<TITLE: QCD and Hadron Structure
ACADEMIC DOMAIN: natural sciences
DISCIPLINE: physics
EVENT TYPE: lecture
FILE ID: ULEC090
NOTES: poor sound quality

RECORDING DURATION: 88 min 22 sec

RECORDING DATE: 23.10.2006

NUMBER OF PARTICIPANTS: 10

NUMBER OF SPEAKERS: 2

S1: NATIVE-SPEAKER STATUS: Swedish; ACADEMIC ROLE: senior staff; GENDER: male; AGE: 51-over

S2: NATIVE-SPEAKER STATUS: Finnish; ACADEMIC ROLE: masters student; GENDER: male; AGE: 24-30

SU: unidentified speaker>


<S1> <WRITES ON BLACKBOARD THROUGHOUT THE PRESENTATION> since last time i'm sorry i was away (xx) this week of course in principle then the lectures , is (xx) last week , so let me just remind you er er last time i went through er rather quickly a a a (xx) serial issue which is this question of bound states (xx) area and er in some sense . <SIGH> this is <SIGH> , quite essential from the point of view of of of of the topic we are discussing because after all we do want to understand hadrons and QCD hadrons are bound states of QCD so er er one might spend more <SIGH> time on this but again er er i have thought i would give a course which covers many things so we , will not go that much into this but the the principles at least i hope that you would recall and those where that <COUGH> they appear as poles <WRITING ON BLACKBOARD, P:06> in the S-matrix scattering amplitudes , and i will not be very accurate here @@ er , and where how do you find these poles the poles occur at the mass of of of the of the bound state you don't find them in any a a a a finite number or some in the finite number of diagrams this is what makes this so different in a sense from ordinary calculations of er scattering amplitudes or so you can sort of understand (core it) has to be like that because a bound state is something which stays bound for a long time stable stage for ever er and so it must be able to ext- er interact in infinite number of (times) . so this is <WRITING ON BLACKBOARD, P:07> so when we are first starting to think that (xx) infinite sum so now , which sum i mean i think we can classify all this time and so on but we cannot easily calculate , so w- we must somehow find some way to identify which diagrams are more important than others and as usual er , er the main tool we have at our hands is the coupling constant so if we talk about Q-E-D we have a fine structure constant (xx) 137 which is small , and then , we can access essentially non-relativistic bound states , which is what we would do for atoms positron and so on , er it's not good enough fo- for hadrons which we know are relativistic but on the other hand we also know that the nature for some reason which we want to understand actually allows us to think of hadrons as non-relativistic bound states as in the (xx) , but keeping to now what we can really do er when we have this situation with non-relativistic bound states and we take them at rest so that th- th- the constituents move only very slowly we can of course think of a bound state which is in a frame where it's moving fast and that's an interesting problem not often looked at but er er tha- that's just in (xx) , the bound state (xx) so you have typically then er diagrams like this you have say an E plus and minus (positronium) and you note by looking at these diagrams that in this situation diagrams that you need to sum which are the most more important than the others are just as simple er ladder , some ladder diagrams , and essentially the the reason is that even though this diagram is (xx) alpha this is (xx) alpha squared alpha cubed and so on so alpha being very small i would think that this diagram is small i mean usually it is however in a situation where you have a a nearly (xx) propagation here so , the the the constituents are almost on the mass shell then you have these propagators here which are have a pole and they are very close to the pole sufficiently close the pole to to er give back er an enhancement which which compensates for the small alpha in the same (field) then er repeats its (xx) alpha , you get couple more propagators and and so all of these are actually in the same order and the sum then diverges to geometric sum and at the point of diverges you can look for pole , so that's how it's sort of can identify <COUGH> er the bound state and in fact this is what (this is what) gives us the schrödinger equation this is a non-relativistic er approximation and you know from , early course in quantum mechanics and that's er the , basic starting point , and then one can ma- make in put in corrections which are (xx) alpha so which are really small corrections an- and they come then through various you know you can have various (xx) photon and (xx) many corrections and it's a whole , whole art in itself to sort of make the the systematic corrections and that has to do with the with the dyson-schwinger equation (xx) which allows you to organise all your corrections in a (common) way , so that is how er er er one can proceed when we have a small coupling constant and if what if the coupling constant is large then very little is known there's some special cases in one plus one dimensions where you can solve things exactly and you can see that the , the the the situation can be very <WRITING ON BLACKBOARD, P:07> they look very different so so fo- for a famous example is what's called the schwinger maybe even massive schwinger model <WRITING ON BLACKBOARD, P:11> where you are in one space one time dimension that helps a lot because in that case you don't have er er real photons propagating real photons are always transversal if you don't have any transverse er er er dimensions then er you have just the (cooling) photons so (xx) potential and , second , special thing is that in this case the mass of the electrons go to zero , and then you find that the only , er particles so you start with a Q-E-D i should have said this is Q-E-D <P:06> so er you do Q-E-D one plus one with the (mass of electrons) and that can be solved exactly , and er in the spectrum then you don't see anything that looks like permanent like electrons only state is a massive photon . exactly (xx) , non-interactive <WRITING ON BLACKBOARD, P:10> that is the , the mass squared i think it is squared (by five) here's the charge er and in one plus one dimension the charge is actually dimension full so makes sense to have mass proportion to u- so you have just this one state you see not no trace of any electrons or anything else er , i- in this limit so i th- i th- often think about this because it it shows that what you really have er er er it's like a strong strong coupling b- because the the the , the (detecting) coupling is er <WRITING ON BLACKBOARD, P:07> it is the E pole and E which is as i mentioned , er the charge as dimension of mass this gives you a dimension less quantitive which (xx) reflective coupling and once we put M-E to the zero you see that goes to infinity and then you have this situation if you go to the other limit where er er this is a small number then you come back actually to er schrödinger equation and e- you can see sort of er ha- hadrons and can identify the quantum numbers of y- y- you tie together E plus and E minus and make all the others , so you have sort of two er limits a weak coupling and a strong coupling and the two are totally different so this is a sobering e- example that er , that er things can , what what you see in the actual physical er spectrum can be very very different from the things you see in the (xx) so there's some other sim- sort of typically one plus one dimensional theories where you can also , the dispersals of (exactly) , and this is quite instructive and of course has been looked at to great detail but er , i think it is still a very special case in the one plus one potential which gives er automatically linear (xx) instead of one over R or proportional to R so you kind of put in confined (xx) right away okay er this was a little diversion here it helps you a little bit to understand the second question which we er er asked ourselves is (connected in) so where does the dirac equation come in , which is what we learn in the second course in quantum mechanics er we say well how about relativity can we write down something which has relativistic invariants erm in the schrödinger equation and then note that dirac proposed this nice equation how does that come about to this and er an- and you can derive that equation first of all you you you recall it's still a a sort of one particle equation you have a you have a er er a wave function , you know it has four components and depends on X and X is the (quaternion) of your particles and moves in er an external field so which you present as something you (xx) gives you (xx) potential point of the particle so you can get er er a dirac equation of from feynman diagrams (xx) in this way that you you take a case where you have one particle which is very heavy so that doesn't really move at all sits there and it gives you the potential in this case of a (xx) potential in this sense a especial case and then you have the other particle which interacts with this now <COUGH> , here we sort of say (relativistics) so any alpha should be good enough good enough in the dirac equation alpha can be if we put alpha to be small then we are back in the schrödinger equation so mhm but with this we have a solution for any alpha and if you look at where this comes from you have the one particle electron let's say mass M and another particle which has a very large mass and sits there and these then interact via photons and er then you can figure out what which are the diagrams you have to sum in order to get to th- the bound state equation in the dirac equation an- and that was as i mentioned last time you have to add to what we have in the schrödinger equation , all crosses between photons you have all those things but then you have er . also all 'cause each time you add one more exchange you have all crosses between , this is what you need to er make this really boost invariance , in terms of the dyson-schwinger equation this is quite radical because the kernel that you usually first approximate with a single photon exchange and then maybe you add two photon exchange to get the dirac equation you actually have to have the kernel itself a sort of interaction having infinite numbers of of exchanges because they have to all be able to cross each other it doesn't happen unless they all sit in the , in the kernel itself so this is you can just look work out what you get from these diagrams it's not very hard you see that they reproduce order by order the dirac equation , now <SIGH> . i often thought about whether and is this (ion) improvement how do we know it's an improvement some diagrams are left out w- we have an arbitrary alpha so alpha is no longer small w- we don't really know i think er what diagrams can be left out so y- you for example you do not have in the dirac equation what you have (xx) w- with diagrams , i- it's like a quenched approximation you have just one line a- and you go and produce (xx) , (starting with) the dirac equation , so <SIGH> since you have relativity alpha is not small it means that the particles have relativistic energies but it also means that you can produce then fluctuating extra er particles there an- and so here you sort of (take) one set of diagrams and you leave out another and er it it may be that there is some arguments to show in one sense that is more important than that but i i certainly haven't seen and as i mentioned it's very frequent that people improve on er er er bound state equations er going from the schrödinger equation which we used to (xx) in non-relativistic (quaternion) in relativistic (quaternion) by starting from the dirac equation er but then you are still sort of faced with the fact that you are still having only let's say a (meson) to particles er er you do not have , particle production at all so so , it's not (offering you much to) see if this is an improvement . <COUGH> okay maybe i should , leave this bound state business as that there are many more things i could , recall and maybe you have already thought of something that you want but it's , it's maybe er a er sort of general comment er er wh- what you realise when when y- you look into this is how complicated are relativistic bound states you combine relativity and quantum mechanics you know it- it's it's non-trivial er as long we s- stay on an- and think just of electrons or quarks and sort of real degrees of freedoms and freedom (xx) when you start to ask <SIGH> how are they bound together what are the bound states and we have these moving bound states which of course we have to have in in QCD you know these scatter hadrons they must be moving then extremely little is done on this you know so <SIGH> so it is sobering (xx) and on the other hand you could say now we have sort of a clue some vague sense that er nature actually , er er er tells us the experiments (there are) that er er hadrons can be regarded in the first approximation to the non-relativistic bound states of (xx) or free quarks and because that's such a remarkable statement in in view of what we know of field theory bound states er er it's sort of tells you this cannot happen in many ways this , if that's true then th- there must be some er clever way to do it for example er you (xx) again perturbation theory must be relevant you can think of hadrons as being er looking at atoms with all the (L and S) (xx) and all this specification of states er it's hard to believe that that would not would not come about unless you still have perturbation theory so (xx) strong in the confined (xx) , i would say and i'm not alone is not very much this is actually you can still use perturbation theory this is one indication of that and this sort of er serves as a counter example that you know so in this case if you (go to strong coupling area) you change you have no trace left of your fundament- your sort of (xx) , so maybe it is a clue . er as to how to proceed , with things we are working on in this (xx) okay <SIGH> but maybe i have to leave these bound states at this point and er i have a few things er er i would want to take up next which are er properties of hadronic scattering that we believe are exact and we understand completely even though we do not are not able to describe what a proton is w- we think we know something about scattering amplitude what pro- two protons or (xx) whatever so these are are the unitarity and then (xx) of amplitude and you can use those constrains to make sure you check them against data and so on , so i will going to those things next for little while and then er thinking the , (in the scheme) comes or there will be b- becoming then this question of of a dual resonance model and things like that so art of of discussing hadronic scattering amplitudes and after that we go er a little bit into the question of spontaneous (xx) very important part of this QCD and then come to the some kind of (collisions) that's a rough outline , so , the next <WRITING ON BLACKBOARD, P:05> thing we return a little bit to something we already have looked into was in the problem sessions and i think in the <WRITING ON BLACKBOARD, P:06> lectures as well , er unitarity and unitarity applies then to the to the scattering matrix , which you may recall , was the matrix which it's a very big matrix tells you <COUGH> er what happens if you come in with the elements are given er th- the physical states you have which are the states you see in the actual detector come in with some state and you go out with some state and those are the matrix elements of the S matrix , and we saw earlier that this is unitary so S (xx) S is equal to to (xx) matrix , it followed from this definition and (xx) theory it has to be like that and then we said well usually we actually remove the unit matrix from S because that's the one where nothing happens and it's kind of trivial so let's talk about this er T matrix instead and then it's not very difficult to see what the T matrix in (the earlier) condition was , you have this , now we say , non-linear condition like this on the T matrix and er and we know that in general that if we have er a (hermitian) conjugate matrix element , then er that's the same as the element of the original or the T matrix without <SIGH> hermitian conjugation but y- you transpose and you take complex conjugate , and this means that if you now take matrix elements of the general unitary relation which is a very er er special case so you take just er matrix elements which are the same so let's say there's two particles to make life a little bit simpler , or it comes on this side i'm sorry , that they are the same two particles in and out so in this case , i would say they are the same S is equal to I and you see then that er the T dagger , forward matrix element as we often say it's just a complex conjugate of the same matrix element and this means that when you take the different between T and T dagger forward matrix elements you get just the imaginary part of (xx) matrix so you get minus I and then er , two I imaginary part of , T now i know that you got this , this is just a matrix element of T <P:07> so this on the left-hand side , and on the right-hand side i have T dagger T same forward matrix element so this then is , the sum i have to , i don't have to but i i will put in here complete set of states , it's called N so then i get T matrix elements for all these Ns and i have a sum over a N so this means summing all and overall numbers of particles and er and also . integrating over all the momenta that the particles have in the given <WRITING ON BLACKBOARD, P:13> yeah okay and , on this side i have again used matrix rotation , so you bare with me <WRITING ON BLACKBOARD, P:15> and so here er you see the normalisation here we could just remind ourselves why we should integrate over all the momenta particles are always on the mass shell because these states are as important states things you see in the detector so we have to be on the shell the particular normalisation here you may recall stems from the normalisation that we use or for the states so you can shuffle <P:05> <COUGH> this factor here into the normalisation of those states but typically we use the so-called relativistic n- er where is that <WRITING ON BLACKBOARD, P:08> we use a normalisation of state which , this , the Q by Q (xx) of the (xx) <WRITING ON BLACKBOARD, P:12> as you recognise Q by Q two E is the same thing you see over there and the reason of course for using the two elec- two energies is that then this becomes possible to write in a relativistic i- invariant forms and we can now say <WRITING ON BLACKBOARD, P:14> if you (xx) (the delta) function here to say this is positive energy , okay this should be just a er er simple identity so you see the D three P (and the) Y over E is equivalent to getting something which is used with relativistic invariant a- and so it's just a nice thing because it means that actually it is , amplitudes here are also relativistically invariant . okay <GOING THROUGH NOTES, P:08> and then the next thing one does here <COUGH> is to er , take out from T another factor , so one one takes out the momentum conserving delta function and you say T is equal to <WRITING ON BLACKBOARD, P:05> two P five four for conventions <WRITING ON BLACKBOARD, P:09> this may be done <WRITING ON BLACKBOARD, P:05> for momentum conserving delta function times some other amplitude which here i call M , since we have always we practice er , translation invariants we have momentum conservation or four components and so er these er are non-zero only when momentum is concerned so it's convenient to take out that and to define M to be that there is multiplied that , and when you do that <SIGH> you get then this delta function coming here on the left-hand side so we should have (xx) this as er , as this , and you get it twice , here , and one of them you can then er roughly cancel so you have on the right-hand side you have delta four P it is P one plus P two minus sum of all the Q (xx) and that comes in square , 'cause this is a forward matrix element so , same P one plus P two and this is very heuristically , er so for P one plus P two minus P one minus P two is a delta zero that's what i'm trying to say (we have) delta squared and the other one , keeps track of , momentum conservation of the Q-Is (that's a momentum) , so being a little sloppy here (xx) this and then you recognise this delta four or momentum conservation is what you will have on the left-hand side because you could then to be forward anyway so you get rid of that as the (forward) clear off very nice and er and you get the , equation to be in the , form that pi is the imaginary part of this newly defined amplitude M , is this sum over all the intermediary states very well the <WRITING ON BLACKBOARD, P:10> integrate over all the momenta and then you have this . momentum conserving delta function . it was left over on the right-hand side and then you have er actually the matrix we're looking at this (matrix) <WRITING ON BLACKBOARD, P:05> this comes in square because there is er , T dagger then (turned) around and then becomes the complex conjugate to the other one . so now we have er th- the good news is that by choosing the forward remember this was a very special case we took the forward matrix element of the general unitarity (xx) which is valid for any non-forward matrix elements but we insisted on taking the forward one because now on the right-hand side and on the left-hand side we have something which is er measurable and this looks exactly , like the expression for the total cross-section , look up and you'll find the rules but it is th- that i i know the- the amplitude and i have to find the the cross-section and in this case because the sum over all possible states and which you can end up in when you start with P one P two , this is really a total cross-section there is only a little factor which is called the flux factor which is again if you look at the (xx) rules you see the total cross-section has er <WRITING ON BLACKBOARD, P:05> flux factor which you can write , in various ways but the one way which we have discussed also during the problem sessions with all those who who were there is this lambda function <WRITING ON BLACKBOARD, P:06> so the lambda function (xx) trying to <WRITING ON BLACKBOARD, P:11> recall what it is , so the square of all the three parameters you have and then minus two times all the cross (xx) so you can also write it like this a little bit shorter and then you have the correct form wrong sign on this (xx) between one and two , okay er this is just a number which depends on the energy and the mass (xx) this is something which you can measure , and on the left-hand side er we have the imaginary part of a scattering amplitude which is what two goes to two , so it came from here , P one P two into P one P two forward direction and er so that is , close to what we can measure as well . that if we can measure the cross-section you see for the (wholeness) or we can extrapolate and get to the zero angle case of course typically we measure then the absolute value of the forward matrix element squared not its imaginary part so how can we actually measure the , imaginary part an- and here we sort of er have a funny coincidence that er on the one hand we needed to take as i mentioned the the forward matrix element to be able to get on the right-hand side something which we can accept experimentally er on the left-hand side the forward direction is also helpful because we can measure the imaginary part with a special trick , (xx) you will know , (xx) just raise your hand how many of you know , okay we'll learn something when you come back in 15 minutes </S1>
<15 MINUTE BREAK>
<S1> okay i think we are <P:11> ready to , continue so i i mentioned that the optical theorem was what we meant to do and this is the optical theorem i should have said that relating the imaginary part of the forward amplitude to the total cross-sections with a simple kinematical </S1>
<S2> why is it called optical theorem </S2>
<S1> er it's a good question because it really comes already some optics in some sense <S2> mhm-hm </S2> er , i was thinking about the same thing and i don't have the answer <SIGH> . anybody else maybe who remembers from the optics course er i can't er i can see it (sometimes) bu- but it's it's a very general business there are aspects of hadron scattering which er <SIGH> are pretty much like optics for example diffraction is used in this special kind of process where there's an analogy to optical diffraction so (they use) the same terminology er yes i was going to also thank you for the question er just rub it in a little bit this really is remarkable in the sense if you think you are for example in a er L-A-C or some very high energy accelerator and you want to know about the total cross-section so the obvious thing is to you- smash the protons into each other an- and measure . how often something happens and this way you get the total cross-section it contains all of the possible processes including (the quartics) and whatever production but now according to this which is an exact statement you can learn about all of these processes not individually but to summed up and there is contribution from all of (them) by looking instead at the case where the protons just pass each other i mean forward direction nothing happens and er and this contains all the information all all the things that can be produced so this is somewhat remarkable , everything is connected in this way in principle y- you could say hopeless to ever think that we could get an exact expression for for one of these amplitudes because it would have to know about everything else that happens , at any energy so <P:05> so that's a sobering thought but of course in practice (the hick's) contribution to total cross-section is very very small part and er w- we probably don't have to consider how it's exact same to the elastic , forward amplitude but what i promised er was er to discuss a little bit about how we can learn about the imaginary part in the forward amplitude and er that's a a a special case where you can do it because as you will recall when you scatter , electromagnetically particles so you could call them electrons now doesn't matter what charge you you use , er , there is a contribution from a single photon exchange , and again er alpha is pretty small for Q-E-D so makes sense to just talk about single photon exchange and it couples then with a coupling constituent here and here which is the charge of the particle and er , so it's called A gamma , it is essentially th- the coupling divided by T T being the m- invariable momentum transfer from er and you see this thing er er blows off as T goes to zero you go to forward direction , er so if we try to form to calculate the total cross-section for the contribution from photon exchange it's actually infinite because we have to square this to get one of the T squared and er it's not sum (over) if you integrate over T you get single range so this is really nothing new the problem if you like with the electromagnetic interaction is that it has infinite range and therefore the total cross-section is a is , just a fact but this we can now use for our present application because typically when we scatter protons then er we don't even start to think about the electromagnetic interaction , they are a fairly small correction and i know where er good enough understanding and er and how deep the scattering goes to (worry about) photon exchange . but there is this photon exchange and because it blows up in the forward direction actually becomes quite large so it i- it is a sufficiently small T if it's er er of the same size the sum range where it's of the same size as the strong scattering cross-section this does not blow up because the the range of the strong force is typically one or (N part) one for , we have of course gluon exchange but that that's not reliable with long distance so the length the range of the interaction is one (xx) so as you go er more and more forward the amplitude does not generate similarity in the strong sector what it does in the electromagnetic one so they become comparable at some point the- then we know also that this one is here <WRITING ON BLACKBOARD> this is just like a (xx) cross-section so er absolutely (xx) as the two begin to interfere with the full amplitude is the sum of the strong and the electromagnetic so when they start to interfere it's the real part with real part so you can you can see from the way they interfere you can get information about this and and there is nothing unknown er here we would typically have a er er a fall factor for the proton but because it goes to such small T er th- the fall factor is just one just cease for full charge of the proton so this er er is as i said somewhat fortuitous here we have to go forward to into optical theorem to to to get on the right-hand side something which was measurable and now we realise because of this Q-E-D business that's also a very good place to be measuring the phase . so i'm just just going to show you a couple of couple of things where this is done this is just a total cross-section er , it's not . maybe you could comment a little bit o- on the total cross-section here measure then say for P-P scattering er , as a historical note er when i started er in this game one was , yeah in the strong belief that the s- the scattering cross-section (fell) and then would come to constant and i'm 40 38 years (xx) it was just there and they created a big er stir because actually i- in russia in serpukhov where they first managed to get the just the boundary to see a slight increase lots and lots of papers people were very excited and er and then of course we want measure it and high energies and those . at least as a log maybe as a log squared there are strong mhm , there are theorems saying we should not go stronger than log squared and this is a log (plots) and this is where we are going to be somewhere around 100 millibar stand there , and here you see , so er but this is not no big deal anymore it was a big deal then it just shows how and what's important and that climate changes all the time this is the real to the imaginary part this second one exactly what er i'll tell you a little bit more about and er , again there are <SIGH> general constrains such that if er for example the cross-section is is actually a constant then the real part has to be zero because it's not quite constant , the real part is is not too big than some ten per cent 15 per cent of the imaginary part <P:06> this is all in the forward direction so here is actually data showing er the interference the me- measurements that have been done <COUGH> so this is centre of mass energies 20 to 30 G-E-V and you see here the scale this is a momentum transfer T that i was discussing and you have to go down to er (control of) five G-E-V squared so in order for the photon exchange to become on the same size the the you can even try it if you fit this you know what the electromagnetic one is doing and then you say well let's assume there's an imaginary and a real part for the strong one which has more this exponential dependence on T which is rather flat in this T range and then you just look at the shape here , and you figure out what the ratio of real imaginary , part is and this is then another way of measuring the total cross-section you can see what i mean <P:06> y- you get both the real and the imaginary part fr- from that curve in the forward direction which means there's a total cross-section so here is well some kind of a maybe <NAME> was actually involved in this at CERN as a some excuse @@ you make a (good) detector on several hundred metres or something beyond the interaction point in order to catch the protons which er scatter ever so slightly and y- you are at 14,000 G-E-Vs the centre of mass of the 7,000 G-E-V proton scatters then this er basically nothing , and then you see the central this this is a strong manifesto of proton (that should) go up here and then this the Q-E-D anti-proton , (xx) (proton) <SU> (nothing) </SU> mhm-hm <SU> mhm </SU> anyway , it's er it's interesting that we can measure the total cross-section from a reaction where nothing happens . but it has been done both ways and before and not just to people have done it both this way and then looked at everything that happens , convince themselves that unitarity is indeed , in force <GOING THROUGH NOTES, P:05> . okay anything more about the this i think i don't need the <ROLLING UP THE SCREEN, P:06> so is- it's this totem collaboration with LHC which er we'll er do this measurement and people in the next corridor <P:05> and all (xx) a little inside information you can have okay that's all for , the optical theorem unitarity at this point and then er i was going to talk a little bit i i jump a little bit in my lecture notes because they were made some time ago and the first time i give this course so it's not always , but i i i though i should say something about resonance continuations we we actually have discussed that <P:06> to some extent before but er <WRITING ON BLACKBOARD, P:09> you should know of the breit-wigner , distribution (what this) er , distribution is so er , so from the beginnings of quantum mechanics er you know that the states which er are eigenstates of the hamiltonian and which have some er energy they . have a time development and (xx) in the schrödinger picture yes which is just a phase , if you want the the state at a later time it's the same state as in the beginning it's just a phase because this is an eigenstate of the hamiltonian so now , if this is unstable . er state or particle in our case will be mostly . er particles then we actually want that the state goes away with time and er that means that er this gets shifted i think this shift also (xx) it (xx) applies to whatever atomic levels an- and so on the energy becomes complex and it has to have a specific sign of the imaginary part such that when you put in this er complex er <WRITING ON BLACKBOARD, P:07> phase it goes the right way that is falls down into the minus gamma over two T times this oscillating phase . so now , if time increases this will fall down exponentially and this is describes then the decay of the state and if you now go to er four-momentum space so you go y- y- you fully transform your time into energy so Q-E-T E to the I three T on this , state then because of that time dependence defined (you get the) denominator E minus E nought plus (xx) alpha two <WRITING ON BLACKBOARD, P:09> so we see this is what we have discussed before that you will typically see the resonance as a pole in the amplitude and the (point) i- if the resonance is not stable in addition to the , the the real energy and to the imaginary part so it goes into the complex plane the pole and how far away from the new axis it is depends on how broad it is that is how big is the width or the inverse lifetime <P:09> so when we do this thing in a a relativistic framework where we can have particles moving both backwards and forwards and have positive and negative energy , then . we get a contribution which is quite similar to that so if we have er two particles collide and they make er a resonance we can think of high five or some will go to rho resonance this would be an amplitude , then . the resonance contribution to this and here er i- , i've given so let's call this A this is called B and this is called R in general that would take the form , some factors on top and then the breit-wigner form . and this breit-wigner form it's not so er unexpected because you'll see that er it's just , positive times negative energy to to make the thing relativistically invariant (in which you) work <WRITING ON BLACKBOARD, P:09> right this should be S , time is then squared plus I and R that is what we have down there where S is E squared , i'm thinking that we are (fitting the) rest here quite easy so S is E squared minus S squared plus plus , I-M gamma then you have dropped away gamma squared because this is in the narrow width approximation , which (xx) next (xx) , so if you agree with the non-relativistic form there it's not too hard to see that you expect something like this and the factors you see here the fact that they factorise it's again u- a consequence of unitarity but on the other hand it's very er natural and this R-A is somehow the vortex coming from there just as when you write <COUGH> write er final draws we have contribution from this vortex with f- from that vortex and then we have the propagator which in this case would be that one so you think that you form the resonance through some channel here called A and the amplitude for doing that is scaled by R-A and then this resonance propagates and that's essentially the nominator and then it couples to some other state B through R-B so it's a reasonably natural , er here or spin there resonance if you would have er i- if this has spin it remembers something about how it was formed in which spin state an- and and that also reflects on how it decays but that is kind of a technical adjustment for what for now what i want to just use and it has this forward i think it's being zero now we will combine this with unitarity so we say close to the pole close to S equals M squared this is the important contribution to the amplitude and er therefore we have er er i concentrate on this term in writing the unitarity relation so i managed to delete it but of course you remember , and i am going to actually use now to optical theorem so look in the forward direction so B has changed into A and and they not only have the same same particles but actually also going to the same direction so this . if i just look at this expression i have here becomes R-A squared because B is equal A and then the imaginary part of the denominator o- or coming from this nominator is <WRITING ON BLACKBOARD, P:07> like so that's the or two times M gamma yeah , two numbers left-hand side and that was supposed to be , equal to sum over all possible final states , this , integral which where the (D three Q-I) or two pi Q is two P and so on just you notice here (xx) pi , and then this amplitude it wasn't like the total cross-section but now we are saying any amplitude which we end up in some final state N will be dominated by the resonance pole so it will say this applies B is equal to N here you all of those amplitudes will be dominated by the resonance pole so we should have this kind of contribution <P:05> and so we again apply our formula here on the right-hand side and , here M squared so we get R-A squared which is a common term and then we have the denominator absolute value <WRITING ON BLACKBOARD, P:05> like that and then the rest has to do to with this R-B , which is now contains B is equal to N so <WRITING ON BLACKBOARD, P:07> so N squared it's the decay amplitude , this vortex which so now we can compare both sides , this expression with that expression , and we see fortunately that er the denominators containing the peak et cetera they are the same on both sides and R-A squared can also be eliminated , and then we are left here we have two M gamma and so we what what we sort of find from this is an expression for . gamma in terms of the amplitudes for the decay which are the stems , so it is reminds pretty much of the total cross-section for the collision of two particles but now we have a particle which is decaying <WRITING ON BLACKBOARD, P:05> specimen R goes to M amplitude squared er this contains also the momentum conserving delta function , so the thing looks pretty much like that the total cross-section the only thing we did a difference is we have a slightly different flax type but because we have now just a particle which is decaying in this case it's two M and this is (part of) the final rules also just the rule for you is you how an amplitude for the decay of particle you can find the width and er this could remain B an atom if you like and er you calculate the transition from from the state and this could tell how you but if you know the amplitude from the transition you will be able to calculate the width , through that , erm so this i should maybe emphasise this is th- the total width <WRITING ON BLACKBOARD, P:05> characteristic of a resonance it always is what you see down down there total width of- the width the energy uncertainty if you like of the state if something has to do with this total lifetime has nothing to do with whe- where it is going into , the the strength for going into some specific channel that's determined by these amplitudes but the shape of the breit-wigner <WRITING ON BLACKBOARD, P:05> what breit-wigner advertised is really this shape (xx) just a name for you to recognise if somebody mentions , so that gamma was the total width one also talks about partial widths <WRITING ON BLACKBOARD, P:09> you could take an example E-G gamma for rho goes to plus E minus , so the rho resonance sometimes decays into , nought pi plus pi minus most of the time it does but occasionally it goes into E plus E minus and that you would then calculate from the branching ratio , so w- this really gives you the fraction of the time <WRITING ON BLACKBOARD, P:09> rho into E plus E minus limited (xx) and if the actual number here i looked it up it's 4.5 ten to the minus four ten to the minus five so a few times into 100,000 decays of the rho it will not go into pi pi but it will be going to E plus E minus , and then the partial width you would say is the total width to characteristic lifetime times the fraction of time it goes into this and then you get something which has the <WRITING ON BLACKBOARD, P:08> dimension of of the width and in the case of the rho this is the number so the the full width total width is a 150 or so M-E-V and this partial width into E plus E minus which is from the order of (xx) is very small , but it's definitely not zero and er , and and for example when you look at this E plus E minus colliders we have looked at the the the er cross-section , so import E plus E minus goes into hadrons and to Q bar you see bumps in particular you see the rho bump i will not <SIGH> take the picture again we had but er but th- you do see it's a function of S y- y- you see it's er rho and then you see other things . and rho squared and you see from the width of the bump the gamma rho or so if the rho did not couple with E plus E minus you wouldn't see any trace of it in in E plus E minus , we wanted to have this . what you do and so from from the size of this , bump you can figure out what's the (coupling) E plus E minus , it's one way and this is the opposite way where you form the rho and of course you can also produce the rho and do it a few hundred thousand times and see how often you get E plus E minus . okay this was about unstable particles and resonances no questions on this . so then er the next important er thing we know about er scattering amplitudes is the analyticity and it can be used to- er together with the optical theorem to , get all kinds of interesting relations and i will describe just , first the analytic structure of the amplitude and then the analytication <WRITING ON BLACKBOARD, P:10> now <SIGH> here the one one one typically one restricts oneself , to two goes to two , the- er what you have of course amplitude is describing two goes to three or one goes to four or whatever and they depend on more variables more invariants and so they become complex functions of many variables and er that's a kind of tricky business <SIGH> so er in practice they are limited to two to two or or just a little bit more than that , so one thinks then of of er some scattering two to two so A and B scatter into C and D and this is described by sum amplitude and that amplitude depends on typically two , er <SIGH> , quantities you could say it's the energy that you collide these guys with and then the scattering angle that they go into , it's assumed that they are spinless particles so you don't have to worry about some as you move the angle you can just think of er colliding A and B head-on and er we are in the centre of mass we can always ('cause of the) lorenz invariants we can allow ourselves to choose this special coordinate system so we have the total energy and the scattering angle and of course we have discussed it's er <WRITING ON BLACKBOARD, P:05> convenient in particular when we discuss . analyticity to define the invariant quantities S and T which are related to the centre of mass in the S channel and the scattering angle through formulas you you know and , for symmetry reasons , it's also frequent to define the U variable which then this to this and you see you put a minus sign whenever you have a particle particles going in different directions so S and S A and B are both incoming so that's a plus A is incoming C is outgoing so it's minus and er if these are three and there are two <WRITING ON BLACKBOARD, P:06> two only er independent variables th- there's a relation some of them is (skewed) but some of them the mass is squared <P:08> and then the physical region , we actually do a scattering experiment , there are restrictions on what values of S and T we can er measure we can measure the amplitude of certain values of S and T , er <P:05> if i can show you . they're different masses it would be that the minimum energy , this is actually called this is E centre of mass squared and so so the minimum centre of mass energy you must have then collide A and B (as given) by their masses if they've moved or moved on , and T on the other hand is er ranging from er . over negative values b- but for now er now i think i i'll for clarity i put all the masses to be the same . because the formulas are a little involved i- in er different masses and it's not important for for this thing so . the T is below no it , yes , T is like this . er , four pi centre of mass squared so pi is centre of mass is just the momentum with which you are colliding , this guy can be expressed with those S again s- so this the physical region that you can make measurements in for , A plus D going into C plus D which we usually refer to as the S channel <P:05> and the point with analyticity is that A (of) S and T which is the scattering amplitude again let's think of these as being scalar particles so we don't have to worry about (spin or S-U-N-V) or anything like that technical complications then it's described by just a single amplitude because there's no spin degree of freedom and that is that amplitude in (analytic) function , of (S-E-T) <WRITING ON BLACKBOARD, P:06> which has certain singularities which are er these continuities which are imposed by unitarity , er , which we'll go come back to but the the the sort of more statement here is that this is (xx) function of S and T and it's the same amplitude and also describes <WRITING ON BLACKBOARD, P:06> er the crossed channels . by which we mean , A plus C bar going into B bar plus D this is the T channel <WRITING ON BLACKBOARD, P:05> and , A plus D bar going into B bar plus C , (xx) channel , so these are different quite different scattering processes so for example if you had B pi going to pi pi then this would be pi pi bar going into pi pi bar quite different interactions and yet er this same amplitude describes both the S channel process and the T channel process and the U channel process that's called crossing symmetry . and that having said , the best sort of to make you more comfortable with why it is like that er you can just think of the final rules so if we had simplest I cubed process again we would say draw the lowest order diagram for A-B going into C-D it would be something like that and the direction of the arrows here wouldn't really be very important if you would be asked to write a final diagram for for the process in the T channel A-C bar going into B bar plus D it would be the same diagram 'cause you are just supposed to put together maybe other diagrams but but this particular diagram would be there and it would be the same er <WRITING ON BLACKBOARD, P:07> here if you would have defined S to be like we just said here you would define the corresponding variable er this channel to be D-A minus pi B bar squared because this is going in different direction so therefore you put a minus sign equals S when pi B bar equals minus pi B so the crossing symmetry says you get the same amplitude A for the process where you reverse some arrows so instead of B going in you have B bar going out instead of proton going in anti-proton going out and at the same time you have to reverse the four momentum all four components energy and (xx) , so you realise if you if you if you just do that if just have a physical scattering for in the S channel and you start to change the four momentum with one of the parts you get negative energy this is not a physical particle in these so that's why er they are different physical regions and it's the same amplitude but (valued at) different values of S and T this is the S channel region and if you go to T channel region then S is negative and T is positive but it's the same amplitude so it's you know the analytic continuation if you know this amplitude precisely for one process then you can continue it in S and T to get into the values which describe the T channel process also if you know again exactly the amplitude for for something then you can deduce everything about the amplitude for a different process , okay i i i'm running out of time here so we have stop here but this crossing symmetry analyticity together with unitarity can all be combined then to write er cauchy type integrals which allow you to er express er the amplitude in arbitrary values of S and T in terms of measure quantities like total cross-sections (xx) sections , so i have to do that next week and you are very welcome also welcome to the problem session if you but you don't have to @@ okay </S1>
