Convergence of the hole-line expansion in nuclear matter

Volume 71 B, number 1

CONVERGENCE

PHYSICS LETTERS

7 November 1977

O F T H E H O L E - L I N E E X P A N S I O N IN N U C L E A R M A T T E R e B.L. FRIMAN • and E.M. NYMAN #

Department of Physics, State University of New York, Stony Brook, New York 11 794, USA Received 7 July 1977 Revised manuscript received 24 August 1977 We discuss the convergence of the hole-line expansion in nuclear matter. Explicit results for ring diagrams, summed to arbitrary orders, indicate that this series is, in fact, semiconvergent at nuclear densities. In Laudau's Fermi-liquid theory, the Fermi-liquid parameters must satisfy conditions of the form (see Pomeranchuk [1 ] ):

Fl/(2l + 1) i> - 1,

(1)

in nuclear matter, similar conditions holding for F i,

G t and G~. If the condition (1) is violated, the system is unstable against deformations of the Fermi surface. Such an instability would cause perturbative expansions for the ground-state energy to diverge. In the diagrammatic (Goldstone) perturbation expansion it exhibits itself already in the random-phase approximation. Thus, if ( 1) is violated, the ring-diagram summation diverges. The point of the argument is that the ring-diagram summation, as we shall see below, has the convergence properties of a power series. Thus a necessary condition for convergence of this power series is IFt/(2l + ])1 ~< 1,

(2)

similar conditions again holding for the other Landau parameters. (When (2) is an equality, there may or may not be convergence.) If (2) is violated, ring diagrams increase in high orders. In this case, one must apparently conclude that the entire hole-line expansion diverges. It appears that at least one of the Fermi-liquid parameters, G~ (the coefficient o f ~ . a~.~) exceeds unity already at nuclear density, both theoretically [2] and experimentally [3]. It therefore follows that the

hole-line expansion, which contains the ring-diagrams, should not be expected to converge. This observation is not new, nor were the methods of Brueckner and others intended to account for contributions to the energy from these low-lying virtual excitations. In fact, e.g., Brandow [4] states that the theory "almost certainly" diverges; his small parameter ~ (= 0.14) has nothing to do with this divergence. In this letter we give some new results which we hope will help clarify the situation. Our discussion is rigorous only when all exchange terms are neglected. While one does not expect these to be important here, we note that they would, if included, modify our results as well as the stability conditions. The tensor force requires, when included in exchange terms, that eq. (1) be replaced by a more complicated set of inequalities, as found in recent work (still in progress) by B~ckman, Jackson and Sj5berg. We consider first a simple and explicit example, the ring-diagram summation with the long-range part of the one-pion-exchange potential. We can use old electron-gas results of Gell-Mann and Brueckner [5] to obtain a closed-form expression for the sum of the ring diagrams, as well as the individual terms in the expansion. The contribution to the energy density E/~2 and its series expansion are E

3.

d3q dw

3

=-~i ~ (-1)n•n/n n=2 ¢' Work supported in part by USERDA under contract No. E(11-1)-300. * On leave of absence from Physics Department,/~bo Akademi, JLbo, Finland.

[Gm(q,w)]n d3q dw/(2rr)4 , m

"

(3) where m is a spin-projection quantum number. Here, we use the abbrevation 31

Volume 71B, number 1

PHYSICS LETTERS

(4)

am(q, co) = - Vm(q) U(q, co)

G(q,O)~

where Vm(q) is the Fourier transform of the potential and U the Lindhard function. We have [e.g. 6]

÷I

U(q, co) = 4 ,f)d{3~k3 O([q +kl-kf)O(kf- k)

1

1

co+ek-ek÷q+ig- cO+ek÷q-ek --i6 '

05

Vm(q) = (ml~q e-Ur° {[al "~2 + Sl2(q)]

qcosroq+lasinroq q2 + U2

^ l+Pro , ) 3S12(q)72-~-0 ll(roq) j

lm) (6)

where r 0 is the cut-off radius, inside which the potential is assumed to vanish. The isospin dependence was 32

I

15

20

,

q(fm")

kF:lO

___"_~~__

(5)

where approximate the single-nucleon energies by kinetic energies, ek = k2/2M. In nuclear-matter calculations, it has become customary to use a different approximation for hole lines, using a spectrum with a gap at the Fermi surface. Such an approximation would be inadmissible in the present application [7]. We have multiplied the interaction by a parameter ?, in order to facilitate the discussion of the convergence properties, which are those of a power series in X. Should the series diverge, say, at the physical value X = 1, eq. (3) nevertheless provides the correct answer as an analytic continuation outside some smaller circle of convergence. In order to study the analytic properties of the sum, we perform a Wick rotation and observe that there is a singularity if the argument of the logarithm becomes negative within the intergration volume. This determines the radius of convergence in the complex k plane. In other words, if the sum of integrals converges, it must converge at each q and co separately. As long as we only study the convergence we need therefore not perform the integrations implied by eq. (3). It follows from general principles that the Lindhard function for symmetric (N = Z) nuclear matter does not depend on the sign of co, and further that it decreases when co2 increases. We therefore study only the case co = 0. The Fourier transform of the cut-off one-pionexchange potential is (see Dahlblom et al. [8] ).

X

7 November 1977

G(q,

Fig. 1. Shown is the function (see text) 0) plotted against the momentum transfer q, for the long-range (r > 0.8 fm) part of the one-pion exchange potential at three values of the Fermi momentum given in fm -1 in the graph. The ring-diagram summation diverges if I G(q, 0)l > 1 at some value ofq.

already included above, in eq. (3). We use r 0 = 0.8 fin. Using eqs. (5) and (6), we obtain the functions Gin(q, 0), which we have, in fig. 1, plotted for m = 0 (zero spin projection along q). We observe a maximum at q = 0 and a minimum at some finite q. Whenever either one of these is outside the strip [G(q, 0)[ ~< 1 we know that the ring-diagram series diverges. The case when G(q, 0) reaches - 1 is, in fact, the result of a well-known physical singularity: it signals the onset of pion condensation. On the other hand, the maximum at q = 0 is the Fermi-liquid parameter G~), given in the present model by the equation r

G 0 = G i n ( 0 , 0 ) = ~0r2//2 2) (1 +/Jr0) e - ' t o

2Mkflzr 2,

(7)

which equals 0.8 for kf = 1.4 fro- 1 and r 0 = 0.8 fro. (All spin projections give the same value at q = 0.) In this simple example, the convergence properties are determined by the minimum at q ~ 0 and the ring summation converges up to the threshold density for pion condensation. However, the value given by eq. (7) for the Landau parameter is too small, and this model therefore should be improved. Especially, the exchange o f # mesons gives an important short-range repulsion. We shall remedy this by giving results from a more ambitious calculation, which will be described in detail in a longer paper. In our calculation we use, instead of a vertical cutoff at r = r0, a pair-distribution function calculated using Fermi-hypernetted-chain methods [9]. The dis-

Volume 71B, n u m b e r 1

PHYSICS LETTERS

7 November 1977

piing is proportional to g2 = gv2(1 + Kv)2,

(fro -I G(q,O)=-I

Nucl. mort.

G(O,O)" +I

%

i

;

i

i

I

Fig. 2. Shown are the m a x i m u m densities at which the ring-

diagram summation converges, as functions of the ratio g~/g~ where go is the coupling constant of the p meson. The two curves correspond to G(q, 0) = -1 and G(0, 0) = l, as indicated on the graphs. The crosses correspond to the parameters used in fig. 3.

tribution function includes short-range correlations due to the central part o f the Reid 3S 1 potential, as well as many-body effects t . Although a comparison is not entirely meaningful, we note that the correlation function used in [2] is much larger in the important region around r = 0.8 fin. Finite-size effects in the m e s o n - n u c l e o n vertices are represented by form factors F of Yukawa type, Fi(q ) = [(A 2 _ m2)/(A 2 + q2)] 1/2,

(8)

with A = 1 GeV for rr mesons and 1.5 GeV for p mesons. These cut-off masses are consistent with recent calculations of Durso et al. [10] as well as a calculation of 7r+ induced break-up o f deuterium by Brack etal.[ll]. We introduce the virtual A isobar in the intermediate states by including an isobar-hole Lindhard function, calculated using methods developed by Brown and Weise [12], using the coupling fN,x" = 2fNN,. The p-nucleon coupling parameters are not known with certainty. In the non-relativistic case, the cou¢ Strictly speaking, the distribution function should, in our application, not be calculated with the full nuclear force; the ~r and p contributions should be removed to avoid double counting. However, the spin and isospin dependence o f these channels was ignored in the FtlNC calculations. Using more appropriate distribution functions would change our results insignificantly.

(9)

where g2/47r = 0.55 and quoted values for Kv vary from 3.6 (vector-dominance model) to 6.6 (HShler and Pietarinen [13] ). This putsg2/g 2 in the range 0.8 p rr to 2.2. We prefer the large value; it gives a reasonable Landau parameter, G~ = 1.7 at kf = 1.4 f m - 1, when used in our calculation. The convergence criteria can be formulated as conditions on the density. In fig. 2 we show the maximum density which allows convergence, as a function ofgp. The value g2/g2 = 1.44 gives the phenomenological p 7r t Landau parameter G O = 1.45 (ref. [3] ) and causes divergence when p > 0.085 fm - 3 , which is well below the central density of heavy nuclei where one would like to use a m a n y b o d y formalism. We also see from fig. 2 that the q = 0 divergence (G = +1) is the one which in practice limits the convergence, rather than the pion-condensation threshold (G = 1). Having established the divergence o f the hole-line expansion, we must now comment on results obtained by including a limited number of terms from the beginning of the series. (Of special interest, of course, is the case where the series is included only to second order.) To illustrate this, we consider the contribution to the energy per particle in nuclear matter up to nth order of the expansion (3) as well as the conE/A

E/A ( MeV )

(MeV)'

.~t

Exoct -60

-40

e=,

A[

A

^^An/-,¢

/, v

-

-20

-2C

kF= 1.4 fm-I I 5

I I0

=

I 15

i 20

O

,

I .5

-

/ I0

I I~

ORDER OF CALCULATION

r.ig. 3. Contribution to the energy per particle o f nuclear matter from ring diagrams up to n t h order at two densities, indicated on the graphs. Shown is also the exact result of.the formal sum of the ring diagrams.

33

Volume 71B, number 1

PHYSICS LETTERS

tribution from the formal sum of the series (continuing analytically to ~, = 1). In fig. 3, we give the partial sums of the series at two Fermi m o m e n t a . One sees clearly that the s u m m a t i o n is, in fact, semiconvergent at nuclear density; taking a few terms in the beginning of it does improve the accuracy of the answer. Thus, there remains no objection towards the B r u e c k n e r - B e t h e - G o l d s t o n e theory of nuclear matter as far as current calculations of the b i n d i n g energy at normal density (kf ~ 1.4 f m - 1 ) goes. However, the situation quickly deteriorates as the density increases, and the semiconvergent aspect soon disappears. Thus, already at kf = 1.6 fm -1 , there is little point in going b e y o n d lowest (second) order in the expansion. With the parameters used here pion condensation sets in at kf = 1.68 fm -1 , and b e y o n d this point the whole calculation is inapplicable. We are grateful to Professor A.D. Jackson for an illuminating discussion.

34

7 November 1977

References [l ] i.la. Pomeranchuk, J. Expt. Theoret. Phys. 35 (1958) 524, (Sovj. Phys.) JETP 8 (1959) 361. [2] G.E. Brown, S.O. B~'ckman, E. Oset and W. Weise, Nucl. Phys., to be published. [31 J. Speth, L. Zamick and P. Ring, Nucl. Phys. A232 (1974) 1; P. Ring and J. Speth, Nucl. Phys. A235 (1974) 315. [4] B.tl. Brandow, Phys. Rev. 152 (1966) 863. [51 M. Gell-Mann and K.A. Brueckner, Phys. Rev. 106 (1957); D.M. Clement, Nucl. Phys. A205 (1973) 398. 161 A. Fetter and J.D. Walecka, Quantum theory of many particle systems (McGraw-Hill, New York, 1971) p. 158. [7] S.-O. B~ckman, Nucl. Phys. A130 (1969) 427. [8] T. Dahlblom, K.G. Fogel, B. Qvist and A. TSrn, Nucl. Phys. 56 (1964) 177. [9] R.A. Smith, private communication. [10] J.W. Durso, A.D. Jackson and B.J. VerWest, Nucl. Phys. A282 (1977) 404. [ 11 ] M. Brack, D.O. Riska and W. Weise, preprint. [121 G.E. Brown and W. Weise, Phys. Reports 22C (1975) 279. [131 G. 11Shier and E. Pietarinen, Nucl. Phys. B95 (1975) 210.