---
title: The Hierarchy Problem and Naturalness
module: Beyond the Standard Model
moduleNumber: 12
lessonNumber: 4
order: 1204
summary: >
  The electroweak scale sits sixteen orders of magnitude below the Planck scale,
  and nothing in the Standard Model protects that gap. The Higgs mass squared picks
  up quadratic corrections proportional to the highest scale in the theory, so
  keeping it at the observed value requires the bare mass and its counterterm to
  cancel to some thirty significant figures. Naturalness treats that cancellation
  as a symptom of missing physics. Supersymmetry, compositeness, and extra
  dimensions each remove the quadratic sensitivity, but the LHC has found none of
  them at the predicted scale.
topics: [Beyond the Standard Model]
draft: false
sources:
  - book: Thomson
    ref: "Ch. 18 — The Standard Model and beyond (the hierarchy problem)"
  - book: Griffiths
    ref: "Ch. 12 — Afterword (open problems, naturalness)"
---

The Standard Model contains two vastly separated mass scales with no dynamical
connection between them: the electroweak scale $v \approx 246$ GeV, set by the Higgs
vacuum expectation value, and the Planck scale $M_{\text{Pl}} \approx 1.2\times10^{19}$
GeV, where gravity becomes strong. Their ratio is

$$
\frac{M_{\text{Pl}}}{v} \approx 5\times10^{16}.
$$

For every fermion and gauge mass this hierarchy is stable, protected by a symmetry
that forbids the mass when it is set to zero. The Higgs mass has no such protection,
and quantum corrections drag it toward the highest scale in the theory. Keeping it
light then demands a cancellation of extraordinary precision. This is the
**hierarchy problem**, and the demand that no such cancellation be needed is the
**naturalness** criterion.

## Why scalar masses are unprotected

Fermion and gauge-boson masses are shielded by symmetries.

- A fermion mass term $m\bar\psi\psi$ couples left- and right-handed fields. Setting
  $m = 0$ restores a **chiral symmetry** under which the two chiralities rotate
  independently. Radiative corrections respect the symmetry, so they can only be
  proportional to $m$ itself: $\delta m \propto m\,\ln(\Lambda/m)$, logarithmic and
  multiplicative. A small fermion mass stays small.
- A gauge-boson mass is forbidden by gauge invariance and generated only through
  spontaneous symmetry breaking, so it is tied to $v$ and inherits the same
  protection.

A scalar mass term $m_H^2\,|H|^2$ is invariant under no symmetry that is restored at
$m_H = 0$. There is no scalar analogue of chiral symmetry. Corrections are therefore
**additive** rather than multiplicative, and they grow with the cutoff.

## The quadratic correction

The one-loop correction to the Higgs mass squared from a fermion of Yukawa coupling
$\lambda_f$ is

$$
\delta m_H^2 = -\frac{\lambda_f^2}{8\pi^2}\,\Lambda^2 + \big(\text{logarithms}\big),
$$

where $\Lambda$ is the scale up to which the Standard Model is valid. The top quark,
with $\lambda_t \approx 1$, dominates. If the theory holds up to the Planck scale,
$\Lambda = M_{\text{Pl}}$ and

$$
|\delta m_H^2| \sim \frac{1}{8\pi^2}\,(10^{19}\ \text{GeV})^2 \approx 10^{36}\ \text{GeV}^2,
$$

thirty-two orders of magnitude above the physical value
$m_H^2 = (125\ \text{GeV})^2 \approx 1.6\times10^{4}\ \text{GeV}^2$.

$$
% caption: The electroweak scale and the Planck scale on a logarithmic ruler. The
% Higgs mass sits at the bottom, but its quantum corrections are set by the top of
% the ruler, sixteen orders of magnitude higher.
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  \node[below, font=\scriptsize] at (6.0,-0.2) {$10^{14}$};
  \node[below, font=\scriptsize] at (9.0,-0.2) {$10^{19}$ GeV};
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  \node[acc, above, font=\scriptsize] at (4.5,1.0) {sixteen orders of magnitude};
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$$

$$
% caption: The top-quark loop correcting the Higgs mass squared. Its magnitude is
% set by the square of the cutoff, so if the Standard Model holds to the Planck
% scale the correction dwarfs the observed Higgs mass.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[thick, dashed] (-1.4,0) -- (-0.5,0);
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  \node[right, font=\scriptsize, align=left] at (4.8,0) {grows as the\\scale squared};
\end{tikzpicture}
$$

## Fine-tuning

The physical Higgs mass is the sum of a bare parameter $m_0^2$ and the correction,

$$
m_H^2 = m_0^2 + \delta m_H^2 .
$$

Both terms are of order $\Lambda^2 \sim 10^{36}\ \text{GeV}^2$, yet their sum is
$10^{4}\ \text{GeV}^2$. The two must cancel to

$$
\frac{m_H^2}{\delta m_H^2} \sim \frac{10^{4}}{10^{36}} = 10^{-32},
$$

so $m_0^2$ and $\delta m_H^2$ agree in their first thirty-two significant figures and
differ only in the thirty-third. Nothing in the theory relates the bare parameter,
fixed at the cutoff, to the loop correction, computed from the low-energy couplings.
The cancellation is therefore an unexplained coincidence.

> **Definition (Naturalness).** A parameter is natural if setting it to zero
> increases the symmetry of the theory, so that its smallness is protected against
> radiative corrections. A theory is fine-tuned when a physical quantity is small
> only because of a precise cancellation among terms that are individually far
> larger, with no symmetry enforcing the cancellation.

The degree of tuning is quantified by the sensitivity of an observable to the
fundamental parameters; a cancellation to one part in $10^{32}$ corresponds to a
tuning of that order. Naturalness does not forbid such a cancellation — nature may
simply be tuned — but treats it as evidence that new physics enters near the
electroweak scale to cut off the quadratic growth before it reaches $M_{\text{Pl}}$.

$$
% caption: The observed Higgs mass squared is the difference of a bare term and a
% loop correction, each of order the cutoff squared. Their near-perfect cancellation
% to thirty-two digits, with no symmetry enforcing it, is the fine-tuning.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0,
  bar/.style={draw, minimum width=13mm}]
  \definecolor{acc}{HTML}{4A6FA5}
  % two tall bars nearly equal
  \draw (0,0) rectangle (1.3,3.8);
  \node[below, font=\scriptsize, align=center] at (0.65,0) {bare\\$m_0^2$};
  \draw (2.2,0) rectangle (3.5,3.78);
  \node[below, font=\scriptsize, align=center] at (2.85,0) {loop\\correction};
  \node[font=\normalsize] at (4.3,1.9) {$=$};
  % tiny remainder bar
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  \node[acc, right, font=\scriptsize, align=left] at (6.7,0.5) {residue $= m_H^2$\\(1 part in $10^{32}$)};
  \node[above, font=\scriptsize, align=center] at (1.75,3.95) {each of order\\the scale squared};
\end{tikzpicture}
$$

## Proposed solutions

Every natural resolution removes the quadratic sensitivity by introducing new
physics near the TeV scale.

- **Supersymmetry.** Each fermion loop is paired with two scalar loops of opposite
  sign, cancelling the $\Lambda^2$ term exactly and leaving a logarithm proportional
  to the superpartner mass splitting. Naturalness then requires superpartners near a
  TeV, the version now in tension with LHC limits, treated in the
  [supersymmetry lesson](/particle-physics/beyond-standard-model/supersymmetry).
- **Compositeness.** The Higgs is not elementary but a bound state of new strongly
  interacting constituents, with a size $\sim 1/\Lambda_{\text{comp}}$. Above the
  compositeness scale the Higgs dissolves and the loop integral is cut off
  physically, as the finite proton size cuts off its electromagnetic self-energy.
  The Higgs is then a pseudo-Goldstone boson of a broken global symmetry, naturally
  lighter than the compositeness scale.
- **Extra dimensions.** If space has additional dimensions in which gravity
  propagates, the fundamental gravitational scale can be near a TeV, and the apparent
  weakness of gravity — the large $M_{\text{Pl}}$ — is a geometric dilution rather
  than a true scale. The hierarchy is then removed because there is no large gap: the
  cutoff $\Lambda$ sits at the TeV scale.

$$
% caption: The hierarchy problem and its three leading resolutions, each cutting off
% the quadratic Higgs-mass growth at a TeV by a different mechanism: superpartner
% cancellation, a composite Higgs, or a lowered gravitational scale.
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$$

## Status after the LHC

The LHC was built at the scale where naturalness predicted new physics, and it found
the Higgs boson at $125$ GeV but no superpartners, no composite resonances, and no
extra-dimensional signatures up to a few TeV. The direct searches push the cutoff
$\Lambda$ upward, which reintroduces a residual tuning: even if new physics enters at
$2$ TeV rather than $M_{\text{Pl}}$, the Higgs mass is now tuned at the percent level,

$$
\frac{m_H^2}{\delta m_H^2}\bigg|_{\Lambda = 2\,\text{TeV}} \sim \frac{(125)^2}{(2000)^2} \sim 4\times10^{-3},
$$

a "little hierarchy" between the Higgs mass and the scale of the physics that is
supposed to stabilize it. Three responses divide the field: that supersymmetry or
compositeness lies just beyond current reach, that naturalness is not a reliable
guide and the electroweak scale is set anthropically or by a landscape of vacua, or
that the resolution takes a form not yet imagined. The hierarchy problem remains the
central unresolved tension of the Standard Model, unlike the neutrino masses it is a
problem of theoretical consistency rather than of direct observation.[^th-hier][^gr-hier]

## Summary

The electroweak scale lies $16$ orders of magnitude below the Planck scale, and the
Higgs mass, alone among the Standard Model masses, is unprotected by any symmetry.
Its quadratic correction $\delta m_H^2 \sim \Lambda^2$ reaches $10^{36}\ \text{GeV}^2$
if the theory holds to $M_{\text{Pl}}$, forcing the bare mass and the correction to
cancel to one part in $10^{32}$. Naturalness reads that cancellation as a signal of
new physics near a TeV. Supersymmetry, compositeness, and extra dimensions each
remove the quadratic sensitivity, but the LHC has excluded the simplest versions,
leaving a residual little-hierarchy tuning and an open problem.[^th-hier][^gr-hier]

[^th-hier]: The hierarchy problem, the quadratic Higgs-mass correction, the fine-tuning estimate, and the menu of solutions (supersymmetry, compositeness, extra dimensions) are set out in Thomson, Ch. 18. The naturalness criterion — that a small parameter should be protected by an enhanced symmetry — is due to 't Hooft (1980).

[^gr-hier]: Griffiths, Ch. 12 (Afterword), lists the naturalness of the Higgs mass among the Standard Model's open problems and the post-LHC status of the proposed extensions.