动态AdS/QCD模型中的各向异性弦张力与磁场诱导脱约束效应
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"这篇研究论文探讨了动态AdS/QCD模型在存在背景磁场情况下的各向异性弦张力和逆磁催化脱约束现象。作者通过引入背景磁场来扩展之前的工作，构建了一个包含爱因斯坦-麦克斯韦-狄拉顿系统的解析解。他们特别关注了在磁场影响下解除约束的转变过程，以及逆磁催化如何改变约束状态。研究发现，弦张力沿磁场方向和垂直磁场方向表现出不同的特性，这对理解和模拟重夸克偶素的行为具有重要意义，并且与实验中的晶格数据相吻合。文章发表在《Physics Letters B》上，为开放获取资源。"
本文的核心知识点包括：
1. **动态AdS/QCD模型**：AdS/QCD是强相互作用量子色动力学（QCD）的一个有效理论框架，它利用了AdS/CFT对应原理，即反德西特空间/共形场论对偶性，将高维引力问题转化为四维强相互作用问题，从而简化了对QCD的计算。
2. **背景磁场**：引入背景磁场是为了研究磁场如何影响QCD中的关键物理现象。磁场可以改变夸克和胶子的行为，尤其是在高密度或高温环境下，如极端天体物理中的中子星内部。
3. **爱因斯坦-麦克斯韦-狄拉顿系统**：这是包含引力、电磁场和标量场的统一理论模型，用于描述在磁场背景下的时空结构和物质相互作用。
4. **各向异性弦张力**：在存在磁场时，弦张力会受到磁场方向的影响，表现为沿磁场方向和垂直磁场方向的不同。这种各向异性对于理解重夸克偶素的性质至关重要，因为它们的结合能受弦张力控制。
5. **逆磁催化脱约束**：在某些条件下，磁场可以促进原本束缚的物质转变为无约束状态，这个过程称为逆磁催化脱约束。这与磁场诱导的相变有关，可能影响QCD相图。
6. **重夸克偶素**：由一个重夸克和一个反重夸克组成的束缚态，如J/ψ和Υ。它们的性质可以作为理解QCD的探针，因为它们对环境变化非常敏感。
7. **晶格QCD**：这是一种数值模拟方法，用于在量子水平上研究QCD。研究结果与晶格QCD的发现一致，表明理论模型与实验数据有良好的符合性。
这项工作深化了我们对QCD在磁场环境下的理解，特别是关于约束状态和弦张力的各向异性，这对于未来研究强相互作用物理和极端条件下的物质行为具有重要的理论和实验意义。
H. Bohra et al. / Physics Letters B 801 (2020) 135184 3
can only be incorporated indirectly from the background metric,
see also [51]. Therefore, contrary to real QCD, the backreaction of
the chiral condensate on the quark-antiquark free energy and vice
versa will be completely ignored in the holographic model (2.1).
This is a major disadvantage of all probe-approximated AdS/QCD
models, like eq. (2.1), where no explicit interplay between the chi-
ral
condensate and Polyakov loop exists. A more accurate and real-
istic
holographic QCD model, that incorporates the backreaction of
the chiral field on the spacetime geometry from the start, would—
although
very interesting—be extremely challenging to construct
analytically. We are thus working with a kind of holographic ana-
logue
of the quenched QCD approximation known from lattice QCD
(no dynamical quarks). As such, one might question how to even
couple the magnetic field to the theory if there are no dynamical
charge carriers. Here, we follow the pragmatic approach of e.g. [88,
89],
in itself magnetic field-dependent generalizations of the sem-
inal
works [90,91], and we thus consider our engineered boundary
model capable of mimicking some essential QCD features, after
which it can be used to describe, without further input, other QCD-
ish
properties. Notice that Einstein-Maxwell-dilaton models have
been used throughout literature as an effective way to describe
QCD in presence of electromagnetic background fields, as it is evi-
dent
from our extensive reference list.
By
varying the action (2.1)one can derive the equations of mo-
tion
for Einstein, Maxwell and dilaton fields. Since, in this work we
are mostly interested in a magnetised black brane solution with
running dilaton, we consider the following Ansätze for the metric
g
MN
, field strength tensor F
(i)MN
and dilaton field φ,
ds
2
=
L
2
S(z)
z
2
−
g(z)dt
2
+
dz
2
g(z)
+
dy
2
1
+e
B
2
z
2
dy
2
2
+dy
2
3
,
φ = φ(
z), A
(1)M
= A
t
(z)δ
t
M
, F
(2)MN
= Bd y
2
∧dy
3
, (2.2)
where S(z) is the scale factor, L is the AdS length scale and g(z)
is the blackening function. z is the usual holographic radial coordi-
nate,
and in our coordinate system it runs from z = 0(asymptotic
boundary) to z = z
h
(horizon radius), or to z =∞for thermal AdS
(without horizon). We introduced a background magnetic field B
in
the y
1
-direction. Because of this background magnetic field, the
system no longer enjoys the SO(3) invariance in boundary spatial
coordinates (y
1
, y
2
, y
3
), and we precisely chose the metric An-
sätze
such that as soon as we switch off the magnetic field the
SO(3) invariance is recovered.
Using
the Ansätze of eq. (2.2)we get four Einstein equations of
motion,
g
(z) + g
(z)
2B
2
z +
3S
(z)
2S(z)
−
3
z
−
z
2
f
1
(z) A
t
(z)
2
L
2
S(z)
=
0 . (2.3)
B
2
ze
−2B
2
z
2
f
2
(z)
L
2
S(z)
+
2B
2
g
(z)
+
g(z)
4B
4
z +
3B
2
S
(z)
S(z)
−
4B
2
z
=
0 . (2.4)
S
(z) −
3S
(z)
2
2S(z)
+
2S
(z)
z
+ S(z)
4B
4
z
2
3
+
4B
2
3
+
1
3
φ
(z)
2
=
0 . (2.5)
g
(z)
3g(z)
+
S
(z)
S(z)
+
S
(z)
7B
2
z
2S(z)
+
3g
(z)
2g(z)S(z)
−
6
zS(z)
+
g
(z)
5B
2
z
3g(z)
−
3
zg(z)
+
2B
4
z
2
+
B
2
z
2
e
−2B
2
z
2
f
2
(z)
6L
2
g(z)S(z)
−
6B
2
+
2L
2
S(z)V (z)
3z
2
g(z)
+
S
(z)
2
2S(z)
2
+
8
z
2
= 0 . (2.6)
Similarly we get the following equation of motion for the dilaton
field,
φ
(z) + φ
(z)
2B
2
z +
g
(z)
g(z)
+
3S
(z)
2S(z)
−
3
z
+
z
2
A
t
(z)
2
2L
2
g(z)S(z)
∂
f
1
(φ)
∂φ
−
B
2
z
2
e
−2B
2
z
2
2L
2
g(z)S(z)
∂
f
2
(φ)
∂φ
−
L
2
S(z)
z
2
g(z)
∂
V (φ)
∂φ
=
0 , (2.7)
and the equation of motion for the first gauge field,
A
t
(z) + A
t
(z)
2B
2
z +
f
1
(z)
f
1
(z)
+
S
(z)
2S(z)
−
1
z
=
0 . (2.8)
One can explicitly check that the equation of motion for the second
Maxwell field is trivially satisfied and hence it will not give any
additional equation. Therefore, we have in total six equations of
motion. However, only five of them independent. Below we will
choose the dilaton equation (2.7)as a constrained equation and
consider the rest of the equations as independent. In order to solve
the latter, we impose the following boundary conditions,
g(0) = 1andg(z
h
) = 0,
A
t
(0) = μ and A
t
(z
h
) = 0,
S(0) = 1,
φ(
0) = 0 , (2.9)
where μ is the chemical potential of the boundary theory which is
related to the near boundary expansion of the zeroth component
of the first gauge field and, as mentioned before, z
h
is the location
of the black hole horizon. Apart from these boundary conditions,
we will also assume that the dilaton field φ remains real every-
where
in the bulk. As we will see later, this condition will severely
restrict our analytic solution for a finite magnetic field.
In
order to solve eqs. (2.3), (2.4), (2.5), (2.6) and (2.8)simulta-
neously,
we adopt the following strategy.
1. We
first solve eq. (2.8)and obtain the solution for A
t
(z) in
terms of f
1
(z) and S(z).
2. Using
A
t
(z), we then solve eq. (2.3) and find the solution for
g(z) in terms of f
1
(z) and S(z).
3. Using
g(z), we then solve eq. (2.4)to obtain f
2
(z).
4. Next,
we solve eq. (2.5)and find φ
(z) in terms of S(z).
5. Finally,
we solve eq. (2.6) and obtain the dilaton potential in
terms of S(z) and g(z).
Applying
the above strategy and solving eq. (2.8), we get the fol-
lowing
solution for A
t
A
t
(z) = K
1
z
0
dξ
ξ
e
−B
2
ξ
2
f
1
(ξ)
√
S(ξ )
+
K
2
. (2.10)
Applying the boundary condition (eq. (2.9)), we get
K
2
=μ, K
1
=−
μ
z
h
0
dξ
ξe
−B
2
ξ
2
f
1
(ξ )
√
S(ξ )
, (2.11)
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