PHYSICAL REVIEW D 100, 034507 (2019) Phase ambiguity of the measure for continuum Majorana fermions Maarten Golterman1,2 and Yigal Shamir3 1 Department of Physics and IFAE-BIST, Universitat Autònoma de Barcelona E-08193 Bellaterra, Barcelona, Spain 2 Department of Physics and Astronomy, San Francisco State University, San Francisco, California 94132, USA 3 Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, 69978, Tel Aviv, Israel (Received 20 May 2019; published 9 August 2019) Integrating over a continuum Majorana fermion formally yields a functional Pfaffian. We show that the phase of this Pfaffian is ambiguous, as it depends on the choice of basis. This ambiguity is naturally resolved within a nonperturbative lattice definition, allowing us to discuss the relation between the phase of the lattice Pfaffian and the effective θ angle of the theory. We also resolve an apparent paradox regarding the induced θ angle when a theory of N Dirac fermions in a real representation of the gauge group is reexpressed in terms of 2N Majorana fermions. We discuss how all this is reflected in chiral perturbation theory. DOI: 10.1103/PhysRevD.100.034507 I. INTRODUCTION showing that each entry of M has a Uð1ÞA phase equal to π=2. Now let us apply a Uð1ÞA rotation that turns the mass In a QCD-like theory with Dirac fermions, the measure matrix into a positive matrix, M ¼ m12N . Because of the of the Euclidean functional integral is positive when all fermions have a positive mass, and, as a consequence, there anomaly, this generates a topological term eiθQ , where is no topological term induced by the fermionic part of the Z theory. This generalizes to all types of fermion irreducible g2 Q¼ d4 xtrðFF̃Þ; ð1:3Þ representations (irreps): complex, real, and pseudoreal. 32π 2 If the theory contains N Dirac fermions in a real irrep, we may reformulate it in terms of 2N Majorana fermions. We is the topological charge, and will be using Majorana fields each of which packs together a Weyl fermion and its antifermion.1 Assuming an equal θ ¼ −πNT=2; ð1:4Þ positive mass m > 0 for all Dirac flavors, the mass matrix M of the Majorana formulation is then given by M ¼ mJ S, where T is the index of the Dirac operator for the fermion with the 2N × 2N matrix irrep in a single instanton background.   Let us consider the consequences of this topological 0 1N JS ¼ ; ð1:1Þ term. T is always even for a real irrep.2 If NT is divisible by 1N 0 4 then eiθQ ¼ 1, and the topological term drops out. If NT is not divisible by 4, we have eiθQ ¼ ð−1ÞQ . Hence, it where 1n is the n × n unit matrix. A nonanomalous chiral appears that the Majorana measure will be positive for even rotation can then be used to bring the mass matrix to a Q, but negative for odd Q. This is puzzling, because the flavor-diagonal form M ¼ mJ rot S where measure of the original Dirac theory is positive for any Q, and, obviously, the Dirac and Majorana formulations S ¼ iγ 5 12N ; Jrot ð1:2Þ should represent the same theory. The paradox would be resolved if the very transition to 1 The precise definition is given in Eq. (2.8) below. the Majorana formulation would somehow generate a “compensating” topological term eiπNTQ=2 . The additional Published by the American Physical Society under the terms of topological term induced by the Uð1ÞA rotation would then the Creative Commons Attribution 4.0 International license. cancel against the compensating topological term. We would Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. 2 We will recover this result in Sec. II. 2470-0010=2019=100(3)=034507(17) 034507-1 Published by the American Physical Society MAARTEN GOLTERMAN and YIGAL SHAMIR PHYS. REV. D 100, 034507 (2019) end up with a positive mass matrix and with no topological where term, as in the original Dirac theory. The purpose of this paper is to show that this is indeed 1 L ¼ F2 þ ψ̄ðD þ M† PL þ MPR Þψ þ iθQ; ð2:2Þ what happens. In reality, it turns out that the paradox 4 described above arises because in the argument we ignored a phase ambiguity of the Majorana measure which is with PR;L ¼ ð1  γ 5 Þ=2, and M is a complex N × N present in the formal continuum theory. The existence of matrix. The topological charge Q was introduced in this ambiguity allows us to require agreement between the Eq. (1.3). We will specialize to a mass matrix of the form Dirac and Majorana formulations. When the Majorana mass matrix involves J S or Jrot S , this requirement implies the M ¼ mΩ ¼ meiα=ðNTÞ Ω̃; Ω̃ ∈ SUðNÞ; ð2:3Þ existence of the compensating topological term in the path integral. Going beyond formal arguments, we demonstrate with real m > 0 and a real phase α. Upon integrating out the the presence of the compensating topological term through fermions the dependence on Ω̃ drops out thanks to the a fully nonperturbative lattice derivation of the transition invariance under nonsinglet chiral transformations, and from the Dirac to the Majorana formulation. Finally, we discuss the implications for the chiral effective theory. detðD þ M† PL þ MPR Þ ¼ eiαQ detN ðD þ mÞ; ð2:4Þ This paper is organized as follows. In Sec. II we show how, in the continuum, a phase ambiguity arises in the where, on the right-hand side, D þ m is the one-flavor choice of a basis for a gauge theory with Majorana Dirac operator. This result can be derived using the spectral fermions. We explain how this ambiguity can be resolved representation of the Dirac operator; see Appendix A. As in a theory with an even number of Majorana fermions by mentioned earlier, T is the index of the Dirac operator in a comparison with the same theory formulated in terms of single instanton background. The measure μðAÞ of the path Dirac fermions. Then, in Sec. III, we show that the lattice formulation implies a natural choice of basis, thus fixing integral is thus the phase consistently, both in the formulations with 1 2 Wilson and with domain-wall fermions. This allows us μðAÞ ¼ e−iθeff Q e−4F detN ðD þ mÞ to discuss the θ angle induced by the lattice fermion action, ¼ e−iθeff Q μ̃ðAÞ; ð2:5Þ reviewing and generalizing the earlier work of Ref. [1]. We consider separately a standalone gauge theory of Majorana where fermions, and a theory of 2N Majorana fermions obtained by reformulating a theory of N Dirac fermions. We then 1 2 μ̃ðAÞ ¼ e−4F detN ðD þ mÞ; ð2:6Þ revisit the precise form of the condensate in the presence of a fermion-induced θ angle, both in the gauge theory as well is positive, and the effective topological angle is as in chiral perturbation theory. This is done in Sec. IV for a theory with Dirac fermions in a complex irrep of the gauge group, and in Sec. V for a theory with Majorana fermions in θeff ¼ θ − α: ð2:7Þ a real irrep of the gauge group. Section VI contains our summary and conclusion. There are six Appendixes deal- ing with technical details. B. Majorana fermions A theory of N Dirac fermions in a real representation of II. MAJORANA FERMIONS AND the gauge group G can be reformulated in terms of 2N THE PHASE AMBIGUITY Majorana fermions. The N Dirac fermions are composed of In this section, we first review some useful standard 2N Weyl fermions. From these Weyl fermions, we con- results for Dirac (Sec. II A) and Majorana (Sec. II B) struct Majorana fermions each of which packs together a fermions. We then discuss the phase ambiguity that is Weyl fermion and its antifermion, which is possible encountered in defining the continuum path integral for because the fermion and the antifermion belong to the Majorana fermions (Sec. II C). same representation of G. The mapping between Dirac fermions (on the right-hand A. Dirac fermions side) and Majorana fermions (on the left-hand side) is Consider a Euclidean gauge theory with N Dirac fermions in some irrep of the gauge group. The partition ΨL;i ¼ ψ L;i ; function for the most general choice of parameters is ΨR;i ¼ CSψ̄ TL;i ; Z  Z  ΨR;Nþi ¼ ψ R;i ; 4 Z ¼ DADψDψ̄ exp − d xL ; ð2:1Þ ΨL;Nþi ¼ CSψ̄ TR;i ; ð2:8Þ 034507-2 PHASE AMBIGUITY OF THE MEASURE FOR CONTINUUM … PHYS. REV. D 100, 034507 (2019) where i ¼ 1; …; N. Here C is the charge-conjugation In the following, we ask the question of how this paradox matrix, and S is the group tensor satisfying the invariance may be resolved in the continuum. In Sec. III we will show property gT Sg ¼ S for all g ∈ G. We recall the basic how it is avoided, by introducing a nonperturbative properties, C−1 ¼ C† ¼ CT ¼ −C, and S−1 ¼ S† ¼ ST ¼ S. regulator. We also introduce To start, let us consider a single Majorana fermion with Lagrangian Ψ̄ ≡ ΨT CS: ð2:9Þ 1 1 L ¼ Ψ̄DΨ ¼ ΨT CSDΨ; ð2:14Þ Thus, Eq. (2.8) determines all the components of the 2 2 Majorana fermions in terms of the original Dirac fermions, or, equivalently, in terms of the corresponding Weyl fields. D ¼ D þ meiαM γ5 : ð2:15Þ Other mappings between Dirac and Majorana fermions are possible, and we give an example in Appendix B. What is The differential operator CSD is antisymmetric, and the special about Eq. (2.8) is that it respects the natural result of formally integrating out the Majorana fermion is mapping between Weyl and Majorana fields. pfðCSDÞ, the Pfaffian of CSD. Proceeding to the lagrangian, for the kinetic term we In the Dirac case, detðDÞ is simply equal to the have (regulated) product of all eigenvalues; see Appendix A. What about Pfaffians? X N 1X2N Introducing the abbreviation A ¼ CSD, the effect of a LK ¼ ψ̄ Dψ ¼ Ψ̄ DΨI : ð2:10Þ i¼1 2 I¼1 I unitary change of basis for Majorana fermions is For the mass term we have A → A0 ¼ U T AU; ð2:16Þ Lm ¼ mψ̄eiαD γ5 ψ ¼ m iαD γ5 Ψ̄e JS Ψ; ð2:11Þ where A and thus A0 are antisymmetric. We will be looking 2 for a change of basis so that A0 will have a skew- diagonal form. where the 2N × 2N matrix J S was introduced in Eq. (1.1), For a real representation, the eigenvalues of the Dirac and αD ¼ α=ðNTÞ is the phase introduced in the Dirac case operator have a twofold degeneracy. Because its Hermitian in Eq. (2.3). We have set Ω̃ ¼ 1, since the SUðNÞ part of part is equal to m cos αM times the identity matrix, the Dirac the original Dirac mass matrix does not play a role in the operator (2.15) is normal, ½D; D†  ¼ 0. Consider an eigen- following. vector χ with eigenvalue λ. By normality, Dχ ¼ λχ implies The flavor symmetry is as follows. In the massless limit, D† χ ¼ λ χ. Hence the theory is invariant under SUð2NÞ transformations Ψ → ðPL h þ PR h ÞΨ; DCSχ  ¼ CSDT χ  ¼ CSðD† χÞ ¼ CSðλ χÞ ¼ λCSχ  : Ψ̄ → Ψ̄ðPL hT þ PR h† Þ; ð2:12Þ ð2:17Þ with h ∈ SUð2NÞ. When the mass term (2.11) is turned on, It follows that CSχ  is an eigenmode with the same the SUð2NÞ symmetry is explicitly broken to SOð2NÞ. The eigenvalue as χ. The eigenmodes χ and CSχ  are orthogo- Dirac formulation of the same theory obviously has nal, ðCSχ  Þ† χ ¼ −χ T CSχ ¼ 0, where we used that the the same global symmetry, but the full symmetry is matrix CS is antisymmetric. manifest only in the Majorana formulation.3 The skew-diagonal representation A0 is achieved by transforming to a basis in which each eigenvector χ is C. Pfaffian phase ambiguity followed by its companion eigenvector CSχ . Selecting arbitrarily one eigenvector from each pair, and labeling the There exists a nonanomalous SUð2NÞ chiral rotation that resulting subset as χ 1 ; χ 2 ; …, we consider the unitary brings the Majorana mass term (2.11) to a diagonal form change of basis generated by the matrix U whose columns m m are comprised of the ordered pairs of eigenvectors, Lm ¼ Ψ̄iγ 5 eiαD γ5 Ψ ¼ Ψ̄eiðαD þπ=2Þγ5 Ψ: ð2:13Þ 2 2 U ¼ ðχ 1 ; eiϕ1 CSχ 1 ; χ 2 ; eiϕ2 CSχ 2 ; …Þ: ð2:18Þ We see that we have an extra Uð1Þ phase of π=2, leading to an apparent paradox, as explained in the Introduction. Notice that, for each pair, we have allowed the second eigenvector to have an arbitrary U(1) phase relative to the 3 For a discussion of how the global symmetry is realized in the original eigenvector. These arbitrary phases play a pro- Dirac formulation, see Ref. [2]. found role, as we will now see. 034507-3 MAARTEN GOLTERMAN and YIGAL SHAMIR PHYS. REV. D 100, 034507 (2019) The 2 × 2 subspace of A0 associated with a pair ϕþ ¼ ϕ− ¼ 0; ð2:24Þ χ; eiϕ CSχ  with eigenvalue λ has the explicit form we achieve agreement between the corresponding factors for     χ T 0 −1 the Dirac and two-Majorana cases.4 iϕ  CSDð χ e CSχ Þ ¼ e λ iϕ : iϕ † Proceeding to the zero modes, in the Dirac case the e χ SCT 1 0 contribution of a pair of zero modes, χ 0 , eiϕ0 CSχ 0 , is just ð2:19Þ ðmeiαD Þ2 ; ð2:25Þ The Pfaffian of A0 factorizes as the product of Pfaffians for the 2 × 2 subspaces, where the Pfaffian of the above 2 × 2 depending on the chirality. In the Majorana case, the subspace is, by definition, equal to −eiϕ λ. Explicitly, corresponding contribution to pfðA0 Þ from each Majorana fermion is Y pfðA0 Þ ¼ ð−eiϕi λi Þ: ð2:20Þ i −meiϕ0 eiαM ¼ −meiϕ0 eiðαD þπ=2Þ ; ð2:26Þ This Presult exhibits a phase ambiguity, represented by the where on the right-hand side we have substituted sum i ϕi . αM ¼ αD þ π=2. The extra phase of π=2 arises during In retrospect, the phase ambiguity can be traced to the the transition from the Dirac to the Majorana formulation, elementary property pfðA0 Þ ¼ pfðAÞ detðUÞ. This relation as we have seen in Eq. (2.13). The contribution from two implies that the phase of the Pfaffian depends on the choice Majorana fermions is thus of basis for the Majorana field on which the differential operator A acts. The basis is represented by the unitary ðmeiϕ0 eiðαD þπ=2Þ Þ2 : ð2:27Þ matrix U, and detðUÞ is, thus, a basis-dependent phase. The rigorous resolution of the phase ambiguity requires a It follows that the Dirac result (2.25) will only be nonperturbative treatment in order to specify the basis, reproduced provided we make the nontrivial choice which we will give in Sec. III. In the rest of this section we restrict ourselves to an even number of Majorana fermions, ϕ0 ¼ π=2 mod π: ð2:28Þ and discuss how the phase may be fixed by appealing to the corresponding theory defined in terms of Dirac fermions, III. NONPERTURBATIVE CALCULATION where no such phase ambiguity exists. As reviewed in Appendix A for the Dirac case, let us In the previous section, we showed that the definition of consider separately the zero modes and the nonzero modes. a theory with Majorana fermions has an intrinsic phase Starting with the nonzero modes, and following Appendix A, ambiguity, which can be used to resolve the apparent the eigenvectors χ  now each have a companion, eiϕ CSχ  , paradox introduced in Sec. I. However, the question of where we have allowed for arbitrary relative U(1) phases. whether, and how, the theory “chooses” the proper phase The contribution of these two pairs of eigenvectors to was left open. In order to address this question, we need a pfðAÞ is properly regulated nonperturbative definition of the theory, which is provided by the lattice. ð−eiϕþ λþ Þð−eiϕ− λ− Þ ¼ eiðϕþ þϕ− Þ ðλ2 þ m2 Þ; ð2:21Þ The lattice action for a Majorana fermion will always have the generic form 12 ΨT AΨ for a suitable antisymmetric operator A. Integrating over the lattice Majorana field where, as in Appendix A, λ2 is the eigenvalue of the second- yields pfðAÞ, which is now well defined. There is no room order operator −D2 PR. For a single Dirac fermion in the for any (phase) ambiguity, because, on any finite-volume same real representation, the contribution of the eigenvectors lattice, A is a finite-size matrix, and the lattice selects the χ  and eiϕ CSχ  to detðDÞ is simply a factor of coordinate basis to define A. Our first result concerns a single Majorana fermion with ðλ2 þ m2 Þ2 : ð2:22Þ no chiral angle(s), and a positive bare mass m0 > 0. Using domain-wall fermions, we show in Appendix C that pfðAÞ The determinant is independent of the arbitrary U(1) phase of is strictly positive in this case. The domain-wall fermion each eigenvector. If we now take two Majorana fermions, the measure is then strictly positive for any number of corresponding contribution to pfðA0 Þ will be Majorana fermions, and in all topological sectors. In this section, we discuss in detail the transition from ½eiðϕþ þϕ− Þ ðλ2 þ m2 Þ2 : ð2:23Þ the Dirac to the Majorana formulation. In Sec. III A, we We see that, by making the choice It is, in fact, sufficient to choose ϕþ þ ϕ− ¼ 0 mod π. 4 034507-4 PHASE AMBIGUITY OF THE MEASURE FOR CONTINUUM … PHYS. REV. D 100, 034507 (2019) regulate the theory using Wilson fermions, and in Sec. III B, Zðθ; θF Þ ¼ Zðθ þ NTθF ; 0Þ: ð3:4Þ using domain-wall fermions. While in the case of Wilson fermions, there is a lacuna in the argument (which we This implies that the relative Uð1Þ phase of the Wilson term discuss in some detail in Appendix D), this is not the case for and the mass term turns into the familiar θ angle in the domain-wall fermions. We establish that the compensating continuum limit. In Eq. (3.4) we have written down the topological term alluded to in the introduction indeed arises generalization of the result of Ref. [1] to N Dirac fermions when needed, thus resolving the paradox. As in Appendix C, in an irrep with index T. In the case that a topological term it proves easier to work with the five-dimensional formu- with θ ≠ 0 is already present in the gauge action, NTθF lation of domain-wall fermions, rather than directly with gets added to θ. any Ginsparg-Wilson operator that arises in the limit of an We pause here to note that the argument given in infinite fifth dimension. We also remark that staggered Ref. [1] is not complete as it stands, because of a subtlety fermions always lead to a fourfold taste degeneracy in the related to renormalization. While it is beyond the scope continuum limit, and so they cannot be used here, given that of this paper to complete the proof, Appensdix D outlines the apparent paradox only arises when NT is even, but not a conjecture on the interplay of the observation of Ref. [1] divisible by 4.5 We summarize the results of this section in and renormalization. However, this subtlety does not Sec. III C. affect the rest of this paper. In particular, in the next A. Wilson fermions subsection we provide an argument analogous to the one given here based on domain-wall fermions, where the If we formulate the theory using Wilson fermions, the subtlety does not arise. resolution of the puzzle relies on the observation, made in Next, let us work out the transition from the Dirac to the Ref. [1], of how the θ angle can be realized within this Majorana case. We start with a single Dirac fermion in a fermion formulation. The starting point of the discussion is a real irrep, where the Wilson fermion operator DW ðθF Þ is one-flavor Wilson operator with both the Wilson and mass given by Eq. (3.1), taking θW ¼ θF and θm ¼ 0. In the terms chirally rotated by angles θW and θm , respectively, Majorana formulation, the 4 × 4 matrix in spinor space becomes an 8 × 8 matrix which mixes the two Majorana DW ðθW ; θm Þ ¼ DK þ eiθW γ5 W þ eiθm γ5 m0 : ð3:1Þ species. In terms of 4 × 4 blocks, the Wilson operator in the Here DK is the naive lattice discretization of the (massless) Majorana formulation is Dirac operator. W is the Wilson term, which eliminates the   fermion doublers, and is chosen for definiteness to be real DK eiθF γ5 W þ m0 DMaj ðθF Þ ¼ ; ð3:5Þ and positive; m0 is the bare mass. The partition function eiθF γ5 W þ m0 DK takes the form (2.1), but with the fermion part of the Lagrangian replaced by6 where we have used Eqs. (2.8) and (2.9). The Lagrangian becomes LF ¼ ψ̄DW ðθW ; θm Þψ: ð3:2Þ 1 First, only the difference θW − θm can be physical, as can be LF ¼ Ψ̄DMaj ðθF ÞΨ: ð3:6Þ seen by applying the transformation 2 The key feature of Eq. (3.5) is that, because of their ψ → eiηγ 5 ψ; ψ̄ → ψ̄eiηγ 5 : ð3:3Þ identical chiral properties, the Wilson and mass terms In the lattice-regulated theory, the determinant of this occur in the same places. Applying an SUð2Þ flavor transformation is unity, and hence it provides an alternative transformation, i.e., using Eq. (2.12) for N ¼ 1 with h ¼ representation of exactly the same theory. It is easily checked expð−iπσ 2 =4Þ ¼ h , and using that hT σ 1 h ¼ σ 3 , the that this transformation leaves the DK part invariant, while Majorana Wilson operator gets rotated into the angles undergo the transformation θW → θW þ 2η,   θm → θm þ 2η. By choosing η ¼ −θm =2 we eliminate the DK þ eiθF γ5 W þ m0 0 DMaj ðθF Þ ¼ : phase of the mass term, while the phase of the Wilson term 0 DK − ðeiθF γ5 W þ m0 Þ becomes θF ≡ θW − θm . With only the angle θF left in the fermion action, and ð3:7Þ with θ as the explicit vacuum angle [see Eq. (2.2)], what Ref. [1] claimed is that, in the continuum limit, When θF ¼ 0, the relative phase of the Wilson and mass terms is zero, for both of the Majorana species. This 5 implies that DMaj ð0Þ is the Wilson operator for two Interpreting the staggered tastes as physical flavors, it is possible that reduced staggered fermions can be employed [3,4]. Majorana fermions with the same bare mass m0 (as We have not explored this further. opposed to the case where one Majorana fermion would 6 We will not need the lattice form of the gauge action. have a mass þm0 and the other −m0 ). 034507-5 MAARTEN GOLTERMAN and YIGAL SHAMIR PHYS. REV. D 100, 034507 (2019) We prove this assertion by applying the transformation Domain-wall fermions are not exactly massless for finite (3.3) with η ¼ π=2 to the second Majorana fermion only.7 N 5 when m0 ¼ 0. The mass induced by a finite fifth Explicitly, it reads Ψ2 → iγ 5 Ψ2 . The Majorana-Wilson direction, usually referred to as the residual mass, is operator transforms into reminiscent of the additive mass renormalization of Wilson fermions. However, the residual mass vanishes in DMaj ðθF Þ the limit N 5 → ∞, which we will take before the continuum   limit. Following this order of limits, the mass term intro- DK þ eiθF γ5 W þ m0 0 duced in Eq. (3.9) renormalizes multiplicatively. Thus, the → 0 iγ 5 ðDK − ðeiθF γ5 W þ m0 ÞÞiγ 5 complications of the additive mass renormalization of the   Wilson case, that we encountered in Sec. III A, are avoided. DK þ eiθF γ5 W þ m0 0 ¼ : ð3:8Þ Our aim in this subsection is to recast the argument given 0 DK þ eiθF γ5 W þ m0 in Sec. III A in terms of the domain-wall formulation of the lattice regularized theory. The first step is to prove The fermion operator for each Majorana fermion is now an analogous result to Eq. (3.4), thus rederiving the theorem exactly the same as in the Dirac case. The corresponding of Ref. [1] in terms of domain-wall fermions. For this, we basis for the Majorana fields is given in Appendix B. It need to define an axial transformation. We take N 5 ¼ 2K follows that the fermion measure of the two-Majorana even, and define the axial transformation as [7] formulation is equal to pf 2 ðCSDW ðθF ÞÞ, and thus equal to the Dirac measure detðDW ðθF ÞÞ. We have proved that the δψðsÞ ¼ eiη ψðsÞ; δψ̄ðsÞ ¼ ψ̄ðsÞe−iη ; 1 ≤ s ≤ K; fermion measure in the Majorana formulation is unchanged −iη δψðsÞ ¼ e ψðsÞ; δψ̄ðsÞ ¼ ψ̄ðsÞe ;iη K þ 1 ≤ s ≤ 2K: from the Dirac formulation. Equation (3.8) shows that we can choose the mass matrix ð3:10Þ to be proportional to the unit matrix, instead of to JS [Eq. (1.1)] or Jrot S [Eq. (1.2)]. Unlike in the formal continuum Following Ref. [7], we define the five-dimensional currents treatment of the previous section, no phase ambiguity, nor 1 any “excess” phase of π=2, arises when the transition to jμ ðx; sÞ ¼ ðψ̄ðx; sÞð1 þ γ μ ÞUμ ðxÞψðx þ μ; sÞ Majorana fermions is done in the lattice-regulated theory. 2 − ψ̄ðx þ μ; sÞð1 − γ μ ÞU †μ ðxÞψðx; sÞÞ; B. Domain-wall fermions j5 ðx; sÞ ¼ ψ̄ðx; sÞPR ψðx; s þ 1Þ − ψ̄ðx; s þ 1ÞPL ψðx; sÞ: In this subsection, we revisit the argument of the previous ð3:11Þ subsection, but now using domain-wall fermions [5] instead of Wilson fermions. As we will see, in the case of domain- The four-dimensional axial current corresponding to the wall fermions, the argument is complete, allowing us to axial transformation (3.10) is conclude that a lattice regularization can indeed be invoked to settle the ambiguity we found in Sec. II. X K X 2K The starting point is the domain-wall fermion action [6] jAμ ðxÞ ¼ − jμ ðx; sÞ þ jμ ðx; sÞ: ð3:12Þ s¼1 s¼Kþ1 for a massive Dirac fermion with bare mass m0 and domain-wall height M, It satisfies the Ward-Takahashi identity X N5 ∂ −μ jAμ ¼ 2j5 ðKÞ þ 2mðψ̄ð2KÞPR ψð1Þ − ψ̄ð1ÞPL ψð2KÞÞ; S¼ ψ̄ðsÞðDK þ M − 1 − WÞψðsÞ s¼1 ð3:13Þ X N 5 −1 þ ðψ̄ðsÞPR ψðs þ 1Þ þ ψ̄ðs þ 1ÞPL ψðsÞÞ where ∂ − − μ is the backward lattice derivative, ∂ μ fðxÞ ¼ s¼1 fðxÞ − fðx − μÞ. − m0 ðψ̄ðN 5 ÞPR ψð1Þ þ ψ̄ð1ÞPL ψðN 5 ÞÞ; ð3:9Þ Analogous to Eq. (3.1), we can now introduce two angles, through the combinations SW ðθW Þ and Sm ðθm Þ, where where ψ is the five-dimensional fermion field ψðx; sÞ, SW ðθW Þ ¼ eiθW ψ̄ðKÞPR ψðK þ 1Þ þ e−iθW ψ̄ðK þ 1ÞPL ψðKÞ; s ¼ 1; …; N 5 . In Eq. (3.9), only the dependence on the fifth coordinate is made explicit. The mass term couples the Sm ðθm Þ ¼ −m0 ðe−iθm ψ̄ð2KÞPR ψð1Þ þ eiθm ψ̄ð1ÞPL ψð2KÞÞ: fields on opposite boundaries. ð3:14Þ 7 Note that the transformation (3.3) is consistent with the SW ðθW Þ replaces the s ¼ K term on the second line of Majorana condition (2.9). Eq. (3.9), and Sm ðθm Þ replaces the mass term (third line) in 034507-6 PHASE AMBIGUITY OF THE MEASURE FOR CONTINUUM … PHYS. REV. D 100, 034507 (2019) Eq. (3.9). Once again, under an axial transformation with S and C as in Sec. II, and R5 is a reflection in the fifth [Eq. (3.10)], θm;W → θm;W þ 2η, and hence only the differ- direction: ence θF ¼ θW − θm is physical. Slightly generalizing the discussion of the previous R5 Ψðx; sÞ ¼ Ψðx; N 5 − s þ 1Þ: ð3:19Þ subsection, here we will keep both θW and θm arbitrary. If we now differentiate the fermion partition function with The reason for adding the reflection is that charge con- respect to θW , the result is hj̃5 ðθW Þi, where h·i indicates jugation (in four dimensions) interchanges left- and right- integration over the fermion fields, and we have defined handed fermions. Here the right- and left-handed modes emerge near the boundaries s ¼ 1 and s ¼ N 5 , respectively, j̃5 ðθW Þ ¼ eiθW ψ̄ðKÞPR ψðK þ 1Þ − e−iθW ψ̄ðK þ 1ÞPL ψðKÞ: and they need to be explicitly interchanged to match the ð3:15Þ four-dimensional picture. The domain-wall fermion action (3.9) in terms of two massless Majorana fermions Ψ1;2 We will prove that in the theory with a nonzero θW , defined by the continuum limit of hj̃5 ðθW Þi yields the axial anomaly. By integrating with respect to θW , it then follows that ΨL;1 ðsÞ ¼ ψ L ðsÞ; Zðθ; θW ; θm Þ ¼ Zðθ þ NTθW ; 0; θm Þ; ð3:16Þ ΨR;1 ðsÞ ¼ R5 SCψ̄ TL ðsÞ ¼ SCψ̄ TL ðN 5 − s þ 1Þ; ΨL;2 ðsÞ ¼ R5 SCψ̄ TR ðsÞ ¼ SCψ̄ TR ðN 5 − s þ 1Þ; where now the path integral is defined with the domain- wall fermion action instead of the Wilson fermion action, ΨR;2 ðsÞ ¼ ψ R ðsÞ; ð3:20Þ and we have again allowed for N Dirac fermions in an irrep with index T. Equation (3.16) generalizes Eq. (3.4) of the is then given, for m0 ¼ 0, by preceding subsection. The proof turns out to be quite straightforward. Let 1X N 5 GðθW ; θm Þ be the inverse of the domain-wall Dirac operator SMaj ¼ ΨT ðN 5 − s þ 1ÞCSDK ΨðsÞ 2 s¼1 DðθW ; θm Þ, with angles θW and θm introduced as in Eq. (3.14). Using Eq. (3.15), and writing j̃5 ðθW Þ ¼ 1X N 5 ψ̄J5 ðθW Þψ, we have þ ΨT ðN 5 − s þ 1ÞCSσ 1 ðM − W − 1ÞΨðsÞ 2 s¼1 hj̃5 ðθW Þi ¼ −TrðJ 5 ðθW ÞGðθW ; θm ÞÞ 1X 5N −1 þ ðΨT ðN 5 − s þ 1ÞCSσ 1 PR Ψðs þ 1Þ ¼ −TrðJ 5 ð0ÞGð0; θm − θW ÞÞ; ð3:17Þ 2 s¼1 where in the second step we used the axial transformation þ ΨT ðN 5 − sÞCSσ 1 PL ΨðsÞÞ; ð3:21Þ (3.10) with η ¼ θW =2 to move the angle θW to the mass term. We now take the limit K → ∞, in which the where σ 1 is again the first Pauli matrix acting on the flavor propagator in Eq. (3.17) becomes translationally invariant index i ¼ 1; 2 of Ψi . Using Eq. (3.19) and Eq. (3.20), the in the fifth dimension. In particular, the propagator Majorana form of Eq. (3.14) is becomes independent of the boundaries, and thus of m and θm (or θm − θW after the axial rotation). It follows that 1 hj̃5 ðθW Þi ¼ hj̃5 ð0Þi for any θW and θm , and the anomaly is SW ðθW Þ ¼ ðeiθW ΨTR ðK þ 1ÞSCσ 1 ΨR ðK þ 1Þ 2 recovered as in Ref. [6]. þ e−iθW ΨTL ðKÞSCσ 1 ΨL ðKÞÞ; With the domain-wall equivalent of Eq. (3.4) in hand, we m now return to the equivalence between one Dirac fermion in Sm ðθm Þ ¼ − 0 ðe−iθm ΨTR ð1ÞSCσ 1 ΨR ð1Þ a real irrep of the gauge group and two Majorana fermions, 2 in the domain-wall regularization. As we will see, the þ eiθm ΨTL ð2KÞSCσ 1 ΨL ð2KÞÞ: ð3:22Þ argument follows similar steps as that for the Wilson- fermion case given in Sec. III A. Sm ðθm Þ gets added to the massless Majorana domain-wall We begin by mapping the action (3.9) into an action for action (3.21), while SW ðθW Þ replaces the s ¼ K term in the two Majorana fermions, denoted as Ψi , i ¼ 1; 2. We again third and fourth lines of Eq. (3.21). make use of Eq. (2.8), but now with a Majorana condition As in Sec. III A, the flavor matrix σ 1 in Eq. (3.22) can be adapted for domain-wall fermions. Analogous to Eq. (2.9), rotated into σ 3 . If we then perform a phase transformation8 we will require that [8] 8 Again, this phase transformation is not anomalous on the Ψ̄ ¼ ðR5 ΨÞT CS; ð3:18Þ lattice. 034507-7 MAARTEN GOLTERMAN and YIGAL SHAMIR PHYS. REV. D 100, 034507 (2019) Ψ2 ðx; sÞ → iΨ2 ðx; sÞ; 1 ≤ s ≤ K; The key observation is that, after the transition from a Dirac fermion to two Majorana fermions, the mass term and Ψ2 ðx; sÞ → −iΨ2 ðx; sÞ; K þ 1 ≤ s ≤ 2K; ð3:23Þ the Wilson term (or its domain-wall fermion counterpart) are proportional to the same matrix in flavor space. As we on the Majorana field Ψ2 , while leaving Ψ1 alone, this have shown, by a sequence of nonanomalous lattice rotates σ 3 into the identity matrix in flavor space. The end transformations (meaning that the Jacobian of each lattice result is that σ 1 is removed from Eqs. (3.21) and (3.22) transformation is equal to one), we may bring the two (while leaving the kinetic term invariant), thus proving that Majorana fermions to a diagonal form, with the same the theory has two Majorana fermions with equal positive phases as for the original Dirac fermion [see, e.g., Eq. (3.8) mass m and the same θ angle as the Dirac theory. Again, for the Wilson case]. using that pf 2 ðAÞ ¼ detðAÞ for any antisymmetric A, we Alternatively, we may elect to apply only SU(2) trans- conclude that the Majorana measure is identical to the Dirac formations to the Majorana fermions. These can bring the measure. Wilson and mass terms, that originally point in the σ 1 direction in flavor space, first into the σ 3 direction, and then C. Summary into the iγ 5 direction [see Eq. (1.2)]. In this situation we We summarize the main results of this section. The starting again obtain two identical Majorana fermions, except with point is a lattice-regularized theory with Wilson or domain- new phases that are shifted by the same amount, θ0W ¼ wall fermions, and with chiral angles θm and θW introduced θW þ π=2 and θ0m ¼ θm þ π=2. In the continuum limit the in Eqs. (3.1) or (3.14), respectively.9 We also allow for an explicit topological phase becomes θ0shf ¼ θ þ Tθ0W ¼ θ þ explicit topological term in the gauge action, with angle θ TðθW þ π=2Þ. Because the difference θ0W − θ0m ¼ θW − θm [see Eq. (2.2)]. is unchanged, when we substitute the new phases into Consider first the case of N identical Dirac fermions. Eq. (3.26) we see that the effective vacuum angle θeff is As first observed in Ref. [1], in the continuum limit an unchanged as well. additional vacuum angle IV. VACUUM ANGLE AND THE CHIRAL θind ¼ NTθW ; ð3:24Þ CONDENSATE: COMPLEX IRREP Our nonperturbative study in the previous section has is induced by the fermions. Introducing the “shifted” angle implications for the chiral expansion of fermions in a real irrep, and, in particular, for the interplay between the θshf ¼ θ þ θind ; ð3:25Þ vacuum angle and the U(1) phase of the fermion mass matrix within the chiral expansion. These will be discussed the operational meaning of this statement is that all in Sec. V below. As a preparatory step, in this section we observables will be reproduced in the continuum limit if review the role of the vacuum angle in the more familiar we set θW ¼ 0, and, at the same time, replace θ by θshf as case of fermions in a complex irrep. We first consider the the angle multiplying the explicit topological term in the chiral condensate in the underlying theory in Sec. IVA, (lattice) Lagrangian. As for the phase of the fermion mass paying special attention to its U(1) phase in the light of the matrix, we trivially have α ¼ NTθm [recall Eq. (2.3)]. results of the previous section. In Sec. IV B we then discuss Substituting this into Eq. (2.7) we conclude that, after how the same features are reproduced in the effective integrating out the fermions, the effective vacuum angle in theory, i.e., in chiral perturbation theory. the gauge field measure is θeff ¼ θshf − α ¼ θ þ NTðθW − θm Þ: ð3:26Þ A. Microscopic theory We begin with a continuum derivation. Starting from In the case of N maj identical Majorana fermions, the same Eqs. (2.1), (2.2) and (2.3), the left-handed and right-handed result holds, with N ¼ N maj =2. fermion condensates are defined by The interesting case is an even number 2N of Majorana fermions, which we have shown to be equivalent to N 1 ∂ log Z ΣL;ij ¼ hψ̄ j PL ψ i i ¼ − ; ð4:1aÞ Dirac fermions, as they should be. This has resolved the V ∂Mij apparent paradox described in Sec. I. We conclude this section by summarizing the result in the case of a single 1 ∂ log Z ΣR;ij ¼ hψ̄ j PR ψ i i ¼ − ; ð4:1bÞ Dirac fermion, N ¼ 1. V ∂Mji 9 Or, to be more precise, in Eq. (D2) in the case of Wilson where V is the volume, and i; j ¼ 1; …; N are flavor fermions. indices. Standard steps using the identity 034507-8 PHASE AMBIGUITY OF THE MEASURE FOR CONTINUUM … PHYS. REV. D 100, 034507 (2019) ðD þ mðΩ† PL þ ΩPR ÞÞð−D þ mðΩPL þ Ω† PR ÞÞ Returning to the general case of Eq. (4.10) we see that the orientation of the condensate is determined by the ¼ −D2 þ m2 ; ð4:2Þ “normalized” mass matrix M=m and by θeff. In retrospect, this pattern is a consequence of Eq. (4.1), which defines the give rise to the expressions condensates via derivatives of the partition function with respect to the mass matrix, together with the fact that the ΣL ¼ −ða1 − a5 ÞΩ; ð4:3Þ partition function itself is invariant under non-Abelian chiral rotations of the mass matrix, and depends on θ (or ΣR ¼ −ða1 þ a5 ÞΩ† ; ð4:4Þ θshf ) and α through their difference θeff only, as we proved rigorously in Sec. III [see, in particular, Eq. (3.26)]. These where are the only features of the condensate that we will need in m the following. a1 ¼ hTr½ð−D2 þ m2 Þ−1 i; ð4:5Þ 2V B. Effective low-energy theory m a5 ¼ hTr½γ 5 ð−D2 þ m2 Þ−1 i: ð4:6Þ We now turn to the effective theory for the Nambu- 2V Goldstone pions associated with the spontaneous breaking of chiral symmetry. As noted above, at this stage the discussion The Tr symbol indicates a trace over spacetime, color and is restricted to QCD-like theories in which the fermions Dirac indices.10 By applying a parity transformation we belong to a complex irrep. The dynamical effective field is may express these quantities more explicitly as pffiffiffi Z ΣðxÞ ¼ Σ0 UðxÞ; UðxÞ ¼ expði 2ΠðxÞ=fÞ; m a1 ¼ DAμ̃ðAÞ cosðθeff QÞTr½ð−D2 þ m2 Þ−1 ; ð4:7Þ NY 2 −1 2V Z ΠðxÞ ¼ Πa ðxÞT a ; ð4:13Þ im a¼1 a5 ¼ − DAμ̃ðAÞ sinðθeff QÞTr½γ 5 ð−D2 þ m2 Þ−1 : 2V where UðxÞ is the SUðNÞ-valued pion field, and Σ0 ∈ Uð1Þ ð4:8Þ is a constant phase factor.11 The leading-order potential is It follows that a1 is real, while a5 is imaginary. Both a1 and f2 B a5 are functions of θeff , defined in Eq. (2.7). Introducing V¼− trðM† Σ þ Σ† MÞ; ð4:14Þ 2 z ¼ a1 − a5 ; ð4:9Þ where we recall that M ¼ meiα=ðNTÞ Ω̃, with Ω̃ ∈ SUðNÞ. We remind the reader that the product Bm is renormaliza- we arrive at tion-group invariant, and depends only on the chiral-limit value of the condensate.12 ΣL ¼ Σ†R ¼ −zðθeff Þeiα=ðNTÞ Ω̃ As we have seen in Sec. II A, the partition function of the microscopic theory depends on α and θ only through their ¼ −½zðθeff Þe−iθeff =ðNTÞ eiθ=ðNTÞ Ω̃: ð4:10Þ difference θeff ¼ θ − α, and the same must thus be true in the effective theory: the Lagrangian of the effective theory must In the special case θeff ¼ θ − α ¼ 0, a5 vanishes while be a function of θeff only, order by order in the chiral a1 ¼ r is real and positive. Hence, in that case, z ¼ r > 0, expansion, starting with the tree-level potential V. Evidently, and V will be a function of only θeff if we set ΣL ¼ −rΩ ¼ −reiα=ðNTÞ Ω̃ ¼ −reiθ=ðNTÞ Ω̃: ð4:11Þ Σ0 ¼ eiθ=ðNTÞ : ð4:15Þ Finally, in the limit m → 0 we recover the Banks-Casher In Appendix E we use the power counting and the relation, symmetries of the effective theory to prove that Eq. (4.15) π provides the unique solution to the requirement that the r ¼ ρð0Þ; ð4:12Þ 2 Any constant SUðNÞ-valued part of Σ can be absorbed 11 where ρðλÞ is the spectral density of the massless Dirac into the pion field. Σ0 may be regarded as a remnant of the η0 operator. field (see, for instance, Refs. [9,10]). 12 In particular, the leading-order chiral Lagrangian is insensi- tive to the quadratic divergence proportional to m=a2 that is 10 When the Dirac operator occurs inside the Tr symbol, by present in the bare lattice condensate away from the chiral limit in convention it carries no flavor indices. any fermion formulation. 034507-9 MAARTEN GOLTERMAN and YIGAL SHAMIR PHYS. REV. D 100, 034507 (2019) tree-level potential (4.14) depends on α and θ only through A. The condensate for a real irrep their difference θeff . We also prove that a similar statement We begin with a general theory of N maj Majorana applies to the next-to-leading-order Lagrangian. fermions, where N maj can be either even or odd. In the effective theory, the tree-level condensate now Allowing N ¼ N maj =2 to be a half integer in Eq. (2.12), takes the form the flavor symmetry of the massless theory is SUðN maj Þ,  which we will assume to be spontaneously broken to ∂V  f 2 B iθ=ðNTÞ ΣL ¼ ¼ − e U0 ; ð4:16Þ SOðN maj Þ. We will consider a mass term of the general form ∂M U¼U0 2 1 where U0 ∈ SUðNÞ is the global minimum of the potential. Ψ̄ðM† PL þ MPR ÞΨ; ð5:1Þ 2 For this to be consistent with Eq. (4.10), the global minimum U 0 must be equal to U n ¼ e2πin=N Ω̃, for some where now 0 ≤ n < N, as we will see next. Substituting Un into Eq. (4.14) gives M ¼ MT ¼ mΩ ¼ me2iα=ðN maj TÞ Ω̃; Ω̃ ∈ SUðN maj Þ; ð5:2Þ VðU n Þ ¼ −f 2 BNm cosðθeff =ðNTÞ þ 2πn=NÞ: ð4:17Þ and we assume m > 0. Formally, the fermion path integral is In Appendix F we prove that the global minimum is a Pfaffian. However, as we have seen in Sec. II C, the phase reached when θeff þ 2πnT is closest to zero. Denoting of this Pfaffian is ambiguous in the continuum. The rigorous the corresponding value of n by nðθeff Þ, the tree-level solution to this problem is to define the Pfaffian via a lattice condensate is thus regularization. For the mass matrix in Eq. (5.2), this gives rise to the following relations in the continuum limit: f 2 B iðθ=ðNTÞþ2πnðθeff Þ=NÞ ΣL ¼ − e Ω̃: ð4:18Þ 2 pfðD þ M† PL þ MPR Þ ¼ eiαQ pf N maj ðD þ mÞ This result for ΣL is consistent with Eq. (4.10), and thus ¼ eiαQ detN maj =2 ðD þ mÞ: ð5:3Þ demonstrates the need to introduce the constant Uð1Þ- valued phase Σ0 into the effective theory. Without Σ0 , the The second equality implies that pfðD þ mÞ is strictly effective theory would have led to a value for ΣL in SUðNÞ. positive, as follows from Appendix C. One way to derive This would have been inconsistent, as, for example, can be Eq. (5.3) is to start from a lattice theory of domain-wall seen in the case θ ¼ α ≠ 0, by comparison with Eq. (4.11). Majorana fermions with θW ¼ 0 and θm ¼ 2α=ðN maj TÞ, and We comment that in exceptional cases there is a take the continuum limit. Defining ΣL and ΣR as in competition between the leading- and next-to-leading- Eqs. (4.1a) and (4.1b), but replacing ψ → Ψ and ψ̄ → Ψ̄, order potentials [11,12]. In that case the discussion the rest of the discussion of Sec. IVA carries over.13 leading to Eq. (4.18) does not apply. But the functional We next consider the case where N Dirac fermions are form of Eq. (4.18) remains valid: it must remain true that traded for 2N Majorana fermions. In the initial Dirac- ΣL is oriented in the direction of eiðθ=ðNTÞþ2πn=NÞ Ω̃ for fermion lattice formulation we again set θW ¼ 0. As some n, where again n depends on θeff only, as can again follows from Sec. III, this choice implies that θshf ¼ θ, be seen by comparison with Eq. (4.10). and thus the angle θ that multiplies the lattice-discretized topological term in the gauge action is set to the same value as in the target continuum theory. As usual, the Uð1Þ phase V. VACUUM ANGLE AND THE CHIRAL of the lattice mass matrix is the same as in the con- CONDENSATE: REAL IRREP tinuum, θm ¼ α=ðNTÞ. In this section we turn to real irreps. In Sec. VA we The key point is that the values of θm and θW in any discuss the condensate, and elaborate on the differences equivalent Majorana formulation are constrained by their between the complex case (discussed in Sec. IV) and the values in the initial Dirac formulation, and, in particular, by real case. We deal separately with a standalone theory of the choice θW ¼ 0 we have initially made. The basic Majorana fermions, and with a theory of 2N Majorana transition to Majorana fermions [using Eq. (2.8) in the fermions that was obtained by reformulating a theory of Wilson case, or Eq. (3.20) in the domain-wall case] gives N Dirac fermions, where the apparent paradox described in rise to a mass term and a (generalized) Wilson term that are the Introduction arises. We then discuss the implications for both oriented in the direction of the matrix JS of Eq. (1.1). the chiral effective theory. In Sec. V B we give a diagram- In itself, JS has an axial Uð1Þ phase of π=2. As a result, the matic proof that, when θeff is held fixed, different orienta- tions of the mass matrix give rise to the same physical 13 The definition of parity is somewhat more subtle with observables. Majorana fermions; see Ref. [13]. 034507-10 PHASE AMBIGUITY OF THE MEASURE FOR CONTINUUM … PHYS. REV. D 100, 034507 (2019) phases of the mass term and the (generalized) Wilson term the pion field UðxÞ is a Z2N element which again depends both get shifted by π=2, becoming θ0m ¼ α=ðNTÞ þ π=2, only on θeff . and θ0W ¼ π=2. In the continuum limit, the new phase of the The situation is similar if we apply an SUð2NÞ trans- mass matrix is α0 ¼ NTθ0m ¼ α þ NTπ=2. The phase θ0W formation that rotates the Majorana mass matrix to M ¼ 0 gets traded for an additional vacuum angle, so that the new meiðα=ðNTÞþπ=2Þ 12N ¼ meiα =ðNTÞ 12N [this corresponds to Jrot S vacuum angle is θ0 ¼ θshf ¼ θ þ NTπ=2. As expected, of Eq. (1.2)]. If we choose to apply the same SUð2NÞ rotation 0 both phases were shifted by the same amount, so that to Σ0 , it becomes Σ0 ¼ eiðθ=ðNTÞþπ=2Þ 12N ¼ eiθ =ðNTÞ 12N . the effective vacuum angle, which is their difference, is Finally, if in the lattice-regularized theory we have applied unchanged, θeff ¼ θ − α ¼ θ0 − α0 . a further U(1) axial transformation that simultaneously Alternatively, we may perform an additional (nonanom- brings the mass matrix to M ¼ meiα=ðNTÞ 12N , and the alous) lattice transformation that brings back the phases to (shifted) vacuum angle of the continuum-limit theory back their original values, θm ¼ α=ðNTÞ and θW ¼ 0, so that to θshf ¼ θ, then in the chiral theory we can correspondingly θshf ¼ θ [for the Wilson case, see Eq. (3.8)]. Once again, set Σ0 ¼ eiθ=ðNTÞ 12N . In all of these examples, the constant θeff is unchanged. mode of the pion field UðxÞ will be a Z2N element determined We next turn to the chiral effective theory, focusing on by θeff only. the case N maj ¼ 2N, with the mass matrix M of Eq. (5.2). The nonlinear field Σ is now an element of the coset B. Chiral expansion for a real irrep SUð2NÞ=SOð2NÞ. It is symmetric, ΣT ¼ Σ, and transforms as Σ → hΣhT under the chiral transformation (2.12), just In the case of a complex irrep, studied in Sec. IV, we have demonstrated that the condensate can be expressed as like M (when elevated to a spurion). Instead of Eqs. (4.13) a function of θ and θeff via Eq. (4.10). We then determined and (4.15), which we had in the case of a complex irrep, the the θ dependence of the chiral Lagrangian by requiring coset field for a real irrep is parametrized as that the effective theory reproduce this result. When we ΣðxÞ ¼ UðxÞT Σ0 ¼ Σ0 UðxÞ; ð5:4Þ expand the chiral Lagrangian around the classical solution in terms of the pion field, the expansion is then manifestly a where now function of θeff only, and not of θ and α separately. It follows that physical observables, such as the decay Σ0 ¼ eiθ̃=ðNTÞ J; ð5:5Þ constant and the pion mass, depend only on θeff as well. In the case of a real irrep, we again expect that the chiral and where J is a real symmetric SOð2NÞ matrix. Once expansion for any physical observable will depend on α and again, the phase θ̃ is to be chosen so that the chiral theory is θ only through their difference θeff . However, establishing a function of θeff only. We will discuss examples of this this result is now more subtle. Let us consider two simple shortly. Equations (5.4) and (5.5) provide a generalization examples, both of which can be parametrized as M ¼ mJ, of the results of Ref. [14], where the role of the U(1) phase Σ0 ¼ J, for the same J. The two cases are then defined by was not discussed, and of Ref. [15], where the discussion taking J ¼ JS , for which α=ðNTÞ ¼ θ=ðNTÞ ¼ π=2, or was limited to θ ¼ α ¼ 0, and J ¼ 12N . J ¼ 12N , for which α ¼ θ ¼ 0. Notice that θeff ¼ 0 in both For simplicity, in the rest of this section we again set cases. Now, using Eq. (5.4), and noting that in both cases Ω̃ ¼ 1 in Eq. (5.2).14 Let us consider the construction of J2 ¼ 12N , it is easy to see that J drops out of the product the chiral theory in the case we have just discussed, where Σ† ðxÞM. However, unlike in the case of a complex irrep, N Dirac fermions get traded for 2N Majorana fermions. In this does not immediately imply that the perturbative the initial Dirac formulation we take the mass matrix to be expansion is independent of the choice of J. The reason meiα=ðNTÞ 1N , and we allow for an arbitrary vacuum angle is the constraints imposed on the pion field: this field is θ. After the transition to the Majorana formulation, the Hermitian, traceless, and satisfies mass matrix is M ¼ meiα=ðNTÞ JS , which is equivalent to a π ¼ Jπ T J: ð5:6Þ Uð1Þ phase α0 =ðNTÞ ¼ α=ðNTÞ þ π=2. Correspondingly, the vacuum angle of the continuum-limit theory becomes Thus, even though J drops out of the tree-level Lagrangian, 0 θ0 ¼ θ þ NTπ=2. A possible choice for Σ0 is eiθ =ðNTÞ 12N . the pion field still depends on it, through the above An alternative, equivalent choice, which involves the constraint, and the pion propagator [14,15] same U(1) phase, is Σ0 ¼ eiθ=ðNTÞ JS . For the latter choice, the factors of JS cancel out between the mass matrix and hπ ij ðxÞπ kl ðyÞi Z   the nonlinear field when the latter is expanded in terms of d4 p eipðx−yÞ 1 1 the pion field. Studying the classical solution as we did in ¼ ðδil δjk þ Jik Jjl Þ − δ δ ; ð2πÞ4 p2 þ M2 2 2N ij kl Sec. IV B, we similarly find that the expectation value of ð5:7Þ 14 The generalization to arbitrary Ω̃ is similar to Sec. IV. depends on the choice of J explicitly. 034507-11 MAARTEN GOLTERMAN and YIGAL SHAMIR PHYS. REV. D 100, 034507 (2019) Let us consider the case N ¼ 1. For J ¼ J S, and Here we also used that all terms in the exponent are even choosing a basis where JS ¼ σ 3 , the constraints translate (and, thus, n is even as well). into π 11 ¼ π 11 ¼ −π 22 , and π 12 ¼ −π 12 ¼ −π 21 . For Next, we discuss the general case of N Dirac fermions in J ¼ 12 , the diagonal elements remain the same as before, a real irrep, comparing the cases J ¼ J S , with J S in whereas for the off-diagonal elements we have Eq. (1.1), and J ¼ 12N . The matrix JS can now be brought π 12 ¼ π 12 ¼ π 21 . Stated differently, for J ¼ σ 3 the expan- into a form in which σ 3 appears N times along the diagonal. sion of the pion field is π ¼ π 3 σ 3 þ π 2 σ 2 , whereas for The constraints on the pion field are now, in this basis, J ¼ 12 it is π ¼ π 3 σ 3 þ π 1 σ 1 . The tensor structure of the nonvanishing propagators is X N −1 π NN ¼ − π ii ; 1 1 1 i¼1 hπ 11 ðxÞπ 11 ðyÞi∶ ðδ11 δ11 þ J 11 J11 Þ − δ11 δ11 ¼ ; 2 2 2 π ij ¼ ð−1Þiþj π ji : ð5:12Þ J ¼ σ 3 ; 12 ; In addition, π ii is real for all i, and π ij ¼ π ji for all i ≠ j. 1 1 A minus sign in the pion propagator hπ ij ðxÞπ ij ðyÞi, hπ 12 ðxÞπ 12 ðyÞi∶ ðδ12 δ12 þ J 11 J22 Þ − δ12 δ12 2 2 cf. Eq. (5.8), occurs when i is even and j is odd, or the  1 − 2 ; J ¼ σ3; other way around, because Jii Jjj ¼ −1 only in this ¼ 1 ð5:8Þ 2; J ¼ 12 : situation. Since minus signs in a field redefinition like Eq. (5.9) do not affect our arguments, we can choose Using a hat to distinguish the pion field for the case J ¼ σ 3 , we see that it will transform into the pion field of the J ¼ 12 π̂ ij ¼ iiþj π ij : ð5:13Þ case if we substitute Now let us consider a diagram with pij π̂ ij propagators, nij π̂ 11 ¼ π 11 ; π̂ 12 ¼ iπ 12 ; ð5:9Þ π̂ ij external lines, and vk;ij vertices with kij π̂ ij lines attached to it. Note that because of Eq. (5.12) we can which corresponds to the replacement of σ 1 by σ 2 in the always take i ≤ j (and i ≠ N if i ¼ j, but this is not expansion of the pion field. Of course, nonperturbatively, important). We have that the redefinition (5.9) is not allowed, but in (chiral) X perturbation theory the only question is whether it leads 2pij ¼ nij þ kij vk;ij : ð5:14Þ to the same order-by-order diagrammatic expansion for any kij correlation function with a prescribed set of external pion legs. We will now prove that This relation implies that a correlation function with nij external π̂ ij lines equals i−ðiþjÞnij times the correlation ð1Þ ðmÞ ð1Þ ðnÞ function in terms of the unhatted meson field π ij , using hπ̂ 11 ðx1 Þ…π̂ 11 ðxm Þπ̂ 12 ðy1 Þ…π̂ 12 ðyn Þi ð1Þ ðmÞ ð1Þ ðnÞ that i−2pij ¼ i2pij , and Eq. (5.14). The full correlation ¼ in hπ 11 ðx1 Þ…π 11 ðxm Þπ 12 ðy1 Þ…π 12 ðyn Þi; ð5:10Þ function changes by the product to all orders in chiral perturbation theory, for any m and n. Y P −ðiþjÞnij − ij ðiþjÞnij A vertex with k π 12 lines attached to it also changes by a i ¼i ; ð5:15Þ ij factor ik after the field redefinition (note that k is always even, so that taking i or −i does not matter). Also, for any where the product and sum are over all pairs ij present in diagram, the number of π 12 external lines n, the number of the correlation function. The sum in the exponent on the π 12 propagators p and the number vk of vertices with k π 12 right-hand side of Eq. (5.15) always has to be even, because lines attached to it are related by every index has to appear an even number of times in the X correlation function for it to not vanish. This means we can 2p ¼ n þ kvk : ð5:11Þ drop the minus sign in this exponent, and we thus find the k desired result. Note that, unlike in the N ¼ 1 example, we do not always It follows from this relation that, for all diagrams, the field redefinition (5.9) indeed leads to the factor in in Eq. (5.10), have that nij is even. A simple counter example is the thus proving this result. Each π 12 propagator flips its sign, correlation function hπ 12 π 23 π 34 π 41 i, which does not vanish, and p such propagators thus lead to a factorPð−1Þp ¼ i2p. but has n12 ¼ n23 ¼ n34 ¼ n14 ¼ 1. However, clearly, ð1 þ 2Þn12 þ ð2 þ 3Þn23 þ ð3 þ 4Þn34 þ ð1 þ 4Þn14 ¼ 20 In addition, the diagram changes by a factor i k kvk because is even. of the vk vertices with k π 12Plines, and thus the diagram A similar type of argument was used in Ref. [16] to show changes by a total factor i2pþ k kvk ¼ in, using Eq. (5.11). the equivalence of “standard” quenched chiral perturbation 034507-12 PHASE AMBIGUITY OF THE MEASURE FOR CONTINUUM … PHYS. REV. D 100, 034507 (2019) theory [17] with “nonperturbatively correct” quenched in one case, and to 12N in the other, giving a diagrammatic chiral perturbation theory. proof that all physical observables are equal in the two cases, as required by the common value of θeff . VI. CONCLUSION In QCD-like theories it is well known that physical ACKNOWLEDGMENTS observables depend only on the effective vacuum angle θeff , We like to thank Steve Sharpe for useful discussions. which is the difference between the explicit angle θ We also like to thank Jan Smit for comments and discussion multiplying the topological term in the gauge-field action, on the first version of this paper, which led to the addition and the (properly normalized) Uð1ÞA angle α of the fermion of two new appendices. The work of M. G. is supported by mass matrix. the U.S. Department of Energy, Office of Science, Office When N Dirac fermions belong to a real irrep of the of High Energy Physics, under Award No. DE-FG03- gauge group, the theory can be reformulated in terms of 2N 92ER40711. Y. S. is supported by the Israel Science Majorana fermions. The integration over a Majorana field Foundation under Grant No. 491/17. yields a functional Pfaffian. As we discussed in the Introduction, the phase of this Pfaffian appears to lead to a paradox: in certain cases, θeff changes by an odd multiple APPENDIX A: SPECTRAL REPRESENTATION of π relative to its value in the initial Dirac theory. Tracing OF THE DIRAC OPERATOR the origin of this phenomenon we showed that, in the continuum, the phase of the functional Pfaffian is in fact Consider the one-flavor Dirac operator for a general inherently ambiguous, as it depends on the choice of basis complex mass, for the Majorana field. A partial solution is that, in the case of 2N Majorana fermions, one can fix the ambiguity by D ¼ D þ meiγ5 α̃ ; ðA1Þ appealing to the corresponding theory of N Dirac fermions where in such a way that the apparent paradox is avoided. A nonperturbative lattice definition of Majorana fer-   † 0 σ̄ μ Dμ mions is free of the phase ambiguity: on any finite-volume D ¼ −D ¼ ; ðA2Þ lattice, the (real-irrep) Dirac operator becomes a finite- σ μ Dμ 0 size matrix, and, moreover, the lattice automatically with σ μ ¼ ð12 ; i⃗σ Þ and σ̄ μ ¼ ð12 ; −i⃗σ Þ. Let us derive the selects the coordinate basis to define the Dirac operator, spectral representation of detðDÞ (see, for example, and, hence, its Pfaffian. We reviewed the work of Ref. [1] Ref. [19]). For a zero mode, depending on its chirality, who argued long ago that, if the Wilson term in the Wilson the eigenvalue is simply meiα̃ ≡ m1  im2. Turning to the lattice action for Dirac fermions is rotated by a phase, that nonzero modes we start with the right-handed spectrum of phase induces a topological term in the continuum limit. the second-order operator, We observed that there is a subtlety with this argument associated with renormalization, which leads to a con- −D2 ψ R ¼ −ðσ̄ μ Dμ Þðσ ν Dν Þψ R ¼ λ2 ψ R ; ðA3Þ jecture (first made in Ref. [18]) on how to complete the argument of Ref. [1], described in Appendix D. We where we take λ real and positive. We consider the generalized this result to domain-wall fermions, where following ansatz for an eigenvector of D: this subtlety does not arise, as well as to the case of    Majorana fermions. This allowed us to unambiguously meiα̃ σ̄ μ Dμ ψR determine the effective vacuum angle, finding consistent σ μ Dμ me−iα̃ cσ ν Dν ψ R results between the Dirac and Majorana formulations in   all cases. ð−cλ þ meiα̃ Þψ R 2 ¼ ; ðA4Þ As an application, we discussed how chiral perturbation ð1 þ cme−iα̃ Þσ μ Dμ ψ R theory reproduces the correct dependence on the explicit (θ) and effective (θeff ) vacuum angles. This behavior has where the components of each column vector correspond to been long known (even if maybe not widely known) for the the two chiralities. Requiring that the column vector on the effective theory for a gauge theory with Dirac fermions, left-hand side is an eigenvector gives rise to a quadratic but, to our knowledge, this is the first detailed study of this equation for c, with the two solutions issue for the effective theory for a gauge theory with sffiffiffiffiffiffiffiffiffiffiffiffiffiffi Majorana fermions. As such, our results fill in a lacuna in im i m2 the discussion of Ref. [14], and resolve a question that was c ¼ 22  1 þ 22 : ðA5Þ λ λ λ left open in Ref. [15]. In particular, we considered the chiral expansion for 2N Majorana fermions in two cases that We denote the resulting eigenvectors by χ . The corre- share θeff ¼ 0, while the mass matrix is proportional to JS sponding eigenvalues are 034507-13 MAARTEN GOLTERMAN and YIGAL SHAMIR PHYS. REV. D 100, 034507 (2019) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ ¼ m1 ∓ i λ2 þ m22 : ðA6Þ We first consider the Dirac case. We write the fermion action as S ¼ ψ̄DDW ðm0 Þψ, where the explicit form of DDW ðm0 Þ may be read off from Eq. (3.9). It was proved in The product of the two eigenvalues is λþ λ− ¼ λ2 þ m2 . Ref. [7] that the partition function of a domain-wall Remembering that there are T zero modes per instanton, it fermion, detðDDW Þ, is strictly positive when m0 > 0. follows that the determinant of the one-flavor Dirac Turning to the case with one Majorana fermion, and operator (A1) is introducing the antisymmetric ADW ðm0 Þ¼R5 CSDDW ðUÞ, Y our task is to prove that pfðADW ðm0 ÞÞ is strictly positive detðD þ meiα̃γ5 Þ ¼ ðmeiα̃ ÞTQ ðλ2 þ m2 Þ: ðA7Þ as well. λ>0 The argument uses elementary calculus. We consider The first factor on the right-hand side is the contribution of a finite lattice, so that configuration space is compact. the zero modes, where Q is the topological charge of the Since detðDDW ðUÞÞ > 0 for any gauge-field configuration (multi)instanton background field. The second factor gives U, it follows that there exists μ > 0 such that the contribution of the nonzero modes in terms of the detðDDW ðUÞÞ ≥ μ2 , for all U. If this were not true, we eigenvalues of the second-order operator. For the N-flavor could find an infinite sequence of configurations Ui , such case, substituting α̃ ¼ α=ðNTÞ gives rise to Eq. (2.4). that lim detðDDW ðUi ÞÞ → 0. Because of compactness, that sequence would have a convergent subsequence U0i → U lim , where U lim is a gauge configuration too. It APPENDIX B: MAJORANA BASES would then follow that detðDDW ðUlim ÞÞ ¼ 0, contrary to the result that detðDDW ðUÞÞ > 0 for all configurations. Let us consider for definiteness the case of a single Dirac Next, we have pf 2 ðADW ðUÞÞ¼detðDDW ðUÞÞ. Therefore, fermion. If we follow the basis transformations that lead to either pfðADW ðUÞÞ ≥ μ or pfðADW ðUÞÞ ≤ −μ. Moreover, Eq. (3.7) and then to Eq. (3.8) we arrive at the following pfðADW ð1ÞÞ is positive for the free case U ¼ 1. relations: It follows that pfðADW ðUÞÞ ≥ μ for all configurations U. pffiffiffi If this were not true, there would be a configuration U0 for Ψ1 ¼ ðΨ01 − iγ 5 Ψ02 Þ= 2; which pfðADW ðU0 ÞÞ is negative. Now choose a smooth pffiffiffi Ψ2 ¼ ðΨ01 þ iγ 5 Ψ02 Þ= 2; ðB1Þ path UðtÞ such that Uð0Þ ¼ 1 and Uð1Þ ¼ U0 . Along this path, pfðADW ðUðtÞÞ must change continuously from pos- where the new Majorana fields Ψ01;2 correspond to the itive to negative, and thus go through zero. But, this is diagonal form of the Dirac operator in Eq. (3.8). The impossible, because we have seen that jpfðADW ðUÞÞj ≥ μ. original Dirac field can be expressed as pffiffiffi APPENDIX D: DISCUSSION OF ψ ¼ PL Ψ1 þ PR Ψ2 ¼ ðΨ01 þ iΨ02 Þ= 2; THE RESULT OF REF. [1] pffiffiffi ψ̄ ¼ ðΨ̄01 − iΨ̄02 Þ= 2; ðB2Þ As was shown long ago in Ref. [20] in the theory with θW ¼ θm ¼ 0, the bare mass m0 in Eq. (3.1) renormalizes where we have used Eq. (B1). Equation (B2) is reminiscent additively. This is related to the fact that there is no of the original notion of Majorana fields in Minkowski symmetry distinguishing between the Wilson term W space as the real and imaginary parts of a Dirac field, for and the single-site term proportional to m0 . This fact a real irrep. As we explained in Sec. II B, in this paper was not considered in Ref. [1]. The proof of the observation we prefer the basis (2.8), because it respects the natural quoted in Eq. (3.4) was given for a theory with Wilson correspondence between Weyl and Majorana fields. fermions in the semiclassical limit, i.e., in the presence of a We stress that the existence of the above basis, in which smooth background gauge field. The additive renormaliza- the Majorana mass matrix is diagonal [see Eq. (3.8)], does tion thus does not arise, as it is caused by quantum not resolve the puzzle we discussed in the Introduction; that fluctuations of the gauge field. In this appendix we describe puzzle must find its resolution when the basis (2.8) is used, a conjecture on the interplay of the observation of Ref. [1] and indeed it does, as we showed in Sec. III. and renormalization. Our discussion here largely overlaps with Ref. [18]. With quantum effects thus “mixing” the Wilson and APPENDIX C: POSITIVITY OF THE single-site mass terms in Eq. (3.1), the question arises DOMAIN-WALL PFAFFIAN FOR whether two angles θW and θm can be unambiguously POSITIVE MASS introduced, and, if so, how this should be done. Here, we In this Appendix we prove that the Pfaffian of a single will discuss the issue, and formulate a natural conjecture domain-wall Majorana fermion is positive for a positive answering this question. A rigorous proof of our conjecture bare mass, i.e., in the absence of chiral phases. is outside the scope of this paper. 034507-14 PHASE AMBIGUITY OF THE MEASURE FOR CONTINUUM … PHYS. REV. D 100, 034507 (2019) First, consider a lattice gauge theory with Wilson condition for DW . It follows that, in Eq. (D2), and for fermions without any θ angles which leads to massless general values of θW and θm , Z is a function of the fermions in the continuum limit. It follows from Ref. [20] difference θm − θW only. We comment that the universal, that in order to construct such a theory, the bare mass m0 logarithmic part of the Z factor is actually independent of needs to be tuned to a critical value mc that depends on the θW and θm . However, this Z factor also has a finite part, bare coupling, i.e., the massless theory is obtained from and that part will in general depend on θW and θm , but, as a lattice theory with fermion operator DK þ W þ mc.15 we have just argued, only through their difference. Using an axial rotation of the form (3.3), we can introduce Our conjecture is that in the fully dynamical theory an angle θW in this theory, turning the fermion operator into Eq. (3.4) holds, with the fermion operator as defined in DK þ eiγ5 θW ðW þ mc Þ. Of course, in the massless theory, Eq. (D2). We observe that this conjecture is natural, in the this angle has no physical consequence, consistent with sense that, in the continuum limit, the mass mAWI is the what one expects in a massless continuum theory. This fermion mass m to be used in Secs. IV and V. construction does imply that if one starts with a theory APPENDIX E: θ DEPENDENCE OF with Wilson term eiγ5 θW W, the critical mass mc introduced THE CHIRAL THEORY to obtain a massless continuum limit also needs to be multiplied by eiγ5 θW . As in Sec. IV B we consider here a gauge theory with N Next, we may introduce a physical mass, which here we Dirac fermions in some complex irrep. For θ ¼ 0, the chiral will take to be the axial-Ward-identity (AWI) mass mAWI , Lagrangian is constructed using the nonlinear field by choosing UðxÞ ∈ SUðNÞ; see Eq. (4.13). We will prove that, at both leading order (LO) and next-to-leading order (NLO), the m0 ¼ mc þ ZmAWI ; ðD1Þ chiral Lagrangian for θ ≠ 0 is obtained via the replacement UðxÞ → ΣðxÞ, where ΣðxÞ ¼ UðxÞΣ0 [see Eq. (4.13)], where we also introduced the multiplicative renormaliza- and where Σ0 is given by Eq. (4.15).17 As before, M is tion constant Z relating the bare subtracted lattice mass given in Eq. (2.3). m0 − mc and the renormalized mass mAWI [20,22]. We may We start at tree level. The requirement that the Lagrangian now introduce another angle θm by considering the operator of the chiral theory depend on θ and α only through their difference θeff is satisfied if the potential admits the form DW ðθW ; θm Þ ¼ DK þ eiγ5 θW ðW þ mc Þ þ eiγ5 θm ZmAWI : f2 B ðD2Þ V¼− trðeifðθeff Þ M† Σ þ H:c:Þ: ðE1Þ 2 We recall that mc has already been determined for This amounts to multiplying M† Σ in Eq. (4.14) by the phase mAWI ¼ 0, and that it is independent of both θW and θm . factor eifðθeff Þ, where fðθeff Þ is a priori an arbitrary (real) The question arises how the Z factor depends on these function of its argument. angles. In order to address this question, we first apply an We first invoke the chiral power counting, which implies axial rotation (3.3) with η ¼ −θW =2 to remove the phase of that the tree-level Lagrangian should be linear in M or M†. the Wilson term, arriving at This dependence is already explicit in Eq. (E1), and so D0W ðθW ; θm Þ ¼ DK þ ðW þ mc Þ þ eiγ5 ðθm −θW Þ ZmAWI : fðθeff Þ ¼ fðθ − αÞ ¼ fðθ − Im log det MÞ; ðE2Þ ðD3Þ must in fact be independent of M. This allows us to set At tree level, the Wilson-Dirac operator D0W is now a fðθeff Þ ¼ c in Eq. (E1), where c is some constant. Next we consider the special case where M ¼ m1N , function of the difference θm − θW , and, by necessity, the with m > 0, and θ ¼ 0. Now Σ ¼ U and the tree-level same applies to the Z factor, order by order in perturbation Lagrangian must be invariant under the (internal) parity theory. The last step is to undo the axial rotation, going from D0W back to DW . Assuming that our renormalization transformation UðxÞ → U † ðxÞ. This invariance is respected condition transforms covariantly under axial rotations,16 only for eic ¼ 1, which completes the argument.18 the same Z factor that we have determined for D0W The reasoning at NLO is similar. A priori, M† Σ can will continue to satisfy the corresponding renormalization again be multiplied by a phase factor eifðθeff Þ, where fðθeff Þ is a new arbitrary real function for each occurrence of 15 M† Σ. But, as before, the power counting restricts every The precise definition of mc is subject to ambiguities of order 2 a [21]. However, we will assume that we are in the scaling 17 region, where these ambiguities can be ignored. We conjecture that a similar statement applies to all orders in 16 This includes as a special case any renormalization condition the chiral expansion. which is invariant under axial rotations of the fields. 18 The choice eic ¼ þ1 is conventional. 034507-15 MAARTEN GOLTERMAN and YIGAL SHAMIR PHYS. REV. D 100, 034507 (2019) such fðθeff Þ to a constant. In the last step we consider the It remains to prove that this solution is in fact the global most general constant phase factors consistent with parity maximum of V over the entire set of saddle points. invariance of the M ¼ m1N , θ ¼ 0 theory, finding that this What complicates matters is that Eq. (F5) can be satisfied does not give rise to any new operators not already present by ϕk ¼ ϕN, or by ϕk ¼ π − 2θ̃ − ϕN. In the former case in the standard NLO chiral Lagrangian. we have cosðθ̃ þ ϕk Þ ¼ cosðθ̃ þ ϕN Þ, whereas in the latter case we have cosðθ̃ þ ϕk Þ ¼ − cosðθ̃ þ ϕN Þ, so that APPENDIX F: PROOF OF EQ. (4.18) cosðθ̃ þ ϕk Þ þ cosðθ̃ þ ϕN Þ ¼ 0. ð1Þ Let us prove, algebraically, that the global minimum of Let us denote by V max the maximal value of V when the tree-level potential (4.14) is given by ΣL of Eq. (4.18), ϕ1 ¼ π − 2θ̃ − ϕN , while the remaining N − 2 independent with n ¼ nðθeff Þ as described in Sec. IV B. phases are equal to ϕN . It follows immediately from the P We begin by writing the SUðNÞ matrix U of Eq. (4.13) as above discussion that in this case V ¼ Nk¼3 cosðθ̃þϕk Þ¼ U ¼ Ω̃ Ũ, so that the potential becomes ðN −2Þcosðθ̃þϕN Þ, leading to the upper bound ð1Þ f 2 Bm V max ≤ N − 2: ðF7Þ V¼− trðeiθeff =ðNTÞ Ũ þ H:c:Þ: ðF1Þ 2 Similarly, if exactly two independent phases are equal to We may assume without loss of generality that Ũ is π − 2θ̃ − ϕN , then the corresponding maximal value is diagonal, ð2Þ bounded by V max ≤ N − 4, and so on. We also need a lower bound on the maximum in Eq. (F6). Ũ ¼ diagðeiϕ1 ; eiϕ2 ; …; eiϕN−1 ; eiϕN Þ; ðF2Þ The maximum value the angle θ̃ þ 2πnðθeff Þ=N can take is equal to π=N. Since sinðxÞ ≤ x, this implies where ϕ1 ; …; ϕN−1 are the independent real variables, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕN ¼ 2πn − ðϕ1 þ    þ ϕN−1 Þ; ðF3Þ cosðθeff =ðNTÞ þ 2πnðθeff Þ=NÞ ≥ 1 − ðπ=NÞ2 : ðF8Þ ð1Þ where n is an arbitrary integer. Introducing the shorthand It follows that V max of Eq. (F6) is larger than V max if θ̃ ¼ θeff =ðNTÞ we need to find the global maximum of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 1 − ðπ=NÞ2 ≥ N − 2; ðF9Þ 1 XN V ¼ trðeiθ̃ Ũ þ H:c:Þ ¼ cosðθ̃ þ ϕk Þ: ðF4Þ which is true for N ≥ 4. 2 k¼1 It remains to check explicitly the cases N ¼ 2; 3. For The saddle-point conditions are N ¼ 2, choosing ϕ1 ¼ π − 2θ̃ − ϕ2 gives V ¼ 0, which is smaller than the maximum in Eq. (F6).19 sinðθ̃ þ ϕk Þ ¼ sinðθ̃ þ ϕN Þ; k ¼ 1; 2; …; N − 1: ðF5Þ For N ¼ 3, if we choose ϕ1 ¼ π − 2θ̃ − ϕ3 and ϕ2 ¼ ϕ3 ð1Þ then V max ≤ 1 according to the upper bound (F7). By First consider a solution with all phases equal. Equation (F3) contrast, for the solution with ϕ1 ¼ ϕ2 ¼ ϕ3 , the maximum then implies that ϕk ¼ 2πn=N, k ∈ f1; …; Ng, for some n, value of the angle θ̃ þ 2πnðθeff Þ=N is now π=3; hence, and V ¼ N cosðθ̃ þ 2πn=NÞ. The global maximum over this N cosðθeff =ðNTÞ þ 2πnðθeff Þ=NÞ is bounded from below set of solutions is obtained for n ¼ nðθeff Þ, defined as before by 3=2, making V max again the true global maximum. as the value of n for which θ̃ þ 2πn=N is closest to zero. The value of this maximum is An exception is the case θ̃ ¼ π=2, for which Eq. (F6) 19 V max ¼ N cosðθeff =ðNTÞ þ 2πnðθeff Þ=NÞ; ðF6Þ vanishes, too. In fact, the tree-level potential is identically zero in this case [11]. For θ̃ close to π=2 there is competition between which reproduces Eq. (4.17). 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