Polyelectrons and Exotic Atoms
e+
e-
e-

0
2
4


1
3
5

e+
Séminaire de physique expérimentale
LAPP, 3 juillet 2009
Andrzej Czarnecki
University of Alberta
Exotic atoms today (simple examples)
Muonic hydrogen
µ-p+
Proton radius
from Lamb shift
Electroweak formfactors
from capture
Lepton flavor violation
(muon-electron conversion)
Exotic atoms today (simple examples)
Muonic hydrogen
µ-p+
Proton radius
from Lamb shift
Electroweak formfactors
from capture
Hydrogen-like ions
and hydrogen
Rydberg and α
Electron mass
Lepton flavor violation
(muon-electron conversion)
Exotic atoms today (simple examples)
Muonic hydrogen
µ-p+
Proton radius
from Lamb shift
Electroweak formfactors
from capture
Hydrogen-like ions
and hydrogen
Rydberg and α
Electron mass
Lepton flavor violation
(muon-electron conversion)
Antiprotonic helium
Antiproton mass
Exotic atoms today (simple examples)
Muonic hydrogen
µ-p+
Proton radius
from Lamb shift
Electroweak formfactors
from capture
Hydrogen-like ions
and hydrogen
Rydberg and α
Electron mass
Lepton flavor violation
(muon-electron conversion)
Antiprotonic helium
Antiproton mass
Here we focus on simple systems with positrons
Outline
Positronium, its ion, and the molecule
1951
1981
2007
Hyperfine splitting in Ps: new measurement
(Orthopositronium decay: new value)
Ps- ion: theoretical prediction for the decay
rate. Comparison with experiment.
Molecule: spectrum. Dipole transition vs.
annihilation.
Positronium
e+
Martin Deutsch
1917 – 2002
_
e
Very similar to hydrogen, except
• no hadronic nucleus
• annihilation
me
m

• reduced mass reduced
e
2
Two spin states:
singlet (para-Ps; short-lived, 0.1 ns)
triplet (ortho-Ps; long-lived, ~150 ns)
All properties can be described by QED, using one parameter:  
1
137.036
Positronium spectrum: discrepancy with QED
Tree-level QED prediction for the hyperfine splitting (HFS)
Recoil effect
e
e
Annihilation effect


e

e



     11   

 HFS
7
 me 4
12
204 GHz
Quantum corrections to the HFS: one-loop
O  5 
me 5  8 ln 2 

 
  9 2 
1005.5 MHz   0.5%
Quantum corrections to the HFS: two-loop
O  6 
Recoil
me 6 1367 5197 2  1 221 2 
53
5 2 1

  
  ln 2    3   ln 
2 
  648 3456
32
24

 2 144 
11.8 MHz  0.006% (Experimental error 0.7 MHz)
Recoil effect and log(α)
Hard photons, k~me, generate
a δ3(r) potential. Its coefficient
can be found from scattering of
free electrons:
it is IR divergent.
Conversely, soft photons, through the iterated Breit
interaction, give a UV divergence.
The trick is to find a common regularization scheme for both.
Pineda and Soto suggested (1998) dimensional regularization,
 m 
d 3
md 3 d 3


 ln 
d 3
d 3
HFS theory vs. measurements
 theory  203391.69 16 MHz
from S. Asai
3.4σ off!
exp
 HFS
 203387.5 1.6 MHz
exp
 HFS
 203389.10  74 MHz
Theory error
magnified 4X
But where does the mysterious value of α come from?
Kabbalah and α
Kabbalah and α
+ 100
Kabbalah and α
+ 2 + 100
Kabbalah and α
+ 30 + 2 + 100
Kabbalah and α
5 + 30 + 2 + 100
= 137
Back to positronium...
Positronium ion: predicted in 1946
e
-
e
e
John Wheeler
(1911-2008)
+
-
Positronium ion: a new test of bound-state QED
e
-
e
Observed 1981 Mills
2
4


1
3
5

e
+
New positronium-ion source
at the FRM II reactor in Munich:
measurements of branching ratios.
Unique for the ion
but very suppressed
Theory of the Ps ion: the wavefunction
r
  r1 , r2  
r1
  
  r1 , r2 , r 
2
r2
The wave function is not known analytically,
but can be found using the variational method.
The ion resembles a positronium “shell”
and a loosely-bound electron.
Now we can easily estimate the dominant decay channels.
Recent measurement of the ion decay rate
 exp  Ps   =2089 15   s 1
7 103
(factor 4-5
improvement
expected)
Recent measurement of the ion decay rate
 exp  Ps   =2089 15   s 1
7 103
(factor 4-5
improvement
expected)
We wanted to
improve this
Positronium ion decay: refinements
Corrections O(α)
single hard photon loops
Corrections O(α2); challenge: divergences  ln 
O(k2) corrections to the amplitude M
soft
Breit hamiltonian → correction to ψ(r=0)
Short-distance two-loop photon exchange
hard
Real photon radiation
Decay rate prediction
+ corrections
  Ps    2.087 963 12  ns 1
with M. Puchalski and S. Karshenboim
PRL 99, 203401 (2007)
Dipositronium Ps2
e+
e-
e-

0
2
4


1
3
5

e+
Discovery of dipositronium 2007
Molecule formation kills
long-lived positronia.
At higher temperature,
fewer atoms on the surface,
fewer molecules formed.
Indeed: at high-T, more long-lived
positronia observed.
Cassidy & Mills, Nature 2007
Spectrum of the molecule Ps2
Non-molecular states of e- e- e+ e+
From Suzuki & Usukura, 2000
Molecular states
From Suzuki & Usukura, 2000
A direct signal of the molecule: transition line.
Autodissociation forbidden
by Bose-Einstein statistics
From Suzuki & Usukura, 2000
A direct signal of the molecule: transition line.
Observable UV transition
Questions about this transition:
What is its accurate energy?
E  EP  ES  0.1815867(8) a.u.
4.9eV
Similar to atomic positronium,
but softer (dielectric effect?):
3 1
EP  ES   a.u.=0.1875a.u.
4 4
with Puchalski,
PRL 101, 183001 (2008)
Questions about this transition:
How often does
radiative transition appear
(before annihilation)?
BR  P  S  
dip  P  S 
annih  P   dip  P  S 
 0.191 2 
with Puchalski,
PRL 101, 183001 (2008)
Competition: dipole transition vs. annihilation
S-state annihilates quite rapidly.
Assume it consists of two weakly-interacting Ps atoms,
with random spins:
1 1
There is a para-positronium pair with probability 2  
4 2
The decay rate:
1 1
1
S


2  pPs 0.25ns
P-state: half of this rate:
 annih 
1
0.5 ns
Competition: dipole transition vs. annihilation
In an isolated P-excited Ps atom lives ~3.2 ns;
In Ps2 the decay is about twice faster
(and the same as in atomic hydrogen)
datom  S d P
d molecule
PS+SP
 SS d
 2datom
2
 dip  P  S  
Branching ratio
BR  P  S 
 
2  dip  atomic Ps  
2
 dip  P  S 
1
1.6 ns
 annih  P    dip  P  S 
~
1
1.6
1
1

1.6 0.5
~ 19%
Summary
Methods of bound-state QED developed largely with
positronium.
New states of positronium-like systems, ions and
molecules: opportunity to test three- and fourbody QED.
The Coulomb wave functions not known analytically
for e+ e- e- and e+ e- e+ e- .
Variational methods very accurate. O(alpha2)
corrections computed for the ion decay rate and
the Ps2 P-S interval. Both are being measured.
The “kinematic” method of finding alpha
Rydberg constant is extremely well measured,
me c 2 2 1
R 
2 hc
13.6eV
hc
We can find alpha if we measure the quotient
of the Planck constant and the “electron” mass,
2h R 
 
me c
2
In practice
heavier particles
are better:
neutrons or atoms.
2R  m p mCs h

c me m p mCs
Atom interferometry
Consider an elastic photon-atom collision:

Cs
'
h    '
2h 2 2
mCs c 2
h
 2
 2c
mCs 2
Preliminary Cesium determination of alpha
1 / 137. 035 999 71 (60)
(4.4 ppb)
Energy levels: ground state and P-excitation
M. Puchalski
Relativistic corrections: perturbations.
Annihilation dominates.
Interval P-S determined with 5 x 10-6 accuracy
(slightly smaller than in Ps, “dielectric effect").
Hylleraas & Ore, 1947
Wave function determined variationally,
using Coulomb potential;
Breakdown of corrections to the Ps ion width
Ratio of three- to two-photon annihilation
Back to positronium...
... and its decays
-
e
ortho-Ps
+
e
para-Ps



exp  o-Ps  =7.0401 7  s
-1
1 10 4
exp  p-Ps  =7.9909 17  ns
2 104
-1
Theory of positronium decay
Corrections O(α)
single hard photon loops
Corrections
soft
hard
O(α2)
Para: AC, Melnikov, Yelkhovsky
Ortho: Adkins, Fell, Sapirstein
O(k2) corrections to decay amplitude
Breit hamiltonian → correction to ψ(r=0)
Short-distance two-loop photon exchange
finite
together,
but give
ln
m
1
 ln
m

History of oPs lifetime measurements
Older:
Latest ones, enlarged scale:
Large discrepancy
5-9 sigma (crisis!)
Kataoka et al., 2008
Asai et al, 2001
Vallery et al, 2003