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Electroweak Physics
Mark Pitt
Virginia Tech
th
16 Summer School in Nuclear Physics
Electroweak physics is a broad subject. I will limit these lectures to:
• Low energies/momentum transfers  Q2 < 1 (GeV/c)
2
• Elastic scattering only (mostly e + N reactions but some  + N and e + e)
These lectures will cover the majority of the electroweak physics
going on at electron accelerators in the nuclear physics category.
Lecture 1
What are we going to cover?

e
Z
N
e + N  e + N N = nucleon
Electromagnetic Form Factors
(GEp , GMp , GEn , GMn )
• GEp , GMp ratio
• 2 photon physics
• improved knowledge of GEn
e
N
e + N  e + N N = nucleon
Parity-violating electron scattering
Neutral Weak Form Factors
(GEZ,p , GMZ,p , GEZ,n , GMZ,n , GAe)
• Strange vector form factors
• Nucleon's anapole moment
Low energy Standard Model Tests
• Weak charge of the electron
• Weak charge of the proton
• Weak charge of the neutron
We will also cover the experimental techniques unique to the
parity-violating electron scattering types of experiments.
Some Useful Resource Material on this Topic
Good recent review articles:
K.S. Kumar and P.A. Souder, Prog. Part. Nucl. Phys. 45, S333 (2000)
D.H. Beck and B.R. Holstein, Int. J. Mod. Phys. E10, 1 (2001)
D.H. Beck and R.D. McKeown, Ann. Rev. Nucl. Part. Sci. 51, 189 (2001)
And two very recent topical workshops
(talks posted online at both sites)

http://www.krl.caltech.edu/~subZ/meet/index.html
http://lpsc.in2p3.fr/congres/pavi2004/
Outline of Lectures
1. Develop the formalism of parity-violating electron scattering
with stops for:
• electromagnetic form factors
• QCD and nucleon "strangeness"
2. Experimental aspects unique to all parity-violating electron scattering
experiments
3. Review of experiments devoted to strange form factor measurements
(including new results just reported last week)
4. Motivation for low energy Standard Model tests
5. Review of experiments devoted to low energy Standard Model tests
Kinematics of Elastic Electron-Nucleon Scattering

p' , E'


v, q
e

p,E
  E - E'
  
q  p  p'
N
energy tra nsfer to recoiling nucleon
3 - momentum transfer to recoiling nucleon
2
q   - q  -Q 2
2
2
 
Q 2  4 E E ' sin 2  e 

squared 4 - momentum transfer
Lorentz invariant
Recall the Dirac Equation and Currents
Dirac equation for free electron:

(i    m)   0
where:


1 0 
  0 

0 
0

 
   ,
  
   
  0 
 0  1
with:
  0 time,   1,2,3 space


leads to electron four-vector current density:


j   e 
where the adjoint is:
satisfies the continuity equation:

    0
 j  0
Bilinear Covariants and Their Symmetry Properties
We describe physical processes through interacting currents
 need to construct most general form of currents consistent with
Lorentz invariance
Terms of the form
 (4  4) 
Scalar

Pseudoscal ar   5
Vector
  
Axial Vector     5
Tensor
  
P
1
1
 1

  1


 1  1
T
1
1
 1

 1


  1  1
where  5  i 0 1 2 3
P: parity operator (spatial inversion)
T: time reversal
C: charge conjugation
Note: P (V*V) = +1
P (A*A) = + 1
P (A*V) = -1
C
1
1
1
1
1
Relation Between Cross Sections and Matrix Elements
For a process
A+B  C+D
the differential cross section is
d
d
cm
1 pf

M
2
64 s pi
2
The physics is all in the matrix element
M
Electromagnetic and Weak Interactions : Historical View
EM: e + p  e + p

J EM ,e
EM , p
J
e
p
-
p
J weak ,e
J weak , N
e
n
e
M  J
EM , p
elastic scattering
 e 2   , EM ,e
  2  J
  p  p 
 Q 
 e2 
  2   e  e
 Q 

V
Weak: n  e- + p + e
x

V
neutron beta decay
Fermi (1932) : contact interaction, form inspired by EM

M  J weak , N GF J  , weak ,e   p  n  GF  e  e
V
x

V
Parity Violation (1956, Lee, Yang; 1957, Wu): required modification to
form of current - need axial vector as well as vector to get a parity-violating
interaction



 

 
M  J weak , N GF J  , weak ,e   p  1   5  n GF  e  1   5  e
(V - A)
x
(V - A)
Note: weak interaction process here is charged current (CC)
But Zel'Dovich Suggests - What About Neutral Weak
Currents ?
The Neutral Current, Zel'Dovich continued
Standard Model of Electroweak Interactions (1967)
Weinberg-Salam Model (1967): electroweak - unified EM and weak
 SU(2) x U(1) gauge theory with spontaneous symmetry breaking
fermions:
Leptons: e-, e -,  - , 
+ anti-particles
Quarks: u , d s , c
b,t
+
anti-particles
gauge bosons:
EM:  (m = 0)
weak: W+,- (mW = 80 GeV/c2)

f
Z0 (mZ = 91 GeV/c2)
W+,f
f'
electromagnetic interaction:
charged fermions participate
Z0
f'
f
charged current weak interaction:
all fermions participate
Neutral weak currents first observed at CERN in 1973
in reactions like

 e
 
 e
f'
neutral current weak :
all fermions participate
Feynman Rules for Calculating M in the Standard Model
The fundamental parameter of the Standard Model is the weak mixing
angle - W sin   e
where e and g are the electromag netic and weak couplings
W
g
Feynman rules:
e 
1
Q2

g
1
Q 2  M W2
2 2


   5

1
Q 2  M Z2
+,-
W
g
2 cos W
Z0
 - only couples to electromagnetic vector current
W, Z - couple to both weak vector and axial-vector currents
e2
g2
M EM  2
M weak  2
Q
Q  M W2 , Z
For Q 2  M Z2
M weak
g 2Q 2
~ 2 2
M EM e M W , Z
2 g2
Note : GF 
8 M W2
is the Fermi coupling constant
c 
f
V

 c Af    5

Electromagnetic e- p Elastic Scattering

From the Feynman rules, the matrix element is:
M  J EM , p
 e   , EM ,e
  2  J
 J EM , p
 Q 
2


 e 
  2   e  e
 Q 
2


e
But the proton (unlike the electron) is not a point-like Dirac particle
(need to introduce form factors to characterize its structure):
  2
i  q 


2
 uN
N | J  | N  u N  F1 (Q )   F2 (Q )

2M N 


Pauli
Dirac
Another way to write the form factors is the Sachs definition:
GE  F1  F2 GM  F1  F2
Q2

4M N2
The cross section for e-p elastic scattering is then given as:
(Rosenbluth formula)
2
2
d
 d   GE  GM
2
2
2 e 
 M 

2

G
tan
 
M

d
d






 Mott 
N
Proton and Neutron EM Form Factors: Measurements
GpE (Q2)
Q 2 ~ 0  2 GeV 2
GEp (0)  q
 e
G
p
M
(Q2)
All follow (appear) to follow dipole form:
 
GD Q 2 
1
 

Q2

1  
2 
0
.
71
(
GeV/
c
)

 
2
In Breit frame
GMp (0)   p
 2.79  N
Fourier transform yields spatial
distribution
(R) = o exp(-R/Ro) where R o ~ 0.25 fm
GnM (Q2)
E  spatial charge distribution
GMn (0)   n M  spatial magnetization distribution
 1.91  N
Nucleon Spacelike (q2 < 0) Electromagnetic Form Factors




J  F1    F2
i  q


Dirac Pauli
2M N
Sachs: GE  F1  F2
GM  F1  F2
Q2

4M N2
• 1960’s – early 1990’s : GpE , GpM, GnE , GnM measured using Rosenbluth
separation in
e + p (elastic)
and
e + d (quasielastic):
d  d   GE2  GM2
2
2 e 

 2GM tan
 

d  d  Mott    

• early 1990’s – present: Polarization observables and ratio techniques
used
e  N  e' N '
e  N  e' N '
d
 ...(GE2  ...GM2 )  ...Pe PNGE GM  ...Pe PN|| GM2

 
 
 

d 
( d / d )unpol
A
A||
Nucleon Spacelike EM Form Factors, World Data - 1993
GpM/pGD
GnM/nGD
Relative
error
~ 2%
~ 5-10%
GpE/GD
~ 10-20%
(GnE)2/G2D
~ 50-100%
Knowledge of nucleon spacelike EM form factors in 1993:


GpE , GpM, GnM follow dipole form GD = (1 + Q2/0.71)-2 at ~20% level
GnE ~ 0 (from quasielastic e-d data)
p
Proton Electromagnetic Form Factor Ratio: G E / G
p
M
Older data: Rosenbluth separation
JLab 2000: M. K. Jones, et al.
JLab 2002: O. Gayou, et al.
using measurements of recoil
proton polarization in Hall A with
e+p  e +p
GEp
Pt E e  E e'
 e 


tan
2 
GMp
Pl 2M

Difference in the spatial distribution of charge and magnetization
currents in the proton
p
p
Proton EM Form Factor Ratio F 2 / F 1 : pQCD predictions
pQCD prediction: As Q2  
Fp1  1/Q4
Fp2  1/Q6
Q2 Fp2 /Fp1  constant
 not being reached yet
Ralston, et al. suggested
different scaling behavior:
Fp2 /Fp1  1/Q
when quark orbital angular
momentum included
Comparison of Polarization Transfer and Rosenbluth Techniques
Recent work on Rosenbluth:
• reanalyis of old SLAC data (Arrington)
• reanalysis of old JLAB data (Christy)
• new "Super-Rosenbluth" measurement
in Hall A (Segel, Arrington)
Conclusion:
• No problem with Rosenbluth
• No problem with polarization transfer
What about radiative corrections?
Have 2-photon graphs been underestimated
in the past?
M. Vanderhaeghen and others say YES.
Using the GEp and GMp from polarization
transfer and improved calculation of
two photon graphs, they can reproduce
the Rosenbluth results.
Still an active area, more later if time...
Neutron Electric Form Factor
Data from:
beam-target asymmetries
recoil polarization
in:
d (e, e’ n)
GnE (Q2)
d (e, e’ n)
3
He (e, e’ n)
at:
Mainz MAMI
Jefferson Lab
NIKHEF
MIT-Bates
Neutron Electric Charge Distribution
__
 (u d)
p (u u d)
n (u d d)
ND3 DNP Polarized Target Apparatus of JLAB E93-026
Microwave
Input
NMR
Signal Out
Frequency
Refrigerator
To Pumps
To Pumps
LN2
LN2
Liquid
Helium
Liquid
Helium
Magnet
e–
Beam
4-94
Target
(inside coil)
1° K
NMR Coil
B
5T
7656A1
Neutron’s Magnetic Form Factor G
n
M:
Current Status
The most precise recent data
comes from ratio measurements:
(d(e,e’n))
(d(e,e’p))
at NIKHEF/Mainz (Anklin, Kubon,
et al.)
and ELSA at Bonn (Bruins, et al.)
(JLAB 95-001)
Large (8-10%) systematic
discrepancy between the two data
sets : likely due to error in neutron
detection efficiency
Newest data: JLAB 95-001 (Xu, et al. 2000) 3He(e, e’) in Hall A
agrees with NIKHEF/Mainz data at Q2 = 0.1, 0.2 GeV2
 more data exists (Q2 = 0.3 - 0.6 GeV2) but requires improved nuclear
corrections (relativistic effects need to be included)
How can we get the nucleon form factors theoretically?
Quantum chromodynamics (QCD): believed to be the correct theory of
strong interactions
• quarks (3 colors for each) inteacting via exchange of
• gluons (8 types)
Until recently only stable objects were mesons (2 quark) and
baryons (3 quark)
proton
u
u
d
gluon
u
u
s
s
valence quarks
“non-strange” sea (u, u, d, d) quarks
“strange” sea (s, s) quarks
So we know the constituents of the proton, we have a quantum field
theory for their interaction  why can't we solve for its structure?
Calculation of Electron’s Magnetic Moment in QED
Quantum Electrodynamics (QED):
theory of interacting electrons and photons
perturbation expansion in  ~ 1/137
((g-2)/2)theory =
(115965230 10) x 10-11
((g-2)/2)experiment = (115965219 1) x 10-11
agreement at 1 part in 108 level
Non-perturbative QCD
Quantum Chromodynamics (QCD):
theory of interacting quarks and gluons
QCD(running of s)
For quarks inside the nucleon,
typical momenta q ~ 0.3 GeV/c
 s ~ 1 cannot solve perturbatively
(unlike QED where  ~ 1/137)
s
Eventually, lattice QCD should provide
the solution;
In meantime we can measured welldefined nucleon properties that will
serve as benchmarks for lattice QCD
like the sea of strange quarks, for example!
Do Strange Quarks Contribute to Nucleon Properties?
Deep inelastic scattering,
contributions of constituents (partons)
to total momentum of proton:
valence quarks: uV 21%
dV 9%
sea quarks:
(u+u) : 7%
(c+c): 3%
gluons: 46%
(d+d): 8%
(b+b): 1%
(s+s): 5%
 + s  c + X + 
+ + 
Proton spin:
measured in polarized deep
inelastic scattering
1 1
 u  d  s   Lq  J g
2 2
u  d  s  0.30  0.10
s  0.1  0.03
 “Proton spin crisis”
What role do strange quarks play in nucleon properties?
proton
u
u
d
u
gluon
u
s
1
s
 x(s  s)dx
Momentum:
valence quarks
“non-strange” sea (u, u, d, d) quarks
“strange” sea (s, s) quarks
~ 4% (DIS)
0
Spin:
 N | s  s | N  ~  10% (polarized DIS)
Mass:
 N | ss | N  ~ 30% (N  - term)
Charge and current:
 N | s   s | N   ??  GEs GMs

Main goal of these experiments : To determine the contributions of
the strange quark sea (s s) to the electromagnetic properties of the
nucleon ("strange form factors").
The complete nucleon landscape - unified description
Elastic scattering:
transverse quark distribution
in coordinate space
Deep exclusive scattering (DES):
Generalized parton dist. (GPD):
fully-correlated quark distribution
in coordinate and momentum space
Electric and magnetic form
factors well - measured
GE GM for p, n
BUT quark flavor decomposit ion
of these form factors is not yet known
u
G E ,M
G
d
E ,M
G
s
E ,M
Deep inelastic scattering (DIS):
Measured nucleon
momentum fractions
(Q 2  2 GeV 2 ) :
 u u ~ 37%
 d  d ~ 20%
 s  s ~ 4%
 glue ~ 39%
longitudinal quark distribution
in momentum space
The question to be answered by this research:
How does the sea of strange quarks (ss pairs)
inside the proton (or neutron)
contribute to its electromagnetic properties:
p
p
n
n
G E , G M, G E , G M ?
 Let's measure the strange form factors
s
s
G E, G M
directly and find out.
JLAB “contracted” to understand nuclei
Nucleon form factors measured in elastic e-N scattering
Nucleon form factors
• well defined experimental observables
• provide an important benchmark for testing non-perturbative
QCD structure of the nucleon

e


N|J |N
e
 N | J | N 
Z
N

GE , GM
electromagnetic form factors
N
Z



Z
E
Z
M
G ,G
neutral weak form factors
• Measured precision of EM form factors in 0.1 - 1 GeV2 Q2 range ~ 2 - 4%
• Projected precision of NW form factors in 0.1 - 1 GeV2 Q2 range ~ 10%
from the current generation of experiments (for magnetic)
where the nucleon wa vefunction is :
| N   | uud   | uudg   | uudss   | uuduu  ...
How to Measure the Neutral weak form factors
 
2

Ne
e
N
2


Z
+
e
 he
N
Z
e
N
2

Z
e
N
e N
(elastic scattering )
 R  L
A
 R  L



e

p e

p
2
 - G FQ 2 
5
6

  form factors   10  10
 4  
e
N
Derive the Parity-Violating Asymmetry (hand-waving)

J EM ,e  Qe e    e  QeVEM ,e

J NC ,e   1  4 sin 2 W  e  e   e 5  e  gVe VNC ,e  g Ae ANC ,e
J EM , N  VEM , N
J NC , N  VNC , N  ANC , N
 1 
M EM ~  2 QeVEM ,eVEM , N
Q 
G
M NC ~
gVe VNC ,eVNC , N  g Ae ANC ,eVNC , N  gVe VNC ,e ANC , N  g Ae ANC ,e ANC , N
2 2


M  M EM  M NC
Cross section proportion al to :
*
M  M EM  2 Re( M EM
M NC )  M NC
2
  L
A R
 R  L

2
2
*
PV
2 Re( M EM
M NC
)
M EM  ...
2

EM ,e EM , N e NC ,e NC , N
g A A V
 QeVEM ,eVEM , N gVe VNC ,e ANC , N
GF Q 2 QeV V

2
4 2
QeVEM ,eVEM , N



Derive the Parity-Violating Asymmetry (hand-waving), cont.
  L
A R
 R  L
GF Q 2 QeV

4 2
VEM , N g Ae ANC ,eVNC , N  QeVEM ,eVEM , N gVe VNC ,e ANC , N 
EM ,e
Q V
VEM , N 
EM ,e
e 
2
 R   L   GF Q 2  AE  AM  AA
A

 R   L  4 2  2 unpol

AE   ( ) GEZ (Q 2 )GE (Q 2 )

AM   (Q 2 ) GMZ (Q 2 )GM (Q 2 )

AA  (1  4 sin 2 W )  G Ae (Q 2 )GM (Q 2 )
 = Q2/4M2
 = [1+2(1+)tan2(/2)]-1
Now how do the neutral weak form factors GEZ and GMZ give us
information about the strange form factors?
First some notation
Recall, we defined the nucleon Dirac and Pauli form factors through :


i

q

N | J  | N  u N  F1 (Q 2 )   F2 (Q 2 )

2M N


 uN


Define the nucleon form factors associated with a given quark current
q as :

 q
i

q

N | q   q | N  u N  F1    F2q

2M N


 uN


The Sachs form factors are then :
GEq  F1q  F2q
GMq  F1q  F2q
Neutral weak form factors  strange form factors
STANDARD
MODEL
COUPLINGS
e
q
1
u
+2/3
1  8/3 sin2W
1
d
s
1/3 1 + 4/3 sin2W
1/3 1 + 4/3 sin2W
+1
+1
qZ
1 + 4 sin2W
J    Qi qi  qi
ELECTROWEAK
CURRENTS
aZ
+1
i
J Z   QiZ qi  qi
i
2 u, p 1 d , p 1 s, p

 ,p
Flavor decomposition of nucleon E/M  p | J  | p : GE , M  GE , M  GE , M  GE , M
3
3
3
form factors:
2
1
1
 n | J  | n : GE ,,nM  GEu ,,nM  GEd ,,Mn  GEs ,,nM
3
3
3
4
4
 8





 p | J Z | p : GEZ,,Mp  1  sin 2 W GEu ,, Mp    1  sin 2 W GEd ,,Mp    1  sin 2 W GEs ,,pM
3
3
 3





Invoke proton/neutron charge symmetry
G
,p
 ,n
Z, p
,
G
,
G
E ,M
E ,M
E ,M


G
3 equations, 3 unknowns
u
d
s
,
G
,
G
E ,M
E ,M
E ,M

Validity of charge symmetry breaking assumption
u  d
GEu ,,Mp  GEd ,,Mn
GEd ,,Mp  GEu ,,nM
GEs ,,pM  GEs ,,nM
Size of charge symmetry breaking effects in some n,p observables:
• n - p mass difference  (mn - mp)/mn ~ 0.14%
• polarized elastic scattering n + p, p+n A = An - Ap = (33 ± 6) x 10-4
Vigdor et al, PRC 46, 410 (1992)
• Forward backward asymmetry n + p  d + 0
Opper et al., nucl-ex 0306027 (2003)
Afb ~ (17 ± 10)x 10-4
 For vector form factors theoretical CSB estimates indicate < 1%
violations (unobservable with currently anticipated uncertainties)
(Miller PRC 57, 1492 (1998)
Lewis and Mobed, PRD 59, 073002(1999)
Parity Violating Electron Scattering Probe of Neutral Weak Form Factors

polarized electrons, unpolarized target
e
 R   L   GF Q  AE  AM  AA
A

 R   L  4 2  2 unpol
2
AE   ( ) GEZ (Q 2 )GE (Q 2 )
AM   (Q 2 ) GMZ (Q 2 )GM (Q 2 )
AA  (1  4 sin 2 W )  G Ae (Q 2 )GM (Q 2 )
p e

 GEs
 GMs
 G Ae
At a given Q2 decomposition of GsE, GsM, GeA
Requires 3 measurements for full decomposition:
Forward angle e + p (elastic)
Backward angle e + p (elastic)
Backward angle e + d (quasi-elastic)

p
2
Strange electric and magnetic
form factors,
+ axial form factor