LETTERE AL NUOVO CIM-ENTO VOL. 22, 2. 14 5 Agosto 1978

Hagedorn's Equation of State for Dense Matter i n a n E x t e r n a l M a g n e t i c F i e l d ( ' ) .

RAMESH CtIAND (**), D. J. GREEN and G. SZAMOSX

Department o] Physics, University o] Windsor . Windsor, Ontario N9B 3P4, Canada

(rieevuto il 7 Marzo 1978)

In this note, we present the Hagedorn equation of state of dense matter in a strong external magnetic field. In an earlier work, Hagedorn (1) used his bootstrap model to treat the early Universe as a black body containing all particles with B(total baryon number) = Q (total charge) = S(total strangeness) ~ 0. In view of the possibility of a strong primordeal magnetic field (2.s), it seems to be useful to extend Hagedorn calcula- tions to take into account the effect of the magnetic field and put the equation of state on record.

The description of the Hagedorn theory may be found elsewhere (1), therefore, only a few sentences will be devoted hero to sketch the basic features of this theory. These are: 1) there are no elementary hadrons, each one consists of all others including re- sonant states (at least for m-~ c~), 2) there exists a l imit ing temperature To, 3) par- ticles can be created or absorbed spontaneously and instantaneously according to the local conditions; thus energy need not dissipate via kinetic-energy transport but can propagate via vir tual particles of all kinds including the resonant states. This elimi- nates strong interactions from the part i t ion function and hence one can deal with un- determined numbers of all kinds of free hadrons. Therefore, the calculation of the par- t i t ion function and of the equation of state depends on a knowledge of the mass spec- t rum only; let 0 ( m ) d m = number of states between m and m-F din. For low.lying states, the mass spectrum is reasonably well-known experimentally. For these known states, ~(m) can be replaced by delta functions. For states with m > M , we shall take the expression of the bootstrap model of Hagedorn:

(~) e(m) = (me' + m')~ exp [m/Tel,

( ') Work suppor ted by the Nat ional Research Council of Canada. ('*) Present address: 1545 Cherboneau, Detroit , Mich. 48207, U.S.A. (z) R. HAGEDORN': Thermodyna~nlcs o! strong inleractions, CERN Report, CERN-71-12 (1971); .dslron. and /lstrophys., 5, 184 (1970). Other re levant references may be found in these papers. ( ') L. W. JONES: in Proceedings o] the International Conference oN SMmrnetrles and Quark Models, edi ted by RAMESH CHAI~D (New York, N. Y., 1970), p. 225; J. C. WHEELER: in Physics of Dense /Ifa/ter, edi ted by C. J. HAI~'SEN (Boston, Mass., 1974), p. 77. (s) P. J. PEEBLF~: Physical Cosmologll (Princeton, N. J., 1971), p. 244.

582

I I A G E D O R N ' 8 ~ Q U A T I O N O F 8TAT~. F O R D ~ N 8 ~ MAT T ~ .R ~ T C . 5 ~

where

a = 2 . 6 3 . 1 0 4 M o V ! ,

m o = 500 M e V ,

T O = 160 M o V .

We shall choose a sui table va lue for the cut-off mass M. Express ion (1) p rov ides a good fit to the known mass spec t rum (1).

Next , we shall ca lcula te the e q u a t i o n of s ta te . W e shal l use t h e sys tem of uni t s in which h = c = k (Bol t zmann ' s cons tant ) = 1 and shall express e v e r y t h i n g in powers of MeV.

The knowledge of the mass spec t rum al lows one to ca lcula te t he p a r t i t i o n func t ion Z ( V , T, B) in a s t r a igh t fo rward manner . W e shall t ake the ex te rna l magne t i c field B to be along the x3-direetion and shal l a ssume t h a t i t is homogeneous in space and con- s tan t in t ime.

The energy e igenvalues for a charged pa r t i c l e in field B are well known (4) and are g iven by

where

E~ = (p] + ~ )~ ,

~ = 21elB(j + �89 + m ~ , j = O, 1, 2 . . . . .

The degeneracy fac tor for a g iven Pa is

go = l e l B r t l 2 ~ ,

with the usual nota t ion . W e now fol low the procedure of Hagedo rn and e x t e n d i t to the case where B is present and to t he separa te cases of charged and neu t ra l par t ic les . Af t e r in tegra t ion in t he respec t ive phase spaces we ob ta in for t he p a r t i t i o n func t ion the fol lowing express ions :

where

In Z ( V , T, B) = In [Zc~Z~,Zco~] ,

- - [grc(-- 1) + gBc] Kj.( 'a~j /T) , 2y~2 t J--0 n--1

in z ~ = ]-5 VT~ + ~ Z g'~,(- ~)"§ + ~'. g'~ ~O~,~ , /T / , t~'l ' a

r

. . . . K m . 2 Z 2 Q ( m ) m 2 2 d in .

M

( ' ) 3I . H . J O H N S O ~ a n d B . A . LIPP3IANN: Phys. P~v., 76, 8 2 8 ( 1 9 4 9 ) .

5 8 4 RAM~SH CHAND, D. J . GR~.~N and G. 8ZAMOSI

In these expressions CH and N stand for the discrete charged and neutral particles up to the cut-off mass M. CON stands for the continuum, for mass m from M to infinity. Subsequently, F and B refer to Fermion and Boson respectively. The index i indicates the discrete-mass spectrum. K x and K 2 are the Hankel functions as defined in Q) and Q(~n) is given in eq. (1) Finally, the g~'s are given by

gFC = gBO ~

where s t refers to the spin of the particle and x ~ = 0 if the particle is its own anti- part icle and x~-- - 1 otherwise. ~ and v stand for photon and neutrino whose contribu- t ion are in the first term of In Z~.

One obtains the pressure P and the average energy density e by the standard expres- sions

P = T ~ v l n Z ( V , T , B ) ,

1 = V T2 _~ in z ( v , T, B).

For numerical computation, we choose M = 1200 MeV. This somewhat low value simplified the numerical work. The influence of the value of M on the equation of state is, however, negligible.

One notices that the In Zc~ goes to zero as B-~ 0. This is due to the degeneracy factor gr Physically speaking, it describes the ordering effect of the magnetic field or the fact that the free-particle case cannot simply be regained by putt ing B = 0. This is, of course, well known. In the numerical computation the B = 0 case was treated separately namely, the charged particles were lumped together with the neutral particles.

TABLE I. - Pressure p . 10-* (MeV)*.

B (G) T (MeV)

l e o 110 120 130 140 150 159

0 0.1942 0.2925 0.4290 0.6208 0.9037 1.370 2.181

10 is 0.1857 0.2777 0.4042 0.5812 0.8423 1.278 2.050

101~ 0.1942 0.2925 0.4289 0.6207 0.9035 1.370 2.181

l0 Is 0.1939 0.2922 0.4287 0.6205 0.9035 1.370 2.181

10 l~ 0.1793 0.2749 0.4088 0.5982 0.8787 1.343 2.152

1020 0.1141 0.1756 0.2661 0.4022 0.6195 1.011 1.745

1022" 0.1088 0.1631 0.2395 0.3512 0.5290 0.8596 1.516

1025 0.1088 0.1631 0.2395 0.3512 0.5290 0.8596 1.516

The numerical results for pressure p and energy density e as a function of T and B are presented in tables I and II . In addition, we have plot ted the p-T-B surface (fig. 1). The surface e-B-T looks very similar and has not been reproduced here.

}IAGEDORN'S EQUATION OF STATE FOR DENSE ]~IATTER ECC.

TABLE I I . - Energy Density e. 10 -~ (MeV) 4.

585

B (O) T ()IeV)

100 110 120 130 140 150 159

0 0.6340 0.9762 1.495 2.353 4.023 8.352 21.16

10 le 0.5969 0.9108 1.385 2.175 3.744 7.928 20.55

1017 0.6339 0.9762 1.495 2.353 4.023 8.352 21.16

10 TM 0.6342 0.9765 1.496 2.354 4.024 8.354 21.16

1019 0.6220 0.9649 1.485 2.344 4.015 8.346 21.15

102o 0.3917 0.6390 1.048 1.783 3.320 7.509 20.18

1022 0.3472 0.5402 0.8527 1.430 2.724 6.560 18.80

1025 0.3472 0.5402 0.8527 1.430 2.724 6.560 18.80

L /

2.1!5 - "

' " 1

15

N,O \

T(MeV)

12c~"--.. " . . - . ( - . " ,.-. ' . " ' . . . .<~2o B ( G )

100

)22

One no t i ces f r o m t h e t a b l e s t h a t t he v a l u e of p r e s su re a n d e n e r g y d e n s i t y d e v i a t e s v e r y s l igh t ly f r o m t h e f r ee -pa r t i c l e cases as long as B is less t h a n a b o u t 10 '~ G. T h i s m a y be u n d e r s t o o d s ince a t such B fields tho m a s s t e r m d o m i n a t e s . As B inc reases a n d

5 8 6 I{,AMESH CHAND, D. ft. GREEN and G. SZA,~IOSI

approaches 10 2o G (which roughly corresponds to the order of M) the value of p and decrease to their l imiting values. This is again due to the ordering effect of the magnetic field. This can also be seen by noting that the entropy density S = (p + e) /T decreases with increasing magnetic field.

The contribution of the continuum increases at a given B as T increases. The same contribution becomes infinite at T ~ T 0, which is a general consequence of the Hage- dorn model.