 Mae'r cyfnodd yn ymwybr i'r cyfnodd. Ydi'r cyfnodd yma sy'n gyweithio'r cyfnodd o un o'r cyfnodd, o'r cyfnodd o'r cyfnodd o'r cyfnodd neu oed gyllwynwys ymwys yw oedol. Felly, ymwys ymwys cyfnodd o'r cyfnodd yr OSM-tyri, sy'n amser cynlluniaeth CUCID, erbyn sy'n amser oedd yn gweld o'r cyfnodd, sy'n amser cael y cyfnodd a'r cyfnodd yn y gwybod. y cwstio aroedd yn ystod o'r cwstio cyfnod Cp sydd fydd eithaf eitem ni'n fydd y fath o'r gwaith sydd yma yma ymol yn y bwrth oedd ymweld i ddechrau'n ceisio yma mwy o'r cyfrifoedd cyffredinol. Felly, rhai bod hynny'n gobeithio'n gobeithio ar y cwstio, wrth gwrs, yn y cyfrifoedd yn y bwysig? A'r ystafell, rwy'n gweithio yma yw'r cyfrifoedd ar y cyfrifoedd yn y cyfrifoedd sydd wedi'i cyfrifoedd through their higher dimensional operators and the standard model. But with new Light States there are a different way of doing things and we can be much more sensitive to extremely tiny couplings. So broadly speaking we're either going to be looking for virtual effects or real effects. Either off shall particles so mediating forces o'r cyfnod o'r newid ymddangos. A'r ystod y gallwn y cyfnod ymddangos, yn ymddangos ar y cyflin o'r cyflin o'r cyflin o'r cyflin. Felly, yn y model ymddangos, mae'n gweithio'r gwaith o'r gwaith graffitio'r cyflin yn y gwaith graffitio'r gwaith graffitio. Mae'n gweithio'r gwaith graffitio'r gwaith graffitio. Wyddech chi, wnaeth yw ymddangos nid ymweld neu yw'n unrhyw ydw i'w gweithio'r cyflin o'r cyflin, sy'n ddwyfodol. Ymddangos yw'r cyflin o'r cyflin. Mae'n byw peth i gyflinnig, yn ddechrau o'r cyflin. Mae'n ddweud aethaf o'r cyflin o'r cyflin? Mae yw ddwy'n o'r cyflin o'r cyflin o'r cyflin. Felly, yn ein cyflin o ddwyfodol, A ddim allan o hyfforddiad hyfforddiad i fynd, ond ymddangos hon yn iawn i gynnwys i'r ystyried i gyda'r leisgrun yn y bach ar y cyf hart. Yn nesaf, ond langydd, a'r parwch cw Gam i'r model ysgrifennie mor mwyfyniaid o ffynu'r newydd, ac mae'r cwestiynau am gwaith yn mynd i ysgol yna i'r dros fylwydol. ein bod ni'n gwybod cyngorol gwennau. Y dyn ni'n gwybod cyngorol gwybod cyngorol, a wnaeth ni'n gweld gwyddog organiad yn gwybod gwyddog a fydd y gweld gwyddog cyngorol gwybod gwyddog credu y byddwyd, a yna yw'r Ameg extensionsеж. Mae'r Ameg tossed yn ei cyfrifol gwybod gwyddog yn pwyng o'r rôl yma, neu mae'n gwybod gwyddon oedd yn gwybod gwyddog. Y chyfrifol gwyddog.ечwymodd wrth gael, oedd mor gweithio'r cyffredin ar omlyniad a'i ceisio'r gwneud cyffredin o'r gael, yn ei eistedd? Mae'r gyffredin cyffredin wedi cyffredin o'r cyffredin newydd pethau, felly cyffredin y cyffredin yn yr awnach aujourdaeth. Adegwramatica, a cyfentwys o agwyswyr, o gyfentwys o gyferwysar, adegwyswyr cyfrwysar o gyrwysir, o gyfentwysar o gyfrwysar o gyfrwysar. Ac o gyfrwysar drafodag gyfrwysar o ganai, d roundedwys a gyrwysar, o gyfrwysar o trauma. Mae'r ddau llwyllau yn unrhyw sydd yn cael ei cofwynt yn staff iawn. Fy ydy'r cwpwnt yn ddweud sydd yn gwybod y gallai'r ddau llwyllau yr un ddau cael eu cyfnodol sy'n ddweud mae'n gwybod yna'r ddau. Yn y modd, ysgrifwng ysgrifwng ysgrifwng ysgrifwng, ysgrifwng ysgrifwng ysgrifwng ysgrifwng ysgrifwng a g tilde o'r ddweud o Fhermion. Felly mae'r ffster yn eich leon. Rhywbeth yw'r propatio parwysau, mae'r ddweud o'r sphin, ond dwi'n credu ffeydd o'r ddweud o'r sphin o'r ddweud o'r dweud, ddweud o'r ffordd o'r cyfnodd o'r mage o'r ddweud, If you don't do that, then this won't couple coherently to a big object, like the spins of past fermions on the earth are generally pointing in random mis directions, modulo the sort of magnetic core, etc. But these kind of particles, unless you have some coherent spin thing, won't have a volume scaling coupling to matter. Okay, so the easiest things to look for, certainly if they're long range, are going to be these kind of couplings. We'll focus on the scalars because these are somewhat easier to treat. Okay, so let's say that we have some scalar. So we've got our scalar phi. We've got our usual code term. We've got some mass for it. And then we've got some coupling, say to, I don't know, to nucleons or whatever. Then as we'll go over in the problem sheet, if we have some source, so some sets collection of nucleons or whatever of the origin, then the phi that we set up will be of the eukala form. So you'll have a e to the minus u r over r. So this will be, yeah, we're going to have the problem sheet. It's, of course, also very standard. Small radii, so it's r significantly less than the Compton wavelength of our scalar. Then this looks just like an inverse square plus some constant term plus small corrections. If we are significantly larger than the Compton wavelength of our particle, then the thing that dominates is the e to the minus mu r part, and we're exponentially suppressed. So the potential that we get, we have a part that looks like a 1 over r potential. We show that in terms of force, we'll give us a usual inverse square law type force. And then once we get past, so this is a, this say is the inverse mass scale. And for a massless particle, this would continue out to extremely small, sorry, extremely large radii. But once we get past this scale, we go very quickly to zero. So this is poorly drawn, but you get the picture, inverse square and then very quickly to zero. Okay, so the obvious, and also the point is, since this equation of motion is, the equation of motion from this Lagrangian is linear, if we take lots of small objects, we can superpose the fields from all of them linearly. So the fields from some complicated objects are just summing up the Yukawa type field we get from each little bit of those objects. Okay, so how do we actually say look for this? Well the obvious kind of thing is to look for some, if the mass corresponds to a long range for this force, such that we can actually probe in some reasonable laboratory or beyond scale, then we can look for the deviation from the inverse square law. It'll be inverse square law at distances shorter than this, but then die away faster. So the biggest fractional, so deviation from 1 over r squared force, which is what we get if we just had gravity. So fractionally largest, distances of order, the inverse mass scale of whatever a new particle is. So people have done tests at a whole range of mass scales, and again this is something that you'll derive in the problem set, but if we look at the constraints on coupling, then an experiment at some given scale say whatever, some length scale, so this is length scale r, will be exponentially suppressed in terms of its constraints at larger masses because you're in this exponential expression region, and then will be parallel suppressed at lower masses. So you get some constraint like that from some experiment which is testing say the force between things separated at some characteristic scale r. And there's a whole set of these things which by now sort of cover a whole range of things. So they go below gravitational strength forces at some distance, so this is around 1 over m plank say, at a distance of around 10 to the minus 4 metres, and the largest scale tests of this kind are on solar system scales. So of order, I don't know, light hours or similar. I can't remember the exact numbers there. But so over a very wide range of scales, if you have some force which is even much, much weaker than gravity, around like of order 10 to the minus sort of 10-ish, every m plank, sorry, 10 to the minus 5 ish or whatever, 10 to the minus 10 at the force, then you can see the deviation from the inverse square law by basically just doing the naive thing of taking, yeah. OK, so well, a scale mediator will generally make it so that everything is attractive or everything repulsive. A vector can have positive and negative charges. That's one feature in terms of, so if you just have normal relativistic matter, all of the same kind, and you are looking to say the difference between some additional scalar and vector components, then that's not going to be the easiest thing to check directly from that, I think. Certainly in terms of its behavior, in terms of radiation and stuff, and in terms of relativistic differences, but yeah, come back to that one. OK, so yeah, so over a very wide range of length scales, so down here in the lab, you go from doing things like you put a very small sphere of order size microns next to some wall, you trap this optically, and you look for forces on it as you vary its distance from the wall. So this is on the 10 to the minus 5 meter scale and so on. Up to larger distance scales, you take origin pension experiments where you have two disks which you're able to spin. You do the kind of thing of cutting holes in the disks such that the geometry is when you turn this one, the torque that you exert on the bottom one through the force that was purely inverse square cancels just geometrically. You arrange the holes such that you don't get any torque if it was purely inverse square. Then you look for, you turn this one, you have a shield in the way between them to make sure that any extraneous charges or whatever don't give you some electromagnetic forces, and then you see if the bottom one turns when you turn the top one, which would indicate some non-inverse square force. Then the very large length scales, you look for how planets, the moon, satellites fall. So you have this thing over a wide range of length scales where we're looking for deviations from an inverse square lot. Okay, so one point, which again, so the first part of the lecture is sort of introduction to the problem set. When the mass gets very small, or equivalently the range of this force gets very long, we have a bit of a problem because there the deviations from inverse square behaviour are rather minor. You're in the part where it looks very much like an inverse square. Okay, so you will get an additional effect as well as gravity, and this will be our GMM M1 M2 over R squared, plus approximately our scalar coupling squared times like our charge 1 under this scalar over R squared and then some small corrections. But in terms of measurements, this will just look like giving G newton a slightly different value. So it's rather hard to tell that something different is going on there. So this then comes back to the question of how can we tell that our force has actually got a part of it comes to a spin 0 and part of it comes to a spin 2? Yes, sorry, that was just the intro to this. So the point about, exactly as Jed was saying, is that the spin 0 part, the scalar, doesn't actually couple to the full GR version, which you won't see from just writing out the normal flat spacetime couplings, it doesn't end up coupling to the gravitational field. So, in particular, you've got, in the full GR setup, we could write down terms which look like if we just look at stress energy tensor, our scalar field times, say, the stress energy tensor, so this would be like G mu nu T mu nu phi. So those are the kind of things that we're able to write down there, but the matter stress energy tensor does not get contributions from the gravitational energy of whatever an object is. In the most extreme case, if we have a black hole, then the stress energy tensor is 0 everywhere, but it still, of course, sources a big gravitational field, so it doesn't outsource this phi field at all. So the best way to look for, well, one of the best ways to look for whether you have some scalar component to your force is to look at differences in terms of where the gravitational field of objects gravitates. So that's what's called the strong equivalence principle. Well, people use the words in somewhat different ways in somewhat different contexts, which is slightly unfortunate, that gravitational binding energy gravitates. So you'll work out one of the consequences of this in the problem sheet, but another would be that if we had some scalar component of the force, then let's say we have the sun sitting there, we have the earth, and we have the moon. In the normal gravitational setting, then the earth and the moon have the same ratio of gravitational to inertial masses. They fall on the same way around the sun, so everything's okay, and it looks like the moon can just orbit the earth happily without having to worry about all this. Modulotidal effects. If you had, now the earth, though, has a different ratio of gravitational binding energy to its total mass, then the moon does. It's more tightly gravitationally bound. So if you had some scalar force that didn't actually get sourced by gravitational binding energy, then this is called the Nordveld effect, and the earth would fall differently towards the sun, because they would have a different, effectively, ratio of gravitational and inertial masses according to this scalar force. So what you'd get is you'd get some deviation of the moon's orbit due to it feeling a different force, a different force ratio, than the earth when it's falling around the sun. Deviations of moon's orbit. Effect works. As long as our force has got a range, which is, so if our inverse mu is longer than of order, an astronomical unit, you'll see the effect of that, and this is constrained to be significantly weaker than gravity. So even in this case, if you go to very large masses, you find that, I can't remember where the exact thing is, that somewhere around here, the strong equivalence principle tests rule out larger couplings than gravity, even at extremely small masses. Okay, but there are also constraints that are usually even stronger from something called the weak equivalence principle. So the strong equivalence principle was that gravitational energy, gravitates in the same way as normal energy. The weak equivalence principle is that the ratios of gravitational to inertial, so I'll see if there's a better chalk anywhere, to inertial mass is the same for all objects, material objects here. So whereas the strong equivalence principle is only to violate that, we need to couple differently to gravity, to violate the weak equivalence principle. What we need is, say, we take a sphere of one element, like aluminium or whatever, we take a sphere of some other element, of the same inertial mass, but then they would feel different forces in the field of some distant body. So they would have different attractive forces experienced towards, say, the sun. And if you have this kind of thing happening, then you can do very sensitive tests to look for it. The instrument which, over a very wide set of length scales, is the best way to look for this, is the torch imbalance. And this is something quite clever. So naively what you do is, you try and make two spheres with exactly the same inertial mass, they move in the same way if you push them, and then look at what the gravitational force on each of them is. The mass is going to be a pretty error-prone procedure. You need to try and fabricate things exactly the same mass. There's a lot of experimental awkwardness there. The torsion pendulum works in the following way. So let's start over here, I guess. So it's basically what you might think. You have some fibre, and then mounted to the fibre is some arm, and at the end of the arm are your two test masses. This is a sort of very naive description. The real ones are somewhat more complicated. Now let's say we have forces on this test mass, and some different force on this test mass, because it's a different mass, because it's got some equivalence principle violation, whatever. Then the combined force, we'll assume this thing is light, the combined force on the object is F1 plus F2. So assuming that the fibre and the support are light compared to the objects, this thing will just hang in the direction set by F1 plus F2. So now we want to ask what's the torque about the axis of the fibre. What we're going to sense is whether or not this fibre twists. So if we do that, we want to look at F1 plus F2 over F1 plus F2. That is the direction here, and we want to adopt that with the torque here. So this one is in position R1, this one is in position R2, then the torque from this mass is R1 cross F1, and the torque from this mass is R2 cross F2. So we can work this out. So expanding this, we've got our F1 dot this component vanishes, because you have two F1s and this perpendicular F1, F2 similarly, so this is F2 dot R1 cross F1 plus F1 dot R2 cross F2, and then exchanging the order of these things, and rearranging, we get that this is equal to R2 minus R1, because you get a sine flip when you flip the order of this thing, dotted with F1 cross F2. So what we see is that, so this is the torque, we only get a non-vanishing torque about the fibre axis, so the thing only twists if the forces on our two objects are parallel. They can be different in magnitude, like say we accidentally made this mass a bit heavier than this mass or whatever, so they could have different magnitudes, but if they were both pointing downwards, if they were both just being attracted to the centre of the earth or whatever, then you wouldn't get any torque about here, it would just make the thing hang slightly differently. If they're feeling forces in different directions, then you can get a torque. So effectively, what this is sensitive to is different directions of down. Okay, so that's cool, but how do we actually get that? Because if we were just sitting above a spherical earth, then even if they are attracted differently to it, because you have some weak equivalence principle violating force, then they'll both just be attracted downwards, you won't actually have this directional difference. So what you need to do in order to get this, you need some kind of asymmetric source distribution. If your force is short-ranged, you can get this very simply by putting your experiment at the bottom of a cliff. So here's our topography, here's the heart where we're doing the experiment, and here's our experiment. So let's say that our force has some range where it's feeling stuff, so this scale is the inverse mass of our force. Then we are seeing stuff in this little circle here, so the net force will be in the direction which is basically that way. So let's say that one of our objects feels this force more strongly than the other, then one of the objects will feel gravity at the centre of the earth, plus a little bit of this force, a noble force like that, so this is say an aluminium thing, a brilliant thing, will feel the force from the centre of the earth plus more of this thing. And that means we get an overall different direction, these things feel different directions of down, and the pendulum will be torqued. And so as you go up to longer-length scales, you can play the same trick. On scales of miles and whatever, you can use, so miles, you can use natural features like mountains, you want a few mountains nearby. On scales of the earth, you can use the fact that features like the sort of asymmetries of the crust, etc. The fact that earth is not a perfect sphere. And once you're at distances greater than an astronomical unit or so, you can use the fact that we're here, on the earth, here's our experimental hut, and the sun is sitting out here, not the scale. So the force from the sun is directed this way, force from the earth is directed this way. So if these two things are attracted differently to the sun, you'll have different horizontal components, again you'll have different directions of down. So across length scales all the way from of order metres to however long you want, you do have in the neighbourhood of us an asymmetric source distribution, so if you have some weak principle violation, this is an extremely sensitive test of it. So the constraints you get, so metre or so hill, metres, so the force should be less than of order 10 to the minus 7 of gravitational strength at short distances, and at long distances should be less than of order 10 to the minus 11. So you have extremely strong bounds on how large such a force can be. It must, the difference between these two things and the resulting force must be significantly weaker than gravity. OK, so these constraints and also fifth force constraints and strong equivalence principle constraints are, yes. Oh, exactly. So that, well, yes. So given the model, you need to work out how it couples differently to protons versus newtons versus electrons, et cetera, and then go from there. But the point is that this happens somewhat generically. So let's say that we just coupled it to something like my coupling to Higgs. That's the sort of lowest dimensional coupling of a scalar to the standard model that we can write down. Then the Higgs is responsible for contributing some of the mass of particles, but not all of it. You have contributions from EM, you have contributions from QCD, et cetera. So since this is only talking to the Higgs part, then the other contributions to the mass won't contribute to the five coupling, so you'll naturally just get a different ratio. The only way to arrange such that you don't get that is to make it so effectively it couples to the stress energy tensor. It's got to couple to exactly that. Talk a bit about whether that's actually a sort of quantum mechanics-y sensible thing in the problem sheet. But genetically, if you do anything other than that, if you just couple it to some random operators or whatever, then through effectively renormalisation, the fact that you've got to add up all the contributions to the mass from all the different sources to get to your nucleus or atom or whatever, then you will have different couplings to different objects. Exactly. Yes, so I expect that they're chosen due to different ratios of protons and neutrons. I don't have those numbers in my head at the moment for the common isotopes, but yeah, you want it so it's got some different number, say, of different ratio of protons and neutrons. Yeah, of course. These quotes are for, yeah, so EP violation at, I think, let's try and remember this, for nucleons, I think somewhere around the 10 to the... OK, I'd need to look this up. So these quotes, of course, depend on how much a couple is different to protons and neutrons. For these constraints... OK, let me get back to that. The number I have in my head is around 1 level, but I should check that, and I'll come back to that in discussion. OK, so these constraints tell us that if our force is long-ranged, pretty much that it's got a longer range than less than a millimetre, then it should couple more wiki than gravity. It's got to be a very, very weak force. And that's something that various people have been somewhat sort of disappointed by because they want to have new forces operating at astrophysical scales doing interesting things in the universe, in cosmology, et cetera. So how robust are these constraints and what kind of model building would you need to do to get round them? Do they apply to models which are simpler than just the sort of extremely simple scalar plus linear coupling model? So I'll talk a little bit about that. And the answer is not always. In some circumstances, you can have what's called screening. OK, so the general idea here is we're going to add some non-linear part to the equations of motion. So previously, if we just wrote down our thing and we have some coupling, let's just simplicity say it's coupling to mass density here. This would be like our M neutron like coupling to fermion masses or whatever. We'll just put a row in here which we understand to me and the standard model sourcing for it. And so this was the model we had in mind in all the discussion so far. And this is linear in phi. So the equation of motion would just be that our d squared phi plus M squared phi is equal to G. But we can also put in an additional potential term here. I mean strictly speaking, this is part of the potential. But anyway. And if this potential has terms such as lambda phi to the 4, then in addition to this, we'll get a non-linear term in the equations of motion. So whereas previously, we could just take our sources, we could take the Eukawa field due to one source. So this was an E to the minus mu minus x1 over x minus x1. We take the field from this one. E to the minus mu x minus if this is a position x1. This is a position x2. x2 minus x over x minus x2. And then we could just add these up to get the field from both particles or both little bit of elements of matter or whatever. Now we can no longer do this because we have non-linear term here. So given an object, we need to actually solve this equation and figure out what's the overall field. Okay. So for a complicated object or even for a circular object, this is not sort of a problem with a sort of trivial solution you can just write down. But effectively, what's going on is that if we have a value of the field phi, then instead of just a half m squared phi squared, we've got lambda over 4 phi to the 4 term. So this is if we m squared plus phi squared over 2 phi squared. So if phi has some overall value, then that effectively looks like oh sorry, lambda of course. Then it sort of looks like we've given it some effective mass squared, which is the original mass squared plus lambda phi squared over 2. So roughly what it's going to be is that in areas where phi has got big, it looks like it's got some larger effective mass. And as we wrote down earlier, a larger effective mass means that it drops off faster. So let's see what that sort of implies in action. So, yeah. Let's do some example of a source which is of radius r. So if we had a very small lambda, which is small enough to be negligible, then the phi here will be our source, which is g rho r squared over, well, we'll ignore 4 phi and stuff, over r outside this thing. So it'll have this 4. So if unscreened. And the question we want to answer is how big does lambda have to be in order that we change this? Okay. So we'll say that the lambda, the phi value just outside the object is of order g rho r squared. So if what you wrote down earlier makes sense, so we're assuming now that the mass, assume that the bare mass, m is very small. So we're not going to worry about that making the four short range to start with. So we've got our effective mass here. If you believe this, which is of order lambda phi just outside the thing squared, which is, goes as lambda, g squared, rho squared, part of the four. And parametrically, we expect this to matter if, so matters if the effective mass squared is larger than of order the scale of variation in this field. If the mass is small compared to the scale of variation, it's not really going to make any real difference. So this condition is the same as saying that we want the size of the object to be, so rearranging this, the same as saying that we want the size of the object to be larger than 1 over g to the third, rho to the third, lambda to the 1 over 6. And if we do a sort of similar question, we say we have some large region of constant density, then if we just look at the mass, then the sort of energy in this thing goes as minus g rho phi plus the half m squared phi squared. So the value, so dh by d phi equals 0, implies that phi is of order g rho over m squared. If instead, what's setting the value of which phi, so this is saying that the value in the middle of a big object when the mass is small compared to the size of the object saturates the value of order of this, the larger mass, the smaller the value saturate at. So instead, if lambda is big, then we saturate when the new h, so the h, the important part is now our lambda over 4, phi to the fourth thing, and making that stationary gives a saturation value, which scales as g rho to the third over lambda to the third. So comparing to this one, this is g rho over g to the two thirds, rho to the two thirds, lambda to the third. So comparing that to the effective mass, we see that we have the same kind of effective mass that we were getting earlier. It's just this thing squared. So in both of these cases, and in general, it acts like we have some effective mass which is set by this parameter. So the larger that lambda is, the larger the effective mass will be, and the more will be suppressed by the fact it looks like the force is shorter range than the bare mass would have told us. OK, so what kind of values for this parameter can we get? How much screening can we get with physical densities? So let's say that we've got our M effective, which is the relevant value for an object of density, whatever, of rho, is of order g to the third, rho to the third, lambda to the sixth. So if we plug in some numbers, let's say that we look at e to be of order 1 over M plank for gravitational strength coupling, and let's say we take rho to be, like, grams per centimeter cubed for a pretty normal material, and we take that all to the power of one-third, that value is around 10 to the minus 3 EV, which in distance scales is 1 over about 0.2 millimeters. So that's a pretty small scale. That's telling us that if we have an object which is of normal density, then whereas you might think that a long-range force where the range of the force is larger than the size of the object would result in the whole object sourcing the field, actually it will act as though it's only sourced by a very thin shell of the outside of the object. The field that you would have got from inside just saturates and the only, it's as if you just had the force coming from a shell of radius inverse M effective. So if you say we're in this circumstance where you thought that your bare mass was of order much larger than solar system scales, so the force would be extremely long-ranging to do all these tests if instead it has a self-coupling and this self-coupling can actually be tiny if all we want to do is take this, because it's lambda 1 over 6, if we want to take this 0.2 millimeters and make it like a few meters or a kilometer or something like for an astrophysical object, even for very, very small lambda we can significantly suppress the field sourced by large objects and so make it that these tests don't actually apply anymore. Okay, so this is a cool idea, so this is called the thin shell effect and it means that, yeah, our tests won't work as we thought they would. Similarly in a torgen pendulum setup, if instead of the whole like kilogram sphere of aluminium and beryllium feeling the force of the earth or the sun, you only have 0.2 millimeters of the outside of the sphere and a little bit of the rock nearby sourcing the field, then you're not going to get a big signal. However, this is still, if you sort of take the theory at face value, not enough to sort of make it easy to get very strong forces and the reason is that due to this thin shell effect we no longer, things no longer look inverse square. So let's say that, still difficult to get forces, forces stronger in gravity, i.e. g larger than of sort of the inverse plank scale as we were writing it. And like said, the reason for that is that if we take our, let's say we take a small object, then that, the switch is smaller than the inverse mass scale associated with its density, it will source the field from throughout, it'll look pretty much like an inverse square potential. If we have some larger object where the size is of order, the effective mass squared, then the field will just starting to feel these non-linear effects so it'll have some more complicated profile. Drawing these things out is, so this is sort of, that was our R and V, R and phi axes. It'll have some more complicated profile where it'll do something, and then flat inside. It'll do something slightly odd inside, then we'll do something non-inverse square cos it's still non-linear in the non-linear regime just outside. So you'll get some funny profile which doesn't look inverse square. So around distances of order the inverse effective mass force violates the inverse square law strongly. So whereas before, we might have thought that a long, force with a small bare mass, so a long range, wouldn't give a strong effect in inverse square law tests at short distances because you're still in the inverse square bit of the potential. If it's got this screening effect, then actually it can because it's got a different effective mass around materials. So, okay, numerically, what kind of bounds do we get here? So if we plug in that number there, if we take lambda as large as we can, so we take the same parameters for we want a gravitational strength coupling and we want, let's say, 1 over 0.2 millimetres times lambda is bigger as it can be to avoid strong coupling issues, so around 4 pi to the sixth, this comes out to be, again, of the scale, so we just make this a bit smaller, we're around 10 to the minus 4 metres again. And remember that this was around the scale where the inverse square law, which is often abbreviated ISL, tests are sensitive to gravitational strength. So what you'd want to try and do is make the screening so strong that everything except tiny objects were screened and you could get away with it because tests of the forces from tiny objects are bad. But what this estimate is telling you is that even if we make the screening as strong as we can, we make lambda all the way up to the strong coupling limit, then the amount of screening we can get for a normal density object is only enough to screen a sort of 0.1 millimetre-sized thing. And though that sounds good, that's still only just enough to get you in the regime of gravitational coupling. So it's pretty hard and this is actually a more generic thing. So this was doing the case of a potential which was lambda 5 to the 4. We could put other potentials in here and see what happened, but you get the same kind of effect. You are always in the regime where a screening doesn't work. A distance is smaller, a larger, sorry, smaller than about 10 to the minus 4 metre. OK, so, yeah. So we're in a situation where we have all of these tests of short-range forces are able to constrain models where both the field just behaves linearly and where the field has some more complicated self-interaction because you get these interesting variations across different distance scales. OK, so, yeah. Any questions on that part of things with forces in general? OK, well, if not, then I'll switch to Axe a bit and I'll talk about the other side of things that I mentioned at the start, production of particles. So this has all been about the forces that they mediate between standard model objects. Now we'll talk about producing particles through standard model processes and seeing evidence for their production and detection. So it depends on the kind of vectors. So, for vectors, there is one particular kind of vector where none of these things apply and that's a dark photon. So if couple to em, so we have our a' mu and then j mu, which is actually the electromagnetic current, so we're just coupling, like the standard model photon, but with some reduced strength, then our bulk objects are neutral, so we don't get some kind of fifth force between big neutral objects. The tests you have to do to look at this are very different. You can look for, say, deviations from Coulomb's law. If you have some, make some charged shell, you can look for production such as astrophysical situations or laboratory situations, et cetera. But, yeah, everything's different if you do this. But this is special. Another kind of coupling. Let's say we couple the b-cell current. So we couple, let's say we have our... So this is dark photon. Let's say we took the other current that's easy to have a light vector for, which is b-cell. So this is also... The point is that these are the two conserved things within the standard model. It definitely should be because it's a gauge symmetry. B-cell is possible to gauge, so it's potentially conserved. So there we have some g, some x-mu, and we've got some j-mu b-cell. So if we write this out, then this is for protons and neutrons. Protons... Sorry. For protons and electrons, that's the same as em, because an electron is minus 1L, plus 1B. So this is just like the em current plus the current for neutrons. So this tells us that if we have some matter, then the em thing is neutral because the protons and electrons cancel, but we get some overall thing from the neutron current, so we do get bulk matter couplings. So if we have a long-range b-cell vector where long range actually only means longer range than sort of microns or so, then we will have fifth force limits. Going beyond this, the situation is actually somewhat more complicated because we'd be coupling to a non-conserved current. Now, you all know that a gauge boson has to couple to a conserved current. If it doesn't, then everything goes to hell. The sort of defamation of that is that if we have the x-mu couples to j-mu like sm and j-mu sm not conserved, then there are processes where we have a load of standard model stuff colliding or whatever, or decaying or whatever, and we produce the longitudinal mode of the x-gauge boson. Then the rate of this production of this gauge boson will be proportional to the coupling squared times the energy of the process squared over the mass of the x-squared times whatever the else is going on in this process. So the point is that for a very light vector, this production, this rate will grow very, very large. This is if it's just got some mass which we're assuming is a Sturkelberg mass, so it's from some high scale. If you have some peaks mechanism or something at some low scale, then you need to worry about that and worry about the production of all the other particles. But if we just have a vector and it's just got some Sturkelberg mass, then you will have production which increases as you try and make the mass of your vector smaller and smaller. This tells you why the gauge thing can't work because as we take the mass to zero, the production rates for all these things go to infinity so everything blows up and everything's bad. So that's sort of the quantitative version of why you can't couple a massless particle like a photon to a non-conserved current. Okay, but then that means that if we look at sort of the plot of mass of our particle and a coupling of our particle, we've got all of our fifth force things which have their usual behaviour and whatnot, but we also have processes happening at high energies. So this is like LAC or something where we have some very high energy and now we have some enormous ratio of energy over the mass of this vector if we say down here. So these processes, instead of just very high masses and high couplings like they do for say a B-L vector or whatever, actually look like this in that case. They get better and better in terms of coupling as the mass goes down because it depends on G over Mx. The point is that for an extremely light vector which couples to a non-conserved current, the situation is actually somewhat different because the theory is kind of a bit sick. So you need to make the couplings incredibly light in order not to have high energy stuff messed with you. So these are the two examples that make sense and give you nice sort of long range of monology and this one doesn't and this one does give you 5th forces. Beyond that everything goes a bit weird. So yes, this implies that your thing is an EFT and has to break down at some scale set by the mass of the thing divided by the G. So yeah, everything has to be completed at that scale. Which is not the case for the dark photon or the B-L, those can just be good up to those scales. So yeah, that's why the situation for vectors is somewhat more complicated, theoretically scalars are somewhat simpler. Yeah, no no, self-interacting vectors are non-abelian gauge theories. People certainly consider those and people consider dark sector versions of them as well. People consider what would happen if you had dark QCD or something like that and there's enormous literature on that and that's a thing. But in terms of having you might worry if you're going to make the mass very light there. Having some say dark QCD with some extremely long confinement scale like cosmologically things might get a bit weird there. But people have certainly considered this and there are many papers on it. Okay. Sorry, a bit of an aside there but useful to understand these things. All right. So let's back to where we are going. It's like I said, all of this stuff was to do with forces mediated by this new particle. But in addition we can look for the effects that come from producing this new particle and then detecting that it's being produced. Either like Tracy was saying the other thing by seeing that some energy has gone missing and the consequence of that energy going missing or by actually just producing it and then hitting something else and you detect physically it hitting something else. Okay, so I mean all of these things have been used and we'll talk about a particular thing that I very briefly went over yesterday which is stellar cooling. So the idea there is that we have a star and we have a very hot inner core and in the same way that if we produce neutrinos in this hot inner core then they can just with very high probability make it all the way through the outer layers and after infinity if we produce new particles they can do the same thing. Okay, so let's try and do more than we did last time and do some quantitative estimate of this kind of thing. So let's take the example of an axion like you've heard about many times by now and let's look at the photon coupling like we were doing for direct detection. So we're going to say that we have our GA gamma gamma our A coupling to the electromagnetic field strength FF tilde. Okay, so we won't be able to do a proper estimate because this actually involves a lot of messy integrals over all some distributions and things but extremely roughly the process that we're looking at is photon coming in hitting another photon and producing an axion so this photon here might be coming from like scaffolding off an electron so creamer cough process. The rate for this parametrically so we just have two photons axion if we're looking at the rate for a photon to convert to an axion then dimensionally if our core of our star isn't some temperature T core which we'll just call T here then this can only depend on the G squared from there and then we need a T core cubed in order to make it up and we'll have like plies and stuff going on. Okay, so then that means the production rate so the number of axions that we've produced per unit time per unit volume will be approximately the number density of photons times the rate at which they get converted to axions number density of photons is of order T cubed so this will be our G squared T to the 6 over like T squared or whatever and then that means that the energy that we lose from the star so the energy per unit time that we lose will be of order the volume of the core times G squared over 16 pi squared times our T core to the 7 because this is the rate of axion production times the energy that each axion carries away again there's only one scale the temperature certainly will be about T to the 7 so we see that you have a very high power of the temperature okay so that's all good it's going to be some big number because the sun is big and the sun is hot but what's it, but G is quite small here G is some 1 over 10 to the 10GV or something like that the thing models that we might care about so how can this compete with all the rest of the energy that the sun is losing so parameters that we've got are for the sun the temperature of the core is around KV and the radius of the core is about 0.5 light seconds okay how much energy is it losing to other means fairly obviously the main mechanism by which the sun loses energy is just emitting photons from its surface so you have black body photons being emitted from the surface and streaming up in space so the power that the sun is losing like that is just set by the Stefan Boltzmann law so the power lost from the surface is of order of the A of the surface times the temperature of the surface to the 4 times the Stefan Boltzmann constant which is in natural units pi squared over 60 because that's what it is the T surface is around 6000 Kelvin for the sun so Kelvin is 10 sorry, that's right so so 10,000 Kelvin is around an EV so this is somewhere in the regime of EV which makes sense because it's emitting optical photons so it's much much colder than the core and the radius of the sun is around 5 seconds so we can compare the energy loss from the core due to axions to the energy that the sun is losing all the time just through photons at its surface so the luminosity in axions over the luminosity at the surface goes like putting all these things together we get that it's 4 times 10 to the 9 g squared seconds times t quarter to the 3 just doing all the algebra and putting in units this comes out to 10 so g divided by 10 to the minus 8 gv inverse all squared so actually that was just putting in all the numbers we had earlier is actually around 10 times this because we were extremely cavalier with all of our approximations here but it gives you the right kind of order of magnitude so it tells you that for a coupling to photons which is somewhere in the regime of 10 to the 8 1 over 10 to the 8 gv then you get a luminosity in axions which is comparable to the luminosity in photons now that would fairly obviously be bad because we have very good models of the sun and we can check that it's behaving as it should it's emitting the neutrinos that it should we detect those the it's wobbling around as it should which is helioseismology and all of that agrees with there not being this extra energy loss channel so measurements so neutrinos and helioseismology which is like I said just looking at the wobbles imply that the luminosity in new stuff over the luminosity in photons should be less than of order sort of 10% which is somewhere around the neutrino luminosity otherwise things definitely go bad you can push it a bit further below and then it's probably still ruled out but there are a few lingering discrepancies of solar models and neutrinos don't quite agree with the helioseismology so it's a bit hard to say but anyway it certainly can't be bigger than about that otherwise everything goes bad it actually goes the other way it goes faster because what's happening in the sun is that you have all of the gravitational so all of the stuff is wanting to collapse the center and what's stopping it so you have all the gravitational force which is making it want to go inwards and what's stopping it is thermal pressure from the middle so you have all of the kinetic energy of these things moving around and the kinetic energy of these things which is maintained by fusion reactions giving you energy stops it from collapsing if you take energy out then it's not able to do that as well so it can't resist the gravity as much so it contracts a bit but this is the counterintuitive fact with gravitational systems when you lose energy it contracts and everything speeds up and gets hotter and in a temperature sense so what actually happens is it runs faster the center becomes a bit hotter and it goes through things quicker than it would that would be extremely drastic if you had something like that of course the limits that we have it can only be a few percent effect but yes so it's often called stellar cooling so people often call this stellar cooling which is a misnomer the core heats up because of the gravitational contraction which it can no longer resist as well so that's the effect ok so what does this mean for constraints on axions so remember from last time that we had in a QCD axion model say we had the coupling to photons was somewhere around 10 to the minus 3 of the 1 over FA in untuned-dish models so this these things would be so solar sun would imply that GA gamma gamma less than of order 1 over 10 to the 9 PV so that would imply that FA is greater than of order 10 to the 6 PV which corresponds to so remember that previously axion mass range that was misalignment was FA of order 10 to the 11 GV and axion masses which was somewhere around 10 to the minus 5 EV so here we're a few we're five orders of magnitude heavier axions and much more strongly coupled so this is telling us one thing that the QCD axion parameter space that our directitation experiments are searching for was safe as we saw in the plot last time but it's telling us that we can actually place a useful constraint on axions of it's also telling us that the QCD axion you'd be at a mass which was still lower than the temperature of the sun so it all makes sense, the axion is light enough that you could actually omit it okay so that's one thing but this is not so much an experiment this is just astrophysical observations though it is informed by Helios seismology and neutrino experiments which give us very precise models of the sun you can also try and do a direct detection of these things and see if it's better than just letting them stream by so the sun is a missing energy thing argument here so we have that these things are missing in the sun and just go out but we can also see if you can produce them in the sun and then make some detector on earth like ADMX but different and see if we can see the results of them actually hitting us so can this is this useful and can it let's do better than the naive energy loss arguments so it's certainly easy but it's a tunic so effectively if we have so our GA gamma gamma is our sort of numbers times 1 over F so it's like 1 over FA times some number which you get just from QCD stuff so if you just had our AGG coupling you get it from that and then if you have some contribution from the anomaly so you have you have fermion axion fermion fermion fermion fermion on the loop so if you have charge fermions which also carry PQ charge then you get contributions from the anomaly plus some some anomaly contribution and this can get contributions from high scale fermions as well which is coming in so Giovanni has a very nice paper where they do this calculation get this number appropriately and for different values for this anomaly coefficient which can take some integer value take some integer ratio then you get different numbers here but you'd need to tune it so you have some cancellation between these two things in order to get this to be very small so you have a naturalist value which is if these things don't cancel very much and then for various values of this you get smaller and smaller numbers but you need to go to more and more tuned ratios here in order to get very small couplings photons so ask Giovanni if you want to know more he is the expert on this alright so yeah so back to the question of can we do better than just letting these things escape into space can we try and directly detect them on earth and the experiment that tries to do that is called cast so it's the CERN axion solar telescope I think so what it is is well like EMX what we are searching for is axion come in we want to convert it to a photon because that's the easiest thing so we want to make a big B field and we want to convert the axion to a photon in a big B field so what they have is they have a big tube which is actually part of an AAC magnet they have a big B field across it so B0 of order 10 Tesla this thing is about 10 meters long and the thing is about the area is quite small because it was LHC so this is about 10 centimetre squared at the end but it's a very big B field which is good so what you'll be searching for is extend this an axion comes in in the B field it converts to a photon the sun is emitting things around a KEV so we'll produce a KEV-ish photon which is an x-ray so we have some x-ray optics at the end we take these photons put them on to an x-ray detector and our x-ray detector will see it and go bing okay so that's the idea but what's the rate for this thing to actually happen so if the axion is very light then we can convert coherence photons and the probability for an axion to convert to a photon is dimensionally so it's going to depend on the coupling squared it's going to depend on the B field squared and the only other thing it's going to depend on is the length that we give it to do the conversion so that gives us length squared so then the other thing we need to know is how many axions are hitting us so given the calculation before axion flux number of axions per second per unit area is if we take g to be 10 to the 10 minus 10 gvimbus then this comes out as 4 times 10 to the 11 per centimeter squared per second from this kind of calculation done a bit more properly okay so how many x-rays do we get so that means x-ray flux gives us dn gamma by dt is our axion flux times our area times our probability that our axion converts to a photon so putting all those things together we get around 10 to the minus 4 per second for that coupling so we see that even if we're at so the limit from the sum was about 1 over 10 to the minus 9 sorry 1 over 10 to the 9 gv even for couplings which are an order of magnitude oh sorry this should have been squared of course even for couplings an order of magnitude smaller we can still get a rate which is like if you do it for a good few weeks you'd expect to see some x-ray photons whereas x-ray photons don't just randomly pop up so cast actually sets the best bounds on this axion photon photon coupling through doing this production in the sun and detection on earth thing the bound it actually sets is as you'd expect from this calculation ga gamma gamma less than the order of 0.7 times 10 to the minus 10 inverse gv and in the future there's going to be an even bigger better experiment called IACSO doing much the same kind of thing but just on a larger scale so IACSO which is the international axion observatory which is a bit of a stretch but anyway so this will be able to get to less than the order a few times 10 to the minus 12 gv inverse by basically using a larger area a bigger b field better x-ray collecting objects optics just scaling cast up but will give significantly the best bound on the axion photon coupling not in a regime where it's going to be ruling out dark matter that AGMX etc will be searching for because this remember implied that ga gamma gamma was somewhere of order 10 to the minus 1 over 10 to the 14 but it is an interesting thing at the higher mass range if you have other production stories and is also intrinsically interesting just because the QCD axion is not the only target other axion light particles are certainly something we are interested in and might want to find so this is a nice example of an experiment which is using both astrophysics and precision detection techniques on earth to put constraints on light particle models so I think now is question time but here you don't because the axion is emitted relativistically so that's the point because the energy of the axion here is of order kev and down here the mass of the axion is say well we are looking at 10 to the minus 5 EV so the gamma factor for the axion is so this is 10 to the 3 10 to the 5 is around 10 to the 8 so it looks if we plot an omega k thing like it's almost on the light cone it's like somewhere here and we want to convert to a photon we had a problem before because the axion was not relativistic here it's super relativistic so it's just fine exactly why we can use a big thing and still get coherent conversion so it's easier to see relativistic axions because of that anyone cares about or do they drop off at some point so it depends what you mean by care about for the QCD axion pretty much yes but towards the higher end of the mass range so here's our MA here's our GA gamma gamma this is our 10 to the 10 gv also the bounds are like that now we can work out where this wraps out because what we want is that so the momentum we've got that our omega squared is equal to k squared plus m squared so we've got our k squared is equal to omega squared minus m squared is approximately equal to omega squared so our k approximately equal to omega 1 minus a half m squared over omega squared k axion omega axion for our photon we've just got that k photon is equal to omega photon so if we look at these things we want that k gamma minus k axion times length is smaller than 1 in order not to have destructive interference we want them to still be in phase throughout the whole length okay so this if they're at the same energy which they are because the conversion doesn't give it energy it's a static B field so this tells us that this is of order our half m axion squared over omega axion times length okay m axion we can take to be we want to solve this for m axion so we want to take that so m axion squared is of order omega axion over length kev over say 10 meter so doing this out sorry this is getting a bit silly so if we do that so we've got this is 10 to the 3 over so 10 meter is 10 times 10 to the 6 microns and a micron is about an inverse EV so this comes out to so we've got 10 to the 3 minus 10 to the 7 so this is about 10 to the minus 4 EV squared this comes out to about 10 to the minus 2 EV all squared so if the axion mass is greater than about 10 to the minus 2 EV then we'll have a problem we'll no longer get coherent conversion so what you'll see here is that this isn't exactly right because there's a different energy distribution etc but somewhere around a bit less than an EV the cast line goes up what they do then is they can actually fix that by introducing some gas into the system to give the photon a different dispersion relation so if you modify the photon dispersion you can make momentum match up again so there's some complicated sequence of bumpy stuff where they've done some gas tricks to make the photon on resonance with the axion again momentum wise but those crap out at some stage as well and then you go right up so around like somewhere around 10 to the minus 2 EV up to some small fraction of an EV and then it doesn't work anymore because you lose the momentum match any other questions? okay let's take it up again