Randall-Sundrum模型：无重影无超驰的稳定区域探索

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"Randall-Sundrum模型参数空间的无重影和无超驰区域" Randall-Sundrum（RS）模型是由Lisa Randall和Randy Sundrum提出的，它是一种五维弦理论或膜理论的变体，旨在解决引力与量子力学之间的矛盾。在这个模型中，我们的四维宇宙被设想为一个在五维弯曲时空中移动的四维“膜”（或者称为brane）。这个五维时空，被称为Randall-Sundrum II模型，包括一个正曲率的五维空间（Bulk）和两个平行的四维膜，其中一个（称为TeV brane）具有较高的能量尺度，通常与标准模型粒子相关联，另一个是Planck brane，对应着普朗克尺度的物理。在RS模型的构建中，标准模型的粒子被假设存在于Bulk中，而不是局限于四维膜上。这些粒子可能拥有非零的brane-localized kinetic terms (BLKTs)，即在brane上的局域动能项，以及可能的bulk mass parameters，即Bulk中的质量参数。这些额外的自由度引入了新的物理效应，但也可能导致理论不稳定性的新问题，例如幻影（Ghost）和测速不稳定性（Tachyonic instability）。幻影是一种负能量密度的虚粒子，它的存在会破坏理论的稳定性。而测速不稳定性则是指粒子的有效速度超过光速，这违反了相对论的基本原理。在这篇论文中，作者G.N. Wojcik, J.L. Hewett和T.G. Rizzo深入研究了RS模型参数空间，明确了哪些区域可以避免这两种不稳定状态。他们通过数学分析和物理推理，确定了允许大量质量项和BLKT存在而不导致不稳定性的参数范围。这个无重影和无超驰的参数空间对于构建现实且在现象学上可行的RS模型至关重要。因为只有在这些区域内构建的模型才能避免理论的内在矛盾，并且能够解释观测到的物理现象。例如，RS模型被用来探讨如何在额外维度中解决自然性问题，如hierarchy problem，以及可能对粒子物理的标准模型给出新颖的解释和预言。关键词：高维场论的物理现象学这篇论文是在2018年由G.N. Wojcik等人发表在JHEP07上的，并被Springer出版。它被归类为Open Access，意味着公众可以免费获取全文，进一步促进了科学知识的传播和研究交流。ArXiv上的预印本编号为1802.05731，表明了该研究在正式出版前已经在学术社区中进行了分享和讨论。

JHEP07(2018)117

where J

(x) and Y

(x) are the order-q Bessel functions of the ﬁrst and second kind, respec-

tively. The variable z

≡

is a function of φ and the mass of the KK mode described

by the index n, given by m

. The normalization constant N

is given by

−π

dφ(1 + ∆

,τ

(n)

(φ)f

(m)

(φ) =

−π

dφe

(n)

(φ)f

(m)

(φ) = δ

. (2.10)

Finally, the constants α

and β

in eq. (2.9) are given by boundary conditions on the UV-

brane (determined by integrating eqs. (2.7) over an inﬁnitesimal interval of φ about φ = 0)

≡ Y

−

−ν

(x

) + τ

x

−ν

(x

), (2.11)

≡ −

−

−ν

(x

) + τ

x

−ν

(x

)

Here, it is convenient to employ the value of z

evaluated at the TeV-brane (φ = π), i.e.,

. Then, z

evaluated at the UV-brane is given by x

, where  ≡ e

−kr

. To ﬁnd the set

of allowed values of x

, and hence the masses of KK tower modes, we must ﬁnd the roots

of the TeV-brane boundary condition equation

−

−ν

) − τ

−ν

) = 0, (2.12)

with x

being the roots of this equation. The masses of the particles in the KK tower

are then m

= x

k. It should also be noted that, in the absence of spontaneous sym-

metry breaking, the even ﬁeld also possesses a massless zero-mode solution, given by the

(normalized) wave equation,

(0)

(φ) =

1 + 2ν

(1 + (1 + 2ν)τ

(1+2ν)kr

− (1 − (1 + 2ν)τ

)

νσ

. (2.13)

In this work, we will determine which values of τ

, τ

, and ν are permitted based on

a set of physical conditions. First, all solutions of eq. (2.12) (i.e., the roots x

), which

yields the mass spectrum of the Kaluza-Klein tower, must be real. Otherwise, the theory

would predict the existence of phenomenologically unviable states (tachyons, for purely

imaginary solutions, or fermions with complex masses squared, for general complex solu-

tions). Secondly, we require the absence of so-called “ghost” states, which are states with

negative probability, as indicated by negative values for the square of the Kaluza-Klein

mode’s normalization. As is standard practice in the literature [19–21, 25, 26], we limit our

discussion of the existence of ghost states to the zero-mode f

(0)

, which yields the condition

1 + 2ν

(1 + (1 + 2ν)τ

)

−

(

+ν

)

− (1 − (1 + 2ν)τ

)

(

+ν

)

> 0, (2.14)

to avoid ghosts.

3 Analysis

Having set up the basic machinery, and in particular established the conditions in eqs. (2.12)

and (2.14) to judge the physicality of a point in parameter space, we begin our analysis

– 5 –

JHEP07(2018)117

by addressing the speciﬁc case that frequently bedevils bulk ﬁelds in theories of extra

dimensions, namely the existence of tachyonic (purely imaginary) Kaluza-Klein masses [19].

In the following sections, we address the conditions under which tachyonic modes do not

appear while the no-ghost condition of eq. (2.14) is simultaneously satisﬁed. Later, we

demonstrate that in the absence of spontaneous symmetry breaking, Kaluza-Klein modes

can only appear with purely real or purely imaginary masses, indicating that our analysis

here, where ghosts and purely imaginary masses are avoided, produces a complete picture

of the allowable parameter space of the model.

3.1 Study of the boundary value equation

First, in an eﬀort to simplify the algebra, we introduce a slightly more convenient fermion

localization parameter, η, by deﬁning

η ≡ −



+ ν



. (3.1)

The no-ghost condition eq. (2.14) then becomes

≡

−2η

(1 − 2ητ

)

− (1 + 2ητ

)

−η

> 0. (3.2)

Meanwhile, the boundary value equation in eq. (2.12) evaluated on the imaginary line

becomes (where we have taken x → ix in eq. (2.12), implying that x in the expression

below is real)

J(η, τ

, ix)Y(η, −τ

, ix) − Y(η, τ

, ix)J(η, −τ

, ix) = 0,

J(η, τ, ix) ≡ J

(ix) − ixτJ

1+η

(ix), (3.3)

Y(η, τ, ix) ≡ Y

(ix) − ixτY

1+η

(ix).

If this equation has a root at some ix, then, it denotes the existence of a KK mode with

a tachyonic mass proportional to this value of ix. The expression can be expanded in a

double power series using the identities,

(ix) =





∞

k=0





k!Γ(1 + k + η)

, (3.4)

(ix) = cot(ηπ)J

(ix) − csc(ηπ)J

−η

(ix),

and then takes the general form (which we deﬁne as f(x))

f(x) ≡

πη

∞

k=0

j=0













(1 + 2(k − j − η)τ

)(1 − 2jτ

)Γ(1 − η)Γ(1 + η)

Γ(1 + k − j − η)Γ(1 + j + η)

−



−η

(1 − 2(j −η)τ

)(1 + 2(k − j)τ

)Γ(1 − η)Γ(1 + η)

Γ(1 + k − j + η)Γ(1 + j − η)



= 0. (3.5)

We now assume that x is not hierarchically large (i.e., x  

−1

). This is motivated

by the fact that the RS model is assumed to be a low-energy approximation of some UV-

complete theory, and hence subject to an ultraviolet cutoﬀ. Otherwise, a hierarchically

– 6 –

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