Calculation of the threshold for π− condensation in neutron matter

Volume 55B, number 1

PHYSICS LETTERS

CALCULATION

OF THE THRESHOLD

~r- C O N D E N S A T I O N S.-O.

20 January 1975

FOR

IN NEUTRON MATTER

BJ~CKMAN* and W. WEISE**

Department of Physics, SUNY, Stony Brook, New York 11794, USA Received 9 December 1974 The threshold conditions for a phase transition to a negative pion condensate in neutron matter are studied. In the nucleon-nucleon particle-hole interaction it is essential to take into account the off-diagonality of the nucleonnucleon interaction due to finite n- meson momentum. A reaction matrix calculated from the Reid soft core potential is used. Allowing the A isobar to enter as an intermediate particle in the polarisation bubbles of the curentcurrent correlation function and including s-wave pion-nucleon interactions, we obtain a critical density around two times normal nuclear matter density. First predictions o f ~r- condensates in neutron star matter [1] and rt° condensates in nuclear and neutron star matter [2] have located the threshold density in a region o f the order of nuclear matter density, Pc ~'Po = 0.17 fm -3 . Since then a considerable amount of work has been devoted to this problem [ 3 - 8 ] . In particular, effects of short range nucleon-nucleon correlations were intercorporated ref. [4], which tend to raise the critical density for the phase transition. Here we study, in an estimative way, the importance of the off-diagonality in the nucleon-nucleon interaction due to the n meson momentum k, together with the inclusion of the A isobar as an excitation channel of the system. In addition, s-wave interactions of the pion will be considered. We shall restrict ourselves to an investigation of art- condensate in neutron matter; except when defining chemical potentials [5] for numerical calculation, we shall consider a normal Fermi liquid consisting of neutrons alone. A phase transition occurs through an instability against the process n --, p + It-. The corresponding vertex operator for a p-wave pion in m~rlfX/-2gl~ozCgi where we take the direction of pion momentum k as the z axis and denote by qzi,f the nucleon field operators in the initial and final state. This vertex creates particle-hole excitations o f the t y p e IpfO = (1 [ V ~ [It {) - l J, 5)] ,(t = particle with spin up)(l) * Work supported by A.E.C. Contract AT(11-1 )3001 ; on Leave from Abo Akademi, 20500 Abo 50, Finland. ** Supported by Deutsche Forschungsgemeinschaft; on leave from University of Erlangen, W. Germany.

where holes have been denoted by a bar. The threshold for n- condensation is signalled by a singularity in the current-current correlation function ( J , J ) (see fig. 1) at a pion frequency coc --/a~- =/an - #p,

(2)

where/a n and #p are the chemical potentials of neutrons and protons in the normal state, respectively. The function (J',J) is related to the pion propagator in the medium, D(k, co), by

D(k, co)=Do(k,w)+Do(k,w)(J,J)Do(k, co),

(3)

where Do(k, co)= [co2-k2 - m 2 +ie] -1 is the free pion propagator. The straight lines in fig. 1 represent the particle-hole interaction in the relevant channel, which we denote by ~, where it is to be noted that in general we face a non-static (co =/=0) situation. The interaction is given by ,1 rll 1

-- 4" ~' 11

t W/700

ms=l. 1 ~tr/s--O

- ~ F10

* Z

P'I0

1

-- ~ ]701 ,

(4)

where F~7 is the particle-particleinteraction in states of total spin S, spin projection m s and isospin 7'.*2 Exchange diagrams are included in eq. (4). As nucleon-nucleon interaction we use Brueckner's )I At k=O and for symmetric nuclear matter, this would be g' in Landau's theory of normal Fermi liquids [ 11 ]. *~ The difference in sign between different spin projections enters because the pion acts as a test field for the spin polarizability of the system.

Volume 55B, number 1

PHYSICS LETTERS

20 January 1975

where

~~ ~ - - ~

~x~ £ ~~, '~k

•r vo',

v j

_

gOPE(k, co) = k2/(co 2 - k 2 -

", ÷ ,,

•

Fig. 1. Representation of the current-current correlation function (J,J) as a series of polarization diagrams. Straight lines

between bubbles represent the full particle-holeinteraction. Upgoing lines denote protons; in our inclusion of isobar excitations these can be replaced anywhere by a A-isobar. reaction matrix [9] calculated from Reid's soft core potential [10]. The pion energy ~ is treated as an excitation energy of the nuclear medium and thus enters only in the energy denominator of the reaction matrix; all non-static effects will be ascribed solely to the one-pion exchange part, as in ref. [4]. The reaction matrix elements of interest are of the form


~

X (Pfl

+m s)

where, with the notation of fig. 1, p _1 _1 i,f - ~ (P - - P ' ) +5 k,

(5)

(6)

are the initial and final relative momenta, respectively, and (Pfl G///'msIPi ) are the reaction matrix elements in momentum space for definite initial and final state partial waves l and/'and total angular momentum J. (For further details, see ref. [9] .) In general we obtain an integral equation for (J,J), since the reaction matrix depends on both Pi and pf. In our calculation we included in eq. (5) all states with/, l' ~< 3. Since the truncation of the partial wave expansion of the onepion exchange introduces an appreciable but spurious degree of non-locality in the reaction matrix, we have subtracted the Born approximation of the one-pion exchange from the reaction matrix. We denote the remainder, due to short range correlations, by gc and have, for the non-static reaction matrix in proper units gNN(k, co) = (1/2f 2) G - gOPE(k, 0) +gOPE(k, co) =go + 1/3 +gOPE(k,w),

the 1/3 comes from subtracting the 8-function piece of OPE. We have neglected the small contribution from the exchange of It°'s in the exchange diagrams of fig. 1. The Lindhard function part that corresponds to eq. (7), reduces to UN -- 2 f 2 p/w (with f2 = 47t X 0.08 in units of m~r2) in the limit of static nucleons. In eq. (7) we have now split the particle-hole interaction into a piece depending only on k and a piece depending on Pi, Pf and k. The latter can be summed to any order by an integral equation. However, we have approximated gc for each value of k, by its value for an average holemomentum (p2) = ~p'2) _3 _ _g k2F (k F is the Fermi momentum of pure neutron matter), and ( p - p ' ) perpendicular to k. As long as no A-isobars are included, (J,J) now becomes a geometric series

(J'J) = -

ll'ra

G~tmslPi) Yrm (]~"/~f) Y/m (/¢"/~i),

(8)

k 2 UN(k, w )

(4rt) 2 i t-r

× (l'mSmslJm +ms)(lmSmslJm

m2);

(7)

I+gNN(k, w)

UN(k, w)'

(9)

with gNN treated as described above. In fig. 2 we present the quantity ~OPE(k, 0) = gOPE(k, 0) + 1/3 together with gc calculated at a density p = 1.6 Po- The reaction matrix elements were obtained as described in ref. [12] by solving linear equations on a set of Gauss points. The kinetic energies of the nucleons were used as their self energies.*a Although the contributions from various channels in gc vary considerably as functions of the momentum transfer k (for example, the contribution from the (3S 1 + 3D1)ms--1 channel changes from 0.03 at k=0 to -0.1 at k=3m~), the sum is remarkably constant. We think that this is due to cancellation of an increasing attraction (with growing pion momentum k) from the core in S-waves, a switch in sign (from repulsion to attraction) at roughly k ~ 2 m~ of the contribution of the second order tensor force (see ref. [18] ) and an increasing net repulsion from P-waves. Based on estimates gc should be good to within a factor of two, and so in any event smaller than 1/3, the subtracted 8-function piece. We finally note 'that a rho meson exchange potential, with the 8-function piece omitted, would give a constant contribution of roughly the size of go" *s The potential energies of protons and neutrons are fairly small, since both are off the energy shell by roughly mrt, and the proton momenta involved are on the average fairly large.

Volume 55B, number 1

PHYSICS LETTERS

Table 1 Threshold parameters for n- condensation in neutron matter. The chemical potentials of Sj6berg [16] (for a 2.5% supplement of protons to the neutron sea) have been used throughout. First row: contributions of nucleon-hole excitations only, including the full particle-hole interaction gNN" Second row: isobar-hole excitations have been added, but only onepion exchange isobar-nucleon interactions are in'cluded. Third row also takes into account isobar-nucleon correlations due to rho meson exchange and a short distance cutoff (d = 0.4 fm), as described in the text. Numbers without parentheses are obtained including s-wave pion-nucleon interactions. For comparison, numbers in parentheses show results if s-wave interaction is omitted. Critical densities are given in units of Po = 0.5 m~ (nuclear matter density).

0,3

3.2 0,1 o; \

-0, I

'

'

;o

;,

-0.2

-0,3 [ -0,4 • -0,5 -0,6 Fig. 2. Plot of the static one-pion exchange interaction ~'OPE(k,0) in Reid's soft core potential, together with additional contribution gc due to short range correlations, in dimensionless units. The correlation part gc is presented for a density p = 1.6 Po here; at p = 2.0 Po, it turns out to be smaller by between 5 and 8%. In table 1 results for the rr- condensation threshold using only the nucleon-nucleon part are given. If the isobar is allowed to substitute for any proton in the polarization bubbles of fig. 1, calculation of the current-current correlation function becomes a coupled channel problem, where now the poles of (J,J) are determined by a secular equation: 1 +gNN UN

gNA UA

det

= 0,

g~NUN

(10)

l +gaaUA

in obvious notation. The isobar Lindhard function part UA becomes

{ 1

, I ) (11) employing the resonant 7r-n amplitude in both the direct and crossed channel, with 6o4 = 2.3 m,r, the position o f the (3.3)-resonance in the rrN lab frame, and ] '*2 = 4/.2 according to Chew-Low theory [19]. As in gNN, we ascribe all nostatic effects in the interactions gNA and gaa to the one-pion exchange, i.e. we write, for example:

ZNZ (k,

= ZNa(k, 0) +gOPE(k,

- gOPE(k, 0). (12)

We take the isobar-hole interaction to be composed of one-pion and rho-meson exchange,

gN&(k, O) = G~A (k ) + G~&(k).

20 January 1975

(13)

Pc/Po

nucleons only

>3

(1.9)

t°c[mlr]

kc[mlr]

-

(1.0)

-

(2.5)

incl. isobars OPEonly

0.5 (0.45)

0.5

(0.46)

2.4 (2.2)

ind. isobars + correlations

2.0 (1.35)

1.05 (0.87)

2.9 (2.2)

The generating potentials are

V~A(k)-- - f f * °l"kS2"k rl. T2,

(14)

V~a(k) -

(15)

m2

k2 + m2

fPffP ( ° l X k ) ' ( S 2 X k ) m2 k2.m 2

r l ' r 2,

where S a n d T are transition spins (isospins). In addition, we take into account repulsive short range correlations (mainly arising from the exchange o f 6o mesons in higher order) together with vertex corrections, which will effectively cut down these potentials in ordinary space at a small distance d of the order o f 0.4 fm. It is to be noted that this procedure affects differently the tensor and spin-spin pieces contained in both eqs. (14) and (15), the minimal effect (for a "zero range" hard core) being to cut out the 8-function p a r t s , - -~ ff*/m 2 and - ] fp f•/m 2, of the OPE and rho exchange potential, respectively. Upon carrying out spin-isospin sums in the relevant particle-hole channel, the explicit expression for eq. (13) becomes Cr~zx(k) = [gOPE(k,0) + ~]

h,r(k),

G~(k) ffi} f p f ; m2 hp(k).

I f m2

(16) (17)

Volume 55B, number 1

PHYSICS LETTERS

These are dimensionless quantities, appropriately adjusted to the Lindhard function parts UN and Ua which already include pion coupling constants (and spin-isospin factors) for convenience. The functions hi(k ) are cutoff factors. A smooth exponential cutoff of the form 1 - exp (--r/d) in ordinary space leads to (A + mi)2

- mr2 hi(k)=(A+mi) 2+k 2 ,

A=l/d,

(18)

a form also used in ref. [14]. For the coupling constants, the quark model [13, 15] gives the relationship fp*/fp =f*/f. In this picture, we simply end up with gAa =gNz~ =gaN in eq. (10). The relation betweenfp and/'is f 2 m 2 (l+rv)2g2NN -- -- ~- 0.65, f 2 m2 g2NN

(19)

where we have used g2NN = 4n × 0.43, .4 g2NN = 4rr X 14.7 and x v = 3.7 for the isovector magnetic moment. The inclusion of A isobars generally moves the threshold density downward, as table 1 shows. As in gNN, repulsive correlations in gNA and g/,A turn out to be crucial. So far, only p-wave interactions of the pion have been discussed. The s-wave interaction is taken into account following ref. [5] (BF) by adding a contribution [Is = 2wp F~ 2,

(20)

(with F~r = 1.36 m~r) to the pion self energy occurring in (J,J). The s-wave part of eq. (20) supplies considerable repulsion and tends to move the critical density upward again. Including all effects, we end up with a critical density of about twice the density of nuclear matter. We note that the critical momentum is bigger (k c ~ 3mrr ) than the results of earlier calculations [ 1 - 4 ] . Inclusion of the s-wave pion-nucleon interaction is the main source for this enhancement of k c. If we neglect the s-wave interaction as well as the isobar contribution, use Sj6berg's chemical potentials [16] we obtain a critical density of twice Po, as in ref. [4]. However, the use of free Fermi gas chemical potentials would lead to a Pc slightly above normal nuclear matter density, indicating that, although we ,4 According to recent data analysis, as communicated to us by A.D. Jackson. 4

20 January 1975

obtain more attraction in gNN than in ref. [4], this is compensated by using more realistic chemical potentials. We have neglected various corrections. Most of them would probably tend to increase the critical density Pc- Since the critical momentum turns out to be large, nucleon recoil effects in the Lindhard function ought to be included. Furthermore, the phase transition point depends strongly upon the treatment of isobar-nucleon correlations, which remain as a major source of uncertainty. Omitting isobar-nucleon interactions other than one-pion exchange would clearly be wrong, leading to a Pc well below nuclear matter density, as obtained by Migdal et al. [7]. The importance of the repulsive rho meson exchange is obvious from table 1. Presumably, with our choice of parameters in eqs. (16-19), using a relatively small rho coupling constant and a relatively short cutoff distance, we have gone as far as we can in favour of a condensate. Increasing the parameters would move Pc higher up. Kinematic corrections to the IrNA vertex [17] which tend to decrease the isobar coupling constant f * , will also point into this direction. We thank G.E. Brown and A.D. Jackson for many stimulating and helpful discussions. We are grateful to A.B. Migdal for, via G.E. Brown, drawing our attention to ref. [7]. We also acknowledge helpful discussions with S. Barshay, M. Johnson, V. Pandharipande and M. Rho and thank O. Sj6berg for detailed unpublished information on his calculations of Fermi liquid parameters.

References

[1] R.F. Sawyer, Phys. Rev. Lett. 29 (1972) 382; D.J. Scalapino, Phys. Rev. Lett. 29 (1972) 386; R.F. Sawyer and D.J. Scalapino, Phys. Rev. D7 (1973) 953; R.F. Sawyer and A.C. Yao, Phys. Rev. D7 (1973) 1579. [2] A.B. Migdal, Soy. Phys. JETP 34 (1972) 1184; Phys. Lett. 45B (1973) 448; Soy. phys. Lisp. 14 (1972) 813; Phys. Rev. Lett. 31 (1973) 247; Nucl. Phys. A210 (1973)421. [3] S. Barshay, G. Vagradov and G.E. Brown, Phys. Lett. 43B (1973) 359; S. Barshay and G.E. Brown, Phys. Lett. 47B (1973) 107. [4] W. Weise and G.E. Brown, Phys. Lett. 48B (1974) 397. [5] G. Baym, phys. Rev. Lett. 30 (1973) 1340; G. Baym and E. Flowers, Nucl. Phys. A222 (1974) 29;

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[6] [7] [8] [9] [10] [i 1]

PHYSICS LETTERS

C.K. Au, G. Baym and E. Flowers, Phys. Rev. Lett. 51B (1974) 1. G.F. Bertsch and M.B. Johnson, Phys. Lett. 48B (1974) 397, and preprint (to be published). A.B. Migdal, O.A. Markin and I.N. Mishustin, Zh. Eksp. Teor. Fiz. 66 (1974) 443. V. Pandharipande, to be published. K.A. Brueckner and J.L. Gammel, Phys. Rev. 109 (1958) 1023. R.V. Reid, Ann. Phys. (N.Y.) 50 (1968) 411. A.B. Migdal, Theory of finite Fermi system and applications to finite nuclei (Interscience, London 1967).

20 January 1975

[12] S.-O. B~ekman and O. Sj6berg, Acta Academiae Aboensis, Ser.B, Vol. 33, Nr. 8 (1973). [13] P. Haapakoski, Phys. Lett. 48B (1974) 307. [14] K. Bongardt, H. Pilkuhn and H.G. Schlaile, Phys. Lett. 52B (1974) 271. [15] M. Ichimura, H. Hyuga and G.E. Brown, Nucl. Phys. A196 (1972) 17. [16] O. Sj6berg, Nucl. Phys. A222 (1974) 161. [17] S. Barshay and G.E. Brown, Phys. Lett. 16 (1965) 165. [18] G.E. Brown, Unified theory of nuclear models and forces, 3rd Ed., (North-Holland 1971) pp. 235. [19] G.F. Chew and F.E. Low, Phys. Rev. 101 (1956) 1570.