 Zato sem tudi počekno pošličnjem, da smo vseznih teori, zato je to počekno vseznačno, da je to, da je je zelo, da je zelo počekno, da je izgleda izgleda izgleda izgleda izgleda izgleda, da sem počekno pričen, da je to, da je to, da je to, da je to. In tudi se način je, da je to, da je to, da je to, da je to. In becomes able to solve the case, the current decision to solve the situation and it will start with a very brief review on perturbation theory at large orders, and how you borrow some, under what condition you can do that and so on. What we know, this is not clearly a very well known subject. Then I will show how you can take a geometric picture to this problem, summarize fats known since decades, I would say. Then this geometric picture is something which I think came out more recently. And then this concept of exact perturbation theory very recently. And then based on this geometric approach, which we will never need unless to have the grand picture, we will just show how to do computation so using computing loops in pi to the 4 and then res�Japanese. And I would show you how our approach is different from previous historical approaches to the same problem. So approaching phy2-d4 in using perturbation theory is clearly not new. At least the epsilon expansion which has since decades. And also there is another approach at fixed dimension. So these are the historical approach. We will take another right with different approach. And then finally, I will show you mit my laptop and the some results.단 So since this is a working progress, takes me just made the disclaimer that some of the plots I will show to you, will not be the final one, but hopefully the grand picture should be correct. So this is basically the work in progress as I said, hopefully will appear at some point with Gabriel Espada, PhD student at SISA and Giovanni Viraloro, which is stuff at ICTP. Vse moraš predstaviti vse prejnje vse vse. Vse sem pripravila, da se hvalača izgovorila vse ko je zelo, in v kventu vse veči, in v kventu, sem željo noh redu z takovime konversje. Zelo sem se, da sem pripravila, da tako, istim, čistin kaj naživaj na konverzijstvu, z vsem radijos. Če, nekaj. If this radius is exactly zero, means that you are expanding around a non analytic point. So this can be very easily seen by a sort of modern argument of Dyson historical very nice argument. So the modern Dyson argument is very general and very nice and very simple, so take an arbitrary theory, so correlation function connected or 1pi, whatever. konektor, koralizacija, vsek, kofikuracija, v momentu, v spesji, ne zelo. I here, I want to emphasize the dependence on h bar. So, this is going to be in a Euclidean version, this is a pat integral on your field, e to the minus the action over h bar, and then with the insertion of your fields, phi 1 x 1, phi n x n. So, this is the standard textbook things that we know. So, here what do we notice here? So, loopwise expansion means that you started this in a semi classical limit in which h bar is supposed to be very small. Actually, you expand around h bar equal to zero. So, the leading configuration is the classical one, then the loop correction, you have the one loop and the higher loop. But then you see here that the point h bar equal to zero is singular. You immediately see that, because in the limit, if I take h bar at zero minus, this exponent becomes positive and everything blows up. So, this correlation function at zero minus is simply infinite. So, I'm assuming here that this is positive definite, so this is a good exponential suppression, so configuration are well defined, but if they are well defined for zero plus, they are clearly ill defined for zero minus. So, you don't need to do big thinking to see that h bar equal to zero as an exact function, this cannot be analytic at zero. And then you see that if we consider coupling constant expansion, which can be related to loopwise expansion, which are what we are going to do today, and it's particularly true for simple theories like phi to the four, this will also all true for the coupling constant expansion. So, the point coupling g equal to zero, say in the phi to the four, is going to be a non analytic point, okay? So, this implies that if you expand in perturbation theory, you will get an asymptotic series, as you know you will not get a convergent series, and then the point is what we can do then with perturbation theory, right? So, with perturbation theory, you can do two things still. So, let me be brief because I guess you know these things. So, one thing that you know, so if I have a variable, say z over g, I write z because I remind some sort of a partition function, but this can be any correlation function, and g is the coupling constant, h bar or the coupling related to h bar. So, when you do perturbation theory, you write this as an infinite series, well, infinite, as some point you have to tronkate, but in principle it's an infinite series of this four. I don't put to the quality because I just said that this doesn't make any sense, this object mathematically, so this cannot be equal to this object that is your physical observable. So, we write this sort of symbol just to denote that this is the asymptotic expansion of that variable. So, since this doesn't make any sense mathematically, you may wonder why we do perturbation theory, but then it turns out that if the theory is asymptotic for small values of the coupling, you still can get meaningful result. So, in particular, if this z n grows at large n, has some factorial power, and then with the coefficient a to the n, which is typically what happens both in quantum mechanics and quantum field theory, this is the behavior for parametrically large n of the coefficient of the expansion. So, if this is the case, then you can simply see that here you can do what is called an optimal tronkation. So, you look at when, at some point this z n is so big that no matter how big is g small, you will start to get worse and worse result. So, if you do this with a trivial sterling formula, you find that the n best, n best is the value in which you should tronkate here this sum. This is of the order one over the modulus of this a times g. So, at the given g, given a theory, there is the best number of loops, so to say, after which you shouldn't go on, because if you go on, it's becoming worse and worse. Clearly depends on the coupling. So, you see when the coupling is very small, ordinary perturbation theory n best is very large, which means that you can reliably use perturbation theory to many, many orders. So, in Q and D we will never worry about, you know, this asymptotic nature of the theory. We had to worry about other things, but. In principle, this whole thing break down even cautiously before you reach n best. Given that the function, given that you don't know what this function does, couldn't you couple function z of g, which starts disagreeing even for smaller, you know. Everything is possible because, you know, this is not, this is no longer mathematical precise. Well, I will argue now in a second that this is unlikely because this behavior will tell you how the Borel sum and form applies, but I tell you this is not a regular circle. So, what you say can happen, although it has not ever been seen as far as I know. So, I mean, of course, if you have more and more particles, you have other, other combinator factors and things get fewer and less. Yeah, yeah, I mean, the situation in realistic theories is much more complicated, but the general picture is typically, yeah, okay. I mean, you see in four dimensional theories there are many other problems. So, whatever I'm saying applies to few systems, in particular, to the five to the four. That's why I'm telling you, but if you go to, say, QCD or QAD, we have other problems that I will briefly mention here. So, there is a limited, at the moment, applicability of all these ideas to large orders. So, renormal ones and so on. So, there are big issues here that... So, I have in mind, you know, simple quantum field theory, simple system, low dimensions. Otherwise, we have other problems. So, anyhow, this is what you can do, and then once you do that, you see that there is an intrinsic error, which is simply what is the tail of your truncated sum that you can also estimate, and then you will see that this goes like e to the minus one over A times G. So, you see that there is an ambiguity. You will never be able to get a result more accurate than this, but the simple fact that, as you know, asymptotic series are not uniquely defined. If I have one asymptotic, they do not uniquely define a function. If I have an asymptotic series, there are an infinite number of functions that have the same asymptotic series. So, this sort of ambiguity is related to this fact that you will never be able to have accurate description. Okay, simply because of the asymptotic nature of the expansion. So, the other things you can do is try to resum the function. As you know, this is also a possibility. So, infinite sums, divergent sums can be given a meaning with analytic continuation and other means. So, the best, the most powerful method to resum a series is the Borel one. So, what you do essentially is the following. You define another function. So, given that, I trivially define another function by simply dividing by the factorial, my original coefficient. So, it's a pretty silly thing. And then I can redefine a function, which I call it ZB of G, which is simply the Laplace transform of this BZ. So, if you do that, it's a trivial exercise, you can see by naked eye that if I do this and you expand this function back, you recover precisely the asymptotic series that you found. But if somehow you are able to find this function in a closer form, this is going to give you a well-defined function, which I call ZB of G. Sorry, this is GT, otherwise there is no G dependence, right? So, now this is, okay. Now, this BZ of T, as an analytic function in T, you see that in order to this to have sense, it has to be analytic on T plus, because we are integrating over this positive semi-real axis. If it is not analytic on T plus, we say that this series is not borrowed summability. So, the notion of not borrowed summability means that the analytic property of this function do not allow to do this. And, okay. In general, even if this function exists, namely, BZ is borrowed summable, there are no singularities on T plus, in general, this is not guaranteed to be equal to the original one, okay? I give you a very simple instance of this, which is in quantum mechanics, if you consider the supersimeric double well, so this is an example of this simple fact, probably one of the simplest example you can imagine, physically interesting. In quantum mechanics, if you consider the supersimeric double well, so you might know that this is a famous model, it undergoes spontaneous symmetry bragging, dynamical spontaneous symmetry bragging, which means that the ground state energy is zero, as a function of the coupling, is zero to all orders in perturbation theory. But then, it actually, is non-zero, non-perturbatively, because of instantons. So, this has been studied by Whiten in his famous paper on dynamical supersimeric bragging. So, in particular, E0 of G goes like E to the minus constant over G plus other times, I mean, it's a non-perturbative effect. So, you see here that if I blindly consider borrors, zero is a borrors summable, so this is a series in which I have all zeros, which means that BZ is zero, and then ZB is zero, and then I will conclude that E0 of G is zero, and clearly, this is not true, because E0 is not zero. So, you see, I have a trivial borrors summable function, which does not reproduce the full result. So, there are some analytic conditions that should be fulfilled. So, whenever you have, you should know analytic properties of the observables, under some assumption, then this becomes an identity. So, you need to know some extra information in order to establish that this is true, and when this is true, we say that the function is, your ZB, your observable Z, is a borrors summable to the exact result. So, you have to add this to the exact result, meaning that not only you, you can borrors sum your Cs, but actually you are not missing any non-perturbative effect. So, this is what we are interested in, not really to borrors summability in itself, but when we are insured that borrors summability gives the full result, with no other missing non-perturbative pieces around, because if this is the case, what's the point to go to strong coupling? Okay? So, you might ask, okay, but this is not easy because we had to know the analytic properties, the exact analytic properties of your system, and this is typically a complicated question. I mean, we don't have an easy answer to that. Luckily, this has been done for some simple system in the past, so we do know a bit on simple system and the analytic properties of this Z, and then there are some result about whether or not you can be borrors summable to the exact result. Now, let me just, to give you some intuition, tell you a bit more on this BZ of T, because we are assuming here that this is analytic on T plus, but let me just shortly give you some intuition on which condition this can be analytic on T plus, okay? BZ. So, the simplest things to do is just to look at it, to consider a theory that has, no, this unsyntotic behavior, as I told you, this is a very general situation, then you see that if I apply to this specific case, if I compute BZ of T, okay, this is roughly given by sum over n of A to the n, A, T to the n, at large n, no, because the n factorial is removed by hand, and then this is simply power series, it goes like one over one minus AT, so very simple. So, you see that there are two situations, depending on the sign of A, so if A is positive, namely means that you have a series with all same sign series. If A is positive, you have a singularity here that is at one over A on the T plus, so A positive, same sign series, it's an indication of not bothersome ability. Straight away, I mean, you should stop at the beginning, okay? So, if A is negative, in more complicated situation, A can be complex, but okay, I don't want to enter into that. So, if A is negative, oscillatory series, things are much better, in particular, you have here that the singularities is on the positive, on the negative real axis, so this is in principle bothersome ability. Why do I say in principle? I say in principle, because in general, this is only the large order behavior of your function, and then what you are able to do with using the large order behavior is having the access only at the close and singularities. So, if I have a T-plane here, and then you have your borel function, so I'm able to explore only the asymptotic region close to T equal to zero, because next to leading correction, here that, of course, in realistic theories, you will be there, one over n correction, will give you possibly other possible singularities in display. So, this is not enough. So, sometimes in the literature, people now claims that, knowing this, you can claim a bothersome ability, but that's not true. This is only telling you that the leading singularities is on the correct side, so it's not on T plus, but then who knows about others, okay? So, in principle, this is borel, and still you have to consider the rest possible, the emergence of other singularities in this plane. Of course, the farther distance is the singularity, and the milder is the singularity, the milder it is, because with this exponential suppression, at some practical level, if the singularities appears very, very far away, you might not even care with some precision, okay? The closer you are, the more important is the source of singularities. So, genetically, then what can happen, just let me give you one minute introduction just for pedagogical purposes on this other approach that I'm not taking. So, typically, what can happen is that you can either have A greater than zero, so direct singularities already at the leading level, or some other singularities that you can find by doing a better analysis here. If you have a singularity at some point on this, on the positive axis, still you shouldn't really give up, you can still keep going, and the way in which you can keep going is the following. So, if this is the zero, the point of singularities, you can deform your contour, let's say I plus, passing up or down of the singularities, so I up or I down, I plus, sorry, and I minus. So, this is what is called a lateral border summation. You avoid the singularities by trivially deforming your contour. So, that's fine, but clearly, it's obvious that there will be an ambiguity because this difference is non-zero, and for this particular case, it's simply the residue of the pole of this singularity. The residue of the pole is something which has to do with this e to the minus one over g that I wrote, it's an unperturbative effect. So, then what can you hope, a hope that you can have is that, okay, you do this, and then you have some ideas because the sum is more or less, it's well-defined that the difference is not, and then you hope that maybe, if this singularity has to do with some non-perturbative correction, then I should add to my series another series, which is now the series around that is non-perturbative correction. This will give me rise to another side of the point, if you wish, with its own perturbative series, and hopefully, there will be another ambiguity there that will cancel this one. And then by playing, then there will be, yet maybe another ambiguity as a bleeding level on this instant on, let's say, instant on sector that might require to add another one, and then you keep going in an infinite chain, and if this succeeds, you actually have reproduced in non-ambiguous result for your observable. So, this goes under the name of resurgence, and it's a possibility to reconstruct using this saber-tubation theory around multisettles your final result. So, yeah, I'm not taking this approach. But that, by itself, only guarantees that you found some prescription ordered by order that's borel resumable. You still do not know that that... Yeah, yeah, there are many... Yes, yeah, yeah, yeah. There are many, many issues that I'm not... Yeah, you see, I'm in a few minutes, I get rid of an entire... So, there are many aspects that I cannot... Well, many in which I simply do not know, I'm not an expert enough, and some I don't have time to review. This is still we are a bit far away from our... So, just to tell you that there is this other approach, okay? So, the best... The ideal approach instead is when you have simply something which is borel resumable to begin with, no instant on contribution, just one series will give you the full result. So, this is the goal. This is the dream, right? So, the... In 75 Ekman, Manjen and Senior, the mathematical physicist, I have proved mathematically, axiomatically, that five to the four theory, well, first of all, by building on some other work by Glim and Jaff and others, and Spencer, they have proved that this theory makes sense. Namely, exist at the nonperturbative level, which is, usually, we take it for granted, but it's not obvious. So, there is a well-defined continuum limit. There is a well-continuum limit in the UV. There is a well-defined infrared limit, a large infinite volume. So, this theory seems to make sense at infinite volume and knock it off. Not only the five to the four, but also in two dimensions, but also in three dimensions. While they were not able to prove the existence of the five to the four, and in fact, no, we believe that this theory doesn't exist, if not for the three theory. So, this gentleman, not only proved that this theory makes sense, but they actually proved that all correlation functions are borre-sommable to the exact result. So, they proved borre-sommability. Borre-sommability, again, I always use in the sense of to the exact result, not on this technical sense of not analyticity. So, they proved that there are no instant on contribution. If you are able to do perturbation theory, you will find all the result. However, they proved as some assumptions. So, the first assumption is that this is in the broken phase, and then they were able to prove, axiomitically they are very powerful, but then you have a lot of conditions. So, they have been able to prove this for an infinitesimal disk close to zero. So, for the values of g, which is, say, less than some epsilon and clearly the real part of g has to be greater than zero. So, g is the coupling of the phi to the fourth. So, g equal to zero just said that it's always an analytic point. So, they say that there is an analyticity here. In this regime here, which is an infinitesimal disk around epsilon, sorry, around zero, on the positive real sides, all the green function are borre-sommable. So, I am confused here. So, first of all, as far as I know, it is proven that this theory exists for energy. So, the theory exists for... No, there are two things. The theory exists for energy. Borre-sommabilities has been established in this regime. But what does it mean, sir? This function, this function bz, it should exist and be analytic, right? Even if you want to have borre-sommability at energy, no matter how small, you will have to establish the existence of this function bz. Yeah. Which they must have done. Yes, but you... They don't... They bypass the problem of b of z, because there are conditions for which, you know, if I tell you what are the analytic properties of my z, not b of z, some analytic condition, and then you can prove indirectly the existence of a borre-representation in terms of an asymptotic series. I haven't seen... No, but what is the problem? The problem is in the existence of the integral or in the improvement that integral agrees with the exact result, which they prove by other means. No, no, they prove, as I told you, they won't go through this explicitly to this integral. They don't need to do that. So, they prove that there are mathematical theorems that tells you that whenever a function has this asymptotic form series with some coefficient in a domain, provided that there are some analyticity properties of this function, which is, I can tell you, it's... Marko, perhaps I did not familiarize the question. Before you told us that what it means for function to be borre-resonable, now you are telling us a different phenomenon for function to be borre-resonable somewhere, but not everywhere. This you did not explain. What does it mean for function to be borre-resonable somewhere, but not everywhere? No, I'm telling you, what they prove it. I will show you, in fact, that this regime will extend at least up to the g critical. I don't understand how, given what you explained before, how does what they prove agree with what you explained before? Before you said, either the function is borre-resonable? No, no, sorry, I mean, this was... I'm sorry, I was too quick. I was too quick. It's understood here that there is a domain, which is the domain of analyticity of your original function, where this makes sense. It's not everywhere. So you start with a theory that has this representation, some domain. So everything I said is always in the domain. It's not for energy. It's not the whole complex plane. This will be too much. There is always a domain behind, in which I'm able to establish this. So domain, perturbation theory, is typically a little disk close to the origin, because I'm simply exploring the function there. Nobody is able to know whether this function here, this asymptotic series, makes sense for large values of the coupling. So that's why it's always, whatever I said, applies locally in a domain. I repeat my question, sorry Marko. I don't understand, if I don't understand the problem, other people don't understand. Consider that integral. You said they did not consider that integral, but now you explain us in terms of this integral, so you have to show how what you are telling now agrees with your previous... Yes, I'm simply telling you that... Does that integral exist for any g? It's not for any g. It is not for any g. This exists for the values of g, let's say dg, whatever, in a domain in which this... I told you that this is not true. This is only true whenever this function is in the properties. These analyticity properties have been established by these people in a domain. So I can tell you they can show us that the Boretsome ability and if we wish the existence of this function is in a domain. This is... Which function? This z is a function which has to be defined on... But they never... You don't need to construct this function. You don't need, but you went through that. We will go through that, but now we are asking the question to everyone. Does this function exist for this scale? Now it's a concrete question. In order to construct Boretsome ability even for one value of g, take your favorite value of g, even for that value of g, first of all I have to construct this function. Does this function exist? Yes, it does exist. It does exist. I will show you... I should take that function for any g. Given that the function... Because this function does not... I will not be able to prove that this function exists for any g. I will be able to... This function that you show, first of all I construct the function bz. If the function bz exists, I just substitute the argument by t by tg and here it goes. I can consider this integral. You can do that. You can do that, but then you do not know what are you computing. You can certainly do that. But then whether this has to do with your original function is to me unclear. In fact, this is what I am going to tell you. What you are saying you can always do, but then the relation with the original z is unclear. OK? OK. So this is what it has been proven and this is certainly... it's an important result. By the way, this has been proved also by the same authors in another paper for phi 3 to the 4. Sorry, just to clarify, epsilon is a concrete number or epsilon... No, epsilon is more. Of course, let me tell you that unless you do not expect any pathologies this can be also order one. But the statement is that there exists an epsilon in such fact. This is the mathematical statement. I must say that I haven't followed the full proof in detail because it's a bit involved. But yes, that's the final outcome of the result. Now I think I'm doing very badly with time. So... If I wanted to look at higher polynomials like phi to the 6, does this tell me anything about phi to the 6 or is that like an entire new... OK, good. In three dimensions I think it's completely unknown what goes on and I believe there is little hope that you have borrowed some ability also because the theory doesn't really exist if you wish. Phi to the 6, yes. But yeah, with that phi not. But here in two dimensions I believe they've been established for a genetic polynomial that you can have borrowed some ability. OK, so sorry, maybe I'm slowing down too much. Let me now come to... So this is the end of the brief of review. Now let me go on to the geometric picture. Hopefully it will be quicker. So geometric picture is the following. So what is perturbation theory? Perturbation theory is essentially nothing else that a way of doing saddle point approximation in theory point of view. So that's what you do. You take the classical configuration and you expand. This is what we call perturbation theory. So then we should ask we should understand. So in general given a system there are many saddles. So I should consider all of them. So what does it mean that the theory is like the phi to the 4 is borrowed or summable and perturbative contribution. So this is clearly a fundamental outstanding question in which we don't have in general an answer in realistic say gauge theories in four dimensions of things like that. But we can have partial answer to simpler systems, in particular up to the fight, up to two-dimensional systems. So yeah, don't have time to properly discuss this question. This was what we have been doing in our previous papers and actually ordinary integrals. But I can give you just a bit of a feeling of what is the key point that will apply also for the fight to the 4. So the key point here can be understood by looking at ordinary integrals. So the final answer will be essentially the following that you should see a system that have actually only one real solution to the question of motion. So at the end of the day everything boils down to this recipe. Solution of motion, which is typically the trivial one, then you are insured that bothersome ability will be guaranteed if there are more than one non-trivial solution then you have to consider other sector like the instanton sectors. When I mean one solution with finite action. So with ordinary integrals so the idea is essentially the following. We can consider an ordinary integral and I can show you that and you can even bypass this analysis of analyticity that Manj and Senor and Ekman were doing because there is a way in which you can hope that you can rewrite your correlation function directly in a borrowable way. So the only point at some point in quantum field theory that you had to do that you had to assume the existence of this function as Slava was saying and this relies on the existence also of the theory. So this is something that would be simple to prove yet to rely on some other results like the one by Ekman et al. So that is the following. If I have an ordinary integral so z of g now this is just an integral one over square root of g integral minus infinity of less infinity dx it is very simple e to the minus I am trying to mimic this is the mimicking of our part integral now with the ordinary integrals as good property so this is convergent so you see that roughly speaking you can do the following so suppose that there is this f as one saddle point which has zero action then you see that you can take you can simply trivially change variables like this now I will be very sketchy because I don't have time to enter into the details here so this integral essentially becomes an integral from minus zero to infinity in the t of an integral so let me exchange let me be cavalier in exchanging of the orders here so you can rewrite this in the following way sorry I mean this is e to the minus t and then an integral here dx delta f over g minus t so simple in nothing I have added an integral and the delta function now you can recognize that this looks like so this is actually an identity it's not really so this is I can somehow look at this as our gt bz no I mean the way in which I written it's actually an identity if you with a delta if you substituted this as a Jacobian so it's not over t so yeah the integral in t of this will give me back the original so now again I don't have time to enter into the details so the point is that you can look if this makes sense you can look at this as just the Borel function because the idea is that if this z is Borel summable then this should have this parameterization here and then by matching so to say this should be some Borel function so when you change the order of integration you have to assume convergence or right so this is a convergent well defined there are some assumptions of x yeah you just have to exchange the integration again I understand that there will be many things here but I am a bit far away from what I should yes you can probably do this and then this is going to be the Borel function so I might ask ok let's see whether this has a singularity we say that the whole point in our language whether this function well first of all it exists in this case it's very simple to see that it exists it's a triviality and then whether it is a singularity on t plus or not and from here you see that apparent possible singularity on t plus can only arise when f prime is equal to 0 because when you take the Jacobian so if you are going to encounter another saddle point you will get a problem if not fine so essentially if your contour from minus infinity to plus infinity you only cross say 0 say x0 equal to 0 which is the only saddle point and there are no others then you will be guaranteed to be Borel sum up so this will be then essentially you will identify this as we did with the Borel sum up function and we will be guaranteed that there will be no singularities on t plus and then here you see that since I simply manipulated my original function I don't need to establish using Watson criteria and so on so mathematical criteria that tells me that zb is equal to z because they serious and considering are serious already with an integral it's not a generic series and then I manipulate to re-write it in this form so if this is you do you do correctly you automatically establish that zg is equal to zb of g it's automatic ok so this is telling you that whenever you take a saddle point approximation you are careful enough that you do cross more than one saddle point then you are guaranteed that you are Borel sum up to the exact result so this thing is actually true also for ok so let me show you because this is now important a given example in the ordinary integral so the simplest of a case like this is actually our 5 to the 4 dimensionally reduced to 0 dimension which is becomes this integral ok if you reach this 5 to the 4 so this object here you can show it's the simplest systems of asymptotic series so it's when you expand the perturbation theory it's asymptotic but then you can re-sum Borel sum and you will get the exact result which you can get automatically for mathematical in this case you see that there are 3 saddles one at the origin and two which are complex conjugate on the imaginary act which are purely imaginary at plus or minus i so there are other saddle points in this complex they shouldn't care you don't need to deform your integral to pass of all possible saddle points that's not correct there is a prescription to which saddle points you should consider in this case just the one at zero is enough you shouldn't deform your contour then there is another possibilities prototypical possibilities which is the broken phase if you wish where you take the mass times to be negative so in this case the saddles are now all real and now we are in troubles because our original integration crosses all of them so my argument doesn't hold anymore so I should deform the contour to make it regular so here the situation now I don't I cannot enter into how to do this this is a little instance of Picard leftist theory so we have to deform this integration at the end of the day it turns out that while here this result is given by a single series you reproduce the exact result by one perturbative series around the origin here you should consider all three saddles we have to deform the contour such that they will cross all three points in a particular way and then at the end this result will be written as some zb zero so the borer sum at zero one plus zb minus one where now with a proper choice of signs which is complicated such that I am redefining this with all pluses so in the ordinary way you will say that this integral is not borer summable but instead it's like a simple instance of resurgence you have not borer summable there will be an ambiguity that is considered by adding to other series so it's a very simple instance of resurgence where with ordinary integrals this front series is very small so you need the three asymptotic series so now the point of this geometric perspective that I was here is very nice because it can give you a perspective now you could change the situation even for ordinary integrals to see that there is a way in which I can reproduce this result without invoking trans series but just using one single perturbation theory so the point is that the the saddle points are always given by the function f so whatever multiplite in the exponent which is weighted by g this is the procedure here if I would have had some other function f zero of x without a multiplication of g still the saddle point will be given by f and not by this f zero f zero is a sort of a spectator simply because I should consider it's a sort of a loop effect it is not weighted by g and the saddles are always given by f so this is the key point here this is the key observation here so given this you see that I can retake this z minus situation in the form in the following way so I define a z hat object which is now a function of g and an arbitrary variable g zero which is defined to be x square over two plus x four over four plus x square over g zero so this is in general for generic g and g zero this is another quantity so it's no longer equal to my original one but for the specific value g zero equal to g oh, thanks for the specific value g zero equal to g is manifestly equal to my original variable so you see that this is manifestly z hat g at g zero equal to g is equal to z minus so what is here, the idea is very very simple so I look at the system at fixed g zero so g zero is supposed to be fixed now at fixed g zero so this term is no longer three level becomes quantum and now here despite my original problem was a double well I went back to the case of the single convex potential because this is plus plus so in other words if I want to do semi-classical expansion of this object z hat the only set that will contribute I'm guaranteed only x equal to zero it's the only set and then there will be of course a different perturbative expansion because now I have another term with respect to the other one clearly this is another object so but this is the crucial point you do perturbation theorem only x equal to zero you resummit with the borosomabilis guaranteed you do the computation and then eventually you reproduce exactly this so you do this computation in g zero and then after you borrow the sum you get an expression which is valid for any g and g zero and then you set g zero is equal to g after borrow transforming and after you do that you reproduce exactly z minus with a single perturbative series no need of transient series or nothing just by this little deformation ok so this is the key point what we call the exact perturbation theory ept exact perturbation theory means that there are the situations in which you are guaranteed by these arguments that perturbation theory does not miss first of all is well defined in the Borel sense and moreover you are guaranteed not to miss any non perturbative contribution but then you have to do things not perturbatively in g naught yeah but g naught g naught before doing at this level you set the g naught equal to g and then at the end at the end clearly the perturbative series here is more complicated and what were supposed to be non perturbative contribution in the original say expansion here becomes little change in the usual perturbative contributions so there will be a different large n behavior they will have different coefficients magically all the non perturbative contribution so these things here actually the ways to do this for example if you had some small term not of this form but just to push very good yeah there are in infinite ways of doing this there is an entire family of possible deformation that you can consider here for instance you can imagine doing there are only some prescriptions so you don't want to deform the term that is convergence you should deform the subleading term but of course there is a vast area of you can play as you wish as long as you are guaranteed at the 3 level what we call the 3 level as only one center so now this is yeah indeed this is the subject of our previous work so with this way you can show that you can reproduce result in quantum mechanics without instantons in models where there are instantons okay but is it important for you to chose x squared because then it is a Gaussian integral and we can always do Gaussian integrals well this is x to the 4 here yeah no well this is not necessary there are situations you could have expanded this around the minimum then you would have an x cube and then you play with the x cube so you can play with any term and the only point is that you don't play with the highest degree term but once you do quantum field theory you are not going to be able to do x cube exactly well now you will see if I will manage we will come to the actual case so the point is that there are a bit of possibly unfamiliar subject topics so yeah so you see that this is this might look trivial in ordinary integrals because after all we are able to do this computation in all possible ways and now you see that this is also true in quantum mechanics and in quantum mechanics becomes challenging because this idea of doing resurgence in quantum mechanics becomes complicated you no longer have a finite number of sedals we have an infinite number of sedals and for each sedal we have to compute the whole quantum mechanical expansion around an instant and this is highly not trivial where loop are much much easier than in quantum field theory so this is a challenging task in fact has not been done for simple system using this resurgence so we are not able to do this aside from very specific systems that enjoys some properties this is a complicated task because you have to do this infinite resumption here with this trick you reproduce this by a single perturbative series so numerically for instance by doing at very large order loops you can do that in incredible accuracy but I don't have time and this is not the main point of my talk the main point of my talk is that having this picture in mind is very useful because first of all it gives if you wish a more modern perspective on the original result by Ekman et al so we understand this result using this trick that you can also extend in quantum field theory as I said you have now to assume that this system is function because this is highly non-trivial so assuming that this function exist we understand this border summability by the simple fact that the action s of phi of this phi to the fourth theory does not have non-trivial classical real solution for the coupling so these are going to be our convention for this theory these systems doesn't have a non-trivial classical solution with finite action non-trivial solution being phi zero equal to zero so this guarantees with our argument that this border function exist and at least you should have a border summability the extension of this border summability should be the largest as possible so g zero so as you know very well this theory has interesting phase structures when g zero is big enough there is a phase transition so what we can say is that this up to the phase transition point g zero star this should be border summable to the exact result if you which is another twist I mean this was already a k-manetar result but with this other point of view you can at least establish that you should not since this absence of solutions is true for energy this should be at least true until this point at this point correlation function become singular then you no longer know so you can keep going so as I was saying here once I construct the function I can evaluate for energy but then what am I evaluating it's something which is an open question actually for you so I don't know what am I evaluating but I am able to keep going at any arbitrary energy but mathematically speaking I am not sure I am computing something physical I will tell you I will show you some plots of this ok so I fear that I will not have time to discuss properly the previous approaches to the phi to the 4 so I will just sketch you this so we can talk in private so the previous approaches to the phi to the 4 using perturbation theory were based on the following ideas so one is epsilon expansion which I guess you know very well I don't need to tell you the only point I want to tell you in epsilon expansion is that the boredomability of the epsilon expansion is a conjecture it's simply a conjecture because it's axiomatic theories we don't have an understanding of whether these theories exist in epsilon dimension there is only an analytic continuation involved so axiomatic people were not able to to study for minus epsilon theories so we really don't know which is the mathematical meaning so the natural of the epsilon expansion is conjectural so there are some empirical reasons to believe that this should be borrowed sum up to the exact result but nobody knows certainly things seems to work so if you do computation you borrow some of the epsilon expansion things seems to work to a good accuracy so this is what people have been doing you do these many loops but remember that you should always borrow some in particular to reach to the dimensions because epsilon is equal to 2 significant number of loops and to do this archaeo with some numerical approximation a borel function which reproduces your perturbative approach and then there is another one developed by Parisi mainly by Parisi, Bresen, Zinjastan and Legilou which usually work at fixed D so D equal to 2 or D equal to 3 then you set up a sort of a Kalan-Simanski equation and in the particular scaling limit there is a beta function where now there is no clearly sliding scale we are in two dimension there are no logs, there is nothing so it's a different beta function so to say it's simply how the covering changes when you change the correlation length or the physical mass and then you do you look for fixed point of this beta function and once you find a fixed point then you evaluate the critical exponents at the fixed point so all these people add a statistical approach so this community was to find the critical exponent at the fixed transition and there are some results the results are pretty poor in two dimensions because of various technical reasons so this works pretty well in three dimensions so before the bootstrap they were almost competitive with until a few years ago with Monte Carlo analysis the epsilon expansion and this fixed dimension now of course bootstrap took over but in three dimensions works pretty well because the accuracy is very very poor so this is the old approach now our approach is slightly different so our approach is more closer to the one of Hamiltonian truncation in fact we were using deliberately the notation a convention of Slava Lorenzo paper so you see this theory is one of the simplest quantum field theory you can imagine the coupling is superenormalizable there is only one divergence to the mass the wave function is finite you don't need to renormalize the field you don't need to renormalize the coupling there is only mild mass renormalization that you can get rid with normal ordering so it says almost a finite theory one of the simplest non integrable theories so in what we do it's very simple is again in the spirit we compute the free energy so the vacuum energy lambda is a function of g0 and then we compute the mass, the physical mass defined as usual in this way so we work in the Euclidean so here this will be the we will take an imaginary momentum here but okay so this physical mass and then we will study the physical mass as a function of the bare parameters and here we will take m0 just we will set the scale in which m0 is equal to 1 the dimensionless parameter here is clearly g0 of m0 squared so this is the actual dimensionless parameter when you do loopwise expansion and then this is equal to g0 when you in this simple trivial system and then we look at the value we call the g0 star when this is equal to 0 when the physical mass goes to 0 I think these are pretty pretty obvious things so I don't have time to show you we computed the many loops so up to 8 loops so this is a world record and I wanted to acknowledge our students that entirely on his own did this so we were just watching his progress on this side of the loop computation it was an amazing task I can tell you it's not only sitting down you need quite good skills to do this multiplicity is computation of the integrals but it's a non-trivial task and then with all these loops what you can do, you find an approximate it's clearly now it becomes a bit dirty numerical recipe you have to find a good approximate for your Borel function this is done with, there are various midots the one we have been using is one is called the conformal mapping you map the complex plane into disk so this is reminiscent of the rock ordinance for conformal people here and another is simply Pade you take the Pade approximate of the Borel function and then you do the computation it's a very very poor game, at some point unless you do computation up to in larger order loops you have to do this you have to go through these numerical steps now if we can have I will show you now the results I'm sorry for being a bit late so this is just an example of the diagrams that Gabriele computed which is amazing so the topology are so complicated so these are some value management graph this is some two point function graphs and one point function we have to study now unfortunately because of lack of time I couldn't also appreciate I told you that with our perspective we also have now it's clear that even in the broken phase, the fight to the fourth theory should be borrorsammable so this is a bit of an extension of Hekman et al results even in the broken phase and there using our technique with this deformation you should get sensible results so these are new territory there was no results whatsoever in the literature all these previous approaches were studying so I will show you some results in the broken phase in the broken phase there are also other complications particularly to follow the tadpoles so that's why we have been computing also this one point function so let me show you the result first in the broken phase which are clearly a bit more accurate so this is just for fun I will show you what you do if you simply are dumb and you do blind perturbation theory you get meaningless results and you clearly see that with asymptotic series it will be completely crazy to do perturbation theory a strong coupling this G is what I call the G0 so that doesn't make any sense so this is instead of the vacuum energy computed with borrorsummation using this conformal mapping mapping method with AS means asymptotic subtraction so this is yet another refinement but don't let's not worry about this so these are the errors that we have and again the normalization here is the same also of Andrea's last talk so the critical value just to give a number is around 2.7, 2.8 so this is the what is the error bar? the error bar is the error because of course there are errors when you do this numerical determination of the borror function so in fact it's pretty determining the error in these games is not trivial because it's also error determination is not rigorous here, I should say that so we try to be conservative because whenever you are not rigorous you should be conservative but you never know so this is our best attempt to find a good error bar it's not trivial issue here to find the error so this is just to show you that optimal truncation works very well for reasonably good coupling once we have up to G to the 8 you see that you can explore up to relatively strong coupling with this normalization 0.4 is not so longer weak coupling and you see that optimal truncation works very well up to almost G 0.5 then of course you go away because at that value you are only considering the one term because you shouldn't rely on all other loops so you should clearly stop but it's nice to see that makes some sense so this is now the mass so this is the mass by the way this is something important so this is also related to probably slava confusion at the beginning so you see that in fact whenever I compute the vacuum energy at the least at this level of accuracy I don't see nothing occurring beyond the phase transition you see I can keep going this is infinite volume of course it's not finite volume but at this level of accuracy everything seems to be continuous I pass through the phase transition analytically and I go to some other whatever that's it so this is the result similarly for the mass so if I compute the mass just I resemble the mass not some other powers just the mass that's what you get you start from 1 and then at some point there which is more or less the one we see that with the good accuracy at the phase transition then you go negative at some point this function is analytic doesn't care just keep going so that's what it is but this is a very nice result you see there is a very good agreement actually an excellent agreement with the Hamiltonian truncation methods although as you will see our error bars are a bit larger so here also just to be completely honest I want to say that whenever you resum M if you wish in back of my mind I knew that the critical exponent in this theory is equal to what the critical exponent nu of the icing and this is why you resum M if I didn't know that then why M and not M squared M cubed or even a fractional power you wouldn't know what is the best choice this is a zoom of the sorry I should have said this in a minute this is our result this is the zoom close to the transition and this is our best estimate for the transition you see that our error bars are a bit larger so it's compatible I guess with previous result this is M this is M I'm plotting M how do you know? Is the theory another function of M squared? Yeah, yeah, yeah I simply take the square root of the series and then I borrow some the series in M squared of course I took the I know that at 1 at g equal to 0 is to be 1 so I took that branch, yeah I actually didn't understand that comment about M squared versus M so if you resum of squared and if everything worked nicely you would expect to see a curve which touches 0 exactly but the error are very very big because you approach to 0 and then there the error are big and you will have much bigger errors so this is the best choice that either you try or because you know the critical response so that's One question about the the errors again so you are dominated by errors in the numerical calculation of higher loop integrals No, yeah, they are good points and such of the polar function the error is dominated by the sensitivity to irrelevant parameters so when you do this border resum mention a good way to keep track of how good you are estimating the border functions to introduce parameters for which your actual physical variable should not depend while the truncated one would depend like some scheme dependence in ordinary in ordinary perturbation theory and this scheme dependence is actually the dominant error then we have also the convergence error that we are at finite number of loops error is the fact that our computation of the highest loop is done numerically and this also has a little error but this is the list of these three So the dominant is the answer for the border function Yes so this cannot be reduced in the future I mean as you keep going increasing the number of loops the sensitivity drastically decreases so if you are able to do a few loops more things will be much better so there is a strong sensitivity on the number of loops so this is a zoom and this is what I was telling you since in principle I do not know the critical exponent I want to see how I could actually predict the critical exponent so this you can do by simply looking at this log derivative of m then you know that the zero of this will give me the critical exponent straight away and then in so doing this is now for technical reasons this works best with the padé way not with this conformal mapping the two by the way are always in full agreement so we have these two different methods just to compare and to see whether things make sense and you see that we have a slightly larger error and then but here we have a good reasonably good estimate for the critical exponent no so again this is optimal tronkation on the mass so you see now on the mass things are not as good as the vacuum energy but still optimal tronkation somehow makes sense up to 0.3 but then of course then it doesn't make any sense so this is now the broken phase this is the vacuum energy in the broken phase in the broken phase again here I am using the normalization for which the phase transition will be very very soon so at 0.25 and you see that I can keep going up to very large values and nothing changes I mean you have an analytic function apparently so this is probably the most preliminary graph I write very preliminary which means that this is really preliminary so the others more or less should be okay but this is really can change in the future so this is the value of the mass in the broken phase as a function of the coupling so here I mean I will be happy to maybe to talk during the break with you but this mass shouldn't at some point even exist as a single particle states because of the decaying in the kick and the kick that should happen here this is the normalization of these people so in this normalization if I am not wrong should happen around 1.5 and apparently we don't see the error are big but we don't see any we don't understand here what's going on so something should happen at those values of the coupling but we don't have enough precision in this stage to see what's going on but aside from that this behavior here is in good agreement with these people and also with some work by Lorenzo and Slava on the broken phase so now let me just end with the most juicy parts which are called curious facts so curious facts are the following so you know that there is a chunk duality in this system so this is one of the main motivations for us to study this quantum field just to see if our ideas here made sense in quantum field theory and chunk duality provides some sort of a check so this is the vacuum energy the blue line is the vacuum energy in the unbroken phase which are now analytically continue up to even values here so this is beyond the phase transition point you see and then I overimposed here the vacuum energy computed in the broken phase and in the broken phase there are two vacuum energies because in the broken phase by chunk duality there are two branches so there are two values of the coupling that gives rise to the dual to the same unbroken theory there are two values of the coupling one is weakly coupled when the other is strongly coupled and this is the green line here and the other is strongly coupled with the coupling that goes like the unbroken phase which is this red line now if this analytic this function here should have nothing to do with actual physical observable I do not understand why this tree overlap so you see that the fact that this tree overlap seems like to say that the my vacuum energy computed in the broken phase analytically continue to beyond G star still is capturing the actual vacuum energy in the broken phase these functions they should agree at least at the physical coupling they will necessarily agree also in some region around this that's good point this is in fact a possible explanation if true which is in fact a possible explanation would require a better precision to see whether this with this precision we have at the moment this is not enough to see whether and also this is maybe this is due to the easing duality maybe this is also can be understood so this is the last curious fact so this is now a bit of a different variable so here this is the coupling I forgot to write so this is the mass of the single particle state as a function of the coupling now this is the coupling in the broken phase normalized as in this paper so in particular with the normalization of this paper the phase transition of course here we are talking about the broken phase here and these are the points which we stole from this paper which are the Hamiltonian tronkesion results without the error I mean we didn't steal the error but the error I think with the error bar will reach this point this particular one so there will be some error bar in this points that unfortunately we didn't get at this stage this black line is the kink mass semi-classical kink mass that for reasons that is not totally clear seems to be in very good agreement even at value of the coupling because it's strong the couple here and here this is the curious fact that this is the mass this is the mass I showed you before in the broken phase where analytically continued evaluated for values of the coupling which are beyond the transition and then rescaled in terms of the broken variable so you see that the two seems to give the correct result and not only that but it seems that you get back transition so this is in the dual coupling so you should see this in reverse so this is stronger and stronger in terms of the broken phase but you see that the trend is that this mass function seems to reproduce once you go beyond the g star the kink mass as I said I will conclude here we don't have an understanding so this is if you wish we don't have an understanding of why is that or if this is trivial because of some duality or it's wrong or it's an accident I just wanted to finish by showing you this curious fact so thank you very much