Longitudinal Spin-Transfer in Λ Production at
HERMES
H. C. Chiang, for the HERMES collaboration
University of Illinois, 1110 W Green St, Urbana, IL, USA 61801-3080
Abstract. Spin transfer in deep-inelastic Λ electroproduction has been studied with the HERMES
detector using the 27.6-GeV polarized positron beam in the HERA storage ring. The longitudinal
spin transfer DLL from the virtual photon to the Λ has been extracted as a function of z, the fraction
of the virtual photon energy carried by the Λ. The observable D LL is sensitive to both helicity
conservation in the fragmentation process and to hyperon spin structure. Including all data taken
at HERMES during the years 1996–2000, a preliminary average value of D LL 004 009 is
obtained in the current fragmentation region x F 0. These results are explained by a Monte Carlo
simulation based on the Lund string model. The principal conclusion of these studies is that even in
the forward-production region x F 0, Λ production in medium-energy deep-inelastic scattering is
complicated by the influence of the target remnant.
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¼
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BACKGROUND AND MOTIVATION
It has been proposed that longitudinal spin-transfer through the fragmentation process
may be studied by examining Λ hyperons produced in the current fragmentation region.
It is of interest to determine the degree to which Λ hyperons “remember” the spin of
their parent quarks, since the Λ is potentially useful as a polarimeter for probing the spin
structure of the nucleon. Lambda hyperon production was studied at HERMES using
deep-inelastic scattering (DIS) of polarized 27.6-GeV positrons off unpolarized gas (H,
D, He, Ne, Kr) targets. The longitudinal spin-transfer coefficient DLL , defined as
¼
PqΛ
P
beam
Dy DLL z
(1)
¼
was measured as a function of z, the fraction of the available energy carried by the Λ.
Here PqΛ is the polarization of a Λ containing struck quark q; this quantity is accessible
via the self-analyzing weak decay Λ p π . The variable Pbeam represents the
beam polarization, and Dy is the photon depolarization factor, calculable from the
relative energy transfer y ν E. The combination Pbeam Dy provides a measure of
the polarization of the struck quark, thus D LL may be interpreted as the fraction of the
struck quark polarization retained by the Λ.
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MODEL PREDICTIONS OF DLL
¼
The spin-transfer coefficient DLL is primarily sensitive to Λ spin structure and the degree
of helicity conservation in the fragmentation process, and may be predicted by several
¼
CP675, Spin 2002: 15th Int'l. Spin Physics Symposium and Workshop on Polarized Electron
Sources and Polarimeters, edited by Y. I. Makdisi, A. U. Luccio, and W. W. MacKay
© 2003 American Institute of Physics 0-7354-0136-5/03/$20.00
543
DLL’
DLL’
1
Ma SU(3)
Ma pQCD
Ma quark-diquark
0.8
1
0.8
0.6
0.4
0.6
0.2
0.4
0
-0.2
0.2
-0.4
0
-0.2
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-0.6
HERMES data 1996-1997
-0.8
HERMES data 1996-2000 (preliminary)
-1
1
z
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z
(a)
(b)
FIGURE 1. Model predictions [2, 3] for the spin-transfer coefficient D LL as a function of z (a), and
HERMES spin-transfer data (b). The models agree that D LL rises with z, but the data, with an average of
004 009, hover around zero and appear to decrease slightly at high z.
¼
¼
models. The models presented here differ in their predictions of the Λ spin structure
but assume the same ideal conditions: (1) all Λ hyperons are produced directly from
the struck quark, and (2) helicity is perfectly conserved throughout fragmentation. The
most basic model is the naïve constituent quark model (NCQM), which predicts that
∆u ∆d 0 and ∆s 1 in the Λ. Because Λ production in DIS is dominated by
scattering off up quarks, we expect to see essentially zero spin transfer. The NCQM
does not predict DLL as a function of z; DLL is simply a constant. A second, more
sophisticated way of predicting Λ spin structure is to obtain it from an SU(3) rotation
of the experimentally determined proton spin structure [1]. This calculation yields a
spin-transfer coefficient of about -0.2 and, like the NCQM, gives no information about zdependence. Finally, there are several phenomenological models that attempt to describe
the z-dependence of DLL ; a few of the predictions [2, 3] are shown in Fig. 1(a). The
behavior of the curves at high z arises from the high-x behavior of the quark polarization
∆qq in the Λ in the various models. All of the phenomenological models agree that
DLL rises as a function of z.
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¼
¼
¼
HERMES SPIN-TRANSFER DATA
Lambda candidates were identified by examining events containing at least three tracks:
a positron track, and two oppositely-charged hadron tracks. Several kinematic requirements were imposed to ensure that the events were in the DIS region: Q 2 1 GeV2 ,
W 2 GeV, and y 085. A positive value of xF was also required, to restrict the data
sample to hyperons produced in the forward direction. Details of the analysis may be
found in Ref. [4], which presents the DLL result from the 1996–1997 HERMES data.
Fig. 1(b) shows the new, preliminary results for D LL as a function of z obtained from
all HERMES data collected in the years 1996–2000. The spin-transfer coefficient is
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small—average value DLL 004 009—and appears to decrease at high z. This trend
contradicts the predictions of the phenomenological models previously described. On
average the data appear to be consistent with the small values of D LL predicted by the
NCQM and the SU(3) model; however, the shape of the data is poorly described by these
models.
There are several possible reasons for the discrepancy between the data and the
models. As previously described, the models assume ideal conditions for Λ production.
In reality, not all Λ hyperons are produced directly from the struck quark—Λ’s may be
produced in the decays of heavier hyperons (Σ , Σ0 , Ξ0 , and Ξ ), or may not contain
the struck quark at all. The total spin-transfer coefficient DLL thus depends on the
fractional contribution of each subprocess to Λ production and the degree of spin transfer
within each of those subprocesses. Spin transfer to Λ hyperons has previously been
studied by the OPAL [5] and ALEPH [6] collaborations, both of which observed large
Λ polarizations. Using several basic assumptions and a Monte Carlo model accounting
for the different Λ production mechanisms, they were able to successfully explain their
data. A similar Monte Carlo model, using the same assumptions, has been developed for
the HERMES DLL data.
¼
¼
¼
¼
MONTE CARLO MODELS FOR DLL
¼
Lambda production mechanisms may be classified in the following way: (1) direct
production from the struck quark, (2) hyperon parent (Σ , Σ0 , Ξ0 , Ξ ) that contains
the struck quark, and (3) Λ or hyperon parent that does not contain the struck quark. The
total spin-transfer coefficient DLL is calculated using
¼
DLL
¼
∑ fqY CqY
(2)
Y
where fqY is a subprocess’s fractional contribution to Λ production, CqY is the individual
spin-transfer coefficient for a particular subprocess, and the sum is over all Λ production
mechanisms. The subprocess fractions f qY are obtained from a Monte Carlo simulation,
and the subprocess spin-transfer coefficients CqY are calculated using several different
spin structure models.
We make two assumptions about quarks in the fragmentation process when calculating CqY : first, the helicity of the struck quark is perfectly conserved, and second, quarks
created during fragmentation have random spin directions [7]. Thus hyperons that do
not contain the struck quark have no net polarization, and CqY 0 for subprocess (3)
described above. For the other subprocesses, the average Λ polarization is determined
by the probability of a polarized quark fragmenting into a hyperon Y with spin SY MY ,
and the probability for that hyperon to decay into a Λ with spin 12 M Λ. The decay
probabilities can be calculated using simple angular momentum conservation arguments
and Clebsch-Gordan coefficients [8]. The production probabilities, on the other hand,
are model dependent.
It can be shown that the average polarization of a hyperon PqY produced directly
from a polarized struck quark is ∆qY qY and 59 ∆qY qY for spin-1/2 and spin-3/2
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TABLE 1. Subprocess spin-transfer coefficients CqY in the NCQM, Burkardt–Jaffe SU(3) model,
and Ashery-Lipkin SU(3) model
NCQM
Λ
Σ
Σ0
Ξ0
Ξ
B-J SU(3)
A-L SU(3)
u
d
s
n
u
d
s
n
u
d
s
n
0
5/9
-2/9
-1/3
0
0
5/9
-2/9
0
-1/3
1
5/9
1/9
2/3
2/3
0
0
0
0
0
-0.17
0.38
-0.12
-0.43
-0.10
-0.17
0.38
-0.12
-0.10
-0.43
0.63
0.55
0.14
0.42
0.42
0
0
0
0
0
-0.07
0.38
-0.16
-0.33
0.00
-0.07
0.38
-0.16
0.00
-0.33
0.73
0.55
0.11
0.47
0.47
0
0
0
0
0
hyperons, respectively [8]. Three different models were used to calculate ∆qY and qY :
the NCQM, Burkardt–Jaffe SU(3) model (in which the proton spin structure is used
to deduce the spin structure of other members of the baryon octet) [1], and Ashery–
Lipkin SU(3) model [8], a modified version of the Burkardt–Jaffe SU(3) model in which
∆q from only the valence quarks is considered. The spin-transfer coefficients CqY , as
calculated by these three models, for Λ, Σ , Σ0 , Ξ0 , and Ξ hyperons either containing
a struck quark (u d s) or not containing a struck quark (n) are listed in Table 1.
MONTE CARLO RESULTS
Fig. 2(a) shows the fractions of Λ’s produced from various direct sources as a function
of z: direct Λ production from string fragmentation, and Λ’s resulting from the decays of
Σ , Σ0 , and Ξ. The plot shows that, on average, 40%-60% of Λ’s observed at HERMES
are decay products of heavier hyperons, thus it is imperative that D LL models consider
the effects of these intermediate decays. Fig. 2(b) shows the fractions of Λ hyperons
containing various flavors of the struck quark as a function of z; on average, about 90%
of Λ’s do not contain the struck quark. Of the remaining 10%, most contain struck u
quarks, as expected from the semi-inclusive DIS cross section. Strange quark dominance
appears when z 08, and can likely be explained by the fact that at high z, there is more
available energy for the creation of s quarks.
Given the extremely small fraction of Λ hyperons that contain the struck quark, it is
not surprising that the observed D LL is close to zero. Fig. 2(c) shows the Monte Carlo
predictions for DLL in comparison with HERMES data. The solid, dashed, and dotted lines represent results from the NCQM, Burkardt–Jaffe SU(3), and Ashery–Lipkin
SU(3) models, respectively. These different methods used to calculate CqY produce very
similar predictions for DLL . We therefore conclude that DLL is sensitive primarily to the
subprocess fractions f qY and that HERMES spin-transfer data cannot be used to determine which model most accurately describes hyperon spin structure.
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¼
¼
¼
¼
546
Lambda Direct
0
Sigma
*
Sigma
Cascade
0.8
0.7
0.6
1
DLL
Subproc fractions f
Subproc fractions f
1
0.9
1
0.9
0.8
0.8
0.6
no struck q in Y
struck u in Y
struck d in Y
struck s in Y
0.7
0.6
0.4
0.2
0.5
0.5
0
0.4
0.4
-0.2
0.3
0.3
-0.4
0.2
0.2
-0.6
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
NCQM
Burkardt-Jaffe SU(3)
Ashery-Lipkin SU(3)
HERMES data 1996-2000 (preliminary)
-0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
z
1
-1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
z
(a)
(b)
1
z
(c)
FIGURE 2. Fractions of Λ’s originating from various sources as a function of z (a), fractions of
hyperons containing different flavors of the struck quark (b), and Monte Carlo predictions for D LL (c)
(where the solid, dashed, and dotted lines indicate the NCQM, Burkardt–Jaffe SU(3), and Ashery–Lipkin
SU(3) models, respectively).
¼
FUTURE WORK
The Monte Carlo models for DLL are successful in that they predict essentially zero spin
transfer to Λ hyperons up to about z 07; this is qualitatively consistent with HERMES
data. However, the shape of the data is rather poorly matched by the Monte Carlo,
suggesting that the spin-transfer models used in this study are too simple. Since the
Monte Carlo indicates that many of the Λ hyperons observed at HERMES are produced
in the target fragmentation region, it may be of interest to reconsider the polarization of
Λ’s originating from the target remnant instead of assuming the polarization is simply
zero. This study primarily provides a rough explanation for HERMES spin-transfer
data; with more sophisticated models, it may be possible to achieve a more detailed
understanding in the future.
¼
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
Burkardt, M., and Jaffe, R. L., Phys. Rev. Lett., 70, 2537 (1993).
Ma, B.-Q., Schmidt, I., and Yang, J.-J., Phys. Rev., D61, 034017 (2000).
Ma, B.-Q., Schmidt, I., Soffer, J., and Yang, J.-J., Phys. Rev., D65, 034004 (2002).
Airapetian, A., et al., Phys. Rev., D64, 112005 (2001).
Ackerstaff, K., et al., Eur. Phys. J., C2, 49 (1998).
Buskulic, D., et al., Phys. Lett., B374, 319 (1996).
Gustafson, G., and Häkkinen, J., Phys. Lett., B303, 350 (1993).
Ashery, D., and Lipkin, H. J., Phys. Lett., B469, 263 (1999).
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