Impact of electromagnetism on phase structure for Wilson and twisted-mass fermions including isospin breaking
IImpact of electromagnetism on phase structure for Wilson and twisted-mass fermionsincluding isospin breaking
Derek P. Horkel ∗ and Stephen R. Sharpe † Physics Department, University of Washington, Seattle, WA 98195-1560, USA (Dated: November 5, 2018)In a recent paper we used chiral perturbation theory to determine the phase diagram and pionspectrum for Wilson and twisted-mass fermions at nonzero lattice spacing with nondegenerate upand down quarks. Here we extend this work to include the effects of electromagnetism, so that it isapplicable to recent simulations incorporating all sources of isospin breaking. For Wilson fermions,we find that the phase diagram is unaffected by the inclusion of electromagnetism—the only effect isto raise the charged pion masses. For maximally twisted fermions, we previously took the twist andisospin-breaking directions to be different, in order that the fermion determinant is real and positive.However, this is incompatible with electromagnetic gauge invariance, and so here we take the twistto be in the isospin-breaking direction, following the RM123 collaboration. We map out the phasediagram in this case, which has not previously been studied. The results differ from those obtainedwith different twist and isospin directions. One practical issue when including electromagnetism isthat the critical masses for up and down quarks differ. We show that one of the criteria suggestedto determine these critical masses does not work, and propose an alternative.
I. INTRODUCTION
The phase diagram of lattice QCD (LQCD) can con-tain unphysical transitions and unwanted phases due todiscretization effects. A well known example is the Aokiphase that can be present with Wilson-like fermions [1]. Unphysical phases occur when the effects of physical lightquark masses are comparable to those induced by dis-cretization, specifically m ∼ a Λ , with a the latticespacing. This can be shown by extending chiral pertur-bation theory ( χ PT) to include the effects of discretiza-tion [2]. Understanding the phase structure is necessaryso that LQCD simulations can avoid working close to un-physical phases, so as to avoid distortion of results andcritical slowing down.Recently, we extended the analysis of the phase dia-gram to the case of nondegenerate up and down quarksfor Wilson-like and twisted-mass fermions [3]. This wasprompted by the recent incorporation of mass splittingsinto simulations of LQCD. We found a fairly com-plicated phase structure, in which, for example, theAoki phase was continuously connected to Dashen’s CP-violating phase [6, 7].A drawback of our analysis was that it did not in-clude the other major source of isospin breaking in QCD,namely electromagnetism. For most hadron properties,electromagnetic effects are comparable to those of themass nondegeneracy (cid:15) q = ( m u − m d ) /
2. For example, inthe neutron-proton mass difference these two effects leadto contributions of approximately − . ∗ e-mail: [email protected] † e-mail: [email protected] “Wilson-like” refers to both unimproved and improved versionsof Wilson fermions. The choice will not matter in this work. For recent reviews of such simulations see Refs. [4, 5]. respectively. Furthermore, the recent LQCD simula-tions alluded to above have included both mass nonde-generacy and electromagnetism. Thus, to be directly ap-plicable to such simulations, we must extend our analysisto include electromagnetism. This is the purpose of thepresent note.We work in Wilson or twisted-mass χ PT (both ofwhich we refer to as W χ PT for the sake of brevity) us-ing a power-counting to be explained in Sec. II. At theorder we work, it turns out that the inclusion of elec-tromagnetism can be accomplished in most cases simplyby shifting low-energy coefficients (LECs) in the resultswithout electromagnetism. Thus we can take over manyresults from Ref. [3] without further work.One new issue concerns the simultaneous inclusionof electromagnetism and quark nondegeneracy withtwisted-mass fermions. The approach we used in the ab-sence of electromagnetism in Ref. [3] (following Ref. [9])was to apply the twist in a different direction in isospinspace ( τ ) from that in which the masses are split ( τ ).This leads to a real quark determinant, and is the methodused to simulate the s and c quarks using twisted-massfermions (see, e.g., Ref. [10]). This does not, how-ever, generalize to include electromagnetism in a gauge-invariant way. Here, instead, we follow Ref. [11], andtwist in the τ direction. When doing simulations, thishas the disadvantage of leading to a complex quark deter-minant, but there are no barriers to studying the theorywith χ PT.The remainder of this paper is organized as follows.We begin in Sec. II with a brief discussion of our power- These results are from the recent LQCD calculation of Ref. [8],and use the convention of that work for the separation of elec-tromagnetic and (cid:15) q effects. This is avoided in Refs. [11, 12] by expanding about the theorywith degenerate quarks and no electromagnetism. a r X i v : . [ h e p - l a t ] S e p counting scheme and a summary of relevant results fromRef. [3]. We then explain, in Sec. III, how electromag-netism changes the results of Ref. [3] for the case ofWilson-like fermions. Section IV describes how to si-multaneously include isospin breaking, electromagnetismand twist, while Sec. V gives our corresponding resultsfor the phase diagram, focusing mainly on the case ofmaximal twist. We conclude in Sec. VI.Two technical issues are discussed in appendices. Thefirst concerns the renormalization factors needed to re-late lattice masses to the continuum masses that appearin χ PT. This issue is subtle because singlet and nonsin-glet masses renormalize differently. This point was notdiscussed in Ref. [3], and we address it in Appendix A,except that we do not include all the effects introducedby electromagnetism.The second appendix concerns the need for charge-dependent critical masses in the presence of electromag-netism. These must be determined nonperturbatively,and various methods for doing so have been used in theliterature. One of these methods, proposed in Ref. [11],can be implemented using partially quenched (PQ) χ PT,and thus checked. This is done in App. B. We find thatthe method only provides one constraint on the up anddown critical masses and must be supplemented by anadditional condition in order to determine both.Appendix B requires results from a χ PT analysis ofa theory with twisted nondegenerate charged quarks atnonzero lattice spacing and at nonvanishing θ QCD . Weprovide such an analysis in a companion paper [13].
II. POWER-COUNTING AND SUMMARY OFPREVIOUS WORK
In order to study the low-energy properties of LQCD,we must decide on the relative importance of the com-peting effects. The power counting that we adopt is m ∼ p ∼ a ∼ α EM > (cid:15) q > ma ∼ a ∼ aα EM ... , (1)where m represents either m u or m d . This is the powercounting adopted in Ref. [3], except that electromagneticeffects are now included. This scheme only makes sense ifdiscretization errors linear in a are absent, either becausethe action is improved or because the O ( a ) terms can beabsorbed into a shift in the quark masses (as is the casein W χ PT [2]).The explanation for the choice of leading order (LO)terms in this power-counting is as follows. Present sim-ulations have 1 /a ≈ QCD ≈
300 MeV we find a Λ QCD ≈ .
1. Thussecond order discretization effects are of relative size( a Λ QCD ) ≈ .
01. This is comparable to α EM , m u / Λ QCD and m d / Λ QCD (given that m u ≈ . m d ≈ m u − m d ∼ m u ∼ m d ). The choice of (cid:15) q as the dominant subleading contri-bution is less obvious, and is discussed in some detail inRef. [3]. The essence of the argument is that, while the (cid:15) q terms are not necessarily numerically larger than generic m terms, they give the leading contribution from quarkmass differences to isospin breaking in the low-energy ef-fective theory. For example, these contributions give riseto the CP-violating phase in the continuum analysis. In this note we keep only terms up to and includingthose proportional to (cid:15) q , so that we have the leading orderterm of each type. We refer to this as working at LO + indicating that it goes slightly beyond keeping only LOterms.We now collect the relevant results from Ref. [3] con-cerning the phase diagram of Wilson-like fermions in thepresence of nondegeneracy. We work entirely in SU(2)W χ PT, in which the chiral field is Σ ∈ SU(2). The LO + chiral Lagrangian for Wilson-like fermions (whether im-proved or not) is L χ = f (cid:2) ∂ µ Σ ∂ µ Σ † (cid:3) + V χ (2) V χ = − f χ † Σ + Σ † χ ) − W (cid:48) [tr( ˆ A † Σ + Σ † ˆ A )] + (cid:96)
16 [tr( χ † Σ − Σ † χ )] , (3)where ˆ A = 2 W a is the spurion field used to introducelattice artifacts. This Lagrangian contains several LECs: f ≈
92 MeV and B from LO continuum χ PT, W and W (cid:48) introduced by disretization errors, and (cid:96) . The latter,though of next-to-leading order (NLO) in standard con-tinuum power-counting, leads to contributions propor-tional to (cid:15) q and thus we keep it in our LO+ calculation. (cid:96) is not renormalized at one-loop order, and matchingwith SU(3) χ PT leads to the estimate [16] (cid:96) = f B m s , (4)indicating that (cid:96) is positive.The final ingredient in Eq. (3) is χ = 2 B M , whichcontains the mass matrix M = diag( m u , m d ), with m u,d renormalized masses in a mass-independent scheme.Since L χ is supposed to represent the long-distancephysics of a lattice simulation close to the chiral and con-tinuum limits, to use it we need to know the relationshipbetween bare lattice masses and the renormalized masses.This relationship is nontrivial when using nondegeneratequarks, and is discussed in Appendix A. This point wasoverlooked in Ref. [3]. A further justification for this choice, also discussed in Ref. [3],is that in SU(3) χ PT such terms are of LO, since they are pro-portional to ( m u − m d ) /m s . To determine the vacuum of the theory, we must mini-mize the potential V χ . Writing (cid:104) Σ (cid:105) = e iθ ˆ n · (cid:126)τ , the potentialbecomes V χ = − f (cid:0) (cid:98) m q cos θ + c (cid:96) (cid:98) (cid:15) q n sin θ + w (cid:48) cos θ (cid:1) , (5)where (cid:98) m q = B ( m u + m d ) , (cid:98) (cid:15) q = 2 B (cid:15) q ,c (cid:96) = (cid:96) f , w (cid:48) = 64 W (cid:48) W a f . (6)Assuming c (cid:96) > | cos θ | = 1. The shaded(pink) phases violate CP with | n | = 1 , cos θ = (cid:98) m q c (cid:96) (cid:98) (cid:15) q − w (cid:48) ) . (7)The boundaries between continuum-like and CP-violating phases lie along the lines | (cid:98) m q | = 2( c (cid:96) (cid:98) (cid:15) q − w (cid:48) ),and are second order transitions. The boundary betweenthe two continuum-like phases with opposite cos θ is afirst order transition. Within the continuum-like phasesthe pion masses are m π = | (cid:98) m q | − c (cid:96) (cid:98) (cid:15) q − w (cid:48) ) , m π ± = | (cid:98) m q | + 2 w (cid:48) , (8)while within the CP-violating phase m π = 2( c (cid:96) (cid:98) (cid:15) q − w (cid:48) ) sin θ , m π ± = 2 c (cid:96) (cid:98) (cid:15) q . (9)The neutral pion mass vanishes along the second or-der transition lines. Plots of these masses are given inRef. [3]. III. CHARGED, NONDEGENERATE WILSONQUARKS
We now add electromagnetism, so that we are consider-ing Wilson fermions with charged, nondegenerate quarks.Precisely how electromagnetism is added at the latticelevel is not relevant; all we need to know is that electro-magnetic gauge invariance is maintained by coupling toexact vector currents of the lattice theory. We work hereonly at LO in α EM , which in terms of Feynman diagramsmeans keeping only those with a single photon propa-gator. We also work at infinite volume, thus avoidingthe complications of power-law volume dependence thatoccur in simulations [8, 17, 18]. A. Induced shifts in quark masses
The dominant effect of electromagnetism is a chargedependent shift in the critical mass, as noted in Refs. [8,11, 19]. Here we discuss this shift from the viewpoint (a) Aoki scenario ( w (cid:48) < w (cid:48) > FIG. 1: Phase diagrams from Ref. [3] including effectsof both discretization and nondegenerate quarks. CP isviolated in the (pink) shaded regions. The (blue) linesat the boundaries of the shade regions are second-ordertransitions (where the neutral pion mass vanishes),while the (yellow) line along the (cid:15) q axis joining the twoshaded regions in (1b) is a line of first order transitions.The analytic expression given for the shaded region in(1a) holds also for that in (1b). As discussed below inSec. III B, these phase diagrams apply also in thepresence of electromagnetism.of the Symanzik low-energy effective Lagrangian [20, 21].It arises from QCD self-energy diagrams in which oneof the gluons is replaced by a photon, and leads to the (a)(b)(c) FIG. 2: Examples of LO contributions fromelectromagnetism to quark self-energies. Diagrams withadditional gluons and quark loops are not shown. Thesethree types of diagram lead, respectively, to the threeoperators listed in Eq. 10. Only the first operator ispresent in the “electroquenched” approximation.appearance of the operators( a ) α EM a ( (cid:88) f e f f f ) , ( b ) α EM a ( (cid:88) f (cid:48) e f (cid:48) ) (cid:88) f e f f f , ( c ) α EM a (cid:88) f (cid:48) ( e f (cid:48) ) (cid:88) f f f , (10)where f = u, d , e u = 2 / e d = − /
3. Examples ofthe corresponding Feynman diagrams are shown in Fig. 2These operators are allowed because electromagnetismbreaks isospin, while Wilson fermions violate chiral sym-metries. Their contributions are smaller than those ofthe (cid:80) f ¯ f f /a operator that leads to the dominant shift in the critical mass. However, because α EM ∼ a ∼ m in our power-counting, α EM /a effects are proportional to a ∼ m / , and thus dominate over physical quark masses.They must therefore be removed by appropriate tuningof the bare masses. Since the combined effect of the threeoperators is independent O ( α EM /a ) shifts in m u and m d ,removing these shifts requires independent tuning of the u and d critical masses.Different methods for doing this tuning have been usedin the literature. The most straightforward, used inRef. [8], is to determine the bare quark masses directlyby enforcing that an appropriate subset of hadron massesagree with their experimental values (keeping all isospinbreaking effects). This avoids the need to directly deter-mine the critical masses, but is the most challenging nu-merically. An alternative approach, proposed in Ref. [11],makes use of a partially-quenched extension of the theory.In Appendix B we check this method by showing how itcan be implemented in χ PT. We find that it cannot de-termine both critical masses, but instead only providesa single constraint between them. We then introduce anadditional tuning criterion which, together with that ofRef. [11], does allow both critical masses to be deter-mined.For the rest of the main text, we assume that thecharge-dependent critical masses have been determinedin some manner, such that O ( α EM /a ) self-energy effectscan be ignored. This leaves electromagnetic correctionsproportional to α EM , which we must keep in our powercounting, as well as higher-order effects proportional to α EM × m etc., which we can ignore.Examples of the latter effects are the bilinears α EM (cid:88) f e f m f ¯ f f and α EM (cid:88) f e f ¯ f /Df . (11)These arise as O ( am ) corrections to the operators ofEq. (10), and are also present directly in the continuumtheory. We stress that, in the Symanzik Lagrangian, onehas no dimensionful parameters other than m and 1 /a ,so bilinears proportional to α EM Λ QCD are not allowed.Factors of Λ
QCD arise when we move from the SymanzikLagrangian to χ PT.The only effect of electromagnetism that is simplyproportional to α EM —and thus of LO in our powercounting—is that arising from one photon exchange be-tween electromagnetic currents. This is a continuum ef-fect, long studied in χ PT. It leads to he following addi-tional term in the chiral potential [22, 23]: V EM = − f c EM tr(Σ τ Σ † τ ) . (12)Here c EM is an unknown coefficient proportional to α EM .All that is known about c EM is that it is positive [24]. Contributions from the isoscalar part of the photon coupling leadto the same form but with one or both τ ’s replaced by identitymatrices. In either case the contribution reduces to an uninter-esting constant, and is thus not included in V EM . B. Phase diagram and pion masses
The competition between electromagnetic effects anddiscretization errors for two degenerate
Wilson fermionshas been previously analyzed in Ref. [25]. Here we add inthe effects of nondegeneracy. This turns out to be verysimple. Using the SU(2) identity4 tr(Σ τ Σ † τ ) = (cid:2) tr(Σ+Σ † ) (cid:3) − (cid:2) tr([Σ − Σ † ] τ ) (cid:3) − , (13)together with χ = (cid:98) m q + (cid:98) (cid:15) q τ , (14)we find that V EM can be absorbed into V χ [given inEqs. (3) and (5)] by changing the existing constants as w (cid:48) −→ w (cid:48) + c EM , and c (cid:96) (cid:98) (cid:15) q −→ c (cid:96) (cid:98) (cid:15) q + c EM . (15)This allows us to determine the phase diagram and pionmasses directly from the results presented in the previoussection. We first observe that, at the order we work, the phasediagram is unchanged by the inclusion of EM —the resultsin Fig. 1 still hold. This can be seen from the form of thepotential in Eq. (5), which, since | n | = 1, depends onlyon c (cid:96) (cid:98) (cid:15) q − w (cid:48) . This combination is, however, unaffectedby the shifts of Eq. (15) and so the phase boundariesand values of θ throughout the phase plane are also un-changed.Similarly, from Eqs. (8) and (9) we see that the neutralpion masses are unchanged throughout the phase plane.In particular, the second-order phase boundaries are (asexpected) lines along which the neutral pion is massless.The only change caused by electromagnetism is to thecharged pion masses, which are increased by the sameamount throughout the phase plane: m π ± −→ m π ± + 2 c EM . (16)One implication is that, for (cid:98) (cid:15) q = 0, the charged pions areno longer massless within the Aoki phase (if present).This is because they are no longer Goldstone bosons,as the flavor symmetry is explicitly broken by electro-magnetism. Also, as noted in Ref. [25], the chargedpion can be lighter than the neutral one inside the CP-violating phases. This is not inconsistent with Witten’sidentity [24] because the latter did not account for dis-cretization effects. Plots of the pion masses are shown inFig. 3.It is perhaps surprising that electromagnetism, whichcontributes at LO in our power-counting, has no effect onthe phase diagram, whereas the subleading contributionsproportional to (cid:98) (cid:15) q have a significant impact. We can For (cid:98) (cid:15) q = 0 our results are in complete agreement with those ofRef. [25]. (a) Aoki scenario with w (cid:48) < − c EM (b) First-order scenario with w (cid:48) > c (cid:96) (cid:98) (cid:15) q (c) First-order or Aoki scenario with − c EM < w (cid:48) < c (cid:96) (cid:98) (cid:15) q FIG. 3: Pion masses for nondegenerate untwistedWilson fermions including electromagnetism. The threepossible behaviors along vertical slices through phasediagrams of Fig. 1 are shown. Solid (blue) lines show m π , while dashed (red) lines show m π ± . Expressionsfor masses are given in the text.understand this by noting that the CP-violating phaseis characterized by a neutral pion condensate, which re-mains uncoupled to the photon until higher order in χ PT(where form factors enter).The implications of these results for practical simula-tions (such as those of Ref. [8]) are unchanged from thediscussion in Ref. [3]. In particular, for the Aoki scenario( w (cid:48) <
0) discretization effects move the CP-violatingphase closer to the physical point than for degeneratequarks, so one must beware of simulating too close tothis transition.
IV. NONDEGENERACY,ELECTROMAGNETISM AND TWIST
When using twisted-mass fermions one must decide onthe relative orientation in isospin space both of the twistand the isospin-breaking induced by quark mass differ-ences and electromagnetism. In the absence of electro-magnetism, the standard choice is to align these two ef-fects in orthogonal directions. For example, one usuallytakes τ for isospin-breaking, as in the continuum, whiletwisting in the τ direction. This is the choice usedin simulations of the strange-charm sector using twisted-mass fermions [26]. It ensures that the fermion determi-nant is real, and (subject to some conditions) positive [9].This was the choice whose phase structure we determinedusing W χ PT in Ref. [3].This approach does not, however, allow for the inclu-sion of electromagnetism. One problem is apparent al-ready in the continuum limit, where the twisted-massquark action is (in the “twisted” basis) [27] ψ ( /D + m q c ω + iγ τ m q s ω + (cid:15) q τ ) ψ . (17)Here /D is the gluonic covariant derivative, m q is the av-erage quark mass, and ω the twist angle with c ω = cos ω and s ω = sin ω . This action is not invariant under fla-vor rotations in the τ direction, so there is no conservedvector current to which the photon can couple. In otherwords, there is no global flavor transformation availableto gauge.To avoid this problem, we recall that twisting is, inthe continuum, simply a nonanomalous change of vari-ables that does not effect physical quantities. Thus weshould start with the standard action including electro-magnetism ψ ( /D − ie /AQ + m q + (cid:15) q τ ) ψ , (18)with A µ the photon field coupling via the charge matrix Q = 16 + 12 τ , (19) Any linear combination of τ and τ is equivalent; τ is the stan-dard choice. and then perform a chiral twist ψ −→ e iωγ τ / ψ, ψ −→ ψe iωγ τ / . (20)This leads to the quark action of Eq. (17) with the addi-tion of the photon coupling ψ /A (cid:20) + 12 ( c ω τ − s ω τ γ ) (cid:21) ψ . (21)Thus the photon couples to a linear combination of vectorcurrents and to an axial current in the τ direction. Inthe continuum, this combination is conserved [given thetwisted mass matrix of Eq. (17)] and the action remainsgauge invariant.We conclude that the correct fermion action to dis-cretize is the sum of Eqs. (17) and (21). This, however,is not possible in a gauge invariant way using Wilson’slattice derivative (except for s ω = 0). The Wilson termbreaks all axial symmetries, so the τ γ part of the pho-ton coupling is to a lattice current that is not conserved.To avoid this problem, and obtain a discretized twistedtheory that maintains gauge invariance, one needs totwist in a direction that leaves the photon coupling toa conserved current. The only choice is to twist in the τ direction. Then the twisted form of the continuumLagrangian is ψ ( /D − ie /AQ + m q c ω + τ (cid:15) q c ω + iγ τ m q s ω + iγ (cid:15) q s ω ) ψ . (22)This is discretized by adding the standard Wilson term.Since the photon is coupled to vector currents that areexact symmetries of both the Wilson term and the fullmass matrix, gauge invariance is retained.This form of the twisted isospin-violating action (with ω = π/
2) is used in the recent work of Refs. [11, 12].It has one major practical disadvantage—the quark de-terminant is complex for nonzero twist. This is true fornondegenerate masses alone, as explained in Ref. [28].Adding electromagnetism only makes the problem worse,since at the least it induces further nondegeneracy in themasses. Because the action is complex, direct simula-tion with present fermion algorithms is challenging. Thisproblem is avoided in Refs. [11, 12] by doing a perturba-tive expansion in powers of (cid:15) q and α EM . The expectationvalues are then evaluated in the theory with no isospinbreaking, for which the fermion determinant with twist-ing is real and positive.In the following section we study the phase diagram ofthe theory with the discretized form of the Lagrangian(22). To our knowledge, this form of the twisted theoryhas not previously been studied in W χ PT either withnondegeneracy alone or with electromagnetism. V. χ PT FOR CHARGED, NONDEGENERATEQUARKS WITH A τ TWIST
The conclusion of the previous section is that thetwisted-mass theory whose phase diagram is of interestis that with lattice fermion Lagrangian ψ L [ D W + m + τ (cid:15) + iγ τ µ + iγ η ] ψ L . (23) ψ L is a lattice fermion field and D W the lattice Diracoperator including the Wilson term (and possibly im-proved). D W is coupled to both gluons and photons,with the latter coupling to the τ vector current. Theaction differs from that considered (implicitly) in Sec. IIIonly by the addition of the two mass parameters µ and η .The four bare mass parameters in (23) are related inthe continuum limit to the renormalized up and downmasses, the twist angle (which is a redundant param-eter) and the QCD theta angle, θ QCD . The aim is totune the bare parameters so that the dimension 4 partof the quark contribution to the Symanzik effective La-grangian is given by Eq. (22) with the desired physicalquark masses, for some choice of ω . As for untwisted Wil-son fermions the dominant effect of electromagnetism isto cause separate O ( α EM /a ) shifts in the (untwisted) upand down masses. These shifts depend on twisted massesonly at quadratic order, so that, to the order we work,they are identical to those for Wilson fermions. They canbe determined by the methods discussed in Sec. III A andAppendix B. They are equivalent to independent shiftsin m and (cid:15) .After the additive shift in m and (cid:15) , all four massesin (23) must be multiplicatively renormalized in orderto be related to the continuum masses in Eq. (22). Asdiscussed in Appendix A, this requires different renor-malization factors for all four masses. We assume herethat these renormalizations have been carried out, so thatthe dimension four term in the Symanzik effective La-grangian is given by Eq. (22) and described by the threeparameters m q , (cid:15) q and ω .We stress that this tuning and renormalization mustbe carried out with sufficient accuracy. If not, instead ofEq. (22), one ends up with a similar form having differ-ent twist angles for the m q and (cid:15) q parts. The parity-oddparts can then only be removed by a combined flavornonsinglet and flavor singlet twist. Since the latter isanomalous, this corresponds to a theory with nondegen-erate quark masses, electromagnetism, a twist angle and a nonvanishing θ QCD . In other words, the theory notonly has the unphysical parity violation due to twisting(which can be rotated away in the continuum limit) butalso the physical parity violation induced by θ QCD . In-deed, to analyze the tuning in χ PT one needs to includea nonvanishing θ QCD , an analysis we carry out in a com-panion paper [13].Assuming that the dimension-four quark Lagrangian isEq. (22), we next investigate which higher-dimension op-erators are introduced into the Symanzik Lagrangian bytwisting. Those operators present for Wilson fermions re-main, but, as discussed in Sec. III, are all of higher orderthan we consider. The dominant operators introducedby twisting will violate parity, because they are linear inthe parity-violating mass terms µ and η . Examples of the new operators are aη G µν (cid:101) G µν , aη ψ (cid:101) G µν σ µν ψ , and aµ ψτ (cid:101) G µν σ µν ψ . (24)Since we generically treat am terms as being beyond LO + [see Eq. (1)], we should be able to ignore these opera-tors. However, because η ∼ (cid:15) q and we are treating (cid:15) q assomewhat enhanced, one might be concerned about drop-ping aη terms. In fact, the aη operators in (24), whenmatched into χ PT, pick up an additional factor of m or p , and thus are unambiguously suppressed. The reasonfor the extra factors is that the LO representation of aflavor-singlet pseudoscalar in χ PT, tr(Σ − Σ † ), vanishesidentically. For the induced θ QCD term, one can also seethis result by noting that it can be rotated into the isos-inglet mass term, leading to a contribution proportionalto mθ QCD ∼ a(cid:15)m .Proceeding in this fashion, we find that all other newoperators induced by the parity-breaking masses are be-yond LO + in our power counting. Thus, once the requi-site tuning has been done, the LO + chiral effective theoryfor τ twisted fermions with isospin breaking is given bythe same result as for Wilson fermions, i.e. f (cid:2) ∂ µ Σ ∂ µ Σ † (cid:3) + V χ + V EM (25)[see Eqs. (3) and (12)], except that the quark mass matrixis now twisted χ = ( (cid:98) m q + (cid:98) (cid:15) q τ ) e iωτ . (26)We analyze the phase structure of this chiral theory inthe next two subsections. A. Phase diagram and pion masses at maximal τ twist We begin working at maximal τ twist, which is thechoice used in Refs. [11, 12]. In this case χ = i (cid:98) m q τ + i (cid:98) (cid:15) q , (27)and the chiral potential becomes − V χ +EM f = (cid:98) m q n sin θ − ( c (cid:96) (cid:98) (cid:15) q + w (cid:48) ) sin θ + c EM (cos θ + n sin θ ) , (28)up to an irrelevant constant. Since c EM >
0, the right-hand side is maximized always with | n | = 1, and we seethat the c EM term becomes a constant. Thus, once again,electromagnetism has no impact on the phase diagram. The first of these corresponds to an induced value of θ QCD pro-portional to aη . This is one way of seeing that the lattice action(23) leads to a complex fermion determinant. (a) Aoki scenario ( w (cid:48) < w (cid:48) > FIG. 4: Phase diagrams including effects ofdiscretization and nondegeneracy for maximally τ -twisted quarks. Electromagnetism has no impact onthe phase diagram. Notation as in Fig. 1. The neutralpion is massless along the second-order phase boundarybetween shaded (CP-violating) and unshaded phases.We also see that the effect of nondegeneracy can be de-duced from the results for the degenerate case (studiedin Refs. [29–31]) simply by shifting w (cid:48) .The resulting phase diagrams are shown in Fig. 4.Comparing to the untwisted results of Fig. 1, we see thatthe role of the Aoki and first-order scenarios has inter-changed. Without loss of generality, we can take n = 1throughout the phase plane. Then, in the continuum-like (unshaded) phases we have sin θ = sign( (cid:98) m q ), cor-responding to the condensate aligning or antialigningwith the applied twist. Second order transitions occurat | (cid:98) m q | = 2( w (cid:48) + c (cid:96) (cid:98) (cid:15) q ). For smaller values of | (cid:98) m q | thecondensate angle is sin θ = (cid:98) m q / (2[ w (cid:48) + c (cid:96) (cid:98) (cid:15) q ]), with two degenerate minima having opposite signs of cos θ . If oneswitches to the “physical basis” in which the twist is puton the Wilson term, then one finds that this phase vio-lates CP, just as in the Wilson case.These results differ significantly from the phase struc-ture for nondegenerate quarks with a maximal τ twist,shown in Fig. 3 of Ref. [3]. In particular, an additionalphase found for w (cid:48) > τ twist is absent here.We stress again that only the theory with a τ twist, i.e.that discussed here, can incorporate electromagnetism.For the pion masses we find the following results.Within the continuum-like phases we have m π = | (cid:98) m q | − c (cid:96) (cid:98) (cid:15) q + w (cid:48) ) , m π ± = | (cid:98) m q | + 2 c EM , (29)while within the CP-violating phase m π = 2( c (cid:96) (cid:98) (cid:15) q + w (cid:48) ) cos θ , m π ± = 2( c (cid:96) (cid:98) (cid:15) q + w (cid:48) + c EM ) . (30)As expected, only the charged pion masses are affectedby electromagnetism. Plots of these results along verticalslices through the phase diagram are shown in Fig. 5.It is interesting to compare to the results with un-twisted fermions, which are given in Eqs. (8) and (9)together with the shift (16) of m π ± by 2 c EM induced byelectromagnetism. We see that the neutral pion massdiffers only by the change of sign of w (cid:48) (which also im-plies the interchange sin θ ↔ cos θ ). This means thatthe results in the two scenarios interchange exactly. Forthe charged pion masses, apart from the interchange ofscenarios there are also overall shifts proportional to w (cid:48) .The implications of these results for present simula-tions are as follows. If one could simulate the theorydirectly (somehow dealing with the fact that the actionis complex) then one would need to avoid working in ornear the CP-violating phase. This is now more difficult inthe first-order scenario than the Aoki scenario—oppositeto the situation with untwisted Wilson fermions. Thisqualitative result is the same as for τ twisting (with-out electromagnetism), although the area taken up byunphysical phases is larger in that case [3]. As notedabove, actual simulations done to date at maximal twistuse perturbation theory in (cid:98) (cid:15) q and α EM , and so evaluateall expectation values in the theory with (cid:98) (cid:15) q = α EM = 0.Clearly, if w (cid:48) >
0, these simulations must be careful tohave ˆ m q large enough to avoid the CP-violating phase. B. Nonmaximal τ twist We have also investigated the phase structure for gen-eral τ twist, i.e. nonvanishing and nonmaximal. One In addition, if these simulations are done close to the onset ofthe CP-violating phase, one would expect the expansion in (cid:98) (cid:15) q to be poorly convergent. This is probably not a problem for themethod of Ref. [27], however, since they take the continuum limitof the term linear in (cid:98) (cid:15) q , and in this limit w (cid:48) = 0 and the latticeartifacts discussed here vanish. (a) Aoki scenario with c (cid:96) (cid:98) (cid:15) q + w (cid:48) < − c EM < − c EM < c (cid:96) (cid:98) (cid:15) q + w (cid:48) < c (cid:96) (cid:98) (cid:15) q + w (cid:48) > FIG. 5: Pion masses for nondegenerate maximally τ -twisted fermions including electromagnetism. Thethree possible behaviors along vertical slices throughphase diagrams of Fig. 4 are shown. Solid (blue) linesshow m π , while dashed (red) lines show m π ± .Expressions for masses are given in the text. motivation for doing so is that twisted-mass simulationscannot achieve exactly maximal twist; another is to seehow the phase diagrams of Fig. 1 change into those ofFig. 4.Expressions are simplified if we define θ relative to atwist ω , i.e. if we use (cid:104) Σ (cid:105) = e iωτ / e iθ ˆ n · (cid:126)τ e iωτ / . (31)Then we find (dropping constants) − V f = (cid:98) m q cos θ + c (cid:96) (cid:98) (cid:15) q n sin θ + w (cid:48) (cos θ cos ω − n sin θ sin ω ) + c EM (cos θ + n sin θ ) . (32)This is not amenable to simple analytic extremization,and we have used a mix of analytic and numerical meth-ods. One can show analytically that the minima alwaysoccur at | n | = 1. This implies that, once again, theelectromagnetism does not play a role in determining thephase structure.The sign of n can always be absorbed into θ , so we canagain choose n = 1 without loss of generality. The po-tential can then be written (up to θ -independent terms)as − V f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n =1 = (cid:98) m q cos θ + cos θ (cid:2) − c (cid:96) (cid:98) (cid:15) q + w (cid:48) cos(2 ω ) (cid:3) − w (cid:48) θ ) sin(2 ω ) . (33)A numerical investigation of this potential finds that, fornonextremal ω , and for all nonzero w (cid:48) , there is a first-order transition as (cid:98) m q passes through zero, irrespectiveof the value of (cid:98) (cid:15) q . At this transition θ jumps from π/ − δ to π/ δ , with δ (cid:54) = 0 depending on the parameters.Thus, unlike at the extremal points ω = 0 , π/
2, thereare no second-order transition lines. Correspondingly,there are no values of the parameters for which any ofthe pion masses vanish. This is very different from thetheory with a τ twist, where we found a two-dimensionalcritical sheet [3] in (cid:98) m q , (cid:98) (cid:15) q , ω space.The absence of critical lines at nonextremal twist canbe understood in terms of symmetries. For ω = 0 and π/
2, the potential has a θ → − θ symmetry, and this Z symmetry is broken by the condensate in the CP-violating phase, leading to a massless pion at the tran-sition. For nonextremal twist, however, the potential ofEq. (32) has no such symmetry. Lacking this symmetry,one expects, and finds, only first-order transitions. At ω = π/ θ and ˆ n are taken into account. VI. CONCLUSIONS
This work completes our study of how isospin breakingimpacts the phase structure of Wilson-like and twisted-mass fermions. The main results are the phase diagramspresented in Figs. 1 and 4, together with the correspond-ing pion masses. These results show how the combina-tion of discretization errors and nondegeneracy can bringunphysical phases closer to (or further away) from thephysical point.The inclusion of electromagnetism into the analysisturns out to be very straightforward, aside from the needto introduce independent up and down critical masses.Electromagnetism has no impact on the phase diagramsat leading order, because the condensates in the CP-violating phases involve neutral pions. The only impactis to uniformly increase the charged pion masses.We have investigated within W χ PT the conditionsused in Ref. [11] to determine the two critical masses inthe presence of electromagnetism. We find that, unlessone makes the electroquenched approximation, the twoconditions are in fact not independent. To determineboth critical masses one needs an additional condition,and we have presented one possibility in Appendix B. Ourcondition requires simulating at nonzero (though small) θ QCD , and thus will be difficult to implement in prac-tice, but provides an existence proof that an alternativecondition exists.Our analysis has been carried out in infinite volume.For the finite volumes used in lattice simulations onemight be concerned about significant finite-volume ef-fects on the electromagnetic contributions. The impacton the results presented here, however, should be mini-mal. The phase diagram will remain unaffected by elec-tromagnetism, while the shifts in critical masses are dom-inated by ultraviolet momenta, themselves insensitive tothe volume. The only significant effect will be on elec-tromagnetic mass shifts, with c EM picking up an effectivepower-law volume dependence [8, 17, 18, 32]. ACKNOWLEDGMENTS
This work was supported in part by the United StatesDepartment of Energy grant de-sc0011637.
Appendix A: Relating lattice masses to those in χ PT In this appendix we describe how bare lattice massesused in simulations with Wilson-like fermions are relatedto the masses m u and m d appearing in χ PT (containedin the mass matrix M ). This discussion draws heavilyfrom the results of Ref. [33]. We do not consider theimpact of electromagnetism here; this is discussed in thesubsequent appendix.We must assume that the number of dynamical quarksin the underlying simulations is N f = 3 (up, down and strange) or N f = 4 (adding charm). Working with upand down quarks alone turns out not to be sufficient, butin any case this is not the physical theory. We must alsohave that am f (cid:28) f , so that an expansionin these quantities makes sense. This condition is met bystate-of-the-art simulations. Note that this condition ismuch weaker than the requirement that the quarks arelight in the sense of χ PT, which is m f (cid:28) Λ QCD . In themain text, we assume the latter condition holds only forup and down quarks.Let m ,f be the bare dimensionless lattice mass forflavor f (i.e. the mass appearing in the lattice action).Because of the additive renormalization induced by ex-plicit chiral symmetry breaking, unrenormalized quarkmasses are given by (cid:101) m f = m ,f − m cr a , (A1)where m cr is the (dimensionless) critical mass for thegiven number of dynamical flavors. Methods to de-termine m cr are described below. Then, as shown inRef. [33], renormalized masses are given by m f = Z m (cid:20) (cid:101) m f + ( r m − (cid:80) f (cid:101) m f N f + O ( a (cid:101) m ) (cid:21) . (A2)Here Z m is the renormalization constant for flavor non-singlet mass combinations such as (cid:15) q = ( m u − m d ) / Z m r m is the corresponding constant for the aver-age quark mass. r m − O ( g ) in perturbation theory. By implementing contin-uum Ward-Takahashi identities, one can determine r m nonperturbatively for N f = 3 and 4, although not for N f = 2 [33]. This is the reason for the restriction on N f noted above. We assume here that r m has been calcu-lated in this way.Equation (A2) shows that the renormalized mass m f does not vanish when ˜ m f = 0 if other flavors are massive.Specifically, for the up and down quarks we have m u + m d = Z m r m (cid:101) m u + (cid:101) m d )+ Z m r m −
12 ( (cid:101) m s + (cid:101) m ch ) , (A3) m u − m d = Z m ( ˜ m u − ˜ m d ) . (A4)(Here we we have chosen N f = 4 for definiteness; the re-sult for N f = 3 is similar.) Thus the two-flavor masslesspoint receives an overall additive shift due to the strangeand charm quarks, and we also see explicitly the differ-ence between singlet and nonsinglet renormalizations.These results imply that, in terms of unrenormalizedmasses, the phase diagrams of Fig. 1 would be trans-lated in the vertical direction (due to the additive mass The correction terms of O ( a ˜ m ) in (A2) are subleading in ourpower-counting and will be dropped henceforth. different factors in the vertical andhorizontal directions. The respective stretch factors are B Z m (1 + r m ) / B Z m . If, however, r m is known,then the two stretch factors can be made equal by ap-plying a finite renormalization to remove the (1 + r m ) / Z m is, however, not useful, since italways appears multiplied by the unknown LEC B .We would like to be able to remove the additive massshift in Eq. (A3). To do so we consider how the criticalmass m cr is determined. The expressions above assumethat it has been obtained by doing simulations with N f degenerate quarks of mass m , and equating m cr to thevalue of m at which the “PCAC mass” vanishes. This isequivalent to imposing (cid:104) π + | ∂ µ (¯ uγ µ γ d ) | (cid:105) (cid:12)(cid:12)(cid:12) m = m cr = 0 . (A5)If, instead, one imposes this condition by varying m = m u = m d , with m s and m ch held fixed at their physicalvalues, then the m cr so obtained automatically includesthe shift due to loops of strange and charm quarks. Thisis because one is enforcing a consequence of chiral sym-metry in the two-flavor subsector. With this new choiceof m cr , and with the adjustment of stretch factors de-scribed above, the phase diagrams of Fig. 1 apply directlyfor lattice masses ˜ m f .This new choice of m cr has a second advantage: it re-moves an additional shift of O ( a ) in the relation betweenbare quark masses and the masses appearing in χ PT.As explained in Ref. [2], this shift is caused by the O ( a )clover term in the Symanzik effective action (and is thusabsent for nonperturbatively improved Wilson fermions).In the main text it is assumed that this shift has beenremoved.Since we include O ( a ) terms in the main text, we mustdetermine how they impact the considerations above.There is no further shift in the quark masses at thisorder—this next occurs at O ( a ) [34]. However, as il-lustrated by Fig. 1, the O ( a ) terms do impact the phasediagram. This means that, in general, one cannot usethe vanishing of the PCAC mass to determine m cr withuntwisted Wilson fermions. For example, if one is in thefirst-order scenario [Fig. 1(b) along the ˆ m q axis], thenthe PCAC mass simply does not vanish for any ˆ m q . In-stead, one must introduce a twisted component to themass, µ ∼ O ( a ), and then enforce the vanishing of thePCAC mass (in the so-called “twisted basis”). Extrapo-lating the result linearly to µ = 0 yields a result for m cr that has errors of O ( a ), which is sufficiently accurate forour analysis. For a detailed discussion of this point seeRef. [34].In summary, by determining r m from Ward identities,and the critical mass from the PCAC mass condition withtwisted-mass quarks, one can obtain lattice quark masseswhich are proportional to those appearing in χ PT at the order we work. Specifically, we find (cid:98) m q B Z m = 1+ r m m u + ˜ m d ) and (cid:98) (cid:15) q B Z m = ( ˜ m u − ˜ m d ) , (A6)where (cid:98) m q and (cid:98) (cid:15) q are the quantities appearing in the chiralpotential of Eq. (5).This analysis can be straightforwardly extended to ar-bitrary twist. We begin with maximal twist, for whichthe mass matrix in χ PT is given by Eq. (27), and therelevant bare masses are µ and η of Eq. (23). In thiscase there is no additive renormalization, but the pres-ence of different renormalization factors for singlet andnonsinglet masses remains. Using the results of Ref. [33],we find (cid:98) m q B Z m = Z S Z P r P r P µ and (cid:98) (cid:15) q B Z m = Z S Z P η . (A7)Here Z S /Z P and r P are finite constants, both of whichcan be determined from Ward identities for N f = 3 and4, but not for N f = 2 [33]. Like r m , r P begins at O ( g )in perturbation theory.At arbitrary twist one has four bare masses, andthey are related to the corresponding four renormalizedmasses using the same renormalization factors as givenin Eqs. (A6) and (A7).Finally, we stress that the analysis presented here doesnot include electromagnetic effects. The dominant sucheffect is that the critical mass m cr has to be chosen differ-ently for the up and down quarks, and is discussed in thefollowing appendix. A subdominant, but still important,effect is that the renormalization factors now depend notonly on α S but also on α EM . The latter dependence canpresumably be adequately captured using perturbationtheory. The formulae given above still hold if one usesthe new critical masses and renormalization factors. Appendix B: Determining the critical masses in thepresence of electromagnetism
The analysis of the previous appendix must be ex-tended when electromagnetism is included, due to thepresence of charge-dependent self energy corrections pro-portional to α EM /a . This implies that the critical massesfor up and down quarks differ, and we label them m cr,u and m cr,d , respectively. In Ref. [11] two methods for anonperturbative determination of these critical massesare proposed. One of these (the method used in prac-tice in Ref. [11]) involves only up and down quarks, andthus can be implemented, and therefore checked, withinSU(2) W χ PT. We do so in this appendix, finding that themethod does not fix both critical masses , but rather con-strains them to lie in a one-dimensional subspace of the Specifically, we have used Z m = 1 /Z S and r m = 1 /r S . m cr,u — m cr,d plane. We then provide an additional con-dition that does completely determine m cr,u and m cr,d .The tuning conditions require the use of twisted-massquarks, although the resulting values of m cr,u and m cr,d apply for both Wilson and twisted-mass quarks. Thusthe lattice quark Lagrangian is given by Eq. (23). Wecan write the mass matrix in two useful forms m + τ (cid:15) + iγ τ µ + iγ η = (cid:18) m ,u + iγ µ ,u m ,d − iγ µ ,d (cid:19) . (B1)The tuning proceeds by first choosing bare twistedmasses µ ,u and µ ,d such that, when multiplicativelyrenormalized as described in the previous appendix, theygive rise, respectively, to the desired physical up anddown quark masses. The negative sign multiplying µ ,d is chosen to correspond to a τ twist. The second stepis to tune the untwisted masses m ,u and m ,d to theircritical values such that the (additively) renormalized un-twisted masses vanish.The method of determining m cr used in the previoussection is no longer useful—the vanishing of the PCACmass is a condition based on the recovery of the chiralSU(2) group, but this group is explicitly broken by elec-tromagnetism. The workaround proposed in Ref. [11]is to add to the sea quarks (labeled u S and d S ) a pairof valence quarks, u V and d V , each of which has thesame charge and untwisted mass as the corresponding seaquark, but has opposite twisted mass. Thus ( u S , u V )and ( d V , d S ) each form a twisted pair. The key pointis that, within each pair, the O ( α EM /a ) shift in the un-twisted mass is common. Therefore it is plausible thatone can determine the critical mass for each pair by en-forcing the recovery of the corresponding valence-sea chi-ral SU(2). Specifically, m cr,u is determined by (cid:104) π uSV | ∂ µ (¯ u S γ µ γ u V ) | (cid:105) (cid:12)(cid:12)(cid:12) m ,u = m cr,u = 0 , (B2)while m cr,d is determined by the analogous conditionwith u → d : (cid:104) π dSV | ∂ µ ( ¯ d S γ µ γ d V ) | (cid:105) (cid:12)(cid:12)(cid:12) m ,d = m cr,d = 0 . (B3)Here π uSV and π dSV are sea-valence pions composed, re-spectively, of up and down quarks. In fact, the tuning can be done using any values of the twistedmasses which respect our power counting. The critical massesdo not depend on the twisted masses at the order we work. This description is equivalent to that of Ref. [11], but differs tech-nically in two ways. First, we find that one need only introducetwo valence quarks to describe the method, rather than the fourused in Ref. [11]. This does not impact the method itself, onlyits description. Second, we work in the twisted basis, rather thanthe physical basis used in Ref. [11].
When using a partially quenched theory, one also needsto add ghost fields, ˜ u V and ˜ d V , to cancel the valencequark determinants. Thus the full softly-broken chi-ral symmetry is the graded group SU (4 | L × SU (4 | R .This raises the question of whether complications arisingfrom partial quenching, or from discretization effects, canlead to corrections to the tuning criteria of Eqs. (B2) and(B3). This is one of the issues we address here by map-ping these conditions into χ PT.We begin by mapping the mass matrix in the un-quenched sector into χ PT. The four parameters ofEq. (B1) map into χ = (cid:18) ˆ m u e iω u
00 ˆ m d e − iω d (cid:19) (B4)= (cid:18) ( (cid:98) m q + (cid:98) (cid:15) q ) e i ( ω + ϕ )
00 ( (cid:98) m q − (cid:98) (cid:15) q ) e i ( − ω + ϕ ) (cid:19) . (B5)The choice of sign for ω d is such that it is positive with a τ twist. χ contains the additional parameter ϕ comparedto the mass matrix analyzed in the main text, Eq. (26). ϕ is a measure of the difference between up and downtwist angles, ω u = ω + ϕ , ω d = ω − ϕ . (B6)As discussed in Sec. V, such a difference corresponds tothe introduction of a nonzero θ QCD —the explicit relationis ϕ = θ QCD / χ in Eq. (B5)—which canbe worked out along the lines of the previous appendix—are not needed here. All we need to know is that, if m ,u = m cr,u and m ,d = m cr,d , then both up and downmasses are fully twisted. Thus the twist angles in χ are ω u = ω d = π/
2, implying maximal twist with no θ QCD term: ω = π/ ϕ = 0. Reaching this point in pa-rameter space is the aim of tuning.When considering the PQ extension of this theory, wewill focus mainly on the quark sector, since the ghostsdo not play a significant role. Collecting the four quarkfields in the order ψ (cid:62) P Q = ( u S , u V , d V , d S ) , (B7)the extended quark mass matrix is χ P Q = (cid:18) ( (cid:98) m q + (cid:98) (cid:15) q ) e iω u τ
00 ( (cid:98) m q − (cid:98) (cid:15) q ) e iω d τ (cid:19) . (B8)The factors of τ arise because, by construction, valencequarks have opposite twisted masses to the correspond-ing sea quarks. We stress that the O ( α EM /a ) shifts areincorporated into the parameters (cid:98) m q and (cid:98) (cid:15) q , along with For reviews of partially quenched theories and the corresponding χ PT, see Refs. [35, 36]. O (1 /a ) shifts. We can also include the O ( a )shifts in the same fashion.To implement the conditions (B2) and (B3) in the PQtheory, we need the extension of Σ to this theory. Thisis a 6 × SU (4 | L × SU (4 | R . In fact, as we only need matrix el-ements for states composed of quarks, and since we knowfrom Ref. [37] that there are no quark-ghost condensates,we can focus on the 4 × P Q . We now argue that the expectation value of Σ
P Q has the form (cid:104) Σ P Q (cid:105) = diag( e iθ , e − iθ , e iθ , e − iθ ) . (B9)This is based on the following results. First, the un-quenched 2 × P Q (i.e. that involving thefirst and last rows and columns) is just the unquenchedΣ field. This is unaffected by partial quenching [38, 39],and its expectation value is given by an unquenched χ PTcalculation. This calculation must include not only non-degeneracy, electromagnetism and twist, but also nonva-nishing θ QCD . To our knowledge such an analysis has notpreviously been done, so we carry it out in a companionpaper [13]. The result is that the unquenched conden-sate (cid:104) Σ (cid:105) only rotates in the τ direction—there are nooff-diagonal condensates such as (cid:104) ¯ u S d S (cid:105) . This fixes thefirst and last entries in Eq. (B9) to have opposite phaseangles.This is an important result for the following, so weemphasize its key features. Although θ QCD (cid:54) = 0 leads toan overall phase in the mass matrix [ e iϕ in Eq. (B5)], itseffect on the condensate (cid:104) Σ (cid:105) is qualitatively similar tothat of a twist ω , despite the fact that the latter leads toopposite phases on u and d quarks. This happens becauseΣ is constrained to lie in SU (2), and so has no way tobreak parity other than rotating in the τ direction. Anoverall phase rotation would take it out of SU (2) into the U (2) manifold.The second result needed to obtain Eq. (B9) is the ex-istence of relations between valence and sea-quark con-densates. In particular, one can show that (cid:104) ¯ u V u V (cid:105) = (cid:104) ¯ u S u S (cid:105) and (cid:104) ¯ u V γ u V (cid:105) = −(cid:104) ¯ u S γ u S (cid:105) , (B10)to all orders in the hopping parameter expansion. Theminus sign in the second relation follows from the op-posite twisted mass of sea and valence quarks. Theresult (B10) holds on each configuration and thus alsofor the ensemble average, even though the measure iscomplex for θ QCD (cid:54) = 0. Since the additive and multi-plicative renormalizations of these condensates are thesame for valence and sea quarks, the result (B10) impliesthat valence and sea up-quark condensates have opposite“twists”, e ± iθ . The same argument applies to the down-quark condensates, and taken together these argumentsdetermine the form of the second and third diagonal el-ements in Eq. (B9).The final result needed to obtain the form (B9) is thevanishing of off-diagonal condensates involving one or more valence quarks, e.g. (cid:104) ¯ u V d V (cid:105) and (cid:104) ¯ u V d S (cid:105) . These dif-fer from the diagonal condensates in that there is no massterm in the quark-level Lagrangian that can serve as asource for such condensates. Thus to determine whetherthey are nonzero one must add a source, e.g. ∆ ¯ d V u V ,calculate the resulting condensate, send the volume toinfinity, and finally send the parameter ∆ →
0. Thisanalysis has been carried out in Appendix A of Ref. [40]in a theory with twisted-mass quarks, although, unlikeour situation, the quarks were degenerate and θ QCD = 0.The general lessons from Ref. [40] are (i) that to ob-tain a nonvanishing condensate one needs a source ofinfrared divergence to cancel the overall factor of ∆, and(ii) that nonvanishing twisted masses cut off such diver-gences. These lessons apply also for all the off-diagonalcondensates that we consider here. However, the argu-ment as given in Ref. [40] assumes that the measure isreal and positive, which does not hold here. Neverthe-less, since we are tuning to θ QCD = 0, we expect theimpact of having a complex measure to be small. Fur-thermore, we know from Ref. [13] that the correspond-ing sea quark condensates, e.g. (cid:104) ¯ u S d S (cid:105) and (cid:104) ¯ u S γ d S (cid:105) ,vanish even when θ QCD (cid:54) = 0. These condensates differfrom those containing valence quarks only by changingthe signs of some of the twisted masses. Since it is thepresence of these masses, and not their detailed proper-ties, that leads to the vanishing of the condensate, weexpect the result holds for all off-diagonal condensates.With the form (B9) in hand, we can now apply the tun-ing conditions (B2) and (B3) in χ PT. We do so by gen-eralizing the analysis of Ref. [41], where the twist anglefor unquenched twisted-mass fermions was determined in χ PT by applying a PCAC-like condition. The requiredextension is from the SU (2) sea-quark sector alone to thefull valence-sea SU (4) symmetry. Much of the analysiscarries over with minimal changes from Ref. [41], so weonly sketch the calculation.The first step is to obtain the pion fields that coupleto external particles in the tuning conditions. FollowingRef. [41], we obtain these by expanding the chiral fieldabout its vacuum value asΣ P Q = ξ P Q e i Π /f ξ P Q , (B11)Π = (cid:88) a =1 π a λ a , (B12) ξ P Q = (cid:113) (cid:104) Σ P Q (cid:105) = diag( e iθ/ , e − iθ/ , e iθ/ , e − iθ/ ) . (B13)Here λ a are the generators of SU(4), with π a the cor-responding pion fields. These are the pions in the PQtheory that are composed of quarks alone, with no ghostcomponent. The pions needed for tuning, π uSV and A similar form to Eq. (B13) holds for the full 6 × SU (4) block, since the pions we leaveout in this way are those containing one or more ghost fields. π dSV , are contained in the upper and lower diagonal 2 × χ PT, of theaxial currents appearing in the tuning conditions. Thesecan be obtained by introducing sources into derivativesusing standard methodology. Since, by definition, ourchiral potential does not contain derivatives, at LO + onlythe LO kinetic term [shown in Eq. (2)] enters into thedetermination of the currents. We do not display theform of the currents, however, as the calculation neededfor each of the tuning conditions is exactly the same asthat carried out in Ref. [41]. This is because each tun-ing condition involves a separate, nonoverlapping SU (2)subgroup of SU (4) (upper-left or lower-right 2 × (cid:104) π uSV | ∂ µ (¯ u S γ µ γ u V ) | (cid:105) ∝ cos θ , (B14) (cid:104) π dSV | ∂ µ ( ¯ d S γ µ γ d V ) | (cid:105) ∝ cos θ . (B15)Thus enforcing either (B2) or (B3) has the effect of set-ting θ = ± π/ (cid:104) Σ P Q (cid:105) = ± diag( i, − i, i, − i ) , (B16)For our choices of signs of the twisted masses µ ,u and µ ,d in Eq. (B1), the ± signs are in fact plusses, i.e. θ = π/ m u in turnchanges ϕ and ω and this impacts the d condensatethrough the quark determinant. One might, therefore,wonder how the two tuning conditions have been suc-cessfully applied in Ref. [11]. To understand this, we notethat this work makes two approximations. First, isospin-breaking effects are evaluated only through linear orderin an expansion in m u − m d and α EM . Second, insertionsof m u − m d or photons on sea-quark loops are dropped(the “electroquenched approximation”). The latter ap-proximation has the effect of disconnecting the two tun-ing conditions—all quark loops in both conditions areevaluated with uncharged, degenerate sea-quarks, so the u -quark condensate cannot be impacted by changes in m d and vice versa. Since χ PT predicts that there is a tightcorrelation between the condensates, it appears to us thatthe electroquenched approximation is theoretically prob-lematic. However, from a purely numerical viewpoint,the dropped contributions may well lead only to smallcorrections.The lack of independence implies that the tuning con-ditions cannot determine both m u,c and m d,c —only oneconstraint on these two critical masses is obtained. Interms of the parameters of mass matrix (B5), the condi-tions determine only a relation between ω and ϕ . Thus,after enforcing either (B2) or (B3) the theory is known to lie along a line in the ω — ϕ plane. In terms of thebare masses, the theory lies along a line in the m ,u — m ,d plane (with, recall, µ ,u and µ ,d fixed at the valuesleading to physical quark masses when m ,u = m ,d = 0).We do know that this one-dimensional subspace includesthe point we are trying to tune to, namely that with( ω, ϕ ) = ( π/ , ϕ = 0, the twist inthe condensate is also maximal, i.e. θ = π/
2. The onlycaveat is that the values of the twisted masses must besuch that one lies in the continuum-like phase, ratherthan the CP-violating phase (see Fig. 4).To complete the tuning we need an additional condi-tion that forces us to the desired point along the allowedline. At first blush one might expect that it would besimple to find an additional tuning condition, since theo-ries with θ QCD (cid:54) = 0 have explicit parity violation. This isin contrast to the parity violation induced by a nonzerotwist ω which, in the continuum limit, can be removedby a chiral rotation. This suggests that one should lookfor quantities that vanish when parity is a good symme-try. The flaw in this approach is that parity is broken by ω (cid:54) = 0 away from the continuum limit—the chiral rota-tion required to obtain the parity-symmetric form is nota symmetry on the lattice. Thus the distinction between ϕ (cid:54) = 0 and ω (cid:54) = 0 no longer holds.The only choice that we have found for a second con-dition involves using the pion masses. Specifically, wefind that, along the line picked out by setting θ = π/ ϕ = 0. This assumes only that we are in thecontinuum-like phase for the physical values of µ ,u and µ ,d .The details of the calculation are presented in Ref. [13].Working at LO + , we find that the constraint θ = π/ m d ˆ m u = − (cid:18) − c (cid:96) (ˆ µ u − ˆ µ d )1 + c (cid:96) (ˆ µ u − ˆ µ d ) (cid:19) . (B17)As noted above, this line passes through the desired point m u = m d = 0. The slope is − c (cid:96) = 0, and in-creases in magnitude as c (cid:96) increases (assuming the phys-ical situation ˆ µ u < ˆ µ d ). There is no singularity when theslope reaches infinity—this simply means that the con-straint line is the m u = 0 axis. For larger c (cid:96) the slope ispositive. It decreases with increasing c (cid:96) , though it alwaysremains greater than unity. The pion masses along theconstraint line are m π = ˆ µ u + ˆ µ d − c (cid:96) (cid:18) ˆ µ u − ˆ µ d (cid:19) + 2 c (cid:96) (cid:18) ˆ m u − ˆ m d (cid:19) − w (cid:48) , (B18) m π ± = ˆ µ u + ˆ µ d c (cid:96) (cid:18) ˆ m u − ˆ m d (cid:19) + 2 c EM . (B19)Thus we see that both masses are minimized along theconstraint line when m u = m d = 0. If one were to imple-5ment this tuning condition in practice, then one wouldapply it for the charged pion masses, since these have noquark-disconnected contractions.This analysis breaks down when c (cid:96) gets too large, be-cause the theory with m u = m d = 0 then lies in thethe CP-violating phase. This can be seen from the result(B18)—for large enough c (cid:96) the squared neutral pion massbecomes negative. This happens sooner for the first-orderscenario, w (cid:48) > χ PT. Because of such terms, even if one perfectly implements our two tuning conditions—namely either Eq. (B2) or (B3) and minimizing the pionmasses—one will not have precisely tuned to m u = m d =0. This can be seen, for example, from the analysis ofRef. [41], where terms of O ( ap , am ) lead to maximaltwist occurring at untwisted masses of O ( aµ ), with µ the twisted mass, rather than zero. Shifts of this sizeoccur also in the presence of isospin breaking, althoughthe detailed form of the corrections will differ. Withinour power-counting, however, µ ∼ a so that the shiftsin the untwisted masses are ∼ O ( a ), beyond the orderthat we consider. [1] S. Aoki, Phys.Rev. D30 , 2653 (1984).[2] S. R. Sharpe and J. Singleton, Robert L., Phys.Rev.
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