Extraction of spin observables in baryon-baryon scattering, sensitive to gluon- and quark-exchange effects
Spin-dependent characteristics of baryon-baryon scattering are discussed, in which QCD dynamics is expected to display itself in most simply treatable regimes: 1). being dominated by semi-classical gluonic field without dynamical quarks; 2). as perturbative scattering. It is argued that peripheral s...
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irk-123456789-1101622017-01-01T03:02:23Z Extraction of spin observables in baryon-baryon scattering, sensitive to gluon- and quark-exchange effects Bondarenco, M.V. Ядерная физика и элементарные частицы Spin-dependent characteristics of baryon-baryon scattering are discussed, in which QCD dynamics is expected to display itself in most simply treatable regimes: 1). being dominated by semi-classical gluonic field without dynamical quarks; 2). as perturbative scattering. It is argued that peripheral semi-classical gluon exchange can make a major contribution to the pseudovector scattering amplitude, whereas one of the tensor amplitude components must be seeded by quark exchange, thus containing a hard scale and allowing for perturbative treatment. Expressions for absolute values of the promising amplitudes are derived in terms of double spin asymmetries in NN-scattering. Options for realization of corresponding polarization measurements in strange hyperon collisions with nucleons are analysed, along with consistency tests supplied by additional spin amplitudes emerging in non-identical. Обговорюються спінові характеристики баріон-баріонного розсіяння, в яких очікуються найменш замасковані прояви КХД ступенів свободи: 1) у квазикласичному глюонному режимі без динамічних кварків; 2) в пертурбативному режимі. Показано, що квазикласичний глюонний обмін найбільш виразно проявляється у псевдовекторній амплітуді розсіяння, тоді як тензорні спінові амплітуди дозволяють виділити процеси обміну кварками, що привносять відносно жорсткий імпульсний масштаб. Запропоновано вирази для абсолютних величин амплітуд, що обговорюються, через двоспінові асиметрії в пружному NN-розсіянні. Обговорюються можливості проведення відповідних поляризаційних вимірів у зіткненнях дивних гіперонів з нуклонами, а також інформація, що надається додатковими амплітудами у цих процесах. Обсуждаются спиновые характеристики барион-барионного рассеяния, в которых ожидается наименее замаскированное проявление КХД степеней свободы: 1) в квазиклассическом глюонном режиме без динамических кварков; 2) в пертурбативном режиме. Показано, что квазиклассический глюонный обмен наиболее отчетливо проявляется в псевдовекторной амплитуде рассеяния, тогда как тензорные спиновые амплитуды позволяют выделить процессы обмена кварками, содержащие относительно жесткий импульсный масштаб. Предложены выражения для абсолютных величин обсуждаемых амплитуд через двуспиновые асимметрии в упругом NN-рассеянии. Обсуждаются варианты проведения соответствующих поляризационных измерений в столкновениях странных гиперонов с нуклонами, и информация, предоставляемая появляющимися в этих процессах дополнительными амплитудами. 2007 Article Extraction of spin observables in baryon-baryon scattering, sensitive to gluon- and quark-exchange effects / M.V. Bondarenco // Вопросы атомной науки и техники. — 2007. — № 5. — С. 31-39. — Бібліогр.: 32 назв. — англ. 1562-6016 PACS: 11.80.Cr, 13.75.Cs, 13.85.Dz, 13.88.+e http://dspace.nbuv.gov.ua/handle/123456789/110162 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы Bondarenco, M.V. Extraction of spin observables in baryon-baryon scattering, sensitive to gluon- and quark-exchange effects Вопросы атомной науки и техники |
| description |
Spin-dependent characteristics of baryon-baryon scattering are discussed, in which QCD dynamics is expected to display itself in most simply treatable regimes: 1). being dominated by semi-classical gluonic field without dynamical quarks; 2). as perturbative scattering. It is argued that peripheral semi-classical gluon exchange can make a major contribution to the pseudovector scattering amplitude, whereas one of the tensor amplitude components must be seeded by quark exchange, thus containing a hard scale and allowing for perturbative treatment. Expressions for absolute values of the promising amplitudes are derived in terms of double spin asymmetries in NN-scattering. Options for realization of corresponding polarization measurements in strange hyperon collisions with nucleons are analysed, along with consistency tests supplied by additional spin amplitudes emerging in non-identical. |
| format |
Article |
| author |
Bondarenco, M.V. |
| author_facet |
Bondarenco, M.V. |
| author_sort |
Bondarenco, M.V. |
| title |
Extraction of spin observables in baryon-baryon scattering, sensitive to gluon- and quark-exchange effects |
| title_short |
Extraction of spin observables in baryon-baryon scattering, sensitive to gluon- and quark-exchange effects |
| title_full |
Extraction of spin observables in baryon-baryon scattering, sensitive to gluon- and quark-exchange effects |
| title_fullStr |
Extraction of spin observables in baryon-baryon scattering, sensitive to gluon- and quark-exchange effects |
| title_full_unstemmed |
Extraction of spin observables in baryon-baryon scattering, sensitive to gluon- and quark-exchange effects |
| title_sort |
extraction of spin observables in baryon-baryon scattering, sensitive to gluon- and quark-exchange effects |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2007 |
| topic_facet |
Ядерная физика и элементарные частицы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/110162 |
| citation_txt |
Extraction of spin observables in baryon-baryon scattering, sensitive to gluon- and quark-exchange effects / M.V. Bondarenco // Вопросы атомной науки и техники. — 2007. — № 5. — С. 31-39. — Бібліогр.: 32 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT bondarencomv extractionofspinobservablesinbaryonbaryonscatteringsensitivetogluonandquarkexchangeeffects |
| first_indexed |
2025-07-08T00:11:25Z |
| last_indexed |
2025-07-08T00:11:25Z |
| _version_ |
1837035434462412800 |
| fulltext |
EXTRACTION OF SPIN OBSERVABLES IN
BARYON-BARYON SCATTERING, SENSITIVE TO GLUON-
AND QUARK-EXCHANGE EFFECTS
M.V. Bondarenco∗
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received May 11, 2007)
Spin-dependent characteristics of baryon-baryon scattering are discussed, in which QCD dynamics is expected to
display itself in most simply treatable regimes: 1) being dominated by semi-classical gluonic field without dynamical
quarks; 2) as perturbative scattering. It is argued that peripheral semi-classical gluon exchange can make a major
contribution to the pseudovector scattering amplitude, whereas one of the tensor amplitude components must be
seeded by quark exchange, thus containing a hard scale and allowing for perturbative treatment. Expressions for
absolute values of the promising amplitudes are derived in terms of double spin asymmetries in NN-scattering. Options
for realization of corresponding polarization measurements in strange hyperon collisions with nucleons are analysed,
along with consistency tests supplied by additional spin amplitudes emerging in non-identical particle collisions.
PACS: 11.80.Cr, 13.75.Cs, 13.85.Dz, 13.88.+e
1. INTRODUCTION
Accepting QCD as a fundamental theory of strong
interactions, the ultimate goal is to obtain in terms
of quark and gluon fields a spatial picture of all
known hadrons, and find their dynamical suscepti-
bilities (polarizabilities and instabilities to finite per-
turbations), that could serve as a ground for consis-
tent calculations of the variety of hadronic processes.
Systems, which may be simpler to try building such
a detailed picture for, are mesons, especially those
which contain heavy quarks (quarkonia). However,
because quarkonium beams are not experimentally
available (their ranges are typically of order millime-
ters), those particles can only be studied by their de-
cays immediately after production. At that, in ra-
diative and leptonic decays, supplying the cleanest
and most direct information on quark distributions,
that can be obtained only on the meson own mass
scale, around a few GeV . Meanwhile, most detailed
and diverse data relate to baryons - lepton scatter-
ing experiments provide knowledge of single quark
distributions, and the subsequent solution of evolu-
tion equations offers incoherent gluon distributions
at Bjørken x not too close to 1 and 0. But in order
to probe spatial and temporal correlations between
quarks and gluons in a nucleon, it is mandatory next
to engage data on hadron-hadron scattering, at dif-
ferent momentum transfer and energy scales.
Qualitatively, the origin of empirical properties
of baryon-baryon interactions allows for interpreta-
tion in terms of quark and gluon exchanges. Sug-
gestions were made that NN-attraction can be de-
scribed in terms of quark exchanges, similarly to
hydrogen bonding in molecules due to electron ex-
change [1]. The repulsive core of NN-interaction of
a few hundreds MeV in magnitude can be explained
by effects of Fermi-degeneracy between constituent
quarks. But in numerical accuracy, the quark model
description of NN-scattering lags far behind the best-
fit meson exchange models. The cause for greater suc-
cess of meson models may be attributed to a weaker
interaction between pions under non-resonance con-
ditions, than that between quarks and gluons. That
pion interaction with nucleons is strong, makes no
further problem if one constructs a NN-potential1 as
a superposition of one-meson-exchange potentials [2],
with the account, where necessary, of resonances in
pion pairs, etc. [3] (see also [4]).
On the other hand, with the knowledge on the mi-
croscopic level that momentum transfer to an hadron
can well result from exchange of gluons and quarks,
despite that those particles do not exist in a free state,
and besides that the quark distribution in baryons
has a formidable spatial dimension, it becomes doubt-
ful that local meson field theories are in principle
capable of explaining all the properties of baryon-
baryon interaction. In fact, at bringing the meson
theory into correspondence with data, there exists a
number of controversies, which stand unresolved over
years.
1. In attempts to describe the spin-orbit part
of interaction between nucleons at low energies via
∗Corresponding author E-mail address: bon@kipt.kharkov.ua
1The ”potential”, generically, is non-local, because it depends not only on positions, but also on momenta, however, that
dependence is relatively small, and speaking of a ”potential” is common.
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2007, N5.
Series: Nuclear Physics Investigations (48), p.31-39.
31
exchange of known mesons, there has been no suc-
cess to explain the large magnitude of the analyzing
power Ay at q ≤ 70 MeV/c (Ay-puzzle; see recent re-
view [5]). The problem is that both π and 2π ex-
changes (including exchanges of ρ and ε viewed as
2π resonances) do not contribute so strongly to spin-
orbit interaction, whereas the interaction through ω-
exchange is too short-range and its coupling constant
is bound by data on ω production. The physical ori-
gin of the given discrepancy may be due to substantial
influence in the peripheral region of gluons, which are
vector quanta. On the other hand, at highest energies
(RHIC) inelastic pp cross-section makes up 10 fm2,
i. e. inelastic processes occur at impact parameters
as large as 2fm between proton centers. At that, re-
lying on the observed stable growth of cross-sections
with energy, one may expect their further increase
at least by a factor of 1.5 − 2. That would already
mean that inelastic processes occur intensely at in-
teraction of nucleon peripheral regions. Because me-
son exchange can not provide the non-vanishing value
of the cross-section at asymptotically high energies,
since its probability falls off as a power law (even
at vacuum quantum numbers - just due to existence
of internal momentum distribution of constituents),
such observations can give direct evidence of an es-
sential role played by the gluon component in the
region 1 fm < r < 2 fm.
2. The meson core of inter-nucleon potential is
not consistently defined. There is a discrepancy be-
tween the core width in the potential and that in the
measured πNN formfactors - it is embarrassing to see
that in meson potentials the core is about twice nar-
rower than quark distribution in the nucleon as mea-
sured by direct EM and weak probes. It should also
be recalled that the working assumption of a point-
like nucleon is perhaps not reliable even in the limit
Nc → ∞, when it is correct for mesons [6]. In gen-
eral, without making reference to the notion of con-
stituent quarks, it is hard to explain the very πNN
formfactors, steeply decreasing with the growth of
the momentum transfer, the empirical law of nucleon-
antinucleon pair suppression, etc. A detailed critique
of meson exchange models may be found in [7].
Thus, without the account for spatial QCD struc-
ture of nucleons, description of their interactions
meets felt difficulties both at large and at short dis-
tances, and moreover, in the relativistic domain. A
logical approach for improvement of the correspon-
dence with data might be to introduce QCD correc-
tions into already approved meson exchange models.
However, it is by now far from clear, how to build
hybrid meson-chromodynamic models and in so do-
ing to avoid double counting. Although some such
attempts were made, they did not manage to refine
general correspondence with data [8].
Perhaps, a more consistent, and yet not despair-
ingly hardly realizable approach would be to apply
pure chromodynamics, but only in those kinematic
domains and for those variables, where meson ex-
change models do not offer an appropriate descrip-
tion. At that, one should look for maximal sim-
plification of QCD equations under favorable kine-
matic conditions. Such conditions are the soft do-
main, where it is legitimate to use semi-classical
Yang-Mills equations without dynamical quarks, or
the hard domain, in which perturbation theory is
applicable. It is natural to expect, that gluon ex-
change in the peripheral domain has the best chance
to be semi-classical, whereas exchange by constituent
quarks between baryons should introduce a relatively
hard scale into the process, and so gives a chance for
perturbative description.
The first objective, thus, is to extract a scatter-
ing amplitude, which is dominated by interaction in
the peripheral domain and does not contain contri-
butions from quark exchange. As we had noted in
general, the peripheral pion exchange contributes to
all possible forms of spin interaction but to spin-orbit.
That means that one should consider a pseudovector
scattering amplitude, which is most closely related to
spin-orbit interaction. Procedure of extraction of the
absolute value of this amplitude from scattering data
will be described in Sec. 2.
Secondly, it is of importance to find amplitudes of
processes, which necessarily require quark exchange,
and thereby are not obscured by soft gluon exchange
effects. Such amplitudes then might allow for per-
turbative treatment, though, possibly, with essen-
tial vertex renormalization, account for a spatially-
inhomogeneous, dynamical condensate, etc. Note,
that in general it is desirable for the process to in-
volve a minimal number of degrees of freedom. In
the case of hadron scattering, that means involving
less quarks into violent dynamics, in order for the
color confinement mechanism we poorly understand
not to fully come to action. These conditions seem
to be best satisfied by the nucleon charge exchange
process, which will be discussed in Sec.3.
In Sec.4 we touch upon advantages gained for the
named purposes with measurement of nucleon colli-
sions with strange hyperons and discuss spin ampli-
tudes which additionally arise in that case.
For simplicity of the presentation, in capacity of
transition matrices between on-mass-shell spin states
we will use Pauli matrices, though applications will
be assumed in the relativistic domain. For the lat-
ter case the matrices may be understood as in [9], or
it just could be noted, that upon transition to the
4-component covariant formalism [10], all relations
remain qualitatively unchanged.
2. EXTRACTION OF GLUON
EXCHANGE EFFECTS IN PERIPHERAL
NN SCATTERING
Consider the matrix amplitude of two dynami-
cally identical (within the unitary symmetry group)
baryons of spin 1/2,
1 + 2 → 3 + 4.
32
In the simplest experimental setting, such baryons
are nucleons. At description of the process in the
c.m.s. (p1 + p2 = p3 + p4 = 0), with the use of the
orthogonal basis
q = p3 − p1, r = p3 + p1 = −p4 − p2,N = [qr],
parametrization of the matrix amplitude reads [11]
M = S31,42I31I42 + AN
31,42
1
2
(
σN
31I42 + I31σ
N
42
)
+BNN
31,42σ
N
31σ
N
42 + Bqq
31,42σ
q
31σ
q
42 + Brr
31,42σ
r
31σ
r
42. (1)
Here σα
31 are Pauli matrices, operating from the spin
space of particle 1 to the spin space of particle 3,
and σβ
42 is the same for 2 → 4. The isospin indices
are suppressed. The amplitude normalization will be
supposed such that the differential cross-section aver-
aged over initial and summed over final polarizations
of particles expresses as
〈
dσ
dt
〉
=
1
2
Sp
1
2
SpMM†
= |S31,42|2 +
1
2
∣∣AN
31,42
∣∣2
+
∣∣BNN
31,42
∣∣2 +
∣∣Bqq
31,42
∣∣2 +
∣∣Brr
31,42
∣∣2 .
(2)
The representation (1) may be called covariant, in
contrast to, e. g., treatment of the amplitude in a
fixed spin basis. It is the covariant representation for
the amplitude, which proves convenient for extraction
of exchanges with specific quantum numbers.
In order to separate in the given scattering process
effects, which might reasonably be regarded as domi-
nated by gluon exchange, let us approach the nucleon
from large distances and consider the gradual actu-
ation of contributions to amplitude (1) from known
mesons. The longest range is ascribed to one-pion ex-
change, which contributes mainly to Bqq
31,42 and, upon
iteration, to S31,42 [12]. At a closer approach to the
nucleon, correlated 2π exchanges become noticeable,
which in the simplest treatment [13], yield
AN
31,42 ≈ 0,
Bqq
31,42 ≈ 0,
S31,42 ≈ BNN
31,42 ≈ Brr
31,42 6= 0.
In other versions of 2π-exchange calculations (see
their comparison in [14]) departures from that sce-
nario are possible, but at any rate, as translated to
potentials, the pion-theoretic spin-orbit interaction
may be regarded as vanishing at r > 1 fm (whereas
the scalar and tensor potentials retain considerable
magnitude down to as far as r ≈ 1.5− 2 fm). Thus,
if the gluon cloud does extend to distances ∼ 2 fm,
its effect will be most unambiguously discernible on
the pion cloud background in the pseudovector am-
plitude2 AN
31,42.
Separating the scattering at impact parameters
1 fm < r < 2 fm from the region r < 1fm would
be sufficiently feasible in case if the scattering pro-
ceeded as a semi-classical deflection in a strong po-
tential field. However, NN-interaction in the region
1 − 2 fm, where the nuclear potential has values of
the order 20 MeV , may be regarded as strong only
at energies lower than that scale, whereas for spa-
tial resolution of 1 fm distances one needs momenta
> 150 MeV/c, i. e. nucleon energies > 10 MeV .
So, it appears like there is no way to avoid quan-
tum effects and intermixture of central and peripheral
region contributions. Nonetheless, when considering
diffractive scattering at E > 1 GeV , the contribution
to the spatial profile of elastic scattering from the
impact parameter region b < 1 fm may prove to be
rather small due to high inelasticity (for elastic am-
plitude equivalent to opaqueness). Also, specifically
in the case of the amplitude AN
31,42, with the account
for quark structure of the nucleon, it may happen
that the spin-orbit interaction in the region between
quarks in average is small. That conjecture finds con-
firmation, for instance, in the Isgur-Karl constituent
quark model [15], [16], in which the observed spec-
trum of P-baryons is nicely reproduced with the ne-
glect of LS-interaction between quarks.
The observable characteristic of AN
31,42 in the case
of NN → NN is usually thought to be single-spin
asymmetry 3
PN =
dσ↑ − dσ↓
dσ↑ + dσ↓
=
1
2Sp 1
2SpMσN
11M
†
1
2Sp 1
2SpMM† , (3)
where the spin projection direction is supposed to be
N. However, through the spin amplitudes present in
(1) the asymmetry PN expresses as4
PN =
〈
dσ
dt
〉−1
Re
(
S31,42 + BNN
31,42
)
AN∗
31,42,
and despite being proportional to AN
31,42, it also con-
tains other amplitudes. Hence, this asymmetry quan-
titatively tells little about AN
31,42 [17].
A non-trivial task, therefore, is to measure AN
31,42
separately from other spin amplitudes. In fact, one
may expect only to determine
∣∣AN
31,42
∣∣2 and the com-
plex phase of AN
31,42 relative to other amplitudes. The
next grade in measurement complexity is double-spin
asymmetries. Those are
Cβα =
〈
dσ
dt
〉−1 1
2
Sp
1
2
SpMσβ
22σ
α
11M
†,
Dβα =
〈
dσ
dt
〉−1 1
2
Sp
1
2
Spσβ
33Mσα
11M
†,
2Occasionally, we will refer to AN
31,42 as to pseudovector amplitude, though it is understood in fact to be just the only
component of the pseudovector amplitude, which is not bound to be zero due to some symmetry reasons or other.
3The quantity PN also (by Madison convention [18]) is denoted as Ay; thereunder it was mentioned in the Introduction.
4For the Hermitean conjugate matrix M† the notation concordant with (1) is
M = S∗31,42I13I24 + AN∗
31,42
1
2
(
σN
13I24 + I13σN
24
)
+ BNN∗
31,42σN
13σN
24 + Bqq∗
31,42σq
13σq
24 + Brr∗
31,42σr
13σr
24,
with particle order in matrix subscripts inverted.
33
Kβα =
〈
dσ
dt
〉−1 1
2
Sp
1
2
Spσβ
33Mσα
22M
†,
spin correlation parameters (otherwise denoted as5
Aβα); depolarization and spin rotation parameters,
and polarization transfer parameters. In the basis
(1), non-zero are 9 diagonal asymmetries
CNN
〈
dσ
dt
〉
=2Re
(
S31,42B
NN∗
31,42
)
+
1
2
∣∣AN
31,42
∣∣2 − 2Re
(
Bqq
31,42B
rr∗
31,42
)
,
Cqq
〈
dσ
dt
〉
=2Re
(
S31,42B
qq∗
31,42
)− 2Re
(
BNN
31,42B
rr∗
31,42
)
,
Crr
〈
dσ
dt
〉
=2Re
(
S31,42B
rr∗
31,42
)− 2Re
(
BNN
31,42B
qq∗
31,42
)
,
DNN
〈
dσ
dt
〉
=|S31,42|2 +
1
2
∣∣AN
31,42
∣∣2 +
∣∣BNN
31,42
∣∣2 −
∣∣Bqq
31,42
∣∣2 −
∣∣Brr
31,42
∣∣2 ,
Dqq
〈
dσ
dt
〉
=|S31,42|2 −
∣∣BNN
31,42
∣∣2 +
∣∣Bqq
31,42
∣∣2 −
∣∣Brr
31,42
∣∣2 ,
Drr
〈
dσ
dt
〉
=|S31,42|2 −
∣∣BNN
31,42
∣∣2 − ∣∣Bqq
31,42
∣∣2 +
∣∣Brr
31,42
∣∣2 ,
KNN
〈
dσ
dt
〉
=2Re
(
S31,42B
NN∗
31,42
)
+
1
2
∣∣AN
31,42
∣∣2 + 2Re
(
Bqq
31,42B
rr∗
31,42
)
,
Kqq
〈
dσ
dt
〉
=2Re
(
S31,42B
qq∗
31,42
)
+ 2Re
(
BNN
31,42B
rr∗
31,42
)
,
Krr
〈
dσ
dt
〉
=2Re
(
S31,42B
rr∗
31,42
)
+ 2Re
(
BNN
31,42B
qq∗
31,42
)
,
(4)
and 3 antisymmetric non-diagonal asymmetries
Dqr〈dσ
dt
〉 = −Drq〈dσ
dt
〉 = Im
[
AN
31,42(S
∗
31,42 −BNN∗
31,42)
]
εqrN ,
Cqr〈dσ
dt
〉 = −Crq〈dσ
dt
〉 = Im
[
AN
31,42(B
qq∗
31,42 −Brr∗
31,42)
]
εqrN ,
Kqr〈dσ
dt
〉 = −Krq〈dσ
dt
〉 = Im
[
AN
31,42(B
qq∗
31,42 + Brr∗
31,42)
]
εqrN .
(5)
The 9 relations (4) may be regarded as a complete
system of equations for determination of 5 complex
numbers S31,42, AN
31,42, BNN
31,42, Bqq
31,42, Brr
31,42 up to a
common for them phase factor. This nonlinear sys-
tem of equations can be explicitly solved. Firstly,
through the measured asymmetries it is possible to
express 3 quantities
4
∣∣Bqq
31,42
∣∣2 =
〈
dσ
dt
〉
(1−DNN + Dqq −Drr) , (6)
4
∣∣Brr
31,42
∣∣2 =
〈
dσ
dt
〉
(1−DNN −Dqq + Drr) , (7)
4Re
(
Bqq
31,42B
rr∗
31,42
)
=
〈
dσ
dt
〉
(KNN − CNN ) , (8)
related to the pair of complex amplitudes Bqq
31,42 ,
Brr
31,42. Those amplitudes may be used as a basis
set in the 2d Euclidean space, to which the com-
plex plane turns upon adoption in it a scalar product
(a, b) = Re (a∗b) = ReaReb + ImaImb. Then, quan-
tities (6-8) serve as Gram matrix elements in the ba-
sis formed by Bqq
31,42 and Brr
31,42. Next, projections of
S31,42 on Bqq
31,42 and Brr
31,42 can be expressed as
4Re
(
S31,42B
qq∗
31,42
)
=
〈
dσ
dt
〉
(Kqq + Cqq) ,
4Re
(
S31,42B
rr∗
31,42
)
=
〈
dσ
dt
〉
(Krr + Crr) ,
and thereupon |S31,42|2 is reconstructed by formula
|S31,42|2 =
{[
Re
(
S31,42B
qq∗
31,42
)]2 ∣∣Brr
31,42
∣∣2 +
[
Re
(
S31,42B
rr∗
31,42
)]2 ∣∣Bqq
31,42
∣∣2
−2Re
(
S31,42B
rr∗
31,42
)
Re
(
S31,42B
qq∗
31,42
)
Re
(
Bqq
31,42B
rr∗
31,42
)}
(9)
·
{∣∣Brr
31,42
∣∣2 ∣∣Bqq
31,42
∣∣2 − [
Re
(
Bqq
31,42B
rr∗
31,42
)]2}−1
.
5To be precise, the initial state parameters are denoted as Aβα , and final state ones - as Cβα. However, under T-invariance
conditions those quantities coincide, so we shall use a more convenient notation Cβα.
34
Finally,
∣∣AN
31,42
∣∣2 can be expressed from the equation
4 |S31,42|2 +
∣∣AN
31,42
∣∣2 =
〈
dσ
dt
〉
(1 + DNN + Dqq + Drr)
by substitution of (9 ). As a result, we arrive to the expression
∣∣AN
31,42
∣∣2
〈
dσ
dt
〉−1
= 1 + DNN + Dqq + Drr
−
{
(Kqq + Cqq)
2 (1−DNN −Dqq + Drr) + (Krr + Crr)
2 (1−DNN + Dqq −Drr)
−2 (Krr + Crr) (Kqq + Cqq) (KNN − CNN )}
×
{
(1−DNN )2 − (Dqq −Drr)
2 − (KNN − CNN )2
}−1
. (10)
This expression employs all 9 diagonal double-
spin asymmetries and the non-polarized differential
cross-section. Note that at small q the r direction
is equivalent to commonly used in experiments di-
rection l of initial particle collision, or final particle
emergence, and direction q is equivalent to the side-
ways direction s in the scattering plane, with respect
to momentum of either of the particles.
Having reconstructed the amplitude AN
31,42 (q) ab-
solute value in the diffractive domain q < 700 MeV/c,
one can subsequently try to estimate its absolute
phase as well, exploiting analyticity properties and
the behavior of single-spin asymmetry (3). If suc-
cessful, it would be worth further to make transfor-
mation from momentum transfers to impact param-
eters, in order to check the conjecture that there is
a suppression in the central region. Should it be the
case, and the profile AN
31,42 (b) of amplitude AN
31,42 (q)
is built up in the region 1 fm < b < 2 fm, then cal-
culation of amplitude AN
31,42 behavior on the basis of
gluon exchange without that of quarks may be sen-
sible. Once more it should be emphasized that for
applicability of the impact parameter representation
the collision must proceed with velocity greater then
those of mechanical oscillations and color circulation
in the nucleon. To this end, perhaps, it suffices to
have E > 1 GeV .
3. AMPLITUDES DETERMINED BY
QUARK EXCHANGE
Turning to the problem of separation of amplitudes
governed by quark exchange, the first point that
needs to be clarified in this business is which quarks
had actually participated in exchange. Baryon beams
available in experiments may contain only quarks of
flavors u, d and s, so in colliding baryons some identi-
cal quarks are always present. For a flavor exchange
reaction, which by itself requires exchange of differ-
ent quarks, it remains to secure absence of additional
exchange of identical quarks. For elastic scattering,
amplitudes are needed which may differ from zero
only if exchange of identical quarks had occurred (the
identity of exchanged quarks here is required by the
elasticity).
In the first case the most practical variant is to
consider nucleon charge exchange np → pn reaction
in the forward direction at high energy. The non-
polarized differential cross-section for this process in
the vicinity of 0◦ direction (that is 180◦ for elastic
scattering np → np) features a peak fittable by a
sum of two Gaussians [19, 20],
〈
dσ
dt
〉
≈ const
s2
(
e−50|t|/GeV 2
+ 0.8e−4.5|t|/GeV 2
)
.
Note here energy dependence dσ/dt ∝ 1/s2, whereas
at other fixed angles the differential cross-section falls
off as dσ/dt ∝ 1/s10.
Historically, the observation that the width of the
first peak is GeV 2/50 ∼ m2
π inspired modeling of this
process in terms of one-pion exchange. At practice,
however, OPE gives in the backward direction zero
instead of a peak, so a pion plus pomeron exchange
has rather to be considered, to produce a cut with
special interference conditions [21]. (And still, as [22]
notes, approach of [21] fails to reproduce behavior of
all polarized observables).
From the standpoint of QCD, one may imagine
the mechanism of the peak formation, in which a
quark from the first nucleon upon knocking a quark
in the second nucleon substitutes it in the very same
state (except flavor), whereas the knocked quark, vice
versa, is sent into the freed position of the quark in
the first nucleon. Since no color neutralization is re-
quired thereat, no qq̄ pair needs to be created, so,
most of the probability can be retained within the
2-hadron channel. The differential cross-section of
two relativistic quark scattering to 180◦ through one-
gluon exchange is proportional to 1/s2, thus no im-
mediate contradiction with the experiment emerges.
Manifestation of some other, harder scale may be
attributed to collective effects between constituent
quarks. As a whole, the given process should be sen-
sitive to quark wavefunction in a nucleon.
In the case of elastic pp scattering the only am-
plitude, which is obliged to be zero in the event of
exchange exceptionally by gluons, is Bqq
31,42. Indeed,
if a diagram of two-fermion scattering can be cut only
through lines of vector quanta, T-invariance forbids
35
appearance in the total amplitude associated with
each half of the diagram of matrices σq. To get as-
sured that the difference of Bqq
31,42 from zero is caused
by exchange of only one quark pair, it is sufficient,
again, to consider scattering at high energy to small
angles.
It would be interesting to compare the ampli-
tude Bqq
31,42 in elastic scattering pp → pp, the am-
plitude Bqq
31,42 in scattering np → np, and Bqq
31,42 in
the reaction np → pn. Naive quark counting shows
that in pp → pp scattering there are 5 possibilities
of exchange by a pair of identical quarks, whereas
in pn → pn processes such possibilities are 4, and
in np → pn there are 4 possibilities of exchange
d(n) ↔ u(p). One has thus two points to check:
1). whether forward
∣∣Bqq
31,42 (θ)
∣∣2 for pp → pp and
pn → pn are indeed in the ratio 5:4, and 2). does
forward Bqq
31,42 for pp → pp and pn → pn, if indeed
governed by quark exchange, develop a small peak
of width ∼ m2
π, similarly to what is known for the
np → pn process?
Realization of the suggested program is hindered
by the circumstance that, as formula (6)6
∣∣Bqq
31,42
∣∣2 =
1
4
〈
dσ
dt
〉
(1−DNN + Dqq −Drr)
≡ 1
4
〈
dσ
dt
〉
(1−DNN + Dss −Dll)
(11)
indicates, for determination of
∣∣Bqq
31,42
∣∣2, final nucleon
polarization measurements are required. By now,
such data at small angles remain too scarce (in con-
trast to differential cross-sections, analyzing powers
[23] and initial-state double-spin correlations). Cus-
tomary complications of re-scattering measurements
after scattering to small angles are aggravated by the
fact that as θ → 0, each of the three parameters DNN
, Dss , Dll gets within 10% from 1, and it is this devia-
tion which needs to be measured. The highest energy
where all D parameters are available, at small scat-
tering angles, is 0.8 GeV (s − 4m2 = 1.5 (GeV/c)2).
Combining data [24] for unpolarized pp → pp dif-
ferential cross-sections with data [25], [26] for D pa-
rameters, we can reconstruct
∣∣Bqq
31,42
∣∣2. Data [27],
[26] allow to do the same for pn → pn. Finally, for
np → pn quantity
∣∣Bqq
31,42
∣∣2 can also be obtained from
data [28], [29], [30], if in 11 Dβα are substituted by the
spin-transfer parameters Kβα. The results together
are shown in Fig. 1.
The obtained dependences make a two-fold im-
pression. On the one hand, values for pp → pp and
pn → pn are uniformly small, as it could be expected
based on their origin from quark scattering to large
angles. The ratio for processes pp → pp and pn → pn
is close to 5/4, though under significant errors (es-
pecially in processes involving neutrons) and small
number of points in the considered domain of small
angles, it is impossible to claim a definite correspon-
dence.
0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07
0
5
10
15
20
25
pp->pp [Barlett et al. 1984, 1985]
pn->pn [Barlett et al. 1985]
np->pn [McNaughton et al. 1991, 1993]
|B
31
,4
2qq
|2 ,m
b/
(G
eV
/c
)2
-t, (GeV/c)2
Fig.1. Behavior of
∣∣Bqq
31,42
∣∣2 for pp and pn elastic
scattering, and for np → pn, at 0.8 GeV
On the other hand, in behavior of
∣∣Bqq
31,42
∣∣2 no
peak is seen on the scale |t| ∼ 0.01(GeV/c)2, although
for np → pn such a peak is present, and even quite
large, reaching one third from the unpolarized cross-
section, though it in principle can not be larger then
its half, since BNN
31,41 ≈
θ→0
Bqq
31,41 must make an equal
contribution.
The cause for the strong difference in behavior of∣∣Bqq
31,42
∣∣2 for pp → pp and np → pn about θ → 0
is not clear. Meanwhile, though with a small likeli-
hood, it can not be excluded that the leftmost point
for pp → pp is inaccurate, e. g., due to a fallacy
brought in by phase shift analysis predictions. In up-
dated data [26], as compared to [25], that point is not
present.
4. ADDITIONAL AMPLITUDES IN
HYPERON-NUCLEON COLLISIONS
For measurement of baryon polarization in the fi-
nal state a valuable advantage may come from ob-
servations of nucleon collisions with strange baryons
(Y N → Y N), due to the fact that hyperon final po-
larizations are detectable by kinematics of their non-
leptonic weak decays7 Λ → Nπ, Σ+ → Nπ. The
lifetime for hyperons is ∼ 10−10s, so their range at
E ≥ 1GeV exceeds 4cm, which is sufficient for con-
duction of scattering experiments, though small an-
gle measurements are difficult. Presently, differen-
tial cross-sections only to angles θ > 20◦ were mea-
sured. At high energies, with the use of gas tar-
gets, there is potential for improving the situation
[31]. Should measurements of hyperon diffractive
scattering become practical, determination of quan-
tity
∣∣Bqq
31,42
∣∣2 and its comparison with the counter-
6Passage from the first line of (11), which is formula (6) to the second line, which implements directly measured quantities
Dss, Dll, related to initial and final particle momenta directions, instead of a universal frame, proves to be exact, not just
approximate.
7Transverse polarization of hyperons is easily measurable by the azimuthal asymmetry of pion emergence. The longitudinal
component of polarization is harder to detect, but with the use of Dalitz diagrams, is also possible.
36
part in pp → pp will be of primary interest. Besides
that, Y N -scattering is interesting by additional am-
plitudes, which emerge in it due to hypercharge sym-
metry violation. Those amplitudes will be discussed
hereafter.
In the process Y N → Y N such an additional am-
plitude comes as
A
[N ]
31,42
1
2
(
σN
31I42 − I31σ
N
42
)
. (12)
This amplitude may be regarded as characteristic of
spin-orbit dependence on quark mass. Analogously to
what has been argued about AN
31,42, amplitude A
[N ]
31,42
as well may be subject to essential cancelations in the
central region.
Representative of the order of magnitude of am-
plitude (12) as compared to AN
31,42 is the ratio
2Re
[(
S31,42 −BNN
31,42
)
A
[N ]∗
31,42
]
2Re
[(
S31,42 −BNN
31,42
)
AN∗
31,42
]
=
1
2Sp 1
2SpMσN
11M
† − 1
2Sp 1
2SpMσN
22M
†
1
2Sp 1
2SpMσN
11M
† + 1
2Sp 1
2SpMσN
22M
† .
(13)
This ratio is convenient for its independence on the
non-polarized cross-section magnitude, which is usu-
ally the source of significant experimental errors.
Since we expect A
[N ]
31,42 to be small, it does not es-
sentially influence formulas (6), (10). Reproduction
of the correct order of magnitude of the ratio (13)
may serve as a test of correctness of the spin-orbit
interaction calculation on the basis of quark models.
Exchange processes can also be studied in Y N
collisions8, even without measurement of the hyperon
polarization. It is natural to expect that in reaction
Σ+p → pΣ+, just like in np → pn, a forward peak
must develop. However, between those two cases two
differences are to be minded. First of all, in the pro-
cess Σ+p → pΣ+ only s quark from Σ+-hyperon and
d-quark from proton may be engaged into exchange,
so there are no Fermi-degeneracy effects in this case,
and hence a harder scale in the peak shape may dis-
appear. Secondly, if a baryon loses s-quark, and a
d-quark comes in its place, the substitution is not ex-
act, so there must occur some rearrangement of the
wave function. Thus, examination of the difference
between peaks in reactions np → pn and Σ+p → pΣ+
can provide comparison between wave functions of
nucleons and hyperons.
Further it may be noted that absolute normal-
ization of cross-sections, necessary for the processes
np → pn and Σ+p → pΣ+ poses a formidable exper-
imental problem, but that can be avoided, if for the
same collisions, for example, Λp, different channels of
charge exchange are simultaneously observed, such as
Λp → Σ+n and Λp → nΣ+. The first channel here
must be similar to np → pn, whereas the second -
to Y N → NY . Therefore, in the forward direction
peaks in Σ+ and n particle distributions, originating
form Λp collisions, should exhibit difference.
For processes of the mentioned type, when all
4 initial and final particles are different, it should
be emphasized that the matrix amplitude represents
sum of (1) and (12) only if in particle numeration
one tracks strangeness, i. e. particles 1 and 3 both
are hyperons. Dynamically, though, it may be conve-
nient rather to track two quarks of the three. In the
latter case, in the place of (12) there must stand an
amplitude of the form9
B
[qr]
31,42
1
2
(σq
31σ
r
42 − σr
31σ
q
42) , (14)
as is obvious from the spin crossing relation
iA
[N ]
31,42 = εNqrB
[qr]
41,42,
derived from Fierz identities for the SU(2) ⊗ SU(2)
group. The corresponding difference of single-spin
asymmetries reads as
2Im
[(
Bqq
31,42 + Brr
31,42
)
B
[qr]∗
41,42
]
εNrq
=
1
2
Sp
1
2
SpMσN
11M
† − 1
2
Sp
1
2
SpMσN
22M
†,
(
A
[N ]
31,42 = 0
)
.
As a relative measure for amplitude B
[qr]
41,42 a ratio
2Im
[(
Bqq
31,42 + Brr
31,42
)
B
[qr]∗
41,42
]
εNrq
2
(∣∣Bqq
31,42
∣∣2 − ∣∣Brr
31,42
∣∣2
)
=
1
2Sp 1
2SpMσN
11M
† − 1
2Sp 1
2SpMσN
22M
†
1
2Sp 1
2Spσq
33Mσq
11M
† − 1
2Sp 1
2Spσr
33Mσr
11M
† .
(15)
can be considered. If we are poised to describe basic,
diagonal tensor spin amplitudes based on pQCD and
baryon wave functions, the framework should be able
to reproduce the correct order of magnitude of the
non-diagonal amplitude B
[qr]
41,42 as well.
5. SUMMARY
In this paper it was argued that when issuing from
covariant representation (1) for the scattering am-
plitude, the amplitude AN
31,42 (q) at relativistic en-
ergies and small scattering angles is dominated by
gluon exchange contributions, and, presumably, from
the peripheral region. The amplitude Bqq
31,42 (q), also
at relativistic energies and small scattering angles,
is seeded by quark exchange processes. As was dis-
cussed, those two amplitudes have best chances to be
treatable by simplest approximations of QCD.
8Experiments of that kind are to be conducted at colliding beams, in order for hyperons after head-on collisions not to stop
and decay too close to the scattering point.
9If the isospin symmetry violation is taken into account, too, as for example it should be in the region of Coulomb-nuclear
interference, then both amplitudes . . . and . . . must be present, so in total there are 7 amplitudes, not 6.
37
For determination of absolute values of the men-
tioned complex amplitudes, expressions were derived,
which contain the experimentally observable double-
spin asymmetries of NN scattering process. At that,
necessary ingredients are the depolarization and spin
rotation parameters DNN , Dqq, Drr, which require
measurement of polarization of one of the final nucle-
ons. Currently, in connection with the development
of nuclear polarized beam and target techniques, the
main emphasis in experiments is laid on measure-
ments double-spin correlations between initial par-
ticles. Those correlation show behavior, which still
receives no satisfactory explanation in field-theoretic
terms [32], and that may be the reason why more la-
borious measurements of final polarizations are not
considered as urgent. Let us stress, however, that of
particular value would be measurements which might
be directly converted to indications concerning profile
shapes or interaction amplitudes. Such unambiguous
indications can be gained only upon the knowledge of
spin correlations between initial and final particles.
Meanwhile, in the absence of sufficient set of po-
larization data, investigations of quark exchange pro-
cesses can be carried out using non-polarized dif-
ferential cross-sections of charge exchange processes,
such as np → pn. There are good reasons to hope
that in processes of that kind the account for final
state quark interactions may be made with the use
of baryon wave functions, between which a perturba-
tive QCD matrix element of quark exchange is eval-
uated. At practice, however, the question remains
open, whether it is possible to use for this goal wave
functions of potential or of bag models, which are es-
sentially limiting cases corresponding to high and low
gluonic transparency within a nucleon.
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ВЫДЕЛЕНИЕ СПИНОВЫХ НАБЛЮДАЕМЫХ В БАРИОН-БАРИОННОМ
РАССЕЯНИИ, ЧУВСТВИТЕЛЬНЫХ К ЭФФЕКТАМ ОБМЕНА ГЛЮОНАМИ И
ОБМЕНА КВАРКАМИ
Н.В. Бондаренко
Обсуждаются спиновые характеристики барион-барионного рассеяния, в которых ожидается наи-
менее замаскированное проявление КХД степеней свободы: 1) в квазиклассическом глюонном режиме
без динамических кварков; 2) в пертурбативном режиме. Показано, что квазиклассический глюонный
обмен наиболее отчетливо проявляется в псевдовекторной амплитуде рассеяния, тогда как тензор-
ные спиновые амплитуды позволяют выделить процессы обмена кварками, содержащие относительно
жесткий импульсный масштаб. Предложены выражения для абсолютных величин обсуждаемых ам-
плитуд через двуспиновые асимметрии в упругом NN-рассеянии. Обсуждаются варианты проведения
соответствующих поляризационных измерений в столкновениях странных гиперонов с нуклонами, и
информация, предоставляемая появляющимися в этих процессах дополнительными амплитудами.
ВИДIЛЕННЯ СПIНОВИХ СПОСТЕРЕЖНИХ У БАРIОН-БАРIОННОМУ
РОЗСIЯННI, ЧУТЛИВИХ ДО ЕФЕКТIВ ОБМIНУ ГЛЮОНАМИ ТА КВАРКАМИ
М.В. Бондаренко
Обговорюються спiновi характеристики барiон-барiонного розсiяння, в яких очiкуються найменш
замаскованi прояви КХД ступенiв свободи: 1) у квазикласичному глюонному режимi без динамiчних
кваркiв; 2) в пертурбативному режимi. Показано, що квазикласичний глюонний обмiн найбiльш вираз-
но проявляється у псевдовекторнiй амплiтудi розсiяння, тодi як тензорнi спiновi амплiтуди дозволяють
видiлити процеси обмiну кварками, що привносять вiдносно жорсткий iмпульсний масштаб. Запропо-
новано вирази для абсолютних величин амплiтуд, що обговорюються, через двоспiновi асиметрiї в
пружному NN-розсiяннi. Обговорюються можливостi проведення вiдповiдних поляризацiйних вимiрiв
у зiткненнях дивних гiперонiв з нуклонами, а також iнформацiя, що надається додатковими амплiту-
дами у цих процесах.
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