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Sidlerstr. 5, CH-3012 Berne, Switzerland

arXiv:hep-ph/9306275v1 15 Jun 19931Thermal PionsBUTP-93/16Ulf-G. Meißnera∗aInstitute for Theoretical Physics, University of Berne,Sidlerstr. 5, CH-3012 Berne, SwitzerlandI discuss the absorption and dispersion of pions in hot matter.

A two-loop calculation inthe framework of chiral perturbation theory is presented and its result is compactly writtenin terms of the two- and three-particle forward ππ scattering amplitudes.At modesttemperatures, T ≤100 MeV, the change in the pion mass is small and its dispersion lawclosely resembles the free space one. At these temperatures, all quantities of interest aregiven to a good degree of accuracy by the first term in the virial expansion which is linearin the density.1.

INTRODUCTIONThe properties of pions in hot hadronic matter are encoded in the pion propagator, inparticular in the mass shift due to the influence of the heat bath,(p0)2 = ⃗p 2 + M2π + Π(p0, ⃗p )(1)where Mπ denotes the pion mass, p0 its energy, ⃗p its three-momentum and Π its self-energy, which is in general a complex quantity. Its real part is related to the dispersionand the group velocity while its imaginary part encodes the information about the pionabsorption.

Chiral perturbation theory (CHPT) allows to systematically calculate thebehaviour of Π at low and modest temperatures as will be spelled out below.Thiscalculation is done under the following assumptions. I assume a hadron gas in a state ofthermal equilibrium which mostly consists of pions.

Therefore, T should not be largerthan approximately 100 MeV because at higher temperatures the massive states start todominate [1]. In particular, there should not be any sizeable baryon contamination in thegas.

Therefore, the results which will be presented below are mostly relevant for futureexperiments and facilities which will have more pions and reach lower temperatures thanthe present ones. It is instructive to briefly recall the scattering of light on a dilute gasof N molecules in a volume V .

The index of refraction will in general be complex and toleading order in the density is given by [2]:n + iκ ≃1 + 2πNk2V f(2)∗Heisenberg fellow.Address after Sept.1, 1993: CRN, Physique Th´eorique, B.P. 20 CR, F-67037Strasbourg Cedex 2, France.Work supported in parts by Deutsche Forschungsgemeinschaft andSchweizerischer Nationalfonds.

Plenary talk, ”Quark Matter 93”, Borlaenge, Sweden, June 1993.

2with f the forward scattering amplitude of photons on a single molecule.Makinguse of the optical theorem, it follows that κ is directly proportional to the total forwardscattering cross section. In complete analogy, we will see that the absorptive and dispersiveproperties of the pions are related to the forward ππ and πππ scattering amplitudes (thelatter is related to the effects of second order in the density).2.

EFFECTIVE FIELD THEORY OF QCD AT FINITE TEMPERATUREAt low energies, the QCD Green functions are dominated by the (almost) masslessGoldstone bosons related to the spontaneous breakdown of the chiral symmetry, SU(3)L×SU(3)R →SU(3)V . Furthermore, to a good first approximation the quark masses can beset to zero (which defines the chiral limit of QCD).

The consequences of these featurescan be worked out in a systematic fashion by making use of an effective field theory,LQCD = Leff[U, ∂µU, M](3)where U = exp[iπ/Fπ] embodies the Goldstone (pion) fields and M is the quark massmatrix. A fundamental scale of the strong interactions at low energies is set by the piondecay constant Fπ, which is defined via < 0|Aiµ|πk >= iδikpµFπ, Fπ ≃93 MeV, with Aiµ theaxial current.

One can systematically expand all Green functions around the chiral limit.This is a simultaneous expansion in small momemta, small quark (pion) masses and smalltemperatures, the corresponding small parameters being p/Λ, M/Λ2 and T/Λ. Here,Λ ≃Mρ is the scale where the non-Goldstone excitations become important.

As shownby Weinberg [3], in the effective Lagrangian framework this amounts to an expansion inpion loops. The effective Lagrangian consists of a string of terms with increasing numberof derivatives and/or quark mass insertions,Leff= L(2) + L(4) + .

. .

(4)To lowest order, one calculates tree diagrams using L(2). This leads to the venerablecurrent algebra results.

At next-to-leading order, one has to calculate one-loop diagramsusing L(2) to perturbatively restore unitarity and tree diagrams with exactly one insertionfrom L(4). The coefficients of the latter terms are not restricted by symmetry but haveto be determined from phenomenology [4].At finite temperatures, only a few modifications occur.

First, if one works in a real-timeapproach, one has to choose a proper path for the time integration in the action. Thispath is called C and extends from −∞to +∞above the real axis, returns below thereal axis and goes down to −∞−iβ parallel to the imaginary axis [5].

Here, β = 1/T.Second, the pion propagator is modified according to well-known rules and finally, theconventional time-ordering at T = 0 is substituted by path-ordering along C for T ̸= 0,e.g.< 0|T(AµAν)|0 > →< 0|TC(AµAν)|0 >(5)More details on CHPT and its application to finite temperature physics can be founde.g. in the review [6] or in Gerber and Leutwyler [1].

33. TWO-LOOP CALCULATION OF THE PION SELF-ENERGY3.1.

One-loop calculation and virial expansionIt is instructive to recall how one calculates the pion properties in CHPT at T = 0.One particularly useful choice of the interpolating pion field is the axial current due tothe PCAC relation. The correlator of two axial currents reads [4]iZdx eipx < 0|T(Aiµ(x)Akν(0))|0 >= δik( pµpνF 2πM2π −p2 + gµνF 2π + .

. .

)(6)This can straightforwardly be extended to the finite temperature case. Another methodis the virial expansion to which I will come back later.

The calculation of the pion self-energy to one loop order has been done by Goity and Leutwyler [7], Shuryak [8] andSchenk [9]. One splits the pion self-energy as Π = Π0 + ΠT , where ΠT vanishes at T = 0and describes the modification due to the heat bath.

Instead of Re Π and Im Π, oneconventionally writesp0 = ω(p) −i2γ(p)(7)with ω(p) the frequency of the pionic waves and Mπ(T) = ω(p = 0) the effective mass.The damping coefficient γ(p) is the inverse time within which the intensity of the wave isattenuated by a factor 1/e. In the one-loop approximation, one simply findsω(p) =q⃗p 2 + M2π(T),γ(p) = 0(8)This is a quite common phenomenon in CHPT calculations - to get to the imaginarypart of a certain quantity with the same accuracy as the real part, one has to work harder.It is simply related to the fact that to lowest order one calculates tree diagrams, whichare real.

The temperature-dependent pion mass Mπ(T) has e.g. been given by Gasserand Leutwyler [10] in the framework of CHPT.

Notice that there is no absorption to oneloop, the only effect of the interaction with the heat bath is that the pion mass and decayconstant become T-dependent. There is another way of presenting the one-loop result.If one inspects the corresponding Feynman diagrams, one sees that one effectively dealswith the ππ scattering amplitude in forward direction, Tππ(s) with √s the cm energy ofthe collision.

To this order, the pion self-energy is linear in the density and one can writeΠT (ωp, ⃗p ) = −Zd3q(2π)32ωqnB(ωq) Tππ(s)(9)with ωk =q⃗k 2 + M2π and n−1B = (exp[βω] −1) is the canonical Boltzmann factor. Thisis also the first term in the virial expansion.

For a general proof, see [11]. This lastformula is very powerful.

If one uses empirical input for Tππ(s), this form allows one togo beyond the strict one loop CHPT result. In particular, the important effect of the ρresonance can thus be implemented fully, whereas in the one-loop representation of Tππ(s)one only sees the tail of the ρ (for more details see e.g.

[9]). One result of employing thisprocedure is that the pion mass depends only very weakly on the temperature [9].

43.2. Two-loop calculationSchenk [12] has extended the above to two loops.

This calculation is much more tedious- there are not only more diagrams, but one also has to account for three-particle scatteringas inspection of some Feynman diagrams reveals. In particular, one finds thatΠT = Π(1) + Π(2)(10)which means that one has contributions which are linear and quadratic in the Boltzmannfactor (density).

The diagrams which contribute to ΠT involve insertions from L(2)effandL(4)eff, the contribution from L(6)effis temperature-independent. Therefore, the final resultfor Π(1) is entirely fixed in terms of Mπ, Fπ and four low-energy constants determinedalready in [4].

The term of second order in the density involves an integration over threepion momenta and a function which is of fourth order in Mπ and the pion momenta.The result is correct up to and including order p6 in CHPT. The explicit formulae aregiven in [12].

Instead of presenting the rather lengthy expressions, let me rather discussa more compact form involving S-matrix elements of two- and three-pion scattering. Oneexpects that at two-loop order the thermal distribution will depend on the effective mass,i.e.

on nB(ωTp ) with (ωTp )2 = ⃗p 2 + M2π(T). In addition, there are the diagrams related tothree-particle scattering.

Now the forward limit of the 3 →3 scattering amplitude doesnot exist due to diagrams where one pion is exchanged between clusters and is close toits mass shell. One thus defines a proper amplitude ˆT33 [12]ˆT33 = T33 +Xq′ T221q2 −M2πT22(11)which has a well-defined forward limit (T22 is the two-particle scattering amplitude).For a more detailed discussion see [12].

Consequently, one can express the result of thetwo-loop calculation asΠT(ωp, ⃗p ) = −Zd3q(2π)32ωTqnB(ωTq ) Tππ(s)−12Zd3q(2π)32ωqd3k(2π)32ωknB(ωq) nB(ωk) ˆT Rπππ(p, q, k)(12)with ˆT Rπππ the proper retarded 3 →3 forward scattering amplitude. Notice again thatin the first term on the r.h.s.

of ΠT the effective pion mass appears, i.e. the thermaldistribution of the pions actually depends not longer on the bare mass as in the one-loopcase.

The second term is the new one, it is believed to be the most general term quadraticin the density (which is, however, not strictly proven). Notice that one could also useT-dependent energies in the n2B term, but this goes beyond the accuracy of the two loopcalculation.

Let me stress again that on can not further simplify this result and expresse.g. in terms of a T-dependent two-particle scattering amplitude only.

For details, see[12]. The damping rate γ(p) follows using unitarity form eq.(12).

If one neglects Bosecorrelations in the initial and final states, it can compactly be written asγ(p) = ω−1pZd3q(2π)32ωqnB(ωq)qs(s −4M2π)σππ(s)(13)

5Indeed, the r.h.s. of this formula represents the collision rate for pions with momentump moving through a pionic target whose momenta are distributed according to the Bosefactor nB(ω).

The mean damping rate is defined by< γ(p) >=Zd3p nB(ωp)γ(p)/Zd3p nB(ωp)(14)3.3. Choice of the ππ scattering amplitudeBefore presenting results, a few remarks on the ππ scattering amplitude are in order.One can rewrite eq.

(9) in terms of a thermal weight function, W(p, s). An elementaryconsideration leads one to conclude that for temperatures T < 200 MeV the exponentialbehaviour of W(p, s) suppresses the scattering contributions with cm energies √s ≤1GeV.

In this energy range, only the S and P partial waves are non-negligible, so that onecan write the forward ππ scattering amplitude asTππ(s) = 32π3T 00 + 9T 11 + 5T 20(15)where the upper index denotes the isospin. As it is well-known, while the isospin zeroS-wave and the P-wave are attractive, the smaller exotic S-wave is repulsive and thusthere are cancellations in Tππ(s).

As is discussed in detail in ref. [13], the one-loop CHPTresults are reliable up to energies of √s ≃500 MeV.

Beyond this energy, the correctionsbecome too large and also, one misses completely the peak due to the ρ in T 11 . Finally,unitarity is violated above 700 MeV.

Therefore, to make full use of the virial expansion,one can either use a semi-phenomenological parametrization of the phase shifts imposingCHPT constraints at low energies [9,12] or use an extended effective Lagrangian includingalso the low-lying resonace fields R, Leff[U, R]. Such an approach has been shown to besuccessful in the description of ππ and πK scattering data [14].

In the following section,I will therefore present results based on the exact two-loop CHPT result (which is correctto order p6) and also making use of the virial expansion and a parametrization of the ππscattering amplitude which extends to energies of about 1 GeV.4. RESULTS AND DISCUSSIONHere, I will only present the most prominent features of the extensive study presentedin ref.[12].

These can be summarized as follows:First, let us consider the temperature dependence of the effective mass Mπ(T). If oneuses the full CHPT two loop result, one finds that for T = 100(150) MeV the mass islowered by 2.5 (14) percent.

Also, the effects of second order in the density are tiny, theyamount to a shift of 2.5 percent at 150 MeV. These n2B corrections stem almost entirelyfrom three-body collisions, the effects of the mass shift in the Bose factor are negligible.The full two loop CHPT result is also not very different from the first term in the virialexpansion making use of a better description of the forward ππ scattering amplitude.Notice that the current algebra result for the ππ forward scattering amplitude leads to anincreasing pion mass because the momentum-dependent part cancels in eq.

(15) and theremainder is constant and negative.

6For the quasi-particle energy ω(p) one finds that the two-loop CHPT result and theanalysis based on the phenomenological description of the ππ scattering amplitude agreewithin 20 per cent at temperatures and momenta below 150 MeV. At higher momenta,the influence of the ρ becomes more and more pronounced and the straightforward CHPTresult is not reliable any more.

Most important, however, is the fact that temperatureeffects on the dispersion are small.The dispersion law of pions in a medium closelyresembles the one in free space. At T = 150 MeV, the scattering with the gas modifiesthe fequency by less than 20 per cent.

This means that in the low T, long-wavelengthlimit one does not find any indication for a substantial change in the dispersion law assuggested by Shuryak [15]. Again, the effects of order n2B are small, i.e.

the first term inthe virial expansion dominates.Let us now consider the mean damping rate. The effects of Bose correlations are ofthe order of a few per cent, indicating again that the main contribution to the dampingrate stems form the terms of first order in the density.

It is important to notice thatsince the damping rate is proportional to the imaginary parts of the partial waves (whichare positive due to unitarity), no cancellations occur betwen the S- and P-waves. Asa consequence, using the imaginary parts of the one loop CHPT prediction for the ππscattering amplitude leads to a result which agrees within 30 per cent with the virialexpansion at T < 100 MeV.

Using the full one-loop CHPT result [13], i.e. the real andthe imaginary parts of the partial waves, the mean damping grows much too strongly asT increases.

This is due to the fact of the unitarity violation in the I = 0 S-wave above600 MeV cm energy (notice that at T = 100 MeV, both collision partners have energiesof about 300 MeV). For temperatures between 100 and 200 MeV, the leading term in thevirial expansion predicts a decrease of the mean damping rate by a factor of two.

Forthese temperatures, however, one has to account for the massive states like the K, ρ, η,. .

.. These increase the damping rate.

For temperatures above 100 MeV, one expects that< γ(p) >∼T 5/12F 4π [16].To summarize, I have shown that the propagation properties of pions in hot matter aredetermined by the pole position of various Green functions, like e.g. < 0|TC(AA)|0 >.

Thepion energy is in general complex, p0 = ω(p) −iγ(p)/2. At low temperatures, the virialexpansion of ω(p) and γ(p) is appropriate.

To first order in the density, the frequencyω(p) and the damping rate γ(p) are determined by the forward ππ scattering amplitude,Tππ(s). To second order in the density, one has additional contributions related to theproper three-particle scattering amplitude ˆTπππ and the effective pion mass enters theBoltzmann factor in the first term of the virial expansion.

However, the effects of ordern2B are small. The main contributions to ω(p) and γ(p) stem from the first term in thevirial expansion.

Furthermore, the temperature effects on the energy and mass are small.The dispersion law of pions at finite temperatures closely resembles the one in the vacuum.5. ACKNOWLEDGEMENTSI am grateful to Andreas Schenk for patient tuition on the material presented here.REFERENCES

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