Investigating Lorentz Invariance Violation in Muon Decay

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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^ndcorrector: prof. dr. R. G. E. Timmermans

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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 general 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 magnitude 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.

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) coordinate system.

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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







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R_y(θ) =







1 0 0 0

0 cos θ 0 sin θ

0 0 1 0

0 − sin θ 0 cos θ







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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_x²

β² (γ − 1)β_xβ_y

β² (γ − 1)β_xβ_z β²

−γβ_y (γ − 1)β_xβ_y

β² 1 + (γ − 1)β_y²

β² (γ − 1)β_yβ_z β²

−γβ_z (γ − 1)β_xβ_z

β² (γ − 1)β_yβ_z

β² 1 + (γ − 1)β_z² β²







, (4)

where β = (β_x, β_y, β_z) = v/c with c the speed of light, β = |β| and γ = 1/p1 − β². 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)

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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 symmetry 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 spectrum 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 x²

"

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, parameterized in spherical coordinates as dΩ = sin θ dθ dφ, and x ∈ [0, 1] is the fractional

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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 energy x = E/E_max is defined in terms of the electron energy E and its maximum E_max= (m²_µ + m²_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

rm

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.

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The total decay rate is given by W₀ = G²_Fm⁵_µ/192π³, where G²_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 W₀ 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 propagator 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 = W₀ π x²

"

3(1 − x) +1

2(4x − 3) ∓ (ˆs · ˆp)

1 − x + 1

2(4x − 3)

−(1 − x)

t₁+ v₁· ˆp ± v₂· ˆs ± v₃· (ˆs × ˆp) +T₁^mlpˆ^mpˆ^l± T₂^mlpˆ^msˆ^l± T₃^mlpˆ^m(ˆs × ˆp)^l

−(4x − 3)

z₁+ u₁· ˆp ± u₂· ˆs ± u₃· (ˆs × ˆp) +H₁^mlpˆ^mpˆ^l± H₂^mlpˆ^msˆ^l± H₃^mlpˆ^m(ˆs × ˆp)^l

±(ˆs · ˆp)(1 − x)(t2+ v₄· ˆp) + (4x − 3)(z₂+ u₄· ˆp)

,

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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.

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

t₁ = z₁ = z₂ = 1 2χ⁰⁰_rs t2 = 5

2χ⁰⁰_rs v^l₁ = χ^0l_rs+ 2χ^0l_ra− 2^lmnχ^mn_ia

v^l₂ = 1

2χ^0l_rs+ 7

2χ^0l_ra+5

4^lmnχ^mn_ia v^l₃ = 3

2χ^0l_ia+ 5 2χ^0l_is v^l₄ = 3

2χ^0l_rs+ 3

2χ^0l_ra− 3

4^lmnχ^mn_ia

u^l₁ = −1

2^lmnχ^mn_ia u^l₂ = 1

2χ^0l_rs+1

2χ^0l_ra+ 1

4^lmnχ^mn_ia u^l₃ = 1

2χ^0l_ia+1 2χ^0l_is u^l₄ = 1

2χ^0l_rs+1

2χ^0l_ra− 1

4^lmnχ^mn_ia T₁^ml = −3

2χ^ml_rs T₂^ml = 7

2χ^ml_rs + 1 2χ^ml_ra T₃^ml = 3

2χ^ml_is

H₁^ml = −1 2χ^ml_rs H₂^ml = 1

2χ^ml_ra H₃^ml = 1

2χ^ml_is .

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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 ²₃(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) + H₁^mlpˆ^mpˆ^l± H₂^mlpˆ^msˆ^l± H₃^mlpˆ^m(ˆs × ˆp)^l

.

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Therefore any experiment that measures ρ necessarily measures these χ^µν parameters as well, and a significant deviation from ρ = ³₄ 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

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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 transformations, 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 parameters 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 ¹₃^◦ 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.

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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 (x⁰y⁰z⁰, in gray), which has its z⁰-axis pointing parallel to the Earth’s polar axis and x⁰-axis parallel to the equatorial plane.

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Fig. 4: The primed frame (x⁰y⁰z⁰, in black) fromFigure 3has its z⁰-axis parallel to the Earth’s polar axis and its x⁰- and y⁰-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 z⁰-axis, takes the coordinate axes to the double primed frame (x⁰⁰y⁰⁰z⁰⁰, in gray), which has its z⁰⁰-axis parallel to the polar axis, it’s x⁰⁰-axis pointing towards the Greenwich meridian, and it’s y⁰⁰-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.

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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 − β². 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.

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pm

a a

pm

p^¢_m M rm

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 → p⁰_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.

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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 considered. This reduces the equation to

g(x, ˆp) · W0 = dW

dx dΩ = W₀ π x²

(1 − x)

3 − t1− v1· ˆp − T₁^mlpˆ^mpˆ^l

+(4x − 3) 1

2− z₁− u₁· ˆp − H₁^mlpˆ^mpˆ^l

# .

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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

t₁ = z₁ = 1 2χ⁰⁰_rs

v^l₁ = χ^0l_rs+ 2χ^0l_ra− 2^lmnχ^mn_ia u^l₁ = −1

2^lmnχ^mn_ia T₁^ml = −3

2χ^ml_rs H₁^ml = −1 2χ^ml_rs,

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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.

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3.2 Boosting to Moving Muon Frame

As a first step towards the laboratory frame, the equation is boosted with a relativistic 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− m²_e pˆ^∗), (17) where the Minkowski norm for the four-momentum E²−|p|² = m² 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_νp^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.

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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 coordinates 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,

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where 1/E_max^∗ is the Jacobian determinant of the boost coordinate transformation.[A]

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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)

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EmaxHΒL 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.

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3.3 Transforming the χ

^µν

Parameters

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

χ^µν = T^µ_ρT^ν_σX^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_ρL^ν_σZ^ρσ

Z^µν = R^µ_ρR^ν_σX^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 parameters 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.

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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 function

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) 4x³− 9x²+ 5

6β . (29)

To define f_SME – the SME contribution – first the following polynomials must be defined:

f₀(x) = 1 15(β − 1)βx²

h−3(β − 1)²+ 2(9β²− 3β − 1)x⁵

+ (−15β²+ 10β + 5)x⁴− 10x²− 10(β − 1)xi f₁(x) = −1

60(β − 1)β²x² h

12(β − 1)²+ (−15β³− 77β²+ 39β + 13)x⁵

+ 10(2β³+ 7β²− 6β − 3)x⁴ − 10(β²− 5)x²− 5(β³− β²− 9β + 9)xi f₂(x) = −1

60(β − 1)β²x² h

12(β − 1)²+ (45β³− 57β²− 21β − 7)x⁵

+ 10(−6β³+ 3β²+ 2β + 1)x⁴+ 10(3β²+ 1)x²+ 5(3β³− 3β²+ 5β − 5)xi f₃(x) = 1

120(β − 1)β³x² h

6(β − 1)²(β²− 3) + (64β⁴− 33β³− 3β²+ 9β + 3)x⁵ + 20β²(−3β²+ 2β + 1)x⁴− 10(β⁴+ 3)x²+ 5(β³− β²− 9β + 9)xi f₄(x) = −1

6β²x(β + 1)(x − 1)(2β + 2βx²+ x²+ 2βx + x − 2) f₅(x) = − 1

60β³x² h

(β + 1)(6(β − 1)²+ (64β² − 3β − 1)x⁵− 60β²x⁴

− 10(β²− 1)x²+ 15(β − 1)x)i f₆(x) = 1

60(β − 1)β³x²

h−6(β − 1)²(β²+ 1) + (−64β⁴− 27β³+ 127β²+ 3β + 1)x⁵ + 20β²(3β²+ 2β − 5)x⁴+ 10(β⁴− 4β² − 1)x²− 5(5β³− 5β²+ 3β − 3)xi

,

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Now f_SME is given by

f_SME = − f₁(x)(Z_r⁰¹cos α + Z_r⁰²sin α) − f₂(x)(Z_r¹⁰cos α + Z_r²⁰sin α) + f3(x)

(Z_r¹²+ Z_r²¹) sin 2α + 1

2(Z_r¹¹− Z_r²²) cos 2α

+ f₄(x) (Z_i²³− Z_i³²) cos α + (Z_i³¹− Z_i¹³) sin α + f₀(x)Z_r⁰⁰+ f₅(x)Z_r³³+ f₆(x)1

2(Z_r¹¹+ Z_r²²),

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where r and i stand for real and imaginary parts respectively, and α is the azimuthal 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 − x³

1 − β, (31)

which f₀, f₁, f₂, f₃ and f₆ 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 f₀ (dotted gray) for an example γ = 29.3, β = 0.9994.

The other functions f₁, f₂, f₃ and f₆ are visually indistinguishable from f₀ at this value of β, so this graph can be taken to represent them all.

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For experimentation it is interesting to look at the behaviour of f when integrated 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 f₀, f₁, f₂, f₃ and f₆ 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(x²_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_cHx_tL

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

Integrating f₄ and f₅ directly and then taking the limit β → 1 gives F_4c(x_t) = lim

β→1

Z 1 xt

f₄(x) dx = 1

3(x_t− 1)²(x_t+ 2) (33) F_5c(x_t) = lim

β→1

Z 1 xt

f₅(x) dx = 1

6(x_t− 1)²(3x²_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− x²_t). (35)

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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_tL

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 functions 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_cHx_tL F_c,SMHx_tL

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

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4.2 Varying Detector Orientation

In the limit β → 1, f₄ and f₅ 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_r⁰⁰− (Z_r⁰¹+ Z_r¹⁰) cos α − (Z_r⁰²+ Z_r²⁰) sin α + 1

2Z_r¹¹(1 + cos 2α) + 1

2Z_r²²(1 − cos 2α) + (Z_r¹²+ Z_r²¹) sin 2α

.

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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_r⁰¹, Z_r⁰², Z_r¹⁰ and Z_r²⁰, 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,SMplus the integral of (36):

F (x_t) = Fc,SM(x_t) + Fc(x_t)

2Z_r¹¹(1 + cos 2α) + 1

2Z_r²²(1 − cos 2α) + (Z_r¹²+ Z_r²¹) sin 2α

.

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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

2Z_r¹¹(1 + cos 2α) + 1

2Z_r²²(1 − cos 2α) + (Z_r¹²+ Z_r²¹) sin 2α

.

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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

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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^µνparameters were transformed to the laboratory frame as well – appearing as Z^µν – and a coefficient function was found for each parameter of Z^µνthat 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 orientation 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 experiment 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.

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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 cos²θ^∗+ sin²θ^∗ = 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)

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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).

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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)

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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).

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Appendix C Tables for the X

^µν

Parameters Real Part of X

^µν

X_r⁰⁰= Z_r⁰⁰

X_r⁰¹= Z_r⁰²sin + cos (Z_r⁰¹cos φ − Z_r⁰³sin φ) X_r⁰²= Z_r⁰²cos + sin (Z_r⁰³sin φ − Z_r⁰¹cos φ) X_r⁰³= Z_r⁰³cos φ + Z_r⁰¹sin φ

X_r¹⁰= Z_r²⁰sin + cos (Z_r¹⁰cos φ − Z_r³⁰sin φ)

X_r¹¹= Z_r¹¹cos²φ − (Z_r¹³+ Z_r³¹) sin φ cos φ + Z_r³³sin²φ cos² + sin ((Z_r¹²+ Z_r²¹) cos φ − (Z_r²³+ Z_r³²) sin φ) cos + Z_r²²sin² X_r¹²= (Z_r¹²cos φ − Z_r³²sin φ) cos² + sin² (Z_r²³sin φ − Z_r²¹cos φ)

+ sin −Z_r¹¹cos²φ + (Z_r¹³+ Z_r³¹) sin φ cos φ − Z_r³³sin²φ + Z_r²² cos X_r¹³= cos Z_r¹³cos²φ + (Z_r¹¹− Z_r³³) sin φ cos φ − Z_r³¹sin²φ

+ sin (Z_r²³cos φ + Z_r²¹sin φ)

X_r²⁰= Z_r²⁰cos + sin (Z_r³⁰sin φ − Z_r¹⁰cos φ)

X_r²¹= (Z_r²¹cos φ − Z_r²³sin φ) cos² + sin² (Z_r³²sin φ − Z_r¹²cos φ)

+ sin −Z_r¹¹cos²φ + (Z_r¹³+ Z_r³¹) sin φ cos φ − Z_r³³sin²φ + Z_r²² cos X_r²²= Z_r²²cos² − sin ((Z_r¹²+ Z_r²¹) cos φ − (Z_r²³+ Z_r³²) sin φ) cos

+ sin² Z_r¹¹cos²φ − (Z_r¹³+ Z_r³¹) sin φ cos φ + Z_r³³sin²φ X_r²³= sin −Z_r¹³cos²φ − (Z_r¹¹− Z_r³³) sin φ cos φ + Z_r³¹sin²φ

+ cos (Z_r²³cos φ + Z_r²¹sin φ) X_r³⁰= Z_r³⁰cos φ + Z_r¹⁰sin φ

X_r³¹= cos Z_r³¹cos²φ + (Z_r¹¹− Z_r³³) sin φ cos φ − Z_r¹³sin²φ + sin (Z_r³²cos φ + Z_r¹²sin φ)

X_r³²= sin −Z_r³¹cos²φ − (Z_r¹¹− Z_r³³) sin φ cos φ + Z_r¹³sin²φ + cos (Z_r³²cos φ + Z_r¹²sin φ)

X_r³³= Z_r³³cos²φ + (Z_r¹³+ Z_r³¹) sin φ cos φ + Z_r¹¹sin²φ

Investigating Lorentz Invariance Violation in Muon Decay