Meson-current corrections to the magnetic moment of the deuteron within an extended Lee model

Nuclear 0

Physics

North-Holland

A399 (1983) 275-286 Publishing

Company

MESON-CURRENT CORRECTIONS OF THE DEUTERON WITHIN

TO THE MAGNETIC MOMENT AN EXTENDED LEE MODEL

P. ERBS and D. SCHtiTTE Inslitut fir Theoretisrhe Kernphysik der Universitiii Bonn, D-5300 Bonn, West Germany Received

26 October

1982

Abstract: The magnetic moment of the deuteron is calculated within an extended Lee model which allows the use of realistic deuteron wave functions. The meson-current corrections which are due to the dynamical effects of the n--meson are computed rigorously and analysed in detail. Conclusions are drawn about the validity of perturbation theory and about the importance of norm and off-shell renormalization corrections.

1. Introduction The observation of meson-current corrections (m.c.c.) to e.m. properties is considered to be one of the most direct evidences for the “presence” of mesons in nuclei [see e.g. ref. ‘)I. The quantitative calculation of these effects, however, have involves the to rely on perturbative arguments ’ -3) since a rigorous treatment unknown field theoretical wave functions of a nucleus (including mesonic degrees of freedom). In order to have some information on the validity of this approach it should be, therefore, interesting to know at least in an example the value of these m.c.c. within a non-trivial field-theoretical model which allows an exact treatment of these effects. A model of this kind is given by the Lee model 4,5). If one identifies the elementary particles (V, N, 0) of this model with neutron, proton and TI-, a rigorous determination of the m.c.c. to e.m. nuclear properties due to ~--exchange becomes Possible. It is the purpose of this paper to present the results of calculations of the m.c.c. to the magnetic moment of the deuteron within this frame. The following questions can be discussed making use of these results: (i) The validity of the perturbative treatment of m.c.c. with the help of (renormalized) OBE diagrams ‘v3) can be checked. Especially, the dependence of the results on the wave function renormalization constant Z (i.e. on the relative magnitude of the neutron-pion cloud determined by the nN vertex form factor) can be investigated. (ii) The magnitude and importance of off-shell contributions of self-energy diagrams can be estimated. Such corrections have been neglected in the literature up to now. 275 May 1983

276

P. Erbs, D. Schiitte 1 Meson-current corrections

(iii) The question

of the cancellation

of the nucleon-recoil-exchange

norm corrections “) can be investigated. (iv) The validity of the static approximation

for the evaluation

term with

of the m.c.c. can

be checked. We have organized our paper as follows. In sect. 2 we shall define our tieldtheoretical model. We shall use an extension of the standard (non-static) Lee model allowing the introduction of a realistic deuteron wave function. We shall work within the Galilei-invariant formulation of the model(j) in order to guarantee a consistent relativity behaviour of nuclear properties. The coupling to the e.m. field (fixed by gauge invariance and the correct description of one-nucleon data) will be specified in sect. 3. In sect. 4 we will elucidate the formal structure of the resulting m.c.c. to the magnetic moment (m.m.) of the deuteron. Using different assumptions for the deuteron wave function, the numerical results for the m.c.c. will be presented and discussed in sect. 5.

2. The Lee model 2.1. DEFINITION

OF THE MODEL

We shall use for the investigation of the m.c.c. a Lee model with specilication. We introduce fermion operators V’, Ng(a = (p,. m,), for the (bare) neutron and proton, respectively, and a boson operator the rc- @,m denoting momentum and spin). The model hamiltonian

the following /l = (pp, ma) a,(k = pk) for then reads

H = Hg+W+&

where H;

= CE,” b;+ v, + &Np+

N, +&&z,,

W = c W$ V,’ Npa, + h.c.

are the standard ingredients [see ref. “)I, and neutron interaction of the structure d = xi&B, Also with the inclusion

of the potential

(equivalent

to baryon

number

0, H commutes

2.2. GALILEI

In order

by

proton-

with the operators

Qz =CC+I/,+C4+a,

and charge)

defined

0 is an additional

v,+N; N,. v,,.

Q1 =C":v,+CNs'Np 4 the eigenvalue problem operators Qi, Q2.

where

the

leading

to the usual

eigenvalues

(ql,

sector structure of symmetry

q2) of the

INVARIANCE

to have an unambiguous

description

of nuclear

properties

(especially

P. E&s, D. Schiitte

/ .~eson-current corrections

277

the m.m.) we shall use the Galilei-invariant version 6, of the Lee model throughout this paper. Only within this formulation can relativity (in the sense of nonrelativistic quantum mechanics) be incorporated leading to a consistent transformation behaviour of any nuclear quantity when changing the frame of references by a boost. This restricts the functions occuring in the hamiltonian H leading to 6, Ez = pt j2m, + A’, in, = mp+m,,

E, = p,$m,,

ok = p,$mx9 W$k =.f%f)(m@b’

dmp>s3(Pa-Pp-Pk)3

mnq

=

mppk -‘%Pp

(2.1)

The two-body potential 8 has to be Galilei invariant in the standard sense of non-relativistic theories. We take mP = 940 MeV, m, = 140 MeV; the quantities A0 and fo(qj are specified in such a way that the interaction W generates a realistic (relativistic) one-pion-exchange potential (see subsect. 2.5) so that we expect to be able to compute reliable predictions of the n--part of the m.c.c. of e.m. properties. 2.3. THE NEUTRON

The introduction of the potential fi does not change the properties of the proton, the pion (remaining trivial) and the neutron. Thus for $,, E,, the’eigenstate and the energy of the neutron, the same formulas hold as for 0 = 0 [see e.g. refs. “*‘)I leading in the Galilei invariant case to ‘)

A=A”s

rl/, = II/,’+*,‘>

(2.2)

where St, $2 represent one-particle (bare neutron) and two-particle states given in momentum space by am

= ~3(p,-p1)LIfna)r G

$3~~~ PA = 63(~,-~2 m,e =

g(q) = z =

I+

(proton-pion)

-p&h)(a.

4)im,h

mpP4-mnP2r

-_f%&/%?2/2~A,I.,

s

~{q)12qz/(q2/2~

-

B2d3q.

(2.3

278

P. Erbs, D. Schiitte / Meson-current

We use here and in the following proton and 4 for the pion. of

We assume the pion

the indices

A < 0. Note that g(q)(a. cloud of the neutron.

corrections

1 for the neutron,

2 and

3 for the

q)[m,) describes the relative wave function The normalization of rb, is such that

The relative strength of the meson cloud, i.e. the norm of the ($,I$,,> = d3(P, -P:,P,,,,. two-particle part of $, relative to that of the one-particle part, is given by l-l/Z.

2.4. THE DEUTERON

of the q1 = 2, q2 = 1 sector have the structure

The eigenstates

where the coefficients c+ dPafkhave to be determined by the eigenvalue equation H$, = E,$,. It has been shown in ref. ‘) - and the same derivation holds for the case when the additional potential 0 is taken into account - that the wave function $, can be written in the form (2.4) where the operator

q(E) is given by ‘)

cp(E)1/,+N;IO) = [Z/r2(E - E, - EJ] ~‘Ns’lO),

s

[(&-~y(y-A+

4p is the solution

of a standard two-body

rm2(y)= lThe wave function

5

If0(q)12q2d3q/

(Ho + U(E,)+

UE,M,

= E,6,,

$)I.

(2.5)

equation (2.6)

where H, = 1 E,I/,+c+

C E,N;N,,

U(E) = (cp(E))-3~(cp(E))-f, and V(E) is a one-pion-exchange potential given by eq. (3.3) of ref. 5). Due to the Galilei invariance, E, is related to the binding energy E, of the deuteron through E, = (p2/2M)+ E,+ A (M = m, +m,), where p is the total momentum of the deuteron. The standard deuteron wave function 4, has the momentum space structure 4,(Pb

P2) = d3(P-P,

-P2)(hl(k12)+hZ(k12)S12(klZ))X,

h,, h, denote the S-, D-state wave functions, where Mk l2 = m,p, -m,,p2, . a2)k2 is the tensor operator and x is the S = 1 spin part. S,,(k) = 3(a. k)(a,. k)-(a, The Lee-model deuteron wave function tj, is according to eq. (2.4) a

P.

combination deuteron)

Erbs, D. Schiitte

of a two-particle

/ Meson-current

219

corrections

and a three-particle

state

(the meson

cloud

of the

which we write in the form

4%= *;+d%;923*;,,

(2.7)

where P,, is the exchange operator of the two protons. With the assumptions eq. (2.1). these functions obtain the explicit form in momentum space:

$;(PIA) = 63(P-~,

of

-p3)(~,(k,2)+~2(kl2)S12(k12))IX, Jz -

6(k) = h(k)r(E,-k2/2p,,), $;(P,,

1

= L+L

plz

m,

~3, ~4) = .f”(kz4)(az.

D(p,,p,,p,)

mP’

k24hfqP2

+P4r

P3)lD(PZ? P3? P4)?

1 1 &+A-;;k:,-$, I 2 P12

=

m&z4 = mnP2- m,p4, M9 = It is convenient

M(p2+p4)-m,p3.

to define the simplified lp; = $; = iI/;,

lb;,

state IF, = IF; + IF; related

to ibP by

but set r = 1

but set r = 1 and D = A-

&

kg4.

(2.9)

The normalization of ~,b,, is then chosen such that the state IF, obeys (IJ,\$,~) = d3(p -p’)(xx’). Note that $, has the structure $, = zF,SNl$S without the proton-proton exchange term, when 4P is given by @P = ~c?~~N; V,’ IO). The norm of $P is almost equal to that of $,, the difference is-quite analogous to the m.c.c. ~ given by offshe11 normalization of the hamiltonian exchange terms.

2.5. THE CHOICE

corrections and pion-exchange effects 3), see sect. 5. For our choice these corrections are 1.2 ” o, where the major part (0.8 ‘I:,)is due to the

OF THE PARAMETERS

In the static approximation 5), corresponding in our case to the assumption = 1, E = E, + E, in the expression for V(E), the quasi-potential V(E) has the form V(E) N f 2% If’(k)12bl~ kb,. k) k2-2m,A ’ Z m,/m,

where k = p1 -pz

is the momentum

transfer.

This suggests

the choice

A = -$m,

for the

binding

energy

of the

neutron

thus

guaranteeing

the

correct

Yukawa

280

character

P. Erbs, D. Schiitte / Meson-current corrections

of the OPE

potential.

Correspondingly,

we choose

in accordance

with

ref. ‘) f(k) =?$

= &*$&,

.$ = 14.4, It

(2.10)

P

so that the potential I/(E) is a realistic OPEP. The cut-off parameter A, usually of the order of n z 1500 MeV, will be varied in order to get an idea of the dependence of the results from the strength of the interaction W.f(k) and A are to be interpreted as renormalized vertex function and neutron energy ‘). respectively, which are fixed >‘by experiment” and which determine the “unrenormalized” quantitiesp and A0 occuring in the definition of the hamiltonian. Finally, the still arbitrary potential 0 is chosen such that 4p is a given (realistic) deuteron wave function. If 4,, is generated from a NN potential V,, fi may be calculated from eq. (2.6) setting VN = V+ U. Since 0 does not enter into the explicit form (2.8) of r,Gp,a more detailed specification of 0 is not needed.

3. The coupling to the e.m. field The postulate of gauge invariance gives an interaction of nucleons and pions with the e.m. field through minimal coupling.-Within our Lee model, this yields for a small, constant magnetic field in addition to the standard L. B terms a particle “seagull” interaction (symbolized by “sea”). Together with the non-conserving usual cr. B terms which we add in order to account for the anomalous m.m. of the nucleon in a phenomenological way, the interaction with a small, constant magnetic

field becomes

Hint =

then *)

j&

CM’ B),

P

M = L,+L,+g,(az+a,)+g,Oa,-

?L,+M,,,, n

Msea=

c s3(p,-Pp-Pk)~+CfO(q)(uxQ)+2

dk

x (q x Qk . q)lN&, + h.c. mnq

=

mppk

-

mnP&%

(3.1) Since we are interested in the effects of m.c.c. due to the dynamical effects of the x-, we disregard possible contributions to the m.m. operator from 8, i.e. we assume this potential to be local. As in standard (non-relativistic) theory, the m.m.

P. Erbs. D. Schiitre / Meson-current correcrions

281

operator contains two parameters gpr gz, which we choose such that the m.m. of the nucleons agree with experiment. This yields the usual result for the proton ~gp =

whereas

the neutron

m.m. obtains

9 sea

m.c.c. from the meson

of the seagull

=

22 (m,--m,) n

This fixes the “bare” and A to be given).

neutron

cloud

(l- $)[(I-~)i-~,9,+9rm]~

9” = ;.92+

where the contribution

2.79,

g-factor

term is given by

s (If(

+f’(&q2 %(qHd3q. dq2

gz by postulating

gn = - 1.91 (assumingf(q)

4. The magnetic moment of the deuteron Having given the the m.m. (eq. (3.1)), the deuteron as the spherical symmetric

wave function of the deuteron (eq. (2.8)) and the operator of it is now straightforward to evaluate the m.m. ,u,, = gde/2m, of expectation value of the operator M, with respect to a space wave packet cp gd

=

(4.1)

(~i”,b)/(~b),

where

=

(4.2)

(P2+&P3-p23(P3)

is decomposed according to eq. (2.7) into two- and three-particle states. cp is supposed to have a spin orientation coinciding with the z-axis. Of course, the value of gd is independent from the choice of-the function W(p). In order to interpret the result for gd, it is convenient to compare it with the standard expression for the m.m. of the deuteron

m, = gp+gn+ z -(.9p+.9”)

(

tpD >

(4.3)

Here, ‘p” = Jd,W(p)d3p is a two-particle wave packet constructed with the help of the standard wave function #I~. The “standard” operator MS’ corresponds to M of the with the modification gf = g,. L, = M,,, = 0, P, is the D-state probability

282

P. Erbs, D. Schiitte / Meson-current corrections

deuteron. An important structure is now given by the fact that ,gd reduces to .yi when $i and $i are replaced by IF; and $i, respectively (see eq. (2.9)), and when exchange terms are neglected, i.e. we have that 9: = [((P21Kl@2)

+<@31M:l@3)

JI(~21Mzl@3> +c.c.]l[(@21~2) +((P31@3)]3

+

s

Gk= IEW(i4d3p,

k = 2, 3.

(4.4)

The proof of the equality of eqs. (4.3) and (4.4) is given in ref. 8). As a consequence, the full expression for the m.c.c. to the m.m. of the deuteron, defined by dg = gd -.qi, is naturally split up into the following terms : 4

= A,+A,+A,,,+A..,,+&j+&,,,,

A, =

((cpkl~,l~k>-(~kl~:l~‘))/(~J~),

A,,, = fi A “OrIll= &3

%,a

K(P~IWIV~)-(~21~,l@3Md~)

k = 2, 3, +CJG

.d(((PI@)-(cpl~)Kcpl~)

=

(~3iM3P23b3>/(d&

=

4

((P21M3P23b3>/((PI~)

+c.c.

(4.5)

The diagrammatic structure of these different m.c.c. is displayed in fig. 1. The interpretation of these terms is the following: A,, A,, A,,, represent off-shell renormalization corrections to the m.m. originating from one-body operators. They vanish in the one-particle case due to the choice of the parameters g,O. gP occuring in the m.m. operator. The corrections are produced by the “dressing factor” r and ~ in the case of A, and A,,, involving the three-particle part of the deuteron wave function ~ by the change of the energy denominator D when $i is compared to $z operator contributions (eq. (2.9)). The norm correction A,,,, contains two-particle (~plP,~lq~~) as discussed in ref. 3), but also an off-shell one-body operator correction, since

,? one-pion-exchange contributions given in a static >3 and c,,, are the standard approximation by the expectation value of ‘p” (the standard deuteron wave function) with respect to a suitable (renormalized) two-body operator 3). Within our model, the different contributions to the m.c.c. of the m.m. of the deuteron can be computed without any approximation. The three-body integrals can be reduced to a three-dimensional integration making use of the symmetry properties of the problem. The details of the analytical expressions to be integrated are given in ref. *).

P. Erhs, D. Schiitte 1 Meson-current corrections

E3 = E, + E,

En=

283

El+ .

Fig. I. Diagrams corresponding to the different meson-current corrections of the magnetic moment of the deuteron, see eq. (4.5). --denotes a proton, === a neutron, and ~~- a pion line. The wave functions ‘p’, ‘p3. @J*.Cp’ are defined in eqs. (4.2), (2.8) and (2.9).

5. Results and discussion Numerical results for the m.c.c. to the m.m. of the deuteron are displayed in figs. 2-5. Fig. 2 shows the sum of all contributions to the m.c.c. dg for deuteron wave functions 4; generated from the Reid potential lo) and the Bonn potential OBEPS [ref. “)I. For the second case, this result is compared to the standard approximation

where E, is the pion-current-exchange diagram (lig. 1). We observe that this approximation is quite good, despite the fact that the individual additional terms, displayed in fig. 3, are of the same order of magnitude as Ag,,. Thus there is a cancellation of the nucleon recoil term cp (see fig. 1) with the sum of the norm and the off-shell renormalization corrections. This cancellation, however, seems to be rather fortuitous. If one assumes e.g. a pure S-wave character for the deuteron

284

P. Erbs, D. Schiitte / Meson-current CUT-OFF-MASS 40

770

1210

1750

_---

(ME'/1

2710

2210

corrections

3140

3640

4100

4510

___------

_----

1

~0.0~2 0.0

1.0

0.5 l-l/Z

Fig. 2. The meson-current corrections dg (see eq. (4.5)) for deuteron wave functions generated from the Reid ( - ) and OBEPS (- -) potentials. For OBEPS, also the standard approximation dg,, = A,,, + A, is displayed (- - - - -). The result for A.9 in the case of the S-wave part of the OBEPS deuteron wave function is denoted with ---. (In this case dg,, = 0).

/iI-.-. -.-. ---. -.--.--.-CUT-OFF-MASS

140

770

1210

1750

2710

2210

,,--

0.00

3140

3640

4100

,-i--

_---

___________----------_

1

______________------_

=zy=_‘,

.’

‘\ gy

;/

\

-0.02

0.0

__ _ --_

.’

\\

-0.01

4570

I

i 0.01

CMEVI

\

\

\ \

‘.

---_

---------__________

I 0.5

1 .o

l-l/Z

wave Fig. 3. Different contributions to the meson-current correction dg ior the OBEPS deuteron function (see eq. (4.5)): A, = -, A, = - -, A,,, = - -, A,,,, = -----, csS.= -.-.-, ~3= - - -.

P. Erbs, D. Schiitte

/ Meson-current

CUT-OFF-MASS 140

??O

1210

1750

2210

285

corrections

(MEVI

2710

3140

3640

4100

4570

I

01

--

T;__t

_._~_~_._ ----.

-

-

-.-.-.-.-.-

--

-

I--.

-\

---. ---+__

_\ \ -_ 01

____________--_-----

-

I

0.5

0.0

1 .o

l-l/L Fig. 4. The same as fig. 3 for the S-wave part of the OBEPS

CUT-OFF-MASS 140

170

1210

1750

2210

2710 I

deuteron

wave function.

(MEVI 3140

3640

4100

4570

0.01 -

r

o.oo-

Fig. 5. Comparison

-

-4

of the rigorous

evaluation

of some contributions

to dg to the static approximation

exact ; - - - = Ag,,, static. - - = E,,,, exact ; - - - - - = E,,,,,static. = Ag,, = E,+ &sear

:

286

P. Erbs, D. Schiitte 1 Meson-current

corrections

wave function, the distribution of the m.c.c. over the terms of eq. (4.5) is quite different (fig. 4) and the estimate dg _Ydg,, fails badly (fig. 2) since dg,, = 0 in this case. In no case do we observe a cancellation of the nucleon-recoil term and the norm correction. A most striking feature of the results is their relative insensitivity with respect to the strength of the nN coupling, parametrized by the cut-off mass n (eq. (2.10)) or the magnitude of the pion cloud given by 1 - l/Z. Even for Z -+ co, where the neutron becomes a pure ~-proton state, the m.c.c. remain stable and very small (5 1.5 %). This is because the e.m. coupling is - for any value of Z - chosen such that g,,, the “renormalized” value of the m.m. of the neutron, coincides with the experiment. This guarantees that eq. (4.4), i.e. the lowest order equality of the m.m. of the deuteron to the standard value, holds independently of Z. The additional contributions given by dg (eq. (4.5)) are then small higher order corrections in the sense of a cluster expansion of the expectation value of the operator M [see e.g. ref. ’ ’ )]. We have also tested the validity of the static approximation (consisting in setting m,/m, = 1, D = A + E, -q2/2m,) for the calculation of the exchange terms E, and E,,, (tig. 5). It is seen that especially the seagull term is overestimated by about 25 % which is due to the simplification for the energy denominator D. Concluding we may say that the standard procedure of evaluating m.c.c. perturbatively with the help of renormalized OBE diagrams seems to be very reliable. However, all contributions, especially also off-shell one-body terms should be taken into account since the validity of the approximation dg Y E,,, + E, seems to depend on a fortuitous cancellation of the other terms. Also a static evaluation of the diagrams may introduce errors of the order of 25 ‘7;.

References 1) See Mesons in nuclei, vol. II, ed. M. Rho and D. H. Wilkinson, 2) 3) 4) 5) 6) 7) 8) 9) IO) II)

11979) F. Villars, Phys. Rev. 72 (1947) 257 M. Chemtob and M. Rho, Nucl. Phys. Al63 (1971) I T. D. Lee, Phys. Rev. 95 (1954) 1329 D. Schiitte and J. da Providencia, Nucl. Phys. A338 (1980) 463 J. M. Levy-Leblond, Commun. Math. Phys. 4 (1967) 157 M. Sawicki and D. Schtitte, 2. Nat. 340 (1981) 1261 P. Erbs, diploma thesis (1982) Bonn, unpublished K. Hollinde, Phys. Reports 68 (1981) 3 R. V. Reid, Ann. of Phys. 50 (1968) 411 T. Mizutani and D. S. Koltun, Ann. of Phys. 109 (1977) 1

(North-Holland,

Amsterdam,