New approach to short range repulsion in pion condensation

Nuclear Physics A287~(1977) 461-494; © North-Xollaxd Publllhlap Co ., Mutendant Not to be roproduoed by photoprlat or miabfilm wlthoat writtan parmi~ion >~ ths yabli~har

NEW APPROACH TO SHORT RANGE REPULSION IN 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 of nucleon repulsion at short distances in a nuclear medium. The change in the critical density for pica condensation from the above effect is computed . Also, we investigate the behavior ofthe critical demity on the inclusion of d-isobars and of S-wave pica-nucleon repulsion .

1. Introdtaction We look afresh at the physics of the pica propagator in dense nuclear matter and determine the changes in the cxitical density for pica condensation from a variety of effects. A new method to examine the effects of the spin dependence of the pionnucleon 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 and co-workers e . s)], is presented This is better known in the literature as the EricsonEricson tt ) (Lorentz-Lorentz) effect in pica propagation in nuclear matter. The method consists of summing up a set of Feynman graphs for the proper selfenergy, to all orders in the S-matrix expansion, in terms ofa contact Hamiltonian Hn defined below. T'he condensation conditions are then fdlrmulated in terms ofthis sum. The non-exchange scattering (defined in secrt. 2), tl~ only one considered by 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 and p(x) = n(x~e(x) where fix) is the neutron field operator tt . This follows from a theory that is relativistic only for the picas and constrains the nucleons to be static (i.e. the infinite 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 of the general correlation formalism . The extemion to a more realistic model with 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 with the nucleon excitation energies. In the case of neutron stars the above constraints are largely valid when looking for significant changes in the equation of state due to pion condensation. We then find that the S-matrix elements reduce to integrals over density correlation functions where is our ground state comprising a correlated Fermi sea of neutrons. We take these correlation functions to be those for a free Fermi gas multiplied by a product of space correlation functions
r>f

~F)

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

where the R(Ex,-x~~) pertains to the hard core and will be specified later. The nonexchange term for the "nH point correlation function isjust p"C(xl, x2 . . . x"). For these terms we show that the replacement of the pion emission and absorption by the contact Hamiltonian is exactly oorrecrt . The main part of the present work will be concerned with summing the effects of these terms to all orders . There are, in addition, "exchange" terms in the correlation function
the contribution of which in the theory with H~ is not the complete exchange contribution one would have obtained from the theory of single pion emission or absorption . This effect has been remedied to fourth order in an expansion in powers of f The 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 put in 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~D for each pair of nucleons, as stated earlier, where RQxi-x~)D = {0,

for fix,-x1~ < a for ~xi -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 density and the d-isobars, essentially transplanted from earlier works ~"6), and estimate their 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 of 0.6 fin Also we get a substantial lowering due to the d-isobars and some increase from the S-wave contribution .

PION CONDENSATION

46 3

Simultaneous to this work there have been several other efforts, with much the same 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 theory of Fermi liquids although there are certain important differences which we pursue in the discussion. This was followed by the work of Weise and Brown who use a G-matrix approach and show that simply the removal of the delta function attraction, a component of the one-pion exchange potential, which ofnecessity must be excluded in the presence of any hard core, gives a large effect . They also extend their analysis to a finite core, though, in so far as the this effect is concerned, they use a static potential approach. A detailed comparison with their results follows in the discussion. Baym and Flowers a) in their treatment use a Hartree approximation to 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 to Migdal's in the sense that it amounts to renormalising the pion-nucleon vertex to account for the effect of nucleon-nucleon interactions and secondly the effects of finite oorrelations cannot be directly evaluated but have to be put in by hand as an effective 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 who discussed qualitatively how the effect of nuclear oorrelations enters in a chiral symmetry approach andwho in collaborationwith Baym s) worked on a model which includes these effects as well as those of the S-wave and the d-isobars obtaining results similar to ours. Secondly the work of Baym and Brown ~a) which in a sense is closer to ours in spirit and content than any of the previous ones. They include, in addition, the effects of p-meson exchange . t ~ 9)

t6 )

t

2. Proper self~nergy in the aew fornwlism t The "contact" Hamiltonian, H~ = p(xx~/u~)V~p* ~ V~p describes the nucleon pole term for the scattering of a ~_ off a single neutron ; with obvious modifications, scattering off isobars can also be included (sect. 9). The contact Hamiltonian can also be used to describe pion scattering from a correlated medium; in this case the necessary informationabout the nuclearwave functions is containedin the correlation functions
The iRh order S-matrix element taken between outgoing and incoming pious and the free filled Fermi sea for the neutrons IFo~ is given by 11~"~ _

(1~

ni

_~ ~

"

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

x V,~p(xi) . . . V~p*(x")o~(xJln+"> "

+>~, +

where R has been defined in the previous section. Let us now look at the neutron part ~1=


(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 terms between them . [For the second order where there are no interference terms; see (A.8a) for the definitions of "direct and "exchange", and also figs. l a and b.] PION (t~, k )

PION(w,k) Fig. l a . 17îe second order "direct" graph.

Fig . l b . The second order "exchange" graph,

The "exchange is considered separately to second order in sect. 8 (this includes the 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 " " x VA'~(xJV~xJ}In+~> ~ - (1-R(Ix + -x~l)~ +>>. ~

(2 .4)

465

PION CONDENSATTON

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

V ~"`k;° 4co co

P" ir ~âi = ~ ~. . . d4x1 . . . d~x J x exp (ik'° ~ x) exP ( - il~°"` . x,)ai ôj dg(xl -xs)~akdF1xz - xa) ~ . . (2.5)

where

_

i

ea~' xd`l~

the pion propagator and ô; signifies ô, operating on xl. Writing exp (iki° ~ x") exp ( - ili°"` ~ xl)

as

exp [i(k'° - k°°`) ' ](x1 + x,~] exp [ - d~° + )t°"~ ~ ](xl - xZ)] . . . and changing variables to xi = x 1 - xZ, . . . x~ _ 1 = x _1 - xp and x = ](x1 + xZ) we find Miâi = V

4a~ m

~~ -

~"~(2n)~-

x exp (- ik ~ x~ . . . exp (-

li-1 ~ ~ J ~ ik ~ x~ _

xô~;-1~_1dF(x~_l,cu) ~ i>~. i

Here

d~{x, u,j =

. . d~xi . . . d4x',_ i

J

1)k,k~ôi'ô~ldp(xi, co) . . .

(1-R(~xi-x~l)) .

- i ~ exp (ik' ~ x~31~ (2a) 3 ~k'Is+m2 -m2 -ie'

(2 .6)

k=k°"`=k~°.

Now we can sift out the phase-space factors and write out the contribution of the nth order "direct" term i~ to the proper self~nergy II. Thus -- (-~~-1 ~~. . . d'x', . . . d3x~_,k,k1 x exp(x

~

(1-

J

ik ~ xi) . . .

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

R(Ixi - x~l)).u

~.~.e~. er~nmr

466

V . SONI

The restriction to all correlations (proper) means that in the expansion i>1

(1 - R(Ixi- x1U)

only those terms are to be included which cannot be generated by taking two terms in lower order and connecting them by a single propagator. Alternatively, a more formal 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 set to the right hand set for every "~ where "r'' goes from 1 to n-1. 3. Tûe direct zero-range hard core correlation We 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 order in f, that is to second order in the contact Hamiltonian H~ (ie. ~,z) . In future all reference to orders is to be taken with respect to H~ . The contribution to the proper self-energy in this order, lIZ, comes only from the , R(~xl - xzD part of the factor 1-R in eq. (2.6) (n = 2~ We have, for the "direct part, from (2.~ with n = 2 IIZ = ( +1) z~zz

J

d3 xexp(-ik~x)k~k~V,Vjd F(x,u~x-R(~x~)).

(3.1)

For dix, ~) we have the relation Vzd fjz,u~) _ ( -u~z +mz)d F +i8 3 (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 function exp (- ik ~ x) remains constant and equal to 1 in the relevant range of integration and may be dropped : ~ IIZ = ~zz (+1) d3 xk,k~VtO1d F{x, cox- R(IxD)~ J Averaging over k we get ~ d3 xkzVz dF (x, wx-R(~x~)~ 3 mzz J which reduces to [from eq . (3.2)] I72 = (+1)

ni

=

3 ~Wz~ J d3x83(x~

,

(3 .3)

(3 .4)

(3.5)

PION CONDENSATION

467

17z = 3Zzpzkz/coz .

(3.6)

It is fairly simple to do the foregoing calculation to any order using the same basic artifice of retaining only the singular tenors. This simplification is exceptional to the zero-range case. For the proper self~nergy in the rtth order, II; we get from (2.~ 17

_ ( - ~~-1

~

P ~ rf 3 r . . . d 3 x_ i t ct>"~ J J d xl

x exp(-k' xi) . . . exp(-ik' x~_ t )kik, ~. . .

J

d 3 k 1 . . . d 3 k" - tktkjkjkk . . . k;-1

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 Bt 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 . . . d3 x~_ t exp(-ik' xi) . . . exp (-ik' xp_ t ) x kik~Vz dt,(xi, co)Vzd,{xi, m) . . . VzdF{x~_ t , ~) ~ ( 1- R)ca~rou~>on.crwe~r

(3.8}

We now point out an interesting feature ofthis method On doing the right counting we find that only one particular set of correlations is significant for the proper polarization . The rest, taken together, make no contribution . The only graph that contributes is the one with n-1 propagators and n-1 correlations between adjacent space points (see fig. 2).

Fg . 2 . The only oorrelations (between adjacent apaco-time points) that contribute for the nth order zero 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 the gem-range hard core (direct graphs Using the critical criteria of Bertsch and Johnson 3), we have to satisfy the following conditions :

where II' is the total contribution to the proper polarization, i .e. the sum of the hard core 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') and Bertsch and Johnson 3). We get w

"_ 1

1 +~p~./w .

This is the particular form for the proper polarization, obtained by us on the inclusion, 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 exact particle-hole bubble contribution to the pion self-energy and reduces in the limit w > 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 pion exchange 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 effect of 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 parameters corresponding to minimum of the critical density (units m; = 1 = h = c): k = 1.73,

pa~

= 1.5,

w

1.0,

PION CONDENSATION

469

when ~ = 2fz/m~. This gives an increase of 50 ~ over the no-hard-core value of 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 zERO RANGE HaRO ~

MINIMIaN pc =0 .24 F - s

I

0

k = I .73 (m  = I ) F-s pc (NO FIARD CORE) =0.16 I I I I 2 4 k2 (m,=1)

6

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. unity for Ixl 5 a and gem outside ofthis range. The computations now become much more complicatedas the aero-range hard core a8'orded us the latitude of ignoring functions in the integrand which were non-singular at the origin. As usual our starting expression 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 the proper polariTation II"~z~, ( )

Wz

ff

Yd

I~Iz_~z+mz-ie x (1- ik ~ y- 2(k ~ Y>= . . .xk ~ ~_( -R(hD~

(4.2)

47 0

V . SOHI

Of the exp (-ik ~ y) expansion the "1" part is the monopole, the k ~ y part does not contribute (on averaging over k it goes to zero) and so the -(~k ~ y)z part is the next term 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 the monopole, which we write as z z 17"~z~ _ - 2a)_s d P rds 3~ ezp(ik'' yXk' k')z o ~ ~z ~ Ik'Iz-mz+mz-is L_R(IYU] .

(4.3)

Averaging over k we get l7ocz>-(2n)-s

kz~zpz 3

C

rd3~ak~ ezp

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

x 1This finally reduces to t 0

(mz

- c~z) I~IZ-u~z+mz-~~~

1 kz z~z 3 wz

(4 .4)

(4.5)

Nezt we treat the -~kz ~ yz contribution to the proper polarization and obtain z z x ( (k y) (k . ~zcz> _ -(2n)-3 ~~ (4.6) k~ 2 R(IYU } . {I~Iz ~ z+mz)-i~ 2 JJ d3~3~ Writing oos (&; ~) = x and oos (&', ~) = x' we get ei~Z~~r~=' ~z z I7Z~ z~ _ -(2a)-s p ~k4 1YI 2~1~1 2 z z CU JJ d3~3~ Ik'I . - CU + m - iE (4 x{xx'-~ 1-x' oos(gv -~~}2 R(IYI~ This divides into two parts,

(4.8)

where ollzcz~

lnzczl _

-(mzwx2~)_

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

_

.~

(2n)_

- 3 ~z Pz ,etlt,llrl=ikaYzxzp~ cu JJ d3~3~ -3 ~~z

JJ d3Yd3k'

Ik'I z e~zl +mz - ~ zk4 lYIzxzP;

PION CONDENSATION

47 1

On performing the algebra we find

ka ~ z P z z ls mz

z

Similarly we get for the other part, ~upz_ 4~ i~72(z) vz Jf"dyCe_°v~3y3+ = 30 o

(4.9)

+ ~Z/ - vZ] .

(4.10)

We now carry out a perturbation analysis of the 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 relevant variables (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 ofdropping exp[-(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.

The finite ore second-order component is 17~(z)+772( 2) _ l7"(z ) : z_2 a z z ~i,(2) = 1 kZ P JLZe_ra[1+Va]+ k  P az+ljj"(z)~ z 3 coz 1s mz

(4.12)

(4.13)

where 'Î72(z) is given by (4.10) and goes to zero as v = 0 or a~ = 1 . The düference between 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 the following : -2co~~ .eo,,~bco+

ôn' ân' I Sco+ I bb+S1T~a.,~.,~ = 0, ôco ~, bo.~ ôb ~. bo.

(4.16)

472

V. SONI

â~(a~I II~.bo.~aW+ ab(â1 11~.~ .bab+a(a(a~,~l ~.~.bo = °~ (4.1~ Ifweevaluate an"~z~ = all' at the values of m, k and b wen earlier, we can compute the "changesn aco and S6 from the above equations. Due to (3.15) we can write +

(3.16) as

an' 8b ~.~.,~Sb = -arl. 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 ab

0.16,

J0.4 (m,~ = 1) for a = l 0.56 fm,

(4,18)

giving a 10 ~ change in critical density and

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 what extent we can treat higher orders . This will also check the procedure of sect 4 and the perturbation expansion around w = m~. If the results are roughly equivalent then we can use the methods developed in sect. 4 for computations in cases where an exact analysis is too complicated. From (2.~ we get 2 zk~lh 2'e-ik2fl= ~n(s) - -(2n)_3 ~up (5.1 ) ~I~I Zp, JJ d3~3~ I ~I +W +m -ie where x = cos (~, ~) and x' ~ = oos (~', ~). This separates into two parts Ihz~ and Zz z p II~z~ _ - ~2 (2n)-s

tlk,lhlse-+~hl= ~d3~ JJ d3

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

IIâz~ _

(5 .2)

~z z e +Ik'llslxe -tklrl~ (2n)-a(cos-ms) ~d3 Yd3 ~ ~ 2 z z a~z I I -cu +m -is

f

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

(5.3)

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

PION CONDENSATION

473

On doing the angular integration and some algebra we obtain = kz Z~z (2n)-'

Ihz>

ady'

J0

~+~dk'

C~4

sin ky

k

( -i~+

8 cos ky ~

(-i)

_ 8 sin ky k' 8 sin ky 24 oos ky 24 sin ky (-t k (kY) (kY)z 3 ~~ + ~ kY + (kY)z + 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 1 + °° d 2n ~ e~yik'dk' = 2n ~y)" 2n dY

The next three teams of (5.4) are accompanied by a S(y) and in that limit give no contribution . The last three terms of the integrand (5.4) are easily evaluated, and thus the whole of (5.4) is z z IJ~cz~ = kz ~ P ~~- ~daC l ~ (5.5) co J0 where C and a are defined later. Hy similar methods we get for nHCZ~ z ~(z~

~~

+v-1

~(vz)

kkY

~o a~Cs_

Y-1) u~kz + Y

us kkY - kY)1 3Yz

2 sin ky + 6(oos ky-1) - 6(sin ky-ky)~ e_ .r C kY (kY)z (kY) 3

_ ~C2(sin ky-ky) 6(cos ky-1 +~(ky) z) -v z + kyz kzy3

'

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

At this point we observe that we shall consider only a~ < m,~ for otherwise the proper self-energy becomes imaginary. We shall comment in the discussion on the plausibility ofthis. The contribution to the proper polarization from the second order finite hard core is evaluated as n~~~z> where

-

~z~~kz ~~- ~~ ~daAe-`°~k+

J0

k J 0~daBe-~+ J 0~dox- ~/kÇ~J ~

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

~5.~ (5.8)

474

V. SOIYI

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

6(oos a- 1+ZOCZ)

v = (~ - cuz)~,

(5.9)

6(sin a -a+6ac3)

a=ky .

We note here that all the above expressions reduce to those of the last section in the appropriate limits. 6. Higher-order direct hard core (Snits) cald~latioos We now see how fàr we can carry out this analysis of the finite hard core in the third and higher orders. From the complexity of the second order it seems inevitable that we will have to make approximations. Using (2.~ and putting in the explicit expressions for the propagators we get the proper self-energy II"~3).. exp(~ki - k)' Yi) 1 17 ~3) = Z3P3 ~3 (2~)6 ~~d3 Yi d3k 1 -W2+Ik1I2fmZ-lE X JJ

d3yzd3kz

P ~ z) ly . ki k k ~ k z k i ~ kz w + kzlz+mz X

{( 1- R(~Yi~)xl -R(IYzI)xl - R(~Y1+Yz~))}~, roa ~p

Now we see that it is only if we retain the leading or "1" (monopole) term in the expansions ofexp (-ik ~ yi) and exp (-ik ~ yZ) that the ki and kz integrals separate. Keeping just the monopole, we average over k : Â3 P 3 exp(iki Yi) ~ii(3) -kz 1 s ffd3 Yi d3k 1 -~ z +~ki ~ z ~ (~)zltz (2n) ~3 +mz -ie 1 X

X

exp.(ik z ~ yz)~ f d3 Yz d3kz -WZ+~kz~z+mz-i~ J {(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 + yz U or consider only those oorrelations which are accompanied by propagators between the adjacent space time points (as in sect . 3). Now we can straightaway use the expression 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

47 5

Although there is no clinching motivation for the monopole approximation it is simple, it reproduces the zero-range result in the limit a = 0 and also takes account of the co-dependence. It is reliable only for small "d' and around w x m i.e. close to 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 right approximation. The replacement l+va for e - "°(1+va) could conceivably be more consistent in terms of "d' dependence but would have the wrong "co" dependence. So we proceed to use the monopole approximation in the following section ; a comparison with other candidates follows in the discussion. 7. Final calculations for critical d~sity Now 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~+ ~Bdae-~"ik~+ dace-~"~k~C. k J0 Jo 0 These expressions are too complicated to deal with as before. The idea of doing the finite hard core to second order exactly and the higher orders in the monopole approximation is to see ifthe neglect ofthe rest of theseries 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 an imaginary contribution to the proper self-energy when co > mx ; we defer comment on it until the discussion . Here we list the expressions we get for the two critical conditions 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 W

O+ b?k N+ b? k2 M+ v

W

1 1 k2

+C'

Wa

2

3WZ

V

(1 +va}w2 +~be-'° k

e - '°(1 +av)

aW

3

b3k2 ae-'°

3 2 -? -~Oe-°`(1+aY~+W3~be-'°(1+aY)a Y 3 W -~6e -°°

1

Wa W2 Y

k

-C'

-~CU

W Ib2k2

b3k Wv

[1+av]Ne -'°

1 b3k2

Wa

3be ;° a +~Wbe-'°(1+Va) _ :

V

[e - '°(1 +av)

1]

C

b2k2 3W 2

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

aW

e- .°

Y

(7.5)

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

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

~~ '

~W

v

YI Ir° m ) v ~ N = ~-~ daAae - c"it>a+ - ~ daBae - c'~k>a+ ~ daec- '~k~°aC} ; 2 k o k o 0 These expressions were used to do a computer analysis for a double root With a = 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~ore value of 15 ~. The plot for this is shown in fig. 5. 1 .91 FINITE RANGE HARD CORE RANGE a =Q40 (m* = I ) .564 F =0

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

L 1 .5

k (m*= I )

I 2A

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

PION CONDENSATION

477

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

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

2~z

(-

X

x-

) ~~~~ 0 J

>ndexdax~ 0

V

4coinwout

exp ( - ik°ut . x)

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

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

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

_ _ WZ

d4xd4x'd3gd34 4eoio coaut ~~~ ~ x exp (i(- k°°t+q +q' + k~. . x) exp (~- q -q'-k'+ ki°) ~ x') o exp (- ~ki - ~o)~) exP ( - ~~o - k°o"`)r) d4 x l~l~zk~° ~ /r°ut Ik'IZ -IkolZ +mz- ie

(2x)io y

(8.2)

This gives â

~-

1 ff~ f ~d 4_ ~d3 ,~gg r 4mmmo"t ,l,lo ,lo

-i 1 ~ co (27[~ V

x g(-k°"t+q+q'+k~S(-k'-q-q'+ki°~(kô -Is'o~(l~o-k°o"~ ~ . kov
(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 and handle this perturbatively to first order in mz -a~z. The contribution to the proper polarization is IYd~ 0

0

{k-(9+9~}z

x 1-(mz-m )

{k-(q+4~}z

d

~

4'u~z . (8.5)

478

V. SONI

We get for the proper polarization (m ~â~ ° 2 ~zwzPz - z- wz )B,

(8.6)

where B is an integral arising from the second term in (4.5). Analysing the hard-core part essentially as in sect. 3 gives z

~HCt

Pr

vr

wz ~0 ,~0 d3 N"3 Y /(2~) 1 ~zkzpz llcz~ xc-~ - - _2 wz .

6

This exactly cancels the first term in the non-hard-core part. Doing a perturbation analysis (see sect. 4) on the remaining term (with a mz - wz factor) we get no change in the critical density due to exchange. It thus seems that the exchange term may not prove too important in det?rmining the critical density. 9. S-wave and isobar ei%cts Now to make this analysis more complete we put in the S-wave nN interaction in the Weinberg form and estimate its effects. The contribution to pion self-energy is easily derived:

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

(9.1)

The Hamilton equations are -SH -

z

__ 8H _ Sa* - ~

i

pz

_i 2~z P4~~

(9.2)

Then from the second equation above i i i i pz = n-~P~=v2~-~~-2~PC~P+2~P~} -2,~P~-4~~, or

~ -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 opposed to the singular contributions from the Oqv* ~ V~ part of H~ in sect . 2. Thus, the S-wave does 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) and S-wave+isobar+hard core (zero range Secondly we put in the effect of the d-isobars in the p-wave, which have, according to Migdal 6), important modifications in the pion propagator. Here we follow the analysis of Brusca and Sawyer `). We first give an argument as to why we can treat the 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 intermediate d cannot occur explicitly due to the vacuum for the d being the normal one. Thus the only rotationally invariant H~ is the scalar Cp(x)V~*(x) ~ V~(x) where C is a constant which carries the coupling, the appropriate Clebsch-Gordon coefficients, 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 that the spins of the d and nucleon are aligned in the same direction . Thus we can consider the contal Hamiltonian as being given exally by what Sawyer and Hrusca have in their e4. (28). In other words, the whole analysis for the protons goes over without modification to the d. This enables us to treat the aero-range hard core with the d and protons to all orders . Values of critical parameters for variom canes Value of k2 fm

2 3 4 S 6

7.ero-raage hard core and isobar Pe ~~-~~

~

0.137 0.119 0.112 0.11 0.104

0.6 0.067 0.72 0.76 0.78

Pe

~~-3~

0.076 0.072 0.0685 0.0683 0.0685

Isobar S-wave and aero-range Lard core

Isobar and

Isobar

~~~

~ 0.7 0.76 0.79 0.82 0.84

Pe

~~ -3~

0.083 0.076 0.072 0.072 0.0685

~

Po ~~-3~

.d

0.74 0.78 0.81 0.63 0.86

0.169 0.133 0.122 0.115 0.112

0.68 0.73 0.T7 0.79 0.82

2~~ran®e . Lard ocre and S-wave Pa

~~ -3 ~ 0.54 0.54 0.493 . 0.442 0.432

1 .33 1 .49 1.6 1.66 1 .76

It seems from the above table that Luger and Luger k will lead to a k -" oo pLase transition, i.e. tLe density B~ asymptotically. to its lowest value ask -. oo . This would certainly not happen wLen roonil effects are rigorously taken into aaoount ; Luge enough k will then force the density up . However, tLroughout this work we assume that tLe nuclei are nonrelativistici.~ k=/211f < 1 and in such cases the changes in minimum critical demity on putting in roooil will be quite small: thus the above values of 9a ate quite reasonable in spite of tLe anomalous k-behavior.

480

V. SONI

The entire proper polariTation to first order is,from Sawyer and Brusca °) [eq. (32)] (9.5) where cvR = 1236-938 MeV is the energy difference between the d and the nucleon and 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 in table 1 for various cases. 10. Dl

on

There are indeed a number of questions that arise from this treatment of pica condensation . The assumption of the static limit is a valid one as can be seen from appendix B, provided the pica energy c~ in the medium is greater than the nuclear excitation energies. This condition is fulfilled in the present theory . In sect. 6 it was pointed out that the monopole approximation was used for the sake of simplicity. However, for the values of a and v encountered, the differences between the monopole approximation and the alternative oné suggested in sect. 6, or for that matter the simple zero-range hard core approximation, are extremely small. The difference in using e -"~(1+va) instead of l+va was found to be of the order of two to three per cent. Thus it seems, for the purposes of our calculation, the choice of the approximation is only academia Expressions similar to our calculations for the zero-range hard wre have been obtained by Migdal e) and oo-workers from the theory of Fenni liquids. They introduce the hard core correlation as a vertex correction fig. 6 to the pica nucleon vertex ; 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) vertex the expression (fig. 6)

To by

T = Tp +l'°°AT,

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

PION CONDENSATION

48 1

(g+g'ss~d where s and s' are the isospin matrices, Q and Q' are the Pauli spin matrices, m is the nucleon mass, and p° 2 [a constant, see re£ e)]. The diagram does 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 correlated with 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 particlehole interaction. This is a non-overlapping set of contributions. We first remark on a serious omission of certain graphs in the work of Midgal et al. These graphs are of exactly the same order, but differ in the ordering of interaction vertices 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 nearly cancel 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 is surprising that only g' and notgappears in the answer for therenormalised 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 nonvertex correction graphs (figs. 7and 8). In the case ofvertex graphs we cannot conserve charge and baryon numbers at all vertices . Only ifwe include the n° are vertex graphs possible. 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 theory of charged pious. The energy of the pion a~ (chosen as Rt just for illustration) goes through the proton in 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.

48 2

V. SONI

causally aeoount for the correlation we must include a meson, which carries the energy w between the space time points, in the particle-hole channel t (fig. 8). Thus the approach of Migdal et aI . does not suffice if we include the causal interaction as opposed to a static potential . We should point out that w and k are not small, conditions necessary for the validity of their approach . Though Midgal et al. assert that g'(w, k) is a function of cu and k, in the Limit of infinite nucleon mass the ~-dependence will disappear in their picture, whereas, it remains if a virtual pion in included in l'°'. 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 from the parameters of nuclei and thus reflects the effect of a phenomenological hard core, whereas, in our case the factor ~ is a consequence of the fact that we consider a hard core 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 part and (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

N d3YL~yx-R)]-(mz-cvs)

J

('exp(ik~y~3k'd3Y Jk~ - fA+m-lE

(-R) .

This separation has two obvious interpretations, a mathematical and a 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 can contribute and this has no imaginary part. Physically, the first term is a static or noncausal . part ofthe propagator i.e. it does care for "u~. Ofcourse, since this part alone contributes 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 dependence on the range a, e.g. mint ~ m,~+ 1/a The zero-range correlation seems to fit this kind of 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 is a direct consequence of the fact that we use the frce pion propagator . Since we are doing perturbation theory in the interaction picture we must use some form of free propagator. Later on we examine the possibilities of going to other interaction pictures 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 a time 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 of the 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 the nucleons are static, that all the energy in the intermediate state is carried by the pion and thus when w Z m we can form a free (in this picture) pion as opposed to one belonging to the wndensate, which immediately gives rise to an imaginary part. Now we examine the possibilities of modifying the interaction picture by including the Swave interaction in Ha. The S-wave is particularly amenable to a modification of 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 the imaginary part arose from the function or from the square root of the negative of the inverse propagator at k = 0. The new expression would be m -cv +pco/n . This will not go imaginary untill cu = (p/2»~)+ m,~+p /4mR delaying the appearance of the imaginary part. The S-wave may be looked upon as giving a new effective mass n>g = m,~ - cv + pug/m,~, the counterpart of m,~ -co in the absence of the S-wave. It is thus obvious that we get a higher effective mass with the repulsive S-wave interaction than without. Since the force due to simple pion exchange is attractive and that due to the S-wave repulsive, we are lead to believe 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 threshold of pion condensation. Thus, to establish that the S-wave really delays the imaginary threshold we must satisfy cvß -cot < pc~/m=, where cvs is. the new value of co at that point where condensation first set in, in the presence of the S-wave interaction. This is 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 interaction 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 offthe attraction near the origin instead of at the outer reaches of the potential, resulting therefore in no new threshold It only causes w to 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 of a hard core of variable range and the virtual pion with the exchange of a massive particle whose mass depends in some simple way on the core radius (or correlation length as has been mentioned earlier). This is so as the hard core always acts in conjunction with the virtual pion exchange to give the causal interaction. It follows that the imaginary threshold is detenmined entirely by the physics we include in the pion propagator . The pion chemical potential can, then, never exceed cvo where mo > 0, i$ determined as the appropriate pole (corresponding to the given initial conditions) of the inverse pion propagator. These remarks also underline the importance of keeping the "m» dependence or causal 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 which would imply that the ground state of the system be modified till there is no imaginary part. 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 the neutron matter might avoid this eventuality. Now, it is expected that for free pious were cu to exceed m,~, the formation of a k = 0 mode would lower the energy . In the presence of other interactions (e.g. S-wave) the k = 0 mode will occur at a sew threshold in w which is determined by the interaction. That this happens and co does not exceed this threshold value is demonstrated in appendix C in a calculation which uses the mean field approach of Sawyer and Scalapino'). We can also lower the chemical potential by the inclusion of electrons t. These processes would give an admixture ofcondensate and k = 0 pious (electrons) over a certain range in the lower density regime and will modify the ground state, in principle, whenever faced with an imaginary polarization . However, both these processes are not significant in neutron star calculations, as the amount of condensation and the effect on the equation of state is small until one reaches the critical densities given by the Bertsch and Johnson conditions. 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 subtracted hard core, i.e. the full finite hard core less the zero-range hard-core effect, whereas in our treatment the co~ependence is kept and enters dynamically in the determination of the critical conditions. Now, since the contribution of the subtracted part of the hard core is small the differences do not show up at all dramatically . [The separation distance of Weise and Brown, eq. (10~ is roughly equivalent to the range a of our correlation.] For ~ < m,~ we do not get a double root for values of the range exceeding 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 future with 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 small range, so we can use the same methods to treat the two cases ; ours with co = m and that of Weise and Brown with co = 0. However, if there are large changes in density the perturbation around fixed values of the critical parameters become questionable.

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

PION CONDENSATION

48 5

The subtracted oontributiôn of the finite hard core to the proper self~nergy is 1 1 SII = - kZ PA ~

-

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

The resWts for m = 0 and cu = mR for two separation distances Separatidiet on . d or a

R+

mR

to 2nd order as in sect . 4

exact 2nd order and monopole to all orders

to 2nd order as in sect . 4

exact 2nd order and monopole all orders

0 .262 9~

0 .283 16 ~

0 .294 22 ~

0.2 7 24

0.25 4~

0 .259 8~

0 .266 10 ~

0.266 10

d = 0 .85 fm p~ (fm -3) increase over um range d = 0 .564 fm (fm - ') increase over zero range pR

The changes in the critical density, from perturbing about the zero-range critical values (secrt. 4), for u~ = nt; somewhat exooed those for co = 0. However, we sce that the change from a zero-range hard core to a finite range of 0.56 fin, of almost throe times the Compton wavelength of a nucleon will cause only minor changes in the critical.density whether we take u~ = 0 or a~ = m,~ The changes fora range of 0.85 fin, which approximates the same parameter in rei t) are slightly larger but the perturbation about a fixed point also becomes less reliable. Even for this value of the range the effects are somewhat smaller than in re£ ~). Given the approximation of using the monopole in the third and higher ôrders, the point this analysis emphasizes, is, that increasing the range of the hard core does not give rise to any dramatic changes in 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 of the authors mentioned in the introduction. There isa further increase, though smaller, on including a correlation with a range of 0.56 fim. On increasing the range further

486

V. SONI

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

Appendix A NEW FORMALISM IN TERMS OF THE CONTACT HAMILTONIAN' VYe begin by checking if in the case of static nucleons we can substitute the contact Hamiltonian H~ _ (~ ./co)p(x)Vq~*(x) ~ Vq~(x) instead of working with the usual p-wave pion-nucleon non-relativistic coupling f(n(x~p(x) ~ Vcp(x)+h.a) where n(x) and p(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

~. . .

J

d~x l . . . 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 Fermi sea IFo) for the neutrons, corresponding to pure neutron matter. We can write 1M 4~ as follows, since each of the six terms of (1 .1) are equivalent, ~a~

4

f dxidxs`dx3dx4~FoIT{n~(xi)n,~(xz)n~(xsM P(x4)}IFo)

x <~~IT{ar
(A .2)

Factoring and calculating the proton part lMP4~ we get the expected space delta functions and time theta functions: Mv4~ _
PION CONDENSATTON

48 9

exp ( Q iqo(t~t - t1)) ~ ~a~ - i exp (i4 ' (xt - xz)~pxd a4 (2n)a (2 n)a r daq' exp(- ~ô(ts - ta)) exp (i9 ' (x3 -xa))Sa~ qo+ie +same with xl, tt ~-. x 3 , t 3 , Saz -" Sby Sb~ --~ _

SR~,

1 fexp( igo(tt - tz)) _ _ - SßzS°"~x' -xz~x3-xa) 2nJ d4o 2n 4o +itl (exp ( - ~o(ts - ta)) _ x dqo S8xS6~~x t -xa~xz - xs) qo+ie J

x

1 (exp ( - igo(ta - t1)) 1 fexP ( - i4ô(tt - ta)) , + ~ dqo +~ dqo~ 2n 2n qo qö

(A .3)

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 nonrelativistic Fourier expansion for the field operator, ct ex

tk ~ x

kkr

where ä~ and ~ are annihilation operators for ~k~ < kf and ~k~ > kT respectively and a is the spin index, we get tt mal = < Folna(x~(xhi°(x~~p(x~IF) _
~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),

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

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

a l1 1 p< Môa'.= P Sa,,S°pd3gexp(i9' (x-~) ;S,pS ~+S(x-x~apS,~ s `(2n) Jo I C21

(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~ .

48 8

V. SONI

The above follows from converting the ~ to an j and using ~,1lV = ~p (as we have excluded spin). Lastly, we do the pion matrix element M;~a~ to give the full expression. Here we have carried out the oombinatorics associated with all the T-products : M~a~ _ - fa ~. . . .dtldaxzdax3dta(2n)-z J

exP

(- ~o(ti - tz)) 8ez d40 40 + ~1

x (exp (- iq~(t3 - ta)) d4ô8av
(A .7)

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

2i (-

~ no"~ IT{01~ *(xz)ON~xz) ~J daxzdaxs x V,cp*(x3)V,~cp(xs)}In,~i
W~/ \- ~ "c/

(A .

were !cô = Wm and I~"` = w~, are used interchangeably for the energy of the incoming and outgoing pion respectively . Finally we do the spin sums f saaaex(Q~~Qr)z,~Q~oa(~~~n~a~aev(~p~ -Az S~a.b + P~xz -x3~apSrY~ _ ~~12p) 2ai1"kl - 2

exp (~ ' (x2 3 ~(2~~ JO~d34

- 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 three terms 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 and corresponds to fig. lb. The final term has a 8(xz - xs) in it which automatically reduces the pion part M!~a~ to zero and so is inoonsoquential and may be dropped. This gives us the following sum : Mta~ _ _fa ~- ~~- ~ ~ ~. . . daxzdax,<~,nIT(o~*(xz)O~xz)

J

t Here A

a (I/(Za)')fS~d'getP(i!' (~z-=s))~

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

PION CONDENSATION

x

C~zP~ -2

{(2~)s J

489

~d34 exp (iq ' (xz - xa)) }z + (zP)~xz - xs)J

. . d4xzd~x3 C(zP)~xz-xs) ( -) ~ w~ 1 ~ 1 ~ ,~~ J 1 ~ z exp (i9 ' (arz xs))} 3 -2 {(2n) o d34 +f

~

f

+ 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 the contact Hamiltonian H~ _ ~,(1/co)p(x)Vtp(x) ~ Vqv*(x) [p(x) = n(x)n(x)]. The second term is the residue . It is thus obvious the entire direct part and part of the exchange come 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 the direct part is identical to that given by H~ i.e. the rtth order direct term is " Mia daxi (1rP_ . . . v~p*(x,w~p(~}I~~>~ -d4x"~~ou~I T{oN~*(xi)o~(xi) ~ n ~ \W/ "J J 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 the original 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 part survives the contour integration. The part of the neutron time-ordered product a~ Mô _
J

dEexp(- iE(tz - ti))exP(iP' (xz - xi)M3P

x ~

1 +2nib(E)B(pr -P) j E+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 Jdtld`xzd4x3dt4exp (- iE(t2 - tl))d3P~Â(~xi~Pr-P) J

x dE'd 3p'exp (- iE'(t~ - t3)~p,S(E~2ni6(pr -p~ J

x x

(~

J

d4o

exp ( - igo(ti - tz)) 40

V ki ~~`

+

~

apz

, exp (- igô(ts - t4)) d4o a°" 9ô + iE

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

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

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

M°~4)

(A.10)

4 2i

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

~~- ~

~~

~~

Woutl

d4xzd4x3'

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

(A.11)

The "exchange" graph follows from the other permutation in thè relevant part of the Wick expansion, for the neutron time ordered product. Thus contracting no(xl) 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 1 ~daP~~~ +2nib(E)B(pr -P) 8~ exP (- iE'(tz - t3)) x [E+ ib J (2nd x exp (ip' ~ (xz -xa))

[E' +iA

+2~i8(E~9(pr -p~ ba. J

(A.12)

Performing the same algebra as before we recover the expression ~~) - f` 1 o= 2( ~~- ~-

Wio

-

1 moot/

(-~

1 ex P (iP. xb3p~pr -P) C(2rz)3 ,~

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

PION CONDENSATION

49 1

where the residue is R

__ _f __ ~~- ~ l ~ _~ 2~ Win

l ~ ( -~ ~(2~)3 1

Wont

X
J0

exp (iP- xM3PB(pr - P) Zd4z2d4 xa J

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~ Appendix B

We digress here to look at the structure of the time dependence of the neutron part and understand the conditions governing the static limit for the nucleons. We start with Mô`~ _ < FoIT{n(x, tt)dx,

This is broken up to give t M~nt '

,

tz~x , r3M(x', t,)}IFo).

(H.1)

_
(H.2)

Now we introduce a complete set of eigenstetes ~ IN)
= ~ exP f ~~.- ~Hxtt -rs)) exP ( t ~~.- Wirxta - ra)) A!, N X

-


N,N

X
(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 Wide contraction which are not ~devant to this calciilation as their indusion does not aher the time depeadenae of the 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)) , dtldtzdt3dt~ dqo dqc (2n)z J ~ qo + pl ,~ qo + ie x exp (icoo,t4) exp ( -~mti)

x {exp ( f ~N(ti - tz) t ~zr(ts - ta)) or exp ( f ieN (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). Doing the "tl and "t4" integrations, and then integrating the consequent 8-ftwctions over qo and qô we have dtzdt3 or I I dtzdt3 JJ

1

1

1 exp ( - Wmtz) exp (~outt3), f - Wout EN Wio f EN 1

- WouttEN -Wint Eèl

x exA ~- it~ (Win f(EN- EAl))) exP ( + it3 (cuut -( t IIEN - 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 time dependence ignored provided eN ~ Wm.o~t. As this happens to be the case we are justified in making the assumptions above. Appendix C We 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 will never exceed m,,~ (e = 0) or rrç (E = 1) (for the case that includes the S-wave pica nucleon 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 the fractional number of a- mesons in the momentum k mode, Y is the fractional number of R ~ mesons in the k = 0 mode, Mk = kf/V}askm,~ (f z = 1.1 the strong coupling constant) and e = 1 for the case that includes the S-wave pica-nucleon interaction and.is otherwise 0.

PION CONDENSATION

493

The expression for the energy follows from the same calculation as in re£') with the new charge constraint X + Y = B2 = fractional number of protons. To get the n- 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) Also (C.4) We could say right here that ~_ -- Px _ ° and so ~_ must also be m~ but we go ahead and show this directly . Writing (G1) as

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

At threshold

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 thus we 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 of x (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 exactly the content of (C.8); until ~r.,~- < m;~ we have no ~-(k = Oj component but as soon as ~- exceeds ~ this component appears in such a way as to keep the chemical potential 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 density with and without 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). References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)

w. weise and G. E. Brown, rhya. Lett. 29 (1974) 386

G. Baym and E. Flowers, Nucl. Phya. A232 (1974) 29 G. Hertach and M. Johnson, Phys . Lett . 4ßB (1974) 397 D. Bruaca and R. F. Sawyer, Nucl . Phys . A236 (1974) 470 R F. Sawyer and A. C. Yao, Phys. Rev. D7 {1973) 1579 A. B. Migdal, IPhya Rev. Lett . 31 (1973) 257 ;1?hys . Lett. 4SB (1973) 448 ; 47B (1973) 96 R F. Sawyer and D. J. Swlapino, 1?hys. Rev. D7 (1973) 957 A. B. Migdal, O. A. Markin and I. M. Mishushtin, ZhETF (USSR) 66 (1974) 443 S. Barahay and G. E. Brown, Phys. belt. 47B (1973) 107 C. K. Au and G. Baym, Nucl. Phys . A236 (1975) 300 M: O. Ericson and T. E. O. Ericaon, Ann. of IPhys. 36 (1966) 323 S. O. Hackman and w. weise, Phys . Lett . S5B (1975) 1 G. Bettach and M. B. Johnson, 1?hys. Rev., in press G. Haym and G. E. Brown, Nucl. Phys. A247 (1975) 395 G. Baym, D. Campbell, R. Dachen and J. Manassah, Phys. Lett ., in press D. Campbell, R Dachen and J. Manessah, Phya Rev., in press