QED corrections to polarized deep-inelastic and semi-inclusive deep-inelastic scattering
Leading-log model-independent QED corrections in DIS of unpolarized electron off tensor-polarized deuteron are considered. Same approach was used for investigation of semi-inclusive DIS of electron by nucleus with detection of hadron and scattered electron. Calculations are based on covariant parame...
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irk-123456789-794182015-04-02T03:02:04Z QED corrections to polarized deep-inelastic and semi-inclusive deep-inelastic scattering Gakh, G.I. Merenkov, N.P. Shekhovtsova, O.N. Quantum electrodynamics Leading-log model-independent QED corrections in DIS of unpolarized electron off tensor-polarized deuteron are considered. Same approach was used for investigation of semi-inclusive DIS of electron by nucleus with detection of hadron and scattered electron. Calculations are based on covariant parametrization of polarization and use of Drell-Yan like representation to describe radiation by initial and scattered electron. Applications to polarization transfer from polarized electron to detected hadron and to scattering by polarized target are considered. DIS of unpolarized electron on tensor-polarized deuteron with tagged collinear photon radiated from initial-state electron are investigated. 2001 Article QED corrections to polarized deep-inelastic and semi-inclusive deep-inelastic scattering / G.I. Gakh, N.P. Merenkov, and O.N. Shekhovtsova // Вопросы атомной науки и техники. — 2001. — № 6. — С. 35-39. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 12.20.-m, 13.40.-f, 13.60.-Hb, 13.88.+e http://dspace.nbuv.gov.ua/handle/123456789/79418 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Quantum electrodynamics Quantum electrodynamics |
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Quantum electrodynamics Quantum electrodynamics Gakh, G.I. Merenkov, N.P. Shekhovtsova, O.N. QED corrections to polarized deep-inelastic and semi-inclusive deep-inelastic scattering Вопросы атомной науки и техники |
| description |
Leading-log model-independent QED corrections in DIS of unpolarized electron off tensor-polarized deuteron are considered. Same approach was used for investigation of semi-inclusive DIS of electron by nucleus with detection of hadron and scattered electron. Calculations are based on covariant parametrization of polarization and use of Drell-Yan like representation to describe radiation by initial and scattered electron. Applications to polarization transfer from polarized electron to detected hadron and to scattering by polarized target are considered. DIS of unpolarized electron on tensor-polarized deuteron with tagged collinear photon radiated from initial-state electron are investigated. |
| format |
Article |
| author |
Gakh, G.I. Merenkov, N.P. Shekhovtsova, O.N. |
| author_facet |
Gakh, G.I. Merenkov, N.P. Shekhovtsova, O.N. |
| author_sort |
Gakh, G.I. |
| title |
QED corrections to polarized deep-inelastic and semi-inclusive deep-inelastic scattering |
| title_short |
QED corrections to polarized deep-inelastic and semi-inclusive deep-inelastic scattering |
| title_full |
QED corrections to polarized deep-inelastic and semi-inclusive deep-inelastic scattering |
| title_fullStr |
QED corrections to polarized deep-inelastic and semi-inclusive deep-inelastic scattering |
| title_full_unstemmed |
QED corrections to polarized deep-inelastic and semi-inclusive deep-inelastic scattering |
| title_sort |
qed corrections to polarized deep-inelastic and semi-inclusive deep-inelastic scattering |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2001 |
| topic_facet |
Quantum electrodynamics |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/79418 |
| citation_txt |
QED corrections to polarized deep-inelastic and semi-inclusive deep-inelastic scattering / G.I. Gakh, N.P. Merenkov, and O.N. Shekhovtsova // Вопросы атомной науки и техники. — 2001. — № 6. — С. 35-39. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT gakhgi qedcorrectionstopolarizeddeepinelasticandsemiinclusivedeepinelasticscattering AT merenkovnp qedcorrectionstopolarizeddeepinelasticandsemiinclusivedeepinelasticscattering AT shekhovtsovaon qedcorrectionstopolarizeddeepinelasticandsemiinclusivedeepinelasticscattering |
| first_indexed |
2025-07-06T03:28:21Z |
| last_indexed |
2025-07-06T03:28:21Z |
| _version_ |
1836866602772987904 |
| fulltext |
QED CORRECTIONS TO POLARIZED DEEP-INELASTIC AND
SEMI-INCLUSIVE DEEP-INELASTIC SCATTERING
G.I. Gakh, N.P. Merenkov, O.N. Shekhovtsova
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
Leading-log model-independent QED corrections in DIS of unpolarized electron off tensor-polarized deuteron
are considered. Same approach was used for investigation of semi-inclusive DIS of electron by nucleus with detec-
tion of hadron and scattered electron. Calculations are based on covariant parametrization of polarization and use of
Drell-Yan like representation to describe radiation by initial and scattered electron. Applications to polarization
transfer from polarized electron to detected hadron and to scattering by polarized target are considered. DIS of unpo-
larized electron on tensor-polarized deuteron with tagged collinear photon radiated from initial-state electron are in-
vestigated.
PACS: 12.20.-m, 13.40.-f, 13.60.-Hb, 13.88.+e
1. INTRODUCTION
The purpose of this paper is developing a unified ap-
proach to computation of the radiative corrections (RC)
for inelastic scattering of polarized electron beam in the
inclusive and semi-inclusive (SI) setups. We investigat-
ed the deep-inelastic scattering (DIS) of unpolarized
electron beam off the tensor-polarized deuteron target (a
process with tagged collinear photon, radiated from the
initial-state electron, has also been investigated). The
ELFE project provides a good opportunity for the mea-
surement of some hadron tensor structure functions [1],
which could give clues to physics of non-nucleonic com-
ponents in spin-one nuclei and study the tensor structure
on the quark-gluon level. The use of the tensor-polarized
deuteron target at HERMES allows investigating the nu-
clear binding effects and nuclear gluon components [2].
As stated above, we considered also inelastic scattering
of polarized electrons in the coincidence setup, namely,
when one produced hadron is detected in coincidence
with the scattered electron. A broad range of measure-
ments falls into the category of coincidence electron
scattering experiments. It includes deep-inelastic SI lep-
toproduction of hadrons, (e, e' h), as well as quasielastic
nucleon or deuteron knock-out processes, (e, e' N) or (e,
e' d). The former class of experiments gives access to
the flavour structure of quark-parton distributions and
fragmentation functions. It is in focus of experimental
programs at CERN, DESY, SLAC and JLab. Some ex-
periments have already been completed and some are
being in preparation. Quasielastic nucleon knock-out
processes allow to study single-nucleon properties in nu-
clear medium and probe the nuclear wave function.
We calculated the QED RC to the above mentioned
processes by means of the electron structure function
method [3] which allows to treat the observed cross sec-
tion including both the lowest order and higher order ef-
fects by the same way. As a result we can obtain clear
and physically transparent formulae for RC. In this re-
port we restrict our consideration to leading accuracy. It
allows us to avoid an attraction of any model for the
hadron structure functions and, as a result, to obtain
some general formulae for quite wide class of the physi-
cal processes.
2. THE TENSOR-POLARIZED TARGET
In present section we give the covariant description
of the cross section of DIS of unpolarized electron beam
off the tensor-polarized deuteron target
).()()()( 211 xT pXkepdke +→+ −− (1)
We use approach which is based on the covariant
parametrization of the deuteron quadrupole polarization
tensor in terms of the 4-momenta of the particles in pro-
cess (1) [4] and use of the Drell-Yan like representation
[5] in electrodynamics, which allows to sum the leading-
log model-independent RC in all orders.
To begin with, we define the DIS cross section of the
process (1), with accounting RC, in terms of the leptonic
µ νL and hadronic Hµν tensors contraction
,)(,
11
211
4
2
2 pk
kkpyHL
VqdydQ
d −== µ νµ ν
π ασ
(2)
where q is the 4-momentum of the intermediate heavy
photon that probes the deuteron structure. Note that only
in the Born approximation (without accounting RC)
q=k1-k2.
The model-independent RC exhibits themselves by
means of the corrections to the leptonic tensor. In the
framework of the leading accuracy this tensor can be
written as a convolution of two electron structure func-
tions D and the Born form of the leptonic tensor BLµ ν
that depends on the scaled electron momenta
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 35-39. 35
∫ ∫= ),(),(),( 212
21
21
21 LxDLxD
xx
dxdx
kkLµ ν
,,,ln,),(
2
2
21112
2
21 x
k
kkxk
m
QLkkLB ===
µ ν
,)(, 2
21
2
21 kkQqkk −−==−
(3)
where m is the electron mass.
The limits of integration on the right side of Eq. (3)
can be derived from the condition that the DIS process
(1) takes place. It is possible if the final undetected
hadron system consists, at least, of a deuteron and a pi-
on. In this case we have
,)(2/,1 211
2
2121 kkpQxxxyxyxx −=≥−−+ δ
,2,/))(( 11
22 pkVVMmM =−+= πδ (4)
where M (mπ) is the deuteron (pion) mass. This inequali-
ty defines the integration limits as follows
,)1/()1(1 1 xyyx −+−≥≥ δ (5)
.)/()1(1 112 δ−+−≥≥ xxyxyx
The Born leptonic tensor is (for the case of the longi-
tudinally-polarized electron beam)
+++−= )(22),( 21212121 µννµµ νµ ν kkkkgkkkkLB
,212 σρµ ν ρ σλ ε kki+ (6)
where quantity λ is the degree of longitudinal polariza-
tion of the electron beam.
The hadron tensor has polarization-independent and
polarization-dependent parts. We consider only the case
of the tensor polarization
)/()~~())(
/(~~)/(~
1
2
~1~13
2
1
2
11121
)(
qpMQpQpqBqp
MppqpaBgaBH T
++
++=
αµνανµα
νµµ νµ ν
−=−= µµνµµ νµ νµ ν 11
2
4
~,/~,~ ppqqqggQB
where Qµν is the deuteron quadrupole polarization ten-
sor. In general all the hadron structure functions Bj
(j=1,2,3,4) depend on two independent variables: q2 and
x′=q2/(2p1q) (within the chosen accuracy
)1/(' 211 xxyxyxxx +−== ). We used the notation of
Ref. [4].
Because the polarization-independent part of the
hadronic tensor depends on the scaled electron momenta
only (by means of 21 kkq
−= ), we can write the respec-
tive contribution to the cross section in the form of the
Drell-Yan representation in the electrodynamics that
takes into account the leading part of the radiative cor-
rections
∫ ∫= ),(
),(
12
2
21
2
21
)(
LxD
x
dxdx
dydQ
kkd uσ
,
),(
),( 2
21
)(
2 ydQd
kkd
LxD
u
B
σ
(7)
where 2
2
1
2 / xQxQ =
, 2121 /)1( xxxxyy +−= and u
means unpolarized Born cross section.
As concerns the polarization-dependent contribution
to the cross section dσ(T) the situation is somewhat differ-
ent. In general we cannot use for it the representation (7)
with simple substitution
dydQddydQd Tu 2)(2)( // σσ → (8)
in both sides of Eq. (7). The reason is that the axes, re-
spect to which the components of the deuteron
quadrupole polarization tensor are defined, can change
their directions at the scale transformation of the elec-
tron momenta: 2,12,1 kk
→ . But substitution (8) can be
useful and applicable if all axes remain stabilized under
this transformation.
Therefore, first we have to find the set of stabilized
axes and write them in covariant form in terms of 4-mo-
menta of the particles participating in the reaction. If we
choose the longitudinal direction l along the electron
beam and the transverse one t in the plane (k1, k2) and
perpendicular to l, then
,/)2( 11
)( MpkS l
µµµ τ −=
,/])([ 112
)( dxypkxybkS t
µµµµ τ −−−=
,,)(2 211
1)( VxybdkkpVdS n == −
σρνµ ν ρ σµ ε
./,1 2 VMxyyb =−−= ττ (9)
One can verify that the set ),,( ntlS µ remains stabilized
under the scale transformation and
.,,,,0, 1
)()()( ntlpSSS ==−= βαδ µ
α
µα β
β
µ
α
µ
If to add one more 4-vector MpS /1
)0(
µµ = to the
set (9), we receive the complete set of orthogonal 4-vec-
tors with the following properties
.,,,0,,, )()()()( ntlnmgSSgSS mn
nmmm === µµµ ννµ
This allows expressing the deuteron quadrupole polar-
ization tensor in general case as follows
,
)()()()(
α β
β
ν
α
µνµµ ν RSSRSSQ mn
nm ≡=
0, == α αβ αα β RRR (10)
because the components 0000 ,, αα RRR identically
equal to zero due to the condition .01 =νµ ν pQ
So, if the components of the deuteron polarization
tensor are defined in the coordinate system with the axes
along the directions l, t and n, the polarization-depen-
36
dent contribution to the cross section of the process (1)
with accounting leading RC can be written in the same
way as polarization-independent one
∫ ∫= ),(),(),(
212
2
21
2
21
)(
LxDLxD
x
dxdx
dydQ
kkd Tsσ
.),(
2
21
)(
ydQd
kkd Ts
B
σ
(11)
Symbol Ts indicates that components of the quadrupole
polarization are defined with respect to stabilized set
(9). The simple calculation gives
+−+= )([2),(
4
2
2
21
)(
nnttttllll
Ts
B RRSRS
yQdydQ
kkd π ασ
+++−=+ 1(2])21(2[,] 2 bGxyxbSRS llltlt ττ
yyxbSBxybBx lt 2[2,)()3 1
43
−=−++ τττ
,])42()21( 43 yBBbyGx +−−++ τ (12)
.)/(,)(2 213 BybxyBGBGxbStt −=+−= τ
3. SEMI-INCLUSIVE DIS WITH POLAR-
IZED FINAL PARTICLE
Here we clarify the question how to calculate QED
RC to the cross section and polarization observables in
the following process (within the considered approach)
.)()()()( 2211 XppkepAke ++→+ −−
(13)
We use the following definition of the cross section
of the process (13) with definite spin orientation of the
proton in terms of the leptonic and hadronic tensors
,
2 2
2
3
2
2
3
4 E
pdkd
q
HL
Nd k ε
σ µ νµ ν= (14)
where 132 ])2()12[( −+= πα VSN Ak , AS is the target
spin, )( 22 Eε is the energy of the scattered electron (de-
tected proton) and q is the 4-momentum of the virtual
photon that probes the hadron block. Hadronic tensor is
defined by the standard way.
The hadronic tensor in general case can be written as
,)()( pu HHH µ νµ νµ ν += (15)
61
)(
215214
2231121
)(
)[(,]~~[)~~(
~~~~~
hSpHppihpph
pphpphghH
p
u
=++
+++=
µ νµ νµ ν
νµνµµ νµ ν
928171 )~(]~[)~( ihNphNpihNp +++ µ νµ νµ ν
+++ µ νµ νµ ν ]~[)~()[(]]~[ 1111102 NpihNphSqNp
++++ µ νµ νµ ν ghSNNpihNph ~)[(]]~[)~( 14213212
++++ µ ννµνµ )~~(~~~~
211722161115 pphpphpph
,,]]~~[ 212118 σρνµ ν ρ σµµ ν ε qppNppih =+
,][,)( µννµµ νµννµµ ν babaabbabaab −=+=
where Sµ is the 4-vector of the proton spin that satisfies
conditions: 0)(,1 2
2 =−= SpS , and ih (i=1-18) are the
hadron SI structure functions which depend in general
on four invariants. These invariants can be taken as
.)(,)(,)(, 2121
2 ppqpqpq
To completely describe this process we will use the
following set of invariant variables
,,2/,/)(2 211
2
211 kkqqpqxVkkpy −=−=−=
./2,/2,/2 22221121 VpkzVpkzVppz ===
The set of stabilized 4-vectors can be chosen as
,/,/)2( 2
11122
)( VMmdpzpS l =−= ττ µµµ
+−+== µµµ ττ 2111
2
1
)(2
2 )2([,/ pzzkdSVm t
νµ ν ρ σµµ ετ 1
)(
21112 2,/])2( kSVddpzz n =−+
,4,/ 21
22
1
3
221 ττσρ −= zdVdpp (16)
,)(,1
2
121
2
2 ji
ij SSzzzd δττ −=−−=
where M (m) is the mass of the target nucleus (detected
proton).
Now we can write down the spin-independent (we
bear in mind that it means independent on the proton
spin only) and spin-dependent parts of the cross section
of the process (13) as
∫ ∫= ),(),( 212
2
21
2
3
2
3
,,),(
22 LxDLxD
x
dxdx
pdkd
d
E ntluσ
ε
,
2
3
2
3
,,),(
22 pdkd
d
E
B
ntlu
σ
ε (17)
where Bdσ , with any low index, denotes the corre-
sponding Born cross section given at shifted values of
2,12,1 kk
→ . The corresponding shifted dimensionless
variables, introduced earlier, read
,)1(,
)1( 21
21
21
1
xx
xxyy
xxy
xyxx +−=
+−
=
./,,/, 21221111 xxzzzzxzzVxV ====
The spin-independent part of the cross section for
longitudinally-polarized electron beam is expressed in
terms of the hadron structure functions as
,
2 14
2
2
3
2
3
)(
22 H
q
VN
pdkd
d
E k
B
u =
σ
ε (18)
−+−−+−= −
21211
1
1 ()1(2 zzhxyyhxyVH τ
,))1(() 541232 hhxyzyzzhxy λ ητ −−−++−
−−++−= ))1(([2)( 12
2
1
2 yzzzxyxydη
.))1((])1(22 2
122121 yzzyzz −−−−−− ττ
If the proton spin is directed along )(lS µ then the
spin-dependent part of the Born cross section reads
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
37
−+−= −
1
2
124
1
4
2
3
2
322 ([[
8
zzdH
mq
dVN
pdkd
dE k
B
l ησε
,]]2) 322 Hyz τ−−
++++−= 7182162 )(/()()2( hhzzhyH ηηλ
−−=→=+ + 1([,)(,) 12142392 zzyhhHHh ii ηη
−−=++−−− 2212121 )((,])(2)2() zzzzzxyxzy ητ
.)2(2)())1( 2211 τyxyzzxyzyz −−++−−
In the case of transverse orientation of the proton
spin (along )(tSµ ) we have
,][
8 4
2
2
1
3
1
4
3
2
3
2
322 H
d
dH
dq
VVN
pdkd
dE k
B
t −= ψ
ησε
−+−−+= 12111
2
1 ())(2([ zzzzzzxyd τψ
.)(,/])2 131422 +→=− ii hhHHdyτ
At last, for the normal orientation of the proton spin
(along )(nS µ ) the spin-dependent part of the cross section
of the process (13) reads
.][
8 3
2
2
44
3
2
3
2
322 H
d
H
q
VVN
pdkd
dE k
B
n ηψσε +−=
4. SEMI-INCLUSIVE DIS
ON POLARIZED TARGET
In this section we consider the polarization phenom-
ena in SI DIS off polarized nucleus
,)()()()( 2211 XpHkepAke ++→+ −−
(19)
where H is arbitrary hadron and nucleus A has definite
vector polarization P. In this case the leptonic tensor is
as before, and the hadronic tensor has the same structure
as defined by Eq. (15), where one needs to use polariza-
tion of the nucleus P instead of the proton spin S and
write )( 2Pp instead of )( 1Sp . Besides, we will use the
notation 181 gg − for the corresponding hadron struc-
ture functions.
As a stabilized set we can use the 4-vectors given in
Eq. (9), where it is necessary to do the substitution
1ττ → . The simple calculation gives
.14
2
2
3
2
3
)(
22 G
q
VN
pdkd
d
E k
B
u =
σ
ε (20)
Note that numerical coefficient in front of G1 is twice
as much as compared with that on the right side of Eq.
(18) in front of H1. The reason is that in this case we do
not fix the spin state of the final hadron H.
The polarization-dependent part of the cross section
for the longitudinal polarization is
−−−= 2114
4
2
3
2
322 )2[(
4
Gzz
Mq
VN
pdkd
dE k
B
l τησε
,]2)21( 4131 GGxy ττ ++−
where the functions Gi (i=1-4) can be derived from Hi by
replacement the hadron structure functions gj instead of
hj.
The corresponding part of the cross section in the
case of the transverse polarization can be written as
−−= −
2
1
14
1
3
2
3
2
322 ()[(
4
zxyb
q
VxybVN
pdkd
dE k
B
t ησε
+++−−− − 1()(2))( 1
132111 xbGGxybzxyz τ
.1,])2 1141 ττ xyybGx −−=+
For the normal polarization the spin-dependent part
of the cross section is
+−= 1(([
4 22
2
1
4
3
2
3
2
322 zyG
xybq
VVN
pdkd
dE k
B
n ησε
.]))2()21()2 4111 Gyxzxyzx −−−−−+ ττ
5. DIS FROM TENSOR POLARIZED TAR-
GET WITH TAGGED PHOTON
The initial-state collinear radiation is very important
in certain regions of DIS at HERA kinematics domain. It
leads to reduction of the projectile electron energy and
therefore to a shift of the effective Bjorken variables in
the hard scattering process as compared to those deter-
mined from the actual measurement of the scattered
electron alone. That is why the radiative events in the
DIS process
)()()()()( 211 xpXkkepdke ++→+ −− γ (21)
have to be carefully taken into account.
In this section we investigate events for the process
(21) with unpolarized electron and tensor-polarized
deuteron. We suggest that the hard photon is emitted
very close to the direction of the incoming electron
beam )1,( 00 < <≤ θθθ γ , where γθ is the angle be-
tween 3-momenta of the initial electron and hard photon.
Besides, the photon detector (PD) measures the energy
of all photons inside the narrow cone with the opening
angle 02θ around the electron beam (the scattered-elec-
tron 3-momentum is also measured).
A set of the kinematics variables, that is especially
adapted to the case of the collinear-photon radiation, is
given by the shifted Bjorken variables
,)(2/,)( 211
22
21
2 kkkpQxkkkQ −−=−−−=
,)(2/)(2 11211 kkpkkkpy −−−=
)(2 11 kkpV −=
,
and the energy fraction of the electron after the initial-
state radiation of a collinear photon
1111 /)(/)(2 εϖε −=−= Vkkpz , where 1ε is the
initial electron energy and ϖ is the energy deposited in
PD.
The relation between the shifted and standard
Bjorken variables reads
38
.,1,
1
,22 zVV
z
yzy
yz
xyzxzQQ =−+=
−+
==
The simple calculation gives
,),,(),(
2
2
0 QyxLzP
dzydxd
d
Σ=
π
ασ
(22)
−+=Σ −
ttttllll RSRSQVQyx ([2),,( 422 π α
,)/ln(,]) 22
0
2
10 mLRSR ltltnn θε=+−
,)1/(]2)1[(),( 0
2
0 zzLzLzP −−+=
where the quantities ltttll SSS
,, can be obtained from the
quantities ltttll SSS ,, , given by the Eq. (12), using the
following substitution ττ ,,,,,, byxbyx → . Note that
components of the quadrupole polarization tensor are
defined with respect to the set of 4-vectors described by
Eq. (9). We restrict ourselves to the model-independent
RC related to the radiation of the real and virtual pho-
tons by leptons. Our approach to the calculation of RC is
based on the account of all essential Feynman diagrams
that describe the observed cross section in framework of
the used approximation. To get rid of cumbersome ex-
pressions we retain in RC the terms that accompanied at
least by one power of large logarithms:
)4/ln(,)/ln(, 2
0
22
0 θ== θLmQLL Q . Besides, in chosen
approximation we neglect the terms of the order of
222
0
2
1
22
0 /,/, Qmm θεθ in the cross section.
The total RC to the Born cross section (21) is given
by the sum of the virtual and soft photon corrections and
the hard-photon emission contribution. The last one is
different for the exclusive and calorimeter event selec-
tion. In the considered approximation it is convenient to
write this RC in the form
)(
4 2
2
fi
RC
dzydxd
d Σ+Σ=
π
ασ
.
The first term is independent on the experimental se-
lection rules for the scattered electron and reads
+−
−
++=Σ zz
z
zzPLLi
2
2
)2(
00 ln
2
1ln3[
1
1)(
2
1{ θ
−−−−++ )1ln(2)1ln(ln2)/ln(ln 2 zzzzYY
+
−
+++−+−
z
zcLizLi
1
ln)]
2
1(2)1(2
3
2
22
2π
+Σ
−
−−+
−−
+ ),,(}
)1(2
161)
1
ln(
1
4 2
2
Qyx
z
zz
z
z
z
z
∫ −
−
−+
0
0
)1(
2
0
0 )1(
1
])1(2ln[),(
u
uP
u
ducyLzP
θ
,,
1
1),,( max1
00
2
21
z
xuZyL
z
zQyxy tttt =
−
++Σ−
where the quantities ,,,,)(,)( max1
)2()1(
txxczPxP θ
YQy tt ,, 2 and the definition of the function )(2 xLi can
be found in Ref. [6]. The expression for the Z term is rather
cumbersome and it will be published elsewhere.
The second term, denoted by fΣ , explicitly de-
pends on the rule for the event selection. It includes the
main effect of the scattered-electron radiation. In the
case of exclusive event selection, when only the scat-
tered bare electron is measured, and any photon,
collinear with respect to its momentum direction, is ig-
nored, this contribution is
∫ −+=Σ
max1
0
10 )1ln[(),(
y
Q
excl YLdyLzPy
,),,(]
1
)
1
1( 21
1
1
1
)1(
ssss Qyxy
y
y
y
P Σ
+
+
+
−
where the quantities max1
2 ,,, yQyx sss can be found in
Ref. [6]. Note that the mass singularity that is connected
with the scattered-electron radiation, exhibits itself
through QL term.
The situation is quite different for the calorimeter
event selection, when the detector cannot distinguish be-
tween the events with a bare electron and events where
the scattered electron is accompanied by a hard photon
emitted within a narrow cone with the opening angle
'
02θ around the scattered-electron momentum direction.
For such experimental setup we derive
∫ +
−=Σ
max1
0 1
)1(
12'
0
0 )
1
1(])1(2ln[)[,(
y
cal
y
PdycyLzP
θ
)],,(
2
1),,( 221 QyxQyxy ssss
Σ+Σ− .
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
39
G.I. Gakh, N.P. Merenkov, O.N. Shekhovtsova
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
1. INTRODUCTION
2. THE TENSOR-POLARIZED TARGET
3. SEMI-INCLUSIVE DIS WITH POLARIZED FINAL PARTICLE
4. SEMI-INCLUSIVE DIS
ON POLARIZED TARGET
5. DIS FROM TENSOR POLARIZED TARGET WITH TAGGED PHOTON
REFERENCES
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