# Quantum Gravity Corrections to Neutrino Propagation

## Transcript Of Quantum Gravity Corrections to Neutrino Propagation

VOLUME 84, NUMBER 11

PHYSICAL REVIEW LETTERS

13 MARCH 2000

Quantum Gravity Corrections to Neutrino Propagation

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 ͑ p¯ ᐉP͒L͞c ഠ 104 s. To next order in p¯ ᐉP, the correction has the form of the Gambini and Pullin effect for photons. A dependence L2os1 ~ p¯ 2ᐉP 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) originate at cosmological distances (ഠ1010 light years͒ [1], together with time resolutions down to submillisecond scale achieved in recent GRB data [2], suggests that it is possible to probe fundamental laws of physics at energy scales near to Planck energy EP 1.3 3 1019 GeV [3,4]. Furthermore, sensitivity will be improved with HEGRA and Whipple air Cherenkov telescopes and by AMS and GLAST spatial experiments. Thus, quantum gravity effects could be at the edge of observability [3,4]. Now, quantum gravity theories imply different spacetime structures [4,5] and it can be expected that what we consider flat spacetime can actually involve dispersive effects arising from Planck scale lengths. Such tiny effects might become observable upon accumulation over travels through cosmological distances by energetically enough particles like cosmological GRB photons.

The most widely accepted model of GRB, the so-called fireball model, predicts also the generation of 1014 1019 eV neutrino bursts (NB) [6,7]. Yet, another GRB model based on cosmic strings requires neutrino production [8]. Present experiments to observe high energy astrophysical neutrinos such as AMANDA, NESTOR, Baikal, ANTARES, and Super-Kamiokande, for example, will detect at best only one or two neutrinos in coincidence with GRB’s per year. The planned neutrino burster experiment (NuBE) will measure the flux of ultrahigh energy neutrinos (.10 TeV) over a ϳ1 km2 effective area, in coincidence with satellite measured GRB’s [9]. It is expected to detect ഠ20 events per year, according to the fireball model. Hence, one can study quantum gravity effects on astrophysical neutrinos that might be observed or, the other way around, such observations could be used to restrict quantum gravity theories.

In this Letter, the loop quantum gravity framework is adopted. In this context, Gambini and Pullin studied light propagation semiclassically [10]. They found, besides

departures from perfect nondispersiveness of ordinary

vacuum, helicity depending corrections for propagating

waves. In the present work, the case of massive spin-1͞2

particles in loop quantum gravity is studied also semiclas-

sically. They could be identified with the neutrinos that

would be produced in GRB. Central ideas and results are

presented, whereas details will appear elsewhere [11].

Loop quantum gravity [5] uses a spin networks basis,

labeled by graphs embedded in a three dimensional space

S. Physical predictions hereby obtained are a “polymer-

like” structure of space [12] and a possible explanation of

black hole entropy [13]. A first attempt to couple spin-1͞2

fields to gravity, along these lines, was made in [14], and

a generalization to the spin networks basis has been devel-

oped in [15]. A significant progress in the loop approach to

quantum gravity was made by Thiemann, who put forward

a consistent regularization procedure to properly define the

quantum Hamiltonian constraint of the full theory, which

includes the Einstein plus matter (leptons, quarks, Higgs

particles) contributions [16]. It is based on a triangulation

of space with tetrahedra whose sides are of the order of

ᐉP. The cornerstone of Thiemann’s proposal is the incor-

poration of the volume operator as a convenient regulator,

since its action upon states is finite. Having at our disposal

a regularized version of the quantum Hamiltonian describ-

ing fermions coupled to Einstein gravity, we will further

need a loop state which approximates a flat 3-metric on S,

at scales L much larger than the Planck length. For pure

gravity this state is called weave [17]. A flat weave jW ͘

is characterized by a length scale L ¿ ᐉP, such that for

distances d $ L the continuous flat classical geometry is

regained, while for distances d ø L the quantum loop

structure of space is manifest. The stronger the inequal-

ity d ¿ ᐉP holds, the more isotropic and homogeneous

the weave looks. For example, the metric operator gˆab

satisfies

͗W jgˆ abjW ͘

dab

1

O

͑

ᐉP L

͒.

Now,

a

generaliza-

tion of such an idea to include matter fields is required.

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VOLUME 84, NUMBER 11

PHYSICAL REVIEW LETTERS

13 MARCH 2000

For our analysis it suffices to exploit the main features that a flat weave with fermions must have: in particular, it must reproduce the Dirac equation in flat spacetime, and this is just the basis of our approximation scheme. It is denoted by jW, j͘, has a characteristic length L , and is referred to simply as a weave.

The use of Thiemann’s regularization for Einstein Dirac theory naturally allows the semiclassical treatment here pursued; expectation values with respect to jW, j͘ are considered thereby. They are expanded around relevant vertices of the triangulation and a systematic approximation is given involving the scales ᐉP ø lD ø lC, the last two corresponding to, respectively, De Broglie and Compton wavelengths of a light fermion. Corrections come out at this level.

The Hamiltonian constraint for a spin-1͞2 field coupled to gravity consists of a pure gravity contribution, a kinetic fermion term, namely,

H͑1͒ : Z d3x 2pdEeiat͑g͒ ͑ipT tiDaj 1 c.c.͒ , (1)

and other terms [16] whose contribution is summarized

in (7) below. Their analysis is an extension of the one

for H͑1͒ given here and will be spelled out in [11]. We

use

t

2

i 2

s ,

the

latter

being

the

standard

Pauli

ma-

trices. The fermion field is a Grassmann valued Majo-

rana spinor CT ͑cT , ͑2is2c͒ءT ͒. The two component

spinor c has definite chirality and it is a scalar under gen-

eral coordinate transformations. Hence, (1) is not parity invariant. The configuration variable is j ͓det͑q͔͒1͞4c,

which is a half density. The corresponding momentum is,

with this choice, p ij ;ءsimilarly as in flat space. The

gravitational canonical pair consists of Eia and the SU͑2͒-

connection

j

Ab

(D ),

where

͑detg͒gab

EiaEib .

Upon regularization [16], the expectation value of (1)

with respect to the weave becomes

͗W , jjHˆ ͑1͒jW , j͘ 2 h¯ X 8 eijk X eIJK

Ω4ᐉP4 y[V͑g͒ E͑y͒

sI >sJ >sK y

3 ͗W , jjjˆB͓y 1 sK ͑D͔͒ ≠ ͓tkhs ͑D͔͒ABwˆ iID͑y͒wˆ jJD͑y͒ jW , j͘

≠jA͑y͒

K

æ

2 ͗W , jj͑tkjˆ ͒A͑y͒ ≠ wˆ iID͑y͒wˆ jJD͑y͒ jW , j͘ 2 c.c. .

(2)

≠jA͑y͒

Here an adapted triangulation of S to the graph g of the

weave state jW, j͘ is adoptedp. Auxiliary quantities used are wˆ kID Tr͑tkhsI͑D͓͒hs2I͑1D͒, Vy͔͒, where Vy is the volume operator restricted to act upon vertex y. hs͑D͒ are holonomies along segments, s, of edges forming tetrahedra

in the triangulation D [16]. VP͑g͒ stands for the set of vertices of g. The second sum, sI>sJ>sKy, involves triples of segments sI , sJ, sK intersecting at y. Notice that one ac-

tually averages over E͑y͒ ny͑ny 2 1͒ ͑ny 2 2͒͞6 pos-

sible triangulations (one for each triple of edges) when the

vertex y is reached by ny edges (the valence) of the graph.

To estimate (2) we associate to it c-number quantities

respecting the index structure, together with appropriate

scale factors arising from dimensional reasons and, most

important, in line with the weave state approximating flat

space with fermions. This amounts to an expansion of ex-

pectation values around vertices of the weave. The explicit

form is taken from the expansion the involved operators would have in powers of the segments sa, jsaj ϳ ᐉP, a pro-

cedure justified for weave states. Useful quantities coming in by expanding wˆ iID͑y͒ sIawˆ ia 1 sIasIbwˆ iab 1 . . . , for instance, are

wˆ ia 1 ͓Aia, pVy͔, 2

wˆ iab 1 eikl͓Aka, ͓Alb, pVy͔͔ ,

8

(3)

whose contribution to the average inp the weave is estimated by considering that of Aia and Vy to be of the order of ϳ1͞L and ϳᐉP3͞2, respectively. To proceed with

the approximation we think of space as being made up of

boxes of volume L 3, whose center is denoted by x . Each

box contains a large number of vertices of the weave, but

is considered infinitesimal in the scale where the

space can be regarded as continuous, so that we take L 3 ഠ d3x. Let Fˆ ͑y͒ be a fermionic operator which

produces the slowly varying (inside the box) function

F͑x ͒, i.e., L ø lD. Also let ᐉ13P Gˆ ͑y͒ be a gravitational

operator with averPage within the box G͑x ͒. The weave is

sPuch

that P

y[V ͑g͒

8 E͑y͒

͗W

,

jjFˆ ͑y͒Gˆ ͑y͒jW

,

j͘

R Box͑x͒ F͑x ͒

y[Box͑x ͒ ᐉ3P

8 E͑y͒

͗W , jj ᐉ13

Gˆ ͑y͒jW, j͘

P

S d3x F͑x ͒G͑x ͒. Notice that the tensorial and

Lie-algebra structure should come out from flat spacetime

quantities exclusively, i.e., 0Eia, tk, ≠b, ecde, eklm, where dab 0Eia 0Eib .

In order to regain the flat spacetime kinetic term of the

fermion Hamiltonian, we demand jW , j͘ to fulfill

͗W , jjjˆB͑y͒ ≠j≠A͑y͒ ͑tkDc͑j͒͒ABwˆ ia͑y͒wˆ jb͑y͒jW , j͘

∑

∏

ഠ i jB͑y͒pA͑y͒ᐉPL 2

h¯∑

∏

3 Lᐉ3P2 ͑tk≠͑cj͒͒AB 0Eia͑y͒ 0Ejb͑y͒ . (4)

The second parenthesis here dictates the overall structure:

(3)

indicates

that

each

wˆ ia͑y͒

contributes

a

factor

of

, ᐉP3͞2

L

since the connection scales with 1͞L (large length limit

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VOLUME 84, NUMBER 11

PHYSICAL REVIEW LETTERS

13 MARCH 2000

) flat spacetime), and pVy contributes a factor of ᐉ3P͞2. Independence on L of the final form of (4) gives the structure of the first parenthesis. The notation ͑j͒ stands for acting only upon j. By expanding (2) at different

orders in powers of s and using (4), one can systematically

i L2 X 8

1

4 ᐉ3P y[V͑g͒ E͑y͒ eijk eIJK 3! sKa sKb sKc sIdsJe

3 p͑y͒tk≠a≠b≠cj͑y͒ ͗W , jjwˆ id͑y͒wˆ je͑y͒ jW , j͘ Z

! 2ik8ᐉP2 d3x p͑x ͒tk 0Ekc≠c=2j͑x ͒ . (6)

determine all possible contributions. Some examples of correction terms are

A similar treatment can be performed for every contribution to the Einstein-Dirac Hamiltonian constraint [11].

i L2 X 8 4 ᐉP3 y[V͑g͒ E͑y͒ eijk eIJK sIasIb sJcsKd pA͑y͒≠djB͑y͒

It is important to stress that the prediction of the values of the corresponding coefficients ki would require a precise definition of the flat weave (or even better, a Friedman-

3 ͗W , jj͑tk͒AB͕wˆ iab, wˆ jc͖ jW , j͘

! k5 ᐉP Z d3x i p͑x ͒tk 0Ekd≠dj͑x ͒ ,

L

2

Lemaitre-Robertson-Walker weave), together with a de-

tailed calculation of the matrix elements. Instead, within (5) the present approach, the neutrino equation, up to order ᐉ2P,

becomes

∑ ≠

Cˆ ∏

ih¯ 2 ih¯Aˆ s ? = 1

j͑t, x ͒ 1 m͑a 2 bih¯s ? =͒is2j͑ءt, x ͒ 0,

≠t

2L

∑

µ∂

∏

µ

∂

Aˆ 1 1 k ᐉP 1 k ᐉP 2 1 k3 ᐉ2 =2 , a 1 1 k ᐉP ,

(7)

∑ 1L

2 Lµ ∂ 2 P

∏

8L

Cˆ h¯ k4 1 k5 ᐉLP 1 k6 ᐉLP 2 1 k27 ᐉ2P=2 ,

b k9 ᐉP . 2h¯

Notice that k4 would produce an additional Dirac mass for the neutrino. Since we are considering particles with a Majorana mass m, we take k4 0. In contrast to [10], we have found no additional parity violation arising from

the structure of the weave. The dispersion relation corre-

sponding to (7) is

E62 ͑p, L ͒ ͑A2 1 m2b2͒p2 1 m2a2 µ ∂2

C

1 µ 2L

6 Bp, ∂

B A C 1 2abm2 ,

(8)

L

where A, B, C have been expressed in momentum space and depend on L . The 6 in Eq. (8) stands for the two neutrino helicities. Let us emphasize that the solution j͑t, x ͒ to Eq. (7) is given by an appropriate linear combination of

plane waves and helicity eigenstates, given that the neutri-

nos considered are massive. Typically, for neutrinos, lD ø lC and our approxima-

tion is meaningful only if L # lD. In this way we make sure that Eq. (7) is defined in a continuous flat spacetime. From here on, h¯ c 1. To estimate the corrections let us consider a massive neutrino with momentum p p¯ . A lower bound for them is obtained by taking 1͞L ഠ 1͞lD jp¯ j p¯ . Up to leading order in ᐉ2P, we get

m2 E6͑p¯ ͒ : E6͑ p, L ͒jpp¯ ,L 1͞p¯ ഠ p¯ 1

2p¯ 1 ᐉP͓͑u2 6 u4͒p¯ 2 1 ͑u1 6 u3͒m2͔

1 ͑u5 6 u6͒ᐉP2 p¯ 3,

(9)

where we are assuming that all ui are numerical quantities of order 1. Besides, these are known functions of the

original parameters ki [11]. To leading order in p¯ , the velocities are

y6͑ p¯ ͒ ≠E6͑ p, L ͒ jpp¯ ,L 1͞p¯

≠p

m2

͑ᐉPp¯ ͒2

1 2 2p¯ 2 1 k1͑ᐉPp¯ ͒ 7 k7 2 . (10)

The order of magnitude of the corrections arising from the

present analysis is calculated using the following values:

m 1029 GeV, p¯ ϳ 105 GeV, L 1010 light years

0.5

3

1042

1

GeV

(distance

traveled

by

the

neutrino

from

emission to detection on Earth). Let us observe that in

Eq. (9) the ratio between second and first order contribu-

tions in ᐉP behaves like ͑p¯ ᐉP͒ ഠ 10214. Now consider the

gravitationally induced time delay of neutrinos traveling at

velocities y6 with respect to those traveling at the speed

of light: Dtn jL͑1 2 y6͒j jk1jL͑p¯ ᐉP͒. Notice that

this expression, though helicity independent, is of the same

form as the one in Ref. [10] for photons. In our case we

obtain Dtn 0.3jk1j 3 104 s. Besides, this correction dominates over the delay due to the mass term 2mp¯22 which is ഠ10210 s. The second interesting parameter is the time

delay of arrival for two neutrinos having different helici-

ties: Dt6 Lj͑y1 2 y2͒j jk7jL͑p¯ ᐉP͒2 ഠ 1.5jk7j 3 10211 s. This correction is suppressed by a factor of ͑p¯ ᐉP͒

with respect to the former and it is comparable to the time

delay caused by the mass term. Finally, consider the char-

acteristic length Los corresponding to two-flavor neutrino oscillations, given by Los ͑Ea22pEb͒ ϵ D2pE , where Ea,b denotes the energy corresponding to the mass eigenstates of

the neutrinos with masses ma,b, respectively. As usual, we

assume that neutrinos are highly relativistic (p¯ a ¿ ma)

and also that p¯ a ϳ p¯ b p¯ ϳ E. The phase Fos describing the oscillation is Fos pLL , where L is the distance

os

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PHYSICAL REVIEW LETTERS

13 MARCH 2000

traveled by the neutrino between emission and detection. The energy difference for the corresponding two flavors is

DE D2mp¯ 2 1 Dr1p¯ 2ᐉP 1 D͑r2m2͒ᐉP ഠ ͑10226 1 Dr1 3 1029 1 10240͒ GeV . (11)

This result could yield bounds upon Dr1, which measures a violation of universality in the gravitational coupling for different neutrino flavors. For the above estimation, Dm2 ഠ 10221 GeV2, and D͑ r2m2͒ ഠ Dm2 were used. Here ri are flavor dependent quantities of the order of 1. To conclude, we notice that (11) implies Lo2s1 ~ p¯ 2ᐉP, seemingly an effect not considered previously [18].

We thank R. Gambini for useful discussions about [10] and T. Thiemann for illuminating comments on the regularization problem. Partial support is acknowledged from CONICyT-CONACyT E120-2639, DGAPA IN100397, and CONACYT 3141PE. The work of J. A. is partially supported by Fondecyt 1980806 and the CONACyT-CONICyT project 1997-02-038. We also acknowledge the project Fondecyt 7980018.

*Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] [1] J. van Paradis et al., Nature (London) 386, 686 (1997); P. J. Groot et al., IAU Circ. No. 6676, 1997; M. L. Metzger et al., Nature (London) 387, 878 (1997). For a review, see, for example, G. J. Fishman and C. A. Meegan, Annu. Rev. Astron. Astrophys. 33, 415 (1995). [2] P. N. Bhat, G. J. Fishman, C. A. Meegan, R. B. Wilson, M. N. Brock, and W. S. Paclesas, Nature (London) 359, 217 (1992).

[3] G. Amelino-Camelia, J. Ellis, N. E. Mavromatos, D. V. Nanopoulos, and S. Sarkar, Nature (London) 393, 763 (1998).

[4] J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, Gen. Relativ. Gravit. 31, 1257 (1999).

[5] For a recent review of the loop approach to quantum gravity, see, for example, C. Rovelli, “Livings Reviews in Relativity,” Vol. 1 [http://www.livingreviews.org/Articles].

[6] E. Waxman and J. Bahcall, Phys. Rev. Lett. 78, 2292 (1997).

[7] M. Vietri, Phys. Rev. Lett. 80, 3690 (1998). [8] R. Plaga, Astrophys. J. Lett. 424, L9 (1994); D. N. Spergel

et al., Nucl. Phys. B291, 847 (1987). [9] M. Roy, H. J. Crawford, and A. Trattner, astro-ph /9903231. [10] R. Gambini and J. Pullin, Phys. Rev. D 59, 124021 (1999). [11] J. Alfaro, H. A. Morales-Técotl, and L. F. Urrutia (to be

published). [12] C. Rovelli and L. Smolin, Nucl. Phys. B442, 593 (1995);

B456, 753 (1995); A. Ashtekar and J. Lewandowski, Classical Quantum Gravity 14, A55 (1997); Adv. Theor. Math. Phys. 1, 388 (1998). [13] C. Rovelli, Phys. Rev. Lett. 77, 3288 (1996); A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, Phys. Rev. Lett. 80, 904 (1998). [14] H. A. Morales-Técotl and C. Rovelli, Phys. Rev. Lett. 72, 3642 (1994); Nucl. Phys. B451, 325 (1995). [15] J. Baez and K. Krasnov, J. Math. Phys. 39, 1251 (1998); T. Thiemann, Classical Quantum Gravity 15, 1487 (1998). [16] T. Thiemann, Classical Quantum Gravity 15, 839 (1998); 15, 875 (1998); 15, 1281 (1998). [17] A. Ashtekar, C. Rovelli, and L. Smolin, Phys. Rev. Lett. 69, 237 (1992). See also J. Zegwaard, Phys. Lett. B 300, 217 (1993); R. Borissov, Phys. Rev. D 49, 923 (1994); J. Iwasaki and C. Rovelli, Int. J. Mod. Phys. D1, 53 (1993); Classical Quantum Gravity 11, 1653 (1994); N. Grot and C. Rovelli, Gen. Relativ. Gravit. 29, 1039 (1997); J. Iwasaki, gr-qc /9807013. [18] G. L. Fogli, E. Lisi, A. Marrone, and G. Scioscia, Phys. Rev. D 60, 053006 (1999).

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PHYSICAL REVIEW LETTERS

13 MARCH 2000

Quantum Gravity Corrections to Neutrino Propagation

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 ͑ p¯ ᐉP͒L͞c ഠ 104 s. To next order in p¯ ᐉP, the correction has the form of the Gambini and Pullin effect for photons. A dependence L2os1 ~ p¯ 2ᐉP 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) originate at cosmological distances (ഠ1010 light years͒ [1], together with time resolutions down to submillisecond scale achieved in recent GRB data [2], suggests that it is possible to probe fundamental laws of physics at energy scales near to Planck energy EP 1.3 3 1019 GeV [3,4]. Furthermore, sensitivity will be improved with HEGRA and Whipple air Cherenkov telescopes and by AMS and GLAST spatial experiments. Thus, quantum gravity effects could be at the edge of observability [3,4]. Now, quantum gravity theories imply different spacetime structures [4,5] and it can be expected that what we consider flat spacetime can actually involve dispersive effects arising from Planck scale lengths. Such tiny effects might become observable upon accumulation over travels through cosmological distances by energetically enough particles like cosmological GRB photons.

The most widely accepted model of GRB, the so-called fireball model, predicts also the generation of 1014 1019 eV neutrino bursts (NB) [6,7]. Yet, another GRB model based on cosmic strings requires neutrino production [8]. Present experiments to observe high energy astrophysical neutrinos such as AMANDA, NESTOR, Baikal, ANTARES, and Super-Kamiokande, for example, will detect at best only one or two neutrinos in coincidence with GRB’s per year. The planned neutrino burster experiment (NuBE) will measure the flux of ultrahigh energy neutrinos (.10 TeV) over a ϳ1 km2 effective area, in coincidence with satellite measured GRB’s [9]. It is expected to detect ഠ20 events per year, according to the fireball model. Hence, one can study quantum gravity effects on astrophysical neutrinos that might be observed or, the other way around, such observations could be used to restrict quantum gravity theories.

In this Letter, the loop quantum gravity framework is adopted. In this context, Gambini and Pullin studied light propagation semiclassically [10]. They found, besides

departures from perfect nondispersiveness of ordinary

vacuum, helicity depending corrections for propagating

waves. In the present work, the case of massive spin-1͞2

particles in loop quantum gravity is studied also semiclas-

sically. They could be identified with the neutrinos that

would be produced in GRB. Central ideas and results are

presented, whereas details will appear elsewhere [11].

Loop quantum gravity [5] uses a spin networks basis,

labeled by graphs embedded in a three dimensional space

S. Physical predictions hereby obtained are a “polymer-

like” structure of space [12] and a possible explanation of

black hole entropy [13]. A first attempt to couple spin-1͞2

fields to gravity, along these lines, was made in [14], and

a generalization to the spin networks basis has been devel-

oped in [15]. A significant progress in the loop approach to

quantum gravity was made by Thiemann, who put forward

a consistent regularization procedure to properly define the

quantum Hamiltonian constraint of the full theory, which

includes the Einstein plus matter (leptons, quarks, Higgs

particles) contributions [16]. It is based on a triangulation

of space with tetrahedra whose sides are of the order of

ᐉP. The cornerstone of Thiemann’s proposal is the incor-

poration of the volume operator as a convenient regulator,

since its action upon states is finite. Having at our disposal

a regularized version of the quantum Hamiltonian describ-

ing fermions coupled to Einstein gravity, we will further

need a loop state which approximates a flat 3-metric on S,

at scales L much larger than the Planck length. For pure

gravity this state is called weave [17]. A flat weave jW ͘

is characterized by a length scale L ¿ ᐉP, such that for

distances d $ L the continuous flat classical geometry is

regained, while for distances d ø L the quantum loop

structure of space is manifest. The stronger the inequal-

ity d ¿ ᐉP holds, the more isotropic and homogeneous

the weave looks. For example, the metric operator gˆab

satisfies

͗W jgˆ abjW ͘

dab

1

O

͑

ᐉP L

͒.

Now,

a

generaliza-

tion of such an idea to include matter fields is required.

2318

0031-9007͞00͞84(11)͞2318(4)$15.00 © 2000 The American Physical Society

VOLUME 84, NUMBER 11

PHYSICAL REVIEW LETTERS

13 MARCH 2000

For our analysis it suffices to exploit the main features that a flat weave with fermions must have: in particular, it must reproduce the Dirac equation in flat spacetime, and this is just the basis of our approximation scheme. It is denoted by jW, j͘, has a characteristic length L , and is referred to simply as a weave.

The use of Thiemann’s regularization for Einstein Dirac theory naturally allows the semiclassical treatment here pursued; expectation values with respect to jW, j͘ are considered thereby. They are expanded around relevant vertices of the triangulation and a systematic approximation is given involving the scales ᐉP ø lD ø lC, the last two corresponding to, respectively, De Broglie and Compton wavelengths of a light fermion. Corrections come out at this level.

The Hamiltonian constraint for a spin-1͞2 field coupled to gravity consists of a pure gravity contribution, a kinetic fermion term, namely,

H͑1͒ : Z d3x 2pdEeiat͑g͒ ͑ipT tiDaj 1 c.c.͒ , (1)

and other terms [16] whose contribution is summarized

in (7) below. Their analysis is an extension of the one

for H͑1͒ given here and will be spelled out in [11]. We

use

t

2

i 2

s ,

the

latter

being

the

standard

Pauli

ma-

trices. The fermion field is a Grassmann valued Majo-

rana spinor CT ͑cT , ͑2is2c͒ءT ͒. The two component

spinor c has definite chirality and it is a scalar under gen-

eral coordinate transformations. Hence, (1) is not parity invariant. The configuration variable is j ͓det͑q͔͒1͞4c,

which is a half density. The corresponding momentum is,

with this choice, p ij ;ءsimilarly as in flat space. The

gravitational canonical pair consists of Eia and the SU͑2͒-

connection

j

Ab

(D ),

where

͑detg͒gab

EiaEib .

Upon regularization [16], the expectation value of (1)

with respect to the weave becomes

͗W , jjHˆ ͑1͒jW , j͘ 2 h¯ X 8 eijk X eIJK

Ω4ᐉP4 y[V͑g͒ E͑y͒

sI >sJ >sK y

3 ͗W , jjjˆB͓y 1 sK ͑D͔͒ ≠ ͓tkhs ͑D͔͒ABwˆ iID͑y͒wˆ jJD͑y͒ jW , j͘

≠jA͑y͒

K

æ

2 ͗W , jj͑tkjˆ ͒A͑y͒ ≠ wˆ iID͑y͒wˆ jJD͑y͒ jW , j͘ 2 c.c. .

(2)

≠jA͑y͒

Here an adapted triangulation of S to the graph g of the

weave state jW, j͘ is adoptedp. Auxiliary quantities used are wˆ kID Tr͑tkhsI͑D͓͒hs2I͑1D͒, Vy͔͒, where Vy is the volume operator restricted to act upon vertex y. hs͑D͒ are holonomies along segments, s, of edges forming tetrahedra

in the triangulation D [16]. VP͑g͒ stands for the set of vertices of g. The second sum, sI>sJ>sKy, involves triples of segments sI , sJ, sK intersecting at y. Notice that one ac-

tually averages over E͑y͒ ny͑ny 2 1͒ ͑ny 2 2͒͞6 pos-

sible triangulations (one for each triple of edges) when the

vertex y is reached by ny edges (the valence) of the graph.

To estimate (2) we associate to it c-number quantities

respecting the index structure, together with appropriate

scale factors arising from dimensional reasons and, most

important, in line with the weave state approximating flat

space with fermions. This amounts to an expansion of ex-

pectation values around vertices of the weave. The explicit

form is taken from the expansion the involved operators would have in powers of the segments sa, jsaj ϳ ᐉP, a pro-

cedure justified for weave states. Useful quantities coming in by expanding wˆ iID͑y͒ sIawˆ ia 1 sIasIbwˆ iab 1 . . . , for instance, are

wˆ ia 1 ͓Aia, pVy͔, 2

wˆ iab 1 eikl͓Aka, ͓Alb, pVy͔͔ ,

8

(3)

whose contribution to the average inp the weave is estimated by considering that of Aia and Vy to be of the order of ϳ1͞L and ϳᐉP3͞2, respectively. To proceed with

the approximation we think of space as being made up of

boxes of volume L 3, whose center is denoted by x . Each

box contains a large number of vertices of the weave, but

is considered infinitesimal in the scale where the

space can be regarded as continuous, so that we take L 3 ഠ d3x. Let Fˆ ͑y͒ be a fermionic operator which

produces the slowly varying (inside the box) function

F͑x ͒, i.e., L ø lD. Also let ᐉ13P Gˆ ͑y͒ be a gravitational

operator with averPage within the box G͑x ͒. The weave is

sPuch

that P

y[V ͑g͒

8 E͑y͒

͗W

,

jjFˆ ͑y͒Gˆ ͑y͒jW

,

j͘

R Box͑x͒ F͑x ͒

y[Box͑x ͒ ᐉ3P

8 E͑y͒

͗W , jj ᐉ13

Gˆ ͑y͒jW, j͘

P

S d3x F͑x ͒G͑x ͒. Notice that the tensorial and

Lie-algebra structure should come out from flat spacetime

quantities exclusively, i.e., 0Eia, tk, ≠b, ecde, eklm, where dab 0Eia 0Eib .

In order to regain the flat spacetime kinetic term of the

fermion Hamiltonian, we demand jW , j͘ to fulfill

͗W , jjjˆB͑y͒ ≠j≠A͑y͒ ͑tkDc͑j͒͒ABwˆ ia͑y͒wˆ jb͑y͒jW , j͘

∑

∏

ഠ i jB͑y͒pA͑y͒ᐉPL 2

h¯∑

∏

3 Lᐉ3P2 ͑tk≠͑cj͒͒AB 0Eia͑y͒ 0Ejb͑y͒ . (4)

The second parenthesis here dictates the overall structure:

(3)

indicates

that

each

wˆ ia͑y͒

contributes

a

factor

of

, ᐉP3͞2

L

since the connection scales with 1͞L (large length limit

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PHYSICAL REVIEW LETTERS

13 MARCH 2000

) flat spacetime), and pVy contributes a factor of ᐉ3P͞2. Independence on L of the final form of (4) gives the structure of the first parenthesis. The notation ͑j͒ stands for acting only upon j. By expanding (2) at different

orders in powers of s and using (4), one can systematically

i L2 X 8

1

4 ᐉ3P y[V͑g͒ E͑y͒ eijk eIJK 3! sKa sKb sKc sIdsJe

3 p͑y͒tk≠a≠b≠cj͑y͒ ͗W , jjwˆ id͑y͒wˆ je͑y͒ jW , j͘ Z

! 2ik8ᐉP2 d3x p͑x ͒tk 0Ekc≠c=2j͑x ͒ . (6)

determine all possible contributions. Some examples of correction terms are

A similar treatment can be performed for every contribution to the Einstein-Dirac Hamiltonian constraint [11].

i L2 X 8 4 ᐉP3 y[V͑g͒ E͑y͒ eijk eIJK sIasIb sJcsKd pA͑y͒≠djB͑y͒

It is important to stress that the prediction of the values of the corresponding coefficients ki would require a precise definition of the flat weave (or even better, a Friedman-

3 ͗W , jj͑tk͒AB͕wˆ iab, wˆ jc͖ jW , j͘

! k5 ᐉP Z d3x i p͑x ͒tk 0Ekd≠dj͑x ͒ ,

L

2

Lemaitre-Robertson-Walker weave), together with a de-

tailed calculation of the matrix elements. Instead, within (5) the present approach, the neutrino equation, up to order ᐉ2P,

becomes

∑ ≠

Cˆ ∏

ih¯ 2 ih¯Aˆ s ? = 1

j͑t, x ͒ 1 m͑a 2 bih¯s ? =͒is2j͑ءt, x ͒ 0,

≠t

2L

∑

µ∂

∏

µ

∂

Aˆ 1 1 k ᐉP 1 k ᐉP 2 1 k3 ᐉ2 =2 , a 1 1 k ᐉP ,

(7)

∑ 1L

2 Lµ ∂ 2 P

∏

8L

Cˆ h¯ k4 1 k5 ᐉLP 1 k6 ᐉLP 2 1 k27 ᐉ2P=2 ,

b k9 ᐉP . 2h¯

Notice that k4 would produce an additional Dirac mass for the neutrino. Since we are considering particles with a Majorana mass m, we take k4 0. In contrast to [10], we have found no additional parity violation arising from

the structure of the weave. The dispersion relation corre-

sponding to (7) is

E62 ͑p, L ͒ ͑A2 1 m2b2͒p2 1 m2a2 µ ∂2

C

1 µ 2L

6 Bp, ∂

B A C 1 2abm2 ,

(8)

L

where A, B, C have been expressed in momentum space and depend on L . The 6 in Eq. (8) stands for the two neutrino helicities. Let us emphasize that the solution j͑t, x ͒ to Eq. (7) is given by an appropriate linear combination of

plane waves and helicity eigenstates, given that the neutri-

nos considered are massive. Typically, for neutrinos, lD ø lC and our approxima-

tion is meaningful only if L # lD. In this way we make sure that Eq. (7) is defined in a continuous flat spacetime. From here on, h¯ c 1. To estimate the corrections let us consider a massive neutrino with momentum p p¯ . A lower bound for them is obtained by taking 1͞L ഠ 1͞lD jp¯ j p¯ . Up to leading order in ᐉ2P, we get

m2 E6͑p¯ ͒ : E6͑ p, L ͒jpp¯ ,L 1͞p¯ ഠ p¯ 1

2p¯ 1 ᐉP͓͑u2 6 u4͒p¯ 2 1 ͑u1 6 u3͒m2͔

1 ͑u5 6 u6͒ᐉP2 p¯ 3,

(9)

where we are assuming that all ui are numerical quantities of order 1. Besides, these are known functions of the

original parameters ki [11]. To leading order in p¯ , the velocities are

y6͑ p¯ ͒ ≠E6͑ p, L ͒ jpp¯ ,L 1͞p¯

≠p

m2

͑ᐉPp¯ ͒2

1 2 2p¯ 2 1 k1͑ᐉPp¯ ͒ 7 k7 2 . (10)

The order of magnitude of the corrections arising from the

present analysis is calculated using the following values:

m 1029 GeV, p¯ ϳ 105 GeV, L 1010 light years

0.5

3

1042

1

GeV

(distance

traveled

by

the

neutrino

from

emission to detection on Earth). Let us observe that in

Eq. (9) the ratio between second and first order contribu-

tions in ᐉP behaves like ͑p¯ ᐉP͒ ഠ 10214. Now consider the

gravitationally induced time delay of neutrinos traveling at

velocities y6 with respect to those traveling at the speed

of light: Dtn jL͑1 2 y6͒j jk1jL͑p¯ ᐉP͒. Notice that

this expression, though helicity independent, is of the same

form as the one in Ref. [10] for photons. In our case we

obtain Dtn 0.3jk1j 3 104 s. Besides, this correction dominates over the delay due to the mass term 2mp¯22 which is ഠ10210 s. The second interesting parameter is the time

delay of arrival for two neutrinos having different helici-

ties: Dt6 Lj͑y1 2 y2͒j jk7jL͑p¯ ᐉP͒2 ഠ 1.5jk7j 3 10211 s. This correction is suppressed by a factor of ͑p¯ ᐉP͒

with respect to the former and it is comparable to the time

delay caused by the mass term. Finally, consider the char-

acteristic length Los corresponding to two-flavor neutrino oscillations, given by Los ͑Ea22pEb͒ ϵ D2pE , where Ea,b denotes the energy corresponding to the mass eigenstates of

the neutrinos with masses ma,b, respectively. As usual, we

assume that neutrinos are highly relativistic (p¯ a ¿ ma)

and also that p¯ a ϳ p¯ b p¯ ϳ E. The phase Fos describing the oscillation is Fos pLL , where L is the distance

os

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PHYSICAL REVIEW LETTERS

13 MARCH 2000

traveled by the neutrino between emission and detection. The energy difference for the corresponding two flavors is

DE D2mp¯ 2 1 Dr1p¯ 2ᐉP 1 D͑r2m2͒ᐉP ഠ ͑10226 1 Dr1 3 1029 1 10240͒ GeV . (11)

This result could yield bounds upon Dr1, which measures a violation of universality in the gravitational coupling for different neutrino flavors. For the above estimation, Dm2 ഠ 10221 GeV2, and D͑ r2m2͒ ഠ Dm2 were used. Here ri are flavor dependent quantities of the order of 1. To conclude, we notice that (11) implies Lo2s1 ~ p¯ 2ᐉP, seemingly an effect not considered previously [18].

We thank R. Gambini for useful discussions about [10] and T. Thiemann for illuminating comments on the regularization problem. Partial support is acknowledged from CONICyT-CONACyT E120-2639, DGAPA IN100397, and CONACYT 3141PE. The work of J. A. is partially supported by Fondecyt 1980806 and the CONACyT-CONICyT project 1997-02-038. We also acknowledge the project Fondecyt 7980018.

*Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] [1] J. van Paradis et al., Nature (London) 386, 686 (1997); P. J. Groot et al., IAU Circ. No. 6676, 1997; M. L. Metzger et al., Nature (London) 387, 878 (1997). For a review, see, for example, G. J. Fishman and C. A. Meegan, Annu. Rev. Astron. Astrophys. 33, 415 (1995). [2] P. N. Bhat, G. J. Fishman, C. A. Meegan, R. B. Wilson, M. N. Brock, and W. S. Paclesas, Nature (London) 359, 217 (1992).

[3] G. Amelino-Camelia, J. Ellis, N. E. Mavromatos, D. V. Nanopoulos, and S. Sarkar, Nature (London) 393, 763 (1998).

[4] J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, Gen. Relativ. Gravit. 31, 1257 (1999).

[5] For a recent review of the loop approach to quantum gravity, see, for example, C. Rovelli, “Livings Reviews in Relativity,” Vol. 1 [http://www.livingreviews.org/Articles].

[6] E. Waxman and J. Bahcall, Phys. Rev. Lett. 78, 2292 (1997).

[7] M. Vietri, Phys. Rev. Lett. 80, 3690 (1998). [8] R. Plaga, Astrophys. J. Lett. 424, L9 (1994); D. N. Spergel

et al., Nucl. Phys. B291, 847 (1987). [9] M. Roy, H. J. Crawford, and A. Trattner, astro-ph /9903231. [10] R. Gambini and J. Pullin, Phys. Rev. D 59, 124021 (1999). [11] J. Alfaro, H. A. Morales-Técotl, and L. F. Urrutia (to be

published). [12] C. Rovelli and L. Smolin, Nucl. Phys. B442, 593 (1995);

B456, 753 (1995); A. Ashtekar and J. Lewandowski, Classical Quantum Gravity 14, A55 (1997); Adv. Theor. Math. Phys. 1, 388 (1998). [13] C. Rovelli, Phys. Rev. Lett. 77, 3288 (1996); A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, Phys. Rev. Lett. 80, 904 (1998). [14] H. A. Morales-Técotl and C. Rovelli, Phys. Rev. Lett. 72, 3642 (1994); Nucl. Phys. B451, 325 (1995). [15] J. Baez and K. Krasnov, J. Math. Phys. 39, 1251 (1998); T. Thiemann, Classical Quantum Gravity 15, 1487 (1998). [16] T. Thiemann, Classical Quantum Gravity 15, 839 (1998); 15, 875 (1998); 15, 1281 (1998). [17] A. Ashtekar, C. Rovelli, and L. Smolin, Phys. Rev. Lett. 69, 237 (1992). See also J. Zegwaard, Phys. Lett. B 300, 217 (1993); R. Borissov, Phys. Rev. D 49, 923 (1994); J. Iwasaki and C. Rovelli, Int. J. Mod. Phys. D1, 53 (1993); Classical Quantum Gravity 11, 1653 (1994); N. Grot and C. Rovelli, Gen. Relativ. Gravit. 29, 1039 (1997); J. Iwasaki, gr-qc /9807013. [18] G. L. Fogli, E. Lisi, A. Marrone, and G. Scioscia, Phys. Rev. D 60, 053006 (1999).

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