Electroweak precision constraints on the Lee-Wick Standard Model

We perform an analysis of the electroweak precision observables in the Lee-Wick Standard Model. The most stringent restrictions come from the S and T parameters that receive important tree level and one loop contributions. In general the model predicts a large positive S and a negative T. To reproduce the electroweak data, if all the Lee-Wick masses are of the same order, the Lee-Wick scale is of order 5 TeV. We show that it is possible to find some regions in the parameter space with a fermionic state as light as 2.4-3.5 TeV, at the price of rising all the other masses to be larger than 5-8 TeV. To obtain a light Higgs with such heavy resonances a fine-tuning of order a few per cent, at least, is needed. We also propose a simple extension of the model including a fourth generation of Standard Model fermions with their Lee-Wick partners. We show that in this case it is possible to pass the electroweak constraints with Lee-Wick fermionic masses of order 0.4-1.5 TeV and Lee-Wick gauge masses of order 3 TeV.


I. INTRODUCTION
The Standard Model (SM) describes the electroweak (EW) interactions with an incredible precision. However, the instability of the Higgs potential under radiative corrections signals our ignorance over the real mechanism of electroweak symmetry breaking (EWSB) and has lead to many extensions beyond the SM. The upcoming LHC era is likely to provide us the tools to check some of the proposed solutions to this problem.
Recently, Grinstein, O'Connell and Wise proposed a new extension of the SM [1], based on the ideas of Lee and Wick [2,3] for a finite theory of quantum electrodynamics. The building block of the Lee-Wick proposal is to consider that the Pauli-Villars regulator describes a physical degree of freedom. In the Lee-Wick Standard Model (LWSM), this idea is extended to all the SM in such a way that the theory is free from quadratic divergences and the hierarchy problem is solved. Every SM field has a LW-partner with an associated LW-mass, these masses are the only new parameters in the minimal LWSM. A potential problem in this model is that the LW-states violate causality at the microscopic level due to the opposite sign of their propagators. However the authors of Ref. [1] argued that there is no causality violation on a macroscopic scale provided that the LW-particles are heavy and decay to SMparticles. The LWSM can be thought as an effective theory coming from a higher derivative theory. However, to insure perturbative unitarity, higher dimension operators cannot be of any type, only those compatible with a LW effective Lagrangian are acceptable [4]. In ref. [1] it was shown with a specific example that unitarity is preserved due to the unusual sign of the LW-particles width. Further considerations on the unitarity of the theory have been presented extensively in the previous literature [5,6,7,8,9], the non-perturbative formulation has been discussed in [10,11,12,13,14] and the one-loop renormalization of LW-gauge theories has been discussed in [15].
Recent work discussed the suppression of flavor changing neutral currents [16], gravitational LW particles [17] and the possibility of coupling the effective theory to heavy particles [18]. On the phenomenological side, the implications for LHC [19,20] and ILC [21] have also been discussed.
The LWSM does not provide any information on the origin of the LW-masses. However, in order to solve the hierarchy problem these masses should not be heavier than a few TeV.
On the other hand, the LW particles can not be too light without getting in conflict with EW precision observables [22]. Therefore the aim of our work is to carry out an analysis of the electroweak precision tests (EWPT) and derive bounds on the masses of the LWSM.
Since the main motivation to introduce the LWSM was to solve the hierarchy problem, large LW-masses will be a source of fine-tuning and will partially spoil the original motivation. In this way the success of the LWSM is associated to its efficiency to pass the EWPT without introducing a severe fine-tuning in the theory.
On the experimental side, determining the parameter space allowed by the EWPT is crucial to know whether the LWSM could be tested or not in the next experiments, in particular at the LHC.
With these motivations we have performed an analysis of the EW observables in the LWSM. As in the original formulation [1], we have assumed the principle of minimal flavor violation (MFV) to simplify the flavor structure of the model. The most stringent constraints come from the S and T Peskin-Takeuchi parameters [23]. We present our results as lower bounds on different combinations of the LW-masses of the gauge and quark fields. If we assume degenerate LW-masses for all the fields, the LW-scale allowed by the EWPT is of order 5 TeV and there is little chance to test this model at the LHC. Relaxing the constraint on equal masses, it is possible to find configurations in the parameter space where one of the masses can be as low as 2.4 − 3.5 TeV, at the price of rising the other masses to be 5 − 8 TeV. This situation is more favorable from the experimental side and it might be accessible at the LHC.
Concerning the fine-tuning of the model, a heavy Higgs gives further contributions to S and T pointing in the same direction as the contributions from the LW-fields, and for this reason is strongly disfavoured. Thus, in order to obtain a light Higgs one has to cancel the rather large contributions to the Higgs mass from the LW-particles running in the loops, that are proportional to the LW-masses squared. We estimate the needed tuning of the model to be at least of order a few per cent.
A possible way to relax the constraints from the EWPT would be to generate an extra positive contribution to T without increasing at the same time the S parameter. By including a fourth generation of fermions of SM-type, with their corresponding LW-partners, it is possible to generate a large T , without generating a too large S. To obtain a T parameter of the needed size one has to assume an approximate custodial symmetry for the Yukawas of the fourth generation. We show that for vector LW-masses of order 3 TeV and Yukawa masses of the fourth generation in the range 0.2 − 1.2 TeV, it is possible to have all the fermionic LW-masses in the range 0.4 − 1.5 TeV and pass the EWPT.
The paper is organized as follows. In Sec. II we give a very brief description of the LWSM, in Secs. III and IV we compute the tree and radiative contributions to the EW precision parameters. In Sec. V we scan over the parameter space of the model and present a detailed analysis of the allowed regions. We consider the extension of the LWSM by including a fourth generation in section VI. We conclude in Sec. VII and show the details of some of the calculations in the Appendices.

II. THE LWSM
The LWSM was originally formulated introducing a higher derivative term for each of the SM-fields. The theory contains one new parameter for every SM-field, the LW-mass corresponding to the dimensional coefficient of the associated higher derivative term. The authors of Ref. [1] introduced new LW-fields and showed that it is possible to reformulate the theory in terms of these fields in such a way that there are no higher derivative terms in the Lagrangian. In this formulation the LW-masses are the masses of the LW-fields and, although the LW-fields mix with the SM ones, the particle content of the theory is more transparent. The LW-fields have the same quantum numbers as their SM partners and the couplings between the SM and LW-fields are the same as the SM couplings, although the signs of the interactions are not always the usual ones. It is possible to consider even higher derivative terms (e.g. six-derivative terms) that will in general lead to additional LW-states, however we will not consider this case. We refer the reader to Ref. [1] for the details and quote here some specific interaction terms that are useful to understand the contributions to the EW observables. We will denote the fields associated to the LW-states with a tilde.
The quadratic Lagrangian for the gauge SM and LW-fields, after setting the Higgs to its vacuum expectation value (VEV), is: where W µν = ∂ µ W ν − ∂ ν W µ , W µ = W a µ T a and similar for the other fields, and g 1,2 are the hypercharge and weak couplings. The sign of the kinetic and mass terms of the LW-fields are opposite to the usual ones. This sign is responsible for the cancellation of the quadratic divergences as well as the microscopic causality violations by the LW-particles.
The quadratic Lagrangian for the fermionic fields after setting the Higgs to its VEV is: where a generation index is understood, q t = (q u , q d ) denotes the SU(2) L doublet, m u,d = λ u,d v/ √ 2 and for simplicity we have omitted the leptonic sector. Different to the SM chiral fermions, the LW-fermions combine into Dirac spinors of masses M q,u,d . We will assume that the LW-fermions transforming in the same representation of the gauge symmetries have the same mass, this is compatible with the MFV principle [24]. Then the matrices M ψ of Eq. (2) are proportional to the identity and we will trade M ψ → 1lM ψ , with M ψ a scalar parameter.
For effects on FCNC when MFV is not satisfied see Ref. [16].
The quadratic Lagrangian for the Higgs field is: where H t = (h + , v+h+iP √ 2 ) andH t = (h + ,h +iP √ 2 ), m 2 h = λv 2 /2 and M h is the LW-mass. Only the physical Higgs field h and its LW-partnerh mix.
The gauge fermionic interactions are: Note that the LW-fermions couple to the gauge fields with the opposite sign compared with the SM-fermions. Ref. [20].
Finally we want to comment on the naturalness of the model. The authors of Ref. [1] showed explicitly that the gauge one loop contributions to the Higgs mass are only logarithmically sensitive to the cut-off of the theory. We want to stress that this contribution is proportional to the square of the vector LW mass, , in such a way that the quadratic divergence is recovered when the LW-mass is divergent. The same effect is present in the fermionic contribution to the Higgs mass, , with M f the fermionic LW-mass. Therefore, to have a light Higgs in a natural way, the LWvectors (fermions) should be lighter than ∼ 2 TeV (∼ 600 GeV), with a mild dependence on the cut-off Λ.

III. TREE LEVEL CONTRIBUTIONS TO THE EW PRECISION PARAMETERS
We discuss in this section the tree level contributions to the EW precision parameters.
We will show that the only parameters that are important in the LWSM are the oblique parameters S and T . In the next section we will compute the 1-loop corrections to S and T and show that the radiative contributions can be as large as the tree level ones.
In the LWSM the mixings between the gauge bosons and their LW partners induce noncanonical couplings for the SM fermions 1 . A shift in the gauge fermion couplings can be reabsorbed into the oblique parameters. Therefore, to correctly define the oblique parameters S, T, U it is necessary to properly normalize the couplings between the fermions and the gauge bosons 2 . We find it useful to work in the effective theory obtained after integrating out the heavy LW fields at tree level. Setting the Higgs field to its VEV, the interactions in the effective theory that are important to normalize the gauge fermion couplings are: where J i µ are the usual currents of SM fermions, and we have considered that the momentum of the LW-vectors is small, p 2 ≪ M 2 i . Since the coefficients in Eq. (5) are the same for all the generations, the same redefinition of the gauge fields leads to canonical gauge couplings for all the SM fermions: The gauge kinetic and mass terms induce contributions to the oblique parameters after the gauge field redefinitions. The tree level contributions to S and T are: Eq. (8) is valid to all order in v in the tree level approximation. Moreover, notice that the sign difference between Eq. (8) and the result of Ref. [1] is due to the additional contribution coming from the redefinition of the gauge fields mentioned above. We can see that for M 1 → ∞ the tree level T parameter cancels, as expected since in this limit we partially recover a custodial symmetry in the LW-gauge sector.
and will be neglected in our analysis.
The effective Lagrangian also includes four fermion operators generated by exchange of LW vector fields, with coefficients of order g 2 i /(2M 2 i ). The constraints from these operators are weaker than the constraints from the oblique parameters.

IV. RADIATIVE CONTRIBUTIONS TO S AND T
In this section we compute the one-loop contributions to the oblique parameters S and T . The most important contributions come from the third generation of LW-fermions.

A. Contributions to T from the gauge-Higgs sector
The one-loop Feynman diagrams involving the LW-Higgs fieldH are shown in Fig. 1. We discuss first the contributions to T . There is no custodial symmetry in the LWSM protecting the T parameter. Thus there is no reason to expect finite radiative contributions to T . As expected from the general arguments of Ref [1] there are no quadratic divergences, however, we obtain corrections from the LW-Higgs sector that are logarithmically sensitive to the UV cut-off of the theory.
We consider the different diagrams of Fig The diagrams of Fig. 1(b) with one SM-Higgs and one LW-gauge field can be explicitly computed. For M 1,2 ≫ m 2 W we can perform an expansion in Higgs-VEV insertions. The leading contribution comes from the zeroth order term, i.e.: we neglect the mixings of the LW-gauge fields due to the mass insertions. In this limit the Feynman diagrams give . A brief explanation of this result is as follows: there is a factor g 1 g 2 v/2 for each vertex, the factor 1/(16π 2 ) comes from the loop and the sign is different from the SM-Higgs contribution because the LW propagator has an extra minus sign. Again, this contribution to T cancels for infinite M 1 .
From the previous result we obtain T ≃ , that is logarithmically divergent with the cut-off. However, for a light Higgs and LW-gauge masses larger than ∼ 2 TeV, these contributions are smaller than the tree level ones, Eq. (8), even in the limit of Λ ∼ M P l . As we will show in the next section, they are also smaller than the fermionic contributions.
There are contributions to T from the diagram of Fig. 1(c), replacing the LW-Higgs propagator by a LW-gauge one. At leading order in a mass insertion expansion, this contribution exactly cancels because the SU(2) L LW-gauge fields are degenerate. There is a non-vanishing contribution at second order but it is suppressed by a factor m 2 W /M 2 2 , and can be neglected. This gives a small S ≃ The Feynman diagram of Fig. 1(b) gives a small contribution also, This contributions to S can be neglected compared with the tree level one, Eq. (7). this method are valid for m u,d ≪ M u,d,q . To obtain a non vanishing T one has to consider at least four mass insertions, this implies that we have to expand S u,d L,R to that order. The result is almost independent on M d because the down Yukawa is much smaller than the up Yukawa for the third generation.
For small LW-fermionic masses (M q,u m t ) the convergence of the mass insertion series is rather slow. In fact, for masses M q,u 1 TeV we have checked that the first non-trivial contribution in the perturbative expansion has large deviations from the non-perturbative one, and can not be trusted. For this reason we will consider also the resummation of the mass insertion series. The diagram of Fig. 2(c) gives the first non-trivial contribution to T in the flavor basis in the mass insertion expansion. For Π 33 , the fermionic propagators (at zeroth order in mass insertions) attached to the gauge vertex could be either q L orq, and the fermionic propagators between the Yukawas could be u R orũ (d R ord) for the up (down) contribution. There is a similar diagram for Π 11 . Using the results of Appendices B and C it is possible to obtain the fermionic vacuum polarizations to all orders in the mass insertions. The result for Π 11 (0) − Π 33 (0) is: where m u,d stand for the masses of the third generation. Eq. (9) includes the contribution from the SM-fermions, that must be subtracted to obtain T . This term is obtained by taking the limit of infinite LW-masses.
The resulting T parameter is negative and it increases for small LW-masses. We make an analysis of the results and its consequences for the LHC in section V. To obtain a better understanding of the importance of the S and T parameters in constraining the model we show in Fig. 3 the 68% and 95% Confidence Level contours in the (S, T ) plane, as obtained from the LEP Electroweak Working Group [26], together with the LWSM predictions for several values of the LW-masses. It is clear from Fig. 3 that there is no region in the parameter space lying within the 68% Confidence Level contour (we have considered LW-masses not larger than 10 TeV). There is however a small but sizable region of the 95% Confidence Level contour that is covered. Choosing all the LW-masses to be equal corresponds to the large yellow points in Fig. 3. In this case only masses above 5 TeV enter into the allowed region. The other coloured points in Fig. 3  In Fig. 4 we show the LW-fermionic masses M q , M u allowed by the EWPT for fixed values of M 1,2 . We have considered the 95% confidence level constraints on the S, T parameters.
The lines divide the parameter space in an upper region allowed by the EWPT and a lower region that does not pass the EWPT.
In order to obtain a rather light LW-fermion, for example an SU(2) L singlet,ũ, with a mass of order 2.5 − 3 TeV, we are forced by the EWPT to have heavy LW-vectors and also a heavy LW-fermion doublet,q, with masses larger than ∼ 5 TeV.
In Fig. 4 we show also the LW-vector masses M 1 , M 2 allowed by the EWPT for fixed values of M q,u . The regions above (below) the lines (do not) pass the EWPT. To obtain a light vector the other vector and the fermions are forced to be heavy, with masses larger than ∼ 5 − 6 TeV. In any case the LW-vector masses can not be lighter than 3 TeV. The lightest vectors are slightly heavier than the lightest fermions, as they give larger contributions to S. In general the LW-vectorB can be lighter thanW . This is because, given a positive S, the EWPT prefer a positive T , that is generated byB and not byW . fourth generation with their LW-partners, without constraints in the isospin violation, will be in general of order 1, much larger than the needed T . We have checked that this is indeed the situation in the present proposal. Therefore, to generate a positive T of the appropriate size, it is necessary to constrain the isospin splitting.
It is immediate to extend the results of sections IV C and IV D to include the one loop effects of a fourth generation. We have scanned over the parameter space fixing the vector LW-masses to be of order ∼ 3 TeV, in order to suppress the large tree level contributions to S and T . It is found that a heavy M q ∼ 3 − 4 TeV is preferred by the data, but a lower M q ∼ 1.5 TeV is still consistent with the EWPT for light m 4 u,d and M u,d . In Fig. 5   In Fig. 6 we show the regions in the plane (m 4 u , m 4 d ) preferred by the EWPT. We find that the isospin violation due to the Yukawas of the fourth generation has to be rather small, satisfying the approximate relation TeV. Whether these heavy states could be produced and detected at the LHC deserves a careful study, some analysis has been done in Refs. [19,21].
The only possible exception to the previous bounds may be the down LW-fermion, whose mass is not well constrained. Since the bottom Yukawa is small, the EWPT do not give important restrictions on M d . Although the model does not explain the origin of the LWmasses, we can expect that the same mechanism gives masses to all the LW-fermions. In We consider the diagonalization of the fermionic mass matrix of the third generation. We collect the up fermions into a three dimensional vector in the following way: and similarly for the down fermions. We adopt the same basis as [20], but with a different notation. Using Eq. (A1) we can write the quadratic fermionic Lagrangian (2) as: where the dots stand for the down sector, η = diag(1, −1, −1) and The mass matrix M u can be diagonalized by independent left and right symplectic rotations S L,R satisfying: where M u,phys is the physical mass matrix, which is diagonal.
To obtain explicit analytic expressions we expand the solutions in powers of Yukawa insertions m u,d . Thus our results are well approximated by the first terms in this expansion in the limit ǫ q,u = mu Mq,u ≪ 1. For the elements of the diagonal matrix M u,phys we obtain the following: For the matrices S L,R we obtain The solution for the down-sector can be obtained from the up-sector simply by changing The authors of Ref. [20] considered the case M q = M u . Their solutions can not be obtained from the case M q = M u that is singular in the limit M q → M u . There is a singularity because in that limit there is a degenerate eigenspace of dimension two, with no preferred eigenvectors. and S respectively. We illustrate the method for resumming the mass insertion series with a particular example. Other cases are simple variations of the one discussed below and they can be obtained by using the same procedure.
Consider the case of the resummed propagator (S q u ) of a SM up-fermion in a SU(2) L doublet. The first three terms of the series are shown in Fig. 7. The first term corresponds to the zeroth order propagator (S q ) which is obtained from the SM kinetic term in the Lagrangian given by Eq. (2): and it takes the form:S where P L = (1 − γ 5 )/2. The following two terms in the expansion, containing at least two mass insertions, can be derived by taking into account the mixing mass term: The first and second order propagators in mass insertions are found to be respectively: where A u and A q , and the zeroth order propagators for u R ,ũ R andq (S, Sũ and Sq respectively) are given by: with P R = (1 + γ 5 )/2. Notice the extra minus sign and the absence of any chirality projector in Sq and Sũ. From the results in Eq. (B4), it is not difficult to infer the generic n-th term and the sum of the series can be obtained: Note thatS q u has only an even number of mass insertions.
where S q , S, Sũ, Sq, A u and A q have been defined in Eqs. (B2) and (B5).M ji has a similar expression toM ij , only the place of the zeroth order propagators must be interchanged as in the case ofS q uqu andSqu q u . Note that the series forM ij andS ij start from one and two mass insertions respectively since the kinetic terms are flavor diagonal.
The fermionic spectrum of the up-sector is given by the poles ofS q u .
Flavor propagators for the down-sector are obtained from those above by changing u → d in Eqs. (B7).

IZATION
We show in this Appendix the fermionic contributions to the vacuum polarization associated to the S and T parameters: with Π µν = g µν Π + (q µ q ν terms).
We consider first the perturbative expansion of Π 33 from Fig. 2(c). Since u R , d R and their LW-partners are singlets of SU(2) L , and the gauge interactions do not mix SM and LW-fermions, the fermionic legs attached to one of the gauge vertices correspond either to q or toq. However, it is possible to have q-legs attached to one of the vertices and either q or q-legs attached to the other vertex, and similar forq. Using the propagators of Appendix B we can write the up contribution to Π 33 to all orders in the mass insertion expansion as: tr d 4 p (2π) 4 (γ µS q u γ νS q u + γ µSq u γ νSq u − 2γ µS q uqu γ νSq u q u ), and a similar contribution from the down sector. The minus sign in the last term is because the LW-fermions couple to the SM-gauge fields with a sign flip compared with the SMfermions, see Eq. (4).
The contributions to Π 11 can be obtained in a similar way, and is given by: The S parameter is proportional to Π ′ 3B (0). The fermionic contribution to Π 3B is more involved because the fermions u R , d R and their LW-partners couple to hypercharge. Therefore we have to consider the diagrams of Figs. 2(b) and (c). The contribution from Fig. 2(c) is similar to Π 33 , with the appropriate charges: and a similar contribution from the down sector with a minus sign due to the different weak charge. The contribution from the down sector can be obtained from Eq. (C5) by changing the factor 1 3 by 1 6 and changing the indices u → d.