at fixed θ. On the macroscopic scale Eq. (27) requires regularization, but in the ϵ domain it is valid as it stands.

We now split the condensate into zero-mode, quenched, and dynamical parts, obtained by replacing ρ in Eq. (27) by ρzm, ρq, and ρd, respectively. This leads to Σ(m,M)=Σzm(m,M)+Σq(m,M)+Σd(m,M)(28)at either fixed ν or fixed θ. As in Eq. (21) we have Σ(m,M,θ)=∑νPν(M,θ)Σν(m,M),(29)and this holds separately for all three contributions.

All equations in this section can be translated to the microscopic scale by the replacements λ=x/VΣ and M=u→/VΣ. In Eq. (27), the prefactor 1/V is then replaced by Σ. For the valence mass on the microscopic scale we will use the notation u^=mVΣ.

In this section we show, for any number of flavors, that the exponentially increasing contribution of the zero modes to the chiral condensate is canceled by an exponentially increasing contribution from the quenched part of the spectrum. The expressions for the contribution of the zero modes are valid without any assumptions, but the other calculations are performed in the ϵ domain of QCD.

Using Eqs. (21) and (24), the zero-mode part of the microscopic spectral density at fixed θ is given by ρzm(x,u→,θ)=δ(x)Z(u→,θ)∑νeiνθ|ν|Zν(u→).(30)Using the Fourier transform of Eq. (4), Zν(u→)=∫-ππdθ2πe-iνθZ(u→,θ),(31)the sum on the right-hand side (RHS) can be rewritten in the form ∑νeiνθ|ν|Zν(u→)=⨍-ππdφ2πA(φ)[Z(u→,θ-φ)-Z(u→,θ)],(32)where A(φ) is the Fourier transform A(φ)=∑νeiνφ|ν|(33)and the symbol ⨍ stands for a principal-value integral. Several comments are in order. (i) The definition (33) is to be understood as a distribution acting on test functions that are twice differentiable and vanish at φ=0, where the sum over ν is divergent. We always assume φ∈[-π,π] because of the integral (32). (ii) Since the integral over A(φ) vanishes, we could subtract Z(u→,θ) in (32) to end up with a test function that indeed vanishes at φ=0. (iii) In the following we always deal with test functions that are twice differentiable and vanish at φ=0. This justifies the introduction of a regulator ϵ>0 which results in a sum that is pointwise convergent except for φ=0, A(φ)=limϵ→0∑ν=-∞∞|ν|eiνφ-|ν|ϵ=limϵ→0∑ν=1∞ν(e-(ϵ+iφ)ν+e-(ϵ-iφ)ν)=limϵ→0∂∂ϵe-ϵ-cosφcoshϵ-cosφ=-12sin2φ2(valid for φ≠0).(34)Note that due to the pointwise convergence, the integral of A(φ) in the form (34) no longer vanishes so that the subtractions have to be made before the limit ϵ→0 is taken. For a function f that is twice differentiable at zero we therefore have ⨍-ππdφA(φ)[f(φ)-f(0)]=-⨍-ππdφf(φ)-f(0)2sin2φ2.(35)Finally, we note that -1/2sin2φ2=∂φcotφ2 is a total derivative which can be integrated by parts on test functions that vanish at φ=0.

The zero-mode part of the spectral density thus becomes ρzm(x,u→,θ)=δ(x)⨍-ππdφ2π12sin2φ2[1-Z(u→,θ-φ)Z(u→,θ)].(36)The zero-mode contribution to the chiral condensate follows trivially from Eq. (27): Σzm(u^,u→,θ)Σ=1u^⨍-ππdφ2π12sin2φ2[1-Z(u→,θ-φ)Z(u→,θ)].(37)In the derivation of this formula we did not make any assumptions on the scaling behavior of the quark masses, and we stress that this result is valid in general.

From the discussion of the sign problem in Sec. III we know that the ratio Z(u→,θ)/Z(|u→|,0) takes its maximum at cosθ=sign detM. Hence, for detM>0, the integral (37) is dominated by the region around φ=θ, and for detM<0 it is dominated by the region around φ=θ-π. For cosθ≠sign detM, the partition function Z(u→,θ) is exponentially smaller (in u→) than Z(|u→|,θ=0), resulting in a contribution to the chiral condensate that increases exponentially with the volume.

The quenched part of the spectral density at fixed θ is ρq(x,u→,θ)=1Z(u→,θ)∑νeiνθρνq(x)Zν(u→)=∫-ππdφ2πρ˜q(x,φ)Z(u→,θ-φ)Z(u→,θ)(38)with ρ˜q(x,φ)=∑νeiνφρνq(x)=J1(2|x|sinφ2)2sinφ2.(39)The last equality in Eq. (39) was obtained from Eq. (26) using Eq. (A7). To avoid confusion, we note that ρ˜q(x,θ) is the spectral density of the true quenched theory at fixed θ. It is convoluted with Z(u→,θ-φ)/Z(u→,φ) as shown in Eq. (38) to obtain what we have defined as the quenched part of the spectral density. The convolution introduces the quark masses into ρq(x,u→,θ), while ρ˜q(x,φ) is independent of the quark masses.

Equation (38) results in the “quenched” contribution to the chiral condensate Σq(u^,u→,θ)Σ=∫0∞2dxu^x2+u^2∫-ππdφ2πJ1(2xsinφ2)2sinφ2Z(u→,θ-φ)Z(u→,θ)=∫-ππdφ2π[12u^sin2φ2-K1(2|u^sinφ2|)sign(u^)|sinφ2|]Z(u→,θ-φ)Z(u→,θ).(40)In the last line we have used Eq. (C8) to obtain the modified Bessel function of the second kind K1. Note that the poles at φ=0 cancel so that the integral can be evaluated as an ordinary integral. Also for the quenched contribution, the integral is dominated by the region around φ=θ or φ=θ-π (depending on the sign of the product of the quark masses) where the effective θ angle vanishes. For θ+argdetM≠0, this again leads to contributions that increase exponentially. For Nf=0 the ratio of partition functions in Eq. (40) is equal to unity. The integral over φ of the expression in square brackets is equal to the “quenched” part of the condensate at topological charge zero, i.e., Σν=0q(u^), which follows by writing the RHS of Eq. (39) as a Fourier sum.

We now show that the two exponentially increasing contributions to the chiral condensate cancel. The sum of zero-mode and quenched contributions is given by Σzm(u^,u→,θ)Σ+Σq(u^,u→,θ)Σ=⨍-ππdφ2π[12u^sin2φ2-K1(2|u^sinφ2|)sign(u^)|sinφ2|Z(u→,θ-φ)Z(u→,θ)]=Σν=0qΣ-⨍-ππdφ2πK1(2|u^sinφ2|)sign(u^)|sinφ2|[Z(u→,θ-φ)Z(u→,θ)-1].(41)This integral converges as a principal-value integral.

The partition function is given in Eq. (9). We write U∈SU(Nf) in the form U=Vdiag(eiφ1,…,eiφNf)V† with V∈U(Nf)/U(1)Nf and ∑kφk=0 and perform a change of variables from U to V and φ→=(φ1,…,φNf). The Jacobian of this transformation is independent of V. If we assume degenerate quark masses, the integrand is also independent of V so that the integration over V simply gives an irrelevant constant. For nondegenerate quark masses the integral over V is of the Harish-Chandra–Itzykson-Zuber type [25] and leads to Z(u→,θ)∼∫∏kdφk2π|Δ(eiφ→)|2δ(∑kφk)×det[exp[ukcos(φℓ+θ/Nf)]]Δ(u→)Δ(cos(φ→+θ/Nf)),(42)where Δ(x→)=∏a<b(xb-xa) denotes the Vandermonde determinant. Unless indicated otherwise, a product or a sum over k is understood to run from 1 to Nf. The symbol ∼ indicates that we have suppressed the normalization constant.

The integrand in (42) is symmetric under permutations of the φk. We denote the nonexponential terms by f(u→,φ→+θ/Nf)=|Δ(eiφ→)|2/(Δ(u→)Δ(cos(φ→+θ/Nf))) and expand the determinant of exp[ukcos(φℓ+θ/Nf)] to obtain Z(u→,θ)∼∫∏kdφk2πf(u→,φ→+θ/Nf)×e∑kukcos(φk+θ/Nf)δ(∑kφk).(43)After shifting each φk by -θ/Nf and performing the integration over φ1 using the δ function, Eq. (43) becomes Z(u→,θ)∼∫∏k≥2dφk2πf(u→,φ¯→)×exp[u1cos(θ-∑k≥2φk)+∑k≥2ukcosφk],(44)where φ¯→=(θ-∑k≥2φk,φ2,…,φk). We will use this representation for the partition function in the denominator of Eq. (41).

Next, we consider the integral in the last line of Eq. (41). For large u^, the Bessel function behaves as Kν(2|u^sinφ2|)=π4|u^sinφ2|e-2|u^sinφ2|[1+O(1/u^)].(45)Therefore the -1 term in the last line of Eq. (41) cannot result in exponentially large contributions, but this term regularizes the integral at φ=0.

We now combine Eq. (43) with θ replaced by θ-φ and Eq. (45) and consider the contribution to the chiral condensate that gives the exponentially large terms. After shifting each φk by (φ-θ)/Nf, we obtain 1Z(u→,θ)∫dφ2π∫∏kdφk2πδ(∑kφk-θ+φ)f(u→,φ→)×exp[-2|u^sinφ2|+∑kukcosφk]=1Z(u→,θ)∫∏kdφk2πf(u→,φ→)×exp[-2|u^sin12(θ-∑kφk)|+∑kukcosφk].(46)We now set the valence mass u^ equal to one of the sea quark masses, say u1. Using the trigonometric identity cosα-cosβ=2sin[(β+α)/2]sin[(β-α)/2], the exponent of the last line in (46) can be written as -2|u1sinθ-∑kφk2|+u1cosφ1-u1cos(θ-∑k≥2φk)+u1cos(θ-∑k≥2φk)+∑k≥2ukcosφk=-2|u1sinθ-∑kφk2|+2u1sinθ+φ1-∑k≥2φk2⁢sinθ-∑kφk2+u1cos(θ-∑k≥2φk)+∑k≥2ukcosφk≤u1cos(θ-∑k≥2φk)+∑k≥2ukcosφk,(47)where in the last line we used the fact that the absolute value of the second term is always smaller than the absolute value of the first term. The RHS of this estimate is exactly the exponent of the integrand for Z(u→,θ); cf. (44).

We have thus found that in Eq. (41) the exponent of the numerator is always smaller than or equal to the exponent of the denominator. Therefore the sum of the zero-mode and quenched contributions to the chiral condensate does not increase exponentially for large rescaled masses uk. The preexponential terms may have (even strong) effects on the integral. They can be zero, can diverge at the saddle points, or can prevent us from reaching some of the saddle points. However, none of these effects can lead to an exponential increase because the integral in (41) is well defined and finite at fixed u→.

The exponential cancellation is illustrated in Fig. 5 for Nf=1 at θ=π/2 and in Fig. 6 for Nf=2 at θ=0, where both Σ1zm and Σ1q increase exponentially in the quadrants where u1u2<0. The sum of the two contributions remains finite. Also shown in Fig. 6 is the dynamical part of the chiral condensate; see Eq. (82).

Finally, we give a heuristic argument why the cancellation between the zero-mode contribution and the quenched contribution to the chiral condensate takes place. When we have |ν| zero modes, the nonzero eigenvalues on the microscopic scale are, on average, shifted from the origin by |ν|; see Ref. [26] [Sec. VIII A]. To find a definitive gap one needs to assume 1≪ν≪VΛQCD4. Explicitly, one can easily show that the asymptotic behavior of the microscopic quenched density is given by ρνq(x)≈|ν|≫11πx2-ν2|x|θ(|x|-|ν|).(48)The corresponding chiral condensate Σνq(u^) can be calculated along Eq. (27), i.e., Σνq(u^)Σ≈|ν|≫12u^π∫|ν|∞dxx2-ν2x(x2+u^2)=|ν|u^(1+u^2ν2-1).(49)For large masses at large, fixed topological index, the quenched chiral condensate has the approximate form Σνq(u^)Σ≈|u^|≫|ν|≫1sign(u^)-|ν|u^.(50)The second term shows that the nonzero-mode spectrum has been depleted by |ν| modes so that the total number of eigenvalues does not depend on ν. The contribution of the zero modes is given by Σνzm(u^)=|ν|/u^.

At fixed (large) ν we thus find that for large u^ the 1/u^ contribution to the quenched part of the chiral condensate exactly cancels the contribution from the zero modes. At fixed θ the contribution of the fermion determinant leads to exponentially large terms (instead of 1/u^ terms) at large u^, but as shown in Sec. V we again have a cancellation between the zero-mode and quenched parts. This cancellation is deeply rooted in topology and spectral flow, which guarantee that the total number of eigenvalues around zero remains the same. For chiral random matrix theory, this can be shown at the technical level [26], but the argument is much more general: the Dirac spectrum near zero is depleted by exactly the same number of levels as we have zero modes. One could argue that the spectral density at fixed θ angle mainly involves very large ν so that in the thermodynamic limit the spectrum acquires a gap at zero. However, since the topological susceptibility is finite in the thermodynamic limit, the number of zero modes is of order V, while the eigenvalue density is of order 1/V, resulting in a gap with a width of order 1/V.

In this section we derive a number of explicit results for one-flavor QCD. In the first subsection we compute the contributions to the spectral density at fixed θ angle. In the second subsection we use these results to compute the dynamical contribution to the chiral condensate and show that a mass-independent total chiral condensate is obtained.