Transverse Momentum of partons · CLAS data, arXiv:0809.1153 [hep-ex] F. Giordano and R. Lamb,...

Transverse Transverse Momentum Momentum of partonsof partons

Mariaelena BoglioneMariaelena Boglione

Torino

TMDs - Lecture 2 M. Boglione - HUGS 2012 2

We will be talking about ...We will be talking about ...

Deep Inelastic scattering

Drell- Yan

e+e- scattering processes

Parton model

Transverse momentum of partons

3D kinematics (i.e. kinematics with partonic transverse momentum)

Integrated distribution and fragmentation functions (transversity and all that ...)

Transverse Momentum Dependent distribution and fragmentation functions

Phenomenology of TMD's: extracting Transverse Momentum Dependent distribution functions from experimental data

Scale dependence and TMD evolution scheme

Lecture 2Lecture 2

Lecture 1Lecture 1


Introducing Spin and Introducing Spin and Intrinsic Transverse MomentumIntrinsic Transverse Momentum


Plenty of theoretical and experimental evidence for transverse motion of partons within nucleons,

and of hadrons within fragmentation jets

Intrinsic Transverse MomentumIntrinsic Transverse Momentum

We cannot learn about the spin structure of the nucleon without taking into rigorous account the transverse motion of the partons inside it !!

Transverse motion is usually integrated over, but there are important spin-k┴ correlations !!

P

xP

k⊥

k


pT distribution of hadrons in SIDIS

Hadron distribution in jets in e+e– processes

Uncertainty principle: Δx ~ 1 fm => Δp ~ 0.2 GeV/c

Gluon radiation±1

± ±k┴



Hadron distribution in jets in e+e– processes

Gluon radiation±1

± ±k┴


Uncertainty principle: Δx ~ 1 fm => Δp ~ 0.2 GeV/c



pT distribution of hadrons in SIDIS


OPEN QUESTIONS:

How do TMD’s depend on the intrinsic transverse momentum ?Gaussian behaviour in the central region …Power law decrease at large transverse momentum…

Does the partonic intrinsic transverse momentum k⊥ (p

⊥)

depend on x (z) ?



Distributions and parton model Distributions and parton model

Definition of parton distribution

Fourier transform from coordinate to momentum space

Quark field operator

Proton state vector


Distribution functions Distribution functions (collinear - integrated)(collinear - integrated)


Very good knowledge of unpolarized distribution functions, q(x,Q2) and g(x,Q2)

Fairly good knowledge of longitudinally polarized, partonic distributions, Δq(x,Q2)

Poor knowledge of longitudinally polarized gluons Δg(x,Q2)

NO direct information on transversely polarized partonic distributions, Δ

Tq(x,Q2), from DIS



There is no gluon transversity distribution function

Transversity cannot be studied in deep inelastic scattering

because it is chirally odd

Transversity can only appear in a cross-section

convoluted to another chirally odd function

TransversityTransversity


TransversityTransversity


Semi Inclusive Deep Inelastic ScatteringSemi Inclusive Deep Inelastic Scattering


'ˆ),(),( →⊥⊥− ⊗⊗= qq

qqYanDrell kxfkxf σσ

Drell Yan ScatteringDrell Yan Scattering


TMD distribution and fragmentation TMD distribution and fragmentation functions in SIDIS processesfunctions in SIDIS processes


Courtesy of Aram Kotzinian

Correlation between spin and

transverse momentum

TMD distribution functionsTMD distribution functions


TMD distribution functionsTMD distribution functions


Mulders and Tangermann, NP B461 (1996) 197, Boer and Mulders, PR D57 (1998) 5780

Leading twist correlatorLeading twist correlator

TMDs - Lecture 2 M. Boglione - HUGS 2012 21M. Boglione

General Formalism with Helicity AmplitudesGeneral Formalism with Helicity Amplitudes

X

Aa

M. Anselmino, M. Boglione, U. D-Alesio, S. Melis, F. Murgia, A. Prokudin PRD 2011

The distribution function fa/A is the number density of partons of type a inside hadron A


22



TMD Distribution FunctionsTMD Distribution Functions


-

ΔNDh/q(z,k⊥)

Collins fragmentation function

Unpolarized fragmentation function

Dh/q(z,k⊥)

TMD Fragmentation FunctionsTMD Fragmentation Functions

pq

φh Sqp┴

Collins functionCollins function


Unpolarized SIDIS cross sectionUnpolarized SIDIS cross section


SIDIS FACTORIZATIONSIDIS FACTORIZATION

TMD-PDFHard Scattering

TMD-FF

All three fundamental blocks contain phases. The most general cross-section is obtained when all of them are kept into account.


TMD’s in SIDISTMD’s in SIDISTrento

conventions SIDIS in parton model with intrinsic k

┴

Factorization holds at large Q2,

and

(Ji, Ma, Yuan)

QCDT kP Λ≈≈ ⊥

);,();,(ˆd);,(d 222 QzDQyQxf hq

lqlq

q qlhXlp

⊥⊥→

⊥→ ⊗⊗= ∑ pkk σσ

Unpolarized Cross Section


TMD’s in SIDISTMD’s in SIDIS


TMD in unpolarized SIDIS TMD in unpolarized SIDIS → Cahn Effect→ Cahn Effect

EMC data, µp and µd, E between 100 and 280 GeV

Azimuthal dependence induced by quark intrinsic motion

CLAS data, arXiv:0809.1153 [hep-ex] F. Giordano and R. Lamb, arXiv:0901.2438 [hep-ex]

W. Käfer, COMPASS collaboration, talk at Transversity 2008, Ferrara


Parametrization of the Unpolarized TMDParametrization of the Unpolarized TMD

Assume a simple, factorized form for the TMD distribution and fragmentation functions, with a gaussian dependence on the intrinsic transverse momentum

Normalized Gaussian(one free parameter)

x and z dependence from world data fits


Determine the value of the two free parameters by fitting experimental data

A cosφ dependence is also generated by Boer-Mulders⊗Collins term, via a kinematical effect in dΔσ , ̂not included in this fit.

At further dependence on cos(2φ) is generated

EMC data, µp and µd, E between 100 and 280 GeV

M.Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin, Phys. Rev. D71:074006,2005.

TMD in unpolarized SIDIS TMD in unpolarized SIDIS → Cahn Effect→ Cahn Effect

(Gaussian widths)


PPTT dependence of data in agreement with a dependence of data in agreement with a

Gaussian kGaussian k⊥⊥ dependence of unpolarized TMDs dependence of unpolarized TMDs

Hint of a z-dependence at small z values

No hint of x dependence in the explored region

CLAS data, arXiv:0809.1153 [hep-ex] CLAS data, arXiv:0809.1153 [hep-ex]


Summary of Lecture 2Summary of Lecture 2

The proton is not simply a bag of marbles … It is a bag of spinning and orbiting marbles !

Intrinsic transverse momentum of partons cannot be neglected … when one takes it into account a completely new 3D world opens up.

8 TMD distribution functions and 2 TMD fragmentation functions embed all possible correlations between spin and transverse momentum.

They are non perturbative, phenomenological objects which need to be parameterized and extracted from experimental data.

The best processes to study TMDs are polarized SIDIS, e+e- and DY processes.

Even the unpolarized SIDIS cross section shows signs of the parton transverse momentum.

Transverse Momentum of partons · CLAS data, arXiv:0809.1153 [hep-ex] F. Giordano and R. Lamb,...

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Transcript of Transverse Momentum of partons · CLAS data, arXiv:0809.1153 [hep-ex] F. Giordano and R. Lamb,...