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SLAC-PUB-
March 1992 T/E
Universal Properties of the Electromagnetic
Interactions of Spin-One Systems*
STANLEY J. BRODSKY
Stanford Linear Accelerator Center Stanford University, Stanford, California 94309
and
JOHN R. HILLER Department of Physics, University of Minnesota-Duluth, Duluth, Minnesota 55812
ABSTRACT
The dominance of helicity-conserving amplitudes in gauge theory is shown imply universal ratios for the charge, magnetic, and quadrupole form factors spin-one bound states: Gc(Q2) : GM(Q2) : GQ(Q2) = (1 - $7) : 2 : -1. These ratios hold at large space-like or time-like momentum transfer in the case of composite systems such as the p or deuteron in &CD. They are also the ratios predicted’for the electromagnetic couplings of the W* for all Q2 in the standard model at tree level. In the case of the deuteron, the leading-twist perturbative QCD predictions are valid at Q2 = jq21 >> AQCDM~, but do not require the kinematical
ratio 77= Q2/4AIj to be large. These results provide new all-angle predictions for the leading power behavior of the tensor polarization Z’2s(Q2, 0) and the invariant ratio B(Q2)/A(Q2). W e a so 1 use a generalization of the Drell-Hearn Gerasimov sum rule to show that the magnetic and quadrupole moment of any composite spin-one system take on the canonical values ,Y = e/M and Q = -e/M2 in the strong binding limit of zero bound-state radius or infinite excitation energy. This allows new empirical constraints on the possible internal structure of the 2’ and W* vector bosons. Simple gauge invariant and covariant models and null zone theory are used to illustrate these results. Complications that arise when the Breit frame is used for form factor analyses are also pointed out.
Submitted to Physical Review D
- Work supported in part by the Department of Energy under contract number DEAC03-76SF00515.
I. INTRODUCTION
The low-energy theorem [l] for the forward Compton amplitude at threshold, and the helicity selection rules [2] of perturbative quantum chromodynamics (QCD) for exclusive scattering amplitudes at high-momentum transfer, indicate that many properties of a system bound by a gauge theory are universal and are the same as those of a corresponding elementary particle of the same spin and charge. In this paper, we shall explore this universality for the case of spin-one bound states in &CD, including both the p meson and the deuteron. In particular, we shall focus on the behavior of the electromagnetic form factors of composite spin-one systems at large momentum transfer, and on the fundamental constraints on the magnetic and quadrupole moments of hadronic and nuclear states imposed by Compton-scattering sum rules. In order to motivate the notion of universality, we first discuss the application of the Drell-Hearn-Gerasimov (DHG) sum rule^ [3-51 to the anomalous^ magnetic^ moment of a spin-one state. We then show how one can use Tung’s [6] extension of this analysis to obtain a new sum rule for the anomalous quadrupole moment of a general spin- one system. Together these sum rules show that in the limit of zero radius or large excitation energies, the magnetic moment ~1 and quadrupole moment Qr approach canonical values:
PI d-,^ Ql=--&^ (1-l)
where e is the total charge and M is the mass of the spin-one system. These are the same values obtained [7] for th e intermediate vector bosons W* in the tree approxi- mation to the standard model. It should be emphasized that the sum rule constraints on Qr and ~1 do not rely on perturbation theory, but only on the existence of unsub- tracted dispersion relations for the relevant helicity-flip Compton amplitudes. The deviation of the observed magnetic and quadrupole moments from the canonical val- ues thus define the “anomalous” moments of a general spin-one system: II,, s /.~r- & and &a - Qr + $, dynamical contributions which must be strictly due to internal structure. Various theoretical and experimental constraints have already been suggested for the magnetic and quadrupole moments of the W. The electromagnetic couplings of the intermediate vector boson are constrained by renormalizability and tree-level unitarity [8] to be those of the standard model. Experiments that have or will place bounds on nonstandard couplings include measurements of g - 2 for the muon 191, ~115+ W -yX [lo], the decay p --) ey^ [ll],^ heavy-ion^ collisions^ [12] and^ e+e-^ annihila- tion processes [ 131. The definition of the three parity-conserving and time-reversal invariant electro- magnetic form factors of a spin-one object is well known [14]. We will discuss the implications of perturbative QCD and helicity selection rules for these form factors at high momentum transfer in terms of the ratio B/A of Rosenbluth form factors
I
as fast as t/M2. However, a more complete analysis [22] finds that the ratio becomes a constant. We elaborate on this in a later section. A critical issue is the determination of the momentum-transfer scale at which perturbative QCD can make meaningful predictions for quantities such as B/A and the tensor polarization for spin-one targets 2’20. Because of kinematic factors, this scale is different from the scale at which A can be predicted. We discuss an appro- priate choice of scale and obtain predictions for B/A and Z”20. In the latter case, our predictions for the deuteron differs significantly from what is usually quoted [23] for the experimentally accessible region [24].
.-
With regard to perturbative calculations at large momentum transfer, we would like to draw attention to the work of Farrar, Huleihel, and Zhang [25] on the helicity- zero to zero deuteron form factor. They find that hidden-color degrees of freedom in the deuteron wavefunction may be important in obtaining the correct perturbative QCD predictions for normalization of the deuteron form factors at experimentally accessible momentum transfers. In addition, as shown in Ref. [26], the evolution of the deuteron’s distribution amplitude leads to the dominance of hidden color state contributions in the asymptotic domain of very large momentum transfer. Our anal- ysis will be independent of the existence of the relative normalization of hidden-color states. To confirm the generality of the form factor analysis, we also consider a simple model in which the composite spin-one system is constructed in a Lorentz-invariant and gauge-invariant way from two spin-l/2 constituents in a zero-binding limit of one-boson exchange. The analysis of the electromagnetic interactions in this model gives a simple demonstration of the connection between radiation null zones [27] and the natural magnetic and quadrupole moments of spin-one systems. An outline for the remainder of the paper is as follows. In Section II we discuss the sum rules and anomalous moments of spin-one systems. Form factors and their ratios are analyzed in Section III. The zero-binding model is presented in Section IV; this. includes discussion of null zones in radiative processes. Finally, Section V contains a brief summary.
II. SUM RULES
The low momentum transfer properties of both elementary particles and compos- ite systems can be related by general principles to integrals over scattering amplitudes. The best-known of these relationships is the Drell-Hearn-Gerasimov (DHG) sum rule [3,4] for the anomalous magnetic moment of spin-l/2 systems. It can be obtained by using an unsubtracted dispersion relation and a low-energy theorem [l] for the helicity-flip Compton amplitude. The generalization to arbitrary spin has been made [4,5]; the form for the spin-one case is
d
=- 1 u; ff b&) - u&)I 7
J (2.1)
where pa = p1 - fi is by definition the anomalous magnetic moment for the spin- one system, bp (0~) is the total cross section for absorption of a photon with spin parallel (antiparallel) to the spin of the target, and w is the photon energy, with && the threshold energy. Although an experimental verification of the DHG sum rule for nucleons has been carried out [28], ‘ 1 t would be interesting to verify this result for deuterons.
The extension of the DHG sum rule analysis to include the quadrupole moment of a spin-one system requires a low-energy theorem to second-order in the photon energy. At this order, the polarizability enters the forward Compton amplitude [29] in addition to the quadrupole moment. However, Tung [6] has shown that one can obtain the following sum rule for the non-forward Compton amplitude: 2 (zm fP(%t) - 1n-i fA(%t)) ,
(2.2)
where M is the mass, Qa = Qr + -& defines the^ anomalous^ quadrupole^ moment,^ v is (s - U)/% and f~ (fA) is the^ helicity^ amplitude^ for^ parallel^ (antiparallel)^ photon and target spins. The standard Mandelstam variables s, t and u are used. The normalization of the helicity amplitudes given by Bardeen and Tung [30] is used to derive (2.2). With this normalization, the optical theorem takes the form
Zm fP,A = 2voP,A. (^) (2.3)
In the forward direction, (2.2) reduces to (2.1), with use of w = v/M. A sum rule that relates Qa to total cross sections does not exist [6]. It is interesting to apply (2.2) t o composite spin-one systems which become point- like in some limit. In such a case the photoabsorption cross section and the integrals that appear in (2.1) and (2.2) vanish as the size R + 0 or the excitation energy z&, -+ po. Thus in this limit, Qa + 0 and pa --) 0. Therefore ~1 = & and Qr = -+ are the canonical moments of a spin-one system. Note that this analysis is non- perturbative. In the case of the standard model, the integrals in (2.1) and (2.2) are higher order N 0(02); thus again pw = 5 and Qw = --&, up to Schwinger- like radiative corrections of order o/r. Specific models for compositeness of leptons and intermediate vector bosons are discussed by Brodsky and Drell [31], Abbott and Farhi [32], and Claudson, Farhi, and Jaffe [33]. The DHG sum rule has also been used to place constraints on quark and lepton compositeness and excited states in the strong-coupling standard model [32] by Jaffe and Ryzak [34]. Note that any spin-one system is required to satisfy the extended DHG sum rule (2.2). This implies universal behavior for the properties of spin-one particles in the zero-radius limit. In the next section we explore a complimentary universality for the form factors of such particles at large momentum transfer in gauge theory.
1 Gc = 2p+(217 + 1)
GM=
2 2P+(277 + 1) 1 GQ = 2p+(217 + 1)
16 --CT0 277- 3
ce
TG& + $21, - l)G;- I GTO (277 - l)z + G&, - G;- I
2’:0--$8^ G+^00 -- v+lG+‘I^ +-’ 1
P-7)
In contrast, in the Breit frame, where <A = 0, p’= -f{= -p”, an x component of a current matrix element is needed to extract GM:
Gc=
2 G”+o GM = 2MJm ,/%j
Go=
-1 G&, + G;-
2MJm 277 ’
Predictions of the behavior of the matrix elements as functions of momentum transfer can then be used to extract the Q” dependence of form factors.
B. Asymptotic forms
Perturbative QCD predicts [36]. that the helicity-zero to zero matrix element G&, will be the dominant helicity amplitude at large Q2 for lepton scattering on a spin- one bound state. This follows since quark helicity is conserved in the hard-scattering quark-gluon amplitude, and the dominant wave function coefficient, or distribution amplitude, has L, = 0. However, it is important to distinguish two scales in the form- factor analysis. The primary scale for the validity of perturbative QCD predictions is set by the requirement that the momentum transfer through the hard-scattering amplitude and the propagators be large compared to the QCD scale AQCD. Since the current value of Am lies between 120 MeV and 200 MeV [37], we take AQCD to be of order 200 MeV. From estimates by Carlson and Gross [35] we conclude that the LCF helicity-flip amplitudes Gzo and G$- are suppressed by factors of 9 and (q)2, respectively. The second scale is a purely kinematic one. In order to control the kinematic factors in (3.7), and thereby retain dominance of G&,, one needs
Q>>+fAqco. (^) w-
This follows from the assumptions Gzo N *G& and Gz- N (^) ( * > 2 G&,. For
-, _. the^ dimensionless^ ratio^ 17=^ &,^ this^ requirement^ translates^ to^ 77>> %.^ Thus^ for
M > AQCD, the validity of perturbative QCD predictions does not depend on taking r) >> 1. The present data for the deuteron form factor [38] a, and also for the photodis- integration of the deuteron at large momentum transfer [39], appear to be consistent with perturbative QCD dimensional counting rules [20] and reduced amplitude scaling [21,40]. The scaling of the dominant form factor fi can be seen in Fig. 1, where the data is plotted in terms of the reduced form factor [21] fd(Q’) G A(Q2)/F$(Q2/4), with FN the dominant nucleon form factor. Perturbative^ QCD^ predicts^ asymptotic scaling for Q2fd(Q2) up t o calculable^ logarithms^ [26].^ Thus,^ optimistically,^ one could expect that the dominance of G$, begins at Q2 N 1 GeV2. We emphasize that the kinematic quantity 7 can be small in the perturbative QCD regime: for example for the deuteron Q2 = 5 GeV2 corresponds to 17 N 0.35, whereas we only require ?j > * 2! 0.05. Thus the domain for leading-power perturbative QCD predictions for the deuteron form factors is Q” >> ~M&!QcD N 0.8GeV’. In this domain, one obtains [22] from (3.7),
B (^) 477h + 1) v-h - i + (7 + 1) tan2 51 -;I li 72 + 77+^ 314 ’^
7’2~40) N -&’ ~2+7]+~+4~(7)+l)tan2~’
In the extreme limit q >> 1, these reduce to
B -~4, Tzo- --l/
1 + tan2 s 2 A 1+4tan2t’^
The asymptotic value of -fi usually quoted [23] f or^ T^ 20 only^ applies^ when^ 8 is zero, and when Q2 is much larger than 4M2, which is nearly 8 GeV’ for the deuteron. For 77<< 1, which is relevant to experiment, we obtain
B-N (^245) %, A- 3 T20 = --3+l^ -^ 2tan^ ‘B)^2. (3.12)
The essential assumption made in all of these results is that the G& amplitude is dominant. The expressions derived for B/A and T20 are compared with experiment [38,41, 241 in Figs. 2 and 3. Clearly, the presently available data do not come close to the prediction for B/A. However, for T 20 the^ trend^ of^ the^ data^ is not^ inconsistent^ with the prediction. Data at larger momentum transfer is clearly needed. It would also be useful to compare elastic electron and positron deuteron scattering to check the size of the two-photon exchange interference contribution to B(Q2) in the dip region. At lower Q2, where perturbative QCD is inapplicable, the behavior of the p and deuteron form factors can have completely different properties. For example, the deuteron quadrupole moment is measured to be [42] Qd = eGQ(0)/M2 = (25.84 f
These follow directly from the photon-absorption vertex in the standard model. At Q2 = 0 this corresponds to [7] th e canonical magnetic moment of e/Mw and quadrupole moment of -e/M&. For comparison, the form factors of a composite spin-one object are, in the LCF, assuming helicity-zero to zero dominance,
G C= (l- f”)2p+c;fft 1> 7 G M = 22p+($ I) 3 G Q= -2p+(Tt q -
(3.18) -.
Notice that the ratios of the three electromagnetic form factors Gc : GM : GQ = (1 - $v) : 2 : -1 are identical for elementary spin-one W ’s and for composite spin-one hadrons in QCD when G&, is dominant. In particular, B/A and TX, for the IV+ are given by (3.10). Th us at large Q2, perturbative QCD predicts that the ratio of form factors for deuterons, pi, etc. become identical to those of the point-like spin-one fields of the standard model. We will see explicit realization of these results in the next section. In the Breit frame analysis, the ratios of form factors do not match those for an elementary W: We therefore conclude that the LCF analysis is the correct approach.
IV. ZERO-BINDING MODEL
As a test of the correctness of the LCF analysis we shall study the form factors of a spin-one system in very simple gauge-invariant, Lorentz-invariant model, in which the composite system corresponds to two lightly-bound spin-i constituents interacting via boson. exchange. Results will be extracted in the zero-binding limit only. The analysis is similar to that required for constructing the hard scattering amplitude TH in perturbative QCD analyses of mesonic form factors [36]. This model is thus directly applicable to the form factors of the p meson at large momentum transfer and, in the context of the reduced amplitude approach [40], is also applicable to the deuteron. The wave functions used are generalizations of the vertex functions employed by Bagger and Gunion [45]. Th e f unctions factor into a spin-dependent part
XJh = c Nit, (^) -“(PI,’ 4qp2, S2)) 6182 pF
(4.1)
where J is the total spin and h the helicity, and a spin-independent part 4(x). In the zero-binding limit, the constituents are collinear, p; = m;p, and the distribution amplitude d(x) becomes 6(xi - mi/M). The spin wave functions then reduce to
Xlh= $ h(d-4, x00=+1(-M), (4.2)
with ch a polarization vector given by
ei ‘(0,^ 1, fi,^ 0))^ co
=FJz
= ~wl>0,0, JiiF=3 (^) (4.3)
in any frame where pi = 0. The reduction can be done in a variety of ways, which differ by the choice of basis for the spinors. One can use the light-cone helicity basis of Brodsky and Lepage [36], a standard helicity basis [46], or the Weyl basis discussed by Hagiwara and Zeppenfeld [47]; except for different phases, they yield the same results. To obtain form factors, we compute the matrix elements for the transition from p, h to p’, h’ in a one-boson exchange approximation. The corresponding diagrams are presented in Fig. 4. For a spin-one boson, with mass A, the usual Feynman rules yield
el (^) Abh BP G;th cc (^) 22y2Q2 + x2 - (x2 - y2)2M2 + h’h z2Q2 + ml - @f2 y2Q2+ m:- y?M2I
+(1 4-b 2) 7
(4.4) where the proportionality constant is determined by the boson-fermion coupling, among other things, and where the numerators inside the brackets are Dirac traces. These traces are
and
A;,/, = Tr{7d?mY"[$ - 52 ri + mll7'x~hl (4.5)
Bh = Trb%d%h $- ji+ ml]y”xJh7v}. They can be reduced to
W-
A;,h = -4M2{&b’P + (z - x2)$‘] + q.d($ + 52)~’ - qd*‘}. (^) (4.7)
and
-. B;,h = -4M2{d[pP + (2 - y2)p’p] - q. ~(2 + Y2)dfi + q.e’*ep}. (^) (
With the restriction to x; = y; = mi/M, the matrix element is found to be
G;,,, 0; -1 M xi&2 + X2 &2 “’ “C’ x2(P” + p’P) + &.d*e’ - q.ee”@) +^ (1 t)^ 2). 52 1 (4.9) Form factors can then be extracted by comparison with (3.2). We obtain
el GIM 2 2 --^ M2 x1^ e2^ M2^ x
x2& + A2 Q2 $2 xfQ2^ +^ X2^ Q2^ xl^ ’
G20c e-1 M21 + e2 M x;Q” + X2 Q2 x2 x;Q2 + X2 Q’ xl ’
G3=0.
sum rules discussed in Section II also are the moments that preserve null zones. In particular, all helicity amplitudes for the subprocess uz + W+r vanish at cos 8 = ed/eW provided that the W+ and the quarks have natural moments. A discussion of bounds on the W anomalous moment that can be obtained from present pjj + WrX data has been recently given by Samuel et al. [lo]. In the crossed reaction e+e- + r* --) VP [49], where V is any massive vector particle with charge or charged constituents, one again predicts dominance of the (0,O) helicity amplitude, so that the cross section g is proportional to sin2 0. This agrees with the perturbative QCD prediction of Ref. [2] for e+e- + p+p- and the standard-model prediction for e+e- -+ W+W- by Alles et al. [50]. Notice, however, that e+e- + W+W- receives contributions from v exchange which do not appear in the above analysis. That the vector particle need not itself be charged implies that the sin2 0 behavior of the cross section should hold at large s = q2 for processes such as e+e- -+ K”Eo. In each case, the time-like form factor ratios should satisfy Gc : GM : GQ = (1 - $J) : 2 : -1, where now 7 = s/4M2.
V.SUMMARY
We have provided two new non-perturbative arguments for the selection of e/M and -e/M2 as the natural magnetic and quadrupole moments of a spin-one particle. These are the canonical values that emerge in the strong binding limit of zero bound- state-radius or infinite excitation energy of a composite spin-one system. The first argument, presented in Section II, is based on an extension [6] (2.2) of the DHG sum rule (2.1). Th e second uses the requirement that radiation null zones of composite particles must be the same as those of point-like particles; this is discussed near the end of Section IV. Arguments that have been given previously in the literature are perturbative in nature. They include the requirement of renormalizability and tree- level unitarity [8] which 1’lmits terms allowed in the interaction Lagrangian, and a perturbative analysis of the DHG sum rule [7]. In the case of the of the 2’ and W* vector.bosons, any deviation from these canonical values beyond that predicted from radiative corrections in the Standard model would provide empirical constraints on the possible internal structure of the gauge particles [lo]. We have also established natural ratios for the electromagnetic form factors of spin-one systems in gauge theory:
Gc(Q2) : GM(&~) : GQ(Q~)= (1 -^ $)^ : 2:^ -1,^ (5.1)
where 17= Iq2j/4M2. Th ese ratios hold at tree level for the W+ in the standard model, and at large momentum transfer for hadrons in perturbative &CD. These results are most easily derived in the light-cone frame assuming the dominance of the helicity- zero to zero amplitude. In the Breit frame one has the complication of evaluating non-leading transverse current matrix elements.
.**
The ratios of the fundamental form factors (5.1) also determine the ratio B/A of Rosenbluth form factors [15] and T 20, the tensor polarization [16]. Both have been measured for the deuteron [38,41,24] out^ to^ momentum^ transfers^ where^ one might have thought perturbative QCD would apply. However, the expressions for these quantities contain kinematic factors that depend on 7 = Q2/4M2, which introduces M as a dimensional parameter in addition to the intrinsic QCD mass scale AQCD. We have argued that the perturbative QCD predictions for B/A and T20 become valid for momentum transfers large compared to $%&&, not AQCD. For the deuteron, this difference is significant and does postpone applicability of perturbative QCD. The predictions made for B/A and T20 are given in (3.10). Comparisons of (3.10) with the deuteron data are shown in Figs. 2 and 3. The often-quoted prediction of -fi for T20 applies only for momentum transfers so large that 7 is much larger than one. Such a large momentum transfer is not actually necessary for a prediction to be made. The general perturbative QCD prediction (3.10) should start to be valid at moderate momentum transfer Q2 >> ~M,&cD. As a result of the universality of electromagnetic form factors, one can conclude that any seemingly point-like spin-one particle could actually be composite. This possibility has received considerable attention with respect to the W [32, 331. In Section IV we explored a simple model for a composite spin-one particle and found that the expected form factor ratios can be obtained. It should also be emphasized that the analysis presented here also applies to any spin-one bound state in gauge theory, for both space-like and time-like electromagnetic processes. In particular, the helicity zero amplitude should dominate and the form factor ratios (5.1) should hold for crossed reactions at large s = q2 such as e+e- + p+p-. It is clearly very important to verify the perturbative QCD predictions for this type of exclusive annihilation process. Although we have considered only spin-one particles, one could imagine consid- ering any class of composite systems with any fixed total spin. In particular, the general analysis of the spin-l/2 case can be applied to the form factors of the helium- 3 nucleus in order to determine the scale where the underlying quark-gluon structure of more general nuclei becomes important.
ACKNOWLEDGMENTS
The work described here was supported in part by the Department of Energy un- der contract DEAC03-76SF00515. Th e authors thank the Aspen Center for Physics for its hospitality. One of us (J.R.H.) thanks the Minnesota Supercomputer Institute, the Theoretical Physics Institute in Minneapolis, the Stanford Linear Accelerator Center, and the Max Planck Institut fiir Kernphysik in Heidelberg for their hospi- tality during a sabbatical leave when much of this work was done; he also thanks F. Coester for stimulating discussions. We also thank S. Drell for helpful comments.
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+++
n Arnold eta/. 0 Cramer eta/. 0 Platchkov et a/.
1
0 2..^4 6
Cl2 (GeV 2,
Figure 1
1
z - iz z
,o d
r n^ Arnold^ eta/. 4 4
, (^) 1 : b?zE::a,, 1
t
ii T
Q2 (GeV2)
Figure 2
