One-loop inert and pseudo-inert minima

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 Most general SU(2) × U(1) scalar potential: m212, λ5, λ6andλ7complex - seemingly 14 independent real parameters Most frequently studied model: model with a Z2 symmetry, Φ2 → -Φ2, meaning m12, λ6, λ7 = 0. It avoids potentially large flavour-changing neutral currents.

Z2-symmetric model • SEVEN real independent parameters. • The symmetry must be extended to the whole lagrangian, otherwise the model would not be renormalizable. Coupling to fermions MODEL I: Only Φ2 couples to fermions. MODEL II: Φ2 couples to up-quarks, Φ1 to down quarks and leptons. . . . Inert: Only Φ1 couples to fermions.

TheINERTminimum, • Since only Φ1 has Yukawa couplings, fermions are massive – “OUR” minimum. • ThePSEUDO-INERTminimum, • Since only Φ1 has Yukawa couplings, fermions are massless. Inert vacua – preserve Z2 symmetry 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. D73 077301 (2006); R. Barbieri, L.J. Hall and V.S. Rychkov Phys. Rev. D74:015007 (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-inertvacua. • Compute ALL SQUARED MASSES at both vacua and demand they are all positive – COEXISTING MINIMA. • Demand that, at the inert vacuum, v = 246 GeV, mh = 125 GeV. • Compare depths of potentials at both minima.

As of 14/02/2015... Do one-loop corrections “invert” the depths of the potential? Inversion of minima for 3% of scanned parameter space : Tree-level expected relative depths : One-loop computed relative depths

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

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

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

One-loop inert and pseudo-inert minima