Fattaneh Abdolghafari ( Yazd University)

Transcription

FATTANEH ABDOLGHAFARI
Yazd University
Sara Taheri Monfared, Seyed Mohammad Moosavi Nejad
• Deep Inelastic Scattering (DIS)
• Introduce structure functions in DIS
•Longitudinally polarized structure functions
•Transversely polarized structure functions
•The effect of Target Mass Correction for three scaling
variables in g1
•The effect of TMC and Higher Twist in g1 and g2
• conclusion
1
• Deep Inelastic lepton-hadron scattering is one of the best- investigated short- distance processes and is
a testing ground for perturbative QCD.
• The scattering of electrons of hadronic target has playes an essential role in our understanding of the
proton as composite particle made up of partons: quarks and gluons.
• Data from electron and neutrino scattering at large momentum transfers, are used to determine the
parton distribution functions (PDFs).
• These PDFs allow us to predict scattering cross sections at particle colliders. A good knowledge of
PDFs is of prime importance for the success of the physics program.
• The basic lepton–nucleon inelastic scattering process :
arxiv: 0709. 17757v2 [hep-ph] 1 Apr (2008)
2
Important quantities in DIS
P
target nucleon momentum
px
momentum of the final hadronic stste x
E
energy of the initial lepton
E'
energy of the final lepton
M
target nucleon mass
q= k − k '
with
p2 = M 2
four- momentum transferred from the lepton to the nucleon
Q 2 = −q 2
invariant mass squared of the final hadronic state
Fattaneh Abdolghafari (Yazd University)
3
structure functions
• Structure functions are a measure of the partonic structure of hadrons, which is important for any process
which involves colliding hadrons. They are key ingredient for deriving PDFs in nucleons.
• Structure functions is depends on:
Hadronic tensor
Leptonic tensor
The leptonic tensor where summed over the unobserved final
lepton spin is determined by electrodynamics
fattaneh Abdolghafari (Yazd University)
4
Hadronic tensor

The unknown hadronic tensor describes the interaction between the virtual photon and the nucleon
and depends upon four scalar structure functions, the unpolarized functions and the spin- dependent
functions.
 These functions must be measured and can then be studied in theoritical models.
5
Polarized lepton-nucleon DIS general aspect
• Polarized DIS, involving the collision a longitudinally polarized lepton beam with a longitudinally
and transversly polarized target, provides complementary and equally important insight into the
structure of the nucleon.
• At first sight the theorical treatment of the polarized case seems to mimic the unpolarized
case, with the structure functions F1,2 ( x ) replaced by g1,2 ( x ) and parton densities q ( x )
replaced by polarized densities ∆q ( x ) .
• But it turns out that the polarized case is much more subtle: there is an
anomalous gluon contribution to g1 ( x ) and g 2 ( x) has no interpretation at
all in purly partonic language.
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Longitudinally polarized structure functions
In the quark parton model, to leading order (LO) in QCD, g1 can be written as a linear
combination of δ q and
δ q:
δ q and δ q Quark helicity distribution for quarks and antiquark of flavour q
eq are the electric charges of the quark-flavors q=u,d,s
In NLO:
Cq and CG are the NLO spin-dependent Wilson coefficient functions.
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Contributions of TMC and HT in g1
 In QCD the spin structure function g1 can be written in the following form:
 Where LT denote the leading twist contribution to g1, while HT denotes the contribution to
arising from QCD operators of higher twist .
 The higher twist power cotributions can be divided in two parts:
 The first term is dynamical higher twist corrections, which are related to multiparton correlation.
 The second term is calculable kinematic target mass corrections.
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Neutron and Deutron polarized structure functions
Because of isospin symmetry the polarized structure functions for proton and neutron related
by exchange of up and down quarks and anti quarks:
Neutron
Deuteron
ω takes in to account the D-state admixture to the deuteron wave function:
Which covers most of the available estimates .
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Transversely polarized structure functions
 The spin structure function g 2 , unlike g1 , does not have an intuitive interpretation in the simple
quark-parton model.
 To understand g 2 properly, it is best to start with the Operator Product Expansion method.
 This measurment demand high precision and large luminosity, since the factors multiplying g 2
tend to be relatively small in DIS Kinematics.
 In the OPE, neglecting quark masses, g 2can be cleanly separated into a twist-2 and a higher
twist terms.
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Transversely polarized structure function
Regardless of the difficulties with a partonic interpretation:
Consists of twist-2 contribution
Consists of twist-3 contribution
The twist-2 contribution is so called Wandzura-Wilczek
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Scaling Variables
Bjorken Variable
Nachtmann Variable
Bloom-Gilman Variable
Wiezmann Variable
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The equation of variables
The Bjorken variable is a standard variable.
In OPE, this variable replaces the Bjorken variable.
Bloom- Gilman variable could make the data scale at Q 2
Which are not large compared to M
2
Weizmann variable seemed to produce scaling even down
to
Q2 = 0
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Comparison of the diagrams
Nachtmann variable
Bloom-Gilman variable
Weizmann variable
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The effect of TMC in g1 for variables
As you can see here, for these variables g1 has difference with entering quark and target mass
corrections. So these variables are important in our diagrams.
2
Polarized structure functions for variables at Q = 5
16
15
Polarized structure function as a function of x in NLO approximation. The solid curve is our model which is compared
with the others.
Fattaneh Abdolghafari (yazd University)
16
Target Mass Correction for g1
 One may follow the method proposed by Georgi and Politzer, to get the target mass correction to
the spin structure functions. The recent calculations show that the explicit twist-2 expression of g1
With the TMC is:
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Target Mass Correction for g2
 The spin structure functions g2 with twist-2 contribution and with the target mass
correction is:
 The above term satisfies the well-known Wandzura-Wilczek relation:
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Operator Product Expansion
• The Operator Product expansion is a rigorous and powerful technique, which provides a very
systematic and complete analysis of inclusive DIS,.
• In this approach the factorization of the hadronic tensor into hard and soft parts is achieved by a
formal expansion of the product of hadronic in coordinate space, which is based on a systematic analysis
of its light-cone behaviour.
• The separation of distances is a key point in the OPE, so we need to have scale :
Λ ₌ 243 � 62 (exp) MeV
For Q²<Λ² the QCD perturbation theory is invalid and we must use the non perturbation theory.
.
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Operator Product Expansion
• The product of two electromagnetic (or weak) currents relevant to DIS is writen as a series expansion
called OPE which enable us to extract a short distance piece in the scattering cross sections.
• The OPE allows one to expand the hadronic matrix element in the Forward scattering amplitude in a
complete set of operators:
coefficient functions
represents the hard scattering of the boson from the parton.
τ
denote the twist of the operator O.
i
represents different operators with the same twist.
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Nachtmann variable in OPE
2
• In addition to the virtuality of exchanged boson, Q , inelastic scattering is also characterzed by the
Bjorken scaling variable X :
• In the massless target and quark limits, ( Q → ∞ ), x
fraction of the target curried by the interacting parton.
2
is equivalent to the light—cone moment
2
• At finite Q , the effects of the target and quark masses modify the identification of the Bjorken x
variable with the light-cone momentum fraction.
• In the OPE, For massless quarks:
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Second Correction Higher Twist
The LSS formulation is closer in structure to the OPE, so it is more likely to be correct way to
implement HT corrections :
The unknown parameters in this formula are determined by fit:
unknown
parameters
magnitude
x
h(x)
h1
0.0493
0.028
0.0489464
h2
0.1175
0.200
0.0483101
h3
0.3406
0.350
0.0470943
h4
0.2115
0.700
0.0272193
h(x)
So that we can conclude the h(x) parameter is nonzero especially at low values of x.
E. Leader, A.V. Sidorov, and D.B. Stamenov, Phys. Rev. D 75, 074027 (2007).
E. Leader, A. V. Sidorov and D. B. Stamenov, Phys. Rev. D 73, 034023 (2006) [hep-ph/0512114].
E. Leader, A.V. Sidorov, and D.B. Stamenov, Phys. Rev. D 67, 074017 (2003).
22
The effect of HT +TMC on xg1 for proton
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The effect of HT +TMC on xg1 for Neutron at Q 2 = 5
Neutron
xg1
x
Duteron
xg1
x
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The effect of HT +TMC on xg2 for n and d at
Neutron
xg2
duteron
xg2
Q2 = 5
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Conclusion
 In the our studies, we unerstod which Nachtmann variable is better than Bjorken variable,
Because it can give certain corrections to bjorken scaling and Next two variables are
phenomenologically better than the Nachtmann variable.
 We have found that the QCD fit to the present data on the spin-structure functions of
proton,neutron and deutron is essentially improved, when the TMC and H-T corrections
are included in the analysis.
 Our results for polarized structure functions are in good agreement with available
experimental data
26
Thanks for your paying attention
28

Fattaneh Abdolghafari ( Yazd University)

Transcription

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