Download Tutorial material for weak interactions and more

Tutorial material for weak interactions and more
Olga Bessidskaia Bylund
February 26, 2017
1
Number of generators
The generators of SU(n) can be represented by a n x n complex hermitian
matrices and any SU(n) transformation can be built from these generators.
The hermiticity condition implies that knowing the elements above and on
the diagonal is enough to know all the elements in the matrix. We have
n(n − 1)/2 elements above the diagonal, each with a real and an imaginary
part. We also have n elements on the diagonal, which need to be real for
the matrix to be hermitian. The condition that the determinant is one can
be shown to be equivalent to that the trace of the matrix is zero. This
constraint on the elements on the diagonal reduces the number of degrees
of freedom by one. So, we get n2 − n + n − 1 = n2 − 1 degrees of freedom
in SU(n) theory. Weak interactions are described by SU(2) theory and thus
have 3 generators. This corresponds to three physical particles: W + , W −
and Z 0 . For SU(3) theory for strong interactions, we have 8 degrees of
freedom.
1.1
SU(3) generators
There are eight 3x3 matrices λi that act as SU(3) generators; these are
given in Eq. (11b) in Mandl and Shaw, where the Gell-Mann representation
is chosen. For example:
λ1 = [0, 1, 0; 1, 0, 0; 0, 0, 0]
(1)
λ2 = [0, −i, 0; i, 0, 0; 0, 0, 0]
(2)
λ3 = [1, 0, 0; 0, −1, 0; 0, 0, 0]
(3)
The λi matrices are related to the colour operators F̂i by F̂i = 12 λi . They
act on color triplets in the Lagrangian, rotating them. Quark wavefunctions
can be factorised into a space+spin part and a colour spinor part χc . The
colour spinors are:
1
r = [1; 0; 0]
(4)
g = [0; 1; 0]
(5)
b = [0; 0; 1]
(6)
From local phase transformations, each generator has a gauge field (gluon
field) associated with it:
Ψf (x) → exp(igs λj ωj (x)/2Ψf ),
µ
f
µ
D Ψ (x) = (∂ +
(7)
igs λj Aµj (x)/2)Ψf ,
j = 1, 2, .., 8
(8)
In QCD, the colour charges are exactly conserved. Each quark carries a
colour charge, while a gluon carries two. Quarks don’t exist in isolation,
but form hadrons, held together by gluons. The hadrons are always colour
neutral (e.g. a red, green and blue quark forming a baryon or a blue quark
and an anti-blue anti-quark forming a meson). Hadrons are described by
product wavefunctions χch = χc1 χc2 .
Colour charges
The colour charges for the colour operators can easily be computed by applying them on the colour spinors, e.g.:
1
1
F̂1 r = g, F̂1 g = r, F̂1 b = 0,
2
2
i
i
F̂2 r = g, F̂2 g = − r, F̂2 b = 0,
2
2
1
1
F̂3 r = r, F̂3 g = − g, F̂3 b = 0.
2
2
(9)
(10)
(11)
Structure constants
The commutator is known to give ifijk Fk , note the sum over k. The right
hand side can be expressed as a linear combination of the eight matrices
F̂k . However, most structure constants are zero. It turns out that the
commutator of any pair of matrices F̂i , F̂j can be expressed as a linear
combination of at most two matrices F̂k . Evaluate the commutators for
some Fi and Fj to find the structure constants fijk .
Let’s evaluate [F̂1 , F̂2 ]
1
[F̂1 , F̂2 ] = ([i, 0, 0; 0, −i, 0; 0, 0, 0] − [−i, 0, 0; 0, i, 0; 0, 0, 0]) =
4
1
= [i, 0, 0; 0, −i, 0; 0, 0, 0] = iF̂3
2
2
(12)
So the structure constants f123 = 1, while f121 = f122 = f124 = f125 =
f126 = f127 = f128 = 0. Also,
[F̂2 , F̂1 ] = −iF3
(13)
showing that f123 = −f213 . The structure constants are in facts antisymmetric with respect to all indices. Note that evaluating the commutator
of e.g. [F̂4 , F̂5 ] will give an expression that is a linear combination of F̂3 and
F̂8 , that is both f453 and f458 are non-zero.
2
V-A theory
Weak interactions have V-A currents, that is vector minus axial vector (axial
vector = pseudovector). Under a parity transformation, the vector part
changes sign but the axial part does not. Therefore, parity is not a symmetry
of the weak force. The V-A structure was deduced from observation of parity
violation of the weak force, as mentioned below.
• Parity violation of the weak force was demonstrated by observing β decays of Cobolt into Nickel with the emission of electrons and anti- neutrinos, in an experiment by Wu et al. If parity was conserved, the rate
of electron emission would be the same in the direction parallel and
anti-parallel to the spin of the atom. This was found not to be the case,
parity was broken. A combination of a vector and axial vector current
was needed. Additional reading: http://www.hep.phy.cam.ac.uk/
~thomson/partIIIparticles/handouts/Handout_9_2011.pdfp298
• The correct decay rates of charged pion1 decays are predicted from
amplitudes derived from V-A theory, but not from vector or axial
only.
• In V-A theory, the weak interaction only acts on left-handed particles and right-handed anti-particles. Only left-handed neutrinos and
only right-handed anti-neutrinos have been observed. In summary, the
observations of parity violation require a theory that is not invariant
under parity transformations. This is achieved by having a vector and
an axial part in the current.
3
Fermi vs IVB theory
The muon can decay into an electron and two neutrinos:
µ− → e− + ν̄e + νµ
(14)
In this way, both the lepton number within each lepton family (e, µ, τ ) and
the electric charge is conserved. A naive picture of this interaction would
3
νµ
µ
νµ
ν̄e
W
µ
ν̄e
e
e
Figure 1: Muon decay in four-fermi theory (left) and in intermediate vector
boson theory (right).
be a four-point interaction with no propagator as in Fermi theory. The
interaction is shown in the Fermi picture in Fig. 1 (left). A more complete
picture of the interaction is given by Intermediate Vector Boson theory (IVB
theory) with the appearance of an intermediate off-shell charged W boson:
µ− → W −∗ + νµ → e− + ν̄e + νµ
(15)
This picture is illustrated in Fig. 1 (right).
In four-fermi theory, the decay width of the muon is given by
Γ(µ− → e− + ν̄e + νµ ) = G2F
m5µ
192π 3
(16)
The lifetime of the muon τµ is the inverse of this value, τ = 1/Γ. This
prediction of the muon life time agrees to the level of a few parts in a million.
More on this in 16.6.1 in Mandl and Shaw. While four-fermi theory gives a
good approximation for muon decays, for weak processes at higher energies,
the IVB model is required. The W boson was discovered in 1983, once there
was a particle accelerator (at CERN) powerful enough to produce it. The
Fermi theory can be seen as an effective theory that is valid at exchange of
energy much lower than the W mass (80 GeV). A yet more complete theory
is electroweak theory, in which also neutral Z bosons can mediate the weak
force.
There are reasons to believe that the Standard Model is an effective
theory that describes physics well at currently available energy, but should
be modified in order to also describe physics at higher energies. Many
specific models, e.g. of supersymmetry and extra dimensions, address some
of the issues with the Standard Model. If one makes the assumption that the
new particles predicted by such theories have an masses above the reach of
our experiments, one can search for new physics in a more general framework
of Effective Field Theory. One assumes that the new particles are too heavy
to be produced, but that at high enough energies, sizable corrections to the
Lagrangian from new physics will occur, affecting the production rates and
distributions of different Standard Model-like processes.
4
Additional reading on the 4-fermi prediction for muon decay: Schwartz
chapter 29.4.
Additional reading on Effective Field Theory: http://arxiv.org/abs/
1303.3876 and http://arxiv.org/abs/1505.00046
4
Neutrino oscillations
The calculations below are meant to fill in the gaps in the part on neutrino
oscillations in Mandl and Shaw. It gives a general form of the mass term
for neutrinos that can be added to the Standard Model. Neutrino mixing:
the interaction states νe , νµ , ντ of neutrinos are not mass eigenstates (1,2,3),
but rather linear combinations of them:
The free field Lagrangian for neutrinos should be expressed in term of
the mass eigenstates:
Inverting the previous equation and using that U is a unitary matrix
= U †:
U −1
In the last step, it is used that γ 0 and Ulj commute (they act on different
spaces). So,
The expression
defines the mass matrix for neutrinos. Here mj is a diagonal matrix
and one gets the same index three times. The free field Lagrangian can be
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written as:
The second term in this expression includes neutrino oscillations - the
change of flavour l of neutrinos over time. These oscillations are possible
because the different neutrinos have different masses.
It is not known by what mechanism neutrinos acquire mass and why
this mass is very small. The see-saw mechanism is a popular model. It is
also not known whether neutrinos are their own anti-particles (Majorana
neutrinos) or not.
Additional reading on the see-saw mechanism: arxiv.org/pdf/hep-ph/
0109264v2.pdf.
Additional reading on measurements of β decays to test whether neutrinos are Majorana particles: https://arxiv.org/pdf/1411.4791.pdf. The
idea is that if the neutrino is its own anti-particle, one can join a neutrino
and an antineutrino into a neutrino propagator and get a double β decay
without neutrino emission (Fig 3 in the link). If the neutrino is not it’s own
anti-particle, this can not be done, and only a double β decay accompanied
by two neutrinos is possible.
4.1
Probability to remain in the same interaction state
Here it is derived what the probability of a neutrino produced as a an electron neutrino to remain in that state some time T later. It is assumed
that only electron neutrinos and muon neutrinos can oscillate to each other,
which is reasonable for comparatively short times.
The starting point is the expression for the mass eigenstates in terms of
the interaction eigenstates as a function of time:
|ν1 (t)i = (cos θ |νe i − sin θ |νµ i)e−iE1 t
(17)
−iE2 t
(18)
|ν2 (t)i = (sin θ |νe i + cos θ |νµ i)e
Inverting the equations at t = 0, one gets:
|νe i = cos θ |ν1 i + sin θ |ν2 i
(19)
Then the probability of the neutrino to be in the same state some time T
later is:
hνe | |νe (T )i = cos2 θe−iE1 T + sin2 θe−iE2 T .
The probability of it remaining an electron neutrino is then
6
(20)
| hνe | |νe (t)i |2 = cos4 θ + sin4 θ + sin2 θ cos2 θ(e−i(E1 −E2 )T + ei(E1 −E2 )T ) =
= cos4 θ + sin4 θ + sin2 θ cos2 θ2 cos[(E1 − E2 )T ] =
= cos4 θ + sin4 θ + 2 sin2 θ cos2 θ{1 − 2 sin2 θ[(E1 − E2 )T /2]} =
= 1 − 4 sin2 θ cos2 θ sin2 [(E1 − E2 )T /2] =
= 1 − sin2 2θ sin2 [(E1 − E2 )T /2](21)
4.2
Rewriting in terms of mass difference
The neutrinos are produced in a flavour eigenstate with some fixed momentum p. When produced, they are not in a mass eigenstate or energy
eigenstate, but in a superpostition of two energies E1 and E2 . Making a
Taylor expansion:
q
1 m21
,
(22)
E1 = m21 + p2 ≈ p +
2 p
q
1 m22
E2 = m22 + p2 ≈ p +
.
(23)
2 p
Then
E1 − E2 =
m21 − m22
.
2p
(24)
In natural units and approximating that the neutrinos travel at the speed
of light, T = L, where L is the distance travelled. Then, writing E in place
of p:
| hνe | |νe (T )i |2 = 1 − sin2 2θ sin2 [(m21 − m22 )L/4E].
(25)
By observing the number of neutrinos that have changed flavour at different distances L from the neutrino source, one can deduce the difference
of the masses squared for the mass eigenstates.
4.3
Best place for seeing oscillations
Assuming E = 4 MeV and ∆m2 = m21 − m22 = 7.5 · 10−5 eV−1 , the largest
oscillation effect would be seen when
∆m2 L
π
= .
4E
2
(26)
or
L=
2πE
= 335 · 109 eV−1 .
∆m2
(27)
Converting from natural units, eV−1 = 1.97 · 10−5 m. So L = 66 km away
from the neutrino source, the effects of the oscillation can be best observed.
7
5
Peskin and Schroeder 4.2
The Lagrangian
describes the free fields and interactions of two hypothetical bosons
(Φ, φ). Provided that the Φ boson is at least twice as heavy as φ, it can
decay into two φ-s. We are to find the mean lifetime τ of Φ. The lifetime
τ of a particle is related to its decay rate ? by Eq. (16.38) in Mandl and
Shaw:
For the Lagrangian (13), we only have one decay mode Φ → φφ, thus
only one Γ to consider. The expression for the differential decay rate is,
from Eq. (16.36) in Mandl and Shaw:
To find M, we do a perturbation expansion in first order of the interaction part of the Lagrangian. We consider the first order term:
Keeping in mind that both fields are real, we call the annihilation and
creation operators for Φ: a, a† and for φ: b, b† . Our initial state is |ii =
a† (p) |0i. Our final state is |f i = b† (p01 )b† (p02 ) |0i, so hf | = h0| b(p01 )b(p0 2) is:
A factor 2 has been inserted because there are two ways of matching the
final state momenta to the two ? fields. The integral over the x-dependent
terms gives (2π)4 δ 4 (p01 + p02 − p), which is already included in the equation
for dΓ. We get:
and
Inserting this into the equation for dΓ:
Choosing the rest frame of the decaying particle, the δ functions impose
that:
8
Integrating over p02 :
Using spherical coordinates: d3 p01 = |p01 |2 d|p01 |dΩ and making a variable
substitution |p01 |2 = E102 − m2φ , 2|p01 |d|p01 = 2E 0 dE 0 :
So, the mean lifetime of Φ is given by:
We see that this only has a real value if MΦ ≥ 2mφ .
6
Madgraph
There was a brief demonstration of how to make predictions from the Standard Model using the event generator MadGraph https://launchpad.net/
mg5amcnlo. We specifically looked at the predictions for producing a lepton
pair e + e− from colliding protons
at 13 TeV collision energy. In the invarip
ant mass spectrum minv = (E10 + E20 )2 − |p01 + p02 |2 a peak was seen at
the mass of the Z boson (91 GeV). At low minv , there was instead a large
contribution to e+ e− production from off-shell photons γ ∗ . The interference
between Z and γ ∗ was found to be small for this process.
9