# Presentation: 571937

## Di-hadron Fragmentation Functions

**Marco Radici**

**Pavia**

- BNL, 7-8 October 2005

- In collaboration with:    A. Bacchetta  (Univ. Regensburg !  DESY)

- RHIC II  Science Workshop

**Workshop on Physics with Polarized Protons at RHIC II**

## Motivation

- mechanism of spin – asymmetry to isolate transversity
- even after integrating ? momenta away

- transfer q**"** to orbital motion of *h *!  Collins effect

- asymmetry in
- sin C   /   **k** £ **P**h ¢ **S**T

- ! depend on **k**T ! **P**hT

- SIDIS : e p**"** ! e’ h X

## Superposition of effects

- _Superposition of effects_

- take                            reaction : **P**T  0 but also **k**T  0 independently

- observe intrinsic **k**T momenta

- _in final state_

- Collins effect

- _in initial state_

- Sivers effect

- _another possibility : in initial state_

- Boer  ‘99

- or Qiu-Sterman effect ?

## search for effects ! SSA , but surviving sdkT

- proof for Drell-Yan and low-pT SIDIS only (Ji, Ma, Yuan,  P.L. **B597 **(’04) 299 )
- universality still under debate      (Collins,  P.L. B**536 **(’02) 43
-                                                             Metz,    P.L. B**549 **(’02) 139
-                                                             Boer, Mulders, Pijlman,  N.P. **B667 **(’03) 201
-                                                             Bomhof, Mulders, Pijlman,  P.L. B**596** (’04) 277
-                                                             Collins, Metz, P.R.L. **93 **(’04) 252001)
- evolution      (Boer,  N.P. **B603 **(’01) 195
-                          Henneman, Boer, Mulders, N.P. **B620 **(’02) 331
-                          Kundu, Metz, P.R. D**65 **(’02) 014009
-                          Idilbi, Ji, Ma, Yuan, P.R. D**70** (’04) 074021)
- elementary cross section suppressed ~ 1/|**P** T|
-                                                        (Anselmino, Murgia, P.L. B**442** (’98) 470
-                                                                Anselmino et al., P.R. D**71** (’05) 014002)

- search for effects ! SSA , but surviving sd**k**T

- _Problems with __**k**__T__ dependence_

## double polarized Drell-Yan (Ralston-Soper ’79) : 

- only transversity involved, no other unknowns,  but

*A*TT small (NLO) by Soffer inequality  (Martin, Schaefer, Stratmann, Vogelsang  ’98
-                                                                    Barone, Calarco, Drago,  ‘97)

- presumably  h1 for antiquarks in p  is small  ! use antiprotons: HESR@GSI

- (A. Bianconi and M.R.,   hep-ph/0504261 , Phys. Rev. D in press)

- MonteCarlo simulation !  doable at   s 1⁄4 200 (GeV)2

## 2 hadron semi-inclusive process

- Collins effect

- e p**"** ! e’ (****1 ****2) X      p p" ! (****1****2) X  ..

- ! asymmetry in the azimuthal orientation of pair plane
-     with respect to some reference plane

- survives sd**k**T

- suggested for the first time by **Collins, Heppelmann & Ladinski, 1994**  but no twist
- analysis nor quantitative calculations  (see also **Ji  1994**)
**Jaffe, Jin, Tang 1998** ! suggestion of SSA from interference of () partial waves
**Bianconi, Boffi, Jakob, M.R., 2000 ****!**  complete twist-2 analysis and first model calc.
**Bacchetta, M.R.** !  partial wave expansion ()L ; twist-3 ; pp collisions...

## Interference Fragmentation Functions for q ! (h1,h2) X  with unpolarized h1,h2

_**I**__nterference __**F**__ragmentation __**F**__unctions_ for q ! (h1,h2) X  with unpolarized h1,h2

- functions of  ( z, **** =z1/z1+z2 ,  Mh2, **k**T2,  **k**T¢**R**T  )     !  ( z, ****, Mh2 )

- (LM)

- Ph=P1+P2
- R=(P1-P2)/2

- (L’M’)

- ( _twist-2_  Bianconi, Boffi, Jakob, M.R., P.R. D**62** (2000) 034008 ;
-   _twist-3_  Bacchetta, M.R., P.R. D**69** (04) 074026; _partial waves_ P.R. D**67** (03) 094002 )

## SIDIS

***R***T

- e p" !  e’ (h1h2) X

- leading-twist d****

- - no specific weight for

- - easier factorization proof ; universality

- -  no admixture with other effects

- most general !

**  ****R**T soft scale: evolution similar to
- transversity ?

- ( see  de Florian, Vanni, P.L. B**578** (’04) 139)

- _SIDIS_

## e+e- ! (+-)jet 1 (+-)jet 2 X

- e+e- ! (******+********-**)jet 1 (******+********-**)jet 2 X

- leading twist

- (Boer, Jakob, Radici,
-    P.R. D**67 **(2003) 094003
-  also Artru, Collins,
-    Z. Phys. C**69** (’96) 277)

- “Artru-Collins” azimuthal asymmetry

- same as in  SIDIS
- universality verified
- at leading twist !

- unknown from BELLE

- N.B. production of collinear pairs ! reduced background ?

## pp collisions

- A + B(" ) ! (C1 C2)C + X

- ab(" ) ! c(" ) d

- pa = xa PA
- pb = xb PB
- pc = PC / zc

**P**C=PC1+PC2 k jet axis  **P**CT=0
-       but
**P**C¢**P**A / **P**C?  large
**P**C?  hard scale
- analysis at leading twist
*o*(1/|**P**C?|)

- N.B. Sb around PB defined as SB
-         Sc around PC defined as RC
-             Sc = Sb

## 0 b,c= - b,c

- p p(" ) ! (****) X

- rapidity

- a+b! c+d
- 0 = 

- a+b"! c"+d
- 0 b,c= - b,c

- quark

- quark

- gluon

- gluon

## Single Spin Asymmetry

- p p" ! (****) X

- _Single Spin Asymmetry_

- unpolarized

- polarized

- (Bacchetta, Radici, P.R. D**70** (’04) 094032)

## cross check with e+ e- : f1 = (1-x) and

- unknown? Consider

- p p ! (****)C (****)D X

- _unpol._

- a+b! c"+d"
- 0 c,d= - c,d

- cross check with e+ e- : f1 = (1-x) and

## Partial wave expansion

- FSI  from interference of
- (****) @ L=0 and (****) @ L=1

**IFF**(z, ****(cos),Mh2)   =  n   **IFF**n (z,Mh2)  Pn (cos)

- _2h c.m. frame_

****  =  a(M1,M2,Mh) + b(M1,M2,Mh)  cos

******X** |****,Xih****,X|
**~**** **|(****)L=0ih(****)L=1| + |(****)L=1ih(****)L=0|** **

- _Partial wave expansion_

- (LM)

- (L’M’)

- q

- q’

## Spin Asymmetry in SIDIS

- _Spin Asymmetry in SIDIS_

- leading twist  e p**"** ! e’  X

- spin asymmetry    (d**"** – d**+**) / (d**"** + d**+**)

- measured ad HERMES

- (Bacchetta, Radici,
-   P.R. D**67** (03) 094002)

## Spin Asymmetry in pp collision

- more general than Tang, hep-ph/9807560

- analogous to pp" ! " X (de Florian et al, ’98) :   S ! RT  cos(’s) ! sin(’s)

- connection to SIDIS: f1a =  (1-x) , lq" ! q" l , zd =1

- evolution similar to H1 (" )  (Stratmann, Vogelsang, ’02)

- (Bacchetta, Radici,
-   P.R. D**70** (’04) 094032)

## Models:  ep" ! e’ (+-) X   at leading twist

- _Models:__  ep__**"**__ __!__ e’ (__****__**+**__****__**-**__) X   at leading twist_

- (Jaffe, Jin, Tang, P.R.L. **80** (98) 1166)

-  no calculation of ****qI (z)
-  ******, ****** stable particles
-  interference from ****-**** phase shifts only

- (M.R., Jakob, Bianconi, P.R. D**65** (02) 074031 )

- uncertainty band from:

-  different  fp / fs strength ratio
-  f1(x), h1(x)   from spectator model
-  f1(x), h1(x)=g1(x)   from GRV98 & GRSV96
-  f1(x), h1(x) = (f1+g1)/2   from  “   “

- spectator model

- spectator model

- Trento conventions ! reverse sign!

**Notes:**
Published results: Jaffe= no model, just figure of merit from known pi-pi elastic scatt. Us=model of interference between different production amplitudes. Comment on: Mh range, sign, width of uncertainty band. Constant (positive) sign in accordance with HERMES data (no sign change). Try to shrink band by upgrading model.

## Refined model

- Breit-Wigner
- m****, ********,  m****, ********
- m**K0** , ******K0**
- from PDG

- form factor

- fit parameters

- +  h.c.

- + same
-     for K**0****S**

****

****

- _Refined model_

- spectator model in  **** :

- off-shell quark

**Notes:**
Upgraded model: for pure p channel coherent sum of rho->2pi and omega->2pi and incoherent sum of omega->3pi. For pure s channel use coherent sum of K0s->2pi and direct production (background). Masses and widths from PDG. Form factors with cutoffs and strength => fitting parameters (respect PDG ratio of omega->2pi or 3pi). Interference s-p gives chiral-odd IFF. From approximations it turns out flavor independence.

##  ́

** ́**

## fit

- [GeV]

- PRELIMINARY

- removes all elastic,
- single and double
- diffractive events
- ! only semi-inclusive

- _fit_

**Notes:**
HERMES “private” communication on partial wave decomposition of D1(Mh) in 4pi (no acceptance corrections). Cuts to grant only semi-inclusive production. Fit these histograms.  Then recall A_UT expression.

## predict

- PRELIMINARY

- f1, h1 from spectator
-                  model

- f1, h1=g1 from
- GRV98 & GRSV96

- _predict_

**Notes:**
From slide 16: Dpol (contains |S_T| and B(y)/A(y)). Flavor sum on PDFs only. Cuts in z. Band due to different PDF parametrizations (much more narrow than before). Result encouraging.

## Conclusions

-  interpretation of upcoming HERMES 2**** semi-inclusive data in terms of
-     collinear fragmentation via IFF seems reasonable and feasible;
-     same analysis in progress for COMPASS and possible also at eRHIC

-  extraction of transversity via IFF more convenient with respect to
-     Collins effect

-  HESR@GSI  will probably offer another tool: collinear fully polarized
-     Drell-Yan with antiprotons (but when ..? .. 2018 .. ?!)

- Transverse spin physics without transverse momenta
- is a real option

-  pp(" ) collisions at RHIC allow for disentangling at the same time
-     transversity and related IFF looking for one / two pion pairs

-  pp! () () X allows to extract information on transverse linear
-           polarization of gluons using spin 1⁄2 targets

## Backup   slides

## double polarized DY (Ralston-Soper ’79) :   p" p" ! l+ l- X

_**how to extract **_
_**transversity **_
_**(at leading twist) ?**_

- double polarized DY (Ralston-Soper ’79) :   p" p" ! l+ l- X

- no inclusive DIS !

- initial state ! polarized Drell-Yan

- chiral-odd partner in

- final state !  semi-inclusive (SIR)

- only transversity involved, no other unknowns,  but

*A*TT small (NLO) by Soffer inequality  (Martin, Schaefer, Stratmann, Vogelsang  ’98
-                                                                    Barone, Calarco, Drago,  ‘97)

- presumably  h1 for antiquarks in p  is small
- ! use antiprotons: HESR@GSI
-     doable with s=200 GeV2

- (A. Bianconi and M.R., P.R. **D71 **(2005) 074014
-                              and   hep-ph/0504261 )

- MonteCarlo simulation

## extract it at e+e- facilities  (Belle@RHIC)

****production : e p**"** ! **"** X ,  p p**"** ! **"** X (E704,RHIC)

- which mechanism ?!

**? **polarization of quark ! intuitively search for **?** polarized final hadrons

**D**ouble **S**pin **A**symmetry (DSA)

- HESR@GSI can help in selecting models

- SIR

- {pq, q, kq}
- not all collinear

- transfer q**"** to orbital motion of *h *!  Collins effect

- asymmetry in
- sin    /   **k** £ **P**h ¢ **S**T

- chiral-odd Collins function  H1**?**  :

- extract it at e+e- facilities  (Belle@RHIC)

## Technical slide !   SIDIS : e p" ! e’ h X  

- Technical slide !   SIDIS : e p**"** ! e’ h X

- Collins effect

- leading twist:

-  ****S  0,****

- convolution

- keep d**** enough
- differential

- (****C = ****h+* *****S)  to break the convolution  [...]

**S**ingle **S**pin **A**symmetry needs

- (Boer & Mulders  ’98)

## insert all unsuppressed A- gluons 

- _Color gauge invariance ?_

- (Boer, Mulders, Pijlman, 2003)

- insert all unsuppressed A- gluons
- and AT gluons at n+=-1 makes
- the nonlocal *q*-*q* correlator color
- gauge invariant
- insert also all AT gluons makes the
- nonlocal *q*-*g*-*q* correlator color
- gauge invariant

- leading-twist projections are semipositive
- definite in Dirac space !  probabilistic
-                                          interpretation

- (Bacchetta and M.R., 2004)

## or extracted self consistently also from p-p collisions (Bacchetta, M.R. 2004 )

- or extracted self consistently also from p-p collisions (Bacchetta, M.R. 2004 )

- p p" ! (****) X

- p p ! (****)C (****)D X

- contains also

- same as            for gluons

- available for spin 1⁄2 hadron
- otherwise chiral-odd *****g* for spin  ̧ 1

- (back)

## collinear ep" ! e’ (+-) X

- (Jaffe, Jin, Tang, ’98)

******X** |****,Xih****,X| **~**** **|(****)L=0ih(****)L=1| + |(****)L=1ih(****)L=0|** **

- FSI  from interference of (****) @ L=0 and (****) @ L=1

- fragmentation in helicity basis

- collinear ep" ! e’ (******+********-**) X

- not general (z,Mh2) dependence !

**IFF**(z, ****(cos),Mh2)   =  n   **IFF**n (z,Mh2)  Pn (cos)

- _2h c.m. frame_

- |**R**T|   =  |**R**|(M1,M2,Mh)   sin 

****  =   a(M1,M2,Mh)   +   b(M1,M2,Mh)  cos

## (Jaffe)

|  |  |
| --- | --- |
|  |  |

|  | 0 |  | 0 |
| --- | --- | --- | --- |
| 0 |  | 0 | 0 |
|  | 0 |  | 0 |
| 0 | 0 | 0 |  |

| 0 | 0 | 0 |  |
| --- | --- | --- | --- |
|  | 0 |  | 0 |
| 0 | 0 | 0 |  |
| 0 | 0 | 0 | 0 |

**s**

**p,+1**

**p,0**

**p,-1**

**s**

**p,+1**

**p,0**

**p,-1**

**spin 1**
****** ******** ******** ******

- (Bacchetta
-  Mulders)

## events for +- Drell-Yan pairs distributed with cross section

- Monte-Carlo simulation

- (A. Bianconi and M.R., hep-ph/0504261)

- events for ******+********-** Drell-Yan pairs distributed with cross section

- NLLA QCD
- corrections
- (compensate
-   in DSA?)

- factorized leptons
- qT dependence
-          (Anassontzis et al. 1988)

- /  d0
- PDF(x,M2) da

- violation of Lam-Tung sum rule

**Collins-Soper frame:  q****T****(*************) in (xz) plane**

- N.B. S1, S2 randomly distributed
-         in Collins-Soper frame

## Double polarized Drell-Yan with antiprotons

- c.m. energy

- invariant mass

- parton momenta

- DIS  regime

- factorization

- for specific M,
- leading twist

- low **** ! high s
- collider mode for HESR@GSI
- at present:

- 2 M ranges explored:
-    Y  9     ̧ M  ̧ 4    J/    (GeV)
- J/  2.5  ̧ M  ̧ 1.5           ( “ )

## cuts:  s=200 GeV2 ;  4<M<9 GeV ; qT>1 GeV/c ; 60o< <120o ; ST=50%

- sample: 80.000 events   !   40.000   !  17.000

- (A. Bianconi and M.R.,
-    hep-ph/0504261)

- 0.08. **** . 0.4

## unambiguous extraction 

- unambiguous extraction
- of h1(x) seems possible

- statistical error bars from 20 repetitions

- cuts:  s=200 GeV2 ;  4<M<9 GeV ; qT>1 GeV/c ; 60o<**** <120o ; ST=50%

## cuts:  s=200 GeV2 ;  1.5<M<2.5 GeV ; qT>1 GeV/c ; 60o< <120o ; ST=50%

- sample: 80.000 events   !   40.000   !  17.000

- (A. Bianconi and M.R.,
-    hep-ph/0504261)

- 0.01. **** . 0.03

## unambiguous extraction 

- unambiguous extraction
- of h1(x) seems possible

- cuts:  s=200 GeV2 ;  1.5<M<2.5 GeV ; qT>1 GeV/c ; 60o<**** <120o ; ST=50%

- statistical error bars from 20 repetitions