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Perturbative unitarity of fractional field theories and gravity

Gianluca Calcagni, Fabio Briscese

Abstract

Motivated by quantum gravity on spacetimes with multi-scale geometry, we analyze quantum field theories with a self-adjoint fractional power $(\Box^2)^{γ/2}$ of the d'Alem\-bert\-ian in the kinetic term, for any real $γ>0$. Selecting a particularly simple version of the kinetic term which we call hermitian polynomial, we study the spectral decomposition of the propagator and, when $γ>1$, obtain the standard mass singularity $-k^2=m^2$. This is the only mode in the perturbative spectrum of asymptotic states, since the only other content of the theory is a cloud of purely virtual particles with complex masses. We also show that other versions of the self-adjoint fractional kinetic term lead to a different distribution of the virtual complex modes but to the same physical spectrum for $0<γ<3$, thus addressing the issue of uniqueness in this class of nonlocal theories. The non-hermitian version of the theory has the $-k^2=m^2$ particle plus a continuum of standard massive modes. Finally, we prove that unitarity of scalar, gauge and gravity models is respected at all perturbative orders if, in the hermitian cases, one adopts the fakeon prescription on scattering amplitudes or, in the non-hermitian case, $0<γ<1$ or $2<γ<3$ with the standard Feynman prescription. These results drastically simplify previous characterizations of fractional quantum gravity, which is super-renormalizable for $γ>2$.

Perturbative unitarity of fractional field theories and gravity

Abstract

Motivated by quantum gravity on spacetimes with multi-scale geometry, we analyze quantum field theories with a self-adjoint fractional power of the d'Alem\-bert\-ian in the kinetic term, for any real . Selecting a particularly simple version of the kinetic term which we call hermitian polynomial, we study the spectral decomposition of the propagator and, when , obtain the standard mass singularity . This is the only mode in the perturbative spectrum of asymptotic states, since the only other content of the theory is a cloud of purely virtual particles with complex masses. We also show that other versions of the self-adjoint fractional kinetic term lead to a different distribution of the virtual complex modes but to the same physical spectrum for , thus addressing the issue of uniqueness in this class of nonlocal theories. The non-hermitian version of the theory has the particle plus a continuum of standard massive modes. Finally, we prove that unitarity of scalar, gauge and gravity models is respected at all perturbative orders if, in the hermitian cases, one adopts the fakeon prescription on scattering amplitudes or, in the non-hermitian case, or with the standard Feynman prescription. These results drastically simplify previous characterizations of fractional quantum gravity, which is super-renormalizable for .

Paper Structure

This paper contains 39 sections, 24 theorems, 164 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Fractional QFT: renormalizability and unitarity (so far)
  3. Renormalizability
  4. Unitarity (so far)
  5. Tree-level Green's function
  6. Branch point and discontinuity
  7. Poles
  8. Tree-level spectrum and unitarity
  9. Arcs at infinity
  10. Complex poles
  11. Confined modes
  12. Unstable modes
  13. Purely virtual modes
  14. Branch point
  15. Discontinuity
  16. ...and 24 more sections

Key Result

Theorem 1

When there are no complex-conjugate pairs $(z_n^\pm,z_n^{\pm*})$ in any Riemann sheet $S_{l}^\pm$.

Figures (5)

  • Figure 1: Contour of the Cauchy representation of the Green's function (\ref{['propzuv']}).
  • Figure 2: Contour of the Cauchy representation of the Green's functions (\ref{['propzuv2']}) and (\ref{['propz']}). $\Gamma_+$ and $\Gamma_-$ are mutually disjoint pieces covering, respectively, the $\text{Re}\,z>0$ and the $\text{Re}\,z<0$ half-plane. The vertical lines $L_\pm$ run along the discontinuity at $\text{Re}\,z=0$, while $\Gamma_\varepsilon^\pm$ circle around the branch point at $z=0$.
  • Figure 3: Contour of the Cauchy representation of the Green's functions (\ref{['Gw']}) and (\ref{['propw2']}) in the plane spanned by $w=-z^2$.
  • Figure 4: Mapping from the $w$-plane (figure \ref{['fig3']}) to the $z$-planes with $z=\pm\sqrt{-w}$ and their combination into the disjoint contour of figure \ref{['fig2']}.
  • Figure 5: Examples of pole distribution in $S_0=S_0^+\cup S_0^-$ symmetric in $S_0^-$ (left, $\gamma=21/2$) and asymmetric (right, $\gamma=31/3$) for the theory with HS operator.

Theorems & Definitions (48)

  • Theorem 1: HP: complex-conjugate pairs ---real $\bm{0<\gamma<2u+1}$
  • proof
  • Theorem 2: HP: empty sheets ---real $\bm{0<\gamma<2u+1}$
  • proof
  • Theorem 3: HP: complex-conjugate poles ---irrational $\bm{\gamma>2u+1}$
  • proof
  • Theorem 4: HP: complex-conjugate poles ---rational $\bm{\gamma=p/q>2u+1}$
  • proof
  • Theorem 5: HS: real poles ---irrational $\bm{\gamma>0}$
  • proof
  • ...and 38 more