Energy Correlators - Part 2 - Ian Moult

[00:00:14]Um so just to remind ourselves where we were last time. So the the goal of these lectures was to try and develop a kind of formalism for understanding observables um like this where we have some kind of collision and then we kind of measure or an observer at infinity measures a bunch of kind of fluxes that are coming off um to infinity. Um and so last time we started just kind of developing the formalism to describe this and in particular we said that in the case where we really have just particles or asmtoic states it was very easy to write down a specific example of such a detector operator um which was the energy flow operator and I'll just remind ourselves of this um and we wrote down it or we defined it by its action on some set of um asmtoic states. So remember these are just some fox states um like this. Um and we said that it acted as a kind of passive detector by essentially just waiting the states um by energy. So it just takes the states that are going in some direction weights by an energy if they're in the appropriate direction like this and returns the state. Um and so this was simple enough that if one wanted to do things in perturbation theory um where you can just act on the external states um like this or in um uh collisions where or theories where you actually have particles, this then enabled you to um try and compute examples like this um by just sticking in um complete set of states in each of the locations here.

[00:01:52]um like this and then you can just use the action on the states um and compute things. Um and so we had this kind of diagrammatic picture of this where we produce some set of states.

[00:02:04]We have some cut at infinity where we perform the measurement like this and then we can put in the different kind of detectors that we want. So for example, if we put two of them, we just put them at these particular directions um like this.

[00:02:24]Um and so that was very um simple. And so then what we were trying to do was to extend this definition to write it not in terms of just the action on um free particle states, but to write down some actual kind of operator definition which would enable us to extend this for example to the case of um conformal field theories. Um, and so what I wanted to do is just pick back up there, write down these definitions, and then we'll start actually working out some kind of more interesting examples other than just setting up the formalism. Um, so what we were doing at the end of last time was we were writing down this average null energy um, condition operator. And so we were first defining this just on some generic light sheet.

[00:03:02]So in a second, we'll show how it relates um to these measurements. Um so I wanted to write this just a bit more generally in ly cone coordinates. Um and so you'll know that for any uh vector we can decompose it um into two ly cone coordinates which I'll generically call n and n bar. And so just to be very specific.

[00:03:33]So I can write it like this. And then in my null plane, I imagine kind of having a null direction like this parameterized by this vector n. Um so along the null plane, I'll then have my orthogonal or it doesn't have to be orthogonal, but I'll just kind of draw it as orthogonal.

[00:03:48]Um other light direction nar, which in this case is coming um out of this light sheet. Um and then I'll have some perpendicular um component x muerp. Um [snorts] and so we defined last time some class of operators which are smeared um or they're the stress tensor smeared along a particular line at a fixed value of x per at a fixed value of n bar but they're smeared in this direction n um like that. Um and so just in equations I can write this.

[00:04:25]So it's a function of this x per and it's just an integral from minus infinity to infinity along this um direction of when written in these like coordinates dx - t - of the so I integrate only one of the coordinates and then I fix the other two and so importantly I don't have to smear in the other two um directions. So the important point is that these two coordinates, it doesn't really matter what they are um but they're fixed and I just smear in this one um like cone coordinate um like this.

[00:05:04]Um and so then the claim which I won't go through the proofs of but you can find um links in the material which I um uploaded. So the claim is that this object actually has all the properties that we wanted for a detector operator.

[00:05:19]So the claim um is that this object is positive in any state.

[00:05:29]And so we'll use this a bunch um later on is positive and in particular it's positive without having to smear in these additional extra directions. Um so it's positive if I just integrate it at a fixed value of xer um and of this x minus. Um [snorts] so the second property which we wanted was that it annihilates the vacuum um like this. Um and finally um that at different um positions on the light sheet um that it commutes um like this. Um, and so this satisfies all the properties that we kind of originally wanted of our detector. Um, and so now what I want to do is to show why this actually is the detector object. Um, and that we this is just a particular choice of frame, but that this is really the object which one is measuring. um at least when one studies um null radiation um um so now we want to go back to thinking about this um original picture but kind of now having this um average null energy condition operator in mind and so We just want to draw a little bit more what's actually happening um in this picture. Um and so if I now draw in my kind of Penrose diagram, as we said last time, we studied a bunch this source and I'm going to imagine that it produces so it produces radiation going in every direction. But I want to focus in particular on radiation going in some um direction n [snorts] um and so I can set up my null coordinates. I'll do this more carefully in a second.

[00:07:31]U and V like this. And I can do these where I can do always choose these to align with the particular direction of my detector.

[00:07:44]Um and so now or so this is just my particular source which is producing radiation um going out. And now I want to think what I'm actually doing when I have some detector for example like at the large um Hadron Collider. what is that actual calerimeter cell doing? And so that calerimeter cell, it exists for all values of time and it's just at very large um distances and it's kind of moving along. So the one can draw the kind of trajectory of some particular calorimeter cell um like that. Um and so it just kind of goes along um at a very large R trajectory.

[00:08:20]And so the first thing we can do is to just give kind of a not very nice definition of the energy correlator which will reproduce um this definition up above and then we'll try and make this more kind of coariant. Um so the simplest definition to write down what the action of this operator is. So again we just imagine this is some stress tensor which is some idealization of the calerimeter cell which kind of moves along that trajectory and it just picks up the energy that's going in some particular direction. So it's easy to write down a kind of definition of this operator.

[00:08:57]So we know that it's going to be some limit as I bring this detector off to infinity.

[00:09:02]Um for dimensions I have some factor of r that comes there and then I'll have some just integral. So my detector is just kind of riding along this trajectory and it can pick up energy at any particular time. And so I just integrate this over all times of the particular projection um of my energy correlator. So it just wants or of my stress tensor. So it's just picking up flux that's going in some particular direction. Um and then this is just at some location um like this.

[00:09:40]Um and so it's a nice exercise that one should do. So you can just take this for example in free field theory. So you really just write this in terms of um creation annihilation operators. So this is like a kind of peskin um style exercise. You just write this in terms of creation annihilation operators for example for a free scaler. And it's very easy to check that if you act on fox states that this has exactly this action above. So a good exercise I want you to do is to check um that if you act with this operator like this it gives exactly um what I set above like this. Um and so for this to be true you'll see that it's very important that you really take this limit as r goes to infinity. So of course if the operator is at some finite um distance it will inject some energy when it performs the measurement. Um but when you really take this limit you'll find that it behaves exactly as you would want um for a passive detector. And so it's a good exercise to just work this out um um in this very simple case of like a free scaler and just check that this actually um works. Um good check bring this stuff down.

[00:10:59]But so this is kind of a and so this relates to the questions that were asked last time. So in the form that this is written, this holds whether the particles are massive or massless. And so it's a completely general expression.

[00:11:09]Um but it's also very hard to kind of work with because it has no um particularly nice symmetries. Um and it's also not manifestally clear why it relates to this particular definition um of the anc here. Um and so in the case that we have only massless radiation or for example in a conformal field theory, we can actually simplify further um this particular um definition and we'll relate it exactly back to this um anc operator here. Um and so this is again very easy to do um by just choosing a kind of nice set of coordinates.

[00:11:45]Um so in particular in that choice of coordinates which I put up there for V and U as my two choices of the light cone coordinates if I'm measuring the radiation in some particular direction.

[00:11:54]One thing one should so you can write down or in Lyon coordinates um and it's very easy to show. So it's a good it's just a simple manipulation which is again worth trying that if you write it in terms of these u and v coordinates defined with respect to this um particular vector n and you consider just the case where you only have massless radiation. So then intuitively what you expect is that instead of having massive particles which will um for example come off to time like infinity all the radiation will just move in a very nice um it will all move with the speed of light. So you'll just have a bunch of radiation which kind of goes um like this. Instead of having to define the detector in this particular limiting way, you can kind of first take the detector to just be a null integral along this particular direction like this. Um, and then move the detector off like this.

[00:13:04]Um so if we again define these U and V coordinates like this you can check that under um this assumption it's easy to manipulate the above um representation to write the action of this detector in the following way where you now just have this limit as this light cone coordinate goes to infinity. So you're really just taking this thing and moving it off um like this of again just for dimensions you have this v ^2 and then you get exactly this null um anac integral um like this.

[00:13:53]And so this kind of makes sense. Um and so we see that this detector that we have and in particular the kind of idealization of this detector in this picture here where we're trying to measure energy is really just a nice example of this um average null energy condition operator. So this is exactly this anac operator um but it's just placed um at asmtoic infinity or on um scribe plus and so it just sits there and measures all the null radiation um that comes through it.

[00:14:28]Um and so this is very nice because it now allows us to use all these nice properties which we understand about the anc that they also hold for detector operators. Um at least if we're just measuring um energy um and this is also as we'll see later this object when written in this form has a lot of symmetries um which if I just write it in this limiting um form up there are much harder um to understand.

[00:14:52]um good so just as so this is now as written is in some very specific um choice of coordinates and so you can also um coariantize this so I'll just write a slightly more general definition which we'll sometimes use um later so it's easy to rewrite this in the more general um x plus and x minus coordinates um and you'll find that this takes this again this is just a rewriting But this makes it a little bit more coariant like this.

[00:15:44]Um and so in the particular case of a conformal field theory, there's actually a nice trick one can do to relate. So instead of having to take this limit to relate these detector operators which are placed at infinity with ones which are placed on a generic um light sheet.

[00:15:57]Um so we can draw kind of two different configurations um for these detectors.

[00:16:08]So the first is the one I drew before where I have some particular light sheet.

[00:16:13]Um and these detectors or these anc operators are smeared along um these position or along this light sheet and they have some se um transverse separation which is a coordinate um separation.

[00:16:31]the other. If I draw now in kind of a threedimensional Penrose diagram, I can put these two detectors at some different positions on the celestial sphere and now they're separated by some um angle um theta.

[00:16:53]And so it's a nice exercise to show. So you can easily relate these in the case of a conformal field theory by a form of null inversion. So you can just do a kind of um inversion which brings one um or one set of operators to the other.

[00:17:07]And so I'll just write it explicitly.

[00:17:10]So you just invert this x plus coordinate um like this.

[00:17:34]And so this gives a nice relation so that these are in fact in a conformal field theory um exactly related. on in particular you can work out that the kind of relation between this transverse separation is related to the angle between the two detectors. And so in particular in the small limit as you bring these together it's the same as kind of bringing these um together on the celestial sphere. Um um and so these two different setups are actually useful. So I'll just sweep this up here. So these two different setups are both useful for um different um physics applications.

[00:18:15]And so even though we'll mostly focus on the case where the detectors are placed at infinity, they're both actually very interesting um observables. Um and so the first if I kind of align these with these pictures up here. Um so this is sometimes called um the distribution frame. And so this is when the this anc operator is in the bulk.

[00:18:40]Um and this from a physical perspective is actually appears in um what are called parton distribution functions.

[00:18:58]And so in this case what I have or in this null sheet is I have some particular state. So again I'll have some of these light ray operators um like this and then I'll have some particular state which I can compute it expectation value in which you can think of some as some kind of like blob like this which moves through the the detectors. Um and on the theory side um the these anac operators placed on this particular kind of light sheet in the bulk um can be used to prove for example the A and C theorems for normalization group flows.

[00:19:39]Um so when they're placed in the bulk you can use them the positivity um properties which I've now erased but you can use their positivity properties um to prove the A and C theorems. Um so this other configuration um shown up here is this so-called detector frame.

[00:20:03]Um yes.

[00:20:06]Yeah.

[00:20:11]But so here this is for so these are for two 2D and 4D for the A and C. Yeah. Um and so so these so I'll I'll just contrast it with this in a second and then I'll comment on that. Um but so when you so these when you use the positivity of the ANC and you do it on this light sheet in the bulk this allows you to prove properties of um theories which are not conformal.

[00:20:37]Um and so in the case where you really place these detectors at infinity, so this is like essentially making measurements in the infinite IR. Um and so this makes it harder to probe properties about the reormalization group flow. Um but if you're in a conformal field theory, this allows you to prove interesting constraints which are called like Maldoscina Hoffman um or conformal collider bounds um which I'll go through in a second. Um um and so in the detector frame these characterize um asmtoic fluxes.

[00:21:17]Um and these can also be used as I said to prove so-called conformal collider bounds which we'll go through in a second.

[00:21:33]um on either kind of OP coefficients or on certain properties of um conformal field theories. Um and so in both cases you have both a kind of physical application of putting these average null energy operators and computing them in some particular state. Um so you can either view it as this blob here or in that case it's this analog of this kind of radiation going off to infinity. Um and in both cases by using the kind of positivity of the anic but in these different states you can prove um different things. Um and so in this um lecture series I'll mostly focus on these conformal collider bounds because they're a little bit easier. Um but there's been a lot of work on also understanding these um not um at the boundary um but in the kind of bulk and using these for studying reormalization group flows. Yes.

[00:22:24]No, no. So parton distribution functions you can think of as essentially an expectation value of these um in or it's kind of an expectation value of this light operator but in some proton states, right? Because the the the partial distribution functions describe the kind of properties of the of the proton deep inside the bulk. Um and so they're really described by in this kind of distribution frame. Um and then so they they kind of collide something happens in the bulk and then you get radiation off to infinity. Um um and so I won't go through them um in this lecture but I can certainly talk more after but those are are properties in the bulk of of space time. [snorts] Yes.

[00:23:08]>> So they can certainly distinguish dynamics and I'll show how that appears in a number of cases um later on. Yes.

[00:23:14]So hope so one of the things which we'll try and describe um as we build so so far this is kind of completely generic but as we start doing kind of one and twopoint functions we'll see how they actually tell you um properties of the underlying system and in particular so the we'll show first for kind of onepoint correlations it will give very um essentially global properties that are fixed by symmetries but then for higher point functions there'll be kind of a relation between the angles and the scale of the system and so they can't get you out everything um but will you can actually get quite a bit out um from these Um [snorts] yes. Any more questions?

[00:23:48]Um perfect. Um great. So before we go on um to actually do something. So now so far we've just been setting these up. It's a little bit boring. Um but I just want to kind of recap what we now um have and then we'll start actually seeing how these can be used um in more interesting ways. Um so what we now have is kind of two perspectives.

[00:24:20]on these detector operators. So we have this correlator perspective.

[00:24:32]Um and so this applies for example in a conformal field theory or when I don't have asmtoic states. And so here we wrote the detector operator um in terms of this um anic which I'll just rewrite.

[00:24:53]Um and so this is very nice because if I happen to have correlation functions so let's say from Simone's lecture you learn how to compute for example some correlation function involving the stress tensor then I can easily convert this by exactly taking this correlation function involving the stress tensor we do the fora transform that we did for the twopoint function and then you also take this stress tensor and integrate it and do this limit in a particular way. So if someone gives you for example any of these correlation functions you can get out um detector correlators.

[00:25:27]Um and so this is very nice because it in some sense gives a non-perturbative definition in a conformal field theory for how to or what these um observables at infinity are. Um and so in particular because there's been a lot of progress on the conformal bootstrap, if you happen to be able to get these higher point functions, you can use um these definitions to convert them into um detector correlators which are kind of physical observables that you could actually um measure um in some system for example in the lab. Um the other definition that we had was this form factor or um amplitude perspective.

[00:26:09]Um and so here we defined um the action of this operator by its action on um asmtoic states um like this. And so he said that this one was useful if we happen to have. So if you have form factors or amplitudes um then we can convert them um into these detector correlators by essentially doing this um integral of and I'll be a little bit schematic.

[00:27:14]Um and so in both ways this gives kind of a relation to or objects with that objects that we are more familiar with.

[00:27:21]So fortunately there are a lot of nice um correlation functions um lying around and there's also a lot of nice form factors or scattering amplitudes lying around. Um and so this will allow us to compute these in a bunch of different theories. Um and so what I'll now start trying to do is to kind of go through and actually um show you a little bit what these look like um in different theories. starting with the simplest theories where we'll often do it from this um correlator perspective and then when we do it in the case of gravity um we'll use um this definition over here in terms of scattering um amplitudes um yes >> this is just some generic object which creates the state. So remember that the way we were we were creating um the state here was just acting with some local operator and so this could be any local operator that you wanted so maybe so this will be the operator that creates your state and then if I want to do these measurements of energy I need the stress tensor and so essentially what I want are correlation functions which involve some operator I don't care what it is that will just produce different states for me and then a whole bunch of different stress tensors. Um and if I have those um then I can compute these energy correlator objects.

[00:28:36]Um and so in a second what we'll do when we give some explicit examples is I'll give cases where for example this operator is like a a scalar a spin one a stress tensor you can just choose whatever it is and it will produce some different state and then one can measure the energy correlator in that state and it will tell you something about that underlying operator. Um So as a as just a local correlator this is generically for generic positions not positive. Um um so as Simone said if you put in for example like reflection positive configurations then it will be um but so what we do is we let's say you happen to have some such correlator um then what I can do is I can essentially do this integral over these stress tensors and take them off to infinity.

[00:29:23]it will turn them into detectors and then I do a fora transform on these two operators to put them into momentum space and then what you'll get out of that is an actual in this case it would be a twopoint um correlation function um that you could measure in a collider um [snorts] and so I'll show this in a in a couple examples we'll start with the onepoint functions and then move to the two-point functions and then hopefully this will become um clearer but you can really take so Simone I think worked out some three-point functions you can really take those you just do this integral and it will turn it into a correlator um or a detector operator.

[00:29:57]Yeah, [snorts] perfect. Um so are there any more questions about this kind of general setup before hopefully it will become um more concrete? Um perfect.

[00:30:14]Okay, so that was just the setup which is a little bit um boring. Um and so now what I wanted to do is to start actually looking at what some of these correlators look like. And hopefully this will give you kind of a a bit of a feel. Um and so the thing we'll start with are the absolute simplest things.

[00:30:34]So they may seem very boring, but they're actually kind of fun. There's a lot of um interesting physics one can do with them, which are just onepoint functions of these detector correlators.

[00:30:43]Um and so what we want to do is to study in different possible states correlation functions um of this anc um operator.

[00:30:56]Um and so we'll do this in a couple different states. So the states which are simplest are when these states are just produced by some local operator. Um and so in those cases as I just wrote this will reduce to um or this will be computable from threepoint functions involving some um operator the stress tensor and then some other operator um over here um and so as was said in Simone's lecture so threepoint functions so in a conformal field theory these three-point functions are um fixed up to just some um discrete set of numbers which parameterize for example different tensor structures. Um and so we'll see that this kind of manifests in the fact that these have very simple properties that are essentially just fixed um by symmetries. Um so in general what we'll see is that these are parameterized by a set of numbers for different tensor structures.

[00:32:00]Um and so in our conformal field theory, these set of numbers will really be just some constants. Um and we'll be able to kind of put interesting bounds on these constants um which will come from the positivity of this um anc operator. And so this will even though these will be very very simple, we'll be able to put interesting um conformal collider bounds on these coefficients.

[00:32:24]In the case when it's not a conformal field theory, these coefficients will depend on the energy scale of the collision.

[00:32:33]So remember I denoted in general this energy which I inject in as Q. Um and so in general these coefficients will depend on um the energy which I put in.

[00:32:42]Um and so one of the reasons why um this is a little bit interesting is that in the real world for example in QCD it depends strongly on scale. And so I'll show some examples um and you can guess how it kind of depends on scale. And so I'll really try and emphasize when or the distinction between a CTF um and a nonCFT. Um and so what I want to go through is just some very different or a few very simple examples. So I want to consider first the case where this is a scalar then the case where it's some conserved current.

[00:33:25]then the stress tensor and then we'll consider where fi is some colliding gravitons um like this. And so hopefully this will give you some intuition for how they encode the underlying um dynamics.

[00:33:51]The other thing which I'll try and emphasize is how simple these functions actually are compared to the kind of offshell um three-point functions which Simone um wrote down.

[00:34:02]Good.

[00:34:04]So we'll start with the just completely um trivial case. So the the simplest case is a scalar source.

[00:34:17]like this. Um, and so what we want to compute is just the onepoint function in some scalar source. And so just to be very explicitly um like this um and so there's two ways of computing this. So one is slightly it's still very easy but you should do as a um homework um is to actually take the structure. So as I said you can always compute these for the from the underlying threepoint function. And so one way of computing this is to take the threepoint function for a scalar operator a stress tensor and another scalar and then actually perform this forier transform and this limiting uh procedure. Um and so this is a a good exercise to do um but you'll get a very simple answer which you can also just immediately guess. Um and so in this case because my operator is a scalar nothing it's kind of SO3 invariant and so I can't write down so it's just invariant on the celestial sphere and I also know that this object if I take this energy um flow operator so remember that this is at some particular location on the celestial sphere but if I integrate it over all um angles it has to just reproduce the total energy operator right so if I take this and integrate this. This just gives me back the total energy or this Q0 of my source. So then it really becomes just the standard um topological operator and it just receives all the energy flux which is coming off to infinity. Right?

[00:36:06]So what we're doing is measuring it at some particular angle. But if I just smear this with unit weight over the entire future infinity, it of course just receives the total energy. And so this is a kind of ward energy constraint which is just saying that energy is conser um and so in this case it's very easy to write down the total answer because it has to be uniform and it has to integrate when I integrate over the sphere um back to the total energy. Um and so this is just essentially a trivial answer that it gives a uniform distribution on the sphere um with the total amount of energy. Um and so this is completely trivial um to write down but it's a good exercise to also get it from this threepoint function. Um and so again it's this is very easy to do. And so you know that or from Simone's lecture that for this there's one tensor structure.

[00:36:57]So there's one a priority unknown coefficient. Um but then you have the ward identity um which fixes this one unknown coefficient which is of course just also um a statement of the conservation of um energy. And so if you work through this um carefully, you'll of course get back um this trivial result. Um um and so this is the simplest and kind of most boring example of an energy correlator which is just a kind of uniform distribution um on the sphere.

[00:37:25]Um any questions about that?

[00:37:28]Um and so I should also say of course that this does not depend on whether this is a conformal field theory or not.

[00:37:33]This is just completely fixed by conservation of energy um and it being uniform.

[00:37:43]Good. So now we'll move on to something which is slightly more um interesting.

[00:38:00]Um so the next case we can do is when our source is a conserved current. And so again I'll I'll restrict to four dimensions JMU. Um and so this will be much more um yes doesn't >> in this particular case. Yeah.

[00:38:28]>> Yeah. So if you so if you take the conformal um threepoint function and take this limit you will get this. And so this is a kind of exercise in the CTF just to show that it gives the same um result. But of course you can also just directly argue this um from symmetries um without any recourse to conformal symmetry. Um and so essentially in this case the reason why is there's only a single um tensor structure. So in all of these you for the energy cor you'll always have this one integral constraint. So you'll always have this ward energy constraint, right? That when I integrate it over the sphere, I have to just pick up all the energy that comes out. Um, and so that's true as long as I have energy conservation. And then if I only have one tensor structure that I can write down by Loren variance, then that just completely fixes it. So in these next cases where things will become more interesting is when you have multiple tensor structures and then this ward entity will fix one of those coefficients and then I'll have some unknown coefficient um and that's why it will tell us you it's either some property of the underlying dynamics or by demanding positivity you can learn something um from it um right because so for example the anc we said is positive and so this answer if I say that this is positive it tells me nothing interesting um but in the next case when When we do this, we'll show that it actually tells us something um interesting.

[00:39:48]Great. Any more questions? Yes.

[00:39:55]>> Yes.

[00:39:58]>> Absolutely. Absolutely. Yes. Yes. Yes.

[00:40:00]Um and so in that case what will happen is you'll get some matrix which is positive semidefinite. Um and that is something one can certainly do. Um but here I'll always take the two just to be the same for simplicity. Um but of course you can take uh you can take interesting mixes so all the sources which I give you you can take all their cross combinations and make some matrix and if you do that and demand that it's positive you get very interesting constraints um and so this is something yeah that has been done for um by Maldosina and Cordova um for instance um great so this generalization to the conserve current will be much more um interesting and so there's two reasons so one on the theory side as I just mentioned mentioned we'll get these nice conformal collider bounds um in the case when it's conformal. And so the other reason why this is interesting is in the real world one happens to have a nice source of um spin one currents. Um so we can actually on the phological side we can actually measure these And as I said, the reason for this is that one can use um the photon um with some particular energy to excite my um system and then I can measure things in that state. Um and so what we say so we'll we'll go through and understand what it form should be and then you can actually go out and test that this works um in the real world. Um so in the case that I have some current I now have a polarization vector um which I'll denote by um zed um just to make sure that it's not the same as the epsilon for the anc um and so what we want to compute is this onepoint function of the energy flow in this um state produced by the current. And so just to be very explicit and then I'll again normalize it just so this thing has dimensions of energy.

[00:42:26]And so now physically the reason why this is slightly more interesting um than that case is I now have um two different vectors right? So I have one which is specifying the kind of spin of my source. So I can really think of this as a physical like um spin um and then I have another which is where I'm making my measurement off infinity.

[00:42:51]Um and so now this thing can depend on the relative angle um between um these two but again it will be essentially completely fixed by symmetry up to one single number. Um and so again there's two ways of computing this. So one way is by taking so the harder way um is to take the JT J3oint function and again once you do this so you'll know that this has two tensor structures plus one ward identity um so we expect one unfixed um coefficient Um, and so we can also see that just by writing down the answer in the same way that we did over here. And so that now there's two possible structures we can write down. So I can always pull out some Q0 over 4 pi. And then can I can have some part which is just uniform on the sphere plus some coefficient which in the general case depends on the Q ^2. So this will just be some function of the collision energy if it's not a conformal field theory. Um and then I have some structure which is just some bilinear in the polarization vectors [snorts] and I can choose this particular normalization so that when I integrate this over the sphere this term just goes to zero and this term just gives me um Q0 right so this satisfies um the ward identity I can rewrite this just in terms of this physical angle here Um like this.

[00:45:02]So by construction um this satisfies this ward identity um but now it has one unknown parameter.

[00:45:14]So before um but now we have one dynamical parameter and so this tells us something about the underlying dynamics of the theory. So as compared to the scalar case we have one dynamical parameter and so importantly so in a conformal field theory right this is just some number so in general it's a function of the energy but in the case of a CTF t this is just some um constant which is often called um a2 um and so this of course these onepoint energy correlators have a very um simple physical um picture right they're really just if I have my um sphere they're just a kind of picture you can think of them as some kind of heat map of where the energy is kind of hot or where it's fold. Um so they really just have some simple you can just plot this as some very simple just kind of picture on um your sphere which for simple states will always just kind of have this very simple so if you have spin and it can just have kind of coast um to the 2n um and so there's a very interesting observation which was made um by Hoffman and Malisena um which is called the conformal collider bound um and so it's it's essentially a completely um trivial statement um but it's ends up being very powerful Um and so what they said was that the energy which I observe on the on the sphere has to always be positive. Um so they just demanded that in this particular state that this is greater than equal to zero.

[00:47:27]Um and it's very easy to work out the consequences of this. So if you solve um this equation, you'll find that a2 as a function of q ^2 is bounded between minus3 halves and three. Um and so this applies whether the theory is um conformal or not. So remember if it's conformal this is just a constant and it's between these two values. If the theory depends on some scale um for example in real world QCD then it still has to be bounded between these for all values of the scale. Um and so it's very easy to come up with theories that saturate either of these bounds. So to i.e that this is the best that one can possibly do. Um and so write down those two scalers and then we can or those two cases and then we can guess what it looks like um in real world QCD. Um [snorts] so an exercise that you should do and again this can be done just very simply with tree level um climate diagrams is to check that this is saturated by free scalers and free firm. Um so you should check just by computing.

[00:48:36]So it's really computing like a tree level. So it's just two particles which come out um back to back. And so if you do that for a free scaler, you can just write down the current and you can work out the final rules um and compute it and you'll find that this gives you um a2 equals 3 um and if you do this for a free firm on.

[00:49:19]So with the standard sibar gam mu si um this will give you a2= minus3 halves um like this um and so the thing which is kind of interesting is by in this so you can also prove this um using standard bootstrap techniques um from the four-point function in uklidian signature but by looking at this particular class of observables you can essentially get this constraint completely trivially right so you just say that this is positive And it's not very hard to you don't need some fancy semi-positive semi-definite programming to extract out that these are the um the kind of bounds. Um and so you can check again as I said that these are you can't do better than this um that this is the best you can do and it's saturated um by free theories. Um yes, >> I think in this case this is just a coincidence because this is a little bit about how I've just normalized it. Yes.

[00:50:18]Um perfect. So now what I wanted to do um or yeah sorry well so the conformal collider bound is just if I remove this um Yeah, I mean I think this is just partly a nomenclature. So in the paper by Maldiscino and Hoffman, they focus primarily on the case of conformal field theories, but for these things it it applies completely generically. Um which is also one of the reasons why it's um interesting. Um yes, >> the best saturates the >> Yeah. So in this case it just saturates the bound. Yeah. Yeah.

[00:51:09]And you can work out so you can give a slightly more refined reasoning by so for this twopoint function we projected it onto different partial waves. Um and so you can also so instead of just taking this and kind of looking when it's positive, you can also project it onto the spin zero and spin one partial waves. Um and it's it's actually the the free formula and the free scaler. They saturate these two different partial waves. Um so you can you can if you kind of work through and draw the FL diagrams it's there's a little bit better way of understanding it. Um but it's essentially just yeah they saturate this.

[00:51:41]>> Yeah.

[00:51:45]>> Yeah. Just massless. Yeah.

[00:51:48]>> Mass will contribute. Yes. Yeah. Yeah.

[00:51:51]Good.

[00:51:53]And so now what I want to do so these have all been in very simple um conformal field theories. And so now to just to have some fun with the real world, one can guess what this looks like um in real world um QCD. Um um so now in real world QCD. So one of the reasons why this is interesting is that of course I can actually excite my my vacuum with a photon at some energy Q ^2 and I can see what comes out, right? And it better obey if I measure the onepoint function it should obey um these constraints. And so in the real world one can go out and measure as a function of Q ^2 A2 um coefficient.

[00:52:43]And so as we said from this it better be bounded from below by minus three halves. So it can't be um below that. um and it has to be bounded by above by the value three. Um and so can anyone so in a CTF it's of course independent of the scale. Um but can anyone guess what this will look like if so if I actually so with a collider I can just measure this because this is really just a photon. So this is an experiment that has been done. Um and so one can check that those bounds are obeyed and it also will tell you a lot about the underlying dynamics. Um, and so can anyone guess what this looks like in real world QCD?

[00:53:31]So maybe yeah. So at very high values of Q ^2 if I hit some photon into the QCD vacuum, what do I produce?

[00:53:43]Yeah. Good. So exactly. So at asmtoic freedom I should have a essentially free firmian and so it should be minus minus three halves. Yeah. Um, so perfect. So up here, it should essentially saturate. So at very high energies when I send in an E plus E minus pair on the vacuum when I produce our quark and anti-quark and as you said by asmtoic freedom, they're essentially a free um quark and anti-quark. And so I should see that it asmtotes at infinity to minus three halves. Um, so as I move this down, so if I do this at very low energies, so if I go back to like the original E plus E minus colliders that were at like point8 GEV, um, what would they see or what do I produce?

[00:54:33]Yeah, pions. And are pions bzons or Yes.

[00:54:38]Very good. And so at this energy or at very low energies, if I collide my E plus E minus pair, I'll produce two pons um which are bzons. And so this will be um saturated um up here.

[00:54:56]Um and then what you do in the real world if you plot it, it has some kind of complicated structure with some resonances, but then it actually looks exactly um like this. Um and so you can actually do this with real data. Um and you'll see that it's really is just two pions which are bzons at very low energies um and then two firmians at very high energies and it really kind of asintotes um you need to go to quite high but even by 100 or so GV you really asmtote um to this and so this hopefully kind of answers a little bit the question so even though you're not getting kind of time resolution dynamics this tells you a lot about for example the structure of the underlying um particles in these examples um um and so of course as we go to higher point functions they'll now be actual functions of of more kinematics and they'll tell you um a lot more um good.

[00:55:42]So are there any more questions about this? Um yes >> so I'll show you so in QCD so the yeah so the value at which it hits zero in like super symmetric theories so it will always be zero because it's kind of some balancing um in QCD I'm not exactly like precisely sure of the exact value but it's some it's it's kind of where there's some mixture of firmians and bzzons it's it's essentially becoming quasi super symmetric at that point um um but unfortunately Yeah. So in in this particular case we have control kind of here and um here and so what goes on in the middle there's a lot of kind of resonances and you see all the like omega fi and stuff and so you really can't compute it um in here and so at the place where it crosses zero you don't have good control because so for free firmians it's bounded by minus three halves and so when you compute perturbative corrections I think the first one it's like plus 9 halves of alpha s or something. So the problem is to get from minus three halves to one is non-perturbative. So you just you lose any control but it's also it's far from where you can use let's say kyrronian and it's far from where you can use perturbation theory and so you have essentially no no real control and so you can kind of say that it's behaving kind of super symmetrically um but but in reality it's less clear what's happening. Um [snorts] excellent. Um any more questions about this um example? Um >> this one I forget. This is maybe like five or six GV. I can I can show you after a plot. Um it's relatively low. So I think I think the J you can see on it the JSI and I forget which side of of the J side it is. Um yes. Um great.

[00:57:41]So in the so we'll go to the anomaly coefficient in a second. So in the case that the theory is super symmetric right so the anomaly coefficients will come from the tttT3oint function which we'll do in a second. So if the theory is super symmetric then I can relate it to the ttt 3 threepoint function but in general this is just some some object.

[00:57:57]Um >> yeah yeah yeah.

[00:58:08]Yes.

[00:58:08]>> Yeah.

[00:58:13]>> Good. So in the in the 3D case, this still gives you some bound and it's just harder to So we'll we'll do the TTT in a in a second where it gives some bound on A over C. And in the in the 3D case, it's just some bound the OP coefficients that appear in that um threepoint function. And so in some cases you can interpret them as I'll discuss in terms of certain nice known quantities.

[00:58:30]Otherwise like in in QCD I don't know how to give an intrinsic definition to this object. It's just the the kind of the coefficient that appears in the in this threepoint function. Um >> exactly. So in in in in yeah so in even dimensions we can relate these to nice known quantities. Um otherwise it's really just some bound on the on the size of the OP coefficients. Um yeah which is is is non so in both cases it's it's interesting. Um yeah so in one case it bounds anomaly coefficients in the other it's just some bound on OP coefficients. Yes. [snorts] Great. Um, so now I'll swap this.

[00:59:11]And so I'll pull this down just to keep that.

[00:59:16]So I want to do one more example with the um with the current because again the current is kind of interesting because we can actually do it in the real world and so it's fun to make um real world plots.

[00:59:31]So the other case which I wanted to do also just to illustrate that you can measure things other than energy um was is to consider measuring a charge correlator um but also in the state produced by the current. And so again the reason for that is that we can do it um in the real world.

[00:59:56]So the second example um in charge or in so as he said if we measure the energy the reason why this is nice is this related to this JTJ um threepoint function. But the other thing which you can do which ends up being quite um fun is you can also measure charge.

[01:00:35]And so this will relate it to a three-point function of three currents um which can in general um be different currents. And so this will be the kind of fun case that we'll do in a second to relate it to anomalies. Um and so just like how we constructed this um anch operator to measure um the energy flow, we can also construct an operator which measures charge. So if I imagine that I have some U1 charge with some current J.

[01:01:07]Just like I did for the stress tensor, I can also compute a charge um detector which lives at some um position on my celestial sphere. And again I just construct this in exactly the same way by integrating my current um like this. And so if you want you can also take this limit to make this directly in terms of a light ray operator. Um and so one important thing so this object will not be positive um um but nevertheless we'll see that we can actually get um something interesting from it. Um so what we want to try and do is to compute the onepoint function.

[01:02:04]So I want to compute the onepoint function for some charge in the background of some other um spin or some other conserved current. So I want to consider here the case where I have one conserved current which produces my state and then where I have one conserved current which I measure here um like this and so the reason why this will be interesting so or just as one um constraint sorry so just like how when I took the energy correlator and integrated over the celestial sphere I got back the total um energy. So if I take this detector and I integrate it, this is again just the total charge. So this is just like um recovering the total um charge flux and will just give me the total charge of my operator.

[01:03:03]So there's a there's a kind of fun case of this um of this correlator which is very interesting.

[01:03:14]So what I want to do is to consider the case where I have two currents. So I'll call them JV and JQ.

[01:03:27]And this will be the one that I make the detector out of. Um, and I want to consider the case where these have a mixed anomaly like this.

[01:03:44]And so in this case, the threepoint function like this has a very simple interpretation. Right? So this is exactly the standard triangle diagram um that you compute to understand the anomaly um like this um and so in particular so this anomaly at least in a conformal field theory the anomaly or the coefficient of this anomaly completely fixes this um three-point function here. So it has one it has some tensor structure which one has to work out but then the coefficient is exactly related to this anomaly coefficient. Um and so then one can ask what happens when we turn this into a detector operator and try and measure this charge flux um on the celestial sphere. And so again there's two ways of doing this. So one is just to take this three-point function and um perform this kind of standard limiting procedure and compute the correlator. But we can also just write down um the answer.

[01:04:57]So again what I want to do is to consider the state produced by these two um background currents um and then measure in this state this charge correlator like this. And so again in general this will depend just like in the case we did before of the energy.

[01:05:20]So I'll have some polarization vector which tells me about the spin of this operator and then I'll have some angle at which I'm measuring um my detector operator.

[01:05:33]And so because of the anomaly what you actually get out is an object which is odd um under um as n or it's kind of odd as I move this around. So instead of being this kind of co squared it will actually be odd in the angle theta. Um and so you can write it down. And so what it is is it's exactly the anomaly coefficient um between these particular currents.

[01:05:58]And now it has some parody odd epsilon jk structure like this. And so if you go just into this frame as we did before where I really write this in terms of the angle um like this, this goes exactly like um the coine of the angle. Um, and so what this says is that if I really do like a kind of heat map like this, what it means is I'll get kind of let's say depending on the sign of this, I'll get all the charge going kind of in one direction. So it essentially produces kind of a current where I'll have like positive charges going along the direction of the underlying spin and then I'll have kind of negative charges going in this direction. And so it really kind of produces an asymmetry of the kind of charges moving in your detector um from the underlying anomaly.

[01:07:09]Um and so it's some kind of fun kind of macroscopic consequence of having this um anomaly. Um and you can also understand very cleanly where this anomaly comes from using this approach of just weighting the asmtoic states. Um and so you can kind of think of what this triangle diagram. So the way you would compute this using the fment diagram approach right for example in electron positron colliders like this um like this. So then we put our cut um shown here and then we wait um with the charge operator like this.

[01:07:55]And so we see that what we're actually measuring when we measure this J or this onepoint function of the charge is we're exactly getting essentially a unitarity cut of the standard triangle diagram.

[01:08:06]And so it's exactly giving the kind of anomaly which is giving us this kind of asymmetry um in the charge flux. Um and so it's kind of a fun way of using these um um detector operators to relate to kind of things that you normally work with um for kind of local correlators.

[01:08:22]And you can really kind of see macroscopically um in your detectors that you have some kind of underlying anomaly um in your theory. Um um are there any questions about that? Um perfect.

[01:08:38]So again, just like how we did the case of um I'll do it. I'll sweep these. So just like how we did the case of real world um QCD for the onepoint energy function, we can also give an example in the real world um where we actually have this um um so there's kind of a cool example of this in the real world.

[01:09:16]So in real world colliders, I can collide for example um two Z bzons like this and then I can consider what would happen if I measure electromagnetic charge. And so of course one knows that um the electromagnetic current because it's a gauge symmetry is of course not anomalous otherwise we would have trouble.

[01:09:52]Um but in the standard model or in a kind of collider experiment I can essentially split this electromagnetic current into the electromagnetic current um carried by barons and the electromagnetic current carried by leptons. Um so if I measured the full current which is not anomalous I would get zero um asymmetry in my detector. Um but if I write this electromagnetic current and again this will be very schematic as the electromagnetic current on lepttons plus the electromagnetic current on experimentally I can actually go out and measure this just the the kind of charge coming on my lepttons and the charge coming on my berons. Um and so just if you do you happen to know if in the standard model if I just take the electromagnetic current on lepttons if this is anomalous or not.

[01:10:56]So it is it is anomalous. So famously in the standard model right the anomaly cancels between the leptons and the baronss or between the quarks and the leptons. Um and so in the standard model both this electromagnetic current restricted to lepttons and restricted to barons are anomalous. And so if you go out and measure this charge correlator on just lepttons you'll see that you get some kind of flux an anomalous kind of current going in one direction of the lepttons you'll get it going in the other direction from the berons but if you add them up you get a perfectly um zero distribution because there's no underlying anomaly. Um and so it's kind of a fun real world um application of this anomaly um which you may think that you don't kind of see these microscopically but you can see them very nicely um in colliders. Yes.

[01:11:47]only states that are >> if I only have states that are well so part of this will depend a little bit.

[01:11:59]So, so remember that in doing this we've in this case we've chosen like a state which is charge neutral. So we'll always have this constraint and I erased it.

[01:12:08]But when we integrate um like this that this is the total charge and then I can just surround my operator. So this is just the charge of my original operator, right? And so if I only have positively charged states, I would therefore also have to have a positively charged operator. And so then what I could have on top of this is I could have some kind of DC offset. And so in this case when I've said it's just cos theta that's because my um like total charge of my operator is zero but more generally I can have like some it will just be like q plus um that cos theta. Yeah if it has an anomaly. Yes.

[01:12:50]Perfect.

[01:12:52]Any other questions about um this setup?

[01:12:55]Um yes. So in Yes. So this DJJQ is just the anomaly coefficient. Um and so it's just a constant. Yeah. Which is you really just computed. It's just the standard like um kind of peskin calculation of this triangle diagram for the particular group theory factors that you have.

[01:13:16]Yeah.

[01:13:20]Great.

[01:13:22]Okay.

[01:13:24]So that covers the um spin one case. And how much time do I have left? Uh okay. I think I have one minute left. So I think so what I want to do um next time is exactly what was asked about is to just quickly go through this um for the case where this source is not spin one but it's spin two. And so this is where you can relate um the kind of analog of the bound um that we drew um I'll just pull it down. So up above when we did this bound on A2 and studied it um in QCD. So what we'll do next time, this is harder.

[01:14:03]Yeah. So in QCD we did this nice um bound in the case where we have a spin one current. And so next time we'll quickly go through this for the case that it's a stress tensor um which gives another very nice uh bound on OP coefficients and then we'll move on to kind of twooint functions of these energy flow operators which get into a bit more dynamics and relate them to kind of um some or the the kind of spectrum of twist two operators um in your theory. Um but we'll do that uh yeah tomorrow.