Intriguing aspects of meson condensation

# Intriguing aspects of meson condensation

## Abstract

We analyze various aspects of pion and kaon condensation in the framework of chiral perturbation theory. Considering a system at vanishing temperature and varying the isospin chemical potential and the strange quark chemical potential we reproduce known results about the phase transition to the pion condensation phase and to the kaon condensation phase. However, we obtain mesonic mixings and masses in the condensed phases that are in disagreement with the results reported in previous works. Our findings are obtained both by a theory group analysis and by direct calculation by means of the same low-energy effective Lagrangian used in previous works. We also study the leptonic decay channels in the normal phase and in the pion condensed phase, finding that some of these channels have a peculiar nonmonotonic behavior as a function of the isospin chemical potential. Regarding the semileptonic decays, we find that that they are feeding processes for the stable charged pion state.

1

## I Introduction

The properties of strongly interacting matter in an isospin and/or strangeness rich medium are relevant in a wide range of phenomena including the astrophysics of compact stars and heavy-ion collisions. It is known that depending on the value of the isospin chemical potential, , and on the value of the strangeness chemical potential, , three different phase can be realized: the normal phase, the pion condensed () phase and the kaon condensed () phase Migdal:1990vm (); Son:2000xc (); Kogut:2001id (). The realization of a mesonic condensate can drastically change the low energy properties of matter, including the mass spectrum and the lifetime of mesons.

Previous analyses of the meson condensed phases by QCD-like theories were developed in Kogut:1999iv (); Kogut:2000ek (). Pion condensation in two-flavor quark matter was studied in Son:2000xc (); Son:2000by () and in three-flavor quark matter in Kogut:2001id (). In particular, the phase diagram as a function of and was presented in Kogut:2001id (). Finite temperature effects in chiral perturbation theory (PT) have been studied in Loewe:2002tw (); Loewe:2004mu (); He:2005nk (); Xia:2014bla (). One remarkable property of quark matter with nonvanishing isospin chemical potential is that it is characterized by a real measure, thus the lattice realization can be performed with standard numerical algorithms Alford:1998sd (); Kogut:2002zg (). The phase and the phase have been studied by NJL models in Toublan:2003tt (); Barducci:2004tt (); Barducci:2004nc (); Ebert:2005wr (); Ebert:2005cs () and by random matrix models in Klein:2004hv (). All these models find results in qualitative and quantitative agreement, and in particular, the phase diagram of matter has been firmly established. However, regarding the low energy mass spectrum in three-flavor quark matter, we found that it was only studied in Kogut:2001id (). Our results are in disagreement with those of Kogut:2001id (), the most relevant difference is in the mixing between mesonic states. Regarding the pion decay, previous works focused on density and temperature effects in standard decay channels Barducci:1990sv (); Dominguez:1993kr (); Loewe:2011tm (), but not all the decay channels have been considered.

In the present paper we analyze the c phase and the c phase in a realization of PT Gasser:1983yg (); Leutwyler:1993iq (); Ecker:1994gg (); Scherer:2002tk (); Scherer:2005ri () that includes only the pseudoscalar mesons. Therefore, the considered chiral Lagrangian approach is valid for MeV, MeV, and MeV. These bounds come from the masses of the proton, the rho meson and the omega baryon, respectively. Moreover, PT is valid in the energy range GeV, corresponding to the breaking scale of the theory. For definiteness we take the following values of the mesonic masses in vacuum: MeV, MeV and MeV. Unless explicitly stated, we will assume that in vacuum all the pion masses and all the kaon masses are equal. By this model we discuss the mixing and the masses of the pseudoscalar mesonic octet and the most relevant pion decay channels in the normal phase, in the c phase and in the c phase. Regarding the mesonic mixing, we discuss the disagreement with the results of Kogut:2001id () by theory group analysis and by explicit calculation using the PT Lagrangian. Regarding the decay channels, since the masses of the mesons strongly depend on and , by changing these chemical potentials some decay channels can become kinematically forbidden and/or other channels that are not allowed in vacuum can be opened.

As we shall formally see, the presence of a baryonic chemical potential is immaterial for the chiral Lagrangian, because mesons have no baryonic charge. However, it is clear that at large values of we expect a transition between hadronic matter and a different phase, presumably a color superconducting phase Rajagopal:2000wf (); Alford:2007xm (); Anglani:2013gfu (). In principle we should limit ourselves to considering , however since the effective Lagrangian is blind to the baryonic chemical potential, we can assume that such inequality is always satisfied. Although we will consider the range of values of MeV, it is worth emphasizing that at asymptotic the system can be studied by perturbative QCD and the ground state is a Fermi liquid with Cooper pairing of quarks Son:2000xc (); Son:2000by ().

One interesting topic that to the best of our knowledge has not been previously discussed in the pion and kaon condensed phases is the screening mass of the photon. By the Nishijima-Nakano-Gell–Mann (NNG) formula

 Q=T3+Y2, (1)

it is possible to relate the electric charge, , to the third component of isospin, , and hypercharge, . In particular, if the vacuum carries isospin and/or strangeness charges, then it will be a superconductor because the gauge group will be broken. Thus, by the Higgs-Anderson mechanism the photon will acquire a Meissner mass. We evaluate the tree-level screening masses finding that in the two meson condesed phases they have the same formal expression. Moreover, the Debye and Meissner masses are equal. In principle, any quark chemical potential breaks the Lorentz symmetry, therefore the Debye and Meissner masses of the photon can be different. However, we will show that the tree-level Lagrangian has to lead to equal Debye and Meissner masses.

The present paper is organized as follows. In Sec. II we briefly review the aspects of PT that are relevant for our work. In Sec. III we consider two-flavor quark matter. We discuss pion condensation driven by an isospin chemical potential reviewing known results and generalizing the study of the low-energy Lagrangian. In Sec. IV we consider three-flavor quark matter, determining the mixing angles and the masses of the pseudoscalar octet. In Sec. V we discuss the pion decay channels in the normal phase and in the phase. In Sec. VI we summarize our results. In the Appendix A we discuss some details about the - vertex factor relevant for pion decays.

## Ii General setting

In this section we briefly review the aspects of chiral symmetry that are relevant for meson condensation. The general Lorentz invariant Lagrangian density describing the pseudoscalar mesons can be written as

 L=F204Tr(DνΣDνΣ†)+F204Tr(XΣ†+ΣX†), (2)

where corresponds to the meson fields, describes scalar and pseudoscalar external fields and the covariant derivative is defined as

 DμΣ=∂μΣ−i2[vμ,Σ]−i2{aμ,Σ}, (3)

with and the external vectorial and axial currents, respectively. The Lagrangian has two free parameters and , related to the pion decay and to the quark-antiquark condensate, respectively, see for example Gasser:1983yg (); Leutwyler:1993iq (); Ecker:1994gg (); Scherer:2002tk (); Scherer:2005ri ().

The Lagrangian density is invariant under provided the meson field transform as

 Σ→RΣL†, (4)

and the chiral symmetry breaking corresponds to the spontaneous global symmetry breaking . The combination of the Nambu-Goldstone bosons (NGBs), with , corresponding to mass eigenstates can be identified with the pseudoscalar mesons fields. In standard PT, the mass eigenstates are charge eigenstates as well. Thus mesons are particles with a well defined mass and charge. The presence of a medium can change this picture. In particular, if the vacuum carries an electric charge, then the mass eigenstates will not typically be charge eigenstates. The presence of a medium can be taken into account by considering appropriate external currents in Eq. (2).

At vanishing temperature the vacuum is determined by maximizing the Lagrangian density with respect to the external currents. The pseudoscalar mesons are then described as oscillations around the vacuum. We use the same nonlinear representation of Kogut:2001id () corresponding to

 Σ=u¯Σuwithu=eiT⋅ϕ/2, (5)

where are the generators and is a generic matrix to be determined by maximizing the static Lagrangian. The reasoning behind the above expression is that under mesons can be identified as the fluctuations of the vacuum as in Eq. (4) with .

In the following we will assume that , , , where is the diagonal quark mass matrix and is a constant, that with these conventions is equal to . Moreover, we will assume that , meaning that the vectorial current consists of the electromagnetic field and a quark chemical potential, with a matrix in flavor space. We first study the case and then the case. Since the two-flavor case is simpler to treat mathematically, it will allow us to establish a number of results that are useful for the description of the three-flavor case.

## Iii Two-flavor case

In two-flavor quark matter the vacuum expectation value of the fields can be expressed as

 ¯Σ=eiα⋅σ=cosα+in⋅σsinα, (6)

where corresponds to the energetically favored direction in space. Assuming equal light quark masses, , the Lagrangian can be written as

 L=F204Tr(DνΣDνΣ†)+F20m2π2Tr(Σ+Σ†), (7)

where is the pion mass for vanishing isospin chemical potential. Expanding the covariant derivative we obtain

 L= F204Tr(∂νΣ∂νΣ†)+F20m2π2Tr(Σ+Σ†) −F2016Tr[vμ,Σ][vμ,Σ†]−iF204Tr ∂μΣ[vμ,Σ], (8)

and considering the quark chemical potential

 μ=diag(μu,μd)=μB3+μIσ32, (9)

we can write

 vν=−2eQAν−2μδν0=−~AνII−~Aν3σ3, (10)

with

 ~AνI=13(eA0+μB,eA), (11) ~Aν3=(eA0+μI,eA). (12)

Given that in Eq. (III) the interaction terms between and are proportional to commutators of these two fields, the only relevant term in is the one proportional to , and this is consistent with the fact that mesons have no baryonic charge. Note that both and explicitly break chiral symmetry giving mass to the (pseudo) NGBs. Equal light quark masses leave invariant ensuring that pions have equal masses. The isospin chemical potential induces a further symmetry breaking, such that with the effect of removing the pion mass degeneracy with a contribution proportional to the isospin charge. Since pions are an isotriplet, it follows that the contribution of the isospin chemical potential to the mass vanishes and the contributions to the is a Zeeman-like splitting, thus

 mπ0 =mπ, (13) mπ± =mπ∓μI; (14)

clearly the condensation of charged pions happens at . The only symmetry of the Lagrangian in Eq. (7) is ; when it is spontaneously broken it leads to a massless NGB, corresponding to one of the two charged pions depending on the sign of the isospin chemical potential.

At the microscopic level, the breaking pattern induced by the isospin chemical potential and the light quark masses is

 SU(2)L×SU(2)R×U(1)B⊃[U(1)Q]→U(1)L+R×U(1)B⊃[U(1)Q], (15)

where corresponds to the electromagnetic gauge symmetry. In the broken phase one of the two charged pions condense, spontaneously breaking the symmetry, meaning that the system becomes an electromagnetic superconductor. Formally, can be expressed as a combination of the generator of and of , thus the breaking of leads to a screening mass for the photon by the Higgs-Anderson mechanism.

Regarding the Lorentz symmetry, the isospin chemical potential explicitly breaks boost symmetry, however by expressing the isospin chemical potential as the expectation value of the field we can formally consider a Lorentz invariant Lagrangian. To formally preserve Lorentz symmetry we will as well employ the Lorenz gauge .

### iii.1 Ground state

For vanishing mesonic fluctuations the Lagrangian is a functional of and ; upon substituting Eq. (6) in Eq. (7) we obtain

 L0(α,n3,~Aμ)=F20m2πcosα+F202sin2α~Aμ3~A3μ(1−n23), (16)

which is a function of the parameters and and a functional of . For vanishing external electromagnetic field and for , the global maximum is at and is independent of , meaning that the ground state has an global symmetry. In this case the custodial is still present and only the curvature of the potential (the pion masses) are affected by the isospin chemical potential. In other words, the isospin chemical potential is not sufficient to tilt the vacuum in one direction, thus the vacuum is the same obtained with .

The stationary point of corresponds to and , which is a global maximum for . In this case the vacuum is tilted by an angle and the ground state has only a residual symmetry (isomorphic to ) for rotations and ; the angle cannot be determined maximizing the ground state Lagrangian and is signaling the existence of a massless NGB.

The ground state Lagrangian can be easily determined and is given by

 ¯L0=⎧⎪⎨⎪⎩F20m2πfor μImπ. (17)

Regarding the screening masses of the electromagnetic field, they can be inferred from Eq. (16). The electromagnetic field has both a Debye mass and a Meissner mass, which are equal and given by

 M2D=M2M=F20e2(sinα)2. (18)

The screening masses vanish in the unbroken phase and are equal to in the broken phase, signaling the breaking of . In principle, the Debye and Meissner masses could be different, because the Lorentz symmetries is explicitly broken by . However, from the fact that the isospin chemical potential can be introduced as in Eq. (12) it is clear that both tree-level screening masses must be equal.

#### Generic chemical potential

To properly understand the previous results regarding the ground state configuration we consider a more general setting with

 μ=12μ⋅σ, (19)

corresponding to a quark chemical potential pointing to an arbitrary direction in isospin space. The ground state Lagrangian is obtained maximizing

 L0=F202(sinα)2(|μ|2−|μ⋅n|2)+F20m2πcosα, (20)

as a function of and . It is clear that , thus is in the plane perpendicular to . This leads to the residual symmetry for rotations around . For the ground state is tilted by an angle . The ground state Lagrangian is the same reported in Eq. (17), but with .

The leading order Lagrangian describing the in medium pions can be obtained expanding Eq. (III) at the second order in the fields. For definiteness we consider and in the vev in Eq. (6). We decompose the Lagrangian at the second order in the fields as follows

 Leff=LK+LM+LL, (21)

where is the kinetic term, is the mass term and is the term linear in the derivatives.

The kinetic part of the Lagrangian can be written as

 LK=F202(δab(cosα)2+nanb(sinα)2)∂νϕa∂νϕb=F202∂νϕaKab∂νϕb, (22)

that manifestly shows meson mixing. Since is a symmetric matrix, it can be diagonalized. By the transformation

 ϕ1 =1F0(n1~ϕ1−n2~ϕ2cosα), (23) ϕ2 =1F0(n1~ϕ2cosα+n2~ϕ1), (24) ϕ3 =~ϕ3F0cosα, (25)

we obtain the canonical kinetic term

 LK=12∂ν~ϕa∂ν~ϕa. (26)

One of the peculiar aspects of the field redefinition above is that in the phase for the terms proportional to diverge. In other words, for vanishing light quark masses the above field renormalization does not seem to work. The correct prescription for handling this issue seems to be to consider the limit only in the physical results.

Regarding the electric charge eigenstates, we find that

 π∓=e±iθF0√2(~ϕ1±i~ϕ2cosα), (27)

where . Note that the standard definition of the charge eigenstates is obtained for , as expected.

For the mass term we find

 LM= −m2π2F20cosαϕaϕa +F202~Aμe~A3μ[cos2α(ϕ21+ϕ22)−sin2α(n⋅ϕ)2], (28)

that in the rotated basis turns out to be

 LM= −m2π2cosα⎛⎝~ϕ21+~ϕ22+~ϕ23cos2α⎞⎠ (29) +~Aμ3~A3μ2[~ϕ21(cos2α−sin2α)+~ϕ22].

The term with a linear dependence on the derivative is given by

 LL=iF202Tr~Aμ3[Σ†,∂μΣ]=−2F20~Aμ3ϕ1∂μϕ2cos2α, (30)

that by the rotated basis redefinition turns in

 LL=−2~Aμ3~ϕ1∂μ~ϕ2cosα. (31)

An interesting aspect is that in the phase this is the only mixing term between the fields. Since it scales as , it vanishes for .

Note that no term of the quadratic Lagrangian depends on , thus the symmetry has been absorbed in the redefinition of the fields. Assuming that no electromagnetic field is present, we can replace in all the Lagrangian terms , obtaining in momentum space

 Leff =(~ϕ1~ϕ2~ϕ3)⎛⎜ ⎜⎝k2−m2πcosα+μ2Icos(2α)−2ik0μIcosα02ik0μIcosαk2−m2π/cosα+μ2I000k2−m2π/cosα⎞⎟ ⎟⎠⎛⎜ ⎜⎝~ϕ1~ϕ2~ϕ3⎞⎟ ⎟⎠ =~ΦS−1~Φt, (32)

where and is the inverse propagator. The energy spectrum is obtained from the poles of the propagator and in the phase we find

 Eπ0 =√p2+μ2I, (33) E~π+ =1√2μI√3m4π+μ4I+2p2μ2I−√(3m4π+μ4I)2+16m4πμ2Ip2=p ⎷μ4I−m4π3m4π+μ4I+O(p2), (34) E~π− =1√2μI√3m4π+μ4I+2p2μ2I+√(3m4π+μ4I)2+16m4πμ2Ip2=√3m4π+μ4IμI+μIp22√7m4π+μ4I(3m4π+μ4I)3/2+O(p4), (35)

where are the two mass eigenstates (note that in this case the subscript does not indicate the electric charge). The dispersion law of the massless mode is linear in momentum with a velocity that tends to the speed of light for and that vanishes for . The charge eigenstates can be expressed as a linear combination of the fields as follows:

 π∓=ie±iθ(a+−a−)√2[√1+a2−(1±a+cosα)~π−−√1+a2+(1±a−cosα)~π+], (36)

where

 a±=μ4I−m4π±√(μ4I−m4π)2+16k20m4πμ2I4k0m2πμI. (37)

Given the particular expression of the coefficients, the propagating particles oscillate between the two electric charge eigenstates with a mixing angle depending on the energy. This is a rather peculiar behavior because in the Standard Model one typically has mixing angles that are not energy/momentum dependent. Note that this oscillation also means that the propagating particles oscillate between isospin eigenstates. This is possible because in the condensed phase the vacuum carries isospin charge which is related to the electric charge by the NNG formula (1), thus nor the electric charge nor the isospin charge are conserved.

Eq. (36) can be inverted to obtain

 (~π+~π−)=(U11U12U21U22)(π+π−), (38)

and defining

 s12=2m2πμIM22=−m4π−μ4Iμ2I, (39)

we find that

 U=⎛⎜ ⎜ ⎜ ⎜⎝1n−1n−M22−√M42+4k20s122ik0s121n+1n+M22+√M42+4k20s122ik0s12⎞⎟ ⎟ ⎟ ⎟⎠, (40)

with

 n±=8k20s212+2M42∓2M22√M42+4k20s2124k20s212. (41)

This result is important for the determination of the width of the pion decays discussed in Sec. V.

The mixing between the charged pion states can be simply understood in two-flavor quark matter. The states are the only states having a non vanishing value of the third component of isospin, and since the vacuum has a nonvanishing , these states can mix. In the three-flavor case things become a little more involved.

## Iv Three-flavor case

In three-flavor quark matter besides the isospin chemical potential one has to consider the strange quark chemical potential. Microscopically, strange quark states can be occupied by electroweak processes if the light quark chemical potential exceeds the strange quark mass. The formal expression of the in medium effective chiral Lagrangian is given by Eq. (7) in which the mesonic octet is introduced by replacing

 u=eiϕaλa/2, (42)

in Eq. (5), where are the Gell-Mann matrices.

The isospin and strange quark chemical potential can be introduced by considering

 μ=diag(μu,μd,μs)=diag(13μB+12μI,13μB−12μI,13μB−μS)=μB−μS3I+μI2λ3+μS√3λ8, (43)

where is the so-called strange quark chemical potential. Note that the actual strange quark chemical potential is , however the diagonal contribution of the baryonic chemical potential is immaterial for mesons.

For three-flavor quark matter the spontaneous symmetry breaking pattern is the following

 SU(3)L×SU(3)R⊃[U(1)Q]→SU(3)V⊃[U(1)Q], (44)

and the corresponding NGBs are identified with the mesonic pseudoscalar octet. The quark masses explicitly break the chiral symmetry, giving mass to the pseudo NGBs. A similar effect is produced by the isospin chemical potential and the strange quark chemical potential, with the additional fact of breaking Lorentz symmetry. The symmetry of the Lagrangian is thus reduced to

 U(1)L+R×U(1)L+R⊃[U(1)Q]. (45)

The breaking of this symmetry leads, at most, to the appearance of one NGB and of the screening masses for the electromagnetic field.

For the external vector current we can write

 vν=−2eQAν−2μδν0=−23(μB−μS)Iδν0−~Aν3λ3−~Aν8λ8, (46)

where

 ~Aμ3 =(eA0+μI,eA), (47) ~Aμ8 =(eA0+2μS,eA), (48)

are the relevant components of the electromagnetic field.

### iv.1 Ground state

In the three-flavor case the most general vev, , depends on parameters, corresponding to the possible orientations in space. However, in the two-flavor case we have found that rotations around the direction of the chemical potential leave the vacuum invariant. We assume that the same is true in the three-flavor case and therefore the vacuum Lagrangian only depends on two angles, and , corresponding to the angles between the vacuum and third component of isospin, and between the isospin and the hypercharge, respectively. This is exactly the same assumption used in Kogut:2001id (), in which it was found that there are three different vacua:

• Normal phase:

 μI

characterized by

 αN=0,βN∈(0,π),¯ΣN=diag(1,1,1). (51)
• Pion condensation phase:

 μI >mπ, (52) μS <−m2π+√(m2π−μ2I)2+4m2Kμ2I2μI, (53)

characterized by

 cosαπ=(mπμI)2,βπ=0, (54)
 ¯Σπ =⎛⎜⎝cosαπsinαπ0−sinαπcosαπ0001⎞⎟⎠ (55) =1+2cosαπ3I+iλ2sinαπ+cosαπ−1√3λ8.
• Kaon condensation phase:

 μS >mK−12μI, (56) μS >−m2π+√(m2π−μ2I)2+4m2Kμ2I2μI, (57)

characterized by

 cosαK=⎛⎝mK12μI+μS⎞⎠2,βK=π/2, (58)
 ¯ΣK= ⎛⎜⎝cosα0sinα010−sinα0cosα⎞⎟⎠= (59) 1+2cosαK3I+cosαK−12√3(√3λ3−λ8)+iλ5sinαK.

Note that the kaon condensation can only happen for

 μS>¯μS=mK−mπ2, (60)

and MeV for our parameter choice.

#### Screening masses

As in the two-flavor case we can determine the screening masses of the electromagnetic field. Remarkably, we find that the screening masses are independent of the angle and have the same expression obtained in the two-flavor case

 M2D=M2M=F20e2(sinα)2. (61)

The nonvanishing value of the Meissner mass implies that in both mesonic condensed phase the system is an electromagnetic superconductor. Note that across the first order phase transition between the two condensed phases the screening masses are discontinuous, because is discontinuous.

### iv.2 Mixing

In the presence of background isospin-rich matter or strangeness-rich matter, the Hamiltonian carries the third component of the isospin charge and of the strangeness charge (or hypercharge). The corresponding charges are explicitly broken, meaning that states with different third component of isospin and different hypercharge can mix. Indeed, the effect of a nonvanishing and is not only to produce a Zeeman-like splitting of the masses but also to tilt the vacuum in a certain direction in the isospin space corresponding to one of the nondiagonal generators of . Let us discuss this issue more in detail. Given that the Hamiltonian has terms proportional to and , the symmetry is explicitly broken. However, for labeling the mesonic states we can use -spin, -spin and -spin quantum numbers (actually only two of them are independent, indeed is one of the Casimir operators), because , and commute with and . In Fig. 1 we report the weight diagram of the pseudoscalar mesonic octet. In the top panel the axes correspond to , , and and the values of the -spin, -spin and -spin multiplets are reported. In the lower panel, mesonic states that can mix are marked with a different symbol. These diagrams are valid both in the phase and in the phase. For example, from the top panel we see that charged pions can mix because they both have , however charged pions cannot mix with kaons, because they all have . In Table 1 we report the and quantum numbers of the mesons with well defined -spin and -spin. In turn, the only allowed mixings are the followings: , , , corresponding to , , mixing. Regarding the and the , they have no well-defined -spin or -spin, thus a different reasoning must be used for understanding whether they mix or not. We will see that their mixing will depend on the particular spontaneously induced charge of the vacuum.

One of the important aspects is that the third component of isospin and the hypercharge form the Cartan subalgebra of , thus the associated charges cannot directy induce mixing between different states. In other words, the and charges can induce Zeeman-like mass splittings, but whether mixing between states will happen or not depends on the spontaneously induced charge of the vacuum. Note that the operator associated to this induced charge can be described in terms of lowering and raising operator of one of the subgroups of .

Let us first consider the normal phase. In the normal phase there is no operator that can induce the mixing of the mesonic states, thus the mesonic states remain unchanged but the and charges will induce Zeeman-like mass splittings.

In any of the condensed phases, there is an additional charge that is spontaneously induced, and the corresponding operator will lead to mixing.

Let us first focus on isospin (or -spin). We have to consider two cases. Suppose that the vacuum has a charge that commutes with , as in the phase, say the charge corresponding to , see Eq. (IV.1). The operators can induce mixing among the charged pions and among the kaons. On the other hand, -spin conservation does not allow the to mix with the .

Suppose that the vacuum has a charge that does not commute with as in the phase, see Eq. (59). Any operator that does not commute with isospin will commute with -spin or with -spin. In the phase , then the vacuum is not invariant under this charge. However, since it follows that -spin is conserved. The lowering and raising operator inducing the mixings will be . Regarding the and the , in this case we have that and do not mix. Since and , these will be the mass eigenstates.

In Kogut:2001id () it was found a different mixing in both condensed phases, with just two blocks and , meaning that states with different -spin, -spin and -spin mix. As we will see in the next section, using the tree-level Lagrangian we find agreement with the mixing reported in Table 1.

### iv.3 Mesonic mass spectrum

For vanishing chemical potentials and for equal light quark masses, the tree-level values of the mesonic octet masses in PT are known to be given by

 m2π =2Gm/F20, (62) m2K =G(m+ms)/F20, (63) m2η0 =2G(m+2ms)/3F20=(4m2K−m2π)/3, (64)

where is the strange quark mass. In the normal phase the effect of the isospin and strange quark chemical potential is a Zeeman-like mass splitting by contribution proportional to the isospin charge and strangeness,

 mπ0 =mπ, (65) mπ± =mπ∓μI, (66) mη0 =√(4m2K−m2π)/3, (67) mK± =mK∓12μI∓μS, (68) mK0/¯K0 =mK±12μI∓μS. (69)

Below we will discuss the masses of the scalar mesons in the condensed phases. To obtain the eigenstates we follow the same procedure used in the two-flavor case, thus we first expand in order to obtain the quadratic terms in the fields then, if necessary, we rescale and rotate them to have canonical kinetic terms. We will denote the mass eigenstates with a tilde.

In Fig. 2 we report the obtained results for the pseudoscalar mesonic octet masses as a function of for three different values of the strange quark chemical potential. In the top panel we take MeV that is smaller than MeV, in the middle panel we take MeV and in the bottom panel MeV, which is the largest possible value of the strange quark chemical potential that can be considered in the present realization of PT. The solid vertical lines correspond to the second order phase transitions between the normal phase and a condensed phase. The dashed vertical lines correspond to first order phase transitions between the phase and the phase. The top panel and the middle panel should be compared with the corresponding results reported in Kogut:2001id () in Fig. 4 and in Fig. 5, respectively. In the normal phase our results agree with those of Kogut:2001id (), but in both condensed phases we disagree with the results reported in Kogut:2001id (). As already discussed, this is due to the fact that we find a different mixing pattern, even if we use the same model of PT of Kogut:2001id (). For this reason we discuss our results in detail.

#### Pion condensation phase

We find mixing within the following pairs of states: and , while the and fields do not mix. Thus, in agreement with the discussion in Sec. IV.2, we do not find mixing between the and the . The reason is that in the phase -spin is conserved and therefore the state and the state cannot mix.

Since the Lagrangian can be organized in a block diagonal form, we can treat separately the various sectors. By the field rescaling

 ϕ1,3 →~ϕ1,3=ϕ1,3cosαπ, (70) ϕ4,5,6,7 →~ϕ4,5,6,7=ϕ4,5,6,7cos(απ2), (71)

we obtain canonical kinetic terms. As in Sec. III.2 it is useful to turn to momentum space, so that one can absorb the terms linear in energy in the propagator. In this way we obtain the canonical Lagrangian

 L=~Φtdiag(S−112,S−145,S−167,S−13,S−18)~Φ, (72)

where