[Paper Review] Living Without Supersymmetry -- the Conformal Alternative and a Dynamical Higgs Boson

Philip D. Mannheim|arXiv (Cornell University)|Jun 1, 2015

Cosmology and Gravitation Theories4 citations

TL;DR

This paper proposes a conformal, non-supersymmetric alternative to the Standard Model in which the Higgs boson arises as a dynamical fermion-antifermion bound state rather than an elementary scalar, resolving the hierarchy problem without quadratic divergences. Through a conformally invariant gauge theory with anomalous dimensions and fermion condensates, the theory generates a narrow Higgs resonance just above threshold, with a calculable width that could distinguish it from an elementary Higgs, while conformal gravity resolves the vacuum energy problem.

ABSTRACT

We show that key results of supersymmetry can be achieved via conformal symmetry. We propose that the Higgs boson be a dynamical bound state rather than an elementary scalar, so that there is no quadratic divergence self-energy problem for it and no need to invoke supersymmetry to resolve it. We study a conformal invariant theory of interacting fermions and gauge bosons, in which there is scaling with anomalous dimensions and dynamical symmetry breaking, with the dynamical dimension of $\\bar{\\psi}\\psi$ being reduced from 3 to 2. With this reduction we augment the theory with a then renormalizable 4-fermion interaction with dynamical dimension equal to 4. We reinterpret the theory as a renormalizable version of the Nambu-Jona-Lasinio (NJL) model, with the gauge theory sector with its now massive fermion being the mean field and the 4-fermion interaction being the residual interaction. It is this residual interaction that generates dynamical Goldstone and Higgs states, states that, as noted by Baker and Johnson, the gauge theory sector itself does not possess. The Higgs boson is found to be a narrow resonance just above threshold. We couple the theory to conformal gravity, with the interplay between conformal gravity and the 4-fermion interaction taking care of the vacuum energy problem. With conformal gravity being a consistent quantum gravity theory there is no need for string theory with its supersymmetric underpinnings. With conformal gravity fits to galactic rotation curves and the accelerating universe not needing dark matter, there is no need to introduce supersymmetry for either the vacuum energy problem or to provide a potential dark matter candidate. We propose that it is conformal symmetry rather than supersymmetry that is fundamental, with the theory of nature being a locally conformal, locally gauge invariant, non-Abelian NJL theory.

Motivation & Objective

• To resolve the hierarchy problem without relying on supersymmetry by eliminating quadratic divergences in the Higgs self-energy.
• To demonstrate that a dynamical Higgs boson can emerge from a conformally invariant gauge theory with fermion condensates.
• To show that the Higgs can be a narrow resonance just above the fermion pair production threshold, with a width that could distinguish it from an elementary Higgs.
• To address the cosmological vacuum energy problem via conformal gravity, avoiding the need for supersymmetry or dark matter.
• To propose conformal symmetry as the fundamental symmetry of nature, replacing supersymmetry, with a locally conformal, non-Abelian Nambu-Jona-Lasinio model as the underlying theory.

Proposed method

• Construct a conformal invariant gauge theory of interacting fermions and gauge bosons, with anomalous scaling in the ultraviolet to maintain conformal symmetry.
• Implement dynamical symmetry breaking via fermion bilinear condensates in the infrared, reducing the dynamical dimension of $\bar{\psi}\psi$ from three to two.
• Use the reduced dimension to render a four-fermion interaction renormalizable, reinterpretating the theory as a renormalizable Nambu-Jona-Lasinio model.
• Identify the residual four-fermion interaction as the source of dynamical Goldstone and Higgs states, not the mean-field gauge theory sector.
• Perform a full one-loop calculation of the scalar self-energy $\Pi_{\rm S}(q^2, M)$, including Wick-ordered, compositeness, and cut contributions, with careful analytic continuation to timelike momenta.
• Solve the gap equation $\hat{\Pi}_{\rm S}(q^2, M) = 0$ numerically to find the pole position $q^2 = (2.189 - 0.051i)M\mu$, yielding a resonance with a small width $\Gamma = 0.017i(M\mu)^{1/2}$.

Experimental results

Research questions

• RQ1 Can a dynamical Higgs boson emerge in a conformally invariant, non-supersymmetric gauge theory without quadratic divergences?
• RQ2 Can the Higgs boson be a narrow resonance just above the fermion pair threshold, with a calculable width that distinguishes it from an elementary Higgs?
• RQ3 Can conformal symmetry alone resolve the hierarchy problem and the vacuum energy problem without requiring supersymmetry?
• RQ4 Does the interplay between conformal gravity and a four-fermion interaction naturally cancel the cosmological constant?
• RQ5 Can the Nambu-Jona-Lasinio model be made renormalizable via dynamical dimension reduction from conformal symmetry?

Key findings

• The Higgs boson is found to be a narrow resonance with a complex pole at $q^2 = (2.189 - 0.051i)M\mu$, corresponding to a mass of $q_{ m R} = 1.480(M\mu)^{1/2}$ and a width $\Gamma = 0.017i(M\mu)^{1/2}$.
• The imaginary part of the self-energy vanishes at threshold due to cancellation between singular terms, ensuring no unphysical divergences and confirming the resonance lies above threshold.
• The full scalar propagator exhibits a Breit-Wigner form with a width $\Gamma \approx 0.017(M\mu)^{1/2}$, consistent with unitarity and a narrow resonance.
• The six contributions to $\hat{\Pi}_{\rm S}(q^2, M)$—Wick low, Wick high, gap region, compositeness, cut, and imaginary part—sum to zero at the pole, confirming self-consistency.
• The resonance mass is close to the threshold value of $1.414(M\mu)^{1/2}$, indicating it is just above the fermion pair production threshold.
• The width $\Gamma$ is small and positive, with the sign correctly chosen by unitarity, and is fixed by the cancellation between real and imaginary parts of the self-energy.