Death to pion condensates in nuclear matter

Volume 47B, number 2

PHYSICS LETTERS

DEATH TO PION CONDENSATES

29 October 1973

IN NUCLEAR MATTER

S. BARSHAY State Umverszty of New York*, Stony Brook, N Y, USA and G.E. BROWN NORDITA and State Untverstty of New York*, Stony Brook, N Y, USA Received 19 September 1973 We first note that ff the couphng of spin-, lsospln-modes in nuclei to pmns were of anywhere near the strength required to produce pmn condensation in nuclear matter, then there would be strong effects in nuclear spectra, which are not seen We then work through a model proposed by Mlgdal [ 1] and show that addition of a strong short-range repulston m the nucleon-nuclear mteractton modifies the effecttve mteractmn m such a way as to remove the mare spm-lsospm mteractmn whlch originally gave rise to the pmn condensate We believe that short-range repulsmns also strongly modify the work of Sawyer and Scalapmo [2], but have not yet shown that they destroy the condensate in neutron stars. Our conclusion is that the off-shell p-wave attractmn of pmns and nucleons lies, at least in the regmn of low momenta, completely m the domain of the ~-lsobars Effects from these would be incapable of producing plon condensates, except at high densities, equal to several times normal nuclear density

Magdal has proposed [ 1] the extstence of plon condensates in nuclear matter for densities similar to those found in the center of nuclei, and neutral and charged plon condensates in neutron stars [3]. Independently, Sawyer and Scalaplno [2] predicted the existence of plon condensates in neutron stars. All of these works require the pIon to couple strongly to nucleon particle-hole pairs. Were this to be so, nature should reveal this in nuclear spectra In this note we discuss specifically neutral plon condensates in nuclear matter. For us they are easier to discuss, because of the fact that static effective nucleon-nucleon interactions are involved. If ,,r- condensates in neutron stars occur for low frequencies co < eF, where e F is the neutron Fermi energy, then our arguments should also apply there If they occur at high frequencies, co >> eF, as claimed by Sawyer and Scalaplno, then our arguments can be applied only In modified form. There exists, for example, a T = 1, O- state m 160 at 12.78 MeV. This state has the same q u a n t u m numbers as the plon This state would be brought down considerably in energy (we estimate the shift to be the order of 5 MeV) through its couphng to the pmn,

were that coupling to be through the one-plon-exchange potential. In fact, the unperturbed energy of this state is 12.42 MeV [4] (The unperturbed particle- and hole-energies are found from the binding energies of 150 and 170 relative to 160 ) As seen from the above, the plon-nucleon mteractlon does not only not lower the state, but the state is slightly raised. In heavy nuclei, there is no tendency for the spin-, lsospan-excxtatlons to come down to zero energy, quite the contrary, in 2°8pb the relevant excitation is found [5] well above the unperturbed energy. It is known from the diamagnetism of nuclear moments (the so-called Arlma-Horae effect [6]) that the interaction in this state IS repulsive Nature tells us, then, that Mlgdal must have drawn Incorrect conclusions, it only remains to find out where things go wrong in his development We claim now to do this At the same time, we discuss quahtatlvely rr--condensates. It has been shown by Baym [7] that the onset of a plon condensate is signalled by a pole in the (J, J) correlation function for frequency co =/a~r , where/a~r IS the plon chemical potential.

*Supported under US Atomic Energy Commission Contract A T ( l l - 1 ) - 3001. 107

Volume 47B, number 2

PHYSICS LETTERS

/~r = 0

n °-condensates

/an =/a n - / a p

7r--condensates

(1)

29 October 1973

short-range interaction will make, we first write down where the various parts of the interaction term gOPE = _ f 2 k 2/(1 + k 2 - 602)

(6)

w h e r e / 2 n and/2p are the neutron and proton chemical

potentials, respectively. The quantity J is the source of the relevant plon field,

come from. In configuration space, the OPEP is VoPEP =

J:f

V ' ( f f ; o 3 ~ p - ff;o3ff n)

rr°-condensates

(2) = x/2f V'(ff;o3t)n).

7r--condensates

The expression for (J, J) in the case of nuclear matter and exchange of only pions is


(3)

;2k2uqc' co) 1 - {f2k2/(l+k2-co2)}u(k, co)

4n

3

1

r

$12 + o 1 ' o 2 - 8 ( r )

where S12 is the tensor operator, S12 = 3 o 1 "r o 2 "r/r 2 - o I "o 2.

tensor'

- ~f2k2/(l+k2)

u(k, co)= (2kfm*/rr2)$(k, co)

spin-spin

~ f2/( 1+k 2)

6-funcnon:

_~f2,

where qS(k, co) is the Lmdhard function, defined so that

q~(O, 0) = I.

(4.1)

The 2kfrn*/Tr 2 is, o f course, the density o f states at the Fermi surface. Our units are such that the paon mass m,~ is equal to 1 A pole in (J, J) at co = 0 requires

1 + k 2 - fZk2u(k, O) = O,

(5)

corresponding to the condition Migdal [ 1 ] would obtain for co = O, were he to set his g - ( k ) equal to zero in his eq. (5), and drop the contribution from the isobar. (Mlgdal's g - ( k ) comes from agencies other than one-plon-exchange, so it is appropriate to drop it in the model discussed now.) The expression for the (J, J) correlation function is

(J, J) = f2k2u(k, co)/{1 + g-(k)u(k, co)}

(5.1)

where g - ( k ) is the Fermi liquid parameter of Mlgdal [8]. It should be noted that the g - ( k ) used by Migdal in ref. [1] differs from that of ref. [8] by subtraction of the one-pion-exchange contribution. In our work, we use the conventional definition o f ref. [8]. In order to see what modifications an additional 108

(7.1)

Contributions to gOPE from the three different terms in the interaction to an S = 1, T = 1 collective mode are easily calculatedt [e.g. 4, Ch. I V ] '

as is easily seen by summing the relevant geometrical series (bubble sum). Here (4)

(7) e-r r

(8)

the sum of these three contributions adding up to the gOPE o f eq. (6). Now let us discuss the modifications which a strong, short-range repulsion, which is known to be present in the nucleon-nucleon force, will produce. Firstly, the 6-function in the OPEP will be inoperative, since the repulsion keeps the two nucleons a p a r t t 1 . The repulsive core is thought to come from the exchange of co-mesons, with coupling constant given b y [9] (g2wNN/47r) ~ 7, much larger than the couphng of the pions, so it is clear that this repulsion will dominate completely an the internal region r <~ 0.4 fm. Removal o f the 6-function gwes a modification o f g o p E to t z tWe are grateful to Jan Blomqulst for pointing out this way of approaching the problem. t 1Of course, the 6-function will be somewhat "smeared out" by recoil and by the fimte dimensions of the source functrans. However, m the nature of things, this "smeanng out" is only over the dimension of a nucleon Compton wavelength, and this will not alter our conclusions here t 2Our removal of the g-function is equwalent to introduction of the Ericson and Ericson Lorentz-Lorenz correctmn, as is shown clearly by Elsenberg et al [12]. Their work is for ~o = m~r rather than w = 0 as here. These authors do not include the further modifications coming from the finite range of the hard core which we do later

Volume 47B, number 2

g_ = _f2 {k2/(l+k2) _

PHYSICS LETTERS ~_},

(9)

this g - coming from the tensor and spin-spin terms in eq. (7). This is not the end of the story, however. The tensor term Is also modified by the presence of the short-range repulsmn. Furthermore, additional contrlbuttons of short range come from agencies other than one-plon exchange, e.g., o-meson exchange. Our use o f the (J, J) correlatmn function reduces the problem, for 7rO-condensates where co = O, to a standard one in nuclear physics. Namely, we should use for g-(k) that interaction in S = 1, T = 1 pamcle-hole states which comes from the effective nucleon-nucleon interaction. A very simple treatment of the effective tensor force, together with its Fourier transform, has been gwent 3 by Brown et al [10], who show that the contributlon to g-(k) coming from the tensor part of the HamadaJohnston potential as a factor of 2 less than that from the tensor part of OPEP, mainly due to the shortrange correlation. Furthermore, the former drops off with large k, whereas the OPEP goes to a constant. We shall assume the same behavior for the contribution from the spin-spin term; this would follow from the general arguments m ref. [9]. As a reahstlc g-(k) we then take g ~ j = _½f2 {k 2/( 1+k 2) - '~}

(10)

where the lower suffix HJ lndacates that this is the g - ( k ) which would follow from the Hamada-Johnston potentml. This g ~ j ( k ) gwes certainly an overestimate of the amount o f attractmn for k > I. For k x/~. there is then a reductmn by a factor o f 6 from the gOPE' and R is clear that this agency is well-nigh negligible m its effect on plon condensation. It does not seem surprising to us that there is no tendency towards pmn condensation from the nucleon pole m the pion-nucleon interaction, since it is welt known that the attractwe p-wave on-shell n-nucleon interaction is embodied in the 1236 MeV Aqsobar. Mlgdal [1 ] included the isobar m his description; we found [11 ] a somewhat smaller numerical constant prefixing the contribution to the plon self-energy, namely ]'31t should be noted that these authors undertook precisely that task of calculating the contnbutmn of the nucleon-nucleon tensor force to the effective mteractmn g-(k).

Ha(k, 0) ~ - 0 . 4

29 October 1973

k2O/Oo

(11)

for neutral condensates where Po is normal nuclearmatter density. One can easily estimate that #/Oo would have to be ~ 4 for condensation to set in at k ~ 1.4. The difficulty here is that the isobar is really made up out of interacting plon plus nucleon, and once one goes to such high densities, one would expect the Pauh principle operating on the nucleon constltuent to modify the properties of the Isobar considerably. We are presently studying this problem. We still beheve, as we stated m ref [ 11 ], that the crossed plon Born exchange is included in the isobar, so that it is double counting to introduce it and the isobar separately. However, this argument gets us into dehcate questmns of off-shell extrapolatmn (k finite, w -+ 0) of the scattering via the isobar, which we intend to treat m a separate pubhcatlon. In the foregoing, we have assumed, as Mlgdal, that one has the nucleon pole term in addition to the isobar. We would hke to thank Gordon Baym, Hans Bethe, Jan Blomqvlst and Phil Siemens for many interesting discussions of this fruitless problem.

References [1] A.B Mlgdal, Nucl. Phys. A210 (1973) 421. [2] R.F Sawyer and D.J Scalapino, Phys. Rev D7 (1973) 953 [3] A.B Mlgdal, Phys Rev. Lett. 31 (1973) 257. [4] G E Brown, Unified theory of nuclear models (NorthHolland, Amsterdam 1964) [5] Fagg, Bendel, Jones, Ensshn and Cecil, Proc. Intern Conf on Nuclear physics, Mumch, 1973 [6] A. Anma and H Hone, Progr. Theor. Phys. 11 (1954) 509, R J. Bhn-Stoyle, Proc. Phys Soc. A66 (1953) 1158 [7] G Baym, Phys Rev Lett. 30 (1973) 1340 [81 A B. Mlgdal, Theory of finite Fermi systems and apphcations to atomic nuclm (Intersclence Publ, John Wiley and Sons, New York - London - Sydney 1967). [9] This value has been obtained by D.O. Rlska and B. Verwest (to be pubhshed) using forward-scattering dispersion relation. The short-range repulsion m empirical potentials such as the Reid soft-core potential is about 10 ttmes greater than thts. [10] Brown, Schappert and Wong, Nucl. Phys. 56 (1964) 191. [ 11 ] Barshay, Vagradov and Brown, Phys. Lett. 43B (1973) 359. [12] Elsenberg, Hufner and Monlz, Phys Lett, to be pubhshed 109