L]~TT]~RE AL NI~OVO CIlYfENTO VOL. 5, N. 6 7 0t tobre 1972

Correlations in Protons (*).

G. PAPINI

Department of Physics and Astronomy University o/ Saskatchewan Regina Campus - Regina, Sask.

(ricevuto il 16 Agosto 1972)

I t is at present widely held among theoreticans that nucleons, and more generally hadrons, are composite particles consisting of numbers of partons. Yet, very strikingly, so complex objects display phenomena quanti tat ively very precise like, for instance, charge and spin quantization. I t is indeed tempting to at t ibute the existence of these quantization phenomena for the proton in particular to correlation effects among its constituents. In two recent papers (1.2) we have already shown that the proton charge can be identified with quantized magnetic flux in the experimental region well described by form factors of the type ~ ~i/(fl~ ~ q2). Conversely, we have also shown that if

magnetic flux and charge coincide, then charge and magnetic form factors satisfy the following relation in the Breit frame of the proton (s):

(1) hF~(q2) -- 1 : - =- 2~c~m{q~} (V~q~+~+~)~ 1+ (~/~+~+~)~ {1-cos~0).

In (1) p is the proton mass and 0 the angle between q and the proton spin. Relation (1) predicts deviations from scaling which are in good agreement with recent experi- ments (4). Incidentally, if Fm is of the type ~ o~i/(fl~ § q2), then V2F~(r) oc F~(r), and

the corresponding metric field also satisfies a similar relation, which ensures the presence of a l~Ieissner effect. We should like to retain this feature in the model to be discussed below.

We wish to show in this paper how the results of refs. (1.2) can be obtained in the frame of a model for composite protons in which correlations are assumed from the very beginning. The model is extended to include the spin which then appears as the

(*) W o r k s u p p o r t e d b y t h e N a t i o n a l R e s e a r c h Counci l of C a n a d a . (1) G. PAPINI: Lett . Nuovo Cimento, 2, 1370 (1971); 3, 384 (1972). (3) G. Pxei~rI: Lett . Nuovo Cimento, 3, 657 (1972). (s) R e l a t i o n (0) of re f . (~) c o n t a i n s a n e r ro r . The t e r m 2 cos ~ 0 in i t shou ld b e d r o p p e d . (4) CH. BERGER, V. BURKERT, G. KlgOP, B. LAI~GENBECK a n d K . RITH: Phys . Left . , 35 B, 87 (1971); L. E. PRICE, J . R . DUNNIN(~ j r . , M. GOITEI~, K . HAI~SO~, T. KIRK a n d R . WILSOn: Phys . Rev. D, 4, 45 (1971).

478

CORRELATIONS IN PROTONS 479

flux q u a n t u m of the iner t ia l field genera ted by the sys tem of part icles t ha t make up the proton. I t should also be po in ted out t h a t a l ready in the present form the mode l contains features one should l ike to see in a more ambit ious theory of e l emen ta ry particles.

We assume the pro ton to have the topology of a r ing (1) and to be m a d e of a n u m b e r of charged par t ic les in te rac t ing among themse lves to form spinless bosons. I t is also assumed t h a t bosons of one charge p r e d o m i n a t e over those of opposi te charge. W e wish to calculate, in a self-consistent way, the e lec t romagnet ic field (a/2n)At, genera ted by the bosons. The Lagrangian of the model is (s)

(2)

where M and 5r~ represent mass and to ta l number of bosons in s ta te s respec t ive ly and is the f ine-s t ructure constant . Var ia t ion of (2) w i th respect to ~* and Ag yields t he equat ions

(3)

and

(4) ~x ~ Nse . [ e )

W e are interested, in par t icular , in calculat ing the Loren tz - invar ian t quan t i t y

= ~ A, dx ~ . P

For s implic i ty we perform all calculat ions in the res t sys tem of the p ro ton for a space p a t h F. U n d e r s ta t ionary condit ions eq. (4) is equ iva len t to the fol lowing equat ions :

(5)

(6)

2z s --~lcc ~ P - - A ~ = 0 ,

o: 2ccN~e (E eA4 i ,

where we have used the relat ion

~ ( r , t) = ~p~(r) exp [-- iE/h] t .

In cyl indrical co-ordinates and in the gauge in which Ao(r, z) is the only nonvanish ing component of the vec tor potent ia l , the re la t ionship be tween magne t ic flux and field is

q~(r, z) = arAo(r, z) .

(~) The signature of the metric is --, +, +, +. Indices preceded by a comma indicate differentia- tion. Summation over repeated indeces is understood. Greek indices take the values 1, 2, 3, 4.

4 8 0 G. PAPIN:

F r o m (5) we obta in the following equa t ion :

(7) ~ er \~ o~! + ~z2 T ~ \ c~ c l

I f we wr i te % in the form

1 W, = ~ exp [i(kz-- lO)]/qt(r),

where V is the proton vo lume and b, l, q the q u a n t u m numbers of s ta te s, then we obtain f rom (7)

(s) + e T + .Z.~ ~ - ~0 f.%)= 0.

W e now conjee ture tha t the boson sys tem at ta ins s tabi l i ty if all bosons are in a s ta te of equal t and k ~ q = 0, and indicate the corresponding radia l w a v e funct ion by ](r). We also assume tha t there is a domain X inside the p ro ton in which ]2 ~ const = 1. An en t i re ly s imilar eondi t ion in the Ginzburg-Landau theory of superconduc t iv i ty assures the existence of a Meissner effect ("). F o r the same reason we have chosen a z-dependence for ~ of the form exp [ikz]. E q u a t i o n (8) then yields

(9) r~r ~r -k ez 2 ~ - ~ ( e l - - q ~ ) = O

whose solut ion wi th in X is

[ (10) ~ = l e + e e x p - - ,

where

1 4~ze 2 42 Me2 '

is the average number of bosons per uni t volume, and the cons tant of in tegra t ion has been taken equal to the p ro ton charge. Thus, in the model the flux appears as ful ly quant ized when

In an ent i re ly analogous way we obtain f rom (6)

(11) ~ [1 ~ ~r~A4~ ~A4] 2aue(E_eA , ) ]2= O

whose solution is

e

(e) g. M. BL&TT: Theory o! Superconductivity (New York, 1964).

CORRELATIONS IN PROTONS ~ 8 1

except for those values r and z for which ]2=fi 0. Outside the p ro ton the po ten t i a l is therefore of the usual Coulomb type. In E eq. (4) has solutions of the form

F r o m the equation

we then obtain

(12)

a exp [-- (1/,~)R] 2 ~ A4 = e R

- - V2A~ = - - 4~0e 2 ~

1 exp [-- (1/,~)R] 4rr~ 2 R

I~ now is possible to calculate Fr and ffm in the Brei t f rame of the proton. F r o m (12) we obtain

1/~) [darA4(r) exp [iq .r] -- (13) Fch(q 2) : ~ - - ,

.~e 3 1/,~2 + q2

while f rom (10) and the equat ion (~)

2~ H~(q) = et~,~ %u (1 - - cos 2 0) ,

where the spin has been taken along the z-axis, we obtain

(14) , 1 / ~ q 2 - - Y , , ( q ) 1 - - ( V q : ~ _ ~ / ~ + p ) ~ l + ( V q ~ 2 + p ) 2 ( 1 - - c o s : 0 ) .

In calculat ing (14) we have used the spin-up solutions of the Dirae equat ion. I f we impose the nmmal iza t ion condi t ion

h (15) 2".,(0) = 2 ~ '

from (13) we get

] / Mc 2 h 4zue 2 4pc

which sets a va lue for the rat io M/~. F r o m (14) and (15) it also follows tha t the magnet ic moment of the proton has the correct value (7). F r o m (13), (14) and (15) we obta in eq. (1).

The extension of the model to include form factors of the type ~ , , /(fl ,+ q2) presents no par t icu lar difficulty.

The results so far obta ined actual ly corresponds only to the Landau solution (~) of eqs. (3) and (4). We hope to soon be able to gain more insight into the i r physical content . I51 the meant inle , we wish to ex tend the model to include the spin of the proton.

( ') F. ft. ERNST, IR. G. SACHS and K. C. WALl: Phys. Rev., 119, 1105 (1960).

4 8 2 G, PAPINI

We accomplish this by means of the inertial field h~ generated by the boson system. The field h4g behaves, under stationary conditions, in a way entirely similar to , ~ . I t must also be calculated in a self-consistent way and corresponds to the first-order correction to the Minkowski metric. Use of the linearized version of the general theory of relativity can be justified in this case. The Lagrangian (2) must be modified by replacing in it Pg--(e/c)A~ by Pg--(e/c)A,--Mch4g and by adding the term

(X (~4

16~t G (h4v I'-- h'g. v)2 '

where G is the Newtonian gravitational constant. Variation of the new Lagrangian with respect to hag yields the additional equation

(16)

If we write

) 2G '

2X ~44 - -

0.

then from (16) we obtain an equation for the gravitational potential Z generated by the boson system, and the mass distribution can therefore be calculated. In the region where ]3= 0, X has the familiar 1/R behaviour. From the definition of the flux of h 4

M c (17) r z ) - - ~ - rh4o(r, z )

and eq. (16) it can be provided that ~ appears fully quantized in units of h/2 at distances larger than X. We therefore identify the spin of the proton with (17). I t then follows that the spin has a structure which manifests itself for certain values of R, in complete analogy to the electric charge. I t is also easy to prove from (5) (with P-- (e / c )A replaced by P-- (e /c )A- -Mch4) and (16) that the quant i ty

q~= 2nr(eAo(r, z )+ Mch4o(r, z))

is quantized. For /~ > X we have in fact

where n is an integer. We then obtain

(18) q~ = eq) Jr 2(f = (1 + 21')h = nh . CC~

The stationary states of the boson system may therefore be labelled by means of the quantum nmnbers l and l' which refer to charge and spin respectively, i n particular, states with different values of 1 and l' but the same value of n would appear to correspond to particles of different charge and spin. The implications of (18) for Regge theory and elementary particles will be studied in a forthcoming paper.