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

IN istic C

shun A bongtMent 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 showthat the dominant relativistic effect in the deuteron matter radius r,,, comes from the coupling tothe lower Dirac component in essentiallythe same way as thecoupling to the D-state and higher-massstates in the Schr5dinger equation. The resulting relativistic correction to r,,, turns out to be sensitiveto dynamics. In the limit of weak binding, it gives a result in good agreement with the traditionalkinematical approach for both local and nonlocal potentials.

1 . Ink uction

r.22

2h=rm+r2+r 2 +r 2e,+rP

n

r

MEC 9

0375-9474/92/SO5 .00 ©1992 - Elsevier Science Publishers B.V . All rights reserved

YSICS

he theory of elastic ed scattering has been under study for many years `-a). Atvery small momentum transfers, the deuteron mean square (m.s.) charge radius r2ch

can be extracted from experiment. This in turn yields information on the structureof the deuteron since it can be related to the m.s . radii of its constituents 3.4) :

where r 2 = 0.743 (2 1 ) fm' is theproton ms. charge radius, while r2 = -0.1192 (18) fM2P

nis the neutron m.s . charge radius as measured in thermal ne scattering :, ) . Severalyears ago, Marsfeld et aL ') have shown that the combination r2ch- rP- can be deduceddirectly from the experimental ratio of ed to ep cross sections with much greaterprecision than each quantity separately . When used with the relativistic correction 3.4)

2 _

3=

fm' ,rrel -- 4 A12

0133

(2)

where M is the nucleon mass, and the correction r;mEC= 0.014 fm- from meson-exchange currents (M EC) 7 L they obtain the result r,,, =1 .950 (3) fm forthedeuteronmatter radius which describes the distribution of nucleon centers of mass in thedeuteron .

Correspondence to : Professor C.W. Wong, Department of Physics, UCLA, Los AngelesCA 90024-1547,USA.

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 thetriplet scattering length of 5.24 fm [ref. s)] . Since that time, another realistic potential,the full Bonn potential 9), hasbeen constructed which gives amuch larger theoreticalvalue of 2.0016 fm. It has now been understood that this large value is due to theenergy dependence of the potential arising from the coupling to abnormal states ofmasses higher than the normal two-nucleon states '®). In particular, the 3.78%admixture of abnormal states contained in the full Bonn potential as been shownto give just the matter-radius excess seen in this potential. Moreover, these abnormalstates are much less spread out in space and will give a correspondingly smallercontribution to the deuteron matter radius if they appear explicitly in the wavefunction (w.f.) . The resulting reduction has been shown to bring the Bonn result torough 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 ourpresent understanding of both relativistic andMEC corrections appearing in eq. (1).The purpose of this paper is to help in this reexamination by putting the relativisticcorrection in a somewhat more general conceptual framework, namely that just asin the case of the D-state and higher-mass states, it is a consequence of couplingto an abnormal channel '°). The abnormal channel in this case is the lower, ornegative-energy, component of the Dirac equation . In addition, we shall show thatfor two models of the deuteron based on the Dirac equation, our relativisticdynamical correction differs significantly from the purely kinematical correctiongiven 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 component4 1 . Its explicitform can be obtained by reducing a relativistic four-component interaction into thenonrelativistic (NR) two-component form with the help of the Foldy-Wouthuysen(FW) transformation "). (An elementary derivative ofeq. (2) is given in the appendixfor the reader's convenience.) An important advantage of this transformation is thatit is unitary, and therefore produces a phase equivalent NR potential from arelativistic one, or vice versa. In particular, eq. (2) owes its simplicity to the factthat it is the leading term independent of the potential V in an expansion in powersof V/M. It is not clear however that the higher-order dynamical terms '2) dependenton V are not needed . In fact, they have been found to be important at largemomentum transfers 12) .

Reexamination of relativistic effects in the deuteron radius haie recently beenmade by a number of authors. As summarized by Toyama et al. '3) (which containsreferences to earlier work), available calculations seem to suggest that they are inthe wrong direction and tend to worsen the discrepancy with experiment . In par-ticular, they have studied relativistic effects in the ratio rm/a in the form of arelativistic version of an effective-range (ER) expansion for rr, ref. 14).

e

1 86

Chun 11'a Wang / Relafivistir channeJ®roupJing effect

cession has been glen ") that this relativistic correction might have to be madein 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 inobtaining a more intuitive understanding of the physical mechanisms at work. Wedo not use the relativistic ER expansion of ref. ") because a w.f. integral appearingin this expansion still has to be calculated . We use instead exactly soluble potentialmodels: local square-well potentials in sect. 2, and the nonlocal separable potentialsof 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-mately 1% lower component of the Dirac w.f. [This is in contrast to the FWtransformation ") which gives rise to a momentum-dependent, but energy-indepen-dent, potential. The difference is immaterial, however, as the equation solved inboth 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 studiedin ref. "'), the lower Dirac component has a m.s . radius r2 not much smaller thanthe value rc ; in the upper component The result is that the energy-dependent increaseof r,,, persists in the final result, thus giving a positive contribution in the wrongdirection, in agreement with eq. (2) and ref. 1-1 ) .

alike eq. (2) however, we find considerable model dependence in sect. 2 forlocal square-well potentials even when they fit the same scattering length . It istherefore of some interest to find that this model dependence becomes quite weakfor potentials with reasonably strong attractive scalar components . For these models,

is about 0.39%, in good agreement with the result ofthe relativistic effect in r ..0.44% from eq.(2) .

oth these results are in significant disagreement with the relativistic correctionsof () 20-0.35% found by van Iii ik ' 5 ) for the NvD separable potentials . To understandIthe discrepancy, we repeat our calculations in sect. 3 with these separable potentials .

e find relativistic corrections of 0.36-0.39% instead over thesame range of potentialparameters, again in rough agreement with en . (2) .

ur relativistic correction is found in sect. 4 to have an interesting dependenceon the nucleon mass M appearing in the Dirac equation . It varies as M-2 for thenordocal separable NvD potentials, just as in eq. (2), but for the local square-wellpotentials, 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 jM2 =2r,,,Jr", as a functionof the binding energy B. Our results are found to approach the kinematical limitas B approaches zero . The dynamical effects not included in eq. (2) are found tobe 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

Checn Wa Wong / Relativistic channel-coupling g

explicitly include dynamical effects if the potential is very

ilocal potentials .

deuteron),

where

E=v(k2$A 2)=j,$T

d$rKdr

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

where b = c =1 are used. Here jA is the reduced mass (

esults for local

are-well

te ti Is

erent r~

387

the usual

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

d-rK

dr

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

9.45926 e

(3)

for the

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

As a consequence, dG/dr has a discontinuty proportional to V�- S(, at r= b, andthe eigenvalue matching condition for a bound state with k`' _ -a2 and energyE. =

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.

d~- K(K-F1)$k,-ZI1Ve4fi+

1 d

(V-S)

d $KIG(r ®-® .

(5)dr` r` E-V$Iu,$S dr dr P,

The effective potential which appears is1

Veff=EV$ (S`'-V2)$S=Keff .

(6)2tL 2M,

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) . (8)dr

(IL 2-a 2 ) becomes

xcotx=-ab-E~g $

(1$ab), (9)~

x=96, q2=-2.u Veff -a2 . (10)

388

Ithe exs

(5) is ol.' course a little more complicated than an ordinary Schr® fingerequation because of the presence of the last term. However, this term disappearswhen So = K. The derivatives of G(r) then become continuous at r = b like aSchr6dinger w.f. even for finite ji. The upper component is thus similar to aSchr8dinge-.- wS, with a similar matching condition [eq. (9) with S", = VJ, the onlydifference 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 andthe 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 potentialI 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 canbe 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% .

euterra

trying tct e

to be shown here .otential obtained in ref.A Table I gives its potential parameters, E

aters, deute7on matter radius r. and its shape-independent approximant 14)

understand the physical contents of this model, we shall not examineressions for various interesting quantities because they are too cumber-

s start instead with the NR "comparison" square-well

CU11 ng / Relativistic channel-coupling effect

TABLE I

rt.0

+Waro ) 21 .2 2

0 1)

Deuteron properties from the one-body Dirac equation for several local square-well potentials

Potential

#1

#2

#3

#4

#5

#6

h (AM

1157

2.1V8

2.1678

2.1678

2.0247

2.4265So (Mev)

0.00

16.04

100

-100

100

-100VO (Mev)

33.067 a)

16.044

-66.416

130.181

-62.481

125.160

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

(jA/,uo )( VO+ SO) .

9bl InIr

araP

0.58355A2391.7856

-0.0405

0.58485A2391.7792

-0-0,403

15055.51081 .8982

-0.0479

0.63165.29561.5969

-0-0193

0.53995.42391 .7834

-0.0473

0.63905.42391 .7678

-0.0206FT 0.00 1.05 1 .08 1.01 1 .16 0.89P111 M) 37.50 3794 35.48 4139 3335 4177r, (fm) 1 .9525 1.9612 1 .9952 1.9124 1 .9602 1 .9653r,,,o (fm) 1.9586 1.9582 1 9953 1.9042 1 .9584 1 .9577r~ (fm2 ) 3.8122 148552 3.9913 3.6642 3.8547 3.8665j (02 ) - 2.9968 3.0217 2.9645 2.7821 3.4007-4r,n/rm M) - 0.57-IFrml rrn (°I®) - -0.11 -0.12 -0.10 -0.14 -0.06-3rn/rm (°`®) - 0.45 2.19 -2.05 0.39 0.66(Ar ../ (%) - 0.46 0.40 0.70

C'hun Wa

ong/ Relativistic channel-coupling eAccording 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 aenergy-independent potential in the following way:

-_ ®arm-~-

Frm

rm rm .

-la Vef

o V®PG_ ,aT d

rm (01YA i arm i

,

rmaT

d rmaPabn

2r_ /

M

That is, there are contributions from both the energy dependence in the

-channeland from the explicit appearance of the F-channel. Now eq. (6) gives

(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,

(14)rm Pabn

we obtain 0.57% for the first, or energy-dependent, term of eq. (12).

e secondterm can be calculated from the entries of table 1, and gives -0.11% for the changein rm when 1 .05% of the deuteron appears in the lower component with its somewhatsmaller 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

RSchr6dinger equation, eq. (13)does not give the abnormal-channel probability Pabn because the relativistic propa-gator 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 varyingSo ( 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 para-meters of two potentials (#3 and #4) along this curve are given in table 1 to allowthe 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 atr = 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) negativethan -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 .

Fig 1 . Dependence of the deuteron matter radius r,,, on the strength -S,, of the scalar potential forpotentials with constant square-well radius h 8 dashed curve) or constant triplet scattering length a (solidcurve) . 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 aconstant, but also increases manotonicAy as & decreases, as shown in table Low we know from eq. 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 parametersare varied to refit a, r,,, would have remained roughly the same. The exact valuescalculated 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) aregiven in table I as potentials #5 and #6.

he approximate phase equivalence between the relativistic potentials #2, #5and #6 and the 1' potential A# I can be improved by correcting the: Muesli cttfferrrwein 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 attractivestrength of the scalar potential . Now S,, is known to be about 200-400 MeV for thenucleon-nucleus interaction at intermediate energies 17

), but it should be weaker inthe deuteron which has a much smaller scalar density. Fortunately, knowledge ofthe exact value of S. is not needed because fig . I shows that rn, is essentiallyindependent of 56 for values stronger than 100 MeV (potential #5) . (There is a veryshallow minimum at S. == 150 MeV.) Table I shows that for potential # 5, r," hasincreased 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.

M

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

05

1- cost b4

~01,41Irmo

NP440

10",

cost 0

11525 fam

- O00

-150

900

0

100- so (Mejv)

3. esults for the seoarable NoThe 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 tut 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 scatterilength . This turns out to be very similar to the square-well potential # I except thatr,n is now smaller by 0.83% . (This change alone would have solved the problem ofdiscrepancy with experiment but for the fact that a separable potential like the Nvpotential 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

Chun Wa Wong / elativisfic channel-roupfing~ffea

391

_ (aV.11) =A .,,21 .

cl T

d

4.u :!

TABLE 2

jk potentials

(15)

in the notation of ref. 16) . When used with eq. (14) (for the square-well potential),it gives the result -iErjr,,, shown in table 2 . The last line Ar ..I r .. of table 2 is theexact result, from which we deduced that (arjaPjt,,j1r,, for the NvD potentials isin 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 lowercomponent probability P[_ is larger in the NvD potentials leading to larger values

Demeron properties from the one-body Dirac equation for several nonlocal separable Nogami-vanD-jk ") potentials

Potential #7 #8 #9 *10 #11

S - -5.0 _1 .0 1 .0 5.0/3 !NOV) 274.42 288.40 2'1 ...̂J '.64.06 252.30A, (106 MeV3 2.8494 3.4428 2.7849 2.5512 2.1930X, 1 N1eV, " 12) 0.07166 0.06548 0.07152 0.07413 0.07876a (Q) 5.4239 5 .4239 5.4239 5.4239 5.4239ri , (fm) 1 .7759 1.7705 1 .7690 1 .7680 1 .7658P -0.0244 -0.0270 -0.0233 -0.0217 -0.0188F; ( °/®1 0.00 1 .53 1 .40 1 .35 1 .26r,,, 1 fm1 1 .9363 1 .9429 1 .9432 1 .9435 1 .9444rMjj (fM) 1 .9582 1 .9621 1 .9619 1 .9617 1 .9615r,', (fm-) 3.7492 3.8012 3.7989 3.7984 3.7985rF2 (fm2 ) - 2.0273 2.1621 2.2211 2.3277-1u rm/ r n (%) - 0.74 0.71 0.70 0.68IFr,,,/ rm M) - -0.36 -0.30 -0.28 -0.24.arm/r°,M - 0.34 0.36 0.37 0.42(Jrm/rm ),e (%) - 0.36 0.37 0.39 0.44

3

th .ALr,/r,, and au r.lr, in addition,

is proportionally larger becauser2 is smadler. 71The bottom line is that the total relativistic correction Jr,,Jr. or

is only a little smaller than that in the local square-well potentials oftable 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)%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 ofa minor programming error. This error has been corrected in the results of ref.' s)(van Dijk, private communications) . Hence the difference between our results andthose of van Dilk must have come from the different procedures used for enforcingbase equivalence between the relativistic cases and their NR limits. The large

differences involved in some of the cases must be considered surprising and notunderstood at the present time. However our results are supported by the fact, tobe shown explicitly in sea. 5, that they agree with the kinematical result of eq. (2)in the limit of zero binding.

f

r

ArMon

Chun Wa Wang / Relativiwir channel-coupfing effect

0_3410(MO )2

I

n the nucleon

to

I aq~ +(: A40)2 IP,

;_)

_IQA12pi)Q 0A4

2AX

r

a P~h.

M

2r2M

( r'j-r)(M2PF)r,

0M

m2

rm c)P.h ,,

2r 2

11

0M

he relativistic effect on r� , studied here has an interesting dependence on thenucleon mass M = 2iu. For local square-well potentials, we note that since the resultis 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

0.57%( o) -0.11%

A4

2

(16)Imp

where we have used the fact that Pu is proportional to Al -2, V, r2 i and r2 are allC

Uinsensitive 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

(17)

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

1 .

x 1 4

e

2 forthe

potential

Ian

1.48

1.3)

1

4

e

2 fort

relativisticpotential

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 2e percentage appearing in

of

e s. (16) a

(17) should

e slightly larger ifas s

i

the bottom liesestimate of 0.449/o (M,,IM)2 given by

e .e mathematically permissible range of relativistic canes i

r. is

coursemuch larger than this.

e can see from either eq. (12)

ere or the coupled-channelunitary transformation defined

yeq. (19) of ref. "') that r,,, cat be reduced by eitherdecreasing the lower-state probability or making the lower component less eaten ein space. It is possible however that physically interesting cases might not

e vedifferent from those discussed here.Theclose conceptual and structural similarities between the description o eq. (12)

based on the idea of effective energy-dependent potential for a bound state an themore familiar one based on the unitary Foldy-Wouthuysen transformation suggestthat these are both valid approaches to the problem of relativistic correction.FW transformation is the more elegant method which recognizes the fact that thelower component might be present in the Dirac spinor even in the absence of apotential. The present approach emphasizes instead the universal features of channelcoupling which are independent of the detailed mechanisms involved, and inaddition it also takes into account dynamical erects not included in eq. (2).

Chun

a Wong/

phase equivalence is

ore strictly enforcetales. These results a ree with the existi

elativistdc° channel-coupling e eco

393

5. yna ica effects

e

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 localsquare-well potentials . The comparison is always made to the corresponding NRresult of the Schrodinger equation with a calculated with NR kinematics . We seethat 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 toexamine ®r2 = 2rm®rm also . Fig. 2 gives re,= Are� as functions of the binding energyB and a potential parameter. For the local square-well potential, the latter is takento be the scalar well depth S,,, while for the nonlocal NvD-potential, it is theparameter s or ref. '6) which is proportional to the ratio of the strength of the s-waveterm to that of thep-wave term in the separable potential . (Other potential parametersare determined by fitting B and the scattering length a. For example, as the scalarpotential becomes increasingly attractive with increasing S(,, the vector potentialmust become more repulsive in order to fit the same B.) e see that as decreasesto zei o, the calculated relativistic correction re, approaches the kinematical limitcf 0.033 fm2 given by eq. (2), and denoted by the horizontal arrows in fig. 2.

394

Chun lVa lVong / Relativistic channel-raupling gffeci

Dependence ofdeuteron properties from the one-body Dirac equation on the binding energy andparameters for local square-well potentials.

Potential

to 12

-13

#14

8 1 NICV1

(Q

ZVOhnemmw% NR

NRh q fin)

2A2,7

2.1958

NOS

jW7

=02S" 1 Nlev ~

0

U

IN

0

0t"' I MeV I

2N76

23142

7LIM

0. 157

51296

11 1

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

n the other hand, there is a clear dependence on dynamics, i.e. on the detailsof 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 thescattering length away from its experimental value, even when the binding energyis kept constant . For example, for S,, = 100 MeV and B = ,,p� the relativistic

0.06~

002

TAM+ 3

0 50 100

8 MY

002

Beip

20

01 .11

# 16

#17

#18

M21 1,671lilt)-21505

Local square well potential

Nonlocal NvD potential

I

I

I

i

5 10

Scalar well depth So (MeV)

Dimensionless parameter s

2-111534

tential

ODE

0.04

0.033

0.02

Fig . 2 . The total relativistic correction r

for different potential parameters and different binding energiesfor 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.

a t1 nin I An)PP, i% 1P_ 1 1.1

1147702,0071_60520.00

1 .1 .55

15A7702.(K)44

_0019031314,13

1547702.0051

-0J)4250.3612 .09

2S87411 .1919

-0.0466OMO72.40

2,587913702

-0.03772.50

74.19

158791.4

-0.05363.14

67.16

158791 .4137

-0.0-5973.75

60.37r An 1 5.4846 5.4877 5,4876 41 .9871 1 .0133 0.9977 0.9911r 4 CM 5AN52 5A852 5A852 1 .0221 1 .0163 1 .0208 1.02280 1 An' 30AN! 3MI87 W194 01743 1 .0149 0.9853 0.9786r I An') S.>738 ?Am - 1 .4940 1 .3083 1.18-52-1, F11 " r", i"') - _Wll _WV - (158 R51 (139J", % I OM6 0.05 2.66 1.08 M

%) 0.1-K) 0.05 1.01 0.98 0.33

where

Chun Wa

an

correction r2 l is 0.034 fm2 when a = 4.83 fm (b = 1 .0 fm for the NR potential),rne

-OA07 fm2 when a = 6.45 fm (b =4.0 fm for the NR 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 localsquare-well poten!ial, and 0.027-0-040 fm :! for the nonlocal NvD potential

AN ELEMENTARY DERIVATION OF EQ. (2)

(r2) = I iPYO d3 r =

/ Relativi.-vic ch

nelusions

pendix

S1=--PU-P2Am

01*(e

nnel-coupfing effect

393

The relativistic effect on the deuteron matter radius r. studied here arises frthe coupling to the lower component of the Dirac equation . As such, it is genericallythe same as the traditional one given in eq. (2) -1.4 .12) . Indeed, we have showexplicitly that our effect agrees with the traditional result given by eq.(2) in the limitof zero binding. However, when the binding is not small, or when the scalar avector components of the potential are not both weak, our effect is also sensitiveto dynamics. Although the traditional relativistic trinematical effect on the deuteronmatter radius is only about 0.4% of r,,,, i.e . only a fraction of the discrepancy of1% between theory and experiment for rm , it might be necessary to go beyond itand include relativistic dynamical erects when discussing the deuteron matter radiuscalculated for realistic potentials. Many of the past calculations of deuteron formfactors for realistic potentials (for example, ref. "')) have already included suchdynamical effects. But the actual values cannot be read accurately from publishedfigures. We would therefore like to appeal to all groups to report their calculatedvalues of relativistic effects in the deuteron radius .

e -is, )00 d-; r

The Foldy-Wouthuysen transformation ") in the presence of a potential cannotbe constructed exactly. It can however be obtained in a NR expansion in powersof 11M, where M is the nucleon mass. If the original wave function is 0, thetransformed one is

41 , = . . . eis, eis241,

(ALI)

(A.2)

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

r2+'

AW

I

Cr-rXpWd3r .

(A.3)4 2 2M`

396

Curs Ima Wong / Relativistic channel-coupling effect

e(2) comes from the second term of (A.3). As is well known '"') dynamical

cots are contained in the higher-order transformations S,2 , - . . . They can readilye extracted from eq. (A.3) by continuing the calculation to higher orders.

I ) VZ Jankus, Phys . Rev. 102 (1956) 15862) M . Gourdin . Nuovo Giro, 28 (1963) 533 ; 32 (1964) 4933) B .M . Casper and F. Gross, Phys . Rev . 155 (1967) 16074) U. Friar, Ann, of Phys. 81 (1973) 3325) L. Koester, W. Mailer and W. Waschkowski, Phys . Rev. Lett . 36 (1976) 10216) S. Klarsfeld et aL, Nucl. Phys . A456 (1986) 3737) M. Kohno, I of Phys . G9 (1983) L858) S . Klarsfeld, I Martorell and D.W.L. Sprung, J. of Phys . GIO (1984) 1659) R . Machleidt, K. Holindt and Ch. Aster, Phys . Reports 149 (1987) 110) C .W. Wone, Nucl . Phys . A536 (1992) 26911) L.L. Foldy and S.A. Wouthuysen, Phys . Rev. 78 (1950) 29;

J.D. Bjorken and 5.13 . Drell, Relativistic quantum mechanics (McGraw-Hill, New York, 1965),sect . 4.3

12) F . Coaster and A. Ostabee, Phys. Rev . C11 (1975) 183613) F. M. Toyama et off, Phys. Rev. C44 (1991) 6714) R.K . Bhadui et al., Phys . Rev. C42 (1990) 1867 ;

D.W.L Sprung, Hua Wu and J . Martorell, Phys . Rev . C42 (1990) 863 ;M.W . Kermode et al., Phys . Rev . C43 (1991) 416

15) W. van Dijk, -Relativistic effect on the radius of a model deuteron", Phys . Script ., to by published16) Y. Nogami and W. van Dijk, Phys . Rev. C34 (1986) 185517) J.A . McNeil, J.R . Shepard and S.J . Wallace, Phys. Rev. Lett. 50 (1983) 143918) R.G. Arnold, C.E. Carlson and F. Gross, Phys. Rev. C21 (1980) 1426 ;

P.L. Chung et al., Phys . Rev. C37 (1988) 2E . Hummel and J.A. Tjon, Phys. Rev. C42 (1990) 423

oferencofte