IFIC/18-004 CFTP/18-004
Flavour Conservation in Two Higgs Doublet Models Francisco J. Botella
a,1
, Fernando Cornet-Gomez
a,2
, Miguel Nebot
b,3
a
arXiv:1803.08521v1 [hep-ph] 22 Mar 2018
Departament de F´ısica Te`orica and IFIC, Universitat de Val`encia-CSIC, E-46100, Burjassot, Spain. b Centro de F´ısica Te´orica de Part´ıculas (CFTP), Instituto Superior T´ecnico (IST), U. de Lisboa (UL), Av. Rovisco Pais, P-1049-001 Lisboa, Portugal.
Abstract In extensions of the Standard Model with two Higgs doublets, flavour changing Yukawa couplings of the neutral scalars may be present at tree level. In this work we consider the most general scenario in which those flavour changing couplings are absent. We re-analyse the conditions that the Yukawa coupling matrices must obey for such general flavour conservation (gFC), and study the one loop renormalisation group evolution of such conditions in both the quark and lepton sectors. We present some new Cabibbo like solution in the quark sector and we show that gFC in the leptonic sector is one loop RGE stable. At a phenomenological level, we obtain the regions for the different gFC parameters that are allowed by the existing experimental constraints related to the 125 GeV Higgs.
1
[email protected] [email protected] 3
[email protected] 2
1
Introduction
Two Higgs Doublet Models (2HDM) [1–3] are a simple and popular class of extensions of the Standard Model (SM). Besides the original motivation, in particular the possibility of having spontaneous CP violation [1], extending the SM scalar sector with a second doublet allows a number of interesting phenomenological consequences. To name a few generic ones: the appearance of new fundamental scalar particles, nonstandard properties of the “quite Higgs-like” scalar discovered at the LHC with a mass of 125 GeV [4, 5], and, related to them, a number of potential deviations in low energy processes with respect to SM expectations. They have been the focus of intense scrutiny before and after the 2012 discovery [6–28]. Additional aspects, including dark matter candidates [29, 30] or sources of CP violation in addition to the Cabibbo-KobayashiMaskawa matrix [31–33], of interest for baryogenesis [34–36], provide further interest in 2HDM. In the SM, concentrating on quarks, a single Yukawa structure in each sector – up and down – is both responsible for: (i) the generation of mass upon spontaneous breaking of SU (2)L ⊗ U (1)Y into U (1)EM , and (ii) the couplings of the quarks to the only fundamental scalar leftover, the Higgs boson, after associating the three would-be Goldstone bosons to the longitudinal polarizations of the massive Z and W ± gauge bosons. As a consequence, there are no tree level Flavour Changing Neutral Couplings (FCNC) of the Higgs to quarks. With two independent Yukawa structures available in each sector, the situation is dramatically changed in the general 2HDM, and FCNC couplings of quarks do arise at tree level. To which extent they appear in the couplings of the different physical neutral scalars depends then on the details of the scalar potential [37]: if the 125 GeV scalar is a mixture of the true-but-unphysical Higgs and the additional neutral scalars, FCNC “leak” into its couplings through that mixing. At the end of the day, as with many New Physics avenues, the presence of FCNC is a double edged feature: since the competing SM gauge mediated contributions to FCNC processes are loop induced, those transitions pose severe constraints while, on the same grounds, provide immediate opportunities to discover deviations from the SM picture. The study of different ways to dispense without problematic too large FCNC couplings and the conditions for their appearance or absence, has drawn sustained attention over the years. As analysed in [38, 39], the absence of FCNC is guaranteed by forcing each right-handed fermion type to couple to one and only one scalar doublet; this absence of FCNC, backed by a Z2 symmetry, is a popular option, and several implementations of this Natural Flavour Conservation (NFC) idea, namely 2HDM of types I, II, and of types X, Y (when the lepton sector is also considered) have been thoroughly explored. The general conditions for the absence of FCNC, that is, that the mass matrix and the remaining Yukawa coupling matrix can be diagonalised simultaneously, were identified early [40–43]. The interplay of how a symmetry requirement could enforce that general NFC and shed some light into the structure of the resulting CKM matrix was thoroughly addressed in [40, 44–51] with interesting consequences. On a different line of thought, stepping back from right out forbiddance, suppression of FCNC in other “natural” manners has also attracted significant interest, including suppression given by masses like in the Cheng-Sher ans¨atz [52], suppression obtained 1
from broken/approximate flavour symmetries [53–56], and symmetry controlled FCNC scenarios [57–61]. Among the later, Branco-Grimus-Lavoura (BGL) models are worth mentioning in particular, since this suppression is simply given by products of CKM matrix elements [62, 63] (see also related extensions [64–66]). In a more recent popular scenario, the Aligned 2HDM [67], the absence of FCNC is a priori achieved (and parametrised) with simple requirements on the Yukawa couplings (for an early mention of this kind of possibility, although in the context of real Yukawa couplings and spontaneous CP violation, see also [68]). The possibility of having effective aligned scenarios has been studied in [69, 70]. Radiative effects and the interplay of tree level FCNC with the Renormalization Group Evolution (RGE) have also been addressed by and large in the literature [9, 47, 71–77]. The aim of this work is to explore different facets of scenarios with general flavour conservation (gFC), i.e. generalised flavour alignment, in 2HDM; in other words, analysing relevant aspects of the most general 2HDM scenarios where tree level FCNC are, a priori, absent. An analysis of FCNC induced in this context by the RGE has been recently presented in [78]. On a purely phenomenological basis, a scenario of this type restricted to the lepton sector was also considered in [79, 80]. The paper is organised as follows. In section 2, we revisit some generalities of 2HDM, fix the notation for the discussion to follow, and recall the most relevant aspects of the conditions leading to gFC. They are then analysed attending to the Renormalization Group Evolution that they obey in section 3, leading to the full set of conditions required to have RGE-stable gFC. The well known type I and type II cases are briefly revisited in section 3.3; section 3.5 is devoted to a particular solution which arises when the CKM matrix is reduced to a single Cabibbo-like mixing. The gFC stability of the lepton sector is discussed in section 3.4. In section 4, we discuss the most relevant experimental constraints on gFC arising from flavour conserving Higgs-related observables, leading to the analysis and results of section 4.4. Appendix A shows that the conditions for the simultaneous diagonalisation of the Yukawa couplings discussed in [40–43] can be drastically reduced to a weaker requirement. Appendix B provides details omitted in the discussion of section 3.
2
Yukawa Couplings and General Flavour Conservation
The Higgs doublets (j = 1, 2) of 2HDM are ϕ+ iθj j √ Φj = e (vj + ρj + iηj )/ 2
(1)
where vj , θj are real numbers, ρj , ηj , are neutral (hermitian) fields and ϕ± j are charged fields. Equation (1) anticipates the assumption that the scalar potential V (Φ1 , Φ2 ) [2,3] is such that V (hΦ1 i, hΦ2 i) has an appropriate minimum at 0 √ hΦj i = iθj . (2) e vj / 2 2
In the “Higgs basis” [7, 37, 81], only one linear combination of Φ1 and Φ2 , H1 , has a non-vanishing vacuum expectation value, −iθ v 0 H1 e 1 Φ1 0 , hH1 i = √ = Rβ −iθ2 , hH2 i = , (3) Φ2 H2 e 0 2 1 with v =
p v12 + v22 , cβ = cos β ≡ v1 /v, sβ = sin β = v2 /v and cβ sβ Rβ ≡ , [Rβ ]−1 = [Rβ ]T . −sβ cβ
The expansion of H1 , H2 around that minimum of the potential reads G+ H+ √ √ H1 = , H2 = , (v + H0 + iG0 )/ 2 (R0 + iI0 )/ 2 where
0 + 0 + ρ η1 H ϕ1 G G = Rβ 1 . = Rβ , = Rβ + , 0 0 + R ρ2 I η2 H ϕ2
(4)
(5)
(6)
The would-be Goldstone bosons G0 and G± provide the longitudinal degrees of freedom of the Z and W ± gauge bosons; furthermore, while H± is already a physical charged scalar field, the physical neutral scalars {h, H, A} are real linear combinations of {H0 , R0 , I0 }, 0 h H α)]−1 = [R[3] (~ α)]T , (7) R[3] (~ α) H = R0 , [R[3] (~ 0 I A with [R[3] (~ α)] a real orthogonal rotation described by three real mixing angles, α ~ = {α12 , α13 , α23 } (cx ≡ cos x, sx ≡ sin x), cα12 −sα12 0 cα13 0 −sα13 1 0 0 0 0 cα23 −sα23 . [R[3] (~ α)] = sα12 cα12 0 0 1 (8) 0 0 1 sα13 0 cα13 0 sα23 cα23 When there is no CP violation in the scalar potential, CP-even H0 , R0 , and the CP-odd I0 , it is customary to h s α cα ρ1 sβα = = H −cα sα ρ2 −cβα
i.e. no mixing connecting the introduce the mixing angle α 0 cβα H (9) sβα R0
where sβα = sin(α + β) and cβα = cos(α + β) (that is, α13 = α23 = 0 and α12 = π/2 − (α + β) in Eq. (8)). Since a ± sign can be included in the definition of the scalar fields without changing their kinetic terms, different conventions for Eqs. (8)–(9) are used in the literature, which may be relevant when comparing expressions.
3
2.1
Quark Yukawa Couplings in 2HDM
The Yukawa couplings of the quarks – SU (2)L doublets Q0L and singlets d0R , u0R – with the scalar doublets read [q] ¯ 0 [Γ1 Φ1 + Γ2 Φ2 ]d0 − Q ¯ 0 [∆1 Φ ˜ 1 + ∆2 Φ ˜ 2 ]u0 + H.c. LY = −Q L R L R
(10)
˜ j = iσ2 Φ∗ . Following Eqs. (3)–(5), with Φ j √
[q] LY
√ 2 ¯0 2 ¯0 0 0 0 ˜ 1 + Nu0 H ˜ 2 ]u0R + H.c. QL [Md H1 + Nd H2 ]dR − Q [M0u H =− v v L [q] [q] [q] = Lm[q] + LG + LCh + LN [q]
(11) (12)
[q]
with mass terms Lm , would-be Goldstone boson couplings LG , and Yukawa couplings [q] [q] to charged and neutral scalars, LCh and LN : Lm[q] ⊃ − d¯0L M0d d0R − u¯0L M0u u0R , (13) √ 2 + 0 0 0 [q] LG ⊃ − G u¯L Md dR + iG0 d¯0L M0d d0R − G− d¯0L M0u u0R − iG0 u¯0L M0u u0R , (14) √v 2 + 0 0 0 [q] H u¯L Nd dR − H− d¯0L Nu0 u0R , (15) LCh ⊃ − v 1 0 ¯0 0 0 [q] LN ⊃ − H dL Md dR + (R0 + iI0 )d¯0L Nd0 d0R + H0 u¯0L M0u u0R + (R0 − iI0 )¯ u0L Nu0 u0R . v (16) The mass matrices are i v h M0d = √ eiθ1 cβ Γ1 + eiθ2 sβ Γ2 , 2
i v h M0u = √ e−iθ1 cβ ∆1 + e−iθ2 sβ ∆2 , 2
(17)
and the second linear combinations of Yukawa matrices which encode the potential FCNC are i i v h v h (18) Nd0 = √ −eiθ1 sβ Γ1 + eiθ2 cβ Γ2 , Nu0 = √ −e−iθ1 sβ ∆1 + e−iθ2 cβ ∆2 . 2 2 For the usual bi-diagonalisation of the mass matrices M0d , M0u , the quark mass eigenstates (without “0” superscript) read dL = Ud†L d0L , dR = Ud†R d0R , uL = Uu†L u0L , uR = Uu†R u0R ,
(19)
with Md = Ud†L M0d UdR = diag(md , ms , mb ) , Mu = Uu†L M0u UuR = diag(mu , mc , mt ) , (20) Nd = Ud†L Nd0 UdR , Nu = Uu†L Nu0 UuR .
(21)
The CKM matrix is V = Uu†L UdL . When both Nd and Nu are diagonal, tree-level FCNC are absent. 4
Expressing Eq. (11) in terms of quark and scalar mass eigenstates (as a shorthand we use [R[3] (~ α)]ij = Rij ), Lm[q] = − d¯L Md dR − u¯L Mu uR + H.c. , (22) √ 2 + [q] LG = − G u¯L V Md dR + iG0 d¯L Md dR + G− d¯L V † Mu uR − iG0 u¯L Mu uR + H.c., v (23) (24) [q]
LCh =
√ n h io 2 − H+ u¯L V Nd dR − u¯R Nu† V dL + H− d¯R Nd† V † uL − d¯L V † Nu uR , (25) v
[q]
LN = − − − − − −
h ¯ d [R11 Md + R21 Hd + iR31 Ad ] d + d¯[R21 Ad + iR31 Hd ] γ5 d v h {¯ u [R11 Mu + R21 Hu − iR31 Au ] u + u¯ [R21 Au − iR31 Hu ] γ5 u} v H ¯ d [R12 Md + R22 Hd + iR32 Ad ] d + d¯[R22 Ad + iR32 Hd ] γ5 d v H {¯ u [R12 Mu + R22 Hu − iR32 Au ] u + u¯ [R22 Au − iR32 Hu ] γ5 u} v A ¯ d [R13 Md + R23 Hd + iR33 Ad ] d + d¯[R23 Ad + iR33 Hd ] γ5 d v A {¯ u [R13 Mu + R23 Hu − iR33 Au ] u + u¯ [R23 Au − iR33 Hu ] γ5 u} , v
(26)
where
Nq + Nq† Nq − Nq† , Aq ≡ , q = u, d, 2 2 are the hermitian and anti-hermitian combinations of Nq and Nq† . With no CP violation in the scalar sector, sβα −cβα 0 sβα 0 , R = cβα 0 0 1 Hq ≡
(27)
(28)
and Eq. (26) reduces to [q]
h ¯ ¯ d γ5 d d [sβα Md + cβα Hd ] d + cβα dA v h − {¯ u [sβα Mu + cβα Hu ] u + cβα u¯Au γ5 u} v H ¯ ¯ d γ5 d − d [−cβα Md + sβα Hd ] d + sβα dA v H − {¯ u [−cβα Mu + sβα Hu ] u + sβα u¯Au γ5 u} v A ¯ ¯ d γ5 d + i A {¯ dAd d + dH uAu u + u¯Hu γ5 u} . −i v v
LN = −
5
(29)
2.2
Lepton Yukawa Couplings in 2HDM
The Yukawa couplings of the lepton SU (2)L doublets L0L and singlets `0R with the scalar doublets are √ 2 ¯0 [`] 0 0 ¯ L (M0` H1 + N`0 H2 )`0R + H.c. (30) LY = −LL (Π1 Φ1 + Π2 Φ2 )`R + H.c. = − v L where, similarly to the quark sector in the previous section, i i v h v h M0` = √ eiθ1 cβ Π1 + eiθ2 sβ Π2 , N`0 = √ −eiθ1 sβ Π1 + eiθ2 cβ Π2 . 2 2
(31)
The mass eigenstates, without “0” superscript, correspond to `L = U`†L `0L ,
`R = U`†R `0R ,
(32)
and M` = U`†L M0` U`R = diag(me , mµ , mτ ),
N` = U`†L N`0 U`R .
(33)
Notice that we do not include right-handed neutrinos νR0 and thus, unlike in the quark sector, there is only one set of Yukawa coupling matrices and we work in the massless neutrino approximation. The leptonic analogs of the Yukawa couplings in Eqs. (25)(26) are √ n o 2 [`] LCh = − H+ ν¯L N` `R + H− `¯R N`† νL , (34) v [`]
h ¯ ` [R11 M` + R21 H` + iR31 A` ] ` + `¯[R21 A` + iR31 H` ] γ5 ` v H ¯ − ` [R12 M` + R22 H` + iR32 A` ] ` + `¯[R22 A` + iR32 H` ] γ5 ` v A ¯ − ` [R13 M` + R23 H` + iR33 A` ] ` + `¯[R23 A` + iR33 H` ] γ5 ` , v
LN = −
with H` ≡
2.3
N` + N`† , 2
A` ≡
N` − N`† . 2
(35)
(36)
General Flavour Conservation
As anticipated in the Introduction and shown in appendix A, the necessary and sufficient conditions obeyed by the quark Yukawa coupling matrices Γα , ∆α , α = 1, 2, in order to have gFC, are that each of the sets {Γα Γ†α }, {Γ†α Γα }, {∆α ∆†α }, {∆†α ∆α },
α = 1, 2,
(37)
is abelian, that is, their elements commute: h i i h i h i h † † † † † † † † Γα Γα , Γβ Γβ = 0, Γα Γα , Γβ Γβ = 0, ∆α ∆α , ∆β ∆β = 0, ∆α ∆α , ∆β ∆β = 0, (38) 6
with α, β = 1, 2. In that case, {Γ1 , Γ2 } are simultaneously bi-diagonalised, and {∆1 , ∆2 } too. In references [40–43], rather than the sets in Eq. (37), the abelian nature of each set {Γα Γ†β }, {Γ†α Γβ }, {∆α ∆†β }, {∆†α ∆β },
α, β = 1, 2,
(39)
was analysed, that is i h i i h i h h Γα Γ†β , Γγ Γ†δ = 0, Γ†α Γβ , Γ†γ Γδ = 0, ∆α ∆†β , ∆γ ∆†δ = 0, ∆†α ∆β , ∆†γ ∆δ = 0, (40) with α, β, γ, δ = 1, 2. As we show in appendix A, Eq. (38) implies Eq. (40), and thus it is the simpler Eq. (38) that we will need to consider. A crucial corollary to these necessary and sufficient conditions is the fact that the simultaneous diagonalisability is intrinsic to the Yukawa coupling matrices themselves, independently of the spontaneous symmetry breaking vacuum characterised by the VEVs v1 , v2 . In other words, the property is independent of β in Eqs. (17), (18); the simultaneous bi-diagonalisability of {M0q , Nq0 } is equivalent to the simultaneous bidiagonalisability of the Yukawa couplings matrices or of any other independent linear combinations of them. Of course, the actual values of the eigenvalues of both M0q (the masses) and Nq0 do depend on the particular linear combinations. For leptons, similarly, {Πα Π†α } and {Π†α Πα } must be abelian in order to have gFC, and the previous corollary applies equally to them. A very relevant consequence follows [40, 44–51]: if gFC is due to the Lagrangian in Eq. (10) being invariant under a (symmetry) transformation of quarks and scalars, the CKM mixing matrix cannot be related to the values of the masses; for example, predictions being made at the time (late 70’s)4 for the Cabibbo angle, like tan θc = md /ms [82, 83], could not lead simultaneously to gFC. Moreover, the resulting mixings are unrealistic (for example, no mixing or a permutation times a complex phase) and radiative corrections cannot be invoked to yield realistic mixings [47]. The most general parameterisation of tree level couplings of fermions to scalars obeying gFC is, quite trivially, nd 0 0 nu 0 0 ne 0 0 Nd = 0 ns 0 , Nu = 0 nc 0 , N` = 0 nµ 0 , nj ∈ C, (41) 0 0 nb 0 0 nt 0 0 nτ which we use in the rest of the paper: in section 3 for the study of the renormalization group evolution and in section 4 for a phenomenological analysis. Notice that, while for the flavour changing couplings the simultaneous presence of scalar and pseudoscalar terms in fermion-scalar Yukawa interactions is not necessarily CP violating, in the diagonal, flavour conserving ones, on the contrary, it is CP violating (see for example [84]). With the flavour conserving matrices Nf in Eq. (41), the hermitian and antihermitian couplings in Eqs. (29) and (35) are, respectively, their real and imaginary parts. For example, for a CP conserving scalar sector with non-zero 4
In the context of SU (2)L ⊗ U (1)Y gauge theories; the literature is richer in examples for SU (2)L ⊗ SU (2)R ⊗ U (1)Y scenarios.
7
mixing cβα 6= 0, if Nf are not real, they constitute new sources of CP violation in neutral couplings. For the couplings to the charged scalar, without entering into details, if Im(nui ndj ) 6= 0, the combination of scalar and pseudoscalar terms in the coupling H+ u¯i dj is CP violating.
3
Renormalization Group Evolution and Flavour Conservation
3.1
Evolution of the Quark Yukawa Coupling Matrices
The one loop evolution of the Yukawa couplings under the renormalization group [73– 75, 85] is (with D ≡ 16π 2 d lnd µ and µ the energy scale): DΓα = ad Γα +
n=2 X
d Γρ Tα,ρ
ρ=1
D∆α = au ∆α +
n=2 X
n=2 X 1 1 † † † † + −2∆ρ ∆α Γρ + Γα Γρ Γρ + ∆ρ ∆ρ Γα + Γρ Γρ Γα 2 2 ρ=1 d with Tα,ρ ≡ 3 tr Γα Γ†ρ + ∆†α ∆ρ + tr Πα Π†ρ , (42)
u Tα,ρ ∆ρ
+
n=2 X ρ=1
ρ=1
1 1 † † + + Γρ Γρ ∆α + ∆ρ ∆ρ ∆α 2 2 d∗ ≡ 3 tr ∆α ∆†ρ + Γ†α Γρ + tr Π†α Πρ = Tα,ρ , (43)
−2Γρ Γ†α ∆ρ
u with Tα,ρ
∆α ∆†ρ ∆ρ
where
5 9 au = ad − g 02 , (44) ad = −8gs2 − g 2 − g 02 , 4 12 with gs , g, g 0 the gauge coupling constants of SU (3)c , SU (2)L and U (1)Y , respectively. Introducing ΓL =
n=2 X
Γρ Γ†ρ ,
ΓR =
ρ=1
n=2 X
Γ†ρ Γρ ,
∆L =
ρ=1
n=2 X ρ=1
∆ρ ∆†ρ ,
and ∆R =
n=2 X
∆†ρ ∆ρ , (45)
ρ=1
Eqs. (42)–(43) read DΓα = ad Γα +
d Tα,ρ Γρ
n=2 X 1 1 ∆ρ ∆†α Γρ , + Γα ΓR + ΓL Γα + ∆L Γα − 2 2 2 ρ=1
(46)
u Tα,ρ ∆ρ
n=2 X 1 1 + ∆α ∆R + ∆L ∆α + ΓL ∆α − 2 Γρ Γ†α ∆ρ . 2 2 ρ=1
(47)
n=2 X ρ=1
D∆α = au ∆α +
n=2 X ρ=1
Equations (46)–(47) are the starting point to analyse the one loop stability of the necessary and sufficient conditions for gFC. For that, one needs to know i h i h i h i h † † † † † † † † D Γα Γβ , Γγ Γδ , D Γα Γβ , Γγ Γδ , D ∆α ∆β , ∆γ ∆δ , D ∆α ∆β , ∆γ ∆δ , (48) 8
under the assumption that Eq. (38) or Eq. (40) hold. With that objective in mind, some simplifications are worth mentioning. Starting with Γα Γ†β , we first notice that D(Γα Γ†β ) = (DΓα )Γ†β + Γα (DΓβ )† = fαβL (Γ) + gαβL (Γ, ∆) , [d ]
[d ]
(49)
with5 fαβL (Γ) = 2ad Γα Γ†β + [d ]
n=2 h X ρ=1
i 1 1 d∗ d Γα Γ†ρ + 2Γα ΓR Γ†β + ΓL Γα Γ†β + Γα Γ†β ΓL , Tα,ρ Γρ Γ†β + Tβ,ρ 2 2 (50)
and [d ] gαβL (Γ, ∆)
n=2 h i X 1 1 † † ∆ρ ∆†α Γρ Γ†β + Γα Γ†ρ ∆β ∆†ρ . = ∆L Γα Γβ + Γα Γβ ∆L − 2 2 2 ρ=1
(51)
[d ]
The relevant property of the decomposition in Eq. (49) is that fαβL depends only6 , [d ]
in terms of matrices, on ΓΓ† and ΓΓ† ΓΓ† terms, while gαβL collects the remaining dependence on ∆’s, which has terms ΓΓ† ∆∆† and ∆∆† ΓΓ† . Then, h i h i h i D Γα Γ†β , Γγ Γ†δ = D(Γα Γ†β ) , Γγ Γ†δ + Γα Γ†β , D(Γγ Γ†δ ) = h i h i [d ] [d ] [d ] [d ] fαβL (Γ) + gαβL (Γ, ∆) , Γγ Γ†δ + Γα Γ†β , fγδL (Γ) + gγδL (Γ, ∆) . (52)
3.2
Evolution with gFC Matrices
It is clear that, if there is gFC, i.e. with Eq. (38), h i h i [d ] [d ] fαβL (Γ) , Γγ Γ†δ = Γα Γ†β , fγδL (Γ) = 0 ,
(53)
and thus h i h i h i [d ] [d ] D Γα Γ†β , Γγ Γ†δ = gαβL (Γ, ∆) , Γγ Γ†δ + Γα Γ†β , gγδL (Γ, ∆) .
(54)
After the simplication brought by Eq. (53), the next step is to trade Eq. (54) for conditions expressed in terms of the physical parameters entering in the matrices M0d , Nd0 , M0u , Nu0 . It is convenient to introduce the following notation 0 Y[d]1 = M0d , 5
0 Y[d]2 = Nd0 ,
0 Y[u]1 = M0u ,
0 Y[u]2 = Nu0 ,
(55)
The superscript [dL ] is chosen in correspondence with the Γα Γ†β matrix combinations; similarly [u
]
fαβR and gαβR will appear in D(Γ†α Γβ ), and fαβL,R in D(∆α ∆†β ) and D(∆†α ∆β ), but we concentrate [d ]
[d ]
for the moment on D(Γα Γ†β ). 6 d Although Tα,ρ do depend on ∆α ’s, there is no matrix depence, only C numbers; this also applies to the leptonic Yukawa couplings Πα .
9
which allows us to rewrite Eqs. (17)-(18) compactly (with summation over repeated indices understood): v 0 √ Γα = Wαi Y[d]i , 2 v 0 ∗ √ ∆α = Wαi Y[u]i , 2 where
v 0 ∗ Y[d]i = √ Γα Wαi , 2 v 0 Y[u]i = √ ∆α Wαi , 2
−iθ e 1 0 cβ −sβ W= , 0 e−iθ2 s β cβ
WW† = W† W = 1 .
(56)
(57)
For completeness, notice that v v v v 0† 0† 0† 0† ∗ ∗ √ Γ†α = Wαi Y[d]i , √ ∆†α = Wαi Y[u]i , Y[d]i = √ Γ†α Wαi , Y[u]i = √ ∆†α Wαi , 2 2 2 2
(58)
i.e. the Hermitian conjugate † (in the space of flavour indices) only gives a complex conjugate in W. One can then write h i h i 4 0 0† 0 0† ∗ ∗ Γα Γ†β , Γγ Γ†δ = 4 Wαi Wβj Wγk Wδl Y[d]i Y[d]j , Y[d]k Y[d]l , (59) v and thus D
h
Γα Γ†β
,
Γγ Γ†δ
i
=D
h i 4 0 0† 0 0† ∗ ∗ W W W W Y Y , Y Y [d]i [d]j [d]k [d]l v 4 αi βj γk δl h i 4 0 0† 0 0† ∗ ∗ + 4 Wαi Wβj Wγk Wδl D Y[d]i Y[d]j , Y[d]k Y[d]l . (60) v
With gFC, the first commutator vanishes, and we just have a linear combination of 0 0† 0 0† different D[Y[d]i Y[d]j , Y[d]k Y[d]l ]. One can indeed invert Eq. (60), i h i 4 h 0 0† 0 0† † † ∗ ∗ D Y Y , Y Y = W W W W D Γ Γ , Γ Γ αi βj γk δl α β γ δ [d]i [d]j [d]k [d]l v4
(61)
0 0 and express the right-hand side of Eq. (61) in terms of Y[d]i , Y[u]j :
i v 2 h 0 0† 0 0† D Y[d]i Y[d]j , Y[d]k Y[d]l = 2 0 0† 0 0† 0 0† 0 0† 0 0† 0 0† Y[d]i Y[d]j Y[u]h Y[u]h Y[d]k Y[d]l − Y[d]k Y[d]l Y[u]h Y[u]h Y[d]i Y[d]j h i h i 0 0† 0 0† 0 0† 0 0† 0 0† 0 0† − 2 Y[u]h Y[u]i , Y[d]k Y[d]l Y[d]h Y[d]j − 2Y[d]i Y[d]h Y[u]j Y[u]h , Y[d]k Y[d]l h i h i 0 0† 0 0† 0 0† 0 0† 0 0† 0 0† + 2 Y[u]h Y[u]k , Y[d]i Y[d]j Y[d]h Y[d]l + 2Y[d]k Y[d]h Y[u]l Y[u]h , Y[d]i Y[d]j .
(62)
As expected from the discussion in section 2.3, having a gFC scenario is related to the Yukawa coupling matrices themselves, it does not hinge on the particular EW vacuum configuration that determines which particular combinations of them are the mass matrices M0d , M0u and the matrices Nd0 , Nu0 (the vacuum configuration is “encoded” 10
in W, which does not appear in Eq. (62)). The last step is to transform into the mass eigenstate basis with UdL in Eq. (19): i v 2 † h 0 0† 0 0† U D Y[d]i Y[d]j , Y[d]k Y[d]l UdL = 2 dL † † † † † † Y[d]i Y[d]j V † Y[u]h Y[u]h V Y[d]k Y[d]l − Y[d]k Y[d]l V † Y[u]h Y[u]h V Y[d]i Y[d]j h i h i † † † † † † − 2 V † Y[u]h Y[u]i V , Y[d]k Y[d]l Y[d]h Y[d]j − 2Y[d]i Y[d]h V † Y[u]j Y[u]h V , Y[d]k Y[d]l h i h i † † † † † † † † + 2 V Y[u]h Y[u]k V , Y[d]i Y[d]j Y[d]h Y[d]l + 2Y[d]k Y[d]h V Y[u]l Y[u]h V , Y[d]i Y[d]j , (63) where the CKM matrix V = Uu†L UdL appears together with the diagonal matrices Y[d]1 = Md , Y[d]2 = Nd , Y[u]1 = Mu , Y[u]2 = Nu .
(64)
In this generic notation – Eq. (55) –, d d d d Y[d]i = diag(yi,j ), {y1,1 , y1,2 , y1,3 } = {md , ms , mb }, d d d {y2,1 , y2,2 , y2,3 } = {nd , ns , nb }, u u u u Y[u]i = diag(yi,j ), {y1,1 , y1,2 , y1,3 } = {mu , mc , mt }, u u u {y2,1 , y2,2 , y2,3 } = {nu , nc , nt }.
(65)
The previous derivation concerns the set {Γα Γ†β }; the evolution equations for {Γ†α Γβ }, {∆α ∆†β } and {∆†α ∆β } are given in appendix B. In order to have a gFC scenario stable under the one loop RGE, one needs that the 0 0 simultaneous diagonalisability of {Y[q]1 , Y[q]2 } is preserved, that is h i h i 0 0† 0 0† 0 0† 0 0† D Y[d]i Y[d]i , Y[d]j Y[d]j = 0 , D Y[u]i Y[u]i , Y[u]j Y[u]j = 0, h i h i 0† 0 0† 0 0† 0 0† 0 D Y[d]i Y[d]i , Y[d]j Y[d]j = 0 , D Y[u]i Y[u]i , Y[u]j Y[u]j = 0.
(66)
With Eqs. (63)–(65), the conditions expressed by the matrix equations in (66) are formulated in full generality, for fixed mass matrices Md , Mu , and CKM mixings V , in terms of the 6 complex parameters nj in Eq. (41). For example, element (a, b) of the 0 0† 0 0† first stability condition in Eq. (66), D[Y[d]i Y[d]i , Y[d]j Y[d]j ] = 0, reads 0=
3 X 2 X
n d 2 d 2 u 2 d 2 d 2 | |yi,a | |yj,b | − |yi,b | |yj,a | Vqa∗ Vqb |yh,q
(67)
q=1 h=1 u u∗ d d∗ u u∗ d d∗ yh,b yi,b + yi,q yh,q yi,a yh,a − 2 yh,q yi,q
u u∗ d d∗ u∗ d d∗ u + yj,q yh,q yj,a yh,a + 2 yh,q yj,q yh,b yj,b
d 2 d 2 | |yj,b | − |yj,a
d 2 d 2 |yi,b | − |yi,a |
o .
The complete set of conditions is given in appendix B. For each set in Eq. (66) there is only one choice i = 1, j = 2 in 2HDM, which gives, as shown in the appendix, 3 11
independent complex equations. It is clear that, with 12 complex conditions overall, in terms of the 6 complex parameters nj , the system is largely overconstrained. In section 3.3 below, we check that the known stable solutions with Nf ∝ Mf are recovered. It is however beyond the scope of this work to address if other solutions could a priori exist for the general one loop RGE stability conditions of gFC. The lepton sector is discussed in section 3.4. Finally, in section 3.5, we present some particular solutions which arise when the CKM matrix reduces to a Cabibbo-like block diagonal mixing.
3.3
Stable gFC with Nf ∝ Mf
When one substitutes Nq = αq Mq , αq ∈ C, in the conditions for one loop RGE stability of gFC given in appendix B, solving them for αu , αd , reduces to finding solutions of (1 + αd αu )(αu∗ − αd ) = 0 ,
(68)
that is αu = −αd−1 or αu∗ = αd . In both cases, there is a basis for the scalars [75] 0 1 H1 1 αd H1 , (69) =p ∗ 0 H2 H2 1 + |αd |2 −αd 1 with H1 and H2 in Eq. (3), such that in Eq. (11) H10 couples only to d0R while H20 couples only to u0R for αu = −αd−1 ; for αu = αd∗ , H10 couples to d0R and u0R , while H20 does not. These cases are none other than the 2HDM of type II and I respectively. For the particular case αu = αd∗ = 0, the scalar doublet which has a zero vacuum expectation value has vanishing Yukawa couplings: this is the Inert 2HDM [29].
3.4
Stable gFC in the Lepton Sector
The one loop RGE of the lepton Yukawa couplings in Eq. (30) reads [85, 86] DΠα = a` Πα +
n=2 X
` Tα,ρ Πρ
ρ=1
where a` = − 94 g 2 −
15 02 g . 4
n=2 X 1 † † + Πα Πρ Πρ + Πρ Πρ Πα 2 ρ=1
With ΠL =
DΠα = a` Πα +
Pn=2 ρ=1
Πρ Π†ρ , ΠR =
` d with Tα,ρ = Tα,ρ ,
Pn=2 ρ=1
(70)
Π†ρ Πρ ,
n=2 X
1 ` Tα,ρ Πρ + Πα ΠR + ΠL Πα . 2 ρ=1
(71)
The crucial difference in the leptonic sector is that, following Eq. (71), D(Πα Π†β )
=
2a` Πα Π†β
n=2 X `∗ ` + Tα,ρ Πρ Π†β + Tβ,ρ Πα Π†ρ + 2Πα ΠR Π†β ρ=1
+ 12
1 ΠL Πα Π†β + Πα Π†β ΠL , (72) 2
and thus it is clear that, if {Πα Π†α }α=1,2 is abelian, then i i h i h h † † † † † † D Πα Πα , Πβ Πβ = D(Πα Πα ) , Πβ Πβ + Πα Πα , D(Πβ Πβ ) = 0 .
(73)
Similarly, D(Π†α Πβ )
=
2a` Π†α Πβ
+
n=2 X
`∗ ` Tα,ρ Π†ρ Πβ + Tβ,ρ Π†α Πρ + ΠR Π†α Π†β + Π†α Π†β ΠR + Π†α ΠL Πβ ,
ρ=1
(74) and thus, if {Π†α Πα }α=1,2 is abelian, then h i h i h i D Π†α Πα , Π†β Πβ = D(Π†α Πα ) , Π†β Πβ + Π†α Πα , D(Π†β Πβ ) = 0 .
(75)
That is, if the Yukawa couplings of leptons are gFC, as in Eq. (41), this is not altered by the RGE: general flavour alignment is one-loop stable in the lepton sector. This can be directly traced back to the absence of right-handed neutrinos and Yukawa couplings involving them in Eq. (30), in clear contrast with the quark sector. This result represents a generalization of previous results restricted to the so called aligned case and pointed out in [77], in agreement with the findings of [80,87]: at one loop level the charged lepton sector remains general Flavour Conserving in full generality without any additional constraint. To be specific and going to the simplest aligned cases, type I, II, X and Y models are defined in the quark sector by Nd = cotβ Md , Nd = −tanβ Md , Type I,X Type II,Y (76) Nu = cotβ Mu , Nu = cotβ Mu . The fact that the leptonic sector alignment was known to be stable under RGE implies that one could analyse the experimental data with previous equation together with the more general leptonic structure (Π2 = ξ` e−iθ Π1 ) − tan β + ξ` M` (77) N` = cotβ cotβ + ξ` in the framework of a model one loop stable under RGE. This would include in a single analysis both type I and X or type II and Y. Note that with the appropriate limits ξ` → 0 or ξ` → ∞ one recovers the four models. Equation (75) implies the new more general result that the models implemented by Eq. (76) together with an arbitrary diagonal N` (not just with Eq. (77)) are one loop stable under RGE.
3.5
Stable gFC with Cabibbo-like mixing
The CKM matrix has a hierarchical structure; keeping only the largest mixing, it has the form cos θc sin θc 0 Vθc = − sin θc cos θc 0 (78) 0 0 1 13
with θc ' 0.22 the Cabibbo mixing angle. It is interesting to analyse the question of one loop RGE stability of gFC conditions with V → Vθc in Eq. (78). First, it is interesting on its own to know if this simplified mixing allows for some stable gFC scenario; second, if that is the case, in terms of those Nq matrices, the deviations of gFC produced by the RGE would be controlled by the initial deviations of the complete CKM matrix from Vθc , the subleading mixings. One should first notice that, since Vθc decouples the third quark generation, nb and nt are expected to remain free parameters. Then, since the only remaining stability conditions concern elements (a, b) = (1, 2) or (2, 1), all the mixing combinations Vqa∗ Vqb , Vaq Vbq∗ equal either cos θc sin θc or − cos θc sin θc , and thus the dependence of the stability conditions on θc disappears. Two classes of stable gFC scenarios follow from the discussion in section 3.3. The first, with Nd = diag(α md , α ms , nb ), Nu = diag(α∗ mu , α∗ mc , nt ), (79) corresponds to a type I 2HDM for the first two generations, while nb and nt are free (and thus M−1 q Nq 6= αq 1). Some particular limit – the extreme chiral limit – of Eq. (79) was already obtained in [77] to justify V ' 1. The second is Nd = diag(α md , α ms , nb ),
Nu = diag(−α−1 mu , −α−1 mc , nt ),
(80)
which corresponds instead to a type II 2HDM for the first two generations (with free nb , nt and M−1 q Nq 6= αq 1 too). In addition to Eqs. (79)–(80), one can check that Nd = diag(eiϕd ms , eiϕd md , nb ),
Nu = diag(eiϕu mc , eiϕu mu , nt ),
(81)
with arbitrary real ϕd , ϕu (and again, arbitrary complex nb and nt ), gives indeed another stable gFC scenario where Nq and Mq are not even proportional in the first two generations sector.
4
Phenomenology
4.1
General considerations
The Yukawa interactions in Eqs. (26) or (29), together with the absence of tree level FCNC parameterised in Eq. (41), have interesting phenomenological consequences in different observables, since they may produce deviations from SM expectations. Those windows on New Physics in different observables are, of course, related: they are controlled by the parameters nj in Eq. (41), by the values of the masses mH± , mH , mA , and by the mixings in the scalar sector, Rij in Eq. (26). In the following we consider for simplicity the CP conserving case in Eq. (28). Our interest lies on the parameters nj in Eq. (41). Among the observables of interest, those that (i) involve the lowest number of new non-SM parameters and (ii) provide direct constraints from existing measurements, are the following. • Observables probing the couplings of the 125 GeV Higgs-like scalar, that we identify with h, that is (i) production mechanisms and (ii) decay modes. In 14
addition to the nj parameters, they involve one extra parameter, the mixing cβα if there is no CP violation in the scalar sector; in the general case, two independent mixings are involved. • Observables probing the couplings of the charged scalar H± , in particular effects of H± in flavour changing processes where the SM contributions involve virtual W ± exchange like (i) tree level decays, modifying for example the expected universality of weak interactions, and (ii) one loop FCNC processes like neutral meson mixings and rare decays. These observables, besides the nj parameters, depend on the mass mH± (and no dependence on the neutral scalar mixings). We concentrate in the rest of this work on the flavour conserving observables related to h: besides probing the gFC matrices in Eq. (41), the bounds they impose also apply to the same flavour conserving couplings of a general 2HDM. Before addressing the different constraints related to experiment, one can formulate a first theoretical requirement on the perturbativity of the Yukawa couplings: |nj | ≤ O(1). v
(82)
√ The precise value adopted in Eq. (82), for example O(1) → 1 or 4π, is not expected to be specially relevant: other phenomenological requirements will be, typically, more restrictive. There is, however, an exception: the “decoupling limit” [88] of the 2HDM, in which sβα → 1 (cβα → 0) removes the non-SM effects from the h couplings (while mH± v suppresses H± mediated non-SM effects), leaving the perturbativity requirement as the only effective constraint. One may further argue that having either mj |nj | or mj |nj |, involves fine tuning between quantities of very different nature: both mj and nj are linear combinations, controlled by β, of Yukawa couplings (times v), but β originates in the scalar potential, meaning that very disparate values of mj and nj involve significant cancellations in one or the other, unless β → 0 or β → π/2. For the sake of clarity, we will only consider Eq. (82) and ignore the previous concerns about eventual fine tuning.
4.2
Production and decay of h
For the observables related to h, one should consider constraints on nj and cβα arising from h production and decay processes at the LHC [89]. In connection to them, additional attention should be paid to the decays of h into light fermions since enhanced decays into light fermions can increase the total width and modify the precise SM pattern of branching ratios. The cross sections for direct q q¯ → h production is also important, since large couplings of h to light quarks, in combination with the luminosities given by the parton distribution functions, could significantly increase them. Before addressing the Yukawa couplings themselves, we recall that, owing to the mixing in the scalar sector, the couplings hV V (V = W, Z) are modified with respect to the SM as hV V, SM : mV 7→ gFC-2HDM : sβα mV . (83) 15
These couplings are involved in vector boson fusion (VBF) and associated production mechanisms, and in decays h → V V ∗ . For the different couplings to fermions Lhf f = −hf¯(af + ibf γ5 )f in Eq. (29), we have a scalar term af , straightforward to compare with the SM one, af :
SM : mj /v 7→ gFC-2HDM : (sβα mj + cβα Re(nj ))/v ,
(84)
and a pseudoscalar term bf absent in the SM, bf :
SM : 0 7→ gFC-2HDM : cβα Im(nj )/v .
(85)
We now discuss in turn decay and production processes. 4.2.1
Decays of h
The decay width h → f¯f , for a generic Yukawa interaction Lhf f = −hf¯(af + ibf γ5 )f , is, at tree level, s m2f m2f mh 2 2 ¯ Γ(h → f f ) = Nc (f ) 1−4 2 1 − 4 2 |af | + |bf | , (86) 8π mh mh with Nc = 3 for quarks and Nc = 1 for leptons; neglecting 4m2f /m2h 1, mh |af |2 + |bf |2 . Γ(h → f¯f ) = Nc (f ) 8π
(87)
With Eqs. (84)–(85), Nc (f ) mh 2 m 7→ Γ(h → f¯f )SM : 8π v 2 f Nc (f ) mh 2 2 2 2 s m + 2s c m Re(n ) + c |n | . (88) Γ(h → f¯f )gFC-2HDM : βα βα f f f βα f βα 8π v 2 The decay h → γγ, central in the discovery of the Higgs, has an amplitude controlled in the SM by two interfering contributions, the one loop triangle diagrams with virtual W ’s and top quarks. The former is modified according to Eq. (83). The later is the only relevant one involving quarks in the SM because of the large ht¯t coupling: mt /v; this amplitude is modified according to Eq. (84). With a pseudoscalar coupling now present, Eq. (85), there is an additional contribution which, however, does not interfere with the SM-like top(scalar coupling)+W . Furthermore, there are other contributions that one may consider: one due to diagrams with virtual H± ’s, and the ones due to other fermions with enhanced couplings to h due to sizable nj . For the charged scalar, they cannot be neglected if H± is relatively light, and thus, barring that possibility, we do not consider them. For the remaining fermions, the values of cβα that h W W decay and production require are typically small (|cβα | ≤ 0.1), and thus the values of nj that one would need for their contributions to be relevant would be at least nj ∼ mt : they would produce huge contributions to the width Γ(h) or to q¯q → h production cross sections (see the discussion in section 4.2.2), in addition to the perturbativity and fine 16
tuning concerns on the Yukawa couplings already mentioned: we thus ignore them altogether, since they will be rendered negligible once other constraints are considered. The width of h → γγ reads α2 m3h Γ(h → γγ)gFC-2HDM = × 256π 3 v 2 2 2 X bf v ˆ af v X AF (xf ) + sβα AV (xW ) + gH± AS (xH± ) + Nc (f )Q2f AF (xf ) , Nc (f )Q2f m m f f f f (89) with xX = 4m2X /m2h . The sum over fermions f includes up and down type quarks, with Qf = 2/3 and −1/3 respectively, and charged leptons with Qf = −1. The contribution 2m2
of the charged scalar H± corresponds to an interaction LH+ H− h = −gH± vH± H+ H− h. gH± depends on the details of the scalar potential that we do not address since this contribution can be safely neglected for mH± > v. The decay into gluons h → gg proceeds through similar diagrams, with the ones mediated by leptons and by W and H± bosons absent: 2 2 3 2 X X af v bf v ˆ αS mh AF (xf ) . AF (xf ) + (90) Γ(h → gg)gFC-2HDM = 3 2 128π v mf mf f
f
The couplings af and bf in Eqs. (89)–(90) appear divided by fermion mass mf since the AF and AˆF functions are defined including the mass factor of the SM hf¯f vertex. The loop functions are [90] AF (x) = −2x [1 + (1 − x)f (x)] , AˆF (x) = −2xf (x), AV (x) = 2 + 3x + 3x(2 − x)f (x), AS (x) = x(1 − xf (x)), where
√ arcsin2 (1/ x), x ≥ 1, i2 h √ √1−x − iπ , x < 1. − 14 ln 1+ 1− 1−x
(91)
( f (x) =
(92)
The dominant contribution in h → γγ comes from AV (xW ) = −8.339. Other representative values of the functions are shown in Table 1. It is important to stress that, while QCD corrections to Eq. (89) are small, that is not the case for Eq. (90) (see for example [91]): we account for them by using Γ(h → gg)gFC-2HDM →
Γ(h → gg)gFC-2HDM × Γ(h → gg)SM ref. , Γ(h → gg)SM
(93) Γ(h→gg)
gFC-2HDM with Γ(h → gg)SM ref. = 0.351 MeV the SM reference value from Table 2, and Γ(h→gg) SM a v computed according to Eq. (90) (for the SM denominator mff = 1, bf = 0). For completeness, reference values of the SM Higgs decays [92–95] are reproduced in Table 2.
17
f AF (xf ) AˆF (xf ) f AF (xf ) AˆF (xf )
t
b 4.75i)10−2
−(4.37 + −(4.78 + 4.76i)10−2
1.3796 2.1010
τ −(2.30 + 2.09i)10−2 −(2.46 + 2.09i)10−2
c
s
µ
−(4.87 + 3.29i)10−3 −(5.07 + 3.29i)10−3
−(8.99 + 3.89i)10−5 −(9.15 + 3.89i)10−5
−(2.53 + 1.20i)10−4 −(2.59 + 1.20i)10−4
Table 1: Values of AF and AˆF for charged fermions of the 2nd and 3rd generations; running masses at µ = mh [96] are used. Channels f¯f BR
¯bb 0.577
τ¯τ 6.32 · 10−2
c¯c 2.91 · 10−2
µ ¯µ 2.19 · 10−4
s¯s 2.46 · 10−4
Channels V V BR
gg 8.57 · 10−2
W W (∗) 0.215
ZZ (∗) 2.64 · 10−2
γγ 2.28 · 10−3
γZ 1.54 · 10−3
Table 2: Reference SM Higgs decay branching ratios for mh = 125 GeV; the total width is Γ(h) = 4.1 MeV.
4.2.2
Production of h
In addition to the decay widths, production mechanisms are also modified. Besides VBF and associated production, already commented ( Eq. (83)), the most relevant one is gluon-gluon fusion (ggF) gg → h [97]. The elementary process is the reverse of the decay h → gg, which is then convoluted with the gluon distribution functions in the proton (in the narrow width approximation production and decay are related straightforwardly). As in the case of the decay, Eq. (93), we incorporate QCD corrections by normalizing the SM prediction to the reference value in Table 3, which shows reference cross sections for different production mechanisms [92–95]. ggF VBF WH ZH ttH bbH 8 TeV 19.27 1.578 0.7046 0.4153 0.1293 0.2035 13 TeV 43.92 3.748 1.380 0.8696 0.5085 0.5116 14 TeV 49.47 4.233 1.522 0.9690 0.6113 0.5805 Table 3: Reference SM production cross sections for mh = 125 GeV (in pb).
We now turn to the direct q¯q → h production mechanism shown in Figure 1. The motivation to consider this production mechanism is that, when |nq | mq , the corresponding cross section may become inappropriately large; one is considering light quarks q 6= t, b. Sensitivity to enhanced Yukawa couplings of light quarks at the LHC has also been discussed, for example, in [98–101]. For a generic Yukawa interaction Lhqq = −h¯ q (aq + ibq γ5 )q, the tree level cross 18
section for direct production pp(¯ q q) → h is, in the narrow width approximation, (94) σ[pp(¯ q q) → h] = |aq |2 + |bq |2 σ0 (E) Lq¯q (E) , where 2 π TeV σ0 (E) ≡ 2 101.8 pb , = 8Nc E 2 E Z 1 1 Lq¯q (E) ≡ dx fq¯p (x, Q2 )fqp (x0 /x, Q2 ) , x x0
(95)
with E the center of mass proton energy, fyp the distribution function of parton y in the proton, x0 =
m2h 4E 2
and Q is the factorization scale.
p q¯ q
h
p
Figure 1: q¯q → h process. In Table 4 we collect the values of σ[pp(¯ q q) → h] and Lq¯q computed [102] for different 7 2 2 quarks and setting |aq | + |bq | = 1 in Eq. (94): this value of the couplings is obviously too large since it effectively corresponds, with respect to the SM, to the change mj /v 7→ v/v, but it allows for easy use. Consider, for illustration, that for the LHC at 8 TeV ¯ dd
u¯u s¯s c¯c Ldd σ L σ L σ L σ ¯ u ¯u s¯s c¯c 8 TeV 14.56 16.60 21.53 24.53 4.41 5.02 2.65 3.01 13 TeV 74.57 29.17 105.28 41.18 27.70 10.84 17.92 7.01 14 TeV 95.49 31.53 133.90 44.21 36.46 12.04 23.83 7.87 Table 4: σ[pp(¯ q q) → h] (×103 ) in pb and Lq¯q (E 2 ) (×103 ) for different q¯q. σ[pp(¯ uu) → h] ∼ 10 pb: one can readily obtain σ[pp(¯ uu) → h] ∼ 10 pb ⇔ |aq |2 + |bq |2 ∼ 7.3 × 10−5 . 7
(96)
Eq. (94) is obtained using the tree level partonic cross section; furthermore, the results in table 4 are obtained multiplying these simple predictions by a common O(1) factor (one for each LHC energy case), chosen such that σ[pp(¯bb) → h] in Eq. (94) reproduces the improved reference values in [92–95]. We also take Q = E/2.
19
Although considering σ[pp(¯ uu) → h] ∼ 10 pb may be unrealistic (the total production cross section in Table 3 for 8 TeV is ∼ 22 pb), from Eq. (96), Γ(h → u¯ u) ∼ 1 MeV: even if it is a significant contribution to the width Γ(h), it might still be compatible with the overall pattern of Higgs signal strengths. To the knowledge of the authors, there are no dedicated analyses of q¯q → h (q 6= b, t) from which experimental input can be used in this manner. However, it is reasonable to expect that this kind of production potentially “contaminates” the analyses of gluon-gluon fusion: in that case, one should add all σ[pp(¯ q q) → h] contributions for light q to the gluon-gluon fusion cross section when analysing Higgs signal strengths. It is then clear that bounds more stringent than Eq. (96) would follow for the sum over all the different channels involved. The simple connection among the decays h → q¯q and the q¯q → h production mechanism – in the narrow width approximation – that follows from Eqs. (88) and (94), is σ[pp(¯ q q) → h] / 1pb = 6.825 Γ(h → q¯q) / 1MeV
TeV E
2
Lq¯q (E) 103
,
(97)
which allows for easy comparison of the relative strengths of the constraints imposed by q¯q → h production and h → q¯q decay for light quarks q. 4.2.3
Constraints
The main source of experimental constraints that we use is the combined analysis of LHC-Run I data from the ATLAS and CMS collaborations in [89], which provides detailed information on (production) × (decay) of the 125 GeV Higgs h for • production: ggF, VBF, associated W h, Zh, and tth; • decay: h → γγ, ZZ, W W , τ τ¯ and b¯b. Results from LHC-Run II in specific channels like (associated V h) × (h → b¯b) [103,104], or (ggF+VBF) × (h → τ τ¯) [105] are starting to improve over [89]. One should also consider off-shell (ggF+VBF)→ h(∗) → W W (∗) constraints on the total width Γ(h) [106], even if they are still weak [107,108]. Finally, dedicated studies like [109,110] put useful bounds on h → µ+ µ− , e+ e− .
4.3
Electric dipole moments
As discussed at the end of section 2, non-real Nf matrices are a source of CP violation in scalar-fermion interactions, which can induce electric dipole moments (EDMs). Consider for example an electron-Higgs coupling Lhee = −h e¯(ae +ibe γ5 )e; the one loop diagram in Figure 2 gives a contribution to the electron EDM de : 2 me 3me ae be 1+O . (98) de = 2 2 16π mh m2h It is to be noticed that, for ae ∼ be ∼ me /v, Eq. (98) gives de ∼ 10−34 e·cm. When |ae |, |be | me /v are a priori allowed, up to the effect of other constraints, a significant 20
enhancement in de can be expected. For current experimental bounds |de | < 10−27 e ·cm, considering only this contribution gives ae be < 8 × 10−5 ,
(99)
or, with Eqs. (84)–(85) and neglecting me with respect to cβα ne , c2βα Re(ne )Im(ne ) < 5 GeV2 .
(100)
Anticipating results from section 4.4, in particular Figure 4(g), it is clear that the bounds imposed by the LHC results are more stringent than Eq. (100). It should also be noticed that including contributions analog to Figure 2 with h → H, A, gives 2 m2h 2 2 mh Re(ne )Im(ne ) cβα + sβα 2 + 2 < 5 GeV2 , (101) mH mA and does not change this conclusion. Furthermore, one loop contributions with virtual H± and neutrinos are suppressed. It is well known that two loop “Barr-Zee” [111–116] contributions can be signifih e
e ae + ibe γ5
ae + ibe γ5
γ
Figure 2: h mediated contribution to de at one loop. cant: studies such as [117,118] address such constraints on CP violating Higgs-fermion couplings. However, those contributions involve different nf couplings simultaneously, together with the masses of the different scalars, preventing a simple translation into bounds on a single parameter. It is to be noticed too that cancellations among different diagrams in that class may occur [65, 119]. Including such kind of analysis is beyond the scope of this work; in any case one should keep in mind that the analysis of EDMs may have some impact on the results of section 4.4. The previous discussion also applies to the EDMs of the u and d quarks and the experimental constraints that the neutron EDM bounds impose, including, in addition, the impact of QCD effects [120].
4.4
Analysis
With the deviations with respect to the SM of the couplings of h and their implications for decays and production mechanisms, one can impose the experimental constraints of section 4.2.3 and explore the allowed values of cβα and the gFC parameters nf in Eq. (41). For the results presented in the following we consider the most conservative 21
25
250
20
20
200
15
10
5
0 −1
|nt| (GeV)
25
|nc| (GeV)
|nu| (GeV)
situation, i.e. all parameters are free to vary simultaneously. Compared to restricted situations where not all parameters are considered simultaneously, this offers a safer interpretation of excluded regions (they are excluded whatever the values of the parameters not displayed) at the price, of course, of larger allowed regions. Figure 3 shows nf vs. cβα for all quarks and leptons. Some comments are in order.
15
10
5
−0.5
0
0.5
0 −1
1
cβα
−0.5
0
0.5
0 −1
1
cβα
50
20
20
40
10
5
|nb| (GeV)
25
15
15
10
0
0.5
0 −1
1
cβα
−0.5
(d) d
0
0.5
0 −1
1
cβα
8
40
|nτ | (GeV)
8
|nµ| (GeV)
50
0 −1
6
4
2
−0.5
0
cβα
(g) e
−0.5
0.5
1
0 −1
0
1
0.5
1
cβα
(f) b
10
2
0.5
20
10
4
1
30
(e) s
6
0.5
10
5
−0.5
0
cβα
(c) t
25
0 −1
−0.5
(b) c
|ns| (GeV)
|nd| (GeV)
100
50
(a) u
|ne| (GeV)
150
30
20
10
−0.5
0
cβα
0.5
1
0 −1
(h) µ
−0.5
0
cβα
(i) τ
Figure 3: |nf | vs. cβα for the different fermions f ; darker to lighter regions correspond to 68, 95 and 99% CL.
• As expected, for cβα → 0, the constraints on nf disappear. 22
• For u, c, d and s quarks, the allowed regions are almost identical, as one could anticipate from their irrelevant role, within the SM, in the available production × decay Higgs signal strengths. The corresponding nf ’s appear to be effectively limited by the contributions to the Higgs width. • Surprisingly, the allowed size of |nt | appears to be independent of cβα : this will be discussed in connection with Fig. 4(c) below. • The nb and nτ cases are also similar, with allowed regions differing from the u, c, d, s cases for |nq |’s below 10-15 GeV and not small cβα . • For ne and nµ , the allowed regions are much more constrained owing to the bounds set by dedicated pp → h → e+ e− , µ+ µ− analyses such as [109, 110]. Although Fig. 3 shows absolute bounds on |nf |’s, it does not give information on arg(nf )’s and cannot be directly read in terms of the scalar and pseudoscalar couplings of h in Eqs. (84)–(85). Considering that, Figure 4 shows ¯bf vs a ¯f with a ¯f ≡ sβα mf + cβα Re(nf ),
¯bf = cβα Im(nf ) .
(102)
Furthermore, to maintain some information on cβα , allowed regions corresponding to |cβα | < 0.01, to 0.01 < |cβα | < 0.1 and to 0.1 < |cβα | are displayed. One can notice that • for the first and second fermion generations, there is no dependence on arg(nf ), since only decays, with rates proportional to |¯ af |2 + |¯bf |2 , are relevant. For quarks, the allowed region for |cβα | < 0.01 is smaller: this is simply due to the perturbativity requirement in Eq. (82). • For the top quark, two separate regions are allowed: this is also expected since independent sign changes in both a ¯t and ¯bt (together with sign changes in cβα , sβα ) do not alter the predictions. For |cβα | < 0.01 the allowed regions are quite reduced and placed around (¯ at , ¯bt ) = (±mt , 0); with 0.01 < |cβα | < 0.1 their size increases and only for |cβα | > 0.1 the interplay of (i) pseudoscalar contributions to gg → h and h → γγ, and (ii) W -top(scalar) interference in h → γγ gives rise to larger regions. • For b and τ , the regions for not too small mixing, |cβα | > 0.01, are ring-shaped; mb and mτ set the radii of such regions, as could be expected from the agreement of h → b¯b and h → τ τ¯ signal strengths with SM expectations. For small mixing, |cβα | < 0.01, the perturbativity requirement on |nb |, |nτ | limits the allowed departure from (¯ af , ¯bf ) = (±mf , 0), giving in fact, for the b case, two disjoint patches. To close this section we recall the discussion on q¯q → h production in section 4.2.2: as commented there, values of nf in agreement with the SM-like Higgs signal strengths
23
5
5
0
5
a¯u (GeV)
0
−5 −5
0
a¯c (GeV)
(a) u
−250 −250
0
a¯d (GeV)
5
5
¯bb (GeV)
¯bs (GeV)
−5 −5
0
−5 −5
0
a¯s (GeV)
(d) d
5
0
−5 −5
5
0.4
|cβα | < 0.01, 0.01 < |cβα | < 0.1, |cβα | > 0.1
¯bτ (GeV)
¯bµ (GeV) 0
5
(f) b
|cβα | < 0.01, 0.01 < |cβα | < 0.1, |cβα | > 0.1
0
0
a¯b (GeV)
0.4 |cβα | < 0.01, 0.01 < |cβα | < 0.1, |cβα | > 0.1
−0.4 −0.4
|cβα | < 0.01, 0.01 < |cβα | < 0.1, |cβα | > 0.1
(e) s
0.4
250
(c) t
|cβα | < 0.01, 0.01 < |cβα | < 0.1, |cβα | > 0.1
0
0
a¯t (GeV)
5 |cβα | < 0.01, 0.01 < |cβα | < 0.1, |cβα | > 0.1
¯bd (GeV)
5
0
(b) c
5
¯be (GeV)
|cβα | < 0.01, 0.01 < |cβα | < 0.1, |cβα | > 0.1
¯bt (GeV)
0
−5 −5
250 |cβα | < 0.01, 0.01 < |cβα | < 0.1, |cβα | > 0.1
¯bc (GeV)
¯bu (GeV)
|cβα | < 0.01, 0.01 < |cβα | < 0.1, |cβα | > 0.1
0
−0.4 −0.4
0
a¯e (GeV)
a¯µ (GeV)
(g) e
(h) µ
0.4
0
−5 −5
0
a¯τ (GeV)
5
(i) τ
Figure 4: Allowed regions at 99% CL for pseudoscalar vs. scalar couplings for the different fermions f with Lhf¯f = − hv f¯(¯ af + i¯bf γ5 )f .
could potentially give production cross sections not far from the dominating SM ones. Figure 5 shows ! . X σ[q q¯h] ≡ σ[pp(q q¯) → h] σ[pp(gg) → h]SM , (103) σ[pp(gg) → h]SM q=u,c,d,s vs. the total Higgs width and vs. the gluon-gluon fusion production cross section in two different analyses: in Figures 5(a) and 5(b), σ[q q¯h] is added to the ggF production 24
cross section, while in Figures 5(c) and 5(d) it is not (and therefore, in the analysis, it does not affect directly observables constrained by experiment). Comparing 5(a)-5(b) with 5(c)-5(d), one can notice that the constraints from Higgs signal strengths are able to bound the size of σ[q q¯h], even if there is room for an overall q q¯ → h cross section which is quite sizable, not far from the complete SM Higgs production cross section. Furthermore, when σ[q q¯h] is added to the ggF production cross section, the agreement with the observed Higgs signal strengths allows for a smaller amount of q q¯ → h, and, for sizable q q¯ → h, it is achieved at the cost of (i) reducing the ggF production cross section and (ii) increasing the total width Γ(h), as the shape of the allowed regions in Figures 5(a) and 5(b) shows. For the results in Figures 3 and 4, the bounds on the the different a ¯f , ¯bf do not differ in both analyses. It should be finally mentioned that, in connection with the previous comments and the size of σ[pp(q q¯) → h], it might be interesting to analyse, for the remaining neutral scalars H and A, the cross sections for pp(q q¯) → H, A at the LHC.
25
1.5
1.5
σ[q q¯h]/σ[pp(gg) → h]SM
σ[q q¯h]/σ[pp(gg) → h]SM
SM
1
0.5
0
1
0.5
0 0
2
4
6
8
10
0
0.5
1
1.5
2
σ[pp(gg) → h]/σ[pp(gg) → h]SM
Γ(h) (MeV)
(a) σ[q q¯h] vs. Γ(h), σ[q q¯h] ⊂ ggF.
(b) σ[q q¯h] vs. σ[pp(gg) → h], σ[q q¯h] ⊂ ggF
1.5
1.5
σ[q q¯h]/σ[pp(gg) → h]SM
σ[q q¯h]/σ[pp(gg) → h]SM
SM
1
0.5
0
1
0.5
0 0
2
4
6
8
10
0
0.5
1
1.5
2
σ[pp(gg) → h]/σ[pp(gg) → h]SM
Γ(h) (MeV)
(c) σ[q q¯h] vs. Γ(h), σ[q q¯h] 6⊂ ggF
(d) σ[q q¯h] vs. σ[pp(gg) → h], σ[q q¯h] 6⊂ ggF
Figure 5: Effect of including q q¯ → h production in ggF in analyses of Higgs signal strengths; darker to lighter regions correspond to 68, 95 and 99% CL.
26
Conclusions In this paper we analyse the question of general Flavour Conservation in extensions of the SM with additional scalar doublets, in particular the 2HDM. We show that the actual necessary and sufficient conditions for gFC are only a subset of the conditions addressed in early references [40–43]. The effect of the one loop Renomalization Group Evolution of the Yukawa coupling matrices on gFC scenarios is discussed in detail. In particular it is to be stressed that in the absence of Yukawa couplings with righthanded neutrinos, gFC in the lepton sector is stable. For the quark sector, some one loop RGE stable scenarios are discussed, including the case of a Cabibbo-like quark mixing matrix. At a phenomenological level, we discuss the constraints that existing data on flavour conserving processes, in particular the ones related to the Higgs, impose on the parameters describing gFC in the different fermion sectors, including a detailed numerical analysis of that parameter space. Direct q q¯ → h production is also considered in detail: although it is completely negligible in the SM, that might not be the case in scenarios such as 2HDM, and it may even be relevant for the production the additional non-SM neutral scalars.
Acknowledgments The authors thank Luca Fiorini for discussions. This work is partially supported by Spanish MINECO under grant FPA2015-68318-R, FPA2017-85140-C3-3-P and by the Severo Ochoa Excellence Center Project SEV-2014-0398, by Generalitat Valenciana under grant GVPROMETEOII 2014-049 and by Funda¸ca˜o para a Ciˆencia e a Tecnologia (FCT, Portugal) through the projects CERN/FIS-NUC/0010/2015 and CFTP-FCT Unit 777 (UID/FIS/00777/2013) which are partially funded through POCTI (FEDER), COMPETE, QREN and EU. MN acknowledges support from FCT through postdoctoral grant SFRH/BPD/112999/2015.
27
A
Simultaneous diagonalisation
The necessary and sufficient conditions for the simultaneous diagonalisation of a set of matrices A1 , . . . , An were first discussed in [40–43]; Theorem 1 of [43] states that for a set A1 , . . . , An of complex m × m matrices, unitary matrices L and R such that L† Ai R is diagonal for all i = 1, . . . , n exist if and only if both sets {Ai A†j }i,j=1,...,n are abelian, that is i h Ai A†j , Ak A†l = 0 and
h
and {A†i Aj }i,j=1,...,n
i A†i Aj , A†k Al = 0,
(104)
i, j, k, l = 1, . . . , n .
(105)
We show now that a weaker requirement on the sets {Ai A†j }i,j=1,...,n and {A†i Aj }i,j=1,...,n is already necessary and sufficient. Consider instead the sets {A1 A†1 , . . . , An A†n },
{A†1 A1 , . . . , A†n An },
(106)
that is, the subsets of Eq. (104) with i = j: each set in Eq. (106) has n elements, all hermitian, rather than the n2 elements in Eq. (104). If both sets in Eq. (106) are abelian, complete sets of orthonormal eigenvectors {~uj }, {~vj }, j = 1, . . . , m, exist [121]: Ai A†i ~uj = λ(i)j ~uj ,
A†i Ai ~vj = λ(i)j ~vj , λ(i)j ∈ R, λ(i)j ≥ 0,
~u∗i · ~uj = ~vi∗ · ~vj = δij . (107)
The matrices in Eq. (106) are simultaneously diagonalised, for ↑ † † U Ai Ai U = diag λ(i)1 , . . . , λ(i)m , U = ~u1 ↓ ↑ V † A†i Ai V = diag λ(i)1 , . . . , λ(i)m , V = ~v1 ↓ and
q q U Ai V = diag λ(i)1 , . . . , λ(i)m . †
i = 1, . . . , n: · ↑ · ~um , · ↓ · ↑ · ~vm , · ↓
(108)
(109)
(110)
With Eq. (110), it follows trivially that Eq. (105) is fulfilled. That is, the requirement that the sets in Eq. (106) are abelian, implies that the (larger) sets in Eq. (104) are abelian too, and is sufficient for the simultaneous bi-diagonalisability of A1 , . . . , An .
B
RGE details
The analysis of the RGE of the quark Yukawa couplings and the stability of the gFC scenario in Eq. (41) has been presented in detail for the set {Γα Γ†β } in section 3.1 28
and 3.2. We reproduce in this appendix the equations relevant for {Γα Γ†β } and also for {Γ†α Γβ }, {∆α ∆†β } and {∆†α ∆β }, omitted for conciseness in section 3. In correspondence with Eqs. (49), (50) and (51), D(Γα Γ†β ) = fαβL (Γ) + gαβL (Γ, ∆) , [d ]
[d ]
D(∆α ∆†β ) = fαβL (∆) + gαβL (Γ, ∆) ,
[d ]
[d ]
D(∆†α ∆β )= fαβR (∆) + gαβR (Γ, ∆) ,
D(Γ†α Γβ ) = fαβR (Γ) + gαβR (Γ, ∆) ,
[u ]
[u ]
[u ]
[u ]
(111)
with n=2 h X
fαβL (Γ) = 2ad Γα Γ†β + [d ]
ρ=1 [d ] gαβL (Γ, ∆)
[d ] fαβR (Γ)
i 1 1 d∗ d Γα Γ†ρ + 2Γα ΓR Γ†β + ΓL Γα Γ†β + Γα Γ†β ΓL , Γρ Γ†β + Tβ,ρ Tα,ρ 2 2
n=2 h i X 1 1 † † † † † † = ∆L Γα Γβ + Γα Γβ ∆L − 2 ∆ρ ∆α Γρ Γβ + Γα Γρ ∆β ∆ρ , 2 2 ρ=1
=
2ad Γ†α Γβ
(112)
n=2 h i X d∗ † d † Tα,ρ Γρ Γβ + Tβ,ρ Γα Γρ + Γ†α ΓL Γβ + ΓR Γ†α Γβ + Γ†α Γβ ΓR , + ρ=1
[d ] gαβR (Γ, ∆)
=
Γ†α ∆L Γβ
n=2 h i X −2 Γ†ρ ∆α ∆†ρ Γβ + Γ†α ∆ρ ∆†β Γρ ,
(113)
ρ=1
[u ] fαβL (∆)
=
2au ∆α ∆†β
n=2 h i X 1 1 u d∗ Tα,ρ ∆ρ ∆†β + Tβ,ρ + ∆α ∆†ρ + 2∆α ∆R ∆†β + ∆L ∆α ∆†β + ∆α ∆†β ∆L , 2 2 ρ=1
n=2 h i X 1 1 [u ] Γρ Γ†α ∆ρ ∆†β + ∆α ∆†ρ Γβ Γ†ρ , gαβL (Γ, ∆) = ΓL ∆α ∆†β + ∆α ∆†β ΓL − 2 2 2 ρ=1
(114)
and [u ]
fαβR (Γ) = 2au ∆†α ∆β +
n=2 h i X u∗ † u Tα,ρ ∆ρ ∆β + Tβ,ρ ∆†α ∆ρ + ∆†α ∆L ∆β + ∆R ∆†α ∆β + ∆†α ∆β ∆R , ρ=1
[u ] gαβR (Γ, ∆)
=
∆†α ΓL ∆β
−2
n=2 h X
i ∆†ρ Γα Γ†ρ ∆β + ∆†α Γρ Γ†β ∆ρ .
(115)
ρ=1
The RGE of the commutation relations of Eq. (38) reads i h i h i h [d ] [d ] D Γα Γ†β , Γγ Γ†δ = gαβL (Γ, ∆) , Γγ Γ†δ + Γα Γ†β , gγδL (Γ, ∆) , i h i h i h [dR ] [dR ] † † † † D Γα Γβ , Γγ Γδ = gαβ (Γ, ∆) , Γγ Γδ + Γα Γβ , gγδ (Γ, ∆) , h i h i h i [u ] [u ] D ∆α ∆†β , ∆γ ∆†δ = gαβL (Γ, ∆) , ∆γ ∆†δ + ∆α ∆†β , gγδL (Γ, ∆) , h i h i h i [u ] [u ] D ∆†α ∆β , ∆†γ ∆δ = gαβR (Γ, ∆) , ∆†γ Γδ + ∆†α ∆β , gγδR (Γ, ∆) , 29
(116)
which, following the discussion in section 3.2, lead to (summation over h = 1, 2 understood) {dL } ≡
i v 2 † h 0 0† 0 0† Y[d]l UdL = UdL D Y[d]i Y[d]j , Y[d]k (117) 2 † † † † † † Y[d]i Y[d]j V † Y[u]h Y[u]h V Y[d]k Y[d]l − Y[d]k Y[d]l V † Y[u]h Y[u]h V Y[d]i Y[d]j h i h i † † † † † † † † − 2 V Y[u]h Y[u]i V , Y[d]k Y[d]l Y[d]h Y[d]j − 2Y[d]i Y[d]h V Y[u]j Y[u]h V , Y[d]k Y[d]l h i h i † † † † † † + 2 V † Y[u]h Y[u]k V , Y[d]i Y[d]j Y[d]h Y[d]l + 2Y[d]k Y[d]h V † Y[u]l Y[u]h V , Y[d]i Y[d]j ,
i v 2 † h 0† 0 0† 0 (118) {dR } ≡ UdR D Y[d]i Y[d]j , Y[d]k Y[d]l UdR = 2 h i h i † † † † † † Y[d]i V † Y[u]h Y[u]h V Y[d]j , Y[d]k Y[d]l − Y[d]k V † Y[u]h Y[u]h V Y[d]l , Y[d]i Y[d]j h i h i † † † † † † † † − 2 Y[d]h V Y[u]i Y[u]h V Y[d]j , Y[d]k Y[d]l − 2 Y[d]i V Y[u]h Y[u]j V Y[d]h , Y[d]k Y[d]l h i h i † † † † † † + 2 Y[d]h V † Y[u]k Y[u]h V Y[d]l , Y[d]i Y[d]j + 2 Y[d]k V † Y[u]h Y[u]l V Y[d]h , Y[d]i Y[d]j ,
{uL } ≡
i v 2 † h 0 0† 0 0† UuL D Y[u]i Y[u]j , Y[u]k Y[u]l UuL = (119) 2 † † † † † † Y[u]i Y[u]j V Y[d]h Y[d]h V † Y[u]k Y[u]l − Y[u]k Y[u]l V Y[d]h Y[d]h V † Y[u]i Y[u]j i h i h † † † † † † † † − 2 V Y[d]h Y[d]i V , Y[u]k Y[u]l Y[u]h Y[u]j − 2Y[u]i Y[u]h V Y[d]j Y[d]h V , Y[u]k Y[u]l h i h i † † † † † † + 2 V Y[d]h Y[d]k V † , Y[u]i Y[u]j Y[u]h Y[u]l + 2Y[u]k Y[u]h V Y[d]l Y[d]h V † , Y[u]i Y[u]j ,
and {uR } ≡
i v 2 † h 0† 0 0† 0 UuR D Y[u]i Y[u]j , Y[u]k Y[u]l UuR = (120) 2 h i h i † † † † † † Y[u]i V Y[d]h Y[d]h V † Y[u]j , Y[u]k Y[u]l − Y[u]k V Y[d]h Y[d]h V † Y[u]l , Y[u]i Y[u]j h i h i † † † † † † − 2 Y[u]h V Y[d]i Y[d]h V † Y[u]j , Y[u]k Y[u]l − 2 Y[u]i V Y[d]h Y[d]j V † Y[u]h , Y[u]k Y[u]l h i h i † † † † † † + 2 Y[u]h V Y[d]k Y[d]h V † Y[u]l , Y[u]i Y[u]j + 2 Y[u]k V Y[d]h Y[d]l V † Y[u]h , Y[u]i Y[u]j .
In order to compute the matrix elements of Eqs. (117)–(120), we notice that h
† † V † Y[u]i Y[u]j V , Y[d]k Y[d]l
i
= ab
3 X
u u∗ d∗ d d∗ d Vqa∗ Vqb yi,q yj,q (yk,b yl,b − yk,a yl,a ),
(121)
u∗ u u∗ d d∗ u yj,q (yk,b yl,b − yk,a yl,a ). Vaq Vbq∗ yi,q
(122)
q=1
and h
V
† Y[d]i Y[d]j V†
,
† Y[u]k Y[u]l
i
= ab
3 X q=1
30
Then, with the parameters in Eq. (65), the matrix elements (a, b) of Eqs. (117)–(120) read {dL }ab =
3 X n=2 X
n d d∗ d d∗ d d∗ d d∗ u 2 yj,a yk,b yl,b − yi,b yj,b yk,a yl,a Vqa∗ Vqb |yh,q | yi,a
(123)
q=1 h=1 u u∗ d d∗ u u∗ d d∗ − 2 yh,q yi,q yh,b yj,b + yj,q yh,q yi,a yh,a
+2
d∗ u∗ d u yh,b yl,b yk,q yh,q
+
d∗ u u∗ d yh,q yk,a yh,a yl,q
d d∗ d d∗ yk,b yl,b − yk,a yl,a d d∗ yj,b yi,b
−
o .
d d∗ yi,a yj,a
{dR }ab = 3 X n=2 X
(124)
Vqa∗ Vqb
n
d∗ d d∗ d yk,b yl,b − yk,a yl,a
u 2 d∗ d u u∗ d∗ d u u∗ d∗ d |yh,q | yi,a yj,b − 2yi,q yh,q yh,a yj,b − 2yh,q yj,q yi,a yh,b
q=1 h=1 d∗ d d∗ d − yi,b yj,b − yi,a yj,a
{uL }ab =
3 X n=2 X
Vaq Vbq∗
u 2 d∗ d u u∗ d∗ d u u∗ d∗ d |yh,q | yk,a yl,b − 2yk,q yh,q yh,a yl,b − 2yh,q yl,q yk,a yh,b
n d 2 u u∗ u u∗ u u∗ u u∗ |yh,q | yi,a yj,a yk,b yl,b − yi,b yj,b yk,a yl,a
o ,
(125)
q=1 h=1 d d∗ u u∗ d d∗ u u∗ − 2 yh,q yi,q yh,b yj,b + yj,q yh,q yi,a yh,a
d d∗ u u∗ d d∗ u u∗ + 2 yh,q yk,q yh,b yl,b + yl,q yh,q yk,a yh,a
u u∗ u u∗ yk,b yl,b − yk,a yl,a
u u∗ u u∗ yi,b yj,b − yi,a yj,a
o ,
and {uR }ab = 3 X n=2 X
(126)
Vaq Vbq∗
n
u∗ u u∗ u yk,b yl,b − yk,a yl,a
d 2 u∗ u d d∗ u∗ u d d∗ u∗ u |yh,q | yi,a yj,b − 2yi,q yh,q yh,a yj,b − 2yh,q yj,q yi,a yh,b
q=1 h=1 u∗ u u∗ u − yi,b yj,b − yi,a yj,a
d 2 u∗ u d d∗ u∗ u d d∗ u∗ u |yh,q | yk,a yl,b − 2yk,q yh,q yh,a yl,b − 2yh,q yl,q yk,a yh,b
o .
For diagonal elements, a = b, the right-hand sides of Eqs. (123)–(126) are identically zero. Attending to the discussion in appendix A, for the RGE stability of gFC we only need Eqs. (123)–(126) with i = j and k = l, in which case, by construction, we have in addition {qX }ba = −{qX }∗ab (q = u, d, X = L, R). For 2HDM, the necessary and sufficient conditions only involve i = j = 1, k = l = 2, for which {dL }da db =
3 X
n Vu∗q da Vuq db (m2uq + |nuq |2 ) m2da |ndb |2 − m2db |nda |2
(127)
uq =1
∗ ∗ − 2 |ndb | − |nda | muq mdb (muq mdb + nuq ndb ) + muq mda (muq mda + nuq nda ) o + 2 m2db − m2da n∗uq n∗db (muq mdb + nuq ndb ) + nuq nda (muq mda + n∗uq n∗da ) , 2
2
31
{dR }da db =
3 X
Vu∗q da Vuq db
n
(128)
uq =1
|ndb |2 − |nda |2 (m2uq + |nuq |2 )mda mdb − 2 |ndb |2 − |nda |2 muq mdb (muq mda + n∗uq n∗da ) + muq mda (muq mdb + nuq ndb ) − m2db − m2da (m2uq + |nuq |2 )n∗da ndb o + 2 m2db − m2da nuq ndb (muq mda + n∗uq n∗da ) + n∗uq n∗da (muq mdb + nuq ndb ) ,
{uL }ua ub =
3 X
Vua dq Vu∗b dq
n (m2dq + |ndq |2 ) m2ua |nub |2 − m2ub |nua |2
(129)
dq =1 2
2
n∗dq n∗ua )
− 2 |nub | − |nua | mdq mub (mdq mub + ndq nub ) + mdq mua (mdq mua + o + 2 m2ub − m2ua n∗dq n∗ub (mdq mub + ndq nub ) + ndq nua (mdq mua + n∗dq n∗ua ) ,
and {uR }ua ub =
3 X
Vua dq Vu∗b dq
n
(130)
q=1
|nub |2 − |nua |2 (m2dq + |ndq |2 )mua mub − 2 |nub |2 − |nua |2 mdq mub (mdq mua + n∗dq n∗ua ) + mdq mua (mdq mub + ndq nub ) − m2ub − m2ua (m2dq + |ndq |2 )n∗ua nub o + 2 m2ub − m2ua ndq nub (mdq mua + n∗dq n∗ua ) + n∗dq n∗ua (mdq mub + ndq nub ) . The formal generalisation of the conditions in this appendix and in section 3 to the case of models with n Higgs doublets instead of 2 is almost straightforward.
32
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