Flavor Changing Neutral Currents in the Lee-Wick Standard Model

Recently an extension of the standard model (the Lee-Wick standard model) based on ideas of Lee and Wick (LW) was introduced. It does not contain quadratic divergences in the Higgs mass and hence solves the hierarchy puzzle. The LW-standard model contains new heavy LW-resonances at the TeV scale that decay to ordinary particles. In this paper we examine in more detail the flavor structure of the theory. We integrate out the heavy LW-fermions at tree level and find that this induces flavor changing Z boson couplings. However, these flavor changing neutral currents are acceptably small since they are automatically suppressed by small Yukawa couplings. This is the case even though the theory does not satisfy the principle of minimal flavor violation. New couplings of the charged W bosons to quarks and leptons are also induced. We also integrate out the LW-Higgs and examine the four-fermion operators induced.


I. INTRODUCTION
The minimal standard model (with dimension five operators to give neutrino's mass) is consistent with almost all experimental observations and possesses many attractive features. For example, it automatically has no renormalizable couplings that violate baryon number and it has a GIM mechanism that suppresses flavor changing neutral currents. Despite the many successes of the standard model, most elementary particle theorists believe, new physics, beyond the minimal standard model, will emerge at the TeV scale. This is because of the hierarchy problem. In the minimal standard model quadratic divergences in the Higgs mass make it difficult to understand how it can be kept so small compared with the Planck scale.
Extensions of the standard model that solve the hierarchy puzzle usually contain new degrees of freedom at the TeV scale that can give rise to unacceptably large flavor changing neutral currents. This happens for example in dynamical symmetry breaking and some versions of low energy supersymmetry. One way to be sure that new degrees at the TeV scale are not in conflict with experimental constraints on flavor changing neutral currents is to impose the principle of Minimal Flavor Violation (MFV) [1]. In MFV only the standard model Yukawa couplings break the Recently ideas proposed by Lee and Wick [2,3] were used to extend the standard model so that it does not contain quadratic divergences in the Higgs mass [4]. Higher derivative kinetic terms for each of the standard model fields are added that improve the convergence of Feynman diagrams and so there are no quadratically divergent radiative corrections to the Higgs mass. The higher derivative terms give rise to propagators with new poles that are massive resonances. These resonances have wrong-sign kinetic terms which naively cause unacceptable instabilities. Lee and Wick propose altering the energy integrations in the definition of Feynman ampli-tudes so that the exponential growth does not occur. It appears that this can be done order by order in perturbation theory in a way that preserves unitarity. However, there is acausal behavior caused by the unusual location of poles in the propagators.
Physically this acausality is associated with the future boundary condition needed to forbid the exponentially growing modes. As long as the masses and widths of the LW-resonances are large enough, this acausality does not manifest itself on macroscopic scales and is not in conflict with experiment. The possibility of distinguishing LW-resonances from ordinary causal-resonances at the LHC was briefly discussed in Ref. [5]. The Lee-Wick photon was searched for experimentally in Ref. [6].
The model of Ref. [4] was extended to include right handed neutrinos with masses much above the weak scale. It was shown that the heavy right handed neutrinos do not give rise to large radiative corrections for the Higgs mass [7].
In [4] minimal flavor violation was imposed to simplify the flavor structure of the theory. In this paper we drop this assumption by allowing for the most general couplings. Integrating out the heavy LW-fermions at tree level we find that there are new flavor changing neutral currents and charged currents, involving both the quark and lepton fields. We calculate the flavor changing Z and W ± couplings and argue that they are acceptably small. The Lee-Wick standard model introduced in Ref. [4] contains higher derivative terms (e.g. the Higgs boson's kinetic term contains two-derivative and four-derivative terms). However there is no principle that forbids the appearance of even higher derivative terms. In fact a Lee-Wick version of the Higgs sector with six-derivative terms was introduced in Ref. [8]. If even higher derivative terms are present then there are additional LW-resonances. However if the additional LW-resonances occur at substantially higher masses the model presented in Ref. [4] may, for LHC physics, be a reasonable approximation to a theory with even more complicated kinetic terms.
It has been shown that gravity coupled to massless fermions leads to a Lee-Wick theory for gravity [9]. Recently it was argued that gravitational radiative corrections induce higher derivative terms of the LW-type, leading to a Lee-Wick vector field in the auxiliary field formulation of the theory [10].

II. THE FLAVOR SECTOR
The kinetic terms for the quarks and leptons in the LW-standard model are [4]: Here we have explicity displayed the generation indices i, j. The 3 × 3 matrices r Q , r L , r U , r D and, r E are hermitian. In the higher derivative formulation, the fermion Yukawas are where repeated generation indices in the above equations are summed over 1, 2, 3.
For simplicity we have neglected the neutrino masses.
The higher derivative terms can be eliminated by adding massive auxiliary LWfermion fields:Q L ,Q R ,L L ,L R ,ũ R ,ũ L ,d R ,d L ,ẽ R andẽ L . We adopt the notation that the fields that create and destroy the LW resonances are denoted with a "tilde".
Then the "kinetic" Lagrange density takes the form (suppressing generation indices for simplicity), where It is convenient to shift the hatted gauge and fermion fields so that their derivative kinetic terms are diagonalized. Using the notationÂ µ = g 3Â Then the fermion kinetic and LW-fermion mass terms become, With these redefinitions and the changeĤ = H −H the Yukawa interactions become, where the generation indices have been suppressed. The LW-field mass matrices are diagonalized by the unitary transformations, These transformations do not, via the kinetic terms, introduce any gauge boson couplings to the heavy LW-fermions that change generation. The coupling matrices from the higher derivative theory can be written in terms of these transformations, for example, The Yukawas take the form, This completes this formulation of the version of the theory with auxiliary fields and without higher derivatives.

III. NEW W AND Z COUPLINGS
When the higher derivative terms are treated as a perturbation one can apply the equations of motion for the quark and the lepton fields to get, As we shall see, this yields new flavor changing Z boson couplings and new charged W ± boson couplings.
It is instructive to also derive the above result from the version of the theory with auxiliary fields by integrating out the LW-fermions at tree-level. This gives rise to an effective theory where the LW-fermions are absent. In figure (1, 2) In the above expression, when there is possible ambiguity about how the SU (2) singlets are formed, we have used square brackets to denote the contraction of SU (2) indices. The covariant derivative is understood to act on all fields to the right of it. When the Higgs doublet gets its vacuum expectation value the standard model fermions get their masses in the usual way, These mass matrices are diagonalized by the following transformations, and the following new W boson couplings The effects can be larger in cases where there is a standard model contribution to interfere with. For example, the change in the rate for b → sνν is of order, The new couplings of the charged W -bosons to the right-handed quark fields are similarly suppressed.
The new couplings of the W ± bosons to the left-handed quark and lepton fields are not necessarily suppressed by the ratio of the tree-level standard model fermion masses participating in the interaction to the TeV scale masses of LW-resonances.
However, these couplings are still small. For example, inclusion of these currents results in a change of the effective, or observed, matrix element V cb of, Similarly there are small corrections to the other effective CKM matrix elements.
Note that the effective CKM matrix is not unitary.

LW-HIGGS
For completeness in deriving the low energy effective field theory and to give the operators that contribute at tree-level to interesting processes, we now integrate out the LW-partner of the Higgs boson at tree-level. Note that in this section we are using In the equation above we use square brackets to denote Lorentz scalars and to make clear how the generation indices are being summed. There is no ambiguity about how SU (2) singlets are formed. The effective Lagrangian in Eq. (24) gives very small contributions to standard model processes (e.g. B −B-mixing, K → πe + e − , etc.).

V. CONCLUSION
In this paper we have considered the Lee-Wick (LW) extension of the minimal Standard Model proposed by [4]. Rather than imposing the principle of Minimal Flavor Violation (MFV), forcing all LW-fermions in the same representation of the gauge group to have the same mass, we explored the most general flavor structure of the theory by imposing no constraints (beyond hermiticity) on the higher derivative couplings. We present the resulting auxiliary field formulation of the theory, involving heavy LW auxiliary fields and showed explicitly the relation to the higher derivative theory. From the auxiliary field formulation involving LW-fields, we derive a low energy effective field theory by integrating out the heavy auxiliary fields at treelevel. We show that new flavor changing neutral currents (FCNCs) and charged currents (FCCCs) are present in the theory.
One would naively expect that these FCCCs and FCNCs would lead to unacceptably large mixings (at odds with experimental results), but we show that these mixings are appropriately suppressed if the LW auxiliary fields are at the TeV scale.
This suppression is a general feature of the theory even though the theory does not satisfy the principle of MFV. We show further that when the principle of MFV is applied that the FCNCs vanish and the FCCCs simply reduce to small additional mixings of the usual Standard Model CKM type, involving both left and right-handed quark fields (recovering the result stated in [4] where MFV was applied).
For completeness, we also found the four-fermion operators resulting from integrating out the LW-Higgs. They are all suppressed by the LW-Higgs mass, and give very small contributions to standard model processes.