Automated Phenomenological Calculations for
High Energy Electroweak Neutrino Scattering
on a Fixed Target
Michael Jay Chernicoff
Advisor: Professor William A. Loinaz
April 20, 2009
Submitted to the
Department of Physics of Amherst College
in partial fulfillment of the
requirements for the degree of
Bachelors of Arts with honors
c 2009 Michael Jay Chernicoff
Abstract
Many theories of particle physics predict the appearance of new phenomena at energy scales at or above a TeV. This is no less true in the neutrino
sector than anywhere else. We wished to test the statistical sensitivity of
neutrino experiments at this scale, specifically beamline experiments firing
on fixed target electrons.
We have, with the help of some outside software, successfully written a
Mathematica based program that automates most of the calculation procedure for neutrino scattering. Our program takes a description of a particular
interaction and calculates the differential cross section for that event at the
tree level. The matrix element method of calculating the cross section from
Feynman diagrams was our basis for these calculations.
The program then uses the cross section and certain user defined beam
parameters to calculate the sensitivity of the event to statistical uncertainties in the vector and axial neutrino-electron coupling constants. This was
done by assuming the Standard Model values to be correct and performing
a regression analysis with our regression parameters (the deviation from the
Standard Model coupling constants) set to zero in the predefined dependent
variables. The resulting uncertainty in the fit for those parameters then gave
us a measure of how statistically limited the measurements taken by the
experiment for that interaction are. By performing these calculations on a
computer, we allowed for better adaptation of our procedure for processes
not examined directly by us.
We performed this procedure for all possible events involving a neutrino
or antineutrino of each of the three known flavors scattering off an electron.
Our principal focus was on muon neutrinos, as those are the most likely to
be used in future beamline experiments.
Acknowledgments
I would like to thank Professor Loinaz for advising my thesis, teaching me
the basics of theoretical particle physics, and putting up with my “erratic”
work schedule.
I would like to thank as well Tadeusz Pudlik, ’09 for working with me
during the past summer. I think the different perspectives with which we
approached the early days of our these really complemented each other, and
learning the basics would not have been the same without you.
I would like to thank Andy Anderson, for installing the Mathematica
packages in romulus.
A special thanks goes out to Devindra Hardawar and the network staff
at the IT department for saving my thesis in its final week.
I would also like to thank the other members of the physics department
faculty. Those of you whom I have had the honor of being taught by have all
been wonderful instructors, and I do not think that I could have written this
thesis without all of the things I learned from you. Professor Zajonc, with
whom I have, unfortunately, not taken a class, has been an excellent adviser
who understands that there is more to a liberal arts education than physics,
even if physics is more important than everything else.
i
I would like to thank all of my fellow majors, especially those in the class
of 2009: Adam, Dylan, Jeff, Matt, Max, Melissa, Tim, the aforementioned
Ted, and Shaw (who will always be one of us). I feel that we have developed
a really strong bond as a class together, and an incredible part of being a
physics major to me has been knowing that I had such a great group of fellow
majors to depend on, whether it was working on problem sets or surviving
the horrors of Intermediate Lab. Thank you all.
I would like to thank Ellen, for keeping the department from imploding
and for being willing to accept late payslips (I hope).
I would like to thank Professor Kevin McFarland of the University of
Rochester for introducing me to the wonderful world of neutrino physics,
even if it was from the experimental side of things.
I would like to thanks the Dean of Faculty’s office for the generous summer
research funding.
I would, of course, like to thank all of my friends here at Amherst. You
have all been the best group of friends, and in some cases roommates, I
have ever had, and my sanity is very grateful for your presence in my life. I
sincerely hope that the end of my senior year will not in any way diminish
the strength of our friendship.
Shaylon: tha mi gad ghr`dh.
Finally, I would be greatly remiss if I did not thank my parents for providing me with so much in life, including this wonderful education. Also thanks
are due to my brother, for providing me with the confidence that there is
some physics out there that I can always do well.
ii
Contents
1 Introduction
1
2 Background
2.1 The Standard Model . . . . . . . . . . . . . . . . . . .
2.1.1 A Brief Introduction to Particle Physics . . . .
2.1.2 A Menagerie of Particles . . . . . . . . . . . . .
2.1.3 A Gauge Theory . . . . . . . . . . . . . . . . .
2.2 Some Important Concepts from Quantum Field Theory
2.2.1 Dirac Gamma Matrices . . . . . . . . . . . . . .
2.2.2 Dirac Spinors . . . . . . . . . . . . . . . . . . .
2.3 Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Introduction to Neutrinos . . . . . . . . . . . .
2.3.2 Weakly Interacting Particles . . . . . . . . . . .
2.3.3 Handedness . . . . . . . . . . . . . . . . . . . .
2.3.4 Flavors . . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Neutrino Oscillation . . . . . . . . . . . . . . .
2.4 Cross Sections . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Measurements . . . . . . . . . . . . . . . . . . .
2.4.2 The Scattering Cross Section . . . . . . . . . .
2.4.3 Accounting for Spin . . . . . . . . . . . . . . . .
2.5 Feynman Diagrams . . . . . . . . . . . . . . . . . . . .
2.5.1 Drawing the Diagrams . . . . . . . . . . . . . .
2.5.2 The Basic Setup . . . . . . . . . . . . . . . . .
2.5.3 Deriving the Coupling Terms . . . . . . . . . .
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3 Calculations
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3.1 Summary of Possible Events . . . . . . . . . . . . . . . . . . . 36
3.1.1 Neutrino/Electron Elastic Scattering . . . . . . . . . . 36
iii
3.2
3.1.2 Inverse Muon Decay . . . . . . . . . . .
3.1.3 Antineutrino/Electron Elastic Scattering
3.1.4 Electron and Tau Neutrino Events . . .
Example Calculations . . . . . . . . . . . . . . .
3.2.1 νµ e− → νµ e− . . . . . . . . . . . . . . .
3.2.2 νµ e− → µ− νe . . . . . . . . . . . . . . .
3.2.3 ν̄µ e− → ν̄µ e− . . . . . . . . . . . . . . .
3.2.4 νe e− → νe e− . . . . . . . . . . . . . . . .
4 External Software
4.1 FeynArts . . . . . . . . . . . . . . .
4.1.1 Description . . . . . . . . .
4.1.2 Create Topologies . . . . . .
4.1.3 Insert Fields . . . . . . . . .
4.1.4 Create Feynman Amplitude
4.1.5 Models . . . . . . . . . . . .
4.2 FeynCalc . . . . . . . . . . . . . . .
4.2.1 Description . . . . . . . . .
4.2.2 FeynCalc Objects . . . . . .
4.2.3 Useful Functions . . . . . .
4.3 Compatibility . . . . . . . . . . . .
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5 Our Software
5.1 The CouplingSM Model . . . . . . . . . . .
5.2 FeynArts to FeynCalc Preprocessor . . . . .
5.2.1 Regular Expressions in Mathematica
5.2.2 Implementation . . . . . . . . . . . .
5.3 Cross Section Calculator . . . . . . . . . . .
5.4 Flux Integrator . . . . . . . . . . . . . . . .
5.5 Linear Regression . . . . . . . . . . . . . . .
5.5.1 Regression Goals . . . . . . . . . . .
5.5.2 Regression Preprocessor . . . . . . .
5.5.3 Regression Implementation . . . . . .
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6 Data
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6.1 Input Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
iv
7 Conclusion
7.1 Interpretation of Results . . . . . .
7.2 The Next Steps . . . . . . . . . . .
7.2.1 More User Friendly . . . . .
7.2.2 More Outgoing Particles . .
7.2.3 Deep Inelastic Scattering . .
7.2.4 Beyond the Standard Model
A Properties of Spinors and Gamma
A.1 Dirac Spinors . . . . . . . . . . .
A.1.1 u Spinors . . . . . . . . .
A.1.2 v Spinors . . . . . . . . .
A.2 Gamma Matrices . . . . . . . . .
A.2.1 Gamma Matrix Relations
A.2.2 Trace Theorems . . . . . .
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Matrices
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B Relativistic Kinematics
B.1 Four-Vectors . . . . . . . . . . . . .
B.1.1 Position-Time Four-Vectors
B.1.2 Generalized Four-Vectors . .
B.1.3 Four-Momenta . . . . . . .
B.2 Neutrino Fixed Target Scattering .
B.2.1 Momenta . . . . . . . . . .
B.2.2 Scalar Products . . . . . . .
B.2.3 Finding θ . . . . . . . . . .
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Generalized Single Variable Linear Regression
Introduction . . . . . . . . . . . . . . . . . . . . . . .
A Simple Linear Regression . . . . . . . . . . . . . .
Multiple Regression Parameters . . . . . . . . . . . .
C.3.1 Setup . . . . . . . . . . . . . . . . . . . . . .
C.3.2 The Matrix Method . . . . . . . . . . . . . .
C.4 Confidence Intervals . . . . . . . . . . . . . . . . . .
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C The
C.1
C.2
C.3
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D Code
105
E Constants
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v
List of Figures
2.1
The Feynman Diagram for muon decay . . . . . . . . . . . . . 21
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
νµ e− → νµ e−
νµ e − → µ − νe
ν̄µ e− → ν̄µ e−
νe e− → νe e− .
ν̄e e− → ν̄e e− .
ντ e − → ντ e − .
ν̄τ e− → ν̄τ e− .
ντ e − → τ − νe .
4.1
4.2
CreateTopologies[0,2→2] . . . . . . . . . . . . . . . . . . . . . 50
Topology from Fig. 4.1 with νj e− → νk `−
l fields added . . . . . 51
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Incoming neutrino beam flux
νµ e− → νµ e− . . . . . . . .
ν̄µ e− → ν̄µ e− . . . . . . . .
νe e− → νe e− . . . . . . . . .
ν̄e e− → ν̄e e− . . . . . . . . .
ντ e − → ντ e − . . . . . . . . .
ν̄τ e− → ν̄τ e− . . . . . . . . .
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B.1 2 to 2 scattering diagram . . . . . . . . . . . . . . . . . . . . . 92
vi
Chapter 1
Introduction
Neutrino Physics
The construction of the Large Hadron Collider has opened up a new era in
high energy physics. As the energies of particle collisions reach and surpass 1
TeV in magnitude, new phenomena not predicted by the Standard Model may
emerge. Experimental verification (or contradiction) of theoretical models
may soon be within reach of the physics community. While the LHC will
(someday) serve an important role in the possible discovery of the Higgs
boson mass and other previously unobserved properties of modern particle
physics, it, and experiments like it, do not focus on a rich subfield: the
physics of neutrinos and their interactions.
Neutrinos, neutrally charged, colorless particles with very small masses,
are of interest to us for a number of reasons. Because neutrinos have mass,
1
they are already in violation of the Standard Model. Any theory that moves
beyond the Standard Model must therefore take the existence of neutrino
masses, and the consequent flavor oscillations, into account. Investigations
into the specifics of neutrino interactions will be important in testing these
new theories.
Neutrinos are unique among elementary particles; they are only influenced
by the weak force, as they have no electric or color charge. Any experiments
probing the fundamental parameters of the weak interaction that use neutrinos do not need to account for the influence of the electromagnetic and
strong forces, which tend to dominate interactions when they are present.
Phenomenology
Before neutrino experiments designed to operate at higher energy scales can
be designed and built, it is necessary to theoretically predict the results of
such experiments. This application of theoretical particle physics to high
energy experiments is known as particle physics phenomenology.
One facet of phenomenology is the prediction of an experiment’s inherent statistical uncertainties. If the Standard Model is correct, a smaller
uncertainty will allow us to make higher precision measurements of various
physical constants. If not, a smaller uncertainty will make it easier to detect
deviations from the model.
The demonstration that an experiment has smaller uncertainties in its
2
dependent variables serves as an important justification for the significant
investment in time and capital involved in building high energy physics experiments. An understanding of an experiment’s statistical dependence on
various factors also allows us to fine tune those factors to improve the experiment.
Neutrino/Electron Scattering
We have specifically examined the statistical uncertainty of experiments that
involve the scattering of ˜100GeV neutrinos off of stationary electrons. Neutrinos and antineutrinos of all three flavors were examined in the course of our
work. We were most concerned with muon flavored neutrinos, as these are
the neutrinos most likely to be used in the next generation of experiments,
such as NuSOnG (Neutrino Scattering On Glass).[10]
To simplify our calculations, we only concerned ourselves with the tree
level contributions to all of the events we examined. This means that we
ignored all process that involved internal loops, particles that are created
and annihilated during the process and thus appear as neither incoming or
outgoing particles. Calculations involving these loops can prove notoriously
difficult, often involving the canceling of infinities through renormalization.
Since we were more concerned with order of magnitude calculations, ignoring
these additional contributions was an acceptable approximation.
Another simplification was to only look at scattering events that produced
3
two outgoing particles. This allowed us to not be overly concerned with the
kinematics and dynamics of the scattering. As each additional outgoing
particle significantly decreases the probability that an event will occur, the
most probable events are the simple two body scattering ones. We were
therefore justified in focusing our attention on those particular problems.
Data Analysis
We calculated the statistical uncertainty for the events we examined by performing a linear regression on the number of outgoing (charged) particle
counts per kinetic energy bin in a simulated detector. The beam flux, and
run time were taken from a hypothetical description of NuSOnG, but they
could apply just as easily to any similar experiment.
Unlike many experimental regressions, our regression parameters were
predetermined. Only the uncertainty in the regression was of interest to us,
as that determined the statistical sensitivity the event to the experimental
variables. To simplify matters, we used the first order deviations of those
variables from their Standard Model values as the regression parameters,
rather than the parameters themselves. This way, any statistical dependence
on the central values (which may not be as predicted by the SM) was removed
from our analysis.
The variables of interest to us were the vector and axial vector neutrino/electron electroweak coupling constants, gVνe and gVνe respectively. These
4
parameterize the coupling between neutrinos and charged leptons via the neutral current weak interaction. We also examined the statistical dependence
on the more fundamental Standard Model parameters ρ and sin2 θW , which
measure the relative strength of the charged current to neutral current interactions and the relative strength of the electromagnetic to weak forces,
respectively.
The fit to sin2 θW was of particular interest to us, as the most precise
measurement of this parameter as of the writing of this paper, is 3σ away
from its predicted Standard Model value. This unexpected result, called
the “NuTeV anomaly” after the experiment in which it was measured, has
yet to be reproduced experimentally, so acquiring better experimental fits to
sin2 θW is a significant priority.[6]
Automation
While most of the calculations we performed, with the possible exception of
the regression analysis, could have been performed by hand, we were interested in devising a means of significantly increasing their automation. This
reduced the work required of us when we actually performed the calculations,
allowed us to easily modify our calculations as necessary, and reduced the
possibility of simple algebraic errors (e.g. incorrect minus signs) on our part.
This automation was accomplished by combining two freely available
Mathematica packages, FeynArts and FeynCalc, with our own Mathemat-
5
ica code. FeynArts generates an algebraic expression for the amplitude of
any tree level event. FeynCalc provides a suite of functions that allowed
us to perform basic quantum field theory operations within Mathematica’s
computerized algebra system. Our own code allowed these two pieces of software to work in concert, and gave us the means to take the output generated
by FeynArts and calculate the counts per bin data necessary to perform the
aforementioned linear regression.
6
Chapter 2
Background
2.1
2.1.1
The Standard Model
A Brief Introduction to Particle Physics
The goal of particle physics (often called high energy physics) is to describe
all of the fundamental properties of the universe through the properties and
interactions of elementary particles. A particle is elementary if it contains
no known substructure or constituent particles, making it in a sense a “basic
building block” of the universe. For instance, an electron is considered (for
now, at least) to be an elementary particle, while a proton, now known to be
a bound state of quarks, is not.1
Elementary particles are divided into two types, “matter” and “force
1
Much of the information presented is based on large portions of [3] and [5]. Anything
derived from another source will be cited appropriately.
7
carriers”. The former make up all known matter, both conventional and
exotic, in the universe, while the latter are responsible for the interactive
forces between the former. Any observed force is described in the realm
of particle physics as an interaction between two particles mediated by one
or more additional “virtual” particles. Such particles cannot be directly
observed, but in terms of the quantum field theory underling particle physics,
are identical to their observed counterparts, except that they do not need to
obey the equation E 2 = m2 + p2 . For instance, a virtual photon exchanged
between two electrons, perhaps representing scattering of the electrons via
Coulombic repulsion, is described by the same fields as an observed photon.
2.1.2
A Menagerie of Particles
Since approximately 1978, three of the four known fundamental forces, electromagnetism and the strong nuclear force, and the weak nuclear force, have
been described in particle physics by a single theory, the Standard Model
(SM). According to the Standard Model, all of the elementary “matter” particles are fermions and the force carriers are bosons.
The SM force carriers, also called gauge bosons, are the photon, which
carries the electromagnetic force, eight gluons, which carry the strong nuclear
force, and the W− , W+ , and Z bosons, which carry the weak nuclear force.
There exists an additional predicted boson, the Higgs, which is theorized to
be responsible for the masses of the particles within the SM.
The remaining fundamental force, gravity, is not described by the Stan8
dard Model. SM and general relativity, which describes the gravitational
force on the macroscopic scale, are simply incompatible without a comprehensive theory of quantum gravity. While there are a number of such theories
presently in existence, they are presently without verification.
This was not an issue for us, as the masses of subatomic particles are so
small that any gravitational force they experience is much weaker than even
the titular weak force. It is therefore always an acceptable approximation
to ignore gravity, and its hypothetical carrier particle, the graviton, when
examining high energy events in nonexotic situations (i.e. not in a black hole
or during the earliest moments of the universe).
Returning to the description of particles within the Standard Model, the
elementary fermions can be further divided into two categories, quarks, those
which interact via the strong force, and leptons, those which do not. As a
consequence of quantum chromodynamics, the theory within the SM describing the strong interactions, quarks can never be found in isolation, but always
exist within bound states of quarks and gluons.
There are six known flavors of quarks. They are, in order of increasing
mass, the up, down, strange, charm, bottom, and top quarks. The u, c, and t
quarks all have a charge of +2/3 e, and the d, s, and b quarks have a charge of
-1/3 e, where e is the (absolute) charge of the electron. The former are often
designated “up-type” quarks and the latter “down-type” in the language of
quantum field theory, because quarks of each type share common interaction
properties.
9
Fermions
Gen I
Neutrinos
νe
Charged Leptons
e“Up-Type” Quarks
u
“Down-Type” Quarks
d
Gen II Gen III
νµ
ντ
µ−
τ−
c
t
s
b
Gauge Bosons
γ
W±
Z0
g (× 8)
Table 2.1: The Standard Model
Leptons can be either charged or uncharged; the latter are referred to as
neutrinos. The three charged leptons are, again in order of increasing mass,
the electron, the muon (or mu lepton), and the tau lepton (rarely tauon).
There are three neutrinos, each one corresponding to a specific charged lepton: νe , νµ , and ντ . The nature of this correspondence, and much more about
neutrinos, is discussed in the §2.3.4.
The fermions are often often grouped into three generations by arranging
the “up type” quarks, “down type” quarks, charged leptons, and neutrinos
in order of mass. The particles of each type with the lowest mass belong
to generation I, those of the middle mass to generation II, and those of the
highest to generation III.
Finally, each elementary fermion in the SM has a corresponding antiparticle, a particle of identical mass and spin but opposite charge.2 The various
particles of the standard model (except for the fermion antiparticles) are
summarized in Table 2.1.
The W+ is the antiparticle of the W− . Each gluon’s antiparticle is one of other gluons,
possibly itself.
2
10
2.1.3
A Gauge Theory
The Standard Model is more than simply a catalog of particles; it is, more
formally, a gauge theory of the group SU(3)×SU(2)× U(1). A gauge theory
is a field theory whose Lagrangian is invariant under a local gauge transformation, one in which the transformation can vary with position.
The gauge group of the Standard Model has three contributing groups.
U(1) is the set of all 1 × 1 unitary matrices. Transformations in this group
correspond to rotations about the unit circle (i.e. multiplication by a factor
of eiφ(x) ). The electromagnetic interactions are invariant under this gauge.
SU(2) is the group of all 2 × 2 unitary matrices with unit determinant.
This group contains the transformations of the classic two element quantum
mechanical spinor. It can thus be represented as the transformations caused
by multiplying by the Pauli matrices. In the case of the Standard Model,
spin is replaced by “weak isospin”, T .
Left-handed fermions have T = 21 , and right-handed fermions have T = 0
(handedness is discussed in §2.3.3). Among the left-handed fermions, neutrinos and up-type quarks have T3 = + 21 , and charged leptons and down-type
have T3 = − 12 . The weak isospin transformations are part of the symmetry
of the weak interactions, which in total form a SU(2)×U(1) group.
Finally, SU(3) is the group of all 3 × 3 unitary matrices with unit determinant. The transformations of this group affect the three element color (in
the chromodynamics sense) vectors. Instead of the three 2 × 2 Pauli matrices, the eight 3 × 3 Gell-Mann matrices form the representation of the SU(3)
11
algebra. These transformations correspond to the strong force and are thus
of little relevance to this paper.
The requirement of local gauge symmetry forces us to add extra terms to
the Lagrangian. These fields are the gauge bosons of the Standard Model.
For instance, the eight gluons emerge from the local SU(3) invariance.
Each of terms in the Standard Model Lagrangian correspond to an interaction between three or more particle fields. In §2.5.3 we explain how these
relate the actual field interactions.
2.2
Some Important Concepts from Quantum
Field Theory
2.2.1
Dirac Gamma Matrices
Paul Dirac introduced the gamma matrices in his attempt to formulate a
relativistic version of the Schrödinger equation. He found that several coefficients in the resulting equation could not be expressed as scalars, but as 4×4
matrices. While the derivation is unimportant here, the matrices themselves
are ubiquitous in quantum field theory.
The four basic gamma matrices, in the Bjorken and Drell convention, are
12
defined as follows:


0

1 0
0


0 −1 0 


0 0 −1

i
 0 σ
γi = 
 ,
−σ i 0
1

0

0
γ =
0


0

0
0
,
(2.1)
(2.2)
where σ i is the ith Pauli matrix for i = 1, 2, 3.
It can be easily shown that
γ0
2
=1 ,
(2.3)
γi
2
= −1 .
(2.4)
(Throughout this paper, 1 represents the appropriately sized identity matrix.)
In addition, γ 0 is Hermitian, and the γ i ’s are antihermtian.
At times, it is necessary to transform a scalar or a vector into a pseudoscalar or a pseudovector (also called an axial vector). To do this, we
13
introduce a fifth gamma matrix,

0

0

γ 5 ≡ iγ 0 γ 1 γ 2 γ 3 = 
0


1
0 0
0 1
1 0
0 0

1

0


0


0
.
(2.5)
A product with γ 5 in it acquires a minus sign under a parity transformation,
which is why multiplying by it turns vector into pseudovectors.
Note that
γ5
2
=1 ,
(2.6)
so multiplication by a second γ 5 restores parity symmetry. γ 5 , like γ 0 , is
Hermitian.
All of the gamma matrices (including γ 5 ) anticommute with each other.
For more properties of the gamma matrices, see Appendix A.
Finally, we introduce the additional Feynman slash notation:
6a ≡ γ µ aµ
,
where a is any four-vector as defined in Appendix B.
14
(2.7)
2.2.2
Dirac Spinors
Now that we have defined the gamma matrices, we may write out the Dirac
Equation:
iγ µ ∂µ ψ ∓ mψ = 0 ,
(2.8)
with
∂µ ≡
∂
∂xµ
.
(2.9)
This equation is valid for wave functions, ψ, that represent spin
1
2
particles.
The sign of the mass term goes from negative to positive if ψ is an antiparticle.
ψ is a four element vector (not a four-vector) called a “Dirac spinor”.
Note that we have chosen our units so that c = 1 and h̄ = 1. In this
convention, momentum, and energy all have the same units, and time and
√
length have the same units. The fundamental charge e now has a value of α.
α is the fine structure constant, a dimensionless constant equal to
e2
h̄c4π0
has the same value of 7.2973525377 × 10−3 , or approximately
regardless
of the system of units employed.
15
1
,
137
that
There are four solutions to the Dirac equation:


u(1) =
 1 


 0 
p


|E| + m 

 pz 
 E+m 


,
(2.10)
,
(2.11)
,
(2.12)
.
(2.13)
px +ipy
E+m

u(2) =

 0 


 1 
p


|E| + m 

 px −ipy 
 E+m 


−pz
E+m


px −ipy
 E+m 
v (1) =
 −p 
z 

p
 E+m 
|E| + m 

 0 




1


pz
v (2)
 E+m 
 px +ipy 


p
 E+m 
= − |E| + m 

 1 




0
The u spinors correspond to particles and the v’s to antiparticles. The
(1) and (2) superscripts do not necessarily designate spin up and spin down
16
states, as the Dirac spinors are not, by default, eigenstates of the matrix


1 σ 0 


2 0 σ
.
However, if the z axis is oriented along the direction of motion, then px =
py = 0 and the spinors are eigenspinors of Sz . In this case, u(1) and v (1) are
spin up, and u(2) and v (2) are spin down.
Additional properties of the Dirac spinors are provided in Appendix A.
2.3
2.3.1
Neutrinos
Introduction to Neutrinos
As explained in §2.1.2, neutrinos are leptons (fundamental fermions which do
not interact strongly) with no charge. The existence of a neutrino was first
proposed in 1930 by Wolfgang Pauli to explain the the seemingly “missing”
energy in beta decay, which must have been carried away by some other
undetectable (and therefore chargeless) particle.
Since the outgoing β particle (a high energy electron) would sometimes
have all of the energy predicted by the pre-neutrino model to within the
measurement capabilities at the time, it was assumed that the neutrino had
no rest mass. Thus the neutrino could carry away a negligible amount of
energy from the decay process, unconstrained by the need to have a minimum
17
amount of energy to create its mass. It is now known that neutrinos do have
a small, nonzero mass orders of magnitude smaller than the mass of their
corresponding charged leptons.[6]
2.3.2
Weakly Interacting Particles
Because neutrinos have no charge, they cannot interact via the electromagnetic force. Since, being leptons, they do not interact strongly either, they
can only experience weak interactions (i.e. only those processes mediated by
W and Z bosons).
Neutrinos thus offer the scientific community an excellent opportunity to
study the weak force, whose contributions to various events are usually much
less significant than the strong or electromagnetic forces. Unfortunately,
neutrinos only interact very rarely. For example, billions of neutrinos created
in the sun pass through every person on the sunward surface of the Earth
(and then through the planet and back out into space) every second without
any effect.[6]
This does make neutrino beams easy to filter, as little other than neutrinos
can pass through very thick panels of metal, lead, or concrete (or in some
cases miles of solid earth), but it also means that the events that are recorded
in the detector are quite infrequent. The higher the energy of the incoming
neutrino is, the more likely it is to interact with another particle it encounters.
Higher energy neutrino beams thus allow more events to be recorded over
the lifetime of the experiment, yielding higher statistical fits for the data.
18
2.3.3
Handedness
Handedness (or chirality) is a property of subatomic particles. The lefthanded component of a particle’s spinor is the component projected out by
the
1
2
(1 − γ 5 ) operator, and its right-handed component the one projected
out by
1
2
(1 + γ 5 ).
Handedness acquired its name because in the relativistic limit of a massless particle, handedness is the same as helicity. A left-handed photon, for
example, has left helicity: its spin points in the opposite direction of its motion. Because a massless particle moves at the speed of light, it will retain
the same helicity regardless of reference frame.
If the particle is massive, however, it is possible to perform a Lorentz boost
into a reference frame where the particle is moving in the opposite direction
but with its spin unchanged. This reverses the helicity of the particle. The
particle’s handedness, however, can remain unchanged.
Neutrinos are rather unique in that only left-handed neutrinos have been
observed. Conversely, anti-neutrinos have all been observed to be righthanded.
When neutrinos were considered massless, we could conclude that only
left-handed neutrinos (and right-handed neutrinos) exist. Apparently the
coupling between all neutrinos and the W or Z bosons includes only the
left-handed projection operator. Therefore only left-handed neutrinos are
produced by any process. A massless particle’s handedness eigenstate is constant, so a left-handed neutrino will remain a left-handed neutrino forever.[6]
19
However, neutrino masses have recently been detected. The handedness
of a massive particle is not preserved over time, since the helicity and handedness eigenstates are no longer equal. An initially left-handed neutrino will
therefore exist in a superposition of left and right-handedness as it propagates. While only left-handed neutrinos will be observed (as only the lefthanded states can still couple to the weak gauge bosons), right-handed neutrinos could very well exist. These right-handed neutrinos, and left-handed
antineutrinos, are referred to as sterile, because they cannot participate in
any Standard Model interaction.[6]
Another possibility is that the neutrino is a Majorana fermion, a spin 1/2
particle that, unlike a Dirac fermion, is its own antiparticle. If this is the
case, what we have been calling neutrinos and antineutrinos would simply be
the left and right-handed states, respectively, of the same particle. As of the
writing of this paper, no experimental conclusion has been reached on this
issue.[6]
Although neutrinos are massive, their masses are very small compared to
their total energy, and they therefore travel at speeds very close to that of
light. So even though a left-handed neutrino can have a right-helicity, the
probability of observing such a state is so small that we can assume that all
neutrinos and anti-neutrinos have only one possible helicity.
20
ν
µ
µ
Wν
e
e
Figure 2.1: The Feynman Diagram for muon decay
2.3.4
Flavors
While the neutrino was originally proposed as a single particle, we have now
identified three different flavors of neutrino and three flavors of antineutrino,
for a total of six general neutrinos in the Standard Model. The names of the
flavors represent the charged lepton to which that particular neutrino couples
via a W boson. For example, in Fig. 2.1, which represents muon decay, the
muon and the W couple to a muon neutrino, while the electron and the W
couple to an electron antineutrino.
A simple device for keeping track of the different neutrinos in an event
is to consider the three lepton number, Le , Lµ , and Lτ , to be conserved
quantities in weak interactions. Each lepton of a particular flavor carries 1
unit of the corresponding lepton number, and each antiparticle -1. In the
case of muon decay, µ− → e− ν̄e νµ , the incoming muon has a lepton number
of Lµ . The outgoing electron, electron antineutrino, and muon neutrino have
lepton numbers of Le , -Le , and Lµ , respectively, for a total lepton number of
21
Lµ . The net lepton numbers is conserved for this event.
2.3.5
Neutrino Oscillation
Lepton number is not completely conserved if neutrinos propagate between
two different interactions. It has been observed that neutrinos’ flavors oscillate over time. This is caused by the neutrino flavor and mass eigenstates
not corresponding directly.3
A neutrino of a given flavor exists in a superposition of three different
mass states when it is created. Because neutrinos with same total energy
and different masses move at different velocities, the probability of measuring the neutrino with each mass changes over time. This means that the
superposition of mass eigenstates must change as well.
Because each mass eigenstate is also a superposition of the different flavor
eigenstates, a change in the mass probability distribution of a neutrino also
changes the flavor probability distribution. A neutrino that was initially one
flavor has a nonzero probability of being observed with a different flavor over
time.
The probability distributions for the mass and flavor eigenstates are sinusoidal, and so these distributions will oscillate in and out of phase with each
other. The total flavor probability distribution will therefore also oscillate
over time, hence the phrase “neutrino oscillation”.
If neutrinos were massless, there would be no mass eigenstates to mix,
3
Most of the information in this section can be found in [6]
22
and no flavor oscillation would occur. The observation of oscillations in solar
and atmospheric (i.e. cosmic ray) neutrinos only confirms that, in violation
of the Standard Model, neutrinos have mass.
2.4
Cross Sections
2.4.1
Measurements
In order to examine the electroweak coupling between neutrinos and electrons, an experiment records the number of outgoing charged leptons over
the course of its runtime, N . Because different particle detectors are placed
at different points in the detector, it is possible to approximate the distribution of the number of counts over the solid angle of the outgoing particles to
calculate
dN
.
dΩ
If the incoming particles are in a beam of uniform luminosity, L, then N
can be related to the cross section, σ, by
dN = Ltdσ
,
(2.14)
where t is the runtime of the beam. From this it is apparent that
dσ
dN
= Lt
dΩ
dΩ
dσ
dΩ
.
is a term that can be calculated theoretically for a given event.
23
(2.15)
More often the number of detected particles is sorted by kinetic energy.
The number of counts per kinetic energy bin is
dN
∂Ω dσ
= Lt
dT
∂T dΩ
= 2πLt sin θ
= Lt
∂θ dσ
∂T dΩ
(2.16)
dσ
dT
(2.17)
Equation (2.17) assumes that the energy distribution of the incoming
beam is uniform. This is almost never true. Instead, the beam will have
some given energy weighted flux per target,
dΦ
,
dEν
where Eν is the energy
of the incoming beam (in our case the incoming neutrino energy). In this
case,[1]
dN
=t
dT
2.4.2
Z
dΦ dσ
dEν
dEν dT
.
(2.18)
The Scattering Cross Section
Fermi’s “Golden Rule” says that the quantum mechanical transition rate for
a given process is given by
transition rate = 2π |M|2 × (phasespace)
(2.19)
in terms of the matrix element (or Feynman amplitude), M, and the phase
space. The former is related to the dynamics of the specific process and is
dealt with below. The latter is entirely dependent on the kinematics of the
24
process and will vary depending on the nature of the interaction (i.e. whether
it is a 1-to-3 decay, a 2-to-3 scattering event, etc.).
For a scattering event with two incoming particles and two outgoing particles, the Golden Rule allows us to write the cross section as
Z
2
|M|
dσ =
S
q
4 (p1 · p2 )2 − m2p1 m2p2
× (2π)4 δ 4 (p1 + p2 − k1 − k2 )
.
d3 k1
16π 3 Ek1
d3 k 2
16π 3 Ek2
(2.20)
The p’s are the four-momenta of the incoming particles, and the k’s are
those of the outgoing ones. For an explanation of four-vector notation, see
Appendix A. S is a statistical factor of
1
j!
for each group of j indistinguishable
outgoing particles. Notice that the delta function enforces the conservation
of four-momentum.
When modeling fixed-target experiments, it is often optimal to work
within the “lab” reference frame, where p2 = 0. Applying this condition
and breaking up the delta function into energy and momentum components
yields
S |M|2
δ (Ep1 + mp2 − Ek1 − Ek2 ) δ 3 (p1 − k1 − k2)
dσ =
64π 2 mp2 |p1 |
3 d |k1 | dΩ
d k2
,
(2.21)
Ek1
Ek2
Z Z
25
where dΩ = sin θdθdφ and θ is the angle between k1 and p1 .
After integrating over |k1 | and k2 and making the appropriate kinematic
substitutions, we finally arrive at the equation for the differential cross section,
dσ
,
dΩ
for 2-body scattering in the lab frame:
dσ
S |M|2
|k1 |2
=
dΩ
64π 2 mp2 ||k1 | (Ek1 + mp2 ) − |p1 | Ek1 cos θ|
.
(2.22)
If p1 and k1 correspond to neutrinos, whose mass is much smaller than
their momentum, we can approximate |p1 | and |k1 | as Ep1 and Ek1 . This
allows us to write
dσ
= S |M|2
dΩ
Ek1
8πmp2 Ep1
2
.
(2.23)
As every event examined in this paper had both an incoming and outgoing
neutrino, this formula for finding
2.4.3
dσ
dΩ
was used exclusively in our calculations.
Accounting for Spin
For most experiments, including the ones on which our calculations were
based, the spins of the incoming and outgoing particles are not specified.
When this is the case, we must average |M|2 over all initial spins and sum
it over all final spins before inserting the matrix element term into the cross
section equation.
While this can be done by computing M for every possible spin config-
26
uration and then summing and averaging the results, such a procedure can
soon become complicated. Instead we take advantage of the fact that most
matrix elements contain terms of the form
G = [ū (k) Γ1 u (p)] [ū (k) Γ2 u (p)]∗
,
(2.24)
where k and p represent arbitrary spins and momenta, and Γ1 and Γ2 are
4 × 4 matrices.
We begin simplifying this expression by taking the complex conjugate
(which can be written as the conjugate transpose of a 1 × 1 matrix):
h
i†
† 0
[ū (k) Γ2 u (p)] = u (k) γ Γ2 u (p)
∗
= u (p)† Γ†2 γ 0† u (k)
= u (p)† γ 0 γ 0 Γ†2 γ 0 u (k)
= ū (p) Γ̄2 u (k)
,
(2.25)
where
Γ̄2 ≡ γ 0 Γ†2 γ 0
.
(2.26)
Now we sum over the the spin orientations of p using Equation (A.5):

X
pspins
G = ū (k) Γ1 
2
X

u(sp ) pū(sp ) p Γ̄2 u (k)
sp =1
= ū (k) Γ1 (6p + mp ) Γ̄2 u (k)
27
.
(2.27)
Doing the same for the spins of k yields
X X
2
X
G=
ū(sk ) kΓ1 (6p + mp ) Γ̄2 u(sk )k
.
(2.28)
sk =1
kspinspspins
If we write out the matrix multiplication explicitly, we have
2
X
ū(sk ) (k)i Γ1 (6p + mp ) Γ̄2
ij
u(sk )(k)j
" 2
#
X
= Γ1 (6p + mp ) Γ̄2 ij
u(sk ) kū(sk ) k
sk =1
sk =1
ji
= Γ1 (6p + mp ) Γ̄2 ij [6k + mk ]ji
= T r Γ1 (6p + mp ) Γ̄2 (6k + mk )
.
(2.29)
If any of the u spinors are replaced by v’s, the corresponding mass acquires
a minus sign. This follows from the Equation (A.10).
The technique used here is sometimes referred to as Casimir’s trick. After
applying it, all of the spinors are removed from the equation for M. The only
remaining steps necessary to sum over spins is to take the trace of a product
of matrices, a much easier problem. Rules for simplifying the traces of gamma
matrices (and therefore slashed four-vectors) can be found in Appendix A.
To average over initial spins, we simply divide the sum over all spins by
2k , where k is the number of incoming particles that can be in two different
spin states. For example, for electron positron annihilation, we divide by 4,
but for neutrino/electron elastic scattering, we only divide by 2. As explained
in §2.3.3, neutrinos are always left-handed particles and travel at relativistic
28
velocities, so their helicites can be approximated as always being left as well.
Thus the spin of an incoming neutrino will always be oriented in one direction
(opposite its direction of motion). The same applies for antineutrinos, which
are always right-handed.
The final summed and averaged square of the matrix element is written
as |M|2 .
2.5
Feynman Diagrams
2.5.1
Drawing the Diagrams
All of the above is useless if we do not know how to calculate M. For that,
we turn to Feynman diagrams. These diagrams, whose rules are derived from
perturbative expansions of QFT, provide an intuitive method of calculating
the contributions to the matrix element from different particle interactions.
Feynman diagrams have two axes, one representing time and the other
displacement. Our convention is to draw time on the horizontal axis and
displacement on the vertical, but the opposite is not unheard of. Particles
are represented by lines drawn in this time-space plane.
There are eight different kinds of lines that can be drawn in a tree level
Feynman diagram. Incoming fermions are drawn as solid lines that go from
the left edge of the diagram to a vertex, with an arrow pointing toward
the vertex. Outgoing fermions are drawn from a point to the right side of
the diagram, with an arrow away from the vertex. Incoming antifermions
29
(i.e. antiparticle fermions) are drawn as outgoing fermions and outgoing
antifermions as incoming fermions, but with the arrow pointing from right
to left. Incoming bosons and outgoing bosons are drawn similarly to fermions,
except that the lines are curvy (and do not always require arrows).
Internal propagators, whether fermionic or bosonic, are drawn in the same
fashion as external fields, but have both ends connected to a vertex. The arrow, if any, is drawn to maintain continuity of the particle paths and preserve
conservation of charge.
Note that antiparticles are drawn as moving backwards in time. While
this is not actually the case, treating an antiparticle as a time reversed particle does work mathematically and is a helpful tool to keep track of appropriate minus signs and spinors.
If it is possible to draw more than one diagram for a specific event, each
must be drawn. The total matrix element for the event is then the sum of
the matrix event calculated for the individual diagrams.
For examples of Feynman diagrams, see §3.1.
30
Particle
Spin 0
Incoming Spin 1/2 Particle
Outgoing Spin 1/2 Particle
Incoming Spin 1/2 Antiparticle
Outgoing Spin 1/2 Antiparticle
Incoming Spin 1
Outgoing Spin 1
Term
1
u
ū
v̄
v
µ
µ∗
Table 2.2: External Line Contributions to Feynman Diagrams
2.5.2
The Basic Setup
Once a diagram has been drawn, M is determined by setting up the following
integral:
− iM (2π)4 δ 4 (p1 + p2 + . . . + pm − k1 − k2 − . . . − kn ) =
4 4 4 Z
d q1
d q2
d q`
...
external lines × vertex factors × propagators
4
4
(2π)
(2π)
(2π)4
(2.30)
for a diagram with m incoming lines, n outgoing lines, and ` internal propagators.
The external line terms are determined by the incoming and outgoing
lines of the diagram. Their contributions are shown in Table 2.2. the “bar”
operator indicates an adjoint spinor, defines as
ū ≡ u† γ0
31
(2.31)
,
Particle
Spin 0
Spin 1/2
Spin 1 (Massless)
Spin 1 (Massive)
Propagator
i
q 2 −m2
i
6q −m
−igµν
q2
−i(gµν −qµ qν /m2 )
q 2 −m2
Table 2.3: Feynman Diagrams propagator terms
The ’s are polarization vectors, which are the spin 1 analog of Dirac spinors.
These played no role in our calculations, and so no detailed explanation is
provided for them in this paper.
The propagators are determined similarly by the internal lines of the diagram. These are shown in Table 2.3, where q and m are the four-momentum
and mass of the corresponding particle. If q << m for a spin 1 boson, the
massless form of the propagator may be used as a valid approximation.
The vertex factors consist of two parts. First there is a conservation of
four-momentum term in the form of a delta function:
(2π)4 δ (p1 + p2 + p3 )
.
Each of the p’s is the four-momentum of a line going into the vertex (outgoing
lines thus have a minus sign). The other part of the vertex factor is the term
associated with the coupling between the three (or sometimes four) particles
at the vertex. These are derived from the Lagrangian.
32
2.5.3
Deriving the Coupling Terms
Let Lint be the term in the total Lagrangian that describes the interaction
between the particles at the vertex. To calculate the coupling, write Lint in
momentum space (i.e. turn any i∂µ terms back into pµ ), divide by i, and
simply remove any of the fields. Whatever remains is the appropriate part
of the vertex factor for that interaction.
For example, Lint for the “neutral current” interaction between two fermions
and a Z boson in the Standard Model is
Lint = −
gZ X µ i
ψ̄i γ gV − gAi γ 5 ψi Zµ
2 i
,
(2.32)
where ψi is the fermion field for some particle i, and Zµ is the gauge boson
field for the Z.[2] gZ is defined in terms of the fundamental charge, e, and
the Weinberg mixing angle θW by
gZ ≡
e
gW
≡
cos θW
sin θW cos θW
.
(2.33)
Note that sin θW is defined as the ratio between the photon and W coupling
constants. On a more fundamental level, θW represents the degree to which
electroweak mixing has occured in the Standard Model.
The vertex factor term for this interaction is thus
−
igZ
γµ gVi − gAi γ 5
2
33
.
(2.34)
The “vector coupling” gV and “axial coupling” gA vary for different
fermions. Explicitly they are
gVi =
√
ρ Ti3 − 2Qi sin2 θW
gAi =
√ 3
ρTi
,
,
(2.35)
(2.36)
where Ti3 is the third component of the weak isospin of the fermion i, and
Qi is that fermion’s charge divided by e. ρ, which represents the relative
strength of the W and Z coupling, is formally defined as
ρ≡
2
MW
MZ2 cos2 θW
,
(2.37)
and has an experimental value of 1.4 The values for all four of these quantities
for different particles are listed in Table 2.4.
The fact that all three neutrinos and all three charged leptons have the
same coupling constants is part of an assumption known as lepton universality, which states that the coupling of leptons to gauge bosons is independent
of flavor. While this has been confirmed experimentally up to present day
limits, there are a number of beyond the Standard Model theories which
posit that it does not hold. Since our calculations were all done in a Standard Model framework, we assumed lepton universality throughout.
The other coupling important to us was the “charged current” interaction
between two fermions and a W boson. Applying the same procedure to that
4
ibid. 299
34
i
νe , νµ , ντ
e, µ, τ
u, c, t
d, s, b
Ti3
1√
ρ
2
1
−2
Qi
0
-1
− 12
− 31
1
2
2
3
gVi
1√
gAi
ρ
1
− + 2 sin2 θ
√ 21 4 2 W ρ 2 − 3 sin θW √
ρ − 12 + 23 sin2 θW
2
1
2√
− 12 ρ
1√
ρ
2 √
− 12 ρ
Table 2.4: Electroweak Coupling Constants for Fermions of type i
particular interaction Lagrangian,
gW X µ
Lint = − √
ψ̄i γ 1 − γ 5 T + Wµ+ + T − Wµ− ψi
2 2 i
(2.38)
yields a vertex factor of
igW
− √ γµ 1 − γ 5
2 2
.
(2.39)
The T + and T − terms are isospin raising and lowering operators, which
indicate that that two fermions must have opposite weak isospin signs (e.g.
an incoming electron coupled to a W also couples to an outgoing neutrino,
and vice versa).[2]
35
Chapter 3
Calculations
3.1
3.1.1
Summary of Possible Events
Neutrino/Electron Elastic Scattering
The first event of interest to us was muon neutrino/electron elastic scattering,
νµ e− → νµ e− . This interaction is very sensitive to new physics at energy
scales approaching a TeV.[10]
3.1.2
Inverse Muon Decay
Conversely, the process of “inverse muon decay”, νµ e− → µ− νe , which has
the same incoming particles, is already well understood thanks to a number
of high precision muon decay experiments. Since a 100 GeV scale νµ beam
incident on electrons will result in both events, examining both will help
36
ν
ν
µ
µ
Z
e
0
e
Figure 3.1: νµ e− → νµ e−
ν
ν
µ
e
W-
µ
e
Figure 3.2: νµ e− → µ− νe
to eliminate any systematic errors in the experiment, increasing the overall
sensitivity.[10]
In order for inverse muon decay to occur, the energy of the incoming
neutrino must be sufficient to create the muon mass. This threshold value is
approximately 10.9 GeV, and any simulation of an inverse muon decay event
must generate a cross section of zero if the incoming neutrino energy is below
this value.[10] Our calculations were done at a high enough Eν values that
this was not a concern.
37
ν
ν
µ
µ
Z
e
0
e
Figure 3.3: ν̄µ e− → ν̄µ e−
3.1.3
Antineutrino/Electron Elastic Scattering
Another event worth examining is muon antineutrino/electron elastic scattering, ν̄µ e− → ν̄µ e− . Not only will this occur if the incoming beam can be
run in an antineutrino mode, but some antineutrino contamination is likely
to occur even when operating in neutrino mode. The combination of the two
elastic scattering events can also be used to cancel out systematic errors, but
some of the more interesting new physics may be lost in the process.[10]
3.1.4
Electron and Tau Neutrino Events
While the hypothetical experiments we examined will likely use muon flavored neutrino or antineutrino beams, there are several compelling reasons
to examine the effects of electron or tau (anti)neutrino scattering as well.
Many electron neutrino experiments are already ongoing, and solar neutrinos are, barring oscillation, electron flavored.[1] There is also considerable
interest in in building tau neutrino experiments, as there are presently very
38
ν
ν
e
ν
e
Z
0
e
ν
e
e
W-
e
e
e
Figure 3.4: νe e− → νe e−
ν
ν
e
e
ν
e
Z
0
W-
e
e
ν
e
e
e
Figure 3.5: ν̄e e− → ν̄e e−
few recorded tau neutrino interactions.[7]
Oscillation is unlikely to occur in most beamline neutrino experiments
that take place within a single facility. There are, however, a number of
neutrino oscillation experiments with similar beam and target characteristics,
but with beam distances on the order of hundreds of kilometers.
Finally, any “pure” muon neutrino beam will contain some muon antineutrinos and electron neutrinos and antineutrinos as contamination. Likewise a
muon antineutrino beam will have muon neutrino and electron (anti)neutrino
39
ν
ν
τ
ν
τ
Z
e
ν
τ
τ
Z
0
e
e
Figure
3.6:
−
−
ντ e → ντ e
0
e
Figure
3.7:
−
−
ν̄τ e → ν̄τ e
ν
ν
τ
e
W-
τ
e
Figure 3.8: ντ e− → τ − νe
contamination as well. The flux for these unintended particles can be predicted based on the characteristics of the beam, and is therefore of significant
interest to us to model the statistical effects of this beam contamination.[10]
Like inverse muon decay, inverse tau decay also has a threshold energy,
in this case at approximately 3.09 TeV. This is a higher energy value for
incoming neutrinos than can likely be produced in the near future. We have
therefore ignored the event ντ e− → τ − νe in this paper.
40
3.2
Example Calculations
νµ e− → νµ e−
3.2.1
We now wish to calculate the differential cross sections for muon neutrino/electron
elastic scattering by hand. This example serves as a pedagogical exercise in
how differential cross sections are calculated. It also provides a means of
double checking our automated results.
Following the Feynman rules as described above, we arrive at the following
equation for the matrix element:
4
Z 4
−igZ µ ν
ν 5
γ gV − gA γ u (p1 )
ū (k1 )
2
−igZ ν e
igµν
e 5
γ gV − gA γ u (p2 )
ū (k2 )
MZ2
2
−i (2π) Mδ (p1 + p2 − k1 − k2 ) =
(2π)4 δ 4 (p1 − k1 + q)
(2π)4 δ 4 (p2 − k4 − q)
d4 q
(2π)4
.
(3.1)
After integrating over one of the delta functions and simplifying, we have
M=
√
2GF ū (k1 ) γ µ gVν − gAν γ 5 u (p1 ) ū (k2 ) γµ gVe − gAe γ 5 u (p2 )
,
(3.2)
where
1
GF ≡ √
4 2
gW
MW
2
1
= √
4 2
41
gZ
MZ
2
.
(3.3)
This quantity, called the “Fermi Coupling Constant”, is well defined from
muon decay experiments.1
We then square the absolute value and apply Casimir’s trick:
2
2 |M|2 = 2G2F ū (k1 ) γ µ gVν − gAν γ 5 u (p1 ) ū (k2 ) γµ gVe − gAe γ 5 u (p2 )
(3.4)
X
|M|2 = 2G2F Tr γ µ gVν − gAν γ 5 6p1 γ ν gVν − gAν γ 5 6k1
spins
Tr γµ gVe − gAe γ 5 (6p2 + me ) γν gVe − gAe γ 5 (6k2 + me )
.
(3.5)
After a significant amount of calculation, these traces simplify to the following:
|M|2 = 32G2F (gVν + gAν )2 (gVe + gAe )2 (p1 · p2 ) (k1 · k2 )
+ (gVν + gAν )2 (gVe − gAe )2 (p1 · k2 ) (p2 · k1 )
−m2e (gVν + gAν )2 (gVe )2 − (gAe )2 (p1 · k1 )
,
(3.6)
where we have divided by two to average over the initial spins of the target
electrons, as explained in §2.4.3.
Note that in the case of the neutrino coupling constants only the “lefthanded coupling” term, (gLν )2 ≡
1
s (gVν
4
+ gAν )2 , remain, while the left and
right-handed electron coupling terms, and their admixture, are present.
1
Griffiths 307
42
,
After making the appropriate kinematic substitutions as detailed in Appendix B and applying “Fermi’s Golden Rule” for 2-to-2 scattering, we arrive
at the differential cross section for this process, cSimple application of the
chain rule gives us
∂θ dσ
∂ cos θ dσ
dσ
= 2π sin θ
= 2π
dT
∂T dΩ
∂T dΩ
me
dσ
= 2π
,
(Eν − T )2 dΩ
(3.7)
(3.8)
where we have employed the expression for cos θ derived in Appendix B. Thus
2G2F me ν e
dσ
=
(gL gV + gLν gAe )2
dT
π
2
T
ν e
ν e 2
+ (gL gV − gL gA ) 1 −
Eν
ν e 2
ν e 2 me T
− (gL gV ) + (gL gA )
Eν2
.
(3.9)
For notational convenience, the following substitutions are conventionally
made,
gVνe ≡ 2gLν gVe
,
(3.10)
gAνe ≡ 2gLν gAe
.
(3.11)
43
Equation (3.10) now becomes
dσ
G2 me νe
= F
(gV + gAνe )2
dT
2π
2
T
νe
νe 2
+ (gV − gA ) 1 −
Eν
νe 2
νe 2 me T
− (gV ) + (gA )
Eν2
.
(3.12)
Other than the energy terms, T and Eν and the electron mass me , the
only parameters in this equation are GF , gVνe , and gAνe . As the Fermi Coupling
constant is well determined by muon decay experiments, we can fit to the
remaining two coupling constants in our regression once we have integrated
over the flux. These constants can themselves be expressed in terms of SM
constants according to the definition of the coupling constants in §2.5.3:
gVνe
gAνe
1
2
= ρ − + 2 sin θW
2
1
=− ρ .
2
,
(3.13)
(3.14)
Any sensitivity to deviations from the SM values for the coupling constants
can therefore be translated into deviations from the SM values of these more
fundamental quantities.
44
νµ e− → µ− νe
3.2.2
We now carry out the beginning of the same calculation for inverse muon
decay in order to illustrate some differences between neutral current and
charged current interactions. Writing out the Feynman rules, we have
4
4
−igW µ
5
ū (k2 ) √ γ 1 − γ u (p1 )
2 2
igµν
−igW ν
5
ū (k1 ) √ γ 1 − γ u (p2 )
2
MW
2 2
Z −i (2π) Mδ (p1 + p2 − k1 − k2 ) = −
(2π)4 δ 4 (p1 − k1 + q)
(2π)4 δ 4 (p2 − k4 − q)
d4 q
,
(2π)4
(3.15)
which simplifies to
GF M = − √ ū (k2 ) γ µ 1 − γ 5 u (p1 ) ū (k1 ) γµ 1 − γ 5 u (p2 ) .
2
(3.16)
The minus sign in front is the result of interchanging the outgoing momenta
in the diagram. Its presence here is purely pedagogical, as the minus sign
will vanish once we take the absolute value of M.
After following the appropriate procedure the following equation for the
45
averaged, summed, squared matrix element emerges:
|M|2 = 64G2F (p1 · p2 ) (k1 · k2 )
1
2
= 64GF me Eν me Eν + m2e −
2
1 2
2
≈ 64GF me Eν me Eν − mµ
2
(3.17)
1 2
m
2 µ
,
(3.18)
(3.19)
where we have taken advantage of the fact that me << mµ .
We could then proceed from here to calculate
dσ
dT
as in §3.2.1. The im-
portant thing to notice here is that there is no dependence on the coupling
constants in any of the above expressions. As they were the only variables
not considered constant in our analysis,
dN
dT
can be numerically predicted for
inverse muon decay. This would allow any experiment with a muon neutrino beam to account for any inverse muon decay contributions to the total
recorded
dN
.
dT
3.2.3
ν̄µ e− → ν̄µ e−
As an example of a process involving antiparticles, we now calculate the differential cross section for muon antineutrino/electron elastic scattering. As
shown in Fig. 3.3, the only difference between this event and muon neutrino/electron scattering is the exchange of p1 with k1 . Performing this sub-
46
stitution on Equation (3.6) gives us
|M|2 = 32G2F (gVν + gAν )2 (gVe + gAe )2 (p1 · k2 ) (k1 · p2 )
+ (gVν + gAν )2 (gVe − gAe )2 (p1 · p2 ) (k1 · k2 )
− m2e (gVν + gAν )2 (gVe )2 − (gAe )2 (p1 · k1 )
.
(3.20)
This is equivalent to flipping the sign in the (gVe ± gAe ) terms. Applying
this substitution to Equation (3.7), we derive the differential cross section
for this process:
dσ
G2 me νe
= F
(gV − gAνe )2
dΩ
2π
2
T
νe
νe 2
+ (gV + gA ) 1 −
Eν
νe 2
νe 2 me T
− (gV ) + (gA )
.
Eν2
3.2.4
(3.21)
νe e− → νe e−
Unlike the other tree level events examined here, νe e− scattering has two
different contributing process, as shown in Fig. 3.4. One is the neutral current
process equivalent to νµ e− elastic scattering, and the other is the charged
current process equivalent to inverse muon decay. The matrix elements for
each of these are identical to their muon flavored cousins (Equations (3.2)
and (3.17)), keeping in mind that all of the mass terms will be me once they
appear explicitly:
47
MN C =
√
2GF ū (k1 ) γ µ gVν − gAν γ 5 u (p1 ) ū (k2 ) γµ gVe − gAe γ 5 u (p2 )
;
(3.22)
GF MCC = − √ ū (k2 ) γ µ 1 − γ 5 u (p1 ) ū (k1 ) γµ 1 − γ 5 u (p2 )
2
.
(3.23)
In order to compute the differential cross section for this event, we must
add the neutral and charged current processes. This can be simplified by
using the Fierz reordering theorem,[5] which allows us to rewrite Equation
(3.24) as
GF MCC = √ ū (k1 ) γ µ 1 − γ 5 u (p1 ) ū (k2 ) γµ 1 − γ 5 u (p2 )
2
. (3.24)
Because the momenta are now in the same order in both contributing
matrix elements, it is possible to apply Casimir’s trick to all four terms in
|M|2 = MN C MN C
∗
+ MN C MCC
∗
+ MCC MN C
∗
MCC MCC
∗
.
(3.25)
If we wished, we could then compute the traces and go on to calculate
dσ
.
dT
48
Chapter 4
External Software
4.1
4.1.1
FeynArts
Description
FeynArts is a Mathematica package that allows the user to generate Feynman
diagrams and their corresponding amplitudes. FeynArts was initially developed in 1990 by Hagen Eck and Sepp Küblbeck, and its development was
taken over by Thomas Hahn in 1998.The latest iteration of FeynArts, version
3.4, can be found at http://www.feynarts.de and requires Mathematica 3
or above to run properly.1
FeynArts operates through three sequential processes: generating topologies, inserting fields, and creating Feynman amplitudes. FeynArts also comes
equipped with a robust Paint function, capable of displaying and manipulat1
All information in this section can be found in [4]
49
2
T1
®
2
T2
T3
T4
Figure 4.1: CreateTopologies[0,2→2]
ing the diagrams generated during this overall process.
4.1.2
Create Topologies
The first step when using FeynArts is to generate topologies for a given
number of incoming particle, outgoing particles, and loops. This creates
a list of all of the possible Feynman diagrams for those specifications. For
example, entering the command “Paint[CreateTopologies[0,2→2]]” generates
the output shown in Fig. 4.1. Specific diagrams can be generated by directly
inputing the arrangement of propagators and vertices, but this is generally
unnecessary if a specific event is to be simulated.
50
e
Νi
® Νk el
Νk
Νk
Νi
Νi
V
Νk
Νi
Z
V
el
el
e
e
T1 G1 N1
el
e
T1 C1 N2
T2 G1 N3
Νk
Νi
W
el
e
T2 C1 N4
Figure 4.2: Topology from Fig. 4.1 with νj e− → νk `−
l fields added
4.1.3
Insert Fields
The next step is to specify which particles, or, in the language of FeynArts,
fields, should be inserted into a generated set of topologies. Anything from
specific particles to entire families of particles can be selected in this fashion.
It is also at this point in the program that a model (e.. the Standard Model)
is selected and any intermediate particles are excluded. As shown in Fig. 4.2,
which represents events of the form νj e− → νk `−
l with Higgs like particles
excluded, both generic and specific fields are inserted simultaneously at this
point, a feature apparently intended to speed up calculation.
The actual InsertFields function has three required input fields: the topol51
class
F[1]
(neutrinos)
self-conj. indices
members
no
Generation F[1, {1}]
F[1, {2}]
F[1, {3}]
F[2]
no
Generation F[2, {1}]
(massive leptons)
F[2, {2}]
F[2, {3}]
F[3]
no
Generation F[3, {1,o}]
(up-type quarks)
Color
F[3, {2,o}]
F[3, {3,o}]
F[4]
no
Generation F[4, {1,o}]
(down-type quarks)
Color
F[4, {2,o}]
F[4, {3,o}]
V[1]
yes
V[1]
V[2]
yes
V[2]
V[3]
no
V[3]
νe
νµ
ντ
e
µ
τ
u
c
t
d
s
b
γ
Z
W
mass
0
0
0
ME
MM
ML
MU
MC
MT
MD
MS
MB
0
MZ
MW
Table 4.1: Abridged list of FeynArts fields for the “SM Model”
ogy, the incoming and outgoing particles, and the model to be applied. The
topology is the output of CreateTopologies. The particles are entered in the
form {Field1,Field2}→{Field3,. . . ,FieldN}. The Field terms are defined by
the model that is implemented. For instance, if the “SM” model is used,
Table 4.1 describes the symbols for the particles that were of use to us. The
model is declared by typing the name of a .mod file after Model→.
An important option for InsertFields is ExcludeParticles. As its name
suggests, this optional command tells InsertFields to not use any designated
particles (or classes of particles) in the Feynman diagrams it creates. In
our case, we used this option to require FeynArts to only consider W and Z
mediated interactions.
52
4.1.4
Create Feynman Amplitude
If all that is needed is a description or illustration of all possible Feynman
diagrams for a given event, then it is possible to simply use the output of
the InsertFields function. FeynArts is capable of going one step further,
however, by actually generating a set of matrix elements corresponding to
those diagrams using the CreateFeynAmp function, whose input is the direct
output of InsertFields.
The output of CreateFeynAmp contains a description of the process in
question, an analytic expression of the matrix element, and a list of substitutions. The substitutions are necessary because FeynArts expresses the matrix element in terms of spinors, Dirac matrices, metric tensors, and generic
masses and coupling factors. The specific values for those masses and constants are then listed in the form of substitutions. While this allows FeynArts
to efficiently calculate many different Feynman diagrams at and beyond the
tree level quickly, it is inconvenient for our purposes, as additional software
must be written to automatically substitute these constants into the matrix
element.
4.1.5
Models
FeynArts comes prepackaged with several particle physics models, including
the Standard Model, a QED only version of the SM, and the Minimal Supersymmetric Standard Model. It is also possible for the user to create his or her
53
own models, either by modifying one of the preexisting models or creating a
new one from scratch; the former is far easier.
The two most important features of a FeynArts class model are the lists
M$ClassesDescrption, which contains information on all of the particles in
the model, and M$CouplingMatrices, which describes the coupling between
those particles. Other features, such as particle number conservation laws
and number of generations round out the model.
4.2
4.2.1
FeynCalc
Description
Where FeynArts allows us to generate matrix elements for any tree level process in which we are interested, it is in a language unknown to Mathematica
in general and thus cannot be immediately used to find differential cross
sections. Fortunately, there exists another Mathematica package, FeynCalc,
which permits algebraic manipulation in the formalism of particle physics.
FeynCalc was originally created by Rolf Mertig and now exists as open
source software. FeynCalc 5.1, the latest version, likewise requires Mathematica 3 or higher and can be found at http://www.feyncalc.org.2
2
All information in this section can be found in [8]
54
4.2.2
FeynCalc Objects
The FeynCalc package contains several functions that represent the algebraic
objects of quantum field theory. The simplest of these is “FourVector[p,µ]”,
which creates a four-vector of the form pµ or pµ (FeynCalc keeps track of
the distinction between contravariant and covariant vectors based on their
placement within an expression when evaluating functions).
“Spinor[p,m]” is a Dirac spinor with momentum p and mass m. The
second argument is unnecessary if the particle is massless. The presence of
a minus sign in front of p indicates a v spinor, its absence a u. Whether the
spinor is an adjoint or not is determined by the position of the spinor relative
to any gamma matrices.
Dirac gamma matrices themselves are denoted by “DiracMatrix[µ]”, where
µ is the Lorentz index of the matrix. “DiracMatrix[5]” is γ 5 . “DiracMatrix[6]” is shorthand for 21 (1 + γ 5 ), and “DiracMatrix[7]” is likewise short for
1
2
(1 − γ 5 ). “DiracSlash[p]” is a contraction of “FourVector[p,µ]DiracMatrix[µ]”
(i.e. 6p).
“MetricTensor[µ,ν]” is the metric tensor, gµν . Finally, “LeviCivita[µ,ν,κ,τ ]”
is the input function for a Levi-Civita epsilon with Lorentz indices in the order entered.3
3
ibid.
55
4.2.3
Useful Functions
FeynCalc also comes with a number of functions that allow the user to easily
perform a number of common tasks in particle physics, including the calculations performed in Chapter 3. The simplest of these is just an overwrite of
the Mathematica “.” operator to perform non-commutative multiplication
of the various matrices described above.
“ScalarProduct[p,q]” is the input for the dot product of vectors p and q.
This function is not evaluated by default, and it is recommended that the
user define each scalar product based on the kinematics of the problem.
The Contract function contracts pairs of Lorentz indices of four-vectors
and metric tensors. For example, “Contract[MetricTensor[µ,ν]FourVector[p,µ]]”
outputs pν (i.e. “FourVector[p,ν]”). Likewise, “Contract[FourVector[p,µ]FourVector[q,µ]]”
evaluates as p · q. EpsEvalutate[expr] is similar but, as its name suggests,
contracts Levi-Civita epsilons.
ComplexConjugate[expr] is a new function that correctly complex conjugates an expression with gamma matrices or other Lorentz indices. This
should be used instead of the default Mathematica Conjugate[expr] function. ComplexConjugate is a necessary function for the implementation of
FermionSpinSum[expr], which only works on expressions of the form “expr1
* ComplexConjugate[expr2]” (where expr1 can be equal to expr2).
FermionSpinSum is a critically important function for our purposes, as
it effectively performs what we have been calling Casimir’s trick, turning
a squared Feynman amplitude into the product of traces. The traces will
56
be left as explicit functions of the form tr(. . . ), but they can be evaluated
by appending “/.DiracTrace→Tr” to the end of an expression containing
FermionSpinSum.
4.3
Compatibility
It is clear that FeynArts and FeynCalc can and should be used in conjunction
with each other. The former can construct matrix elements for any given
event, and the latter can then calculate |M|2 from them. Unfortunately,
there are some compatibility issues.
The packages were initially designed in such a way that the output of
FeynArts could be entered as input in FeynCalc, but the changes made to
FeynArts in version 3 have significantly impaired this feature, as the output
of CreateFeynAmp is no longer formatted identically to the objects used
by FeynCalc. FeynArts does have two functions that can ameliorate this
problem.
InputForm reformats the output into a form that can be easily reentered into another function at the price of legibility. More useful is the
ToFA1Conventions function. This rewrites various spinors and matrices that
FeynArts outputs into the older forms, which are identical to the FeynCalc
representations of those same objects.
FeynCalc is just as incapable of directly performing the substitutions
provided by the CreateFyenAmp’s output as FeynArts is, so we were required
57
to write a preprocessing function in Mathematica that will be called when
moving from one program to the other. The details of this software are
provided in the next chapter.
Another compatibility issue that we had to deal with was the fact that,
even though the two programs are not mutually compatible, they have enough
functions with the same names that both packages cannot operate within
Mathematica simultaneously. Once output was generated in FeynArts, it
was saved as a file within Mathematica. This file was then opened once the
Mathematica kernel had been reset for use with FeynCalc.
58
Chapter 5
Our Software
Note: All of the code described within this chapter can be found in Appendix
D.
5.1
The CouplingSM Model
The FeynArts model “SM” contains all of the field and coupling information from the Standard Model, including the Higgs interactions. All of the
electroweak coupling constants of interest to us are rendered as their SM
values in terms of sin θW , with ρ equal to 1. As this expresses six (or more
if lepton universality does not hold) variables in terms of one, extracting
the coupling constants from the matrix elements created by FeynArts would
prove difficult.
Instead, we created a modified version of the “SM” model, dubbed “Cou-
59
plingSM”, in which the coupling constants are treated as their own unassigned variables. To do this, we wrote a Mathematica function, NoTheta,
which when applied to the M$CouplingMatrices object of the “SM” Model,
rewrites four of the coupling matrices and leaves the rest unchanged.
The neutrino/neutrino/Z coupling matrix is modified so that the first
order left hand coupling constant is multiplied by “2gnuL”. The charged
fermion/charged fermion/Z coupling matrix is modified so that the first order
” and the right by “ geV−geA
”,
left hand constant is multiplied by “ 2geV+geA
2
(SW2 − 12 )
where “SW” is the FeynArts symbol for sin θW . The corresponding terms
for the quark/quark/Z coupling matrices are similarly modified, in case we
would have chosen to use them at a later time. NoTheta leaves all of the
other coupling matrices unchanged.
The “SM” model is then loaded, M$CouplingMatrices is rewritten by NoTheta applied to each element in the initial M$CouplingMatrices, and the
now modified model is dumped into the newly created file “CouplingSM.mod”.
This model was then entered into the appropriate field whenever the InsertFields function was called.
5.2
5.2.1
FeynArts to FeynCalc Preprocessor
Regular Expressions in Mathematica
As explained in §4.3, it was necessary to write a function capable of making
FeynArts output readable as input by FeynCalc. This was accomplished
60
through the extensive use of regular expressions.
A regular expression is a computational tool for recognizing patterns in
strings. Rather than representing a specific substring, a regular expression
can stand for a number of different possible substrings based on programmed
criteria. Several features of regular expressions allow for this flexibility.
Regular expressions support boolean OR operations applied to substrings
within the regular expression. Regular expressions also have quantification
operators that will search for zero to one, zero or more, one or more, or
a designated number of a particular substring in a row. Finally, regular
expressions can search for categories of characters, such as any digit or any
character that is not a letter of the alphabet.
A simple, if slightly inefficient, example of a regular expression, using the
syntax preferred by Mathematica, is “(bb)|([∧b]{2})”. This regular expression corresponds to the substring of two “b”s in a row or the substring of
any two characters that are not “b” in a row.
Mathematica provides full support for all standard regular expression
constructs though the function RegularExpression[“regex”], where “regex”
is a properly formatted regular expression.
Two additional Mathematica functions are necessary for editing strings
with regular expressions. The function StringCases[“string”,patt] gives a
list of all substrings within “string” that match patt, which may either be
another string or a regular expression. StringReplace[“string”,s→sp] replaces
any instance of the string or regular expression s within the string “string”
61
with the string sp.
5.2.2
Implementation
The first step for our preprocessor, after converting the input data to a string,
is to separate out individual matrix elements from the total output of CreateFeynAmp, in case more than one diagram contributes to the particular event
(e.g. ve e− scattering). The Convert function considers each substring that is
bracketed by the strings “GraphName” and “FeynAmp”, or “Graph Name”
and the end of the string, to be an individual matrix element. These particular strings were chosen purely on the basis of FeynArts standard formating
of the output of CreateFeynAmp. This selection mechanism has the added
bonus of removing the unnecessary prefix information that accompanies that
output.
At the present time, Convert requires the user to specify how many contributing diagrams exist for the event. This is unfortunate in that it cuts
down on the overall automation of our software, it does make the overall
implementation easier. Additionally, it is of pedagogical value to require the
user to understand all of the tree level processes involved in an interaction.
After the initial amplitudes are processed by Convert, each individual
matrix element is passed to the Convert0 function, which handles the rest of
the preprocessing. First, any remaining prefix data is removed. The postfix
substitution information is then separated from the rest of the string. Two
lists are generated from the postfix, one for the substrings to be replaced
62
and another with their corresponding replacements. Those replacements are
then substituted into the matrix element string.
The FeynArts PropDenom function, which represents the denominator
associated with a propagator, is then rewritten in terms of simple Mathematica operations (e.g. division, squaring, etc.). It is at this point that the
small q approximation for massive spin 1 bosons is made by simply rewriting
the denominator.
Finally, the tail of the now corrected matrix element string is adjusted to
contain the appropriate number of closing brackets for a proper Mathematica
expression. The string is then converted back to an expression as the output
of Convert0.
This entire procedure is managed by the FullConvert function, which
takes the initial amplitude data and the number of diagrams as input. Both
of these are sent to the Convert function, whose output will be a list of
the amplitudes of each diagram. Each element in this list is then input
into Convert0, with the set of outputs forming a new list to be output by
FullConvert. This list is then sent to the next portion of our software, which
implements the functions provided by FeynCalc.
5.3
Cross Section Calculator
The FullCalculate function employs all of the FeynCalc operations we wish
to apply to the matrix elements following their preprocessing. Its input is
63
the list of matrix elements and the masses of the target particle and the
two outgoing particles for the event. These masses can be either entered
numerically or (more preferably) entered as undeclared variables using the
same notation employed by FeynArts in Table 4.1, for the sake of consistency.
Before FullCalculate is called, the scalar products of each of the momenta
with each other are defined as described in Appendix B in terms of E1 (i.e.
Eν ), T, and the masses. The mass terms are defined as unassigned variables
with m2 corresponding to mp2 , m3 to mk1 , and m4 to mk4 . Note that m3 is
not assumed to be 0.
FullCalculate’s first instruction is to call the function Calculate, which has
only the list of matrix elements as input. This function calculates |M|2
using the FeynCalc functions described in Chapter 4.
Calculate accounts for the interference of two or more contributing diagrams by applying FermionSpinSum to
N
N X
X
Mi M∗j
,
i=0 j=0
where N is the number of matrix elements in the list. As noted in §4.2,
complex conjugation is implemented using FeynCalc’s ComplexConjugation
function. The rest of Calculate simply uses the appropriate FeynCalc functions to sum |M|2 over the final spins and divides by 2 to average over the
initial electron spin.
Once |M|2 is returned to FullCalculate, that function then calls Cross-
64
Section, a function which calculates
dσ
dT
algebraically as described in §3.2.
FullCalculate then swaps in the user defined masses for m2, m3, and m4
using string replacement. Finally, after some simple cleanup, it returns the
differential cross section in terms of SM constants, the coupling constants,
E1, and T.
Although it could have been done earlier, it is at this point in the code
that the SM constants, but not the coupling constants, are defined numerically as given in Appendix E, with the masses in units of eV and the rest
dimensionless. The SM values of ρ and sinθW are defined here as well, but
since they do not appear in the present representation of
dσ
,
dT
they have no
immediate effect on the coupling constants.
5.4
Flux Integrator
At this point, it is necessary to calculate
dN
dT
before any regression analysis
can be performed. As shown in Equation (2.18) this requires us to integrate
over the incoming neutrino beam flux. This is accomplished by the function
BinCruncher. Its input is the differential cross section, running time, and
dΦ
the flux. The latter is written in terms of a table of ordered pairs, ( dE
(Eν ),
ν
Eν ).
BinCruncher uses these points to generate a piecewise function of
dΦ
dEν
in
terms of E1 with each specified point at the mean E1 value of each piece.
However, the actual piecewise function is never constructed explicitly by the
65
function. Instead, the product of each piece and the differential cross section
are integrated over E1 individually, and those integrals are then summed and
multiplied by t× to yield
5.5
5.5.1
dN
.
dT
Linear Regression
Regression Goals
Once we have an expression for
dN
dT
for a given event and beam, the next step
is to determine its sensitivity to deviations from the SM of the electroweak
coupling, in the form of gVνe and gAνe . We accomplished this by means of a
linear regression. For the details of linear regression analysis see Appendix
C.
Unlike most regressions, ours were run with the regression parameters
known a priori. If a regression is modeled as y = f (x), then our y would
simply be equal to f (x) for each value of x with f predefined. This way,
the central values of the regression parameters are simply their SM values,
and their uncertainty reflects the sensitivity of the experiment to statistical
uncertainties.
We took this one step further by rewriting the coupling constants in the
66
following way:
gVνe ⇒ (gVνe )SM + δgVνe
,
gAνe ⇒ (gAνe )SM + δgAνe
.
δgVνe and δgAνe represent the deviations of the coupling constants from their
SM values, while (gVνe )SM and (gAνe )SM are the SM values for the coupling
constants, expressed in terms of the values of ρ and sin θW found in Appendix
E.
When this substitution was made, any products of two or more δ terms
were set equal to 0, as the deviations are only likely of significance to first
order. Thus,
(gVνe )2 ⇒ ((gVνe )SM )2 + 2 (gVνe )SM δgVνe + O (δgVνe )2
,
(gAνe )2 ⇒ ((gAνe )SM )2 + 2 (gAνe )SM δgAνe + O (δgAνe )2
,
gVνe gAνe ⇒ (gVνe )SM (gAνe )SM + 2 (gAνe )SM δgVνe + 2 (gVνe )SM δgAνe + O (δgVνe )2 + O (δgVνe )2
After this substitution was made, gVνe and gAνe were set equal to their SM
values. The regression was then made with δgVνe and δgAνe as the regression
parameters. The central values of our regression parameters were therefore
predetermined by the regression to have a value of 0, and their uncertainty
would represent solely the statistical sensitivity to measurements of deviations from the Standard Model.
67
.
5.5.2
Regression Preprocessor
String manipulations were again used to implement this procedure. The
function DataTable constructs a table of values to be used in the regression.
Its input is
dN
dT
and a list of values of T at which data points are to be taken
for the regression. DataTable, after declaring some preliminary constants,
constructs a table of x values by calling, in sequence, the functions TSub,
MakeXData, and PrepX for each value of T.
TSub calculates
dN
dT
at the specific value of T by substituting that number
for T. This streamlines the process of trying to pass a function through
another function, which can cause errors in Mathematica.
MakeXData performs the substitutions explained in §5.5.1. The resultant
mathematical expression is of the form
a + bδgAνe + cδgVνe
.
PrepX converts this expression into a three element list by converting the
+ or - operators into commas and placing brackets around the expression.
The order of elements in this list is a, b, c.
DataTable then creates a table of y values by calling TSub and MakeYData for each value of T. MakeYData is similar to MakeXData, but does not
add the δ terms; its output is simply a.
The third and final table that DataTable produces contains the standard
deviations associated with each of the y values. As we are only concerned
68
with statistical uncertainties in our analysis, the number of hits in each energy
bin is assumed to follow a Poisson distribution. Therefore, each σ is set equal
to the square root of its corresponding y value.
A list of all three tables is then exported to the function MultilinearRegression, which implements the linear regression itself.
5.5.3
Regression Implementation
MultilinearRegression performs a linear regression using the matrix method
as described in Appendix C. The first column of the x data (the a term) is
subtracted from the y data to fit the remaining two columns of the x data to
a zero vector of the same length. The uncertainties are given by the σ table,
and no correlations are assumed between each of the measurements.
The output of MultilinearRegression gives the central values of the regression parameters; the χ2 value of the fit; and the uncertainties, correlation,
and a graph displaying the 1σ and 90% confidence ellipses for δgVνe and δgAνe .
The uncertainties, correlation, and confidence region graph are then generated for δρ and δ sin2 θW through standard methods of error propogation.
The scales of the graphs are given in the input to MultilinearRegression and
are the only inputs the functions accepts apart from
dN
.
dT
The central values
are provided as a means of doublechecking the regression. If all has gone
properly, they should be close to 0. The χ2 value should also be close to 0,
as the regression has been engineered to be a perfect fit.
69
Chapter 6
Data
6.1
Input Flux
We used the flux profile shown in Table 6.1 and Fig. 6.1 for all of the events
we examined. This flux is similar, but not identical to, the fluxes of proposed
high energy neutrino experiments, such as NuSOnG[10]. We simulated an
experiment with 3 × 1019 protons on target, a measure of the number of
neutrinos created for the beam, per year run over 5 years.[10] Because the
differential flux was given per 106 protons on target, we simply multiplied
dN
dT
to put it in the correct units.
Six hundred kinetic energy bins were used in our regression, with T rang-
ing from 0 to the maximum Eν of the incoming beam.
70
Eν (GeV)
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
360
380
400
420
440
460
480
500
520
540
560
580
600
dΦ
/106
dEν
Protons on Target
771.3764
5625.51
10624.99
12953.9
9974.87
5504.07
3429.59
3298.634
4063.61
4395.66
3619.162
2955.796
2288.316
1254.01
885.2
521.167
170.3828
168.5529
40.72519
39.9989
6.0909
6.05385
3.564138
1.333776
1.304137
0.222296
0.2371159
0.4149531
0.5409204
0.51869
Table 6.1: Incoming Neutrino Beam Flux
71
106 POT
dF
dEΝ
104
1000
100
10
1
100
200
300
400
500
600
EΝ HGeVL
Figure 6.1: Incoming neutrino beam flux
Event
−
νµ e → νµ e −
ν̄µ e− → ν̄µ e−
νe e − → νe e −
ν̄e e− → ν̄e e−
ντ e − → ντ e −
ν̄τ e− → ν̄τ e−
δgVνe
0.001958
0.009403
0.006217
0.04511
0.001958
0.009403
Correlation
δgAνe
δρ
0.007339
0.6576
0.01468
0.01244
0.8764
0.02488
0.01694
0.08365
0.03388
0.04715
0.9925
0.09431
0.007339
0.6576
0.01468
0.01244
0.8764
0.02488
δ sin2 θW Correlation
0.006333
0.8109
0.03360
0.9360
0.02584
0.2884
0.1632
0.9943
0.006333
0.8109
0.03360
0.9360
Table 6.2: Uncertainties in Deviations from the SM Value of Electroweak
Parameters
6.2
Results
The uncertainties and correlations for δgVνe , δgAνe , δρ, δ sin2 θW are shown in
Table 6.2.
The 1σ and 90% confidence regions for gAνe vs. gVνe and sin2 θW vs. ρ are
shown in Fig. 6.2 through 6.7, where we have used the SM values of the
constants for the origin of the plots.
No uncertainty data is available for νµ e− → µ− νe , as there is no dependence on the electroweak coupling constants in the cross section for that
72
-0.48
0.24
-0.49
sin2 ΘW
gA
0.23
-0.50
0.22
-0.51
0.21
-0.52
-0.07
-0.06
-0.05
-0.04
0.96
0.98
1.00
1.02
1.04
1.02
1.04
1.02
1.04
Ρ
gV
Figure 6.2: νµ e− → νµ e−
-0.48
0.24
-0.49
sin2 ΘW
gA
0.23
-0.50
0.22
-0.51
0.21
-0.52
-0.07
-0.06
-0.05
-0.04
0.96
0.98
1.00
Ρ
gV
Figure 6.3: ν̄µ e− → ν̄µ e−
-0.48
0.24
-0.49
sin2 ΘW
gA
0.23
-0.50
0.22
-0.51
0.21
-0.52
-0.07
-0.06
-0.05
-0.04
0.96
0.98
1.00
Ρ
gV
Figure 6.4: νe e− → νe e−
73
-0.48
0.24
-0.49
sin2 ΘW
gA
0.23
-0.50
0.22
-0.51
0.21
-0.52
-0.07
-0.06
-0.05
-0.04
0.96
0.98
1.00
1.02
1.04
1.02
1.04
1.02
1.04
Ρ
gV
Figure 6.5: ν̄e e− → ν̄e e−
-0.48
0.24
-0.49
sin2 ΘW
gA
0.23
-0.50
0.22
-0.51
0.21
-0.52
-0.07
-0.06
-0.05
-0.04
0.96
0.98
1.00
Ρ
gV
Figure 6.6: ντ e− → ντ e−
-0.48
0.24
-0.49
sin2 ΘW
gA
0.23
-0.50
0.22
-0.51
0.21
-0.52
-0.07
-0.06
-0.05
-0.04
0.96
0.98
1.00
Ρ
gV
Figure 6.7: ν̄τ e− → ν̄τ e−
74
event (see §3.2.1). We were able to generate the differential cross section for
this event, which agreed with the predicted value derived from the by-hand
calculations. If we wished, we could have gone on to integrate over the flux
and determine
dN
,
dT
but a number of singularities and discontinuities exist in
the resultant function which are heavily dependent on the flux profile. A
graph of this function was therefore considered to be not particularly useful.
dN
dT
should therefore be calculated numerically for each individual experiment.
75
Chapter 7
Conclusion
7.1
Interpretation of Results
The most important result is that our procedure generated data. We were
able to take an event, muon neutrino/electron elastic scattering, and with
our software generate its matrix element, calculate the differential cross section, integrate over a beam flux to determine the distribution of counts in
kinetic energy bins, and perform a linear regression to determine the statistical uncertainty of the event to deviations from the Standard Model values
of the neutrino/electron electroweak coupling constants. While most of this
calculation could have been done by hand, the versatility of our software
allows for easy modification of either the model or the event.
Muon neutrino/electron elastic scattering generated the smallest uncertainties of all of the fits. This agrees with the assertion that, at least for a
76
beam operating under the parameters we used, this event is optimal one for
studying neutrino interactions.
The orders of magnitude of our actual numbers for νµ /e scattering are
important as well. They demonstrate that next generation neutrino scattering experiments will have low statistical uncertainties. This will allow those
experiments to either better refine the known values of SM constants or be
more likely to find values that lie outside of the Standard Model. The better
statistical fit may also help to confirm or reject the NuTeV anomaly.
Although we cannot use the inverse muon decay data to directly examine
sensitivity to deviations in gVνe and gAνe , simply knowing
to derive
dN
dT
dσ
dT
and being able
from it allows us to filter out the effects of this event from the
more meaningful νµ /e elastic scattering when an actual experiment is run.
By comparing the measured bin counts to the predicted values for inverse
muon decay, we should be able to determine the experimental uncertainty
of the bin counts for the elastic scattering events. Also, since both elastic
scattering and IMD contain a factor of GF , we may even be able to use this
event to improve our measurements of the Fermi coupling constant, which
would ultimately improve our fits for gVνe and gAνe .
Knowing
dN
dT
for muon antineutrino/electron elastic scattering and elec-
tron (anti)neutrino/electron scattering also allows us to more easily filter
out any beam contamination if the flux of the wrong kind of neutrinos in
the beam can be determined. As these numbers are usually known for any
experimental beam, even prior to construction, this task should not prove
77
difficult.
As explained in §3.1, future experiments may also wish to use any one
of the neutrino or antineutrino scattering events to probe the electroweak
sector. In particular, a muon neutrino beam can likely be reconfigured to run
in antineutrino mode, providing the experimenters with more than one check
of their data for the same experiment. The calculation of the differential cross
sections for the two electron neutrino events also demonstrates our software’s
capability to handle events with more than one contributing diagram.
The tau and muon neutrino events produced the same statistical uncertainties. Assuming lepton universality, this was in agreement with our theoretical predictions, as the muon and tau mass terms do not appear in the
elastic scattering equations. The construction of a tau neutrino experiment
would therefore provide a strong test of lepton universality by comparing the
results of that experiment with the results of muon neutrino experiments.
The only difficulty there (aside form producing a sufficient number of tau
neutrinos) is that it is impossible to normalize the scattering data through
inverse tau decay at present energy scales, as that event will not occur.
7.2
7.2.1
The Next Steps
More User Friendly
While our software accomplished all of its design goals, there is much room
for improvement. For instance, the automation is still incomplete, and some
78
of the information entered by the user is superfluous. Because FeynArts
and FeynCalc cannot both be active packages at the same time, there must
be more than one user input step. Still, it would not be unreasonable to
combine every function called after the matrix element is created into one
master function. The only real inconvenience to doing that would be the
large number of input parameters that the user must enter.
Most of these inputs are necessary, but the variables representing the
number of matrix elements and the masses of three of the particles are all
found somewhere in the output of CreateFeynAmp. The programs would
require much less of the user if these quantities were taken out of this output
and used by the other functions.
7.2.2
More Outgoing Particles
Another flaw with our program is that it only calculates cross sections for
scattering events with two outgoing particles. While FeynArts can easily
calculate the Feynman amplitudes for an event with any number of incoming and outgoing particles, the kinematics become far more difficult as the
number of particles increases.
The simplest 2 to 3 scattering events we could model would be the emission of a single additional photon during the processes we have already examined. Because the extra outgoing particle is guaranteed to be massless,
the kinematics would be simplified. Modeling these events is useful, as their
presence will affect the bin counts of an experiment.
79
7.2.3
Deep Inelastic Scattering
Our entire project has focused solely on neutrinos scattering off of electrons.
Any conceivable fixed target experiment will use stationary atoms for these
targets. These atoms will also contain nucleons, and the neutrinos will scatter
off of them as well. At the energy scales examined in this paper, the incoming
neutrinos will be energetic enough to scatter off of the constituent quarks of
the protons and neutrons. This process is known as deep inelastic scattering
(DIS).[2]
Because the three valence quarks inside of a nucleon each carry some
momentum even when the nucleon itself is at rest, the calculations involved
in DIS must factor this momentum distribution through the inclusion of a
parton distribution function. In addition, DIS must account for the fact that
there is a probability of hitting one of two different flavors of quark, which
each interact differently and may have one of three different colors. If all of
that were not complicated enough, the incoming particle may even scatter off
of one of the sea quarks or antiquarks that exist within the nucleons, which
can be one of four different flavors.[5]
It is also very likely that there will be more than one outgoing particle in
DIS. Therefore, any analysis of DIS must use also use the more complicated
kinematics associated with such a scattering event.
All of this may seem daunting, especially once the equations of quantum
chromodynamics begin to play a role. However, several simplified models of
DIS do exist. Not only must neutrino/nucleon scattering be accounted for as
80
a byproduct of the neutrino/electron scattering, but DIS itself can contribute
experimental data on the quark electroweak coupling constants.[10] Developing a modified version of our software that supports these calculations is
therefore a priority.
7.2.4
Beyond the Standard Model
Our software can also be used to examine scattering experiments with models
other than the Standard Model. This change can be easily implemented by
writing a new model file in FeynArts.
Not only could we then test the statistical sensitivity of these new models,
but we would also be able to examine how a new model affects
dN
.
dT
This
would help us tailor an experiment to best look for experimental verification
of another model.
81
82
Appendix A
Properties of Spinors and
Gamma Matrices
A.1
A.1.1
Dirac Spinors
u Spinors
The following properties are true of the Dirac spinor u and its adjoint ū =
u† γ 01 :
(γ µ pµ − m) u = 0 ,
(A.1)
ū (γ µ pµ − m) = 0 ,
(A.2)
ū(1) u(2) = 0 ,
(A.3)
ūu = 2m ,
(A.4)
2
X
u(s) ū(s) = (γ µ pµ + m)
s=1
83
1
All information in this Appendix can be found in [3]
.
(A.5)
A.1.2
v Spinors
The following properties are true of the Dirac spinor v and its adjoint v̄ =
v†γ 0:
(γ µ pµ + m) uv = 0 ,
(A.6)
v̄ (γ µ pµ + m) = 0 ,
(A.7)
v̄ (1) v (2) = 0 ,
(A.8)
v̄u = −2m ,
(A.9)
2
X
v (s) v̄ (s) = (γ µ pµ − m)
.
(A.10)
s=1
A.2
A.2.1
Gamma Matrices
Gamma Matrix Relations
The following properties are true of the gamma matrices defined in §2.2:
[γ µ , γ ν ] = 2g µν
,
(A.11)
γµ γ µ = 4 ,
(A.12)
γµ γ ν γ µ = −2γ ν
γµ γ ν γ λ γ µ = 4g µν
,
(A.13)
,
γµ γ ν γ λ γ σ γ µ = −2γ σ γ λ γ ν
84
(A.14)
.
(A.15)
By applying the above relations to products involving the Dirac Slash
notation, 6a ≡ γ µ aµ , we derive the following:
A.2.2
[6a, 6b] = 2a · b ,
(A.16)
γµ 6aγ µ = −26a ,
(A.17)
γµ 6a6bγ µ = 4a · b ,
(A.18)
γµ 6a6b6cγ µ = −26c6b6a .
(A.19)
Trace Theorems
There are also a series of “theorems” that apply to the traces of gamma
matrices. The first of these is that the trace of any odd number of gamma
matrices is always 0. The rest of these relations are:
T r (γ µ γ ν ) = 4g µν ,
T r γ µ γ ν γ λ γ σ = 4 g µν g λσ − g µλ g νσ + g µσ g νλ
(A.20)
,
(A.21)
T r (6a6b) = 4a · b ,
(A.22)
T r (6a6b6c6d) = 4 [(a · b) (c · d) − (a · c) (b · d) + (a · d) (b · c)]
.
(A.23)
Because γ 5 is the product of an even number of gamma matrices, the
85
trace of γ 5 times an odd number of gamma matrices is 0. Otherwise,
T r γ5 = 0 ,
T r γ 5 γ µ γ ν = 0,
T r γ 5 γ µ γ ν γ λ γ σ = 4iµνλσ ,
T r γ 5 6a6b = 0 ,
T r γ 5 6a6b6c6d = 4iµνλσ aµ bν cλ dσ
(A.24)
(A.25)
(A.26)
(A.27)
.
The four index Leviv-Civita epsilon is defined by
µνλσ



−1 if µνλσ is an even permutation of 0123,



≡
1 if µνλσ is an odd permutation of 0123,




 0 if any two indices are equal.
86
(A.28)
Appendix B
Relativistic Kinematics
B.1
B.1.1
Four-Vectors
Position-Time Four-Vectors
Special relativity says that temporal and spatial coordinates are dependent on
each other when transforming from one inertial reference frame to another.
It is therefore convenient to abandon the traditional, purely spatial three
vector for a new notation: four-vector.1
The simplest of these is the position-time four-vector, xµ , where µ runs
from 0 to 3. As a rule, Greek indices begin at 0, while Roman ones begin at
1. x1 , x2 , and x3 are the familiar spatial coordinates x, y, and z. x0 is equal
to the time, t. Technically, x0 = ct, but remember that we have set c equal
to 1.
1
All information in this section can be found in [3]
87
Recall from relativity the existence of the invariant,
I = x0
2
− x1
2
− x2
2
− x3
2
.
(B.1)
The invariant is so named because it remains unchanged under a Lorentz
transformation. It would be convenient if the invariant could be written as a
sum over the indices of a four-vector, but the symmetry between the spatial
and temporal coordinates breaks down when it comes to the minus signs in
the invariant.
To deal with this, we define the metric tensor, gµν , as follows:


0
0
1 0


0 −1 0

0


g=

0 0 −1 0 




0 0
0 −1
.
(B.2)
I can now be written as the double sum
I = gµν xµ xν
,
(B.3)
where, as is the case throughout this paper, we use the Einstein summation
convention of summing over any index that is both raised and lowered within
a product.
As the metric tensor will frequently appear in calculations, we can simplify
88
things by defining the covariant four-vector,
xµ = gµν xν
.
(B.4)
The invariant can now be written as
I = xµ x µ
B.1.2
.
(B.5)
Generalized Four-Vectors
The concept of a covariant four-vector is more than just a means for more
easily writing out the time space invariant. The same concept can be easily
applied to any four vector as they can to a position-time four-vector. We
define the scalar product of two four vectors the product of covariant and
contravariant (the technical term for a four-vector with a raised index) fourvectors:
a · b ≡ aµ b µ = aµ b µ = a0 b 0 − a2 b 2 − a3 b 3 − a4 b 4
,
(B.6)
where aµ and bµ are arbitrary four-vectors.
This scalar product can be rewritten in terms of the the classical threevector dot product,
a · b = a0 b 0 − a · b .
(B.7)
Just as with the position-time four vector, the scalar product of any four-
89
vector with itself will return an invariant quantity. This is denoted by
a2 ≡ a · a = a0
2
− a2
.
(B.8)
Note: Do not confuse the square of a four-vector with its number 2 component. While they are, unfortunately, notationally indistinguishable, context should make it clear which quantity is intended. For instance, the 2
component will rarely appear without the 1 and 3 components near by.
B.1.3
Four-Momenta
A four-vector that is of particular interest to us is the energy-momentum
four-vector, also known as the four-momentum, pµ . The components of pµ
are
p0 = E,
p 1 = px ,
p2 = py ,
p3 = pz
.
(B.9)
E is the relativistic energy, and the p’s are the components of the relativistic
momentum. The relativistic kinetic energy of a free particle, T is simply
E − m.
According to relativity,
E2 =
p
m2 + p2
,
(B.10)
where m is the rest mass of the particle whose four-momentum is represented
90
by pµ . Thus,
p2 = E 2 − p2 = m2
,
(B.11)
which is obviously an invariant quantity.
Furthermore, for two four-momenta p and k,
p · k = Ep Ek − p · k = Ep Ek − |p| |k| cos θ
,
(B.12)
where θ is the angle between p and k. This form is useful if one wishes to
derive a value for θ in terms of momenta, masses, and energy.
The laws of conservation of energy and conservation of momentum can
be combined into a single law of conservation of four-momentum. The total
four-momentum of the particles going into an event is equal to the total fourmomentum of those coming out. This is a very useful condition for solving
the kinematics of particle scattering events.
B.2
B.2.1
Neutrino Fixed Target Scattering
Momenta
We now wish to derive the kinematic relations for 2-to-2 scattering of a neutrino off of a fixed target. As shown in Fig. B.1, there are two incoming and
two outgoing particles, with four-momenta p1 , p2 , k1 , and k2 . The subscripts
here are for labeling, and do not represent components of the four-vectors.
91
p
k
1
1
q
p
k
2
2
Figure B.1: 2 to 2 scattering diagram
Such component labeling is dropped for this section of Appendix B.
p1 describes the incoming neutrino. While it is now known that neutrinos
have mass, the upper bounds on the three neutrino masses are small enough
that that they may be ignored as negligible. Therefore, mp1 = 0 and Ep1 =
p1 .
p2 describes the target object. As we are calculating our kinematics in
the lab frame (as opposed to the center of mass frame), p1 = 0. Therefore
Ep2 = mp2 .
While k1 is massless in all of the events we examined, we did not make
this assumption in deriving the kinematic expressions. This allows us greater
flexibility for adapting our work in the future.
B.2.2
Scalar Products
The scalar products of the various momenta appear when solving for |M|2 .
Since these are independent of the particles and interactions involved in the
92
specific event, we solved them here in terms of masses and energies and
entered them into our software as part of the FullCalculate function.
Before doing this, we first rewrote the energies to be purely in terms of
the masses, the incoming neutrino energy, Eν , and the kinetic energy of the
k2 particle (i.e. the one that was not an outgoing neutrino), T . Obviously,
Ep1 = Eν
,
Ep2 = mp2
(B.13)
.
(B.14)
Simply by applying the definition of relativistic kinetic energy allowed us to
see that
Ek2 = mk2 + T
.
(B.15)
Solving for the remaining energy term required us to invoke conservation of
energy. Namely,
Ep1 + Ep2 = Ek1 + Ek2
⇒ Ek1 = Eν + mp2 − mk2 − T
.
(B.16)
The scalar products of the momenta with themselves were trivial. By
93
invoking Equation (B.11), we found that
p21 = 0 ,
(B.17)
p22 = m2p2
,
(B.18)
k12 = m2k1
,
(B.19)
k22 = m2k2
.
(B.20)
Any scalar product involving p2 was likewise easily computed, as that
particle is at rest. Thus,
p1 · p2 = mp2 Eν
,
(B.21)
p2 · k1 = mp2 (Eν + mp2 − mk2 − T )
p2 · k2 = mp2 (mk2 + T )
,
.
(B.22)
(B.23)
To solve for the remaining scalar products, we used conservation of fourmomentum,
p1 + p2 = k1 + k4
(B.24)
and the following relation: for any two four-vectors p and k,
p·k =±
1
(p ± k)2 − p2 − k 2
2
94
.
(B.25)
For example, when solving for p1 · k2 , we wrote
p1 · k2 = −
1
(p1 − k2 )2 − p21 − k22
2
.
(B.26)
The minus sign is a result of subtracting the momenta before squaring, instead of adding. We then plugged in the appropriate mass terms and substituted p2 − k1 for p1 − k2 to get
1
− (p2 − k1 )2 + m2k2
,
2
1 2
=
−p2 − k12 + 2 (p2 · k1 ) + m2k2
,
2
1
= mp2 (Eν + mp2 − mk2 − T ) + m2k2 − m2p2 − m2k1
2
p1 · k2 =
.
(B.27)
Following the same procedure yielded the following results for the remaining scalar products:
1 2
mk1 − m2p2 − m2k2
2
1 2
k1 · k2 = mp2 Eν + mp2 − m2k1 − m2k2
.
2
p1 · k1 = mp2 (mk2 + T ) +
B.2.3
,
(B.28)
(B.29)
Finding θ
As seen in Chapter 3, it was at times necessary to have an explicit equation
for the cosine of the angle between p1 and k1 , which we have designated θ.
95
As shown in Equation (B.12),
cos θ =
Ep1 Ek1 − (p1 · k1 )
|p1 | |k1 |
.
(B.30)
With the appropriate substitutions, this becomes
Eν (Eν + mp2 − mk2 − T ) − mp2 (mk2 + T ) − 21 m2k1 − m2p2 − m2k2
q
cos θ =
,
Eν (Eν + mp2 − mk2 − T )2 − m2k1
(B.31)
which simplifies to
cos θ = 1 −
mp2 T
Eν (Eν − T )
(B.32)
when m2=m4 and m3 = 0, as is the case in neutrino/electron elastic scattering.
96
Appendix C
The Generalized Single
Variable Linear Regression
C.1
Introduction
A linear regression is a means of fitting a set of N data points {xi , yi }N
i=1
to some to a given model function, f (x). This function must be a linear
combination of M arbitrary basis functions of x. The linear coefficients of
the combination are often referred to as the regression parameters.1
In order to perform a regression analysis, the following conditions must
be met. The data must be overdetermined relative to the model function,
i.e. M < N . The uncertainty in y for each i, σi , must be either known or,
if unknown, must be uniform. Furthermore, σi is assumed both to be the
1
This Appendix follows the derivation outlined in [9]
97
standard deviation of a normally distributed error and to be independent of
xi .2
It is possible to apply this procedure to sets of data with uncertainty in
both the x and y values, but since this was not the case in our analysis, we
do not describe the necessary modifications here.
The linear regression gives the best fit for the regression parameters by
minimizing the chi-square merit function
2
χ =
2
N X
yi − f (xi )
σi2
i=1
(C.1)
with respect to the regression parameters, provided that the errors in y are
indeed normally distributed. We will demonstrate how this is done by means
of examples: first a simple case and then the one that applied to our calculations.
C.2
A Simple Linear Regression
The simplest model function, aside from the trivial case of a constant, is the
straight line:
y = a + bx .
2
Press et al.
98
(C.2)
In this case, the chi-square is given by
2
χ (a, b) =
2
N X
yi − a − bxi
σi2
i=1
.
(C.3)
To minimize this function, we set its partial derivatives with respect to a
and b equal to 0:
N
X
∂χ2
yi − a − bxi
0=
= −2
∂a
σi2
i=1
0=
,
N
X
∂χ2
xi (yi − a − bxi )
= −2
∂b
σi2
i=1
(C.4)
.
(C.5)
,
(C.6)
.
(C.7)
Solving for a and b yields
N
N
X
X
x2 2
y
a=
i
j
σi
σj
N
N
X
X
x2 2
−
k
σk
j=1
i=1
N
X
1
σm
k=1
N
X
2
x2 2
−
n
σn
12
σi
i=1
N
X
xj yj 2
σj
−
j=1
N
X
1
σm
m=1
x` y` 2
σ`
`=1
N
X
x
!2
p2
σp
p=1
n=1
m=1
N
X
b=
2
N
X
x2k 2
σk
k=1
N
X
2
x2 2
n
σn
−
n=1
N
X
y` 2
σ`
`=1
N
X
x
!2
p2
σp
p=1
We also wish to determine the uncertainty of a and b. Simple propagation
of errors tell us that
σf2
2
N
X
∂f
2
=
σi
∂yi
i=1
99
,
(C.8)
where f equals a or b. Summing over i for a and b yields
N
X
x2 2
i
σi
σa2 =
i=1
N
X
1 2
σm
N
X
x2n 2
σn
n=1
m=1
!2
1 2
σm
m=1
(C.9)
.
(C.10)
p2
σp
12
σi
i=1
N
X
,
p=1
N
X
σb2 =
−
N
X
x
N
X
x2n 2
σn
−
n=1
N
X
x
!2
p2
σp
p=1
By definition, taking the square root of these variances gives us the standard
deviations. If we once again assume a normal distribution of errors, we may
uses the standard deviations as uncertainties in the regression parameters.
The final important quantity that can be derived from this linear regression is the covariance between a and b, σab , given by
−
σab =
N
X
xi 2
σi
i=1
N
X
m=1
1 2
σm
N
X
x2n 2
σn
n=1
−
N
X
x
!2
.
(C.11)
p2
σp
p=1
The covariance allows us to calculate the statistical correlation between a
and b through the following formula:
Cor (a, b) =
100
σab
σa2 σb2
.
(C.12)
The linear regression has now produced the central values, uncertainty,
and correlation of the regression parameters a and b. While the numerous
sums can seem daunting, they are a simple matter to resolve using a computer.
C.3
C.3.1
Multiple Regression Parameters
Setup
The notation becomes a lot messier when fitting to functions more complicated than a straight line. Here the chi-square merit function is given by
"
#2
M
N
X
X
1
yi −
aj Xj (x)
χ2 =
2
σ
i
j=1
i=1
,
(C.13)
where {Xj (x)} is the set of basis functions of x, and {aj } are the corresponding regression parameters.
C.3.2
The Matrix Method
To clean up our expressions, we now define the N × M matrix A and the
length N vector b,
Xj (xi )
σi
yi
bi ≡
.
σi
Aij ≡
101
,
(C.14)
(C.15)
Equation (C.13) can now be rewritten as
χ2 =
N
X
bi −
i=1
M
X
!2
aj Aij
.
(C.16)
j=1
Minimizing the chi-square function now requires
0=

N
X

bi −
M
X
i=1
!2
Xk (xi )
aj Aij
M

,
(C.17)

j=1
k=1
which we can rewrite as
M
X
αkj aj = βk
,
(C.18)
j=1
where
α ≡ AT · A , and
(C.19)
β ≡ AT · b
(C.20)
are an M × M matrix and length M vector respectively.
If we let a be the length M vector of the regression parameters, Equation
(C.18) can now be written in matrix form as
α·a=β
.
(C.21)
Solving this system of linear equations then yields the elements of a.
Through the use of basic error analysis, as in part 2, it can be shown
102
that the variances and covariances of the regression parameters are simply
the elements of the symmetric matrix C ≡ α−1 :
σa2i = Cii
,
σai aj = Cij = Cji
C.4
(C.22)
, i 6= j
.
(C.23)
Confidence Intervals
Once these values have been calculated, we may wish to select two of the
regression parameters and plot a region of confidence around their central
values. The value of one of the parameters, designated am , is measured on
the x axis, and that of the other, an , is measured on the y. This region takes
the form of the interior of an ellipse in which the values for the two regression parameters are likely to fall to within a specified number of standard
deviations or percent.
While there are multiple methods of graphing a confidence region, we
chose to solve this problem by instructing the computer to draw the region
that keeps the function ∆ (u, v) less than a given number. u and v are the
x and y coordinates, respectively, translated to put the origin at (am , an ). ∆
is defined by
∆ (u, v) ≡ u · P−1 · u ,
103
(C.24)
Confidence Level ∆ <
1σ
2.30
90%
4.61
2σ
6.18
99%
9.21
3σ
11.8
99.99%
18.4
Table C.1: Number which ∆ is less than for a given confidence level
where
 
u 
u ≡   , and
v


Cmm Cmn 
P≡

Cnm Cnn
(C.25)
.
(C.26)
This number is determined by the confidence level of the region, as shown in
Table B1.
104
105
Appendix D
Code
Rewrite "SM" Coupling To Include Elctroweak Coupling Constants
NoThetaAIlhs : C@- F@1, 8j1_<D, F@1, 8j2_<D, V@2DDM Š rhs_E :=
lhs Š 88HI EL gnuL IndexDelta@j1, j2DL  HCW SWL,
I EL HHdZZZ1  H4 CW SWL + HdZe1 + HdSW1 HSW ^ 2 - CW ^ 2LL  HCW ^ 2 SWLL  H2 CW SWLL IndexDelta@j1,
j2D + 1  H2 CW SWL H1  2 Conjugate@RowBox@8dZfL1, @, RowBox@81, , , j1, , , j1<D,D<DD
IndexDelta@j1, j2D + 1  2 dZfL1@1, j1, j1D IndexDelta@j1, j2DLL<, 80, 0<<;
NoThetaAIlhs : C@- F@2, 8j1_<D, F@2, 8j2_<D, V@2DDM Š rhs_E :=
lhs Š 88HI EL HgeV + geAL IndexDelta@j1, j2DL  H2 CW SWL,
I EL HHdZAZ1  2 + HHdSW1  HCW ^ 2 SWL + dZe1L SWL  CW + HdZZZ1 HSW ^ 2 - 1  2LL  H2 CW SWL HdZe1 + HdSW1 HSW ^ 2 - CW ^ 2LL  HCW ^ 2 SWLL  H2 CW SWLL IndexDelta@j1, j2D + 1  HCW SWL
HSW ^ 2 - 1  2L H1  2 Conjugate@RowBox@8dZfL1, @, RowBox@82, , , j1, , , j1<D,D<DD
IndexDelta@j1, j2D + 1  2 dZfL1@2, j1, j1D IndexDelta@j1, j2DLL<,
8HI EL HgeV - geAL IndexDelta@j1, j2DL  H2 CW SWL, EL
HHdZAZ1  2 + HdZZZ1 SWL  H2 CWL + HHdSW1  HCW ^ 2 SWL + dZe1L SWL  CWL IndexDelta@j1, j2D +
1  CW SW H1  2 Conjugate@RowBox@8dZfR1, @, RowBox@82, , , j1, , , j1<D,D<DD
IndexDelta@j1, j2D + 1  2 dZfR1@2, j1, j1D IndexDelta@j1, j2DLL<<;
NoThetaAIlhs : C@- F@3, 8j1_, o1_<D, F@3, 8j2_, o2_<D, V@2DDM Š rhs_E :=
lhs Š 88HI EL HguV + guAL IndexDelta@j1, j2DL  H2 CW SWL,
I EL HHH1  2 - H2 SW ^ 2L  3L HConjugate@RowBox@8dZfL1, @, RowBox@83, , , j2, , , j1<D,D<DD 
2 + 1  2 dZfL1@3, j1, j2DLL  HCW SWL +
H- HdZAZ1  3L - H2 HdSW1  HCW ^ 2 SWL + dZe1L SWL  H3 CWL + HdZZZ1 H1  2 - H2 SW ^ 2L  3LL 
H2 CW SWL + HdZe1 + HdSW1 HSW ^ 2 - CW ^ 2LL  HCW ^ 2 SWLL  H2 CW SWLL IndexDelta@j1, j2DL
IndexDelta@o1, o2D<, 8HI EL HguV - guAL IndexDelta@j1, j2DL  H2 CW SWL,
I EL HH- HdZAZ1  3L - HdZZZ1 SWL  H3 CWL - H2 HdSW1  HCW ^ 2 SWL + dZe1L SWL  H3 CWLL IndexDelta@
j1, j2D - H2 SW HConjugate@RowBox@8dZfR1, @, RowBox@83, , , j2, , , j1<D,D<DD  2 +
1  2 dZfR1@3, j1, j2DLL  H3 CWLL IndexDelta@o1, o2D<<;
NoThetaAIlhs : C@- F@4, 8j1_, o1_<D, F@4, 8j2_, o2_<D, V@2DDM Š rhs_E :=
lhs Š 88HI EL HgdV + gdAL IndexDelta@j1, j2DL  H2 CW SWL,
I EL HHHSW ^ 2  3 - 1  2L HConjugate@RowBox@8dZfL1, @, RowBox@84, , , j1, , , j2<D,D<DD  2 +
1  2 dZfL1@4, j1, j2DLL  HCW SWL +
HdZAZ1  6 + HHdSW1  HCW ^ 2 SWL + dZe1L SWL  H3 CWL + HdZZZ1 HSW ^ 2  3 - 1  2LL  H2 CW SWL HdZe1 + HdSW1 HSW ^ 2 - CW ^ 2LL  HCW ^ 2 SWLL  H2 CW SWLL IndexDelta@j1, j2DL
IndexDelta@o1, o2D<, 8HI EL HgdV - gdAL IndexDelta@j1, j2DL  H2 CW SWL,
I EL HHSW HConjugate@RowBox@8dZfR1, @, RowBox@84, , , j1, , , j2<D,D<DD  2 +
1  2 dZfR1@4, j1, j2DLL  H3 CWL + HdZAZ1  6 + HdZZZ1 SWL  H6 CWL +
HHdSW1  HCW ^ 2 SWL + dZe1L SWL  H3 CWLL IndexDelta@j1, j2DL IndexDelta@o1, o2D<<;
NoTheta@
other_D :=
other
106
FeynArts to FeynCalc Preprocessor
H*Takes the FeynArts amplitude output and
creates a list of matrix elements compatible with FeynCalc*L
FullConvert@amp_, number_ D := Module@8matrices, list<, matrices = Convert@amp, numberD;
list = Table@Convert0@matrices@@iDD@@1DDD, 8i, Length@matricesD<DD
H*Seperates out the individual matrix elements from the output amplitude of FeynArts*L
Convert@amp_, number_ D := Module@8samp, M, j, matrices, list<,
samp = ToString@ampD; H*Converts amplitude to a String*L
M = number; H*Number of matrix elements*L
H*Creates 1‰M table of empty Strings*L
matrices = Table@"", 8i, M<D;
H*Fills the table with the matrix
elements by looking for appropriate prefixes and postfixes*L
For@j = 1, j £ M, j ++, If@j < M, matrices@@jDD =
StringCases@samp, RegularExpression @"GraphName.*FeynAmp"DD;
samp = StringReplace @samp, matrices@@jDD ® ""D;,
matrices@@jDD = StringCases@samp, RegularExpression @"GraphName.*"DD;DD;
matrices H*Outputs table of matrix elements*L
D
H*Defines the "small q" approximation propogator denominator*L
PropDenom@x_D := 1  x ^ 2;
H*Makes a single matrix element compatible with FeynCalc*L
Convert0@amp_D := Module@8samp, matrix, denom, newdenom, replacements , truematrix,
replacementsa , replacementslista , replacementsb , replacementslistb <,
H*Converts the matrix elements to a string if it has not been already*L
samp = ToString@ampD;
H*Deletes FeynArts prefix*L
matrix = StringCases@samp, RegularExpression @"Relative.*"DD;
H*Removes the postfix insertion data from the matrix element*L
replacements = StringCases@matrix@@1DD, RegularExpression @"8Mass.*"DD;
truematrix = StringReplace @matrix, replacements ® ""D;
H*Generates a list of the terms to be replaced*L
replacementsa = StringCases@replacements @@1DD, RegularExpression @"8.*RelativeCF<"DD;
replacementslista = ToExpression @replacementsa @@1DDD;
H*Generates a list of the terms that will
replace the ones in the previous list,in the same order*L
replacementsb = StringCases@StringReplace @replacements @@1DD, replacementsa ® ""D,
RegularExpression @"8.*<"DD;
replacementslistb = ToExpression @replacementsb @@1DDD;
H*Replaces the terms in the matrix element itself*L
For@j = 1, j £ Length@replacementslista D, j ++,
truematrix = StringReplace @truematrix, ToString@InputForm@replacementslista @@jDDDD ®
ToString@InputForm@replacementslistb @@jDDDDDD;
H*Converts the denominator of the matrix element to an expression
recognizable by Mathematica as division*L
denom = StringCases@matrix@@1DD, RegularExpression @"PropagatorDenominator \W@^,D*,"DD;
truematrix = StringReplace @truematrix, denom ® "PropDenom@"D;
H*Fixes the tail of the matrix element to be readable by Mathematics*L
truematrix = StringReplace @truematrix, RegularExpression @"\W,"D ® "D"D;
truematrix = StringReplace @truematrix, "DD" ® "D"D;
H*Converts the matrix element back to a Mathematica expression*L
ToExpression @truematrixD
D
107
Cross Section Calculator
H*Calculates the differential cross section with respect to T from the matrix elements*L
FullCalculate @matrixlist_ , M2_, M3_, M4_D :=
Module@8matrixsquared , S, mass2, mass3, mass4, crosssection , swap, finalcrosssection <,
H*Takes the output of the preprocessor ,the matrix elements,
and calculates the spin averaged and summed absolute square of their sum*L
matrixsquared = Calculate@matrixlistD;
S = 1; H*Fix the number of indistinguishable incoming particles*L
H*Assigns the input values of the masses*L
mass2 = ToString@M2D;
mass3 = ToString@M3D;
mass4 = ToString@M4D;
H*Calculates the differential cross section from the matrix element data*L
crosssection = CrossSection @matrixsquared , S, mass2, mass4D;
crosssection = Simplify@crosssection D;
H*Substitutes in the appropriate masses*L
swap = ToString@InputForm@crosssection DD;
swap = ToString@StringReplace @swap, "m2" ® mass2, IgnoreCase ® FalseDD;
swap = ToString@StringReplace @swap, "m3" ® mass3, IgnoreCase ® FalseDD;
swap = ToString@StringReplace @swap, "m4" ® mass4, IgnoreCase ® FalseDD;
H*Converts back to a Mathematica expression*Lfinalcrosssection = ToExpression @swapD
D
H*Takes the output of the preprocessor ,the matrix elements,
and calculates the spin averaged and summed absolute square of their sum*L
Calculate@matrixlist_ D := Module@8INITIAL, elements, totalamplitude , spinsum, spinaverage<,
INITIAL = 2; H*Assumes a 2 to x scattering event*Lelements = Contract@matrixlistD;
H*The matrix elements from each diagram*Ltotalamplitude = 0;
H*Explicitly determines the absolute value squared of the sum of the matrix elements*L
For@j = 1, j £ Length@elementsD, j ++, For@k = 1, k £ Length@elementsD, k ++,
totalamplitude += elements@@jDD * HComplexConjugate @elements@@kDDD . 8li1 ® li2<LDD;
H*Sums over final and initial spin.*L
spinsum =
Simplify@EpsEvaluate@Contract@FermionSpinSum @totalamplitude D . DiracTrace ® TrDDD;
spinaverage = spinsum  INITIAL H*Averages over initial spin*L
D
H*Calculates the differential cross section from the matrix element data*L
CrossSection @matrixsquared_ , S_, mass2_, mass4_D := Module@8crosssection , newcrosssection <,
H*Decides whether to apply the simplifying assumption for elastic scattering*L
If@StringMatchQ @mass2, mass4D,
H*Calculates dӐdW*L
crosssection = S matrixsquared HE3  H8 Π m2 E1LL ^ 2;
H*Calculates dӐdT*L
newcrosssection = 2 Π m2  HE1 - TL ^ 2 crosssection ,
H*Calculates dӐdW*L
crosssection = S matrixsquared HE3 ^ 2 - m3 ^ 2L  H64 Π ^ 2 m2 E1
Abs@Sqrt@E3 ^ 2 - m3 ^ 2D HE1 + m2L + HE3 ScalarProduct @p1, k1DL  Sqrt@E3 ^ 2 - m3 ^ 2DDL;
H*Calculates dӐdT*L
newcrosssection = - 2 Π D@ScalarProduct @p1, k1D  HE1 Sqrt@E3 ^ 2 - m3 ^ 2DL, TD crosssection D;
newcrosssection H*Returns dӐdT*L
D
108
Flux Integrator
BinCruncher@crosssection_ , flux_, targets_ , time_D := Module@8s, N, Flist, F, f, i, k<,
s = crosssection ; H*Takes the input differential cross section*L
N = Length@fluxD; H*Determine number of flux data points*L
Flist = Table@0, 8N<D; H*Create an empty table of length N*L
H*Assign the first flux to an energy range,starting at E1=1*L
Flist@@1DD = 8flux@@1DD@@1DD, 0, Mean@8flux@@1DD@@2DD, flux@@2DD@@2DD<D<;
H*Assign all of the middle fluxes to energy ranges*L
For@i = 2, i < N, i ++, Flist@@iDD = 8flux@@iDD@@1DD,
Mean@8flux@@i - 1DD@@2DD, flux@@iDD@@2DD<D, Mean@8flux@@iDD@@2DD, flux@@iDD@@2DD<D<D;
H*Assignthe last flux to an energy range*LFlist@@NDD :=
8flux@@NDD@@1DD, Mean@8flux@@N - 1DD@@2DD, flux@@NDD@@2DD<D,
2 flux@@NDD@@2DD - Mean@8flux@@N - 1DD@@2DD, flux@@NDD@@2DD<D<;
Flist;
H*Integrate in pieces to get dNdT*Lf@T_D = targets * time *
Sum@Integrate@Flist@@kDD@@1DD s, 8E1, Flist@@kDD@@2DD, Flist@@kDD@@3DD<D, 8k, N<DD
109
Regression Preprocessor
H*Constructs the input data tables for the regression*L
DataTable@f_, bins_D :=
ModuleB8size, function, xdatafunction , ydatafunction , xstring, x, y, Σ<,
size = Length@binsD; H*Determines length of tables*L
H*Assigns SM values to coupling constants*L
gV = Ρ H- 1  2 + 2 SW ^ 2L;
gA = Ρ H- 1  2L;
H*Constructs data table for the fitting function with independent variables*L
x = Table@PrepX@InputForm@
Simplify@MakeXData@ToString@InputForm@Expand@TSub@f, bin@@jDDDDDDDDDD, 8j, 1, size<D;
H*Constructs data table for the dependent variables*L
y = Table@
Simplify@MakeYData@ToString@InputForm@Expand@TSub@f, bin@@kDDDDDDDD, 8k, 1, size<D;
H*Constructs data table for the uncertainties of the dependent variables*L
Σ = TableB
F
8x, y, Σ<
Abs@y@@mDDD , 8m, 1, size<F;
H*Reassigns the coupling constants to their SM values plus first order deviation terms*L
MakeXData@function_ D := Module@8errorfunction , xdatafunction <,
errorfunction = StringReplace @function, "gV2^2" ® "HgV^2+2*gV*∆gVL"D;
errorfunction = StringReplace @errorfunction , "gA2^2" ® "HgA^2+2*gA*∆gAL"D;
errorfunction = StringReplace @errorfunction , "gA2*gV2" ® "HgA*gV + ∆gV*gA + ∆gA*gVL"D;
errorfunction = StringReplace @errorfunction , "gV2" ® "HgV+∆gVL"D;
errorfunction = StringReplace @errorfunction , "gA2" ® "HgA+∆gAL"D;
xdatafunction = Simplify@ToExpression @errorfunction DD
D
H*Reassigns the coupling constants to their SM values*L
MakeYData@function_ D := Module@8regularfunction , ydatafunction <,
regularfunction = StringReplace @function, "gA2" ® "gA"D;
regularfunction = StringReplace @regularfunction , "gV2" ® "gV"D;
ydatafunction = Simplify@ToExpression @regularfunction DD
D
H*Seperates the different addends of dNdT with deviations into their own subsets*L
PrepX@xdatafunction_ D := Module@8xstring, xdata<,
xstring = ToString@xdatafunction D;
xstring = StringReplace @xstring, "+" ® ","D;
xstring = StringReplace @xstring, "-" ® ",-"D;
H*Removes commas from negative exponents*L
xstring = StringReplace @xstring, "^," ® "^"D;
xstring = StringReplace @xstring, RegularExpression @"^,"D ® ""D;
xstring = StringReplace @xstring, "∆gV" ® "1"D;
xstring = StringReplace @xstring, "∆gA" ® "1"D;
xstring = "8" <> xstring <> "<";
xdata = ToExpression @xstringD
D
H*Substitutes a numeric value for T for some function of T*L
TSub@f_, x_D := Module@8fstring, xstring, fnew<,
fstring = ToString@InputForm@fDD;
xstring = ToString@InputForm@xDD;
fnew = ToExpression @StringReplace @fstring, "T" ® xstringDD;
fnew = ToExpression @StringReplace @ToString@InputForm@fnewDD, "I" ® "0"DD
D
110
Linear Regression Implementation
H*Performs the regression analysis*L
MultilinearRegression @data_, xscale_ , yscale_ , x2scale_ , y2scale_ D :=
ModuleA8x, y, uncert, corri, Σ, M, N, Α, p, Β, c, a, Γ,
Χ2, corrf, rho, sw2, corr2, c2, D, ellipse, D2, p2, ellipse2<,
N = Length@data@@2DDD; H*Sets the length of the data tables*L
M = 2; H*Sets the number of regression parameters*L
H*Defines the y matrix*L
y = Table@data@@2DD@@jDD - data@@1DD@@j, 1DD, 8j, N<D;
H*Defines the x matrix*L
x = Table@data@@1DD@@i, jDD, 8i, N<, 8j, 83, 2<<D;
uncert = data@@3DD; H*Sets the uncertainty table*L
H*Defines the correlation between the two columns in the x matrix Hi.e. noneL*L
corri = IdentityMatrix @ND;
H*Sets up the Σ matrix for matrix multiplication *L
Σ = Table@corri@@i, jDD * uncert@@iDD * uncert@@jDD, 8i, N<, 8j, N<D;
Α = [email protected]@ΣD.x; H*Defines Α*L
Β = [email protected]@ΣD.y; H*Defines Β*L
H*Solves for the regression parameters*L
a = LinearSolve@Α, ΒD;
c = Abs@Inverse@ΑDD; H*Defines c*L
H*Calculates the uncertainties for the coupling constants*L
Γ = Map@Sqrt, Diagonal@cDD;
H*Calculates the chi-squared value of the fit*L
Χ2 = Hy - x.aL.Inverse@ΣD.Hy - x.aL;
H*Calculates the correlation between the coupling constants*L
corrf = Table@c@@i, jDD  Sqrt@c@@i, iDD * c@@j, jDDD, 8i, M<, 8j, M<D;
H*Calculates the uncertainty for Ρ*L
rho = Abs@- 2 Γ@@2DDD;
H*Calculates the uncertainty for sin2 Θ*L
sw2 =
SW ^ 2 * Sqrt@HΓ@@1DD  gVL ^ 2 + HΓ@@2DD  gAL ^ 2 - 2 HΓ@@1DD  gVL HΓ@@2DD  gAL corrf@@1DD@@2DDD;
H*Calculates the correlation between Ρ and sin2 Θ*L
corr2 = Sqrt@Abs@H- 8 gV  HgA ^ 2LL Γ@@2DD ^ 2 + H1  H2 gALL corrf@@1DD@@2DDDD;
H*Constructs the c matrix for Ρ and sin2 Θ*L
c2 = 88rho ^ 2, corr2 * Hrho * sw2L<, 8corr2 * Hrho * sw2L, sw2 ^ 2<<;
H*Draws the confidence regions for the coupling constants*L
D@u_, v_D = Simplify@8u, v<.p.8u, v<D;
p = Inverse@Table@c@@i, jDD, 8i, 81, 2<<, 8j, 81, 2<<DD;
ellipse =
RegionPlot@8D@u - gV, v - gAD < 2.30, D@u - gV, v - gAD < 4.61<, 8u, gV - xscale, gV + xscale<,
8v, gA - yscale, gA + yscale<, PlotPoints ® 50, FrameLabel ® 8"g V ", "g A "<,
PlotStyle ® None, Epilog ® 8PointSize@SmallD, Point@880, 0<, 8gV, gA<<D<D;
H*Draws the confidence regions for Ρ and sin2 Θ*L
D2@u2_, v2_D = Simplify@8u2, v2<.p2.8u2, v2<D;
p2 = Inverse@c2D;
ellipse2 = RegionPlotA8D2@u2 - Ρ, v2 - SW ^ 2D < 2.30, D2@u2 - Ρ, v2 - SW ^ 2D < 4.61<,
8u2, Ρ - x2scale, Ρ + x2scale<, 8v2, SW ^ 2 - y2scale, SW ^ 2 + y2scale<,
PlotPoints ® 50, FrameLabel ® 9"Ρ", "sin2 ΘW "=, PlotStyle ® None,
Epilog ® 8PointSize@SmallD, Point@880, 0<, 8Ρ, SW ^ 2<<D<E;
E
H*Outputs results*L
8a, Γ, corrf@@1DD@@2DD, 8rho, sw2, corr2<, Χ2, ellipse, ellipse2<
111
Appendix E
Constants
The following were used as the values of constants in our calculations:[2][10]
c
h̄
e
α
me
mµ
mτ
GF
ρ
sin2 θW
1
√1
α
−3
7.2973525377 × 10
0.510998910 MeV
105.658367 MeV
1776.84 MeV
1.6637 × 10−5 GeV−2
1
.2227
112
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[2] C. Amsler et al. (Particle Data Group), Physics Letters B 667 (2008),
no. 1.
[3] Griffiths, David, Introduction to Elementary Particles, John Wiley &
Sons, New York, 1987.
[4] Hahn, Thomas, FeynArts 3.4 User’s Guide, 2008.
[5] Halzen, Francis and Alan D. Martin, Quarks and Leptons: An Introductory Course in Modern Particle Physics, John Wiley & Sons, New
York, 1984.
[6] J.M. Conrad, Neutrino Experiments, arXiv:0708.2446v1 [hep-ex], 2008.
[7] Loinaz, William A., Apr 2009, Personal Interview.
[8] Mertig, Rolf, The FeynCalc Book, 1999.
113
[9] Press, William H., Saul A. Teukolsky, William T. Vetterling, and Brian
P. Flannery, Numerical Recipes: The Art of Scientific Computing, Third
ed., Cambridge University Press, 2007.
[10] T. Adams et al., Terascale Physics Opportunities at a High Statistics,
High Energy Neutrino Scattering Experiment: NuSOnG, Int. J. Mod.
Phys. A 24 (2009), no. 4, 671–717.
114