FFK-2017
Dimitar Bakalov,
INRNE-Sofia, assoc. of INFN-Trieste
Presenting results obtained with:
• The FAMU collaboration
• P. Danev, V. Korobov, and S. Schiller
Topics to be discussed
1. Modelling the FAMU experiment
2. Spectroscopy of H2+
3. Nuclear size effects in --atom spectrum
The FAMU experiment
(work in progress)
The FAMU experiment
- Measuring the Hyperfine Splitting (HFS)
of -p with accuracy 10-5
- Extract the Zemach radius of the proton
accurate to 1% or better
Currently: 3 independent experiments plan
to measure RZ
Motivation
charge radius rch
e--p
scattering &
spectroscopy
rch = 0.8775(51)
m-p
spectroscopy
rch=0.84089(39)
Zemach radius RZ
RZ=1.037(16) Dupays&al’03
RZ=1.086(12) Friar&Sick’04
RZ=1.047(16) Volotka&al’05
RZ=1.045(4) Distler&al’11
Either confirm a e-p value
or admit: e-p and m-p differ
???
A. Vacchi INFN - Trieste (I)
6
Modeling the FAMU experiment
A few complements to Dr. Mocchiuti’s talk:
• Efficiency of the experimental method
• Tools for modeling the target geometry
• Multi-pass optical cavity
The experimental idea
• The time distribution
of the events of muon
transfer as signature
of resonant excitation
of F=1 spin state.
• Resonant wavelength
recognized by the
difference between
counts in a time gate
Theoretical estimates of pO
Experimental data on pO(T)
pO(T)
pO(T)=M(E;T) pO(E) dE
Calculated vs. measured rates
Enhancement of magnetic field
• t<p
• t>p
d =d/c
Enhancement of efficiency
Optimized efficiency
S/N~0.01
n
Possible next target: p-p HFS
• Different, but very
clear signature
• High S/N
• Existing laser sources
• Difficult system for
theory
• No theoretical results
beyond leading order
E2 electric quadrupole
spectrum of H2+
(work in progress)
Precision spectroscopy of H2+
Possible transitions:
• E1 (electric dipole)
• Forbidden E1
• 2-photon electric
• E2 electric quadrupole
• ...
• ...
Precision spectroscopy of H2+
Our current interest:
• ...
• E2 electric quadrupole
• ...
A paper in preparation
(with P.Danev, V.Korobov, S.Schiller)
Precision spectroscopy of H2+
QED effects of
high order taken
into account.
Theoretical calculation
of E1, E2, forbidden E1
and 2-photon spectra
of H2+, including the
Zeeman, AC-DC Stark
and light shift, etc.
Transitions with
lowest response
to external fields
selected.
Determining the electron-toproton(deuteron) mass ratio;
other fundamental constants
Designing molecular clocks
....
Precision H2+ spectroscopy
Theoretical calculation
of E1, E2, forbidden E1
and 2-photon spectra
of H2+, including the
Zeeman, AC-DC Stark
and light shift, etc.
QED etc.
effects
The
effects
of high order are
be taken into
oftoaccount.
laser
light
polarization
Transitions with
needed
for
lowest response
to external
fields
line
pattern
to be selected.
recognition
Determining the electron-toproton(deuteron) mass ratio;
other fundamental constants
Designing molecular clocks
....
Hyperfine structure of E2-lines
J=5/2
L=2
L=0,J=1/2
J=3/2
Hyperfine structure of E2-lines
J=5/2
L=2
L=0,J=1/2
J=3/2
HF + Zeeman splitting
Jz=3/2
Jz=1/2
Jz=1/2
J=3/2
Jz=-1/2
J=1/2
Jz=-1/2
Jz=-3/2
E2 transition rate
dPif /dt=
t (/2h) (Ev’L’-EvL)
(e.c.)
 v’L’,vL2 v’L’||Q||vL2
(nrel)
 FI’S’J’,ISJ
(hfs)
 J’J’z|JJz,2q2 |Tq|2, q=J’z-Jz
(pol)
where T=(Ak)(2)
Parameterization of T
k
z
B
R(a,b,g)
z’
Ay
Ax
x
q
y
x
A
A=|A|(cos q, sin q eij, 0)
Varying , =/4
Varying , =0, =/4
Nuclear size corrections to
muonic hydrogen spectrum
(just a topic to discuss)
An old paper from 1988…
“Эффекты электромагнитной
структуры ядер в мезомолекулах”.
Ядерная физика т.48 (2), с.335 ) (1988)
Estimates of the corrections to
E=E(mol)-E(at)
for the hydrogen isotope muonic atoms
and molecular ions in the leading order.
Breit Hamiltonian of -p
Point-like proton approximation
Breit Hamiltonian of -p
Point-like proton approximation (contd.)
Breit Hamiltonian of -p
For the static EM structure of the proton, the
Dirac & Pauli formfactors F1,2(q2) were used.
Gn(q2), n=1,2,…: linear combinations of F1,2
Define:
Breit Hamiltonian of -p
The r-dependence of U(1) is modified:
Breit Hamiltonian of -p
Closed expressions for U(1) 
Pohl et al.,Nature(2010)
No logarithmic terms in r2; no r terms
Contribution of
(RM)
U
p
to LS in

Contribution of
(RM)
U
d
to LS in

Where is the contribution of U(D)?
Two alternatives:
• Use U(FSZ) with r2G and U(D) with point p
• Use U(FSZ) and U(D) with F1 and F2
Both ways lead to the same energy levels
No easy / straightforward demonstration
…If using F1,2 instead of GE? [1]
•
•
Take rG=0.862 fm  rF=0.79 fm
U(D) generates corrections involving
rF
U(D)F= U(D)point(1-2 rF+O
r2)
In general, r and r2 are
independent
•
E(D)=20|U(D)|20point(-2 rF)~0.08
…If using F1,2 instead of GE? [2]
•
Moreover, if accounting for FSZ effects in
20(r) at r~0 (as in Zemach case):
E(D)0.115 meV
• To compare with E(th)LS=209.9779(49)
• In -d: rG=2.125 fm  rF=2.122
fm
E(D)0.252 (0.368) meV,
to compare with E(th)LS=228.7766(10)