PHYS 5326 – Lecture #3

Monday, Jan. 29, 2007 PHYS 5326, Spring 2007Jae Yu

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PHYS 5326 – Lecture #3

Monday, Jan. 29, 2007Dr. Jae Yu

1. Neutrino-Nucleon DIS2. -N DIS Formalism3. Proton Structure Function and Parton Distribution

Functions (PDFs)


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Neutrino Nucleon Deep Inelastic Scattering• DIS (Deep Inelastic Scattering) of lepton-nucleon are

traditionally used to probe nucleon structures• Neutrinos (especially ) are excellent probes

– Extremely light– Structureless– Weak interaction only Probes helicity

• are normally used for these experiments. Why?

• Nucleons consist of partons– Structure of nucleon is described by parton distribution

functions (PDF) Constituents’ probability distributions in fractional momentum space

• DIS are viewed as neutrino-parton elastic scattering


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Structure Function Measurements• A complete set of Lorentz scalars that parameterize the

unknown structure of the proton• Properties of the SF lead to parton model

– Nucleon is composed of point-like constituents, partons, that elastically scatter with neutrino

• Partons are identified as quarks and gluons of QCD• QCD theory itself does not provide parton distributions

within the proton– Then what does QCD provide?– The dynamics of strong interactions using color quantum

numbers • QCD analysis of SF provides a determination of nucleon’s

valence and sea quark and gluon distributions (PDF) along with the strong coupling constant, s


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Kinematics of -N CC Interactions• DIS is a three dimensional problem• Three kinematic parameters provide full

description of a DIS event are– p: Muon momentum

: Angle of outgoing muon

– EHad: Observed energy of outgoing hadrons

• Neutrino energy becomes– E=EHad+E+Mp


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DIS Kinematics

, 0, 0, k E E

, 0, 0, 0Pp M

Leptonic System

Hadron System

k k’

W+(W-)

p,

} EHadP

q=k-k’

q, (q)xP

' , sin cos , sin sin , cosk E p p p

' 'p p q p k k Why isn’t NC used for SF?


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DIS Lorentz Invariant Variables

2s p k CMS Energy

Energy transferred to the hadronic System

HadP

p qE E E

M

Four momentum transfer of the interaction

22 2 ' 2 2 cosQ q k k m E E p Invariant mass of the hadronic system

2 22 ' 2 22P PW p p q M M Q

2 2P PM M E


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DIS Lorentz Invariant Variables cont’dBjorken Scaling Variable = Fractional Momentum of the Struck parton within the nucleon

x

Inelasticity y

*11 1 cos

2y where * is CMS

scattering angle of

2

2

q

p q

2

2 p

Q

M

p q

p k

HadE

E

E


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DIS FormalismMatrix element for -N interaction

' '52 2

1, 1 , ,

1 /2F

CCW

Gu k s u k s X J N p s

Q M

M

Weak Coupling Constant

W (CC) Propagator Lepton Hadron

Inclusive Spin-Averaged Cross section

22

22 2

1

21 / 2

NF

W

m Ed G mL W

d dE E EQ M

Leptonic Tensor

Hadronic Tensor


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-N DIS Cross Sections for SF Extraction

22 2

2 12

23

1 , 2 ,2 22

1 ,2

P

F P

M xy yy F x Q xF x Q

Ed G M E

dxdy yy xF x Q

2 2 2 22

222

23

1 4 /1 ,

2 2 1 ,2

1 ,2

P P

F P

M xy y M x Qy F x Q

E R x Qd G M E

dxdy yy F x Q

22 2

21

, 1 12 2

L

T P

F QR x Q

xF M x

Using ratio of absorption xsec for longitudinal and transversely polarized boson, R


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Structure Functions and PDF’s

12 2TT T

xF xq x xq x

• Assuming parton model, -N cross section can be rewritten in terms of point-like particle interactions that exchange a intermediate vector boson

2 22

22 21 2 1

1 /

TTT TF

W

d G xsq x y q y k x

dxdy Q M

• Comparing the parton-neutrino to proton-neutrino SF and PDF’s are related as

Spin 0 partons

If no spin 0, 2xF1=F2

Parity violating components

2 22

22 21 2 1

1 /

TT T TF

W

d G xsq x y q y k x

dxdy Q M

2 2 2

TT T TF xq x xq x xk

3 2

TT TxF xq x xq x


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• -N scattering resolves flavor of constituents– CC changes the flavor of the struck quark– Charge conservation at the vertex constraints

• Neutrinos to interact with d, s, u, c• Anti-neutrinos to interact with d, s, u, c

• For parton target, the quark densities contribute to SF are

Linking to Quark Flavors

p p pq d x s x 12 NxF x

3 2 2NV VxF x xu x xd x xs x xc x

;Vu x u u

p p pq u x c x

p p pq u x c x

p p pq d x s x

12 NxF x

xu x xu x xd x xd x

xs x xc x xc x xc x

3 2 2NV VxF x xu x xd x xs x xc x

Vd x d d


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How Are PDFs Determined?• Measure -N differential cross sections, correcting for target• Compare them to theoretical x-sec• Fit SF’s to measured x-sec• Extract PDF’s from the SF fits

– Different QCD models could generate different sets of PDF’s– CTEQ, MRST, GRV, etc

Fit to Data

for SF


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Comparisons of Neutrino and Antineutrino cross sections


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What can PDF’s depend on?• Different functional forms of PDF and SF’s• Order of QCD calculations

– Higher order (NLO or NNLO) calculations require higher order PDF’s

• Different assumptions in the protons– No intrinsic sea quarks– Fixed flavors only

• Approximation at non-perturbative regime– Different method of approximating low x behavior


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Homework Assignments• Provide a method to measure the average valence quark

distributions in a -N scattering experiment• Derive the Lorentz invariant variables of -N scattering,

s, Q2, W2, x and y on pages 6 and 7 of this lecture.• Make at least 8 plots (two - 1d histograms, two - 2d

scatter plots, four composite variables) using root• All these are due Monday, Feb. 12


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Homework Assignments• Draw a few additional Feynman diagrams for

higher order GSW corrections to -N scattering at the same order as those on pg 7 in this lecture– Due: One week from today, Wed., Feb. 5

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PHYS 5326 – Lecture #3