1. VACUUM STRUCTURE OF N HIGGS DOUBLET MODELS Pedro Ferreira ISEL and CFTC, Universidade de Lisboa PASC Winter School, December 20 th, 2007

2. The Standard Model: one Higgs doublet, Φ. The state of least energy of the theory corresponds to a vacuum expectation value for Φ of ONE HIGGS… This spontaneous symmetry breaking gives mass to the fermions and W and Z bosons, leaving the gluons and photons massless. CP is not spontaneously broken in the SM. The Higgs potential is very simple:

3. The Two-Higgs Doublet Model (2HDM): one of the simplest extensions of the standard model, where one considers two Higgs doublets, Φ 1 and Φ 2. TWO HIGGS… The Higgs potential can now have as much as 14 independent parameters… V = µ 1 x 1 + µ 2 x 2 + µ 3 x 3 + µ 4 x 4 + λ 11 x 1 2 + λ 22 x 2 2 + λ 33 x 3 2 + λ 44 x 4 2 + λ 12 x 1 x 2 + λ 13 x 1 x 3 + λ 14 x 1 x 4 + λ 23 x 2 x 3 + λ 24 x 2 x 4 + λ 34 x 3 x 4 with

4. The NORMAL minimum, for which The CHARGE BREAKING (CB) vacuum, with The CP vacuum, with α has electric charge! δ breaks CP (for a CP conserving potential)! TWO HIGGS… The vacuum structure is now more elaborate. We have:

5. The difference in the values of the potential at the Normal and CB stationary points is given by The difference in the values of the potential at the Normal and CP stationary points is given by TWO HIGGS… P.M. Ferreira, R. Santos e A. Barroso, Physics Letters B 603 (2004) 219 A. Barroso, P.M. Ferreira and R. Santos, Physics Letters B 632 (2006) 684 It may be shown that:

6. TWO HIGGS… Thus, if there is a (Normal) charge-and-CP conserving minimum, the global minimum of the theory is “a” Normal one. The Normal minima are thus completely stable against tunneling to a deeper CB or CP one. The masslessness of the photon is guaranteed by the theory itself. It can actually be shown that if the Normal vacuum is a minimum, then the CB stationary point is in fact a saddle point. There are currently some doubts whether that also applies to the CP vacuum – we say a normal minimum can have a CP minimum above it, Ivanov’s most recent paper claims otherwise. Stay tuned…

7. TWO HIGGS… I said that if there existed a Normal minimum, then the global minimum of the 2HDM would also be Normal. But WHICH Normal minimum? They are not unique… Because the Normal minimum results from solving a pair of coupled cubic stationarity equations, there can be nine Normal stationary points… … but of these we can only obtain a maximum of two non-degenerated Normal minima. These different Normal minima may coexist in the model! We may have the situation of a Normal minimum with the standard values for the W and Z masses (N 1 minimum), for which v 1 2 + v 2 2 =(246 GeV) 2, lying above another one (N 2 minimum), for which the sum of the squares of the vevs is completely different.

8. TWO HIGGS… A. Barroso, P.M. Ferreira and R. Santos, Physics Letters B 652 (2007) 181

9. *MANY* HIGGS… We were able to semi-generalise these results to theories with an arbitrary number N of Higgs doublets (N-Higgs doublet model, NHDM). The potential is given by where the indices range from 1 to N, N > 2, and the { μ, λ} coefficients may be complex.

10. *MANY* HIGGS… The possible vacua of the NHDM are of the same type as the 2HDM: Normal, Charge Breaking and CP breaking (when CP is defined). They are now, however, much more complicated, with N fields having independent vevs. Amazingly enough, it is possible to perform successive unitary transformations on the fields (basis changes) so that the most general vacua reduce to extremely simple forms, written on the same field basis:

11. *MANY* HIGGS… It has been shown that: A. Barroso, P.M. Ferreira, R. Santos and João P. Silva, Physical Review D74 (2006) 085016 and thus, again, if the normal vacuum is a minimum it will be deeper than any CP stationary point!

12. *MANY* HIGGS… What about the charge breaking stationary point? It so happens that A. Barroso, P.M. Ferreira, R. Santos and João P. Silva, Physical Review D74 (2006) 085016 Numerical calculations show that already for N = 3 one may have a CB minimum lying below a Normal one. There is thus a substantial qualitative difference between the 2HDM and the NHDM – in the latter, a normal minimum may, in principle, tunnel to a dangerous CB minimum and is thus unstable. This raises the possibility of imposing bounds on the parameters of the NHDM to avoid such instability…

13. NO HIGGS – JUST CONCLUSIONS The 2HDM Normal minimum, when it exists, is stable against tunnelling to a CB or CP vacuum. The theory is stable. The Normal minimum in the 2HDM is not unique. For some combinations of parameters, lower Normal minima with different values for gauge boson and fermion masses can exist. For a model with an arbitrary number of Higgs doublets, a normal minimum, if it exists is guaranteed to be deeper than any CP vacuum of the model… … but for the NHDM this does not imply that the global minimum of the theory is Normal: Charge Breaking may still occur as CB minima deeper than the normal one may exist.