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Higher multipole polarizabilities of hadrons from compton scattering amplitudes
ANNALS
OF PHYSICS
120, 145-174 (1979)
Higher Multipole from Compton
Polarizabilities of Hadrons Scattering Amplitudes I. GUIASU
Institute
of Physics
and Nuclear
Engineering,
P.O.
Box 5206,
Bucharest,
Romania
AND
E. E. RADESCU* International
Centre
for
Theoretical
Physics,
Trieste,
Ital)
Received August 6, 1978
Starting with partial wave projections of the Compton scattering invariant amplitudes free of kinematical singuIarities or zeros, generalized electric and magnetic multipole polarizabilities of spin-0 and spin-l/2 hadrons are introduced as threshold limits of the non-Born parts of the corresponding electric and magnetic Compton multipoles. These objects, which enter the higher low-energy theorems for elastic y-hadron scattering, are shown to be acceptable quantum fieId theoretical generalizations of the usual static multipolar polarizabilities encountered in classical electrodynamics or non-relativistic quantum mechanics. Using the analyticity properties of the amplitudes, sum rules for the generalized electric and magnetic quadrupole polarizabilities of the proton and the pions are written down and evaluated numerically within the two-particle unitarity approximation.
I. INTRODUCTION While first introduced in particle physics by Klein [I], Baldin [I], Goldanski [3] and Petrunkin [4] in the context of Compton scattering on hadronic targets and used also by Alexandrov and others [5] in the study of slow neutron scattering by the strong Coulomb fields of heavy nuclei, the concept of hadron polarizability has attracted much interest in the last years due to its usefulness in other domains of research, such as that concerning hadronic atoms [6, 71. The theoretical investigation of the hadron polarizability phenomenon is somewhat facilitated in those cases (hadronic atoms, for instance) when non-relativistic quantum mechanics can be applied, because then the static polarizabilities of the hadrons may easily be introduced, according to wellknown procedures, in complete analogy with the atomic or nuclear polarizabilities. The situation is not so simple in Compton scattering off hadronic targets, because * On leave of absence from: Institute of Physics and Nuclear Engineering, P.O. Box 5206, Bucharest, Romania. 145 OOO3-4916/79/070145-3O$O5.00/0 All
Copyright 0 1979 by AcademicPress, Inc. rights of reproduction in any form reserved.
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what really matters in the low-energy region are some structure-dependent coefficients, which, although strongly correlated to the polarizability properties of the target, do not, in general, coincide with the usual static polarizabilities. This paper is not devoted to the problem of establishing exact connections between the static polarizabilities and the parameters (other than the target’s mass, charge and anomalous magnetic moment) expressing the low-energy behaviour of the Compton S-matrix element. Tn this respect we refer the reader to the basic work of Petrunkin [4, 81 and to some other more recent results [9]. Our purpose is merely of a pragmatic nature and consists in introducing certain threshold limits of the Compton scattering partial waves (with the s- and zl-channel Born pole contributions taken off) as “generalized” electric and magnetic multipole polarizabilities of the corresponding hadron. Through the combined use of dispersion relations and the unitarity condition, sum-rules for these objects can be more or less straightforwardly derived from which their interpretation as polarizabilities can be made quite transparent. Indeed, it will be shown that if the target particle under consideration behaves like a non-relativistic quantum system, some integral contributions in the obtained sum rules would be negligible and one would thus be left with the usual non-relativistic quantum mechanical formulae for the static polarizabilities. Other analogies (specifically in the electric-quadrupole case) are made directly on the basis of the classical electrodynamical expression of the cross-section for radiation scattering off a system of charges possessing an induced multipole polarizability. Identifying the generalized polarizabilities in terms of Compton scattering multipoles rather than in terms of threshold values or threshold derivatives with respect to energy and momentum transfer of the continuum parts of the invariant amplitudes (as previously done in the literature for the dipole [4, l&13] or quadrupole cases [14, 151) has the advantage of facilitating a more direct contact with facts already known from atomic or nuclear physics. This article is organized as follows: In Sec. II the simpler case of spin-0 hadrons is considered. Generalized multipole polarizabilities are defined after establishing the threshold behaviour of the spin-0 Compton multipoles on the basis of the lowenergy theorems directly obtained in view of the good kinematical properties (i.e. absence of kinematical singularities or zeros) of the two invariant ampiltudes one is working with. The interpretation of the objects introduced is particularly discussed in the dipole and quadrupole cases in the framework of the sum rules written down. In Sec. 111analogous results are presented for the spin-l/2 case. Sec. IV is devoted to some numerical evaluations of the pions and the proton generalized quadrupole polarizabilities. We think the estimation of the proton generalized quadrupole electric polarizability is worth performing in view of a comparison with recent calculations of the static correspondent of this quantity [14]. Appendix A contains some of the lengthy formulae necessary in Sec. III, while in Appendix B the connections between the static polarizabilities and photoabsorption cross-sections in nonrelativistic quantum mechanics are recalled to help to understand the analogies invoked in the text. We now display some kinematical relations regarding the Compton scattering,
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POLARIZABILITIES
which will be used throughout this paper. The photon (hadron) momenta before and after the collision are denoted by k(p), k’(p’); c”(k) and c,+(k’) are the initial and final photon polarization vectors. The Mandelstam invariants S, t, u are defined as usual in terms of the quadrivectors K = +(k + k’),
P = XP + p’),
Q
= k’ - k =p
s = (P + I’C)~ = (p + k)2 = M2 + 2P . K - $Q”;
t _
(k
u E
s+
t
(p
-
K)”
=
(p’
-
k)”
---_ M2
-
2J’
. K
-
&Q2;
--I, -
k’)”
(1.1) :=
+ u = 2M”;
Q2;
(1.2)
v G p . K = i(s - u),
where M denotes the target’s mass. The kinematics of the process in the centre-of-mass system are established by the relations s = w2 2 11= M2 - 20, * (E + oJ,x) , t
= -2w,“(l
(1.3)
- x),
W = the total energy, E
=
s -t M2 2(s)l/"
= the hadron energy,
s - M2 w c -- 2(s)W = the photon energy, x = cos 8, = 1 + -!2CO,2’
6, = the centre-of-mass scattering angle.
The kinematics in the laboratory system are given by s=M2+2Mw,
t = 2ww’(z
z = cos 8,
-
‘1 = -
2w2(1 - 2) , + (u,,j,f)(, _ z) )
(1.4)
u=M”---Mu-t,
OJ = the energy of the incident photon, w’ = the energy of the emergent photon, WI =
1 + (w,C)(l
- z) .
We shall also need the expressions of u and t in terms of the incident lab. photon energy o and x (the cosine of the centre-of-mass scattering angle (9,):
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GUIASU
t
zcz
-2w2
AND
RADESCU
(1 - X) 1 + 2(4M)
V:;Mw[l-;&)
’
(1.5)
&$)I.
II. SPIN-O CASE 2.A. Basic Formulae for Spin-O Compton Scattering The S matrix for Compton scattering off spin-0 hadronic targets (referred to from now on as pions in this section) is Sfi = afei + i(2n)-2 (lSk,,kGp,,~@-~l~ am+
T&k).
(2.1)
The tensor T,,( p’, k’; p, k) can be specified in terms of two scalar invariant amplitudes, free of kinematical singularities, zeros or constraints, through the following gauge invariant decomposition [ 16, 171: T,,(P’,
k’;
p, 4
=
A(k . k’g,, - k,,k:) - B[k . k’P,P,
- (P eK)(P,k:
+ P,k,) + g,,(P . IY)~]. (2.2)
The s- and u-channel Born pole structure of the amplitudes even under s - u crossing) is
A, B (which are both
(2.3)
where e2/4sr N l/137 is the fine structure constant and p denotes the target’s mass for the spin-0 case considered in this section. The kinematical relations Eqs. (l.l)-(1.5) will be used with the obvious replacement A4 -+ p. The two independent helicity amplitudes and the corresponding projections into partial waves (electric and magnetic multipoles) are: fi,h,
1 t) = 4- [(s - p2j2 + st] B
= 874~)~‘~ i (J%(W)+ M,(w)) &W, kl
firds, t>= - $[2A + (a- pp)B] &&)
= -87r(s)l12 f (E,(w) - M,(w)) d;,-,(x). z-1 stand for the usual rotation group functions [18].
(2.4)
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In the above normalizations
the optical theorems read
“T
Imf,,,(s,
149
POLARIZABILITIES
=
2 (GE, 1=1
+
(2.5)
%,),
t = 0) = (’ v4p2)z Im B(s, t = 0) = (S - $7 Or,
where aEtMjl and ur denote respectively the multipolar and total cross sections for photoabsorption on hadron. Further we shall also need the inversion formulae: J%(W) + M,(w) = &
(2z Tl:‘2’ ~2 j_:’ dx &(x)(1
+ x) B(s, t),
4(o) - M&J)= - & (2z;$,1’a’ .c:’ dx d;,,(x)(l x [W,
t>4 (+ - pp)B(s,01.
- X) (2.6)
In the following we shall be particularly interested in the continuum parts of the invariant amplitudes AC, BC,i.e., in those parts which remain after the s - u-channels Born poles have been taken of? A(& t) = ABoyS, t) + AC(S, t),
(2.7)
B(s, t) = BBorn(s, t) + Bc(s, t).
To lowest order in electromagnetism, employing the usual analyticity assumptions, we have at our disposal a general dispersion representation [16] for the invariant amplitudes, which will prove helpful in further considerations, especially when numerical evaluations will be performed. It consists of an unsubtracted fixed momentum transfer dispersion relation for the B amplitude and an unsubstracted fixed u dispersion relation for a particular combination of A and B: Be@, t, u) = $ s,rz ds’ BtS)(s’, t) ($-s --s 2AC(s, t, u) + ($ s- l -ds’ n J42
-
pz) BE& 1, u)
2/P(s’,
+ $j4;/t’
+ A), s -u
u) + [(t(s’, u)/4) s’ - s
2/F)@‘, u) + ((f/4) t’ -
t
$1 B(*)(s’, u)
p2) BW’,
u)
, t(s’, u) = 2p2 - s’ - u. (2.9)
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GUIAQU
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The weight functions A(@, BtS) and Att), P in the above formulae denote, respectively, the absoprtive parts of the invariant amplitudes A, B in the s and t channels. The dispersion representation given by Eqs. (2.8) and (2.9) does not conflict with the normal t- and u-channel Regge pole model prediction for the asymptotical behaviour. Unfortunately Regge asymptotics at large s and fixed t does not allow for a fixed t unsubtracted dispersion relation for the A amplitude too; this is why one has to resort to other dispersion lines, such as that leading to Eq. (2.9) (which brings in the unpleasant t-channel absorptive parts) in order to avoid the appearance of unknown subtraction functions. We close this subsection mentioning that the quantities introduced here should bear the labels (Ch) or (N) corresponding, respectively, to Compton scattering on charged or neutral hadronic targets. This specification will be made occasionally explicit in the course of this paper where also sometimes a labelling (in the pion case) in terms of t-channel isospin (I = 0 on I = 2) will be used through y-(=0'
= y-‘“h.’ + @VW , (2.10)
2.B. Low-Energy
Theorems and Generalized Multipole PolarizabiIities
We are now prepared to analyse in terms of which parameters occurring in Compton scattering on spin-0 hadrons can one conveniently characterize the way in which the hadron will respond to the presence of the electric or magnetic fields. To this purpose we begin with the consideration of the low-energy theorems (i.e. expansions of helicity amplitudes or directly measurable quantities like the differential crosssection in powers of the photon frequency at fixed scattering angle) existing for the process we are interested in [I91 and note first [17] that once one has at one’s disposal (as we here have) a set of invariant amplitudes simultaneously free of kinematical singularities and zeros, one can automatically derive all wanted low energy theorems starting from the double Taylor series developments (in u2 and t) of the continuum parts of the amplitudes: AC($, t) = .,p.o + .2/414 + L40.1 + .a.,
(2.11) Bc(,J~, t) = B”.” + ,2Bl.“ + tBO.1 + . . . .
subsequently changing the variables from v2, t to, say, w, , cos 0, and finally ordering everything in powers of wc . The structure dependent constants Ai*j, BQ (i, j = 0, I,...) thereby introduced represent intrinsic characteristics of the considered particle in terms of which many of its properties (like the one we are concerned with-the ability of the hadronic cloud of getting polarized in external electric or magnetic fields) can be formulated. As an illustration of this statement, we mention now that generalized dipole electric (&El) and magnetic (I?,,> polarizabilities of the hadron can be identified as linear
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POLARIZABILITIES
combinations of the s channel threshold values (V = 0, t = 0) of the continuum A”, 3” of A, B (i.e. ADso, ISon0from Eqs. (2.11)) according to it& + KM1 = E
l?q -
(2.12)
P(V = 0, t = O),
Kh.fl = - &
parts
[2A”(V = 0, t = 0) - $B’(v
= 0, t = O)].
(2.13)
This identification is easily provided by a second-order low-energy theorem for the unpolarized Compton scattering differential cross section, given, in the centre-of-mass system, by do
k) d-2 c.m.
=&
[IA,1 I2+ IL1 I”].
(2.14)
Doing the necessary algebra and subsequently going to the laboratory frame to keep as much contact as possible with the classical electrodynamics formula for scattering of electromagnetic waves on polarizable (charged) objects, one finds (-g),*&
= ($)”
; g-j”
(I -
0J2
2 @(V = 0, t = 0) 87-r
x [ (l$z) -
](l + z”) - 9
z)’
G4” -
P2m(~ ST
= 0, t = 0)
II
+ o(w3)
2 (2*
15)
which justifies Eqs. (2.12) (2.13). Indeed, essentially the same angular dependence in the Rayleigh scattering (the piece of order w2 in the photon frequency in Eq. (2.15)) is obtained classically [20] when radiation scatters off polarizable targets with static electric (a) and magnetic (Is) polarizabilities. The connection between the induced electric (magnetic) dipole moments d~$$~~(magnetic) and the corresponding inducing electric (magnetic) fields E(B) is (in rationalized units) (2.16) The static polarizabilities 01, p appearing in classical calculations and the quantities are obviously KEl > KMl , which have been called generalized dipole polarizabilities, not to be identically assimilated with each other. As a matter of fact, in the classical (or even in the non-relativistic quantum mechanical) analogue of Eq. (2.15) there appears, for instance, instead of I?E1not just a but also a term containing the mean squared electrical charge radius of the target (arising classically from an average of the electric field over the target’s charge distribution, necessary to obtain the effective electric field to be put in the formulae for the dipole radiation). We stress here that the analogies with the classical (or quantum mechanical) calculations represent only a guide to assign physical interpretation to parameters otherwise unambiguously
152
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defined in terms of well established quantities describing the Compton scattering on hadronic targets within a framework based on general principles of quantum field theory, such as relativistic (and gauge) invariance. This is why we use the attributive “generalized” for the polarizabilities introduced and we postpone for other parts of this paper any further discussion of such analogies. In this respect we only mention now that particular attention will be paid to those situations in which, in certain limits and under certain assumptions, the parameters introduced as generalized polarizabilities will show themselves quite similar to their non-relativistic quantum mechanical analogues. At this point let us explain why we attached the dipole indices El , M, to the generalized polarizabilities GE1, I?~, introduced through Eqs. (2.12), (2.13). For this purpose, all we need are low-energy theorems for the multipoles El , MI and these can be obtained simply with the aid of the inversion formulae (2.6). Separating the Born parts in EL, M& according to El = EF’=’ + E,c,
(2.17)
Mz = Mforn + M,G,
(where EForn, M For” are calculated from Eqs. (2.6) with the invariant amplitudes A, B replaced by their s- and u-channel Born poles (Eqs. (2.3)) while ErC, MIC are similarly obtajned with A, B replaced by AC, Bc) and retaining onIy the first coefficients A”eo, go,0 in the developments (2.1 I), one arrives at the desired new identification of cE1 , I?.,,1 directly in terms of the first multipolar partial waves E1 , M,: KEl
=-&-
-h’(w)
,
ZM1
w=o
Ml”(W) =-ix-
w=o
*
(2.18)
Low energy theorems similar to Eqs. (2.18) can be deduced in the same way for all higher multipoles Et , M, in terms of higher coefficients AiJ, I&j entering the expansions (2.1). So one can obtain the threshold behaviour (w + 0) of EIC, MLC in terms of corresponding partial derivatives with respect to v2 and t of AC(v2,t), Bc(v2, t) evaluated at v = 0, t = 0. We thus introduce through the following definitions some other new structure constants, KEI , f?M (expressible as linear combinations of Ai#i, I&j), which will be called generalized higher multipole polarizabilities of the target: Kq =
(21)[(2Z -
I)! !I” E&o) OP 0 + 1)
I0-o ’
I = 1, 2,... . (21)[(21 - l)! !I” Mzc(o) kMz z (I + 1)
CP
(2.19)
Io*9
The Z-containing factors in the above relations are included to ensure the necessary connections with analogous parameters describing the polarizability properties of charge systems in classical electrodynamics or non-relativistic quantum theory, and we shall discuss this point in more detail later on.
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153
2.C. Sum Rules for Dipole Generalized Polarizabilities
For the sum of generalized electric and magnetic dipole polarizabilities one immediately finds the known sum rule [21,2,22]
KEl $ KM1
(2.20) which involves the total cross section for photoabsorption on the hadron. w0 denotes the inelastic threshold determined by the target’s first excited state in the continuum; for pions
Eq. (2.20) follows from the identification (2.12), the unsubtracted dispersion relation for the B amplitude Eq. (2.8) at t = 0 and the optical theorem (last of Eqs. (2.5)). The question of an analogous sum rule for the difference I?~, - zAr, is much more delicate, since writing down an unsubtracted forward dispersion relation for the relevant combination of invariant amplitudes A, B which appears in Eq. (2.13) would conflict with the normal t-channel Regge poles model prescription. Under these circumstances one can either look for additional dynamical assumptions guaranteeing for the more or less accidental disappearance at t = 0 of the bad terms in the Regge asymptotics of the relevant combination of amplitudes, or simply abandon the forward direction, using, for instance, Eq. (2.9) or other possible unsubtracted nonforward dispersion representations, and thus accept the unavoidable appearance of the t-channel absorptive parts as a fact of life. The first possibility, while based on a more drastic set of hypotheses, has at least the obvious aesthetical advantage (as only s-channel absorptive parts will then come in) that it leads to sum rules which render the physical interpretation of the hadron polarizability phenomenon quite easy in as far as analogies with the better studied cases of molecules, atoms or nuclei are concerned. This line has been followed recently in Refs. [23]. In order to get a forward sum rule for & - KM , the authors of Refs. [23] start with the difference between the amplitudes 2i - $B for charged and neutral pion Compton scattering which, having isospin Z = 2 in the t-channel (see Eqs. (2.10)), can be expected to have a better asymptotic behaviour for large v than each of the amplitudes (2A - pzB)(Ch)*(N) separately. Indeed, the t-channel Regge poles with Z = 2, if they exist, probably have a negative intercept 01(o) < 0, so that a possible fixed pole (Jp = 0+) would be the one which mainly governs the asymptotics. Therefore (2A - p2B)(‘=2) would tend to a constant for v + co; consequently, writing down a forward dispersion relation for this quantity with a subtraction at infinity, working out the (s-channel) absorptive parts through partial wave-projections and optical theorems (see, for instance, Eqs. (2.9) and (2.5)) and joining Eq. (2.20) (for charged and neutral pions) to the sum rule so obtained for (
154
GUIASU
AND
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IA
OEI
-
4
I=2 u’Mz --
I
(2.21;
‘4
14 - UML I=2 GE& 4 1 I. (2.22)
The superscript I = 2 on aEl, uM1 on the r.h.s. of Eqs. (2.21) and (2.22) obviously means the difference between the quantities corresponding to + and 7~” targets. More restrictive sum rules (in the sense that the superscript I = 2 in Eqs. (2.21), (2.22) can be dropped, the relations now holding directly for the charged and neutral cases separately) have also been obtained in Refs. 23 under the supplementary assumption of s-channel helicity conservation in high energy photon scattering [24]. In such forward sum rules the unknown subtraction constants K, (proportional, with known simple factors, to the asymptotic value (V ---f co) of the forward amplitude 2A - p2B) still remain as a serious problem, even though one can again appeal to assumptions or models to dispose of them (in the parton model, for instance [25], A -+ 0 for v + cc so that Kg+) = Kg’) = Kz=” = 0). In turn, Eqs. (2.21), (2.22), or more exactly, their mentioned analogous without the superscript I = 2, have the virtue of maintaining close contact with known facts in molecular, atomic or nuclear physics [23]. Indeed, when K, and the contributions to the continuum integral from higher excited states can be disregarded (i.e. if the term with w/p and terms with I > 1 are small, as it happens, for instance, in nuclear physics where the actual integration threshold w0 is much smaller than the target’s mass CL), one gets for the quantities cE1 , K~ introduced as “generalized” dipole electric and magnetic polarizabilities, the usuallnon-relativistic expressions (2.23) (2.24)
Conversely, for light hadrons, when the term with the supplementary energy factor w/p can be thought of as the most prominent one, one finds, even without neglecting K, the known current algebra [26] or chiral theory [27] result I?& e! -KMl.
(2.25)
After this short discussion on forward sum rules for the dipole generalized polarizabilities, we go on now to present some results within the other framework mentioned, e.g. that involving annihilation-channel exchanges through the explicit appearance in the new sum rules for zE - b, of the t-channel absorptive part of the relevant combination of Compton scatiering ifivariant amplitudes.
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POLARIZABILITIES
We first start with the fixed u dispersion relation Eq. (2.9), which at s = $, t = 0, gives us the desired sum rule
where
For the s-channel absorptive part using the partial wave projections (Eqs. (2.4)) and optical theorems (Eqs. (2.5)) one can write the exact expression
[2A f (+$
- P2j‘$” (5 u) = 4’;&f’ $ &,(.~)(a~,- q,,& (2.29)
t(s, 21) = 2p” - s -
l,,
I-X __
2
= -
45 (s -
4 p2)2 ’
while for the t-channel absorptive part, if only two particle (two pion) states are considered in the t channel, one has [2‘4
i- ($ - P2jq:‘, (t,u)
= 4JZen(2J +
1) g+“(t) (t(t ;;@)j””
h?(t) PJCOSyq,
(2.31)
where P, are the usual Legendre polynomials, 4 expresses the angle in the centre-ofmass of the t-channel reaction 7r~ 4 yy, h,(t) denote the TUT partial waves, cos 4 =
t - 2(p2 - u) = (t(t - 4pL2))l/z ’
(t(t “4&’
(2.32)
IzJ(t) = eisJ(‘) sin 8,(t),
(2.33)
stand for the TT phase shifts), and g+“(t) are the ~7r -+ yy partial waves related to the helicity amplitude f+.+ corresponding to transitions with both photon helicities equal to + I. For t in the elastic unitarity region ghJ has the phase SJ of the corresponding TX -+ rrrr channel. For completeness, we display immediately below the partial wave projections of both t-channel helicity amplitudes f++, f+-, describing the annihilation into two photons:
(6,
==
c J,even
(2J
+
1)
g+“(t)
[‘(’
;;p2)]J’2
PJ(cos
$),
(2.34)
156
GUIASU
f+-
AND
= - ; [(s - $)”
RADESCU
+ st] B
= J Zen (2J + 1) g-J(t) [ t(t ;64P2)]“in
&(cos
$).
Such projections and other details about the theoretical investigation of the ~TZ--+ yy process can be found in the basic paper of Gourdin and Martin [28]. The sum rule given by Eqs. (2.26)-(2.29) can then be written as
- &
jar2 $
c
(25
+
1)
g+“(t)
[ ‘(’
fffp2’]J’z
k,*(t)
PJ(cos
#)
J.even
+ higher (than Z-T states) annihilation
channel contributions.
(2.36)
If instead of the fixed u dispersion relation Eq. (2.9), one starts with a fixed angle representation, one can get another version of Eq. (2.36), the direct and annihilation channel contributions balancing each other according to the dispersion line chosen. The (formally) simplest sum rule is obtained if one works in the backward direction (0 = 180”) [lo, 121: ;El
-
%U1
= $1: - &
$ (1 + $)
b(Ye4 - dNo)l
s,12 $ J Zen (2J + 1) pr ;p2)]J’2
+ higher (than rr7~states) annihilation u(Yes) and o(No) denote the photoabsorption the parity flip and non-parity flip multipoles: a(Yes) =
Jf
a(No)
=
2
channel contributions.
cross sections containing
UE~ +
l=odd
g+“(t) mt>
C
(2.37)
respectively
OM~,
l=even
GE1+
c
Z=even
(2.39)
UMz.
l=odd
2.D. Sum Rules for Quadrupoie Generalized F’olarizabilities
Along the same lines as in the previous subsection one can also establish sum rules for the generalized electric and magnetic quadrupole polarizabilities of the spin-0 hadrons KEs
, w=o
EM2
=
12
M2Yw) --J--
(2.40) CO=”
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157
Through the inversion formulae (2.6) and with the aid of the expansion (2.11) we shall express them in terms of the derivative with respect to t of the continuum parts of the invariant amplitudes A, B evaluated at v = 0, t = 0:
As seen from Eq. (2.41) for the sum (IzE1 f KM1) a forward sum rule can be obtained without problems in view of the good high-energy asymptotic behaviour of the B amplitude. Using Eq. (2.8) and expressing the s-channel absorptive part B(S)(s, t) in terms of multipolar photoabsorption cross-sections, one easily gets
In the particular case in which w0 < p and terms with the supplementary factor w/p as well as terms with I > 2 are small, one finds from Eq. (2.43) the usual nonrelativistic expression
For the difference z.Ez- K&,, , as happened also for tag - iM1 , the bad asymptotic behaviour of the A amplitude in the forward directionlprevents us writing down a sum rule like (2.43) unquestionably free of unknown subtractions. One may then again use Eq. (2.9) and, working in the same manner as in the corresponding dipole case, one ends up with the following sum rule of the same type as Eq. (2.36):
X
-
s (t(t
X
(2.45) LA+ G - p2)BYct,tL(tT cos~)/eo~i-,‘,l,(~-~,P)‘,~~.
m dt
au2t
1 a - 4@))l/2 a cos tj
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GLJIAFjU
III.
AND
RADESCU
SPIN-;
HADRONS
3. A. Basic Formulae for Spin- 112 Compton Scattering
The S-matrix element for the Compton effect on hadronic spin l/2 targets (referred to as nucleons from now on) Y(k) + N(P) -
Y’(k’) + WP’)
is Sfi = 6f,i - i(27r)” 6(p’ + k’ - p - k) m(4kOkipop@-1/z x dk’)
(2~)~~
J(P’) T,,(P’, k’; P, k) U(P) dk).
(3.1)
k(p), k’(p’) stand for the photon (nucleon) momenta before and after the collision; E,(k)(u(p)) and l ,+(k’)($p’)) describe the photon (nucleon) polarization states; m denotes the nucleon mass. The kinematical relations Eqs. (l.l)-(1.4) will be used this time with the corresponding replacement A4 ---f m.
To make this paper self-contained we now display the specification of the tensor Ai , known to be free of kinematical zeros or singularities: Tsy in terms of the Bardeen and Tung [17] invariant amplitudes
TAP’,
k’; P, k) = f =Gfi(s,
t, u),
(3.2)
i=l
Pl = K21111,
g2 = K211,1,
92 = I[mI, - (P . K) I1 - K21,]I, g4 = I[K2&, - mK21, + (P ’ K) 1,]Z, Z5 = I[K”l,
(3.3)
- +(P2K2 - (P . K)2) Z,]Z,
Z6 = I[/, + $(P . K) ml, - iP212 - $(P * K) I5 + $mK21, + &(P . K) ,,]I,
where I,” = t?,” - (k,k;/k
. k’),
(3.4)
and 4 = gdy
4 = gw 2 14
=
PAY
. K>
P,,
. 9,
4 = pup, ,
4 = Gwv - YYYUI~ (3.5) 15= $IY~(Y. W TV- Y~Y*K) ~ul* >
4
=
PLLY"
+
YUPY
3
We do not give here the explicit form of the s-u-channels Born pole structure of the amplitudes Ai but refer the interested reader to Ref. [17]. The amplitudes A1,2,4,j are even while AS.6 are odd under s - u crossing. The introduction of helicity amplitudes, as well as the partial wave projections and the
HIGHER
MULTIPOLE
159
POLARIZABILITIES
needed low-energy theorems, due to the cumbersome aspect of the relations in the spin-l/2 case, have been relegated to Appendix A. Defining the following combinations of + and - multipoles (see Appendix A) El = $[l&-- + (/ + 1) El+], (3.6)
MI = +ph!&- + (E + 1) Ml+],
in terms of which the (partial wave) optical theorems read Im El = 2
a(El), (3.7)
Im MI = 2
0(M),
33 fJT = c (Q(E1)+ wfd
(34
I=1
(c+ is the total cross section for photoabsorption on nucleon), one can introduce again generalized electric and magnetic multipole polarizabilities (this time for spin l/2 systems) through the same kind of formulae as Eqs. (2.19) from the previous section: i?EI =
(21)[(21 - l)!!]’ (1 + 1)
. Elc(w) cfJzz Iw-0 ’
(21)[(21 - l)!!]Z M”(w) tT,&fl= -7 (I+ 1)
(3.9)
(,)+ *
The identification of the dipole and quadrupole generalized polarizabilities as defined in Eqs. (3.9) in terms of more manageable objects (as far as dispersion relations are concerned) like derivatives (with respect to y2and t evaluated at threshold (V = t = 0)) of the (crossing symmetrized) continuum parts of the invariant amplitudes AiC = A. - AForn, can be worked out with the aid of the formulae from Appendix A and lo;ks like [12] KEl + K&jM1= -4”, [ cpso + T Cpq)
(3.10)
for the dipole case, and (3.12) (3.13)
for the quadrupole polarizabilities. 595/120/1-11
160
GUIASU
AND
RADESCU
To discuss the interpretation of these first multipolar “generalized” polarizabilities, we now give the lengthy form of the low enregy theorem for the unpolarized proton Compton scattering differential cross-section valid to the order w5 in the photon frequency. In Ref. [15] this low-energy theorem has been established by means of the threshold derivatives 0’ (i = l,..., 6; k, 1 = 0, l,... ), while we now give it in an alternative form in terms of the coefficients expressing the developments of the non-Born part of the multipoles in powers of w (Eqs. (A.23)-(A.25)). One gets in a straightforward manner:
kalab. =(43,+(a,,,,,)- $& x [1 - 4f
(1 - z)] ( y-)”
[A + Bz + Cz2 + Dz3]
(3.14)
where (da/&), represents the Powell cross-section (undeveloped in powers of w), ) is wdn)kE1.~M71u1 the generalized dipole polarizabilities (17~~, I?~,) contribution and the last term expresses the new w4(w5) frequency orders (i.e. quadrupolar terms, mixed terms of order 1 and terms containing the first two derivatives of the dipoles, Et, E,2W,1, K?): ($),
= ; (&),
IL1 + (Q,;);;’
1, (3.15) [(l - z)” +f(z)l [1 + (4m>(l - 41”
z)12 + (g
f(z) = a0+ qz + a2z2, a, = 2h + $X2 + 3X3 + fh4, a, = -4h - 5h2 - 2X3, a2 = 2X + +A2 - A3 - &X4, (h = 1.793 denotes the proton anomalous magnetic moment)
=
(1 - z) + 6 (+)’ X2
12 [Cl -
(
- y-$ x
I
(1 - z)” - 10 (+)3
(1 - z)~]
4” @El- hf3 + (1 + 4” GE1+ hIM31
[1 - 4 (5)
(1 - z)](Z)
(l ; z)2 [(2 - 2x - X2) - z(2 + 2h + X2)1(& - i&+1)
- F
(1 - 2)” (&
- I?.&2 -
- F
(1+ 2)”(G1+ ZMl)$
(* + zy
- z, (X2 + 2h)(i?El + f&J (3.16)
HIGHER
A
m3 -S(Zfi,-
MULTIPOLE
+ 2&+)J
+ 3fl?&+)9O+ 8&-
1
+ 8 d5 (A + -g)
161
POLARIZABILITIES
(C,E - c p ) ,O- 4 VT (2 + 2A + P) (cy + c,*y
- (4 -+ 16h + 6h3@-
- A&-)*1 + (4 + 16h + 6h2)(&+ - &&+-,J
+ (16 + Sh + 6h2)@- + @l-)*1 + (8 - 8h - 6X2)(&+ + A?!,-)J], B
m3[16(al-
+ 2al+)s2 + 8 ~“5 (1 + 3X + P)(C,E -
CIM),O
t 4 d/3(2 i- 2h t 2P)(C,E + CIM)*O + 4(1 + 4x -j- h2)(&
- A&-).1
- 4(4 + 4x + X2)(& + - ml+)*l + 4(3 + X2)(8,- + II&-y + 4(6 - A’)(&+ + II?&+)J], Lfl = r?r3[24(2aZ- + 3i@2+)so + 8(&i,- + 2&+)*2 - 8 2/3(2 + 5/j + 92) X (c? - clM).’
+ 2 ~‘3 (4 + 4h + 6X2)(C,E + CIM).” + 2h2(&- - j$&->J
t (12 - 2X2)(&‘,+ - Ml+)-1 -
+ (16 + 8h + 2h”)&+ D = 16m3[(2&-
+ 3E2+)~o -
(4 + 8h + 2X2)&-
+ q-y
+ A?Il+)J], (I + A) qf
(3.17)
~~01.
(,@I~, aFYi are defined in Appendix A.) Let us now isolate the contribution of the electric quadrupole in Eq. (3.14):
(E2) to (&/&)rab
du (__ dQ >
(E2)
(3.18) This form agrees with the result [ 151 of a simple classical electrodynamics calculation of the interference between the Thompson scattering and the scattering due to an induced quadrupole polarizability K E2 = zE2 (note that R,, from Eq. (3.18) stands for &K of Ref. [I 51, due to a difference in the definition of K) and gives once more support to the interpretation of ~~~ as quadrupole polarizability. 3.B. Sum Rules for the Generalized Quadrupole Polarizabilities
of the Proton
As sum rules for the generalized dipole polarizabilities i?Er , 17,, in the nucleon case have already been quite extensively presented and analysed in the liter-mare, here we shall only derive and discuss sum rules for the quadrupole polarizabilities i7E2and iM2 . We begin with the treatment of GE2+ ZM2 , as again in this case one can take
162
GUIAt$U
AND
KADESCU
advantage of the good high energy asymptotic behaviour of the combination of invariant amplitudes defining, according to Eq. (3.12), this sum, to write down without difficulty a fixed-t dispersion relation and avoid in this way the occurrence of any undesirable annihilation channel contribution. Starting with unsubtracted fixed momentum transfer (t) dispersion relations for the amplitudes &(s, t) and A&, t) (which is possible at least as far as the normal Regge behaviour is concerned), using Eq. (3.12) and the optical tehorem Im
(
A,+TAs
)I t=o = k
dw,,
where uT denotes the total cross-section for photoabsorption immediately arrives at the sum rule
3 m* + 2rr2m sug w
Ap)(w,
on protons,
one
t = 0).
w. = ~(1 + ~/2m) denotes the single pion photoproduction threshold and the superscript (s) indicates the absorptive part in the s-channel. Using the projection in multipoles listed in Appendix A to work out the last two terms of Eq. (3.20) one can put the obtained sum rule in the equivalent form:
+ *
jl$
(1 + 2 +)1’2
Tm I( 2)’
(&-
+ Ml-)
+ f pz++ Mzf) ((I+w2 + 1- 1) Z=l
+
2(1 + 2)(12+
-
Z(1 + 2) (1 + G)(
+ (E,,
+ Al,-,,)
I -
1)
(=)
1 + 2 $)liP
(1(Z2 + 31 +
+ Z(Z + 2) (1 + 2 -y
+ (I + 1) (+)z)
1) + 2w2
(1 + 5)
+ 31+
1) (C)
+ (I + 1) (e)2)
HIGHER MULTIPOLE
163
POLARIZABILITIES
+ 2(Z(Z-I- 2))“” (Cf $ Cpf) x (I +2+-p
+2g2(1
If, in the above sum rule, contributions to would be negligible (i.e. if u//n could be put I > 2 disregarded), one would find from Eq. usual non-relativistic formula connecting the over the photoabsorption cross-sections of the
(3.21)
+x-),1/.
the integral from higher excited states
Irrespective of whether in the actual case of the proton it is permissible or not to make such neglections, the formal aspect of Eq. (3.22) again gives support to the interpretation of ZEI , r~-,, (defined in Eqs. (3.9)) as generalized multipole polarizabilities. Eqs. (2.23), (2.24), (2.44) from Sec. II as well as Eq. (3.22) above are obtained in a dispersion theoretical framework with subsequent approximations which would be understandable or justifiable for non-relativistic quantum systems. In Appendix B we present the usual quantum mechanical procedure to express the static multipole polarizabilities (of not necessarily spinless systems) in terms of (completely analogous) formulae involving the corresponding photoabsorption cross-sections such that the reader has at his disposal the needed material. For the difference ~~~ - KMz a sum rule can be established starting with Eq. (3.13) and using a fixed t dispersion relation for the A, c - AsCamplitude once subtracted at u = m2, with the subtraction function expressed further as an unsubtracted dispersion relation (in the t variable). So one finds:
(6” -
cM2)(1)
= - $ s,I:+.,? ds’
[A,‘“‘@‘, u = m”) - A;S)(s’, ZI = rn”)] (s’ - I??)”
(3.24)
3 m ds, AgS)(S’,t = 0) --I 79 (rnlu)2 (s’ - m2y ’ [A;“(t’,
24 == m2)
-
At’@‘,
t’”
IV. NUMERICAL
u
=
,,I’)]
(3.25)
ESTIMATES
In this section we shall compute some of the contributions to the quadrupole generalized polarizabilities of the proton and the pions from the sum rules written
164
GUIAfjU
AND
RADESCU
down in the previous sections. We do not attempt a thorough numerical analysis, through the use of all presently available information on the quantities entering the sum rules envisaged, but rather concentrate on the most reliable pieces and only discuss within simple models some of the less reliable ones. We begin with the evaluation of the sum of the electric and magnetic generalized quadrupole polarizabilities kE2 + ii,, as given by Eq. (3.21). One of the integrals over the total cross-section for photoabsorption on protons (the one related to &;I + K~& is known from Ref. [29],
while the other has been obtained integrating numerically reference, 3 ;D d dw> w __ 2?r%l s*,). w3
the data given in the same
E 0.52 x 1O-3 fm5.
(4.21
The third integral in Eq. (3.21) has been evaluated (retaining only waves with .Z = 4, g, 9) using the single pion photoproduction data of Refs. [30] for the energy region 180 MeV < w < 250 MeV and the data of Ref. [31] for 250 MeV -=cw < 1210 MeV. In this way one gets the estimate (KE2 + KMz)proton N 0.8 x lO-4 fm”.
(4.3)
It should be stressed that this number results from surprising compensations between different contributions and therefore the inclusion of the double-pion photoproduction yield may be quite relevant. In the evaluation of the difference I& - /T,+,,we are faced with the difficult problem of the annihilation channel contribution (the integral over t-channel absorptive parts in Eq. (3.25)). While the s-channel contributions (Eq. (3.24)) can be treated in terms of known photoproduction multipoles, as mentioned before (disregarding possible complications from extrapolations to the unphysical region in the s-channel absorptive parts at u := m2), the t-channel piece is highly model-dependent and all we intend to do now is to provide an estimate for it, which should not be taken too seriously, based on a simple model previously used in connection with cEl - ifiI1 [12, 131. The contribution of the rrrr states to (I?~, - ii,,)(,) , retaining only s-waves, is as follows:
e 3.8 x 1O-3 fm5.
(4.4)
where the Frazer-Fulco [32] Nn + nii amplitudefF’o(t) and the Gourdin-Martin [28] nm -+ yy amplitude gy’o(t) are computed in an N/D model for the XT + T-i-m(Z = J = 0) partial wave as in Refs. 12 and 13.
HIGHER
MULTIPOLE
POLARIZABILITIES
165
Adding to Eq. (4.4) the number obtained for the s-channel piece (I&, - iW2)t1) = 0.6 x 1O-3 fm5
(4.5)
I?,, - KM1 N 11.4 x 1O-4 fm3,
(4.6)
and taking
as given by the latest direct experimental extraction [33], a total value for the difference of the generalized quadrupole electric and magnetic polarizabilities of the proton, (&
- ~~~~~~~~~~N 4.25 x 1O-3 fm”,
(4.7)
would result. We note that the resulting value (from Eqs. (4.3) and (4.7)) of & appears to be roughly three times larger (and opposite in sign) than the value found by Schroder [13, 141 for the static correspondent of this quantity. We now start presenting the results of a numerical evaluation of the sum of the quadrupole electric and magnetic polarizabilities of charged and neutral pions on the basis of the sum rule Eq. (2.43). The corresponding partial wave photoabsorption cross-sections uEI, GF~~are calculated in Breit-Wigner form including p”, w”, 4 states in the rr” case and pf, A, +, A,+ states in the vi- case. The needed widths are taken from Ref. [35]; for FAL+.ny, F,,,-,n, we take the same calculated values as those used in Refs. [23] (PAI,,,, N 0.57 MeV, PA,+,, - 0.91 MeV). After integrating the BreitWigner expressions over the resonance widths, one gets (KE2 + ZM2)(n~)N -0.31
x lOA fm5,
(ZE2 + KM2)(*+) N $0.4 x 1O-5 fm5.
APPENDIX
(4.8) (4.9)
A
In this appendix we shall display the multipole decomposition of the six independent helicity amplitudes describing the Compton scattering on spin 112 targets (nucleons) following the procedure and results originally given in Ref. [36]:
166
GU1ASI.J
AND
RADESCU
(A.3)
(A.41
(A-5)
64.6)
Normalizations are such that the two-particle (pion + nucleon) unitarity connection between the imaginary parts of the Compton partial waves and the Chew-GoldbergerLow-Nambu [37] photoproduction multipoles looks like Im -G+ = d(l + 1) I Eo+~)- 12, Im Et- = ql(l + 1) I EQ-,)+ j2, Im M,* = qZ(/ + 1) / Al,* jp, Im CIE = -q(1 + I)(&/ + ‘W2
(A-7)
Re(E(z+l)- ’ M$+l~-h
Im CtM = q(Z + 1)(1(1 + 2))‘j2 R&E,+ * Nf+), q = pion momentum
in the centre-of-mass system. The needed relationship between the Bardeen and Tung [17] invariant amplitudes Ai and the six independent helicity amplitudes is:
f+-1,+1= sin2 -2 cos -2 (s y2j2 fI-l.-+l
1
(A, + A3),
0, (s - 117‘q2 &3,2 [(s +
= 2m sin3 z
m”>A, + m(s - m2)A2 +
64.9) 2m2A3],
(A.lO)
HIGHER
f+1,+1 = &
MULTIPOLE
167
POLARIZABILITIES
cos + $ (2[(s - m*)* + WA] A, - m(su - 7774)A5 (A.ll)
- [(s - HP)* - m*t] A,S,
i e f-t1.:1 = _znl sin 22 cos2 2“2 ‘s,,:‘:” f-lr.-.*l
= &I co9 -+f(s
[--2n1A, - $(s + m2) A, - m&1,
- m”)” [2A4 +
!??A,
+ A,].
(A.12) (A.13)
Defining (A. 14) (x = cos 8, is the c.m. scattering angle introduced in Eq. (1.3)) and using known expressions of ni:jL (x) in terms of Legendre polynomials, one gets: [W+
- M,+) - U + NE,, --
- M,,)
+ 2(1(1 + 2))l” (GE - C,“)]
(s - 7x*)* . (1 + 1) 64~~s” (21 + 1)(21 + 3) x {--1(21 + 3)[(S + 772”)A;-l - 177(s- 777?)‘4-l] + (21 + 1)(3Z + 4)[(S + m”) AIE -
nr(s
- 777’)A,y
- (21 + 3)(3Z + 2)[(S + 777’)A4+l - m(s - 777”)Ay] + (I + 2)(2Z + l)[(S -t- m”) A4+2 - m(s - 177”)AT] - 2s[Z(2Z + 3) A;-l + (21 + 1)(51 + 8) A3 - (21 + 3)(5Z + 2) p
- (I + 2)(2Z + 1) A?*]},
[VU + 2))1’2 (Es+ - Mz+) + (41 + 2))l’” (E,,
- (21 + 3)(A4+1 + A?
- Ai-1 - f$l)],
[(I + 2)@,+ - Mz+) - @,-,, - KJ = (s - m2j2 , 64~s~
- M,,)
I 2 0,
(A.15)
+ 2(ctE - c,“)]
121
(A.16)
- W(Z + 2N1” CC; - C,“)]
(I + 1) (21 + 1)(21 + 3)
x ((I + 2)(2Z + 3)[(S t m’) ‘4:-l ?- 177(s- 7772)/i-l
+ 2n?&l]
168
GUIAt,XJ
AND
RADESCU
- 3(21 + I)(1 + 2)[(s + m”) A,’ + m(s - m’) Azz + 2m2A,‘] + 31(21 + 3)[(s + m2) A4fl + m(s - ~11’)AFfl + 2m2Ap] - Z(21 + l)[(s + rn’) p
+ m(s - 1722)Ay
u@,+ + MI+) + (1 + 2)(f%., + KL)
+ 2m%y]),
+ 2(E(l + w2
12
1 (A.17)
(CF + c,q
= (s - m2)2 . cl+ 1) 647rs(s)~~” (21 + 1)(21 + 3, x {m2[l(21 + 3)(2A1;’
+ Al,-‘) + (I + 2)(21 + 1)(2Ay
+ Al,f2)]
+ 2(21 + 1)[(2s - m2)(21 + 3) + m2(l + I)] A,’ + 2(21 + 3)[(2s - m”)(21 t- 1) + m2(1 + l)] AF’ + (21 + l)[-(2s
+ m’)(21 + 3) + m”(1 + l)] A,
+ (21 + 3)[-(2s
+ m2)(21 -t- 1) + m2(l + l)] A”s’
+ ms[(31 + 2)(21 + 3) A5+l + l(21 + 3) A;-’ f (31 + 4)(21 + 1) A,7f,
+ (1 + 2)(21 + 1) A?’
120
(A.18)
[(Kl+ 2)Y2 (Ez+-/-Ml+)- (l(1+ 2y2(E,;,+ Ml,,) + 2(GE+ GM)1 =
(s -
m2Y . (1 +
64~s
1x4~ + 2w2 ((21 + 1)[244
(21 f
I)(21 + 3)
4
z _ Az+z) 4
+ +(s + m2)(A5’ - AF2) + m(A,1 - A:+2)] + (21 + 3)[2m(A;-1 - A;+l) + $(s + n?)(A;-’ w + 2)(&+ + ML+) + w,,
- A?‘) + m(Al,-l - A$+‘)]},
(A.19)
+ M&3 - 2(1(1+ 2))1/2(C$E+ czq
= (364.rrsf,2 - m212. (21: Ii::+ +3w
131
3) ((1 + 2x21+ 3)(2Aki1 + mAk’ + A;-?
+ 1)(1+ 2)(2A,I + mA,1 + A;) + 31(21+ 3)(2A;+l + mALiI + A;+‘)
+ 1(21+ 1)(2Aj+’ + mAkf2 + A:+‘)}.
(A.20)
HIGHER MULTIPOLE POLARIZABILITIES
169
As the invariant amplitudes we are working with are free from kinematical problems, to find low-energy theorems for the multipoles El*, Ml*, CfcM1 valid to the order W* in the photon frequency, all one must do is to start with the double series expansions
use also the expansions
of the multipoles
in powers
of w (A.23)
(A.24)
cp’“‘+)
= (A!g+’ (2c)
f (Cf(M).C),~ (5)“. ?L=O
and identify the corresponding w-powers In this way one finally arrives at (El- -
M1-)‘O = (El+ -
(Eli
+ M,‘)*”
(El-
-
through Eqs. (A.15)-(A.20).
M,+)*O = ; C,op”,
= (El+ + Ml+)so = $ tn (Cl*’ -j- T C;.‘),
,Ml-)pl = 5 [-3C,oso - mC,oao - 2~~22C,o~Q] z (El;;,- - j$&-)*l - 2c,o*q
(Eli
-
M,+)*l = ; [-3C:‘.O
+ T ($0 + ,&;-“]
c (Eli,’ - i&+).1 - 2c;J3,
(A.29
170
GUIASU
2 d3(ClE
-
cl”)*”
2 d/5 (CIE + cy)~”
(E,- - M2-)”
AND
RADESCU
= -i??C,O*O, = -mcy,
= & [16n~~C,0’~- 6wz2C;*’ - 5C,oSo] G (E2- - @2-),” - ; c,o.o.
(E,+ - &f,+)‘* = & [16t&,0*1 + 4/r&;“]
(E,- + M2-)‘o = & [16rn3 (C4o.l + 7 C;*‘)
+ lo,c:‘O
GE (g2+ -
lB2+yo
- 5m (Ct.’
+ y C;-“)
- 3m2Cyq
z
- !f (c,“*” + T cp.“),
(E,+ + i14,+)~~ = -& [16m3 (Cigl + $ CT!*‘) + z~~c~*~],
etc.
(A.26)
So the threshold behaviour of the non-Born part of the spin-l/2 Compton multipoles appears to be known and specified in terms of partial derivatives with respect to v2 and t (evaluated at the s-channel physical threshold v = t = 0) of the continuum parts of the invariant amplitudes A, .
APPENDIX
B
This appendix is intended to provide some of the links allowing for the interpretation of f?,, , KMM1as “generalized” multipole polarizabilities of the hadron. For this purpose we start recalling briefly the way in which the static multipole polarizabilities KEt , ~~~ are introduced (see [7, 141) in the non-relativistic quantum mechanics as well as the existing connections between these objects and corresponding integrals over the total cross sections for photoabsorption of EZor MI photons on the hadron. We limit the illustrative considerations presented here to electrical transitions only. Let us consider the case of a polarizable hadron in the Coulomb field of a nucleus with charge number 2. The interaction of the induced electric multipole of order I with the external Coulomb field is described by the potential (in rationalized units e2/457 = l/137) dr p(r) r2P~(cos9).
CB.1)
HIGHER
MULTIPOLE
171
POLARIZABILITIES
9 denotes the angle between the vectors R and r, p(r) is the charge density and the r integration runs over the hadron exposed to the external field. Perturbation theory in the static approximation (which implies neglection of the atomic excitation energies with respect to the excitation energies of the hadron) leads, using the closure relation for the atomic states, to the energy shift due to the hadron polarizability phenomenon 1 = - 3 Km EfdT
&.h,.z,)
(pu+l))n,,~,,
03.2)
where l(QoH KEz
=
2 ($)
c
a
I .I” dr
f(r) ,F,H
rzpz(cos _ ,fT,H
T>
I @aH>/2 ’
(B.3)
is the usual definition of the static 2E-multipolar electric polarizability of the hadron (energy E and wave functions of the hadrons bear the superscript H; the sum is extended over all excited states (cx) of the hadron, the subscript 0 referring to its unexcited state; (n, , I,) in Eq. (B.2) obviously specify the known principal and orbital quantum numbers). To establish the relationship between K El and the total cross-section for photoabsorption of El photons, one begins with [38] (B-4)
which represents the integral over the line of the total cross section for photoabsorption of a photon with polarization Ek,n, (A = il), in terms of the matrix element of the electromagnetic current j(r). Performing a multipole decomposition, in the limit kr < 1 one has for an El transition
(where Yhz,j, are, respectively, spherical and Bessel functions) and therefore
s
273 l)(Z +1)e2
CT&dE, = --
w
line
Asforkrg
- (z+
/ s
I
YA”(&)
I a> dr /*. (B.7)
1, h&r)
kzrz
~~~“~ (21 + l)!!
03.8)
172
CXJIA$U
AND
RADESCU
one can put, for a spin J, target,
x 1 dr p(r) ~‘YA’(&)
NJ,, W, a)/’ W&J - E, - w)
(B.9)
with qw; 1) = 27r”W + l)(Z + 1) &&-l 1[(21 + 1)!!]2 .
(B.10)
Averaging also over the photon helicity, one has
u&(w) = qw; I)
e2 c” X2Ja + 1) h,M,
x I<(Jd, P 111dr p(r) &V (&)
11(J,), a>[’ %J%- Em- ~1. (B.11)
But
so that we can write (B. 11) in the form: u~yw)
= qw;
Z)
(21 + 1) 4T
e2;
(2J,:
1)
(B.12)
x 6(E, - E, - w).
One then finds
from where the desired connection
1 dw&Y4 T&P swo WZZ
(I+ 1) I)!!]2
= 2i[(21 -
KEz ’
(B.14)
HIGHER MULTIPOLE
POLARIZABILITIES
173
finally results. Eq. (B.14) explains the Z-containing factors in those formulae of the text which serve as definitions of the generalized multipoIe polarizabilities. ACKNOWLEDGMENTS
The authorswould like to thank ProfessorG. Furlan for readingthe manuscriptand Dr. C. Pomponiufor hisaid in the numericalevaluations.Oneof them(E.E.R.) isalsogratefulto Professor Abdus Salam,the InternationalAtomic EnergyAgencyand UNESCOfor hospitalityat the TnternationalCentrefor TheoreticalPhysics,Trieste. REFERENCES 1. A. KLEIN, Phys. Rev. 99 (1955),998. 2. A. M. BALDIN,Nucl.Phys. 18 (1960), 310. 3. V. I. GOLDANSKY, 0. A. KARPUKHIN, A. V. KUTSENKO, AND V. V. PAVLOVSKAYA, Nucl. Phys. 18 (1960),473. 4. V. A. PETRUNKIN, Zh. Eksp. Teor. Fiz. 40 (1961),1148(in Russian). 5. Y. A. ALEXANDROV AND I. I. BONDARENKO, Zh. Eksp. Teor. Fiz. 31 (1956),726(In Russian; V. S.BARASHENKOV, I. P. STAKEMNOV, AND Y. A. AIEXANDROV, Zh. Eksp. Teor. Fit 32 (1953, 154(in Russian);Y. A. ALEXANDROV, Zh. Eksp. Teor. Fiz. 33 (1957),294(in Russian);G. BREIT AND M. L. RUSTGI, Phys. Rev. 114 (1959), 830; Y. A. ALEXANDROV, G. S. SAMOSVAT, J. SEREETER, AND TSOY GEN SOR, Zh. Eksp. Teor. Fit. Pis’ma 4 (1966),196(in Russian);Sov. Phys. JETP
Len. 4 (1966), 134. 6. F. JACHELLO AND A. LANDE, Phys. Lert. B 35 (1971),205. 7. T. E. 0. ERICSON AND J. H~~FNER, Nucl. Phys. B 47 (1972), 205. 8. V. A. PETRUNKIN, Proc. Lebedev Phys. bzst. 41 (1968), 165. 9. S. RAGUSA, Phys. Rev. D 11 (1975), 1536. 10. J. BERNABEU, T. E. 0. ERICSON, AND C. FERRO FONTAN, Phys. Lett. B 49 (1974), 381. 11. P. S. BARANOV AND L. V. FILKOV, Fiz. Elem. Chasrits At. Yadra 7 (1976),108(in Russian); D. M. AKHMEDOV AND L. V. FILKOV, Nucl, Phys. B 125 (1977), 530. 12. I. GUIASU AND E. E. RADESCU, Phys. Rev. D 14 (1976), 1335; Phys. Rev. D 18 (Comments and
Addenda),(1978),1728. 13. I. GUIA~U, C. POMPONIU, AND
E. E. RADESCU, Ann. Phys. (N.Y.) 114 (1978), 296-331. Fortschr. Phys. 24 (1976), 85. 15. I. GUIASU AND E. E. RADESCU, Phys. Rev. D 18 (1978),651. 16. H. AEIARBANEL AND M. L. GOLDBERGER, Phys. Rev. 165 (1968), 1594. 17. W. A. BARDEEN AND Wu-Kr TUNG, Phys. Rev. 173 (1968), 1423. 18. A. EDMONDS, CERN Report55-26,1955. 19. F. E. Low, Phys. Rev. 96 (1954), 1428; M. GELL-MANN AND M. L. GOLDBERGER, Phys. Rev. 96 (1954), 1433; S. RAI CHOUDHIJRY AND D. 2. FREEDMAN, Phys. Rev. 168 (1968), 1739; V. M. SHEKTER, Yad. Fiz. 7 (1968), 1272. 20. V. S. BARASHENKOV AND H. J. KAISER, Fortschr. Phys. 10 (1962), 33. 21. M. GELL-MANN, M. L. GOLDBERGER, AND W. THIRRING, Phys. Rev. 95 (1954), 1612. 22. L. I. LAPIDUS, Zh. Eksp. Teor. Fiz. 43 (1962),1358(in Russian).
14. U. E. SCHR~DER,
174 23. A. (III 24. F. Y. 25. S. 26. M. 27. V. 28. M. 29. M. 30. P. 31. 32. 33. 34. 35.
36. 37. 38.
GUIA@J
AND
RADESCU
I. Lvov, V. A. PETRUNKIN, AND S. A. STARTSEV, Lebedev Phys. Inst. Report No. 173, 1976 Russian); A. 1. Lvov AND V. A. PETRUNKIN, Lebedev Phys. Inst. Report No. 170, 1977. J. GILMAN et al., Phys. Lett. B 31 (1970), 387; J. BALLAM et al., Phys. Rev. D 5 (1972), 545; EISENBERG et al., Nucl. Phys. B 104 (1976), 61. J. BRODSKY, F. E. CLOSE, AND J. E. GUNION, Phys. Rev. D 6 (1972), 177. V. TERENTEV, Yad. Fiz. 16 (1972), 162 (in Russian). N. PERVUSHIN AND M. K. VOLKOV, Phys. Lett. B 55 (1975), 405; 58 (1975), 177. GOURDIN AND A. MARTIN, Nuovo Cimento 17 (1960), 224. DAMASHEK AND F. J. GILMAN, Phys. Rev. D 1 (1970), 13 19. NACELLE, W. PFEIL, AND D. SCHWELA, Nucl. Phys. B 26 (1971), 461; P. NACELLE AND W. PFEIL, Nucl. Phys. B 31(1971), 1. R. G. MOORHOUSE, H. OBERLACK, AND A. H. ROSENFELD, Phys. Rev. D 9 (1974), 1. W. H. FRAZER AND J. R. FULCO, Phys. Rev. 117 (1960), 1603. P. BARANOV et al., Phys. Lett. B 52 (1974), 122. U. E. SCHROEDER, Phys. Rev. D 12 (1975), 896. Review of Particle Properties, Rev. Mod. Phys. 48 (1976). V. I. RITUS, Zh. Eksp. Tear. Fiz. 33 (1957), 1264; 32 (1957), 1536. G. F. CHEW, M. L. GOLDBERGER, F. E. Low, AND Y. NAMBU, Phys. Rev. 106 (1957), 1345. J, M. EISENBERG AND W. GREINER, “Excitation Mechanisms of the Nucleus” North-Holland, Amsterdam/London, 1970.
OF PHYSICS
120, 145-174 (1979)
Higher Multipole from Compton
Polarizabilities of Hadrons Scattering Amplitudes I. GUIASU
Institute
of Physics
and Nuclear
Engineering,
P.O.
Box 5206,
Bucharest,
Romania
AND
E. E. RADESCU* International
Centre
for
Theoretical
Physics,
Trieste,
Ital)
Received August 6, 1978
Starting with partial wave projections of the Compton scattering invariant amplitudes free of kinematical singuIarities or zeros, generalized electric and magnetic multipole polarizabilities of spin-0 and spin-l/2 hadrons are introduced as threshold limits of the non-Born parts of the corresponding electric and magnetic Compton multipoles. These objects, which enter the higher low-energy theorems for elastic y-hadron scattering, are shown to be acceptable quantum fieId theoretical generalizations of the usual static multipolar polarizabilities encountered in classical electrodynamics or non-relativistic quantum mechanics. Using the analyticity properties of the amplitudes, sum rules for the generalized electric and magnetic quadrupole polarizabilities of the proton and the pions are written down and evaluated numerically within the two-particle unitarity approximation.
I. INTRODUCTION While first introduced in particle physics by Klein [I], Baldin [I], Goldanski [3] and Petrunkin [4] in the context of Compton scattering on hadronic targets and used also by Alexandrov and others [5] in the study of slow neutron scattering by the strong Coulomb fields of heavy nuclei, the concept of hadron polarizability has attracted much interest in the last years due to its usefulness in other domains of research, such as that concerning hadronic atoms [6, 71. The theoretical investigation of the hadron polarizability phenomenon is somewhat facilitated in those cases (hadronic atoms, for instance) when non-relativistic quantum mechanics can be applied, because then the static polarizabilities of the hadrons may easily be introduced, according to wellknown procedures, in complete analogy with the atomic or nuclear polarizabilities. The situation is not so simple in Compton scattering off hadronic targets, because * On leave of absence from: Institute of Physics and Nuclear Engineering, P.O. Box 5206, Bucharest, Romania. 145 OOO3-4916/79/070145-3O$O5.00/0 All
Copyright 0 1979 by AcademicPress, Inc. rights of reproduction in any form reserved.
146
GUIAQU
AND
RADESCU
what really matters in the low-energy region are some structure-dependent coefficients, which, although strongly correlated to the polarizability properties of the target, do not, in general, coincide with the usual static polarizabilities. This paper is not devoted to the problem of establishing exact connections between the static polarizabilities and the parameters (other than the target’s mass, charge and anomalous magnetic moment) expressing the low-energy behaviour of the Compton S-matrix element. Tn this respect we refer the reader to the basic work of Petrunkin [4, 81 and to some other more recent results [9]. Our purpose is merely of a pragmatic nature and consists in introducing certain threshold limits of the Compton scattering partial waves (with the s- and zl-channel Born pole contributions taken off) as “generalized” electric and magnetic multipole polarizabilities of the corresponding hadron. Through the combined use of dispersion relations and the unitarity condition, sum-rules for these objects can be more or less straightforwardly derived from which their interpretation as polarizabilities can be made quite transparent. Indeed, it will be shown that if the target particle under consideration behaves like a non-relativistic quantum system, some integral contributions in the obtained sum rules would be negligible and one would thus be left with the usual non-relativistic quantum mechanical formulae for the static polarizabilities. Other analogies (specifically in the electric-quadrupole case) are made directly on the basis of the classical electrodynamical expression of the cross-section for radiation scattering off a system of charges possessing an induced multipole polarizability. Identifying the generalized polarizabilities in terms of Compton scattering multipoles rather than in terms of threshold values or threshold derivatives with respect to energy and momentum transfer of the continuum parts of the invariant amplitudes (as previously done in the literature for the dipole [4, l&13] or quadrupole cases [14, 151) has the advantage of facilitating a more direct contact with facts already known from atomic or nuclear physics. This article is organized as follows: In Sec. II the simpler case of spin-0 hadrons is considered. Generalized multipole polarizabilities are defined after establishing the threshold behaviour of the spin-0 Compton multipoles on the basis of the lowenergy theorems directly obtained in view of the good kinematical properties (i.e. absence of kinematical singularities or zeros) of the two invariant ampiltudes one is working with. The interpretation of the objects introduced is particularly discussed in the dipole and quadrupole cases in the framework of the sum rules written down. In Sec. 111analogous results are presented for the spin-l/2 case. Sec. IV is devoted to some numerical evaluations of the pions and the proton generalized quadrupole polarizabilities. We think the estimation of the proton generalized quadrupole electric polarizability is worth performing in view of a comparison with recent calculations of the static correspondent of this quantity [14]. Appendix A contains some of the lengthy formulae necessary in Sec. III, while in Appendix B the connections between the static polarizabilities and photoabsorption cross-sections in nonrelativistic quantum mechanics are recalled to help to understand the analogies invoked in the text. We now display some kinematical relations regarding the Compton scattering,
HIGHER
MULTIPOLE
147
POLARIZABILITIES
which will be used throughout this paper. The photon (hadron) momenta before and after the collision are denoted by k(p), k’(p’); c”(k) and c,+(k’) are the initial and final photon polarization vectors. The Mandelstam invariants S, t, u are defined as usual in terms of the quadrivectors K = +(k + k’),
P = XP + p’),
Q
= k’ - k =p
s = (P + I’C)~ = (p + k)2 = M2 + 2P . K - $Q”;
t _
(k
u E
s+
t
(p
-
K)”
=
(p’
-
k)”
---_ M2
-
2J’
. K
-
&Q2;
--I, -
k’)”
(1.1) :=
+ u = 2M”;
Q2;
(1.2)
v G p . K = i(s - u),
where M denotes the target’s mass. The kinematics of the process in the centre-of-mass system are established by the relations s = w2 2 11= M2 - 20, * (E + oJ,x) , t
= -2w,“(l
(1.3)
- x),
W = the total energy, E
=
s -t M2 2(s)l/"
= the hadron energy,
s - M2 w c -- 2(s)W = the photon energy, x = cos 8, = 1 + -!2CO,2’
6, = the centre-of-mass scattering angle.
The kinematics in the laboratory system are given by s=M2+2Mw,
t = 2ww’(z
z = cos 8,
-
‘1 = -
2w2(1 - 2) , + (u,,j,f)(, _ z) )
(1.4)
u=M”---Mu-t,
OJ = the energy of the incident photon, w’ = the energy of the emergent photon, WI =
1 + (w,C)(l
- z) .
We shall also need the expressions of u and t in terms of the incident lab. photon energy o and x (the cosine of the centre-of-mass scattering angle (9,):
148
GUIASU
t
zcz
-2w2
AND
RADESCU
(1 - X) 1 + 2(4M)
V:;Mw[l-;&)
’
(1.5)
&$)I.
II. SPIN-O CASE 2.A. Basic Formulae for Spin-O Compton Scattering The S matrix for Compton scattering off spin-0 hadronic targets (referred to from now on as pions in this section) is Sfi = afei + i(2n)-2 (lSk,,kGp,,~@-~l~ am+
T&k).
(2.1)
The tensor T,,( p’, k’; p, k) can be specified in terms of two scalar invariant amplitudes, free of kinematical singularities, zeros or constraints, through the following gauge invariant decomposition [ 16, 171: T,,(P’,
k’;
p, 4
=
A(k . k’g,, - k,,k:) - B[k . k’P,P,
- (P eK)(P,k:
+ P,k,) + g,,(P . IY)~]. (2.2)
The s- and u-channel Born pole structure of the amplitudes even under s - u crossing) is
A, B (which are both
(2.3)
where e2/4sr N l/137 is the fine structure constant and p denotes the target’s mass for the spin-0 case considered in this section. The kinematical relations Eqs. (l.l)-(1.5) will be used with the obvious replacement A4 -+ p. The two independent helicity amplitudes and the corresponding projections into partial waves (electric and magnetic multipoles) are: fi,h,
1 t) = 4- [(s - p2j2 + st] B
= 874~)~‘~ i (J%(W)+ M,(w)) &W, kl
firds, t>= - $[2A + (a- pp)B] &&)
= -87r(s)l12 f (E,(w) - M,(w)) d;,-,(x). z-1 stand for the usual rotation group functions [18].
(2.4)
HIGHER
MULTIPOLE
In the above normalizations
the optical theorems read
“T
Imf,,,(s,
149
POLARIZABILITIES
=
2 (GE, 1=1
+
(2.5)
%,),
t = 0) = (’ v4p2)z Im B(s, t = 0) = (S - $7 Or,
where aEtMjl and ur denote respectively the multipolar and total cross sections for photoabsorption on hadron. Further we shall also need the inversion formulae: J%(W) + M,(w) = &
(2z Tl:‘2’ ~2 j_:’ dx &(x)(1
+ x) B(s, t),
4(o) - M&J)= - & (2z;$,1’a’ .c:’ dx d;,,(x)(l x [W,
t>4 (+ - pp)B(s,01.
- X) (2.6)
In the following we shall be particularly interested in the continuum parts of the invariant amplitudes AC, BC,i.e., in those parts which remain after the s - u-channels Born poles have been taken of? A(& t) = ABoyS, t) + AC(S, t),
(2.7)
B(s, t) = BBorn(s, t) + Bc(s, t).
To lowest order in electromagnetism, employing the usual analyticity assumptions, we have at our disposal a general dispersion representation [16] for the invariant amplitudes, which will prove helpful in further considerations, especially when numerical evaluations will be performed. It consists of an unsubtracted fixed momentum transfer dispersion relation for the B amplitude and an unsubstracted fixed u dispersion relation for a particular combination of A and B: Be@, t, u) = $ s,rz ds’ BtS)(s’, t) ($-s --s 2AC(s, t, u) + ($ s- l -ds’ n J42
-
pz) BE& 1, u)
2/P(s’,
+ $j4;/t’
+ A), s -u
u) + [(t(s’, u)/4) s’ - s
2/F)@‘, u) + ((f/4) t’ -
t
$1 B(*)(s’, u)
p2) BW’,
u)
, t(s’, u) = 2p2 - s’ - u. (2.9)
150
GUIAQU
AND
RADESCU
The weight functions A(@, BtS) and Att), P in the above formulae denote, respectively, the absoprtive parts of the invariant amplitudes A, B in the s and t channels. The dispersion representation given by Eqs. (2.8) and (2.9) does not conflict with the normal t- and u-channel Regge pole model prediction for the asymptotical behaviour. Unfortunately Regge asymptotics at large s and fixed t does not allow for a fixed t unsubtracted dispersion relation for the A amplitude too; this is why one has to resort to other dispersion lines, such as that leading to Eq. (2.9) (which brings in the unpleasant t-channel absorptive parts) in order to avoid the appearance of unknown subtraction functions. We close this subsection mentioning that the quantities introduced here should bear the labels (Ch) or (N) corresponding, respectively, to Compton scattering on charged or neutral hadronic targets. This specification will be made occasionally explicit in the course of this paper where also sometimes a labelling (in the pion case) in terms of t-channel isospin (I = 0 on I = 2) will be used through y-(=0'
= y-‘“h.’ + @VW , (2.10)
2.B. Low-Energy
Theorems and Generalized Multipole PolarizabiIities
We are now prepared to analyse in terms of which parameters occurring in Compton scattering on spin-0 hadrons can one conveniently characterize the way in which the hadron will respond to the presence of the electric or magnetic fields. To this purpose we begin with the consideration of the low-energy theorems (i.e. expansions of helicity amplitudes or directly measurable quantities like the differential crosssection in powers of the photon frequency at fixed scattering angle) existing for the process we are interested in [I91 and note first [17] that once one has at one’s disposal (as we here have) a set of invariant amplitudes simultaneously free of kinematical singularities and zeros, one can automatically derive all wanted low energy theorems starting from the double Taylor series developments (in u2 and t) of the continuum parts of the amplitudes: AC($, t) = .,p.o + .2/414 + L40.1 + .a.,
(2.11) Bc(,J~, t) = B”.” + ,2Bl.“ + tBO.1 + . . . .
subsequently changing the variables from v2, t to, say, w, , cos 0, and finally ordering everything in powers of wc . The structure dependent constants Ai*j, BQ (i, j = 0, I,...) thereby introduced represent intrinsic characteristics of the considered particle in terms of which many of its properties (like the one we are concerned with-the ability of the hadronic cloud of getting polarized in external electric or magnetic fields) can be formulated. As an illustration of this statement, we mention now that generalized dipole electric (&El) and magnetic (I?,,> polarizabilities of the hadron can be identified as linear
HIGHER
MULTIPOLE
151
POLARIZABILITIES
combinations of the s channel threshold values (V = 0, t = 0) of the continuum A”, 3” of A, B (i.e. ADso, ISon0from Eqs. (2.11)) according to it& + KM1 = E
l?q -
(2.12)
P(V = 0, t = O),
Kh.fl = - &
parts
[2A”(V = 0, t = 0) - $B’(v
= 0, t = O)].
(2.13)
This identification is easily provided by a second-order low-energy theorem for the unpolarized Compton scattering differential cross section, given, in the centre-of-mass system, by do
k) d-2 c.m.
=&
[IA,1 I2+ IL1 I”].
(2.14)
Doing the necessary algebra and subsequently going to the laboratory frame to keep as much contact as possible with the classical electrodynamics formula for scattering of electromagnetic waves on polarizable (charged) objects, one finds (-g),*&
= ($)”
; g-j”
(I -
0J2
2 @(V = 0, t = 0) 87-r
x [ (l$z) -
](l + z”) - 9
z)’
G4” -
P2m(~ ST
= 0, t = 0)
II
+ o(w3)
2 (2*
15)
which justifies Eqs. (2.12) (2.13). Indeed, essentially the same angular dependence in the Rayleigh scattering (the piece of order w2 in the photon frequency in Eq. (2.15)) is obtained classically [20] when radiation scatters off polarizable targets with static electric (a) and magnetic (Is) polarizabilities. The connection between the induced electric (magnetic) dipole moments d~$$~~(magnetic) and the corresponding inducing electric (magnetic) fields E(B) is (in rationalized units) (2.16) The static polarizabilities 01, p appearing in classical calculations and the quantities are obviously KEl > KMl , which have been called generalized dipole polarizabilities, not to be identically assimilated with each other. As a matter of fact, in the classical (or even in the non-relativistic quantum mechanical) analogue of Eq. (2.15) there appears, for instance, instead of I?E1not just a but also a term containing the mean squared electrical charge radius of the target (arising classically from an average of the electric field over the target’s charge distribution, necessary to obtain the effective electric field to be put in the formulae for the dipole radiation). We stress here that the analogies with the classical (or quantum mechanical) calculations represent only a guide to assign physical interpretation to parameters otherwise unambiguously
152
GUIAljW
AND
RADESCU
defined in terms of well established quantities describing the Compton scattering on hadronic targets within a framework based on general principles of quantum field theory, such as relativistic (and gauge) invariance. This is why we use the attributive “generalized” for the polarizabilities introduced and we postpone for other parts of this paper any further discussion of such analogies. In this respect we only mention now that particular attention will be paid to those situations in which, in certain limits and under certain assumptions, the parameters introduced as generalized polarizabilities will show themselves quite similar to their non-relativistic quantum mechanical analogues. At this point let us explain why we attached the dipole indices El , M, to the generalized polarizabilities GE1, I?~, introduced through Eqs. (2.12), (2.13). For this purpose, all we need are low-energy theorems for the multipoles El , MI and these can be obtained simply with the aid of the inversion formulae (2.6). Separating the Born parts in EL, M& according to El = EF’=’ + E,c,
(2.17)
Mz = Mforn + M,G,
(where EForn, M For” are calculated from Eqs. (2.6) with the invariant amplitudes A, B replaced by their s- and u-channel Born poles (Eqs. (2.3)) while ErC, MIC are similarly obtajned with A, B replaced by AC, Bc) and retaining onIy the first coefficients A”eo, go,0 in the developments (2.1 I), one arrives at the desired new identification of cE1 , I?.,,1 directly in terms of the first multipolar partial waves E1 , M,: KEl
=-&-
-h’(w)
,
ZM1
w=o
Ml”(W) =-ix-
w=o
*
(2.18)
Low energy theorems similar to Eqs. (2.18) can be deduced in the same way for all higher multipoles Et , M, in terms of higher coefficients AiJ, I&j entering the expansions (2.1). So one can obtain the threshold behaviour (w + 0) of EIC, MLC in terms of corresponding partial derivatives with respect to v2 and t of AC(v2,t), Bc(v2, t) evaluated at v = 0, t = 0. We thus introduce through the following definitions some other new structure constants, KEI , f?M (expressible as linear combinations of Ai#i, I&j), which will be called generalized higher multipole polarizabilities of the target: Kq =
(21)[(2Z -
I)! !I” E&o) OP 0 + 1)
I0-o ’
I = 1, 2,... . (21)[(21 - l)! !I” Mzc(o) kMz z (I + 1)
CP
(2.19)
Io*9
The Z-containing factors in the above relations are included to ensure the necessary connections with analogous parameters describing the polarizability properties of charge systems in classical electrodynamics or non-relativistic quantum theory, and we shall discuss this point in more detail later on.
HIGHER
MULTIPOLE
POLARIZABILITIES
153
2.C. Sum Rules for Dipole Generalized Polarizabilities
For the sum of generalized electric and magnetic dipole polarizabilities one immediately finds the known sum rule [21,2,22]
KEl $ KM1
(2.20) which involves the total cross section for photoabsorption on the hadron. w0 denotes the inelastic threshold determined by the target’s first excited state in the continuum; for pions
Eq. (2.20) follows from the identification (2.12), the unsubtracted dispersion relation for the B amplitude Eq. (2.8) at t = 0 and the optical theorem (last of Eqs. (2.5)). The question of an analogous sum rule for the difference I?~, - zAr, is much more delicate, since writing down an unsubtracted forward dispersion relation for the relevant combination of invariant amplitudes A, B which appears in Eq. (2.13) would conflict with the normal t-channel Regge poles model prescription. Under these circumstances one can either look for additional dynamical assumptions guaranteeing for the more or less accidental disappearance at t = 0 of the bad terms in the Regge asymptotics of the relevant combination of amplitudes, or simply abandon the forward direction, using, for instance, Eq. (2.9) or other possible unsubtracted nonforward dispersion representations, and thus accept the unavoidable appearance of the t-channel absorptive parts as a fact of life. The first possibility, while based on a more drastic set of hypotheses, has at least the obvious aesthetical advantage (as only s-channel absorptive parts will then come in) that it leads to sum rules which render the physical interpretation of the hadron polarizability phenomenon quite easy in as far as analogies with the better studied cases of molecules, atoms or nuclei are concerned. This line has been followed recently in Refs. [23]. In order to get a forward sum rule for & - KM , the authors of Refs. [23] start with the difference between the amplitudes 2i - $B for charged and neutral pion Compton scattering which, having isospin Z = 2 in the t-channel (see Eqs. (2.10)), can be expected to have a better asymptotic behaviour for large v than each of the amplitudes (2A - pzB)(Ch)*(N) separately. Indeed, the t-channel Regge poles with Z = 2, if they exist, probably have a negative intercept 01(o) < 0, so that a possible fixed pole (Jp = 0+) would be the one which mainly governs the asymptotics. Therefore (2A - p2B)(‘=2) would tend to a constant for v + co; consequently, writing down a forward dispersion relation for this quantity with a subtraction at infinity, working out the (s-channel) absorptive parts through partial wave-projections and optical theorems (see, for instance, Eqs. (2.9) and (2.5)) and joining Eq. (2.20) (for charged and neutral pions) to the sum rule so obtained for (
154
GUIASU
AND
RADESCU
IA
OEI
-
4
I=2 u’Mz --
I
(2.21;
‘4
14 - UML I=2 GE& 4 1 I. (2.22)
The superscript I = 2 on aEl, uM1 on the r.h.s. of Eqs. (2.21) and (2.22) obviously means the difference between the quantities corresponding to + and 7~” targets. More restrictive sum rules (in the sense that the superscript I = 2 in Eqs. (2.21), (2.22) can be dropped, the relations now holding directly for the charged and neutral cases separately) have also been obtained in Refs. 23 under the supplementary assumption of s-channel helicity conservation in high energy photon scattering [24]. In such forward sum rules the unknown subtraction constants K, (proportional, with known simple factors, to the asymptotic value (V ---f co) of the forward amplitude 2A - p2B) still remain as a serious problem, even though one can again appeal to assumptions or models to dispose of them (in the parton model, for instance [25], A -+ 0 for v + cc so that Kg+) = Kg’) = Kz=” = 0). In turn, Eqs. (2.21), (2.22), or more exactly, their mentioned analogous without the superscript I = 2, have the virtue of maintaining close contact with known facts in molecular, atomic or nuclear physics [23]. Indeed, when K, and the contributions to the continuum integral from higher excited states can be disregarded (i.e. if the term with w/p and terms with I > 1 are small, as it happens, for instance, in nuclear physics where the actual integration threshold w0 is much smaller than the target’s mass CL), one gets for the quantities cE1 , K~ introduced as “generalized” dipole electric and magnetic polarizabilities, the usuallnon-relativistic expressions (2.23) (2.24)
Conversely, for light hadrons, when the term with the supplementary energy factor w/p can be thought of as the most prominent one, one finds, even without neglecting K, the known current algebra [26] or chiral theory [27] result I?& e! -KMl.
(2.25)
After this short discussion on forward sum rules for the dipole generalized polarizabilities, we go on now to present some results within the other framework mentioned, e.g. that involving annihilation-channel exchanges through the explicit appearance in the new sum rules for zE - b, of the t-channel absorptive part of the relevant combination of Compton scatiering ifivariant amplitudes.
HIGHER
MULTIPOLE
155
POLARIZABILITIES
We first start with the fixed u dispersion relation Eq. (2.9), which at s = $, t = 0, gives us the desired sum rule
where
For the s-channel absorptive part using the partial wave projections (Eqs. (2.4)) and optical theorems (Eqs. (2.5)) one can write the exact expression
[2A f (+$
- P2j‘$” (5 u) = 4’;&f’ $ &,(.~)(a~,- q,,& (2.29)
t(s, 21) = 2p” - s -
l,,
I-X __
2
= -
45 (s -
4 p2)2 ’
while for the t-channel absorptive part, if only two particle (two pion) states are considered in the t channel, one has [2‘4
i- ($ - P2jq:‘, (t,u)
= 4JZen(2J +
1) g+“(t) (t(t ;;@)j””
h?(t) PJCOSyq,
(2.31)
where P, are the usual Legendre polynomials, 4 expresses the angle in the centre-ofmass of the t-channel reaction 7r~ 4 yy, h,(t) denote the TUT partial waves, cos 4 =
t - 2(p2 - u) = (t(t - 4pL2))l/z ’
(t(t “4&’
(2.32)
IzJ(t) = eisJ(‘) sin 8,(t),
(2.33)
stand for the TT phase shifts), and g+“(t) are the ~7r -+ yy partial waves related to the helicity amplitude f+.+ corresponding to transitions with both photon helicities equal to + I. For t in the elastic unitarity region ghJ has the phase SJ of the corresponding TX -+ rrrr channel. For completeness, we display immediately below the partial wave projections of both t-channel helicity amplitudes f++, f+-, describing the annihilation into two photons:
(6,
==
c J,even
(2J
+
1)
g+“(t)
[‘(’
;;p2)]J’2
PJ(cos
$),
(2.34)
156
GUIASU
f+-
AND
= - ; [(s - $)”
RADESCU
+ st] B
= J Zen (2J + 1) g-J(t) [ t(t ;64P2)]“in
&(cos
$).
Such projections and other details about the theoretical investigation of the ~TZ--+ yy process can be found in the basic paper of Gourdin and Martin [28]. The sum rule given by Eqs. (2.26)-(2.29) can then be written as
- &
jar2 $
c
(25
+
1)
g+“(t)
[ ‘(’
fffp2’]J’z
k,*(t)
PJ(cos
#)
J.even
+ higher (than Z-T states) annihilation
channel contributions.
(2.36)
If instead of the fixed u dispersion relation Eq. (2.9), one starts with a fixed angle representation, one can get another version of Eq. (2.36), the direct and annihilation channel contributions balancing each other according to the dispersion line chosen. The (formally) simplest sum rule is obtained if one works in the backward direction (0 = 180”) [lo, 121: ;El
-
%U1
= $1: - &
$ (1 + $)
b(Ye4 - dNo)l
s,12 $ J Zen (2J + 1) pr ;p2)]J’2
+ higher (than rr7~states) annihilation u(Yes) and o(No) denote the photoabsorption the parity flip and non-parity flip multipoles: a(Yes) =
Jf
a(No)
=
2
channel contributions.
cross sections containing
UE~ +
l=odd
g+“(t) mt>
C
(2.37)
respectively
OM~,
l=even
GE1+
c
Z=even
(2.39)
UMz.
l=odd
2.D. Sum Rules for Quadrupoie Generalized F’olarizabilities
Along the same lines as in the previous subsection one can also establish sum rules for the generalized electric and magnetic quadrupole polarizabilities of the spin-0 hadrons KEs
, w=o
EM2
=
12
M2Yw) --J--
(2.40) CO=”
HIGHER
MULTIPOLE
POLARIZABILITIES
157
Through the inversion formulae (2.6) and with the aid of the expansion (2.11) we shall express them in terms of the derivative with respect to t of the continuum parts of the invariant amplitudes A, B evaluated at v = 0, t = 0:
As seen from Eq. (2.41) for the sum (IzE1 f KM1) a forward sum rule can be obtained without problems in view of the good high-energy asymptotic behaviour of the B amplitude. Using Eq. (2.8) and expressing the s-channel absorptive part B(S)(s, t) in terms of multipolar photoabsorption cross-sections, one easily gets
In the particular case in which w0 < p and terms with the supplementary factor w/p as well as terms with I > 2 are small, one finds from Eq. (2.43) the usual nonrelativistic expression
For the difference z.Ez- K&,, , as happened also for tag - iM1 , the bad asymptotic behaviour of the A amplitude in the forward directionlprevents us writing down a sum rule like (2.43) unquestionably free of unknown subtractions. One may then again use Eq. (2.9) and, working in the same manner as in the corresponding dipole case, one ends up with the following sum rule of the same type as Eq. (2.36):
X
-
s (t(t
X
(2.45) LA+ G - p2)BYct,tL(tT cos~)/eo~i-,‘,l,(~-~,P)‘,~~.
m dt
au2t
1 a - 4@))l/2 a cos tj
158
GLJIAFjU
III.
AND
RADESCU
SPIN-;
HADRONS
3. A. Basic Formulae for Spin- 112 Compton Scattering
The S-matrix element for the Compton effect on hadronic spin l/2 targets (referred to as nucleons from now on) Y(k) + N(P) -
Y’(k’) + WP’)
is Sfi = 6f,i - i(27r)” 6(p’ + k’ - p - k) m(4kOkipop@-1/z x dk’)
(2~)~~
J(P’) T,,(P’, k’; P, k) U(P) dk).
(3.1)
k(p), k’(p’) stand for the photon (nucleon) momenta before and after the collision; E,(k)(u(p)) and l ,+(k’)($p’)) describe the photon (nucleon) polarization states; m denotes the nucleon mass. The kinematical relations Eqs. (l.l)-(1.4) will be used this time with the corresponding replacement A4 ---f m.
To make this paper self-contained we now display the specification of the tensor Ai , known to be free of kinematical zeros or singularities: Tsy in terms of the Bardeen and Tung [17] invariant amplitudes
TAP’,
k’; P, k) = f =Gfi(s,
t, u),
(3.2)
i=l
Pl = K21111,
g2 = K211,1,
92 = I[mI, - (P . K) I1 - K21,]I, g4 = I[K2&, - mK21, + (P ’ K) 1,]Z, Z5 = I[K”l,
(3.3)
- +(P2K2 - (P . K)2) Z,]Z,
Z6 = I[/, + $(P . K) ml, - iP212 - $(P * K) I5 + $mK21, + &(P . K) ,,]I,
where I,” = t?,” - (k,k;/k
. k’),
(3.4)
and 4 = gdy
4 = gw 2 14
=
PAY
. K>
P,,
. 9,
4 = pup, ,
4 = Gwv - YYYUI~ (3.5) 15= $IY~(Y. W TV- Y~Y*K) ~ul* >
4
=
PLLY"
+
YUPY
3
We do not give here the explicit form of the s-u-channels Born pole structure of the amplitudes Ai but refer the interested reader to Ref. [17]. The amplitudes A1,2,4,j are even while AS.6 are odd under s - u crossing. The introduction of helicity amplitudes, as well as the partial wave projections and the
HIGHER
MULTIPOLE
159
POLARIZABILITIES
needed low-energy theorems, due to the cumbersome aspect of the relations in the spin-l/2 case, have been relegated to Appendix A. Defining the following combinations of + and - multipoles (see Appendix A) El = $[l&-- + (/ + 1) El+], (3.6)
MI = +ph!&- + (E + 1) Ml+],
in terms of which the (partial wave) optical theorems read Im El = 2
a(El), (3.7)
Im MI = 2
0(M),
33 fJT = c (Q(E1)+ wfd
(34
I=1
(c+ is the total cross section for photoabsorption on nucleon), one can introduce again generalized electric and magnetic multipole polarizabilities (this time for spin l/2 systems) through the same kind of formulae as Eqs. (2.19) from the previous section: i?EI =
(21)[(21 - l)!!]’ (1 + 1)
. Elc(w) cfJzz Iw-0 ’
(21)[(21 - l)!!]Z M”(w) tT,&fl= -7 (I+ 1)
(3.9)
(,)+ *
The identification of the dipole and quadrupole generalized polarizabilities as defined in Eqs. (3.9) in terms of more manageable objects (as far as dispersion relations are concerned) like derivatives (with respect to y2and t evaluated at threshold (V = t = 0)) of the (crossing symmetrized) continuum parts of the invariant amplitudes AiC = A. - AForn, can be worked out with the aid of the formulae from Appendix A and lo;ks like [12] KEl + K&jM1= -4”, [ cpso + T Cpq)
(3.10)
for the dipole case, and (3.12) (3.13)
for the quadrupole polarizabilities. 595/120/1-11
160
GUIASU
AND
RADESCU
To discuss the interpretation of these first multipolar “generalized” polarizabilities, we now give the lengthy form of the low enregy theorem for the unpolarized proton Compton scattering differential cross-section valid to the order w5 in the photon frequency. In Ref. [15] this low-energy theorem has been established by means of the threshold derivatives 0’ (i = l,..., 6; k, 1 = 0, l,... ), while we now give it in an alternative form in terms of the coefficients expressing the developments of the non-Born part of the multipoles in powers of w (Eqs. (A.23)-(A.25)). One gets in a straightforward manner:
kalab. =(43,+(a,,,,,)- $& x [1 - 4f
(1 - z)] ( y-)”
[A + Bz + Cz2 + Dz3]
(3.14)
where (da/&), represents the Powell cross-section (undeveloped in powers of w), ) is wdn)kE1.~M71u1 the generalized dipole polarizabilities (17~~, I?~,) contribution and the last term expresses the new w4(w5) frequency orders (i.e. quadrupolar terms, mixed terms of order 1 and terms containing the first two derivatives of the dipoles, Et, E,2W,1, K?): ($),
= ; (&),
IL1 + (Q,;);;’
1, (3.15) [(l - z)” +f(z)l [1 + (4m>(l - 41”
z)12 + (g
f(z) = a0+ qz + a2z2, a, = 2h + $X2 + 3X3 + fh4, a, = -4h - 5h2 - 2X3, a2 = 2X + +A2 - A3 - &X4, (h = 1.793 denotes the proton anomalous magnetic moment)
=
(1 - z) + 6 (+)’ X2
12 [Cl -
(
- y-$ x
I
(1 - z)” - 10 (+)3
(1 - z)~]
4” @El- hf3 + (1 + 4” GE1+ hIM31
[1 - 4 (5)
(1 - z)](Z)
(l ; z)2 [(2 - 2x - X2) - z(2 + 2h + X2)1(& - i&+1)
- F
(1 - 2)” (&
- I?.&2 -
- F
(1+ 2)”(G1+ ZMl)$
(* + zy
- z, (X2 + 2h)(i?El + f&J (3.16)
HIGHER
A
m3 -S(Zfi,-
MULTIPOLE
+ 2&+)J
+ 3fl?&+)9O+ 8&-
1
+ 8 d5 (A + -g)
161
POLARIZABILITIES
(C,E - c p ) ,O- 4 VT (2 + 2A + P) (cy + c,*y
- (4 -+ 16h + 6h3@-
- A&-)*1 + (4 + 16h + 6h2)(&+ - &&+-,J
+ (16 + Sh + 6h2)@- + @l-)*1 + (8 - 8h - 6X2)(&+ + A?!,-)J], B
m3[16(al-
+ 2al+)s2 + 8 ~“5 (1 + 3X + P)(C,E -
CIM),O
t 4 d/3(2 i- 2h t 2P)(C,E + CIM)*O + 4(1 + 4x -j- h2)(&
- A&-).1
- 4(4 + 4x + X2)(& + - ml+)*l + 4(3 + X2)(8,- + II&-y + 4(6 - A’)(&+ + II?&+)J], Lfl = r?r3[24(2aZ- + 3i@2+)so + 8(&i,- + 2&+)*2 - 8 2/3(2 + 5/j + 92) X (c? - clM).’
+ 2 ~‘3 (4 + 4h + 6X2)(C,E + CIM).” + 2h2(&- - j$&->J
t (12 - 2X2)(&‘,+ - Ml+)-1 -
+ (16 + 8h + 2h”)&+ D = 16m3[(2&-
+ 3E2+)~o -
(4 + 8h + 2X2)&-
+ q-y
+ A?Il+)J], (I + A) qf
(3.17)
~~01.
(,@I~, aFYi are defined in Appendix A.) Let us now isolate the contribution of the electric quadrupole in Eq. (3.14):
(E2) to (&/&)rab
du (__ dQ >
(E2)
(3.18) This form agrees with the result [ 151 of a simple classical electrodynamics calculation of the interference between the Thompson scattering and the scattering due to an induced quadrupole polarizability K E2 = zE2 (note that R,, from Eq. (3.18) stands for &K of Ref. [I 51, due to a difference in the definition of K) and gives once more support to the interpretation of ~~~ as quadrupole polarizability. 3.B. Sum Rules for the Generalized Quadrupole Polarizabilities
of the Proton
As sum rules for the generalized dipole polarizabilities i?Er , 17,, in the nucleon case have already been quite extensively presented and analysed in the liter-mare, here we shall only derive and discuss sum rules for the quadrupole polarizabilities i7E2and iM2 . We begin with the treatment of GE2+ ZM2 , as again in this case one can take
162
GUIAt$U
AND
KADESCU
advantage of the good high energy asymptotic behaviour of the combination of invariant amplitudes defining, according to Eq. (3.12), this sum, to write down without difficulty a fixed-t dispersion relation and avoid in this way the occurrence of any undesirable annihilation channel contribution. Starting with unsubtracted fixed momentum transfer (t) dispersion relations for the amplitudes &(s, t) and A&, t) (which is possible at least as far as the normal Regge behaviour is concerned), using Eq. (3.12) and the optical tehorem Im
(
A,+TAs
)I t=o = k
dw,,
where uT denotes the total cross-section for photoabsorption immediately arrives at the sum rule
3 m* + 2rr2m sug w
Ap)(w,
on protons,
one
t = 0).
w. = ~(1 + ~/2m) denotes the single pion photoproduction threshold and the superscript (s) indicates the absorptive part in the s-channel. Using the projection in multipoles listed in Appendix A to work out the last two terms of Eq. (3.20) one can put the obtained sum rule in the equivalent form:
+ *
jl$
(1 + 2 +)1’2
Tm I( 2)’
(&-
+ Ml-)
+ f pz++ Mzf) ((I+w2 + 1- 1) Z=l
+
2(1 + 2)(12+
-
Z(1 + 2) (1 + G)(
+ (E,,
+ Al,-,,)
I -
1)
(=)
1 + 2 $)liP
(1(Z2 + 31 +
+ Z(Z + 2) (1 + 2 -y
+ (I + 1) (+)z)
1) + 2w2
(1 + 5)
+ 31+
1) (C)
+ (I + 1) (e)2)
HIGHER MULTIPOLE
163
POLARIZABILITIES
+ 2(Z(Z-I- 2))“” (Cf $ Cpf) x (I +2+-p
+2g2(1
If, in the above sum rule, contributions to would be negligible (i.e. if u//n could be put I > 2 disregarded), one would find from Eq. usual non-relativistic formula connecting the over the photoabsorption cross-sections of the
(3.21)
+x-),1/.
the integral from higher excited states
Irrespective of whether in the actual case of the proton it is permissible or not to make such neglections, the formal aspect of Eq. (3.22) again gives support to the interpretation of ZEI , r~-,, (defined in Eqs. (3.9)) as generalized multipole polarizabilities. Eqs. (2.23), (2.24), (2.44) from Sec. II as well as Eq. (3.22) above are obtained in a dispersion theoretical framework with subsequent approximations which would be understandable or justifiable for non-relativistic quantum systems. In Appendix B we present the usual quantum mechanical procedure to express the static multipole polarizabilities (of not necessarily spinless systems) in terms of (completely analogous) formulae involving the corresponding photoabsorption cross-sections such that the reader has at his disposal the needed material. For the difference ~~~ - KMz a sum rule can be established starting with Eq. (3.13) and using a fixed t dispersion relation for the A, c - AsCamplitude once subtracted at u = m2, with the subtraction function expressed further as an unsubtracted dispersion relation (in the t variable). So one finds:
(6” -
cM2)(1)
= - $ s,I:+.,? ds’
[A,‘“‘@‘, u = m”) - A;S)(s’, ZI = rn”)] (s’ - I??)”
(3.24)
3 m ds, AgS)(S’,t = 0) --I 79 (rnlu)2 (s’ - m2y ’ [A;“(t’,
24 == m2)
-
At’@‘,
t’”
IV. NUMERICAL
u
=
,,I’)]
(3.25)
ESTIMATES
In this section we shall compute some of the contributions to the quadrupole generalized polarizabilities of the proton and the pions from the sum rules written
164
GUIAfjU
AND
RADESCU
down in the previous sections. We do not attempt a thorough numerical analysis, through the use of all presently available information on the quantities entering the sum rules envisaged, but rather concentrate on the most reliable pieces and only discuss within simple models some of the less reliable ones. We begin with the evaluation of the sum of the electric and magnetic generalized quadrupole polarizabilities kE2 + ii,, as given by Eq. (3.21). One of the integrals over the total cross-section for photoabsorption on protons (the one related to &;I + K~& is known from Ref. [29],
while the other has been obtained integrating numerically reference, 3 ;D d dw> w __ 2?r%l s*,). w3
the data given in the same
E 0.52 x 1O-3 fm5.
(4.21
The third integral in Eq. (3.21) has been evaluated (retaining only waves with .Z = 4, g, 9) using the single pion photoproduction data of Refs. [30] for the energy region 180 MeV < w < 250 MeV and the data of Ref. [31] for 250 MeV -=cw < 1210 MeV. In this way one gets the estimate (KE2 + KMz)proton N 0.8 x lO-4 fm”.
(4.3)
It should be stressed that this number results from surprising compensations between different contributions and therefore the inclusion of the double-pion photoproduction yield may be quite relevant. In the evaluation of the difference I& - /T,+,,we are faced with the difficult problem of the annihilation channel contribution (the integral over t-channel absorptive parts in Eq. (3.25)). While the s-channel contributions (Eq. (3.24)) can be treated in terms of known photoproduction multipoles, as mentioned before (disregarding possible complications from extrapolations to the unphysical region in the s-channel absorptive parts at u := m2), the t-channel piece is highly model-dependent and all we intend to do now is to provide an estimate for it, which should not be taken too seriously, based on a simple model previously used in connection with cEl - ifiI1 [12, 131. The contribution of the rrrr states to (I?~, - ii,,)(,) , retaining only s-waves, is as follows:
e 3.8 x 1O-3 fm5.
(4.4)
where the Frazer-Fulco [32] Nn + nii amplitudefF’o(t) and the Gourdin-Martin [28] nm -+ yy amplitude gy’o(t) are computed in an N/D model for the XT + T-i-m(Z = J = 0) partial wave as in Refs. 12 and 13.
HIGHER
MULTIPOLE
POLARIZABILITIES
165
Adding to Eq. (4.4) the number obtained for the s-channel piece (I&, - iW2)t1) = 0.6 x 1O-3 fm5
(4.5)
I?,, - KM1 N 11.4 x 1O-4 fm3,
(4.6)
and taking
as given by the latest direct experimental extraction [33], a total value for the difference of the generalized quadrupole electric and magnetic polarizabilities of the proton, (&
- ~~~~~~~~~~N 4.25 x 1O-3 fm”,
(4.7)
would result. We note that the resulting value (from Eqs. (4.3) and (4.7)) of & appears to be roughly three times larger (and opposite in sign) than the value found by Schroder [13, 141 for the static correspondent of this quantity. We now start presenting the results of a numerical evaluation of the sum of the quadrupole electric and magnetic polarizabilities of charged and neutral pions on the basis of the sum rule Eq. (2.43). The corresponding partial wave photoabsorption cross-sections uEI, GF~~are calculated in Breit-Wigner form including p”, w”, 4 states in the rr” case and pf, A, +, A,+ states in the vi- case. The needed widths are taken from Ref. [35]; for FAL+.ny, F,,,-,n, we take the same calculated values as those used in Refs. [23] (PAI,,,, N 0.57 MeV, PA,+,, - 0.91 MeV). After integrating the BreitWigner expressions over the resonance widths, one gets (KE2 + ZM2)(n~)N -0.31
x lOA fm5,
(ZE2 + KM2)(*+) N $0.4 x 1O-5 fm5.
APPENDIX
(4.8) (4.9)
A
In this appendix we shall display the multipole decomposition of the six independent helicity amplitudes describing the Compton scattering on spin 112 targets (nucleons) following the procedure and results originally given in Ref. [36]:
166
GU1ASI.J
AND
RADESCU
(A.3)
(A.41
(A-5)
64.6)
Normalizations are such that the two-particle (pion + nucleon) unitarity connection between the imaginary parts of the Compton partial waves and the Chew-GoldbergerLow-Nambu [37] photoproduction multipoles looks like Im -G+ = d(l + 1) I Eo+~)- 12, Im Et- = ql(l + 1) I EQ-,)+ j2, Im M,* = qZ(/ + 1) / Al,* jp, Im CIE = -q(1 + I)(&/ + ‘W2
(A-7)
Re(E(z+l)- ’ M$+l~-h
Im CtM = q(Z + 1)(1(1 + 2))‘j2 R&E,+ * Nf+), q = pion momentum
in the centre-of-mass system. The needed relationship between the Bardeen and Tung [17] invariant amplitudes Ai and the six independent helicity amplitudes is:
f+-1,+1= sin2 -2 cos -2 (s y2j2 fI-l.-+l
1
(A, + A3),
0, (s - 117‘q2 &3,2 [(s +
= 2m sin3 z
m”>A, + m(s - m2)A2 +
64.9) 2m2A3],
(A.lO)
HIGHER
f+1,+1 = &
MULTIPOLE
167
POLARIZABILITIES
cos + $ (2[(s - m*)* + WA] A, - m(su - 7774)A5 (A.ll)
- [(s - HP)* - m*t] A,S,
i e f-t1.:1 = _znl sin 22 cos2 2“2 ‘s,,:‘:” f-lr.-.*l
= &I co9 -+f(s
[--2n1A, - $(s + m2) A, - m&1,
- m”)” [2A4 +
!??A,
+ A,].
(A.12) (A.13)
Defining (A. 14) (x = cos 8, is the c.m. scattering angle introduced in Eq. (1.3)) and using known expressions of ni:jL (x) in terms of Legendre polynomials, one gets: [W+
- M,+) - U + NE,, --
- M,,)
+ 2(1(1 + 2))l” (GE - C,“)]
(s - 7x*)* . (1 + 1) 64~~s” (21 + 1)(21 + 3) x {--1(21 + 3)[(S + 772”)A;-l - 177(s- 777?)‘4-l] + (21 + 1)(3Z + 4)[(S + m”) AIE -
nr(s
- 777’)A,y
- (21 + 3)(3Z + 2)[(S + 777’)A4+l - m(s - 777”)Ay] + (I + 2)(2Z + l)[(S -t- m”) A4+2 - m(s - 177”)AT] - 2s[Z(2Z + 3) A;-l + (21 + 1)(51 + 8) A3 - (21 + 3)(5Z + 2) p
- (I + 2)(2Z + 1) A?*]},
[VU + 2))1’2 (Es+ - Mz+) + (41 + 2))l’” (E,,
- (21 + 3)(A4+1 + A?
- Ai-1 - f$l)],
[(I + 2)@,+ - Mz+) - @,-,, - KJ = (s - m2j2 , 64~s~
- M,,)
I 2 0,
(A.15)
+ 2(ctE - c,“)]
121
(A.16)
- W(Z + 2N1” CC; - C,“)]
(I + 1) (21 + 1)(21 + 3)
x ((I + 2)(2Z + 3)[(S t m’) ‘4:-l ?- 177(s- 7772)/i-l
+ 2n?&l]
168
GUIAt,XJ
AND
RADESCU
- 3(21 + I)(1 + 2)[(s + m”) A,’ + m(s - m’) Azz + 2m2A,‘] + 31(21 + 3)[(s + m2) A4fl + m(s - ~11’)AFfl + 2m2Ap] - Z(21 + l)[(s + rn’) p
+ m(s - 1722)Ay
u@,+ + MI+) + (1 + 2)(f%., + KL)
+ 2m%y]),
+ 2(E(l + w2
12
1 (A.17)
(CF + c,q
= (s - m2)2 . cl+ 1) 647rs(s)~~” (21 + 1)(21 + 3, x {m2[l(21 + 3)(2A1;’
+ Al,-‘) + (I + 2)(21 + 1)(2Ay
+ Al,f2)]
+ 2(21 + 1)[(2s - m2)(21 + 3) + m2(l + I)] A,’ + 2(21 + 3)[(2s - m”)(21 t- 1) + m2(1 + l)] AF’ + (21 + l)[-(2s
+ m’)(21 + 3) + m”(1 + l)] A,
+ (21 + 3)[-(2s
+ m2)(21 -t- 1) + m2(l + l)] A”s’
+ ms[(31 + 2)(21 + 3) A5+l + l(21 + 3) A;-’ f (31 + 4)(21 + 1) A,7f,
+ (1 + 2)(21 + 1) A?’
120
(A.18)
[(Kl+ 2)Y2 (Ez+-/-Ml+)- (l(1+ 2y2(E,;,+ Ml,,) + 2(GE+ GM)1 =
(s -
m2Y . (1 +
64~s
1x4~ + 2w2 ((21 + 1)[244
(21 f
I)(21 + 3)
4
z _ Az+z) 4
+ +(s + m2)(A5’ - AF2) + m(A,1 - A:+2)] + (21 + 3)[2m(A;-1 - A;+l) + $(s + n?)(A;-’ w + 2)(&+ + ML+) + w,,
- A?‘) + m(Al,-l - A$+‘)]},
(A.19)
+ M&3 - 2(1(1+ 2))1/2(C$E+ czq
= (364.rrsf,2 - m212. (21: Ii::+ +3w
131
3) ((1 + 2x21+ 3)(2Aki1 + mAk’ + A;-?
+ 1)(1+ 2)(2A,I + mA,1 + A;) + 31(21+ 3)(2A;+l + mALiI + A;+‘)
+ 1(21+ 1)(2Aj+’ + mAkf2 + A:+‘)}.
(A.20)
HIGHER MULTIPOLE POLARIZABILITIES
169
As the invariant amplitudes we are working with are free from kinematical problems, to find low-energy theorems for the multipoles El*, Ml*, CfcM1 valid to the order W* in the photon frequency, all one must do is to start with the double series expansions
use also the expansions
of the multipoles
in powers
of w (A.23)
(A.24)
cp’“‘+)
= (A!g+’ (2c)
f (Cf(M).C),~ (5)“. ?L=O
and identify the corresponding w-powers In this way one finally arrives at (El- -
M1-)‘O = (El+ -
(Eli
+ M,‘)*”
(El-
-
through Eqs. (A.15)-(A.20).
M,+)*O = ; C,op”,
= (El+ + Ml+)so = $ tn (Cl*’ -j- T C;.‘),
,Ml-)pl = 5 [-3C,oso - mC,oao - 2~~22C,o~Q] z (El;;,- - j$&-)*l - 2c,o*q
(Eli
-
M,+)*l = ; [-3C:‘.O
+ T ($0 + ,&;-“]
c (Eli,’ - i&+).1 - 2c;J3,
(A.29
170
GUIASU
2 d3(ClE
-
cl”)*”
2 d/5 (CIE + cy)~”
(E,- - M2-)”
AND
RADESCU
= -i??C,O*O, = -mcy,
= & [16n~~C,0’~- 6wz2C;*’ - 5C,oSo] G (E2- - @2-),” - ; c,o.o.
(E,+ - &f,+)‘* = & [16t&,0*1 + 4/r&;“]
(E,- + M2-)‘o = & [16rn3 (C4o.l + 7 C;*‘)
+ lo,c:‘O
GE (g2+ -
lB2+yo
- 5m (Ct.’
+ y C;-“)
- 3m2Cyq
z
- !f (c,“*” + T cp.“),
(E,+ + i14,+)~~ = -& [16m3 (Cigl + $ CT!*‘) + z~~c~*~],
etc.
(A.26)
So the threshold behaviour of the non-Born part of the spin-l/2 Compton multipoles appears to be known and specified in terms of partial derivatives with respect to v2 and t (evaluated at the s-channel physical threshold v = t = 0) of the continuum parts of the invariant amplitudes A, .
APPENDIX
B
This appendix is intended to provide some of the links allowing for the interpretation of f?,, , KMM1as “generalized” multipole polarizabilities of the hadron. For this purpose we start recalling briefly the way in which the static multipole polarizabilities KEt , ~~~ are introduced (see [7, 141) in the non-relativistic quantum mechanics as well as the existing connections between these objects and corresponding integrals over the total cross sections for photoabsorption of EZor MI photons on the hadron. We limit the illustrative considerations presented here to electrical transitions only. Let us consider the case of a polarizable hadron in the Coulomb field of a nucleus with charge number 2. The interaction of the induced electric multipole of order I with the external Coulomb field is described by the potential (in rationalized units e2/457 = l/137) dr p(r) r2P~(cos9).
CB.1)
HIGHER
MULTIPOLE
171
POLARIZABILITIES
9 denotes the angle between the vectors R and r, p(r) is the charge density and the r integration runs over the hadron exposed to the external field. Perturbation theory in the static approximation (which implies neglection of the atomic excitation energies with respect to the excitation energies of the hadron) leads, using the closure relation for the atomic states, to the energy shift due to the hadron polarizability phenomenon 1 = - 3 Km EfdT
&.h,.z,)
(pu+l))n,,~,,
03.2)
where l(QoH KEz
=
2 ($)
c
a
I .I” dr
f(r) ,F,H
rzpz(cos _ ,fT,H
T>
I @aH>/2 ’
(B.3)
is the usual definition of the static 2E-multipolar electric polarizability of the hadron (energy E and wave functions of the hadrons bear the superscript H; the sum is extended over all excited states (cx) of the hadron, the subscript 0 referring to its unexcited state; (n, , I,) in Eq. (B.2) obviously specify the known principal and orbital quantum numbers). To establish the relationship between K El and the total cross-section for photoabsorption of El photons, one begins with [38] (B-4)
which represents the integral over the line of the total cross section for photoabsorption of a photon with polarization Ek,n, (A = il), in terms of the matrix element of the electromagnetic current j(r). Performing a multipole decomposition, in the limit kr < 1 one has for an El transition
(where Yhz,j, are, respectively, spherical and Bessel functions) and therefore
s
273 l)(Z +1)e2
CT&dE, = --
w
line
Asforkrg
- (z+
/ s
I
YA”(&)
I a> dr /*. (B.7)
1, h&r)
kzrz
~~~“~ (21 + l)!!
03.8)
172
CXJIA$U
AND
RADESCU
one can put, for a spin J, target,
x 1 dr p(r) ~‘YA’(&)
NJ,, W, a)/’ W&J - E, - w)
(B.9)
with qw; 1) = 27r”W + l)(Z + 1) &&-l 1[(21 + 1)!!]2 .
(B.10)
Averaging also over the photon helicity, one has
u&(w) = qw; I)
e2 c
x I<(Jd, P 111dr p(r) &V (&)
11(J,), a>[’ %J%- Em- ~1. (B.11)
But
so that we can write (B. 11) in the form: u~yw)
= qw;
Z)
(21 + 1) 4T
e2;
(2J,:
1)
(B.12)
x 6(E, - E, - w).
One then finds
from where the desired connection
1 dw&Y4 T&P swo WZZ
(I+ 1) I)!!]2
= 2i[(21 -
KEz ’
(B.14)
HIGHER MULTIPOLE
POLARIZABILITIES
173
finally results. Eq. (B.14) explains the Z-containing factors in those formulae of the text which serve as definitions of the generalized multipoIe polarizabilities. ACKNOWLEDGMENTS
The authorswould like to thank ProfessorG. Furlan for readingthe manuscriptand Dr. C. Pomponiufor hisaid in the numericalevaluations.Oneof them(E.E.R.) isalsogratefulto Professor Abdus Salam,the InternationalAtomic EnergyAgencyand UNESCOfor hospitalityat the TnternationalCentrefor TheoreticalPhysics,Trieste. REFERENCES 1. A. KLEIN, Phys. Rev. 99 (1955),998. 2. A. M. BALDIN,Nucl.Phys. 18 (1960), 310. 3. V. I. GOLDANSKY, 0. A. KARPUKHIN, A. V. KUTSENKO, AND V. V. PAVLOVSKAYA, Nucl. Phys. 18 (1960),473. 4. V. A. PETRUNKIN, Zh. Eksp. Teor. Fiz. 40 (1961),1148(in Russian). 5. Y. A. ALEXANDROV AND I. I. BONDARENKO, Zh. Eksp. Teor. Fiz. 31 (1956),726(In Russian; V. S.BARASHENKOV, I. P. STAKEMNOV, AND Y. A. AIEXANDROV, Zh. Eksp. Teor. Fit 32 (1953, 154(in Russian);Y. A. ALEXANDROV, Zh. Eksp. Teor. Fiz. 33 (1957),294(in Russian);G. BREIT AND M. L. RUSTGI, Phys. Rev. 114 (1959), 830; Y. A. ALEXANDROV, G. S. SAMOSVAT, J. SEREETER, AND TSOY GEN SOR, Zh. Eksp. Teor. Fit. Pis’ma 4 (1966),196(in Russian);Sov. Phys. JETP
Len. 4 (1966), 134. 6. F. JACHELLO AND A. LANDE, Phys. Lert. B 35 (1971),205. 7. T. E. 0. ERICSON AND J. H~~FNER, Nucl. Phys. B 47 (1972), 205. 8. V. A. PETRUNKIN, Proc. Lebedev Phys. bzst. 41 (1968), 165. 9. S. RAGUSA, Phys. Rev. D 11 (1975), 1536. 10. J. BERNABEU, T. E. 0. ERICSON, AND C. FERRO FONTAN, Phys. Lett. B 49 (1974), 381. 11. P. S. BARANOV AND L. V. FILKOV, Fiz. Elem. Chasrits At. Yadra 7 (1976),108(in Russian); D. M. AKHMEDOV AND L. V. FILKOV, Nucl, Phys. B 125 (1977), 530. 12. I. GUIASU AND E. E. RADESCU, Phys. Rev. D 14 (1976), 1335; Phys. Rev. D 18 (Comments and
Addenda),(1978),1728. 13. I. GUIA~U, C. POMPONIU, AND
E. E. RADESCU, Ann. Phys. (N.Y.) 114 (1978), 296-331. Fortschr. Phys. 24 (1976), 85. 15. I. GUIASU AND E. E. RADESCU, Phys. Rev. D 18 (1978),651. 16. H. AEIARBANEL AND M. L. GOLDBERGER, Phys. Rev. 165 (1968), 1594. 17. W. A. BARDEEN AND Wu-Kr TUNG, Phys. Rev. 173 (1968), 1423. 18. A. EDMONDS, CERN Report55-26,1955. 19. F. E. Low, Phys. Rev. 96 (1954), 1428; M. GELL-MANN AND M. L. GOLDBERGER, Phys. Rev. 96 (1954), 1433; S. RAI CHOUDHIJRY AND D. 2. FREEDMAN, Phys. Rev. 168 (1968), 1739; V. M. SHEKTER, Yad. Fiz. 7 (1968), 1272. 20. V. S. BARASHENKOV AND H. J. KAISER, Fortschr. Phys. 10 (1962), 33. 21. M. GELL-MANN, M. L. GOLDBERGER, AND W. THIRRING, Phys. Rev. 95 (1954), 1612. 22. L. I. LAPIDUS, Zh. Eksp. Teor. Fiz. 43 (1962),1358(in Russian).
14. U. E. SCHR~DER,
174 23. A. (III 24. F. Y. 25. S. 26. M. 27. V. 28. M. 29. M. 30. P. 31. 32. 33. 34. 35.
36. 37. 38.
GUIA@J
AND
RADESCU
I. Lvov, V. A. PETRUNKIN, AND S. A. STARTSEV, Lebedev Phys. Inst. Report No. 173, 1976 Russian); A. 1. Lvov AND V. A. PETRUNKIN, Lebedev Phys. Inst. Report No. 170, 1977. J. GILMAN et al., Phys. Lett. B 31 (1970), 387; J. BALLAM et al., Phys. Rev. D 5 (1972), 545; EISENBERG et al., Nucl. Phys. B 104 (1976), 61. J. BRODSKY, F. E. CLOSE, AND J. E. GUNION, Phys. Rev. D 6 (1972), 177. V. TERENTEV, Yad. Fiz. 16 (1972), 162 (in Russian). N. PERVUSHIN AND M. K. VOLKOV, Phys. Lett. B 55 (1975), 405; 58 (1975), 177. GOURDIN AND A. MARTIN, Nuovo Cimento 17 (1960), 224. DAMASHEK AND F. J. GILMAN, Phys. Rev. D 1 (1970), 13 19. NACELLE, W. PFEIL, AND D. SCHWELA, Nucl. Phys. B 26 (1971), 461; P. NACELLE AND W. PFEIL, Nucl. Phys. B 31(1971), 1. R. G. MOORHOUSE, H. OBERLACK, AND A. H. ROSENFELD, Phys. Rev. D 9 (1974), 1. W. H. FRAZER AND J. R. FULCO, Phys. Rev. 117 (1960), 1603. P. BARANOV et al., Phys. Lett. B 52 (1974), 122. U. E. SCHROEDER, Phys. Rev. D 12 (1975), 896. Review of Particle Properties, Rev. Mod. Phys. 48 (1976). V. I. RITUS, Zh. Eksp. Tear. Fiz. 33 (1957), 1264; 32 (1957), 1536. G. F. CHEW, M. L. GOLDBERGER, F. E. Low, AND Y. NAMBU, Phys. Rev. 106 (1957), 1345. J, M. EISENBERG AND W. GREINER, “Excitation Mechanisms of the Nucleus” North-Holland, Amsterdam/London, 1970.