I think it's sort of like.
So good morning, everyone. Thank you very much for coming. Yes, and we continue basically our journey to go from here to here.
So Lucy, I'm just told you that's roughly speaking now that I see we have this snapshot of protons after the protons,
we get out of gluons and we collide balloons, OK? And then the question is,
how do gluons and quirk and gentle particles interact with each other to give us the Higgs or to
give us any kind of complicated interaction that we measure at the LHC and this guy was discussing,
So how does this work? Now let me start from the very beginning.
So what do we want to discuss? So which kind of system we are talking about now?
We're talking about subatomic particles at very, very high energy.
OK, so you are know that if you talk about something which is very small subatomic,
you need quantum mechanics and you know that if you want something, it is very, very highly energetic.
You cannot use nuclear dynamics. You need to use special relativity like this.
You know, what you may not know is that is what happens when you have to combine the two.
Now you can start to say, OK, I want the theory that respects the laws of quantum mechanics and special reality.
Now, one of the big achievements scientific achievements of the last century is that if you ask for theory which behaves like this,
then there's only one theory that works. So you cannot just combine them randomly.
If you have a theory with respect to the laws of quantum mechanics and special relativity,
at least as a case of energy scale that we're interested in, there's only one way to put them together coherently.
So this way, it's called one theory was discussed already with myself, already by Lucien.
And the interesting thing of this theory is that it's really constrained by the principle of quantum mechanics overactivity.
So they're not theories that we can tweak according to our wishes.
So are completely fixed. So if I have a particle content of my theory, for example, I know that my theory works once a week.
So and I know the symmetry. So Gavin was mentioning, this gets at this.
I know how which kind of. Basically, charge is going south in this theory.
Then this completely fixes to what I what I can get them.
She's extremely nice. Because this allows TSA to do calculation for four percent from first principles.
So we do not model stuff. We really do compute them starting from the basic laws of quantum mechanics and special relativity.
So in order to understand what how quarks and gluons interact and how they interact with a Higgs,
so we need to understand the quantum theory of the strong force.
And this was already introduced up is called quantum dynamics. So it's what that.
How what's influence interact together? And going back to Gavin's T-shirt them is the upper part of the T-shirt.
So this is what the government was saying, that this part is very well established.
So we know that is the right theory. We do not need to check whether this is correct.
We know what is correct. It is well-defined theory. So it then I hear no ambiguities.
Everything is here. It is very, very well established. So is it good news?
Yes, it is good news. Unfortunately, he's also extremely, extremely difficult.
So we cannot really do much with it in general, even with the largest supercomputers,
we can not just to put this theory on a computer and predict LHC rates doesn't work.
So then what can we do? Well, Suzanne was mentioning it already that there is a very,
very nice feature about the wonderful tourism, which is that they behave differently at different case.
And the theory that describes strong interaction is a very, very, very interesting feature.
So at loss case. So the Protoss case, let's say.
It is very strongly interacting. We cannot do much with them from first principles.
We need to go through all the machinery that Alison was discussing mattered very high energy.
So for example, here around the Higgs Mathur, then this year becomes very, very weakly.
Carports in this plot just show you the strength of the coupling of strong interaction goes very,
very large at low scale and becomes very small at high scale.
So that loss gets very large. You cannot do anything from first principles, almost anything from first principles at large energies.
It becomes quite small. So why is it nice? Well, if you're which is weakly coupled is much easier to deal with, and I say what?
You strongly Typekit why? Well, because if the theory is weakly cupholder in the first approximation,
we can imagine that interactions are actually not really there so that we have a theory of free quarks and gluons,
and the interaction is only slightly changing this picture. So in other words, if we have a small parameter, we can compute that this better.
But if expansions that Gavin Anderson were mentioning before,
so I want to compute some observable in a sphere which is with a couple the first I switch off Rick optioning that I can see the first correction.
A second correction is also fourth as much as as I need. Why is this nice square?
Because while one capital is very difficult, we don't know how to deal with them in general.
We have very good techniques, at least in principle, to compute the coefficients and do this function.
These are what we call Feynman diagrams. I am pretty sure you heard the name some, at least at some point.
But let me go through a little bit of more detail for what these objects are because.
This hour, I will need this in the following. So what on the items?
Well, just imagine I want to computer some theoretical prediction for scattering reactions at the collider.
I have some initial state, for example,
to glue on some miniature state and this some interaction happens how to get to some final stage for a couple of weeks up with many other particles.
And I'm interested in this. I mean, see what I can do with this.
This calculation in and if I thought about this expansion now already more than 50 years ago, five month old, I have to do this.
So I said, look for any given theory.
There are only a few set of basic building blocks, for example, for QCT that building blocks, which involve the glue ones, which are this one.
And thus similar ones for wasn't. And he will show me some of that.
Now, why is this important this building block? What do we do with them? Well, just imagine I want to compute the first time in this expansion.
OK, now what? My mentality is that you have to do the following. So take up your initial state and take your final state.
Now, just to make a very simple example, I don't even discuss that.
He's just imagine we want to consider what happens if I have to laws that scatter into Tougaloo so that one of the simplest processes you can imagine.
OK, so I have to glue on in to go to close out.
OK, fine. So what if you are interested in this stem?
What you need to do is to use this building block to connect the initial and the final state in all possible ways, avoiding close close loops.
And I would come back to this one. I think meanwhile, I have this two building blocks.
How can I connect this with this in all possible ways? What I can do with like this, like this, like this crossing and like this.
And that's it. So I have this four different objects.
And the interesting part is that the five months also tell us that each of these objects has a unique expression associated with them.
So if I sum up all this object that we call Feynman diagram,
I get an object that we call scattering amplitude that contains all the non-trivial dynamical information about the theory.
So all the dynamics can be computed in perturbation theory just by summing this bunch of objects.
OK, now this is the first step. What happens if I won the Hyatte, then what?
The punch line is absolutely identical. That's what happens.
Is that every time I move one of them apart, I have to take my original object them and drive them with extra particles.
So either real particles. So for example, I take this object here and emit areal particle, or I can form loops.
So I have this object here. I can create a virtual particle that connects this to and get something like this.
And the rule is that every time I jump one order, I have to add one extra factor five particles, either here or taught in all possible ways.
Now this is what Feynman said Larson told us a long, long time ago.
And as I said, so the interesting part is that this is not just cartoons.
So there are well-defined sets of rules to associate to each diagram in a lengthy form.
And these are my take form when I encodes all the properties of the dynamics of your reaction.
OK? There are two kind of picture that I can draw picture that don't have closed loops up.
So these are called tresa. Because they branch out like a tree.
And they're very nice, because if I use this rule that turned out to be very simple function, a simple, rational function of you off momentum.
And I feel polarisation. So we don't have polarised like photos. So rational functional propositions that if I have close loops is more complicated
because I also have to do something in the grass and I got more complicated functions.
This is a punch line, and he's absolutely not new. So this was very well understood in the sixties.
It's chapter one of every book in almost every book on quantum theory.
And the interesting part is that it is fully algorithmic. So I want to compute the Higgs boson decaying sorry produce a Higgs boson with two tops.
I, like Gavin, was showing you at the seventh saw. Then perturbation theory just drove a bunch of times.
I'm so happy with them. That's it. And I know in principle how to do it. OK.
The question is, how many times do I need in this series to do physics and then we see how many diagnoses to compute?
Now. Just imagine we want to do this expansion.
And I told you that the strong coupling had the Higgs at the scale of elements like physics is more or less point one,
which means that if I only take the first time, I'm not going to get a very precise result.
If I get the second time in Visa, I get something which is more or less 10 percent corrections.
OK. So we cannot neglect these kinds of corrections with showing you we want to go to one percent.
So this is not enough. This is not enough. And we really need to go through very, very,
very high orders in this expansion to get them to be actually able to proper peaks physics at the fundamental level without saying a more or less OK.
So we need to go to very high orders the same principle. We know how to do it.
I thought, you know, we compute a bunch of diagrams with extra some and extra looks.
OK, I'll do this or we need to do. Can we do it, sir?
Well, let's again forget about the Higgs for it for the time being and focus on this very simple example.
I have two choices and two balance out. OK, I want to compute one x doesn't one.
I want to compute this. OK, so add one x w one.
Now, if you just try to draw it on a piece of paper, you will find that that's not so nice because we have that quite a lot of diagrams.
One X doesn't, and you've got at the beginning, we had four.
Glad for you dressed that we want to warn you that 22 and the expression I you get out of I'm on time is not particularly nice,
so it's a rational function, so I want you to read it, but it's rational function.
So it's relatively simple, but it's long, and that's not the end of the story.
So this is a piece of the answer so that it is pretty.
So the simplest possible process to getting one of the most you can imagine you got to take formal notice was the 98 page long.
OK. And then you can imagine if you start adding the Higgs stands at the heart of the particles, how this can behave?
OK. So for example, we want to add one or two hire one to add one more, one more glue on how they look like.
When I was trying to plot this, but my computer kept crashing when I was trying to print, it cannot do it.
But what I can tell you is how many Feynman diagrams contribute.
So if I have two balloons and if I stay three sites for three, I told you twenty five,
if I have two hundred twenty two thousand thirty four thousand and so on, so forth.
And each of these diagrams come with more and more complicated. So if you have twenty five diagrams of 90 pages, you can imagine what happened here.
Okay. So for a long time, this was thought to be a showstopper. So this is a report that was written.
Is that describing the fees potential of future colliders more or less 30 years ago?
And is this my sense? This is the conclusion to say the cross section four elementary two to four processes have not been calculated,
and that complexity is such that they may not be evaluated in the feasible and foreseeable future.
And we're not talking about the precision of cross-sections here, but talk about the most simple, rough estimate of what these are.
OK. Now you can say, OK, great, we cannot even think about doing this.
Can you at least finish these calculations as of yet? We have to add the one real articles and one the other one.
So how does it look? We are told why don't let you got something similar. There's a problem.
I told you that the one you have loops, you have to compute integers, so you have this object integrated over.
So good luck with that. So you get very difficult to integrate.
And just imagine that you are extremely good in doing integral.
So you sit down, you spend three years your computer's integrator and there's not that you find is infinity.
OK. So it's not particularly nice. So in principle, we know everything so five-month-old that you don't get everything we want in practise, basically.
We cannot go very far. OK. And that is why, for a long time, people thought that doing precision physics that it is impossible in principle.
We know what to do with them in practise. Nothing works. OK, now understand your story.
So this was my very first interaction with the Nobel prise winner, so I was a fresh out of Ph.D.
I just left Italy for the first time in my life. I was in the USA, was first week of my postdoc and this very,
very famous Nobel prise comes home and want to talk with all the young people about physics.
And then most of my friends say, What are you working on? I work on X and Y on X and Y.
At that time was a permutation of supersymmetry or extra dimensions. OK.
And he was super happy. Very fascinating. Very, very nice. I super happy.
I want it up with this guy. Then I go in and what are you thinking?
What I'm trying to do? Positional fidelity. And I only became blank. And he was really worried to say, You cannot do it.
I mean, there is no point. So don't don't waste your life doing this.
So and it is messy. Well, yes, you are going to find supersymmetry and we are going to find that sentimental.
But we were never will never be able to do precision physics. OK?
And he was far from stupid. OK, so now he had to point to some extent because of what I just told you.
So if we kept using this Feynman diagrams and whatever the techniques that we use in the 70s, then yes, it was a it was hopeless.
But in the meantime, people actually didn't lose hope and start looking at this horrible formula as more entities.
And they found something which was, on the one hand, the missing and extremely without a hand in the fighting longer.
So you think 96 98 pages of formula, some you spent the two ESM simplifying it them and the result that you've got is one time.
So this ninety eight pages collapse to either zero or two times, according to how you select whatever edition.
It's not that I'm doing something else. I mean, this is the same identical formula.
So again, so this formula here is absolutely identical.
So it's literally I mean, is it simplification? Get get get to this one.
And this is fantastic, but it's also very annoying, right?
I mean, you spent three years of in this and then you find this OK, so it's not particularly nice.
And then people say, what can we try to do it with more armed with more particles?
So, for example, six particle technologies at one time, 20 Feynman diagrams impossible to computer back then.
And actually, they computed it and they found again want them or a few times, according to how you select.
But this is very simple. And they even tried to say, Well, what happens if I have infinite number of particles?
Can I get the same kind of simplicity? You say this from what I hear, and this number here are very similar.
I don't they didn't tell you what these objects are, but basically it's just simple.
Simple point is that that's how the momentum is flowing into diamond. OK.
So it's very nice the final result is good, but what is going on?
I mean, I don't want to spend three years to get the results, which is one line, right? And this is for a long time not understood.
For a long time we saw this, that people saw this. These are this this simplification.
But you understand why this happened. She's got it bad because if you don't ask them what is happening, you cannot use it.
So you have to compute the horrible results and then simplify. You start now.
At the beginning, I was telling you all the story about quantum mechanics, special relativity, what happens in the past decade or so?
Let's people actually took what I said before very, very seriously.
So what happens, I told you that the structure of one of these here is extremely rigid them.
It's very much constrained by specialising in quantum mechanics and to, it seems, just thwarts.
But if you look, if you walk out the consequences, these are not just words.
So if you look at special relativity, what is the underlying core concept of special activity?
Why does everything has to happen locally in space and time? Because I do not want to violate causality.
And this is nice once again,
but it's very easy to work out that this has an extremely interesting consequence on which kind of singularities your your family values,
your amplitudes can happen to some of them and that can.
It turns out that at least a three level there are the final results can have only a very simple set of singularities.
Well, if I just copy that off the spelling diagrams, I find all kind of messy singularities at the end.
I can only have very, very few of them and very, very simple.
OK, it's nice also, especially if it me that they have only some kind of sunglasses.
But if I don't tell you what happens on this singularities, this is not particularly useful and this is what quantum mechanics comes into in.
So I don't know if you remember quantum mechanics,
as this principle of unity to quantum mechanics takes probability and then evolves capability in a deterministic way.
OK. This has also extremely important consequences because it can, but it can be used to tell you what happens to close to this single point.
And what happens is something which is almost a miracle.
So it happens that the if you go close to a single point of an amplitude with, for example, six final state gluon,
you'll find that these amplitude breaks down into pieces, which are exactly the same kind of complex with less number of schools.
Why is this useful? Well, because if I can use this breakdown systematically, I can build the amplitudes recursively, so I have a complicated.
Either that or not, I look at all the singularities and I see that in all the singularities, this becomes an umbrella with less rules.
So if I start from the more simple amplitude, which is actually simple,
then I can use it as a seed to do recursion and I can get the basically almost for free.
I can add legs almost for free. If I do, if I'm a lighter, I'm sorry, I h legs cost me a lot of a lot of effort.
For example, this 220 Feynman diagrams that give this a two to four scattering.
How many times do I need in the recession?
You don't even need a computer to do this calculations in your head because there are two thousand so to have two hundred and twenty diagrams.
Thanks to this, symmetries collapses into two taps that you can complete in your head and you
can use in this kind of principle computer all kind of the level that you want.
So now we understand this very well and we can use it.
And so three level computations is a problem solver so you can take your phone download and not that it is online compute.
And amplitude for four protons to suffer for four gluons.
That could not be done by Supercomputer 30, I think that's improved, but still OK.
So this is a done. What about loops? My look is much more complicated.
But on the other hand, you can imagine applying similar ideas. So what is a loop at the end and the loop?
But you can always think it as I have a bunch of three that I joined together.
OK? And again, this is more or less water.
But if you take this water deeply, you can try to use the same tricks that allow to get very nice sea level amplitudes to get look one.
I need it work now. I don't want to show the details,
but it is possible to cut open a loop using a game principle of quantum mechanics that simplifies dramatically and then put it back together.
And at the end, you end up with much simpler expression, which are also simpler to integrate at one loop,
the expression becomes so simple at the end, and no matter what the process, you have the computer, you just need to compete for intercourse.
So no matter how complicated the problem is. So this is completely solved them so.
Same same app on your phone can also give you one look, maybe not on your phone or your laptop.
OK, but still high a loop is much more delicate than this nice simplification doesn't happen.
This is where we're stuck more or less now. So we do not really know how to do them, but we find this a long time ago.
We just computed them. We find all kinds of horrible functions.
Now we have a much better understanding of which kind of function happens to appear in amplitudes.
We see, for example, that all the complicated integers that lead to higher loop computations are live in very, very nice space.
So all the one loop amplitude leaves on a sphere, so you can only think them of doing this.
Interact is moving around on the sphere. And if you add loops, this fee become more and more complicated.
The first thing you can do, you can punch the sphere and get it done later, and then you can get more complicated picture.
Probably if you if you ever want to somehow string some string theory tarka,
you saw similar features when people say you need to compact DeFi from the time dimension of string theory to the fourth dimension, as we see.
And it is just not. The pictures look the same, so it seems that we find exactly the same structure and we don't understand it the better.
We start the seeing structure appearing, and there is a hope that we can get a breakthrough also in this kind of calculation.
So I cannot say that this is problem solved, but I can say that there is a lot of progress.
OK. So we have the new understanding and you're thinking of how we deal with petabytes of quantum theory that allows us,
that allows us to do stuff which are unimaginable only a few years ago.
And the question is, OK, what can we do with that? So this is a plot that I think I took from Gavin that tells you, well, how,
how long, how many calculations and this order in perturbations here are available.
So remember, we want to get to a few percent of our fastest point one. So a few percent we need at least the first two steps.
OK, so long time ago, it was very difficult. Do these calculations?
Because we do not know how to compute amplitudes are complicated,
we got infinity and indeed going from one calculation to the other took significant amount of time.
Everything was scattered. And as the processes we can see that became more and more difficult, we had to wait more and more time.
So at some point, there was a couple of years when nothing happened. It was just too difficult.
You could not do it. OK, but then something happened around this time.
So what happened here? What I told you is infinities in the in the in the amplitudes I didn't tell you.
Why do they come from? What happens is that around this time, we started understanding exactly how to deal with them.
And then all these ideas that I was just mentioning about the higher latitudes started moderating.
Yeah, OK. So there was some kind of better understanding.
And then the question is, was it useful? Or we could keep getting one calculation every 10 years?
So after this happened? So this is the situation now. OK, so now we can basically computer a lot of processes to extremely high precision.
So these are many, many processes done by very different people.
Many of them involve either the Higgs or the background. So now we really can do precision physics for the Higgs.
The LHC and getting from colliding protons gluons collected working laws and get actual precision studies of the Higgs.
Now you may wonder, is it all necessary? I was giving you this argument about the, well, strong coupling at this point one.
So I need another for the stuff that is the computer that one can check whether it is useful to compute them or not.
OK. So what can we look at what we can look at? The simplest possible observable, which is roughly speaking, how many things we produce,
isn't going what we call the total cross-section something we can measure. So already now, so this is the measurement that we have.
Now, how can you be better than this now? But so you can count how many weeks you have and say, OK, experimentally,
I see this many things and then you want to predict this number in theory from the
tactical side to see whether your standard model prediction agrees with your data or not.
Because if it's not, it means that the Higgs is not the standard model. It's not like the standard model.
We found something. As I said, go on and the computer perturbation theory, First Order is what you get.
So this is experimental value. OK. This is a theoretical prediction.
OK. So if we believe that this is a reliable approximation of reality, it means that the Higgs that we measure is not the standard model for sure.
OK, but then we know that this is not a reliable way to have with the higher the higher order.
So what happens if we do it? Wow. Next order. And getting closer.
Next door, there we are basically there, but there is a problem.
So if I tell you that this is a battle of ideas serious about what you should tell me that the outlook is not converging.
OK, so that that dam could be here or could be here so we can just overshoot.
And then because of this, even if this looks kind of OK, people do that.
A hairy calculation that was completed only last year and could be that even one more than you do with them and you get exactly here.
So you see that everything converge very nicely and you are absolutely.
Compatibility doesn't mean that Hicks is asking the standard model error of the larger,
but at least it means that can be found on one file or as far as we know.
But it also tells you that in order to do this study, understand this actually makes for real.
You really need to control by sesion feet. You have a very large, a very good approximation.
Now these are the total cross-section. What can you do when you have this kind of procedure in the last few minutes?
I want to discuss just one example of how you can use this procedure to do something that for a long time was thought to be impossible.
Now, Gavin was telling you that we have established to some extent the coupling of the hexa to the third generation of feminism.
But it is impossible that it's to directly test the Coppins to second generations,
because that's too difficult to imagine, the impossible to measure too many Americans.
Now what you can imagine doing is, OK, I cannot measure it directly. But Gavin was also telling you that you can have an indirect process of coupling.
So if you remember he was showing you this virtual Thompson or real stops,
and so you want to indirect constraints and that'll give you direct constraints. Now you can get.
Can we get direct, indirect constraints about. Flavour you can accomplish for the second generation, you can imagine.
But in the same game that Gavin played with Tulsa, with the Champ Walk.
Okay. We cannot do it. We cannot measure peaks and charm directly.
But there is a negligible probability for two balloons to become a couple of virtual charms that he lays back into the Higgs.
OK. And if we now think a gluon and use it as a problem solution was telling you that the NDIS,
what you're doing is take a photon and you use it as a probe to look inside the proton.
Yeah, we can do exactly the same. Only we can use a gluon to loop inside to look inside this loop here and see if we can probe the structure of that,
at least indirectly, the structure of the interaction to the second generation.
OK, now people studied this and this is what they found. So in this a plot,
what they show is how the Higgs behaves either in the standard model or if it's coupled
with time is different from someone a more because not like the standard model. So one is a standard model.
So if you have a standard model, see this. This is some of the Higgs.
It's not important what it is, but it's a property of the Higgs that you can measure. And if they exist on some the money, you should measure this.
And then they studied some theories to see me as a son of a mother, but not quite the same and see which kind of deviations you get.
Now, if you get more or less crazy theory, you get large deviations fine.
But this capacity for serious of zoonotic. So what happens?
You have theories which are similar to the standard model, but not quite.
Then you'll get kind of deviations that if you look here, which are the two percent level,
but there is no way to distinguish standard model from these kind of theories
if we cannot control these properties at this kind of precision at the LHC.
But I just told you that fortunately we can. And then.
People use this technique, and actually it's not just in theory is actually measured.
So this is what the experimental collaboration's measures right now for that this is called kappa,
but is roughly speaking is what Gavin was telling you.
Basically, what Gavin was putting that carbon at the bottom more or less?
So how this deviates from the standard model and this then you come with the champ, look now this deviation from a standard model, right?
This that's worse than the model is and what the order are inside this circle is what experimental measurements allow right now.
Now, right now, it's not particularly constraining.
So right now it's telling you that we can measure deviation from the charm and bottom of what the 40 and 60,
so eventually want to see whether these are the standard model or not.
So 60 times on the models, not a great measurement them. But I also show you showing you are telling you that and it is just the beginning of the
full programme going on in the future and already had this high luminosity that they'd see,
which is a proof that this is going to happen no matter what.
This is the kind of constraint that you can get on the time you have a copy and you can really look at the at.
Valuation close to the standard model and constrain them and use this procedure in direct measurements
and precision predictions to try to pin down something which is impossible to look at directly.
Now why there are different costs and different Corazza. Why does it seem that what Gavin was showing with this onSaturday plaza?
So when you do this, you assume that your error.
So I think our error will go down by significant fraction and different culture means a different assumption of what is going to happen to the error.
So really, we need to work harder to make by season's precision predictions to fully use this, this, this, this kind of potential.
That added, see, this is just an example of plenty of them.
But let me conclude that now.
So I hope to have convinced him that we have convinced that it's very important to study the Higgs and to do this at the LHC.
Really, we need very good control on UCD because it is a very well known theory in principle.
We know how to do it in practise is very difficult to barter.
And there's been a lot of research on your progress and a lot of your theoretical developments in how we understand the combination
of quantum mechanics and special activity that allow us to to really create a new generation of theoretical predictions.
And now in this talk,
I cover a tiny fraction of what actually is going on in the field and the tiny fraction of what is actually needed to go to precision physics.
There are many other interesting avenues. But the interesting part is that no matter what we do, we keep looking at UCD.
And it's not just that we compete with a bunch of more,
have a more accurate prediction that we keep learning about the new structures in one theory and how to apply them for real life, real life physics.
Already now we can do a lot better the season programme, the more data you have, the better you can go.
So the best is yet to come. We'll have more data, better your understanding. And that it's, as Gavin said, is only the beginning of this story.
And with precision, we can really explore the new hicks interactions at the very core.
Now was there a time in place in time when people said, OK?
Doing physical as an act, this is not particularly interesting, that's it.
So if we succeed over work, yes, at the beginning of the of the last century, so what not, Calvin said, basically is complete.
That's it. So all we need to do is to measure some long quantities to a great degree of precision.
This is the kind of prediction, to be fair, educated in Cambridge. So I don't know that five years later, there was special activity at less than 30.
At least that's where quantum mechanics activity. So sometimes go into a great detail of investigation is what is required to move forward.
So let's keep exploring. And thank you very much.