.-
POLARIZED t INTRINSIC
AND GLUON
UNPOLARIZED DISTRIBUTIONS*
STANLEY
J. BRODSKY
Stanford Linear Accelerator Centela, Stanford University, Stanford, Califomin 945' 09 and .
IVAN
SCHMIDT
Stanford Linear Accelerator Center.., Stanford University, Stanford, California 51430.9 and
I
Universidad
Casilla
Federico Santa Adaria 1 TO-V, ValparaiYso, Chile
ABSTRACT Theoretical constraints on both the polarized and unpolarized intrinsic gluorl distribution functions are developed. In pa.rticular, we study tllc beha\;ior of the gluon polarization aaymmetry in the nucleon in the sma.ll and la.rge s regiolls, a.ud relate the intrinsic distribution to the retarded part of the spin-depenclent I~ound-state potential. A simple model for the polarized and unpolarized intrinsic distriljutious is proposed which incorporates the QCD constraints. The model predicts t,llat, t,he spill carried by intrinsic gluons in the nucleon is approximately 0.5.
Submitted
to Physics Letter-s
* Work supported in part by Department (SLAG), and by Fundacibn Andes, Chile.
of Energy contract
DE-ACO;I-
76S1;` 00515
-
The intrinsic
-.
1.
gluon distribution
INTRODUCTION
GS,H(x, Qi) d escribes the fractional light-coue ol' t,l~e
momentum
distribution
of gluons associated with the bound-sla.te to the extrinsic distribution, Given the intrinsic distribution, equations
dynamics
hadron H, in distinction processes or evolution. distribution scale Qo. In principle, to compute
.
which is derived from ra.cliative one cau obtain the eslriusic starting at the boulld-st,ate
by applying
the QCD evolution
one must solve the non-perturbative gluon distribution. calculate
bound sta.te equation ot'mot ion ill
c~uaiitu~ii
the intrinsic
In the case of positronium the photon clistributiou,
electrodynamics
one can readily
a,t, least to first
u-
order in the fine structure plitudes limit
consta,nt cx. The analysis requires coherence betweeu and positron couple to the photous.
in which the electron
IIL the iul' ra4
this coherence in the neutral
atom ensures 2~finite photou clistribut~iol~. gluon distribut,ion of a haclrou is \vhicll
In the QCD case, the analysis of the intrinsic essentially limit non-perturbative.
However, there are several theoretica
constraints
its form: of fragmentation functions, distribution I' llllctJious
1. In order to insure positivity
Gu,b(x) must behave as an odd or even power of (1 - x) at, :L t the relative statistics have the behavior: to the fragmentation helicity of a and b!" Thus the gluon distribution G&c) function -
1 according to
must
of a. ullcleon
(1 - x)` at z --+ 1 to ensure correct, crossing ,' DN,~( 2). Tl iis result holds inclivitl\lallj~ for ea.ch
of the gluon a.nd the nucleon. of quarks to gluons telids to match the sign of tile quark llelicity in the large 2 limit!" 2 We define the llelicity-aligllecl and
2. The coupling
to the gluon helicity
-
--
c .-
anti-aligned
gluon distributions:
G+(x)
= G,,,,,(x)
and G-( .c) = C:,, I,\. 1(x).
The gauge theory couplings imply
hll
G-(x)/G+(x)
--+ (1 - x)".
(1)
3. In the low z domain the quarks in the hadron radiate g1u011s coherc-:lltly, one must compute emission of gluons from the qua.rk lines t?l.liillg interference between amplitudes. + G-(z). We define AG(x) = G' (x) iill,
and
account and
- (F(x)
G(z) = G+(z)
1 W e sh a 11slow that the asymmetry
ra.tio AG(cc)/G(.c) fro111
vanishes linea.rly with 2; perhaps coincidentally, Reggeon eschange!31 The coefficient functions; constituents, however, for equal pa,rtition we will show that
this is a.lso the prediction
at 2 -+ 0 depends on the haclro~lic waveof the ha,dron' momentum s among its
where NQ is the number of valence quarks. 4. In the z + 1 limit, the stuck quark is far off-shell so that one can use perturlxk the threshold dependence of the structure f' ullc.t.ions.
tion theory to characterize We find for three-quark
bound states
lim G+(x)
X-+1
t
C(l - ,)sNqP2 = c(1 - x)",
(:1)
Thus G-(Z)
-+ C(l-
x)~ at J: N 1. This is equivalent
to the spect,a.t~or-coulltillg
rule developed in Ref. 4.
-
c
We can write down a simple analytic the nucleon which incorporates
model for the intrinsic
gluon distribution
in
all of the above constraint,s:
LAG(X) = F[(l
- x)~ + (1 - x)" - 2(1 - :@I
(4)
a.nd 1 - ")4 + (1 - X)5 + 2( 1 - X)j]
In this model the momentum < xg >= -. J; dxxG(x)
fraction
carried by intrinsic and the helicity
gluons in the nucleon is gluons
= (137/210)N, = S/lSN.
ca.rriecl lay the intrinsic
is AG = so] &AG(z) gluon distribution imply
Th e ratio AG/
< zy >=
112/137 for the intrinsic aual~~ses <
is independent
of the normalization
N. Phenomello1ogic;l.l
that the gluons carry aqproximately
one-half of the protjon' momentuln: s of the intrinsic
xq/N >Z -
0.5. We shall assume that this is a good characterization The momentum
gluon distribution.
sum rule then implies N - 0.9 and AC - 0.5. A and theoretical limits on gluon and
quark spin
review of the present experimental the nucleon is given in ref.5. -` .In the following trinsic
in
sections we will ana,lyze both the pola.rized aad unpolarizccl functions using both perturba,tive
ill-
gluon distribution
a.ntl iloll-l' (~rtrlrl)a.t,i\ict (unpolarized over po-
methods. larized
First we study the behavior of the gluon asymmetry
distributions)
in the small 5 region where it turus out to be al' rosi111a.tel?; l-` wavefunction. ` The loga.rithliiic ultriL-
independent
on the details of the bound-state
violet cut-off dependence of the intrinsic of the extrinsic distribution;
distribution
matches with t,he lon:cer cumoff distribution is studied
the Q" evolution
of the extrinsic
in detail in Ref. 6.
-
-. c .-.
In section 3 we shall show that the intrinsic retarded part of the spin-dependent bound-state
gluon distribution potential ( $$
is rclat.ed t,o t,he u > 12fJ. ` liis a.llo\vs l'
us to derive sum rules for the difference of gluon distribution functions for hadrons with different potential.
(and fla.glllelltation) part of the
spin in terms of the spill-depelldel~l
bound-state
2.
INTRINSIC
GAUGE
wa.vefunction
FIELD
DISTRIBUTIONS
A general bound-state of definite number
(72)
can be espa.nded in terms of (FocIi) states We defin(~ the
FOCIi
of elementary
free fields.
espansiotl
at t,lle
equal "time" corresponding
7 = t + z in the light cone gauge AS = A" + A3 = 0. Labelling renormalized amplitudes a.s GTalB(z&J), tQ)
tl le cI' t ri`3u t' IS I 1011I' uirction is
for a constituent given by:
u B - in the bound state - (see Ref. 7 for deta.ils and definitiolls)
We first consider positronium function of photons
as an example, .and calculate
t,he iut,rillsic
dist2ril)ut1ion
Gy,positrorliz~llL. To leading order in the billding pair production, and higher pxticlc
energ!' we can
neglect pair annihilation, The distribution
number FocIi states. f' r01u Ille t,he
function
for positive helicity
photons G+ is calculated
diagrams of Fig. l(a) for th e case of Jz = $1 ortho-positrollium corresponding
(~1.51). Similarly,
dia,grams for negative helicity photons are shown in Fig. 1( II), \vl~c~:re an helicity. 11-li,he diagrallls, t,lle ~1~1x1
arrow up 1 (d own L) indicates positive (negative) fermion (positron). line corresponds to a particle (electron),
and the lower to an antiparticle pa.ra.meteriza,tion of momenta t,lla.t
We have also indicated
the light-cone 5
-
c ..^.
we will use when the photon
couples to an electron or positron. by (x, il)
With
this clloice,
the photon is always parameterized in all cases. The appropriate listed in Table I.[` ] The calculation bound-state matrix
and the fina,1 state has the same form are
elements for the various helicit,y trmsitiolls
is now straightforward.
If we denote by 7,b(y, L71) the, tjwo-body the results a.~:
valence wavefunction
(lowest Foclc state amplitude),
where
D = nlgp (.fflil)" p--X +m2 -$ +772'-it
l-y
.c
. ` ll(, illtrimic l'
Here m and MB are the electron and bound state masses, respectively. gluon distribution defined in Eq. 6 is obtained by integrating
these expressions over
-
-. c
the transverse momentum
up to the cut-off Qi. The sa.me a,pproa,ch gives
(9)
Gg;olt)ioJz=O= G,/ortl,o J,=O .
The polarized and unpolarized AG(x, zl) photon distribution z functions iL) are given I~)-: ,
G+(x, CL) - G-(x,
G(x, CL) = G+(x, zL) + G-(x, gl,
. Let us now consider the sma.11 x limit 2 = 0, we readily obtain: for these functions. Espaudillg arori~rcl
AG(x N o,Q
= -!?.7r2q
0
and
G(X -0,l;;)
= ---g
w"ktx
0
The infrared singularity It should ultraviolet
at Q1 --+ 0 is elimina.ted beca,use of the neutrality the singularity in G' cl) (z, at :I' +
of the at,om. a11
be noted that
0 is actually
singularity
for any non-zero value of Gl since .C = (k" + 1;2)/(l~)0 + 11' cau ) -co. By definition, parton invariant the intrinsic mass M: distribution M" = xi C(;L., (2;) refers to
only be zero if k, t
Fock states with limited restriction regularizes
iL 1
" Irn2 i
< ` Cji. his I'
the x -+ 0, il
# 0 region.
On the other hand, the extrinsic
c .-
--
contribution Physical
is derived from
Fock states exceeding this cut-off, of the intermediate cut-off
Q$ <
Jbt'
<
Q' .
quantities
are independent
Qo; the logarithmic structure functiolls. pl~otous,
-.
dependence on Qo cancels in the sum of intrinsic Note that the integral of xG, the momentum is always well-defined. function $(y,zl)
aad extrinsic fraction
carried 1~~7 intrinsic
In order to proceed further,
we sha.ll assume tha,t the wave
is peaked at y r" l/2.
We then obtain (12)
for the polarization ..
asymmetry.
We have found that this result is numerically. ;-I.ccu t,e ra wavefunctions.
for a large range of positronium
The opposite region (x -+ l), where the fermions emit hard photons, CLUIbe also readily studied. After changing variables (1 - y) = (1 - x)(1 - T), a.nd expallcling
around (1 - x) + 0, we obtain:
G+ (x&)
=
(1-x)
27r2 2(27r)3 s
l d7
X
1 (j+L-l;` +n12 ,)` if:+m2 +- 1-T T [
1
1
2
'
(X3)
G- (x,k7,) =
27r2 2(27r)3 s
0
(' - 'I3
d7
8
c .-
-.
Thus the x + 1 behavior $(y,zl).
depends on the endpoint
beha,vior of the \Z' ei' a,\` U1lctiol-l 1. If
Let us assume that $(y,zl)
- y" for y -+ 0, and N (1 - y)" for y + / $(y,?l) I2 d ominate at z t
p > 4, then the terms that contain regime corresponds
1 since y > .u. This momentum of the - kll) Ii
to the photon taking most of the longitudinal
bound state from the electron. -. will dominate, positron. Then GS = G-
If p < (I, the terms that conta.in 1 ,$(y - :c,&
which corresponds to the photon taBking its large momentum
from t,lle
constant constant
(1 - x)1+21,. (231) :
(P)
=
(1 - z)3+2h 0, y + 1) bcha.vior ol' ,$(:y, ?l). powers itre the s;~me.
where h = If $(y,iJ
mi72(p,
s) is the lowest endpoint power (y t under y + (1 -y),
is invariant
th en the two endpoint
-` .-
In any case:
aG(x, ZL, t 1 G(x, i.d
i.e., the helicity of the photon
(x --+ 1)
;
tends to be aligned with positronium
that of the bound state at
large 2. In the case of relativistic
h=
l.[" A perturbative analJ;sis is of tire
We now extend this analysis to QCD bound states. certainly fermion t justified for heavy quark systems
IlO1
.
Since the general structure
fermion plus gluon vertices given in Table I is dictated conservation, we will assume that this perturbalive
by Lore~~t,z iuvaristructrl1.e is also
ante and parity
9
-
--
C
-.
applicable retaining
to light-quark
systems.
We thus analyze the intrinsic
gluon distribut,ion color
only first order corrections by the replacement endpoint behavior
to the valence Fock state. The appropriate of (a) by (CF~,)
factor is obtained We find similar ticular,
where CJ- = 4/3 for NC = 3. 111par-
to that found in the abelian calculation. e< l/y
the gluon asymmetry
at z -+ 0 is nG(z)/G(x)
> 2 N IR\;~x where 1 behaviol~ for the
Np is the number of fermions in the valence Fock state. The :c t three-quark proton can also be determined["'
G+ .G-
-
(1 -X)4 (x4) (1 - x)6 .
w
3.
CONNECTION
WITH
THE
BOUND
STATE
POTENTIAL
On general grounds we expect a connection (distribution function
between the proba.bility interaction
for emission part of t,he
of photons or gluons) a.nd the hyperfine
bound state potential
since both depend on the exchange of t,ransverse ga.uge quant' a. to the transverse potential function. 1la.sa. corresponding
In fact, each diagram that contributes -` ` cut-diagram
in the expression for the distribution differ by just a denominator
In the a,ctua.l ca.lculation,
these quantities
D. Thus
1 s
0
dz GglB (x,Q;)
=
-
>
U' i)
where G,,B
is the unpolarized
distribution
function
of gauge fields y in the I~ouncl ~IICI 1\4u
state B, V is the potential is the bound-state
due to gluon exchange and self-energy correckiolls, (non-retarded)
mass. Note that the instantaneous
piece clots not
depend on MB, so it does not contribute.
As discussed in section 2, these quant,ities
-
c
are regulated singularity
at z +
0 by the ultraviolet
cutofF Qi in the invariant
I' ma.ss. ` llis
cancels in the hyperfine
splitting:
j dx[ Gy/ortlml~ /pars = - (s)/,,. (4 - % (:I:)]
0
L
where ( )hfS refers to the spin-dependent
part of the bound stnte potential.
In the case of gluons in QCD bound states, we obta.in a.nalogo~~s results:
for mesons (p and r), and
1 J dx
0
[ G!d, (x) -
G,/*
(x)]
= -
(20)
for baryons (p and A). These expressions mentation hyperfine functions splitting can be analytically continued, relating the diff' erence of hagspin to t,he
-` ` -
of gluons DHls (z, Q2) into hadrons H of diKerelIt piece of the bound state potential.
4.
The gluon distribution evolution
CONCLUSIONS
of a hadron is usually assumed to be generat;ed from QCD functions beginning at an initial scale Qi."" In such a
of the quark structure
model there are no gluons in the hadron at a resolution is completely incoherent;
scale below Qo. The evolution
i.e., each quark in the hadron ra,diates intlcpendelltly. 11
-
-. c .-.
In the approach presented here it is recognized that the bound state wavefunction itself generates gluons. the gluon distribution This is clear from the relationship, Eq.
17, which com~ects
poteutial. ` o tl~e esl' gluon
to the transverse part of the bound-state
tent that gluons generate the binding, distribution. another
they also must appear in the iutrillsic
We emphasize that the diagrams in which gluons connect o11e quark to equations. Evolution contribu-
are not present in the usual QCD evolution in the bound-state scales M2 > Qi. equation
tions correspond lines at resolution
to self-energy corrections
to the qua' .rk
Eqs. 4 and 5 give model forms for the polarized . distributions
and unpolarized
iutriusic
gll1011
in the nucleon which take into account coherence a.t low :c and pert urbaat high Z. It is expected that this should be a good chara.c:teriza.tioll at the resolution scale Qi N "4;.
tive constraints
of the gluon distribution It is well-known
-
that the leading power at z - 1 is increased wheu QCD e\` olut,iou The change in power is [II
is taken into account.
(21)
where CA = 3 in QCD.
For typical
values of Qo -
1 GeV,
Am
-
0.2 Gel/" the
change in power is moderate: determination
Aps(2 Gel/")
= 0.28, A~~(10 GeV' ) of the proton
= 0.78. .q recent a.t Q' = :! GeV"
of the unpolarized
gluon distribution
using direct photon and deep inelastic data has been given in ref. 13. 7' best fit o\xx 11e the interval 0.05 5 z 5 0.75 assuming the form zG(z, Q" = 2 Gel;.` ) = A(1 - L)` I~
gives qs = 3.9 f 0.11(+0.8 - 0.6), w h ere the errors in parenthesis allow for sJlst,ematic uncertainties. This result is compatible with the prediction
12
qy = 4 for t,lrcJ intrinsic
-
-. z
gluon distribution to evolution.
at the bound-state
scale, allowing for the increase i n the power clue
Acknowledgements We wish to thank J. Gunion, M. Karliner IS also thanks the kind hospitality and P. G. Lepa,ge for helpful discussions. and the theory group at SLAC.
of R. Blankenbecler
REFERENCES
1. V. N. Gribov
and L. N. Lipatov, Phys. Rev. Dl,
Sov. J. Nucl. Phys. 15, 438 and 675 (1972).
2. J. D. Bjorken,
1376 (1970). J. Ellis, M. Karliner, Whys. Lett. B206, 3U9
3. See, for example, (1988). 4. R. Blankenbecler,
Phy~.
S. J. Brodsky,
S. J. Brodsky
Phys. Rev. DlO, 2973 (1974); J. F. Gunion, and J. F. Gunion, l' hys. Rev. DIY,
Rev. DlO, 242 (1974); S. J. Brodsky
1005 (1979), w h ere coherence effects are also discussed. 5. M. Karliner, 6. M. B. Einhorn 7. S. J. Brodsky 8. J.D. Bjorken, 24th Rencontre de Moriond, Mas 1989; preljrint I' AUP 1730-m59.
and J. Soffer, Nucl. Phys. B274, 714 (1986). and G. P. Lepage, Phys. Rev. D22, 2157 (1980 ) J. Kogut, D. Soper, Phys. Rev. D3, 1382 (1971).
9. 2. F. Ezawa, Nuovo Cim. 23A, 271 (1974). 10. G. P. Lepa.ge, Proceedings, SLAC S ummer Institute (19Sl).
11. S. J. Brodsky .and G. P. Lepage, Proc. of the Int. Symp. on High Energy Physics with Polarized Beams and Polarized Targets, Lausanne, Switzerland
13
(1980).
-
-. t ..
12. See, e. g., F. Martin, Vogelsang, Dortmund 13. P. Aurenche,
Phys. Rev. D19, 13S2 (1979); M. Gliick, University preprint DO-TH-S9/3
E. K.ey+ a,ntl W.
(19SY).
et al.,Phys.
Rev. D39, 3275 (19S9).
14
t -.
m
-kl
+ ik:! + ([I - k1) - i(!p - k2)
2
Y-X
>
ICI + ikz
X
-
(!I - ICI) + i(tz - X-2)
Y-X
>
kl + ih el+
2
222
Y
-ICI + ik2 + .tl - it2
X 8415A2
Y
>
Table I Photon/gluon and negative
-` '
emission vertices helicities, ,/j
(eux, c;,qux,)
for particles
with
positive
(T)
(I)
in light-cone
coordinates. for fermions
An overall
fact,or A = re-
zk2& spectively.
d=
multiplies
each result,
and anti-fermions
For gluon emission cr is replaced by 4/3 cr,.
-
G+=
$2
..
ut t I +=&-i 45 t t vt
2
+
2
+
(a)
ut t2 +4 I sii-t
1.W
Fig. 1 Diagrams that contribute
to the distribution
function
for positive polarized pho-
tons (a), and for negative polarized photons (b), for J, = $1 ortho-positronium bT' i?T).