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Alexei ProkudinAlexei ProkudinJefferson LaboratoryJefferson Laboratory
HUGS 2011, Jefferson LabHUGS 2011, Jefferson Lab
Theory and PhenomenologyTheory and Phenomenology
of of TTransvers ransvers MMomentum omentum DDependentependent
distributionsdistributions
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Distributions and parton model
Alexei Prokudin – Lecture II
This diagram is called “handbag diagram”
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Distributions and parton model
Alexei Prokudin – Lecture II
This diagram is called “handbag diagram”
- parton distribution
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Distributions and parton model
Alexei Prokudin – Lecture II
Why quarks are on mass-shell?
This one is virtual! However the main contribution comes from
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Distributions and parton model
Alexei Prokudin – Lecture II
Definition of parton distribution
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Distributions and parton model
Alexei Prokudin – Lecture II
Definition of parton distribution
Fourier transform from coordinate to momentum space
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Distributions and parton model
Alexei Prokudin – Lecture II
Definition of parton distribution
Quark field operator
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Distributions and parton model
Alexei Prokudin – Lecture II
Definition of parton distribution
The proton state vector
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Distributions and parton model
Alexei Prokudin – Lecture II
Definition of parton distribution
Position of the field in coordinate space
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Distributions and parton model
Alexei Prokudin – Lecture II
Definition of parton distribution
This matrix element is called “bilocal”
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Distributions and parton model
Alexei Prokudin – Lecture II
Definition of parton distribution
This matrix element is called “bilocal”
Why quark fields are separated?
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Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about quark momentum? Suppose that protonis moving along Z direction with a high momentum, then
“Big”component
is a new variable called lightcone momentumfraction
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Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about quark momentum?
“Big”component
“Small” component
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Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about quark momentum?
“Big”component
“Small” component “Small” component
“Transverse” component
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Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about quark momentum?
“Big”component
“Small” component “Small” component
“Transverse” component
Not always small and can be observed!
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Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about hadronic tensor?
Quarks are “probed” at exactly value of
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Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about hadronic tensor?
The integration can be performed now! We will also use new variable
See lectures of Alberto Accardi
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Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about distributions?
It is a matrix, we need to project it in order to find distributions
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Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about distributions?
It is a matrix, we need to project it in order to find distributions
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Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about distributions?
Unpolarized quark distribution
Our matrix, that is called quark-quark correlator
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Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about distributions?
Lightcone momentum fraction
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Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about distributions?
Lightcone momentum fraction
Measured in the process
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Gauge invariance
Alexei Prokudin – Lecture II
QCD is invariant under gauge transformations
It means that all observables are also gauge invariant
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Gauge invariance
Alexei Prokudin – Lecture II
QCD is invariant under gauge transformations
It means that all observables are also gauge invariant
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What we forgot?
Alexei Prokudin – Lecture II
We forgot that quark and remnant are colored thus they interact via gluon exchanges!
This object is called Wilson line
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Gauge link
Alexei Prokudin – Lecture II
Wilson line restores gauge invariance!
so that
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Gauge link
Alexei Prokudin – Lecture II
Wilson line restores gauge invariance!
so that
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Gauge invariant definition of distributions
Alexei Prokudin – Lecture II
means that is integrated out,
but we want TMD and transverse momentum dependence!
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How can we observe quark transverse momentum?
SIDIS If produced hadron has transverse momentum
it will be sensitive to quark transverse momentum
Alexei Prokudin – Lecture II
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SIDIS: variables
6 variables
Analogue of Bjorken x for fragmenting quark
Orientation of the spin of the proton
Alexei Prokudin – Lecture II
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SIDIS: variables
cm energy
Relation (show it!)
Alexei Prokudin – Lecture II
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SIDIS: frames
There are two convenient frames to studySIDIS
• frame
• frame
Relation:
Alexei Prokudin – Lecture II
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Experimentally we measure
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Theoretically we assume
Alexei Prokudin – Lecture II
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Theoretically we assume
Alexei Prokudin – Lecture I
Distribution
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Theoretically we assume
Distribution
Fragmentation
Alexei Prokudin – Lecture II
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Theoretically we assume
Distribution
Fragmentation
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Structure of the cross-section
Distribution
Fragmentation
Alexei Prokudin – Lecture II
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Structure of the cross-section
Alexei Prokudin – Lecture I
Distribution
Fragmentation
Not observed = integrated over
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Structure of cross-section
Distribution
Fragmentation
Alexei Prokudin – Lecture II
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Structure of cross-section
Alexei Prokudin – Lecture I
Distribution
Fragmentation
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Structure of cross-section
Distribution
Fragmentation
Momentum conservation
Alexei Prokudin – Lecture II
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Structure of cross-section
Momentum conservation
“Big”components “Small” components
“Transverse” components
Now we cannot neglect them!Alexei Prokudin – Lecture II
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Structure of the cross-section
Momentum conservation
Observed transverse momentum of hadron “Transverse” components = intrinsic
transverse motion of quarks
We “observe” transverse motion of quarks in SIDIS!
Alexei Prokudin – Lecture II
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Gauge invariance
Alexei Prokudin – Lecture I
We sum all these gluons
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Gauge invariance
Alexei Prokudin – Lecture I
We sum all these gluons
And the gauge link is now
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Gauge invariance
Alexei Prokudin – Lecture I
We sum all these gluons
Fields are not only separated in direction, but also in this makes TMDs sensitive to gauge invariance
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TMDs
8 functions in total (at leading twist)