Download Very Special Relativity as a background field theory

Intro
VSR
SFQED
VSR/QED
Very Special Relativity as a background field theory
Anton Ilderton
Plymouth University, UK
2016-10-05
1605.04967 [hep-th], Phys. Rev. D 94 (2016) 045019
Outro
Intro
VSR
SFQED
VSR/QED
Outline
1. Intro: Lorentz violation and Very Special Relativity
2. Intro: Strong Field QED.
3. Particle motion in VSR and in background fields
4. SF-QED Ø SIM(2)-QED correspondence
Outro
Intro
VSR
SFQED
VSR/QED
What is VSR?
New physics from violation of spacetime symmetries?
Liberati, Class. Quant. Grav. 30 (2013) 133001
VSR
1. Replace Lorentz group (6 dim) Ñ subgroup (2,3,4 dim.)
2. Keep translation invariance.
3. Build (Lorentz invariance violating) QFTs.
Cohen & Glashow, PRL 97 (2006) 021601
Original motivation: allows for neutrino masses.
Cohen & Glashow, hep-ph/0605036
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Intro
VSR
SFQED
VSR/QED
Outro
SIM(2)
Largest subgroup: 4-dimensional SIM(2).
J3 ,
K3 ,
T1 “ K 1 ` J 2 ,
T2 “ K 2 ´ J 1
c constant, same dispersion.
Ñ p2 “ constant.
SIM(2) + parity = Lorentz.
Ñ Parity violation.
Bucher et al, PRL 116 (2016) 112503
B. Bucher et al, PRL 116 (2016) 112503
Null hyperplanes invariant...
Ñ Lightlike stability group
Preferred lightlike direction
Ñ From an æther?
nµ “ p1, 0, 0, 1q
But no scale?
Gibbons et al, PRD 76 (2007) 081701
Cheon et al, PLB 679 (2009) 73
Direction, but no velocity
Intro
VSR
SFQED
VSR/QED
SIM(2) and mass terms
SIM(2) permits nonlocal terms of the form
Dirac equation. . .
nµ
n.B
. . . in SIM(2)
δm2
n
{ “m
2n.p
p{ “ m
ÝÑ
p{ ´
p2 “ m2
ÝÑ
p2 “ m2 ` δm2 .
Mass-shell:
δm Ø small neutrino mass.
Cohen & Glashow PRL 97 (2006) 021601
Outro
Intro
VSR
SFQED
VSR/QED
Outro
Particle motion in background fields
Background plane wave field (basic laser model):
`
˘
eAµ pxq “: aµ pn.xq “ ma0 0, cospωn.xq, sinpωn.xq, 0
nµ “ p1, 0, 0, 1q
x0
Lightlike wave vector nµ .
x−
x+
Frequency ω.
Intensity (coupling) a0 .
x3
Integrable model
x+ = 0
(Lorentz, KG, Dirac all solvable)
Sengupta, Bull. Math. Soc. Calcutta 44 (1952) 175, Volkov, Z. Phys. 94 (1953) 250
Intro
VSR
SFQED
VSR/QED
Classical motion
σ, rates involve averages over many field oscillations.
Nikishov & Ritus 1963
Example: classical momentum.
πµ pn.xq “ pµ ´ aµ pn.xq `
average
Ó
2p.apn.xq ´ a2 pn.xq
nµ
2n.p
(linear in a Ñ 0, quadratic ­Ñ 0)
qµ :“ xπµ y “ pµ `
a20 m2
nµ .
2n.p
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Intro
VSR
SFQED
VSR/QED
VSR connection
a20 m2
nµ
2n.p
ÝÑ q 2 “ m2 ` a20 m2
πµ ÝÑ qµ “ pµ `
π 2 “ m2
quasimomentum
effective mass
T.W.B. Kibble, Phys. Rev. 138 (1965) B740,
Harvey, Heinzl, Ilderton, Marklund PRL 109 (2012) 100402
Frequency scale drops out
Averaged motion in plane waves described by VSR?
a20 m2 Ø δm2
Outro
Intro
VSR
SFQED
VSR/QED
Outro
QED in background fields
QED in a background field / BSM photon background.
`
˘
eAµ pxq “: aµ pn.xq “ ma0 0, cospωn.xq, sinpωn.xq, 0
nµ “ p1, 0, 0, 1q
x0
‹ Very high frequency scale ω.
x−
x+
‹ Cannot be probed by typical processes.
x3
‹ Effective physical observables: averages.
x+ = 0
Intro
VSR
SFQED
VSR/QED
Averaging
Basic idea.
Obtain an “effective” theory by:
1. Killing rapidly oscillating terms
2. Keeping slowly varying terms
e.g. aµ pn.xq
e.g. a2 pn.xq “ ´δm2
Classically: gives quasi-momentum/VSR momentum.
Quantum: consistent, gauge invariant truncation of QED?
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Intro
VSR
SFQED
VSR/QED
Dirac equation
Dirac equation in the background plane wave:
`
˘
iB{ ´ a{ ψ ´ mψ “ 0
Volkov solutions:
ż
ψ“
n.x
˙
„

ˆ
ż
n
{ a{ pn.xq
2
up exp ´ip.x´i 2p.a´a bp `. . .
dp 1`
2n.p
Average: ψ Ñ ψav . The averaged field obeys:
ˆ
iB{ ´
˙
δm2
n
{ ψav ´ mψav “ 0
2in.B
The Dirac equation in SIM(2) VSR!
a20 m2 Ø δm2
Outro
Intro
VSR
SFQED
VSR/QED
Outro
Averaging in QED
Fermion propagator in a plane wave.
D.M. Volkov, Z. Phys. 94 (1935) 250
H. Mitter, Acta Phys. Austr. Suppl. 14 (1975) 397
ż
SVolk px, yq “
´i...
dp e
˙
ˆ
˙ δm2 n{
ˆ
p{ ` m
n
a{ n
{ a{ pn.xq
{ pn.yq 2n.q
1`
1`
2n.p
p2 ´ m2 ` i
2n.p
q2
Intro
VSR
SFQED
VSR/QED
Outro
Averaging in QED
Fermion propagator in a plane wave.
D.M. Volkov, Z. Phys. 94 (1935) 250
H. Mitter, Acta Phys. Austr. Suppl. 14 (1975) 397
average
SVolk px, yq ÝÑ Svsr px´yq “
ż
δm2 n
q{ ` m ´ 2n.q{
d4 q
e´iq.px´yq
p2πq4 q 2 ´ m2 ´ δm2 ` i
Restores translation invariance, as in VSR. X
Cohen & Glashow PRL 97 (2006) 021601
Poles shifted to VSR mass. X
Spin– 12 propagator of VSR. X
Intro
VSR
SFQED
VSR/QED
The VSR vertex and the Ward Identity
"
: However . . .
Extra slowly-varying terms at vertices from products of fields.
ż
˙ ˆ
˙
ˆ
a{ pn.xq{
n µ
n
{ a{ pn.xq
... 1 `
γ 1`
Gµν . . .
2n.p1
2n.p
Gives correction to the QED vertex:
γ µ Ñ Γµ :“ γ µ `
δm2 nµ
2n.pp ` kqn.p
Outro
Intro
VSR
SFQED
VSR/QED
The VSR vertex and the Ward Identity
"
: However . . .
Extra slowly-varying terms at vertices from products of fields.
ż
˙ ˆ
˙
ˆ
a{ pn.xq{
n µ
n
{ a{ pn.xq
... 1 `
γ 1`
Gµν . . .
2n.p1
2n.p
Gives correction to the QED vertex:
γ µ Ñ Γµ :“ γ µ `
δm2 nµ
2n.pp ` kqn.p
!
: The three-point vertex of VSR. X
Dunn & Mehen, hep-ph/0610202
Outro
Intro
VSR
SFQED
VSR/QED
The VSR vertex and the Ward Identity
Provided all slowly varying terms are retained
Ilderton, PRD 94 (2016) 045019
a(n.x)
!
γµ
Gµν
a(n.x)
γµ −
"
a2 nµ
n
/ Gµν (k)
2n.(q + k)n.q
SIM(2)–QED
1. VSR 3-point vertex recovered.
2. Ward Identity is preserved:
k.Γ “ 0 X
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Intro
VSR
SFQED
VSR/QED
Higher-order vertices
"
: However. . . VSR has n-photon vertices for all n.
Dunn & Mehen, hep-ph/0610202
Back to the propagator:
ˆ
˙
ˆ
˙
ż
n
a{ pn.yq{
n
{ a{ pn.xq
SVolk px, yq “ . . . 1 `
. . . p{ . . . 1 `
2n.p
2n.p
n.x ­“ n.y ùñ apn.xqapn.yq rapidly oscillating.
Outro
Intro
VSR
SFQED
VSR/QED
Higher-order vertices
"
: However. . . VSR has n-photon vertices for all n.
Dunn & Mehen, hep-ph/0610202
Back to the propagator:
ˆ
˙
ˆ
˙
ż
n
a{ pn.yq{
n
{ a{ pn.xq
SVolk px, yq “ . . . 1 `
. . . p{ . . . 1 `
2n.p
2n.p
n.x ­“ n.y ùñ apn.xqapn.yq rapidly oscillating.
What happens if n.x “ n.y?
Ñ Short distance behaviour of the propagator.
Outro
Intro
VSR
SFQED
VSR/QED
Higher-order vertices
Singular term in Volkov propagator:
ż 4
d p n
{ ´ip.px´yq
SVolk Ą i
e
9 δpn.x ´ n.yq{
n.
4
p2πq 2n.p
‘Instantaneous lightfront propagator’.
Brodsky et al, Phys.Rept. 301 (1998), Bakker . . . A.I. et al, Nucl.Phys.Proc.Suppl 251 (2014) 165
Outro
Intro
VSR
SFQED
VSR/QED
Higher-order vertices
Singular term in Volkov propagator:
ż 4
d p n
{ ´ip.px´yq
SVolk Ą i
e
9 δpn.x ´ n.yq{
n.
4
p2πq 2n.p
‘Instantaneous lightfront propagator’.
Brodsky et al, Phys.Rept. 301 (1998), Bakker . . . A.I. et al, Nucl.Phys.Proc.Suppl 251 (2014) 165
p!
q!
k!
k!
n.x
SIM(2)
QED
p
k
q
k
Generates precisely the higher-order VSR vertices. X
Ilderton, PRD 94 (2016) 045019
Outro
Intro
VSR
SFQED
VSR/QED
VSR QED Ø SFQED
VSR: an effective description of QED in a high frequency wave
What is the VSR ‘æther’?
Background EM wave.
Preferred direction?
Wave vector.
Nonlocal. . .
Interaction with the wave.
. . . but trans. invariant.
Restored by averaging.
Outro
Intro
VSR
SFQED
VSR/QED
Conclusions
Summary
SIM(2) QED as an effective theory of SFQED.
New connections between
§ BSM physics with/without Lorentz violation.
§ SFQED and Lorentz invariance violation
Work in progress
Other VSR groups?
Other sectors of the standard model?
Neutrino mass in this picture? (From e.g. loop effects.)
Laser physics as “VSR in the lab”?
Heinzl and Ilderton, to appear
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