View
218
Download
0
Category
Preview:
Citation preview
JAM global QCD analysis of spin-dependent partondistributions
Nobuo Sato
In Collaboration with:Alberto Accardi, Jacob Ethier, Wally Melnitchouk
1 / 26
Motivations for extracting spin PDFs (SPDFs)
I Spin sum rule:
1
2=
1
2∆Σ + ∆g + Lq + Lg
I Large x behavior of SPDFs:
q↓ ∼ (1− x)2q↑ → ∆q/q → 1 as x→ 1
I Testing the Burkhardt-Cottingham sum rule:∫ 1
0
dx g2(x) = 0
2 / 26
How to extract SPDFs?
I SPDFs are extracted from cross section asymmetriesbetween different initial state polarizations.
I Inclusive DIS: e+N → e′ +X (In this talk)
I Semi-Inclusive DIS: e+N → e′ + h+X
I Inclusive Jet production: p+ p→ J +X
I Inclusive π0 production: p+ p→ π0 +X
I W boson production: p+ p→ W → l +X
3 / 26
How to extract SPDFs
I In practice the polarized data covers a small range in Q2
in comparison with unpolarized data.
I In order to maximally utilize the data, the JAM analysisset the cuts to Q2 ≥ 1GeV2 and W 2 ≥ 3.5GeV2.
I For such cuts the extraction of g1 and g2 is potentiallysensitive to:
I Finite Q2 corrections: TMC, HT
I Nuclear corrections for nuclear targets (especially for high-x).
4 / 26
Polarized inclusive DIS
I The measured DIS asymmetries can be described interms of g1 and g2 structure functions
A|| =σ↑⇓ − σ↑⇑σ↑⇓ + σ↑⇓
= D(A1 + ηA2)
A⊥ =σ↑⇒ − σ↑⇐σ↑⇒ + σ↑⇐
= d(A2 − ξA1)
A1 =(g1 − γ2g2)
F1A2 = γ
(g1 + g2)
F1
I In the JAM analysis g1 and g2 are parametrized as
g1(x,Q2) = gT2+TMC
1 + gT31 + gT4
1
g2(x,Q2) = gT2+TMC
2 + gT32
5 / 26
Polarized inclusive DIS
g1(x,Q2) = gT2+TMC
1 + gT31 + gT4
1
g2(x,Q2) = gT2+TMC
2 + gT32
I Leading twist @ NLO without TMC
gT21 (x) =1
2
∑q
e2q [∆Cqq ∗∆q(x) + ∆Cqg ∗∆g(x)]
I with TMC (Blumlein, Tkabladze NPB 553 427)
gT2+TMC1,2 (x) = ζ11,2 g
T21 (ξ) + ζ21,2
∫ 1
ξ
dz
zgT21 (z) +ζ31,2
∫ 1
ξ
dz
zgT21 (z) log(z/ξ)
I ζ1,2,31,2 = ζ1,2,31,2
(x, M
2
Q2
)and ξ = ξ
(x, M
2
Q2
)6 / 26
Polarized inclusive DIS
g1(x,Q2) = gT2+TMC
1 + gT31 + gT4
1
g2(x,Q2) = gT2+TMC
2 + gT32
I Twist-3 part of g2 (PRD 83,094023) with free parameters ti
gT32 (x) = t0
[log x+ (1− x) +
1
2(1− x)2
]+
4∑i=1
ti(1− x)i+2
I Twist-4 part of g1 (spline parametrization)
gT41 (x) =h(x)
Q2
I Twist-3 part of g1 can be obtained from gT32 (NPB,553)
gT31 (x) =4M2x2
Q2
[gT32 (x)− 2
∫ 1
x
dy
ygT32 (x)
]7 / 26
Nuclear corrections in polarized DIS
I To achieve flavour separation (at least between ∆u and∆d) polarized data with deuterium and 3He are used.
I Ignoring Fermi motion the “nuclear” structure functionsare given by
gAi (x,Q2) = Pp/A gpi (x,Q
2) + Pn/A gni (x,Q2)
I This is called Effective Polarization Approximation(EPA). (no dependence on x)
8 / 26
Nuclear corrections in polarized DIS
I At large x the nuclear smearing plays an important role(PRC 88,5)
I The nuclear corrections can be implemented asconvolutions between the nuclear smearing functions(fNij ) and nucleon structure functions gNi
gAi (x,Q2) =∑N
∫dy
yfNij (y, γ)gNj (x/y,Q2)
9 / 26
SPDF parametrization
I For inclusive DIS we only need to parametrize∆q+ = ∆q + ∆q for q = u, d, s and ∆g
I We use the following parametrization
∆q+(x) = Nxa(1− x)b(1 + dx)
I In practice the inclusive DIS data is insensitive to ∆s+
and ∆g. We fix their shape parameters using the resultsfrom DSSV collaboration except for the normalizations.
10 / 26
The fitting
In this work two fits are considered
I RUN0: Global data of inclusive DIS without JLab data.We include the new COMPASS data (arXiv:1503.08935)
I RUN1: RUN0 + JLab data.
I In both runs we fit 26 parameters
- 10 shape parameters for SPDFs- 4+4 parameters for proton T3 and T4- 4+4 parameters for neutron T3 and T4
11 / 26
World Data kinematics (Q2 > 1GeV 2, W 2 > 3.5GeV 2)
10−3 10−2 10−1 100
x
100
101
102
Q2
EMC(npts = 10)
10−3 10−2 10−1 100
x
100
101
102
Q2
SMC(npts = 28)
10−3 10−2 10−1 100
x
100
101
102
Q2
SLAC(npts = 579)
10−3 10−2 10−1 100
x
100
101
102
Q2
HERMES(npts = 103)
10−3 10−2 10−1 100
x
100
101
102
Q2COMPASS
(npts = 81)
(2015)
10−3 10−2 10−1 100
x
100
101
102
Q2
JLab(npts = 675)
12 / 26
World Data kinematics
0
100
200
300
400
500
600
700
800
#p
ts
RUN0 proton
deuteron
helium
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9x
0
50
100
150
200
250
300
#p
ts
RUN0
RUN1
0
100
200
300
400
500
600
700
800
#p
ts
RUN1
0 2 4 6 8 10
Q2
0
50
100
150
200
250
300
#p
ts
RUN0
RUN1
13 / 26
RUN0 fit
npts = 801 χ2 = 667.90 χ2DOF = 0.83
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
χ2DOF
−1.0−0.5 0.0 0.5 1.0
Npts(T > D) Npts(T < D)
HERMES data1 A1n
HERMES data1 Apap
EMC A1p
SMC data1 A1p
SMC data2 A1d
COMPASS data 1 A1d
COMPASS data 2 A1p
SLAC E142 A13He
SLAC E142 A23He
SLAC E80130 Apap
COMPASS data 1 A1p
SLAC E143 Apap
HERMES data1 Apad
SLAC E155x Atped
SLAC E143 Apep
SLAC E143 Aped
SLAC E155 Apep
SLAC E154 Ape3He
SLAC E143 Apad
SLAC E155 Apad
HERMES data1 A2p
SMC data2 A1p
14 / 26
RUN1 fit
npts = 1476 χ2 = 1405.3 χ2DOF = 0.95
0.0 0.5 1.0 1.5 2.0 2.5
χ2DOF
−1.0−0.5 0.0 0.5 1.0
Npts(T > D) Npts(T < D)
JLabHallA E99117 1 Apa3HeHERMES data1 A1n
EMC A1pHERMES data1 Apap
SLAC E142 A13HeSMC data2 A1dSMC data1 A1p
SLAC E80130 ApapSLAC E142 A23He
COMPASS data 1 A1pCOMPASS data 2 A1pCOMPASS data 1 A1d
JLabHallA E06014 1 Ape3HeJLabHallA E06014 2 Apa3He
SLAC E143 ApapJLabHallA E06014 2 Ape3He
SLAC E155x AtpedSLAC E143 Apep
HERMES data1 ApadSLAC E143 Aped
JLabHallB eg1 dvcs 1 ApapSLAC E154 Ape3He
SLAC E155 ApepJLabHallB eg1b ApaDd
SLAC E155 ApadSLAC E143 Apad
HERMES data1 A2pJLabHallB eg1 dvcs 1 Apad
SMC data2 A1pJLabHallB eg1 dvcs 2 Apap
JLabHallA E99117 1 Ape3HeJLabHallA E06014 1 Apa3He
15 / 26
Error Analysis
I We use the Hessian Method to propagate the parameteruncertainties.
I The Hessian (H) and Covariance matrix (C):
χ2(~p) =∑i
[Datai − Thyi(~p)]2
σ2uncor,i + σ2
cor,i
Hij =1
2
∂χ2(~p)
∂pi∂pj
∣∣∣∣~p=best
, C = H−1
16 / 26
Correlation matrices for RUN0 and RUN1
upN
upa
upb
upd
dpN
dpa
dpb
dpd
spN
gN
T3p
t0T3
pt1
T3p
t2T3
pt3
T4p
h1T4
ph2
T4p
h3T4
ph4
T3n
t0T3
nt1
T3n
t2T3
nt3
T4n
h1T4
nh2
T4n
h3T4
nh4
up N
up a
up b
up d
dp N
dp a
dp b
dp d
sp N
g N
T3p t0
T3p t1
T3p t2
T3p t3
T4p h1
T4p h2
T4p h3
T4p h4
T3n t0
T3n t1
T3n t2
T3n t3
T4n h1
T4n h2
T4n h3
T4n h4
RU
N0
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
upN
upa
upb
upd
dpN
dpa
dpb
dpd
spN
gN
T3p
t0T3
pt1
T3p
t2T3
pt3
T4p
h1T4
ph2
T4p
h3T4
ph4
T3n
t0T3
nt1
T3n
t2T3
nt3
T4n
h1T4
nh2
T4n
h3T4
nh4
up N
up a
up b
up d
dp N
dp a
dp b
dp d
sp N
g N
T3p t0
T3p t1
T3p t2
T3p t3
T4p h1
T4p h2
T4p h3
T4p h4
T3n t0
T3n t1
T3n t2
T3n t3
T4n h1
T4n h2
T4n h3
T4n h4
RU
N1
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
I Correlation matrix ρ is defined asρi,j = Ci,j/
√Ci,iCj,j
I The inclusion of the JLab datainduces higher decorrelationbetween the HT (neutron)parameters and SPDF parameters.
I The uncertainties on thedistributions are estimated bydiagonalizing the covariancematrix.
17 / 26
The eigen-directions
I The shifts of the parameters from their best values canbe parametrized with scale factors {ti} in the eigen-basisof the covariance matrix {ei}:
∆~p = ~p− ~p0 =∑i
tiei
I The Hessian method assumes that
P(∆~p) ≈∏i
Pi(ti) ∝∏i
exp
[−1
2χ2(ti)
]I The probability distribution function P for the shifts
decorrelates in the eigen-space18 / 26
Distributions along eigen-directions
−0.03 −0.02 −0.01 0.00 0.01
t0
0
10
20
30
40
50
P(t
0)
t−t+
−4 −3 −2 −1 0 1 2 3 4
t12
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
P(t
12)
normal− dist
68%
95%
δO2 ' 1
2
∑i
[O(t+i )−O(t−i )
]2
I t±i are defined as the edges of the confidence region.19 / 26
RUN1 parameters relative to RUN0
−3
−2
−1
0
1
2
3
4
5
RU
N1/
RU
N0
N a
b
d N a
b
dN
N
t0 t1 t2t3
t0
t1 t2t3
h1
h2
h3
h4
h1h2
h3
h4
∆u+
∆d+
∆s+
∆g
T3p
T3n
T4p
T4n
RUN1 uncertanties
20 / 26
The extracted g1 structure function at Q2 = 1GeV2
RUN0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.02
0.00
0.02
0.04
0.06
xg1
Total
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.02
0.00
0.02
0.04
0.06
T2
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.02
0.00
0.02
0.04
0.06
T3
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.04
−0.03
−0.02
−0.01
0.00
0.01
0.02T4
proton 95%
proton 68%
neutron 95%
neutron 68%
RUN1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.02
0.00
0.02
0.04
0.06
xg1
Total
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.02
0.00
0.02
0.04
0.06
T2
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.02
0.00
0.02
0.04
0.06
T3
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.04
−0.03
−0.02
−0.01
0.00
0.01
0.02T4
proton 95%
proton 68%
neutron 95%
neutron 68%
I Significant reduction of the uncertainties in gTOTAL1
I Most of the uncertainties are due to HT contributions.21 / 26
The extracted g2 structure function at Q2 = 1GeV2
RUN0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.04
−0.03
−0.02
−0.01
0.00
0.01
0.02
0.03
0.04
xg2
Total
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.04
−0.03
−0.02
−0.01
0.00
0.01
0.02
0.03
0.04T2
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.04
−0.03
−0.02
−0.01
0.00
0.01
0.02
0.03
0.04T3
proton 95%
proton 68%
neutron 95%
neutron 68%
RUN1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.04
−0.03
−0.02
−0.01
0.00
0.01
0.02
0.03
0.04
xg2
Total
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.04
−0.03
−0.02
−0.01
0.00
0.01
0.02
0.03
0.04T2
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.04
−0.03
−0.02
−0.01
0.00
0.01
0.02
0.03
0.04T3
proton 95%
proton 68%
neutron 95%
neutron 68%
I problem: large variations in the fitted HT → need tocheck goodness of fit.
22 / 26
Goodness of fit
I We use the likelihood-ratio test to compare the goodnessof fit between fits with and without HT.
I The statistics t is defined as
t = −2 ln
(L(without HT)
L(with HT)
)= χ2(without HT)− χ2(with HT)
I t is assumed to be distributed as the Chi-squareddistribution with
DOF = DOF (without HT)−DOF (with HT)
23 / 26
Goodness of fit
I For a given observed value of t we compute the p-valueof its corresponding Chi-squared distribution.
I interpretation: The smaller the p-value, the larger theprobability that the data can discriminate against thepresence of HT.
#pts #params χ2 χ2DOF p-value
RUN0 801 26 667.90 0.8620.58
RUN0 w/o HT 801 10 682.11 0.862
RUN1 1476 26 1405.3 0.9698.4× 10−12
RUN1 w/o HT 1476 10 1492.41 1.018
I HT is only relevant if we include Jlab data.
24 / 26
The extracted SPDFs at Q2 = 1GeV2
RUN0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
x∆u
+
Q2 = 1GeV2
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.20
−0.15
−0.10
−0.05
0.00
0.05
x∆d
+
JAM
DSSV
no HT
95%
68%
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
0.6
0.8
1.0
1.2
1.4
rati
oto
cent
ral
∆u+
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
0.6
0.8
1.0
1.2
1.4
rati
oto
cent
ral
∆d+
JAM
DSSV
no HT
RUN1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
x∆u
+
Q2 = 1GeV2
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
−0.16
−0.14
−0.12
−0.10
−0.08
−0.06
−0.04
−0.02
0.00
0.02
x∆d
+
JAM
DSSV
no HT
95%
68%
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
0.6
0.8
1.0
1.2
1.4
rati
oto
cent
ral
∆u+
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
x
0.6
0.8
1.0
1.2
1.4
rati
oto
cent
ral
∆d+
JAM
DSSV
no HT
25 / 26
...
Conclusions
I An error propagation for global fits based on the Hessianmethod was discussed.
I The new JLab data conclusively favors the extraction ofthe polarized structure functions with HT contributions.
TODO:
I Inclusion of SIDIS and inclusive polarized Jets/pions.
I Extension of the global fit to include unpolarized PDFsfits.
26 / 26
Recommended