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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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coming from the integral representation [10]
Ω
iν
(B) =
ν
2
4π
3
Z
d
2
z
ρ
ρ
2
+ (b − z)
2
!
1+iν
¯ρ
¯ρ
2
+ z
2
1−iν
,
we may evaluate the cross section σ(s, ρ, ¯ρ) to be
σ(s, ρ, ¯ρ) '
ρ¯ρ
2π N
2
Im
Z
dν β(ν) (sρ¯ρ)
j(ν)−1
¯ρ
ρ
−iν
. (11)
2 The Cross Section Deep into Saturation
At fixed AdS energy squared S, the phase ∆(S, B) will in general vanish in the
limit B → ∞. On the other hand, as we approach smaller and smaller impact
parameters, ∆ will in general grow and reach saturation at B ' B
s
(S), where
∆ is of order one.
We will be mostly interested in weakly coupled gauge theories, where the
phase is predominantely imaginary [12].
4
In this case, saturation is reached at
5
2 Im ∆
S, B
s
(S)
' 1 .
A typical plot of the saturation line in the
B, ln S
plane is given in Figure 2.
In particular, for large S we have the linear relation
B
s
(S) ' ω ln S + ··· , (12)
where ··· represents subleading terms in S. This can be shown, as customary
[17], by approximating the integral in (9) at the saddle point iB = j
0
(ν) ln S.
Saturation is then reached when the phase in (9) vanishes at the saddle – i.e.
when (1 + iν
s
) B
s
= (j(ν
s
) −1) ln S. These conditions imply that
ω = −i j
0
(ν
s
)
where the saturation saddle point ν
s
is defined in terms of the Regge trajectory
j(ν) by
(1 + iν
s
) ω = j(ν
s
) −1 .
The cross section σ(s, ρ, ¯ρ) near saturation |ln(¯ρ/ρ)| & B
s
(sρ¯ρ) exhibits ge-
ometric scaling [18]. More precisely, the integral (11) has a leading behavior
given by
σ(s, ρ, ¯ρ) ∼ ¯ρ
2
τ
−(1+iν
s
)
(1−ω)
2
, (13)
4
On the other hand, at strong coupling and for large impact parameters the phase shift is
predominantly real and is given by the gravi-reggeon exchange in AdS. Studies of DIS in this
regime include [14]. Saturation effects at strong coupling have also been analyzed in [15], and
a conjectured relation to black hole formation was put forward in [16].
5
Note that it is usually believed that, when Im ∆ ' 1, non–linear BK corrections to ∆
of order N
−4
due to fan diagrams also become relevant [24]. As long as those corrections
are negligible for impact parameters larger then the saturation line, they are irrelevant in the
discussion which follows, since they will predominantly affect the phase shift in the black disk
region.
5
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