F Giordano Collins Fragmentation for Kaon
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Transcript of F Giordano Collins Fragmentation for Kaon
September 17th, 2014, TorinoFrancesca Giordano
COLLINS MEASUREMENTS @ BELLE
Francesca Giordano 2
Fragmentation processor how do the hadrons get formed?
Francesca Giordano 2
Fragmentation processor how do the hadrons get formed?
Dqh
h
q
Fragmentation function describes the process of hadronization of a parton
D h q (z) is the probability that an hadron h with energy z is generated from a parton q
z Eh
Eq=
Eh
Eb=
2Eh
Q
Francesca Giordano 2
Fragmentation processor how do the hadrons get formed?
Strictly related to quark confinement
Dqh
h
q
Fragmentation function describes the process of hadronization of a parton
D h q (z) is the probability that an hadron h with energy z is generated from a parton q
z Eh
Eq=
Eh
Eb=
2Eh
Q
Francesca Giordano
Dqh
h
q
3
e+
e- q
Fragmentation function describes the process of hadronization of a partonStrictly related to quark confinementCleanest way to access FF is in e+e- qq
Fragmentation processor how do the hadrons get formed?
D h q (z) is the probability that an hadron h with energy z is generated from a parton q
z Eh
Eq=
Eh
Eb=
2Eh
Q
Francesca Giordano
Dqh
h
q
3
e+
e- q
Fragmentation function describes the process of hadronization of a partonStrictly related to quark confinementCleanest way to access FF is in e+e- qq
Fragmentation processor how do the hadrons get formed?
e+e!hX /X
q
e+e!qq(Dhq +Dh
q )
D h q (z) is the probability that an hadron h with energy z is generated from a parton q
z Eh
Eq=
Eh
Eb=
2Eh
Q
Francesca Giordano
AhUT = " "
" +"
⇒⇒
⇒⇒
/ Dhq
Dqh
h
q
4
e+
e- q
Fragmentation function describes the process of hadronization of a partonStrictly related to quark confinementCleanest way to access FF is in e+e- qqUniversal: can be used to study the nucleon structure when combined with SIDIS and hadronic reactions data
Fragmentation processor how do the hadrons get formed?
/ H?h1q
D h q (z) is the probability that an hadron h with energy z is generated from a parton q
z Eh
Eq=
Eh
Eb=
2Eh
Q
Francesca Giordano
Collins Fragmentation
5
Collins mechanism: correlation between the parton transverse spin and the direction of final hadron
left-right asymmetry
H1⊥q
H1⊥ PTq
quark
PT
Francesca Giordano
Collins Fragmentation
5
Collins mechanism: correlation between the parton transverse spin and the direction of final hadron
left-right asymmetry
H1⊥q
H1⊥ PTq
quark
PT
strictly related to the outgoing hadron tranverse momentum
TMD!
Francesca Giordano
Collins Fragmentation
5
Collins mechanism: correlation between the parton transverse spin and the direction of final hadron
left-right asymmetry
H1⊥q
H1⊥ PTq
quark
PT
strictly related to the outgoing hadron tranverse momentum
TMD!
chiral even
Chiral odd!
chiral oddchiral oddX H1
⊥
Francesca Giordano
KEKB
BELLE @ KEKB
Francesca Giordano 6
Francesca Giordano
BELLE @ KEKB
Good tracking ϴ [170;1500] and vertex resolution
Good PID: ự(π) ≳ 90%ự(K) ≳ 85%
7
Belle spectrometer:4𝛑 spectrometer optimized for CP violation in B-meson decay
On resonance: √s = 10.58 GeV (e+ e- → Y(4S) → BB)
Off resonance √s = 10.52 GeV (e+ e- → qq (q=u,d,s,c))
Total Luminosity collected:1000 fb-1!!!
KEKB:Asymmetric e+ e- collider
(3.5 / 8 GeV)
Francesca Giordano
Collins Fragmentation
8
e+e!hX /X
q
e+e!qq(H?1,q +H?
1,q)h
e-
e+ q
q
e+e ! q q ! hX
Francesca Giordano
Collins Fragmentation
8
e+e!hX /X
q
e+e!qq(H?1,q +H?
1,q)h
e-
e+ q
q
e+e ! q q ! hX
chiral even
Chiral odd!
chiral oddchiral oddX H1
⊥
Francesca Giordano 9
In e+e- reaction, there is no fixed transverse axis to define azimuthal angles to, and even if there were one the net quark polarization would be 0
Collins Fragmentationh1
h2Back-to-Back jets
q
q
e-
e+
Francesca Giordano 9
In e+e- reaction, there is no fixed transverse axis to define azimuthal angles to, and even if there were one the net quark polarization would be 0
But if we look at the whole event, even though the q and q spin directions are unknown, they must be parallel
-
e+e ! q q ! h1 h2 Xh = , K
Collins Fragmentationh1
h2Back-to-Back jets
q
q
e-
e+
Francesca Giordano
Reference frames
10
𝜙0 method:hadron 1 azimuthal angle with respect
to hadron 2
reference plane (in blue) given by the e+e- direction and one of the hadron
e+e ! q q ! h1 h2 X h = , K
reference plane (in blue) given by the e+e- direction and the qq axis-
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
7
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
9
𝜙1+𝜙2 method:hadron azimuthal angles with respect
to the qq axis proxy-
Thrust axis= proxy for the qq axis-
Francesca Giordano
Reference frames
11
e+e ! q q ! h1 h2 X h = , K
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
7
𝜙1+𝜙2 method:hadron azimuthal angles with respect
to the qq axis proxy-
Thrust axis= proxy for the qq axis-
h1
h2 Back-to-Back jets
q
q
e-
e+
Francesca Giordano
Reference frames
11
e+e ! q q ! h1 h2 X h = , K
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
7
𝜙1+𝜙2 method:hadron azimuthal angles with respect
to the qq axis proxy-
Thrust axis= proxy for the qq axis-
h1
h2 Back-to-Back jets
q
q
e-
e+
Francesca Giordano
Reference frames
11
e+e ! q q ! h1 h2 X h = , K
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
7
𝜙1+𝜙2 method:hadron azimuthal angles with respect
to the qq axis proxy-
Thrust axis= proxy for the qq axis-
h1
h2 Back-to-Back jets
q
q
e-
e+
Francesca Giordano 12
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
7
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
9
𝜙0 method:hadron 1 azimuthal angle with respect
to hadron 2
𝜙1+𝜙2 method:hadron azimuthal angles with respect
to the qq axis proxy-
Reference frames
F [X] =X
Z[2h · kT1h · kT2 kT1 · kT2]
d2kT1d2kT2
2(kT1 + kT2 qT)X
kTi = zi pTi
M0
1 +
sin
2 21 + cos
2 2cos(20)F
hH?1 (z1) ¯H
?1 (z2)
D?1 (z1)
¯D?1 (z2)
i M12
1 +
sin
2 T1 + cos
2 Tcos(1 + 2)
H?[1]1 (z1) ¯H
?[1]1 (z2)
D[0]1 (z1) ¯D
[0]1 (z2)
F [n](zi) Z
d|kT |2h |kT |Mi
i[n]F (zi, |kT |2)
D. BoerNucl.Phys.B806:23,2009
Francesca Giordano 12
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
7
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
9
𝜙0 method:hadron 1 azimuthal angle with respect
to hadron 2
𝜙1+𝜙2 method:hadron azimuthal angles with respect
to the qq axis proxy-
Reference frames
F [X] =X
Z[2h · kT1h · kT2 kT1 · kT2]
d2kT1d2kT2
2(kT1 + kT2 qT)X
kTi = zi pTi
M0
1 +
sin
2 21 + cos
2 2cos(20)F
hH?1 (z1) ¯H
?1 (z2)
D?1 (z1)
¯D?1 (z2)
i M12
1 +
sin
2 T1 + cos
2 Tcos(1 + 2)
H?[1]1 (z1) ¯H
?[1]1 (z2)
D[0]1 (z1) ¯D
[0]1 (z2)
F [n](zi) Z
d|kT |2h |kT |Mi
i[n]F (zi, |kT |2)
D. BoerNucl.Phys.B806:23,2009
Francesca Giordano 12
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
7
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
9
𝜙0 method:hadron 1 azimuthal angle with respect
to hadron 2
𝜙1+𝜙2 method:hadron azimuthal angles with respect
to the qq axis proxy-
Reference frames
F [X] =X
Z[2h · kT1h · kT2 kT1 · kT2]
d2kT1d2kT2
2(kT1 + kT2 qT)X
kTi = zi pTi
M0
1 +
sin
2 21 + cos
2 2cos(20)F
hH?1 (z1) ¯H
?1 (z2)
D?1 (z1)
¯D?1 (z2)
i M12
1 +
sin
2 T1 + cos
2 Tcos(1 + 2)
H?[1]1 (z1) ¯H
?[1]1 (z2)
D[0]1 (z1) ¯D
[0]1 (z2)
F [n](zi) Z
d|kT |2h |kT |Mi
i[n]F (zi, |kT |2)
D. BoerNucl.Phys.B806:23,2009
Francesca Giordano
Product of 2 Collins FFs
13
Like-sign couples
Francesca Giordano
Product of 2 Collins FFs
13
Like-sign couples
Fav Ł UnFav Fav Ł UnFav
Francesca Giordano
Product of 2 Collins FFs
14
Like-sign couples
Fav Ł UnFav Fav Ł UnFavUnlike-sign couples
All charges couples
Fav Ł Fav UnFav Ł UnFav
Fav + unFav Ł UnFav + Fav
Francesca Giordano 15
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
7
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
9
e+e ! q q ! h1 h2 X h = , K𝜙0 method:
hadron 1 azimuthal angle with respect to hadron 2
𝜙1+𝜙2 method:hadron azimuthal angles with respect
to the qq axis proxy-
Reference frames
R0 =N0(0)
hN0iR12 =
N12(1 + 2)
hN12i
Francesca Giordano 16
But! Acceptance and radiation effects also contribute to azimuthal
asymmetries!
Double-ratios
Francesca Giordano 16
But! Acceptance and radiation effects also contribute to azimuthal
asymmetries!
Double-ratios
Simulation: unlike-sign, like-sign
Francesca Giordano 16
But! Acceptance and radiation effects also contribute to azimuthal
asymmetries!
Double-ratios
Simulation: unlike-sign, like-signData: unlike-sign, like-sign
Francesca Giordano 16
But! Acceptance and radiation effects also contribute to azimuthal
asymmetries!
Double-ratios
Simulation: unlike-sign, like-signData: unlike-sign, like-sign
To reduce such non-Collins effects:divide the sample of hadron couples in unlike-sign and like-sign (or All-charges),
and extract the asymmetries of the super ratios between these 2 samples:
Unlike-sign couples / Like-sign couples Unlike-sign couples / All charges
Dh1h2ul = RU/RL Dh1h2
uc = RU/RC
Francesca Giordano 17
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
7
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
9
𝜙0 method𝜙1+𝜙2 method
Double-ratios
M0
1 +
sin
2 21 + cos
2 2cos(20)F
hH?1 (z1) ¯H
?1 (z2)
D?1 (z1)
¯D?1 (z2)
i
Fitted by
B12(1 +A12 cos(1 + 2)) B0(1 +A0 cos(20))
M12
1 +
sin
2 T1 + cos
2 Tcos(1 + 2)
H?[1]1 (z1) ¯H
?[1]1 (z2)
D[0]1 (z1) ¯D
[0]1 (z2)
Dh1h2uc = RU/RCDh1h2
ul = RU/RL
Francesca Giordano 17
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
7
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
9
𝜙0 method𝜙1+𝜙2 method
Double-ratios
M0
1 +
sin
2 21 + cos
2 2cos(20)F
hH?1 (z1) ¯H
?1 (z2)
D?1 (z1)
¯D?1 (z2)
i
Fitted by
B12(1 +A12 cos(1 + 2)) B0(1 +A0 cos(20))
M12
1 +
sin
2 T1 + cos
2 Tcos(1 + 2)
H?[1]1 (z1) ¯H
?[1]1 (z2)
D[0]1 (z1) ¯D
[0]1 (z2)
A0 =
sin
2
1 + cos
2 FhH?
1 (z1) ¯H?1 (z2)
D?1 (z1)
¯D?1 (z2)
iA12 =
sin
2
1 + cos
2
H?[1]1 (z1) ¯H
?[1]1 (z2)
D[0]1 (z1) ¯D
[0]1 (z2)
Francesca Giordano
Published results:
18
PRD 78, 032011 (2008)PRD 78, 032011 (2008)
𝜙0 method𝜙1+𝜙2 method
𝜋𝜋Belle publications
PRL 96,232002, (2006)PRD 78, 032011 (2008)
Francesca Giordano
Belle vs. Babar
19
Francesca Giordano
SIDIS
20
DF
σFF
chiral even
Chiral odd!
chiral oddchiral oddX H1
⊥
Francesca Giordano
SIDIS
20
DF
σFF
H?1h1 ⊥
H?1h1
chiral even
Chiral odd!
chiral oddchiral oddX H1
⊥
Francesca Giordano
[Airapetian et al., Phys. Lett. B 693 (2010) 11-16]
Collins amplitudes in SIDISH?
1h1 AUT /
π+/−
Κ+/−
Deuterium
21
𝜋 Collins FF
Francesca Giordano
[Airapetian et al., Phys. Lett. B 693 (2010) 11-16]
Collins amplitudes in SIDISH?
1h1 AUT /
π+/−
Κ+/−
Deuterium
21
𝜋 Collins FF
Anselmino et al.Phys.Rev. D75 (2007)
transversity
Francesca Giordano
DY 2011, BNL - May 11th, 2011Gunar Schnell
quark pol.
U L T
nucl
eon
pol.
U f1 h1
L g1L h1L
T f1T g1T h1, h1T
Twist-2 TMDs
significant in size and
opposite in sign for charged
pions
disfavored Collins FF large
and opposite in sign to
favored one
leads to various cancellations
in SSA observables
21
Transversity distribution
(Collins fragmentation)
+
u
[A. Airapetian et al, Phys. Rev. Lett. 94 (2005) 012002]
2005: First evidence from HERMES
SIDIS on proton
Non-zero transversity
Non-zero Collins function
-0.1
0
0.1
0.2
-0.2
-0.1
0
0.1 0.2 0.3
2 !
sin
(" +
"S)#
UT
$
$+
x
$-
z
0.3 0.4 0.5 0.6 0.7
Wednesday, May 11, 2011
H?,unfav1 H?,fav
1
[Airapetian et al., Phys. Lett. B 693 (2010) 11-16]
Collins amplitudes in SIDISH?
1h1 AUT /
π+/−
Κ+/−
Deuterium
21
𝜋 Collins FF
Anselmino et al.Phys.Rev. D75 (2007)
transversity
Francesca Giordano
Collins amplitudes in SIDIS
22
H?1h1 AUT /
AUU / h?1 H?
1
Francesca Giordano
Collins amplitudes in SIDIS
22
H?1h1 AUT /
K+ amplitudes larger than 𝜋+?
AUU / h?1 H?
1
Francesca Giordano
More recently...
23
z qT sin2𝜭/(1+cos2𝜭) pT
New!
New!
New!
New!New!New!
New! New! New!
z qT sin2𝜭/(1+cos2𝜭) pT
New!
New!
New!
New!New!New!
New! New! New!
Duc = RU/RC
Dkuc = RUk/RCk
Dkkuc = RUkk/RCkk
Dkkul = RUkk/RLkk
Dkul = RUk/RLk
Dul = RU/RL
Francesca Giordano
More recently...
23
z qT sin2𝜭/(1+cos2𝜭) pT
New!
New!
New!
New!New!New!
New! New! New!
z qT sin2𝜭/(1+cos2𝜭) pT
New!
New!
New!
New!New!New!
New! New! New!
Word of caution: this analysis is mainly aimed at kaons, so kinematic cuts and binning are optimized for kaons, and the same values used for pion too.
𝜋𝜋 results cannot be compared directly to published results
Duc = RU/RC
Dkuc = RUk/RCk
Dkkuc = RUkk/RCkk
Dkkul = RUkk/RLkk
Dkul = RUk/RLk
Dul = RU/RL
Francesca Giordano 24
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
7
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
9
𝜙0 method𝜙1+𝜙2 method
M0
1 +
sin
2 21 + cos
2 2cos(20)F
hH?1 (z1) ¯H
?1 (z2)
D?1 (z1)
¯D?1 (z2)
i M12
1 +
sin
2 T1 + cos
2 Tcos(1 + 2)
H?[1]1 (z1) ¯H
?[1]1 (z2)
D[0]1 (z1) ¯D
[0]1 (z2)
More recently...
Francesca Giordano 24
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
7
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
9
𝜙0 method𝜙1+𝜙2 method
M0
1 +
sin
2 21 + cos
2 2cos(20)F
hH?1 (z1) ¯H
?1 (z2)
D?1 (z1)
¯D?1 (z2)
i M12
1 +
sin
2 T1 + cos
2 Tcos(1 + 2)
H?[1]1 (z1) ¯H
?[1]1 (z2)
D[0]1 (z1) ¯D
[0]1 (z2)
Both interesting: different integration of FFs in pTi, might provide information on the Collins pT
dependence
Technically more complicated: require the determination of a qq proxy (Thrust axis)-
More recently...
Francesca Giordano 24
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
7
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
9
𝜙0 method𝜙1+𝜙2 method
M0
1 +
sin
2 21 + cos
2 2cos(20)F
hH?1 (z1) ¯H
?1 (z2)
D?1 (z1)
¯D?1 (z2)
i M12
1 +
sin
2 T1 + cos
2 Tcos(1 + 2)
H?[1]1 (z1) ¯H
?[1]1 (z2)
D[0]1 (z1) ¯D
[0]1 (z2)
Both interesting: different integration of FFs in pTi, might provide information on the Collins pT
dependence
Technically more complicated: require the determination of a qq proxy (Thrust axis)-
More recently...
Francesca Giordano
N j,raw = PijNi
i = ,K
25
PID correction
Francesca Giordano
Perfect PID j = i
N j,raw = PijNi
j = e, µ,,K, p
i = ,K
25
PID correction
Francesca Giordano
Perfect PID j = i
ự(π) ≳ 90% ự(K) ≳ 85%
N j,raw = PijNi
BUT!!
j = e, µ,,K, p
i = ,K
25
PID correction
Francesca Giordano
Perfect PID j = i
ự(π) ≳ 90% ự(K) ≳ 85%
90%
N j,raw = PijNi
BUT!!
j = e, µ,,K, p
i = ,K
25
PID correction
Francesca Giordano
Perfect PID j = i
ự(π) ≳ 90% ự(K) ≳ 85%
e, µ,K, p
90%
10%
Pij =
0
BBBB@
Pe!e Pe!µ Pe! Pe!K Pe!p
Pµ!e Pµ!µ Pµ! Pµ!K Pµ!p
P!e P!µ P! P!K P!p
PK!e PK!µ PK! PK!K PK!p
Pp!e Pp!mu Pp! Pp!K Pp!p
1
CCCCA
N j,raw = PijNi
BUT!!
j = e, µ,,K, p
i = ,K
25
PID correction
Francesca Giordano
Perfect PID j = i
ự(π) ≳ 90% ự(K) ≳ 85%
e, µ,K, p
90%
10%
Pij =
0
BBBB@
Pe!e Pe!µ Pe! Pe!K Pe!p
Pµ!e Pµ!µ Pµ! Pµ!K Pµ!p
P!e P!µ P! P!K P!p
PK!e PK!µ PK! PK!K PK!p
Pp!e Pp!mu Pp! Pp!K Pp!p
1
CCCCA
N j,raw = PijNi
BUT!!
j = e, µ,,K, p
i = ,K
N i = P1ij N j,raw
25
PID correction
Francesca Giordano
ijHow to determine the Pij?
26
Francesca Giordano
ijHow to determine the Pij?From data!
26
Francesca Giordano
ijHow to determine the Pij?D D0
+slow
From data!
26
Francesca Giordano
ijHow to determine the Pij?D D0 K
+slow
+fastFrom data!
26
Francesca Giordano
ijHow to determine the Pij?D D0 K
+slow
+fast
mD m0
D
Negative hadron = .
(no PID likelihood used)K
From data!
26
Francesca Giordano
ijHow to determine the Pij?D D0 K
+slow
+fast
mD m0
D
Negative hadron = .
(no PID likelihood used)K
Negative hadron identified as .
mD m0
D
From data!
26
Francesca Giordano
ijHow to determine the Pij?D D0 K
+slow
+fast
mD m0
D
Negative hadron = .
(no PID likelihood used)K
Negative hadron identified as .
mD m0
D PK!
From data!
26
Francesca Giordano
ijHow to determine the Pij?D D0 K
+slow
+fast
mD m0
D
Negative hadron = .
(no PID likelihood used)K
Negative hadron identified as .
mD m0
D PK!
K
PK!K
From data!
26
Francesca Giordano
ijHow to determine the Pij?D D0 K
+slow
+fast
mD m0
D
Negative hadron = .
(no PID likelihood used)K
Negative hadron identified as .
mD m0
D PK!
K
PK!K
PK!p
p
From data!
26
Francesca Giordano
ijHow to determine the Pij?D D0 K
+slow
+fast
mD m0
D
Negative hadron = .
(no PID likelihood used)K
Negative hadron identified as .
mD m0
D PK!
µ
PK!µ
K
PK!K
PK!p
p
From data!
26
Francesca Giordano
ijHow to determine the Pij?D D0 K
+slow
+fast
mD m0
D
Negative hadron = .
(no PID likelihood used)K
Negative hadron identified as .
mD m0
D PK!
eµ
PK!e
PK!µ
K
PK!K
PK!p
p
From data!
26
Francesca Giordano
2D correctionDetector performance depends on momentum
and scattering angle!
27
Francesca Giordano
2D correctionDetector performance depends on momentum
and scattering angle!
Pij Pij(p, )
27
Francesca Giordano
2D correctionDetector performance depends on momentum
and scattering angle!Bin
#plab [GeV/c] bin ranges
0 [0.5,0.65)1 [0.65,0.8)2 [0.8,1.0)3 [1.0,1.2)4 [1.2,1.4)5 [1.4,1.6)6 [1.6,1.8)7 [1.8,2.0)8 [2.0,2.2)9 [2.2,2.4)
10 [2.4,2.6)11 [2.6,2.8)12 [2.8,3.0)13 [3.0,3.5)14 [3.5,4.0)15 [4.0,5.0)16 [5.0,8.0)
Bin #
cos𝜽lab
bin ranges0 [-0.511,-0.300)1 [-0.300,-0.152)2 [-0.512,0.017)3 [0.017,0.209)4 [0.209,0.355)5 [0.355,0.435)6 [0.435,0.542)7 [0.542,0.692)8 [0.692,0.842)
Pij Pij(p, )
27
Francesca Giordano
2D correction
no PID cut e/μ PID cut π PID cut
K PID cut p PID cut Unsel.
K from D* decay for plab in [1.4,1.6) and cos𝜽lab in [0.209,0.355)
0
BBBB@
Pe!e Pe!µ Pe! Pe!K Pe!p
Pµ!e Pµ!µ Pµ! Pµ!K Pµ!p
P!e P!µ P! P!K P!p
PK!e PK!µ PK! PK!K PK!p
Pp!e Pp!mu Pp! Pp!K Pp!p
1
CCCCA
28
Francesca Giordano
2D correction
no PID cut e/μ PID cut π PID cut
K PID cut p PID cut Unsel.
K from D* decay for plab in [1.4,1.6) and cos𝜽lab in [0.209,0.355)
0
BBBB@
Pe!e Pe!µ Pe! Pe!K Pe!p
Pµ!e Pµ!µ Pµ! Pµ!K Pµ!p
P!e P!µ P! P!K P!p
PK!e PK!µ PK! PK!K PK!p
Pp!e Pp!mu Pp! Pp!K Pp!p
1
CCCCA
28
Francesca Giordano
2D correction
no PID cut e/μ PID cut π PID cut
K PID cut p PID cut Unsel.
K from D* decay for plab in [1.4,1.6) and cos𝜽lab in [0.209,0.355)
0
BBBB@
Pe!e Pe!µ Pe! Pe!K Pe!p
Pµ!e Pµ!µ Pµ! Pµ!K Pµ!p
P!e P!µ P! P!K P!p
PK!e PK!µ PK! PK!K PK!p
Pp!e Pp!mu Pp! Pp!K Pp!p
1
CCCCA
pπ, K -> j from D* decay pπ, p -> j from ờ decay
pe, µ -> j from J/ψ decay
28
Francesca Giordano
2D correction0
BBBB@
Pe!e Pe!µ Pe! Pe!K Pe!p
Pµ!e Pµ!µ Pµ! Pµ!K Pµ!p
P!e P!µ P! P!K P!p
PK!e PK!µ PK! PK!K PK!p
Pp!e Pp!mu Pp! Pp!K Pp!p
1
CCCCA
pπ, K -> j from D* decay pπ, p -> j from ờ decay
pe, µ -> j from J/ψ decay
29
Francesca Giordano
2D correction0
BBBB@
Pe!e Pe!µ Pe! Pe!K Pe!p
Pµ!e Pµ!µ Pµ! Pµ!K Pµ!p
P!e P!µ P! P!K P!p
PK!e PK!µ PK! PK!K PK!p
Pp!e Pp!mu Pp! Pp!K Pp!p
1
CCCCA
pπ, K -> j from D* decay pπ, p -> j from ờ decay
pe, µ -> j from J/ψ decay
29
Francesca Giordano
PID correction
K
< 15%
30% >30
Francesca Giordano
uds-charm-bottom-tau contributions
31
𝜋𝜋 couples
Published 𝜋𝜋 studied a charm enhanced data and found charm contribute only
as dilution=> charm contribution corrected out
Francesca Giordano 32
𝜋K couples
uds-charm-bottom-tau contributions
Francesca Giordano 33
KK couples
uds-charm-bottom-tau contributions
Francesca Giordano 33
KK couples
uds-charm-bottom-tau contributions
For the moment charm contributionis not being corrected out
in any of the samples (𝜋𝜋, 𝜋K, KK)
Francesca Giordano
yCollins asymmetries
34
Francesca Giordano 35
𝜋𝜋 => non-zero asymmetries, increase with z1, z2
𝜋K => asymmetries compatible with zero
KK => non-zero asymmetries, increase with z1,z2
similar size of pion-pion
𝜙0 yasymmetries
Francesca Giordano 36
yasymmetries𝜙0𝜋𝜋 => non-zero asymmetries,
increase with z1, z2
𝜋K => asymmetries compatible with zero
KK => non-zero asymmetries, increase with z1,z2
similar size of pion-pion
But! charm have different contributions, we need to account for it!
Francesca Giordano 37
versus sin2𝜭/(1+cos2𝜭)
A0 =
sin
2
1 + cos
2 FhH?
1 (z1) ¯H?1 (z2)
D?1 (z1)
¯D?1 (z2)
i
linear in sin2𝜭/(1+cos2𝜭),
go to 0 for sin2𝜭/(1+cos2𝜭) 0
fit form: p0 + p1 sin2𝜭/(1+cos2𝜭)
𝜋𝜋
𝜋K
KK
QCD test?
Francesca Giordano 38
Collins fragmentation: u-d-s contributions(vs 0.1, 27th Sept, 2012)
1 Definition and assumptions
u, d → π (ud, ud)
Dfav = Dπ+
u = Dπ−
d = Dπ−
u = Dπ+
d(1)
Ddis = Dπ−
u = Dπ+
d = Dπ+
u = Dπ−
d(2)
s → π (ud, ud)
Ddiss→π = Dπ+
s = Dπ−
s = Dπ+
s = Dπ−
s (3)
u, d → K (us, us)
Dfavu→K = DK+
u = DK−
u (4)
Ddisu,d→K = DK−
u = DK+
u = DK+
d = DK−
d= DK−
d = DK+
d(5)
s → K (us, us)
Dfavs→K = DK−
s = DK+
s (6)
Ddiss→K = DK+
s = DK−
s (7)
In the end we are left with 7 possible fragmentation functions:
Dfav , Ddis, Ddiss→π, D
favu→K , Ddis
u,d→K , Dfavs→K , Ddis
s→K (8)
2 Pion-Pion
e+e− → π±π∓ (unLike sign)
NUππ ∝
!
q
e2q(Dfav1 D
fav2 +Ddis
1 Ddis2 ) (9)
e+e− → π±π± (Like sign)
NLππ ∝
!
q
e2q(Dfav1 Ddis
2 +Ddis1 D
fav2 ) (10)
3 Pion-Kaon
e+e− → π±K∓ (unLike sign)
NUπK ∝
!
q
e2q (11)
e+e− → π±K± (Like sign)
NLπK ∝
!
q
e2q (12)
1
Assuming charm contribute only as a dilution
Fragmentation contributions
Francesca Giordano 39
For pion-pion couples:
For pion-Kaon couples:
For Kaon-Kaon couples:
Fragmentation contributions
Francesca Giordano 39
For pion-pion couples:
For pion-Kaon couples:
For Kaon-Kaon couples:
Not so easy! A full phenomenological study needed!
Fragmentation contributions
Francesca Giordano
ySummary & outlook
40
𝜙0 asymmetries present similar features for 𝜋𝜋 and KK couplesvery small/compatible with zero for 𝜋K couplesfor 𝜋𝜋 and 𝜋K the sin2𝜭/(1+cos2𝜭) dependence of asymmetries are not inconsistent with a linear dependence going to zeroKK show a more convoluted sin2𝜭/(1+cos2𝜭) dependence
Francesca Giordano
ySummary & outlook
40
𝜙0 asymmetries present similar features for 𝜋𝜋 and KK couplesvery small/compatible with zero for 𝜋K couplesfor 𝜋𝜋 and 𝜋K the sin2𝜭/(1+cos2𝜭) dependence of asymmetries are not inconsistent with a linear dependence going to zeroKK show a more convoluted sin2𝜭/(1+cos2𝜭) dependence
𝜙12 asymmetries with Thrust axis in progress study using jet algorithm instead of Thrust in progress
Francesca Giordano
ySummary & outlook
40
𝜙0 asymmetries present similar features for 𝜋𝜋 and KK couplesvery small/compatible with zero for 𝜋K couplesfor 𝜋𝜋 and 𝜋K the sin2𝜭/(1+cos2𝜭) dependence of asymmetries are not inconsistent with a linear dependence going to zeroKK show a more convoluted sin2𝜭/(1+cos2𝜭) dependence
𝜙12 asymmetries with Thrust axis in progress study using jet algorithm instead of Thrust in progress
Stay tuned!
Francesca Giordano
http://abstrusegoose.com/342
41
Francesca Giordano 42
Backups
Francesca Giordano
Kinematic variables
43
hadron energy fraction with respect to parton
qT component of virtual photon momentum transverse to the h1h2 axis in the frame
where h1 and h2 are back-to-back
z 0.2 0.25 0.3 0.42 1
pT12 0 0.13 0.3 0.5 3
pT0 0 0.13 0.25 0.4 0.5 0.6 0.75 1 3
qT 0 0.5 1 1.25 1.5 1.75 2 2.25 2.5
sin2𝜭/(1+cos2𝜭) 0.4 0.45 0.5 0.6 0.7 0.8 0.9 0.97 1
z1, z2
pT component of hadron momentum transverse to reference direction1. 𝜙1+𝜙2 method: the thrust axis
2. 𝜙0 method: hadron 2
pT1, pT2
pT0
Francesca Giordano 44
𝜙0 yasymmetries
Francesca Giordano 45
𝜙0 yasymmetries
Francesca Giordano
yMore asymmetries
46
𝜙0