SPIN PHYSICS with SPIN-0 HADRONS 1 Transversity
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hep-ph/9805410 VUTH 98-16
NIKHEF 98-012 FNT/T-98/04
SPIN PHYSICS WITH SPIN-0 HADRONS
R. JAKOB
Universita di Pavia and INFN, Sezione di Pavia,
Via Bassi 6, I-27100 Pavia, Italy, e-mail:[email protected]
D. BOER
National Institute for Nuclear Physics and High Energy Physics (NIKHEF)
P.O.Box 41882, NL-1009 DB Amsterdam, the Netherlands
P.J. MULDERS
Department of Physics and Astronomy, Free University,
De Boelelaan 1081, NL-1081 HV Amsterdam, the Netherlands
+
We discuss various azimuthal asymmetries in semi-inclusive DIS and e e !
h h X whichinvolvechiral o dd quantities like the transversity distribution h and
1 2 1
? ?
a fragmentation function H . For the fragmentation describ ed by H azimuthal
1 1
angular dep endence has to b e measured, but no p olarization vector of a nal state
hadron. We present rst results on asymmetries including the electroweak currents
in one-hadron inclusive DIS.
1 Transversity Distribution h (x)
1
Besides the quite well-known parton distribution f (x) there are two di erent
1
spin distributions: the longitudinal spin distribution g (x), on whichanum-
1
1
b er of new exp erimental results have b een rep orted at this conference ; the
a
second one, the transversity distribution, h (x), remains completely unknown
1
b
as far as exp erimental data are concerned. The transversity distribution h is
1
equally imp ortant for the description of quarks in nucleons as the more familiar
function g ; their information is complementary.
1
The reason why h is not determined yet, is the fact that it is a chiral o dd
1
function, and consequently suppressed in simple pro cesses like totally inclusive
3
DIS . The quark contentof h (x) in terms of left-handed and right-handed
1
Presented at the workshop on "Deep Inelastic Scattering and QCD, DIS98", Brussels
(1998)
a 2
The function h (x)was rst discussed by Ralston and Sop er in Drell-Yan scattering.
1
b
The alternative notations q (x), q (x) and q(x) (or sometimes q (x) for the latter)
T
are also in use instead of f , g and h , resp ectively.
1 1 1 1 f = g = - h = -
1 1 1
Figure 1: Schematic interpretation of the three distribution functions f (x), g (x) and h (x)
1 1 1
in terms of (di erences of ) probabilities. The longitudinal direction is assumed to b e along
the horizontal axis.
quarks (de ned via pro jection op erators P = (1 )=2) is of the form
5
R=L
RL LR , i.e., contains a transition from R to L and vice versa. Since QCD
interactions conservechirality, h (x) cannot o ccur alone in a pro cess (case a in
1
the gure b elow), but has to b e accompanied by a second chiral o dd quantity,
like e.g. the fragmentation function H (z ) (case c b elow) which is the p endant
1
to h for the fragmentation; obviously also the fragmentation function H can-
1 1
+
not o ccur alone (as in case b), i.e., is not accessible in e e ! hX (see also a
4
recent article by Ja e ).
R ∆ L
R L R ∆ L R L
R L R L R L
R Φ L R L R Φ L
a) b) c)
Figure 2: Chiral o dd functions can hardly b e prob ed in the simple pro cesses a) (DIS) and
+
b)(e e !hX ), where they contribute only via (suppressed) quark mass e ects. But they
0
are accessible when combined with another chiral o dd quantity like e.g. in c)(`H ! ` hX ).
2 Azimuthal Asymmetry in SIDIS / `Collins E ect'
To measure transversity in one-hadron inclusive DIS one can lo ok for parts
of the cross section involving h and H . It has the disadvantage that the
1 1
p olarization vector of an observed spin-1/2 hadron in the nal state has to b e
determined. Although not imp ossible in principle, for instance the decay of
5
, it is exp erimentally dicult. 's would o er this p ossibility
Alternatively, one can measure azimuthal angular dep endences in the pro-
duction of spin-0 or (on average) unp olarized hadrons. This pro duction is
describ ed by the transverse momentum dep endent fragmentation function
?
H (x; k ) which is also chiral o dd and, moreover, \naive time-reversal o dd",
T
1
i.e., non-vanishing only due to nal state interactions. For a more detailed dis- 2
? 6
cussion of H (x; k ) see elsewhere . The use of a \naive time-reversal o dd"
T
1
observable and azimuthal angular dep endence to obtain information on h (x)
1
7
was prop osed by Collins . A particular useful reformulation of this prop osal
0
for `H ! ` hX can b e given in terms of appropriately weighted cross sections
Z
0 `
d (eH ! e hX ) d
def
2
hW i = d q W : (1)
T
` 2
2 dx dz dy d d q
B h
T
` `
We nd ( , are the azimuthal angles of h and the target spin, resp ectively)
h S
2
X
Q 4 s
T
?(1)a
2 a ` `
jS j (1 y ) e x h (x ) H (z ) (2) sin( + ) =
T B B h
a 1 h S
1
4
M Q
h
a;a
containing the ( avor summed) pro duct of the transversity distribution with
2
?
a k -momentof H
T 1
Z
n
2
k
def
?(n)
T
2 2 ?
H (z) = z d k H (z; zk ): (3)
T T
1
1
2
2M
h
? + +
3 H from e e Annihilation (e e ! h h X )
1 2
1
2
Recently we prop osed a way to obtain information on the rst k -moment
T
? +
of H from a cos(2 ) angular dep endence in two-hadron pro duction in e e
1
8
annihilation. The two hadrons have to be in the quark and anti-quark jet
of an almost back-to-back jet event, resp ectively, and their momenta should
?
?
H b e reconstructed. Since the fragmentation functions involved are H and
1 1
(the corresp onding one for the anti-quark) the e ect is indep endent of the spin
of the pro duced hadrons; to lo ok for a pair of pions is the most obvious choice,
c
since they are abundantly pro duced. Again the appropriate way to pick out
the cos(2 )-dep endent part of the di erential cross section is a weighted cross
section
Z
` +
d d (e e ! h h X )
def
1 2
2
h W i = d q W : (4)
T
2
2 d dz dz d q
1 2 T
2 2
At the Z-p ole, i.e., Q ' M , where most of the LEP data are available, the
Z
?(1)a
?(1)a
H annihilation via a Z b oson dominates and the pro duct of H with
1
1
d
can b e obtained from the weighted cross section
P
?(1)a
?(1)a
2 2 ` a
2
(z ) 3 Q y (1 y ) c c H H (z )
Q 2 1
1 1 2
1
a;a
T
cos(2 ) = (5)
2 2 2 2 2
2 2
4M M 4 s c [(Q M ) + M ]
1 2
W W Z Z Z
c
This is spin physics with spin-0 hadrons !
d
Strictly, this gives information for the quantities only at the scale of the Z mass; evolution
has to b e taken into account for usage in other hard pro cesses at a di erent scale. 3 P ⊥ P ^ k' 1 1 h1 φ
P 2 ^z k
lepton scattering plane (cm)
+
Figure 3: Kinematics for two-hadron pro duction in e e annihilation.
2 2 2 2
where s (c ) is the sin(cos) of ; the factors c = g + g , c = g g are
W W W 1 2
V A V A
relevant combinations of (axial-)vector couplings (for later use: c =2g g ).
3 A V
0
4 Electroweak `H ! ` hX
During this conference wehave heard that high energy exp eriments reach the
9
precision where e ects from electroweak currents have to b e considered . We
have calculated the di erential cross section for one-hadron inclusive lepton-
hadron scattering thereby taking into account
the exchange of a photon or a Z b oson (including interference terms)
p olarized b eam/target/ nal state hadron (spin-1/2)
dep endence on the transverse momentum of the pro duced hadron.
10
The full results will be presented elsewhere . Here we give some examples
in form of weighted cross sections listed in Table 1. The electromagnetic part
in the cross section of the rst line is the asymmetry discussed in Sec. 2.
` `
Additionally, from the interference term there is a cos( + ) dep endence
S
h
involving the same combination of distribution and fragmentation functions
(second line). In principle, it provides an indep endent piece of information for
a avor decomp osition. But since, the term is probably dicult to measure ac-
curately, one may think of a more mo dest usage. Together with the simplifying
assumption of only one avor contributing dominantly to the fragmentation,
+
e.g. u-quark fragmenting to a in the region of large x and large z , this
B h
?(1)
may b e used for an indep endent cross-checkon h (x)H .
1
1 4
The electromagnetic part of the cross-section in the third line was discussed
11
b efore . Also here we nd an orthogonal angular dep endence resulting from
the interference of electromagnetic and weak interaction (fourth line).
Particularly interesting is the p ossibility of non-zero \naive time-reversal
o dd" distribution functions, whichwould showupinaweighted cross section
like in the fth line. The electromagnetic term was discussed by Bo er and
12
Mulders . 5
h i
1
2
4 s
W hW i jS j (1 y ) x
4 T B
Q
P
Q
` ` 2 ` a ` a
T
sin( + ) e +8g e g +16c c
a 1 2
a 1 2
S V V
h
a;a
M
h
?(1)a
a
h (x )H (z )
B h
1
1
P
?(1)a
Q
M
` ` ` a a
T
Z Z
cos( + ) 8g e g h (x ) H (z )
2
a 1 B h
2
1
1
h S V A
a;a
M
Q M
h
Z
3
P
Q
2 ` a ` a ` `
T
e +8g e g +16c c sin(3 )
2 a 1 2
a 1 2
V V S
h
a;a
6M M
h
?(2)a ?(1)a
h (x ) H (z )
B h
1
1T
3
P
Q ?(2)a ?(1)a
` ` ` a
T
cos(3 ) 8 g e g h (x ) H (z )
2 a 1 B h
1
h S V A
1T
a;a
6M M
h
P 2
Q 2 2y +y
2 ` a ` a ` `
T
e +8g e g +16c c sin( )
a 1 2
a 1 1
V V S
h
a;a
M 2 2y
h
?(1)a
y(2 y )
` a ` a a
+ 8g e g +16c c f (x )D (z )
a 1 2 B h
3 3 1
A A
1T
2 2y
2 2 2
2
Q (Q M )
Q
1 1
Z
= =
2 2 2 2 2 2 2 2
1 1 2
2 2 2
16 s (Q M 16 s Q M c ) + c M
W Z W Z W Z W Z
Table 1: Weighted cross sections in semi-inclusive DIS.
Acknowledgments
This work is part of the research program of the foundation for Fundamental Researchof
Matter (FOM), the National Organization for Scienti c Research (NWO) and the TMR
program ERB FMRX-CT96-0008.
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