# Homogeneously Modified Special relativity (HMSR)

## Abstract

This work explores a Standard Model extension possibility, that violates Lorentz invariance, preserving the space-time isotropy and homogeneity. In this sense HMSR represents an attempt to introduce an isotropic Lorentz Invariance Violation in the elementary particle SM. The theory is constructed starting from a modified kinematics, that takes into account supposed quantum effects due to interaction with the space-time background. The space-time structure itself is modified, resulting in a pseudo-Finsler manifold. The SM extension here provided is inspired by the effective fields theories, but it preserves covariance, with respect to newly introduced modified Lorentz transformations. Geometry perturbations are not considered as universal, but particle species dependent. Non universal character of the amended Lorentz transformations allows to obtain visible physical effects, detectable in experiments by comparing different perturbations related to different interacting particles species.

## 1 Introduction

Most of Lorentz Invariance Violating (LIV) theories are characterized by a modification of the free particle kinematics. This effect is supposed to be caused by the interaction of the free propagating particle with the space-time quantized background structure. In fact one expects that Planck-scale interactions could manifest themselves in a “low” energy scenario as tiny residual effects, that can modify standard physics. Several candidates theories, such as Standard Model Extension (SME) [1, 2, 3, 4, 5], Double Special Relativity (DSR) [6, 7, 8, 9, 10, 11, 12, 13, 14, 15], Very Special Relativity (VSR) [16, 17], have been proposed. All of these theories share the feature of considering modified dispersion relations for free particles, with the amended form \(E^2-(1-f(p))p^2=m^2\) and \(f(p)=\sum _{k=1}\alpha _{k}(E_{P})p^k\). The resulting space-time geometry acquires an energy-momentum dependence, that can be described resorting to Finsler geometry [18, 19, 20, 21, 22, 23]. Some proposed scenarios main feature consists in providing background structures that introduce preferred directions to violate Lorentz Invariance, as in SME [1]. This characteristic implies that the space-time is no more isotropic and therefore an inertial observers privileged class must exist. Quantum Gravity effects could emerge in a symmetry breaking fashion. In fact, spontaneous symmetry breaking is a useful concept, employed for example in particle physics. One, therefore, can suppose that, even in the high energy limit, the Planck scenario, quantum gravity could present the same mechanism, breaking the Lorentz symmetry. The introduction of a privileged reference frame might not be a real problem, even if it might result conceptually difficult. Some studies, for example, attempt to correlate this privileged reference frame with the natural one, used for the description of the Cosmic Microwave Background Radiation (CMBR). But there are apparently no physical reasons that can justify any connection between a supposed quantum phenomenon of the Planck scale, with the CMBR classical physics description. In fact, nowadays, it is not clear how to introduce and justify these preferred inertial observers.

In this work, to preserve the idea of space-time isotropy, a possible way to introduce a LIV theory, without a preferred class of inertial observers, is explored. The Lorentz symmetry is therefore only modified, as in DSR theories [10, 11, 12] and the idea of space-time isotropy results restored respect to the amended Lorentz transformations. Lorentz symmetry perturbations are not introduced in an universal way, instead every particle species presents its personal modification, as suggested in [4]. This corresponds, for the high energy limit, to a redefinition of the maximum attainable velocity, different for every particle type, as for example in [16]. Moreover, in this way, it results possible to predict detectable physical effects, without the introduction of any exotic mechanisms.

## 2 Modified dispersion relations

*f*perturbation function preserves the rotational invariance of the MDR, the

*g*one instead breaks this symmetry, introducing a preferred direction in space-time. It is even important to stress that the lack of distinction between particles and antiparticles in MDR means that one is dealing with a CPT even theory, in fact the dispersion relations do not present a dependence on particle helicity or spin. In order to contemplate a CPT odd model extension, it is sufficient to include this kind of dependence in MDR formulation. Since the publication of the Greenberg paper [26], it is well known that LIV does not imply CPT violation. The opposite statement is supposed true in the same work [26], but this point is controversial. The idea that CPT violation automatically implies LIV is widely debated in literature [27, 28, 29, 30] and it was confuted, for example, in [31].

*F*(

*p*), in fact the perturbation function homogeneity hypotheses permits to write the MDRs (1) as:

*f*and

*g*, one obtains that the function

*F*, defined in (3), results homogeneous of degree 1, condition to be a candidate Finsler pseudo-norm. It is important to underline the difference between a Finsler structure and a pseudo-Finsler one. The first geometric structure is constructed using a positively defined metric to pose the norm, instead the second one resorts to a not positive one. Here the pseudo-Finsler geometry is used, because one is dealing with the space-time structure, where the underlying global metric is the Minkowski one (\(\eta _{\mu \nu }\)), with signature \(\{+,\,-,\,-,\,-\}\).

*g*is posed equal to zero \((g=0)\) in order to try to construct an isotropic LIV theory. Hence only MDR, preserving rotational symmetry, will be considered:

*f*, that preserves rotational symmetry can be rewritten in the form:

*p*as a function of \(\xi \), that permits to fix the expansion coefficients \(\alpha _{k}\) in order to satisfy the equation itself. In literature, for most physical cases, the

*h*(

*p*) series terminates at first or second order. In these eventualities, it is always possible to find an approximation series for \(p(\xi )\). This procedure is analogous to map a function of the variable \(p=|\overrightarrow{p}|\) on a function of the variable \(\xi \), noting that \(\xi \) and

*p*have a biunivocal correspondence.

*f*is well posed, it is necessary to verify that the energy solution of the equation, obtained using (2) and (1):

*n*value. Dividing the previous equation by \(E^{2}\), it becomes:

*n*degree polynomial. If the magnitude of this polynomial remains limited, that is this function represents a tiny perturbation, compared to the magnitude of

*p*, the solution of the equation is \(\xi \simeq 1\). So, posing the correct constrain on the coefficients of the series (2), one can obtain a real value energy

*E*from Eq. (9) and:

Now, fixed the MDR general form it is possible to investigate the space-time induced modified geometrical structure.

## 3 The Finsler geometric structure of space-time

*Hamiltonian*is defined, using the modified metric (14), as:

*Legendre*transformation, as:

*Lagrangian*, that is:

Now it is useful to deal with the obtained pseudo-Finsler metric structure of the space-time, introducing the Cartan formalism, that is resorting to the *vierbein* or *tetrad*. In this work the vierbein acquires an explicit dependence on energy-momentum, because one is dealing with an energy depending Finsler pseudonorm defined space-time.

*vierbein*are equivalent if they originate the same metric:

^{1}

*thetrad*elements equivalence classes will be considered, identifying every class with one representative. To originate the (14) metric, the

*vierbein*must have the following expression:

*affine*connection:

*Cartan*or

*spinorial*connection, as:

## 4 Modified Lorentz transformations

Using the *tetrad*, it is possible to construct the explicit form of the modified Lorentz group. The obtained representation preserves the form of the MDR and the homogeneity of degree 0 of the perturbation functions.

*vierbein*it is possible to define projection from a tangent (local) space, parameterized by the metric \(g_{\mu \nu }(x,\,v)\) to another local space, identified by a different metric tensor \(\overline{g}(x',\,v')_{\mu \nu }\) as summarized in the following graph: From now on, the dependence of thetrad and the metric tensor will be generalized from the space-time coordinates (

*x*) to the coordinates of the phase space \((x,\,p)\). In this way a dependence on the momentum is included, like in Finsler geometry [34]. The dependence on the position is supposed trivial [24, 25] and therefore will be neglected to preserve the space homogeneity. Only the dependence on momentum (velocity) is maintained and all the physical quantities are generalized, acquiring an explicit dependence on it. The graph of the transition from one tangent (local) space to the other becomes: where is indicated the explicit dependence of the metric from momentum.

The Modified Lorentz Transformations, introduced in (40), are therefore the isometries of the Modified Dispersion Relations.

*f*in the MDR:

*f*. Therefore the action of the modified Lorentz group preserves the homogeneity of the perturbation function

*f*, preserving the MDR form (4).

## 5 Very Special Relativity (VSR) correspondence

*p*functions. These functions can be written in a perturbative fashion as \(f_{i}=1-h_{i}\), where \(h_{i}\ll 1\) are the velocities modification parameters. From this relation, it is possible to derive an explicit equality for the energy:

*f*is the homogeneous perturbation, introduced in this work (2), if its magnitude remains negligible, compared to the momentum, the ratio \(\frac{|\overrightarrow{p}|}{E}\) have a finite limit for \(p\longrightarrow \infty \) (\(\frac{|\overrightarrow{p}|}{E}\longrightarrow 1+\delta \)). Consequently, even the

*f*function admits finite limit, \(f(1+\delta )\longrightarrow \epsilon \). In this way the perturbation \(f_3\), for \(p\longrightarrow \infty \), tends to \( \lim _{p\longrightarrow \infty } \, f_3^{\;2} = 1- f(1+\delta ) = 1-\epsilon \, \). Therefore it is possible to obtain the Coleman and Glashow’s “Very Special Relativity” (VRS) scenario as a high energy (high momentum) limit [16, 17]. In this case it is possible to recover from equation (46) a massive particle “personal”

*maximum attainable velocity*\(c'\):

*p*and \(p'\) are:

## 6 Relativistic invariant Mandelstam variables

In HMSR every particle species has its own metric, with a personal maximum attainable velocity. Moreover every species presents its personal Modified Lorentz Transformations (MLT), which are the isometries for the Modified Dispersion Relation (MDR) of the particle. The new physics, caused by LIV, emerges only in the interaction of two different species. That is every particle type physics is modified in a different way by the Lorentz symmetry violation. Therefore, to analyze the interaction of two particles, it is necessary to determine how the reaction invariants – that is the Mandelstam relativistic invariants – are modified.

*vierbein*to project the particles momenta on the Minkowski tangent space:

*e*indicates the tetrad related to the first particle and with \(\tilde{e}\) the vierbein related to the second one. With this internal product it is now possible to define the Mandelstam variables

*s*,

*t*and

*u*, remembering that: and considering the

*p*and

*q*momenta as belonging in general to different particle species.

If the two interacting particles belong to the same species, the internal product and therefore the Mandelstam variables present no differences from standard Physics. That is the momenta of particles of the same kind, live in the same tangent (local) space, constructed with the Finsler metric. Instead, if the particles belong to different species, the definition of the internal product requires the necessity to correlate different local tangent spaces.

The complete physical description of interactions can be made using the formalism of the *S* matrix, which results to be an analytic function of the Madelstam variables. Since these quantities result covariant, respect to the amended Lorentz transformations (MLT), the concept of isotropy is restored. In this way the necessity of introducing a privileged class of inertial observers disappears.

*g*and

*e*are related to the first particle species, \(\widetilde{g}\) and \(\widetilde{e}\) are related to the second one and \(\overline{g}\) and \(\overline{e}\) to the third one. All the processes, introduced for the two particles interaction, can be generalized in this way for generic n-particles (n-momenta) interactions. Finally the amended Lorentz group acquires the explicit form:

## 7 Modified momenta composition rules and Double Special Relativity (DSR) correspondence

*modified composition rule*for the momenta:

Therefore HMSR model, proposed in this work, presents an analogy with DSR theories [10, 11], but it is important to underline a difference. In fact the modified composition rule does not present an universal character, instead it is species dependent, and moreover it is associative and abelian. The new physics emerges by the comparison of different particle species that have different Modified Lorentz Transformations. The construction of the modified physics can, therefore, predict physical effects, experimentally detectable. Instead in case of a physics modification with universal character, independent from the particle species, new physical effects correspond just to a redefinition of the units of measure, that is the speed of light [39].

## 8 Standard model modifications

*e*is the coupling constant (the electric charge),

*p*represents the incoming spinor field momentum and \(p'\) the outgoing one, and the generalized metric has been introduced:

*virerbein*element) correlated to the gauge field and the index \(\mu \), even if greek, represents a coordinate of the Minkowski space-time \((TM,\,\eta _{\mu \nu })\). Under the gauge fields Lorentz covariance assumption, the \(\overline{e}\) thetrad reduces to the form \(\overline{e}_{\mu }^{\;\nu }=\delta _{\mu }^{\;\nu }\) and the gauge bosons live in the flat space-time. The term, that in (80) and (83) multiplies the conserved current, is a generalization of the analogous term borrowed from curved space-time QFT, where its explicit form is given by: \(\sqrt{|\det {[g]}|}\) [41]. With the previous definitions it is possible now to write the interaction Lagrangians of the LIV perturbed theories, that become for QED:

*g*(

*p*) represents the metric computed in (14) that coincides with that used in (67).

It is important to underline that the modified Dirac matrices (68) are used to write the kinetic part of the Lagrangian, describing the free fermion propagation. This part determines the form of the propagator of the particle and, as consequence, the dispersion relation, that is the MDR (4).

The same generalization can be applied to the SM Lagrangian, using again the vierbein correlated to different fermions to project them on the common support Minkowski space-time. Even if one is not dealing with mass eigenstates, it is possible to consider the perturbation again as function of momentum and energy ratio (4).

*f*doublets can opportunely be defined in the usual fashion, as:

*f*doublets as:

*Y*and \(\tau ^{0}\) are the usual matrices, in diagonal form, correlated respectively with the

*U*(1) and the

*SU*(2) gauge symmetries. \(\widetilde{e}(B)_{\mu }^{\nu }\) and \(\widetilde{e}(W^{0})_{\mu }^{\nu }\) are the vierbein correlated respectively with the gauge fields \(B_{\mu }\) and \(W^{0}_{\mu }\). As for QED, in order to guarantee that the neutral currents live in the tangent space \((TM,\,\eta _{\mu \nu })\), the interaction terms must be written using the generalized \(\widetilde{\varGamma }\) matrices (79), that depend on the particle \(\psi \).

*g*dependents only on the particle flavor and not on the particle chirality.

*f*, with momentum

*p*. The generalized metric, generated by these modified matrices, takes the form:

*SU*(3) gauge symmetry group, with

*i*and

*j*representing the colour indices of the quark fields, \(g_{s}\) the strong coupling constant and \(\overline{e}(G)_{\mu }^{\,\nu }\) represents the projector (tetrad) correlated with the \(G_{\nu }^{a}\) gauge field. Even in this case the Lagrangian can be rewritten in the simpler form:

*f*species considered, and \(\{G_{\mu \nu }\), \(W_{\mu \nu }\), \(B_{\mu \nu }\}\) represent the gauge fields strength. The similarity with [1] is given by the tensor that appears in the perturbation term \(k^{(f)}_{\mu \nu \alpha \beta }=g^{(f)}_{\mu \nu }\,g^{(f)}_{\alpha \beta }-\eta _{\mu \nu }\eta _{\alpha \beta }\). Supposing the gauge field interaction with the background negligible, this Lagrangian term reduces to the standard form:

## 9 Allowed symmetries

The SM modifications, introduced in HMSR, are conceived in order to preserve space-time homogeneity and isotropy, but even the standard physics interactions and. As consequence, the same \(SU(3)\times SU(2)\times U(1)\) internal symmetries are preserved. To prove this statement it is possible to verify that the Coleman–Mandula theorem [42] is still valid. In this way it results that the allowed symmetries are restricted to the direct product of internal ones with those generated by the modified Lorentz group, introduced before. A less rigorous proof can be obtained generalizing a Witten argument [43] about the fact that any additional kinematic and non internal symmetry would overconstrain the scattering amplitude. Therefore any further symmetry generator beyond Lorentz group would allow nontrivial scattering amplitude only for a discrete set of scattering angles.

## 10 Coleman–Mandula theorem generalization

- 1.
Lorentz invariance respect to the Modified Lorentz Transformations

- 2.
Particle number finitness: \(\forall M>0\)\(\exists n<\infty \) number of particles with mass \(m<M\)

- 3.
Elastic scattering is an analytic function of the modified Mandelstam variables

- 4.
Nontrivial scattering happens for almost all energies

- 5.
\(\forall g\in G\), where

*G*is the symmetry group, the element \(g\in U(1)\) is representable in a identity neighbourhood via an integral operator, with distribution kernel

*S*matrix is expressed as a function of the modified Mandelstam variables and is therefore invariant under the action of the modified Lorentz group.

*i*index is related to the different particle species taken into account. These operators therefore satisfy the relation:

*i*particle species. Since the operators commute with the Poincaré group generators, as in the classical case, they satisfy the Lie algebra commutation rules:

*S*matrix, function of the new defined Mandelstam variables. Moreover the computation of the particle number with mass lower than a given number is:

*i*index represents the particle species, taken into account. Now, as in the classical version of the theorem, it is possible to find operators:

*i*particle species. The last statement is true because \([P_{\mu }^{\,(i)},\,J^{(i)}(p)]\) is given by a linear combination of \(P_{\mu }^{\,(i)}\) momenta, so the Jacobi identity:

*i*represents the particle species.

## 11 SME correspondence

## 12 Conclusions

A possible way to introduce a standard model extension, that preserves the idea of isotropy, is the key idea of HMSR and this work. As already highlighted, this is possible taking into account some concepts of the SME [1] with some ideas borrowed from DSR [10, 11]. The key point consists in constructing the space-time starting from a pure kinematical modification, that results depending on the particle species. This leads to a new space-time structure, that depends on the propagating material body momentum, the Finsler geometry. The Lorentz invariance is not broken, but modified, introducing an amended Lorentz group, in order to redefine the concept of spatial symmetries and reconcile the introduced perturbations of space-time with the idea of symmetry conservation. Moreover the perturbation considered have only a kinetic character, so the dynamic is not affected and new exotic interactions are not introduced, preserving the internal \(SU(3)\times SU(2)\times U(1)\) standard model symmetry. In this way it is simple to generalize the concept of isotropy, respect to the new generalized personal Lorentz transformations. The physical effects of such a theory can emerge only in interaction processes where different particle species are involved. In fact a universal modification would generate, for example, a redefinition of the measure units, effect very difficult to be detected. Instead, as shown in [24, 25], processes like GZK cut-off and neutrino oscillations, where different particle species interact, can be affected by this LIV model.

## Footnotes

- 1.
A simple example is \(e'^{a}_{\;\mu }(x)=-e^{a}_{\;\mu }(x)\).

## Notes

### Acknowledgements

A special thanks to prof. Alan Kostelecky for reading the draft of this work, suggesting some relevant improvements, and thanks to all the partecipants of the Third IUCSS Summer School and Workshop on Lorentz- and CPT-violating Standard-Model Extension, for the interesting and useful physics conversations.

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