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Equation of motion approach to the meson-meson interaction in a nuclear medium
Volume 217~ number 4
PHYSICS LETTERS B
E Q U A T I O N O F M O T I O N A P P R O A C H TO T H E M E S O N - M E S O N IN A NUCLEAR MEDIUM
2 February 1989
INTERACTION
M.N. S 1 N H A - R O Y Department of Physics, Bidhannagar College, Salt Lake, BF- I42, Sector II, Calcutta 700 064, India
and S.S. M A N D A L Department ~?fPh.vsics, Maulana Azad College, 8. R.A.K. Road, Calcutta 700 013, India
Received 14 August 1988
We present a method for calculating the effective interaction between two mesons in a nuclear medium by solving the equation of motion of a double-time retarded commutator, defined to represent the propagation of two mesons. The effective interaction is found to be non-local and repulsive.
The construction of a potential model for h a d r o n s based on the strong coupling flux tube limit of Q C D has recently generated considerable interest in the study o f m e s o n - m e s o n scattering [ 1 ]. This strong coupling limit ( r e g i m e ) describes the long range p h e n o m e n a o f interest in nuclear physics. The flux tube rearrangement terms [2 ] which are assumed to be responsible for m e s o n - m e s o n (g-re) scattering act as a non-local potential between mesons. This non-local potential is attractive but numerically very much smaller than the typical strong interaction term [2] and fails to predict anything about the existence o f a two-meson b o u n d state. This situation suggests that the m e s o n - m e s o n scattering be e x a m i n e d using different approaches. One possibility is to assume meson exchange between two interacting mesons [2,3 ]. According to this model the ~ - ~ scattering takes place via the f o r m a t i o n and decay or exchange o f a 9-meson. Kokski and Isgur [3 ] have o b t a i n e d a detailed description of such processes in terms of the string breaking mechanism. The c o m p a r i s o n o f the results o b t a i n e d from the flux tube rearrangement approach with that o b t a i n e d from mesonic exchange and decay via the string breaking m e c h a n i s m is not conclusive. Therefore, we feel that it will be interesting to reexamine the m e s o n - m e s o n scattering again from a different angle and to check whether the m e s o n - m e s o n effective interaction is attractive and non-local, a n d whether the two-meson b o u n d state can exist in nuclear matter. In this letter, we calculate the effective m e s o n - m e s o n interaction associated with the exchange o f a n u c l e o n hole pair [4,5] in a nuclear m e d i u m . This process yields a repulsive and non-local effective coupling between two mesons for both m o m e n t u m d e p e n d e n t a n d point type interactions between the two-meson and the nuc l e o n - h o l e p a i r channel. We define a d o u b l e - t i m e r e t a r d e d c o m m u t a t o r to represent the p r o p a g a t i o n o f two mesons and solve the equation of m o t i o n ( E O M ) it satisfies. We have used a decoupling procedure devised by one of us [ 6 ], according to which the higher-order meson G r e e n ' s functions that a p p e a r when writing the equation o f m o t i o n for the retarded c o m m u t a t o r o f the two mesons, are set equal to lower o r d e r through a certain p r o p o r t i o n a l i t y constant. This p r o p o r t i o n a l i t y constant is d e t e r m i n e d by conserving the frequency m o m e n t s to infinite order under the 369
Volume 217, number 4
PHYSICS LETTERSB
2 February 1989
free particle approximation. Using the solution of the retarded commutator we obtain an expression for the twomeson correlation function F(Q, o9) which can be written in the form
Fo(Q, ~o) F(Q, co) = 1 -ga(Q, o)) Fo(Q, ~o) '
( 1)
where Fo ( Q, o)) is the bare two-meson propagator and g4 (Q, o9) is the effective strength of the meson-meson interaction. The system of interacting pions and nucleons is described by the following hamiltonian:
H= ~
EpC~Cp~+
pa
~, o)(k)(b~bk+½)+Hl+H~,-,
(2)
k < kc
where Ep = p z~ 2m is the free nucleon energy, do(k) = (m~ + k 2)~/~ is the usual pion energy and m is the nucleon mass, with h=c= 1, in these units. The nucleon creation (annihilation) operators are denoted by C ~ (C~). The pion field is represented by operators b~ (bk) corresponding to the creation (annihilation) of a pion. The interaction term//i describing the bilinear coupling of pions to nucleons is given by
I+, =
y=
~ v4(k,
[kl,lk[
k')(b_.+bU (b_~. +b~.)CLC,+~+,.~,
(3)
p~
(k+k')~O
where ~:4 describes the basic interaction strength between the two mesons and the nucleon-hole pair channel. The residual short range interaction between nucleons is denoted by H~ and is not of immediate interest in the present discussion, although its role may be important in the calculation of the nucleon propagator. The scattering of two pions coupled by the Ht term in eq. (3) is diagrammatically shown in fig. 1. This process yields a nonlocal effective interaction g4 between two mesons resulting from the exchange of a nucleon-hole pair. In eq. (3) k~ is an appropriately chosen pion cut-off momentum for this calculation. Now we introduce the two-pion correlation function F(Q, ~o) as
F(Q, o~)= ~ Fp,p~(e, oJ) , pip2
where Fp,p2(Q, o~) is the Fourier transform ofF,,p2 (Q, t) defined by
Fp~,~(Q, t)=-i0(t)([{~z~(t)~p'+Q(t)-F~'(t)FÈ'+~(t)}~{~z~2+~(~)~zÈ2(~)-F~+Q(~)FÈ~(~)}])~
(4)
where ~zk(t) = b_ k(t) + b~ (t) and F, (t) = b_r (t) - b~ (t). This has been introduced in the same spirit as defined for the fermion propagators in refs. [6,7 ]. The angular brackets denote the expectation value over the exact ground state of the coupled system. The function Fr,p_~(Q, t) represents the propagator of two pions. Using the relation
idO/dt=~O/~t+ [0, H I , the equation of motion for Fr,p,_(Q, t) is written as
i2(d2/dt2)Fp,p~(Q, t) 86(t)(fi~,v2 +6p2._v,_Q)[co(pL)+~o(p L+ Q ) ] + [~o(p~)+o~(p~ +Q) ]ZFp~p~(Q, t~+ F <~ co t)
(5)
Fig. 1. The equation for the scattering of two mesons, Mesons are denoted by wavylines and nucleons by solid lines. The heavy dots represent meson-nucleoninteraction. 370
Volume 217, n u m b e r 4
PHYSICS LETTERS 13
2 February 1989
where
F ~p Ip2 > t\ o~-'~ t)=-32iO(t)
2
~"4(Pl , k ' ) ~ ) 4 ( - - P l - Q ,
k")
Ik' I.Ig' I
(p) + k ' ¢ 0 )
(--pj -Q+k"
× ~
¢o)
([~Zk(t)~*"(t)C+~.(t)Cp+.,+k'..(t)C~'~'(t)Cp'-~,-(2+k"..'(t),
pa,p' ~7'
{~L+Q(0)Tr~(o) -FL+Q(0)F.2(0)}] ) .
(6)
Taking the Fourier transform on both sides of eq. (5) and summing overp2, we obtain
{092- [co(p,)+co(p, +Q)]2}Fp,(Q, co)=16[co(pt)+co(p, +Q)]+F~,')(Q, co) ,
(7)
where
F~,,(Q, co)= Z G~.2(Q, co), F~)(Q, o))= Z F~L(Q, co)" P2
(8)
P2
We now introduce the decoupling procedure, i.e. express the higher-order pion Green's function F ~ ~(Q, co) as a linear combination of all F's:
F~; )(Q, co)=A,,(Q, co) ~ F~(Q, o9) ,
(9)
$
where Ap, (Q, co) is the proportionality constant. Substituting eq. (9) in eq. (7) and using the fact that 2~ F,.(Q, co) = F ( Q, co) we obtain
F.~ (Q. co) =
16 [co(p~ ) +CO(PL+Q) ] +A,, (Q, co)F(Q, co) co?-- [co(t,, ) + co(.o, + Q)]2
(lO)
Summing both sides ofeq. (10) overpl and solving for F(Q, co) we get
F(Q, co)= ~p' 16[o.~(p,)+co(p, +Q)]/{co2
[co(p~)+co(p|+Q)]2} 1 -- ~,, Am (Q, co)/{co2_ [co(p, ) +co(Px + Q ) ]2}
(11)
Comparing eq. ( 1 ) and eq. ( 11 ) the effective pion-pion interaction is given by g 4 ( Q , o9 ) _~.
1 x" A.~(Q, co) Fo(12, co) ~ co2_ [co(p,)+co(t,~ +12)12
(12)
with
Fo(Q, c o ) : ~ { [co-o)(p~)-o)(p~ + Q ) ] - ~ - [co+o)(p,)+co(p, + Q ) ] - ' } .
(13)
Pl
Following eq. (9) Ap, ( Q, co) may be written as
Ap, (Q, co) = F~'I)(Q' co) F(Q, co)
(14)
which, with the help of the Fourier transform, can be expressed as
f ~ F ~r~) ( Q, t)ei~tdt Ap,((2, c~) = J~_~ F( Q, t )ei°~'dt '
(15)
where the expression for F~,~) (Q, t) can be represented by 371
Volume 217, number 4
PHYSICS LETTERS B
F~,'>(Q,t)=-32iO(t)
~
2 February 1989
74(p,,k')y4(-p~-Q,k") ~
Ik" [,I k': I
<[a(t),p(0)]>,
(16)
p(7,p'(Y"
where
c~(t) =~k (t)~,,(t)C~,(t)C~+,,+~,,~(t)C~,~,(t)C~,_~,_Q+k,,,~,(t), 3o = zc~2+Q(0)n,2(0) - F ~ 2 + Q ( 0 ) F , 2 ( 0 ) .
(17)
The commutator in eq. (16) can be expanded into power series using the well-known expansion of the Heisenberg operator O(t) = e ~ " O e - ~ ' ' =
~ ~(it)" [/4,
[/4,...t/4, 01...11.
(~8)
tl=O
With the help of eq. (18) we obtain (it) 2 ([¢x(t), 3 ( 0 ) ] ) = ( [ a ( 0 ) , / 3 ( 0 ) ] ) + i t ( [ [H, a ( 0 ) 1,/3(0) ] ) + -~.i- ( [ [H, [H, c~(0) ] 1,/3(0) ] ) + .... (19) As one can see from eq. ( 16 ) Fp(l ~(Q, t) depends quadratically on 74. Since, in the present case our aim is to generate m e s o n - m e s o n scattering in terms of 74 we shall extract the effects up to y ] only from eq. (19). For this reason we will replace all the H's in eq. (19) by//o'S. By doing this we are able to evaluate the commutators in eq. (19) without any difficulty. The evaluated results form a geometric series and can be summed. Similarly using eq. (18) the commutators in eq. (4) can be easily evaluated to obtain an expression for F(Q, aO. This is equivalent to conserving the frequency moments up to infinite order [ 6 ] under the free particle approximation. Using these expressions for F), 1~ (Q, t) and F ( Q , t) in eq. (15) and performing the integrations over time we obtain Ap, (Q, o)). We have g4 (Q, e) ) from eq. ( 12 ) o f the form 128
1
g4(Q'°9)=8[fo(Q, oo)]2 ,,,~k, ~02-[09(p~)+og(p,+Q)] 2?4(p''k')74(-p'-Q'Q-k') Ipt + Q I < kc
(
,
x oo-~o(k')2~o(Q-k')
-
,
o)+~)(~')+co(Q-k')
<20,
,~
This is our final expression for the effective m e s o n - m e s o n interaction g4 (Q, 0)) in a nuclear medium. The important aspect of the problem can now be elucidated with great simplification by considering the case of zero total m o m e n t u m Q = 0 o f the meson pair and focussing on the two-pion energy threshold ~0= 2m~ region where a bound state may manifest itself for an attractive g4. To examine this possibility the effective interaction g4 will be evaluated for the situation mentioned above and for the cases when the basic interaction strength 24(k, k' ) in eq. (20) becomes non-local and of point-type, i.e.
74(k,k')=Z(k.k')/lk[lk'[,
y4(k, k ' ) = Z o ,
(21)
where Z and 2 o are constant parameters. Combining eq. ( 20 ) and (21 ) and then performing a change of variable PL + k' ~P2, the expressions for g4 (Q = O, co = 2rn~) are given by g'= and 372
g4(Q=O,o)=2m,~) 22
-
4m,~ Fg
~
l'l < k¢ Ip2--pl [
[p,.(p2--Pl)]2 ( 1 [Pl [ 4[p2--Pl
]~7 (p2_~pt)2 +
1
)
(P2--P,)'q-4m~ ×Suv(P2),
(22)
Volume 217, number 4
PHYSICS LETTERS B
gm(Q=O, ~o=2m=) g0
2~
4rn= -
1(
V2
2 February 1989
1
1
(P2-P~)2 + ( p 2 _ p l ~ 2 + 4 m ~
;,,~
) XSnF(P2),
(23)
{P2--Pl ]
where
(fix/m~/k~ +
F o ( Q = 0, o)= 2m=) = - - ~ \-
+
In
m ~/ kv
)
(24)
and
Shy(k)= ~ n,~(1--np+k.~),
nk=l
Ikl
n,=0
Ikl>kv.
(25)
per
Here fl=k~/kv, kv is the Fermi m o m e n t u m of the nuclear m e d i u m and SHy denotes the usual H a r t r e e - F o c k ( H F ) structure factor. After analysing the restrictions in eqs. (22) and (23) the angular integrations are performed analytically. The r e m a i n i n g integrals over p and p' are done numerically for various values ofkc and kF. Our results are presented in table 1. F r o m our results, it appears that the effective m e s o n - m e s o n interaction g~,o( Q = 0, ¢o= 2m~) in nuclear matter is repulsive for both types of basic interaction strength. In particular, it is less repulsive when the basic interaction strength (Y4) is non-local. It should also be noted that for any value of rk,.I the repulsive nature of the effective interaction increases gradually with increasing value of Ikvl and it attains a final value when [kv] becomes large. On the other hand, when the value of IkFI is kept fixed and the magnitude of the pion cut-off m o m e n t u m kc decreases, then also gl.o is repulsive but the magnitude of the repulsion increases very rapidly. Since the effective m e s o n - m e s o n interaction gLo is repulsive the possibility of the formation of a two-meson b o u n d state in a nuclear m e d i u m becomes unreal. This conclusion is quite consistent with the observation of Migdal [4 ]. Although our approach is f u n d a m e n t a l l y different from that pursued in refs. [2,3 ] the EOM method provides detailed i n f o r m a t i o n about the dependence of the effective interaction (gl,0) on kc and kv. This information is essential for the proper u n d e r s t a n d i n g of m e s o n - m e s o n scattering.
Table 1 The effective interaction g~/gois given in units of 1/m,kv. Each quantity in the first, second and third columns should be multiplied by a factor 10- 2, those in the fourth, fifth and sixth columns by 10- J. kF
1.42 1.82
2.62 3.42 4.22 5.02
kc
5.0
4.0
3,0
2.0
1.5
1.0
0.5
0.082 0.255 O,1O0 0.317 0.129 0.4I 5 O.149 0.477 O.161 0.513 O.169 0.535
O.149 0.470 0, 179 0.571 0.220 0.707 0,249 0.778 0.258 0.815 0.265 0.836
0.307 0.983 0,355 1.140 0.411 1.308 0.436 1.378 0.448 1.412 0.455 1.431
0.0751 0.241 0,0816 0.260 0.0872 0,275 0.0893 0.281 0.0904 0.284 0.0909 0.286
O.127 0.407 O,133 0.424 0.138 0,437 O.1403 0.442 O.14l 0.445 O.l 41 0.446
0.237 0,754 0.242 0.767 0,245 0,777 0.247 0.781 0,247 0.783 0.248 0.784
0.0558 O.177 0.0561 O,178 0.0563 O.1785 0.0564 O.1787 0.0564 O.1784 0.0564 O.1789
373
Volume 217, number 4
PHYSICS LETTERS B
2 February 1989
T h e a u t h o r s are grateful to Dr. J.N. D e o f V E C P , Calcutta, and Dr. G. G h o s h o f S I N P , Calcutta, for t h e i r v a r i o u s a c a d e m i c help.
References [ 1] N. Isgur and J. Paton, Phys. Lett. B 124 (1983) 247. [2] G.A. Miller, Phys. Rev. D 37 (1988) 2431. 13] R. Kokoski and N. lsgur, Phys. Rev. D 35 ( 1987 ) 907. [4] A.B. Migdal, Rev. Mod. Phys. 50 (1978) 107. [ 5 ] A.M. Dyugayev, Zh. Eksp. Theor. Fiz. Pisma 22 ( 1975 ) 181; S. Weinberg, Phys. Rev. 166 ( 1966 ) 1568. [6] D,N. Tripathy and S.S. Mandal, Phys. Rev. B 16 (1977) 231. [7] D.N. Zuberev, Usp. Fiz. Nauk 7l (1960) 320; Soy. Phys. Usp. 3 (1960) 320; V.L. Bonch-Bruevich and S.V. Tyablikov, The Green function method in statistical mechanics (North-Holland, Amsterdam, 1962 ).
374
PHYSICS LETTERS B
E Q U A T I O N O F M O T I O N A P P R O A C H TO T H E M E S O N - M E S O N IN A NUCLEAR MEDIUM
2 February 1989
INTERACTION
M.N. S 1 N H A - R O Y Department of Physics, Bidhannagar College, Salt Lake, BF- I42, Sector II, Calcutta 700 064, India
and S.S. M A N D A L Department ~?fPh.vsics, Maulana Azad College, 8. R.A.K. Road, Calcutta 700 013, India
Received 14 August 1988
We present a method for calculating the effective interaction between two mesons in a nuclear medium by solving the equation of motion of a double-time retarded commutator, defined to represent the propagation of two mesons. The effective interaction is found to be non-local and repulsive.
The construction of a potential model for h a d r o n s based on the strong coupling flux tube limit of Q C D has recently generated considerable interest in the study o f m e s o n - m e s o n scattering [ 1 ]. This strong coupling limit ( r e g i m e ) describes the long range p h e n o m e n a o f interest in nuclear physics. The flux tube rearrangement terms [2 ] which are assumed to be responsible for m e s o n - m e s o n (g-re) scattering act as a non-local potential between mesons. This non-local potential is attractive but numerically very much smaller than the typical strong interaction term [2] and fails to predict anything about the existence o f a two-meson b o u n d state. This situation suggests that the m e s o n - m e s o n scattering be e x a m i n e d using different approaches. One possibility is to assume meson exchange between two interacting mesons [2,3 ]. According to this model the ~ - ~ scattering takes place via the f o r m a t i o n and decay or exchange o f a 9-meson. Kokski and Isgur [3 ] have o b t a i n e d a detailed description of such processes in terms of the string breaking mechanism. The c o m p a r i s o n o f the results o b t a i n e d from the flux tube rearrangement approach with that o b t a i n e d from mesonic exchange and decay via the string breaking m e c h a n i s m is not conclusive. Therefore, we feel that it will be interesting to reexamine the m e s o n - m e s o n scattering again from a different angle and to check whether the m e s o n - m e s o n effective interaction is attractive and non-local, a n d whether the two-meson b o u n d state can exist in nuclear matter. In this letter, we calculate the effective m e s o n - m e s o n interaction associated with the exchange o f a n u c l e o n hole pair [4,5] in a nuclear m e d i u m . This process yields a repulsive and non-local effective coupling between two mesons for both m o m e n t u m d e p e n d e n t a n d point type interactions between the two-meson and the nuc l e o n - h o l e p a i r channel. We define a d o u b l e - t i m e r e t a r d e d c o m m u t a t o r to represent the p r o p a g a t i o n o f two mesons and solve the equation of m o t i o n ( E O M ) it satisfies. We have used a decoupling procedure devised by one of us [ 6 ], according to which the higher-order meson G r e e n ' s functions that a p p e a r when writing the equation o f m o t i o n for the retarded c o m m u t a t o r o f the two mesons, are set equal to lower o r d e r through a certain p r o p o r t i o n a l i t y constant. This p r o p o r t i o n a l i t y constant is d e t e r m i n e d by conserving the frequency m o m e n t s to infinite order under the 369
Volume 217, number 4
PHYSICS LETTERSB
2 February 1989
free particle approximation. Using the solution of the retarded commutator we obtain an expression for the twomeson correlation function F(Q, o9) which can be written in the form
Fo(Q, ~o) F(Q, co) = 1 -ga(Q, o)) Fo(Q, ~o) '
( 1)
where Fo ( Q, o)) is the bare two-meson propagator and g4 (Q, o9) is the effective strength of the meson-meson interaction. The system of interacting pions and nucleons is described by the following hamiltonian:
H= ~
EpC~Cp~+
pa
~, o)(k)(b~bk+½)+Hl+H~,-,
(2)
k < kc
where Ep = p z~ 2m is the free nucleon energy, do(k) = (m~ + k 2)~/~ is the usual pion energy and m is the nucleon mass, with h=c= 1, in these units. The nucleon creation (annihilation) operators are denoted by C ~ (C~). The pion field is represented by operators b~ (bk) corresponding to the creation (annihilation) of a pion. The interaction term//i describing the bilinear coupling of pions to nucleons is given by
I+, =
y=
~ v4(k,
[kl,lk[
k')(b_.+bU (b_~. +b~.)CLC,+~+,.~,
(3)
p~
(k+k')~O
where ~:4 describes the basic interaction strength between the two mesons and the nucleon-hole pair channel. The residual short range interaction between nucleons is denoted by H~ and is not of immediate interest in the present discussion, although its role may be important in the calculation of the nucleon propagator. The scattering of two pions coupled by the Ht term in eq. (3) is diagrammatically shown in fig. 1. This process yields a nonlocal effective interaction g4 between two mesons resulting from the exchange of a nucleon-hole pair. In eq. (3) k~ is an appropriately chosen pion cut-off momentum for this calculation. Now we introduce the two-pion correlation function F(Q, ~o) as
F(Q, o~)= ~ Fp,p~(e, oJ) , pip2
where Fp,p2(Q, o~) is the Fourier transform ofF,,p2 (Q, t) defined by
Fp~,~(Q, t)=-i0(t)([{~z~(t)~p'+Q(t)-F~'(t)FÈ'+~(t)}~{~z~2+~(~)~zÈ2(~)-F~+Q(~)FÈ~(~)}])~
(4)
where ~zk(t) = b_ k(t) + b~ (t) and F, (t) = b_r (t) - b~ (t). This has been introduced in the same spirit as defined for the fermion propagators in refs. [6,7 ]. The angular brackets denote the expectation value over the exact ground state of the coupled system. The function Fr,p_~(Q, t) represents the propagator of two pions. Using the relation
idO/dt=~O/~t+ [0, H I , the equation of motion for Fr,p,_(Q, t) is written as
i2(d2/dt2)Fp,p~(Q, t) 86(t)(fi~,v2 +6p2._v,_Q)[co(pL)+~o(p L+ Q ) ] + [~o(p~)+o~(p~ +Q) ]ZFp~p~(Q, t~+ F <~ co t)
(5)
Fig. 1. The equation for the scattering of two mesons, Mesons are denoted by wavylines and nucleons by solid lines. The heavy dots represent meson-nucleoninteraction. 370
Volume 217, n u m b e r 4
PHYSICS LETTERS 13
2 February 1989
where
F ~p Ip2 > t\ o~-'~ t)=-32iO(t)
2
~"4(Pl , k ' ) ~ ) 4 ( - - P l - Q ,
k")
Ik' I.Ig' I
(p) + k ' ¢ 0 )
(--pj -Q+k"
× ~
¢o)
([~Zk(t)~*"(t)C+~.(t)Cp+.,+k'..(t)C~'~'(t)Cp'-~,-(2+k"..'(t),
pa,p' ~7'
{~L+Q(0)Tr~(o) -FL+Q(0)F.2(0)}] ) .
(6)
Taking the Fourier transform on both sides of eq. (5) and summing overp2, we obtain
{092- [co(p,)+co(p, +Q)]2}Fp,(Q, co)=16[co(pt)+co(p, +Q)]+F~,')(Q, co) ,
(7)
where
F~,,(Q, co)= Z G~.2(Q, co), F~)(Q, o))= Z F~L(Q, co)" P2
(8)
P2
We now introduce the decoupling procedure, i.e. express the higher-order pion Green's function F ~ ~(Q, co) as a linear combination of all F's:
F~; )(Q, co)=A,,(Q, co) ~ F~(Q, o9) ,
(9)
$
where Ap, (Q, co) is the proportionality constant. Substituting eq. (9) in eq. (7) and using the fact that 2~ F,.(Q, co) = F ( Q, co) we obtain
F.~ (Q. co) =
16 [co(p~ ) +CO(PL+Q) ] +A,, (Q, co)F(Q, co) co?-- [co(t,, ) + co(.o, + Q)]2
(lO)
Summing both sides ofeq. (10) overpl and solving for F(Q, co) we get
F(Q, co)= ~p' 16[o.~(p,)+co(p, +Q)]/{co2
[co(p~)+co(p|+Q)]2} 1 -- ~,, Am (Q, co)/{co2_ [co(p, ) +co(Px + Q ) ]2}
(11)
Comparing eq. ( 1 ) and eq. ( 11 ) the effective pion-pion interaction is given by g 4 ( Q , o9 ) _~.
1 x" A.~(Q, co) Fo(12, co) ~ co2_ [co(p,)+co(t,~ +12)12
(12)
with
Fo(Q, c o ) : ~ { [co-o)(p~)-o)(p~ + Q ) ] - ~ - [co+o)(p,)+co(p, + Q ) ] - ' } .
(13)
Pl
Following eq. (9) Ap, ( Q, co) may be written as
Ap, (Q, co) = F~'I)(Q' co) F(Q, co)
(14)
which, with the help of the Fourier transform, can be expressed as
f ~ F ~r~) ( Q, t)ei~tdt Ap,((2, c~) = J~_~ F( Q, t )ei°~'dt '
(15)
where the expression for F~,~) (Q, t) can be represented by 371
Volume 217, number 4
PHYSICS LETTERS B
F~,'>(Q,t)=-32iO(t)
~
2 February 1989
74(p,,k')y4(-p~-Q,k") ~
Ik" [,I k': I
<[a(t),p(0)]>,
(16)
p(7,p'(Y"
where
c~(t) =~k (t)~,,(t)C~,(t)C~+,,+~,,~(t)C~,~,(t)C~,_~,_Q+k,,,~,(t), 3o = zc~2+Q(0)n,2(0) - F ~ 2 + Q ( 0 ) F , 2 ( 0 ) .
(17)
The commutator in eq. (16) can be expanded into power series using the well-known expansion of the Heisenberg operator O(t) = e ~ " O e - ~ ' ' =
~ ~(it)" [/4,
[/4,...t/4, 01...11.
(~8)
tl=O
With the help of eq. (18) we obtain (it) 2 ([¢x(t), 3 ( 0 ) ] ) = ( [ a ( 0 ) , / 3 ( 0 ) ] ) + i t ( [ [H, a ( 0 ) 1,/3(0) ] ) + -~.i- ( [ [H, [H, c~(0) ] 1,/3(0) ] ) + .... (19) As one can see from eq. ( 16 ) Fp(l ~(Q, t) depends quadratically on 74. Since, in the present case our aim is to generate m e s o n - m e s o n scattering in terms of 74 we shall extract the effects up to y ] only from eq. (19). For this reason we will replace all the H's in eq. (19) by//o'S. By doing this we are able to evaluate the commutators in eq. (19) without any difficulty. The evaluated results form a geometric series and can be summed. Similarly using eq. (18) the commutators in eq. (4) can be easily evaluated to obtain an expression for F(Q, aO. This is equivalent to conserving the frequency moments up to infinite order [ 6 ] under the free particle approximation. Using these expressions for F), 1~ (Q, t) and F ( Q , t) in eq. (15) and performing the integrations over time we obtain Ap, (Q, o)). We have g4 (Q, e) ) from eq. ( 12 ) o f the form 128
1
g4(Q'°9)=8[fo(Q, oo)]2 ,,,~k, ~02-[09(p~)+og(p,+Q)] 2?4(p''k')74(-p'-Q'Q-k') Ipt + Q I < kc
(
,
x oo-~o(k')2~o(Q-k')
-
,
o)+~)(~')+co(Q-k')
<20,
,~
This is our final expression for the effective m e s o n - m e s o n interaction g4 (Q, 0)) in a nuclear medium. The important aspect of the problem can now be elucidated with great simplification by considering the case of zero total m o m e n t u m Q = 0 o f the meson pair and focussing on the two-pion energy threshold ~0= 2m~ region where a bound state may manifest itself for an attractive g4. To examine this possibility the effective interaction g4 will be evaluated for the situation mentioned above and for the cases when the basic interaction strength 24(k, k' ) in eq. (20) becomes non-local and of point-type, i.e.
74(k,k')=Z(k.k')/lk[lk'[,
y4(k, k ' ) = Z o ,
(21)
where Z and 2 o are constant parameters. Combining eq. ( 20 ) and (21 ) and then performing a change of variable PL + k' ~P2, the expressions for g4 (Q = O, co = 2rn~) are given by g'= and 372
g4(Q=O,o)=2m,~) 22
-
4m,~ Fg
~
l'l < k¢ Ip2--pl [
[p,.(p2--Pl)]2 ( 1 [Pl [ 4[p2--Pl
]~7 (p2_~pt)2 +
1
)
(P2--P,)'q-4m~ ×Suv(P2),
(22)
Volume 217, number 4
PHYSICS LETTERS B
gm(Q=O, ~o=2m=) g0
2~
4rn= -
1(
V2
2 February 1989
1
1
(P2-P~)2 + ( p 2 _ p l ~ 2 + 4 m ~
;,,~
) XSnF(P2),
(23)
{P2--Pl ]
where
(fix/m~/k~ +
F o ( Q = 0, o)= 2m=) = - - ~ \-
+
In
m ~/ kv
)
(24)
and
Shy(k)= ~ n,~(1--np+k.~),
nk=l
Ikl
n,=0
Ikl>kv.
(25)
per
Here fl=k~/kv, kv is the Fermi m o m e n t u m of the nuclear m e d i u m and SHy denotes the usual H a r t r e e - F o c k ( H F ) structure factor. After analysing the restrictions in eqs. (22) and (23) the angular integrations are performed analytically. The r e m a i n i n g integrals over p and p' are done numerically for various values ofkc and kF. Our results are presented in table 1. F r o m our results, it appears that the effective m e s o n - m e s o n interaction g~,o( Q = 0, ¢o= 2m~) in nuclear matter is repulsive for both types of basic interaction strength. In particular, it is less repulsive when the basic interaction strength (Y4) is non-local. It should also be noted that for any value of rk,.I the repulsive nature of the effective interaction increases gradually with increasing value of Ikvl and it attains a final value when [kv] becomes large. On the other hand, when the value of IkFI is kept fixed and the magnitude of the pion cut-off m o m e n t u m kc decreases, then also gl.o is repulsive but the magnitude of the repulsion increases very rapidly. Since the effective m e s o n - m e s o n interaction gLo is repulsive the possibility of the formation of a two-meson b o u n d state in a nuclear m e d i u m becomes unreal. This conclusion is quite consistent with the observation of Migdal [4 ]. Although our approach is f u n d a m e n t a l l y different from that pursued in refs. [2,3 ] the EOM method provides detailed i n f o r m a t i o n about the dependence of the effective interaction (gl,0) on kc and kv. This information is essential for the proper u n d e r s t a n d i n g of m e s o n - m e s o n scattering.
Table 1 The effective interaction g~/gois given in units of 1/m,kv. Each quantity in the first, second and third columns should be multiplied by a factor 10- 2, those in the fourth, fifth and sixth columns by 10- J. kF
1.42 1.82
2.62 3.42 4.22 5.02
kc
5.0
4.0
3,0
2.0
1.5
1.0
0.5
0.082 0.255 O,1O0 0.317 0.129 0.4I 5 O.149 0.477 O.161 0.513 O.169 0.535
O.149 0.470 0, 179 0.571 0.220 0.707 0,249 0.778 0.258 0.815 0.265 0.836
0.307 0.983 0,355 1.140 0.411 1.308 0.436 1.378 0.448 1.412 0.455 1.431
0.0751 0.241 0,0816 0.260 0.0872 0,275 0.0893 0.281 0.0904 0.284 0.0909 0.286
O.127 0.407 O,133 0.424 0.138 0,437 O.1403 0.442 O.14l 0.445 O.l 41 0.446
0.237 0,754 0.242 0.767 0,245 0,777 0.247 0.781 0,247 0.783 0.248 0.784
0.0558 O.177 0.0561 O,178 0.0563 O.1785 0.0564 O.1787 0.0564 O.1784 0.0564 O.1789
373
Volume 217, number 4
PHYSICS LETTERS B
2 February 1989
T h e a u t h o r s are grateful to Dr. J.N. D e o f V E C P , Calcutta, and Dr. G. G h o s h o f S I N P , Calcutta, for t h e i r v a r i o u s a c a d e m i c help.
References [ 1] N. Isgur and J. Paton, Phys. Lett. B 124 (1983) 247. [2] G.A. Miller, Phys. Rev. D 37 (1988) 2431. 13] R. Kokoski and N. lsgur, Phys. Rev. D 35 ( 1987 ) 907. [4] A.B. Migdal, Rev. Mod. Phys. 50 (1978) 107. [ 5 ] A.M. Dyugayev, Zh. Eksp. Theor. Fiz. Pisma 22 ( 1975 ) 181; S. Weinberg, Phys. Rev. 166 ( 1966 ) 1568. [6] D,N. Tripathy and S.S. Mandal, Phys. Rev. B 16 (1977) 231. [7] D.N. Zuberev, Usp. Fiz. Nauk 7l (1960) 320; Soy. Phys. Usp. 3 (1960) 320; V.L. Bonch-Bruevich and S.V. Tyablikov, The Green function method in statistical mechanics (North-Holland, Amsterdam, 1962 ).
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