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
Distributions and parton model
Alexei Prokudin – Lecture II
This diagram is called “handbag diagram”
Distributions and parton model
Alexei Prokudin – Lecture II
This diagram is called “handbag diagram”
- parton distribution
Distributions and parton model
Alexei Prokudin – Lecture II
Why quarks are on mass-shell?
This one is virtual! However the main contribution comes from
Distributions and parton model
Alexei Prokudin – Lecture II
Definition of parton distribution
Distributions and parton model
Alexei Prokudin – Lecture II
Definition of parton distribution
Fourier transform from coordinate to momentum space
Distributions and parton model
Alexei Prokudin – Lecture II
Definition of parton distribution
Quark field operator
Distributions and parton model
Alexei Prokudin – Lecture II
Definition of parton distribution
The proton state vector
Distributions and parton model
Alexei Prokudin – Lecture II
Definition of parton distribution
Position of the field in coordinate space
Distributions and parton model
Alexei Prokudin – Lecture II
Definition of parton distribution
This matrix element is called “bilocal”
Distributions and parton model
Alexei Prokudin – Lecture II
Definition of parton distribution
This matrix element is called “bilocal”
Why quark fields are separated?
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
Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about quark momentum?
“Big”component
“Small” component
Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about quark momentum?
“Big”component
“Small” component “Small” component
“Transverse” component
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!
Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about hadronic tensor?
Quarks are “probed” at exactly value of
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
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
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
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
Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about distributions?
Lightcone momentum fraction
Distributions and parton model
Alexei Prokudin – Lecture II
What do we know about distributions?
Lightcone momentum fraction
Measured in the process
Gauge invariance
Alexei Prokudin – Lecture II
QCD is invariant under gauge transformations
It means that all observables are also gauge invariant
Gauge invariance
Alexei Prokudin – Lecture II
QCD is invariant under gauge transformations
It means that all observables are also gauge invariant
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
Gauge link
Alexei Prokudin – Lecture II
Wilson line restores gauge invariance!
so that
Gauge link
Alexei Prokudin – Lecture II
Wilson line restores gauge invariance!
so that
Gauge invariant definition of distributions
Alexei Prokudin – Lecture II
means that is integrated out,
but we want TMD and transverse momentum dependence!
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
SIDIS: variables
6 variables
Analogue of Bjorken x for fragmenting quark
Orientation of the spin of the proton
Alexei Prokudin – Lecture II
SIDIS: variables
cm energy
Relation (show it!)
Alexei Prokudin – Lecture II
SIDIS: frames
There are two convenient frames to studySIDIS
• frame
• frame
Relation:
Alexei Prokudin – Lecture II
Experimentally we measure
Theoretically we assume
Alexei Prokudin – Lecture II
Theoretically we assume
Alexei Prokudin – Lecture I
Distribution
Theoretically we assume
Distribution
Fragmentation
Alexei Prokudin – Lecture II
Theoretically we assume
Distribution
Fragmentation
Structure of the cross-section
Distribution
Fragmentation
Alexei Prokudin – Lecture II
Structure of the cross-section
Alexei Prokudin – Lecture I
Distribution
Fragmentation
Not observed = integrated over
Structure of cross-section
Distribution
Fragmentation
Alexei Prokudin – Lecture II
Structure of cross-section
Alexei Prokudin – Lecture I
Distribution
Fragmentation
Structure of cross-section
Distribution
Fragmentation
Momentum conservation
Alexei Prokudin – Lecture II
Structure of cross-section
Momentum conservation
“Big”components “Small” components
“Transverse” components
Now we cannot neglect them!Alexei Prokudin – Lecture II
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
Gauge invariance
Alexei Prokudin – Lecture I
We sum all these gluons
Gauge invariance
Alexei Prokudin – Lecture I
We sum all these gluons
And the gauge link is now
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
TMDs
8 functions in total (at leading twist)
