One-loop inert and pseudo-inert minima Pedro Ferreira ISEL and CFTC, UL, Portugal Toyama, 14/02/2015 Preliminary results, with Bogumila Swiezewska, Univerity of Warsaw

The Two-Higgs Doublet potential m 2 12, λ 5, λ 6 and λ 7 complex - seemingly 14 independent real parameters Most general SU(2) × U(1) scalar potential: Most frequently studied model: model with a Z 2 symmetry, Φ 2 → -Φ 2, meaning m 12, λ 6, λ 7 = 0. It avoids potentially large flavour-changing neutral currents.

Z 2 -symmetric model Coupling to fermions MODEL I: Only Φ 2 couples to fermions. MODEL II: Φ 2 couples to up-quarks, Φ 1 to down quarks and leptons.... SEVEN real independent parameters. The symmetry must be extended to the whole lagrangian, otherwise the model would not be renormalizable. Inert: Only Φ 1 couples to fermions.

Inert vacua – preserve Z 2 symmetry The INERT minimum, Since only Φ 1 has Yukawa couplings, fermions are massive – “OUR” minimum. The PSEUDO-INERT minimum, Since only Φ 1 has Yukawa couplings, fermions are massless. WHY BOTHER? In the inert minimum, the second doublet originates perfect Dark Matter candidates! Inert neutral scalars do not couple to fermions or have triple vertices with gauge bosons.

Tree-level vacuum solutions INERT: PSEUDO-INERT: E. Ma, Phys. Rev. D (2006); R. Barbieri, L.J. Hall and V.S. Rychkov Phys. Rev. D74: (2006); L. Lopez Honorez, E. Nezri, J. F. Oliver and M. H. G. Tytgat, JCAP 0702, 028(2007); L. Lopez Honorez and C. E. Yaguna, JHEP 1009, 046(2010); L. Lopez Honorez and C. E. Yaguna, JCAP 1101, 002 (2011)

These minima can coexist in the potential, which raises a troubling possibility... Local minimum - INERT Global minimum – PSEUDO-INERT

Tree-level relations between the depth of the potential at minima Let v be the VEV at the INERT minimum, and v’ the PSEUDO-INERT VEV. Then: QUESTIONS: HOW DO THESE RELATIONS CHANGE WITH LOOP CORRECTIONS? CAN THERE BE AN “INVERSION” OF INERT AND PSEUDO-INERT MINIMA DEPTHS AT ONE-LOOP?

The one-loop inert minima were studied by Gil, Chankowski and Krawczyck in Phys. Lett. B717 (2012) 396, but the emphasis here is in comparing the relations between depths of tree-level and one-loop potentials. One-loop effective potential Field-dependent mass eigenvalues - PAIN. TOY MODEL – only scalar sector (so far), global SU(2)×U(1). Scalar masses calculated using (so far) the effective potential approximation (second derivatives of the potential).

Compute one-loop effective potential. Minimize it, requiring SIMULTANEOUS EXISTENCE of inert and pseudo-inert vacua. Compute ALL SQUARED MASSES at both vacua and demand they are all positive – COEXISTING MINIMA. Demand that, at the inert vacuum, v = 246 GeV, m h = 125 GeV. Compare depths of potentials at both minima.

As of 14/02/ Do one-loop corrections “invert” the depths of the potential? : Tree-level expected relative depths : One-loop computed relative depths Inversion of minima for 3% of scanned parameter space

Now here’s something interesting... : Tree-level expected relative depths : One-loop computed relative depths Inversion of minima for 0.5% of scanned parameter space! Tree-level obtained formula seems almost exact...

Now here’s something EVEN MORE interesting... IMPOSSIBLE TO HAVE SIMULTANEOUS MINIMA IN THIS REGION AT TREE-LEVEL!

NO CONCLUSIONS – ONLY FUTURE HARD WORK! - Recover from jet-lag...