Lorentz invariance violation and chemical composition of …Many of these QG models have led to...

Lorentz invariance violation and chemical composition of UHECRs

A. Saveliev, L. Maccione, G. Sigland in collaboration with several other people

TAUP - Munich - 05.09.2011

The problem of QG

The problem of QG

Why do we need quantum gravity?Philosophical intuition: reductio ad unumLack of predictability by GR (singularities, BH entropy)

The problem of QG

Why do we need quantum gravity?Philosophical intuition: reductio ad unumLack of predictability by GR (singularities, BH entropy)

What do we need to understand quantum gravity?Observe phenomena that “have to do” with QGExtract testable predictions from the theory(ies)

Why LV?

Homogeneity

Principle of relativity

Isotropy

Pre-causality

Lorentz invariance

Implies linearity of coordinate transformations

Implies the group structure

Implies reciprocity togetherwith Principle of Relativity

Implies a notion of past and future

Known theories of gravity rest on Einstein’s equivalence principle local Lorentz invariance

von Ignatowski (1910-1911)

Challenging Lorentz invariance

lPl =�

�GN/c3 ⇥ 1.6� 10�35 mMPl =

��c/GN ⇥ 1.22� 1019 GeV

Lorentz invariance relates, through homogeneity, short to long distances.

What happens if we have a minimum length scale?

Homogeneity likely to be broken

Boost invariance likely to be broken

Modified dispersion relations

€

M ≡ spacetime structure scale, generally assumed ≈ MPlanck =1019 GeV

Assuming rotation invariance we can expand this as

From a purely phenomenological point of view, the general form of Lorentz invariance violation (LIV) is encoded into the dispersion relations

Many of these QG models have led to modified dispersion relations

E2 = p2 + m2 + �(p, M)

…

5

UHECRs can be probes of Lorentz symmetry violation, e.g. induced by Quantum Gravity

Lorentz violation is expressed through modified dispersion relations

An estimate of the critical momentum at which LV effects are important is

Exotic physics and UHECRs

E2 = c2p2�

1 +m2c2

p2+

⇤

n

�(n)pn�2

Mn�2Pl

⇥

€

m2

p2≈pn−2

Mn−2⇒ pcrit ≈ m

2Mn−2n

UHECRs can be probes of Lorentz symmetry violation, e.g. induced by Quantum Gravity

Lorentz violation is expressed through modified dispersion relations

An estimate of the critical momentum at which LV effects are important is

Exotic physics and UHECRs

E2 = c2p2�

1 +m2c2

p2+

⇤

n

�(n)pn�2

Mn�2Pl

⇥

€

m2

p2≈pn−2

Mn−2⇒ pcrit ≈ m

2Mn−2n

n pcrit for νe pcrit for e- pcrit for p+

2 p ≈ mν~1 eV p≈me=0.5 MeV p≈mp=0.938 GeV

3 ~1 GeV ~10 TeV ~1 PeV

4 ~100 TeV ~100 PeV ~3 EeV

CR Spectrum

A View of the All Particle SpectrumKASCADE-Grande collaboration, arXiv:1009.4716

Ultra-High Energy Cosmic Rays

2Freitag, 3. Juni 2011

KASKADE-Grande collaboration arXiv:1009.4716

LIV effects on UHECR spectra

Propagated (simulated) LIV spectra LM, Taylor, Mattingly, Liberati, JCAP 0904 (2009)

modified GZK effect

LV in the spectrum

Final constraints in case n=4

red/blue regions:allowed by absence of VC up to ~1020 eV

green points/black crosses:in agreement with observed spectrum within 95% and 99% CL resp.

LM, Taylor, Mattingly, Liberati, JCAP 0904 (2009)

Also nuclei?

E [eV]

1810

1910

]2

> [

g/c

mm

ax

<X

650

700

750

800

850 proton

iron

QGSJET01QGSJETIISibyll2.1EPOSv1.99

E [eV]

1810

1910

]2

) [g

/cm

ma

xR

MS

(X

0

10

20

30

40

50

60

70 proton

iron

Figure 1.6: Recent results for the chemical composition of cosmic rays at ultra high ener-gies, showing ⇥Xmax⇤ (left) and RMS (Xmax) (= �x) (right) for measurements (filled datapoints) and simulations (lines and unfilled data points). Upper panel: Pierre Auger Col-laboration; Lower panel: High Resolution Fly’s Eye Collaboration; taken from [25] and[26]

These cascades start by the hadronic interaction of a cosmic ray nucleuswith atoms of the atmosphere. In the first order only pions, either chargedor neutral, are produced. The neutral pions almost instantly decay into twophotons which induce an electromagnetic cascade. Charged pions for theirpart either decay (at low energies) into muons and muon neutrinos or re-interact (at high energies), producing both kinds of pions again (for furtherreference see e.g. [24]).

The two main quantities to deduct chemical composition from the cascadeare the average value and the fluctuation of Xmax (the (longitudinal) depthof the shower at which the number of secondary particles is maximal), called⇥Xmax⇤ and Xmax, respectively. Using a simplified air shower model based onthe ideas of Heitler, one can show that ⇥Xmax⇤ is smaller for heavier nucleiwith the same energy, approximately given by [7]

⇥Xmax⇤ = � (ln E � ⇥ln A⇤ + ⇥) , (1.21)

� and ⇥ being some hadronic, interaction-specific parameters, E the energyof the nucleus and A its mass number. On the other hand, RMS (Xmax) ispredicted to decrease with A.

12

PAO Coll, PRL 104 (2010)

HiReS Coll, PRL 104 (2010)

LIV in heavy nuclei

Transformation of a nucleus (A,Z) into one or more nuclei (A’,Z’).

Dispersion relation

2 new processes: - emission of Cherenkov radiation in vacuum- spontaneous decay

1 existing process: photodisintegration

LIV for UHECR Nuclei

Ultra High Energy Cosmic Rays (UHECRs) are the particles withthe highest energies ever observed æ Candidates for observing LIVe�ects

I The main reaction for UHECR nuclei is photodisintegration, inthe simplest case:

A

Z

N + “ æ AÕZ

ÕN Õ + BW

N ÕÕ

with AÕ = A ≠ B and Z Õ = Z ≠ W .However, due to LIV and the MDR for composite particles[Jacobson et al., 2003],

E 2A,Z = p2A,Z + m2A,Z +

÷

A2pn+2

A,Z

MnPl,

two new reactions may appear:

14 A. Saveliev

Lorentz invariance violations and UHE nuclei

A. Saveliev, LM, G. Sigl, JCAP 2011LIV for UHECR Nuclei

-102 -100 -10-2 -10-4 -10-6 -10-818.5

19.0

19.5

20.0

20.5

21.0

21.5-102 -100 -10-2 -10-4 -10-6 -10-8

h

logp thrdec êeV

4He

16O

56Fe

10-8 10-6 10-4 10-2 100 10218.5

19.0

19.5

20.0

20.5

21.0

21.510-8 10-6 10-4 10-2 100 102

h

logp thrVCêeV

4He

16O

56Fe

Constraints from Spontaneous Decay and VC radiation

Emax

= 1019.6 eV Emax

= 1020 eV4He ≠3 ◊ 10≠3 . ÷ . 4 ◊ 10≠3 ≠7 ◊ 10≠5 . ÷ . 1 ◊ 10≠416O ≠7 ◊ 10≠2 . ÷ . 1 ≠2 ◊ 10≠3 . ÷ . 3 ◊ 10≠256Fe ≠1 . ÷ . 200 ≠3 ◊ 10≠2 . ÷ . 4

[Saveliev et al., 2011]

Constraints from spontaneous decay and VC emission

LIV for UHECR Nuclei

-102 -100 -10-2 -10-4 -10-6 -10-818.5

19.0

19.5

20.0

20.5

21.0

21.5-102 -100 -10-2 -10-4 -10-6 -10-8

h

logp thrdec êeV

4He

16O

56Fe

10-8 10-6 10-4 10-2 100 10218.5

19.0

19.5

20.0

20.5

21.0

21.510-8 10-6 10-4 10-2 100 102

h

logp thrVCêeV

4He

16O

56Fe

Constraints from Spontaneous Decay and VC radiation

Emax

= 1019.6 eV Emax

= 1020 eV4He ≠3 ◊ 10≠3 . ÷ . 4 ◊ 10≠3 ≠7 ◊ 10≠5 . ÷ . 1 ◊ 10≠416O ≠7 ◊ 10≠2 . ÷ . 1 ≠2 ◊ 10≠3 . ÷ . 3 ◊ 10≠256Fe ≠1 . ÷ . 200 ≠3 ◊ 10≠2 . ÷ . 4

[Saveliev et al., 2011]

Lorentz invariance violations and UHE nuclei

18.5 19.0 19.5 20.0 20.5 21.0log pêeV

0.01

0.1

1

10

100

1000

10000

105lMFPêMpc

h=-10-2

h=10-2

h=-10-4

h=10-4

h=0

A. Saveliev, LM, G. Sigl, JCAP 2011

Fe mean-free-path

Conclusions

UHECRs can serve as probes for Planck scale physics: spectrum and chemical composition

Constraints coming from protons are very strong, BUT they might be invalid if heavy nuclei are present at UHE

Constraints from nuclei are becoming interesting and can be made stronger by computing photodisintegrated spectra (to be done!!)

Backup slides

LV in the spectrum

Propagated (simulated) LIV spectra

• Effect of LIV: modify absorption of protons on the CMB (increases/decreases the photon energy needed to interact for ηp < 0/>0)

• Recovery of flux at high energy, due to reduced inelasticity

• Effect of sources: where is the closest UHECR source? We don’t know, but the effect is different from the one due to LIV

LM, Taylor, Mattingly, Liberati, arXiv:0902.1756

LV in the photon spectrumGalaverni, Sigl, PRL 100, 021102 (2008) LM, S. Liberati, JCAP 0808, 027 (2008)Galaverni, Sigl, PRD 78, 063003 (2008)

LV in the photon spectrumGalaverni, Sigl, PRL 100, 021102 (2008) LM, S. Liberati, JCAP 0808, 027 (2008)Galaverni, Sigl, PRD 78, 063003 (2008)GZK photons are produced by the decay of π0s due to pion production


In LI theory they are attenuated mainly by pair production onto CMB and URB leading to a theoretically expected photon fraction < 1% at 1019 eV and < 10% at 1020 eV.



Present limits on photon fraction: 2.0%, 5.1%, 31%, 36% (95% CL) at 10, 20, 40, 100 EeV




LIV strongly affects the threshold of this process: lower and also upper thresholds.





If kup < 1020 eV then photon fraction in UHECR much larger than present upper limits

-18-16-14-12-10-8-6-4-2 -18 -16 -14 -12 -10 -8 -6 -4 -2

-18-16-14-12-10-8-6-4-2

-18-16-14-12-10-8-6-4-2

Biref.

log

log

Crab

n=3

-10-8-6-4-202 -10 -8 -6 -4 -2 0 2

-10

-8

-6

-4

-2

0

2

-10

-8

-6

-4

-2

0

2

log

log

n=4






LIV also introduces competitive processes: γ-decay

-18-16-14-12-10-8-6-4-2 -18 -16 -14 -12 -10 -8 -6 -4 -2

-18-16-14-12-10-8-6-4-2

-18-16-14-12-10-8-6-4-2

Biref.

log

log

Crab

n=3

-10-8-6-4-202 -10 -8 -6 -4 -2 0 2

-10

-8

-6

-4

-2

0

2

-10

-8

-6

-4

-2

0

2

log

log

n=4







If photons above 1019 eV are detected then γ-decay threshold > 1019 eV

-18-16-14-12-10-8-6-4-2 -18 -16 -14 -12 -10 -8 -6 -4 -2

-18-16-14-12-10-8-6-4-2

-18-16-14-12-10-8-6-4-2

Biref.

log

log

Crab

n=3

-10-8-6-4-202 -10 -8 -6 -4 -2 0 2

-10

-8

-6

-4

-2

0

2

-10

-8

-6

-4

-2

0

2

log

log

n=4

LV in the photon spectrum

-10-8-6-4-202 -10 -8 -6 -4 -2 0 2

-10

-8

-6

-4

-2

0

2

-10

-8

-6

-4

-2

0

2

log

log

-18-16-14-12-10-8-6-4-2 -18 -16 -14 -12 -10 -8 -6 -4 -2

-18-16-14-12-10-8-6-4-2

-18-16-14-12-10-8-6-4-2

Biref.

log

log

Crab

Galaverni, Sigl, PRL 100, 021102 (2008) LM, S. Liberati, JCAP 0808, 027 (2008)Galaverni, Sigl, PRD 78, 063003 (2008)GZK photons are produced by the decay of π0s due to pion production






If photons above 1019 eV are detected then γ-decay threshold > 1019 eVn=3 n=4

LV in the neutrino sector

Effects on oscillations

Ecr ⇡ MPl✓

�m2

M2Pl⌘n⌫

◆1/n 0.2 GeV (n=3)20 TeV (n=4)

Strong constraints already from neutrino experiments!

Oscillations: �c/c . 10�27

Also quantum decoherence effects alter oscillation patterns

LIV: prospects for the UHE neutrino sector

Neutrino Cherenkov emissionvery low rate, irrelevant on Hubble scales

Neutrino splittingvery fast rate above 1019 eV

Neutrino decaycan mimic a Z-burst effect --> effect at only 5% on the UHECR spectrum, not visible yet

Mattingly, LM, Galaverni, Liberati, Sigl, JCAP 1002:007,2010

⇥ ! ⇥�(⇥g)

⌫ ! ⌫⌫⌫̄

� ! �qq̄

Possible cosmogenic neutrino constraints on Planck-scale Lorentz violation 10

This width can be turned into a decay length as

Lννν̄ =c

Γννν̄∼ 1.7× 10−3 Mpc η−4ν

(

E

1019 eV

)−13

. (18)

This makes clear we need to push the required energies above 1018.5 eV (with ην = 1)for the rate to be appreciable. As a final remark, we notice that the decay length in

Eq. (18) strongly depends on both the energy and ην . Therefore, the error about its

actual magnitude we might have made in our estimate will reflect in very small errors

in the determination of the energy at which LV effects start to be relevant as well as of

the constraint on ην .

4.1.2. Z boson resonance At such high energies the Z could be real - i.e. there is a

resonance in the matrix element. Even in this regime, however, the neutrino decay

time can be computed easily, as the only hypothesis one has to relax is that the Z 4-

momentum r satisfies r2 # M2Z . The magnitude of r2 can be easily computed exploitingthe kinematic equations. We obtain r2 = 16/27ηνE4ν/M

2Pl. The final decay length is then

Lννν̄ ∼ 1.7× 10−3 Mpc η−4ν(

Eν1019 eV

)−13

×

[

(

1−16

27ην

E4

M2Pl M2Z

)2

+

(

ΓZMZ

)2]

. (19)

A comparison between the two different decay lengths, Eq. (18) and Eq. (19), can be

found in Fig. 1. The Z resonance is hit at E4 = 27/16 (η−1ν M2PlM

2Z). We notice that even

(E/eV)10

log18 18.5 19 19.5 20 20.5 21 21.5 22

(L/M

pc)

10lo

g

−20

−15

−10

−5

0

5

10 Full computation

Approximate computation

Figure 1. Comparison between computations of the decay length without(Eq. (18)/red dashed line), and with (Eq. (19)/black solid line) the Z boson resonance.

though the two computations lead to very different results above the resonance, they

will not lead to any appreciable effect in the neutrino spectra, as at such energies the

decay lengths are anyway much smaller than the propagation distance of cosmological

neutrinos.

Possible cosmogenic neutrino constraints on Planck-scale Lorentz violation 12

we have to resort to full MonteCarlo simulations of the UHECR propagation from

sources to the Earth.

We simulated then the propagation of UHECR protons in the Inter Galactic

Medium using the Monte Carlo package CRPropa [65], suitably modified to take into

account LV in the neutrino sector. The simulation parameters are the following: we

simulated unidimensional UHECR proton propagation, with source energy spectrumdN/dE ∝ E−2.2, from a spatially uniform distribution of sources located at redshiftz < 3 according to the Waxman & Bahcall (WB) distribution used in [66]. The injection

proton spectrum was tuned to fit AUGER data [59].

5.1. Results for flavor bind LV scenario

Figure 2 shows the outcome of the simulations for different values of the LV parameter ηνin the best case scenario, together with experimental sensitivities from some existing and

planned observatories, as well as the Waxman & Bahcall bound [68, 69] for reference.+

Results are in agreement with qualitative expectations previously discussed: Above a

(E / eV)10

log13 14 15 16 17 18 19 20 21

)−1

sr

−1 s

−2dN

/dE

(eV

cm2 E

−210

−110

1

10

210

(E / eV)10

log13 14 15 16 17 18 19 20 21

)−1

sr

−1 s

−2dN

/dE

(eV

cm2 E

−210

−110

1

10

210

Experimental sensitivitiesARIANNA − A phaseARIANNA − fullANITAWB limitAUGER

= 0η = 1η

−1 = 10η−2 = 10η−3 = 10η−4 = 10η

Figure 2. Evolution of the predicted LV neutrino spectra varying ην in the “best casescenario”. Sensitivities of main UHE neutrino operating and planned experiments areshown, as found in [51, 47, 67]. The Waxman & Bahcall limit [68, 69] in the interestingenergy range is shown for reference.

+ This limit is in fact an estimate of the neutrino luminosity of sources of UHE Cosmic Rays andγ-rays, in the hypothesis that the sources are optically thin to the escape of UHE particles and thatboth γ-rays and neutrinos are originated from UHECR interactions with radiation backgrounds. It isworth mentioning that this bound might be strongly affected by QG effects, as shown in [70].

�(4)� .✓

Eobs

6⇥ 1018 eV

◆�13/4

LIV: prospects for the UHE neutrino sector

Mattingly, LM, Galaverni, Liberati, Sigl, JCAP 1002:007,2010

The Greisen-Zatsepin-Kuzmin effect

p + � ! N + ⇡ Eth =2mpm⇡ + m2⇡

4✏⇠ 4 · 1019 eV on CMB

9:1 L51321+PY*62143+PZ/[)3+ HLYZK$1;;176

F73"-0,1 3%, ;$0573- ;(0,180,8#*- 3012(3 2(3$06%P- 9%3_)$07,5

,73"-0, !

"&$-10,%,3-

27"#(&;(0, ;$0573#(0,

;%($8;$0573#(0,8-,-$). "011

;(0,8;$0573#(0,8-,-$). "011

;(0,8;$0573#(0,8$%#-

!107$3-18271#89- (,830120"0)(3%" 9%3_.%$5?,". W0$-,#e81.22-#$. 9$-%_(,) %# !dSRSS

307"5 %P0(5 #*(1 30,3"71(0,I

!"#$%%

& #'&

()#$

!"

%&& !!!" #

9:1 L51321+PY*62143+PZ/[)3+ HLYZK$1;;176

F73"-0,1 3%, ;$0573- ;(0,180,8#*- 3012(3 2(3$06%P- 9%3_)$07,5

,73"-0, !

"&$-10,%,3-

27"#(&;(0, ;$0573#(0,

;%($8;$0573#(0,8-,-$). "011

;(0,8;$0573#(0,8-,-$). "011

;(0,8;$0573#(0,8$%#-

!107$3-18271#89- (,830120"0)(3%" 9%3_.%$5?,". W0$-,#e81.22-#$. 9$-%_(,) %# !dSRSS

307"5 %P0(5 #*(1 30,3"71(0,I

!"#$%%

& #'&

()#$

!"

%&& !!!" #

UHECRs at E>1020 eV must be produced within 100 Mpc from us. OR there is violation of Lorentz symmetry for boosts > 1011.

Lorentz invariance violation and chemical composition of …Many of these QG models have led to...

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