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NISTIR 88-3877

Scattering of Polarized

Photons By Protons

Leonard C. Maximon

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

National Measurement Laboratory

Center for Radiation Research

Gaithersburg, MD 20899

October 1988

# 88-3877

1988

c-2

NISTIR 88-3877

Scattering of Polarized

Photons By Protons

Leonard C. Maximon

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

National Measurement Laboratory

Center for Radiation Research

Gaithersburg, MD 20899

October 1988

National Bureau of Standards became the

National Institute of Standards and Technology

on August 23, 1988, when the Omnibus Trade and

Competitiveness Act was signed. NIST retains

all NBS functions. Its new programs will encourage

improved use of technology by U.S. industry.

U.S. DEPARTMENT OF COMMERCE

C. William Verity, Secretary

NATIONAL INSTITUTE OF STANDARDS

AND TECHNOLOGY

Ernest Ambler, Director

f w ^

SCATTERING OF POLARIZED PHOTONS BY PROTONS

Leonard C. Maximon

Center for Radiation Research

National Institute of Standards and Technology*

Gaithersburg, MD 20899

ABSTRACT

We derive the differential cross section for the scattering of plane

polarized photons by protons below pion threshold. The contribution from the

charge, spin, and anomalous magnetic moment are calculated with no

approximation in the photon energy, w . The contribution from the proton

structure is included only to lowest order (viz., 0( (w/M) 2

) ) , and comes from

the electric and magnetic dipole polarizabilities.

I. INTRODUCTION

The determination of the electric and magnetic pol ari zabi 1 ity of the

proton by the scattering of low energy photons (below pion threshold) has been

of interest, both theoretically and experimental ly , since the measurements of

Gol'danskii et_ aj_. , [1] in 1960. Further progress in this subject has been

detailed in a number of reviews [2-4], the latest being that of Petrun'kin

[5]

, which is both comprehensive and excellent. More recently, there has been

renewed activity in view of experimental measurements with MAMI A at Mainz

[6]

, and a number of theoretical studies have appeared, discussing relativistic effects, methods of measuring polarizabilities and quark models — L'vov

[7]

, Petrun'kin et al., [8], Friar [9] and references cited therein.

*Formerly National Bureau of Standards
The subject is of particular concern because of the apparent conflict

between the sum rule determination of the sum of the electric and magnetic

dipole pol ari zabi 1 i ti es , a + $, and the values of a. and a derived from

experimental measurements of the cross section for photon scattering from the

proton [10], We comment further on this in the Summary of this article. (See

also [11] and [12]). for a discussion of a and 6 see [8] , [9] ,[13] ,[14].

For photon energies below the pion threshold the cross section for

Compton scattering from the proton is dominated by the cross section from a

point particle with charge, spin, and magnetic moment equal to those of the

protoru This cross section, summed over polarizations of the initial and

final photons as well as over the spins of the initial and final protons, was

first given by Powell [15] in the appendix of a short article; it constitutes

the largest contribution to the cross section [5] , [1 6] , [1 7]. If, in addition,

we include the contribution of lowest order in w/M coming from the electric

and magnetic dipole pol ari zabi 1 i ti es (viz., of order (w/M) 2

) we have the cross

section for the scattering of unpolarized photons in the lab system.

da _ da;

d Q ciQ point Me 2

" ^ ^ ul"~

^ wu)

1 [a(l +COS 2 0 ) + 2£cose] (1

)

where

a) to - sin 2

e

to da

d fi! poi nt

(la)

2
is the cross section for a "point proton," given by Powell [15],

Here k_, _k‘ are the monenta of the initial and final photons in the lab

system

a) = | k_| , oa

1 = |_k

' |

0 = $

' )

( 2

M = proton mass

k = proton anomalous magnetic moment = 1.79

and initial and final photon energies oo and oo' are related by the Compton

condi ti on

“ 1 + ^

(l-cose)

Here and throughout this note we use Gaussian units for the electric charge,

so that e

2 /-hc = 1/137. Except for this, our notation conforms to that of

Rjorken and Orel 1 [18].

In (1) we have written the cross section in a form such that various

analytic aspects are manifestly clear. The factor (oj'/co) 2 comes from phase

space and kinematical factors [18]. The remaining expression is the absolute

square of the amplitude, and hence, from crossing symmetry, is invariant under

the substitution u + -V. Further, we have written the contribution from the

point particle without any approximation. On the other hand, since the terms

connected with the electric and magnetic polarizabilities are relatively small

corrections for these energies, only their contribution of order (ai/M ) 2 has

been included. This comes from the cross term between the amplitude (to order

( co/M

)

2

) connected with polarizability and the Thompson amplitude. Higher

3
order terms, of order (oj/M) 4

, are given in [16] and [17]. They include

contributions from the square of the dipole polarizabilities, the cross terns

between the order (io/M) 2 terms in the dipole amplitudes and the order (w/M) 2

terms in the Powell cross section (la), contributions from higher multipoles,

and, most importantly, the contribution from t-channel exchange of a single

neutral pion. For more detailed discussion of the importance of the n°

exchange term, see [5] and [9]„

In the light of proposals to make polarized photon scattering

measurements using some of the high duty cycle electron accelerators currently

available, we extend, in this work, the cross section (1), (la), to include

the case of plane polarized photons. The expression obtained is given in

eqs. (13)-(16) o Again it is written in a form having all the character!' sti cs

just described in connection with (1), (la).

4
II. SCATTERING CROSS SECTION FOR POLARIZED PHOTONS

The basic expression for the amplitude for the scattering of a photon

with initial and final four-momentum and polarisation k, and k

1 ,

1 is given

in 1 [18], p. 127 (7.67)

S

fi V

2 / E^E

i /2to2ai'

^ 2lr ^ 5 ‘^ P

f

+k ‘~ P i~

l< ^'*' ^

where

= 4,re 2 u(p

f)

s f

) [(-if) ^ (-V) + (-if)

) X Hf)] u(P

i ,s

i

) . (7)

Here p.,p^ are four-momenta of the initial and final protons with spins s^ and

Sf.

In order to include the anomalous magnetic moment k, we replace, as in

[18], p.245 (10.88), the proton current ey^ by er^, where

er

n

= % F l^ 2

) + io

pv

qV

2M

F

2 (q2) • (8)

Here q is the photon four momentum. Since both the initial and final photons

are real, k

2 = k' 2 = 0, and the argument of F

1 and F

2 is zero, for which

F i

( 0 ) = F

2 (0) = 1. We thus have

er

y

ey +

y

v e<

q 2M ( 8 ')

x We have modified their expression only in that e

2 has been replaced by 4ne2

, as noted following (3).

5
Thus we must make, in (7), the replacements

¥ + ¥ - ¥ & for q=k (9)

¥'*¥'- fm¥’X' for q

= k ' (9a)

(in view of k«e - k

1 -e

' = 0).

It should be noted that the proton current given in (8) or (S') is

derived under the assumption of an on-shell vertex for a free nucleon. Thus

it cannot be assumed automatically to be applicable to Compton scattering,

even on a f ree-nucleon, since the intermediate nucleon is necessarily

off-shell. We will nonetheless use the current given by (8

1 ). For further

discussion on this point we refer the reader to a recent paper by Naus and

Koch [19]* We therefore substitute (9) and (9a) in (7), square, and integrate

over the final proton direction as in [18], and obtain

da

dfi

! poi nt

(ffw2 • ( 10)

Summing this cross section over final proton spins and averaging over initial

spins by taking traces (but not over photon polarization) we obtain our final

result* The algebra is carried out in the lab system (jd. = 0) with the

transverse gauge for the photon:

e = (O.e) U

2 - 1)

«’ * (O.e') (e' 2 = 1)

(ID

6
Writing the result in terms of the polarizability vectors _e and _e' and the

unit vectors in the direction of the photon three-momenta.

n = k / | k | n' = k_' / | k_*

| ( 12 )

we have, for the contribution from a "point proton,"

da(e,e' )

d n

poi nt

= - ( )

2

r i 2

4 1 Me 2 ' ^

aj

'

f f M_ + a- . 4( e . E ' )2 . 2 )

03 (JL) — —

i

<

00 00 M

2

l

4k[(1-cosq) 2 - ((£xn)*(e'xn') - z>e' ) 2

]

+ ^

2

1

( 7 - 3 cose)

(

1 -cose)

- ( 7 (_£xn_) • ( £_' xn_' ) - 3_£«e_' ) ((_exn_) •(_£* xn_‘ ) - _e*_e'

)

+ 4 K

3 (1 - cose)

- (£x n_) •(_£' xjV ) ((_£xn) •(_£' xn_' ) - _£•_£' )

+ <

4

[i - C (_e. xh) * (_e_* xn' ^

)

2

]

( 13 )

/

7
In deriving (13) we have made frequent use of the Compton condition (3), which

may also be written in the form M(w-co') = k • k

1 . We have also used the

rel ati on

(_e ' *_n ) 2 +

1 ) 2 = 1 - (_£•£*) 2 - (jl'Jl

1 ) 2 + [ (_£X

J2.) * (_E

' X

J2

' ) ] 2

• ( 14 )

The cross section including the contribution of lowest order in (w/M ) 2

connected with the proton structure is given by the square of the amplitude

for the point proton (leading to (13)) and the cross term between the (u/M ) 2

terms from the dipole poi ari zabi 1 i ti es , a and $, and the Thompson amplitude.

To this order, the a and 3 terms do not involve the anomalous magnetic moment

k. We thus have [5]

d 0 ( e , e

1 ) da

e

2

^ ^ oj

1 2

dipole

(“) mo)' [a (_£•£_') 2 + 3 (£x_n) .(_e‘ xjn' )(_£.£') ]

. (15)

The cross section for photon scattering, not summed over either the initial or

final photon polarizations is then given by (13) and (15)

da( e ,

1 ) da( e ,e

1 )

dft da point

dcr(_e ,_e' ) dQ

d i pole

( 16 )

If the polarization of the final photon is not observed, then the cross

section (16) should be summed over e'. The sums are all of the form
For the point cross section we obtain

r, + — - 2(e*n')2) V

U) 0)

'

t

2< (l-cose

)

2

da(_e

)

da

poi nt

1/ r - e. 2

.> 2ritL.^

2 ^ + IMc 2 J '

oaoo

1+ K

2 [5(l-cose) + y2sin 2

e - 2(_e«_n' ) 2

] ,

,

i I

!

M2 I

i >(18)

1

+ <

3 [2(1-COS0) + 2sin 2 e - 2(£«_n ' ) 2

]

+ k

4

[ 1 + sin 2 e - (_e*jn

1 ) 2

] i t v

The contribution from the dipole polarizabilities, summed over final photon

pol ari zati on, is, from (15) and (17),

da( e)

da

di pol

coo)

1 [ a( 1 - (_e»_n1 ) 2

) + 6 n_* n_'] (19)

(Note here that a and $ have the dimensions of L

3

; the dimension of woo' should

therefore be taken to be 1 /

2

.

)

The cross section for the scattering of plane polarized photons, summed

over final polarizations is then given by (18) and (19):

da(_e) da(_e)

da ~ da

da( e )

poi nt

da dipole

(20)

9
Finally, defining the scattering cross sections for photons with initial

polarization perpendi cul ar to and parallel to the photon scattering plane,

dc^/dfi and dc^/dft, we have from (IS)- (20)

,

r f^r +

V

03 03

'

2<(l-cose) 2

1

da

d

n"

9 9 i 9

1/ r e <; , ww

h

MC 2 J J , + M

2

+ k

2

[ 5(l-cose) +-^sin2 e] < !v

l \ + k

3 [2(1-cos9) + 2s i2 e]

j \

( 21 )

+ k

4 [1 + si n

2 e]

v

e

2 ,03* 2 " I'Mc 2

'^ wa3 '{ 2a + 2 3 cose}

i j V

and

r

(—r + — - 2s i n

2 e)

^u3 03

J

2x(l-cose) 2

d^

dft

e

2

, 2

, a3^ 2

03

0303M

2

+ K

2 [5(l-cose) - sin 2 e]

j

+ <

3 [2(l-cose)

]

+ I K

4

2

' 2 -

(jy| c

2

') 0303* {2a cos 2 0 + 20 cose}

•u

( 22 )

10
As can be seen either from these expressions or directly from (18), the

difference does not depend on the magnetic polarizability:

k

2 (1 + k) 2

] si n

2

6

-2 r e

2

> cow a sin2

e ( 23 )

III. SUMMARY

A number of observations which may help provide a guide for experimental

measurements using plane polarized photons follow from eq. ( 21 ), ( 22 ), and

(23). First, from eq. (23) it is obvious that the cross sections da /d£2 and

da /dn are equal for forward and backward scattering (e = 0 ° and 0 = 180°)

(see Fig. 1). Therefore, if Compton scattering of polarized photons is to be

used to determine the electric polarizability, a, scattering angles in the

region near 90° should be chosen. Moreover, we note from eqs. (21) and (22)

that at 0 = 90°, the cross section da /dn is independent of both a and 3 ; only

the cross section for the "point proton" remains (see Fig. 1). Thus, separate

measurements of da

(| /dft and dc^/dft at 90° could provide both a check of the

validity of neglecting (w/M ) 4 terms by comparison of the measured da^/dft with

eq. (22), and a determination of a by comparison of da^/dft with eq. (23). (We

note that the maximum value of da^/dft - da^/dn does not occur at precisely 90°

since w' is a function of 0 , as given in eq. (3).)

Second, looking at the terms in a and 3 in eqs. (21) and (22), one sees

that, in the forward direction, each of the cross sections, dcr^/dft and

da /dn, depends on a and 3 only in the form of the sum a + 3 . Therefore, if

one considers the sum of the dipole polarizabilities to be fixed by the sum

rule [9], [20],

11
a + 8 (24)

- _L qtot (ui)dM

2 7T 2

a) 2

U)q

then it follows that one must make measurements for angles e > 90° in order to

determine the individual values of a and $. This may be seen for the cross

section for unpolarized photons shown in Fig. 4 of [9] and in Fig. 1 of this

article for the cross sections for polarized photons. Thus for both of these

reasons, measurements of da

(| /dft and da^/dft with polarized photons would appear

to be most useful in the region 9 > 90°,

Although a measurement of the difference of the cross sections, given in

eq. (23), is of special interest in that it depends only on a, the asymmetry.

A(u),e)

da

da^ + (25)

is often of preferred experimental ly since many of the experimental

uncertai nti es cancel. In Fig. 2 we plot A(w,e) as a function of the

scattering angle for a number of values of 3, with fixed a + 3. It should be

noticed there that in the region of large asymmetry, 60° < e < 110°, a change

from 3 = 0 to 3 = 2 x 10~ 4 fm 3

, produces at most a 6 % change in the asymmetry.

ACKNOWLEDGMENTS

We have benefited greatly from numerous discussions with J. L. Friar,

W. C. Parke, and J. Ahrens on many aspects of this question and are indebted

to them for the insights they provided. We thank J. S. O'Connell for an

insightful critique and D, R. Lehman for bringing reference [19] to our

attention. We owe a very special note of appreciation to F. V. Hayward, who

suggested the problem to us and provided the constructive dialogue which

enabled its completion.

12
REFERENCES

[1] V. I. Gol'danskii, 0. A. Karpukhin, A. V. Kutsenko, and V. V.

Pavlovskaya, Zh. Eksp. Teor. Fiz. ]38, 1695 (1960) [Sov. p hys. JETP 11

,

1223 (I960)], and Nucl. Phys. IS, 43 (1960).

[2] The Nucleon Compton Effect at Low and Mediim Energies, Proc. (Trudy) of

the P. N. Lebedev Physics Institute, Vol. 41, ed. by D. V. Skobeltsyn

(tr. Consultants Bureau, New York, 1969 ).

[3] H. Rollnik and P. Stichel, Springer Tracts in Modern Physics 79_, 1 (1976).

[4] P. S. Baranov and L. V. Fil'kov, Fiz. El em. Chastits At. Yadra. 108

(1976) [Sov. J. Part. Nucl. 7_, 42 (1976)].

[5] V. A. Petrun'kin, Fiz. El em. Chastits At. Yadra. _12 , 692 (1981) [Sov. J.

Part. Nucl. 12, 273 (1981)].

[6] Physics with MAMI A 1983-1987, Institut fur Kernphysi k/SFB 201, Johannes

Gutenberg-llniversitut, Mainz, May 1988. See, in particular, J. Ahrens

et_ _al_. , Compton scattering off the proton, pp. 1-4, and A. Zieger,

et _al_. , Electric and Magnetic polarizability of the proton, pp. 5-7.

[7] A. I. L'vov - Preprint 344, P. N. Lebedev Physical Institute, Moscow,

1987.

[8] V. A. Petrun'kin, A. I. L'vov, and P. S. Baranov, Preprint 12, P. N.

Lebedev Physical Institute, Moscow 1988.

[9] J. L. Friar, Los Alamos National Laboratory preprint LA-UR-88-2845, 1988.

[10] P. Baranov, G. Buinov, V. Godin, V. Kuznetzova, V. Petrun'kin,

L. Tatari nskaya, V. Shirthenko, L. Shtarkov, V. Yurtchenko, Yu. Yanulis,

Phys. Lett. 52B , 122 (1974); see also ref. [9] above.

[11] T. E. 0. Ericson, Interaction Studies in Nuclei, ed. by H. Jochim and

B. Ziegler, North Holland Pub. Co. (1975); see also J. Bernabeu,

T.E.O. Ericson and C. Ferro Fontan, Phys. Lett. 49B , 381 (1974).

[12] I. Caprini, I. Guia^u, and E. E. Radescu, Phys. Rev. D25, 1808 (1982).

[13] V. A. Petrun'kin, Sov. Phys. JETP _13 , 804 (1961) and Nucl. Phys. _55, 197

(1964). Note in particular the comment on the first of these articles

given on p. 198 of the second article.

[14] J. L.‘ Friar, Ann. Phys. (N.Y.) 95, 170 (1975).

[15] J. L. Powell, Phys. Rev. 75, 32 (1949).

[16] I. Guia^u, C. Pomponiu, and E. E. Radescu, Annals of Physics (N.Y.) 114 , 296 (1978).

[17] I. Guia^u, and E. E. Radescu, Annals of Physics (N.Y.) 120 , 145 (1979).

[18] J. 0. Bjorken and S. 0. Orel!, Relativistic Quantum Mechanics, McGraw

Hill Book Co. , N.Y. (1964).

[19] H.W.L. Naus and J. H. Koch, Phys. Rev. C^6, 2459 (1987).

[20] M. Oamashek and F. J. Gilman, Phys. Rev. 01, 1319 (1970).

13
FIGURE CAPTIONS

Fig. 1. Cross sections da^/dft and da /dft plotted as a function of photon

scattering angle, for an incident photon energy oo = 100 MeV. The

relatively flat curves all pertain to da^/dft; the curves with a pronounced dip all pertain to da/dft. The solid curves are for

it

cc = 6 = 0. The remaining curves all have a + B = 14.3 x 10~ 4 fm 3

. Dotted curves have $ = 2 x 10~ 4 fm 3

, long dashed curves have B = 0;

dash-dot curves have B = 3 x 10- 4 fm 3

.

Fig. 2 . Asymmetry A ( os , 0 ) plotted as a function of photon scattering angle,

for an incident photon energy w = 100 MeV. The solid curve is for

a = B = 0. The remaining curves all have a + B = 14.3 x 10 -4 fm 3

. Dotted curve has B = 2 x 10 -4 fm 3

, long dashed curve has B = 0;

dash-dot curve has B = 2 x 10" 4 fm 3

.

14
(JS/7

UJO yc —O 0 uojpss SSOJQ
OG)(X)N(Din^fOCNlv-0 v-OOOOOOOOOO

XjpuuuuXsv

50

60

90

120

150

180

Photon

scattering

angle

(deg)
FORM NBS-1 1 4A ( REV 11 -84)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Pu bl i cation D ate

SHEET (See instructions) NISTIR 88-3877 OCTOBER 1988

4. TITLE AND SU BTITLE

SCATTERING OF POLARIZED PHOTONS BY PROTONS

5. AUTHOR(S)

Leonard C. Maximon

6. PERFORMING ORGANIZATION (If joint or other than NBS, see in struction s) 7. Contrac^Grant No.

NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE 8 - Type of Report & Period Covered

GAITHERSBURG, MD 20899

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

10.

SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11.

ABSTRACT iA 200-word or less factual summary of most significant i n formation . If document includes a significant

bibliography or I i teratu re survey, mention it here)

Vie derive the differential cross section for the scattering of plane

polarized photons by protons below pion threshold. The contribution from the

charge, spin, and anomalous magnetic moment are calculated with no

approximati on in the photon energy, uj. The contribution from the proton

structure is included only to lowest order (viz., 0 ((uj/M) 2 )), and comes from

the electric and magnetic dipole polarizabilities.

12. KEY WORDS /Six to twelve entries; alphabetical order; capital! ze only proper names; and separate key words by semicolon sj

el ect romagneti c interaction of protons; photon-proton scattering; polarized photons;

proton anomalous moment; proton Compton scattering; proton dipole polarizabilities

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