1. Transversity and 2 pion production
Transversity and inclusive 2π production
M. Radici (Pavia)
with A. Bacchetta (DESY)
(LM)
(L’M’)
Dihadron Fragmentation Functions (DiFF)
chiral-odd partner
DiFF=probab. of q->(h1 h2) X ; happen in various S.I. reactions: DIS, collisions, annihilations
In particular, DiFF=spin analyzer of fragmenting q ; chiral-odd partner to extract transversity of q in h in SSA -> exp. interest -> new data (HERMES, COMPASS)
Avantage (th. and exp.) w.r.t. Collins effect: azimuthal asymmetry even with situation collinear to jet
2h asymmetry
P1 £ P2 ¢ ST
Collins effect
k £ Ph ¢ ST
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Transversity and 2 pion production

2. Transversity and 2 pion production
Spin asymmetry in e p" ! e’ (h1h2) X
at leading twist
following Trento conventions, spin asymmetry is ?
SIDIS kin. defined as Trento conventions ; cross sect differential in x,y,z,Mh,phiR,phiS,th ; th -> expansion in (LM) ; then int dcos(th)
Twist-2 cross sect very simple : unpol (with f1 & D1) + T-pol (with h1 & H1angle) -> SSA defined as Trento conventions.
N.B. int dcos(th) gives pi/4 ; previous paper: moment also in sin(phiR+phiS) -> additional factor ½ ; HERMES takes coefficients of Fourier (phiR,phiS) expansion + Legendre (th) expansion -> factor 4/pi (now) or 8/pi (previous). Extraction questionable:
[ A + B sin(th) sin(phiR+phiS) ] / [ C + D cos(th) + D (cos^2(th)-1) ] -> take B/C ?! Anyway, HERMES has problems in taking int dcos(th) because of incomplete phase space…
B(y)=(1-y-y^2 g^2/4)/(1+g^2) A(y)=(1-y+y^2/2+y^2 g^2/4)/(1+g^2) where g^2=(2Mx)^2/(xy(s-M^2)) target mass corrections. end N.B.
DiFF unknown from e+e- ; or from pp collisions (self consistently -> later details). Meanwhile, need models! ? LM
from e+e- → (h1h2) (h1’ h2’) X (BELLE)
or models ! 1 2
check universality in SIDIS and e+e- at twist 2
(Boer, Jakob, Radici P.R. D67 (03) )
partial wave (LM)
then d
or in pp collisions (Bacchetta & Radici P.R. D90 (04) ) see later…
Bacchetta & Radici
P.R. D67 (03)
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Transversity and 2 pion production

3. Transversity and 2 pion production
published models
ep" ! e’ (+-) X
(Jaffe, Jin, Tang, P.R.L. 80 (98) 1166)
no calculation of δqI (z) → “figure of merit”
s-p interference from - elastic scattering
phase shifts only; sign change from Re[]
(Radici, Jakob, Bianconi, P.R. D65 (02) )
spectator model
interference ~ (s) – Im[]
uncertainty band from
different fp / fs strength
ratio
PDF input
Jaffe: fully collinear, twist=2 SIDIS, factorize z and Mh^2 dependence, the latter all in pi-pi phase shifts as in elastic pi-pi scattering.
But pi-pi -> pi-pi , while here is q -> R -> pi-pi -> sign change in F(d0,d1) not necessarily reflects in SSA.
No calculation of z dependence (R=sigma for s-wave, R=rho for p-wave; sigma, rho are STABLE particles).
Pavia: new idea of asymmetry from FSI (s-p interference) ; full z,Mh^2 dependence in calc. of interf. diagram with resonant rho (p-wave) and
background direct production (s-wave) -> SSA from Im [ T(s) x T*(p) ].
Model parameters (couplings) fixed with arguments (no fits); uncertainty also from other inputs (pdf’s).
Reshuffling according to Trento conventions: 8/pi and change sign -> not so bad with preliminary HERMES data!
P. van der Nat
DIS2005
Trento conventions ! £ (– 8/)
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Transversity and 2 pion production

4. Transversity and 2 pion production
improved model
( Bacchetta & Radici, hep-ph/ )
ep" ! e’ (+-) X
PYTHIA output on HERMES
kinematics:
0.1<y< Q2 > 1 GeV2
0.023<x< W2>4 GeV2
excludes elastic and
single/double diffractive events
→ semi-inclusive Mh
2. q →  X2 → +- X (14.81%)
events
3. q →  X3 → +- X (0.31%)
Pavia-DESY: improve old Pavia model ; advantage of fitting parameters on MC output which reproduces unpolarized data.
To give idea of complication, Mh spectrum of (pi+pi-) pairs with cuts to exclude everything but semi-inclusive pairs in deep-inelastic kin.
Various channels: excluded because give the narrow bump at 0.48 which would make life harder in fitting.
1. = all - ( )
events
4. q →  X’4 → +- (0 X’4) (8.65%)
5. q → η X’5 → π+π- (X X5) (2.05%)
Total
(π+π-) pairs
6. q → K0 X6 → π+π- X (3.41%)
1. All – (2+..+6) = background (70.77%)
events
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Transversity and 2 pion production

5. Transversity and 2 pion production
spectator model approximation
1. background ≡ q → π+π- X1 no resonance → real s-wave channel
2. q → ρ X2 → π+π- X2
X1 = X2 = X3 = X4 = X
p-wave channel = coherent sum | |
3. q → ω X3 → π+π- X3
4. q → ω X’4 → π+π- (π0 X’4)
Warning: ω → [(π π)L=1 π]J=1 X4
max number of (π+π-) pairs in s-p interference ~ Im [ p-wave channel ]
parameters
s-wave
p-wave
Essential features of model (previous and present):
spectator approx. (quantum # of quark) with (pi+pi-) in some (LM): s-wave with no resonance -> real channel
p-wave for rho/omega -> 2pi; p-wave also for omega->3pi.
same spectator (X1=X2=X3=X4) for all channels -> max interference (not realistic, gives upper bound in SSA); p-wave from | |^2.
Hence s-p interference is: (1.)-| |* ~ Im [ ]
2) channel 4. is overemphasized, because we know (OBELIX, Asterix, etc..) that omega (J-PC=1- - and I-G = 0-) decays in all 3
cyclic [ [pi pi]_L pi]_J combinations with same probability and L=J=1 for isospin reasons-> how many (pi+pi-) pairs in p-wave survive after
pi0 integrated away? Are there (pi+pi-) pairs in s-wave, after rearranging angular momenta? Question for second step evolution of model.
3) for each vertex coupling x exp. form factor; for p-wave also Breit-Wigner with mass-width from PDG. For omega->3pi mass is the invariant mass of (pi pi pi), then int dpi0 approximated with resonant contribution for m_omega = M3 (narrow width approx => channel only imaginary and active for Mh<M_omega – m_pi ~ GeV). Cut-off different for s- and p-channels. Total of 11 parameters.
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Transversity and 2 pion production

6. Transversity and 2 pion production
fit of PYTHIA distributions in invariant mass and z
p-wave
1. background s-wave
Total
Total Mh
Fit of D1(Mh) and D1(z) as it is output from PYTHIA MC in HERMES kin. p-p interference [omega->3pi] x [rho->2pi] is Mh~0.6.
Dimensions of parameters to grant [D1(z,Mh)]= # of pairs at z,Mh =int dzeta dvec kT dvec RT D1(..) with [D1(..)] = GeV^(-4).
Chi^2/d.o.f. = 25 due mainly to Mh~0.6 disagreement because of rho-omega constructive interference (which is absent in exp. Histograms). Anyway, best performance with couplings with all relative sign. Absolute sign undetermined (plots are moduli squared) -> need SSA data to fix absolute sign; also couplings determined modulo an overall factor (<-> luminosity) but inessential in SSA
Back to Asymmetry: once fitted D1-> predict H1angle; then get pdf’s from models -> predict SSA z
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Transversity and 2 pion production

7. Transversity and 2 pion production
models for transversity
Strat
Soffer, Stratmann, Vogelsang
P.R. D65 (02)
Kor
Korotkov, Nowak, Oganessian
E.P.J. C18 (01) 639
Schw
Schweitzer et al.
P.R. D64 (01)
Wak
Wakamatsu
P.L. B509 (01) 59 u d x
from GRV98-LO
@ Q2 = 2.5 GeV2
Models for h1: Strat -> saturated Soffer bound on GRV98+GRSV00 LO (standard 0.26 GeV^2 + evolution
Kor -> simulation of HERMES Collins effect using 0.4 GeV^2 with GRV94+GRSV96 + evolution
Schw -> chQSM in large Nc from Bochum Wak -> again chQSM.
Flavor symmetry for DiFF: u->(pi+pi-) = d->(pi-pi+). Then flipping pair gives R->-R, zeta->-zeta.
Asymmetry integrated in x,y with HERMES kin cuts: Q=1/sxy > 1 -> xmin ; W^2>4 -> xmax
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Transversity and 2 pion production

8. Transversity and 2 pion production
spin asymmetry
0.023<x<0.4 ; 0.1<y<0.85
Q2>1 GeV2 s=56.2 GeV2
HERMES % scale
PRELIMINARY uncertainty
ρ → π+π-
ω → (π+π-) π0 Mh
bins
Shape in Mh^2 is ok. But overall factor ~5 more -> max interference (all pairs from all channels with channel 1. because same spectator X).
Then also omega->3pi (~ 0.5 GeV) overestimated because all (pi+ pi-) supposed in relative p wave (see slide 5)
Fit asymmetry reducing number of active pairs in interference plus additional reduction for omega->3pi. To make reasonable comparison, use same exp bins (particularly 3rd one around our minimum, where Jaffe -> sign change): asym for i bin from int dMh^2 H1angle upon bin width, then int dMh^2 D1 upon same bin width, and then take ratio.
Fit from factor 40% reduction overall, plus factor 60% for omega->3pi -> 24% on omega->3pi Mh
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Transversity and 2 pion production

9. Transversity and 2 pion production
spin COMPASS
deuteron
0.1<y<0.9 Q2>1 GeV2 s=604 GeV2
0.004<x<0.4 Mh
Joosten – DIS2005
0.03<x<0.4
With same previous fit, look at deuteron in COMPASS.
In deuteron, isospin symmetry=1/9[4u-d-4ubar+dbar +4d-u-4dbar+ubar] plus h1d ~ -1/4 h1u and f1d = ½ f1u ; also int dx with xmin= > overall factor ~10 with respect to proton target at HERMES. Also, sign reversed because COMPASS does not follow Trento conventions.
If int dx with xmin=0.03 (~ HERMES), gain a factor ~2, but still small asym. (10% < COMPASS/HERMES < 22%) Mh z
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Transversity and 2 pion production

10. Transversity and 2 pion production
proton
0.1<y<0.9 Q2>1 GeV2 s=301 GeV2
spin COMPASS
0.004<x<0.4
….. Mh
0.03<x<0.4
Sign change because of “anti” Trento convention. For int dx with xmin=0.004 results ~ ½ of HERMES.
Changing limits in int dx with xmin=0.03 (~ HERMES) makes some difference: + ~20-40% (55% < COMPASS/HERMES < 96%). Mh z
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Transversity and 2 pion production

11. Transversity and 2 pion production
p-p collision
possible at RHIC, GSI and JPARC
Bacchetta & Radici P.R. D90 (04)
p p" ! (π π) X
self-consistent extraction of
p p ! (π π)C (π π)D X
Kin. of pp collisions: Pc.Pa propto Pc┴ (PcT=0) hard scale -> leading order in 1/Pc┴ ; phiR angle of RcT w.r.t. xc ; phiS of p pol. (all phi’s w.r.t. scatt.plane). All cross sect. are convolutions! dsigma diff. in rapidity, cos(th)c , Pc┴ , phiR and phiS.
El. cross sect. unpol. ab->cd (no helicity flip): qq->qq qq’->q’q and all combinations with also qbar; then qg->qg gg->qqbar qqbar->gg gg->gg
“ “ “ pol. ab↑->c↑d (with “ “ ): qq↑->q↑q qq’↑->q’↑q and all combinations until qbar↑; gq↑->q↑g ; no others because spin½ target and no gluon transversity.
Unpol. two jets: dsigma diff. also in variables for d El. cross section unpol. as before
Coeff. B = pol. ab->c↑d↑ (2 helicity flips): qq->q↑q↑ .. as before with also qbar until gg->q↑qbar↑ but no Compton scatt.
Coeff. C = “ “ “ “ : qqbar->g↑g↑ and gg->g↑g↑ possibility of extracting transv. pol. of g using spin½ target
Cross checks: 1jet f1a=δ(1-x) , zd=1 and el. cross sect. lq↑->l’q↑  SIDIS Similar to Λ↑ production: SΛ = RT cos(phiR)
2jet f1=δ(1-x) and el. cross sect. e+e- ->q↑qbar↑  e+e-
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Transversity and 2 pion production