RADIATIVE CORRECTIONS IN ELASTIC AND DEEP-INELASTIC ...

RADIATIVE CORRECTIONS IN ELASTIC AND DEEP-INELASTIC
ELECTRON-PROTON SCATTERING
A.V. Afanas’ev a) , I.V. Akushevich a,b) , N.P Merenkov
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
a)
Nord Carolina Central University, Durham and TJNAF, Newport News, USA
b)
On leave of absence from National Centre of Particle and High Energy Physics,
Minsk, Belarus
The electron structure function method is applied to calculate model-independent radiative corrections to an
asymmetry of electron-proton scattering. The representations for both spin-independent and spin-dependent parts of
the cross section are derived. Master formulae take into account the leading correction in all orders and the main
contribution of of the second order next-to-leading ones and have accuracy at the level of one per mille.
PACS: 12.20.-m, 13.40.-f, 13.60-Hb, 13.88.+e
1. MASTER FORMULA
Precise polarization measurements in both inclusive
and elastic scattering are crucial for understanding the
structure and fundamental properties of a nucleon.
One important component of the precise data analysis
is radiative effects which always accompany the processes
of electron scattering. The first calculation of radiative
corrections (RC) to polarized deep-inelastic scattering
(DIS) was done in [1,2] where the first order correction
has been found.
In present work we apply method of the electron
structure function for calculation of RC. Within this approach
DIS in one photon exchange approximation can
be considered as the Drell-Yan process [3]. The corresponding
cross section with accounting of RC can be
written as an contraction of two electron structure functions
and the hard part of the cross section [4,5]. Traditionally
these RC include main effects caused by loop
corrections as well soft and hard collinear photons and
e + e
− - pairs. But it was shown in [5] how one can improve
this method by inclusion also effects due to radiation
of one non-collinear photon. The corresponding
procedure concludes in modification of the hard part of
cross section that leads to exit beyond the leading approximation.
A straightforward calculation based on the quasi-real
electron method [6] can be used to write cross section of
DIS process
−
−
e ( k1 ) + P( p1
) → e ( k2) + X ( p x
)
(1)
in the following form
1 1
dσ
( k1, k2 ) dz
dσ
(
2
h
k% 1,
k%
2
)
= dz
2 1
D( z1, L) D( z2, L) ,
2
dQ dy
т т
z
dQ% dy%
2
z1m
z2m
2
L = ln Q , (2)
2
m
where m is the electron mass and
2 2 2 p1 ( k1 − k2
)
Q = − ( k1 − k2
) , y = , V = 2 p1k1
.
V
The reduced variables which define the hard cross section
in the integrand are
k
k%
= z k , k% 2 1 2
= , Q = Q
1 1 1 2
z2
z2
%
z
,
1 − y
y%
= 1 − .
z z
The electron structure function D( z, L) includes contributions
due to photon emission and pair production
+ − + −
γ e e e e
D = D + DN
+ DS
.
For the photon contribution into the structure function
one can use iterative form
1 2
е
Ґ
k
1 жα
L ц
Д k
1
k = 1 k ! 2π
(3)
γ
D = δ (1 − z) + з ч P ( z) ,
и ш
k
P ( z) =Д L Д P ( z) P ( z) Д ,
1 1 1
k
z dt
P ( z) P ( z) P ( t)
P ж ц
=Д т з ч
и t ш t
1 1 1 1
z
1
2
1 + z
3
P1
( z) (1 z ) (1 z) 2ln
1 z θ
δ ж ц
= − − ∆ + − з ∆ + ч
− и 2 ш , ∆ → 0 .
The nonsinglet part of the structure function due to
real and virtual pair production can be included in the iterative
form of D ( z. L)
by replacing α L / 2π on the
γ
right side of Eq. (3) with the effective electromagnetic
α L α
eff 3 α L
coupling → = − ln ж 1
ц
2π 2π 2
з −
3π
ч that is (within the
и ш
leading accuracy) the integral of the running electromagnetic
constant.
The lower limits of integration with respect to z 1 and
z 2 in the master equation (2) can be obtained from the
condition of existence of inelastic hadronic events which
reads in terms of dimentionless variables as
z1z2 + y − − xyz1 Ј z2
z th
1 ,
2
2 2
Q
M
th
− M
x = , zth
= ,
2 p1 ( k1 − k2
)
V
which leads to
1 − y + xyz1
1 + zth
− y
z2m
= , z1m
= . (4)
z1
− zth
1 − xy
The matrix element squared of the considered process
in one photon exchange approximation is proportional
to contraction of leptonic and hadronic tensors.
The representation (2) reflects the properties of leptonic
one. Therefore, it has universal nature and can be applied
to processes with different final hadronic states. In
particular, we can use the electron structure function ap-
40
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 40-44.

proach to compute RC to the elastic and deep-inelastic
electron-proton scattering cross sections.
On the other hand, the straightforward calculation in
the first order with respect to α [1,2,6] and the recent
calculation of the leptonic current tensor in the second
order [7-10] for longitudinally polarized initial electron
demonstrate that in the leading approximation spin-independent
and spin-dependent parts of this tensor are the
same for so-called nonsinglet channel contribution. The
latter corresponds to photon radiation and e + e - -pairs production
through single-photon mechanism. The difference
appears in the second order due to possibility of
pair production by double-photon mechanism [10].
Therefore, the representation (2), being slightly modified,
can be used for calculation of RC to cross sections
of different processes with longitudinally polarized electron
beam.
In our recent work [11] we applied this method to
compute RC to the ratio of the recoil proton polarizations
measured in CEBAF Jefferson Lab [12], where the
proton electric form factor G e was measured. In the
present work we use the electron structure function
method for calculation of model-independent part of RC
to asymmetry in scattering of longitudinally polarized
electrons on polarized protons at the level of per mile
accuracy for elastic and deep-inelastic hadronic events.
The cross section of the scattering of the longitudinally
polarized electron by the proton with given longitudinal
(l) or transverse (t) polarization for both elastic
and deep-inelastic events can be written as a sum of its
spin-independent and spin-dependent parts
l,
t
dσ ( k1 , k2 , S) dσ ( k1, k2 ) dσ
( k1, k2
, S)
= + η
2 2 2
, (5)
dQ dy dQ dy dQ dy
where S is 4-vector of the target proton polarization and
η is the product of the electron and proton polarization
degrees. Herein after we assume η = 1.
The master equation (2) describes the RC to spin-independent
part of the cross section on the right side of
Eq. (5) and the corresponding equation for the spin-dependent
part reads
l,
t
dσ
( k1, k2, S)
2 =
dQ dy
1 1
( p)
1 2 1 2
z1m
z2m
т dz т dz D ( z , L) D( z , L)
ґ (6)
l,
t
dσ hard
( k% 1, k%
2
, S) 2 ,
( p) e e e e ( p
dQ% D D γ
+ − + −
= + D ) ,
dy%
N
+ DS
and [10]
2 2
+ −
e e ( p)
α L й 2(1 − z)
щ
DS
= (1 z)ln z θ (1 z 2 m / ε )
2
4π
к
+ + − −
л 2
ъ
.
ы
The representation (6) is valid if radiation of
collinear particles does not lead to change polarization
4-vectors. In general case it is not so [13] but in this paper
we will use just such polarizations which satisfy this
conditions (see next Section)
The asymmetry in elastic scattering and DIS processes
is defined as the ratio
l,
t
l, t dσ
( k1, k2
, S) A =
,
dσ
( k , k )
1 2
therefore, RC to asymmetry requires the knowledge of
RC to both spin-independent and spin-dependent parts.
2. THE LEADING APPROXIMATION
Within the leading accuracy (taking into account
terms of the order (αL) n ) the electron structure functions
can be computed in all orders of the perturbation theory.
In this approximation we have to take the Born cross
section as hard part in integrands of master Eqs. (2), (6).
We express the Born cross section in terms of leptonic
and hadronic tensors as follows
2 2
dσ
4 π α ( Q ) B
2 4 L
µ ν H
µ ν
= ,
(7)
dQ dy VQ
where α(Q 2 ) is the runing electromagnetic constant that
takes into account for the effects of vacuum polarization
and
F M Sq
H = − F g% + p% p%
− iε
q [( g + g ) S − g ],
p q p q p q
2
µ ν 1 µ ν 1µ 1ν µ ν λ ρ λ
1 2 2
1 1 1
Q
L g k k k k i q p
2
2
B
µ ν
= − %
µ ν
+
1µ 2ν +
1ν 2 µ
+ ε
µ ν λ ρ λ 1 ρ
,
qµ qν
p1q
g% µ ν
= gµ ν
− , p% 2 1µ = p1
µ
− .
2
(8)
q
q
In Eqs. (8) we assume the proton and electron polarization
degrees equal to unity. The proton structure functions
F 1, F 2 and g 1, g 2 depends on variables x′=-q 2 /2p 1q,
q 2 =(p x-p) 2 . In Born approximation, x′ =x but they differ
in general case, when radiation of photons and electronpositron
pairs are allowed.
It is convenient to express the 4-vector of proton polarization
in terms of 4-momenta of particles in process
up + Vk − [2 uτ
+ V (1 − y)]
k
t
Sµ
=
2 2 2
− uV (1 − y)
− u M
1µ 2 µ 1µ
2 2
2M k
l 1µ − Vp1 µ
2 M
Sµ = , u = − Q , τ = .
MV
V
,
(9)
As normalization is chosen, the elastic limit can be
reached by a simple substitution in hadronic tensor
G + λ G
F x G F x
2 1 + λ
2 2
1 2
1 (1 ) ,
2 (1 )
e m
→ δ − ў
m
→ δ − ў ,
1 й ( G )
1 (1 )
m
Ge Gm
g x GmG λ −
щ
→ δ − ў
e
2 к
+
1 + λ ъ
л ы ,
2
1 λ ( G )
g1
→ − δ (1 − xў
)
m
− Ge Gm
, q
λ = −
2
2
1 + λ
4M
,
where G m and G e are magnetic and electric proton formfactors
which depend on q 2 .
A simple calculation gives the spin-independent and
spin-dependent parts of the cross section in the form
B
2 2
dσ
4 π α ( Q ) 2
=
2 4 [(1 − y − xyτ
) F2 + xy F1
] , (10)
dQ dy Q y
41

B
2 2
dσ l 8 π α ( Q ) йж
2 − y ц 2τ
щ
=
2 2 к з τ − ч g
1
+ g
2 ъ
dQ dy V y ли
2xy
ш y
, (11)
ы
dσ
Q M
dQ dy V y Q
(12)
B
2 2 2
t 8 π α ( ) ж 2
= − (1 − y − xyτ
) g
2 2 2
1
+ g2
ц
з
y
ч ,
и ш
Thus, within the leading accuracy, the radiatively corrected
cross section of the process (1) is defined by
Eq. (2) for its spin-independent piece with (10) as a hard
cross section into integrand and by Eq. (6) with (11) or
(12) as the hard part for its spin-dependent piece.
It is useful to extract the leading first order correction
to the Born approximation as defined by master
Eqs. (2), (6). For this purpose we can use the structure
function D γ with L → L − 1 and
2( ∆ ε )
∆ → ∆
1
= τ + z+ , z+
= y(1 − x),
V (1 − xy)
for D(z 1,L) and
2( ∆ ε )
2( ∆ ε )
∆ → ∆
2
= τ + z+
,

Pˆ
f ( r, xў) = f ( r , x ), Pˆ
f ( r, xў) = f ( r , x ),
t 1 t s 2 s
xyr
xyr
xt
= , x = .
xyr z xyr z
1 2
s
1
+
2
+
Note that quantity r 1(r 2) coinsides with z 1(1/z 2) for radiation
of a single collinear photon. The hard cross section
(14) has neither collinear nor infrared singularities. The
different terms on the right-hand side of Eq. (14) have
singularities at r=r 1, r=r 2 and γ=1. Singularities at first
two points are collinear and at third one is unphysical
that arise at integration. Collinear singulariries vanish
due to action of operators P
ˆt
and P ˆs on the terms containing
N. The unphysical singulariry cancels because in
the limiting case r → 1 we have
r2 − r r1
− r
= 1, = − 1, Tt
+ Ts
= 0.
| r − r | | r − r |
2 1
The corresponding result for spin-dependent hard
cross section can be written by very similar form
l,
t
B
z+
dσ dσ
hard
l, t ж α ц α l, t м 1 − r1
ˆ l,
t
= 1 U dz P
2 2 з + δ ч + 2
4
н
t
Nt
dQ dy dQ dy π Q
т
и ш о 1 − xy
1 −
−
1 −
r2
P ˆ l,
t
s N
s
z +
+
+
r
т
r−
dr
l,
t
2W
+
2
y + 4xyτ
dr й 1 − P ж 2( r − r)
ц
Ρ + + ч −
л и ш
+
s
2 l, t 2 l,
t
т
з (1 r ) N
s
Ts
1 − r
к
| r − r2 | (1 − z )
r
r− +
2
1 − Pˆ
t
ж (1 + r ) N
з
| r − r1
| и 1 − xy
where
2 l,
t
t
+
zth
2 2
l,
t
1
− r Tt
3
2 r( r )
x αў
ы
) щ}
1 2
1 2
( rQ )
, (15)
r
2
l t M
− 1 l
U = 1, U = (1 − y − xyτ
) , W = 4 yτ
W,
2
Q
t 2
W = 2 y (1 + 2 xτ
) W, W = (1 + r) xg1 + x g2
,
l
N = 2[2 r − z − xy( r + 2 τ )] g − 8 xўτ
g ,
t
l
N = 2[2 − z − xyr(1 + 2 τ ) g − 8 xўτ
g ,
s
t
N = 2[1 − y − z + r − xy( r + 2 τ )]( xyg + 2 xўg
),
t
1 2
t й
1 − z щ
Ns
= 2
к
1 − y − xy(1 + 2 τ ) + ( xyrg1 + 2 xўg
2
),
л
r ъ
ы
Tt l = 2rg1 − 4x ′ τ g2, Ts l = 2( z − 1)( g1 − 2x ′ τ g2
),
t
T = 2xyrg + 2 xў(1 − y + r − 2 xyτ
) g ,
t
1 2
t
й
ж 1
ц щ
Ts
= 2( z − 1) к xyg1 + xўз
1 − y + − 2 xyτ
ч g2
.
r
ъ
л
и
ш ы
The polarized hard cross section (15) as well is free
from any singularities. Note that radiation of photon at
large angles by the initial and final electrons increases
the region of variation for quantity r in (14) and (15),
because for collinear radiation r 1 < r < r 2 and now we
have r - < r 1 and r + > r 2. It may be important if the hadron
structure functions are large in these additional regions.
ў
The function δ ( 1 − x ′ ) is used to perform the integration
with respect to inelasticity z .
The final result for unpolarized case has the following
form (we do not introduce special notation for the
elastic cross-section)
B
dσ hard dσ ж α ц α м 1 − r1 ˆ
1 − r2
= 1 P ˆ
2 2 з + δ ч + 2
2 н t N − Ps
N
dQ dy dQ dy и π ш V о 1 − xy 1 − z+
+
r
r+
ˆ
2
2 W dr й 1 − Pt
(1 + r ) N
+ т dr
+ Ρ ( ( r1
2
т к
+
4 1 r | r r1
| 1 xy
r y + xyτ
− − −
−
r−
л
2 2 2
1 − Pˆ
s
ж (1 + r ) N ц щьп
α ( rQ )
− r) Tt
) − з + ( r2
− r) Ts
ч ъ э ,
| r − r2
| и 1 − z+
шыюп
r
where
2 2
2 2(1 − y − xyτ
)( Ge
+ λ Gm
)
N = Gm
+
2 2
,
x y r(1 + λ )
T
t
=
2 2
2[ r(1 − y) − 1]( Ge
+ λ Gm
)
2 2
,
x y r(1 + λ )
(16)
2 2 2 2
2[ r + y − 1]( Ge + λ Gm ) 2 2 τ ( Ge + λ Gm
)
Ts
= , W = G
.
2 2
m
−
x y r(1 + λ )
xyλ
(1 + λ )
The Born cross section on the right-hand side of
Eq. (16) is defined as
B
2 2
dσ
2 π α ( Q )
= ґ
2 2
dQ dy V
2 2 2
й
2 2[1 − y(1 + τ )]( Ge
+ λ Gm
щ ж Q ц
кGm
+ δ y .
2
ъ з − ч
л
y (1 + λ )
ы и V ш
When writing this last equation we take into account that
δ(1-x)=yδ(1-Q 2 /V).
The spin-dependent hard cross section for elastic
hadronic events can be written in the very similar to (16)
form
l,
t
B
dσ dσ
hard
l, t ж α ц α l, t 1 − r1
ˆ l,
t
= 1 U { P
2 2 з + δ ч +
t
Nt
+
dQ dy dQ dy и 2 π ш V 1 − xy
1 −
1 − Pˆ
t
+ + Ρ [ − ґ
1 − z 1 − r | r − r |
ж
з
и
+
r
l,
t
r+
r2
ˆ l,
t W dr
Ps
N
s т dr
2
т
+
r y + 4xyτ
−
r−
+ r N ц
− ˆ
s
+ − + P ґ
1 − xy ш | r − r | (1 − z +
)
(1
2 l,
t
)
t
l,
t
2 r( r1
r)
Tt
ч
1
2
2 2
ж 2 l, t 2( r2
− r) l,
t ц α ( rQ )
з (1 + r ) N
s
+ Ts
ч]}
,
2 2 2
и
r2
ш (4 M + Q r)
r
l
t 2
W = 4 yτ
W , W = 2 y (1 + 2 xτ
) W,
й 4τ
щ
W = r x + r − G + к r + + r G G
y
ъ
л
ы
2
[ (1 ) 1]
m
(1 )
m e
,
й
ж 2 ц щ
T r к r G з ч G G ъ
л
и xy ш ы
l
2
t
= ( + 2 τ )
m
+ 2τ
− 1
m e
,
1
4. HARD CROSS SECTION
FOR ELASTIC HADRONIC EVENTS
To describe the hard cross section for elastic hadronic
events we use the replacement given after formulae
(9) in expressions (14) and (15). For Born cross sections,
which enter in these equations, see Eqs. (10-12).
43

l
2 ж 2 ц
t
Ts = − r(1 + 2 τ ) Gm − 2 τ з − r ч GmGe , Tt
= r{ − [ r(1 − xy)
+
и xy ш
1
− y − xy G + − y − xy + r + G G T = r − r
2
t
1 2 τ )]
m
(1 2 τ 4 τ )
m e}, s
[
xy y G xyr r y G G
2
(1 + 2 τ ) + 1 − ]
m
− [2 τ (2 − ) + 1 + (1 − )]
m e,
й 1 − xy щ
N r r xy G к r G G
xy
ъ
л
ы
l
2
t
= (2 τ + )(2 − )
m
+ 8 τ − τ
m e,
l
N = r(2τ
+ 1)(2 − xyr)
G
s
2
m
2
[2 r(1 + 2 τ ) GmGe − r(2 − xyr) Gm
].
й 1
щ
+ 8 τ к r(1 + τ ) GmGe
,
xy
ъ
л
ы
t
N = [1 − y + r − xy( r + 2 τ )][2( r + 2 τ ) G G −
t m e
2 t 1
r(2 − xy) Gm
, Nt
= [1 − y + − xy(1 + 2 τ )] ґ
r
Note that argument of electromagnetic formfactors in
2
Eqs. (15) and (16) is − Q r .
The Born cross sections on the right-hand side of
Eq. (16) have the following form
B
2 2
dσ
l 4 π α ( Q ) й ж 1 ц
= 4 1 (1 2 )
2 2 2 к τ + τ − Gm
Ge
− + τ
dQ dy V(4 M + Q )
з
y
ч
л и ш
2
ж y ц 2 щ ж Q ц
−ґ з 1 G 2
ч m ъ δ −з y
V
ч
и ш ы и ш
for longitudinally polarized of the target proton and
B
2 2 2
dσ
t 8 π α ( Q ) M
= [1 − y(1 + τ )] ґ
2 2 2 2
dQ dy V (4 M + Q ) Q
2
йж
y ц 2
щ ж Q ц
к з 1 − ч Gm − (1 + 2 τ ) GmGe
δ з y − ч ,
2
ъ
л и ш
ы и V ш
for transverse one. The argument of form factors in the
last two formulae is –Q 2 .
In this paper we consider model-independent QED
radiative correction to the polarized DIS and elastic
electron-proton scattering. Our calculations are based on
the electron structure function method which allows to
write both the spin-independent and spin-dependent
parts of the cross section with accounting RC to the leptonic
part of interaction in the form of well-known
Drell-Yan representation. The corresponding RC includes
explicitly the first order correction as well as the
leading-log contribution in all orders of perturbation
theory and the main part of the second order next to
leading-log one. Moreover, any model-dependent RC to
the hadronic part of interaction can be included in our
analytical result by insertion it as an additive part of the
hard cross section in integrand in master equations.
To derive RC, we take into account radiation of photons
and e + e - -pairs in collinear kinematics which produces
a large logarithm L in the radiation probability (in
D-functions) and radiation of one non-collinear photon
that enlarges the limits of variation of the hadron structure
function arguments. It may be important that these
functions are sharp enough. In this case the loss in radiation
probability (the loss of L) can be compensated by
the increase in the value of the hard cross section.
On the basis of our analytical result we constructed
Fortran code ESFRAD (http://www.jlab.org/~aku/RC) and
perform some numerical estimations [15] for kinematical
conditions of current and future experiments. We
found two regions where the higher order radiative corrections
can be important. These are the traditional region
of high y and the region around the pion threshold.
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