Nuclear A-dependence near the Saturation Boundary

13 Pages·0.11 MB·English

by  A. H. Mueller

CU-TP-1082
3 Nuclear A-dependence near the Saturation Boundary1
0
0
2 A.H. Mueller

n Physics Department, Columbia University
a
New York, N.Y. 10027 USA
J

5
1
Abstract

1
The A-dependence of the saturation momentum and the scaling behav-
v
9 ior of the scattering of a small dipole on a nuclear target are studied in
0
the McLerran-Venugopalan model, in fixed coupling BFKL dynamics and
1
1 in running coupling BFKL dynamics. In each case, we find scaling not too
0
far from the saturation boundary, although for fixed coupling evolution the
3
0 scaling function for large A is not the same as for an elementary dipole. We
/ find that Q2 is proportional to A1/3 in the McLerran-Venugopalan model
h s
p and in fixed coupling evolution, however, we find an almost total lack of
-
p A-dependence in Q2 in the case of running coupling evolution.
s
e
h
:
v 1 Introduction
i
X
r The study of high density QCD, states or systems where gluon occupation
a
numbers are large compared to one, has become one of the central topics
in QCD. There is a considerable literature whose focus is explaining much
of small x and moderate Q2 HERA data[1-4] as well as the general features
of hadron production in ion-ion collisions at RHIC[5-11] using high density
wavefunctions as the basic ingredient. In any such description the central
parameter is the saturation momentum, Q , the scale at and below which
s
occupation numbers are as large as 1/α in the light-cone wavefunction.
The energy dependence of the saturation momentum has been widely
studied[12-17] and it now appears that one has pretty good control over that
1This researchis supported in part by the US Department of Energy.
1
dependence in the context of resummed next-to-leading order BFKL dynam-
ics. Recently there has also appeared some discussion of the A-dependence
of the saturation momentum for large nuclei[11-18] beyond the McLerran-
Venugopalan model[19] model. The conclusion has been pretty much the
same as for the McLerran-Venugopalan model, that is Q2(A) A1/3.
s ∼
Our purpose here is to revisit the question of the A-dependence of the
saturationmomentumimplementing theBFKLdynamicsmorefullythanhas
been done so far. In case fixed coupling BFKL dynamics is used we confirm
that the primary A-dependence of Q2(A) is proportional to A1/3. The result
s
is given in (26). We also confirm that the scattering of a small dipole, of size
1/Q, on the nucleus gives a scattering amplitude which is only a function of
Q2/Q2(A) so long as one is not too far from the saturation boundary. We
s
note, however, that the scaling function is not the same as for the scattering
of a dipole on another elementary dipole. (In the fixed coupling problem
it does not appear possible to sensibly talk of dipole-proton scattering in a
perturbative context.) The results for the scattering amplitude are given in
(27) and (29).
In case a running coupling BFKL dynamics is used the situation changes
radically[16,17]. Now the saturation momentum becomes almost completely
independent of A at very large rapidities. Eq.(44) gives our result here for
the weak A-dependence which is present.
In both the fixed coupling and running coupling situations we have found
itusefultoviewtheBFKL-QCDevolutioninanunusualway. Thetraditional
way to view the scattering of a small dipole on a larger dipole, or on a hadron
ornucleus, istoviewevolutionaspartofthewavefunction ofthelargerobject
which is then probed by the smaller dipole. In the present problem it seems
more convenient to view the BFKL evolution as part of the wavefunction of
the smaller dipole, which evolution produces gluons (dipoles) of larger scale
which eventually interact with the hadron or larger scale.
Finally, because Q2 is not very large we do not feel confident in deciding
s
which picture, McLerran-Venugopalan, fixed coupling BFKL dynamics or
running couplng BFKL dynamics is more appropriate for RHIC energies
and for the LHC heavy ion regime. It might well be that the McLerran-
Venugopalan model along with a modest amount of fixed coupling evolution
is dominant at RHIC energies so that Q2 A1/3 is the appropriate behavior
s ∼
of the saturation momentum. We would guess that the running coupling
regime is likely dominant for protons and small nuclei at HERA energies,
2
and is perhaps important for large nuclei at LHC energies, but this is by no
means certain. It would be very useful to have some numerical calculations
which are reliable near the saturation region to try and see what the A-
dependence of Q actually is in the energy regions covered by the different
s
accelerators.
2 The semiclassical region (McLerran-
Venugopalan model)
We begin our discussion of the A-dependence near the saturation boundary
with a brief review of the McLerran-Venugopalan model[19]. Let T be the
N
amplitude for the scattering of a quark-antiquark dipole of size ∆x 1/Q
⊥ ≡
on a nucleon. Using a normalization where the cross-section for scattering is
σ = 2T (1)
N N
one has
π2α
T = x2xG (2)
N N
2N
c ⊥
where xG is the gluon distribution of the nucleon evaluated at scale Q2.
N
Let T (b) be the corresponding amplitude for the scattering of a dipole of
A
size ∆x on a nucleus at impact parameter b. Then, the dipole-nucleus cross-
⊥
section is
σ = 2 d2bT (b) (3)
A A
Z
with[19,20]
Q¯2(A)
T = 1 exp[ s ] (4)
A − − 4Q2
where
C
Q¯2(A) = 2σ 2√R2 b2 ρQ2 = FQ2(A) (5)
s N · − N s
c
with, as usual[21]
3
4π2αN
Q2(A) = c 2√R2 b2ρxG. (6)
s N2 1 −
c −
ρ is the nuclear density, Q¯ is the quark saturation momentum and Q the
s s
gluon saturation momentum. It should be emphasized that Q 1 defines
≡ ∆x⊥
Q. Q is not a true momentum variable connected to x dependence by a
⊥−
Fourier transform.
Eq.(4) has all the basic properties which we wish to note in this section.
First of all (4) exhibits geometric scaling[3]. That is T is only a function
A
of Q2(A)/Q2.T exhibits saturation so that T 1 when Q¯2/(4Q2) 1.
Finalsly, when 4AQ2/Q¯2 1,T is additive in thAe v≃arious nucleosns. Tha≫t is
s ≫ A
σ
TA ≃ N 2√R2 b2 ρ, (7)
Q2/Q2s≫1 2 −
so that there are no non-trivial nuclear effects present when Q2/Q2 1.
s ≫
3 Fixed coupling BFKL dynamics
Nowweshallconsiderthescatteringofadipoleofsize∆x = 1/Qonadipole
of size ∆x = 1/µ (where Q2/µ2 1) and on a “nucl⊥eus” made up of A
⊥ ≫
such dipoles distributed uniformly in a sphere of radius R and having density
ρ = A/(4πR3). Of course this is not a realistic nucleus, since it is not made
3
up of realistic nucleons, but it is not possible to deal with a realistic nucleon
in the fixed coupling regime. Since our object is to study A-dependence we
must build our nucleus out of individual objects which do make sense for
fixed coupling, and there are small dipoles.
In the vicinity of the saturation boundary the amplitude for scattering of
a dipole Q on a dipole µ is given by[16]
Q2 1−λ0
T (µ,b,Q) = T (bµ) s ℓnQ2/Q2 (8)
µ 0 Q2 s
(cid:18) (cid:19)
where b is impact parameter and
1 exp[2αNc χ(λ0)Y]
Q2s(b,µ,Y) = a(bµ)µ2[ℓnαα2]1−1λ0 π 13−λ0 . (9)
[αY]2(1−λ0)
4
T and a are of order one when bµ is not large. Eq.(8) is valid in the region
0
4αN
1 ℓnQ2/Q2 cχ (λ )Y (10)
≪ s ≪ π ′′ 0
r
4π
αY ℓn21/α (11)
≫ (1 λ )2N χ (λ )
0 c ′′ 0
−
9π
αY ℓn2(αY) (12)
≫ 16(1 λ )3N χ (λ )
0 c ′′ 0
−
where λ is defined by χ(λ ) = χ(λ0), and χ(λ) = ψ(1) 1ψ(λ) 1ψ(1 λ)
0 ′ 0 −1 λ0 −2 −2 −
withψ thelogarithmicderivativeof−theΓ-function. Forsimplicity ofnotation
we now use Q to represent the quark saturation momentum. Since the issue
s
in this andthe next section is the functional dependence of Q thedistinction
s
between Q and Q¯ is not important.
s s
The conventional way of looking at (8) is to view the dipole Q as mea-
suring the gluon distribution of dipole µ so that
dxGµ(x,b,Q2) 1Q2 Q2s 1−λ0 ℓnQ2/Q2 (13)
d2b ∼ α Q2 s
(cid:18) (cid:19)
when Q2/Q2 > 1. (We note that when Q2/Q2 is on the order of one, dxGµ
s s d2b ∼
Q2/α.) However, one may view the QCD-BFKL evolution in the opposite
s
sense so that one considers the dipole Q to evolve to lower momentum
scales,and then (8) gives the probability of finding a gluon (or dipole) at
scale µ in the parent dipole Q. More precisely
d N˜ (Q,b,Y,µ) 1 µ2 1−λ0 ℓn(Q˜2/µ2) (14)
dY g ∼ α Q˜2
(cid:18) s(cid:19)
is the number of gluons per unit rapidity at scale µ in the wavefunction of a
dipole of size ∆x = 1/Q. In the large N limit dNg would also represent the
c dY
⊥
number of dipoles of size ∆x 1/µ in a parent dipole of size 1/Q. In (14)
we have introduced Q˜ which⊥is≥defined by
s
Q2 µ2
s = . (15)
Q2 Q˜2
s
That is if one writes
5
Q2(b,µ,Y) = µ2f(b,Y) (16)
s
then
Q˜2(b,Q,Y) = Q2/f(b,Y), (17)
s
and
dN˜ (Q,b,Y,µ)
g
T α . (18)
µ
∼ dY
In Fig.1, the two directions of evolution are pictured. In the left-hand part
of the picture the shaded saturation region for a dipole of size µ and rapidity
Y is exhibited. Only the part of the saturation region where ℓnk2/µ2 > 0
is shown. In the right-hand part of the picture the part of the s⊥aturation
region having ℓnQ2/k2 < 0 is shown for a dipole of size Q and rapidity Y.
For the µ-dipole, evol⊥ution proceeds from y = 0 up to y = Y, while for the
Q dipole evolution goes from y¯ Y y = 0 to y¯= Y.
− ≡ −
Now consider the scattering of a dipole Q on a nucleus made up of dipoles
of size µ and nuclear density ρ. We view the process in the rest frame of the
nucleus and in two steps. In the first step one takes the number density, dNg,
dy
of dipoles of size K and having rapidity y in the wavefunction of dipole Q,
and in the second step one takes the scattering amplitude for a dipole of size
K on the nucleus. Thus
Y Q dN (Q,y,K) dK
g
T (µ,b,Q) dy t (K,b,µ) (19)
A A
∼ dy · K
Z0 Zµ
where
dN (Q,y,K) dN˜ (Q,b,y,K)
g = d2b g ′ . (20)
′
dy dy
Z
TherapidityofthedipoleK in(19)shouldbegreaterthanℓnA1/3 sothatthe
dipole is coherent over the size of the nucleus, but we suppose that ℓnA1/3
is small enough that it can be neglected in setting the upper limit of the
y integral in (19). Now
−
6
Q Q
s
Y
Y-Y
0
y y
Y
0
µ ~
Q 2
s ln k µ2
Figure 1:
t (K,b,µ) α2(1/K2)ℓn K2/µ2 ρ2√R2 b2 (21)
A
∼ · −
so long as t is small, where we have taken the gluon distribution of a dipole
A
µ in the nucleus to be α ℓn K2/µ2 at scale K. Using (14), (20), and (21) in
(19) one finds that the K integration (19) diverges in the infrared. But, this
integration should be cut off at the value, K , when t , given in (21) reaches
0 A
one[21-24]. Thus
1 K2
T ( 0)1 λ0ℓn(Q˜2/K2) (22)
A ∼ α2 Q˜2 − s 0
s
where, from (21),
K2 Q2(MV)
0 s α2ℓn(K2/µ2)ρ2√R2 b2 /µ2. (23)
µ2 ≡ µ2 ∼ 0 −
or
K2
0 α2cA1/3ℓn(α2cA1/3) (24)
µ2 ∼
with
2ρ√R2 b2/µ2 = c(b,µ)A1/3.
−
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(K2 is the quark saturation momentum in the Mclerran-Venugopalan model
0
of a nucleus made of dipoles of size 1/µ.) We shall always assume that
α2cA1/3 1 in order to have non-trivial nuclear effects.
≫
Now use (24) to eliminate K in (22). This gives
0
Q2 1 Q2
T [α2cA1/3ℓn(α2cA1/3) s]1 λ0 [ℓn ℓn(α2cA1/3) ℓnℓn(α2cA1/3)]
A −
∼ Q2 α2· Q2− −
s
or
Q2Q2(MV) 1 µ2Q2
T ( s s )1 λ0 ℓn[ ] (25)
A ∼ µ2Q2 − α2 Q2Q2(MV)
s s
The saturation momentum for the nucleus is defined as the value of Q2 at
which T becomes of order one. This gives
A
Q2(A) Q2 Q2(MV) 1 1
ℓn( s ) = ℓn( s)+ℓn( s )+ [ℓn +ℓnℓn1/α2]+const (26)
µ2 µ2 µ2 1 λ 2
0
−
where Q2(MV) is the saturation momentum in the McLerran-Venugopalan
s
model. We note however that T does not quite have the usual scaling form
A
(8) since
Q2(A) 1−λ0 ℓn(Q2/Q2(A)
T s 1+ s . (27)
A ∼ Q2 ℓn[ µ2Q2s(A) ]!
(cid:18) (cid:19) Q2Q2(MV)
s s
Eq.(27) is only valid for Q2/Q2(A) > 1. When Q2/Q2(A) < 1 there is more
s s
than one dipole of scale K in the parent dipole Q and unitarity effects will
0
further suppress the T given in (27)[22-24]. One can express the scaling
A
behavior of T more clearly by defining ariable
A
Q¯2(A) = Q2cα2A1/3ℓn(cα2A1/3) = Q2Q2(MV)/µ2 (28)
s s s s
in terms of which, from (25),
1 Q¯2(A) 1λ0
T s ℓn Q2/Q¯2(A) (29)
A ∼ α2 Q2 s
(cid:18) (cid:19)
(cid:0) (cid:1)
where, now,(29) can be used when
8
1
Q2 1 1 1−λ0
> ℓn . (30)
Q¯2(A) α2 α2
s (cid:18) (cid:19)
While Q¯ (A) gives the simpler looking scaling behavior it is Q (A) which is
s s
the actual saturation momentum of the nuclear light-cone wavefunction.
Finally, we evaluate T /T . From (8) and (29) one finds
A µ
1 Q¯2(A) 1−λ0 ℓn(Q2/Q¯2(A))
T /T s s . (31)
A µ ∼ α2 Q2 ℓn(Q2/Q2)
(cid:18) s (cid:19) s
Using (28) gives
cA1/3 ℓn(cα2A1/3
T /T ℓn1 λ0(cα2A1/3) 1 . (32)
A µ ∼ (cα2A1/3)λ0 − − ℓn(Q2/Q2)
(cid:18) s (cid:19)
Thus not too far from the saturation boundary there is significant shadowing
with the A-dependence of T proportional to (A1/3)1 λ0. It is this shadowing
A −
which, according to Ref.11, causes particle production in heavy ion collisions
to scale,roughly, like N . We note that far from the scaling region, when
part
ℓnQ2/µ2 >> 1, BFKL dynamics will replace λ by 0, and an A1/3 behavior
αY 0
will again emerge for T . This is the region where double leading logarithmic
A
behavior is valid.
4 Running coupling BFKL dynamics
In this section we revisit our discussion in the last section, but now using
running coupling BFKL dynamics. We shall see that saturation looks quite
different when running couplings effects are present.
We begin by considering the scattering of a dipole Q on a dipole µ where,
as before, Q/µ 1. As in the last section[16-17]
≫
Q2 1−λ0 1
T (µ,b,Q) T s ℓnQ2/Q2 + . (33)
µ ≃ 0 Q2 s 1 λ
(cid:18) (cid:19) (cid:20) − 0(cid:21)
T is still a function of bµ but now Q is given by
0 s
4Nχ(λ ) 3 a 1/3 1
ℓn[Q2/Λ2] = 0 Y + ξ Y1/6 (34)
s sπb(1 λ0) 4 c 1 − 1 λ0
− (cid:16) (cid:17) −
9
where
N (1 λ )[χ (λ )]2
a = c − 0 ′′ 0 (35)
s 4πbχ(λ0)
and
1 λ
0
c = − (36)
2
with b = 11N1c2−π2Nf, and where Λ is the usual QCD parameter. ξ1 is the first
zero of the Airy function, A (ξ). Of course, the form given in (33) is valid
i
only when ℓnQ2/Q2 1 so that the constant terms in (33) and (34) are
s ≫
arbitrary, and unimportant.
Theremarkablethingabout(34)isthatthereisnoµ dependencepresent[16].
−
As noted earlier the µ dependent corrections to (34) can be of size ℓn2(µ2/Λ2)
− √Y
so that in case µ2/Λ2 1 there is a transition region between a low-Y re-
≫
gion where fixed coupling dynamics occurs and a high-Y regime where run-
ning coupling dynamics occurs. This transition value of Y is[16] Y
trans
≃
π(1−λ0) 1 . Eqs. (33) and (34) apply well above this transition regime.
2bNcχ(λ0) α2(µ)
The lack of a µ dependence in Q at large Y indicates the insensitivity of
s
−
Q to the nature of the target probed by the dipole Q. Thus, (33)-(36) apply
s
equally to a proton as to a dipole. And, of course,these equations must apply
also to nuclei indicating that Q , at large Y, has no A dependence. This is
s
−
in striking contrast to A1/3-dependence of Q2 found both in the McLerran-
s
Venugopalan model and in fixed coupling BFKL evolution as given by (6)
and (26), respectively. It is the purpose of the present section to try and
explain why there is no A dependence in Q2 in running coupling evolution.
− s
It is useful to consider an expression for ℓnQ2/Λ2 which interpolates be-
s
tween fixed coupling evolution and running coupling evolution. To that end
consider
4N χ(λ )
ρ (Y,µ) = c 0 Y +ℓn2(µ2/Λ2). (37)
s
sπb(1 λ0)
−
When Y/ℓn2(µ2/Λ2) 1
≪
10