S-matrix bootstrap for resonances

We study the $2\rightarrow2$ $S$-matrix element of a generic, gapped and Lorentz invariant QFT in $d=1+1$ space time dimensions. We derive an analytical bound on the coupling of the asymptotic states to unstable particles (a.k.a. resonances) and its physical implications. This is achieved by exploiting the connection between the S-matrix phase-shift and the roots of the S-matrix in the physical sheet. We also develop a numerical framework to recover the analytical bound as a solution to a numerical optimization problem. This later approach can be generalized to $d=3+1$ spacetime dimensions.Comment: Minor typos corrected, matches published versio

Ideals of general forms and the ubiquity of the Weak Lefschetz property

Let $d_1,...,d_r$ be positive integers and let $I = (F_1,...,F_r)$ be an ideal generated by general forms of degrees $d_1,...,d_r$, respectively, in a polynomial ring $R$ with $n$ variables. When all the degrees are the same we give a result that says, roughly, that they have as few first syzygies as possible. In the general case, the Hilbert function of $R/I$ has been conjectured by Fr\"oberg. In a previous work the authors showed that in many situations the minimal free resolution of $R/I$ must have redundant terms which are not forced by Koszul (first or higher) syzygies among the $F_i$ (and hence could not be predicted from the Hilbert function), but the only examples came when $r=n+1$. Our second main set of results in this paper show that further examples can be obtained when $n+1 \leq r \leq 2n-2$. We also show that if Fr\"oberg's conjecture on the Hilbert function is true then any such redundant terms in the minimal free resolution must occur in the top two possible degrees of the free module. Related to the Fr\"oberg conjecture is the notion of Weak Lefschetz property. We continue the description of the ubiquity of this property. We show that any ideal of general forms in $k[x_1,x_2,x_3,x_4]$ has it. Then we show that for certain choices of degrees, any complete intersection has it and any almost complete intersection has it. Finally, we show that most of the time Artinian ``hypersurface sections'' of zeroschemes have it.Comment: 24 page

S-matrix bootstrap for resonances

We study the $2\rightarrow2$ $S$-matrix element of a generic, gapped and Lorentz invariant QFT in $d=1+1$ space time dimensions. We derive an analytical bound on the coupling of the asymptotic states to unstable particles (a.k.a. resonances) and its physical implications. This is achieved by exploiting the connection between the S-matrix phase-shift and the roots of the S-matrix in the physical sheet. We also develop a numerical framework to recover the analytical bound as a solution to a numerical optimization problem. This later approach can be generalized to $d=3+1$ spacetime dimensions.Comment: Minor typos corrected, matches published versio

THE MITE ORNITHONYSSUS SYLVARIUM (C. AND F.) (ARACHNIDA: ACARINA-MACRONYSSIDAE) ATTACKING FOWL IN PUERTO RICO

THE MITE ORNITHONYSSUS SYLVARIUM (C. AND F.) (ARACHNIDA: ACARINA-MACRONYSSIDAE) ATTACKING FOWL IN PUERTO RIC