SUSY and Particle Physics, Part 1 - Nima Arkani-Hamed

one of the great things about being an organizer of the schools that Igor is one of the organizers of the school so thanks for calling me an organizer Igor I appreciate it so this is the plan so I'm talking about supersymmetry at low energies now the school as you know is is mostly focused on more formal properties of supersymmetric theories and so I will be talking about about how supersymmetry can make contact with the with with the real world at the TV scale but the lectures will will also be be focused to some extent on how the same sorts of ideas and techniques that are widely applicable across studying theories and equals one N equals two in higher end the same basic theoretical structures that work even with N equals one phenomenologically in n equals zero as well so I just wanted so so what I want to do is is is talk about the the phenomenology of low energy Susy in a way that's as coherently as possible continuously connected to everything else about supersymmetry that you are learning in the school ah so as you all know people have been eagerly anticipating the discovery of supersymmetry at at colliders especially at the LHC as you also all know the LHC is now actually on and running it used to be a time translational invariance statement when they gave these lectures that the LHC was going to turn on in five years that was a true statement for many many years but it's all of a sudden no longer a true statement it's on running nothing's going to explode again and so that's that's that's really enormous ly exciting and for almost as long as people have been anticipating thinking about LHC physics the prime candidate for new physics beyond the standard model that people have been talking about has been supersymmetry now if you think about it the discovery of supersymmetry would be something really really rather dramatic it would be the first extension of our notions of space since Einstein the discovery of a new space time symmetry the first example of a broken space time symmetry even in the vacuum spontaneously broken space time symmetry so it's something it's something very dramatic you know bells are going off it's a it's it's a very very big deal and you might wonder why should we be so lucky why should it be our generation which is the general you know the hundred years went by since Einstein all those poor Schmucks in between okay discovered quantum mechanics too but after that after that it was all the same old stuff quantum mechanics and relativity into quantum field theory into this gauge group it does this it does that it's all great it's wonderful we got the standard model it's wonderful but but but they didn't run into these profound new symmetries of space-time and all of this stuff why should we run into it what's what's what's the reason why we why we might expect something like this something as dramatic as this is actually going on so that's that's part of the what I want to do is spend maybe the first half-hour of today's lecture telling you what that should be the case um and and it is it is something remarkable and it is associated with something which is actually a big shock it's unfortunately something which even though it is a big shock is presented in most books like it's some boring thing and alternatives are exciting but in fact it's the other way around and the big shock is that the Higgs mechanism will the dumb scalar field is likely the correct way the lecture weak symmetry is broken in nature okay that's a very very big shock we've never seen something like that before in a state of nature I'll make that point precise and we've seen all sorts of other ways that symmetry breaking can happen but not that particular way and so it's really something new now the argument for why that's that's the case of the relatively detailed one that has to do with the the structure of electric precision test and what I want to do is spend like I said maybe half an hour just setting that up because it's it's very important and also to orient you properly to a realizing that that get a light pigs perturb with the pigs lying around is something actually shocking it's not something it's not something boring it's something shocking and taking it seriously is what motivates something equally shocking to happen something like this large extension of a notion of this large extension of space-time symmetries is offered by heating so that's what I want to do spend the first part going over this story I'm going to go over it relatively rapidly if you heard the story before it's always fun to hear something you understand well explained again if you haven't heard it before you probably won't grab all the details as we as we fly by but I want you to remember all of the words the words are correct and if they sound surprising or funny just remember the words are right go back I think I think I gave them more detailed lectures on this topic at the at the P ICP school back in 2007 which I suspect are online and there I think there was two lectures devoted to going through the subject in in more detail so but that's what we're going to start off doing and then having motivated why we need something as dramatic as Susy we'll talk about we'll talk about at least a particle content of the MSM and some of the and and it's to really its two main qualitative successes things that we want to preserve as we keep keep going so that's the plan for today tomorrow we will start talking about the way the super symmetry breaking shows up at around the week scale in terms of the structure of supersymmetric theories perturbed by so-called soft terms we'll talk about the soft terms are and we'll talk about how to think about them in a supersymmetric way really picking up a theme that that that not see mentioned in his lectures that we can think of all coupling constants is background super fields and in fact we can think about soft Susy breaking in terms of just turning on higher theta components of these background super fields but that allows us a very powerful way of thinking about what soft terms are and all sorts of aspects of their physics without ever leaving the supersymmetric world so even though we'll be doing tons of not for symmetric calculations we'll even do one two four five loop non super symmetric calculations we'll never leave the comfort and safety of the super symmetric world and we'll manage to get these remarkable results through the power of holography as well as not these stress and Edward stressed as well because this this helps remarkably to solve and control theories in the super symmetric limit but we'll see in the second lecture that it also helps greatly to control and understand the structure of softly broken theories and depending on how we're doing are in the third lecture I will I will go up to even higher energies and try to come up with and try to tell you about the various ways people have thought about that these soft terms can arise from an underlying theory noxee and i have have divided responsibilities to some extent not he's going to tell you that the general theory for how supersymmetry is broken and in the end of the day he's going to talk about a wonderful chiral super field x that contains all the information about susie breaking in it he will tell you where x comes from from theories that break susie I will take X and I will use it and so it's a tag-team action ok ok very good so let's let's let's start off with why we should be so lucky well let me just let's just back up a lot and just talk about what we've seen in the world so what we've seen is a bunch of spin 1/2 particles and a bunch of spin 1 particles some of these spin out some of these spin 1/2 and spin 1 particles are massive like the W in the Z or mass of the top quark is massive all the fermions are massive um but I want to ignore the masses and first approximation so whatever the masses are I want to imagine that ah that I want to describe what the physics looks like it energies way above the masses so intuitively we should be able to ignore the masses so let's imagine everything is massless at the moment some hypothetical world where energy they don't know 10 times the mass of the top quark we want to just talk about what's going on at energies 10 times the mass of the top quark I'm a dumb guy I haven't opened up okay I'm a dumb guy who opened up asking Schroeder enough to flip through and see there's such a thing as the lagrangean but I'm too lazy I don't know about gauge theory I don't know about anything like that so ah so by the way you shouldn't take this as a as a template for how to behave okay but uh but I'm just telling you how a dumb guy would discover the correct way of thinking about the world this is I think a very important point you should you should appreciate I'll try not to stop with stupid philosophical asides too much but it's very very important thing about physics to understand how how you can be dumb and figure out what's going on by which I mean but by Dom I don't mean willfully dumb I mean not clever okay physics isn't about being clever it's about the principles telling you how things work so if you understand something properly it should not be for some separate random clever reason that you understand every little things should things follow from general principles and you should be able to see that if you do something the dumb way that's the first best way to try to do something if it works yay but if not you'll learn something and you should learn what the dumbest thing doesn't work try the second dumbest things for the third dumbest thing until you get it okay that's really the number one rule of progress in in science you don't have to be a abon ischial genius to figure everything out as I will describe okay so I'm a dumb guy I don't know anything about gauge theories I just know I'd be spin half spin one particles but I've read enough of tasking the schroeder to know that I should write down interactions for them okay so here's some kinetics very in general some kinetic terms for the gauge field from the fermions note I'm not writing F squared I don't know anything about a PI just write D mu a nu squared I don't know maybe there's other ways I could contract the indices I also know enough to know that that since everything is massless here I want to ignore massive I just want to talk about that well I'm ignoring masses so any interactions that I should write here so be nice dimensionless interactions anything which is dimension full would have a negative mass dimension and presumably that's something that irrelevant small I can I can neglect it has to do with much much much much higher energy shorter distances I so I just want to write down every possible dimensionless interaction so I could write down interactions like this I could write down interactions between the gauge fields I could things like this I have to write all sorts of interactions like that okay back this is about all I can write down which is dimensionless okay great so given the particle content a bunch of spin 1/2 and spin 1 particles if I want to have a theory that's local I write down the Lagrangian that's why I write down the Lagrangian everything's nice and local there are Simon rules that can calculate there all these different coupling constants all these coupling constants just by engineering dimensions are dimensionless so there's an invariant sense in which they're weak or strong just a dimensionless number is small ok very good all right so that's it so that's all that I could possibly describe the world that low energies if this is the particle content we have but I start having a problem when I think about what I really physically want to describe see I have these massless spin-1 particles and the massless spin-1 particles I know only have the physical states are only two polarizations so they only have whole isset e ok so I can write down if I wanted to write down some plane wave solution for one of these massless spin-1 particles I would write down something like a mu of X is epsilon mu e to the IPX and there's naively for component 2 epsilon zero one two three and so that's too many this this Lagrangian is describing four degrees of freedom never mind the fact that if you look in more detail some of one of the wrong sign kinetic terms and all of this stuff there's worse things going on but minimally I have four degrees of freedom and not two I know I know enough to know that there's only holistic I'm only trying to describe two degrees of freedom there's four of them this is the price that I pay to be able to write down the Lagrangian and describe the physics in a way that makes locality manifest okay so I'm trying to make locality manifest but what's occurring in the Lagrangian has way too many degrees of freedom okay well just at the level of these of polarization vectors what could I do I could say well those polarization vectors don't have to be random they can satisfy a constraint everything should be Lorentz invariants or the constraint could be something like P mu epsilon mu equals zero that's fine that's one constraint that knocks for the three degrees of freedom but I still got three degrees of freedom not two so I'm stuck I'm totally screwed okay I'm trying to describe massless spin-1 particles and I can't do it I have three degrees of freedom lying around you can say oh come on don't give me a hard time let me just say look there's two polarization vectors I'll go to some frame and I'll say the polarization vectors are like you know what you read in Jackson okay that's what the polarization vectors are there's two polarization vectors everything's fine that's true that's perfectly fine you can define those to be the polarization vectors in some frame of reference with the photon moving in the axis it's true that I can't force it to have that form in the Lorentz invariant way right I could add anything proportional to P mu to epsilon and it would continue to satisfy the first constraint but I can choose not to do it I can say that there's only these these two guys however the price I pay is if I do a Lorentz transformation in another frame it won't have that form and another frame the photon will be moving in some new direction and written in terms of that direction it will have time and the prime component okay so we're stuck one way or another either there's no Lorentz invariant way of talking about the two polarization breakfasts or you can declare that your two polarization vectors but the price is that what you think we think these polarization vectors transform nicely under Lorentz transformations but they don't okay so one way or another our difficulty is that epsilon mu is not uniquely determined even by that constraint but epsilon mu and epsilon mu plus anything any function of P times P mu the only thing we can do to talk about two degrees of freedom the tool elicit ease is declared this is our declaration declare that these are to be identified in other words we have to declare that these two configurations for epsilon correspond to exactly the same state so this is the new sort of thing that we don't run into when we talk about the spin 1/2 and spin zero particles I'm repeating things that many of you have seen in many different guises but I'm saying in the language it's going to be useful for everything else I'll be talking about um so we have to declare that these are to be identified so this is a new idea the idea that in order to describe a massless spin-1 particle in a manifestly local way we have to introduce a redundancy into our description of the physics we have to say that these two polarization vectors correspond to the same state now we're down to two states which is correct ok now this has real teeth ok this has this has real consequences it means the following a remarkable thing I can take the theory the Lagrangian that I just erased a very general Ranjan that I race and I can start computing scattering amplitudes with it ok compute amplitudes Fineman diagrams I compute all of them and in the end of the day to get a physical amplitude for particles that are Felicity h1 up to HM let's say photons of hoods to the h1 of HN I take those things that I compute from priming diagrams and I contract them into these polarization vectors but if the states are to be identified then something remarkable has to be the case it must be the case that if I do this shift on the external polarization vectors this amplitude doesn't change so something remarkable has to happen it must be the case that for example if I take P mu and doesn't - mu with any other indices here that this should vanish I can't write down any old random Lagrangian I have to write down the Lagrangian that somehow makes sure that this remarkable thing happens I have to ensure that this redundancy is is is there realized now this redundancy is otherwise known in old textbooks of gauge invariance or gauge symmetry but it's not a symmetry to redundancy its the declaration that a mu so if I say the statement in position space its that mu and mu plus B mu something are to be identified and only if you have a Lagrangian that has this property do you even have a prayer in hell of getting amplitudes that satisfy this property this is really what you need this is really really what you need but in order to ensure that redundancy you need to start off from Lagrangian that has this sort of gauge identity okay so the gauge redundancy is not fundamental it's not there it's all in our head it's the way we choose to describe the physics of massless spin-1 particles because we want to make locality manifest it's an incredibly convenient way of doing it because it'll does allow us to make locality manifest and use all of the machinery of quantum field theory and when you have a standard field theory Fineman diagrams and so on and when you have when you have many spin one particles then clearly from the arguments I told you just this polarization vector argument you should allow for this possibility and when you look in a little more detail if you allow self interactions between those spin one particles you actually have to augment this a little bit by further allowing a rotation on the a index okay so that completes that completes it into a non abelian transformation on the epsilon then the corresponding non abelian transformation on the other side again this is something this is something that one could spend a lot more time talking about but I just wanted to stop it at this point that describing the massless spin-1 particle is because they only have olicity only two degrees of freedom forces us to have redundancy and the most convenient way of doing it is the gauge redundancy that we all know and love okay and the correct answer ends up being that you write down a gauge invariant lagrangian plus gauge fixing terms and that thing gives you amplitudes that satisfy this property and therefore correctly describe the scattering of the whole isset ease you want okay so again if we want to talk about physics at very high energies then compared to the masses of all these particles now we have a standard apparatus and when we say that the standard model gauge group is su 3 cross s to do cross C 1 this is what we mean okay so neglecting all the masses of the particles the useful redundancy to describe nature at short distances is SB 3 color plus SC to left across d1 hyper-charged so there are gauge bosons in the adjoint representation of that group and the matter fields are vaio fermions three copies of them because there's three generations I'm going to use this notation for the fermions and also later we'll the corresponding super fields but if this is a table that you don't know off by heart then you should learn it that's something else that you should learn in this school okay so the representation okay that's the that's the matter representation for the standard model okay so these ones adjust at these guards of singlets and the corresponding group these are the hyper charges okay so if we want to describe what's cool I said in the limit where we neglect the masses I just want to know what's going on at for scattering these particles at high energy is a good way of thinking about it is in terms of that gains redundancy with the matter particles transforming in that way under the gauge group and so you might think where does something exciting happen in this picture of the world okay we have you know we measure things at around 100 GeV 200 GV we discovered the top quark it weighs 175 GV the wnz way about 100 GeV I go to very short distances at this nice beautiful perturbative theory that makes sense where does something exciting happens where does something new happen well something new happens at the plunks okay the pong scale we have no idea what's going on gravity becomes strong non-normalizable effects nonda normalizable higher dimension operators that might be suppressed by the pond seal becomes important all hell breaks loose space-time is emergent blah blah blah okay all that stuff is going to happen uh because of this very weak irrelevant interaction gravity that becomes strong at around the park energies that's what you very naively think but it's wrong it's wrong because actually in the theory that we just described there was a little sleight of hand which is very important and the actual fact of the matter is we take you see I've done something ultra conservative here I just took the particle as we'd seen I didn't say the word Higgs I didn't say anything like that yeah just taking particles we've seen and I want to ask if anything goes goes wrong in that theory naively if we make it the skates Theory everything is fine up to arbitrarily high energies maybe until you hit the punk scale but there is something wrong in fact the theory of the particles we know and love and nothing else predicts its own demise it breaks down it cannot be consistent already at one TeV and the difference between one TV and 100 GeV is significant literally 11.2 TV something like that okay so let's go through that argument so why is that the reason is I made a mistake right from the get-go and I said well at very high energies we can ignore the mass of the particles again if it'll spin one particles that's misleading that's misleading because the spin one particle has three degrees of freedom when it's massive and everything we just talked about has to do with the behavior of the to Felicity is for the massless spin-1 particle okay there is one more degree of freedom that survives even at high energies that we have to talk about roughly speaking we've got to talk about the longitudinal component of this massive spin one particle so let's figure out how to do that now I'm not going to do this quite in the full glory of the standard model uh I'll do it in something extremely close and then I'll give you the relevant modification that just converts it to the right answer at the end but let's let's talk about a slightly toy theory of a massive well actually let's before doing that let's talk about something simpler let's talk but a massive photon okay so let's say I saw a nature somehow I saw that there was a just a long-distance Coulomb long then I saw that it turned into you callosum okay so I wanted to try what's going on well ah I would write down the Lagrangian that looks like a quarter F mu nu squared plus M Squared mu a mu let's say there's even a Fermi on and it has some interaction the only dimensionless interaction it could have anyway looks looks looks like this and let me as convention will put a G here so let me call this quantity V for no good reason right now other than to note that the mass would just be G D if I want the canonical normalization for this field and that this guy this side by virtue of that coupling have some coupling to the photon right so this would on the face of it describe a theory that gives me a what looks like it's short distances Coulomb interaction between these fermions in a very long distances it turns into you kaua this there is in gauge invariant who cares there's no such thing as gauge symmetry arm so I'm just writing it down it has the correct has the correct degrees of freedom where physicists we're not philosophers so this is just right now you might say but come on where is that that gate symmetry surely as we go to very high energies compared to the mass of the particle I have to I have three degrees of freedom I have the two transverse components and I have the spring zero component and surely I need to see that at short distances the transverse guys are interacting with nice with the nice gauge redundancy and everything okay so surely I should be able to just in order to study that physics at very very high energies I want to have this gauge Anansi around I'm sorry I want to have it around because it's a useful thing to study those two trans those two velocities of the massless spin-1 particle that's fine in fact if you haven't seen this little exercise nothing will convince you more that gauge symmetry isn't the symmetry but a redundancy but the little magic act we're about to do which is going to convert this crappy awful non gauge invariant Lagrangian to a beautiful nice gauge invariant Lagrangian okay this is possible because there's ain't no such thing as gauge symmetry how do we do it well it's not gauge invariant so let's do a gauge transformation when we do in gauge transformation it's not gauge invariant so mu goes to a Miu plus D mu theta so you say that's what I've always been told big master miss and gauge invariant look how terrible it is that's fine the theory that we started with had no redundancies right that's what it does under gauge transformation so there's a very nice trick imagine doing a gauge transformation but elevating the gates transforming parameter to a new field okay so I'm going to change my Lagrangian now new Lagrangian new start start over okay the new Lagrangian is this where theta is now a field okay and everything else is the same as it was very good now the old Lagrangian didn't have any gauge redundancy by construction the new Lagrangian has a gauge redundancy now I have a symmetry a gate redundancy under mu goes to a mu plus d mu lambda and theta goes to theta minus lambda you might complain this looks cheap that looks stupid it's this is called the Higgs mechanism okay as we'll see in a moment okay so it's not cheaper stupid it's actually a really good idea notice that I can always eliminate fixit's redundancy to the gauge which eliminates theta I can always choose lambda of X to equal theta of X that's fine theta is gone and I have a gauge fixed description which gets me back to the original Lagrangian but what's useful about this way of talking about things is that at high energies for the purposes of questions involving high-energy scattering I choose a different gauge I choose a more standard gauge Fineman gauge or you want to be a little fancier arc see gauge to eliminate the bit of mixing between theta and a but the really important point is that what this is making manifests is that there are three degrees of freedom the two degrees of freedom associated with a mu and one more associated with theta that's the longitudinal mode and we've isolated who it is okay in fact if I want to study the physics at very short distances there's even a nice limit I can take I can take a limit where I hold the V fixed and I send g20 let's say I do that I hope B fix and I send g20 then the dynamics of this data completely decoupled from everything else okay and what do I have I just have an ice-free theta field that's it so at very high energies compared to Gd compared to the mass I have what looks like just QE d plus an additional scalar degree of freedom which is the longitudinal mode in this case nothing goes wrong just free at high energies everything is great this corresponds to the ancient fact known to people like Feltman and others in the 60s that massive QED is Rijn or malai Zabaleen way of saying it but in this way of saying it it's just clear the physics makes sense at arbitrarily high image I can just put in a mass term by hand no where as I say the word Higgs mechanism nothing everything is fine okay that's very special to a case of you want one of the exercises I want you to do after we go through the case of su to presently is for you to go back to this u1 case and see and try to understand why this limit is allowed to arrive in the U one case and not in the su two case by seeing how we arrive at the limit that we talked about starting from a Higgs Theory upstairs okay but anyway let me just continue with the let me continue with the narrative so if we're talking about so now let's talk about something more exciting I want to talk about a triplet of vectors that are massive okay so once again at very high energies if they're to interact nicely with each other just their longitude four components I'm going to make an su to gauge redundancy theory out of them okay but I'm going to add something that gives them a mess [Applause] so mu here is some a mu a sigma a okay just a triplet of guides now I've chosen to do something here it's a choice which is simple and happens to match something important in the standard model I've chosen choice to make this match term invariant under an su to global symmetry don't confuse it with the gauge symmetry it's under an su to global symmetry there is no gauge symmetry the mass term is just broken it okay but there's a global symmetry which is saying that all three of these guys have the same mass I'm just doing that for for a convenience again it'll take place in the standard model okay great this theory describes a triplet of massive particles whose masses GV once again I want to isolate the longitudinal degree of freedom so I do exactly the same trick that trick now tells me to take a Mew and they do again transformation on it so it turns in a mew turns into that under a gate transformation so you hear some eetu the i theta a sigma egg okay and once again i do exactly the same thing i elevate you to a field okay now this is a nice new exciting gauge invariant theory whose Lagrangian is negative 1/4 trace x squared plus trace u inverse covariant derivative u squared V squared now let me do exactly the same thing I want to study these new degrees of freedom the longitudinal degrees of freedom of this massive spin one particles in the theta is now there's three Thetas there are three Thetas that have conveniently grouped into that matrix u okay the gay transformations act on youth obviously act on you this theory is gauge invariant but now let me do exactly the same thing let me imagine keeping D fix and sending g20 okay so again I'm imagining scattering it very very very high energies compared to the mass of these particles so then I can essentially neglect the gauge coupling the theory breaks into two sectors that dot don't talk to each other the gate sector that talks to itself and this you stuff let's focus on the Lagrangian just for the use it's just given by B squared trace u inverse DM u u u inverse mu u well beforehand when you was abelian u was e to the I theta it's same manipulation and this gave us just B theta squared it was a free theory but now U is not an alien it's a it's either the I theta a TA and you can recognize this as a nonlinear signal model and this is an N interacting theory fact if we just expand it out what do I get I get B squared D theta squared all the three Thetas but then I get terms that look like theta squared D theta squared plus dot dot get all sorts of terms and if I canonically normalized this field so if I say D theta is a no-no pi then this is d pi squared plus this is all schematic here B PI squared PI squared D PI squared over V squared plus dot dot dot this is an interacting non-normalizable theory ok something physical happens there is non-trivial scattering between the pi particles so I look at the scattering between PI particles there's an amplitude but just by dimensional analysis has to go like energy squared over V squared and therefore at an energy comparable to V there's amplitude becomes big I mean this theory needs an ultraviolet completion the logic is identical for why quantum gravity needs an ultraviolet completion exactly isomorphic logic it breaks down it has a maximum UV cut off of around V and if you're more careful and if not he doesn't yell at me I'll put a 4pi in front of it as well okay that's not 2pi than expansion parameter we can fight about it again that tell you all right I'm very good all right so so let's oh I should have said something I said there's an not nonlinear signal model um in fact so what are the symmetries that that it manifests of Lu has an su 2 left cross su 2 right symmetry okay just a u goes to L you are if L n our arbitrary 2 by 2 matrices and you see when L is equal to R so the piece with l equals r u goes to L u L just infinitesimally sends Thetas shifts faders goes to theta plus constant plus dot dot dot ok so that symmetry is just linearly realized ok that symmetry is the one that's spontaneously broken so X into L cross su 2 R is broken to the diagonal you too so that's the nonlinear signal model it describes the breaking of su to left cross su right down to the diagonal su to when l equals R dagger that's an honest su 2 transformation the unbroken diagonal su 2 when l equals r it's the thing that shifts it and moves you from one vacuum state to another vacuum state if you think about these things as being also in those aren't they are those some roses okay this is the effect of Lagrangian that describes also impose on we didn't say the word symmetry breaking we didn't we're just studying what happens to the longitudinal components of this massless spin-1 particle we've discovered that it's described by a non linear signal model okay and again that nonlinear signal model breaks down at high energies it needs an ultraviolet completion so let's draw a picture of the energy scales in the theory okay so here's four PI D that's the lambda Max and here's G V which is where the mass of the three gauge balls on this so you see there's something very nice there's a huge range of energies parametrically where it makes sense to talk about this theory of massive spin one particle there's nothing wrong i can go well above the mass of this guy everything is fine if a perfectly well nice perturbative theory everything is great the only thing that's wrong with this theory is that if you care that energy is a border for pi d it needs a replacement it has to come from somewhere else okay needs an ultraviolet completion now in the real world the G's aren't all that small and this number is around 100 GeV and this number is around 1.2 TeV or 1.5 PV or 1.7 PV depending on who you talk about depending how seriously take those pi okay but it's something which is parametrically above to the sense that the couplings are weak it's primarily above the mass of the gauge boson now we haven't been there yet we haven't been to 200 GeV we haven't been to one 1.2 TV sorry okay this is how we've managed to have this wonderful spectacular successful theory agrees with data everything is fine we call it the standard model we call it the standard model um despite the fact latency in the game have we done it this is a theory that we've largely been testing okay but we've also learned something rather important that something new if I just take the particles I know in love something new doesn't just have to happen at the Planck scale something new has to happen way way way way way before then something new has to happen before 1.2 TVs before 4 PI Z is that clear any questions about that ok very good so what could that new thing be well what we learn in books as the standard model is getting this low energy Theory out of a linear signal model in the ultraviolet that's the Higgs mechanism that's the that's the usual sorry that's the usual perturb ative Higgs mechanism that that we talked about by introducing a fundamental scalar field okay so for example I can write down I can introduce the higgs doublet H I can write down the Lagrangian which is a BH dagger BH and have a potential arm which is lambda H dagger H minus V squared squared notice just just accidentally in this case H if I think of it as an H 0 and then H 1 let's say oh these are both complex and notice that both this kinetic term and the potential if I write each one of these as something real put something imaginary it just so happens that H dagger H R is a naught squared plus a 1 squared plus B naught squared plus B 1 squared which is so4 invariant so this is an accident there's a bigger symmetry of this theory just accidentally that's an fo for symmetry which I can think of as su 2 cross su 2 and so it has the phenomena that we talked about before there is in global su 2 symmetry normally called the custodial symmetry which survives which happens in this theory with this particular you be completion with this particular this particular model you can see that more nicely by grouping that doublet into a two by two matrix which is H naught H 1 and H 1 star negative H naught this is H naught and epsilon times H naught and now now H this is invariant under H goes to L H are the SU 2 gauge symmetry is L is gauging the su 2 L part of this global symmetry and the Higgs getting a dev giving the real part of a zero component is just saying that H is equal to V times the identity to IQ identity matrix and so you see if you take so that means that I can parameterize a general point of general vacuum by V times some two by two unitary matrix you just acting on this front by the left and the right okay and so shoving this back into that Lagrangian just gives us this nonlinear signal model okay now the advantage of this theory is that this is now a very very high energies this is the theory that's perfectly well-defined again only a dimensionless couplings it has the gauge redundancy everything interacts with itself everything is fine it has a UV it is a UV completion of this picture actually it's only a partial UV completion of that picture but let's just talk about it for a second so what happens in this theory what happens is that there's a physical Higgs particle okay there's four degrees of freedom here three of them become the longitudinal component we talked about there's a physical Higgs particle the mass of that Higgs particle M Squared is around lambda times V squared which you can distribute see by staring at the the quadratic piece in this potential so what happens in this theory is here was the scale for pi v here's a scale GB but somewhere in here there is a scale square root lambda V this is where the physical Higgs particle lives down here we don't see the physical eggs particle down here we just have this nonlinear signal model that we talked about the amplitudes are getting big big big big big but then something new happens you see that there's a Higgs around okay you see there are gigs around and then the ultras and then the theory above that is just this linear signal model everything is nice and fine and you get to run to shorter distances we'll talk about some constraints in a moment going to shorter distances very well you certainly get to go for quite a while let's see that there's a zeroth-order consistency of this picture the amplitudes down here we're growing like energy squared over V squared just as we said so right around the scale how big are the amplitudes getting well it's a root lambda V squared over V squared so they reach they saturate at around lambda that makes perfect sense because above that scale you all of a sudden see all my three Goldstone bosons of the physical Higgs particle are unified into some bigger guy everything is going fine as I go to shorter distances what are the elementary four point interaction between these scalars it's just that Lambda Phi or the fourth coupling right lambda x of the fourth coupling and the size of that interaction is lambda so everything makes perfect sense okay if lambda is small I have fixed this theory I get to go March to shorter distances with it so that's what the Higgs mechanism is buying for you it's not beauty it's not that the symmetry is nice we don't want to mess with symmetry everything in physics has teeth anything which does something for you is buying something for you the teeth here is that it's allowing the theory can make sense to much much much higher energies than it would have otherwise okay now that coupling constant is not in general asymptotically free I'll tell you what the situation is in a general theory but just I'll tell you what the situation is in standard model more precisely in a second but it's not in general asymptotically free so that means that if lambda is perturbative if it's weakly coupled okay something new happens pretty early and then I go to higher energies maybe I had a landau Pole exponentially far away goes like e to the 1 over lambda ok so there's some land on pole scale up here that you might worry about lambda land oh okay but I'm winning exponentially thanks pushing new things that have to happen exponentially far away what happens is they crank up lambda as I cranked up lambda from the Lord as you point to you start getting worried because oh I'm not being saved oh I just got saved it's okay well nothing say oh I just got saves okay until I get up to around here you're not getting saved right from the low energy point of you are not getting saved not you're the high energy point of view we're not getting saved because that lambda pole scale comes down right on top of you okay so this whole picture of a Higgs mechanism only is useful to the extent that that quarter coupling is perturbative okay you can write down a theory if you like with a big quarter coupling at low energies is indistinguishable from not adding a Higgs and kind of juice industries but from not having MiG's because the theory doesn't make sense of shorter energies anyway is that clear okay so this is what we learn in books okay this is this is this is what we learn the books and as I said it lulls you into thinking that this is the boring thing and that other things strong dynamics technicolor maybe supersymmetry we'll come back to persimmon tree in a second are exciting and really it's reread the other way around because while this looks very simple from the point of view writing down the field theory we've never seen anything like this in nature in a state of nature I will make that more precise in a second that's the entire content of the hierarchy problem is that we've never seen anything like this in nature before or we've never seen anything like this in nature before unlesss will describe there's someone out there fine-tuning with a desire to make a phase transition happen or something exciting to happen but there's always an external agents that's finally adjusting things to engineer this effective theory this effective theory in a precise sense is not natural and we've never seen it before we've seen this phenomenon of symmetry breaking all over the place but never like this what you be completed is always something different for example we don't have to look far tcd itself gives us PI on there effectively Ranjan is exactly the same one that we're talking about and what UV completes the pion effect of Lagrangian QCD the theory does get strongly coupled and you go over into a completely different phase at short distances there's no PI ons there's no elementary scalars because of TCD things can find chiral symmetry breaks there's a wonderful story but that agrees the freedom of the long distance non-normalizable theory bear no resemblance whatsoever to the degrees of freedom that that are there at short distances that's the typical situation so when we see things like that happen in a state of nature naturally occur it's always been so far always been because that basic phenomenon has taken place that's the conservative thing that could happen okay we could run exactly the same story and say where can we get this effect of lagrangian from let's get it from something that looks like QCD okay just mimic the PI on story and that theory is called Technicolor and I'll repeat myself for the third time that in my mind would have been by far the most conservative thing that could be going on it would also be by far the most that could be going on from the point of view of finding something novel and exciting and new I'd be like oh that damn thing again yeah yeah yeah we got a conventional transmutation we generate a scale you make gold stones blah blah blah great wonderful but we've seen it before it wouldn't be it's not something essentially new so the important thing that happened when our experimental colleagues uh when the left one experiment in the very late 80s and early 90s was that we got strong evidence that this picture is incorrect that this conservative picture is incorrect and instead something like a perturbative light Higgs is really preferred by the data so let me just quickly tell you what that story is again this is something uh I think you'll be able to find online in more detail from my 2007 Vipp lectures let me just give you the rough structure in the argument the rough structure of the argument is that that well this is a nice non-normalizable theory um and we've written down its T derivative terms but of course because this Ananda normalizable theory we should expect all sorts of higher-order terms as well okay now many of these higher-order terms come from shorter distance physics and we don't know we don't know what they are oh I should have mentioned something quickly um I should have just mentioned mention at this point very quickly uh we can having having having gauged it here the unitary gauge in terms of you is putting you equal to one I can always choose the gauge where u equals one and then when u equals one this just turns into a Mew and this is the master okay so this kinetic term for the Goldstone bosons always turns into the master term for the gauge field well that's where we got it from okay and in general unless I tell you what this higher order what the full theory is upstairs I can't compute what the coefficient are so let's say we really do have a theory which gets strongly coupled up here there's some scale lambda around 4 PI V there's some there's some UV Theory up here ok at the scale for pi V if I knew what it was I could integrate everything out and I will get an effective Lagrangian that has this as a leading term and we have correction better suppressed by powers of V okay that's what I'd expect but as a scale to low energies you see we're not at V so the mass of the gauge boson is down at GD so let's say I want to study things near the mass of the gauge boson that's what the left one experiment did it sat on the V Pole and it collected well between left 1 and slack there roughly 5 million Z's were collected and studied sitting on the pole ok so we're interested in physics at the V Pole so the usual effective Lagrangian is not sitting up there at 4 PI V it's somewhere down near G V so I should just run I run from that scale down so I have to compute loops in this non-normalizable theory that's fine that's not scary everything is perfectly okay there are contact terms at short distances that I don't know but I can generate radiatively and I'll just give the two relevant examples for example oh and I should have said in in the standard model I should have just I promised I'd tell you the proviso the nonlinear signal model the standard model is su - L cross su 2 R but you gauge this is gauged and aun hyper-charged is the just the t3 subgroup of that s you two are okay so that's how the standard model arises in this picture it's not just an su - its su 2 cross u 1 and the you once it's inside the T 3 component of the hyper charge anyway there is for example this diagram that you can compute well before computing a diagram there's a completely invariant operator that you could have in the theory nothing stops me from having this operator in in the theory it involves W and B explicitly so this is probably proportional to G 1 G 2 and I can generate it as a radiative effect if nothing else it could be sitting there already at high energies but there's also a low energy contribution to it that I can compute explicitly from a loop of these Goldstone particles ok this is a loop calculation than Ananda normalizable theory but I can I can get a reliable answer for it precisely because it gets a contribution all the way from the hi scale down to the low scale it has a logarithmic dependence on the external momentum here every time you have a logarithm that's something that you can compute in the long distance theory ok so there's a piece here that goes like log of the scale that I started wherever it was maybe it's not 4 PI V the first scale where I get this effectively Ranjan so let me call that lambda nonlinear signal model ok I get I get a correction to this operator there's an unknown piece so I get this operator how do you write it like this there's an unknown UV piece but there's a piece that I can compute that goes like g1 g2 over 16 v squared log of lambda nonlinear signal model where I began over well where I end which might be G V the mass of the particle so somewhere in that vicinity but the point is if there's a piece that's immense by this logarithm gets a running effect of a low energy theory I can compute it reliably in this effective theory and it has a weak but nonetheless it has some dependence on what that scale lambda nonlinear signal model is it's logarithmically dependent on that scale so of course I don't know what this unknown contribution is but there's at least a contribution to this object which is which grows logarithmically now the overall coefficient of this operator is normalized with some alphas that they won't be called about but it's called the S parameter okay and that's something that you can measure by doing precise experiments on the Z the reason is that if I go to unitary gauge this is affecting it's giving me some mixing between between the w3 component and the B component okay and so it shifts the coupling of the vector to the standard bottle fermions okay it's shifted by how much shifted by something which is about a percent which is the size of this rate of effect times the logarithm so let's degenerate say about part per thousand could part for 100 type of Corrections of the strength of the coupling of the Z to the fermions this is something that was measured experimentally the whole reason these guys went the 5,000,000 Z's is so that the statistical error on any measurement they could make would be 10 to the minus 3 and what started approaching 10 to the minus 4 so people were measuring these couplings to the 1 per mil in a few times 10 to the minus 4 levels because they knew that you could use that to probe something that was sitting there at the part for 102 apart 4,000 level okay and it's true that we don't know the unknown piece but if this is if this is a part in 100 there would have to be some decent conspiracies if this whole thing ended up being around a part in a thousand or part in 10,000 okay so this is one Operator it turns out that there is another one which is a which is which looks like the kinetic term except it has powers of p3 in it I should say the reason why these two operators us spring to mind is something to look at I haven't I haven't stressed it but this breaking the fact that we're gauging this hyper-charged is the subgroup of s due to right breaks this su to right symmetry that we're talking about before this custodial symmetry that we're talking about is broken by by these effects so what's special about the two operators that are written down there the leading things involving loops of these Goldstone bosons that are sensitive to this breakage so that's why they are special anyway is also normalized alpha is called the T parameter similarly gets radiative contribution from from a loop of the W of the B gauge field it similarly has a log in it so there are two logarithmic lis enhanced effects okay which are important now the punchline of the story is that Smt have been measured experimentally I won't even bother putting the actual sizes and units here but anyway if something looks like T and s there are some to Sigma experimental ellipse these things are of order point one so that the size of these scales are roughly 0.1 and 0.05 in in these in these directions that's as advertisers because we've gotten down to around the per mil level in accuracy on these coupling constants but the point is that you can figure out there's the unknown part that you don't know but you can figure out what you would get from these calculable parts in the low energy theory and what you find is if lambda nonlinear signal models at what TV you're out here somewhere like I said it doesn't hit you in the face you have to do a lot of work the experiments have to do a lot of work the fears have to do less work if you calculate some one loop diagrams but but you get a really remarkable result okay there is a contribution that's sitting there if we have this picture where these dulcimer zones emerge from an underlying strongly coupled theory you are beginning life way outside over three sigma really more because you're combining two of them outside the experimental error bars okay so this is the sum total but very strong but circumstantial evidence that independent of saying what it is Technicolor this that just a property of this long distance theory we've really taken the theory of just the massive spin one particles that we've seen ultra super duper seriously if we take that theory ultra super duper seriously we see that it's telling us that wants to break down not just at one TV where it absolutely has to it must break down at one TV right but this is telling us something more detailed it wants to break down earlier something new will have to happen earlier and you crosses the lips around lambda nonlinear signal model now roughly around 100 GeV so for example if we had gotten this nonlinear signal model from an effective Higgs Theory at low energies then what would land and only a civil model be it will just be the mass of the physical higgs particle that will be the first scale at which the low-energy theory looks like these Goldstone boson fields and so that tells us in that particular theory if it's a Higgs that the Higgs should be like it should be around 100 GeV but more importantly it's telling us something else independent of any particular theory this nonlinear signal model needs to be replaced by something before it gets strongly coupled way before it gets strongly coupled almost right next door okay so something new and perturbative has to be happening right above our heads and in all the years that people have thought about it the only new perturbative thing that can happen right above your heads is a Higgs particle or a collection of Higgs particles getting this nonlinear signal model out of a linear signal model okay I'm going through this in so much detail because this is the reason why we expect that this most conservative picture is wrong and that's something rather striking is going to happen we have to see something we've never seen before we have to see an effective theory with light scalars light perturbative scalars lying around we've never seen that before now in fact I want to have time to go on this into much more detail but let me just say that if you take this Higgs effective theory if you take now the standard model so that's what standard model is right so the standard model is just adding the Higgs so to our table of particles we can add a Higgs which is a scalar and whose quantum numbers are exactly the same as L okay and now everything everything makes sense in addition in addition to the kinetic term and the potential we also have these new cowell couplings so this is a well defined theory as you go to the shorter and shorter distances ah nothing is nothing funny happens around four five or five years a total Mirage scale now nothing's going on there it gets replaced by this theory before any of that happens and now it turns out that the only dangerous thing now as we go to shorter and shorter distances the only thing you've got to worry about for the this theory making sense all by itself is that this V has a squirt a coupling and at this court a coupling gets too big the theory will hit a landau pole at relatively low energies let's say you want this theory to make sense all the way up to the Planck scale having it make sense all the way up to the Planck scale tells you that lambda should be smaller than something that in turn makes the physical Higgs particle lighter than something and that something is around 180 GED okay so plague physical hits particles lighter than 180 GeV this theory that allows us to sail past four PI V also allows us to sail all the way up to the park scale nothing wrong okay if Mantha gets too small something else happens you have to look at it in more detail in the renormalization group but not only does it not grow as you go to high energies but it actually gets driven negative as you go to high enough energies okay gets driven negative by the effect of the su to gauge gauge coupling and the normalization group equations and that means that the effective potential did another lower minimum than our minimum at much larger values of the field strength and if you want to demand that our universe hasn't tunneled from our nice current vacuum into that one that puts a lower bound on lambda lambda can't be too small and that number turns out to be around 115 GeV rather remarkably if the Higgs is right at 115 GeV and it's just a standard model and nothing else all the way up to the Planck scale then we would cuddle out of our safe vacuum into these low energy into these other vacuum on a timescale comparable to the age of the universe okay so there's a relatively narrow window between 115 GeV in 100 80 GeV but it's not that narrow anywhere in that window this is a perfectly fine theory makes sense through energies up to where we care to the Planck scale where something new has got to happen anyway so so in this range everything is just swell okay so we can go all the way up to the Planck scale without a problem now imagining that this theory makes sense to very very high energies all by itself without any anything new happening also immediately explains a number of other striking facts about the world and explains them just as accidental symmetries of this theory you see if there is new physics coming in at some mass scale Capital m you would expect there to be in addition to these normalizable interactions higher order higher dimension operators suppressed by the mass scale m and well we can just write many many of them down but if we ignore all of them and if we just look at these interactions which are going to control the ones at low energies this theory has approximate symmetries fact exact symmetries in the limit where you ignore these and ignore instant ups but there is very odd number lepton number okay so these are just automatic accidental symmetries of this theory you don't have to do anything you just have to declare that the next new thing doesn't happen for a long time from now okay so I can write down de Max and six operators that would violate baryon number but as long as the scale that suppresses them is above ten to the fourteen Givi everything is fine okay similarly with lepton number I can write down dimension five operators that give me neutrino masses and if I put those dimension five operators at 10 to the 14 GeV everything is fine so in fact everything looks like ten of the 14 GB is a great place for new stuff to happen okay but if nothing new happens up to there everything is great we totally understand all of these wonderful approximate symmetries of the standard model there is more detailed things the more detailed things are unlimited you turn off these Niccolo couplings if we turn them off the theory gets a very large chiral symmetry I have three generations of fermions of three different types Q u BL and E the Q deletes as I like to call them and there's a separate u3 symmetry that acts on each one of those q delese okay you three that rotates the Q's use these elves and ease those cuter leaves those that you three times you three that you three to the fifth symmetry is broken explicitly by these u comma matrices in a very well-defined way these are calamy trees or spore eons for the breaking of that u 3 to the fifth symmetry exactly the same idea not see what's talking about these lectures we can think about the landers of background fields that transforms under the u 3 cross 3 3 the u 3 to the fifth symmetry and so they just have not nonzero values which is which is breaking them but we can understand it's broken in a very specific way it's broken up the quantum numbers of these guys and as is well known all of the phenomenology of flavor flavor violations in the standard model are completely consistent with this very basic picture for where this flavor breaking comes from I could write down higher dimension operators suppressed if I write down random higher dimension operators suppressed by even a thousand TeV or ten thousand TeV I would find that all of that great success of the standard model in explaining all these more detailed things KK bar mixing CP violation decay on system B physics there's a whole lot of flavor phenomenology that's been studied experimentally in the last 30 35 years all of that is wonderfully consistent with this picture with no new sources of flavor violation no new things that break this u3 to the test symmetry until you get the scales of let's say 100 or TV so in fact this effective Lagrangian is not only theoretically self-consistent will let you go all the way up to the Planck scale it also explains these remarkable symmetries that we no longer have to think of being fundamental or engineered but are simply a consequence of the picture that this is the world up to energies way way above anywhere we either are or about to go so what's wrong why isn't that it why aren't we done what's wrong is very closely related to why we've never seen this effective Lagrangian in nature before I said that we haven't seen this sort of part of the for effect of old Ranjan in nature before but of course if you've read anything about phase transitions in condensed matter systems this effect of Lagrangian describes second-order phase transitions almost all the time okay there's a good reason why Ken Wilson one is Nobel Prize so what am I talking about well that sir that effect of the garage and describes phase transitions your second-order phase transitions but if you you know pick up a random hunk of material it isn't poise of the edge of a second order phase transition okay you have to do something to it right you hand it to an experimental as preferably so you don't do anything they go and they're grungy basements they put on their goggles whatever they they do I hope this is being recorded right oh sorry I love all you experiments are awesome totally awesome of course particle smart metals don't put on goggles so I'll reserve my respects for non particle expert models that does know I'm kidding I'm kidding anyway so and they do something they want to bring a closer second order phase transition so they cool it or they heat it or they do whatever the heck they have to do to tune the temperature to put it really close to that second order phase transition they have to finally adjust something and they do it because they care or somebody cares right but it just doesn't sit there randomly being poised right close to the edge of a second order phase transition why do they have to do that it's because naturally the value of these M squared parameters wants to be somewhere near where the ultraviolet cutoff of the theory is and not down at some random energy low-energy scale that we might care about the usual way of describing this so that it hits you in the face is that people like to compute these radiative Corrections so you compute in the standard model the biggest effect comes from a loop of the top quark and we can we can compute if I just compute that one loop diagram I get a correction to the mass of the top cork so I encourage you to do this little exercise in more detail so the three is because there's three colors actually there's just anyway and it's quadratically divergent okay and numerically even though this looks like a loopback do this whole thing is really not all that small it's around 0.3 lambda u V squared as a number so there is a correction to the higgs mass that gets bigger and bigger and bigger as I make the ultraviolet cutoff higher and higher and higher now you might complain and say that if you computed this in dim reg these power divergences are absent and that's true uh in fact power differences are completely meaningless things right the fact that there's a quantity of power diversion is telling you that its value of SuperDuper sensitive on what you think is going on at ultra short distances you can't compute it not only you can't compute it can't even computer design you can't compute anything about it right it's totally dominated by what's going on in short distances if you have a power divergence what you should do is you completely reabsorb the effect of all of that stuff into a redefinition of what you mean by the by the bare parameters but there it's completely totally dominated at short distances that's the beauty of dim reg then drag allows you to do that say yeah I did all of that and now the only things that I care about are logarithmic divergences which are sensible things logarithmic divergences get equal contributions from all scales high below and there are something that you can reliably talk about in a long distance affection theory as we've just seen one example of earlier ok but what this is indicative of is that you can invent any physics you want in high energies and imagine this huge couple suet in any old way now computer in dim red computer anyway the heck that you want there will be a contribution to the mass that goes like 0.3 times the mass of those new heavy states squared ok do the computation anyway you want in a more complete picture of what the theory is and heavy particles imagine things are coupled up together in some reasonable way you'll always get some big correction to the mass of this particle so you can I just want to give you a variety of elementary ways of approaching it you can just explicitly evaluate timing diagrams you can compute the woman Weinberg potential so uh so there's this very there's a very pretty way of thinking at least at one loop about what the effective potential is for any scalar which is another formula you should carry around in your head because it's very useful if you don't want to if for some quick and dirty reason you don't want to compute fine line diagrams and the idea is to and and the point is that we can think of the energy density coming from the 1/2 H bar Omega something that happens for Omega energies for all the harmonic oscillators that are out there in the world put the world in a big box every mode of the harmonic oscillator each one of those harmonic oscillators is 1/2 H bar Omega 0 0 point energy add up all those zero point energies you get something which is the sum over all the modes let's say I have a bunch of bosons of the square root of sum over all the caves the square root of K squared plus the mass squared of that particle and if I convert the sum to an integral it's the volume of the box times the integral D cubed K square root K squared plus M squared with a 1/2 so I get an energy density so that's so this is e but I get an energy density V which is the integral D cubed K root K squared plus M squared for bosons with a half I'm going to minus 1/2 root K squared we'll temper me on squared with the fermions with the fermions of the opposite sign for this for the vacuum energy and now imagine I turn on an expectation value for the Higgs or for any other scalar field then in the spectrum of the theory a bunch of particles are going to become massive in response to the fact that I've turned that on and so all of these things depend on H now we can just do this integral and you see there's a dominant piece which goes like which goes like DK K cubed that's the overall vacuum energy that's the cosmological constant that's a big problem but you won't talk about it right now and there's a lambda U V to the fourth there's a sub leading term that goes like lambda U V squared I forget if it's 64 or 32 or whatever PI squared but then it's the super trace of M squared in the Higgs background the bosons - fermions and it's and a finally a logarithmic piece which goes like a super trace from the next order piece which is M Squared squared and the Higgs background log M Squared to the lambda squared okay so this is another way of thinking about it oh and there it is once again there's this all quadratically divergent contributions to the potential which is proportional to the super trace of whatever the spectrum of the theory looks like in the background of the heat so if I turn on an expectation value for the Higgs I get a mass for the top quart for example which is lambda top times the expectation value of the Higgs I get masses for the W's and Z's let's go like G times expectation value of the Higgs and so on but as you can see if I just put in the particles of the standard model there's no special cancellation here and in fact the the dominant piece is the one that we talked about coming from that guy so I'll just take five more minutes and then just just just to finish this part of the story so so this is the this is this is the problem there is no reason we just said that this theory seems to make sense all the way up to the Planck scale that's fine it can make sense as a consistent theory all the way up to the point scale but a crucial feature of this of the theory is that there's a there's a scale for the expectation of the Higgs around 200-250 GED which is very very different than any other short distance scale forget gravity it's different than any of the other short distance scales that we talked about which was one of the good features of this picture right allow the separation we didn't have to talk about new things for a long time to explain all these accidental symmetries everything was great but that very thing also seems to make it very very hard to understand why this Higgs of sitting around and is so life a more invariant way of saying this and also also of distinguishing it why this doesn't happen to us for spin 1/2 and spin 1 is the following for massless spin 1/2 and spin 1 particles there is really a discontinuous difference between masses and masses and just counting degrees of freedom if you have a mass to be more precise if you have a massless spin 1/2 particle which carries some charge then it only has 1 whole isset e right you have a you have something a part of the velocity and it's antiparticle carrying its opposite charge would have negative velocity but as a single whole city there's one bio formula that's associated the fact that the massless particle is a chiral symmetry acting on it but even before talking about specific things about symmetries just the degrees of freedom of that there's one degree of freedom for the particle and one for the antiparticle right whereas the massive spin 1/2 particle with the same symmetries doubles the numbers of degrees of freedom so if I start with something that's massless I can't just all of a sudden make it massive because the degrees of freedom have to change discontinuously the normal way we say that is that the theory has an extra symmetry in the limit as it becomes massless okay and so that symmetry makes it impossible to relatively generate a correction that's also a true statement but it's closely related to this degree of freedom issue that I mentioned similarly for spin one why don't I generate a quadratically divergent correction to the mass of the photon anybody what is the reason this is a trick question the reason is not gauge invariant because because there's no such thing as gauge invariant gauge invariance with this thing we introduced by hand to talk about the massless spin-1 particle but it's because there's a discontinuous difference between Vassalo spin 1 and massive spin 1 you go from two to three degrees of freedom if I have a theory with the two degrees of freedom just sitting there for the massless spin-1 particle I can't all of a sudden by computing rative Corrections get three degrees of freedom the gauge redundancy is something that we put in by hand to help us study that physics but it's not actually an invariant property of what it is the reason there isn't the correction is because of this degree of freedom issue but that degree of freedom issue was simply not there for the scalar okay the degrees of freedom in the scale are the same whether its masses or massive that's why we have this problem that's why in nature when this thing happens to us someone's got to finally adjust something there's no reason why the parameters of a hunk of metal sitting there are such that it naturally puts it right at the edge of a phase transition you have to do something to it similarly if someone just handed me this theory and I put it on I don't know I put on a lattice I computer I simulated it if I so desired I could adjust the parameters to make the expectation value of the Higgs as small as I want but I'd have to adjust the parameters very very finely I'd have to adjust them to cancel out that huge contribution from quantum Corrections and so in order to get it down to a much lower scale be I have to fine-tune parameters to an accuracy v divided by that scale squared so if I want to imagine this theory makes sense all the way up to the Planck scale if I were putting this theory on my home computer I would find that I have to choose the parameters in that theory the input bear Lagrangian I have to choose the parameters incredibly incredibly incredibly carefully and adjust them to thirty two decimal places until I managed to get something that reproduces what desired at long distances once I do that everything is fine there's nothing wrong theory makes sense there's nodes it's not like you could all the time there are these big Corrections you can't compute anything not at all perfectly sensible one fine tuning but it looks very funny the only way I could accomplish it on my computer is by tuning the parameters the only way we've seen it in nature is when some grubby experimentalists has got the finger on the dials and has tuned to the parameters and yet this is the shock this is why I say it's shocking and yet all evidence is that that's the theory that describes our world in the sector that it breaks electric symmetry okay so that's what we'll um uh return to next time and I thought it was going to go 50% faster but um but oh let me just make one final statement this will be very very quick so ah so this means that if we want to avoid having this tuning just I didn't get to the big punchline we have to have new physics around the corner okay we have to have new physics around the corner because this lambda UV shouldn't be very far away from the TV scale it was very far away from the TV scale never mind the Planck scale those 100 TV up softly fine-tuning to a part in 100,000 or partner million okay if there is actually natural something new has got to happen right away right above her head right at the weak scale if I don't want to imagine there's any of this crazy fine adjustment of the parameters going on now you might worry that this is a kind of a fight it's a I mean is this argument really right if we ever see anything like this before in fact the 20th century saw three examples of exactly this phenomenon three example that the phenomenon of something that appeared finely tuned this kind of argument could be run this kind of argument would predict the scale where new physics has to happen to new physics happen exactly where it should have okay I won't go through all three examples but I'll go through the one that looks essentially identical to ours which is the story of the charged ion so if we take the PI on and we imagine the quark masses are zero so the PI on should have been exact Goldstone bosons they should have been massless well they're not quite massless because there's a quadratically divergent correction to the mass of the PI plus coming from a photon loop exactly the same story as you would draw the Higgs in a W loop in the standard model there's a quadratically divergent correction to the mass of the pylon come from a photon loop okay so they're two you could have said what could be going on if we didn't know anything else sitting from low energies you say something new has got to happen at a low enough energy that I can understand what the mass of the pion is natural and you would make a guess for what that mass scale is it's around two GeV if you just put in the numbers and ask that the quadratically divergent correction isn't too big the same argument we make here you would predict that something new has got to happen in 2gb something new did happen so all the rest of QCD all the rest of the resonances everything else does come in in fact it comes in even before it needs to come in to solve this problem what solves the problem in QCD is the Roma's on exchange is what's canceled if you like the contribution from the photon and the whole theory becomes of course at high energies we have quarks that go on that's not even that affected theory anymore but the Roma's on comes in at 770 MeV even before this TGV that it had to say that this is a generic to generic fact that in the natural theories one they work the new physics comes in when there's a weak coupling factor the new physics comes in that we coupling factor before it absolutely has to to solve the problem in the case of QCD doesn't seem like there is one but it's n so that difference between 770 MeV and 2 GeV is actually a square root of n factor ok there are many other examples which where there's two other examples where exactly this issue arose this logic is correct it makes a correct prediction that something you've got to happen the new thing happens ok I'll leave it to you to think about what the other two are ah but so this is a very very very important argument that tells us that we have to see new physics at ve D scale there's got to be new particles around the corner and not random new particles those new particles have to have the property that they get rid of that they explain why this problem isn't there and the final comment is if there are new particles lying around then let me I want to make the following quick comment it won't be a focus of the lectures but we know that bearing on the lepton number of very good symmetries in the standard model if we're going to say that there is new particles right around the corner of the TVV now unlike the case of the standard model we have to worry about whether bearing lepton number or good symmetries because now we could write down higher dimension operators that would violate these symmetries so there's got to be some extra structure that guarantees that barren and lepton number are pretty good symmetries ok but if there isn't the theory is dead anyway so we should we might as well talk about theories your Baran lepton number continued to be pretty good symmetries at the TV scale perhaps exact symmetries at the TV scale now in the standard model every particle the carries baryon number or lepton number is a fermions and that means that this guy negative 1 to the barrel number negative 1 to the lepton number negative 1 to the fermion number is a symmetry all of those asymmetry this object is the symmetry under which every particle in the standard level is neutral okay that's just a simple consequence of the fact that everything in the standard model the carries barren and lepton number is the Fermi so if in this zoo of new particles we have at the TV scale to solve the hierarchy problem if there exists any particle which so this should also be a good quantum number of that theory right because Barrowman lepton number had better be good quantum numbers if there's any particle which carries a non-zero charge under the symmetry there's any seven particles that have nonzero values of these charges the lightest of those particles is necessarily stable okay does the camp decay to anything everything just in the standard model is neutral under it this is a very general argument it's true for huge class of extensions of the standard model the fact that barring a lepton number symmetries are good and the fact that that object is that all the channel model particles are neutral under that gives a very good opportunity for there to exist essentially stable new particles in the new physics that comes in all that has to happen is that one of the zoo of particles has got to carry a charge under this under the symmetry okay so that means that we should generically expecting any extension of the standard model that addresses that addresses a hierarchy problem at the TV scale there should be some new TV ish particles that are stable now you might say Baron number might be broken a little bit okay that's true it might be broken a little bit we can even imagine that it is broken by dimension six operators we can even imagine that whatever breaks it also breaks well breaks this symmetry to the symmetry that would protect the particle that we're talking about but even if it's a dimension six operator ah the scale of that operators such that well you know that that the lifetime of the proton has got to be bigger than 10 to the 34 years you know that that scales like the mass of the proton to the fifth divided by whatever the scale of that dimension six operator is to the fourth just by dimensional analysis and if it's the same sorts of interactions which are even breaking very odd number uh if those same sorts of things make this new particle decay just by dimensional analysis you can figure out that the lifetime of that particle would be roughly the ratio of its mass of the proton mass so te d2m proton a thousand roughly to the tip so it would be ten to the fifteen times shorter lived in the proton which is ten to the minus 15 times 10 to the 34 years which is 10 to the 19 years which is a billion times longer than the age of the universe okay so I'm saying anything we do at the week scale that guarantees that the proton is long-lived enough even maybe in these symmetries aren't exact maybe they are we'll see sometimes they can be but even if they're not whatever we do as long as we insure the proton is long-lived enough if there's anything charged under this thing there's definitely some particle out there that's cosmologically long-lived okay at the weak scale and something men you have you have heard is that if that particle is neutral then it's an excellent candidate for the Dark Matter delivers okay that's a really remarkable little computation that maybe someone can persuade me to do later you can also do it yourself but it's a really remarkable fact totally different argument from a completely different part of physics that also argues for the existence of new particles sitting around there at the weak scale with these kinds of with these with normal perturbative interactions with everything else but I wanted to stress this origin of it because this is one of the immediate consequences of saying there is something new there's a lot that we have to worry about right away because we don't have this beautiful standard model explanation for why all these approximate symmetries are good but there's one opportunity to that there's a good reason why so many of these theories have Dark Matter candidates because this should be a very good symmetry near the TEG scale so those are general things that you should expect in a theory that goes beyond the standard model that addresses the hierarchy problem and starting next time we will talk about why supersymmetry is such a wonderful way to go to do this Thanks

Channel: Institute for Advanced Study

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Id: ndKx5Jd3OPg

Length: 92min 40sec (5560 seconds)

Published: Mon Jul 10 2017