I Nuclear Physics A141 (1970) 369--374; (~) North-HollandPublishing Co., Amsterdam 2.L I

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C O U L O M B E F F E C T O N T H E P R O T O N - P R O T O N

L O W - E N E R G Y S C A T T E R I N G P A R A M E T E R S

THIEN VO-DAI and Y. NOGAMI Physics Department, McMaster University, Hamilton, Ontario, Canada t

Received 13 October 1969

Abstract: The effect of the Coulomb force on the protort-proton low-energy scattering parameters is examined when the nuclear i~teraction is represented by a separable potential. Naqvi's two- term potential as well as Yamaguohi's oae-term potential are considered, following Harring- ton's method. After removing the Coulomb effect, the "nuclear" scattering length and effective range are both found to be somewhat larger (in absolute values) than those previously obtained by using hard-core local nuclear potentials.

1. Introduction

A sensitive test o f charge symmetry as well as charge independence o f the nuclear interaction is provided by measurements o f the low-energy nucleon-nucleon scattering

parameters in the 1S state, namely, the scattering length and effective range. In order to compare the scattering parameters for pp, pn and nn, however, one has to remove

the effect o f the Cou lomb interaction for pp. This is usually done by first assuming a pp nuclear potential Vs which together with the Cou lomb potential Vc reproduces the observed pp scattering data, and then, solving the Schrrdinger equat ion for Vs without including the Cou lomb potential to obtain the scattering parameters t*

Thus, for the pp system there are two sets o f scattering parameters, say, asc and

rsc, and as and r s. The parameters asc and r~c are directly determined f rom the ob- served phase shift and are supposed to be reproduced by Vs + Vc while a S and r s are the "nuclear" scattering parameters obtained f rom V~ alone. The suffix sC implies tha t the relevant quanti ty comes f rom both strong and Coulomb interactions, whereas the quanti ty with the suffix s is due only to the strong nuclear interaction.

As is obvious f rom the definition o f as and r s for pp, they depend on the nuclear potential Vs. Al though this shape dependence is very small, we now require a very precise knowledge o f a s and r~ because all that concerns us about change symmetry or change independence is very small differences between quantities for pp, pn and nn. Also the experimental data now available are so accurate that even a very slight dependence o f a s and r s on V S can be meaningfully detected.

t Work supported bY the National Research Council of Canada. *t To be more precise, one has to consider the effect of vacuum polarization. But this is so small an

effect that it is unimportant in our analysis.

369

3 7 0 Trl I~N VO-DAI AND Y. NOGAMI

The shape dependence of a s a n d r s has been examined for various local potentials that fit the pp scattering data by Heller, Signel and Yoder 1). They found, by using hard-core potentials,

a s = - (16.6 to 16.9) fm (1.1)

to be the spread of the probable value of as. Incidentally our as corresponds to Heller et aL's a. , , which they call the nn scattering length. For clarity, we distinguish as and a . . . In addition to the effect of the Coulomb interaction there are many dif- ferences between pp and nn interactions 2, 3). Hence, removing the Coulomb effect in the pp scattering does not give the nn scattering. It only gives quantities which can be compared with corresponding quantities for nn.

The purpose of this note is to estimate a S and r s when the nuclear interaction is represented by a nonlocal, separable potential. This problem was investigated by Harrington 4) who showed that the solution to the Lippmann-Schwinger equation can be given in a closed form even if the Coulomb potential is added to a separable potential. As an illustration, Harrington considered a simple one-term potential of Yamaguchi's type 5), and obtained as = -18 .0 fm and r S = 2.93 fro. Note that this value o fa s is well outside the range given by (1.1).

The Yamaguchi-type potential, however, does not reproduce the experimental pp phase shift at higher energies because it has no short-range repulsion. Hence, it will be of interest to reexamine this with a more realistic separable potential which fits the pp phase-shift up to about 300 MeV. As an example of such a potential we have chosen Naqvi's one 6) which consists of two terms representing a short-range repulsion and a long-range attraction, respectively. We have found that the results with Naqvi's potential are only slightly different from Harrington's ones.

For the experimental values for asc and rsc, we take 7)

asc = -7.786 fro, rse = 2.840 fro, (1.2)

which are very sfightly different from those taken by Harrington ( -7 .81 fm and 2.80 fm).

2. Harrington's work

Let us summarize the relevant part of Harrington's work 4). The nuclear potential for pp in the 1S state in the momentum space is assumed to be

Vs(k', k) = Z ~,o,(k')o,(k). (2.1) i

Then the total phase shift is t h e sum of the Coulomb and nuclear phase shifts, 6(k) = 6c(k) + 6sc(k), and

cot rsc(k ) = - R e D / I m D with D = det [l +AI(Ek)]. (2.2)

Here the matrices A and I are respectively defined by

Aij = 6~j2~ (2.3)

and

COULOMB EFFECT 371

1 fo ° q2dqf,(q)f j(q) I,j(Ek) = ~ E-~q_Ek_i----~ ,

where E k = k2(2/~) - i. The functionfi(k) is defined by

k ( k ) = rdrF o kr 2dfg, q o qr ,

with

(2.4)

(2 .5)

(2.15)

The experimental values of asc and rsc (L2) are consistent with

a s = -17.597 fm, r s = 2.969 fm

Yo(kr ) = (2 0-1Co(t/)M~, ' t(2ikr), (2.6)

where M(z) is the Whittaker function, t/ = #e2/k, and Co(t/) is the conventional Coulomb barrier-penetration factor:

Co(t~) = {2rct/[exp (2~t / ) - 1]-1} ~. (2:7)

If one assumes a Yamaguchi-type single-term potential:

V ~ ( k ' , k ) = 2 g ( k ) g ( k ' ) with g ( k ) = (f12-l-k2)-1, (2.8)

one obtains

f ( k ) = g(k)Co(t/) exp [2,/arctg (k/fl)]. (2.9)

It is difficult to obtain the integral (2.4) exactly, but a good approximation is achieved by expanding it with respect to pe2/fl =- a << 1. Ignoring terms of the order of a2 In or higher, Harrington obtained

kC~(tl) cot 6sc(k ) + 2aflh(t/)

= k cot 5s(k)[1-4aflk -~ arctg (k/f l)]-2afl[In (4af12/(f12+k2))+~], (2.10)

where the function h(t/) is defined in terms of the ~h (digamma)function as h ( ~ ) = Re 0 ( - i t / ) - l n t/, and 7 = 0 .5772 . . . is Euler's constant. The "nuclear" phase-shift 6s(k) is given by

k cot 5s(k) = - 27:(;tp)-1 (f12 + k2)2 _ (2fl)- 1(fl2 _ k2). (2.11)

From eqs. (2.10) and (2.11) one can obtain the scattering length and effective range as follows:

1/a,c = ( l -4a) /as+2~f l ( ln 4~+y), . ~ (2.12)

rsc = (1 --4=)r, + 4eft-111-2(3flas)- 1], (2.13)

where as -- - 1 +½B, rs --- - 1 + B (2 .14)

372 THIEN VO-DAI AND Y. NOGAMI

and (4zr)-lA# -- -1 .1439 fm -3, fl = 1.0813 fm -1 (2.16)

Then 43 ~ 0.06; so that the expansion in a is justified.

3. Naqvi's potential

The simple Yamaguchi-type potential discussed in the preceding section does not fit the pp scattering phase-shift at higher energies. This can be remedied by introducing another separable term which represents a short-range repulsion. As an example let us consider the potential which was proposed by Naqvi 6):

V~(k', k) = ~. 2,g,(k')gi(k), (3.1) i = 1 , 2

where

gl(k) = (fl :+kZ) -1, g2(k) = k2(fl2+k:) -2. (3.2)

The function f l (k) is the same a s f ( k ) of (2.9), whilef2(k) is given by

f2(k) = f~ (k) + fl:f~(k), (3.3)

wheref~(k) ~- dfl(k)/dfl z. Some more details are given in the appendix. In evaluating Iij(k), we again expand it ' with respect to a and ignore terms of the order of ez In c~ or higher. After a straightforward but lengthy manipulation we obtain the following:

1/asc = (1 - 4~)/a s + 2~fi(In 43 + 7), (3.4)

r~c = (1 - 4e)r o + 4eft -~ [1 - 2(3flao)- t], (3.5)

where ao and ro are respectively given by t

I+ (i-u)

x i 1 + 3 2 2 + 2z ]-I 62---~ 3 - ~ ( 1 - 6 3 ) ' (3"6/

4fl2 [ 4 - 332' + 3 2 @ ( 1 + - 6

- 2-L [3+ 722 (1- la--~t) + 16~ (1+~)] } 4fla

x l + e 2 z + 2z ( 1 - 6 e ) (3.7) 62-7

If one put e = 0 in the above expressions, one obtains as = ao and r s = ro.

t The parameters 2i in eqs. (3,6) and (3.7) should be replaced by 2t/t[~.

COULOMB EFFECT 373

The parameters 2~, 2 2 and fl are determined so that asc and rsc in eq. (1.2) are reproduced and also the phase shift 6s changes its sign at the lab energy 240 MeV. Thus we arrive at the following results:

and a s --- -18.151 fm, r S = 2.930 fm (3.8)

2 t 1 4 : : /22 = I l i + f l -/ l l+~fl - / i i+ R,

(A.3)

(A.4)

(4rc)-121/z = -2.6949 fm -2, 2z/21 = -2.6370,

fl = 1.36891 fm -1 (3.9)

In this case, 4c~ ~ 0.04. The above values of a s and rs are only slightly different from those of eq. (2.16),

4. Concluding remarks

We have shown that the "nuclear" scattering length as determined by using the Naqvi-type separable potential is somewhat larger, in its absolute value, than those obtained by Heller et aL ~) by using hard-core local potentials. The effective range r s for the non-local potential also seems to be larger than that for hard-core local potentials. Presumably, these differences are due to the softness of the short-range repulsion of the separable potential. A velocity dependent potential also seems to yield a larger value for las[ [ref. 8)]. Recently, Sprung and Srivastava 9) proposed new local potentials with very soft repulsive cores. Their pp potentials SSC-PP-1 and SSC-PP-2 respectively give the following results

= / -17.443 fm t2.843 fm (4.1) as ~ - 17.428 f ro ' rs = t2.847 fm"

Again, lasl is larger than those obtained from hard-core potentials. It is interesting to compare as with the nn scattering length ann. However, the ex-

perimental value of ann seems to be still quite uncertain 10).

We are grateful to Mr. M. K. Srivastava for supplying us with the results quoted in (4.1).

Appendix

The matrix elements Iij for the Naqvi-type potential are obtained as follows. First, I l l is the same as in sect. 2, and is given by 4)

# (f12 _ k 2 Fh(~/) + In 4~fl2 q/ R e l i l ( k ) - zc(flZ+kZ) t 4fl +0~fl L ~ +YJ~ , • (A.1)

Im I~ ~(k) = ½C~(rl)k(fl z + k2) - 2 exp [4t/arctg (k/fl)]. (A.2)

Then others are obtained in terms of I~l or f l :

-/12 = -/21 ---- I l l+½f l2 - / l l ,

374 T H I E N V O - D A I A N D Y . N O G A M I

where

R(k) = f14 f~ 2d 2f;Z(q)_fl(q)f~,(q) (A.5) 7-~ q q - - - - , 6~z Jo E q - E ~ - ie

which is reduced to

R(k) = ~f14 ( q2 dq[f l(q)o l(q)]2 6 - ~ 3 ~ + O ( a 2 In a). (A.6)

A crucial step in the manipulat ions is to note that the singular funct ion h(t/) can be isolated as on the 1.h.s. o f eq. (2.10). Let us introduce the following notat ions:

R e l l l = Ao+aA~, I m l l l = B, (A.7)

where A o = [Re/11 L= o- Note that h(~) is contained only in A~. Then one can derive

cot 6~o = R e D _ /(Re D)o A,} I m D t ~ m D + a , (A.8)

where (ReD)o is obtained by replacing A by Ao in ReD. One can easily identify the term AJB with the second term on the 1.h.s. of eq. (2.10).

Note added in proof: Very recently two papers 11) have appeared in which ann is

calculated. Miller et al. obtained ann = --(17.3 to 17.6) fm using various local

potentials and also velocity dependent ones. Kermode and Sprung obtained - ( 1 6 . 9 to 17.6) fm for a variety of local potentials including those with very soft cores 9).

Here, ann differs f rom our a~ due to the effect o f the ph mass-difference which is 0.3

to 0.5 fm2); namely as = ann+(0.3 to 0.5) fro. I t is not clear to us if the pn mass- difference has been considered in ref. 1).

References

1) L. tteller, P. Signell and N. R. Yoder, Phys. Rev. Lett. 13 (1964) 577 2) E. M. Henley, in Isospin in nuclear physics (North-Holland, Amsterdam, 1969) 3) J. S. Letmg and Y. Nogami, Nucl. Phys. B7 (1968) 527 4) D. R. Harrington, Phys. Rev. 138 (1965) B691 5) Y. Yamaguchi, Phys. Rev. 95 (1954) 1628 6) J. H. Naqvi, Nucl. Phys. 59 (1964) 289 7) R. J. Slobodrian, Phys. Rev. Lett. 21 (1968) 438 8) I. Slaus, Rev. Mod. Phys. 39 (1967) 575, footnote 4 9) D. W. L. Sprung and M. K. Srivastava, Nucl. Phys., in press

10) R. P. Haddock, R. M. Salter, M. Seller, J. B. Czirr and D. R. Nygren, Phys. Rev. Lett., 14 (1965) 318; R. Wilson, Comments on Nucl. Part. Phys. 2 (1968) 141; P. G. Butler, N. Cohen, A. N. James and J. P. Nicholson, Phys. Rev. Lett. 21 (1968) 470

11) M. D. Miller, M. S. Sher, P. Signell, N. R. Yoder and D. Marker, Phys. Lett. 30B (1969) 157; M. W. Kermode and D. W. L. Sprung, Nucl. Phys. A135 (1969) 535