Abstract
For an exact chiral symmetry which is spontaneously broken at zero temperature, we show that at nonzero temperature, generally pions travel at less than the speed of light. This effect can first appear at next to leading order in an expansion about low temperature. When the chiral symmetry is approximate, we obtain two formulas, like that of Gell-Mann, Oakes, and Renner, for the static and dynamic pion masses.
BNL Preprint BNL- PT-961, February, 1996. Pions are light because they are (almost) Goldstone bosons: in QCD, quarks have an (approximate) chiral symmetry of SU (2) × SU (2) r which is spontaneously broken to the usual isospin symmetry of SU (2) V by the dynamical generation of a quark condensate [1] [2] [3] . Notably, the pion mass squared is proportional to the up and down quark masses through the formula of Gell-Mann, Oakes, and Renner [4, 5] .
In this paper we consider how pions propagate in a thermal bath; our results apply as well to pions propagating in a Fermi sea of nucleons. In the limit of exact chiral symmetry, pions are true Goldstone modes and so massless. At zero temperature relativistic invariance then requires pions to travel at the speed of light. Our basic point is elementary: since the presence of a medium, either thermal or a Fermi sea, provides a privileged rest frame, relativistic invariance no longer applies, and so typically pions travel at less than the speed of light. We also derive how the formula of Gell-Mann, Oakes, and Renner generalizes to nonzero temperature [6] . Because the pion's velocity is less than c, in a medium the pion dispersion relation, as a function of momentum, is "flattened" from that at zero temperature. Such a flattening has been found in a wide variety of models [7] [8] [9] . It is often described as the results of detailed properties of these models, such as p-wave interactions and the like. Our results show that at least some of the flattening arises on very general grounds, as a consequence of chiral symmetry breaking in a medium.
While we speak of pions throughout, our conclusions apply to Goldstone bosons in any system which is relativistically invariant at zero temperature. Indeed, our results for the changes in the pion dispersion relation have exact analogies with spin waves in antiferromagnets [10, 11] . We think that our manner of derivation -in terms of the pion decay constants -is novel and illuminating. The detailed calculations which we perform to demonstrate this effect in the linear σ model extend and complement previous results by Itoyama and Mueller [12].
We begin with a heuristic derivation, in the limit of exact chiral symmetry. At zero temperature, the matrix element of the axial vector current, A µ a , sandwiched between the vacuum and a pion of momentum P µ = (p 0 , p) is:

Pions are light because they are (almost) Goldstone bosons: in QCD, quarks have an (approximate) chiral symmetry of SU (2) × SU (2) r which is spontaneously broken to the usual isospin symmetry of SU (2) V by the dynamical generation of a quark condensate [1] [2] [3] . Notably, the pion mass squared is proportional to the up and down quark masses through the formula of Gell-Mann, Oakes, and Renner [4, 5] .
In this paper we consider how pions propagate in a thermal bath; our results apply as well to pions propagating in a Fermi sea of nucleons. In the limit of exact chiral symmetry, pions are true Goldstone modes and so massless. At zero temperature relativistic invariance then requires pions to travel at the speed of light. Our basic point is elementary: since the presence of a medium, either thermal or a Fermi sea, provides a privileged rest frame, relativistic invariance no longer applies, and so typically pions travel at less than the speed of light. We also derive how the formula of Gell-Mann, Oakes, and Renner generalizes to nonzero temperature [6] . Because the pion's velocity is less than c, in a medium the pion dispersion relation, as a function of momentum, is "flattened" from that at zero temperature. Such a flattening has been found in a wide variety of models [7] [8] [9] . It is often described as the results of detailed properties of these models, such as p-wave interactions and the like. Our results show that at least some of the flattening arises on very general grounds, as a consequence of chiral symmetry breaking in a medium.
While we speak of pions throughout, our conclusions apply to Goldstone bosons in any system which is relativistically invariant at zero temperature. Indeed, our results for the changes in the pion dispersion relation have exact analogies with spin waves in antiferromagnets [10, 11] . We think that our manner of derivation -in terms of the pion decay constants -is novel and illuminating. The detailed calculations which we perform to demonstrate this effect in the linear σ model extend and complement previous results by Itoyama and Mueller [12] .
We begin with a heuristic derivation, in the limit of exact chiral symmetry. At zero temperature, the matrix element of the axial vector current, A µ a , sandwiched between the vacuum and a pion of momentum
with a and b isospin indices. The pion decay constant f π ∼ 93MeV ; whenever we write f π , we mean its value at zero temperature. At nonzero temperature, because of the presence of the medium, we expect that there are two distinct pion decay constants, one for the timelike component of the current, f
and one for the spatial, f
Both matrix elements are computed at a temperature T in the imaginary time formalism. Implicitly the timelike component of the momentum, p 0 , is analytically continued from euclidean values (pions, as bosonic fields, have p 0 = 2πnT for integral n) to Minkowski values, The possibility of two distinct pion decay constants is familiar from nonrelativistic systems, such as discussed by Leutwyler [11] ; in this context it was recognized previously by Kirchbach and Riska [13] and by Thorsson and Wirzba [6] .
By assumption the chiral symmetry is exact, and only broken spontaneously by the vacuum. Consequently, while the axial current acts nontrivially on the vacuum, it nevertheless is conserved on the pion mass shell. At zero temperature this is trivial: the divergence of the matrix element in (1) is 0|∂ µ A µ |π ∼ f π P 2 , which vanishes when P 2 = −ω 2 + p 2 = 0, as expected for a massless, relativistically invariant field.
At nonzero temperature, however, the condition that the axial current is conserved on the pion mass shell leads to interesting restrictions on the two pion decay constants. The divergence of the axial current in (2) vanishes when
At nonzero temperature each pion decay constant, f t π and f s π , has a real and an imaginary part. The pion mass shell then lies in the complex plane, at p 0 = −iω − γ. Equating the real parts of (3) gives
The requirement that pions travel at less than (or equal to) the speed of light, v ≤ 1, implies Re f s π ≤ Re f t π . To obtain this, we assume that the imaginary parts can be neglected relative to the real parts,
Physically, v ≤ 1 is most familiar: pions move through a medium as if it has an index of refraction greater than or equal to one. The imaginary part of the mass shell is given by
The requirement that pions are damped, and not anti damped, fixes γ to be semi positive definite; using (4), this then constrains the real and imaginary parts of f t π and f s π . Our analysis only applies to soft, "cool" pions. Soft implies that the momenta of the pions, ω and p, are small relative to the real parts of f t π and f s π . By "cool" we mean that the temperature is well below that for the restoration of chiral symmetry. If the chiral symmetry is restored at a temperature [14] . Hence as T → T χ , the region over which we can speak of soft pions vanishes. The physical basis for this is familiar: as T → T χ , a new mode, the σ meson, is becoming massless; if the chiral phase transition is of second order, then over large distances the physics is controlled by an O(4) critical point.
Assuming that the imaginary parts of f t π and f s π are nonzero at zero momentum, from (6) γ ∼ p as p → 0. This follows from the chiral symmetry, as a consequence of Goldstone's theorem [1] : at zero momentum, the complete inverse pion propagator must vanish at zero momentum, for both the real and imaginary parts. This doesn't preclude the imaginary parts of f t π and f s π from vanishing at zero momentum, in which case γ goes to zero faster than linearly. For example, in the linear σ model considered below, the imaginary part vanishes exponentially, (26). For spin waves in antiferromagnets, γ ∼ p 2 [10] . In any case, that γ vanishes at zero momentum implies that even with nonzero damping, we can rigorously extract the pion contribution -and so f t π and f s π -from the two point correlation function of axial vector currents.
To make our conclusions rigorous, and to extend them to an approximate chiral symmetry, we employ the chiral Ward identities for QCD. We follow the analysis of Shore and Veneziano [3] , who use a formalism due to Zumino [16] . We assume that there are two flavors of quarks, each with a (current) quark mass = m. A chiral Ward identity between the form factors, and the propagators, for the quark composite operator φ
where qq T is the quark condensate, and 0|T
the inverse propagator for the φ a 5 field, each at a temperature T . The Ward identities have the same structure at nonzero temperature as at zero temperature; the only change at T = 0 is that thermal expectation values of matrix elements, denoted by the subscript T , enter. Assume that φ 5 is directly proportional to the pion field,
The normalization constant "b" is a function of both temperature and momentum. The temperature dependence follows from our analysis. The momentum dependence is neglected about zero momentum, even when m π = 0. As emphasized by Shore and Veneziano [3] , this is the standard assumption which allows for the partial conservation of the axial current.
Using the normalization (8) and the definitions (2) in the Ward identity (7) given above, we obtain
For the pion propagator ∆ −1 π (P ) we take the following form:
Similar forms of the pion propagator have appeared previously [7] [8] [9] ; for us this form is motivated by the need to satisfy the chiral Ward identity in (9) . In the pion propagator, we insist that the coefficient of p 2 0 is unity; that is, that the field is canonically normalized. This is required by the analysis of ref. [3] . Secondly, we allow for a pion velocity which is less than one by introducing the velocity "v". Since the quark mass m = 0, we introduce a pion mass, m π . Lastly, we introduce an imaginary part of the pion self energy, ImΠ(P ), which is a function of momentum. This form of the propogator should be valid in an expansion about zero momentum.
The chiral Ward identity of (9) shows that the assumption used to derive (3) is correct: the divergence of the axial current vanishes when ∆ −1 π = 0, which defines the pion mass shell. Since the chiral Ward identity holds for arbitrary momentum, however, we can derive several identities by matching the coefficients of p 2 0 , p 2 , and 1, for both the real and imaginary parts.
Equating the terms ∼ p 2 0 fixes the constant of proportionality because the quark operator and the pion field to be
Since both terms on the right hand side of (11) change with temperature, so does the factor "b". Matching the terms ∼ p 2 fixes the velocity as in (4) . Lastly, matching the imaginary parts in (9) gives ImΠ(P ) = 2ωγ, with γ as in (6) .
Away from the chiral limit, we match the real parts at zero momentum, p 0 = p = 0, to obtain the generalization of the relation of Gell-Mann, Oakes, and Renner to nonzero temperature:
This is the same expression as Dashen [5] found at zero temperature, except that instead of f π , at nonzero temperature the real part of f t π enters. A relation like (12) was obtained by Thorsson and Wirzba [6] ; they did not recognize, however, that in general f t π has an imaginary part, and so wrote just f t π instead of Re f t π . We note that the pion mass in (12) is the dynamic pion mass. That is, it is the position of the singularity in the pion propagator in the complex p 0 plane at p = 0. One can alternately ask about the static pion mass, which is the position of the singularity in the pion propgator for p 0 = 0 in the complex p plane. From the form of the pion propagator, m static π = m π /v, and so by (4) and (12) 
Obviously, v ≤ 1 implies that m static π ≥ m π . We now consider where these effects first appear in an expansion about zero temperature. Using either a nonlinear [17] or a linear [18] σ model, to leading order in
Hence to leading order in low temperature, pions move at the speed of light and are undamped. This was established by Dey, Eletsky, and Ioffe [20] , who showed that to ∼ T 2 /f 2 π , the thermal average of the two point function of either vector or axial vector currents is directly proportional to a linear combination of those at zero temperature. Since these two point functions are lorentz covariant at zero temperature, they remain so to ∼ T 2 /f 2 π . The first place where the effects which we are discussing can enter is at next to leading order, ∼ T 4 . The pion self energy has been computed to this order using a nonlinear σ model [21, 22] . In particular, we believe that that v < 1 may be implicit in the results of Schenk [22] .
To illustrate the effect we instead follow Itoyama and Mueller [12] , and calculate at weak coupling to one loop order in a linear σ model. In euclidean space time the lagrangian is
where φ = (σ, π) is an O(4) isovector field. We introduce a background magnetic field h which is proportional to the current quark mass m. For h = 0, the vacuum expectation value of the σ is σ 0 = µ 2 /λ, where we then shift σ → σ 0 + σ; for two flavors, f π = σ 0 . We compute terms of order ∼ T 4 /(f [18] , so all we have to do is extend the calculation of f t π and f s π to this order [19] . Consequently, we merely sketch the simplest way of performing the calculations. For the real parts, the terms of interest arise from diagrams involving a virtual σ and a π in a loop, such as
where
, it suffices to approximate this integral by its value at zero momentum, neglecting the K dependence in the σ propagator, so (16) becomes
In the integral we have ignored apparent ultraviolet divergences to concentrate on the term ∼ T 2 . Of course renormalization is taken care of as usual at zero temperature.
To compute terms of ∼ T 4 , it is necessary to expand the integral in (16) to ∼ P 2 , including both terms ∼ P 2 and terms ∼ P µ P ν . Besides the integral in (17), we also need
where n µ = (1, 0). The imaginary part of expressions cannot be extracted so easily. We evaluate the imaginary part only near the pion mass shell, which is for ω ∼ p. In this region, the only contribution to the imaginary part of (17) is from
In this expression E 1 = k is the energy of the pion, E 2 = (p − k) 2 + m 2 σ is the energy of the σ, and n 1 = n(E 1 ), n 2 = n(E 2 ) are the corresponding Bose-Einstein distribution functions. This result can be obtained in various ways, such as following [23] . In all there are four possible δ-functions in energy which contribute to Im I(P ). For ω ∼ p T only that in (19) contributes, and corresponds to Landau damping. In this region, the δ-function requires
We assume that k m σ , and then expand the energies accordingly; this is justified, since from (20) , when m σ ω, p, then k m σ . The result for the imaginary part is
Because the fields being scattered have large momentum, the Bose-Einstein distribution functions are essentially Boltzman, which generates the exponential suppression seen in (21) . These integrals are sufficient to reproduce the results of Itoyama and Mueller [12] for the pion self energy. To evaluate the corresponding terms for the pion structure constants, we need the axial current in the linear σ model,
The diagrams which contribute at one loop order to f t π and f s π are given in fig. (5) of [18] . Besides the pion self energy, there is a contribution from a σ-π loop at the vertex for A µ a . These contributions can be evaluated expanding integrals like (16) and using (18) . For the imaginary parts, we need the integrals
and
The results of the computations are as follows. At one loop order the quantity
typically enters. To the order we work, we also need
Then at weak coupling in the linear σ model, to
By (4) the pion velocity is
while from (6) the pion mass shell is
Including the one loop self energy computed to this order, the pion propagator is
To ensure that ∆ −1 (P ) has canonical normalization we introduce a factor for wave function renormalization of the pion,
It is elementary to check that the zero of (30) agrees with (29). We have included the results to leading order in the external field h, when the pion mass is nonzero. Assuming that the quark condensate is proportional to the vacuum expectation value of the σ field,
we also verify our generalization of the formula of GellMann, Oakes, and Renner in (12) for the dynamic pion mass:
We conclude with some general comments. We admit that the effects we have computed at low temperatures in (25) -(33) are small. This does not mean, however, that they necessarily remain small for intermediate temperatures of physical interest [7] - [9] . Secondly, we remark that the form of the inverse propagator in (10) with v ≤ 1, and the corresponding difference between dynamic and static masses, applies not just to Goldstone bosons, but to any scalar field at nonzero temperature. For example, in numerical simulations on the lattice in euclidean spacetime, typically what is measured is only the static mass, not the dynamic.

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