Transcript Document

Elementarteilchenphysik
Antonio Ereditato
LHEP University of Bern
Lesson on:
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Electroweak interaction and SM (8)
1
Unification of fundamental interactions
•At the moment of Big Bang energy density was very large, interactions happened at really high
energy, of the order of the Planck scale (1019 GeV), at such energy the four fundamental forces had
the same intensity, the coupling constant of gravitational, strong, electromagnetic and weak
interactions were equal.
•When the forces acquired different values? At which scale the unification breaks down?
•The first force to decouple was gravitation, followed by strong force and then electromagnetic and
weak forces. Electromagnetic and Weak force decoupled at low energies high q2 (~104 GeV).
•The breaking of the symmetry makes three of the four mediating bosons of the EW theory very
massive (W+, W- and Z0) determining the short range of the weak interaction and thus its
“weakness”.
•Historically the first “unification” was made by Maxwell unifying electrostatic and magnetic
formalisms. By the end of the 60’ Glashow, Salam and Weinberg worked-out a theory predicting the
unification of the weak and the electromagnetic interactions into the electro-weak (EW) force
•Differently from electromagnetic unification, the
GWS theory predicted a unique coupling
constant, the electric charge e and only one
parameter (sin2W), to be measured in
experiments.
•The EW theory is very well established today
and its success contributes to the success of
the so-called Standard Model of particles and
interactions
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Glashow
Salam
Weinberg
2
Energy density affect the coupling constants (i.e. the intensity) of the forces, during time in
our universe the coupling constant changed, or as it is said they are “running coupling
constants”.
Time
The electroweak unification (1 and 2)
Time
The EW + strong force unification
In many SUSY models all coupling converge to the same value at high
energies, one of the goals of LHC and high energy accelerators is to
explore energies present at the very first moment after the big bang
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The Glashow, Weinberg, Salam Model
As energy rises electromagnetic and weak forces present the same coupling, that is the two forces have
the same intensity. This energy is well within present accelerator reach. GWS theorized a model based on
the SU(2)L x U(1)Y. From the group algebra one would expect four massless mediating bosons, arranged
into a SU(2)L weak isospin triplet (I = 1) and a singlet (I = 0) of the weak hypercharge group U(1)Y.
SU(2)
æa b ö æ a * c * ö æ1 0 ö
=ç
ç
÷×ç *
÷
*÷
c
d
b
d
è
ø è
ø è0 1ø
a b
= +1
c d
group of unitary 2 x 2 matrices
U(1)
group of unitary 1 x 1 matrices
Nb: These matrixes operate on spinors!!
The three isospin triplet bosons are: W W(1), W(2) , W(3) while B is the the isospin singlet The physical
(massive) states are the W +, W -, Z 0 and photon A. The latter (neutral bosons) are related to the
massless bosons via mixing (rotation):
B0
g
W is called the Weinberg angle
Nb: in GWS theory all bosons are massless,
For the sake of renormalizability
W3
QW
Z0
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Experimental approach
Weak interactions have been always been observed to have a weaker coupling than EM decays, e.g. beta
decays presented longer decay times. Many hypotheses were made. Fermi made a first effective theory on
weak decays.
GF= 10-5 GeV-2
GF=g2/MW 2
GF is not dimensionless so it is not a fundamental quantity
And a point like interaction do not allow renormalization. Then
the point like interaction should be replaced by a mediator.
We could substitute a point like interaction with a propagator,
the point like interaction would held if q2<<MW2, the mass is
needed to make the propagator weak at low energies.
Massive propagators can not be renormalized, so without any
other theory we could not make amplitude calculation (i.e.
solve Feynman diagrams) of weak interactions.
Higgs theory (shown later) solve this problem, by adding a
new potential to the electroweak lagrangian which make it
possible to have massive W and Z bosons, and also gives
mass to fermions.
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Weak boson masses
We remember that:
G
g2
=
2 8MW2
Now, since
e = gsinqW
it follows that:
æ g 2 2 ö1/ 2 æ e 2 2 ö1/ 2 37.4
MW = ç
(GeV )
÷ =ç
÷ =
2
8G
8Gsin
q
sin
q
è
ø
è
W ø
W
MW
MW2
2
while, in the most general case of Higgs scalars: M Z =
It can be shown that: M Z =
cosqW
r cos2 qW
The Weinberg angle measured in weak interaction processes (neutrino interactions) and the masses
of the weak bosons measured at the colliders agree well:
Measured masses of the weak bosons:
MW = 80.398 ± 0.025 GeV
MZ = 91.1876 ± 0.0021 GeV
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Theoretical approach
We define the weak hypercharge, analogously to the strong hypercharge (apart from a factor 2 difference):
Ystrong = 2(Q - I3strong isospin )
Yweak = (Q - I3weak isospin )
One has then:
JmY = JmEM - Jm(3)
Performing the calculations, recalling the mixing relation, and setting g’/g = tanW , one obtains:
L=
g - +
g
(JmW m + Jm+W m- ) +
(Jm(3) - sin 2 qW JmEM )Z m + gsin qW JmEM Am
cosqW
2
weak charged current
weak neutral current
EM neutral current
e = g × sin QW
i.e. unification of weak and EM interactions:
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In other words, we have expressed the two experimentally observed neutral currents (the EM and the
weak) in terms of the currents belonging to the symmetry groups SU(2)L, and U(1)Y which depend on
the coupling g and a free parameter W
These two coupling constants are replaced by e and by the sin2W parameter, the latter to be measured
in experiments:
JmEM = J m(3) + J mY
JmNC = Jm(3) - sin 2 qW J mEM
Just as Q generates the group U(1)EM , which regulates the EM currents, the Y operator generates the
symmetry group U(1)Y, and takes into account both the EM and neutral weak component. The enlarged
group SU(2)L x U(1)Y takes into account the incorporation of weak and EM interactions.
The SU(2)L x U(1)Y group was introduced by Glasgow (1961) before the discovery of neutral currents and
extended by Weinberg (1967) and Salam (1968).
Another important result obtained with the EW unification theory is that one naturally eliminates the
divergencies occurring in calculating graphs in the original weak interaction theory.
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Electroweak couplings
As the EW bosons, also fermions get weak isospin and hypercharge. In doing so the GWS model has to
take into account that the weak charged-current interaction violates parity, while the EM interaction is
parity-conserving. Remember that Y = Q - I3
Fermion multiplets
I
I3
Q/e
Y
Leptons
Quarks
eL
eL
L
L
L
L
½
+½
-½
0
-1
-1/2
eR
R
R
0
0
-1
-1
uL
d´L
cL
s´L
tL
b´L
½
+½
-½
+2/3
-1/3
+1/6
uR
cR
tR
0
0
+2/3
+2/3
dR
sR
bR
0
0
-1/3
-1/3
Recalling the expression of the weak neutral current:
JmNC = Jm(3) - sin2 qW JmEM
one obtains the corresponding relations for the coefficients of the LH and RH couplings of fermions to the Z:
gL = I3 - Q sin 2 qW
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gR = - Q sin2 qW
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Examples of couplings
e - +n e ®
W - ® e - + ne
e - +n e
1 1
1 1
+
®
+
2 2
2 2
1 1
1
1
TI33 = - + ® - +
2 2
2
2
TI =
e = g × sin Q =
g ×g¢
g 2 + g ¢2
TI = 1
TI33 = -1
g (f ) =
g
cos QW
1 1
+
2 2
1 1
®- 2 2
®
Q
ì
ü
× íTI33 - sin2 QW ý
e
î
þ
(g’/g = tanW )
Z0
g (d ) =
g
cos QW
1
ì
ü
ïï 1 - 3 e
ïï
2
× í- sin QW ý = -0.423 × g
2
e
ï
ï
ïî
ïþ
d
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Spontaneous symmetry breaking
Wμ1,2,3, Bμ
W+, W-, Z0, γ
This transformation is the result of the
phenomenon
of
Spontaneous
Symmetry
Breaking (SSB). In the case of the electroweak
force, it is known as the Higgs Mechanism.
Spontaneous symmetry breaking (SSB) occurs in a situation where, given a symmetry of the equations
of motion, solutions exist which are not invariant under the action of this symmetry without any
explicit asymmetric input (hence the attribute “spontaneous”).
A situation of this type can be illustrated by means of a simple (classical physics) example. Consider the
case of a linear vertical stick with a compression force applied on the top and directed along its axis. The
physical description is obviously invariant for all rotations around this axis. As long as the applied
force is mild enough, the stick does not bend and the equilibrium configuration (the lowest energy
configuration) is invariant under this symmetry. When the force reaches a critical value, the symmetric
equilibrium configuration becomes unstable and an infinite number of equivalent lowest energy stable
states appear, which are no longer rotationally symmetric but are related to each other by a rotation. The
actual breaking of the symmetry may then easily occur by effect of a (however small) external asymmetric
cause, and the stick bends until it reaches one of the infinite possible stable asymmetric equilibrium
configurations.
In substance, what happens is that when some parameter reaches a critical value, the lowest energy
solution respecting the symmetry of the theory ceases to be stable under small perturbations and
new asymmetric (but stable) lowest energy solutions appear. The new lowest energy solutions are
asymmetric but are all related through the action of the symmetry transformations. In other words, there is
a degeneracy (infinite or finite depending on whether the symmetry is continuous or discrete) of distinct
asymmetric solutions of identical (lowest) energy, that maintain the symmetry of the theory.
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Spontaneous symmetry breaking (cont.)
The same picture can be generalized to quantum field theory (QFT), the ground state becoming the vacuum state. This
means that there may exist symmetries of the laws of nature which are not manifest to us because the physical world in which
we live is built on a vacuum state which is not invariant under them. In other words, the physical world of our experience
can appear to us very asymmetric, but this does not necessarily mean that this asymmetry belongs to the fundamental laws
of nature. SSB offers a key for understanding (and utilizing) this physical possibility.
The application of SSB to particle physics in the 1960s and successive years led to profound physical consequences and
played a fundamental role in the edification of the current Standard Model of elementary particles. In the case of a global
continuous symmetry, massless bosons (known as “Goldstone bosons”) appear with the spontaneous breakdown of the
symmetry according to a theorem by J. Goldstone in 1960. The presence of these massless bosons, first seen as a serious
problem since no particles of the sort had been observed, was in fact the basis for the solution, by means of the so-called
Higgs mechanism, of another similar problem. This is the fact that the 1954 Yang-Mills theory of non-Abelian gauge fields
predicted unobservable massless particles, the gauge bosons.
According to the “mechanism” established in a general way in 1964 independently
by P. Higgs and others, in the case that the internal symmetry is promoted
to a local one, the Goldstone bosons “disappear” and the gauge bosons acquire a
mass. The Goldstone bosons are “eaten up” to give mass to the gauge bosons,
and this happens without breaking the gauge invariance of the theory.
Note that this mechanism for the mass generation for the gauge fields is also what
ensures the renormalizability of theories involving massive gauge fields
(such as the Glashow-Weinberg-Salam electroweak theory.
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The Higgs mechanism
Peter Higgs
In the Higgs model particle masses arise in a beautiful, but complex, progression. It starts with a particle that
has only mass, and no other characteristics, such as charge, that distinguish particles from empty space. We
can call his particle H. H interacts with other particles; for example if H is near an electron, there is a force
between the two. H is of a class of particles called bosons. It is also a scalar particle (s = 0).
The parameters in the equations for the field associated with the particle H can be chosen in such a way that
the lowest energy state of that field (empty space) is one with the field not zero. It is surprising that the field
is not zero in empty space, but the result is: all particles that can interact with H gain mass from the
interaction.
The picture is that of the lowest energy state, "empty" space, having a crown of H particles with no energy of
their own. Other particles get their masses by interacting with this collection of zero-energy H particles. The
mass (or inertia or resistance to change in motion) of a particle comes from its being "grabbed at" by the
Higgs particles.
The Higgs particle (field) permeates the whole Universe at any space-time point
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Search for the Higgs as a free particle at the LHC
The strength of the Higgs coupling is proportional to the mass of the
particles involved so its coupling is greatest to the heaviest decay
products which have mass < mH/2. For example, if mH > 2Mz then
the couplings for decay to the following particle pairs:
Z0Z0 : W+W- : τ+τ- : pp : μ+μ- : e+e- are in the ratio
1.00 : 0.88 : 0.02 : 0.01 : 0.001 : 5.5 x 10-6
Negative searches have been conducted at LEP and TEVATRON. The present bounds on
its mass is [114 GeV, ~1000 GeV].
The Higgs discovery will be one of the main goals of the LHC with the LHC ATLAS and
CMS experiments.
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Backup
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Neutrino scattering
Neutrino electron scattering gave the first evidence for the correctness of the EW theory and
allowed the first estimates of sin2W (remember the Gargamelle experiment).
This was followed by a series of neutrino scattering experiments: off hadrons (for the
measurement of structure functions) and off electrons (to study purely leptonic, EW processes).
For example, the CHARM II experiment at CERN (1984-1991), that collected the largest world
statistics of
and
interactions, for the measurement of sin2W
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Neutrino scattering (cont.)
For neutral current reactions:
Where:
s is the cms squared
energy
y is the elasticity Ee/E
LL, RR, RL, LR indicate
the helicities of the
scattering leptons
For charged current reactions:
gL2 =1 and gR2 = 0
For the electron
coupling to the W
Lower cross-section for antineutrinos: it would be the same as for neutrinos if the scattering
occurred on RH electrons (forbidden by the V-A structure of the weak charged current)
1
gL = - + sin2 qW gR = sin 2 qW (NC only)
2
1
gL = 1- + sin 2 qW gR = 0 + sin 2 qW (NC + CC)
2
n m + e or n m + e
ne + e
Measurement of cross-sections for different reactions (ambiguities) allows determining the coupling constants
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Examples of purely leptonic neutrino interactions
Muon-neutrino off electron scattering can only proceed via NC reactions


Z0
e-
e-
Electron-antineutrino off electron scattering can proceed via NC and CC reactions
e
e
+
Z0
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e
e-
-
e
W
e-
e-
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