Quantum Gravity Corrections to Neutrino Propagation

VOLUME 84, NUMBER 11 PHYSICAL REVIEW LETTERS 13MARCH 2000

Jorge Alfaro,1,* Hugo A. Morales-Técotl,2,† and Luis F. Urrutia3,‡ 1Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile 2Departamento de Física, Universidad Autónoma Metropolitana Iztapalapa, A.P. 55-534, México D.F. 09340, México 3Departamento de Física de Altas Energías, Instituto de Ciencias Nucleares Universidad Nacional Autónoma de México, A.P. 70-543, México D.F. 04510, México (Received 27 September 1999) Massive spin-1͞2 fields are studied in the framework of loop quantum gravity by considering a state approximating, at a length scale L much greater than Planck length ᐉP, a spin-1͞2 field in flat spacetime. The discrete structure of spacetime at ᐉP yields corrections to the field propagation at scale L . Neutrino bursts (p¯ ഠ 105 GeV) accompanying gamma ray bursts that have traveled cosmological distances L are considered. The dominant correction is helicity independent and leads to a time delay of order 4 ͑p¯ ᐉP ͒L͞c ഠ 10 s. To next order in p¯ ᐉP, the correction has the form of the Gambini and Pullin effect 21 2ᐉ for photons. A dependence Los ~ p¯ P is found for a two-flavor neutrino oscillation length.

PACS numbers: 04.60.Ds, 14.60.Pq, 96.40.Tv, 98.70.Rz

The fact that some gamma ray bursts (GRB) origi- departures from perfect nondispersiveness of ordinary nate at cosmological distances (ഠ1010 light years͒ [1], vacuum, helicity depending corrections for propagating together with time resolutions down to submillisecond waves. In the present work, the case of massive spin-1͞2 scale achieved in recent GRB data [2], suggests that it is particles in loop quantum gravity is studied also semiclas- possible to probe fundamental laws of physics at energy sically. They could be identified with the neutrinos that 19 scales near to Planck energy EP ෇ 1.3 3 10 GeV would be produced in GRB. Central ideas and results are [3,4]. Furthermore, sensitivity will be improved with presented, whereas details will appear elsewhere [11]. HEGRA and Whipple air Cherenkov telescopes and by Loop quantum gravity [5] uses a spin networks basis, AMS and GLAST spatial experiments. Thus, quantum labeled by graphs embedded in a three dimensional space gravity effects could be at the edge of observability [3,4]. S. Physical predictions hereby obtained are a “polymer- Now, quantum gravity theories imply different spacetime like” structure of space [12] and a possible explanation of structures [4,5] and it can be expected that what we black hole entropy [13]. A first attempt to couple spin-1͞2 consider flat spacetime can actually involve dispersive fields to gravity, along these lines, was made in [14], and effects arising from Planck scale lengths. Such tiny effects a generalization to the spin networks basis has been devel- might become observable upon accumulation over travels oped in [15]. A significant progress in the loop approach to through cosmological distances by energetically enough quantum gravity was made by Thiemann, who put forward particles like cosmological GRB photons. a consistent regularization procedure to properly define the The most widely accepted model of GRB, the quantum Hamiltonian constraint of the full theory, which so-called fireball model, predicts also the generation of includes the Einstein plus matter (leptons, quarks, Higgs 1014 1019 eV neutrino bursts (NB) [6,7]. Yet, another particles) contributions [16]. It is based on a triangulation GRB model based on cosmic strings requires neutrino pro- of space with tetrahedra whose sides are of the order of duction [8]. Present experiments to observe high energy ᐉP. The cornerstone of Thiemann’s proposal is the incor- astrophysical neutrinos such as AMANDA, NESTOR, poration of the volume operator as a convenient regulator, Baikal, ANTARES, and Super-Kamiokande, for example, since its action upon states is finite. Having at our disposal will detect at best only one or two neutrinos in coinci- a regularized version of the quantum Hamiltonian describ- dence with GRB’s per year. The planned neutrino burster ing fermions coupled to Einstein gravity, we will further experiment (NuBE) will measure the flux of ultrahigh need a loop state which approximates a flat 3-metric on S, energy neutrinos (.10 TeV) over a ϳ1 km2 effective at scales L much larger than the Planck length. For pure area, in coincidence with satellite measured GRB’s [9]. It gravity this state is called weave [17]. A flat weave jW͘ is expected to detect ഠ20 events per year, according to is characterized by a length scale L¿ᐉP, such that for the fireball model. Hence, one can study quantum gravity distances d $ L the continuous flat classical geometry is effects on astrophysical neutrinos that might be observed regained, while for distances d øL the quantum loop or, the other way around, such observations could be used structure of space is manifest. The stronger the inequal- to restrict quantum gravity theories. ity d ¿ ᐉP holds, the more isotropic and homogeneous In this Letter, the loop quantum gravity framework is the weave looks. For example, the metric operator gˆ ab ᐉ ͗ ͘ ෇ ͑ P ͒ adopted. In this context, Gambini and Pullin studied light satisfies Wjgˆ abjW dab 1 O L . Now, a generaliza- propagation semiclassically [10]. They found, besides tion of such an idea to include matter fields is required.

Z a For our analysis it suffices to exploit the main features that ͑1͒ 3 pEi T H :෇ d x ͑ip tiDaj1c.c.͒ , (1) a flat weave with fermions must have: in particular, it must 2 det͑g͒ reproduce the Dirac equation in flat spacetime, and this is just the basis of our approximation scheme. It is denoted and other terms [16] whose contribution is summarized by jW, j͘, has a characteristic length L , and is referred to in (7) below. Their analysis is an extension of the one ͑1͒ simply as a weave. for H given here and will be spelled out in [11]. We ៬ ෇ i ៬ The use of Thiemann’s regularization for Einstein Dirac use t 2 2 s, the latter being the standard Pauli ma- theory naturally allows the semiclassical treatment here trices. The fermion field is a Grassmann valued Majo- ء pursued; expectation values with respect to jW, j͘ are con- rana spinor CT ෇ ͑cT , ͑2is2c ͒T ͒. The two component sidered thereby. They are expanded around relevant ver- spinor c has definite chirality and it is a scalar under gen- tices of the triangulation and a systematic approximation eral coordinate transformations. Hence, (1) is not parity ෇ ͓ ͑ ͔͒1͞4 is given involving the scales ᐉP ø lD ø lC, the last two invariant. The configuration variable is j det q c, corresponding to, respectively, De Broglie and Compton which is a half density. The corresponding momentum is, ء wavelengths of a light fermion. Corrections come out at with this choice, p ෇ ij ; similarly as in flat space. The a ͑ ͒ this level. gravitational canonical pair consists of Ei and the SU 2 - j ab a ib The Hamiltonian constraint for a spin-1͞2 field coupled connection Ab (D ), where ͑detg͒g ෇ Ei E . to gravity consists of a pure gravity contribution, a kinetic Upon regularization [16], the expectation value of (1) fermion term, namely, with respect to the weave becomes X X h¯ 8 ͗W, jjHˆ ͑1͒jW, j͘ ෇ 2 eijk eIJK ᐉ4 E͑y͒ Ω4 P y[V͑g͒ sI >sJ >sK ෇y ≠ AB 3 ͗W, jjjˆ ͓y1s ͑D͔͒ ͓t h ͑ ͔͒ wˆ ͑y͒wˆ ͑y͒ jW, j͘ B K A͑ ͒ k sK D iID jJD ≠j y æ ≠ 2 ͗W, jj͑t jˆ ͒A͑y͒ wˆ ͑y͒wˆ ͑y͒ jW, j͘ 2 c.c. . (2) k ≠jA͑y͒ iID jJD

Here an adapted triangulation of S to the graph g of the the approximation we think of space as being made up of weave state jW, j͘ is adopted. Auxiliary quantities used boxes of volume L 3, whose center is denoted by x៬. Each 21 p w ෇ ͑t h ͑ ͓͒h V ͔͒ V box contains a large number of vertices of the weave, but are ˆ kID Tr k sI D sI ͑D͒, y , where y is the volume operator restricted to act upon vertex y. hs͑D͒ are is considered infinitesimal in the scale where the holonomies along segments, s, of edges forming tetrahedra space can be regarded as continuous, so that we take in the triangulation D [16]. VP͑g͒ stands for the set of ver- L 3 ഠ d3x. Let Fˆ ͑y͒ be a fermionic operator which tices of g. The second sum, sI >sJ >sK ෇y, involves triples produces the slowly varying (inside the box) function ͑៬͒ 1 ˆ ͑ ͒ of segments sI , sJ , sK intersecting at y. Notice that one ac- F x , i.e., LølD. Also let ᐉ3 G y be a gravitational tually averages over E͑y͒ ෇ n ͑n 2 1͒͑n 2 2͒͞6 pos- P y y y operator with averageP within the box G͑x៬͒. The weave is sible triangulations (one for each triple of edges) when the 8 such that ͑ ͒ ͗W, jjFˆ ͑y͒Gˆ ͑y͒jW, j͘ ෇ vertex y is reached by n edges (the valence) of the graph. P P y[V g E͑y͒ y ͑៬͒ ᐉ3 8 ͗ 1 ˆ ͑ ͒ ͘ ෇ Box͑x៬͒ F x y[Box͑x៬͒ P ͑ ͒ W, jj ᐉ3 G y jW, j To estimate (2) we associate to it c-number quantities R E y P 3 ͑៬͒ ͑៬͒ respecting the index structure, together with appropriate S d xF x G x . Notice that the tensorial and scale factors arising from dimensional reasons and, most Lie-algebra structure should come out from flat spacetime 0 ia k cde klm important, in line with the weave state approximating flat quantities exclusively, i.e., E , t , ≠b, e , e , where ab 0 ia 0 b space with fermions. This amounts to an expansion of ex- d ෇ E Ei . pectation values around vertices of the weave. The explicit In order to regain the flat spacetime kinetic term of the form is taken from the expansion the involved operators fermion Hamiltonian, we demand jW, j͘ to fulfill a a would have in powers of the segments s , js j ϳ ᐉP, a pro- ≠ cedure justified for weave states. Useful quantities coming ͗W, jjjˆ ͑y͒ ͑t D ͑j͒͒ABwˆ ͑y͒wˆ ͑y͒jW, j͘ ͑ ͒ ෇ a a b B ≠jA͑y͒ k c ia jb in by expanding wˆ iID y sI wˆ ia 1 sI sI wˆ iab 1 ...,for ∑ ∏ instance, are i 2 ഠ jB͑y͒pA͑y͒ᐉPL 1 p 1 p h¯ wˆ ෇ ͓A , V ͔,ˆw ෇ e ͓A , ͓A , V ͔͔͔ , ∑ ∏ ia 2 ia y iab 8 ikl ka lb y ᐉ3 (3) 3 P ͑t ≠͑j͒͒AB 0E ͑y͒ 0E ͑y͒ . (4) L 2 k c ia jb whose contribution to the average in the weave is The second parenthesis here dictates the overall structure: p ᐉ3͞2 P estimated by considering that of Aia and Vy to be of the (3) indicates that each wˆ ia͑y͒ contributes a factor of , 3͞2 L order of ϳ1͞L and ϳᐉP , respectively. To proceed with since the connection scales with 1͞L (large length limit 2319 VOLUME 84, NUMBER 11 PHYSICAL REVIEW LETTERS 13MARCH 2000 p X ᐉ3͞2 i L 2 8 1 ) flat spacetime), and Vy contributes a factor of P . eijkeIJK sa sb sc sdse ᐉ3 ͑ ͒ K K K I J Independence on L of the final form of (4) gives the 4 P y[V͑g͒ E y 3! structure of the first parenthesis. The notation ͑j͒ stands 3p͑y͒tk≠a≠b≠cj͑y͒͗W, jjwˆ id͑y͒wˆ je͑y͒ jW, j͘ for acting only upon j. By expanding (2) at different Z ᐉ2 3 ͑៬͒ 0 kc 2 ͑៬͒ orders in powers of s and using (4), one can systematically ! 2ik8 P d x p x tk E ≠c= j x . (6) determine all possible contributions. Some examples of A similar treatment can be performed for every contri- correction terms are bution to the Einstein-Dirac Hamiltonian constraint [11]. X i L 2 8 It is important to stress that the prediction of the values of eijkeIJK sasbscsd p ͑y͒≠ j ͑y͒ the corresponding coefficients k would require a precise 4 ᐉ3 E͑y͒ I I J K A d B i P y[V͑g͒ definition of the flat weave (or even better, a Friedman- AB 3 ͗W, jj͑tk͒ ͕wˆ iab,ˆwjc͖ jW, j͘ Lemaitre-Robertson-Walker weave), together with a de- ᐉ Z tailed calculation of the matrix elements. Instead, within P 3 i 0 kd ᐉ2 ! k5 d x p͑x៬͒tk E ≠dj͑x៬͒ , (5) the present approach, the neutrino equation, up to order P, L 2 becomes ∑ ∏ ء ˆC ≠ ih¯ 2 ih¯Aˆ s?=1៬ j͑t, x៬͒ 1 m͑a2bih¯s?=៬ ͒is2j ͑t, x៬͒ ෇ 0, ≠t 2L ∑ µ ∂ ∏ µ ∂ ᐉ ᐉ 2 ᐉ ˆ ෇ P P k3 ᐉ2 2 ෇ P A 1 1k1 1k2 1 P= , a 1 1k8 , (7) ∑ L Lµ ∂ 2 ∏ L ᐉ ᐉ 2 k k Cˆ ෇ h¯ k 1k P 1k P 1 7 ᐉ2 =2 , b ෇ 9 ᐉ . 4 5 L 6 L 2 P 2¯h P

Notice that k4 would produce an additional Dirac mass original parameters ki [11]. To leading order in p¯ , the for the neutrino. Since we are considering particles with velocities are a Majorana mass m, we take k4 ෇ 0. In contrast to [10], ≠E6͑p, L ͒ we have found no additional parity violation arising from y6͑p¯ ͒ ෇ jp෇p¯ ,L ෇1͞p¯ the structure of the weave. The dispersion relation corre- ≠p m2 ͑ᐉ p¯ ͒2 sponding to (7) is ෇ 1 2 1k͑ᐉ p¯ ͒ 7k P . (10) 2¯p2 1 P 7 2 E2 ͑p, L ͒ ෇ ͑A2 1 m2b2͒p2 1 m2a2 6 µ ∂ The order of magnitude of the corrections arising from the C 2 present analysis is calculated using the following values: 1 6 Bp, ෇ 29 ϳ 5 ෇ 10 ෇ µ 2L ∂ m 10 GeV, p¯ 10 GeV, L 10 light years C 0.5 3 1042 1 (distance traveled by the neutrino from B ෇ A 1 2abm2 , (8) GeV L emission to detection on Earth). Let us observe that in Eq. (9) the ratio between second and first order contribu- where A, B, C have been expressed in momentum space tions in ᐉ behaves like ͑p¯ ᐉ ͒ഠ10214. Now consider the and depend on L . The 6 in Eq. (8) stands for the two neu- P P ͑ ៬͒ gravitationally induced time delay of neutrinos traveling at trino helicities. Let us emphasize that the solution j t, x velocities y with respect to those traveling at the speed to Eq. (7) is given by an appropriate linear combination of 6 of light: Dtn ෇ jL͑1 2y6͒j ෇ jk1jL͑p¯ ᐉP͒. Notice that plane waves and helicity eigenstates, given that the neutri- this expression, though helicity independent, is of the same nos considered are massive. form as the one in Ref. [10] for photons. In our case we Typically, for neutrinos, lD ø lC and our approxima- 4 obtain Dtn ෇ 0.3jk1j 3 10 s. Besides, this correction tion is meaningful only if L #lD. In this way we make m2 dominates over the delay due to the mass term 2 which sure that Eq. (7) is defined in a continuous flat space- 2¯p ഠ 210 time. From here on, h¯ ෇ c ෇ 1. To estimate the correc- is 10 s. The second interesting parameter is the time delay of arrival for two neutrinos having different helici- tions let us consider a massive neutrino with momentum 2 ៬ ෇ ៬ ties: Dt6 ෇ Lj͑y1 2y2͒j ෇ jk7jL͑p¯ ᐉP͒ ഠ 1.5jk7j 3 p p¯ . A lower bound for them is obtained by taking 211 ៬ 2 10 s. This correction is suppressed by a factor of ͑p¯ ᐉP͒ 1͞L ഠ 1͞lD ෇ jp¯ j ෇ p¯ . Up to leading order in ᐉP,we get with respect to the former and it is comparable to the time delay caused by the mass term. Finally, consider the char- m2 E6͑p¯ ͒ :෇ E6͑p, L ͒jp෇p¯ ,L ෇1͞p¯ ഠ p¯ 1 acteristic length Los corresponding to two-flavor neutrino 2¯p 2p 2p oscillations, given by Los ෇ ͑ ͒ ϵ , where Ea,b de- ᐉ ͓͑ ͒ 2 ͑ ͒ 2͔ Ea2Eb DE 1 P u2 6u4 p¯ 1 u1 6u3 m notes the energy corresponding to the mass eigenstates of ͑ ͒ᐉ2 3 1 u5 6u6 Pp¯ , (9) the neutrinos with masses ma,b, respectively. As usual, we assume that neutrinos are highly relativistic (p¯ a ¿ ma) ϳ ෇ ϳ where we are assuming that all ui are numerical quantities and also that p¯ a p¯ b p¯ E. The phase Fos describ- of order 1. Besides, these are known functions of the ing the oscillation is F ෇ pL , where L is the distance os Los 2320 VOLUME 84, NUMBER 11 PHYSICAL REVIEW LETTERS 13MARCH 2000 traveled by the neutrino between emission and detection. [3] G. Amelino-Camelia, J. Ellis, N. E. Mavromatos, D. V. The energy difference for the corresponding two flavors is Nanopoulos, and S. Sarkar, Nature (London) 393, 763 (1998). Dm2 [4] J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, Gen. DE ෇ 1Drp¯ 2ᐉ 1D͑r m2͒ᐉ 2¯p 1 P 2 P Relativ. Gravit. 31, 1257 (1999). [5] For a recent review of the loop approach to quantum grav- 226 29 240 ഠ͑10 1Dr1 3 10 1 10 ͒ GeV . (11) ity, see, for example, C. Rovelli, “Livings Reviews in Rela- tivity,” Vol. 1 [http://www.livingreviews.org/Articles]. This result could yield bounds upon Dr1, which measures [6] E. Waxman and J. Bahcall, Phys. Rev. Lett. 78, 2292 a violation of universality in the gravitational coupling (1997). for different neutrino flavors. For the above estimation, [7] M. Vietri, Phys. Rev. Lett. 80, 3690 (1998). 2 221 2 2 2 Dm ഠ 10 GeV , and D͑r2m ͒ഠDm were used. [8] R. Plaga, Astrophys. J. Lett. 424, L9 (1994); D. N. Spergel Here ri are flavor dependent quantities of the order of et al., Nucl. Phys. B291, 847 (1987). 21 2ᐉ [9] M. Roy, H. J. Crawford, and A. Trattner, astro-ph/9903231. 1. To conclude, we notice that (11) implies Los ~ p¯ P, seemingly an effect not considered previously [18]. [10] R. Gambini and J. Pullin, Phys. Rev. D 59, 124021 (1999). We thank R. Gambini for useful discussions about [11] J. Alfaro, H. A. Morales-Técotl, and L. F. Urrutia (to be published). [10] and T. Thiemann for illuminating comments on [12] C. Rovelli and L. Smolin, Nucl. Phys. B442, 593 (1995); the regularization problem. 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