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A1 11 009901 16

i Danos, MIchael/Relativlstic many-body bo QC100 .U556 V147;SUPP1;1977 C.I NBS-PUBCO NBS MONOGRAPH 147

Supplement 1

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards.

elativistic Many-Body

Bound Systems:

ectromagnetic Properties
NATIONAL BUREAU OF STANDARDS

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" Located at Boulder, Colorado 80302.
Relativistic Many-Body Bound Systems:

Electromagnetic Properties .

Michael Danos

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

and

Vincent Gillet

Departement de Physique Nucleaire

Centre d'Etudes Nucleaires de Saelay

BP n°2-91190 Gif-sur-Yv$tte-France

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

U.C, NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued April 1977
Library of Congress Cataloging in Publication Data

Danos, Michael.

Relativistic many-body bound systems.

(NBS monograph ; 147, suppl. 1) Supplement to the authors' Relativistic many-body bound systems.

Supt. of Docs, no.; C 13.44: 147/Suppl. 1.

1. Field theory (Physics) 2. Quantum field theory. 3. Problem

of many bodies. 4. Particles (Nuclear physics) I. Gillet, Vincent,

joint author. II. Title III. Series: United States. National Bureau of Standards. Monograph ; 147, suppl. 1. QC100.U556 no. 147, suppl. 1 [QC793.3.F5] 602M [530.r44] 77-1469

National Bureau of Standards Monograph 147, Supplement 1

Nat. Bur. Stand. (U.S.), Monogr. 147, Suppl. 1, 52 pages (Apr. 1977)

CODEN: NBSMA6

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1977

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price $1.10

Stock No. 003-003-01764-0
CONTENTS

Page

Chapter I - ELECTROMAGNETIC INTERACTION OPERATORS 1

1.1 - Introduction 1

1.2 - Form of the interactions 1

1.3 - Vector dominance ^

Chapter II - ELASTIC ELECTRON SCATTERING FORM FACTORS 6

II. 1 - Spin 0 field 6

II. 1.1 - Scattering terms 6

II. 1.2 - Pair creation and pair annihilation terms 10

11. 2 - Spin 1/2 field 12

11. 3 - Spin 1 field 16

II. 3.1 - The current operators 16

II. 3. 2 - Form factor matrix elements 17

11. 4 - Center of mass and recoil corrections 21

Chapter III - QUADRUPOLE AND MAGNETIC STATIC MOMENTS 26

III. 1 - Quadrupole moments 26

III. 1.1- Spin 0 field 26

III. 1.2 - Spin 1/2 field 27

III. 1.3 - Spin 1 field 28

III. 2 - Magnetic moments 28

111. 2.1 - Spin 0 field 28

111. 2. 2 - Spin 1/2 field 28

111. 2. 3 - Spin 1 field 29

Chapter IV - VECTOR DOMINANCE 30

IV. 1 - Form factors 30

IV. 2 - Quadrupole moment 32

IV. 3 - Magnetic moment 35

Appendix - ADDENDUM AND ERRATA TO NBS MONOGRAPH 147 38

iii

ABSTRACT

The formulae for the calculation of the electron scattering form factors, and

of the static magnetic dipole and electric quadrupole moments, of relativistic

many-body bound systems are derived. The framework, given in NBS Monograph 147,

is relativistic quantum field theory in the Schrodinger picture ; the physical

particles, i.e. the solutions of the interacting fields, are given as linear com- binations of the solutions of the free fields, called theparton fields. The parton-photon interaction is taken as given by minimal coupling, p p - eA ; in

addition the contribution of the photon-vector meson vertex of the vector dominance model is derived.

Key words: Electrcxnagnetic properties; form factors; magnetic moments;

quadropole ironients; quantijm field theory; relativistic many body systems;

vector dominance.

V

I.l

CHAPTER I

ELECTROMAGNETIC INTERACTION OPERATORS

1.1 - INTRODUCTION

In a previous publication^'-^, NBS Monograph 147, the description of a nucleus has been formulated as a stationary problem of relativistic quantum field

theory. In that report, only the problem of the stationary state energies and wave functions has been considered.

In the present Monograph, we shall derive the formulae needed for computation of the electromagnetic properties of these states, i.e. the electron scattering form factors, the magnetic and quadrupole static and transition moments.

To this end we shall give the explicit form of the one-body matrix elements in

terms of the single particle discretized basis states used in NBS Monograph 147.

These matrix elements can then be used in a well-known fashion to calculate the

properties of the many-body solutions.

From now on, we shall refer to NBS Monograph 147 as I. In this report, we

shall use all the definitions and notations introduced in I. We shall refer to a formula from that book as for example 1(4.123) and to a page number as for example I p. 105. In particular, we refer to Chapter II for the phase conventions and

the angular momentum diagramatic coupling techniques and to Chapter III for the

definition of the metric. As in I, we shall treat successively the cases of spin

0, spin 1/2, and spin 1 fields. For the spin 1 field, we employ the formulation

of R. Hayward, ref . [2]

1.2 - FORM OF THE INTERACTIONS

As interaction with the electromagnetic field we consider only the minimal

coupling, i.e. the forms obtained by the replacement

- 1 -
1.2

p

-V p - eA

in the Lagrangian. In other words no anomalous moment is ascribed to the spin

1/2 and the spin 1 fields. However, we shall add explicitly the photon-vector

meson interaction of the vector dominance model. These two prescriptions are introduced since in the present framework on the one hand a certain part of the

masonic contributions to the magnetic moments is treated explicitly while on

the other hand the configuration spaces contemplated in I-Chapter V are truncated at rather low energies (about 1 GeV) . In particular, they do not contain any

baryon-antibaryon components which would presumably contribute the bulk of the

PY vertex.

Since we consider photon emission or absorption processes, we need only

terms linear in the photon field . Thus we can write in general

d-^r J A . (1.1) interaction

The currents J are of the form, for spin 0 fields

y

J = ie((^* 1^ T 4 - |— c^"" T *) , (1.2)

y dx Z dX z y y

for spin 1/2 fields

and for spin 1 fields [2]

J = ie ^ Y„ (1/2 + T /2)y^ , (1.3)

V- V- ^

J = ie(p Y Y/ T IT + ir Y/ Y t p) • (1^4)

y y 4z 4 yz

The electron scattering form factors, after factorizing out the photon propagator, are given by (see for example refs. [s] and

d\ J^(?) e"-"^^ . (1 .5)

The quadrupole moment is obtained by evaluating the quadrupole operator

q(?) = 2z^ - x^ - y^ , (1.6)

with the charge density = - i

Q = - i d\ J,(^) q(^) . (1.7)

This form follows from the application of Siegert's theorem, which we briefly

recall here. Let us note first that in the limit q 0 the electric field of the

magnetic multipolarities goes to zero faster than that of the electric and longitudinal multipolar ities. Furthermore, for real photons the longitudinal multipolarities are absent. Then, one can apply Siegert's theorem which is based on

- 2 -
1.3

the identity for the electric field

Ij^ = - V + (q^) fj^(qr) ,

where V^^^ denotes the scalar multipole

V^(r) = Y^m j^(qr) .

(1.8)

(1.9)

Thus for q

-> 0 and combining the proper time dependence in the real fields A

and J in order to obtain a real energy

j3 , . i d r J A = —

y y to J

3

->-> d r J.g

1

0) d-^r J.^V

1^ d r VJ.V d-^r p V =

o d r p^V ,

which yields the form (1.7) for the quadrupole part of V.

The magnetic moment operator is in general of the form, ref . [2]

^ = y (1 + a) o

where a is the spin operator of the field. Therefore we have for spin 0

for spin 1/2

and for spin 1

y = - ly.

y = - ly.

y = - ly.

d\ t ,

d\ 3,(2 + s) ,

d r J

(1.10)

(1.11)

(1.12)

(1.13)

Note that these expressions contain no anomalous part in agreement with the introductory remarks. Note also that the form (1.10) is valid for arbitrary spin

in the formulation of R. Hayward, ref. f2] . In the cases s = 0 and s = 1, eqs.

(1.11) and (1.12) thus yield a particularly simple result since the magnetic

moment is proportional to a conserved quantity, namely the total angular momentum of the particle. Finally, the use of the fourth component of the conserved

electromagnetic current in eqs. (1.11) to (1.13), as is well-known, arises from

a quasi-Siegert theorem. Namely from the relation

= rot % ,

we get the identity in the limit q

1 -> X = -jrx'S+qxr f (qr) , (1.14)

where B is a constant (independent of space coordinates) . Herewith we have for

the interaction

- 3 -
1.4

d r J. A =

,3 -> d r p. A^

= ^2in d r p. (r X B)

e2m

d^r d\ (1.15)

where the integral over Z means the expectation value of the orbital angular

momentum of the system. The replacement of the current J by the velocity v must

be carried out in a formulation of the Hamiltonian in which the spin-orbit cou- pling has been eliminated, e.g. for the spin 1/2 particle by going from the

first order to the second order Dirac equations, see ref.[5], eq. (12.11), or more generally for any spin, ref.[2], section 4. In these formulations the interaction of the spin with the magnetic field is already m the form s.B.

The expectation value of (1.15) is evaluated by integration over a nomalized

quantity which has the sign of the charge, i.e. the charge density

Po^^^ = - i J^(^).

1.3 - VECTOR DOMINANCE

In the vector dominance model, one assumes that the most important interaction of hadrons and photons arises by the process in which a photon is converted into a neutral vector meson which then interacts with the hadronic system

via the strong interactions. Consequently the direct interaction of the photons

with the hadronic current via the term J A is assumed to be negligible and is

y y frequently dropped. We shall not make here this assumption. We shall however

add a term to the interaction Lagrangian describing the photon-vector meson interaction and call it the vector dominance term.

The simplest gauge invariant Lorentz scalar which is linear in both the

photon and vector meson fields is

^int ^ d X F (x) 4> (x) yv^ ' yv

(1.16)

where

and

F (x) = 3 A (x) - 3 A (x) yv^^ yv vy^'

$ (x) = 3 (jj (x) - 3 0) (x)

\iv y V V y^ '

(1.17)

(1.18)

in terms of the photon vector potential A^(x) and the field of the u-meson

to^(x). A similar expression holds for the neutral p-meson. Since we shall use

the Lorentz gauge

- 4 -
1.5

(1.19)

for the vector mesons, eq. (1.16) can be simplified by integrating by parts

3 to = 0,

mt d x(A 3 $ - A 3 $ )

^ V y yv y V yv'

= - 2g d X A (3 3 to - 3 3 , to ) V y y V y V y

= - 2g m d X A^(x) u^(x) . (1.20)

In the first term we have used the Klein-Gordon equation, see 1(3.149), to

replace the d' Alembertian by the meson mass, while the second term vanishes

owing to the Lorentz condition (1.19).

Thus the vector dominance interaction for the to and the (neutral) p° fields

are of the form

VD

^. = - G

int yto d X A^(x) u^(x)

YP

d^x A^(x) p°(x) (1.21)

Note that the replacement of the d ' Alambertian by the mass in (1.20) is

correct since (1.21) will be used to evaluate matrix elements and our basis

functions for the vector mesons indeed obey the Klein-Gordon equation. Of

course, it is not implied that the Klein-Gordon equation comprises the complete

equations of motion. Also recall that (1.16) is only an ad hoc effective

interaction which is not contained in the original Lagrangian.

- 5 -
II.

CHAPTER II

ELASTIC ELECTRON SCATTERING FORM FACTOR

II. 1 - SPIN 0 FIELD

II . 1 . 1 - Scattering terms

In the case of pions, we must consider separately the photon absorption

terms, figure 2.1. a (which we shall call "scattering terms") and the pair cre- ation and annihilation terms, figures 2.1.b and 2. I.e. They differ by the form

of the isospin operator and the coupling adopted between the particle states and

the transferred multipoles.

/ 71+ IT \ IT /

/ \ /

/ \ /

/ S /

\

(a) fb)

'3\

^ I \

(c)

Fig. 2.1

We treat first the scattering terms figure 2.1. a. Only the charged pions

contribute, thus we introduce the charge operator

(2.1)

- 6 -
II.

with

^b] = - 16 (2.2) m mO

Polarization in ordinary space is added to polarization in isospin space

with an overall space and isospin amplitude denoted by c

^'

'-^ and c^^'-^ for the

space vector and space scalar parts respectively :

J = ie T2[ct'l](^,-(vW - (vW ^.-),D]^.)j[00]

^ (2.3)

J, = ie T[ct°'J(r^|^xW _ (|_,.-),[1]^.)][00] . (2.4)

4 4

These forms are constructed to express the fact that the current is an invariant, see chapter I of ref. [sj and chapter II of ref . [l] .

The complex fields t^'"*^ and ' are obtained in their discretized forms from

the real fields, expressions 1(3.24), by selecting the proper creation and annihilation parts. In the case of the scalar current one may introduce the ir' fields of 1(3.25) with the relations 1(3. 2a) and 1(3. 2b),

I— = - i tt'"^ and I— (^'"^ = - i tt' . (2.5)

dX, oX,

4 4

Finally in eq. (1 .5) we shall make use of the multipole expansion in tensorial form of the plane wave

e^^ = 4. I (i)^ £[qW fM][0] j^(qr) . (2.6)

a

In all the terms of the expressions (2.3) and (2.4), the isospin parts

yield the same contribution according to the recoupling diagram of figure 2.2,

namely

TS^aW ^D]][o] ^Ji] ,[i]j[oJ ^Ji] ,[iJj[oJ

= - [i|t|i][aW aW ,[i]][0]

. (2.7)

- 7 -
II.

/

1 I \/

1 —

1 0 X

-4 rxT

[1]

TO

Fig. 2.2

The first term of the vector current contribution, eq. (2.3), is calculated according to figure 2.3. The adopted coupling corresponds to the change of

the initial angular momentum unit, L = - ^> after application of

the gradient operator. The second term of eq. (2.3) yields the recoupling diagram of figure 2.4. The final result for the vector form factor for the spin 0

field is thus

le

F(q) = ^ 4Tr (-[Ihll])

^2^2

+L V V

aj^^j£,|L|£>,^j^,,^(q)

- (-)

Z +\+L V V

^ilh^^l^l^a TOO (q)

(2.8)

where we have defined

- 8 -
- 9 -
II.

Next we evaluate the scalar form factor due to the scalar current, eq. (2.4)

Owing to eq. (2.5), the two terms give equal contributions, the space part of

which is evaluated in figure 2.5 . Together with the isospin contribution of eq.

(2.7) we get

F,(q) = - ie 4Tr ^ (i) (-[i|t1iJ)[^J«.2I^J

^2^2 (2.10)

2,

r dr g. (r) j.(qr) h, , (r) [A A 2

J qL JJL J.

n i

—h

1^

Fig. 2.5

II. 1.2 - Pair creation and pair annihilation terms

In the case of pair production or annihilation, figures 2.1.b and 2.1.C, we

must supply in isospin space the operator

wi th

(2.11)

[xT]

since only charged pions contribute. Thus, with polarization amplitudes c

which include both ordinary space and isospin space, the invariant form of the

current for the pair diagram is (we write V^^*^-^ to indicate that it is a unit

- 10 -
II.

scalar in isospin space ; conversely for

->- J = ie

+ same term with (f)'^ in place of (() ' , (2.12)

J, = ie

4

(2.13)

+ same term with (j)'^ in place of (J)' :

We consider first the pair creation term of figure 2.1.b.

The isospin part yields according to figure 2.6 the following contribution,

both for the scalar and vector form factors

1 1 T

1 1 T

1 1 0. [llx|,]2 [aW lb] eW]M . (2.14)

' /[iMi]

Fig. 2.6

- 11 -
II.

We turn to the space part. We consider first the vector form factor. The

recoupling geometry is the same as in figures 2.3 and 2.4, except for a phase

^2

(-) . The final expression is thus

F(q) = — 4-n- ) ) (i) ^ ' (1 1 1 T

VjJlj T ILl 1 0.

v,v

(-)

1 2 aLj^^[^llLUJs^5,\:„ (q)

£ +1+L V V - (-) Vi^zl^l^l^"'/"'

(2.15)

Likewise for the scalar form factor, see figure 2.5 together with figure

2.6

r V ^2 ^l"^^ 1 F, (q) = - ie 4Tr [ L

(i) ' (i

^ VjJlj T

1

1 1 T

1 1 T

.1 1 0.

^2^2 (2.16)

^dr g, „ (r) j/qr) h, , (r) [A

^ri ^2^2

[^0 -CS^^ Jot] .M^[oo]

For the pair annihilation terms these expressions hold upon replacement of

the creation operators A by the annihilation operators A and upon multiplication

by the phase (-1)

II. 2 - SPIN 1/2 FIELD

Owing to the limitation of our configuration space, see Chapter VI ref . [ij

we here have to consider only the scattering terms. Furthermore, since we assume

the neutron to have no electromagnetic interactions we introduce the charge pro-

- 12 -
II.

jection operator

with, as in eq. (2.2)

= 1 (, .T[cW .WjM). (2.17) ^ 2

= - i 6 ^ . (2.18)

m mO '

2

Introducing again polarization amplitudes c^'*'^-^ in ordinary (index x) and

isospin (index y) spaces the currents are

.[-^ JIO] ^[10] ^ ,^^^[n] ^[10] ^[01] ^j[00j ^ (2.19)

= i|

Ojr

JOO] ^ , r ^ ct°'] T^°^^ ^] . (2.20)

The isospin part is treated in figure 2.7 and yields

-[l/2]j[0] 1 , ^^JlJ ,[l]j[0]^(l) ^,[1/2] Jl/2]j[0]

= |Y (-)y[l/2lxWll/2j[Bt^/2] Jl/2] Jyjj[0] ^ (^.21)

2 ^=0,1

with

cl^°^ = 1 . (2.22)

[l/2] 1/2 .0

Jl/2] 1/2

1/2.

n[l/2j _LZ2.

1/2

1/2

1/2 1

1/2

1/2 ,0

1/2

1/2'

1/2

Figure 2.7

12.

1/2

[i/21tWIj/2J

- 13 -
II.

We shall evaluate first the vector current contribution to the form factor.

The spinor field with its large and small components

s-

' (2.23)

is given in its discretized form in 1(3.98). Expressing the Y matrices in terms

of the Pauli matrices (J, the expression (2.19) for the vector current becomes

3 = i£ [ T 5^(-)y[c^'y](L" M T^°y] s . at'o] M D] M

y=l,0

(2.24)

The vector part of the form factor has thus a geometry given by figure 2.8

together with figure 2.7 for the isospin

F(q) = - ie 2tt i y (-)y[l/2lTM|,/2j[i/2|ar'l|l/2]

y=l,0

'< I I (-)

'1/2

1/2 h

1 L .

1/2 X, jj

1/2 ^2 h

L 1 Z L .

,(q)

^2^1 ^

]

(2.25)

where

^1^2

(q) = r dr j^(qr) ^^H^X^^""^ (2.26)

- 14 -
II. 10

[jl] Ji

1/2

[j2] V

Fig. 2.8

The scalar current (2.20) is in terms of the large and small components

eq. (2.23),

-^4 2 y=0,l

(2.27)

The evaluation of the scalar form factor for the space geometry is given in

figure 2.9 and the final result together with the isospin part (2.21) is

F^(q) = ie 27r I 1 5; (-)y [l /2 1 x ^ | 1 /2j 1?2

( y=1.0

I (i)

v,£,j,(X,)

1/2 jj

1/2 ^2 h

L 0 5, £ .

V2j2(^2)

'1/2

jj

1/2 22

L 0 £ £ -

[X,|X2I£]U

^1^2

^1^1^2^2^

(q))[B.^^'^^^ ^^^^'^^

v,£, ^2^2

[oyj^r^olj [oojj

(2.28)^

with the definitions

(a) = r dr u (r) u (r) j (qr) . 11 ^2 2 (2.29)

- 15 -
11.11

V V

U ( ) ^1^1 ^2^2^

B

Fig. 2.9

II. 3 - SPIN 1 FIELD

II. 3.1 - The current operators

The invariant form of the spin 1 current operator (1.4) is obtained by

introducing explicitly the y matrices for spin 1, 1(3.118), and the p' fields of

1(3.131), 1(3.132) and 1(3.133). These fields are already broken into their scalar and vector components. Thus, noting that in the Lorentz gauge, see 1(3.134)

IT- = = 0 ,

we get for the vector current

^2(-.^^[llJ ^[10] JOIJJCOO]

,^i,[n] ^[10] Joi]j[oo]

J = ie

- le T2(p.[,[n] ^[10] ^[oi]j[oo]

.-.-[JnJ ^[10] ^[oi]j[oo] p..) ^ (2.31)

where the charge projection operator of eq. (2.1) has been introduced and where

c^''-^ denotes both space and isospin polarization amplitudes.

- 16 -
II. 12

Likewise the scalar current is given by

• Tz-^.r [oi] roili[oo] ^, ^ ^ +r [oi] [oili [oo] -^,+.

J^ = iel(p'Lc'- -'t'- -JJ"- -"ir'+Tr' [c^ -'x'- -'J'-.-'p') .

(2.32)

II. 3. 2 - Form factor matrix elements

We have to consider only scattering terms of the type shown in figures

2.1. a owing to the limitation of our configuration space to at most one spin 1 vector meson, see Chapter VI ref . [l] .

The evaluation of the isospin matrix elements in (2.31) and (2.32) is

identical to the spin 0 case and is given in eq. (2.7), figure 2.2.

For the vector part of the current and from the explicit form of the y matrices, one sees that after acting on the fields, i.e. in the step from the first

to the second form of equation (2 . 31) , the remaining parts of y^-'^-^ are simply unit

vectors e^"^-^. We therefore will have to evaluate terms of the form

(2.33)

according to figure 2.10, where the symmetry properties of the invariant triple

product have been used. We also have introduced the space polarization amplitudes

dW.

Fig. 2.10
11.13

When evaluating the several contributions to the vector current (2.31)

from the multipolarities k =M, S and 5? we note that for the field only the

longitudinal multipole 5? exists. Furthermore, we note that the second term in

eq. (2.31) is the hermitian conjugate of the first one. We therefore calculate

explicitly its contribution only to the vector form factor, the space geometry

of which is given in figure 2.11 and yields

J,W2 KV 'J q' (2.34)

J2

Z2

J?

J^l^

K

q

p[j,i£2r^]

Fig. 2.11

The complete vector form factor with the isospin geometry of figure 2.2

thus is, denoting again by c^'^-^ both the space and isospin amplitudes

- 18 -
11.14

f(q) = ie 4Tr(- [iItWii]) [ (i)"^

J^-J,-l+«.

^1^

(J2+I) 12^2

T T T +1 P>^'' l-'"<P>, L'^CPM J q J

—-± W

'

(J2-1) "r2-2

^^2 .-[^,0 ^[,i]JV' 0 r,o]^oo]

+ h> c • •

(2.35)

with the definitions

^ (q) (2.36)

12 2

where the radial functions U , 9C and Z are given in eqs. 1(3.236), 1(3.233) and

1(3.235) respectively. As an example of the hermitian conjugate contributions, we

obtain for the second term of (2.35) (recall eq. 1(2.28)),

- le 4Tr(-H [llrt'^Il]) K-i)

2 J,-J„-l+£ J

(Jj+1)

(2.38)

J2J,Jj + U"

- 19 -
II. 15

We now turn to the scalar current, eq. (2.32). Here the two terms give

the same contribution and we have

J,(q) = 2ie T^'[ct°^] ,[01]] [00] ^. ^ (2.39)

The scalar form factor thus is, see figure 2.12

F^(q) = 2ie (- [i|t['^|i]) 4tt I

(i)

^

^1^1 J1J2

^2-^2

^2^2

1 Jj

1 X2 ^2

.0 £ I -

2 "l^l ^2''2

X [I^'''^ A^'^'^ c[°'] q[^°]]rOOl

, (2.40)

with (see eqs. 1(3.232) through 1(3.236))

(2.41)

4x<^> = ^XJ-1 i^<jJ-l(^) - ^Xj+1 iv5^'lW^jj^j(r) . (2.43)

and

- ^XJ i ^^vjj(^> ' (2.44)

4x(-> = ^J.l^^v'jJ.l^^) ^ ^Xj-l^^^?JJ-l(^) ' (2.45)

4'(^> = - \j.l^<jJ.l(^> ^ ^XJ-l^<JJ-l(^)- (2.46)

- 20 -
11.16

A,

K2

y[J2] —\— A]

^2 X

^2

J.

ir I n

K2

Fig. 2.12

II. 4 - CENTER OF MASS AND RECOIL CORRECTIONS

We must distinguish the pseudo-center of mass motion introduced into the

solutions by the use of the center of mass pseudo-Hamiltonian, eq. 1(1.41), and

by the use of a truncated basis, from the relativistic recoil of the target nucleus in the electron scattering process. We shall discuss successively the extraction of the center of mass motion and the corrections required to account for

the relativistic recoil.

We assume that the pseudo-Hamiltonian, 1(1.50) and 1(1.41)

5if = H + i 5(P^ + fi^R^), (2.47)

has solutions of the form 1(1.55) for the ground state

. ^.s. =*o(^> ^g.s.^V' (2.48)

where <I'q(R) is the normalized Os center of mass wave function and xCC^) is the

physical ground state wave function of the relative intrinsic particle coordinates 5^. From the relation

y i> \e^'^''^\^ y = \e^'^'^\<^ y<y [e^^-^lx ^, (2.49)

<. g.s.' ' g.s.'^ ^ o' '^o^^g.s.' "^g.s./

we see that the physical elastic form factor is of the form

- 21 -
11.17

2 2 F,(q) = F (q) . , . (2.50)

y corrected calculated

->• where F (q) calculated is the form factor calculated from the solution lii y g.s

with the one-body matrix elements given in the previous sections. Note that the

correction resulting from eq. (2.50) can be important even for small values of

the transferred momentum q depending on the magnitude of the parameter f^.

We now evaluate the effect of the relativistic recoil of the target nucleus at large momentum transfer. The following remarks are limited to the case of

elastic scattering. It is best to use the brick-wall coordinate system in which

the recoiling composite object moves with momenttam - q/2 before collision and

+ q/2 after collision. The velocity is

V = ±(q/2)(M^ + (q^/4))"'/^ , (2.51)

and the Lorentz factor

Y = (1 + (2^)^)^^^' (2.52)

where M is the mass of the composite particle. In the brick-wall system the

Lorentz contraction is the same in the initial and final states. Thus, in the

many-body matrix elements, the overlaps associated with the constituents which do

not interact with the electromagnetic field are unity. We must however evaluate

the effect of the boost on the value of the one-body matrix element involving the

interacting constituent particle.

For example, for a spin 0 constituent in the basis state <f) „ (x) of the solut ion 4' calculated in the laboratory system, we get in the brick—wall system

F(q) =<<)>;^(q/2)Ie^^^|<^;^(- q/2)> (2.53)

where we have assumed the transferred momentum to be along the z direction. In

eq. (2.53), <j)\(± q/2) is the state boosted with the velocity v, eq. (2.51). For

the spin 0 case, the boosted functions (f)^j^ differ from the laboratory functions

(j)^^ firstly by a Lorentz contraction, secondly by a change of normalization. The

Lorentz contraction is of course the same for the initial and final states. The

change of normalization is given by the Lorentz factor Y' Therefore we have

*v£(^» y. z) = y, Yz) , (2.54)

- 22 -
II. 18

and eq. (2.53) becomes

F(q) = dx dy Ydz (x, y, yz) e^'^^ (j)(x, y, yz) . (2.55)

A change of variables yields finally

F(q) =

j dx dy dS <|>*(x, y, O e''^'^^^^^ <|)(x, y, O • (2.56)

Thus the effect of the relativistic recoil for a spin 0 constituent is fully

taken into account by scaling of the transfer momentum

F(q) . = F j(q/Y) • ^ physical corrected ^ ' (2.57)

However, for a constituent of non-zero spin, in addition to compensating

for the Lorentz contraction as in eq. (2.52), we must also evaluate the effect

of the Lorentz transformation on the spinors. This way eq. (2.56) becomes

F(q) = i dx dy d5(f(x, y, 5) + 6ijr(x, y, 5)) e i(q/Y)5

X ('/'(x, y, O + 6ip(x, y, ?)) , (2.58)

JV = <(ij; + 64)|i|; + 6^} . (2.59)

We outline the calculation of the change 6i(j in the wave function of the

constituent, using as an example the case of a spin 1/2 particle. There only the

small components are modified by the boost which entails the substitution

a.V -» a 9 + a 8 + ya^ 8^ = a.V^ + (y - l)o^3^ • (2.60) XX yy55 5 5?

The term a.^^ contributes to }p as in eqs. 1(3.72) and 1(3.73) which have been

evaluated to yield eqs. 1(3.88) and 1(3.89). In order to evaluate the correction

term we introduce

(y - l)a,3, = T2[htO aC'^jM [JO vWj[o] (2.61)

with

Jl] =-i(y- l)'/2fi^^ . (2.62)

- 23 -
II. 19

.[1] 1 p'- - X

K K

1/2 X

Jl/2]

- la

Fig. 2.13

From the recouplings of figure 2.13 we obtain the field expansion

1 1 k"

•51^(5) = 4l (i)^ (-)^"" ^ 1/2 A i /6 a

v2.j

NXK

1/2 £ j

1/2 X Nj

where

[Jj 1/2] Jl/2]

ttJlJJl]j[Kj^[Nj^,e)6v^,,(0]t^'J

]

[oo]

dp(2E(E + m))"'/^ f^j^(p) j^(p?)

(2.63)

(2.64)

We note however that for the relatively low transfer momentum region

covered by the truncated configuration space of Chapter VI, ref . [l] , we expect

the recoil corrections to be small, in contrast to the center of mass correction discussed above, since <Si|i is of the order of

-24 -
11.20

Y

- 1 %

2 q (2.65)

2

• 8M

We finally note for completeness that for the case of inelastic scattering the calculation is most easily performed in the generalized brick wall system in which the yf^ctors are the same in the initial and the final states.

Then, considering the excitation of the system of rest-mass M by the energy B,

-y -> so that the rest-mass of the final state is M„ = M + B, we have P. = - Myv,

P„=(M+B)Yv, E,=MV, E„=(M+B)y and the momentum transfer four-vector is

q = ((2M+B)Yv, iBy) . With obvious modifications the formulae of this Chapter

then hold also for inelastic form factors. We shall not go into detail here.

- 25 -
III.l

CHAPTER III

QUADRUPOLE AND MAGNETIC STATIC MOMENTS

III.l - QUADRUPOLE MOMENTS

The quadrupole moment operator the form of which is derived in Chapter I section 2

Q(r) = 2z - X - y = ^J-^ r Y^^Cr) , (3.1)

reads in tensorial form

Q(

[2] where the amplitudes corresponding to the definition (3.1) are

h^^'^ = - <S - . (3.3)

• m mO

Furthermore, the complete quadrupole operator must of course include the charge

projection operators defined in eqs. (1.2), (1.3) and (1.4) for the spin 0, 1/2

and 1 cases respectively.

III. 1.1- Spin 0 field

The scattering terms, figure 2.1. a give the contribution to the quadrupole

moment

- 26 -
III. 2

Q = -ieI (i)''"''[lh^'^|l][<^ll2|g

(3.4)

[2,r dr ^_g^^^^(r) /re? ,,2^rh^^^^(r)l_A . >rl^'''^ A

[01]J J20].[00] JJL J,

where the recoupling is similar to that used in figure 2.5 for eq. 2.10. The definitions of the amplitudes are given by eqs. (3.3) and (2.2). Likewise for the

pair creation contribution, figure 2.1.b, we get from figures 2.5 and 2.6 and

from eq. (2.16)

^1 1 Tl

^1^ T

^2^-2

1 1 T

1 1 0.

(3.5)

^^,[^0 JVJ Jot] j2o]^[oo]^

with the definition of eq. (2.11). The pair annihilation term, figure 2. I.e.

gives the same contribution as (3.5) except for the additional phase (-)

III. 1.2 - Spin 1/2 field

The quadrupole matrix elements for the spin 1/2 scattering term is derived

according to the recoupling of figure 2.9 and, in similar fashion as in eq.

(2.28) , we get

Q = ^ (-)^[i/2|tW|i/2] (i)

I

(i)

^ r y=i.o ^ ^^^j]

^2^2^2^2

1/2 jj

1/2 ^2 h

.0 2 2.

1/2 j/

1/2 X2 22

.022

[^1^212]

[^,1^212]

2- r dr

2, r dr

16Tr 2 , . , v ^ r u (r) u (r)

5 VjZj V2X2

15? 2 , . . .

[j 1/2] [j 1/2]

'< B „ c

[oy] j2o]^[oo]

(3.6)

- 27 -
III.

III. 1.3 - Spin 1 field

For the spin 1 field figure 2.12 and eq. (2.40) yield

Q = -2ie[l|Tf']|l]^ a)'^''' jl-l

VjJ, "^1^2 K,X

V2J2

11

^"2^2

1 X, J,

1 X2 J2

_0 2 2 .

f 2, re 2 /l^K - /2^2, , ^

^•^2'^ [01] J20].[00] , r dr l-r-r K (r) L (r) x U ^, ^, J JJL J » J,X, J2X2 <,Vj K2V2

(3.7)

where the functions K and L are given in eqs. (2.41) through (2.46). The summations Kj and (3.7) extend of course over all the multipolarities of the

particle.

III. 2 - MAGNETIC MOMENTS

III. 2.1 - Spin 0 field

From the remarks of Chapter I section 2 we get immediately for the spin 0

"scatering" term,

VjAj 12 12

^2^2

. t^^'^M JOI] Jl0]][00]

^ V V

1 2

(3.8)

and similar terms for pair creation and annihilation contributions with the

1 £]+£2

added phase (-) and (-) respectively.

III. 2. 2 - Spin 1/2 field

For spin 1/2 the j term of (1.12) is immediate since j is a good quantum

-> . , number, while the spin term s requires a simple recoupling.

- 28 -
III.

p=if 1^ (-)y[u2\M\u2]m . (i)

y=l,0

'2 ^1

v,i^,(X,)j.

jjj2 ^1^2 '^ri '

+ 6 ^1^2 ^1

1/2 a, j,

1/2 Jl, j2

10 1.

r dr u „ (r) u . (r)

+ 6

1/2 X, jj

1/2 Xj

1 0 1

r^dr V (r) V , (r)|(- [1/2131^'^ I 1/2])

X [b

[j,l/2j [j,l/2]

(3.9)

III. 2. 3 - Spin 1 field

For spin 1 the expression is as simple as in the spin 0 case since J is a good quantum number

y = le

VjJjKj 12 12 12

'^I'^l ''2^2

where the svmmation extends over all multipolarities Kj and K2 of the particle.

Note that in expressions (3.8), (3.9) and (3.10) we have set = 1 , cf.

eqs. (1.11), (1.12) and (1.13) in confomity with our basic assumption of no anomalous moment. Furthermore, tue truncated space of ref. [l]. Chapter VI, does

not allow for pair creation or annihilation terms in the static moments of

spin 1/2 and spin 1 fields.

- 29 -
IV. 1

CHAPTER IV

VECTOR DOMINANCE

IV. 1 - FORM FACTORS

The contribution to the elastic scattering form factor of the u meson

through the vector dominance graphs of figure 4.1. a and 4.1.b is simply of the

form, according to Chapter I section 3

F^'^^q) = G d\ e^'l'' a,^(?) , (4.1)

-y where <^jj(r) is the field, the expansion of which is given in eqs. 1(3.222)

through 1(3.225).

Fig. 4.1

The analogous expression for the p meson must include a projector on the

0

neutral component p

-30 -
IV. 2

F(P>(t) = G

y ^ YP

= G

f d\ei^T[cW nWjM - , (4.2)

TP ,

with the isospin amplitudes c^'^ given by eq. (2.2) and where in (4.2) the dot

implies a scalar product in isospin space. Thus the isospin wave function in

each term of the p expansion is replaced as follows

(4.3)

Apart from this isospin coupling of the creation (or annihilation) operators

to the amplitude c^'-^, the oi and p mesons yield identical expressions.

Furthermore, the Fourier-Bessel transforms of the radial part of the field

which appear in the matrix elements are, for example, of the general form (see

expressions 1(3.232) through 1(3.236))

r dr j^(qr)w:^j,(r) 6 2,r dr P% jj^(qr) j^(pr) /f f^j(p)

2, TT 5(q-p ) . 'V /2 . ,

1 t f ( ^ 2 V2E vj'-^'' ' (4.4)

while the space geometry is obtained, introducing space polarization amplitudes

d^^^^ as in eq. (2.33)

fafKW ;[«]][o] [,W ,[.] ;W][o]

(4.5)

The vector form factor for the (u-meson dominance term is thus given by

KVJZ

\/2E NJ£

KV KV

where

(4.7)

- 31 -
IV. 3

and

M = 0 if K = JW

,

n = ] if K = 8 , <£ . (4.10)

The scalar form factor is

F. (q) = iG 4Tr ./^^ Y C-)"^ J >£J 4^ -S- f ^(q) Y(o v2 ''^ ^ ' V2E m vJ^

vJ

.[(aW.(-/*'aW)5W]M . (4.,.)

For the p field, these expressions hold with a replacement according to 4.3,

for example

IV. 2 - QUADRUPOLE MOMENT

The evaluation of the quadrupole moment requires a special treatment, since

the usual procedure, followed in Chapter III, involves an interaction of the

electromagnetic field with a conserved vector current. In contrast, here the

interaction energy arises from the emission or absorption of a neutral particle.

Thus we shall start from the defining equation for the energy of a quadrupole

moment (rank two tensor denoted by Q) interacting with an electric field gradient

E = qQ}V8 H Q Q0Vg , (4.13)

where the tensor product implies contraction of the tensor indices.

We have here introduced the unit tensor of rank 2, Q. Eq. (4.13) is valid as

long as the gradient of the field is constant over the volume of the system. On

the other hand the vector dominance interaction energy is given by (1.21). Hence

we shall recognize the vector dominance quadrupole moment Q by equating these

energies >

G

f 3 - ^

d rA 0) = Q Q0V<g" for q -> 0 . (4.14)

y y <w —

32
IV. 4

For the photon vector potential we use a plane wave in the Lorentz gauge

l = lei^ = T[al^^J etOjM . (4.15)

-> The electric field S thus is

• .-^

I = - A = iql e^'''^ , (4.16)

and the electric field gradient

,->-> Vl = - q qQ^e'-"''' , (4.17)

where the wavy line denotes the construction of rank two tensors according to

^_ q2 5^^.[1] ^[l]j[2] ^Jl] Jl]j[2]^[0] ^i^

^ ^^^j3^

Let us write eq. (3.2) as Q=2Q[bM q[2]][0] , (,.,9)

-[21 with Q the unit spherical tensor of rank 2. We get with a simple recoupling

=U„ , . 0

= - <i 2 .'t^'^ ^UtWjM (4.20)

where we have made use of the defining relation

[q[2] etl] eW][0] = 2 . (4.21)

On the other hand the left hand term of eq. (4.14) is calculated with the

multipole J = 2 of the u field which in the limit q 0 has the dominant components (of space character), see eqs. 1(3.222) through 1(3.225)

s=>i j[AW[ji] ,[]] r2ij[o] ^,^,^ (,,)

)

We substitute the plane wave expansion (2.6) for the vector potential A in

^0 L ^ J J - %21

+ h.c. ... . (4.22)

" 33

~
IV.

eq. (4.14) together with the field expansion (4.22). The recoupling geometry of

figure 4.2 yields in the limit q =>0

V (0

, (4.23)

where we have used eqs. 1(3.232), 1(3.215) and, for the integration over r the

relation

[1] J2

i

4l] £ 0

I

rhr j,(pr) j,(qr) = L ME^I . (4.24)

2 A.t2]

/

[1] _1.

•['] 1

i

15

I]

Fig. 4.2

We finally compare the expressions (4.23) and (4.20) noting that (see 1(3.31))

(4.25)

7/2 _ .5/2... 2

which yields the complete contribution of the vector dominance term to the static quadrupole moment from the u field

V

(4.26)

with

4 .3/2 J/2 5/2 /r^

2 v2 V '

^ Jm^ Y<^

' (4.27)

(4.28)

- 34 -
IV. 6

since in (4.22) the field energy E equals the u mass m^.

[2]

In eq. (4.26) the quadrupole polarization tensor d"--" has been substituted

according to

,[2] ^ ^.[,].[1]3[2]

^ (4.29)

where a"-i"- iss the polarization vector of the real photon field; it is coupled

with the momentum direction amplitude q^'^ of the real photon. The values of

[21 . . d^ are associated with the preparation of the system according to well known

rules, for example see ref. [_5\.

As in the case of section IV. 1, one goes from the contribution of the (o meson to the contribution of the p*^ meson by a substitution analogous to eq.

(4.12), namely

[(aW _ iM) dMjM . r[(At20 - J20] J0l]j[00] ^ KV KV

In order to calculate eq. (4.27) the following values are useful

L^/^0)

V

r = 1 if V = 0 , = 5/2 if V = 1 , = 63/8 if V = 2 ,

(4.31)

IV. 3 - MAGNETIC MOMENT

We shall follow a similar path as in the previous section. On the one hand

the magnetic coupling energy is of the form

E = y B . (4.32)

With a plane wave potential A, eq. (4.15)

1 = rotX= l/2[et'] vW iM] M

= T/2 iq[et'] qtO t^Jj M ,

see 1(2.38). Hence with a simple recoupling

E = iq T/2 M t[y^'J qt'^ 1^] M '

where p is the unknown magnetic moment and y^'^ a unit tensor of rank 1.

On the other hand we shall use the expression of the energy in terms of the

(4.33)

(4.34)

" 35

"
IV. 7

interacting fields, limiting the expansion of w^, 1(3.222), to the magnetic dipole and taking the limit q =>0. Thus

i'^^.Aqr) T[[eW fW] W (.W - A W

)J

M . (4.35)

V

Utilising the recoupling of figure 4.3 corresponding to the substitution of

-) Its expansion for the plane wave real photon potential A

E = G f r^dr j , (qr) W*', , (r) i[(aW - a W ) gDl ^H] M .

We note that

vll MV M\>

(4.36)

r^dr j,(qr)W^„(qr) = ^S-f,,(q) .

f , (q) „

= a^/^ C , q L^''^(0) ,

(4.37)

(4.38)

' = 1 if V = 0 , = 3/2 if V = 1 ,

. = 35/8 if V = 2 . (4.39)

Thus the contribution of the to field dominance term to the magnetic moment

is equal to

y = Ip T[(aW ,b]][o]

, (4.40)

with from comparing eq. (4.34) to eqs. (4.36), (4.37) and (4.38)

.,/: 3/2 5/2 ^ t3/2.^>, rr - y = - 1/2 i: 01 C , L (0) — G

V vl V Im^

As before we have introduced the system polarization amplitudes

,[i] = t5[iJlWj[i] .

(4.41)

- 36 -
IV. 8

4'^

jti

[i] e

,[.]

q[«]

16

Si] Fig. 4.3

[1]

1 1 —1

1

0 X 1 i

2Z

X 1

a

... 1

I

Finally, the p field contributions are similar to eq. (4.39) with the substitution of eq. (4.12), namely

~ 37 ~
A.l

APPENDIX

ADDENDUM AND ERRATA TO N.B.S. MONOGRAPH 147

A - DIRECT CALCULATION OF THE CM. QUADRATIC COORDINATE MATRIX ELEMENTS

In N.B.S. Monograph 147, ref. [l] , the matrix elements of the operator

E + e (A.l)

were calculated by expanding twice on complete basis vector sets. This leads to

complicated numerical steps. Furthermore, since by necessity any functional space must be truncated, the expressions 1(4.43) and 1(4.44) may yield poor accuracy. This is of course not the case for the bi-linear terms 1(4.45) which involve

no intermediate sums.

We shall give here an alternate way for calculating the matrix elements

(A.l), which is direct and free from the limitations of intermediate expansions.

From the result

(A.2)

we get the general expression

-38 -
A.

r T I 1 /^2 -zE „2 ^ -zE „2 ^2. i ^ 7 [Vj Jj K

[-J (x e E + e Ex )|v2 J2 "^J

P?

I P2 '^P2 ^K^^l'V

^ ^2

JjJ2 1

1 2 "^^1 2

"^^2

TT f-«"(p,-P2) .j/'S(prP2>

-A P1P2

+ A I ^ + 6 (PJ-P2) —2—T)\ ' ^ Pj P, Pn 'J

1 ^^2' 2 2 P, ?2

(A. 3)

where F (E, is the norm factor for the various cases k and A is a weight.

K 1 ' 2 K ° Namely for the ir -meson, k = 1 , J = £

Fl(Ej,E2) (A. 4)

J _ £(£+1)

^1 2~ (A.5)

for the w-and p-mesons, magnetic multipolarities , k = 2, J = £

F = F

2 I and A-^ B.2 = A-^j (A.6)

for the (o-and p-mesons, electric multipolarities, k

17)

= 3, J = £ ± 1

h = Met " \

(A.7)

J JCJ:^^ j^ (j.i)(jH.2) ^ ^2 ; 3 -2 2 ^2 2

(A. 8)

for the (jo-and p-mesons, longitudinal multipolarities, k = 4, J = £ ± 1

^4 = m

/e,E2

(A.9)

A-^ = A"^ \ ^3 (A. 10)

for the nucleon, k=5, J=£±l/2, X=J±l/2

- 39 -
A.

^5 = 'Ej+m /E2+ni

(A. 11)

A^ =^ £a+i) +Irsx(x+i) , (A. 12)

where in A^ we have already taken the limit Ej ~ ^2'

The evaluation of eq. (A. 3) is carried out with the change of variables.

P]-P2 = P Pl'^P2 " ^ ^^2 =

-J

«iP .

setting in the integrand

<S'(p) FXp) = - F'(p)/

p=0 '

5"(p) F(p) = F"(p)/p^Q

the limit p = 0 being taken after differentiation.

We thus obtain the general result

r ^ I 1 .->2 -zE „2 ^ -zE ^2 ->-2. 1 ^ 1

L

1 "^1 '2'^^ ^ E + e E X )|v2 J2 kJ

(A. 13)

(A. 14)

(A. 15)

2 ' 2 4 2 4

1 1 P _^ zPy^ 3 zP _ z P^

^ ("k - 2- 2^ ^ E" 4E E^ 4e2

P DPE^ e"^^(f" .(P)f .(P)+2f' .(P)f' „(?)+f „(P)f'; p(P))

(A. 16)

In this expression Z = for the meson cases and £ = for the nucleon case.

The factor (6. . ) ej

^1^2

the following values

The factor (5„ „ ) exists only for the nucleon case. The coefficient N takes

Jij X,2 1^

K = 1 , TT-meson

N = J (J +1) - —

4E^

K =2, to- and p-mesons, magnetic multipolarities

(A. 17)

N2 =Nj (A. 18)

- 40 -
A.

K = 3, u- and p-mesons, electric multipolarities

,4

N„ = J,(J, + 1) + 2 - ; (A. 19)

^ ' 4E^

K = 4, 0)- and p-mesons, longitudinal multipolarities

p2 p4

N = J (J +1) + 2 + (-^ - -^)m ; (A. 20)

^ ' 4E^ 2E^

K = 5, the nucleon

^ K£a-M)(E-Hn) ^ X(X.l)(E-m)^ - ^ ^ 4 • (A.21)

^ 2 E E 4e'^ 4e2

->2 Finally the factor 1 in eq. (A. 16) comes from the expression of X in tensorial

form. No isospin coefficient (which wDuld be t) is included. .

B - DIRECT CALCULATION OF THE CM. LINEAR COORDINATE MATRIX ELEMENTS

When computing the matrix elements of the operator Sx, 1(4.31), an expansion

on a complete set has been used, see 1(4.32), with the above inconveniences. A

more direct expression can be obtained starting from 1(4.31) and carrying out

the steps 1(4.35) through 1(4.37) with the full integrand gx. One notes that

S= e e'^ is a symmetric function of p^ and When integrating by part the

^'(P]~P2) term, see 1(4.36), the (p) factor goes to zero when p = P]~P2 goes

to zero. Consequently the expression for 1(4.32) is directly (see 1(4.37))

P dP g(P)

^r2

(B.l)

The same applies to 1(4.38), 1(4.39), 1(4.41) and 1(4.42).

C - NEW PHASE DEFINITION FOR THE PION FIELD

Finally. note that the way the pion field is defined p. 42, eq. (3.8), the

pion nucleon matrix elements p. 98, eq. (5.16), are purely imaginary. It is

- 41 -
A.

therefore worthwhile, for computational purposes, to redef ine the pion. field as

^-i(??-Et)^(l)+ ^[1]+.

Using that definition the complex phase of (5.16) becomes the real number

-a +1 +1+1

(i) ^

'

D - ERRATA TO N.B.S. MONOGRAPH 147

The errata are given in the form of the complete corrected formula or of a fraction thereof such that they may be directly pasted over the wrong parts in

the Monograph 147

p. 7, eq. (1.8;

^^^(x) = (Uj(x)bj + vj(x)c^)

p. 7, eq. (1.12)

^^^^^(x.t) = (e "^^ Uj(x)bj + e "'jV(x)Cj

j

-ie.t 1 E.t

p. 19, eq. (2.4)

m m -m

p. 37, eq. (2.79)

T 1 T+l-| Oil = 0 , t = 1 . T t T J ^n+1

(T) p^(T-l)

T

0

LT

1 T-i

1 1

t T .

a (T+1) n

p. 46, eq. (3.37)

se=-Y(^yd^) + m(t Ai ) K

- 42 -
p. 73

Utilizing the foraulas (2.60) and (2.61), it can be checked that

p. 75, second line of eq. (3.238)

p. 76, add to the end of the section the comment :

We list the parities and the number V of quantas for the lowest energy discretized multipole solutions of the spin 1 fields

g M ^

0 0

1 1

2

3

+ - +

- + -

p. 80, eq. (4.13)

p. 83, last line of eq. (4.18)

1 j,-l J, + / (J^+1)J2 1 J2+I

0 1 IJ

°J^-1,J2+1

^"i^i-'^

^^2"^2'^^

p. 90, second line of eq. (4.41)

/J^(J2+1)

1 J^+1

1 J2-I ^2

.0 1 1.

^1^2

^J^+1,J2-1^P) + ^V2

1

1 J2+I ^2

0 1 1.

v^V2

^J^+1,J2+1^P^

- 43 -
p. 94, first line of eq. (5.8)

+1/2

if/ V"

j'j"

p. 95, Exchange in figure 5.3.b the boson and fermion lines

A.

(v3/3j3l/2)

/

/

(vi^^l) {y^f^i^lID

\

\

(a) (b)

p. 96, first line of eq. (5.15)

p. 98, fourth line of eq. (5.16)

,^3 j3 1/2 X,3 j3 1/2

p. 100, first line of eq. (5.26)

G

(4n) I

^^«Py5). %%Q55^

R=0,2

p. 102, first line of eq. (5.27)

X

(4n)

(oH-p+Yf5) L 5

- 44 -
A.

p. 102, first line of eq. (5.28)

G

(4n)

p. 102, first line of eq. (5.29)

^ L o(B vf

p. 102, first line of eq. (5.30)

p.]25, eq. (A. 5)

X = X + V, (t- 1 ) i O 1 o

'

p. 125, eq. (A. 6)

p. 135, add at the bottom of the page the missing equation :

^ = 47r I i^ A F^.(pr)[pW fWjW

I z J [b w ,

a.)] M [bpi (-;> x[;] (t.)] M (C.4, X

p. 136, eq. (C.7)

- 45 -
ACKNOWLEDGMENTS

The authors gratefully acknowledge the help of the documentary Group of the

Departement de Physique Nucleaire at Saclay in preparing the text. They are particularly indebted to Mrs. Paulette Gugenberger who prepared and edited the

manuscript and its final typed version and to Mrs. Eliane Thureau who carried

out the typing in a form suitable for direct reproduction.

- 46

-
R.l

REFERENCES

1 - DANOS M. and GILLET V., Relativistic Many-Body Bound Systems, N.B.S.

Monograph 147 (1975).

2 - HAYWARD R.W. , The Dynamics of Fields of Higher Spin, NBS Monograph 154

and private communication.

3 - DE FOREST T. and WALECKA J.D. , Advances in Physics KS, 1 (1966).

4 - DONNELY T.W. , Annual Review of Nuclear Sciences (1975).

5 - FANO U. and RACAH G. , Irreducible Tensorial Sets, Academic Press, New York

(1959)

6 - BETHE H.A. and SALPETER E.E., Quantum Mechanics of One and Two Electron

Systems, in Handbuch der Physik, S. Flilgge Ed., Springer (1957).

- 47 -
NBS-1MA (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS MN 147, Suppl. 1

2. Gov't Accession

No.

3, Recipient's Accession No.

4. TITLE AND SUBTITLE

Relativistic Many-Body Bound Systems;

Electromagnetic Properties

5. Publication Date April 1977

6. Performing Organization Code

7. AUTHOR(S)

Michael Danes and Vincent Gillet

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2401104

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as 9.

13. Type of Report & Period

Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 77-1469

16. ABSTRACT (A 200-word or less factual summary of most sigfiidcant information. If document includes a significant

bibliography or literature survey, mention it here.)

The formulae for the calculation of the electron scattering form factors, and of

the static magnetic dipole and electric quadrupole moments, of relativistic manybody bound systems are derived. The framework, given in NBS Monograph 147, is

relativistic quantum field theory in the Schrodinger picture; the physical

particles, i.e., the solutions of the interacting fields, are given as linear

combinations of the solutions of the free fields, called the parton fields. The

parton-photon interaction is taken as given by minimal coupling, p p

- eA; in

addition the contribution of the photon-vector meson vertex of the vector dominance

model is derived.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

)

Electromagnetic properties; form factors; magnetic moments; quadropole moments;

quantum field theory; relativistic many body systems; vector dominance.

18. AVAILABILITY 2] Unlimited

I For Official Distribution. Do Not Release to NTIS

I X' Order From Sup. of Doc, U.S. Government Printing Office

Washington, D.C. 20402, SD Cat. No. C13. 44:147 Suppl. 1.

I

! Order From National Technical Information Service (NTIS)

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