 this talk would be application of light from methods mostly to the, yeah, to the 5-4 theory on overview. First, I will a little bit remind this light front couple cluster methods and application this to the 5-4 theory as then some light from Fox state expansion results using symmetrical polynomial basis function. Also we're talking about this difference between light front and equal time, critical coupling value, then some sector dependent calculations result and of course some summary. So 5-4 theory, just some quick remind of course Lagrangian for the light front one plus one Hamiltonian does not have this one plus one there is no this kinetic term from the transverse direction. So we have just mass term and interaction term for the light front X plus equal zero mode expansion would be look like this. Whereas of course this X minus is this light cone momentum and P plus is a conjugate X minus is a longitudinal direction and P plus is a light cone momentum and of course creation and annihilation operator commutator. For the Hamiltonian we have this four terms kinetic energy term, three to one annihilation of three particles creating one particle term, one to three annihilation of one creation of three and of course two to two. Now application of this I love CC to this 5-4 theory so we have this light front eigenvalue problem whereas the light front Hamiltonian applying to this eigenstate give us this mass squared expression. And without doing the four states truncation we applying this ansatz so whereas the state is really application of this exponentiation of the operator T to this valent state and this valent states usually include very small amount of particle so for the 5-4 it would be really only because we will consider only odd sector it would be really only one particle. It's differ from this really origin of this couple cluster methods from the nuclear physics and nuclear chemistry whereas really valent states are much larger and really for them they did not create more particles they just created excitations and this operator T is conserved or required quantum number included light front momentum and of course increases particle number. So if we really multiply both part of the eigenvalue problem by this e to the minus t we will obtain this Baker-Hausdorff expression of the light front Hamiltonian on the left hand side applied to the valent state and on the right hand side we have this expression and this guy really gain eigenstate psi. So eigenvalue becomes this effective light from Hamiltonian applied to the valent state and so from both side right now we have the simple valent state and now we do projections we project this on the valent state this equation effective Hamiltonian and we also project on the orthogonal to the valent state. And this second so-called auxiliary equations will determine us operator T in that functions in the operator T which is really showing distribution of the momentum and how much we will truncate we really of course will use truncation right now up to this point everything exact except of course higher for state wave function connecting to the law of state function through the sum soon again as soon as we do this truncation operator T then this connection between a lower state and higher state becoming already just some special way but it's still infinite amount of terms but now we also truncate this projection so for example if the valent state would be one particle yeah for the 5-4 theory here this if we keep infinite amount of term and we consider only odd sector so it would be time three, five, seven and so on so but because our real time we will truncate as many as we need so if we have I don't know time five function in the T operator we will truncate to the time three, five, seven, nine, 11 we of course will do much just to the one equation and yeah this effective Hamiltonian could be calculated from this Baker-Hausdorff expansion and it would be of course we don't keep infinite amount of term we terminate this when this increase in the particle number matches the truncation of this projection and in the previous talk yeah we were talked about the spectator depends on council divergences but we don't have here okay so for the 5-4 application of LFCC the valent state again it would be just one particle, one boson state and we take this simple contribution to the T operator which is really one to three so annihilation of one particle creation of three so it's really increase particle number by two and it's exactly what we want for the Laplace order because we wish we don't want to mix odd and even sectors and we really will consider odd sector function T2 will be symmetric in its argument and this projection on this auxiliary state this would be truncated to just the state of the three bosons because again it should be odd sector and we only need one function so we truncate to only one equation excuse me so far you started with this light quantization and then you propose an answer for the vacuum for the ground state that's what you've learned right no really we wish not to do a regular Fox space truncation so we still truncating but we truncating on the way how this higher Fox state connected to the lower Fox state but we still keeping infinite amount of the Fox state but you decided, well you started by deciding that you are quantizing your theory in the light form yes of course, yeah it's a light round sure, but did you say why you wanted to do that because I'm trying to follow, you know you can propose an answer for the vacuum then but it's not the vacuum, it's the massive state the vacuum is trivial I keep hearing that, I keep hearing that but is that, can you say something about that why is it trivial? In the light front? Because light front momentum, yeah light front momentum P plus it's always positive so we don't have, we never have positive we never have negative momentum so you cannot create vacuum from the I don't know, plus five and minus five, yeah there is no such thing, it's always on the zeros so the vacuum is empty state and that's it yes, of course, no I mean and now you're trying to populate it so now you're trying to obtain it no, no, balance states, these five states it's not vacuum state, it's just this state with a minimum amount of particles right, but this is an answer for what? you're trying to build a wave function yeah, we try to do five four without full space truncation in calculating a mass of states yes not the vacuum, the vacuum is trivial so just calculate the mass of states the one particle excitation one particle excitation so the vacuum being trivial you directly try to identify the side of the states for five four yes, yeah, so so again, because we truncated this projection only to the three states we terminate this Baker-House-Dorff expansion for this effective Hamiltonian also just one to three particles so creation, yeah we only keep these terms really who only create two particles, no more and this is really more efficient ways and instead to generate all these Baker-House-Dorff expansion terms just try to wait just to look what terms we really need okay, so how Valen's state would look so this kinetic term from the Hamiltonian add to this and this is projection on the Valen state yeah, we will have only one particle state so it's only two terms contribute kinetic energy and three to one and T2 operator this is again from the Baker-House-Dorff because this guy minus two particle and this is plus two particle so just really direct one to one and auxiliary equation again, which is cut to the only three states it has many more terms but point is that every term just create two particles yeah, two particles here, nothing here but this two particle here, again, nothing here and so we have really only two equations the equations for one function for this function, what is there known here? the T2 function we don't know T2, we extract T2 from this auxiliary equation exactly no to extract T to find T2 and as soon as we know T2, yeah, we can solve our eigenvalue problem we know T2, we can find physical mass okay, so this is Valen state equation after integration, so we have this T2 function from the T operator and it's really rescales again, these guys is a fraction of the Alangent-Hugh-Geno momentum so we rescales this and in order to express integral in this fraction and G is right now lambda or 4 pi mu squared so it's depending on the bare mass, yeah mu is a bare mass, capital M is a physical mass and this leads to the definition of this dimensional mass shift delta which will be talk about meaning of this shift later and it's of course through this shift exactly you also could say this mass renormalization is because it's physical mass how physical mass connected with the bare mass okay and this auxiliary equation it has so again, this is from the one to three term this is really wave function renormalization term and this is from some other terms in the Hamiltonian again from the Baker-Hausdorff expression and here we can see this non-linearity yeah T2 tilde squared no second order in T2 so this is so called loop correction so this is exactly coming from this infinite amount of fork state here because if we really expand this in the series it would be infinite series over coupling G starting from the first order in G so it's exactly how this LFCC methods implement this correction from all high order state okay and without really showing result for this light front couple cluster methods yet right away we wish to talk about just to compare with this fork state truncation where again we can see the odd sector and we just truncate up to three particles so again we keep one particle and three particle states then if we apply this light front eigenvalue problem in the one plus one direction then we will have in this case only two it would be coupled integral equation depending on the how many fork state we keep in here here we have only two fork states so it would be only two equations and this is really yeah it's already expressed in the ratio of the psi three and psi one and we can right away see that this definitely does not have loop correction in this second equation which is really coming from the three body sector it definitely does not have a loop correction which light front couple cluster has so it's simpler that does not include physics from the higher state and there's some description of the structure of the equation first is the same as a valence equation so this exactly the same as a valence state and LFCC but auxiliary equation it's for the LFCC it's contained five terms kinetic energy two to two time wave functionalization this vertical looks the most important and in comparison this fork state truncation to up to three particles only it has only three terms and does not have this loops correction also kinetic terms are not the same in the LFCC physical mass would be for all constituents but for this fork state truncation uses bare mass in three body sector unless of course we apply the sector dependent normalization and I already told that of course LFCC contains this vertical loops which is partial resummation of all orders now in order to solve really both with fork state truncated I give many problem and LFCC equations we use this fully symmetrical polynomials so they are this is multivariate polynomials of order n and x1 and x2 they are really domain is this triangle for the three particle yeah and of course because of the conservation of momentum yeah I remember that this x i's their momentum fraction so conservation of momentum it means that they all sum to one so we really have only two independent variables also it could be more than one polynomial of the given order so we're using second subscript to differentiate this possibility and again of course we don't take all to the infinity we truncate to some order n and for example this T2 tilde function could be expressed through this polynomials and this coefficient which we need to find and this is just for the convenience because equations both really LFCC and truncated they all contain this factor so we write a way to get rid from this factor okay so and this polynomials we did this work with our students so it could be shown that really all of them could be constructed from the second order and third this C guys right now is monomials and some general monomial could be constructed from the second and third order whereas they look this way and for example six order polynomial could be made to two ways either second order cube to the cube or this third order polynomial squared and it's of course we also really by hand using ground Schmidt we make them orthonormal for this truncated case or LFCC case and again it's much better than as again was mentioned in the previous talk much better than DLCQ okay so now if we take this life from couple cluster methods auxiliary equation and project them on this basis function expressed through these symmetrical polynomials we will obtain really matrix equation whereas what we are looking for is this coefficients A and this delta it's again the same shift it's expressed also could be expressed through this matrices and coefficients I will show this matrices on the next slide they are completely mostly computed by Gauss-Ligandre quadratures which is of course if you remember they give you exact answer up to some order and so really give exact answer for our needs so this is mostly a really overlap of two polynomials this matrices all these matrices all these numbers are finite there's no back originals no no everything fine yeah and now finally result so again we find we will find these coefficients A's and then we will find T2 this function T from the operator T and then we will put to this shift which is connected physical mass and bare mass and from here we can find again of course we first fix some G and then we will find this ratio and then we take another G and find another ratio okay so here really it result for the LFCC folk space truncation up to the three particles and also sector dependent mass did not talk yet but results here anyway and we can see that this the view is doing exactly this folk space truncation up to three particles it's doing the views LFCC coming really approximately 1.5 for the G and sector dependent doing better closer to LFCC so how expensive is this to perform these calculations for example if you were to include one more term into this the five particle term in LFCC would it be feasible or is it becoming quickly LFCC would be yeah because of this non-linearity T2 squared term yeah it's difficult so this is the reason why we really did not expand LFCC we did a folk state truncation except instead of just to the three we did up to nine I will show next but yeah LFCC expensive but doable so it's exactly really what would be my summary is that because LFCC really we will see later doing views then truncation to for example time 579 state folk state truncation exactly because we included only one this T operator just annihilation of one particle creation of three particles no really just adding two particles so we exactly thinking about adding another term to see how much better it will do so summary for LFCC at this point so again it's models relatively simple but require numerical techniques in comparison with folk state truncation shows introduce physical mass for kinetic energy terms without use of sector dependent renormalization and again it's doable yeah particularly using this symmetrical polynomials and these folk states this low folk state truncation up to the three particles definitely are doing views now what about extended folk state expansion so here for the odd sector we will go up to the ninth state and for the even sector up to the eighth state so now Eigen state of this fully Hamiltonian with interaction of course would be expressed through this, through this folk state and so really every physical state we include time ideally infinite amount of the folk state and of course it's normalized and now just again this Eigen value light front Eigen value problem just yield us this coupled system of equation where we really can see that wave function of the sector M it's connected with itself also with the two up states and two down states which is of course nothing surprising so it's preserved with either odd sector or even sector okay and for this kind of to do numeric here we needed polynomials extended to the not just to the C body but to the N body and right now this polynomials it's again it's product of these monomials whereas this powers of course summing up up to N and it started from the second power because of course first power because of the constraint it's really equal to one yeah because again domain okay so now using these multivariate polynomials we again obtain just matrix equation whereas what we're looking for is this coefficient C and we have this different matrices here like overlap of non orthogonal basis function in time given sector and here we don't do grand Schmid because it's produced so terrible around of error so we really did this using single value decomposition as usually through this U matrices and diagonal matrix matrix of the eigenvectors of the B and diagonal matrix D and also in U we only kept column associated with eigenvalues above some positive threshold according to the Wilson and of course we can define some new coefficients and new matrices. So finally we have this again ratio of the physical mass to B mass squared relative to this critical coupling and this is of course interpolation again odd, oh no maybe this for this without sector no without sector dependent really odd we went only up to the seven particle sector and for the even to the eight and here we also plot in four times odd why because there is no binding state for the even sectors yeah so really two odd will give us one even so this is some sort of like check and we can see really that even and for odd they relatively close to each other no except near critical coupling which is nothing surprising. So here as they intersect mass equals zero and few different points which is really taken as the error estimation and we take in your critical coupling is 2.1 plus minus five hundredths. How does this result compare with LFCC? LFCC was going to the 1.5 yeah it was going this way no I mean a little bit higher than 1.5 so this truncation up to the seven and up to the eight for odd and even really doing better again we- I thought that if the LFCC mass is smaller it means that it is a rational approach it means that LFCC is better is LFCC a rational approach? Yeah LFCC giving you physical mass yes for the constituents but because we're including only one T operator only what this what adding two particles T operator one to three this is not enough but it's still a very rational approach right? Yes yeah you're adding all this contribution from the high order So what I'm going to do my question is the follow-up I do a calculation I see two values for the mass one from this approach and one from LFCC and I see that LFCC value is smaller Shouldn't I conclude that I should prefer LFCC way function because since it's smaller and if the method is variational whatever is smaller means that I'm doing a better job. Why are you saying that this is better than LFCC can you explain? No, I'm thinking- Well the LFCC calculation there breaks down close to the critical coupling. Well I don't understand what does it mean that it breaks down it gives you some answer for the way function No, there are no solutions to the non-linear It's becoming he at this point it's becoming complex Okay but let's look at the last point which is not complex It's lower than the Fox state location point Doesn't that mean that I should prefer the LFCC point or is there some other problem which I'm not aware of? I would just add more terms to the LFCC Why should I add more terms? It's already a calculation it gave you a way function If normally if I'm doing if there are two different groups which provide two different variational answers I look at whichever answers gives a smaller energy I say well that group is doing better period You can of course add more terms but even without adding more terms they are already doing better So is this correct logic or not? You jump ahead to the plot that combines light front cover poster and the- But I think this without combining this question should have an answer Another one, there So we have everything light front and Fox space truncation and also sector dependent Fox state truncation So but this is the last point of the light front So up to this point it's just yeah it's numerically just becoming complex numbers here Okay so it's not much different up to that point The goal was really to obtain critical coupling value We did not look for the best wave function we looked for the critical coupling value so we were doing this Fox state truncation high and higher order in order to see this convergence and for example we can see that five and seven almost identical and sector dependent require really nine but Yeah I understand but this whole curve not just the critical coupling this whole curve is an interesting observable We heard in the talk of Marko Cirogna that he can compute using Barreira summation this whole curve so this whole curve contains a lot of information not just the critical coupling so it would be interesting to know eventually not just the critical coupling but the whole curve as a general comment Okay so in any case so this light front extended Fox space truncated give us critical coupling at 2.1 Now let's compare for the equal time again we really have to do this rescaling because different people using different for the G so now it's all in this G bar and we can see that light front guys always has lower critical coupling than equal time there is a systematic difference and so these brought us to the idea I mean we were talking with some patriarchal the light cone Masai Burkut and in 1993 he did this work which connected BMS of the light front with the equal time BMS through these really tadpole contributions if calculated in the equal time again light front there is no vacuum to vacuum contribution in the light front but in the equal time there is this tadpole contributions and of course clear why it couldn't be in the light front because we go from zero to four particles and then from four particles to zero which is impossible unless one of the light front momentum would be negative and we cannot have We asked maybe a stupid question but I could use a little bit about the sign maybe just a question of convention why is the plus on this side of the equation and not on the other side of the equation because I would have thought that precisely because you're missing this on the light front you should be shifting the light front to give you the equal time and not get it around probably better question to directly to the Masai who has much higher there are two minus signs in there actually this represents the negative of the tadpole oh ok maybe there are some yeah at some point it would be because you're missing a piece that's positive no these backing expectation values are proportional to the tadpole contribution that's Matias' work to show that correspondence and that's where they're at the minus sign it would be negative yeah it would be negative so it would add exactly because after adding this stuff to the light front we will come approximately close to the equal time oh so basically the way that you've defined some prevention this thing is actually negative yeah yes yeah yeah yeah yeah yeah of course yeah yeah no we would not present this if it would be these graphs are calculated in which theory the equal time Masai is Matias Burkart he did this in equal time and we did this in the light front and this is some sort of like that they equal so for the equal time is the tadpole for the light front it's c2 p3 no from 3 to 1 so some sort of agreement that it would be the same in both equal time and light front so equal time so he was calculating these guys we were calculating these guys yeah we're together so now the question was how to calculate this so this is the expectation value vacuum expectation value of the square of the field in the interaction full theory with the direction and this is in the free theory so in order to calculate this like in any time quantum field theory book we're doing this point splitting so we're shifting light front time and light front momentum by the epsilon so we separate from the zero and we also inserting using fullness completeness of the state of the Hamiltonian we insert this one here inside and so now we have to calculate two matrix element right one and left one and of course a joint representation for this operator shifted by this in this joint representation so right this sandwich could be calculated and of course left also could be calculated and for the free state and by the way because here we have only time creation annihilation operator we really because remember this is the eigenstate of the full Hamiltonian so each eigenstate including all infinite number of the Fox state but it's really for each state only one particle will contribute because of course here only one operator in the five field and for the one particle free state because really it would be right now free theory free Hamiltonian so we really just have this annihilation operator here and for the left creation operator here so it all could be calculated and this final expression for these two sandwiches and we also can insert in this free guy just to have the same because again our free theory really can't each only contained one body state we now can really come to representation of the modified Bessel functions for this difference and argument of course has to go to zero so when it's go to zero it's really have this logarithmic form and also this earlier factor and so really this difference equal to this sum and as you can see it's negative there is a really probability of one body state in every eigenstate and this is of course would be logarithmically divergent if we approaching critical coupling when physical mass goes to zero and this how we define this shift so shift really become positive because we put in this negative side here so we either can say that life around it's equal time minus some shift or we can express this as a ratio or we can express this as a coupling constant for the equal time through the light front coupling constant or we can express this as a ratio of the physical and bare masses for the equal time through this ratio in the light front whatever necessary just to see that there is some convergence we looked at the relative probabilities for this odd sector again it's only up to seven body but what we see right away is that each next sector becoming ten times at least order smaller than previous one so there is convergence definitely contribution from the highest sector they becoming less and less probable so this is a good point but the bad thing that near critical coupling at two point one there is no kick because has to be divergence there because again physical mass goes to zero but this guy doesn't go to zero unfortunately so we don't see this kick growth we would expect that probability near critical coupling somewhere here has to suddenly increase particular near critical coupling higher foxtade we expecting would be more and more important but delta should remain finite unit critical coupling yes but it's unfortunately they will yeah because when this goes to zero this guy has to go to zero and in our calculation it didn't go to zero didn't go enough to zero so this is the reason why we started to do sector dependent hoping that invariant mass I mean higher foxtade will not be suppressed by this invariant mass so what is your interpretation of this I mean your interpretation is that you just you just need more states is that you need higher particle number states to be able to actually perform this procedure that is the problem is that you just don't have enough particles we see that in this regular foxtade truncation because invariant mass still quite high so this higher foxtade they still suppressed as a contribution of this higher more excited foxtade has to grow near critical coupling it has to become more and more important you would expect near critical coupling and it means that really probability of the one body state has to go to zero no I mean it has to become smaller and smaller near critical coupling and we don't see this unfortunately but from convergence point at least this plot of probability some sort of encouragement that each higher foxtade sector becomes less and less probable and now this is a plot of shift relative to the critical coupling whereas the points obtain extrapolation in the basis side and these two curves this is a linear and this is a quadratic fit which is really view constructed only up to g equal no I mean taking data from up to critical coupling equal one and I will explain why this into the next slide but in any case shift is relatively good because when we add this shift to our light front mass critical coupling we have equal time critical approximately coupling or do you have agreement in the full mass curve because also the full mass curve has to be ratio of physical mass to equal time mass we can see that there is a problem arise after g equal one so this is the reason why we only used instead to go down down it started to grow and this is the reason why we used for this shift calculation only point up to g equal one but before g equals one you get good agreement yes, yeah so this is again the suppression of this higher foxtade they don't exhibit themselves well enough because it's still probability again for the one body sector still staying up high does not allow them to expand to give more probability for the higher foxtade I don't understand this point because the m squared has to go to zero and here it goes to point 84 and then starts shooting up yes, yes, yes so this is we consider this is why for example foxtade truncation is a bad thing to do or we have to include much more many state not just on 7, 9, 8, but more no, I didn't say before you show the plot where the mass was going through zero before previously you had a plot before you started discussing this mass change you had a plot which had a mass reaching zero yeah, but it was a light front yeah, and now it's equal time this is using the correction this is divided by the bare mass of the light front so the correction between equal time and light front is diverging once you get to near the critical coupling and so the connection is breaking down that's why this swings up here at the larger G yeah, so this is the bare mass for equal time well, this is straight here we have some unpublished results about how this behaves when you compute the correction in equal time and there you can swipe yeah, there's clearly something wrong in the light front calculation near the critical coupling that's the bad news in this because of this yeah, so again this probability of the one body sector in the really some sort of lower state doesn't go to zero as we wish so then this expression which is really expression for the shift will diverge but if we use points up to G equal 1 to estimate shift then we would obtain this kind of value and from the which is half equal time value for the critical coupling with some sort of relatively close to each other keeping in mind this error again, if we don't use data too close to the critical coupling which is 2.1 and this is why we try to use this sector dependent scheme in order again to avoid this because sector dependent in the higher so if we truncate up to the three particles you will have only one body and three bodies so there is a self energy in the one body but there is no self energy in the three body sector and because of this I mean every constituent will have physical mass here so it means that higher fork state is not suppressed by this invariant mass invariant mass again this is really result of the application of the kinetic energy operator on the state this is what is so called invariant mass so I will not go to the detail of the sector dependent calculation in any case you are giving physical masses for this higher state whatever it is up to three particles or if we really did this up to nine particles and each previous for that particular truncation we calculate the masses we just use some special scheme from the sector dependent calculation but in case there is this plot is the result with this new scheme sector dependent scheme which is supposed to give more room for this higher fork state and again because convergence is slower here we have to go for the odd case up to the nine body sector and really this is a good thing I mean this is what is expected and again estimation for the critical coupling about 2.1 they really only three some sort of little bit different but five, seven and nine relatively agree with each other and now this is a combined result whereas this open symbols it just for state truncation from three to seven and then close them dark figures it's the sector dependent scheme up to the nine truncation and also LFCC result for this critical coupling and what is there could be extracted from this so the sector dependent and standard fork state truncation they some sort of agree with each other and critical coupling from both cases coming to approximately 2.1 sector dependent converge more slowly and this is really expected because we expect that higher fork state should become more important so it doesn't converge so quickly and also there is this plot of these probabilities again we will hoping maybe here we will have this increasing of the probabilities near critical coupling again for the but at least we see that sector dependent probability is higher which is again dark figures higher than the regular space fork space truncation except of course I mean when we go to the highest state they almost the same but we still don't see this a growth of probabilities near critical coupling and some sort of summary out of this there is convergence we definitely see that probability for the highest state is smaller than probability for the lower states and that also this sector dependent probability they are higher than regular fork state truncation probabilities which is again good indication that in the sector dependent approach higher fork state becoming more important but we expected more rapid increase near critical coupling we did not see this so this hypothesis that this high invariant mass suppressing this higher fork state it's really was some sort of incorrect so we wanted to go to some coherent state approach or to add another term in the LFCC just hope it will resolve this issue to see this growth really of this higher fork states near critical coupling thank you for your attention I have a question I did not understand it seems to me you are saying that you are expecting to see that as you are approaching the critical coupling the high occupation numbers will play more and more important role and you don't see is this a message that I should this we consider as a problem so something wrong here because of this because we expect critical coupling is exactly the phase transition so this higher fork state they have to play more important role so their probability has to be higher in equal time this happens so something wrong with the light front something we don't take into account one slide we forgot to add is that LFCC does show an increase but because the calculation breaks down before you reach the critical coupling it's not clear this was another positive point about LFCC I just did not put this slide LFCC probability would grow only they started to grow somewhere okay LFCC we stop some sort of a 1.5 so they grow near this guy so really we just wanted to add more term to LFCC and see what happened because maybe exactly the main point it's exactly truncation and LFCC I mean truncated but just truncation of the way not the truncation in the states one possible problem with the LFCC calculation is that we start with the valence state just as the one particle state and that kind of implies you're keeping that as too important and you probably need to expand the valence sector to include more states so that the one particle state can disappear it can't disappear completely from the current on-sauce because that destroys everything but if you include a more complicated valence sector we have more freedom and then presumably we could reach the critical couple in recalculation that's one of the things you want to try thanks more questions from the two or three people in the room who actually did two light front perhaps some questions well then let's postpone questions to lunch and thanks again