IMPLICATIONS OF THE NEW PROTON
CHARGE rms RADIUS FROM
SPECTROSCOPY OF MUON HYDROGEN
ATOM
Cyril Adamuscin, Stanislav Dubnicka, Anna Zuzana Dubnickova
INTRODUCTION
The primary building block of the visible Universe the positively charged proton
- is compound of (u,u,d)-quarks
- ⇒ non-point-like with some nonzero charge distribution
- and in EM interactions it manifests (similarly the
neutron to be compound of (d,d,u)-quarks) EM structure
1
- the size of charge distribution in a specific reference frame is described by the charge root-meansquare (rms) radius
r
2 >.
rp = < rEp
(1)
The charge distribution inside of the proton is defined:
as the Fourier transformation of the proton electric form factor (FF) GEp(t), t = q 2 = −Q2
1 Z
ρ(r) =
GEp(Q2)eiQrd3r
3
(2π)
(2)
The inverse Fourier transformation then gives
2
GEp(Q ) =
Z
ρ(r)e−iQrd3r
(3)
If ρ(r) is spherically symmetric distribution
⇒ the previous relation can be rewritten into the
2
spherical coordinates and by the integration over
θ and φ angles one gets
2
GEp(Q ) = 4π
Z ∞
0
sinQr
ρ(r)r2dr
Qr
(4)
For the case of Qr << 1
(Qr)3
sinQr ' Qr −
+ ...
6
(5)
and
2
GEp(Q ) = 4π
Z ∞
0
4πQ2 Z ∞ 4
r ρ(r)dr −
r ρ(r)dr + ...
6 0
(6)
2
Now, taking into account the charge distribution
density normalization
Z ∞
0
ρ(r)4πr2dr = 1
(7)
and the definition of charge-mean square radius
<
2
rEp
>=
Z ∞
0
r2ρ(r)4πr2dr
(8)
one finally gets
Q2
2
GEp(Q ) = 1 −
< rEp
> +...
6
2
3
(9)
from where the relation between charge meansquare radius and proton electric FF is obtained
2
< rEp
>= 6
dGEp(t)
|t→0.
dt
(10)
There are two completely different sources of
experimental information on charge rms radius rp of the
proton:
• elastic electron-proton scattering
• the precision spectroscopy of atomic hydrogen
The newest Review of Particle Physics
N.Nakamura et al (Particle Data Group), J. of Phys.G37(2010)
075021
gives the value
r
2 > = 0.8768(69)f m
rp = < rEp
4
(11)
based mainly on precision spectroscopy of electron hydrogen atom and calculation of bound state in
QED.
DETERMINATION OF PROTON CHARGE
rms RADIUS BY ELASTIC ELECTRON-PROTON
SCATTERING
First of all - two types of elastic processes are
used for obtaining of experimental information on GEp(t):
• unpolarized elastic scattering e−p → e−p
• the longitudinally polarized electron beam
→
→
in the polarization transfer process −
e p → e−
p
from which proton charge rms radius can be determined.
5
Unpolarized e−p → e−p scattering
The corresponding differential cross-section in
Lab. system in one-photon-exchange approximation,
with E0 - electron beam energy and θ - electron scattering
angle, is
dσ(e−p → e−p)
dσ
=
[A(t) + B(t)tan2θ/2] (12)
dΩ
dΩM ott
where A(t), B(t) - proton structure functions and
dσ
α2 cos2θ/2
1
=
dΩM ott 4E02 sin4θ/2 1 + (2E0/mp)sin2θ/2
(13)
is the relativistic Rutherford cross-section of massless electron scattering from point-like proton.
The structure of the proton (similarly of the neutron) is completely described by electric GEp(t) and
magnetic GM p(t) FFs (spin 1/2 particle) and
G2Ep(t) − t/4m2pG2M p(t)
A(t) =
;
1 − t/4m2p
6
B(t) = −2.t/4m2pG2M p(t).
(14)
Note, that the ratio
dσ(e−p → e−p) dσ
/
dΩ
dΩM ott
(15)
when evaluated for a fixed squared momentum
transfer −t and plotted against tan2θ/2 yields a
straight line.
The slope and intercept of this line yields information on GEp(t) and GM p(t) - Rosenbluth technique.
Figure 1: Experimental data on proton electric and magnetic form factors.
However - in the differential cross-section (12)
7
Figure 2: Experimental data on neutron electric and magnetic form factors.
with (14) G2M p(t) is multiplied by (−t/4m2p) - factor, ⇒
with −t increased the measured cross-section (12)
becomes dominant by G2M p(t)-part contribution - making the extraction of G2Ep(t) more and more difficult.
Therefore the first method of a determination of
GpE (t) is considered to be unreliable!
In despite of this fact, the data on unpolarized
elastic electron-proton scattering for very low
8
values of −t have been used
I.Sick, Phys. Lett B576 (2003) 62
P.G.Blunden, I.Sick, Phys. Rev. C72 (2005) 057601
for determination of the proton charge rms radius.
A sophisticated data analysis, based on a continuedfraction expansion of GEp(t), was carried out and
very large value
rp = 0.895(18)f m
(16)
of the proton charge rms radius has been found.
However, this result was criticized by
A.De Rujula, Phys. Lett. B693 (2010) 555
because it is based on an extrapolation of data
with a large spread and a poor χ2 per degree of
freedom.
9
→
→
Polarization transfer process −
e p → e−
p
Utilization of the longitudinally polarized elec→
tron beams in the polarization transfer process −
ep→
→
e−
p - makes possible to measure GEp(t) with very high
precision.
−
→
The polarization P of the recoil proton has only
two nonzero components
A.I.Akhiezer, M.P.Rekalo, Sov. J. Part. Nucl. 4
(1974) 277.
• perpendicular Pt to the proton momentum in scattering plane
r
h
Pt = (−2) τ (1 + τ )GEpGM ptanθ/2
I0
10
(17)
• parallel Pl to the proton momentum in the scattering plane
h(Ee + Ee0 ) r
Pl =
τ (1 + τ )G2M ptan2θ/2
I0mp
(18)
where h is the electron beam helicity,
I0 = G2Ep +τ [1+2(τ +1)tan2θ/2]G2M p and τ = Q2/4m2p.
⇒
GEp
Pt (Ee + Ee0 )
=−
tanθ/2
GM p
Pl 2mp
(19)
Recently in Jlab
M.K.Jones et al, Phys. Rev. Lett. 84 (2000) 1398
O.Gayon et al, Phys. Rev. Lett. 88 (2002) 092301
V.Panjabi et al, Phys. Rev. C71 (2005) 055202
have measured simultaneously Pt and Pl of the recoil pro→
→
ton in the polarization process −
e p → e−
p.
11
The new data on the ratio µpGEp(Q2)/GM p(Q2)
for 0.49GeV 2 ≤ Q2 ≤ 5.54GeV 2 have been obtained by (19) (see Fig. 3)
Figure 3: New JLab polarization data on the ratio µp GEp (t)/GM p (t)
They confirm the Rosenbluth method in a determination of GEp(Q2) space-like behavior from the e−p →
e−p process to be unreliable !.
12
In order to find the corresponding behavior of GEp(t)
in extended space-like region, we have analyzed all
existing nucleon FF data by our 10 resonance Unitary and Analytic model
S.Dubnicka, A.Z.Dubnickova, P.Weisenpacher, J.
Phys. G29 (2003) 405
of the nucleon EM structure
- excluding the Rosenbluth space-like GEp(t) data
- and replacing them by JLab proton polarization
data on the ratio µpGEp(t)/GM p(t).
The results are presented in Figs. 4 and 5 by
full lines.
13
Figure 4: Theoretical behavior of proton electric and magnetic form factors.
Figure 5: Theoretical behavior of neutron electric and magnetic form factors.
Description of that ratio is presented in Fig. 6a.
14
Figure 6: a)JLab polarization data, b)Charge distribution behavior
The corresponding charge distribution inside of
the proton
C.Adamuscin, S.Dubnicka, A.Z.Dubnickova, P.Weisenpacher,
Prog. Part. Nucl. Phys. 55 (2005) 228
is presented on Fig.6b by full line, generating the value
of the proton charge mean-square radius
¿
rp2
À
= 0.7208(117)fm2.
Just the square-root of the latter gives
15
the proton charge rms radius
rp = 0.8490(69)f m,
(20)
however, different from the value
r
2 > = 0.8768(69)f m
rp = < rEp
(21)
given by the newest Review of Particle Physics
(2010).
A couple of years we have been disappointed with
this disagreement, but only until to the muon hadrogen atom experiment in 2010.
16
DETERMINATION OF PROTON CHARGE
rms RADIUS BY PRECISION SPECTROSCOPY
IN MUON HYDROGEN ATOM
This method is based on precision measurement
of Lamb shift
W.E.Lamb and R.C.Retherford, Phys. Rev. 72
(1947) 241
i.e. the energy difference measurement between 2S1/2 and 2P1/2 states in hydrogen atom
and its comparison with a theoretical formula
for energy difference, dependent explicitly on the
charge rms radius rp of the proton, to be calculated in
the framework of QED.
17
The proton charge rms radius
r
2 > = 0.8768(69)f m,
rp = < rEp
given by the newest Review of Particle Physics
K.Nakamura et al (Particle Data Group), J. of Phys.
G37 (2010) 075021
has been extracted from the electron hydrogen
spectroscopy.
r
2 >
However, a much better determination of rp = < rEp
is expected to be obtained by measuring the Lamb shift in
muon hydrogen atom - an atom formed by a proton
p and a negative muonµ−, where the total theoretically predicted energy difference
18
F =1
F =2
between measured 2S1/2
and 2P3/2
states takes the
form
∆E = [209.9779(49) − 5.2262rp2 + 0.0347rp3]meV (22)
r
2 > given in f m.
with rp = < rEp
In the muon hydrogen atom:
• the muon is about 200 times heavier than the electron
• the atomic Bohr radius is correspondingly about 200
times smaller in µp in comparison with ep hydrogen
atom
• in this manner effects of the finite size of the
proton on the muon hydrogen atom spectroscopy
are thus enhanced
19
A measurement of the muon hydrogen atom Lamb
shift has been considered to be one of the fundamental experiments in atomic spectroscopy more
than 40 years, but only recent progress in muon
beams and laser technology made such an experiment possible.
The experiment was performed
R. Pohl et al, Nature Vol. 466 (2010) 213
at the πE5 beam-line of the proton accelerator at the
Paul Scherrer Institute (PSI) in Switzerland.
F =1
F =2
The measured transition 2S1/2
− 2P3/2
has been
found to be 49881.88(76)GHz, which corresponds to the
energy difference ∆E = 206.2949(32)meV .
20
By a comparison of this result with the previous theoretically estimated energy difference (22) one finds 10
times more precise proton charge rms radius
rp = 0.84184(67)f m
(23)
disproving the value of Review of Particle Physics (2010)
rp = 0.87680(690)f m,
obtained by an electron hydrogen atom spectroscopy, but
coinciding with our proton charge rms radius
rp = 0.84895(688)f m,
predicted 5 years earlier
C.Adamuscin, S.Dubnicka, A.Z.Dubnickova, P.Weisenpacher,
Prog. Part. Nucl. Phys. 55 (2005) 228
by the analysis of the nucleon electromagnetic
form factors data in the framework of the U nitary&Analytic
model of nucleon EM structure.
21
Nevertheless, there was some criticism
A.De Rujula, Phys. Lett. B693 (2010) 555
A.De Rujula, Phys. Lett. B697 (2011) 26
of the proton charge rms radius value
rp = 0.84184(67)f m
obtained by measuring Lamb shift in the muon hydrogen
atom, as it can be strongly dependent on the chosen
model form of GEp(t).
The problem is concerned of the third coefficient
at the theoretically predicted energy difference
∆E between measured 2S1F =1/2 and 2P3F =1/2 states,
which depends on the third Zemach moment
3
< r >(2)=
Z
d3rr3ρ(2)(r)
(24)
with
ρ(2)(r) =
Z
→
−
d3r2ρcharge(| −
r −→
r2 |)ρcharge(r2).
22
(25)
If Fourier transform of GEp(t) for ρ is inserted and integrated by parts, one finds the third Zemach moment
to be
48 Z ∞ dQ 2
Q2 2
2
< r >(2)=
[G (Q ) − 1 +
r]
π 0 Q4 Ep
3 p
3
(26)
By integration of various shapes of G2Ep(Q2) one gets
values
M.O.Distler, J.C.Bernauer, T.Walcher, Phys. Lett.
B696 (2011) 343
shown in
Table
GEp
dipole
2
< rEp
> < r3 >(2) f =< r3 >(2) /rp3
0.6581
2.023
3.789
Friar-Sick 0.801
2.71
3.78
De Rujula 0.771
36.2
53.5
non-dipole 0.7207
2.38
3.89
23
2
< rEp
> is determined from the slope of G2Ep(Q2)
at Q2 = 0.
De Rujula has suggested a Toy model to be excluded
by experiment.
We have completed the results by a realistic non-dipole
behavior of GEp(t), obtained in a global analysis of
all existing nucleon FF data by our U nitary&Analytic
model of nucleon EM structure, which gives results compatible with other authors, besides De Rujula.
So, we can conclude that muon hydrogen experiment represents a very sophisticated result beyond
any doubt and its coincidence with our result confirms
non-dipole behavior of GEp(t) in the space-like region, revealed first time by JLab proton polarization data,
and increases our belief in existence of the zero of GEp(t)
at t ≈ −13GeV 2, hoping to be confirmed experimentally.
24