We critically assess to what extent it makes sense to bound the Wilson coefficients of dimension-six operators. In the context of Higgs physics, we establish that a closely related observable, <math><msub><mi>c</mi><mi>H</mi></msub></math>, is well defined and satisfies a two-sided bound. <math><msub><mi>c</mi><mi>H</mi></msub></math> is derived from the low momentum expansion of the scattering amplitude, or the derivative of the amplitude at the origin with respect to the Mandelstam variable <math><mi>s</mi></math>, expressed as <math><mi>M</mi><mo>(</mo><msub><mi>H</mi><mi>i</mi></msub><msub><mi>H</mi><mi>i</mi></msub><mo>→</mo><msub><mi>H</mi><mi>j</mi></msub><msub><mi>H</mi><mi>j</mi></msub><mo>)</mo><mo>=</mo><msub><mi>c</mi><mi>H</mi></msub><mi>s</mi><mo>+</mo><mi>O</mi><mo>(</mo><msub><mi>g</mi><mi>SM</mi></msub><mo>,</mo><msup><mi>s</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>)</mo></math> where <math><msub><mi>g</mi><mi>SM</mi></msub></math> represents all Standard Model couplings. This observable is nondispersive and, as a result, not sign-definite. We also determine the conditions under which the bound on <math><msub><mi>c</mi><mi>H</mi></msub></math> is equivalent to a bound on the dimension-six operator <math><msub><mi>O</mi><mi>H</mi></msub><mo>=</mo><mo>∂</mo><mo>|</mo><mi>H</mi><msup><mo>|</mo><mn>2</mn></msup><mo>∂</mo><mo>|</mo><mi>H</mi><msup><mo>|</mo><mn>2</mn></msup></math>.

Italy

Joan Elias Miró

Canada

Andrea L. Guerrieri

Mehmet Asım Gümüş

Extremal Higgs couplings

We find the complete set of conditions satisfied by the forward <math><mrow><mn>2</mn><mo>→</mo><mn>2</mn></mrow></math> scattering amplitude in unitary and causal theories. These are based on an infinite set of energy dependent quantities (the arcs) which are dispersively expressed as moments of a positive measure defined at (arbitrarily) higher energies. We identify optimal finite subsets of constraints, suitable to bound effective field theories (EFTs), at any finite order in the energy expansion. At tree level arcs are in a one to one correspondence with Wilson coefficients. We establish under which conditions this approximation applies, identifying seemingly viable EFTs where it never does. In all cases, we discuss the range of validity in both energy and couplings, where the latter has to satisfy two-sided bounds. We also extend our results to the case of small but finite <math><mi>t</mi></math>. A consequence of our study is that EFTs in which the scattering amplitude in some regime grows in energy faster than <math><msup><mi>E</mi><mn>6</mn></msup></math> cannot be UV completed.

France

CERN

Brando Bellazzini

Switzerland

Riccardo Rattazzi

Marc Riembau

Francesco Riva

Positive moments for scattering amplitudes

We bootstrap the <math><mi>S</mi></math> matrix of massless particles in unitary, relativistic two dimensional quantum field theories. We find that the low energy expansion of such <math><mi>S</mi></math> matrices is strongly constrained by the existence of a UV completion. In the context of flux tube (FT) physics, this allows us to constrain several terms in the <math><mi>S</mi></math> matrix low energy expansion or—equivalently—on Wilson coefficients of several irrelevant operators showing up in the FT effective action. These bounds have direct implications for other physical quantities; for instance, they allow us to further bound the ground state energy as well as the level splitting of degenerate energy levels of large FTs. We find that the <math><mi>S</mi></math> matrices living at the boundary of the allowed space exhibit an intricate pattern of resonances with one sharper resonance whose quantum numbers, mass, and width are precisely those of the world-sheet axion proposed by Athenodorou, Bringoltz, and Teper and Dubovsky, Flauger, and Gorbenko. The general method proposed here should be extendable to massless <math><mi>S</mi></math> matrices in higher dimensions and should lead to new quantitative bounds on irrelevant operators in theories of Goldstones and, also, in gauge and gravity theories.