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ADS1293CISQX/NOPB 其他数据使用手册 - TI（德州仪器）
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usual impact parameter. The phase shift δ(s, b) is itself given by
e
2iδ(s,b)
=
Z
dρ
ρ
3
f
1
(ρ) f
3
(ρ)
Z
d¯ρ
¯ρ
3
f
2
(¯ρ) f
4
(¯ρ) e
2i∆(S,B)
, (3)
with ∆(S, B) the phase shift in AdS, which depends on the AdS energy squared
and impact parameter S and B, according to
1
S = ρ¯ρs ,
cosh B =
ρ
2
+ ¯ρ
2
+ b
2
2ρ¯ρ
. (4)
In particular, B is the geodesic distance between the points ρ, x and ¯ρ, ¯x in H
3
,
with b = x − ¯x. These represent the impact points of the operators O
1
and O
2
in the transverse space. Finally, the functions f
i
are the radial wave functions
for the scattering states. For scalar operators O
1
, O
2
of dimension ∆
1
, ∆
2
they
are given by f
i
∝ Q
i
ρ
2
K
∆
1
−2
(Q
i
ρ) for i = 1, 3, and by f
i
∝ Q
i
¯ρ
2
K
∆
2
−2
(Q
i
¯ρ)
for i = 2, 4 [11]. We normalize the wavefunctions so that
Z
dρ
ρ
3
f
1
(ρ) f
3
(ρ) =
Z
d¯ρ
¯ρ
3
f
2
(¯ρ) f
4
(¯ρ) = 1 . (5)
As shown in [5], the impact parameter representation (2) and (3) approx-
imates the conformal partial wave decomposition of the correlator (1) in the
channel O
1
O
2
→ O
?
1
O
?
2
, with intermediate states of conformal dimension and
spin respectively given by
√
S cosh(B/2) and
√
S sinh(B/2). In analogy with
the usual results for scattering in flat space, we then expect that AdS unitarity
implies [9, 10]
Im ∆(S, B) ≥ 0 ,
even though the phase shift δ(s, b) does not satisfy a simple unitarity constraint.
We shall focus, for concreteness, on the very relevant and simple case of
vanishing momentum transfer q = 0 and equal virtualities for the incoming and
outgoing states Q = Q
1
= Q
3
and
¯
Q = Q
2
= Q
4
. It is then natural to construct,
from the correlator (2), the following effective cross section
Σ
s, Q,
¯
Q
= 2
Z
d
2
b Re
1 −e
2iδ(s,b)
.
Using (3), the cross section Σ can be conveniently written as
2
Z
dρ
ρ
3
f
1
(ρ)f
3
(ρ)
Z
d¯ρ
¯ρ
3
f
2
(¯ρ) f
4
(¯ρ) σ(s, ρ, ¯ρ) , (6)
where we have defined the unintegrated cross sections
σ(s, ρ, ¯ρ) =
Z
d
2
b σ(s, ρ, ¯ρ, b) , (7)
σ(s, ρ, ¯ρ, b) = Re
1 −e
2i∆(S,B)
.
1
We take the AdS quantities S and B to be dimensionless, measured in units of the AdS
radius.
3
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