CP violation in extended Higgs sectors

Transcription

1 CP violation in extended Higgs sectors Corfu Sep 7 Per Osland University of Bergen Work with O. M. Ogreid, M. N. Rebelo arxiv:7.4768, JHEP (censored) (and work in progress)

2 Consider the potential of a 3HDM V = Y ab a b + Z abcd( a b)( c d) a, b, c, d run over values,,3 3diagonal(real)Y s: Y, Y, Y 33 3o -diagonal ones (complex) lots of Z abcd, some real, some complex Z badc = Z abcd, Z cdab = Z abcd, Y ba = Y ab ll counted: 54 parameters (not all independent)

3 Consider the potential of a 3HDM V = Y ab a b + Z abcd( a b)( c d) we may rotate: ll counted: 54 parameters 3 = U. diagonalize bilinear part (- 6 parameters). remove relative phases (- parameters) Remaining: 46 parameters (linearly independent) Olaussen et al, (general formula) Compare HDM: parameters 3

4 How do we identify CP violation? If all coefficients and all vevs are real, then CP is conserved First guess: otherwise violated But phases of vevs may be modified by a phase rotation on the field. The coefficients in the potential would pick up such phases. The most general CP transformation allows (Branco, Lavoura, Silva, 999) i CP! U ij j U unitary, arb.

5 How do we identify CP violation? Certain reparametrization invariants should vanish, for CP to be conserved In the HDM, conditions expressed in terms of 3 invariants Branco, Rebelo, Silva-Marcos, 5 Gunion, Haber, 5 in terms of potential parameters These conditions can also be written in terms of masses and physical couplings. Lavoura & Silva, 994; Botella & Silva 995 Grzadkowski et al, 4 In the 3HDM, more invariants

6 Spontaneous CP violation? ssuming there is no explicit CP violation. Could there be spontaneous CPV? Branco, Gerard, Grimus, 984: If a unitary matrix U exists, that is a symmetry of the Lagrangian, satisfying also U ij h j i = h i i then the vacuum is CP invariant No spontaneous CP violation either

7 HDM case Suppose CPV is established Is it explicit or spontaneous? Gunion & Haber defined 4 invariants (in terms of potential parameters) that can be used to exclude spontaneous CPV Equivalently (Grzadkowski et al, 6):. Charged Higgs mass takes a particular value, in terms of neutral Higgs masses and couplings. Quartic Charged Higgs coupling takes a particular value, in terms of neutral Higgs masses and couplings If satisfied, CPV is spontaneous

8 Back to the NHDM In some cases, finding (or excluding!) a transformation U satisfying U ij h j i = h i i may be difficult Other approach: Transform to the Higgs basis : Only one doublet has non-vanishing vev illustrated for 3 doublets

9 S3 symmetric 3HDM Consider the potential V = V + V 4 V = µ h S h S + µ (h h + h h ), V 4 = (h h + h h ) + (h h h h ) + 3 [(h h h h ) +(h h + h h ) ] + 4 [(h S h )(h h + h h ) +(h S h )(h h h h )+h.c.]+ 5 (h S h S)(h h + h h ) + 6 [(h S h )(h h S )+(h S h )(h h S )] + 7 [(h S h )(h S h )+(h S h )(h S h )+h.c.] + 8 (h S h S). h S h, h doublet under S3 singlet under S3 Term with 4 plays a special role 6 different complex vacua with vevs (w,w,w S )

10 IRF (Irreducible Rep.) RRF (Reducible Rep.) w,w,w S ρ, ρ, ρ 3 C-I-a ŵ, ±iŵ, x, xe ± πi 3, xe πi 3 C-III-a, ŵ e iσ y, y, xe iτ C-III-b ±iŵ, x + iy, x iy, x C-III-c ŵ e iσ, ŵ e iσ, xe iρ y, xeiρ y,y C-III-d,e ±iŵ, ϵŵ xe iτ, xe iτ,y C-III-f ±iŵ,iŵ re iρ ± ix, re iρ ix, 3 re iρ reiρ C-III-g ±iŵ, iŵ re iρ ± ix, re iρ ix, 3 reiρ re iρ C-III-h 3ŵ e iσ, ±ŵ e iσ xe iτ,y,y C-III-i y, xe iτ,y 3(+tan σ ) +9 tan σ ŵ e iσ, x, ye iτ,ye iτ ±ŵ e i arctan(3 tan σ) ye iτ, x, ye iτ C-IV-a ŵ e iσ, re iρ + x, re iρ + x, x C-IV-b ŵ, ±iŵ re iρ + x, re iρ + x, re iρ + re iρ + x C-IV-c + cos σ ŵ, re iρ + r 3( + cos ρ)+x, ŵ e iσ re iρ r 3( + cos ρ)+x, re iρ + x C-IV-d ŵ e iσ, ±ŵ e iσ r e iρ + x, (r r )e iρ + x, r e iρ + x C-IV-e sin σ sin σ ŵ e iσ, re iρ + re iρ ξ + x, re iρ re iρ ξ + x, C-IV-f IGNORE THIS PRT ŵ e iσ re iρ + x + cos(σ σ ) cos σ ŵ e iσ, re iρ + re iρ ψ + x, ŵ e iσ re iρ re iρ ψ + x, re iρ + x C-V ŵ e iσ, ŵ e iσ xe iτ,ye iτ,z Table. Complex vacua. Notation: ϵ = and for C-III-d and C-III-e, respectively; ξ = 3 sin ρ / sin ρ, ψ = [3 + 3 cos(ρ ρ )]/( cos ρ ). With the constraints of table 4 the vacua labelled with an asterisk ( ) are in fact real.

11 Important: constraint 3 constraints Each vacuum, C-I-a, C-III-a, etc, is accompanied by a set of constraints on the coefficients of the potential These arise from the minimisation conditions, and their mutual consistency

12 Consider requires 4 = IRF (Irreducible Rep.) RRF (Reducible Rep.) w,w,w S ρ, ρ, ρ 3 C-I-a ŵ, ±iŵ, x, xe ± πi 3, xe πi 3 C-III-a, ŵ e iσ y, y, xe iτ C-III-b ±iŵ, x + iy, x iy, x C-III-c ŵ e iσ, ŵ e iσ, xe iρ y, xeiρ y,y C-III-d,e ±iŵ, ϵŵ xe iτ, xe iτ,y C-III-f ±iŵ,iŵ re iρ ± ix, re iρ ix, 3 re iρ reiρ C-III-g ±iŵ, iŵ re iρ ± ix, re iρ ix, 3 reiρ re iρ C-III-h 3ŵ e iσ, ±ŵ e iσ xe iτ,y,y C-III-i y, xe iτ,y 3(+tan σ ) +9 tan σ ŵ e iσ, x, ye iτ,ye iτ ±ŵ e i arctan(3 tan σ) ye iτ, x, ye iτ C-IV-a ŵ e iσ, re iρ + x, re iρ + x, x C-IV-b ŵ, ±iŵ re iρ + x, re iρ + x, re iρ + re iρ + x C-IV-c + cos σ ŵ, re iρ + r 3( + cos ρ)+x, ŵ e iσ re iρ r 3( + cos ρ)+x, re iρ + x C-IV-d ŵ e iσ, ±ŵ e iσ r e iρ + x, (r r )e iρ + x, r e iρ + x C-IV-e sin σ sin σ ŵ e iσ, re iρ + re iρ ξ + x, re iρ re iρ ξ + x, C-IV-f st example IGNORE THIS PRT ŵ e iσ re iρ + x + cos(σ σ ) cos σ ŵ e iσ, re iρ + re iρ ψ + x, ŵ e iσ re iρ re iρ ψ + x, re iρ + x C-V ŵ e iσ, ŵ e iσ xe iτ,ye iτ,z Table. Complex vacua. Notation: ϵ = and for C-III-d and C-III-e, respectively; ξ = 3 sin ρ / sin ρ, ψ = [3 + 3 cos(ρ ρ )]/( cos ρ ). With the constraints of table 4 the vacua labelled with an asterisk ( ) are in fact real.

13 Consider C-III-c st example (w,w,w S )=(ŵ e i, ŵ e i, ) Real potential, complex vevs Is CP violated (spontaneously) or not? It is not violated But there is no simple way to find the matrix U Go to the Higgs basis: h h h S = v ŵ ŵ ŵ ŵ v e i e i v =(ŵ +ŵ ) The coefficients of the potential remain real h h h S So CP is conserved normalisation (check via invariants)

14 w e ˆ h = (8) v v = (wˆ + wˆ ) w ˆ h hs st example hs h v (9 h = wˆ w ˆ 3 3 v v = ( w ˆ + w ˆ ) v = ( w ˆ + w ˆ ) Construct the matrix U in this form: cos sin hvs = (wˆ +wˆ ) vv = (wˆ + wˆ CP h S ) i( + ) U cos Usin e i!cos ij sin cos j sin = co + wˆ ) sin sin cos cos U = ei( + ) sin cos cos sin i( +(9) ) cos i( = e ) s U + U = ei( + ) sin cos U = e sin = h sin sin j i i i cos sin Uij h cos cos i( Choose + ) such into ( rotate cos that vevs sin cos = e sin i i i(ae, ae, ) i V = V+ Vi4 i i i (ae, ae, ) same (ae, ae, ) (aemodulus!, ae, ) ˆ h h wˆ wˆ w e i h w h ˆ ˆ = w i h wˆ h = wˆ wˆ v h = w ˆ w ˆ e h = U v h v v hs hs S v exp[ i( + )/] hs further rotation by i i exp[ i( + )/], ) h, h (ae, ae, ) i i (ae, ae, ) and we obtain the vevs in the form + exp[ i( (aei, ae (w, w, ws ) )/], ) i + )/], ) Under a CP transformation (complex conjugation, (w, w, ws ) = (w ˆ ei, w ˆ ei, ) accompanied by h $ h ) the potential is invariant! symmetry since 4 = (

15 Consider requires 4 = IRF (Irreducible Rep.) RRF (Reducible Rep.) w,w,w S ρ, ρ, ρ 3 C-I-a ŵ, ±iŵ, x, xe ± πi 3, xe πi 3 C-III-a, ŵ e iσ y, y, xe iτ C-III-b ±iŵ, x + iy, x iy, x C-III-c ŵ e iσ, ŵ e iσ, xe iρ y, xeiρ y,y C-III-d,e ±iŵ, ϵŵ xe iτ, xe iτ,y C-III-f ±iŵ,iŵ re iρ ± ix, re iρ ix, 3 re iρ reiρ C-III-g ±iŵ, iŵ re iρ ± ix, re iρ ix, 3 reiρ re iρ C-III-h 3ŵ e iσ, ±ŵ e iσ xe iτ,y,y C-III-i y, xe iτ,y 3(+tan σ ) +9 tan σ ŵ e iσ, x, ye iτ,ye iτ ±ŵ e i arctan(3 tan σ) ye iτ, x, ye iτ C-IV-a ŵ e iσ, re iρ + x, re iρ + x, x C-IV-b ŵ, ±iŵ re iρ + x, re iρ + x, re iρ + re iρ + x C-IV-c + cos σ ŵ, re iρ + r 3( + cos ρ)+x, ŵ e iσ re iρ r 3( + cos ρ)+x, re iρ + x C-IV-d ŵ e iσ, ±ŵ e iσ r e iρ + x, (r r )e iρ + x, r e iρ + x C-IV-e sin σ sin σ ŵ e iσ, re iρ + re iρ ξ + x, re iρ re iρ ξ + x, C-IV-f nd example IGNORE THIS PRT ŵ e iσ re iρ + x + cos(σ σ ) cos σ ŵ e iσ, re iρ + re iρ ψ + x, ŵ e iσ re iρ re iρ ψ + x, re iρ + x C-V ŵ e iσ, ŵ e iσ xe iτ,ye iτ,z Table. Complex vacua. Notation: ϵ = and for C-III-d and C-III-e, respectively; ξ = 3 sin ρ / sin ρ, ψ = [3 + 3 cos(ρ ρ )]/( cos ρ ). With the constraints of table 4 the vacua labelled with an asterisk ( ) are in fact real.

16 nd example Important difference wrt st example: ŵ S 6= n overall phase rotation would make Transform to the Higgs basis: h h h S = N (ŵ ŵ wˆ S ) N (ŵ ŵ ) N 3 (ŵ ŵ X) w S e i e i complex h h h S X is chosen to make lines and 3 orthogonal are normalisations N,N,N 3

17 nd example This vacuum is more constrained (4 constraints) ŵ, ŵ i are related and rotating to it follows that (be i, be i ) + = gain by the h $ h symmetry, we see that CP is conserved lternatively, in the Higgs basis, we can rotate the phases of the fields that have vanishing vevs and note that the potential is real, CP is conserved

18 3rd example, T. D. Lee potential HDM with real coefficients: V ( )= + ( ) + B( ) + C( )( )+ C( )( ) + [( )(D + E + F )+h.c.]. vevs: ( e i, ) Higgs basis reached via transformation = v e i e i with (normalisation) v = +

19 3rd example, T. D. Lee potential Transformation generates ( ) and( ) termsproportionalto ± ( ) sin v. Even putting this to zero, coefficients of quartic terms would in general be complex. CP is violated (spontaneously)

20 Summary Powerful methods exist to check for CP conservation in multi-higgs-doublet models Invariants can be related to physical couplings and masses Transforming to a Higgs basis offers a simple way to check for spontaneous CP violation The latter approach is useful when vevs are complex

21 Summary Powerful methods exist to check for CP conservation in multi-higgs-doublet models see also talk by Ivo Medeiros Varzielas Invariants can be related to physical couplings and masses Transforming to a Higgs basis offers a simple way to check for spontaneous CP violation The latter approach is useful when vevs are complex