PHYS 5326 – Lecture #3
Monday, Jan. 29, 2007
Dr. Jae Yu
1. Neutrino-Nucleon DIS
2. n-N DIS Formalism
3. Proton Structure Function and Parton Distribution
Functions (PDFs)
Monday, Jan. 29, 2007
PHYS 5326, Spring 2007
Jae Yu
1
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
• nm 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
Monday, Jan. 29, 2007
PHYS 5326, Spring 2007
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 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, as
Monday, Jan. 29, 2007
PHYS 5326, Spring 2007
Jae Yu
3
Kinematics of nm-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. 29, 2007
PHYS 5326, Spring 2007
Jae Yu
4
DIS Kinematics
nm,`nm
k
k’
W+(W-)
Leptonic
System
Hadron
System
pm, qm
q=k-k’
q, (`q)
xP
P
m, m+
}
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. 29, 2007
PHYS 5326, Spring 2007
Jae Yu
Why isn’t NC used for SF?
5
DIS Lorentz Invariant Variables
CMS Energy
s   p + k   M + 2M P En
2
2
P
Energy transferred to the 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
W   p    p + q   M + 2M Pn  Q
2
Monday, Jan. 29, 2007
' 2
2
2
P
PHYS 5326, Spring 2007
Jae Yu
2
6
DIS Lorentz Invariant Variables cont’d
Bjorken Scaling Variable = Fractional Momentum of the
Struck parton within the nucleon
Q
q

x
2 p  q 2 M pn
2
2
p  q EHad n

Inelasticity y 

p  k En
En
1
y  1  1 + cos q * 
2
Monday, Jan. 29, 2007
PHYS 5326, Spring 2007
Jae Yu
where q* is CMS
scattering angle of m
7
DIS Formalism
Matrix element for n-N interaction
GF
1
' '
M
u
k
, s   a 1   5  un  k , s   X J CC N  p, s 
m

2
2
2 1 + Q / MW
Weak
Coupling
Constant
W (CC)
Propagator
Lepton
Hadron
Inclusive Spin-Averaged Cross section
2
m
E
d
1
GF mn m
m
a

L
W
a
d  m dEm 1 + Q 2 / M W2  2 En Em  2 2
2
nN
Monday, Jan. 29, 2007
PHYS 5326, Spring 2007
Jae Yu
Leptonic
Tensor
Hadronic
Tensor
8
n-N DIS Cross Sections for SF Extraction
2


M P xy  n n 
y
n n 
2
2
 1  y 
 F2  x, Q  + 2 xF1  x, Q  
2 n n 
2 En 
2
d
2GF M P En 


  y  n n 

dxdy

  y 1   xF3  x, Q 2 

  2 

Using ratio of absorption xsec for
longitudinal and transversely
polarized boson, R
2



F
Q
2
R  x, Q   L  2  1 
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 En
2 1 + R  x, Q  
d
2GF M P En 





dxdy

  y  n n 

2

y
1

n
F
x
,
Q

  2  3 



Monday, Jan. 29, 2007
PHYS 5326, Spring 2007
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 that exchange a
intermediate vector boson
Spin 0
nT
d 2 n T
GF2 xs
n
T
2
n
T
q  x  + 1  y  q + 2 1  y  k  x partons



dxdy  1 + Q 2 / M 2 2 
W
nT
d 2 nT
GF2 xs
2
nT
nT



q
x
+
1

y
q
+
2
1

y
k
x








2 

2
2

dxdy  1 + Q / M 
W
n n T
n n T
n n T

2 xF1
 2  xq  x  + xq  x 


• Comparing the partonn n T
neutrino to protonn n T
n n T
n n T 

F2
 2  xq  x  + xq  x  + 2 xk

neutrino SF and PDF’s

are related as
n n T
n n T
n n T


xF

2
xq
x

xq
x




3
Parity violating


components
Monday, Jan. 29, 2007
PHYS 5326, Spring 2007
Jae Yu
If no spin 0, 2xF1=F2
10
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
qn p  d p  x  + s p  x 
np
 x + c  x
qn p  u p  x  + c p  x 
q
np
q
u
d
p
p
p
 x + s  x
p
Monday, Jan. 29, 2007
2 xF1n N  x   2 xF1n N  x 
 xu  x  + xu  x  + xd  x  + xd  x 
+ xs  x  + xc  x  + xc  x  + xc  x 
xF3n N  x   xuV  x  + xdV  x  + 2xs  x   2xc  x 
xF3n N  x   xuV  x  + xdV  x   2 xs  x  + 2 xc  x 
u  x   u  u; dV  x   d  d
PHYS 5326, SpringV2007
Jae Yu
11
•
•
•
•
How Are PDFs Determined?
Measure n-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
Monday, Jan. 29, 2007
PHYS 5326, Spring 2007
Jae Yu
12
Comparisons of Neutrino and
Antineutrino cross sections
Monday, Jan. 29, 2007
PHYS 5326, Spring 2007
Jae Yu
13
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
Monday, Jan. 29, 2007
PHYS 5326, Spring 2007
Jae Yu
14
Homework Assignments
• Provide a method to measure the average valence quark
distributions in a n-N scattering experiment
• Derive the Lorentz invariant variables of n-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
Monday, Jan. 29, 2007
PHYS 5326, Spring 2007
Jae Yu
15
Homework Assignments
• Draw a few additional Feynman diagrams for
higher order GSW corrections to n-N scattering
at the same order as those on pg 7 in this lecture
– Due: One week from today, Wed., Feb. 5
Monday, Jan. 29, 2007
PHYS 5326, Spring 2007
Jae Yu
16