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

Revision 2

Accepted · core change · record

Changes against revision 1:

Title · changed
Revision 1: How negative can forward-limit coefficients be once gravity is included
Revision 2: 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 · changed · core
Revision 1: 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.
Revision 2: 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 $c_2 \ge -\operatorname{O}(G \Lambda^4/M^2)\log(M b_{\mathrm{IR}})$, 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)

Decisions