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PROFESSOR: What is gamma?

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Well, it seems to have
a beta in there, alpha.

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It's a little unclear
what gamma means,

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so let's try to
make sense of it.

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Gamma has units of energy,
because this term has

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units of energy, and
therefore the other term

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must have units
of energy as well.

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So gamma has units of energy.

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And from a unit of energy
you could get a time.

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So tau, you can define
it as h over gamma.

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That has units of time.

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h bar has units of
energy times time,

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so if you divide by energy,
you get units of time.

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And what is this number?

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So this is some time.

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And it's equal to m
over 2 alpha beta h bar.

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We have this guy.

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So the natural
thing, of course, is

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to compare to the time delay
associated due to this process

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that we've been studying.

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So all the time delay,
delta t, was 2 h bar dE--

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no, d delta dE.

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And this is 2 h bar dk
dE times d delta dk.

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Now, d delta dk, we calculated
it up on the blackboard.

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So this was equal to--

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this had been calculated as
1 over beta at resonance.

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At the resonance, this time
delay is calculated above,

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and it's equal to 1 over beta.

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This thing is also
easily calculated,

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because this is 1
over dE dk, which

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is h bar squared k over m.

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Therefore, what do we get?

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2 h bar divided
by h bar squared.

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k, you're at rest so that's
alpha over m, m here.

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And d delta dk is a beta.

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So what do we get here?

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Delta t is 2m over
h bar alpha beta.

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And compare with this
one, it's 4 times tau.

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So h over-- so
the end conclusion

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is that h bar over
gamma, which is a time,

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is tau, which is equal
to delta t over 4.

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So that's the intuition
for the half width.

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So in the distribution of
this scattering amplitude,

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there's a half width.

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It's an energy distribution.

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And there's a time
associated with it, which

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is h bar over the half width.

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And, being a time,
it must be related

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to some physical time
that has a period,

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and there's nothing
else than the delay.

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So if the delay is
large, gamma is small,

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and the width is small.

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It's a narrow resonance.

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So a narrow resonance is one in
which the width is very small.

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So this has enormous
applications.

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It's used in nuclear
physics all the time.

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It's used in particle
physics as well.

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The Higgs-Boson was
discovered over a year ago,

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it's a particle
but it's unstable.

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It decays very fast.

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If you thought in
terms of that, it just

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gets created by
a reaction, stays

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in the well for a little
while, and poof, disappears.

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So its mathematics
in cross-sections

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is governed by resonances.

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So we call the Higgs
particle a particle,

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but anything that is
unstable it's more like, more

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precisely viewed as an object
that represents a resonance.

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The Higgs particle
doesn't live too long.

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Lives about 10 to
the minus 22 seconds.

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Very little time.

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And then it decays and it goes
into b-bbar, bottom bottom bar

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quark.

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As it goes into z's, it
goes into tau lambda.

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So it goes into two photons.

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It can decay into
all these things.

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So in the case of
the Higgs-Boson,

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the center energy
of the resonance

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is observed in scattering
amplitudes at a center energy E

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alpha of 125 GEv.

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The width is very small.

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In fact, the width is
about gamma, it's about 4

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MEv plus minus 5%.

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4 MEv, that's very
little, because an MEv

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is 1/1000 of a GEv.

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So it's a very narrow resonance.

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And from this gamma,
you can get a time.

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And the time is lifetime.

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It's about 10 to the
minus 22 seconds.

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So this is the language
that we use to describe

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any unstable particle.

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We think of it as resonance.

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It's sometimes called resonance.

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And the title of papers
that appear at that time

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is the resonance observed in
the data, the Higgs-Boson.

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And the answer was yes.

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This was observed
as a resonance.

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It is a scattering
amplitude that is uniform,

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and it has some energy,
has a little peak.

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You can imagine
people can't quite

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measure the width directly.

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It's too narrow.

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So the plots don't
show the width,

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but the width is implied, and
there's related calculations

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over here with the lambda.

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So it's all a nice story.

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OK.

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So look what we've done.

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We began by discussing
that we could not get

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time delays that were negative.

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Times advance, then we
get long time delays,

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and that's a resonance.

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We showed that they
correspond to changes,

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rapid changes of delta in
the positive direction.

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And we've modeled them
so that in general, we

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have a general
description of what's

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happening near a resonance.