# 2) Introduction of quark spin in the recursive jet model. Application to ...

2) Introduction of quark spin in the recursive jet model. Application to ...

IWHSS10, VeneziaII - Introduction of quark spinin the recursive jet model.Application to quark polarimetry(jet-handedness and Collins effects)Xavier ARTRUIPNL, Lyon, FranceReference: DSPIN09 (Sept. 1-5, 2009, Dubna)

Content• Recall about models without spin• Recall about Collins effect and Jet Handedness• Collins effect in the string model• The model :multiperipheral model with quark exchanges• Collins and jet-handedness effects in this model• Possibility of Monte-Carlo generation• Decay of the spin information• Inclusion of vector mesons

Why consider quark spin in fragmentation models ?Quark polarimetry :• q↑→ Λ↑ +XOK for helicity and transversity, but statistics is low.• q↑→ π (or 2 π) + X, with Collins effect (for transversity)= asymmetry in φ(π) - φ Spin ( or φ(π−π) minus φ S )• q↑→ 2 π (or 3 π) + X, with jet handedness (for helicity)= asymmetry in φ(π 1 ) − φ(π 2 )Many variables, many different situations… Could be treated in aunified way with a jet simulation model incorporating the quark spindegree of freedom.

Present jet models (without spin).1) the recursive modelField & Feynman, Peterson, …q 0 → h 1 + q 1 ,q 1 → h 2 + q 2 , etc.q 3q 2h 4h 3h 2h 1q 1Momentum conservation :q 0k n = p n+1 + k n+1Splitting function of q n :dW = f n (ζ n , k nT ) dζ n d 2 k nT ; ζ n = k n,z / k n-1,z

Note :-The splitting functionis not the inclusive spectrumf n (ζ n , k nT )F n (z n , p nT )of the n th rank meson. We have :z n = (1- ζ n ) ζ n-1 ζ n-2 … ζ 1k nT = - (p nT + p (n-1)T … + p 1T )- A cuttof in k nT of f n (ζ n , k nT ) avoids large quark virtualitiesand insures Local Compensation of Transverse Momentum(Krzyvwicky & Peterson)

2) the string modelMassive stringq 0−k 2T+k 2T −k 1T+k 1Tq 1⎺q 2q 2 ⎺q 1q 0h 2h 1

Space-time viewth Nq Nh 1h 2q 1q 2q 0z

COLLINS EFFECT1) The single-particle Collins effectTransversely polarised fragmentation function:F(z,p T ;S T ) = F 0 (z,p T ) [ 1+ A T (p T ×S T ) z /p T ]Squarkhadronp TXzsin(φ S - φ h )A T = analysing power = H ┴ (z,p T)/ D (z,p T)−1 < A T < +1

Defects of the single-particle Collins effect- requires a good definition of the jet axis- gluon radiation smears out the effect : at high Q 2C.E. with respect to q’ (yellow ellipse) can be strong,but C.E. with respect to q is weak.- Q 2 evolution more complicated than DGLAP (Sudakov factor)q = jetmomentumq’p Tp’ T

COLLINS EFFECT - 2) the 2-particle relative Collinseffect, or “interference fragmentation”Replace p T by the relative transverse momentum of two particles,r T = < z 2 p 1T – z 1 p 2T > /(z 1 +z 2 ) = P jet θ 12 z 1 z 2 /(z 1 +z 2 )Not p 1T – p 2T !Integration is made over P T = p 1T + p 2T .Advantages: - r T is insensitive to errors on the jet axis and to gluon radiation- it is enhanced by local compensation of transverse momentumAnalysing power: A(z 1 ,z 2 , r T ) = H

Jet handednessNachtman 1977 ; Efremov, Mankiewicz and Tornqvist 19921) minimal version, with 2 particlesLongitudinally polarised fragmentation function:F( z, p T ; z’, p’ T ; S L ) =F 0 ( z, p T ; z’, p’ T ) × [ 1+ A L S.(p T ×p’ T ) / |p T ×p’ T | ]−1 < A L < +1sin( φ’ h - φ h )Same defects as the single-particle Collins effect : sensitiveto error on the jet axis ; smeared by gluon emission

2) safe version of handedness : with 3 particlesRecipe : Take a third particle in the jet and replace the jet axis bythe direction of P 123= p 1+ p 2+ p 3.An estimator of quark helicity is thenE = < (p 1× p 2) . p 3> / | P 123|(this is more or less the original proposal of Efremov et al)

String mechanism for the Collins effecta) The Lund 3 P 0 mechanismWhen the string breaks, a quark-antiquark pair iscreated in the 3 P 0 state (= vacuum quantum numbers)stringquark+k TSL−k TSantiquarkstringr L = - r×k T→ correlation between k T and S quark

) application to the Collins effectSpin-zero mesons :Scalar3P 0Pseudo-scalar1S 0String decay into pseudoscalar mesons :−k 2T+k 2T −k 1T+k 1T+k 3Tq 3 ⎺q 2 q 1L 3L 2L 1⎺q 3 q 2 ⎺q 1q 0−k 3Th 3h 2h 1

Predictions of the string + 3 P 0 mechanism- alternate Collins effects (for pions or kaons)- strong C.E. of the subleading particle. This may explain the large C.E.of « unfavored » pions. Another cause is that unfavored pions havesmaller p L , therefore smaller smearing Δp T = p L Δθ, where Δθ is thejet axis uncertainty.- a ρ-meson has a C.E. opposite to the pion ones [Czyzewski](in fact, this applies to a longitudinal ρ).- 〈p T2 〉 should be larger for pions and kaons than for scalar mesonsand longitudinal vector mesons (independently on quark polarisation)• The Schwinger mechanism of quark pair creation is different : it givesno Collins effect [X.A. & Czyzewski]

A candidate : the multiperipheral modelh Nh N-1h 3q N-1 q 3 q 2 q 1h 1h 2q Nγ, Z 0 , W ± q 0

« Interrupted » multiperipheral amplitudePreliminary simplifications :• Pauli spinors• disentangle k L and k T• Gaussian propagatorsp 3p 2k 3k 2 k 1p 1M 123 (k 1T , k 2T , k 3T ) = M 3 (k 3T ) M 2 (k 2T ) M 1 (k 1T )k 0analogue of m+γ.kM n (k nT ) = exp{- (k nT ) 2 } (µ + σ . k nT ) σ zpropagator pseudoscalar vertexµ = complex “mass” ( analogue of γ 5 )

Joint p T - distributions• Focus on the transverse momentum variable• neglect the mass-shell constraintsJoint p T - distributions of the n leading particlesJ 12…n (p 1T , p 2T ,… p nT, ) = trace R 12…n (k 1T , k 2T ,… k nT ; S)R 12…n = cross section matrix (or « correlator ») of the n firstemissions= M(k nT , k nT ) … M(k 1T , k 2T ) ρ 0 M † (k 1T , k 2T ) … M † (k 1T , k 2T )interruptedamplitudedensity matrixof the initial quark = (1+ σ.S)/2

1 rst rank Collins effectJ 1 (p 1T ) = exp{- p 1T2} ×{ |µ| 2 + p 1T2+ 2 Im(µ) S.(z×p 1T ) }(z = unit vector along z-axis)Collins effectAnalysing power :A T = 2 Im(µ) |z × p 1T | / (|µ| 2 + p 1T2 )The Collins effect is predicted on the same sideas in the string model if Im(µ) > 0

Joint p T spectrum for the 1 rst and 2 nd ranksJ 12 (p 1T ,p 2T ) = exp{- p21T - p22T } × {(|µ| 2 + k 21T ) (|µ| 2 + k 2T2 ) - 4 k 1T . k 2T Im 2 (µ)+ 2 Im(µ) S T . (z × k 1T ) (2 k 1T . k 2T - |µ| 2 - k 2T2 ) ← Collins+ 2 Im(µ) S T . (z × k 2T ) (|µ| 2 - k 1T2 ) ← Collins- 2 Im(µ 2 ) S L . (k 1T × k 2T ) } ← handednessThe Collins effects in k 1T and k 2T can be re-grouped in- a global C.E. in k 1T + k 2T- a relative C.E. in r T = (z 2 p 1T - z 1 p 2T ) / (z 1 +z 2 )

Properties-This model reproduces qualitatively the results of theLund + 3 P 0 mechanism, concerning the tranverse spin effects(alternate C.E., strong unfavored C.E., etc.)- In addition, it has longitudinal spin effects (jet handedness).- Positivity : A L2+ A T2≤ 1 (A L : Collins ; A T : handeness)

Implementation in a Monte-Carlo generatorDensity matrix ρ n and polarisation S n of the n th left-over quark :ρ n = (1+ σ.S n )/2 = R 12…n / trace{R 12…n }Recursion relation :R 12…n = M(k nT , k nT ) R 12…(n-1) M † (k nT , k nT )Next two slides : explicit formula giving S n from S n-1 and k nT

Recursion relation 1) spin normal to the emission planetake x-axis along k nT :S y n = C -1 { 2|k nT | Im(µ) - (|µ| 2 + k nT2 ) S y n-1 }with C = trace{M(k nT , k nT ) ρ n-1 M † (k nT , k nT )}= |µ| 2 + k nT2- 2 |k nT | Im(µ) S y n-1S y n can be different from zero even if S y n-1 is zero :Transverse polarisation can be generated in the quarkchain (analogue of Boer-Mulders effect)

2) - spin components in the emission planeS x n k nT2-|µ| 2 - 2|k nT | Re(µ) S x n-1= C -1S z n - 2|k nT | Re(µ) |µ| 2 - k nT2S z n-1helicity is partly converted into transversity and vice-versa (likea rotation in the emission plane).Handedness appears as the result of two effects :1) partial conversion of S z (q 0 ) into S T (q 1 ) along k 1T ,= analogue of h 1L⊥( a « worm gear » effect)2) Collins effect in q 1 ↑ → h 2 + q 2 ,

Decay of spin informationIf we integrate over the transverse momenta, the polarisationof the n th left-over quark decreases geometrically with n.S z n = D LL Sz n-1 |D LL | and |D TT | ≤ 1S T n = D TT ST n-1D jj = w jj / w 0w0 = ∫ d(k T2 ) exp{- k T2} (|µ| 2 + k T2 ) >0w LL = ∫ d(k T2 ) exp{- k T2} (|µ| 2 - k T2 )w TT = - ∫ d(k T2 ) exp{- k T2} |µ| 2

Inclusion of spin-1 mesonsVector meson can be introduced, replacing the pion vertex σ z byΓ = G L V z + G T σ.V T σ zV = vector of the meson spin state(real for linear polarisation)G L , G T = complex coupling constants (generally different).

p T - spectrum of a linearly polarised ρ mesonJ 1 (p T ,V) = |G T | 2 exp(-p T2 ) × {(|α| 2 V z2 + V T2 ) (|µ| 2 +k T2 ) - 4 Im(µ) Im(α) V z V T .k T(α = G L /G T )(a)+ 2 Im(µ) |α| 2 V2z S T .(z×k T ) (b)+ 2 Im(α) (|µ| 2 +k T2 ) V z V T .(z×S T ) (c)+ 2 Im(µ) [ V T .(z×k T ) V T .S T + V T .k T V T .(z×S T ) ] (d)+ 4 Re(α) Im(µ) V z S z V T .(z×k T ) } (e)Line (a) : unpolarised quark ; gives tensor polarisation.

p T - spectrum of ρ meson (continued)Line (b) : global Collins effect, for longitudinal ρ only(agree with Czyzewski)Line (c) : oblique linear polarisation of ρ , perpendicularto S T , resulting in a relative π - π Collinseffect (function h 1LT )Line (d) : asymmetry in sin(2 φ V - φ p - φ S )analogous to pretzelosityLine (d) : oblique linear polarisation of ρ , perpendicularto p T , resulting in a π × π handednessWe obtain qualitatively the same effects as for twosuccessive direct pions (there could be a duality…)

Other mechanisms of Collins effect•Interference between a spin zero and a spin-1 resonance decay•between a resonance and a background amplitude•Qiu-Sterman diagram :orThe gluon « tells the quarkwhere is the z-axis »(othewise it would « decay »isotropically).This role can be equallyplayed by a string or amultiperipheral quark chain(previous slides)

Main results• a very simple model, inspired from the old multiperipheralmodel of Amati, Fubini and Stanghelini, can implement thespin degree of freedom in quark jet simulation• It makes a quantum realisation of the spin effectspredicted by the classical string + 3 P 0 model• Jet-handedness and Collins effects are obtained for pionseither emitted directly or coming from vector meson decay.• In the direct mechanism, handedness appears as theresult of two successive effects :- the first emission rotates the quark spin ,- a Collins effect takes place in the second emission

Main results (continued)• In the resonant case, Collins and handednesseffects are associted to oblique linearpolarisations of the vector meson.• successive emissions gradually erase the memoryof the initial transverse and longitudinalpolarisations, but at different rates D TT and D LL .

Discussion• The model is somewhat over-simplified. The mostcriticable approximation was to ignore the massshellcontraints.• A justification for using Pauli spin instead of Diracspin would be welcome• Our method of including the quark degree offreedom could be implemented in Monte-Carlo jetgenerators like Pythia.THANK YOU !

Conclusion• Transversity and helicity are very beautifull, yet differentsisters• Transversity is related to the nucleon magnetic moment, atleast via the Burkardt connexion, may be also via thetensor charge• It is strongly correlated to the intrinsic k T (Sivers, Collins,Boer, Mulders, Kotzinian). For this subject, the hydrogenatom is a useful pedagogical tool.• Quark polarimetry is in the infancy (therefore promising)• I apologize for this a very incomplete overview and too fewreferences• Thank you for attention !

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