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Hybrid models at finite temperature and deconfinement
Physics Letters B 300 ( 1993 ) 278-282 North-Holland
P H YSIC S 1.EI T ER S B
Hybrid models at finite temperature and deconfinement H. Falomir Departamento de Fisica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, c.c. 67, 1900 La Plata. Buenos Aires, Argentina
M. L o e w e a n d J.C. R o j a s Facultad de Fisica, Pontificia Universidad Cat61ica de Chile, Casilla 306, Santiago 22, Chile
Received 4 September 1992
We consider a chiral bag model at finite temperature, with a thermal skyrmion of Eskola and Kajantie modelling the exterior sector. Supposing that the "Cheshire Cat" scenario takes care of all zero temperature effects, we show the existence of a critical temperature over which there can only exist a quark-gluon plasma phase, since the bag is no longer stable. This fact is interpreted as the occurrence of a deconfinement phase transition.
During the last years a considerable amount o f work has been done in connection with the deconfining phase transition, induced by thermal effects, in the hadronic world. A m o n g the different techniques employed to discuss or analyze such an effect, we can mention Q C D on the lattice [ 1 ], chiral perturbation theory [2], Q C D sum rules [3], and also some discussions inspired by the old M I T bag model [ 4 ]. The critical deconfining temperature moves between 120 and 160 MeV approximately, depending on the model. Some authors have conjectured that the critical deconfining temperature and the critical temperature associated to chiral symmetry restoration should be the same [5 ]. In this letter we want to reconsider this i m p o r t a n t problem from a different perspective, namely invoking the hybrid models [ 6 ] in which we have as ingredients a chiral bag model with a non-perturbative external pion field configuration, the skyrmion. This approach has been particularly successful in order to analyze different phenomenological properties o f the hadrons at zero t e m p e r a t u r e [ 7 ]. As far as we know the problem of thermal deconfinement in this frame has not been discussed previously in the literature. It is i m p o r t a n t to remark here that the pure skyrm i o n model provides a remarkably successful description o f the nucleon and other baryons and me278
sons after a suitable modification o f the basic model [8]. Recently, K. Eskola and K. Kajantie ( E K ) [9] were able to generalize the S U ( 2 ) skyrmion field configuration to a finite temperature situation, extending the construction o f skyrmions, due to Atiyah and M a n t o n ( A M ) [ 10 ] starting from instantons in ~4. F o r a suitable scale of the instanton, the ansatz o f A M leads to a good a p p r o x i m a t i o n o f the numerical solution o f the variational equation obeyed by the skyrmion [ 11 ]. The consistent construction o f chiral bag models requires a careful analysis o f the zero point energies at the one loop order and o f the cancellation o f ultraviolet divergences occurring clue to the presence o f the bag wall [ 12 ]. The c o m b i n a t i o n of these effects with an external skyrmionic tail provides, as is well known, a highly interesting "Cheshire Cat" description of the nucleon, where all low energies physical observables, as for example the mass, the R M S radius, the isoscalar and vector magnetic moments, etc., turn out to be essentially i n d e p e n d e n t o f the bag radius [ 8,12]. F r o m this point o f view the bag radius does not play any role in determining physical quantities. It corresponds only to a demarcation between two different descriptions of the relevant degrees o f freedom: quarks and gluons inside the bag and, outside, pions and the skyrmion.
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 300, number 3
PHYSICS LETTERS B
The aim of this note is to have a qualitative discussion of the effects of temperature on equilibrium between the two phases, using the thermal skyrmions of EK for the non-perturbative sector. It is well known in finite temperature field theory that no new ultraviolet singularities appear due to the effects of temperature. For example, in Q E D or the 2 0 4 theory we can have a consistent theory renormalizing at zero temperature [ 13 ]. As a second example, we can mention that the usual axial anomaly, responsible for the n o decay, does not receive any temperature dependent modification [14]. Recently, it has also been shown [15] that the ultraviolet divergences in the calculation of the free energy in the chiral bag model do not depend on the temperature. These considerations allow us to overcome the problem of ultraviolet divergences in our case. Essentially, we assume here that all zero point effects have been taken into account at zero temperature, giving rise to the "Cheshire Cat" scenario [ 12 ]. So, if there is a particular equilibrium radius at finite temperature, it should be determined by the new thermal contributions. If for a certain temperature the model is unable to produce an equilibrium radius, we will interpret this fact as the occurrence of thermal deconfinement. In our qualitative approach we will neglect the finite size effects of the bag on the quark and gluon states, considering them as a gas of free particles. We will also assume, that the existence of this populated thermal vacuum inside the bag does not have any influence on the baryonic number of the hadron, neglecting the chemical potential. The description of the external vacuum will be a superposition of the thermal skyrmion and a gas of free massless pions. As we will see, the crucial contribution for deconfinement comes from the non-perturbative configuration. The existence of a defined radius at finite temperature comes from an equilibrium between the external and internal variations of pressures due to thermal effects. From the literature [ 16 ] we have that the internal pressure of the quark gluon plasma is given by Ppl . . . . ( T )
37 --2"r'4
(1)
where we have considered two flavors and three colors. The external pressure of the pion gas is
11 February 1993
Ppions( T ) = 3
7~2T 4 .
(2)
The variation of the pressure due to the skyrmion, APsk(T) = P s k ( T ) - - P s k ( 0 ) ,
(3)
can be calculated from the dependence of the energy of the thermal skyrmion as a function of the bag volume. The energy (mass) of the skyrmion outside the bag, in units of FJ4e (where e is the constant in front of the stabilizing term in the Skyrme lagrangian and F~ is the pion decay constant) is given by M = 4x
[ dr L2 (0I 2 +2 sin2/] -
R
\ Or)
[L
sin~SI-2r (~/] ~ + sin2f]} +~TL \Or/
(4)
where R is the radius of the bag measured in units of 2/eF,~. In the previous expression f d e n o t e s the profile function of the thermal skyrmion of EK [ 9]. It is given by f ( r ) =z~ (1 -
r+½22[l~c°th(IJr)-l/r] dr2+ llzZ24+~r22coth(l~r) ] .
(5)
The constant/z in this formula is 27rT, where the temperature has been expressed in units of ½eFt, and 2 is the size (in the same units as R) of the instanton which gives rise to the skyrmion. Now, the pressure of the skyrmion at radius R and temperature T, PSk= OM/OVbag,is given by
I(R, T) Psk(R, T ) =
47rR 2
(6)
In this formula I(R, T) denotes the integrand in eq. (4), including the 47r factor. At a given temperature a state of equilibrium between the internal and external phases can exist if there is a radius at which the pressures are equal. The equilibrium condition is 2 4
34_2"T4 e F~ [ I ( R , O ) - I ( R , T ) ] 76Jr i = 1287cR 2
(7)
For the numerical analysis of this expression we have used the constant values for F , and e according to ref. [ 1 1 ], fitted from the proton and A masses. In ref. [ 9 ] it is shown that the size 2 varies between 1.45, for/~=0, and 2.5 in a reasonable range of variation of T. It is important to remark that the value o f F , we 279
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PHYSICS LETTERS B
have used, 129 MeV, is much smaller than the experimental one, 186 MeV, using the normalization ofref. [ 11 ]. We find that eq. (7) in general has two solutions (i.e. two possible values for the radius for each t e m p e r a t u r e ) , up to a point where both values coincide. Beyond this point there is no solution. This is shown in fig. 1. F o r our estimates we have taken 2 = 1.45. F o r the parameters written above, we find that this critical temperature is Tc = 109.1 MeV. We interpret this fact as the occurrence o f a deconfining phase transition, since for higher temperatures only the q u a r k - g l u o n plasma phase does exist. O u r value for Tc is smaller than those obtained in other models [ 1,2,3 ], but this is due to the small value o f F= we have taken. The critical t e m p e r a t u r e grows linearly with F=. This is shown in fig. 2. The variation o f the critical t e m p e r a t u r e as a function o f the p a r a m e t e r e, for a constant F~, turns out to be much insensitive as the dependence on F . . In fact, we see in fig. 3 that Tc suffers a variation o f less than 10% in a wide region a r o u n d the value given in ref. [ 11 ]. In general, according to EK, the value o f 2 grows with temperature, if we m i n i m i z e the mass of the thermal skyrmion for each value o f T. This is indicated in fig. 4. If we choose this criteria, the critical temperature diminishes, for every value ofF=, but its dependence on the p a r a m e t e r e turns out to be much less sensitive than in the case with a constant value o f 2. This is shown in fig. 5. F r o m our discussion, it
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F~Fig. 2. Here we show the linear dependence of Tc (MeV) as a function of F, (MeV) (continuous line: e=5.45; dotted line: e=7.5). 120.00
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Fig. 1. This figure shows the right hand term of eq. (7) for three different temperatures (dashed line: 83.2 MeV; continuous line: 109.1 MeV; dotted line: 139.8 MeV). The horizontal lines are the left term. We see that 109.1 MeV is the critical temperature. 280
Fig. 3. We show Tc (MeV) as a function of the parameter e with F== 129 MeV. turns out that F , is the most relevant p a r a m e t e r in order to fix the critical temperature. Finally we explore another possibility, namely the existence of a temperature dependence o f F~ using the well known results of Gasser and Leutwyler [ 17 ] according to chiral perturbation theory ( x P T ) at finite temperature. Although this is a different theoretical framework, we think, it is interesting to explore in this scenario the consequences of a temperature dependent pion decay constant. We know, also from xPT [ 18 ], that the nucleon and pion masses are quite stable with respect to the temperature, at least in the
Volume 300, number 3
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F2(T)=F2(O)
6.75
x
1
4 T 2 (2g)2 F ~ ( 0 )
i dxJX2-#~/T2~ 7---1 / '
(8)
,u~tl T
A
4.50
where we have taken only two flavors. It is possible to c o m p u t e this integral exactly by going to the chiral limit w h e r e / z . = 0. In this way we get
2.25
FZ(T)=F2(O) ( 1 -
0.00 0.00
1.00
2.00 2u.
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Fig. 4. The behavior of k, obtained by minimizing the mass of the skyrmion for each temperature, is plotted against/z= 2nT (2 and T dimensionless).
95.00
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(9)
The t e m p e r a t u r e dependence of the p a r a m e t e r e is chosen exactly in the same way, i.e. eo multiplying the same factor o f the previous equation. The result from this analysis is shown in fig. 6. Here we show the d e p e n d e n c e o f Tc on F~ (0). The value o f the critical t e m p e r a t u r e is found using eq. ( 7 ) , where the and are used for different F , ( 0 ) as an input, eo being constant, eo= 5.45. We conclude that in this case the critical t e m p e r a t u r e also rises in a quasi-linear way with F ~ ( 0 ) . The numbers are smaller, as we could have predicted, c o m p a r e d with the constant F , case. The m e c h a n i s m responsible for the occurrence o f this phase transition can be viewed as a consequence o f the concentration o f the skyrmion a r o u n d the origin due to t e m p e r a t u r e [ 9 ]. This has the effect o f in-
F,~(T) 120.00
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e(T)
45.00 ;70.00[ 20.00 0.90
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e
Fig. 5. The dependence of T¢ (MeV) as a function ofe is shown for a temperature dependent 2 (continuous line) compared with the constant 2 case (dotted line) of fig. 3. range o f temperatures we consider here. Invoking this fact, for our analysis we have also introduced a temperature d e p e n d e n t skyrmion p a r a m e t e r e, imposing that the total mass o f the nucleon, described as a skyrmion, remains constant as a function o f temperature. The p a r a m e t e r 2 could have, as previously, a d e p e n d e n c e on temperature, but for our discussion we have taken it constant, 2 = 1.45. The expression we use here is the following:
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Fig. 6. It is compared the dependence of T¢ on F=( T= 0) (MeV) for a model where F= is constant (dotted line) and where it is assumed that F= depends on T according to chiral perturbation theory (continuous line). 5.45 ).
(eo=
281
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creasing the pressure on the bag, due to the tail o f the s k y r m i o n , c o m p a r e d to the z e r o t e m p e r a t u r e pressure. In o t h e r words, the tail o f the s k y r m i o n always " s u c k s " the bag to the outside, the pressure b e i n g dep e n d e n t on the height o f the profile f u n c t i o n at R. T h e t e m p e r a t u r e d i m i n i s h e s this height giving rise to an increase o f the external pressure. T h i s can c o m pensate the c o r r e s p o n d i n g increase o f the i n t e r n a l pressure only up to To. N o t e that this critical t e m p e r ature is smaller t h a n 1 / 2 ~ 2 3 4 MeV. In fact, 1/2 is a natural b o u n d for the t e m p e r a t u r e in o r d e r to a v o i d i n s t a n t o n interactions. We w o u l d like to n o t e that the lowest ( d i m e n s i o n less) e q u i l i b r i u m radius R takes values a r o u n d 0 . 8 1.1, b e i n g essentially i n s e n s i t i v e to the t e m p e r a t u r e . So, the b e h a v i o r o f the d i m e n s i o n a l r a d i u s is determ i n e d by F,~(T). F o r e x a m p l e , R ~ 0 . 5 f m for a constant F ~ = 129 MeV. ( T h i s v a l u e should not necessarily be i d e n t i f i e d with the h a d r o n ' s size, since the tail o f the s k y r m i o n is part o f the h a d r o n i c structure in this picture. ) Finally, it should be stressed that the m a i n result o f o u r analysis, n a m e l y the existence o f a critical t e m p e r a t u r e o f the o r d e r o f 100 MeV, r e m a i n s true o v e r a wide range o f v a r i a t i o n s o f the p a r a m e t e r s o f the model. T h e a u t h o r s a c k n o w l e d g e the s u p p o r t f r o m F U N DACION ANDES and FUNDACION ANTORC H A S , u n d e r G r a n t Nr. C-11626. M.L. a c k n o w l edges partial s u p p o r t f r o m F O N D E C Y T ( C h i l e ) , u n d e r G r a n t Nr. 0 7 5 1 / 9 2 C . J.C.R. a c k n o w l e d g e s the support from CONICYT (Chile), under Grant 47/ 92 a n d H.F. acknowledges partial s u p p o r t f r o m CONICET (Argentina).
References
[ 1] For a review, see e.g.F. Karsch, preprint CERN-TH-5489/ 89 (1989).
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11 February 1993
[2]J. Gasser and H. Leutwyler, Phys. Lett. B 184 (1987) 83; B 188 (1987) 477; P. Gerber and H. Leutwyler, Nucl. Phys. B 321 ( 1989 ) 387. [3] A.I. Bocharev and M.E. Shaposnikov, Nucl. Phys. B 268 (1986) 220; H.G. Dosch and S. Narison, Phys. Lett. B 203 ( 1988 ) 155; H. Leutwyler and A.V. Smilga, Nucl. Phys. B 342 (1990) 302; C.A. Dominguez and M. Loewe, Phys. Lett. B 233 (1989) 201;Z. Phys. C 51 (1991) 69. [4] R.D. Pisarski, Phys. Lett. B 110 (1982) 155; M.I. Gorenstein, S.I. Lipskikh and G.M. Zinovjev, Z. Phys. C22 (1984) 189. [ 5 ] A. Barducci, R. Casalbuoni, S. De Curtis, R. Gatto and E. Pettini, Phys. Lett. B244 (1990) 311, and references therein. [6] A. Chodos and C.B. Thorn, Phys. Rev. D 12 (1975) 2733; G.E. Brown, A.D. Jackson, M. Rho and V. Vento, Phys. Lett. B 140 (1984) 285; V. Vento, M. Rho, E.B. Nyman, J.H. Jun and G.E. Brown, Nucl. Phys. A 345 (1980) 413. [ 7 ] For a review, see e.g.R.F. Alvarez-Estrada, F. Fern~indez, J.L. Sfinchez-G6mez and V. Vento, Models of hadron structure based on quantum chromodynamics (Springer, Berlin, 1986). [8] For a review, see e.g. Skyrmions and anomalies, eds. M. Jezabek and M. Praszatowicz (World Scientific, Singapore, 1987 ), and references therein. [9] K.J. Eskola and K. Kajantie, Z. Phys. C 44 (1989) 347. [10] M.F. Atiyah and N.S. Manton, Phys. Lett. B 222 (1989) 438. [ 11 ] G.S. Adkins, C. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552. [12] L. Vepstas and A.D. Jackson, Phys. Rep. 187 (1990) 109 and references therein. [13] H.A. Weldon, Phys. Rev. D 26 (1982) 1393; J.F. Donoghue, B.R. Holstein and R. Robinett, Ann. Phys. (NY) 164 (1985) 233; G. Barton, Ann. Phys. (NY) 200 (1990) 271; P. Fenley, Phys. Lett. B 196 (1987) 175, and references therein. [14] H. Itoyama and A.H. Mueller, Nucl. Phys. B 218 (1983) 349; C. Contreras and M. Loewe, Z. Phys. C 40 (1988) 253. [ 15 ] M. De Francia, H. Falomir and E.M. Santangelo, Phys. Rev. D45 (1992) 2129. [ 16] J. Cleynmans, R.V. Gavai and E. Suhonen, Phys. Rep. 130 (1986) 217. [ 17 ] J. Gasser and H. Leutwyler, Phys. Lett. B 184 ( 1987 ) 83. [ 18 ] H. Leutwyler and A.V. Smilga, Nucl. Phys. B 342 (1990) 302.
P H YSIC S 1.EI T ER S B
Hybrid models at finite temperature and deconfinement H. Falomir Departamento de Fisica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, c.c. 67, 1900 La Plata. Buenos Aires, Argentina
M. L o e w e a n d J.C. R o j a s Facultad de Fisica, Pontificia Universidad Cat61ica de Chile, Casilla 306, Santiago 22, Chile
Received 4 September 1992
We consider a chiral bag model at finite temperature, with a thermal skyrmion of Eskola and Kajantie modelling the exterior sector. Supposing that the "Cheshire Cat" scenario takes care of all zero temperature effects, we show the existence of a critical temperature over which there can only exist a quark-gluon plasma phase, since the bag is no longer stable. This fact is interpreted as the occurrence of a deconfinement phase transition.
During the last years a considerable amount o f work has been done in connection with the deconfining phase transition, induced by thermal effects, in the hadronic world. A m o n g the different techniques employed to discuss or analyze such an effect, we can mention Q C D on the lattice [ 1 ], chiral perturbation theory [2], Q C D sum rules [3], and also some discussions inspired by the old M I T bag model [ 4 ]. The critical deconfining temperature moves between 120 and 160 MeV approximately, depending on the model. Some authors have conjectured that the critical deconfining temperature and the critical temperature associated to chiral symmetry restoration should be the same [5 ]. In this letter we want to reconsider this i m p o r t a n t problem from a different perspective, namely invoking the hybrid models [ 6 ] in which we have as ingredients a chiral bag model with a non-perturbative external pion field configuration, the skyrmion. This approach has been particularly successful in order to analyze different phenomenological properties o f the hadrons at zero t e m p e r a t u r e [ 7 ]. As far as we know the problem of thermal deconfinement in this frame has not been discussed previously in the literature. It is i m p o r t a n t to remark here that the pure skyrm i o n model provides a remarkably successful description o f the nucleon and other baryons and me278
sons after a suitable modification o f the basic model [8]. Recently, K. Eskola and K. Kajantie ( E K ) [9] were able to generalize the S U ( 2 ) skyrmion field configuration to a finite temperature situation, extending the construction o f skyrmions, due to Atiyah and M a n t o n ( A M ) [ 10 ] starting from instantons in ~4. F o r a suitable scale of the instanton, the ansatz o f A M leads to a good a p p r o x i m a t i o n o f the numerical solution o f the variational equation obeyed by the skyrmion [ 11 ]. The consistent construction o f chiral bag models requires a careful analysis o f the zero point energies at the one loop order and o f the cancellation o f ultraviolet divergences occurring clue to the presence o f the bag wall [ 12 ]. The c o m b i n a t i o n of these effects with an external skyrmionic tail provides, as is well known, a highly interesting "Cheshire Cat" description of the nucleon, where all low energies physical observables, as for example the mass, the R M S radius, the isoscalar and vector magnetic moments, etc., turn out to be essentially i n d e p e n d e n t o f the bag radius [ 8,12]. F r o m this point o f view the bag radius does not play any role in determining physical quantities. It corresponds only to a demarcation between two different descriptions of the relevant degrees o f freedom: quarks and gluons inside the bag and, outside, pions and the skyrmion.
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 300, number 3
PHYSICS LETTERS B
The aim of this note is to have a qualitative discussion of the effects of temperature on equilibrium between the two phases, using the thermal skyrmions of EK for the non-perturbative sector. It is well known in finite temperature field theory that no new ultraviolet singularities appear due to the effects of temperature. For example, in Q E D or the 2 0 4 theory we can have a consistent theory renormalizing at zero temperature [ 13 ]. As a second example, we can mention that the usual axial anomaly, responsible for the n o decay, does not receive any temperature dependent modification [14]. Recently, it has also been shown [15] that the ultraviolet divergences in the calculation of the free energy in the chiral bag model do not depend on the temperature. These considerations allow us to overcome the problem of ultraviolet divergences in our case. Essentially, we assume here that all zero point effects have been taken into account at zero temperature, giving rise to the "Cheshire Cat" scenario [ 12 ]. So, if there is a particular equilibrium radius at finite temperature, it should be determined by the new thermal contributions. If for a certain temperature the model is unable to produce an equilibrium radius, we will interpret this fact as the occurrence of thermal deconfinement. In our qualitative approach we will neglect the finite size effects of the bag on the quark and gluon states, considering them as a gas of free particles. We will also assume, that the existence of this populated thermal vacuum inside the bag does not have any influence on the baryonic number of the hadron, neglecting the chemical potential. The description of the external vacuum will be a superposition of the thermal skyrmion and a gas of free massless pions. As we will see, the crucial contribution for deconfinement comes from the non-perturbative configuration. The existence of a defined radius at finite temperature comes from an equilibrium between the external and internal variations of pressures due to thermal effects. From the literature [ 16 ] we have that the internal pressure of the quark gluon plasma is given by Ppl . . . . ( T )
37 --2"r'4
(1)
where we have considered two flavors and three colors. The external pressure of the pion gas is
11 February 1993
Ppions( T ) = 3
7~2T 4 .
(2)
The variation of the pressure due to the skyrmion, APsk(T) = P s k ( T ) - - P s k ( 0 ) ,
(3)
can be calculated from the dependence of the energy of the thermal skyrmion as a function of the bag volume. The energy (mass) of the skyrmion outside the bag, in units of FJ4e (where e is the constant in front of the stabilizing term in the Skyrme lagrangian and F~ is the pion decay constant) is given by M = 4x
[ dr L2 (0I 2 +2 sin2/] -
R
\ Or)
[L
sin~SI-2r (~/] ~ + sin2f]} +~TL \Or/
(4)
where R is the radius of the bag measured in units of 2/eF,~. In the previous expression f d e n o t e s the profile function of the thermal skyrmion of EK [ 9]. It is given by f ( r ) =z~ (1 -
r+½22[l~c°th(IJr)-l/r] dr2+ llzZ24+~r22coth(l~r) ] .
(5)
The constant/z in this formula is 27rT, where the temperature has been expressed in units of ½eFt, and 2 is the size (in the same units as R) of the instanton which gives rise to the skyrmion. Now, the pressure of the skyrmion at radius R and temperature T, PSk= OM/OVbag,is given by
I(R, T) Psk(R, T ) =
47rR 2
(6)
In this formula I(R, T) denotes the integrand in eq. (4), including the 47r factor. At a given temperature a state of equilibrium between the internal and external phases can exist if there is a radius at which the pressures are equal. The equilibrium condition is 2 4
34_2"T4 e F~ [ I ( R , O ) - I ( R , T ) ] 76Jr i = 1287cR 2
(7)
For the numerical analysis of this expression we have used the constant values for F , and e according to ref. [ 1 1 ], fitted from the proton and A masses. In ref. [ 9 ] it is shown that the size 2 varies between 1.45, for/~=0, and 2.5 in a reasonable range of variation of T. It is important to remark that the value o f F , we 279
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have used, 129 MeV, is much smaller than the experimental one, 186 MeV, using the normalization ofref. [ 11 ]. We find that eq. (7) in general has two solutions (i.e. two possible values for the radius for each t e m p e r a t u r e ) , up to a point where both values coincide. Beyond this point there is no solution. This is shown in fig. 1. F o r our estimates we have taken 2 = 1.45. F o r the parameters written above, we find that this critical temperature is Tc = 109.1 MeV. We interpret this fact as the occurrence o f a deconfining phase transition, since for higher temperatures only the q u a r k - g l u o n plasma phase does exist. O u r value for Tc is smaller than those obtained in other models [ 1,2,3 ], but this is due to the small value o f F= we have taken. The critical t e m p e r a t u r e grows linearly with F=. This is shown in fig. 2. The variation o f the critical t e m p e r a t u r e as a function o f the p a r a m e t e r e, for a constant F~, turns out to be much insensitive as the dependence on F . . In fact, we see in fig. 3 that Tc suffers a variation o f less than 10% in a wide region a r o u n d the value given in ref. [ 11 ]. In general, according to EK, the value o f 2 grows with temperature, if we m i n i m i z e the mass of the thermal skyrmion for each value o f T. This is indicated in fig. 4. If we choose this criteria, the critical temperature diminishes, for every value ofF=, but its dependence on the p a r a m e t e r e turns out to be much less sensitive than in the case with a constant value o f 2. This is shown in fig. 5. F r o m our discussion, it
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F~Fig. 2. Here we show the linear dependence of Tc (MeV) as a function of F, (MeV) (continuous line: e=5.45; dotted line: e=7.5). 120.00
110.00
I00.0(
90.0[
6020 0.90
80.00
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3.15
5.40
7.65
9.90
e
................ I
55.00 30.00
,
<3
~_._._-__ 5.00
-
_
~
..............
, II
/)' -20.00 0.70
1.20
1.70 R
2.20
2.70
Fig. 1. This figure shows the right hand term of eq. (7) for three different temperatures (dashed line: 83.2 MeV; continuous line: 109.1 MeV; dotted line: 139.8 MeV). The horizontal lines are the left term. We see that 109.1 MeV is the critical temperature. 280
Fig. 3. We show Tc (MeV) as a function of the parameter e with F== 129 MeV. turns out that F , is the most relevant p a r a m e t e r in order to fix the critical temperature. Finally we explore another possibility, namely the existence of a temperature dependence o f F~ using the well known results of Gasser and Leutwyler [ 17 ] according to chiral perturbation theory ( x P T ) at finite temperature. Although this is a different theoretical framework, we think, it is interesting to explore in this scenario the consequences of a temperature dependent pion decay constant. We know, also from xPT [ 18 ], that the nucleon and pion masses are quite stable with respect to the temperature, at least in the
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F2(T)=F2(O)
6.75
x
1
4 T 2 (2g)2 F ~ ( 0 )
i dxJX2-#~/T2~ 7---1 / '
(8)
,u~tl T
A
4.50
where we have taken only two flavors. It is possible to c o m p u t e this integral exactly by going to the chiral limit w h e r e / z . = 0. In this way we get
2.25
FZ(T)=F2(O) ( 1 -
0.00 0.00
1.00
2.00 2u.
3.00
4.00
Fig. 4. The behavior of k, obtained by minimizing the mass of the skyrmion for each temperature, is plotted against/z= 2nT (2 and T dimensionless).
95.00
70.00
3F~]
(9)
The t e m p e r a t u r e dependence of the p a r a m e t e r e is chosen exactly in the same way, i.e. eo multiplying the same factor o f the previous equation. The result from this analysis is shown in fig. 6. Here we show the d e p e n d e n c e o f Tc on F~ (0). The value o f the critical t e m p e r a t u r e is found using eq. ( 7 ) , where the and are used for different F , ( 0 ) as an input, eo being constant, eo= 5.45. We conclude that in this case the critical t e m p e r a t u r e also rises in a quasi-linear way with F ~ ( 0 ) . The numbers are smaller, as we could have predicted, c o m p a r e d with the constant F , case. The m e c h a n i s m responsible for the occurrence o f this phase transition can be viewed as a consequence o f the concentration o f the skyrmion a r o u n d the origin due to t e m p e r a t u r e [ 9 ]. This has the effect o f in-
F,~(T) 120.00
4T2"~
e(T)
45.00 ;70.00[ 20.00 0.90
,5.15
5.40
7.65
9.90
142.50I
e
Fig. 5. The dependence of T¢ (MeV) as a function ofe is shown for a temperature dependent 2 (continuous line) compared with the constant 2 case (dotted line) of fig. 3. range o f temperatures we consider here. Invoking this fact, for our analysis we have also introduced a temperature d e p e n d e n t skyrmion p a r a m e t e r e, imposing that the total mass o f the nucleon, described as a skyrmion, remains constant as a function o f temperature. The p a r a m e t e r 2 could have, as previously, a d e p e n d e n c e on temperature, but for our discussion we have taken it constant, 2 = 1.45. The expression we use here is the following:
115.00 87.50 60.00 100.00
125.00
t50.00 F.
175.00
200.00
Fig. 6. It is compared the dependence of T¢ on F=( T= 0) (MeV) for a model where F= is constant (dotted line) and where it is assumed that F= depends on T according to chiral perturbation theory (continuous line). 5.45 ).
(eo=
281
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creasing the pressure on the bag, due to the tail o f the s k y r m i o n , c o m p a r e d to the z e r o t e m p e r a t u r e pressure. In o t h e r words, the tail o f the s k y r m i o n always " s u c k s " the bag to the outside, the pressure b e i n g dep e n d e n t on the height o f the profile f u n c t i o n at R. T h e t e m p e r a t u r e d i m i n i s h e s this height giving rise to an increase o f the external pressure. T h i s can c o m pensate the c o r r e s p o n d i n g increase o f the i n t e r n a l pressure only up to To. N o t e that this critical t e m p e r ature is smaller t h a n 1 / 2 ~ 2 3 4 MeV. In fact, 1/2 is a natural b o u n d for the t e m p e r a t u r e in o r d e r to a v o i d i n s t a n t o n interactions. We w o u l d like to n o t e that the lowest ( d i m e n s i o n less) e q u i l i b r i u m radius R takes values a r o u n d 0 . 8 1.1, b e i n g essentially i n s e n s i t i v e to the t e m p e r a t u r e . So, the b e h a v i o r o f the d i m e n s i o n a l r a d i u s is determ i n e d by F,~(T). F o r e x a m p l e , R ~ 0 . 5 f m for a constant F ~ = 129 MeV. ( T h i s v a l u e should not necessarily be i d e n t i f i e d with the h a d r o n ' s size, since the tail o f the s k y r m i o n is part o f the h a d r o n i c structure in this picture. ) Finally, it should be stressed that the m a i n result o f o u r analysis, n a m e l y the existence o f a critical t e m p e r a t u r e o f the o r d e r o f 100 MeV, r e m a i n s true o v e r a wide range o f v a r i a t i o n s o f the p a r a m e t e r s o f the model. T h e a u t h o r s a c k n o w l e d g e the s u p p o r t f r o m F U N DACION ANDES and FUNDACION ANTORC H A S , u n d e r G r a n t Nr. C-11626. M.L. a c k n o w l edges partial s u p p o r t f r o m F O N D E C Y T ( C h i l e ) , u n d e r G r a n t Nr. 0 7 5 1 / 9 2 C . J.C.R. a c k n o w l e d g e s the support from CONICYT (Chile), under Grant 47/ 92 a n d H.F. acknowledges partial s u p p o r t f r o m CONICET (Argentina).
References
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[2]J. Gasser and H. Leutwyler, Phys. Lett. B 184 (1987) 83; B 188 (1987) 477; P. Gerber and H. Leutwyler, Nucl. Phys. B 321 ( 1989 ) 387. [3] A.I. Bocharev and M.E. Shaposnikov, Nucl. Phys. B 268 (1986) 220; H.G. Dosch and S. Narison, Phys. Lett. B 203 ( 1988 ) 155; H. Leutwyler and A.V. Smilga, Nucl. Phys. B 342 (1990) 302; C.A. Dominguez and M. Loewe, Phys. Lett. B 233 (1989) 201;Z. Phys. C 51 (1991) 69. [4] R.D. Pisarski, Phys. Lett. B 110 (1982) 155; M.I. Gorenstein, S.I. Lipskikh and G.M. Zinovjev, Z. Phys. C22 (1984) 189. [ 5 ] A. Barducci, R. Casalbuoni, S. De Curtis, R. Gatto and E. Pettini, Phys. Lett. B244 (1990) 311, and references therein. [6] A. Chodos and C.B. Thorn, Phys. Rev. D 12 (1975) 2733; G.E. Brown, A.D. Jackson, M. Rho and V. Vento, Phys. Lett. B 140 (1984) 285; V. Vento, M. Rho, E.B. Nyman, J.H. Jun and G.E. Brown, Nucl. Phys. A 345 (1980) 413. [ 7 ] For a review, see e.g.R.F. Alvarez-Estrada, F. Fern~indez, J.L. Sfinchez-G6mez and V. Vento, Models of hadron structure based on quantum chromodynamics (Springer, Berlin, 1986). [8] For a review, see e.g. Skyrmions and anomalies, eds. M. Jezabek and M. Praszatowicz (World Scientific, Singapore, 1987 ), and references therein. [9] K.J. Eskola and K. Kajantie, Z. Phys. C 44 (1989) 347. [10] M.F. Atiyah and N.S. Manton, Phys. Lett. B 222 (1989) 438. [ 11 ] G.S. Adkins, C. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552. [12] L. Vepstas and A.D. Jackson, Phys. Rep. 187 (1990) 109 and references therein. [13] H.A. Weldon, Phys. Rev. D 26 (1982) 1393; J.F. Donoghue, B.R. Holstein and R. Robinett, Ann. Phys. (NY) 164 (1985) 233; G. Barton, Ann. Phys. (NY) 200 (1990) 271; P. Fenley, Phys. Lett. B 196 (1987) 175, and references therein. [14] H. Itoyama and A.H. Mueller, Nucl. Phys. B 218 (1983) 349; C. Contreras and M. Loewe, Z. Phys. C 40 (1988) 253. [ 15 ] M. De Francia, H. Falomir and E.M. Santangelo, Phys. Rev. D45 (1992) 2129. [ 16] J. Cleynmans, R.V. Gavai and E. Suhonen, Phys. Rep. 130 (1986) 217. [ 17 ] J. Gasser and H. Leutwyler, Phys. Lett. B 184 ( 1987 ) 83. [ 18 ] H. Leutwyler and A.V. Smilga, Nucl. Phys. B 342 (1990) 302.