Electromagnetic polarisabilities of the proton in an independent particle potential model

ELSEVIER Nuclear Physics A 605 (1996) 433-457

N U C L E A R PHYSICS A

Electromagnetic polarisabilities of the proton in an independent particle potential model

N. Barik a, B.K. Dash a,1, p. Das b, A.R. Panda b a P.G. Department of Physics, Utkal University, Bhubaneswer-751004, India

b Department of Physws, Kendrapara College, Kendrapara-754211, India

Received 2 August 1994; revised 1 February 1996

Abstract

We consider the electric and magnetic polarisabilities of the proton including the valence quark as well as pion dressing effects in an independent quark model with an effective scalar-vector harmonic potential which renders the solvability of relativistic Dirac equations for confined quarks, and has in fact acted as an altemative to the Cloudy Bag Model (CBM). This model which has been applied successfully to a variety of hadronic problems is also observed here to yield the electromagnetic polarisabilities of the proton satisfactorily without any free parameters. The electric and magnetic polarisabilities, including valence quark core and pion cloud effects, obtained here as ~ p = 14.074× l0 -4 fm 3 and ~ p = 3 . 1 5 5 × l0 - a f m 3, are well within the uncertainties of their experimental measurements of (10.9 __ 2.2 _+ 1.4) × 10 -4 fm 3 and (3.3 _ 2.2 -I- 1.4) × 10 -4 fm 3, respectively and are also in agreement with other model estimations.

PACS: 12.40 Aa; 12.40 Qq; 14.20 Dh

Keywords: Electromagnetic polarizability, Bound-quark-orbitals; Quark-Core; Pion-cloud; Pion-polanizability

1. Introduction

The study of the electromagnetic polarisabili ty of a nucleon is quite fundamental in

understanding its intrinsic structure. The empirical values [1] of electric and magnetic

polarisabil i t ies of proton when observed individually were found to have large uncertain-

ties [2], but their sum in fact is well determined in a model independent manner [3]

through a dispersion sum rule and is found to be

~v + ~ P = (14.2 _ 3) × 10 -4 fm 3, (1 ,1)

Present address, Department of Physics, Government College, Sundergarh-770002, India.

0375-9474/96//$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PII S0375-9474(96)00103-0

434 N. Barik et a l . / Nuclear Physics A 605 (1996) 433-457

where the proper electric and magnetic polarisabilities are [4]

OLp = ~ p - - - - ( r 2 ) c h , 3Mp

Ot

(1.2a)

and

tip = (10.9 + 2.2 _ 1.4) × 10 -4 fm 3

~p ~-" (3.3 ___ 2.2 + 1.4) × 10 -4 fm 3,

which has also been confirmed by a more recent measurement [7]. Such measurements have generated fresh interest in this area of study resulting in a number of recent

publications [8]. Like any other low energy hadronic phenomena, electromagnetic polarisability cannot

be investigated in a straightforward manner by the first principle application of Quantum ChromoDynamics (QCD). Therefore, various phenomenological models for hadrons [9-13] incorporating the basic features of QCD, have been tried to estimate the electromagnetic polarisabilities. For example, calculations [9-11] based on the original version of the MIT bag model [14] predict etp = 7.1 × 10 -4 fm 3, which is in reasonable agreement with experiment, whereas in a Chiral Quark Model (×QM) [15] the prediction for up = 16 × 10 -4 fm 3 which is higher than the corresponding experimental value.

But if additional contributions due to pion cloud effects are taken into account in xQM-calculations with a cubic potential [13], one obtains up = (7-9) × 10 -4 fm 3.

In the present investigation, we calculate the electromagnetic polarisabilities of the proton in a potential model of relativistic independent quarks which has provided a useful alternative to the Cloudy Bag Model (CBM) [16]. Such a model has been quite successful in estimating the leptonic decay widths and decay constants of light neutral vector mesons [17]. It also explains successfully the radiative transitions between low lying vector and pseudoscalar mesons [18]. Recently the model has also been applied to weak leptonic decays of pseudoscalar mesons [19] to produce the corresponding weak decay constants quite in agreement with experiment. Further, the present potential model has been used earlier in the study of several low energy phenomena in the baryonic

[3p = ~p - - - (rZ)ch, (1.2b) 2Mp

where (rZ)eh is the mean square charge radius, Mp is the mass of the proton, (x = (1/137) and the negative terms in (1.2) are due to the retardation effects. From an analysis of observed data [5] it is found that

~ e - ~P = (10 + 5) × 10 -4 fm 3. (1.3)

Thus, from (1.1) and (1.3) along with experimental value of (~2\1/2 - /oh =0.83 fm: one finds that ~e = (8.6 + 2.6) × 10 -4 fm 3. But the magnetic polarisability, when deduced

in the same manner, is found to be very small with quite large experimental uncertainties. However, there is a recent measurement of individual electromagnetic polarisabili-

ties of the proton [6] reporting the values as

N. Barik et aL /Nuclear Physics A 605 (1996) 433-457 435

sector such as octet baryon masses [20], magnetic moments [21], weak electric form

factors [22], as well as nucleon electromagnetic form factors and charge radii [23] with a reasonable amount of success. Also application of this model in the ordinary light meson sector to estimate the (q~) pionic mass in consistency with that of the PCAC-pion [24] and (p-w) as well as (p-o~) mass splittings [25] has achieved remarkable success. In

view of the success of the model in such wide ranging phenomena in baryon as well as

meson sectors, we use it here for the estimation of electromagnetic polarisabilities of the proton. It has also been observed in Chiral Perturbation Theory (xPT) [8] estimation that the pionic contribution towards the magnetic polarisability of proton is quite significant

and so for the sake of completeness we also estimate in the present investigation the electromagnetic polarisabilities from the valence core as well as the pionic cloud effects.

The present paper is organised as follows. In Section 2 we describe the potential

model and fix up our notations to be used in the later sections. In Sections 3 and 4 we calculate the core contributions towards electric and magnetic polarisabilities of the proton, respectively. In Section 5 we calculate the pionic contributions towards the electric and magnetic polarisabilities of the proton. In Section 6 we discuss our results along with a comparison with other model calculations and experimental data.

2. Potential model

We now describe the potential model and its applications which we use for our calculations in later sections. In nucleons, for low energy phenomena, the long range non-perturbative multigluonic interaction plays a dominant role compared to the short- range Coulomb-like one-gluon exchange interaction. In the absence of adequate knowl- edge about long range quark-quark interaction, we choose from a phenomenological point of view a flavor-independent potential U(r) confining the constituent quarks inside the hadron core. Such a potential may be expressed as an admixture of equal scalar and vector parts in harmonic form for later simplifications and is written as

with

U(r) = ½(1 +~l°)V(r) (2.1)

V(r)=ar2+Vo; a > 0 ,

where a and V 0 are potential parameters. The zeroth order quark dynamics in core level is described by the independent quark Lagrangian density

.~qO(r) = ½i~( r)~'~q( r) - q( r ) (mq + U( r) )q( r) (2.2)

which subsequently leads to Dirac equation for individual quarks of m a s s mq as

["/°Eq - ~' "p - mq - U ( r ) ] * q ( r ) = 0, (2.3)

436 N. Barik et al./Nuclear Physics A 605 (1996) 433-457

where qq(r) is quark wave function written in two component form as

(2.4

We may note here that G,.,(r) and +B(r), the two components of *$r), have opposite p&y witi respect to each other. Further, these two components are the eigenfunctions

of operators J2, Jz, (a .L + 1) and L* with eigen values xj+ 1), mj, +K, l,(f, -I- 1)

and 1,(Z, + l), respectively, with I, and I, as the angular momenta of the states +*(r) and *n(r). For decoupling of the Dirac equation and separating the radial and angular

parts, we take

f(r) h(r) = - r ‘zq(f) (2.5)

and

where yjr(i) are normalised spin angular functions formed by Pauli spinors with spherical harmonics of order 1, which for example are written for j = I + f as:

andfor j=Z-4 as:

$&;:;“,(?) = _ \/(17;;,=::2)yTI-Il*(i)XT

(2.6a)

(2.6b)

where

1 XT= 0 ( 1

and x1 = 0

( 1 1 .

Eliminating the spin-angular parts in the conventional manner, one obtains the coupled equations for the reduced radial functions f(r) and g(r) as

[IT,-m,-V(r)]f(r)+q+$g(r)=O

and

(2.7)

df( r) --:f(r)-(Eq+mq)g(r)=O.

dr (2.8)

N. Barik et al./ Nuclear Physics A 605 (1996) 433-457 437

The eigen values K here are explicitly given by

K = [ ( j + 1 / 2 ) f o r l A = ( j - 1 / 2 ) a n d l B = ( j + I / 2 ) ,

- ( j + 1 /2) for l A = ( j + 1 /2) and 1B = ( j - 1 /2 ) .

Now, the evaluation of g(r) from Eq. (2.8) and its substitution in (2.7) yields,

d2f ( r ) [ 1 ( l+ 1) ] dr-----T-+ (Eq + m q ) ( E q - m q - V ( r ) ) r2 f ( r ) = 0 . (2.9)

Defining

( E ' q + m ' q ) = ( E q + m q ) = h q ,

with

E ' q = ( E q - V o / 2 ) and t~q=(mq +go /2 ) ,

the reduced radial equation (2.9) can be expressed in general form as

f,n,u( r) + [hq( E,q_ mq) _ C2r 2 , l ( l + 1) ] r 2 L l j ( r ) = 0, (2.10)

where c 2 = hqa. Using the asymptotic behaviour as well as the behaviour at the origin one can express the solution of Eq. (2.10) as

f ( r ) = r `+' exp( -cr2 /2)~(r ) , (2.11)

which subsequently leads to a well behaved and conventional solution as

f ( r ) =Ar l+1 exp( -cr2 /2)F(a , b, cr2). (2.12)

The asymptotic behaviour of the solution leads to

a = - n r = - ( n - I ) ; n = 1 , 2 , 3 ,

which as a result gives rise to the bound state conditions for the individual quarks as

(E'q - re'q) = (4n + 2 l - 1). (2.13)

Further, expressing the hypergeometric function in terms of the associated Laguerre polynomial, we obtain the solution for individual quark state as

fn,(r) = Ar t+' exp( - r 2/2 ~2 ) ( 1 + 1/2) !( n - 1 ) ! L t+ l /2/ ' rZ/r2 ] (2.14) ( n + l - 1 / 2 ) ! n-I k / n l ] '

where ?.t = (a~-q)-1/4= 1/ggC ", a state dependent length parameter. Now introducing an over all normalisation

- 1 N.t=A ( l + l / 2 ) ! ( n 1)! (rm) ' (,, + t - 1/2)!

Eq. (2.14) reduces to

r / / 2 - 2 1+1/2 2 -2 f . t ( r )=N. t ~ 1 r e x p ( - r /2r . t )L ._ , (r /r,,,). (2.15)

438 N. Barik et al. /Nuclear Physics A 605 (1996) 433-457

Next for the sake of completeness our evaluation of g,Jr) for two cases i.e. j = (1+ l/2) and j = (I - l/2) are written as follows for j = (1+ l/2):

g,,(r)= -F f ( i /+2

exp(-r2/27,2,)[L~‘j2(r2/F~,) +21C’,+_~[*(r~/Yi,)] nl nl

(2.16)

for j=(I- l/2):

g,,(r)=? f ( i I

exp( -r*/27:,) nl nl

x [ ( 12 + I - l/2) LLY’/2( ‘2/F,Z[) + nL’,_ I/* (‘2/Y;,)] . (2.17)

Thus, using normalisation condition

/ qq( ~-)‘?~(r) dr= 1,

along with the previously defined functions for q&r) one obtains the over all normalisation constant more specifically as

iV;=l 4r( n) x nl

r,‘, r( n + 1+ l/2) (3Eb + mb) * (2.18)

Now the positive and negative energy quark orbitals for definite quantum numbers n, I, j are also written as

*:$)( ‘I= L ifnlj( r> /r (u. ;) g,,j( r),r qljm,( ‘1 1

and

where the angular parts are explicitly written as

[ [ /~Y,.,-r/2(i)X,

(2.19)

N. Barik et aL / Nuclear Physics A 605 (1996) 433-457 439

We may note here that the above solutions form a complete set of orthogonal states leading to the appropriate wave functions for the quarks and hence will be used in our investigation in the later sections.

3. Electric polarisability of the proton due to quark core

With the model described in the section before, we consider now the calculation of the valence quark core contribution towards the electric polarisability of the proton. To do so, one considers the proton in an external electric field which affects the quark orbitals in the core and subsequently gives rise to a shift in the quark binding energy of the second order. From such a consideration it is straightforward to obtain an expression for the electric polarisability due to the valency quark core of the proton [14] as:

1 I<~, -+ I~qrq cos 010)12 at+-= 2rr ~-" ~-~ ( E v ~ - E o ) ' (3.1)

q v÷0

where c~_ and ~_ are respectively the core contributions towards polarisability from the positive (Fig. la) and negative frequency (Fig. lb) intermediate states Iv-+). The effective quark charge ~'q = (eq - ½Qe), with Q, e and eq as the charge of the proton, magnitude of the electron charge and charge of the quark, respectively. Such a definition is in fact necessary to avoid the spurious centre of mass motion. Further, rq is the position coordinate of the corresponding quark and (E,_+- E 0) is the appropriate energy denominator in the second order with E,_+ being the binding energies of positive and negative frequency quark orbitals.

The total electric polarisability due to core contribution is the sum total of positive and negative frequency contributions and is written as

O~pC __-- ~+c ..[_otc, (3.2)

with a complete set of intermediate states; a~_ and aL explicitly become,

1 I (nlj + [ eq rq cos 0 I t S 1 /2) ] 2 a~_= ~ E E ' , (3.3)

q nljm I Enl j+-- EIs i /2

and

1 ] (n l j - leqrq cos 011S 1/2) 12 = 3--4 E E '

q nljmj E n l j - - - EIS I/2 (3.4)

1 I l / 2 ( r ) ( r cos 0) d r l 2

E' (3.s) q nljmj Enlj+-- Els 1/2

where the symbol ]E'nl},nj indicates that (n, l, j) being (1, O, ½) is excluded from the summation. Eq. (3.3) can be written explicitly in terms of the positive frequency quark orbitals obtained in Eq. (2.19) as,

440 N. Barik et al . / Nuclear Physics A 605 (1996) 433-457

(a)

Fig. 1. (a) Contribution towards electromagnetic polarisabilities from quark core positive energy intermediate states; (b) contribution towards electromagnetic polarisabitities from quark core negative energy intermediate states.

The integral in the numerator of (3.5), when factorised into radial and angular integration parts, becomes,

f dr *'+..tj)t, r)tWl s l /2(r ) ( rq cos O) = l(n, 1, j )F(I , j, mj), (3.6)

where

de l(n, 1, j ) = fo [fmJ(r)flsl/2(r) + g"tj(r)glSl/2(r)]r dr, (3.7)

and

F(l, j, mj)= fo 2~r d~bfo~r dO cos 0 sin 0 q~1~r, fl P)tq~tj,,fl P)"

Thus, with (3.6) one writes (3.5) as

(3.8)

One may note here from Eq. (3.8) that only the odd /-quark orbitals contribute towards the o~_ in Eq. (3.9). However, higher I contributions are suppressed due to the

1 12(n, 1, j)F2(l, j, rnj) a~.= ~ E~ 2 E' (3.9)

nljmj Enlj+-- EIs 1/2

N. Barik et a l . / Nuclear Physics A 605 (1996) 433-457 441

corresponding large energy denominators and hence taking the leading contribution from l = 1, i.e. P state, one writes (3.9) explicitly as

e 2 [ I2 (n , 1, 1 /2 )F2(1 , 1 /2 , mj) O/'¢ ~ - - E

+ 31T [ Enp n>~ 1 I /2 -- EIS 1/2

12(n, 1, 3 / 2 ) F 2 ( 1 , 3 / 2 , mj) ]

+ Enp 3/2 - EIS 1/2 ] ' (3.10)

2 2 where in fact ~qe~q 2 =-~e has been taken. Next using the standard orthogonality of spherical harmonics when one evaluates the angular parts F(1, 1/2, m j) for m: = _+ 1 /2 and F(1, 3 /2 , mj) for m j = _+3/2 and _+ 1/2, one finds the only contributing terms F(1, 1/2, 1/2) and F(1, 3 /2 , 1/2) as 1/3 and v/2/3, respectively. When substituted

C in (3.10), one obtains et+ as

40L

n>~l

I2( nP 1/2)

EnP I/2 -- EIs 1/2

I2 (np 3/2) ] + 2 ( G 3)S-- el i:) " (3.11)

The present model having no spin-orbit splitting amounts to a degeneracy of P 1 /2 and P 3 / 2 energy levels. Thus taking this and effectively the leading-order contribution with n = 1, one obtains Eq. (3.11) to be

4~ 1 a + = 2--7- E i p - E , s [ I2(1p 1/2) +212(1P 3 / 2 ) ] . (3.12)

Using Eq. (3.7) one can evaluate I2(1p 1/2) and 12(1p 3 /2) as

[:? ]2 I 2 ( 1 P 1 / 2 ) = flpl/2(r)flsl/2(r)+glPl/2(r)glsl/2(r)}rdr

(3.13a)

and

[So ]2 12(1p 3 /2 ) = {flP 3/2(r)fls l /2( r ) + g l p 3/2(r)gls 1/2(r)} r dr .

(3.13b)

Substituting the functions f and g from Eqs. (2.15)-(2.17), the integrals in (3.13a) and (3.13b) can be obtained in the form,

9 x 26~klp 1 I2(1P 1 /2) 2

(3E'ms + m'q)(3E]p + m'q)(E'ls - m'q) "~p "qsP 1/2, (3.14a)

IZ( 1P 3 / 2 ) = 12( 1P 1/2)'q2sp 3/2/Xl~p 1/2, (3.14b)

442 N. Barik et al./ Nuclear Physics A 605 (1996) 433-457

where;

and

Rp = ( ~ l p / r l S "]- r l S / ~ l p ) ,

5/~k ip 3/hns n. P 1 . = 1 + X,s( s +

(3.14c)

(3.14d)

2 [ 3/hW ] 'lf]sP3/2 ~-- "qgPl/2 + ) t l S ~ S ]" (3.14e)

Similarly, we consider the negative frequency contribution towards polarisability which can be expressed as

1 ]j'e2,tj)(r)**,sl/2(r)l, " "" cos O) drl 2 a~-= Ee'~ E (3.15)

2rr q nljmj Enlj-- Els 1/2

Here again factorising the integral in the numerator of (3.15) into radial and angular integration parts we obtain,

f dr W,(~)(r)t*,s l/2(r)(rq cos 0) = K(n, 1, j)F'(l, j, mj), (3.16)

where

K(n, I, j)= £ {g.o(r)f, sl/2(r) +fmj(r)glsl/2(r)}r dr.

and

(3.17)

F'(l, j, mj)=(-1)J+mF' ftp,j_mj(~)*(cr.P)tpo,/2,/2(~) d~. (3.18)

In Eqs. (3.17) and (3.18) also the solution for quark orbitals of (2.20) has been used. Thus, with (3.16) one writes

-1 K2(n, l, j)F'2(1, j, mj) -2 • (3.19) ore- = E eq Enlj +

2"rr q nljmj EIS 1/2

Here also a scrutiny of Eqs. (3.17) indicates that only even l wave function contributes towards aL and further higher even l contributions are damped out. Thus, taking the leading contributions from S and D orbitals only one obtains

1 -2 ~ [ K2(n' O, 1/2)F'2(0, 1/2, mj) o~C = -- - - E eq [ E.s + g s

27i" q n>~ 1 1/2 1/2

K2(n, 2 .3 /2 )F '2 (2 , 3 /2 . mj) ] (3.20)

+ EnD 3/2 + Els 1/2 ]"


Using orthogonality of spherical harmonics one can evaluate the contributing angular parts F'(0, 1/2, - 1/2) and F'(2, 3 /2 , - 1/2) as 1/3 and ~ - / 3 , respectively. Now

2 2 O~c_ substitution of Eq ~q2 = 3e leads to as

e 2 KZ(nS 1/2) K2(nD 3 /2) ] a t = - - - E + 2 . (3.21) ] - 27w Ens + Els E~D + Els n>~l 1/2 1/2 3/2 1/2

Now ignoring the contributions due to states with n > 1 which are very small and taking effectively the leading order contribution with n = 1, one can obtain,

4a K2(IS 1/2) KZ(1D 3/2) ot ~_ + 2 (3.22)

27 2Ejs I/2 EID 3/2 q'- EIS ~/2

Further as in the positive frequency case a calculation of K2(1S 1/2) and K2(1D 3/2) gives

36 K2(1S 1/2)

3E'Is -t- m'q

and

(3.23)

t l /2 15 × 27 his 1 2 KZ(1D 3 /2 ) = (3E, D +m,q)(3E,i s +re'q) -~ID] "~"5-'qSDg/2,.,D (3.24)

where

R D = ( r lD/ r l s -4- r l s / r lD) , (3.25)

and

= 11 (, + t ' )//, + (3.26)

In fact, et~. and ed_ calculated in Eqs. (3.12) and (3.22) when substituted in Eq. (3.2) lead to the expressions for the total core contribution towards electric polarisability of

c We may note that we have considered here the negative energy the proton Cte. intermediary contribution et c_ explicitly from the quark orbitals of negative binding energy of the model. However, the same could also have been obtained from the consideration of the baryonic intermediaries of negative parity such as N *- [15]

4. Magnetic polarisability of proton due to quark core

In the presence of an external magnetic field, the proton can also undergo shifts in its energy levels which on parity considerations leads to second-order energy correction as

I(ml,,~l I n> 12 diem= E , (4.1)

m,'. E,~ --E.

444 N. Batik et a l . / Nuclear Physics A 605 (1996) 433-457

where "~l = IxzB is the magnetic interaction with the magnetic field B = (0, 0, B) and the z-component of the magnetic dipole moment is IX,. Thus,

AEm= ~ I(mll~zBIn) 12 (4.2)

m-~n Em - E n

Further, when explicitly written the dipole moment operator in quark space becomes,

1 3

tXz= -~ E ei~lo(i)(rqX V(i)),. , (4.3) i = 1

where i stands for the quarks in the proton core. Thus, with Eq. (4.3), Eq. (4.2)

becomes,

[ ( m l ~ ~., i i AEm= ~, " ' r q × ~ l ' i " z B ' n - ' 2 (4.4) )( ( ) ] I)1

m 6 n E m - E,,

Taking m to be the ground state, one obtains the energy shifts as

1 l(Oleq~o(i)(rqXV(i)) n l u ) l 2 ae0=- E E

q v~O Ev - Eo

On indentification of AE 0 as -2"rrf3~,B with 13~, as the core contribution towards the magnetic polarisability of the proton, one obtains

1 I(Oleq~o(i)(rqXV(i))z lv)[ 2

q v#O Ev-Eo (4.5)

Again as in the earlier case, to remove the center of mass effect one defines some effective charge ~q = ( e q - ~Qe), which subsequently yields to the above magnetic

polarisability as

1 I(Ol~q~lo(i)(rqXV(i))zlu)l 2 ~3~ = ~ E ~- E~ - E 0 (4.6)

q v ~ 0

Further, considering separately the positive and negative energy intermediate states, one

obtains

1 I(Ol~q~o(i)(rqX~l(i))zlv+-)[2 13~= ~ E E E~±-E0 , (4.7)

q v ~ 0

when the total quark core contribution towards magnetic polarisability is expressed as,

13~, = 13% + [3 c • (4.8)


To calculate 13+ one rewrites Eq. (4.7) including all the quantum numbers of the intermediate states as

1 J(1S 1/2lEq[Xa2-yctl][nlj +)12 13+= ~ ]~ E ' , (4.9)

q nljmj Enl j + - EIS 1/2

where the prime indicates exclusion of the 1S 1/2 state as intermediate one. Again the expectation value being taken here is that of a pseudoscalar operator requires one to consider the intermediate states to be of even parity such as I1D 3 /2 +) and similar other higher states. However, in the present investigation of [3+ the contribution from higher states is expected to be small and so we restrict ourself to only lID 3 /2) intermediate state. Then we get,

1 eq21 (1S 1/21(xaz-ya,) l lD3/2)[ 2

q EID 3/2 -- Els 1/2

which when further factorised into radial and angular parts becomes

et K2(1D 3/2) . ,~2(2 , 3/2, mj) 13+ = ~ Y', , (4.10)

rnj EjD 3/2 -- Eis 1/2

where

f ^'( ~ ( 2 , 3 / 2 , m~)= ~p2,3/z,m~(r) sin - s i n 2 0 sin 0 cos 0 e- i~ ' /

) 0 cos 0 e ~ sin20

)'( t~0 I/2 1 /2(~) d ~ . (4.11)

Further, the orthogonality of spherical harmonics yields E,,j~'2(2, 3/2, m j ) = 2/9, which when substituted in Eq. (4.10) gives

2et KZ(ID 3/2) c _ (4.12)

13+ 27 EID -- Els

Thus, with KZ(1D 3/2) as in Eq. (3.24), 13+ can be explicitly expressed in terms of the model parameters.

Similarly, one writes the contributions from negative frequency intermediate states for 132 from Eq. (4.7) as

1 ](nlj-]Eq~o(i)(rq×~l(i))z]lS 1/2) t 2 132= ~ y ' y" (4.13)

q n,l , j - - E n l j - - E1s I/2

However, in the present case the appropriate negative frequency intermediate states in 1- 3 leading order being ] 1P 2 ) and I 1P 2- ), one writes from Eq. (4.13) as

~[ 1- IIS 1/2)12 1 ](1Ps- [xetz-Yetl W-- q~ 8w [ ElF 1/2 + EIS 1/2

I(1Ps- }xe t2 -ye t l l lS 1/2) 12 . ( 4 . 1 4 )

4 E1p 3/2 + Els 1/2

446 N. Barik et al . / Nuclear Physics A 605 (1996) 433-457

On further simplification of Eq. (4.14) one gets

Ot [3c_

3 ~ Ele +E1s

"if'2(1, 1/2, mj)I2(1P 1/2) +~"2(1, 3/2, my)12(1p 3/2) ],

(4.15)

where

~"(1, 1 / 2 , mj) = ( - 1) "~+ 1/2 ,/2 o (P)*Mq 01/21/2( ) sin 0 d a (4.16)

and

~'(1, 3/2, mj) =(--1)"J+l/2 ftPl3/2_m~( ?)Mq~o l/21/2( P) s in0d l2 (4.17)

with matrix M being written as

M = ( 0e i* e-i*)0 " (4.18)

A thorough scrutiny of ~"(1, 1/2, m ) and ~"(1, 3/2, rn i) for all possible values of mrs indicates that only ,~'(1, 1/2, - 1/2) and ,~'(1, 3/2, - 1/2) are the contributing ones which on calculation gives

if ' (1, I / 2 , - 1/2) = - 2 / 3 (4.19)

and

~ " ( 1 , 3 / 2 , - 1 / 2 ) = ~ / 3 . (4.20)

Thus using these values of ~" 's we obtain

2ct 13 c_= - - - [ 2 1 2 ( 1 P 1/2) + 12(1p 3 / 2 ) ] / ( E w +Els ). (4.21)

27

Substituting already evaluated 12(1p 1/2) and I2(1p 3/2) in Eqs. (3.13a) and (3.13b), respectively, [3 C_ can be determined.

Thus, using Eqs. (4.12) and (4.21) for [3+ and [3~, respectively, in Eq. (4.8) one can obtain the total core contribution towards magnetic polarisability [3~. As in the case of electric polarisability, we may also note that instead of considering positive energy contributions as has been done here from the explicit solutions for the quark orbitals of the present potential model the same could have been obtained from the consideration of positive parity baryonic intermediary such as the A ++ state [15].

5. Pionic contributions towards electric and magnetic polarisabilities

As mentioned earlier in xPT it has been observed that the pionic effect is quite relevant as far as the magnetic polarisability of proton is concerned [15]. So, keeping

N. Barik et aL / Nuclear Physics A 605 (1996) 433-457 447

this in mind in the present investigation we also estimate the additional contributions to the electric and magnetic polarisabilities of the proton due to its associated pion cloud. To do so, we consider the effective Lagrangian within the framework of Chiral Potential Model, written as

~ e f f ( x ) - - ~ q ° ( x ) ff',~-~:(x) "{'-..~ll"r (x ) "Jt-.~lem (X), (5.1)

where the independent quark Lagrangian density .~q0(x) is as described in (2.2). The free pionic Lagrangian density _C~°(x) is

- im.~q~ ( 5 . 2 )

and the Lagrangian density corresponding to the quark-pion interaction is taken to be linear in isovector pion field ~(x) as

i 2 , ~ ( x ) = - ~-~G( x)77( x )vs ( ' r , q~)q(x) , (5.3)

with f~ as the phenomenological pion decay constant and G ( x ) = (mq + V(x)/2). Further, the electromagnetic interaction . ~ m ( x ) is also written as

.~'alem (X) = --eeq~( x)'y~a~q(x) - e(q~ × 01a.~o)3 Ate(x)

+ eZtp+q~_A~(x) A~(x) , (5.4)

where A~(x) is the photon field. Further, the description of the pionic effect requires the dressed proton state up to second order in pion-quark coupling which is also written as [211

I IP) + Y'.cB (5.5) B

where ]P) is the bare 3-quark core state of the proton and IB~r) is the combination of the valency quark core state and pion. Here CB~, the coefficient of the baryon-pion state [BTr), can also be explicitly expressed in terms of the vertex functions as

CBIr=FBP r J dk / I V * " ~ I \ (2~)3/2 / B'IT ~ 1 ~ ]

where B is the intermediate baryon and can be any one among neutron, proton, A ++, A ÷ and A ° for which the corresponding Clebsch-Gordon (C.G.) coefficients FBp are -- V~/3 , 1/V~', -- 1/¢2-, 1/¢3- and 1/vr6, respectively. Further, Zp and I CB~ 1 2 are the probabilities of finding the state IP) and IBm), respectively, and thus Zp + EI CB~ 12 = 1. Here Zp is related to the nucleon self-energy contribution E(Ep) by the relation

Zp= [ 1 - {0 Y'~( Ep)/OEpI] e,=M,]-' (5.6)

ThUS, we will be using Eqs. (5.1) to (5.6) to obtain the expression for electric and magnetic polarisabilities due to the pion dressing of the proton.


o o )

(b) " " " r %

I I

0 o )

(c) " ' - - - " (d) ~ - q ' " . . . . " / I '

/ \ \ Fig. 2. (a) and (b) Contribution towards electric polarisabilities due to scattering of a photon from the pionic cloud; (c) contribution towards magnetic polarisability from the virtual excited core states in the presence of a pion; (d) contribution towards magnetic polarisability from a mixture of core and pion cloud.

5.1. Pionic contribution towards electric polarisability

To obtain the pionic contribution towards electric polarisability one usually considers the proton-proton scattering amplitude to second order where in fact one requires an explicit evaluation of the diagrams as shown in Figs. 2a to d. The contribution from Fig. 2a can be evaluated to yield the expression for pionic contribution towards polarisability as,

1 I (x~rl d~ I ~ ) I 2 ~ a ~ = _ E , (5.7)

2 ~r E x - Ep x ~ P

where d r is the electric dipole moment operator due to the pionic cloud and is written as

ie 2 dk dk' d: = - - E ,~,~f f f - ~ 7 ( w , , / w , ) " ~ [ ( a , ~ + at , , ) (-a_, ,~ + ~,,~)]

2 i , j = l

× e x p [ i ( k - k ' ) . x ] ( x cos 0) dx. (5.8)

Here the annihilation and creation operators a and a* arise due to the component pion field expansion with w k as the energy of the pion field of momentum k. Eq. (5.7), with the dressed proton state of Eq. (5.5) and the dipole moment operator as described the Eq. (5.8) yields

1 dk I d k 2 r V PB 1 ~<a>= f f f a v e [ F E B (~r(k2) [ d7 2~, (2~) ~ ~ I ~ ( v ) > B = n,A + + OJp,

O~kIv*BP] × ( - ~ ( p ' ) IdT* I ~ ( k , ) > ~ . (5.9)

N. Barik et al. / Nuclear Physics A 605 (1996) 433-457 449

In Eq. (5.8) the baryon-pion vertex fucntion vjaa'(k) is written as [23]

i:'""l VJaW(k)=i 4VC~- t m• ] 2 ~ k

with the vertex form factor of the present potential model as [23]

ro.k (5.11) u ( k ) = 1 2 h . ( 5 E . + 7 m ' . ) exp 4 "

Here fNN~ and fNa~ are the bare pseudoscalar coupling constants and ~Ba' and 'r BB' are the corresponding baryon-baryon spin and isospin operators. Thus an estimation of Eq. (5.9) with the integration of the relevant momenta yields

2~ 'r) NN 'r)Na] I~1, (5.12)

"(•' =-3- [I~N~<" ">NN<" +I~=<" " ">N~<" where the integral I~) is written as

[ ('(',> I] (u<',> 1 1 fk 4 dk.(cos'O) d(cos0) ~q3~ k ..3f. 1,,.,=..-.~ o.,<-----r). 1"#1 = "rrm~ Jl q-~0 ~ k, I

(5.13)

where q is the momentum transfer due to the photon interaction with the pion. Thus, differentiating and putting the limit q ---) 0 along with angular integration in Eq. (5.13) we obtain I~l as

2 / 5 I~1 = rrm~ f k4 dk u (k ) [u(k )F , (k ) + u'(k)F2(k ) +d ' (k )F3(k)] , (5.14)

where u'(k) and u"(k) are, respectively, the first and the second derivatives of u(k) with respect to k. Further, we also write explicitly the expression for Fl(k), F2(k), F3(k) as,

Fl( k) = 4(04 2~ 2 /w'~, (5.15a)

F2(k) = to~ 3~% /w4, (5.15b)

and

F3(k) = 1/w 4. (5.15c)

Thus substituting (5.14) in Eq. (5.12) along with the SU(6) ratios of the pseudo-vector coupling constants and with the corresponding spin-isospin factors we obtain the contribution towards electric polarisability as

342a o~(a) 2 = f ~ N ~ I ~ I . (5.16)

25

4 5 0 N. Barik et aL /Nuclear Physics A 605 (1996) 433-457

Next, proceeding in similar manner we estimate the contribution towards electric polarisability from Fig. 2b. However, here the part of the electromagnetic Hamiltonian obtainable canonically from the electromagnetic Lagrangian, as

__£,2

f ' ( q ~ ( x ) + q~2(x)). A20(x) dx 2

contributes. This being of second order in the electromagnetic coupling e, yields the

expression for the electric polarisability as,

f + cos2 o) dxl/5>, (517)

where tPi's are the components of the pseudoscalar pion field. Thus, substituting the dressed proton states and the vertex contributions along with the appropriate momentum integrals and a rearrangement of terms in Eq. (5.17) as in the earlier case one obtains the contribution towards the electric polarisability as

1712 2

a(b)= 25 fNN~I~'" (5.18)

Next, an estimation with respect to the diagram Fig. 2c yields to a contribution as

1 1(/51Eq~q(rq cos 0) 1S~) l 2 ~ ( ~ ) = - Y'. , ( 5 . 1 9 )

2"rr Ep - E x X~P

which when simplified with the dressed proton state and the appropriate quark orbitals, gets factorised into a quark core part and the pionic vertex function part so as to be written as

~(c)=oICp 3 2 ( ( v j P N ( k ) V j *NP(k ) q - ~ ' J \ VJ *aP(k) ' (2~) w,

is the valence core contribution towards proton's electric polarisability as where a p reported in Eq. (3.2). Again with a substitution of the vertex functions from Eq. (5.10) along with some simplification Eq. (5.20) becomes,

~(~) ~ c ( 5 . 2 1 ) = "~-Jb~N ~r Ot p 1,rr 2 ,

where

1 f k 4 dk z L,2 ,rim 2 J wk3 (k ) . (5.22)

Thus, in similar manner when one tries to estimate the contribution due to Fig. 2d, one obtains,

1 < / 5 t d ~ l X ' r r > < X ~ l E q ~ q ( r q c o s O ) l / 5 > = - - (5.23)

et(~) 2"rr ~" Ep - E x X~P

However, the matrix element </51 dz ~ I XTr> in Eq. (5.23) vanishes due to the vanishing angular integral and hence,

a(2 ) = 0. (5.24)


where

Thus, the net pionic cloud contribution towards the electric polarisability can be

expressed simply as an algebraic addition of the contributions obtained in Eqs. (5.12), (5.18), (5.21) and (5.24) i.e.

~ = ~a~ + ~ + ~ . (5.25)

With the net polarisability due to the pion cloud obtained in this manner as a~ together C with that due to the valency quark core obtained earlier as ct p, one can realize the proton

polarisability in the present model as [15]

ete = Zpct~, + (1 - Zv)a ~ . (5.26)

In a recent article by Holstein and Nathan [26] it has been observed that the contribution due to Fig. 2b can be expressed in terms of pion's electric polarisability ~ with the constituent structure of the pion. Such an observation can in fact be useful in

calculating the pion's polarisability ~ within the framework of the present model. To do so, we consider the interaction Hamiltonian in the local form expressed in terms of pion's polarisability as [27]

'¢~eff (X) = - - ¼F~v ( x ) F~V(x) [4Tr ~2m~q~ + ( x)q~- ( x ) ] , (5.27)

which with Fig. 2b leads to the contribution towards the electric polarisability of proton Afi~ (say) as,

A~ P = - - m ~ ( ~ l f : W~l(X) + w~(x): dx I ~>. (5.28)

However, Eq. (5.28), with the dressed proton state, as described in Eq. (5.5), yields to

1 . d k • - ~ ~ . , F Z B p I ~ 3 v j P B ( k ) V j B P ( k ) , (5.29) A ~ P = mlretE (2'rr) 3 B ~ Wj,

which with the vertex functions of Eq. (5.10) becomes,

--~ k 4 A ~ - °~------L-F 2 2 3~m. f dk~.~( k ) E F , p f ~ p ~ ( O" . o" )PB('r • "r ) P". (5.30)

Thus, again using the ratios between pseudovector coupling constants and the SU(6) spin and isospin factors along with the CG coefficients F2e one obtains the pion's polarisability contribution towards electric polarisability of proton as

3 4 2 ~ 2 25 fNN~I~3" (5.31)

1 k 4

I~3- ~m~ f dk~u2(k). (5.32)

(b) In fact the contribution to the proton polarisability due to Fig. 2b in terms of et~ as provided in Eq. (5.18) is realized through Eq. (5.26) as

A~ P = (1 -- Zp)a ~b). (5.33)


This may be compared with the expression in Eq. (5.31) to obtain an explicit expression of the pion's polarisability in the framework of the present model as:

~ = (1 - Zp)etI~j/2I~3. (5.34)

5.2. Magnetic polarisability

Next as in the case of electric polarisability, we also consider the effect due to the pionic dressing of the proton towards the magnetic polarisability. To do so, one requires the consdideration of diagrams such as shown in Fig. 2b, c and d. Indentifying the contribution towards magnetic polarisability from Figs. 2b, c and d as 13~ ), [3(~ ) and 13~ ) respectively, we obtain the net pion cloud effect towards magnetic polarisability as,

= 13 )+ + (5.35)

However, it has been found [26] that the estimation of magnetic polarisability due to the pionic cloud effects of Fig. 2b is opposite to the electric polarisability estimated due to it. Thus, taking this into account one obtains,

13~ ) = - a ~b). (5.36)

The e.m. interaction effecting the transition as shown in Fig. 2c and d, is in fact due to the first two terms of the total e.m. Lagrangian as described in Eq. (5.4). For the estimation due to Fig. 2c, the quark current being relevant, one requires the interaction Lagrangian described in the first term of Eq. (5.4). Thus, the interaction Hamiltonian canonically derived from this interaction Lagrangian can be written with a little simplification as

Heff = ~pB ]~ eqtrq 2 . (5.37) q

where B is the magnetic field and ~v is the magnetic moment of the bare proton at the quark core level obtainable in the present model as

o~

~ P = - - ~ f o drrf(r)g(r)

4Mp - nm. (5.38)

3E', + m',

Then from the earlier definition one obtains the expression for the magnetic polarisability of the dressed proton due to Fig. 2c as;

ix__Loz ~ (/51Eqeqtrql X~r)(X~rlEp Ex Y'.qeqtrq[/3) (5.39)

2~r x

Taking the dressed proton state as described earlier and with the appropriate SU(6) matrix elements of the operators, we obtain the expression for the magnetic polarisability [3(~ ) in Eq. (5.39) as,

644 1 13~) = 25 (1 - ¢~')2~2 Ma - Mefr~s.I,~ 2, (5.40)


where the energy denominator has been approximated by M a - M p in the soft photon or static limit.

Next we consider Fig. 2d to estimate its contribution towards magnetic polarisability i.e. [3(0). To do so, we will be using here the Hamiltonian obtained canonically from the first and the second terms of the em interaction Lagrangian as described in Eq. (5.4). Such a contribution is obtained by the expression

13(d) (2,rr)7 / ¢y - + f dk, dk 2 drto3/2 k2

V *ap 1 ×exp[i (k2-k , ) ' r][rX(kz-k , )]z co3/2 . (5.41)

k, M a - - M p

This can be further simplified to be written as

13(d)= ie~zP (1--vr6-)(l+vr~-) ( VPA ) (2rr)-------- ~ ~ - ( M a _ M p ) f dk I dq

× q2.-:----(8(q))-q,--(B(q)) dq2 ~ kz ] Ik2=kL+q

where again q is the momentum transfer due to the photon interaction. Eq. (5.42) when integrated with respect to the momentum variable q becomes zero. Thus,

~ ) = 0. (5.43)

Hence the net contribution towards the magnetic polarisability due to the pion cloud effects is obtained from the substitution of the Eqs. (5.40) and (5.43) in Eq. (5.35) as,

(1-1,/-6-) 2 1 17let Mp).f2N~i~2 2 - - - f~N~I~l . (5.44) t3,, = 2 9 ~r ~2 ( Ma _ 25

We will use the expression (5.44) for the magnetic polarisability of the proton due to its pion dressing over and above the valency quark core contribution as [15]

13r, = Zp[3~, + (1 - Zp)[3,~. (5.45)

6. Results and discussion

With the formalism developed in the earlier sections, we are now in a position to evaluate the valency quark as well as pionic cloud contributions to the electric and magnetic polarisabilities of proton (eta, eta) and ([~, [3~) which in fact with proper substitutions in the form of Eqs. (5.26) and (5.45) yield the proper electric and magnetic polarisabilities of the proton as,

Otp = Zpot~, + (1 - Zp) ot ~ (6.1a)

454 N. Barik et al . /Nuclear Physics A 605 (1996) 433-457

and

[3p = Zp[3~, + (1 - Zp)13.~. (6.1b)

However, to compare our estimations with the experimental measurements in terms of

tip and [3p, the retardation effects [15,28] as described in Eqs. (1.2a) and (1.2b) are to be taken into account. To do so, we take the flavor independent potential parameters and the quark masses in accordance with Ref. [17] as

(a , V0) = (0.017166 GeV 3, -0 .1375 GeV) (6.2)

and t (mu = m'd) = 0.01 GeV.

These parameters would lead to quark binding energies relevant to our present calculation as

E'ls 1/2 = 0.540223 GeV,

E'Ip 1/2 = E'~p 3/2 = 0.758107 GeV, (6.3) t El t) 3/2 = 0.947784 GeV.

Now using the model quantities to estimate the valence quark contributions we evaluate Iz(1P 1/2), I2(1p 3/2) , K2(1S 1/2) and K2(1D 3 /2 ) as follows:

12(1P 1 /2) = 5.9888 GeV-2; 12(IP 3 / 2 ) = 18.1251 GeV -2,

K2(IS 1 /2) = 13.5384 GeV -2, KZ(1D 3 / 2 ) = 0.9505 GeV -2. (6.4)

Then from Eqs. (3.12) and (3.22) we evaluate the core contribution towards the proper electric polarisability of the proton as

c~], = 8.039 × 10 -4 fm 3 (6.5)

Next, to estimate the total contribution towards proper electric and magnetic polarisabilities of the proton as reported in Eqs. (6.1a) and (6.1b) we obtain the contribution due to pionic dressing of proton using Eqs. (5.25) and (5.35), respectively. For this we use the experimentally measured values of pion mass and f~N~ = 0.08 [29]. Further, using the vertex form factor as described in Eq. (5.9) we estimate the integrals described in (5.14) and (5.24) numerically and obtain them as

I~l = 14.183577 Gev -3, I~2 = 0.725664 and I,~3 = 0.286779. (6.6)

However, the values of the integrals in Eq. (6.6) when substituted in Eq. (5.25) it yields the proper electric polarisability of the proton due to pionic cloud effects as,

a n = 16.053 × 10 -4 fm 3. (6.7)

Then, using Eq. (6.1a) with the model values of Zp = 0.64131 [21], we obtain the net proper electric polarisability of proton as

a e = 10.774 × 10 -4 fm 3, (6.8)

which with the retardation corrections as described in Eq. (1.2a) and the experimental proton charge radius of 0.82 fm [30] becomes,

~ p = (10 .774+3 .3) × 10 - 4 = 14.074× 10 -4 fm 3. (6.9)

N. Barik et a l . / Nuclear Physics A 605 (1996)433-457 455

This is found to lie within the uncertainties of the recent experimental measurement of ~p as (10.9 + 2.2 + 1.4) × 10 -4 fm 3 [6,7]. We may note here that in the present work we have attempted a calculation taking the contribution of the quark core as well as pion dressing of proton into consideration. The same has also been done in M.I.T. bag model [9-11] obtaining the core contribution etp = 7.1 × 10 -4 fm 3 which in fact is not very different from our result in Eq. (6.5). When one evaluates the same in a Chiral quark

model [13] one obtains ap = 16 × 10 -4 fm 3 which is much higher than ours. However, a significant additional correction due to the pionic contribution is required in this model to obtain ap within (7-9) × 10 -4 fm 3 [15] to be comparable with the experiment. We may further note here that in M.I.T. bag model [9-11 ] as well as in Chiral quark model [13] one obtains ~p using the corresponding model values for the proton charge radius as 0.6 fm which is significantly different from its experimental value of (0.82 + 0.02) fm

[30]. However, in the present potential model when one calculates the proton charge radius one obtains ( r 2 ) ~/2= 0.79 fm [21] which is quite close to the experimental measurement [30]. Further a most recent calculation [31] based on Chiral perturbation theory and Compton amplitude analysis obtains ~p = 13.6 × 10 -4 fm 3 which is very

close to the present estimation. The integral in (6.6) when substituted in Eq. (5.31) of the earlier section along with the other parameters as mentioned above one obtains,

A ~ = 0 . 3 1 4 ~ . (6.10)

This may be compared with the estimated values of the 0.8fi~ according to Ref. [26], 0 . 5 1 ~ according to a model independent analysis of Ref. [32] and 0.5~'~ according to the Hedgehog estimation of Ref. [33]. We observe that the present estimation is closer to the results obtained in Refs. [32,33].

As discussed in Eq. (5.34) one can make a direct estimation of fi~ in the present model as;

~ = 4.985 × 10 -4 fm 3. (6.11)

This result may be compared with the available experimental measurements (in 10 -4

fm 3 units) as,

'2.2 + 1.1 (~/~/~ "rr~) [34],

~ = 6.6 + 1.4 (radiative w-scattering) [35], (6.12)

~20 + 12 (radiative "rr-photoproduction) [36].

One can observe here that while the present estimation of pion's polarisability in the lowest order has a reasonable agreement with radiative pion scattering data [35], it is higher than "V"/~ ~r w data [34]. It may be noted here that QCD via Chiral perturbation theory requires pion polarisability given in terms of the experimental axial coupling constant in radiative pion decay to be of the order of 2.4 × 10 -4 fm 3.

We also find from Eqs. (4.8), (4.12) and (4.21) the proper magnetic polarisability of the proton as contributed by the valence quarks only to be

[3~, = -0 .8653 × 10 -4 fm 3. (6.13)


which again to be appropriately combined with the corresponding pionic contribution 13~ to yield the net proper magnetic polarisability of proton. Using the experimental masses

of p, A and "tr and the model estimation for ix ° = 2.3 nm, we can calculate from Eq.

(5.29)

[3.~ = -3 .458 x 10 -4 fm 3. (6.14)

Then, using (6.1b) along with the model estimation [21] of Zp = 0.64131 we obtain the

net proper magnetic polarisability of the proton as

[3p = - 1.795 x 10 -4 fm 3. (6.15)

Taking into account the retardation correction, the magnetic polarisability of the proton

becomes

~p = 3.155 x 10 -4 fm 3. (6.16)

which tums out to be higher when compared with the recently calculated value of ~p = 1.4 x 10 - 4 f m 3 of Ref. [31]. However, it may be pointed out here that the present

estimation is very close to the central value of the recent measurements [6,7] giving ~p ~- (3.3 _+ 2.2 _ 1.4) X 10 - 4 fm 3.

We may note that in the present investigation we have only considered proton polarisabilities. In a similar manner the polarisabilities of the neutron can also be considered. In that case we find that like in other quark models here also they do not differ from those of the proton at the core level. The degeneracy at the proper polarisability level between the proton and neutron is not removed even when the pionic contributions up to second order are taken into account. Thus, the present potential model in the lowest order calculates the same proper electric and magnetic polarisabilities for neutron and proton leading to a larger ~ p and ~p values for proton than those for the neutron which is contrary to the experimental observation [6]. Although it has been felt in Ref. [37] that the pionic cloud effect can account for the experimentally observed difference in the proton neutron polarisability, the leading order calculation in the present model fails to realise the same.

Thus, inspite of the limitations of the present potential model in realising the correct value for the difference in neutron and proton polarisabilities, it gives quite a reasonable explanation of proton's electromagnetic polarisabilities considering the contributions due to valency quark core as well as the pion cloud dressing. The present model with the effective potential in scalar-vector harmonic form has yielded Gaussian solutions for quark orbitals which in fact has given rise to much simplified analytical expressions for the polarisabilities.

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Electromagnetic polarisabilities of the proton in an independent particle potential model