New term in effective field theory at fixed topology

A random matrix model for lattice QCD which takes into account the positive definite nature of the Wilson term is introduced. The corresponding effective theory for fixed index of the Wilson Dirac operator is derived to next to leading order. It reveals a new term proportional to the topological index of the Wilson Dirac operator and the lattice spacing. The new term appears naturally in a fixed index spurion analysis. The spurion approach reveals that the term is the first in a new family of such terms and that equivalent terms are relevant for the effective theory of continuum QCD.

A random matrix model for lattice QCD which takes into account the positive definite nature of the Wilson term is introduced.The corresponding effective theory is derived to next to leading order.It reveals a new term proportional to the topological index of the Wilson Dirac operator and the lattice spacing.We conjecture that this term is the first in a new family of such terms.Within the effective field theory we derive the universal parametric correlation relevant for the axial anomaly and show that, the contribution from the measure to the axial anomaly, is completely dominated by the would be zero modes of the Wilson Dirac operator.This effective theory also naturally explains the effect of order a-improvement observed in the spectrum of the Wilson Dirac operator of lattice QCD simulations.
The duality between random matrix theory (RMT) and low energy effective field theory (EFT) has revealed a plethora of insights in physical systems as diverse as quantum chromo dynamics (QCD) [1,2] and topological solid state systems which realise Majorana fermions [3].Most results obtained address average spectral properties of a central operator for the system in question, such as the Hamiltonian or the Dirac operator, but also universal parametric correlations can be obtained from RMT and EFT [4][5][6][7][8][9].The duality between the two approaches is highly valuable as some questions may be technically easier to address in one of the two frameworks.In addition some questions only have a natural formulation in one approach.One example of this is the effect of the Hermiticity-properties of the operator in question: In the RMT formulation the Hermiticity properties are obvious since the operator is directly present, on the contrary in the EFT approach these properties are hidden in the low energy constants (LEC) [10].In this work we investigate how the positive definite nature of an operator explicitly appearing in RMT affects the dual EFT.Remarkably this will allow us to extend the duality and show that random matrix theory can be used to discover new terms in the low energy effective theory.The new term found is intimately linked to the topological properties of the theory.
The physical realization we study here is lattice regularised QCD.In particular, we focus on the Wilson term which is essential in order to remove the Fermonic doublers from lattice QCD, see e.g.[11].The Wilson term is a covariant Laplacian and thus positive definite.The explicit symmetry breaking of the Wilson term is well understood in EFT [12][13][14], however, the effect of the positive definite nature of the Wilson term on the EFT is studied here for the first time.We will introduce a new random matrix model (RMM), which takes into account the fact that the Wilson term is positive definite.A general method to derive the next to leading order terms in the EFT from the RMM is then developed and used.The resulting EFT uncovers a new term in the effective action for fixed topology.The term which is linear in the lattice spacing and the topological index is similar to an axial mass term.We conjecture this term to be the first in a family of such new terms in effective actions at fixed topology.
In addition, we use the EFT obtained to derive a new universal parametric correllator for the QCD Dirac operator: The distribution of the chiralities over the spectrum of the Hermitian Dirac operator.This new statistics is highly relevant for the axial anomaly.As we demonstrate explicitly it shows how, despite an apparent logarithmic divergence, the contribution from the measure to the axial anomaly, is completely dominated by the would be topological modes of the Dirac operator.
Finally, we use the EFT to explain why an order aimprovement of lattice actions does not only move the Dirac eigenvalues closer to the origin but at the same time also decreases the width of the distributions, as was observed in lattice QCD simulations [15].
While the results presented here are applied to lattice regularised QCD, we stress that they only rely on the global symmetries and general nature of the operator in question.The results thus apply to all systems with the same properties.Moreover, the method to obtain higher order terms in the effective theory from random matrix theory introduced here may be applied directly to other symmetry classes.
The new RMM with a positive definite analogue of the Wilson term is defined as The matrix D W is the RMM analogue of the Wilson Dirac operator.We use a chiral basis and the diagonal term proportional to a corresponds to the Wilson arXiv:2004.10420v1[hep-lat] 22 Apr 2020 term.The parameter a > 0 is the analogue of the lattice spacing.The complex matrix W ∈ C n×(n+ν) and the positive definite Hermitian matrices A ∈ Herm + (n) and +TrA+TrB) .The exponents ν A , ν B ≥ 0 are unconstrained by symmetry and will combine into two LEC's in the EFT.Note that contrary to the RMM introduced in [16] it is not possible to absorb the sign of a into the random matrix it is multiplied by, and as we show below this introduces odd terms in a in the effective theory.The fermion masses M R , M L ∈ C N f ×N f also comprise axial quark masses, that will be exploited as source variables.
The partition function has fixed index ν of the Dirac operator where |ψ j are the eigenvectors of D W .This RMM is invariant under ν → −ν, ν A ↔ ν B , and then n → n+ν, which corresponds to swapping right-and left-handed modes.Though it changes the sizes of W , A and B like n ↔ n + ν the overall size, N = 2n + ν, of D W is fixed.As the original block structure can be retrieved by a unitary transformation, this "reflection" leaves the partition function invariant.Additionally ν A − ν B is odd under this transformation while ν A + ν B is even.This allows us to chose ν A − ν B = w t ν + ν and ν A + ν B = w M ≥ 0, where w t and w M become LEC's in the dual EFT.
The dual EFT is obtained from the RMM in two steps.The first step is exact.We express the determinants as an integral over Fermionic variables Ψ (R/L) and then average over A, B and W .After using the superbosonization formula [17] to exchange the dyadic matrices Ψ (R/L) † Ψ (R/L) with the unitary matrices U (R/L) we obtain In the second step we introduce the counting The shifted mass matrix M (a) R/L = M R/L + a + w m a/N results from the positivity of the Wilson term.All terms apart from the last in the action also appear in the EFT approach [12][13][14] and this allows us to identify N as the dimensionless volume where the dimension is set by a LEC's multiplying each term in the effective action.However, the last term in (4), which takes the form of an axial mass proportional to νa, has not appeared previously.
Let us try to understand why the new term has not appeared in effective actions previously.From the RMT side the term could not be generated by the previous model [16] since this by construction was even in a. From the EFT side [12][13][14], one writes down the most general effective action consistent with the symmetries order by order in a of the Symanzik action at fixed θ-angle.However, one discards all terms that are total derivatives [18,19].Here, we consider the theory in a sector with fixed index of the Dirac operator, therefore in the associated continuum expansion the topological density, which is a total derivative, will integrate to ν.Including the total derivatives associated with the fixed topology opens for possible new terms in the Symanzik expansion, which in turn gives rise to new terms in the effective theory.The term w t νaTr(U − U −1 )/2 has just the right structure to be generated from such a total derivative term in the Symanzik expansion.This motivates us to conjecture that the new term found here is the first in a family of such terms.
A word on the counting before we proceed: We here make use of the p-regime counting for m and derivative terms therefore also enters at the leading orders in the EFT.Since it is zero dimensional the RMM does not generate dynamical terms.However, the virtue of the RMT approach used here is that we can explicitly derive the corresponding EFT from the RMT and in this way generate terms in the EFT that may have been overlooked in the EFT approach.The use of the p-counting is motivated below where we discuss the effect of improving the lattice action.There we will also connect to theregime where the zero-mode of the pion field dominates at leading order [20].
In order to check that the new term in the EFT is physically valid we now derive the partition function at fixed vacuum angle θ.From the EFT approach no new terms are expected in Z θ , since for fixed θ we may follow the standard path and safely neglect total derivatives.As we now show by explicitly deriving Z θ this is indeed the case.
We use the relation to define the partition function at fixed θ for non zero a.Note that despite the new term proportional to ν in the EFT we still have Z ν = Z −ν , as follows by using the invariance of the measure under U → U −1 , and thus Z θ = Z −θ .For notational simplicity we keep N f = 1 where U = e i θ and set the mass matrix to L = m a = m + a + w m a/N ∈ R, meaning without axial mass.Using that a ∼ 1/ √ n, the sum over ν in (5) imposes the constraint θ + θ + w t a sin( θ) = 0 We obtain for the partition function up to Despite the new term at fixed ν, the partition function at fixed θ has no new terms.Z θ is, as it should be, perfectly consistent with the standard EFT approach going through the Symanzik expansion and neglecting boundary terms.The effective theory with fixed θ just obtained allows us to compute the topological susceptibility Therefore, we have ∆ν 2 ∼ √ n in the counting considered, and hence the new term is in fact typically enhanced by a factor n 1/4 .We now turn to the axial anomaly.As shown by Fujikawa [21] the response of the Fermionic measure to an axial transformation includes a nontrivial Jacobian J. Here, we extend the argument of Fujikawa to non-zero lattice spacing by using the eigenvalues and eigenvectors of the Hermitian Wilson-Dirac operator, D 5 = γ 5 (D W + m).We follow the proof of Fujikawa and obtain Fujikawa regulates the infinite sum as where Λ is the width of the regularization.Subsequently, he reformulates the regulator to show that the sum equals the topological index ν of the gauge field configuration.We follow a different path and consider the quantity which allows us to turn the sum in the exponent of J into an integral.To understand the integrand in the exponent, we introduce the ensemble averaged distribution of the chiralities over the spectrum of D 5 We have the relation where the Green function of Hence we may compute ρ 5,χ from the EFT by the supersymmetry technique.For simplicity, we consider the quenched case employing the relation where the quenched partition function, Z 1|1 , comprises a single valence fermion and boson and is given by a supersymmetric integral Here Str and Sdet refer to the supersymmetric versions of the trace and the determinant, see [23].
The EFT may also be used to derive properties of the real eigenvalues λ k of the original Wilson-Dirac operator.
For example, we may get the distribution of the chiralities over the real eigenvalues of the Wilson Dirac operator [9] ρ through the relaion [9] ρ The counting a ∼ m ∼ 1/ √ n employed here is relevant as the order a term will push the real eigenvalues of the Wilson Dirac operator out to order a.Hence, the Fermion mass m, which in the quenched supersymmetric partition function becomes the eigenvalue, must be of the same order.Order a improvement of the lattice action will primarily reduce the LEC of the order a term in the EFT action and thus correspondingly the magnitude of the eigenvalue.If the order a improvement is so accurate that the LEC of the order a term in the EFT becomes of order 1/ √ n, then the relevant counting for the quark mass becomes m ∼ 1/n.Thus accurate order a improvement will connect back to the standard -counting where m ∼ 1/n and a ∼ 1/ √ n.The results derived here, in this way, allow us to monitor the effect of the order a improvement.Moreover, the results derived here explains an open problem: Order a-improvement of the Wilson Dirac operator naturally moves the eigenvalues of the Wilson Dirac operator closer to the origin, since the order aterm acts as a mass.Surprisingly, however, as observed in [15] the distribution of the real eigenvalues also becomes more narrow when the Wilson Dirac operator is order a-improved.The action of (15) offers a natural explanation: The width of the distribution of the real modes comes from the order 1 and 1/ √ n terms in the action which, we note, includes m.As the action is order a improved the relevant eigenvalue and hence the relevant m used in the supersymmetric method decreases.This in turn suppress the order 1 and 1/ √ n terms resulting in a narrower (and hence more continuum like) distribution of the real modes of the Wilson Dirac operator.
To summarize, we have introduced a new RMM with a positive definite analogue of the Wilson term.Subsequently we derived and analyzed the dual effective action.
The new term found in the dual effective action for fixed topology is conjectured to be the first in a family of such new terms.To study this further it would be most interesting to solve the new RMM at finte N e.g. using the method of biorthogonal polynomials.This would allow to fully understand the effect of the NLO terms in the unquenched case where the supersymmetric approach of the present letter becomes computationally challenging.