Fermionic Supersymmetry in Time-Domain

Felipe A. Asenjo,1, ∗ Sergio A. Hojman,2, 3, 4, † Héctor M. Moya-Cessa,5, ‡ and Francisco Soto-Eguibar5, § 1Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago 7491169, Chile. 2Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Santiago 7491169, Chile. 3Departamento de Física, Facultad de Ciencias, Universidad de Chile, Santiago 7800003, Chile. 4Centro de Recursos Educativos Avanzados, CREA, Santiago 7500018, Chile. 5Instituto Nacional de Astrofísica, Óptica y Electrónica. Calle Luis Enrique Erro No. 1, Santa María Tonantzintla, Puebla 72840, Mexico. A supersymmetric theory in the temporal domain is constructed for bi-spinor fields satisfying the Dirac equation. It is shown that using the Dirac matrices basis, it is possible to construct a simple time-domain supersymmetry for fermion fields with time-dependent mass. This theory is equivalent to a bosonic supersymmetric theory in the time-domain. Solutions are presented, and it is shown that they produce probability oscillations between its spinor mass states. This theory is applied to the two-neutrino oscillation problem, showing that the flavour state oscillations emerge from the supersymmetry originated by the time-dependence of the unique mass of the neutrino. It is shown that the usual result for the two-neutrino oscillation problem is recovered in the short-time limit of this theory. Finally, it is discussed that this time-domain fermionic supersymmetric theory cannot be obtained in the Majorana matrices basis, thus giving hints on the Dirac fermion nature of neutrinos.

PACS numbers: Keywords:

I. INTRODUCTION tum mechanics for fermions described by the Dirac equation. In this case, we show that the simplest T-SUSY Supersymmetry is one of the cornerstones of theoreti- theory can be constructed for Dirac fermions with time- cal physics [1], being ubiquitous in almost every branch dependent mass, i.e, using the Dirac matrices basis. of physics [2–24]. We completly develop this fermionic T-SUSY theory, finding solutions for different possible time-dependent The standard way to proceed is to construct super- masses. This theory is equivalent to its bosonic partner, symmetric theories in a space-domain, where the super- allowing us to obtain a massive fermion field behavior partners and supercharge operators take into account the which is analogue to a light-like one. spatial variations of, for example, external potentials. In Besides, this theory produces probability states oscilla- this work, it is not our aim to focus in those spatial super- tions, and thus can be used to study the neutrino oscilla- symmetries, but instead, to inquire if a temporal version tion problem. In this way, we give a different perspective of such theories is possible for massive fields [25]. to the origin of neutrino oscillations through supersym- Recently, García-Meca, Macho Ortiz and Lorrente metry in the time-domain. Sáez [25] have introduced the concept of supersymmetry We end our proposal by showing the impossibility to in time-domain (T-SUSY) for Maxwell equations. This implement this temporal supersymmetry in the Majo- is a supersymmetry occurring in the temporal part of the rana basis. Therefore, this theory discerns between Dirac massless field dynamics, completely uncoupled from the and Majorana neutrinos. spatial evolution of the field. They studied the appli- cations of T-SUSY in the realm of optics for dispersive media, showing the novel capabilities of this theory to II. T-SUSY FOR DIRAC FERMIONS arXiv:2012.11024v1 [quant-ph] 20 Dec 2020 introduce hypothetical materials with new optical fea- tures. Besides, they show that this time-domain super- Let us consider a bi-spinor field (satisfying the Dirac symmetry applies to any field described, in principle, by a equation in flat spacetime) with the form d’Alambertian equation. In other words, they developed the bosonic version of the T-SUSY theory. ψ e Ψ = + + , (1) It is the purpose of this manuscript to show that a ψ−e− T-SUSY theory can also be obtained in relativistic quan- in terms of wavefunctions ψ+ and ψ−, and the spinors 1 0 e = , e = , (2) + 0 − 1 ∗Electronic address: [email protected] †Electronic address: [email protected] † † † ‡Electronic address: [email protected] with the properties e+e+ = 1 = e−e−, and e+e− = 0 = § † Electronic address: [email protected] e−e+. 2

The simplest form of a T-SUSY theory can be obtained where E± are complex time-dependent functions that from field propagation in one spatial dimension. In this satisfy the Ricatti equation form, the Dirac equation in flat spacetime for a massive dE fermion field Ψ reads i ± − E2 + k2 = W , (11) dt ± ± 0 1 iγ ∂0Ψ + iγ ∂1Ψ = mΨ , (3) such that ψ±(0) = 1. Whenever dm/dt 6= 0, then E+ 6= where m is the mass of the field. Besides, ∂0 is a time E−. In this way, for a specific time functionality of the derivative and ∂1 is a space derivative in an arbitrary di- mass, solutions for E± can be found in order to satisfy rection. As always, Dirac matrices fulfill γµγν + γν γµ = this T-SUSY description. µν µν 2η 14×4, where the metric is η = (1, −1, −1, −1), and All the framework for a standard supersymmetric the- 14×4 is the identity matrix in this four-dimensional space- ory in quantum mechanics can be straightforwardly uti- time. lized for this T-SUSY theory. For example, its algebra is In order to evaluate explicitly the T-SUSY theory for defined by the matrix operators fermions, let us choose the following Dirac matrices H 0 0 0 0 Q H = + , Q = , Q† = − ,  1 0 0 0   0 0 0 1  0 H− Q+ 0 0 0 0 1 0 0 0 0 1 0 γ0 =   , γ1 =   . (4) (12)  0 0 −1 0   0 −1 0 0  that have the closed algebra H = {Q, Q†}, [H, Q] = 0 = 0 0 0 −1 −1 0 0 0 [H, Q†], {Q, Q} = 0 = {Q†, Q†}, for the bosonic H and fermionic Q and Q† operators. Now, in order to decouple the space dynamics, let us as- Finally, we notice that this fermionic T-SUSY theory sume that ∂ Ψ = ikΨ with an arbitrary constant k. This 1 can be put in analogue fashion to its bosonic counterpart k plays the role of the constant momentum of the fermion developed in Ref. [25]. We can rewrite Eq. (7) in the propagation. In this way, under this representation, we form find that Eq. (3) becomes simply d2 1 Q±ψ± = k ψ∓ , (5) 2 + 2 ψ± = 0 , (13) dt n± in terms of supercharge operators where d 2 −1/2 Q± = i ∓ m , (6) n (t) = k − W . (14) dt ± ± where now d/dt is a total time derivative. Eqs. (5) corre- Eq. (13) is equivalent to the one describing the propaga- spond to a set of supersymmetric equations of quantum tion of light in a medium with supersymmetric refraction mechanics, now obtained in the Dirac matrices basis (4). indices n±. In this T-SUSY theory, both refraction in- Notice that, then, the simplest T-SUSY theory requires dices are related by that the mass be the responsible for the origin of the su- −1/2 perpotential of this supersymmetry. This occurs when 1 dm n+ = 2 + 2i . (15) the mass becomes time-dependent, m = m(t). n− dt Eqs. (5) can be used to calculate the equations that This relation shows the equivalence of fermionic T-SUSY each wavefunction ψ± satisfies. They are with bosonic T-SUSY discussed in Ref. [25] 2 H±ψ± = k ψ± , (7) where we have defined the super-Hamiltonians H± in III. SIMPLE SOLUTIONS terms of superpotentials W± as d2 We can get simple solutions for fermion fields with H = Q Q = − + W , (8) ± ∓ ± dt2 ± time-dependent mass. Let us first consider the simple solution for the massive where case, when k ≪ m(t). For this case, from Eq. (11), we dm find W = ∓i − m2 , (9) ± dt E±(t) ≈ ∓m(t) . (16) such that W− − W+ = 2i dm/dt. Thereby, the difference between states lies in the time dependence of the mass. Therefore, the two mass states have different behavior In general, from Eq. (7), we can write the wavefunc- stemming from just one time-dependent mass. tions (1) as Similarly, another simple solution can be obtained in the light-mass (ultra-relativistic) case, when k m(t). Z For this case, k corresponds (approximately) to the en- ψ±(t) = exp i dt E± , (10) ergy of the particle. Let us assume that m(t) = m0 +(t), 3

p 2 2 with a constant m0 , and E±(t) = k + m0 + δ± ≈ supersymmetry due to the temporal dependence of the k + δ±(t), such that δ±(t) k. In this case, Eq. (11) mass. becomes Considering the bi-spinor (1), let us define a bi-spinor with mixed states dδ± d i − 2kδ± = ∓i − 2m0 . (17) Φ dt dt Φ = a = U Ψ , (22) Φb By writing δ± = δR± + iδI± (with δR± and δI± reals), with the spinor wavefunctions Φa and Φb defining new then we find dδR±/dt − 2kδI± = ∓d/dt, and dδI±/dt + 2kδ = 2m . This set gives the equation states. Here, U is a unitary matrix with role of mixing R± 0 states. It can be written as [26–29] 2 2 d δR± d 2 cos θ 12×2 sin θ 12×2 2 + 4k δR± = ∓ 2 + 4km0 , (18) U = , (23) dt dt − sin θ 12×2 cos θ 12×2 which allow us to find the solutions where θ is called the mixing angle in vacuum, and 12×2 is the 2 × 2 identity matrix. (t) = 0 cos (Λt) , With all the above, the amplitude of mixed states 2 ±Λ + 4km0 change, from Φa(t = 0) to Φb(t), can be calculated to be δR±(t) = −0 cos (Λt) , ∗ ∗ Λ2 − 4k2 Amp (Φa → Φb) = sin 2θ ψ+(0)ψ+(t) − ψ−(0)ψ−(t) /2. 2Λk Finally, the probability of mixed states change is given δ (t) ≈ ± 0 sin (Λt) . (19) by I± Λ2 − 4k2 2 P (Φa → Φb) =|Amp (Φa → Φb) | for arbitrary constants m and Λ. 0 0 2 −α 2 2 Finally, we can find another interesting solution. Let =sin 2θ e sinh β + sin ρ , (24) us consider the propagation of the fermion field in a where medium with constant complex refractive index n+ = Z n+R − in+I (n+R and n+I are real constants). In this α = dt (Im {E−} + Im {E+}) , case, the fermion field behaves analogously to light prop- agating in a dispersive and absorbing medium, with so- 1 Z β = dt (Im {E−} − Im {E+}) , lution of Eq. (13) given by E+ = 1/n+, or ψ+ = 2 exp(it/n+). This can only occur for a complex time- 1 Z ρ = dt (Re {E } − Re {E }) . (25) dependent mass 2 − + 2 m(t) = −i3/2ζ tan i1/2ζ t , (20) The terms proportional to exp(−α) and sinh β enter in the probability due to the contribution of the amplitude 2 2 2 of wavefunctions (10). Their phases only contribute to when n+R = n+I + 1/k , and for a constant ζ defined 2 √ 2 2 the oscillation probability through sin ρ. by ζ = 2n+Rn+I /(n+I − n+R). This mass (20) al- In this way, the superoscillations between the mixed lows to find the complex non-constant supersymmetric states, given by the above transition probability, are due 2 2 2 1/2 −1/2 refractive index n− = k + iζ − 2i sec (i ζt) . only to the different solutions of Eq. (11). This occurs if From Eqs. (5), we can also obtain that ψ− = the fermion field has a non-constant mass. 0 R 0 0 R 0 0 R t 00 00 −ik exp(i dt m(t )) dt exp(it /n+ − i dt m(t )). As an example, let us evaluate the above probability Complex mass (20) violates current conservation of the (24) with the previous simple example for the solution Dirac equation. However, this can be re-interpreted in of a massive fermion, namely k ≪ m, and E± ≈ ∓m. terms of the effective absorbing media. An interesting For this case, Im{E±} ≈ 0, and α = 0 = β. Then the feature of mass (20) is that it becomes real for a short probability (24) reduces to time interval t 1/|ζ|, where Z 2 2 P (Φa → Φb) ≈ sin 2θ sin dt m . (26) m(t) ≈ ζ2t . (21) Thus, the superoscillation for the massive case occurs This solution is the equivalent to the bosonic propagation because the mass can evolve in time. in a medium studied in Ref. [25].

V. APPLICATION TO THE TWO-NEUTRINO IV. OSCILLATIONS IN T-SUSY OSCILLATION PROBLEM

The above T-SUSY theory allows now to consider the We now invite the reader to re-think the two-neutrino phenomenon of oscillation of states in a different fashion. oscillation problem in terms of the T-SUSY theory devel- These superoscillations have their origin in the subyacent oped above. Neutrino oscillations have been studied in 4 the context of supersymmetry [30–42], but not in a theory for fermions in time-domain, with time-dependent mass. The electron and muon flavour eigenstates can now be identified by the bi-spinor (22). In terms of the T-SUSY theory, the probability that an electron neutrino be ob- served later as a muon neutrino is given by Eq. (24). Un- der this T-SUSY theory, the oscillation occurs due to the supersymmetric nature of the neutrino in time-domain, as its mass is not constant. It is always one neutrino, with a unique mass, that it is oscillating between supersymmetric mass states. Let us model the neutrino oscillation considering the previous simple solution (19) for an ultra-relativistic fermion, with m k. By choosing Λ = k, and 2 (a) 0 = 3m0/(4k), we fulfill the conditions m0 and 2 δ k. In this case, Re{E±} ≈ k ± (m0/4k) cos(kt), and 2 Im{E±} = ∓(m0/2k) sin(kt). Thereby, α = 0, β(t) = 2 2 2 2 (m0/2k ) [1 − cos(kt)], and ρ(t) = −(m0/4k ) sin(kt). In this way, the total probability for flavour states change becomes 2 P (νe → νµ) ≈ sin 2θ m2 sinh2 0 [1 − cos(kt)] 2k2 m2 + sin2 0 sin(kt) . (27) 4k2

This probability is general under the assumption m0 k. However, for any time scale t 1/k , the probability (27) for the flavour states change becomes simply (b) 2 2 2 m0t P (νe → νµ) ≈ sin 2θ sin , (28) ˆ 2 −8 t1/k 4k FIG. 1: (a) Probability P = P/ sin 2θ (in units of 10 ) of flavour change (24) for neutrino superoscillations, in −2 which is equivalent to the standard result for the two- terms of mˆ = m0/k (in units of 10 ), and tˆ= kt. (b) neutrino oscillation problem [26–29], for neutrinos trav- Ratio r = P/Pt1/k (between probabilities (27) and elling a distance t ∼ L, and identifying k as the neutrino (28)) as a function of tˆ for mˆ = 10−3. The inset plot energy. shows this ratio (in units of 10−3) for 20 < tˆ < 30. The above neutrino superoscillations are not produced by different mass states, but because the appearing of the supersymmetry in time-domain due to the single neutrino non-constant mass. The behavior of this neutrino super- VI. DISCUSSION oscillation for larger times, described by probability of flavour change (27), is depicted in Fig. 1(a). We show When the mass of a fermion field is time-dependent, the probability for superoscillations Pˆ = P/ sin2 2θ as it then a supersymmetric in time-domain theory can be evolves in time tˆ = kt, increasing in amplitude as the consistently defined. This T-SUSY is the time analogue value of mˆ = m0/k 1 grows. of any non-relativistic supersymmetric quantum mechan- Probability (27) differs from the approximate probabil- ical theory. We have presented several solutions for dif- ity (28) in a notorious form as the time increases. This ferent time-dependent masses. It is remarkable that this is shown in Fig. 1(b), where the ratio between probabil- fermion theory contains solutions that mimic the light- ities (27) and (28), say r = P/Pt1/k, is presented as it like behavior studied for the bosonic T-SUSY theory. evolves in time, for mˆ = 10−3. The oscillation of this One of the main results extracted from this theory is ratio r shows that the complete probability neutrino su- the possibility for oscillation between states due to the peroscillation (27) can be larger on times 0 < tˆ < 3.7, to temporal changes of fermion mass. These superoscilla- later decreases. For long times, it is expected a depar- tion was applied to the two-neutrino oscillation problem, ture from the standard probability oscillation (28). In obtaining that the oscillation between flavour states can this way, the effect of neutrino flavour oscillation due to be explained by invoking only the time-dependence of a a T-SUSY theory could be tested for neutrinos travelling single neutrino mass (not different masses for different long times or large distances. states). In this way, this fermionic T-SUSY theory in- 5 troduces a different and interesting way to understand of a Majorana fermion. In this way, Majorana super- the neutrino oscillation problem under the new light of symmetry is possible only in space-domain. Therefore, supersymmetry. if T-SUSY theory can be used to explain the neutrino In the above framework for the T-SUSY theory, we oscillations, then it would work as an indicator of the have used the Dirac matrices basis. One can ask what Dirac fermionic nature of neutrinos, instead of Majorana happens in other bases. We remark that this T-SUSY fermions [43–46]. theory cannot be constructed in the Majorana basis. In fact, considering for example the Majorana matrices γ0 Similarly, the T-SUSY theory presented here is its sim- and γ1 in Eq. (3), then we obtain for the bi-spinor (1) plest, yet not trival, possible version. The extension to the equations other types of physical interactions is straightforward, and it is currently under investigation. However, in the dψ± = (ik ∓ m) ψ∓ , (29) specific case of electromagnetism, through minimal cou- dt pling, no supersymmetric modifications are found to the that do not allow us to construct the T-SUSY version cases presented here.

[1] P. G. O. Freund, Introduction to Supersymmetry, Cam- [23] S. C. Huot et al., JHEP12(2006)015. bridge Monographs on Mathematical Physics (Cam- [24] S. Chatrchyan et al. (CMS Collaboration), Phys. Rev. D bridge University Press, Cambridge, 1986). 88, 052017 (2013). [2] J. L. Feng, J.-F. Grivaz and J. Nachtman, Rev. Mod. [25] C. García-Meca, A. Macho Ortiz and R. Lorrente Sáez, Phys. 82, 699 (2010). Nature Commun. 11, 813 (2020). [3] L. J. Hall and L. Randall, Phys. Rev. Lett. 65, 2939 [26] B. Kayser, Neutrino mass, mixing and ﬂavor change, in (1990). Review of particle physics, Particle Data Group collabo- [4] Z. Chacko et al JHEP01(2000)003. ration, S. Eidelman et al., Phys. Lett.B 592, 145 (2004). [5] H. E. Haber and G. L. Kane, Phys. Rep. 117, 75 (1985). [27] S. Mondal, arXiv:1511.06752 (2015). [6] I. Aitchison , Supersymmetry in Particle Physics (An [28] G. Fantini, A. Gallo Rosso, V. Zema and F. Vissani, Elementary Introduction) (Cambridge University Press, Introduction to the Formalism of Neutrino Oscillations, 2007). The State of the Art of Neutrino Physics, 37 (2018), [7] S. A. Hojman and F. A. Asenjo, Phys. Rev. D 92, 083518 [29] S. T. Petcov, Advances in High Energy Physics 2013, (2015). article ID 852987 (2013). [8] J. I Díaz, J. Negro, L. M. Nieto and O. Rosas-Ortiz, J. [30] E. Ma, Supersymmetry and Neutrino Masses, PoS, Phys. A: Math. Gen. 32, 8447 (1999). corfu98, 047 (1999). [9] P. D. D’Eath, Supersymmetric Quantum Cosmology, [31] J. C. Romão and J. W. F. Valle, Nuclear Phys. B 381, Cambridge Monographs on Mathematical Physics (Cam- 87 (1992). bridge University Press, Cambridge, England, 1996). [32] O. C. W. Kong, Physics of Atomic Nuclei 63, 1083 [10] P. Vargas Moniz, Quantum Cosmology - The Supersym- (2000). metric Perspective - Vol. 1: Fundamentals (Springer, [33] O. C. W. Kong, Mod. Phys. Lett. 14, 903 (1999). Berlin, 2010). [34] R. Dermíšek and S. Raby, Phys. Rev. D 62, 015007 [11] P. D. D’Eath and D. I. Hughes, Phys. Lett. B 214, 498 (2000). (1988). [35] J. W. F. Valle, AIP Conference Proceedings 1200, 112 [12] V. Dzhunushaliev, J. Math. Phys. 49, 042108 (2008). (2010). [13] R. Allahverdi, K. Enqvist, J. Garcia-Bellido and A. [36] A. Faessler, Progress in Particle and Nuclear Physics 48, Mazumdar, Phys. Rev. Lett. 97, 191304 (2006). 151 (2002) [14] J. L. Hewett and J. D. Wells, Phys. Rev. D 55, 5549 [37] O. Haug, J. D. Vergados, A. Faessler and S. Kovalenko, (1997). Nucl. Phys. B 565, 38 (2000). [15] S. Yu, X. Piao, J. Hong and N. Park, Nature Communi- [38] T.-F. Feng and X.-Q. Li, Phys. Rev. D 63, 073006 (2001). cations 6, 8269 (2015). [39] N. Arkani-Hamed, L. Hall, H. Murayama, D. Smith and [16] F. Cooper, A. Khare, R. Musto, and A. Wipf, Ann. Phys. N. Weiner Phys. Rev. D 64, 115011 (2001). (N.Y.) 187, 1 (1988). [40] R. Adhikari and G. Omanovi`c,Phys. Rev. D 59, 073003 [17] F. Cooper, A. Khare, and U. Sukhatme, Supersymme- (1999). try in Quantum Mechanics (World Scientiﬁc, Singapore, [41] P. Ghosh and S. Roy, JHEP 04 (2009) 069. 2001). [42] F. Takayama and M. Yamaguchi, Phys. Lett. B 476, 116 [18] M. de Crombrugghe and V. Rittenberg, Ann. Phys. (2000). (N.Y.) 151, 99 (1983). [43] P. B. Pal, Am. J. Phys. 79, 485 (2011). [19] F. Correa and M. S. Plyushchay, Ann. Phys. 322, 2493 [44] S. M. Bilenky, J. Hošek and S. T. Petcov, Phys. Lett. (2007). 94B, 495 (1980). [20] K. B. Efetov, Advances in Physics 32, 53 (1983). [45] L. F. Li and Frank Wilczek, Phys. Rev. D 25, 143 (1982) [21] G. Junker, Supersymmetric Methods in Quantum, Statis- [46] T. Han and B. Zhang Phys. Rev. Lett. 97, 171804 (2006). tical and Solid State Physics (IOP Publishing, 2019). [22] M. Carena et al., Nucl. Phys. B 503, 387 (1997).