Thank you. It's on now, right? Great. So, I was trying to reproduce roughly where we left it off yesterday afternoon. So, we talked about measuring the time-dependent CPA symmetry in B2JP-psi-k short, which is sort of this third kind of CP violation that cannot be absorbed entirely in BB by mixing, but it's also not something that is attributed to the decay amplitudes, but it's really something that is connected to this phase difference between a neutral meson, say, B0 decaying to some final state and first oscillating into its antiparticle B0 by and then decaying through another pass. And we sort of wrote down or derived or I encourage you to derive the expression for the time-dependent CPA symmetry. So there's a term that goes like sine of the mass difference times time and cosine delta mt. And so, of course, this lambda parameter depends on the final state. So there's a piece which describes BB by oscillations and there is a part which connects to the weak phases in the decay to the final state. And B2JP-psi-k short is a particularly nice example because we said that in this case lambda psi-k short is to a very good approximation just given by e to the minus 2i beta, where beta is one of the phases of the, sorry, one of the angles of the standard unitary triangle over here. I'm wrong, that's all for, sorry. And so the case is where interpreting the data is theoretically clean. So we know empirically we said that q over p, the magnitude of q over p deviating where not from one relates to CP violation in mixing. And we know experimentally that in the K-on system that's a 2 times 10 to the minus 3 effect in the B system. It's much smaller in the standard model, but we know experimentally that even in the presence of new physics the current constraints are at the percent level. And so if the magnitude of lambda is equal to 1, then this time-dependent CPA symmetry becomes much simpler, this second term vanishes. And if moreover you can convince yourself that the decay amplitudes are such that terms with one weak phase dominate. So in the case of B to J psi-k short, that's B to C, C by S, and we went through these 3 and ping-win or so-called 3 and ping-win diagram contributions, which is a little bit schematic, but it's, I think it's good enough for our purposes here. Then you see that this first term, which goes like the Kobi-Bo angle squared, this term goes like the Kobi-Bo angle to the 4s. So, and then also these matrix elements, because it starts out as a 3 plus some ping-win matrix elements, whereas this only has loop contributions. So one expects, in addition to the CKM enhancement of the first term, also the hadronic matrix element of the first term to be bigger than the second term. And sort of this is the, so if we neglect this second term, which is expected to be good to sort of at the one degree level, sorry, let's just say one percent level, then sort of this is one of the cleanest examples where the decay is dominated by one weak phase, and therefore this time-dependent CPA symmetry measures some complex phases in the, in the Lagrangian, which in the standard model just relate to the phase in the CKM matrix. But for example, if you had some new physics, so new physics could easily contribute to BB biomixing in a way that the magnitude of Q over P could stay very near unity, but its phase could easily differ from the standard model prediction, which just comes from the box diagrams for BB biomixing. And so experimentally the measurement of, of, of sine 2 beta now, and this is done by the E plus E minus B factories, and recently by LHCB, something like 0.68 plus minus 0.02, and I just want to put this up on the blackboard to illustrate that this really has become a very precise measurement. And it's kind of interesting because I said in the first lecture that both LHCB and the Japanese B factory will have sort of almost two orders of magnitude more data than what the current analysis are based on. So these kind of theoretical questions of what the hadronic uncertainties related to QCD processes are about to become important, and they will be important on the next five to a new time scale, because for the anticipated LHC upgrade and belt two statistics, we can no longer wave our hands that we know that these terms are small with that there needs to be some quantitative estimate of that. And if, and in fact in the literature over the last couple of years there are more and more, there are a number of papers with different approaches and using different theoretical methods to try to come up with strategies how to bound these kind of terms. So maybe I should just go permanently to the laptop why am I getting this. So I apologize. So in the usual fits, this is the measurement of sine to beta, and that is, except for the Khabibu angle itself, it's like one of the best known flavor parameters experimentally. There is a very important similar process. You can ask the question what is, you can just just take this decay so it was a clean, theoretically very clean measurement in the B sub D sector. What happens if you look at the same process in B sub S decay, so you want to look at some B sub S decay to see, so a big decay into C C by S final state. So the simplest example of that is J psi phi. So the phi is a meson which at the quark level is an SS bias state and J psi is still the C C bias state. And much of the same, actually the exact same analysis goes through. The difference is that Q over P in the B sub S case is of course a different thing because here we were talking about B sub D mixing. In this case whatever the time dependent CPA symmetry relates to Q over P now in the B sub S sector and some different A by over A relevant to this final state. But the whole theoretical understanding of it that to a very good approximation there is no CP violation in mixing, to a very good approximation there is no direct CP violation. The relative phases of these different terms are very different in the B sub S sector and in the B sub D sector and the reason is that, so B sub S mixing comes from, so instead of, so now you get VTB, VTS style, so instead of VTD which in the standard parameterization has a large phase now the mixing involves VTS which does not have a large phase in the standard model and therefore actually you can make a prediction that the time dependent CPA symmetry in this process in the B sub S sector has to be approximately 20 times smaller than in the B sub D sector. But in terms of the theoretical uncertainties it's exactly the same analysis and of course unfortunately the notation is somewhat different because instead of talking about two better people talk about five sub S, some people talk about minus two better sub S and they are all the same quantities it's really just the same kind of phase difference between the direct decay and mixing and decay and this is now measured by LHCB to be something like, so the central value as expected is much smaller than there and within statistical uncertainties there is no evidence yet for this type of CP violation being non-zero in the B sub S system but the interesting thing is that the uncertainties just from run one at the LHC have become comparable to what the B factor is could measure so let me see if I did not have this problem the last several days. Sorry I thought that I put a plot in here but I did not so never mind. So that's one class of measurements which will certainly continue both in the B sub D and the B sub S systems. We know that the theoretical uncertainties is at least a factor of few times smaller than the current measurements but if you are thinking forward sort of ten years in the future these measurements could become just by scaling with statistics and order of magnitude more precise and that level there's certainly a lot to do for a series to really be able to utilize the experimental sensitivity that's going to be achieved and in some sense there are lots of ways to recycling the same ideas and by doing different measurements which are sensitive to different new physics so here the point was that these are three level decays so new physics in these measurements could enter BB by mixing but it's exceedingly unlikely to enter in the decay amplitudes because these are three level decays in the standard model the CKM elements are not particularly small so it's very unlikely for a new physics to be able to compete with the standard model description of the decays themselves they could affect the mixing. Now one can do similar kind of measurements also in final states where these decays themselves A and A bar are dominated by penguin diagrams by loop processes and the goal of that is to try to search for new physics now in the decay amplitude so if I if I look at some final state which is which is dominantly coming from a B meson by some loop process then in that CPA symmetry measurement Q over P would be the same as it is in B to J psi K short but if I compare this result with some other result where now new physics could enter the decay amplitudes then I get some interesting constraints on possible new physics contributions and again the simplest example and lots of different cases is B sub D goes to phi K short so I just replaced the J psi by a phi and so the quirk level process here is B to S S by S right and then the K short just inherits the spectator quirk from the B meson and you cannot do a three level diagram for this process so again I'm kind of a little bit loosely speaking but now the dominant contributions to these decays are coming from penguin diagrams so you again I have up chime and top in these loops with the W so this can make a so this is the deep work so this can make a phi K short final state and the analysis is exactly the same as here except that I don't have the tree contribution so and of course this penguin amplitudes for this process are totally different the matrix elements are different than they are here but I'm nevertheless going to use the same notation just because because that's what people do in the literature and because we are running out of letters unfortunately that's kind of a problem in the field so I can still use CKM uniterity to write VTB times VTS star which comes with PT as minus VC BVC S time minus VU BVU S time so I'm going to end up with basically the exact same expression let me put a prime on this just so not to confuse you so these are different piece than it they are over there but the structure is exactly the same and again I can still tell you the same story that the first term is is of whether the cobibu angle squared in absolute value this is of where the sinus tetacobibu to the force now in that case I could argue that the first term had a larger matrix element because it had a tree contribution and loop contributions now both of these come only from some loop level process so the naive expectation would be that this matrix element which is not the same as that guy but it should be comparable there shouldn't be a large hierarchy between between the mate between between this matrix element and that matrix element and then again I find so this is the amplitude for this process whatever it's I shouldn't write equal it's really proportional to the matrix element of some terms that have this structure in the effective Hamiltonian and so this term is still expected to be dominated over the second term by a factor of 20 or so so one over all signs squared tetacobibu and so everything we said here so q over p of course it's exactly the same as in this case and now a by over a instead of the second term being of order 1% or less compared to the first term here I would expect this to be of order for a 45% or less so still everything we said over there is going to be valid over here so this time-dependent CPA symmetry should also measure sign to better and the only difference is that let me just write it this way that the corrections will go like the cubby boy angle squared so they are expected to be something like 45% instead of being at the sub percent level and and again the reason this is interesting is because what you are testing here so there's the same BB biomexing process but now these loop amplitudes could easily receive new physics contributions that would introduce new new phases in the decay amplitudes that could introduce a difference between these CPA symmetries and sign to better measured in B2J sci-k show it so this has been done for a number of final states and that's the heavy-flavor averaging groups average of all the different measurements so B2 5k 0 so it's written as 5k 0 because people do this or so experiment let's do this both for 5k show it and 5k long and that's averaged and that's what's called 5k 0 and you see that this very narrow band that's the sign to better measurement from J psi k show it and in many modes which in the standard model I dominated by these kind of penguin amplitudes you see that the results are completely consistent with all of these measurements measuring the same CPA symmetry that is the same phase difference between these different kind between the two different ways of getting from an initial B0 a B0 by to this final state and that gives you constraints on actually both on I mean the overall consistency gives you constraints on new physics in BB biomexing and the relative spread of these measurements constrain new physics contribution in this type of penguin decays any questions excuse me and sort of the possibilities in in in sort of don't want to say playing these games because people I mean experimentally spends years of their lives doing these measurements but there are really lots of final states where you can use the same method to get interesting information about short distance physics so maybe I just want to say a few words that if you could do a measurement which is not B2 C C by S or B2 SS by S but if you take a process which is B2 which is dominated by B2 U U by D so then you would get a phase of VUB VUD star and this could be achieved for example if you studied the time-dependent rate of B sub D goes to pi plus pi minus where rho plus rho minus now then these measure these measurements would now pick up both the phase from Q over P in the standard convention this phase the dominant phase here the dominant weak phase here so in the standard convention I hate being convention dependent but it's a it's it's simple to think about it that way that the weak phase of this is very small in the in the in the in the convention that everyone uses if you have some process which is dominated by B2 U decay then in addition in addition to the mixing phase you would pick up the phase of VUB so this was the side which is VUB VUD star divided by VCB VCD star it's complex conjugate I can never remember which but up to assign the phase of VUB in the standard convention is gamma so if you do the same measurements in these decays then you would get a measurement of actually better which is connected to the phase of Q over P plus gamma which is connected to the phase of A by over A and this is kind of opening a Pandora's box because what happens for these processes is that you have B2 U there's your D by a quag so U by a D so this would be a Pi plus this would be a Pi minus and the ping Vietnam diagrams which is B2 D I think that is probably a 1% chance that I will get this right so this is a Pi minus that's a Pi plus so the subtlety in this case is that the the tree diagram now comes with a phase which is VUB VUD and the weather of magnitude of that is lambda KB4 cubed the penguin diagrams go like VTB VTD and the weather of magnitude of that is again the KB4 angle cubed so the big difference between these kind of processes and these ones is that there is no hierarchy in the CKM elements between the contributing amplitudes so as a result there was so you can no longer say that this is a clean measurement of something because now you have several contributions to the amplitude which have different weak phases and there is really a huge amount of literature using isospin symmetry and other theoretical methods how you can nevertheless disentangle the strong interaction physics from the weak interactions here so if any of you have seen that or want to ask me questions about it then we should talk about it later I don't think I want to say more about it that there is sort of a smooth transition between theoretically very clean processes well at least in the last 10-20 years there was very little for a series to do because you know from first principles that the theoretical uncertainties were negligible compared to the experimental reach to measurements where even 10-15 years ago people knew that you have to work harder to understand sufficiently I come up with some tricks to understand processes which at first sight seem like untractable because there is no hierarchy between the contributing amplitudes another case which is very important and I won't have time to explain it but I want to mention it so there's something special about direct measurements of the CKM angle gamma so if you look at the definition of gamma it is the only angle of the unitary triangle which does not involve the top quack and as a result of that so if you look at the definition of what better is there is whatever will be CKM elements involving the top quack enter here and so if you want to measure either alpha or better you can sort of immediately see from the definitions of those angles that the top quack is somehow directly involved and that tells you that you can in any process involving B mesons or K mesons or D mesons you can only access that through loop diagrams which can in principle be affected by new physics on the contrary gamma only depends on quacks which can be on shell in a BDK and in fact the ways to measure gamma entirely from three level processes and that's going to be important for what I want to say next that so so so getting a three level determination of the CKM matrix is going to play a special role in in in in constraining new physics and I'll I'll say a few more words about that in a minute so I just want to give you the idea that what happens here is that so so so the measurements of gamma contrary to those cases come primarily for from charged B decays so you're looking at B plus or B minus decays to particular final states and it's really a different so yeah so hmm so let me just draw things for a moment so these are W's so this is like just a forefoot usual for me interaction so this is B minus D0 by K minus so that the basic idea is that so if you look at for example B minus decays then there's a B to C transition where you and then you can end up with a final state that is D0 K minus and you can take the suppressed B to U transition and you can end up with a B minus decaying to D0 by and K minus now these are different final states so what I'm talking about but nevertheless they can interfere because D0 and D0 by also mix just like K0 and K0 by gives you K short and K long where B0 and B0 by gives you B light and B heavy those are just labels of the mass eigenstates being mixtures of the CP eigenstates so similarly here the the D0 and the D0 by mix with one another and therefore these two different processes can in fact end up with the same final states giving in interference which between these two diagrams which will allow you to measure the CKM angle gamma and I am happy to take any questions but I wasn't going to say more about it because the details are really kind of quite complex so the punchline is that there are several methods in this domain as well to control to have the theoretical uncertainties under very good control but it's kind of different than over there but it's very important that you can measure gamma cleanly from three level processes without any involvement of neutral meson mixing any questions so why am I telling you all this it's a good question the reason is that so you go back to the first plot I had so if you look at all these constraints on the CKM matrix so we said that you can measure a gamma which is this gray wedge from three level processes you can determine VUB the magnitude of the B2U CKM element from three level processes that's done from semi-laptonic decays but if you look at everything else on this plot the measurement of sine 2 beta epsilon k the CP violation parameter in KK biomexing where the measurement of alpha from this type of processes so all the other things besides gamma and VUB come from loop level processes where if you have TV scale new physics that could certainly influence the result of these measurements so if you're asking how well do these measurements constrain new physics then the fit is really not this fit that you should be performing this fit assumes the standard model and it assumes in particular that all loop diagrams as well as three diagrams are dominated by just the standard model contributions that is integrating out the top quack and the W boson the Z boson etc so so one of the interesting questions you can ask which is actually so in a lot of new physics models it will remain true that what shall I write lots of models have the properties that the three by three CKM matrix remains unitary and three level processes meaning a three level in the standard model are dominated by the standard model contributions themselves and that's just from the sort of the well justified conventional wisdom that you expect new physics not to be a new physics to be at a heavier mass scale than the weak scale and therefore it would be new much much easier for new physics to compete in loop processes than in three level processes and one very simple way to parametrize the effect of such new physics on these measurements is just to assume that M12 so whatever in here in each meson system so M12 separately for the K, D, B sub S and B sub D system I could parametrize it as M12 for that particular system in the standard model so I should put some other index on it but we'll get dizzy of indices and so this is the standard model piece and the new physics piece will have a magnitude that I will call H and it has a phase which I will call sigma so I have been completely general we saw in the first lecture that when you describe neutral meson mixing to determine the mass difference and the width difference and to determine the phase of Q over P which affects all these CP violation measurements what really matters is this off diagonal element M12 if you remember that relied just on the assumption that M12 is much less than gamma 1 2 sorry the other way around and certainly you would expect new physics to maintain this because new physics is again unlikely to contribute to gamma 1 2 so so one of the interesting things where you can say fairly general things about a large class of models is that if I just assume that new physics essentially comes as an effective modification of BB by B sub S B sub S by DD by KK by mixing and I can ask the question how well can I constrain that so what happens in that case is that the measurements of gamma would still measure essentially the phase in this combination of CKM elements the magnitudes of the CKM elements would be unaffected because they come mostly from semi-latonic decays and however BB by mixing which gives this constraint with these rings and epsilon K and sine 2 better and also they can all be affected so if you do that then you end up with a much more complicated fit not surprisingly there are new parameters in the fit this H and sigma parameters and therefore you expect the constraints to be less good because you have more parameters to fit and the result of that exercise so this is a some plots from the CKM fitter group is that for example if you ask how big can this new physics contribution in B sub D mixing B then sort of just ignore the plot on the left that's just for historical interest that you know 10 years ago new physics could be bigger than the standard model in BB by mixing right now the constraints are that new physics that so the vertical axis is the phase of new physics this parameter sigma the horizontal axis is the magnitude of new physics in BB by mixing that is H and the constraints are kind of at the 20 30 percent level and not surprisingly the value of sigma effects how good the constraints are on on on on the magnitude itself and you can directly translate these kind of fit results to the possible scale of new physics so if you assume if you assume that new physics just contributes with an operator that I can parameterize as some Wilson coefficient over some scale of new physics time to be by left gamma mu the left so this is the same as the standard model operator responsible for BB by mixing but it has a some arbitrary coefficient coming from new physics and it is suppressed by new physics some new physics scale rather than the weak scale then you can translate these constraints on H to the possible combination of this coupling over the scale of new physics square and again depending on whether the new physics has the same kind of CKM suppressions and loop suppressions you find that the result is at the several TV scale and even in the case even if you assume that new physics has the same CKM suppression and loop suppression as the standard model you still find that sort of you're in the ballpark of for example so so one way in for example in the MSSM how BB by mixing could be affected is like square gluino box diagrams and the mass scale that you are indirectly probing here is at the TV scale which is kind of comparable to the gluino mass is probed by the LHC yes yes so I am paging on the wrong one sorry yes so robot and etobai refer to a modification of the usual Wolfenstein parametrization well you don't have it anyway so the usual Wolfenstein parametrization is this expansion in the angle to sell the order and robot and etobai is a modification of the parametrization such that it maintains unitarity exactly so it's very close to row and data the difference is starts at where the lambda car be able to the force and higher orders and it's used in these fits by both the CKM fitter and the ut fit group which are the two big groups performing this CKM fits because then you don't have to worry about approximations of the CKM matrix so you just get another parametrization which is exactly unitary and and that's nicer because you don't need to think about whether the neglected terms are really negligible or not so does any other questions so this is approximately how far I wanted to get yesterday and what I wanted to so I'm going to sort of slightly switch topics and use the remainder of today to tell you about some basically about heavy-quark effective theory and some ways how to address some of the hadronic physics which is relevant for understanding this phenomena and yeah so these are the different sine to beta measurements right so this may be mission impossible but I would like to get so recently there is quite some excitement about so let me just tell you about what that plot is and then maybe in half an hour later we'll get to understand why those predictions are kind of fairly reliable which is what I would like to do so about three years ago the Boboyer collaboration reported some interesting measurements of this quantity so I is defined as the decay rate of B to D tau nu divided by the average of B to D and either an electron or a mu one so let me just call it I'm going to call it like so it's the average of electron and mu on neutrino and similarly our star is the same quantity where D star tau nu divided by again the average of the electron the mu on and the neutrino and so this generated quite some excitement because the Boboyer measurement was the black ellipse which was reported to be more than 3 sigma away from the standard model which is roughly that green box and it was in less than a month ago that the Bell experiment published their measurement at a conference which is the blue ellipse and at the same time LHCB could also do one of these measurements so now you see that you end up for these observables with a funny situation that there seems to be a kind of significant tension with the standard model again I don't want to jump to conclusions because obviously these measurements will be done a lot better but it's kind of intriguing and historically the interest in these decays were motivated by the fact that for example if you are in the 2x doublet model the sort of the type 2x doublet model which you have in the MSSM for example then you could have contributions to the numerator that kind of I enhanced by tangent beta square that tangent beta is the ratio okay well I didn't is anyone talking about what I'm talking about or should I assume that all of you know about the MSSM type so so in the MSSM there has to be 2x doublets tangent beta is the ratio between the two webs that are motivated regions of parameter space where tangent beta is large and the final state with the tau mass could be enhanced by charged Higgs contributions so sort of just these diagrams well of course now this coupling would be proportional to the tau mass and this coupling could be proportional to the bottom mass or the bottom you cover so there is a term which goes like mb m tau over the charged Higgs mass square and it has this possible enhancement by tangent beta and historically people have talked about tangent beta possibly being quite large 20 30 or more so even if you know even if the charged Higgs mass is at several hundred gv this was an interesting scenario that people have talked about for a long time that could manifest itself by giving different results than the standard model because in the standard model you only have this diagram with the W exchange and of course the W couples universally to the tau and the electron and the mu one so so that's just the motivation of one of the things that that hopefully we can understand how the theoretical predictions are made and and and should you believe any of this you know is that is that green box as big as it's drawn in that plot or is it ten times larger and to be honest I think that the other reason I wanted to tell you a little bit about heavy quack symmetry is because really it's one of the sort of the nicest examples well these effective theory ideas both work in practice and and they are useful experimentally and also there is sort of a hand waving understanding of what it is that simplifies the QCD dynamics and it's not even cheating when one waves their hands and that's kind of a unusual case so so you want to know that if you plot the strong coupling alpha S of mu as a function of the scale mu then the coupling is strong at low energies or long distances and and and strong interactions become better but even high energy or short distances and you could ask the question in what cases do we know at least if I ask myself the question in principle what are the ways to make sort of rigorous predictions from from from QCD then of course there is the in some ways the simplest example well where perturbation theory holds and that's a lot of things that Michelangelo talked about that if you are calculating some hundred GV or TV scale process of some decay of some particle then you can just go to higher weather in perturbation theory and that calculate higher and higher loops in the perturbation expansion and because the coupling is small it should give you a good approximation to the result of course you should remember that all of these expansions are asymptotic and actually that's an important story that is in the background for example when Michelangelo talked about the relationship between the top quark pole mass and the MS biomass and he mentioned that for the big quark mass once you go to three loop and four loop then the perturbation series doesn't appear to converge anymore so you know that all asymptotic series at some point start to blow up and for the top quark that doesn't appear to be an issue yet at four loop so that even in perturbation theory one should be aware of subtleties that this is none of these are convergent expansions they are at best asymptotic and so one of the other cases that we saw is using symmetries and I would put this example of B to J psi K show it that we spent a lot of time on into this example into this category because what really happens is that if you what we were using and I'm sorry I'm just using I can't write J psi I always write psi I'm sorry so so in some sense all the simplifications that we have talked about just come from the simple from the CP invariance of QCD that we could derive that the matrix element of B zero to Psi K short is the same as B by zero to Psi K short with some quantifiable accuracy that goes like on the C square times what we called penguin over three but essentially what we have used is some symmetry of QCD namely CP invariance of QCD and there is this other example of when you can use effective theories like heavy quirk effective theory which is which is which is which is some limit of QCD and we'll see that if you take the limit that MB and MC I treated as much greater than lambda QCD then there will be some new symmetries of strong interactions that emerge that can be used to understand some matrix elements in a model independent manner that will be relevant for that story it's also relevant for the determination of for example the CKM element VCB that VCB can be determined model independently because you can use HQET to derive relations for the decay rates and you can quantify the uncertainties in some cases to be suppressed by two powers of the hedronic scale over the heavy quirk mass and there's another important point that I so in each of these cases except for in perturbation theory in this in some sense the higher the term sort of your uncertainty goes like all for us to some power and you need to worry whether the scale is really the nominal scale of some of the highest energy scale that occurs in a process or somehow the physics gives rise to lower scales that could make this expansion worse but in all of these cases the name of the game is quite often how you estimate the next order term that you have not computed what is the theoretical uncertainty in any of these predictions and I think it's very important to remember that in many that that you know having an expansion in some parameter that is small is almost never enough there's always some experimental guidance that's needed to see how well each of these expansions work and and the reason is because we always kind of just guess what is the scale of the suppression of the next term and sort of a nice example is that if you look at the pion decay constant that's 140 MeV the row mass is 770 MeV and if I take mk square over ms so the k on mass over the strange quark mass that's also of where the lambda QCD but numerically it's 2 GeV so and okay so this is you know that the k on is so the the pseudo scalar octets are pseudo goldstone bosons right of the spontaneously broken chiral symmetry and whatever mk-square formally goes like weather ms times lambda QCD and that's why mk-square over ms is formally of whether lambda QCD which you would expect to be a few hundred MeV but sometimes you can get large enhancements and unless you understand the physics really to one higher order than we talk about it's it's very often only experimental data which can tell you whether your expansion is better behaved or whether there is some fluke enhancement of something that we would really have not expected from just the power counting of the expansion any questions so so heavy quark symmetry or so if you have a B meson so we are trying to use the argument whatever the the separation of scales that the B quark mass is much greater than the typical scale of QCD interactions so a B meson if you wish has a characteristic size which is one of the lambda QCD and that's the scale on which strong interactions take place at the same time the Compton wavelengths of the B quark itself is of where the one over mb so this physical picture that there's this heavy B quark which sort of sits in this B meson is really a good approximation even though they sort of very complicated dynamics that takes place because of confinement so here you have QQ by pearls you have the spectator quark you have gluons and really don't understand in detail the dynamics of what's happening but you know that the scale and the characteristic wavelengths of the exchange gluons is very different than the scale of the heavy quark mass and so the physical picture is really very much similar to atomic physics that you can think of the B quark in the B meson rest frame the B quark has to be almost at rest and it's really not perturbed in any significant way by these strong interactions between the light degrees of freedom that confine this to a B meson and there's a way to formalize all of this so one usually so it's convenient to introduce the four velocity of the B meson which is just B over M and so because of this physical picture and there's actually so formal ways to derive it that so there are two important predictions is that that the spin of the B quark interacting with the light degrees of freedom so this has to be a suppressed interaction and so if you think about the hydrogen atom there is an exact analogy to this that the hyperfine splitting which comes from the interactions of the nucleon spin with the electron spin in the hydrogen atom that's also a subleading interaction and the other so this is usually called spin symmetry because it tells you that at leading order whether the B quark spin in some arbitrary direction points upward points down does not change the dynamics of this system if you have more than one heavy quarks then I can imagine having some V current which changes this B quark to a charm quark some different mass which is still from the point of view of this system is just like a point like source of color of QCD interaction just like in the hydrogen atom the the nucleus is a static electric source this is a static color source so again at leading order if you change the heavy quark mass so for example you change mb to mc by having a weak interaction giving rise to as for example a semi-leptonic B decay to a B quark to C quark and the lepton and the neutrino then to a good approximation these light degrees of freedom are not affected by the change in the heavy quark mass in some sense the only thing that they care about is the velocity of this heavy quark and again sort of the the the the atomic physics analog of this is that isotopes have a very similar chemistry because I just put one more neutron in the so yeah so just look at the different isotope and the electron wave functions to a leading order are not affected by this and so so far I am mostly waving my hands but there is a way to to derive an effective theory that formalizes all of this and that actually becomes quite predictive for for example for semi-leptonic B decays it also allows you to understand the spectroscopy of these B mesons so so what happens in these systems is okay so so if you look at what is the so the total angular momentum of course of a headron is a conserved quantity right that and so that commutes with the Hamiltonian and in these systems you can write the total angular momentum a sort of the spin of the B quark plus the spin of the light degrees of freedom where the spin of the light degrees of freedom for me contain the spins of the quarks and it also contain a possible orbital angular momentum whatever and I argued here that the spin of the heavy quark is not going to affect the dynamics of this system so at leading order the spin of the heavy quark is also a conserved quantity and because of the total angular momentum is just the spin of the B quark plus the light degrees of freedom that also tells you that the light degrees of freedom in these systems have a total spin that is conserved and so this looks like a totally trivial set of statements but the result is that I can so you know you have you have I have an arbitrary I don't know what is the so at the quark model there is the B quark and the spectator quark but the qq by pairs the gluons there is orbit angular momentum so the spin of the light degrees of freedom can be a whole there is a whole tower of possibilities but for any value of the spin of the light degrees of freedom there will be two states coming from the B quark spin which is one half either adding or subtracting from that so so in the B system there will be the two lighter states the B and the B star so the B star is an angular momentum one state this is an angular momentum zero state and they will correspond to the spin of the light degrees of freedom being a half so you get a spin zero state and the spin one state and there's sort of a tower of such states well there are some levels which are split to two states because you either when you combine the light degrees of freedom with the B quark you can you can you can get one of two levels so the next one is usually called in the B system it's called so this is the B which is whatever its mass is 5.20 AGV this guy is at 5.32 then there is an excited state of a B which is a B1 5.72 so this is a spin one state there's a spin two state which is observed at 5.74 and the point is that this spacing which is the different light quark spins are of order lambda qcd but the level splitting in each of these doublets is of order lambda qcd square divided by the heavy quark mass and similarly here the exact value is not the same but the power accounting is such that these are several hundred MEV and these splitting are tens of MEV and you can immediately make several predictions because you can look at the same thing in the D system in the D system the ground state is whatever the D meson which is a pseudo scaler there is the D style which is the analog of the B style and the next doublet of states is the D one which is 2.42 AGV it will become in a clear in a moment why I'm writing all this down the D2 style is 2.46 so now this splitting just like there it's of order lambda qcd over m chime in this case so you see that for example one prediction from heavy quark symmetry will be that mb star minus mb divided by md star minus md should be the same as the ratio of the chime quark to the bottom quark mass and experimentally that seems to work quite well I mean here you have a mass difference of something like close to 50 MEV here you have a mass difference of 140 MEV and the ratio of of a bit of 50 MEV to 140 MEV is indeed in the ballpark of mc over mb and of course this relation has corrections which are now of order of the qcd scale over either m chime or m bottom so the prediction so you have an expansion in some at least in some formal limit small parameter lambda qcd over mb is something like 0.1 lambda qcd over m chime is not so small it's of order of 0.3 and this is what I mentioned before that one has to test empirically whether whether an expansion in a parameter like 0.3 does that make sense or does that not make sense and it seems that in these cases that works quite well so there is immediately some understanding of so there's essentially a big tower of these states and of course these heavier states decay with strong interaction but it's important that from the point of view of heavy quark symmetry there are these doublets of states with spins one unit apart that have strong interaction properties which are closely related to each other by heavy quark symmetry any questions so the way you can make all of this formal and this is really just the hand waving arguments at leading or at all that you can write down an effective theory which allows you to do actual calculations it also allows you to go to sub leading or at all and sort of heavy quark effect heavy quark effective theory was really the example that was then later sort of recycled for a soft collinear effective theory and it's the foundation of many of the factorization based methods in in heavy quark physics so I can write the so let's just take instead of the label B I will use a label capital Q for a heavy quark so I can write so again let me just say a few words about how heavy quark effective theory works or how what what the effective theory is so I can write the momentum of the heavy quark as mbv plus some residual momentum and the goal of this exercise is that I want to separate the physics that scales like this large parameter the heavy quark mass in the problem and somehow I want to remove that piece and be left with sort of the dynamics only depend on this residual momentum and the and parts of the physics which in the heavy quark limit is constant and and and does not scale with the heavy quark mass so I want to separate out the pieces which scale somehow with the heavy quark mass from the terms that do not because I know that soft qcd interactions will relate so the part of the physics which is non perturbative cannot scale with the heavy quark mass and I somehow want to get a handle on that so pq square is obviously you square this plus 2mq v dot k plus k square and if you look at what is the propagator of a heavy quark so it's just i will pq slash minus mq and you can write this as so I multiply and divide by pq slash plus mq that is mq v slash plus k plus mq right so in the denominator mq square disappears and therefore and in the numerator I have used pq slash plus mq and pq slash is just this so again to separate out the terms which are big and which are not big in the heavy quark limit this is so you look at what are the terms that go like mq so this is just one plus v slash over two times i over v dot k and I can neglect the terms which are subleading in the heavy quark limit and what you see is that instead of a usual quark propagator which has some dirac structure and depends on the heavy quark mass you end up with this heavy quark propagator which is now manifestly independent of the heavy quark mass mq has disappeared from the propagator of this heavy quark and similarly so one usually differs so this this let me go one step further if you ask what is the coupling of a gluon to a heavy quark then of course in full qcd this is g strong gamma mu times the generator lambda u over two and this one plus v slash over two it's it's a project on a projection operator so if I define p plus as one plus v slash over two then it has the property that p plus square is equal to p plus that's what project to us are and and sort of the the interpretation of this is that if you are in the b quark rest frame that then in the b quark rest frame one plus v slash over two is nothing else but a half times one plus gamma zero and and and and a half one plus gamma zero projects in the rest frame of a quark on the particle rather than the anti particle components of the four components pinoyer so so this propagator has so this one plus v slash over two has the interpretation of projecting out at leading order the particle rather than the anti particle components of the of the heavy quark field so because the propagator has this structure you can also show that so in in this coupling this interaction is surrounded by this heavy quark propagator that sometimes people use a double line notation to distinguish from a light quark propagator and you can show that p plus gamma mu p plus is the same as v mu times p plus so you just have to commute through the gamma mu with the v and that means that the gluon coupling to a heavy quark simplifies so the Feynman rules for that is just going to be i g slash v mu lambda a over two and now there's again something funny happening that there's no Dirac matrix in the coupling of a gluon to a heavy quark at leading order in the in this effective theory and this is just a manifestation of the fact that that the gluons sort of soft gluons are independent of the heavy quark spin what is the spin of this fermion does not affect at all the interaction with soft gluons and so one can make all of this very formal I mean from the QCD Lagrangian which is just i d slash minus m q q you can introduce new heavy quark fields which take the large momentum components out of this full QCD fields and you can write down the HQET Lagrangian and if and if you want there is like a whole formalism that is worked out to sub leading order but the physics is just what we said that you want to separate the large and small momentum components of what's happening in in in these interactions now how does that affect semi-laptonic decays and we care about semi-laptonic decays because of this current anomaly in in in one case and also because historically decays like b2d or b2d style just electron neutrino well very important for determining the ckm element vcb it's also a testing ground for this effective theory and so what happens it so i wanted to explain that this heavy quark limit gives very important simplifications for this semi-laptonic decays as well well again a priori it is some non-perturbative dynamics that determine the matrix elements for these transitions so let me just give you an example so if you look at b2d decay then the hadronic matrix element you are interested in is a d meson with some momentum and you have some current so you are interested in this matrix element right this is a this is the b2c semi-laptonic current the part of the current which involves the electron and the neutrino that does not have and that does not participate in strong interactions so in some sense it's trivial and you want to parameterize these matrix elements and it turns out that for this particular case where it's b2d it's only the vector part of the current which has a non-zero matrix element and in general that can be parametrized by some function f plus and the only thing which is Lorentz invariant so there's an initial momentum and the final momentum and you can whatever q square is always defined as p minus p prime square and that's the only Lorentz invariant quantity you can construct from p and p prime other than just p square and p prime square which are just the masses of these particles so there are going to be some functions which can depend on whatever they are allowed to depend on which in this case is only q square and the the sort of Lorentz invariance tells you that there are two possibilities it either depends on the p mu plus p prime mu or there could be another term which is f minus of q square times p mu minus p prime mu and you can write down sort of a similar similar parametrization of the matrix element for b2d star transition as well which is more complicated because in that case was the vector and the axial current contribute and the physics of these decays is that if you think about it so there's this b meson which has the beak work in the in the b meson rest frame there's the beak work and it's surrounded by this light degrees of freedom that sometimes people refer to as brown muck and there's this weak current which instantaneously changes the at a on a time scale much shorter than the characteristic time scale of qcd interactions changes the beak work to a charm quark and heavy quark symmetry gives you a number of relations between these form factors that describe these matrix elements so in b2d there are these two form factors in full qcd in b2d star lepton neutrino case the i'm not going to write it out explicitly the four form factors that parametrize the hadronic physics and i will think about it whether i come back to this tomorrow or not the point is that that these six functions you can show from heavy quark symmetry to be related to be given by just one universal function of q square so instead of six functions of q square there is only one function that that that all the hadronic physics only depends on one function instead of six function and that's directly comes from heavy quark symmetry because it's a consequence of the fact that the d and the d star from the point of view of heavy quark symmetry they are part of the same doublet and so whether you have a vector current or an axial current whether the final state or d or a d star in some sense there are symmetry relations between these matrix elements and that has been historically the one of the ways how the vcbckm element was determined and it is also crucial for understanding these predictions because so in in b2d star there are four form factors okay i should say one more thing that in the case of the electron or the mu one you can immediately see that this piece which goes like p minus p prime vanishes for a massless lepton right because if you are if you are if if if if if if p minus p prime is between the electron and the neutrino spinner then for a massless leptons that term vanishes so in fact the only difference between massive and massless leptons is that in the b2d case one of the form factors does not contribute for an electron or a mu one and similarly here for the d star for the electron and mu on final state when the lepton mass is negligible there are only three form factors that contribute but the point is that all of this is very very precisely measured by so so so so so experiments have done extremely precise measurements with the b2d star electron neutrino and the d electron neutrino form factors which by heavy quirk symmetry are related to the same function this is what people call the isguivice function and there are one over m corrections to this description that violate heavy quirk symmetry and it can be parametrized but these four form factors describe this full angular distribution of the decay to this final state and you see that the theoretical description and the data is completely consistent with one another so the point is that this way heavy quirk symmetry allows us so it gives us relations that the pieces that you need to know to make a prediction for the tau final state which is so in the b2d case you measure this form factor and you are not sensitive to this one but the form factor that is not measured is related by heavy quirk symmetry to the one that is measured and similarly in the d star case and as a result of that to make a prediction for these ratios the theoretical uncertainty is really quite small because in some sense the hadronic physics is measured by doing the measurements of the decay distributions of these terms in the denominator for an electron and the muon final state and the extra ingredient the extra extra form factor the extra hadronic matrix element that contributes to the numerators of these terms is related by heavy quirk symmetry to the things that that people have already measured so so that's really the basis for making theoretical predictions for these quantities and the fact and the result of that is actually people have gone through this exercise and put in the subleading terms which are somehow parametrized and also constrained from the data and that yield the predictions that this ratio for the d case is supposed to be near 0.3 and for the d star case it's supposed to be near 0.25 and even though I call these theoretical predictions to a large extent it's really just a function of measured objects already and you know where this will take us in the future I think that that's an open question that I certainly don't know it it it it appears like a very unusual place for a new physics to show up this is certainly not how people expected new physics to show up and if I had to guess it's premature to do any significant conclusions about this nevertheless it's kind of an intriguing thing that this is what the data says right now and and I just wanted to give you a sort of a glimpse of what the theoretical background is to to make these predictions I should say one more thing that this is also something which lattice qcd can do very precisely and also for the lattice qcd calculations actually this heavy quirk effective theory ideas are extremely useful because it also simplifies the lattice qcd calculations so I apologize this I realized was not very well organized but I'm happy to answer any questions you have


Hello and welcome to this session. In this session we will discuss a question which says that if a plus b iota is equal to 3 k minus iota whole square whole upon 9 a plus iota then claim that a square plus b square is equal to 9 a square plus 1 whole square whole upon 8 to 1 a square plus 1. Now before starting the solution of this question we should know a result. And that is if z is equal to x plus y a iota be a complex number then the conjugate of z is equal to x minus y a iota or you can write the conjugate of z is equal to x minus y a iota. This means that the conjugate of a complex number is found by changing the sign of the imaginary part. Now this result will work out as a key idea for solving out this question. And now we will start with the solution. Now this is given to us so given 3 a minus iota whole square whole upon 9 a plus iota. Now using this result which is given as a key idea the conjugate of a plus b iota is equal to the conjugate of 3 a minus iota whole square whole upon 9 a plus iota a plus b iota is equal to the conjugate of iota whole square whole upon the conjugate. Now using my minus b iota is equal to 3 a plus iota whole square 9 a minus iota. Now let us write this equation as equation number 1.  Now this implies a square minus b square is equal to minus 1 equals 0. Now where I will just b the whole into a minus b the whole which is equal to a square minus b square whole upon 9 a whole square minus iota square this will be equal to plus so it will be plus b square is equal to iota square is equal to minus 1 so it will be here minus 1 so it will be minus 1 here. Now this is b square is equal to 9 a square plus 1 whole square whole upon 2 a square plus 1. That is all for this session. Hope you all have enjoyed this session.