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SLAGPUB
May 1984
(T/E)
ERRATUM
REDUCED NUCLEAR AMPLITUDES IN
QUANTUM CHROMODYNAMICS*
STANLEY J. BRODSKY AND JOI-INR.HILLER
Stanford Linear &celerator_Center
Stanford University, Stanford, Calqornia $
The 2-body phase space factor in Eq. (3.2) should read as follows: da dfl c.m. yd-rnp
This correction gives minor quantitative changes in Figs. 3 and 4. The new Fig. 3 is shown on next page. The ordinates of Fig. 4 should be multiplied by fi. Because these changes are not qualitative in nature, the related discussion in the text is unchanged.
Published in Physical Review m, 475 (1983)
* Work supported by the Department of Energy, contract DEAC03-76SF00515.
, SLAC-PUB- PURD-TH-83- January 1983 T/E
REDUCED NUCLEAR AMPLITUDES IN
QUANTUM CHROMODYNAMICS
Stanley J. Brodsky Stanjord Linear Accelerator Ceded Stanford University, Stanford, California 94805 and The Institute for Advanced Study Princeton, New Jersey 08540
John R. Hiller Departkent of Physics, Purdue University§ Weat Lafayette, Indiana 47907 and The Institute for Advanced Study Princeton, New Jersey 08540
Submitted to Physical Review C
- Work^ supported^ in part^ by the Department^ of Energy^ under^ contracts^ DE AC03-76SFOO515, DEAC02-76ER02220 and DEAC02-76ER01428; and also by an Albert Einstein Memorial Fellowship.
t Permanent Address.
5 Present Address.
freedom. A method for computing large momentum transfer exclusive scatter- ing amplitudes for hadrons and nuclei, starting with a Fock state wave function expansion on the light-cone (equal r = t + z), has been developed.2 At large momentum transfer one can readily derive QCD predictions for the leading fixed angle power-law scaling behavior and spin structure of hadronic and nuclear scat- tering matrix elements. However, the explicit evaluation of the multiquark and gluon hard scattering amplitudes needed for predicting the normalization and angular dependence for a nuclear process, even at leading order in a8, requires the consideration of millions of Feynman diagrams. Beyond leading order one must include contributions of non-valence Fock states, wave function and binding corrections, and a rapidly expanding number of radiative corrections and loop diagrams. In this paper we will discuss a new definition of nuclear scattering amplitudes which provides a simple method for identifying the dynamical effects of nucleon substructure, consistent with QCD and covariance. Although this technique cannot replace a full QCD calculation, it does provide a basis for constructing models for “reduced” nuclear scattering amplitudes consistent with QCD scaling laws and gauge invariance. The basic idea for this method was given by Brodsky and Chertok.3 Consider the deuteron form factor as measured in electron-deuteron elastic scattering. In general, a form factor F(Q2 = -q2) is the probability amplitude that the target remains intact after absorbing four-momentum q. To the extent that we can neglect its binding energy, the deuteron can be represented as two nucleons, each with an equal portion of the nuclear momentum. Therefore the deuteron form factor contains the probability that each nucleon remains intact after absorbing one-half of the momentum transfer. We thus define the “reduced” deuteron form factor FdQ2) fd(Q2) = Fp(Q2/4) Fn(Q2/4) (1.1) which effectively removes the fall-off of the measured form factor due to the internal degrees of freedom of the nucleons. It is defined separately for each
helicity form factor.
The reduced form factor must still be a decreasing function of Q2 since it still contains the probability that the scattered nucleons reform into the ground state deuteron. An important prediction of &CD is that, module logarithmic factors that come from the running coupling constant and anomalous dimensions of the hadronic distribution amplitudes, the large Q2 behavior is
fd(Q2) - G. (^) (l-2)
Thus the reduced deuteron form factor and meson form factors (for helicity x = 0 to x’ = 0) have the identical (monopole) scaling law. After removing the nucleon form factors, the nucleons are effectively reduced to point-like spin
l/2 fermions, so the reduced deuteron and meson form factors have the same
dimensional scaling behavior f - (1/Q2)“-‘, basically the slowest possible for
two-particle composites. Similarly, if one.defines for-A = 3:
_- (^) fHs(Q2) = [FN(Q2/g)]3Fk4Q2) t (' = 'I2 to " = 'i2) (1.3)
then QCD predicts that the reduced He3 (and triton) form factor scales at large Q2 in the same way as a nucleon form factor:
f~$(&~) - h(Q2) - P/Q212 - (^) (1.
A comparison of data5 with the QCD prediction
(1 + Q2/m;) fd(Q2) 21 cod. (^) (1.5)
is shown in Fig. 1. (Here rn$ = 0.3 GeV2, as predicted in Ref. 3, although any
value rni < 1 GeV2 is irrelevant for the comparison.) The results show that
QCD works remarkably well down to scales of order Q2 - 1 GeV2!
One can compare the definition (1.1) for the reduced deuteron form factor with the standard “impulse approximation” form
Fd(Q2) = F~TQ~) F~Q~) (^) (1.6)
2 , (2.3b) with pn, pp and Pd the momenta of tively.
the neutron, proton and deuteron, respec-
The nominal fixed-angle scaling behavior of the reduced amplitude is pre- dicted by dimensional counting rules6 Modulo logarithms they give
m - ~$7 (^) f(invariants (^9) > (2.
where p$ = tu/s is the transverse momentum and n is the number of “elemen- tary” fields in the external state (ingoing and outgoing photons, leptons, gluons, quarks or reduced nucleons). Thus for deuteron photodisintegration the reduced amplitude scales as
myd-+np -^ P$^ f(&7n.)^ I^ (2.5) theangle ecern. being that of the proton direction with respect to the beam direction in the c.m. frame. This is the same QCD scaling as that for M7~+ai52;
- here M is a meson with constituents q1 and q2.
We can motivate the definition of the reduced amplitude by returning to the basic definition of hadronic matrix elements in r-ordered perturbation theory:’
where the 9 are the equal 7 = t+z wave functions and T is the momentum-space quark-gluon scattering amplitude. A sum over the Fock state amplitudes and quark and gluon helicities is understood. In the zero nuclear binding energy limit the nuclear Fock state wave function reduces to the product of wave functions for collinear nucleons with the nuclear momentum partitioned among the nucleons in proportion to each nucleon mass. Thus one is evidently neglecting corrections of order 2rnNAcBE/y2 where rnN is the nuclear mass, ABBE the nuclear binding energy and p2 a hadronic scale parameter, as well as contributions from higher Fock states in the nucleus, e.g. the hidden-color six-quark configurations.
At this stage of approximation one must compute the corresponding multi- nucleon scattering amplitude, e.g., the amplitude for the elastic electron-deuteron scattering process
e+p (’p^ )^ +n (’p^ )-+e’+p’(i^ p+i^ q)+n’tip+t^ q)^.^ (2.7)
If the momentum transfer occurs rapidly compared to the scale of hadronic bind- ing then one can argue (as in the Chou-Yang model of elastic scatteringa) that the probability amplitude for transfering the required momentum 23 to each nu- cleon is proportional to it,s elastic form factor. Since Sudakov effects always sup- press near on-shell (long-distance) momentum transfer mechanisms from pinch singularities9 and endpoint regions of phase space,1oj11 one can argue that large momentum transfer is always local in &CD. Thus this assumption is justified, with corrections of order p2/q2. A specific diagram which explicitly exhibits the factorization intrinsic to the reduced deuteron formfactor is shown in Fig. 2.
_-As^ an application^ of^ nuclear^ amplitude^ reduction,^ we consider^ deuteron^ dis- integration. The reduced amplitude is defined in (2.2). Both the scaling behavior (2.5) and a model for the angular dependence are discussed in-the next section. Some other processes12 that might be pro fitably treated with our reduction method are pp -+ d?r+,13 pd + H3n+ and nd (^) -+ rd. The reduced amplitudes have the same QCD scaling behavior as the amplitudes for qp -+ AIn, qqq + BT and aM + ~FM, respectively, where J3 represents a baryon. From (2.4) we find the scaling to be
mpp+dn+ -^ PT2 fW)^9
“pddH3n+ -^ PT4 f(e)^9
“‘nd+nd -^ PT4 fW)^ *
(2.8a)
(2.8b)
(2.8~)
zero. A computation of the squared amplitude summed over final spins (see the Appendix) then givesll’
j2(ec.,.) = N (ue1 Lte2)2 ( iiuz
transverse XT’ --iongitudinal^ (3- where N is a normalization constant with dimensions GeV2/srad and “trans- verse” indicates an average over the two possible helicities. In the limit of & >> md we find
/2(&.m.) = N
[(2el - 1) + co8&.m.]2 1 Y transverse 1 - cos2&.m. (^) k (I+ cos2ec.m.) , longitudinal
with the charges normalized by el -e2 = 1. This, when combined with (3.2) pro- vides a one-parameter model for the asymptotic behavior of deuteron photodis- integration away from the beam axis. The actual angular distribution predicted by_QCD from the coherent sum over the many diagrams of t,he type illustrated in Fig. 2 is undoubtedly more complicated than that given by the above model. Nevertheless Eq. (3.5) should be representative of the scaling and functional dependence predicted by QCD for the reduced photodisintegration amplitude. The simple model given in (3.5) makes apparent the need for data at higher energies. The points plotted in Fig. 4 were extracted by inspection from the data in Fig. 3 under the assumption that scaling had begun. The error bars reflect the range of values that would be consistent with the data. The empirical form sin40,.,. fits the points fairly well but does not agree with (3.5). In particular, (3.5) is unbounded at one or both endpoints. Of course the physical cross section is not unbounded at either endpoint; its rise is curtailed by mass terms dropped in our approximations. However, the sin4tJ,.,. behavior of the data is not com- patible with any rise at all. If the yM -+ q ij model is a good guide, then a sign that experimental energies are approaching the true scaling limit would be that the value of f2(&,.) near the backward or forward direction has become large relative to the values at wider angles.
B. Dibaryon Resonances
An interesting feature of QCD is the possible occurence of resonances in the dibaryon system corresponding to six-quark Fock states which are dominantly hidden color, i.e., orthogonal to the usual n-p and A-A-configurations. Signals for such resonances could appear in photo- or electrodisintegration of the deuteron at fixed i = M2 in a specific partial wave in the full amplitude. The virtual phot,on probe may enhance the signal since it is sensitive to off-shell configurations in the nuclear target. Analysed6 of deuteron photodisintegration data have suggested the presence of dibaryon resonances with masses at 2.26 GeV and 2.38 GeV, although definitive results have been elusive. The isolation of possible dibaryon contributions from the hard-scattering background is clearly interesting and important. It would be useful to have a specific model of the hard-scattering continuum since this would permit a more precise separation of the resonance and background contributions. Given the correct kinematic regime, the reduced amplitude technique leads directly to just such a mddel. -As an application of this approach we treat deuteron electrodisintegration. We have alrea,dy discussed photodisintegration, but for that-process the reso- nances occur at energies where the asymptotic form (3.2) does not apply. In electrodisintegration, however, the kinematics of resonance production are con- sistent with large transfers of momentum for the nucleons. The methods of the previous sections should then be applicable. We write the full disintegration amplitude as the sum of a dibaryon resonance amplitude MDB and a background amplitude MBG:
M ed-+ epn =MDB+MBG.
As discussed in Sec. 2, the hard-scattering background amplitude factorizes into a reduced amplitude mgG and the appropriate nucleon form factors,
MBG =^ mBG^ Fp(ip)^ &(h)^.^ (3.7)
From (2.4) we find that the nominal scaling behavior for the reduced amplitude
Upon subsitution of (3.10) and (A.4) Eq. (3.9) becomes, in the limit i < 8,
C I~BGI~ -^ -^ Fl^ + (5 + 2c + c2}^ 8(1-8 F2 c) +^ {a2 -^ b2}^ 2(1 -c)’ F
+ ((1 -^ c)(a -^ b) [2b +^ a(1 +^ c) +^ (a-^ b)(l -^ c)]
- 2(a - b)2(1 + c) - ab(l -c)“] q;2c)
where the Fi are given in (A.14),
a = (Ep - gp. fie)/s1i2 , (3.12a)
b = (Ep - ap. &)/td2 (^) , (^) (3.12b)
C =ie*$k 9 (3.12~)
fie is the beam direction, $k the direction. of the outgoing electron and (Ep, &) the four-momentum of the proton, all in the c.m. frame. The invariants used to define the Fi are, in the same limit, given by
Q2 -N f ~(1 - c) , (3.13a)
i 21 -f ~(1 - c) - s(a - b) ,
ii “Y s(a- 6).
(3.13b)
(3.13c)
The expression in (3.11) should describe the background near a resonance. For
a transverse deuteron the background amplitude is suppressed by additional factors of (5 /s)l12 that come from angular momentum effects.lt
4. CONCLUSION
The reduced amplitude method discussed in Sec. 2 is very general. The prin- cipal formulas, Eqs. (2.1) and (2.4), give an accurate estimate of the leading QCD behavior of hadron-helicity conserving amplitudes. Comparison with experiment
should provide a new test of &CD. These formulas also imply constraints on low energy models since one expects a synthesis4 of QCD and nuclear physics. Our results suggest the possibility that fully analytic nuclear amplitudes can be constructed which at low momentum transfer fit standard electromagnetic and chiral boundary conditions and low energy theorems, while satisfying the scaling law and anamolous dimension structure predicted by QCD at high momentum transfer.
An application to deuteron disintegration and a model for its angular de- pendence were described in Sec. 3. The prediction for the photodisintegration differential cross section is contained in (3.2) and (3.5). The general form for the square of the electrodisintegration amplitude is given by (3.9), (A.4) (A.13) and (A.14). This latter result provides a new means for understanding the back- ground to dibaryon resonances. Equation (3.11) supplies a specific prediction for this background. The predictions made for deuteron disintegration apply to an energy do- main that is as yet uncharted by coincidence experiments. With the advent of intermediate-energy cw electron beams l7 this should-soon notbe the case. Some other nuclear processes that are of interest in the context of the reduced am- plitude method are mentioned at the end of Sec. 2. We urge experimentalists to pursue the acquisition of data at the largest possible energy and momentum transfer in order to test the scaling behavior predicted by &CD.
ACKNOWLEDGEMENTS
This work was supported in part by the Department of Energy under con- tracts DEAC0376SF00515, DEAC02-76ER02220 and DEAC02:76ER01428. One of us (J.R.H.) was^ also supported^ in^ part^ by^ an Albert^ Einstein^ Memorial Fellowship established by t.he Federal Republic of Germany.
with
and
[dx] = dxl dx2 6(1- x1 - x2)
X Jh =^ c Nit2^ x;lf2^ x;lJ2^ 4w Sl) 0 (x224 52) - 81182 (A-7) For the massless case considered here, one can use”
where cf = F( I/ &)(O, 1, fi, 0) in a frame with p = (I 3 1,O,O, 1i; I). In writing
the final formulas we will assume that the wave function Q[,obeys the symmetry
It is useful to define the integrals
*=lq$!--$q
and
(A. 10)
(A.ll)
where ,/i is the c.m. energy of the process for which (A.l) is a subprocess and
Q2 =o,^ 9=^ i^ is the^ photodisintegration^ limit.^ These^ integrals^ appear^ in^ the
following combinations:
11 = III2 9 12 =^ I’T^ +^ II’-^ II^ Q*=O+^0 ,--
I3 = 11’I2 -;^ III2^ Qdo=^0 f
Id = m-II’* (^) Q*=O--+ 0.
In the transverse case we find
Fi =o^ ,^ i#
(A.12a) (A.12b)
(A.12~)
(A. 12d)
(A. 13a)
(A. 13b)
and in both the longitudinal and scalar cases we obtain
A Fl -; (2ir+i-&2)55^ -(ii-Q2$]+^ 12(i+Q2@g)}^9 (A. 14a)
zi-+(ir-3i-2Q2)%$+2(if2Q2)$^4 e i
- 12(y-3i(;- ti+2Q2)y+i(t+Q2);]
(A. 14b)
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