sign in sign up
<<
Home , CP violation

Physics Letters B 565 (2003) 27–32 www..com/locate/npe

Matter–antimatter generated by loop

Gaetano Lambiase a,b, Parampreet Singh c

a Dipartimento di Fisica “E.R. Caianiello”, Università di Salerno, 84081 Baronissi (Sa), Italy b INFN, Gruppo Collegato di Salerno, Italy c Inter-University Center for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411007, India Received 3 April 2003; received in revised form 6 May 2003; accepted 16 May 2003 Editor: M. Cveticˇ

Abstract We show that provides new mechanisms through which observed – asymmetry in the can naturally arise at temperatures less than GUT scale. This is enabled through the introduction of a new length scale L, much greater than Planck length (lP), to obtain semiclassical weave states in the theory. This scale which depends on the of the particle modifies the dispersion relation for different helicities of and leads to asymmetry.  2003 Published by Elsevier B.V. Open access under CC BY license.

PACS: 04.60.Pp; 11.30.Fs; 98.80.Cq

1. Introduction Loop quantum gravity which is based on the quantization of spacetime itself, results in a polymer like structure of quantum spacetime. The classical Many theories of quantum gravity are expected spacetime is a coarse grained form of underlying to bring non-trivial modifications to the underlying discrete quantum spacetime and one of the important spacetime near Planck scale. Loop quantum gravity, issues in loop quantum gravity is to understand the which is one of the candidate theories of quantum transition from discrete quantum spacetime to smooth gravity, predicts a discrete spectrum for geometrical classical spacetime. Though the low energy sector of operators [1]. However, inaccessibility of Planck scale loop quantum gravity and the transition to the classical in laboratories poses a challenge to test such predic- spacetime is yet to be completely understood, there tions. It is hence desired that a contact be made with have been some attempts in this direction to obtain the classical world through some semi-classical tech- the semi-classical states in the theory which include niques. This might also open a window to see the sig- construction of weave states which can approximate natures of quantum gravity at the level of effective the- 3-metrics [2] and via coherent states which peak ories which may differ from conventional low energy around classical trajectories [3]. For more discussion theories. on related issues we refer the reader to a recent review [4]. E-mail addresses: [email protected] (G. Lambiase), The coherent state approach to understand low [email protected] (P. Singh). energy sector is based on finding quantum states which

0370-2693  2003 Published by Elsevier B.V. Open access under CC BY license. doi:10.1016/S0370-2693(03)00761-5 28 G. Lambiase, P. Singh / Physics Letters B 565 (2003) 27–32 give minimum dispersion for the observables in the asymmetry is hence highly desirable. The origin of theory, whereas the weave state approach involves a matter–antimatter asymmetry in the Universe thus re- new length scale L lP such that for distances d L mains one of the unsolved puzzles whose resolution polymer structure of quantum spacetime becomes may be possible through some new aspects of physics manifest and for d  L one recovers continuous arising through a fundamental theory. There have been flat classical geometry. This approach was extended earlier proposals based on quantum gravity framework to study the construction of weave states describing to generate matter–antimatter asymmetry, like from gravity coupled to massive spin-1/2 Majorana fields primordial spacetime foam [14], quantum gravity de- in a series of important papers by Alfaro, Morales- formed uncertainty relations [15] and string based sce- Técotl, and Urrutia (AMU) [5–7]. This new length narios [16]. In this Letter, we would like to present scale modifies the dispersion relation for different another interesting scenario arising out from quan- helicities of fermions and breaks Lorentz invariance tum gravity through weave states. We would show in the theory. that weave states of spin-1/2 fields provide a nat- Apart from loop quantum gravity, deformations ural mechanism to generate matter–antimatter asym- of the Lorentz invariance manifest by means of a metry at temperatures of the order of reheating tem- slight deviation from the standard dispersion relations perature of inflation, far below the GUT tempera- of particles propagating in the vacuum have been ture. suggested in various ways, see, for example, [8,9]. The modifications to dispersion relation may arise if the underlying spacetime is non-commutative [10]. 2. Lepton asymmetry in AMU formalism These theories which are characterized by a non- commutativity parameter of dimensions of square of In the weave state approach, the goal is to find a length, may serve as a description for foamy structure loop state which approximates a classical geometry of quantum spacetime. Similar modifications have also at a scale much larger than lP. A semi-classical been studied in the framework of [11, weave state corresponding to Majorana fermions is 12]. These approaches foresee a dispersion relation in characterized by a scale length L such that lP L  vacuo of particles of the form (we shall use natural λD = 1/p,whereλD is the de Broglie wavelength units c = 1 = h¯) of the and p its corresponding momentum. For the Dirac equation with quantum corrections to E2 ≈ p2 + m2 + f(M,pl ), (1.1) P be properly defined on a continuous flat spacetime where f(x) is a model dependent function, M fixes arising through weave state construction, it is required a characteristic scale not necessarily determined by that the scale length L  1/p [5]. Such a scale is −19 −1 Planck length lP ∼ 10 GeV ,andplP 1. As a known as mobile scale [7] which is different for consequence of Eq. (1.1), the quantum gravitational different fermionic species and the upper bound on medium responds differently to the propagation of L corresponding to the weave state of a particular particles of different energies. fermion is set by the momentum of that fermion. One With the breakdown of the Lorentz invariance in the may also treat this scale as a universal scale and obtain theory, CPT violation is expected and so new mecha- bounds on it by observations [17–19], however, in this nisms to generate observed matter–antimatter asym- Letter we would restrict to the case of L as a mobile metry in the Universe. The matter–antimatter asym- scale. metry in the Universe is conventionally understood, The introduction of scale L leads to modifications for example, through processes occur- in dispersion relation which have been studied in var- ring at GUT or electroweak scales. However, most of ious interesting contexts [20–22]. Similar phenomena these conventional mechanisms to generate this asym- have also been studied for photons [6,23,24]. The dis- metry in or extensions of it run into persion relation with leading order terms in lP and L one or another problem [13], like inflation would sig- can be written as [18] nificantly dilute the asymmetry produced during GUT 2 = + 2 + 4 ± + 2 era. Any low temperature mechanism to generate this E± (1 2α)p ηp 2λp m (2.1) G. Lambiase, P. Singh / Physics Letters B 565 (2003) 27–32 29 with particles and achieve the thermodynam- ical equilibrium.     l 2 l GUT theories offer an ideal setting for Sakharov’s α = κ P ,η= κ l2,λ= κ P , conditions to be satisfied [26]. Baryon number viola- 1 L 3 P 5 L2 2 tion occurs in these theories since gauge bosons me- (2.2) diate interactions that transform into and antiquarks. C is maximally violated in the electro- where κ1,κ3 and κ5 are of the order of unity. Here ‘+’and‘−’ refer to two helicity states of the fermion. weak sector, and CP violation follows by making the We would specialize to the limiting case of Majorana coupling constants of lepto- gauge bosons com- fermions with vanishingly small mass, m → 0. In plex. Finally, out of equilibrium condition is achieved this way we can treat the fermions as Weyl particles. by the expansion of the universe when the reaction 2 rates become lower than the Hubble expansion rates We would further neglect lP terms in comparison to other dominating terms in the above dispersion at some freeze-out temperature. Such a temperature relation. Thus, the dispersion relation can be rewritten is characterized by the decoupling temperature Td , which, in the GUT baryogenesis scenario, is given by as 16 Td ∼ 10 GeV.However, GUT baryogenesis runs into problems because inflation occurring at similar tem- 2 2 E± = p ± 2λp. (2.3) perature dilutes the . For baryons to be produced after inflation it is necessary to re- It should be noted that though the above modifica- heat the Universe to the scale of MGUT which is un- tion to the dispersion relation is of linear in momen- realistic in inflationary scenarios. In fact, bounds on tum, the correction term effectively behaves as the production give the reheating temperature 8 10 one cubic in momentum. This is because L is a mo- TR of the order of 10 –10 GeV [27], whereas in bile scale and its upper bound scales as 1/p. Thus, SUSY inflation models this may be raised to 1012 GeV the above correction, which is similar to other cu- [28]. Similarly, processes like electroweak baryogen- bic in momentum modifications to dispersion rela- esis and suffer from problems like very tion [4], dies out rapidly at low momenta. We would small region of parameter space which can yield asym- now discuss the implications of this dispersion rela- metry and lack of direct measurement of relevant pa- tion for the case of , where the helicity dis- rameters [13]. persion can be casted in terms of the difference be- It is worth to quote some alternative mechanisms tween energy levels of particle and states proposed in literature which are not based on quan- which leads to a net difference in their number den- tum gravity. As observed in Refs. [29,30], if the CPT sities and hence lepton asymmetry in this frame- and the baryon number is violated, a baryon work. asymmetry could arise in thermal equilibrium. This Matter–antimatter asymmetry is generally under- mechanism to generate baryon asymmetry has been stood through Sakharov conditions who in his sem- applied in different contexts: the spontaneous break- inal paper [25], showed that to generate the non- ing of CPT induced by the coupling of baryon number zero baryonic number to entropy ηB ∼ (2.6–6.2) × current with a scalar field [29]; baryogenesis asym- 10−10 from a baryonic symmetric universe, the fol- metry generated from primordial tensor perturbation lowing requirements are necessary: (1) baryon num- [31] and matter–antimatter asymmetry through inter- ber processes violating in particle interactions; (2) action between gravitational curvature and fermionic C and CP violation in order that processes gener- spin [32]. For other mechanisms related to the lepton ating B are more rapid with respect to B¯ ; (3) out asymmetry, see Ref. [33] and reference therein, as well = of the equilibrium: since mB mB¯ , as follows from as Ref. [34]. CPT symmetry, the equilibrium space phase density In loop quantum gravity, the different dispersion re- of particles and antiparticles are the same. To main- lations of particles having different helicity determines tain the number of baryon and antibaryon different, a deviation from thermal equilibrium between neutri- = = ¯ i.e., nB nB¯ , the reaction should freeze out before nos and antineutrinos, n(ν) n(ν),wheren(ν) and 30 G. Lambiase, P. Singh / Physics Letters B 565 (2003) 27–32 n(ν)¯ are the number density of positive helicity neu- It vanishes as λ = 0.1 Similar results hold for anti- trinos and negative helicity antineutrinos, respectively. neutrinos: We would further assume that there are no additional χ √ √ gT 3 x( x2 + z + z)2 1 mechanisms which give rise to asymmetry. In n(ν)¯ = dx √ , 2 x + such a case, the deviation from the chemical equilib- 2π √ x2 + z e 1 rium, which generates the baryon asymmetry, occurs 2 z   only due to loop quantum gravity effects and the ex- λ 2 F(ν)¯ = 2 HT3 pansion of the universe. If neutrinos are produced with T energy E, then the dispersion relation (2.3) gives  χ √ √  d g x( x2 + z + z)2 1 × dx √ .  2 x + dz 2π √ x2 + z e 1 E2 = p2 + 2λp ⇒ p = E2 + λ2 − λ, (2.4) 2 z (2.8) for neutrinos, and Thus the net neutrino asymmetry generated via loop quantum gravity effects would become

 "n =|n(ν) − n(ν)¯ | 2 = 2 − ⇒ = 2 + 2 + E p 2λp p E λ λ, (2.5) χ 2gλT 2 x gT 3 = dx + I(z) for antineutrinos. Note that energy dispersion relation π2 ex + 1 2π2 forbids antineutrinos to have p ∈[0, 2λ], whereas no 0  ∞ such restriction arises for neutrinos. This is purely a 2gλT 2 π2 (− exp(χ))n ≈ + 12 loop quantum gravity effect induced through quantum 2 2 π 12 = n structure of spacetime which seemingly favors one n 1  12 χ 6 helicity over another. + ln(1 + e ) − , (2.9) The number density of neutrinos at the equilibrium LT (LT)2 for a given temperature T is (for kB = 1) where √ 2 z √ √ 2 + − 2 χ √ √ x( x z z) 3 2 2 I(z)≡ dx √ . (2.10) gT x( x + z − z) 1 2 + x + n(ν) = dx √ , (2.6) x z(e 1) 2π2 x2 + z ex + 1 0 0 In evaluating (2.9) we have neglected the contribution coming from the I(z)-term since at low temperatures where χ = 1/(LT), x = E/T , z = (λ/T )2, T satisfies with respect to Planck’s one it is expected to be very ˙ =− =˙ the relation T HT,andH a/a,beinga(t) the small compared to other terms. If we note that L  λD scale factor of the universe, [30]. The dot stands for the derivative with respect to the cosmic . The 1 departure from the chemical equilibrium caused by the In absence of loop quantum gravity corrections, the deviation from the chemical equilibrium occurs only if particles are massive. expansion of the universe is encoded in the quantity In fact, being F = 3Hn+˙n [30], which turns out to be ∞ gT 3 x2 dx n =  , 2   2π e x2+(m/T )2 + 1 2 0 λ 3 F(ν) = 2 HT F T the function becomes [30]    ∞  2   χ √  m d g x2 dx  √ F = 2 HT3  , (2.7) 2 + − 2 2 d g x( x z z) 1 T dy 2π e x2+y + 1 × dx √ . 0 dz 2π2 x2 + z ex + 1 0 where y = (m/T )2 and it vanishes as m = 0 [30,35]. G. Lambiase, P. Singh / Physics Letters B 565 (2003) 27–32 31 and use the upper bound L ∼ 1/p¯ ∼ T ,wherep¯ is This leads to asymmetry between matter and antimat- the de Broglie momenta of neutrinos at a particular ter species and yields the observed value at around re- temperature, we can estimate the neutrino asymmetry heating temperatures. Our proposal introduces a way arising at that temperature. for generation of matter–antimatter asymmetry via Near GUT temperatures T ∼ 1016 GeV, the ratio loop quantum gravity, whose complete analysis would of neutrino asymmetry to entropy density, "n/s (s ∼ require relaxing the massless limit and secondly tak- 0.44g∗T 3, with g∗ ∼ 102 [26]), turns out to be of ing into account various standard model interactions the order of 10−5 which would, however, be washed in unison with loop quantum gravity. Then we shall out by inflation. Interesting temperatures would be be able to know how the above mechanism to gener- near reheating temperatures2 of the order of 1010– ate matter–antimatter asymmetry contributes relative 1011 GeV where this ratio would become of the to other processes. This opens up a new arena to make order of 10−10. At lower temperatures the amount phenomenological studies in loop quantum gravity in of asymmetry generated would keep on decreasing future. till it becomes negligible, though the asymmetry It is a remarkable phenomena that quantum struc- generated at reheating temperature would hold till ture of spacetime itself may generate matter–anti- the neutrinos finally decouple. This lepton asymmetry matter asymmetry in the universe. In fact, this might would lead to the baryon asymmetry through various be a generic feature of theories of quantum gravity. It GUT and electroweak processes and thus contribute to reflects that quantum gravity may lead to effects occur- the existing mechanisms to produce matter–antimatter ring at lower energy scales, specially in the desert be- asymmetry in the Universe. tween electroweak and Planck scale, which may pro- vide natural answers to some unsolved problems.

3. Conclusion Acknowledgements An intriguing prediction of modern approaches to quantum gravity is a slight departure from Lorentz’s P.S. thanks Council for Scientific and Industrial invariance, which manifests in a deformation of the Research for a research grant. The authors thank the dispersion relations of photons and fermions. Such referee for constructive comments. results have been indeed suggested in loop quan- tum gravity [5,6,24], string theory [11,12] and non- commutative geometry [10]. The former is endowed References with a scale length characterizing the scale on which new effects are non-trivial, thus to wonder if there exist [1] C. Rovelli, Living Reviews, Vol. 1, Loop Quantum Gravity, at different scenarios where these effects become testable http://www.livingreviews.org/articles; (see [17,36]) is certainly of current interest. T. Thiemann, gr-qc/0110034; T. Thiemann, gr-qc/0210094; In this Letter we have shown that such modifica- R. Gambini, J. Pullin, Loops, Knots, Gauge Theories and tions induced by loop quantum gravity might help Quantum Gravity, Cambridge Univ. Press, Cambridge, UK, to put some light on unsolved problems like matter– 1996; antimatter asymmetry in standard model. Application J. Pullin, hep-th/9301028; of weave states for Majorana fermions naturally leads C. Rovelli, Class. Quantum Grav. 8 (1991) 1613; A. Ashtekar, Lectures on New Perturbative Canonical Gravity, to difference in energies for different chiralities which World Scientific, Singapore, 1991. may be interpreted as difference in particle and an- [2] A. Ashtekar, C. Rovelli, L. Smolin, Phys. Rev. Lett. 69 (1992) tiparticle energies for the case of massless neutrinos. 237. [3] T. Thiemann, gr-qc/0110034; M. Varadarajan, J.A. Zapata, Class. Quantum Grav. 17 (2000) 2 In this case λ/T ∼ lP/L and the corrections to the neutrino 4085; 3 asymmetry due to I(z) termwouldgoas(lP/L) which for the H. Sahlmann, T. Thiemann, O. Winkler, Nucl. Phys. B 606 − temperature range of 1011 GeV would be of the order of 10 24. (2001) 401; Hence, our approximation in Eq. (2.9) is justified. T. Thiemann, gr-qc/0206037. 32 G. Lambiase, P. Singh / Physics Letters B 565 (2003) 27–32

[4] L. Smolin, hep-th/0303185. [20] F.W. Stecker, S.L. Glashow, Astropart. Phys. 16 (2002) 97; [5] J. Alfaro, H.A. Morales-Téctol, L.F. Urrutia, Phys. Rev. O.C. De Jager, F.W. Stecker, Astrophys. J. 566 (2002) 738. Lett. 84 (2000) 2318. [21] D. Sudarsky, L.F. Urrutia, H. Vucetich, Phys. Rev. Lett. 89 [6] J. Alfaro, H. Morales-Técolt, L.F. Urrutia, Phys. Rev. D 65 (2002) 231301. (2002) 103509; [22] T.J. Konopka, S.A. Major, New J. Phys. 4 (2002) 57. J. Alfaro, H. Morales-Técolt, L.F. Urrutia, in: R.T. Jantzen, [23] R.J. Gleiser, C.N. Kazameh, Phys. Rev. D 64 (2001) 083007. V. Gurzadyan, R. Ruffini (Eds.), Proceedings of the Ninth [24] R. Gambini, J. Pullin, Phys. Rev. D 59 (1999) 124021. Marcel Grossmann Meeting on General Relativity, World [25] A.D. Sakharov, JETP Lett. 5 (1967) 24. Scientific, Singapore, 2002. [26] E.W. Kolb, M.S. Turner, The Early Universe, Addison– [7] J. Alfaro, H. Morales-Téctol, L.F. Urrutia, Phys. Rev. D 66 Wesley, Reading, MA, 1989. (2002) 124006. [27] S. Sarkar, Rep. Prog. Phys. 59 (1996) 1493. [8] G. Amelino-Camelia, J. Ellis, N.E. Mavromatos, D.V. [28] J. McDonald, Phys. Rev. D 61 (2000) 083513. Nanopoulos, S. Sarkar, 393 (1998) 763; [29] A. Cohen, D. Kaplan, Phys. Lett. B 199 (1987) 251; For a review, see: G. Amelino-Camelia, Lect. Notes Phys. 541 A. Cohen, D. Kaplan, Nucl. Phys. B 308 (1988) 913. (2000) 1. [30] A.D. Dolgov, Ya.B. Zeldovic, Rev. Mod. Phys. 53 (1981) 1. [9] D.V. Ahluwalia, hep-ph/0212222. [31] S. Mohanty, B. Mukhopadhyay, A.R. Prasanna, hep- [10] G. Amelino-Camelia, S. Majid, Int. J. Mod. Phys. A 15 (2000) ph/0204257. 4301; [32] P. Singh, B. Mukhopadhyay, Mod. Phys. Lett. A 18 (2003) A. Matusis, L. Susskind, N. Toumbas, JHEP 0012 (2000) 02; 779; G. Amelino-Camelia, Int. J. Mod. Phys. D 11 (2002) 35. B. Mukhopadhyay, P. Singh, gr-qc/0301002; [11] V.A. Kostelecky, S. Samuel, Phys. Rev. D 39 (1989) 683; B. Mukhopadhyay, P. Singh, gr-qc/0303053. D. Colladay, V.A. Kostelecky, Phys. Rev. D 58 (1998) 116002; [33] A.D. Dolgov, Phys. Rep. 370 (2002) 333. D. Colladay, V.A. Kostelecky, Phys. Rev. D 55 (1997) 6760; [34] M. Claudon, L.J. Hall, I. Hinchliffe, Nucl. Phys. B 241 (1984) V.A. Kostelecky, R. Lehnert, Phys. Rev. D 63 (2001) 065008; 309; O. Bertolami, C.S. Carvalho, Phys. Rev. D 61 (2000) 103002. K. Yamamoto, Phys. Lett. B 168 (1986) 341; [12] J. Ellis, K. Farakos, N.E. Mavromatos, V.A. Mitsou, D.V. O. Bertolami, G.G. Ross, Phys. Lett. B 183 (1987) 163; Nanopoulos, Astrophys. J. 535 (2000) 139; A. Linde, Phys. Lett. B 70 (1977) 306; J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, G. Volkov, Gen. V.A. Kuzmin, V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. Relativ. Gravit. 32 (2000) 1777; B 191 (1987) 171. J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Phys. Rev. D 61 [35] D. Toussaint, S.B. Treiman, F. Wilczek, A. Zee, Phys. Rev. (2000) 027503. D 19 (1979) 1036. [13] M. Dine, A. Kusenko, hep-ph/0303065. [36] L.F. Urrutia, Mod. Phys. Lett. A 17 (2002) 943; [14] D.V. Ahluwalia, M. Kirchbach, Int. J. Mod. Phys. D 10 (2001) J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Phys. Rev. D 63 811. (2001) 124025; [15] G. Amelino-Camelia, Mod. Phys. Lett. A 12 (1997) 1387. J. Ellis, E. Gravanis, N.E. Mavromatos, D.V. Nanopoulos, gr- [16] O. Bertolami, D. Colladay, A.V. Kostelecky, R. Potting, Phys. qc/0209108; Lett. B 395 (1997) 178. T. Jacobson, S. Liberati, D. Mattingly, Phys. Rev. D 66 (2002) [17] J. Alfaro, G. Palma, Phys. Rev. D 65 (2002) 103516. 081302; [18] J. Alfaro, G. Palma, hep-th/0208193. T. Jacobson, S. Liberati, D. Mattingly, hep-ph/0209264; [19] G. Lambiase, Mod. Phys. Lett. A 18 (2003) 23; G. Lambiase, Gen. Relativ. Gravit. 33 (2001) 2151; G. Lambiase, gr-qc/0302053; S. Sarkar, Mod. Phys. Lett. A 17 (2002) 1025. The general formalism can be found in: G. Lambiase, Phys. Lett. B 560 (2003) 1.