# Two-flavor chiral perturbation theory at nonzero isospin: pion condensation at zero temperature

## Abstract

In this paper, we calculate the equation of state of two-flavor finite isospin chiral perturbation theory at next-to-leading order in the pion-condensed phase at zero temperature. We show that the transition from the vacuum phase to a Bose-condensed phase is of second order. While the tree-level result has been known for some time, surprisingly quantum effects have not yet been incorporated into the equation of state. We find that the corrections to the quantities we compute, namely the isospin density, pressure, and equation of state, increase with increasing isospin chemical potential. We compare our results to recent lattice simulations of 2 + 1 flavor QCD with physical quark masses. The agreement with the lattice results is generally good and improves somewhat as we go from leading order to next-to-leading order in \(\chi \)PT.

## 1 Introduction

Quantum chromodynamics (QCD), the fundamental theory of strong interactions, has a rich phase structure, particularly at finite baryon densities relevant for a number of physical systems including neutron stars, neutron matter and heavy-ion collisions among others [1, 2, 3]. However, finite baryon densities are not accessible directly through QCD since the physics is non-perturbative and lattice calculations are hindered by the fermion sign problem. Though it is worth noting that some progress has been made in circumventing the sign problem through the fermion bag and Lefschetz thimble approaches [4]. There is also the additional possibility of solving QCD at finite baryon density with quantum computers since the sign problem is absent in quantum algorithms [5].

While finite baryon density is inaccessible through lattice QCD, finite isospin systems in real QCD can be studied using lattice-based methods, see Refs. [6, 7] for some early results. The most thorough of these studies were performed only recently [8, 9, 10] even though finite isospin QCD was first studied over a decade ago using chiral perturbation theory (\(\chi \)PT) in a seminal paper by Son and Stephanov [11]. \(\chi \)PT [12, 13, 14, 15] is a low-energy effective field theory of QCD that describes the dynamics of the pseudo-Goldstone bosons that are the result of the spontaneous symmetry breaking of global symmetries in the QCD vacuum. Being based only on symmetries and degrees of freedom, the predictions of \(\chi \)PT are model independent.

It is agreed through both lattice QCD and chiral perturbation theory studies that at an isospin chemical potential equal to the physical pion mass there is a second-order phase transition at zero temperature from the vacuum phase to a pion-condensed phase. With increasing chemical potential there is a crossover transition to a BCS phase with a parity breaking order parameter, \(\langle {\bar{u}}\gamma _{5}d\rangle \ne 0\) or \(\langle {\bar{d}}\gamma _{5}u\rangle \ne 0\), that has the same quantum numbers as a charged pion condensate. Furthermore, for large temperatures of approximately 170 MeV, the pion condensate is destroyed due to thermal fluctuations. Various aspects of \(\chi \)PT at finite isospin density can be found in Refs. [11, 16]–[23]]. Finite isospin systems have also been studied in the context of QCD models including the non-renormalizable Nambu–Jona–Lasinio model [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], and the renormalizable quark-meson model [39, 40, 41, 42], with the results found there being largely in agreement with lattice QCD. A very recent review of meson condensation can be found in Ref. [43].

In addition to the study of pions at finite isospin chemical potential there has also been recent interest in the study of pions in the presence of external magnetic fields, which are relevant in the context of neutron stars with large fields (magnetars) and possibly in RHIC collisions, which generate magnetic fields due to accelerated charged beams of lead and gold nuclei. In neutron star cores, an isospin asymmetry is present since protons are converted into neutrons and neutrinos through electron capture. However, in the presence of a magnetic field, finite isospin systems are difficult to study due to the fermion sign problem on the lattice QCD that arises as a consequence of flavor asymmetry between up and anti-down quarks for electromagnetic interactions. The complex action problem is tackled by studying finite isospin densities for small magnetic fields, where the sign problem is mild. The lattice observes a diamagnetic phase [44], while studies in chiral perturbation theory valid for magnetic fields \(eB\ll (4\pi f_{\pi })^{2}\) suggests that pions behave as a type-II superconductor [45].

More recently, due to the accessibility of the equation of state (EoS) of pion degrees of freedom through lattice QCD, there has been a lot of interest in the possibility of pion stars [22, 46], a type of boson star that does not require the hypothesized axion, which was initially proposed as a solution to the strong CP problem in QCD. Pion stars, on the other hand, only require input from QCD and it is conjectured that pion condensation takes place in a gas of dense neutrinos [47]. Recent work shows that pion stars are typically much larger in size than neutron stars due to a softer equation of state and that the isospin chemical potentials at the center of such stars can be as high as 250 MeV for purely pionic stars and smaller for pion stars electromagnetically neutralized by leptons [46].

The goal of this paper is to revisit the equation of state for finite isospin QCD in the regime of validity of \(\chi \)PT, where we expect \(\mu _{I}\ll 4\pi f_{\pi }\). The equation of state (at tree level) was originally calculated in Ref. [11] of QCD. In this paper, we calculate the equation of state within \(\chi \)PT and incorporate leading order quantum corrections.

We begin in Sect. 2 with a brief overview of chiral perturbation theory and discuss how to parametrize the fluctuations around the ground state. We derive the Lagrangian that is needed for all next-to-leading order (NLO) calculations within \(\chi \)PT at finite isospin chemical potential allowing for a charged pion condensate. In Sect. 3, we use this NLO Lagrangian to calculate the renormalized one-loop free energy at finite \(\mu _I\). In Sect. 4, we calculate the isospin density, the pressure, and the equation of state in the pion-condensed phase. Our results are compared to those of recent lattice simulations. We summarize our findings in Sect. 5 and present some calculations’ details in Appendices A–E.

## 2 \(\chi \)PT Lagrangian at \({\mathscr {O}}(p^4)\)

For massless quarks, the global symmetries of QCD are \(SU(2)_L\times SU(2)_R\times U(1)_B\), which is reduced to \(SU(2)_V\times U(1)_B\) for nonzero quark masses in the isospin limit, i.e. for \(m_u=m_d\). If \(m_u\ne m_d\), this is further reduced to \(U(1)_{I_3}\times U(1)_B=U(1)_u\times U(1)_d\). Adding a quark chemical potential \(\mu _q\) for each quark, the symmetry is \(U(1)_{I_3}\times U(1)_B=U(1)_u\times U(1)_d\) irrespective of the quark mass. In the pion-condensed phase, the \(U(1)_{I_3}\) symmetry is broken. In the remainder of the paper, we work in the isospin limit.

*f*is the (bare) pion decay constant and

*m*is the (bare) pion mass. The relation between the physical pion mass \(m_{\pi }\) and

*m*, and between the physical pion decay constant \(f_{\pi }\) and

*f*are briefly discussed in Appendix B. The covariant derivatives at finite isospin are defined as follows

### 2.1 Ground state

### 2.2 Parametrizing fluctuations

^{1}We discuss this briefly in Appendix C.

*U*is an

*SU*(2) matrix that parameterizes the fluctuations around the ground state:

### 2.3 Leading-order Lagrangian

### 2.4 Next-to-leading order Lagrangian

^{2}

## 3 Next-to-leading order effective potential

The order-\(p^2\) contribution to the effective potential is given by minus the static part of the Lagrangian \({\mathscr {L}}_2\). The one-loop contribution which is of order \(p^4\) is given by a Gaussian path integral and is ultraviolet divergent. The ultraviolet divergences must be regularized and we choose dimensional regularization. Dimensional regularization sets power divergences to zero and logarithmic divergences show up as poles in \(\varepsilon \), where \(d=3-2\varepsilon \) is the number of spatial dimensions (see below). The divergences are cancelled by renormalizing the coupling constants appearing in the static part of the Lagrangian \({\mathscr {L}}_4\), which is also of order-\(p^4\).

### 3.1 Vacuum phase

### 3.2 Pion-condensed phase

## 4 Thermodynamics

*d*-wave scattering lengths, while the coupling constant \({\bar{l}}_3\) has been estimated using three-flavor QCD [13]. Finally, the coupling \({\bar{l}}_4\) is related to the scalar radius of the pion and has also been estimated to the value quoted above.

*m*and

*f*. The central values \(m_\mathrm{cen}\) and \(f_\mathrm{cen}\) are obtained by using the central values of \({\bar{l}}_i\), \(m_{\pi }\) and \(f_{\pi }\). The minimum and maximum values of

*m*and

*f*denoted by \(m_{\min },\ f_{\min }\) and \(m_{\max },\ f_{\max }\) respectively are obtained by combining the maximum and minimum values of the \({\bar{l}}_i\)s, \(f_{\pi }\), and \(m_{\pi }\). The values for the bare pion mass and decay constant are

*m*,

*f*and the low-energy constants, cf. Eqs. (B.11)–(B.12). This result holds to all orders in perturbation theory and is also in agreement with the lattice simulations of [8, 9, 10]. Moreover, if \(a_4(\mu _I^c)>0\), the transition is second order. The coefficient \(a_4(\mu _I)\) can be read off from Eq. (E.36). Evaluated at \(\mu ^c_I=m_{\pi }\), we find

*m*and

*f*. The NLO band is obtained by varying the parameters of

*m*and

*f*as given in Eqs. (66)–(68). We also show the lattice results for the pressure from Ref. [46]. The pressure increases steadily with the chemical potential. The NLO pressure increases faster than the LO pressure and is in good agreement with the lattice results.

The red curves shows the tree-level result and the blue curve shows the one-loop result using the central values of the parameters *m* and *f*. The band is obtained by varying the parameters *m* and *f* as given by Eqs. (66)–(68). We also show the lattice points from Ref. [46]. There is no pion condensate in the vacuum up to the critical isospin chemical potential \(\mu _I^c=m_{\pi }\). Hence \(n_I\) is independent of \(\mu _I\), which is an example of the Silver-Blaze property, namely that thermodynamic functions do not depend on \(\mu _I\) all the way up to its critical value [50]. For \(\mu _{I}\) larger than the critical isospin chemical potential \(\mu _I^c=m_{\pi }\), the density increases steadily. The isospin density as a function of \(\mu _I\) increases as one goes from LO to NLO, and the latter is in better agreement with the lattice results of Ref. [46].

*m*and

*f*. The blue band is obtained by varying the parameters of

*m*and

*f*as given by Eqs. (66)–(68). The black dashed line shows the lattice results from Ref. [46]. We notice that the NLO equation of state is stiffer than the LO one and that the difference increases steadily with the pressure \({\mathscr {P}}\). Moreover, the NLO EoS is in better agreement with the lattice results for small values of \({\mathscr {P}}/m_{\pi }^4\) than the LO EoS, while for larger values it is the other way around.

## 5 Summary

In conclusion, we have derived the \(\chi \)PT Lagrangian which is necessary for all NLO calculations at finite isospin. We have applied this Lagrangian calculating the pressure, isospin density, as well as the equation of state. Our predictions are in good agreement with the lattice results of Ref. [46] and improves as one goes from LO to NLO. This is the first test of \(\chi \)PT in the pion-condensed phase beyond leading order. The Lagrangian we have derived can be used to calculate e.g. the one-loop corrections to the quasiparticle masses in the pion-condensed phase. Here a nontrivial check would be to show that one of the branches is a massless Goldstone boson. The Lagrangian for three-flavor QCD can be derived in the same way and opens up the possibility to study quantum effects in phases that involve pion or kaon condensation. In the case of pion condensation, one can again compare with the lattice results of Ref. [46], as well as between those of the two and three-flavor calculations. This will give us an idea of the effects of the strange quark. Work in this direction is in progress [54].

## Footnotes

- 1.
Consider e.g. a theory with an

*SO*(3) symmetric Lagrangian with the ground state picking up a vev say in the*z*-direction. If the vev is rotated to the*y*-direction, then the (un)broken generators must be rotated accordingly. - 2.
There are additional operators with couplings \(l_5\)–\(l_7\) and \(h_2\)–\(h_3\) which are not relevant for the present calculation.

## Notes

### Acknowledgements

The authors would like to thank B. Brandt, G. Endrődi and S. Schmalzbauer for useful discussions as well as for providing the data points of Ref. [46]. The authors would also like to thank the Niels Bohr International Academy for hospitality during the later stages of this work. P. A. would like to acknowledge the Faculty Life Committee at St. Olaf College and the Nygaard Study in Norway Endowment for partial travel support.

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