- Email: [email protected]
Pion dispersion relation at finite temperature in the linear sigma model from chiral Ward identities
20 April 2000
Physics Letters B 479 Ž2000. 156–162
Pion dispersion relation at finite temperature in the linear sigma model from chiral Ward identities Alejandro Ayala a , Sarira Sahu a , Mauro Napsuciale a
b
b
Instituto de Ciencias Nucleares, UniÕersidad Nacional Autonoma de Mexico, Aptartado Postal 70-543, ´ ´ Mexico Distrito Federal 04510, Mexico ´ Instituto de Fısica, UniÕersidad de Guanajuato, Loma del Bosque 103, Leon ´ ´ Guanajuato 37160, Mexico Received 6 January 2000; received in revised form 31 January 2000; accepted 1 March 2000 Editor: W. Haxton
Abstract We develop one-loop effective vertices and propagators in the linear sigma model at finite temperature satisfying the chiral Ward identities. We use these in turn to compute the pion dispersion relation in a pion medium for small momentum and temperatures on the order of the pion mass at next to leading order in the parameter mp2 r4p 2 fp2 and to zeroth order in the parameter Ž mprms . 2. We show that this expansion reproduces the result obtained from chiral perturbation theory at leading order. The main effect is a perturbative, temperature-dependent increase of the pion mass. We find no qualitative changes in the pion dispersion curve with respect to the leading order behavior in this kinematical regime. q 2000 Elsevier Science B.V. All rights reserved. PACS: 11.10.Wx; 11.30.Rd; 11.55.Fv; 25.75.-q
The propagation properties of pions in a thermal hadronic environment are a key ingredient for the proper understanding of several physical processes taking place in dense and hot plasmas. The scenarios include dense astrophysical objects such as neutron stars – where the inclusion of mesonic degrees of freedom is essential to determine the equation of state of matter in the inner core – and the evolution of the highly interacting region formed in the aftermath of a high-energy heavy-ion collision. In the latter context, knowledge of the pion dispersion relation would help to properly account for the compo-
nent of the dilepton spectrum originated in the hadronic phase, below the critical temperatures and densities for the formation of a quark-gluon plasma ŽQGP. w1x, and to shed light on possible collective surface phenomena which are speculated to occur as a consequence of the mainly attractive interaction between pions and nuclei w2x. In order to account for the hadronic degrees of freedom at temperatures and densities below the QCD phase transition to a QGP, one needs to resort to effective chiral theories whose basic ingredient is the fact that pions are Goldston bosons, originated in
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 2 9 8 - 7
A. Ayala et al.r Physics Letters B 479 (2000) 156–162
the spontaneous breakdown of chiral symmetry. Chiral perturbation theory Ž x PT. is one of such effective theories that has been successfully employed to show the well known result that at leading perturbative order and at low momentum, the modification of the pion dispersion curve in a pion medium is just a constant, temperature dependent, increase of the pion mass w3x. Any deviation from this leading order behavior in perturbation theory can only be looked for at higher orders. Second order calculations of the pion dispersion curve can be found in Refs. w4,5x. The simplest realization of chiral symmetry is nevertheless provided by the much studied linear sigma model which possesses the convenient feature of being a renormalizable field theory, both at zero w6x and at finite temperature w7x. This model has lately received a boost of attention in view of recent theoretical results w8x and analyses of data w9x that seem to confirm, though not without controversy w10x, that a broad scalar resonance, with a mass in the vicinity of 600 MeV – that can be identified with the s-meson – indeed exists. In addition to the above mentioned characteristics, the linear sigma model also reproduces – as we will show – the leading order modification to the pion mass in a thermal pion medium Žfor temperatures on the order of the pion mass., when use is made of a systematic expansion in the ratio Ž mprms . 2 at zeroth order, where ms , mp are the vacuum sigma and pion masses, respectively. This important point, often missed in the literature w11x, provides the connection between the linear sigma model and x PT at finite temperature. In this work, we use the linear sigma model to construct effective vertices and propagators to one loop. We use these in turn to compute the one loop modification to the pion propagator in a pion medium. We require that the effective vertices and propagators thus constructed satisfy the chiral Ward identities w12x. The net result is a next to leading order calculation in the parameter mp2 r4p 2 fp2 , where fp is the pion decay constant and to zeroth order in the parameter Ž mprms . 2 . From the effective pion propagator, we explore the behavior of the pion dispersion curve at small momentum and for temperatures on the order of the pion mass. Though the momentum dependence of the correction term is non-trivial, we find no qualitative difference from the leading
157
order result in this regime. Possible effects due to a high nucleonic density as well as general details of the calculation are left out for a future work. The Lagrangian for the linear sigma model, including only the mesonic degrees of freedom and after the explicit inclusion of the chiral symmetry breaking term, can be expressed as w6x 2
2
L s 12 Ž Ep . q Ž Es . y mp2 p 2 y ms2 s 2 y l2 fp s Ž s 2 q p 2 . y
l2 4
2
Ž s 2 q p 2 . , Ž 1.
where p and s are the pion and sigma fields, respectively, and the coupling l2 is given by
l2 s
ms2 y mp2 2 fp2
.
Ž 2.
From the Lagrangian in Eq. Ž1., one obtains the Green’s functions and the Feynman rules to be used in perturbative calculations, in the usual manner. Thus, in particular, the bare pion and sigma propagators Dp Ž P ., Ds Ž Q . and the bare one-sigma two-pion and four-pion vertices G 12i j, G 04i jk l are given by Žhereafter, capital letters are used to denote four momenta. i Dp Ž P . d i j s
i Ds Ž Q . s
i 2
P y mp2
d ij
i 2
Q y ms2
i G 12i j s y2 i l2 fp d i j i G 04i jk l s y2 i l2 Ž d i jd k l q d i kd jl q d i ld jk . .
Ž 3.
These Green’s functions are sufficient to construct the modification to the pion propagator, both at zero and finite temperature, at any given perturbative order. An alternative approach consists on exploiting the relations that chiral symmetry imposes among different n-point Green’s functions. These relations, better known as chiral Ward identities Ž x WIs., are a direct consequence of the fact that the divergence of the axial current may be used as an interpolating field for the pion. For example, two of the x WIs satisfied
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158
Here v s 2 mp T and v n s 2 np T Ž m, n integers. are discrete boson frequencies and q s < q <. From Eq. Ž5. we obtain the time-ordered version I t of the function I, after analytical continuation to Minkowski space. The imaginary part of I t is given by
´ Ž qo .
Im I t Ž q0 ,q . s Fig. 1. Dominant contribution to the sigma self-energy at one loop.
yI Ž i v ™ q o y i e ,q . 1
– order by order in perturbation theory – by the functions Dp Ž P ., Ds Ž Q ., G 12i j and G 04i jk l are w6x
sy 16p
fp G 04i jk l Ž ;0, P1 , P2 , P3 . s G 12k l Ž P1 ; P2 , P3 . d i j q G 12l j Ž P2 ; P3 , P1 . d i k
=ln
q G 12jk Ž P3 ; P1 , P2 . d i l fp G 12i j Ž Q;0, P . s Dsy1 Ž Q . y Dpy1 Ž P . d i j ,
3
I Ž v ,q . s T Ý n
=
d k
H Ž 2p .
1 3
v n2 q k 2 q mp2 1
2 2 Ž v n y v . q Ž k y q . q mp2
.
Ž 5.
½
aŽ Q 2 . q
2T q
1 y ey v q Ž q 0 , q . rT
ž
1 y ey v y Ž q 0 , q . rT
/5
=Q Ž Q 2 y 4 mp2 . ,
Ž 4.
where momentum conservation at the vertices is implied, that is P1 q P2 q P3 s 0 and Q q P s 0. Therefore, any perturbative modification of one of these functions introduces modifications in other, when the former are related to the latter through x WIs. At one loop and after renormalization, the sigma propagator is modified by finite terms. At zero temperature this modification is purely imaginary and its physical origin is that a sigma particle, with a mass larger than twice the mass of the pion, is unstable and has a Žlarge. non-vanishing width coming from its decay channel into two pions. At finite temperature the modification results on real and imaginary parts. The real part modifies the sigma dispersion curve whereas the imaginary part represents a temperature dependent contribution to the sigma width. After renormalization, the leading one-loop contribution to the sigma self-energy, in an expansion in the parameter Ž mprms . 2 , can be shown w13x to arise from the diagram depicted in Fig. 1. In the imaginary-time formalism of Thermal Field Theory ŽTFT., the expression for this diagram is given as 6 l4 fp2 I Ž v ,q ., where the function I is defined by
I Ž i v ™ q o q i e ,q .
2i
Ž 6.
where Q 2 s q02 y q 2 , ´ and Q are the sign and step functions, respectively, and the functions a and v " are given by aŽ Q 2 . s
(
1y
v " Ž q0 ,q . s
4 mp2 Q2
< q0 < " a Ž Q 2 . q 2
,
Ž 7.
whereas the real part of I t at Q s 0 is given by Re I t Ž 0 . s y
1 8p
` 2
H0
dk Ek
1 q 2 f Ž Ek . ,
Ž 8.
(
where Ek s k 2 q mp2 and the function f is the Bose–Einstein distribution f Ž Ek . s
1 e
Ek r T
y1
.
Ž 9.
Therefore, the one-loop effective sigma propagator becomes i Dsw Ž Q . s
i Q
2
y ms2 q 6 l4 fp2 I t
Ž Q.
.
Ž 10 .
The temperature-independent infinities are absorbed into the redefinition of the physical masses and coupling constants by the introduction of suitable counterterms in the usual manner.
A. Ayala et al.r Physics Letters B 479 (2000) 156–162
159
Pc Ž P . d i j s yl2d i j
i G 12w i j
2
ij
2 t
Ž Q; P1 , P2 . s y2 i l fp d 1 y 3 l I Ž Q .
i G 04w i jk l Ž ; P1 , P2 , P3 , P4 . s y2 i l2 1 y 3 l2 I t Ž P1 q P2 . d i jd k l q 1 y 3 l2 I t Ž P1 q P3 . d i kd jl q 1 y 3 l2 I t Ž P1 q P4 . d i ld jk 4 .
Ž 11 .
It can be shown w13x that the functions in Eq. Ž11. arise from considering all of the possible one-loop contributions to the one-sigma two-pion and fourpion vertices, when maintaining only the zeroth order terms in a systematic expansion in the parameter Ž mprms . 2 . We now use the above effective vertices and propagator to construct the one-loop modification to the pion self-energy. The leading contributions come from the diagrams depicted in Fig. 2. These are, explicitly
P a Ž P . d i j s l2d i j
d4K
Hb Ž 2p .
1 4
6 l2 fp2
ž / ms2
=
d K
Hb Ž 2p .
K 2 y mp2
° ~
=
1 y 3 l2 I t Ž 0 .
¢ 1y
6 l4 fp2 ms2
2
K y mp2
¶ 1 y 3 l2 I t Ž P q K .
~
=
6 l4 fp2
¢ 1y
ms2
I tŽ PqK . y
2
Ž PqK .
2
•,
ß
ms2
Ž 12 . where the subindex b means that the integrals are to be computed at finite temperature. We now expand the denominators in the second and third of Eq. Ž12., keeping only the leading order contribution when considering mp , T and P as small compared to ms . Adding up the above three terms and to zeroth order in Ž mprms . 2 where
ž
l2 1 y
2 l2 fp2
/
ms2
f
mp2 2 fp2
,
Ž 13 .
the pion self-energy can be written as
P Ž P. s
mp2
ž / ½ ž ž / 2
fp2
TÝ
= 5q2
y
mp2
2 fp2
n
d3k
H Ž 2p .
P2qK 2 mp2
1 3
2
K q mp2
/
9I t Ž 0 . q 6 I t Ž P q K .
5
,
where we carry out the calculation in the imaginarytime formalism of TFT with K s Ž v n , k . and P s Ž v , p .. The pion dispersion relation is thus obtained from the solution to P 2 q mp2 q Re P Ž P . s 0 ,
1 4
1 4
Ž 14 .
K y mp2
4
Hb Ž 2p .
2
= 5 y 9I t Ž 0 . y 6 l2 I t Ž P q K . 4
P b Ž P . d i j s yl2d i j
ž / ms2
d4K
°
Fig. 2. Dominant contributions to the pion self-energy at one loop in the effective expansion. The heavy dots denote the effective vertices and propagators.
In order to preserve the x WIs expressed in Eq. Ž4., the corresponding one-loop effective one-sigma two-pion and four-pion vertices become
4l2 fp2
t
2
I Ž 0.
¶ •
ß
Ž 15 .
after the analytical continuation i v ™ po q i e . As it stands, Eq. Ž14. contains a temperature-dependent infinity coming from the product of the vacuum piece in Re I t Ž0. and the temperature-dependent piece in the indicated integral, as well as vacuum infinities. However, it is well known that finite temperature does not introduce new divergences and that whatever regularization and renormalization is
A. Ayala et al.r Physics Letters B 479 (2000) 156–162
160
needed at zero temperature, it will also be necessary and sufficient at finite temperature. Let us recall that at one loop, renormalization is carried out by introducing appropriate counterterms into the original Lagrangian and that at this level, no temperaturedependent infinities arise. Going to two loops, one will always encounter temperature-dependent infinities in integrals involving only the bare terms of the original Lagrangian. The remarkable property of renormalizable theories is that these temperaturedependent infinities are exactly canceled by the contribution from the loops computed by using the counterterms that were introduced at one loop. The above was explicitly shown for the case of the self-energy in the f 4 theory by Kislinger and Morley w14x and for the sigma vacuum expectation value in the linear sigma model by Mohan w7x. For this reason, here we omit the renormalization procedure and consider only the finite, temperature-dependent terms throughout the rest of the calculation and refer the reader to the cited references for details. Let us first look at the dispersion relation at leading order. After analytical continuation all the terms are real. The integrals involved are 3
TÝ n
d k
1
H Ž 2p .
3
K 2 q mp2
™
'
TÝ n
d3k
`
dk k
H0
mp2 2
g Ž Trmp .
2p
Ž 18 .
which coincides with the result obtained from x PT w3x. We now look at the next to leading order terms in Eq. Ž14.. The first of these is purely real and represents a constant, second order shift to the pion mass squared y9
mp2
2
ž /
TÝ
2 fp2
n
d3k
H Ž 2p .
I t Ž 0. 3
K 2 q mp2
™ 92 j 2 g Ž Trmp . h Ž Trmp . mp2 ,
`
h Ž Trmp . '
H0
dk Ek
f Ž Ek . .
Ž 20 .
The remaining term in Eq. Ž14. shows a non-trivial dependence on P. It involves the function S defined by
n
d3k
H Ž 2p .
I tŽ PqK . 3
K 2 q mp2
.
ReS r Ž p 0 , p . ' 12 w S Ž i v ™ p 0 q i e , p . q S Ž i v ™ p 0 y i e , p . x
mp2 2
sy P
d3k
ž / 2p
g Ž Trmp . ,
Ž 16 .
dimensionless function of the ratio arrows indicate only the temperature the expressions. Thus, the dispersion from
1 q 2 j g Ž Trmp .
Ž
p 02 y p 2
.
y 1 q 3 j g Ž Trmp . mp2 s 0 ,
Ž 17 .
with j s mp2 r4p 2 fp2 < 1. For T ; mp , g ŽTrmp . ; 1, therefore, at leading order and in the kinematical
`
dk
`
dk
X
H Ž2p .3 Hy` 2p0 Hy` 2p0
=w1q f Ž k 0 . q f Ž kX0 . x X
where g is a Trmp and the dependence of relation results
Ž 21 .
The sum is performed by resorting to the spectral representation of I t and Ž K 2 q mp2 .y1 . Thus, the real part of the retarded version of S, after analytical continuation is
K 2 q mp2
3
Ž 19 .
where h is a dimensionless function of the ratio Trmp defined by
f Ž Ek .
2p 2
Ek
p 02 s p 2 q mp2 1 q j g Ž Trmp . ,
SŽ P . 'T Ý
2
K2
H Ž 2p .
™ y mp2
1
regime that we are considering, Eq. Ž17. can be written as
X2
2p ´ Ž k 0 . d w k 0 y Ž k y p . 2y mp2 x 2 Im I t Ž k 0 ,k .
=
X
p0 y k 0 y k 0
,
Ž 22 . where P represents the principal part of the integral. The integration in Eq. Ž22. can only be performed numerically. Fig. 3 shows a plot of the temperaturedependent terms of the function S˜Ž p˜ 0 , p . s y Ž24p 4 .ReS r Ž p˜ 0 , p . for T s mp and p˜ 0 taken as the solution of Eq. Ž15. up to the P-independent terms of Eq. Ž14.. From this figure we see that S˜ is
A. Ayala et al.r Physics Letters B 479 (2000) 156–162
Fig. 3. S˜Ž p˜ 0 , p .r mp2 as a function of pr mp for T s mp . The function shows a mild monotonical increase in the kinematical region considered. This increase is not enough to change the qualitative behavior of the dispersion curve, given that S˜ contributes at second order in the expansion parameter j .
non-vanishing at p s 0 and thus it also contributes to the perturbative increase of the pion mass. We also notice that S˜ is a monotonically increasing function of p in the kinematical range considered. However, this increase is not strong enough to cause a qualitatively significant change on the shape of the dispersion curve, given that this function contributes multiplied by the perturbative parameter j 2 . Including all the terms, the dispersion relation up to next to leading order, for T ; mp and in the small momentum region is obtained as the solution to
½
p 02 s p 2 q 1 q j g Ž Trmp .
j2 q 2
g Ž Trmp . 9h Ž Trmp .
y4 g Ž Trmp .
5
mp2 q j 2 S˜Ž p 0 , p . .
161
tices only in the limit where all the external momenta are zero w6x. Nevertheless, we have also checked that these functions arise from considering all of the possible contributions to the vertices of interest, when maintaining only the zeroth order terms in an expansion in the parameter Ž mprms . 2 w13x. We have used these objects to compute the next to leading order correction to the pion propagator in a pion medium for small momentum and for T ; mp . We have shown that the linear sigma model yields the same result as x PT at leading order in the parameter j s mp2 r4p 2 fp2 when use is made of a systematic expansion in the parameter Ž mprms . 2 at zeroth order. This result was to be expected since the kinematical regime we consider is that where the temperature, the pion momentum and the pion mass are treated as small quantities, such as in the case of x PT. The main modification to the pion dispersion curve in the considered kinematical regime is a perturbative increase of the in-medium pion mass. The shape of the curve is not significantly altered. For a sigma meson with a mass on the order of 600 MeV and in the mentioned kinematical regime, corrections in powers of the parameter Ž mprms . 2 are small. If a lighter sigma meson exists, corrections of order Ž mprms . 2 could be important. It remains to include possible effects introduced by a high nucleonic density. Finally, it is interesting to note that the formalism thus developed could be employed to explore the behavior of the pion dispersion curve in the large momentum region where, in principle, the lowest order x PT Lagrangian cannot be used. These issues will be treated in a following up work.
Ž 23 .
Fig. 4 shows the dispersion relation obtained from Eq. Ž23. for T s mp where we also display the solution without the term j 2 S˜Ž p 0 , p .. As mentioned before, inclusion of this last term does not alter the shape of the dispersion relation in this kinematical regime. In conclusion, we have shown that in the linear sigma model at finite temperature, the one-loop modification of the sigma propagator induces a modification in the one-sigma two-pion and four-pion vertices in such a way as to preserve the x WIs. Strictly speaking, chiral Ward identities fully constrain ver-
Fig. 4. Pion dispersion relation obtained as the solution to Eq. Ž23. for T s mp Župper curve.. Shown is also the dispersion relation obtained by ignoring the term S˜Ž p 0 , p . Žlower curve..
162
A. Ayala et al.r Physics Letters B 479 (2000) 156–162
Acknowledgements Support for this work has been received in part by CONACyT Mexico under grant numbers I27604-E, ´ 29273-E and 32279-E. References w1x w2x w3x w4x
C. Gale, J. Kapusta, Phys. Rev. C 35 Ž1987. 2107. E.V. Shuryak, Phys. Rev. D 42 Ž1990. 1764. J. Gasser, H. Leutwyler, Phys. Lett. B 184 Ž1987. 83. A. Schenk, Phys. Rev. D 47 Ž1993. 5138; R.D. Pisarski, M. Tytgat, Phys. Rev. D 54 Ž1996. R2989; D. Toublan, Phys. Rev. D 56 Ž1997. 5629. w5x The two loop computation of the imaginary part of the self energy was carried out by J.L. Goity, H. Leutwyler, Phys. Lett. B 228 Ž1989. 517. w6x B.W. Lee, Chiral Dynamics, Gordon and Breach, 1972.
w7x L.R.R. Mohan, Phys. Rev. D 14 Ž1976. 2670. w8x N.A. Tornqvist, M. Roos, Phys. Rev. Lett. 76 Ž1996. 1575; 77 Ž1996. 2333; 78 Ž1997. 1604; for a recent discussion on the status of the sigma meson see for instance What do we know about sigma?, Workshop on Hadron Spectroscopy session III, Frascati Italy, March Ž1999.: Frascati Physics Series XV, 75–252 Ž1999., T. Bressani, A. Feliciello, A. Fillipi ŽEds... w9x M. Svec, Phys. Rev. D 55 Ž1997. 5727. w10x N. Isgur, J. Speth, Phys. Rev. Lett. 77 Ž1996. 2332; M. Harada, F. Sannino, J. Schechter, Phys. Rev. Lett. 78 Ž1997. 1603. w11x See for example C. Contreras, M. Loewe, Int. J. Mod. Phys. A 5 Ž1990. 2297. w12x This approach has been previously used to study low energy pion scattering at zero temperature, see J.L. Lucio, M. Napsuciale, M. Ruiz-Altaba, The linear sigma model at work: Successful postdictions for pion scattering, hep-phr9903420. w13x Work in progress. w14x M.B. Kislinger, P.D. Morley, Phys. Rev. D 13 Ž1976. 2771.
Physics Letters B 479 Ž2000. 156–162
Pion dispersion relation at finite temperature in the linear sigma model from chiral Ward identities Alejandro Ayala a , Sarira Sahu a , Mauro Napsuciale a
b
b
Instituto de Ciencias Nucleares, UniÕersidad Nacional Autonoma de Mexico, Aptartado Postal 70-543, ´ ´ Mexico Distrito Federal 04510, Mexico ´ Instituto de Fısica, UniÕersidad de Guanajuato, Loma del Bosque 103, Leon ´ ´ Guanajuato 37160, Mexico Received 6 January 2000; received in revised form 31 January 2000; accepted 1 March 2000 Editor: W. Haxton
Abstract We develop one-loop effective vertices and propagators in the linear sigma model at finite temperature satisfying the chiral Ward identities. We use these in turn to compute the pion dispersion relation in a pion medium for small momentum and temperatures on the order of the pion mass at next to leading order in the parameter mp2 r4p 2 fp2 and to zeroth order in the parameter Ž mprms . 2. We show that this expansion reproduces the result obtained from chiral perturbation theory at leading order. The main effect is a perturbative, temperature-dependent increase of the pion mass. We find no qualitative changes in the pion dispersion curve with respect to the leading order behavior in this kinematical regime. q 2000 Elsevier Science B.V. All rights reserved. PACS: 11.10.Wx; 11.30.Rd; 11.55.Fv; 25.75.-q
The propagation properties of pions in a thermal hadronic environment are a key ingredient for the proper understanding of several physical processes taking place in dense and hot plasmas. The scenarios include dense astrophysical objects such as neutron stars – where the inclusion of mesonic degrees of freedom is essential to determine the equation of state of matter in the inner core – and the evolution of the highly interacting region formed in the aftermath of a high-energy heavy-ion collision. In the latter context, knowledge of the pion dispersion relation would help to properly account for the compo-
nent of the dilepton spectrum originated in the hadronic phase, below the critical temperatures and densities for the formation of a quark-gluon plasma ŽQGP. w1x, and to shed light on possible collective surface phenomena which are speculated to occur as a consequence of the mainly attractive interaction between pions and nuclei w2x. In order to account for the hadronic degrees of freedom at temperatures and densities below the QCD phase transition to a QGP, one needs to resort to effective chiral theories whose basic ingredient is the fact that pions are Goldston bosons, originated in
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 2 9 8 - 7
A. Ayala et al.r Physics Letters B 479 (2000) 156–162
the spontaneous breakdown of chiral symmetry. Chiral perturbation theory Ž x PT. is one of such effective theories that has been successfully employed to show the well known result that at leading perturbative order and at low momentum, the modification of the pion dispersion curve in a pion medium is just a constant, temperature dependent, increase of the pion mass w3x. Any deviation from this leading order behavior in perturbation theory can only be looked for at higher orders. Second order calculations of the pion dispersion curve can be found in Refs. w4,5x. The simplest realization of chiral symmetry is nevertheless provided by the much studied linear sigma model which possesses the convenient feature of being a renormalizable field theory, both at zero w6x and at finite temperature w7x. This model has lately received a boost of attention in view of recent theoretical results w8x and analyses of data w9x that seem to confirm, though not without controversy w10x, that a broad scalar resonance, with a mass in the vicinity of 600 MeV – that can be identified with the s-meson – indeed exists. In addition to the above mentioned characteristics, the linear sigma model also reproduces – as we will show – the leading order modification to the pion mass in a thermal pion medium Žfor temperatures on the order of the pion mass., when use is made of a systematic expansion in the ratio Ž mprms . 2 at zeroth order, where ms , mp are the vacuum sigma and pion masses, respectively. This important point, often missed in the literature w11x, provides the connection between the linear sigma model and x PT at finite temperature. In this work, we use the linear sigma model to construct effective vertices and propagators to one loop. We use these in turn to compute the one loop modification to the pion propagator in a pion medium. We require that the effective vertices and propagators thus constructed satisfy the chiral Ward identities w12x. The net result is a next to leading order calculation in the parameter mp2 r4p 2 fp2 , where fp is the pion decay constant and to zeroth order in the parameter Ž mprms . 2 . From the effective pion propagator, we explore the behavior of the pion dispersion curve at small momentum and for temperatures on the order of the pion mass. Though the momentum dependence of the correction term is non-trivial, we find no qualitative difference from the leading
157
order result in this regime. Possible effects due to a high nucleonic density as well as general details of the calculation are left out for a future work. The Lagrangian for the linear sigma model, including only the mesonic degrees of freedom and after the explicit inclusion of the chiral symmetry breaking term, can be expressed as w6x 2
2
L s 12 Ž Ep . q Ž Es . y mp2 p 2 y ms2 s 2 y l2 fp s Ž s 2 q p 2 . y
l2 4
2
Ž s 2 q p 2 . , Ž 1.
where p and s are the pion and sigma fields, respectively, and the coupling l2 is given by
l2 s
ms2 y mp2 2 fp2
.
Ž 2.
From the Lagrangian in Eq. Ž1., one obtains the Green’s functions and the Feynman rules to be used in perturbative calculations, in the usual manner. Thus, in particular, the bare pion and sigma propagators Dp Ž P ., Ds Ž Q . and the bare one-sigma two-pion and four-pion vertices G 12i j, G 04i jk l are given by Žhereafter, capital letters are used to denote four momenta. i Dp Ž P . d i j s
i Ds Ž Q . s
i 2
P y mp2
d ij
i 2
Q y ms2
i G 12i j s y2 i l2 fp d i j i G 04i jk l s y2 i l2 Ž d i jd k l q d i kd jl q d i ld jk . .
Ž 3.
These Green’s functions are sufficient to construct the modification to the pion propagator, both at zero and finite temperature, at any given perturbative order. An alternative approach consists on exploiting the relations that chiral symmetry imposes among different n-point Green’s functions. These relations, better known as chiral Ward identities Ž x WIs., are a direct consequence of the fact that the divergence of the axial current may be used as an interpolating field for the pion. For example, two of the x WIs satisfied
A. Ayala et al.r Physics Letters B 479 (2000) 156–162
158
Here v s 2 mp T and v n s 2 np T Ž m, n integers. are discrete boson frequencies and q s < q <. From Eq. Ž5. we obtain the time-ordered version I t of the function I, after analytical continuation to Minkowski space. The imaginary part of I t is given by
´ Ž qo .
Im I t Ž q0 ,q . s Fig. 1. Dominant contribution to the sigma self-energy at one loop.
yI Ž i v ™ q o y i e ,q . 1
– order by order in perturbation theory – by the functions Dp Ž P ., Ds Ž Q ., G 12i j and G 04i jk l are w6x
sy 16p
fp G 04i jk l Ž ;0, P1 , P2 , P3 . s G 12k l Ž P1 ; P2 , P3 . d i j q G 12l j Ž P2 ; P3 , P1 . d i k
=ln
q G 12jk Ž P3 ; P1 , P2 . d i l fp G 12i j Ž Q;0, P . s Dsy1 Ž Q . y Dpy1 Ž P . d i j ,
3
I Ž v ,q . s T Ý n
=
d k
H Ž 2p .
1 3
v n2 q k 2 q mp2 1
2 2 Ž v n y v . q Ž k y q . q mp2
.
Ž 5.
½
aŽ Q 2 . q
2T q
1 y ey v q Ž q 0 , q . rT
ž
1 y ey v y Ž q 0 , q . rT
/5
=Q Ž Q 2 y 4 mp2 . ,
Ž 4.
where momentum conservation at the vertices is implied, that is P1 q P2 q P3 s 0 and Q q P s 0. Therefore, any perturbative modification of one of these functions introduces modifications in other, when the former are related to the latter through x WIs. At one loop and after renormalization, the sigma propagator is modified by finite terms. At zero temperature this modification is purely imaginary and its physical origin is that a sigma particle, with a mass larger than twice the mass of the pion, is unstable and has a Žlarge. non-vanishing width coming from its decay channel into two pions. At finite temperature the modification results on real and imaginary parts. The real part modifies the sigma dispersion curve whereas the imaginary part represents a temperature dependent contribution to the sigma width. After renormalization, the leading one-loop contribution to the sigma self-energy, in an expansion in the parameter Ž mprms . 2 , can be shown w13x to arise from the diagram depicted in Fig. 1. In the imaginary-time formalism of Thermal Field Theory ŽTFT., the expression for this diagram is given as 6 l4 fp2 I Ž v ,q ., where the function I is defined by
I Ž i v ™ q o q i e ,q .
2i
Ž 6.
where Q 2 s q02 y q 2 , ´ and Q are the sign and step functions, respectively, and the functions a and v " are given by aŽ Q 2 . s
(
1y
v " Ž q0 ,q . s
4 mp2 Q2
< q0 < " a Ž Q 2 . q 2
,
Ž 7.
whereas the real part of I t at Q s 0 is given by Re I t Ž 0 . s y
1 8p
` 2
H0
dk Ek
1 q 2 f Ž Ek . ,
Ž 8.
(
where Ek s k 2 q mp2 and the function f is the Bose–Einstein distribution f Ž Ek . s
1 e
Ek r T
y1
.
Ž 9.
Therefore, the one-loop effective sigma propagator becomes i Dsw Ž Q . s
i Q
2
y ms2 q 6 l4 fp2 I t
Ž Q.
.
Ž 10 .
The temperature-independent infinities are absorbed into the redefinition of the physical masses and coupling constants by the introduction of suitable counterterms in the usual manner.
A. Ayala et al.r Physics Letters B 479 (2000) 156–162
159
Pc Ž P . d i j s yl2d i j
i G 12w i j
2
ij
2 t
Ž Q; P1 , P2 . s y2 i l fp d 1 y 3 l I Ž Q .
i G 04w i jk l Ž ; P1 , P2 , P3 , P4 . s y2 i l2 1 y 3 l2 I t Ž P1 q P2 . d i jd k l q 1 y 3 l2 I t Ž P1 q P3 . d i kd jl q 1 y 3 l2 I t Ž P1 q P4 . d i ld jk 4 .
Ž 11 .
It can be shown w13x that the functions in Eq. Ž11. arise from considering all of the possible one-loop contributions to the one-sigma two-pion and fourpion vertices, when maintaining only the zeroth order terms in a systematic expansion in the parameter Ž mprms . 2 . We now use the above effective vertices and propagator to construct the one-loop modification to the pion self-energy. The leading contributions come from the diagrams depicted in Fig. 2. These are, explicitly
P a Ž P . d i j s l2d i j
d4K
Hb Ž 2p .
1 4
6 l2 fp2
ž / ms2
=
d K
Hb Ž 2p .
K 2 y mp2
° ~
=
1 y 3 l2 I t Ž 0 .
¢ 1y
6 l4 fp2 ms2
2
K y mp2
¶ 1 y 3 l2 I t Ž P q K .
~
=
6 l4 fp2
¢ 1y
ms2
I tŽ PqK . y
2
Ž PqK .
2
•,
ß
ms2
Ž 12 . where the subindex b means that the integrals are to be computed at finite temperature. We now expand the denominators in the second and third of Eq. Ž12., keeping only the leading order contribution when considering mp , T and P as small compared to ms . Adding up the above three terms and to zeroth order in Ž mprms . 2 where
ž
l2 1 y
2 l2 fp2
/
ms2
f
mp2 2 fp2
,
Ž 13 .
the pion self-energy can be written as
P Ž P. s
mp2
ž / ½ ž ž / 2
fp2
TÝ
= 5q2
y
mp2
2 fp2
n
d3k
H Ž 2p .
P2qK 2 mp2
1 3
2
K q mp2
/
9I t Ž 0 . q 6 I t Ž P q K .
5
,
where we carry out the calculation in the imaginarytime formalism of TFT with K s Ž v n , k . and P s Ž v , p .. The pion dispersion relation is thus obtained from the solution to P 2 q mp2 q Re P Ž P . s 0 ,
1 4
1 4
Ž 14 .
K y mp2
4
Hb Ž 2p .
2
= 5 y 9I t Ž 0 . y 6 l2 I t Ž P q K . 4
P b Ž P . d i j s yl2d i j
ž / ms2
d4K
°
Fig. 2. Dominant contributions to the pion self-energy at one loop in the effective expansion. The heavy dots denote the effective vertices and propagators.
In order to preserve the x WIs expressed in Eq. Ž4., the corresponding one-loop effective one-sigma two-pion and four-pion vertices become
4l2 fp2
t
2
I Ž 0.
¶ •
ß
Ž 15 .
after the analytical continuation i v ™ po q i e . As it stands, Eq. Ž14. contains a temperature-dependent infinity coming from the product of the vacuum piece in Re I t Ž0. and the temperature-dependent piece in the indicated integral, as well as vacuum infinities. However, it is well known that finite temperature does not introduce new divergences and that whatever regularization and renormalization is
A. Ayala et al.r Physics Letters B 479 (2000) 156–162
160
needed at zero temperature, it will also be necessary and sufficient at finite temperature. Let us recall that at one loop, renormalization is carried out by introducing appropriate counterterms into the original Lagrangian and that at this level, no temperaturedependent infinities arise. Going to two loops, one will always encounter temperature-dependent infinities in integrals involving only the bare terms of the original Lagrangian. The remarkable property of renormalizable theories is that these temperaturedependent infinities are exactly canceled by the contribution from the loops computed by using the counterterms that were introduced at one loop. The above was explicitly shown for the case of the self-energy in the f 4 theory by Kislinger and Morley w14x and for the sigma vacuum expectation value in the linear sigma model by Mohan w7x. For this reason, here we omit the renormalization procedure and consider only the finite, temperature-dependent terms throughout the rest of the calculation and refer the reader to the cited references for details. Let us first look at the dispersion relation at leading order. After analytical continuation all the terms are real. The integrals involved are 3
TÝ n
d k
1
H Ž 2p .
3
K 2 q mp2
™
'
TÝ n
d3k
`
dk k
H0
mp2 2
g Ž Trmp .
2p
Ž 18 .
which coincides with the result obtained from x PT w3x. We now look at the next to leading order terms in Eq. Ž14.. The first of these is purely real and represents a constant, second order shift to the pion mass squared y9
mp2
2
ž /
TÝ
2 fp2
n
d3k
H Ž 2p .
I t Ž 0. 3
K 2 q mp2
™ 92 j 2 g Ž Trmp . h Ž Trmp . mp2 ,
`
h Ž Trmp . '
H0
dk Ek
f Ž Ek . .
Ž 20 .
The remaining term in Eq. Ž14. shows a non-trivial dependence on P. It involves the function S defined by
n
d3k
H Ž 2p .
I tŽ PqK . 3
K 2 q mp2
.
ReS r Ž p 0 , p . ' 12 w S Ž i v ™ p 0 q i e , p . q S Ž i v ™ p 0 y i e , p . x
mp2 2
sy P
d3k
ž / 2p
g Ž Trmp . ,
Ž 16 .
dimensionless function of the ratio arrows indicate only the temperature the expressions. Thus, the dispersion from
1 q 2 j g Ž Trmp .
Ž
p 02 y p 2
.
y 1 q 3 j g Ž Trmp . mp2 s 0 ,
Ž 17 .
with j s mp2 r4p 2 fp2 < 1. For T ; mp , g ŽTrmp . ; 1, therefore, at leading order and in the kinematical
`
dk
`
dk
X
H Ž2p .3 Hy` 2p0 Hy` 2p0
=w1q f Ž k 0 . q f Ž kX0 . x X
where g is a Trmp and the dependence of relation results
Ž 21 .
The sum is performed by resorting to the spectral representation of I t and Ž K 2 q mp2 .y1 . Thus, the real part of the retarded version of S, after analytical continuation is
K 2 q mp2
3
Ž 19 .
where h is a dimensionless function of the ratio Trmp defined by
f Ž Ek .
2p 2
Ek
p 02 s p 2 q mp2 1 q j g Ž Trmp . ,
SŽ P . 'T Ý
2
K2
H Ž 2p .
™ y mp2
1
regime that we are considering, Eq. Ž17. can be written as
X2
2p ´ Ž k 0 . d w k 0 y Ž k y p . 2y mp2 x 2 Im I t Ž k 0 ,k .
=
X
p0 y k 0 y k 0
,
Ž 22 . where P represents the principal part of the integral. The integration in Eq. Ž22. can only be performed numerically. Fig. 3 shows a plot of the temperaturedependent terms of the function S˜Ž p˜ 0 , p . s y Ž24p 4 .ReS r Ž p˜ 0 , p . for T s mp and p˜ 0 taken as the solution of Eq. Ž15. up to the P-independent terms of Eq. Ž14.. From this figure we see that S˜ is
A. Ayala et al.r Physics Letters B 479 (2000) 156–162
Fig. 3. S˜Ž p˜ 0 , p .r mp2 as a function of pr mp for T s mp . The function shows a mild monotonical increase in the kinematical region considered. This increase is not enough to change the qualitative behavior of the dispersion curve, given that S˜ contributes at second order in the expansion parameter j .
non-vanishing at p s 0 and thus it also contributes to the perturbative increase of the pion mass. We also notice that S˜ is a monotonically increasing function of p in the kinematical range considered. However, this increase is not strong enough to cause a qualitatively significant change on the shape of the dispersion curve, given that this function contributes multiplied by the perturbative parameter j 2 . Including all the terms, the dispersion relation up to next to leading order, for T ; mp and in the small momentum region is obtained as the solution to
½
p 02 s p 2 q 1 q j g Ž Trmp .
j2 q 2
g Ž Trmp . 9h Ž Trmp .
y4 g Ž Trmp .
5
mp2 q j 2 S˜Ž p 0 , p . .
161
tices only in the limit where all the external momenta are zero w6x. Nevertheless, we have also checked that these functions arise from considering all of the possible contributions to the vertices of interest, when maintaining only the zeroth order terms in an expansion in the parameter Ž mprms . 2 w13x. We have used these objects to compute the next to leading order correction to the pion propagator in a pion medium for small momentum and for T ; mp . We have shown that the linear sigma model yields the same result as x PT at leading order in the parameter j s mp2 r4p 2 fp2 when use is made of a systematic expansion in the parameter Ž mprms . 2 at zeroth order. This result was to be expected since the kinematical regime we consider is that where the temperature, the pion momentum and the pion mass are treated as small quantities, such as in the case of x PT. The main modification to the pion dispersion curve in the considered kinematical regime is a perturbative increase of the in-medium pion mass. The shape of the curve is not significantly altered. For a sigma meson with a mass on the order of 600 MeV and in the mentioned kinematical regime, corrections in powers of the parameter Ž mprms . 2 are small. If a lighter sigma meson exists, corrections of order Ž mprms . 2 could be important. It remains to include possible effects introduced by a high nucleonic density. Finally, it is interesting to note that the formalism thus developed could be employed to explore the behavior of the pion dispersion curve in the large momentum region where, in principle, the lowest order x PT Lagrangian cannot be used. These issues will be treated in a following up work.
Ž 23 .
Fig. 4 shows the dispersion relation obtained from Eq. Ž23. for T s mp where we also display the solution without the term j 2 S˜Ž p 0 , p .. As mentioned before, inclusion of this last term does not alter the shape of the dispersion relation in this kinematical regime. In conclusion, we have shown that in the linear sigma model at finite temperature, the one-loop modification of the sigma propagator induces a modification in the one-sigma two-pion and four-pion vertices in such a way as to preserve the x WIs. Strictly speaking, chiral Ward identities fully constrain ver-
Fig. 4. Pion dispersion relation obtained as the solution to Eq. Ž23. for T s mp Župper curve.. Shown is also the dispersion relation obtained by ignoring the term S˜Ž p 0 , p . Žlower curve..
162
A. Ayala et al.r Physics Letters B 479 (2000) 156–162
Acknowledgements Support for this work has been received in part by CONACyT Mexico under grant numbers I27604-E, ´ 29273-E and 32279-E. References w1x w2x w3x w4x
C. Gale, J. Kapusta, Phys. Rev. C 35 Ž1987. 2107. E.V. Shuryak, Phys. Rev. D 42 Ž1990. 1764. J. Gasser, H. Leutwyler, Phys. Lett. B 184 Ž1987. 83. A. Schenk, Phys. Rev. D 47 Ž1993. 5138; R.D. Pisarski, M. Tytgat, Phys. Rev. D 54 Ž1996. R2989; D. Toublan, Phys. Rev. D 56 Ž1997. 5629. w5x The two loop computation of the imaginary part of the self energy was carried out by J.L. Goity, H. Leutwyler, Phys. Lett. B 228 Ž1989. 517. w6x B.W. Lee, Chiral Dynamics, Gordon and Breach, 1972.
w7x L.R.R. Mohan, Phys. Rev. D 14 Ž1976. 2670. w8x N.A. Tornqvist, M. Roos, Phys. Rev. Lett. 76 Ž1996. 1575; 77 Ž1996. 2333; 78 Ž1997. 1604; for a recent discussion on the status of the sigma meson see for instance What do we know about sigma?, Workshop on Hadron Spectroscopy session III, Frascati Italy, March Ž1999.: Frascati Physics Series XV, 75–252 Ž1999., T. Bressani, A. Feliciello, A. Fillipi ŽEds... w9x M. Svec, Phys. Rev. D 55 Ž1997. 5727. w10x N. Isgur, J. Speth, Phys. Rev. Lett. 77 Ž1996. 2332; M. Harada, F. Sannino, J. Schechter, Phys. Rev. Lett. 78 Ž1997. 1603. w11x See for example C. Contreras, M. Loewe, Int. J. Mod. Phys. A 5 Ž1990. 2297. w12x This approach has been previously used to study low energy pion scattering at zero temperature, see J.L. Lucio, M. Napsuciale, M. Ruiz-Altaba, The linear sigma model at work: Successful postdictions for pion scattering, hep-phr9903420. w13x Work in progress. w14x M.B. Kislinger, P.D. Morley, Phys. Rev. D 13 Ž1976. 2771.