The Lee-Wick Standard Model
We construct a modification of the standard model which stabilizes the Higgs mass against quadratically divergent radiative corrections, using ideas originally discussed by Lee and Wick in the context of a finite theory of quantum electrodynamics. The Lagrangian includes new higher derivative operators. We show that the higher derivative terms can be eliminated by introducing a set of auxiliary fields; this allows for convenient computation and makes the physical interpretation more transparent. Although the theory is unitary, it does not satisfy the usual analyticity conditions.
The extreme fine-tuning needed to keep the Higgs mass small compared to the Planck scale (i.e., the hierarchy puzzle) has motivated many extensions of the minimal standard model. All of these contain new physics, beyond that in the minimal standard model, which might be observed at the Large Hadron Collider (LHC). The most widely explored of these extensions is low energy supersymmetry. In this paper we introduce another extension of the standard model that solves the hierarchy puzzle.
Our approach builds on the work of Lee and Wick Lee:1969fy (); Lee:1970iw () who studied the possibility that the regulator propagator in Pauli-Villars corresponds to a physical degree of freedom. Quantum electrodynamics with a photon propagator that includes the regulator term is a higher derivative version of QED. The higher derivative propagator contains two poles, one corresponding to the massless photon, and the other corresponding to a massive Lee-Wick-photon (LW-photon). A problem with this approach is that the residue of the massive LW-photon pole has the wrong sign. Lee and Wick argued that one can make physical sense of such a theory. There is no problem with unitarity since the massive LW-photon is not in the spectrum; it decays through its couplings to ordinary fermions. However, the wrong sign residue moves the poles in the photon two point function that are associated with this massive resonance from the second sheet to the physical sheet, introducing time dependence that grows exponentially. Lee and Wick and Cutkosky et al. CLOP () propose a modification of the usual integration contour in Feynman diagrams that removes this growth and preserves unitarity of the S matrix111The consistency of this approach is controversial Nakanishi:1971jj ().. This was further discussed in Lee:1971ix (); Coleman ().
The theory of QED that Lee and Wick studied is finite. In this paper we propose to extend their idea to the standard model, removing the quadratic divergence associated with the Higgs mass, and thus solving the hierarchy problem. In the LW-standard model, every field in the minimal standard model has a higher derivative kinetic term that introduces a corresponding massive LW-resonance. These masses are additional free parameters in the theory and must be high enough to evade current experimental constraints. For the non-Abelian gauge bosons the higher derivative kinetic term has, because of gauge invariance, new higher derivative interactions. Hence the resulting theory is not finite; however, we argue that it does not give rise to a quadratic divergence in the Higgs mass, and so solves the hierarchy puzzle. A power counting argument and some explicit one loop calculations are given to demonstrate this. For explicit calculations, and to make the physics clearer, it is useful to remove the higher derivative terms in the Lagrangian density by introducing auxiliary LW-fields that, when integrated out, reproduce the higher derivative terms in the action.
The LW-standard model222LW extension of the standard model would be more precise. has a new parameter for each standard model field, which corresponds physically to the tree-level mass of its LW-partner resonance. Explicit calculations can be performed in this theory at any order in perturbation theory, and the experimental consequences for physics at the LHC, and elsewhere, can be studied. The nonperturbative formulation of Lee-Wick theories has been studied in Kuti (); Boulware:1983vw (). Lee-Wick theories are unusual; however, even if one does not take the particular model we present as the correct theory of nature at the scale our work does suggest that a further examination of higher derivative theories is warranted. Some previous work on field theories with non-local actions that contain terms with an infinite number of derivatives can be found in Ref. nlocal ().
Ii A Toy Model
To illustrate the physics of Lee-Wick theory Lee:1969fy (); Lee:1970iw (); Boulware:1983vw () in a simple setting, we consider in this section a theory of one self-interacting scalar field, , with a higher derivative term. The Lagrangian density is
so the propagator of in momentum space is given by
For , this propagator has poles at and also at . Thus, the propagator describes more than one degree of freedom.
We can make these new degrees of freedom manifest in the Lagrangian density in a simple way. First, let us introduce an auxiliary scalar field , so that we can write the theory as
Since is quadratic in , the equations of motion of are exact at the quantum level. Removing from with their equations of motion reproduces in Eq. (1).
Next, we define . In terms of this variable, the Lagrangian in Eq. (3) becomes, after integrating by parts,
In this form, it is clear that there are two kinds of scalar field: a normal scalar field and a new field , which we will refer to as an LW-field. The sign of the quadratic Lagrangian of the LW-field is opposite to the usual sign so one may worry about stability of the theory, even at the classical level. We will return to this point. If we neglect the mass for simplicity, the propagator of is given by
The LW-field is associated with a non-positive definite norm on the Hilbert space, as indicated by the unusual sign of its propagator. Consequently, if this state were to be stable, unitarity of the matrix would be violated. However, as emphasized by Lee and Wick, unitarity is preserved provided that decays. This occurs in the theory described by Eq. (4) because is heavy and can decay to two -particles.
In the presence of the mass , there is mixing between the scalar field and the LW-scalar . We can diagonalize this mixing without spoiling the diagonal form of the derivative terms by performing a symplectic rotation on the fields:
This transformation diagonalizes the Lagrangian if
A solution for the angle exists provided . The Lagrangian (4) describing the system becomes
where and are the masses of the diagonalized fields. Notice the form of the interaction; we can define and then drop the primes to obtain a convenient Lagrangian for computation.333In the following, we will always assume that so that .
Introducing the LW-fields makes the physics of the theory clear. There are two fields; the heavy LW-scalar decays to the lighter scalar. At loop level, the presence of the heavier scalar improves the convergence of loop graphs at high energy consistent with our expectations from the higher derivative form of the theory. We can use the familiar technology of perturbative quantum field theory (appropriately modified CLOP ()) to compute quantum corrections to the physics.
It is worth pausing for a moment to consider loop corrections to the two point function of the LW-field. Using the one loop self energy, the full propagator for the LW-scalar is given, near , by
Note that, unlike for ordinary scalars, there is a plus sign in front of the self energy in the denominator. This sign is significant; for example, if one defines the width in the usual way (i.e., near the pole the propagator has denominator ) then, from a one loop computation of the self energy , the width of the LW-field is (for Im )
This width differs in sign from widths of the usual particles we encounter. With this result in hand, we can demonstrate how unitarity of the theory is maintained in an explicit example. Consider scattering in this theory. From unitarity, the imaginary part of the forward scattering amplitude, , must be a positive quantity. Near , the scattering is dominated by the pole and therefore the imaginary part of the amplitude is given for Im by
The unusual sign of the propagator is compensated by the unusual sign of the decay width.
As another consequence of this sign, the poles associated with these LW-particles occur on the physical sheet of the analytic continuation of the matrix, in violation of the usual rules of matrix theory. These signs are also associated with exponential growth of disturbances, which is related to the stability concerns alluded to earlier. Lee and Wick, and Cutkowsky et al argued that one can nevertheless make sense of these theories by modifying the usual contour prescription for momentum integrals. The Feynman prescription can be thought of as a deformation of the contour such that the poles on the real axis are appropriately above or below the contour. The Lee-Wick prescription is equivalent to imposing the boundary condition that there are no outgoing exponentially growing modes. It is well known that such future boundary conditions cause violations of causality. In the Lee-Wick theory the acausal effects occur only on microscopic scales, and show up as a peculiar time ordering of events; for example, the decay products of a Lee-Wick particle appear at times before the Lee-Wick particle itself is created. It is believed that this theory does not produce violations of causality, or any paradoxes, on a macroscopic scale Coleman ().
Iii The Hierarchy Problem and Lee-Wick Theory
In this section, we consider a scalar in the fundamental representation interacting with gauge bosons. We find the Lagrange density for the LW version of such a theory and show by power counting appropriate to the higher derivative version of the theory that the scalar mass is free of quadratic divergences. We then show by an explicit one loop calculation that the ordinary scalar and the massive LW-fields do not receive a quadratically divergent contribution to their pole masses.
iii.1 Gauge Fields
The higher derivative Lagrangian in the gauge sector is
where , and with the generators of the gauge group in the fundamental representation. We can now eliminate the higher derivative term by introducing auxiliary massive gauge bosons . Each gauge boson is described by a Lagrangian
where . To diagonalize the kinetic terms, we introduce shifted fields defined by
The Lagrangian becomes
Note that for a gauge boson all the commutators vanish, there are no traces and an extra overall factor of .
To perform perturbative calculations, we must introduce a gauge fixing term. We could introduce such a term in the higher derivative Lagrangian, Eq. (12), in terms of the Lagrangian involving and , Eq. (15), or even in the Lagrangian with mixed kinetic terms for and , Eq. (13). As is usual in gauge theories, all of these choices will yield the same results for physical quantities, but they may differ for unphysical quantities. Different gauge choices can differ on how divergent unphysical quantities are. Therefore, we will only compute physical pole masses below. In these computations, we introduce a covariant gauge fixing term for the gauge bosons, , in the two field description of the theory given in Eq. (15). In this choice of gauge, the propagator for the gauge bosons is given by
while the propagator for the LW-gauge field is
iii.2 Scalar Matter
Let us move on to consider scalar matter transforming in the fundamental representation of the gauge group. In ordinary field theory, such a scalar field has a quadratic divergence in its pole mass. The higher derivative Lagrangian is given in terms of the scalar field by
We eliminate the higher derivative term by introducing an LW-scalar multiplet . Then the Lagrangian is given in terms of the two fields and by
where the covariant derivative is
For simplicity we take the ordinary scalar to have no potential at tree level, . It is not hard to include a potential for in the analysis, and to show that the potential does not change our results.
We diagonalized the pure gauge sector by shifting the gauge fields; in terms of the shifted gauge fields the hatted covariant derivative is
where is the usual covariant derivative. To diagonalize the scalar kinetic terms, we again shift the field
The scalar Lagrangian becomes
iii.3 Power Counting
Having defined the higher derivative and LW forms of the theory, we present a power counting argument for the higher derivative version of the theory which indicates that the only physical divergences in the theory are logarithmic. Since the power counting argument depends on the behaviour of Feynman graphs at high energies, we only need to consider the terms in the Lagrangian which are most important at high energies.
For the perturbative power counting argument in the higher derivative theory, it is necessary to fix the gauge. We choose to add a covariant gauge fixing term to the Lagrange density and introduce Faddeev-Popov ghosts that couple to the gauge bosons in the usual way. Then the propagator for the gauge field is
We work in gauge. Note that corresponds to Landau gauge and that the gauge boson propagator scales as at high energy. The propagator for the scalar in the fundamental representation is
At large momenta the scalar propagator scales as while the Faddeev-Popov ghost propagator scales as , as usual. There are three kinds of vertices: those where only gauge bosons interact, vertices where gauge bosons interact with two scalars, and vertices where two ghosts interact with one gauge boson. A vertex where vectors interact (with no scalars) scales as while a vertex with two scalars and vectors scales as . The vertex between two ghosts and one gauge field scales as one power of , as usual.
Consider an arbitrary Feynman graph with loops, internal vector lines, internal scalar lines, internal ghost lines, and with or vertices with vectors and zero or two scalar particles, respectively. We also suppose there are ghost vertices. Then the superficial degree of divergence, , is
We can simplify this expression using some identities. First, the number of loops is related to the total number of propagators and vertices by
while the total number of lines entering or leaving the vertices is related to the number of propagators and external lines by
where is the number of external scalars, is the number of external vectors, and is the number of external ghosts. Finally, because the Lagrangian is quadratic in the number of scalars and ghosts, the number of scalar lines and ghost lines is separately conserved. Thus,
With these identities in hand, we may express the superficial degree of divergence as
Gauge invariance removes the potential quadratic divergence in the gauge boson two point function. Scalar mass renormalizations have , so that . Consequently, the only possible quadratic divergence in the scalar mass is at one loop. However, gauge invariance also removes the divergence in the scalar mass renormalization, because two of the derivatives must act on the external legs. To see this, note that the interaction involves
Since we are working in Lorentz gauge, . We may ignore the term compared to the term, as it is less divergent. Thus the most divergent terms in the interaction are or , where the acted on by the derivatives is an internal line. But by integration by parts and use of the gauge condition, we see that, at one loop, we can always take one of the derivatives to act on the external scalar. Thus the theory at hand is at most logarithmically divergent.444It may seem that adding operators with more than four derivatives could yield a finite theory, but that is not the case. These theories are still logarithmically divergent.
iii.4 One loop Pole Mass
The power counting argument above was presented in the higher derivative version of the theory. As a check of the formalism we show, in the LW version of the theory, that the shift in the pole masses of the ordinary scalar, the LW-scalar and the LW-gauge boson do not receive quadratically divergent contributions at one loop. It is important to compute a physical quantity since it is for these that the higher derivative theory and the theory with LW-fields give equivalent results555We have fixed different gauges in our discussion of the power counting argument in the higher derivative theory and our explicit computations in the LW version of the theory. Consequently, we can only expect agreement between these theories for physical quantities.. We perform the computations in Feynman gauge, using the propagators in Eqs. (16) and (17), and regulate our diagrams where necessary using dimensional regularization with dimension .
iii.4.1 The normal scalar
At one loop, there are four graphs contributing to the scalar mass, as shown in Figure 1. We find
We see that the quartic and quadratic divergences in this expressions cancel in the sum, so that the mass is only logarithmically divergent.
iii.4.2 The LW-scalar
At one loop the shift in the pole mass is determined by the self energy evaluated at . The Feynman graphs are shown in Figure 2.
Once again, the quartic and quadratic divergence cancel in the sum of the graphs, so that there is only a logarithmic divergence in the mass of the LW-scalar.
iii.4.3 The LW-vector
For the LW-vectors the self energy tensor has the form
The shift in the pole mass is determined by . The relevant graphs are shown in Figure 3. They are very divergent. There are individual terms in Figure 3(c) that diverge as the sixth power of a momentum cutoff. However these cancel. There is also a quartic divergence in diagrams (b), (c) and (d) that cancels between them. To check that the quadratic divergence cancels we regulate the diagrams with dimensional regularization. In dimensions, a quadratic divergence manifests itself as a pole at . Hence, we set , expand about and extract the part of .
We find that
As expected, the pole cancels in the sum. Finally, we note that there are quadratic divergences in . Only the gauge invariant physical quantity must be free of quadratic divergences.
Iv Lee-Wick Standard Model Lagrangian
Now that we have understood why the radiative correction to the Higgs mass cancels in these higher derivative theories, we move on to discuss the Lagrangian which describes the standard model extended to include a Lee-Wick partner for each particle. The gauge sector is as before.
iv.1 The Higgs Sector
A higher derivative Lee-Wick Higgs sector was considered previously in Kuti (). We take the higher derivative Lagrangian for the Higgs doublet to be
where the covariant derivative is given by
while the potential is
We can then eliminate the higher derivative term by introducing an LW-Higgs doublet . As before, we then diagonalize the Lagrangian by introducing the shifted field . To diagonalize the gauge field Lagrangian, we introduced Lee-Wick gauge bosons , , and as well as the usual gauge fields , and . In terms of these fields the covariant derivative is
is the usual standard model covariant derivative. We introduce the notation
for the LW-gauge bosons. The Lee-Wick form of the Higgs Lagrangian is then
where is given by the expression
In unitary gauge, we write
With this choice, the mass Lagrangian for the Higgs scalar, its partner, the charged LW-Higgs and pseudoscalar LW-Higgs fields is
There is mixing between the usual Higgs scalar and its partner; this mixing can be treated perturbatively. It is possible to diagonalize the mass matrices of these particles via a symplectic rotation, which preserves the diagonal form of the kinetic terms.
The Higgs vacuum expectation value induces masses for the gauge bosons. First, we focus on the mass Lagrangian for the LW-gauge bosons. In terms of the and LW-gauge fields, the Lagrangian is
There is mixing between the and LW-gauge fields. We can diagonalize this Lagrangian by writing
where the mixing angle is given by
We expect that lie in the TeV range, so that is a small angle.
There is also mixing between the gauge fields and the LW-gauge fields. We will treat this mixing perturbatively. The Lagrangian describing this mixing is
where is the Weinberg angle and are the usual tree level standard model masses for the and gauge bosons. One consequence of the mixing is that there is a tree level correction to the electroweak parameter
The current experimental constraint on this parameter is pdg () which leads to .
iv.2 Fermion Kinetic Terms
For simplicity, we discuss explicitly the case of a single left-handed quark doublet . It is straightforward to generalize this work to the other representations, and to include generation indices.
The higher derivative theory is
Naive power counting of the possible divergences in this higher derivative theory shows that there are potential quadratic divergences in one loop graphs containing two external gauge bosons and a fermionic loop. However, gauge invariance forces these graphs to be proportional to two powers of the external momentum so that the graphs are only logarithmically divergent. In this case, this cancellation is most easily understood in the LW description of the theory, which we now construct.
We eliminate the higher derivative term by introducing LW-quark doublets , which form a real representation of the gauge groups. The Lagrangian in this formulation becomes
Eliminating the LW-fermions with their equations of motion
reproduces the higher derivative Lagrangian, Eq. (51).
To diagonalize the kinetic terms, we introduce the shift , and the Lagrangian becomes
Note that and combine into a single Dirac spinor of mass .
Now let us return to the issue of potential quadratic divergences in the theory. Inspection of the Lagrangian, Eq. (54), shows that the only one loop graphs involving fermionic loops are the graphs of Figure 4. Figure 4a is a one loop correction to the gauge boson propagator, and consequently is proportional to , where is the momentum flowing into the graph. Thus, the graph is logarithmically divergent, as is well known. Figure 4b is a one loop correction to the LW-gauge boson propagator. One might think that this graph could introduce a quadratic divergence of the LW-gauge boson mass. However, the vertices between the fermions and the gauge bosons are equal to the vertices between the fermions and the LW-gauge bosons, as can be seen in Eq. (54). Thus, Figure 4b is logarithmically divergent. Higher loop graphs in the theory are at most logarithmically divergent by power counting.
iv.3 Fermion Yukawa Interactions
To simplify the discussion in this section, we will neglect neutrino masses. In the higher derivative formulation, the fermion Yukawas are
where repeated flavor indices are summed. In the formulation of the theory in which there are no higher derivatives, and in which the kinetic terms are diagonal, this becomes
The presence of the LW-fields in this equation improves the degree of convergence at one loop. For example, consider a one loop correction to the Higgs two point function coming from the first term of Eq. (56). Various degrees of freedom can propagate in the loop: the and quarks, and also the and LW-quarks. The presence of the LW-quarks cancels the quadratic divergence in the loop with only the quarks. The sum of these four graphs reproduces the result one would find by computing the corresponding correction in the higher derivative formulation of the theory, Eq. (55).
To simplify the flavor structure of the theory, we adopt the principle of minimal flavor violation MFV (). This forces all LW-fermions in the same representation of the gauge group have the same mass. Now the Yukawas can be diagonalized in the standard fashion. For notational brevity, we choose to use the same symbol for the weak and mass eigenstates. In terms of the mass eigenstate fields666They are mass eigenstate fields when mixing between the normal and LW-fields is neglected. This mixing can be treated as a perturbation.,
Here is the usual CKM matrix. The LW-fermions decay via the Yukawa interactions; for example, . LW-gauge bosons can decay to pairs of ordinary fermions. All the heavy LW-particles decay in this theory, so the only sources of missing energy in collider experiments are the usual standard model neutrinos.
In this paper we have developed an extension of the minimal standard model that is free of quadratic divergences. It is based on the work of Lee and Wick who constructed a finite version of QED by associating the regulator propagator in Pauli-Villars with a physical degree of freedom. Our model is a higher derivative theory and as such contains propagators with wrong sign residues about the new poles. Lee and Wick, and Cutkosky et al. provide a prescription for handling this issue. The LW-particles associated with these new poles are not in the spectrum, but instead decay to ordinary degrees of freedom. Their resummed propagators do not satisfy the usual analyticity properties since the poles are on the physical sheet. Lee and Wick (see also Cutkosky et al.) propose deforming integration contours in Feynman diagrams so that there is no catastrophic exponential growth as time increases. This amounts to a future boundary condition and so LW-theories violate the usual causal conditions. While the Lee Wick interpretation is peculiar it seems to be consistent, at least in perturbation theory, and predictions for physical observables can be made order by order in perturbation theory.
Since the extension of the standard model presented here is free of quadratic divergences it solves the hierarchy problem. Our theory contains one new parameter, the mass of the LW-partner, for each field. We reduced the number of parameters by imposing minimal flavor violation to simplify the flavor structure of the theory. To make the physical interpretation clearer and the calculations easier we introduced auxiliary LW-fields. The Lagrangian written in terms of these fields does not contain any higher derivative terms. When the LW-fields are integrated out the higher derivative theory is recovered.
This paper focused on the the structure of the Lagrange density for the Lee-Wick extension of the standard model. We constructed the Lagrange density, examined the divergence structure and showed how to introduce auxiliary fields to clarify the physical interpretation. For the future, a more extensive discussion of the phenomenology of the theory including its implications for LHC physics is appropriate.
Acknowledgements.DOC would like to thank Stephen Adler for a helpful discussion and for pointing out a useful reference. The work of BG was supported in part by the US Department of Energy under contract DE-FG03-97ER40546, while the work of DOC and MBW was supported in part by the US Department of Energy under contract DE-FG03-92ER40701.
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