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Nonstandard order parameters and the origin of CP violation

The consideration of chirality-preserving 2-fermion order parameters may shed new light on the strong CP problem and the breakdown of flavor symmetries. We describe two situations, one having the standard KM picture for weak CP violation and another having new sources of weak CP violation.

flavor interactions on roughly 1000 TeV scales. The latter is a typical flavor scale needed to account for the light quark masses. The manner in which the phases are transmitted to the quark mass matrix and other observables can differ substantially from models with scalar fields. We are motivated by the strong CP problem, but we will also describe the resulting picture for weak CP violation. We will discuss how one choice of the order parameters can lead to the standard KM picture, and how another choice implies that new sources of CP violation are likely to exist.
We are exploring the possibility that various flavor symmetries and CP are both violated dynamically on the scale of strong flavor interactions. Above these scales we assume that there is nothing but a CP conserving, massless gauge theory. Naively a dynamical breakdown of CP by strong interactions would sprinkle CP violating phases everywhere in the low energy theory, including the effective operators responsible for the quark masses, and result in a θ much too large. We will suggest that the proliferation of CP violating phases can be constrained by the pattern of symmetry breaking.
In particular we suppose that CP violation originates in the phases of a pair of order parameters which are also responsible for breaking some U (1) flavor symmetry.
We will be considering mass matrices such that some of the elements respect a U (1) symmetry while other elements do not. We will describe how the former elements are protected from CP violating phases, while the latter pick up a phase from one of the CP violating order parameters. At the end we will discuss how the U (1) can be part of a gauged flavor symmetry.
Before describing the order parameters we first consider three possible sets of mass matrices, each having a vanishing argdetM. In each case we also give the charge of the flavor symmetry we are considering, where N i is the ith family number. Each # denotes in general a different real number.
The key property of these matrices is that they are form invariant: addition and multiplication of matrices of the form of M U will again yield matrices of the same form with the same phase, and the same for M D . This is necessary since sums of masstype diagrams involving multiple insertions of mass matrices (and/or flavor changing order parameters) are expected to contribute to the final mass matrix. Although the phases appear in a rather trivial way, for φ U − φ D = nπ we will have a CP violating phase appearing in the KM matrix.
We would now like to identify the flavor-violating order parameters which could introduce phases into the mass matrices in this way. First, the order parameters must respect electroweak symmetry since we are supposing that they arise at the ≈ 1000 TeV flavor scale. Second, to guarantee a unique phase in each mass matrix, it is clear that there must not be two or more order parameters in either the up-or down-sectors having independent phases. We might expect for example two order parameters with opposite U (1) flavor charge, but we must insist that these be hermitian conjugates of each other so that their phases are equal and opposite.
An order parameter with these properties is one which is not normally considered, namely the chirality-preserving part of the quark propagator. In particular we suppose that the origin of CP violation lies in the following amplitudes coupling right-handed quarks of different families. We assume that the pattern of symmetry breaking produces these order parameters for only one pair of families labeled by indices (f 1 , f 2 ), with f 1 = f 2 , which are such as to break the U (1) flavor symmetry.
The U (1) symmetry implies that φ U +φ D is freely adjustable, leaving only φ U −φ D as a physical phase. Note that our symmetry is vectorial; the corresponding purely righthanded transformation is not a symmetry of the rest of the theory. If there were no weak interactions, and no other physics caused transitions between the up and down sectors, then there would be two vector U (1) symmetries and then no physical phase would remain. This emphasizes the role of weak interactions in producing CP violation of any kind in this picture.
We expect other independent order parameters generated by the strong flavor interactions, which signal the breakdown of various other symmetries. For example there may be 4-fermion order parameters, since they can also respect electroweak symmetry, but which break other flavor and/or chiral symmetries. Such 4-fermion order parameters may play a crucial role in generating the quark masses (in the presence of some fermions with TeV scale masses) [6].
Our central assumption is that the order parameters in (4,5) are the only source In this way we are drawing a distinction between the dynamical breakdown of flavor and CP as reflected by the order parameters, and the subsequent appearance of flavor and CP breaking in other amplitudes of the theory.
U (1) breaking and CP violation in a general amplitude can then be deduced from the effects of all possible insertions of the order parameters. Ignoring weak corrections for the moment, the result is that the CP violating phase is correlated with the flavor charge of the amplitude. This is evident in the mass matrices (1-3). Consider also the chirality-preserving terms for the right-handed quark fields, which in the low energy theory are of the form The Hermitian matrices C U and C D , and the matrices in the nonrenormalizable terms, will have the same phase structure as the mass matrices M U and M D respectively.
That is, they have the same phases in the same elements.
In the low energy theory one can transform the renormalizable kinetic terms to conventional flavor-diagonal form by redefining the fields, The H U and H D matrices can be chosen to be Hermitian and they again have the same phase structure as the original M U and M D respectively. Under this transformation the mass matrices change, but they retain their original phase structure. Thus in the low energy theory we have quarks with normal kinetic terms, and mass matrices with a CP violating phase which preserves θ = 0. The order parameters in (4,5) are affecting the mass matrices both through direct insertions in mass generating diagrams and through the field redefinition we have just described.
This discussion has ignored weak interaction corrections, and in fact these generate the well known small contributions to θ [9]. In addition to a safely small finite contribution, the standard model generates "infinite" contributions occurring at high orders in the weak coupling. In our picture such effects simply represent the running of θ below the flavor scale; above the flavor scale θ is trivially zero if the fundamental theory is a CP conserving, massless gauge theory. It is the softness of the dynamically generated quantities at the flavor scale, as in (4,5), which turns the "infinite" θ of the standard model into a calculable and safely small quantity.
Any new physics which, like the weak interactions, couples together the up and down sectors would have to be carefully considered as a source of additional contributions to θ. One example is SU (2) R interactions. Another would be nonperturbative operators of the form Q Lj Q Rk Q Lk Q Rj where j and k denote different families. These operators could connect an off-diagonal element of the up mass matrix to one in the down mass matrix, and thus could produce contributions to both mass matrices of a form different than we have been considering. We assume that such effects are either nonexistent or sufficiently small.
The flavor physics will generate nonrenormalizable terms suppressed by the large flavor scale, such as those in (6,7), but these do not lead to corrections which affect the phase structure of the mass matrices. They may however produce contributions to weak CP observables, in addition to the phase in the KM matrix; we will return to such effects later.
We will now consider the matrices presented in (1-3) in more detail. Case 1 turns out to be of the most interest if our goal is to obtain a standard KM picture for weak CP violation. This may also be possible in Case 2, but we will explain why it is more likely to have additional sources of CP violation in this case. In Case 3, for any choice of (f 1 , f 2 ), we are not able to obtain realistic masses and mixings, and we will not consider this case further.
For realistic mass matrices in Case 1 we are forced to choose (f 1 , f 2 ) = (2, 3), so that transitions occur between U 2R and U 3R and between D 2R and D 3R in the original flavor basis. For the resulting mass terms (Q Lj M U jk U Rk + Q Lj M D jk D Rk ) + h.c. we specialize somewhat from the matrices in (1) and consider the following, where all parameters are real.
We have used a common proportionality constant ρ U in M U , and ρ D in M D , although this need not be strictly true. This choice reflects how these elements can arise from the neighboring mass elements in the same row, via the right-handed transitions. Zeros have been placed in the (3, 1) elements since they are likely to be small and unimportant to the results; they would arise from diagrams involving multiple insertions of the mass matrices.
By studying the transformation from the present weak eigenstate basis to the mass eigenstate basis we can deduce properties of the KM matrix V . We find for example that, up to small corrections, where In addition we find that the bulk of the contribution to |V cb | must come from X U (so as to obtain realistic values for |V ub |) which then implies that ρ U ≈ .9. Thus the effects of the right-handed transitions must be large. For two of the angles of the unitarity triangle we find The possibility that φ U − φ D = π/2 (corresponding to a relative factor of i between the order parameters in (4,5)) is certainly consistent with a realistic α.
We illustrate these results with the following values. Note that V ud and V us are always real, no matter what the value of φ U and φ D . The next case we consider will be different.
For Case 2 we are forced to choose (f 1 , f 2 ) = (1, 2), which leads to the following matrices.
For Cabibbo mixing we find where The Y D term dominates and thus ρ D is determined. For the angle β appearing in the unitarity triangle we find If ρ U is much smaller than .95 the unitarity triangle is very thin, even if φ U −φ D ≈ π/2, and it is then not possible for the KM matrix to account for the observed CP violation.
Independent of this we are able to obtain the other three real mixing parameters of the KM matrix.
We illustrate the case of a fat unitarity triangle with the following values.
1 : R b : R t = 1 : .30 : 1.03 (29) Note that in contrast to Case 1 (with the same phase convention as used there) the phases in the KM matrix now appear mostly in the 2 × 2 block involving the lighter quarks.
.974 − .0140i .0398 + .000044i .00910 + .000018i −.0388 + .000004i .999 In the standard model it is conventional to absorb phases into the quark fields so as to move phases in V into more standard positions, and in particular remove phases from the V ud and V us elements. But in our case the new flavor physics generates additional nonrenormalizable operators which are not invariant under these phase redefinitions. In particular let us consider the flavor changing chromomagnetic moment operators in the down sector, which have been considered as a new physics contribution to ε ′ /ε [8], The origin of these operators should be closely associated with the origin of the down- We may estimate the Λ required for this effect to account for the observed ε ′ /ε [1]. Assuming that Im(C D ds ) ≈ √ m d m s β/Λ 2 we deduce from the analysis in [8] that Λ/ √ β ≈ 5 TeV. Since this is above the electroweak breaking scale, this new contribution to ε ′ can easily be large enough, even for values of β which make the KM contribution tiny.
To account for ε itself, we need a superweak, ∆S = 2, 4-fermion operator. Such operators emerge naturally from the exchange of new flavor gauge bosons, which can lead for example to a 1 M 2 (D 1R γ µ D 1R −D 2R γ µ D 2R ) 2 operator in the original flavor basis (M characterizes the flavor scale). In the mass eigenstate basis this will produce a ∆S = 2 operator, which then picks up a phase due to the phase redefinitions we have just discussed. Accounting for large mixing between D 1R and D 2R , due to a large ρ D , we obtain roughly M/ √ β ≈ 10 4 TeV. Since this is above the expected scale of new flavor physics, the superweak contribution to ε can easily be large enough, even for values of β which make the KM contribution tiny.  The four-fermion interactions respecting the U (1) would generate mass matrices of the following form (with the ordering (Q 1 , Q 2 , Q 3 , Q 4 )), for Case 1 and Case 2 respectively, To this we must add the effects of the right-handed transitions, which involve the (f 1 , f 2 ) = (2, 3) flavors in Case 1 and the (f 1 , f 2 ) = (1, 2) flavors in Case 2. In both cases it is natural for the mixing between the fourth family and the lighter families to be small, in which case our previous discussion will continue to hold.
Finally, suppose that the fourth family mass also breaks the U (1) flavor symmetry.
This would occur in the case of complex conjugate SU (2) representations if the fourth family quark mass term was Q 4L Q 3R + h.c. (Then of course the Q 3 and Q 4 families in the original flavor basis are no longer close to being the mass eigenstates.) By reordering the right-handed fields, (Q 1R , Q 2R , Q 3R , Q 4R ) → (Q 2R , Q 1R , Q 4R , Q 3R ), it then turns out that the mass matrix takes the same form as the second matrix in (32). We only mention this case here to make contact with the model in [6]. 1 In conclusion we have related CP violation to a phase mismatch in certain flavor changing order parameters involving right-handed quarks. The absence of strong CP violation is related to the very particular way these order parameters feed phases into the mass matrices. We described two cases, one with a standard KM picture of weak CP violation, and the other where additional sources of CP violation in the K system are likely. In the latter case there may be smaller than expected CP violation in the B system. In both cases we related the angles of the unitarity triangle to the phases in the flavor changing order parameters. We cannot claim a final resolution of the strong CP problem without a complete and unambiguous theory of quark masses, and thus we await the results from the B factories for further guidance.