One-loop inert and pseudo-inert minima

Pedro Ferreira

ISEL and CFTC, UL, PortugalToyama, 14/02/2015

Preliminary results, with Bogumila Swiezewska, Univerity of Warsaw

The Two-Higgs Doublet potential

m212, λ5, λ6 and λ7 complex - seemingly 14 independent real parameters

Most general SU(2) × U(1) scalar potential:

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

Coupling to fermions

MODEL I: Only Φ2 couples to fermions.

MODEL II: Φ2 couples to up-quarks, Φ1 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 Φ1 couples to fermions.

Inert vacua – preserve Z2 symmetry

• The INERT minimum,

• Since only Φ1 has Yukawa couplings, fermions are massive – “OUR” minimum.

00

and 0

21 2

11 v

• The PSEUDO-INERT minimum,

• Since only Φ1 has Yukawa couplings, fermions are massless.

221

02

1 and 00

v

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

MeVmGeVmGeVmGeVv

e

t

W

511.017380

246

!0!080

246

e

t

W

mm

GeVmGeVv

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, 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?

: 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...