 Okay, good morning. So I'll Pick up where I left off, but let me just make some remarks. I Realize that yesterday's lecture covered a lot of ground and it was probably very difficult to follow everything I Understand that I don't I don't really want to apologize for that the the point of point is that I absolutely Cannot pour knowledge into your head. I wish that I could I'm not able to do that my aim was to to to To cover this material in the sense that you can at least get a feeling for what is involved to really Understand the details and for those of you who are interested I have actually given you enough to work out the details for yourself It will take longer than the lecture took obviously And another remark I want to make is that today we're going to finish up this more formal material and then we're going to start with a more phenomenological subject and You can absolutely understand the rest of the lectures without understanding all the details there So you can sort of put it in a in a black box as all discussed. Okay, so those are just some preliminary remarks I Do want to Make one point. I need some chalk I'm going to back up just a few steps from where I left off to make a very important point that I went much too fast over even by the Standards standards of the rest of the lecture. So let's come back to the gauge covariant kinetic term For chiral multiplets. So without gauge fields. We just have Phi dagger Phi with gauge fields We have this factor here and here's the component expression. It involves the scalar fields Phi their fermion partners Psi this is the okay. This is the normal gauge covariant derivative. There's a D term Those are all things that I mentioned last time. I completely Forgot to mention and I left it off the slides. I've now put this back this very very important term here This is a this is an interaction between the scalar from the chiral multiplet the fermion from the chiral multiplet and the Fermion from the gauge multiplet the gauge, you know, so in Feynman diagrams We know that when we have a fermion coupled to a gauge field we have a vertex like this Right, we have a fermion fermion gauge field So in the in supersymmetry these come from the chiral multiplet this guy comes from the gauge multiplet And so what that vertex looks like is like this Okay, here's the here's the scalar from them from the Chiral multiplet the fermion from the chiral multiplet and we write it's often useful to write the gauge, you know In this way that it's a line with a wiggly thing through it Okay, and so you can see that the and these diagrams both This guy looks like g and this guy looks like g. Okay, there's a factor of square root to here, right? Which is an actual factor. Okay, but you can see that what these these two vertices are related by supersymmetry and obviously this is a very important Effect, okay All right Any questions on this? Okay, so then the next thing is to write gauge invariant kinetic terms for the gauge multiplet itself, okay, and We have this this combination right here. We talked about this last time. This is a chiral super field This is the thing whose lowest component is the gaugino field lambda. Okay, and We can work out write a gauge invariant kinetic term by taking a d2 theta integral of this product of w since these w's are chiral Okay, and we end up with a gauge invariant Super symmetry and gauge invariant kinetic term for the gauge Multiplet that looks like this. Okay, so it has the standard kinetic term for amu the standard kinetic term for lambda and This the square of the auxiliary field term. Yes This is the case for a u1 gauge field and so the gaugino is neutral So just like the gauge goes on is neutral But when we have a non-Abelian case that will absolutely be a covariant derivative there okay, so in the Now this Auxiliary field D looks a lot like the auxiliary field F for the chiral multiplet It appears quadratically and without derivatives. And so once again, we can Integrate it out right so to understand the structure of that Let's generalize things a little bit and consider the case where we have n chiral super fields, okay, and let's do this rescaling v to gv so that g's are in the normal Standard place that we usually use for phenomenology Okay, and then what we then what we have is that the Lagrangian just is Generalized by having this sort of sum here and each of the fields can have its own charge q, okay and then the quadratic terms they look like there's a d multiplying the sum of Quadratic terms for each scalar plus the one-half d squared from the kinetic term Okay, and if we integrate out d we can do that by completing the square we see that we get an extra term To the potential which is again a perfect square right because that's the way it works And what's inside the square is the sum of phi dagger phi weighted by the charges okay, so if we have charges plus minus one say in a Susie version of QED this would be the difference of the phi dagger phi's for the two multiplets Okay, okay, so and as with the case of F This completing the square and doing the path integral is the same as imposing the equation of motion So we can write oops. That should really be a D Okay, the D term potential, okay is one-half d squared where D is what you get by imposing the equations of motion Right, and you can see again that That that that D the auxiliary field is a good order parameter for breaking supersymmetry supersymmetry is unbroken if and only well if D is if if Supersymmetry is broken if D is non-zero right unbroken supersymmetry requires D to be zero Okay questions Okay All right, so then there's some exercises here if you want to work out Do you want to see if you understand these formulas you can try it out on some some some Some things here. I'll post this on the on the slides. Okay these slides will be posted Okay But now now to generalize this to a non-Abelian gauge theory. Okay, it's In it's the same exact idea for the Abelian case except that now we have one We have the we have the the the chiral super fields Transforming under some non-Abelian transformation. So capital T. Capital A are the generators say of SUN Okay, and oh maybe there's one gauge parameter, which is now a chiral super field for each generator Okay, so this is the usual generalization of a non-Abelian UN trans SUN transformation and as we as before the kinetic term is not invariant under this right because we can't Phi dagger and omega dagger and omega are different objects and we use exactly the same kind of trick to make it covariant We introduce one gauge Super field for each generator as we would expect. Okay, and these VA's are are are Real and then this object here is gauge invariant, but now we can't You know that you the these are now matrices omega and V are all matrices So we can't sort of write this exponential as the exponential of the sum of these things anymore, right? But otherwise, it's just the same thing as before. Okay, if we were to expand for small omega We would see that V at first order shifts just like it did before but then there are these higher order terms that come from the non-Abelian nature and we can define components with simple gauge transformation Transformations if we observe that this object right here, okay If you're a mathematically inclined person, this is sort of a spinner version of the moro carton form But if not, it's just this what it is and this thing has a simple transformation property Okay, it looks like it has a piece that transforms like e to the omega e to the minus omega on both sides and this derivative right here Okay, so this should remind you of a formula from non-Abelian gauge theory Which is that an ordinary non-Abelian gauge field Transforms in the same way except that this except its index is a space-time index and this derivative is a Space-time derivative, so this is kind of like a spinner version of a gauge field Okay, that's exactly what it is and so that you can sort of follow the same steps I'm not going to give all the details because your mind is probably numb by this point with all these formulas But you you can you can now sort of follow your nose because you know What to do with this and you sort of do the same thing with this except with spinner derivatives, okay, so and so in particular You wanted to find the component fields in terms of covariant derivatives of this combination to get things that have simple gauge transformation properties, okay, so here are the component fields okay, and these have I Actually didn't write it down. Maybe I should have but This thing here transforms like an ordinary gauge field and these two things transform under gauge transformations like As an adjoint, okay, I'll add that to the slides, but okay, you're like I said your Remind is numb. Hope I'm hoping to wake you up in about a few minutes, okay The other components vanish in West Semino gauge Let's not even write them down and now if you write down the gauge invariant kinetic term Wow, I left off a bunch of stuff here too, okay So the gauge invariant kinetic term here for this We again have the the the the covariant versions of the the kinetic term for the skaters. We have this Yeah, no, it's fine. It's fine, right we have this term right here which involves the The Gay-Geno field coupling this is what I wrote down on the board And then we have the auxiliary field da and the da there's one for every gauge generator And it's Phi dagger ta Phi that multiplies that Okay. Yeah, this is correct okay and We again use the same kinds of ideas to the right the gauge kinetic term We have this object where again we're using our favorite dude here the moral carton form and this thing transforms in the adjoint representation with chiral super fields and so we can write again a The kinetic term here, okay now the theta term really does do something Okay, we're not going to use it in these lectures But it's actually very important plays an important role even in the renormal in the non-renormalization theorems Okay, but we won't have time to talk about that and so if you look at the the complete Lagrangian for a non-Abelian gauge theory with some matter field Phi Chiral superfield Phi we have a gauge invariant kinetic term. We have a theta term. We have a Kinetic term for the gay genomes with a covariant derivative And this is the usual adjoint covariant derivative and we have the d squared terms and the kinetic term here Okay, so this is the normal adjoint kinetic term for gauge invariant covariant derivative for for lambda okay Integrating out the chiral superfields we get again in the abelian case We had a the thing that was squared We get a perfect square the thing that was squared in the abelian case was Phi dagger Phi is weighted by their charges In this case, it's Phi dagger Phi weighted by what else the generators, right? Which are the charges of a non-Abelian gauge group? Okay, and I gave an exercise for for the devout Okay Questions you can't wait for it to end okay, so That was sort of the end of the this more formal part of the lectures. Okay, I hope that it was Useful or at least a bit inspiring to some of you that you really can you can do this for yourself, but Frankly when you're actually you're like me I'm actually more a phenomenology guy when you're doing phenomenology this formalism is very useful to keep track of Symmetries in a manifest way. It's very very important as you know that we understand What the most general thing you can write that's invariant under symmetries is that's extremely important to do that I don't spend my time calculating, you know anti-commutators of Susie derivatives, right? You just use the basic results. Okay, so you can think of this somewhat as being a black box You could just say look chiral superfield is some object that I call Phi that is Includes as its components a scalar field a vial fermion field and a complex scalar a vial fermion and another complex auxiliary field The a gauge super field. Let's take the u1 case contains a gauge enol vial fermion and a real gauge field and a real auxiliary field Okay, and then there are magic formulas, right? You're just there's some operation called d2 theta that if you take w it gives you this in Components and is an operation called d4 theta that if you do this it gives you that Etc. Okay, so if you just take these are the basic formulas that you need to understand everything in the rest of the lecture Okay, so you can treat it a bit like magic even though it's true. Okay All right So Super symmetry is broken in nature. It's a beautiful symmetry Right just there. There are people who spend their whole lives in Unbroken super symmetry their whole lives as physicists, but we're not going to do that because we're interested in making contact with the real world Okay, now. This is again a vast subject and I cannot do it justice, but I will try to give you a few highlights and And hopefully get a bit of an overview of this subject. So obviously we don't see soup We don't live in a supersymmetric world Now, let me just make some very general remarks about Super symmetry breaking The idea of super symmetry is that at some level. It's an exact symmetry of nature, right? Well, let's say this Super symmetry is is a is a space-time symmetry as I kept saying over and over again So if I want to include gravity, I need to gauge it, right ordinary Einstein gravity involves gauging the Lorentz group the space-time symmetries and supersymmetry forces you then to gauge the full supersymmetry algebra actually, okay, and That means that it doesn't just like in an ordinary gauge theory We don't have the option to do explicit breaking. We have to break Gauge symmetry spontaneously if you want to have a consistent description, okay? Just a parenthetical remark if you don't understand this then wipe it from your memory It's not actually quite true what I've said It's actually the case that explicit breaking is the same as spontaneous breaking for gauge theories But if you didn't understand that forget about it, okay? Anyway, we want to break things spontaneously and that means that what we're looking for is a super symmetry in variant Lagrangian whose ground state breaks supersymmetry. That's what we must have okay, and So we don't have the option just to say well supersymmetry. It's like it's like the approximate Isospin invariance in the strong interaction It's it's it's an approximate symmetry, but it doesn't have any it never becomes an exact symmetry in any sense Doesn't have to anyway supersymmetry has to become an exact symmetry if it plays a role in nature, so We need to look at spontaneous breaking we need to find a Susie invariant Lagrangian whose ground state breaks supersymmetry and We know that that happens if and only if the vacuum energy is positive, right? we know that from general principles and At the classical level the potential is just a sum of squares Okay There's a there's a square of an F term for every chiral super field and there's a square of a D term for every gauge super field Right, that's the classical potential and each of these terms is positive And so we need we see that supersymmetry is unbroken if and only if all these F's and D's are zero and Conversely if we want supersymmetry to broken be broken at least one of them must be non zero So this tells us what to look for we're looking for theories that have vebs for some of these auxiliary fields okay It's really not very hard to write down a theory like this, okay So I'm only going to talk about the very simplest example. This is the Poloni model and The model is excruciatingly simple. In fact, this is a bit of a lie as we'll see but let's consider something Whoops, this is a d4 theta. Okay d4 theta the ordinary single chiral super field ordinary kinetic term and the linear Superpotential term Okay, only a linear term. Well, what does this actually do? Well, if I look at the F term It's the derivative of omega which is just the coefficient of the linear term. So it's non zero So according to my my my thing there supersymmetry is broken Okay Now that seems a little bit cheap because how can I get that from just a quadratic plus linear Lagrangian? How can that do anything and the answer is it can't really? Because if I just work out what this thing does in component fields, it's just a free theory Okay, it's just that somehow I've added a non-zero vacuum energy to it Okay, so this only breaks supersymmetry in a very formal way It is true that the generators Q are broken, but it's not there's no interesting Susie breaking dynamics There's no interesting dynamics at all. The VEV is undetermined. So you might think this is just silly But it's not actually silly because a minor modification of this actually gives me a theory that genuinely breaks supersymmetry so Give it some non-trivial interactions, but put these interactions into the Kaler potential into the d4 theta term so up till now we've been talking about Renormalizable theories and that's perfectly fine But we know that when we include gravity the low energy theory certainly contains non-renormalizable Interactions if we have any sort of heavy fields and integrate them out we expect effects like this Okay, in fact this Poloni model is universally the low energy effective theory that arises from the or a ferdy Mechanisms so the rafferty mechanism. I'm not going to talk about but it's the thing that most reviews of supersymmetry talk about and This exact thing happens you generate a term like this from integrating out heavy things In any case Let's just add this term Okay, and so the same machinery that we use to work out the component fields of this can be used for that and What we find is that instead of just having f dagger f From this term right here. We have an f dagger f Minus we have this term right here. Okay, and here. I've written only the terms There are lots of other terms from this involving fermions all kinds of other stuff But here I only care about the scalar potential so I'm only writing the terms that involve no derivatives and Involve the scalar fields, okay Now we can eliminate f using its equation of motion and now we find a non-trivial potential Okay, and this potential has a minimum at phi equals zero Okay, so it has a non-trivial minimum you can work out the masses and so on right Okay, so if you work out the masses just taking the second derivative of the potential you find that the scalar mass is non-zero That makes sense the potential has fixed it so it has some second derivative and I haven't shown you about the fermions But the fermion does not get a mass from this Okay, from this it's actually not hard to see you can just easily check that there is no Psi psi term anywhere in that in that in that pit thing. Okay All right, so there's a model that that breaks supersymmetry. No no problem Okay, now the fact that we found that this theory has a massless fermion Is extremely general okay because we have spontaneously broken a global symmetry here namely supersymmetry Right and whenever we have an ordinary global symmetry a u1 or SU n or something that breaks We know that we have one Nambu Goldstone boson for every broken generator And if you have a massless Nambu Goldstone boson for every broken generator, right? And so in the case of supersymmetry what we're breaking is some fermionic symmetries q alpha where they're global symmetries We're breaking them spontaneously and so we should expect a massless fermion Goldstein oh, it's called right. We expect a Goldstein oh To come arise from this okay, so it's at least it shouldn't be surprising Okay, and in fact you can prove this in great generality just like there's a sort of an operator level proof for the Goldstone theorem in In an ordinary for ordinary symmetries. There's a similar proof for For for supersymmetry as well. So this is exactly as general as the ordinary Existence of ordinary Goldstone bosons, so we're not I'm not going to show that but let me just show it to you in the very Simplest case which is that well, let's consider a general theory of an arbitrary Super potential only chiral fields and an arbitrary super potential then the potential looks like this, right? It's the square of f which is this okay summed over these guys And so if we minimize the potential Minimizing the potential if we treat phi and phi dagger as independent fields we have to satisfy this equation right here Okay, but if you look at this you see that supersymmetry breaking in this theory means that the f term has to be non-zero So this is non-zero Right, okay, this is non-zero But that means that this piece right here has a zero eigenvalue a non-trivial zero eigenvector Sorry, right. That's exactly what this equation says, but this thing right here is exactly the fermion mass matrix Okay, voila There's a there's a similar proof only slightly more complicated if you take include D terms Okay, so there's always going to be a gold steen oh now again by our analogy with gauge theories If we have an ordinary if we're sorry ordinary symmetries ordinary Bosonic symmetries if we gauge the symmetry if we gauge a symmetry that spontaneously broken We know what happens right what happens is that if the generator is gauged and instead of having a massless Nambu Goldstone boson the name there is a Nambu Goldstone boson becomes the longitudinal component of the gauge Field a massive gauge field via the Higgs mechanism Right, and so we might very well expect that when we have supergravity Okay, supergravity contains a graviton a spin-2 particle and the gravitino a spin three halves particle that when we since that is gauging super Symmetry if we spontaneously break super symmetry we might expect that the the the the the golds the gold steen oh Becomes the longitudinal component of the gravitino and gives us a massive spin three halves field Okay, and that is indeed what happens. Okay, so what happens in the context of? Of supergravity is that we get a massive spin three halves field Okay, and the its mass is proportional to the gravitational coupling times the VEV the gravitational coupling is 1 over m plank And the breaking term is is f the breaking parameters f which has dimension 2 Okay, so this has dimensions of a mass, okay, and this is indeed what happens Okay, so this is again a vast subject, which I'm only alluding to here But the point here is to be aware of the fact that of course m plank is the biggest scale that we think we can talk about with a straight face in in in in particle physics right and so f can be at most if we imagine that Susie's broken at the plank scale f would Be m plank squared now Susie's already broken at the highest scale possible in nature. It's probably not Useful symmetry so we're always interested in the case where f is small compared to m plank That means we always have a light gravitino We're often interested in the case where f is much smaller than m plank for example You might think that the most natural value for f would be Tev squared in which case this would be a very very small number So these very light gravitinos can have they're very light They're certainly light enough that we need should perhaps worry about them and they do in fact have interesting consequences for phenomenology Clire phenomenology and Cosmology, okay, and I will allude to that to that a little bit later Okay, but obviously I'm not going to go into the formalism here Any questions on this? Okay now as we talked about at the beginning of the first lecture and as Andrea waltzer also reviewed very nicely the Motivation for extending the standard model the main motivation for extending it is the the hierarchy problem or the only motivation that points to new physics specifically at the Tev scale and If supersymmetry is going to solve the hierarchy problem Therefore the super partners of the observed particles must have masses of order the Tev scale Okay with a tiny with a little caveat that we'll talk about later Okay So you might think It's very natural to think from what you've heard so far that the most natural Possibility is that there's some sort of a Higgs sector Maybe not maybe something like that Poloni model that I talked about maybe something fancier Maybe something we normalizable that's easy to write theories like that and that Higgs sector some fields gets vev's Just like the Higgs gets a vev and they get vev's of order a Tev or hundreds of GEV and that breaks supersymmetry And so just like we discovered the Higgs in 2012 the Higgs that breaks an electroweak symmetry We can hope to discover this super Higgs sector that breaks supersymmetry at the Tev scale Okay, that would be fantastic Unfortunately, it's not possible and the reason it's not possible is because if you search through your black box of formulas for the for the the couplings of In supersymmetric theories, you will never find a scalar Gay-Geno-Gay-Geno coupling Okay, there's no coupling like that Okay, and that's what you need to give the Gay-Geno a mass from a theory like this Right, you would need to write find some coupling of the Higgs to the Gay-Geno because for example we certainly have not seen the Glue Eno the colored super partner of the glue on at the fermionic that would be an octet fermion We certainly haven't seen a light version of that Okay, and so there isn't any way to gen to generate a mass for the Gay-Geno at tree level So the Gay-Geno mass is just way too small in a scenario like that. So this just doesn't work Okay, there are a number of possibilities that you can use that do work. There are plenty of things that do work There's so many of them I Barely going to scratch the surface again Okay Some of the possibilities are that we say okay, you know if tree level doesn't work We need to have effects that are loop level effects, but they need to be very big So basically we need to have strong Super symmetry breaking at the TEV scale that would certainly be very exciting It's very hard to make models like this although some have tried including me Another thing you can do that works is to have super partner effects from super partners coming from loop effects So the loops make them smaller, but all you need to do is make the super symmetry breaking scale a bit higher So this for example the most famous version of this is is is gauge mediation, but there's also anomaly mediation Yeah Yes Yeah, something like a technicolor sector that instead of breaking Electro-weak symmetry it breaks super symmetry Okay, you know It sounds sounds reasonable. I mean it's and it may be reasonable. It may actually be reasonable Okay So the But there are very there aren't any concrete models because we don't know very much about these kinds of theories Okay, actually we know enough about these kinds of theories to know it's not so easy to make these first models Okay, so however we can make successful models of the second kind For example gauge mediation or anomaly mediation or two two versions of this Okay, or we just have perturbative loop effects that generate the observed super partner masses from an initially higher scale of Super symmetry breaking and then there's hidden sector Susie breaking which is what I'm going to Talk about a little bit more detail Okay So what is hidden sector super symmetry breaking? Here the idea is that the Lagrangian of the world sort of Approximately should have written this down but the that we have some hidden fields x and then we have the standard model fields and then the Lagrangian of the world splits up into a A piece that involves just the standard model fields and a piece that involves just the hidden sector plus some higher-dimension operators Okay, so that's that's what I mean by a hidden sector Okay, and these higher-dimension operators can be suppressed by some large scale one over M so the typical size of effects in that that that The connection is small right okay And then this hidden sector can break super symmetry For example this Poloni model or something. Let's say there's some hidden field x hidden factor chiral super field And it has an F term that is non-zero. Okay, if there are many x's in the hidden sector that have An F term I can just take a linear find the linear combination that has an F term and the other ones Have zero F term kind of like the Higgs basis and electroweak symmetry breaking Okay, so I can always shift the lowest component of x to make the VEV zero So that as far as low energy physics is concerned the effects of x are The most important effect of x are just given by its VEV, which is just an F term okay, and The F term is non-zero because of the the dynamics in the hidden sector So now let's think about what kinds of higher-dimension operators here. Can we write down? Okay Obviously we haven't talked about this sector yet. How do we supersymmetrize the standard model? We're getting to that Okay What kinds of effects can we write down? Okay, well we want to write down effects which are Basically powers of x over m Okay, and remember so so what we do is we basically wherever we had some Coupling before some some coupling in the Lagrangian we can now That gave us some supersymmetry invariant. We can now just put powers of x over m Okay, so for example the ordinary one and type of invariant we can write is the gauge Kinetic term and we had some coefficient here, but now we can just write x over m times this right? This is a d2 theta interval So we have to put in a chiral super field but x does the job perfectly and now because x has a theta squared component That is that theta squared component is picked out by this d2 theta term here and as far as the potential terms go We get a gay genome mass Okay, so right away Before when we tried to if we tried to write down some renormalizable theory the big problem is that we couldn't write a gay genome mass Immediately we see we can write a gay genome mass which is of order f over m okay, and Similarly for the scalars the kinetic term for the scalars is x has as phi dagger phi We can take its coefficient to be x dagger x over m squared and we find a contribution to the scalar mass Now we have two f's that can be eaten by these d4 theta's and so we get immediately a scalar mass so we get a scalar mass which is f over m scalar mass squared which is f over m squared and We see that this is successful in the sense that these guys are naturally of the same order of magnitude Right if we have the same scale m suppressing both of these operators then the scalar mass and the gay genome mass So the same order of magnitude Okay, there are more things that you can write down. Okay, so scalar and gay genome masses are well that You may think that's enough because that that's sort of all the super partner Mass kinds of mass terms we would want but there are there are actually more things you can write down For example We can write down for when we have super potential terms for example a cubic super potential term We could write down Some x over m here and we could get cubic scalar interactions Okay, we could also write down Terms like this where we put in just an x in the d4 theta integral. We could get terms like this It's a little bit less clear what these terms are because it involves an auxiliary field of phi So to see what this does let's integrate out the auxiliary field phi in the presence of this term And what we see is that it modifies the f term Basically the f term is the square of the coefficient of the linear term of f And so what we get is a new term like this which is again Like this over here is actually a cubic self interaction in the case of a cubic super potential okay, and again all of the Suzy breaking mass terms we get from this are of order f over m So this very very simple prescription which has just include all the higher dimension operators With x suppressed by the same scale m gives us a whole host of different kinds of Super symmetry breaking masses all of the same order of magnitude So this very generic this looks very generic and looks like it's very easy to get all the super partner masses of the same size Okay Okay, so that's just what I'm saying here Okay, that is that the message I'm trying to give you is that that it's very now Of course we could put in higher order terms suppressed by additional powers of x over m Nobody tells you to stop okay, and those just give you smaller effects Okay, so life is good okay Now what I want you to note is that if the if the Suzy breaking if the scale f For example, if the scale m here then what is the scale m well Let's suppose the scale m was m-plank because we certainly know that there are higher dimension operators suppressed by powers of m-plank from super gravity Okay, there may be even bigger effects, but we could we could imagine that situation Then if you work out what f needs to be in order to get Tev scale masses. It's about 10 to the 11 gv all squared Okay, and so it's some enormous scale So we would expect the dynamics that break supersymmetry is it inaccessibly large scales in a model like that, okay, and Therefore we could integrate out all the heavy particles all the dynamics that actually break supersymmetry could be integrated out at that very high scale But what's left what's important to us at low energy is the vev fx, right? Okay, and so for us down at you know down at the Tev scale It looks just as if supersymmetry is explicitly broken Okay, but explicitly broken by the f component of some chiral super field Okay, and this is Okay, this looks just like explicit breaking, okay, and so this actually leads to a Phenomenological approach, okay, we can say okay look is since we don't know whether we're ever going to be able to Undersee in experiments the Susie breaking dynamics the sort of Higgs sector that breaks supersymmetry It may be at inaccessibly high scales So given the fact that we haven't seen any sign of supersymmetry. Let's just take a very phenomenological approach Okay, and let's just break supersymmetry explicitly Okay, and just assume that all the super partners are at the Tev scale Okay, and we can ask from that phenomenological approach a more bottom-up approach What are the allowed breakings that we could have what are the allowed breakings that we could imagine? Okay, so if we don't want to use any particular UV model to see what we can have Well, the basic bottom-up constraint is that when we break supersymmetry Of course one way to break supersymmetry is just to write the standard model. Well, that's supersymmetric Okay, but we know that doesn't solve the problem We know that what we're trying to do is we're trying to introduce the super partners and also Introduce sub Z breaking but in such a way that we don't introduce any new Quadratic sensitivity to UV physics. That's the hierarchy problem We want to have supersymmetry and have it broken exactly in such a way that it Solves the hierarchy problem. We'd like to find out what that is. What's the most general way? Okay, and by the way these two things that I This is actually linked to what I've just talked about namely the hidden sector Susie breaking because you would certainly hope That the hidden sector breaking that I just talked about has this property All right, you would want to hope you would hope that this spontaneous breaking by the hidden sector This hidden sector dynamics, which I talked about Doesn't Doesn't if I have Susie broken at some very high scale that that still solves the hierarchy problem Okay, so all of this the technical term we say is that we need supersymmetry breaking to be soft okay Questions on this that clear all right, and so let's understand this constraint Let's understand what is the most general super explicit supersymmetry breaking which is soft okay, and let's explore this in the context of This sort of simple model here I just have a single chiral super field and a single gauge field, okay, and You can easily generalize this to more indices, but this is this is this will be this will be enough for us. Okay, and Let's understand what we can do. Okay, so notice that I have written the couplings here Okay, in particular the coefficients of the kinetic terms. I've written them as s and z. So these are super fields Okay, so I'm allowing the couplings to be super field this trick is going to be useful again Okay, as it was before we used this to prove the non-renormalization theorem Right, and the idea was that we can promote these guys to to super fields S is a chiral super field Z is a real super field and these super potential couplings are chiral super fields okay So the Susie non-renormalization theorem that we that we we proved last lecture Shows that this Lagrangian that I've written down has only logarithmic Renormalization and it only has renormalization of z and s I didn't talk about the renormalization of s It's sad because it's such a beautiful subject, but we don't have time, but trust me It's log I mean it's you don't have to trust me actually you know in components This is a gauge coupling and a gauge coupling is only logarithmically renormalized. Okay, so you don't have to know the beautiful theory, but Okay, so they're only logarithmic divergences in the Susie limit right if I don't break supersymmetry. They're only logarithmic Sensitivity to UV physics and now we want to preserve that feature when we include Susie breaking right But there's a really simple way to do that an incredibly simple way to do that Okay, which is just to take all of these super field couplings and turn on theta dependent Terms in them right so I'm changing the Lagrangian I'm explicitly breaking supersymmetry by allowing higher terms higher theta dependent terms in these super fields Okay, so you agree that that breaks supersymmetry Okay, I hope the question is Does it break supersymmetry softly and the answer is of course Why of course? Because I just the whole point of the non-renormal the the whole way I thought about the non-renormalization theorem was to say that look you can do all your Calculations treating all these things z's and kappa's and all these things you can treat them as Superfields okay as a theorist and all the results just go through Okay, so if I can treat them as super fields I can treat them as super fields with non-zero theta components Which is what I'm now doing Okay So a little more concretely if I imagine well, what kind of what kind of logarithmically divergent What kind of logarithmic divergences do I get if I write them in super space? I find for example a logarithmic Divergence in the kinetic term phi dagger phi right and it's coefficient is lambda dagger lambda Okay, and there's a z right now if z lambda and Z and little lambda are super fields with non-zero theta components You can just expand this out and you can see what divergences you get that involve the The Susie breaking terms This is very simple, but it's very profound Okay, any questions on this There's there is one subtlety in this argument was a there's a few little subtleties. I'm gonna tell you know they're They're minor things they have to do with basically classifying all possible effects. They're not that subtle but one you have to be a little one thing you have to be a little careful about is you can also have a Logrithmic divergence of this kind right here Okay, so this thing right here is D4 theta times a logarithmic divergence times phi a single power of phi Now, why didn't we include such a term in our original Lagrangian? Why didn't I write D4 theta times phi? Okay, well the answer is because if phi is a Chiral super field then D4 theta of phi is a total derivative. So it doesn't matter in the action Okay, so it's logarithmic divergence also doesn't matter unless these guys have non-zero higher components Okay, then this thing including in particular the mu dagger right here is got an anti chiral bit in it And this is not a total derivative. Okay, and so This effect is linear in phi. So it only happens if I have a Gauge singlet phi and it happens actually from this one from a one-loop diagram like this that comes from the cubic coupling from From lambda. Okay. All right, but this is but this is the basic idea the big idea is that that if since there are logarithmic divergences When you when these coupling super fields are just pure numbers I have the same logarithmic divergences in a sense when they are super fields Okay, and you can work out many properties of Suzy loops and super Suzy breaking theories using these ideas Okay, any questions? Yes. I'm sorry say again The question is am I saying that in this setup the super potential is still not renormalized the answer is absolutely Okay, absolutely when before I said that you could You could do the renormalization theory with the couplings Treated as super fields. I really meant it. I really meant it now. There really are super fields Okay, that's all I'm saying Okay Other questions Yes, the question is before I was treating these couplings as super fields Spurrions, they didn't have kinetic terms now if they come from say a hidden sector Suzy breaking they do have kinetic terms and so it doesn't that change things. Okay, I believe that's the question, right? Well, the answer is it changes things but only a tiny tiny tiny tiny tiny bit Okay, if the scale of supersymmetry breaking is very high Okay, then all of the physical excitations of the sector that breaks supersymmetry except for the Goldstein oh Okay, but that's a fermion, but all the scalars okay have Masses which are much larger which are at the scale f which say is 10 to the 11 gv or some very high scale And so I can integrate them out when I integrate them out. I write a connect an effective theory that has no Hidden sector fields at all in it. So that's this theory There it's true that there is the Goldstein oh so I lied to you just a little bit because the Goldstein oh survives Or maybe the Gravitino it can be much lighter But that means that I have to add some terms to this that just depend on this extra fermion They are not going to change the soft Suzy breaking Okay That makes sense In fact, let me just say that you can even think of the auxiliary field in this way The auxiliary field notices a term that has a mass but no kinetic term Right, I can think of it as having a very large mass and an ordinary kinetic term Right, so you can think of the auxiliary field is sort of an infinitely heavy field if you want so Well, I don't know if that helped or hurt, but anyway Okay All right, so you can ask okay everything that I get in this way is soft Okay, everything I get by taking a coupling constant and turning on some higher Components of it is soft breaking. Is there anything else that's soft? okay, and actually the answer is well Okay, here you have to be a little bit careful I I feel howie-haber looking over on my shoulder and worrying about subtleties, but essentially The answer is no, okay The point is that that you can write even the things that are not That are not gotten in this way you can analyze them by thinking about Thinking about them as higher components of some super field So for example, let's suppose notice that the cubic terms that I've been able to write so far which I've said are soft breaking are Holographic they always look like Phi cubed plus Hermitian conjugate suppose my aim in life is to write non-holomorphic cubic terms That's my aim in life. Okay, don't ask why but it is okay Then I can write that in super space by just introducing some new in a really cheap way by just writing d4 theta x Times Phi dagger Phi squared where x has got a non-zero highest component like this Okay, now I can ask is that soft and the answer is no, it's not soft Why because x now has dimension Whatever 3 right no that doesn't sound right No h has dimension one x actually has dimension minus one sorry x has dimension minus one remember theta has dimension minus a half Okay, so this x should have written this down x has dimension minus one Okay, so since it has a negative mass dimension it generally has Power if I think of it as a coupling with a negative mass dimension it has power divergences So I can get a quadratic divergence like this. Okay, which looks like that Okay, now the reason that how we would would object is because you see I only have this kind of divergence of Phi is a gauge Singlet if I don't have a gauge singlet. I don't have this and I can claim this is soft Okay, however however I didn't say it here. Okay, however There's no good UV remember there's a sort of synergy between these two kinds of Ideas one is this hidden sector Susie breaking where I found that there were actual chiral super fields or actual super fields It got vevs and that was what broke super symmetry and now I've shown that that's what gives you soft braking Right, if you turn on sort of non-standard soft braking like this There's no good UV model for for these kinds of effects Okay, so we will when we talk about soft braking will always talk about these things you can get from turning on higher components of superfields Okay Questions on this Okay, so the summary is that for our purposes Soft Susie braking is equivalent to turning on higher components of super fields Okay, these are exactly the effects that we would expect if Susie's broken in hidden sector And this is a great starting point for phenomenological treatment of super symmetry Okay Okay, so now we're going to talk about the The how to apply super symmetry to the real world to Particle physicists the minimal supersymmetric standard model is the mssm to the rest of the world I guess it's the main school of science and mathematics where maybe they just I think they actually Study the minimal supersymmetric standard model here because their symbol is a bunch of penguins sitting around waiting for something to happen Okay, okay Okay, so even this Is a vast subject the minimal supersymmetric standard model, okay, so there are many things I'm not going to be able to cover I recommend for more details. I recommend this reference. It's a standard reference. It's very nice It was originally written in 97, but he's updating it all the time So it's it's current you don't have to worry that it's out of date Okay, it unfortunately uses the non-standard metric, but what can you do? Okay okay, so Now let's try to imagine how the real world could be supersymmetric our world Okay, so we have to start with the standard model and The first thing is let's translate the standard model into our language of left-handed vial fermions. Okay, so we don't go nuts We are you know, we have to use left-handed vial fermions for everything in a supersymmetric theory So it's very simple. I had the standard model wallet card before and I had What I had for the fermions with both left and right-handed fermions But I need to convert the left hand the right-handed guys to be left-handed And so for example, whereas before I had a you right. I now have a left-handed Vial fermion you which I call you see which has gauge conjugate quantum numbers compared to what I had before Okay, so this is just a rewriting here Okay, so for example, you see is just you are dagger times an epsilon thingy Okay All right, and the Q the Q's for example are just a lowercase Q's Okay, so this is the standard model wallet card We just write down the most general interactions compatible with the gauge symmetries of these fields and that is the standard model Okay So now to embed this in a supersymmetric theory. It's quite simple, right? I every fermion in the standard model now needs to be part of a chiral super field So for every for example for the left-handed for the left-handed quark doublet I need this this is here. It is it's now the theta term in a chiral super field And so there must be a complex scalar which I denote by Q tilde That is its partner Okay, and it's traditional to name these guys by putting an S in front of the name So this is a quark. This is a squark Right and so on and you know an up quark becomes an up squark So one complex scalar for every vial fermion Okay, and the gauge fields for every gauge field I have s u3 cross s u2 cross u1 gauge fields of the standard model for each gauge field. I need to put it in its own Gauge supermultiple it and so for every gauge field for every generator for all eight of these guys for all three of these guys I need to introduce a gauge of multiple it which has in addition to the gauge field a Vial fermion so now there's one vial fermion for every gauge field that's been added okay, and These are named by putting eno at the end of the name So a gauge boson becomes a gauge eno glue on a glue eno and a w boson becomes a we know not a y no Okay, what about the Higgs doublet? Okay, so now let's talk about the Higgs the Higgs is the one scalar that we've seen Right the fact that we've seen what appears to all as far as we can tell as an elementary scalar is One of the main motivations for supersymmetry after all we should remember that in this C of all this formalism and You might briefly think for about a nanosecond you might realize you might go back and say hey wait a minute You know if I go back here I can see that actually this Higgs scalar has the same quantum number as this L Well, okay the hypercharge is opposite But that's really just a convention because I could just use H dagger right or H tilde whatever I could conjugate H and get something that has a minus hypercharge right so hey Why why not put H in the same you know make H a super partner of L? Okay, you might think about that for a nanosecond But that's a really terrible idea because then when the Higgs gets a VEV I'm going to break lepton number and let's just not go there, okay That's a really bad idea, so we really need the Higgs needs its very own It's in Higgs needs its very own Chiral super field where the Higgs doublet is the lowest component, and then we have a vial fermion the Higgs eno Okay So we have we have we have essentially doubled the number of degrees of freedom in the standard model We've added one super partner for every Observed particle just by embedding these things in these super multiplets that we've been talking about Okay, so now the idea is great now. We know what the particle content is Let's just start writing down the allowed super symmetric interactions, and let's see what we've got okay so We have kinetic terms for everybody. That's a good start, okay, but I hope I don't need to write them down We know what they are we have kinetic terms for everybody good What kinds of interact the old that automatically gives us all of the standard model gauge interactions? Those are coming for free Okay, so all the standard model gauge interactions are present just because we have a supersymmetric gauge theory So the only non-trivial thing we have to worry about is the super potential Okay, and now things get Interesting okay, if we if we just say we're just following the wall the wallet card approach Right, I just write down everything that's allowed by the gauge quantum numbers From what I've said so far. I have this these guys right here. I have a whole bunch of terms Okay, now this first term right here. This is a core a left-handed quark a Higgs doublet and a Right-handed up quark. Okay, so this has it contains Contains in it you cow a couplings right if I remember the component expression for this if I work out the components This contains an ordinary you cow a coupling so that's great Okay, that's fantastic. What are all these other terms? What are all these other terms? Well, this is a coupling of three quarks Okay, this is an operator that has the quantum numbers of a neutron Right an up quark and two down quarks. It's a color singlet because I've contracted I haven't shown it here, but I've contracted the color indices using a three index epsilon symbol Okay This is bad Because it violates Baryon number okay, one of the Really beautiful features of the standard model is that I cannot write down Anything that violates Baryon number at renormalizable ever right Andrea Voltser Reviewed this very nicely yesterday in yesterday's lecture. Okay, but here I can So how is it that I'm imposing more symmetry and yet I can you know Super symmetry and yet I can write down terms that violate these symmetries Well, because if I work out what this really is in the Lagrangian it involves say a quark quark Squark coupling it's a you cow a coupling between two quarks and a squark Okay, and so because I've introduced the scalars squarks, which I didn't have in the standard model I can write down a renormalizable theory that breaks Baryon number Okay, that's not so good right and Similarly here this term violates Baryon number and lepton number this one violates lepton number this one violates lepton number Okay, not so good. Now. There's um And also I do not have any you cow a couplings to that involve the the right-handed down quark or the right-handed electron Okay, so I have too many of these violators and not enough of these guys that I want Okay, and here again the holomorphy of the super potential is critical because in the standard model the way I get with one Higgs field you cow a couplings for all the fermions is I make use of the fact that I can use H and H dagger to write to write You cow a couplings but in the super potential because of the restrictiveness of holomorphy I cannot write things like q dagger h dagger dc That's not super symmetric Okay, so super symmetry doesn't allow me to do that now of course I could also have chosen to put the Higgs The scalar Higgs doublet into a field H tilde that has the opposite quantum numbers And then I could have written you cow a couplings involving D and e but not you cow a couplings involving you Right can't have both and I can still write all this stuff here actually sorry I couldn't write this so this one would be gone in that case because there's this this is H tilde now So this one would be gone Okay All right, but it still has the same problems. I mean it has essentially the same problems Okay, so now let's deal we have to deal with these problems one at a time Okay, so one thing is we need to have both you cow a couplings for all the fermions so what we do is we introduce two Completely different chiral super field eights up and h down so we're forced to introduce two Higgs's two Higgs super fields. Okay, they're usually called h up and h down Okay, so let's do that. Let's extend the model even further and then what we have is we have this We now have everything and sorry. I guess this is h up here Okay, this should be an h up here. Okay, so we have now we have everything We have all the you cow a couplings that we want namely here. We can also write a Quadratic term h up h down and we still have all this stuff that we don't want so Now at least we have all the stuff that we want We can worry about the stuff that we don't want in a little bit and this gives us the wallet card for the mssm in other words the each everything here is now a chiral super field, right has both Scalers and fermions and here are the gauge quantum numbers and we have two Higgs doublets that have opposite hyper charge questions on this so now We're just going to assume that all these terms that violate baryon number and lepton number are just absent Okay, we'll come back to that assumption in a little bit because it turns out that that assumption has very far-reaching implications beyond even the That you know the violation of baryon and lepton numbers. We'll see okay But we'll just make that assumption for now For reasons that I refuse to explain right now these terms are usually called our parody violating terms Okay, I absolutely refuse to explain why that is right now because I think that at this point in the logical order We should just say look we can easily those symmetry those guys could be forbidden just because for example baryon and lepton number are good Approximate symmetries of nature. That's just how it works. Okay, we don't have to introduce something as exotic as our parody just yet okay All right, so without assumption here is the the the super potential that we have and I've written out all the flavor Indices here so I and J go from one to three there the flavor indices to emphasize the fact that we have the full three-generation Flavor structure of the the standard model and we have this mu term right here I have not written the gauge indices, but they are contracted in the obvious way Q is a three for example and you see is a three bar So this one would have an upper color index and this a lower color index There are also s u2 indices that would be contracted between this guy and that guy and so on okay, but that's just like in the standard model So this looks pretty good. We have all three generations of yukawa couplings. We have a sort of a Higgs mass term Looks pretty good, right and all we've had to do is to impose some global symmetries to get rid of some of these Terms that violate lepton number and baryon number Okay, now you sometimes have people hear people say oh ha ha ha. Yeah, supersymmetry is so minimal It doubles the number of particles. Yeah, that's really minimal ha ha ha right But I think that's completely the wrong way to look at things Okay, because all these extra particles that we've added are related by in fact a space time Symmetry to the particles we already know we have With the exception of that we need one additional Higgs Doublet, so I think a fair way to say it is that we've added one Higgs doublet to the theory and We've had to impose some global symmetry or symmetries to get rid of some terms that are otherwise allowed Okay, so if you compare that to any other sort of Extension of the standard model believe me. I'm a model builder. This is fantastic. This is absolutely amazing Okay, fact it's sort of hints that maybe we're moving in the right direction. We could be going towards simplicity Unfortunately, this is not going to this this assessment is not going to survive when we look at supersymmetry breaking Okay, so let's look at it all right so We need to break supersymmetry It's crucial that we break supersymmetry because none of these additional particles that we've added have we seen in Nature, so we need them to be sufficiently heavy to avoid Detection So let's take this phenomenological approach of just breaking supersymmetry softly Okay, then we know that we can have gaugino masses Okay for all of the the gauge gauginos and I've introduced a notation here We have the bino the we know and the Gluino I don't know why this is up here. It should be down here Okay We also have scalar masses for all of the scalars Okay, and so and again, I've shown the generation indices so I and J run from 1 to 3 they run over the generations We have we have masses for the left-handed squarks Okay, the right-handed up squarks the right-handed down squarks now Everybody knows that scalars cannot be left-handed and right-handed, but I'm using this language to emphasize You know which is commonly used to emphasize what the scalar is a super partner of right, okay? If you're keeping track if you give details you'll notice that I've suddenly started using capital H's for the scalar fields I cannot stand to use small H's for the scalar fields. I'm sorry. I can't stand it So I'm going to be a little inconsistent and use capital H's Then we can also write these holomorphic cubic terms, right? So for every super potential coupling we could write a three scalar coupling and notice see the tilde means the super partner So the H's don't have a tilde on them because they are the things that you know They are the scalars one of linear combination of H up and H down has actually been seen So the tilde's are the mythical beasts that we haven't seen yet Okay, so these are three scalars here three scalars three scalars. I've indicated the the flavor indices So it's very important It's an extremely important point that we'll return to later that all of these parameters have Flavor indices that if we want to break supersymmetry we have to say something about the flavor structure of all of these terms and Then finally we have because we have one quadratic term in the super potential. We have a holomorphic Quadratic term as a soft breaking term as well, which is H up H down Okay, so there are other Terms that you can write down which are technically soft that do not arise. This is what I said earlier They do not arise from higher components of super field couplings So these are unlikely to arise from UV complete models of Susie breaking So we're not going to worry about them in any case. We certainly have a lot to worry about already Okay, I think this is pretty much where I'll stop yeah next time Unfortunate we're stopping here because we're right on the threshold of being able to talk about the Higgs potential But I think I'll I'll stop here and take some questions. Thank you