By examining the symmetry constraints on the semi-inclusive pseudoscalar
particle production in unpolarized inelastic lepton-hadron scattering, we
present a complete, exact Lorentz decomposition for the corresponding hadronic
tensor. As a result, we find that it contains five independent terms, instead
of the four as have been suggested before. The newly identified one is odd
under the naive time reversal transformation, and the corresponding structure
function is directly related to the single spin asymmetry in the semi-inclusive
pseudoscalar meson production by a polarized lepton beam off an unpolarized
target.Comment: This manuscript is of no use at its present form, so the author
  withdrew it. Sorry

Lu, Wei

English

arXiv

By examining the symmetry constraints on the semi-inclusive pseudoscalar particle production in unpolarized inelastic lepton-hadron scattering, we present a complete, exact Lorentz decomposition for the corresponding hadronic tensor. As a result, we find that it contains five independent terms, instead of the four as have been suggested before. The newly identified one is odd under the naive time reversal transformation, and the corresponding structure function is directly related to the single spin asymmetry in the semi-inclusive pseudoscalar meson production by a polarized lepton beam off an unpolarized target

Lu, W

CERN Document Server

H
EP
-P
H
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50
63
43
Symmetry analysis of the hadronic tensor for the semi-inclusive
pseudoscalar meson leptoproduction from an unpolarized nucleon target
Wei Lu
CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China
and Institute of High Energy Physics, P.O. Box 918(4), Beijing 100039, China

By examining the symmetry constraints on the semi-inclusive pseudoscalar par-
ticle production in unpolarized inelastic lepton-hadron scattering, we present a com-
plete, exact Lorentz decomposition for the corresponding hadronic tensor. As a
result, we nd that it contains ve independent terms, instead of the four as have
been suggested before. The newly identied one is odd under the naive time rever-
sal transformation, and the corresponding structure function is directly related to
the single spin asymmetry in the semi-inclusive pseudoscalar meson production by a
polarized lepton beam o an unpolarized target.
PACS Number(s): 11.30.Er, 13.85.Ni, 13.85.+e
Typeset Using REVTEX

Mailing address
1
In the particle physics, the symmetry analysis plays a very important role, since it can forbid or
allow for the existence of physical quantities before we set about the details of dynamics. Although
the principles and methods involved in the symmetry analysis are usually not complicated, in some
circumstances it is a highly nontrivial matter to arrive at a complete result. As the time reversal
(T ) invariance of interactions is involved, this is even the case. In fact, most mistakes associated
with the symmetry analysis can be traced to the confusion the so-called naive T transformation
with the full T transformation. In other words, the constraints due to time reversal invariance are
often not properly considered.
In this paper, we clarify an unsatisfying fact [1] [2] existing in the literature, i.e., the early
suggested Lorentz decomposition of the hadronic tensor for the spinless particle leptoproduction
from unpolarized nucleon target is incomplete. In principle, one may make eorts to attend the
more general case in which the target nucleon is polarized and the spin state of the detected
particle is specied in the case it has spin. However, such an analysis will be tedious and the
phenomenological implications of the corresponding structure functions are not easy to exemplify.
Therefore, we choose to study the simplest case in which a pseudoscalar meson is semi-inclusively
detected in the deeply inelastic lepton scattering o an unpolarized target. Obviously, the result
can also apply to the semi-inclusive baryon production if one does not measure the baryon spin.
Nevertheless, we assume the lepton beam is polarized so that the phenomenological consequence
of the newly identied structure function can be conveniently demonstrated.
To be specic, we consider
l(P
l
; S
l
) + p(P
p
)! l(P
0
l
) + (P

) +X; (1)
where the particle momenta in the brackets is self-explanatory. We normalize the spin vector in
such a way that S
l
 S
l
=  M
2
l
and S
l
 P
l
= 0 for a pure lepton state.
In the one-photon exchange approximation, the proton structure is probed by a space-like
photon with momentum q = P
l
 P
0
l
. The invariant cross section for the process considered can be
written as a contraction of the leptonic and hadronic tensors:
2
E0
l
E

d(S
l
)
d
3
P
0
l
d
3
P

=

2
16
3
M
p
E
l
Q
4
L

(P
l
; S
l
; q)W

(q; P
p
; P

); (2)
where M
p
is the target mass, E
l
the beam energy in the laboratory frame, and Q =
p
 q
2
.
Conventionally, the leptonic and hadronic tensors are dened as
L

(P
l
; S
l
; q) =
1
2
Tr[/P
0
l


/P
l


1 + 
5
/S
l
2
] =
q
2
2
(g

 
q

q

q
2
) + 2(P
l
 
q

2
)(P
l
 
q

2
) + i"

q

S

l
;
(3)
and
W

(q; P
p
; P

) =
1
4
X
X
Z
d
4
ye
iqy
< P
p
jj

(0)j(P

); X >
out out
< (P

); X jj

(y)jP
p
>; (4)
respectively. The electromagnetic current is dened to be j

=
P
f
e
f

 
f


 
f
, with f being the avor
index and e
f
being the quark charge in unit of the electron charge. In dening W

(q; P
p
; P

), the
out-state property of the nal states has been labelled explicitly.
The hadronic tensor contains all the information about the nucleon structure and pion pro-
duction. Because the fundamental vertex of the deeply inelastic scattering in the one-photon
approximation is electromagnetic, W

(q; P
p
; P

) should be subjected to the hermiticity, current
conservation, parity conservation and time reversal invariance. To write out the general expression
of W

(q; P
p
; P

) in terms of structure functions, one ought to exhaust all the possible candidate
terms by imposing all the symmetry constrains that the electromagnetic interaction satises.
Concerning W

(q; P
p
; P

), its Lorentz decomposition has ever been suggested by Mulders and
collaborators [1] [2]. Here we simply recapitulate their result in our language. Choosing z^ axis
to be along the direction of the virtual photon momentum, one can dene a covariant transverse
vector P
?
, which is
P

?
= (0; P
x
; P
y
; 0) (5)
in the target rest frame. Then, the Mulders decomposition for W

(q; P
p
; P

) reads
W

(q; P
p
; P

) =W
1
( g

q
2
+ q

q

) +W
2
(P

p
 
P
p
 q
q
2
q

)(P

p
 
P
p
 q
q
2
q

)
+W
3

(P

p
 
P
p
 q
q
2
q

)P

?
+ P

?
(P

p
 
P
p
 q
q
2
q

)

+W
4
P

?
P

?
; (6)
3
whereW 's are called structure functions, dependent on q
2
, P
p
q, P

q and P
p
P

. Our ndings to
be presented are that this Lorentz expansion is incomplete, and in a more complete decomposition
there exists another term, which is imaginary, antisymmetric under  $ , and odd under the
naive T transformation.
Let us examine in turn the constraints that all the electromagnetic symmetries impose on the
hadronic tensor. First, the electromagnetic interaction is gauge invariant, as is reected by the
current conservation condition:
q

W

(q; P
p
; P

) = q

W

(q; P
p
; P

) = 0: (7)
Second, the hermiticity j
y

= j

of the electromagnetic current leads to
[W

(q; P
p
; P

)]

= W

(q; P
p
; P

): (8)
Thirdly, the electromagnetic interaction is parity conserved. For a generic Lorentz coordinate
vector x

, we follow Itzykson and Zuber [3] and dene ~x

= x

. Then, under a parity (P)
transformation, the momentum P

and spin vector S

behave in the fashion
P

P
!
~
P

; S

P
!  
~
S

: (9)
On the other hand, the electromagnetic current satises
j

(x)
P
! j

(~x): (10)
As a result, the parity conservation of the electromagnetic interaction yields
W

(q; P
p
; P

) = W

(~q;
~
P
p
;
~
P

): (11)
Fourthly, the fundamental electromagnetic vertex is T invariant. Under a T transformation,
one has
P

T
!
~
P

; S

T
!
~
S

: (12)
and
4
j
(x)
T
! j

( ~x): (13)
In addition, the complex conjugation should be operated and an in state changed into the corre-
sponding out state, or vice versa. In general, an in state is related to its corresponding out state
by S matrix (operator) [4]:
j >
in
= Sj >
out
; (14)
where S can be written as
S = 1 + iT (15)
with T being the transition matrix (operator). Unless the state is composed of an individual
particle or a set of non-interactive particles, the in state diers from its corresponding out state.
Experiences tell us that there exist violent multi-interactions in the hadron system, so we cannot a
priori identify the semi-inclusive hadron out state with its corresponding in state. As a consequence,
T invariance can only tell us
W

(q; P
p
; P

) =
"
1
4
X
X
Z
d
4
ye
iqy
<
~
P
p
jj

(0)j(P

); X >
in in
< (P

); X jj

(y)j
~
P
p
>
#

: (16)
If the nal-state interactions are neglected, the in state reduces to its corresponding out state.
The resulting simplied time reversal transformation can be called the naive time reversal trans-
formation, under which one has
W

(q; P
p
; P

) = [W

(~q;
~
P
p
;
~
P

)]

: (17)
Examining Eqs. (11) and (17), one can see that it is more convenient to use the complex PT
transformation instead of the individual discrete transformations P and T . This is even the case
in discussing the single spin asymmetry, which is odd under the naive PT transformation. In our
case, a PT transformation gives rise to
W

(q; P
p
; P

) =
"
1
4
X
X
Z
d
4
ye
iqy
< P jj

(0)j(P

); X >
in in
< (P

); X jj

(y)jP >
#

: (18)
5
Substituting Eqs. (14) and (15) into (18), one has
W

(q; P
p
; P

) = [W

(q; P
p
; P

)]

+ T

(q; P
p
; P

); (19)
where T

(q; P
p
; P

) incorporates the dierence between the in and out states. Equation (19)
implies that W

(q; P
p
; P

) can be decomposed into two parts, one even and another odd under
the naive PT transformation.
We note that the metric tensor g

gets into work in our decomposition. As for the full
antisymmetric rank-four tensor "

, it is not involved because we are dealing with the unpolarized
target. Keep in mind thatW

(q; P
p
; P

) satises the current conservation condition, which greatly
constrains the forms in which momenta and spin vectors contribute. Regarding the metric tensor,
its only independent, gauge invariant combination is g

q
2
 q

q

. In our case, we can construct two
independent momentum combinations P

p
 q

(P
p
q)=q
2
and P


 q

(P

q)=q
2
, whose contractions
with the photon momentum q

vanish. Another combination P

p
 P


(P
p
q)=(P

q) can be expressed
in terms of P

p
  q

(P
p
 q)=q
2
and P


  q

(P

 q)=q
2
. Exhausting all the possible candidate terms,
we attain
W

(q; P
p
; P

) = F
1
(g

q
2
  q

q

) + F
2
(P

p
 
P
p
 q
q
2
q

)(P

p
 
P
p
 q
q
2
q

)
+ F
3
(P


 
P

 q
q
2
q

)(P


 
P

 q
q
2
q

)
+ F
4

(P

p
 
P
p
 q
q
2
q

)(P


 
P

 q
q
2
q

) + (P


 
P

 q
q
2
q

)(P

p
 
P
p
 q
q
2
q

)

+ i
^
F

(P

p
 
P
p
 q
q
2
q

)(P


 
P

 q
q
2
q

)  (P


 
P

 q
q
2
q

)(P

p
 
P
p
 q
q
2
q

)

: (20)
If the Mulders parameterization is used, one has equivalently
W

(q; P
p
; P

) =W
1
( g

q
2
+ q

q

) +W
2
(P

p
 
P
p
 q
q
2
q

)(P

p
 
P
p
 q
q
2
q

)
+W
3

(P

p
 
P
p
 q
q
2
q

)P

?
+ P

?
(P

p
 
P
p
 q
q
2
q

)

+W
4
P

?
P

?
+ iW

(P

p
 
P
p
 q
q
2
q

)P

?
  P

?
(P

p
 
P
p
 q
q
2
q

)

: (21)
The newly identied term, associated with
^
F in Eq. (20) or with
^
W in (21), is imaginary, anti-
symmetric under  $ , and odd under the naive T transformation. Physically, it incorporates
6
the distinction between the inclusively detected hadron state and its corresponding in state. Since
such a dierence is caused by the nontrivial nal-state interaction described by the transition op-
erator T [see Eqs. (14){(16), (18) and (19)], it can be said that the new term reects the eects of
nal-state multi-interactions.
To exemplify the phenomenological consequences, we call for the polarized incident lepton
beam, as has been specied in our kinematics. Obviously, when the hadronic and leptonic tensors
contact, their imaginary parts co-work to make a contribution to the cross section. Using our
parameterization for W

(q; P
p
; P

), we have
E
0
l
E

d(S
l
)
d
3
P
0
l
d
3
P

=

2
16
3
M
p
E
l
Q
4

Q
4
F
1
+M
2
p
(2E
l
E
0
l
 
Q
2
2
)F
2
+

2(P
l
 P

)(P
0
l
 P

) 
1
2
M
2

Q
2

F
3
+ M
p
(2E
0
l
P
l
 P

+ 2E
l
P
0
l
 P

  E
0

Q
2
)F
4
+ 2M
p
S
l
 q  P

^
F
o
: (22)
Consequently, structure function
^
F can be related to the correlation among the lepton polariza-
tion vector, the momentum of the virtual photon and that of the semi-inclusively detected pion.
Experimentally, of more interest are the single spin asymmetries. Considering that large spin asym-
metries of the order of ten or more percent have been observed in inclusive pion production [5] [6]
from hadron-nucleus xed target experiments, we discuss briey in the following the feasibilities
to measure the longitudinal and transverse spin asymmetries in the process we are considering.
We rst discuss the case in which the lepton beam is longitudinally polarized. Neglecting the
lepton mass eects, we may replace S
l
with H
l
P
l
, where H
l
is the lepton helicity. The lepton
scattering plane is symmetric under the rotation about the beam axis, so there are only ve
independent degrees of freedom. We work in the target rest frame, with the z^ axis along the beam
direction and the x^ axis in the lepton scattering plane. Call the outgoing angle of the scattered
lepton 
0
l
. When the detected pion falls into the solid angle d


= d cos 

d

, the dierential cross
section reads
E

d(H
l
)
d
3
P

dE
0
l
d cos
0
l
=

2
8
2
M
p
Q
4
E
0
l
E
l

Q
4
F
1
+M
2
p
(2E
l
E
0
l
 
Q
2
2
)F
2
+

2(P
l
 P

)(P
0
l
 P

) 
1
2
M
2

Q
2

F
3
+ M
p
(2E
0
l
P
l
 P

+ 2E
l
P
0
l
 P

  E
0

Q
2
)F
4
  2H
l
M
p
E
l
E
0
l
jP

j sin 
0
l
sin 

cos

^
F
o
: (23)
The longitudinal spin asymmetry for the inclusive pion production is dened as
7
AL
=
Z
E

d(+)
d
3
P

dE
0
l
d cos 
0
l
dE
0
l
d cos
0
l
 
Z
E

d( )
d
3
P

dE
0
l
d cos
0
l
dE
0
l
d cos 
0
l
Z
E

d(+)
d
3
P

dE
0
l
d cos 
0
l
dE
0
l
d cos
0
l
+
Z
E

d( )
d
3
P

dE
0
l
d cos
0
l
dE
0
l
d cos 
0
l
; (24)
which can be cast into
A
L
=  2E
l
M
p
jP

j sin 

cos

Z
dE
0
d cos
0
l
^
F
E
02
l
Q
4
sin 
0
l


Z
dE
0
l
d cos 
0
l
E
0
l
Q
4

Q
4
F
1
+M
2
p
(2E
l
E
0
l
 
Q
2
2
)F
2
+

2(P
l
 P

)(P
0
l
 P

) 
1
2
M
2

Q
2

F
3
+M
p
(2E
0
l
P
l
 P

+ 2E
l
P
0
l
 P

 E
0

Q
2
)F
4

 1
: (25)
Notice that the integrations over E
0
l
and sin 
0
l
cannot be completed at present unless we have
known how structure functions depend on them.
Now we attend the transverse polarization case of the incident lepton beam, in which there are
six independent degrees of freedom. Again, we work in the target rest frame, letting the z^ axis
be along the beam direction but specifying the x^ axis as the lepton polarization direction. Then,
the spin vector is S

l";#
= M
l
(0; 1; 0; 0) for the lepton polarized parallel and antiparallel to the
x^ axis, respectively. Provided the scattered lepton and the detected pion y into the solid angles
d

0
l
= d cos 
0
l
d
0
l
and d


= d cos 

d

respectively, the cross section can be written as
E

d(S
l";#
)
d
3
P

dE
0
l
d

0
l
=

2
16
3
M
p
Q
4
E
0
l
E
l

Q
4
F
1
+M
2
p
(2E
l
E
0
l
 
Q
2
2
)F
2
+

2(P
l
 P

)(P
0
l
 P

) 
1
2
M
2

Q
2

F
3
+M
p
(2E
0
l
P
l
 P

+ 2E
l
P
0
l
 P

 E
0

Q
2
)F
4
2M
l
M
p
jP

j [E
l
sin 

cos

+E
0
l
(sin 
0
l
sin
0
l
cos 

  cos 
0
l
sin 

cos

)]
^
F
o
: (26)
The transverse spin asymmetry is dened as
A
T
=
Z
E

d(S
l"
)
d
3
P

dE
0
l
d cos
0
l
dE
0
l
d

0
l
 
Z
E

d(S
l#
)
d
3
P

dE
0
l
d cos 
0
l
dE
0
l
d

0
l
Z
E

d(S
l"
)
d
3
P

dE
0
l
d cos
0
l
dE
0
l
d cos

0
l
+
Z
E

d(S
l#
)
d
3
P

dE
0
l
d cos 
0
l
dE
0
l
d cos

0
l
: (27)
From Eq. (26), one has
A
T
=  2M
l
M
p
jP

j
Z
dE
0
l
d cos

0
l
E
0
l
Q
4

E
l
sin 

cos

+E
0
l
(sin 
0
l
sin
0
l
cos 

  cos 
0
l
sin 

cos

)

^
F


Z
dE
0
l
d cos

0
l
E
0
l
Q
4

Q
4
F
1
+M
2
p
(2E
l
E
0
l
 
Q
2
2
)F
2
+

2(P
l
 P

)(P
0
l
 P

) 
1
2
M
2

Q
2

F
3
+ M
p
(2E
0
l
P
l
 P

+ 2E
l
P
0
l
 P

  E
0

Q
2
)F
4
	
 1
: (28)
8
Comparing Eqs. (25) and (28), we observe that the single transverse spin asymmetry is O(M
l
=E
l
)
suppressed relative to the longitudinal spin asymmetry. So measuring the transverse spin asymme-
try is infeasible in practice. At moderate energies, however, we can anticipate that the longitudinal
spin asymmetry is be measured.
In summary, we examined the hadronic tensor for the semi-inclusive pion leptoproduction in
the case of unpolarized target. It is found that a term has been missed in the early suggested
Lorentz decomposition of the hadronic tensor. Such an incompleteness is simply due to the fact
the constraint of time reversal invariance on the inclusive one-particle production is not properly
considered. By carefully examining the contents of time reversal transformation, we suggest a
complete Lorentz decomposition for the hadronic tensor considered, which contain ve independent
terms instead of the four in the literature. The newly identied term is imaginary, antisymmetric
under  $ , and odd under the naive time reversal transformation. Moreover, the possible
phenomenology pertinent to the new structure function is discussed.
9
REFERENCES
[1] P.J. Mulders, Phys. Rep. 185, 83 (1990).
[2] J. Levelt and P.J. Mulders, Phys. Rev. D 49, 96 (1994).
[3] C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980).
[4] J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields (McGraw-Hill, New York, 1965).
[5] J. Antille et al:, Phys. Lett. 94B, 523 (1980).
[6] A.L. Adams et al:, Phys. Lett. 261B, 201 (1991); 264B, 462 (1991).
10


Symmetry analysis of the hadronic tensor for the semi-inclusive pseudoscalar meson leptoproduction from an unpolarized nucleon target

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