Jung-Wook Kim - A first look at Magnusian bootstrap

Um okay so uh thank you for the organizers for bringing together uh this wonderful workshop and um I must say that uh this is my first time in Saklay and I've been enjoying the region quite a So uh today's talk will be based on uh these uh series of papers um that I've written over the uh last two years with um these fabulous collaborators. Um you'll find some familiar names and uh this is topic that uh I've started thinking uh very recently um about so we may call it the magnesium bootstrap. So uh let's get started. So uh the first thing that uh you'd ask is what do I mean by the magnesium bootstrap? So um um since this is something that could be new uh let's break try to break it down a bit. So that's the first thing uh that we recognize is the word bootstrap. Um if you don't know what a bootstrap is I've drawn a picture of it here. Um you can see that uh I'm not very fond of AI. Uh but in the physics context uh bootstrap means that you start from a set of uh consistency conditions for example symmetry unitarity and then you try to uh constrain um the dynamics that you can get from these uh consistency conditions and um the next thing that you recognize is the word magnusian. Um um you may see it for the first time but this just means that you're looking at the logarithm of uh the s matrix. So the logarithm of the Smatrix uh we could call it the Magnusian and the matrix elements of this log uh we call it the Magnus amplitudes and um the kind of questions that I'll be trying to answer is um twofold. So first um we want to compute magnus amplitudes. Uh can you actually compute them without uh directly invoking in the lranji or um the action of the field theory that you're starting with. The second is uh more close to what people usually call uh the bootstrap in the modern usage which is positivity bounds. So can we constrain um for example the uh values of Wilson coefficients um using magnus amplitudes and um this is a very uh we've just started scratching the surface of this uh program and uh what I'll show you today is that we can actually generalize u the for example the BCFW recursion relations to Magnus amplitudes and uh one thing that kind of surprised me is that the the time delay as a consistency condition on um the causality of your effective field theory is actually more natural to consider uh using magnus amplitudes. So um the motivation for uh thinking about the log of the s matrix comes from uh gravitational waves and you can see that um this is the reason I was put on Friday. This is the gravitational weight day and uh if so Stephano did a good job of explaining what post meanovkin dynamics is but uh if you forgot uh it's just starting from free particle theory of special relativity and you're starting to add corrections from GR and here what you wanted to do was to take the classical limit and it turns out that uh when you want to take the classical limit of the Smatrix it makes more sense sense to talk about the log of the Smatrix because uh one of the things that we know about the Smatrix elements is that they have the so-called hyperclassical terms. So uh there are a class of um Fman diagrams which resums into a phase. They're usually known as iconal amplitudes. So this is what it is. And if you resum this uh two two to two scattering amplitudes um you get a leading term and a subleing term which you can expand in h bar but it comes with one over h bar at the front and uh because of this when you expand out uh this exponential you get something like 1 / h bar to the uh second power 1 / h bar to the third power and these are the so-called hyperclassical terms and these are terms that uh makes taking the classical limit a bit more tricky. So what people uh realize is that we can try to look at the log of the symmetric and one thing that uh people later on realize is that um this is not actually equal to what we call iconal which is the face up here because um it the taking the log of the Smatrix element is not equal to taking the matrix element of the lock and Uh so how are they used? Um they're used to generate observables. So this is the uh so-called kok formalism combined with this exponential form of the s matrix and what you get is nested commutators like this exponentiated commutators. And when you take the classical limit um you just substitute these uh commutators by pos brackets and this is why we can call them um the generator of observables and uh they're called the magnesian or magnus amplitudes because they are computed by the magnus series and uh this formula here how we compute the observables from magnus amplitudes um they'll come in handy a bit So uh the next question that we can ask is uh why do you want to do bootstrap using magnus amplitudes? Um the smetric bootstrap is already hard. Uh you're throwing in an extra log into it. It it which makes it even harder and my motivation here is IR divergences. So one thing that we know about IR divergences is that um they tend to be very closely tied to exponentials. So we have the vineber phase which is uh a phase. So it's sits on exponential. Uh we also have these uh five of kishing operators which also takes a form of exponentials and there are a lot of other things for example there's this uh cuspenolous dimensions which also sits on the top of an exponential and other things. And another thing that we know about IR divergences is that um they seem to obscure some of the uh positivity constraints. Um for example, Julia has worked on some and this is the uh the reason that I started thinking about this uh program. Of course, um we're only starting to investigate uh this direction and unfortunately I wouldn't have anything uh deep to talk about these cutoffs but let's hope for the future. So I've talked about magnus amplitudes which are matrix elements of the log of this matrix. So what are they? So these are definitions. So the scattering amplitudes they are matrix elements of the S matrix itself which are computed by the Tyson series and we have a diagrammatic uh representation [clears throat] of scattering amplitudes which are known as Fman diagrams. Uh much of it uh follows for Magnus amplitudes. It's just a matrix element of the log and it's computed by the Magnus series. And one difference is that instead of finement diagrams, they're computed by so-called Magnus diagrams which has directed edges here. And um the meaning of direct edges uh will come clear in a bit moment. So what's the Magnus series? Um so let's imagine that uh we are back in our first year of PhD uh which is a bit of a miserable time and uh what we learn in the first class on quantum field theory is the Smatrix. So it's comput uh it's defined as uh by the transition amplitude given here overlap between instate and out states which is computed as a matrix element of this s and in practice it's computed as uh the asintoic limit of the time evolution operator given like this and we compute it using the Dyson series where uh we have the exponentiated uh exponential of time uh time ordered exponentials. Uh when we solve uh for the log of this matrix, we're doing most the same thing. It's just that uh we're using a slightly different um expansion where we instead of using the uh u time evolution operator, we write it as an exponential of omega. Something like this here. And this gives us a formal series expansion of the kai or um the log of this matrix. And uh this might be your first time looking at a magnet series. And there is a good reason for it because if you're doing the computations by hand which is typically uh typical for first year PhD students you don't want to do this because um the integration domain is non-uniform. So you have orderings between time integrals. There are also nested commutators like this. And although uh the numbers that sit in front look like expon uh one over factorials they are actually related to perui numbers and so on. So um although the topic the magnus wrote this series solution in 1954 uh it didn't really fly until very recently because uh you don't want to do these calculations by hand. Now uh we know that perturbative quantum view theory you can just define it as diagrammatic expansion. So uh this is what's done for the magnet series uh done by the magnus diagrams and the main difference uh from fment diagrams is that you have slightly different propagators. So because the magnus series is an expansion in commutators and time orderings the relevant twooint functions or propagators are green functions and the so-called hotmeric cut functions which are just uh anti-commutators of the fields and um these hot functions uh appear when you start considering the loop expansion. So uh they'll only appear like uh very tangentially in this talk. So let's compare fman diagrams versus magnus diagrams. Um the fman diagrams uh we just draw all possible diagrams. Uh the edges correspond to fment properators and uh these diagrams uh are summed up into something like this. So you divide by the symmetry factor and then uh the integrant is computed by just multiplying all the edge uh edges and uh the vertices. Magus diagrams you're it's more or less the same. It's just that these edges have directions related to uh causal um prescriptions of the properators. And one difference here is that um you do have the same symmetry factor and the same uh integrant that comes from multiplying all these vertex rules. You have an extra weight factor uh which is called Mura coefficients. So these are weight factor associated to different portions. And just to give you some examples, uh this is uh one of the diagrams that you draw for five point amplitudes. Um the integrant is just a product of a green functions and vertex rules and the symmetry factor uh because of external legs it's one. Now if you consider a slightly different uh magnus diagram where you now have directed edges because they correspond to propagators uh well propagators. Um the difference is in the propagators as seen here and you could reverse uh one of the directions and um the grid function has changed um the direction and you'll see that the the weight has changed a bit. So before it was 1 over 6, now it's 1 over 3. And the symmetry factors um they actually compute uh they are computed uh with the directions. So uh for showing non-trivial symmetry factors uh I've included these uh vacuum magnets diagrams. So for on the top um you don't really have a symmetry factor. for the bottom uh you can uh reflect left and right. So you have symmetry factor of two. And one of the things that uh we worked out is that these coefficients can actually be computed just by uh working with diagrams uh which relies on um solving differential equation corresponding to uh the solution for the Magnus series diagrammatically uh which is what MAS formula is doing and also there's a Hoff algebraic uh approach uh using graphs and what it turns out is that uh um rather than using these recursive definitions, it actually is a bit easier to uh bootstrap the con uh the coefficients using consistent conditions and these uh coefficients actually seem to know uh something about the physics. So uh one interesting relation that's uh satisfied by these coefficients is what we call edge contraction rule. So here is the uh ma coefficient for the original diagram. Um now you take one edge of the original diagram and reverse the edge and then when you add them up it becomes the moral coefficient uh of the same graph where that edge has been uh contracted to a vertex. So these are some examples. Um I'm not sure if you can see the numbers but uh here it's 1 over3rd. This is 1 over 6. And this edge when you shrink it, it becomes 1 over2. And when you add 1 over 6 and 1 over 3, uh you get 1/2. Uh the same is true for uh this this one here. So in the middle um this one is contracted. So the left one has 1 over 4. Uh this one is 1 over 12. And you get 1/3. Um some non-trivial example is this one where um this this diagram has 1 over 60. This one has minus 1 over 60. So when you add them up you get zero. And this has a very nice uh physical interpretation. So when you're doing when you're approaching the physics from a Wilsonian EFD point of view, you can consider this uh uh edge that has been contracted to a point as a degrees of freedom of a heavy degrees of freedom which gets integrated out into an effective vertex. [sighs] Uh another interesting property that's satisfied by Mura coefficients is what we call um fman reduction. So it's uh uh relation satisfied by uh tree diagrams. I think there should be a loop generalization but um you haven't managed to uh actually write down one yet. So what happens here is that you first fix a topology of the tree graph and you consider uh all edge orientations uh of that graph. So given like there and when you add all the relevant uh coefficients that actually go into magnets diagrams what you end up with is uh one of our symmetry factor of the fman diagram. So these are some examples. Um when you add these four different uh uh edge uh direction precisions uh you get one over three factorial which is the symmetry factor of a fman diagram which looks something like this. And on the bottom uh it's more or less the same. you have four uh uh edges uh attached to this uh center vertex. The only difference is that you have some terms which are negative but all you also get one over four factorial and this means uh when you're computing magnus amplitudes uh you can just consider them as uh scattering amplitude with uh additional absent friction given by the mor coefficients at least for three graphs and this is why you can just generalize uh the the the recursion relation ations or recursive methods that has been developed for scattering amplitudes to uh magnus diagram magnus amplitudes for yeah so for example you start with magnus amplitudes up to endpoint amplitudes what [snorts] you first do you just erase off all predictions and consider them as all uh fman propagators you do a recursion for example bcfw to get uh n.1 n plus one uh amplitude And then you just reconstruct uh the absent impression. >> Sorry, [snorts] this is not obvious to me because there's the pole but there's a combinatorics. So for fment diagrams you just have the symmetry factor of the graph. >> Mhm. >> So it's clear that factorization will work. But here you have this mural coefficient. So why why is this obvious that when you put something on shell split as a product of two magnet amplitudes with their correct coefficient? Um no so it's more like uh you can consider magnus amplitudes as scattering amplitudes with special absent frictions that's what I'm >> but there's also the combinatorics >> um >> shell on a magnus diagram >> mhm >> then the weights also factoriize this ma coefficient factoriize or >> factoriize um >> it's not only the pole but it's the fact that the numerator That's the right thing. But >> right, but the numerators are the same for >> except except the symmetric except the coefficient is this combinatoral. Well, well, the the coefficient um yeah the combinatoric coefficient is what you're adding over the >> but does it factoriize I guess is my question like >> does the symmetry >> I have a graph and I and I say I go on shell does [snorts] now the moral coition become a product of the things on the left and the things on the right I mean if the recussion works it must be the case that it's not obvious at Well, I'd say that the ISO frictions are going to be slightly different because you're reconstructing the IPS from the the scattering amplitudes. So, uh it's probably because you're going through scattering amplitudes. You're not just directly using the magnus amplitudes here. So, I think that's one difference there. So the the basic uh picture that I have here is that um when you consider scattering amplitudes as just sum over fment diagrams then um the mor coefficients are just there to add uh how you should uh give the absent frictions. So uh if you think in terms of fman diagrams then um I think this should be obvious. >> Okay, maybe I can ask you later. >> Okay. Right. Right. So, so that was the the onshell recurs uh on shell bootstrap that um I've uh alluded to uh as the first target. Uh the second target is positivity bounds. Um if you're uh new to positivity bounds, it's just uh trying to find inequalities on coefficients uh to get uh to carve out the space of consistent EFDs. And the target uh positivity bound will be the positivity of the time delay uh which is related to um causality of your uh theory. And this is a nice uh observable because uh time delay is an observable which uh actually fits into this uh exponentiated uh bracket structures computed from uh Magnus diagrams and uh it has been argued that uh if you have a resolvable um time advance uh it means uh you have uh occasional behavior in the effective field theory. um and that's been argued in uh using different uh arguments. So there's the gal theorem. Um also you could argue that you can form a closed tight curve or um you can start from microality or uh analyticity on the oper plane to uh derive this result. And our interpretation will be that if you get something that seems to be a casual then it means that you need some completion of the theory. So you have to add new operators or new degrees of freedom. And uh for technical reasons uh we'll adopt what's known in the literature as IR causality where you look at the the difference of the time delay with EFT effects and without EFT effects and uh it's allowed to have small negativity because uh this only this casuality uh only applies when you have actual resolvable uh negativity. So um the time delay uh in uh relativistic context is usually computed using iconal amplitudes. Um our proposal is to generalize this you uh to magnus amplitudes. So first thing that you do is to take the expectation value uh using wave packets. It's more or less similar to um going to impact parameter space. And uh our proposed definition is something like this. So you have the exponentiated uh pos brackets. And when you work out uh exponentiated brackets acting on the time coordinates and when you work out uh that the leading term is exactly this differential. So this literally means you're computing the shift of the uh the time coordinates of the world line of your particle. And one nice thing about this uh generalization is that um there's in principle um no uh obstruction uh in going to uh multiparticle kinematics for example 323 scattering and uh one of the problems known for iconal resumation is that um nobody really knows how to do it beyond 222 scattering. So uh when we talk about time delay um we should also talk about how it's experimentally measured. Um so this um the slides that I prepared to show how it could be measured in practice. So up there we have the source here we have the observer and uh because uh and here's the black hole which gives you the the scattering potential. And what you see first uh what happens here is that um so the source bursts and after a bit of time uh you'll see um the the burst from the trajectory where uh the the light has traveled uh far away from the black hole. A little while later you'll see um the the burst for the trajectory that's closer to the black hole. And what you're comput uh and what you will measure is the difference between the times and uh this will be computed as the difference of delta t's computed using this um time delay uh uh proposal and uh one thing that I'd like to mention here is that this delta t is actually divergent in uh four dimensions. So that's why uh we are using this definition here. >> [sighs and gasps] >> So uh the the class of theories that we'll be considering is Einstein Maxwell plus FFR plus so something like this where you have this EFT scale um with EFT cutoff G given by lambda um these are the uh the Laurens invariance and uh because it's easier to work with dimensionless Wilson coefficients uh and also it's easier to work with length scale Um this is the normalization that we'll be taking. So the EFT length scale is just one over lambda. Uh it also has extra factors of one / h bar and dimensionless coefficients will be uh written uh using C. So uh we are doing EFD. So first thing that we have to do is to analyze the scales of the problem. So uh what the the problem that uh we'll be considering is um wave scattering on black holes spinning black holes uh in the geometric optics approximation. So this is just a spinning plus magnesian generalization of uh what Stephano and Andy and Gab worked on a few years ago. So there are basically five scales. Um the first is the Compton wavelength of the black hole background. Now for astronomical uh astrophysical black holes um this turns out to be astronomically small than the plank length. So you can just set it to be zero. The second is the EFT length scale and the the wavelength of the scattered gra uh gravitational or electromatic wave. And there's the scale because you want to have the energy of the scattered photon or graviton to be smaller than the EFG scale. Next there's the so-called uh spin length of the black hole uh which is angular momentum of the black hole divided by its mass. Uh this is related to the horizon scale of the black hole and the largest scale is given by the impact parameter which is uh inversely proportional to the transferred momentum onto the electromatic wave. And from these parameters uh you can con uh construct dimensionless numbers used for the expansion. So uh we're throwing off um this component wavelength. So we have uh yeah three expansion parameters. So oops yeah so EFT uh wave wave optics corrections and um spin corrections. Well, there's also the the gravitational correction known as the postman coughing correction and um this there's this uh extra cutoff because uh we don't want to we don't want the impact parameter to be too small because uh the rays will just fall in into the black hole. Now one thing that sort of surprised me is that um the EFT scale is smaller than the wave optics scale. So it means that EFT corrections are less important. Well, the wave optics corrections are more important than EFT corrections which means you have to include information of the polarization of uh the gravitational wave or electromatic wave that you're scattering and this means uh we have to consider um 2x2 matrices which uh encode uh all scattering uh information. So you can have a polarization preserving scattering plus to plus minus to minus or polarization flipping uh scattering plus to minus and minus to plus and uh because this system has been studied uh already for non- spinning case. Uh what we'll be focused on is um whether uh if you add spin to black hole um does it actually give you better positive bounds and uh this is how we set up the parameters for um one thing that you only need to remember is this combination here. So sin theta um if it's plus it's retrograde uh scattering. So if your black hole is spinning like this then in the the retrograde orbits uh you go against the spin. So it's going around like that. For prograde uh you go along with the spin. So when you compute the time delay uh for photons uh this is what you get and uh if you apply the IR katic condition you just take off uh the first line which is the GR contribution and this is what you get as uh the time delay and you can see that the only place where uh spin of uh the black hole enters is uh in this part. So we'll focus first focus on uh the uh R cube corrections being zero. So this is what you get as the causality condition. Uh it's given in terms of absolute values because uh you can see that there's plus and minus oops. And uh in this setup you can just set your dimensionless Wilson coefficient to be one to fit fix the uh the EFT scale by beta. And this is what you get as uh the the constraints on your Wilson coefficient. And it turns out that uh you do indeed get a slightly tighter bounce from uh including spin. So one thing that you see here is that uh for retrograde orbits um although you have greater [clears throat] time delay for fixed values of impact parameter um this doesn't really give you a better uh bound because for retrograde orbits um it's there is a limit on how close you can put the trajectories close to the horizon and uh it you actually get better bounds for prograde orbits because although the time delay is smaller um you can actually put the uh the trajectories closer to the horizon and the refinement or the uh the improvement in this uh factor here is about uh 30%. Oops. Yes. Uh we can do the same thing for gravitons. Um this is what you get uh when we set uh our fourth uh corrections. So uh our fourth corrections to zero and um the same thing. So you just set the Wilson coefficient to one absolute value of the Wilson coefficient to one and this is the bound you get. It's a two-sided bound and uh more or less the same pattern appears here. So for alpha squar greater than zero uh you have retrograde orbits which gives you greater time delay but uh because the uh the range of impact parameters that you can put is uh more limited um you don't actually get refinement uh you actually get refinement from the prograde orbits where you can place your orbits closer to the horizon and this is the same for alpha prime squared smaller and zero. Now uh something interesting happens uh when you consider the R4th corrections. So if you turn off R cube corrections, this is what you get as um the positivity bounds and um this small negativity that's allowed on the right hand side just invalidates any meaningful constraints. But um our question was do spin effects improve the bounds? So um just as a toy example we'll consider this strict positivity instead where we set um this spin factor here uh to be one. So uh this is plus one for retrograde orbits. Um you can just flip it to minus uh for prograde orbits. That's what we'll do. And this is the uh uh bound that you get if you don't have spin. So you have this um opening here. Uh it's given by the cone and uh you do get uh somewhat uh refinement um if you add spin effects. So this is for B uh impact parameter uh equal to 7GM. This is the uh critical impact parameter for retrograde orbits for extreal occur. So that's why we choose uh chose seven. But um this cone gets uh this cone is actually from prograde orbits. So you can be a bit more brave and uh set B to be a bit smaller. In the extreme case you can set it to be 4GM and then this cone becomes a straight line. So for extreme occur prograde orbits uh the critical impact parameter is 2GM. So it is possible that this is actually allowed but um we didn't want to uh be too brave and >> doesn't this tell you that >> setting the right hand side to zero this doesn't make sense because for inance in string theory we have r cubed zero and actually the fourth not zero right so they only on this line [snorts] >> y so it is likely that this is going to be modified >> by even at level >> well this is actually one loop but Um yeah >> one loop >> uh right so we're considering two to2 scattering where one is the the black hole so it's massive massive scattering and um this bound really comes in when you consider one loop because uh R4 correction >> one classical >> yeah yeah yeah classical one loop yeah so even if you uh turn on negativity um it's its effect is to just bring down this tip to uh the diagonal line. So um this straight line persists but uh I think it's probably going to be modified when you add higher loop corrections. Yeah, >> I'm surprised because I would suggest that the difference between the R R cub the fourth needs to be suppressed by GM [snorts] but that's not the case, right? It's the first color by not by order one. So in string theory order one >> it's just two different za values or something >> right >> one of them is zero the other one is >> zeta 2 or the four I forget or three >> okay so there the uh ef scale would be the string length scale um let me think yeah well we can talk about >> [laughter] >> Oh, no, no, no. So, sorry. Um, yeah, sorry. So, this is the Okay. Uh, no. This is the different uh Yeah, I should go back to the action. Yeah. And now I see what you mean. Yeah. Right. So this is the coefficient for these two when you set >> a is it cosmon >> or or >> no no no no it's going to be r to the four because >> r to the fourth >> yeah yeah >> so the par even and par o >> uh it's actually par even pieces so one is the just reman squared one is the one with the r r star squared yeah so it's considering between This one and this one. Yeah. So, this tells us that you have to turn on these two at the same time. Right. Um, okay. So, I should start wrapping up, I think. Okay. Right. So I've tried to um convey that uh looking at the log of the symmetric could be an interesting thing to study. So uh it was originally motivated by uh studying the iconal approximation or the iconal phase of scattering amplitudes but it turns out that it's actually something different. Um so the magnesian refers to the log of the S matrix and uh magnus amplitudes are the matrix elements of the magnusian and they're different from uh scattering amplitudes uh because they use different properators and you have to add extra uh weight factors for these properators. And uh we've did very elementary exploration of how known bootstrap methods could be generalized to Magnus amplitudes and uh we've seen that the onshoke recursion relation for example BCFW recursion relation could be generalized um generalized because we are really going through scheduling amplitudes and also that uh some of the positive bounds uh especially the positivity of the time delay which is unobservable seems to be more naturally defined uh using magnus amplitudes. Now uh there are some future directions. Uh the first obvious one is to extend this on bootstrap to uh loop level. So can we actually generalize uh unitarity methods that's used to construct uh loop integrants? Can they be generalized to magnus amplitudes? That's one question that I'm thinking about. Another thing uh that we should think about is uh can we actually understand the analytic structures of magnus amplitudes. So this would mean that we have to um develop uh the so-called lambda analysis um that has been used for scattering amplitudes to magnus amplitudes and this will also be important for understanding the IR diver structures and uh if you think about it and this was actually my motivation for starting to consider this uh bootstrap in the first place and of course this would uh require us to have a better understanding of these moral coefficients. So um this is what I prepared for today and thank you [applause] [applause] >> okay plenty of time for questions. Yeah, be a general question. Uh I mean you mentioned about the IR the bet maybe the better properties uh regarding the IR. So would would these be better defined for a CFD where you know the uh like say n equals 4 young ms or something uh where the usual s matrix is not very well defined. And um is there a possibility that these amplitudes can be made more? So kind of general just >> well um we don't talk about the matrix and compromis. >> Exactly. So but maybe there is something like whether something like this could make sense for some reason. Um >> I think for CFDs uh it's a bit different because um so the log of this matrix uh um so in CTF bootstrap we really consider uh correlation functions which are uh directly um observables on their own. uh here we're uh going through we're um computing magnet samples which are generator of uh observables. So we are not directly looking at the observables themselves and that's what I would think we are doing. Um [snorts] does it make sense to talk about log of correlation functions? Um maybe but uh also uh here we're what we have in mind is that uh we have a uh weak coupling expansion so we can make sense of the perturbative expansion and uh in the usual case where CTF bootstrap is very strong uh we really work with a strong coupling so um I think there is a gap uh to be addressed there but uh for n equals 4. Uh we could also consider we uh we coupling. Yes, that's true. Well, one thing that might be interesting here is um so we have uh Wilson lines which are uh path ordered exponentials in n equals 4 for example. And uh if you think about the mathematics the time ordering and path ordering is not really different. So it is possible that we can apply these magnet series tools to Wilson lines. That is one direction that I've been thinking about. But um I think I'm not sure if this is the direction that >> that could be interesting because in in strong coupling there's also a semiclassical picture. >> Mhm. >> Okay. Here you're doing recoupling but we know that in some regime those lines also give you exponential something. >> Right. Right. Yeah. >> Right. So let me just try to make it a little sharper. I mean at least there are some infrared divergences that exponentiate and therefore when you take the log you might think that there would be something additive and therefore maybe up to some additive piece that you can drop >> you might have some well definfined objects. >> Yeah. Yeah. So um I think uh for the current understanding the the IO divergence has to be like multiplicatively uh subtracted and that's one of the uh maybe you can correct me but yeah that's one of the uh the subtleties in actually applying these positive bounds uh when you have high divers maybe we can avoid uh this complications and that was like my motivation for going into thinking about this ction. Yeah, >> maybe I have a question. This hop algebra that shows up, is it the same hop algebra that is involved in this PHD proof of ability or is it different? usual story of the fact that you can reormalize by adding capital terms in sub graphs and like the kind of non Wilsonian proof of >> okay that I'm not familiar but um for reormalization the half algebra considered there is the the so-called CK half algebra and uh con Kramer's I think that's the name so it's a different half algebra yeah >> I think they're closely related algebra the original minimization >> they're related. >> I I think yeah I think this is kind of a generalization. [snorts] >> Well the the paper that we first saw well not first but uh the the set of papers that uh we first saw this half algebra they do mention the relation between the two but um I think the the the mathematics behind the half algebbras are a bit different. So my understanding is that this one is really uh related to composition of functions uh concramers um I don't have a good picture but yeah uh at least in the context of remalization it's related to picking out the the divergent pieces. Yeah, >> I think there's like a secret relation between composition of functions and picking up >> could be. Yeah, >> more questions. >> Okay, not let's thank again [music]