The physical nucleon in static source meson theory

The physical nucleon in static source meson theory

ANNALS OF The PHYSICS: 7, Physical 154-173 (1959) Nucleon in Static F. R. HALPERN Palmer Physical Laboratory, Department of Physics, of...

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ANNALS

OF

The

PHYSICS:

7,

Physical

154-173

(1959)

Nucleon

in Static

F. R. HALPERN Palmer

Physical

Laboratory,

Department

of Physics,

of Physics,

L.

Meson

Theory*

SARTORI

TTniversity,

Princeton,

New

Jersey

NISHIMURA

Rutgers

University,

R. Department

AND

Princeton K.

Source

New

Brunswick,

iVew Jersey

SPITZER

Colzmlbia

(Jniversity,

New

IYork,

Xew

York

A wave function for the physical nucleon in static source meson theory has been constructed by the method of moments. The wave function includes contributions from states containing as many as five mesons, and is shown to be a good approximation to the ground state of the Hamiltonian. Because of the restrictive nature of the static model, the probability that the meson cloud have a particular angular momentum or isotopic spin turns out to be almost independent of the dynamics of the system. Since the matrix elements of the nucleon spin and isotopic spin operators can be expressed in terms of these probabilities, such matrix elements are det,ermined essentially by the kinematics of the meson cloud. This fact renders unreliable any predictions of the static model which depend on matris elements of the nucleon operators. The consistent failure of t,he static model t,o explain satisfactorily the scalar part of the magnetic moment may be ascribed to this cause. The quantities which depend on meson operators are more closely related to the dynamics of the system. The vector part of the magnetic moment is in agreement with experiment, and the electron-neutron interaction turns out much too large; these results differ little from those obtained by the Chew-Low one-meson approximation. However, the contributions of the many meson states are not negligible, and the mean number of mesons is found to be closer to two than to one. Evaluation of the charge renormalization enables us to test the scattering amplitudes predicted by the static model against the sum rules of Cini and Fubini. It is found that, for the conventional value of renormalized coupling constant, the sum rules are strongly violated. This result, together with the substantial many meson component of the wave function, casts doubt on the validity of the one-meson approximation in the static model. * This Research Research.

work was supported and Development

in part Command,

by the Atomic 151

Air

Force Energy

Office of Scientific Commission, and

Research, Air Office of Saval

PHYSICAL

NUCLEON

155

I. INTRODUCTION

Static source meson theories have been extensively studied since Yukawa originall:y introduced the meson as an explanation of nuclear forces. There have been two principal motivations for studying static models rather than covariant theories: by restricting the nucleon to at most several discrete degrees of freedom an immense simplification is attained in the calculations, and by employing a cutoff the problem of divergences may be completely avoided. The extent to which a static m.odel might represent the behavior of actual pions and nucleons has never been completely clear, although it has been hoped that low-energy phenomena might be satisfactorily explained within the framework of such a model. Recently, Chew and Low have given a treatment of the static model which yields results in good agreement with some of the low-energy phenomena. Lowenergy pion-nucleon scattering (1) meson photoproduction (2) and the vector part of the nucleon anomalous magnetic moment (3) have been successfully calculated in the one-meson approximation. However, the scalar part of the magnetic moment (3) and the neutron-electron interaction (4) are not adequately explained. Furthermore, it has been pointed out by Cini and Fubini (5) that certain sum rules, which are a rigorous consequence of the static model, are not satisfied by the experimental mesorl-nucleon scattering amplitudes if t)he coupling constant has the value which is necessary to obtain the agreements mentioned. There is, therefore, some question as to whether the one-meson approximation adequately represents the predictions of the static model. In the present work an approximat,ion to the ground state of the system (i.e., the physical nucleon) has been constructed. This approximate state contains terms with as many as five virtual mesons. It is a member of a sequence which has been shown to be convergent. For the conventional values of coupling constant and cutoff, convergence appears to have been almost attained at t,he point to which the calculation has been carried; thus, it is to he expected t,hat many of the propert,ies of the exact solution may be inferred by extrapolation. It was hoped that a knowledge of the structure of even this one state might give some more reliable insight into the correspondence of the stat,ic model with reality. To a certain extent it has been possible to achieve this goal and unfort’miately the results argue primarily against the validity of the model. There are two types of operators which appear in the st,atic model. These are the nucleon spin and isotopic spin operat,ors 0 and r, and the meson creation and annihilat#ion operators. Our results suggest that the values of mat’rix element,s of the nucleon operators between physical nucleon states, as calculated in the static model, are very unlikely to be relevant to the description of any physical process. The reason is not, as might appear at first, that these matrix elements are closely related to the renormalization constants and are therefore extremely cutoff-dependent quantities. This is in fact not true, and these matrix elements may be sensibly defined even for infinite cutoff. The difficulty is, on the contrary, that

1X

HALPERT,

SARTORI,

NISHIMURA,

AND

SPITZER

6he matrix elements in question are almost’ exclusively determined by the kinematics of the system and are, therefore, almost independent of both the cut,off and coupling constant. They in fact approach their st’rong-coupling values monotonically and rtlpidly; as far as the matrix elements of the nucleon operators are concerned, we find that strong coupling begins when the product of the unrenormnlized coupling constant and the cutoff is of t,he order of magnitude of two meson masses. But the coupling constant renormalizat,ion itself involves t,he matrix element of UT (1) which is precisely of the type WC are discussing. It. turns out t,hat the conventional value jr2 = 0.08 derives from an unrenormnlized coupling c011stn11t .fo2 z 0.6 (see Sect,ion 3). Thus, even for a cutoff - nucleon mass and jr’ = 0.08, strong coupling has already begun for the nucleon operators. [See Eq. (18) and Table I.] The matrix elements of the second type, namely t’hose of the meson operators, do depend 011 the dynamics of the system. These matrix elements appear in the calculation of the meson contribution to the electromagnetic properties of the physical nucleon. It turns out that the vector part of the anomalous magnetic moment is correctly predicted from our wave function. However, the calculat,ed values for the scalar part, of the moment and the charge radius of the neutron are in disagreement with experiment, and differ little from the results of the ChelvLow treatments. Insofar as the electromagnetic properties are concerned, therefore, it appears that the predictions of the static model are adequately represented by the one-meson approximation. The difficulties with the Cini-Fubini sum rules persist., however. As has already been mentioned, one of our results is an accurate relationship between unrenormalized and renormalized coupling constants. The value fr” = 0.08 requiresfo’ = 0.6, and this combination is in sharp disagreement with the sum rules (23) if the experimental values are used for the cross sections. The sum rules follow from the static model with no further assumptions, and therefore ought certainly to be satisfied by the scattering amplitudes calculated from the model. But if the scattering amplitudes were to change in a more accurate treatment so as to come into agreement’ with the sum rule, it is difficult to see how the agreement with experiment could cont,inue. In such a case the agreement of the onemeson approximation with experiment would have t,o be considered fortuitous, as claimed by Cini and Fubini. The method employed in the present calculation is the method of moments (6). That this is indeed a convergent method has been shown previously in a discussion of the polaron problem (7). The convergence in the present problem is in fact stronger than for the polaron, and it is easy to show, using similar arguments, not only that matrix elements which are linear in the Hamiltonian approach their correct, values, but even that those which are quadratic do so. 111 particular, if J/n is our nth approximation to the physical nucleon state, we can

PHYSICAL

show that the deviation

157

NUCLEON

defined by

(1) approaches zero. This quantity of course vanishes for an exact solution and its magnitude measures how close any given approximation is to convergence. In the present calculation ra2, measured in square meson masses, decreased from 170 in the zeroth approximation t’o 7 in the final order calculated (for fl = 0.6). The succeeding sections describe the construction of our wave function, and the calculation of the matrix elements of the nucleon operators, the anomalous moments and the mean square charge radius of the neutron. II.

THE

I\;UCLEON

WAVE

FUNCTION

The state vector for the physical nucleon was calculated with the method of moments (6). This method consists of constructing a trial vector of the form P,(H)4, where P, is a polynomial of degree n in the total Hamiltonian of the system, and 4 is an arbitrary vector. The degree of t’he polynomial determines the order of approximation; the coefficients in the polynomial are determined variatiolydlly so as to minimize some quantity. It t,urns out to be simplest t.o minimize the energy, and this was the procedure followed. As a check, the first few orders were also calculated by minimizing the deviation (T,~defined in Eq. (l), but, the results differed very little. The initial function 4 was taken to be the bare nucleon state. 50 advantage appears to result) from st,arting with a more complicated wave function. As described in Ref. 6, the solution in nth order involves the calculation of the matrix elements of t.he first 2n powers of the Hamiltonian, taken wit,h respect to the init,ial function. These m&ix elements are designated as H, :

Hi = (4 IHi14).

(2)

The result of t,he variational principle is that, the energy eigenvalue is given by the lowest root of the determinantIn equation 1

fi

E”

Ho HI Hz Sri(E) = H, Hz Hs Hn-3, The stat*e vector

H,+I

. ..

jj?”

.-a H, .-- H,+l

= 0.

(3)

. . . Ha--1

in nth order is then given by

h=

fn(W 4 H--E'

(4)

158

HALPERN,

SARTORI,

NISHIMURA,

AND

SPITZER

where ,!? is the root of (3). Since fn (H) /( H - 2?) is an (n - 1 )st-degree polynomial and the interaction creates and destroys mesons singly, our wave function in l&h approximation contains contributions with up t,o n - 1 mesons. Three different methods were used to calculate the moments Hi . The first method was to use interaction diagrams. This has been discussed for t,he polnron problem (8) and nothing essentially new occurs in the present work. For calculations beyond H, (i.e., fourth order) t.he use of diagrams becomes inefficient, and tedious because of the very large number of diagrams. I’or this reason the diagrams have played a minor role in the present work except as a check. The other two methods involve the observation that the matrix element H, may be written as Hi = (H”4 ( If%),

a+lJ=i.

(5)

Thus, after writing out the state vectors Hc#J,Hz+, H3+, . . . , explicitly, one can evaluate the moments by taking the inner product of the appropriate pair of states.’ The distinction between the last two methods of calculation lies in the manner in which the many-meson states are described. As has been pointed out in t,he preceding article (9) (to be referred to as I), this can be done in a variety of ways. The simplest way is t,o specify the charge and z component of angular momentum of each individual meson, together with its wave function. Such a descript,ion is diagonal only in I3 and JI, . A second method, described in I, is to define states which are diagonal in 12, J2, I, , Jz , and various permut’ation operators, The first method was the one utilized in most of the calculations; it is more clumsy than the second in that it requires a larger number of states. However, it exhibits more clearly the contributions of each of the four possible states of the bare-nucleon core and is thus suggestive of certain resu1t.s about the matrix elements of nucleon operators mentioned in the introduction. The second method is more compact and furnishes an appropriate form in which to present our results. The Hamiltonian for the static model may be written as a four by four matrix in the space of the nucleon spin and isotopic spin. The kinetic energy is diagonal in this representation and the total Hamiltonian has the form H

=Ho+Hr,

(6)

(7) HI = c 1 The a useful

fact that check.

a number

of different

(VA+ + Vk*&*), combinations

determine

(8) the same

moment

furnishes

PHYSICAL

159

NUCLEON

where Al; is the matrix

r aoo(~),di&+(k), lha+o(~), 2a++(,47) 7 4%-(l~), -amI( 2a+-(k), - \fiU,“(k) d&(k), (

2a--(k),

au-+(k), -d%-o(k),

-alo(k), -42Uo-(Ii),

- &zo+(k)

.

(9)

Go(li) J

In this representation the a’s are operators which annihilate mesons of definite charge and J, . The first index on these operators refers to isotopic spin and the second to angular momentum, although because of the complete symmetry of the theory between space and isotopic spin it is not really necessary to make these identifications. A,* is the t,ransposed matrix in which the destruction operators are replaced by t,he corresponding creation operators.” The basis vect,ors

r1 (10) 11 0 0 0 1

represent the states of proton wit,h spin up, proton with spin down, neutron wibh spin up and neutron with spin down, respectively. Ikally, VA-is given by ill)

where f0 is the unrat’ionalixed, unrcnormalized coupling constant and z!(k) is the usual cutoff function of the static theory. When one operates with H” on any of the bare-nucleon states (IO), one obtains a linear combination of these states, each multiplied by the sum of a variety of products of creation and destruction operators and a t,otal of n. fact)ors 17, Jr*, and w. (‘The W’Scome from Ho , and the V’s from HI .) The annihilation operators may be commuted out until only creat’ion operators remain. In practice one derives the j-meson part of H”4 by operat’ing with Ho on the j-meson part of H”-‘4, n,nd with the creat’ion and destruction parts of HI on t,he (j - 1) and (j + I)-meson parts of HrLp’+, respectively. There are several possible ways of symmetrizing. We have found it most convenient t,o regard only mesons with the same charge and .I, as identical particles. As an example we exhibit t,he first 2 We w;e a spherical wave decomposition of the meson field to emphasize the symmetry between the angular momentum and the isotopic spin. See for example, \Tick (2) or Sachs (10).

160

HALPERS,

SARTORI,

NISHIMURA,

AND

SPITZER

three powers of H operating on the bare proton stnt)edl (it is evidently immaterial which of the four bare nucleon stabe is used as the initial state): Ho41 = 91 , Ii&l

= Ck Vk*(a~o(~)~l + d- 2ao+(lc)da * + dzo(W

HZ+1 = Ck,k2 V$c2([b,*,)’

+ hEa,*+ + 2a*oa:o

+ 22/2[u-*+do - u*ou:+]& + 245[&uo*t

(12a) + 2a:+m4,

+ 4a*-u:+lh - uo*-u:+]~3

+ @o*oa:+- ~o*+dohl + Ck VE*ah&#+

+ d%ao*+dz+ ddo93

(1211)

(12c) + 2a:+a

+ 9Ck / Vk I”~1. Aft,er (12b) the arguments of the creation operators have been dropped. In Dhe first term of (12c), for example, (cL~*~)~ actually st,ands for a:o(kl)u,*,(kz). The states exhibited in (12a)-( 12~) are sufficient to calculate all the moments up to Hq . For example, Hd is just the norm of the state Hz+. Since states containing different mesonsare orthogonal, as are the four bare-nucleon states, the expression (Hz+ 1H’c$) will have ten contributions from the two-meson part, four from the one-meson part and one from the no-meson part. From the structure of the Hamiltonian it may be seen that all the contributions in all orders can be expressed in terms of integrals 1, defined by (13) For H4 , one obtains 9012” from the two-meson part’, 914 from the one-meson part and 811z2from the no-meson part.3 With the square cutoff function which has been employed, the integrals I, may all be evaluated in terms of elementary functions, and the moments thus obtained. However, it is evident that the number of contributions grows large quite rapidly: for example, there are about 150 different five-meson states which eventually appear. Thus, the utility of this method is exhausted about one or two orders after the diagrams fail. As has already been mentioned, in this scheme one can observe directly the contributions to the moments which come from the four bare-nucleon states. We here mention several rehrtionships which we have noticed among these numbers. Some of these statements are susceptible to proofs which will be given shortly; the remainder merely appear to hold true empirically in all orders to 3 The index m in Eq. (13) was chosen so that I, shall have the dimensions of (energy)m. It follows that in any contribution to H, the sum of the indices of the I’s must equal n, the order of the moment.

PHYSICAL

NUCLEON

161

which we have calculabed. In the first place the coefficients of the I’s are quite independent of t’he parameters of the model (coupling constant and cutoff), but depend only on its structure. Furthermore, with (proton, spin up) as the initial function., it was observed that, the sum of the coefficients of all the terms associated with (neut’ron, spin down) was exactly twice the sum of the coefficients of terms associated with (neutron, spin up) or (proton, spin down). The latter t’wo must of course be equal on account of the symmet,ry of the theory. Finally, it appears that the sums of the coefficients associated with (proton, spin up) and (neutron, spin down) are approximately equal. The latter observat’ion is one which must be considered at this point as empirical. The other method of calculating the moments was again to calculate H”+, but this t’ime using as a basis the states described in I. To accomplish this we introducl- a set of functions F~,,~(axs’ such that H”c$ = c

F&(aXB’(I;~

, . . . , k,) U,Y(aXP)(kl , . . . , Ii,)+,

(14)

where the sum is taken over y, q, o(, /c?and the k’s and the U’s are the states described in I. It is not difficult to establish difference equations which relate the F(,, to the F+l, , and which may then be used t’o calculate the F’s from the initial condition Ho+ = 4. It’ will he observed that there are many fewer states in this descript,ion than in the preceding one. The functions F are tabulated in Appendix I. There are four possible values for the angular momentum and isotopic spin eigenvalues of the meson cloud : (0,O) , ( 1,O) , (0,l) , and ( 1 ,I ) . We wish to show that the two for which the angular momentum and isotopic spin are different make very small contributions. The functions F cIL)are essentially polynomials in the meson energies. If the F’s belong to a symmetric representation of the permutation group all the coefficients in the polynomials will have t,he same sign. For any ‘other representation the sum of the coefficients will vanish; consequently the contributions of the nonsymmetric representations to any matrix element are likely to be very small since t,here exist many possibilities for cancellations. In particular it will be recalled t’hat the moments are reducible t)o sums of products of the integrals I, , with numerical coefficients. Each state U,Y’““’ makes its own contribution to the moment and the sum of the coefficients of the 1’s in this contribution is just the square of the sum of Dhe coefficients of the corresponding F, y(a>
162

HALPERN,

SARTORI,

NISHIMURA,

AND

SPITZER

contained a factor e in the coefficient used to define the st’ate U. Thus, the states of angular momentum zero and one never belong to the same representat’ion of the permutation group. From a state that belongs to angular momentum zero and isotopic spin one, or vice versa, it is never possible to form a symmetric state and therefore, the states (0,l) and (1,O) make much smaller contributions than do (0,O) and (1,l). Our observation that the sum of the coefficients for (neutron, spin up) and (proton, spin down) is equal to the coefficient for (neutron, spin down) is readily justified by the result that the sum of the coefficients for (0,l) and (1,0) is zero. In fact the contributions to (neut,ron, spin up) and (proton, spin down) are ,24 (1,l) + 3; (OJ) + ?,6 (1,l) + ?i (1,O) = $6 (1,l) which is of course the contribution of (neutron, spin down). The moments may be calculated with some effort from the functions F and are tabulated in Appendix II. The self energy of the nucleon (which is just the energy eigenvalue) is also tabulated, with the deviation c defined in Eq. (1) . III.

MATRIX

ELEMEBTS

OF

THE

NUCLEON

OPERATORS

The probabilities PO0, PO1 , PI0 , and PII that the cloud is in one of the four possible states determine the matrix elements of the nucleon operators. Since PO0 + PI0 + PO, + Pll = 1

(15)

PllJ = PO1 w 0,

(16)

and, as we have seen,

it is only necessary to determine the relative magnitudes of PII and PO0to determine completely all the nucleon operators. We can determine PII and PO0approximately from our earlier observation t,hat the contributions of (neutron, spin down) and (proton, spin up) are roughly equal. The (proton, spin up) contribution is PO0 + 93’ POT + 23’ PM + ,16 PI1 z PO0 + $6 PII and this equals the (neutron, spin down) contribution $9 PI1 . Thus PO0 = $4 PI1 or P 00* 44,

P 11

The matrix elements of the nucleon terms of these probabilities:

w

34.

operators

may be simply

obtained

(a j uirx 1/I> = (& / uin / &s)(Poo - $6 PI0 - 35 PO1 + $9’ Pll), (a)

I ui

1 P)

=

@a

I ui

I ~o)[poo

+

PO1

-

45

(PlO

+

ml,

in (17)

PHYSICAL

163

NUCLEON

where t,h’e 1CX) and 1 /?) are real-nucleon states and 1$a) and 14~) are the corresponding bare-nucleon states (10, 11). From t)he above relations and the previously obtained value of the P’s we can find an approximate expression for t.he charge renormalization:

which is indeed the strong-coupling limit (12). ‘3 , imilarly, the scalar part of the anomalous magnetic moment is (3) A = ?i(p 1U?- 1 / p) = -?:i(P10 + PII) = -15,

(19)

where 1p) is the physical proton with spin +js. These values are approximately true for rail large values of the coupling constant and cutoff. Large in this case means th.nt f& - 2, where K is the cutoff. Table I contains a summary of our results for the ratio of bhe renormalized to the unrenormalieed coupling constant. The ratio jr/f0 has also been calculated with an exponential cutoff and the results are in substantial agreement with the values in Table I. Our values should be compared with t)he value jr/j0 = 0.65 for ft = 0.19 obt’ained by Miyazawa (3), who used an exact sum rule that relates the scattering cross section to the ( p). Friedman et al. (12) have also calculated t.his ratio matrix element, (P j (~~7~ and obtain the value jr/f0 = 0.38 for an unrenormalized coupling constant fo’ = 0.712 and a cut’off at K = 6.13~. Table II lists the scalar part of the anomalous magnetic moment for several values of the coupling constant. It should be noted that t,he very large disagreement between the theory and the experimental value pLs= -0.06 is inherent in the kinematics of the static model. Sachs (10) has pointed out, that the total P wave part of the cloud must be about 9 % to give the correct scalar moment, TABLE

I

THE RATIO OF THE RENORMALIZED COUPLISC CONSTANT TO THE UNRENORMALIZED COUPLIKG COSSTANT FOR SEVERAL VALUES OF THE UNRENORMALIZED COUPLING CONSTANT The cutoff is square our approximation.

at K = 6~. The numbers

at the top

of the columns

indicate

the order

Order

fu’ 0 0.2 0.4 0.6

1

2

3

4

5

1 0.690 0.653 0.636

1 0.554 0.496 0.471

1 0.505 0.534 0.405

1 0.492 0.417 0.386

1 0.488 0.412 0.383

of

lb4

HALPERN,

SARTORI,

NISHIMURA,

TABLE THE

SCALAR PART COUPLING

OF THE ANOMALOUS CONSTANT fo AND

AND

SPITZER

II

MAGNETIC MOMENT As A FUNCTIOS THE ORDER OF APPROSlM.4TION 1~

OF THE

Order

fo2

1

0.2 0.4 0.6

2

3

-0.334 -0.378 -0.396

-0.232 -0.260 -0.272

4

-0.371 -0.423 -0.446

TABLE

-0.381 -0.437 -0.460

5 -0.383 -0.440 -0.462

III

THE

PROBABILITIES FOUR AVAILABLE

OF THE MESON CLOUD ANGULAR MOMENTUM

OF THE NUCLEOX BEING IN THE AND ISOTOPIC SPIN STATES

fo"

Pll

Pm

PI0

0.2 0.4 0.6

0.575 0.660 0.693

0.00045 0.00048 0.00045

0.00045 0.00048 0.00045

PO0 0.424 0.339 0.306

whereas our kinematic arguments show that for all but the weakest coupling the probability of finding the cloud in a P state is about 75 %. Our results are in close agreement with those of Miyazawa (3). The mesic renormalizat’ion of P-decay, because of t,he symmetry between angular momentum and isotopic spin, involves essentially the same matrix element as the scalar moment and hence is also unsatisfactorily predicted by the static model (11). For completeness we list in Table III the probabilities of the four states of the cloud for our best (5th) approximation. The bare nucleon state is included in the (0, 0) state. It is not clear from our data whether or not there is a maximum in the (0,l) state at fo2 = 0.4. From the rapid convergence of these probabilities as evidenced in Tables I and II, our values for POOand PII are probably correct to within several parts in a thousand. IV.

MATRIX

ELEMENTS

OF

THE

MESON

OPERATORS

In this section we consider the meson contribution to anomalous magnetic moment, the mean square charge radius of the neutron and the probability of finding a given number of mesons in the cloud. The first two quantities may be directly compared with experiment. The third is not of this type, but it has a certain intuitive appeal that makes its calculation worthwhile. All three quantities are directly expressible as bilinear expressions in the meson creation and an-

PHYSICAL

165

NCCLEON

nihilation operators. The expressions for the anomalous moment are given by Sachs (119) and Miyazawa (3), those for the charge radii by Sachs (IO), Salzman (4): and Treiman and Sachs (4). The probability of finding n mesons in the cloud is t’he norm of the n meson component of a normalized nucleon wave funct,ion. For the calculat’ion of the meson matrix elements there is no simple unifying idea analogous to our kinematic considerations of the previous section. We are therefore obliged to calculat,e these quantities by more direct methods. Our wave function in nth order is of the form

where the coefficients ai have been determined the matrix element of an operator Q is

by the variational

principle.

Thus,

(J/ I Q I ti) = C aiajQii I where

Qij = (4 / Hi&H’

(21)

/ 4).

The Qij may be computed directly, although the effort required increases rspidly with the order of approximation. All these quantit,ies may be expressed in terms of the integrals I, defined in Eq. (13). We shall not give the details of the calculations for the Q’S, but only state the results for the quantities of physical interest. Unlike the matrix elements of the nucleon operators, those of the meson operators are unfortunately quite slowly convergent. After five steps in our approximation procedure all of them are clearly converging, but a further change as large as I O-20 percent, before ultimat’e convergence, appears probable. However, since t,he dependence of these quantities on the order of approximation is quite smooth, we feel justified in extrapolating the curves to estimate the “value at The vector part of t,he anomalous moment has two contributions convergence”. [Rliyazawa’s ccl and ~2 (S)], one similar to the coupling constant renormalization and the ot,her to the contribution of the meson current. Table IV lists t’he value TABLE IV CALCULATED AND EXTRAPOLATED VALVES OF THE VECTOR PART OF THE NUCLEON ANOMALOUS MAGNETIC MOMENT f"

So'

1

2

3

4

5

0.0476 0.0678 0.0880

0.2 0.4 0.6

0.068 0.076 0.080

0.392 0.433 0.450

0.617 0.724 0.770

0.759 0.965 1.070

0.839 1.146 1.332

IXxtrZLp. 0.95 1.45 2.00

lG(i

HALFERN,

SAHTORI,

NISHIMURA,

ASD

SPITZER

of t*he sum of these two contjrihutions in each of the orders we have calculated and the extrapolated value for three values of the coupling constant. We est,imate that t.he extrapolat.ed values represent. t.he predictions of the t.heory to within 15 %. The estrapolnt.ion has been performed by eye, by fitt.ing a curve of the form p(n) = ~[l - CCrL-‘)] + p(l), and by using a polynomial. For a value ft = 0.55 which corresponds to fi” z 0.08 the experimental value of the magnetic moment EL, = I .85 is obtained. The electron-neutron interaction can be calculated with our wave function by using Eq. (16) of Treiman and Sachs (4). The calculation is very similar to that of the magnetic moment,. The result is given either in terms of R”, t)he mean square charge radius of t.he ueut.ron, or of Tie , an energy which is conventionally used to measure the electron-neutron interaction:

V”=go0z27

(22)

where r. is the classical electron radius. This calculation was carried only to fourth order, and the extrapolated values are therefore much less reliable. The results are given in Table V. Perhaps the most interesting aspect of these results is that the charge radius appears to be ext,remely insensitive to the value of the coupling constant. The interaction V, seems to be converging to a value somewhat smaller than the 10 kev calculated by Treiman and Sachs (4)) but still very much larger than the experimental value, which is essentially zero when account is taken of the Foldy term. Finally, we consider the probability p, that the cloud contain r mesons. Here again the results are not definitive. However, p, as a function of the order of approximation has a simple behavior and it is possible to state someinequalities. In particular it may be observed empirically that p, as a function of the order of approximation has a single maximum, and when this is passedit is a decreasing function. In Table VI below are listed our results for p,(n), the probability of r mesons in the nth order wave function. TABLE ELECTROK-NEUTRON INTERACTION. AS DEFINED IN Eq.

I’

THE QUANTITY (22), IN UNITS

TABULATED OF kev

Is

VO ,

Order

fo” 0.2 0.4 0.6

Extrap. 1

2

3

4

0.82 0.91 0.96

1.98 2.01 2.00

2.76 2.82 2.78

3.40 3.60 3.57

-5 -6 -6

PHYSICAL

TABLE THE

!?ROBABILITY

p,(n)

OF r ~~ESONS IN THE OF J'?. ? Is THE MEAX

167

NUCLEON

VI nth APPRoSIMATIOS NUMBER OF MESONS

FOR

SEVERAL

VUIJES

1 n

0.2

0.4

0.6

f 0

1

1

0.652

2 3 4 5

0.4% 0.409 0.379 0.367

0.348 0.424 0.416 0.404 0.399

1

0.609

0.391

2 3 4 5

0.406 0.312 0.262 0.235

0.460 0.425 0.387 0.361

1 2 3

0.590 0.374 0.269 0.207 0.170

0.410

4

5

0.473

0.420 0.360 0.314

2

3

4

5

0.09i

0.150 0.166 0.172

0.026 0.044 0.050

0.019 0 .Oll

0.002

0.35 0.62 0.79 0.89 0.95 0.39

0.133 0.214 0.239 0.245

0.153 0.218 0.273 0.273

0.048 0.093 0.115

0.019 0.038

0.007

0.73 1.00 1.22 1.38 0.41 0.78

0.063 0.130 0.163

1.11

0.031 0.066

0.015

1.42 1.68

There are a variety of conclusions that may be drawn from Table VI. For example, it clearly gives upper bounds on ~0 and pl for all three values of the coupling constant, as well as lower bounds on 7, the average number of mesons, We have made no detailed attempt to extract all possible information from Table VI. However, it is worth noting that there are probably more mesons in the cloud than might be suggested by the success of the one-meson approximation of Chew and Low (1). There is, of course, a difference between real and virtual mesons. It does appear that states with as many as four virtual mesons make . . significant contributions to the nucleon wave function. This conclusion is in agreement with the argument of Cini and Fubini (5). V.

COSCLUSIONS

By constructing a good spproximatSion to the ground state of static source symmetric pseudoscalar meson theory, we have been able to examine some of the consequences of this theory. It has been found that those quantities which can be described by matrix elements of the nucleon operators appear to give artificial results in this theory. The matris elements are determined to a larger extent by the kinematic restrictions of the model than by the dynamical properties. It is possible t.o understand, in this context, the consist.ently bad agreement

I68

HALPERN,

SARTORI,

NISHlMURA,

AND

SPITZER

of t’he scalar part of nucleon anomalous magnetic moment with the experimental values. On the other hand, the model is quite successful in predicting the vector part of the nucleon anomalous magnet’ic moment,. To what extent this is fortuitous is by no means clear. Unfortunately, there are very few experimental numbers wit’h which the ground state wave funct,ion may be tested, and thus it is hard to draw any very firm conclusion. Slightly more information may be obtained by comparing the matrix elements of the nucleon operators calculated directly by the method of Section III, with the values obtained by integrating experimental cross sections. These sum rules have been derived by several authors and we use two of them in a form given by Miyazama (3). They are:

and

(23)

The right-hand sides have b 0th beent calculated by Miyazawu using the experimental cross sections and are 0.11 and 0.068, respectively. If it is assumed that the static model predicts the experimental cross sections, then the first rule implies that fl - fi-’ = 0.11. However, we have already obtained a relation between fl and f: (Table I). This relation suffices to determine uniquely both fo and fr2 if their difference is known. For f 02 - jr2 = 0.11 it turns out thatfo2 = 0.15 and fr” = 0.04. The latter is just half of the accepted value. On the other hand if jr2 is set equal to 0.08, then f~’ = 0.53. The integral must then be 0.45 or about four times its value as calculated with experimental cross sections. From the second sum rule it is again possible to calculat’e ft and jr”. If the experimental cross sections are assumed to be the predictions of the model the result is that fez = 0.14 and jr* = 0.039. On the other hand one may again insist that f: = 0.08 and calculate the left-hand side. This gives a value of 0.39 for the integral. Both sum rules thus lead to an inconsistency if the experimental cross sections are taken to be the predictions of the static model. We are led to the conclusions, already mentioned in the introduction, which cast doubt on the validity of the model. Other similar tests may be employed but these are less sensitive if the Born approximation term is the bulk of the effect as occurs, for example, for t.he vector moment.

PHYSICAL

169

NUCLEOK

APPENDIX

I

The constant coefficient of the bare nucleon term in H”4 is just the moment H, ; that is H”c$ = H,+ + . .- .

(Al)

These are tabulated in the next appendix. There is a single one-mesonst,ate. The wave function in this state will be called F(,, jk) ; that IS, H”+ = H,+ + c a:(k)F&k)criw$ F(ll,(X:)

= 0,

F(l)(k)

= v*(k),

F(,)(k)

= v*(k)w,

+ . .- ,

)

(AZ)

C-43)

F(3)(k) = v*(k)cLk2 + lYl*) F(J)(k) = V*(k)cLQ3 + 38wJz + 2013) FcS)(li) = V*(&,”

+ 66~~~12+ 5OwJ3 +

2114

+

487(L)‘.

The two, three and four meson states are described in I and the labeling and normalizations are the same as that given there. Subscripts on W’Sand V’s will indicate the k of which they are a function, e.g., 8, = V(k2). F(O) = F’(l) = 0, F@,2(2 :< 2) = 3v1*v2*, F&P

x 2) = F&2

x P) = 0,

F@,2(1’ x 1’) = -2vI*v?*, F@J2(2 x 2) = 9~vl*v2*(w F&l”

x 2) = F&2

+ w2), x 1”) = - 2

F@,2(1” ‘X 1”) = -3vL*Vs*(wI Fc4)2(2 >< 2) = V1*Vz.*[lliIp

V1*vP*(w

- WP),

+ COP), + (6~012+ Sum + (iwz’)],

F~4)12(1’X 2) = Fc4,1”(2 X 1’) = -i<6v,*V2*(w~

- wz’),

Fc4,2(12 X 1”) = - V1*Vz*[6012 + 4w12+ 6w1w2+ 4w2’], F,,2(2

>: 2) = l”l*V7.*[17413 + 285(03 + b&)12+ 15/2wl” + 15w1’wz+ 15w,0~~+ 15/2w2”j,

170

HALPERN,

F~J2(1”

SARTORI,

NISHIMURA,

AKD

SPITZEH.

X 2) = F~5,12(2 X P) iv% -:- --2

h2(1”

Vl*Vz*[46(

Wl - wz)lz + 3W13+ 2wl”wr - 2w,w,2 - 3w,3] )

x 1”)

= -V1*V,*[681a

+ 15O(w1 + wz)I2 + 50113+ lOq”wz + lOwlo;” + 5,:].

In what follows those states with zero amplitude are not listed, and when two states have the same amplitude by spin-isotopic spin symmetry only one is noted. The obvious factor VI*V2* . . . is also omitted. F,3,3(3

x

3)

=

,5&

F
F&(21

x

3)

=

ly;(wl

+

x 21) = y

Fw3(13

x 13) = -1qwl

Fw21(13

2d’ x 21) -$-

w2

+

w3),

(Wl + wz + w3) ) + 02 + w3), i(w,

+ w2 -

2w3) 7

F,,,,21(13 x 21) = 21/Zi(WI - UP), F@,3(3 x 3) = 29gr, + 596( WI2+ 02’ + wz2)+ 2?3( w1w*+ wzw3+ W3Wl)) F,,,3(21 X 21) = .\/“[36f%iIz

f

*Y~$(wI”

+

w??

+

(A5)

We)

+ 40,. /3(WlW2 F&(13

Fc5)121(13X 21) = 48~i(w1

+

x 21) = 51/2i(a

Fc,,,21(3 X 21) = - g F,,z21(3

wzw3

+

WOWl),

X 1”) = -[3001* + 2O(w;L + w*3+ WJZ> +

F&w13

+

X 21) = -

F[6)121(21 x 21) = y

w2

w3)(w1

+

w2

+

w2w3

-

2w3),

+

3wzW3

+

w3wl)],

+ wz + oh) (WI - a), (w12+ w,’

I&

w;-

9

+

:3o(WlWZ

(

wz”

-

2~3~

+

-

3w3w2

6~1~2

f

3~1~3)

,

- 3WIW3))

(w12+ w; - 2~3’ - 6~1~2+

3~1~3

+

3~203)

,

PHTSKAL

171

NUCLEON

(A(3)

F@,,31(31

x 2”) = 4LgB

F@,231(31 x a”> = y F(b)331 (31 x a”> =

2i&G

y

FC6)l31(4 x 31) = -~ “yy Fc,,z3l(4

X 31) = - 9

F@,,31(4

x 31) = -*

(w1 + wl - 2W3)) (WI - W2))

(WI

+

w2

+ wa -

3WJ)

(WI + w2 - 2wd ) (WI - W2)) (WI + W2+ w3 - w4) )

To construct the nucleon wave function it is necessary to specify the coefficients czi of Ey. (20). For the three values of the coupling constant f~” that have been studied these are listed in Table VII.

172

HALPERN,

SARTORI,

NISHIMURA,

TABLE COEFFICIENTS

OF THE

ANI)

VII

POLYNOMIALS

OF EQ.

0.2

fu”

a5

I

a4

-9.9162

CL3 a2 a1 a0

II.

THE

(20)

0.4

0.6

1

1

-7.7500 17.9025 -9.4504 -6.2861 +3.2569

32.1745 - 37.7525 9.1223 3.5269

APPENDIX

SPITZER

MOMENTS,

SELF-ENERGIES,

-6.7257 12.3696 -1.9942 -7.8868 1.9632

AND

DEVIATIONS

H,, = 1, HI = 0, Ha/9 = Iz , Hd9

= Is ,

H4/9 = 19Iz” + I4, H,,‘B = 58IJ:! + Is , Hs/9

= 4872; + 871417 + 5OI,” + Is ,

H,/9

= 2977IJ;L

+ 125IJ2 + 1711413 + IT,

H,,‘Q = 15,315I; HJ9

+ 5954141: + 6712I;Iz

= 157,8041&

+ ll,WlI&

+ 1721612 + 28OIJz + 1631: + Is,

+ 29,804IJJz

+ 5,624Ia

+ 2281712 + 4361,jIs + 59OIsI., + 19 , TABLE THE

SELF-ENERGY

AND Q IN UNITS CONSTANT f~*

OF MESON AKD ORDER

VIII 31~~~s AS FUSCTIONS OF CALCTJLATION n

OF THE

COUPLING

T

0

0.2 0.4 0.6

';' g _-___~~0 7.7 0 0

-I

11.0 13.4

1

-E 5.7 8.8 11.2

2

(T 4.9 7.3 9.1

-E 7.3 11.7 15.1

3

,J 2.9 4.8 6.3

-E _____ 7.8 12.8 10.8

0 ______

PHYSICAL

Hm’9

173

NUCLEON

572,0651,, + 394,510Z4Zz3 + 668,J60Z3”Z2’ + 19,350Z612’ + 61,560ZJJz

+ 36,255ZfZe + 41,41214Z3” + 293Z8Z2 + 64817Z3

+ l,OlOZ,jZl + 58215”ZlO, 8,965,8851:<1; + 901,170ZJ;

+ 3,651,450Z~Z~Z22

+ l,390,216Zp”Z~ + 31,726Z7Z;

+ 117,884ZJ3Z2

+ 168,058ZJ4Z2 + 93,66OZ$: + 110,79lZ&

+ 3671913 + 9251$3

+ 1,642Z7Z4 + 2158ZtjZb + Zu , 24,617,051Z26 + 26,897,655Z41: + 1,885,4601&

+ 61,333,64OZ~“Z2”

+ 9,237,6-161$&

+ 5,217,8491tZp

+ 12,404,03OZ4Z33Z2 + 1,196,OOSZ; + 49,477IJz + 211,2161,13Ze + 338,2601$4Zz + 191,84OZ&” + 2,551IJd

+ 37841j15 + 2,150&

+ 197,996Z5”Z2

+ ZE .

Table VIII contains the self-energies and deviations for several values of the coupling constant as functions of t,he order of approximation. RECEIVED:

February

20, 1959 REFERENCES

1. 6.

F. CHEW AND F. 15. Low, Phys. f2ev. 101, 1570 (1956); G. C. WICK, new. Modern Phys. 27, 339 (1955). 2. G. F. #CHEW AND F. E. Low, Phys. Rev. 101, 1579 (1956). 3. H. MIYAZAWA, Phys. Rev. 101, 1564 (1956). 4. G. SALZMAN, Phys. Rev. 99,973 (1955); S. B. TREIMAN AND R. G. SACHS, Phys. Rev. 103, 435 (1956). 5. M. CI.VI AND S. FUBINI, Phys. Rev. 102, 1687 (1956); Nuovo cimento 3, 764 (1955). 6. F. R. HALPERN, Phys. Rev., 107, 1145 (1957). 7. F. R. HALPERN, Phys. Rev. 111, 1 (1958). 8. F. R. HALPERN Phys. Rev. 109, 1836 (1958). 9. F. R. HALPERN, preceding paper [Annals of Physics 7, 146 (1959)]. IO. R. G. SACHS, Phys. Rev. 87, 1100 (1952). 11. M. Ross, Phys. Rev. 104, 1736 (1956). 12. &I. H. FRIEDMAN, T. D. LEE, AND R. CHRISTIAN, Phys. Rev. 100, 1494 (1955).