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ADS1293CISQX/NOPB 其他数据使用手册 - TI（德州仪器）
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低功耗，三通道， 24位模拟前端的生物电位测量 Low Power, 3-Channel, 24-Bit Analog Front End for Biopotential Measurements
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ADS1293CISQX/NOPB数据手册
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In this language, σ(s, ρ, ¯ρ, b) is the natural object which automatically satisfies
the unitarity bound 0 ≤ σ ≤ 2 due to AdS unitarity Im ∆ ≥ 0. Moreover, for a
black disk region we have σ → 1, corresponding to a phase shift ∆ with large
imaginary part.
In general, we cannot evaluate the integral over the impact parameter b.
We may, on the other hand, use the relation (4) between b and the AdS impact
parameter B to rewrite the cross section σ(s, ρ, ¯ρ) in (7) as
σ(s, ρ, ¯ρ) = 2πρ¯ρ
Z
∞
|ln(¯ρ/ρ)|
dB sinh B Re
1 −e
2i∆(sρ¯ρ,B)
. (8)
It is now apparent that we are probing the phase ∆(S, B) for fixed S = sρ¯ρ and
for B ≥ |ln(¯ρ/ρ)|. Finally, note that the unintegrated cross sections satisfy, due
to conformal invariance, non trivial relations under the transformation ρ → ¯ρ
2
/ρ
with s → s
ρ
2
/¯ρ
2
, b → b (¯ρ/ρ), which leave invariant S and B. More precisely,
σ(s, ρ, ¯ρ, b) is invariant whereas
ρ
2
¯ρ
2
σ
s
ρ
2
¯ρ
2
,
¯ρ
2
ρ
, ¯ρ
= σ(s, ρ, ¯ρ) .
The phase shift ∆(S, B) depends both on the number of colors N and on
the ’t Hooft coupling ¯α
s
= α
s
N/π of the theory. For large energy squared
and impact parameter S and B, the phase ∆ will be dominated by the leading
Regge pole of the planar diagrams of the theory [9, 10], and will have a general
representation of the form
2
∆(S, B) =
1
N
2
Z
dν β(ν) S
j(ν)−1
Ω
iν
(B) , (9)
where the Regge spin j(ν) and residue β(ν) depend implicitly only on the ’t
Hooft coupling ¯α
s
and are even functions of ν. The function Ω
iν
(B) computes
radial Fourier transforms in H
3
, satisfies
H
3
+ ν
2
+ 1
Ω
iν
= 0 and is given
explicitly by
Ω
iν
(B) =
1
4π
2
ν sin νB
sinh B
.
Whenever the AdS phase satisfies |∆| 1, the full cross section is well
approximated by a single Reggeon exchange, and we may write
σ(s, ρ, ¯ρ, b) ' 2 Im ∆(S, B). (10)
In this case, the integral over the impact parameter b can be explicitly per-
formed. In fact, using the Regge representation (9) for the phase shift, together
with
3
Z
d
2
b Ω
iν
(B) =
1
2π
ρ¯ρ
¯ρ
ρ
−iν
,
2
In this paper, in order to have a uniform notation, we use slightly different conventions
then in [9, 10]. In particular, ∆ = −πΓ
there
and β
here
= −πβ
there
.
3
In order to correctly compute the normalization of this Fourier transform, as well as of
the ones in the sequel of the paper, it is safest to compute at non–zero momentum transfer
and take the limit q → 0.
4
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