Abstract
We demonstrate two properties of the trace of the energy-momentum tensor $T^{\mu}_{~ \mu}$ in the flat spacetime. One is the decoupling of heavy degrees of freedom; i.e., heavy degrees of freedom leave no effect for low-energy $T^{\mu}_{~ \mu}$-inserted amplitudes. This is intuitively apparent from the effective field theory point of view, but one has to take into account the so-called trace anomaly to explicitly demonstrate the decoupling. As a result, for example, in the $R^{2}$ inflation model, scalaron decay is insensitive to heavy degrees of freedom when a matter sector ${\it minimally}$ couples to gravity (up to a non-minimal coupling of a matter scalar field other than the scalaron). The other property is a quantum contribution to a non-minimal coupling of a scalar field. The non-minimal coupling disappears from the action in the flat spacetime, but leaves the so-called improvement term in $T^{\mu}_{~ \mu}$. We study the renormalization group equation of the non-minimal coupling to discuss its quantum-induced value and implications for inflation dynamics. We work it out in the two-scalar theory and Yukawa theory.
Highlights
The energy-momentum tensor Tμν is an important object in quantum field theory [1,2]
This m2-term should be canceled by other contributions so that the amplitude is insensitive to ultraviolet physics
The m2-term in Mtree diverges as one takes the heavy limit of ψ. This originates from the fact that the scalar mass squared is sensitive to ultraviolet physics and one needs fine-tuning to realize Mphys ≪ mphys
Summary
The energy-momentum tensor Tμν is an important object in quantum field theory [1,2]. It provides generators of spacetime symmetry (Poincaresymmetry in the flat spacetime). The decoupling of heavy degrees of freedom from low-energy Tμμ-inserted amplitudes is not obvious at first sight This is because μ μ consists of mass terms (classical breaking of scale invariance). The quantum-induced value of η is studied in the λφ theory [20,25,26]: Δη 1⁄4 −λ3=ð864ð4πÞ6Þ at the leading (three-loop) order It appears from the renormalization of trace-anomaly terms (i.e., composite operators [37,38,39,40]) and is related with an inhomogeneous term of the β function of η. We find that Δη appears at the oneloop order in the two-scalar theory and Yukawa theory, as a threshold correction when additional degrees of freedom are heavy and decouple from the low-energy dynamics.
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