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MARKUS KLUTE: Hello.

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So in the last video, we
looked at ranges of forces.

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So we already saw one
aspect, and wanted

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to learn about what
kind of interaction

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happens when you study
a specific force.

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You learn about
the force carrier

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from noting the range in
which forces interact.

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In this class, we
talk about decays.

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But in more general
terms, when we

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want to measure
properties of forces,

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we have basically
three concepts at hand

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which can be
experimentally determined.

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The first one is
masses of bound states.

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| you might remember
from atomic physics

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that you learn a lot about
electromagnetic interaction

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by studying, for example, the
hydrogen atom, where you have

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an electron circling
around the program,

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and you can study
in detail aspects

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of the electromagnetic
interaction.

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The second aspect is decay
rates of unstable particles,

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or the width of an
unstable particle.

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So in quantum mechanics,
the lifetime of the particle

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is related to the width.

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And so that's what we're going
to discuss in this video.

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And then lastly, we can look
at the reaction rates expressed

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as cross sections.

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So that's the topic of
the next video after this.

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Let's talk about decays.

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So we can define this new
symbol, the decay rate lambda,

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and s as a function
of time, probability

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that a particle will survive
at least until some time t.

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We can now discuss this,
and say, the probability

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at some time t relates to
the probability at some time

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t plus delta t by the likelihood
1 minus the decay rate

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times the time
interval, delta t.

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And so from this, we find that
the change of the probability

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for the particle to survive
is proportional to that

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probability and decay rate.

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So now if we integrate
this, we find

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that the log of this probability
is equal to some constant times

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lambda decay rate times time.

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So now, if you simply assume
that the particle existed

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at the initial time t, we
said we set k equal to 0.

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We find this very
famous exponential decay

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law, e to the power
of minus lambda t.

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And this is shown here in this
picture as this exponential.

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So far, so good.

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We can now define and look at
this distribution a little bit

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more.

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We can, for example,
look at the average time

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for that a particle lives,
because the average time

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tau is simply given by the
integral from 0 to infinity.

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So we basically integrate
over this distribution

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to get the average
time for the particle.

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And that's equal to 1 over
the decay rate, 1 over lambda.

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You can do the algebra,
yourself, in fact,

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to follow this.

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So if you now express this
probability for the particle

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to survive until some time
t through the lifetime,

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you find this is equal
to e to the minus t

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over the lifetime
of the particle.

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So you might not want
to look at one particle,

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but a sum of particles
and look, at the time

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dependence of the number of
particles which survived.

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Because it's equal to
the number of particles

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as a function of time is
equal to the initial number

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of particles times
the probability

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that any given
particle survives.

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And that's, again, given
by this exponential.

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In nuclear physics,
one often talks

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about the half life, half
life time of the particle

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or of [INAUDIBLE].

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And that's given,
as you would assume,

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by the time it takes for half
of the particles to decay.

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So N of tau 1/2 is equal to the
number of initial particles, N0

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over 2.

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And you find, then,
that this half life

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is related to the
lifetime of the particle

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with a factor of about 2/3.

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This leads to some confusion
in numerical values

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sometimes when you ask for
specific answers from up here,

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in experiments.

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All right, so there's another
aspect of decays which arises,

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which comes from a fundamental
property of quantum mechanics.

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So if you have an
unstable state,

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or any unstable state does
not have exact energy state,

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it just follows, if you want,
from the uncertainty principle.

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So the width of the
particle is quantized,

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and it's quantized
with this lambda here.

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And if you can see,
lambda relates, again,

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to the decay rate or the
lifetime of the particle.

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Another complication
can occur when

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there's multiple ways for
the particle to decay.

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For example, you
have a Higgs boson

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as we see on the
next slide, which

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might decay into multiple--

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has a way to decay into
multiple particles.

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Here, we define a partial
width, where the partial width

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is defined as half the
width of the particle

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to decay into a specific mode.

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And then the total
width of the particle

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is given by the sum
of the partial widths

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of all possible ways for
the particle to decay.

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Using this, you
can also calculate

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the likelihood of a particle
to decay in a specific way.

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That's called the
"branching fraction."

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And that's given by the
partial width divided

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by the total width, or the
partial decay weight divided

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by the total decay rate.

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Again, the total has to
be 1, the probability

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for a particle to
decay in any mode is 1.

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Therefore, the sum of the
branching ratios is 1 as well.

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All right, so looking
at a specific example,

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the Higgs boson is probably
my favorite example

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in this entire class.

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You find here, given
branching fractions or ratios

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of specific decay modes,
because it's not always in which

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the Higgs boson decay,
but the most dominant one.

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The most prominent one is the
one with Higgs boson decays

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into a pair of b quarks.

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We will later see, maybe
even as an exercise,

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why the distribution
function of branching ratios

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is the way it's
being shown here.

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The Higgs boson has been
measured with a mass

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of 125 gV and a little bit.

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And you see at this mass
here, the prominent decay mode

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is simply the b
bar, but it's also

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possible for the exponential to
decay into a pair of W bosons,

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where an interesting
loop diagram into gluons.

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Even so, gluons are massless,
or tau, charm, Z bosons, and so

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on.

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And we just showed you in a
paper which was submitted today

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to the arXiv, the
Higgs boson also

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can decay into a pair of muons
with a branching ratio of 2

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times 10 to the minus 4.

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So it's rather rare,
but it's possible.