Constraints and SM input

ρ₀^Lρ₀^L†+κ₀^Lκ₀^L†

ρ^L₀+ρ₀^L

ρ^L†₀ ρ^L₀+κ^†L₀ κ₀^L

. (IV.27)

Before ending this section we note that thetan β dependent terms in the evolution equations for the Yukawa couplings disappear in the real case. In the CP-violating caseρis no longer basis-independent and therefore there is a residual dependence ontan βin this case. For a thorough discussion of basis independent quantities in the CP-violating case we refer to [16].

IV.3 Constraints and SM input

IV.3.1 Low-energy constraints on λ

In the recent review of 2HDM [11], the authors have given a comprehensive overview on the latest constraints on theλ^F_ij. The most stringent ones are in the quark sector, coming from the neutral meson mixing, and we will therefore limit ourselves to these constraints in the following.

The master formula forF⁰−F^¯0mixing mediated by tree level Higgs scalars in the vacuum insertion approximation can be found in [17]:

∆M_F = (ρ_ij^F)² M_F

"

S_F c²_β−α m²_h +^s

2β−α

m²_H

! + ^PF

m²_A

#

(IV.28)

S_F = ¹ 6B_Ff_F²M²_F

"

1+ ^M2F (m_i+m_j)²

#

P_F = ¹ 6B_Ff_F²M²_F

"

1+ ^11M

2F

(m_i+m_j)²

#

Here M_F and ∆M_F are the mass and mass difference of the neutral mesons respectively, and f_F is the corresponding pseudo-scalar decay constant. The parameterBFis defined as the ratio of the actual matrix element compared to its value in the vacuum insertion approximation [17]. The numerical values of the parameters we use are listed in TableIV.2.

To calculate the limits onλ_ij^F, we require that the sum of the SM and 2HDM theoretical predictions for∆M_F does not exceed the experimental value by more than 2 standard deviations:

∆M^SM_F +∆M^2HDM_F ≤^∆Mexpt_F +2σ (IV.29) whereσ=^qσ_expt² +σ_SM² is a combination of the experimental and theoretical uncertainties. For theK⁰−K^¯0 and D⁰−D^¯0 mixing, the non-perturbative in-teractions make the SM calculation very difficult. Here we therefore simply

IV.3Constraints and SM input 183

Meson M_F(GeV) B_F f_F(GeV)

K⁰(d ¯s) 0.4976 [18] 0.75±0.026[19] 0.1558±0.0017[19]

D⁰(uc¯ ) 1.8648 [18] 0.82±0.01 [20] 0.165 [20]

B⁰_d(d¯b) 5.2795 [18] 1.26±0.11[19] 0.1928±0.0099 [19]

B⁰_s(s¯b) 5.3663 [18] 1.33±0.06[19] 0.2388±0.0095 [19]

TableIV.2: Parameters of the neutral mesons K⁰, D⁰,B⁰_dand B⁰_s.

assume that the 2HDM contribution is not larger than the experimental value by more than 2 standard deviations. This corresponds to setting the SM con-tribution to zero in Eq. (IV.29) as was done in [15]. The experimental and SM values we thus use are listed below.

1. K⁰−K^¯0:

∆M^expt_K0 = (3.483±0.006) ×10⁻¹⁵GeV [18]

∆M^SM_K0 = 0 2. D⁰−^D¯0

∆M_D^expt₀ = 1.57^+0.39_−0.415×10⁻¹⁴GeV [18]

∆M^SM_D0 = 0 3. B⁰_d−B^¯0_d

∆M^expt_B

d = (3.344±0.0197±0.0197) ×10⁻¹³GeV [18]

∆M^SM_Bd = 3.653^+0.48_−0.30×10⁻¹³GeV [21]

4. B⁰_s−B^¯0_s

∆M^expt_B

s = (116.668±0.270±0.171) ×10⁻¹³GeV [22]

∆M^SM_B

s = 110.6^+17.1_−9.9 ×10⁻¹³GeV [21]

The 2HDM contribution is then calculated using Eq. (IV.28). We note that the quark masses appearing in Eq. (IV.28) are the low energy ones defined more or less at the scale of the respective meson masses. For internal consis-tency we use the following values from ref. [23] (in GeV):

mu(2GeV) = 2.2×10⁻³, mc(mc) =1.25 ;

m_d(2GeV) = 5.0×10⁻³, m_s(2GeV) =0.095 , m_b(m_b) =4.2 .

IV

184 Constraining General 2HDM by the Evolution of Yukawa Couplings

However, the impact of the actual quark masses used is very small since the masses appearing in (ρ_ij^F)² and the dominant pseudo-scalar matrix element M²_F/(m_i+m_j)²essentially cancel, and we get similar results using the masses defined atm_Zinstead.

From Eq. (IV.28) we can see that the main uncertainty of this estimate is due to the unknown masses of the CP-even and CP-odd Higgs bosons. It is also clear that the contribution to the mixing from the CP-odd exchange is much larger due to the extra factor 11 inP_Ffor the dominant pseudo-scalar matrix element. We will consider three different representative cases. We also remind the reader that in some cases there is an extra factor of√

2in the definition of λ_ij^F. With all this in mind we get the following constraints onλ_ij^F:

• m_h=m_H=m_A=120GeV λuc . _{0.13 ,}

λ_ds . 0.08, λ_db.0.03, λ_sb.0.05 .

• m_h=mH=m_A=400GeV λuc . _{0.44 ,}

λ_ds . 0.27 , λ_db.0.12 , λ_sb.0.18 .

• ^mh=m_H=120GeVm_A=400GeV λuc . _{0.30 ,}

λ_ds . 0.20 , λ_db.0.08 , λ_sb.0.12 .

The first and second cases are examples of typical low and intermediate masses for the Higgs bosons, whereas the last case illustrates that the main re-striction comes from the exchange of the CP-odd Higgs. All in all we conclude from these different cases that a representative value for these constraints is given byλ^F_i6=j .0.1and this is the generic value we will use when analyzing the effects ofZ₂ breaking in the running of the Yukawa couplings in the next section.

IV.3.2 General input

For the RGE evolution towards high scales we need a set of input parameters at the low scaleµ=mZ=91.186GeV. The experimental input we have are the masses and the measured parameters of the CKM-mixing matrix as well as the gauge couplings. We have neglected constraints coming from the neutrino sector. The quark and charged lepton masses at the scale m_Z we take from

IV.3Constraints and SM input 185

Ref. [23], their values are (in GeV)

mu = 1.29×10⁻³, mc=0.619 , mt=171.7 ; m_d = 2.93×10⁻³, ms=0.055 , m_b=2.89 ; m_e = 0.487×10⁻³, m_µ=0.103 , m_τ =1.746 . For the3×3CKM matrix we use the PDG [18] phase convention

V_CKM=





c₁₂c₁₃ s₁₂c₁₃ s₁₃e^−iδ

−^s12c₂₃−^c12s₂₃s₁₃e^iδ c₁₂c₂₃−^s12s₂₃s₁₃e^iδ s₂₃c₁₃ s₁₂s₂₃−^c12c₂₃s₁₃e^iδ −^c12s₂₃−^s12c₂₃s₁₃e^iδ c₂₃c₁₃



 , (IV.30)

where s_ij = sin θ_ij and c_ij = cos θ_ij. We will also use this convention for the phases at the high scale. The values for the angles and the phase follow from [18]

s₂₁ = λ , s₂₃=Aλ², s₁₃e^iδ = ^Aλ3(¯ρ+i ¯η)√

1−^A2λ4

√1−λ²

1−A²λ⁴(¯ρ+i ¯η)^{ .} (IV.31) with

λ=0.2253 , A=0.808 , ¯ρ=0.132 , ¯η=0.341 . (IV.32) There is of course still a large freedom in how one chooses the remaining freedom at the weak scalem_Z. We chose to put the CKM-mixing always in the down quark sector and have thus at the EW scale

V_L^U = V_R^U=I

V_L^D = V_CKM^† V_R^D=I V_L^L = V_R^L=I .

The last two are a consequence of our neglecting neutrino masses and mixings.

The Yukawa couplings at the EW scale are thus:

(κ^U₀)_ij = κ^U_ij =

√2m_i

v , (ρ^U₀)_ij=ρ^U_ij (i, j=u, c, t) (κ₀^D)_ij = V_CKMκ_ij^D=V_CKM

√2m_i

v , (ρ^D₀)_ij=V_CKMρ^D_ij (i, j=d, s, b) (κ₀^L)_ij = κ_ij^L=

√2m_i

v , (ρ₀^L)_ij=ρ_ij^L (i, j=e, µ, τ) At any energy higher than the EW scale, the Yukawa couplingsκ0andρ0in general become non-diagonal and complex. Thus they need to be transformed

IV

186 Constraining General 2HDM by the Evolution of Yukawa Couplings

to the mass eigenstates by the bi-diagonalization defined in Eq. (IV.13) in or-der to giveκandρ. The latter can then be used together with the diagonal ele-ments of the former to calculateλ^F_i6=j. When performing the bi-diagonalization we always keep to the PDG conventions for how to write the CKM matrix.

For the electroweak VEV we usev² = 1/(√

2G_F) withG_F = 1.16637·10⁻⁵ GeV⁻²from PDG [18] and for the phase difference between the two VEVs we start fromθ=0such that there is no spontaneous CP-violation. For the gauge couplings we use the PDG [18] values: α = 1/127.91, αs = 0.118and for the weak mixing angle we use the on-shell valuesin²θ_W=0.2233.