Transcript

Ofer Aharony: How Does Confinement Begin? On the Possible End-Points of Conformal Windows

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· 1 h 1 min · 12,375 words · 96 timestamped sections

This is the full, timestamped transcript of "Ofer Aharony: How Does Confinement Begin? On the Possible End-Points of Conformal Windows", an episode of Simons Foundation published January 22, 2026 (1 h 1 min, about 12,375 words). It mentions 19 people, works and ideas, including QCD, RG flow, CFT fixed point, CFT scenario, CFT merger.

What this episode is about

The discussion focuses on the phase structure of Quantum Chromodynamics (QCD) in various dimensions, particularly the concept of conformal windows, which determine the boundaries of conformal symmetry in gauge theories. It explores the possible endpoints of these windows through specific examples, aiming to deepen the understanding of how conformal windows can terminate and their implications for theories with massless fermions.

Key takeaways

  • Conformal windows: the ultimate phase transition party, where theories either dance or disband as flavors deplete.
  • In the game of QCD, too many flavors spoil the broth—cross the critical line and it's confinement time!
  • Who knew? A mere integer tweak to flavor can send conformal theories spiraling into the abyss of confinement.
  • Large N, large flavors: a recipe for continuous mysteries in quantum physics—not your average smooth transition!
  • From weakly to strongly coupled: a thrilling ride through QCD's phase diagram, where the flavors never end... until they do.

Who speaks in this episode?

What is discussed, and when

  1. – 10:00: QCD, QCD3, SU3, Jang Shrian
  2. – 20:00: RG flow, BKT scaling
  3. – 30:00: RG flow, CFT merger, CFT vacuum, BKT scaling
  4. – 40:00: RG flow
  5. – 50:00: RG flow, Maxwell term, QED, Turn Simons term
  6. – 1:00:00: RG flow, CFT instability, CFT merger, CFT

Notable quotes from "Ofer Aharony: How Does Confinement Begin? On the Possible End-Points of Conformal Windows"

  • While the two loop beta function is positive so that they can cancel and we have a non-trivial fixed point of the minmalization group at some gauge coupling G which is G star. So in this regime of NF slightly smaller than 11 halves nc we.
    about CFT fixed point · at
  • It has been investigated in many lattice simulations. So I won't have space here to give credit to all the lattice studies of discussion and I think the consensus.
    about SU3 · at
  • Regime? So that question is the main motivation for this talk. So of course we're particularly interesting interested in this question for 3 plus one dimensional QCD but…
    about QCD · at
  • Window side and then ask what could go wrong when we change this continuous parameter NF over NC. Of course, one possibility is that the theory could just become free. I mean…
    about QCD4 · at
  • End and in particular if X is slightly smaller than XC we have this very small RG flow which is some a slow RG flow which is sometimes called Miransky or BKT scaling or in…
    about RG flow · at
  • Fixed point, we expect that C should be positive because of course the CFT vacuum should be a stable vacuum. But it turns out that one thing that can happen when we just follow…
    about CFT vacuum · at
  • Transition but we just start flowing to some other minimum of the effective potential. So this is a second possibility for how the control window can end in general.
    about CFT scenario · at
  • Okay. So in this CFT merger scenario, what happens as we approach this critical point? So we above X critical, we just flow to this fixed point. But this flow becomes slower and.
    about CFT merger · at

Who and what is mentioned in this episode?

People

  • Thomas Dumitresco — Person · Referenced collaborator (in-audience) · 1 mention · first at
    Named as a collaborator on the work underlying the talk; also noted as being present in the audience.
    Directions. And this talk is based on work that should hopefully appear very soon in collaboration with Thomas Dumitresco who's in the audience and with my student Jang Shrian.
  • Jang Shrian — Person · Referenced student collaborator · 1 mention · first at
    Identified as the speaker's student and a collaborator on the work the talk is based on.
    Soon in collaboration with Thomas Dumitresco who's in the audience and with my student Jang Shrian.

Concepts & topics

  • QCD — Concept · Core topic · 1 mention · first at
    Quantum Chromodynamics is the main subject of the talk, discussing its phase structure and conformal windows.
  • RG flow — Concept · Key concept · 16 mentions · first at
    Renormalization Group flow is essential for understanding the transition between different phases in QCD.
  • CFT fixed point — Concept · Key concept · 4 mentions
    Fixed points in CFT are central to the discussion of how conformal windows can end.
  • CFT scenario — Concept · Key concept · 4 mentions
    Different scenarios for how the conformal window can end are discussed, including the CFT scenario.
  • CFT merger — Concept · Key concept · 2 mentions · first at
    The merger scenario is one of the ways the conformal window can end, discussed in detail.
  • CFT instability — Concept · Key concept · 2 mentions · first at
    The instability scenario is another way the conformal window can end, where the CFT becomes unstable.
    Phase resist where for x smaller than xc we would have a different vacuum somewhere nearby. So in this scenario unlike the previous scenario the CTF does continue to exist…
  • CFT parameter space — Concept · Key concept · 4 mentions
    The parameter space of CFTs is discussed in the context of continuous parameters in the large N limit.
    Operators and correlation functions of light operators are believed to depend smoothly on this parameter X. So in this limit, we really do have a continuous parameter that…
  • QCD4 — Concept · Key concept · 4 mentions
    QCD in four spacetime dimensions is a specific case discussed in the context of conformal windows.
  • CFT vacuum — Concept · Key concept · 2 mentions · first at
    The vacuum state of a CFT is discussed in the context of stability and phase transitions.
  • SU3 — Concept · Key concept · 1 mention · first at
    SU(3) is mentioned as a specific case within the broader discussion of QCD and its conformal windows.
  • QCD3 — Concept · Key concept · 1 mention · first at
    QCD in three spacetime dimensions is discussed as another example with a conformal window.
    One dimensional QCD but similar questions occur also in many other theories. For example in 2 plus one dimensional QCD there's also a similar story. In that case NF could be as…
  • QED — Concept · Key concept · 1 mention · first at
    Quantum Electrodynamics is discussed as an example theory with a conformal window.
    General we could have any five squ and five six couplings of these scalar fields. So this theory is weakly coupled if either this sims coupling k is large or the number of…
  • CFT — Concept · Key concept · 1 mention · first at
    The plural form of CFT, referring to multiple conformal field theories in the discussion.
    Really continuously related to each other. For instance, they have different flavor symmetries. Each one has an SUNF cross unf flavor symmetry, which of course does very…
  • CFT dual — Concept · Key concept · 4 mentions
    Dual descriptions of CFTs are mentioned in the context of super symmetric QCD.
    Happen in some cases is that the theory could become free in some other set of variables and we know that's what happens in super symmetric QCD where we have some dual…
  • BKT scaling — Concept · Key concept · 2 mentions · first at
    Berezinskii-Kosterlitz-Thouless scaling is mentioned in the context of slow RG flows near the end of the conformal window.
    Beta function is very slow here. And in this case we actually have this very large gap forming between the infrared scale say the scale of fpi and the car run that you would…
  • Maxwell term — Concept · Key concept · 2 mentions · first at
    The Maxwell term is part of the Lagrangian discussed in the context of QED with a Chern-Simons term.
    Of RG flows then we start at high energies by taking this f 6 coupling to zero. So since we have a maxwell term then at high energies the gauge interactions go to zero. So we…
  • Turn Simons term — Concept · Key concept · 1 mention · first at
    The Chern-Simons term is introduced as an additional parameter in the discussion of QCD3.
    Case we can ask that when we approach this transition in various different directions I should mention that if K is non zero then we have to fine-tune the firmian mass because…

Full transcript with timestamps

So this talk will be a continuation of a talk that I gave here last year but I won't assume that you were here or that you remember anything from that talk. Um so the talk last year was about the phase structure of QCD in in 2 plus one space-time dimensions and in particular about how the control window ends there and my talk today will be a more general talk about how control what conformal windows are and and how they can end. And I apologize that this year I will not be giving a blackboard talk because I have too many graphs that I want to show you. But still do please do feel free to stop me with questions at any time even though this is not a blackboard talk. So I'll begin by outlining the question of conformal windows motivated by the conformal windows of QCD in 3+ 1 and in 2 plus one dimensions. I'll describe the general setup and then I'll discuss what are all the possibilities uh for how conformal

windows can end and how this can be identified. And uh the second part of my talk will be devoted to describing in detail three very specific examples where we can really uh control how conformal window ends and illustrate uh these different scenarios for how conformal windows can end. And I'll end with a summary and some future directions. And uh this talk is based on work that should hopefully appear very soon in collaboration with Thomas Dumitresco who's in the audience and with my student Jang Shrian. So uh the mo big the main motivation is understanding confirmal windows like the ones that appear in QCD. So we're we're interested here in QCD which is essentially the same lagen as we have for the strong nuclear interactions in nature except that we'll take the quirks to be massless to make things more interesting and we'll analyze general values of the number of colors and c and the number of flavors nf. So we have

this very simple lranian describing the theory and we have since we'll use massless firmians we have a carol symmetry s1f cross s1 and also a barian number symmetry and then with these symmetries this is really the most general reormalizable action so it's a well- definfined question to ask what happens if we begin with this action flow to low energies what do we get at low energies and of course it's known for more than 50 years uh that this theory is asically free if the number of colors is less than 11 halves times the number of flavors. So if an f is bigger than this then the theory is just inferred free and does doesn't do anything interesting at low energies. uh but below this bound the coupling starts becoming larger as we go down in energy and then if an f is only slightly smaller than 11 halves in C. So for instance for SU3 if the number of flavors is 16 then the one loop beta function is negative and very small

while the two loop beta function is is positive uh so that they can cancel and we have a non-trivial fixed point of the minmalization group at some gauge coupling G which is G star. So in this regime of NF slightly smaller than 11 halves nc uh we know from perturbation theory that the theory flows to a weekly couple of the fixed point which is some non-trivial confirmal field theory and then uh it's believed that uh as we decrease the number of flavors this confiral theory becomes more and more strongly coupled but it still continues to exist for some range in the number of flavors. Uh however if we then uh continue going down in the number of our flavors eventually uh there is a transition and we go over to a phase similar to what we see in nature for SU3 with three late flavors where we have confinement and carl symmetry breaking. So there is this uh phase diagram uh where there is some

range in the number of flavors where we flow to a conformal filter and another range where we have confinement at a car symmetry breaking and uh it would be nice to understand uh what is this critical number of flavors where we transition from one behavior to the other and what this transition behaves like. Unfortunately uh since this transition happens at strong coupling we do not know how to compute analytically what is this critical number of flavors. It has been investigated in many lattice simulations. So I won't have space here to give credit to all the lattice studies of discussion and I think the consensus from the lattice papers is that for SU3 this transition happens somewhere between eight and and 10 flavors. So the main topic of this talk will be to ask what happens uh when we approach uh the end of the second window. How does it transition from a conformal theory to become a a confining and and car

symmetry breaking theory in this case? Now of course for SU3 this is not a very sharp question because NF is an integer. So we have one value of NF where we flow to a CFT then another value where we have confinement but of course this transition is is not I mean it's a discrete parameter. So this is not a sharp question. However we can make this into a sharp question by taking the large end limit. So we we'll take a large number of colors and also a large number of flavors while keeping the ratio x uh constant in the larger limit.

So so in this limit this parameter x becomes almost continuous. We still have inferred freedom when x is bigger than 11 halves and a weekly coupled cft slightly below this range. And then eventually we have this transition to carol symmetry baking. Now uh we have in the middle we have this family of control theories. Of course, it's important to emphasize that these control theories along this line are not really continuously related to each other. For instance, they have different flavor symmetries. Each one has an SUNF cross unf flavor symmetry, which of course does very strongly when we go move along this direction. But in the large unlimited properties of this do behave continuously and smoothly as a function of X. For instance, the anomalous dimensions of various operators and correlation functions of light operators are believed to depend smoothly on this parameter X. So in this limit, we really do have a continuous

parameter that labels this family of CFTs and more generally this family of reization group flows from the weekly coupled QCD at high energies to this low energy behavior. And it becomes a well- definfined question to ask what happens at this transition at this critical value of X. How do we go over from this conformal field theory regime to the confining and carl symmetry breaking regime? So that question is the main motivation uh for this talk. So of course we're particularly interesting interested in this question for 3 plus one dimensional QCD but similar questions occur also in many other theories. For example in 2 plus one dimensional QCD there's also a similar story. In that case NF could be as large as we want. And if an F becomes very large then again we have a weakly coupled confirmal theory that's controlled in this case by not by perturbation theory but by the expansion in a large number of flavors. So when an

F is much bigger than in C again we have a weakly coupled confirmal theory. Well we know that for a small number of flavors uh I guess 40 years ago buff proved that in this regime uh we have Carol symmetry breaking uh to unf two. So we have a very similar phase diagram to the one that we have in 3 plus one dimensions. And again in the larger limit we can ask what controls this transition from the conformal filtery regime to this regime where we have confinement and flavor symmetry breaking in the two plus one dimensional case. We can also add an extra discrete parameter by adding also turn signals coupling K for this C gauge theory and then we have actually have two discrete or three discrete parameters K and F and NC. And it's interesting to analyze the theory in the limit where all three of these discrete parameters become large but with some fixed ratios. So then this becomes a two-dimensional phase diagram

where in one region we we know that we flow to a confirmal theory which is weekly coupled if either K or NF becomes large but when both of them are small then we know that we have this other phase and it's interesting to ask how does this confirm window end in this case we can ask that when we approach this transition in various different directions I should mention that if K is non zero then we have to fine-tune the firmian mass because it's no longer protected by by parody invariance as it is without the the the trans Simons term but we can still always fine-tune the farm mass to the fixed point and then what ask what happens when we do that finetuning and and change uh these uh these continuous parameters uh we still expect to have some transition where the control window ends and we'd like to understand what that transition behaves like so the more general setup that I'll analyze in this talk is a family of

generalization group flows from some high energy gauge theory to low energies and we want to have some set of discrete parameters which in the large and C limit may become continuous parameters that parameterize the flow. We'd like to have some region in this parameter space when we flow to a conformal fixed point perhaps with some fine-tuning. And so in that region we have what we call the conformal window where we flow to a conformal fixed point and we want to have some other region where we do not flow to a fixed point and we want to ask how does the transition between these two regions behave. So that will be the general question that I'll try to address in this talk. So

exactly yes so so in the storage unlimited becomes a continuous parameter and yes so the analysis will indeed be very analogous to the study of of of phase transitions and that's what I'll mention in my next slide I should also mention that we discussed it here for learn theories but there are similar issues that come up when we have realization group flows and boundaries or defects and so on so there's an obvious generalization to that case now these ralization group flows may in addition to the discrete parameters that I mentioned they could also involve continuous parameters and if the high energy theory has more than one relevant operators then we have our gene invariant ratios of of these couplings.

So we might have extra continuous parameters and the behavior of the control window could also depend on the values of these continuous parameters. Now in principle we could also just fix the discrete parameter and ask what happens when we only vary the continuous parameters and then as Tim was implying then this becomes exactly the usual question in phase transitions of what happens when you change uh continuous RG parameters uh in in in some in some theory and then what typically happens is that we have a second order phase transition where we flow to a control theory and often we have a line of second order phased transitions that ends at some multicritical point and becomes a line of first order phase transitions. So that's obviously one thing that can happen even without having all these discrete parameters.

But in the context of control windows, we do want to have also these discrete parameters which means in some sense that the CTF along this line of second order phase transitions also varies. Uh so that will give us a richer uh array of possibilities compared to that just this usual case which appears in the study of of standard phase transitions. So that's uh the setup that I'll be discussing here. So are there any questions about the general setup? Yes, please.

Uh, so that breaks some of the symmetry, right? Uh, yeah. Yes. Sure. Sure. So, so and so, right? So, one possibility is that yeah, that you could have a control theory with some extra free free fields at lower energies. So, that's indeed one type of of possibility that that happens. Uh, but I'm Yeah. So, yeah. So this will not happen in any of my examples but yes in general one could have some but I'll just refer to that as as a general CF it's still a CFT in some sense with some free sector in it. So yeah but definitely that's also possible

well without large n then it's really just a discrete family of theories. I mean I don't know of any way to really make sense of questions like this in the context of like non- integer numbers of flavors. Uh

I understand but flavor right but in the massive theory there's no transition I mean so so so then I mean yeah so I'm not sure

if you're one sorry just to turn it on there's a button yes turn on yeah this yes this

if you have if you're one flavor in the conformal window if if If you remove that flavor, you're going to be in a confined. Then if you add a mass to that flavor,

right, you're going to induce that phase transition um by adding a changing a continuous parameter somehow.

Um would it be the same nature? I guess no, I don't think it's going to be of the same nature. Well, we can discuss that in each of the scenarios that I'll describe, but uh there's no I don't know. Well, yeah, it will depend a bit on on the precise way that the control window ends. But yeah, definitely each scenario for the end of the control window will also map into some scenario for what happens in this m in this massive theory. But I think we may as well just study it in the massless case first. Yes, please.

Well, not really because of course for non integer values of an f of course you have phase problems and so on. I mean so it's there there are much more subtle issues. I mean so yeah in some senses it is discreet but yeah but there are all these phase problems that you have to overcome and so on. So yeah, so so it's for for some questions that's true, but I'm not sure if for for this complete setup if one can really treat as an F as a continuous parameter. Okay, so let me move on to describe what are all the possibilities for how conformal windows of this type can end.

So we'd like to start from the conformal window side and then ask what could go wrong when we change this continuous parameter NF over NC. Of course, one possibility is that the theory could just become free. I mean that's what happens in QCD4 at the upper end of the control window. The theory just becomes free. But of course in QCD4 at least we know that that cannot happen at a smaller number of flavors just because the theory has a non-trivial beta function. Then another thing that can happen in some cases is that the theory could become free in some other set of variables and we know that that's what happens in super symmetric QCD where we have some dual description of the same CFT in some other set of variables that becomes free at this critical value but at least for QCD and for the other examples that we'll discuss we don't know any candidate for what this alternative free theory could be so we

believe that that is not what happens at the end of the control window but in general that is always another possibility for what could lie at the other end of this conformal window and of course in that case the this transition is well understood is just the transition of turning on a small beta function here but I'll focus on other cases here so what else could go wrong from the pl of this confirmal field theory so this confirmal field theory only has irrelevant operators of course that's why we can flow to it in QCD in massless QCD without any fine-tuning so the lowest dimension operator is just trace of fu squ which starts at a dimension slightly above four which you can comput in perturbation theory. And then the next operators are for Fmy operators whose dimension starts around six here. And then as we change this parameter X, all of these dimensions continuously uh are continuously changed and all these

operators mix together. And then uh one thing that could happen uh at the end of this confirm window is that possibly one combination of these operators could become marginal. So its dimension could go to four in the 3+ one dimensional case. And uh one could ask what happens then and the scenario was uh suggested by many people. It's about the end of the confl window in particular uh in this paper by by Capan at all. So what happens if an operator goes down in its dimension and becomes marginal. So of course as long as the operator is irrelevant uh then the beta function just looks like this with a positive slope and we have an attractive fix point where wherever we start we just flow to that fixed point uh where the slope of the beta function is as usual related to the dimension of this operator minus d. So that's the usual story of of the infrared stable fix point that we believe we have for larger

values of x. And then naively what one may think is that when we cross this uh place where the operator becomes marginal perhaps the dimension of this operator would become negative and then all that would change is that we have this negative slope of the beta function and then this theory would become repulsive at low energies. So that would explain why we don't flow to that fixed point anymore. So that's a priori one possible scenario. uh but in fact uh this scenario is not what generically happens when an operator becomes marginal and the reason for that is that uh before it's the dimension of the operator goes down below d it hits the marginal value delta equals to d and what the beta function actually looks like there generically is is is of this form because we know that at that point the beta function vanishes because we have a fixed point and its derivative also vanishes because delta is equal to

dh but generically the second derivative of the beta function will not vanish at that point unless there are some extra symmetries uh that ensure that. But in QCD and in the other examples I'll describe there are no such extra symmetries. Uh so in those examples we expect uh that if an operator becomes marginal the beta function uh should behave like this and this completely changes the story about what happens when the dimension of the operator goes down. So because when the dimension hits d we have this shape of the beta function then by continuity what this implies is that when we're slightly above this point at x bigger than xc then we do still have this stable fixed point uh where the operator has a dimension that's slightly above d. uh but uh again by this continuity we know that in this range we actually have two different fixed points one infrared stable one and one infrared unstable one

uh where the dimension of the analogous operator is is less than d and the full uh phase structure of the theory looks like this where we have these these two separate fixed points and uh we we still float to this fixed point if we land close to it but we have also this other fixed point and then what happens if we continuously vary our parameters in particular this parameter x is that at the critical value we would have this form of the beta function and then as we continue modifying it we move over to this shape of the beta function where these two fixed points merge and annihilate and we no longer have any fixed points of of the beta function. So instead of having some unstable fixed point as one would natively expect. we actually see that there are no fixed points at all. And this is what's known as the confirmal field theory merger scenario for how confirmal windows can end by an annulation of two fixed

points. And this is one generic scenario uh for how confirmable windows can end. And of course in this case when we go for a smaller than C, we don't have a fixed point anymore. And we just continue flowing until we reach some uh well presumably the flow all the way to minus infinity implies that we have this other confining symmetry breaking vacuum. So this is a a wellstudied scenario for how confirmable windows can end and in particular if X is slightly smaller than XC we have this very small RG flow uh which is some a slow RG flow which is sometimes called Miransky or BKT scaling or in other contexts a walking RG flow. So this is obviously one viable scenario for how confirmal windows can end. But what I'd like to stress in this talk is that there are also other possible scenarios which are equally generic. So one should not focus just on the scenario but one should keep one's mind open and and really test

which scenario actually happens for the end of the confirmal window in a specific theory like QCD. So this is one scenario. Now one could ask what else could go wrong from the point of the confir theory. What else could cause this uh confir window confir theory to stop describing the infrared behavior and from the point of view of just the confir window it sorry just the confirmal field theory it turns out that there's one other possibility for how a confirmal window can end so in addition to looking at the beta function we could also look at the effective potential of our theory as the function of the vacuum expectation values of various operators so in the QCD for example we can look at the effective potential for the mezzans now we can compute this effective potential in the full QCD theory but we can also compute it just in the fixed point that we flow to and at the fixed point the form of this effective

potential is actually almost fixed by confiral invariance. So if the mezison operator has some dimension delta m then the global symmetries and confiral invariance tell us that the effective potential for this parameter has to be proportional to the mezzan raised to some specific power times an arbitrary dimensionless constant c. Now, of course, typically when we flow to this fixed point, we expect that C should be positive because of course the CFT vacuum uh should be a stable vacuum. But it turns out that one thing that can happen when we just follow a family of conformal field theories as we change these continuous parameters is that this parameter C could become negative as we continue following this family and we'll see an example of this in a few minutes.

And of course if that happens then what that implies is that for x smaller than xc we just have this stable vacuum at the origin but then at x equals to xc uh this this vacuum becomes almost unstable and as we go beyond for x smaller than x this vacuum becomes unstable. Of course in the full the in the full QCD theory we don't have just this specific power law we have also higher uh dimension operators that affect this effective potential. So the effective potential would eventually stabilize and we have a situation that resembles a second order phase resist where for x smaller than xc we would have a different vacuum uh somewhere nearby. So in this scenario unlike the previous scenario the CTF does continue to exist albeit as a non-unitary theory even beyond this transition but we just start flowing to some other minimum of the effective potential. So this is a second possibility for how the control window

can end in general. Now there are actually two additional possibilities uh for how this control window can end if we consider the full rization group flow from the weekly couple theory at high energies to to low energies. So one thing that can happen in the context of this flow is that uh this alternative controlal theory could start playing a role. So I already mentioned that if we have a CTF merger then this implies that at least we get close to the transition there's actually another unstable fixed point appearing in in the full space of of theories. uh and in the control merger scenario that fixed point is actually pretty close to the original control theory that we flow to. And then one thing that can happen as we change our parameter x is so if we believe that so so this uh so so there is some operator uh who's which is the the the lowest dimension operator which controls the RG flow. So if you imagine first

doing the RG flow in all the other operators and then considering just the RG flow as a function of this coupling which is the the most relevant coupling then we can imagine that for some range of values of X we land here and continue flowing to the original fixed point that we had but then perhaps there could be some critical value of X where we actually flow to this other fixed point instead this unstable fixed point. Of course that requires one finetuning but as we change the parameter X we exactly can accommodate such a finetuning of one parameter that would lead us to flow to this alternative fixed point and then as we continue flowing then the RG flow will just lead us to flow away just like in the CTF merger scenario. So this is what we call the alter the alternative control filter scenario where the critical value of X even though the CTF continues to exist. Uh we don't we stop flowing to it and and we start flowing

somewhere else. And the scenario seems at least as generic as the merger scenario in the sense that if we have a merger and we approach this merger then it's possible that we're on the right hand side of this and then we would continue flowing to the original CFD. It's also possible that we're on the left hand side of this and then even before we hit the merger, we would hit this other conformal field theory first and and flow away. So the scenario is definitely seems at least as generic as as a merger scenario. And of course it's also possible that the south of CFD could be far away. So in some sense this is maybe even more generic. Uh so this is a third scenario that can happen and we'll see that it does happen in some examples.

A microphone sorry. is more like a property of uh boundary conditions on your flow or

that's right. So this scenario is is we're discussing flow from a very specific UV completion of just the UV free QCD to low energy. So indeed this scenario is not intrinsic to the low energy theory but we're following a family of RG flows from a specific UV starting point and then in that specific family of RG flows we could have this alternative CTF scenario.

Uhhuh. Because usually there are many cases where both are perturbative and you have some domain from which you flow.

Exactly. There's always places where you flow away to massive phase. Exactly. Right. But the question is say specifically for QCD what happens when we change the number of flavors and then it could happen that in that very specific RG flow from QCD we would have this thing instead of having the merger. So we would first even before the fixed points merge if they do merge would flow to this alternative CFD and then start flowing as well. And this has very different predictions for uh for measurements on the lettuce as I'll describe in a moment.

Okay. So a final scenario is analogous to what would be a first order phase transition in statistical mechanics. namely we have this effective potential say for the mezzans and another thing that can always happen and that is very difficult to rule out is that in this effective potential we could just develop some other vacuum that's far away from our CTF vacuum so even if the CTF is perfectly fine for a large range of values of x it could be that it's some specific value of this parameter x the effective potential develops another vacuum say with mevs of order of the QCD scale and then this other vacuum could start having lower energy and then of course we'll just have the analog of a first order phase resist where instead of flowing to the CTF we will just flow to this other vacuum. So this is another scenario which can happen and does happen in some examples and of course it could also happen in principle at least

in QCD. So we have these four different scenarios for how a conformal window can end. Two of them are intrinsic in the sense that they depend on properties of the CTF that it flows to and can be analyzed just from the lower energy point of view. while the other two are extrinsic in the sense that they depend on the full RG flow that we're studying whether that full RG flow leads us to the CTF or if it just leads us somewhere else and uh the scenarios can be distinguished as I'll described in the next few slides by the behavior of these RG flows as we approach uh this end of the control window. So I'll describe in a in the next few slides exactly how we would distinguish these scenarios by say lattice simulations of this family of of theories or this family of generalization group flows and a request to the lattice people is to try to measure then of the control window and QCD and and related theories precisely

enough so that we can distinguish these four different scenarios and know and figure out what actually happens in QCD. Of course that requires doing these simulations at large values of n because we we need to be close to this transition and this parameter x really only becomes continuous at large n. So this is not an easy request but I hope that still lattis technology will enable uh this to be studied. So in order to do that let's describe what would be the implications of these four different scenarios for measurements that one could do say in in lattice simulations of QCD. So again the traditional scenario which as far as I know is more or less the only one that's actually analyzed in the lattice literature is this scenario that we would have this CTF merger. So in this case uh when we simulate these theories uh sorry

yes yeah of course the same thing can be analyzed in many other I mention a QCD because maybe such theories could be studied on the latis with these twisted auchi kawawaii which have access to very very large end

right and maybe that is a better place to start for latis simulations I don't know I That's a good question. But if we if you only have adjints, then this is not a continuous parameter. So you want to have adjints and also some small number of flavors so that you can really have this continuous parameter. And that's of course so forget about my

Yes. So thank you. Okay. So in this CFT merger scenario, what happens as we approach this critical point? So we above above X critical, we just flow to this fixed point. But this flow becomes slower and slower as we approach the end of the control window because the dimension of this operator approaches D. So the flow becomes slower and slower. So that's what we would see if we analyze X approaching XC from above. If we have X approaching X from below then as I mentioned we have this again very slow reormalization group flow because the beta function is very slow here. And in this case we we actually have this very large gap forming between the infrared scale say the scale of fpi and the car run that you would get at low energies and the high energy scale. So they're related in this case by this famous exponential factor because of this very slow RG flow. So this is what's known as Madsky and BKT scaling which of course

eager also mentioned this BKT transition in his talk. So this is the maybe more traditional scenario but the other scenarios are also possible. So if we have this CTF instability scenario, so in this case as we approach this critical value from above, we wouldn't really see anything special going on. I mean so this effective potential develops an almost flat direction. But okay, in the full theory, it's not really a flat direction. So nothing too special would happen when X approaches XC from above in particular, the G flow would not really slow down when when this happens. Now when x approaches xc from below in this case then we would have this other vacuum and the infrared scale say the scale of f pi and the low energy theory in this case would scale as a power law of xc minus x. So that would be controlled by the dimension of the mezan operator and by the dimension of the sleeping correction to the

effective potential. So it's easy to see that these parameters would control the relation between the infrared and the UV scale. And in principle on the lattice it should be possible to measure this ratio between the IR scale and the UV scale and so to compute this well to see if we are in this scenario and to measure this parameter alpha. So this is what we would have in this CTF and stability scenario. The third scenario I described is this alternative CTF scenario where at some critical value of X we flow to this other fixed point instead of to the original fixed point that we wanted to have. So in this scenario the CTF the flow to the CTF again slows down when we approach the critical point but now it slows down for a completely different region. It's not that the dimension of some operator becomes small but just that we happen to land close to this other fixed point and and of course that slows down the RG

flow but in a different way. I mean we'd first have a slow RG flow then a faster RG flow as we go to this fixed point and then again we would just end up at this other fixed point. So here the the flow flows down at some intermediate range of scales instead of at low energies. So that's something that again can hopefully be seen in lattice simulations as we approach x= xc from below. Then again we have this relatively slow flow and then we uh flow somewhere else and one can compute that also in this case uh the ratio between the RG the IR scale and the UV scale would be given by some power of XC minus X where in this case that power would be determined by just the dimension of the leading relevant operator at this alternative fix point.

So again this is something that hopefully can be measured in lattice simulations. And finally in this fourth scenario where we have another vacuum. So in this scenario like in first order phase transitions nothing really special happens when we approach the transition from either side. Uh but we just first flow to this fixed function and then for x smaller than x we would just flow to this other vacuum with the non-zero expectation value that immediately jumps from zero to a non-zero value without going to zero as x approaches xc. So again this is something that should be easy to identify in lattice simulations.

So those are all the scenarios and how to identify them. Of course unfortunately we don't know how to identify which of these scenarios happens in QCD neither in 3 plus one dimensions nor in 2 plus one dimensions because the theory there is strongly coupled and hard to control. Uh but hopefully a lot of simulations can actually simulate this theory very close to this transition point and and see which of these scenarios actually happens. Of course, it's challenging on the lettuce because we have to go to large n in order to be close to this transition and of course to be with massless firmians and in some of these scenarios we also have to investigate a very large range of scales which is also in practice difficult on the lettuce but at least in principle this question can be answered on the lattice and uh so far that's really the best tool that we have for for testing these scenarios. Yes,

please. Yes. Why are they not lice action you use? That is couldn't different lattice actions which all flow to QCD and a give you different initial conditions in the so in general. Yes. But so I'm imagining that I'm I'm taking my lattice theory so that it but so that does get to the continuum flow already at a high enough energy so that all the other operators die die off. So indeed you you'd want to get to the continuum region at at a very high. So that's also another reason why we you need this very large range of scales. So of course if you don't have a large range of scales between the lattice scale and the QCD scale then indeed the results could also depend on on the precise uh precisely what happens at the lattice scale which other irrelevant operators you have and so on. But so but this question becomes well defined if you have if there's a large if one / a is much bigger than lambda QCD and then again lambda QCD has

to be much bigger than this infrared scale. So you really need this very large range of scales and then it becomes a well- definfined question that's intrinsic to QCD4 but yeah otherwise on the lattice indeed you could be sensitive also to irrelevant operators at the lattice scale. So yeah thank you. Now another way to possibly test these scenarios could be by the doing a bootstrap analysis of this confirmal theory the the fixed point of of QCD. So so far this this uh no bootstrap analysis has has found these fixed points. But in principle it should be possible to find these fixed points by a bootstrap analysis. But of course a bootstrap analysis just of the CTF cannot really tell us which scenario happens because that depends as we discussed on the full RG flow from QCD from the weekly coupled high energy theory uh to to low energies. However in the bootstrap we could in principle see if there is a merger so if delta goes to

D. So if there is a merger from the CTF point of view then we could have the merger scenario or or we could have some other scenarios. But of course if no operator goes down to D then that would rule out the merger scenario and it would be interesting to see if by bootstrap methods one could also test this CFT and stability scenario or not. Again in principle this should also be a minimal uh to some bootstrap type analysis but again that will only tell us that can only rule out scenarios. It cannot really tell us by itself which scenario actually happens in QCD. So in the rest of this talk uh instead of discussing QCD I'll describe three other examples of control windows where in these examples the control window will be weakly coupled so that we can actually compute everything that I described and see which of these scenarios actually happened and in some of these examples we'll also have extra

parameters to tune and then we'll see that for some values of these other parameters we have one scenario and for other values we have another scenario illustrating that all of these scenarios are are generic. So are there any questions before I move on to specific examples?

Okay, so my first example will be naively very similar to QCD will just be 3 plus one dimensional QCD where we add also some number of massless scalers in addition to the massless firmians that we know and love. So what happens if we have some number of scalers? Well, natively not too much changes. I mean so of course we have this diagram. If we add some number of scalers, the theory would still become asmtotically free for some critical value of the number of scalers plus the number of firmians and we would still expect to have this control window which is weakly coupled and then becomes strongly coupled. So naively one would think that nothing changes but in fact that's not true because when we have also scalers then at least in the we coupling region we also have extra ph of the fourth couplings that are marginal and that can and will be turned on and then one has to analyze what happens to the flow of

these extra fight of the fourth coupling. So we're forced to analyze not just the naive QCD theory but a theory where we also have these phyto the fourth coupling. So the len so and the yeah and this and this was analyzed in in in several papers a few years ago by medinal and by applequist at all. So uh the theory we're forced to analyze now is a theory where we have young mills plus these massless fmians and these massless scalers whose whose mass of course we have to fine-tune to zero but so we just fine-tune that to zero. But then in addition to that we have these two types of four scalar couplings either a sort of a single trace in flavor space to the fourth coupling or a double trace in flavor space to the fourth coupling. So we have these two independent couplings that uh when we approach we coupling are both marginal and we have to follow what happens to their randomization group flow. So the

limit that I'll be considering now is a limit where we have ns scalers nf femians and we'll take ns nf and nc all to go to infinity but with fixed ratios that I'll denote by xs and xf and then we also have these three couplings g h tilda and f tilda. So in the large end limit these are the combinations that we need to keep fixed in order to have a nerve theory. So we have these three parameters and there is some range of initial values where all three of these parameters are asmtotically free so that we have a well- definfined starting point in the UV and then we have these three parameters we can turn on starting from the UV. So we have two realization group invariant combinations of these three parameters and so the question about the control window now becomes uh in in some sense a question about a fourdimensional space of theory. We have these two discrete parameters of the ratios of these numbers and we have two

extra parameters coming from seeing which randomization group flow we're following uh from the UV theory and we'd like to ask what do we get in the IR as a function in general of all four of these parameters. So that is a well- definfined question and again we'll see that we have a control window in some range and no control window in some other range and we'd like to see how it ends. Uh now the surprising thing is that in this case we'll see that the control window can actually end at recoupling when we still have control.

So first of all in order to be at coupling of course we want to be close to the edge of the usual control window. So we want the number of scalers plus the number of firmians to be close to this asmtotic freedom point. Uh so in that case what happens is that the flow of the gauge coupling is basically independent to one up and also two leorters of these other fight the fourth couplings. So the gauge coupling just flows to some weakly coupled fixed point independently of the other parameters and then along that RG flow of the gauge coupling we can study how the other two parameters these two four scalar parameters H and F flows and it turns out that if we start from a very small number of scalers smaller than 0.073 times the number of colors then the randomization group flow of these other two parameters actually has four fixed points. So one of these fixed points is completely stable so that we would

generically flow to it. That's this fixed point F3. And then there are two other fixed points where we have to fine-tune one parameter and one other fixed point that where we would have to fetune two extra parameters to flow to it. So this is what the RG flow looks like. Again this is all weakly coupled and in this range where XS is smaller than 0.073. But it turns out that when we increase the number of of uh of scalers to beyond 0.073 073 times the number of colors. Again, still staying very close to this asmtotic freedom bound. The flow changes in the space of these two five to the fourth couplings. Two of these fixed points merge. So they just continuously merge as we change this parameter excess. They merge and analate just like we discussed in the CTF merger scenario.

And we're left only with two fixed points both of which are unstable. So that generically in this case of course we would no longer flow to any fixed point. So that can all be seen just by direct perturbative analysis. So we can now consider a sequence of theories where we gradually increase the number of scalers and also decrease the number of firmians so that we stay say at the same distance from that asic freedom point and then we choose some specific values of of these phth couplings in the UV or more precisely of the RG invariant combinations of these f the fourth couplings and the gauge coupling and then we'd like to ask when we increase this parameter xs how do the flows behave so one option is that we continue flowing to this infrared stable fixed point uh f3 all the way until that fixed point merges and disappeared. So that's one logical option and and in that case we would have what we call this CT

safety merger scenario. But another scenario is that when we continuously change this parameter, we could flow to one of these other fixed points at some fixed value of X. Say we could flow either to this fixed point or in this fix fixed point which both require just one finetuning. And then we would have an example of this other alternative CTF scenario. And of course and the other two scenarios would imply that we actually don't flow that the flow actually ends by forming another vacuum somewhere else. So that's also a possibility and all of these possibilities in this example can just beam tested analytically because we can compute exactly the beta functions. We can compute exactly the effective potential. So in this case the analog of them as an expectation value is just a an expectation value of phag which can just be computed perturbatively and we can check which scenario happens in this

example. So I I won't bore you with actual computations but when analyzing the effective potential you can show that actually it does not develop any vacuum elsewhere. So we don't have either of these scenarios where we have an alternative vacuum but we can realize both of the other scenarios. So again it's we we have now a two dimensional family of RG flows depending on these continuous parameters which we can take to be say the values of H and F when the RG flow of the gauge coupling reaches some specific point. So these are RG invariant combinations of the UV parameters. So what's drawn here? So there is some so the range bounded by these blue dots is the range where the theory is asymmettoically free just by computing the beta function. So that's the range where we have a well- definfined UV starting point in the case that the number of scalers is zero and then we gradually increase the number of

scalers as compared to the number of firmians until we approach uh this uh this value where the fixed points merge which is approximately 0.072. So in that range we have a smaller uh so so the range of asmtotic freedom is is the range so the range of these black dots is where the theory stops becoming asmtotically free. So, so that's not an interesting range from this point of view. But in in the other range, we still have an asmtotically free randomization group flow. And then it turns out that in some region of this parameter space, we continue flowing to the CTF all the way until the merger.

But in some other range region of this parameter space, we actually flow to one of the alternative CFTs before we hit this merger of the fixed points. So we realize these alternative CFT scenarios and actually here we can realize flows to any to either one of these other fixed points that can exist. So in in this bottom region we flow to one of these fixed points and there's a very small region here which is hardly visible in the plot where we flow to the other fixed point. So this is an example where we can see that both of these scenarios the CTF merger and this alternative CFD scenario are both realized in a measure in a non-zero measure region of of the parameter space. So any questions about this example?

That's right. So you need an extra this parameter x until we hit at some value we hit this other fixed point and then we go to its other side so that we just flow away. So uh so in some sense when we describe the other parameters we we we we land here but then as as we as we change x we at some point we we start we land on this point and then we just have the flow from the other side of it. So that's exactly this extra parameter of the relative number of scalers which we're changing and then we can tune to this value where we had this alternative CTF

exactly aligned. Uh no I don't think so. Aligned in what sense? I mean you're parallel to each other. Uh

for uh no I don't I don't think so. I mean they're approximately aligned but yeah but not exactly. I mean of course this is trated to leading order and the larger limit and so on but yeah but yeah they're not exactly aligned I think. Any other questions about this example?

Yeah, sorry. Yeah, inside the blue. So, so the here the is asmtotically free inside this bigger region the the region that's not black in this region there is a well- definfined RG flow and we can just ask what happens to the RG flow as we change X. So inside this so for small values of X in this whole region we flow to the fixed point but then when we increase X eventually we only flow to the CTF in the region that's bounded by these red dots. So outside of these red dots somewhere in the middle we we flow to this other CTF and then we start flowing far away.

What was X now? X is the number of scalers. So we're changing the number of scalers while still keeping the the one beta function fixed. So we're decreasing the number of FMIOians and increasing the number of scalers.

Oh sorry. So you're both changing number. Yeah and keeping the changing them keeping the one beta function fixed. Just of course we could also examine various other directions but this is just some specific direction in this four parameter space of theories that that I mentioned. Any other questions? Okay. So my second example uh I'll discuss a completely different theory just QED in two plus one dimensions with a T Simon's term. So we have a maxual term we have a turn simons term with some level k and then we'll couple that to just an f scalar fields and then in general we could have any uh five squ and five six couplings of these scalar fields. So this theory is weakly coupled if either this sims coupling k is large or the number of flavors nf is large and then we have a one over an f expansion and in particular in order to have a continuous parameter we'll again take the limit. So in this case the number of

colors will stay one. It does not go to infinity. But these two discrete parameters K and an F can be taken to infinity with a fixed ratio kapa. And in order to have a well- definfined family of RG flows then we start at high energies by taking this f 6 coupling to zero. So since we have a maxwell term then at high energies the gauge interactions go to zero. So we can just have a free theory at high energies which is well defined for h equals z. And then as we go down in energies eventually we hit the scale. So for large enough f the relevant scale is E^2 times NF. So at that scale the gauge interactions start becoming important and we could also generate F and a mass at that scale. And then if we fine-tune these parameters we can land at various interesting fix points and we like to ask do we end at or for what range of values of this parameter do we actually end the RG flow at a fixed point or not?

So so that's the question we want to analyze. So if we only if we only fine-tune the mass term h then this randomization group flow so for very large k this is a very weakly coupled so we would just end up at the usual critical fix point which is just the critical un model and the limit that k goes to infinity uh but we're interested in analyzing this also for small values of k but if we just fine-tune one parameter then these flow to what are called turn smons matter fixed points so these are sometimes called the critical bzon turn s matter fixed points h and but uh for the purposes of this talk we're actually not interested in this urgy flow but we're interested in the flow where we fine-tune both the f squ and the f of the fourth coupling to end up at at a tritritical point so in this so this turn simmon's matter theory so I should have said that formally for energies below the scale lambda the

maxwell term becomes unimportant and we just have a turn simon's matter theory that can be exactly solved in this limit of large k and large nf and then that theory has also another fixed point which is a triric critical point for which we have to fine-tune both the f squ and the f of fourth theory that theory is also weakly coupled in the limit of large k and large enough f of course if k goes to infinity that theory is just the free scalar point but in general there is a family of theories connected to it and we like to ask if we follow this family of gflows uh always fine-tuning these two parameters so that we try to end at an entreal fixed point then what happens can we still flow to a fixed point also when we change this parameter k over an f or do we stop flowing to a fixed point at some point.

So the issue in this case is once again that uh because we have a weekly couple theory we have this extra parameter that is close to being marginal which is the coefficient of this 5 to the six cup and actually if we go all the way to infinite number of flavors then this parameter is actually exactly marginal. So here we we'll be interested in taking an f large but not all the way to infinity and if it's large but not all the way to infinity then this parameter does have a small beta function that can be exactly computed and so we can analyze exactly what this ph of the six coupling does and so this is the exact form of the beta function which depends on this continuous parameter kapa which I remind you is just the ratio of the trans samus level to the number of flavors and it turns out that for some range of values of this kapa up to very specific value then there is an an infrared stable fix point for this ph of

the six coupling and we have a flow from our theory to this fixed point in the context of just the churn seven's matter theory now it's much I mean in that case it can this flow can be exactly solved it it's not known how to solve it exactly if we also have the maxwell coupling um so in that range the turn sims matter theory that we have at low energies has this non-real fixed point and then when we get to this boundary value this fixed point again if we just analyze the low energy CFTs this fixed point merges with another fixed point at a double zero of this beta function that I had on the previous slide and disappears and our question is again if we start from this very specific high energy theory which is just this free uh Maxwell turn salmon's matter theory uh when we start from that theory and vary this parameter kappa do we continue flowing to this fixed point all the way until it merges

with another fixed point which is what we call this CTF merger scenario or do we have one of the others scenarios. So to check that again we have to carefully follow these urgy flows in the full theory and we also need to compute the effective potential uh for this operator. So in this case the lowest dimension operator is again just fagger fi. So we'll call it aa this normalized pheri operator and we have to compute the effective potential uh for this operator. Now this effective potential was computed almost three years ago in the lower energy trans matter theories by dpodal. So that is in essence a computation of the effective potential at this try critical fix point that we want to flow to. And what they found is that at low energies and for positive vacuum expectation values of this fi operator the effective potential takes this form. So it's just proportional to the this six coupling which is just in

some sense just the classical form of of the effective potential. Uh so navly one may think that this means that the flow is always stable but actually it turns out that the value of this phix 6 coupling at the fixed point actually changes sign when we change kapa. So that there's a critical value of kapa which is close to this value where we have a merger but it's below it. So there's this critical value 0.0361 where as we cross that point the value of this ph the six coupling get the fixed point goes over from being positive to being negative. So from just the CFT point of view, this means that we can have this CFT instability scenario where the CFT just develops a negative effective potential so that we do not expect to flow to it anymore. But again our question is what happens in this case for the full RG flow starting from the max turns theory where we could also have interesting things going on at

the scale of of the gauge coupling. So for that we have to analyze the full effective potential in this theory to see if maybe we would have some other long vacuum. So yeah, just a second. I'm almost done with this example and then I'll I'll take questions. So we computed the full effective potential in the theory and the larger limit. And what we find is that indeed the full effective potential it doesn't have a vacuum somewhere else. It only has this vacuum that when we're close to the fixed point is very close to the origin. So this is the log of the effective potential essentially and it has this one minimum uh close to the origin which is the one that would be expected in this CTF and stability scenario and it does not have a minimum anywhere else. So for this family of VG flows, we do realize this CFT instability scenario where up to this critical value of kapa we flow to the CTF but below it we flow somewhere

else even though there is still a notreal fixed point of the normalization group. So any questions about this other example Victor

why was there no quadratic term in this? So so I just fine tuned to to zero. So I fine tune to zero at low energies both the five squared and the five of the fourth term. So, so at at high energies I fine tune them to some value that depends on the gauge coupling but I fine-tune them so that at low energies I flow exactly to this non-trivial triitical fix point. So I I can always make that fine-tuning and I continue doing the same fine-tuning all the way until something interesting happens. So I try my best to still flow to that CTF by fine-tuning this to the best of my ability. But then after we cross this transition, we see that no matter how hard I fetune, I cannot float to this fix point anymore, but I float to this other vacuum that's nearby. operator no other operator becomes

indeed no other operator becomes marginal but the effective potential does just become negative at at this at this critical value so again the effective potential so okay I don't didn't draw this on this slide but just the effective potential on the CTF becomes negative so the CTF in some sense becomes non-unitary uh but it just does that continuously and uh we stop flowing to this

Well, it's the same thing because and at the CTF I'm I'm I'm tuning all the relevant operators to zero. So, so that's the same as saying that in my effective potential I would not have these terms. I mean, so that's the same statement just from the UV and and from the IR points of view. Yeah,

that's right. It is existent. It's just non unitary anymore. So, the CTF continues to exist even below this transition. They just don't flow to it from the UV and it is an intrinsic property indeed.

Wait, does the CF no in this case? No. It's the vacuum of the CTF that becomes unstable. It's not some specific operator, but the vacuum becomes unstable because we have this effective potential becoming negative.

But often in in CTFs where there is an intrinsic non-unitarity, let's say fi cube theory, right?

Okay. Some ex scaling dimensions become complex for instance. That's right. So also in this case what happens because because the vacuum becomes unstable. I mean

but but that what Victor said should be true that's that like a signal of this potential becoming negative could be that some scaling exponents become imaginary.

Oh yeah sure definitely but but they would start developing a very small imaginary part when we cross this transition. That's right. But again it's that's from the CFT point of view. But from the point of view of RG flow we just don't flow there anymore. We flow somewhere else. And that's what I'm interested in for this talk. Okay. So I'm afraid I'll have to rush a bit through this final third example. So in this third example, we're discussing again just to have a wide range of examples. We're discussing a family of theories that has a dual holographic description which is weekly coupled and weekly curved. And to simplify things, we're just taking the holographic dual to have just gravity plus a single scalar field. So in that case whenever the scalar potential is minimum and it has some extraum then we know that that describes in general a dual fixed point in in the dual field theory uh which

again which has some central charge depending on on the value of the potential there. And we can discuss randomization group flows between these fixed points just by looking at classical solutions to this gravity action uh where the scale factor of this of the changes in this way and where also this scalar field changes as as we move in in the radial direction. uh so these types of families of CF so uh to be for simplicity I I won't discuss any top down examples of this but we'll just imagine choosing by hand some family of of potentials in the bulk and then it turns out that in some cases as analyzed by cures at all in some cases so so this this is the scalar potential here it's always negative but we could have say a maximum of the scalar potential which in this case means we have a re relevant operator and then we could turn on that relevant operator and float to some other CTF where the same operator would

become irrelevant So this is like the standard picture of holographic realization group flow. Uh and in order to discuss the possibility of safety mergers in this scenario for instance we can imagine a scenario where we modify the potential such that a minimum and an adjacent maximum of the potential join together and and and analate just as we change the shape of of this potential. And in that case we can ask if we have an energy flow starting from some other CFT flowing there do we continue flowing to that theory all the way until we hit this merger or not. So I'm afraid I don't have time to discuss the details but yeah if we have a CFT merger then that would mean that we flow to this theory all the way until the fixed points merge and then we start flowing somewhere else in princip in principle we might imagine that we could also have an alternative CTF scenario in this holographic scenario where we flow to a

maximum instead of to a minimum but it's easy to show that that's actually impossible but what can happen is that we could have this alternative vacuum scenario where we have one energy flow landing at this theory But we could have for the same UV theory. It turns out that in the solraphic setup even though it's somewhat surprising we can have for the same UV theory two separate uh solutions uh to the bulk field equations which end up in different uh CTFs and IR with different uh central charges and operator dimensions. And then the question just becomes which of these RG flows has a has a lower energy so that we would know if our original CFT continues to dominate until we reach this merger or do we develop another vacuum that has a lower energy and which describes a flow to completely different CTF and it turns out that again I won't have time to discuss any of the examples but in many examples we find that indeed

there is this other RG flow which takes over before we hit the merger. So even though in principle the CFTs we could have had the merger scenario we actually flow somewhere else before we hit the merger but again for a different range of potentials in the bulk we do have the CTF merger scenario. So in this case both of these examples are also possible. So let me just summarize since I'm out of time. So I discussed in this talk four possible scenarios for how confirmal windows can end. And all of these scenarios seem to be generic in the sense that we don't need any finetuning uh to get any of these scenarios. So of course I mean there may be a fine tuning that's needed to flow to these fixed points to begin with. So that I'm sort of taking for granted but beyond that in the space of theories we don't need any fine-tuning to get any of these scenarios. So they all seem equally generic. And I reviewed three

examples uh where we saw that in some regions of the parameter space we realize each of these four different scenarios and of course many additional examples can also uh be analyzed and of course what we'd like to do is to understand which of these scenarios it's actually realized in interesting theories like QCD4 and and QCD3 and I described how one would distinguish these if one knew how to simulate these theories and the larger limit as we approach the end of the control window but unfortunately we don't have any ideas for how to do that analytically.

In some sense the best thing that we can do analytically is to analyze things in this weekly coupled regime and then in that regime we can have some weak indications for which scenario actually takes place. So for instance if at weoupling we see that a dimension of some operator goes down then that would give some weak indication in favor of the merger scenario. If we see that the coefficient of the effective potential goes down that would give a weak indication of the CTF and stability scenario. And similarly for the other scenarios we can find some weak indications of them in the we coupling region but we don't we don't have anything uh more than that in the special case of QCD3 there is some so in that case as I mentioned we have this two dimensional control window parameterized by x over n and by f overn and in that case in some regime of the parameter space where we're in the planer limit we can show that neither of

the cf related scenarios can happen because in this case we control the infrared fixed points very well in the planer limit and we know that they do not merge and also they do not become unstable but we still can show that there is a transition in this case. So in this case this transition cannot be governed by a merger or an instability and we must have uh one of the other uh scenarios. Uh but again we still don't know uh which one and that will require lattice simulations. Uh so thank you

Questions about this episode

Who speaks in "Ofer Aharony: How Does Confinement Begin? On the Possible End-Points of Conformal Windows"?

Ofer Aharony (Host), Thomas Dumitresco (Guest), Jang Shrian (Guest).

What is "Ofer Aharony: How Does Confinement Begin? On the Possible End-Points of Conformal Windows" about?

The discussion focuses on the phase structure of Quantum Chromodynamics (QCD) in various dimensions, particularly the concept of conformal windows, which determine the boundaries of conformal symmetry in gauge theories. It explores the possible endpoints of these windows through specific examples, aiming to deepen the understanding of how conformal windows can terminate and their implications…

What is mentioned in this Simons Foundation episode?

19 people, works and ideas are mentioned, including QCD, RG flow, CFT fixed point, CFT scenario, CFT merger, CFT instability. Each one on this page links to its own page with every episode that mentions it.

How long is "Ofer Aharony: How Does Confinement Begin? On the Possible End-Points of Conformal Windows"?

The episode runs 1 h 1 min; the transcript is about 12,375 words in 96 timestamped sections.

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About this transcript: the episode's spoken words as captioned on YouTube, split into timestamped sections. Reclip.Me locates every person, work and idea in the text, keeps the time it is first named, and quotes the words around it verbatim. Last updated January 26, 2026.