What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs...
Transcript of What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs...
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What is the Higgs boson?José Miguel Figueroa-O’Farrill
School of Mathematics
Friday 22 February 2013(Innovative Learning Week)
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Suddenly, last summer...
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Physics: it’s where the action is!
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F = ma = md2x
dt2
Newton’s second law inertial mass
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x(t) is the path followed by the object
x(t)
v =dx
dt
F = ma = md2x
dt2
Newton’s second law inertial mass
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F = 0 =)
Newton’s first law
object moves at aconstant velocity
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Conservative forces
F = �rV
V (x) potential function
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Conservative forces
F = �rV
V (x) potential function
E = 12m|v|2 + V (x)Energy is conserved
E(t1)
E(t2)
E(t1) = E(t2)
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derived Newton’s second law from a principle!
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derived Newton’s second law from a principle!
Nature uses as little aspossible of anything.
Johannes Kepler(1571-1630)
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x(t1)
x(t2)
The principle of least action
action I =
Z t2
t1
�12m|v|2 � V (x)
�dt
| {z }
lagrangian
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The paths with the least action satisfy Newton’s second law... and conversely!
x(t1)
x(t2)
The principle of least action
action I =
Z t2
t1
�12m|v|2 � V (x)
�dt
| {z }
lagrangian
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The paths with the least action satisfy Newton’s second law... and conversely!
x(t1)
x(t2)
The principle of least action
action I =
Z t2
t1
�12m|v|2 � V (x)
�dt
| {z }
lagrangian
Actions change the way we think about Physics!!!
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Symmetries
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A transformation x 7! x
0 is called a symmetry if the action does not change :
I(x) = I(x0)
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A transformation x 7! x
0 is called a symmetry if the action does not change :
I(x) = I(x0)
Symmetries take a path with least action to another path with least action (perhaps even to the same path!).
So symmetries take solutions of Newton’s equation to solutions of Newton’s equation.
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• a translation
For example,
is a symmetry provided that
V (x+ a) = V (x)
x 7! x+ a
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• a translation
For example,
• a rotation x 7! Rx
is a symmetry if V (Rx) = V (x)
i.e., V (x) = f(|x|)
is a symmetry provided that
V (x+ a) = V (x)
x 7! x+ a
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Of course, even though the action has symmetries, solutions of Newton’s equations may break some or all of that symmetry.
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Of course, even though the action has symmetries, solutions of Newton’s equations may break some or all of that symmetry.
Consider free motion: V (x) = 0
The trivial trajectoryx(t) = x0
is invariant under arbitrary rotations about the point x0
x0
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If we give it a little push so that that particle is now moving at constant (nonzero) velocity
x(t) = x0 + tv
we have broken the rotational symmetry to those with axis of rotations v
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Summary
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Summary
➡Dynamical systems can be described by an action principle
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Summary
➡Dynamical systems can be described by an action principle
➡ Solutions have least action
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Summary
➡Dynamical systems can be described by an action principle
➡ Solutions have least action
➡ Symmetries of the action take solutions to solutions
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Summary
➡Dynamical systems can be described by an action principle
➡ Solutions have least action
➡ Symmetries of the action take solutions to solutions
➡A given solution might break some (or all) of the symmetry of the action
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Relativistic fields
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And Maxwell said...
and there was light!
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In vacuo, Maxwell equations are
where
c
electric field
magnetic field
speed of light
r ·E = 0 r⇥E = �@B
@t
r ·B = 0 r⇥B =1
c2@E
@t
E(x, t)
B(x, t)
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An immediate consequence is that the electric and magnetic fields obey the (massless) wave equation :
1
c2@2E
@t2�r2E = 0
1
c2@2B
@t2�r2B = 0
The wave equation can also be derived from an action.
(In fact, that is what light, radio waves, X-rays,... actually are.)
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Let us consider the scalar wave equation:
1
c2@2�
@t2�r2� = 0
I =
Zdt
Zd
3x
1
c
2
✓@�
@t
◆2
� |r�|2!
Solutions of the wave equation minimize the action:
This action has “hidden” symmetries which mix space and time!
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In fact, if we introduce a 4-vector :
x
µ = (x0,x) x
0 = ct
the action becomes
⌘ =
0
BB@
�1 0 0 00 1 0 00 0 1 00 0 0 1
1
CCAwhere
µ = 0, 1, 2, 3
I =
Zd
4x
X
µ,⌫=0,1,2,3
⌘
µ⌫ @�
@x
µ
@�
@x
⌫=
Zd
4x⌘
µ⌫@µ�@⌫�
@µ :=@
@x
µ
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The action has relativistic symmetry:
�(x) 7! �(x0) (x0)µ = ⇤µ⌫x
⌫ ⇤T ⌘⇤ = ⌘
Lorentz transformations
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x
µ coordinates in Minkowski spacetime.
“The views of space and time that I wish to lay before you have sprung from the soil of experimental physics , and therein lies their strength. They are radical . Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of u n i o n o f b o t h w i l l r e t a i n a n independent reality.”
Hermann Minkowski (1908)
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Maxwell’s equations are also relativistic!
Fµ⌫ =
0
BB@
0 E1 E2 E3
�E1 0 B3 �B2
�E2 �B3 0 B1
�E3 B2 �B1 0
1
CCA Fµ⌫ = �F⌫µ
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Maxwell’s equations are also relativistic!
Fµ⌫ =
0
BB@
0 E1 E2 E3
�E1 0 B3 �B2
�E2 �B3 0 B1
�E3 B2 �B1 0
1
CCA Fµ⌫ = �F⌫µ
@µF⌫⇢ + @⌫F⇢µ + @⇢Fµ⌫ = 0
Maxwell’sequations {() ⌘µ⌫@µF⌫⇢ = 0
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Maxwell’s equations are also relativistic!
Fµ⌫ =
0
BB@
0 E1 E2 E3
�E1 0 B3 �B2
�E2 �B3 0 B1
�E3 B2 �B1 0
1
CCA Fµ⌫ = �F⌫µ
@µF⌫⇢ + @⌫F⇢µ + @⇢Fµ⌫ = 0
Maxwell’sequations {() ⌘µ⌫@µF⌫⇢ = 0
They can also be derived from an action, but for an “electromagnetic potential”!
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Aµ = (�,A)
Fµ⌫ = @µA⌫ � @⌫Aµ
The Bianchi identity @µF⌫⇢ + @⌫F⇢µ + @⇢Fµ⌫ = 0
can be solved by
for some
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Aµ = (�,A)
Fµ⌫ = @µA⌫ � @⌫Aµ
The Bianchi identity @µF⌫⇢ + @⌫F⇢µ + @⇢Fµ⌫ = 0
can be solved by
for some
determined up to a gauge transformation
Aµ 7! Aµ + @µ✓
which is only
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Aµ = (�,A)
Fµ⌫ = @µA⌫ � @⌫Aµ
The Bianchi identity @µF⌫⇢ + @⌫F⇢µ + @⇢Fµ⌫ = 0
can be solved by
for some
determined up to a gauge transformation
Aµ 7! Aµ + @µ✓
which is only
Aµ describes the photonRelativistic fields correspond to particles and
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The Maxwell action (for !)Aµ
is unchanged under
➡gauge transformations
➡Lorentz transformationsAµ(x) 7!
�⇤�1
�µ
⌫A⌫(⇤x)
Aµ(x) 7! Aµ(x) + @µ✓(x)
I =
Zd
4x ⌘
µ⇢⌘
⌫�Fµ⌫F⇢�
![Page 53: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/53.jpg)
Mass
![Page 54: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/54.jpg)
For relativistic theories, the mass appears as quadratic terms in the action :
I =
Zd
4x
�12⌘
µ⌫@µ�@⌫�� 1
2m2�
2�
![Page 55: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/55.jpg)
For relativistic theories, the mass appears as quadratic terms in the action :
I =
Zd
4x
�12⌘
µ⌫@µ�@⌫�� 1
2m2�
2�
![Page 56: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/56.jpg)
For relativistic theories, the mass appears as quadratic terms in the action :
I =
Zd
4x
�12⌘
µ⌫@µ�@⌫�� 1
2m2�
2�
I describes a massive scalar field obeying the Klein-Gordon equation :
or
� 1
c2@2�
@t2+r2�+m2� = 0
⌘µ⌫@µ@⌫�+m2� = 0
![Page 57: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/57.jpg)
There is also a massive version of Maxwell equations, described by the Proca action:
I =
Zd
4x
�� 1
4⌘µ⇢⌘
⌫�Fµ⌫F⇢� + 1
2m2⌘
µ⌫AµA⌫
�
![Page 58: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/58.jpg)
There is also a massive version of Maxwell equations, described by the Proca action:
I =
Zd
4x
�� 1
4⌘µ⇢⌘
⌫�Fµ⌫F⇢� + 1
2m2⌘
µ⌫AµA⌫
�
which yields the Proca (=massive Maxwell) equation:
⌘µ⌫@µF⌫⇢ +m2A⇢ = 0
![Page 59: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/59.jpg)
There is also a massive version of Maxwell equations, described by the Proca action:
The Proca action is no longer gauge invariant!
I =
Zd
4x
�� 1
4⌘µ⇢⌘
⌫�Fµ⌫F⇢� + 1
2m2⌘
µ⌫AµA⌫
�
which yields the Proca (=massive Maxwell) equation:
⌘µ⌫@µF⌫⇢ +m2A⇢ = 0
![Page 60: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/60.jpg)
Summary
![Page 61: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/61.jpg)
Summary
➡ The wave equation is relativistic
![Page 62: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/62.jpg)
Summary
➡ The wave equation is relativistic
➡Maxwell’s equations are also relativistic
![Page 63: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/63.jpg)
Summary
➡ The wave equation is relativistic
➡Maxwell’s equations are also relativistic
➡ They are both obtained from Lorentz-invariant actions defined on Minkowski spacetime
![Page 64: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/64.jpg)
Summary
➡ The wave equation is relativistic
➡Maxwell’s equations are also relativistic
➡ They are both obtained from Lorentz-invariant actions defined on Minkowski spacetime
➡ The Maxwell action depends on a field Aμ which is determined only up to gauge transformations
![Page 65: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/65.jpg)
Summary
➡ The wave equation is relativistic
➡Maxwell’s equations are also relativistic
➡ They are both obtained from Lorentz-invariant actions defined on Minkowski spacetime
➡ The Maxwell action depends on a field Aμ which is determined only up to gauge transformations
➡Mass is the quadratic term (without derivatives) in the action
![Page 66: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/66.jpg)
The Higgsmechanism
![Page 67: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/67.jpg)
Actions can be combined in order to “couple” fields.
![Page 68: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/68.jpg)
Actions can be combined in order to “couple” fields.
The abelian Higgs model couples the Maxwell field to a complex scalar: � = �1 + i�2
![Page 69: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/69.jpg)
Actions can be combined in order to “couple” fields.
The abelian Higgs model couples the Maxwell field to a complex scalar: � = �1 + i�2
Zd
4x
�� 1
2F2 + 1
2 |D�|2 � V (�)�
Dµ� = @µ�+ ieAµ�
F 2 = 12⌘
µ⇢⌘⌫�Fµ⌫F⇢�
|D�|2 = ⌘µ⌫Dµ�D⌫�
![Page 70: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/70.jpg)
Actions can be combined in order to “couple” fields.
The abelian Higgs model couples the Maxwell field to a complex scalar: � = �1 + i�2
electric charge
Zd
4x
�� 1
2F2 + 1
2 |D�|2 � V (�)�
Dµ� = @µ�+ ieAµ�
F 2 = 12⌘
µ⇢⌘⌫�Fµ⌫F⇢�
|D�|2 = ⌘µ⌫Dµ�D⌫�
![Page 71: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/71.jpg)
Zd
4x
�� 1
2F2 + 1
2 |D�|2 � V (�)�
![Page 72: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/72.jpg)
The first term is just the Maxwell action, which is still unchanged under gauge transformations
Aµ 7! Aµ + @µ✓
Zd
4x
�� 1
2F2 + 1
2 |D�|2 � V (�)�
![Page 73: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/73.jpg)
The first term is just the Maxwell action, which is still unchanged under gauge transformations
Aµ 7! Aµ + @µ✓
The second term also remains unchanged if, in addition,
� 7! e�ie✓�
Zd
4x
�� 1
2F2 + 1
2 |D�|2 � V (�)�
![Page 74: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/74.jpg)
The first term is just the Maxwell action, which is still unchanged under gauge transformations
Aµ 7! Aµ + @µ✓
And this is also a symmetry of the third term, provided that
V (�) = f(|�|)
The second term also remains unchanged if, in addition,
� 7! e�ie✓�
Zd
4x
�� 1
2F2 + 1
2 |D�|2 � V (�)�
![Page 75: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/75.jpg)
The Maxwell equation is modified by an “electric” current term :
⌘�µ@�Fµ⌫ = J⌫ = 12e(�@⌫�� �@⌫�)� e2A⌫ |�|2
![Page 76: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/76.jpg)
The Maxwell equation is modified by an “electric” current term :
J⌫ = (⇢, j)
⌘�µ@�Fµ⌫ = J⌫ = 12e(�@⌫�� �@⌫�)� e2A⌫ |�|2
![Page 77: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/77.jpg)
|�|
Potential
V (�) = f(|�|)
v
![Page 78: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/78.jpg)
|�|
Potential
V (�) = f(|�|)
unstable critical point|�| = 0
v
![Page 79: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/79.jpg)
|�|
Potential
V (�) = f(|�|)
unstable critical point|�| = 0
stable critical points|�| = v
v
![Page 80: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/80.jpg)
|�|
Potential
V (�) = f(|�|)
unstable critical point|�| = 0symmetric
stable critical points|�| = v
v
![Page 81: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/81.jpg)
|�|
Potential
V (�) = f(|�|)
unstable critical point|�| = 0symmetric
stable critical points|�| = v
symmetry is broken!
v
![Page 82: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/82.jpg)
|�|
V (�) = f(|�|)
![Page 83: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/83.jpg)
|�|
V (�) = f(|�|)
� = |�|ei✓
![Page 84: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/84.jpg)
|�|
V (�) = f(|�|)
� = |�|ei✓ V (�)
�
![Page 85: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/85.jpg)
|�|
V (�) = f(|�|)
� = |�|ei✓
( )
V (�)
�
![Page 86: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/86.jpg)
Imagine a ball rolling on this potential:
![Page 87: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/87.jpg)
Imagine a ball rolling on this potential:
![Page 88: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/88.jpg)
Imagine a ball rolling on this potential:
![Page 89: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/89.jpg)
Imagine a ball rolling on this potential:
![Page 90: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/90.jpg)
Expanding about the unstable critical point, we see a massless photon and a “tachyonic” scalar.
imaginary mass
m2 = f 00(0) < 0
Nature dislikes unstable critical points and quantum/thermal fluctuations will move the system away from them and towards a stable critical point.
![Page 91: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/91.jpg)
Let’s expand about a stable critical point with |�| = v
![Page 92: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/92.jpg)
Let’s expand about a stable critical point with |�| = v
� = (v + h(x))eie✓(x)/v
![Page 93: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/93.jpg)
Let’s expand about a stable critical point with |�| = v
� = (v + h(x))eie✓(x)/v
and let’s apply a gauge transformation
� 7! e�ie✓� = v + h Aµ 7! Aµ +1
v@µ✓
![Page 94: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/94.jpg)
Let’s expand about a stable critical point with |�| = v
� = (v + h(x))eie✓(x)/v
and let’s apply a gauge transformation
� 7! e�ie✓� = v + h Aµ 7! Aµ +1
v@µ✓
we obtain...
| {z }
Bµ
![Page 95: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/95.jpg)
I =
Zd
4x
�� 1
2F2 + 1
2e2v
2⌘
µ⌫BµB⌫
+ 12⌘
µ⌫@µh@⌫h� 1
2V00(v)h2 + · · ·
�
![Page 96: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/96.jpg)
| {z }massive “photon” M2 = e2v2
I =
Zd
4x
�� 1
2F2 + 1
2e2v
2⌘
µ⌫BµB⌫
+ 12⌘
µ⌫@µh@⌫h� 1
2V00(v)h2 + · · ·
�
![Page 97: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/97.jpg)
| {z }massive Higgs boson!
m2 = V 00(v) > 0
| {z }massive “photon” M2 = e2v2
I =
Zd
4x
�� 1
2F2 + 1
2e2v
2⌘
µ⌫BµB⌫
+ 12⌘
µ⌫@µh@⌫h� 1
2V00(v)h2 + · · ·
�
![Page 98: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/98.jpg)
The photon “ate” one of the two scalars and became massive!
| {z }massive Higgs boson!
m2 = V 00(v) > 0
| {z }massive “photon” M2 = e2v2
I =
Zd
4x
�� 1
2F2 + 1
2e2v
2⌘
µ⌫BµB⌫
+ 12⌘
µ⌫@µh@⌫h� 1
2V00(v)h2 + · · ·
�
![Page 99: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/99.jpg)
Summary
![Page 100: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/100.jpg)
Summary
➡ The abelian Higgs model couples a Maxwell field with a charged complex scalar subject to a “mexican hat” potential
![Page 101: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/101.jpg)
Summary
➡ The abelian Higgs model couples a Maxwell field with a charged complex scalar subject to a “mexican hat” potential
➡ Expanding around the unstable critical point, one sees a massless photon and a “tachyonic” charged scalar
![Page 102: What is the Higgs boson? - School of Mathematicsjmf/Teaching/Lectures/HiggsILW.pdfWhat is the Higgs boson? José Miguel Figueroa-O’Farrill School of Mathematics Friday 22 February](https://reader035.fdocuments.in/reader035/viewer/2022070609/5ae072f57f8b9af05b8da809/html5/thumbnails/102.jpg)
Summary
➡ The abelian Higgs model couples a Maxwell field with a charged complex scalar subject to a “mexican hat” potential
➡ Expanding around the unstable critical point, one sees a massless photon and a “tachyonic” charged scalar
➡ Expanding around any of the stable critical points (and thus breaking the symmetry) one finds a massive “photon” and a Higgs boson!
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Virtually the same mechanism gives masses to all the (massive) elementary particles in the standard model: intermediate vector bosons, quarks and leptons.
The discovery of a Higgs boson in CERN last summer confirms the Higgs mechanism after almost 50 years since it was first proposed and after more than 20 years of experimental search!
It is remarkable that Nature continues to show mercy at this collective of evolved primates inhabiting a blue planet orbiting a yellow sun and to reveal little by little its innermost secrets to those who strive to discover them.
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“Das ewig Unbegreifliche an der Welt ist ihre Begreiflichkeit.”
"The most incomprehensible thing about the universe is that it is comprehensible."