Объяснение поляризационных данных в рамках модели эффективного цветового поля

Научная сессия-конференция секции ЯФ ОФН РАН «Физика фундаментальных взаимодействий» 21-25 ноября 2011, ИТЭФ

Объяснение поляризационных данных в рамках модели эффективного цветового поля

В.В.Абрамов

ГНЦ Институт физики высоких энергий

План доклада

Введение Происхождение поляризационных эффектов Глобальный анализ поляризационных данных Поляризация Λ ̃in pA соударениях (E766)A-зависимость поляризации Λ в e+A соударениях (HERMES) Зависимость AN от множественности для π+ в p↑p (BRAHMS) Осцилляция AN(xF) для p↑p(A) соударений (ФОДС и BRAHMS) Оценки масс и аномальных хромомагнитных моментов составлящих кварков из глобального анализа данных Заключение

Происхождение поляризационных эффектов

1) Генерация хромомагнитного и хромоэлектрического полей КХД струн после первоначальной цветовой перезарядки. 2) Микроскопический механизм Штерна-Герлаха в цветных полях КХД струн для генерации поляризационных явлений. 3) Прецессия спинов тестовых кварков наблюдаемой частицы в цветных полях, приводящая к осцилляции сил Штерна-Герлаха. 4) Правила кваркового счета для эффективных цветных полей (поля, создающиеся движущимися кварками и антикварками из налетающей частицы и из мишени являются линейными функциями их числа с соответствующими весами). 5) Круговое хромомагнитное поле (де)фокусирует тестовые кварки, что может приводить к зависимости от энергии √s резонансного типа для поляризационных наблюдаемых AN, PN.

Глобальный анализ данных

Глобальный анализ данных: AN, PN, ρ00 & α = (σ T – 2σ L)/(σT + 2σL).

Всего 86 инклюзивных и эксклюзивных реакций для hh, hA, AA & lN-взаимодействий, более 5500 экспериментальных точек. A↑ + B → C + X {Анализирующая способность для С, AN(pT, xF,√s) }.

A + B → C↑ + X {Поляризация для C, PN(pT, xF,√s) }.

В т.в. КХД односпиновые эффекты малы: AN SmQ/EQ 1%.

Наблюдаемые поляризационные эффекты много больше 1%.Модели: Sivers; Collins; Szwed; орбитальное движение кварков (Liang Zuo-tang, C. Boros, Трошин, Тюрин и др.); полу-классические механизмы (Anderson, De-Grand, Рыскин и др.). Многие из наблюдаемых явлений не находят своего объяснения в рамках существующих моделей.

Цветное поле между кварком и антикварком

Зависимость поля от расстояния r от оси струны:E(3)

Z = -2αsνA /ρ2 exp(-r2/ρ2), (1)

B(2)φ = -2αsνA r/ρ3 exp(-r2/ρ2), (2)

где νA – число кварков, ρ =1.25RC 2.08 ГэВ-1, RC

-1 0.6 ГэВ, RC – радиус конфайнмента, αs = gs2/4π 1.

Имеется продольное хромо-электрическое поле Ea и круговое хромомагнитное поле Ba.

μaQ = sgags/2MQ –

хромомагнитный момент составляющего кварка.

A.B.Migdal, S.B.Khohlachev, JETP Lett. 41, 194 (1985).

Also, Yu.Goncharov, Int.J.Theor.Phys.49, 1155 (2010).

Действие сил Штерна-Герлаха на кварк в цветном поле струн

Спектаторы - кварки, которые не являются составляющими C. Например, в pp→Ξ0+X тестовые s и u кварки из Ξ0 “измеряют”

поле, создаваемое кварками-спектаторами с весом νA= λ, антикварками с νA= 1, и кварками мишени νB= -τλ.

λ = − |ψqq(0)|2 /|ψq q ̃(0)|2 1-e1/8 -0.13315 цветной фактор (5) λ = −0.1332±0.0006, τ = 0.0534±0.0009 для 86 реакций.

fx ≈ μax ∂Ba

x/∂x + μay ∂Ba

y/∂x (3) fy ≈ μa

x ∂Bax/∂y + μa

y ∂Bay/∂y. (4)

СП

ЕК

ТАТОР

Ы

∫Ea ~ ∫Ba ~ [2 + 2λ - 3τ λ ]

Правила кваркового счета

Тестовый кварк Q из наблюдаемого адрона C “измеряет” ∫Ва & ∫Ea.

Эффективное цветное поле:

≡ C

Прецессия спина кварка в поле струны

dξ/dt ≈ a[ξ Ba] + d[ξ [Eav]] (BMT-уравнения) (6)

a = gs(gaQ – 2 + 2MQ/EQ)/2MQ (массы MU ≈ MD ≈ 0.3 GeV) (7)

d = gs[gaQ – 2EQ/(EQ+MQ)]/2MQ (EQ - Q энергия) (8)

ΔμaQ =(ga

Q-2)/2 (аномальный хромомагнитный момент кварка).

Спонтанное нарушение киральной симметрии приводит к дополнительной динамической массе кварка ΔMQ(q) и Δμa

Q(q).

В инстантонной модели: ΔμaQ (0) ≈ –0.2 (Н. Кочелев, 1998);

Δμa

Q (0) ≈ –0.744 (Д. Дьяконов, 2003).

ΔMQ(q), ΔμaQ(q) →0, если q =ρ0pT→∞. ρ0 = 0.0153±.0009

Глобальный анализ дает ΔμaQ (0) ≈ -0.4÷ -0.7 (Q -аромат).

Уравнения для AN, PN и ρ00-1/3

PN ≈ C(√s) F(pT, A)[G(φA) – σG(φB) ], (14)

G(A) = [1 – cos A]/A + εφA, прецессия спина и сила Ш-Г. (15)

C (√s) = v0/[(1 – ER/√s )2+δ2R]1/2, фокусировка кварков (16)

F(pT,A) = {1 – exp[-(pT/p0T)3 ]}(1 – α lnA). Цветной форм-фактор (17)

Всего 8 локальных параметров для каждой из реакций: D, α, σ, E0, ER, f0, a0, p0

T. V.Abramov, Phys. At. Nucl. 72 (2009) 1872. V.V. Abramov 2011 J. Phys.: Conf. Ser. 295 01208643 глобальных параметра для 86 реакций (ε, λ,τ, MQ, Δμa

Q …).

v0 = -D gaQ ξ0

y /2ρ(gaQ –2 ). Магнитуда AN, PN и ρ00-1/3 (18)

Поляризация Λ ̃в pp и pA-соударениях

Для поляризации Λ̃ в pp и pA соударениях большинство данных при высокой энергии, √s > 27 ГэВ и PN нулевая (синие точки). Значительная PN

в Е766 (J. Felix (1995), talk at ICTP, Trieste, Italy) √s = 7.31 ГэВ (красные точки). Большая PN объясняется эффектом фокусировки анти-кварков в круговом хромомагнитном поле с ER=7.2±1.1 ГэВ, δR=0.064

PN ~1/[(1 – ER/√s )2+δ2R]1/2 (19)

А-зависимость PN для Λ в e+A-соударениях

Поляризация Λ PN(A) в e+A соударениях измерена в экспери-менте HERMES. K.Rith, DIS2010.Эффективный вклад кварков, создающий цветное поле νA= 1+λ(3Aeff -2)–τ(λ+1),где λ≈-0.133, τ ≈.053, Aeff ≈0.6A1/3. Поле и PN ~νA уменьшаются с ростом A, и PN ≈ 0 при А ≈120. Красная кривая –предсказание.

e+ A →Λ e+ X , √s =7.26 ГэВ

Зависимость AN(xF) от множественности

AN(xF) для образования π+ в p↑ p соударений измерена в экспери-менте BRAHMS. J.H.Lee, DIS2009.Большее значение множественности

соответствует более сильному цветному полю, поскольку и то и другое пропорционально числу создающих его струн. Вклад qq̃ пар при высоких энергиях в создание поля, fN:

p↑ p → π+ X, √s =200 ГэВ

Rm =Multiplicity/Mean

fN ~ 1 + am(Rm-1), am=0.025±0.004 (24)

Осцилляции AN(xF) для p↑p(A) → p X

AN для p↑ p(A) → p Xизмерена в эксперименте ФОДС-2 в ИФВЭ (красные точки). V.V. Abramov et al. Phys. Atom. Nucl. 70: 1515, 2007. √s = 8.77 GeV. Большие pT и xF.Blue stars are the BRAHMS data for √s = 200 GeV. J.H.Lee, SPIN2006.Осцилляция AN(xF) вызвана прецессией спина кварка и осцилляцией силы типа Штерна-Герлаха в сильном цветном поле.

p↑ p(A) → p X

√ s

Глобальный анализ данных:оценки масс составляющих кварков

Динамические массы, при q = 0. Результаты глобального анализа: MU = 0.2375± 0.0009 ГэВ/с2 MU ≈ mP/4 MD = 0.3004 ± 0.0024 ГэВ/с2 MD ≈ 4/3 MU ≈ mP/3 MS = 0.5225 ± 0.0048 ГэВ/с2 MS ≈ MU + MD ≈ 7mP/12 MC = 1.415 ± 0.062 ГэВ/с2 MC ≈ 3mP/2 MB = 4.413 ± 0.340 ГэВ/с2 MB ≈ 3MC ≈ 9mP/2 Из форм-факторов заряженых пионов: MU ≈ MD ≈ 0.25 GeV/с2; A.F.Krutov, V.E.Troitsky, Eur. Phys. J. C20 (2001) 71. (JLAB data)

MQ = (2/3)1/2πFπ = 0.24 ГэВ/с2; С.Б.Герасимов, ЯФ 29(1979)513.

MU = 0.263 ГэВ/с2; M.Mekhfi, Phys.Rev. D72(2005)114014. 14

Глобальный анализ данных: аномальные хромомагнитные моменты

Аномальные хромомагнитные моменты Δμa =(ga -2)/2 при q=0:

ΔμaU(0) = -0.524 ± 0.004 Инстантонная модель:

ΔμaD(0) = -0.438 ± 0.005 Кочелев: Δμa = -0.2;

ΔμaS(0) = -0.510 ± 0.005 Дьяконов: Δμa = -0.744;

ΔμaC(0) = -0.658 ± 0.025 (использовали разные массы)

ΔμaB(0) = -0.621 ± 0.037

ρ = 4.77 ± 0.10 ГэВ-1 or 0.94 ± 0.02 Фм, поперечный радиусB(2)

φ = -2αsνA r/ρ3 exp(-r2/ρ2) ~ 1/ρ2 ~ 0.04 ГэВ2 ~ 6x1013 T оценка величины хромомагнитного поля.

15

N.I. Kochelev, Phys. Lett. B426(1998) 149.

D. Diakonov, Prog. Part. Nucl. Phys. 51(2003)173.

Заключение

Предложен единый механизм для объяснения значительных поляризационных эффектов.

Десятки реакций (86), инклюзивных и эксклюзивных были проанализированы в рамках модели эффективного цветового поля, в том числе данные ряда реакций, не получивших пока интерпретации в рамках других механизмов.

Глобальный анализ мировых данных позволяет оценить ряд параметров, описывающих взаимодействие кварков, в том числе их динамические массы и хромомагнитные моменты.

Back-up slides

Probe quark focusing in ECF Ba

The dependence of C(√s) = v0/[(1 – ER/√s )2+δ2R]1/2, for AN and PN

is due to focusing properties of circular chromomagnetic field Ba.

The focusing effect is similar to the one used in a Tokamak type thermonuclear reactor to keep plasma away off reactor’s walls.

Focusing Lorentz force F = gs[vBa]Ia leads to the prolongation of probe quark stay in a color field and enhance polarization effects in case of ER > 0. For opposite field direction we have a defocusing effect,

ER < 0 and there is an increase of AN or PN with the rise of energy √s.

Осцилляции AN(xF) для p↑p(A) → p X

Из AN = C(√s) F(pT, A)[GA(φA) – σGB(φB) ] оцениваем функцию GA(φA) = AN /C(√s)/F(pT, A) + σGB(φB). (24)

p↑ p(A) → p X

Те же данные при √s =8.77 ГэВ и √s =200 ГэВ описываются универсальной осциллирующей функцией:GA(A) =[1– cos A]/A + εφA, описывающую результат прецессии спина кварка и действие силы Штерна-Герлаха в цветном поле. Здесь A – средний угол прецессии спина кварка иε = -0.00461 ± 0.00006.

√ s

Dependence of AN & PN on xF & pT

PN ≈ -δPx D; (Ryskin, 1988) (12)

In Ryskin model δPx ≈ 0.1 GeV/с is constant.

In the ECF model we have a dynamical origin of δPx dependence on kinematical variables (√s, pT, xA(B), xF) and on a number of (anti)quarks in hadrons A, B & C, and also on quark color ga

Q–factor and its mass MQ.

This dependence is due to microscopic Stern-Gerlach mechanism and quark spin precession in the ECF.

D ≈ –∂/∂pT ln(d3σ/d3p); D = 5.79 ± 0.07 GeV–1 (13)

Multiplicity dependence of AN(xF)

The π+ production AN in p↑

p collisions is measured in the BRAHMS experiment. J.H.Lee, DIS2009.The data are presented here for three bins of multiplicity, normalized to the mean value (Rm). Higher Rm corresponds to larger ECF value due to correlation of the number of strings and multiplicity. Larger ECF gives in this model higher AN.

p↑ p → π- X

Rm =Multiplicity/Mean

Additional transverse momentum of quark Q is due to Stern-Gerlach type force in ECF

δpx = gaQ ξ0

y [(1 – cosφA)/φA + εφA]/2ρ/(gaQ – 2 + 2MQ/EQ), (9)

φA = ωAxA spin precession angle in the fragmentation region of A.

ωA = gsαsνA S0(gaQ – 2 + 2MQ/EQ)/(MQ cρ2) «frequency» (10)

xA(B) = (xR ± xF)/2 scaling variables (11)

Due to microscopic Stern-Gerlach effect quark Q gets an additional spin-dependent transverse momentum δpx, which causes an azimuthal asymmetry AN or transverse hyperon polarization PN:

S0 ≈ 1.489 ± 0.062 fm (ECF length); ε = -0.00461 ± 0.00006.

ε is small due to subtraction of Thomas precession term from for chromomagnetic contribution to the δpx.

Quark counting rules for ω0A

SPEC

TATO

RS

In case of the reaction p↑p→π+ +X the polarized probe u quark from π+ “feels” field, created by 3 spectator quarks with weight νA= λ, and by 3 target quarks with νB= -τλ, respectively:

νtot = [3λ - 3τ λ ] < 0.

∫Ba ~ ω0A = ω0

U [3λ - 3τ λ ] > 0; AN > 0;

since ω0U = gsαsS0(ga

U – 2)/(MQ cρ2) < 0,

due to (gaU – 2) < 0.

p↑ + p → π+ + X

≡ C

Λ polarization in νμA-collisions

The Λ polarization in νμA collisions is measured in the NOMAD experiment. D.V.Naumov, Acta Phys. Polon. B33:3791-3796, 2002. We assume that W+ interacts with d-quark and produce u-quark, moving forward, in νμ direction. The ECF is created by this u-beam from νμ, and by the two quarks from the target remnant, which are moving in opposite direction in c.m.

xF= -0.27 (target fragmentation region)

νμA →Λ μ- X , √s =6.82 GeV

AN for π+ in e+p-collisions

The π+ production AN in e+p collisions is measured in the HERMES experiment. K.Rith, SPIN2010.J.Phys.Conf.Ser.295:012056,2011.We assume that virtual photon produce q-q-bar pair (vector meson dominance), which interacts with the target quarks and produce π+. The sign of AN and xF are changed to the opposite.

e+ p↑ →π+ e+ X , √s =7.26 GeV

AN for K+ in e+p-collisions

The K+ production AN in e+p collisions is measured in the HERMES experiment. K.Rith, SPIN2010.The not monotonous pT behavior of the AN is due to the dependence of scaling variables yA and yB on polar angle θcm. This leads to the dependence on pT of the quark spin precession angles φA, φB and to the dependence of the AN.

e+ p↑ →K+ e+ X , √s =7.26 GeV

The definition of φA & φB precession angles

Variable yA(B) takes into account the quark motion inside proton and spin precession in the ECF: yA = xA – (E0/√s + f0 )[1 + cosθcm ] + a0[1 – cosθcm ], (22)

yB = xB – (E0/√s + f0 )[1 – cosθcm ] + a0[1 + cosθcm ], (23)

φA = ωAxA ≈ ω0AyA = precession angle A (20)

φB = ωBxB ≈ ω0ByB = precession angle B (21)

where ω0A(B) = gsαsνA(B) S0(ga

Q – 2)/(MQ cρ2) - the limit of ωA(B) at high quark energy EQ.

where a0, f0 & E0 – phenomenological parameters.

Precession angle φA(B) “measures” color field integral in the fragmentation region of hadron A(B).

План доклада

Введение Происхождение поляризационных эффектов Глобальный анализ поляризационных данных Поляризация Λ ̃in pA соударениях (E766) Поляризация Λ в νμ A взаимодействиях (NOMAD) AN для π+ и K+, образующихся в e+p↑ соударениях (HERMES) A-зависимость поляризации Λ в e+A соударениях (HERMES) Зависимость AN от множественности для π± в p↑p (BRAHMS) Осцилляция AN(xF) для p↑p(A) соударений (ФОДС и BRAHMS) Оценки масс и аномальных хромомагнитных моментов составлящих кварков из глобального анализа данных Заключение

Summary

Tenth of reactions (86), exclusive and inclusive, have been analyzed in the framework of the Effective color field model, including those, which are usually not considered or recently measured.

The measured data could be used in a global analysis in order to estimate parameters, describing such phenomena as spontaneously broken chiral symmetry, hadron and quark mass origin, confinement, color quark interaction and its transition to hadrons.

Exclusive reaction π-p↑ → K0 Λ↑

The values of GA(A), estimated from the data in a wide range of c.m. energies, are scattered near the universal model curve (a solid one), which oscillates as a function of the spin precession angle A.

π-p↑ → K0 Λ↑

D.J.Grennell et al. Phys. Rev. D6(1972)1220. √s =3.06-3.49 GeV.

W. Beusch et al. Nucl. Phys. B99(1975)53. √s =3.21 GeV.

I.A.Avvakumov et al. Yad. Fiz. 42(1985)1152. √s =8.72 GeV.

Exclusive reaction K-p↑ → K-p


K- p↑ → K- p

M.Borghini et al. Phys. Lett. 31B(1970)405. √s =3.53 GeV.

M.Borghini et al. Phys. Lett. B36(1971)497. √s =4.46-5.24 GeV.

Exclusive reaction p n↑ → n p


p n↑ → n p

M.A. Abolins Phys. Rev. Lett. 30(1973)1183. √s =2.77-4.74 GeV.

D.G. Crabb et al. Nucl. Phys. B185(1981)1. √s =6.85 GeV.

Exclusive reaction pn↑ → np

At large spin precession angle A the linear term εφA dominates in the experessionGA(A) =[1– cos A]/A + εφA,where ε = -0.00461 ± 0.00006 is a phenomenological parameter, as expected from the ECF model.Large precession angle A values are reached at low energy √s =2.77 GeV.

p n↑ → n p

The relation of local and global parameters

ω0A = ω0

Q νA; (Q = u,d,s,c,b); φA = ω0AyA (32)

For many reactions the local parameters can be expressed via the global ones, that allow to estimate the global parameter values for the data analysis. E0 = rg MQ[1 + (2 - 8d0)/(2-ga

Q)]; (50)

d0 ≡ a0 + f0 = d1(2) – b2exp[-(pT/pd)3]; (49)

a0 ≡ aQ – b1exp[-(pT/pa)3]; (Q = u,d,s,c,b) (49)

rg = sign(ω0A) (51)

ω0Q = gsαsS0(ga

Q – 2)/(MQ cρ2) (51)

ER = 4rg MQ/(2-gaQ); (50)

AN for π- in e+p-collisions

The π- production AN in e+p collisions is measured in the HERMES experiment. K.Rith, SPIN2010,J.Phys.Conf.Ser.295:012056,2011. The data are described well for different xF and reactions. The ECF is described by Quark counting rules for q-q̃ pair, moving forward and uud-quarks, moving in the opposite direction.

e+ p↑ →π- e+ X

AN for K- in e+p-collisions

The K- production AN in e+p collisions is measured in the HERMES experiment. K.Rith, SPIN2010. J.Phys.Conf.Ser.295:012056,2011.

e+ p↑ →K- e+ X

Oscillation of AN for p↑p(A) → π+ X

The π+ production AN in p↑

p(A) collisions is measured in the E704, FODS-2, BRAHMS and many other experiments. The data at √s =200 GeV have negative values of A due to new q-q̃ pair production ~ exp(-√s/W) and change of the ECF sign. The data at √s < 70 GeV have positive A and an approximate scaling for AN(xF,pT). Parameter W=272.7±1.3 GeV.

p↑ p(A) → π+ X

Oscillation of AN for p↑ p(A) → π- X

The data for the p↑ p(A) → π- X reaction are also described well by the ECF model with a universal function GA(A), shown by the solid black curve.

It is very interesting to measure AN at different energies, from √s =70 GeV up to 500 GeV. The data points should move along the curve from positive A region to the negative one.

p↑ p(A) → π- X

Exclusive reaction K+p↑ → K+p

The exclusive reactions are also analyzed in the framework of the ECF model. The values of GA(A), estimated from the data in a wide range of c.m. energies, are scattered near the universal model curve (a solid one), which oscillates as a function of the spin precession angle A.

K+ p↑ → K+ p

Possible origin of single-spin phenomena

The main assumptions of a semi-classical mechanism: Effective color field (ECF, chromomagnetic & chromoelectric) is created during the hadron interaction. The ECF is a superposition of string fields, created by moving spectator quarks & antiquarks after initial color exchange and new quark production. Spectator quarks and antiquarks from a projectile and from the target contribute the ECF with different weights. Spectators are all quarks which are not constituents of the observed hadron C in the reaction A + B →C + X. Quark counting rules describe the ECF. Probe quark Q from the detected hadron interacts with non-uniform color field via its chromomagnetic moment μa

Q and its color charge gS (we call this “microscopic Stern-Gerlach mechanism”).

Possible origin of single-spin phenomena

Microscopic Stern-Gerlach effect in chromomagnetic field and Thomas spin precession in chromoelectric field lead to probe quark polarization. Quarks with different initial spin projections on the quantization axis get different PT-kicks in transverse direction. The ECF is considered as an external with respect to a probe quark Q in observed hadron. The hadron polarization is the average polarization of its constituent quarks.

Quark spin precession (BMT) in ECF is an additional phenomenon, which leads to a specific dependence of polarization (oscillation) as a function of kinematical variables (xF, pT or scaling variables xA(B)=(xR±xF)/2).

Preliminary LHCb data for pp → Λ↑ X

The Λ polarization PN in pp or pA collisions is measured at different energies. The LHCb experiment has preliminary results at √s =7 TeV, which where presented at the IHEP seminar. The xF values are near zero, so the PN is consistent with zero, as expected.

p p → Λ↑ X

√ s

An example of quark focusing in field Ba

p↑ + p(A) → π+ + X,

focusing effect when

0A = 1.85 > 0;

√s < 70 GeV

ER = 3.31 ± 0.09 GeV

1/C(√s) ~ (1-ER/√s );

√s0 = 100 GeV

AN is decreasing to a finite value when √s is increasing.

√s =4.89 GeV, BNL

√s =200 GeV,

BRAHMS

FODS-2

√s =8.77 GeV

E704

√s =19.4 GeV

An example of quark defocusing in field Ba

p +p(A) → Λ↑ + X,

defocusing effect when

0A = −2.41 < 0;

ER = −2.95 ± 0.30 GeV

Au+Au → Λ↑ + X,focusing effect when0

A = +44.78 >0;ER =+4.805 ±0.016 GeV

PN is increasing to a finite value when √s is increasing.

√s =200 GeV,

STAR

√s =4.86 GeV,

BNL

The global analysis

The global analysis of single-spin data allows to reveal general regularities and data trends, which are otherwise not seen. To reveal and explain these regularities and the data trends in the framework of common mechanism, the Effective Color Field model was developed.

Data base for single-spin inclusive and exclusive reactions was created in a unified format. It contains now data for 86 different reactions with more then 5500 data points and continue to grow.

Vector meson polarization (ρ00-1/3, α)

The best studied reactions: Polarization in hр & hA–collisions. 9 reactions № 51÷59, 116 points. High precision data.

Model:

Solid curve:

G(φA) = (1- cosφA)/φA+εφA

K*-φK*+ρ0

Ј/ψ

pCu→Y(S2)

pCu→Y(S1)

pp→Y(S1)

Global analysis: exclusive reactions

Exclusive reactions, in which analyzing power or polarization was measured in hр & hA–collisions. 12 reactions, 3165 points.

№ Reaction Points № Reaction Points123456

π- p↑ → π0 n↑

π- p↑ → K0 Λ↑ π- p↑ → π- p↑ π+ p↑ → π+ p↑

p̃↑ p → π- π+

p̃↑ p → K- K+

13532215203840713

789101112

K- p↑ → K- p↑

K+p↑ → K+p↑

K+ n↑ → K0 p↑

p p↑ → p p↑ p n↑ → n p↑

p A→ Λ↑ K+ n

39288575326348

Appendix A: PN estimate

Hyperon polarization P with respect to the normal to the production plane can be estimated via angular distribution of its decay products:

W (θπ) = const (1 + aPeπ), (A.1)

where eπ - unit vector in the direction of the π- - meson in the rest frame of the hyperon (in case of Λ↑ →p π- decay). The decay parameter a = 0.642 ± 0.013 .

Appendix A: ρij estimate

Vector meson spin matrix density elements can be estimated via angular distribution W (θ, φ) = dN/dΩ of decay products (spin-0 mesons in decay V → h1 + h2, ):

W(θ,φ) = 0.75{cos2θ ρ00 + sin2θ (ρ11 + ρ-1-1) /2 – sin2θ (cos φ Re ρ10 – sinφ Im ρ10)/√2 + sin2θ (cos φRe ρ-10 + sinφIm ρ-10)/√2

– sin2θ[(cos(2φ)Re ρ1-1 – sin(2φ)Im ρ1-1)]}/π. (A.2)

Here θ is the polar angle between the direction of motion of h1 and the quantization axis, φ is the azimuth angle.

Appendix A: ρij estimate

Integrating over the angle φ, we get

W(θ) = 0.75[(1 - ρ00) + (3 ρ00 - 1) cos2θ]. (A.3)

Similarly, integrating over the angle θ, we get

W(φ) = 0.5[1 – 2cos(2φ)Re ρ1-1 + 2sin(2φ) Im ρ1-1]/π . (A.4)

By measuring W(θ), we can estimate ρ00. Other elements, ρ10 and ρ-1-1, can be studied by measuring W(θ,φ).Diagonal elements ρ11, ρ00 and ρ-1-1 for the matrix with unit trace are the relative intensities of the spin meson m to take the values 1, 0 and -1, respectively, which must be equal to 1/3 for the case of unpolarized particles.

Estimate of α = (σT - 2σL) / (σT + 2σL).

Another possibility for measuring the polarization of vector mesons is implemented in their decays to a pair of fermion and antifermion. For example, to measure the polarization of J/-meson we are using the angular dependence of its decay into μ+ μ- in a spiral basis, in which the quantization axis is directed along the direction of the vector meson in the laboratory frame. We define θ* as the angle between the momentum of μ+ in the rest frame of J/ and the quantization axis. The normalized angular distribution of the μ+ is given by

I(cos θ*) = 1.5(1 + α cos2 θ*)/(α + 3 ). (A.5)

For non-polarized vector mesons, we have α = 0, whereas α = +1 or -1 for 100% of the transverse or longitudinal polarization, respectively.

Polarization in nuclei collisions

Au+Au →Λ↑ + XPolarization of Λ in Au+Au–collisions.

Experiment STAR:

√s = 62 и 200 GeV.

There is energy dependent global Λ- hyperon polarization in heavy ion collisions at pT > 2.7 GeV/c. Combine effect of large color fields ~fNA1/3 and correlation of production and reaction planes.

Predictions of AN for √s = 130 GeV, θCM= 4.1°

Solid red curve – predictions √s = 130 GeV, θCM = 4.1°.

p↑ + p → π+ + X

Е704: √s = 19.4 GeV

BRAHMS:

√s = 62.4 GeV √s = 200 GeV

Dashed blue curve – predictions for √s = 200 GeV, θCM = 4.1°.

AN scaling is violated at √s > 70 GeV due to new quark production.

Global data analysis: AN

Inclusive reactions, in which analyzing power was measured in hр & hA–collisions. 23 reactions, 876 points.

№ Reaction № Reaction № Reaction12345678

p↑ p(A) → π+

p↑ p(A) → π-

p↑ p(A) → K+

p↑ p(A) → K-

p↑ p(A) → np↑ p → π0

p↑ p → K0S

p̃↑ p → π+

91011121314

p̃↑ p → π-

p̃↑ p → π0

d↑ A → π+

d↑ A → π-

π+ p↑ → π+

π- p↑ → π0

p↑ p(A) → pπ- d↑ → η

17181920212223

K- d↑ → π0

π- d↑ → π0

p̃↑ p → ηp↑ p → p̃p↑ p → ηp̃ d↑ → π0

π- p↑ → π-

1516

Global data analysis: AN

The best studied reactions: AN in hр & hA–collisions. 14 reactions № 1÷14, 510 points. High precision data.

√s =200 GeV

Model: Solid curve:


p↑p→π±K±

p↑p→n√s <70 GeV

AN(xF) and GA(φA) oscillate due to spin precession in color field.

Global data analysis: PN


p p(A) → Λ↑ p A → Ξ–↑

p A → Ξ0↑

p A → Σ+↑

p p → p↑

p A → Σ–↑

p A → Ω–↑

Σ– A → Λ↑

Σ– A → Σ+↑

3334353637383940

K– p → Λ↑

p̃ A → Λ̃↑

p A → Ξ̃+↑

p A → Σ̃–↑

Λ A → Ω–↑

K– A → Ξ–↑

Λ A → Ξ–↑

p A → Ξ̃0↑

4142

π+ p → Λ↑

K+ p → Λ↑

p A → Λ̃↑

π– p → Λ↑

n A → Λ↑

K+ p →Λ̃↑

Σ– A → Ξ–↑

Σ– A → Λ̃↑

434445464748

Reactions, in which hyperon polarization was measured in hр & hA–collisions. 25 reactions, 916 points.

Baryon polarization oscillation

The best studied reactions: PN in hр & hA–collisions. 19 reactions № 24÷42, 691 points. High precision data.

Model – solid curve:


K- p →Λ↑ + X

We can see oscillation for K- p →Λ↑ + X

Data for 46 most studied reactions, -10 < φA < 40.

Model: Solid curve:


Ξ̃+Σ̃−Ξ0̃

For anti-hyperon production in pp or pA collisions the effective color field and the precession angle φA are high due to large number of spectator anti-quarks. As a result the polarization oscillates as a function of xF or φA.

Dependence of frequency ω0A and ECF

on √s and atomic weights A1, A2

At high energy √s new quark and antiquark production changes the ECF intensity.

In case of ion collisions the effective number of spectator quarks in a projectile nucleus is equal to its number in a tube with transverse radius limited by the confinement:

qA = 3(1+fN)Aeff ~ 3(1+fN)A1/3 (23)

q̃A = 3fNAeff ~ 3fNA1/3 (24)

New quark contribution fN is a suppressed at high pT & xF since fast probe quark leaves the ECF very quickly and is not influenced by it.

fN = nqexp(-W/√s)(1-XN)n, n = 1.11 ± 0.03; nq = 4.28 ± 0.04; (25)

XN = [(pT/pN)2 + xF2 ]1/2; pN = 39.6 ±1.6 GeV/с; W = 273±1 GeV.

The case of A1A2-collisions

In case of А1А2-collisions the new quark contribution fN to ECF & string number νA at a given pT & xF is modified as:

fN = nqexp(-W/√s)(1-XN)n, (26)

XN = [(pT/pN)2 + xF2 ]1/2; (27)

WA = W/(A1A2)1/6 (28)

nA = n(A1A2)1/6 (fractality parameter) (29)

n = 1.11 ± 0.03, W = 273 ± 1 GeV,

nq = 4.28 ± 0.04, pN = 39.6 ± 1.6 GeV/с;

where A1 and A2 are atomic weights of colliding nuclei.

Predictions of AN for √s = 500 GeV, θCM= 4.1°

Solid red curve – prediction √s = 500 GeV, θCM = 4.1°.

Е704: √s = 19.4 GeV

BRAHMS:

√s = 62.4 GeV √s = 200 GeV

Dashed blue curve – predictions for √s = 200 GeV, θCM = 4.1°.

p↑ + p → π+ + X

AN scaling violation at √s > 70 GeV due to new quark production.

Global data analysis : AN, PN, ρ00


Au+Au → Λ↑

Au+Au → Λ̃↑

p A → J/ψ↑

p̃ A → J/ψ↑

p A → Ү(1S)↑

p A → Ү(2S)↑

p̃ p → ρ(770)↑

56575859

p p → φ(1020)↑

n A → K*(892)–↑

n A → K*(892)+↑

p̃ p → Ү(1S)↑

p̃ p → Ү(2S)↑

AuAu→K*(892)0↑

AuAu→ φ(1020)↑

636465

e+ A → Λ↑

e+ A → Λ̃↑

e+ p↑ → π+ e+ p↑ → π– μ– p↑ → h+

μ– p↑ → h–

5152535455

666768

606162

Reactions, in which PN was measured in AuAu–collisions, vector meson polarization, PN & AN in lepton-nucleon collisions. 20 reactions, 308 points.

Quark counting rules for frequency ω0A

General frequency ω0A equations for q и q̃ probes from hadron С:

ω0A(q)= ω0

Q{q̃new +λqnew – q̃used - λqused +λqA + q̃A –τ(λqB+q̃B)} (27)ω0

A(q̃)= ω0Q{λq̃new +qnew – λq̃used - qused +qA + λq̃A –τ(qB+ λq̃B)} (28)

Quarks & antiquarks spectators from projectile contribute to ω0

А, with weights λ & 1 respectively. Spectators from target have additional factor –τ.

Ea ~ Ba ~ ω0A = ω0

U [3λ - 3τ λ ] > 0; AN > 0;

ω0U ~ (ga

U – 2) < 0.

p↑ + p → π+ + X

SPEC

TATO

RS

Thomas precession effect in effective color field

U = s·ωT - an additional term in the effective Hamiltonian (12)

Direction and magnitude of the force F= gsEa is determined by quark counting rule for ECF. FZ ~ -[2 + 2λ - 3τ λ ]<0 for Q=s in pp→Ξ0+X,

FZ =gSEaZ = -2gSαS [1 + λ - 3τ λ ]/ρ2 < 0 for Q=s in pp→Λ+X, (15)

FZ ~ -[3λ - 3τ λ ]>0 for Q=u in pp→π+ +X.

Force FZ is processes dependent! δPN > 0 for Q=s in pp→Λ+X.

Additional Thomas precession term δPN > 0 is opposite in sign to the DeGrand model predicted negative polarization for pp→Λ+X. In ECF model dominates chromomagnetic field contribution with δPN < 0 .

ωT ≈ [F v]/MQ - Thomas frequency for EQ»MQ. (13)

δP = -ωT/ΔE – polarization for pp→Λ+X, where ΔE >0. (14)

Data for 46 most studied reactions, -20 < φA < 20.

11.41

xF > x0, pT > 0.3 GeV/с. 46 reactions, 1427 points.

Model:

Solid cureve:


In instanton model dynamical quark mass MQ & anomalous chromomagnetic moment Δμa

Q depend on momentum transfer q:

Dependence of MQ & ΔμaQ on q

11.22 Data analysis: q0 = 1.5 ± 0.8 GeV/c.

D.I.Diakonov, 2003

(62)

(63)

(64)

Summary-2

A semi-classical mechanism is proposed for single-spin phenomena. Effective color field of QCD strings, created by spectator quarks & antiquarks is described by quark counting rules. Microscopic Stern-Gerlach effect in chromomagnetic field and Thomas spin precession in chromoelectric field lead to large SSA. The energy and atomic weight dependence of effective color fields, combined with quark spin precession phenomenon, lead to oscillating behaviour of AN and PN as a function of kinematical variables. Additional anti(quark) production at high √s > 70 GeV changes the dependence on kinematical variables and violates the approximate AN(xF) or PN(xF) scaling. Quark focusing or defocusing in the effective color field leads to an additional resonance like energy dependence of AN or PN.

Quark counting rules for ω0A

General formulas for q and q̃ probe quarks from hadron С:

ωq A = ω0

Q{q̃new +λqnew – q̃used - λqused +λqA + q̃A –τ(λqB+q̃B)} (1)

ωq̃ A = ω0

Q{λq̃new +qnew – λq̃used - qused +qA + λq̃A –τ(qB+ λq̃B)} (2)

SPEC

TATO

RS

In case of the reaction p↑p→π+ +X the polarized probe u quark from π+ “feels” field, created by 3 spectator quarks with weight νA= λ, and by 3 target quarks with νB= -τλ, respectively:

νtot = [3λ - 3τ λ ] < 0;

∫Ba ~ ω0A = ω0

U [3λ - 3τ λ ] > 0; AN >0;

ω0U = gsαsS0(ga

U – 2)/(MQ cρ2) <0.

p↑ + p → π+ + X

≡ C

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Объяснение поляризационных данных в рамках модели эффективного цветового поля