Unphysical phases in staggered chiral perturbation theory
aa r X i v : . [ h e p - l a t ] M a r Unphysical phases in staggered chiral perturbation theory
Christopher Aubin, ∗ Katrina Colletti, † and George Davila ‡ Department of Physics & Engineering Physics, Fordham University, Bronx, New York, NY 10458, USA
We study the phase diagram for staggered quarks using chiral perturbation theory. In beyond-the-standard-model simulations using a large number ( >
8) of staggered fermions, unphysical phasesappear for coarse enough lattice spacing. We argue chiral perturbation theory can be used tointerpret one of these phases. In addition, we show only three broken phases for staggered quarksexist, at least for lattice spacings in the regime a ≪ Λ . PACS numbers: 11.15.Ha,11.30.Qc,12.39.Fe
I. INTRODUCTION
The fact that unphysical phases may arise in lattice simulations for coarse lattice spacings has been known forsome time [1–5]. Such phases arise when the squared mass of a meson becomes negative in a region of the relevantparameter space. When this occurs we must find the true minimum of the potential so that we can expand aboutthe true ground state of the theory. Doing so for lattice simulations is important as the continuum limit cannot beproperly taken unless they are performed in the unbroken, physical, phase, where the vacuum state has the symmetriesof the action.For staggered quarks, the case of interest here, unphysical phases appear when ca < − m , where m is the lightquark mass, for some parameter c (the specific form will be discussed in Sec. III) arising from the O ( a ) taste-symmetry breaking potential. This implies that these unphysical phases can be studied using rooted staggered chiralperturbation theory (rS χ PT) [3, 4], which requires a to be fine enough such that the low-energy effective theory isvalid. Thus, we are interested in a region such that m Λ QCD < ∼ a Λ ≪ . (1)The first condition assumes the parameter c < ca is large enough that one of the squared meson masseshave become negative, while the second condition is necessary for our low-energy effective theory to be valid.If simulations are performed in the broken phase, one cannot use the numerical results to describe physical systems.As such, understanding where these unphysical phases occur and how to detect them is essential in understandingthe system being simulated. In Ref. [5], one unphysical phase for staggered quarks was studied and an analysis of themass spectrum was performed, noting the possibility of additional broken phases in the system. However, it is clearthat the phase in Ref. [5] is not seen in 2+1-flavor simulations (see Refs. [6, 7] for example).In recent work looking into beyond-the-Standard-Model (BSM) theories by Ref. [8] using 8 or 12 flavors of degeneratestaggered quarks, two broken phases were seen in additional to the standard physical phase. One of the phasesexamined in Ref. [8] shares several features as the phase studied in Ref. [5], as we discuss in this work.In Ref. [8], the authors found three distinct phases appearing in the staggered theory for 12 flavors of staggeredquarks. The first phase, seen at weaker coupling, is the unbroken phase, as it retains the discrete shift symmetry ofstaggered fermiosn, and has the expected mass spectrum, at least approximately. The intermediate phase, at slightlystronger coupling, we argue falls within the window in Eq. (1) so that rS χ PT is applicable, and is the broken phaseseen in Ref. [5]. Finally, the phase that arises at the strongest coupling in Ref. [8] is outside the chiral regime, andthus cannot be studied using the methods of this paper.One can use the replica method for rS χ PT [9] to generalize the results of Ref. [5] for n f degenerate flavors and n t tastes-per-flavor. We define n q ≡ n f n t as the number of quarks in our resulting theory. The phase studied in Ref. [5],which we will refer to as the “ A -phase,” appears when a δ ′ A − < −
43 (2 µm + a ∆ A ) , (2) ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] These would then be 4+4-flavor or 4+4+4-flavor simulations.
FIG. 1: The maximum allowed number of quarks as a function of pion mass to ensure a simulation is in the unbroken phasefor the coarse asqtad MILC ensembles ( a ≈ .
125 fm). The dashed line is n q = 3 and the shaded region shows allowed valuesof n q as a function of the pion mass.FIG. 2: The maximum allowed number of quarks as a function of pion mass to ensure a simulation is in the unbroken phasefor the fine asqtad MILC ensembles ( a ≈ .
09 fm). The dashed line is n q = 3 and the shaded region shows allowed values of n q as a function of the pion mass. where m is the light quark mass, and we are denoting δ ′ A , the hairpin term of Refs. [4, 5], as δ ′ A − . This is assumingthree flavors of degenerate rooted-staggered quarks, but if we generalize this using the replica method to n f flavorsand n t tastes, we can rewrite the condition for broken phase as n f n t = n q > (cid:18) µm + a ∆ A − a δ ′ A − (cid:19) . (3)Given a sample set of parameters in MILC simulations for the a = 0 .
125 fm and a = 0 .
09 fm asqtad ensembles forthese values on the right-hand side [10], we show the maximum number of allowed quarks, n q, max , for the simulationto remain in the unbroken phase as a function of m π (the Goldstone pion mass) in Figs. 1 and 2. The shaded regionshows allowed values of n q as a function of the pion mass. The dashed lines in these figures indicate n q = 3, which iswell below the limit for being in the unbroken phase (except for m π < ∼
128 MeV on the coarse ensemble and m π < ∼ A -phase when we simulate 8 or 12 quarks than when we simulate fewerquarks. More specifically, if the (Goldstone) pion mass is around 500 MeV, the simulation would most likely be in theunbroken phase for 8 quarks (2 flavors, 4 tastes per flavor), while in the A -phase for 12 quarks. Figures 1 and 2 weregenerated using parameters from asqtad MILC ensembles for various lattice spacings. The specific picture will changewith different staggered quarks such as nHYP staggered quarks [11, 12], as are used in Ref. [8], but qualitatively wewould expect similar results.In this paper we study the staggered phase diagram for all values of the rS χ PT parameters that may arise during asimulation. In Ref. [5], a third possible phase was discussed (which we will refer to as the A ′ -phase) and we show thatit cannot occur. Instead, in addition to the A -phase, there are two other broken phases that we label the V -phaseand the T -phase. We also show that one of the two broken phases seen in Ref. [8] is most likely the A -phase discussedin Ref. [5], and suggest other ways to check if indeed this is the case.We organize this paper as follows. In Sec. II we define the staggered chiral Lagrangian for an arbitrary number offlavors, n f , and summarize the results of previous work with the notation we will use in this paper. Then in Sec. III,we find all of the minima of the potential in the revelant region [see Eq. (1)]. We focus on the A -phase as that hasthe features seen in one of the broken phases in Ref. [8]. Finally we conclude in Sec. IV. We include two appendiceswhere we list the masses in the A -phase and the T -phase in Appendix A and Appendix B, respectively. II. THE STAGGERED CHIRAL LAGRANGIAN
The starting point of our analysis is the S χ PT Lagrangian for n f flavors of quarks [4]. The Lagrangian is writtenin terms of the field Σ = exp( i Φ /f ), a 4 n f × n f matrix, withΦ = U π + K + · · · π − D K · · · K − ¯ K S · · · ... ... ... . . . . (4)The elements shown are each 4 × T a = { ξ , iξ µ , iξ µν , ξ µ , ξ I } . (5)In euclidean space, the gamma matrices ξ µ are hermitian, and we use the notations ξ µν ≡ ξ µ ξ ν [ µ < ν in Eq. (5)], ξ µ ≡ ξ µ ξ and ξ I ≡ I is the 4 × SU (4 n f ) L × SU (4 n f ) R symmetry, Σ → L Σ R † . Thecomponents of the diagonal (flavor-neutral) elements ( U a , D a , S a , etc. ) are real, while the off-diagonal (flavor-charged)fields are complex ( π + a , K a , etc. ), such that Φ is hermitian.From Ref. [4], the Lagrangian is given by L = f ∂ µ Σ ∂ µ Σ † ) − µf Tr( M Σ + M Σ † )+ 2 m U I + D I + S I + · · · ) + a V , (6)where µ is a constant with dimensions of mass, f is the tree-level pion decay constant (normalized here so that f π ≈
131 MeV), and the m term includes the n f flavor-neutral taste-singlet fields. Normally, in physical calculations,we would take m → ∞ at the end to decouple the taste-singlet η ′ I , however in a broken phase there is no physicalreason to assume a large value for m , so we will retain that parameter in our calculations. Finally, V = U + U ′ isthe taste-symmetry breaking potential given by − U = C Tr( ξ ( n f )5 Σ ξ ( n f )5 Σ † )+ C X ν [Tr( ξ ( n f ) ν Σ ξ ( n f ) ν Σ) + h.c. ]+ C X ν [Tr( ξ ( n f ) ν Σ ξ ( n f )5 ν Σ) + h.c. ]+ C X µ<ν Tr( ξ ( n f ) µν Σ ξ ( n f ) νµ Σ † ) , (7) −U ′ = C V X ν [Tr( ξ ( n f ) ν Σ) Tr( ξ ( n f ) ν Σ) + h.c. ]+ C A X ν [Tr( ξ ( n f ) ν Σ) Tr( ξ ( n f )5 ν Σ) + h.c. ]+ C V X ν [Tr( ξ ( n f ) ν Σ) Tr( ξ ( n f ) ν Σ † )]+ C A X ν [Tr( ξ ( n f ) ν Σ) Tr( ξ ( n f )5 ν Σ † )] . (8)The ξ ( n f ) B in V are the block-diagonal 4 n f × n f matrices ξ ( n f ) B = ξ B · · · ξ B · · · ξ B · · · ... ... ... . . . , (9)with B ∈ { , µ , µν ( µ < ν ) , µ, I } . The mass matrix, M , is the 4 n f × n f diagonal matrix M = mI n f × n f , as we areonly interested in the degenerate case that is relevant for these BSM studies.As is well known [3, 4], while this potential breaks the taste symmetry at O ( a ), an accidental SO (4) symmetryremains. This implies a degeneracy in the masses among different tastes of a given flavor meson, which is seen in thetree-level masses of the pseudoscalar mesons. We can classify these mesons into irreducible representations of SO (4).The mass for the meson M (composed of quarks a and b ) with taste B , is given at tree-level by m M B = µ ( m a + m b ) + a ∆ B , (10)for mesons composed of different quarks, and m M B = 2 µm a + a ∆ B + n f n t a δ ′ B − , B = V, A , (11)for the flavor-neutral mesons. The ∆ B ’s are given in Ref. [4] and are linear combinations of the coefficients in thepotential U and we have the hairpin terms, δ ′ V ( A ) ± ≡ f (cid:2) C V ( A ) ± C V ( A ) (cid:3) . (12)The difference in Eq. (11) from previous works is that we have the factor n f n t / n f is the number of (degenerate) staggered flavors in our calculation, while n t is the number of tastes per flavor wewish to keep (hence the factor of 1/4). The factor n f n t ≡ n q will be the number of degenerate fermions we have inour theory. Note we do not include the m term here for simplicity. We note that in our calculations, if n t = 4, then we are not rooting the underlying theory, and as such the theory does not correspondto “rooted” staggered quarks. Given that empirically the ∆ B ’s are all positive in simulations, δ ′ V − is consistent with zero, and δ ′ A − < η ′ A meson. It was shown in Ref. [5] thatin current 2 + 1-flavor simulations, it is very unlikely the simulation will be performed in this phase. Instead, therehas been evidence of this phase appearing in BSM simulations [8], and this can easily be understood from Eq. (10).As discussed in the Introduction, and shown in Figs. 1 and 2, assuming that the actual value of δ ′ A − is, to a firstapproximation, dependent only upon the specific fermion formulation and not the number of flavors (or tastes), thenas n q increases, the simulations are more likely to be performed in the phase described in Ref. [5]. As discussed in Ref. [5], in the A -phase, all of the squared meson masses will be positive given the relationshipsbetween the different parameters, except possibly for the tensor taste flavor singlet, η ′ ij . Specifically, we have (rewritingthis expression with our notation), m η ′ ij = − n f n t a δ ′ A − − n f n t a δ ′ V + + 16 m µ (cid:0) a ∆ T − a ∆ A + n f n t a δ ′ V + (cid:1)(cid:0) a ∆ A + n f n t a δ ′ A − (cid:1) . (13)The parameter δ ′ V + in this expression has not yet been measured, and as such, m η ′ ij has the possibility of goingnegative. This new phase, which we denote the A ′ -phase, could in principle arise in the staggered phase diagram. Inthe next section we study the phase diagram in general and find that this is not the case, while additional phasesother than the A ′ -phase do exist. III. GENERAL PHASE DIAGRAM
To find the vacuum state of the theory, we must minimize the potential, W = − µmf Tr(Σ + Σ † ) + a U + a U ′ , (14)where we have already substituted M = mI n f × n f . This calculation is most simply done in the physical basis, whereeverything is written in terms of (for three flavors) π , η , and η ′ instead of the flavor-basis mesons U , D , and S . Fordegenerate quarks, we define the singlet as η ′ B = 1 √ n f ( U B + D B + S B + · · · ) , (15)for any number of flavors/tastes. As these are the mesons most likely to acquire a negative mass-squared, we focussolely on these. From here on we remark that in the degenerate quark mass limit, the octet meson masses of a giventaste have equal masses which we denote with m π , while the η ′ masses are distinct from these. We note that thenumber of flavors n f (so long as it’s greater than 1) will not affect our results; n f will only indicate a greater likelihoodof being in the broken phase at this point.Generally, δ ′ A − and δ ′ V − are the parameters likely to be negative, and they only arise in the η ′ masses. We inferthe symmetry breaking to only occur in the η ′ direction in flavor space. Therefore, we are going to keep only thismeson in our expression for Φ when looking for the minima of the potential in Eq. (14). This will be valid right nearthe critical point, and since we are looking at this perturbatively, we are looking only at small fluctuations about theminimum. So long as no other squared mass goes negative in the phase, our results should give us the correct massspectrum for the broken phase. If a squared mass does go negative, as in Eq. (13) for certain values of δ ′ V + , we arenot near a minimum of the potential, and thus such additional phases are not stable.Keeping only the η ′ B , Φ and Σ are block-diagonal in flavor space. We can write the condensate h Σ i in terms of the16 real numbers σ I , σ µ , σ µν ( µ < ν ) , σ µ , and σ , h Σ i = σ I (cid:0) I n f × n f (cid:1) + iσ µ (cid:16) iξ ( n f ) µ (cid:17) + iσ µν (cid:16) iξ ( n f ) µν (cid:17) + iσ µ ξ ( n f ) µ + iσ (cid:16) iξ ( n f )5 (cid:17) , (16) As these parameters are non-perturbative low-energy constants, they would have a dependence upon the number of quarks in thesimulation, but without knowing that dependence a priori , we take them to be independent of n q as an initial approximation. with the condition that P B σ B = 1. Upon substituting this into the potential, we find the potential (not surprisingly)is only dependent upon the magnitudes of these sets of coefficients, given by σ A = X µ σ µ ! / , (17) σ T = X µ<ν σ µν ! / , (18) σ V = X µ σ µ ! / , (19)so that we have, up to an unimportant constant and for arbitrary numbers of flavors/tastes, W = − f (cid:20) µmσ I − σ A (cid:16) a ∆ A + n f n t a δ ′ A − (cid:17) − σ V (cid:16) a ∆ V + n f n t a δ ′ V − (cid:17) − σ T a ∆ T (cid:21) . (20)When minimizing this potential we find three distinct non-trivial phases: A -phase : a ∆ A + n f n t a δ ′ A − < − µm , (21) V -phase : a ∆ V + n f n t a δ ′ V − < − µm , (22) T -phase : a ∆ T < − µm . (23)The A -phase was discussed in detail in Ref. [5], and the results for the V -phase are identical to those for the A -phasewith the replacement A ↔ V in all of the relevant equations. The T -phase is distinct here, and it is unlikely thata simulation will be performed in this phase. This is because all of the parameters ∆ B are positive in simulations,so these conditions are only likely to hold for the A and V phases because the parameters δ ′ A − and δ ′ V − tend to benegative. Nevertheless, we discuss this phase briefly in the Appendix for completeness. We note that in principle,more than one of the conditions in Eqs. (21) through (23) may hold simultaneously, but in fact we only see thesethree phases. This implies that only one condition will point to the true minimum about which to expand. In theunlikely case that two of the left-hand sides are equal, for example, a ∆ T = a ∆ A + n f n t a δ ′ A − , (24)this would introduce a symmetry between (in this case) the axial- and tensor-tastes, but this does not introduce adistinct phase.We note that none of these three broken phases correspond to the A ′ -phase discussed in the previous section.Approaching this phase from A -phase, we find a saddle point in the potential, and as such this is an unstableequilibrium point. Thus, we will not explore that case further.We have for the A -phase, σ T = σ V = 0 , σ I ≡ cos θ A = − µma ∆ A + n f n t a δ ′ A − , σ A = q − σ I , (25)the V -phase, σ T = σ A = 0 , σ I ≡ cos θ V = − µma ∆ V + n f n t a δ ′ V − , σ V = q − σ I , (26)and for the T -phase, σ A = σ V = 0 , σ I ≡ cos θ T = − µma ∆ T , σ T = q − σ I . (27) (a) (b)FIG. 3: Figure 7 from Ref. [8], showing the masses of the π and π ( π in our notation) as a function of the quark mass. (a)shows the masses in the broken phase discussed here and (b0 shows the masses in the unbroken phase. These break the remnant SO (4) symmetry [to SO (3) for the A and V phases, and to SO (2) × SO (2) for the T phase].The direction of each of the vectors a µ , a µ , and a µν is arbitrary, and we will choose a particular direction in Eqs. (28)and (29).In each of these cases, we have the condensate of the form h Σ i = exp [ iθ B T B ] 0 0 · · · iθ B T B ] 0 · · · iθ B T B ] · · · ... ... ... . . . , (28)where B = A, V, T , and T B = iξ A − phase ξ V − phase iξ T − phase . (29)In each of these cases, the shift-symmetry [3–5] that exists which has the form in the chiral theory,Σ → ξ ( n f ) µ Σ ξ ( n f ) µ , (30)is broken. Thus, as was seen in Ref. [8], one can use the difference between neighboring plaquettes and the differencebetween neighboring links to determine if we are in a broken phase. However, those parameters are sensitive only tothe breaking of the single-site shift symmetry, so they cannot distinguish between the A , V , or T -phase. Thus, for acomplete understanding of the phase seen in the simulation, the mass spectrum should also be studied.The key difference between the unbroken phase and the various broken phases is that the squared-meson masses inthe broken phases have the generic form m M = A + B m . (31)Here A and B are independent of the quark mass but are dependent upon a . Unlike the unbroken phase, the squaredmeson masses are linear in m as opposed to m , and for some mesons B = 0 so that the mesons have a massindependent of the quark mass.Figure 3 is a reprint of Fig. 7 from Ref. [8], which shows the masses for the pseudoscalar taste pion as well as thetaste-45 pion for two lattice spacings. Fig. 3(a) shows one of the two broken phases seen as a function of the inputquark mass for 12 quarks (in our notation we would set n f = 3 , n t = 4). Of the two phase transitions discussed inthat paper, the chiral effective theory seems useful for understanding the second here (that appears at smaller latticespacing around β ≈ . A -phase.The π in Fig. 3(a) has an approximately constant mass as a function of the quark mass, which would be consistentwith the calculation of Ref. [5]: m π = − n f n t a δ ′ A − , (32) m m π π i5 π π i4 π π I π i π π ij FIG. 4: m π vs. the quark mass in the A -phase in arbitrary units. These units are such that when m ≥ m π and the dashed line to m π . (recall δ ′ A − < m π = m π + a ∆ A ¯ µ m , (33)where ¯ µ is defined below in Eq. (A1). Were this the V -phase, the π would still have a constant mass, but the π would have the behavior of Eq. (33) (with ∆ A → ∆ V ). Similarly, were this the T -phase, as can be seen in Appendix B,the π mass is dependent upon the quark mass while the π mass is constant.We can see that as m → m π → m π as rS χ PT predicts. This gives credence to the fact that the intermediatephase seen in Ref. [8] is in fact this A -phase, but a more detailed analysis would require several things. First, oneshould perform a fit to the forms above for the taste-45 and taste-5 pions. More importantly, one should measure of allof the different taste meson masses to see the pattern as predicted in Ref. [5] (and shown in Appendix A). Figure 3(b)shows the other side of the transition (larger β , and thus a smaller lattice spacing), and immediately shows a differentpattern: The four axial-taste pions are nearly degenerate and the difference m π µ − m π ≈ constant as a function of m . This is (roughly) the pattern seen in the physical regime of rS χ PT [3, 4].We show in Figs. 4 and 5 plots of m π vs. m and m η vs. m for the different tastes in the A -phase. The units inthese plots are arbitrary, chosen so that the values m > π and π masses, which are to be compared with the masses shown in the left-hand plotof Fig. 7 in Ref. [8].As for the phase at stronger coupling, it is unlikely that rS χ PT could explain this region. As we have seen in thiswork, rS χ PT shows that there should be at most four phases: the unbroken phase as well as the A , T , and V phases.However, these all are within the regime governed by the constraint in Eq. (1), most importantly that a Λ ≪ χ PT is infact describing the region as we expect it is.
IV. CONCLUSION
We have studied the complete phase diagram for staggered quarks with an arbitrary number of degenerate flavors,at least within the window given in Eq. (1). In this regime, there are only four phases: one unbroken (physical) phase,as well as three phases where the (approximate) O ( a ) accidental SO (4) symmetry is broken. Of these three phases,as seen previously [5], only one phase (the A -phase) is likely to be seen in simulations, but only if one looks at theorieswith 8 or 12 flavors of quarks [8]. The additional possible phase that was suggested to exist in Ref. [5] [when thesquared mass of the η ′ ij , Eq. (13), goes negative] does not correspond to a stable region. m m η η ′i5 η ′5 η ′i4 η ′45 η ′I η ′4 η ′i η ′ij FIG. 5: m η vs. the quark mass in the A -phase in arbitrary units. These units are such that when m ≥ m η = 0 corresponds to the three Goldstone bosons in this phase. While rS χ PT cannot fully explain both broken phases seen in Ref. [8], it can give a picture of broken phases that arelocated close to the unbroken phase (as a function of the lattice spacing). Studying the plaquette & link differencesas well as as the staggered meson mass spectrum would allow one to determine specifically which region of the phasediagram the simulation is in. For BSM studies this is essential as it is more likely to enter these unphysical regimesfor additional quark flavors.
Appendix A: A -Phase In this appendix we list the meson masses that appear in the A -phase [5]. Here we put them in terms of n f n t asabove, and we define µ ≡ − µa ∆ A + n f n t a δ ′ A − . (A1)The first masses we list are constant in the quark mass: m η ′ i = 0 , (A2) m η ′ = m π = − a ∆ A − n f n t a δ ′ A − , (A3) m π i = − n f n t a δ ′ A − , (A4) m π i = m η ′ i = m π + a ∆ T , (A5) m η ′ = m π + a ∆ V + n f n t a δ ′ V − , (A6) m π = m π + a ∆ V . (A7)We note that Eq. (26) in Ref. [5] [corresponding to our Eq. (A6)] has a typo, as the final term in that expressionshould be + a δ ′ V in that paper’s notation, not − a δ ′ V . With the above, we can determine the constants,∆ A , δ ′ A − , ∆ T , ∆ V , δ ′ V − . Then m π = m π + a ∆ A ¯ µ m (A8)0allows us to determine ¯ µ . With m π I = m π + ¯ µ m (cid:0) a ∆ I − a ∆ V (cid:1) + a ∆ V (A9) m η ′ I = n t m + m π − n f n t a δ ′ A + + a ∆ V + ¯ µ m (cid:18) n f n t a δ ′ A + + a ∆ I − a ∆ V (cid:19) (A10)allows us to determine ∆ I and δ ′ A + respectively, and finally m η ′ ij = ¯ µ m m π i − n f n t (cid:0) − ¯ µ m (cid:1) ( a δ ′ A − + a δ ′ V + ) (A11)for δ ′ V + .The following four masses are then determined from those above results, m π i = m π + ( m π − m π )¯ µ m + ( m η ′ i − m π )(1 − ¯ µ m ) , (A12) m π ij = m π i + ¯ µ m (cid:16) m η ′ i − m π i (cid:17) , (A13) m η ′ = m π (cid:0) − ¯ µ m (cid:1) , (A14) m η ′ i = m η ′ i − ( m η ′ i − m η ′ )¯ µ m . (A15)This shows that we have non-trivial relationships between the various masses. Additionally, as seen in Figs. 4 and 5we have several crossings of the meson masses for both the π and the η . While those figures are for a specific set ofparameters, they are indicative of the qualitative features of the A -phase. Appendix B: T -Phase In this appendix we list the masses for the mesons in the T -phase, where just as before, the octet masses are equal.In this case we define ¯ µ ≡ µ − a ∆ T . (B1) Note that if we take the m → ∞ limit seriously we would not examine the η ′ I mass. However, given that we are in an unphysicalphase, there is no reason to assume that this is the case, so we keep this mass in our theory. This expression would allow us to obtain δ ′ A + along with m at the same time as they have different dependencies on the quark mass. m π = m η ′ = − a ∆ T ¯ µ m (B2) m π I = ¯ µ m (cid:0) a ∆ I − a ∆ T (cid:1) (B3) m η ′ I = n t m + ¯ µ m (cid:0) a ∆ I − a ∆ T (cid:1) (B4) m π = m η ′ = − a ∆ T (1 − ¯ µ m ) (B5) m π = m η ′ = (cid:0) a ∆ I − a ∆ T (cid:1) (1 − ¯ µ m ) (B6) m η ′ = m η ′ = a ∆ V − a ∆ T + n f n t a δ ′ V − (B7) m η ′ = m η ′ = a ∆ V − a ∆ T − n f n t − ¯ µ m ) a δ ′ A + + n f n t µ m a δ ′ V − (B8) m η ′ = m η ′ = a ∆ A − a ∆ T + n f n t a δ ′ A − (B9) m η ′ = m η ′ = a ∆ A − a ∆ T − n f n t a δ ′ V + + n f n t µ m (cid:0) a δ ′ A − + a δ ′ V + (cid:1) (B10) m π µ = a ∆ A − a ∆ T (B11) m π µ = a ∆ V − a ∆ T (B12) m η ′ = m η ′ = m η ′ = m η ′ = 0 (B13) m π = m π = m π = m π = 0 (B14)With ∆ T “large and negative,” these are all positive or zero with the exception of those with the δ ′ A + or δ ′ V + terms.As in the A or V phase, if those parameters are such that m η ′ < m π = A + B m persists in this phase, but for one we see a different pattern than in the A or V phases. Additionally, there are no mixings between different taste mesons. Nevertheless, it is unlikely, giventhe empirical evidence, that one would be able to run a simulation in this phase, and as such we will not discuss thisphase further. ACKNOWLEDGMENTS
We would like to thank Claude Bernard, Anna Hasenfratz, and Steve Sharpe for useful discussions. Additionallywe would like to thank Claude Bernard for helpful comments on the manuscript. [1] S. Aoki, Phys. Rev.
D30 , 2653 (1984).[2] S. R. Sharpe and R. L. Singleton, Jr, Phys. Rev.
D58 , 074501 (1998), hep-lat/9804028.[3] W.-J. Lee and S. R. Sharpe, Phys. Rev.
D60 , 114503 (1999), hep-lat/9905023.[4] C. Aubin and C. Bernard, Phys. Rev.
D68 , 034014 (2003), hep-lat/0304014.[5] C. Aubin and Q.-h. Wang, Phys. Rev.
D70 , 114504 (2004), hep-lat/0410020.[6] C. Aubin, C. Bernard, C. E. DeTar, J. Osborn, S. Gottlieb, E. B. Gregory, D. Toussaint, U. M. Heller, J. E. Hetrick, andR. Sugar (MILC), Phys. Rev.
D70 , 114501 (2004), hep-lat/0407028.[7] A. Bazavov et al. (MILC), Rev. Mod. Phys. , 1349 (2010), 0903.3598.[8] A. Cheng, A. Hasenfratz, and D. Schaich, Phys. Rev. D85 , 094509 (2012), 1111.2317.[9] C. Bernard, Phys. Rev.
D73 , 114503 (2006), hep-lat/0603011.[10] C. Bernard, private communication.[11] A. Hasenfratz and F. Knechtli, Phys. Rev.
D64 , 034504 (2001), hep-lat/0103029.[12] A. Hasenfratz, R. Hoffmann, and S. Schaefer, JHEP05