- Email: [email protected]
Effective mass and width of pions at T ≠ 0
16 June 1994
PHYSICS LETTERS B ELSEVIER
Physics Leaers B 329 (1994) 312-316
Effective mass and width of pions at T 4= 0 Chungsik Song t Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA Received 14 March 1994 Editor: M. Dine
Abstract
We calculate the effective mass and thermal width of the pions in hot hadronic matter based on an effective chiral lagrangian. The lagrangian includes the vector and axial-vector mesons explicitly. The effective mass is obtained from the pole position of the pion propagator at finite temperature and the inverse of the screening length. The results obtained from both cases decrease with temperature but are different numerically. The thermal width of the pion is computed from the imaginary part of the pion self-energy in the medium. The width is almost negligible at low temperatures but increases rapidly with temperature at T > 100 MeV.
It is expected that hadronic matter undergoes a phase transition into a new phase of quarks and gluons at very high temperature and/or baryon density [ 1 ]. Mesons and baryons in low temperature phase might be changed in their properties due to the presence of hot matter. Such modifications are very interested because of the possibility that they might provide with information of the state of matter and some signatures of the phase transition [2-6]. The masses of mesons are studied in the connection to the symmetry properties of hot hadronic matter [ 5 ]. However, the concept of the mass is not uniquely defined at finite temperature [ 7] and various definitions are proposed in different context. It has been shown that there is discrepancy between these definitions of mass [6]. This indicates that they represent different physical quantities and one should be very careful to relating whatever definition of the mass to physical ones.
! E-mail: [email protected].
The hadronic matter at low temperature (T < 100 MeV), mainly consists of pions, can be analyzed systematically in the framework of chiral perturbation theory at finite temperature [8,9]. The interactions among the pions generate power corrections, controlled by the expansion parameter where F,~ = 135 MeV. The pions are described by an effective lagrangian which is given at the lowest order by
T2/4F~,
1 1 £.=-~TrO~,UO~'Ut +-~TrM(U+Ut).
(1)
The quark mass matrix M is given by diag(mu,
rod,ms) and r 12i , u = exp [~2i~i ---~jCia=i]exp L~--~-Cj
(2)
where ¢ ' s are pseudoscalar octet and A's are GellMann matrices. The properties of pions are modified in hot matter because of the interactions with particles in a heat bath. The modifications are included in the self-energy,
0370-2693/94/$07.00 ~) 1994 Elsevier Science B.V. All rights reserved SSDI 0 3 7 0 - 2 6 9 3 ( 94 ) 0 0 5 2 0 - H
313
C. Song/Physics Letters B 329 (I 994) 312-316
II (to, k), and the propagation of the collective excitation is determined from the relation to2 = k2 + m 2 + H~r(to, k),
(3)
where to is the pion energy and k = [k[ is the pion momentum. The real part of the self-energy is related to the dispersion relation and the imaginary part determines the absorption of a particle in the heat bath. The self-energy of the pion at finite temperature is obtained from Eq. ( 1 ) as
are included [12]. We extend the previous calculation to larger group symmetry to include the effects of kaons and K* mesons on the properties of pions in hot matter. In the effective lagrangian, the pseudoscalar mesons ( ¢ ) are described by the non-linear o" model while the vector (Vu) and axial-vector (Au) mesons are included as massive Yang-Mills fields of the SU(3)z x SU(3)R chiral symmetry [12]:
1F2TrD~UD~Ut
£0=
l_ Wr( p L l~,LI.t,v
ILr = ~ 2 [4(to 2 - k 2) - m 2] g ( m 2 , T ) .
(4)
-- 2 --'~" pa,--
FR FRI.LV q- -/zv--
)
+ m 'rr(A A L'' + g(m 2, T) is given by p2dp 1 g(m~, T) = 4 ] (2~r) 2 2 t o ~ ( e ~ / r - 1) ' d
- is¢ Tr (D,~ UD,.,U t F Lu'' + D uU t Dv UF Ru'' ),
(5)
/
W~r = ~/pZ + m 2.
(6)
The pole position of the propagator at T 4= 0 has been regarded as an effective mass of the collective mode in hot matter. This pole mass can be obtained from the Eq. (3) in the limit k ---, 0, 092 _ mr2 _ R e [ I I ~ ( w , k ---~0) ] = 0 .
where
D~U = O~U - i g A L U + igUA~, F~L~R = agA L'R - 3vA L'R - ig[ A L'R, A~, L R ].
where
(8)
(9)
We add explicit SU(3)v symmetry breaking terms which are proportional to the quark mass. Since the quark mass matrix M transforms linearly under the SU(3)L xSU(3)R [ 13], the SU(3)v symmetry breaking terms will be breaking = U3 Tr M ( U + U t )
(7)
su(3) __
Eq. (7) can be solved self-consistently. It has been shown that the effective mass is not changed at low temperature but slightly increases as temperature greater than 100 MeV [9] (see Fig. 2). However, the mean energy of pions in a heat bath of T > 100 MeV is already too high for a systematic expansion in powers of temperature and energy to be useful [ 10,11]. Moreover, in this range of temperature the pion properties are significantly affected by contributions from resonances, especially p mesons. To study the pions at temperatures greater than 100 MeV we need to include the resonances explicitly and use an approach which does not rely on the low temperature expansion. In this paper we use an effective chiral lagrangian, which includes vector and axial-vector mesons explicitly, to describe the interaction of the pions in hot matter. In the previous paper we restrict to SUL (2) x SUR (2) x U(1) symmetry and the p, to and al mesons
1 L Ltxv ( MU* + UM) ~771Tr[F~F
+ FRvFR~'"(MU + UtM) ] 1 Tr[F~,UF L Rlzg M + F R []'tFLIZVM] - ~12 _ u, .......
(10)
where v, ~71, 7/2 are constants. For the vector and axialvector mesons we have chosen to induce the symmetry breaking by the kinetic energy mixing terms [ 14]. We consider isospin invariant limit, mu = ma = fn. The parameters of the effective lagrangian have been inferred from the experimental values for the masses of pseudoscalar mesons (Tr, kaon), vector (p) and axial-vector ( a l ) mesons and the decay widths of p, K* and al mesons [ 15]. The pion self-energy at finite temperature is calculated based on this effective chiral lagrangian with oneloop diagrams. In the effective lagrangian approach, it is assumed that the properties of the system are describable at the tree level, where the masses and coupling constants are to be regarded as the physical
C. Song/Physics Letters B 329 (1994) 312-316
314 P
~---~-V
where Ame = m~ -- mZv and Amy = m2v -- m~. n( o~p ) (n(wv)) is the Bose-Einstein disribution function at temperature T for pseudoscalar (vector) mesons and w~ = ~ + m~. The coefficients F's and G's depend on the external momentum and masses. The explicit forms are given by
P I I~"
%%
I
I
x
(a)
(b)
Fig. 1. One-loopdiagrams for the pion self-energy.P indicates pseudoscalar mesons and V does vector mesons. For (a) we consider (~r, p) and (K, K*) mesonloops and (n', K, r/) meson loops for (b). ones. Loop diagrams, which are neglected, produce only renormalization effects on these parameters [ 16]. In finite temperature field theory there are corrections due to the interaction with the particles in the heat bath as well as the vacuum corrections. We use the fact that the hadronic matter is rather dilute at the temperatures we are interested in; for example, the mean free path of the pion is about 2 fm at T ,-~ 150 MeV [ 2,15 ]. In this approximation the leading contribution to the temperature-dependent loop corrections is given by the one-loop diagrams which are interpreted as the scattering processes in the medium. The self-energy of the pion is calculated from diagrams which are shown in Fig. 1. Other diagrams which include the vector and axial-vector mesons in the loop are suppressed by the large mass of these mesons [ 12]. The self-energy is, then, given by II,~ = I I ~ ) + ~''KK*IFI(a)-FII(~"b) +
ll(b)q- l-I~b)"
(11)
Here II~)'s are the contributions from the pseudoscalar (P) and vector (V) meson loops (Fig. 1(a)) and II(eb)'s are terms which come from the pseudoscalar ( P ) "tadpole" diagrams (Fig. l(b)) [15]. To calculate the pole mass we take the limit k --~ 0 for the real part of the self-energy. The contributions from the pseudoscalar-vector meson loops can be obtained from
4 + -~v6v(4 - 6v)(oo 2 + Am2),
&=
--~4S2to 2, - 4 [2(o2 + Am2 + m2 -- 2---~v(w2 _ m2 )2]
+ -~vrv(4--rv)[4m2eo~Z--(w2+Am2)2],
G1 G2
=
-
-
-
mTv(4 - 48v + 8 2) (w 2 + Am2v),
mT(4_4av+a )[44 2 1
=
- - - -
(13) where
6v - g2F2 4m 2 2(g m2v 1 l+~m~
g2F~,~2FgEe~ -1 4m02 j
\-~m02 j
,
with ~7 = m(~l + ~72). The coupling constant gv depends on the vector meson included and is given by
go my gv = v/1 + rl mp"
(15)
The "tadpole" diagrams contribute to the real part of the self-energy as Re[II(pb)(w, k --~ 0)] = - 8
f
dpp2 {n(°~P)
~
2we
× [ B ~ - 2B~ (oJ2+ m 2 ) + 4BroJ2p 2] }. Re[II~e)(eo, k ~ 0)] = ~
× rLPl +
dpp z
G2(0)2-t-Amv)
(16)
2eoe The explicit forms of B f ' s are Bf = be/6F~, where b p = 5m~, 2(m 2 + m2), and m~r 2 for 7r, K, r/respectively. For pion B2 and B3 are
F2p 2 + ( (02 + Am2) 2 _ 4(o2o)2j
+ n(o)v) [~1 +
(14)
] }' B~r = ~ (12)
4m02
'
C. Song/Physics Letters B 329 (1994) 312-316
I
0.20
I
I
I
0.20
0.15
0.15
> v
315
> v
0.10
0.10
E=
0.05
0.00
0.05
=
0.00
I
=
I
0.05
i
0.10
I
=
0.15
I
0.00 0.00
0.20
,
I
0.05
T (GeV)
4
(
1-
g2F~'~ 2
4m~)
'
(17)
B~ = B~I2, B~ = B~I2 for kaon and B~ = B~ = 0 for ri meson. From Eq. (7) the pole mass is obtained and given in Fig. 2 as a function of temperature. As temperature increases the pole mass of the pion decreases, but the change is very small, as expected from the GellMann-Oakes-Renner relations (~m,~(T) ,~ 10 MeV at T = 160 MeV). This behaviour is opposite to the result of chiral perturbation theory, which shows an increase of the pion mass with temperature. The difference appears when T > 100 MeV. At finite temperature, alternative definition of mass can be considered which in general is not equivalent to the one defined from the pole position of the propagator. One can define an effective mass by the inverse of the screening length which is obtained from the static infrared limit of the self-energy. This definition has been widely used in the lattice simulation and deduce the symmetry properties of hot hadronic matter [ 17]. The screening mass is written as ms = ~ / m ~ + rI,~(o~ = 0 , k ~ 0).
(18)
The chiral perturbation calculation at the lowest order gives 2
II~(o)=O,k---*0)=
-
I
~
0.10
I
0.15
,
I
0.20
T (GeV)
Fig. 2. The effective mass (pole mass) of the pions in hot matter. The dotted line is the result of the chiral perturbation calculation at the lowest order.
g6F'~ (1 g(g2 B ~ = ~ , +r/)--~--m---~
,
m;r - ' m 2 T" 3F2gt ~, ).
(19)
Fig. 3. The effective mass (screening mass) of the pions in hot matter. The dotted line is the result of the chiral perturbation calculation at the lowest order.
The effective mass calculated from Eq. (18) and Eq. (19) is not the same as the one obtained from the pole position of the propagator. The screening mass decreases as temperature increases, which is opposite to the pole mass. We get the similar result when we include the vector and axial-vector mesons. The real part of the selfenergy in the static infrared limit is given by R e [ I I ~ ) (m = 0 , k ---*0) ]
4gym P
p2dp n(o)p)
mZv
(27r) 2 2o)e '
22f
-
Re[II(pb) (o9 = 0, k ~ 0) ] = -8(B~" - 2m~B~)
f x
p2dp n(o)e)
(20)
(27r) 2 2we
From Eqs. (18) and (20) we have
ms = m~r 1 - - ~
g(m~, T) + 2g(m2x, T)
+ g(m~,T)}] '/2
(21,
The contributions from Ilev's are exactly canceled by the terms from the tadpole diagrams. The result depends only on the pion mass. Even though there is a slight decrease, the screening mass is almost constant at temperatures we consider (Fig. 3). When we compare the result with that obtained from the pole position, we get different values. However, both
316
C. Song/Physics Letters B 329 (1994) 312-316
masses decrease with temperature and the screening mass decreases more slowly than the pole mass. This result reminds us to be careful of the definition of mass at finite temperature. In the chiral limit where m,r = 0, we can see explicitly that the screening mass, defined from the static infrared limit of the real part for the self-energy, becomes zero in the chiral limit. One can also show that the pole mass goes to zero as the mass of the pion becomes zero. This means the pion mass is independent of temperature in the chiral limit. Since the pions are regarded as the massless Goldstone bosons corresponding to the spontaneously broken symmetry, the pion should remain massless at low temperature as long as the chiral symmetry remains broken. This is consistent with the result obtained from the chiral perturbation calculation: chiral symmetry protects all the masses from picking up a contribution of order T2 [ 10]. The thermal width ~,r(k) of the pion can be obtained from the imaginary part of the self-energy: 7 r (k) - - Im II
(22)
to
Since tadpole diagrams have no imaginary parts, we consider diagrams in Fig. 1 (a). It has been shown that the thermal width has a peak at momentum k ,~ 300 MeV/c and increases with temperature [ 15]. We take the average value of the thermal width as
~,T= [d3k ~T(k) / / d 3 k 1 J ~ ~ 1 e'°/r 1"
(23)
-
The result are shown in Fig. 4 and compared with current algebra result ~,r ~ TS/3F4. The thermal width is almost negligible at low temperatures but increases rapidly with temperature when T > 100 MeV. The difference from the current algebra result shows up as temperature greater than 100 MeV. We have obtained the effective mass and thermal width of the pions in hot hadronic matter based on the effective lagrangian. The effective mass cannot be uniquely defined. We have different values for the pion mass at finite temperature. However, both definitions give us a slight decrease of the mass when T > 100 MeV. When the temperature is lower than the pion mass the effective mass is almost constant in temperature. As temperature increases (T > 100 MeV), the thermal width of the pions rapidly increases. The increase in the thermal width will affect on the dilepton
I
0.20 A
I
I
I
'-
/
0.15 0.10 0.05
j
0.00 i
0.00
I
0.05
i
I
0.10
i
I
i
0.15
I
0.20
Temperature (GeV) Fig. 4. The thermal width of the pions at finite temperature. The dotted line is the result of the current algebra.
emissions from the ~" - 7r annihilations in hot matter. Such an increase in the width implies the melting of the hadronic degrees of freedom in hot matter. This work was supported by the National Science Foundation under Grant No. PHY-9212209 and by the Welch Foundation under Grant No. A-1110.
References [I] L. McLerran, Rev. Mod. Phys. 58 (1986) 1021; E.V. Shuryak, The QCD vaccum, Hadrons, and the Superdense Matter (World Scientific, Singapore, 1988). [2] J. Goity and H. Leutwyler, Phys. Lett. B 228 (1989) 517. [3] P. Gerber, H. Leutwyler and J. Goity, Phys. Lett. B 246 (1990) 513. [4] E.V. Shuryak, Phys. Rev. D 42 (1990) 1764. [5] T. Hatsuda, Y. Koike and Su H. Lee, Phys. Rev. D 47 (1993) 1225. [6] Chungsik Song, Phys. Rev. D 48 (1993) 1375. [71 A.V. Smilga, Nucl. Phys. B 335 (1990) 569. [8] J. Gasser and H. Leutwyler, Phys. Lett. B 188 (1987) 477. [9] P. Gerber and H. Leutwyler, Nucl. Phys. B 321 (1987) 387. [10] H. Leutwyler and A.V. Smilga, Nucl. Phys. B 342 (1990) 302. [11] A. Schenk, Nucl. Phys. B 363 (1991) 97. [12] Chungsik Song, Phys. Rev. D 49 (1994) 1556. [ 13 ] H. Goergi, Weak Interactions and Mordern Particle Theory (The Benjamin/Cummings Publshing Company, Inc., Menlo Park, CA., 1984). [141 S. Gasiorowicz and D. Geffen, Rev. Mod. Phys. 41 (1969) 531. [ 15 ] Chungsik Song, Propagation of pions in hot hadronic matter, Texas A&M University Preprint (Feb. 1994). [16] B.W. Lee and H.T. Nieh, Phys. Rev. 166 (1968) 1057. [17] C. DeTar and J. Kogut, Phys. Rev. D 36 (1987) 2828.
PHYSICS LETTERS B ELSEVIER
Physics Leaers B 329 (1994) 312-316
Effective mass and width of pions at T 4= 0 Chungsik Song t Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA Received 14 March 1994 Editor: M. Dine
Abstract
We calculate the effective mass and thermal width of the pions in hot hadronic matter based on an effective chiral lagrangian. The lagrangian includes the vector and axial-vector mesons explicitly. The effective mass is obtained from the pole position of the pion propagator at finite temperature and the inverse of the screening length. The results obtained from both cases decrease with temperature but are different numerically. The thermal width of the pion is computed from the imaginary part of the pion self-energy in the medium. The width is almost negligible at low temperatures but increases rapidly with temperature at T > 100 MeV.
It is expected that hadronic matter undergoes a phase transition into a new phase of quarks and gluons at very high temperature and/or baryon density [ 1 ]. Mesons and baryons in low temperature phase might be changed in their properties due to the presence of hot matter. Such modifications are very interested because of the possibility that they might provide with information of the state of matter and some signatures of the phase transition [2-6]. The masses of mesons are studied in the connection to the symmetry properties of hot hadronic matter [ 5 ]. However, the concept of the mass is not uniquely defined at finite temperature [ 7] and various definitions are proposed in different context. It has been shown that there is discrepancy between these definitions of mass [6]. This indicates that they represent different physical quantities and one should be very careful to relating whatever definition of the mass to physical ones.
! E-mail: [email protected].
The hadronic matter at low temperature (T < 100 MeV), mainly consists of pions, can be analyzed systematically in the framework of chiral perturbation theory at finite temperature [8,9]. The interactions among the pions generate power corrections, controlled by the expansion parameter where F,~ = 135 MeV. The pions are described by an effective lagrangian which is given at the lowest order by
T2/4F~,
1 1 £.=-~TrO~,UO~'Ut +-~TrM(U+Ut).
(1)
The quark mass matrix M is given by diag(mu,
rod,ms) and r 12i , u = exp [~2i~i ---~jCia=i]exp L~--~-Cj
(2)
where ¢ ' s are pseudoscalar octet and A's are GellMann matrices. The properties of pions are modified in hot matter because of the interactions with particles in a heat bath. The modifications are included in the self-energy,
0370-2693/94/$07.00 ~) 1994 Elsevier Science B.V. All rights reserved SSDI 0 3 7 0 - 2 6 9 3 ( 94 ) 0 0 5 2 0 - H
313
C. Song/Physics Letters B 329 (I 994) 312-316
II (to, k), and the propagation of the collective excitation is determined from the relation to2 = k2 + m 2 + H~r(to, k),
(3)
where to is the pion energy and k = [k[ is the pion momentum. The real part of the self-energy is related to the dispersion relation and the imaginary part determines the absorption of a particle in the heat bath. The self-energy of the pion at finite temperature is obtained from Eq. ( 1 ) as
are included [12]. We extend the previous calculation to larger group symmetry to include the effects of kaons and K* mesons on the properties of pions in hot matter. In the effective lagrangian, the pseudoscalar mesons ( ¢ ) are described by the non-linear o" model while the vector (Vu) and axial-vector (Au) mesons are included as massive Yang-Mills fields of the SU(3)z x SU(3)R chiral symmetry [12]:
1F2TrD~UD~Ut
£0=
l_ Wr( p L l~,LI.t,v
ILr = ~ 2 [4(to 2 - k 2) - m 2] g ( m 2 , T ) .
(4)
-- 2 --'~" pa,--
FR FRI.LV q- -/zv--
)
+ m 'rr(A A L'' + g(m 2, T) is given by p2dp 1 g(m~, T) = 4 ] (2~r) 2 2 t o ~ ( e ~ / r - 1) ' d
- is¢ Tr (D,~ UD,.,U t F Lu'' + D uU t Dv UF Ru'' ),
(5)
/
W~r = ~/pZ + m 2.
(6)
The pole position of the propagator at T 4= 0 has been regarded as an effective mass of the collective mode in hot matter. This pole mass can be obtained from the Eq. (3) in the limit k ---, 0, 092 _ mr2 _ R e [ I I ~ ( w , k ---~0) ] = 0 .
where
D~U = O~U - i g A L U + igUA~, F~L~R = agA L'R - 3vA L'R - ig[ A L'R, A~, L R ].
where
(8)
(9)
We add explicit SU(3)v symmetry breaking terms which are proportional to the quark mass. Since the quark mass matrix M transforms linearly under the SU(3)L xSU(3)R [ 13], the SU(3)v symmetry breaking terms will be breaking = U3 Tr M ( U + U t )
(7)
su(3) __
Eq. (7) can be solved self-consistently. It has been shown that the effective mass is not changed at low temperature but slightly increases as temperature greater than 100 MeV [9] (see Fig. 2). However, the mean energy of pions in a heat bath of T > 100 MeV is already too high for a systematic expansion in powers of temperature and energy to be useful [ 10,11]. Moreover, in this range of temperature the pion properties are significantly affected by contributions from resonances, especially p mesons. To study the pions at temperatures greater than 100 MeV we need to include the resonances explicitly and use an approach which does not rely on the low temperature expansion. In this paper we use an effective chiral lagrangian, which includes vector and axial-vector mesons explicitly, to describe the interaction of the pions in hot matter. In the previous paper we restrict to SUL (2) x SUR (2) x U(1) symmetry and the p, to and al mesons
1 L Ltxv ( MU* + UM) ~771Tr[F~F
+ FRvFR~'"(MU + UtM) ] 1 Tr[F~,UF L Rlzg M + F R []'tFLIZVM] - ~12 _ u, .......
(10)
where v, ~71, 7/2 are constants. For the vector and axialvector mesons we have chosen to induce the symmetry breaking by the kinetic energy mixing terms [ 14]. We consider isospin invariant limit, mu = ma = fn. The parameters of the effective lagrangian have been inferred from the experimental values for the masses of pseudoscalar mesons (Tr, kaon), vector (p) and axial-vector ( a l ) mesons and the decay widths of p, K* and al mesons [ 15]. The pion self-energy at finite temperature is calculated based on this effective chiral lagrangian with oneloop diagrams. In the effective lagrangian approach, it is assumed that the properties of the system are describable at the tree level, where the masses and coupling constants are to be regarded as the physical
C. Song/Physics Letters B 329 (1994) 312-316
314 P
~---~-V
where Ame = m~ -- mZv and Amy = m2v -- m~. n( o~p ) (n(wv)) is the Bose-Einstein disribution function at temperature T for pseudoscalar (vector) mesons and w~ = ~ + m~. The coefficients F's and G's depend on the external momentum and masses. The explicit forms are given by
P I I~"
%%
I
I
x
(a)
(b)
Fig. 1. One-loopdiagrams for the pion self-energy.P indicates pseudoscalar mesons and V does vector mesons. For (a) we consider (~r, p) and (K, K*) mesonloops and (n', K, r/) meson loops for (b). ones. Loop diagrams, which are neglected, produce only renormalization effects on these parameters [ 16]. In finite temperature field theory there are corrections due to the interaction with the particles in the heat bath as well as the vacuum corrections. We use the fact that the hadronic matter is rather dilute at the temperatures we are interested in; for example, the mean free path of the pion is about 2 fm at T ,-~ 150 MeV [ 2,15 ]. In this approximation the leading contribution to the temperature-dependent loop corrections is given by the one-loop diagrams which are interpreted as the scattering processes in the medium. The self-energy of the pion is calculated from diagrams which are shown in Fig. 1. Other diagrams which include the vector and axial-vector mesons in the loop are suppressed by the large mass of these mesons [ 12]. The self-energy is, then, given by II,~ = I I ~ ) + ~''KK*IFI(a)-FII(~"b) +
ll(b)q- l-I~b)"
(11)
Here II~)'s are the contributions from the pseudoscalar (P) and vector (V) meson loops (Fig. 1(a)) and II(eb)'s are terms which come from the pseudoscalar ( P ) "tadpole" diagrams (Fig. l(b)) [15]. To calculate the pole mass we take the limit k --~ 0 for the real part of the self-energy. The contributions from the pseudoscalar-vector meson loops can be obtained from
4 + -~v6v(4 - 6v)(oo 2 + Am2),
&=
--~4S2to 2, - 4 [2(o2 + Am2 + m2 -- 2---~v(w2 _ m2 )2]
+ -~vrv(4--rv)[4m2eo~Z--(w2+Am2)2],
G1 G2
=
-
-
-
mTv(4 - 48v + 8 2) (w 2 + Am2v),
mT(4_4av+a )[44 2 1
=
- - - -
(13) where
6v - g2F2 4m 2 2(g m2v 1 l+~m~
g2F~,~2FgEe~ -1 4m02 j
\-~m02 j
,
with ~7 = m(~l + ~72). The coupling constant gv depends on the vector meson included and is given by
go my gv = v/1 + rl mp"
(15)
The "tadpole" diagrams contribute to the real part of the self-energy as Re[II(pb)(w, k --~ 0)] = - 8
f
dpp2 {n(°~P)
~
2we
× [ B ~ - 2B~ (oJ2+ m 2 ) + 4BroJ2p 2] }. Re[II~e)(eo, k ~ 0)] = ~
× rLPl +
dpp z
G2(0)2-t-Amv)
(16)
2eoe The explicit forms of B f ' s are Bf = be/6F~, where b p = 5m~, 2(m 2 + m2), and m~r 2 for 7r, K, r/respectively. For pion B2 and B3 are
F2p 2 + ( (02 + Am2) 2 _ 4(o2o)2j
+ n(o)v) [~1 +
(14)
] }' B~r = ~ (12)
4m02
'
C. Song/Physics Letters B 329 (1994) 312-316
I
0.20
I
I
I
0.20
0.15
0.15
> v
315
> v
0.10
0.10
E=
0.05
0.00
0.05
=
0.00
I
=
I
0.05
i
0.10
I
=
0.15
I
0.00 0.00
0.20
,
I
0.05
T (GeV)
4
(
1-
g2F~'~ 2
4m~)
'
(17)
B~ = B~I2, B~ = B~I2 for kaon and B~ = B~ = 0 for ri meson. From Eq. (7) the pole mass is obtained and given in Fig. 2 as a function of temperature. As temperature increases the pole mass of the pion decreases, but the change is very small, as expected from the GellMann-Oakes-Renner relations (~m,~(T) ,~ 10 MeV at T = 160 MeV). This behaviour is opposite to the result of chiral perturbation theory, which shows an increase of the pion mass with temperature. The difference appears when T > 100 MeV. At finite temperature, alternative definition of mass can be considered which in general is not equivalent to the one defined from the pole position of the propagator. One can define an effective mass by the inverse of the screening length which is obtained from the static infrared limit of the self-energy. This definition has been widely used in the lattice simulation and deduce the symmetry properties of hot hadronic matter [ 17]. The screening mass is written as ms = ~ / m ~ + rI,~(o~ = 0 , k ~ 0).
(18)
The chiral perturbation calculation at the lowest order gives 2
II~(o)=O,k---*0)=
-
I
~
0.10
I
0.15
,
I
0.20
T (GeV)
Fig. 2. The effective mass (pole mass) of the pions in hot matter. The dotted line is the result of the chiral perturbation calculation at the lowest order.
g6F'~ (1 g(g2 B ~ = ~ , +r/)--~--m---~
,
m;r - ' m 2 T" 3F2gt ~, ).
(19)
Fig. 3. The effective mass (screening mass) of the pions in hot matter. The dotted line is the result of the chiral perturbation calculation at the lowest order.
The effective mass calculated from Eq. (18) and Eq. (19) is not the same as the one obtained from the pole position of the propagator. The screening mass decreases as temperature increases, which is opposite to the pole mass. We get the similar result when we include the vector and axial-vector mesons. The real part of the selfenergy in the static infrared limit is given by R e [ I I ~ ) (m = 0 , k ---*0) ]
4gym P
p2dp n(o)p)
mZv
(27r) 2 2o)e '
22f
-
Re[II(pb) (o9 = 0, k ~ 0) ] = -8(B~" - 2m~B~)
f x
p2dp n(o)e)
(20)
(27r) 2 2we
From Eqs. (18) and (20) we have
ms = m~r 1 - - ~
g(m~, T) + 2g(m2x, T)
+ g(m~,T)}] '/2
(21,
The contributions from Ilev's are exactly canceled by the terms from the tadpole diagrams. The result depends only on the pion mass. Even though there is a slight decrease, the screening mass is almost constant at temperatures we consider (Fig. 3). When we compare the result with that obtained from the pole position, we get different values. However, both
316
C. Song/Physics Letters B 329 (1994) 312-316
masses decrease with temperature and the screening mass decreases more slowly than the pole mass. This result reminds us to be careful of the definition of mass at finite temperature. In the chiral limit where m,r = 0, we can see explicitly that the screening mass, defined from the static infrared limit of the real part for the self-energy, becomes zero in the chiral limit. One can also show that the pole mass goes to zero as the mass of the pion becomes zero. This means the pion mass is independent of temperature in the chiral limit. Since the pions are regarded as the massless Goldstone bosons corresponding to the spontaneously broken symmetry, the pion should remain massless at low temperature as long as the chiral symmetry remains broken. This is consistent with the result obtained from the chiral perturbation calculation: chiral symmetry protects all the masses from picking up a contribution of order T2 [ 10]. The thermal width ~,r(k) of the pion can be obtained from the imaginary part of the self-energy: 7 r (k) - - Im II
(22)
to
Since tadpole diagrams have no imaginary parts, we consider diagrams in Fig. 1 (a). It has been shown that the thermal width has a peak at momentum k ,~ 300 MeV/c and increases with temperature [ 15]. We take the average value of the thermal width as
~,T= [d3k ~T(k) / / d 3 k 1 J ~ ~ 1 e'°/r 1"
(23)
-
The result are shown in Fig. 4 and compared with current algebra result ~,r ~ TS/3F4. The thermal width is almost negligible at low temperatures but increases rapidly with temperature when T > 100 MeV. The difference from the current algebra result shows up as temperature greater than 100 MeV. We have obtained the effective mass and thermal width of the pions in hot hadronic matter based on the effective lagrangian. The effective mass cannot be uniquely defined. We have different values for the pion mass at finite temperature. However, both definitions give us a slight decrease of the mass when T > 100 MeV. When the temperature is lower than the pion mass the effective mass is almost constant in temperature. As temperature increases (T > 100 MeV), the thermal width of the pions rapidly increases. The increase in the thermal width will affect on the dilepton
I
0.20 A
I
I
I
'-
/
0.15 0.10 0.05
j
0.00 i
0.00
I
0.05
i
I
0.10
i
I
i
0.15
I
0.20
Temperature (GeV) Fig. 4. The thermal width of the pions at finite temperature. The dotted line is the result of the current algebra.
emissions from the ~" - 7r annihilations in hot matter. Such an increase in the width implies the melting of the hadronic degrees of freedom in hot matter. This work was supported by the National Science Foundation under Grant No. PHY-9212209 and by the Welch Foundation under Grant No. A-1110.
References [I] L. McLerran, Rev. Mod. Phys. 58 (1986) 1021; E.V. Shuryak, The QCD vaccum, Hadrons, and the Superdense Matter (World Scientific, Singapore, 1988). [2] J. Goity and H. Leutwyler, Phys. Lett. B 228 (1989) 517. [3] P. Gerber, H. Leutwyler and J. Goity, Phys. Lett. B 246 (1990) 513. [4] E.V. Shuryak, Phys. Rev. D 42 (1990) 1764. [5] T. Hatsuda, Y. Koike and Su H. Lee, Phys. Rev. D 47 (1993) 1225. [6] Chungsik Song, Phys. Rev. D 48 (1993) 1375. [71 A.V. Smilga, Nucl. Phys. B 335 (1990) 569. [8] J. Gasser and H. Leutwyler, Phys. Lett. B 188 (1987) 477. [9] P. Gerber and H. Leutwyler, Nucl. Phys. B 321 (1987) 387. [10] H. Leutwyler and A.V. Smilga, Nucl. Phys. B 342 (1990) 302. [11] A. Schenk, Nucl. Phys. B 363 (1991) 97. [12] Chungsik Song, Phys. Rev. D 49 (1994) 1556. [ 13 ] H. Goergi, Weak Interactions and Mordern Particle Theory (The Benjamin/Cummings Publshing Company, Inc., Menlo Park, CA., 1984). [141 S. Gasiorowicz and D. Geffen, Rev. Mod. Phys. 41 (1969) 531. [ 15 ] Chungsik Song, Propagation of pions in hot hadronic matter, Texas A&M University Preprint (Feb. 1994). [16] B.W. Lee and H.T. Nieh, Phys. Rev. 166 (1968) 1057. [17] C. DeTar and J. Kogut, Phys. Rev. D 36 (1987) 2828.