Compton scattering by the proton and nucleon polarizabilities

Nuclear Physics North-Holland

AS26 (1991) 674-684

COMPTON AND

SCATTERING

NUCLEON

POLARIZABILITIES

C.Y. CHEUNG Institute of Physics, Academia

BY THE PROTON

and

Y.S. YEH

Sinica, Taipei, Taiwan 11.529, Republic of China

Received 11 June 1990 (Revised 25 October 1990) Abstract: Low-energy Compton scattering by proton is studied. We include the effects of the A(1232) excitation and also the TT- and q-exchange contributions in this work. Issues concerning nucleon polarizabilities are discussed.

1. Introduction The scattering of low-energy photons by hadrons is qualitatively different from that by leptons. This is because hadrons are composite and participate in strong interactions. It is well known that, to first order in the fine-structure constant cr, leptonic Compton scattering is described by the two Feynman diagrams shown in figure 1. However, for hadrons one must, in addition, include effects due to strong interactions and compositeness, such as meson loops and the excitation of hadronic resonances in the intermediate states. Of course as far as strong interaction physics is concerned, it is these extra mechanisms that make Compton scattering an interesting process to study. The strong interaction corrections actually allow one to use Compton scattering as a tool for studying interesting hadronic effects. The study of Compton scattering has a long history. To order (Y in the scattering amplitude, the relativistic cross section for Compton scattering by a point-like spin-4 object was first derived by Klein and Nishina ‘) in 1929. It was found that in the limit of vanishing photon energy (w), the quantum mechanical amplitude agrees with the classical

Thomson

scattering

formula:

sqW’O)=-%.&‘, m

(1.1)

where (Y= l/ 137 is the fine-structure constant, m is the mass of the charged particle, and E(E’) is the polarization vector of the incident (scattered) photon. The cross

P

0375-9474/91/%03.50

6

P’ Fig. 1. Compton 1991 - Elsevier

Science

scattering: Publishers

Born diagrams. B.V. (North-Holland)

C. Y. Cheung,

section

for Compton

scattering

XS.

Yeh / Compton

by a particle

having

scattering

675

nonzero

anomalous

magnetic

moment was worked out by Powell ‘> in 1949. Since the early 1950’s, a series of low-energy theorems concerning Compton scattering by hadrons have been derived. First of all, it was proven (1.1) is the correct and Goldberger

by Thirring

low-energy

‘) showed

‘) that the Thomson

limit to all orders

that, in a relativistic

scattering

amplitude

eq.

in (Y.Later Low 4), and Gell-Mann and gauge invariant

theory,

to order

cy and up to terms linear in the photon energy, the scattering amplitude of photon by a spin-i particle depends only on the charge, mass and magnetic moment of the particle. Generalization to terms of order quadratic in the photon energy was made by Klein 6), Baldin ‘), and Petrun’kin “). These authors showed that, to the required order, the cross section of Compton scattering by a hadron, with spin 0 or i, depends on two extra structure constants (Ye and pier, such that

where dpt is the point-particle or Powell amplitude, o(w’) is the incident photon energy, and n(n’) is a unit vector in the direction of the incident

(scattered) (scattered)

photon momentum q(q’). Note that eq. (1.2) is valid to order quadratic in photon energies for the scattering cross section, but not the scattering amplitude. In general, the scattering amplitude JZZcontains additional spin-dependent terms of second order, whose contributions to the cross section are, however, of higher orders. (See sect. 2.) The new terms in eq. (1.2) correspond to Rayleigh scattering, and the new respectively as the electric and structure constants LYEand PM can be identified magnetic polarizability of the target particle. The electromagnetic polarizabilities are important properties of hadrons, since they measure the structural responses of hadrons to external electromagnetic perturbations. Theoretically, the parameters (Ye and &, can be used to constrain models of hadronic structure. For instance, in quark models, (Ye and plvl are directly related to quark wave functions and excitation spectra of hadrons; they also get contributions from other strong-interaction corrections such as those from meson clouds. Recent reviews of the experimental and theoretical situations concerning Compton scattering and hadronic polarizabilities can be found in refs. 9-11). Scattering of polarized photons by protons has been recently examined in ref. I’). To date, for obvious reasons, hadronic electromagnetic polarizabilities have been investigated experimentally in some details for protons only. In the photon-energy region below the pion production threshold, it is usually assumed that eq. (1.2) is adequate, and the unknown constants cyE and PM are extracted from experiment by fitting eq. (1.2) to data. From experimental data in the incident photon-energy region of -50-150 MeV, Petrun’kin lo) found that (Ye= 11.3 f 2.5 x 10e4 fm3 ,

pM = 2.9 f 2.5 x 10e4 fm3 ,

(1.3)

676

C. Y. Cheung,

Fig. 2. Compton

On the theoretical

Y.S. Yeh / Compton

scattering

front, a dispersion

which, together with experimental following sum rule lo): a,+

by proton:

integral

scattering

A-contributions.

has been found for the sum (an + &,

total photo-absorption

cross sections,

PM = 14.2 f 0.5 x lop4 fm3 .

gives the

(1.4)

Eq. (1.4) has been used as a constraint in extracting the experimental values of eq. (1.3). Considerable efforts have been made to directly calculate (Yeand &, in various models of the nucleon, namely the constituent quark model 13,14), MIT bag model 15,16), chiral bag model “), and Skyrme model i8). The theoretical results can be summarized as follows: p~=O-3x10-4fm3.

~E=7-11x10-4fm3,

(1.5)

Although the final results obtained from various model calculations seem to be qualitatively consistent with one another, quantitative details in different models, however, could be very different. (See sect. 2.) Needless to say, the extraction of LYEand PM from data is sensitively dependent on our knowledge of the “background” terms in the scattering amplitude. The purpose of this paper is to include higher-order terms by calculating the A-excitation contribution (fig. 2), and also the t-channel rr- and q-exchange contributions (fig. 3), in order to provide a better description of the background. In addition, we also discuss issues concerning the A-contribution to the nucleon paramagnetic polarizability. In this work, we will limit ourselves to the tree approximation for the following reasons: (i) As we are dealing with strong interactions in the low-momentum transfer regime, there is not a workable quantum field theoretical framework at our disposal. Including meson loops in our calculation would have the inherent problem concerning renormalizability or sensitive dependence on cutoffs. Moreover, due to the fact that we are dealing with large coupling constants, expansion in terms of meson loops is by no means a systematic approach. Furthermore, since we will use an effective yNA vertex which is extracted from the analysis of photo-pion production

Fig. 3. Compton

scattering

by proton:

T- and v-exchange

contributions.

C. Y Cheung,

on free nucleons to include reaction

without

Y.S. Yeh / Compton

taking into account

scattering

of meson loops, it would be inconsistent

them in this work. (ii) In a similar but different on free nucleons,

it has been

677

shown

context,

namely

that, with an effective

the (‘y, V)

yNA

vertex,

the tree approximation works reasonably well up to the A-resonance energy region I’). More recent works 20) have emphasized the need to include pion-nucleon final-state similar section mental

interactions,

which

however

does not apply

in this work.

(iii) Using

a

approach, Williams 21) has calculated the photon-proton scattering cross up to the A-resonance region, and found reasonable agreement with experidata. (iv) In principle, one could also use dispersion relation to calculate

the Compton scattering cross section. But then it would be impossible to separate contributions corresponding to different powers of the photon energy. Consequently the dispersion relation approach is not useful for the purpose of extracting the nucleon polarizabilities ((Ye, PM) from Compton scattering data. In the light of the above discussion, the approach we take in this work should be reasonable for estimating terms of higher order in the photon energy. 2. A(1232)

contribution

The A-contribution to Compton scattering by nucleon is given by the two Feynman diagrams shown in fig. 2. It is well known that the excitation of the A-resonance is intimately related to the nucleon paramagnetic polarizability in Compton scattercan easily be seen from the non-relativistic form of the ing 13-18). This connection A-contribution to the Compton scattering amplitude, L%?~.There are two possible yNA couplings, namely the magnetic dipole (Ml) and the electric quadrupole (E2). However experimentally it is found that the E2 multipole is much smaller than the dominant Ml multipole 19). This result is supported by the quark model prediction that the EZ-coupling vanishes altogether 22). Hence we shall neglect the E2 contribution in this work. The Ml-coupling for the y(q)N(p)+ A vertex is conventionally written as 1g723)

(2.1) where g,,, is the effective yNA-coupling constant (isospin factor included), WA is the mass of A (1232), S is the transition spin operator which connects spin-4 spinors to spin-2 ones 24). Using

the following s:sj

where ai are Pauli spin matrices, frame,

-fro.

K q’-

relation

for Cartesian

= s, - $riuj )

it is straightforward

~p’)x..]x(qx.)}+(qct-q’;F~~~),

components,

(2.2) to show that, in the laboratory

(2.3)

C. Y. Cheung, Y.S. Yeh / Compton scattering

618

where p,, = p + q, and for photon

energy

below

the pion production

have neglected the decay width of the A. In eq. (2.3), there proportional to (q’ x E’) . (q x E). This term should be identified to the nucleon

magnetic

polarizability

the coefficient

is energy

dependent.

due to the A excitation The energy-independent

we

(p”,).

Note that here

part is given by

= 12.0 x 10m4 fm3 for gyNA = 3.34, which energy-dependent part m)]‘, indicating that assumption of constant Since pg) has already

threshold,

is a term which is as the contribution

(2.4)

is obtained empirically from the -yN+ VN reaction 19). The is smaller than PM (‘) by approximately a factor of [ CV/( md in general &, varies with the photon energy, so that the polarizabilities makes sense only in the weak field limit. been included in the total magnetic polarizability &,,, in eq.

(1.2), it must therefore be subtracted from &A so as to avoid double phenomenologically obtained A contribution to the paramagnetic agrees with the value obtained in the Skyrme model of the nucleon Pe’(Skyrme)

= 12 x 10e4 fm3.

counting. This polarizability i8): (2.5)

The numerical coincidence of eq. (2.4) and eq. (2.5) is probably fortuitous, nevertheless it indicates that the two results are close. On the other hand, the MIT bag 15) and chiral bag “) model calculations typically give Pg’(bag)

= 2 x 1O-4 fm3.

(2.6)

This large discrepancy is puzzling. Note that both eq. (2.4) and eq. (2.5) are obtained using the empirical yNA-coupling constant, which is typically lo-30% larger than that predicted by SU(6) or the naive quark model 19,23). This could be one of the reasons for the disagreement, but it is not enough to explain the difference between eqs. (2.4) and (2.5) and eq. (2.6). Nevertheless, it should be emphasized that eq. (2.4) has been obtained in a way that is without any assumptions concerning the nucleon structure. It is interesting to note that in the chiral bag model calculation “), the diamagnetic contribution comes from quark-antiquark “pair” graphs. While in the Skyrme model “), it comes from the effective yyNN coupling, which originates from the yynrr vertex in the gauged Skyrme lagrangian. The diamagnetic contribution calculated in the Skyrme model turns out to be much larger than that in the chiral bag model, so that the net magnetic polarizabilities obtained in the two models are comparable. However, our result seems to favor the Skyrme version of the story. The spin-dependent term in eq. (2.3) is also quadratic in o. However, due to the spin dependence, it does not interfere with the Thomson scattering term, SO that its contribution to the scattering cross section is of order w3 or less.

C. Y. Cheung,

Y.S. Yeh / Compton

scattering

679

3. m and q contributions In this section,

we consider

contributions

shown in fig. 3. The effective axial anomaly in the “triangle”

r

from the TP and v-exchange

nyy vertex, which originates diagram, is given by “)

?rYY

(3.1)

=

also be written

constant, tensor with

A is the photon

where gryy is the effective coupling E,~~, is the totally

antisymmetric

E 0123

=

field operator,

1. A similar

and

effective vertex can

down for the T:

r

TYY

(3.2)

=

The coupling constants g,,, and g,,, n+ yy and 17+ yy respectively, viz. r and likewise

mechanisms

from the well-known

n+YY

for the 7. Substituting

can be related

=

to the partial

decay widths

&2nm,g~yY,

of

(3.3)

in the experimental

partial

decay widths 26), we

get g 7rY.Y = -0.037

(3.4)

g ‘)YY= -0.135

(3.5)

coupling where g,,, has been taken to be negative relative to the pion-nucleon constant girNN = 13.45, as determined empirically from photo-production 1oX25).In the case of the T, the sign of g,,, is ambiguous, and we have arbitrarily set it to have the same sign as gmir,,. Moreover, little is known about the precise value of the r]-nucleon coupling constant gSNN. From various one-boson-exchange (OBE) models of the nucleon-nucleon interaction, it is found that 27*28) (3.6) where the lower limit corresponds SU(3) symmetry predicts that 29)

to the latest

&iVN = &Ni.&(l

result.

On the other

hand,

-:a>,

where S/(1 -8) is the ratio of D-type to F-type generally accepted value of 6 =$, we obtain g,,,.,, = 2.75 .

flavor

(3.7) coupling

strength.

Taking

the

(3.8)

However this is hardly a reliable estimation, since we know that SU(3) is rather badly broken, and also eq. (3.8) is very sensitive to the value taken for the parameter

680

C. Y. Cheung,

For example,

8.

change limit

by about

a 10% variation

in the value estimate

with the SU(3) prediction

of 6 around

of the magnitude

$ would

cause

g,,,

of gnlNN, with the sign

for the r-exchange

= 6.80. contribution

(3.9) is given by

~(p’)iy,u(p)&“‘“P”q&E~qpE,, and similarly

for the n-pole

to

we shall take the lower

given by eq. (3.8). That is g,,,

The amplitude

scattering

80%. In the light of the above discussion,

of eq. (3.6) as a rough

consistent

Y.S. Yeh / Compton

contribution

4. Results

(3.10)

d,.

and discussions

Results obtained in this work are displayed in figs. 4-7 for incident photon energy of 100 MeV. From fig. 4, we see that the interference between the Born and the A terms is constructive. At 100 MeV, the size of the A-contribution is roughly 10% of the Born term. As discussed in sect. 2, the part of the A-contribution that is proportional to the magnetic polarizability (&) term of eq. (1.2) has been subtracted so as to avoid double counting. We shall call this remaining A contribution the “residual” A-contribution. The pion-pole contribution vanishes in the forward direction, and is of the same order of magnitude as the “residual” A contribution at backward angles, where the two contributions tend to cancel each other. Due to the larger n-mass and smaller nNN coupling constant, the n-pole contribution is

120

0 Fib.

Fig.

180

Angle

4. Laboratory angular distribution with aE= p M =O. Solid line = Born; long dashed Born+ “residual” A; short dashed line = Born + pion; the 7) contribution is negligible.

line=

C. Y. Cheung, KS.

Yeh / Compton scattering

681

25

20 C

5

15

-0

10

5

Lab. Fig. 5. Laboratory angular distribution: sponds to zero polarizabilities. Curves

40

160

120

60

0

Angle

A + r + TJ+ aE + &, The dashed curve correBorn+“residual” with 1,2,3 correspond to PM - (3,0, -3) x 10e4 fm3, respectively, a,+ &, = 14.2 x 10e4 fm3.

60

Lab.

60

100

Energy

Fig. 6. Energy dependence at @,,, = 90”. Solid Born + “residual” A + r + q + aE + PM, both with an as long-dashed curve, but with aa and PM given by - ref. 31); triangle - ref. ‘*); diamond

120

140

(MeV)

curve = Born + cr,+ &,,, , and long-dashed curve = and &, given by eq. (1.3). Short-dashed curve = same eq. (4.1). Experimental data: stars - ref. ?; square - ref. 33); enclosed region - ref. 34).

682

C. Y. Cheung, Y.S. Yeh / Compton scattering 0.0

-0.1

n -0.2 d

-0.3

-0.4

0

30

C.M. Fig. 7. Fore-back

asymmetry

60

90

Angle

in the center of mass frame. The curves labelled with the same labels in fig. 5.

1 to 3 correspond

to those

only about 10% of the rr-pole contribution; it is not shown in fig. 4. Fig. 5 shows the sensitivity of the differential cross section to the magnitude of the magnetic polarizability PM, where the dispersion relation constraint eq. (1.4) has been applied. Energy dependence of the differential cross section at Olab = 90” is shown in fig. 6 together with available experimental data, where the solid curve corresponds to eq. (1.2) with the experimental values of CX~and PM given by eq. (1.3). This represents the best fit to the data without including the contributions from A, T, and r]. Taking into account of these additional contribution, and again with (Ye and PM given by eq. (1.3), we obtain the long-dashed curve. To bring this curve as close to the solid one as possible

(short-dashed

curve),

we need roughly

crE = 14.2 x 10m4 fm3,

PM = 0.0 x 10m4 fm3 ,

(4.1)

which is about one standard deviation away from the previous values given in eq. (1.3). Clearly more accurate data 35) are needed before a more precise determination of (Ye and PM is possible. In the center-of-mass frame, polarizability

contribution

apart from some relativistic

to the Compton d(Q)

scattering

= c+W:.,.E

corrections,

amplitude

* E’

the electric

is given by (4.2)

so that its interference with the Thomson amplitude, eq. (l.l), is symmetric with respect to the center-of-mass angle 8 =$r. In contrast, the interference of the magnetic polarizability term &PM)

= A4&ll.(n

x a) * (n’x &‘I

(4.3)

C. Y. Cheung,

with the Thomson

amplitude

that the fore-back

asymmetry

KS.

Yeh / Compton

is antisymmetric

$$‘)-~(74’)

scattering

about

683

8 =&r. Therefore,

we expect

I

(4.4)

g(e)+$r-e)

since the quantity A,, is a ratio, is more sensitive to PM than to (Ye. Furthermore, it might be less sensitive to systematic errors than the angular distributiondo/dfi Fig. 7 shows the sensitivity of A,, to the variation of PM, again with the sum rule eq. (1.4) applied. In summary, we have studied several issues concerning the scattering of photons by protons in the energy region below the pion production threshold. The results are summarized as follows: (i) The A contribution to the nucleon paramagnetic polarizability is obtained phenomenologically, which agrees with the Skyrme model result, but is much larger than that obtained in bag model calculations. (ii) After substracting out the paramagnetic polarizability part, the “residual” A-contribution to the Compton scattering cross section is about 10% of the Born term. (iii) The r-pole contribution vanishes in the forward direction, and it tends to cancel the A contribution in backward angles. The T-pole contribution is negligible in the energy region considered here. (iv) The fore-back asymmetry Afb is more sensitive to PM than to (Ye. Since the asymmetry A,, is a ratio, it might be less sensitive to systematic errors than the differential cross section, and hence might be experimentally cleaner to measure. One of the authors (CYC) is grateful to Dr. S.P. Li and Dr. L. Tiator for helpful discussions. This work was supported in part by a grant from the National Science Council of the Republic of China.

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W. Thirring, Phil. Mag. 41 (1950) 1193 F.E. Low, Phys. Rev. 96 (1954) 1428 M. Gell-Mann and M.L. Goldberger, Phys. Rev. 96 (1954) 1433 A. Klein, Phys. Rev. 99 (1955) 998 A.M. Baldin, Nucl. Phys. 18 (1960) 310 V.A. Petrun’kin, Sov. Phys. JETP 13 (1961) 808; Nucl. Phys. 55 (1964) 197 P.S. Baranov and L.V. Fil’kov, Sov. J. Part. Nucl. 7 (1976) 42 V.A. Petrun’kin, Sov. J. Part. Nucl. 12 (1981) 278 J.L. Friar, in Proc. Workshop on electron-nucleus scattering, EIPC, Marciana ed. A. Fabrocini, et al. (World Scientific, Singapore, 1989) L.C. Maximon, Phys. Rev. C39 (1989) 347 F. Schoberl and H. Leeb, Phys. Lett. B166 (1986) 355 D. Drechsel and A. Russo, Phys. Lett. B137 (1984) 294 A. Schafer, B. Miiller, D. Vasak and W. Greiner, Phys. Lett. B143 (1984) 323 P.C. Hecking and G.F. Bertsch, Phys. Lett. B99 (1981) 237

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scattering

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