Exclusive Processes in Quantum Chromodynamics and the Light-Cone Fock Representation

96 Pages·2000·0.78 MB·English

SLAC{PUB{8649
October 2000
Exclusive Processes in Quantum
Chromodynamics and the Light-Cone
Fock Representation(cid:3)
Stanley J. Brodsky
Stanford Linear Accelerator Center
Stanford, California 94309
e-mail: [email protected]
Contribution to the Boris Io(cid:11)e Festschrift
At the Frontier of Particle Physics
A Handbook for QCD
Edited by M. Shifman
(cid:3)
Work supported by the Department of Energy under contract number DE-AC03-76SF00515.
ABSTRACT
Exclusive processes provide a window into the bound state structure of hadrons in
QCD as well as the fundamental processes which control hadron dynamics at the am-
plitude level. The natural calculus for describing bound state structure of relativistic
composite systems needed for describing exclusive amplitudes is the light-cone Fock
expansion which encodes the multi-quark, gluonic, and color correlations of a hadron
in terms of frame-independent wavefunctions. In hard exclusive processes in which
hadrons receive a large momentum transfer, perturbative QCD leads to factorization
theorems which separate the physics of bound state structure from that of the rele-
vant quark and gluonic hard-scattering reactions which underlie these reactions. At
leading twist, the bound state physics is encoded in terms of universal \distribution
amplitudes," the fundamental theoretical quantities which describe the valence quark
substructure of hadrons as well as nuclei. The combination of discretized light-cone
quantization and transverse lattice methods are now providing nonperturbative pre-
dictions for the pion distribution amplitude. A basic feature of the gauge theory
formalism is \color transparency," the absence of initial and (cid:12)nal state interactions
of rapidly-moving compact color-singlet states. Other applications of the factoriza-
tion formalism are brie(cid:13)y discussed, including semileptonic B decays, deeply virtual
Compton scattering, and dynamical higher twist e(cid:11)ects in inclusive reactions. A new
type of jet production reaction, \self-resolving di(cid:11)ractive interactions" provide em-
pirical constraints on the light-cone wavefunctions of hadrons in terms of their quark
and gluon degrees of freedom as well as the composition of nuclei in terms of their
nucleon and mesonic degrees of freedom.
1 Introduction and Overview
Exclusive processes, as de(cid:12)ned by Feynman,[1] are scattering reactions in which the
kinematics of all initialand (cid:12)nal state particles are speci(cid:12)ed. Hadronic exclusive pro-
cesses present an extraordinary challenge to theoretical analysis in quantum chromo-
dynamics. Virtually all the complexities of perturbative and non-perturbative QCD
are relevant toexclusive reactions, from con(cid:12)nement and chiralsymmetry breaking at
low momentum transfers, to the dynamics of quarks and gluons at high momentum
transfers. A complete description of exclusive amplitudes must deal with all of the
complexitiesofthenon-perturbativestructure ofhadrons|notjustthesingle-particle
(cid:13)avor, momentum, and helicity distributions of the quark constituents familiar from
inclusive reactions|but also multi-quark, gluonic, hidden-color correlations, and the
phase structure intrinsic to hadronic and nuclear wavefunctions.
Exclusive hadronic reactions range from the space-like and time-like form fac-
tors measured in electron-hadron scattering and electron-positron annihilation, to
hadronicscatteringreactionssuch asproton-protonscattering, pionphotoproduction,
2
and di(cid:11)ractive vector meson electroproduction. Exclusive amplitudes also govern the
decay of heavy hadrons and quarkonia into a speci(cid:12)c (cid:12)nal states. One of the most
pressing goals is to understand the QCD physics of exclusive B-meson decays at the
amplitudelevel, since the interpretationof the basic parametersof electroweak theory
and CP violation depend on hadronic dynamics and phase structure.
Exclusive processes are particularly challenging to compute in QCD because of
theirsensitivitytotheunknownnon-perturbativeboundstatedynamicsofthehadrons.
However, insomeimportantcases, theleadingpower-lawbehaviorofanexclusive am-
plitude at large momentum transfer can be computed rigorously via a factorization
theorem which separates the soft and hard dynamics. The key ingredient is the fac-
torization of the hadronic amplitude at leading twist. As in the case of inclusive
reactions, factorization theorems for exclusive processes allow the analytic separation
of the perturbatively-calculable short-distance contributions from the long-distance
non-perturbative dynamics associated with hadronic binding.
Other important examples of exclusive processes which are particularly relevant
to QCD analyses are the transition form factor F(cid:13)(cid:25)0 measured in e(cid:13) e(cid:25)0e in which
! +
only a single hadron appears, two-photon reactions such as (cid:13)(cid:13) (cid:25) (cid:25)(cid:0), virtual
! + +
Compton scattering (cid:13)(cid:3)p (cid:13)p, elastic scattering reactions such as K p K p,
! ++ !
photoproduction reactions such as (cid:13)p (cid:25)(cid:0)(cid:1) , quarkonium decay, hard di(cid:11)ractive
0 !
reactions such as (cid:13)(cid:3)p (cid:26) p and nuclear reactions such as deuteron photodisintegra-
!
tion (cid:13)d pn.
!
The main focus of this introductory review will be on the fundamental quark
and gluon processes and hadron wavefunctions which control hadron dynamics at
the amplitude level. Despite much progress, the subject is still in its infancy, and
many important problems, controversies, and puzzles remain unresolved. A number
of excellent specialized reviews of exclusive processes in QCD are available which can
provide further technical details and additional references.[2, 3, 4, 5, 6, 7]
Compton scattering on a proton, (cid:13)p (cid:13)p, the elastic scattering of a real photon
!
on a proton is a primary example of an exclusive amplitude. Covariance and discrete
symmetries reduce the number of helicity amplitudes M(cid:21)(cid:13);(cid:21)p;(cid:21)(cid:13)0;(cid:21)p0((cid:13)p (cid:13)0p0) to
!2
6 analytic functions FI(s;t);(I = 1;:::6) of the invariants s = (k + p) and t =
2
(p p0) . These amplitudes, by s t crossing, also describe pair production in
(cid:0) !
photon-photon collisions(cid:13)(cid:13) ppaswellaspairannihilationpp (cid:13)(cid:13). The challenge
! !
is to compute the scattering of the proton by photons starting from a basis where the
fundamentalcarriersoftheelectromagneticcurrentarecon(cid:12)nedquarks. Furthermore,
in a relativistic quantum (cid:12)eld theory, a bound state has (cid:13)uctuations of arbitrary
number or quanta. A comprehensive review of real and virtual Compton scattering
has recently been given by Vanderhaeghen.[7] The analytic behavior of the Compton
amplitude is constrained by general considerations: see Fig. 1.
(1) The low energy theorem[8, 9] for forward Compton scattering normalizes the
2
proton and photon helicity-conserving amplitude at threshold s = M and t = 0 to
3
Conformal Limit
B= 0
Regge
s>>(–t)
Chiral
Exclusive Process
s,(–t)~L 2
QCD
PQCD
s,(–t)>>L 2
QCD
N 0
e
8-2000
Abelian Limit 8561A16
Figure 1: Domains of exclusive amplitudes in QCD.
the total charged squared of the target proton. Similarly, the helicity-(cid:13)ip amplitude
at threshold is normalizedto the square of the proton’s anomalousmagneticmoment.
The explicit demonstration of these facts for the case of a composite target is highly
non-trivial: there is a remarkable cancelation of the terms proportional to the sum of
quark charges squared fromphotons scatteringon the constituents with contributions
involving the proton intermediate state.[10]
(2) The optical theorem relates the imaginary (absorptive) part of the forward
photon and proton helicity-conserving Compton amplitude to the total photoabsorp-
tion cross section. Given the total cross section, one can use dispersion relations to
derive the s dependence of the entire forward Compton amplitude. Thus all of the
complexities of photoabsorption cross section are implicitlycontained in the exclusive
amplitude. These include multiplescattering Glauber/Gribovprocesses, which in the
case of a nuclear target, lead to shadowing and anti-shadowing on a nuclear target.
(3) The Regge limit: At (cid:12)xed t with s t, the analytic form of any 2 2
(cid:29) (cid:0) !
exclusive scattering amplitude is characterized by a sum of Regge terms,
(cid:11)R(t)
M(s;t) = (cid:12)R(t)s S[(cid:11)(t)] (1)
R
X
where
i(cid:25)(cid:11)(t)
1 e(cid:0)
S((cid:11)) = (cid:6)
sin(cid:25)(cid:11)(t)
is the signature factor determined by s u crossing and unitarity. The Compton
!
amplitude is described in Regge theory by even signature C = + Regge poles and
0
cuts including the Pomeron. At small t, the exchange of a virtual (cid:25) through its
4
0
anomalous (cid:25) (cid:13)(cid:13) coupling can also enter. At large t the Pomeron can be cal-
!
culated in the BFKL formalism[11] which sums multi-gluon exchange. In addition,
both photons can interact locallyon the same quark lineleadingto a (cid:12)xed Regge con-
tribution at (cid:11)R = 0, the \J = 0 (cid:12)xed pole" contribution,[12, 13] thus distinguishing
proton processes from the corresponding processes involving vector mesons. Cross-
ing and Regge theory at t > s from the cross-channel baryonic Reggeon exchange
(cid:0)
trajectories provide further constraints.
(4) The Chiral Correspondence Principle: At long wavelengths where hadron sub-
structure cannot be resolved, QCD must reduce to a dual e(cid:11)ective theory of hadrons,
such as the chiral Lagrangian.[14] This duality between the quark and gluon degrees
of freedom in the propagators of local color-singlet operators is the basis of the QCD
sum rules. [15, 16] In addition, soft exclusive amplitudes must match to the ampli-
tudes calculated in the chiral theories. Since baryons enter as solitons, as described
in the Skyrme model, the annihilation process pp (cid:13)(cid:13) at small relative veloci-
!
ties should can be described in terms of Skyrmion { anti-Skyrmion annihilation.[17]
Conversely, the production process (cid:13)(cid:13) pp amplitude corresponds to Skyrmion
!
{ anti-Skyrmion formation. This duality constraint has profound consequences for
the analytic behavior of the proton Compton amplitude since the soliton picture of
pp annihilation implies the existence of an important scale in the proton Compton
2
amplitude at t 4Mp. [18]
(cid:25)
(5) Asymptotic High Momentum Transfer Constraints: Although the physics of
the Compton amplitude for general kinematics is extraordinarily complicated, per-
turbative QCD provides a simple guide to the physics of the Compton process at
large momentum transfer. The physical picture is as follows: the valence quarks
in a hadron wavefunction are dominated by kinematic con(cid:12)gurations in which its
2 2
constituent quark have small relative momenta k 300 MeV : Thus at high
2 h ?i ’
momentum transfer t;u k with (cid:12)xed t=s or center-of-mass angle, one ex-
(cid:0) (cid:29) h ?i
pects that proton Compton scattering should be driven by the perturbative process
(cid:13)(qqq) (cid:13)0(qqq)0 where the valence quarks scatter from a direction roughly collinear
!
with the initial proton to that of the (cid:12)nal proton. Since the coupling is dimension-
2
less, such a tree amplitude falls nominally as 1=t : Furthermore since quark helicity
is conserved at high energy in QCD, and the dominant valence quark wavefunctions
0
have Lz = 0, proton helicity (cid:21)p = (cid:21)p should be conserved.
Since the coupling is dimensionless, simple power-law scaling [19, 20, 21] implies
that the proton helicity-conserving Compton amplitude will have the nominal fall-
2
o(cid:11) f(t=s)=t . Wavefunction con(cid:12)gurations with more than the minimum number of
constituents are power-law suppressed since more hard interactions are required. The
corresponding (cid:12)xed-cm-angle cross section is thus predicted to fall as
2
f(t=s)
d(cid:27)=dt((cid:13)p (cid:13)p) j j :
2 4
! / s t
This prediction is consistent with the scaling of the large angle Compton scattering
6 2
data [22]: s d(cid:27)=dt((cid:13)p (cid:13)p) const at 4 < s < 12 GeV . As seen in Fig. 2, the
! (cid:24)
5
6
experimental data for Compton scattering indicates consistency with s d(cid:27)=dt scaling
at momentum transfers as small as a few GeV. In fact, the angular dependence of the
data appears to have universal angular function when the di(cid:11)erential cross section is
6
scaled by the nominal s power.
107
2 GeV
3 GeV
4 GeV
106 5 GeV
)
0
1
V 6 GeV
e
G g g
b P P
n
(
105
sd dt

6
s
104
1 0 –1
8-2000 cos Q
8561A10
Figure 2: Scaling of proton Compton scattering at (cid:12)xed (cid:18)cm. The data and from
Shupe et al.[22]
2
Asymptotic freedom impliesthat the magnitude of the e(cid:11)ective coupling (cid:11)s(q ) of
the exchanged gluons which carry the large momentum transfer is suÆciently small
thataperturbativeanalysisoftheunderlyinghardscatteringamplitudeTH((cid:13)(qqq)
!
(cid:13)0(qqq)0) is meaningful. The nominalpower-law fallo(cid:11) given by dimensionalcounting
will then be modi(cid:12)ed by the dependence in the running coupling associated with the
two hard gluon exchanges as well as additional logarithms arising from the evolution
of the proton wavefunction. The crossed channel processes can be obtained by simple
s t crossing.
!
TheasymptoticpredictionsobtainedfromperturbativeQCDalsoimplyimportant
analyticconstraints on the formof the Compton amplitude.[23] Forexample, ifRegge
theory is to be consistent with the power fall-o(cid:11) predictions of perturbative QCD,
then the Reggeon powers (cid:11)R(t) must asymptote at large t to negative integers:
(cid:0)
lim t (cid:11)R(t) = 0; 1. Similarly, the Regge coeÆcient functions (cid:12)R(t) must have
(cid:0)!1 (cid:0)
power-law fall-o(cid:11).
The hard scattering physical picture outlined above for the proton Compton am-
plitudeisthebasisforageneralformalismforanalyzingthebehaviorofhardexclusive
6
processes in which some or all of the interacting particles receive a high momentum
transfer. In fact in many cases, factorization theorems can be derived which at lead-
ing power in the momentum transfer Q allow one to write the full hadron amplitude
as the convolution of the amplitude TH for the hard scattering of the valence quarks,
collinear up to the scale Q~, with the hadron distribution amplitudes (cid:30)(xi;Q~) repre-
senting each of the hadron receiving a hard momentum. [24]
In this introductory review, I will outline many of the developments in the appli-
cation of QCD to exclusive processes. Many technical details will not be given here,
but are available in the original papers. An essential tool will be light-front (light-
cone) quantization which provides a frame-independent representation of relativistic
bound state wavefunctions in quantum (cid:12)eld theory. The quantization procedure and
rules of calculation in light-cone time-ordered perturbation theory are outlined in
the Appendix. The general framework for the applications to QCD are illustrated
schematically in Figs. 3 and 4. In each of the illustrated processes, one sees a
factorization of the hadronic physics in terms of the light-cone Fock wavefunctions
convoluted with perturbatively-calculable hard scattering amplitudes.
The natural calculus for describing bound state structure of relativistic composite
systems inquantum(cid:12)eld theoryisthe light-coneFockexpansion. The light-coneFock
~
wavefunctions n=H(xi;k i;(cid:21)i) thus interpolate between the hadron H and its quark
?
andgluondegreesoffreedom. Thelight-conemomentumfractionsoftheconstituents,
xi = ki+=P+ with ni=1xi = 1; and the transverse momenta ~k i with ni=1~k i =~0
? ? ?
appear as the momentum coordinates of the light-cone Fock wavefunctions. A crucial
P P
~
feature is the frame-independence of the light-conewavefunctions. The xi and k i are
(cid:22) ?
relative coordinates independent of the hadron’s momentum P . The actual physical
transverse momenta are p~ i = xiP~ +~k i: The light-cone Fock representation is an
? ? ?
extraordinarily useful tool for representing hadrons in terms of their quark and gluon
degrees of freedom. It also provides exact representation of the electromagnetic form
factors and other local matrix elements at all momentum transfer. I will also review
this formalism in the following sections.
The principles of factorization of soft and hard dynamics have been recently ex-
tended in a number of other directions:
(1) The deeply virtual Compton amplitude (cid:13)(cid:3)p (cid:13)p has emerged as one of
!
the most important exclusive QCD reactions.[25, 26, 27, 28] The process factorizes
into a hard amplitude representing Compton scattering on the quark times skewed
parton distributions. The resulting skewed parton form factors can be represented as
diagonal and o(cid:11)-diagonalconvolutions of light-cone wavefunctions, as in semileptonic
B decay.[29]Newsumrulescanbeconstructedwhichcorrespondtogravitonscoupling
to the quarks of the proton.[25]
0 2
(2) The hard di(cid:11)ractionof vector mesons (cid:13)(cid:3)p V p at high Q and high energies
!
forlongitudinallypolarizedvector mesons factorizesintoaskewed partondistribution
0
times the hard scale (cid:13)(cid:3)g gV amplitude, where the physics of the vector meson is
!
7
(a) Light Cone Fock Expansion (d) Form Factors p ' p' p' l ' J+ (0) pl
u q
p = y uud uud uud + y uudg dug uudg +... Fll ' (Q2) = Sn x, k^ x, k^ + q^
p qqq : y uud xxx132,, , k kk^^^13,2 ,l, l1l32 p, l y n y n p+q, l '
n n
y n (xi, k^ i, l i) : iS =1xi = 1, iS=1 k^ i = 0 Large Q2 TH
f
f
(b) Distribution Amplitude p, l p+q, l '=l
f (Mx,Q) = (cid:242) d2k^ x, k^ g*
M n 2 < Q2 y 2 1–x, –k^ TH=S xx12 yy12 + ...
x3 y3
a 2
= s
4 f (xi, yi,)
(c) Deep Inelastic p ' X p J+(z) J+(0) p Q
g* g* q q (e) Compton g p g' p' p' J m (z) Jn(0) p
q q = S x x Large s, t k k'
n
p p
p p TH
y n y n f f
xq, k^ 2 p, l p+q, l '=l
q(xBJ, Q) =Sn (cid:242) P d2k^ dx y k k'
M 2n < Q2, xq = xBJn THCompton=S xx12 yy12 + ...
a 2 x3 y3
98–39478A10 = PT4s f (xi, yi, q cm)
Figure 3: Representation of QCD hadronic processes in the light-cone Fock expan-
sion. (a) The valence uud and uudg contributions to the light-cone Fock expansion
for the proton. (b) The distribution amplitude (cid:30)(x;Q) of a meson expressed as an
integral over its valence light-cone wavefunction restricted to qq invariant mass less
than Q. (c) Representation of deep inelastic scattering and the quark distributions
q(x;Q) as probabilistic measures of the light-cone Fock wavefunctions. The sum is
over the Fock states with invariant mass less than Q. (d) Exact representation of
spacelike form factors of the proton in the light-cone Fock basis. The sum is over
all Fock components. At large momentum transfer the leading twist contribution
factorizes as the product of the hard scattering amplitude TH for the scattering of the
valence quarks collinear with the initial to (cid:12)nal direction convoluted with the proton
distribution amplitude. (e) Leading-twist factorization of the Compton amplitude at
large momentum transfer.
8
(f) Virtual Compton g * p g 'p' (g) Vector Meson Leptoproduction g * p V p'
p' l ' Jm (z) Jn (0) p l '
g * (q) V = r , w, f, J/y
Large – q2 = Q2
' '
f
k p p' g * v
x
2 2
q Large –q = Q
V
p p' 1–x
p p'
'
S k = + + +
n q
(h) Weak Exclusive Decay
p p'
y n y n D J+ (0) B W– n
=
' B
+
D
+ –q
k n n
p p' S B b d c + B b c c
y n+2 y n n y n g y n D+ y n+2 d g y n D+
9–97
8348A11
Figure 4: (f) Representation of deeply virtual Compton scattering in the light-cone
Fock expansion at leading twist. Both diagonal n n and o(cid:11)-diagonal n+2 n
! !
contributions are required. (g) Di(cid:11)ractive vector meson production at large photon
2
virtuality Q and longitudinal polarization. The high energy behavior involves two
gluons in the t channel coupling to the compact color dipole structure of the upper
vertex. The bound-state structure of the vector meson enters through its distribu-
tion amplitude. h Exact representation of the weak semileptonic decays of heavy
hadrons in the light-cone Fock expansion. Both diagonal n n and o(cid:11)-diagonalpair
!
annihilation n+2 n contributions are required.
!
9
contained in its distribution amplitude.[30, 31, 32] The data appears consistent with
2
the s;t and Q dependence predicted by theory. Ratiosof these processes for di(cid:11)erent
0
mesons are sensitive to the ratio of 1=x moments of the V distribution amplitudes.
(3) The two-photon annihilation process (cid:13)(cid:3)(cid:13) hadrons, which is measurable in
+ + !
single-tagged e e(cid:0) e e(cid:0)hadrons events provides a semi-local probe of even charge
+ ! 0 0 +
conjugation C = hadron systems (cid:25) ;(cid:17) ;(cid:17)0;(cid:17)c;(cid:25) (cid:25)(cid:0), etc. The perturbative QCD
0
calculation of the simplest channel, (cid:13)(cid:3)(cid:13) (cid:25) , will be discussed in the next section.
+ !
The (cid:13)(cid:3)(cid:13) (cid:25) (cid:25)(cid:0) hadron pair process is related to virtual Compton scattering on
!
a pion target by crossing. Hadron pair production is of particular interest since the
leading twist amplitude is sensitive to the 1=x 1=(1 x) moment of the two-pion
(cid:0) (cid:0)
distribution amplitude coupled to two valence quarks. [31, 33]
2 Calculations of Exclusive Processes in QCD
2.1 The Photon-to-Pion Transition Form Factor
The simplest illustration of an exclusive reaction in QCD is the evaluation of the
2
photon-to-pion transition form factor F(cid:13) (cid:25)(Q ) since only one hadron is involved.
! 0
The transition form factor is measurable in single-tagged two-photon ee ee(cid:25)
!
reactions. The form factor is de(cid:12)ned via the invariant amplitude
(cid:22) 2 2 (cid:22)(cid:23)(cid:26)(cid:27) (cid:25)
(cid:0) = ie F(cid:25)(cid:13)(Q )(cid:15) p(cid:23)(cid:15)(cid:26)q(cid:27) : (2)
(cid:0)
The lowest order contribution is shown in Fig. 5. It is convenient to choose a
+ + +
frame where the virtual photon has zero q : q = (q ;q(cid:0);q ) = (0;2p q=p );p =
+ + 2 + 2 2 2 ? (cid:1)
(p ;p(cid:0);p ) = (p ;M =p ;0 ) with q = Q = q .
? ? ? (cid:0)
0
We can readily compute the (cid:13)(cid:3)(cid:13) (cid:25) amplitude in light-cone time-ordered per-
!
turbation theory using the rules given in the Appendix. The central input will be the
two-particle irreducible light-wave Fock state amplitude:
(x;k ) = F:T: 0 T (z=2) ( z=2) (cid:25) z+=0 : (3)
? j (cid:0) j j
D E
which sums all of the nonperturbative physics associated with the pion. The denom-
inator in Fig. 5(b) associated with the fermion propagator between the two photons
2
is proportional to (k +x1q ) . As in inclusive reactions, one must specify a factor-
? ?
ization scheme which divides the integration regions of the loop integrals into hard
and soft momenta, less or more than the resolution scale Q~. At large Q2 one then
(cid:12)nds [34, 35, 24] (x1 +x2 = 1)
2 2 (Nc)(e2u e2d) 1 dx k?2<Q~2 d2k
F(cid:25)(cid:13)(Q ) = q Q2 (cid:0) 0 x1x2 16(cid:25)?2 (x;k?) (4)
Z Z
where the factorization scale separating the soft and hard domains of integration in
the wavefunction is Q~ = (mini=1;2 xi)Q:
10