AN EXTENSION OF THE STATISTICAL BOOTSTRAP MODEL

TO INCLUDE STRANGENESS


A. S. Kapoyannis*+, C. N. Ktorides* and A. D. Panagiotou*



*Division of Nuclear and Particle Physics, University of Athens,

GR-15771 Athens, Hellas


+Institute of Nuclear Physics, NCSR Demokritos,

GR-15310 Agia Paraskevi Attikis, Hellas





Abstract

We extend the Statistical Bootstrap Model (SBM) in order to describe hadronic systems which

carry strangeness. We establish that the hadronic phase can exist only in a region of the phase

space (T,  
q , s ) , which is bounded by a "critical" surface. The SBM leads to a mass spectrum


with the asymptotic form : ( )
m = -
m e [
xp T T*], where the exponent can take different

values. We first employ the usual version of the SBM ( =2) and impose strangeness neutrality in

order to obtain the relation between the temperature and the chemical potentials  ,  . We find
q s


an unphysical behaviour of  near the critical surface. Subsequently, we study another version
s

of SBM where =4 and we are led to a realistic as well as acceptable behaviour. In this case we

are also able to relate the parameter of the model, T0 , (the critical temperature at zero  ) with
q


the MIT bag constant B . The relation between (T,  
q , s ) can be used to predict the particle

ratios on the critical surface at the limit of the Hadron Gas phase.





1


1. Introduction


The Statistical Bootstrap Model (SBM) [1,2,3] constitutes an effort for a self


consistent thermodynamical description of relativistic, multiparticle systems. The


basic idea is to bypass the employment of interaction at a distance between


particles, in favour of successive levels of organisation of matter into particle-like


entities of increasing complexity known as fireballs. In the context of strong


interaction physics the original set of input particles corresponds to all known


hadrons.


The agent which carries dynamical information in the bootstrap scheme is the


mass spectrum ( )
m of the fireballs. It satisfies an integral equation with the


generic form [4]:

n n
1
( )
m = (m - m (1)
0 ) + m- m
i (mi )dmi
=2 ! =1
n n i i =1


where m0 represents the mass of an input particle and mi , i=1,2,...,, stands for

the fireball masses in ascending order of complexity.


It was within the framework of the SBM that the idea of the existence of a


critical temperature, beyond which the hadron phase of matter gives way to a


different one, was first introduced [5]. Nowadays such a scenario receives concrete


support from QCD, through lattice computations [6]. The emerging picture calls


for a quark-gluon plasma (QGP) phase. Given the potential significance of


strangeness as far as the identification of the QGP phase is concerned, we have


undertaken the extension of the SBM so as to include the quantum number of


strangeness.





2


2. Incorporation of Strangeness into the Bootstrap scheme


The generic form of the bootstrap equation in the presence of strangeness is


the following:


B(
~ p2 ) (~ 2 ~
p , ,
b )
s = g B 2 2 2
- +
bs
(p ) 0(p mbs )
        





input term

n n n
1
4
+ p- p b- b s- s
= = = =
n 2 n! i K i K i
i 1 {b 1 1
i } i {si } i

n

B(
~ p 2 2 4
,
i ) (~p ,b ,s
i i i )d pi (2)
i =1


where (
b )
s stands for baryon (strangeness) number and the g represent the
bs



degeneracy factors.


Let us provide the main ingredients which have been employed in order to



reach the above result. We use a system of units in which = c = k = 1 and

introduce Touscheck's integration measure [7], according to which


V V
2 p 
d 3 p 2 2 4 2 2
, , - , , , (3)
3 (m b s)dm 3 0 ( p m )d p (m b s)dm
h (2 )

where V  is a boosted four-volume, which is parallel to the four-momentum of a


given particle:


V
V  p 
= (4)
m


This relation underlies the specification that each fireball carries its own volume


[8].Note that the expression on the left side of the arrow in (3) counts the number


of available particle states in the rest frame of volume V, which are characterised


by the mass spectrum ( )
m .


We have further converted ( )
m to the spectrum function (m2 ) according

to





3


( )
m dm = (m2 )dm2

and have set


V
2  p V
2 m
(
B p2 ) = ( = . (5)
2 )3 (2 )3

We have also introduced a rearrangement of the form

~
(
B p2 ) ( p2 ,b, s) B( p2 ) (
~ p2
, b, )
s . (6)


The specific choice one makes in using the above relation is of crucial


significance as far as the dynamical description of the SBM is concerned. Next we


carry out three Laplace transformations, one continuous and two discrete, which


lead to the replacements:


( P ,b, s , ,
 ) (  .
B S )


The new set of variables will be vested with a thermodynamical content in


terms of the inverse (four) temperature and the fugacities for baryon and


strangeness quantum number, respectively.


In the centre of mass frame and with the transformation

= 1 3 1 3 1
, , which introduces the quark fugacities (q up-down,
B = -
q s B S

s strange), we arrive at the following form for the bootstrap equation


( , , ) = 2 (
G , , ) - e [
xp (G , , +1 , (7)
q s q s q s)]

where ( , , is the known input term:
q s)


( b s - p
= e g B p2 p2 2 4
0 - m d p
B S ) B   ~
, , S bs ( ) ( bs )
b=- s=-




2 ~
= b s g B m m K m (8)
B S bs ( 2bs) bs
1 ( bs )
b=- s=-




4


and (
G , , is the mass-spectrum containing term:
q s )


-  ~
(
G ) b s
= e p
, , B 2 ~ 2 , , 4
S ( p ) ( p b )sd p
B S B
b=- s=-



2 ~
= mB( 2
m ) (
~ 2
m , , ) 2
K m dm , (9)
B S 1( )
0

which involves the unknown function ~
. In eqs. (8) and (9) K denotes the

modified Bessel function of the second kind.


The bootstrap equation (7) displays, in the - G plane, Fig. 1, a square root

branch point at


(T , , = ln 4 - 1 , (10a)
cr q cr s cr )

(
G T ,  , = 2 . (10b)
 ln
cr q cr s cr )


Eq. (10a) defines a critical surface in the 3-d space (T,  ,  which sets the
q s )


limits of the hadronic phase. Points radially outside the critical surface belong to


unphysical solutions of the bootstrap equation and are thereby assigned to a region


where a new phase of matter, presumably the QGP phase, makes its appearance.


Let us also introduce the temperature T0 according to


(T , = 1, = 1 = ln 4 - 1 , (11)
0 q s )

which constitutes the highest temperature beyond which the Hadron Gas phase


does not exist.


We finally turn our attention to the thermodynamical description of the


system. To this end we take into account the fact that according to the bootstrap


scheme the number of available states in a volume d 3 p around p, baryon number






B and strangeness number S is given, in covariant form, by





5


2V ext p 

2 4
~ p ,b, s d p , (12)
3 ( )
(2 )

where V ext
 is the total external (four) volume available to the system. It is a


constant as far as the integration over d 4 p is concerned. Accordingly, the grand


canonical partition function for the system reads, in covariant form,


V p 
 2 
ln Z( ,V , , ) b s p
2
4 .
3 
= -
S (~p , ,b )
=- =-
b s (2 ) s e d p
B S B




Switching to quark fugacities, choosing the four-vectors V  and  to be parallel


and going to the frame for which  = ( , ,
0 ,
0 )
0 [8] we write

V
ln Z( ,V , , ) = m2 (
~ m2 , , )K 2 . (13)
2 ( )
m dm
q s q s

2 2 0

Our problem now is to express the above partition function in terms of the function

(G , , which contains the bootstrap mass spectrum. Once this is done we
q s )


shall be in position to extract specific results from the extended SBM, via the


inclusion of strangeness.




3. Connection with Phenomenology and Results


Let us return to equation (4). The (fireball) volume to mass constant


provides a quantity that can be related to the MIT bag model [9]. We set


V V
i
= = 1 , (14)
m m 4B
i



where B is the MIT bag constant and where V (m denotes the volume (mass) of
i i )


the fireball. The first equality in (14) comes from the assumption that the volume


(mass) of a given fireball is the sum of the volumes (masses) of the constituent





6


fireballs.


Let us assess the splitting between the (
B p2 ) and ( p2 ,b, )
s in the bootstrap


equation. We start with the "natural" definition of (
B p2 ) as given by (6). Here, we

have a purely kinematical assignment to this quantity so all dynamics of the


bootstrap model are carried by ( p2 ,b, )
s [10]. Setting (
B m2 ) H m2
we find in
0



this case


2
H = . (15)
0 (2 )34B

~
A rearrangement of the factors B and ~
would imply a behaviour of the

~
form B(m2 ) = const md . Any choice for which d 2 entails an absorption of part

~
of the dynamics into B . Traditionally, SBM applications have centered around the

~
choice B(m2 ) ~ m4. Setting (
B m2 ) H m4
we are now obliged to introduce a
2



reference mass scale ~
m in order to relate ~
with :

m2
(
~ m2 ,b, s) 2
= . (16)
2
~ (m ,b, )s
m


We also determine, for this case,


2m2
~
H = . (17)
2 (2 )34B

We stress that for any choice other than the one given by (6), one is forced to enter


a reference mass scale into SBM descriptions.


Given the above remarks, relevant to the phenomenological connection with


QCD, let us turn our attention to the asymptotic behaviour of the mass spectrum


function ( )
m , as m . It can be shown [11,12] that

~
( 2 ) (
~ 2 ,{ }) -


({ }) 3
e [
xp *
B m m C m m T , (18)
m ({ })]



7


where T *({ }
) satisfies the criticality equation, c.f. Eq. (10a). In the above relation

{ }
is a collective index for fugacities, while (
C { }
) is a quantity independent of

mass.

~
For a given choice B(m2 ) = const md we have








( -
   
 

, (19)

 { }) ({ })
[ ({ })]

where c=3+d.


In the literature it is more common to refer to the index = c - 1, which


gives the exponent entering the asymptotic behaviour of (
~ ,
m { }
). Thus the

choices, entailed by relations (15) and (17), correspond to = 4 and = 2,

respectively. These, in fact, happen to be the two cases which facilitate analytic


procedures linking the canonical partition function to the term G , which contains


the mass spectrum, and eventually, through the bootstrap equation, to the input


term . We find, for = 2 [13,14],

V
ln Z(V , , , ) = - 2
G
3
q s ( , ,q s)
(
2 ) H
2



V 1 ( , ,
q s )
= - (20)
3
4 H G( , ,
q s )
2 2 - e

and for = 4 [14],

V
1
ln Z( ,V , , ) = 3 . (21)
3 3 x (Gx, ,
q s )dx
q s
4 H0

Now, hadronic interactions pertain to physical situation where the total


strangeness number is zero. In the thermodynamic context of the SBM the relevant


condition is





8


ln Z(V , , ,
q s )
S = = 0 . (22)
s s (V, ,q)

The above relation determines a surface in the (T,  ,  space. For = 2,
q s )


we determine


G( , , 2
q s )
e ( , , , , , ,
q s ) ( q s) ( q s)
+ = 0 (23)
2
G( , ,
q s )
2 - e s s


and for = 4,

T 1 1 ( y, ,
q s )
dy = 0
5
y . (24)
0 2 - e [xpG(y s
, ,
q s )]

The derivations of the above equations along with numerical approaches to


their solutions will be given elsewhere [14].


Before we present results from our studies of the critical and the S = 0

surfaces for the cases = 2 and = 4 we should comment on the following

potential disparity between them. According to the analysis conducted in [8] these


two values for lie below and above the threshold value of = 7 2 , respectively.

This separates the regime where the energy density of the hadronic system goes to


infinity, as T T < 7 2 and where it stays finite ( 7 2) . On this basis, the
cr ( )

= 4 value appears to be the most desirable one, as far as the existence and

observability of a QGP phase is concerned. On the other hand Hagedorn and


Rafelski [13] have shown that the infinity problem for < 7 2 can be bypassed


through finite volume corrections. In fact, it was on this basis that the = 2 choice

has dominated the theoritical scene as far as the SBM is concerned.


In Fig. 2a we depict the profile of the critical surface for = 2 and = 4 by




9


displaying two characteristic cuts of this surface perpendicular to the  axis. Fig.
s



2b depicts a similar setup with the cuts now occuring perpendicularly to the q


axis. It is observed that the results depicted in the above figures seem to advocate


similar qualitative behaviour between the two choices of = 2 and = 4.

A radical departure between the two values of appears as soon as one

compares the way by which the S = 0 surface meets the critical one. Leaving the

discussion of the numerical methodology which produces the corresponding curves


to our forthcoming paper [14], let us explain the way we have chosen to display


the relevant results. In Fig. 3 we show how, for different values of T0, the S = 0

surface joins the critical surface along the direction  T = 0 4
. in the  - T
q s



plane. The almost vertical lines represent the profile of the critical surface near the

 - T plane (the  axis is not shown) which is, in fact, cylindrical. For = 2
q q



(Fig. 3a) we have the unphysical behaviour of a rising chemical potencial  as we
s



approach the end of the hadronic phase. By juxtaposition the meshing between the


two surfaces for = 4 (Fig. 3b) has a desirable, as well as reasonable behaviour. It

is on this basis that we consider the = 4 version as the most realistic one, as far

as the approach to the QGP phase from the hadronic side is concerned and it is for


this case that we intend to focus our SBM-based analysis on particle ratios.


For comparison we have added to the same diagram the intersection of the


plane  T = 0 4
. with the surface S = 0, which comes from solving the equation
q



F (T) -1 - -1 + 2 - -1 -2 +
K ( s q s q) F (T)
Hyp ( s q s q )
(25)
F (T)( 2 - -2 -1 + 3 - -3
2 3 = 0 .
s q s q ) F (T)( s s )

The above equation is derived for an ideal hadron gas (see for example [15,16]).





10


Fig. 4a(b) depicts the projection of the intersection of the surface S = 0

with the critical surface for different values of T -
0 on the  plane, when
s q

= 2 ( = 4).

Fig. 5 gives a 3-d picture of how S = 0 curves along different directions in

the  - T plane and intersects with the critical surface for a specific value of T
q 0,



when = 4.

Finally in the framework = 4 model one can directly relate T0 and the bag

model parameter B . In table 1 we show how the corresponding values pair up. On


a phenomenological basis one has the option of using as input either a value for T0


or a value for B , an occurrence which further underlines the advantages of the

= 4 model.





T0 (MeV) B1/4 (MeV)


150 154.690


160 177.120


170 201.170


180 226.704


190 253.578


200 281.655


Table 1. Connection between T0 and B.





11


4. Summary and Conclusions.


In this paper, we presented an enlargement of the SBM scheme which


includes the strangeness quantum number. Our efforts were directed towards the


determination, in the 3-dimensional (T,  ,  space, of : 1) the critical surface
q s )


which sets the limits of the hadronic phase and which is determined by the


bootstrap equation and 2) the S = 0 surface, relevant to hadronic processes,

which is determined from the partition function. Finally we compared two specific


SBM versions corresponding to the values = 2 and = 4 of the exponent,

which enters the asymptotic expression for the mass spectrum ( )
m . We found


desicive advantages of the second ( = 4), as opposed to the first version which

has dominated SBM studies in the past. Note that the = 4 version admits, as

direct phenomenological input from QCD, the value of the MIT bag constant and


relates it uniquely to T0 .



Acknowledgement. We wish to thank Prof. R. Hagedorn for useful and interesting discussions.





12


References


[1] R. Hagedorn, Suppl. Nuovo Cimento 3, 147 (1965); R. Hagedorn and


J.Ranft, Suppl. Nuovo Cimento 6, 311 (1968); ibid 6, 311 (1968)


[2] R. Hagedorn, 1985 "Springer Lecture Notes in Physics", 221 ed. K. Kajantie


(Berlin, Heidelberg, New York) p.53


[3] R. Hagedorn, 1995 "Hot Hadronic Matter" NATO-ASI-Series B346 eds.


J. Letessier et. al., p.13


[4] S. Frautschi, Phys. Rev. D3, 2821 (1971)


[5] N. Cabbibo and G. Parisi, Phys. Lett. 59B, 67 (1975)


[6] L. D. McLerran and B. Svetitsky, Phys. Lett. 98B, 195 (1981); J. Kuti,


J. Polonyi and K. Szlachanyi, Phys. Lett. 98B, 199 (1981); J. Engels,


F. Karsch, I. Montvay and H.Satz, Nucl. Phys. B205 [FS5], 545 (1982); ibid


Phys. Lett. 101B, 89 (1981)


[7] B. Touschek, Nuovo Cimento 28B, 295 (1968)


[8] R. Hagedorn, I. Montvay and J. Rafelski, "Hadronic Matter at Extreme


Energy Density", eds. N. Cabbibo and L. Sertorio, Plenum Press, New York,


49 (1980)


[9] A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, V. F. Weisskopf, Phys. Rev.


D9, 3471 (1974)


[10] R. Fiore, R. Hagedorn and F. d' Isep, Nuovo Cimento 88A, 301 (1985)


[11] W. Nahm, Nucl. Phys. B45, 525 (1972)


[12] J. Letessier and A. Tounsi, Nuovo Cimento 99A, 521 (1988)


[13] R. Hagedorn and J. Rafelski, Phys. Lett. 97B, 136 (1980)


[14] A. S. Kapoyannis, C. N. Ktorides and A. D. Panagiotou, paper in preparation





13


[15] J. Cleymans, K. Redlich, H. Satz, E. Suhonen, Z. Phys. C58, 347 (1993)


[16] M. N. Asprouli and A. D. Panagiotou, Phys. Rev. D51 1086 (1995);


A. D. Panagiotou, G. Mavromanolakis and J. Tzoulis, Phys. Rev. C53 1353


(1996)



Figures


Fig. 1. G as a function of .

Fig. 2. (a) Comparison of the intersections of planes of constant s-quark chemical


potential  with the critical surface (T,  , = ln4 - 1 for the values
q s)
s



of = 2 and = 4 and for T0=180 MeV. (b) Comparison of the

intersections of planes of constant q-quark chemical potential  with the
q



critical surface (T,  , = ln4 - 1 for the values of = 2 and = 4
q s)

and for T0=180 MeV.


Fig. 3. (a) Projection on the plane (T,  of intersections of planes of constant
s ) q



(  T = 0 4
. ) with the surface S = 0 for different values of T
q 0 , when



= 2. (b) Projection on the plane (T,  of intersections of planes of
s )


constant (  T = 0 4
. ) with the surface S = 0 for different values of
q q



T0 , when = 4.


Fig. 4. (a) Projection on the plane (  , of the intersection of the critical
q s )

surface and the surface S = 0, when = 2. (b) Projection on the plane

( , of the intersection of the critical surface and the surface S = 0,
q s )

when = 4.

Fig. 5. Intersections of planes of constant with the surface S = 0 and the
q





14


intersection of this surface with the critical one, for T0=185.655 MeV,


when = 4.





15








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