All possible models to explain the hierarchy problem?

[Suekdccia]

TL;DR

All possible models to explain the hierarchy problem?

There is an interesting paper by Arkani-Hamed and collaborators (https://arxiv.org/abs/1607.06821) to address the hierarchy problem.

There, they consider many possible models of fundamental particle physics where they all have an exact copy of the Standard Model but with different Higgs masses.

However, they indicate that this assumption is done for simplicity, but that we could relax these assumptions and consider models with greater differences

At the beginning, they say:

The first step is to introduce N sectors which are mutually non-interacting. The detailed particle content of these sectors is unimportant, with the exception that the Standard Model (SM) should not be atypical; many sectors should contain scalars, chiral fermions, unbroken gauge groups, etc. For simplicity, we imagine that they are exact copies of the SM, with the same gauge and Yukawa structure.

And at the end:

However, it is easy to imagine a broader class of theories that realizes the same mechanism. We can relax the assumption that the Higgs masses are uniformly spaced (or even pulled from a uniformly distribution) or that all the new sectors are exact copies of the SM. It is also possible to construct different models of reheating, with new physics near the weak scale to modify the UV behavior of the theory.All of this made me wonder: Are they saying that we could relax the assumptions of the model so that it would even include all possible UV theories (meaning all possible "microscopic" or "high-energy" physics, such as the different theories of quantum gravity and theories of everything proposed so far)?

 

[ChrisF]

TL;DR: All possible models to explain the hierarchy problem?

There is an interesting paper by Arkani-Hamed and collaborators (https://arxiv.org/abs/1607.06821) to address the hierarchy problem.

There, they consider many possible models of fundamental particle physics where they all have an exact copy of the Standard Model but with different Higgs masses.

However, they indicate that this assumption is done for simplicity, but that we could relax these assumptions and consider models with greater differences

At the beginning, they say:

The first step is to introduce N sectors which are mutually non-interacting. The detailed particle content of these sectors is unimportant, with the exception that the Standard Model (SM) should not be atypical; many sectors should contain scalars, chiral fermions, unbroken gauge groups, etc. For simplicity, we imagine that they are exact copies of the SM, with the same gauge and Yukawa structure.

And at the end:

However, it is easy to imagine a broader class of theories that realizes the same mechanism. We can relax the assumption that the Higgs masses are uniformly spaced (or even pulled from a uniformly distribution) or that all the new sectors are exact copies of the SM. It is also possible to construct different models of reheating, with new physics near the weak scale to modify the UV behavior of the theory.All of this made me wonder: Are they saying that we could relax the assumptions of the model so that it would even include all possible UV theories (meaning all possible "microscopic" or "high-energy" physics, such as the different theories of quantum gravity and theories of everything proposed so far)?

Re: All possible models to explain the hierarchy problem?

The short answer is: not quite, and the reason why is instructive.

Nnaturalness works because it imposes a specific constraint — the SM should not be atypical among the N sectors. This is doing real physical work. The reheaton selects the lightest sector precisely because the distribution of Higgs
masses is bounded and the SM sits at a predictable location within it. If you relax the assumptions all the way to "any possible UV theory," you lose that selection mechanism. The model becomes unfalsifiable — you can always invoke
some sector configuration that produces any observed output. That's not a solution to the hierarchy problem; it's a restatement of it in a larger space.

What Arkani-Hamed et al. mean by "relaxing assumptions" is more modest: you can vary the spacing of Higgs mass parameters, use non-uniform distributions, or modify the reheating mechanism — but the sectors still need to share enough
structure with the SM that the selection principle operates. It's not an invitation to include arbitrary UV completions.

On whether it encompasses all theories of quantum gravity / TOEs:
No — and deliberately so. Nnaturalness is agnostic about quantum gravity precisely because it doesn't need to solve that problem. It solves the weak-scale hierarchy problem without touching the Planck scale. String theory, LQG,
asymptotic safety — none of these are required or excluded by the mechanism. They live at a different energy scale and a different problem.

The broader question worth sitting with:
Every proposed solution to the hierarchy problem — SUSY, extra dimensions, composite Higgs, relaxion, Nnaturalness — assumes the Planck scale is the natural reference point and then asks why the Higgs is so much lighter. But that
assumption is never derived; it's inherited from dimensional analysis in QFT.

An alternative framing: what if the Higgs mass isn't unnaturally light relative to some UV scale, but is instead the natural output of the vacuum's geometric properties — the permittivity and permeability of empty space that determine
how fields propagate and where they stabilize? In that case there is no hierarchy problem to solve, because the Planck scale was never the right reference. The problem dissolves rather than gets explained.

That's a minority position, but it's worth noting that every solution to the hierarchy problem requires new physics that hasn't been observed, while the vacuum already has two directly measurable properties that set every
electromagnetic scale in nature.

Christian Fuccillo

 

[sdoygb]

TL;DR： 所有可能的模型都能解释层级问题吗？

Arkani-Hamed 及其合作者撰写了一篇有趣的论文（https://arxiv.org/abs/1607.06821），探讨了层次结构问题。

在那里，他们考虑了许多可能的基本粒子物理模型，这些模型都与标准模型完全相同，只是希格斯粒子的质量不同。

然而，他们指出，做出这一假设是为了简化模型，但我们可以放宽这些假设，并考虑差异更大的模型。

一开始，他们说：

第一步是引入N个互不相互作用的扇区。这些扇区的具体粒子组成并不重要，但标准模型（SM）不应是异常的；许多扇区应包含标量、手征费米子、未破缺的规范群等等。为简单起见，我们假设它们是标准模型的精确副本，具有相同的规范结构和汤川结构。

最后：

然而，我们很容易想象出一类更广泛的理论来实现相同的机制。我们可以放宽希格斯玻色子质量均匀分布（甚至均匀分布）的假设，或者放宽所有新扇区都是标准模型精确副本的假设。我们也可以构建不同的重加热模型，引入弱相互作用尺度附近的新物理来修正该理论的紫外行为。所有这些都让我不禁思考：他们是不是在说，我们可以放宽模型的假设，使其涵盖所有可能的紫外理论（即所有可能的“微观”或“高能”物理，例如迄今为止提出的各种量子引力理论和万物理论）？

关于 Arkani-Hamed 等人的论文：一旦将假设放宽到“任何具有标量和规范群的紫外理论”，该机制就变成了一种统计上的同义反复。只要有足够多的扇区和足够多的希格斯质量扫描，最终总会偶然发现一个弱标度扇区。这固然精妙，但它用一个微调问题换来了一个人择选择问题。
一个更受约束的替代方案是具有有限乘法预算的单曲面分层生成。考虑一种降维方法，其中有效作用曲面由刚度比参数化：
λ2/λ₁ = 150 = 2 × 3 × 5²
这种质因数分解恰好提供了三个独立的几何生成器。该层次结构是通过对表面积进行七层重新缩放构建的：
A_{n+1} = N_{n+1} · A_n
其中 N_n 是逆电磁耦合系数 α⁻¹ ≈ 137.036... 的十进制数字。七层编码后的总编码容量为 ln M_7 = 97.27，跨越了普朗克尺度到电弱尺度约 17 个数量级的差距。层数固定，因为乘法预算 2 × 3 × 5² 在三个素数生成器之后就已耗尽；不存在第八层。
同一表面预测了无需希格斯势调节的电弱尺度。质量量子为 K = 839.74 keV，由电子质量决定。W 玻色子和希格斯玻色子的质量源于质量映射因子 Ψ_m = √(λ₁·λ₂) :

m_W = 80.35 GeV (CDF 2022: 80.433 ± 0.094 GeV, 偏差 −0.10%)

m_H = 125.4 GeV (ATLAS/CMS：125.11 ± 0.11 GeV，偏差 +0.23%)
与多元宇宙理论的主要区别：

没有多个互不相干的扇区。

没有希格斯质量扫描或均匀分布。

不存在人为选择。弱尺度和电弱质量由具有零自由参数的单个联合作用面的几何形状预测得出。
这是否算作 Arkani-Hamed 意义上的“UV 完备化”？它是一个高能几何框架，但它预测弱尺度是基于有限的乘法预算，而非统计系综。我很好奇是否存在其他 BTSM 机制能够用有限的素因子预算来固定层级结构。
（完整推导：Zenodo DOI：10.5281/zenodo.20153652，第 4.2 节和 6.3.6 节。）