Transcript Document 7457375

Photon splitting in magnetic
fields as a probe of ultralight
spin-0 fields
Emidio Gabrielli
Helsinki Institute of Physics
in collaboration with K. Huitu and S. Roy
Photon splitting
g
+ external magnetic field 
g g
it is due to non-linear photon interactions
induced by vacuum polarization effects
in QED the absorption coefficient K is very
small for magnetic fields of the order of Tesla
K ~ (B/B(crit))6
B(crit)= m2/e = 4.41 x 109 T
new physics could contribute with sizeable
effects
At tree-level there are no photon self-interactions
Quantum effects, induced by interactions of photons
with charged particles (i.e. electrons, positrons, etc.),
generate  photon self-interactions
described by the Heisenberg-Euler effective lagrangian,
obtained in the constant EM field strength limit
observable (but rare) effects:
- photon propagating in constant magnetic fields:
birefringence of vacuum 
ellipticity, rotation of the light polarization-plane
- the light-light scattering g g  g g
Heisenberg-Euler lagrangian
gauge-invariant Lagrangian
non-linear interactions
L(int)  is a function of two
gauge invariant quantities
Heisenberg-Euler lagrangian
( in relativistic unities c=1, h=1)
as derived by J.Schwinger PR 82, 664 (’51)
m = electron mass
Expanding the H-E lagrangian
at order a2
m =electron mass
provides the leading correction to
the non-linear interactions
Dispersions effects:
photon propagating in static and
homogeneous magnetic field B
 refraction index
B
q
k
polarizations of photons
magnetic vectors
parallel

perpendicular 
to (k,B) plane
vacuum polarized by B becomes birefringent
a method to measure vacuum birefringence
E.Iacopini and E.Zavattini
PLB 8, 151 (’79)
from PVLAS
webpage
different polarization vectors will propagate
with different phase velocities
linear polarization  elliptical polarization
out of B . Ellipticity y induced by birefringence
y =
L

( n ||  n  )
Photon splitting (no dispersion)
g(k) + external magnetic field  g (k1) g(k2)
not possible in vacuum, but in presence of external B
according to the Heisenberg-Euler theory, photon
dispersion relations are modified,
refraction index  n > 1
let consider first the case of no-dispersion (n=1)
S.Adler, Ann. Phys. 67, 599 ( ’71)
matrix element can be obtained from the
H-E lagrangian or equivalently from …
resumming the full series of diagrams
+
+ S
+
permutations
only an even number of total vertices can
contribute due to the Furry’s theorem 
Tr[odd-number of Dirac-gammas]=0
no dispersion case
 = 1 + 2
kinematic allowed solution :
all three-momenta parallel
only one light-like
four-momentum
in the case of no dispersions the photon splitting
with only one interaction of the external field is
forbidden
the leading effect is given by three external
insertions, the hexagon diagram
this is due to: gauge invariance and the fact that
there is only one light-like four momentum in the
reaction.
leading order for the
matrix element M(gg g)
P(d) = exp(- k d)
P(d)= survival probability
traveling a distance d
k= absorption coefficient
d k << 1
N (g  gg )  N (g )  d  k


M=matrix element
g g phase-space
no-dispersion
Adler, Ann. Phys. 67, 599 ( ’71)
m =electron mass
parallel and perp. polarization vectors
with respect to the plane (k,B)
/m <<1
phase space integral
energy distribution
Effects of dispersions on photon splitting
momenta of final photons are not anymore
parallel to initial ones  small opening angle.
in the Golden Rule formula for the absorption
coefficient one has to change :
Effects of dispersion on
photon splitting
Kinematical condition
selection rules for
polarized transitions
Adler, Bahacall, Callan, Rosenbluth, PRL 25, 1061 (’70)
Adler ( ’71)
reaction
CP
Kinematic
allowed
forbidden
forbidden
allowed
allowed
allowed
forbidden
forbidden
allowed
forbidden
forbidden
forbidden
conclusions
(QED)
splitting of perpendicular-polarized photons is
absolutely FORBIDDEN by dispersive effects
splitting of parallel-polarized photons is ALLOWED
photon splitting provides a mechanism for
the production of polarized photons
effects of dispersion on matrix element are small
k  0.1  (B/B crit ) 6  ( / m) 5  cm 1
difficult to detect photon splitting in typical
laboratory experiment , too rare event
k  4 1088  (B/T ) 6  ( / ev)5  cm1
one needs very large B ~ Bcrit and/or  > MeV
for  >> 2m ~ MeV, the pair production mechanism
g  e+e- dominates over g gg
Adler (’ 71) provided the exact calculation of
k valid beyond the approximation /m << 1.
neutral ultra-light spin-0 bosons
Neutral scalar/pseudoscalar particles can
have gauge invariant couplings with photons:
L  effective scale of dimension [mass]
F(m,n) EM field strentgh
F~(m,n) = e(m,n,a,b) F(a,b)
known examples are
light: axion boson
pseudo-scalar particle
pseudo-goldstone boson of Peccei-Quinn
symmetry (solve the strong-CP problem in QCD)
mass expected in the range of m ~ O(meV)
heavy: Higgs boson
scalar particle
necessary to provide all masses in the SM
mass expected in the range m ~ 100-800 GeV
coupling H-gg generated at 1-loop
The axion has a very weak coupling
If astrophysical constraints are taken into
account L ~ 106-1011 GeV
G.Raffelt, Phys. Rept. 198, 1 (’90)
Recently, it has been shown that it is possible
to relax astrophysical constraints
E.Masso and J.Redondo, JCAP, 0505, 015 (’05)
decay-width (G) of the axion is VERY small,
G = m3/L2  almost stable particle on
cosmic time scale
Effects of spin0-gg couplings on
photon propagation
in external magnetic/electric fields
replacing g g f  <B> g f gives a mixing massterm in the photon-spin-0 system
the gamma spin-0 conversion is possible in external
EM field (Primakof effect)
it could generate photon  spin-0 oscillations for
photons propagating in magnetic fields
G.Raffelt, L.Stodoslky, PRD 37, 1237 (’88)
angular momentum and 3-momentum not conserved 
3-momentum absorbed by external field
mass, coupling and parity of ultra-light spin0
particle can be determined from measurement
of vacuum birefringence and dichroism
L.Maiani, R. Petronzio, E. Zavattini, PLB 175, 359 (’86)
the birefringence can induce ellipticity on a linearly
polarized Laser beam in external magnetic field
R. Cameron et. al. [BFRT collab.] PRD 47, 3707 (’93)
recently PVLAS collaboration (’05) has measured a
large value for the ellipticity
E.Zavattini et. al. [PVLAS collab.], PRL 96, 110406 (’06)
too large for QED ! New physics effect ?
if interpreted in terms of light axion implies
an axion mass m ~ 10-3 eV and L ~ 106 GeV
Ultralight axions can also be tested in laboratory by
different kind of experiments.
P.Sikivie, PRL 51, 1415 (’83)
After a Laser beam passes through a magnetic field
an axion component can be generated.
Light shining from a wall by using a second magnet
It is possible to check the parameter region
explored by PVLAS data, by using Xray laser facility
R.Rabadan, A.Ringwald, K.Sigurdson, PRL 96, 110407, (’06)
very small effect: P(gg) ~ [P(ga)]2
Photon splitting in magnetic field
induced by gg-spin-0 coupling
E.G., K.Huitu, S.Roy
PRD 74, 073002 (06)
We used the technique of effective photon propagator
+ optical theorem to calculate absorption coefficient
Imaginary part of (pseudo)scalar propagator (width)
gives the leading effect in the photon-splitting
absorption coefficient
the full series of diagrams has been exactly summed
up in the effective photon propagator
A radiation field
F(ext) external field
no physical effect.
absorbed by field
renormalization
tadpoles
mixing term
effective photon propagator
summed up at all orders
full propagator of spin-0 field
including self-energy diagrams
temporal gauge
A0=0
effective photon propagator
(case scalar field + B)
T selects polarizations
with magnetic component
parallel to Plane (B,k)
R, T are projectors
  ext
S B  B
R selects polarizations
with magnetic component
perpendicular to (B,k) plane
effective photon propagator
(case pseudoscalar field + B)
T selects polarizations
with electric component
parallel to (B,k) plane
  ext
P E  B
R selects polarizations
with electric component
perpendicular to (B,k) plane
photon self-energy
(scalar + magnetic field)
 self-energy of scalar field
modifies the photon dispersions for the
polarizations with magnetic component
parallel to (B, k) plane
solutions of photon dispersions
are obtained by looking at the
poles of the propagator
master equation for photon dispersion

k 2   2  | k |2
m = renormalized mass
of spin-0 particle
external electric field
gauge invariant solutions
external magnetic field
hierarchy of scales
of the same order of
characteristic small parameters
D
<< 1
4
m
G
<< 1
m
m3
G=
64  L2
solutions easily found by expanding in D/m4
D
solutions
massless mode
massive mode
in order to have real solutions
 1
critical field
the massive mode can be excited
from the vacuum if

| B |  BCRIT
G
=
m
analogous results for external electric fields
and/or pseudoscalar interactions
residue (Z) at the poles
it is connected to the norm of the quantum state
physical solutions must have positive value
for the residue at the pole of the propagator
M(-) massive solution is unphysical since Z(-) < 0
there are only 2 physical solutions, one massless
and one massive
photon dispersions
refraxion index of massless mode n(=0)
dispersion relation of massive mode
photon absorption coefficient k
massive mode
k=
mg

from optical theorem
Gg
G m
G
k=
+ O( 2 )
m
1  
2
G= width of spin-0 particle
G m
G2
k=
+ O( 2 )
m
1  
when B approaches the critical value
(1) there is a resonant effect
however, unitarity requires Z(+)
< 1
validity of perturbation theory up to
G2
  1 2
m
k
max
G
=
+ O( )

m
m2
same results can be re-obtained by using the
Golden Formula for absorption coefficient,where
the matrix element M is:
k2=M+2
for the massless mode Z+  Z0 ~ 1 and k2=M02
photon absorption coefficient
massless mode
(/m)2 (B/m2)2 <<1
scalar case + B: allowed by kinematic
k=
B6 5
k QED
m16
1
3840
B6 5 a 6

16
me 60 2
m  mL
incidentally, PVLAS data
have central value m ~ 2me
k > kQED
if m < 5 me
pseudoscalar + B: forbidden by kinematic
Numerical results
for Laser frequency 1eV <  < 102eV ,
10-2eV < m < 102eV and 103 GeV < L < 1010 GeV
B=1T, the massive mode gives largest contribut.
photon splitting could be tested in lab
experiments by using high brilliance Lasers
dN/dt ~ 1018 s-1
we assume that in the range of mass explored
the dominant decay channel is in two photons
E.G., K.Huitu, S.Roy, PRD 74, 073002 (06)
colored areas excluded at 95 % C.L.
B= 5 Tesla of L=10m length, dN/dt=1018 /s
(left) 1 day (right) 1 year running time
E.G., K.Huitu, S.Roy, PRD 74, 073002 (06)
colored areas excluded at 95 % C.L.
(1 year running)
Conclusions
two-photon-spin0 coupling can induce photon splitting
on static and homogeneous magnetic fields
the absorption coefficient is much larger than in QED
for typical masses m~10-3 eV, L~106 GeV, and B~O(T)
large areas of the parameter space could be tested
by optical Laser experiments, with B=1-10 T
Lasers with  >> 1eV would allow in principle
to explore regions of smaller couplings.
Not clear how to detect photon splitting in this case