How negative can forward-limit coefficients be once gravity is included? · revision history

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Title

How negative can forward-limit coefficients be once gravity is included

Plain

Without gravity, a certain coefficient in the scattering of two particles must be positive. With gravity it may be slightly negative, and the size of the allowed negative value is not known.

Precise

For a weakly coupled 4D effective field theory of a massless scalar coupled to Einstein gravity, with $2\to 2$ amplitude $M(s,t) = 8\pi G s^2/(-t) + c_2 s^2/\Lambda^4 + ...$, determine the lowest value of $c_2$ consistent with unitarity, analyticity, crossing and a UV completion whose amplitude Reggeizes (grows as $s^{2 + \alpha' t}$) above a scale $M_*$. Under this Regge assumption proposals give $c_2 \ge -\operatorname{O}(\Lambda^4/(M_{\mathrm{pl}}^2 M_*^2))$; smeared impact-parameter sum rules in $D = 4$ instead give $c_2 \ge -\operatorname{O}(G \Lambda^4/M^2)\operatorname{log}(M b_{\mathrm{IR}})$, where $M$ is the EFT cutoff and $b_{\mathrm{IR}}$ an infrared impact-parameter cutoff. An answer is the optimal bound in $D = 4$ with its assumptions stated, saying which form is optimal, or an explicit consistent UV completion that violates it.

Settled by

A derivation from stated axioms of the optimal lower bound on $c_{2}$, checked against explicit string-theory amplitudes.

Kind

well-posed

Answer type

value

References

(empty)

Literature status

contested

Status note

Bounds allowing Planck-suppressed negativity have been derived since 2020, but their size depends on the assumed Regge behavior of the UV completion or on an infrared cutoff.

Posed since

(empty)