Cerenkov's Effect and Neutrino Oscillations in Loop Quantum Gravity

6 Pages·0.1 MB·

by  G. Lambiase

Cerenkov’s Effect and Neutrino Oscillations in Loop Quantum Gravity
G. Lambiasea,b∗
a
Dipartimento di Fisica ”E.R. Caianiello” Universit´a di Salerno, 84081 Baronissi (Sa), Italy.
bINFN - Gruppo Collegato di Salerno, Italy.
(February 7, 2008)
Bounds on the scale parameter L arising in loop quantum gravity theory are derived in the
framework ofCerenkov’seffectandneutrinooscillations. AssumingthatLisanuniversalconstant,
we infer L&10−18eV−1, a bound compatible with ones inferred in different physical context.
PACS No.: 14.60.Pq, 04.62.+v, 95.30.Sf
I. INTRODUCTION
3
0
0
The difficulties to build up a complete theory of quantum gravity has motivated the development of semiclassical
2
approaches in which a Lorentz invariance breakdown occurs at the effective theory level. The deformations of the

n Lorentz invariance manifest by means of a slight deviation from the standard dispersion relations of particles propa-
a
gatinginthevacuum. Suchmodificationshasbeenproposedinthepaper[1],andderivedintwodifferentapproaches:
J
String Theory [2] and Loop Quantum Gravity [3–5].
6
In this work we shall refer to the Alfaro, Morales-T´ecotl, Urrutia (AMU) paper [4,5] which is based on the loop
1
quantum gravity [6]. In this model a new length scale occurs, with l 10−19GeV−1. The scale separates
Pl
1 the distances d that manifest the quantum loop structLure of space (dL≫ ) fr∼om the continuous flat spaLce (d & ).
≪ L L
v As observed in [7], this length scale might give rise to some observable effect of quantum gravity.
8 In the AMU formalism, the dispersion relation of freely propagating fermions and photons have the form
5
0 E2 =A˜2p2+m2, (I.1)
i i i i
1
0 where A˜ encodes the Lorentz invariance deformation -dependent and is different for different species of particles.
i
3 L
More specifically, for Majorana fermions [4], the dispersion relation is given by
0
/
c E2 =A2p2+ηp4 2λp+m2, (I.2)
q ± p ±
- where
r
g
2
v: A =1+κ lPl +κ lPl , η =κ l2 , λ=κ lPl . (I.3)
p 1 2 3 Pl 52 2
Xi (cid:18) L (cid:19) (cid:18) L (cid:19) L
r The signsstandforthehelicityofthepropagatingfermions,andκi areunknowncoefficientsoftheorderκi (1).
a Fo±r photons, the dispersion relation derived in AMU approachis [5] ∼O
ω2 =k2[A2 2θ l k], (I.4)
± γ ± γ Pl
where
2+2Y
l
Pl
A =1+κ . (I.5)
γ γ
(cid:18) L (cid:19)
Again signs stand for the helicity dependence of photons in the dispersion relation, κ (1), θ (1), and
γ γ
± ∼ O ∼ O
Y = 1/2,0,1/2,1,.... It is worth note that a similar result has been obtained by Gambini and Pullin [3] with
−
A =1 and Ellis et al. [2] with the difference that the helicity dependence is absent.
γ
The aim of this paper is to determine bounds on the scale length . Alfaro and Palma [7] applies the AMU theory
L
totheobservedGreizen-Zatsepin-Kuz’min(GZK)limitanomalyandtothesocalledTeV-γ paradox,i.e. thedetecton
of high-energy photons with a spectrum ranging up to 24 TeV from Mk 501, a BL Lac object at a red-shift of 0.034
∗
E-mail: [email protected]
1
( 134Mpc). Assuming that no anomalies there exist, as recent works seem to indicate [8], they find that, with a
∼
specific choice of the κ -parameters,the favorite range for is (Y =0)
i
L
4.6 10−17eV−1 & &8.3 10−18eV−1. (I.6)
× L ×
Bounds on are inferred in two different contexts: i) the emission of radiation by charged particles via Cerenkov
L
effect; ii) neutrino oscillations. The analysis is carried out for universal constant, and k = 0 and k = 0 in (I.1).
3 5
L
Besides, to explore the leading order helicity independent effects of photons, we also put θ = 0. A detailed study
γ
including the helicity term in (I.4) has been recently performed by Jacobson et al. in [9].
II. CERENKOV EFFECT IN THE AMU THEORY
This Section is devoted to study the emission of the Cerenkov radiation by a charged particle whose dispersion
relationismodifiedbythemicroscopicloopstructureofspace. Aswellknown,theCerenkoveffectcausesthecharged
particles to decelerate radiatively, and the spectrum of the radiated energy might be used as a sensitive means for
probingtheirproperties. Thiseffecthasbeenalsostudiedin[9,10],butthemethodtoderiveboundson isdifferent.
L
The dispersion relation of photons (I.4) allows to infer the phase velocity v
γ
ω
v 1+(A 1). (II.1)
γ γ
≡ k ∼ −
For massive particles, the groups velocity v , corresponding to the physical velocity of the particles, is obtained by
p
deriving the energy (I.2) with respect to the momentum p
dE m2
v 1+(A 1) . (II.2)
p ≡ dp ∼ p− − 2p2
The ratio between the phase velocity (II.1) and the group velocity (II.2) is (p E)
≃
v m2
γ
=1+ (A A ) (A 1)(A 1). (II.3)
v 2p2 − p− γ − p− γ −
p
The Cerenkov process occurs if the condition v /v <1 holds, i.e.
γ p
m2
(1+A A 2A )<0. (II.4)
2p2 − p γ − γ
In the limit the above relation reduces to m2 <0, so that the Cerenkov effect is not kinematically allowed.
L→∞
From Eq. (II.4) one defines the threshold energy E
0
m
E = , (II.5)
0
2[1+A (A 2)]
γ p
−
above which a charged particle is faster than the coprresponding phase velocity of light and must then emit Cerenkov
radiation.
The rate of the emitted energy by the charged particle is [11]
dE ω¯ v2
W(E)= =e2 1 γ ωdω (II.6)
dt Z0 − vp2!
(E2 E2)(E E )2
=e2m2 − 0 − 0 ,
2E2E2
0
where the integration is overall the possible emitted frequencies ω¯ = E E and Eqs. (II.3) and (II.5) have been
0
−
used. The distance that charged particles can travel before to relax their energy as Cerenkov radiation is
xi Ef dE
L(E )= dx= (II.7)
f
− W(E)
Zxf ZEi
E (E E )(E +E ) 1 1 1 1
0 ln i− 0 f 0 +6E +2E2
≃ 4e2m2 (E +E )(E E ) 0 E E − E E 0 (E E )2 − (E E )2
(cid:26) i 0 f − 0 (cid:18) f − 0 i− 0(cid:19) (cid:20) f − 0 i− 0 (cid:21)(cid:27)
2
where E denotes the initial energy of the particle, and E is the final energy which differs from the thresholdenergy
i f
E by an infinitesimal quantity, E = (1+ζ)E , with ζ 1 (E > E > E ). In the limit E E , Eq. (II.7)
0 f 0 i f 0 i 0
≪ ≫
reducestoL(E ) E /2e2m2ζ2. Thustheloopquantumgravityeffectsbecomeimportantondistanceofcosmological
f 0
relevance, provide∼d ζ is extremely small. In fact, for L(E ) Mpc and taking E 109GeV as threshold energy, it
f 0
follows ζ <10−8. Notice that E =E corresponds to the as∼ymptotic value L(E )∼ .
f 0 f
→∞
Todetermine aboundonthe parameter ,we shallworkinthe contextofcosmicraysphysics. As widelybelieved,
L
cosmicrayshaveextragalacticorigin[12–14]withenergiesexceeding109GeV,andsourceslocatedatdistancesgreater
than few Mpc. Besides, at the energy greater than 1010GeV, the spectrum should contain mostly protons [13,15].
∼
The large number of observed events with energies exceeding 1010GeV suggests that Cerenkov effect occurs with a
threshold energy expected to be of the order E &109GeV. For protons with m GeV and for Y = 1/2,0,we get
0
∼ −
- Y = 1/2. In such a case A 1+κ (l / ), and we put κ =0 in the expression for A 1+κ (l / ). Being
γ γ Pl 2 p 1 Pl
− ∼ L ∼ L
δκ κ κ (1) (κ >κ ), it follows
1 1 γ 1 γ
≡ − ∼O
2δκ l E2
= 1 Pl 0 &10−8eV−1. (II.8)
L m2
This bound is nearly in the range of nuclear physics, so that it can be discarded since no evidence of loop
structure occurred at this scale.
- Y = 0. The coefficient A reduces to A 1+κ (l / )2, and we put κ = 0 thus A 1+κ (l / )2. Being
γ γ γ Pl 1 p 2 Pl
∼ L ∼ L
δκ κ κ (1) (κ >κ ), it follows
2 2 γ 2 γ
≡ − ∼O
√2δκ l E
= 2 Pl 0 &10−18eV−1. (II.9)
L m
This bound agrees with (I.6) derived in Ref. [7].
III. NEUTRINO OSCILLATIONS IN AMU THEORY
The possibility that neutrinos particles may oscillate in different flavor states is the most discussed problem of
todaytheoreticalandexperimentalphysics. Neutrino oscillationsinthe vacuumoccurowingto the non-degeneration
of the mass-matrix eigenvalues and to the difference of the mass eigenstates ν >, ν > from the weak interaction
1 2
eigenstates ν >, ν >. In the standard scenario, the energy splitting betwee|n two|mass eigenvalues E =m2/2p,
i=1,2, is ∆|Ee =∆|mµ2/2p, where ∆m2 =m2 m2 is the mass squareddifference. The oscillationlengthiMis giveni by1
1− 2
L =2π/∆E.
M
In AMU’s theory, the energy of particles is modified according to Eq. (I.2):
m2
E =p+(Ai 1)p+ i , i=1,2, (III.1)
i p− 2p
where
2
l l
Ai =1+κ(i) Pl +κ(i) Pl . (III.2)
p 1 2
L (cid:18) L (cid:19)
The parameters κ , j =1,2,... are different for all particles. It is hence natural to assume that κ and κ are flavor
j 1 2
depending. In this new setting, the energy splitting is
∆m2
∆E +δA p, (III.3)
p
≈ 2p
1It is worth note that the dispersion relation can be also modified by taking into account the effects of gravitational fields,
which might affect the quantummechanical phaseof massive neutrinos (see for example[16–18], and also [19]).
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in which δA =A(1) A(2). The corresponding oscillation length is therefore
p p p
−
2π L L
M LQG
L= = . (III.4)
∆m2 L +L
M LQG
+δA p
p
2p
Here,L =2π/δA p is the oscillationlength inducedby loopquantumgravitycorrections. Itis worthnoting that
LQG p
in Ref. [4] the oscillation length of neutrinos is p−2l−1, a result obtained by imposing p−1 (see also the recent
∼ Pl L∼
papers[20,21]). Weshalltakeadifferentpointofviewinwhichthescalelength isafree(anduniversal)parameter.
L
The transition probability is
π∆r
Pνe→νµ =sin22ϑsin2 L , (III.5)
(cid:18) (cid:19)
being ∆r the distance travelled by neutrinos and ϑ the vacuum mixing angle. Loop quantum gravity corrections are
relevant when L L , i.e.
M LQG
≈
∆m2
δA . (III.6)
p ≈ 2p2
Let us discuss separately the case κ =0 and κ =0.
2 1
- κ =0. Denoting with δκ =κ(1) κ(2) (1), and using Eq. (III.2), one gets
2 1 1 − 1 ∼O
2l δκ p2
Pl 1
= . (III.7)
L ∆m2
For solar neutrinos, ∆m2 10−10eV2 and p MeV, so that
∼ ∼
10−6eV−1. (III.8)
L∼
In the case of atmospheric neutrinos, ∆m2 10−3eV2 and p GeV, which implies
∼ ∼
10−7eV−1. (III.9)
L∼
According to previous discussion, these values have to be discarded since no evidence of loop structure of space
has been observed at this scale.
- κ =0. By defining δκ =κ(1) κ(2) (1), and using Eq. (III.2), one obtains
1 2 2 − 2 ∼O
√2δκ l p
2 Pl
= . (III.10)
L √∆m2
For solar neutrinos it follows
10−18eV−1. (III.11)
L∼
whereas for atmospheric neutrinos one gets
10−18eV−1. (III.12)
L∼
In both cases, the scale length belongs to the bound derived by Alfaro and Palma (I.6).
L
We conclude noting that loop quantum gravity provides us a scheme in which flavor oscillations can occur even for
massless particles or for massive but degenerate neutrinos.
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IV. CONCLUSION
Despite the concrete difficulties to probe quantum gravity effects occurring at the Planck scale, recently there has
beenanincreaseintheconvictionthatquantumgravityshouldpredictaslightdeparturefromtheLorentzinvariance,
which manifests itself in a deformation of the dispersion relations of photons and fermions. Such results have been
indeedinferredinloopquantumgravity[3–5]andstringtheory[2]. Inbothapproaches,ascalelength, characterizing
the scale onwhichnew effects arenontrivial,appears. The naturalarenafor probingthe occurrenceofthe new scale
is provided by gamma ray bursts physics, and by the observedGZK limit anomaly and related processes [7] (see also
Refs. [22,23] for different contexts).
The aim of this paper has been to infer bounds on the length scale occurring in the AMU theory. Such bounds
L
have been determined in the context of Cerenkov effect and neutrinos oscillations. Our analysis yields
&10−18eV−1.
L
Remarkably,itiscompatiblewithboundsdeterminedinRef.[7]. Thus,correctionsinducedbyloopstructureofspace
becomes relevant for distances bigger than 10−18eV−1, an estimation that gives some hope to bring quantum gravity
effects to the realm of experimental results2.
ACKNOWLEDGMENTS
It is a great pleasure for the author to thank J. Alfaro, H. Morales-T´ecolt,and L.F. Urrutia for reading the paper.
ManythanksalsotoD.V. AhluwaliafordiscussionsonCerenkov’seffectduringhisvisitinSalerno,andtothereferee
of MPLA for comments.
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nucleon mass in the case of Ref. [22]). Results of Ref. [22] seem to be very promising to probe quantum gravity effects as
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