arXiv:hep-ph/9807399v4 12 Feb 1999

HUTP-98/A048
Soft Superweak CP Violation and the Strong CP Puzzle
Howard Georgi∗ and Sheldon L. Glashow† Lyman Laboratory of Physics Harvard University Cambridge, MA 02138 7/98 revised 2/99
Abstract We discuss a class of models in which CP is violated softly in a heavy sector adjoined to the standard model. Heavy-sector loops produce the observed CP violation in kaon physics, yielding a tiny and probably undetectable value for ǫ′. All other CPviolating parameters in the effective low-energy standard model, including the area of the unitarity triangle and θ, are finite, calculable and can be made very small. The leading contribution to θ comes from a four-loop graph. These models offer a natural realization of superweak CP violation and can resolve the strong CP puzzle. In one realization of this idea, CP is violated in the mass matrix of heavy majorana neutrinos.
∗georgi@physics.harvard.edu †glashow@physics.harvard.edu

1 Introduction
We examine the old idea of superweak CP violation [1] from the modern perspective of naturalness. Superweak look-alike models are easy to make: for example, by introducing a second SU(2)-doublet spinless field β, like the Higgs doublet but without a vacuum expectation value. When the couplings of the ordinary Higgs are real but those of β are complex, the exchange of a virtual β is the sole source of CP violation. If its couplings are so tiny that these interactions are negligible except in the neutral K mass matrix, we obtain an effectively superweak model.
However, models of this ilk are hideously unnatural. There is no symmetry to keep the Yukawa couplings of the Higgs boson real. Physically meaningful quantities like the area of the unitarity triangle, or equivalently the Jarlskog J parameter [2], receive infinite contributions from quantum loop effects. A free parameter of the theory — the complex phase in the Kobayashi-Maskawa (KM) matrix — has been set to zero without any justification. We shall describe a class of superweak models that is free of this affliction, and as well, of a strong CP problem [3]. We show how to calculate the area of the unitarity triangle as a finite radiative correction in these models.
In our models, CP is violated softly in a heavy sector adjoined to an otherwise standard and CP-conserving standard model.1 Soft CP violation is an old idea2 We believe that the class of models we describe in this paper is simpler than most models in the literature, and more effective in suppressing non-superweak effects. Our low-energy sector consists solely of the three fermion families, the SU(3) × SU(2) × U(1) gauge bosons, and one relic Higgs boson. Because CP violation is soft, the dimension-4 interactions in the model are naturally real. The only CP-violating phase appears in the mass matrix of the heavy sector. In particular, the KM matrix is real at tree level. CP-violating corrections to the dimension-4 couplings are generated by quantum loops involving the heavy sector, but they are finite and calculable functions of the renormalizable parameters of the model. Observed CP violation in the neutral kaon system arises from a dimension-6 interaction produced by a box diagram involving the heavy particles. Two other important parameters relevant to potentially observable CP-violating phenomena, the area of the unitarity triangle and the strong CP violating parameter θ, are both tiny for a wide choice of parameters: the former undetectable, the latter innocuous. We calculate the leading contributions to both below.
Ours are classic superweak models [1] where the only observable effect of the new interactions is its contribution to the CP-violating part of K0-K0 mass mixing. This simple scheme is disfavored at the one-sigma level by current data on B decay.3 However, by choosing the couplings of the heavy sector to the third family larger than those to the first two families we can implement the “3 scenario” of Barbieri, Hall, Stocci and Weiner (BHSW) [5]. In that case, the superweak interactions can affect the neutral B-meson sector so as to fit the data better. We do not discuss this possibility here.
1Our models can be adapted, with some additional architecture, to violate CP spontaneously rather than softly. Some might find this more elegant, but here we are striving for simplicity.
2For other work on soft CP violation, see [4], and references therein. This reference also describes models which are quite similar to those we construct here. We will discuss the differences below. But our focus is also somewhat different, in that we concentrate on constucting models that are naturally superweak.
3However, see the Note Added on page 11.
1

2 A Simple Model

We append two new and heavy particle species to the the standard model: two multiplets of one sort, ξα, where α = 1 or 2, and one of another, χ. We begin by specifying only two properties of these particles:

1. CP is violated softly in the ξ mass matrix, i.e., by dimension-2 operators if ξ is a boson, or by dimension-3 operators if it is a fermion.

2. There are renormalizable Yukawa couplings by which ξα and χ couple in pairs to the light left-handed quark doublets ψL as in figure 1, where:

ψL =

V † UL DL

.

(1)

V , the tree-level KM matrix, is real because of our assumption of soft CP violation. It

follows

that

ξ

is

a

spinless

boson

and

χ

a

spin

1 2

fermion,

or

vice

versa.

We

will

argue

later that we obtain the maximum suppression of θ if the ξα are colorless fermions, but

for now, we will leave this unspecified.

We denote the Feynman amplitude for the coupling as f oiα, where i = d, s, b is a flavor index. An overall constant f sets the scale of the Yukawa couplings, and we will assume the components of oiα to be of order unity.4 In this basis, the assumption of soft CP breaking requires oiα to be real.
...................................................................................................................χ......................... ψLi f oiα ξα

Figure 1: The coupling of the left-handed quark doublet to the heavy sector (i is a flavor index).
The oiα are real.

....................................................................................................................χ......................... ψL f oiα ηαy ≡ f Oiy Ξy
Figure 2: The coupling of the left-handed quarks to the heavy sector in the mass eigenstate basis
for the Ξ’s. The Oix are complex. 4In the 3 scenario of [5], the i = b components oiα would be larger that the others.

2

Soft CP violation in the ξ mass matrix leads to mass eigenstates Ξy that are related to ξα by a (complex) unitary transformation,

ξα = ηαy Ξy .

(2)

It is convenient to express the new interaction in terms of these mass eigenstates. This is illustrated in figure 2, in which we have defined

Oiy ≡ oiα ηαy .
α

(3)

If the ξs are scalars, the field redefinition (2) can generate phases in their self-couplings. These do not affect low-energy physics directly, and their virtual effects inside heavy sector loops are generally small compared to the effects due to the complex parameters Oiy. Thus, we analyze the CP violation arising exclusively from these parameters, which can be done without considering the details of the heavy sector physics.
Arrows placed on the heavy sector lines in figure 1 and figure 2 represent the flow of a quantum number, “heaviness”, associated with the heavy sector fields. The interactions of figure 1 and figure 2 conserve this quantum number. It could be that there are weaker interactions that violate heaviness, but we will not discuss these. If either Ξx or χ are neutral Majorana particles, heaviness would only be conserved modulo 2. In these special cases (which we discuss separately) there are additional contributions to CP-violating effects. The arrows in figure 1 are important for another reason. In our analysis of the CP violation in these models, we will sometimes treat the x index on Ξ as an additional flavor. The structure of figure 2 is such that this generalized flavor flows along the solid line with the CP-violating coupling f Oix regarded as a matrix in flavor space.

ψ....L..j.....................................χ.........................................................................................Ξ..............y....................................................................................χ..........................................ψ.........Lk ψLk Ξx ψLj

Figure 3: This box graph produces the superweak interaction.

The superweak interaction arises from the box graph shown in figure 3. The coefficient of the effective 4-fermion coupling produced by figure 3 at the scale Mχ is of order:

αf2 Ixy Mχ2

Ojx

Ok∗x

Ojy

Ok∗y

+

x

↔

y

(4)

where (j, k) = (d, s), (d, b), or (b, s) and where αf = f 2/4π. Ixy is an integral depending
on m2Ξx/Mχ2 and m2Ξy /Mχ2. If the dimensional parameters are the same order of magnitude, these mass ratios and Ixy are of order unity. The details of the 4-fermion operator produced

3

depend on the specific properties of ξα and χ but all lead to effective flavor-changing 4fermion interactions. The operator from figure 3 is renormalized by QCD effects at lower energies, but this effect is less than a factor of two in all circumstances we consider. It does not affect our order of magnitude estimates and will be ignored. Our models, as we shall demonstrate, are natural realizations of the superweak model. Thus the interaction corresponding to figure 3, with j = d and k = s (two incoming s quarks and two exiting d quarks), provides essentially all the observed CP violation in the neutral kaon system.

3 The area of the unitarity triangle

The CP-violating corrections to the renormalizable interactions of the low-energy standard model must be proven to be small if figure 3 is to be responsible for all observable CP violation in our model. In particular, radiative corrections will induce finite complex phases in the (initially real) KM matrix. Some of these phases can be removed by field redefinitions and do not represent real CP violation. But the area of the unitarity triangle,

A≡

1 2

Im

Vub Vu∗d Vc∗b Vcd

,

(5)

is an invariant measure of the KM CP violation. In the standard model, and in the standard parameterization used in the particle data booklet, it is

A=

1 2

sin δ13 s12 s13 s23 c12 c213 c23

.

(6)

..........................................................................χ.................................................................................. ψLk Ξx ψLj

Figure 4: This fermion self-energy graph introduces phases into the KM matrix.

The leading contribution to A comes from the self-energy diagram in figure 4, whose

CP-conserving part produces an irrelevant infinite renormalization of the real parameters

of the theory. Its CP-violating part is finite, calculable, and produces real physical effects.

This contribution has the form

ψLj iD iCjk ψLk

(7)

where C is a real, antisymmetric matrix in flavor space. Up to factors of order one (depending on the Ξ − χ mass ratios), it is

Cjk

≈

αf 4π

Im (Oj1 Ok∗1) ≈

αf 4π

(oj1ok2 − oj2ok1)

.

(8)

We must redefine the ψL field to restore the canonical form of its kinetic energy. Hereafter we

assume that αf (and hence C) is small. To lowest order in αf , the required field redefinition

is:

ψL → ψL′ = (I + iC/2) ψL .

(9)

4

The complex field redefinition, (9), also introduces phases into the quark mass matrices, and

thus into the KM matrix. The Yukawa couplings that give rise to the tree level quark masses

have the form

√√

2 v

DR

φ†

MD

ψL

+

2 v

UR

φ˜†

MU

V

ψL

+

h.c.

(10)

In terms of the redefined fields, these Yukawa couplings become

√√

2 v

DR

φ†

MD

(I

−

iC/2)

ψL′

+

2 v

UR

φ˜†

MU

V

(I

−

iC/2) ψL′

+

h.c.

(11)

and lead to the radiatively-corrected mass terms:

DR MD (I − iC/2) DL′ + UR MU V (I − iC/2) V † UL′ + h.c.

(12)

where MD and MU are diagonal matrices. To lowest order, this field redefinition does not affect the quark mass eigenvalues, but it does change the KM matrix.

To estimate A we must determine the correction to V , the tree-level KM matrix. As a

first step, we diagonalize the mass-squared matrices of the left-handed quarks, which from

(12) are:

(I − iC/2) MD2 (I − iC/2) and (I − iC′/2) MU2 (I − iC′/2) ,

(13)

where

C′ = V C V † .

(14)

These mass-squared matrices may be diagonalized by unitary transformations Again to order αf , the appropriate transformations are:

DL′′ = (I + iF ) DL′ and UL′′ = (I + iG) UL′

(15)

where F † = F and G† = G are real symmetric matrices. To lowest order in C they are:

Fjk

=

−

1 2

Cjk

m2j m2j

+ −

m2k m2k

and

Gjk

=

−

1 2

Cj′ k

m2j m2j

+ −

m2k m2k

,

(16)

for j = k and Fjk = Gjk = 0 otherwise. The KM matrix, corrected for the effect of figure 4,

is:

(1 − iG) V (1 + iF ) .

(17)

We proceed to calculate the area of the unitarity triangle. If A were zero, all phases in the KM matrix could be removed by field redefinitions and there would be no CP violation from W exchange. If it were very small compared to its standard-model value, (6), i.e., if

A ≪ |s12 s13 s23| ,

(18)

the remnant phase in the KM matrix would also be very small and CP violation from W exchange could be neglected. Explicit calculation to lowest order in C gives

A

=

1 2

−s12 s213 C12 + s213 s23 C23 +

m2d m2s

−

m2u m2c

s13 s23 C12

−

m2d m2b

−

m2u m2t

s12 s23 C13 +

m2s m2b

−

m2c m2t

s12 s13 C23 + · · ·

5

(19)

where · · · indicate terms that are suppressed by additional factors of small quark mass ratios. The terms in (19) are proportional to elements of the matrix C. Each coefficient is suppressed relative to the right-hand side of (18), either by small quark mass ratios or because s13 is the smallest of the KM angles. Thus the area of the unitarity model is tiny compared to its standard-model value unless αf ≥ 4π. We have succeeded in producing a model in which all CP violation from W exchange is naturally negligible.

4 Strong CP

The basic idea of soft CP violation as a solution to the strong CP problem is that CP

invariance in the absence of soft breaking requires that the QCD θ parameter be zero.

Then all CP violating corrections to θ due to the soft breaking are finite and calculable in

terms of the soft breaking parameters. One can check whether they are small enough avoid

phenomenological problems. In some realizations of our models, we will see that significant

strong CP violation is induced. Others will survive this hurdle.

We first consider contributions to θ in tree approximation. These can arise only if there

are colored fermions that have CP violating mass matrices in tree approximation. But if

the Ξx are colored fermions, it would be unnatural for the phase of the determinant of theie mass matrix should vanish. To avoid a strong CP puzzle at the tree level, we must assume

that the Ξx are not colored fermions.5 We hereafter suppose that the Ξx are either scalars or colorless, and consider loop diagrams that could contribute to θ.

The field redefinitions produced by CP violating self-energy diagrams like that in figure 4

do not induce a non-zero value of θ to any order in αf because they are necessarily hermitian in flavor space (they are associated with hermitian counter-terms in the Lagrangian).

Therefore the determinant of the transformed mass matrix is real. This result applies to the

field redefinitions produced by any self-energy diagram, however complex.

CP-violating loop corrections to the Yukawa couplings can generate non-zero θ. These

corrections begin at the two-loop level. However, we shall see that if we add one additional

restriction, our models require a rather complicated flavor structure to produce a phase in

the determinant of the quark mass matrix, so that the leading contributions to θ are very

small. We shall identify the required flavor structure, then find and estimate those Feynman

graphs giving the largest contribution.

We assume that the heavy-sector masses are large compared to the TeV scale of elec-

troweak symmetry breaking. Thus quark masses appearing in the denominators of Feynman

integrals can be ignored, but explicit quark mass dependence arises from the Yukawa cou-

plings (10) of the Higgs doublet φ which, in our models, are real at tree level. Suppose heavy

sector loops were to produce a complex CP-violating contribution to the Yukawa couplings

of the form:

√√

2 v

DR

φ†

∆MD

ψL

+

2 v

UR φ˜† ∆MU

V

ψL

+ h.c.

(20)

Its lowest-order contribution to the phase of the determinant of the quark masses is

∆θ ≈ Im tr ∆MD MD−1 + ∆MU MU−1 .
5This was emphasized to us by Darwin Chang. See [4].

(21)

6

Because some quark masses are small, the inverse quark mass matrices in (21) threaten to produce large phases. However, no such large effects occur in our model. The heavy-sector particles couple only to the left-handed quarks. The only couplings of the right-handed quarks are the Yukawa couplings of (10), which themselves are proportional to the quark mass matrices. Thus the inverse mass matrices in (21) are always canceled. Had we instead coupled the heavy sector to the right-handed quarks, this cancellation would not occur and there would be a potentially larger contributions to θ, in only two loops. We discuss this contribution in the Appendix.
Another potential two-loop contribution to θ imposes an important constraint on our class of models. If the Ξx are scalars, there are renormalizable couplings of the Ξx to the Higgs doublet, and we can draw the diagram of figure 5 [4]. This contribution is proportional to the unknown coupling constant for the coupling of two Ξs to φ†φ. We could assume that this coupling is small and ignore it, but it seems more elegant to eliminate the diagram entirely by assuming that the Ξx are fermions, which we do in the remainder of this note.
ψ..L.j.......................Ξ............x................χ....Ξ...........y...........................................φ.........ψ..........L.k.............φ...............................D......Rℓ

Figure 5: A contribution to the Yukawa coupling that contributes to θ if the Ξx are scalars.

To simplify our study of other consequences of (21), we make use of the essential notion mentioned earlier, of x as a generalized flavor index allowing us to trace flavor through each Feynman diagram. For example, the graph in figure 4 is proportional to the hermitian flavor matrix Ox Ox†.
We first show that diagrams involving heavy sector loops, but no Higgs loops, do not contribute to θ. These diagrams do not involve quark mass matrices because the inverse mass matrix in (21) cancels the quark mass matrix from the Yukawa coupling. Because ∆θ is a trace in flavor space, the quark flavor indices are always summed over. Thus the trace is a product of objects of the following form:

Kxy =

Oj∗x Ojy

j

(22)

where K11 and K22 are real and K12 = K2†1. Because x flavor is conserved, x indices
appear at each end of every x line: once as the first index of a Kxy factor and once as the second. Thus the trace involves equal numbers of K12 and K21 factors and is necessarily real.
Had we introduced three Ξs rather than two, figure 6 would give a non-zero contribution ∼ αf3αs/(4π)4 proportional to the possibly-complex product K12 K23 K31. The gluon loop in figure 6 is necessary to produce a one-particle-irreducible contribution to the Yukawa
coupling.

7

ψ..L.i.....................................................................Ξ....χ..........x..................................................................................................................ψ..................L.j.....................................................................................................................Ξ.........χ..............y.....................................................................................................................ψ.................L.k......................................................................................................................Ξ........χ...............z.....................................................................................................................ψ.................L..ℓ...........................φ........................................D.........................R.ℓ.............................................................................................D......Rℓ

Figure 6: A contribution to the Yukawa coupling that could produce a contribution to θ were there
more than two Ξs.
..........................................................................φ................................................................................ ψLk UR or DR ψLj
Figure 7: A Higgs loop that introduces dependence on the quark mass matrices.

To find a contribution to θ, we must consider diagrams involving two different non-trivial flavor structures. This is possible in the presence of Higgs loops that introduce quark massmatrix dependence. However, it is not enough to have one Higgs loop and one heavy sector loop. A Higgs loop like that shown in figure 7 produces a contribution with flavor structure:

Bjk ≡ V T MU2 V + MD2 , jk
while a Ξ loop like that shown in figure 4 has flavor structure:

(23)

Wjxk ≡ Ojx Ok∗x

(24)

for x = 1 or 2. Both are hermitian matrices so that the trace of their product is real. To have a complex trace, the diagram must involve the three independent hermitian matrices B, W 1 and W 2. This structure arises from graphs with two Ξ loops, one Higgs loop, and a gluon loop to make them one-particle irreducible, like that shown in figure 8.
The largest contribution to θ from such Feynman diagrams is of order

αs 4π

αf 4π

2

VT

MU2 V + MD2 16π2v2

≈

αs 4π

αf 4π

2

m2t 16π2v2

,

(25)

where the simple form is sufficient because the t quark is so much heavier than the others.

5 Numbers
The ratio αf /Mχ is determined by the requirement that the graph in figure 3 reproduces the observed CP violation observed in neutral kaons. We set equal the contribution from
8

ψ..L.i.....................................................................Ξ....χ..........x..................................................................................................................ψ..................L.j.....................................................................................................................Ξ.........χ..............y.....................................................................................................................ψ.................L.k..................................................................U.......................R............................o........φ.........r........................D...........................R.....................................................................ψ.................L..ℓ...........................φ........................................D.........................R.ℓ.............................................................................................D......Rℓ

Figure 8: A contribution to the Yukawa coupling that produces a non-zero θ.

the four-fermion operator (dLγµsL) (dLγµsL) with coefficient (4) and the product ǫ∆mK, where ǫ ≈ 2 × 10−3 and ∆mK ≈ 3.52 × 10−15 GeV. To evaluate the contribution from the four-fermion operator, we use the vacuum insertion approximation, not because we believe
it, but because it is simple and likely to yield the correct order of magnitude. We find:

ǫ ∆mK

≈

1 2mK

8 3

fk2m2K

αf2 Mχ2

(26)

or

αf Mχ

≈

2 × 10−8 GeV−1 .

(27)

To resolve the strong CP problem, it is necessary that (25), the dominant contribution to θ in our models, is less than the bound of θ < 3 × 10−10 following from searches for the
neutron electric dipole moment [3]. This yields the constraint:

αf < 0.044 .

(28)

Combining (27) and (28), we find:

Mχ < 2 × 106 GeV .

(29)

This upper bound on Mχ is comfortably above the electroweak breaking scale, so our assumption that the heavy sector is far removed from the low-energy sector is justified. Evidently, there is plenty of room for a natural superweak model that solves the strong CP problem.

6 Majorana Fermions
We consider the interesting special case where either χ or the Ξx are neutral Majorana fermions: heavy particles that arise naturally in grand unified theories and may play a role in producing neutrino masses. Both possibilities introduce new wrinkles because the fermion number associated with the Majorana particles is no longer conserved and new classes of diagrams must be considered.
If χ is the Majorana fermion, it does not change our estimates significantly because it does not change the way that flavor flows through the diagrams. The basic flavor backbone
9

of the quark and Ξ lines is unchanged. There are simply more ways to connect the χ lines because of their Majorana character. Our arguments about the flavor structure of CP violating contributions are not affected, and the bounds on αf and Mχ are unchanged.
If the Ξx are Majorana fermions, our estimates are affected. Generalized flavor is no longer conserved: there are no arrows on the Ξ lines. In particular, Higgs loops are no longer needed to obtain a non-zero θ. We find contributions proportional to K122, as defined in (22), from graphs like that shown in figure 9.
ψ..L.j..................................................χ..................Ξ.................x....................ψ....................L.k............................................................................................χ.......................................................................................................................................ψ...................L.ℓ.................................................................D.....................Rℓ...........................................................................D......Rℓ Ξy φ

Figure 9: A contribution to the Yukawa coupling that produces a non-zero θ with Majorana Ξs.

Graphs such as that in figure 9 produce contributions to θ of order

αs

αf

2
.

4π 4π

(30)

This strengthens the bound on αf by a factor of 4πv/mt ≈ 18. For the special case of Majorana ξs, the the bounds on αf and Mχ become:

αf < 0.0024 , Mχ < 1.2 × 105 GeV .

(31)

This model remains viable, although αf must be quite small to suppress θ. Nonetheless we find it quite attractive because the coupling of the light neutrinos to the Majorana Ξs
could produce the neutrino masses seemingly required to explain the recent evidence for oscillations of atmospheric neutrinos [6].6

7 Conclusions
Wolfenstein’s original “superweak model” [1] introduces CP violation in an ad hoc fashion. We have reproduced the essential observable consequences of this model with a minimal addition to the standard model at a large mass scale. Other potential solutions to the strong CP problem have been proposed, such as a massless up quark or the Peccei-Quinn axion and its less visible variants. These solutions are more elegant than ours, but are phenomenologically challenged. The observable consequence of our solution is the absence of large CP violating phases in the Kobayashi-Maskawa matrix. Indeed, if such a model is
6Equation (31) yields too strong a bound for a plausible see-saw origin of neutrino masses. However, the diagram in figure 5 involves two different Ξs. If their masses are hierarchical (with mχ comparable to the lightest Ξ mass), (30) is reduced by a Ξ mass ratio and the upper bound on mχ is correspondingly increased.

10

correct, direct CP violation in the B-sector will be difficult if not impossible to detect and ongoing searches for a non-vanishing ǫ′ parameter are bound to fail.
Note Added: In a recent preprint, S. Mele has argued that in a model like ours in which the constraint from ǫ is removed, a real KM matrix can still fit the data [7]. Checchia et. al [8] come to the opposite conclusion, but they assume that the new physics does not contribute significantly to the neutral B meson mass difference (see [5]).
Acknowledgments
We are very grateful to D. Bowser-Chao and D. Chang for many important comments. One of us (HG) thanks Sheldon Stone for a helpful comment and gratefully acknowledges the support of the Aspen Center for Physics, where some of this work was completed. One of us (SLG) thanks Paul Frampton for his challenge to devise a plausible model of CP violation in which the KM matrix is nearly real. We are also pleased to acknowledge conversations with S. Barr and R. Mohapatra. This work was supported in part by the National Science Foundation under grant PHY-9218167.
A Right Handed Models
Here we exhibit the three-loop diagram that contributes to θ if the CP violating heavy sector couples to the right-handed quarks rather than the left-handed quarks. If, for example, the heavy sector couples to right-handed D quarks, then the diagram in figure 10 gives a nonzero contribution to θ. However, the corresponding diagram in our “left-handed models” that
D...Ri........................................................................Ξ..............χ.......................x...........................................................................................................................D.......................Rj..................................................................ψ.......................L.k.....................................φ................................................U......................R...ℓ...................................................................ψ.....................Lm...............................................................................................................................................ψ......Lm φ
Figure 10: A contribution to the Yukawa coupling in models in which the CP violating heavy
sector couples to the right-handed D quarks.
we consider in this paper (figure 11), does not contribute to θ. The crucial difference is that in our models, the only potentially flavor-changing coupling of the right-handed quarks is the usual coupling to the charged Goldstone boson components of the Higgs doublet, or equivalently, to the longitudinal W s. As mentioned above, all diagrams with an external DR must have a factor of MD associated with the DR external line. Thus the inverse mass matrices in (21) cancel with this explicit fact of the mass matrix, and the contribution of
11

figure 11 to the determinant of the quark mass matrix is the trace of a produce of two hermitian matrices, and is therefore real. However, in figure 10, the factor of MD occurs in the middle of the diagram, rather than on the external line.
The three-loop contributions to θ in right-handed models are not huge. They may not be a phenomenological problem for some range of parameters. But we hope that this discussion helps the reader see why there is no contribution at all at the three loop level in our lefthanded models (unless the Ξx are Majorana particles, as discussed above).
ψ..L.i........................................................................Ξ..............χ.......................x............................................................................................................................ψ................L..j................................................................U.....................R...k....................................φ...............................................ψ....................L.ℓ.................................................................D........................Rm...........................................................................................................................................D.......Rm φ
Figure 11: The corresponding diagram in the “left-handed models” discussed in this paper does
not contribute to θ.
References
[1] L. Wolfenstein, Phys. Rev. Lett. 13 (1964) 562. For a recent review, see L. Wolfenstein, Comments Nucl.Part.Phys.21 (1994) 275. A different approach to the superweak interaction is discussed in B. Holdom, Phys. Rev. D57 (1998) 357, hep-ph/9705231.
[2] C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039 and Z. Phys. C29 (1985) 491.
[3] See: J.E. Kim, Phys. Rep. 150 (1987) 1 and N. Ramsey, Phys. Scripta T59 (1995) 323.
[4] D. Bowser-Chao, D. Chang and W.-Y. Keung, Phys. Rev. Lett. 81 (1998) 2028. See also the useful review in D. Bowser-Chao, D. Chang and W.-Y. Keung, hep-ph/9811258.
[5] R. Barbieri, et al., Phys. Lett. B425 (1998) 119, hep-ph/9712252.
[6] T. Kajita, to appear in the proceedings of the XVIIIth International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, June 1998; Super-Kamiokande collaboration, hep-ex/9807003.
[7] S. Mele, “Indirect Measurement of the Vertex and Angles of the Unitarity Triangle,” CERN-EP/98-133, hep-ph/9810333.
[8] P. Checchia et. al, “A Real CKM Matrix?,” DFPD-99-EP-4, Jan 1999. 7pp. hep-ph/9901418.
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