Constraining Lorentz violation with ultra-high energy
photons.
G.Rubtsov, P. Satunin, S.Sibiryakov,
based on PRD 86 (2012) 085012 (hep-ph/1204.5782) + further work
Institute for Nuclear Research, Moscow
13 December 2012
Satunin P.S. (INR, Moscow)
Constraining LV with UHE photons
December 2012
1 / 14
Outline
Motivation of LV
Model, modified Feynman rules
Perturbative calculations: γN → N e+ e−
Semiclassical calculation: γ → e+ e− in a weak magnetic field.
Conclusions for experiments.
Satunin P.S. (INR, Moscow)
Constraining LV with UHE photons
December 2012
2 / 14
Motivation
Several approaches to quantum gravity predict Lorentz invariance
violation (LV) at high energies (e.g. Horava-Lifshitz).
Horava, 2009
Blas, Pujolas, Sibiryakov, 2010
Theories with violation of the Lorentz Symmetry as an effective theory
Kostelecky, Colladay, 1998
Coleman, Glashow, 1999
Satunin P.S. (INR, Moscow)
Constraining LV with UHE photons
December 2012
3 / 14
Model
Assumptions: C−, P −, T −parity, rotational symmetry, LV operators of
dimension up to 6.
1
L = ψ̄ (iγ µ Dµ − m) ψ − Fµν F µν +
4
+iκ ψ̄γ i Di ψ +
ig
ξ
Dj ψ̄γ i Di Dj ψ +
Fkj ∂i2 F kj
2
M
4M 2
⇓
Dispersion relations:
Eγ2 = k 2 +
Satunin P.S. (INR, Moscow)
ξk 4
,
M2
Ee2 = m2 + p2 (1 + 2κ) +
Constraining LV with UHE photons
2gp4
.
M2
December 2012
4 / 14
Modified spin sums and propagators
X
(a)
ε(a)
µ εν
a=1,2
X
ξ 2 k2
= diag −1 −
, 1, 1, 1
M2
X
us (p)ūs (p) = γ µ p̃µ + m,
s=1,2
v s (p)v̄ s (p) = γ µ p̃µ − m
s=1,2
0
p̃ = E,
i
p̃ = p
i
gp2
1+κ+ 2
M
Propagators
p
k
Satunin P.S. (INR, Moscow)
i (γ µ p̃µ + m)
p̃µ p̃µ − m2 + i
i
ξk 2
=
diag −1 − 2 , 1, 1, 1
ξk2
M
E 2 − k2 1 + M
+ i
2
=
Constraining LV with UHE photons
December 2012
5 / 14
Modified vertices
h
g
= −ieγ µ − ieδiµ κγ i + 2 pi1 (p1 · γ +
M
+pi2 (p2 · γ) − (p1 · p2 )γ i
p1
p2
p1
p2
=
µ ν
ige2 i
j
j
i
ij
γ
(p
−
p
)
+
γ
(p
−
p
)
+
δ
((p
−
p
)
·
γ)
δi δj
2
1
2
1
2
1
M2
p1
p2
Satunin P.S. (INR, Moscow)
=−
i
2ige3 h µ ν λ
µ ν λ
µ ν λ
δ
δ
δ
+
δ
δ
δ
+
δ
δ
δ
i j j
j i j
j j i
M2
Constraining LV with UHE photons
December 2012
6 / 14
Pair production on a nuclei γN 7→ e+ e− N
7
N14
ωLV = −κk +
k |ωLV | m2
−→
ξ
2
−g
σ=
km
k3
,
M2
28 Z 2 α3 2k
ln
9 m2
m
Bethe, Heitler, 1934
k |ωLV | m2
−→
α2 Z 2/3
m2 k |ωLV | LV
σγN
→N e+ e− ∼
3
1
k |ωLV |
k |ωLV | 2
ln
m2
m2
8Z 2 α3
1
2k |ωLV |
1
−→
σ
=
ln
ln
2
2/3
2
3 k |ωLV |
m
α Z
αZ 1/3
m2
k|ωLV |
Satunin P.S. (INR, Moscow)
σ=
4Z 2 α3
LV
· σγN
→N e+ e− — suppression
Constraining LV with UHE photons
December 2012
7 / 14
Pair production in a weak magnetic field. LI case.
Semiclassical method: used to compute Shwinger effect (pp in electric
field)
Affleck, Alvarez, Manton, 1982
Monin, Voloshin, 2008
H = (H, 0, 0); kµ = (ω, 0, ω, 0)
Z
Γ≈
−S[xµ ]
Dxµ e
I
,
S=
q
dφ m ẋ2µ − ieAµ ẋµ − ωT + ωY,
mẍµ
EOM: q = ieFµν ẋν
ẋ2µ
ψ
B
z
A
Satunin P.S. (INR, Moscow)
Solution (the right arc):
p
p2 + m2
θ
p
θ
t=
φ−
, y = −iψ = −i
sh φ −
,
eH
2
eH
2
p
θ
θ
x = 0,
z=−
ch φ −
− ch
eH
2
2
Constraining LV with UHE photons
December 2012
8 / 14
Pair production in a weak magnetic field.
~
ϕ — angle between ~k and H.
8
m3
Γ ∝ exp −
3 ωeH sin ϕ
→
LI case
Klepikov, 1952
Toll, 1952
Robl, 1952
√
3/2 #
2
2 2ω
2m
0
Γ ∝ exp −
− ωLV
.
3eH sin ϕ
ω
"
LV case
→
0
ωLV
= −κω +
0
ωLV
<−
Satunin P.S. (INR, Moscow)
2m2
ω
→
ξω 3
gω 3
−
.
2M 2
4M 2
exponential suppression!
Constraining LV with UHE photons
December 2012
9 / 14
Current bounds on LV parameters
Eγ2 = k 2 +
ξk 4
,
Mp2
Ee2 = m2 + p2 (1 + 2κ) +
2gp4
.
Mp2
|κ| < 10−15 — absence of anomally synchrotron losses at LEP
Altshul, 2009
1013
−5 ·
<ξ<
detected
1011
— photons with energies 50 TeV have been
HEGRA, CANGAROO, HESS
−1011
106
<g<
— photon spectrum of Crab nebula and AGNs
(assumption — SSC model)
Altshul, 2006
subplankian LV is allowed
Satunin P.S. (INR, Moscow)
→
can be suff. more suppressed using UHECR
Constraining LV with UHE photons
December 2012
10 / 14
Possible future bounds
GZK cutoff: p γCM B → p π0 or p γCM B → n π+ ,
π0 → γ γ
2m2
k (simplified – symm. threshold)
Galaverni, Sigl 2008
m2
+
−
on a nuclei γN → N e e
→ ωLV < − k k < 1019.5 eV
0 < − 2m2 k > 1019.5 eV
in magnetic field γ → e+ e− → ωLV
k
0 >
gamma decay γ → e+ e− → ωLV
supp. pp
supp. pp
If UHE photons will be detected, LV will be restricted.
Model-independent bound!
Satunin P.S. (INR, Moscow)
Constraining LV with UHE photons
December 2012
11 / 14
UHECR experiments — 2015 sensitivity estimates
Satunin P.S. (INR, Moscow)
Constraining LV with UHE photons
December 2012
12 / 14
Conclusions
Even small (κ ∼ 10−26 or ξ ∼ g ∼ 10−6 ) LV strongly suppresses pair
production (both in an electric field of a nuclei or in a weak magnetic
field)
Subplankian LV is NOT experimentally closed yet.
If some photons with energy 1018 − 1019 eV will be detected,
subplankian LV (and some models of quantum gravity) will be
forbidden (except fine-tuned case ωLV ≈ 0).
Satunin P.S. (INR, Moscow)
Constraining LV with UHE photons
December 2012
13 / 14
Thank you for your attention!
Satunin P.S. (INR, Moscow)
Constraining LV with UHE photons
December 2012
14 / 14