 Yeah, so good afternoon everyone and first of all thanks to the organizers to have to opportunity to present a talk at this nice conference which I really enjoy and So during the day, we already had several talks about one of the main problems of the standard model Higgs sector, so the triviality problem for instance by Harga or by Francesco So let's close today by talking about another big issue of the standard model Higgs sector, the electroweak vacuum stability problem And parts of this work was done with several collaborations and I think at least four of them attending also this conference So probably you already heard it the LHC works remarkably well in 2016 so at the end of August they already reached an integrated luminosity of 25 inverse femtobon which was basically the target of for the whole year 2016 And of course what you can do if you have such an amount of data you can make a direct search for new physics beyond the standard model And probably you also heard about this 750GV excess where our friends from phenomenology and model building were quite excited about it Unfortunately it seems that this excess was just in statistical fluctuations and is completely deaf by now But what you can of course also do is in direct search for new physics So for instance in the standard model itself you can check whether the predictions of the standard model fit to the experimental data or not For instance you can check the various couplings of the Higgs to the other particles And indeed so the mass of the Higgs is in fact such a parameter which may point towards a scale of new physics So in principle in the classical theory the Higgs mass is a free parameter of the theory However in a quantum theory you can derive bounds on this parameter by our G arguments as you can see in this plot It was depicted here as basically the Higgs mass given over the ultraviolet cut off scale which is basically the scale until the standard model is valid So we view the standard model just as an effective theory and beyond that scale new physics should enter the game And basically in perturbation theory so these mass bounds were first computed of course in perturbation theory The lower bound is given by the vacuum stability problem while the upper bound is given by the London pole or to be more precisely by the triviality problem However the really important lesson from this picture is that for any given finite cut off value you only get a finite infrared window of allowed Higgs masses So if your Higgs is too heavy you basically run into the London pole if the Higgs is too light the vacuum might be not stable due to top fluctuations And you already can see so the standard model Higgs mass of 125 gV causes some tension with this lower bound And if you basically investigate the perturbative effect of potential you get with the measured values of Higgs mass of 125 gV and the top mass of 173 gV An effective potential with this shape So you have here our standard electric minimum of 246 gV But it seems also that top fluctuations can generate a second minimum at very high energy scales And in the worst case the universe basically just can tunnel from this false vacuum state into the true one with rather traumatic consequences for your personal lives However there is an ongoing debate in the literature about this instability picture So for instance the second minimum occurs at some transplank scale and I personally would no longer trust calculations which is done purely in the standard model With a second minimum at I think 10 to the 26 gV or so And of course in toy models there are also discrepancies to letters studies which don't see such a behavior And you can also show in toy models that you only can get a metastability, an instability and effective potential Basically if you choose implicit removalization conditions which are in conflict with a well defined partition function So because the standard model is a rather complicated theory for a moment let us restrict ourselves also to a simple toy model Because you can already read off most of the fluctuation induced properties of this low mass bound in such a simple Eucava toy model Where you have a real scalar field which represents the Higgs and also a direct fermion which stands for the top quark You couple both as in the standard model via Eucava interaction term And we impose here a discrete Z2 symmetry on the potential in order to mimic the electric symmetry group And we also impose a discrete chiral symmetry in order to protect the fermions against the cryoling and mass term And then of course it's a quite easy task for you to calculate the one loop beta function for instance for the running quadratic coupling And then clearly the term arising from the pure top quark loop comes with a negative sign And this basically just means that the integrated quadratic coupling decreases in an infrared to UV perspective Which is also fine so far But now if you construct the effective one loop potential in perturbation theory You make certain assumptions for instance you assume that the potential is basically a 5-4 type Then you impose certain renormalization conditions on the mass term in order to get an electroweak minimum at 246 GV And you take here as a quadratic coupling basically the integrated quadratic coupling from this beta function But you identify the running at G scale with the field amplitude And then clearly if the quadratic coupling drops below zero then you get so in this particular toy model you would get in complete instability That is basically the perturbative picture However in the simple toy model the fermions only appear as bilinears So you basically can integrate them out And you can investigate basically the fermion determinant And what you obtain is that the interaction part of this fermion determinant is basically a strictly monotonic increasing positive function for every field value in this effective field theory framework So in principle it should not be possible that fermion fluctuations can induce such an instability And let me also emphasize in principle the evolution of the scalar potential is a multi-scale problem So the full effective potential in field theory would provide a separate information for all scales involved While in this case you mix up momentum scale information with the field amplitude But however all of us know a quite good method to investigate such a multi-scale problem on a perturbative as well as on a non-perturbative level by the battery equation So for the battery equation you just make a simple derivative answer for the effective average action From that you obtain the flow equation for the scalar potential Depending on both of course the field amplitude as well as the IG scale And you can also derive the flow equations for all other couplings involved in the system as well as the anomalous dimensions and so on and so forth This is now a complicated partial differential equation system which you have to solve And in order to solve you really need a good PDE solver Because you really have to track the potential over a huge range of field amplitudes And you also run over many scales so basically in the ideal case from the Planck scale down to the electric scale And we use your PDE solver based on pseudo-spectrum methods Well unfortunately my numerical abilities are very limited But fortunately we have two clever PhD students in our group So Julia Borchardt as well as Benjamin Knorr Both of them have also a poster and they applied their pseudo-spectrum methods to several problems And if you have time maybe in 10 or 15 minutes during the poster session I strongly recommend approach these guys if you have a hard numerical problem Okay now we have our flow equations, we have a quite good solver But we of course also need some initial conditions on the UV cutoff scale And just as a first test let's just restrict the bare potential to the perturbatively renormalizable operators Which basically means up to 5-4 coupling And just of course this is still a toy model but just in order to come in contact with the standard model We fine-tune the bare mass term or the bare vacuum expectation value depending if we start in the symmetric order The symmetric broken regime in order to get an electric scale of 236 gV And we also tune the bare yukawa coupling in order to get a top mass of 173 gV And this is now an example flow for a potential with a cutoff starting at a cutoff scale of 10 to the 9 gV And of top of the log locked so in the coffee break it worked So basically what you will see, you will see here the RG slicing scale running down towards the infrared And then you will basically see how the effective potential is built up So sorry for the delay I feel free to think about your numerical problem which you will bring to Julia and Benjamin But I also put that I will tell you what should happen What you basically will see is just that effective potential will build up By integrating out the quantum fluctuations And you will basically see at the end of the flow that we just get a single minimum here At 246 gV as it is constructed but we don't observe any instability at all at large scales And this small plot here is just because some of my collaborators don't like this log locked type plot How the electric minimum looks in this log locked type form And this is basically just assumed inversion where you can see then at the end the usual double wave form Okay after we have computed the effective potential or not Of course we can extract from the effective potential the Higgs mass, infrared Higgs mass And this is done here for instance for various values of the cutoff for vanishing a bare-quartic coupling Okay if you now crank up the coupling these are the results for 0.1, 1, 10, 100 And what you basically found is that the Higgs mass is a monotonically increasing function of the bare-quartic coupling As you would expect And therefore you get from this OG perspective running down from the UV to the infrared in an effective field theory approach That natural lower bound emerges just from the fact that the value for the bare-quartic coupling is bounded from below In order to start with a well-defined bare potential in the UV such that you have a well-defined partition function And this is by the way in great agreement with non-perturbative letter simulations This would basically just mean that a Higgs mass outside this mass bounds cannot be connected to any conceivable sets of bare parameters And therefore you get a clear answer for instance the Higgs mass is of course this is not just a toy model But if the Higgs mass would be 200 gv then there would be a scale where new physics should set in at roughly 10 to 9 gv or so Okay if we are already in the framework of effective field theories Of course it's rather expected that there will be also other high-order operators at a standard model cutoff scale just generated by the underlying theory of the standard model And of course clearly you can choose a negative bare-quartic coupling in the UV if the potential is stabilized by high-order operators And for instance these are example Higgs masses where we choose the bare-quartic coupling negative but we choose a fixed bare coupling for the 5 to the 6 term And you clearly see that the lower mass bound no longer holds if also higher-order operators are permitted And the reason for this from an RG perspective is quite simple So basically of course for the potential this is not the full story Because the complete non-perturbed structure will be generated during the RG flow However if you for a moment just consider the running of the bare-quartic coupling as well as the 5 to the 6 coupling You basically understand the main mechanisms why we can diminish the lower bound And the reason is basically that you in principle can map a theory which starts at some UV cutoff scale with vanishing bare-quartic coupling And vanishing 5 to the 6 coupling which would correspond to this black curve here To a theory which has a positive 5 to the 6 coupling in the UV but a negative bare-quartic coupling And so roughly speaking you can view this orange and this red curve here as a horizontal displacement of this black curve Roughly speaking Of course you can also ask yourself what happens if I don't restrict myself to the case of lambda 3 is equal to 3 For instance these are basically the curves for lambda 3 is equal to 1 and lambda 3 is equal to 10 And what you basically see is that you cannot push the scale of new physics to arbitrarily large scales Basically because the running of this high-order operator is basically dominated by its power counting behavior And that is also the reason why you cannot further stabilize the bare potential by a 5 to the 8 coupling or 5 to the 10 coupling or any polynomial coupling We also tested this by generalizing for instance you cover interaction to a 5 to the 3 psi-vap-psi interaction term and so on And the main message is you cannot beat power counting Because what happens so for instance I could also choose here just blindly I enforce lambda 3 is equal to 3 And then I would further decrease this bare-quartic coupling here And what happens then is quite interesting Because as I have told you before in the case of 5 to the 4 type potentials You at the end so the black curve here is the effective potential You get always a stable effective potential at the end of the flow However if you further and further decrease the bare-quartic coupling for any given value of lambda 3 You already seed a meter stability in the bare potential So our results don't show that there is no meter stability at all in the effective potential But if there is one it has to be seeded already and the bare potential has to become from the underlying UV physics And it's not necessarily related to top fluctuations And basically this red curve here shows you the border between you have a stable scalar potential during all the RG scales And below this red curve even in the class of higher order operators you already seed a meter stability from the underlying theory But this is somehow different as this meter stability occurs in the standard model Of course what you can also do in the simple model you can do for the full standard model Higgs sector And this is of course a little bit tedious to deriving all the flow equations but more or less a straightforward task And basically also for the full standard model Higgs sector the same mechanisms are at work You get a lower bound and also an upper bound naturally from the RG flow itself You restrict yourself at the standard model cutoff scale to 5 to the 4 bare potentials And of course if you investigated extended bare potentials on a polynomial type you can diminish this lower bound but not arbitrarily low So the most important point for the phenomenology question is What I really can, so basically what is the difference between this red mass bound within the 5 to the 4 class And also the mass bound derived from higher order operators at the UV scale And at the Planck scale the difference between two bounds is roughly one GV So if the Higgs mass stays at 125 GV and the lower bound I think it's around 128 or 129 GV It strongly depends on the precise value of the top of the top mass So let's say the lower bound is 128 GV then you can diminish this lower bound by higher order operators by one GV So that would mean 127 GV So you see there's still some tension at least if you wish that the standard model is valid up until to the Planck scale But as it was mentioned in the first talk of this parallel session of course you can also go at the standard model cutoff scale To some bare actions or to bare potential which is beyond polynomial order So for instance we tested so far several log type bare potentials or also some rational functions So in principle the dream would be you start with a stable bare potential in the UV You integrate down the flow towards the infrared you get an electric minimum of 246 GV A Higgs mass of 125 GV and you complete a stable potential all the way up It turned out that isn't so easy So far I haven't tried periodic potentials in the UV but I have learned that I should approach poster number I think it was 14 right? So that is really quite a hard task so it's straight forward really to construct such bare potentials on a mean fields level Where you only include the fermion fluctuations and you can handle the problem more or less analytically That is really straight forward but as soon as you include the bosonic fluctuations they have such a huge impact Because you have deformed the bare potential in such a strong non-perturbative way That is quite hard to obtain the correct infrared physics So they can really write down very fancy bare potentials which result in more fancy effective actions So for instance you can construct a bare potential where your effective potential has a minimum at 246 GV But the Higgs mass is as large as the Planck scale That is basically our current task to basically scan possible bare potentials In order to construct effective potentials more or less which have the properties we would like to have to And in order that the German will not kill me Feel free to read the conclusions and I am happy for questions