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IAIN STEWART: [INAUDIBLE] theory graphs calculating
the effective theory graphs using
the same infrared regulator, renormalizing each of them
separately, and then subtracting them
to figure out higher-order corrections to the Wilson
coefficients.
And as part of that discussion, we also
talked about scheme dependence because when
you do that procedure for normalization
in the effective theory, you're making a choice for the scheme.
We picked m s bar, but you could pick other choices.
And at the end of lecture, we talked about the fact
that when you make different scheme choices,
it can affect things.
So it can affect the Wilson coefficients.
It'll affect the matrix elements of operators.
It'll affect your matching coefficients
at one loop and your anomalous dimensions at two loops,
but those scheme dependencies cancel out in observables.
So it's important to take the scheme into account and work
in the same scheme.
But if you consistently use the same scheme for everything,
then you will be OK.
Now, if you're just doing calculations on pen and paper,
it's very easy that you just-- well, we
love the m s bar scheme.
Let's use that.
But remember that sometimes, there
will be things in your result that you
may want to get from elsewhere.
For example, maybe there's some matrix elements
of the operators.
You want to get them from the lattice.
Well, if you look up results in a lattice,
they're not using m s bar.
There's no non-perturbative definition of m s bar.
So they'll be using some other scheme.
And those results will have to be converted
to m s bar if you're going to put them together
with your results.
So that's just something to be aware of.
Before we leave this topic and go on to something else,
I thought I'd say a few words about phenomenology.
What is all this technology good for?
So a nice example of that is beta s gamma.
So beta s gamma--
it's a neutral current process.
It doesn't happen in the Steiner model at tree level.
And therefore, what that means, if it doesn't
happen at tree level, is that it's
sensitive to loop corrections.
And you could go--
since we talked about a channel with b to c, you bar d,
but this is very analogous.
We have a b meson in the initial-- b quark
in the initial state.
So it's the same scales in the problem.
Effectively, the b quark scale is the light scale.
We want to get rid of the things that
would mediate this decay that are inside the loop, which
will be w bosons and top quarks in this example.
And so you draw this in the standard model.
There's various diagrams that contribute.
One is this one.
So here's a b quark changing into a strange quark
through a top quark and a w.
And if we integrate out the w in the top,
then we get some local operator, much as
we were talking about in our examples up here.
It's just that we have different diagrams.
And we can also, from the low-energy point of view,
enumerate the operators.
For some reason, I started calling it Q instead of O.
Let me give you an example of some of these operators.
So here's what you would call a magnetic dipole
operator, couples directly to the photon, which
is in the F mu nu.
And it takes the b to an s bar.
And that's an example of a higher dimension operator.
Remember, this is dimension 2 and then plus 3
there, so that's 5.
There's a factor of the b quark mass that just comes in
because of the chiral structure of the operator, which
you can think of as there needs to be a mass insertion here.
You need one factor of that mass in order for the diagram
not to give 0.
If you really start to do beta s gamma,
and you want to go through the whole story,
then you have to actually think about other operators.
And there's a whole basis of them.
And one thing you could do is you
could take the operator up there and just
replace the photon by a gluon.
That won't give a tree-level contribution to beta s gamma,
but these guys are charged under electromagnetism.
So you could just have a direct coupling
to the b quark or the strange quark,
and then loop up the gluon, and get a correction
from an operator like this one.
And then there's also four quark operators,
like the ones we talked about before,
but with different flavors.
And if you enumerate all of them,
there's nine more different ways of making four-quark operators.
So you could build some basis using the equation of motion
to simplify the operators as much as possible.
And then you get down to these ones.
And you can go through the program
that we talked about, of doing matching and renormalizations
with evolution.
And really what I want to talk about here
is a little bit about phenomenology,
because the interesting thing about this loop
is that if there was nu physics-- since
in the standard model at tree level it doesn't happen,
if there's nu physics, then you can be sensitive to it
here, because you're sensitive to heavy particles
in this loop.
That's why beta s gamma is always
used to constrain nu physics.
So in our effective theory, we have just an operator that does
this, which is this 07 gamma.
And if you think about what C 7 gamma is at lowest order,
let's just calculate that diagram over there.
And the result from doing that will
be some function of m w over m top,
because those are the things that are appearing in the loop.
And if you do that calculation, you get a number like that,
if you stick values for the top m w.
And then you can start doing loop corrections.
The first thing you might think about,
which is actually not even suppressed
by any factors of the coupling alpha s strong,
would be to just loop up the Q1 operator, which I guess I don't
know why I wrote it this way.
This should be b.
You could just loop the Q1 operator, contract the u u var,
and then you can get a beta s transition from this operator.
So this is the u quark here, and then attach a photon.
And there's no factor in that loop of the strong coupling,
because this is an electromagnetic coupling,
and this guy here is just this Q1 operator.
So this doesn't look like it's loop suppressed relative
to that.
And this is a little subtle, but this guy actually is 0.
So you have to use a good scheme for gamma 5,
but it turns out to be 0.
So the first type of loop corrections that you get,
which are suppressed by a factor of alpha s,
come from diagrams like--
well, there's various diagrams, but one of them
is like this, where I take the same diagram there
and I attach an extra gluon on top of it.
This guy diverges.
And this two-loop calculation is order alpha strong,
and it gives the leading order anomalous dimension,
what we were calling gamma 0.
It's not the only diagram.
There's other diagrams, too.
So we could go through that and do the similar type of thing,
just with more diagrams than we had in our example.
In particular, we could construct
from the tree level matching in the one-loop anomalous
dimension, the leading log result. Let me
just write that down for you.
Putting in numbers for the anomalous dimensions, at least
sometimes putting in numbers for them.
So the eta factor here is the ratio of alphas.
So this is similar to what we saw before in our example
where we got a ratio of alphas raised to a power.
And if you want to pick a mu for this process over here,
beta s gamma, then the right mu to think about is m b.
So we want to take mu to the m b.
And so if I plug in numbers--
and this is really what I wanted to emphasize--
if I plug in numbers here for these various things,
this guy here gives a factor of 0.7.
This guy here is this minus 0.2 that we talked about up there.
This factor here is a bit small, 0.085.
0.96.
That's right.
And then this piece here is a substantial correction.
And if I wrote down all these numbers correctly,
then the final result comes out to be,
if I keep three digits, minus 0.3, which you can see
is a fairly substantial change from minus 0.2.
It's a 50% change.
So just taking into account the evolution
gives a 50% correction.
So if you didn't take it into account, and you just said,
well, forget about effective theory.
I just calculate this graph.
That's the standard model, you'd think that there is nu physics.
We've certainly tested beta s gamma at better than 50%,
more like the 10% level.
So you really have to take into account these effects
that we've been talking about, like this leading log
evolution, if you want to look for nu physics, because you've
got to get the right standard model result. 50% larger.
And actually, people go two orders beyond what
I'm talking about here.
They go to the next, the next leading log order,
when they really do precision beta s gamma physics.
So some of the state of the art calculations of multiloop
diagrams have been done exactly for beta s gamma
because these effects are so important for looking
for nu physics.
And it's even worse when you get it put it
in the branching ratio, because a 50% enhancement in a coupling
when you put it in the branching ratio,
you're squaring the amplitude.
So that's a factor of 2.3.
So these are really crucial corrections
to take into account.
So that's what this electroweak Hamiltonian is actually used
for when you do phenomenology.
So that's what I wanted to say about the electroweak
Hamiltonian, just to give you a flavor for it.
There's lots more that you could do with it.
We could talk about more phenomenology,
but let me stop there, since the idea is
to give you an introduction to the concepts
and we've done that.
So now we'll move on to something else,
which is a different concept, unless there's any questions.
So all this business of schemes and stuff
comes in when people are talking about beta s gamma
and making this kind of model prediction.
And there's actually schemes you can pick where you mess this
up, but then, if you're careful, you
get the same answer in the very end.
So the next topic that I want to talk about
is an example of something that's bottom up
effective field theory.
We've talked about top down with this example
of removing heavy particles.
And the classic example of bottom
up is chiral perturbation theory or chiral Lagrangians.
So our purposes here are not--
again, they're not a full exploration
of this topic, which is a very large topic.
So what are our goals?
Bottom up effective theory example.
We will also see in this example the utility
of using something that's called the nonlinear
realization of symmetry, non-linear symmetry
representations, and the kind of connection
to field redefinitions.
Since field redefinitions is something
we've been talking about, we'll talk about that.
Another thing that this is an example of
is an example where loops are not suppressed by the coupling
constant.
Instead, they're actually suppressed by powers
in the power expansion.
So that's kind of totally different than what
we saw when we were integrating all the heavy particles,
where you just put a loop.
You're down by some factor of alpha s,
but you're at the same order in the power expansion
and 1 over large scale.
This is going to be different.
And therefore, in some ways this has a non-trivial power
accounting--
more non-trivial, anyway, than what
we were just talking about.
So we'd like to give this as an example of non-trivial power
accounting, and in fact, prove something that's called
a "power accounting theorem."
What the power accounting theorem
means is that ahead of time, you should
be able to figure out from your effective theory what
order various things are.
If you draw a diagram, you should
know even before you calculate it how many powers in the power
accounting expansion you have.
If you didn't know that, you wouldn't
know which diagrams to compute and which ones not to compute.
You'd just have to compute them all.
That's, of course, way too much work.
So in order to formulate the effective theory--
especially since there's an infinite number of diagrams.
So in order to formulate the theory, remember,
you really have to have power accounting under control.
And that means you need things like this in order
to identify what's leading order.
So we'll talk about that.
So I'm imagining that maybe 50% of the class
has seen chiral perturbation theory before in some form.
I know I teach it in Quantum Field Theory 3,
so if you took QFT 3 with me, you saw a glimpse into it.
The things we're going to emphasize here are a bit
different, but I am going to assume that you have knowledge
of it at some level, and I'll assign you reading if you
don't.
The other thing I'm going to assume
that you have some knowledge of is spontaneous symmetry
breaking, because that's not the topic.
That's not one of the things that I've listed here.
That's not something that I really want to delve into,
but of course, if we talk about the chiral Lagrangian, that's
something that comes in, in particular, when we're
talking about it from QCD.
So I'll remind you of some things that are hopefully
familiar, and anything that's unfamiliar,
you should do additional reading on.
So I know that there are some people that
are taking QFT 3 right now, so this may be something
they haven't got to yet.
So I will do some review, but for further reading,
you should see QFT 3 or some other source.
I've posted some readings on the website.
So if we start with QCD, massless QCD,
we can divide it into a left-handed part
and a right-handed part.
And then we can talk about the transformation
where we take the left-handed field
and transform it by a separate amount
than the right-handed field under a unitary transformation.
And that's a symmetry of the theory.
Now, exactly what symmetry you have
depends on how many components you're
talking about in the psi field.
If I say the psi field is light, then you
might think of up and down, and that's one possibility.
Or you could throw the strange in there,
and say, well, the strange is light, too,
and then you have up, down, and strange.
And that just the difference between SU2 and SU3,
and both of these are viable choices.
So let's make a little table.
We have a group, which is the transformations given
by the left and the right.
And it's going to be broken spontaneously
to some other group.
So it could be SU3.
So there's saying that the left and the right
are in SU3, each of them.
And in QCD, that's broken to the vector subgroup.
So in this case, the psi would be u d s.
And you get Goldstone bosons from that.
And there's eight of them.
And that's the pions, the kaon, and the eta.
And keeping thing from our perspective what
is the expansion parameter, and it's not so great,
because the strange quark mass is not that light.
So you can think that the strange quark mass over lambda
QCD is maybe something like 1/3-ish,
but you're not really getting better than that.
So there was 8 generations here, 8 generators here,
broken to 8 here.
And then you have 8 Goldstones.
You could do better, but make less predictions,
if you just considered SU2 and just maybe up and down light.
Then you just have the pions.
And then you just have MU and MD over lambda QCD,
which is more like 1/50, so a much better expansion.
But then you can't do kaon physics
with this, because the kaon is not
part of the effective theory, and that can be the case.
So in order to construct a theory for the Goldstones,
which are the light degrees of freedom--
so this is we've identified the light degrees of freedom--
you'd like to construct an effective theory for them.
Those are bound states of the particles
in your original theory, and you don't know, on pen and paper,
how to calculate the matching.
And that's characteristic of a bottom up effective theory--
that you don't do the matching, that you just
start from the bottom up.
So in this case, you don't do the matching
because it's non-perturbative.
So what you do is you say, let's construct
a type of field theory based on the fact
that I know what the degrees of freedom are
and I know something about symmetry.
And in particular in this case, you
know something about symmetry breaking.
So one kind of logic is, then what you'll get
is you'll get coefficients times operators again.
The operators will be built out of your pion/kaon/eta fields.
And you'll be able to calculate matrix elements of these.
But you'll get some coefficients and you don't know the value.
So these coefficients you could fit to data,
because you haven't determined them
from some high-level theory.
You just could fit them to the data and do phenomenology.
So it's kind of exactly the opposite
from our high-energy point of view,
from the theory with the electroweak Hamiltonian, where
we thought the matrix elements were
going to be non-perturbative.
Here are the matrix elements.
They're something you calculate.
And the coefficients are including
the non-perturbative physics.
You could also get these from the lattice.
So a lattice QCD calculation can tell you
the values for the C's to use.
And then you could just use the chiral Lagrangian
to do phenomenology.
Now, because we have this bottom-up point of view,
there's something that we don't know.
And that is, we actually don't know precisely what
theory we started with, because all we're
encoding about that theory is the symmetry breaking pattern.
And so if there's a bunch of upper-level theories
that have the same symmetry breaking pattern,
they all look the same.
They'll look like they have the same chiral Lagrangian.
And the thing that distinguishes them
is that they would have different coefficients.
And you wouldn't know that unless you figured out
what the coefficients are.
So that's another way of just saying
that the high-energy physics is being
encoded in the coefficients.
So the same chiral Lagrangian would show up
for different high-level theories
that have the same symmetry breaking pattern.
So this is just one example of chiral Lagrangians.
And we're going to use it as an example, which
is perhaps the most familiar, to illustrate our bullets.
So one of our bullets was related
to non-linear representations in field redefinition,
so let's start with that one.
Problems set's due today.
Problem set 2 is posted.
So let me talk about something called the "linear sigma
model."
And we'll use this as an example of something
that has a symmetry breaking pattern that's
similar to the one we want, same as the one we want.
And we'll construct, from this, the chiral Lagrangian.
And since it's the same symmetry breaking pattern,
it's also a viable chiral Lagrangian directly for QCD.
So what is the linear sigma model?
So I'll talk about fields that I call pi
without a vector symbol on top.
They have two components.
One component that's in the diagonal, then one component--
these are matrices in the off diagonal,
and different entries in the sigma 3.
So I want to think of this is a kind of full theory.
So the full theory for the sigma model is the following--
kinetic term, mass term, interaction term.
And I can couple it to something else
like a fermion to make it more interesting,
with Yukawa couplings.
And this theory has an SU2 left cross SU2 right symmetry, where
I take psi left to L psi left, psi right to R psi right.
And I also transform pi to L pi R dagger.
And if you think of these L's and these R's, you
should think of them as something with some generators.
So L would have something like that, with tau left generators,
and then some parameters alpha L. If I worked infinitesimally
in those parameters, then if I expanded this out,
this would give a linear transformation
to the sigma and the pi vector.
That's what it's meant, that it's linear--
so a linear infinitesimal transformation to pi and sigma.
So this theory has spontaneous symmetry breaking.
And if we write out the potential by taking the traces,
and we can write in a way that makes
that obvious by completing the square.
And we can always throw away irrelevant constants
when we do that.
So we get something like that.
So the minimum is shifted.
And so if we want to shift ourselves over
to describe perturbations around that vacuum,
then we can do that.
So let this guy have a VEV, which I'll call curly
V, square root mu squared over lambda.
Pi vector field has no VEV, because we
want to maintain the vector symmetry.
And we talk about a shifted field,
which describes perturbations around the VEV
when we do quantum field theory in the sigma twiddle.
And then we write our Lagrangian in terms of the sigma twiddle,
just by making a change of variable.
And that makes clear who's getting masses
when we expand around that vacuum
and who's remaining massless.
And of course, the Goldstones are remaining massless,
and the Goldstones, by clever choice of notation,
are called Pi.
Let's see if I can squeeze it all in here.
This is something they tell you in Professor 101 never to do,
but I'm going to do it anyway.
Since if you can't read something,
you can always look at my notes when I them.
So I didn't write all the fermion notes down again,
but I could do the field redefinition in those modes
as well, the shift in those terms.
And the vector subgroup, which remains unbroken,
is no transformation for sigma and the Pi field
has left equal to right, so we just get some matrix V
on both sides.
And if we look at the Lagrangian that we have,
then we can identify the masses of various things
at tree level.
And I didn't write down the fermion part,
but the fermion gets mass 2 from the [INAUDIBLE] coupling,
just like in the standard model.
And the Pi, of course, remain massless.
And the idea of thinking about this is an effective theory
is that we can take these to be large.
And then there's a clear separation
between low-energy degrees of freedom
and heavy degrees of freedom.
So if we want to describe this with an effective theory,
we would like to describe the physics of the pions
without worrying too much about the sigma and the psi.
So we could do that just by thinking about this Lagrangian,
but it turns out that you can make field redefinitions
and think about using different formulations which
are entirely equivalent.
And some of those are more useful than this linear sigma
model.
So we're thinking here of field redefinitions
as an organizational tool.
So you'll see what I mean when we go through this.
So let's consider some different choices.
So there's something called the "square root representation."
So I just make a field redefinition like that--
involves a square root.
And if I expand it, it starts a sigma twiddle,
and then it goes a bunch of other terms.
So it's certainly within the realm
of the things that are allowed by our field redefinition
theorem.
And then I also talk about making a field redefinition
for the Pi as well.
And again, this is Pi plus other terms.
So in this square root representation,
we're going over to these fields S and psi.
So these are our new fields.
So we can make that change of variable
and just write out our linear sigma model again.
And it's the same theory, it's just field redefined.
And it looks kind of Godawful, but it's more beautiful
than it looks.
So that's something you don't want to do more than once.
So that's one possible way that we
could deal with the effective theory,
is in terms of these variables, S and phi.
Let me write down a couple other ways,
and then we'll talk about which we might pick.
So there's something called the "exponential representation,"
also very common.
These are very common in the literature,
where instead of using S and phi, we use S and sigma.
So we take our original fields, and we rewrite it
as v plus S times sigma.
And then we think of sigma as the exponential, and hence
the name, of our fundamental Goldstone field.
So this is not the same Pi.
It's some Pi prime, if you like.
But I'm going to drop the prime.
So you write everything in terms of S and sigma,
and you keep in mind that inside the sigma is the Pi.
So the S part at the beginning is the same.
And this guy's a little bit nicer to write down.
So the terms that are pure S remain the same,
so S cubed, S 4th.
Since there's no funny square roots in our field
redefinition, things are a little simpler.
So that's, again, an equivalent version of the sigma model,
just using different fields.
Everything I can calculate in the original sigma model,
I can calculate these formulations as well.
And the final one is going to be different.
And that's the non-linearity chiral Lagrangian.
And what I'm going to do to get there is I'm
just going to drop the things that
have mass from my previous theory, which are S and psi.
So I take this exponential representation
of the sigma model.
These guys are massive.
I think about-- well, I integrate them out explicitly.
In this case, what we're doing here, which is not QCD,
we can do that, just remove them.
That amounts to the lowest order, just dropping them.
And then we have just the thing involving the pion,
which is just very simple--
that.
So this is a viable thing for low energies.
The first three actions that we wrote down are identical.
They're just field redefined versions of each other.
They're really equivalent theories.
And this final one, L chi, is equivalent
for low-energy phenomenology of the pions.
So the first three are actually equivalent to the last one
if we restrict ourselves to low-energy interactions.
So in order to see that, let's do an example
where we calculate something.
Let's think about calculating Pi plus Pi 0 goes to Pi plus Pi
0, so scattering of Goldstone bosons.
And q will be the momentum transfer.
So in kind of an obvious notation,
it's the difference between the final and the initial,
or the initial and the final for the charge guy
or the neutral guy.
And there's two types of diagrams that can come in.
You could have a direct scattering diagram
or you could have an exchange diagram.
so this is Pi plus coupling to either the sigma or the S,
depending on which representation we're using.
And then the Pi 0 is down here.
And I want to take these, and I want to expand.
And I'm going to expand in q squared over v squared.
That's what I'm going to mean by "low energy".
v was setting the scale of the masses.
So let's see how that works out in the different cases.
If I do it in the linear case, and I
have both of these types of contributions,
this guy just gives me some symmetry factor,
which is 2 minus 2i lambda.
This guy as a propagator, square root looks very different.
And it turns out in the square root,
that you can find out that this guy is order q to the 4th,
so we won't even write it down.
Exponential-- this guy is again ditto and ditto,
and then I don't have room, but I'll put it up here.
In the nonlinear, we don't even have that diagram
because we just dropped it.
And if I take this line, which is the only one that
looks different, I can combine these two terms together.
And once I do that, then I realize that this is also
giving the same answer.
So just rearrange it, put in the relation between mass
and coupling and VEV, which looks like that, and expand.
And you get i q squared over v squared as well.
So if I stop at order q squared, all versions
give the same thing.
The linear and the square root were equivalent.
The linear, the square root, and the exponential
are just different versions of the same thing.
And all that's changed by making the field redefinition
is how I think about these two diagrams.
In the linear, the leading order term is not
coming from just this diagram, it's
also coming from that diagram.
And it's actually coming from a cancellation
between those diagrams.
In the square root and exponential representation,
this diagram alone gives the leading order term
and this diagram gives higher order
terms, which is kind of what you want
if you want to think about these heavy particles
as something that is giving high-order corrections.
So that can depend on exactly how you formulate it.
And from the point of view of just keeping
the low-energy degree of freedom,
the nonlinear just gives you immediately the answer
from very simple Lagrangian.
Of course, it was just the exponential with something
dropped.
So from the point of view of doing calculations,
the linear version here is the one that you don't want to use,
because there's cancellations between diagrams
that you have to figure out.
And those cancellations actually affect
what you call "leading order," because leading
order for this calculation is order q squared,
and you don't see that until there's that cancellation
that's taken effect.
So it's very hard to formulate a power accounting
for the linear theory for that reason.
But if any of these nonlinear theories
we could formulate a power accounting,
then everything is nice and beautiful.
So what happens is that because of chiral symmetry,
we have derivative couplings.
And in the linear version, we don't
see that until we cancel terms.
So if you like, if you think about that
as a property of the symmetry, you'd like to make it explicit,
and the other representations do that.
The version that's most convenient
is the non-linear guy, since it's the simplest.
And we just think about it, forgetting
about the heavy stuff.
And it's what we really want for our bottom-up effective theory
because it only has the low energy sigma
field, which has a pion in it, and it
has the derivative couplings.
If you look at how the symmetry works,
which I didn't talk about when I wrote down
all the different versions, S is for singlet.
That's what's behind the name S, so it doesn't transform.
Sigma transforms on both sides with an L
on the left and an R dagger on the right.
And that comes from looking back, noting
that S doesn't transform, and looking back
at how the Pi field transformed.
It's just exactly that way.
Now, if you write out the sigma, which is transforming linearly,
as the exponential of i tau dot Pi,
and if I put it in a convenient normalization, which
is the VEV, or the v, then you can work out
from this transformation how the Pi field transforms.
And it transforms nonlinearly, and hence, that's
the name for--
that's where the name comes from.
If you do the infinitesimal version of the transformation,
and you work it out for the Pi a field, which
is a-th component of the vector, a being 1, 2, or 3,
part of the transformation is simply a shift.
And then there are additional pieces,
which are things that you keep, that are
order pi squared and higher.
And that's those pi squared and higher terms that they're
saying it's nonlinear.
This term here is what's telling you it better be derivatively
coupled, because if it's derivatively coupled,
if you have a derivative, that kills this constant,
and that was something that was hidden in our linear version
of the theory.
Now, we've constructed it here from this kind of point
of view of integrating out.
So I said QCD, we can integrate it out,
but all we care about is the symmetry-breaking pattern.
So we wrote down a theory, which is this linear sigma model,
had the same symmetry-breaking pattern.
And we could remove the heavy fields in that,
and then the low-energy theory, I claim,
is the same one that you would use to describe QCD,
because it's the same symmetry-breaking pattern.
But that's kind of a roundabout way of getting
to where we wanted to go.
And you'd really like to just get
right to the chiral Lagrangian from the start,
and you can do that.
So rather than going through the linear sigma model,
we could also just directly get to where we want to go.
And here's how you do that.
So go back to what the symmetry-breaking pattern was
and figure out what is it that we're
trying to do with the low-energy effective theory.
So we have G that's broken to H. The Goldstones are
transformations in the coset, which is G/H.
And we'd like to parameterize them.
And what we're doing with this sigma field
is we're parameterizing fluctuations
to take us around the coset.
AUDIENCE: I'm a little confused, because if you don't expand
sigma, that Lagrangian has still the full symmetry as you
do left, as you do right.
But it contains the v, [INAUDIBLE]
parameter for the rest of the symmetry.
That Lagrangian is presenting the unbroken phase or--
IAIN STEWART: So remember, I went over
to the shifted fields, the sigma twiddle,
and then I was expanding around the correct vacuum--
I mean, the one that's the lowest energy vacuum, not
the unbroken.
I went from sigma--
I originally had sigma--
sorry.
Thank you for that.
So really, when I talked about the linear version,
there was two versions of the linear version.
There was the original version where I wrote it down
in terms of sigma and Pi vector, and then I went over
to sigma twiddle and Pi vector.
And when I went over to sigma twiddle,
it was just a shift where I shifted myself
to the proper vacuum.
So I'm really doing the perturbation theory
in the linear here about the proper vacuum.
And all the other versions, I'm also
doing about that proper vacuum, the lowest energy vacuum.
So there was a step at the beginning, which
is the classic step for spontaneous symmetry
breaking, which is--
in some sense, I had to do that to get
started when I went from the sigma
to the sigma twiddle field.
But you should really think about it
as the sigma twiddle field as being
field in this linear case.
That's a sigma twiddle there.
Other questions?
So how would we do this construction of sigma
if we didn't know about this linear sigma model
way of getting there?
So we have a generator, G, which is
in L comma R. That's my notation for the combined SU2 left
cross SU2 right, for example.
And that's going to be broken down to something
that's in the vector sub-group, which I'll denote like this--
V comma V. And then an element of that we can call h.
And if we want to parameterize the coset,
you can think that you have a G left and a G right
in a kind of obvious notation.
And you want some kind of field to parameterize fluctuations,
where we pull out the h.
So I'll call that "cascade."
So let's just think about an example.
So say we had something that looked
like this, which is some set of generators that
is in the original L comma R. And let
me insert in here 1 in the form of gR,
gR dagger, and then write it as a product of generators
that looks like this.
So I just multiply the various-- this guy
multiplies that guy, this guy multiplies that guy.
So this guy here has the same entries in both,
so that's in h.
This is in the v comma v, which is h.
And this guy here, this gR dagger,
is then the thing that's parameterizing the cascade
field.
So this is a matrix that's parameterizing the coset,
and transforms in the way that we said.
So that's the idea behind what the sigma field is doing.
Now, you can ask, well, that just seems like one choice.
And I could think about parameterizing the coset
in many different ways.
And that's true.
And there's actually a nice formalism
that takes that into account.
So if you just talk about broken generators, which I can call X,
then a very general definition of what this cascade field is
is just an exponential involving those broken generators,
and some fields to describe the fluctuations,
and normalized in some way.
And this is due to Callan, Coleman, Wess and Zamino.
So it's the CCWZ prescription for parameterizing a coset.
And the point of their prescription
is that you have choices here for how you represent
the broken generators.
We start out with left generators
and right generators, and then you
go over to the vector generators,
but what ones are broken?
Well, you could take different linear combinations
that are broken.
And those would be equally viable choices.
Doesn't have to be a group, the broken generators,
so you can pick different choices of this X.
And if we pick X, which is the left generators,
that's actually what gives us what
we were talking about before.
Because then we end up parameterizing the cosets
by something as 1 in the second entry, because we
don't have the right.
We're just saying pick the left, the left are broken.
One possible choice, left plus right is unbroken.
Then we end up with an entry here, which is sigma.
That's our sigma.
And also, if you work out the transformation,
how the sigma transforms, you could reproduce--
you can also derive that sigma goes to L, sigma R dagger.
But that's just one choice.
You could do other choices.
So for example, you could pick X a to be tau left
a minus tau right a.
That's another possible choice.
And that would lead to a different field,
but an equally valid parameterization of the coset.
And this was actually also popular.
It's something that people usually denote by C.
I'm going to leave further discussion of that
to your reading.
There's a very nice review by Aneesh Manohar, where
he goes through some of these things for CCSW,
and does a nice job of talking about both of these in more
detail than I've done here.
So further discussion of this point will be in your reading.
You can also look back at the original paper, if you like.
But Aneesh has distilled it nicely.
So any questions about that?
There's a way of thinking--
I'll just keep talking until someone raises their--
there's a way of thinking about field redefinitions,
even from this low-energy point of view.
That's the key here.
And it has to do with the freedom that you have
and how you parameterize the coset.
So even if we're constructing it from the bottom up,
there's different representations.
In fact, Weinberg likes the square root version
of parameterizing the coset.
Maybe he's the only person.
He has very good reasons for being allowed
to stick with his original results,
rather than the exponential representation,
which everyone else seems to favor, including me.
So if we go back to talking about QCD,
then it's common to call this curly V f over root 2.
1/2 is a decay constant.
And so the conventions, then, are
the following-- at least one possible choice of conventions.
There's some choice about where you put the 2's, and then
everything else is fixed.
So this is one possible common choice
for the exponential representation.
And then that gives you the chiral Lagrangian--
looks like that.
And if you expand this out, you get the standard kinetic term
for the phi field.
That's what this normalization factor does.
And then if you keep expanding, you get some interaction terms
like this one, et cetera.
So there's four point interactions in that Lagrangian
as well.
And that's because we had a curved--
our coset is nontrivial.
So part of this story here when you talk about symmetry
breaking has to be about the fact
that symmetry is also broken explicitly
in the standard model.
And so you have to add--
the Goldstones are only pseudogoldstones.
You have to add a mass.
And the way you do that is by doing a spurion analysis.
Remind you how that goes.
You write a mass term like this, and to left and right.
The mass, you could say, is just the diagonal matrix
of up and down for SU2.
And you could pretend, rather than talking about things that
are violating the symmetry like these guys,
you let the mass transform and pretend
that you actually have something that's invariant.
By pretending that the mass transforms,
you're able-- if you pretend the mass transforms
in the chiral theory-- of constructing
things that violate the symmetry in the same way.
You construct invariance in both theories with this kind
of transformation law.
And then when you go back and fix them, and say no, no,
it doesn't transform, it's really a fixed thing,
not a field that could transform-- it's fixed,
the number--
then you violate the symmetry explicitly in the same way
as up here.
It's a trick for how to build things that are
covariant in a certain form.
So you could add the variant operators
of the chiral Lagrangian.
This is a different v, some parameter,
different than the curly one we were talking about
earlier, which is now called f.
So this is the spurion story--
how to break symmetry explicitly in a different theory,
the same way.
And if you expand a quadratic order,
you get mass terms for the pions.
I won't go through that.
But one kind of an important thing about it
is that the mass is proportional to v times the quark
mass, so v0, so the parameter in the Lagrangian, linear
in the quark masses, quadratic in the Goldstone mass.
You can also a couple of currents.
And you can do a similar type of analysis
with spurion-type transformation analysis for the currents.
And I'm not going to go through this
in any gory detail, but just enough
to do what we need to do here.
So here's a left-handed fermion current, standard model.
Could be coupling to the W boson, for example.
And you could think of getting J by taking a functional
derivative with respect to a field that's left handed.
And then I can do the same kind of thing,
where I think about this L and let it transform, just like I
was letting the M transform.
So think about J as this capital J times minus L.
And if I then think about this L mu transforming,
I can build a chiral Lagrangian, and then
use this formula to construct the current
in the chiral Lagrangian.
So you couple a spurion current, L mu a tau a,
and you, in this case, in order to couple it,
you can let it transform like a left-handed gauge field.
And then you get something that's invariant.
So the transformation that will make it invariant
is to think of it as a left-hand gauge field.
So L mu goes to L, L mu, L dagger.
And then that gives you an invariant
in the original theory.
And then you build in a variant of the x theory.
And so if you do this, and you're
replacing, in your chiral Lagrangian,
your partial derivative by a covariate one,
it acts on the left because that's where
the sigma is transforming.
And so it's an easy way of building that in.
And you just get an i L times the sigma on the left.
So you take everywhere you have a derivative, you replace it
by this combination.
So we had two derivatives in our chiral Lagrangian.
Expand it out, and we can figure out
how to put it in the left-handed spurion.
So the spurion is just kind of a general way
of going from one to the other just
by tracking symmetry breaking.
And you'll get some more practice
with some of this on your problem set.
So on your problem set, what I've
asked you to do on problem set 2 is to do a one-loop calculation
in this chiral theory.
So you can really see that this is a quantum field
theory, an effective field theory,
not only a tree-level mnemonic theory, but really
a field theory in its own right where
you have to consider loops, contractions, Wick's theorem.
Everything is the same as a real quantum field theory should be.
It's an effective field theory of certain degrees of freedom.
So you do a one-loop calculation for the decay constant.
And there's two problems in the problem set.
The first one is fairly straightforward.
And then that's the second problem.
That's all I'm asking you to do because it's fairly involved.
It's like maybe two normal problems.
So here's our chiral Lagrangian.
And we're going to count in this chiral Lagrangian,
in terms of power counting, partial squared as being order
v0 ab q.
That means we're going to count p squared
as being order M pi squared.
That's going to be our accounting.
And so we're going to expand in these things as small
and something else that is large.
And we'll figure out what the large thing is in a minute.
So we do the expansion like that, where something
is downstairs that has mass dimensions and the things that
are upstairs are p squared and then Pi squared.
So the type of expansion we have here
is a combination of a derivative expansion, although it's
a bit of a different derivative expansion than before,
and an M q expansion, simultaneously driven
expansion in these two things.
If we look at L0, then it has things in it
like a propagator--
dashed line for the scalar plan propagator.
It has four point interactions.
And if you look at how those scale,
we could write out what the Feynman rule is,
but if you look at how they scale,
they go like p squared over f squared, or M Pi squared
over f squared.
The p squared you could put on shell,
and it would be M Pi over f squared.
And you will actually need this Lagrangian
and this Feynman rule for your homework,
so let me tell you what it is.
And I'll leave it to you to work out the Feynman rule.
So commutators, there's a factor of 6.
And then there's also a term that
would give four-point scattering from the M2 term
in the Lagrangian.
So that also gives an M Pi squared over f squared term,
and that looks like that.
So you could build up a higher point function,
six-point function, et cetera from
the leading-order Lagrangian that has all this in it.
It has two parameters, f and v0.
So what about loops?
And what about this?
What is this scale lambda chi?
So the type of loops that you can have if you have
a four-point interaction, and just draw a solid line since
it's easier than drawing dashed lines--
think of all these as dashed.
So there's cross diagrams as well as the original diagram.
This is not the calculation I'm asking
you to do in the homework.
If we thought about our example of Pi plus Pi 0 scattering,
then we could contract up Pi plus and Pi 0 fields,
and we could have a loop that looks like that.
And in order for our theory to be unitary,
we have to think that these loops make sense,
because there's cutting rules, and all that story
should go over for this quantum field theory.
And it does.
And that means that if you like the imaginary part of this loop
is needed, and you can't get the imaginary part
without the real part, so we have
to also think that the real part is
a physically meaningful thing.
And so we can go ahead and compute it,
and figure out what's going on.
So there's derivative couplings that go like f squared, 1
over p squared over f squared.
L is the loop momenta.
Primes are the final.
Unprimes are the initial.
You can parameterize it however you like.
I'm using here dimensional regularization.
I like to dimensional regularization, and so do you.
Dimensional regularization here preserves chiral symmetry,
which is something that we want to preserve.
And I'll tell you a little later what
would have happened if we'd done this with a cut-off.
So if you do this calculation, just calculate those guys,
you get terms that go like p to the 4th.
You get terms that go like p squared, M Pi squared,
terms that go like M Pi to the 4th.
You get a 4 Pi, because it's a loop,
and then you get an f to the 4th,
because there's four factors of f.
So what this behaves like, it's p squared
or M pi squared over f squared, which
was our tree-level, four-point function, times something
that's causing suppression, which is p squared
or M Pi squared over 4 Pi f squared.
So the loop is actually down by a factor of p
squared over 4 Pi f, all squared.
So just by explicit calculation and not breaking the symmetry,
we see that loops are suppressed by p squared over lambda chi
squared, where lambda chi here is defined to be 4 Pi f.
So the 4 Pi is important because f is like 130 GeV,
and you need this 4 Pi in order to have some range
for your chiral expansion.
So if you plug in numbers, depending
on your conventions for f-- and there's
a couple of common conventions.
You could get numbers that are like 1.6 or 1.2 GeV, so
some scale of order a GeV.
On general grounds, you could also
ask the question, if I just think about it
from the effective theory, what is
a reasonable scale for the denominator,
for this lambda chi?
Remember, you're going to have to construct higher order
operators as well.
And so you need to figure out what's
the right scale for the suppression of those higher
order operators.
And if you did it on physical grounds,
than what you'd expect the lambda chi to be
is about the mass of the rho.
Because that's the lowest energy hadron that you've
left out of this chiral theory.
You might expect lambda chi's of order M
rho, which is sort of 770 MeV.
And so you can't really say whether it's 770 or 1.2,
because the difference between those,
although important numerically, is not
within the realm of the kind of factor 2-ish meaning
of these twiddle symbols.
So these twiddle symbols mean "parametrically of order."
Then there's two symbols-- there's
much greater than or twiddle.
"Twiddle" means factor of 2-ish.
"Much greater" means parametrically [INAUDIBLE]
or much less than.
And just with those two symbols, there's some freedom.
These are different ways of doing an estimate for what
lambda chi are, just saying that there is that freedom,
and you have to do the full calculation
in a particular channel to really figure out--
or a particular piece of phenomenology--
what the lambda is.
Extract it from data if you like.
So that is likely a good place to stop.
And next time, we'll talk a little bit more
about these loops.
And we'll talk about what happens
to these loops in the power counting theorem that you have,
that organizes all diagrams in this theory,
and tells you what you have to do if you want to construct
the Lagrangian order by order--
how you deal with when to put in the loops,
when to put in the counter terms.
We'll talk about those things next time.