Effective lagrangian for a mass dimension one fermionic field in curved spacetime

Journal of High Energy Physics, Feb 2018

R.J. Bueno Rogerio, J.M. Hoff da Silva, M. Dias, S.H. Pereira

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Effective lagrangian for a mass dimension one fermionic field in curved spacetime

HJE ective lagrangian for a mass dimension one fermionic eld in curved spacetime R.J. Bueno Rogerio 0 1 3 J.M. Ho 0 1 Field Theory 0 1 0 09972-270 , Diadema, SP , Brazil 1 Guaratingueta, Departamento de F sica e Qu mica , 12516-410, Guaratingueta, SP , Brazil 2 and S.H. Pereira 3 Universidade Estadual Paulista (UNESP) , Faculdade de Engenharia In this work we use momentum-space techniques to evaluate the propagator G(x; x0) for a spin 1=2 mass dimension one spinor eld on a curved Friedmann-RobertsonWalker spacetime. As a consequence, we built the one-loop correction to the e ective lagrangian in the coincidence limit. Going further we compute the e ective lagrangian in the nite temperature regime. We arrive at interesting cosmological consequences, as time-dependent cosmological `constant', fully explaining the functional form of previous cosmological models. E ective Field Theories; Cosmology of Theories beyond the SM; Thermal - E 1 Introduction 2 3 4 5 6 consequences Final remarks 1 Introduction A short review on the e ective lagrangian formalism One-loop corrections to mass dimension one fermionic elds Cosmological implications 4.1 4.2 Limit H Finite-temperature corrections and its late time cosmological spinor elds has been explored in many areas, as accelerator physics [3{5], cosmology [6{17] and mathematically inclined areas as well [18{20]. In the early days of mass dimension one spinors, the theory was presented in such a way that a breaking Lorentz term taken part in the spin sums. As a net result the associated quantum eld was non-local and a there was preferred axis of symmetry. After all, the theory was shown to be invariant under SIM (2) and HOM (2) transformations [21], being then a typical theory carrying the very special relativity symmetries [22]. Quite recently, important advances the spinor dual theory has opened the possibility of circumvent the Weinberg no-go theorem, proposing a spinor eld of spin 1=2 endowed with mass dimension one, local, neutral with respect to gauge interactions, and whose theory respect Lorentz symmetries [23{25]. We should bring back to the scene the canonical Wigner work on the irreducible representations of the Poincare group [26]. By Poincare group, as usual, it is understood the semi-simple extension of the orthochronous proper Lorentz group e., when not only the orthochronous proper group is considered. This point was also analyzed by Wigner, in a less known paper [27]. Interestingly enough, Wigner found fermionic irrep's whose behavior under1 C; P and T is exactly what is expected for bosonic (quantum of) elds. For concreteness, while conventional wisdom stay that fermions belonging to the standard model (quarks and leptons) obey T 2 = 1 ((CP T )2 = 1) and bosons T 2 = +1 ((CP T )2 = +1), Wigner shown that, in the very realm of full Poincare symmetries, it is possible to have T 2 = +1 for fermions (leading also to (CP T )2 = +1) for fermions. It turns out that the eld taken into account in this work performs a realization of the (indeed odd) aforementioned fermionic representation, from where we can adduce method should be implemented [28]. Therefore, we split the quantum eld excitation from the background (classic) eld as where h stands for the quantum eld uctuation and is the classical background [40]. It is straightforward to see that eq.(2.1) can be written as where 2 m2 + R is the so-called e ective mass. Consequently, the lagrangian induced by one-loop e ects, say L (1), is given by the functional integral over the quantum elds exp i Z ~ dxL(1) = N Z dh exp i Z ~ dxL ; where N is just a normalization factor. Di erentiating both sides of eq.(2.4) with respect to 2, we have ! 1 2 + h; = lim = lim and therefore where G(x x0) is the Green's function that satis es the equation Hereupon, the one-loop e ective lagrangian is obtained after integrating the propagator in 2, in the coincidence limit x ! x0. Accordingly, this is the recipe that will form the foundation to compute one-loop correction for mass-dimension-one fermions, to be developed and presented here. 3 One-loop corrections to mass dimension one fermionic elds Consider the mass-dimension-one eld lagrangian in a curved space-time scenario L0 = 1 p 2 (3.5) (3.6) (3.7) (3.8) (3.9) The covariant derivatives, r , are de ned as r where denotes the spin connection given by 0 = 0 and j = for the Dirac matrices HJEP02(18)45 0(t) = 0 ; and j (t) = 1 a(t) j ; where stands for the Dirac matrices in Minkowsky spacetime in the Weyl representation. Towards to execute the background- eld method presented in refs. [40, 41], rstly it is necessary to split the eld in its classical background and the quantum uctuation , ! + . The one-loop e ects, whose net e ect here is encoded in L (1), will be governed by the functional integral over the elds as in eq.(2.4). Thus, the one-loop contribution to the e ective lagrangian is now related to the Green's function by = lim with the subtlety of taking the trace over the spinor indexes [40]. The appropriate lagrangian for mass dimension one fermionic elds in the curved space reads R : ; where we are ignoring self-coupling terms. The corresponding equation of motion for the quantum uctuation eld can be written as 2 2 = 0; where the e ective mass is given by 2 m2 + R 4 3 H2(t), being H(t) = a_=a the Hubble parameter. Notice the presence of a rst derivative term, coming from the spin sums. Even with a typical scalar eld lagrangian, the spinor character of the eld at hand shows up 2 E GE ( it; ~x; it0; ~x0) = ( g) 1=2 ( it; ~x; it0; ~x0); (3.10) { 4 { E2 is the Euclidean e ective mass. As stated in [43], this momentum-space representation is well-de ned only in a local neighborhood of x x0 = 0 (or better saying x ! x0). However, for the study of ultraviolet divergences in two-point (like Feynman propagator) or bivector quantities (as the energy-momentum tensor) when the coincidence limits are taken, or for the study of systems involving low-order quasilocal variations of the background eld (as we present here), results based on the use of the momentum-space representation technique are be valid. It would not be su cient for the consideration of processes involving rapid changes of the background eld, as, for example, pair productions and topological e ects due to phase transition. In these situations, the method based in the Heat-Kernel [45], which is a similar method providing the same net result [43], would be more pro table since it allows for perturbative treatments of more complicated situations, e.g., when higher derivative of the background eld is included. Now, we shall take advantage of the coincidence limit (the quasi-local situation) to use the momentum space quantization toolkit writing G(p) = Z dxeip (x x0) GE ( it; ~x; it0; ~x0); GE (x ! x0) = Z d4p Z 1 (2 )4 0 dse p2 3iH(it)p0+ a22(iit) jpj+ 2E s: Decomposing the momentum usually as p = (p0; jpj sin cos ; jpj sin sin ; jpj cos ), leading, then, to we have2 GE (x ! x0) = (3.11) (3.12) i Z 1 (2 )4 0 dse Z 2 0 d 2 s Z 1 E 0 d sin e jpj2+ Ha((iitt)) 0 jpj s : (3.14) Even being Gaussian integrals, the matrices present in the exponential makes the integration a bit laborious. The net result can be written in terms of incomplete gamma functions [46, 47] GE (x ! x0) = 4 identity matrix.The Euclidean e ective one-loop lagrangian is obtained by inserting eq.(3.15) into eq.(3.6), after to switch back from the Euclideanized 2As a remark we emphasize that had we working with the rst Elko formulation, whose relativist symmetries are governed by SIM (2) or HOM (2) Lorentz subgroups, then the equivalent momentum space Green function would be GE(x x0) = The one-loop lagrangian can thus be written as dse m2ssl 3 l=0 n=0 1 X d l ( 1)n 9 H2(t) 4 n! (1=2 + n) 1=2+n 0 dse m2ssl+n 5=2: (3.18) form, i. e., writing LE ! L and H2(it) ! H2(t), we have L (1) = ~ ( 1)l ( R 3H2(t))l and write the incomplete gamma function as a power series [47] In the sequel we expand the exponential as e Us = Pl1=0 dlsl, where d l (3.19) # (3.20) (3.21) Taking advantage of the complete gamma function we can express the e ective lagrangian as follows L (1) = ~ investigate its impacts on an e ective cosmological constant and scalar of curvature [ 37, 38 ]. Considering the general gravitational action Z Sgrav = d4xLgrav = 1 lagrangian given by the sum of eqs.(3.1), (3.19) and (3.20), reads In order to study some consequences of the above action into a cosmological context, we derive the FRW equations by means of its Lagrangian formulation, which basically consists in introducing a lapse function N (t) into the metric (3.2) as ds2 = N 2(t)dt2 a2(t)(dx2 + dy2 + dz2). The Euler-Lagrange equations obtained by variations of L with respect to N (t) and a(t) will furnish the two Friedmann equations, and at the end we make N (t) = 1. We assume that the spinorial eld corresponding to the matter content in L0 can be split as (x) = (t) (~x), with satisfying the normalization condition : = 1, a convenient fact justi ed in ref. [17]. Moreover, we shall investigate the case in which the spinor eld is homogeneously lling all the universe, so that ri (~x) = 0, and also the background evolution is smooth and adiabatic. Such condition is naturally satis ed at late time evolution of the universe. With these assumptions, the complete Lagrangian in the presence of the N (t) function is3 HJEP02(18)45 Le The corresponding Friedmann equations are: 2 )~ 64 2 H2 2 3 8 3 8 limits in what follows. p g = N (t)a3(t), a_2(t) a2(t) a_(t)N_ (t) . N(t) Written in this form we recognize the energy density of the eld and its quantum corrections on the right-hand side of (4.2) and the pressure and its quantum corrections on the right-hand side of (4.3). In the limit ~ ! 0 and = 0 we recover the torsion free equations obtained in [48]. The last term on the right of (4.2) corresponds to the quantum correction to H2 while the term proportional to m4~ is the correction to the cosmological constant term. Within the plethora of research possibilities we shall depict two interesting 3Tracing back the lapse function presence consequences, it is fairly simple to see that it amounts to be = a_ 2(t) 3 4 N2(t)a2(t) 1 and j = 2aN_(t()t) 0 j. We also have R = N2(t6)a(t) a(t) + { 7 { mpl H mpl m in ation occur. In this limit we have In order to look for possible consequences of the above equations into early universe, where quantum e ects may be relevant, we analyse the rst Friedmann equation (4.2) in the limit , which corresponds to an universe of about t = H 1 10 43s, before where we have introduced the Planck mass mpl = 1=p G. If the kinetic term is negligible we see that a positive contribution to the cosmological constant can be obtained if > 1=2, indicating that even in the absence of a cosmological constant term, i.e. = 0, a positive contribution proportional to H2 in the last term survives, which can be interpreted as a cosmological term induced just by quantum e ects and could be responsible for the in ationary phase at early universe stages. As the universe expands and H diminish the other terms starts to dominate, as the terms proportional to . 4.2 context in [48]. The Friedmann equations just reduces to one equation: with (t) = A + BH(t)2 and A and B constants given by A = m2pl 8 1 2 + 2 )~ ; Such behaviour is analogous to models having a time varying cosmological term, which are motivated by renormalization group to the quantum vacuum energy [49, 50]. Also, analytical solutions for the Friedmann equation (4.5) have also been obtained for phenomenological models with time varying cosmic terms [36]. The solution for the scale factor is indicating a de Sitter solution for the scale factor, which can indicate an accelerating solution for late time evolution. By supposing and m much smaller than mpl and due to negative contribution of the quantum correction into A, the net e ect in the evolution is to smooth the growth of the scale factor. { 8 { (4.4) consequences The possible extension of the e ective lagrangian to encompass nite-temperature e ects can be obtained from the formalism just applied previously. In order to do so, we shall impose a periodicity condition on the imaginary time y0 in the con guration space Green ! function. Then, performing the shift + n , where = 1=kBT and kB is the Boltzmann constant, and summing over n (see [42] and references therein) one is able to express the propagator as G (y; y0) = G(x + n u; x0); u = (1; 0; 0; 0): 1 X Taking advantage of the delta distribution and the Poisson summation formula [42] one obtain as the thermal Green function for the case at hand the expression 1 X n 1 eip0n = 2 1 X Aiming to extract some physical information about the correction just presented, we consider the expression (5.4) at present time. In such a situation the term containing the Hubble parameter H(t)= become insigni cant when compared to the previous terms. Therefore we are left with (1) L (0) = ~ where the subscript (0) denotes the present time context. Looking for a nite quantum correction, we proceed manipulating the sum using methods of the regularized zeta function and dimensional regularization. Thus, after some manipulations we are able to write L (0) in terms of zeta function as (1) L (0) = ~ or, in a more direct fashion yielding to a nite result. 3 3 4 ( 1)+ (0) + 1 { 9 { (5.1) (5.2) (5.3) (5.4) (5.5) (1) (5.7) HJEP02(18)45 1=kBT0 1= 0 H0 m In order to maintain the approximation in which the last term of (5.4) is negligible today, we must impose 2:7K, the rst term of (5.7) dominates if m 10 4eV, which puts an upper limit to the mass of the eld in order to have a nite 1= 0, which corresponds to 10 33eV m sum from (5.7). With such approximation, we have the mass of the eld constrained to In such limit the quantum correction we are interested is dominated by the rst term of (5.7), which must accompany the potential term of the bare lagrangian, namely the term m2 2, which shall corrects the Friedmann equation accordingly. Thus, looking for the present time slowly varying limit of (4.5), we have the term A corrected to With the upper limit to the mass imposed above and supposing of (5.8) dominates over the second and third one, and additionally A0 m, the last term BH02 from (4.6), showing that the quantum correction at nite temperature act exactly like a cosmological constant term, evolving as which could drive the recent phase of acceleration of the universe. The zero temperature limit corresponds to 0 ! 1, which cancels the contribution to accelerated expansion due to temperature e ects. Also, for larger values of the mass m of the eld (in the range above) the expansion is attenuated, showing the e ect of the gravitational attraction against the repulsion. Finally, in order to reproduce the expected value of 10 47GeV 4 for the energy density of the cosmological constant according to standard CDM model, the mass of the eld must be m ' 10 9eV, in good agreement to the limit range adopted above. 6 Final remarks In this work we completed the program of deriving the e ective lagrangian for a massdimension-one fermionic eld in a curved space-time with slowly varying background eld in a quasi-local situation. We also have computed the one-loop corrections in the nite temperature situation. More than an academic exercise, we highlight some cosmological applications of the present study, at least in the cases of smooth and adiabatic background evolution. In the zero temperature case, we have analyzed two di erent cases, corresponding to an early time universe and a late time universe. In the zero temperature limit we have seen that in the limit H mpl m and > 1=2 a positive quantum cosmological term appears naturally, which may be responsible for the accelerated in ationary expansion after about t H 1 10 43s, where the quantum e ects are to be relevant. For the late time evolution we studied the limit _ H ,  (5.8) (5.9) corresponds to a model with a time varying cosmological term already studied in [48], but here with the corresponding quantum correction, whose solution for the scale factor is also of accelerating type, an exponential growth. A most complete model should, eventually, also include additional matter components, as radiation and baryonic matter. In the case of nite temperature corrections, the cosmological scenery studied was that one corresponding to late time expansion with low temperature limit and also m H0, leading to an interesting contribution of the temperature correction acting like a cosmological constant term, and setting a limit to the mass of the eld of about 10 9eV in order to reproduce the value of the standard model. This contribution comes exclusively from the nite temperature correction. Acknowledgments RJBR would like to thanks Dr. Carlos Hugo Coronado Villalobos for useful discussions during the manuscript writing stage and to CAPES for the nancial support. SHP is grateful to CNPq | Conselho Nacional de Desenvolvimento Cient co e Tecnologico, Brazilian research agency, for nancial support (No. 304297/2015-1; 400924/2016-1). JMHS thanks to CNPq for nancial support (No. 304629/2015-4; 445385/2014-6). Open Access. 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R.J. Bueno Rogerio, J.M. Hoff da Silva, M. Dias, S.H. Pereira. Effective lagrangian for a mass dimension one fermionic field in curved spacetime, Journal of High Energy Physics, 2018, 145, DOI: 10.1007/JHEP02(2018)145