Meson-meson interactions and the equation of state for dense nucleons

Volume 56B, number 2

PHYSICS LETTERS

14 April 1975

MESON-MESON INTERACTIONS AND THE EQUATION STATE FOR DENSE NUCLEONS

OF

V. CANUTO* Institute for Space Studies, Goddard Space Flight Center, NASA, New York, New York 10025, USA and B. DATTA Department of Physics, Oty College of the Oty University of New York, New York, N. Y. 100 31, USA Received 25 November 1974 The effect of the interaction among mesons upon the equation of state for a high-density system of nucleons is studied, and it is found that such interactions do not significantly alter the asymptotic behaviour of the equation of state.

In a recent paper Walecka [1] has computed the equation of state P = p ( 6 ) (P = pressure; 6--- energy density) of a high-density system of nucleons interacting via scalar and vector mesons. The computation was performed after hnearizing the meson field equations by replacing the meson fields by their constant expectation values. However, the interaction among the mesons was not included. In this letter we study the effect of such an interaction on the asymptotic behaviour of P = P ( 6 ). It has recently been suggested by Lee [2] that nucleons interacting via scalar mesons might lead to the existence of abnormal nuclear states. This opens the possibility of the existence of abnormal neutron stars. It is well-known that the exchange of scalar mesons (o) represents attractive N-N potential, and that in order to have a realistic equation of state, vector meson exchange has also to be included. We do not consider vector mesons here as the aim is not to obtain realistic equations of state, but merely to see the effect of the forces among the scalar mesons upon the equation of state. We choose our Lagranglan density for the N-N interaction to be .Q =./~ l(~b, q~) + ~ 2(~b) * Also with the Department of Physics, City College of the City University of New York, New York. 148

where

1(~, ~) = - ~-(%0 + mN)~ + g ~ ./9 2(~) = _ ~_(0/~tb)2 _ !2m2'~2av- k¢ 4 (~b) 2

m2adp2

2(1 + F - 2 ~ 2 ) 2

2(1 + F-2~b 2)

: case (a)

:case (b)

= - l - 4 [ X / 1 +! 4((Ou~)2+m2~2")- 1] : case(c) Here ~ is a nucleon field of mass m N and ~bis the scalar meson field of mass m a. Our notations are such that ~i = 1 = c, and a 4-vector is written as Au = (A, A4) = (,4, iAo). The Lagrangian (a) contains, besides the free field terms, a counter term k~b4. Lagrangian (b) was proposed by Weinberg [3] in connection with 7r-lr interaction. Lagrangian (c) was originally proposed by Born and Infeld [4] for electrodynamics and studied by Heisenberg [5] in connection with high-energy physics. For our discussion, cases (b) and (c) can be considered as purely phenomenological ways of treating the o-o interaction. We compute the stress tensor Tuv following the mean field approximation of Walecka, and obtain the following values for pressure and energy density:

Volume 56B, number 2

P -- g' ~ ~ .rl.

i i -- 2 :1* - ]

PHYSICS LETTERS

14 April 1975

100

1

0 75

c=- r44=r +rl t SP/~

where

///'*"

05O

sss/ 025

0 OClola

kF

*= 9" f r~ (2~) 3 o

I

iOis

I

I

i i

i0 ~r

P (g cm "s)

d k k2%/~-~m~2, Fig. 1. Plot o f 3P/c versus p = C / c 2 for Lagranglan (a). The dashed curve corresponds to k = 0.

m N - mN - gq~o and IO0

T 1 --- ~2 rn2,h2 + X~b4o G~O

:case(a)

i

~

i

T

I ~lll

--

2

O'--0 ~

I

,

I II

i

J

J

,/",,..r.leo.~

3P/~ O7!

_ ! m 2 ~2(i + F-2~2)-1

r

/ I~

:case (b)

--0"

=1-4 ~/1,+14m2~o2 _ 1 0 2!5i

Here q~o is the constant expectation value of ~b, and is obtained by solving self-consistently the meson field equation which for the three cases are given by: mayo2

+ 4 ~,~3o = I

2

: case (a)

tf/~,'/

00010~

i01s p (g cm -S)

mar

(1 +F-2~o2j

tpo

(I+F-2~o2) 2

=I

= IVI +14m2~b2 a o

:ease(c)

Toso/

kF

g~

J

i017

:case (b)

where I=

i

Fig. 2. Plot of 3P/c versus p = ~/e2 for Lagrangian (b). The dashed curve corresponds to the case with no a-a interaction.

I O0

2

J

.. 2 ,::-2.~3

mac o

ma¢o

J

dkm~

3P/~

J~~ , , - ~ ' ~ /

//// //

:

(2.) 3 To see how the interaction among the scalar mesons affects the asymptotic behaviour 3 P = d , we .plot 3P/~ versus O, where 0 -- g / c 2. These plots are shown in figs. 1, 2 and 3. The experimental value of m a is not known for certain. In our calculations we have taken it to be 1.1 5 GeV, for this value gives the correct binding energy for normal nuclear state in the hard-sphere model calculations [2]. We have assumed

°°°o,S

jo,6

'

......

io,r

p (g cm-a)

Fig. 3. Plot of 3P/~ versus p = ~ / c 2 for Lagrangian (c) for I = 1 fm and 1 = 2 fm, respectwely. The dashed curve corresponds to the case with no o-o interaction.

the values o f g and X to be given by the o-model [6], in which if one neglects the pion mass one gets 149

Volume 56B, number 2

PHYSICS LETTERS

g2/4rt = rtN coupling strength .~ 15 x__ g2 m J2 m N 2 ~ 35.4. The parameter F in Lagrangian (b) is about 190 MeV for the case considered by Weinberg [3] ; we have adopted the same value here. The parameter I in Lagrangian (c) corresponds to a 'length' in natural units. Since the value of this 'length' is unknown, we have performed the computations for two values of l: 1 fm and 2 fm. The plots indicate that the interaction among the scalar mesons makes the equation of state softer for case (a) and stiffer for cases (b) and (c).

150

14 April 1975

One of the authors (B.D.) would like to thank Dr. R. Jastrow for his hospitality at the Institute for Space Studies, New York. This research was supported in part by a grant from NASA (RF-05560).

References [1] J.D. Walecka, Ann. Phys. 83 (1974) 491. [2] T.D. Lee, Columbia University preprint: CO-2271-27 (1974). [3] S. Weinberg, Phys. Rev. 166 (1968) 1568. [4] M. Born and L. Infeld, Proe. Roy. Soc., Ser. A144 (1934) 475. [5] W. Heisenberg, Z. f. Physik 133 (1952) 65. [6] B.W. Lee, Chiral dynamics (Gordon and Breach, 1972).