Reaching the Planck scale with muon lifetime measurements

Planck scale modified dispersion relations are one way how 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 loop quantum gravity and Hořava-Lifshitz 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 κ-Poincaré dispersion relation in the bicrossproduct basis with Planck scale sensitivity with help of the muon’s lifetime.


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 E P of order 10 19 GeV, respectively at distance of the Planck length ℓ P of 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][2][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 dependence on the particles' energy [4][5][6][7][8].
Recently, it has been pointed out that in the comparison of lifetimes of particles and antiparticles (in particular for muons) Planck scale sensitivity for κ-deformations of the Poincaré algebra [9][10][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. When dealing with MDRs, one has to be careful with what to call a Lorentz factor, since it makes a difference if it is defined in terms of velocities or in terms of momenta. To distinguish clearly between them we will introduce the velocity Lorentz factor γ = 1 √ 1−v 2 and the momentum Lorentz factorγ = p0 m . The relation between these quantities depends on the MDR in consideration.
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 modified dispersion relations. 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 modified dispersion relation. In general, this will be a Finslerian length measure [14][15][16][17][18], i.e. a function F (x,ẋ) depending on position and the velocity of an observer, which is 1-homogeneous with respect to the 4-velocity argumentẋ. 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 it to several Planck scale modified dispersion relations motivated from the κ-Poincaré algebra, as well as to MDRs inspired from loop quantum gravity (LQG) and Hořava-Lifshitz gravity.
Most interestingly, we explicitly calculate the time dilation of the lifetime of muons in facilities like the Large Hadron Collider (LHC), or the planned Future Circular Collider (FCC), at CERN, from the Finsler length measure induced by the κ-Poincaré dispersion relation in the bicrossproduct basis. We find that the deformation parameter κ could be constrained by 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.
Notational conventions in this article are that η denotes the Minkowski metric with signature (+, −, −, −), the indices a, b, c, ... run from 0 to 3 while the indices i, j, k, ... run from 1 to 3. The symbol∂ a = ∂ ∂pa denotes derivative with respect to momentum coordinates on the cotangent bundle.

II. THE TIME MEASURE FROM GENERAL MODIFIED DISPERSION RELATIONS
We begin our study by reviewing the mathematical procedure to obtain a time measure from modified dispersion relations. Then 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 H(x, p) on the point particle phase space, technically the cotangent bundle T * M of spacetime. To associate a time measure to massive point particles, resp. observers, from the dispersion relation one employs the so called Helmholtz action of free particles [14][15][16][17][18][19][20] given by where µ is an arbitrary curve parameter, "dot" means derivative with respect to this parameter, f is a function such that f = 0 is equivalent to the dispersion relation H(x, p) = m 2 and λ is a Lagrange multiplier. To transform this action into a length measure for massive particle trajectories we use the following algorithm: 1. Variation with respect to λ enforces the dispersion relation.
2. Variation with respect to p a yields an equationẋ a =ẋ a (p, λ) which must be inverted to obtain p a (x,ẋ, λ) to eliminate the momenta p a from the action.

Finally the desired length measure is obtained as
The crucial step in this algorithm is to be able to find p a (x,ẋ, λ), i.e. to invert the relationẋ a =ẋ a (p, λ). If this is globally possible, or 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 f (H, m) = ln(H(x, p m )), for homogeneous Hamiltonians, or, for general Hamiltonians, f (H, m) = ln( H(x,p) m n ) or f = H(x, p) − m n , often with n = 2. We will now apply this algorithm to general first order modifications of the GR dispersion relation. In particular we find that, to first order, the choice of the function f does not play any role as long as ∂ H f = ∂f ∂H = 0.

B. First order modified dispersion relations
Let g be a general Lorentzian metric with signature sign(g) = (+, −, −, −) and h(x, p) a function on the cotangent bundle T * M of spacetime. A general first order modification of the GR dispersion relation is defined by the Hamilton function The term g(p, p) = g ab (x)p a p b defines the GR point particle dispersion relation on a curved spacetime, ǫ is a perturbation parameter, counting the first non-trivial correction to GR, and h(x, p) a perturbation function, which needs to be specified depending on the application in consideration. In the context of quantum gravity phenomenology, ǫ is usually related to the Planck length or Planck energy and h(x, p) can, for example, be obtained from Planck scale modified dispersion relations, such as the κ-Poincaré dispersion relation, whose influence we investigate in Section III A, or others, which we discuss in Section III B. The steps of the previously outlined algorithm can now be performed as follows: 1. Variation of the action (1) with respect to λ yields f = 0, which in turn enforces the dispersion relation 2. Variation of the action (1) with respect to p a and using the perturbative Hamiltonian (2) yieldṡ which can be rewritten as (indices are raised on lowered with the components of the metric g) Explicitly inverting this equation completely to obtain p a (x,ẋ, λ) is not possible, but also not necessary in perturbation theory, as we will see soon. We note the following two relevant relations: where we introduced the notation g(ẋ,ẋ) = g ab (x)ẋ aẋb .
3. Using (7) in the dispersion relation (3), using a first order expansion of the Lagrange multiplier λ = λ 0 + ǫλ 1 and solving the dispersion relation order by order leads to 4. Combining all the results from (6), (7) and (8) in (1) for the Hamiltonian (3) the action becomes At this order the function f and its derivatives all cancel out and so the specific choice is not relevant. The perturbation function h appearing in (9) has to be understood as h = h(x,p(x,ẋ)), withp a (x,ẋ) = mẋ a √ g(ẋ,ẋ) .
We have proven that the Finsler function which governs the massive point particle motion of first order modified dispersion relation is As a concrete example, consider an n-th order polynomial modification which yields In this section, we analyse how the lifetime of a fundamental particle is modified by the assumption that it lives on a Finsler spacetime, see [22,23] for precise mathematical definitions of Finsler spacetimes, induced by a MDR. On these, the clock postulate is implemented in the following way.
The proper time an observer, or massive particle, experiences between events A and B along a time-like curve (her worldline) in a Finsler spacetime (M, F ) is the length of this curve between events A and B: where again "dot" means derivative with respect a parameter µ. We aim to investigate the decay of fundamental particles in accelerators, therefore we shall discard pure gravitational 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 to neglect curvature effects at small coordinate distance around every point on Finsler spacetimes [24]. Thus to zeroth order we consider g(ẋ,ẋ) in Eq.(12) as the norm in Minkowski space, which we shall label η(ẋ,ẋ). In Cartesian coordinates we simply write Since the arc-length is invariant under reparametrizations, we can transform the arbitrary parameter µ to the time coordinate in the laboratory frame, x 0 .
= t in (13). Therefore, using (12), we have the following modification of the proper time between the events with parameters (x 0 ) A = t A to (x 0 ) B = t B (from now on we omit the label "AB" in ∆τ AB ): where, for convenience, we introduced the usual velocity Lorentz factor with v i . = dx i /dt and v 2 = δ ij v i v j . 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 v 2 is roughly a constant, which means that we are able to simplify the above expression, which will be analysed in the next subsection for some cases of interest. In the following, ∆t will be the time measured in the laboratory frame in which the particle is accelerated, while ∆τ is the proper time experienced by the particle, respectively measured by an observer comoving to the particle. As we shall see, the Finsler geometric length measure will lead us to corrections of a particle's lifetime depending on the general perturbation function h under consideration.

A. The κ-Poincaré dispersion relation in bicrossproduct basis
For the κ-Poincaré dispersion relation in the bicrossproduct basis, the first order correction of the quadratic GR Finsler function is a polynomial of degree n = 3. The symbols h a1a2a3 for this case is h a1a2a3 = − 1 3 (δ 0 a1 δ ij δ i a2 δ j a3 + δ 0 a2 δ ij δ i a1 δ j a3 + δ 0 a2 δ ij δ i a1 δ j a3 ) and null otherwise, see for example [26]. Hence the integrand in (15) becomes Moreover, ǫ is 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-Poincaré algebra. Therefore, we express the lifetime of a fundamental particle that probes a Finsler spacetime induced by the κ-Poincaré dispersion relation in the bicrossproduct basis as (we define ∆t This geometric invariant quantity defined by Eq. (13) 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 ∆t, can be related to the proper lifetime of the particle ∆τ , depending on its coordinate velocity v thorugh the factor γ. To first order κ −1 , we find for the laboratory frame lifetime of the particle In order to compare with data from particle accelerators, we need to express the velocity γ factor defined in (16) in terms of the energy p 0 and mass m of the particles. The conversion between these dependencies is non-trivial due to the deviations from the usual relativistic setting. To do so, we derive the zero component of the 4-momentum of the particles, which satisfies the MDR: Solving this relation for γ as function of p 0 yields γ = p0 m + m 2κ 1 − 3 Employing this dependency in (19) gives us the lifetime as a function of p 0 where ∆t SR = p0 m ∆τ is the usual special relativistic dilated lifetime. This result leads us to introduce the momentum Lorentz factorγ = p0 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 γ(γ) = γ + m 2κ 1 − 3γ 2 + 2γ 4 . From (21), we are able to identify the dimensionless quantity δ p0,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: 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] τ µ = (2.1969811 ± 0.0000022) × 10 −6 s = 2.1969811 µs ± σ τ , and its most precise measurement was done for low energy muons in [28]. From (23), we see that the relative uncertainty of this measurement reads 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 uncertainty, 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 anti-muon, for a κ-Poincaré basis in which the Hamiltonian is undeformed, and modifications take place when comparing the lifetimes of particles and anti-particles 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 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 predictions [29], see also Section III B. As we shall see now, we will be able to increase the estimated bound from lifetime observations in two orders of magnitude in comparison to previous approaches [13]. Comparing (22) and (24), we can estimate a lower bound for the κ parameter using the momentum Lorentz factor,γ, achieved in facilities like the LHC (p 0 /m ∼ 10 4 ) or that shall be achieved in the FCC (p 0 /m ∼ 10 5 ) [13] 1 . Using the mass of the muon [27] m µ ≈ 105.6583745 MeV (25) andγ LHC = 10 4 we find the LHC upper bound as which lies three orders of magnitude below the Planck energy E P ≈ 1.22 × 10 19 GeV. This is already an interesting result, since it is two orders of magnitude higher than the bound proposed in [12,13]. Using the optimalγ factor which can be reached by the LHC for muons fromγ LHCopt = 6.5TeV/m µ = 6.1 × 10 4 one even reaches Surely these assumptions of reaching the optimalγ value for muons with the LHC are very optimistic. However, latest with the next generation colliders, such as the FCC, expectingγ factors of ∼ 10 5 , the Planck scale sensitivity should be attained. Besides that, the assumption of the momentum Lorentz factorγ FCC = 50TeV/m µ = 4.7 × 10 5 may alleviate the required precision of the muon lifetime measurement to test the Planck scale.

B. Isotropic MDRs
We extend our analysis of time dilations to general modified dispersion relations which are rotational invariant, i.e. depend only on the norm of the spatial momentum q = δ ij p i p j . The time measuring Finsler function (10) becomes Using the relationp a = mẋ a √ η(ẋ,ẋ) , the reparametrization invariance of the time measure (13) and the notation from the previous section forp 0 (ẋ) = mẋ 0 √ η(ẋ,ẋ) = mγ and q = δ ijp i (ẋ)p j (ẋ) = m δijẋ iẋj η(ẋ,ẋ) = mγv, we obtain the general time dilation formula for this kind of modified dispersion relations where the units of the leading order perturbation parameter ǫ must be adopted depending on the choice of h. Often the leading order terms beyond special relativity are charachterized by a polynomial h = r,s σ rs p r 0 q s , where σ rs are numerical coefficients and r, s are integers. For these modifications the time dilation interms of the velocity Lorentz factor becomes 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 factorγ. Therefore it is necessary to derive the relation between γ and (p 0 , m) case by case, analogously to the derivation we presented it explicitly for the κ-Poincaré dispersion relation in bicrossproduct basis in the previous section before (21). In Table I we list explicitly several modification functions h and their lifetime prediction in terms of the velocity Lorentz factor γ, for several prominent cases in the literature in the context of the κ-Poincaré algebra, loop quantum gravity and Hořava-Lifshitz gravity.
We stress that κ-Poincaré algebra can be cast in different momentum space bases, with modified dispersion relations governed by different powers of the Planck energy, see for instance some cases of undeformed and first order corrections in [37] and second order ones in [32,33]. Therefore, the lifetime of fundamental particles constitutes an example of basis dependence of physical observables in the κ-Poincaré algebra context.
For the dispersion relations just depending on the metric factor η(p, p) = p 2 0 − q 2 , it comes with no surprise that the corrections are independent of the particle velocities, since these are still local Lorentz invariant.
The LQG and Hořava-Lifshitz gravity inspired corrections were discussed in the context related to the running of the spacetime dimension (see, for instance [39] and references therein).

IV. CONCLUDING REMARKS
The high precision measurement of the lifetime of muons in accelerators yields a window to reach Planck scale sensitivity in the near future and to verify the viability of modified dispersion relations employed in quantum gravity phenomenology, as we demonstrated with the κ-Poincaré dispersion relation in the bicrossproduct basis.
The precise mathematical derivation of the, in general Finslerian, proper time measure for massive particles from general modified dispersion relations, presented in equation (10), lays the foundation for this insight. We highlighted a subtlety which appears when working with the velocity description of physical effects of MDRs and one wants to translate these into a momentum dependent description, namely the difference between the velocity Lorentz factor γ = 1 √ 1−v 2 and the momentum Lorentz factorγ = p0 m , which in is not trivial, i.e. γ =γ. Once we obtained the time measure, it is straightforward to derive the muon lifetimes for multiple modified dispersion relations which we summarized in Table I in terms of the velocity Lorentz factor γ. In particular our calculation shows that the muon lifetime is an observable that is sensitive to the different momentum space bases of the famous κ-Poincaré dispersion relation and could be used to discriminate the physical interpretation and viability of some different bases.
With our findings we added another piece to the systematic analysis of Planck scale modified dispersion relations and their predictions of observables. Surprisingly, for first order in Plank energy corrections, the Planck scale sensitivity for muon lifetimes lies in reach under optimistic optimal conditions already with the LHC, but latest with the planned FCC.