Minimal Lee-Wick Extension of the Standard Model

We consider a minimal Lee-Wick (LW) extension to the Standard Model in which the fields providing the most important contributions to the cancellation of quadratic divergences are the lightest. Partners to the SU(2) gauge bosons, Higgs, top quark, and left-handed bottom quark are retained in the low-energy effective theory, which is valid up to approximately 10 TeV; the remaining LW partners appear above this cutoff and complete the theory in the ultraviolet. We determine the constraints on the low-energy spectrum from the electroweak parameters S and T, and find LW states within the kinematic reach of the LHC at the 95% confidence level.


I. INTRODUCTION
The intriguing idea of Lee and Wick (LW) [1] to promote Pauli-Villars regulators to the status of physical fields was recently applied to develop a LW extension to the Standard Model (LWSM) [2]. While the original LW proposal was designed to render QED finite, the purpose of the LWSM is to use the LW opposite-sign propagators in loop diagrams to solve the hierarchy problem. This solution is analogous to the supersymmetric one in that it relies on cancellation between pairs of loops to remove quadratic divergences, but differs in that the LW particles carry the same statistics (and other quantum numbers) as their SM partners. Several recent papers investigate the formal properties and phenomenology of the LWSM [3,4,5,6,7].
In the present work, we consider a version of the LWSM in which only a subset of the full spectrum of LW partners lie within the reach of the LHC. Motivated by Little Higgs models [8], we study the possibility that only the LW partners of the SU(2) gauge bosons, Higgs, t quark, and left-handed b quark appear in the low-energy effective theory. These fields provide the most significant contributions to the cancellation of quadratic divergences in the Higgs sector and render the effective theory natural, provided the cutoff is < ∼ 10 TeV.
The remaining LW spectrum may appear above this cutoff, or the theory may be completed by other, more exotic physics. This minimal LW low-energy theory is distinguished by its simplicity, making it an ideal subject for comprehensive phenomenological investigation.
In this Letter we present the constraints on this model's spectrum that follow from oblique electroweak parameters, in particular the Peskin-Takeuchi S and T parameters [9]: where Π are the usual self-energy functions, s ≡ sin θ W , c ≡ cos θ W parametrize the weak mixing angle and m Z 0 is the measured Z boson mass. Our approach is similar to that of other recent work, in particular Ref. [3] (ADSS), but differs in that we do not assume a complete LWSM spectrum with large sets of mass-degenerate particles. Exact one-loop formulae for S and T , which have not appeared in previous literature, are necessary for a proper treatment of corrections in our model. We agree with the original LWSM work [2] and ADSS [3] that the leading oblique corrections occur at tree level in the LWSM, contrary to the claim in Ref. [5]. Part of this discrepancy is due to differing definitions in the literature of what physics is "oblique" (see the discussion in Sec. II). Moreover, the identification of oblique parameters in the effective Lagrangian of Ref. [5] appears to be fundamentally different from ours and Refs [2,3] so that the numerical results are not easily compared; we do not consider this issue further here.
Our paper is organized as follows: In Sec. II we establish conventions for specifying the spectrum, which take into account potentially substantial mixing between SM and LW particles. We present the one-loop formulae for the S and T in Sec. III. We present our numerical results in Sec. IV, and Sec. V summarizes our conclusions.

II. PRELIMINARIES
We study the S and T parameters in an effective theory obtained by integrating out heavy-mass eigenstates. At tree level, this procedure is equivalent to eliminating the heavy fields from the Lagrangian using their classical equations of motion. Since the gauge sector of our model includes LW partners to only the SU(2) gauge bosons, one finds ∆T tree = 0 , in agreement with the results of ADSS in the limit M 1 → ∞ [The U(1) LW gauge boson contributes to ∆T tree at O(v 2 /M 2 1 )]. Here, M 1 and M 2 represent the unmixed LW U(1) and SU(2) gauge boson masses in the auxiliary field formulation of the LWSM, as defined in Ref. [2], and v is the Higgs vacuum expectation value. The parameter U turns out to be 2 ) and is similarly suppressed in loop effects, so we do not consider it further. Note that the constraints on new physics from oblique parameters are meaningful only if vertex corrections are small. The derivation of Eqs. (3)-(4) includes field redefinitions that force the couplings of the gauge fields to SM currents to match those of the SM at tree level. Thus, the definitions of S and T used here (and in ADSS) subsume the largest vertex corrections.
The one-loop contributions to the self-energies Π AB in Eqs. (1)-(2) arise from the diagrams in Fig. 1. We evaluate these diagrams using mass eigenstates on the internal lines; since SM and LW fields mix, one must first define conventions to specify the spectrum. The following mixing effects are taken into account: The mass matrix in Eq. (5) where subscript 0 here and below indicates mass eigenstates. The mixing angle θ satisfies with mass eigenvalues In addition, the LW sector has pseudoscalarP and charged scalarh + states with masses mh. We work in unitary gauge, where all unphysical scalars are eliminated from the theory.
2. Gauge mixing. The SM SU(2) gauge boson and its LW partner (W , W ) mix via [2] where m W = 1 2 g 2 v is the unmixed SM W mass. The mass matrix is diagonalized by the symplectic transformation  where, using W 1 0 to indicate the charged heavy mass eigenstate, with eigenvalues satisfying In the neutral sector, mixing only occurs between the SM Z boson and the LW W 3 [2]: where m Z = m W /c is the unmixed SM Z mass. The photon decouples as a consequence of electromagnetic gauge invariance. Equation (13) is diagonalized via the symplectic trans- where and the eigenvalues are given by 2. Fermion mixing. Our model includes LW partners to the fields t L , t R , and b L . The mass terms of the third-generation fermions read [3,7] where define our basis for the third-generation fields. The fields t L , t R , b L , and b R are the SM fields with their usual quantum numbers, while the tilded fields are LW. The unprimed LW fields are the partners of the SM fields and hence have the same quantum numbers and chirality. The primed LW fields have the same quantum numbers as the unprimed LW fields of the opposite chirality, in order to permit SU (2)×U (1)-invariant LW mass terms. Thus, for example,t L andt ′ R both transform as a (2, + 1 6 ) under SU (2)×U (1), the same as t L . The matrix η = diag(1, −1, −1) conveniently encodes the opposite signs between SM and LW kinetic terms or mass terms. Then one finds We diagonalize these mass matrices via transformation matrices S a L and S a R , for a = t or b, such that M 0 is diagonal with positive eigenvalues: Additional details regarding the solution to Eqs. (21) will appear elsewhere [10]; for the purposes of this calculation, we simply note that solutions were obtained numerically.

III. LOOPS
A consistent calculation of oblique parameters in a perturbative theory must yield results that are ultraviolet finite, since these parameters describe physical observables. Here we consider the deviation of S and T from their SM values, so one must subtract any purely SM contributions. While individual diagrams can diverge, we find that the final subtracted results are finite and cutoff independent.
First consider the S parameter, which receives contributions from the diagrams of Fig. 1a, b, and d; the diagram in Fig. 1c is not relevant since its contributions to Π 3B is q 2 independent. From the purely Higgs-sector diagram in Fig. 1a we find where Note that the contribution to the self-energy from Fig. 1a vanishes if the LW states are decoupled, so the result must be finite without any SM subtraction, as is indeed the case.
The contribution from Fig. 1b, however, involves a diagram with purely SM particles (the Higgs and Z bosons), with non-SM couplings. In this case, one must subtract the same diagrams evaluated with infinite LW masses. One finds where C 3B is the coefficient of a Fig.1b diagram with internal scalar (S) and vector (V) particles of mass m S and m V , respectively, ξ 1 ≡ m 2 S /M 2 2 , ξ 2 ≡ m 2 V /M 2 2 , and Table I gives the values of C 3B , ξ 1 , and ξ 2 for each term summed in Eq. (24), as well as for the SM subtraction.
To compute the fermionic contribution to S, we first parametrize the gauge-fermion couplings evaluated in the mass eigenstate basis: where P L (P R ) are the left (right)-handed chiral projection operators, and Ψ 0 represents T 0 and B 0 , the transformation of Eq. (18) The last term of Eq. (27) represents the SM subtraction, with m t, SM and m b, SM the t and b masses, respectively, obtained in the decoupling limit of the LW states. The cancellation of logarithmic divergences between various contribution to ∆S 1d requires  (29).
where the matrices E L,R are also evaluated in the mass eigenstate basis. One finds The removal of divergences from ∆T 1d [leading to the finiteness of Eq. (31)] requires delicate cancellations between the t, b, and tb diagrams not only for the LL + RR coefficients of the quadratic divergences, but also between the LL+RR and LR+RL coefficients of the logarithmic divergences. Indeed, these cancellations may be verified [10].

IV. RESULTS
To obtain the constraints on our model, we choose as input parameters M 2 , mh, and a common fermion mass parameter M F = M q = M t . With the lightest gauge boson mass eigenvalues m W 0 , m Z 0 fixed by the measured masses, specifying M 2 fixes the gauge boson spectrum of the model. We choose the Higgs mass parameter m h that appears in the SM Lagrangian to be 115 GeV, which provides our reference mass in defining S and T ; specifying mh then completely fixes the Higgs spectrum of the theory. Finally, we set the lightest fermion mass eigenvalues m t 0 , m b 0 to the physical quark masses, and decouple the LW partnerb R , which is not part of the minimal low-energy theory, by taking M b → ∞; specifying M F then completely fixes the fermionic spectrum of the theory. Note that the choice M F = M q = M t is merely a convenience, although naturalness suggests M q and M t are comparable; the general case provides substantial additional freedom in accommodating current experimental bounds [10]. Finally, note that each set of input parameters (M 2 , mh, M F ) corresponds to a (slightly) different value for the Lagrangian mass parameter m t . This mass corresponds to the SM reference value in the limit of decoupled LW partners, about which deviations in S and T are measured. We shift [9] the model predictions for the oblique parameters for each input parameter set to coincide with the t reference mass 170.9 GeV assumed in the computation of the experimentally allowed region of the S-T plane.  With LW gauge and fermion states typically heavier than this, other low-energy constraints on the model, such as those from flavor-changing processes [6,7], are likely only to be relevant in the lighter LW Higgs sector. A more complete investigation of flavor and 3.5 electroweak constraints on the effective theory of interest will appear in Ref. [10].

V. CONCLUSIONS
We have considered the constraints from oblique electroweak parameters in an extension to the SM that includes LW partners to the SU(2) gauge bosons, the Higgs doublet, and the t L,R and b L quarks. This low-energy theory has the smallest particle content required to cancel the largest contributions to the Higgs quadratic mass divergences, rendering the effective theory natural up to ∼ 10 TeV (similar to Little Higgs models). Above this scale one may uncover the remaining particle content of the LWSM, or perhaps an even more exotic ultraviolet completion. This effective theory is meritorious because its spectrum is simple and allows a more focused and complete study of phenomenological constraints and collider signatures; the electroweak analysis presented here is a necessary first step. Our conclusion that the LW partners in this effective theory can be kinematically accessible at the LHC (as specified in Figs. 2-3) suggests a broad range of interesting phenomenological issues for further study.