扭曲额外维度中的希格斯暗物质：截断惰性双峰模型

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"这篇学术论文探讨了一种基于扭曲额外维度的物理模型，称为"截断惰性双峰模型"，该模型与希格斯粒子和暗物质相关。研究使用了5维(5D)的ℤ2对称结构，通过三个D3骨架构建，其中两个具有负张力的IR（红外）维度位于超维区间的一端，一个具有正张力的UV（紫外）骨架置于区间中间，形成IR-UV-IR模型。在构建过程中，未考虑物质场的后反应影响。文章进一步介绍了如何利用5D SU(2) Higgs双态作为Goldberger-Wise稳定场，通过超势方法解决5D耦合标量重力方程。在这样的几何配置下，研究了5D标准模型（SM）玻色子零模在低能下的有效理论。这里的2ℤ2-偶零模对应于标准模型已知的自由度，而2ℤ2-奇数零模则可能作为暗物质的组成部分。在低能有效的标量扇区中，发现了一个模拟SM希格斯玻色子的标量，以及一个稳定的第二标量粒子（暗希格斯），它是暗物质的潜在候选者。暗希格斯是Z2奇数希格斯双峰的零模成分，这使得模型具备了双峰结构，可能影响暗物质的性质和探测。文章经过四次编辑过程，最终于2015年10月6日发表在JHEP期刊上，由Springer为SISSA出版。作者团队来自华沙大学、加利福尼亚大学戴维斯分校和巴基斯坦的国家物理中心，他们通过电子邮件进行联系并讨论了这个复杂且创新的物理模型，探讨了额外维度如何影响希格斯机制和暗物质的存在及特性。" 此篇论文的贡献在于提出了一种新的理论框架，将标准模型与暗物质理论相结合，通过额外维度的数学模型来解释希格斯粒子和暗物质的性质，这有助于深化对宇宙基本组成和相互作用的理解。此外，它还展示了如何在弦理论或膜理论的背景下，使用高级数学工具来解决物理学中的复杂问题。

JHEP10(2015)033

compact with boundaries, the action contains the Gibbons-Hawking boundary term,

= −

∂M

−ˆgK, (2.2)

where K is the intrinsic curvature of the surface of the boundary manifold ∂M, given by

K = −ˆg

µν

∇

= ˆg

µν

, (2.3)

with n

being the unit normal vector to the surface of the boundary manifold ∂M and ˆg

µν

is the induced boundary metric. For the 5D manifold with 4D Poincar´e invariance (n

= 1

and n

= 0), the intrinsic curvature reduces to

K = −

ˆg

µν

∂

ˆg

µν

. (2.4)

The solution of the Einstein equations resulting from the above action is the RS metric (1.1),

where the AdS curvature k is related to Λ

= −6k

. (2.5)

Since the above setup is compactiﬁed on an interval y ∈ [−L, L], rather than on a

circle as in RS1, one needs to be careful and show that the solution (1.1) is compatible

with the boundaries and that the eﬀective 4D cosmological constant is zero, see also [45].

We will see below that we need a ﬁne tuning between the 5D cosmological constant Λ

and the brane tensions λ

UV,IR

in order to get zero 4D cosmological constant. One can

calculate the eﬀective 4D cosmological constant Λ

from the action (2.1) by integrating

out the extra-dimension,

= −

−L

√

−g



− Λ

− λ

δ(y) − λ



δ(y + L) + δ(y − L)





−ˆgK



−L

(2.6)

where R = −20A

− 8A

and Λ

= −6A

corresponding to the solution (1.1). Using

A(y) = −k|y| we ﬁnd,

= (λ

− 6k) + (λ

+ 6k) e

−4kL

, (2.7)

which can only be zero if

= −λ

= 6k. (2.8)

This result explicitly shows that one needs a positive tension brane at y = 0 and two

negative tension branes at y = ±L in order to obtain zero 4D cosmological constant. This

is the usual ﬁne tuning which appears in brane world scenarios [1, 3, 4]. Hence we have

a 5D geometry with AdS solution (1.1) with negative bulk cosmological constant and a

positive tension brane in the middle and two equal negative tension branes at the end of

the interval, see ﬁgure 2.

We would like to mention here that we are considering a rigid IR-UV-IR geometry

where the distance L is tuned in order to solve the hierarchy problem. To stabilize the

The Gibbons-Hawking boundary term is needed in order to cancel variation of the Ricci scalar at the

boundaries so that the RS metric (1.1) is indeed a solution of the Einstein equations of motion.

– 6 –

JHEP10(2015)033

IR-UV-IR setup one may consider a mechanism like the one proposed by Goldberger and

Wise (GW) [3, 4] introducing a bulk scalar ﬁeld with appropriate brane potentials such that

the energy minimization would set the size of the 5D interval and yields a compactiﬁcation

scale that would solve the hierarchy problem. Since one of our aims is to analyze EWSB

due to a 5D SU(2) Higgs doublet in the IR-UV-IR model, therefore it is natural to consider

the bulk SU(2) Higgs doublet as the GW stabilizing ﬁeld. We explore this option in the

following subsection. A similar analysis for the case of RS1 has been considered in ref. [21].

However, there the full scalar-gravity coupled equations are not solved, the authors adopt

a small backreaction anstaz. In this work, we are using so-called superpotential method to

solve the full scalar-gravity coupled equations analytically.

2.2 The IR-UV-IR model: with backreaction

In this subsection we employ an SU(2) Higgs doublet in the bulk of IR-UV-IR model and

obtain the background solutions for the 5D scalar-gravity coupled theory. Although our

solution generating technique is adopted for a speciﬁc geometric conﬁguration (IR-UV-IR

model), this approach can be used to solve the Higgs-gravity backreaction in any warped

extra-dimensional model (e.g. RS1) with a bulk Higgs. We use the following most general

4D Poincar´e invariant metric ansatz:

= e

2A(y)

µν

+ dy

, (2.9)

where A(y) is a general y-dependent warp-function and η

µν

represents the 4D Minkowski

metric. We consider the following scalar-gravity action for our model,

√

−g



− |D

− V

(H)

− V

(H)δ(y) − V

(H)

δ(y + L) + δ(y − L)



+ S

, (2.10)

where R is the 5D Ricci scalar and H is the SU(2) Higgs doublet. Whereas, V

(H) and

UV(IR)

(H) are the bulk and UV(IR) brane potentials, respectively. Above, the S

is the

Gibbons-Hawking boundary action deﬁned in eq. (2.2).

We can write the y-dependent vacuum expectation value (vev) of the SU(2) Higgs

doublet as:

hHi =

√

φ(y)

. (2.11)

In order to ﬁnd the background solutions we need to solve the coupled scalar-gravity

Einstein equations. The equations for the background ﬁelds A(y) and φ(y) following from

the above action (2.10) and the metric ansatz (2.9) can be written as,

− V

(φ), (2.12)

+ 6A

= −

− V

(φ) − V

(φ)δ(y) − V

(φ)

δ(y + L) + δ(y − L)

, (2.13)

+ 4A

∂V

(φ)

∂φ

∂V

(φ)

∂φ

δ(y) +

∂V

(φ)

∂φ

δ(y + L) + δ(y − L)

. (2.14)

– 7 –

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