## Investigating Lorentz Invariance Violation in Muon Decay

### Bachelor's thesis

August 28, 2014

Student: J. K. Vermijs, N° 1634917 Supervisor: dr. ir. C. J. G. Onderwater

2^{nd}corrector: prof. dr. R. G. E. Timmermans

Abstract

Lorentz invariance testing of weak decay mechanisms has recently been used as a method to examine the assumptions of the standard model of particle physics. Building on a gen- eral expression of Lorentz invariance violating muon decay, that had been formulated in the muon rest frame, an expression for the energy spectrum of the decay electron in the laboratory frame is obtained. Since this expression now contains Lorentz violating parameters, it can be used to design experiments that would put bounds on the mag- nitude of Lorentz violation. To make it possible to relate experimental data between experiments, transformations are given that move these coordinate system dependent parameters between muon rest frame, laboratory frame, and an inertial reference frame fixed in space.

### Contents

1 Introduction . . . 2

1.1 Lorentz Invariance . . . 2

2 Theory . . . 4

2.1 Muon Decay . . . 4

2.2 Standard Model Extension . . . 6

2.3 Coordinate Transformations . . . 8

2.4 From Laboratory Frame to Muon Rest Frame . . . 11

3 Procedure . . . 13

3.1 Reduction of Parameters . . . 13

3.2 Boosting to Moving Muon Frame . . . 14

3.3 Transforming the χ^{µν} Parameters . . . 18

4 Results . . . 19

4.1 Energy Spectrum in Laboratory Frame . . . 19

4.2 Varying Detector Orientation . . . 23

4.3 Transformed X^{µν} parameters . . . 23

5 Discussion & Conclusion . . . 24

5.1 Further Research . . . 24

Appendix A Derivation of Lorentz Boost Change of Variables . . . 25

Appendix B Exchanging Integration Order of E and θ . . . 27

Appendix C Tables for the X^{µν} Parameters . . . 28

Appendix D Mathematica File for Decay Rate in the Laboratory Frame 29 6 Acknowledgements . . . 33

7 References . . . 33

### 1 Introduction

The weak interaction is known to violate the parity (P) and charge conjugation (C) symmetries, as well as the combined CP symmetry. The combined CPT symmetry, which includes time symmetry (T), is fundamental to general relativity and holds in field theories – such as the standard model of particle physics – as long as Lorentz invariance is assumed.[1] Investigating Lorentz invariance provides a way to challenge the assumption of CPT symmetry, like C and P symmetries have been challenged in the past.

Another motivation to investigate Lorentz invariance is provided by models of quantum gravity, which are part of ongoing efforts to unify the standard model of particle physics with the theory of general relativity. Several of these models potentially violate Lorentz invariance at high energies, which could manifest itself in lower energy experiments as well.[2]

To model Lorentz invariance violation in the standard model, extensions to the standard model (SME) have been developed by Kostelecký et al.[3] which have been used in papers on beta, charged pion and neutral kaon decays by Noordmans[4,5]

and Vos[6]. This study is built on a similar development for muon decay in the muon center of mass system.[7]

The objective of this study is to transform the SME muon decay rate to a form that allows experimental observables to be derived, in particular for muons that are moving at relativistic speeds in the laboratory, in an arbitrary horizontal direction. These experiments would then be able to provide bounds on the values of the Lorentz invariance violating parameters that occur in the theory.

### 1.1 Lorentz Invariance

Lorentz invariance is the property that the laws of physics remain unchanged under the group of Lorentz transformations, which is composed of the rotations in three-dimensional space and the Lorentz boosts.

Rotations relate coordinate systems that have different orientations, but coin- ciding origins. Any three-dimensional rotation can be described by a sequence of rotations around the three spatial axes of a four-dimensional (spacetime) coordi- nate system.

The matrices that represent rotations by θ radians around the x, y and z axes are given by

R_{x}(θ) =

1 0 0 0

0 1 0 0

0 0 cos θ − sin θ 0 0 sin θ cos θ

(1)

R_{y}(θ) =

1 0 0 0

0 cos θ 0 sin θ

0 0 1 0

0 − sin θ 0 cos θ

(2)

R_{z}(θ) =

1 0 0 0

0 cos θ − sin θ 0 0 sin θ cos θ 0

0 0 0 1

, (3)

where the rotation is counterclockwise when looking towards the origin from a viewpoint lying on the rotation axis. The inverse transformations are given by taking the matrix transpose, or by substituting −θ in place of θ.

Lorentz boosts are the spacetime transformations that relate one system to another when they have identical orientations, but their origins are offset from each other by a constant velocity v. The general matrix form is given by

Λ(β) =

γ −γβ_{x} −γβ_{y} −γβ_{z}

−γβx 1 + (γ − 1)β_{x}^{2}

β^{2} (γ − 1)β_{x}β_{y}

β^{2} (γ − 1)β_{x}β_{z}
β^{2}

−γβ_{y} (γ − 1)β_{x}β_{y}

β^{2} 1 + (γ − 1)β_{y}^{2}

β^{2} (γ − 1)β_{y}β_{z}
β^{2}

−γβ_{z} (γ − 1)β_{x}β_{z}

β^{2} (γ − 1)β_{y}β_{z}

β^{2} 1 + (γ − 1)β_{z}^{2}
β^{2}

, (4)

where β = (β_{x}, β_{y}, β_{z}) = v/c with c the speed of light, β = |β| and γ = 1/p1 − β^{2}.
The inverse transformation is given by substituting −β for β. In this paper only a
boost in the x direction (β = (β, 0, 0)) is used in particular, for which the matrix
reduces to

Λ(β) =

γ −βγ 0 0

−βγ γ 0 0

0 0 1 0

0 0 0 1

. (5)

### 2 Theory

### 2.1 Muon Decay

Muons are charged leptons with a mean lifetime of 2.2 µs, that decay via the weak interaction, which is mediated by W bosons. In the most common decay mode, a negative muon decays to an electron, a muon neutrino and an electron antineutrino. (Positive muons instead decay to a positron, a muon antineutrino and an electron neutrino.) A Feynman diagram of this decay mode is pictured in Figure 1.

Fig. 1: Feynman diagram of the most common muon decay mode[9]. Mediated by a W-boson, the muon decays to a muon neutrino, an electron neutrino and an electron.

Muon decay is described by the Fermi fermion interaction, in which its symme- try violating properties arise from the V − A (vector minus axial vector) structure of this interaction[10], since a vector changes sign under parity transformations, while an axial vector does not. Because of P violation, the electron emission spec- trum is not independent of orientation, but related to the spin direction of the decaying muons. Because of C violation, the spectrum for positive muons has a spin dependence that is opposite in sign to that of the negative muon decay.

According to the standard model, the differential muon decay rate in the muon rest frame is given by[11]

dW

dx dΩ = W0

π x^{2}

"

3(1 − x) +2

3ρ(4x − 3) ∓ ξ(ˆs · ˆp)

1 − x + 2

3δ(4x − 3)

# , (6) where natural units with c = 1 are used. dΩ is the differential solid angle, parame- terized in spherical coordinates as dΩ = sin θ dθ dφ, and x ∈ [0, 1] is the fractional

energy of the electron (or positron). The direction of the electron momentum is
given by ˆp, and the direction of the muon spin by ˆs. Because neutrinos are very
hard to detect, their spins and momenta have been integrated over. The electron
spin has similarly been integrated over, because it is also difficult to measure. The
electron mass m_{e} = 511 eV is small compared to the muon mass m_{µ} = 105.7 MeV,
and it has been neglected, greatly simplifying the expression. The fractional en-
ergy x = E/E_{max} is defined in terms of the electron energy E and its maximum
E_{max}= (m^{2}_{µ} + m^{2}_{e})/2m_{µ} = 52.8 MeV (which follows from four-momentum conser-
vation when neglecting the small neutrino masses).

A diagram of the muon decay, ignoring the undetectable neutrinos, is shown inFigure 2:

*x*
*z* *y*

**r**m

**p`**

e

**s`**

m

y

Fig. 2: A muon at rµ with spin direction ˆs_{µ} decays. The probability for the electron
to be emitted in the direction ˆp_{e} depends on the angle ψ = ˆs · ˆp between
the electron direction and the muon spin.

That the decay violates P symmetry can be seen in the presence of the ˆs · ˆp term, while C symmetry violation presents itself as the ∓ sign, the upper sign (−) being valid for the negative muon and the lower sign (+) for the positive muon.

(Note: There is no other distinction between electrons and positrons in the theory, so for convenience, only ‘electron’ will be used, where it should be understood that

‘positron’ might be substituted instead for positive muon decay.)

The Michel parameters ρ, ξ and δ are predicted by the standard model to have values of ρ = 3/4, ξ = 1 and δ = 3/4, but as will be seen in the next section, in the standard model extension these values depend on the spatial orientation of the muon decay.

The total decay rate is given by W_{0} = G^{2}_{F}m^{5}_{µ}/192π^{3}, where G^{2}_{F} is the Fermi
coupling constant and m_{µ} again is the muon mass. Integrating over all angles
and energies in (6) reproduces this value, which implies that the differential decay
rate divided by W_{0} gives the probability density for an electron to be emitted with
energy x in the direction ˆp. This probability density function is then parameterized
by the muon spin direction ˆs.

### 2.2 Standard Model Extension

In the standard model extension (SME), Lorentz violation may enter the muon
decay rate in many different ways. In this study only a modified W-boson prop-
agator is considered, originally hW^{µ+}W^{ν−}i = −ig^{µν}, where g^{µν} is the Minkowski
metric. The modification consists of adding a traceless complex tensor χ^{µν} to the
propagator that encompasses underlying SME parameters[7]. The propagator thus
becomes hW^{µ+}W^{ν−}i = −i(g^{µν}+χ^{µν}). Then the SME expression for the differential
muon decay rate in the muon rest frame becomes[7]

dW

dx dΩ = W_{0}
π x^{2}

"

3(1 − x) +1

2(4x − 3) ∓ (ˆs · ˆp)

1 − x + 1

2(4x − 3)

−(1 − x)

t_{1}+ v_{1}· ˆp ± v_{2}· ˆs ± v_{3}· (ˆs × ˆp)
+T_{1}^{ml}pˆ^{m}pˆ^{l}± T_{2}^{ml}pˆ^{m}sˆ^{l}± T_{3}^{ml}pˆ^{m}(ˆs × ˆp)^{l}

−(4x − 3)

z_{1}+ u_{1}· ˆp ± u_{2}· ˆs ± u_{3}· (ˆs × ˆp)
+H_{1}^{ml}pˆ^{m}pˆ^{l}± H_{2}^{ml}pˆ^{m}sˆ^{l}± H_{3}^{ml}pˆ^{m}(ˆs × ˆp)^{l}

±(ˆs · ˆp)(1 − x)(t2+ v_{4}· ˆp) + (4x − 3)(z_{2}+ u_{4}· ˆp)

,

(7)

where again x is the fractional energy of the electron, ˆp is the direction of the electron momentum and ˆs the direction of the muon spin. Once more the neutrinos and electron spin have been integrated over. Again, the ± signs represent equations for the negative muon/electron and positive muon/positron with the upper and lower sign respectively.

In equation (7), the components of χ^{µν} have been replaced by scalar, vector
and tensor parameters according to the following list of equations:

t_{1} = z_{1} = z_{2} = 1
2χ^{00}_{rs}
t2 = 5

2χ^{00}_{rs}
v^{l}_{1} = χ^{0l}_{rs}+ 2χ^{0l}_{ra}− 2^{lmn}χ^{mn}_{ia}

v^{l}_{2} = 1

2χ^{0l}_{rs}+ 7

2χ^{0l}_{ra}+5

4^{lmn}χ^{mn}_{ia}
v^{l}_{3} = 3

2χ^{0l}_{ia}+ 5
2χ^{0l}_{is}
v^{l}_{4} = 3

2χ^{0l}_{rs}+ 3

2χ^{0l}_{ra}− 3

4^{lmn}χ^{mn}_{ia}

u^{l}_{1} = −1

2^{lmn}χ^{mn}_{ia}
u^{l}_{2} = 1

2χ^{0l}_{rs}+1

2χ^{0l}_{ra}+ 1

4^{lmn}χ^{mn}_{ia}
u^{l}_{3} = 1

2χ^{0l}_{ia}+1
2χ^{0l}_{is}
u^{l}_{4} = 1

2χ^{0l}_{rs}+1

2χ^{0l}_{ra}− 1

4^{lmn}χ^{mn}_{ia}
T_{1}^{ml} = −3

2χ^{ml}_{rs}
T_{2}^{ml} = 7

2χ^{ml}_{rs} + 1
2χ^{ml}_{ra}
T_{3}^{ml} = 3

2χ^{ml}_{is}

H_{1}^{ml} = −1
2χ^{ml}_{rs}
H_{2}^{ml} = 1

2χ^{ml}_{ra}
H_{3}^{ml} = 1

2χ^{ml}_{is} .

(8)

Here the subscripts r and i indicate real and imaginary parts, while s and a indicate symmetric and antisymmetric parts. Since the tensor is traceless, there are 15 real and 15 imaginary independent parameters represented here.

The Michel parameters get modified values in this equation. For example,
considering ρ as the factor with which ^{2}_{3}(4x − 3) (without ˆs · ˆp) occurs in the
equation, its value is now

ρ = 3 4− 3

2

z1+ u1· ˆp ± u2· ˆs ± u3· (ˆs × ˆp)
+ H_{1}^{ml}pˆ^{m}pˆ^{l}± H_{2}^{ml}pˆ^{m}sˆ^{l}± H_{3}^{ml}pˆ^{m}(ˆs × ˆp)^{l}

.

(9)

Therefore any experiment that measures ρ necessarily measures these χ^{µν} param-
eters as well, and a significant deviation from ρ = ^{3}_{4} would imply that these
parameters have nonzero values. Because the χ^{µν} parameters are dependent on
the orientation and potentially an absolute velocity of the experiment in space, ρ
and the other Michel parameters now also have values that depend on this rotation
and boost. The detection sensitivity for each of these components can be enhanced
or reduced by selecting for electron directions or energies.

When (7) is integrated over all spatial directions (dΩ) and all valid energies (dx), the total decay rate is obtained, and thus also its reciprocal, the muon

lifetime. Using the tracelessness of the χ^{µν} tensor, it can be shown[4] that the total
muon decay rate is independent of the χ^{µν} parameters and therefore measurements
on the muon lifetime cannot be used to determine Lorentz-invariance due to a
modified W-propagator in this manner. Only measuring the energy and direction
distribution of the decay electrons will allow χ^{µν} to be determined.

### 2.3 Coordinate Transformations

The standard model extension is Lorentz invariant under observer transforma-
tions, but under particle transformations the Lorentz violating χ^{µν} parameters
will change.[4]. This means the χ^{µν} parameters must be transformed to a standard
reference frame to enable the comparison of values from different measurements
and experiments. In general each experiment will have a unique orientation on
Earth, and this orientation changes over time as the Earth rotates. Additionally
to the transformations on the χ^{µν} parameters, the electron momentum must also
be transformed from the muon center of mass frame to the laboratory frame.

To relate the orientations of various experiments, the Lorentz violating param- eters are transformed to an Earth-centered reference frame that is approximately inertial with respect to far away sources of radiation. This inertial reference frame has a fixed orientation in space, defined by having its z-axis parallel to the Earth’s polar axis and its x-axis parallel towards the intersection of the equatorial and ecliptic planes, with both these directions taken at noon UT1 on 2000-01-01. The y-axis is taken to complete a right-handed orthogonal coordinate system.

A detailed description of this coordinate system and of the rotations from a
laboratory frame to the inertial reference frame can be found in [12]. To simplify
analysis, this paper uses the approximate model that is also outlined therein, which
neglects several small deviations between intermediate coordinate systems and will
stay to within ^{1}_{3}^{◦} of the full model for the next 50 years.[12]

The laboratory frame is defined with the x, y and z axes pointing south, east and upwards respectively, forming a right-handed orthogonal coordinate system.

The transformation from laboratory frame to inertial reference frame consists of a rotation by the colatitude φ around the y-axis to align the z-axis with the Earth’s polar axis, a rotation by the longitude λ around the z-axis, and finally another rotation around the z-axis to cancel the Earth’s rotation. SeeFigure 3andFigure 4 for a diagram of the transformations.

Fig. 3: A laboratory is located at rlab, with colatitude φ and longitude λ on the
Earth’s surface. The laboratory coordinate frame (x y z, in black) has its
z-axis pointing upwards, orthogonal to the surface of the Earth, while the
x- and y-axes point towards the south and east respectively, parallel to the
surface. The transformation R_{y}(−φ), which is a clockwise rotation by φ
around the y-axis, takes the coordinate axes to the primed frame (x^{0}y^{0}z^{0},
in gray), which has its z^{0}-axis pointing parallel to the Earth’s polar axis
and x^{0}-axis parallel to the equatorial plane.

Fig. 4: The primed frame (x^{0}y^{0}z^{0}, in black) fromFigure 3has its z^{0}-axis parallel to
the Earth’s polar axis and its x^{0}- and y^{0}-axes parallel to the equatorial plane,
in the southern and eastern direction respectively. (From the point of view
of the laboratory at r_{lab}, they point to the sky due south and due east.)
The transformation R_{z}(−λ), which is a clockwise rotation by the longitude
λ around the z^{0}-axis, takes the coordinate axes to the double primed frame
(x^{00}y^{00}z^{00}, in gray), which has its z^{00}-axis parallel to the polar axis, it’s x^{00}-axis
pointing towards the Greenwich meridian, and it’s y^{00}-axis completing the
right-handed coordinate system. The final rotation (Rz(−Ωt), not pictured
but conceptually identical to R_{z}(−λ)) takes the double primed system to
the inertial reference system.

The full transformation from the laboratory frame to the inertial reference frame is given by

R = R_{z}(−Ωt)R_{z}(−λ)R_{y}(−φ), (10)
where Ω is the angular velocity of the earth, t is time, and φ and λ are the colatitude
and latitude of the laboratory. The Ry and Rz matrices are defined by (2) and
(3). Plugging in the angles, the transformation is represented by the matrices

R =

1 0 0 0

0 cos Ωt sin Ωt 0 0 − sin Ωt cos Ωt 0

0 0 0 1

1 0 0 0

0 cos λ sin λ 0 0 − sin λ cos λ 0

0 0 0 1

1 0 0 0

0 cos φ 0 − sin φ

0 0 1 0

0 sin φ 0 cos φ

,

(11) Because the first two matrices represent a rotation around the same axis, their angles can be taken together as (t) = −λ − Ωt, so that the expression becomes

R =

1 0 0 0

0 cos (t) − sin (t) 0 0 sin (t) cos (t) 0

0 0 0 1

1 0 0 0

0 cos φ 0 − sin φ

0 0 1 0

0 sin φ 0 cos φ

. (12)

### 2.4 From Laboratory Frame to Muon Rest Frame

There are two transformations from the laboratory frame to the muon rest frame:

a rotation to align the muon velocity with the x-axis, and a subsequent boost along the x-axis that cancels its speed to zero. If the muon is travelling horizontally in the laboratory frame with speed β at an azimuthal angle α to the laboratory x-axis (pointing south), its velocity four-vector is

p_{µ} = γm_{µ}(1, β cos α, β sin α, 0),

where mµis the muon rest mass and γ = 1/p1 − β^{2}. To reach pµ = γmµ(1, β, 0, 0),
a clockwise rotation around the z-axis by α is used

M = R_{z}(−α) =

1 0 0 0

0 cos α sin α 0 0 − sin α cos α 0

0 0 0 1

, (13)

where R_{z} is defined by (3). See Figure 5 for a diagram of the transformation.

**p**m

**p**m

a a

**p**m

**p**^{¢}_{m}
*M*
**r**m

ring

*x*

*y*
*z*

Fig. 5: In the laboratory frame, a muon at position rµcircles around a storage ring.

Its momentum p_{µ} is at an angle α to the x-axis. The transformation M –
a clockwise rotation by α around the z-axis – transforms p_{µ} → p^{0}_{µ} = M p_{µ}
so that the momentum is parallel to the x-axis.

When boosted by

Λ =

γ −βγ 0 0

−βγ γ 0 0

0 0 1 0

0 0 0 1

, (14)

the muon rest frame is reached, where p_{µ} = (m_{µ}, 0, 0, 0). The matrices R (12), M
(13) and Λ (14) together fully specify all the transformations needed in this study.

### 3 Procedure

In this section, the mathematical procedures used to obtain the results of this paper are outlined. First a reduction of the scope of the problem is discussed, then the details of transforming the muon decay rate to the laboratory frame are shown, and finally the transformations of the Lorentz invariance violating parameters are mentioned.

### 3.1 Reduction of Parameters

To simplify the problem under consideration, the dependence of (7) on the muon spin direction is ignored, and the electron momentum dependence alone is consid- ered. This reduces the equation to

g(x, ˆp) · W0 = dW

dx dΩ = W_{0}
π x^{2}

(1 − x)

3 − t1− v1· ˆp − T_{1}^{ml}pˆ^{m}pˆ^{l}

+(4x − 3) 1

2− z_{1}− u_{1}· ˆp − H_{1}^{ml}pˆ^{m}pˆ^{l}

# .

(15)

Now an experiment needs only to be sensitive to the electron energy and direction.

This changes the number of objects under consideration from 4 scalars, 8 vectors and 6 tensors in (8) to the following 2 scalars, 2 vectors and 2 tensors

t_{1} = z_{1} = 1
2χ^{00}_{rs}

v^{l}_{1} = χ^{0l}_{rs}+ 2χ^{0l}_{ra}− 2^{lmn}χ^{mn}_{ia} u^{l}_{1} = −1

2^{lmn}χ^{mn}_{ia}
T_{1}^{ml} = −3

2χ^{ml}_{rs} H_{1}^{ml} = −1
2χ^{ml}_{rs},

(16)

which after applying tracelessness of the χ^{µν}_{r} tensor leaves 12 real and 3 imaginary
independent parameters, half of the total number.

As mentioned before, the χ^{µν} parameters as well as the electron properties x
and ˆp need to be boosted to the laboratory frame. This will be described in the
next section.

### 3.2 Boosting to Moving Muon Frame

As a first step towards the laboratory frame, the equation is boosted with a rela- tivistic γ that corresponds to the muon velocity in the laboratory frame.

To avoid confusion between the muon rest frame and the boosted frame, in this
section the quantities in the muon rest frame will be denoted with an asterisk. For
example, p^{∗} denotes the electron four-momentum in the muon rest frame, while p
is the four-momentum in the boosted frame.

Expressing the four-momentum in terms of the energy E and the momentum direction ˆp, it becomes

p^{∗} = (E^{∗}, |p^{∗}| ˆp^{∗}) = (E^{∗},p

E^{∗2}− m^{2}_{e} pˆ^{∗}), (17)
where the Minkowski norm for the four-momentum E^{2}−|p|^{2} = m^{2} was used. Once
again using the approximation m_{e}= 0, and substituting x^{∗} = E^{∗}/E_{max}^{∗} , results in
p^{∗} = (x^{∗}E_{max}^{∗} , x^{∗}E_{max}^{∗} pˆ^{∗}). (18)
The boost is represented by the matrix transformation p^{∗µ} = Λ^{µ}_{ν}p^{ν}, where Λ
is the matrix in (14). The transformation then becomes

p^{∗} = (x^{∗}E_{max}^{∗} , x^{∗}E_{max}^{∗} pˆ^{∗}) = Λ (E, E ˆp). (19)
The electron momentum direction is parameterized using spherical coordinates
with x as the polar axis. The corresponding polar angle is θ, and φ is the azimuthal
angle measured from the y-axis to the z-axis. This parameterization gives

ˆ

p^{∗} = (cos θ^{∗}, sin θ^{∗}cos φ^{∗}, sin θ^{∗}sin φ^{∗})
ˆ

p = (cos θ, sin θ cos φ, sin θ sin φ) , (20) with starred quantities as before. A diagram of the parameterization in spherical coordinates is shown in Figure 6.

*x*
*y*

*z*

**p`**
f

q

Fig. 6: The vector ˆp = (cos θ, sin θ cos φ, sin θ sin φ) is specified in spherical coor- dinates using a polar angle θ measured from the x-axis and an azimuthal angle φ measured from the y-axis. The gray vector shows the projection of ˆp in the yz plane, with the dashed guidelines indicating the y- and z-intercepts.

Applying the boost Λ results in the following change of variables:

φ^{∗} = φ

cos θ^{∗} = cos θ − β
1 − β cos θ
sin θ^{∗} = sin θ

γ(1 − β cos θ)
x^{∗} = E

E_{max}^{∗} γ (1 − β cos θ)
sin θ^{∗}dθ^{∗}dφ^{∗}dx^{∗} = 1

E_{max}^{∗}

1

γ(1 − β cos θ)sin θ dθ dφ dE,

(21)

where 1/E_{max}^{∗} is the Jacobian determinant of the boost coordinate transforma-
tion.[A]

To get an expression for the energy spectrum of the electron in the laboratory frame, (15) needs to be integrated over φ and θ. The integration limits for φ are [0, 2π], and the limits for θ are [0, h(E)], where h is defined by

h(E) =

arccos1 β

1 − E_{max}^{∗}
γE

E > E_{max}^{∗}
γ(1 + β)

π otherwise

, (22)

on the domain E ∈ [0, Emax] where E_{max} = E_{max}^{∗} /γ(1 − β) . See Figure 7 for a
graph of h(E) for various values of γ. See [B] for a derivation of h(E).

When h(E) is expressed in terms of x = E/E_{max}= γ(1−β)E/E_{max}^{∗} , it becomes

h_{x}(x) =

arccosx + β − 1

βx x > 1 − β 1 + β

π otherwise

, (23)

from which it is more readily seen (^{1−β}_{1+β} → 0 as β → 1) that for large β and x, it
reduces to

h_{x}(x) = arccosx + β − 1

βx (24)

*E*maxHΒL *E*
Π

Π 2

Θ

0

Γ 2 Γ 5

Γ 30

Fig. 7: The upper integration limit of θ in the laboratory frame is θ = h(E). The
function h(E) has been graphed for γ ∈ {2, 5, 30}, where the horizontal
axis has been rescaled so that E_{max} = E_{max}^{∗} /γ(1 − β) falls on the same
point for each value of γ. It is apparent that the clipping h(E) = π for low
E becomes irrelevant as β approaches 1.

Integrating (15) over φ ∈ [0, 2π] and θ results in the expression for the energy spectrum in the laboratory frame,

f_{E}(E) =

Z h(E) 0

Z 2π 0

g(x^{∗}, ˆp^{∗}) 1
E_{max}^{∗}

sin θ

γ(1 − β cos θ)dφ dθ, (25) where g is the probability density function defined by (15). See [D] for a Mathemat- ica notebook used to evaluate this integral and calculate the final energy spectrum in the laboratory frame. These results will be presented in a later section.

The energy spectrum at this point still contains the χ^{µν} parameters, which
are defined in the muon rest frame. For the energy spectrum function to be of
use, the parameters also need to be transformed to the laboratory frame, which is
described in the next section.

### 3.3 Transforming the χ

^{µν}

### Parameters

The χ^{µν} parameters are related to the inertial reference frame by

χ^{µν} = T^{µ}_{ρ}T^{ν}_{σ}X^{ρσ}, (26)

where T = ΛM R is the combination of matrices (12), (13) and (14), and X^{µν}
are the Lorentz violating parameters in the inertial reference frame. Note that R
depends on time, as the rotation matrix changes with the rotation of the Earth.

Although the transformation here is from the inertial reference frame to the muon rest frame, and R was defined as the transformation from the laboratory frame to the inertial reference frame, R does not appear inverted here. This is because the components of objects like tensors and vectors transform oppositely to the coordinate axes. Λ and M are not used oppositely here, but they were defined as object transformations in the first place, so also do not need to be inverted.

For mathematical treatment of the decay rate in the laboratory, it initially
suffices to transform to an intermediate set of parameters Z^{µν} in the laboratory
frame, related to χ^{µν} and X^{µν} by

χ^{µν} = L^{µ}_{ρ}L^{ν}_{σ}Z^{ρσ}

Z^{µν} = R^{µ}_{ρ}R^{ν}_{σ}X^{ρσ}, (27)

where L = ΛM is the transformation relating the laboratory frame and the muon
rest frame. Combining (27) with the energy spectrum (25), terms for several
parameters of Z^{µν} can be collected. See [D] for a Mathematica notebook that
implements collecting the Z^{µν} parameters in the energy spectrum equation. In the
next section, these collected parameters in the energy spectrum will be presented.

With the transformed energy spectrum (25) and the transformed χ^{µν} param-
eters Z^{µν}, the latter can be determined in the laboratory. Values for the Z^{µν}
parameters can then be transformed to X^{µν} through application of R. Since R
depends on the sidereal time and date of the experiment because of the rotation
of the Earth, Z^{µν} and χ^{µν} also depend on this time. In addition to measuring
electron energy and direction, it is therefore important to keep track of the time
of each detection.

### 4 Results

### 4.1 Energy Spectrum in Laboratory Frame

The energy distribution for the electron emitted by muon decay is given in the
laboratory frame (in the limit where x > ^{1−β}_{1+β}) by the probability density func-
tion

f (x) = f_{SM}(x) + f_{SME}(x), (28)
where x = E/E_{max} is the fractional energy of the electron in the laboratory frame,
and β is the speed of the decaying muon in the laboratory frame.

The function f_{SM} is the contribution due to the standard model, given by:

f_{SM}(x) = (β + 1) 4x^{3}− 9x^{2}+ 5

6β . (29)

To define f_{SME} – the SME contribution – first the following polynomials must
be defined:

f_{0}(x) = 1
15(β − 1)βx^{2}

h−3(β − 1)^{2}+ 2(9β^{2}− 3β − 1)x^{5}

+ (−15β^{2}+ 10β + 5)x^{4}− 10x^{2}− 10(β − 1)xi
f_{1}(x) = −1

60(β − 1)β^{2}x^{2}
h

12(β − 1)^{2}+ (−15β^{3}− 77β^{2}+ 39β + 13)x^{5}

+ 10(2β^{3}+ 7β^{2}− 6β − 3)x^{4} − 10(β^{2}− 5)x^{2}− 5(β^{3}− β^{2}− 9β + 9)xi
f_{2}(x) = −1

60(β − 1)β^{2}x^{2}
h

12(β − 1)^{2}+ (45β^{3}− 57β^{2}− 21β − 7)x^{5}

+ 10(−6β^{3}+ 3β^{2}+ 2β + 1)x^{4}+ 10(3β^{2}+ 1)x^{2}+ 5(3β^{3}− 3β^{2}+ 5β − 5)xi
f_{3}(x) = 1

120(β − 1)β^{3}x^{2}
h

6(β − 1)^{2}(β^{2}− 3) + (64β^{4}− 33β^{3}− 3β^{2}+ 9β + 3)x^{5}
+ 20β^{2}(−3β^{2}+ 2β + 1)x^{4}− 10(β^{4}+ 3)x^{2}+ 5(β^{3}− β^{2}− 9β + 9)xi
f_{4}(x) = −1

6β^{2}x(β + 1)(x − 1)(2β + 2βx^{2}+ x^{2}+ 2βx + x − 2)
f_{5}(x) = − 1

60β^{3}x^{2}
h

(β + 1)(6(β − 1)^{2}+ (64β^{2} − 3β − 1)x^{5}− 60β^{2}x^{4}

− 10(β^{2}− 1)x^{2}+ 15(β − 1)x)i
f_{6}(x) = 1

60(β − 1)β^{3}x^{2}

h−6(β − 1)^{2}(β^{2}+ 1) + (−64β^{4}− 27β^{3}+ 127β^{2}+ 3β + 1)x^{5}
+ 20β^{2}(3β^{2}+ 2β − 5)x^{4}+ 10(β^{4}− 4β^{2} − 1)x^{2}− 5(5β^{3}− 5β^{2}+ 3β − 3)xi

,

Now f_{SME} is given by

f_{SME} = − f_{1}(x)(Z_{r}^{01}cos α + Z_{r}^{02}sin α) − f_{2}(x)(Z_{r}^{10}cos α + Z_{r}^{20}sin α)
+ f3(x)

(Z_{r}^{12}+ Z_{r}^{21}) sin 2α + 1

2(Z_{r}^{11}− Z_{r}^{22}) cos 2α

+ f_{4}(x) (Z_{i}^{23}− Z_{i}^{32}) cos α + (Z_{i}^{31}− Z_{i}^{13}) sin α
+ f_{0}(x)Z_{r}^{00}+ f_{5}(x)Z_{r}^{33}+ f_{6}(x)1

2(Z_{r}^{11}+ Z_{r}^{22}),

(30)

where r and i stand for real and imaginary parts respectively, and α is the az- imuthal angle of the muon momentum in the laboratory frame.

Now in the limit β → 1, f4 and f5go to zero, and the others to infinity, because of the β − 1 factors in the denominators. First multiplying out this factor, then taking the limit β → 1, and finally multiplying by β − 1 again, results in the function

f_{c}(x) = 2
3

1 − x^{3}

1 − β, (31)

which f_{0}, f_{1}, f_{2}, f_{3} and f_{6} all approach as β → 1. The subscript c is used to
suggest muon velocity approaching the speed of light. Figure 8 shows a graph of
fc and f0 at an example γ = 29.3 as used by the Muon g − 2 experiment[11]. For
high γ, the ability to substitute f_{c} represents a drastic simplification of f_{SME}.

0.2 0.4 0.6 0.8 1.0 *x*

200 400 600 800 1000

*f HxL*

Fig. 8: Plot of fc(black) and f_{0} (dotted gray) for an example γ = 29.3, β = 0.9994.

The other functions f_{1}, f_{2}, f_{3} and f_{6} are visually indistinguishable from f_{0}
at this value of β, so this graph can be taken to represent them all.

For experimentation it is interesting to look at the behaviour of f when inte-
grated from a threshold value x = x_{t} to the maximum x = 1, because often there
is some energy value below which the detector is not sensitive. In the limit β → 1,
the integral of the functions f_{0}, f_{1}, f_{2}, f_{3} and f_{6} will approach the integral of f_{c},
which is given by,

F_{c}(x_{t}) =
Z 1

xt

f_{c}(x) dx = (x_{t}− 1)^{2}(x^{2}_{t} + 2x_{t}+ 3)

6(1 − β) . (32)

A graph of F_{c} is shown in Figure 9.

0.2 0.4 0.6 0.8 1.0 *x*_{t}

200 400 600 800

*F*_{c}*Hx*_{t}L

Fig. 9: Plot of Fc (black) and the integral of f0 (gray dotted) for an example γ = 29.3, β = 0.9994.

Integrating f_{4} and f_{5} directly and then taking the limit β → 1 gives
F_{4c}(x_{t}) = lim

β→1

Z 1 xt

f_{4}(x) dx = 1

3(x_{t}− 1)^{2}(x_{t}+ 2) (33)
F_{5c}(x_{t}) = lim

β→1

Z 1 xt

f_{5}(x) dx = 1

6(x_{t}− 1)^{2}(3x^{2}_{t} + 2x_{t}+ 1). (34)
In the same way, the integral of f_{SM} becomes

F_{c,SM}(x_{t}) = lim

β→1

Z 1 xt

f_{SM}(x) dx = 1

3(x_{t}− 1)^{2}(3 + x_{t}− x^{2}_{t}). (35)

These functions are plotted in Figure 10.

0.2 0.4 0.6 0.8 1.0 *x*_{t}

0.2 0.4 0.6 0.8 1.0

*FHx*_{t}L

Fig. 10: Plot of F_{4c} (black), F_{5c} (gray) and F_{c,SM} (dotted black) for an example
γ = 29.3, β = 0.9994.

As Fc is much greater than Fc,SM, it is not very informative to plot both func- tions in the same graph, so Figure 11 plots their ratio instead.

Since the f_{SME} functions will vary with the sidereal time of the experiment, and
its orientation and muon velocity, by doing experiments where these properties
vary and comparing the results with the ratios between the f_{SME} functions and
f_{SM}, the magnitudes of the Z^{µν} parameters can be measured.

0.2 0.4 0.6 0.8 1.0 *x*_{t}

500 1000 1500

*F*_{c}*Hx*_{t}L
*F*_{c,SM}*Hx*_{t}L

Fig. 11: Plot of Fc/Fc,SM.

### 4.2 Varying Detector Orientation

In the limit β → 1, f_{4} and f_{5} go to zero, and the other functions in (30) are
replaced by f_{c}. What remains is

f_{c,SME}(x) = f_{c}(x)

Z_{r}^{00}− (Z_{r}^{01}+ Z_{r}^{10}) cos α − (Z_{r}^{02}+ Z_{r}^{20}) sin α
+ 1

2Z_{r}^{11}(1 + cos 2α) + 1

2Z_{r}^{22}(1 − cos 2α) + (Z_{r}^{12}+ Z_{r}^{21}) sin 2α

.

(36)

It is readily seen that a substitution α → α + π leaves the sines and cosines of 2α
unaffected, while the sines and cosines of α pick up a minus sign. This means that
a comparison of the decay rate between two opposite detectors would be sensitive
to Z_{r}^{01}, Z_{r}^{02}, Z_{r}^{10} and Z_{r}^{20}, but not the other parameters. Similar observations
can be made for other choices of the angle between two detectors. For example,
with a 90 degree angle (α → α + π/2) the 2α terms pick up a minus sign, while
cos α → − sin α and sin α → cos α. To measure values for each different parameter,
they would need to be isolated from the others. Since most of them have a different
dependence on α, this can be exploited to partially isolate sets of parameters and
gain information about them. In principle, using data for a whole spectrum of α
values, most of the parameters in this equation could be isolated from each other.

The threshold integral of (28) in the limit β → 1 is just F_{c,SM}plus the integral
of (36):

F (x_{t}) = Fc,SM(x_{t}) + Fc(x_{t})

Z_{r}^{00}− (Z_{r}^{01}+ Z_{r}^{10}) cos α − (Z_{r}^{02}+ Z_{r}^{20}) sin α
+ 1

2Z_{r}^{11}(1 + cos 2α) + 1

2Z_{r}^{22}(1 − cos 2α) + (Z_{r}^{12}+ Z_{r}^{21}) sin 2α

.

(37)

Dividing by Fc,SMresults in an expression for the relative magnitude of the SM and
SME terms, here presented for an example energy threshold of 1.8 GeV (x_{t}= 0.58)
as in [11]:

F (0.58)

F_{c,SM}(0.58) = 1 + 1189.72

Z_{r}^{00}− (Z_{r}^{01}+ Z_{r}^{10}) cos α − (Z_{r}^{02}+ Z_{r}^{20}) sin α
+ 1

2Z_{r}^{11}(1 + cos 2α) + 1

2Z_{r}^{22}(1 − cos 2α) + (Z_{r}^{12}+ Z_{r}^{21}) sin 2α

.

(38)

### 4.3 Transformed X

^{µν}

### parameters

Using (27), the transformations of X^{µν}, χ^{µν} and Z^{µν} are known. As a reference
for relating quantities measured in the laboratory frame to the X^{µν} parameters of

the inertial reference frame, a conversion table is provided in [C]. When statistical
analysis on experiments at different times, orientations and/or muon velocities
is done to obtain bounds for the Z^{µν} parameters, converting coordinate systems
can then be done to obtain bounds for the X^{µν} parameters, which are the more
fundamental quantity of interest.

### 5 Discussion & Conclusion

The standard model extended version of the muon decay rate in its rest frame was
transformed to the laboratory frame and integrated to give an energy spectrum
probability density function. The Lorentz invariance violating χ^{µν}parameters were
transformed to the laboratory frame as well – appearing as Z^{µν} – and a coefficient
function was found for each parameter of Z^{µν}that remained in the energy spectrum
function after integrating over spin and electron momentum. In the limit for β
approaching 1, the coefficient functions were found to become delightfully simple.

Knowledge of the form with which the Z^{µν} parameters occur in the energy
distribution, and how this is related to the orientation of the electron detector
(through the momentum direction angle α of the decaying muons) can allow an
experimenter to derive values for the Z^{µν} parameters. By varying detector orienta-
tion and sidereal time (which relates to the orientation of the experiment through
the rotation of the Earth), and potentially even varying the muon velocity, bounds
for the parameters can be obtained though statistical analysis.

To relate experimental data to other experiments around the world, or over
longer periods of time, this study has additionally described how to transform Z^{µν}
to the parameters X^{µν} in the inertial reference frame, which are the fundamental
quantities of interest (unlike the locally valid χ^{µν} and Z^{µν} parameters).

### 5.1 Further Research

There are two clear avenues for further research. First, the depth of the theoretical analysis could be expanded. Summing over all spin contributions that occur in the muon decay rate, while drastically simplifying analysis, also loses a lot of information. Additionally, integrating over the electron direction in the laboratory frame discards the possibility of obtaining extra information from a detector that is sensitive to the electron direction. There is much left that could be analyzed in a more extensive approach which is beyond the scope of this paper.

Second, the formulae presented here have not been tied to a specific exper-
iment for measurement. New experiments could be executed, or existing data
reanalyzed, to obtain values for the Z^{µν} parameters analyzed here, which can then
be transformed to provide data on X^{µν}.

### Appendix A Derivation of Lorentz Boost Change of Variables

Starting with the parameterization from (20) ˆ

p^{∗} = (cos θ^{∗}, sin θ^{∗}cos φ^{∗}, sin θ^{∗}sin φ^{∗})
ˆ

p = (cos θ, sin θ cos φ, sin θ sin φ) , (39) and the Lorentz boost from (14) and (19),

(x^{∗}E_{max}^{∗} , x^{∗}E_{max}^{∗} pˆ^{∗}) =

γ −βγ 0 0

−βγ γ 0 0

0 0 1 0

0 0 0 1

(E, E ˆp), (40)

explicitly it becomes:

x^{∗}E_{max}^{∗}

1
cos θ^{∗}
sin θ^{∗}cos φ^{∗}
sin θ^{∗}sin φ^{∗}

=

γE(1 − β cos θ) γE(cos θ − β)

E sin θ cos φ E sin θ sin φ

. (41)

The boost along the x-axis does not affect the y and z components, so it follows
that φ^{∗} = φ. From the time component of the four-vector equation it is immediately
seen that

x^{∗} = E

E_{max}^{∗} γ (1 − β cos θ) . (42)
Substituting this into the equation for the x component gives

cos θ^{∗} = cos θ − β

1 − β cos θ, (43)

which through the identity cos^{2}θ^{∗}+ sin^{2}θ^{∗} = 1 also implies
sin θ^{∗} = sin θ

γ(1 − β cos θ). (44)

Now by the chain rule,

d sin θ^{∗}

dθ = d sin θ^{∗}
dθ^{∗}

dθ^{∗}

dθ = cos θ^{∗}dθ^{∗}

dθ. (45)

Substituting the previous two equations and then solving for ^{dθ}_{dθ}^{∗}gives

∂θ^{∗}

∂θ = 1

γ(1 + β cos θ). (46)

It is readily seen from (42) that

∂ x^{∗}

∂E = 1

E_{max}^{∗} γ(1 + β cos θ)

∂x^{∗}

∂θ = E

E_{max}^{∗} γ(1 − β sin θ).

(47)

Furthermore, it is trivially seen that ^{∂φ}_{∂φ}^{∗}= 1 and all other partial derivatives
between coordinates are 0. Therefore the Jacobian determinant is

∂x^{∗}

∂E

∂x^{∗}

∂θ

∂x^{∗}

∂φ

∂θ^{∗}

∂E

∂θ^{∗}

∂θ

∂θ^{∗}

∂φ

∂φ^{∗}

∂E

∂φ^{∗}

∂θ

∂φ^{∗}

∂φ

=

1

E_{max}^{∗} γ(1 + β cos θ) _{E}^{E}∗

maxγ(1 − β sin θ) 0

0 γ(1+β cos θ)^{1} 0

0 0 1

= 1

E_{max}^{∗} , (48)

which finally implies that the volume element change of variables is
sin θ^{∗}dθ^{∗}dφ^{∗}dx^{∗}= 1

E_{max}^{∗}

sin θ

γ(1 − β cos θ)dθ dφ dE. (49)

### Appendix B Exchanging Integration Order of E and θ

0 ^{Π}

2 Π

Θ 0

*Emax**
1+ Β
*Emax**

1- Β
*E*

Fig. 12: Integration region of E and θ in the laboratory frame

Integrating the probability density function g(x^{∗}, ˆp^{∗}) defined in (15) in the muon
rest frame is done by

Z π 0

Z 1 0

Z 2π 0

g(x^{∗}, ˆp^{∗}) sin θ^{∗}dφ^{∗}dx dθ^{∗}. (50)
Because E = x^{∗}E_{max}^{∗} /γ(1 − β cos θ) reaches its minimum value at E = x^{∗} = 0 and
its maximum at x^{∗} = 1, E = E_{max}^{∗} /γ(1 − β cos θ), in the laboratory frame this
integral becomes[A]

Z π 0

Z E_{max}^{∗} /γ(1−β cos θ)
0

Z 2π 0

g(x^{∗}, ˆp^{∗}) 1
E_{max}^{∗}

sin θ

γ(1 − β cos θ)dφ dE dθ. (51)
Figure 12 shows a graph of the integration region. When integrating over θ first,
the lower integration limit is seen to be simply θ = 0, but the upper integration
limit is θ = π for E < E_{max}^{∗} /(1 + β), and the inverse of E = E_{max}^{∗} /γ(1 − β cos θ)
otherwise. This leads to the form seen in (22).

### Appendix C Tables for the X

^{µν}

### Parameters Real Part of X

^{µν}

X_{r}^{00}= Z_{r}^{00}

X_{r}^{01}= Z_{r}^{02}sin + cos (Z_{r}^{01}cos φ − Z_{r}^{03}sin φ)
X_{r}^{02}= Z_{r}^{02}cos + sin (Z_{r}^{03}sin φ − Z_{r}^{01}cos φ)
X_{r}^{03}= Z_{r}^{03}cos φ + Z_{r}^{01}sin φ

X_{r}^{10}= Z_{r}^{20}sin + cos (Z_{r}^{10}cos φ − Z_{r}^{30}sin φ)

X_{r}^{11}= Z_{r}^{11}cos^{2}φ − (Z_{r}^{13}+ Z_{r}^{31}) sin φ cos φ + Z_{r}^{33}sin^{2}φ cos^{2}
+ sin ((Z_{r}^{12}+ Z_{r}^{21}) cos φ − (Z_{r}^{23}+ Z_{r}^{32}) sin φ) cos + Z_{r}^{22}sin^{2}
X_{r}^{12}= (Z_{r}^{12}cos φ − Z_{r}^{32}sin φ) cos^{2} + sin^{2} (Z_{r}^{23}sin φ − Z_{r}^{21}cos φ)

+ sin −Z_{r}^{11}cos^{2}φ + (Z_{r}^{13}+ Z_{r}^{31}) sin φ cos φ − Z_{r}^{33}sin^{2}φ + Z_{r}^{22} cos
X_{r}^{13}= cos Z_{r}^{13}cos^{2}φ + (Z_{r}^{11}− Z_{r}^{33}) sin φ cos φ − Z_{r}^{31}sin^{2}φ

+ sin (Z_{r}^{23}cos φ + Z_{r}^{21}sin φ)

X_{r}^{20}= Z_{r}^{20}cos + sin (Z_{r}^{30}sin φ − Z_{r}^{10}cos φ)

X_{r}^{21}= (Z_{r}^{21}cos φ − Z_{r}^{23}sin φ) cos^{2} + sin^{2} (Z_{r}^{32}sin φ − Z_{r}^{12}cos φ)

+ sin −Z_{r}^{11}cos^{2}φ + (Z_{r}^{13}+ Z_{r}^{31}) sin φ cos φ − Z_{r}^{33}sin^{2}φ + Z_{r}^{22} cos
X_{r}^{22}= Z_{r}^{22}cos^{2} − sin ((Z_{r}^{12}+ Z_{r}^{21}) cos φ − (Z_{r}^{23}+ Z_{r}^{32}) sin φ) cos

+ sin^{2} Z_{r}^{11}cos^{2}φ − (Z_{r}^{13}+ Z_{r}^{31}) sin φ cos φ + Z_{r}^{33}sin^{2}φ
X_{r}^{23}= sin −Z_{r}^{13}cos^{2}φ − (Z_{r}^{11}− Z_{r}^{33}) sin φ cos φ + Z_{r}^{31}sin^{2}φ

+ cos (Z_{r}^{23}cos φ + Z_{r}^{21}sin φ)
X_{r}^{30}= Z_{r}^{30}cos φ + Z_{r}^{10}sin φ

X_{r}^{31}= cos Z_{r}^{31}cos^{2}φ + (Z_{r}^{11}− Z_{r}^{33}) sin φ cos φ − Z_{r}^{13}sin^{2}φ
+ sin (Z_{r}^{32}cos φ + Z_{r}^{12}sin φ)

X_{r}^{32}= sin −Z_{r}^{31}cos^{2}φ − (Z_{r}^{11}− Z_{r}^{33}) sin φ cos φ + Z_{r}^{13}sin^{2}φ
+ cos (Z_{r}^{32}cos φ + Z_{r}^{12}sin φ)

X_{r}^{33}= Z_{r}^{33}cos^{2}φ + (Z_{r}^{13}+ Z_{r}^{31}) sin φ cos φ + Z_{r}^{11}sin^{2}φ