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MARKUS KLUTE: Welcome
back to 8.701.

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So we continue our discussion
now of electron and proton

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scattering.

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And we dive deep into
the structure using

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deep inelastic scattering.

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Inelastic here means
that we are destroying

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the structure of the proton
in the scattering process.

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But we have a way to look
at the remnant of the proton

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and also of the
scattered electron,

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and then compare our
theoretic expectation

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for the cross-sections with
the finding in experiments.

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Let me just talk about
this in more general terms.

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The energy of the probing
electron or the photon

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in the scattering
process allows us

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to look at the proton
with varying resolution.

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So at very low
energies, we basically

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see a point-like particle.

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And then the scattering
process looks

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very much like the scattering
of an electron with a muon.

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If we increase the
energy of the electron,

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we can see that there's
an extended charge

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distribution in the proton.

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Further increase allows
us to resolve the fact

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that the proton is made
out of three quarks.

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And if you increase
the energy further,

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we see a lot of new
particles appearing, quarks

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and antiquarks and
gluons, which make up

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the structure of the proton.

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The picture here is--

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I like this very much, I drew
this myself some years ago--

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is what I would like
to have you remember.

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So in deep inelastic
scattering experiments,

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we basically use the
photon scattered--

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radiated off the
electron as a magnifying

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glass for the proton.

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So we can look into the
structure of the proton here,

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and we see the distribution
of the charged particles

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in the proton--

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only the electrically-charged
particles.

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We don't have
scattering between--

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direct scattering between
photons and gluons.

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So in measurements,
what we can do

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is we can test
scattered electron,

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and we can look for the
remnant of the proton

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in our measurement, and then we
do differential cross-section

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measurements, compare
with our theory,

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and can infer information about
the structure of the proton.

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To do this, we have special
kinematic variables which

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turn out to be very useful.

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The most important
one is probably

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this x year, which is called
the Bjorken scaling x or Bjorken

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x, which you can think about--

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so this is q squared,
the momentum transfer

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of the photon, p is the
momentum of the proton,

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q is the momentum transfer,
this q here, factor of 2.

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And what this is basically
the fraction of the momentum

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carried by the parton here
in the scattering process.

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There's a few other
useful variables,

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but I don't want to go into
any of the details yet.

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So there's a number of
very important scattering

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experiments.

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The first one I mentioned
before is SLAC-MIT experiment,

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which led to the discovery
that the proton is made out

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of quarks, and the Nobel
Prize in Physics 1990

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to Jerry Friedman, Henry
Kendall, and Taylor.

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And what they did
is they basically

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had a beam at SLAC of
electrons of 5 to 20 GeV.

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And they scattered this beam
off of hydrogen target protons.

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And they used this
spectrometer here

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in order to then make a
differential measurement

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of the scattered electrons.

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So that's very cool.

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Even higher energies
were available

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at HERA, the
electron-positron collider,

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where the energies
of the electrons

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were in the order of 30 GeV
of the protons up to 830 GeV.

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And so what we find then
in those collisions,

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the differential cross-section
measurements, is shown here.

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And it's very-- not
an easy plot to read.

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So you see our
structure functions

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here in the logarithmic plot.

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Remember, this is the
a log 10 plot here.

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And you see here q squared,
the momentum transfer

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of the photon, so the energy
used in the scattering process.

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When we try to
read this here, we

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can look at a fixed q
squared, for example.

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And at a fixed q squared,
you see that if you probe--

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if you are testing for a
fixed fraction of the partons

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taking away partons, a
fraction of the parton's

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momentum of the proton, you see
that the lower the fractions,

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the more particles you see.

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So at a fixed energy, you
see many, many more particles

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the lower you go
in the fraction.

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So there seem to be like
an increase of particles

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the lower the fraction is.

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If you then check for a fixed
fraction, let's say 0.4--

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it means that the parton carries
40% of the proton's energy--

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you see that it's almost flat
as a function of q squared.

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So it seems like there's 40%--

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the number of particles you
see at 40% of momentum fraction

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is constant, this q squared.

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However, if you look at smaller
energies-- sorry, smaller

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momentum fractions, you
see that higher energies

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seem to show even more particles
at this momentum fraction.

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And the ways to understand
this is two diagrams.

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So the first one
is this one here,

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where you see a quark
radiating a gluon.

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And so what you see is
that the deeper you look,

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you are able to then
resolve this part here.

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And so you see more
quarks and gluons

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which carry even smaller
momentum fractions

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than the initial quark here.

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You also see diagrams
like this, which

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is called gluon splitting,
as a gluon splits

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into a quark/antiquark here.

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And you start resolving those.

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And those also carry
lower momentum.

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The evolution of our--

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of this parton distribution
function, or the structure

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functions, can be
calculated and described

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in the so-called
DGLAP equations.

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And all you do here is
calculate the contributions

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from the so-called quark
and gluon splitting.

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So you calculate the
splitting functions,

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these higher-order corrections
to a very simple quark model.

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And you find that you can
actually very nicely describe

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those curves here.

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So you see this yellowing--

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yellow here is
the QCD fit, which

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basically uses those
splitting functions as input.

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All right.

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So we learned quite a bit
about this proton already.

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If we want to now
calculate a cross-section

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of a proton scattering
with a proton,

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we are actually interested
in the energy distributions

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or the momentum distributions
of the partons in the protons.

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And so we'll come back
to how we use this later.

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But I want to introduce parton
distribution functions which

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do exactly that.

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They're defined
as the probability

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to find a parton in
the proton that carries

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energy between x and x plus dx.

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You can write them using the
structure functions before.

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But they literally
describe this probability.

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And so what you find
inside the protons are now

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the valence quark, the down
quark and the up quarks,

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c quarks and
antiquarks, and gluons.

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So you want to describe
those momentum distributions

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or energy distributions
of those particles.

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There's a number of sum rules.

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If you integrate momentum
fractions from 0 to 1,

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1 being the momentum of the
proton, if you integrate them

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all together, you
have to find 1,

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because that is the momentum
of the proton you start with.

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If you integrate the down
quarks and the up quarks,

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you find 1 or 2,
meaning that those

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are the number of valence
quarks we have available.

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If you integrate
the distributions

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of strange and antistrange
and charm and anticharm,

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you get 0, because
there needs to be

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the same number of
strange and antistrange

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and charm and antistrange.

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And because of
energy conservation,

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the sum needs to-- this sum,
those sums need to be 0.

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And then we can look at those.

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So what's shown here is x
times the Parton Distribution

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Functions, the PDFs,
as a function of f.

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And what you see here
for our valence quarks,

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you see a distribution which
is kind of what you expect--

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it almost peaks at 0.3, a
surge, and has the distributions

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because there's
kinematics involved,

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and also interactions--

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because of the interactions
with the gluons.

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And then you see the c
quarks and antiquarks.

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And you see them increasing
in numbers quite significantly

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here as you go to small fraction
of the momentum carried.

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It's exactly what we just
discussed in the previous plot

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already.

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An interesting way to look at
this very same distribution

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function is if you plot them
proportional to the momentum

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fractions, or the area
proportional to the momentum

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fractions.

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And what you see there is
that a very significant part

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of the momentum of the proton
is carried by the gluon.

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So you see here again, our
valence quarks, our c quarks,

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and the gluons itself.

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So that's all I wanted to say
on the structure of the proton.

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We'll later in a lecture see
how we can use those PDFs,

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those Parton Distribution
Functions, in order

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to calculate a cross-section
in proton scattering.