Reaching the Planck scale with muon lifetime measurements

Abstract

Planck scale modified dispersion relations are one way to capture the influence of quantum gravity on the propagation of fundamental point particles effectively. We derive the time dilation between an observer’s or particle’s proper time, given by a Finslerian length measure induced from a modified dispersion relation, and a reference laboratory time. To do so, the Finsler length measure for general first order perturbations of the general relativistic dispersion relation is constructed explicitly. From this we then derive the time dilation formula for the κ-Poincaré dispersion relation in several momentum space bases, as well as for modified dispersion relations considered in the context of string theory and loop quantum gravity. Most interestingly we find that the momentum Lorentz factor in the present and future colliders can, in principle, become large enough to constrain the Finsler realization of the κ-Poincaré dispersion relation in the bicrossproduct basis as well as a string theory inspired modified dispersion relation, at Planck scale sensitivity with the help of the muon’s lifetime.

Full text

Reaching the Planck scale with muon lifetime measurements
Iarley P. Lobo 1,2,* and Christian Pfeifer 3,4,†
1Department of Chemistry and Physics, Federal University of Paraíba,
Rodovia BR 079—km 12, 58397-000 Areia-PB, Brazil
2Physics Department, Federal University of Lavras, Caixa Postal 3037, 37200-900 Lavras-MG, Brazil
3ZARM, University of Bremen, 28359 Bremen, Germany
4Laboratory of Theoretical Physics, Institute of Physics, University of Tartu,
W. Ostwaldi 1, 50411 Tartu, Estonia
(Received 1 December 2020; accepted 6 May 2021; published 28 May 2021)
Planck scale modified dispersion relations are one way to capture the influence of quantum gravity on
the propagation of fundamental point particles effectively. We derive the time dilation between an
observer’s or particle’s proper time, given by a Finslerian length measure induced from a modified
dispersion relation, and a reference laboratory time. To do so, the Finsler length measure for general first
order perturbations of the general relativistic dispersion relation is constructed explicitly. From this we then
derive the time dilation formula for the κ-Poincar ´e dispersion relation in several momentum space bases, as
well as for modified dispersion relations considered in the context of string theory and loop quantum
gravity. Most interestingly we find that the momentum Lorentz factor in the present and future colliders
can, in principle, become large enough to constrain the Finsler realization of the κ-Poincar ´e dispersion
relation in the bicrossproduct basis as well as a string theory inspired modified dispersion relation, at
Planck scale sensitivity with the help of the muon’s lifetime.
DOI: 10.1103/PhysRevD.103.106025
I. INTRODUCTION
A major difficulty in the search for quantum gravity
effects is that the scale at which they are expected to
become relevant is at the Planck energy EPof order
1019 GeV, respectively at distance of the Planck length
lPof order 10−35 m. Thus, in order to detect Planck scale
effects one either needs to reach very high energies, or,
probe very small lengths scales.
In the absence of a complete theory of quantum gravity,
phenomenological models which shall capture aspects of
quantum gravity often employ Planck scale modified
dispersion relations (MDRs) to effectively capture the
interaction of particles, propagating through spacetime,
with the quantum nature of gravity [1–3]. Such dispersion
relations predict a deviation of particle trajectories from the
general relativistic geodesics, with a leading order term in
powers of the inverse Planck energy. Thus, MDRs lead to
tiny corrections of the predictions of general relativity
(GR), which are in principle detectable. To be able to detect
these effects realistically, they need to be amplified, for
example through accumulation over a long travel time of
the particles. One observable, which meets this requirement
and is accessible, is the time of arrival of high energetic
gamma rays reaching us from gamma ray bursts at high
redshift, for which Planck scale MDRs predict a depend-
ence on the particles’energy [4–8].
Recently, it was pointed out that in the comparison of
lifetimes of particles and antiparticles (in particular for
muons) Planck scale sensitivity for κdeformations of the
Poincar´e algebra [9–11] is at reach [12,13]. In these
considerations the momentum Lorentz factor attained at
particle accelerators plays the role of amplifier of the
Planck scale effect.
Inspired by the promising findings of reaching Planck
scale sensitivity with muons, we study the lifetime of
elementary particles and the time dilation between their rest
frame and a laboratory frame, induced by Planck scale
MDRs. Following the famous clock postulate, that the
proper time an observer measures between two events in its
rest frame is the length of the observer’s worldline between
these events, the first ingredient necessary for our study is
the length measure for wordlines induced by a MDR. In
general, this will be a Finslerian length measure [14–18],
i.e., a function Fðx; _
xÞdepending on position and the
velocity of an observer, which is 1-homogeneous with
*lobofisica@gmail.com,iarley_lobo@fisica.ufpb.br
†christian.pfeifer@zarm.uni-bremen.de
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PHYSICAL REVIEW D 103, 106025 (2021)
2470-0010=2021=103(10)=106025(8) 106025-1 Published by the American Physical Society

respect to the 4-velocity argument _
x. Assuming a “flat”
dispersion relation results in a flat length measure, i.e.,
independent of x, from which the derivation of the proper
time along a worldline as function of the lab coordinate
time can be done explicitly.
We derive the time dilation formula between laboratory
frame and particle rest frame for the most general MDR in
first order perturbations of the GR dispersion relation
explicitly, and apply the formula to several Planck scale
MDRs motivated from the κ-Poincar´e algebra, string
theory, and loop quantum gravity (LQG).
For the derivation we assume that the dispersion relation
and the resulting length measure is universal for all massive
physical objects, so in particular for the muon and the
observer. On a curved spacetime, this ensures the imple-
mentation of the weak equivalence principle, i.e., that
gravity couples universally to all physical objects, since
the dispersion relation encodes the coupling between
gravity and point particles (or the point particle limit of
fields). Moreover, transformations which leave the
dispersion relation invariant then are candidates for
observer transformations. For MDRs, these are modified
Lorentz transformations. Hence, we are not analyzing
Lorentz invariance violating (LIV) scenarios, for which
it is assumed that observers are still related by Lorentz
transformations.
Throughout our calculations we carefully distinguish
between the velocity Lorentz factor γ¼1
ffiffiffiffiffiffiffiffi
1−v2
pand the
momentum Lorentz factor ¯γ¼p0
m, which in general do
not coincide in the context MDRs.
Most interestingly, we explicitly calculate the time
dilation of the lifetime of muons with energies available
in facilities like the Large Hadron Collider (LHC), or the
planned Future Circular Collider (FCC), at CERN, from the
κ-Poincar´e dispersion relation in the bicrossproduct basis
(which is identically to the MDR derived from a D-brane
string theory model). On the basis of a deformed relativity
principle, which is implied by the Finsler geometric treat-
ment of the modified dispersion relation,1we find that the
deformation parameter, κor MQ, could be constrained by
muon lifetime measurements at the colliders at the order of
magnitude of the Planck energy, thus reaching Planck scale
sensitivity for this quantum gravity phenomenology model
with muon lifetimes.
In this article ηdenotes the Minkowski metric
diagðþ;−;−;−Þ, indices a; b; c; …run from 0 to 3 and
indices i; j; k; …run from 1 to 3. The symbol ¯
∂a¼∂
∂pa
denotes derivative with respect to momentum coordinates.
II. THE TIME MEASURE FROM GENERAL
MODIFIED DISPERSION RELATIONS
We briefly review the mathematical procedure to obtain a
time measure from MDRs, before we apply the procedure
to general first order modifications of the GR dispersion
relation.
A. The general algorithm
Point particle dispersion relations are level sets of
Hamilton functions Hðx; pÞon the point particle phase
space, technically the cotangent bundle TMof spacetime.
To associate a time measure to massive point particles,
respectively observers, from the dispersion relation one
employs the Helmholtz action of free particles [14–20]
S½x; p; λH¼Zdμð_
xapa−λfðHðx; pÞ;mÞÞ;ð1Þ
where μis an arbitrary curve parameter, “dot”means
derivative with respect to this parameter, fis a function
such that f¼0is equivalent to the dispersion relation
Hðx; pÞ¼m2, and λis a Lagrange multiplier.
To obtain a length measure for massive particle trajec-
tories from this action we use the following algorithm:
(1) Variation with respect to λenforces the dispersion
relation.
(2) Variation with respect to payields an equation _
xa¼
_
xaðp; λÞwhich must be inverted to obtain paðx; _
x; λÞ
to eliminate the momenta pafrom the action.
(3) Using paðx; _
x; λÞin the dispersion relation, one can
solve for λðx; _
xÞ.
(4) Finally the desired length measure is obtained
as S½x¼S½x; pðx; _
x; λðx; _
xÞÞ;λðx; _
xÞH.
The crucial step in this algorithm is to be able to find
paðx; _
x; λÞ, i.e., to invert the relation _
xa¼_
xaðp; λÞ.Ifthis
is globally possible, only locally, or not at all, depends on
the dispersion relation under consideration and the choice
of the function f[16,21]. Among others, choices
employed in the literature are fðH; mÞ¼lnðHðx; p
mÞÞ,
for homogeneous Hamiltonians [16],or,
f¼Hðx; pÞ−mn, often with n¼2[17,19], for nonho-
mogeneous Hamiltonians. Another suggestions for the
inhomogeneous case is fðH; mÞ¼lnðHðx;pÞ
mnÞ, which is not
yet investigated in detail.
B. First order modified dispersion relations
Let gbe a general Lorentzian metric and hðx; pÞbe a
function on the cotangent bundle TMof spacetime. A
general first order modification of the GR dispersion
relation is defined by the Hamilton function
Hðx; pÞ¼gðp; pÞþϵhðx; pÞ:ð2Þ
The term gðp; pÞ¼gabðxÞpapbdefines the GR point
particle dispersion relation on a curved spacetime, ϵis a
1Although the string theory case is related to a LIV scenario, in
contrast to the κ-Poincar´e one that deforms Lorentz symmetry, we
show that one can promote it to a deformed relativity scenario and
analyze the resulting muon lifetime phenomenology (an issue that
is further discussed in Sec. III).
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perturbation parameter, counting the first nontrivial cor-
rection to GR, and hðx; pÞa perturbation function, which
needs to be specified depending on the application in
consideration. In the context of quantum gravity pheno-
menology, ϵis usually related to the Planck length or
Planck energy and hðx; pÞcan, for example, be obtained
from Planck scale MDRs, such as the κ-Poincar´e dispersion
relation and others, whose influence we investigate in the
course of this paper.
The steps of the previously outlined algorithm can now
be performed as follows:
(1) Variation of the action (1) with respect to λyields
f¼0, which in turn enforces the dispersion relation
gðp; pÞþϵhðx; pÞ¼m2:ð3Þ
(2) Variation of the action (1) with respect to paand
using the perturbative Hamiltonian (2) yields
_
xa¼λ¯
∂aH∂Hf¼λð2paþϵ¯
∂ahÞ∂Hf; ð4Þ
which can be rewritten as (indices are raised on
lowered with the components of the metric g)
pa¼_
xa
2λ∂Hf−
1
2ϵgab ¯
∂bh: ð5Þ
Explicitly inverting this equation completely to
obtain paðx; _
x; λÞis not possible, but also not
necessary in perturbation theory, as we will
see soon. We note the following two relevant
relations:
_
xapa¼λð2gðp; pÞþϵpa¯
∂ahÞ∂Hf; ð6Þ
gðp; pÞ¼ gð_
x; _
xÞ
4λ2∂Hf2−ϵ_
xa¯
∂ah
2λ∂Hf;ð7Þ
where we introduced the notation gð_
x; _
xÞ¼
gabðxÞ_
xa_
xb.
(3) Using (7) in the dispersion relation (3), using a first
order expansion of the Lagrange multiplier λ¼λ0þ
ϵλ1and solving the dispersion relation order by order
leads to
λ0¼ffiffiffiffiffiffiffiffiffiffiffiffiffi
gð_
x; _
xÞ
p
2m∂Hf;λ1¼ffiffiffiffiffiffiffiffiffiffiffiffiffi
gð_
x; _
xÞ
p
4m3∂Hfh−
_
xa¯
∂ah
4m2∂Hf:
ð8Þ
(4) Combining all the results from (6), (7), and (8) in (1)
for the Hamiltonian (3) the action becomes
S½x¼Zdτffiffiffiffiffiffiffiffiffiffiffiffiffi
gð_
x; _
xÞ
pm−ϵh
2m:ð9Þ
At this order, the function fand its derivatives all
cancel and so the specific choice is not relevant.
The perturbation function happearing in (9)
has to be understood as h¼hðx; ¯
pðx; _
xÞÞ, with
¯
paðx; _
xÞ¼m_
xa
ffiffiffiffiffiffiffiffiffi
gð_
x;_
xÞ
p.
We have proven that the Finsler function which governs
the massive point particle motion of first order MDR is
Fðx; _
xÞ¼mffiffiffiffiffiffiffiffiffiffiffiffiffi
gð_
x; _
xÞ
p−ϵffiffiffiffiffiffiffiffiffiffiffiffiffi
gð_
x; _
xÞ
phðx; ¯
pðx; _
xÞÞ
2m:ð10Þ
As an example, consider an nth order polynomial
modification
hðx; pÞ¼ha1a2…:anðxÞpa1pa2…panð11Þ
⇒hðx; ¯
pðx; _
xÞÞ ¼ mnha1a2…:anðxÞ_
xa1_
xa2…_
xan
gð_
x; _
xÞn
2ð12Þ
which yields
Fðx; _
xÞ¼mffiffiffiffiffiffiffiffiffiffiffiffiffi
gð_
x; _
xÞ
p−ϵmn−1ha1a2…:anðxÞ_
xa1_
xa2…_
xan
2gð_
x; _
xÞn−1
2
:
ð13Þ
III. THE MUON LIFETIME FROM MODIFIED
DISPERSION RELATIONS
Next, we analyze how the lifetime of a fundamental
particle is modified by the assumption that it propagates on
a Finsler spacetime [22,23] induced by a MDR. The clock
postulate is implemented in the following way.
The proper time an observer, or massive particle,
experiences between events Aand Balong a timelike
curve (her worldline) in a Finsler spacetime ðM;FÞis the
length of this curve between events Aand B:
ΔτAB ≐m−1ZμB
μA
Fðx; _
xÞdμ:ð14Þ
We aim to investigate the decay of fundamental particles
in accelerators; therefore, we shall discard pure gravita-
tional effects, i.e., the spacetime curvature, and rely
on Finsler-deformations of Minkowski proper time.
Mathematically this is justified by the existence of special
coordinates, which allow one to neglect curvature effects at
small coordinate distance around every point and a given
direction on Finsler spacetimes [24]. Thus to zeroth order
we consider gð_
x; _
xÞin Eq. (13) as the usual Minkowski
metric, which we shall label ηð_
x; _
xÞ. In Cartesian coor-
dinates we simply write
ηð_
x; _
xÞ¼ð_
x0Þ2−δijð_
xiÞð_
xjÞ:ð15Þ
Since the arc length is invariant under reparametriza-
tions, we transform the arbitrary parameter μto the time
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coordinate in the laboratory frame, x0≐tin (14). Using
(13), we have the following modification of the proper time
between the events with parameters ðx0ÞA¼tAto ðx0ÞB¼
tB(from now on we omit the label “AB”in ΔτAB):
Δτ¼ZtB
tA
dtγ−1−ϵ
2mn−2γn−1ha1…an
dxa1
dt …dxan
dt ;ð16Þ
where, for convenience, we introduced the ususal velocity
Lorentz factor
γ¼1
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1−v2
pð17Þ
with vi≐dxi=dt and v2¼δijvivj.
Suppose a fundamental particle has its lifetime dilated in
a circular accelerator, like the LHC or the FCC [25]. In this
case, the norm of the three-dimensional velocity v2is
roughly a constant, which allows for a simplification of the
above expression. In the following, Δtwill be the time
measured in the laboratory frame in which the particle is
accelerated, while Δτis the proper time experienced by the
particle, respectively measured by an observer comoving to
the particle.
A. The κ-Poincar´e dispersion relation in bicrossproduct
basis type
For the κ-Poincar´e dispersion relation in the
bicrossproduct basis, the first order correction of the
quadratic GR Finsler function is a polynomial of degree
n¼3. The symbols ha1a2a3for this case are ha1a2a3¼
−1
3ðδ0
a1δijδi
a2δj
a3þδ0
a2δijδi
a1δj
a3þδ0
a3δijδi
a1δj
a2Þand null
otherwise; see for example [26]. Hence the integrand in
(16) becomes
γ−1þϵm
2γ2v2¼γ−11þϵm
2γðγ2−1Þ:ð18Þ
Moreover, ϵis a parameter expected to be of the order of the
inverse of the energy scale at which quantum gravitational
corrections are expected to take place, which we simply
denote as deformation parameter κ−1, as it is usually done
in the context of the κ-Poincar´e algebra.
Therefore, we express the lifetime of a fundamental
particle that probes a Finsler spacetime induced by the
κ-Poincar´e dispersion relation in the bicrossproduct basis as
(we define Δt≐tB−tA)
Δτ¼Δt
γ1þm
2κγðγ2−1Þ:ð19Þ
This geometric invariant quantity defined by Eq. (14)
measures the proper time a particle experiences, and is
related to the time which passes in the laboratory, with
respect to which the particle is accelerated. Thus the
measured lifetime of a particle in a laboratory, denoted
by Δt, can be related to the proper lifetime of the particle
Δτ, depending on its coordinate velocity vthrough the
factor γ. To first order κ−1, we find for the laboratory frame
lifetime of the particle
Δt¼γΔτ1−m
2κγðγ2−1Þ:ð20Þ
In order to compare with data from particle accelerators,
we need to express the velocity γfactor defined in (17) in
terms of the energy p0and mass mof the particles. The
conversion between these dependencies is nontrivial due to
the deviations from the usual relativistic setting. We
derive the 4-momentum of the particles, which satisfies
the MDR:
p0¼∂
∂_
x0Fðx; _
xÞ¼mγ−m2
2κðγ2−1Þð2γ2−1Þ;
pi¼∂
∂_
xiFðx; _
xÞ¼mγvi−1þm
κγð2−γ2Þ:ð21Þ
Solving the first relation for γas a function of p0yields
γ¼p0
mþm
2κð1−3p2
0
m2þ2p4
0
m4Þ. Employing this in (20) gives
us the lifetime as a function of p0
Δt¼ΔtSR1þm
2κm
p0
−2p0
mþp0
m3;ð22Þ
where ΔtSR ¼p0
mΔτis the usual special relativistic dilated
lifetime expressed in terms of the particle’sp0component.
This result leads us to introduce the momentum Lorentz
factor ¯γ¼p0
m. We would like to emphasize that for MDRs,
in general, the momentum Lorentz factor is different from
the velocity Lorentz factor, as we have demonstrated by the
derivation of the relation γð¯γÞ¼¯γþm
2κð1−3¯γ2þ2¯γ4Þ.
Before we continue we would like to point out a short
general comment on deformed Lorentz transformations. In
general, the 4-momentum defined as pa¼∂
∂_xaFsatisfies
the deformed dispersion relation. This can be proven by
using the Lagrange multiplier (8) in (5), to express paas a
function of _
x, which coincides with pa¼∂
∂_
xaFwhen using
Fas identified in (10). In fact, for a particle at rest (v¼0,
γ¼1), the dispersion relation implies for the momenta
p0¼mand pi¼0;i¼1, 2, 3. If we consider the
momenta as a function of γand apply the transformation
p0¼p0ð1Þ→p0
0¼p0ðγÞ¼ ∂
∂_
x0Fand
pi¼pið1Þ→p0
i¼piðγÞ¼ ∂
∂_
xiF; ð23Þ
then the dispersion relation Hðx; pÞ¼Hðx; p0Þ¼m2
is satisfied. Therefore, the transformation that links
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the 4-momentum of the particle at rest to the
4-momentum ðp0ðγÞ;p
iðγÞÞ represents a deformed
Lorentz transformation.
From (22), we are able to identify the dimensionless
quantity δp0;m, depending on the mass and energy of the
particles attained in accelerators, which is responsible for
an effect beyond special relativity and is the one which we
compare with the uncertainty of the most precise
experimental values of the mean lifetime of fundamental
particles:
δp0;m ¼m
2κð¯γ−1−2¯γþ¯γ3Þ≈m
2κ¯γ3:ð24Þ
In the last approximation we focused on the term which
dominates for high energetic particles.
For a concrete example, let us consider the case
of the muon particle. The muon mean lifetime amounts
to [27]
τμ¼ð2.1969811 0.0000022Þ×10−6
s¼2.1969811 μsστ;ð25Þ
and its most precise measurement was done for low energy
muons in [28]. From (25), we see that the relative
uncertainty of this measurement reads
στ=τμ≈10−6:ð26Þ
In the following, we shall explore the consequences of
assuming that experiments in the LHC or the FCC could
measure the muon lifetime with the same relative uncer-
tainty, which, as we shall demonstrate, would allow one to
set significant constraints on the quantum gravity energy
scale. An analogous prediction was done previously for
the case of the decays of the muon and antimuon, for a
κ-Poincar´e basis in which the Hamiltonian is undeformed,
and modifications take place when comparing the lifetimes
of particles and antiparticles in the context of CPT
violation [12,13].
As a matter of fact, had we used the same basis of [12],
i.e., with an undeformed Casimir operator as a Hamilton
function, we would have derived the standard Minkowski
metric, without Finsler modifications, thus producing no
effect beyond special relativity in the lifetime of particles
depending on their relative velocity. We should stress that
this is a general feature of the use of different coordinates in
curved momentum spaces, i.e., different momentum space
bases lead to inequivalent relativistic theories and predic-
tions [29]; see also when we discuss isotropic dispersion
relations in the next section. As we shall see now, we will
be able to increase the estimated bound from lifetime
observations in 2 orders of magnitude in comparison to
previous approaches [13].
Comparing (24) and (26), we can estimate a lower bound
for the κparameter using the momentum Lorentz factor, ¯γ,
achieved in facilities like the LHC (p0=m ∼104) or that
shall be achieved in the FCC (p0=m ∼105)[13].2Using the
mass of the muon [27]
mμ≈105.6583745 MeV ð27Þ
and ¯γLHC ¼104we find the LHC upper bound as
κLHC ≥mμ¯γ3
LHC
2
τμ
στ
≈5.3×1016 GeV;ð28Þ
which lies 3 orders of magnitude below the Planck energy
EP≈1.22 ×1019 GeV and corresponds to the scale of
some inflationary models [32]. This is already an interest-
ing result, since it is 2 orders of magnitude higher than the
bound proposed in [12,13].
Using the optimal ¯γfactor which can be reached by the
LHC for muons from ¯γLHCopt ¼6.5TeV=mμ¼6.1×104
one even reaches
κLHCopt ≥mμ¯γ3
LHCopt
2
τμ
στ
≈1.2×1019 GeV ∼EP:ð29Þ
We should stress that the assumption of reaching these
optimal conditions, like the precision (26) in the LHC, are
maximally optimistic. For instance, the measurement of
short-lived hadrons has been recently performed at the
CMS with relative uncertainty of order Oð10−2Þ[33]. Some
extra difficulties arise when using the muon decay due to its
very long lifetime.3However, the decay of exotic long-lived
particles (with lifetimes of the order 10 ps to 10 ns) also
have been searched in some LHC experiments [34], and
there is room for improvement in this analysis [35].
However, latest with the next generation colliders,
such as the FCC, the muon lifetime shall be amplified
by the Lorentz factor ¯γFCC ¼4.7×105, which alleviates
the needed relative uncertainty to constrain this effect at the
Planck scale: it could be of the order Oð10−4Þto Oð10−3Þ,
which lies close to current capabilities. Besides that, this
observable represents an unforeseen opportunity for testing
Planck scale physics in prospective facilities like Muon
Colliders [36], where the dilated lifetime could be mea-
sured in longer baselines at the TeV scale.
Before moving on to further MDRs we point out that the
dispersion relation of the type H¼ηðp; pÞþϵp0δijpipj
does not only emerge in the context of the κ-Poincar´e
algebra, but also in the context of propagation of particles
2These estimates are based on energies 6.5 TeV for the LHC
[30] and 50 TeV for the FCC [31].
3Nevertheless, the deformed muon lifetime is a good candidate
effect due to the smallness of the muon’s mass, which works as an
amplifier.
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in a quantum spacetime modeled by D-brane fluctuations in
string theory [37]. In our approach, the LIV nature of this
string theory dispersion relation gets supplemented by
deformed Lorentz transformations induced by (23). The
bounds (28) and (29) then translate into a bound on the
quantum gravity scale ξ
MQG, for this LIV to DSR lifted
model, which is obtained by replacing κby MQG
2ξ. The
bounds do not apply to the original LIV string
theory model.
B. Isotropic modified dispersion relations
We extend our analysis of time dilations to general
MDRs which are rotational invariant, i.e., depend only on
the norm of the spatial momentum q¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
δijpipj
q. The time
measuring Finsler function (10) becomes
Fð_
xÞ¼mffiffiffiffiffiffiffiffiffiffiffiffiffi
ηð_
x; _
xÞ
p1−ϵhð¯
p0ð_
xÞ;¯
qð_
xÞÞ
2m2:ð30Þ
Using the relation ¯
pa¼m_
xa
ffiffiffiffiffiffiffiffiffi
ηð_
x;_
xÞ
p, the reparametrization
invariance of the time measure (14) and the notation from
the previous section for ¯
p0ð_
xÞ¼m_
x0
ffiffiffiffiffiffiffiffiffi
ηð_
x;_
xÞ
p¼mγand
q¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
δij ¯
pið_
xޯ
pjð_
xÞ
q¼mffiffiffiffiffiffiffiffiffiffi
δij _
xi_
xj
ηð_
x;_
xÞ
q¼mγv, we obtain the
general time dilation formula for this kind of MDRs
Δt¼γΔτ1þϵ
2m2hðmγ;m ffiffiffiffiffiffiffiffiffiffiffiffi
γ2−1
qÞ;ð31Þ
where the units of the leading order perturbation parameter
ϵmust be adopted depending on the choice of h. Often the
leading order terms beyond special relativity are charac-
terized by a polynomial h¼Pr;s σrspr
0qs, where σrs are
numerical coefficients and r,sare integers. For these
modifications the time dilation in terms of the velocity
Lorentz factor becomes
Δt¼γΔτ1þ1
2X
r;s
σrsm
EPrþs−2
γrðγ2−1Þs
2:ð32Þ
To compare this dilation formula directly with the lifetime
of particles of a certain energy, one needs to rewrite this
expression in terms of the momentum Lorentz factor ¯γ.
Therefore it is necessary to derive the relation between γ
and ðp0;mÞcase by case, analogously as we presented in
the previous section before (22).
To conclude, we list several prominent modification
functions hand their particle lifetime prediction in terms of
the velocity Lorentz factor γin Table I.
With our findings we added another piece to the
systematic analysis of Planck scale MDRs and their
predictions of observables. Surprisingly, for first order in
Plank energy corrections, the Planck scale sensitivity for
muon lifetimes lies in reach already with the LHC, under
optimal conditions, but latest with the planned FCC for
more attainable requirements.
ACKNOWLEDGMENTS
The authors thank Nick E. Mavromatos, Albert de
Roeck, and Alice Florent for the insightful comments.
C. P. was supported by the Estonian Ministry for Education
and Science through the Personal Research Funding Grant
No. PSG489, as well as the European Regional
Development Fund through the Center of Excellence
TK133 “The Dark Side of the Universe”and was funded
by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation), Project No. 420243324. I. P. L. was
partially supported by the National Council for Scientific
and Technological Development, CNPq Grant No. 306414/
2020-1. The authors would like to acknowledge network-
ing support by the COST Action QGMM (CA18108),
supported by COST (European Cooperation in Science and
Technology).
TABLE I. Time dilation formulas for different MDRs.
Type and theory Perturbation function Time dilation
Monomials
r¼1;s¼2: D-brane recoil [38]
r¼1;s¼2:κ-Poincar´e bicrossproduct [26], D-brane foam [37] h¼αpr
0qsΔt¼γΔτ½1þα
2ðm
EPÞwγrðγ2−1Þs
2
w¼rþs−2
r¼0;s¼3: example from [15], Liouville-String QG [39]
r¼4;s¼0: LQG inspired MDRs [40–42]
p0powers
r¼1:κ-Poincar´e Magueijo-Smolin basis [43] h¼αðp2
0−q2Þpr
0Δt¼γΔτ½1þα
2ðm
EPÞrγr
Metric factor powers
s¼2:κ-Poincar´e preferred basis in [44] h¼αðp2
0−q2ÞsΔt¼γΔτ½1þα
2ðm
EPÞ2ðs−1Þ
IARLEY P. LOBO and CHRISTIAN PFEIFER PHYS. REV. D 103, 106025 (2021)
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