Relativistic channel-coupling effect in the deuteron matter radius

uclear physics A541 (1992) 384-396 North-Holland

IN istic

YSICS

C

shun

A

bong

tMent of Pki-sics, University' ofCalifornia, Los Angeles CA 90024, USA Received 6 August 1991 (Revised 9 December 1991) ct: Local sqtmre-well and ronlocal separable potentials are used in the Dirac equation to show that the dominant relativistic effect in the deuteron matter radius r,,, comes from the coupling to the lower Dirac component in essentially the same way as the coupling to the D-state and higher-mass states in the Schr5dinger equation. The resulting relativistic correction to r,,, turns out to be sensitive to dynamics. In the limit of weak binding, it gives a result in good agreement with the traditional kinematical approach for both local and nonlocal potentials.

1 . Ink

uction

-a he theory of elastic ed scattering has been under study for many years ` ) . At very small momentum transfers, the deuteron mean square (m.s.) charge radius r2ch can be extracted from experiment. This in turn yields information on the structure 3.4) : of the deuteron since it can be related to the m.s. radii of its constituents 2 r +r2 +r 2 +r 2 ,+r2 .2h=rm

P

n

re

MEC 9

2 fM2 where r 2P = 0.743 (2 1 ) fm' is the proton ms. charge radius, while rn = -0.1192 (18)

is the neutron m.s. charge radius as measured in thermal ne scattering :, ) . Several years ago, Marsfeld et aL ') have shown that the combination r2ch- r -P can be deduced directly from the experimental ratio of ed to ep cross sections with much greater precision than each quantity separately. When used with the relativistic correction 3.4) = 2 _ 3 , rrel -- A12 0133 fm' 4

(2)

where M is the nucleon mass, and the correction r;m EC= 0.014 fm- from mesonexchange currents (M EC) 7 L they obtain the result r,,, =1 .950 (3) fm for the deuteron matter radius which describes the distribution of nucleon centers of mass in the deuteron. Correspondence to: Professor C.W. Wong, Department of Physics, UCLA, Los Angeles CA 90024-1547, USA. 0375-9474/92/SO5 .00 ©1992 - Elsevier Science Publishers B.V. All rights reserved

Chun Wa Wong / Relativistic channel-coupling

e

larsfeld et al. ') have further pointed out that this experimental value is sig-

nificantly smaller than the value of about 1 .97 fm from realistic potentials fitting the triplet scattering length of 5.24 fm [ref. s )]. Since that time, another realistic potential, the full Bonn potential 9), has been constructed which gives a much larger theoretical value of 2.0016 fm. It has now been understood that this large value is due to the energy dependence of the potential arising from the coupling to abnormal states of masses higher than the normal two-nucleon states '®). In particular, the 3.78% admixture of abnormal states contained in the full Bonn potential as been shown to give just the matter-radius excess seen in this potential. Moreover, these abnormal states are much less spread out in space and will give a correspondingly smaller contribution to the deuteron matter radius if they appear explicitly in the wave function (w.f.). The resulting reduction has been shown to bring the Bonn result to rough agreement with the other realistic potentials '0). This leaves the original I% discrepancy still to be accounted for. Before this discrepancy can be resolved, we must reassess the reliability of our present understanding of both relativistic and MEC corrections appearing in eq. (1). The purpose of this paper is to help in this reexamination by putting the relativistic correction in a somewhat more general conceptual framework, namely that just as in the case of the D-state and higher-mass states, it is a consequence of coupling to an abnormal channel '°). The abnormal channel in this case is the lower, or negative-energy, component of the Dirac equation . In addition, we shall show that for two models of the deuteron based on the Dirac equation, our relativistic dynamical correction differs significantly from the purely kinematical correction given by eq. (2). The relativistic correction shown in eq. (2) is known to come from the Zitterbewegung motion of the nucleons cue to the lower component 4 1. Its explicit form can be obtained by reducing a relativistic four-component interaction into the nonrelativistic (NR) two-component form with the help of the Foldy-Wouthuysen (FW) transformation "). (An elementary derivative ofeq. (2) is given in the appendix for the reader's convenience.) An important advantage of this transformation is that it is unitary, and therefore produces a phase equivalent NR potential from a relativistic one, or vice versa. In particular, eq. (2) owes its simplicity to the fact that it is the leading term independent of the potential V in an expansion in powers of V/ M. It is not clear however that the higher-order dynamical terms '2) dependent on V are not needed . In fact, they have been found to be important at large momentum transfers 12) . Reexamination of relativistic effects in the deuteron radius hai e recently been '3) (which contains made by a number of authors. As summarized by Toyama et al. references to earlier work), available calculations seem to suggest that they are in the wrong direction and tend to worsen the discrepancy with experiment . In particular, they have studied relativistic effects in the ratio rm /a in the form of a ref. 14). e relativistic version of an effective-range (ER) expansion for rr,

1 86

Chun 11'a Wang / Relafivistir channeJ®roupJing effect

cession has been glen ") that this relativistic correction might have to be made in addition to eq. (2). We would like to clarify in this paper the relation between these more recent studies and the traditional one involving eq . (2). In particular, we are interested in obtaining a more intuitive understanding of the physical mechanisms at work. We do not use the relativistic ER expansion of ref. ") because a w.f. integral appearing in this expansion still has to be calculated . We use instead exactly soluble potential models: local square-well potentials in sect. 2, and the nonlocal separable potentials of Nogami and van Dijk ") (NvD) in sect. 3. We find that for both potential models, the dominant relativistic effect arises from the energy dependence of the effective potential due to the coupling to the approxi.f [This is in contrast to the FW w mately 1% lower component of the Dirac . transformation ") which gives rise to a momentum-dependent, but energy-independent, potential. The difference is immaterial, however, as the equation solved in both cases is the same Dirac cquation.] We find that as in the Schr6dinger equation, this energy dependence causes r,, to increase . Unlike the higher-mass states studied in ref. "'), the lower Dirac component has a m.s. radius r 2 not much smaller than the value rc; in the upper component The result is that the energy-dependent increase of r,,, persists in the final result, thus giving a positive contribution in the wrong direction, in agreement with eq. (2) and ref. 1-1 ) . alike eq. (2) however, we find considerable model dependence in sect. 2 for local square-well potentials even when they fit the same scattering length. It is therefore of some interest to find that this model dependence becomes quite weak for potentials with reasonably strong attractive scalar components . For these models, the relativistic effect in r .. is about 0.39%, in good agreement with the result of 0.44% from eq.(2) . oth these results are in significant disagreement with the relativistic corrections of () 20-0.35% found by van Iii Iik ' 5 ) for the NvD separable potentials. To understand the discrepancy, we repeat our calculations in sect. 3 with these separable potentials . e find relativistic corrections of 0.36-0.39% instead over the same range of potential parameters, again in rough agreement with en . (2). ur relativistic correction is found in sect. 4 to have an interesting dependence on the nucleon mass M appearing in the Dirac equation. It varies as M-2 for the nordocal separable NvD potentials, just as in eq. (2), but for the local square-well potentials, it is made up of a combination of M - ' and M -2 terms. The similarity between our relativistic correction and the purely kinematical result of eq. (2) is made more explicit in sect. 5 by studying j 2M =2r,,,Jr", as a function of the binding energy B. Our results are found to approach the kinematical limit as B approaches zero. The dynamical effects not included in eq. (2) are found to be stronger for the nonlocal NvD potential than for the local square-well potentials . We therefore conclude in sect. 6 that our relativistic treatment of the deuteron matter radius is consistent with the existing treatment, and that there is a need to

387

Checn Wa Wong / Relativistic channel-coupling g

explicitly include dynamical effects if the potential is very i erent r~ local potentials . esults for local

are-well

the usual

te ti Is

The one-body Dirac equation for a local potential made u of a scalar component S(r) and a vector component V(r) reads

- K) F(r)=-[E-V-(jc$S)]G(r), d dr r d $ dr rK

(3)

G(r)=[E- V$ (IA +S)]F(r),

where b = c =1 are used. Here jA is the reduced mass ( 9.45926 deuteron),

e

for the

E=v(k $A )=j,$T 2

2

is the total energy, and K = -1 is used if the upper component G(r) is an S-state. By eliminating the lower component F(r), we see that G(r) satisfies the equation d~- K(K-F

dr`

r`

1 1)$k,-ZI1Ve4fi+

d

E-V$Iu,$S dr

d $ (V-S)

dr

K P,

r® IG(

-® . (5)

The effective potential which appears is Veff= E

V$

1 2tL

(S`'-V2)$S=

Keff . 2M,

(6)

The last term in eq . (5) for the square-well potential, V(r) = - Vod (b - r) ,

S(r)

=

-

S,,e(b - r) ,

(7)

involves

d (V-S)=(VO,-S,,)s(r-b) . dr

(8)

As a consequence, dG/dr has a discontinuty proportional to V - S(, at r= b, and -a2 and energy the eigenvalue matching condition for a bound state with k`' _ (IL E. = -a ) becomes 2

2

xcotx=-ab-

E~g $ ~

(1$ab),

(9)

where x=96,

q

2

=-2.u Veff - a2 .

(10)

We should add that the discontinuity in eq . (8) appears because only the w.f.'s G(r) and F(r), and not their derivatives, are matched at the square-well radiub b. Eq. (5) also shows that the discontinuity vanishes as expected in the NR limit of IL - X .

CU11

388

ng / Relativistic channel-coupling effect

I trying t understand the physical contents of this model, we shall not examine the ex ct e ressions for various interesting quantities because they are too cumbers to be shown here. s start instead with the NR "comparison" square-well otential obtained in ref. A Table I gives its potential parameters, E euter ra aters, deute7on matter radius r. and its shape-independent approximant 14) rt.0

+ Waro ) 2 1 . 2

0 1)

2

(5) is ol.' course a little more complicated than an ordinary Schr® finger equation because of the presence of the last term. However, this term disappears when So = K. The derivatives of G(r) then become continuous at r = b like a Schr6dinger w.f. even for finite ji. The upper component is thus similar to a Schr8dinge-.- wS, with a similar matching condition [eq. (9) with S", = VJ, the only difference being that there is still coupling to the lower component. It is thus of some interest to consider first a deuteron potential (#2 of table 1) with b and S,, = Q chosen to M the deuteron binding energy B = 2.224579 MeV and the triplet scattering length of potential # 1. We see from table I that the resulting paramstars calculated from the Dirac eq. (3) are very close to those for potential I calculated from the Schr6dinger equation for the same reduced mass IAO = 469.45,926 MeV. [The small differences between the two sets of ER parameters can be tnxed to the fact that relativistic kinematics is used to obtain a = 0.231332 fm -, fron, B for eq.(3), but a =0.231607 fm - ' is used in the NR equation.] However,r. has now increased by 0.45% . TABLE I Deuteron properties from the one-body Dirac equation for several local square-well potentials Potential

#1

#2

#3

#4

#5

#6

h (AM So (Mev) VO (Mev) 9bl a ra P FT P111 M) r, (fm) r,,,o ( fm) r~ (fm 2 ) j (02 ) -4r,n/rm M) -IF rml rrn ( °I®) -3 rn/ rm (°`®) (Ar ../ (%)

1157 0.00 33.067 a) 0.5835 5A239 1.7856 -0.0405 0.00 37.50 1 .9525 1.9586 3.8122 -

2.1V8 16.04 16.044 0.5848 5A239 1.7792 -0-0,403 1.05 3794 1.9612 1.9582 148552 2.9968 0.57 -0.11 0.45 0.46

2.1678 100 -66.416 1505 5.5108 1 .8982 -0.0479 1 .08 35.48 1 .9952 1 9953 3.9913 3.0217

2.1678 -100 130.181 0.6316 5.2956 1.5969 -0-0193 1.01 4139 1.9124 1.9042 3.6642 2.9645

2.0247 100 -62 .481 0.5399 5.4239 1 .7834 -0.0473 1 .16 3335 1 .9602 1 .9584 3 .8547 2.7821

2.4265 -100 125 .160 0 .6390 5 .4239 1 .7678 -0 .0206 0.89 4177 1 .9653 1 .9577 3.8665 3.4007

-0.12 2.19

-0.10 -2.05

InIr

-

-

') Potential for the Schri3dingeT equation, or lim ( ,,,

(jA

/,u o)( VO+ SO).

-0.14 0.39 0.40

-0.06 0 .66 0.70

C'hun Wa

ong / Relativistic channel-coupling

e

According to ref. °0 ), the deuteron matter radius for a many-channel euter w.f. differs from that in an energy-independent single-channel e it a energy-independent potential in the following way: rm (01YA i arm i rm aT _ /d rm aPabn 2rM Frm -_ ®arm -~-

rm

rm

,

.

That is, there are contributions from both the energy dependence in the -channel and from the explicit appearance of the F-channel. Now eq. (6) gives - la Vef

aT

o d

V®PG_ ,

(13)

where PG _ is the G-state probability inside the square well (=37.5% for potential #2). With Vo =16 .04 MeV, this yields 1 .28% for eq. (13). When used with the result °®) 1 arm - 0 .0044,

rm

Pabn

(14)

we obtain 0.57% for the first, or energy-dependent, term of eq. (12). e second term can be calculated from the entries of table 1, and gives -0.11% for the change in rm when 1 .05% of the deuteron appears in the lower component with its somewhat smaller radius. The total change of 0.46% agrees well with the actual result of 0.45% also shown in the table. We should add parenthetically that unlike the R Schr6dinger equation, eq. (13) does not give the abnormal-channel probability Pabn because the relativistic propagator is different from the Schrodinger propagator. The full dependence of rm on So is a little complicated, and is best approached in two steps. Starting from potential #2, we shah first study the effect of varying So ( Vo being always adjusted to fit B), without changing other potential parameters. The results a;e plotted against So in fig. 1 and shown as the dashed curve labeled "const b". We see that rn, increases monotonically with increasing S® . [The parameters of two potentials (#3 and #4) along this curve are given in table 1 to allow the reader to make more quantitative comparisons. Note that V® - S® = 32 eV 2So .] The physical origin of this increase in rn, is easy to describe . The w.f. G(r) outside the potential has the model-independent slope of -a. There is in general a kink at r = b, with G(r) kinking down (up) when So is greater (smaller) than its value (16 MeV) in potential #2. That is, G(r) must have a slope less (more) negative than -a just inside b. This is possible only if gb is smaller (larger) and closer to (further from) 6r. This interpretation is confirmed by table 1 .

Chun Wa K'artg / Relafirktic channel-caupfing ~ffect

1- cost b M

4

~0

1,4 NP

Irmo

1

4

4

0

1

0",

cost 0

05

- O00

11525 fam

-150

900

-

s o (Mejv)

0

100

Fig 1 . Dependence of the deuteron matter radius r,,, on the strength -S,, of the scalar potential for potentials with constant square-well radius h 8 dashed curve) or constant triplet scattering length a (solid curve). The shape-independent values r,,,o of eq. (11) for potentials of constant b are also given as open circles.

owever, the scattering length a for these potentials with constant b is not a constant, but also increases manotonicAy as & decreases, as shown in table L ow we know from e q. 111) that r,,,,, also changes with a . The value of for these potentials are shown as open circles in fig. 1 . We can see that the change in r.. comes primarily from the change in This means that if the potential parameters are varied to refit a, r,,, would have remained roughly the same. The exact values calculated with refitted potentials are shown in fig. I by the solid curve labelled "covet a". Two of the potentials on this curve (in addition to potential #2) are given in table I as potentials #5 and #6. he approximate phase equivalence between the relativistic potentials #2, #5 and #6 and the 1' potential A# I can be improved by correcting the: Muesli cttfferrrwe in r,, with the help of the ERformula (11). The result is shown in table I as (err,,/ r,,,),` . e see that for the potentials of table 1, r. is not too sensitive to So, the attractive strength of the scalar potential. Now S,, is known to be about 200-400 MeV for the nucleon-nucleus interaction at intermediate energies 17 ), but it should be weaker in the deuteron which has a much smaller scalar density. Fortunately, knowledge of the exact value of S. is not needed because fig. I shows that rn, is essentially independent of 56 for values stronger than 100 MeV (potential #5) . (There is a very shallow minimum at S. == 150 MeV.) Table I shows that for potential # 5, r," has increased by 0.40% from the value of 10525 fm for the NR potential # I The relativistic correction is larger for the physically less interesting cases with repulsive scalar potentials, reaching 0.70% at S,, = -100 MeV.

Chun Wa Wong /

3.

elativisfic channel-roupfing~ ffea

esults for the seoarable No

391

jk potentials

The relativistic corrections found in the last section for local square-well potentials have the same sign, but are about twice the 0 .20-0.35% effect found by van Dijk '5) for the relativistic separable NvD potentials "). Hence it is of some interest t ut why they differ. To allow a close comparison, we repeat the calculations leading to table I for the NvD potentials . First we find a NR potential (#7) fitting only the scatteri length. This turns out to be very similar to the square-well potential # I except that r, n is now smaller by 0.83% . (This change alone would have solved the problem of discrepancy with experiment but for the fact that a separable potential like the Nv potential is less realistic than the Bonn potential which is essentially local.) Relativistic kinematics, same as that used for the square-well potentials #2-#6, is used in the relativistic NvD potentials #8-# 11 . The first, or energy-dependent, term of eq. (12) is calculated by means of the analytic result _ (aV.11)

cl T

= d

A .,,21 . :! 4 .u

(15)

in the notation of ref. 16) . When used with eq. (14) (for the square-well potential), it gives the result -i Erjr,,, shown in table 2 . The last line Ar .. I r .. of table 2 is the exact result, from which we deduced that (arjaPjt ,,j1r,, for the NvD potentials is in the range 0.0041-0.0043, i.e. a little smaller than the value shown in eq. (14). Tables I and 2 show the model dependence of relativisic corrections . The lower component probability P[_ is larger in the NvD potentials leading to larger values TABLE 2 Demeron properties from the one-body Dirac equation for several nonlocal separable Nogami-van D -jk ") potentials Potential

#7

/3 !NOV) A, (106 MeV3 X, 1 N1eV, " 12)

274 .42 2 .8494 0 .07166 5 .4239 1 .7759 -0.0244 0.00 1 .9363 1 .9582 3 .7492

S

a (Q)

ri, (fm) P F; ( °/®1 r,,, 1 fm1 rMjj (fM) r,', (fm-)

r 2F (fm2 )

-1 u r m/ r n (% ) IF r,,,/ rm M) .arm /r°, M

(J rm/rm ),e (%)

-

-

-

#8

#9

-5 .0 288 .40

_1 .0 2'1 ..^.J

3 .4428 0 .06548 5 .4239 1 .7705 -0.0270 1 .53 1 .9429 1 .9621 3 .8012 2 .0273 0.74 -0.36 0.34 0.36

2 .7849 0.07152 5 .4239 1 .7690 -0 .0233 1 .40 1 .9432 1 .9619 3 .7989 2 .1621 0 .71 -0 .30 0 .36 0 .37

*10

#11

1 .0 .64 ' .06 2 .5512 0 .07413 5 .4239 1 .7680 -0 .0217 1 .35 1 .9435

5 .0 252 .30 2 .1930 0.07876 5 .4239 1 .7658 -0 .0188 1 .26 1 .9444 1 .9615 3 .7985 2 .3277 0.68 -0.24 0.42 0.44

1 .9617 3 .7984 2 .2211 0 .70 -0 .28 0 .37 0 .39

3

Chun Wa Wang / Relativiwir channel-coupfing

effect

is proportionally larger because th .A L r,/r,, and a u r.lr, in addition, The bottom line is that the total relativistic correction Jr,,Jr. or r 2 is smadler. 71 is only a little smaller than that in the local square-well potentials of table I for the more interesting case of attractive scalar potentials. Our results e with those of van Dijk ") who gives a smaller value of 0.20 (0.30, 0.35)% r otential #8 (W9, # 10). Our calculation also gives an effectiw range of ff . O ffmm for the three potentials shown in table I of ref. ") as calculated from their eq. (4.8) or (5.16) instead of the reported 1 .7 fm. The reported 1 .7 fm is incorrect because of a minor programming error. This error has been corrected in the results of ref.' s ) (van Dijk, private communications). Hence the difference between our results and those of van Dilk must have come from the different procedures used for enforcing base equivalence between the relativistic cases and their NR limits. The large differences involved in some of the cases must be considered surprising and not understood at the present time. However our results are supported by the fact, to be shown explicitly in sea. 5, that they agree with the kinematical result of eq. (2) in the limit of zero binding. f

n the nucleon he relativistic effect on r, studied here has an interesting dependence on the nucleon mass M = 2iu. For local square-well potentials, we note that since the result is insensitive to S,, if the latter is positive (i .e. W the scalar potential is attractive), we may use the special case S,, = V,, for which eq. (12) is valid. Used with eqs . (13) and (14), it gives A rM to P, ;_) I aq~ +(: A40 )2 I Q _ IQA12pi) 0 A4

on

2 AX

r

a P~h.

0.57%( o) - 0.11 % Imp A4

2

M

2r2M

(16)

where we have used the fact that Pu is proportional to Al -2, V, r2iC and r2U are all insensitive to M, and that .1rl r, =0.39°/® at  =938.9 MeV (potential # 5) . he M-dependence for the NvD separable potentials turns out to be closer to that of eq. (2) because eq. (15) gives

r,

M 0_3410

rm c)P.h,, 2r 2M

m2 0

( MO

)2 I

(

r'j-r)(M2PF) 0 11 (17)

where we have used the fact (from table 2) that A te; is roughly independent of M and c& the ratio s of potential strengths in the upper and lower components, being

Chun

a Wong/

elativistdc° channel-coupling e eco

393

1 . x 1 4 e 2 for the potential I an 1 .48 1 .3) 1 4 e 2 fort relativistic potential fi (#11). e percentage appearing in eq. (17) is that for potential (wit s = -5) . e values appropriate to other values s are given i table 2 e percentage appearing in of e s. (16) a (17) should e slightly larger if phase equivalence is ore strictly enforce as s i the bottom lies tales. These results a ree with the existi estimate of 0.449/o (M,,IM)2 given by e . e mathematically permissible range of relativistic canes i r. is course much larger than this. e can see from either eq. (12) ere or the coupled-channel unitary transformation defined y eq. (19) of ref. "') that r,,, cat be reduced by either decreasing the lower-state probability or making the lower component less eaten e in space. It is possible however that physically interesting cases might not e ve different from those discussed here. The close conceptual and structural similarities between the description o eq. (12) based on the idea of effective energy-dependent potential for a bound state an the more familiar one based on the unitary Foldy-Wouthuysen transformation suggest that these are both valid approaches to the problem of relativistic correction. e FW transformation is the more elegant method which recognizes the fact that the lower component might be present in the Dirac spinor even in the absence of a potential. The present approach emphasizes instead the universal features of channel coupling which are independent of the detailed mechanisms involved, and in addition it also takes into account dynamical erects not included in eq. (2). 5.

yna ica effects

Dynamical effects are included in our calculations, and can be isolated explicitly. We show first in table 3 the effect of a change in the binding energy B for local square-well potentials . The comparison is always made to the corresponding NR result of the Schrodinger equation with a calculated with NR kinematics . We see that the relativistic correction increases with increasing B, as shown in table 3, becoming rather large for strongly bound states . Our dynamical correction vanishes at B = 0 (or a = oc) where rm becomes infinite and PF vanishes . Since r, becomes infinite in this limit, it would be interesting to examine ®r2 = 2rm®rm also. Fig. 2 gives r e, = Are as functions of the binding energy B and a potential parameter. For the local square-well potential, the latter is taken to be the scalar well depth S,,, while for the nonlocal NvD-potential, it is the parameter s or ref. '6) which is proportional to the ratio of the strength of the s-wave term to that of the p-wave term in the separable potential. (Other potential parameters are determined by fitting B and the scattering length a. For example, as the scalar potential becomes increasingly attractive with increasing S(,, the vector potential must become more repulsive in order to fit the same B.) e see that as decreases to zei o, the calculated relativistic correction r e, approaches the kinematical limit cf 0.033 fm2 given by eq. (2), and denoted by the horizontal arrows in fig. 2.

Chun lVa lVong / Relativistic

394

channel-raupling gffeci

TAM+ 3 Dependence of deuteron properties from the one-body Dirac equation on the binding energy and parameters for local square-well potentials.

to 12

Potential 8 1 NICV1 hnemmw% h q fin) S" 1 Nlev ~ t "' I MeV I

# 14

01 .11

# 16

#17

2 .195 8 U 23142

NOS IN 7 LIM

ZVO NR jW7 0 0 . 157

=02 0 51296

5.4877

5,4876

-13

(Q NR 2A 2,7 0 2N76

a t1 ni n I An) P P, i% 1 P_ 1 1.1

114770 2,0071 _6052 0.00 1 .1 .55

15A770 2.(K)44 _0019 0313 14,13

4 CM

5AN52 30AN!

5A852 3MI87

r

An 1

r

5.4846

0 1 An'

r I An')

-1,

F11 " r",

J",

11

5A852

_Wll

2S8741 1 .1919 -0.0466 OMO 72 .40 41 .9871

2,5879 13702 -0.0377 2.50 74.19 1 .0133

-

(158

1 .4940

1 .3083

1 .0221 01743 -

_WV

OM6 0.1-K)

#18

M21 lilt) -21505 15879 1.4 -0.0536 3.14 67.16 0.9977

W194 ?Am

.>738 S

-

i"') %I %)

154770 2.0051 -0J)425 0.36 12 .09

0.05 0.05

1 .0163 1 .0149

1 .0208 0.9853 R51

2.66 1.01

tential

1.08 0.98

1,671

2 -111534

15879 1 .4137 -0.0-597 3.75 60 .37 0.9911

1.0228 0.9786

1.18-52

(139

M 0.33

1 lim" - , , I A / tl")l V. + So

n the other hand, there is a clear dependence on dynamics, i.e. on the details of the potential model, when B is not too small (_- I or 2 MeV). In a simiiar vein, we find that our relativistic correction changes significantly with changes of the scattering length away from its experimental value, even when the binding energy is kept constant . For example, for S,, = 100 MeV and B = ,,p  the relativistic

0.06~

ODE

8 MY

0.04

002

0.033

Beip 002

20

Local square well potential I

0

I

I

50

Scalar well depth So (MeV)

0.02

Nonlocal NvD potential i

100

5

10

Dimensionless parameter s

Fig . 2 . The total relativistic correction r for different potential parameters and different binding energies for the local square-well potential and the nonlocal Nogami-van Dijk potential . The scattering length a fitted is the same for each curve, but varies with B.

Chun Wa

an / Relativi.-vic ch nnel-coupfing effect

393

correction r rne2 l is 0.034 fm 2 when a = 4.83 fm (b = 1 .0 fm for the NR potential), -OA07 fm2 when a = 6.45 fm (b = 4.0 fm for the N R potential) . The dynamical effect is much smaller if B and a are both fixed at their observe values. Fig. 2 shows that in this case r2,; is in the range 0.031-0.037 fm :2 for the local square-well poten!ial, and 0 .027-0-040 fm :! for the nonlocal NvD potential nelusions

The relativistic effect on the deuteron matter radius r. studied here arises fr the coupling to the lower component of the Dirac equation. As such, it is generically the same as the traditional one given in eq. (2) -1.4.12). Indeed, we have show explicitly that our effect agrees with the traditional result given by eq.(2) in the limit of zero binding. However, when the binding is not small, or when the scalar a vector components of the potential are not both weak, our effect is also sensitive to dynamics. Although the traditional relativistic trinematical effect on the deuteron matter radius is only about 0.4% of r,,,, i.e. only a fraction of the discrepancy of 1% between theory and experiment for rm , it might be necessary to go beyond it and include relativistic dynamical erects when discussing the deuteron matter radius calculated for realistic potentials. Many of the past calculations of deuteron form factors for realistic potentials (for example, ref. "')) have already included such dynamical effects. But the actual values cannot be read accurately from published figures. We would therefore like to appeal to all groups to report their calculated values of relativistic effects in the deuteron radius. pendix AN ELEMENTARY DERIVATION OF EQ. (2)

The Foldy-Wouthuysen transformation ") in the presence of a potential cannot be constructed exactly. It can however be obtained in a NR expansion in powers of 11M, where M is the nucleon mass. If the original wave function is 0, the transformed one is , eis, e is241, (ALI) 41 = . . . where

(A.2)

S1=- PU-P 2Am

is chosen to eliminate "odd" terms in the hamiltonian to order (I/ M)". To this order 2)

(r

= I iPYO d 3 r =

01*(e

e -is, )00 d-; r

' r2 + AW I Cr-rXpWd 3 r . 2 2M` 4

(A.3)

396

Curs Ima Wong / Relativistic channel-coupling effect

(2) comes from the second term of (A.3). As is well known '"') dynamical e cots are contained in the higher-order transformations S,2 , - . . . They can readily e extracted from eq. (A.3) by continuing the calculation to higher orders. oferencofte I) 2) 3) 4) 5)

VZ Jankus, Phys. Rev. 102 (1956) 1586 M . Gourdin . Nuovo Giro, 28 (1963) 533 ; 32 (1964) 493 B.M . Casper and F. Gross, Phys. Rev. 155 (1967) 1607 U. Friar, Ann, of Phys. 81 (1973) 332 L. Koester, W. Mailer and W. Waschkowski, Phys . Rev. Lett. 36 (1976) 1021

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