1. Emidio Gabrielli Helsinki Institute of Physics
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

2. g + external magnetic field  g g
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

3. 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

5. Heisenberg-Euler lagrangian
gauge-invariant Lagrangian
non-linear interactions
L(int)  is a function of two
gauge invariant quantities

6. Heisenberg-Euler lagrangian ( in relativistic unities c=1, h=1)
as derived by J.Schwinger PR 82, 664 (’51)
m = electron mass

7. Expanding the H-E lagrangian
at order a2
m =electron mass
provides the leading correction to
the non-linear interactions

8. photon propagating in static and homogeneous magnetic field B
Dispersions effects:
photon propagating in static and
homogeneous magnetic field B
 refraction index q B k
polarizations of photons
magnetic vectors
parallel 
perpendicular 
to (k,B) plane
vacuum polarized by B becomes birefringent

9. with different phase velocities
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

10. 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 …

11. resumming the full series of diagrams+
+ permutations + S
only an even number of total vertices can
contribute due to the Furry’s theorem 
Tr[odd-number of Dirac-gammas]=0

12. kinematic allowed solution : all three-momenta parallel
no dispersion case
kinematic allowed solution :
only one light-like
four-momentum
all three-momenta parallel

13. matrix element M(gg g)
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)

14. k= absorption coefficient
P(d)= survival probability
traveling a distance d
k= absorption coefficient
d k << 1
M=matrix element
g g phase-space

15. no-dispersion m =electron mass w/m <<1
Adler, Ann. Phys. 67, 599 ( ’71)
m =electron mass
parallel and perp. polarization vectors
with respect to the plane (k,B)
w/m <<1

16. phase space integral
energy distribution

17. 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 :

18. Effects of dispersion on polarized transitions
photon splitting
Kinematical condition
selection rules for
polarized transitions

19. CP Kinematic reaction allowed forbidden forbidden allowed allowed
Adler, Bahacall, Callan, Rosenbluth, PRL 25, 1061 (’70)
Adler ( ’71) CP
Kinematic
reaction
allowed
forbidden
forbidden
allowed
allowed
allowed
forbidden
forbidden
forbidden
allowed
forbidden
forbidden

20. 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

21. one needs very large B ~ Bcrit and/or w > MeV
difficult to detect photon splitting in typical
laboratory experiment , too rare event
one needs very large B ~ Bcrit and/or w > MeV
for w >> 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 w/m << 1.

22. 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)

23. 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 ~ GeV
coupling H-g-g generated at 1-loop

24. The axion has a very weak coupling
If astrophysical constraints are taken into
account L ~ 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

25. Effects of spin0-g-g couplings on photon propagation in external magnetic/electric fields
replacing g g f  <B> g f gives a mixing mass-term 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

26. an axion mass m ~ 10-3 eV and L ~ 106 GeV
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, (’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

27. very small effect: P(gg) ~ [P(ga)]2
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, , (’06)
very small effect: P(gg) ~ [P(ga)]2

28. Photon splitting in magnetic field induced by gg-spin-0 coupling
E.G., K.Huitu, S.Roy
PRD 74, (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

29. tadpoles mixing term no physical effect. absorbed by field
renormalization
A radiation field
F(ext) external field
tadpoles
mixing term

30. effective photon propagator
summed up at all orders
full propagator of spin-0 field
including self-energy diagrams

31. effective photon propagator (case scalar field + B)
temporal gauge
A0=0
effective photon propagator
(case scalar field + B)
T selects polarizations
with magnetic component
parallel to Plane (B,k)
R selects polarizations
with magnetic component
perpendicular to (B,k) plane
R, T are projectors

32. effective photon propagator (case pseudoscalar field + B)
T selects polarizations
with electric component
parallel to (B,k) plane
R selects polarizations
with electric component
perpendicular to (B,k) plane

33. (scalar + magnetic field)  self-energy of scalar field
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

34. master equation for photon dispersion
m = renormalized mass
of spin-0 particle
gauge invariant solutions
external electric field
external magnetic field

35. characteristic small parameters
hierarchy of scales
of the same order of D
characteristic small parameters
solutions easily found by expanding in D/m4

36. in order to have real solutions
massless mode
massive mode
in order to have real solutions
critical field

37. the massive mode can be excited from the vacuum if
analogous results for external electric fields
and/or pseudoscalar interactions

38. 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

39. photon dispersions refraxion index of massless mode n(w=0)
dispersion relation of massive mode

40. photon absorption coefficient k G= width of spin-0 particle
massive mode
from optical theorem
G= width of spin-0 particle

41. validity of perturbation theory up to
when B approaches the critical value
(x1) there is a resonant effect
however, unitarity requires Z(+) < 1
validity of perturbation theory up to

42. k2=M+2 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

43. photon absorption coefficient
massless mode
(w/m)2 (B/m2)2 <<1
scalar case + B: allowed by kinematic
incidentally, PVLAS data
have central value m ~ 2me
k > kQED if m < 5 me
pseudoscalar + B: forbidden by kinematic

44. Numerical results for Laser frequency 1eV < w < 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

45. colored areas excluded at 95 % C.L.
E.G., K.Huitu, S.Roy, PRD 74, (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

46. colored areas excluded at 95 % C.L. (1 year running)
E.G., K.Huitu, S.Roy, PRD 74, (06)
colored areas excluded at 95 % C.L.
(1 year running)

47. 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 w >> 1eV would allow in principle
to explore regions of smaller couplings.
Not clear how to detect photon splitting in this case