Collins fragmentation in Kaon prodution

Francesca Giordano (UIUC), Matthias Grosse–Perdekamp (UIUC), Will Jacobs (Indiana University at Bloomington), Noriaki Kobayashi (TokyoTec), Hairong Li (Indiana University at Bloomington), Yoshiyuki Miyachi (TokyoTec),

Akio Ogawa (BNL), Ami Rostomyan (DESY), Gunar Schnell (University of the Basque Country), Ralf Seidl (RIKEN), Toshi-Aki Shibata (TokyoTec), Charlotte Van Hulse (University of the Basque Country),

Anselm Vossen (Indiana University at Bloomington)

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Francesca Giordano

Double ratios

Ph1

φ1

Ph2

φ2 − πθ

Thrust axis

e−

e+

Ph1⊥

Ph2⊥

Figure 3: Definition of the azimuthal angles φ1 and φ2 − π of the two hadrons, wherethe angles are formed between the scattering plane and their transverse momenta Phi⊥

around the thrust axis n.

the Collins effect can only be visible in the combination of 2 functions being able to createa single spin asymmetry each. Accordingly the combination of a quark and an antiquarkCollins function in opposing hemispheres gives a product of two sin(φ) modulations forthe two azimuthal angles φ1 and φ2, resulting in a cos(φ1 + φ2) modulation (see Fig. 3). Ine+e− these azimuthal angles are defined as

φ1,2 =n

|n|·!

z × n

|z||n|×

n × Ph1,2

|n||Ph1,2|

"

acos

!

z × n

|z||n|·n × Ph1,2

|n||Ph1,2|

"

, (4)

where z is just a unit vector in the z-axis defined by the e+e− axis and n is the thrust axis(defined in section 2.2 below), used as a surrogate for the quark-antiquark axis.

Transverse polarization: Additionally one still needs an average transverse polarizationof both quarks. Since the e+e− process does not exhibit a well defined polarization axisonly an average transverse polarization can yield this property. In fact the virtual photonemitted has to be spin 1 which in the helicity basis of the incoming leptons can be createdby the combinations +− and −+. In the case of creating a quark-antiquark pair underthe CMS angle of θ = π/2 (see Fig. 3) both lepton helicity combinations would be equallycontributing and transverse polarization of the quarks has to average out. Hence the quark-antiquark pair will have antiparrallel spins on average. Under more general scatteringangles the possibility of antiparallel spins will be proportional to sin2 θ.

2.2. Azimuthal asymmetries

Two different azimuthal asymmetries will become important in the course of the analysis.Therefore the calculation of them will be first described before having a closer look at theslightly different cross sections. The method already mentioned in the previous subsectionjust translates the definition of the Collins function (eq. 3) into the e+e− → qq case. This

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e

+e

! qq ! hadrons

Ph1

φ0

Ph2 θ2

e−

e+

Ph1⊥

Figure 4: Definition of the azimuthal angle φ0 formed between the plane defined by thelepton momenta and that of one hadron and the second hadron’s transverse momentumPh1⊥ relative to the first hadron.

The dependence on the transverse momentum and on the fractional energy was omittedin the previous formulas for the sake of clarity. The kinematic prefactors are defined as:

A(y) = (12 − y − y2)

CMS=

1

4(1 + cos2 θ) (11)

B(y) = y(1 − y)CMS=

1

4(sin2 θ) , (12)

where y = (1 + cos θ)/2 is a measure of the forwardness of the hard scattering process.Clearly the measurement of the Collins function itself lies hidden in the convolution inte-gral and could at this stage only be obtained under assumptions on the behavior of theintrinsic transverse momentum pT .The second method stays differential in the both azimuthal angles and thus reads[8]:

dσ(e+e− → h1h2X)

dΩdz1dz2dφ1dφ2=

!

q,q

3α2

Q2z21z

22

"

e2q/4(1 + cos2 θ)Dq,[0]

1 (z1)Dq,[0]1 (z2)

+e2q/4 sin2 θ cos(φ1 + φ2)H

⊥,[1],q1 H⊥,[1],q

1

#

, (13)

where the fragmentation functions appear as the zeroth[0] or first[1] moments in theabsolute value of their corresponding transverse momenta:

F [n](z) =

$

d|kT |%

|kT |M

&n

F (z,k2T ) . (14)

Here one is able to access the first kT moment of the Collins function, which can be differ-ent from the preivious convolution seen in equation 9. Nevertheless these cross sections

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Thrust method: phi0 method:

D

hh12 ' 1 +

sin

2T

1 + cos

2T

cos(1 + 2) D

hh0 ' 1 +

sin

2T

1 + cos

2T

cos(20)

modulations we want to measure

T = Thrust polar angle

Collins asymmetries should present this dependence

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Francesca Giordano

Thrust accuracy

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Francesca Giordano

Thrust versus qqbar opening angle

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Francesca Giordano

Reweighted MCManually add an azimuthal modulation at MC generated level

• Case (1) kinematic independent azimuthal modulation• 10% asymmetries unlike sign couples• -5% asymmetries like sign couples

• Case (2) z-dependent modulation similar to real data. • parametrize real data asymmetries with a simple fit function that simulate the

product of two Collins FF: (p0 + p1 zT1) (p0 + p1 zT2)

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Francesca Giordano

Reweighted MCpion-pion

case (1) case (2)

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Francesca Giordano

Reweighted MCManually add an azimuthal modulation at MC generated level

• Case (1) kinematic independent azimuthal modulation• 10% asymmetries unlike sign couples• -5% asymmetries like sign couples

• Case (2) z-dependent modulation similar to real data. • parametrize real data asymmetries with a simple fit function that simulate the

product of two Collins FF: (p0 + p1 zT1) (p0 + p1 zT2)

• Case (3) same z-dependence as case (2), but p0 = p0 *2

• Case (4) same z-dependence as case (2), but p0 = p0 *5

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Francesca Giordano

Reweighted MCpion-pion

case (3) case (4)

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Francesca Giordano

Reweighted MCpion-kaon

case (1) case (2)

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Francesca Giordano

Reweighted MCpion-kaon

case (3) case (4)

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Francesca Giordano

Reweighted MCkaon-kaon

case (1) case (2)

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Francesca Giordano

Reweighted MC

case (3) case (4)

kaon-kaon

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Francesca Giordano

Linear fit of correction factorsCorrection for Thrust asymmetries

Correction for phi0 asymmetries

Use the correction factors from case (4), which has the smallest uncertainty,and then assign a systematic uncertainty as half the difference of the asymmetries

corrected with the largest and smallest correction factor

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Francesca Giordano

Results

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Francesca Giordano

Corrected results

pT2 pT2

pT2pT2

z2 z2

z2z2

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Francesca Giordano

qT

sin2θT/(1+cos2θT)

Corrected results

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Francesca Giordano

Comparison with pi-pi published

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Francesca Giordano

Comparison with Babar

z

pT

z

pT

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Francesca Giordano

Summary

Understanding difference with babar results

Pid systematics

Belle Note 1346 vs. 0.3 ready

Goal: preliminary by DIS

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