Consistent quantization of internal fermionic (particle-ghost) symmetry as a solution to the gauge hierarchy problem

Kawamura (Int. J. Mod. Phys. A 30, 1550153 (2015)) conjectured that theories with an exact internal fermionic symmetry — a particle-ghost symmetry whose Grassmann-odd generators pair each heavy field with an unphysical ghost partner of the same spin but reversed statistics — are free of the gauge hierarchy problem: Standard Model particles, as symmetry singlets, receive no destabilizing radiative corrections from heavy particles in symmetry doublets, with the cancellations demonstrated at one loop in toy models and a grand-unification prototype. Whether any such theory can be quantized consistently — unitary and causal, with a non-empty physical spectrum despite negative-norm ghost states and non-local star-product ghost self-interactions — and whether the cancellations survive to all perturbative orders without fine tuning, is unresolved. The problem — construct a nontrivial, unitary, causal quantum field theory with exact internal fermionic particle-ghost symmetry and all-orders hierarchy stabilization, or prove it impossible — remains open.

Background

The gauge hierarchy problem asks why the Higgs boson mass $m_H\approx125$ GeV stays far below the highest physical scales: quantum corrections to an elementary scalar's mass-squared grow quadratically with the cutoff $\Lambda$, so with $\Lambda$ as large as the Planck scale $M_{\rm Pl}\sim10^{19}$ GeV, keeping $m_H$ light appears to require fine tuning. Spacetime supersymmetry (SUSY), which pairs each boson with a fermion of equal mass, cancels these corrections order by order, but no superpartner has been observed. Kawamura (2015) reconsidered the problem under the assumption that the Standard Model plus massive neutrinos (called SM+$\alpha$) holds essentially up to a high scale $M_U$ such as $M_{\rm Pl}$, and asked what hidden symmetry $X$ could stabilize the hierarchy: the SM particles should be singlets of $X$, heavy particles should fill non-trivial $X$-multiplets, and $X$ should remain unbroken so that radiative corrections from heavy to light fields cancel.

The proposed $X$ is an internal fermionic symmetry, realized in toy models as a particle-ghost symmetry. Its Grassmann-odd generators $Q_F$, $Q_F^\dagger$ mix an ordinary field with a ghost partner carrying the same spin but opposite statistics — for example a complex scalar $\varphi$ is paired with an anticommuting scalar ghost $c_\varphi$ (a fermionic scalar field). The graded algebra, of $OSp(2|2)$ type (an orthosymplectic supergroup: bosonic $U(1)\times U(1)$ rotations of the particle and ghost sectors extended by fermionic particle-ghost transformations), also contains ghost-number charges and nilpotent fermionic charges. This resembles BRST symmetry in gauge theories — whose nilpotent charge likewise pairs physical with unphysical fields and whose physical states are selected by subsidiary conditions (the Kugo-Ojima construction) — but $Q_F$ is not generated by a hermitian operator and is not tied to a gauge redundancy. Because $c_\varphi$ has the same mass and couplings as $\varphi$ but reversed statistics, its loop contributions to a light field $\phi$ cancel those of $\varphi$: at one loop the mass-squared of $\phi$ receives no correction from the heavy doublet, mimicking the SUSY cancellation while the ghost partner is unphysical (a negative-norm state projected out of the spectrum).

Two structural features make consistency nontrivial. First, quantizing a wrong-statistics field produces negative-norm states; the proposed cure — subsidiary conditions on states projecting out the ghost sector — is known, from the author's follow-up work, to leave a physical spectrum that, for a theory built purely from doublets, is empty (only the vacuum state survives). A realistic theory needs symmetry singlets (the SM fields) coexisting with unphysical doublets. Second, ghost self-interactions cannot be written as ordinary local terms; they are introduced through a non-local star product, raising causality and renormalization questions. Whether a local quantum field theory with finitely many fields can accommodate the required ghost sectors, or whether an extended-object description is essential, is left open, the author suggesting that the fundamental objects may be extended rather than point-like. A prototype grand-unified model illustrates the payoff: heavy gauge bosons made unphysical by the symmetry would not mediate proton decay.

Adjacent proposals must be distinguished from this one. The Lee-Wick Standard Model cancels quadratic divergences using ghost partners of ordinary statistics but wrong-sign kinetic terms, a different mechanism. Free fields with reversed spin-statistics can be constructed mathematically via doubled Lorentz representations (Tóth, 2014), but this says nothing about interacting, symmetry-protected cancellations. What no construction has achieved is the combination the conjecture requires: an exact internal fermionic symmetry, a unitary causal quantum field theory with non-empty physical spectrum, and all-orders cancellation of the dangerous radiative corrections.

Problem Statement

The problem, as posed at the end of Kawamura (2015), is the consistency core of the conjecture: determine whether a quantum field theory with an exact internal fermionic (particle-ghost) symmetry can be formulated that (a) satisfies unitarity, causality, and the usual axioms of quantum field theory despite negative-norm ghost states and the non-local star-product self-interactions of unphysical fields; (b) has a non-empty physical spectrum in which symmetry-singlet fields (the SM+$\alpha$ particles) coexist with unphysical symmetry doublets; and (c) receives no destabilizing radiative corrections to singlet mass parameters from the heavy doublets, with the one-loop cancellations and equal renormalizations of the toy models holding to all orders of perturbation theory, or non-perturbatively, without fine tuning. Whether such a nontrivial theory can be constructed — including a consistent quantization of the non-local ghost interactions and a consistent treatment of gauge bosons and gravitons as unphysical constituents — or requirements (a)–(c) are mutually incompatible is the open determination. The source additionally poses, alongside this same objective of establishing the fermionic-symmetry route to the hierarchy problem, the companion directions of deriving the low-energy fermionic-symmetric pattern from a concrete UV completion and of identifying indirect experimental tests such as parameter relations at the scale $M_U$.

References

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