Supersymmetry & Grand Unification: Lecture 1

Stanford University most of the basic ideas they were going to talk about this quarter originated out of questions having to do with renormalization puzzles and up sometimes puzzles sometimes useful observations about the renormalization of the standard model so I thought it would be a good idea to spend a little bit of time explaining to you we've done this before we've done this a couple of times but still are to describe what we normalization is it's a combination of two things it's the combination of learning how to eliminate out of the description of physics things having to do with distances which are so small that they're irrelevant to the questions that you're asking and it is what the other half of it is very simple it's dimensional analysis in fact it's learning to think about how dimensional analysis tells you the answer to some of the very difficult problems of quantum field theory that have to do with distances much much smaller than you might be then you might be interested in there's some simple examples of renormalization namely getting rid of things which are too small to be interesting to you if you're interested in the nucleus and how the nucleus works you may begin with quarks quarks are the fundamental underlying constituents of the nucleus but for many many purposes the nucleus is really a collection of protons and neutrons and it doesn't matter very much what those protons and neutrons are made out of so a simple step is to solve the theory of quantum chromodynamics to discover and to compute the properties of the protons and neutrons but then once you're finished with that we'll in computing the properties of the protons and neutrons you might mean their mass you might mean their spin a few other properties but also the forces between them also the forces between them in nucleus the protons and neutrons move pretty slowly so there's no real need or no strong need for relativity theory basically the nucleus is roughly speaking described by the nonrelativistic quantum mechanics or protons and neutrons and all you really need from the quark theory is to find out what the properties of protons or neutrons are masses spins and the forces between them once you've got that you can forget where it came from and say protons neutrons and some simple forces between them but then if you're interested in atoms you may not be terribly interested in what makes up the nucleus you use nuclear physics to calculate the properties of nuclei which nuclei exist which nuclei what they're in fact if you're interested in atomic physics you're not even very interested in the mass of the nucleus but still you might want to know the mass of a given nucleus a few of its properties not much basically just its mass and its charge once you have the mass and the charge of the nucleus you can forget completely where it came from you can forget protons and neutrons and you can say there are nuclei there are 92 of them well more than 92 because there are isotopes but you can start over again and say the elementary particles for today's thoughts not today's but I mean if you were doing nuclear physics if you're doing atomic physics you say we start over again with a new set of the elementary particles if you get quark so if you get protons and neutrons and we start with nuclei 92 of them or however many of them there are plus of course electrons electrons are pretty small but we can't get rid of the electrons and still be able to do atomic physics so we calculate the properties of atoms atoms mean nuclei plus electrons we calculate their properties and once we have their properties we can say all right here's the collection of atoms that exist in nature there's cookie atoms and coffee atoms and chocolate atoms ketchup atoms all those various kinds of atoms now once you have them you don't need to ask where they came from anymore well you might want to know a little bit more than that because if you have to calculate we're going to talk a little bit about the properties of we're not gonna talk about the properties of molecules tonight but we're going to talk about how you think about molecules how you go from atoms to molecules but in going from atoms and molecules it's the same kind of deal you figure out how atoms build themselves into molecules and once you have molecules you can more or less forget where they came from and start putting them together and Tinkertoy assemblies and build up anything from there so at each stage you eliminated the things which are smaller than you were really interested in and the result was a more coarse grained description perhaps not as exact a description but a far more useful description for your purposes well the same thing is true in quantum field theory where you have things at all possible scales every possible wavelength are constitutes a degree of freedom waves of arbitrary wavelength arbitrarily small wavelengths but in describing physics at one-lane scale you don't really want to have to deal with all the things at very very small length scales which are uninteresting to you so you invent a way of summing up all of the effects of very small distances and replacing it by effective no parameters in your description that's all that renormalization really is that together with some dimensional analysis let's let me give you an example of renormalization again it's just this example of going from atoms to molecules or from going from the arm from from nuclei plus electrons to atoms and how you deal with how you deal with atomic forces and so forth where atomic forces come from how they come out of electrons protons not not protons and neutrons but nuclei and electrons this is a classic example it's not usually thought of as renormalization but it is renormalization we're getting rid of the very small degrees of freedom and also the very fast degrees of freedom small usually goes with fast the smaller a system is typically the faster its motion is so you're getting rid of not only small things but also very rapid things that's the goal of renormalization so let's start with something we know very well the theory of atoms the theory of atoms is very simple it's just a nucleus and a bunch of electrons and what I'm actually interested in is a pair of atoms it could be a pair of hydrogen atoms but that could be more complicated than a pair of hydrogen atoms just some pair of atoms and what I'm interested in is describing the dynamics of this pair of atoms in a simple way then the complex structure of electrons a cloud of electrons and all that very very complicated stuff what I'd like to describe them by is simply two particles two particles that I call atoms with forces between them how do you understand those forces now the advance that you have here which is an advantage is that the electron motions are very very fast by comparison with the motions of the nuclei nuclei a heavy nuclei are a thousand times or two thousand times and the lightest nucleus the hydrogen nucleus is two thousand times of heavier than an electron and bigger nuclei are even more even more massive so you can almost think of the atom as a heavy bowling ball with a bunch of little flies surrounding it and the flies moving very very much faster than the bowling ball is going to move because the given force is just not going to accelerate the bowling ball very much so you can make an approximation and then your first approximation you say that the bowling ball nuclei are so heavy that they don't move at all what I want to calculate is the effective potential the potential energy between two nuclei or between two atoms in a way that gets rid of the electrons in fact which gets rid of separately the electrons and the nuclei so what do you do you start this is a quantum mechanical problem there is any quantum mechanical problem you start with a Hamiltonian the Hamiltonian is just another expression for the quantum mechanical version of energy so let me write down for you what Hamiltonian I would write down if I really want it to do this problem I would say first of all I'm not interested in Corrections coming from the special theory of relativity from relativity theory that's not interesting for for molecules so you begin with nonrelativistic quantum mechanics and that means the energy first of all consists of the kinetic energy of the nuclei let's take the two nuclei to be the same it could be for example hydrogen I so the momentum of the first nucleus it'll scald P 1 P 1 squared divided by twice the mass of the nucleus that's the kinetic energy of the first nuclear the nucleus just a nucleus not the atom then plus p2 squared that's the kinetic energy of the second nucleus and then of course there is the Coulomb force between the nuclei the Coulomb force between the nuclei is plus e squared and the energy it's plus e squared divided by the distance between them are 1/2 let's just call it r1 - that's the I've left out dimensional you know dimensional for PI's and stuff like that that's not what I'm interested in and then what comes next the electrons so let's sum up everything that goes into the electrons everything that had that involves the electrons first of all is a kinetic energy of all the electrons ok so let's call the electrons let's call the momenta of the electrons Q Q sub I squared over twice the mass of an electron let's call that small mass his big mass for the nucleus his small mass for the electrons and this is a sum over all the electrons sum over all the electrons that's there that's the kinetic energy then there's the forces between the electrons that's plus r e squared divided by let's call it little R IJ that's the distance between the I thin jet electron this is the Coulomb energy between them electrostatic energy and what have I left out oh I have left out the force between the protons and the electrons so that's one more term I guess we can call it minus e squared because they're opposite charge attractive minus e squared divided by the distance between the I thought on and either one of the nucleus I don't know what to call it let's call it capital R 1 I that means the distance between the first nucleus and the ice electron and then another one for the second for the second electron r sub one so r2i all right there are things which involve the electrons and there were things which involve the nucleus here are all the things that involve the electrons now intuitively and I won't try to prove this intuitively there are two timescales here one timescale is long and it has to do with the very slow motions of the nuclei why are the motions of the nuclei slow because the nuclei are very heavy and they move around under modest forces in fact they're never accelerated very much inside a molecule and so we think of them as very very slow by comparison with the very very rapid motions of the electrons electrons much more rapid than the electrons form a blur and we're trying to get rid of that blur or replace it by something else well this is fairly easy to do in principle in concept to actually do it might be more difficult but in concept it's very simple you take everything in the Hamiltonian here that involves the electrons group it together and think of it as the Hamiltonian of the electrons alone but what about the positions of the nuclei for present purposes now we're going to say the nuclei move so slow and the electrons are just their wavefunction so rapidly that in first approximation we can say view the nuclei I'm not moving at all they're fixed then nailed down they're stationary and we just fix them and we think of this as a Hamiltonian for the electrons in a fixed background the fixed background being the nailed down stationary nuclei so we fix the positions of the nuclei once the positions are the figure nuclei are fixed then this becomes the expression for the energy or for the Hamiltonian but it's still a function of the position of the nuclei it's a function of the position of the nuclei but those can just be yup well whose momentum for the moment yes in fact for the moment we don't even need to we just say they're nailed down they're very very heavy okay we're not going to violate the uncertainty principle or anything like that because eventually we're going to take those momenta back into account and the approximation here is just that they move slowly all right then we take this Hamiltonian which only involves the electron coordinates and the fixed positions of the of the nuclei and we solve it what does that mean we solve the Schrodinger equation and we solve the Schrodinger equation for the lowest energy state we find the lowest energy state and in fact more important the lowest energy eigenvalue the energy the ground state energy basically the ground state energy of the electrons in the background of the nuclei let's call that something let's call that let's see let's see let's just call it e e of electrons all the terms that involve electrons a of electrons and what is it a function of the only thing it's a function of is the positions of the nuclei that's all everything else is taken care of this is the lowest energy state the lowest energy associated with the positions of r1 and r2 those are the positions of the V or the nuclei now what do we do next now we say okay let's take all of this and forget about it erase it we can't forget about it of course not completely but we can replace it by the energy of the electrons as a function of r1 and r2 but plus e of electrons of r1 and r2 what do we have now now we have a problem involving only the protons all only the nuclei excuse me only the nuclei a potential energy between them that was originally there plus another term in the energy which also can be regarded as potential energy it's a function of the position of the two nuclei and therefore it's a contribution to the potential energy this is the way that we eliminate out of the problem the electrons and replace it with something else in fact it's only a function of the distance between the electrons and so what we actually find is that the entire arm dynamics of the electrons is summarized we first of all have the Coulomb potential between the nuclei but then another term in the potential energy and the other term in the potential energy when you combine it with this term here gives you a force or gives you a potential energy which looks something like this when the when the nuclei are extremely close together it's completely dominated by the Coulomb term here repulsive when they're far away there isn't much energy between them and in between there's some some attraction and that attraction is due to the electrons however we never have to think about the electrons again for this for this hydrogen molecule or whatever it is we don't have to think about it we just have an effective potential between them and we solve the problem of two protons in a potential like that we know what the solution is going to look like it's going to sit at the bottom here it's going to be some wave function and that's a process of number one eliminating fast degrees of freedom but we can also call it renormalization we started with a potential which was just this potential we got rid of the high frequency degrees of freedom and the entire potential energy was then replaced by another function so the potential was renormalized that's the basic idea all renormalization is a with that idea eliminating very very fast degrees of freedom and replacing them by an effective slow renormalized Hamiltonian Hamiltonian or however you doing your dynamics okay so that's such a renormalization we want to apply that the quantum field theory so I'm going to show you how you apply that the quantum field theory what does it that you what is it that you get rid of what you get rid of you can either think of it as things associated with very small distances or you can think of it as things having to do with either very high frequencies or very very short wavelength short wavelengths entail high frequencies and so the things that we're getting rid of in the problem are the degrees of freedom associated with very high energies very high frequencies let's see yeah before we do that let's just go through some dimensional analysis and very simple dimensional analysis just to take a break from from renormalization theory for a minute and I just want to do some simple dimensional analysis first of all in our physics in general there are three length scales that you need distance time and a mass and we can get rid of two of them in elementary particle physics by taking the two dimensional parameters of elementary particle physics namely h-bar and C and setting them equal to one that's - that's that's a specification of two combinations of the of the scales and the problem of the dimensions in the scale of in the problem but it still leaves one dimension that you have to specify one dimension that you need one quantity with dimensions and that dimension can be taken to be a length scale once you set H bar equal to one and C equal to one there's only one dimensional quantity that has to be specified it could be taken to be a mass it can be taken to be an energy it can be taken to be a momentum or it can be taken to be a length or it can be taken to be a time but you need one specification alright with these notations here then the units of mass the bracket means the units of mass are the same as the units of energy why because a equals MC squared and C is equal to 1 and also equal to the units of momentum also true that the units of length which we call that length are the same as the units of time that's because C is equal to 1 but the units of length and time are not the same as the units of length and our mass and energy in fact they are inverse so one dimensional specification either a unit of mass length of time but once you fix it that's it and all masses have units of inverse length all momenta have units of inverse length and so forth so let's keep that in mind first of all that's a bit of dimensional analysis we have only units of length everything else is determined or mass depending on how we think about okay let's now talk about a typical quantum field theory um let's say I think we'll start with a scalar quantum field theory and I'll show you what's entailed in renormalization we have for simplicity now just a quantum field theory with a single scalar field Phi that's all there is it has a Lagrangian and from the Lagrangian we derive Fineman diagrams we're going to talk about those Fineman diagrams a little bit and even set up roughly what the calculation of refinement diagram would look like some simple ones um okay so the Lagrangian here is the Lagrangian for Phi it's very simple it's just the derivative of Phi with respect to either space or x squared we've done these things before that's the kinetic term in the Lagrangian and then there may be a potential energy in the Lagrangian minus V of Phi which is a function of Phi now what are the units of a Lagrangian the units of a Lagrangian okay what do we do with the Lagrangian we integrate it over we use a Lagrangian in the principle of least action that's where Lagrangian is for its whole in classical physics in any case in quantum physics it's used in the path integral but basically in either case the quantity of real interest is the action itself and what is the action in terms of the Lagrangian it's the integral of the Lagrangian but the integral of Lagrangian over what and space time and space this is Lagrangian of a field theory so the interesting quantity which is usually called s sometimes s sometimes action sometimes I it doesn't matter but for now it's SS for action huh s for sex that's like action isn't it no no no I didn't say that our s for action and that's the integral d4x of the lagrangian and the action is always something yeah question action has units of H bar but H bar has been set equal to 1 so H bar is now dimensionless action is dimensionless has no dimensions at all and that means whatever the Lagrangian is it must have inverse length to the fourth power the Lagrangian must have inverse length to the fourth power do Lagrangian density in here must have inverse length to the fourth power now what I'm really after is the dimensions of the field itself by what is the dimension of Phi itself so let's see we can figure it out the Lagrangian has to be um let's take this term here this term here is derivative of Phi with respect to x squared and it has to have dimensions of inverse length to the fourth inverse length to the fourth because it has to cancel this D 4x here before X or X as units of length this must have units of length to the minus 4 okay well are whatever the units of Phi are the units of the derivative of Phi of the derivative the units of Phi times length to the minus 1 that's the unit of the derivative of Phi e differentiating with respect to X just gives you a length inverse and now the square of this is just these the dimension of v squared times length to the minus 2 and that has to give you length to the minus 4 and that tells you that the units of Phi squared must be length to the minus 2 or to summarize the units of a scalar field the units of a scalar field I'll just inverse length length to the minus 1 good thing to keep in mind very very useful for all kinds of purposes that's an example of that measure of using some simple dimensional analysis now what about the rules of a Fineman graphs let's talk about the rules for finding graphs a little bit first of all the finding graphs all come from VF i VF i might have things like in it what does it have in it VF i has things in it like N squared over 2 that's just a parameter it really is of course secretly the square of the mass of the field divided by 2 times 5 squared and then it can have things involving let's call it G times 5 cubed and then it can have things of 5/4 in it let's call a coefficient of 5/4 lambda that's a traditional notation lambda 5/4 and on and on and on it can have all kinds of terms in it and notice that these terms also have dimensions these terms also have dimensions in fact the 5 squared has units of length of the -2 if I cube this units of length to the minus 3 and so forth and so on and from that we can read off what the dimensions of vfi are okay now let's talk about Fineman diagrams Fineman diagrams are built out of vertices and the vertices come from the potential here G let's see so ya GM must have a unit of mass yeah why is it mass the whole thing has to have units of length to the minus 4 all right here's length to the minus 3 this must be inverse length but inverse length is the same as mass right so G would have units of mass this obviously has units of mass squared what about the dimensions of lambda they mentioned less no dimensions at all that's significant dimensionless coupling constants are especially important in in renormalization theory the other ones have units of either mass mass squared and of course if you introduce more terms up here then you can follow what the dimensions are okay now a quantum field theory or the rules of Fineman diagrams are built out of two elements one element of the vertices and the vertices are read off from the potential here for example g v cubed represents a vertex in which three particles come together and the coefficient of such a vertex and a Fineman diagram is just g v to the fourth that's four particles coming together this one has coefficient G this one has coefficient lambda what about M Phi squared well M Phi square is very simple thing a particle comes in let's put a little cross to indicate and goes back out just the put a little cross there just to indicate that that something happened there namely the particle was absorbed and re-emitted and the coefficient is just M Squared over two so when you're building a Fineman diagram a Fineman diagram has a value the value corresponds to the amplitude for a process to happen or it has a value and in building it up there's the elements of the vertices and the propagators the propagators represent the motion from one point to another alright so let's talk about the propagators the propagate is from one space-time point to another we can just draw them as lines in between of course the particle doesn't really move along a straight line that's not the implication here the particle is a quantum mechanical particle but the motion from one point to another or the emission from one point detection of another point will call a propagator I will just indicate it by a line going from one point to another now I'm going to tell you what the meaning of the propagator is the mathematical meaning of the propagator the mathematical meaning of the propagator is just the amplitude that if you start a particle at one point that if create a particle at one point that you'll detect it at another point but mathematically what corresponds to is to starting with a vacuum applying the field operator Phi at let's call it X X let's go this point X and let's call this point y these are not the x and y coordinates of a point there's a point X and point Y so you've created a particle at X and now you want to remove the particle at Y so that might be some 5y the creation operator creates the creation operator that's in Phi of X creates a particle and you can think of it two ways you can think of let's let's think of it this way the action of Phi of Y on the bra vector zero creates a particle at Y the action of Phi of X here creates a particle at zero but this can be read create a particle at X what's the amplitude that that after a certain amount of time goes by that you find the particle at Y that's that's the meaning that's the meaning of this symbol here the amplitude that if you created a particle at X you would detect it at Y and that's what the propagator is that is what the propagator is it's just the amplitude that if you create a particle at X you'll detect it at Y but now I can ask what is that what are the dimensions of this object here what are the dimensions of this object are the vacuum that's just a state the state of lowest energy vacuums and states don't have dimensions you wouldn't say the state has a certain mass no the state is just a specification of a configuration it has no dimensions but Phi has dimensions Phi has dimensions of inverse length so what's the dimension of this object inverse length squared now if there is no mass for Phi then there is no length scale and the problem other than just the distance between X and ye x and y would be the only thing which would specify any length scale in the problem so can you guess what this answer what the answer to this quantity has to be 1 over the length squared but what length right I don't think it would make sense to put here 1 over x squared now that doesn't look good how about 1 over Y squared now I've got 1 of X Y I don't know what do I even mean by X Y X X is a 4 vector and Y is a 4 vector what do I even mean by that all right the obvious thing is 1 over the distance between them X minus y squared and the distance now meaning the Lorentz invariant distance the proper time or the proper separation between them 1 over X minus y squared so that's something we learn from dimensional analysis that if there is no mass in the problem if the if the particle doesn't have a mass then the propagator is just 1 over the distance between the points squared notice something important it blows up it gets very big when the distance between the two points is close so the amplitude for starting at one point and detecting a particle at a very close point just on dimensional grounds blows up and diverges that's the source of all divergences in quantum field theory that propagators have divergences like this and that can cause infinities and problems just the fact that the propagator becomes so big at small distances ok everything else is just building up Fineman diagrams and calculating them but also of course interpreting them ok so let's now start with the idea of renormalization and how it works in this very simple context let's start with renormalization of the mass and notice that in scalar field theory a parameter that appears is actually the mass squared turns out that the mass rarely appears in scalar field theory it only appears by virtue of taking the square root of the mass squared thing which appears in the dynamics of the theory is typically the square of the mass has to do with the fact that energy is the square root of P squared plus M Squared it's always M Squared which appears so we might as well ask about renormalization of M Squared what does it mean and how do we do it so from this picture over here we see that the that the interpretation of a mass term is just a basic simple Fineman diagram or a simple vertex and a Fineman diagram where a particle is absorbed that at the point and emitted from the same point now does it have to be exactly the same point well if we have a microscope which is only good up to where we built accelerators which study things down to a distance scale of 10 to the minus whatever it is 17 centimeters or whatever it is we're not really interested in the details of what goes on on distance is smaller than that so if we can find a process in nature which would mimic this vertex even if that vertex might be separated and fuzzy over a very very tiny mass of length scale we might not be interested in that if we're blurring our eyes to such fine distinctions anything any diagram which absorbs a particle and re-emits it from a nearby point would be counted as part of the mass term in an effective description in which we don't look too closely in other words in its description where we get rid of all the very short distance and high frequency degrees of freedom so let's let's ask are there any Fineman diagrams that we can build that will mimic just a particle coming in and a part of glue going out let's all give you one first of all let's build it out of this vertex over here a fight to the fourth texe the lambda vertex here's a very simple process a particle comes in goes out but here's a Fineman diagram where there's a little bit crazy but still it's Fineman diagram where a particle is emitted and comes back to essentially exactly the same point okay comes back to exactly the same point really what's happening as a particle is not going anywhere is of course it's just being emitted and absorbed very quickly that's the way to think about it this is a Fineman diagram and if I want to calculate it I'm going to give you and show you how to use the rules for calculating it it's the amplitude for a process in which a particle comes in and goes back out and this whole apparatus here may be on such a small scale that we might not even see it might be too small scale for us to be interested in but what is this propagator this propagator is where did I write the propagators did I erase them already now here it is there's just one of X minus y squared but x and y are the same point in this problem well maybe they're not exactly the same point maybe I am sort of blurring distinctions down to some what's called a cut-off scale down to set some very very small distance scale we might want to ignore separations on that small scale let's call the cutoff and a quantum field theory cutoff meaning to say we don't think about scale smaller than that it's a sort of arbitrary thing to do but we can do it let's say we're not going to be interested in distinctions on scales smaller than smaller than scale Delta all right then what should we put in here we should put in a propagator which is separated by distances no longer than Delta we should put in a propagator which separated by distance Delta if we're interested on length if we're interested in physics on length scales a little bit longer than Delta but we are not interested on length scales smaller than Delta then we might want to say let's calculate the Fineman diagram but smearing this vertex here smearing it a little bit smeary in space-time smearing it over a distance of size Delta then we'll all we get will get one over Delta squared for the propagator one over Delta squared for the propagator we're throwing away all scales smaller than Delta and what are what I leave out I left out the coefficient lambda coming from the five fourth term lambda over Delta squared so what's the amplitude then in this approximation for particle to come in be absorbed and be readmitted well it's got two terms it has the original mass term it can happen in two ways it can happen just by the original mass term absorbing it and remitting it and then it can happen by this more complicated process and the total amplitude then is the original mass term let's give it a new name I'm going to give it a new name over here with a quart M sub zero zero standing for original starting value M sub zero squared over two plus this over here another way to say it is if you're not going to look at distinctions on such small scales that the effective mass term the effective mass and the effective mass term is M naught square as M naught squared over 2 plus lambda over Delta squared so this is an example of the renormalization of mass it's getting rid of all degrees of freedom on scales too small to be interesting to us too fast to be interesting to us and lumping the two together lumping the two together into a single blurry effect which I will just make a blurry cross here stuff comes in and goes out and the amplitude the quantum-mechanical amplitude for it is M naught squared plus lambda over Delta squared all right that's that's mastery normalization are there other in other words the true mass of a part of a mess that you would see in the laboratory for experiments in which nothing moves so rapidly and so fast that you can see where your accelerator simply doesn't expose distances smaller than Delta the effective mass that you see is this yeah and what's quite of it well yeah all right this is the effective mass term in the Lagrangian the effective mass and of course the lambda over Delta squared here there's pies and other things in there there are numerical things in there so it's not just dimensional analysis dimensional analysis plus evaluation of some some integrals but most of it can be done by just the much of it can be done by dimensional analysis okay so this is this is the shift in the mass due to getting rid of very very high frequency fluctuations and summing them up and putting them into an effective description change in mass this would be called the renormalized mass but of course when I finished we haven't evaluated every possible Fineman diagram that can go into this let's do another Fineman diagram for the same thing just to see how it works all right let's see we I'll put it over here this is a Fineman diagram that looks like this again it's using this this is separated over again separate this over here here's another Fineman diagram let's evaluate it it also has a particle coming in and a particle going out but now we have something new we have two points what do we do with those two points what do we do with those two points in order to get rid of things on scale smaller than Delta well we basically evaluate that the diagram when the distance between these two points is again about Delta strictly speaking you might integrate over the position of these points let's even do that let's even think about that that we should think about that let's hold this point fixed this is the point where the first particle is absorbed and the second particle is absorbed or emitted from a nearby point but remember that quantum mechanical amplitudes are always sums over all the possible ways that a thing can happen so if we fix this point where the particle is absorbed we should want to eat the grade or sum over all the places where it could be emitted all the nearby places it could be emitted from so let's let's see what that would give us let's I think we know that the propagator is 1 minus X or over X minus y squared what's that yeah 3 particles that one yeah we're still examining the effect of this term this term has four particles coming together at a vertex so here's four particles coming together at a vertex and here's four particles coming together at a vertex again but this diagram is higher-order in lambda it's to the next order in lambda this one only has one lambda let's move this over here lambda over Delta squared that's what this one gives now we look at this one over here this one will contain two factors of lambda if lambda happens to be a small number you might say this one's smaller if lambda is small then lambda squared is smaller but let's work it out and this will typically be true if lambda is a small number ah but let's see what kind of things we get first of all we're going to get lambda squared so this is a higher order Fineman diagram right it's a second order Fineman diagram instead of a first order fireman diagram contains the coupling constant or the constant lambda quadratically and then it contains three propagators going between these two points let's call a point of absorption here X and let's call the point where the emission takes place let's call it X plus Delta I don't just want to call it why I want to call it X plus Delta to indicate that there's a separation Delta between the two points so what do we have we have three propagators between these two points what are those three propagators each propagator is a one over Delta squared right the distance between these two points is Delta the propagator is one over Delta squared so this means 1 over Delta to the 6 1 over Delta to the sixth but now I'm not finished where does this if the particle is emitted at X it can be a minute of X plus it could be if it's absorbed at X it can be emitted at X plus Delta but we're really interested in the amplitude integrated over Delta all possible ways that this event can happen particle being absorbed by a of X and being emitted from any nearby position any nearby position where we're not going to look too closely about whether this position is the same or not the right thing to do is to integrate this now I know I'm throwing rules at you but you can imagine that these rules make sense quantum mechanical amplitudes are always sums over all the possible ways of going from one place to another all the possible ways of getting from the initial particle to the final particle having been absorbed the X you can be emitted at any point X plus Delta but then you have to integrate over Delta all right what is the integral over Delta how many dimensional integral is it Delta is a four vector they give you a hint we have one over delta to the six but now we have to integrate over delta but what is an integral over delta mean it means an integral over the four coordinates the four space-time coordinates of Delta right you have two Delta is a four vector if you want to integrate over a four vector you have to integrate over its four components so we can write this del D Delta not the Delta at the Delta 1 Delta 2 D Delta 3 and so forth or we can just write D fourth Delta that means that it's an integral over Delta over the four components of Delta ok now where are we going to integrate it between well we might want to integrate it to large distances but it's not going to matter because this is going to be so concentrated at small distances that that's mainly dominated by when these two poor sorry when these two points are close to each other so good so the important thing to know is that the integral goes from small distances of order Delta to larger distances from small distances of order Delta s or the lower end of integration here Delta for each one of the coordinates now you care to guess how big this integral is lambda is a dimensionless number it comes on the outside of the integral lambda squared but over here but how can I evaluate this integral our D fourth lambda over lambda to the sixth lambda to the six means lambda squared cubed the answer is dimensional analysis we are not going to do any integrals in this class we're just going to say dimensional analysis tells us the answer that is integral what does the integral depend on just on the lower end of integration here that's all that we don't take into account distance is smaller than the cut off that's all it means there's a cut off in the field theory and we cut off the distance scales at distance Delta the answer to this integral can only depend on Delta there's nothing else for it to depend on so what's the answer got to be 1 over Delta squared that's all it can be so notice what we get from here we get something that looks very much of course it's a number here you have to sit down and calculate more carefully there's pies and other things so there's some numerical factor here but the numerical factor is not terribly interesting to us and if we're trying to do a high-precision experiment that would be but what are we find for this here this here is the same as this except with a lambda squared lambda squared over Delta squared there are many many other diagrams which will also contribute 1 over Delta squared with higher powers of lambda for example here's another diagram whoops that one will also give but how many powers of lambda one two three four this will give a lambda to the fourth of some kinds of this others is lambda cubes also so the end the answer is that there's an infinite series of terms each with one over Delta squared H with one over Delta squared plus a few other things on for example we haven't used this term here let's see what we would get from this term from the Phi cubed how do we make a diagram okay this is this is interesting more a little more subtle dimensional analysis what do we get if we have an integral that looks like this an integral from Delta to infinity of something which goes at small distances like 1 over Delta let's say a 1 dimensional integral now 1 dimensional integral will be Delta what though what does that look like that's a logarithm right that's a logarithmic integral now on dimensional grounds you would look at this and say this has no dimensions at all so you might think that it's a constant but it's not a constant because it's actually a divergent integral whenever you find the divergent integral e at the Breloom Oh careful and an integral like this you'll just recognize as a logarithm and the answer will be logarithm of Delta so whenever you get an integral which has as many powers in the numerator as it has in the denominator it's always a logarithm that's a rule we're not going to use it very much but it's an interesting rule on dimensional analysis an integral with as many powers in the numerator as in the denominator always becomes a logarithm if there's more powers in the denominator then you just use dimensional analysis you would say that this one sorry one yeah this would just be 1 over Delta ok so let's let's do this diagram here this doesn't have a line running down the middle and it's composed out of two cubic vertices two cubic vertices with three particles coming into each vertex there so what is this one going to be all right there are two propagators that's a 1 over Delta to the 4th each propagator is a 1 over Delta squared we have to integrate it and what about the vertices each vertex is a G so this G squared D fourth Delta again we have to integrate over all the positions of this point keeping this point fixed particle is absorbed here emitted from some nearby point but we have to take into account that that nearby point could be smeared over some some range of locations how about this what do we get for this here we have as many powers and numerators in the denominator right log and again we integrate from Delta to whatever this one gives us G squared logarithm of Delta so we see there are times that we don't get 1 over Delta squared sometimes we get logarithm other possibilities also but the single biggest thing what I mean by big when Delta is small when the cut off distance is small which is bigger 1 over Delta squared or log Delta logarithm is a very wimpy function it doesn't vary very much logarithm is much weaker than any power 1 over Delta squared when Delta is small is very dominant over the logarithm of Delta here nevertheless there are these logarithmic Corrections for the mass but the dominant things when the cut off gets small if you have a very small distance cutoff is these powers these inverse powers of 1 over Delta squared all right so this is called mastery normalization yeah yeah well yeah that is a good question of course and that's because I didn't tell you exactly how to calculate it let's just say log to the base e okay but the problem the point is if you change the base of the logarithm what does it what happens to a log if you change the base the new mirror not an additive constant and numerical multiplicative constant but I didn't tell you what the multiplicative factors here the PI's and all that stuff are anyway so that would be absorbed into the into the multiplicative factor here so in a typical calculation there will be some PI's and ease and other stuff here and by changing the base of the logarithm you'll just change these coefficients here so it doesn't did know these are dimensionless these are completely dimensionless so it just changed the numerical coefficient Oh in this case you're right yeah for this case yes yeah right the point is yeah the G's have units of mass but what are they supposed to be giving they're supposed to be giving a correction to something which is a mass squared so the units of this G squared here is the same as this units of this mass squared that means the rest of it is dimensionless well all I was saying was that you can absorb the which logarithm into your legs you can you can but you can also just notice that this and this have the same units and say look there's a numerical number here that has to be computed by somebody with a little more power than we're exhibiting here but the you're right it can be absorbed into into units okay arm but that's the idea of mass renormalization this gives an amplitude and amplitude squared is a possibility of this it is it is but more important this kind of amplitude represents the mass of a particle that's more important for our purposes now that a diagram in which a particle is absorbed in a location and emitted very nearby is a effective description of the mass of the particle now of course it's not obvious how this translates into the inertia of a particle and so forth for our purposes now it's just a parameter in our grunge in but we're seeing that the renormalized parameter the parameter after you get rid of distances smaller than a certain length scale the corrections to it here they are well it's not just a mess that gets renormalized let's do another example even if you just have the mass term utilization you still have an infinite overturns don't you and there number one a lot of those diagrams yeah so you mean you can keep going oh yes yeah keep going right so it's already infinite now if lambda is small it may be that the series is a series of smaller and smaller terms and it might converge but still there are an infinite number of terms and edit na right depends on ya what happens to land yeah what let's talk about the renormalization of lambda lambda is also something that gets renormalized so let's talk about that it's not just a mass that gets renormalized every term in the Lagrangian can also get renormalized let's see how that works is the link because you might be out so small so when you do talk of it have also to 1,000 hours a log is count that is dimensionless because really a log always means log of Delta over some other scale in the problem and I didn't want to get into that that logs ago confusing the real things that I'm interested in are these things and these are less confusing these are dimensionally just what you would expect I'm sorry I told you about logs often what happens is that as the logarithm is really log of Delta times some energy scale and the problem that you're interested in but let's let's let's get rid of logs logically event low lower case Delta in my face right northeast over what oh you were using have our mixed up lower case Delta and upper case Delta yeah yeah capital Delta was the inside the integral oh yeah yeah and the answer should involve small Delta all right sorry good the answer always involves small Delta big Delta is an integration variable all right let's take another example um from a operational point of view meaning experimental point of view Lambda Phi to the fourth is nothing but the amplitude for particle for two particles to come in and to be go out again is two particles from an operational experimental point view if all of this takes place again on a scale which is too small to see you'll lump it all up into an effective coefficient to fight to the fourth two particles in two particles out by to the fourth alright so for example you could have Fineman diagrams which look like this this will also contribute a renormalization to fight to the fourth I'm not going to work it out the reason I'm not going to work it out is because I know it'll give a log and I don't want to tell you any more about logs it will give you a log but each one of these terms here is something which in itself can be renormalized so what's going on is the effect of very small distance physics when you sum it all up and take it all into account the same way we took the electrons into account when we when we studied molecules for example we take them into account we solve for them we get rid of them we do whatever has to be done and we find some effective description which doesn't involve those microscopic degrees of freedom the result is renormalization of everything everything gets changed everything gets shifted so the parameters of the theory that you measure are not the parameters of the theory that you input into the theory that's main lesson parameters of theory you measure are those which take into account all the short distance physics on scales smaller than you can see you repeat that again you what you measure includes the scale smaller than what you see or nothing yeah what you measure has taken into account by summing all the Fineman diagrams involving length scales smaller than what you can see it has an effective output which is the sum of all that stuff which you lump together you lump it together into a single thing that you call the mass squared over 2 so the physical mass squared the thing you measure when you do an experiment that lets say relatively low energy that doesn't involve really really small distances is this renormalized mass here which is the sum of all that junk so is the conclusion that somehow or another the n0 term must be canceled all of those other terms exactly yeah we're going to come to that shortly but before we do let's talk about the renormalization of some other masses in this is the this is the renormalization of the mass of a scalar particle you might think it works essentially the same way for a Fermi on and it doesn't fermions are better behaved I would call this bad behavior you want bad behavior in the following sense you want it to get out an answer let imagine in your head that the cutoff distance scale is something very very small imagine in your head that the cutoff distance scale might be the Planck scale we might be we might be trying to account for all physics on scales between the experimental scale all the ways down to the Planck scale always down to that very very small plunk scale then what we would put in for Delta here would be the plunk length the plunk length is terribly small it's 19 orders of magnitude smaller than the Saye 20 orders of magnitude smaller than the size of a proton r17 orders of magnitude smaller than the most current the up-to-date accelerator experiments 17 maybe 16 orders of magnitude and so delta squared is a very very small number in ordinary particle physics units for example if we use GeV then the Planck distance would be 10 to the minus 19 inverse GeV and 1 over Delta squared would be what 10 to the 38th in in a 10 to the 38th but what kind of man what kind of answer do we want to get for example for the Higgs boson the Higgs boson is an example of a scalar particle what kind of answer do we want to get we want to get something like about 200 GeV 200 but what are we getting we're getting 10 to the 38th this is terrible we don't want to get 10 to the 38th the only way to avoid it incidentally the signs of all of these things some of them may be plus some of them may be - I didn't track the signs of anything and M naught squared itself can be negative or positive we want to get out a certain answer which is of order 200 GeV but we're getting contributions which are 17 orders of magnitude bigger the only way that this can make sense is if this all of this cancels precisely - 17 digits ok the 17 digits this has to cancel and leave over something in the seventh sorry in the 33rd sorry it has to cancel not the 1717 seven to thirty four digits and leave over something I think it's in the 35th or maybe it's in the 34th digit this is called fine-tuning this is the fine-tuning problem of the Higgs boson let's just recall what we know about the mass term of the Higgs boson arm and where we know it from the Higgs boson has a potential or the Higgs field has a potential which looks like that near the origin it looks like five squared all right but with a negative coefficient I never told you that this M naught squared was positive it can be negative or positive yup Lander is assumed to be a number which is not much bigger than one now why why do we assume that we assume that first of all because this will all be nonsense if the terms I mean you know completely intractable if the numbers got bigger and bigger and bigger but even more than that we know that this kind of quantum field theory becomes mathematically inconsistent when lambda gets big so lambda of order 1 is as big as it could be and it really should be smaller than that so assume that lambda is modestly small okay now let's uh modestly small but not the not humungousaur small there's no reason for it to be very small and in fact experimentally we know that it can't be too small okay we know that it can't be too small we know it's a number of water which is less than once there's still a lot of terms that are in there just because of the deltas well if when do itself was 10 to the minus 30 it's you know 0.9 then you know it falls off but it still has a number of terms oh there's an infinite number of terms and all you know you have to go a long way down this chain before lambda to the high power over Delta is a relatively small number so it means that a whole bunch of them have to conspire you know I'm not sure maybe 50 or something have to conspire to give this ridiculously small answer our small lambda lambda is typically a number of water 1% right let me just remind you what we know about lambda we speculated or I will tell you that there's good reasons we won't go into them why is the potential for the Higgs boson first of all why it has to have this Mexican Hat shape that was in order to facilitate the problem of getting masses spontaneous symmetry breaking and so forth but are the potential typical potential would be of the form - let's call it mu squared over 2 v squared I've called it instead of calling it M naught squared I've called it - mu squared because it's got to be negative we know that it has to be negative they get spontaneous symmetry breaking x' plus again lambda phi to the fourth and there might be more terms there but the other terms are believed not to be important all right what if we where does the minimum occur the minimum occurs we can calculate the minimum by differentiating with respect to Phi and setting it setting the derivative equal to zero and what you'll find is that at the minimum at the minimum find min squared which we also called f squared remember we call this distance here F we also call that F squared that was equal to MU squared over lambda if lambda is a number of order of magnitude one and so forth and F you remember about F how big the def have to be for experimental reasons remember all the masses of Z and the W boson were all controlled by F the known masses of the Z boson told us that F was about 200 GeV all right that tells us that if less lamb there is some absurdly different number than one which it shouldn't be that mu is also a number which is of order 200 GeV in other words this mass term is of order 200 GeV even if it's negative the hell will the fact that it's negative it's of order 200 GeV but yet it is composed out of this ridiculous sum of terms every one of which is 38 orders of matter which can be as big as 38 orders of magnitude bigger than the thing you're trying to get this is known as the fine-tuning problem sometimes called the gage hierarchy problem the hierarchy problem the hierarchy being the hierarchy of mass scales from a plunk scale down to the scale of ordinary particle physics enormous gap enormous ratio of scales and is not just that there's a small number in physics the mass scale of like particles there's this incredibly finely tuned thing which is composed of many pieces all of which have to add up to something small so that's ridiculous I mean you know we've understood that that's a ridiculous idea for oh I don't know sometime in the early 80s or late 70s when this problem was first identified and question is what do you do about it but before I get onto that let me talk about the mastery normalization of other particles the other particles of the standard model and why they are not in themselves a problem it's really just one fine-tuning you don't have to separately fine-tune the mass of the electron and those sorts of things why doesn't the electron have the same kind of problem okay so let's talk about fermions fermions don't have the same sort of problem so let's talk about the mass term of a fermion the mass term of a fermion involves the mass not the mass squared and it's contained in the term in the Lagrangian which is M sidebar sigh sigh are the Fermi on fields that describe the creation and annihilation of fermions and the term in the Lagrangian that controls or that defines the mass of the electron or the quark or whatever it is has just has the form M side bar side now as I told you last time several times ago first of all sidebar SCI is also a term in Lagrangian which creates a process where a fermion comes in and gets really Tsai absorbs a fermion at a position and sidebar emits it so again it's very very much like this term here except it only involves the mass and not the mass square but that's not a big deal we can certainly imagine diagrams which renormalize the mass but I'm going to show you now why it's not as big a deal for the fermions so do you remember I told you a couple of times ago that sidebar psy is a thing which always absorbs a left-handed particle and emits a right-handed particle or absorbs a right-handed particle and emits a left-handed particle it flips you between handedness R of the Fermi on if the Fermi on is moving down and the side bars are of an axis with a right-handed Hill isset a right-handed spin relative to its direction of motion then the mass term will flip it okay that's an important thing to know so this is the thing which takes left to right or right to left now what kind of thing could renormalize the mass of the electron for example one of the things that we put into the standard model was the emission by the fermions of a scalar or the emission of a gauge boson mission of a scaler or the scaler let's represent this way scaler is a valid line the gauge boson this could be quantum electrodynamics and quantum electrodynamics the only thing that can happen to an electron is it emits a gauge boson the photon what happens to the handedness of an electron when it emits a gauge boson the answer is nothing the hand in this doesn't change so emitting a gauge boson takes left - left or right - right you can work that out from the properties of the Lagrangian that the emission of a photon doesn't flip the helicity of a Fermi on what about the emission of a scalar particle well it turns out the scalar particle does flip from left to right or from right to left so gauge bosons don't flip a scalar particles do flip the direction of rotation of the spin of the electron relative to its momentum ok now let's look at processes which could renormalize the mass of the electron and this is fun this is a little bit of a surprise supposing the electron starts out with no mass notice in the boson case where is it that even if the particle started with no mass it gets this huge humongous contribution from renormalization and the renewal the renormalization of the mass is not dependent on whether the electron had order the particle had the mass in the first place so all these terms would be there even if this one weren't there now let's take the case of the electron so what kind of things can we have here's the electron moving along it could emit and absorb a photon now by definition a mass turn practically by definition a mass term is recognized as an amplitude for left to become right or right to become left from an experimental point of view of mass term was recognized as an oscillation between left and right can we get left going to right can that happen and let's suppose the starting mass of the electron is zero let's begin with the electron mass being zero is there any way to go from left to right well you could have made a photon and reabsorb a photon but a photon always takes left - left or right - right and there's no way by a sequence of emissions of absorptions of photons to make a left go to right the conclusion is in quantum electrodynamics or in any theory which only involves the emission of gauge bosons fermions get no mass renormalization at all they get no mastery normalization at all so the electromagnetic self energy of an electron if the electron started massless if it started massless the correction would be zero a massless electron is a consistent thing well in pure quantum electrodynamics not in the standard model but in pure quantum electrodynamics a massless electron is a consistent thing and if it starts out massless the bare or the starting point the input mass it will stay that way why because photons emitted and reabsorbed will not flip from left to right what about supposing now there is a scalar particle that the electron couples to the Higgs field is an example let's suppose that the Higgs field can be absorbed and re-emitted or sorry emitted and reabsorbed right what kind of thing does that make can that make left go to right no it can't because the scalar emission here always takes left whereas it always takes left to right or right to left this is the scalar particle here the scalar particle always involves transition and so here you can go from left to right all right but then in the reabsorption here you're going to go back to left and if you think about it no matter how complicated the Fineman diagram is it's always going to involve an even number of vertices and because it involves an even number whatever gets emitted must get reabsorbed that's all it comes down to if it's going to be just electron goes to electron whatever gets emitted must get reabsorbed the number of vertices must be even and if the number of vertices is even from this then then you can't make a left go to right and so again scalar emission and absorption will have no effect on the mass of the electron if it starts from zero ok can you ever get a shift of the mass of the electron for example in quantum electrodynamics the answer is yes but only if you start with a mass what does a mass do a mass can be thought of as a vertex and a Fineman diagram which does take left to right but the coefficient is proportional to the input mass so let me draw a Fineman diagram which does create which does shift the mass of the electron all you have to do is emit and absorb a photon but in the middle here have one of these mass insertions which itself flips from left to right then you can start with left emitting the photon takes you to right but then the mass term takes you back to left and then oh sorry I didn't do that right left goes to left and then the mass term takes left to right and then right to right the value of this finan diagram contains two factors or three factors it contains the electric charge twice remember the Fineman diagram for the emission of a photon is proportional to the electric charge the coupling constant is the electric charge so this kind of Fineman diagram would have the electric charge squared but it also has the original starting mass original starting mass let's call it M naught so we see that the shift in the mass is proportional to the mass itself so now we can write down that the mass of the electron will consist of M naught plus something of water east squared times M naught plus more complicated diagrams e to the fourth times M naught in such a way that if the starting mass was zero to begin with it would remain zero the implication of that is that not at the starting mass was zero that it would remain zero instead of the starting mass was very small it would remain small you would still ask the question why is the electron so light compared to some fundamental scale but at least you wouldn't have this fine-tuning problem you would not be cancelling out differences between large numbers to make a small number and say for reasons I don't know the mass of the electron may be small and all the corrections to it are even smaller so fermions are in that sense better behaved than scalar particles what about gauge bosons well gauge bosons have the same property that if they start out massless I won't go into it if they start out massless they remain massless to all orders so what that means is that to understand the mass scales of the standard model the fine-tuning problem is really only one fine-tuning problem it's the fine-tuning that's associated with the Higgs boson if you get that one right all the other ones will be okay because all the other masses are direct response is to the shift of the Higgs field where is it done not we erased it I guess yeah to the shift of the Higgs field and if the shift of the Higgs field is controlled and if it doesn't undergo this enormous renormalization and shift off to some huge amount then everything else will be okay all the other there will be no other fine tunings the only fine-tuning in the standard model no really crazy fine-tuning there are only really crazy fine-tuning is the mass term from the Higgs boson which has to be about 38 orders of magnitude smaller than whatever fundamental scale you might envision that's the great puzzle that supersymmetry and other kinds of theories like that are intended to address roughly speaking the question is is there any kind of context where the Higgs boson cannot get a mass correction which is enormous ly large is there any kind of theory where the Higgs boson itself will also be a particle whose mass isn't driven to enormous ly large values by renormalization and so the answer is yes we know some there's one other fine-tuning extreme fine-tuning in physics it doesn't it it only comes into play though when you start thinking about gravity before you think about gravity if you don't think about gravity then it's not a problem it is a problem that's associated with renormalization and it's the problem of the renormalization of vacuum energy renormalization of vacuum energy is also a serious problem it's also a thing of very serious need for fine-tuning and I'll just tell you what it is since we're do since we're dealing with fine-tuning first of all what the energy of the vacuum is for most purposes in physics doesn't matter why not well it's an additive constant they're all energies an empty space has energy so be it but then as long as when you add a particle it adds the right amount of energy that's okay who cares what the what the vacuum energy is it doesn't play any role in physics the real point is that only energy differences are important in physics energy differences so when you think about the mass of the electron don't think about it as the man as the energy of electron think about it as the extra energy that you have to add to the vacuum whatever the vacuum energy is you have to add a little energy when you add an electron so it's energy differences the energy difference between the energy of an electron and the energy of a vacuum which is really what we call the mass of the electron so who cares then if the vacuum gets its energy shifted by a large amount as long as everything else gets shifted along with it so let's let's come back to who cares in a minute but let's just ask what does happen to the vacuum energy and the vacuum energy is something which is also renormalized and it's renormalized by diagrams which have no input and no output particles in other words just vacuum to vacuum the effect of such diagrams is to renormalize the energy of the vacuum but who cares I mean that the vacuum energy is have no physical significance but nevertheless let's ask what it looks like so for example in the v 4 theory and the v 4 theory let's do the v 4 theory so there are 4 yeah let's draw a diagram from the v 4 theory that involves no in particles and no out particles just vacuum to vacuum here it is four particles at the vertex here and that's it how big is that diagram what the numerically how big is it well it contains lambda and then it contains a propagator from here back to the same place that's a factor of one over Delta squared after I it's a factor 1 over Delta squared if this little separation here is of order Delta but then there's another one here which is another 1 over Delta squared so the whole thing is 1 over Delta to the fourth every diagram that you can write down will typically go as 1 over Delta to the fourth times they're more complicated ones but they'll also go as 1 over Delta to the fourth and so the vacuum energy is also something that gives renormalized because it gets renormalized it means that if you really want it to be zero you better fine-tune it to very hard precision but on the other hand who cares what it is we don't care what it is we don't care what it is until gravity becomes important and the reason is that in standard Theory the source of the gravitational field is energy if the vacuum has energy it means that the vacuum itself empty space gravitates we're not going to go into this now this of course has to do with dark energy it has to do with the cosmological constant we'll talk about it another time but yes this is also something that has to be fine-tuned and has to be fine-tuned once again to enormous ly high precision so really strictly speaking there are two fine-tuning problems in physics at least to two that we know about one of them is the vacuum energy which has to be tuned to something like 123 decimal places and the other is then the higgs mass which only has to be tuned to something like 30 or 35 decimal places or something like that these are the two problems in in many respects these are the sort of biggest puzzles in particle physics at the present time because it may turn out that these puzzles are simply due to a misstatement of the problem you know there often happens that that really really thorny puzzles go away when you state the problem correctly but it's been a long time since these problems were were identified and nobody really knows what the solution to them is supersymmetry is a potential solution to the to the fine-tuning of the Higgs boson it is not a potential solution to the fine-tuning of the cosmological constant so the next time I'm getting a little tired I think we'll quit early tonight on the the time for a couple of questions but the next time I'll start to tell you a little bit about what supersymmetry is for the mass of the Higgs dr. was catarah drumset playing distance but could be contacted by visiting so he could be Penn County Park so just does the fine tuning depend we can talk about all the contributions to the mass of the particle which come from fluctuations of wavelengths from degrees of freedom down to some very small distance scale the answer will depend on that scale it will depend on that scale so the real question is ordinary quantum field theory how far down in scale do we expect it to make sense before it becomes nonsense all those modes all those frequencies are real things that we think are really there because quantum field theory makes sense down to those scales that's a sort of lower bound on what these effects are what I'm saying is if there we have some scale where we expect the physical change it could be the Planck scale it could be some unification scale if we have some idea where physics changes then the physics down to that scale contributes some country lucien and it will only be bigger than that that's the point will only be bigger this is kind of a lower bound now what do you choose for Delta and practice in practice you choose for Delta R the distance down to which you think you understand physics at the present time so we have the standard model the standard model appears to make sense to very very small distance scales we talked a little bit about unification of coupling constants didn't we about how the coupling constants seem to track together and they seem to come together at something like about 10 to the 16 GeV well a lot of us think that that's the scale or where quantum field theory continues down to beyond that 10 to the 16 GU fee for even smaller scales it can only get worse but if you believe that quantum field theory in the standard model makes sense to some small length scale let's say 10 to the 16th GeV or whatever it is then you're obligated to say where the effects of all the scales in between what they did to the theory how they renormalize the theory and what canceled them all out and it can only get worse by thinking about even smaller scales so right so we think we have a pretty good understanding of physics down to scales which are many times smaller then then the weak interaction scale in this ordinary weak interaction scale we have a good understanding of it only though if we can explain what happened to the mass contributions and the renormalization contributions from scales between the experimental scale and the smaller scale where we think the theory makes sense so in that sense what I've described is sort of a lower bound on how bad these things can be where Delta we simply take to be the scale at which we still believe the theory makes sense but what's the definition scalar particle it can be give an example of this character Higgs particle it is ste step is a scalar particle is probably more of them we've never actually detected an elementary particle let me call scalar or how do you find how you define is that it doesn't it doesn't transform on the rotation of coordinates it doesn't transform on the rotation the photon is a vector particle it's described by a vector potential which when you rotate coordinates the components of the vector change the gravitational field is a tensor particle fermions simply state let me say it even simpler the scalar particle is a particle with spin zero no angular momentum no spin angular momentum that's a simple saving of it for the scene for immigration see for mental is a gauge boson has been z particles of spin one particle oh it's a spin one particle all these particles has been one W plus W minus or spin and if they're the same as the Z for event that's saying the spin is the same as the Z particle which is the same as a photon all right all these particles called gauge bosons are very similar keep emphasizing they're very very similar in parallel to to the photon they have a Maxwell like description for the vacuum energy how come the the bows ground state doesn't cancel the fermionic on stage okay that would require again a huge amount of fine-tuning the actual answers are a little more delicate they involve various numbers here and fern and so forth the infinitely many such terms some of them positive some of them negative and the ones some of the ones from fermions are negative some of the ones from from bosons are positive but unless the masses of the particles and the coupling constants were very delicately matched they wouldn't cancel matching them is what supersymmetry does supersymmetry does exactly that it cancels out if I will do the question was why fermions and bosons than whether the questioner know it knows something well the question of knew is that the least the simplest Fineman diagrams for the vacuum energy of fermions is- the simplest corresponding diagrams for bosons is positive and in principle if a fermion and a boson had exactly the same mass they could cancel each other in the vacuum energy but if they have slightly different masses then you're going to get again these quantities here with huge coefficients which may be proportional to the mass difference for example of a Fermi on the boson but or different coupling constants you will have to fine-tune it in any case to to get it to work out that's what supersymmetry sometimes does okay long other questions for more please visit us at stanford.edu

Channel: Susskind Lectures

Views: 91,935

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Keywords: 1 Supersymmetry, Grand Unification, 2010

Id: W6srShxBCrk

Length: 101min 21sec (6081 seconds)

Published: Wed Nov 14 2012