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arXiv:0906.1523v1 [hep-ph] 8 Jun 2009

Sequential Flavour Symmetry Breaking Thorsten Feldmann, a Martin Jung b,c and Thomas Mannel b a

b

c

Physik Department T31, Technische Universit¨at M¨ unchen, D-85747 Garching, Germany.

Theoretische Physik 1, Fachbereich Physik, Universit¨at Siegen, D-57068 Siegen, Germany.

Instituto de F´ısica Corpuscular, CSIC-Universitat de Val`encia, Apartado de Correos 22085, E-46071 Valencia, Spain.

Abstract The gauge sector of the Standard Model (SM) exhibits a flavour symmetry which allows for independent unitary transformations of the fermion multiplets. In the SM the flavour symmetry is broken by the Yukawa couplings to the Higgs boson, and the resulting fermion masses and mixing angles show a pronounced hierarchy. In this work we connect the observed hierarchy to a sequence of intermediate effective theories, where the flavour symmetries are broken in a step-wise fashion by vacuum expectation values of suitably constructed spurion fields. We identify the possible scenarios in the quark sector and discuss some implications of this approach.

1

Introduction

The origin of flavour remains one of the main mysteries in modern particle physics, and many attempts have been made to understand the phenomenon of flavour by postulating certain (discrete) flavour symmetries (see e.g. [1–6] and references therein), or by localizing fermions in extra dimensions (see e.g. [7–14] and references therein), to name two popular ideas. While such scenarios can successfully explain some of the issues related to the hierarchies observed in fermion masses and mixings, the origin of the proposed new mechanisms (e.g. from the embedding into a Grand Unified Theory or even String Theory, respectively) still remains an open issue. Alternatively, we may start from a bottom-up approach in which the phenomenon of flavour is just parametrized as in the Standard Model (SM). In fact, the SM has an approximate global flavour symmetry GF (see below), which is broken by the Yukawa couplings, inducing the fermion masses and mixings. Such an explicit symmetry breaking is usually parametrized by introducing spurion fields with a definite behaviour under the symmetry to be broken. In the case at hand, focusing on the quark sector, the Yukawa matrices YU and YD are considered as complex spurion fields [15], transforming non-trivially under GF . A special role is played by the top quark, which has a Yukawa coupling of order one, breaking the original flavour symmetry group GF to a smaller sub-group G′F (see below), which is still a good symmetry as long as the remaining Yukawa couplings are negligible. In a recent paper [16], two of us have shown that in such a case it is convenient to consider a non-linear representation of GF in which the subgroup G′F is linearly realized. In this context, it turned out to be useful to assign a canonical mass dimension to the Yukawa spurion fields, since in this way the top Yukawa coupling could be understood as originating from a dimension-four operator while the remaining Yukawa terms are dimension five, thereby reflecting the hierarchy between the top mass and the lighter quark masses.1 If we take this approach seriously, two immediate implications arise: • The spontaneous breaking GF → G′F induces Goldstone modes who call for a dynamical interpretation. One possibility is to consider local flavour symmetries, where Goldstone modes become the longitudinal modes for massive gauge bosons. Another alternative is to keep the Goldstone modes as physical axion-like degrees of freedom.2 These issues will be discussed in somewhat more detail in a separate publication [19]. • The breaking GF → G′F induced by the top-quark Yukawa coupling can be considered the first step in a sequence of flavour symmetry breaking steps taking place at different physical scales Λ ≫ Λ′ ≫ Λ′′ ≫ . . . Through the VEVs of the spurion fields, the hierarchy of scales should be directly related to the observed hierarchy for quark masses and mixings. 1

A similar construction can be performed in the lepton sector, when the SM is minimally extended by a dimension-5 operator in order to describe non-vanishing neutrino masses [17, 18]. 2 An option to avoid Goldstone modes alltogether is to restrict oneselves to discrete flavour symmetries.

1

In the following, we shall identify the flavour sub-groups for each of the intermediate effective theories in the construction above and identify the corresponding representations for quark fields, spurions and Goldstone modes. We will also briefly discuss the requirements for the spurion potential necessary for such a scenario.

2

Successive flavour symmetry breaking

In this section we identify the sequence of intermediate (residual) flavour symmetries which arise when the original flavour symmetry of the SM gauge sector is broken in a step-wise fashion at different scales, set by the VEVs of the relevant spurion fields and linked to the observed hierarchies in the quark masses and CKM angles. Considering the Yukawa sector for the quarks, ¯L H ˜ UR + YD Q ¯ L H DR + h.c. , −LY = YU Q

(2.1)

we may consider independent phase transformations for the 3 quark multiplets (QL , UR , DR ) and the Higgs field (H). Among these 4 phases, 2 are identified as baryon number U(1)B and weak hypercharge U(1)Y which are not broken by the Yukawa matrices, whereas the 2 remaining U(1) symmetries are broken by hYU i = 6 0 or hYD i = 6 0. We thus define the flavour group in the quark sector as3 GF = SU(3)3 × U(1)4 /(U(1)B × U(1)Y ) = SU(3)QL × SU(3)UR × SU(3)DR × U(1)UR × U(1)DR .

(2.2)

For the Yukawa sector to be formally invariant under GF , we assign the following transformation properties to the spurion fields, YU ∼ (3, ¯3, 1)−1,0 ,

YD ∼ (3, 1, ¯3)0,−1 ,

(2.3)

where the terms in brackets refer to the three SU(3) factors, and the subscripts to the two U(1) factors, respectively. Counting parameters, we have 2 × 18 = 36 entries for the spurions YU,D and 3 × 8 + 2 = 26 symmetry generators, leaving 36 − 26 = 10 physical parameters in the quark Yukawa sector, which can be identified with the 6 quark masses, the 3 CKM angles, and the CP-violating CKM phase (see also [20]).4 3 Our discussion differs from the one in [15] where the independent phase rotations for the Higgs fields have been overlooked. 4 Similarly, considering the U (1) phases in the lepton sector, we obtain the SM flavour group

Glepton = U (3)2 /(U (1)e × U (1)µ × U (1)τ ) F for massless neutrinos, and

˜ F lepton = U (3)2 G for massive neutrinos which are generated by a lepton-number violating dim-5 term in the Lagrangian −LMaj =

1 gν (HℓL )T (HℓL ) . ΛL

2

Table 1: SM values for the quark masses [21], and approximate scaling with the Wolfenstein parameter λ ∼ 0.2. Light-quark masses (u,d,s) are given in the MS scheme at µ = 2 GeV, charm and bottom masses as m ¯ c (m ¯ c ) and m ¯ b (m ¯ b ), and the top mass is evolved down to the scale mb . The evolution between the scales mb and mc is negligible for our considerations.

mq nq = logλ (mq /mt ) mq nq = logλ (mq /mt )

u [1.5 − 4.5] MeV 6−9 c [1.0 − 1.4] GeV 3−4

d [5.0 − 8.5] MeV 6−8 b [4.0 − 4.5] GeV 2−3

s [80 − 155] MeV 4−6 t [250 − 300] GeV 0

In order to specify the sequence of flavour symmetry breaking, we have to identify a hierarchy between the Yukawa entries (YU )ij and (YD )ij . However, before the flavour symmetry is actually broken, the Yukawa matrices can be freely rotated by transformation matrices in GF , and therefore the a-priori ranking of individual entries in the Yukawa matrices seems to be somewhat ambiguous. On the other hand, the right-handed rotations and a common left-handed rotation for up- and down-quarks are not observable in the SM, anyway, leaving the quark masses and CKM angles as the only relevant parameters. We therefore find it sufficient to choose a basis where the right-handed rotations are unity, while for the left-rotations we restrict ourselves to matrices VuL and VdL which scale in the same manner as the CKM matrix. This leaves us with the generic power counting5 n λ u λ1+nc λ3 hYU iij ∼ (VuL )ij (yu )j ∼ λ1+nu λnc λ2 , λ3+nu λ2+nc 1 n λ d λ1+ns λ3+nb hYD iij ∼ (VdL )ij (yd )j ∼ λ1+nd λns λ2+nb , (2.4) 3+nd 2+ns nb λ λ λ where we introduced the scaling for quark Yukawa couplings with the Wolfenstein parameter (λ ∼ 0.2 ≪ 1) as yi ∼ λni (with nt = 0), and inserted the standard power counting

In the first case, we have 18 parameters in the spurion YE and 2 × 8 − 1 = 15 symmetry generators, leaving 3 physical parameters to be identified with the 3 charged lepton masses. In the second case, we have 18 + 12 = 30 parameters from the spurions YE and gν , from which we subtract 2 × 9 = 18 symmetry generators, to obtain 12 physical parameters, which are the 6 lepton masses, the 3 PMNS angles, one Dirac phase, and the 2 Majorana phases. 5 During the sequence of flavour symmetry breaking, some of the entries can actually be set to zero by exploiting the freedom to rotate the VEVs of certain spurion fields with respect to the corresponding residual flavour group.

3

for CKM elements, VuL ∼ VdL ∼ VCKM

1 λ λ3 ∼ λ 1 λ2 . λ3 λ2 1

(2.5)

The scaling of the quark masses can be constrained from the phenomenological information in Table 1, where we assume in the following that renormalization-group effects (in the sequence of effective theories to be constructed) do not change the hierarchies observed at low scales a lot. More precisely, to keep the discussion simple, we restrict ourselves to: • nd > ns > nb > 0 and nu > nc > nt ≡ 0, • nc ≥ nb and ns > nc .

The remaining degree of freedom in choosing values for the ni leads to several options, among which are also cases where one or two spurions receive their VEV at the same scale simultaneously. To be concrete, we focus on three cases with more or less natural and distinct scale separation, (a1) nc < nb + 2 < nb + 3 < ns , (a2) nc < nb + 2 < ns < nb + 3, and (b) nb + 2 < nc < ns < nc + 1, which are summarized in Table 2. A detailed derivation of the various steps in the flavour symmetry breaking can be found in the appendix. Let us discuss some common and distinct features of the different scenarios: • Common to all scenarios is the second step of symmetry breaking which (at least in our set-up with only one electroweak Higgs doublet) is unambiguously induced by the VEV for the (YD )33 element which gives rise to the bottom-quark mass. Below the scale Λ′ ∼ yb Λ, the residual flavour symmetry is G′′F = U(2)QL × U(2)UR × U(2)DR .

(2.6)

At first glance, it appears as just the 2-family analogue of the original flavour group GF . However, there are two important differences: First, it appears one additional U(1) factor compared to GF . Second, it still contains an off-diagonal spurion field χs , which is a doublet of SU(2)QL and the only spurion which is charged under the additional U(1). Only if this spurion field (and the associated breaking of the extra U(1) symmetry) were absent, we would recover an effective 2-family model where, as is well-known, one would have no CP-violation in the quark Yukawa sector. • From an aesthetic point of view, the alternative labeled (a1) in Table 2 is somewhat favoured. It can be realized with a rather natural hierarchy of scales. For instance taking nb = 2 , nc = 3 , ns = 6 , nu,d = 8 , which fits well to the phenomenological mass spectrum, one obtains an equal separation of scales,6 Λ(n) = λ(n+1) Λ . 6

For comparison, scenario (a2) can be realized, for instance, by nb = 2.5, nc = 3.5, ns = 5, nu,d = 7, leading to the tower of scales (λ2.5 , λ3.5 , λ4.5 , λ5 , λ5.5 , λ6 , λ7 ) Λ. Similar, case (b) could be realized by nb = 2, nc = 4.5, ns = 5, nu,d = 7 with (λ2 , λ4 , λ4.5 , λ5 , λ5.5 , λ6 , λ7 ) Λ.

4

Table 2: Three alternative sequences of flavour symmetry breaking, and associated parameter counting for the Yukawa matrices. Notice that the following equalities always hold: # Spurions + # VEVs − # Symmetries = 10 , # Goldstones + # Spurions + # VEVs = 36 , where 10 refers to the 6 quark masses + three CKM rotations + one CKM phase, and 36 refers to the original 2 × 18 real parameters in the Yukawa matrices YU and YD . SU (3)QL SU (2)QL SU (2)QL (a)

(b)

Flavour Symmetry × SU (3)UR × SU (3)DR × U (1)2 × SU (2)UR × SU (3)DR × U (1)3 × SU (2)UR × SU (2)DR × U (1)3

SU (2)DR × U (1)4 (a1) SU (2)DR × U (1)3 SU (2)DR × U (1)2 U (1)2 (a2) SU (2)DR × U (1)3 U (1)3 U (1)2 SU (2)UR × SU (2)DR × U (1)3 SU (2)DR × U (1)3 U (1)3 U (1)2

U (1)2 –

(CP /) (CP /)

GBs 0 9 14

Spur. 36 26 20

VEVs 0 1 2

Symm. 26 17 12

19 20 21 24 20 23 24 17 20 23 24

14 12 11 6 12 8 6 16 12 8 6

3 4 5 6 4 5 6 3 4 5 6

7 6 5 2 6 3 2 9 6 3 2

24 26

4 0

7+1 9+1

2 0

Scale Λ ∼ yt Λ Λ′ ∼ yb Λ

Λ(2a) ∼ yc Λ Λ(3a1) ∼ yb λ2 Λ Λ(4a1) ∼ yb λ3 Λ Λ(5a1) ∼ ys Λ Λ(3a1) ∼ yb λ2 Λ Λ(4a2) ∼ ys Λ Λ(5a2) ∼ yb λ3 Λ Λ(2b) ∼ yb λ2 Λ Λ(3b) ∼ yc Λ Λ(4b) ∼ ys Λ Λ(5b) ∼ yc λ Λ Λ(6) ∼ ys λ Λ Λ(7) ∼ yu,d Λ

Moreover, the smallest non-abelian sub-group for this case is given by SU(2)DR × U(1)2 . This residual flavour symmetry may thus be taken as the simplest non-trivial example to study the dynamics of flavour spurions and its consequences for flavour physics, including the construction of higher-dimensional operators for flavour transitions with minimal flavour violation (or beyond [26]), the dynamics of Goldstone modes, and the construction of realistic scalar potentials. • In all cases, the symmetry is eventually broken down to U(1)2 = U(1)uR × U(1)dR .

The corresponding effective theory now still contains three complex spurion fields, among which one spurion is uncharged under either of the two U(1) groups. Consequently, when the latter acquires its VEV, its phase cannot be rotated away by 5

symmetry transformation.7 At this very step, we therefore generically encounter a CP-violating phase, which in our case is associated with the (YD )12 element. • Finally, the two U(1) symmetries will be broken by the (YU )11 and (YD )11 elements associated with the up- and down-quark mass. Notice that these symmetries are chiral, and the corresponding U(1) anomalies contribute to the effective θ-parameter in QCD. The related spurion fields may serve as a solution to the strong CP problem as in the general Peccei-Quinn setup [23–25]. This will be discussed in more detail in [19].

3

Invariants and Potentials for scalar spurion fields

In this section we consider how the sequential symmetry breaking, described in the last section, could be achieved spontaneously. The question of how an appropriate potential could look like is discussed in many different contexts (see e.g. [27–29]), but no general recipe for constructing a potential that leads to a specific symmetry breaking has been found. In any case, a potential for the spurion fields can only depend on invariants under the flavour symmetry group GF . Due to the form of the potential these invariants should take the appropriate VEVs which finally specify the ten physical parameters (6 quark masses and 4 CKM parameters). Of course we are unable to derive a potential which achieves this complicated symmetry breaking, but we may at least identify ten independent invariants in terms of which we may express the physical quantities. These invariants can be constructed from monomials of the basic scalar spurion fields YU (x) and YD (x), and may thus be classified by their canonical dimension. Before considering the 3-family case, it is instructive to look at the simpler example of two families with the flavour symmetry gF = SU(2)QL × SU(2)UR × SU(2)DR × U(1)2 , first. It exhibits 11 symmetry generators which leaves 5 physical parameters (4 masses and the Cabibbo angle) from the 16 parameters in the Yukawa matrices. Classifying the invariants by increasing canonical dimension, we find (2)

v1 /Λ2 = yu2 + yc2 ,

(2)

(2)

v2 /Λ2 = yd2 + ys2 ,

i1 = tr(U) ,

(2)

i2 = tr(D) , (4)

(2)

(4)

(2) (2)

(4)

(2)

i1 = tr(U 2 ) − (i1 )2 ,

i2 = tr(UD) − i1 i2 , i3 = tr(D 2 ) − (i2 )2 ,

(4)

v1 /Λ4 = −2yu2 yc2 , (4)

v2 /Λ4 = sin2 θ(yc2 − yu2 )(yd2 − ys2) − yc2yd2 − yu2 ys2 , (4)

v3 /Λ4 = −2yd2ys2 ,

(3.1)

where we introduced the combinations

U = YU YU† ,

D = YD YD† ,

7

(3.2)

Alternatively, in a previous step of the construction, one could have identified two spurion fields with the same quantum numbers, whose VEVs in general cannot be made real simultaneously. This mechanism thus gives a particular realization of spontaneous CP-violation [22].

6

(k)

(k)

which transform homogeneously under SU(2)QL , and where we denote with vα = hiα i (m) the VEVs of the 5 invariants. The potential V = V (iα ) may now be expanded around its minimal value in the form XX (m,k) (k) 1 (k) (m) (m) V = , (3.3) iβ − vβ i − vα Mα,β Λm+k−4 α k,m α,β (m,k)

where Λ is a UV-scale which renders the positive semi-definite matrix Mα,β dimensionless. Notice that higher-dimensional operators appear unavoidably if we assign canonical mass dimension to the (scalar) spurion fields YU,D . As already mentioned, the mechanism how such an effective potential could be generated by integrating out some new degrees of freedom in an underlying theory, remains an open issue. In principle we may also invert the relations to obtain the Cabibbo angle and the (k) masses as functions of the vi , however, the above invariants are not yet very suitable for the further discussion: • As we have seen in the previous section, the order of the different symmetry-breaking steps depend on the relative size of the Yukawa entries, which in the 2-family case are characterized by the exponents {nu , nc , nd , ns , (1 + ns )} (in the hierarchical limit). It is therefore desirable to consider invariants that feature the very same exponents. • To put the invariants on a similar footing, they should have the same canonical dimension (i.e. we have to introduce rational functions of the above invariants). (4)

• Instead of i2 it would be desirable to have an invariant that vanishes in the no-mixing case (θ = 0). Such invariants can be constructed from the commutator [U, D], (8)

i1 = det ([U, D]) ,

(8)

v1 /Λ8 =

1 2 (y − yu2 )2 (ys2 − yd2)2 sin2 2θ . 4 c

(3.4)

We therefore modify the above definitions as follows, I1 = tr(U) , I2 = tr(D) , 1 I3 = I1 − tr(U 2 )/I1 , 2 1 I2 − tr(D 2 )/I2 , I4 = 2 det ([U, D]) , I5 = 4 I1 I2 (I1 + I2 )

V1 /Λ2 = yu2 + yc2 , V2 /Λ2 = yd2 + ys2 , y2y2 V3 /Λ2 = 2 u c 2 , yu + yc y2y2 V4 /Λ2 = 2 d s 2 , ys + yd (yc2 − yu2 )2 (ys2 − yd2 )2 sin2 2θ 2 . (3.5) V5 /Λ = 2 (yu + yc2)(yd2 + ys2)(yu2 + yc2 + yd2 + ys2 )

The invariants I1−5 now take their VEVs according to the power-counting for masses and mixing angles. For instance, with our standard case, nc < ns < 1 + ns < nu ∼ nd , we have V1 ∼ λ2nc ≫ V2 ∼ λ2ns ≫ V5 ∼ λ2+2ns ≫ V3,4 ∼ λ2nu,d , 7

which defines the sequence of symmetry breaking. We may then solve (3.5) for masses and mixing angle to obtain p V1 ± V1 (V1 − 4V3 ) n V1 /Λ2 2 ≃ , yc,u = V3 /Λ2 2Λ2 p V2 ± V2 (V2 − 4V4 ) n V2 /Λ2 2 ys,d = , ≃ V4 /Λ2 2Λ2 (V1 + V2 ) V5 V5 sin2 2θ = ≃ , (3.6) (V1 − 4V3 )(V2 − 4V4 ) V2 where the approximate relations refer to the SM hierarchies. We note in passing that models based on texture zeros, which imply relations between the masses and the mixing angles [30], may be mapped onto relations between invariants. In turn, a relation between invariants always characterizes a class of Yukawa matrices which may or may not feature texture zeros in a particular flavour basis. This may be explicitly demonstrated by considering a simple two-family model with one texture zero. We use the basis in which YU is diagonal and 0 a (3.7) YD = a 2b is given in terms of two parameters a and b. This model implies the relation √ p 4 (V1 − 4V3 ) ( V2 V4 − 2V4 ) ≃ 4 V2 V4 , V5 = V1 + V2

(3.8)

which translates into a relation between the Cabibbo angle and the down-type masses, r md (3.9) tan θ ≃ ms which is phenomenologically reasonable. We now turn to the 3-family case, which can be studied along the same lines. We have to identify in total ten independent invariants. The two quadratic and the three quartic invariants are again given by (2)

i2 = tr(D) .

(2)

(4)

(2) (2)

i1 = tr(U) ,

(3.10)

and (4)

(2)

i1 = tr(U 2 ) − (i1 )2 ,

i2 = tr(UD) − i1 i2 ,

(4)

(2)

i3 = tr(D 2 ) − (i2 )2 .

(3.11)

The remaining 5 invariants, which are necessary to specify the physical quark flavour parameters, thus have to be built from even higher-dimensional invariants. For the dimension8

6 terms, we choose (6)

3 (4) (2) (2) i i − (i1 )3 ≡ 3 det(U) , 2 1 1 1 (4) (2) (4) (2) (2) (2) = tr(U 2 D) − i1 i2 − i2 i1 − i2 (i1 )2 , 2 1 (4) (2) (4) (2) (2) (2) = tr(UD 2 ) − i3 i1 − i2 i2 − i1 (i2 )2 , 2 3 (4) (2) (2) = tr(D 3 ) − i3 i2 − (i2 )3 ≡ 3 det(D) . 2

i1 = tr(U 3 ) − (6)

i2

(6)

i3

(6)

i4

(3.12)

Finally, among the dimension-8 invariants only one is linearly independent, and we choose (8)

i1 = tr (U[U, D]D) ,

(3.13)

which completes the list of invariants for the 3 × 3 case. (m) The potential V = V (iα ) can again be expanded as in (3.3). The sequential breaking of GF as proposed in the last section can emerge only through a hierarchy of VEVs for the various invariants. This hierarchy has to be put in by hand in (3.3) and may perhaps find its explanation in an underlying theory above the scale Λ.8 In fact, our choice of VEVs is such, that the first breaking of GF → G′F is obtained, if the potential V generates a (2) (sizeable) VEV for the i1 invariant, only, (2)

(m)

v1 ≃ yt2Λ2 ,

vk

≃ 0 otherwise ,

(3.14)

in which case we obtain a non-vanishing top quark Yukawa coupling, while all other parameters (which give rise to the lighter quark masses and CKM parameters) still (approximately) vanish. The next step is the breaking of G′F → G′′F . Clearly the relevant potential V ′ can (m) only depend on the invariants of G′F , which we denote as jk . As before, we introduce quadratic terms which transform under SU(2)L × U(1)T , namely 2 triplets, (2)

(2)†

(2) (2)† D ′ = Y˜D Y˜D ,

(3.15)

X ′ = Y˜D ξb ,

(2)

(3.16)

Ξ′ = ξb† ξb .

(3.17)

U ′ = YU YU

,

one charged doublet

and one singlet

8

We note, however, that restricting ourselves to the most general set of dimension-4 operators, where X X X (4) (2) (2) (2) λi ii , 2ρij ii ij + m2i ii + V = i

i

i,j

only part of the flavour symmetry will be broken by the minimum of the potential, including the case GF → G′F for a particular sub-set of parameter space.

9

In terms of these, the invariants of dimension-2 can be written as (2)

j1 = tr(U ′ ) ,

(2)

j2 = tr(D ′ ) ,

while the fourth-order invariants are (4) (2) j1 = tr (U ′ )2 − (j1 )2 , (4) (2) j3 = tr (D ′ )2 − (j2 )2 ,

(2)

j3 = Ξ′ ,

(4)

(3.18)

(2) (2)

j2 = tr(U ′ D ′ ) − j1 j2 , (4)

(2) (2)

j4 = X ′† X ′ − j2 j3 .

(3.19)

Finally, there are 2 linear independent invariants of dimension 6, (6)

(6)

j1 = X ′† U ′ X ′ ,

j2 = X ′† D ′ X ′ .

(3.20)

At tree level, the potential V ′ simply follows from the original potential V by expressing (m) (m) the invariants ik by the invariants jα and the VEV for the top Yukawa coupling, see appendix C. Including radiative corrections in the effective theory below the scale Λ (or more precisely, below the mass scale of the scalar degree of freedom related to the VEV yt Λ), the parameters of the effective potential might change accordingly. The general form is thus again given by XX (m,k) (k) 1 (k) (m) (m) ′ . (3.21) jβ − wβ j − wα Nα,β V = (Λ′ )m+k−4 α k,m α,β (2)

The next step in the symmetry breaking, G′F → G′′F , will then be achieved by w3 ≃ yb2 (Λ′ )2 . This scheme can be repeated until the complete flavour symmetry is broken. (m) (m) Note, that the invariants ii and ji introduced above are all real. Therefore, the (m,k) (m,k) parameters Mi,j and Ni,j have to be real as well to yield a hermitian potential. As described above, the CKM phase, corresponding to the SM mechanism for CP-violation, appears when one of the spurion fields receives a complex VEV. The potential allows for spontaneous CP-violation, as soon as an invariant of one of the residual flavour symmetries becomes complex. In the scenarios discussed above, this is the case for (4) (4) † † 3a1 ∗ ∗ GF : L1 = Re χ13 ξd ξs χ23 , L2 = Im χ13 ξd ξs χ23 , (3.22) (3b) ′(4) ′(4) and GF : L1 = Re ξu† ξc ξs† ξd , L2 = Im ξu† ξc ξs† ξd , (3.23) (′ )(4)

where L2 is odd under CP. As in the 2-family example, we again introduce rational functions of the invariants that are convenient for the discussion of power-counting or parameter relations in models with

10

texture zeros. The modified set of invariants for the 3-family case reads V1 /Λ2 = yu2 + yc2 + yt2 ∼ λ0 , V2 /Λ2 = yd2 + ys2 + yb2 ∼ λ2nb , y 2 y 2 + y 2 y 2 + y 2y 2 V3 /Λ2 = u c 2 u 2 t 2 c t ∼ λ2nc , yu + yc + yt 2 2 y y + y 2y 2 + y 2y 2 V4 /Λ2 = d s 2 d 2 b 2 s b ∼ λ2ns , ys + yd + yb y 2 y 2y 2 V5 /Λ2 = 2 2 u 2 c 2 t 2 2 ∼ λ2nu , yu yc + yu yt + yc yt y 2 y 2y 2 V6 /Λ2 = 2 2 d 2 s 2 b 2 2 ∼ λ2nd , yd ys + yd yb + ys yb

I1 = tr(U) , I2 = tr(D) , 1 I1 − tr(U 2 )/I1 , I3 = 2 1 I2 − tr(D 2 )/I2 , I4 = 2 I5 = det(U)/I1 /I3 ,

I6 = det(D)/I2/I4 ,

(3.24)

which determines the 6 Yukawa couplings corresponding to the quark masses, and tr (U[U, D]D) , I1 I2 (I1 + I2 ) 1 det ([U, [U, D]]) , I8 = 2 I12 I2 (I1 + I2 )2 I32 I7 1 det ([[U, D] , D]) I9 = , 2 I22 I1 (I1 + I2 )2 I42 I7 i det ([U, D]) , I10 = − 2 2 2 I1 I2 (I3 + I4 ) I7 =

2 V7 /Λ2 ≃ yb2 θ23 ∼ λ2(nb +2) ,

θ12 θ13 cos δ ∼ λ2(nb +3) + λ2(ns +1) , θ23 2 2 2 θ13 + θ12 θ23 − 2θ12 θ23 θ13 cos δ ∼ λ2(nb +3) ,

2 V8 /Λ2 ≃ yb2 θ13 + ys2

V9 /Λ2 ≃ yb2

V10 /Λ2 ≃ ys2 θ12 θ23 θ13 sin δ ∼ λ4 λ2(ns +1)

(3.25)

which determines the angles and the CP-violating phase in the standard parametrization [31]. Again, the invariants I7−10 are defined in such a way that they vanish in the no-mixing case. Moreover, I10 6= 0 signals CP-violation. We may again solve for the SM parameters to obtain the quark Yukawa couplings 2 2 yt,c,u (V1,3,5 ) and yb,s,d (V2,4,6 ), as well as the (approximate) solutions for the mixing angles

θ12

V7 V8 2 2 , θ13 ≃ , θ23 ≃ V2 V2 2 θ13 V 2V 2 V 2V 2 V9 2 cos δ − − 22 10 , θ12 sin2 δ ≃ 22 10 , ≃ θ23 V7 V4 V7 V8 V4 V7 V8

(3.26)

where we also neglected terms of order λ−4 ys2 /yb2. Finally, we consider again a simple model with texture zeros in the 3 × 3 Yukawa matrices [32], 0 Cu 0 0 Cd 0 YU = Cu∗ 0 Bu , YD = Cd∗ 0 Bd , (3.27) ∗ ∗ 0 Bu |Au | 0 Bd |Ad | 11

which yields the following approximate relations between quark masses and mixing angles 2 mu θ13 |Vub |2 ≃ , ≃ 2 |Vcb |2 θ23 mc

2 2 2 md |Vtd |2 θ13 + θ12 θ23 − 2θ12 θ23 θ13 cos δ ≃ . ≃ 2 |Vts |2 θ23 ms

(3.28)

As before, this can be formulated in a basis-independent way in terms of the following approximate relations between invariants r r V8 V5 V6 V9 ≃ , ≃ . (3.29) V7 V3 V7 V4

4

Conclusions

In this paper we have shown how the hierarchies in quark masses and mixings can be associated with a particular sequence of flavour symmetry breaking. The different scales at which the individual steps of partial flavour symmetry breaking occur are separated among each other by not more than 1-2 orders of magnitude. Depending on the assumed power counting for the quark masses, we have identified different scenarios that are compatible with phenomenology. We have also given some general arguments for the possible form of scalar potentials that may realize the sequence of flavour symmetry breaking and identified the invariants that may be used to expand the potential around its minimum or to classify ans¨atze for the Yukawa matrices involving texture zeros in a basis-independent way. In all cases, the minimal non-abelian flavour sub-group is given by SU(2)DR × U(1)2(3) . Its further breaking eventually leads to an effective theory with a residual U(1)2 flavour symmetry, where one of the spurion fields is uncharged. When this spurion achieves a complex VEV, its phase cannot be rotated away and provides the one and only source for CP-violation in the quark Yukawa sector. The CP-violating phase is thus generated at rather low scales (compared to, say, a GUT scale). A dynamical interpretation of the Goldstone modes, appearing at each step of the (global) flavour symmetry breaking, can be achieved by promoting the flavour symmetries to local ones, where the Goldstone modes become the longitudinal modes of the corresponding massive gauge bosons. One the other hand, the final chiral U(1)2 symmetries are anomalous and the associated Goldstone bosons couple to the QCD instantons. They may thus be used to resolve the strong CP-problem as in the general Peccei-Quinn setup, with the corresponding Goldstone modes appearing as axion fields. Details will be presented in a separate publication [19].

Note added While completing this work, the paper [33] appeared, where a 2-Higgs-doublet scenario with a large ratio of VEVs (tan β ∼ mt /mb ≫ 1) was considered. In this case, the original flavour symmetry is broken in one step as GF → G′′F = U(2)3 , see Eq. (2.6) and the discussion in [16]. The possible enhancement with tan β allows for interesting observable 12

deviations from the SM and from minimally flavour-violating scenarios with minimal Higgs sector. In [33] it has been shown that they can be identified in a very transparent way using the non-linear representation of flavour symmetries suggested in [16]. It is evident, that the related change in the hierarchies of the Yukawa matrices for tan β ≫ 1 would also imply a different pattern for the sequence of flavour symmetry breaking which could be worked out in an analogous way as presented in our work.

Acknowledgements We would like to thank Michaela Albrecht for many helpful discussions. TM wants to thank Aneesh Manohar for a helpful discussion on the invariants. This work has been supported in part by the German Research Foundation (DFG, Contract No. MA1187/101) and by the German Ministry of Research (BMBF, Contract No. 05HT6PSA), by the EU MRTN-CT-2006-035482 (FLAVIAnet), by MICINN (Spain) under grant FPA2007-60323, and by the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042).

A

Sequence of flavour-symmetry breaking in the SM

In this appendix, we present the detailed derivation of the different scenarios for sequential flavour-symmetry breaking as discussed in the text.

A.1

Leading order

Neglecting all terms of O(λ) in YU and YD , only the top-quark Yukawa coupling in (YU )33 survives, due to our general assumption nq > 0. We thus obtain the breaking (which – apart from the additional U(1) factors – coincides with the discussion in [16]) GF → G′F = SU(2)QL × SU(2)UR × SU(3)DR × U(1)T × U(1)U (2) × U(1)DR

(A.1)

R

∼ SU(2)QL × SU(2)UR × SU(3)DR × U(1)Q(2) × U(1)U (2) × U(1)DR , L

(A.2)

R

where the equivalence in the second line arises if we take into account the globally conserved baryon number, implying the relation (2)

(2)

3B = T + QL + UR + DR for the quark charges, where T counts the quark-number for the third generation in QL (2) (2) and UR , and QL and UR for the first two generations (see also appendix B). The decomposition of YU and YD in terms of irreducible representations of G′F and the representation 8 8 of the 9 Goldstone modes (Πa=4..8 L,UR , ΠL = −ΠUR ) remains as in [16], with ! 0 (2) (2) ˜ Y Y D 0 U † (ΠUR ) , YD = U(ΠL ) , (A.3) YU = U(ΠL ) U † ξ b 0 0 y Λ t

and U(Π) = exp [iΠa T a /Λ].

13

A.2

Order Λ′ /Λ

Let us first consider the transformation properties of the residual spurion fields with respect to G′F (here the subscripts refer to the U(1) factors defined in Eq. (A.1)), and their scaling with λ, n λ u λ1+nc (2) YU ∼ (2, 2, 1)1,−1,0 ∝ Λ, λ1+nu λnc n 1+ns 3+nb d λ λ λ (2) Y˜D ∼ (2, 1, ¯3)1,0,−1 ∝ Λ, λ1+nd λns λ2+nb ξb† ∼ (1, 1, ¯3)0,0,−1 ∝ λ3+nd λ2+ns λnb Λ . (A.4) We now assume that at the scale Λ′ ≪ Λ, the next-highest entry in the residual spurion fields gets its VEV. For nc > nb , the spurion ξb† will have the largest eigenvalue, 9 hξb†i = (0, 0, y˜b) Λ ≡ (0, 0, xb ) Λ′ ,

(A.5)

with xb = O(1) such that y˜b ∼ Λ′ /Λ ∼ mb /mt . Similarly as for the discussion of the 2HDM with large tan β in [16], this further breaks the flavour symmetry to G′F → G′′F = SU(2)QL × SU(2)UR × SU(2)DR × U(1)Q(2) × U(1)U (2) × U(1)D(2)

(A.6)

∼ SU(2)QL × SU(2)UR × SU(2)DR × U(1)III × U(1)U (2) × U(1)D(2) ,

(A.7)

L

R

R

R

R

where now U(1)III acts on all quarks in the third generation. The 5 additional Goldstone modes (Π′DR a=4..8 ) are introduced as (2) Y˜D = YD(2) χs U † (Π′DR ) , (A.8) ξb† = 0 0 xb Λ′ U † (Π′DR ) . with U(Π′ ) = exp [iΠ′a T a /Λ′ ].

A.3

Alternative (a1): nc < nb + 2 < nb + 3 < ns

At this stage, the further breaking of the flavour symmetry depends on the details about the assumed power counting for the quark masses. Let us first discuss the scenario (a1): 9

(2)

If we allow for nc = nb , the spurion YU also will get its VEV simultaneously, such that in the scenario (a) discussed below, the scales Λ′ and Λ′′ would coincide.

14

A.3.1

Order Λ′′ /Λ

In the case nc < nb +2, it is convenient to classify the residual spurion fields of G′′F according to (A.7), 1+n λ u λ1+nc (2) YU ∼ (2, 2, 1)0,−1,0 ∝ Λ, λ1+nu λnc n λ d λ1+ns (2) YD ∼ (2, 1, 2)0,0,−1 ∝ Λ, λ1+nd λns 3+n λ b χs ∼ (2, 1, 1)−1,0,0 ∝ Λ, (A.9) λ2+nb

such that the next spurion getting a VEV is 0 0 0 0 (2) hYU i = Λ≡ Λ′′ 0 y˜c 0 xc

(A.10)

(2)

with xc = O(1), implying y˜c ∼ Λ′′ /Λ ∼ mc /mt . The VEV of YU flavour symmetry as (3a1)

G′′F → GF

thus further breaks the

= SU(2)DR × U(1)C × U(1)III × U(1)Q(1) × U(1)U (1)

(A.11)

∼ SU(2)DR × U(1)C × U(1)Q(1) × U(1)U (1) × U(1)D(2) ,

(A.12)

L

L

R

R

R

where U(1)C refers to the second-generation quarks in QL and UR . This implies 5 additional Goldstone bosons (Π′′L,UR a=1..3 , Π′′L 3 = −Π′′UR 3 ), appearing via (1) YU 0 (2) ′′ YU = U(ΠL ) U † (Π′′UR ) , (A.13) 0 xc Λ′′ † χ13 ξd (2) ′′ ′′ , χs = U(ΠL ) . (A.14) YD = U(ΠL ) χ23 ξs† A.3.2

Order Λ(3) /Λ (3a1)

The residual spurions of GF

now scale/transform as

(1)

YU ∼ (1)0,1,−1,0 ∝ λnu , ξd† ∼ (2)0,1,0,−1 ∝

ξs† ∼ (2)1,0,0,−1 ∝

λnd λ1+ns λ1+nd λns

χ13 ∼ (1)0,1,0,0 ∝ λ3+nb ,

,

χ23 ∼ (1)1,0,0,0 ∝ λ2+nb ,

,

(A.15)

where the subscripts refer to the U(1) charges in (A.12). In this case, assuming ns > nb +2, the next spurion to receive a VEV is χ23 , which breaks the U(1)C symmetry, (3a1)

GF

(4a1)

→ GF

= SU(2)DR × U(1)Q(1) × U(1)U (1) × U(1)D(2) . L

R

(A.16)

R

The associated Goldstone boson φ′′′ appears as a simple phase, χ23 = xsb eiφ

′′′ /Λ′′′

Λ′′′

and

with Λ′′′ /Λ ∼ yb λ2 . 15

ξs† → ξs† eiφ

′′′ /Λ′′′

(A.17)

A.3.3

Order Λ(4) /Λ (4a1)

The residual spurions for GF

read

(1)

YU ∼ (1)1,−1,0 ∝ λnu , ξd† ∼ (2)1,0,−1 ∝ ξs†

λnd λ1+ns 1+nd

∼ (2)0,0,−1 ∝ λ ∼ (1)1,0,0 ∝ λ3+nb .

χ13

λ

ns

, , (A.18)

For ns > nb +3, the next spurion to get a VEV is χ13 which breaks another U(1) symmetry, (4a1)

GF

(5a1)

→ GF

= SU(2)DR × U(1)U (1) × U(1)D(2) . R

(A.19)

R

The corresponding Goldstone mode φ(iv) appears again as a phase factor used to redefine (1) the spurions YU and ξd according to their U(1) charge. It should be noted that after the U(1)Q(1) is broken, the residual spurions ξd and ξs have the same quantum numbers with L

(5a1)

respect to GF , and therefore the relative phase of their VEVs will provide the source for spontaneous CP-violation. A.3.4

Order Λ(5) /Λ

Taking hξs i = 6 0, we next break (5a1)

GF

(6a1)

→ GF

= U(1)U (1) × U(1)D(1) R

(A.20)

R

and the remaining spurion fields are given by (1)

YU ∼ (1)−1,0 ∝ λnu ,

(1)

YD ∼ (1)0,−1 ∝ λnd ,

χ12 ∼ (1)0,0 ∝ λ1+ns .

(A.21) (1)

Here we have decomposed the SU(2)DR doublet ξd into two complex singlets YD and χ12 , (v)

(1)

A.4 A.4.1

(v)

ξd† = (YD , χ12 ) U † (ΠR ) .

ξs† = (0, xss Λ(5a1) ) U † (ΠR ) ,

(A.22)

Alternative (a2): nc < nb + 2 < ns < nb + 3 Order Λ′′ /Λ and Order Λ(3) /Λ

These steps are the same as for the alternative (a1) above. A.4.2

Order Λ(4) /Λ

In this case, i.e. for ns < nb + 3, the next spurion to receive a VEV is hξs i ∼ ys Λ, which breaks (4a1)

GF

(5a2)

→ GF

= U(1)Q(1) × U(1)U (1) × U(1)D(1) L

16

R

R

(A.23)

leaving us with 4 singlet spurion fields (1)

YU ∼ (1)1,−1,0 ∝ λnu , (1)

YD ∼ (1)1,0,−1 ∝ λnd , χ12 ∼ (1)1,0,0 ∝ λns +1 , χ13 ∼ (1)1,0,0 ∝ λnb +3 .

A.4.3

(A.24)

Order Λ(5) /Λ

Taking now hχ13 i = 6 0, we break (5a2)

GF

(6a2)

→ GF

= U(1)U (1) × U(1)D(1) R

(A.25)

R

with the remaining spurion fields as for case (a1).

A.5

Alternative (b): nb + 2 < nc < ns < nc + 1

The case nc > nb + 2 may be considered as somewhat less likely, because in order to have yc /yb . λ2 at some high scale we would have to require sizeable renormalization effects in order to recover mc /mb ∼ 0.3 at low scales. A.5.1

Order Λ′′ /Λ

In that case, the next spurion to get a VEV would be 0 0 hχs i = Λ= Λ′′ y23 x23

(A.26)

with x23 = O(1) and thus Λ′′ /Λ = λ2 mb /mt . This leads to the breaking (3b)

G′′F → GF

= SU(2)UR × SU(2)DR × U(1)Q(1) × U(1)U (2) × U(1)D(2) . L

R

Introducing three new Goldstone bosons ((Π′′L )a=1,2,3 ), we parametrize † † ξu ξd (2) (2) ′′ ′′ , YD = U(ΠL ) YU = U(ΠL ) , χs = U(Π′′L ) hχs i . † ξc ξs† A.5.2

(A.27)

R

(A.28)

Order Λ(3) /Λ (3b)

The residual spurions of GF ξu† ξc†

∼ (2, 1)1,−1,0 ∝

∼ (2, 1)0,−1,0 ∝

scale/transform as λnu λ1+nc , ξd† ∼ (1, 2)1,0,−1 ∝ λnd λ1+ns , λ1+nu λnc , ξs† ∼ (1, 2)0,0,−1 ∝ λ1+nd λns .

(A.29)

In this case, the next spurion to receive a VEV is ξc† , which breaks (3b)

GF

(4b)

→ GF

= SU(2)DR × U(1)Q(1) × U(1)U (1) × U(1)D(2) , L

R

introducing three new Goldstone bosons at the scale Λ(3) ∼ yc Λ. 17

R

(A.30)

A.5.3

Order Λ(4) /Λ (1)

(4b)

Decomposing the doublet ξu† into two singlets YU and ϕ12 , the remaining spurions of GF are (1)

YU ∼ (1)1,−1,0 ∝ λnu , ξd† ∼ (2)1,0,−1 ∝

ξs† ∼ (2)0,0,−1 ∝

and

λnd λ1+ns

λ1+nd λns

,

ϕ12 ∼ (1)1,0,0 ∝ λ1+nc ,

.

(A.31)

Notice that the flavour group and the representations of the spurion fields are the same as (4a1,4a2) for GF , only that the role of χ13 is now played by ϕ12 . As in the case of scenario (a2), we assume that the next spurion to get a VEV is ξs , breaking the flavour symmetry at Λ(4b) ∼ ys Λ, (4b)

GF

(5b)

→ GF

= U(1)Q(1) × U(1)U (1) × U(1)D(1) . L

R

(A.32)

R

The remaining steps in the flavour symmetry breaking follow scenario (a2), except for Λ(5b) ∼ hϕ12 i ∼ λyc Λ.

A.6

Order Λ(6) /Λ and Order Λ(7) /Λ (6)

Since in all scenarios the residual spurion field χ12 is uncharged under GF , its VEV will in general be a complex number whose phase cannot be rotated away by flavour transformations. The CP symmetry in the Yukawa sector will thus be broken spontaneously by hχ12 i/Λ ∼ λys ∼ Λ(5) /Λ, if the potential singles out a non-vanishing imaginary part. (1) (1) Finally, the VEVs for YU and YD break the remaining flavour symmetry (6a1)

GF

→ nothing

(A.33)

and give masses to the up- and down-quark, where the order of symmetry breaking is not really important.

B

Various U (1) charges

For convenience, we collect in Table 3 the various U(1) charges appearing in the construction of the flavour symmetry breaking. Notice that some U(1) charges are linear dependent, (2)

(2)

3B = T + QL + UR + DR , = III +

(B.1)

(2) QL

= III + C

(2) (2) + UR + DR , (1) (1) (2) + QL + UR + DR

18

(B.2) .

(B.3)

Table 3: Various U(1) charges appearing in the discussion of the sequential flavour symmetry breaking.

(u, d)L (c, s)L (t, b)L uR cR tR dR sR bR

C

3B 1 1 1 1 1 1 1 1 1

UR 0 0 0 1 1 1 0 0 0

DR 0 0 0 0 0 0 1 1 1

T 0 0 1 0 0 1 0 0 0

III 0 0 1 0 0 1 0 0 1

(2)

QL 1 1 0 0 0 0 0 0 0

(2)

UR 0 0 0 1 1 0 0 0 0

(2)

DR 0 0 0 0 0 0 1 1 0

C 0 1 0 0 1 0 0 0 0

(1)

QL 1 0 0 0 0 0 0 0 0

(1)

UR 0 0 0 1 0 0 0 0 0

(1)

DR 0 0 0 0 0 0 1 0 0

Expressing the GF -invariants through G′F -invariants (m)

The explicit relations between the 10 invariants iα of the full flavour group GF and the (m) 9 invariants iα of the residual flavour group G′F read (2)

(2)

(2)

i1 = j1 + yt2 Λ2 ,

(2)

(2)

i2 = j2 + j3 ,

(C.1)

for the dimension-2 invariants, and (4)

(4)

(4)

(4)

(4)

(4)

(2)

i1 = j1 − 2 yt2Λ2 j1 , (2) (2)

(2)

i2 = j2 − j1 j3 − yt2 Λ2 j2 , (4)

i3 = j3 + 2j4 ,

(C.2)

for the dimension-4 terms, together with 3 (6) (4) i1 = − yt2 Λ2 j1 , 2 (6)

(4)

i2 = −yt2 Λ2 j2 − (6) i3 (6)

i4 and (8) i1

1 (2) (4) j j 2 3 1

1 2 2 (4) (2) (4) (2) (2) (2) (4) = − yt Λ j3 − j1 j4 + j2 J3 − j3 j2 , 2 3 (2) (4) (6) (2) (4) (2) (2) = 3j2 − 3j2 j4 + j2 j3 − j3 j3 , 2 (6) j1

1 (4) (4) (2) (2) 4 4 + yt Λ + j1 = − j4 + j2 J3 2 1 1 (2) (4) (4) (2) (2) (4) (4) (4) (2) (4) + j1 j3 − j2 j2 + j1 j2 + (j1 )2 j3 + (j2 )2 j1 . 2 2

(2) j1

2yt2Λ2

(C.3)

(6) j1

19

(C.4)

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