Compton scattering on protons at low energies

Nuclear Physics B125 (1977) 530-546 © North-Holland Publishing Company

COMPTON SCATTERING ON PROTONS AT LOW ENERGIES D.M. AKHMEDOV and L.V. FIL'KOV P.N. Lebedev Physical Institute o f the Academy o f Sciences o f the USSR

Received 2 November 1976

On the basis of new dispersion relations at fixed t, elastic ~.p scattering was analyzed at photon energies.below 400 MeV. In an energy range higher than the ~rmeson photoproduction threshold theoretical predictions are in agreement with experimental data in the whole energy range considered, except at energies near the a(1236) resonance for an angle 0 = 90°. It is shown that when obtaining the proton polarizability coefficients (~ and ~) from the available experimental data on O'P scattering, it is important to take into account the contribution of the terms with vn (n >14) to the differential cross-section expansion with respect to v. Calculations of these terms lead to values of ~ and ff differing considerably from those obtained earlier. New sum rules for proton polarizability coefficients are constructed and analyzed.

1. Introduction A great number of papers have been devoted to the investigation of Compton scattering on protons in the framework of dispersion relations (d.r.). A thorough analysis of these papers and references is presented in reviews [ 1 - 3 ] . Qualitative agreement with experimental data was obtained already in the early papers. In the energy range of the incident photon ( 1 5 0 - 2 2 0 MeV) and in the region of the A(1236) resonance the theoretical curves ran, however, high above experimental values. As was shown in ref. [4], in the region of the A(1236) resonance the imaginary parts of the ~'p scattering am,~litudes determined from the unitarity conditions through photoproduction amplitudes gave too large a contribution to the cross section thus leaving no space for the real parts of the amplitudes. In connection with the appearance of recent new data on 3'p scattering and more detailed multipole analyses of photoproduction amplitudes, it is of interest to investigate 3'P scattering with the aid of new and more precise d.r. When analyzing Compton scattering on protons one usually uses either d.r. at a fixed angle [5,6] or at fixed t [ 1 - 3 ] . In the first case we are dealing with the functions with complicated analytical properties in a complex plane of the s-variable. This leads to additional assumptions in calculations. This is not the case with d.r. at fixed t but here arises the problem of determining subtractional functions at t 4= 0. In papers on 3'P scattering, d.r. are written as a rule for the amplitudes suggested by Prange [7]. These amplitudes are linearly dependent for the scattering angles 530

D.M. Akhmedov, L. V. Fil'kov / Compton scattering on protons at low energies

5 31

0 = 0 ° and 180 °. The constraints arising between the amplitudes for the abovementioned angles have not been taken into account earlier in constructing d.r., which probably has led to too high values of the theoretical cross sections [4]. In the present paper these constraints between the amplitudes are used to find two out of the four subtractional functions. The remaining subtractional functions are determined with the aid of d.r. at fixed u with subtraction and low-energy limit. The expressions for the amplitudes thus obtained satisfy all the kinematic constraints. Calculations of the cross section of 7P scattering carried out on the basis of these d.r. with the use of modern multipole photoproduction analyses are in good agreement with experimental data in an energy range higher than the photoproduction threshold. The agreement in the interval 1 5 0 - 2 2 0 MeV has improved. However theory still differs from experiment near the A(1236) resonance for the scattering angle 0 = 90 ° . F r o m the comparison of theory with experiment the sign of gnNNF~ < 0 o f the decay amplitude coincides with that found earlier [8]. In the photon energy range lower than the photoproduction threshold, 7P scattering can be used for the determination of the coefficients of generalized electric (a-) and magnetic ( ~ ) p r o t o n polarizability. In this case the differential cross section is usually represented as a series in the photon energy v taking account of the terms up to v 3 inclusive, under the assumption that the contribution of the terms with v n (n ~> 4) is negligibly small. In the present paper, one evaluates the terms omitted in the abovementioned expansion with the aid of the d.r. used here for the analysis of experimental data on 3'P scattering below 400 MeV. Calculations showed that in the energy range 8 0 - 1 1 0 MeV, where the polarizability coefficients were determined with the least errors, the contribution of these terms is large and leads to the values = (20.0 + 1.1) × 10 -43 and ~ = ( - 6 . 1 +- 1.6) × 10 -43 cm 3, which differs greatly from those generally accepted [9,10]. The d.r. obtained in the paper are also used to construct sum rules for the polarizability coefficients. Calculation of these sum rules indicates the importance of taking into account the annihilation channel for the difference d - ~. Sect. 2 of the present paper considers the kinematics of 3'p scattering; tn sect. 3 d.r. are constructed; in sect. 4 7P scattering is analyzed on the basis of these d.r.; in sect. 5 the terms with v n (n ~ 4) are estimated in the expansion of the differential cross section and the coefficients N and ~are obtained from the experimental data on 7P scattering; in sect. 6 sum rules for the coefficients ~ and/3 are constructed and analyzed.

2. Kinematics

Let us designate 4-momenta of the input and output photons by kl and k 2 and the corresponding nucleon momenta by Pl and P2. Starting from these vectors the

532

D.M. Akhmedov, L. V. Fil'kov / Compton scattering on protons at low energies

following invariants can be derived $ = (]31 + kl) 2 ,

t/-- (Pl - k2) 2 ,

t = (k2

-

kl)

2 ,

(l)

which are connected by the relation s+u+t=2m

(2)

2

Introduce the following set of orthogonal basis vectors K -- ~I( k ~ + k2) Q _- I 2 ( k 2 - k D , (PK) P' = P - ~ K,

N u = ieuvxotYvKxQo ,

(3)

where P = ½(Pl + Pz). The requirements of conservation of parity, gauge invariance, invariance with respect to a charge conjugation and the crossing-symmetry conditions make it possible to write the scattering amplitude in the form [5,7,11 ] [ (e2P'),(e le')

~-(P2)Yfiu(Pl) = -u(.P2) [

~

" iT, +/(T2]

(ezN) (elN) IT3 +/~T4] + (ezP') (elN) - (e2N) (elP') N2 (p')2K2 7s Ts

+

/ + (e2P') (elN) + (e2N) (elP') (p,)2K 2 " ")'s/~T6, u ( p D ,

(4)

where el and e 2 are 4-vectors of the polarization oi Lhe initial and final photons. The amplitudes T i are functions of the invariant variables s, u and t only and possess the following symmetry properties with respect to the substitution s *, u: Ti(s, t) = ~iTi(u, t) ,

=/ Hi

1

( -1

(5)

fori =1,3,5,6, for i = 2, 4 .

The differential cross section (in the c.m.s.) for scattering on nucleons of unpolarized photons built with the aid of the amplitudes Ti has the form 1 do 1 m2 r 2 d£2 - 4e 4 14/2 {(E2 + m2 - 602z) (t TI [2 + iT312)

+ 2 Wco2(E + o~z) (I.T21/ + 17"412) + 2 m w ( 2 E + to + coz) X R e ( T i T ~ + TAT+4) + 26o2(1 - z)lTsI 2 + 2W26o2(1 + z)[T612} ,

(6)

533

D.M. A k h m e d o v , L. V. Fil'kov / C o m p t o n scattering on protons at low energies

where m is the n u c l e o n mass, r o = e2/47rm, W = x / s i s the total energy, co = (s m 2 ) 2 x / s i s the p h o t o n energy, E = (s + m 2 ) / 2 x / s i s the n u c l e o n energy, z = cos 0, 0 is the scattering angle. In expression (4) the coefficients of Ti have singularities of the form {t [st + (s - m 2 ) 2 ]}-1. Since the total amplitude Tfi has n o kinematic singularities the a b o v e m e n t i o n e d singularities impose some constraints b e t w e e n the amplitudes T i for 0 = 0 ° and 0 -- 180 ° (a) t = 0:

Tl(s, O) + T3(s, 0) = 2Ts(s, 0 ) ,

(7)

Ts(s , O) = - ( s - m 2) [Tz(s, O) + T4(s, 0 ) ] / 4 m ,

(8)

(b) t = - 4 c o 2 = - ( s - rn2)2 /s!

TI(S, t = - 4 6 o 2) - T3(s, t = - 4 6 o 2) = 2mT6(s, t = --4602), T6(s, t = - 4 J )

(9)

= (s - m z) [Ta(s, t = - 4 c o 2) - T2(s, t = - 4 c o 2 ) ] / [ 2 ( s + m 2 ) ] . (10)

In constructing d.r. for the amplitudes T i in all the previous papers the abovem e n t i o n e d constraints b e t w e e n the amplitudes were n o t considered which, as has been m e n t i o n e d above, leads p r o b a b l y to too high values of the calculated cross sections.

3. Dispersion relations Let us construct d.r. at fixed t w i t h o u t subtraction for the amplitudes Ti odd with respect to the s u b s t i t u t i o n s ~+ u,

Re Ti(s , t) = r i

2

mz- u

S

+

7r - P (

u) 2 ds'A}S)(s ', t) (s' - s) (s , -- u) ,

i=2,4,

(11)

and with one subtraction at the point u = rn 2 for the amplitudes even with respect to the s u b s t i t u t i o n s ~ u

r{

1

Re Ti(s, t) = i~m--W~_s

+

1 ~ u-m mff~u] + 71"

X

(s' . m. z ) (. s ' .

i=1,3,5,6.

u. ) - ( s '

m z+t)(s'-s

2

p

ds'A}S)(s ', t) (m +#)2

+Ti(m 2,t) ,

02)

534

D.M. Akhmedov, L. V. Fil'kov / Compton scattering on protons at low energies

Here A}S)(s, t) are imaginary parts of the amplitudes in the s-channel,/1 is the 7r meson mass, r i are residues in the nucleon pole rl = - e 2 m ( 1 + 7"3), 1 2

r 2 = - f i e (1 + za), r3=O , 1 2 r 4 = e 2 [½(1 + Xp)2 (1 + z3) + ~-Xn( 1 - r3)] ,

rs =%e2m(1 + ~.p)(1 + 7-3), r 6 = -½e2(1 + ;kp) (1 + r3),

(13)

where kp and Xn are anomalous magnetic moments of the proton and neutron, respectively. We represent the subtractional functions in the form

Ti(m 2, t) = Ti(m 2, O) + tcbi(m 2, t ) ,

i = 1, 3, 5, 6 ,

(14)

where Ti(rn 2, 0) are low-energy limits and equal to Tl(m 2, 0) = 0 , T3(m 2, 0)

= e2

[~kp(1 + 7"3) + ~,2]/m ,

Ts(m 2, 0) = e 2 [Xp(1 + 7"3) + ;k2]/2m ,

T6(rn 2, O) = - e 2 k 2 / 2 m 2 ,

k = 1 [~,p(1 + 7"3) + ~kn(1 - 7"3)],

(15)

and the functions ~i(m z, t) depend on t only. It is easy to verify that the amplitudes Ti(s, t) written with the aid of d.r. (11) and (12) satisfy conditions (7) and (8) at t = 0. The constraints between the amplitudes (9) and (10) for backward scattering will be used to determine two (out of the four) subtractional functions cbi(m 2, t) [3]. To this end we substitute into relations (9) and (10) the amplitudes Ti(s, t) expressed in terms of d.r. (11) and (12) at t = - ( s - m2)2/s. After some simple transformations one obtains expressions for the subtractional functions @i(m 2, t) through the integrals of the imaginary parts of the amplitudes in the s-channel • l(m 2, t) -- ~3(rn 2, t) = 1 P f 7?"

dx

(m + u) 2

(X -- S1)(X

--

m4/Sl)

I m2(2XSl -- s2 -- m4) (X - ~/./~+-/.) [AtS)(x,t) - A~S)(x,t)]

X lSl~-~ _ ~ - )

- miA(S)(x, t) - A(S)(x, t)]/ ,

J

(16)

D.M. Akhmedov, L. V. Fil'kov / Compton scattering on protons at low energies

535

dx

1

qb6(m2, t) = ~ P f

(x - Sl)(X -

m4/sl)

(m+#) 2

s 1 = rn 2 - ~ t' + ½ x / ~ 4m2). One determines the remaining subtractional functions q~l(m 2, t) + q~3(m 2, t) and q~s(m 2, t) by writing for the amplitudes Ti(m 2, t) (i = 1, 3, 5) dispersion relations with respect to t and s at fixed u = rn 2 with one subtraction at the point t = 0 [3] where

A~t)(t,,u=m2)+A(30(t,,u

qbl(m 2, t) + qb3(m 2, t) =--1 P J

n

1P f -n-

u (m + u) z

t ' ( t ' - t)

4,u 2

=

2

m ) at'

A~S)(s',u = m 2 ) + A ( S ) ( s ' , u = m 2) (s' - m2)(s ' - m 2 + t) ds',

@s(m 2, t) -glrNNFn + gnNNFn + 1

p~]t

1~2-------[ - ~P f

(18)

~ A ~t)(t'__2,u = m2 ) t ( t ' - t) dt'

4 la2

1p f~

-~-

A~S)(s ', u = m 2) (s' - m s) (s' - m 2 + t) ds',

(19)

(m+bt) 2

where A}t)(t ', u = m 2) are imaginary parts of the amplitudes in the t-channel, F~ = + [16zrF o_.2.r p - 3] 1/2, Fn = + [ 16nFn--,2.rp~ 3] 1/5, Pn is the V meson mass.

4. Analysis of dispersion relations Let us go over the analysis of 7P scattering in the energy range up to 400 MeV with the aid of the above d.r. In these d.r. the imaginary parts of the amplitudes in the s-channel were expressed through the amplitudes of one- and two-rr meson photoproduction. The single n meson photoproduction amplitudes were represented in the form of the retardation term plus an expansion of other parts into a series with respect to partial waves. The latter were determined using multipole analyses by Pfeil and Schwela [12] (up to 250 MeV) and by Moorhouse et al. [13] (from 250 MeV up to 1210 MeV). The contribution from photoproduction of two mesons was considered within the Williams absorption model [14]. The contribution from the annihilation channel was taken into account through the contributions of n ° and rbmeson poles, the e-meson and the diagrams shown in

536

D.M. Akhmedov, L. V. Fil'kou / Compton scattering on protons at low energies

/

\

/ /

/

\

/

\

\

k

\/

I I I I ,1.

j I I

I I I I ,k

I

\/

i I I t .I.

\

Fig. 1. Fourth-order Feynman diagrams contributing to the t-channel of,rp scattering.

fig. 1. The w i d t h of the decay n o ~ 2 7 was taken equal to P "tr0--,2 "/ = 7.92 eV [15] and the sign g=NNF= < 0 [8,16]. For the ~ meson it was assumed that Pn-,2-r = = 324 eV [17] ,g2NN/4zr = 1 [18]. The sign of the residue in the r7 meson pole was taken inverse to that ofgTrNNFTr , i.e. gnNNFn ~ O. Calculations showed that the con•

~

a?Cff--Il~ • \\

/I

'\\

~ 300

I P(Pf~v)

!

\

iI

/

....... 0

50

1 ~00

S I ~50

i 200

i 2~0

Fig. 2. E n e r g y d e p e n d e n c e o f t h e ~ p s c a t t e r i n g d i f f e r e n t i a l cross s e c t i o n f o r an angle 0 c . m . = 70 ' C u r v e 2 c o r r e s p o n d s t o t h e c a l c u l a t i o n s w i t h ~ = 5, t h e d a s h e d l i n e r e p r e s e n t s t h e results o b t a i n e d in refs. [2,3,19]. Designations of experimental points X : [9 ]; ~: [27] ;,~ : [ 3 t ] ; }: [29];

~:[301;~:

[331; ~: I341.

D.M, Akhmedov, L. V. Fil'kov / Compton scattering on protons at low energies

537

tribution of the r~ meson is much smaller than that of the 7r° meson. Assumptions for the e-meson were: mass me = 660 MeV, the total width I"e = 640 MeV [15]. The term P e ~ 2 3 f g e N N = A X 10 - 3

/~

47r

was considered to be u n k n o w n and calculations were carried out for A = 0, 5, 15. Figs. 2--4 represent the results of calculations of energy dependences of the Tp scattering differential cross sections at the angles 70 ° , 90 ° and 135 ° and figs. 5 - 8 present angular dependences of the differential cross sections for the energies 60, 214, 247 and 325 MeV. Curve 1 corresponds to the calculations with A = 0; curve 2

o ~5o

I

200

250

300

350

))(tTev)

Fig. 3. Energy dependence of the ~'p scattering differential cross section for an angle 0c.m. = 90 ~. Curve 1 corresponds to calculations with ,x = 0; curve 5 : on the a s s u m p t i o n o f identically zero real parts o f t h e amplitudes. Designations o f experimental points: @ : [28] ; ~ : [ 32] ; ~ : [35 ] ; the other designations are the same as in fig. 2.

538

D.M. A k h m e d o v , L. V. Fil'kov / C o m p t o n scattering on protons at low energies

~0

-t

d6(t35")

if,T\

t36 °

! / i

,/ i

l I

~38 °

429 °

\

/

+

~132°

//

+ ///2.

0] ~00

~

~3t °

I

I

1

I

I

I

J50

200

250

300

350

"9 (rt,V)

i

Fig. 4. Energy dependence of the "/p scattering differential cross section for an angle Oc.m. = 135 ° . The designations are the same as in figs. 2, 3.

I

0

30 °

I

60 °

J

I

I

gO*

~20 °

e:~

Fig. 5. Angular dependence of the ',/p scattering differential cross section for 60 MeV energy. The designations are the same as in figs. 2, 3.

D.M. Akhmedop, L. V. Pil'kor / Compton scattering on protons at low energies

/

539

/

/

t da(~

/ / /

iI /I I I !

3

I

i//I//

30*

60 °

90 °

t20 °

0 ~,

Fig. 6. Angular dependence of the ~,p scattering differential cross section for 214 M e V energy. Cmwe 3 corresponds to calculations with ~ = ] 5 ; curve 4 with A = 0 and with a disregard o f photoproduction o f two 7r mesons; the dashed-dotted line corresponds to ~ = 5 and to the sign of the residu~ in the 7r0 meson pote,g~rNNF~r > 0. The other designations ate the same as in figs. 2, 3.

to those with A = 5 ; curve 3 to A = 15 ; curve 4 to A = 0 and with a disregard of photoproduction of two 7r mesons. The dashed curve shows the results obtained in refs. [2,3,19] (in ref. [2] curve 2+). The results of the present paper differ strongly from those of refs. [2,19]. This difference is mainly due to the fact that the amplitudes Ti written with the aid of the d.r. presented above satisfy all the kinematic relations ( 7 ) - ( 1 0 ) , and that the subtractional functions are determined more reliably. From figs. 6 - 8 it is seen that the consideration of photoproduction of two 7r mesons decreases the differential cross section in this energy interval for scattering at small angles and increases it for scattering at rather large angles. The dashed-dotted curves in figs. 6 - 7 correspond to predictions of d.r. when the sign of the residue of the 7r° meson pole is taken positive, i.e. g~NNF~> 0. These

540

D.M. Akhmedov, L. V. Fil'kov / Cornpton scattering on protons at low energies

~a-A--

/ /

6

/ /

i I1~

iiii t ,/

3

2

I

I

L

0 30" 60" 90" t20" O~ Fig. 7. Angular dependence of the ~,p scattering differential cross section for 247 MeV energy. The designations are the same as in figs. 2, 3.

curves, as distinct from those w i t h grrNNFrr < O, coincide with experimental data neither in absolute value nor in form, which testifies in favour of the sign gnNNF, r < 0 found earlier in ref. [8]. From the analysis it follows that the contribution of the e-meson should be not large (in agreement with the result of ref. [20]). The most optimal contribution corresponds to A = 5, which imposes limitations on the product of the decay width e -+ 27 with the e meson-nucleon coupling constant

v


(2o)

The contribution of the diagrams o f fig. 1 is considerable for a 3' quantum energy lower than the rr-meson photoproduction threshold, and leads to the increase of the differential cross section. The theoretical cross sections thus obtained differ greatly from the experimental data of ref. [10] for v = 8 0 - 1 1 0 MeV. In an energy range higher than the p h o t o p r o d u c t i o n threshold the contribution o f these diagrams is small. It should be noted that in refs. [1,21] the contribution o f these diagrams was taken into account with a wrong sign. The choice o f the correct sign,

D.M. Akhmedov, L. V. Fil'kov / Compton scattering on protons at low energies

541

t

t2

tO

8 6 /t

I 0

30 °

1 60 °

90*

I t20"

1 0

~,~,

Fig. 8. Angular dependence of the .).p scattering differential cross section for 325 MeV energy. The designations are the same as in figs. 2, 3.

however, leaves the basic conclusions of ref. [1 ] unchanged. In an energy range v ~ 150 MeV the theoretical values of the differential cross sections obtained in the present paper are in better agreement with experimental data as compared to the results of the calculations performed in other papers (see, e.g., ref. [3]). Agreement with experiment has also improved in the energy range 1 8 0 - 2 2 0 MeV, although the difference has not yet been eliminated for the angular distribution at v = 214 MeV. In the energy range of the A(1236) resonance for the angle 0 = 90 °, the difference between theoretical and experimental data remains. Our calculations made on the assumption that real parts of the amplitudes Ti are identically zero (curve 5, fig. 3) confirm the conclusion of ref. [4] that the experimental data in the region bf the A(1236) resonance at 0 = 90 ° are saturated by the contribution of the imaginary parts of the amplitudes only, leaving no space for the real parts. However, before drawing any essential conclusions, new experimental data are needed with smaller errors and with a good energy resolution.

5. Proton polarizability In the low-energy region the 7P scattering amplitude may be represented in the form of a series in the photon energy p. The coefficient of u2 in the expansion of the

542

D.M. Akhmedov, L. V. Fil'kov / Compton scattering on protons at low energies

spin-independent amplitude includes (besides an electric charge and an anomalous magnetic moment) also structural constants which are called generalized electric and magnetic ~ p r o t o n polarizabilities [22]. To obtain ~ and ~ f r o m experimental data on 3'P scattering, one usually uses for the 7p scattering differential cross section an expression in the form of an expansion in v taking account of the terms up to and including v 3 [22,23]: do

[do,

--= dr2

t

~

e2

v

~ -v2[1--3--(1 p 47rm m

ZL)][~(I+z[)+2~ZL]+O(v4),

(21)

where z L = cos 0 L, 0L is the scattering angle of ~' quanta in the lab system, (do/dgZ)p is the differential cross section of the scattering of photons on a structureless particle with spin -1 written in the form of a series in v up to v 3 inclusive. The coefficients ~ and ~ were first found by Gol'dansky et al. [9] from the analysis of 3'P scattering at v = 55 MeV with the help of expression (21) (with the terms up to v 2) using the sum rules for N + ~: = (9 + 2) X

10 -43

cm 3 ,

~ = (2 -+ 2) X 10 -43

(22)

cm 3 .

On the basis of new experimental data in the energy range 8 0 - 1 1 0 MeV for the angles 0 L = 90 ° and 150 ° and using (21), Baranov et al. [10] have obtained = (10.7 -+ 1.1) X

10 -43

cm 3 ,

~ = (--0.7 -+ 1.6) X

10 -43

cm 3 .

(23)

In the abovementioned papers [9,10] the contribution of the omitted terms was assumed to be negligibly small. In ref. [24], however, the contribution of the 7r° meson pole to the differential cross section of ~,p scattering was shown to be considerable in the energy range 80:-110 MeV. This contribution starts from v4 and becomes particularly large for scattering of 3' quanta at large angles, which leads to values of - ~ differing considerably from the predictions of (23). In the present paper we investigate (on the basis of d.r.) the contribution to (21) of all the omitted terms with v n (n >~ 4). With this purpose we obtain the differential cross section using the above d.r. (11) and (12) for the amplitudes Ti(s, t) and expand jt in v taking account of the terms up to v 3, do d~

-(

do ~

) p

e2 +-v 2 [1 - 3-u(1 --ZL) ] 87r2m m

X [qbl(m2 , 0)(1 + z 2) -- 2~3(m 2, 0)ZLl .

(24)

Subtracting this expression from the differential cross section constructed with the aid of the same d.r. but without an expansion in v we find the contribution of the terms omitted in (21). To calculate this contribution we use the same assumptions as were made in obtaining the results presented in figs. 2 8 curve 2, that gave the best agreement with experiment. The results of calculations of the corrections to (at 0L --- 90 °) and to (~ - ~) (at 0L = 150 °) in units of 1 0 - 4 3 c m 3 conditioned by the contribution of the terms with v n (n >~ 4) are presented in table 1. The following designations are used in table 1 : B is the contribution from the Born one-nucleon terms in ( d o / d ~ ) p with v n (n >I 4); A is the contribution of 7r° and 77meson poles

B

0.010 0,043 0.098 0.177 0.283 0,417 0.58 0.78 1.01 1.29 1.61 2.00

10 20 30 40 50 60 70 80 90 100 110 120

90 °

v (MeV)

0 L =

Table 1

0.06 0,25 0.54 0.89 1.30 1.74 2.19 2.65 3.12 3.62 4.19 4.72

A

0.03 0.15 0.30 0.66 0.99 1.28 2.01 2.87 3.53 4,12 4.63 5.30

I 0.01 0.03 0.06 0.11 0.t7 0.23 0.31 0.39 0.48 0.57 0.67 0.78

e 0.04 0.18 0.36 0.77 1.16 1.51 2.32 3.26 4.01 4.69 5.30 6.08

I+e 0.11 0.47 1.00 1.84 2.74 3.67 5.09 6.69 8.14 9.60 11.10 12.80

7,~ 0.04 0.16 0.36 0.68 1.11 1.71 2,49 3.53 4.91 6.78 9,38 13.18

B

o L = 150 °

0.15 0.58 1.21 1.96 2.78 3.64 4.55 5.50 6.58 7.90 9,43 11.54

A -0.06 -0.23 -0.50 -0.82 -1.04 -1.17 -1.61 -2.10 -2.52 -2.98 -2,96 -2.93

I 0.01 0.04 0,08 0.13 0.18 0.23 0.26 0.28 0.27 0.21 0.10 -0.11

E

-0.05 -0.19 -0.42 -0.69 -0.86 -0.94 -1.35 -1.82 -2.25 -2.77 -2.86 -3.04

l+e

0.14 0.55 1.15 1.95 3.03 4.41 5.69 7.21 9.24 11.91 15.95 21.68

"~(~ - ~)

~2

~3

,'..74

y .t...,

544

D.M. Akhmedov, L. V. Fil'kov / Compton scattering on protons at low energies

(the contribution of the latter is small); e is the contribution of the e-meson; I is the contribution from dispersion integrals in s, u and t-channels; 7 ~ and ~,(N -- ~) are summary contributions of all the terms omitted in (21). As is seen from table 1 the contribution from the terms omitted in (21) is very large in the energy range 8 0 - 1 1 0 MeV. The calculations of these terms in the analysis of experimental data [10] in this energy range leads to the following values of the polarizability coefficients:

~=(20.0_+ 1.1) X 10-43cm3,

3 = ( - 6 . 0 - + 1.6) X 10-43 cm3.

(25)

The calculations of the corrections in the analysis of experimental data for 55 MeV and 0L = 90 ° [9] gives = (14 -+ 2 -+ 5) X 10-43 cm 3 ,

(26)

which does not contradict the result (25). In the latter expression -+5 is obtained due the systematic error of the experiment [25]. The values (25) found for ~ and 3 are, generally speaking, model-dependent. This model dependence is determined by the contributions from dispersion integrals in the s, u and t channels and from the e meson. To obtain model-independent values of these quantities one should carry out an experiment on 7P scattering at such values of v where the contributions from dispersion integrals and the e-meson are small, e.g. at v ~< 50 MeV.

6. Sum rules for proton polarizability coefficients Comparing the expressions for 7P scattering differential cross sections (21) and (24), we find that the proton electric and magnetic polarizability coefficients are expressed only through the t-dependence parts of subtractional functions 1

a- = - ~

1

~bl(rn 2, 0 ) ,

~ = ~ ¢I~3(m2, 0 ) .

(27)

If one determines the difference between the subtractional functions dPl(m 2, 0) - ~3(m 2, 0) with the aid of expression (12), it is easily seen that we obtain the usual sum rule [26] connecting ~ + 3-with the total photoabsorption cross section

-

+3 =~

1 ? o(v) J

~-

d v = (14.1-+ 0.3) X 10 - 4 3 c m 3.

(28)

VO

For the difference ~ - 3 we find a new sum rule using expression (18) for the sum of subtractional functions 1

(

I/ f

4~ 2

dr'

[A~t)(t"u=m 2 ) + A ( t ) ( t " u

=m2)]

D.M. Akhmedov, L. V. Fil'kov / Compton scattering on protons at low energies

545

(m+~) 2 One calculates the sum rule (29) with the same assumptions as those used in calculating curve 2 in figs. 2 - 8 . Then taking into account (28) we find the following theoretical values o f ~ and ~: = 2.2 X 10 4 3 cm 3,

if= 11.9 × 10 4 3 cm 3 .

(30)

The main contribution to the sum rule (29) is given by the annihilation chennel, the basic part of the contribution being given by the diagrams in fig. 1. The contribution o f photoproduction makes up ~10%. A large contribution of the annihilation channel in ~ - fl was indicated also in refs. [36,37]. The values o f ~ and fl (30) obtained from these sum rules (29) differ considerably from those of (25) obtained from experimental data [10]. This difference is due to the fact that, as has been mentioned above, our theoretical predictions for the energy range 8 0 - 1 1 0 MeV exceed greatly the experimental data [10]. To clarify the reasons for such disagreement it is necessary that the contribution from the annihilation channel into 7P scattering be further investigated and new experimental data for low energies be obtained. It should be noted that making the basic contribution to the sum rules for fl, the diagrams of fig. 1 make a small contribution into the coefficients of energy decomposition o f d6/d~2 at un ( n / > 4) and thus do not affect greatly the results o f extracting ~ and ~ from the experimental data on 7P scattering with the use o f table 1. In conclusion the authors are pleased to thank Dr. P.S. Baranov, Dr. A.I. L'vov, Dr. V.A. Petrun'kin and Dr. L.N. Shtarkov for fruitful discussions.

References [1 ] [2] [3] [4] [5] [6] [7] [8] [9] [I0] [1 I]

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