The Lee-Wick Standard Model
CALT 68-2643
UCSD/PTH 07-04
The Lee-Wick Standard Model
Benjamı́n Grinstein,1, ∗ Donal O’Connell,2, † and Mark B. Wise2, ‡
1Department of Physics, University of California at San Diego, La Jolla, CA 92093
2California Institute of Technology, Pasadena, CA 91125
(Dated: October 22, 2018)
Abstract
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 interpre-
tation more transparent. Although the theory is unitary, it does not satisfy the usual analyticity
conditions.
∗bgrinstein@ucsd.edu
†donal@theory.caltech.edu
‡wise@theory.caltech.edu
http://arxiv.org/abs/0704.1845v3
mailto:bgrinstein@ucsd.edu
mailto:donal@theory.caltech.edu
mailto:wise@theory.caltech.edu
I. INTRODUCTION
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 [1, 2] 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. [3] propose a modification of
the usual integration contour in Feynman diagrams that removes this growth and preserves
unitarity of the S matrix1. This was further discussed in [5, 6].
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
1 The consistency of this approach is controversial [4].
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 model2 has a new parameter for each standard model field, which cor-
responds 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 [7, 8]. 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 TeV scale our work does suggest that a further examination of higher deriva-
tive 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. [9].
II. A TOY MODEL
To illustrate the physics of Lee-Wick theory [1, 2, 8] in a simple setting, we consider in
this section a theory of one self-interacting scalar field, φ̂, with a higher derivative term.
The Lagrangian density is
Lhd =
∂µφ̂∂
(∂2φ̂)2 −
m2φ̂2 −
gφ̂3, (1)
so the propagator of φ̂ in momentum space is given by
D̂(p) =
p2 − p4/M2 −m2
. (2)
For M ≫ m, this propagator has poles at p2 ≃ m2 and also at p2 ≃ M2. 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 φ̃, so that we can write the theory as
∂µφ̂∂
m2φ̂2 − φ̃∂2φ̂+
M2φ̃2 −
gφ̂3. (3)
2 LW extension of the standard model would be more precise.
Since L is quadratic in φ̃, the equations of motion of φ̃ are exact at the quantum level.
Removing φ̃ from L with their equations of motion reproduces Lhd in Eq. (1).
Next, we define φ = φ̂+ φ̃. In terms of this variable, the Lagrangian in Eq. (3) becomes,
after integrating by parts,
∂µφ̃∂
M2φ̃2 −
m2(φ− φ̃)2 −
g(φ− φ̃)3. (4)
In this form, it is clear that there are two kinds of scalar field: a normal scalar field φ and a
new field φ̃, 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 m for simplicity,
the propagator of φ̃ is given by
D̃(p) =
p2 −M2
. (5)
The LW-field is associated with a non-positive definite norm on the Hilbert space, as indi-
cated by the unusual sign of its propagator. Consequently, if this state were to be stable,
unitarity of the S matrix would be violated. However, as emphasized by Lee and Wick, uni-
tarity is preserved provided that φ̃ decays. This occurs in the theory described by Eq. (4)
because φ̃ is heavy and can decay to two φ-particles.
In the presence of the mass m, there is mixing between the scalar field φ and the LW-
scalar φ̃. We can diagonalize this mixing without spoiling the diagonal form of the derivative
terms by performing a symplectic rotation on the fields:
cosh θ sinh θ
sinh θ cosh θ
 . (6)
This transformation diagonalizes the Lagrangian if
tanh 2θ =
−2m2/M2
1− 2m2/M2
. (7)
A solution for the angle θ exists provided M > 2m. The Lagrangian (4) describing the
system becomes
′∂µφ′ −
m′2φ′2 −
′∂µφ̃′ +
M ′2φ̃′2 −
(cosh θ − sinh θ)3g(φ′ − φ̃′)3, (8)
wherem′ andM ′ are the masses of the diagonalized fields. Notice the form of the interaction;
we can define g′ = (cosh θ − sinh θ)3g and then drop the primes to obtain a convenient
Lagrangian for computation.3
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 [3]) 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 p2 = M2, by
D̃(p) =
p2 −M2
p2 −M2
(−iΣ(p2)) −i
p2 −M2
+ · · ·
p2 −M2 + Σ(p2)
. (9)
Note that, unlike for ordinary scalars, there is a plus sign in front of the self energy Σ(p2) 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 p2 −M2 + iMΓ) then, from a one
loop computation of the self energy Σ, the width of the LW-field is (for Im p2 > 0)
Γ = −
. (10)
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 φφ scattering in this theory. From unitarity, the imaginary part of the forward
scattering amplitude, M, must be a positive quantity. Near p2 = M2, the scattering is
dominated by the φ̃ pole and therefore the imaginary part of the amplitude is given for Im
p2 > 0 by
ImM = −g2
(p2 −M2)2 +M2Γ2
. (11)
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 S matrix, in violation of the usual rules
3 In the following, we will always assume that M ≫ m so that g′ ≃ g.
of S 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 iǫ 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 [6].
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.
A. Gauge Fields
The higher derivative Lagrangian in the gauge sector is
Lhd = −
tr F̂µνF̂
D̂µF̂µν
D̂λF̂λ
, (12)
where F̂µν = ∂µÂν−∂νÂµ− ig[Âµ, Âν ], and Âµ = ÂAµTA with TA the generators of the gauge
group G in the fundamental representation. We can now eliminate the higher derivative
term by introducing auxiliary massive gauge bosons Ã. Each gauge boson is described by a
Lagrangian
L = −
tr F̂µνF̂
µν −M2A tr ÃµÃµ + 2 tr F̂µνD̂µÃν , (13)
where D̂µÃν = ∂µÃν − ig[Âµ, Ãν ]. To diagonalize the kinetic terms, we introduce shifted
fields defined by
Âµ = Aµ + Ãµ. (14)
The Lagrangian becomes
L = −1
trFµνF
DµÃν −DνÃµ
DµÃν −DνÃµ
− ig tr
Ãµ, Ãν
g2 tr
Ãµ, Ãν
Ãµ, Ãν
− 4ig tr
Ãµ, Ãν
DµÃν
−M2A tr
ÃµÃ
. (15)
Note that for a U(1) gauge boson all the commutators vanish, there are no traces and an
extra overall factor of 1/2.
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 La-
grangian involving A and Ã, Eq. (15), or even in the Lagrangian with mixed kinetic terms
for Â and Ã, 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, AAµ , 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
DABµν (p) = δ
ηµν − (1− ξ)
, (16)
while the propagator for the LW-gauge field is
D̃ABµν (p) = δ
AB −i
p2 −M2A
ηµν −
. (17)
B. 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 φ̂ by
Lhd =
D̂µφ̂
D̂µφ̂
D̂µD̂
D̂νD̂
− V (φ̂). (18)
We eliminate the higher derivative term by introducing an LW-scalar multiplet φ̃. Then the
Lagrangian is given in terms of the two fields φ̂ and φ̃ by
D̂µφ̂
D̂µφ̂
+M2φφ̃
D̂µφ̂
D̂µφ̃
D̂µφ̃
D̂µφ̂
− V (φ̂), (19)
where the covariant derivative is
D̂µ = ∂µ + igÂ
A. (20)
For simplicity we take the ordinary scalar to have no potential at tree level, V (φ̂) = 0. It
is not hard to include a potential for φ̂ 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
D̂µ = Dµ + igÃ
A, (21)
where Dµ = ∂µ+ igA
A is the usual covariant derivative. To diagonalize the scalar kinetic
terms, we again shift the field
φ̂ = φ− φ̃. (22)
The scalar Lagrangian becomes
L = (Dµφ)†Dµφ− (Dµφ̃)†Dµφ̃+M2φφ̃†φ̃+ ig(Dµφ)†ÃAµTAφ+ g2φ†ÃAµTAÃBµTBφ
−igφ†ÃAµTADµφ− ig(Dµφ̃)†ÃAµTAφ̃+ igφ̃†ÃAµTADµφ̃− g2φ̃†ÃAµTAÃBµTBφ̃. (23)
C. 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 nec-
essary to fix the gauge. We choose to add a covariant gauge fixing term −(∂µÂAµ)2/2ξ 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
D̂ABµν (p) = δ
AB −i
p2 − p4/M2A
ηµν − (1− ξ)
− ξ pµpν
. (24)
We work in ξ = 0 gauge. Note that ξ = 0 corresponds to Landau gauge and that the
gauge boson propagator scales as p−4 at high energy. The propagator for the scalar in the
fundamental representation is
D̂ab(p) = δab
p2 − p4/M2
. (25)
At large momenta the scalar propagator scales as p−4 while the Faddeev-Popov ghost prop-
agator scales as p−2, 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 n vectors interact (with no
scalars) scales as p6−n while a vertex with two scalars and n vectors scales as p4−n. The
vertex between two ghosts and one gauge field scales as one power of p, as usual.
Consider an arbitrary Feynman graph with L loops, I ′ internal vector lines, I internal
scalar lines, Ig internal ghost lines, and with V
n or Vn vertices with n vectors and zero or
two scalar particles, respectively. We also suppose there are Vg ghost vertices. Then the
superficial degree of divergence, d, is
d = 4L− 4I ′ − 4I − 2Ig +
V ′n(6− n) +
Vn(4− n) + Vg. (26)
We can simplify this expression using some identities. First, the number of loops is related
to the total number of propagators and vertices by
L = I + I ′ + Ig −
(V ′n + Vn)− Vg + 1, (27)
while the total number of lines entering or leaving the vertices is related to the number of
propagators and external lines by
(nV ′n + (n + 2)Vn) + 3Vg = 2(I + I
′ + Ig) + E + E
′ + Eg, (28)
where E is the number of external scalars, E ′ is the number of external vectors, and Eg 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,
Vn = 2I + E, 2Vg = 2Ig + Eg. (29)
With these identities in hand, we may express the superficial degree of divergence as
d = 6− 2L− E − E ′ − 2Eg. (30)
Gauge invariance removes the potential quadratic divergence in the gauge boson two point
function. Scalar mass renormalizations have E = 2, so that d = 4 − 2L. 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
φ†D4φ ∼ φ†(∂2 + ∂ ·A + A · ∂ + A2)2φ. (31)
Since we are working in Lorentz gauge, ∂ ·A = 0. We may ignore the A2 term compared to
the A · ∂ term, as it is less divergent. Thus the most divergent terms in the interaction are
φ†A · ∂∂2φ or φ†∂2A · ∂φ, where the φ 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.4
D. 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 results5. We perform the computations in Feynman gauge, using
4 It 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.
5 We 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.
FIG. 1: One loop mass renormalization of the normal scalar field. The curly line is a gauge field
while the zigzag line is the LW-gauge field. The dashed line represents the scalar field.
the propagators in Eqs. (16) and (17), and regulate our diagrams where necessary using
dimensional regularization with dimension n.
1. The normal scalar
At one loop, there are four graphs contributing to the scalar mass, as shown in Figure 1.
We find
−iΣa(0) = g2C2(N)
(2π)n
(32a)
−iΣb(0) = −g2C2(N)
(2π)n
k2 −M2A
(32b)
−iΣc(0) = −g2C2(N)
(2π)n
(32c)
−iΣd(0) = −g2C2(N)
(2π)n
. (32d)
We see that the quartic and quadratic divergences in this expressions cancel in the sum, so
that the mass is only logarithmically divergent.
FIG. 2: One loop mass renormalization of the LW-scalar field. The dotted line represents the
LW-scalar field while the other propagators are as in Figure 1.
2. The LW-scalar
At one loop the shift in the pole mass is determined by the self energy Σ(p2) evaluated
at p2 = M2φ. The Feynman graphs are shown in Figure 2. We find
−iΣa(M2φ) = −g2C2(N)
(2π)n
(33a)
−iΣb(M2φ) = g2C2(N)
(2π)n
k2 −M2
(33b)
−iΣc(M2φ) = g2C2(N)
(2π)n
k2 − 2p · k
4M2φ − 4p · k
k2(k2 − 2p · k)
(33c)
−iΣd(M2φ) = g2C2(N)
(2π)n
4M2φ − 2p · k
(k2 −M2A)(k2 − 2p · k)
. (33d)
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.
3. The LW-vector
For the LW-vectors the self energy tensor has the form
ΣABµν (p
2) = δAB
Σ(p2)ηµν + Σ
′(p2)pµpν
. (34)
The shift in the pole mass is determined by Σ(M2A). The relevant graphs are shown in
Figure 3. They are very divergent. There are individual terms in Figure 3(c) that diverge
FIG. 3: One loop mass renormalization of the LW-vector field. The propagators are as in Figure 1.
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
n dimensions, a quadratic divergence manifests itself as a pole at n = 2. Hence, we set
n = 2− ǫ, expand about ǫ = 0 and extract the 1/ǫ part of Σ(M2A). We find that
−iΣa(M2A) =
C2(G)
(35a)
−iΣb(M2A) =
C2(G)
(35b)
−iΣc(M2A) =
C2(G)
(35c)
−iΣd(M2A) =
C2(G)
. (35d)
As expected, the 1/ǫ pole cancels in the sum. Finally, we note that there are quadratic
divergences in ΣABµν (p
2). Only the gauge invariant physical quantity Σ(M2A) 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.
A. The Higgs Sector
A higher derivative Lee-Wick Higgs sector was considered previously in [7]. We take the
higher derivative Lagrangian for the Higgs doublet Ĥ to be
Lhd =
D̂µĤ
D̂µĤ
D̂µD̂
D̂νD̂
− V (Ĥ), (36)
where the covariant derivative is given by
D̂µ = ∂µ + igÂ
A + ig2Ŵ
a + ig1B̂µY, (37)
while the potential is
V (Ĥ) =
Ĥ†Ĥ −
. (38)
We can then eliminate the higher derivative term by introducing an LW-Higgs doublet H̃.
As before, we then diagonalize the Lagrangian by introducing the shifted field Ĥ = H − H̃.
To diagonalize the gauge field Lagrangian, we introduced Lee-Wick gauge bosons Ã, B̃, and
W̃ as well as the usual gauge fields A, B and W . In terms of these fields the covariant
derivative is
D̂µ = Dµ + igÃ
A + ig2W̃
a + ig1B̃µY, (39)
where
Dµ = ∂µ + igA
A + ig2W
a + ig1BµY (40)
is the usual standard model covariant derivative. We introduce the notation
Ãµ = gÃ
A + g2W̃
a + g1B̃µY (41)
for the LW-gauge bosons. The Lee-Wick form of the Higgs Lagrangian is then
L = (DµH)†DµH −
DµH̃ +M2HH̃
†H̃ − V (H, H̃) + i (DµH)† ÃµH
− iH†ÃµDµH +H†ÃµÃµH − i
µH̃ + iH̃†ÃµD
µH̃ − H̃†ÃµÃµH̃, (42)
where V is given by the expression
V (H, H̃) = V (H − H̃)
H†H − v
H†H − v
H̃†H̃ − λ
H†H − v
H̃†H +H†H̃
H̃†H̃
H̃†H̃
H̃†H̃
. (43)
In unitary gauge, we write
 , H̃ =
h̃+iP̃√
 . (44)
With this choice, the mass Lagrangian for the Higgs scalar, its partner, the charged LW-
Higgs and pseudoscalar LW-Higgs fields is
Lmass = −
v2(h− h̃)2 + M
h̃h̃+ P̃ P̃ + 2h̃+h̃−
. (45)
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 sym-
plectic 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 SU(2) and U(1)
LW-gauge fields, the Lagrangian is
Lmass =
W̃ aµW̃
g1g2v
W̃ 3µB̃
B̃µB̃
B̃µB̃
W̃ aµW̃
aµ. (46)
There is mixing between the W̃ 3 and B̃ LW-gauge fields. We can diagonalize this Lagrangian
by writing
cosφ sinφ
− sin φ cosφ
 , (47)
where the mixing angle is given by
tan 2φ =
g1g2v
M21 −M22 + (g22 − g21)
. (48)
We expect that M1,2 lie in the TeV range, so that φ 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
Lmix = M2W
W+µ W̃
−µ + W̃+µ W
+M2ZZµ
cos θW W̃
3µ − sin θW B̃µ
, (49)
where θW is the Weinberg angle andMW , MZ are the usual tree level standard model masses
for the W and Z gauge bosons. One consequence of the mixing is that there is a tree level
correction to the electroweak ρ parameter
∆ρ = ρ− 1 = −sin
2 θWM
. (50)
The current experimental constraint on this parameter is |∆ρ| . 10−3 [10] which leads to
M1 & 1TeV.
B. Fermion Kinetic Terms
For simplicity, we discuss explicitly the case of a single left-handed quark doublet QL.
It is straightforward to generalize this work to the other representations, and to include
generation indices.
The higher derivative theory is
Lhd = Q̂LiD̂/ Q̂L +
Q̂LiD̂/D̂/D̂/ Q̂L. (51)
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 propor-
tional 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 Q̃L, Q̃
R which
form a real representation of the gauge groups. The Lagrangian in this formulation becomes
L = Q̂LiD̂/ Q̂L +MQ
Q̃LQ̃
R + Q̃
+ Q̃LiD̂/ Q̂L + Q̂LiD̂/ Q̃L − Q̃′RiD̂/ Q̃′R. (52)
Eliminating the LW-fermions with their equations of motion
Q̃′R = −
Q̂L, Q̃L =
D̂/D̂/
Q̂L, (53)
FIG. 4: One loop graphs involving fermions which are potentially quadratically divergent. The
solid lines represent fermion propagators while the curly and zigzag lines represent gauge bosons
and LW-gauge bosons, respectively.
reproduces the higher derivative Lagrangian, Eq. (51).
To diagonalize the kinetic terms, we introduce the shift Q̂L = QL − Q̃L, and the La-
grangian becomes
L = QLiD/QL − Q̃LiD/ Q̃L − Q̃′RiD/ Q̃′R +MQ
Q̃LQ̃
R + Q̃
−QLγµÃµQL + Q̃LγµÃµQ̃L + Q̃′RγµÃ
µQ̃′R. (54)
Note that Q̃L and Q̃
R combine into a single Dirac spinor of mass MQ.
Now let us return to the issue of potential quadratic divergences in the theory. Inspec-
tion 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 p2, where p 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.
C. Fermion Yukawa Interactions
To simplify the discussion in this section, we will neglect neutrino masses. In the higher
derivative formulation, the fermion Yukawas are
LY = giju ûiRĤǫQ̂
L − g
d̂iRĤ
L − g
L + h.c., (55)
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
LY = giju (uiR − ũiR)(H − H̃)ǫ(Q
L − Q̃
L)− g
(diR − d̃iR)(H† − H̃†)(QjL − Q̃
−gije (eiR − ẽiR)(H† − H̃†)(L
L − L̃
L) + h.c.. (56)
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
uR and QL quarks, and also the ũR and Q̃L 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 [11]. 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 fields6,
miu(u
R − ũiR)(H − H̃)ǫ(QiL − Q̃iL)−mid(diR − d̃iR)(H† − H̃†)(QiL − Q̃iL)
−mie(eiR − ẽiR)(H† − H̃†)(LiL − L̃iL) + h.c.
, (57)
where
 , Q̃L =
V d̃L
 , Q̃′R =
V d̃′R
 . (58)
Here V is the usual CKM matrix. The LW-fermions decay via the Yukawa interactions; for
example, ν̃e → e−h̃+ → e−tb̄. 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.
6 They are mass eigenstate fields when mixing between the normal and LW-fields is neglected. This mixing
can be treated as a perturbation.
V. CONCLUSIONS
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 interpreta-
tion 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 exten-
sion 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.
Acknowledgments
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.
[1] T. D. Lee and G. C. Wick, Nucl. Phys. B 9, 209 (1969).
[2] T. D. Lee and G. C. Wick, Phys. Rev. D 2, 1033 (1970).
[3] R. E. Cutkosky, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, Nucl. Phys. B 12, 281
(1969).
[4] N. Nakanishi, Phys. Rev. D 3, 811 (1971).
[5] T. D. Lee and G. C. Wick, Phys. Rev. D 3, 1046 (1971).
[6] S. Coleman, In “Erice 1969, Ettore Majorana School On Subnuclear Phenomena”, New York,
282 (1970).
[7] K. Jansen, J. Kuti and C. Liu, Phys. Lett. B 309, 119 (1993) [arXiv:hep-lat/9305003];
K. Jansen, J. Kuti and C. Liu, Phys. Lett. B 309, 127 (1993) [arXiv:hep-lat/9305004].
[8] D. G. Boulware and D. J. Gross, Nucl. Phys. B 233, 1 (1984).
[9] D. Evens, J. W. Moffat, G. Kleppe and R. P. Woodard, Phys. Rev. D 43, 499 (1991);
J. W. Moffat, arXiv:hep-ph/0003171.
[10] W. M. Yao et al. [Particle Data Group], J. Phys. G 33, 1 (2006).
[11] R. S. Chivukula and H. Georgi, Phys. Lett. B 188, 99 (1987); L. J. Hall and L. Randall, Phys.
Rev. Lett. 65, 2939 (1990); G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, Nucl.
Phys. B 645, 155 (2002) [arXiv:hep-ph/0207036].
http://arxiv.org/abs/hep-lat/9305003
http://arxiv.org/abs/hep-lat/9305004
http://arxiv.org/abs/hep-ph/0003171
http://arxiv.org/abs/hep-ph/0207036
	Introduction
	A Toy Model
	The Hierarchy Problem and Lee-Wick Theory
	Gauge Fields
	Scalar Matter
	Power Counting
	One loop Pole Mass
	The normal scalar
	The LW-scalar
	The LW-vector
	Lee-Wick Standard Model Lagrangian
	The Higgs Sector
	Fermion Kinetic Terms
	Fermion Yukawa Interactions
	Conclusions
	Acknowledgments
	References
