PHYS 5326 – Lecture #4
Monday, Jan. 27, 2003
Dr. Jae Yu
1. Neutrino-Nucleon DIS
2. Formalism of n-N DIS
3. Proton Structure Functions and PDF
We must meet in a different room
Wednesday, Jan. 29!!! Where? SH 129?
Monday, Jan. 27, 2003
PHYS 5326, Spring 2003
Jae Yu
1
Neutrino Nucleon Deep Inelastic Scattering
• DIS (Deep Inelastic Scattering) of lepton-nucleon are
traditionally used to probe nucleon structures
• Neutrinos are excellent probes
– Extremely light
– Structureless
– Weak interaction only  Probes helicity
• Nucleons consist of partons
– Structure of nucleon is described by parton distribution
functions (PDF)  Fractional momentum distributions of the
constituents
• DIS are viewed as neutrino-parton elastic scattering
Monday, Jan. 27, 2003
PHYS 5326, Spring 2003
Jae Yu
2
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 does not provide parton distributions within proton
• QCD analysis of SF provides a determination of nucleon’s
valence and sea quark and gluon distributions (PDF) along
with the strong coupling constant, as
Monday, Jan. 27, 2003
PHYS 5326, Spring 2003
Jae Yu
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Kinematics of n-N CC Interactions
• DIS is a three dimensional problem
• Three kinematic parameters provide full
description of a DIS event are
– pm: Muon momentum
 qm: Angle of outgoing muon
– EHad: Observed energy of outgoing hadrons
• Neutrino energy becomes
– En=EHad+Em+Mp
Monday, Jan. 27, 2003
PHYS 5326, Spring 2003
Jae Yu
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DIS Kinematics
nm,`nm
k
k’
W+(W-)
Leptonic
System
Hadron
System
pm, qm
q=k-k’
q, (`q)
P
m, m
}
xP
EHad
k  En , 0, 0, En 
k '  Em , pm sin q m cos m , pm sin q m sin m , pm cosq m 
p  M P , 0, 0, 0

p'  p  q  p  k  k '
Monday, Jan. 27, 2003
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PHYS 5326, Spring 2003
Jae Yu
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DIS Lorentz Invariant Variables
s   p  k   M P2  2M P En
CMS Energy
Energy Transferred to Hadronic System
pq
n
 En  Em  EHad
MP
Four Momentum Transfer of the Interaction

Q  q   k  k
2
2

' 2
 m  2En Em  pm cosq m 
2
m
Invariant Mass of the hadronic system
    p  q
W  p
2
Monday, Jan. 27, 2003
' 2
2
 M P2  2M Pn  Q 2
PHYS 5326, Spring 2003
Jae Yu
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DIS Lorentz Invariant Variables cont’d
Bjorken Scaling Variable = Fractional Momentum of the
Struck parton within the nucleon
q
Q
x

2 p  q 2M pn
2
2
p  q EHad n


Inelasticity y 
pk
En
En

1
y  1  1  cos q *
2
Monday, Jan. 27, 2003
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PHYS 5326, Spring 2003
Jae Yu
where q* is CMS
scattering angle of m
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DIS Formalism
Matrix element for n-N interaction
GF
1
' '
M
u m k , s  a 1   5 un k , s  X J CC N  p, s 
2
2
2 1 Q / MW
Weak
Coupling
Constant
W (CC)
Propagator
Lepton
Hadron
Inclusive Spin-Averaged Cross section
nN
d
1

d m dEm 1  Q 2 / M W2
2

Monday, Jan. 27, 2003

2
m
E
GF mn m
m
a
L
W
a
2 En Em 2 2
PHYS 5326, Spring 2003
Jae Yu
Leptonic
Tensor
Hadronic
Tensor
8
n-N DIS Cross Sections for SF Extraction
2

M P xy  n n 
y
2
n n 
2


1

y

F
x
,
Q

2
xF
x
,
Q

2
1
2 n n 


2
E
2
2G M E
d
n 
 F P n 
  y
dxdy

 y1   xF3n n  x, Q 2
  2 


Using ratio of absorption xsec for
longitudinal and transversely
polarized boson, R










2



F
Q
2
L
2 
 1
Rx, Q  

1
2 

 T 2 xF1  2M P x  

M P xy y 2 1  4M P2 x 2 / Q 2  n n 
2


1

y


F
x
,
Q

2
2
2 n n 

2
E
2
1

R
x
,
Q
2GF M P En 
d 
n


  y
dxdy

 y1  nF3n n  x, Q 2
  2 


Monday, Jan. 27, 2003
PHYS 5326, Spring 2003
Jae Yu



9







Structure Functions and PDF’s
• Assuming parton model, n-N cross section can be
rewritten in terms of point-like particle interactions
d 2 nT
GF2 xs

dxdy  1  Q 2 / M W2

d 2 nT
GF2 xs

dxdy  1  Q 2 / M W2

• Comparing the partonneutrino to protonneutrino SF and PDF’s
are related as
Parity violating
components
Monday, Jan. 27, 2003

q

nT
2
x   1  y

2
q
nT
 21  y k
nT

x 
Spin 0
partons

qnT x   1  y 2 qnT  21  y k nT x 
2 


n n T
n n T
n n T

2 xF1
 2 xq x   x q x 


n n T
n n T
n n T

F2
 2 xq x   x q x   2 xkn n T 


n n T
n n T
n n T

xF3
 2 xq x   x q x 



PHYS 5326, Spring 2003
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If no spin 0, 2xF1=F2
Linking to Quark Flavors
 n-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
qnp  d p x   s p x 
q  u x   c x 
np
p
p
np
 u x   c x 
np
 d x   s x 
q
q
p
p
p
p
Monday, Jan. 27, 2003
2 xF1nN x   2 xF1n N x 
 xux   xu x   xd x   x d x 
 xsx   xcx   xcx   xcx 
xF3nN x   xuV x   xdV x   2 xsx   2 xcx 
xF3n N x   xuV x   xdV x   2 xsx   2 xcx 
u x  u  u; dV x  d  d
PHYS 5326, SpringV2003
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•
•
•
•
How Are PDFs Determined?
Measure n-N differential cross sections, correcting for target
Compare them with 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
Monday, Jan. 27, 2003
PHYS 5326, Spring 2003
Jae Yu
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Comparisons of Neutrino and
Antineutrino cross sections
Monday, Jan. 27, 2003
PHYS 5326, Spring 2003
Jae Yu
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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) 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
Monday, Jan. 27, 2003
PHYS 5326, Spring 2003
Jae Yu
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Homework Assignments
• Provide a method to measure the average valence
quark distributions in a n-N scattering experiment
– Due: One week from today, Mon., Feb. 3
• Derive the Lorentz invariant variables of n-N scattering,
s, Q2, W2, x and y on pages 6 and 7 of this lecture.
– Due: One week from today, Mon., Feb. 3
Monday, Jan. 27, 2003
PHYS 5326, Spring 2003
Jae Yu
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