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IAIN STEWART: So
last time, we were
00:00:24.140 --> 00:00:27.530
talking about the chiral
Lagrangian as another example
00:00:27.530 --> 00:00:29.368
of an effective field
theory to illustrate
00:00:29.368 --> 00:00:31.160
some of the tools of
effective field theory
00:00:31.160 --> 00:00:32.660
that we didn't meet
when integrating
00:00:32.660 --> 00:00:33.955
our heavy particles.
00:00:33.955 --> 00:00:36.820
It was an example of a bottom-up
effective field theory.
00:00:36.820 --> 00:00:38.570
So we were constructing
it from the bottom
00:00:38.570 --> 00:00:42.320
up for the most part.
00:00:42.320 --> 00:00:44.810
That's how one should
think about it.
00:00:44.810 --> 00:00:46.563
There was several
things in our program
00:00:46.563 --> 00:00:48.480
to understand from chiral
perturbation theory.
00:00:48.480 --> 00:00:50.240
And we kind of got
halfway through.
00:00:50.240 --> 00:00:52.160
So we want to understand
nonlinear symmetry
00:00:52.160 --> 00:00:53.065
representations.
00:00:53.065 --> 00:00:54.440
And we went through
how you could
00:00:54.440 --> 00:00:56.630
think about linear
representations
00:00:56.630 --> 00:00:59.780
and making changes of
variable and putting
00:00:59.780 --> 00:01:02.000
the Lagrangian in a form
where you could see where
00:01:02.000 --> 00:01:05.910
the nonlinear realization was
coming from very explicitly,
00:01:05.910 --> 00:01:08.540
as well as another way
where we constructed it
00:01:08.540 --> 00:01:12.800
from the bottom up thinking
about parameterizing
00:01:12.800 --> 00:01:14.720
the co-set.
00:01:14.720 --> 00:01:16.650
And then we started
talking about loops
00:01:16.650 --> 00:01:17.900
in chiral perturbation theory.
00:01:17.900 --> 00:01:20.883
And I gave you this
example of this loop here.
00:01:20.883 --> 00:01:22.550
So we constructed the
chiral Lagrangian.
00:01:22.550 --> 00:01:24.260
That's what I've written here.
00:01:24.260 --> 00:01:25.640
In terms of the
sigma field, this
00:01:25.640 --> 00:01:30.110
is the nonlinear version, sigma
field, exponential of M field.
00:01:30.110 --> 00:01:32.690
M field has the
pi n fields in it.
00:01:32.690 --> 00:01:35.360
I rewrote it in a little bit
of a different notation, which
00:01:35.360 --> 00:01:37.550
will be useful today,
where I introduced
00:01:37.550 --> 00:01:39.200
this thing called chi.
00:01:39.200 --> 00:01:41.840
But f squared over
8 chi is just Mq.
00:01:41.840 --> 00:01:44.042
So it's just a rescaling.
00:01:44.042 --> 00:01:46.250
And I just wanted to have
the same kind of pre-factor
00:01:46.250 --> 00:01:47.750
here and here.
00:01:47.750 --> 00:01:49.430
That's why I did that.
00:01:49.430 --> 00:01:54.260
The power counting is in p
squared derivatives or M pi
00:01:54.260 --> 00:01:56.480
squared, which is
the same as Mq.
00:01:56.480 --> 00:01:59.690
And the reason that M pi
squared and Mq count the same
00:01:59.690 --> 00:02:03.260
is there's this relation
between pi squared and the quark
00:02:03.260 --> 00:02:04.910
masses.
00:02:04.910 --> 00:02:06.890
And then we started
talking about loops.
00:02:06.890 --> 00:02:09.710
And we said that, if
we looked at this loop,
00:02:09.710 --> 00:02:12.350
the result for this loop diagram
would be a factor of p squared
00:02:12.350 --> 00:02:14.142
or M pi squared times
a factor of p squared
00:02:14.142 --> 00:02:17.000
or M pi squared
divided by 4 pi f.
00:02:24.140 --> 00:02:26.450
This is the same size
as the lowest order,
00:02:26.450 --> 00:02:28.190
so the suppression
factor is this.
00:02:45.755 --> 00:02:47.380
So we can say that
loops are suppressed
00:02:47.380 --> 00:02:49.600
by p squared over some
scale lambda chi squared.
00:02:49.600 --> 00:02:51.760
So this is scale
of our expansion.
00:02:51.760 --> 00:02:54.460
And we can associate that
scale to 4 pi times f.
00:02:57.590 --> 00:02:59.870
OK, so we're going to
continue with that story
00:02:59.870 --> 00:03:03.970
unless there's any
questions from last time.
00:03:03.970 --> 00:03:08.920
OK, so we want to use
dimensional regularization.
00:03:08.920 --> 00:03:12.670
And if we use dimensional
regularization,
00:03:12.670 --> 00:03:14.710
we can think about
having an MS bar scheme.
00:03:17.620 --> 00:03:19.890
So how do we do that?
00:03:19.890 --> 00:03:21.640
It's a little different
than gauge theory,
00:03:21.640 --> 00:03:23.810
but the idea is
exactly the same.
00:03:23.810 --> 00:03:25.540
So we look at the
dimensions of objects.
00:03:25.540 --> 00:03:30.910
This capital M-- supposed
to be a capital M--
00:03:30.910 --> 00:03:36.310
is like a scalar field, so it
has dimension 1 minus epsilon.
00:03:36.310 --> 00:03:39.520
The decay constant
then, if you look
00:03:39.520 --> 00:03:42.760
at it, which is the coupling,
has to have dimension
00:03:42.760 --> 00:03:43.953
1 minus epsilon 2.
00:03:43.953 --> 00:03:46.120
And you can see that because
this exponential better
00:03:46.120 --> 00:03:47.900
be the exponential of
something dimensionless
00:03:47.900 --> 00:03:49.442
so that, whatever
the dimension of M,
00:03:49.442 --> 00:03:54.220
should cancel the
dimension of f.
00:03:54.220 --> 00:03:56.740
And then that works out with
the look with the measure
00:03:56.740 --> 00:03:57.820
as well, which has a ddx.
00:04:01.180 --> 00:04:07.880
So therefore, just like you
did for a gauge coupling,
00:04:07.880 --> 00:04:11.590
you can say f bare in some mu
to the minus epsilon times f.
00:04:24.770 --> 00:04:27.830
And if you want to do the same
thing for the second term,
00:04:27.830 --> 00:04:30.500
though I've put
the V0 inside the--
00:04:30.500 --> 00:04:33.200
sorry, there's something
missing in my equation.
00:04:33.200 --> 00:04:36.300
There was a V0 here.
00:04:36.300 --> 00:04:38.300
So before last time, we've
written this equation
00:04:38.300 --> 00:04:40.790
as V0 out front
and then Mq inside.
00:04:40.790 --> 00:04:43.940
And I just rescaled
it to this chi.
00:04:43.940 --> 00:04:47.030
OK, so you can think
either in terms of chi.
00:04:47.030 --> 00:04:53.960
Chi is kind of absorbing the
parameter along with the Mq,
00:04:53.960 --> 00:04:55.400
but they're equivalent things.
00:04:55.400 --> 00:04:58.340
So you can think of also
doing a rescaling that
00:04:58.340 --> 00:05:00.410
gets the dimension
of this term right.
00:05:00.410 --> 00:05:03.980
And the thing that
you want is this.
00:05:03.980 --> 00:05:06.290
The P0 would then be mu
to the minus 2 epsilon
00:05:06.290 --> 00:05:08.960
to compensate for the fact that
the first term had an f squared
00:05:08.960 --> 00:05:10.402
and the second
term just had a V0.
00:05:13.097 --> 00:05:14.680
But that's just to
dimension counting.
00:05:32.820 --> 00:05:35.820
There's also no mus in physical
quantities like the pi n
00:05:35.820 --> 00:05:38.850
mass, which is an
observable, which
00:05:38.850 --> 00:05:41.520
has just a definite value.
00:05:41.520 --> 00:05:45.680
And there's also, in
chiral perturbation theory,
00:05:45.680 --> 00:05:48.360
no mus in the quark mass.
00:05:48.360 --> 00:05:51.780
So when you look at
V0 over f squared,
00:05:51.780 --> 00:05:58.491
if you look at it bare, that's
the same as renormalized.
00:06:02.340 --> 00:06:05.340
The other thing that's different
about this than in a gauge
00:06:05.340 --> 00:06:07.470
theory is, in the gauge
theory, you'd have a z.
00:06:07.470 --> 00:06:11.250
You'd have g bare is equal
to e to the epsilon times g,
00:06:11.250 --> 00:06:13.343
but then you'd have a zg.
00:06:13.343 --> 00:06:14.760
But in chiral
perturbation theory,
00:06:14.760 --> 00:06:16.800
the way that the
loops work, the loops
00:06:16.800 --> 00:06:20.940
aren't renormalizing the
leading order Lagrangian.
00:06:20.940 --> 00:06:23.070
The loops are renormalizing
something else
00:06:23.070 --> 00:06:25.560
because they ended
up being suppressed.
00:06:25.560 --> 00:06:29.265
So you don't need counter-terms
here for the leading order
00:06:29.265 --> 00:06:30.840
Lagrangian.
00:06:30.840 --> 00:06:33.510
And that's why I didn't
put a zf or a zV0 here.
00:06:37.680 --> 00:06:39.210
OK, but when you
do this loop, you
00:06:39.210 --> 00:06:41.377
do get ultraviolet divergences.
00:06:53.070 --> 00:06:54.758
So you get 1 over
epsilon divergence
00:06:54.758 --> 00:06:56.550
if you're using
dimensional regularization.
00:06:56.550 --> 00:06:59.460
And it comes along with
the log of mu squared.
00:06:59.460 --> 00:07:01.710
And then it'll be
divided by some scale
00:07:01.710 --> 00:07:05.400
that you have in your loop, like
an external momentum p squared,
00:07:05.400 --> 00:07:11.490
or it could be the pi n mass.
00:07:11.490 --> 00:07:13.862
Both of those will
show up generically.
00:07:24.960 --> 00:07:35.592
And in the way we're doing
counting, the way the loops are
00:07:35.592 --> 00:07:37.800
showing up, if I can bind
together the factors that I
00:07:37.800 --> 00:07:42.480
told you about here, it's either
p to the fourth, 4 powers of p,
00:07:42.480 --> 00:07:46.920
or 2 of p and 2 of M
pi or 4 powers of M pi.
00:07:46.920 --> 00:07:49.020
So to cancel that
1 over epsilon,
00:07:49.020 --> 00:07:50.542
we need a counter-term.
00:07:50.542 --> 00:07:52.500
But it's not a counter-term
in that Lagrangian.
00:07:52.500 --> 00:07:54.843
It's a counter term in the
higher order Lagrangian.
00:08:00.640 --> 00:08:03.010
So we're not done by just
calculating the loop.
00:08:03.010 --> 00:08:05.527
We have to actually include
some higher dimension
00:08:05.527 --> 00:08:07.860
operators that are the same
order in our power counting.
00:08:19.240 --> 00:08:21.130
So we'll talk about
SU(3) a little later.
00:08:21.130 --> 00:08:24.010
I'll start out by
talking about SU(2).
00:08:24.010 --> 00:08:28.240
In SU(2), the form of that
Lagrangian with higher order
00:08:28.240 --> 00:08:32.140
terms is just taking what
we had before and just
00:08:32.140 --> 00:08:33.991
going to a higher dimension.
00:08:37.090 --> 00:08:40.330
So we could have two
more derivatives.
00:08:40.330 --> 00:08:43.809
One way of doing that is taking
what we had at lowest order
00:08:43.809 --> 00:08:44.680
and squaring it.
00:08:51.880 --> 00:08:54.820
Another way of doing it
would be to take and contract
00:08:54.820 --> 00:08:56.420
the indices a little
bit differently.
00:08:56.420 --> 00:09:00.490
So I could do it like this.
00:09:03.520 --> 00:09:07.360
The trace is cyclic, so I can
move things around from back
00:09:07.360 --> 00:09:08.530
to front.
00:09:08.530 --> 00:09:10.330
But I could have
mu nu in one trace
00:09:10.330 --> 00:09:12.902
and mu nu in another trace
rather than just contracting
00:09:12.902 --> 00:09:13.610
within the trace.
00:09:13.610 --> 00:09:17.050
That's another possible term.
00:09:17.050 --> 00:09:18.550
And there's a bunch
more terms which
00:09:18.550 --> 00:09:22.570
I decided to wait and enumerate
when we do SU(3) to give you
00:09:22.570 --> 00:09:24.550
a full enumeration.
00:09:24.550 --> 00:09:32.800
These additional
terms you can build
00:09:32.800 --> 00:09:39.250
from one Mq, which means one
of these chi fields or chi
00:09:39.250 --> 00:09:45.490
objects and two partials.
00:09:45.490 --> 00:09:54.340
That's one possibility or
two Mq's because each Mq
00:09:54.340 --> 00:09:56.222
counts like two derivatives.
00:09:56.222 --> 00:09:58.180
So there's some other
terms that we could build
00:09:58.180 --> 00:09:59.873
that are analogs of this term.
00:09:59.873 --> 00:10:01.540
You could square that
term, for example.
00:10:01.540 --> 00:10:04.420
That would be a possible
higher order term.
00:10:04.420 --> 00:10:07.360
And so these coefficients
of this Lagrangian
00:10:07.360 --> 00:10:10.300
are what we need to cancel
off the divergences.
00:10:10.300 --> 00:10:12.490
So the counter-terms
that we have
00:10:12.490 --> 00:10:14.200
that renormalize
these loop graphs
00:10:14.200 --> 00:10:15.862
come from this Lagrangian.
00:10:38.020 --> 00:10:42.220
So the theory is renormalizable
in a EFT sense, order
00:10:42.220 --> 00:10:44.800
by order in its power
counting expansion.
00:10:44.800 --> 00:10:46.740
When you go to order
p to the fourth,
00:10:46.740 --> 00:10:48.490
you have to consistently
put in everything
00:10:48.490 --> 00:10:50.110
that's order p to the fourth.
00:10:50.110 --> 00:10:53.110
That includes the loops,
which are p to the fourth,
00:10:53.110 --> 00:10:59.260
as well as new local
Lagrangian interactions.
00:10:59.260 --> 00:11:01.580
AUDIENCE: Is that a
choice to not renormalize
00:11:01.580 --> 00:11:03.370
the f bare and [? d ?] bare?
00:11:03.370 --> 00:11:05.620
IAIN STEWART: They just
don't get renormalized.
00:11:05.620 --> 00:11:07.870
AUDIENCE: But why couldn't
I just multiply by 1 plus--
00:11:07.870 --> 00:11:08.620
IAIN STEWART: Oh--
00:11:08.620 --> 00:11:12.072
AUDIENCE: --order p squared
by times delta [INAUDIBLE]..
00:11:12.072 --> 00:11:13.780
IAIN STEWART: Oh, you
want to try to make
00:11:13.780 --> 00:11:15.010
a different scheme for them?
00:11:15.010 --> 00:11:17.302
AUDIENCE: I'd screw up the
power counting [INAUDIBLE]..
00:11:19.930 --> 00:11:25.890
IAIN STEWART: So
effectively, you
00:11:25.890 --> 00:11:27.890
could screw up the power
counting by doing that.
00:11:27.890 --> 00:11:31.690
So let me tell you
what you can do.
00:11:31.690 --> 00:11:33.400
You could put a 2 here, right?
00:11:36.670 --> 00:11:38.530
But you don't really
have much more freedom
00:11:38.530 --> 00:11:40.270
than just multiplying
by a number.
00:11:40.270 --> 00:11:41.895
AUDIENCE: So I can't
multiply by 1 plus
00:11:41.895 --> 00:11:43.798
p squared over s squared.
00:11:43.798 --> 00:11:44.590
IAIN STEWART: Yeah.
00:11:44.590 --> 00:11:46.548
That would be a screwing
up the power counting.
00:11:46.548 --> 00:11:50.945
And effectively, you wouldn't
be multiplying by p squared.
00:11:50.945 --> 00:11:52.570
You'd be putting
derivatives in, right?
00:11:52.570 --> 00:11:53.860
Because there's no p.
00:11:53.860 --> 00:11:58.893
So p is not something that
you're allowed to multiply by.
00:11:58.893 --> 00:12:00.310
AUDIENCE: So that's
the only thing
00:12:00.310 --> 00:12:02.680
that's stopping me-- is what
is derivatives [INAUDIBLE]??
00:12:02.680 --> 00:12:03.850
IAIN STEWART: You could
put derivatives in,
00:12:03.850 --> 00:12:05.440
but then that's equivalent
to something here.
00:12:05.440 --> 00:12:08.160
So you have to ask what you're
doing because you're mixing up
00:12:08.160 --> 00:12:08.660
things.
00:12:08.660 --> 00:12:09.310
AUDIENCE: OK.
00:12:09.310 --> 00:12:09.720
IAIN STEWART: Yeah.
00:12:09.720 --> 00:12:11.030
AUDIENCE: [INAUDIBLE]
definitely a better way to do.
00:12:11.030 --> 00:12:13.180
I was just wondering if
there was sort of freedom.
00:12:13.180 --> 00:12:15.097
IAIN STEWART: Yeah,
there's no freedom really.
00:12:15.097 --> 00:12:15.890
AUDIENCE: Good.
00:12:15.890 --> 00:12:18.432
AUDIENCE: Why you don't get the
renormalization [INAUDIBLE],,
00:12:18.432 --> 00:12:21.312
like from [INAUDIBLE]
[? loop? ?]
00:12:21.312 --> 00:12:23.770
IAIN STEWART: Because all the
loops end up being suppressed
00:12:23.770 --> 00:12:27.980
by p squared or M pi squared.
00:12:27.980 --> 00:12:33.220
So if you write down any diagram
and you think it might give
00:12:33.220 --> 00:12:38.210
something here, even if
it's a two-point function,
00:12:38.210 --> 00:12:40.825
it just doesn't.
00:12:40.825 --> 00:12:43.870
AUDIENCE: [INAUDIBLE]
to 4 [INAUDIBLE]..
00:12:43.870 --> 00:12:45.560
IAIN STEWART: That's right.
00:12:45.560 --> 00:12:48.940
Oh, so if you want, you could
think that some terms here
00:12:48.940 --> 00:12:51.670
are kind of corrections
to the kinetic term
00:12:51.670 --> 00:12:54.200
if you derive the
equation of motion
00:12:54.200 --> 00:13:03.730
and you include L and that L
and this L. But this Lagrangian,
00:13:03.730 --> 00:13:06.130
there's no loop corrections
to this Lagrangian.
00:13:06.130 --> 00:13:09.820
There's no corrections to
this lowest order Lagrangian.
00:13:09.820 --> 00:13:12.460
And that's because the
loops are all suppressed.
00:13:12.460 --> 00:13:13.960
Totally different
then gauge theory,
00:13:13.960 --> 00:13:16.310
loops are suppressed by p
squared or M pi squared.
00:13:16.310 --> 00:13:18.460
So they only renormalize
some high order things.
00:13:18.460 --> 00:13:24.670
And we don't have to worry
about really thinking about f.
00:13:24.670 --> 00:13:27.850
I mean, the way I've
talked about f here
00:13:27.850 --> 00:13:29.650
and the factors of
mu is just in order
00:13:29.650 --> 00:13:32.530
to see where these factors
and mu come out here.
00:13:32.530 --> 00:13:38.702
But what actually happens is
that you get 1 over epsilons
00:13:38.702 --> 00:13:40.660
that are cancelled like
the counter-terms here.
00:13:40.660 --> 00:13:45.700
And there's no need from a
renormalization perspective
00:13:45.700 --> 00:13:47.655
to sort of change
your definitions here.
00:13:47.655 --> 00:13:49.280
People play with
different definitions,
00:13:49.280 --> 00:13:50.905
but it's like they
use f squared over 4
00:13:50.905 --> 00:13:52.420
instead of f squared over 8.
00:13:52.420 --> 00:13:54.700
That's the extent
of what people do
00:13:54.700 --> 00:13:56.965
with playing with the
leading order Lagrangian.
00:13:56.965 --> 00:13:58.420
AUDIENCE: So you have
[INAUDIBLE] the same thing with
00:13:58.420 --> 00:13:59.830
the [INAUDIBLE] that
[INAUDIBLE] [? is not ?]
00:13:59.830 --> 00:14:01.212
[? renormalizing ?] [INAUDIBLE].
00:14:01.212 --> 00:14:02.920
IAIN STEWART: Both of
them are not, yeah.
00:14:02.920 --> 00:14:03.970
Yeah.
00:14:03.970 --> 00:14:07.270
So loops that you build
out of these interactions
00:14:07.270 --> 00:14:16.418
end up renormalizing these
terms, which is kind of neat.
00:14:16.418 --> 00:14:19.830
AUDIENCE: That's just
because [INAUDIBLE]..
00:14:19.830 --> 00:14:22.746
That's just because there's
[INAUDIBLE] [? sort of ?]
00:14:22.746 --> 00:14:24.210
[INAUDIBLE].
00:14:24.210 --> 00:14:25.840
IAIN STEWART: Yeah.
00:14:25.840 --> 00:14:29.160
Well, so we'll prove in a
minute a general power counting
00:14:29.160 --> 00:14:32.570
formula that tells us
how to organize all this.
00:14:32.570 --> 00:14:34.920
So for now, you can think
of it as an observation,
00:14:34.920 --> 00:14:38.970
but we'll build it into a
general formula that tells you
00:14:38.970 --> 00:14:41.573
sort of how you would
organize the power counting
00:14:41.573 --> 00:14:42.990
and renormalization
of this theory
00:14:42.990 --> 00:14:46.810
to all orders in its
expansion in a minute.
00:14:46.810 --> 00:14:49.060
AUDIENCE: So when you write
down all these [INAUDIBLE]
00:14:49.060 --> 00:14:51.680
at [? power ?] [? p4, ?]
does that include all
00:14:51.680 --> 00:14:52.800
the [INAUDIBLE] you need?
00:14:52.800 --> 00:14:53.610
IAIN STEWART: Yeah.
00:14:53.610 --> 00:14:55.530
That's right.
00:14:55.530 --> 00:14:57.270
Once I've enumerated
all the terms,
00:14:57.270 --> 00:14:59.310
which I will do for
you with SU(3)--
00:14:59.310 --> 00:15:02.580
so it includes things like this
and things like this as well--
00:15:02.580 --> 00:15:04.710
then those are all the
possible counter-terms
00:15:04.710 --> 00:15:07.278
that you could actually
need to renormalize
00:15:07.278 --> 00:15:08.820
any of the one-loop
diagrams that you
00:15:08.820 --> 00:15:12.420
could generate with the
leading order Lagrangian.
00:15:12.420 --> 00:15:15.067
And you know that
by power counting.
00:15:15.067 --> 00:15:16.650
Once you know the
loops are that order
00:15:16.650 --> 00:15:19.830
and you've included all local
interactions, then you're done.
00:15:23.160 --> 00:15:28.860
So there's things I didn't
write down there, so just
00:15:28.860 --> 00:15:30.930
comment about that.
00:15:30.930 --> 00:15:34.747
So what is the equation
of motion here?
00:15:34.747 --> 00:15:37.080
It's a little more complicated
because we have the sigma
00:15:37.080 --> 00:15:40.680
field, but you can work out
that the equation of motion
00:15:40.680 --> 00:15:41.460
is the following.
00:15:57.580 --> 00:16:01.440
And I've given you
some reading, so you
00:16:01.440 --> 00:16:06.000
can read about a derivation
of this equation which
00:16:06.000 --> 00:16:07.590
takes a little effort.
00:16:07.590 --> 00:16:09.330
So your leading order
equation of motion
00:16:09.330 --> 00:16:12.120
basically allows you to
get rid of partial squareds
00:16:12.120 --> 00:16:13.590
on the sigma.
00:16:13.590 --> 00:16:16.350
And that's why
I've always written
00:16:16.350 --> 00:16:18.887
one partial on each sigma.
00:16:18.887 --> 00:16:20.970
So that's what you should
think of this guy doing.
00:16:29.860 --> 00:16:31.500
There's also some
SU(2) identities
00:16:31.500 --> 00:16:37.380
that have been used, even
in what I've written,
00:16:37.380 --> 00:16:40.230
because you could think of
having other types of traces
00:16:40.230 --> 00:16:42.300
like this one, for example.
00:16:42.300 --> 00:16:46.925
Instead of having trace squared,
I could have the following.
00:16:46.925 --> 00:16:48.300
Just put everything
in one trace.
00:16:55.110 --> 00:16:57.840
I always have to alternate
sigma sigma dagger,
00:16:57.840 --> 00:17:04.170
sigma sigma dagger because del
mu sigma sigma dagger is 0.
00:17:04.170 --> 00:17:06.990
Because sigma sigma dagger is 1.
00:17:06.990 --> 00:17:11.160
And that provides an identity
that I'm also able to use.
00:17:11.160 --> 00:17:13.035
And so I want to
have sigmas and sigma
00:17:13.035 --> 00:17:14.160
daggers next to each other.
00:17:18.733 --> 00:17:21.150
Sorry, chiral symmetry means
that sigmas and sigma daggers
00:17:21.150 --> 00:17:23.290
should be next to each other.
00:17:23.290 --> 00:17:26.290
This is an identity that I can
also use to simplify things.
00:17:26.290 --> 00:17:28.380
So that allows me to
move things around.
00:17:28.380 --> 00:17:31.230
And this operator,
though, in SU(2)
00:17:31.230 --> 00:17:33.330
is actually related
to these two.
00:17:33.330 --> 00:17:34.740
It's not unique.
00:17:34.740 --> 00:17:37.500
Because of the structure
of Pauli matrices,
00:17:37.500 --> 00:17:40.500
you have a formula
which you can work out,
00:17:40.500 --> 00:17:43.350
which says that
this guy is actually
00:17:43.350 --> 00:17:44.865
half of the trace squared.
00:17:51.240 --> 00:17:54.240
And there's another
one if you did.
00:18:03.550 --> 00:18:09.615
This guy, he also can be related
to those two guys over there.
00:18:09.615 --> 00:18:10.990
So there's a bunch
of things that
00:18:10.990 --> 00:18:13.780
went into even writing down
the level of information
00:18:13.780 --> 00:18:14.860
that I told you.
00:18:14.860 --> 00:18:16.900
You could think
there's more things,
00:18:16.900 --> 00:18:19.130
but then, when you enumerate
all the possible things
00:18:19.130 --> 00:18:21.130
that you can do, you find
that those more things
00:18:21.130 --> 00:18:23.030
are related to these things.
00:18:23.030 --> 00:18:25.030
So you really want to
construct a minimal basis.
00:18:25.030 --> 00:18:27.030
And there's some things
that go into doing that.
00:18:35.280 --> 00:18:41.060
So at order p to the fourth, as
I said, we include both loops.
00:18:41.060 --> 00:18:45.170
And those loops will have
terms like p to the fourth log
00:18:45.170 --> 00:18:46.370
mu squared over p squared.
00:18:52.370 --> 00:18:55.970
And we include terms
like the L1 and L2,
00:18:55.970 --> 00:19:02.150
which are p of the
fourth type interactions.
00:19:02.150 --> 00:19:04.010
Once I take out
the counter-term,
00:19:04.010 --> 00:19:06.520
the renormalized coupling
is mu dependent just
00:19:06.520 --> 00:19:07.520
like the gauge coupling.
00:19:14.075 --> 00:19:15.950
This just comes from a
four-point interaction
00:19:15.950 --> 00:19:17.780
with the Li.
00:19:17.780 --> 00:19:20.960
And the mu dependence,
by construction,
00:19:20.960 --> 00:19:22.640
was canceled between
these two things.
00:19:25.310 --> 00:19:28.400
The divergence from the loop is
cancelled by the counter-term.
00:19:28.400 --> 00:19:30.500
And correspondingly, the
corresponding statement,
00:19:30.500 --> 00:19:32.083
if you want to make
it in terms of mu,
00:19:32.083 --> 00:19:35.600
is that the mu dependence that
is tightly tied to that 1 over
00:19:35.600 --> 00:19:39.065
epsilon is cancelled in the
renormalized quantities.
00:19:50.172 --> 00:19:51.630
So there's these
two contributions.
00:19:51.630 --> 00:19:53.070
They're both mu dependent.
00:19:53.070 --> 00:19:55.853
And that mu dependence cancels.
00:19:55.853 --> 00:19:57.520
So the way that you
should think of this
00:19:57.520 --> 00:20:01.190
is you should think
mu is a cut off.
00:20:01.190 --> 00:20:03.550
It's not a hard cut off,
but it is a cut off.
00:20:08.350 --> 00:20:10.455
And what the cut off
is doing is dividing up
00:20:10.455 --> 00:20:11.830
infrared and
ultraviolet physics.
00:20:16.210 --> 00:20:18.610
In this case, the
low energy physics
00:20:18.610 --> 00:20:26.050
is in the matrix
elements in the loops
00:20:26.050 --> 00:20:28.730
where we have our propagating
low energy degrees of freedom,
00:20:28.730 --> 00:20:31.060
which are the pions.
00:20:31.060 --> 00:20:34.645
And the high energy physics is
in the coefficients, as usual.
00:20:43.730 --> 00:20:44.980
And they're both mu dependent.
00:20:44.980 --> 00:20:47.270
And you can think of that
mu dependent as a cut off
00:20:47.270 --> 00:20:49.310
that divides up how
much of the physics
00:20:49.310 --> 00:20:52.010
goes into those low energy
loops, how much of the physics
00:20:52.010 --> 00:20:55.630
goes into the couplings.
00:20:55.630 --> 00:20:59.050
So the difference between
this and integrating
00:20:59.050 --> 00:21:01.030
on a massive particle
is not the physics
00:21:01.030 --> 00:21:03.550
of where things go because
the low energy physics always
00:21:03.550 --> 00:21:04.717
goes in the matrix elements.
00:21:04.717 --> 00:21:06.790
The high energy physics
goes in the couplings.
00:21:06.790 --> 00:21:10.638
The difference is that, here,
the way that you should think
00:21:10.638 --> 00:21:13.180
of it is that we can calculate
the matrix elements explicitly
00:21:13.180 --> 00:21:16.510
because our theory is in terms
of the right degrees of freedom
00:21:16.510 --> 00:21:21.910
to describe long distance
physics, which are the pions.
00:21:21.910 --> 00:21:24.010
And then the coefficients
are the unknowns.
00:21:24.010 --> 00:21:28.093
That's the way in
which it's different.
00:21:28.093 --> 00:21:29.760
And that's really
kind of the difference
00:21:29.760 --> 00:21:31.500
between bottom-up and top-down.
00:21:42.740 --> 00:21:47.500
So with that in
mind, if you want
00:21:47.500 --> 00:21:51.670
to think of what these
couplings are, if we just
00:21:51.670 --> 00:21:59.470
divide by f squared, then you
can think about dimensionally
00:21:59.470 --> 00:22:01.030
what they are.
00:22:01.030 --> 00:22:02.860
In order for them
to sort of match up
00:22:02.860 --> 00:22:05.260
with what we got
from the loops is
00:22:05.260 --> 00:22:10.800
that they have some dependence
that looks like this.
00:22:10.800 --> 00:22:12.400
You can think of
it as parameterized
00:22:12.400 --> 00:22:14.590
by some coefficients a and b.
00:22:14.590 --> 00:22:17.200
If I want to match up with the
4 pi f squared that comes from
00:22:17.200 --> 00:22:19.750
the loops, then I have to sort
of say that there's a 4 pi
00:22:19.750 --> 00:22:22.150
hiding inside the Li's.
00:22:22.150 --> 00:22:24.200
So I can do that.
00:22:24.200 --> 00:22:26.300
And then there's some
coefficients here.
00:22:26.300 --> 00:22:27.730
There's some mu dependence.
00:22:27.730 --> 00:22:29.770
And the scales that
are in the coefficients
00:22:29.770 --> 00:22:32.350
are the high scales, like
lambda chi and higher
00:22:32.350 --> 00:22:34.970
and rho, those types of things.
00:22:34.970 --> 00:22:36.940
So let me just
call it lambda chi.
00:22:36.940 --> 00:22:38.920
And then there's
some numbers here a
00:22:38.920 --> 00:22:44.650
and b, which encode
the high mass physics.
00:22:44.650 --> 00:22:47.560
Now, the point of thinking
about power counting
00:22:47.560 --> 00:22:49.570
is that thinking about
the fact that the loops
00:22:49.570 --> 00:22:51.610
and these coefficients
are the same size
00:22:51.610 --> 00:22:54.130
actually tells you that
these ai's and bi's should
00:22:54.130 --> 00:22:54.820
be order 1.
00:23:04.760 --> 00:23:07.310
And this actually goes under
the rubric of something called
00:23:07.310 --> 00:23:08.780
naive dimensional analysis.
00:23:21.900 --> 00:23:23.613
So what naive
dimensional analysis says
00:23:23.613 --> 00:23:26.030
is that this cut off that we've
put between the low energy
00:23:26.030 --> 00:23:28.310
physics and the high energy
physics is arbitrary.
00:23:28.310 --> 00:23:29.300
And we could change it.
00:23:29.300 --> 00:23:31.250
We could change it
by a factor of 2.
00:23:31.250 --> 00:23:35.070
And what changing it does is
it moves pieces back and forth.
00:23:35.070 --> 00:23:37.100
But if we could move
pieces back and forth,
00:23:37.100 --> 00:23:39.800
then you wouldn't expect that
the two things that you're
00:23:39.800 --> 00:23:43.130
talking about would be
different in magnitude
00:23:43.130 --> 00:23:45.760
because we're allowed to
move pieces back and forth.
00:23:45.760 --> 00:23:47.840
So you expect that the
size of contributions
00:23:47.840 --> 00:23:51.680
from the coefficients are about
the same size as the loops.
00:23:51.680 --> 00:23:58.320
And that's this naive
dimensional analysis.
00:23:58.320 --> 00:24:00.410
So changing mu moves
pieces back and forth.
00:24:05.450 --> 00:24:14.610
The sum is mu independent, but
each individual thing is not.
00:24:21.330 --> 00:24:26.130
And because we're able to
move things back and forth,
00:24:26.130 --> 00:24:28.860
we expect them to be the
same order of magnitude.
00:24:44.300 --> 00:24:50.030
And that is this statement
here, that the ai's and bi's,
00:24:50.030 --> 00:24:53.300
once I account for the 4 pis--
00:24:53.300 --> 00:24:56.202
so with this argument,
I can figure out
00:24:56.202 --> 00:24:58.160
where there's 4 pis hiding
in the coefficients.
00:24:58.160 --> 00:25:00.108
Because I can identify
4 pis in loops.
00:25:00.108 --> 00:25:02.150
And then if they're supposed
to be the same size,
00:25:02.150 --> 00:25:06.460
I can also identify 4
pis in coefficients.
00:25:06.460 --> 00:25:10.150
So if you had figured this out,
I don't know, 20 years ago,
00:25:10.150 --> 00:25:13.430
then you could have got a PhD
thesis like Aneesh Manohar did,
00:25:13.430 --> 00:25:14.710
which was 25 pages long.
00:25:20.540 --> 00:25:22.400
Short PhD theses do exist.
00:25:27.605 --> 00:25:28.105
OK.
00:25:37.160 --> 00:25:39.110
So what do we do in practice?
00:25:39.110 --> 00:25:41.210
In practice, we have
to pick a value of mu.
00:25:41.210 --> 00:25:43.680
Just like when we were
talking about gauge theory,
00:25:43.680 --> 00:25:44.930
we need to pick a value of mu.
00:25:44.930 --> 00:25:47.780
There we were picking
things like mu equals Mb.
00:25:47.780 --> 00:25:50.630
Here, what people do
is they typically pick
00:25:50.630 --> 00:25:52.700
mus that are high.
00:25:52.700 --> 00:25:54.710
So maybe they would
pick the rho mass.
00:25:54.710 --> 00:25:58.640
Maybe they would
pick lambda chi.
00:25:58.640 --> 00:26:02.390
Or they pick just something
in between, like a GeV.
00:26:02.390 --> 00:26:04.190
These are typical
values that people use.
00:26:06.710 --> 00:26:12.820
So what that means is that
you've removed all large logs
00:26:12.820 --> 00:26:13.930
from your coefficients.
00:26:13.930 --> 00:26:15.930
And that's because you
want dimensional analysis
00:26:15.930 --> 00:26:18.830
to hold for your coefficients
because you don't know them.
00:26:18.830 --> 00:26:20.740
So you would like to
have some power counting
00:26:20.740 --> 00:26:21.430
estimate for them.
00:26:21.430 --> 00:26:22.990
Then you'd like to go
out and fit them to data.
00:26:22.990 --> 00:26:25.115
And you'd like the result
that you get from fitting
00:26:25.115 --> 00:26:27.880
to the data to agree with
your power counting estimate,
00:26:27.880 --> 00:26:29.290
so that you're happy.
00:26:29.290 --> 00:26:33.460
It turns out that it
does, so you are happy.
00:26:33.460 --> 00:26:37.300
And so to avoid large
logs in the story,
00:26:37.300 --> 00:26:39.310
you put the large logs
into the matrix elements.
00:26:54.655 --> 00:26:57.030
So that's different than our
story in gauge theory, where
00:26:57.030 --> 00:26:59.613
we were thinking that we would
re-sum all the large logarithms
00:26:59.613 --> 00:27:00.780
in the coefficients.
00:27:00.780 --> 00:27:02.490
And then our matrix
elements would also
00:27:02.490 --> 00:27:03.750
have no large logarithms.
00:27:03.750 --> 00:27:05.250
Here, we're just
saying, well, let's
00:27:05.250 --> 00:27:07.800
allow for large logs
in the matrix element.
00:27:07.800 --> 00:27:09.992
But the story is also
different from gauge theory
00:27:09.992 --> 00:27:11.700
in another way, which
is that there's not
00:27:11.700 --> 00:27:14.810
an infinite series here
of large logarithms
00:27:14.810 --> 00:27:15.810
that you need to re-sum.
00:27:31.190 --> 00:27:33.710
And that's related to the fact
that the kinetic term didn't
00:27:33.710 --> 00:27:35.237
get renormalized.
00:27:35.237 --> 00:27:37.070
There's no mu dependence
in the coefficients
00:27:37.070 --> 00:27:37.960
of the kinetic term.
00:27:37.960 --> 00:27:41.030
So when the loop
graphs aren't having
00:27:41.030 --> 00:27:44.450
coefficient squared depending
on mu, that doesn't happen.
00:27:44.450 --> 00:27:47.300
We simply have one log
in our loop graphs,
00:27:47.300 --> 00:27:50.060
one log from our
counter-term diagrams.
00:27:50.060 --> 00:27:51.840
They explicitly cancel.
00:27:51.840 --> 00:27:53.060
There's no high order terms.
00:27:53.060 --> 00:27:55.268
The renormalization group
here is completely trivial.
00:27:55.268 --> 00:27:57.900
If you integrate,
you just get one log.
00:27:57.900 --> 00:28:01.900
So there's not really even
a reason to talk about it.
00:28:01.900 --> 00:28:04.790
OK, so there's certainly some
differences in this theory
00:28:04.790 --> 00:28:06.200
than there are in gauge theory.
00:28:29.370 --> 00:28:31.650
And so typically, the
paradigm that you have is you
00:28:31.650 --> 00:28:33.030
can calculate matrix elements.
00:28:33.030 --> 00:28:34.830
You're going to fit
the coefficients.
00:28:34.830 --> 00:28:36.468
You think of enough
experiments such
00:28:36.468 --> 00:28:38.010
that you can get
all that information
00:28:38.010 --> 00:28:39.052
about those coefficients.
00:28:39.052 --> 00:28:40.980
And then you can think
of other experiments
00:28:40.980 --> 00:28:43.833
and make predictions.
00:28:43.833 --> 00:28:45.750
Now, as you go to higher
orders in the theory,
00:28:45.750 --> 00:28:47.292
you get more
coefficients because you
00:28:47.292 --> 00:28:48.780
keep having to go
to higher orders
00:28:48.780 --> 00:28:51.630
and construct higher
dimension operators.
00:28:51.630 --> 00:28:54.480
And so the paradigm of figuring
out how to fit the coefficients
00:28:54.480 --> 00:28:55.800
gets harder and harder.
00:28:55.800 --> 00:28:57.240
And at some point,
you effectively
00:28:57.240 --> 00:28:58.410
lose predictive
power because you
00:28:58.410 --> 00:28:59.993
can't think of enough
observables that
00:28:59.993 --> 00:29:04.110
can be measured in order to
fit all your coefficients.
00:29:04.110 --> 00:29:07.410
But certainly, at
order p to the fourth,
00:29:07.410 --> 00:29:09.502
people can think of
many more observables.
00:29:09.502 --> 00:29:11.085
And actually, people
have worked out p
00:29:11.085 --> 00:29:13.470
to the sixth as well here.
00:29:19.217 --> 00:29:20.800
So two-loop chiral
perturbation theory
00:29:20.800 --> 00:29:22.133
is kind of the state of the art.
00:29:26.933 --> 00:29:28.100
AUDIENCE: I have a question.
00:29:28.100 --> 00:29:30.500
IAIN STEWART: Yeah.
00:29:30.500 --> 00:29:33.464
AUDIENCE: So why allow log in
this case [? is not ?] screwing
00:29:33.464 --> 00:29:35.630
up the [INAUDIBLE]?
00:29:35.630 --> 00:29:38.123
IAIN STEWART: Yeah.
00:29:38.123 --> 00:29:40.290
It doesn't screw it up
because it only happens once.
00:29:44.390 --> 00:29:46.565
So basically, what you
could say is you can say,
00:29:46.565 --> 00:29:47.690
I have this matrix element.
00:29:47.690 --> 00:29:49.880
It's got a large log
and a coefficient.
00:29:49.880 --> 00:29:52.580
The coupling, it's not
like large log times
00:29:52.580 --> 00:29:55.190
the coupling is something that
you could count as order 1
00:29:55.190 --> 00:29:57.260
because that's not
going to reappear
00:29:57.260 --> 00:30:00.140
in the higher order when
you go to the higher order.
00:30:00.140 --> 00:30:02.030
So you still have a large log.
00:30:02.030 --> 00:30:04.460
And you are allowed to
say that large log is
00:30:04.460 --> 00:30:07.070
the most important piece
of my matrix element,
00:30:07.070 --> 00:30:10.130
but it doesn't repeat
itself in the higher orders.
00:30:10.130 --> 00:30:11.900
Because once you
go to higher loops,
00:30:11.900 --> 00:30:13.275
you're actually
getting something
00:30:13.275 --> 00:30:16.960
that's power suppressed,
not loop suppressed.
00:30:16.960 --> 00:30:18.710
So the loops are giving
power suppression.
00:30:18.710 --> 00:30:20.900
And that changes
a lot of things.
00:30:28.575 --> 00:30:29.075
OK.
00:30:32.920 --> 00:30:35.050
What would happen if
we used a hard cut off?
00:30:39.880 --> 00:30:43.900
Maybe some of this story would
be a bit more transparent
00:30:43.900 --> 00:30:46.600
if we'd done that because we'd
see that we were explicitly
00:30:46.600 --> 00:30:49.450
dividing up low energy
and high energy physics.
00:30:49.450 --> 00:30:52.540
But there's a price to
pay for that transparency.
00:30:52.540 --> 00:30:55.450
And here, effectively
everybody decides
00:30:55.450 --> 00:30:56.720
it's too high a price to pay.
00:30:56.720 --> 00:30:59.680
And so they use dim reg.
00:30:59.680 --> 00:31:02.170
So what type of
prices would you pay?
00:31:02.170 --> 00:31:03.610
So if you did that,
this loop here
00:31:03.610 --> 00:31:05.027
would actually
have terms that are
00:31:05.027 --> 00:31:07.675
cut-off to the fourth over
lambda chi to the fourth.
00:31:07.675 --> 00:31:09.175
And that guy breaks
chiral symmetry.
00:31:12.190 --> 00:31:14.710
I mean, it's giving
a constant term.
00:31:14.710 --> 00:31:17.710
And you're not seeing
the derivative coupling.
00:31:17.710 --> 00:31:21.010
And that means, effectively,
that there's no counter-term
00:31:21.010 --> 00:31:23.260
to absorb this
dependence in our L chi
00:31:23.260 --> 00:31:26.690
because our L chi
respected chiral symmetry.
00:31:26.690 --> 00:31:30.190
So that's kind of bad.
00:31:30.190 --> 00:31:31.690
If you had a cut-off,
you'd also get
00:31:31.690 --> 00:31:34.750
terms that went cut-off
squared times momentum squared
00:31:34.750 --> 00:31:37.720
where two powers of the p
are replaced by cut-off.
00:31:37.720 --> 00:31:41.710
And that, in this language
that we used earlier,
00:31:41.710 --> 00:31:43.720
would break the power
counting in the sense
00:31:43.720 --> 00:31:46.060
that we have a
power counting that
00:31:46.060 --> 00:31:48.880
says that the loops should be
suppressed by d to the fourth.
00:31:48.880 --> 00:31:51.370
And it's broken by this.
00:31:51.370 --> 00:31:55.870
And then you would need to
absorb that and renormalize
00:31:55.870 --> 00:31:58.090
you're leading order coupling.
00:31:58.090 --> 00:31:59.810
But effectively, all
the renormalization
00:31:59.810 --> 00:32:01.810
would be doing is restoring
your power counting.
00:32:01.810 --> 00:32:03.940
It's not doing something
that you should interpret
00:32:03.940 --> 00:32:05.380
as really a physical thing.
00:32:08.760 --> 00:32:11.500
So it's just something
to be avoided.
00:32:11.500 --> 00:32:16.860
And then finally, the physics
that you actually want,
00:32:16.860 --> 00:32:23.830
which is the logs of the
cut-off, would also be there.
00:32:23.830 --> 00:32:33.520
And you would do the same
thing as you do in dim reg.
00:32:33.520 --> 00:32:38.530
You would absorb that in
higher dimension operators, OK?
00:32:38.530 --> 00:32:40.990
So we won't use a
cut-off because we
00:32:40.990 --> 00:32:42.640
don't want to think
about things that
00:32:42.640 --> 00:32:45.057
are breaking power counting
or messing up chiral symmetry.
00:32:52.510 --> 00:32:55.000
Another thing that's different
about chiral theories
00:32:55.000 --> 00:32:56.770
versus gauge theories
is the structure
00:32:56.770 --> 00:32:58.390
of infrared divergences.
00:32:58.390 --> 00:33:00.460
And that's because you
have derivative couplings.
00:33:08.480 --> 00:33:11.650
So you have many fewer
infrared singularities
00:33:11.650 --> 00:33:13.160
than you have in gauge theory.
00:33:13.160 --> 00:33:15.700
And in fact, you
usually don't have any.
00:33:19.360 --> 00:33:21.580
And one way of describing
that is that you usually
00:33:21.580 --> 00:33:24.940
have a good M pi
squared goes to 0
00:33:24.940 --> 00:33:28.570
or p squared goes to 0
limit of your results.
00:33:33.230 --> 00:33:36.760
So you can just explicitly
take these limits
00:33:36.760 --> 00:33:39.680
and talk about the
results in those limits.
00:33:39.680 --> 00:33:42.430
And if we look back at
what we were talking about,
00:33:42.430 --> 00:33:44.310
you get p to the
fourth log p squared.
00:33:44.310 --> 00:33:46.770
So there's a p to the fourth
multiplying the log p squared.
00:33:46.770 --> 00:33:48.520
So you're not seeing
log p squared blow up
00:33:48.520 --> 00:33:51.280
because it's got so many
powers of p multiplying it.
00:33:57.100 --> 00:34:00.040
Although our focus
is on formalism,
00:34:00.040 --> 00:34:03.040
I have to at least give you one
example of something predictive
00:34:03.040 --> 00:34:04.780
and phenomenological.
00:34:04.780 --> 00:34:12.219
So pi pi scattering is a nice
example of phenomenology.
00:34:12.219 --> 00:34:14.800
And it's particularly
nice if you look at it
00:34:14.800 --> 00:34:17.770
below an elastic threshold.
00:34:17.770 --> 00:34:19.769
So you just have
elastic scattering.
00:34:25.870 --> 00:34:28.795
So you could look at
pi pi goes to 4 pi.
00:34:28.795 --> 00:34:30.420
And that's something
you could actually
00:34:30.420 --> 00:34:32.760
look at in chiral
perturbation theory.
00:34:32.760 --> 00:34:37.440
But if we look at just
pi pi goes to pi pi
00:34:37.440 --> 00:34:40.739
where we have not enough
energy to produce 4 pions,
00:34:40.739 --> 00:34:44.500
then the scattering is
particularly simple.
00:34:44.500 --> 00:34:48.449
It's just described by
an S matrix, which we can
00:34:48.449 --> 00:34:50.969
enumerate channel by channel.
00:34:50.969 --> 00:35:04.680
And it's just a phase where
this L is a partial wave phase.
00:35:04.680 --> 00:35:05.700
And the i is an isospin.
00:35:08.820 --> 00:35:12.800
So for each isospin and for
each angular momentum state,
00:35:12.800 --> 00:35:13.800
we get different phases.
00:35:13.800 --> 00:35:17.190
But that's encoding
all the scattering.
00:35:17.190 --> 00:35:19.490
That's like doing
non-relativistic scattering
00:35:19.490 --> 00:35:24.230
or very simple quantum
mechanical scattering.
00:35:35.657 --> 00:35:37.740
And when you have an elastic
scattering like that,
00:35:37.740 --> 00:35:44.880
there's something called the
effective range expansion,
00:35:44.880 --> 00:35:47.875
which is a derivative expansion
for the phase shift delta.
00:35:50.540 --> 00:35:52.500
And if you do it for an
arbitrary partial wave,
00:35:52.500 --> 00:35:54.350
this is how it looks.
00:35:54.350 --> 00:35:56.060
So this is, again,
something that you
00:35:56.060 --> 00:36:00.410
would find in the discussion
of scattering theory in quantum
00:36:00.410 --> 00:36:01.418
mechanics.
00:36:04.700 --> 00:36:06.500
And you have a
derivative expansion
00:36:06.500 --> 00:36:09.860
of this quantity p to
the 2L plus 1 cotan
00:36:09.860 --> 00:36:13.070
of that phase shift.
00:36:13.070 --> 00:36:19.790
And the a's here and the
r0 depend on what channel
00:36:19.790 --> 00:36:22.880
you're talking about.
00:36:22.880 --> 00:36:26.150
And if we actually just
take the fact that we can--
00:36:26.150 --> 00:36:28.220
so this is a description
from quantum mechanics.
00:36:28.220 --> 00:36:29.990
This is true irrespective
of what theory
00:36:29.990 --> 00:36:32.750
you're talking about as
long as you don't have
00:36:32.750 --> 00:36:34.802
other channels you can produce.
00:36:34.802 --> 00:36:37.010
If you talk about it from
chiral perturbation theory,
00:36:37.010 --> 00:36:40.790
though, we can just calculate
pi pi goes to pi pi.
00:36:40.790 --> 00:36:46.520
And if you do that,
which [? Weinberg ?] did,
00:36:46.520 --> 00:36:49.175
then you just get results
for these coefficients.
00:36:53.708 --> 00:36:55.500
So I'll just quote a
couple of them to you.
00:37:12.200 --> 00:37:14.470
So you just get
parameter-free results,
00:37:14.470 --> 00:37:16.220
parameter-free in the
sense that they just
00:37:16.220 --> 00:37:18.560
involve M pi and this thing f.
00:37:18.560 --> 00:37:21.490
This thing f is actually
measured by pi on decay,
00:37:21.490 --> 00:37:23.390
so it's not an unknown.
00:37:23.390 --> 00:37:25.490
And then from that,
at lowest order
00:37:25.490 --> 00:37:27.260
in chiral perturbation
theory, you
00:37:27.260 --> 00:37:30.800
have no parameters once
you fixed M pi and f pi.
00:37:30.800 --> 00:37:32.450
And you can just
make predictions
00:37:32.450 --> 00:37:35.690
for these scattering lengths,
and those are the predictions.
00:37:35.690 --> 00:37:44.570
And they're parameter-free in
the sense that I just said.
00:37:47.520 --> 00:37:49.340
OK, so that gives
you some of the idea
00:37:49.340 --> 00:37:51.460
of what chiral perturbation
theory can do for you.
00:37:58.330 --> 00:38:01.968
OK, so let's talk about
back to formalism.
00:38:01.968 --> 00:38:03.760
There's many more
phenomenological examples
00:38:03.760 --> 00:38:06.880
you could do, but
that's not our focus.
00:38:06.880 --> 00:38:12.820
Back to formalism and
back to this general power
00:38:12.820 --> 00:38:25.020
counting discussion, so let's
consider an arbitrary diagram
00:38:25.020 --> 00:38:27.300
in this theory, this
chiral perturbation theory.
00:38:32.340 --> 00:38:34.470
And let's enumerate some
pieces of that diagram.
00:38:37.600 --> 00:38:40.550
So we'll say that it has
some number of vertices.
00:38:40.550 --> 00:38:42.360
We just count how
many times I've
00:38:42.360 --> 00:38:45.660
inserted vertices
from the Lagrangians,
00:38:45.660 --> 00:38:48.360
and I'll call that NV.
00:38:48.360 --> 00:38:49.755
Some number of internal lines--
00:38:55.410 --> 00:38:56.420
call NI.
00:38:56.420 --> 00:38:58.005
Some number of external lines--
00:39:02.730 --> 00:39:04.350
the external lines
are all pions.
00:39:08.700 --> 00:39:10.695
That's what our
theory is describing.
00:39:10.695 --> 00:39:14.920
It could be kaons and etas
if we're doing SU(3)--
00:39:14.920 --> 00:39:16.290
and. then some number of loops.
00:39:18.890 --> 00:39:22.510
So we have an integer associated
with each of these things.
00:39:22.510 --> 00:39:24.000
And when I talk
about vertices, I
00:39:24.000 --> 00:39:25.920
don't want to restrict myself
just to the leading order
00:39:25.920 --> 00:39:26.420
vertices.
00:39:26.420 --> 00:39:29.310
I want to talk about also
these Li's and higher order
00:39:29.310 --> 00:39:31.030
vertices as well.
00:39:31.030 --> 00:39:35.190
So let me have a notation for
that, where I take this integer
00:39:35.190 --> 00:39:37.560
NV, which counts all
vertices, and split it
00:39:37.560 --> 00:39:39.480
into pieces that count
the vertices at each
00:39:39.480 --> 00:39:42.600
of those different orders
in the expansion in p.
00:39:51.280 --> 00:39:54.100
So Nn is the number
of vertices that
00:39:54.100 --> 00:40:01.195
are order p to the n or M pi
to the n or combinations of p
00:40:01.195 --> 00:40:07.230
and M pi to the n, but
n of them, all right?
00:40:07.230 --> 00:40:08.790
So hopefully, that's clear.
00:40:08.790 --> 00:40:11.030
So if I have two insertions
of the leading order
00:40:11.030 --> 00:40:13.580
Lagrangian and one insertion
of the sub-leading,
00:40:13.580 --> 00:40:16.910
then n sub 0 would be 2.
00:40:16.910 --> 00:40:18.710
And N sub 1 would be 1.
00:40:18.710 --> 00:40:20.677
And the total would be 3.
00:40:20.677 --> 00:40:22.010
That's what this notation means.
00:40:40.890 --> 00:40:42.540
So we'll assume we're
using a regulator
00:40:42.540 --> 00:40:43.830
like dimensional regularization.
00:40:43.830 --> 00:40:45.900
So we don't have to worry
about the regulator messing up
00:40:45.900 --> 00:40:46.525
power counting.
00:40:46.525 --> 00:40:48.210
And basically,
that means we could
00:40:48.210 --> 00:40:52.000
ignore the regulator as far as
this discussion is concerned.
00:40:52.000 --> 00:40:52.860
And we just count.
00:40:57.570 --> 00:41:00.462
And effectively, we can
just count mass dimension.
00:41:05.190 --> 00:41:06.755
So we'll count
lambda chi factors.
00:41:09.700 --> 00:41:10.950
Let's think about it that way.
00:41:13.920 --> 00:41:21.450
And we'll think about counting
lambda chi factors for a matrix
00:41:21.450 --> 00:41:27.255
element that I'll call curly
M, which has NE external pions.
00:41:30.080 --> 00:41:32.550
So NE lines are
poking out of it.
00:41:38.740 --> 00:41:42.180
So then if we look
at the vertices,
00:41:42.180 --> 00:41:45.450
we can count how many factors
of lambda chi there is.
00:41:45.450 --> 00:41:47.010
And I'm counting
all dimensionful
00:41:47.010 --> 00:41:49.680
and turning all dimensionful
things into lambda chi.
00:41:55.320 --> 00:42:02.820
And so each different order in
N gets a number of lambda chis
00:42:02.820 --> 00:42:05.910
because we even saw that
already in the examples
00:42:05.910 --> 00:42:08.730
we treated where we got f
squared from the leading order.
00:42:08.730 --> 00:42:11.250
But from the
sub-leading, we got Li.
00:42:11.250 --> 00:42:14.110
And that was dimensionless.
00:42:14.110 --> 00:42:18.540
So for n equals 2, which
is the leading order here,
00:42:18.540 --> 00:42:20.200
leading order was p squared.
00:42:20.200 --> 00:42:23.840
I said that not quite
right a minute ago.
00:42:23.840 --> 00:42:27.100
So lowest order is p
squared n equals 2.
00:42:27.100 --> 00:42:28.020
That's L0.
00:42:30.650 --> 00:42:31.880
That had an f squared.
00:42:31.880 --> 00:42:35.960
And that comes out, if I just
have 4 minus 2, that's 2.
00:42:35.960 --> 00:42:42.250
n equals 4, that was giving our
Li's, which were dimensionless,
00:42:42.250 --> 00:42:42.750
OK?
00:42:42.750 --> 00:42:44.208
So you can see the
formula working.
00:42:44.208 --> 00:42:47.440
And if we went to higher orders
in the derivative and chiral
00:42:47.440 --> 00:42:49.690
expansion, then we'd
start getting lambda chis
00:42:49.690 --> 00:42:50.975
in the denominator.
00:42:53.650 --> 00:42:56.160
So this is just counting
from the vertices, which
00:42:56.160 --> 00:42:57.900
you should think
of as counting just
00:42:57.900 --> 00:43:00.660
from the pre-factors
in the Lagrangian.
00:43:00.660 --> 00:43:13.290
There's also f's that
come with the pions
00:43:13.290 --> 00:43:18.340
because every factor of the pion
field comes with a factor of f.
00:43:18.340 --> 00:43:19.980
You always have pi over f.
00:43:19.980 --> 00:43:25.200
And I'm turning f into 4
pi f for this discussion.
00:43:25.200 --> 00:43:26.855
I'm not worrying
about the 4 pis.
00:43:26.855 --> 00:43:28.230
You could do a
more fancy version
00:43:28.230 --> 00:43:29.897
of this where you
worry about the 4 pis,
00:43:29.897 --> 00:43:33.700
but let's just focus
on the dimensions.
00:43:33.700 --> 00:43:35.890
So if you have an
internal line, that's
00:43:35.890 --> 00:43:37.660
a contraction of 2 pion fields.
00:43:37.660 --> 00:43:38.640
So that gets 2f's.
00:43:38.640 --> 00:43:40.687
And the external line
is just 1 pion field,
00:43:40.687 --> 00:43:41.520
so that just gets 1.
00:43:49.000 --> 00:43:51.910
Topologically, these different
things that we enumerated
00:43:51.910 --> 00:43:55.186
are not unrelated.
00:43:55.186 --> 00:43:57.760
So the order identity
tells us that the number
00:43:57.760 --> 00:44:01.450
of internal lines is
the number of loops
00:44:01.450 --> 00:44:04.330
plus the number of
vertices minus 1.
00:44:06.880 --> 00:44:08.710
And so we use that
to get rid of NI.
00:44:17.190 --> 00:44:19.230
So then we can just put
these things together.
00:44:34.395 --> 00:44:38.520
So if I get rid of NI here
and I replace it by NL--
00:44:38.520 --> 00:44:40.050
so that's that term--
00:44:40.050 --> 00:44:44.940
I replace it by the sum over
the Nn's which are the vertices.
00:44:44.940 --> 00:44:47.340
And then this gives me a plus 2.
00:44:51.600 --> 00:44:52.920
OK.
00:44:52.920 --> 00:44:56.295
Now, that's not the dimension
of the left-hand side.
00:44:56.295 --> 00:44:58.170
That's just the dimension
of the ingredients.
00:44:58.170 --> 00:45:00.555
There's also things that are
coming from factors of M pi
00:45:00.555 --> 00:45:02.940
or factors of p.
00:45:02.940 --> 00:45:07.170
And let me just call that
something, E to the D
00:45:07.170 --> 00:45:10.290
where D is just some
number, integer,
00:45:10.290 --> 00:45:17.420
and then some function of
logarithms of p over mu
00:45:17.420 --> 00:45:18.480
or M pi over mu.
00:45:18.480 --> 00:45:24.030
So E could be M pi or p.
00:45:24.030 --> 00:45:25.772
And for the purpose
of power counting,
00:45:25.772 --> 00:45:26.730
I'm not distinguishing.
00:45:26.730 --> 00:45:30.840
So let's just call it E
just to have a notation that
00:45:30.840 --> 00:45:34.290
could be either M pi or p.
00:45:34.290 --> 00:45:36.040
And then there's one
more thing we can do,
00:45:36.040 --> 00:45:38.010
which is we can look
at the left-hand side.
00:45:38.010 --> 00:45:40.740
And we can say, just by
dimensional analysis,
00:45:40.740 --> 00:45:43.920
the matrix element, what
should be its dimension?
00:45:43.920 --> 00:45:46.320
And depending on how many
bosons you have sticking out,
00:45:46.320 --> 00:45:48.487
your matrix element should
have a certain dimension,
00:45:48.487 --> 00:45:50.290
which you could figure out.
00:45:50.290 --> 00:45:56.190
And that dimension
is just 4 minus NE.
00:45:59.130 --> 00:46:02.820
Two-point function would scale
like p squared, et cetera.
00:46:02.820 --> 00:46:06.060
So now, I have different
things that are giving mass
00:46:06.060 --> 00:46:08.445
dimensions, the lambda
chis and the E's.
00:46:08.445 --> 00:46:11.430
But because I know what the
answer has to be, 4 minus NE,
00:46:11.430 --> 00:46:27.280
I can solve for D.
00:46:27.280 --> 00:46:31.110
And that's what I got.
00:46:31.110 --> 00:46:31.800
OK.
00:46:31.800 --> 00:46:34.770
So the answer for D, in order
to get the dimensions right,
00:46:34.770 --> 00:46:36.930
we have to compensate for
the number of lambda chis
00:46:36.930 --> 00:46:39.690
by factors of M pi and p.
00:46:39.690 --> 00:46:42.600
And when we do that, then
we need this many of them
00:46:42.600 --> 00:46:45.247
in order to get the
dimensions right.
00:46:45.247 --> 00:46:46.830
So one thing you see
from this formula
00:46:46.830 --> 00:46:48.782
is that D is greater
than or equal to 2.
00:46:48.782 --> 00:46:50.490
That's because these
things are positive.
00:46:50.490 --> 00:46:53.370
The Lagrangian starts with n
equals 2 and then goes higher.
00:47:00.520 --> 00:47:05.740
And when I add either of these
terms, I cause suppression,
00:47:05.740 --> 00:47:06.790
or I stay the same.
00:47:16.820 --> 00:47:17.770
So this could be 0.
00:47:17.770 --> 00:47:19.850
This could be 0, or
it could be bigger.
00:47:19.850 --> 00:47:20.845
But it can be smaller.
00:47:27.900 --> 00:47:35.355
So you always get
more E's, which
00:47:35.355 --> 00:47:39.005
are M pis or p's, by adding
vertices or adding loops.
00:47:39.005 --> 00:47:41.380
So when you looked at the loop
graphs that were built out
00:47:41.380 --> 00:47:43.530
of the leading order
Lagrangian, those terms
00:47:43.530 --> 00:47:45.780
had this be 0 because n was 2.
00:47:45.780 --> 00:47:48.780
But then you got suppression
because we built loop graphs.
00:47:48.780 --> 00:47:50.718
So you got suppression
from this term.
00:47:50.718 --> 00:47:53.010
And when we looked at the
higher dimensional operators,
00:47:53.010 --> 00:47:53.820
there was no loop.
00:47:53.820 --> 00:47:55.487
So we didn't have
this term, but then we
00:47:55.487 --> 00:47:57.800
had a contribution
from this guy.
00:47:57.800 --> 00:47:59.250
But those were trading off.
00:48:04.700 --> 00:48:07.283
So having this is,
effectively, part
00:48:07.283 --> 00:48:09.950
of what we need in order to make
sure the theory is well-defined
00:48:09.950 --> 00:48:12.350
because it tells us how
to organize the theory
00:48:12.350 --> 00:48:15.440
and what parts of
the theory we need
00:48:15.440 --> 00:48:18.560
to worry about if we
want a certain accuracy
00:48:18.560 --> 00:48:21.628
and what parts we can ignore.
00:48:21.628 --> 00:48:23.420
We need to know that
we don't need to think
00:48:23.420 --> 00:48:24.740
about two-loop diagrams.
00:48:24.740 --> 00:48:28.490
And this tells us that we don't.
00:48:28.490 --> 00:48:29.422
Yeah.
00:48:29.422 --> 00:48:31.880
AUDIENCE: I didn't understand
[? where you ?] [INAUDIBLE]..
00:48:31.880 --> 00:48:33.547
IAIN STEWART: Yeah,
let me say it again.
00:48:33.547 --> 00:48:38.150
So we figured out the lambda
chis by this stuff up here.
00:48:38.150 --> 00:48:39.950
Then I said, let
there be an arbitrary
00:48:39.950 --> 00:48:44.300
E to the D, some
parameter which we haven't
00:48:44.300 --> 00:48:45.860
figured out anything yet.
00:48:45.860 --> 00:48:49.010
But I know by just
what possibly this
00:48:49.010 --> 00:48:51.840
could depend on that it could
also depend on an M pis or pis.
00:48:51.840 --> 00:48:55.160
So let me put some polynomial
power in and then some function
00:48:55.160 --> 00:48:56.770
that could be non-polynomial.
00:48:56.770 --> 00:48:58.437
AUDIENCE: But where
would that-- can you
00:48:58.437 --> 00:49:00.738
give an example of a
calculation where I
00:49:00.738 --> 00:49:02.030
would see exactly what that is?
00:49:02.030 --> 00:49:02.900
IAIN STEWART:
Yeah, so if you did
00:49:02.900 --> 00:49:04.820
this loop calculation
we did a minute ago,
00:49:04.820 --> 00:49:06.530
we got a p to the fourth.
00:49:06.530 --> 00:49:07.820
And then D would be 4.
00:49:07.820 --> 00:49:10.040
AUDIENCE: And those
p's came from where?
00:49:10.040 --> 00:49:13.770
IAIN STEWART: Oh, they came
from the momentum going--
00:49:13.770 --> 00:49:16.580
so if you looked
at this diagram,
00:49:16.580 --> 00:49:18.060
there's p coming in here.
00:49:18.060 --> 00:49:21.020
And then it goes
into the loop, right?
00:49:21.020 --> 00:49:22.730
And this is
derivatively coupled,
00:49:22.730 --> 00:49:24.545
so you get p's in the numerator.
00:49:24.545 --> 00:49:27.597
AUDIENCE: But that wasn't
taken care of with [INAUDIBLE]..
00:49:27.597 --> 00:49:28.430
IAIN STEWART: Right.
00:49:28.430 --> 00:49:30.380
Because NV is just
counting lambda chis,
00:49:30.380 --> 00:49:33.335
which are constants,
not the p's.
00:49:33.335 --> 00:49:37.088
It's just counting the
constants, the f's.
00:49:37.088 --> 00:49:37.630
AUDIENCE: OK.
00:49:37.630 --> 00:49:38.422
IAIN STEWART: Yeah.
00:49:38.422 --> 00:49:39.008
AUDIENCE: OK.
00:49:39.008 --> 00:49:39.800
IAIN STEWART: Yeah.
00:49:39.800 --> 00:49:44.213
And then you equate it to this,
and then you get the D. Yeah.
00:49:44.213 --> 00:49:45.880
So you could have
tried to set things up
00:49:45.880 --> 00:49:48.730
by thinking about counting p's
instead of counting lambdas.
00:49:48.730 --> 00:49:50.470
But it's-- yeah, anyway.
00:49:54.950 --> 00:49:58.190
All right, so what people
often refer to this
00:49:58.190 --> 00:50:16.450
is they say it's p
counting because you're
00:50:16.450 --> 00:50:18.400
counting momenta.
00:50:18.400 --> 00:50:21.820
And that includes p or M
pi, but sometimes people
00:50:21.820 --> 00:50:24.160
call it p counting.
00:50:24.160 --> 00:50:29.080
And just to do some examples,
when we had the lowest order
00:50:29.080 --> 00:50:33.760
Lagrangian, this guy
comes out as 2 powers of p
00:50:33.760 --> 00:50:35.421
because there's two derivatives.
00:50:38.650 --> 00:50:42.940
And that, in our
formula, is just the fact
00:50:42.940 --> 00:50:46.150
that D is 2 for that.
00:50:46.150 --> 00:50:51.520
When we thought about this loop
with leading order Lagrangians,
00:50:51.520 --> 00:50:56.260
which are scaling
like p squared,
00:50:56.260 --> 00:50:58.183
we ended up having D equals 4.
00:50:58.183 --> 00:51:00.100
And the way that that
comes out of the formula
00:51:00.100 --> 00:51:03.160
is because of this
loop term, which is 1.
00:51:03.160 --> 00:51:06.070
And then there's 2 plus 2 is 4.
00:51:10.270 --> 00:51:15.478
And this guy is 4 because
of the explicit suppression.
00:51:21.330 --> 00:51:22.220
OK.
00:51:22.220 --> 00:51:25.340
So the theory is organized
as an expansion of this p.
00:51:30.520 --> 00:51:32.470
All right, so I
want to come back
00:51:32.470 --> 00:51:35.080
to SU(3) partly
because the problem
00:51:35.080 --> 00:51:37.570
that I gave you is in SU(3).
00:51:37.570 --> 00:51:41.230
So we'll do some discussion
of the SU(3) case.
00:51:53.400 --> 00:51:55.510
I'll go into a few things
in a little more detail
00:51:55.510 --> 00:51:58.570
than we did for
SU(2), where we were
00:51:58.570 --> 00:52:00.070
focusing on more formal things.
00:52:02.670 --> 00:52:06.280
So SU(3) would have gamma
matrices instead of the Pauli
00:52:06.280 --> 00:52:08.470
matrices.
00:52:08.470 --> 00:52:10.270
There's two bases
that you can use.
00:52:10.270 --> 00:52:12.100
You can either use
this basis, which
00:52:12.100 --> 00:52:15.040
is like enumerated
1 to 8, or you
00:52:15.040 --> 00:52:16.820
could use the charged basis.
00:52:16.820 --> 00:52:19.630
And if you use
the charge basis--
00:52:19.630 --> 00:52:22.030
and often you write it
out as a matrix like this.
00:52:28.790 --> 00:52:31.100
On your problem set, you're
free to pick which basis
00:52:31.100 --> 00:52:31.880
you want to use.
00:52:37.220 --> 00:52:39.950
It may be that one or
the other is easier,
00:52:39.950 --> 00:52:41.962
but I can't even tell
you which one is easier
00:52:41.962 --> 00:52:42.920
since I don't remember.
00:52:46.105 --> 00:52:48.185
So sometimes one or the
other is easier to use.
00:52:48.185 --> 00:52:50.060
And you have a freedom
of what basis to pick.
00:52:54.520 --> 00:53:01.960
If you expand, in this case,
the trace of sigma Mq dagger
00:53:01.960 --> 00:53:08.890
plus Mq sigma dagger, which
is that term that had a V0,
00:53:08.890 --> 00:53:10.870
then that gives
mass to the mesons,
00:53:10.870 --> 00:53:11.935
as it did for the pions.
00:53:18.308 --> 00:53:19.850
Because the symmetry
group is bigger,
00:53:19.850 --> 00:53:21.700
you get more predictions.
00:53:21.700 --> 00:53:25.060
Here, you get predictions
for the kaon masses.
00:53:25.060 --> 00:53:35.120
And you get things like the
fact that the neutral kaons have
00:53:35.120 --> 00:53:41.180
massless order Md plus
Ms. And you get things
00:53:41.180 --> 00:53:46.950
like eta pi 0 mixing, where
there's a mixing matrix.
00:53:46.950 --> 00:53:49.270
So if you look at the
masses of eta and pi 0,
00:53:49.270 --> 00:53:51.050
they're actually non-diagonal.
00:53:51.050 --> 00:53:52.890
So for the eta in
the pi 0 system,
00:53:52.890 --> 00:53:54.980
you actually get a matrix.
00:53:54.980 --> 00:53:56.315
So M squared is a matrix.
00:53:59.430 --> 00:54:03.800
So for example, for the pi
0, it was just M up plus Md.
00:54:06.980 --> 00:54:11.060
But then once you're in SU(3),
there's actually a mixing term.
00:54:11.060 --> 00:54:13.895
And there's like an M up
minus M down term here
00:54:13.895 --> 00:54:20.220
that is mixing between, and
then same thing over here.
00:54:20.220 --> 00:54:22.190
So the etas and the pi
0s actually are mixing.
00:54:22.190 --> 00:54:25.310
And there's something
in this n tree.
00:54:25.310 --> 00:54:27.560
And the mixing is isospin
violating in the sense
00:54:27.560 --> 00:54:29.050
that it's M up minus M down.
00:54:29.050 --> 00:54:33.230
So it's a small effect,
but this is something
00:54:33.230 --> 00:54:36.440
that you can predict from
chiral Lagrangian, something
00:54:36.440 --> 00:54:39.560
about because is describes
isospin violating effects
00:54:39.560 --> 00:54:40.550
from the quark masses.
00:54:46.100 --> 00:54:48.320
So the reason that
I mention that is
00:54:48.320 --> 00:54:50.810
because, often when you
do calculations, keeping
00:54:50.810 --> 00:54:53.420
track of Mu's,
Md's, and Ms's all
00:54:53.420 --> 00:54:56.820
as separate independent
parameters is a little much.
00:54:56.820 --> 00:54:58.620
And so you want to
make an approximation.
00:54:58.620 --> 00:55:01.430
And so if you make
an approximation that
00:55:01.430 --> 00:55:03.110
ignores isospin violation--
00:55:09.090 --> 00:55:15.400
so we often ignore
isospin violation.
00:55:15.400 --> 00:55:17.680
Isospin violation is very small.
00:55:17.680 --> 00:55:19.720
And if you remember,
for the SU(3) case,
00:55:19.720 --> 00:55:23.460
you're expanding in Ms
over lambda QCD or Mk
00:55:23.460 --> 00:55:24.180
over lambda chi.
00:55:24.180 --> 00:55:25.440
So it's not a great expansion.
00:55:25.440 --> 00:55:27.840
You have something like a third.
00:55:27.840 --> 00:55:30.180
So ignoring isospin
is perfectly valid
00:55:30.180 --> 00:55:31.820
if you're expanding in a third.
00:55:31.820 --> 00:55:33.570
So basically, because
there is a hierarchy
00:55:33.570 --> 00:55:36.792
between the strange quark
mass and the down and the up,
00:55:36.792 --> 00:55:39.000
you want to focus on places
where you get the largest
00:55:39.000 --> 00:55:39.570
corrections.
00:55:39.570 --> 00:55:41.975
And one way of making
an approximation that
00:55:41.975 --> 00:55:44.100
allows you to do that is
to ignore isospin and take
00:55:44.100 --> 00:55:47.400
Mu equal to Md.
00:55:47.400 --> 00:55:52.200
So if we take Mu at
Md to be some M hat--
00:55:52.200 --> 00:55:55.800
which if you want an exact
definition, you could say
00:55:55.800 --> 00:55:58.185
it's the average.
00:55:58.185 --> 00:55:59.868
And you'd drop the difference.
00:56:04.350 --> 00:56:06.750
And then you can think
of the strange quark mass
00:56:06.750 --> 00:56:08.280
as being somewhat
bigger than M hat.
00:56:14.450 --> 00:56:17.175
That's an approximation that
you can use on your problem set,
00:56:17.175 --> 00:56:17.675
for example.
00:56:23.760 --> 00:56:26.420
So let me write out
here with all the terms
00:56:26.420 --> 00:56:28.130
in the chiral Lagrangian is.
00:56:28.130 --> 00:56:32.120
And I'll do it for
a case where we
00:56:32.120 --> 00:56:37.582
include in our chiral Lagrangian
one other type of coupling,
00:56:37.582 --> 00:56:39.290
which is this left-handed
current that we
00:56:39.290 --> 00:56:42.000
talked about last time.
00:56:42.000 --> 00:56:44.665
So we talked about a spurion
analysis for the chi term.
00:56:44.665 --> 00:56:46.040
And I said you
could do something
00:56:46.040 --> 00:56:49.160
similar to a couple in
a left-handed current.
00:56:49.160 --> 00:56:53.120
And we had this,
where D mu sigma
00:56:53.120 --> 00:56:57.680
was partial mu sigma times
the left-handed current sigma.
00:56:57.680 --> 00:56:59.180
So we thought of
putting it together
00:56:59.180 --> 00:57:00.950
into a coherent derivative.
00:57:00.950 --> 00:57:03.650
And if I do that, it modifies
the leading order Lagrangian
00:57:03.650 --> 00:57:06.465
and just makes these partials
into covariant derivatives.
00:57:15.250 --> 00:57:19.760
Power counting, for
counting purposes,
00:57:19.760 --> 00:57:22.250
you count sigmas of order 1.
00:57:22.250 --> 00:57:27.320
You count D mu sigma as order
p, which means you count
00:57:27.320 --> 00:57:29.700
L mu, the source, as order p.
00:57:29.700 --> 00:57:31.070
This is a left-handed source.
00:57:33.620 --> 00:57:37.513
And you count chis and
Mq's as of order p squared.
00:57:37.513 --> 00:57:39.430
That's just repeating
what we've already said.
00:57:44.640 --> 00:57:46.950
And these higher order
terms, which where L's, we
00:57:46.950 --> 00:57:48.250
can then enumerate.
00:57:52.850 --> 00:57:57.650
And I now just use
D's, covariant D's
00:57:57.650 --> 00:58:00.590
when I write them down.
00:58:00.590 --> 00:58:03.200
I'm now in SU(3).
00:58:03.200 --> 00:58:06.860
And it turns out that one of the
relations that we used in SU(2)
00:58:06.860 --> 00:58:08.960
doesn't carry over to SU(3).
00:58:08.960 --> 00:58:14.930
So if I just think of guys with
four covariant derivatives,
00:58:14.930 --> 00:58:18.820
it turns out that there's
one more operator there.
00:58:22.400 --> 00:58:27.350
So there's those two, which
we talked about, L1 and L2.
00:58:27.350 --> 00:58:30.620
And then there's also L3.
00:58:30.620 --> 00:58:32.180
So we can't get rid of this guy.
00:58:35.540 --> 00:58:37.910
I'm always writing sigmas
next to sigma daggers because
00:58:37.910 --> 00:58:41.240
of the chiral transformation.
00:58:41.240 --> 00:58:43.967
I'm also imposing
parity, though I'm not
00:58:43.967 --> 00:58:45.800
going to spend much
time talking about that.
00:58:49.780 --> 00:58:52.000
And really I want to
enumerate, also, for you
00:58:52.000 --> 00:58:59.090
some of the terms that involve
the quark mass, the chi guy.
00:58:59.090 --> 00:59:01.860
So you could have a guy
that's a cross-term.
00:59:01.860 --> 00:59:06.630
This would take care of the
renormalization of things
00:59:06.630 --> 00:59:09.320
like p squared and pi
squared times 1 of epsilon.
00:59:13.560 --> 00:59:16.290
That's L4, L5.
00:59:41.610 --> 00:59:44.115
We could also have the quark
mass type term just squared.
00:59:48.657 --> 00:59:49.990
So take the trace and square it.
00:59:49.990 --> 00:59:52.560
That's L6.
00:59:52.560 --> 00:59:54.570
It turns out that we
could also build something
00:59:54.570 --> 00:59:56.880
with the right parity by just
having the difference instead
00:59:56.880 --> 00:59:57.380
of the sum.
00:59:57.380 --> 00:59:58.918
And that's a different operator.
01:00:05.610 --> 01:00:06.585
So that's L7.
01:00:10.080 --> 01:00:11.190
We're going to go up to 9.
01:00:16.200 --> 01:00:18.398
Don't be afraid.
01:00:18.398 --> 01:00:20.190
You can have something
with 2 sigma daggers
01:00:20.190 --> 01:00:25.440
and 2 chis, which is another way
of building a chiral invariant.
01:00:25.440 --> 01:00:30.190
And then for parity,
you need the other way.
01:00:30.190 --> 01:00:33.510
And then finally--
something called
01:00:33.510 --> 01:00:39.030
L9, which involves a
trace that involves
01:00:39.030 --> 01:00:43.042
L mu nu, where L mu nu is built
out of this external current.
01:00:43.042 --> 01:00:45.000
So we can take two of
our covariant derivatives
01:00:45.000 --> 01:00:46.477
and take a commutator.
01:00:51.840 --> 01:00:53.826
That's giving the
final operator.
01:01:10.850 --> 01:01:16.190
So there's a complete basis for
SU(3) for a left-handed current
01:01:16.190 --> 01:01:17.428
in the chi.
01:01:17.428 --> 01:01:19.220
AUDIENCE: [? So does ?]
[? L mu ?] [? nu ?]
01:01:19.220 --> 01:01:20.018
[? have a field ?] [INAUDIBLE]
[? associated with ?]
01:01:20.018 --> 01:01:20.180
[INAUDIBLE]?
01:01:20.180 --> 01:01:21.200
IAIN STEWART: Yeah,
I'll write it down.
01:01:21.200 --> 01:01:21.742
AUDIENCE: Oh.
01:01:45.698 --> 01:01:47.240
IAIN STEWART: So if
L mu is something
01:01:47.240 --> 01:01:50.690
that has SU(3) indices, has an
SU(3) matrix hiding inside it,
01:01:50.690 --> 01:01:53.660
then there's a
commutator term as well.
01:01:53.660 --> 01:01:57.590
And then it's just
this combination.
01:01:57.590 --> 01:02:00.998
I also use the
equation of motion.
01:02:00.998 --> 01:02:03.540
And I didn't talk about it, but
I use the equation of motion.
01:02:03.540 --> 01:02:05.370
And I used SU(3) relations.
01:02:05.370 --> 01:02:06.950
Much as I talked
about for SU(2),
01:02:06.950 --> 01:02:09.448
there's some SU(3)
relations that survive.
01:02:09.448 --> 01:02:11.240
And I got rid of some
operators doing that.
01:02:16.200 --> 01:02:19.620
Now, you could say, well, I
have this SU(3) and SU(2),
01:02:19.620 --> 01:02:21.990
so why don't I try
to relate them?
01:02:21.990 --> 01:02:23.340
One of them has a kaon.
01:02:23.340 --> 01:02:24.420
The other one doesn't.
01:02:24.420 --> 01:02:27.000
The one without the kaon
thinks about the kaon
01:02:27.000 --> 01:02:28.298
as a heavy particle.
01:02:28.298 --> 01:02:30.090
The one with the kaon
thinks about the kaon
01:02:30.090 --> 01:02:31.530
as a light particle.
01:02:31.530 --> 01:02:32.520
Those are two theories.
01:02:32.520 --> 01:02:35.820
I could try to make them
match up with each other.
01:02:35.820 --> 01:02:37.890
And that's something that
you can do, actually.
01:02:41.860 --> 01:02:48.300
So there's a little
bit of a lesson there.
01:02:48.300 --> 01:02:50.220
So that was why I
want to mention it.
01:02:56.790 --> 01:02:58.540
So SU(2) and SU(3)
seem like they're
01:02:58.540 --> 01:02:59.690
describing similar physics.
01:02:59.690 --> 01:03:01.560
They both could describe pions.
01:03:01.560 --> 01:03:03.130
They both have pions in them.
01:03:03.130 --> 01:03:04.090
But the SU(3) has more.
01:03:04.090 --> 01:03:05.770
It's got the kaon.
01:03:05.770 --> 01:03:08.500
That means that, in
the SU(2) theory,
01:03:08.500 --> 01:03:13.540
the kaon is in the
coefficients, OK?
01:03:13.540 --> 01:03:16.090
So if you do a
correspondence, you
01:03:16.090 --> 01:03:18.650
get relations that
are like this.
01:03:18.650 --> 01:03:20.260
So the 2 here means SU(2).
01:03:25.380 --> 01:03:28.180
And where I don't put any
subscript, it's SU(3).
01:03:32.080 --> 01:03:35.410
And something
that-- this is a 96.
01:03:44.140 --> 01:03:48.010
And if you really do that, what
I said, compare observables,
01:03:48.010 --> 01:03:50.710
you get relations like
this one where you actually
01:03:50.710 --> 01:03:54.190
see the kaon is
showing up on this side
01:03:54.190 --> 01:03:56.470
and is being encoded in
coefficients in the SU(2)
01:03:56.470 --> 01:03:58.190
theory.
01:03:58.190 --> 01:03:59.868
So what you think of
as the coefficients
01:03:59.868 --> 01:04:01.660
in your chiral theory
depends on the matter
01:04:01.660 --> 01:04:03.783
that you've put in,
including things
01:04:03.783 --> 01:04:05.950
like what particles like
the kaon, what group you're
01:04:05.950 --> 01:04:06.640
talking about.
01:04:09.500 --> 01:04:18.680
This is an explicit example of
the kaon being the coefficients
01:04:18.680 --> 01:04:21.890
if we use SU(2).
01:04:25.690 --> 01:04:31.510
OK, so just like in SU(2),
we have to go through
01:04:31.510 --> 01:04:34.420
a renormalization of the Li's.
01:04:39.280 --> 01:04:41.890
And you can think of
that by writing a formula
01:04:41.890 --> 01:04:42.520
like this one.
01:04:45.410 --> 01:04:47.690
Bare Li is equal to
some renormalized one,
01:04:47.690 --> 01:04:51.800
which I put a bar on top
of and then a counter-term.
01:04:51.800 --> 01:04:55.850
And the counter-term
will be some coefficient,
01:04:55.850 --> 01:05:01.760
some number of 4 pis,
and then some epsilons.
01:05:01.760 --> 01:05:05.570
And just like in
gauge theory, we
01:05:05.570 --> 01:05:09.070
get rid of the Euler
gamma and the log 4 pi.
01:05:09.070 --> 01:05:11.330
We have an Ms bar
type definition.
01:05:11.330 --> 01:05:13.790
And in chiral
perturbation theory,
01:05:13.790 --> 01:05:19.010
people often take an extra
1 along with the ride
01:05:19.010 --> 01:05:21.510
just because they're allowed to.
01:05:21.510 --> 01:05:23.510
Because it tends to show
up in the kind of loops
01:05:23.510 --> 01:05:24.530
that you encounter.
01:05:24.530 --> 01:05:28.940
So sometimes that's
included, sometimes it's not.
01:05:28.940 --> 01:05:35.780
So for example, as an
example of a diagram,
01:05:35.780 --> 01:05:38.060
you could think about this one.
01:05:38.060 --> 01:05:42.022
And that does cause
mass renormalization
01:05:42.022 --> 01:05:42.980
for the physical boson.
01:05:48.840 --> 01:05:51.120
So if you think about
our lowest order relation
01:05:51.120 --> 01:05:57.570
as where we started, let me
write that as M0 squared is
01:05:57.570 --> 01:06:04.320
4V0 over f squared Mu plus Md.
01:06:07.950 --> 01:06:10.060
Sometimes people call this--
01:06:10.060 --> 01:06:18.330
well, making up
some more notation,
01:06:18.330 --> 01:06:21.758
yeah, which I'll
use in a minute.
01:06:21.758 --> 01:06:23.550
And so if you actually
calculate this loop,
01:06:23.550 --> 01:06:25.300
then you get a correction
to that formula.
01:06:25.300 --> 01:06:29.820
This would have be M pi squared,
but here we get a correction.
01:06:29.820 --> 01:06:34.500
If you did it in
SU(2), just to keep
01:06:34.500 --> 01:06:38.760
the formula a little simpler,
then it would look like this.
01:06:43.270 --> 01:06:46.710
So M0 is like the pi n mass, but
the pi n mass at lowest order
01:06:46.710 --> 01:06:47.952
in the chiral expansion.
01:06:47.952 --> 01:06:49.410
So it's not the
physical pi n mass.
01:06:52.320 --> 01:07:01.110
So 2L 4 bar in theory two with
two flavors with two SU(2) plus
01:07:01.110 --> 01:07:04.770
an L5 bar minus 4 n L6 bar--
01:07:04.770 --> 01:07:06.570
so these various
coefficients are coming in
01:07:06.570 --> 01:07:10.883
with some numbers in front,
some combination of them.
01:07:10.883 --> 01:07:12.300
And then there's
some contribution
01:07:12.300 --> 01:07:13.870
from a chiral loop.
01:07:13.870 --> 01:07:19.600
And if I take away that 1,
it's just a chiral logarithm.
01:07:19.600 --> 01:07:28.470
So this is 4 pi f squared log
M0 squared over mu squared.
01:07:31.890 --> 01:07:35.190
There would be an extra term
of just plus 1 times this
01:07:35.190 --> 01:07:38.580
if I hadn't gotten rid of that,
the minus 1 times that, OK?
01:07:38.580 --> 01:07:39.720
So this is from the loop.
01:07:39.720 --> 01:07:43.500
This is from the explicit
dimension operators
01:07:43.500 --> 01:07:46.200
that were of the higher
dimension, some combination
01:07:46.200 --> 01:07:47.700
of them.
01:07:47.700 --> 01:07:51.230
The UV divergence is absorbed
in some combination of them.
01:07:51.230 --> 01:07:53.750
And if you want to figure out
exactly how UV divergences go
01:07:53.750 --> 01:07:56.430
in between this, then you've
got to think of renormalizing
01:07:56.430 --> 01:07:59.470
more than just this diagram.
01:07:59.470 --> 01:08:02.310
And you can generically
think of observables
01:08:02.310 --> 01:08:07.090
as having this kind of
expansion in Mu and Md.
01:08:07.090 --> 01:08:10.440
So if you like, you should think
of M0 here as really just an up
01:08:10.440 --> 01:08:11.462
plus M down.
01:08:11.462 --> 01:08:13.920
This is saying, at lowest order
chiral perturbation theory,
01:08:13.920 --> 01:08:15.720
there's a linear term.
01:08:15.720 --> 01:08:19.439
But then here there's an M0
to the fourth term, which is
01:08:19.439 --> 01:08:21.750
quadratic in the quark masses.
01:08:21.750 --> 01:08:24.779
And there's also a quadratic
term from the loops.
01:08:24.779 --> 01:08:27.660
So you have an expansion
in the quark masses.
01:08:27.660 --> 01:08:32.880
Think of M0 as the quark masses
and M pi as the meson mass.
01:08:32.880 --> 01:08:34.859
I just gave you the
example of the pion mass,
01:08:34.859 --> 01:08:38.910
but this is generically
true for observables
01:08:38.910 --> 01:08:42.210
that you might calculate that
they have this type of result
01:08:42.210 --> 01:08:44.218
where you have an expansion.
01:08:44.218 --> 01:08:47.880
AUDIENCE: Is there any physical
reason for why this mass square
01:08:47.880 --> 01:08:52.390
of the meson space [? would ?]
[? be ?] [INAUDIBLE]??
01:08:52.390 --> 01:08:54.859
IAIN STEWART: So one
way of saying it is--
01:08:54.859 --> 01:08:55.359
yeah.
01:08:55.359 --> 01:08:59.189
So you might think, well, why
is it not M pi equals Mu and Md?
01:08:59.189 --> 01:09:00.910
And the glib way of
saying it is, well,
01:09:00.910 --> 01:09:02.535
if you think about
M pi squared and you
01:09:02.535 --> 01:09:04.450
think about it
having an expansion,
01:09:04.450 --> 01:09:06.990
then it could have a
constant plus linear term
01:09:06.990 --> 01:09:08.117
plus quadratic term.
01:09:08.117 --> 01:09:10.575
You don't have the constant
because of the chiral symmetry.
01:09:10.575 --> 01:09:14.069
So the linear term is the
first thing that's allowed.
01:09:14.069 --> 01:09:16.939
That's one way of saying it.
01:09:16.939 --> 01:09:19.380
I mean, if you think about
it in the bosonic theory,
01:09:19.380 --> 01:09:23.100
we have the Lagrangian
is quadratic in masses,
01:09:23.100 --> 01:09:25.047
right, whereas in
the fermionic theory,
01:09:25.047 --> 01:09:26.130
you have linear in masses.
01:09:26.130 --> 01:09:29.670
And that's also part of
what it had to do with,
01:09:29.670 --> 01:09:33.330
but it's linear combination
of symmetry breaking and that.
01:09:37.779 --> 01:09:41.680
It's allowed, so it happens
is kind of the bottom line.
01:09:41.680 --> 01:09:43.628
But it's allowed
by the symmetries.
01:09:47.319 --> 01:09:47.819
OK.
01:09:47.819 --> 01:09:50.611
And so finally, this is
the example I'm actually
01:09:50.611 --> 01:09:51.569
going to get you to do.
01:09:54.960 --> 01:09:58.410
So I'll have you look on the
problem set at the k constants.
01:09:58.410 --> 01:10:01.270
And they have an analogous
result to that one.
01:10:01.270 --> 01:10:04.470
You'll do the
calculation in SU(3).
01:10:04.470 --> 01:10:07.860
And the kind of result that
you should expect to get
01:10:07.860 --> 01:10:09.450
is something that
looks like this.
01:10:22.340 --> 01:10:24.250
So I'm using this
B0 notation that I
01:10:24.250 --> 01:10:28.900
introduced over there just so I
could write it all on one line.
01:10:33.170 --> 01:10:36.410
And these mus with
the subscript i
01:10:36.410 --> 01:10:39.110
are just some shorthand
for some contributions
01:10:39.110 --> 01:10:39.920
coming from loops.
01:10:51.320 --> 01:10:57.800
And f is the result in L0.
01:11:00.920 --> 01:11:06.940
So f is the parameter
in L0, whereas f pi
01:11:06.940 --> 01:11:10.360
is the physical decay
constant of the pion.
01:11:16.830 --> 01:11:21.240
So one result which is
encoded in this formula
01:11:21.240 --> 01:11:23.160
is that the Lagrangian
parameter is actually
01:11:23.160 --> 01:11:25.200
equal to the decay
constant at lowest order
01:11:25.200 --> 01:11:25.950
in chiral perturbation theory.
01:11:25.950 --> 01:11:27.460
That's something I didn't cover.
01:11:27.460 --> 01:11:29.810
I cover it when I teach
quantum field theory three.
01:11:29.810 --> 01:11:32.880
You can look at my notes
to see that derivation.
01:11:32.880 --> 01:11:36.640
Those of you that have
taken QFT3 from me,
01:11:36.640 --> 01:11:37.902
you've already seen that.
01:11:37.902 --> 01:11:40.110
And basically, what I'm
asking for in the problem set
01:11:40.110 --> 01:11:41.820
is I guide you
with several parts,
01:11:41.820 --> 01:11:45.067
but how to think about
these higher order terms.
01:11:45.067 --> 01:11:46.650
What are the loop
graph contributions?
01:11:46.650 --> 01:11:49.890
How do you get these terms from
the higher order Lagrangians?
01:11:49.890 --> 01:11:51.870
How you put it all together?
01:11:51.870 --> 01:11:55.950
It's a nice example of this use
of chiral perturbation theory.
01:11:55.950 --> 01:11:57.840
And again, we see that
a physical observable,
01:11:57.840 --> 01:12:00.660
which is this decay constant,
has a chiral expansion.
01:12:00.660 --> 01:12:03.330
And the thing that chiral
perturbation theory is actually
01:12:03.330 --> 01:12:06.720
doing is allowing you
to predict both the form
01:12:06.720 --> 01:12:09.390
of that expansion, as well
as these things here, which
01:12:09.390 --> 01:12:12.450
are chiral logarithms.
01:12:12.450 --> 01:12:16.270
So if you ask about predictive
power, the polynomial terms
01:12:16.270 --> 01:12:18.450
in terms of higher order
in the quark masses
01:12:18.450 --> 01:12:21.630
that are polynomial, you end
up having unknown coefficients.
01:12:21.630 --> 01:12:23.550
But the terms with
logarithms have
01:12:23.550 --> 01:12:26.670
coefficients which are fixed
by your lower order Lagrangian.
01:12:26.670 --> 01:12:28.170
So those are things
that you predict
01:12:28.170 --> 01:12:31.800
with the chiral
perturbation theory.
01:12:31.800 --> 01:12:33.360
When people do
lattice calculations,
01:12:33.360 --> 01:12:34.950
they need to do
chiral extrapolations.
01:12:34.950 --> 01:12:36.867
And then they're using
formulas like this one.
01:12:40.140 --> 01:12:43.384
OK, so any questions about that?
01:12:43.384 --> 01:12:45.283
AUDIENCE: Did the
M pi [INAUDIBLE]??
01:12:45.283 --> 01:12:46.700
IAIN STEWART: Yeah,
physical mass.
01:12:46.700 --> 01:12:47.870
This is the physical mass.
01:12:47.870 --> 01:12:49.670
This is the quark masses.
01:12:49.670 --> 01:12:54.610
M0 is just this combination
with M up and M down.
01:12:54.610 --> 01:12:56.690
AUDIENCE: So the mu
dependence cancel between--
01:12:56.690 --> 01:12:58.148
IAIN STEWART: And
the mu dependence
01:12:58.148 --> 01:13:01.040
cancels because these
guys all depend on mu.
01:13:01.040 --> 01:13:03.230
Yeah.
01:13:03.230 --> 01:13:06.600
And same thing here,
these guys depend on mu.
01:13:09.668 --> 01:13:12.190
Here, I have just enough
room to make that explicit.
01:13:15.440 --> 01:13:17.828
OK, so that actually
covers all the goals
01:13:17.828 --> 01:13:19.620
that we had for chiral
perturbation theory,
01:13:19.620 --> 01:13:21.410
though it's a fun topic.
01:13:21.410 --> 01:13:23.870
And there's many more
things we could discuss,
01:13:23.870 --> 01:13:27.420
but that's what we
needed to discuss.
01:13:27.420 --> 01:13:30.000
And so we're going to move on.
01:13:30.000 --> 01:13:31.920
So the next thing I
want to talk about
01:13:31.920 --> 01:13:33.920
as an example of effective
field theory is heavy
01:13:33.920 --> 01:13:34.970
quark effective theory.
01:13:48.160 --> 01:13:50.580
So again, what are our goals
with this effective theory?
01:13:58.730 --> 01:14:00.500
We will see some new
features showing up
01:14:00.500 --> 01:14:01.667
that we haven't seen before.
01:14:06.887 --> 01:14:09.220
We will find out what it means
to take a Lagrangian that
01:14:09.220 --> 01:14:10.280
has labeled fields.
01:14:16.315 --> 01:14:18.190
We'll spend a little
bit of time on symmetry.
01:14:18.190 --> 01:14:21.465
Because in this heavy
quark effective theory,
01:14:21.465 --> 01:14:23.840
there's actually something
called heavy quark's symmetry,
01:14:23.840 --> 01:14:26.140
which is not apparent in
QCD, but becomes apparent
01:14:26.140 --> 01:14:29.110
in this theory.
01:14:29.110 --> 01:14:35.590
And there's a trick known as
using covariate representations
01:14:35.590 --> 01:14:37.900
in order to encode
symmetry predictions.
01:14:37.900 --> 01:14:40.000
And it's a very
powerful thing that's
01:14:40.000 --> 01:14:42.750
not special to this theory, but
you could use it in general.
01:14:42.750 --> 01:14:44.000
And I want to teach it to you.
01:14:44.000 --> 01:14:45.208
So that's one thing we'll do.
01:14:52.558 --> 01:14:54.100
It's kind of like
a spurion analysis,
01:14:54.100 --> 01:14:59.050
but a little bit more powerful.
01:14:59.050 --> 01:15:03.190
And finally, anomalous
dimensions that are functions
01:15:03.190 --> 01:15:09.790
is something that we'll show
up here, not just numbers.
01:15:09.790 --> 01:15:20.180
There's something called
reparameterization invariance,
01:15:20.180 --> 01:15:25.490
which you can think of a kind of
symmetry that we'll talk about.
01:15:25.490 --> 01:15:31.180
And finally, if that
list is not long enough,
01:15:31.180 --> 01:15:36.820
we'll add one more
thing which I hinted at.
01:15:36.820 --> 01:15:40.090
So I said, when we
talked about Ms bar,
01:15:40.090 --> 01:15:42.387
that it wasn't a perfect
scheme for doing things,
01:15:42.387 --> 01:15:43.720
but there were some limitations.
01:15:43.720 --> 01:15:45.512
And we'll come, at the
end of this chapter,
01:15:45.512 --> 01:15:47.080
to what those limitations are.
01:15:54.140 --> 01:15:58.310
The limitations come from
power-like scale separation.
01:15:58.310 --> 01:16:02.210
And that's related to
something called renormalons.
01:16:02.210 --> 01:16:05.030
So we'll learn what
a renormalon is
01:16:05.030 --> 01:16:07.640
and why it has something to
do with the failure of Ms bar
01:16:07.640 --> 01:16:10.893
and how one can get
around that failure
01:16:10.893 --> 01:16:12.560
and how one actually
needs to get around
01:16:12.560 --> 01:16:15.000
that failure in some cases.
01:16:15.000 --> 01:16:17.840
OK, so that's the list of goals.
01:16:17.840 --> 01:16:22.220
It's kind of, in some ways, in
order of importance, actually.
01:16:24.750 --> 01:16:27.000
So when we think about heavy
quark effective theory,
01:16:27.000 --> 01:16:29.137
you shouldn't think about
it as just something
01:16:29.137 --> 01:16:31.470
that you would need to do if
you wanted to do heavy work
01:16:31.470 --> 01:16:34.200
physics because the idea
of what we're doing here
01:16:34.200 --> 01:16:35.915
is much more general than that.
01:16:35.915 --> 01:16:37.290
The idea of what
we're doing here
01:16:37.290 --> 01:16:39.990
is we're saying,
take a heavy particle
01:16:39.990 --> 01:16:43.770
and think about what
happens if I tickle it.
01:16:43.770 --> 01:16:49.120
And that heavy particle could be
a heavy source to some theory.
01:16:49.120 --> 01:16:51.120
We'll talk about it in
the context of them being
01:16:51.120 --> 01:16:52.578
heavy quarks, but
you should really
01:16:52.578 --> 01:16:56.100
think of it as any
heavy particle tickled
01:16:56.100 --> 01:16:57.060
by light particles.
01:17:15.010 --> 01:17:18.370
And what we really
mean is that we want
01:17:18.370 --> 01:17:21.037
to study the heavy particle.
01:17:21.037 --> 01:17:22.870
And since we want to
study it, we better not
01:17:22.870 --> 01:17:23.980
remove it from the theory.
01:17:27.532 --> 01:17:29.740
So we want to tickle it with
light degrees of freedom
01:17:29.740 --> 01:17:32.830
with small momentum
transfer, but we don't want
01:17:32.830 --> 01:17:35.578
to integrate that particle out.
01:17:35.578 --> 01:17:37.120
So another way of
saying this, if you
01:17:37.120 --> 01:17:39.037
don't want to think of
it as a heavy particle,
01:17:39.037 --> 01:17:40.810
is that you have some source.
01:17:40.810 --> 01:17:45.550
And that source can be
tickled and could wiggle,
01:17:45.550 --> 01:17:47.718
but it's sort of mostly
just a static source.
01:17:47.718 --> 01:17:48.760
And then it could wiggle.
01:17:48.760 --> 01:17:51.663
And that's what this is an
effective field theory for.
01:17:51.663 --> 01:17:53.830
So we'll talk about it in
the context of heavy quark
01:17:53.830 --> 01:17:54.460
effective theory.
01:17:54.460 --> 01:17:56.543
And some of the things,
like heavy quark symmetry,
01:17:56.543 --> 01:17:57.892
will be special to that theory.
01:17:57.892 --> 01:18:00.100
But more generally, the kind
of approach we're taking
01:18:00.100 --> 01:18:01.100
is a more general thing.
01:18:04.330 --> 01:18:07.270
So what's heavy quark
effective theory?
01:18:07.270 --> 01:18:08.837
So you have a heavy quark.
01:18:08.837 --> 01:18:11.295
And you have it sitting in a
bound state, which is a meson.
01:18:14.770 --> 01:18:18.070
So that means it's surrounded
by light degrees of freedom,
01:18:18.070 --> 01:18:21.520
one heavy quark,
lots of light junk.
01:18:21.520 --> 01:18:24.430
In the quark model, you'd say
it's a Q bar q, heavy cork
01:18:24.430 --> 01:18:31.270
capital Q and a light cork
little q, like b bar d,
01:18:31.270 --> 01:18:33.490
which is the B0.
01:18:33.490 --> 01:18:36.850
But that's just a quark
model approximation
01:18:36.850 --> 01:18:38.470
for what the degrees
of freedom are.
01:18:38.470 --> 01:18:41.137
And there's many more
pairs of Qq bars and gluons
01:18:41.137 --> 01:18:42.970
that are really forming
this hydronic state.
01:18:45.547 --> 01:18:48.130
So you'd like to be able to make
model independent predictions
01:18:48.130 --> 01:18:51.370
for that without having
to worry about the fact
01:18:51.370 --> 01:18:54.790
that you're not parameterizing
properly that stuff
01:18:54.790 --> 01:18:55.990
that you can't calculate.
01:18:55.990 --> 01:19:00.140
And you can do that using
effective field theory.
01:19:00.140 --> 01:19:01.820
There's two scales
in the problem.
01:19:01.820 --> 01:19:04.540
There's the size of this
thing, the inverse size
01:19:04.540 --> 01:19:05.650
of order lambda QCD.
01:19:05.650 --> 01:19:09.317
That's the hadronization
that scale.
01:19:09.317 --> 01:19:11.650
And that's a scale that's
much less than the quark mass.
01:19:11.650 --> 01:19:14.190
So you're really just expanding
in lambda QCD divided by Mq.
01:19:14.190 --> 01:19:15.940
That's what this
effective theory will be.
01:19:21.610 --> 01:19:27.250
And what you want
to describe are
01:19:27.250 --> 01:19:31.882
fluctuations of the heavy
quark due to the light quarks.
01:19:31.882 --> 01:19:33.340
So you could think
that this guy is
01:19:33.340 --> 01:19:36.550
so heavy he just sits in
the middle of the state,
01:19:36.550 --> 01:19:39.087
and he's static.
01:19:39.087 --> 01:19:41.170
And then he gets tickled
by the light stuff that's
01:19:41.170 --> 01:19:42.295
flying and whizzing around.
01:19:42.295 --> 01:19:43.660
And that's a reasonable picture.
01:19:47.200 --> 01:19:48.760
So what we really
want to do here
01:19:48.760 --> 01:19:52.090
is an example of top-down
effective field theory again.
01:19:52.090 --> 01:19:58.210
So going back to our
like integrating out
01:19:58.210 --> 01:20:02.860
heavy particles, but now we
want to keep the heavy particles
01:20:02.860 --> 01:20:04.540
in the theory, not remove them.
01:20:04.540 --> 01:20:06.790
But still, we want to take
a low energy limit of them.
01:20:09.710 --> 01:20:14.050
So that means that we have to
take a low energy limit of QCD,
01:20:14.050 --> 01:20:19.438
which is id slash
minus Mq Q. And you
01:20:19.438 --> 01:20:20.980
can see that part
of the problem here
01:20:20.980 --> 01:20:23.560
is that the Mq is
upstairs, which
01:20:23.560 --> 01:20:26.530
makes taking the limit
not completely obvious.
01:20:26.530 --> 01:20:28.810
In the case where we were
doing the heavy bosons,
01:20:28.810 --> 01:20:30.892
we sort of saw that it
was always in propagators.
01:20:30.892 --> 01:20:33.100
And we could just think of,
since it's always they're
01:20:33.100 --> 01:20:35.500
internal, we just expand.
01:20:35.500 --> 01:20:37.640
Here, we want to keep
the Q in the theory.
01:20:37.640 --> 01:20:41.930
So we have to really think
about how we take this limit.
01:20:41.930 --> 01:20:48.490
So let's start slowly and
consider the propagator
01:20:48.490 --> 01:20:50.987
for a heavy quark.
01:20:50.987 --> 01:20:52.570
And we'll come back
to the Lagrangian.
01:20:56.890 --> 01:21:06.990
And we'll consider it with some
on-shell momentum, which we'll
01:21:06.990 --> 01:21:09.570
consider an on-shell
momentum to be parameterized
01:21:09.570 --> 01:21:11.560
in the following way.
01:21:11.560 --> 01:21:13.200
So if I have an
on-shell momentum,
01:21:13.200 --> 01:21:17.640
I'll say that p is equal to
MqV, so that p squared is
01:21:17.640 --> 01:21:23.310
equal to Mq squared
and this V is 1.
01:21:23.310 --> 01:21:25.590
So once I pull out
the mass dimension Mq,
01:21:25.590 --> 01:21:28.440
the remaining thing I call V.
And that's just some parameter
01:21:28.440 --> 01:21:32.850
which squares to 1 on-shell.
01:21:32.850 --> 01:21:36.990
Now, if I have kicks, which are
from light degrees of freedom,
01:21:36.990 --> 01:21:39.780
then I don't have
exactly on-shell.
01:21:39.780 --> 01:21:45.240
So P mu is MqV, which is like
on-shell piece plus something
01:21:45.240 --> 01:21:48.480
small, k mu.
01:21:48.480 --> 01:21:52.620
And k mu is of order lambda QCD.
01:21:52.620 --> 01:21:56.280
So this is like saying the
on-shell piece plus the tickle.
01:22:01.380 --> 01:22:03.240
And if I want to
construct the propagator,
01:22:03.240 --> 01:22:09.120
then the propagator is encoding
the optional off-shellness
01:22:09.120 --> 01:22:10.740
of the degree of freedom.
01:22:10.740 --> 01:22:13.510
That's 1 over the off-shellness.
01:22:13.510 --> 01:22:14.760
So it'll depend on the tickle.
01:22:24.710 --> 01:22:33.200
OK, so we could just take QCD
propagator, which is this,
01:22:33.200 --> 01:22:34.530
and just plug in that formula.
01:22:50.100 --> 01:22:52.920
There's Mq squared, which
cancels this Mq squared.
01:22:56.900 --> 01:23:00.650
Then there's some cross-terms
that don't get cancelled.
01:23:00.650 --> 01:23:04.760
So there's one that comes from
the dot product of the V dot M
01:23:04.760 --> 01:23:05.540
with the k term.
01:23:05.540 --> 01:23:06.050
That's this.
01:23:06.050 --> 01:23:07.550
And then there's
the k squared term.
01:23:07.550 --> 01:23:10.130
So I have these.
01:23:10.130 --> 01:23:11.420
And then I can expand that.
01:23:19.700 --> 01:23:22.310
M is a positive
quantity, so it doesn't
01:23:22.310 --> 01:23:25.070
change the sign of the i0.
01:23:25.070 --> 01:23:27.650
And if I expand in M, then the
leading term looks like that.
01:23:27.650 --> 01:23:30.350
It's M independent.
01:23:30.350 --> 01:23:32.750
And then there's some order
1 over M terms, which I could
01:23:32.750 --> 01:23:33.917
also work out what they are.
01:23:38.150 --> 01:23:40.610
OK, so that we could expand
even though we don't, a priori,
01:23:40.610 --> 01:23:42.426
know how to expand
the Lagrangian.
01:23:46.400 --> 01:23:48.562
We could also think about
vertices in this theory.
01:23:48.562 --> 01:23:49.520
This is the propagator.
01:23:49.520 --> 01:23:51.380
What about vertices?
01:23:51.380 --> 01:23:53.090
And again, this is a top-down.
01:23:53.090 --> 01:23:55.820
So we can think about
vertices in QCD.
01:23:55.820 --> 01:23:58.340
So if this is a
heavy particle here,
01:23:58.340 --> 01:24:00.890
we could think about
what would happen.
01:24:00.890 --> 01:24:02.000
How do we expand those?
01:24:06.460 --> 01:24:08.210
And there doesn't look
like there's really
01:24:08.210 --> 01:24:11.390
anything going on here because
there's nothing to expand.
01:24:11.390 --> 01:24:14.540
But because these things
are heavy particles here,
01:24:14.540 --> 01:24:17.870
you realize from the
propagator formula
01:24:17.870 --> 01:24:20.090
here that you're going
to have 1 plus V slashes
01:24:20.090 --> 01:24:23.670
on each side of this guy.
01:24:23.670 --> 01:24:26.880
So for a propagator
on each side,
01:24:26.880 --> 01:24:28.807
you can make a simplification.
01:24:35.350 --> 01:24:38.740
And that's because 1
plus V slash over 2 gamma
01:24:38.740 --> 01:24:44.110
mu 1 plus V slash over 2,
after some Dirac algebra,
01:24:44.110 --> 01:24:48.850
is just V mu times 1
plus V slash over 2.
01:24:51.610 --> 01:24:54.460
So the gamma mu becomes a V mu.
01:24:54.460 --> 01:24:58.520
So the vertice, once you
take that into account,
01:24:58.520 --> 01:25:01.840
is just minus igTA V mu.
01:25:06.700 --> 01:25:08.980
Even if we don't think
about starting with the QCD
01:25:08.980 --> 01:25:10.690
Lagrangian, if this
is our Feynman rule
01:25:10.690 --> 01:25:12.850
and that's our propagator,
we can write down
01:25:12.850 --> 01:25:18.820
an effective theory for that
that gives those Feynman rules.
01:25:18.820 --> 01:25:21.377
And that actually is
the HQET Lagrangian.
01:25:30.640 --> 01:25:37.360
So sorry for going
a little bit over,
01:25:37.360 --> 01:25:40.930
but I wanted to at least
get a little bit into this.
01:25:40.930 --> 01:25:43.990
OK, so by construction
this way, we
01:25:43.990 --> 01:25:45.947
can arrive at this Lagrangian.
01:25:45.947 --> 01:25:47.530
And next time, I'll
come back and I'll
01:25:47.530 --> 01:25:49.660
show you how you
can really properly
01:25:49.660 --> 01:25:52.300
take this limit of
the QCD Lagrangian
01:25:52.300 --> 01:25:55.010
to get the same thing.
01:25:55.010 --> 01:25:56.710
OK, but that is the HQET.
01:25:56.710 --> 01:25:59.170
The lowest order of the
HQET Lagrangian is this.
01:26:02.810 --> 01:26:05.120
So it's a linear V
dot D decoupling.
01:26:05.120 --> 01:26:07.403
V is a parameter,
and so I've put it
01:26:07.403 --> 01:26:08.570
as a parameter on the field.
01:26:08.570 --> 01:26:11.780
We'll see more why that
was done next time.
01:26:11.780 --> 01:26:14.000
And there's a projection
relation on the field.
01:26:14.000 --> 01:26:15.650
If I'm using a
four-component field,
01:26:15.650 --> 01:26:17.192
then I have this
projection relation.
01:26:17.192 --> 01:26:19.590
I could also read it in
a two-component notation,
01:26:19.590 --> 01:26:21.920
and then I wouldn't need that.
01:26:21.920 --> 01:26:24.695
But for four-component
notation, you do need that.
01:26:24.695 --> 01:26:27.610
We'll talk about
that next time, too.