 Okay, so I'm Andrei Davidovich and I'm very glad that I have a possibility to take part in this conference in honor of Dirk and my presentation is on geometrical splitting and reduction of endpoint Feynman diagrams. Well, I want to start with some very simple slides describing what the Feynman diagram is and as an example we can use the quantum electrodynamics where electrons interact using the photons. So elementary interaction vertex is electron emitting a photon and we can also organize interaction between two electrons, electrons using a photon and also we can have more complicated examples of interaction where we have more photons and some of the photons go from one electron line to another electron line and in such way so we can get so-called closed cycles or loops and one of the examples here is a triangle diagram and another example is four-point function which is a box diagram. Now, when we calculate Feynman diagrams, so we are using so-called Feynman rules, so each vertex can have some indices, Lorentz indices, spinar structure and it's proportional to the coupling constant and may depend on the moment of the particles. Then each line connecting two interaction vertices, it's a so-called propagator and it has such a form where we have the four-dimensional momentum in the denominator which basically consists of the energy and of three-dimensional momentum and this is like a green function for D'Alembert-like equation and also each closed loop implies integration over four-dimensional momentum k flowing around this loop but as a result of this integration we may get divergences and so we need some kind of regularization to permit these divergences and one of the most common tools used in loop calculations is dimensional regularization. The idea is to use the space-time dimension, I denote it as a small k is n as a regulator, so we basically introduce some small epsilon to this dimension which is close to zero, so we change four-dimensional integration into n-dimensional integration and then if the integrals are singular then the singularities appear as one over epsilon poles and this is the simple example of dead pole diagram, so for example nu denotes the power of propagator and when nu is equal to one or equal to two, so we have divergences, we have a gamma function of epsilon which basically gives us one over epsilon pole. Now, so I'm going to speak about a little bit more general case when we have one loop n-point function and it looks like that, so we have basically capital n external legs here and in physics this might correspond to the processes say m particles to n minus m particles and in general when we consider arbitrary external momenta, arbitrary internal masses, so such an integral even a scalar integral would depend on half n times times n minus one in momentum invariance of such form and n internal masses, so basically the number of variables grows quadratically with respect to the number of external legs and this is the basically the definition of the integral, scalar integral, scalar frame and integral corresponding to the n-point function and it has n denominators like that and basically we integrate over the loop momentum which is denoted as q in this case. Now, coming back to this birthday, so how Moscow Hobart and Mines got connected through the story began in the early 90s and at that point, so I lived in Moscow, worked on such n-point Feynman diagrams using Mellon-Berns approach, hyperdramatic functions, etc. and Dirk worked in Mines also known as Mayans and studied similar Feynman diagrams using also some hyperdramatic functions but he was using Carlson R functions and in 1992, Dirk invited me to give a seminar at the University of Mines and if I'm not mistaken, this was our first meeting and later on for two years I was a postdoc in Bergen, Norway and Dirk went to Hobart in Tasmania and of course, I mean many of you know that Australia is called Down Under and Tasmania is under Down Under and he worked with Bob Del Borger and then Dirk returned to Europe and next year in 1996, so I was able to continue research work with Bob basically within the same project and this is how our work on a geometrical approach to Feynman diagrams was started and also I spent four years in Mines and of course we've had a lot of useful communications with Dirk, so this is how this presentation is related to Dirk and I have basically another three-point function here so that's how Moscow Hobart and Mines got connected to Dirk. Now back to the integral, so let's consider this endpoint integral with unit powers of propagators, normally these are master integrals that we need and use Feynman parametric representation for this integral, so we have n for integral with one delta function so this and alphas are Feynman parameters. So in the standard Feynman parametric representation, so we have quadratic denominator which contains some quadratic part multiplied by momentum invariance and some linear part multiplied by the squared masses, but we can using the condition that the sum of alphas is equal to one due to this delta function we can make the quadratic form homogeneous in alphas and so basically we multiply linear part by the sum of alphas and then we can rewrite it in slightly different form where we get these quantities Cjl and Cjl can be defined as sum of masses squared minus the corresponding momentum invariant divided by some masses and they can be associated with cosine of some angles and when this cosine is equal to one, so this corresponds to two particle pseudo threshold and when this cosine is equal to minus one then this corresponds to two particles threshold and of course direct geometrical interpretation through the angles and cosines of these angles is possible when the corresponding cosines are between minus one and one, so in this case the angles tau are real but in other regions so we just can continue can consider analytical continuation and instead of trigonometric cosine so you get hyperbolic cosines a simple example of a two-point function so basically we can associate with two-point function where we have external momentum k12 and internal masses m1 and m2 we can associate a triangle and this triangle has sites m1 and m2 and the third site is basically the absolute value of the momentum and also we will use the perpendicular which we call m0 so using this perpendicular so we split our triangle into two rectangular triangles and just to remind so the cosine of this angle has such form of course it's just trigonometric identity and again I repeat that at the pseudo threshold so these sites m1 and m2 are together so the angle tau is equal to zero and at the threshold so they just go in different directions so the angle tau is equal to pi so this is the geometrical picture associated with this simple diagram for the three-point function this situation is a little bit more complicated so we have three independent external momentum variance and we have three masses and the geometrical picture would give us a tetrahedron three-dimensional tetrahedron and we three sites all these tetrahedron are associated with the masses m1 m2 m3 and three other sites shown in red are associated with the absolute value of the external momentum of the three external momentum and here also we can drop a perpendicular from this point on to the triangle made out of these momentum variance and here we have three angles and the definition of cosine so this angle is here if you go to the four-point function so then we get already a simplex and this simplex should be understood in four dimensions so the this slide requires four-dimensional imagination so and we have four masses m1 m2 m3 m4 so the diagram itself is shown here so basically we have four external momentum but we have six internal external momentum variance because we also get mandelstam variables s and t here and okay so in the simplex so we have four masses and we have six momentum variance and what is shown in red is in fact a three-dimensional tetrahedron but the picture itself is four-dimensional because we also have these mass sites and also here basically important quantities are gram determinant which is the determinant of the matrix of these cosines d4 and also the basically the gram determinant corresponding to this red tetrahedron which I call lambda lambda 4 and the hyper volume of this simplex is related to square root of gram determinant and the volume of this red tetrahedron is related to the square root of this lambda which is also gram determinant and if we drop a perpendicular from this point onto the red tetrahedron so the length of this perpendicular m0 would be proportional to the square root of d4 over lambda 4 now using these geometrical pictures so we try to go from fine one parameters to the geometrical picture and on this way we make some substitutions of variables linear and quadratic substitutions and in fact what we do we change the argument of the delta function that now it has this quadratic structure which we used to have in the denominator and this quadratic structure would be equal to one because it's in the argument of the delta function this how we this is how we get rid of quadratic structure in the denominator instead so we put it into the argument of the delta function but while making this transformation so we earn some linear some linear denominator which we have here and capital C is basically a modified matrix well basically we have the same cosines but they are multiplied by some quantities which are square roots of capital F's and capital F's are just partial derivatives of gram determinant multiplied by the corresponding masses and we can see that in fact the vector composed of these square root square roots of F divided by the masses is an eigenvector of this matrix C so and eigenvalue of this vector is just the gram determinant okay so and we also would need the generalizations of lambda which can be easily generalized in n dimensions and also the length of the perpendicular which is still proportional to the square root of these two determinants now we continue to work on the parametric integrals and now we try to make the quadratic form diagonal and we do the corresponding rotation and transform this quadratic form to the diagonal form and if all lambdas are positive then we can just rescale the betas and get the delta which corresponds to hypersphere to the hypersurface of hypersphere and then the only weight in our integral would be one of the parameters gamma to certain power and all the rest all the rest is basically the measure of integration nothing more and all the depend except for the pre-effector all the dependence on the momenta and masses is in these limits of integration which is a dimensional solid angle and amazing thing is that this n-dimensional solid angle is basically the same solid angle as we have here in basic simplex so it's so basically by this transformation so we got the integral where we have the integration over this solid angle and delta function gives us the hypersurface of the unit hypersphere so basically we integrate over this piece of hypersphere which is cut by this solid angle okay so and also from this expression we can see that we have special case when the space time dimension lower case n is equal to the number of external legs so then this factor disappears and basically we have just non-euclidean hyper volume or content how mathematicians call it and this special case it's basically a two point case in two dimensions or three point case in three dimensions or four point case in four dimensions and so and another important question is what happens if some of the lambdas which we get when we make the quadratic form diagonal if some of them are negative it means that in this case so we would need to we would get hyperbolic surface instead of spherical surface and all our equations or all our results for these Feynman diagrams can be obtained just by analytical continuation of the results which we get in another region for example for the spherical case but now again to compare these two approaches or these two representations this is standard Feynman primate representation so we have unfold integral with one delta function and it depends on the masses and momentum invariance basically through these masses and cjl in the quadratic form which is in the denominator of this parametric integral in the geometric representation well again except for this pre-factor all the dependence on the masses and momentum variance is in the n-dimensional solid angle so it's just here so there is no dependence left in the integrand and another important thing is that we can use this geometrical representation for splitting and I will explain what does splitting mean and for the reduction of the number of variables in separate pieces of the of the occurring functions and basically for the simplification of the result now let's continue with the two-point function so this is the basic triangle associated with our two-point function again we have mass m1 m2 and the absolute value of the momentum and our integration will go over a circle unit circle basically over the arc tau12 so this is the angle tau12 but if we use the perpendicular here then this tau12 would consist of tau01 and tau02 so if you put some point 0 here and all the relations I mean we get just using trivial trigonometric formula and the area of course will be proportional to the sign of this tau12 like in usual trigonometric formula but now what we can do here so if we now consider new momentum variance k01 squared and k02 squared and of course square root of k12 squared is equal to these two square roots then each of the resulting each of the resulting triangles will be rectangular triangle and the sides of these triangles will satisfy the Pythagorean theorem of course and we can split our integral which would go over this angle tau12 into two integrals one of these integrals would go over tau01 and another integral would go over tau02 and basically each of these pieces can be associated with new Feynman integral but with different momentum and masses so we split basically our original integral with k12 squared and m1 m2 into two integrals the arguments are shown here but what's important that now these arguments of each of these integrals satisfy the Pythagorean theorem so now not all three of them are independent but just two of them are independent so effectively we reduce the number of variables so if these integral dependent on three variables now both integrals on the right hand side depend only on two variables now of course I mean one of the variables we can just use as the dimension and if we speak about the dimensionless variables this means that in the original integrals integral we had two dimensionless variables and in each of the resulting integrals we have one variable less so starting with two dimensionless variables we end up with one dimensionless variable and if we look at the quadratic form in the Feynman parametric integral so the original one was this one but in each of the resulting integrals we can use this Pythagorean theorem and we get for example in one of the integrals we get just such expression which is which has two terms instead of three terms here and we can easily calculate the integral with such denominator which gives us in arbitrary dimension just the f to one hyper just house Gauss hyper geometric function now the three point case again so let's try well remember we have this special case when the space time dimension is equal to the number of external legs and here basically in this special case our result for Feynman integral is just the area of spherical triangle which is the spherical excess and if we consider these dihedral angles size so this would be just the sum of this size minus pi and this result corresponds to the result obtained by Bernie nickel in 1978 so we just reproduce the result his result in this simple case but in general case when we have arbitrary space time dimension so we basically follow this geometrical procedure so first we drop perpendicular from this point on to the red triangle which is composed of the momentum variance and this is the picture basically this is the spherical triangle or spherical or hyperbolic triangle so basically the integration goes over this spherical triangle but the the solid angle is defined by this tetrahedron so basically we have dual picture now so we drop this perpendicular so let's call this point 0 then we connect it with each of the vertices of this red triangle and we get corresponding connections in this picture as well and basically we split our original tetrahedron into three tetrahedra and in each of these tetrahedra we have extra two conditions on the variables which we get now just take one of them say this lower one along the points one two and zero and split it again into two tetrahedra so now I mean our original tetrahedron is split into six tetrahedra but by dropping this perpendicular so we get an extra condition on the again due to this category and theorem and if we look at the number of variables then in the original one in the original three point function we had six variables but one of them can be extracted as dimensions so we we get five dimensionless variables in the original one after the first splitting into three tetrahedra so we get two relations so now each resulting each of the resulting integrals depends only on three independent variables dimensionless variables and after the second splitting so we get one extra relation so each of the integrals depends only on two dimensionless variables starting from five we get two but we have six pieces and the original quadratic form of in Feynman parametric integral was this one containing six terms and if we take one of these lower tetrahedra so the the quadratic denominator would contain just three pieces so and this is basically an illustration so we started with five basically we reduce the number of variables by three so we go from five variables to two variables and using this representation with quadratic denominator containing just three terms so we can easily calculate it and we get for the general space-time dimension we get the result in terms of apple hypergeometric function of two variables and by the way all these arguments of the occurring hypergeometric functions have very transparent geometrical meaning so basically if you look at these pictures so it's just the length of corresponding line and that's it okay and now for the four point function for the four point function so we have these simplex in four dimensions and we have certain hyper-solid angles so to say at these vertex and and what this solid angle cuts out of hypersphere is a non-euclidean tetrahedron in fact which might be spherical tetrahedron or hyperbolic tetrahedron depending on on our external invariance and so I'm trying to make this picture here which should be also understood in it's a three-dimensional non-euclidean tetrahedron basically either spherical or hyperbolic and now we also begin the splitting to reduce the number of variables because we have too many variables in the general case in the general case we have four masses and six momentum variants so we have 10 variables so we have nine dimensionless variables so let's do the splitting so first we drop a perpendicular from this point on to this red tetrahedron so their intercept is just one point so because this is one-dimensional object this is three-dimensional object in four dimensions so their intercept is just a point and this point corresponds to a certain point inside this non-euclidean tetrahedron now we connect this point to all the vertices of this red tetrahedron and also here we get corresponding lines and now we can so basically we split the original tetrahedron into four tetrahedron now we take one of them for example this one and drop a perpendicular from this point to the base of this tetrahedron now we split it into three tetrahedrons so we split the original one into four now we split this one into three and at each step of splitting we get extra conditions due to this Pythagorean theorem because we always drop the perpendicular so we use this Pythagorean connection many times so now take one of the resulting tetrahedra this one for example and drop another perpendicular from this point to this point to this side so and basically this is what we get so sorry we first we split the original tetrahedron into four then into three and then into two so basically we split it into 24 tetrahedra four times three times two so and now let's look at the number of independent variables in each of the pieces in the original one so we had 10 variables minus one dimension so it's nine dimensionless variables after the splitting the original tetrahedron into four tetrahedra so we get three relations in each of the resulting pieces so we get six independent variable sorry dimensionless variables after the second step we get four dimensionless variables and after the last step we get three dimensionless variables in each of the resulting 24 pieces so starting with nine dimensionless variables sorry and up by the functions which depend only on three dimensionless variables each so we got read basically by six of them and if we look at the quadratic form in Feynman parameter integral so the original one had 10 terms and if we just take one of these two sorry for example this tetrahedron so then the corresponding denominator has just four terms instead of ten terms and you can also see that normally we have partial sums of these Feynman parameters multiplying the corresponding invariance and this looks like a general feature so I will discuss it a little bit in the general table and yes and we can basically for each of these integrals so we can calculate the we can get exact result in arbitrary dimension and it would be a certain certain case of Lorechel's run function so fn so here is the definition of this function but it depends on three dimensionless variables so like we calculated by this Pythagorean theorem and here is the table showing the reduced number of variables for two three and four point function and also it can be easily generalized to arbitrary endpoint function so total number of dimensionless variables is two five or nine and this is the total number for endpoint function the number of pieces in in how many simplices we split our original simplex is two in the two point case six in the three point case 24 in the four point case and one can easily see that each time we just multiply it again so we get n factorial in the general case and the reduced number of variables is one two three and in general case it would be n minus one so we started with the number of variables quadratically growing here and we ended up through this geometrical method by linearly growing number of variables and also if you look at the quadratic forms so again we see that I mean there is a general feature that we get these partial squares of partial sums of Feynman parameters okay so now the summary so well so we suggest you propose this geometrical way to regulate dimensionally regulated Feynman diagrams all our momenta squares and masses require direct geometrical meaning and we also get this hyper solid angle which basically which gives us all the information about the dependence on the momenta squared and the masses and in the one loop endpoint case so we can relate the results to certain volume integrals in the non-euclidean geometry and of course it could be either Lobachevsky case or Schleffly case so we can study this but it depends basically it depends on our momenta variables and the masses and analytic continuation so can be can be done from basically from spherical case to hyperbolic case and in a number of cases so the results all the epsilon expansion so you get hyperdramatic functions but still we need to consider epsilon expansion if you want the results for physical quantities so some of them can be presented in terms of generalized poly logarithms in more complicated cases so we can get multiple poly logarithms and so on but we didn't consider the epsilon expansion in this presentation and geometrical splitting gives us basically straightforward way to reduce general integrals to those with lesser number of independent variables and the resulting integrals can be calculated and can be expressed in terms of hyperdramatic functions and here are the types of hyperdramatic functions that we get here and also maybe the last slide so I want to mention that there are several papers right now which use geometrical methods maybe similar to this one maybe sometimes slightly different from this one so I try to collect some of them but certainly I couldn't get all of them so my apologies if if I forgot something so you can tell me if I forgot something here but there are many people working basically using some of the some of these methods so thank you thank you Andrei thanks very much thanks for all the time you've put into these nice pictures I have one question just quickly you mentioned in the end that you did not discuss the epsilon expansion but what you did say is that you refer to dimensions right you had formulas with the hypergeometrics and arbitrary dimension so I was wondering because at the beginning you said there's something special about the two point function two dimensions the three point function three dimensions and so on these cases where it seemed like the dimension should be something very specific and now you said that you can actually do it in all dimensions I was just wondering if you could explain a little bit well I think in one of these papers maybe in this one this case of spacetime dimension equal to the number of external legs was explicitly considered for arbitrary number I think it probably was this one but I need to refresh my memories but if we look at the expressions here basically let me see yes so basically what we can see here in the geometrical representation when n lower case is equal to the number of external legs so basically what we have this denominator disappears because it's power zero basically and what we did is basically just the hyper volume of this piece of hypersphere which is cut out by our solid angle so say if we consider four point functions so this is the volume of the spherical or hyperbolic tetrahedron without yield factors and this is the property also in higher dimensions so for example for the four point functions it's a three dimensional non-euclidean tetrahedron for the five point function it would be four dimensional simplex but again it would be the hyper volume or content call it and so on so we will not get any weight factors like that so the situation is simpler but the problem to calculate such quantities it's still a complicated problem yeah and you're saying that this geometrical composition applies no matter if this factor is there or not I mean we know what what does it mean geometrically but geometrically even calculation of the volume of three dimensional non-euclidean tetrahedron is a complicated task so it was a non-trivial task to solve it thank you very much there's a question from David yes well really it's a comment to remind you about a rather interesting geometrical paper that you also wrote in Tasmania with Bob D'Oborgo about the volume of three-body phase space which you interpreted geometrically now that is taking us into elliptic territory it's the discontinuity of the two-loop sunrise diagram and since you did that work there's been a lot of interest in three-loop and four-loop sunrise diagrams so I found myself last week trying to help Albrecht Clem understand the four-loop sunrise diagram working out the volumes of the five-body phase space in two dimensions and going back to your work with Bob to remind myself how to do that so not only did you make contributions at the one-loop level where the multiple polar logarithms arise but you also opened up this field of elliptics and beyond which is the subject of many conferences these days okay thank you thank you David yes of course I remember about that paper but I had just 45 minutes for my why I didn't include it thank you very much let's all thank Andre again for the talk