We continue after coffee.

Not a cosmology talk, despite the general theme. Instead, this is about sphalerons, and perturbative but unusual EW phenomena. Sphaleron-like (2 to 10!) processes enhanced at high energies by emission of large numbers of gauge bosons. These processes are non-perturbative, which makes things complicated. So instead consider the simpler case of perturbative processes with large (~ 50) final state particles.

Consider first on-mass tree-level 1 to n cross sections. Recursion relations for amplitudes result in factorial growth of amplitudes in several theories; ordinary φ

Things get technical here. See the notes. It's actually quite clear so far, but I don't know that I can take many notes here. Tree-level, on-shell threshold assumptions convert path integral to ordinary differential equation.

Many questions,

Consider gauge-Higgs theory. Consider 2 to

Conjectured that expression found in non-relativistic limit holds for arbitrary energies in the large-

Loop corrections

More detail at loop level? Only threshold corrections have been calculated in general. Argued but not proved that correction exponentiates. Tree-level growth ~ log

What about Yang-Mills/QCD? Can't use same analysis techniques, as started from threshold limit. This is impossible with massless states. Also, theories with unbroken gauge theories have

Higgs (quartic) inflation with non-minimal gravitational coupling. Non-minimal coupling leads to modified potential in Einstein frame: asymptotically flat for large field values. Need non-minimal coupling > 0.01 to make quartic potential consistent with Planck 2015 data.

However, there is a problem with the Higgs mass. 2008 calculation found need Higgs mass in [140, 185] GeV, obviously inconsistent with data. The problem is the instability in the Higgs potential. The proposal is to solve this problem by adding new physics to stabilise this problem related to neutrino physics and DM. Looks like add minimal DM in form of fermion SU(2) triplet

Different choices: larger fermion multiplicity of fiveplet sufficient to solve stability alone. Type-I seesaw actually worsens stability when right-handed neutrino heavy due to large Yukawa.

Role of top quark mass? Took central value, assumed stability problem.

Gauge couplings? No Landau pole (below Planck scale).

Outstanding Higgs inflation problem; instabilities from Higgs components? Not addressed here... depends on inflaton VEV.

**11:20 am:***New EW Effects at ~100 TeV*, Valya KhozeNot a cosmology talk, despite the general theme. Instead, this is about sphalerons, and perturbative but unusual EW phenomena. Sphaleron-like (2 to 10!) processes enhanced at high energies by emission of large numbers of gauge bosons. These processes are non-perturbative, which makes things complicated. So instead consider the simpler case of perturbative processes with large (~ 50) final state particles.

Consider first on-mass tree-level 1 to n cross sections. Recursion relations for amplitudes result in factorial growth of amplitudes in several theories; ordinary φ

^{4}with or without VEVs, and gauge-Higgs theories as in the SM.Things get technical here. See the notes. It's actually quite clear so far, but I don't know that I can take many notes here. Tree-level, on-shell threshold assumptions convert path integral to ordinary differential equation.

*n*derivatives give factor of*n*! (is this compensated? Apparently not in at least some cases, e.g. φ^{4}can be computed exactly at this level.) Divergence is due to factorial growth of perturbation series, a symptom of its asymptotic nature. Usually this only shows up at high order in perturbation theory. But for many final state particles, this appears at tree level and symbolises the breakdown of perturbativity (?)Many questions,

*e.g.*are these a consequence of the particular assumptions made? No, this behaviour remains away from threshold, for more realistic theories, and including loop corrections.Consider gauge-Higgs theory. Consider 2 to

*n*processes, initial states on-shell. Consider going away from the multi-particle threshold. This gets complicated; equations of motion lead to recursion relations for tree-level amplitudes. Work in non-relativistic limit; Galilean symmetry means that leading corrections proportional to total kinetic energy. Can expand as a series in this, then exponentiate. In large*-n*limit, some suppression arises but factorial growth remains. Can now do phase space integrals and factorial growth remains.Conjectured that expression found in non-relativistic limit holds for arbitrary energies in the large-

*n*limit. Further, notable that his functional form is*same one*(leading terms) seen in sphaleron processes. Some evidence from numerical calculations of cross sections using modern simulation tools.Loop corrections

*hint*that this breakdown happens at lower energies, possibly as low as 35 TeV.__Questions__More detail at loop level? Only threshold corrections have been calculated in general. Argued but not proved that correction exponentiates. Tree-level growth ~ log

*n*; loop correction ~*n*. Thus clearly grows even faster. But obvious question about NNLO. Also, sign of first correction different in broken/unbroken scalar theory. But conclusion holds: perturbative description breaks down, rigorous non-perturbative technique needed.What about Yang-Mills/QCD? Can't use same analysis techniques, as started from threshold limit. This is impossible with massless states. Also, theories with unbroken gauge theories have

*massive*cancellations in amplitudes; despite factorial growth in Feynman diagrams, result does show this property.**12:00 pm:***Higgs Inflation, Seesaw Physics and Fermion Dark Matter*, Nobuchika OkadaHiggs (quartic) inflation with non-minimal gravitational coupling. Non-minimal coupling leads to modified potential in Einstein frame: asymptotically flat for large field values. Need non-minimal coupling > 0.01 to make quartic potential consistent with Planck 2015 data.

However, there is a problem with the Higgs mass. 2008 calculation found need Higgs mass in [140, 185] GeV, obviously inconsistent with data. The problem is the instability in the Higgs potential. The proposal is to solve this problem by adding new physics to stabilise this problem related to neutrino physics and DM. Looks like add minimal DM in form of fermion SU(2) triplet

*and*type-III neutrino see-saw with right-handed neutrino masses of the order 0.25—6 TeV (lower bound from LHC, upper bound from Higgs potential). New fermions so improve vacuum stability. Also consider fermion SU(2) fiveplet and type-I neutrino see-saw. Both can work with non-minimal couplings ~ 10^{4}.Different choices: larger fermion multiplicity of fiveplet sufficient to solve stability alone. Type-I seesaw actually worsens stability when right-handed neutrino heavy due to large Yukawa.

__Questions__Role of top quark mass? Took central value, assumed stability problem.

Gauge couplings? No Landau pole (below Planck scale).

Outstanding Higgs inflation problem; instabilities from Higgs components? Not addressed here... depends on inflaton VEV.

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