Nuclear Physics A287~(1977) 461-494; ©North-Xollaxd Publllhlap Co ., MutendantNot to be roproduoed by photoprlat or miabfilm wlthoat writtan parmi~ion >~ ths yabli~har

NEW APPROACH TO SHORT RANGE REPULSIONIN PION CONDENSATION ~

VIKRAM SONI

Department of Physics, University of Calljornia, Santa Barbara, Calijornla 93106

Received 14 May 1975(Revised I8 March 1977)

A6stra~ti :We develop a new method, using static demity oorreladon functiom, to loot : at the effects ofnucleon repulsion at short distances in a nuclear medium. The change in the critical density forpica condensation from the aboveeffect iscomputed . Also, we investigate the behavior ofthe criticaldemity on the inclusion ofd-isobars and ofS-wave pica-nucleon repulsion .

1. IntrodtactionWe look afresh at the physics ofthe pica propagator in dense nuclear matter and

determine the changes in the cxitical density for pica condensation from avariety ofeffects. A new method to examine the effects of the spin dependence of the pion-nucleon coupling and the oorrelations due to short range repulsion between nucleons,which have been considered by some authors from a different standpoint [e.g.Weise and Brown t .9) and eo-workers, Baym and co-workers ~ " 'o), Migdal andco-workers e. s)], is presented This is better known in the literature as the Ericson-Ericson tt ) (Lorentz-Lorentz) effect in pica propagation in nuclear matter.Themethod consists ofsumming up a set of Feynman graphs for the proper self-

energy, to all orders in the S-matrix expansion, in terms ofa contact Hamiltonian Hndefined below. T'he condensation conditions are then fdlrmulated in terms ofthis sum.Thenon-exchange scattering (defined in secrt. 2), tl~ only one consideredby previous

authors'"), can be reproduced using a contact Hamiltonian

where ~. = 2fi/nç, f2 = 1.1 is the square of the pica-nuceeon coupling co~tant,m is the a - energy in the neutron medium, cp(x) is the pica field operator andp(x) = n(x~e(x) where fix) is the neutron field operator tt . This follows from a theorythat is relativistic only for the picas and constrains the nucleons to be static (i.e . theinfinite nucleon mass limit). Such a theory is valid only in the case of charged pica

f Supported by the National Scdenoè Foundation.rr For simplicity in the equatiom we have usod only the single nucleon pole terms in the s;N scattering

amplitude in our derivation ofthe general correlation formalism . The extemion to a more realistic modelwith isobar intermediate states is immediate and is discussed in sect. 9.

461

462

V. SOM

condensation in pure neutron matter when the pion energy co is large compared withthe nucleon excitation energies . In the case of neutron stars the above constraintsare largely valid when looking for significant changes in the equation of state dueto pion condensation.We then find that the S-matrix elements reduce to integrals over density correlation

functions <F~p(x l ) . . . p(x")~Fi, where ~F) is our ground state comprising a correlatedFermi sea of neutrons. We take these correlation functions to be those for a freeFermi gas multiplied by a product of space correlation functions

(1 -RQxt-x~D = C(xl , xz . . . x,a,r>f

where the R(Ex,-x~~) pertains to the hard core and will be specified later. The non-exchange term for the "nH point correlation function isjust p"C(xl, x2 . . . x"). For theseterms we show that the replacement of the pion emission and absorption by thecontact Hamiltonian is exactly oorrecrt . The main part of the present work will beconcerned with summing the effects of these terms to all orders .

There are, in addition, "exchange" terms in the correlation function

<FIP(xi)P(xz) ~ . . p(x")iFi

the contribution of which in the theory with H~ is not the complete exchangecontribution one would have obtained from the theory of single pion emission orabsorption . This effect has been remedied to fourth order in an expansion in powersoffThe single most important aspect of this work is a dynamical treatment of the

nucleon hard core to estimate its effects on .pion condensation. The hard core is putin successive stages to keep track of the new physics :

(i) The original case of no hard core which has already been studied(ü) The zero-range hard core .(iü) The finite-range hard core .

This is accomplished by putting into the correlation function a factor (1-R(~x,-x~Dfor each pair of nucleons, as stated earlier, where

RQxi-x~)D = {0,for fix,-x1~ < a

for ~x i -x~~ > a,

where (ü) corresponds to the limit a = 0.Apart from the hard core, we put in other significant effects in the last section which

are the S-wave ~-N interaction from the Weinberg interaction Lagrangian densityand the d-isobars, essentially transplanted from earlier works ~"6), and estimatetheir effects on the caitic~l density.The results give a 50 % increase in critical density due to the zero-range hard core.

Thenceforth the increase coming from the finite range is small (15 ~) for a range of0.6 fin Also we get a substantial lowering due to the d-isobars and some increase

from the S-wave contribution .

PION CONDENSATION

463

Simultaneous to this work there have been several other efforts, with much thesame end in view, though with rather different approaches . We briefly review these .Migdal e. e) and co-workers have arrived at somewhat similar results from the theoryof Fermi liquids although there are certain important differences which we pursuein the discussion . This was followed by the work of Weise and Brown t ~ 9) who usea G-matrix approach and show that simply the removal of the delta functionattraction, a component ofthe one-pion exchange potential, which ofnecessity mustbe excluded in the presence of any hard core, gives a large effect . They also extendtheir analysis to a finite core, though, in so far as the this effect is concerned, theyuse a static potential approach. A detailed comparison with their results follows inthe discussion . Baym and Flowers a) in their treatment use a Hartree approximationto an effective four-fermion interaction in the same channel as pion exchange, which,as pointed out by them, is oversimplified . Their approach is somewhat similar toMigdal's in the sense that it amounts to renormalising the pion-nucleon vertex toaccount for the effect of nucleon-nucleon interactions and secondly the effects offinite oorrelations cannot be directly evaluated but have to be put in by hand as aneffective coupling constant .We also comment on other related work which appeared after the completion of

this work. This includes, firstly, the work of Campbell, Dasher and Manassah t6)

who discussed qualitatively how the effect of nuclear oorrelations enters in a chiralsymmetry approach andwho in collaborationwith Baym t s) workedon amodel whichincludes these effects as well as those of the S-wave and the d-isobars obtainingresults similar to ours. Secondly the work of Baym and Brown ~a) which in a senseis closer to ours in spirit and content than any of the previous ones. They include,in addition, the effects of p-meson exchange .

2. Proper self~nergy in the aew fornwlism t

The "contact" Hamiltonian, H~ = p(xx~/u~)V~p* ~ V~p describes the nucleon poleterm for the scattering of a ~_ off a single neutron ; with obvious modifications,scattering offisobars can also be included (sect. 9). The contact Hamiltonian can alsobe used to describe pion scattering from a correlated medium; in this case thenecessary informationabout the nuclearwave functions is containedin the correlationfunctions <F~p(xt) . . . p(~)~Fi which multiply all scattering amplitudes where p(xt)has been defined earlier as the neutron density operator p(xt) = n(x)n(x~ If the pionenergy co is large compared to the nuclear excitation energies we need only considerequal time correlations (see appendix B). For a demonstration that H~ does, indeed,simulate the desired set of graphs in which the single emission and absorption aretaken into account, see appendix A. Here we show how the pion self-energy can bewritten in terms ofthe correlation function in atheory with the n-N interaction givenby H~ .

f The terms ""proper self-energy" ' and "`proper polatiration" are used interchangably in what follows.The author apologises for not having consistently used only one of them .

The iRh order S-matrix element taken between outgoing and incoming pious andthe free filled Fermi sea for the neutrons IFo~ is given by

11~"~ _ (1~_~ "

d~xl . . . d4x"<~"~IT(O~*(xi)ni ~ . . .

xV,~p(xi) . . . V~p*(x")o~(xJln+"><FoIT{p(x~)p(xZ) . . . p(x")}IFo>"

+>~, +

whereRhas been defined in the previous section. Let us nowlook at the neutron part~1= <FoIT{p(xi)P(xs) . . . p(~}IFo)~

(2.2)

As the time ordering can be ignored (appendix B) we write

The above expression can be expanded [as is done for the second order equations(A.4}{A.6)] to give a "direct part, an "exchange» part and the interference termsbetween them. [For the second order where there are no interference terms; see(A.8a) for the definitions of "direct and"exchange", and also figs. la and b.]

PION (t~, k )

PION(w,k)

Fig. l a . 17îe second order "direct" graph.

Fig. lb . The second order "exchange" graph,

The "exchange� is considered separately to second order in sect. 8 (this includesthe part H" does not account ford In the higher orders we confine ourselves to the"dired~ part, to which the relevant contribution from MP~ is Mp(direct~ where

and p is the nucleon density. Thus"

"xVA'~(xJV~xJ}In+~> ~- (1-R(Ix+-x~l)~

(2.4)+>>. ~

This can be simplified to give n-1 propagators and a single N-product. Since thereare n! possible terms of this kind the right combinatorics gives

~âi =P"~

ir ~. . .Jd4x1 . . . d~x�V ~"`k;°

4co co

where

PION CONDENSATTON

465

xexp (ik'° ~ x�) exP (- il~°"` . x,)aiôjdg(xl -xs)~akdF1xz -xa) ~ . .

_

i

ea~' xd`l~

the pion propagator and ô; signifies ô, operating on xl. Writing

as

exp [i(k'° - k°°`) ' ](x1 + x,~] exp [-d~°+)t°"~ ~ ](xl-xZ)] . . .

Here

exp (iki° ~ x") exp (- ili°"` ~ xl)

- i ~ exp (ik' ~ x~31~d~{x, u,j = (2a) 3 ~k'Is+m2-m2 -ie'

-- (-~~-1~~. . .Jd'x', . . . d3x~_,k,k1

x ~(1-

R(Ixi - x~l)).u~.~.e~.er~nmr

(2.5)

and changing variables to xi = x 1 -xZ, . . . x~ _ 1 = x�_1 -xp and x = ](x1+xZ) wefind

Miâi = V

4a~ m

~~-~"~(2n)~-li-1 ~~J ~ . .

Jd~xi . . . d4x',_ i

x exp (- ik ~ x~ . . . exp (- ik ~ x~_ 1)k,k~ôi'ô~ldp(xi, co) . . .

xô~;-1~_1dF(x~_l,cu) ~ (1-R(~xi-x~l)).

(2.6)

i>~. i

k=k°"`=k~°.

Now we can sift out the phase-space factors and write out the contribution of thenth order "direct" term i~ to the proper self~nergy II .Thus

xexp(- ik ~ xi) . . . exp(- ik ~ x~_i~lô~ldr{xi, cu) . . . ô~-'~ - 'dF(a~ _ i, m)

466

V. SONI

The restriction to all correlations (proper) means that in the expansion

For dix, ~) we have the relation

(1 -R(Ixi-x1U)i>1

only those terms are to be included which cannot be generated by taking two termsin lower order and connecting them by a single propagator. Alternatively, a moreformal statement is that the proper part is designated by those terms of

II(1- R(Ixi-xjU)i>1

which have the property that if we divide the sequence of x into the set xl, xz . . . x,and the set x~ + 1 , + . . . x�, there is at least one"R~ function linking the left hand setto the right hand set for every "~ where "r'' goes from 1 to n-1.

3. Tûe direct zero-range hard core correlationWe first compute the proper self-energy contribution from the zero-range hard

core for the direct graph in second order. We are, ofcourse, working to fourth orderin f, that is to second order in the contact Hamiltonian H~ (ie. ~,z) . In future allreference to orders is to be taken with respect to H~. The contribution to the properself-energy in this order, lIZ, comes only from the , R(~xl -xzD part of the factor1-R in eq . (2.6) (n = 2~ We have, for the "direct� part, from (2.~ with n = 2

IIZ = ( +1) z~zzJd3xexp(-ik~x)k~k~V,VjdF(x,u~x-R(~x~)).

(3.1)

Vzdfjz,u~) _ ( -u~z+mz)d F+i83(x).

(3.2)

Since we are interested in the -R(~x, -xz~) part, which is nonzero only for (~x~) = 0,just the singular part of the integrand can contribute . Therefore, the functionexp (- ik ~ x) remains constant and equal to 1 in the relevant range of integrationand may be dropped :

IIZ = ~~zz (+1)

Jd3xk,k~VtO1dF{x, cox-R(IxD)~ , (3 .3)

Averaging over k we get

I72 = (+1) 3~mzz J

d3xkzVzdF(x, wx-R(~x~)~ (3 .4)

which reduces to [from eq . (3 .2)]

~ni = Wz~ d3x83(x~ (3 .5)3 J

_ ( -~~-1 P ~

rf 3 r

3 i17�

~

ct>"~ J J d xl . . . d x�_ t

PION CONDENSATION

467

17z = 3Zzpzkz/coz .

xexp(-k' xi) . . . exp(-ik' x~_ t )ki k, ~ . . .Jd3k 1 . . . d3k"- tktkjkjkk . . . k;-1

(3.6)

It is fairly simple to do the foregoing calculation to any order using the samebasic artifice of retaining only the singular tenors. This simplification is exceptionalto the zero-range case .For the proper self~nergy in the rtth order, II;� we get from (2.~

x(momentum expansion of the propagators) ~ (1 -R)au~o~routto~.(v~~~f

(3'~

Now we can average over all the k to get products like

cos 6, {oos B t cos 9z +'sin Bl sin Bz cos (~ 1 - ~z)} . . . cos B�

which when averaged properly give by means similar to the second order case :

17;, _ (-~)"-l ~i~~~ ~. . . ~d3x'1 . . . d3x~_ t exp(-ik' xi) . . . exp (-ik' xp_ t)

x k ik~Vzdt,(xi, co)Vzd,{xi, m) . . . VzdF{x~_ t , ~)~ (1-R)ca~rou~>on.crwe~r

(3.8}

We nowpoint out an interesting feature ofthis method On doing the right countingwe find that only one particular set of correlations is significant for the properpolarization . The rest, taken together, make no contribution . The only graph thatcontributes is the one with n-1 propagators andn-1 correlations between adjacentspace points (see fig. 2).

Fg . 2 . The only oorrelations (between adjacent apaco-time points) that contribute for the nth orderzero range hard core.

PION(w,k) ~~ PION(~,'k)

Fig. 3 . The pion-self-energy bubble .

468

V. SOTiI

The proper self-energy from (3 .8) is thus given by

We can now estimate the magnitude of the change in the critical density due to thegem-range hard core (direct graphs Using the critical criteria of Bertsch andJohnson 3), we have to satisfy the following conditions :

where II' is the total contribution to the proper polarization, i.e. the sum of the hardcore contributions to all orders and the contribution from the self-energy bubble(sce fig. 3) -p i,k1/w, which has been evaluatod.by Sawyer and Yao')andBertsch andJohnson 3). We get

"_ 1

w

1 +~p~./w .

This is the particular form for the proper polarization, obtained by us on theinclusion, to all orders, of the zero-range hard core. It is typical of the generic form

obtained by all the authors mentioned in the introduction where ~(w, k) is the exactparticle-hole bubble contribution to the pion self-energy and reduces in the limitw > p~ to ~P/w.We pause here to identify gr as the factor ~} in our case, the same factor obtained

by Weise and Bmwn on removing the delta function attraction in the one pionexchange potential Ofcourse physically, what we have done amounts to exactly that .

Other authors have used a g' computed phenomenologically from nuclear data,and of course this number could differ from ~ as the latter only includes the effectof a hard core of zero range. Thus,

P = ~-wZ- l~~p~/w = 0, (3.12)

(3.13)

Solving the cubic in w for a double root we get the following critical parameterscorresponding to minimum of the critical density (units m; = 1 = h = c) :

k = 1.73,

pa~ = 1.5,

w

1.0,

when ~ = 2fz/m~. This gives an increase of 50 ~ over the no-hard-core valueof the critical density which is evaluated from [see refs. a. s)]

P~ = 2(WI~3/3~k z .

(3.14)

The p versus kz curve is shown in fig. 4.

z.o

PION CONDENSATION

469

zERO RANGE HaRO ~

MINIMIaN pc =0.24 F -sk = I .73 (m� = I )

pc (NO FIARD CORE) =0.16 F-s

I

I

I

I

I0

2

4

6k2 (m,=1)

Fig. 4. The k~ versus p~ curve for the zero-range hard core .

4. Seaood-order ßnite hard core (e" a` '' expansion -direct graP~)

Next we go on to the finite hard core. We treat the function R(Ixl) as being. unityfor Ixl 5 a andgemoutside ofthis range. Thecomputations nowbecome much morecomplicatedas the aero-range hard core a8'orded us the latitude ofignoring functionsin the integrand which were non-singular at the origin. As usual our starting expres-sion is (2.~. Putting in the explicit expression for the propagator

u~z ,~Yd

I~h-cuz+mz-is

Doing the time integrations, eta and expanding exp (- ik ~ y), we obtain for theproper polariTation II"~z~,

( ) Wz ff Yd I~Iz_~z+mz-ie

x (1- ik ~ y-2(k ~ Y> = . . .xk ~ ~_(-R(hD~

(4.2)

470

V. SOHI

Of the exp (-ik ~ y) expansion the "1" part is the monopole, the k ~ y part does notcontribute (on averaging over k it goes to zero) and so the -(~k ~ y)z part is the nextterm that must be considered. Hereafter we divide II"~z> into two contributions,IIo~z~ from the monopole and llZ~z~ from the -Z(k ~ y)Z term . We first do themonopole, which we write as

Nezt we treat the -~kz ~ yz contribution to the proper polarization and obtain

~zcz> _ -(2n)-3 ~~z

JJ d3~3~ {I~Izx

~(z+mz)-i~

(k2y)z (k . k~ 2R(IYU} .

(4.6)

Writing oos (&; ~) = x and oos (&', ~) = x' we get

I7Z~ z~ _ -(2a)-s~zpz

JJ d3~3~

zei~Z~~r~ ='

z

~k41YI2~1~12CU

Ik'I . - CU + m - iE

This divides into two parts,

where

t Here v = (nes-m=)'~_ .

x{xx'-~ 1-x' oos(gv -~~}2R(IYI~(4 .~

ollzcz~_

(2n)_

-3 ~zPz

JJ d3~3~,etlt,llrl=ikaYzxzp~cu

lnzczl _ -(mz-wzx2~)_

-3 ~~zJJ d3Yd3k' Ik' Iz e~zl+mz-~ zk4lYIzxzP;

(4.8)

dzPz yXk' k')z-o17"~z~ _ ~2a)_s rds~

3~ ezp(ik''L_R(IYU] . (4.3)~z Ik'Iz-mz+mz-is

Averaging over k we get

l7ocz>-(2n)-s .3kz~zpz

Crd3~ak~ ezp

~lk' Y)~.-R(IYU]

-c~z)x 1-(mz

I~IZ-u~z+mz-~~~(4 .4)

This finally reduces to t1 kz z~z

0 3 wz (4.5)

On performing the algebra we find

PION CONDENSATION

471

Similarly we get for the other part,

ka ~zPzzz

ls mz

i~72(z)=

vz 30~upzz_

f"dyCe_°v~3y3+4~

+ ~Z/ - vZ] .

(4.10)JoWe now carry out a perturbation analysis ofthe contributions coming from these

two terms ofthe exp ( - ik ~ y) expansion to estimate their eûect on the critical density,eta Now, from the zero-range hard core case we have a set ofvalues for all the relevantvariables (in the units m,~ = 1) to perturb about :

ko = 1.7,

coo = 1,

bo = Po~ = 1.5 .

Since we are perturbing around coo = 1 = m~, we make the simplification ofdroppingexp[-(mz -coz)~y] from the integrands in the expressions for 1172(2) and(a/a~,)1n2(z>,The original expression for 1T(k, co) (zero-range hard core) is

co

3 co

The second-order component of (4.11) is

II' (z) _ ~(p~lm)zkz .

(4.12)

The finite ore second-order component is 17~(z)+772(2 ) _

the following :

l7"(z ) :z_2

a z z~i,(2) = 1 kZP JLZe_ra[1+Va]+

k  P az+ljj"(z)~3 co z

1s mz

z

(4.9)

(4.13)

where 'Î72(z) is given by (4.10) and goes to zero as v = 0 or a~ = 1 . The düferencebetween 17�(z) and'I!'(z) is b17"(z) -- 17,.(2)-1T(z) .The equations to be satisfied for pious to condense are as before :

THus, if we perturb the parameters [ie., co and b] about their critical values, we get

-2co~~.eo,,~bco+ôn'

I

Sco+ân'

I

bb+S1T~a�.,~ .,~ = 0,

(4.16)ôco ~,bo.~

ôb

~. bo.

472

Ifweevaluate an"~z~ = all' at the values ofm, kand bwenearlier, we can computethe "changesn aco and S6 from the above equations. Due to (3.15) we can write(3 .16) as

IIâz~ _

V. SONI

+â~(a~III~.bo .~aW+ ab(â111~.~.bab+a(a(a~,~l~.~.bo = °~

(4.1~

an' Sb = -arl.8b ~.~.,~Now using II' _ -(pa,kz/co)/(1 +âp2/w) and arl = all"~z~ we can solve (3.16) and(3.1~ simultaneously for cu and b. We get on doing the algebra

giving a 10 ~ change in critical density and

ab

0.16,

for a

l= J0.4 (m,~ = 1)

(4,18)0.56 fm,

Zz zII~z~ _ -

p (2n)-sJJ d3~d3~

tlk,lhlse-+~hl=~2

5. Exsct calculation for finite core (second order)

Having seen the effect of the "1» and "kz» terms in the perturbation analysis,we calculate the proper self~nergy exactly in second order and investigate to whatextent we can treat higher orders . This will also check the procedure of sect 4 andthe perturbation expansion around w = m~. If the results are roughly equivalentthen we can use the methods developed in sect . 4 for computations in cases wherean exact analysis is too complicated. From (2.~ we get

~n(s) - -(2n)_3 ~up2

JJ d3~3~ ~zk~lh2'e-ik2fl=

~I~IZp,

(

)I I +W +m -ie

5.1

where x = cos (~, ~) and x' ~ = oos (~', ~). This separates into two parts Ihz~ and

x(xx~+~ 1-x~ ~(~Pi-~Pz))2~(-R)~

(5.2)

~z z

e+Ik'llslxe -tklrl~a~z (2n)-a(cos-ms) f~d3Yd3~ ~ 2

z

zI I -cu +m -is

x(xx'+~-~c 1-x' cos(~pl-~z))zkz(-R).

(5.3)

We calculate A~cz~ essentially by the techniques used for calculating II'2 z~ in (4.9).

PION CONDENSATION

473

On doing the angular integration and some algebra we obtain

Ihz> = kz Z~z (2n)-'Jady'

~+~dk'C~4

sin kyk ( -i~+ 8 cos ky ~(-i)

0

8 sin ky_

k' 8 sin ky

24oos ky

24sin ky(kY)zk(-t

~~ + ~

kY

+

(kY)z(kY)3

+ j i8 sin ky(kYx~Y) +

24 cos ky(kY)z~Y -24 sin ky(kY)3(k,Y)~J

e~~~.

(5.4)

The first three terms may be calculated by writing

where C and a are defined later.Hy similar methods we get for nHCZ~

1

+°°

d2n2n ~

e~yik'dk' = 2nd ~y)"YThe next three teams of (5.4) are accompanied by a S(y) and in that limit give nocontribution . The last three terms of the integrand (5.4) are easily evaluated, andthus the whole of (5.4) is

IJ~cz~ = kz ~zPz ~~-J~daC l ~

(5.5)co 0

~(z~

~~

z~(vz)

~ a~Cs_

kkY+ u~kz

Y-1)

us kkY -kY)1o

Y

3Yz

+v-1C2 sin ky + 6(oos ky-1) - 6(sin ky-ky)~ e_ .r

kY (kY)z (kY)3

_ z ~C2(sin ky-ky)

6(cos ky-1 +~(ky) z)

'-v kyz +kzy3

-6(sinky-ky+6(ky)3)lk3y~

At this point we observe that we shall consider only a~ < m,~ for otherwise the properself-energy becomes imaginary. We shall comment in the discussion on the plausi-bility ofthis . The contribution to the proper polarization from the second order finitehard core is evaluated as

n~~~z> -~z~~kz ~~- ~~

J ~daAe-`°~k+ ~daBe-~+~0 k J0 J0

~dox-~/kÇ~J

where

~5.~

A = sin a+2(oos a-1)/a-2(sin a-a)/az, (5.8)

474

V. SOIYI

B = 2sin a/a +6(oos a -1)/az -6(sin a-a)/a 3,

(5.9)

2(sin a-a)

6(oos a-1+ZOCZ)

6(sina-a+6ac3)

v = (~-cuz)~, a=ky.

We note here that all the above expressions reduce to those of the last section inthe appropriate limits.

6. Higher-order direct hard core (Snits) cald~latioosWe now see how fàr we can carry out this analysis of the finite hard core in the

third and higher orders. From the complexity ofthe second order it seems inevitablethat we will have to make approximations. Using (2.~ and putting in the explicitexpressions for the propagators we get the proper self-energy II"~3). .

17�~3) = Z3P3

1

d3

d3k

exp(~ki -k)' Yi)~3 (2~)6 ~~

Yi

1 -W2+Ik1I2fmZ-lE

X JJ d3yzd3kz

wP+~kzlz+mz

z) lyki

. k k ~ kz ki ~ kz

X {(1- R(~Yi~)xl-R(IYzI)xl-R(~Y1+Yz~))}~,roa~pNow we see that it is only if we retain the leading or "1" (monopole) term in theexpansions ofexp (-ik ~ yi) and exp (-ik ~ yZ) that the ki and kz integrals separate.Keeping just the monopole, we average over k :

~ii(3) -- (~)zltzÂ3P3

1

d3

d3k

exp(iki ~ Yi)

kz~3 (2n) s ff

Yi

1 -~z+~ki ~z+mz-ie 1

X

Jfd3

d3k

exp.(ik z ~ yz)~Yz

z -WZ+~kz~z+mz-i~

X {(1-R(IY1 DX1-R(IYzDI(1-R(IYi +YzU)}crrova~r

(6.2)

The next obvious simplification is to drop the third correlation 1- R(~yi + yzU orconsider only those oorrelations which are accompanied by propagators betweenthe adjacent space time points (as in sect . 3). Now we can straightaway use theexpression for II'o~z) (4.5) to get for the above

It is obvious that this can be extended to any order giving the result

PION CONDENSATION

475

Although there is no clinching motivation for the monopole approximation it issimple, it reproduces the zero-range result in the limit a = 0 and also takes accountof the co-dependence . It is reliable only for small "d' and around w x m i.e . closeto zero range. Otherwise the neglect ofthe rest ofthe series (especially the next order)could well make it invalid. However, it is difficult to decide as to which is the rightapproximation. The replacement l+va for e - "°(1+va) could conceivably be moreconsistent in terms of "d' dependence but would have the wrong "co" dependence .So we proceed to use the monopole approximation in the following section ; acomparison with other candidates follows in the discussion.

7. Final calculations for critical d~sityNow we can write the total contribution of the direct finite hard core to the proper

self~nergy, which includes :(i) Monopole or "1» term to all orders (as in sect. 6).(ü) Complete contribution in the second order.This expression is

O =~ I ~daAe-~"ik~+ k J~Bdae-~"ik~+

dace-~"~k~C.Jo

0

0

These expressions are too complicated to deal with as before. The idea of doingthe finite hard core to second order exactly and the higher orders in the monopoleapproximation is to see ifthe neglect ofthe restoftheseries expansion forexp (-ik ~ y)[i.e . 17 as calculated in (4.3)] produces any unanticipated change, and, secondly,if the perturbation around co ~ mR is justifiable. The' question also arises of animaginary contribution to the proper self-energy when co > mx ; we defer commenton it until the discussion . Here we list the expressions we get for the two criticalconditions and look for a double root in the inverse propagator

ôP/âm = Q = 0.

(7.3)When we get coincident roots for P and Q, the critical conditions are satisfied :

-3p i,e -"°(1 +av~ +~p i.e -"°(1 +avkuz, (7 .4)

476

V. SONI

Q = -(1+k2)+3ai2-b21z

O+ b?kN+ b?k2M+ 1b3k

[1+av]Ne-'°W

v

W

3 Wv

1 k21 b3k2 ae-'° 1 b3k2

Wa

3 2-? -~Oe-°`(1+aY~+W3~be-'°(1+aY)a -~CU 3be ;° a +~Wbe-'°(1+Va)

3 W

Y

_ : V

-~6e -°°Wa

(1 +va}w2 +~be-'°Wa W2 -C'

Ib2k2[e -'°(1 +av)

1]

Cb2k2

e-.° aWV

k

Y k W

3W 2 Y

2+C' 3WZ e-'°(1 +av)

aW+ ~WI [e -'°(1 +av)-1][1 +~be''°[a2W]],

(7.5)

where O, v and a have been defined before and b = p~. :

-

M= ~1~ daAe - '~ka(-2W)+ 1 ~ daBe'l'~k~ ~-W~~k2 0 .

k o

v '

YI Ir°

v ~

m

)N= ~-~ daAae -c"it>a+ - ~ daBae-c'~k>a+ ~ daec- '~k~°aC} ;2k o

k o

0

These expressions were used to do a computer analysis for a double root Witha = 0.40 (m,R = 1) = 0.564 fin we get a minimum in p~ versus k at p~ = 0.786 (»t,~ = 1)or 0.277 fm -3 . This gives an increase in critical density, over the aero-range hard~orevalue of 15 ~. The plot for this is shown in fig. 5 .

1 .91

FINITE RANGE HARD CORERANGE a =Q40

(m* = I )=0.564 F

MINIMIIfN pc=0277 F'3k=1.60 (mT=I)

L

IIA

1.5

2Ak (m*= I )

1~~ S. The k venus pa corn for the (mite pane Lard core (a ~ 0.56 fm~

Next we go on to the second-order exchange term. The expression from eq . (A.8)is, after putting in the effect ofthe residue R, i .e., the total exchange contribution tosecond order,

~zi

2~z (-x-

) ~~~~ J>ndexdax~

V

4co w

exp(-ik°ut . x)0 0

in out

On putting in the explicit expression for the propagator we get for the non-hard-corepart [i.e. drop the -R(Ix-x'I) part],

Mae~ _ _WZ (2x)io y

4eo co

~~~ ~d4xd4x'd3gd34

io aut

This gives

Finally

PION CONDENSATION

477

8. Second-order "exchange" with zem-range hard core

x exp (iki° ~ x~(ki° . k°°`x-)OzdF(x-x') exp (i(q +q') ~ (x-x~)

X d3gd3Q'(2n)6 (1-R(Ix-x'D).

xexp (i(- k°°t+q +q'+k~. . x) exp (~-q-q'-k'+ ki°) ~ x')

x exp (-~kio -~o)~) exP ( - ~~o -k°o"`)r) d4l~l~zk~° ~ /r°utIk'I Z -IkolZ+mz- ie

â~ -~

-i

1

1

~

~ 4_

ff

f

3 �,~gg r

co

(27[~ V

4mmmo"t ,l,lo ,lod ~d

x g(-k°"t+q+q'+k~S(-k'-q-q'+ki°~(kô -Is'o~(l~o-k°o"~

Mcz~ _ .i z 2 -i

1

1 ~~-~ut) ~~ ~~

kz{k-(q+9~}zd34d3~iae cu z (2a}z V 2co

0 0 {k-(9+q'}z-WZ+mz-ie .

(8 .2)

~ . kov<Ik~lz

(8.4)

Now we already found that without any hard core co x m,~ at the critical point.So we expand to first order in mz-u~z andhandle this perturbatively to first order inmz -a~z . The contribution to the proper polarization is IYd~

0

0

{k-(9+9~}zx 1-(mz-m ) {k-(q+4~}z

d~ 4'u~z .

(8.5)

478

V. SONI

We get for the proper polarization

~â~ ° 2 ~zwzPz -(mz-wz)B,

where Bis an integral arising from the second term in (4.5). Analysing the hard-corepart essentially as in sect. 3 gives

z Pr vr

~HCt

wz ~

,~d3N"3Y /(2~)

60 0

1 ~zkzpzllcz~ - _

.xc-~ - - 2

wz

(8.6)

This exactly cancels the first term in the non-hard-core part . Doing a perturbationanalysis (see sect. 4) on the remaining term (with a mz-wz factor) we get no changein the critical density due to exchange. It thus seems that the exchange term may notprove too important in det?rmining the critical density .

or

9. S-wave and isobar ei%cts

Now to make this analysis more complete we put in the S-wave nN interactionin the Weinberg form and estimate its effects . The contribution to pion self-energyis easily derived:

iH = n*~+Vtp* . p9~ .~~Q~*~-~p(~*W-~P*~)+~cp*q~.

(9.1)

The Hamilton equations are

-SH -

z

i

pz

__8H _

_iSa* - ~

2~z P4~~

Then from the second equation abovei

i

i

i

pz= n-~P~=v2~-~~-2~PC~P+2~P~} -2,~P~-4~~,

(9.2)

-wz~p = Vzq~ -mx~P-~~pw9~ .

(9.3)

Thus the contribution to the proper polarization from the above interaction is

PION CONDENSATION

479

We go on to estimate the effects of having the S-wave with a aero-range hard core.The H,~, due to the 5-wave contains terms like n*W or its complex conjugate. Now,in the zero-range hard core, we found that only terms which are singular at~x-x'~ = 0 can contribute. However, in this H,~, there are no such terms as opposedto the singular contributions from the Oqv* ~ V~ part of H~ in sect . 2. Thus, the S-wavedoes not contribute at all to the hard core.The results for the S-wave effels are quoted in the tables at the end where we list

the critical parameters for S-wave alone, S-wave plus hard .core (zero range) andS-wave+isobar+hard core (zero range

Secondly we put in the effect ofthe d-isobars in the p-wave, which have, accordingto Migdal 6), important modifications in the pion propagator. Here we follow theanalysis of Brusca and Sawyer `). We first give an argument as to why we can treatthe d exactly as we treated the protons, for the results derived in sect . 2.Given the ground state for the problem and the static nature of the d and the

nucleons, we amve at a contact Hamiltonian like the one in sel. 2, where the inter-mediate d cannot occur explicitly due to the vacuum for the d being the normalone. Thus the only rotationally invariant H~ is the scalar Cp(x)V~*(x) ~ V~(x) whereC is a constant which carries the coupling, the appropriate Clebsch-Gordon coeffi-cients, etc Since we need only the coupling to a single mock of the p-wave pion field,which has ms = 0, we can restrict the d to S= = t~. Also, the coupling dilates thatthe spins ofthe d andnucleon are aligned in the same direction . Thus we can considerthe contal Hamiltonian as being given exally by what Sawyer and Hrusca havein their e4 . (28). In other words, the whole analysis for the protons goes over withoutmodification to the d. This enables us to treat the aero-range hard core with the dand protons to all orders .

Values ofcritical parameters for variom canes

It seems from the above table that Luger and Luger k will lead to a k-" oo pLase transition, i.e. tLedensity B~ asymptotically . to its lowest value ask -. oo . This would certainly not happen wLen roonileffects are rigorously taken into aaoount ; Luge enough k will then force the density up . However, tLrough-out this work we assume that tLenuclei are nonrelativistici.~ k=/211f < 1 and in such cases the changesin minimum critical demity on putting in roooil will be quite small: thus the above values of9a ate quitereasonable in spite of tLe anomalous k-behavior.

7.ero-raage Isobar and Isobar S-wave 2~~ran®e .Value of hard core and Isobar

~~~and aero-range Lard ocre and

k2 fm isobar

Pe ~~-~~ ~ Pe ~~-3~ ~ Pe ~~-3~ ~

Lard

Po ~~-3~

core

.d

S-wave

Pa ~~ -3 ~

2 0.137 0.6 0.076 0.7 0.083 0.74 0.169 0.68 0.54 1 .333 0.119 0.067 0.072 0.76 0.076 0.78 0.133 0.73 0.54 1 .494 0.112 0.72 0.0685 0.79 0.072 0.81 0.122 0.T7 0.493 . 1 .6S 0.11 0.76 0.0683 0.82 0.072 0.63 0.115 0.79 0.442 1.666 0.104 0.78 0.0685 0.84 0.0685 0.86 0.112 0.82 0.432 1 .76

480

V. SONI

The entire proper polariTation to first order is,from Sawyer andBrusca °) [eq. (32)]

(9.5)

where cvR = 1236-938 MeV is the energy difference between the d andthe nucleonand X is a parameter which fits the d-width at the approximate value XZ = 4.38.As before III is given to all orders by

We can use this to get the critical parameters as before. The results are tabulated intable 1 for various cases.

10. Dl

on

There are indeed a number of questions that arise from this treatment of picacondensation . The assumption of the static limit is a valid one as can be seen fromappendix B, provided the pica energy c~ in the medium is greater than the nuclearexcitation energies. This condition is fulfilled in the present theory .

In sect. 6 it was pointed out that the monopole approximation was used for thesake of simplicity. However, for the values of a and v encountered, the differencesbetween the monopole approximation and the alternative oné suggested in sect. 6,or for that matter the simple zero-range hard core approximation, are extremelysmall. The difference in using e -"~(1+va) instead of l+va was found to be of theorder of two to three per cent. Thus it seems, for the purposes of our calculation, thechoice of the approximation is only academia

Expressions similar to our calculations for the zero-range hard wre have beenobtained by Migdal e) and oo-workers from the theory of Fenni liquids. Theyintroduce the hard core correlation as a vertex correction fig. 6 to the pica nucleonvertex ; the shaded figure gives the exact vertex, i.e. including the correlation .

=ro + r,GGI'E

Fig. 6. Vertex oorredion [from ref. ")] for correlation.

They observe that the exact vertex T is given in terms of the free (unshaded) vertexTo by the expression (fig. 6)

T = Tp +l'°°AT,

where A = (GG)p�ta, with G the Green function for nucleons in the medium and Athe pole part of GG. The îador ir°° (scattering amplitude) is given by (mpo/~Z)T°°

PION CONDENSATION

481

(g+g'ss~d where s and s' are the isospin matrices, Q and Q' are the Pauli spinmatrices, m is the nucleon mass, and p° 2 [a constant, see re£ e)] . The diagramdoes not include a meson in the particle-hole channel.We have considered multiple scattering graphs involving charged pions only. These

are graphs of many particle-hole loops connected by charged pions and correlatedwith each other, the correlation being spin and isospin independent. Midgal's graphs,as observed above, are one loop graphs with a spin and isospin dependent particle-hole interaction. This is a non-overlapping set of contributions.We first remark on a serious omission ofcertain graphs in the work of Midgal et al.

These graphs are of exactly the same order, but differ in the ordering of interactionvertices along the loop and thus are not vertex corrections (fig. ~. In any estimate .

Fig . 7 . The non-vertex correction graph corresponding to 6g . 6 (of the same order).

these graphs are of comparable importance . In two interesting limits they nearlycancel the contribution from the vertex correction of Midgal et al. :

(i) when the nucleon-nucleon interaction is spin and isospnn independent;(ü) and when the spin and isospin dependence of the interaction preserves

Wigner SU(4).Also, it issurprising that only g' andnotgappears in the answer fortherenormalised

charged pion vertex [see re£ 8), e4. (10)] .We now point out that in a theory of charged pion exchange in the particle-hole

channel (our case) the only graphs that are possible belong to the category of non-vertex correction graphs (figs. 7and 8). In the case ofvertex graphs we cannot conservecharge andbaryon numbers at all vertices . Only ifwe include the n° are vertex graphspossible.The main point that emphasises the di8'erence in the two approaches is that to

Fig . 8 . The self-energy ootmterpart to 6g . 7 to show that only the non-vertex graph contributes in a theoryof charged pious. The energy ofthe pion a~ (chosen as Rt just for illustration) goes through the protonin the virtual pair, through the A* in the particle-hok channel and again through the virtual proton to

the right. Here p(w) designates a proton particle with energy m.

482

V. SONI

causally aeoount for the correlation we must include a meson, which carries theenergy wbetween the space time points, in the particle-hole channel t (fig . 8). Thusthe approach ofMigdal et aI . does not suffice if we include the causal interaction asopposed to a static potential . We should point out that w andk are not small, condi-tions necessary for the validity of their approach . Though Midgal et al. assert thatg'(w, k) is a function ofcu and k, in the Limit ofinfinite nucleon mass the ~-dependencewill disappear in their picture, whereas, it remains if a virtual pion in included inl'°'. In the case of a zero-range correlation, however, only the static part contributes(see next paragraphs giving similar results. In this case ~(w, k) is determined fromthe parameters of nuclei and thus reflects the effect ofa phenomenological hard core,whereas, in our case the factor ~ is a consequence of the fact that we consider a hardcore of zero range.The crux of the question of imaginary proper self-energy is to account for two

facts : (i) the fact that the zero-range hard core never gives rise to an imaginary partand (ü) from our analysis the imaginary part âppears at the threshold cu = m~ .Further we make rather definite physical speculations on the above problems .The structure of the pion self-energy for the direct hard core goes like

NJd3YL~yx-R)]-(mz-cvs)

('exp(ik~y~3k'd3Y(-R) .

Jk~ -fA+m-lE

This separation has two obvious interpretations, a mathematical anda physical one.The former is that the first term contains that part which is singular at the origin .Fn other words, with the zero-range correlation, only the first or singular .part cancontribute and this has no imaginary part. Physically, the first term is a static or non-causal .part ofthe propagator i.e . it does care for "u~�. Ofcourse, since this part alonecontributes to the zero-range correlation, no imaginary thresholds can appear.One is tempted to associate the correlation, which is characterized by its range,

and the intermediate pion with an intermediate state with some simple dependenceon the range a, e.g . mint ~ m,~+ 1/a The zero-range correlation seems to fit this kindof picture in the sense it gives m,~, = oo, precluding any imaginary part . However,this is not borne out by the facts ; the threshold remains at co = m,~ for all a. This isa direct consequence of the fact that we use the frce pion propagator . Since we aredoing perturbation theory in the interaction picture we must use some form of freepropagator. Later on we examine the possibilities of going to other interactionpictures where more interactions can be wnveniently. incorporated into Ho, the

f From charge conservation the pion can be regarded as a static external field provided we make atime dependent definition of all charged baryon fields in the theory, simultaneously making the pion .field time independent . Ibis, in elFed, just tacks on the extra energy m to that ofthe charged baryon.[3ee eqs. (~ and (ll) is ref. ") .]

PION CONDENSATION

483

free Hamiltonian. Going back to our problem, in this picture it is obvious, since thenucleons are static, that all the energy in the intermediate state is carried by the pionand thus when w Z m� we can form a free (in this picture) pion as opposed to onebelonging to the wndensate, which immediately gives rise to an imaginary part .Now we examine the possibilities ofmodifying the interaction picture by including

the Swave interaction in Ha. The S-wave is particularly amenable to a modificationof the propagator.When properly included the S-wave redefines the interaction picture such that the

inverse propagator is modified to G-1[coZ -(pug/m~)-kZ- m~ +ie]. In sect. 5 theimaginary part arose from the function or from the square root of thenegative of the inverse propagator at k = 0. The new expression would bem�-cv +pco/n . This will not go imaginary untill cu = (p/2»~)+

m,~+p /4mRdelaying the appearance of the imaginary part. The S-wave may be looked uponas giving anew effective mass n>g =

m,~ -cv +pug/m,~, the counterpart of

m,~-coin the absence of the S-wave. It is thus obvious that we get a higher effective masswith the repulsive S-wave interaction than without. Since the force due to simplepion exchange is attractive and that due to the S-wave repulsive, we are lead tobelieve the S-wave repulsion cuts offthe outer reaches ofthe attractive pion potential,increasing the effective mass . However, the S-wave alsô increases w at the thresholdof pion condensation. Thus, to establish that the S-wave really delays the imaginarythreshold we must satisfy cvß -cot < pc~/m=, where cvs is. the new value of co at thatpoint where condensation first set in, in the presence ofthe S-wave interaction. Thisis seen to be the case from our data (table 1~The hard core, introduced earlier simply as a space correlation, however, has

no such feature. It is not amenable to a propagator modification to another inter-action picture, as here we put in a cut-offwhich commences from the origin outwards,in the attractive pion potential . It is a repulsive interaction as before, but snips offtheattraction near the origin instead of at the outer reaches of the potential, resultingtherefore in no newthreshold It only causes wto increase, due to its repulsive nature,till cv = m; (or the S-wave threshold in that picture).

It thus seems unreasonable, on this count, to associate the intermediate state ofa hard core of variable range and the virtual pion with the exchange of a massiveparticle whose mass depends in some simple way on the core radius (or correlationlength as has been mentioned earlier). This is so as the hard core always acts inconjunction with the virtual pion exchange to give the causal interaction.

It follows that the imaginary threshold is detenmined entirely by the physics weinclude in the pion propagator . The pion chemical potential can, then, never exceedcvo wheremo > 0, i$ determined as the appropriate pole (corresponding to the giveninitial conditions) of the inverse pion propagator.

These remarks also underline the importance of keeping the "m» dependence orcausal behaviour. In a static potential approach all this information about thresholds,which involves new physics, is lost. Also it leads to the wrong physics beyond

thresholds as the imaginary self-energy indicates an instability in the system whichwouldimply that the ground state of the system be modified till there is no imaginarypart.Apart from the S-wave modification which will indeed be a significant effect in

delaying the imaginary threshold we shall now see how the other physics of theneutron matter might avoid this eventuality . Now, it is expected that for free piouswere cu to exceed m,~, the formation of a k = 0 mode would lower the energy . In thepresence of other interactions (e.g. S-wave) the k = 0 mode will occur at a sewthreshold in w which is determined by the interaction. That this happens and co doesnot exceed this threshold value is demonstrated in appendix C in a calculation whichuses the mean field approach of Sawyer and Scalapino'). We can also lower thechemical potential by the inclusion of electrons t. These processes would give anadmixture ofcondensate and k = 0 pious (electrons) over a certain range in the lowerdensity regime and will modify the ground state, in principle, whenever faced with animaginary polarization . However, both these processes are not significant in neutronstar calculations, as the amount of condensation and the effect on the equation ofstate is small until one reaches the critical densities given by the Bertsch andJohnsonconditions .We now make a comparison with Weise and Brown t) with whom our approach

is at variance, most notably, in that they neglect the urdependence of the subtractedhard core, i.e. the full finite hard core less the zero-range hard-core effect, whereasin our treatment the co~ependence is kept and enters dynamically in the determina-tion of the critical conditions. Now, since the contribution of the subtracted part ofthe hard core is small the differences do not show up at all dramatically . [Theseparation distance of Weise and Brown, eq . (10~ is roughly equivalent to the rangea of our correlation.] For~ < m,~ we do not get a double root for values of the rangeexceeding 0.56 fm as the proper self~nergy goes imaginary. This can, however,be avoided as the foregoing discussion asserts . (We might do a calculation in futurewith the S-wave etc., although probably 0.56 frn is a reasonable range.)As we see from the results, the perturbation theory used in sect . 4 gives results

which compare rather well with the exact calculations of sect. 5 for the case of smallrange, so we can use the same methods to treat the two cases ; ours with co = m� andthat of Weise and Brown with co = 0. However, if there are large changes in densitythe perturbation around fixed values of the critical parameters become questionable.

T Thin has been pointed out by ßaymand F7owas = ; for then, ody the 6nt ofthe Hertach and Johnsonconditions has to be satisfied, i .e. only G~ ' ~ 0 (G~ ' is the complete pion Green fimdion). This willnaturally lower the pion chemical potential ~ . Also, since Iç ~ R,-Ib, the presence of electrons in theinitial state would imply an equal number ofprotons, bringing ~ up (or Iç down).

The subtracted oontributiôn of the finite hard core to the proper self~nergy is

SII = -kZ PA ~

1

-1

Here we must be careful in putting m = 0 only for those parts which actually comefrom the subtracted finite hard core, i.e . in v = (m2 -w~)~ and in the co-dependenceof O, eq. (7.1~ We quote the results for w = 0 and cv = m� for two separapondistances, 0.564 and 0.85 fin in table 2.

d = 0.85 fm

p~ (fm -3)increase over um range

d = 0.564 fm

PION CONDENSATION

485

T~2

The resWts for m = 0 and cu = mR for two separation distances

Separatidiet.

d or a

R+ mR

pR (fm-')increase over zero range

The changes in the critical density, from perturbing about the zero-range criticalvalues (secrt. 4), for u~ = nt; somewhat exooed those for co = 0. However, we sce thatthe change from a zero-range hard core to a finite range of 0.56 fin, ofalmost throetimes the Compton wavelength of a nucleon will cause only minor changes in thecritical.density whether we take u~ = 0 or a~ = m,~ The changes forarange of0.85 fin,which approximates the same parameter in rei t) are slightly larger but the perturba-tion about a fixed point also becomes less reliable. Even for this value ofthe rangethe effects are somewhat smaller than in re£ ~) . Given the approximation of usingthe monopole in the third and higher ôrders, the point this analysis emphasizes, is,that increasing the range ofthe hard core does not give rise to any dramatic changesin the critical density.To sum up, we find a fairly large (50 /) increase in the critical density for pion

condensation from the zero-range correlation: this is in agrcement with the work ofthe authors mentioned in the introduction. There isafurther increase, though smaller,on including a correlation with a range of 0.56 fim. On increasing the range further

onto 2nd order exact2ndorder to 2ndorder

as in and monopole to as insect . 4 all orders sect . 4

exact 2ndorderand monopole

all orders

0.262 0 .283 0.294 0.2 79 ~ 16 ~ 22 ~ 24

0.25 0.259 0.266 0.2664 ~ 8 ~ 10 ~ 10

486

V. SONI

the system becomes unstable to the formation of zero-momentum pious as the pionchemical potential in the medium exceeds some coo, which is determined by theunperturbed Hamiltonian, Ho, of the interaction picture we work in. The repulsiveS-wave pion-nucleon interaction fiuther raises the critical density but the inclusionof the d-isobar substantially reduces the critical density almost compensating forthe augmentation from the correlation and the S-wave.

Appendix A

NEW FORMALISM IN TERMSOF THE CONTACTHAMILTONIAN'

VYe begin by checking ifin the case of static nucleons we can substitute the contactHamiltonian H~ _ (~ ./co)p(x)Vq~*(x) ~ Vq~(x) instead ofworkingwith the usual p-wavepion-nucleon non-relativistic coupling f(n(x~p(x) ~ Vcp(x)+h.a) where n(x) andp(x) are the neutron and proton field operators respectively .The terms of the S-matrix (in the fourth order of the interaction) which contribute

to the pion self~nergy are

5~4~ =4i~. . .

Jd~xl . . . d4x4f4n,(xix~~appp(xi~~xi)Px(xsxQ1)x.~x2)

x ô~p*(xz)n~x3XQ~QaPa(xa~tW(xs}Pp(xaxQi),vn,(xa~i~'(xa)

(A.1)

plus five other such terms with the space-time coordinates reshu®ed.The matrix element 1M4~ for this process is taken between the outgoing and

incoming pious with the normal vacuum for the protons and the fi~ee, filled Fermisea IFo) for the neutrons, corresponding to pure neutron matter. We can write 1M4~as follows, since each of the six terms of (1.1) are equivalent,

~a~

4fdxidxs`dx3dx4~FoIT{n~(xi)n,~(xz)n~(xsMP(x4)}IFo)

x <DIT(p~(xi)px(x2)Pe(x3)p,~(x4))ID><~~IT{ar<v(xik7,N~*(x~k~k~(x3~~*(x4)}In~.)

x a-terms.

(A.2)

Factoring and calculating the proton part lMP4~ we get the expected space deltafunctions and time theta functions:

Mv4~ _ <OIT{(P~xi)Px(xs)p~xs)Pr(x~)}Io),

We get

or

i

exp(Q iq (t~ -t1)) exp (i4 ' (xt-xz)~pxda4 (2n)a~a~ -~

o tr (2n)a

daq' exp(- ~ô(ts - ta)) exp(i9 ' (x3 -xa))Sa~qo+ie

+same with xl, tt ~-. x3, t3, Saz -" Sby Sb~ --~ SR~,

_

1 exp(_ f

igo(tt -tz)) _-SßzS°"~x' -xz~x3-xa) 2nJ 4o +itl d4o 2n

x (exp (-~o(ts-ta)) dqo_S8xS6~~x t -xa~xz-xs)J qo+ie

1

(exp(-igo(ta -t1))

1

fexP (-i4ô(tt -ta))

,x2n

qo+~

dqo2n

qö+~

dqo~

As we are considering static neutrons, only the zero energy part contributes,hence we can ignore the time coordinate t in the neutron part Mo. Using the non-relativistic Fourier expansion for the field operator,

ct ex

tk ~ xk<krk>kr

where ä~ and ~ are annihilation operators for ~k~ < kf and ~k~ > kT respectively anda is the spin index, we get tt

mal = <Folna(x~(xhi°(x~~p(x~IF) _ <Fo~p(x)P(~IFo).

(A.5)

~al - ~FO~ V2 ~ ~~~(~k~a+Q~,°~Âki +Akfll(~k~° +Llks°)ki k{ k= k~

X(df~.+~°~) exp (d~i -kt)' x) exp (~~z -kz)' x~IFo),

~a) = V2 ~~ ~oktklokikibara°p+oklkçL~k(t2+Stik~kl ta

PION CONDENSATTON

489

(A.3)

x S~a�-ak.k~~kst,a~pa,~) exP (~~i -kt)' x)exp (~~z -kz)' x~,

Pl1

1 p<aMôa'.=

C21Sa,,S°p-

`(2n)s Jod3gexp(i9' (x-~)I

;S,pS�~+S(x-x~apS,~

(A.6)

f See appendix H.tt On usia~ the time independence and the fad that M;4' is multiplied by the apace d-function from

~ts~ .

488

The above follows from converting the ~ to an j and using ~,1lV = ~p (as wehave excluded spin).

Lastly, we do the pion matrix element M;~a~ to give the full expression . Here wehave carried out the oombinatorics associated with all the T-products :

M~a~ _ -fa ~. . .J

.dtldaxzdax3dta(2n)-z

exP(- ~o(ti -tz)) 8

d40+~1

ez 40

x (exp (- iq~(t3 -ta)) d4ô8av<FlneutronslF)qô+ie

mx V

4

`~~�,, exp(- ihôta)exp (ilc$"`t,) exP ( - ik°°` ' x3) exP (ik'° . xz)0 0

x -0;0;dF(x3-xz) x Q-terms,

(A.7)

where 0j is the symbol for the ô, operating on xz. This is equivalent to

Mca~ -_ - 2i (- W~/ \-~"c/ ~J daxzdaxs~no"~IT{01~*(xz)ON~xz)

were !cô = Wm and I~"` = w~�, are used interchangeably for the energy of theincoming and outgoing pion respectively . Finally we do the spin sums f

saaaex(Q~~Qr)z,~Q~oa(~~~n~a~aev(~p~ -AzS~a.b +P~xz -x3~apSrY~

_ ~~12p) 2ai1"kl -2~(2~~3 JO~d34

exp (~ ' (x2 - x3))~2

x(S~~ra-b,~b~,+Si~~+2(~p)b(xz-x3xatla~-a+~li+a,~~]~ (A.8a)

At this point, looking at the structure of the neutron part, we can identify these threeterms with the following graphs.The first term 4(~p)z corresponds to the "direct graph� in fig. la the two bubbles

each contributing a "p". The second term is dubbed the "exchange� graph andcorresponds to fig. lb. The final term has a 8(xz - xs) in it which automaticallyreduces the pion part M!~a~ to zero and so is inoonsoquential and may be dropped.This gives us the following sum :

Mta~ _ _fa~- ~~-~~~. . .J daxzdax,<~,nIT(o~*(xz)O~xz)

t Here A a (I/(Za)')fS~d'getP(i!' (~z-=s))~

V. SONI

x V,cp*(x3)V,~cp(xs)}In,~i<neutrons~(Q-terms~~8a~,

(A.

x v~*(x3)o~(x3))I~>...

PION CONDENSATION

489

x C~zP~-2 {(2~)s J~d34exp (iq ' (xz - xa)) }z +(zP)~xz-xs)J

+f~( -) ~

1 ~

1 ~ ,~~. .

Jd4xzd~x3 C(zP)~xz-xs)

w~

1 ~

z-2 {(2n)3 fo d34 exp (i9 ' (arz -xs))}

+T[V,rp(xz)V~*(x3)~~*(xz)0~(xs)]I~ioi}~

(A.8b)

Thus the first term is what we would get if we worked to second order with thecontact Hamiltonian H~ _ ~,(1/co)p(x)Vtp(x) ~ Vqv*(x) [p(x) = n(x)n(x)] . The secondterm is the residue . It is thus obvious the entire direct part and part of the exchangecome from H~; the residue in the exchange part can be treated separately . Thus,apart from the above qualifications, this reproduces (at least for the direct case)the expression which derives from the contact Hamiltonian H~ to second order.For the higher orders we find, from a similar treatment, that the expression for thedirect part is identical to that given by H~ i.e . the rtth order direct term is

"Mia

~

n~ \W/(1rP__

"J Jdaxi -d4x"~~ou~IT{oN~*(xi)o~(xi) . . . v~p*(x,w~p(~}I~~>~

We now do the two significant graphs, "direct and "exchange", exactly ; that is,retaining the time coordinates for the neutrons and working to fourth order in theoriginal pion-nucleon coupling to see if this tallies with the above expression (A.8b).Introducing propagators it becomes quite evident how only the zero-energy partsurvives the contour integration.The part of the neutron time-ordered product

which corresponds to the direct graph isMô

a~ _ <FIT{na(xi)n.(xz)ne(x3)nP(x~}IFi

Môiâ~ °

1 ,a~ JdEexp(-iE(tz -ti))exP(iP' (xz -xi)M3P(2a)

x ~

1

+2nib(E)B(pr -P) jE+i8

x (2n~ a°° ,~~, exp (-iE'(t4-t3)) exP (iP~ ' (x4-xs)M3P~

490

V. SONI

The entire expression that remains is

Ma(a) _ 1(2a)io .Îa J

Jdtld`xzd4x3dt4exp (- iE(t2 -tl))d3P~�Â(~xi~Pr-P)

xJdE'd3p'exp (- iE'(t~-t3)~p,S(E~2ni6(pr-p~

(~

exp(-igo(ti -tz))

, exp(- igô(ts -t4))Jd4ox

40+~

apz

d4o

9ô+ iE

a°"

x V ki~~`

~~

exp(il~~`t4) exp (- ilcôtl)exp (ik)° ~ xz)m out

x exp ( -ik°"t ~ x3)-V;V1d~(x3-xz) x (Q-terms),

(A.10)

~ca) _2~ ~~~2n) Jo

~Pr-P~3PJz(2~i)z(nomlT{DiW'(xz)~~(xz)( 4

X o~v`(x3)o~(x3)}I~~> x 4(from traces) ~ -~~

Woutld4xzd4x3'

4M°~4)

2i ~~- ~

~~

W1nJPz(M°4+~id4xzd4xs - the direct term.

(A.11)

The "exchange" graph follows from the other permutation in thè relevant partof the Wick expansion, for the neutron time ordered product. Thus contractingno(x l) with nP(x4) and n~(x3) with n,(xz) we get

Mâ=) - .f4 (2x)4 ~d3pdEexP ( - iE(t4-ti))exP(iP'(xa-xi))

1

l 1x [E+ib +2nib(E)B(pr-P)J8~ (2nd

~daP~~~ exP (- iE'(tz -t3))

x exp (ip' ~ (xz -xa)) [E'+iA +2~i8(E~9(pr -p~Jb�a.

(A.12)

Performing the same algebra as before we recover the expression

~~) - f`

1

1

(-~

1

ex (iP. xb3p~pr -P)o=

2( ~~- ~- Wio

- moot/

C(2rz)3 ,~

P

x(nom~TiD~P~xz)~~(xz)~~P*(xs)~~D(xa)}~n~i+R,

where the residue is

PION CONDENSATION

491

R __ _f

__

_~~~- ~

l ~

l ~(-~ ~ 1

Jexp (iP- xM3PB(pr - P)

JZd4z2d4xa2 ~

Win

Wont

(2~)3

0

X <noutl - T{or~*(zz)or~P*(z3)o~P(z2)o~9~(z3)}+ T{V~p(zz)Vi~A*(z3)o~*(z2)o~P(x3)}Inin),

and x = x2-x3 . This is consistent with eq. (A.8b~

This is broken up to give t

Appendix B

We digress here to look at the structure ofthe time dependence ofthe neutron partand understand the conditions governing the static limit for the nucleons.We start with

Mô`~ _ <FoIT{n(x, tt)dx, tz~x ,, r3M(x', t,)}IFo).

(H.1)

M~nt ' _ <FoIT~n(x, ti~dx, tz)]IFo)CFoITLn(x', taM(x', t~~IFo)

- CFoIT~n(x, tiM(x', t4)~IFo)CFoIT(n(x', t3M(x, t~)]IFo)-

(H.2)

Now we introduce a complete set of eigenstetes ~ IN)<Nh betwcen all pairs ofneutron ûeld operators in the T-products above. Then taking co� as the energyeigenvalue for the ground state (or Fermi sea) and tvN, y as the eigenvalues for theexcited intermediate states we get~41 = ~ exP f~~.- ~Hxtt -rs)) exP ( t~~.- Wirxta - ra))A!,N

X <FoIT{n(x, U)IN)<Nln(x, 0)}IFo)<FoIT{n(x', AIM)<Mln(x', U)}IFo)

- ~exp(tdw.-~Nxtt-r4))exP(ti(~.-~irXr3-t2))N,N

X<FoIT{~(x,~)IN)<Nln(?~,~)}IFo)<FoIT{n(x,U)IM)<Mln(x,U)}IFo)- (B.3)

We now combine the factored out time dependence from above with that coming

f We have left out the terms involving normal products in the Widecontraction which are not~devantto this calciilation as their indusion does not aher the time depeadenae ofthe operator.

492

V. SOI~iI

from the proton and pion parts [see (A.~] to obtain the following time structure

1 ('

exp{-~o(ti -tz }) exp { -.f4'o{t3 -t4)) ,(2n)z J

dtldtzdt3dt~~

qo+pl

dqo,~

qo+ ie

dqc

or

xexp (icoo�,t4) exp ( -~mti)x {exp (f~N(ti -tz) t ~zr(ts -ta)) or exp (fieN (t l -t4)t~ar(t3 -tz))},

(B.4)

where sN = m� -coN, ea, = cu� -u~~,, and where for simplicity we use the same ~N),~M) for both terms in (B.3) (which does not change anything in principle). Doingthe "tl� and "t4" integrations, and then integrating the consequent 8-ftwctions overqo and qô we have

dtzdt3

1

1

exp(-Wmtz)exp (~outt3),-Wout fEN - WiofEN

I I dtzdt3

1

1J J

-WouttEN -WintEèl

xexA ~- it~(Winf(EN-EAl))) exP (+it3(cu�ut-(tIIEN -e r))~If en,, eY ~ Wm. cot we can neglect eN,Y completely in the above expression. This

leaves us free to remove the sum over intermediate states so that we recover (2.8).It is thus established that the static limit can be taken and the neutron operator timedependence ignored provided eN ~ Wm.o~t. As this happens to be the case we arejustified in making the assumptions above.

Appendix CWe show, using the mean field thoory approach of Sawyer and Scalapino'), that

with the introduction of a k = 0 mode of the ~- mesons the chemical potential willnever exceed m,,~ (e = 0) or rrç (E = 1) (for the case that includes the S-wave picanucleon interaction) .From eq. (2.16) in reC'),

E = NwkX+NYm,~-2N(X+n~(1-(X+n~u1T~X~Mk +~Np{X+Y),

(C.1)

where E is the energy of the system, N is the total number of baryons, X is thefractional number ofa- mesons in the momentum k mode, Yis the fractional numberof R~ mesons in the k = 0 mode, Mk = kf/V}askm,~ (fz = 1.1 the strong couplingconstant) and e = 1 for the case that includes the S-wave pica-nucleon interactionand.is otherwise 0.

Also

and show this directly . Writing (G1) as

At threshold

PION CONDENSATION

493

The expression for the energy follows from the same calculation as in re£') withthe new charge constraint X+ Y = B2 = fractional number of protons. To get then- ch~micsl potential we write (C.1) as t

E = N,~_u~k+N,~_°mR-2NN~_BMk+

p(Nx_+N,~-°),

(C.2)

where we drop 1- (X+ Y)~ as X and Y are very small at condensation and write(X + Y)~ = B, the fraction of protons, to correctly use the relation âE/âNx- _ ~-,i.e. we must put the charge constraint, B' = X+ Y, only after the derivative is taken :

(C.3)

(C.4)

We could say right here that~_ -- Px _ ° and so ~_ must also be m~ but we go ahead

and minimizing this with respect to A = Y/X, we get

From (C.6) and (C.~ we get

and from (C8) ând (C.3},

The meaning of (C.8) is interpreted as such : A = Y/X cannot be negative and thuswe cannot-have any n-(k = 0) component until apt > m;+m~. The solution for

t Here N,_ is the number of x'(k) in the system (chemical potential s ~.) and N,_o is the number ofx (k ~ 0) in the system (chemical potential = N, _ ~.

494

V. SONT

p,~_ in the absence of any such component is given by ~_ _ ~(cak+sp/2rr~). Thus~_ 5_ m~ until ~eh+sp/2n>x) 5 m~, the equality gives u~k = m,~+m~. This is exactlythe content of (C.8); until ~r.,~- < m;~ we have no ~-(k = Oj component but as soonas ~- exceeds ~ this component appears in such a way as to keep the chemicalpotential always equal to m~. This shows rather elegantly that the chemical potential~_ 5 m~. Just for illustration we compare the value we get for the critical densitywith andwithout n -(k = 0) (for e = 0).The expression for the respective critical densities for the case e = 0 are

This shows that the ~-(k = 0) components always keep the threshold density lower(i.e. in the regime where ~_ would exceed 1 without them).

References1) w. weise and G. E. Brown, rhya. Lett. 29 (1974) 3862) G. Baym and E. Flowers, Nucl. Phya. A232 (1974) 293) G. Hertach andM. Johnson, Phys . Lett . 4ßB (1974) 3974) D. Bruaca and R. F. Sawyer, Nucl . Phys . A236 (1974) 4705) R F. Sawyer and A. C. Yao, Phys. Rev. D7 {1973) 15796) A. B. Migdal, IPhya Rev. Lett . 31 (1973) 257 ;1?hys . Lett. 4SB (1973) 448 ; 47B (1973) 967) R F. Sawyer and D. J. Swlapino, 1?hys . Rev. D7 (1973) 9578) A. B. Migdal, O. A. Markin and I . M. Mishushtin, ZhETF (USSR) 66 (1974) 4439) S. Barahay andG. E. Brown, Phys. belt. 47B (1973) 10710) C. K. Au and G. Baym, Nucl. Phys . A236 (1975) 30011) M: O. Ericson andT. E. O. Ericaon, Ann. of IPhys. 36 (1966) 32312) S. O. Hackman andw. weise, Phys . Lett . S5B (1975) 113) G. Bettach and M. B. Johnson, 1?hys . Rev., in press14) G. Haym and G. E. Brown, Nucl. Phys. A247 (1975) 39515) G. Baym, D. Campbell, R. Dachen and J. Manassah, Phys. Lett ., in press16) D. Campbell, R Dachen and J. Manessah, Phya Rev., in press