Table Of ContentProton electron elastic scattering and the proton charge radius
G. I. Gakh
National Science Centre ”Kharkov Institute of Physics and Technology”
61108 Akademicheskaya 1, Kharkov, Ukraine
A. Dbeyssi, E. Tomasi-Gustafsson,∗ and D. Marchand
CNRS/IN2P3, Institut de Physique Nucl´eaire, UMR 8608, 91405 Orsay, France
V. V. Bytev
Joint Institute for Nuclear Research, Dubna, Russia
Itissuggested thatprotonelastic scatteringonatomicelectrons allows aprecisemeasurement of
2 theprotonchargeradius. Verysmallvaluesoftransferred momenta(uptofourorderofmagnitude
1 smaller than theones presently available) can bereached with high probability.
0
2 PACSnumbers:
n
a
The problem of the proton size has been recently ob- servables for proton electron elastic scattering, in a rela-
J
ject of large interest, due to the recent experiment on tivistic approachassuming the Bornapproximation,was
2
muonic hydrogen by laser spectroscopy measurement of derived. The relations connecting kinematical variables
1
the νp(2S-2P) transition frequency [1]. The result on in direct and inverse kinematics were given. In particu-
] the proton charge radius r = 0.84184(67) obtained in lar,itwasshownthatlargepolarizationeffectsappearat
h c
this experiment is one order of magnitude more precise beam energies around 15 GeV. Moreoverthe transferred
t
- but smaller by five standard deviation compared to the momenta are very small even when the proton energy is
l
c best value previously assumed r = 0.8768(69) fm [2] in the GeV range. In this work we focus on the second
c
u
(CODATA). Previous best measurements include tech- issue and apply to the problem of a precise and consis-
n
[ niques based on H spectroscopy, which are more precise, tent determination of the protonradius. The kinematics
but compatible with electronprotonelasticscattering at of proton-electron scattering is extremely peculiar and
1 small values of the four momentum transfer squaredQ2. interesting in this respect.
v
The most recentresult fromelectron protonelastic scat- In the elastic interaction between a proton and an
2
7 tering, rc =0.879(5)stat(4)syst(2)model(4)group fm, canbe electron, assuming that the interaction occurs through
5 found in Ref. [3]. the exchange of a virtual photon of four momentum
2 While corrections to the laser spectroscopy experi- k = (ω,~k), the observables can be expressed as func-
.
1 mentsseemwellundercontrolinframeofQEDandmay tionsoftwoformfactors,electricGE,andmagneticGM,
0 be estimated with a precision better than 0.1%, in case which are functions of Q2 =−k2 only.
2 ofep elastic scattering the best precisionwhichhas been The electric form factor, G (Q2) in the non relativis-
E
1
achieved is of the order of few percent. Different sources tic limit is related to the charge distribution through a
:
v of possible systematic errors to the muonic experiment Fourier transform. For small values of Q2 one can de-
Xi have been discussed, however no definite explanation of velop GE(Q2) in a Taylor series expansion:
this difference has been given yet (see Ref. [4] and Ref-
r 1
a erences therein). GE(Q2)=1− 6Q2 <rc2 >+O(Q2), (1)
Recent works have been devoted to the scattering of a
where one takes into account the fact that the density
proton projectile on an electron target (see Ref. [5] and
(being the square of the wave function) is an even func-
references therein). The possibility to build beam po-
tionofthe spatialdistancer, whereasthe scalarproduct
larimeters for high-energy polarized (anti)proton beams
~k~r is an odd function. The root mean squared radius is
has been shown [6]. Experiments have been done [7, 8],
the derivative of the form factor at Q2 =0
and are ongoing with the aim to understand the exper-
imental fact that a proton beam circulating through a dG (Q2)
<r2 >=−6 E . (2)
polarized hydrogen target gets polarized [9]. The possi- c dQ2
bility to polarize antiprotons beams would open a wide (cid:12)Q2=0
(cid:12)
domainofpolarizationstudiesatthe GSIfacilityforAn- The value itself of G (Q2 = 0) is gi(cid:12)ven by the normal-
E (cid:12)
tiproton and Ion Research (FAIR) [10, 11]. Assuming ization to the proton charge.
C-invariancein electromagneticinteractions,the (elastic Form factors are derived from unpolarized ep scatter-
and inelastic) reactions p +e− and p¯+e+ are strictly ing through the Rosenbluth separation: measurements
equivalent. at fixed Q2 for different angles allow to extract the elec-
In Ref. [5], the cross section and the polarization ob- tricandmagneticformfactors. Thepolarizationmethod
2
[12] has been recentlyapplied [13] providingveryprecise
measurements of the ratio G /G up to large values of ×10−6
E M
Q2 ≃9GeV2. Thelargerprecisioncomestothefactthat 2]V 0.8
e
in this case one measures a polarization ratio, in which G
radiative corrections (at first order) cancel and the sys- [x 0.7
a
tematics effect related to the beam polarization and to 2Qm 0.6
polarimetry are essentially reduced.
Radiative corrections and Coulomb corrections have 0.5
to be applied to ep scattering experiments in particular
0.4
for unpolarized measurements. Besides the problems re-
latedto the fact thatthere is no modelindependent way 0.3
to calculate those radiative corrections which depend on
0.2
the hadron structure, and that correlations exist in ex-
tracting form factors from the Rosenbluth fit [14], one 0.1
has to face the extrapolation of the data to Q2 = 0 as
discussed in Ref. [3]. The smallest value of Q2 reached 0
0 0.05 0.1 0.15 0.2 0.25 0.3
in that experiment was 0.004 GeV2. E [GeV]
p
The possibility to access much smaller values of Q2 is
offeredby the elastic reactioninduced by a protonbeam
on an electron target. Let us consider the reaction
FIG.1: Maximumfourmomentumtransfersquaredasafunc-
p(p )+e(k )→p(p )+e(k ), (3) tion of theproton beam kinetic energy.
1 1 2 2
whereparticlemomentaareindicatedinparentheses,and
k =k −k =p −p . The expression of the differential where M is the proton mass and E is the proton beam
1 2 2 1
cross section for unpolarized proton-electron scattering, energy. Beingproportionaltotheelectronmasssquared,
in the coordinate system where the electron is at rest, thefourmomentumtransfersquaredisrestrictedtovery
can be written as: small values. In Fig. 1 we report Q2 as a function
max
dσ πα2 D of the proton kinetic energy, in the MeV range. One
= , (4) can see that the values of transferred momenta are very
dQ2 2m2p~2Q4
small: for a proton beam with kinetic energy E = 100
p
MeV, (Q2) =0.2×10−6 GeV2.
max
D = −Q2(−Q2+2m2)G2M +2[G2E +τG2M] From energy and momentum conservation, one finds
1 Q2 2 the following relation between the angle and the energy
−Q2M2+ 2mE− . (5) of the scattered electron:
1+τ 2
" (cid:18) (cid:19) #
(E+m)(ǫ −m)
2
whereτ =Q2/4M2 andG arethe Sachselectric and cosθe = , (7)
E,M |p~| (ǫ2−m2)
magnetic form factors, m is the electron mass, p~ is the 2
momentum of the proton beam. where ǫ2 is the energy of thpe scattered electron. Eq. (7)
Similarlytoepscattering,the differentialcrosssection shows that cosθe ≥ 0 (the electron can never be scat-
diverges as (Q2)2 when Q2 → 0. This is a well known teredbackward). Inthe inversekinematics,theavailable
result,whichisaconsequenceoftheonephotonexchange kinematical region is reduced to small values of ǫ2:
mechanism and allows to reachvery large cross sections. 2E(E+m)+m2−M2
The expression (5) differs from the Rosenbluth formula ǫ2,max =m M2+2mE+m2 , (8)
[15], as additional terms depending on the electronmass
which is proportional to the electron mass. From mo-
can not be neglected. The electric contribution to the
crosssectiondominates,beinginalltheallowedQ2range mentum conservation, on can find the following relation
∼107 times larger than the magnetic one. between the kinetic energy E2 and the angle θp of the
scattered proton (Fig. 2):
LetusconsiderthecasewhenE =100MeV.Thepro-
p
ton energy is under the pion threshold for pp reactions, E±+M = [(E+m)(M2+mE)± (9)
2
which helps in reducing the hadronic background.
m2
The properties of the inverse kinematics has been dis- M(E2−M2)cosθ −sin2θ ]
p M2 p
cussed in Ref. [5]. It has been shown that for a given r
energy of the proton beam, the maximum value of the [(E+m)2−(E2−M2)cos2θp]−1,
four-momentum transfer squared is:
which shows that for one proton angle there may be
4m2(E2−M2) two values of the proton energy, (and two correspond-
(Q2) = , (6)
max M2+2mE+m2 ing values for the recoil- electron energy and angle-, and
3
×10−3
V] V]0.75
e e
M G
[E2 0.2 [e
∆ E 0.7
0.15
0.65
0.1
0.6
0.05
0.55
×10-3
0
0 0.2 0.4 0.5
sinθ
p 0 0.2 0.4 0.6 0.8 1
cosθ
e
FIG. 2: Difference of thekinetic energy of thescattered pro-
tonfromthebeamkineticenergy,Ep=100MeV,asafunction FIG. 3: Kinetic energy of the recoil electron as a function of
of the sine of the proton scattering angle.
the cosine of the electron scattering angle for beam energy
Ep=100 MeV.
for the transferred momentum Q2). The two solutions
coincide when the angle between the initial and final
hadron takes its maximum value, which is determined ] 3
b
bytheratioofthe electronandscatteredhadronmasses, m
sinθh,max =m/M =0.544·10−3. Hadrons are scattered θ[e
fromatomicelectronsatverysmallangles,andthelarger s
o
is the hadron mass, the smaller is the available angular c
d 2
range for the scattered hadron. The difference between σ/
d
thescatteredprotonkinetic energyandthebeamkinetic 6
-0
energy is shown as function of the proton scattering an- 1
gle in Fig. 2. The proton kinematics is very close to
the beam, which makes the detection very challenging. 1
However a magnetic system with momentum resolution
oftheorderof10−4canprovideatleastthemeasurement
of the energy of the scattered proton. This would allow
a coincidence measurement which may help in reducing
0
the possible background. 0.5 1
cosθ
Whiletheprotonisemittedinanarrowcone,theelec- e
tron is scattered up to 90◦. The energy dependence as
function of the cosine of the angle for the recoil electron
is shown in Fig. 3. FIG.4: Differentialcrosssectionasafunctionofthecosineof
In Ref. [5] it was shown that polarization observables theelectron scattering angle for beam energy Ep=100 MeV.
areverysmallatsmallenergy,makingverydifficulttheir
measurement. Therefore,theapplicationofthepolariza-
tion method [12] to inverse kinematics seems very chal- The differential cross section as a function of cosθ is
e
lengingatlowenergy. Nevertheless,theratioofG /G shown in Fig. 4 in the angular range 10◦ ≤ θ ≤ 80◦.
E M e
can be derived from the ratio of two correlation coeffi- It is large when the electron angle is close to 90◦ and
cients, for example C /C . Having a proton beam and monotonically decreasing. The cross section, integrated
tl tt
anelectrontargetbothpolarizedinthedirectionnormal in this angular range, is 25×104 mb. Assuming a lumi-
tothescatteringplane,givesaccesstotheproductofG nosityL=1032 cm−2 s−1 withanidealdetectorwithan
E
and G , once the unpolarized cross section is known: efficiency of 100%, a number of ≃ 25×109 events can
M
be collected in one second. Therefore, the reaction (3)
DC =−4mMQ2G G . (10) allows to reachvery small momenta with huge cross sec-
nn E M
4
tion. The very specific kinematics, however, makes the Cedex
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One of us (A.D.) acknowledges the Lebanese CNRS (2010).
forfinancialsupport. Thisworkwaspartlysupportedby [14] E. Tomasi-Gustafsson, Phys. Part. Nucl. Lett. 4, 281
CNRS-IN2P3 (France) and by the National Academy of (2007).
Sciences of Ukraine under PICS n. 5419 and by GDR [15] M. N.Rosenbluth,Phys. Rev.79, 615 (1950).
n.3034’PhysiqueduNucl´eon’(France). L.Tassan-Gotis
thanked for useful discussions on experimental possibili-
ties.
∗
E-mail: [email protected]; Permanent address:
CEA,IRFU,SPhN, Saclay, F-91191 Gif-sur-Yvette
