Model for the quasifree polarization-transfer measurements in the (p , n ) reaction at 495 MeV

NUCLEAR PHYSICSA

Nuclear Physics A568 (1994) 895-905 North-Holfand

Model for the quasifree polarization-transfer measurements in the (jj, ii) reaction at 495 MeV G.E. Brown Physics Department, State University of New York at Stony Brook, Stony Brook, NY 11794-3800, USA

J. Wambach ’ Institut fiir Kemphysik, Forschungszentrum

Jiilich, D-52425 Jidich, Germany

Received 2.5 February 1993 (Revised 22 July 1993)

Abstract

The recent (6, ii) polarization transfer experiments show the ratio of spin-longitud~nai to spin-transverse response functions R&R,, measured at a three-momentum transfer of 1.72 fm-’ in t2C and 40Ca, to be essentially unity for small projectile-energy loss. This can be expfained theoretically if the nucleon-nucl~n tensor interaction is zero at this momentum transfer. We can partially achieve this by introducing a density-dependent p-meson mass rnz.

1. Introduction Recent experiments on the (ji;, ii) reaction by McClelland et al. [ll have strikingly confirmed earlier indications [2,3], that the ratio of spin-longitudinal to spin-transverse response functions, R,/Z?,, measured at a three-momentum transfer of 1.72 fm-’ in r2C and 40Ca is essentially unity for small projectile-energy losses, decreasing somewhat below unity for energy losses > 70 MeV. These results put in doubt many theoretical predictions that there should be a “softness” in the longitudinal channel due to the pion-exchange tail of the NN interaction. At momentum transfers such as those selected in the experiment this would substantially increase the longitudinal response, lil,, above the transverse one. The experimental findings are consistent, however, with the high-energy Drell-Yan experiments 143 which show no enhancement of the pion cloud in nuclei as compared to that of the free nucleon. ’ Also at: Department of Physics, University of Illinois at Urbana-Champaign, 03759474/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDI 0375-9474(93)E0458-K

Urbana, IL 61801, USA.

G.E. Brown, J. Wambach

896

/ fp7 ii) reaction

We propose a model, and carry out a schematic calculation, which connects the analysis of the quasifree polarization-transfer measurements with the Drell-Yan experiments [4], and, also, with the measurements of the charge and current response in quasielastic electron scattering [Sl. By pushing our assumptions somewhat, possibly to an extreme, we are able to explain the quasifree polarization transfer measurements. After outlining the underlying ideas we give results of a semi-realistic model calculation which bears out most of the features of the schematic treatment.

2. Development The quasielastic (5, ii) polarization transfer experiments are interpreted theoretically in terms of a spin-isospin interaction, usually taken to be (Eq. (1) of ref.

m

+

g’+Cp i

u2-

4* rp2(q, 0) (s2+m;>

I

a,Xq^a,Xq”

71.72.

1

It includes pion- and rho-meson exchange and accounts for short-range correlations via the parameters g’ and C,. The notation suggests that the longitudinal response is determined by pion exchange, except for g’, and that the transverse response is governed by p-meson exchange. As Baym and Brown 161have shown, because of short-range correlations, the p-meson contributes importantly to g’ and the longitudinal response is strongly affected. This argument will be essential for our discussion. When looking at differences in the interaction which could cause a deviation of RJR, from unity, it is more transparent to go back to the decomposition in terms of central and tensor invariants. Ignoring form factors for the moment, one has

G.E. Brown, J. Wumbach / (5 n’)reaction

897

Only this part can cause a difference, because the a, * upI - TV terms contribute equally to the longitudinal and transverse response. For our first schematic calculation we use the ratio of coupling constants from ref. [7], f,2/mi

= 2 f ‘/m:.

(3)

This simple assumption will not greatly change the results of a more realistic treatment (to be presented below) and keeps the discussion transparent. The rough equality for the responses,

(4) for the range q - 1.72-1.75 fm-i and small o, with R, decreasing somewhat below R, for these q and w > 70 MeV, seems to be required consistently by several experiments [l-31. This can be accomplished by making I&&, o = 0) N 0 which implies that

f2/mS (4’+mt) f,"/m;

(q2 +m2,)

e”

(5)

or, since m2, -=zq2 -ez rni,

f,2/mz f ‘/(m$4’) =

l

(6)

for q - 1.72 fm-’ = 2.4m,. Unfortunately for this simple explanation, when using m,,, and mp from the Particle Data book, this ratio is 0.4 instead of unity. One might argue that a number of effects have been left out, but we shall show by more detailed calculations later that our simple estimate is quite realistic. In particular, effects of two-body correlations are not large since the tensor interaction chiefly connects two-nucleon states of relative zero angular momentum to L = 2 states and the centrifugal barrier in the latter suppresses correlation effects. Note that because of this barrier, effects come predominantly from distances larger than r _ h/q, and the meson-exchange model should be adequate. Brown and Rho [8] have suggested that meson masses as well as the nucleon effective mass should decrease with increasing density. The condition (6) can be satisfied if

= 2.5,

G.E. Brown, .I. Wumbach / (5 n’) reaction

898

giving m” -L = 0.8. mP QCD sum rule calculations [12,13] predict a density-dependent

effective mass as

mtb> ___ = 1 - CP/P,, mP where p,, denotes the saturation density of nuclear matter and c is a density-independent constant. Hatsuda and Lee [13] give c = 0.18 as their central value. Hence

m,*(po)=

o

B2 .

7

m&J

(10)

which is close to our estimate 1141l, It should be noted that the (p, n> reaction is surface-dominated due to absorption of the projectile. Therefore the effective density is lower than p. and, as is readily seen from Eq. (9), mF/mp will not be as low as 0.8 in 40Ca. Unless the p-mass drops much faster than the QCD sum rule estimates, this effect can only partially explain the experiment. It is conceivable, however, that additional corrections such as a softening of the rNN vertex function give additional improvements. We shall comment on these later. The pion mass mT, which is generated by explicit chiral symmetry breaking, does not scale as the other masses [8,9]. From the analysis of pionic atoms it is known that the optical potential is slightly repulsive so that the in-medium pion mass actually increases slightly with density [9,10]. It has been argued that the pion-nucleon coupling f decreases slightly in medium [ll]. Both effects would tend to decrease f 2/m i, but they are small, and we ignore them.

3. Choice of interaction Our results will be somewhat sensitive to the choice of the spin-isospin interaction, although, qualitatively, the model calculation performed in the last section should hold for any boson-exchange interaction with reasonable coupling constants to fit the two-body scattering data.

l

Chanfray and Ericson have shown that exchange current field decrease the quark condensate further by (lo-201% this correction will bring m,*(p,)/m, down to - 0.8.

type processes involving the virtual pion over that of Hatsuda and Lee. Including

G.E. Brown, J. Wambach / @, ii) reaction

899

In various versions of the Bonn potential, the cut-off A, in a monopole form factor of the rNN vertex has been fiied at 1.3 GeV or higher. Such values for the cut-off are required to obtain a realistic strength of the tensor force to fit both the scattering data and the deuteron properties. On the other hand, a body of evidence has accumulated suggesting that the pionic form factor is quite soft. Some time ago Thomas [El showed that the data on the sea quark content of the proton can be used to obtain restrictions on the t-dependence of the pion vertex function r,. Frankfurt, Mankiewicz and Strikman [16] extended this analysis, finding A, < 0.5 GeV. Inclusion of more mesons in the nucleon cloud Hwang, Speth and Brown [17] to raise this limit to N 950 MeV. Clearly such values are too low to correctly describe the deuteron. In an effort the reconcile the deep inelastic data with the two-nucleon properties Holinde and Thomas [18] chose A, = 800 MeV but then had to introduce an additional pseudoscalar meson (which they call rr’> of mass 1200 MeV. Assuming a hard form factor of A,, = 2.0 GeV, the r’NN coupling constant was adjusted to fit the NN data. More recently it has been recognized 1191 that there are, indeed, at least two objects with pionic quantum numbers, one an elementary meson and the other the correlated pr system coupled to the quantum number of the pion. While the former is unlikely to yield strong enough coupling to the nucleon as required by the fits to the two-body data, the latter may explain the properties of the Holinde-Thomas r’. A discussion of the situation can be found in ref. [20]. Since we find this scenario quite convincing we will employ the Holinde-Thomas interaction. A more detailed description, in terms of the pr system, can be fed in later as the results [19] become available.

4. Numerical results As outlined above we use the Holinde-Thomas interaction (I-IT) [18] for our quantitative calculation. In the spin-isospin channel this interaction is given by

f2

I/,,(q, @>=

m2,

r,2(q, &g_

y2

w)q2

(q2+mz!)

+

gf

r$(q, +I2 @2_

n

(q2+m3

u1*q^a2-q

i

I

f2 C(q, a2 m;a2- (q2+r$)

t-P

u

x4^a2Xq^

l

I

71.7 27

(11)

with /(f2/m~) = 6.85, = 3.64, A, = 0.8 GeV, A,, = 2 GeV and AP = 1.3 GeV. We include effects of short-range correlations by multiplying the r-space central part of V,, with a correlation function f(r) = 1 j,,(r/r,). A value rC = 0.25 is chosen which corresponds to the inverse Compton

900

G.E. Brown, J. Wumbach / (pf Fi)reaction

--...__ . .

x_

. .._ --.__._

-1oo.c-

------- . . . . ..__._.__

1.2

10

0.8

0.6 ... .. ..___.___._____............. ___...........--...--.--04

:

Fig. 1. Upper part: the spin-longi~dinal and spin-transverse interactions resufting from the HT potential after the inclusion of short-range correlations (dashed lines) and with a medium-modified p-meson mass for mz/mp = 0.8 (fun lines). Lower part: the Fermi liquid parameter g’(q) with the free p-mass (dashed line) and with mp*/mp = 0.8 (full line).

wavelength of the W-meson [211 (in principle, tensor correlations are also present, represented by a Fermi liquid parameter h’. For realistic interactions, however, this parameter is small 17,221 and can be safely ignored here.) The resulting static spin-longitudinal and spin-transverse interactions are displayed as the dashed Iines in Fig. 1. They show the familiar q-dependence. It is instructive the extract the Fermi liquid parameter g’ from this interaction, which is used in phenomenological parameterizations such as as Eq. (1). The result is given by the dashed line in the lower part of Fig. 1. At small q the value is somewhat lower than those extracted from Gamow-Teller systematics (g ‘ - 0.7-0.8). On the other hand, it agrees quite we11with G-matrix calculations [7]. A medium dependence of the p-meson mass is easily incorporated in the HT potential. One can treat mp*/mp as a parameter so as to force the tensor interaction V,,, to zero at the relevant momentum transfer of the (p, n> experiment, as discussed above. It turns out that the resulting value is close to m,*(p,)/m, = 0.8, consistent with our schematic calculation. The resulting interactions are shown as the full lines in Fig. 2. In both, Vr_ and I’,, we find a strong

GE. Brown, J. Wambach / @, n’)reaction

901

2.5

2.0

-

1.5

-

!~ ‘:. i, *...

.z E

1.0

-*

0.5 -

O.O 0.0

M.0

I 1h--i_[.

_

150.0

200.0

100.0

o (MeV)

Fig. 2. Upper part: the nuclear-matter spin-longitudinal and spin-transverse response functions as a function of energy o and at fixed momentum transfer q,,= 1.72 fm-‘. The dashed lines use the free HT interaction while the full lines result from dropping the p-meson mass. The dash-dotted line denotes the free Fermi gas response. Lower part: the ratio R, /RT from the HT interaction (dashed) and with dropping mass Gull). The data were taken from ref. 111.

increase in the repulsion with g’ going from _ 0.56 to a value slightly above unity at q N 2.5 fm-’ (lower part of Fig. 1). At the same time both interactions become weakly repulsive near q. = 1.72 fm-‘, the moments transfer of the e~eriment. More quantitativeiy this is illustrated by the nuclear-matter response functions and the ratio RJR,, both at pa, which are displayed in Fig. 2. As is well known, without medium modifications, R, and R, are quite different (dashed lines in the upper part of Fig. 2) and the ratio is strongly energy-dependent (dashed Iine in lower part). A dropping p-meson mass wipes out this effect (full line) as experiment would suggest. At the same time, both R, and R, are hardened. While a nuclear-matter calculation with dropping p-mass can explain the LAMPF measurement one has to bear in mind the surface nature of the (p, n) reaction. Without performing a sophisticated finite-nucleus DWBA calculation [231 we can still get some realistic estimate for the finite-nucleus effects by using a semi-classical description of the response functions and treating the spin-isospin interaction in a local density approximation (LDA), i.e. including a density depen-

902

G.E. Brown, J. Wambach / (6 id reaction

w (MeV)

Fig. 3. Same as Fig. 2 but for 40Ca. The dotted line gives the result from decreasing the r’ mass in the HT potential as well as softening the rrNN form factor by the same density-dependent factor as for the p-meson. The latter procedure is suggested by a recent model in which the V’ and the vNN form factor originate largely from correlated pp effects.

dence for m;/m,. Following the work of ref. [24] we obtain the finite-nucleus response functions as

where RF,y(q, o, p) are the local nuclear-matter response functions, including longitudinal-transverse mixing through the surface (see ref. [24] for details), and F(r) is a weight factor derived from the projectile distortion function. It can be evaluated reliably in the eikonal approximation. Assuming a linear density dependence for the p-meson mass as m%/m, = 1 - cp/p,, as suggested by available QCD sum rule calculations, the results for 40Ca are shown in Fig. 3. The upper part displays R, and R, with (full lines) and without (dashed line) dropping mass (the free response is given by the dash-dotted line). As for nuclear matter, their difference is drastically reduced and both response functions are hardened somewhat. In the ratio the effect is not as dramatic, however. Nonetheless, there is a significant improvement for lower energies (dashed line: without dropping mass,

G.E. Brown, J. Wambach / @, ii) reaction

903

full line: with dropping mass>. On the other hand, above 100 MeV excitation energy, the description of the data is worse. Here one should bear in mind that, above the quasielastic peak, the contribution from two-step processes becomes quite significant 125-271. Double-scattering contributions to the spin observables, which are not well understood at present, may thus be important. Secondly, one has to be careful with the proper choice of frame when going to higher energies [231. We have also included effects from coupling to isobar-hole excitations which are known to increase the tensor interaction in the nuclear medium. When dropping the p-meson mass these effects are negligible, however.

5. Discussion We have investigated a model with dropping hadron masses to explain the equality of the spin-longitudinal and spin-transverse responses, as inferred from recent quasifree polarization-transfer (5, ii) measurements at LAMPF. A decrease of the p-meson mass with density leads to an increase in its tensor interaction which alters the momentum dependence of the longitudinal and transverse interactions. In nuclear matter and semi-classical finite-nucleus calculations, a value of mz/mp = 0.8 explains the data. Such a value seems low, given the surface nature of the (p, n) reaction. Using a Glauber estimate, we find that the nucleus is probed at u 0.6~~ which yields ( mF/mp> = 0.88, when using a linear density dependence. With such a value RJR, is still reduced but not enough to explain the data. A more detailed LDA calculation confirms this accurately. On the other hand, an effective p-meson mass of 0.88 fits well with values deduced from other experiments. For instance, in an analysis of (e, e’) and (p, p’) scattering data at 318 MeV on the stretched 12~~ and 14- states in *‘*Pb [291 agreement between the electron and proton quenching factors can be found for effective masses

= 0.79-0.86.

(13)

Similarly, from studies of the (5, ii’) polarization transfer at E, = 200 MeV in 160, Stephenson and Tostvin [301 find that lowering rng to = 0.94 * 0.02

(14)

changes the effective isovector spin interaction so as to significantly improve the fit to the data. In i”B@ p”>, at the same energy, Baghaei et al. 1311 find that decreasing tb 0.9 substantially improves their fit to the data.

904

G.E. Brown, J. Wambach

/ @, nf) reaction

There are other mechanisms which should be considered. [SJ implies that A, should scale as mp, i.e.

Brown-Rho

scaling

In calculations of the rNN vertex function by Jansen et al. 1191 this can be understood in terms of intermediate prr states having the quantum numbers of the pion. In the same spirit the r’ meson in the HT potential should represent a pr composite and hence rnk should also scale roughly as mt. To estimate the influence of these two mechanisms on the ratio RJR, we have therefore decreased the mass of the 7~’ meson as well as softened AT in the rrNN form factor by the same density-dependent factor as for the p-meson. The result is indicated by the dotted line in Fig. 3. At excitation energies below 100 MeV there is some improvement but still we fall short of explaining the data. A more thorough investigation based on a full model for the prr effects in the NN potential [19] is in progress. It should be recognized that Brown-Rho scaling [B] is only approximate and incomplete. For example, mP enters into the rNN vertex function nonlinearly, once rrp rescattering effects are taken into account. Thus A, is expected to decrease even more rapidly than mP. We conclude that the inclusion of the medium dependence of the p-meson mass significantly reduces the isovector part of the tensor interaction and removes much of the discrepancy between the ratio of spin-longitudinal and spin-transverse response functions. The medium dependence of the p-mass leads to a very different momentum dependence of the spin-isospin interaction from that with the free mass (see Fig. 1). This implies a different q-dependence of the ratio RJR,. Our predictions are given in Fig. 4. At 4 = 1.1 fm-‘, currently measured at LAMPF [32], one should observe a significant enhancement at low w (dashed

I 3c I

Fig. 4. The ratio R, /RT

50 0

loo.0

w (MeV) for several momentum transfers as predicted by our model.

G.E. Brown, J. Wambach / (6 n’) reaction

905

line) while at q-transfers > 2 fm-’ the ratio should be strongly suppressed since Vr becomes attractive (dash-dotted line). We therefore consider measurements at different q-transfers crucial to our model.

We thank J. McClelland for information on the data and J. Speth for useful discussions. This work was supported in part by DOE grant DE-FG 0188ER40388 and NSF grant PHY-89-21025.

References [l] J.B. McClelland et al., Phys. Rev. Lett. 69 (1992) 582 [2] T.A. Carey et al., Phys. Rev. Lett. 53 (1984) 144 [3] L.B. Rees et al., Phys. Rev. C34 (1986) 627 [4] D.M. Aide et al., Phys. Rev. Lett. 64 (1990) 2479 [5] G.E. Brown, M. Rho and M. Soyeur, Nucl. Phys. A553 (1993) 705~; Acta Phys. Pol. 24 (1993) 491 [6] G. Baym and G.E. Brown, Nucl. Phys. A247 (1975) 395 [7] S.-O. B&man, G.E. Brown and J.A. Niskanen, Phys. Reports 124 (1985) 1 [S] G.E. Brown and M. Rho, Phys. Rev. Lett. 66 (1992) 2720 [9] J. Delorme, M. Ericson and T.E.O. Ericson, Phys. Lett. B291 (1992) 379 [lo] C. Schiitz, K. Holinde, J. Speth and M.B. Johnson, KFA preprint [ll] G.E. Brown and M. Rho, Phys. I.&t. B237 (1990) 3 [12] G.E. Brown, Nucl. Phys. A231 (1991) 397c [13] T. Hatsuda and S.H. Lee, Phys. Rev. C46 (1992) R34 [14] G. Chanfray and M. Ericson, Nucl. Phys. A556 (1993) 427 [15] A.W. Thomas, Phys. Lett. B126 (1989) 97; Prog. Theor. Phys. Suppl. 91 (1987) 204; Prog. Part. Nucl. Phys. 20 (1987) 21 [16] L.L. Frankfurt, L. Mankiewicz and MI. Strikman, Z. Phys. A334 (1989) 343 (171 W.-Y. Hwang, J. Speth and G.E. Brown, Z. Phys. A339 (1991) 383 [18] K. Holinde and A.W. Thomas, Phys. Rev. C42 (1990) R1195 [19] G. Janssen, K. Holinde and J. Speth, in preparation [20] K. Holinde, Nucl. Phys. A543 (1992) 143~ [21] M.R. Anastasio and G.E. Brown, Nucl. Phys. A285 (1977) 516 [22] J. Wambach, A.D. Jackson and J. Speth, Nucl. Phys. A348 (1980) 221 [23] M. Ichimura, K. Kawahigashi, T.S. Jorgensen and C. Gaarde, Phys. Rev. C39 (1989) 1446 [24] G. Chanfray, Nucl. Phys. A474 (1987) 114 [25] G.F. Bertsch and 0. Scholten, Phys. Rev. C25 (1982) 804 [26] D.L. Prout, Ph.D. thesis, University of Colorado, 1992 [27] Th. Sams, Niels Bohr Institute preprint (1993) [28] G.E. Brown, Z.B. Li and K.F. Liu, Nucl. Phys. A555 (1993) 225 [29] N.M. Hintz, A.M. Lallena and A. Sethi, Phys. Rev. C45 (1992) 1098 [30] E.J. Stephenson and J.A. Tostevin, in Spin and isospin in nuclear interactions, eds. SW. Wissink et al. (Plenum, New York, 1991) p. 281 [31] H. Baghaie et al., Phys. Rev. Lett. 69 (1992) 2054 [32] J.B. McClelland, private communication