EINN, Paphos, Cyprus, November 4, 2015
Lepton universality test
in the photoproduction of e-e+ versus "-"+ pairs on a proton target
Vladyslav Pauk
Thomas Jefferson
National Accelerator Facility
Newport News, VA, USA
in collaboration with M. Vanderhaeghen
incoming and outgoing electron laboratory energies. Obtaining the form factors
ntial cross section at a variety of Q2 and scattering angles is termed a Rosenbluth
The proton radius
(3) is an approximation valid if the target’s Compton wavelength is small comproton the latter statement is marginal. Instead we define the form factors from
n above, and for the electric and magnetic proton
radii, weradius
promote the former derived
2 def
RE
=
dGE
6
dQ2
-01
extraction
,
(11)
Q2 =0
2⇡↵
2 2
between RM and the magnetic form factor.
agree for the electron
E
=
|
(0)|
RE
L
2
3
xperiments for obtaining the form factors at low and moderate Q are done at
(uncertainty
~0.6%)
[5]. They quote their results by fitting form
factors using
a variety of di↵erent
atomic spectroscopy
scattering
r measured elastic
di↵erential
cross sections. Their fits lead to
RE = 0.879(8) fm ,
(12)
mµ ⇠ 200 me
[6], several uncertainties are combined into a singlemuon
uncertainty
limit.
vs electron
tain the proton radius using electrons is to measure the energy levels, the Lamb
Lamb shift, in ordinary electronic hydrogen. The proton radius measurements
2
~
ydrogen are quite remarkable because the proton radius
Bohr radius
a0e↵ects
= are 2very small.
can be measured so precisely, and the proton size independentmenergies
can be
ee
hat proton size dependent terms can be isolated.
roton energy spectrum is illustrated in Fig. 1. The Figure
is not
to scale. (That
10x better
stronger
effect
itting between the 2S1/2 and 2P1/2 levels—is about 10% of the 2P fine structure
precision
from the proton size
tive splitting that is about right.)
The puzzle
"H data:
RE = 0.8409 ± 0.0004 fm
Antognini et al.(2013)
Pohl et al.(2010)
-02
ep/eH data:
7 σ difference !?
RE = 0.8775 ± 0.0051 fm
CODATA(2012)
4
(µb/GeV )
Lepton pair production on a proton target -03
E = 0.5 GeV
Measure the ratio of the e vs µ cross sections10 2
d /dt dMll
-
- +
direct access to the ratio of the form factors
the same systematics
ee
2
-
10
depending on the recoil proton Lab angle
and integrate over the lepton phase space
2
-t (GeV )
0.01
0.02
- +
µ µ process
Bethe-Heitler
0.03
no lepton acceptance corrections needed 1
Pauk, Vanderhaeghen
arXiv:1503.01362[hep-ph]
- large cancellation of radiative corrections
q
0.03
0.04
0.05
0.06
0.07
0.08
2
2
⌘ t/(4M
).
Furthermore,
for
a
fixed
value
of
t,
the
with
↵
⌘
e
/4⇡ ⇡ 1/137, where
⌘
1
- increase the count rates
oiling proton lab angle ⇥lab
p is expressed in terms of in2
lepton velocity in the l l+ c.m.-t (GeV
frame,
) with m
ants as : recoil proton Lab angle
60
mass, and where the proton FFs GEp and GM p a
0.03
2
2
50
of
t.
The
weighting
coefficients
multiplying
the FF
M
+
2(s
+
M
)⌧
lab
ll
p
cos ⇥p =
.
(1)
have the following general structure :
2
40
2(s M ) ⌧ (1 + ⌧ )
0.02
p
lab
(deg)
-
30
he differential
crossLab
section
for the dominating BH proproton
momentum
0.01
20
s to the p ! l l+ p reaction has been studied in different
✓
◆
0 lab
p
|p
|
=
(100
174)M
eV
/c
1+
0 lab
texts in the
[19–21].⌧ (1
In +
this⌧ )work, we will con(1)
(2) 1
10
|~
pliterature
| = 2M
CE,M = CE,M + CE,M ln
,
r the cross section differential in the momentum transfer
1
0
2
2
nd invariant mass
of
the
lepton
pair
M
⌧ = t/(4M )
0.04
0.05
0.06
0.07
0.08
ll , and integrated0.03
r the lepton angles, which corresponds with detecting the
2
2
Mll (GeV )
oiling proton only. This cross section can be written as :
cess to the p ! l l p reaction has
2 been studied in different
The differential
cross section for contexts
the dominating
BH proin the literature
q [19–21]. In this work, we will conp
0 lab
4m
2
ansfer t: |~
p |
= 2M ⌧ (1 + ⌧ ), with
+
with
↵
⌘
e
/4⇡
⇡
1/137,
where
⌘
1
is the
M
cess
to
the
p
!
l
l
p
reaction
has
been
studied
in
different
sider
the
cross
section
differential
in the momentum transfer
Furthermore, for a fixed value of t, the
+
lepton velocity in the l l c.m. frame, with m the lepton
2
lab
contexts
in
the
literature
[19–21].
In
this
work,
we
will
cont
and
invariant
mass
of
the
lepton
pair
M
, and(1)integrated
(2)
ab angle ⇥
is
expressed
in
terms
of
inp
mass, and where the proton FFs GEp and GM p are functions
CE,Mll = CE,M + CE,M
sider the cross section differential
inover
the coefficients
momentum
transfer
the
lepton
angles,
corresponds
with detecting the
of t. The weighting
multiplying
thewhich
FFs in Eq.
(3)
2 structure :
2
2
have
the
following
general
E = 0.5 GeV
M
+
2(s
+
M
)⌧
t
and
invariant
mass
of
the
lepton
pair
M
and integrated
recoiling
only. This
ab
ll
ll ,proton
10 2 cross section can be written as :
p
=
.
(2)
2
2(s over
M 2 ) the
⌧ (1lepton
+ ⌧)
angles,
which
corresponds
with
detecting
the production to leading
- +
Two-fold differential cross section of the BH
Bethe-Heitler
order
e
e
3
q
dbe written as : ↵
4
1
recoiling
proton
only.
This
cross
section
can
cross section
for the
dominating
BH
pro2
4m
2
=✓
·2 2
·
↵ ⌘
/4⇡
⇡ in
1/137,
⌘
1 Mdt
2
2
◆
2 is the
2
2
4
10
l+with
p reaction
has ebeen
studied
differentwhere
(GeV )t (M
dM
M -t)where
t) 1term
+ ⌧ expresse
the
second
1(s
+
ll
(1)ll
(2)ll 1
BH we will+con- 3
CE,M 1 ln
,
(4)
erature
[18–20].
In this
E,M =
0.01
d work,
4CE,M
lepton
velocity
in
the l l c.m.↵ frame, Cwith
m
the +
lepton
- + limit of small le
2 hancement
2 µthe
1
in
µ ,
tion differential in the momentum
transfer
⇥
C
G
+
C
⌧
G
(2)
=
·
·
E
M
0.02
Ep
M
p
2
2
2
2
2
4
mass,
and
where
the
proton
FFs
G
and
G
are
functions
(1)
(2)
2
p
dt M
dM
(s MEp) t M
(M
t) 1 + ⌧
ass of the lepton pair
and integrated
ll
ll , ll
The
coefficients
C
,
and
C
0.03
1
E,M
E,
t. The
weighting
the
FFs
in
Eq.
(2)
gles,of
which
corresponds
withcoefficients
detecting the multiplying
2
2
⇥ asstructure
C
(2) 0.03 through
invariants
as :0.07
nly. have
This cross
can begeneral
written
: E GEp
the section
following
: + C M ⌧ GM p ,
0.04
0.05
0.06
0.08
-04
d /dt dMll
2
4
(µb/GeV )
Cross section
2
2
ll
(deg)
4
↵
4
1
·
·
(s Weighting
M 2 )2 t2 (Mll2 coefficients:
t)4 1 + ⌧
where the second term expresses the large logarithmic enhancement in the limit of small lepton mass in2the BH process.
10 60
(1)
(2)
(1)
The coefficients CE,M , and CE,M are found to
be expressed
C
= t s
E
50
through
✓ invariants
◆ as :
(µb/GeV )
3
E = 0.5-t (GeV
GeV2)
form factor parametrization:
2
ow the
⇥ differential cross section d /dt dM⇤ll
p
p
lab
(deg)
d /dt dMll
2
lab
2
2
2
+
M
s
M
M
0.03
ll +
ee
CE G2Ep + CM ⌧ G2M p ,
(3)
2⇥ 2
1+
(1)
(2) 1
2
2
2
⇥
⇤2 +
40
0.02
(1) ln
+
M
t
t
M
+
M
(M
CE,M = CE,M + CE,M
,
(3)
2
2
2
4
2
2
2
2
2
ll 6M t + t +
ll 4m M ll
-tt(GeVM
) ll +
CE = t1 s M
s M
M10ll +
ll
ll
30
(2)
2
2
2
0.01
⇤ M0.01
2⇥ 2
- + 2s
=
t
s
M
M
2
2
2 CE
2
2
2
2
µ µM
ll +
20
⇥
⇤
+
M
t
t
M
+
M
(M
+
t)
+
4m
M
,
0.02
(1)
ll
ll
⇥
CE = t s M 2 s M 2 Mll2 + t Mll4 + 6Mll2llt + t2 + 4m2 Mll2 ll
2
2
4
2
⇥ M42 t2
0.03
1 10
⇥
⇤
(2)
+
M
(M
+
t
2
2
2
2
2
2
Cll2E+ t)2=
s M
Mll0 + t M
Mllll+ 2t ) +2
+ Mll2 t t2 Mll2 + M 2 (M
+ 4m2tM 2sMll2 M
,
(5)llll+ t + 4m
⇥ large logarithmic
(1)0.03 0.040.04
(1) 0.050.05 2
0.03
0.06
where
the second term expresses the
en-2m2 ⇤
(2)
0.06 0.07
0.07 20.08
0.08 ⇤
2 2 ⇥
2
CE = t s M 2 s M 2 Mll2 + t Mll4 + t2 +
4m
M
+
2t
C
=
C
2
2
4
2M
2 E 2 2M (1
2 +2⌧ ) Mll
2
2t
ll
2
2
+
M
t
M
(M
+
t
)
+
2m
t
2M
M
+
4m
M
hancement2 in the
of 4small
lepton
mass
inll the
BH
process.
ll Mll (GeV
⇤ ll
)
2 ⇥limit
2
2
2
2
2
2
2
2
2
+ Mll t
M
(M
+
t
)
+
2m
t
2M
M
+
4m
M
,
(6)
(2)
(2)
-t
(GeV
)
(1)
(2)
2
⇤⌧ ) M 2 t
ll
ll
2 ⇥= 4
(1)
(1) to be
60C
C
+
2M
(1
+
The
coefficients
C
,
and
C
are
found
expressed
2
2
2
2
2
⇥
⇤
ll
E,M
E t + 4m M
2
(1)
(1)
=
C
(1 + ⌧ ) Mll M
t
Mll (7)
+
,
2 E,M
2 C
4
2 E
2 2M
2
ll
M
C
=
C
2M
(1
+
⌧
)
M
t
M
+
t
+
4m
M
,
ll
ll
ll
0.03invariant mass
M
E
FIG.
panel: comparison of the (lepton pair)
through
invariants
as :
50 2: Upper
+
⇥
+
⇥
⇤
p 4process
three
curves)
2
(2)
(2)
dependence
p ! e 2e p(upper
process
curves) vs the
4
2 (2) 2
2 2
2
2γp → 2eofe the
2 (upper
2three
2
CM = CE + 2M 2 (1 + ⌧ ) Mll2 Ct (2)Mll=
+ tC
+ 4m+ M
t
2m
.
(8)
+
2M
(1
+
⌧
)
M
t
M
+
t
+
4m
M
t
2m
ll
llp40! µ µ p process
ll (lowervs
three curves) at ll
E 0.02
= 0.5 GeV, and
M
E
for three values of the momentum transfer t as indicated. The lower
30
theshows
γp the
→corresponding
"-"+p process
(lower
curves)
panel
kinematic
relation between
the lepton
2
In
Fig.
2,
we
show
the
differential
cross
section
d
/dt
dM
Bernauer et al. (2013)
0.01
pair
ll
20invariant mass and the recoiling proton lab angle.
and below ⇡⇡
cross sections
at a
forthreshold,
p ! and
(l comparing
l+ )p which
is accessed
by measuring the recoil-
Experimental feasibility
-05
γp → "-"+ p : 200 000 events produced
o
()
p
100
Compton
90
Eγ = 0.5 GeV
Bethe-Heitler
80
70
60
Com
pton
102
p$ 0
50
10
40
p$0
30
20
p$+$-
10
0
0
p$+$-
20
40
60
Adlarson (2015)
80
100
Energy p (MeV)
1
-06
Radiative
pion
photo-production
cross section at low values of the radiated photon energy E , to the well known p ! ⇡
0
section in a model independent way. The low-energy limit of E 0 ! 0 corresponds with
0 → π0p
MAID
Mll2 ! m2⇡ . The low-energy theorem then
allowsparameterization
us to express the of
p the
! ⇡γp
p different
Low-energy theorem
section as [1]:
2
d
1
e
!
· 2·
2
2
2
dtdMll
(Mll m⇡ ) ⇡ (s
(GeV2 )
⇣
0.02
0.03
M 2)
1/2 (s, M 2 , m2 )
⇡
⌘c.m.
kinematic
function defined
=0 p)
x2 + y 2 + z 2
( p!⇡ 0asp): (x, y,
( z)
p!⇡
d
(µb / sr)
d⌦⇡
( p!µ µ+ p)
Eq. (1), W (v) is defined as :
W (v) =
0.01
s
d
d⌦⇡
◆c.m.
· W (v) · 2⇡
,
3
where m⇡ (M ) denote the pion (nucleon) mass respectively, e is the electric charge, and
pγ$0
t
✓
3.23
with v ⌘
p
1
1.0 ⇥ 10
2
( p!⇡ 0 p)
1+
✓
 1.0
2
v +1
2v
◆
✓
2xy
2xz
Rµ/e  10
2yz. Further
3
◆
v+1
· ln
,
v 1
accuracy
at ~30%
sufficient
4M 2 /t, and the cross section (d /d⌦⇡ )c.m.isdenotes
the c.m. different
section of the p ! ⇡ 0 p process. Using the MAID phenomenological parameterizatio
experimentally has been
2
p ! ⇡ 0 p cross section [2], we are able to estimate
the
M
the p !
ll
achieved dependence
at 10-20% of
level
3.35
3.3 ⇥ 10
2
 0.3
cross section in the low-energy limit, according to Eq. (1), which is shown in Fig. 1 for t
S. Schumann et al (2010)
3.49
9.0 ⇥ 10 2
 0.1
kinematics as considered in Fig.2 of our paper for the p ! (l l+ )p process.
4
l
4
l
2
2
2
2
+ t + 4m
+ t + 4m
Mll2
⇤
Mll2
,
t
2m+
e
(6)
⇤
.
(7)
+
e vs. μ μ cross section ratio
2
Rµ/e
d (µ µ+ + e e+ )
⌘
d (e e+ )
1,
σ(e- e+ +μ- μ+ )/σ(e- e+ )
xed value of t above and below µ µ+ thresholds, it opens
The
cross sectionextraction
ratio of the cross sece possibility for
a high-precision
on ratio :
(8)
-07
Eγ =0.5 GeV
1.14
1.135
1.13
-t=0.03 GeV2
lepton universality
violation
μ
GEp /GeEp =1.01
here d stands for d /dt dMll2 . The potential advantage of 1.125
ch a ratio
measurement
is that absolute
normalization
unσe-e+
fixed by measuring
below
"-"+
1.12
rtainties to first approximation
drop
out.
Indeed,
at
a
fixed
threshold
lepton universality
+
lue of t, the e e cross section can be fixed by measur- 1.115
μ
GEp =GeEp
+
g the cross
section
below
µ
µ
and the corredifference in measured threshold,
proton charge
onding normalization, mainly due to GEp , can be used to 1.11
FF in electron vs muon observables
termine the e e+ cross section above µ µ+ threshold. A
leads
to a 0.2%ofabsolute
effect
for
bsequent
measurement
the sum of
e e+
+Rµ"/eµ+ cross 1.105
0.066 0.068 0.07 0.072 0.074 0.076 0.078
ctions above µ µ+ threshold, then allows to extract the raMll2 (GeV2 )
o Rµ/e , which is displayed in Fig. 3. One sees that in the
nematic range where only the e e+ and µ µ+ channels
e contributing, this ratio varies between 10 to the
13 interference
%. We
between the Compton Born and BH
ke to notice that corrections, notably radiative contributions
corrections,
∼15 smaller than the effect due to the
time-like
Compton
scattering
first order
also drop
out of this
ratio, measured at the same
1% variation in the value of G"Ep
lue of the recoiling proton momentum and angle. An ac-
. In the whole M range of interest, the asymmetry fo
therefore give a clear tool to separate the two channels. I
is undetected, and only the recoiling proton i
+
Besides the unpolarized cross section for the p ! the
lmeasured,
llepton
p pair
the measured asymmetry above µ µ+ threshold i
process, we may also consider the sensitivity of polarization
diluted by the µ µ+ /e e+ ratio, as given by Eq. (10), an
+
observables to distinguish between the e e+ and µ µreaches
pro-values around -5 % as can be seen from Fig. 4.
Photon asymmetry
-08
duction processes. We will consider here the case of the linear Linear photon asymmetry AIU
The linear photon asymmetry
of the γp → (l-l+)p process
photon asymmetry defined as:
AlU
d
=
d
d
k+d
k
?
?
(9)0
,
AlU
n
e
g
n
n
.
y
d
g
e
s
y
n
t
e
to
ll
(black) curve, is an estimate of the physical background due tok+the
µ µ production takes on large values as can be seen from
interference with the timelike Compton process.
Fig. 4. A direct measurement of the µ µ+ asymmetry ma
-0.1
- +
ee
- +
- +
ee +µµ
where d k ( d ? ) stands for the differential cross section for-0.2a
−"+ threshold:
Above
"
photon
with
linear polarization
sum of the asymmetries
of the e parallel
e+ and (perpendicular)
µ µ+ channels:to the
around 2 % will allow to distinguish b
-0.3
plane spanned by the photon and recoiling proton momenta. ton R extractions µfrom
- +
muonic and
µ
E
-0.4
When measuring
the+ recoiling1 proton only, the asymmetry Such an experiment can be performe
+
AlU (e e + + µ µ ) =
-0.5
above
µ µ threshold is 1given
by the following weighted
+ Rµ/e
cilities such as the Mainz Mikrotron
+
+
-0.6
sum of the asymmetries of the+ e e and µ µ channels:
Lab, thus adding a further piece of e
⇥ AlU (e e ) + Rµ/e AlU (µ µ+ ) . (11)
-0.7
derstanding of the “proton
radius puz
1
E = 0.5 GeV
+
+
+ 4µ the
µ linear
) = photon asymmetry in the kine--0.8
lU (eineFig.
We A
show
2
1 + Rµ/e
-t = 0.02 GeV
+
matic range around µ µ threshold.
It is seen that the
linear
++
+
The asymmetry
the(e
" e" for
takes
0.04
0.05
0.06
0.07
0.08
AlU
) production
+the
Rµ/e
(µ
µ on
) How. (10)0.03
Acknowledgeme
photon
asymmetry⇥isfor
very
small
e A
e+lUchannel.
2
2
+
M
(GeV
)
+
e e production:
ll
ever, large
for thevalues
µ µ compared
channel thetoasymmetry
reaches a value
approaching 1 at µ µ+ threshold and decreases in absolute
We asymmetry
like to A
thank
Achim
Denig,
FIG. 4: Linear photon
of
the
p
!
(l
l+ ) p pro
lU
+
- simultaneous
measurements
of R and
A allow
to arises
extract
the
"(blue)
" cross
section
value
by going away
from the threshold.
Such
behavior
Hornidge,
Merkel,
cess. The dashed
curveHarald
corresponds
with e e+Vladimi
production
the the
dashed-dotted
(red) curve
corresponds
with
µ µ+discussio
production
due to an exact cancellation, at µ µ+ threshold, between
cettina
Sfienti
for
useful
The solid
curve is the asymmetry corresponding with the sum
- a complementary tool to ensure the proper account
of (black)
the
corrections
+
analytical and lepton mass logarithmic terms in the analogous
byaccording
the Deutsche
of the e e +supported
µ µ+ channels
to Eq. (10). Forschu
2
Summary
-09
proton puzzle - no solution so far
a complementary experimental test is highly desired
comparing the photoproduction of a lepton pair on a proton, through
detection of the recoiling proton
measurement of the ratio Re/µ with an absolute precision of around 0.2 %
distinguish between e vs µ proton RE extractions
feasible for existing electron facilities: Mainz Mikrotron (MAMI) and
Jefferson Lab