 Pakrat,powers. Vsličo vladi. Vsličo, da udobnijo, da me upunem v prezipce. Povedeš? Povedeš? Predobno. Drža je? kinetic work has long been in making en. It's about non perturbative renormalization group up to 6 order of the derivative expansion and it's done in collaboration with Bartrandel Mojč, Joob Sateh, and Leonie Kane. She's also there. OK, so there are three most important questions we wanted to address in this work. One, the first is does the derivative expansion converge in the non perturbative RG calculations? that we should not converge at all. So this means that if we take higher and higher orders of the derivative expansion, will we improve the determination of the critical exponents or not? The second question is, how do we remove and minimize the regulator dependence in the universal quantities? We know that if we deal with an exact theory, način je nekaj neče nezavrešen in način. Kaj smo priličovati, naredimo način. Taj bi se počutila, da je opešnima? Prejdovalo. Prejdovalo, kaj je to počutil način, da so se počutila način, nekaj pozaj za predstavljenje na numerikale delarje. Wsrednji različno, način da se počutila, Tako, smo počučali do vsej držav, kako je vseči model. Zelo je bilo vsečo vseči. Vsečo, da je bilo vseči vseči, je, da se vseči vseči vseči, kako je vseči vseči vseči vseči vseči, The regulator basically gives a large mass to modes that have way vectors that are smaller than some cut of vector k. And basically, it does very little to the modes that have a vectors that are larger than k. In neselj, da se boljo občutno postavilo, da se je občutno občutno vsega vsega kratiljeva. In pa, da se dovolj, da bi se povedal, da je taj vsega hrednjela, če je gamma in načo vsega zašljena, in da se zašljete vsega vsega vsega vsega, This was done by Wetterishi 93. And this equation is written here. So you can learn a lot from this flow equation, but in order to get some concrete calculations, you need to do some approximations to it. So the derivative expansion is one of such approximation schemes. So it can be considered a natural approximation scheme for studying long distance behavior. Zato komentujemo, da je bilo tako nekaj delanje in nekaj nekaj delanje vživljene kotoče. Vse gačno pospravimo tako, da smo počasni za zvrste, in počasni za svojo delanje. in z so, kako ti da je to? Z nami jo z delovom večnje, ker je lpa, je bilo, ki budem iztajiet, potenčnje laplji, spet počet. Vzbeno šeo tražuje se, ki sem hlajte vzbati, na pravi trajdu, od nekaj prefakte, da je no zatanja, zato je alesko z delovom večnih večnjih večnjih. So the second order of the approximation, you can do a little better than this, is to take the effective potential, and then take this coefficient in front of the gradient term, now to be a function of phi, and this function also so flows, It renormalizes, and basically this is a second order of the approximation. Then you can do a little better for the fourth order, Težko, ki videli, težko sredajemo vse vse mačne funkcioni, ki se potrebil v Kodilji, v Kodila in v momentu. Tukaj museli pašnji funkcioni, ki se zrečili, z kaj je način vsega vsega. In jaz videli, ki je zvržavila v 2003, in kodilji, kaj se zrečili, There are three functions describing the problem of order four. In at order six you play the same game and you find that you need additional functions on top of those at order four, which is 13 functions in all. What you do is you take these ansatz, or any of these ansatz, they enter the exact flow equation with them tudi ni zrešte vsega čavodnih učin v prvom štefnega. Ko bome v vsega v srčju, bilo se tudi 13 učin, Zdaj se sem vsega, vsega, ko dodače od tvojom nomerikom, da we se nane科idovenih vsega in tudi pačenih učin, torevi se bolje vsega. Po njej učin ribon enok đi, so videl mi, spremljenju, da se počecila, da je to pravno. Zdaj smo vzeliči v rolu, kaj je pomečna pomečna, in svojo je pravno izvrjuno potensijo, z katerima rolu, taj je zrst, iz kurdu zrst nekaj, taj je dve oprejvene, da se svoje oprejvene je odvrstik. In taj je tudi nekaj, In vse svarijo, da are eight functions that are additional in necessary at the sixth order of the derivative expansion. Okay, so we solve for fixed points, but this is not without problems. So there are a lot of things to be very careful about when you are working with this. So from the numerical side, we want to have results that are independent of numerical details. we have checked every possibility of numerical inaccuracy. So just to tell you fast which one, so for example grid range you need to have sufficient grid range. so for example we found that using a grid range which is three or more times, minimičnosti je suficjena, da se predstavila preseženje. Zato, da je resolucija, je zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo dole, dole, dole zelo zelo to stor bats tako račov in imposece shouldn't that ash楽 sell misto strizo e Z Questom ozda, da bomo skupati wedi sekund, v živu izgodim z numerikbelih detajov in z checkoutem, bo sortil vseh vsehey, o bananaserama. OK, če, da je kismo so načinu z numerikbelih detajov različiti katerične, godsim süyorim, če nači način u cloči nekaj stajov vizenski, da priješajš arbitročne objezice. Od vse objezice je čojstven objezice v regulatori in drugi, kaj je najbolj povrstveno, je čojstveno z konštena renormalizacija. Na povisku, da smo želi vseh, so smo najbolj prošli, ki so pripravili. skupaj, da zelo tukaj, če je to zelo, če je to začasniti, zelo se je začasniti v potrebranju viziračnih, zelo ovo je nečakaj problem, tako imamo, da bamo vzivno o to, če je to začasniti. OK. Zelo, se zelo tako zelo se zelo, če je to začasniti. Quadra dovi organizacja mali regulatorjo kori streams bland Get Perfect. So we have here the standard exponential regulator, which was introduced in the early works by Wetterisch. And we have also studied this theta-n regulator. So let me just say for one moment about this. Theta-n, so n is the parameter which we can change, it is here. If n is one, for example, this regulator, if n is one and alpha parameter is one, this regulator is a lithium regulator. But in our flow equations, especially at high orders, you need the regulator to be a little more regular. So you see that because you require a certain number of derivatives in q squared of the regulator in the flow equations, this regulator has them, which are smooth. If you have n larger than one, so we use n, for example, 3, 4, 5 or 6. So then the natural question arises is how to minimize, what can we do to minimize the regulator dependence within an approximation scheme. Well, you can change the parameters of the regulator. So we see here that both of these regulators are one parameter families of the regulator. And basically what you can do is you can vary this parameter to obtain some sort of an optimal value for critical exponents. So how does this look? So, for example, for theta 4 regulator, I show you here the evaluation of the exponent nu, independence of the regulator parameter. So I did not show the LPA result here because it's somewhere here. It's 0.85, something, and the curve is very, very wide. And I show you here the second order derivative expansion, the fourth order and the sixth order. And for comparison, I plot you here the conformal bootstrap result. So what you can see, what is important to notice here is that the convexity of these curves is alternating with the order of the expansion. This is one thing. Second thing is that the curvature becomes more and more pronounced as you increase the order of the expansion. And basically this means that you can fail easily in choosing the regulator parameter in a high order of the expansion. Whereas for the low order, you can choose a parameter that's not so optimized, but you can still have an equally not so good result, so to say. And for all these curves, there is an extremum. So we call the optimal value of the exponent, the value of the exponent at the extremum. So here we have a similar figure, but for the exponent eta. And so basically we have the same game of alternating convexities. Now, and I should note that these optimal parameters for eta are not exactly the same as for nu. But also we can determine the optimal exponents. So when you do this business, then you reach a conclusion that the optimal critical exponents, so they have a certain dependence on the regulator. But this dependence becomes smaller and smaller as you increase in orders. So this is a feature that is very, very nice. And this is also true for the critical exponent eta. And so this is something that one should expect if the derivative expansion is to be well-behaved. So here I show you the exponent nu in increasing orders of the derivative expansion. So for different regulators, as we evaluate and you see here how these values converge to the conformal bootstrap result. And here you see in the inset you see the zoom of this region between order 4 and order 6. So it's pretty nice. And the same is true for the exponent eta. So we see also here the optimal exponent eta becomes very close to the conformal bootstrap value at order 6. Ok, so to summarize what we have learned. So the derivative expansion shows some sort of weak convergence. Ok, so, but this convergence is weak in the following sense. So, for example, if we increase the order of the derivative expansion with a fixed regulator parameter, you will most likely obtain nonsense. Because we have seen that the curves become very steep with the increasing order. So it's very easy to get the regulator parameter very wrong. So also exponent nu and omega, they always show a minimum sensitivity point in the parameter. This is important. So this is something that can be counted on. And the convexity of curves nu and eta alternates with increasing order. We do not fully understand why this is true, but it seems to be true. And what is very important is that optimizing the regulator at any order of the derivative expansion gives very consistent values of the exponents. And they become closer and closer as the order of the derivative expansion increases independence of the regulator. So this is a feature that you want. And also one thing that I haven't shown in figures is that the optimal parameter alpha for nu exponents and eta exponent, this difference between this value diminishes as the order of the derivative expansion increases. So this is something that we are also very happy about. OK, so now I have told about good things, and now a little bit about some things that are not so good. So, for example, our exponent omega is a little bit problematic. So you see here the evaluation of the PMS value of omega at order LPA, at order 2, 4, and 6. So at order 4 we are really very close to the conformal goods value, but this might be just a coincidence. Because we see that at order 6 the value goes off. And this is what we have. We think we understand. We have certain clues that something about the omega is different. So, for example, we see that the omega of alpha curve, these curves do not alternate with increasing orders. So what we obtain is that at LPA the optimal value of omega is at the maximum, at order D it's at the minimum, at order D4 it is also at the minimum, and at order D6 it's at the maximum. OK, there is something wrong there. And we also think we have a reason why, OK, we might see a glimpse of a reason why this is so. So usually when you derive the flow equations of the MPRG, you just put in the derivative expansion and you don't worry, for example, at order 2 that you have the terms in the flow equations that have q to the power 4. But this, so when you think about this term of the order, which includes integration of the order q to the fourth, should not be in the first place in the flow equation of the second order. So it should not matter if you take these terms or not. And indeed this does not influence the evaluation of nu or eta, we have checked this, but taking or dropping these terms has a large influence on the evaluation of omega. So basically the equations of the order 6 were truncated in vertices exactly in this way, because if you think about it and you write the flow equation at order 6, you are going to have terms which are in order q to the 36. And these terms make a huge, there is a huge combinatorial explosion of such terms, and basically to compactify the equations in a human form, you need to drop all these terms. And this is what was done in the order 6, but the propagator was not expanded, so we think that maybe something might be there. And this is on our to-do list. And also just to mention briefly the d equals to 2 case. So here the derivative expansion is in a worse situation a priori, because the same model has at a larger dimension it has an eta exponent, which is smaller, and in a certain sense one could say that the derivative expansion is better if the exponent eta is smaller for the same model, which means that in larger dimensions. But we are not much worried about this, because within the non-perturbative Rg there is BMW's heme, which is much more suited. So our preliminary results in d equal to are well off from the exact ones. We obtained just very crude results just to give you a hint of this. It's new, for example, 1.06, and eta 2.238, whereas it should be 1 and 25. And the problem in d equal to is much more demanding, and we are also studying this at the present time. So to conclude, I conclude with this table, which includes the comparison of the present results with the previous precise evaluations of the critical exponents. And it is by my firm belief that omega can be somehow mended in these results, because it's sticking out as worse than the others. So thank you for your attention with this. Kaj če o rematče?