Научная сессия-конференция секции ЯФ ОФН РАН «Физика фундаментальных взаимодействий» 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 Λ̃ in pA соударениях (E766) A-зависимость поляризации Λ в e + A соударениях (HERMES) Зависимость A N от множественности для π + в p p (BRAHMS) Осцилляция A N (x F ) 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 A N, P N.

Глобальный анализ данных Глобальный анализ данных: A N, P N, ρ 00 & α = ( σ T – 2 σ L )/( σ T + 2 σ L ). Всего 86 инклюзивных и эксклюзивных реакций для hh, hA, AA & lN-взаимодействий, более 5500 экспериментальных точек. A + B C + X {Анализирующая способность для С, A N (p T, x F,s) }. A + B C + X {Поляризация для C, P N (p T, x F,s) }. В т.в. КХД односпиновые эффекты малы: A N S m Q /E Q 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 = -2α s ν A /ρ 2 exp(-r 2 /ρ 2 ), (1) B (2) φ = -2α s ν A r/ρ 3 exp(-r 2 /ρ 2 ), (2) где ν A – число кварков, ρ =1.25R C 2.08 ГэВ -1, R C ГэВ, R C – радиус конфайнмента, α s = g s 2 /4π 1. Имеется продольное хромо- электрическое поле E a и круговое хромомагнитное поле B a. μ a Q = sg a g s /2M Q – 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 ppΞ 0 +X тестовые s и u кварки из Ξ 0 измеряют поле, создаваемое кварками-спектаторами с весом ν A = λ, антикварками с ν A = 1, и кварками мишени ν B = -τλ. λ = |ψ qq (0)| 2 /|ψ q q ̃(0)| 2 1-e 1/ цветной фактор (5) λ = ±0.0006, τ = ± для 86 реакций. f x μ a x B a x /x + μ a y B a y /x (3) f y μ a x B a x /y + μ a y B a y /y. (4) СПЕКТАТОРЫ E a ~ B a ~ [2 + 2λ - 3τ λ ] Правила кваркового счета Тестовый кварк Q из наблюдаемого адрона Cизмеряет В а & E a. Эффективное цветное поле: C

Прецессия спина кварка в поле струны dξ/dt a[ξ B a ] + d[ξ [E a v]] (BMT-уравнения) (6) a = g s (g a Q – 2 + 2M Q /E Q )/2M Q (массы M U M D 0.3 GeV) (7) d = g s [g a Q – 2E Q /(E Q +M Q )]/2M Q (E Q - Q энергия) (8) Δμ a Q =(g a Q -2)/2 (аномальный хромомагнитный момент кварка). Спонтанное нарушение киральной симметрии приводит к дополнительной динамической массе кварка ΔM Q (q) и Δμ a Q (q). В инстантонной модели: Δμ a Q (0) –0.2 (Н. Кочелев, 1998) ; Δμ a Q (0) –0.744 (Д. Дьяконов, 2003). ΔM Q (q), Δμ a Q (q) 0, если q =ρ 0 p T. ρ 0 = ±.0009 Глобальный анализ дает Δμ a Q (0) -0.4÷ -0.7 (Q -аромат).

Уравнения для A N, P N и ρ 00 -1/3 P N C(s) F(p T, A)[G(φ A ) – σG(φ B ) ], (14) G( A ) = [1 – cos A ]/ A + εφ A, прецессия спина и сила Ш-Г. (15) C (s) = v 0 /[(1 – E R /s ) 2 +δ 2 R ] 1/2, фокусировка кварков (16) F(p T,A) = {1 – exp[-(p T /p 0 T ) 3 ]}(1 – α lnA). Цветной форм-фактор (17) Всего 8 локальных параметров для каждой из реакций: D, α, σ, E 0, E R, f 0, a 0, p 0 T. V.Abramov, Phys. At. Nucl. 72 (2009) V.V. Abramov 2011 J. Phys.: Conf. Ser глобальных параметра для 86 реакций (ε, λ,τ, M Q, Δμ a Q …). v 0 = -D g a Q ξ 0 y /2ρ(g a Q –2 ). Магнитуда A N, P N и ρ 00 -1/3 (18)

Поляризация Λ̃ в pp и pA-соударениях Для поляризации Λ̃ в pp и pA соударениях большинство данных при высокой энергии, s > 27 ГэВ и P N нулевая (синие точки). Значительная P N в Е766 (J. Felix (1995), talk at ICTP, Trieste, Italy) s = 7.31 ГэВ (красные точки). Большая P N объясняется эффектом фокусировки анти- кварков в круговом хромомагнитном поле с E R =7.2±1.1 ГэВ, δ R =0.064 P N ~1/[(1 – E R /s ) 2 +δ 2 R ] 1/2 (19)

А-зависимость P N для Λ в e + A-соударениях Поляризация Λ P N (A) в e + A соударениях измерена в экспери- менте HERMES. K.Rith, DIS2010. Эффективный вклад кварков, создающий цветное поле ν A = 1+λ(3A eff -2)–τ(λ+1), где λ-0.133, τ.053, A eff 0.6A 1/3. Поле и P N ~ν A уменьшаются с ростом A, и P N 0 при А 120. Красная кривая –предсказание. e + A Λ e + X, s =7.26 ГэВ

Зависимость A N (x F ) от множественности A N (x F ) 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, f N : p p π + X, s =200 ГэВ R m =Multiplicity/Mean f N ~ 1 + a m (R m -1), a m =0.025±0.004 (24)

Осцилляции A N (x F ) для p p(A) p X A N для p p(A) p X измерена в эксперименте ФОДС-2 в ИФВЭ (красные точки). V.V. Abramov et al. Phys. Atom. Nucl. 70: 1515, s = 8.77 GeV. Большие p T и x F. Blue stars are the BRAHMS data for s = 200 GeV. J.H.Lee, SPIN2006. Осцилляция A N (x F ) 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: M U = ± ГэВ/с 2 M U m P /4 M D = ± ГэВ/с 2 M D 4/3 M U m P /3 M S = ± ГэВ/с 2 M S M U + M D 7m P /12 M C = ± ГэВ/с 2 M C 3m P /2 M B = ± ГэВ/с 2 M B 3M C 9m P /2 Из форм-факторов заряженых пионов: M U M D 0.25 GeV/с 2 ; A.F.Krutov, V.E.Troitsky, Eur. Phys. J. C20 (2001) 71. (JLAB data) M Q = (2/3) 1/2 πF π = 0.24 ГэВ/с 2 ; С.Б.Герасимов, ЯФ 29(1979)513. M U = ГэВ/с 2 ; M.Mekhfi, Phys.Rev. D72(2005)

Глобальный анализ данных: аномальные хромомагнитные моменты Аномальные хромомагнитные моменты Δμ a =(g a -2)/2 при q=0: Δμ a U (0) = ± Инстантонная модель: Δμ a D (0) = ± Кочелев: Δμ a = -0.2; Δμ a S (0) = ± Дьяконов: Δμ a = ; Δμ a C (0) = ± (использовали разные массы) Δμ a B (0) = ± ρ = 4.77 ± 0.10 ГэВ -1 or 0.94 ± 0.02 Фм, поперечный радиус B (2) φ = -2α s ν A r/ρ 3 exp(-r 2 /ρ 2 ) ~ 1/ρ 2 ~ 0.04 ГэВ 2 ~ 6x10 13 T оценка величины хромомагнитного поля 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 B a The dependence of C(s) = v 0 /[(1 – E R /s ) 2 +δ 2 R ] 1/2, for A N and P N is due to focusing properties of circular chromomagnetic field B a. The focusing effect is similar to the one used in a Tokamak type thermonuclear reactor to keep plasma away off reactors walls. Focusing Lorentz force F = g s [vB a ]I a leads to the prolongation of probe quark stay in a color field and enhance polarization effects in case of E R > 0. For opposite field direction we have a defocusing effect, E R < 0 and there is an increase of A N or P N with the rise of energy s.

Осцилляции A N (x F ) для p p(A) p X Из A N = C(s) F(p T, A)[G A (φ A ) – σG B (φ B ) ] оцениваем функцию G A (φ A ) = A N /C(s)/F(p T, A) + σG B (φ B ). (24) p p(A) p X Те же данные при s =8.77 ГэВ и s =200 ГэВ описываются универсальной осциллирующей функцией: G A ( A ) =[1– cos A ]/ A + εφ A, описывающую результат прецессии спина кварка и действие силы Штерна- Герлаха в цветном поле. Здесь A – средний угол прецессии спина кварка и ε = ± s

Dependence of A N & P N on x F & p T P N -δP x D; (Ryskin, 1988) (12) In Ryskin model δP x 0.1 GeV/с is constant. In the ECF model we have a dynamical origin of δP x dependence on kinematical variables (s, p T, x A(B), x F ) and on a number of (anti)quarks in hadrons A, B & C, and also on quark color g a Q – factor and its mass M Q. This dependence is due to microscopic Stern-Gerlach mechanism and quark spin precession in the ECF. D –/p T ln(d 3 σ/d 3 p); D = 5.79 ± 0.07 GeV –1 (13)

Multiplicity dependence of A N (x F ) The π + production A N 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 (R m ). Higher R m corresponds to larger ECF value due to correlation of the number of strings and multiplicity. Larger ECF gives in this model higher A N. p p π - X R m =Multiplicity/Mean

Additional transverse momentum of quark Q is due to Stern-Gerlach type force in ECF δp x = g a Q ξ 0 y [(1 – cosφ A )/φ A + εφ A ]/2ρ/(g a Q – 2 + 2M Q /E Q ), (9) φ A = ω A x A spin precession angle in the fragmentation region of A. ω A = g s α s ν A S 0 (g a Q – 2 + 2M Q /E Q )/(M Q cρ 2 ) «frequency» (10) x A(B) = (x R ± x F )/2 scaling variables (11) Due to microscopic Stern-Gerlach effect quark Q gets an additional spin-dependent transverse momentum δp x, which causes an azimuthal asymmetry A N or transverse hyperon polarization P N : S ± fm (ECF length); ε = ± ε is small due to subtraction of Thomas precession term from for chromomagnetic contribution to the δp x.

Quark counting rules for ω 0 A SPECTATORS In case of the reaction p pπ + +X the polarized probe u quark from π + feels field, created by 3 spectator quarks with weight ν A = λ, and by 3 target quarks with ν B = -τλ, respectively: ν tot = [3λ - 3τ λ ] < 0. B a ~ ω 0 A = ω 0 U [3λ - 3τ λ ] > 0; A N > 0; since ω 0 U = g s α s S 0 (g a U – 2)/(M Q cρ 2 ) < 0, due to (g a U – 2) < 0. p + p π + + X C

Λ polarization in ν μ A-collisions The Λ polarization in ν μ A collisions is measured in the NOMAD experiment. D.V.Naumov, Acta Phys. Polon. B33: , We assume that W + interacts with d-quark and produce u-quark, moving forward, in ν μ direction. The ECF is created by this u- beam from ν μ, and by the two quarks from the target remnant, which are moving in opposite direction in c.m. x F = (target fragmentation region) ν μ A Λ μ - X, s =6.82 GeV

A N for π + in e + p-collisions The π + production A N in e + p collisions is measured in the HERMES experiment. K.Rith, SPIN2010. J.Phys.Conf.Ser.295: ,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 A N and x F are changed to the opposite. e + p π + e + X, s =7.26 GeV

A N for K + in e + p-collisions The K + production A N in e + p collisions is measured in the HERMES experiment. K.Rith, SPIN2010. The not monotonous p T behavior of the A N is due to the dependence of scaling variables y A and y B on polar angle θ cm. This leads to the dependence on p T of the quark spin precession angles φ A, φ B and to the dependence of the A N. e + p K + e + X, s =7.26 GeV

The definition of φ A & φ B precession angles Variable y A(B) takes into account the quark motion inside proton and spin precession in the ECF: y A = x A – (E 0 /s + f 0 )[1 + cosθ cm ] + a 0 [1 – cosθ cm ], (22) y B = x B – (E 0 /s + f 0 )[1 – cosθ cm ] + a 0 [1 + cosθ cm ], (23) φ A = ω A x A ω 0 A y A = precession angle A (20) φ B = ω B x B ω 0 B y B = precession angle B (21) where ω 0 A(B) = g s α s ν A(B) S 0 (g a Q – 2)/(M Q cρ 2 ) - the limit of ω A(B) at high quark energy E Q. where a 0, f 0 & E 0 – phenomenological parameters. Precession angle φ A(B) measures color field integral in the fragmentation region of hadron A(B).

План доклада Введение Происхождение поляризационных эффектов Глобальный анализ поляризационных данных Поляризация Λ̃ in pA соударениях (E766) Поляризация Λ в ν μ A взаимодействиях (NOMAD) A N для π + и K +, образующихся в e + p соударениях (HERMES) A-зависимость поляризации Λ в e + A соударениях (HERMES) Зависимость A N от множественности для π ± в p p (BRAHMS) Осцилляция A N (x F ) 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 K 0 Λ The values of G A ( 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 K 0 Λ D.J.Grennell et al. Phys. Rev. D6(1972)1220. s = 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 The values of G A ( 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 M.Borghini et al. Phys. Lett. 31B(1970)405. s =3.53 GeV. M.Borghini et al. Phys. Lett. B36(1971)497. s = GeV.

Exclusive reaction p n n p The values of G A ( 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. p n n p M.A. Abolins Phys. Rev. Lett. 30(1973)1183. s = 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 εφ A dominates in the experession G A ( A ) =[1– cos A ]/ A + εφ A, where ε = ± 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 ω 0 A = ω 0 Q ν A ; (Q = u,d,s,c,b); φ A = ω 0 A y A (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. E 0 = r g M Q [1 + (2 - 8d 0 )/(2-g a Q )]; (50) d 0 a 0 + f 0 = d 1(2) – b 2 exp[-(p T /p d ) 3 ]; (49) a 0 a Q – b 1 exp[-(p T /p a ) 3 ]; (Q = u,d,s,c,b) (49) r g = sign(ω 0 A ) (51) ω 0 Q = g s α s S 0 (g a Q – 2)/(M Q cρ 2 ) (51) E R = 4r g M Q /(2-g a Q ); (50)

A N for π - in e + p-collisions The π - production A N in e + p collisions is measured in the HERMES experiment. K.Rith, SPIN2010, J.Phys.Conf.Ser.295: ,2011. The data are described well for different x F 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

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

Oscillation of A N for p p(A) π + X The π + production A N 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 A N (x F,p T ). Parameter W=272.7±1.3 GeV. p p(A) π + X

Oscillation of A N 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 G A ( A ), shown by the solid black curve. It is very interesting to measure A N 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 G A ( 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 μ a Q and its color charge g S (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 P T -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 (x F, p T or scaling variables x A(B) =(x R ±x F )/2).

Preliminary LHCb data for pp Λ X The Λ polarization P N 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 x F values are near zero, so the P N is consistent with zero, as expected. p p Λ X s

An example of quark focusing in field B a p + p(A) π + + X, focusing effect when 0 A = 1.85 > 0; s < 70 GeV E R = 3.31 ± 0.09 GeV 1/C(s) ~ (1-E R /s ); s 0 = 100 GeV A N 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 B a p +p(A) Λ + X, defocusing effect when 0 A = 2.41 < 0; E R = 2.95 ± 0.30 GeV Au+Au Λ + X, focusing effect when 0 A = >0; E R = ±0.016 GeV P N 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 (ρ 00 -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(φ A ) = (1- cosφ A )/φ A +εφ A K* - φ K* + ρ0ρ0 Ј/ψЈ/ψ pCuY(S2) pCuY(S1) ppY(S1)

Global analysis: exclusive reactions Exclusive reactions, in which analyzing power or polarization was measured in hр & hA–collisions. 12 reactions, 3165 points. ReactionPointsReactionPoints π - p π 0 n π - p K 0 Λ π - p π - p π + p π + p p̃ p π - π + p̃ p K - K K - p K - p K + p K + p K + n K 0 p p p p p p n n p p A Λ K + n

Appendix A: P N estimate Hyperon polarization P with respect to the normal to the production plane can be estimated via angular distribution of its decay products: W (θ π ) = const (1 + aPe π ), (A.1) where e π - 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 = ±

Appendix A: ρ ij 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 h 1 + h 2, ): W(θ,φ) = 0.75{cos 2 θ ρ 00 + sin 2 θ (ρ 11 + ρ -1-1 ) /2 – sin 2 θ (cos φ Re ρ 10 – sinφ Im ρ 10 )/2 + sin2θ (cos φRe ρ sinφIm ρ- 10 )/2 – sin 2 θ[(cos(2φ)Re ρ 1-1 – sin(2φ)Im ρ 1-1 )]}/π. (A.2) Here θ is the polar angle between the direction of motion of h 1 and the quantization axis, φ is the azimuth angle.

Appendix A: ρ ij estimate Integrating over the angle φ, we get W(θ) = 0.75[(1 - ρ 00 ) + (3 ρ ) cos 2 θ]. (A.3) Similarly, integrating over the angle θ, we get W(φ) = 0.5[1 – 2cos(2φ)Re ρ sin(2φ) Im ρ 1-1 ]/π. (A.4) By measuring W(θ), we can estimate ρ 00. Other elements, ρ 10 and ρ -1-1, can be studied by measuring W(θ,φ). Diagonal elements ρ 11, ρ 00 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 = (σ T - 2σ L ) / (σ T + 2σ L ). 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 cos 2 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 + X Polarization 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 p T > 2.7 GeV/c. Combine effect of large color fields ~f N A 1/3 and correlation of production and reaction planes.

Predictions of A N for s = 130 GeV, θ CM = 4.1° Solid red curve – predictions s = 130 GeV, θ CM = 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, θ CM = 4.1°. A N scaling is violated at s > 70 GeV due to new quark production.

Global data analysis: A N Inclusive reactions, in which analyzing power was measured in hр & hA–collisions. 23 reactions, 876 points. ReactionReactionReaction p p(A) π + p p(A) π - p p(A) K + p p(A) K - p p(A) n p p π 0 p p K 0 S p̃ p π p̃ p π - p̃ p π 0 d A π + d A π - π + p π + π - p π 0 p p(A) p π - d η K - d π 0 π - d π 0 p̃ p η p p p̃ p p η p̃ d π 0 π - p π

Global data analysis: A N The best studied reactions: A N in hр & hA–collisions. 14 reactions 1÷14, 510 points. High precision data. s =200 GeV Model: Solid curve: G(φ A ) = (1- cosφ A )/φ A +εφ A p pπ ± K ± p pn s

Global data analysis: P N ReactionReactionReaction p p(A) Λ p A Ξ – p A Ξ 0 p A Σ + p p p p A Σ – p A Ω – Σ – A Λ Σ – A Σ K – p Λ p̃ A Λ̃ p A Ξ̃ + p A Σ̃ – Λ A Ω – K – A Ξ – Λ A Ξ – p A Ξ̃ π + p Λ K + p Λ p A Λ̃ π – p Λ n A Λ K + p Λ̃ Σ – A Ξ – Σ – A Λ̃ Reactions, in which hyperon polarization was measured in hр & hA–collisions. 25 reactions, 916 points.

Baryon polarization oscillation The best studied reactions: P N in hр & hA–collisions. 19 reactions 24÷42, 691 points. High precision data. Model – solid curve: G(φ A ) = (1- cosφ A )/φ A +εφ A K - p Λ + X We can see oscillation for K - p Λ + X

Data for 46 most studied reactions, -10 < φ A < 40. Model: Solid curve: G(φ A ) = (1- cosφ A )/φ A +εφ A Ξ̃ + Σ̃ Ξ̃ 0 For anti-hyperon production in pp or pA collisions the effective color field and the precession angle φ A are high due to large number of spectator anti-quarks. As a result the polarization oscillates as a function of x F or φ A.

Dependence of frequency ω 0 A and ECF on s and atomic weights A 1, A 2 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: q A = 3(1+f N )A eff ~ 3(1+f N )A 1/3 (23) q̃ A = 3f N A eff ~ 3f N A 1/3 (24) New quark contribution f N is a suppressed at high p T & x F since fast probe quark leaves the ECF very quickly and is not influenced by it. f N = n q exp(-W/s)(1-X N ) n, n = 1.11 ± 0.03; n q = 4.28 ± 0.04; (25) X N = [(p T /p N ) 2 + x F 2 ] 1/2 ; p N = 39.6 ±1.6 GeV/с; W = 273±1 GeV.

The case of A 1 A 2 -collisions In case of А 1 А 2 -collisions the new quark contribution f N to ECF & string number ν A at a given p T & x F is modified as: f N = n q exp(-W/s)(1-X N ) n, (26) X N = [(p T /p N ) 2 + x F 2 ] 1/2 ; (27) W A = W/(A 1 A 2 ) 1/6 (28) n A = n(A 1 A 2 ) 1/6 (fractality parameter) (29) n = 1.11 ± 0.03, W = 273 ± 1 GeV, n q = 4.28 ± 0.04, p N = 39.6 ± 1.6 GeV/с; where A 1 and A 2 are atomic weights of colliding nuclei.

Predictions of A N for s = 500 GeV, θ CM = 4.1° Solid red curve – prediction s = 500 GeV, θ CM = 4.1°. Е704: s = 19.4 GeV BRAHMS: s = 62.4 GeV s = 200 GeV Dashed blue curve – predictions for s = 200 GeV, θ CM = 4.1°. p + p π + + X A N scaling violation at s > 70 GeV due to new quark production.

Global data analysis : A N, P N, ρ 00 ReactionReactionReaction 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) p p φ(1020) n A K*(892) – n A K*(892) + p̃ p Ү(1S) p̃ p Ү(2S) AuAuK*(892) 0 AuAu φ(1020) e + A Λ e + A Λ̃ e + p π + e + p π – μ – p h + μ – p h – Reactions, in which P N was measured in AuAu–collisions, vector meson polarization, P N & A N in lepton-nucleon collisions. 20 reactions, 308 points.

Quark counting rules for frequency ω 0 A General frequency ω 0 A equations for q и q̃ probes from hadron С: ω 0 A (q)= ω 0 Q {q̃ new +λq new – q̃ used - λq used +λq A + q̃ A –τ(λq B +q̃ B )} (27) ω 0 A (q̃)= ω 0 Q {λq̃ new +q new – λq̃ used - q used +q A + λq̃ A –τ(q B + λq̃ B )} (28) Quarks & antiquarks spectators from projectile contribute to ω 0 А, with weights λ & 1 respectively. Spectators from target have additional factor –τ. E a ~ B a ~ ω 0 A = ω 0 U [3λ - 3τ λ ] > 0; A N > 0; ω 0 U ~ (g a U – 2) < 0. p + p π + + X SPECTATORS

Thomas precession effect in effective color field U = s·ω T - an additional term in the effective Hamiltonian (12) Direction and magnitude of the force F= g s E a is determined by quark counting rule for ECF. F Z ~ -[2 + 2λ - 3τ λ ]0 for Q=u in ppπ + +X. Force F Z is processes dependent! δP N > 0 for Q=s in ppΛ+X. Additional Thomas precession term δP N > 0 is opposite in sign to the DeGrand model predicted negative polarization for ppΛ+X. In ECF model dominates chromomagnetic field contribution with δP N < 0. ω T [F v]/M Q - Thomas frequency for E Q »M Q. (13) δP = -ω T /ΔE – polarization for ppΛ+X, where ΔE >0. (14)

Data for 46 most studied reactions, -20 < φ A < x F > x 0, p T > 0.3 GeV/с. 46 reactions, 1427 points. Model: Solid cureve: G(φ A ) = (1- cosφ A )/φ A +εφ A

In instanton model dynamical quark mass M Q & anomalous chromomagnetic moment Δμ a Q depend on momentum transfer q: Dependence of M Q & Δμ a Q on q Data analysis:q 0 = 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 A N and P N 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 A N (x F ) or P N (x F ) scaling. Quark focusing or defocusing in the effective color field leads to an additional resonance like energy dependence of A N or P N.

Quark counting rules for ω 0 A General formulas for q and q̃ probe quarks from hadron С: ω q A = ω 0 Q {q̃ new +λq new – q̃ used - λq used +λq A + q̃ A –τ(λq B +q̃ B )} (1) ω q̃ A = ω 0 Q {λq̃ new +q new – λq̃ used - q used +q A + λq̃ A –τ(q B + λq̃ B )} (2) SPECTATORS In case of the reaction p pπ + +X the polarized probe u quark from π + feels field, created by 3 spectator quarks with weight ν A = λ, and by 3 target quarks with ν B = -τλ, respectively: ν tot = [3λ - 3τ λ ] < 0; B a ~ ω 0 A = ω 0 U [3λ - 3τ λ ] > 0; A N >0; ω 0 U = g s α s S 0 (g a U – 2)/(M Q cρ 2 )