So, last class we are discussing the solution of unconstrained optimization problem using Newton's method and we have seen there are two drawbacks are there. One drawback is there that you have to each iteration you have to find out the Hessian matrix or second derivative of the cost function or objective function and that must be at each iteration must be a positive definite matrix. And this is the first drawback and because this condition must be satisfied because we are going over the descent direction. So, that each iteration from kth iteration to k plus 1th iteration the function value should decrease that is the condition that. So, Hessian matrix must be positive definite another drawback is there each iteration you have to do the inverse of this matrix whose dimension is n cross n that inversion Hessian matrix inversion you need it at each iteration. And n is the small n is the number of variables involved in the decision objective functions. Then we have seen that this problem can be overcome by using quasi Newton's method it is similar to Newton's method only the inverse of the Hessian matrix at each iteration is replaced by a some matrix S suffix kth iteration that matrix is updated at each iteration that you have to update. Then how you have to update that one we have explained this one, but now we will derive that expressions. If you recollect this one that we want to replace in the Newton's method this inversion that means Hessian matrix inversion by a positive definite matrix X of k suffix k. And then next is next is X of k how we have what is called updated this one we have seen that this is the updated expression we have shown you that X k plus 1 if you see this one X k plus 1 update X k then this things you have to calculate delta k is nothing but a that successive two iteration that X difference of X values gamma k is the successive two iteration the gradient of objective functions difference of gradient of objective function. This way you have to update how we got it this expression that we have we are going to discuss now. And the derivation is based on that our concept like this way that the function value let us call f is a function which is a function of X and variable the function difference of function value at two successive points carries the information of derivative of the functions. Similarly, the next second derivative that the gradient of the function at two successive points carries the information of the second derivative of the function that means Hessian matrix information carries based on this one we have retained that one that Hessian matrix of the function at k plus 1 instant or the iteration is equal to the difference of two difference of two gradients at successive points two successive points X k plus 1 and X is equal to X superscape k. This difference carries the information of the second derivative of this function that means Hessian matrix information if you multiply it by that change in that is variable change in difference in that variable change if you multiply it then this is equal to expression you can write it from there I rearranged yesterday if you see X k plus 1 minus X superscape k is equal to this matrix inversion then Hessian matrix inversion. So, I mentioned earlier then Newton's method for solving the unconstrained optimization problem this is creating problems first thing delta square f Hessian matrix must be positive definite that is next is this inversion is always exist when the number of variables is very large the composition burden involved here is much. So, this one we are not taking the inverse we are replacing this one by a matrix X k plus 1. So, this we are replacing X k plus 1 now see this one how this X k plus 1 is that X of k each time it is updating. So, that is we will discuss now now look this let us call this expression that is what we will write it this expression you write this equation number 1 plus this is equation number 1. So, now question is given X k then X k plus 1 X k plus 1 can be obtained can be obtained using correction formula correction formula and that correction formula is based on the rank 1 type or rank 2 type correction formula. So, various type of correction formula is available that means 1 may be a correction type of order 1 rank 1 rank 2 another way. So, you can write various correction formulae have been developed in the literature have been developed most of which are of the type 1 or type 2. So, let us see now we can write X k plus 1 is equal to X k. So, this now I am updating X k I know X k plus 1 at kth iteration what is the Hessian matrix inversion at k plus 1 at the iteration that I am finding out by S k plus adding a correction term delta of suffix k where this delta of k is a matrix of same dimension of X k and X k dimension as you know it is n by n this matrix also n cross n matrix and that matrix is a positive definite matrix this. So, that matrix of rank 1 or rank 2 or rank 2 the simplest choice of delta k 1 can take like this way the inner product or the outer product of any two vectors which is not a null vector. So, I can define delta k as a special choice of delta k like this way alpha k u k and u k transpose this is the inner product of vector u k at kth iteration u k any vector you have considered multiplied by u k transpose. So, this product of this outer product of the vector u k and this same vector outer product if this is a matrix of dimension n cross n where the u k dimension is n rho 1 column and alpha k is a scalar quantity whose value is greater than 0 that is scalar quantity scalar greater than 0 this. So, this is the choice of this one and this matrix if you see the this is always positive semi definite matrix. So, if you add it with a positive definite matrix another positive semi definite matrix results will be a at most it will be a positive semi definite matrix of that one or sorry this result will be a positive definite matrix of that one again because this is a positive definite matrix. Now, a positive definite matrix add with a delta k which delta k is positive semi definite matrix results will be a positive definite matrix this matrix is positive definite matrix and this is positive semi definite matrix and that results will be positive definite matrix. So, this with this one let us call this equation is I am considering this equation number 2 and this is the equation number this equation is equation number 3. From 2 one can write it from 2 x k plus 1 that what I mentioned is that is a symmetric matrix symmetric if s k is symmetric. And see when you take the outer product of 2 vectors this results is you will get a symmetric matrix. So, symmetric matrix plus symmetric matrix is a symmetric matrix of this one that is from 2 we can see this one. Then note the what we have defined delta k is nothing but a the change in decision variable value at k plus 1 th instant minus at k th instant. So, that is we are denoted by delta k symbol similarly, gamma k this gamma k is the difference of gradients at 2 successive points.  So, let us call this is equation number last equation we have given the number 3. So, this is 4 and this is 5 again our main job is here now replace that u k and alpha k this u k and alpha k replace in terms of the information delta k and gamma k that is our next step how to replace it. Aim is to replace is to replace that our alpha k and u k in equation 2 with delta k and gamma k this. So, how you do this one now see this equation number 1 this equation number recall the equation number 1. If you recall equation number 1 then I can write it x superscript k and this is what we have denoted by this one gamma k the change in the what is called gradient of the function at 2 successive point change we have denoted by gamma k. So, I can write this is gamma k is equal to delta k see this one if you take this if you take this side it will be a delta k. So, I can write it this expression from equation number 1. So, now what is x k plus 1 this I can write it x k plus delta k that is the what we have consider x k plus 1 we have consider like this way into gamma k is equal to delta k. So, now I can this if I take it that side x k gamma k in the right hand side. So, I can write it delta k gamma k is equal to delta k minus x k into gamma k this is x k. So, that one you can write it and one can write it this one because you know what is delta k I have consider delta k is nothing but a alpha k what a inner product of same matrix u k.  So, it is a vector of dimension n cross 1 and this is also dimension u k transpose dimension 1 cross n. So, this quantity this two product scalar quantity scalar quantity now both side I multiplied by gamma k transpose both sides and alpha k is also scalar. So, if both side you multiplied by gamma k then left hand side alpha k then gamma k transpose u of k then u transpose of gamma k is equal to because I both side I multiplied by gamma k this equal to delta k minus s of k into gamma of k. Now, this you see this is a scalar and inverse of the scalar is that one this is also scalar. So, what is this value is later let us call it is 5 that is also 5 if you take inverse of this scalar quantity scalar.  So, let us write it this is equation number last we have gone up to equation 5 then you can recall that one is this you can consider equation number 6 then you can call this is a equation number 7 that one and there is a identity you can see one can write it this identity without any problem note this identity note this identity what is this identity alpha k u k u k transpose.    So, this is a scalar so this is this scalar this scalar same thing same quantity I am written down so you can cancel it. So, this alpha and this alpha you can cancel it so what is left alpha k u k u k transpose that is same thing, but we have written into this and this are the identity either you write this one or you write this same thing, but this quantity whole because I can divide this is a matrix right hand side this quantity from here to here must be a matrix and this is divided by a scalar quantity you see this is a scalar this is a scalar and alpha is we have considered alpha is a scalar. So, I divide the matrix by a scalar quantity so these are the identity you can say. So, now from 2 what is the equation number 2 if you just say the equation number 2 here this equation what how to update that x k plus 1 k plus 1th iteration. So, 2 if you take that side x k plus 1 minus x k it will be equal to delta k. So, I am writing x k plus 1 minus x k and x k you know what is the inverse of Hessian matrix at k th iteration or k 2 successive points k plus 1 th iteration the inverse of Hessian matrix and this is the k th iteration inverse of the Hessian matrix. The difference of this one is written by you see that one equation number 2 this difference of this delta k and delta k. So, I can write it that the value of delta k now here what is the value of delta k from equation from this equation this expression I can write it the whole expression what is because this nothing but a that one just now we have shown it.   So, this and you can see here this is nothing but a transpose of that one that quantity is transpose of this quantity is transpose of that one or vice versa. So, divided by that scalar quantity and what is this quantity this is a scalar whatever the value of this one and this value are same. So, I can write it square.  So, it is scalar quantity and this quantity you see this is what you can easily verify this one this is a scalar quantity again and this is also scalar this 2 scalar quantity is a scalar quantity u k is a column vector u k is the column vector whose dimension is n row 1 column and this is the transpose of this one that will be a row vector. So, column vector multiply post multiply by same matrix with a row vector that results is a matrix. So, I am getting this one. So, from 6 and 7 that 7 expression you see with this one alpha k u k gamma k square this quantity I can write by replace by this one denominator expression I can replace by that one or replace by that equation number 7.  So, using 6 equation number 6 and 7 using equation number 6 and 7 or from 6 and 7 from 6 and 7 we can write x k plus 1 minus x of k s this is s k is equal to I am replacing from 6 and 7 delta k minus x k gamma k into delta k minus x k this is s k gamma k whole transpose divided by gamma k transpose delta k minus u k and this is gamma k. So, that what is this value I just see this one this gamma k transpose x k this is delta k minus this I am replacing equation see this one here alpha k u k transpose gamma k equation to gamma k transpose delta k minus x k then gamma k x k gamma k. So, this is x k delta k minus x k x k into gamma k s k is a matrix multiplied by vector column vector is a vector. So, vector minus this vector and multiplied by another row vector. So, it is a scalar quantity. So, this I can write it.  So, you see at each iteration now we have derived at each iteration the inversion that is Hessian matrix inversion is replaced by a matrix equivalent to matrix x k and next iteration x k plus 1. That means, inversion of Hessian matrix at k plus 1th iteration we can obtain knowing the previous value of x k and this compute that one and this one you know delta k is the difference of the values of the decision variables x value difference at two successive point k plus 1 and k 2 1 and gamma k you know the difference in gradient of a objective function at two successive points the difference this. So, I can compute this one for k is equal to 0 k is equal to 0 1 2 3 dot dot in this way. So, each iteration one will compute recursively like this way I am avoiding the inversion of a what is called that Hessian matrix inversion when you will compute the solution of optimization problem using the Newton's method. And when you will replace in by a matrix this and recursively we are calculating s k plus 1 in place of inversion of a matrix then we will call it is a quasi Newton's method. It is this is like Newton's method only the different way we are calculating. So, one of the advantages this is look at this point this is a scalar quantity this whole thing is a scalar one must be careful that denominator part should not be very very small or 0 if it is 0 or very very small this quantity will blow I mean very large value will come. So, that is the one disadvantage of this rank one this problem can be this is this derivation is based on the rank one because we have considered if you recollect the delta k we have added with a x k by choosing the delta k matrix like this way a vector outer product of same vectors u k u k transpose and that rank is 1 and that basis we have derived. So, if this is very small or very very small or 0 then you cannot apply these things. So, one can the alternative way is you go for rank 2 for choosing the delta k. So, this is the one disadvantage advantage is there you do not need to check the gradient direction because that direction that is what we are checking the gradient direction which in turn we got that what is called Hessian matrix must be positive definite matrix it is not necessary in this case because it is automatically once you see this one once you have selected at k is equal to 0 s of 0 is positive definite matrix any positive definite matrix of that one.  So, the results always it is guaranteed that this is a positive definite matrix. So, you need not to check the what is called descent direction of a function each iterations. So, that is the one disadvantage of this Newton's method and due to this one it is widely used for optimization what is called unconstrained optimization problem that method quasi Newton method is widely used due to this advantage. So, next we will see so far we have discussed that how to solve that what is called unconstrained optimization problem by using the numerical techniques or directly you can solve it let us call unconstrained optimization how will you solve it first the gradient of this vector u s and 0 and solve it find out the what is called stationary points then you find out the second Hessian matrix of this one and check whether the Hessian matrix is positive definite or negative definite or not check it and you will be able to conclude that function is minimum or maximum. If the Hessian matrix is positive definite then the function is a minimum function value of the function you achieve the minimum at that stationary point. If the Hessian matrix is negative definite by analytically if you solve if you get it negative definite this means at the stationary point corresponding to that stationary point the function value you will get the maximum. So, let us go next is that constrain optimization problems how to solve the constrain optimization problems. So, next is your optimality and once you see knowing this algorithm and one thing I just forgot to tell you that what is called steepest descent method the convergence rate is linear means order 1 whereas in conjugate gradient method the convergence rate is between 1 and 2 order 1 means what it is linearly it is approaching to the what is the error is decreasing or it is approaching to the function value is approaching to the minimum value of this function linearly. Whereas in Newton's method the convergence rate is quadratic in other words the convergence rate order is 2 and similarly that what is called Newton's method is order convergence rate is much faster than the steepest descent method and its order is 2. So, next we will consider the optimality condition for constrain optimization problem optimality conditions for constrain optimization problems. So, let us if you recollect our previous first lecture we have discussed that what is the basic structure or mathematical formulation of optimization problems agree. Now, we are going for constant optimization problem basic structure if you see generalized constraint optimization problem. Most of the practical problems you see there must be constant real time real world problems that must be constraints and not only constraints then equality constraint means and what is called inequality constraints agree in addition to this one there is a side constraints are there some of the variables cannot be negative all these things that we have discussed at length in our first lecture or second lecture agree. So, let us call what is the generalized optimization problems that minimize if you recollect minimize f of x and x dimension is n cross 1 which you can write it with x as a n variable. So, our decision variables are n variables are there we have to minimize these functions. So, let us call this function I am going to equation number 1 agree. So, subject to I just recollect our problem that subject to equality constraint. So, we have a h i which is a function of all decision variables may not be some of the function may be few decision variables x 2 x 3 dot x n and these are the all equality constraints agree and this i varies from 1 to dot dot p. So, there are p equality constraints is there with the optimization problems subject to how many i is equal to 1 h 1 is equal to 0 h 2 is equal to 0 dot dot h p is 0. So, we have a p equality constraint and also inequality constraints g j which is a function of x 1 x 2 x 3 dot dot x n agree x n is equal to 0 and that we have a x 1 2 dot dot m small m. So, we have a m inequality constraint sorry this is inequality we have a m inequality constraints are there. So, our problem is minimize this function subject to this constraints agree. So, this is equation number 2 and this is equation number 3 if the equation number 2 3 1 2 3 are all function of linear functions then we will call linear programming optimization problems. Otherwise it is a non-linear optimization problems any one of this is non-linear that we have discussed earlier also. So, we have a linear optimization problem may be non-linear optimization problem may be let us see about the constraints get some idea about this constraints. Suppose, we have a problem like this way minimize example just to explain the constraint of a minimize f of x and x is a in your case 2 variables dimension is. So, it is a x 1 minus 2.5 whole square plus x 2 minus 2.5 whole square agree our problem is minimize this quadratic functions agree. Subject to g 1 of x which is a function of x 1 and x 2 agree and is equal to 2 x 1 plus 2 x 2 minus 3 is less than equal to 0. Let us call we have a only one inequality constraint. Please note that when there is a equality constraint is there we are using h small h when there is a inequality constraints are there we are using the small g that is we will use this notation throughout our discussion. So, this is the problem and this is you have to optimize this one and if you see the graphically this problem of that one agree. Another constraint also you can see g 2 which is a function of x 1 and x 2 agree is equal to minus x 1 less than equal to 0. Another is g 3 x 1 x 2 is equal to x 2 greater than equal to 0 just you can see this one. So, this what is this one let us see this point g 2 inequality constraints minus x 1 is less than equal to 0. That means it indicates the x 1 value is always positive. So, this in other way I can write it this one I can write it also g 2 of x 1 and x 2 I can write it is equal to x 1 greater than equal to 0. Because this expression is showing that this quantity always less than 0 when it is possible when x 1 is greater than 0 greater than equal to 0. So, this indirect or in other words I can write this one. So, let us say that constant g 2 in place of this one I am writing that one x 1 is greater than equal to 0. That means this indicates where which region that x 1 is always positive. So, I can see the our region of x 1 is that side this side this. So, if you consider this is our x 1 and this is our x 2. So, this indicates that x 1 greater than 0. So, x 2 greater than 0 the third inequality condition g 3 means x 2 value is always positive or 0. So, this indicates that region that region. So, the first quadrant g 2 and g 3 indicates the first quadrant the our design space or design variables must lie in the first quadrant from g 2 and g 3 I can say this. Now, what is this equation that first constraint inequality constraint that you just draw the line that 2 x 1 plus 2 x 2 minus 3 equal to 0 draw the line first. Then I will decide less than 0 in which side of the line or greater than 0 which side of the line. So, if you draw equal to then x 1 is if x 1 is 0 then x 2 is becoming x 1 is 0 2 x 2 minus 3 is equal to 0 means x 2 is 1.5. Let us call this is 1 this is 2 this is 3. So, 1.5 is here this point this is the equation of this straight line this. Then when x 2 is 0 then x 1 is again 1.5 this is 1 2 3 and 1.5 is here 1.5 is here. So, this is equation of straight line. So, I can draw it this one this is I cannot extend because it does not valid because x 1 x 2 value is greater than 0 means in first quadrant. So, you cannot go like this way that valid up to this region design space is only first quadrant. So, this indicates that equal to 0 g 1 that mean g 1 is equal to 0 any point on the line, but we have given that g 1 is less than equal to 0. Again less than equal to 0 that means which region the g 1 less than equal to 0 that should be what is called in this region that is this region is g 1 less than 0 less than equal to 0. When equal to that is when g 1 equal to 0 that is on the line when g 1 less than 0 that means in this region the whole region if you see just straight line if you extend it the whole region g 1 the whole region of that one the whole region g 1 of x 1 x 2 less than equal to 0 either on the line or this, but our g 2 and g 3 are telling that our design variable x 1 and x 2 lies on this first quadrant. So, we cannot go outside this first quadrant. So, our design space or design variables must lie within this triangle. Now, what is our problem or problem is to minimize that one. So, if you see this one this is nothing but a equation of a circle whose center is 2.5 2.5 comma 2.5 this is our center of the circle. Now, it is obvious you see minimum value of the suppose this constants are not there what is the minimum value of the function at centers means 0, but we have a constant. Now, if you increase the size of this that is that with the center if you just the function value you have to increase it. Suppose, I put it a function value this is some value function, but this does not satisfy the our constants agree this three constant. Combinedly we have shown this is the our space where this our x 1 x 2 must lie. So, now next our if you go on increasing. So, our minimum value of the function from geometrical point of view one can say lies on the straight line on the straight line. How from there you draw a circle which will touch this straight line and beyond that if you want to increase the size of this circle it lies no doubt on the design space or the design space, but function value is increased this is not the minimum. Minimum of it will be what which will touch this line let us call that one. So, this is o point this is p point here. So, now you can easily find out what is this what is this point at which this circle will touch and that point you will get the minimum value of the function it is a simply geometric concept. So, when it will touch this one one can find out which from geometric point of view from this point to on this line what is the perpendicular distance you can find out. Once you know the perpendicular distance of this one you can find out what is the coordinate of that one by using some geometry of this one. So, once you know this coordinate immediately you know what is the function value put the value of x 1 and x 2. So, I will just give you this hence that o p you can find out if you write it o p the perpendicular of distance this one by coordinate geometry distance 2 x 1 you see this equation plus 2 x 2 minus 3 divided by the coefficient of a x plus a x plus b y plus c is equal to you know the distance from any point is that value is 2 square minus 4 plus 2 square 4 put the value at x is equal to x 1 is equal to 2.5 from this point we are doing and x 2 is 2.5 if you find out this one that value will come approximately not approximately it come this one. Now, this line this o p and this straight line are perpendicular to each other and you see you can easily find out the slope also if you like the slope of that one also. What is the slope that means if you draw this one this and this coordinate you do not know let us call it is x 1, x 1 bar and then x 2 bar this point. So, immediately you can find out this is from here to here is x 1 bar y x 2 bar this is you know 2.5. So, it is 1.5 divided by this is you know 2.5 this is 2.5 this is x 2 x 1. So, 2.5 minus x 1 this 2 ratio this by this ratio is same means 1. So, the slope I know this one from there you will get it 1 expression x 1 bar and x 2 bar expression. So, I can write it if you see 2.5 minus x 1 x 2 bar minus 2.5 minus x 1 bar is equal to 1 that implies x 1 bar is equal to x 2 bar. So, this coordinate x 1 x 2 are same next what you do this coordinate must satisfy this circle equation. And what is the circle equation if you see the circle equation is our x 1 minus 2.5 whole square plus x 2 minus 2.5 whole square is equal to the distance we got it. If you recollect this one o p distance is 7 by 2 root 2 7 by 2 root 2 7 by 2 root 2 7 by 2 root 2 whole square. So, put this value x is equal to because it satisfy at x is equal to x bar and x 1 bar x 2. So, if you put this value in this case. So, it will be x 1 bar minus 2.5 whole square plus x 2 bar minus 2.5 whole square is equal to this quantity. So, solve this two equations I know x 1 is equal to x 2. So, x 1 we know x 1 is equal to x 1 2 is equal to x 2 bar. Then you can find out x 1 bar from this one you can find out x 1 bar is equal to x 2 is equal to that value will come x 1 bar will become 1.26. Please check it just in this expression you put it will get it. So, and we know and physically this is the what is called this three equations inequality constraints I just drawn in a graph and shown without solving any what is called optimization technique all this thing physical many geometric concept of view. I find out what is the point at what point the function value will be minimum that is one. But next we have to do it through what is called by analytically. Now, look at this expression if you recollect this one in general let us call we have a one expression is there x 1 x 2 that expression and that expression is what in your for this specific problem see that one 2 x 1 plus 2 x 2 minus 3 is less than equal to 0. So, this quantity this quantity I can always write in equality constraint form, but what does it mean this quantity is less than 0. That means if I add some positive quantity with this one then I can make it this is equal to 0. So, I am writing now that our inequality constraints is now converted into equality constraints how see this one plus I am adding one positive term because this is less than 0 means negative term I have to add some positive term to make this is equal to 0. So, any inequality constraints are there I can always write it into a this plus some positive quantity you have to add this quantity that is what quantity is greater than 0 or equal to. Now, look at this one when s is equal to 0 that means this satisfy the equality constraint that satisfy the equality constraints. So, any inequality constraint now I can convert into a equality constraint by adding s 1 square which is a greater than equal to 0. And it cannot be negative if it is a negative s 1 square is negative that means it indicates this quantity is g 1 is always negative this quantity. And if it is coming negative that means this violates our inequality constraint given that constraints. So, that s 1 square this should be always greater than equal to 0. Now, question is that we know at this moment how to solve the unconstrained optimization problems. So, if you have a what is called general problem that we have mentioned it minimize the function subject to equality constraint and inequality constraint. And inequality constraint we know how to convert into a equality constraint. So, now next question is if we can convert the unconstrained sorry a constrain optimization problem with equality and inequality constraints and inequality constraints we can convert into equality constraints. Then the problem is if you can transform that unconstrained optimization problem way what is called a constrain optimization problem into unconstrained optimization problem then we can solve our problems as per our we have considered earlier. The unconstrained optimization problem we know how to solve it either numerically or what is called by analytic method. So, let us see this one how to convert one way conversion is quickly if you see this one minimize same problem f of x 1 x 2 that x 1 minus 2.5 whole square x 2 minus 2.5 whole square subject to h of 1 only one equality constraint is there is equal to minus 2 x 1 plus 2 x 2 minus 3 is equal to 0. So, only one equality constraint is there. So, how to convert this what is called constrain optimization problem how to transform into a unconstrained optimization problem one can do like this way you see I will find out x from here x 2 is equal to what minus 2 x 1 plus 3 if you take this divided by 2 is nothing but a half this 2 to cancel it is a minus x 1 plus 1.5. If you put this value of x 2 put the value of x 2 in this objective function now it is a function of x 1 and x 2 is a function of I can write this is a function of x 1. So, it is a function of x 1 which I can write it now if you just put this value of x 1 minus 2.5 whole square then in place of x 2 I will write it minus x 1 plus 1.5 minus 2.5 1.5 minus 2.5 whole square now you see this constraint I have force the equality constraint force in objective functions. So, now it is become a unconstrained optimization problem there is a unconstrained optimization problem optimization problem. So, this problem we can solve what we have discussed earlier this one similarly inequality constraint is there I have just mentioned how to convert any equality constraint and then proceed in the similar way. So, next class we will discuss more general inequality constraint also along with the equality constraint how to convert what is called equality constraint optimization problem into unconstrained optimization problems. Thank you.


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I observe the Stefan Institute of the University of Juana in Slovenia. He obtained his Ph.D. from the University of Slovenia. After that, he moved to Italy for a postdoc position at the ENFN Frascati. Also, he's a scientific associate at CERN. So the title of his talk is outdated on the LHCD Photonexcess. So if you can hear me, Jeremy, maybe you can start your presentation. Yeah, sure. And you can share your screen and everything. Okay, thank you. Let's see this. Do you hear me? Yes, I can hear you. Okay, good. Okay. So I was kindly asked to, by the organizers, to give you a short update on the recent buzz in the higher-energy phenol community. It's about the 750-day Photonexcess seen at the LHC. So as probably most of you already heard about this, the first LHC 13TV data seemed to exhibit an excess in the diphotone spectrum, which is the excess is seen both by Atlas and CMS. And of course, with any such early hints, it's most likely just a statistical fluctuation. But I think it's timely to start considering what if it's not. And in the following, I will be, I will consider the 750G with the Photonexcess, assuming it's due to some real new physics. And in particular, I will assume there is some new particle. I will commonly denote as S. And I will try to address the following issues. What can we learn about its production and decays using existing data? In particular, I'll deal with the consistency with other measurements and searches. And the second issue I'll try to address is whether the possible theoretical interpretations of the success, in particular, can as be embedded into solutions of some of the most existing puzzles and whether the associated predictions for related phenology. So first, let me try to give you somehow a theorist summary of what the experimental measurements are or searches are about. So what the experiments see is a diphotone peak over a monotonously varying or in particular falling background. And the search and the signature is theoretically very clean and experimentally simple. So the background composition in this search is mostly coming from real diphotone events and also a small fraction of photon jet events where a jet fakes a photon. The purity of the sample, so the fraction of events which actually contain two real hard PT photons is very high and it's measured in the data. The background spectrum as fitted in regions away from the excess actually has been found to be in good agreement with next-to-next-to-limb or acoustic predictions, although these are not actually used to normalize the background or its shape. Furthermore, Atlas in particular and CMS have looked at in more detail at the events in the excess region and they don't seem to exhibit any unusual characteristics. In particular, they don't seem to exhibit considerable missing energy, extra hydronic activity, unusual angular distributions or similar. Of course, this is all with very limited current statistics so it's not comprehensive. So in the following, I will assume, so taking into account this information, I will assume we are dealing with a prompt single production of the resonance S and which decays to a two-body decay to a pair of real photons. Given these assumptions then, using the London Young theorem, we can imply that these resonance has to have been 0, 2 or higher. Of course, other possibilities in this have been proposed in the literature. However, they typically require some tuning of parameters. Particularly, instead of considering a single resonance decay into the photons, one could consider the production of resonance associated with, through the decays of a heavier particle, associated with possibly some dark matter which results generically in some missing energy or alternatively, the resonance could decay not directly to the photons but to some lighter particles which would in turn decay to photons which would result in culminated diphoton events which could, in principle, be interpreted by the experiments as simple diphoton events or one could even consider more complicated kinematics. For example, cascade or three-body decays. So there has been various discussions in the literature about these possibilities. The main issue concerning production of S is actually ensuring or testing compatibility between the data taken at 8TV and 13TV. Now, since the ratio of the expected event rates at 8TV and 13TV depends on the enhancement of the cross-example of the production cross-section between these two energies, this one can use this to infer on the preferred production modes of S. In particular, in these two figures that I'm showing, so the left-hand one assumes a narrow width and the right-hand one assumes a width of roughly 6%. And here what I'm showing here are the chi-squared functions which combine the 8TV and 13TV searches by CMS in blue and atlas in red. Just let me check if there is a problem. No, there is no problem. So in case you don't hear me, just let me know. Or there is another problem. So here I'm showing the combined 8 and 13TV by photon search results of CMS in blue and atlas in red given the two width hypothesis, either narrow or wide width. And in black is the combined chi-squared. And you see on the right-hand side, and you see just for illustration on top of this I've sketched the... So this is all plotted as a function of this enhancement factor of the cross-section between the 8TV and 13TV data. And for illustration on top of it, I'm plotting the expected ratios of these cross-sections if the production of S is dominated by particular initial state partners. The largest enhancement of course expected for heavy-flavor quark-anti-quark annihilation and for gloom fusion. On the other hand, in case this was a pure photon fusion production, the expected production enhancement between 8 and 13TV would be actually very small. Certainly below a factor of three or so. What we see in these two plots is that current combination of 8 and 13TV data already tells us that production from heavy-flavor or gloom fusion is preferred to have better consistency between these two. Of course the conclusion also depends somewhat on the assumed width because the significances of the excesses in 8 and 13TV data depend on the width assumptions. And they are somewhat different in CMS and ATLAS. In particular, the ATLAS data exhibit larger significance of the excess in a large width hypothesis while the CMS data exhibit somewhat larger significance of the excess in both 8 and 13TV assuming a narrow width reasons. Also from this exercise one can already infer a preferred production cross-section of a diphoton resonance at 13TV and this varies between 3 and 10 inverse phantom barns and this wide range covers actually all the possibilities regarding the width and the spin. In particular, a larger width and spin tool imply a larger cross-section. The other conclusion one can draw from this plot is that pure photon fusion, as I said, is disfavored. What this tells us is that this resonance S has to have at least two decay months. So it cannot only couple two photons, which is already very helpful. The second issue as I discussed is the issue of the width of the resonance. In particular, the ATLAS data seem to prefer a sizable width while the CMS excesses in both 8 and 13TV data are more significant if narrow and combined this actually results in no preference for S width if one combines both ATLAS and CMS. This is seen in these two plots where I plot now the chi-square of the significance of the excess over the backward only hypothesis combining CMS and ATLAS and also 8 and 13TV data again as a function of the enhancement of the cross-section between the 8 and 13TV and we see that the combined significance at very high enhancements so when 13TV data dominates is roughly the same for narrow or wide width. Once one combines CMS and ATLAS which have different preferences. Another difference between CMS and ATLAS analysis is that the most sensitive CMS analysis or the CMS analysis which exhibits the largest excess is an inclusive analysis meaning that the PT of the photons is fixed and it's not very high, it's 75TV. On the other hand, ATLAS has two analysis, two public analysis and the one which exhibits a slightly more significant excess actually uses a harder PT cut on the photons. In particular, the leading photon is required to have 0.4 has to have PT of more than 0.4 of the environment mass of the dephoton events and the sub-leading has to have at least 0.3. Combined again, these different significance assuming different photon cuts currently do not exhibit any preference for either spin 0 or spin 2. For those of you who may wonder why one would have this sensitivity is that typically while spin 0 predicts a boosting variant isotropic distribution of photons spin 2 actually prefers a more forward more forward photons so would prefer a more inclusive analysis in principle. Okay, so now we know that S has to couple at least two channels so we can now do the following exercise we can assume that S only couples to production channel and to photons which are the observed decay channel then the system is actually completely constrained and we can plot the preferred parameter regions in terms of these two widths for example we can choose a dominant gluon fusion production modes and the dephotonic decay mode and we can plot the preferred range of parameters in this plane given the current measurements in particular if we assume these two decay modes are the only ones significant for S then we have this range of parameters which are preferred if instead we actually require that the width would be wide as preferred by Atlas data then the range of parameters would be required to lie in this upper band here so anything between these two bands is actually perfectly allowed the current data and the second feature of these plots so the right hand plot is the same one as the left hand one just assuming the dominant production coming from BB bar instead of gluon fusion and basically all the other heavy flavor production modes would lie somewhere between these two these are the extreme possibilities because the gluon fusion has the largest luminosity at 750 gb and the BB bar has the smallest luminosity and so the second feature of these two plots is that of course if the production if the decay width dominating the production mode is lowered then at some point photon fusion production starts becoming important and since photon fusion is disfavored when comparing 8 TV and 13 TV data this tells us that this upper left corner of the parameter space is disfavored simply from compatibility between 8 and 13 TV data so this limits the preferred region to these lower right handed parts of the plots here and furthermore on the uppermost right hand side of these plots one see constraints coming from actually searching for digest resonances and this is important because if this S couples dominant to gluons or quarks then it will also tends to decay to these final states and thus it can be searched for in the form of digest resonances and these are the current constraints still coming from 8 TV data superimposed in particular in the case of gluon fusion it tells us that the gluon fusion and the photon decay modes alone cannot saturate a large width because they would be excluded by digest resonances so this of course tells us on the other hand tells us that there is room for additional decay modes of S so one should look for those and one can make a compilation of current bounds on possible decay modes of S to various thermal final states and most of these current constraints still come from 8 TV LHC data but some have been recently updated with 13 TV searches but the numbers in this table do not change by more than order one factors if one includes the 13 TV results in particular what we see is that constraints coming from 8 TV the searches are more significant in if S was to decay to the left and electric final states in particular Z gamma Z Z or Z Higgs or even die Higgs or W plus W minus the least constrained possible decay modes of S are of course hydronic decay modes namely tt bar br bar or light jets and also the invisible decay width of S so combining all this information one can now play the following game one can use the same technique as before but instead of just leaving the third possible additional possible decay modes of S free one can assume that the total decay width of S is now saturated by a single additional decay mode and this leads to this parameter space plot on the left hand side here where the different colouring regions now refer to saturating the decay width of S with different startable final states and what we observe here is that in particular that almost so any electric or leptonic final states cannot serve to saturate a large width of S so they cannot significantly contribute to decay width of S while this is still allowed if one assumes either an invisible decay width or a decays to heavy flavour quarks so again one can turn this around and say if S does not couple significantly to startable quarks and one wants to saturate a sizeable width then this is most easily done by decays to non-stannable final states in particular it's possible for S to decay invisibly and this is something that actually can be tested with using something similar to VBF tag production so vector boson fusion tag production used in Higgs measurements but due to lack of time I will not discuss this further so but what I would like to explore a little bit deeper is this connection with our implications of a possible invisible decay width in particular on the right-hand side plot we can see that given a possible dominant production mode of S so being either glue glue or heavy flavour which I preferred we can see that the contribution of the possible invisible decay mode of M decay mode of S can easily saturate a sizeable decay width and being in accordance with the current bounds on missing energy signatures as you can see because to decay invisibly then one could speculate that it could have some connections with dark matter and this has been explored in depth in several studies already and what is interesting about this possibility is that one obtains correlated effects in both the S width but also in the dark matter relic abundance and in both directed indirect dark matter detection experiments and here I'm just showing a simple example where I'm assuming S is a scalar and it's coupled to fermionic dark matter and then one can superimpose all these constraints on a single plot namely the required scalar coupling so the effective dark matter yukawa coupling to S as a function of the dark matter mass then the saturating the measured dark matter abundance singles out a preferred region in the parameter space and then this can be matched onto the effective invisible decay width of S and this is of course these two constraints can only be saturated at individual points in the parameter space and this should be contrasted with constraints coming from direct detection which then actually depend on how S is dominant in produced in particular if S couples to light quarks direct detection in the form of here parameterizing in the form of spin independent scattering cross-section on protons already severely constrain such possibilities and would actually disfavor completely basically completely exclude saturating the width of S with the dark matter decays so with such an exercise we'll see that the preferred region, the preferred production mode in this case in this very simplistic models would be coming from blue infusion and possibly the BB bar so just to make clear that the two plots refer to either scalar or pseudo scalar couplings of S to dark matter, otherwise there are no the particularity of the pseudo scalar dark matter coupling is of course that in this case the spin independent dark matter scattering cross-section vanishes so the bounds coming from direct dark matter detection experiments are much much weaker and are not shown in this plot ok so the second more theoretical issue I want to discuss is how generic is such phenology so how generically does one expect a diphoton resonance at the LTC to be a first sign of some new physics and this can be done using effective theory methods and as an example let me consider S as a Starr model singlet scalar now if this is so then all interactions to Starr model except the Higgs portal would be irrelevant interactions so naively one would expect to see a resonance not in diphotons but in the Higgs final states ok so this would be the leading interaction one would expect to see in an effective theory description of Starr model singlet scalar of course the easiest way to forbid such dominant decay modes is to assume an approximate CP conservation in this sector and assume that S is a pseudo scalar then of course these interactions are forbidden by CP and then the next interactions in the power counting of such an EFT are actually and then these can in fact dominate phenology and this is where the interesting discussion enters namely that assuming a Starr model gauge invariance then automatically correlates the diphoton with other so with decay widths to other electric final states in particular looking at these possible operators in the leading power counting on the EFT we see that there are only for a given CP assignment of S one only has two possible operators with two different SU2 quantum numbers contributing which can contribute to the diphtonic width of S in particular if in the limits where only one of these two possible operators is dominant one gets definite predictions for the widths of S into other electric final states in particular one sees that the current bounds coming from ZZ and WW searches resonances in WW decaying to ZZ and WW final states already exclude the possibility of S coupling predominantly SU2 triplet gauge field strengths so more generally so given that there are only two operators we can more generally write a so called decay amplitude sum rules combining the measurements of all the four possible decay as decay widths to electric final states and these decay amplitude sum rules rely only upon SU2 invariance of course they can be phrased in terms of decays or ratios of decays and applying them to the three or the so assuming the so normalizing all the remaining three possible electric final states to the diphtonic to the measure diphtonic rate one sees that all the by so one can then correlate the decays to Z gamma for example with the decays to ZZ or WW and in these correlation plots one sees that depending so irrespective of the possible interferences between these amplitudes which enter these decay modes at least two additional electric modes should be non-vanishing irrespective of the combination of the operators contributing to these modes so this is a clear experimental target that we know that of the four possible electric final states of S given that experiments have observed one of them at least two further more should be non-vanishing so should be observable and so on the so on the left-hand side I'm superimposing the ratios of ZZ and WW and also the possible the current experimental constraints on these ratios so in gray is the current experimental bound on Z gamma in red on ZZ and in blue on WW but what could also what can also do similar plot by just projecting this three-dimensional surface onto a two-dimensional plane and then plot the expected rate of WW decay width or as a function of Z gamma or ZZ widths okay and so these are the contours of constant RWW and again superimposed are the current experimental bounds okay so this the simple amplitude, some rules that I presented two slides before are of course applied to the transverse modes to the transverse modes of the electric gauge bosons so there are slight modifications of possible Higgs operators so coupling S to the Higgs but the numerical plots these two numerical plots that I showed here are actually already including possible contributions of these additional contributions which would induce the case of S to longitudinal decay modes of the WS and ZZ on the other hand some rules of violation of these constraints would indicate a violation would indicate a breakdown of the EFT power counting in particular the leading operator which allows one to completely couple the diphtonic decay width of the rest is of dimension 9 while the leading operator contributing in the EFT to the diphtonic decay width was of dimension 5 so it's a very significant breakdown of the EFT power counting that would be required if no no other electric decay mode of S was found this discussion can of course be generalized for S of larger isospiner representations and also for spin 2 and you can find the details in the publications and I will not discuss it further but we'll move on on another related issue which is the possibility to disentangle the electric nature of S namely until now I've always discussed the example of S being a singlet but this is of course not given and there are possible experimental tests where one could test whether S is in fact a singlet or maybe a neutral component of a larger electric multiplet one example is looking for associated production of S with a longitudinally polarized electric gauge boson and in particular one can show rigorously that a sternum of singlet output to QQ bar will predict a hard QQ bar spectrum a hard spectrum of production of S associated with the longitudinal electric gauge boson when produced from QQ bar interest and this can be seen in the left-hand side plot here where we're using BB bar annihilation production for a given value of the effective coupling of S to B quarks and you see that for S being a singlet the PT spectrum of S or Z or W is much, much harder than in the case where S is a neutral component of electric doublet and it goes back to the fact that the Yukava couplings are irrelevant operators if S is a singlet while there are normalizable operators if S is a part of a doublet what can do a similar argument for a doublet coupled to gluon field strength in this case one can also predict S produced associated with the longitudinal mode of electric gauge boson will be harder in case S is a doublet but of course discerning these two possibilities requires one to disentangle the dominant production mode and this can also be tested by looking at events where S is recoiling against hard hydronic jets or B jets and this has been shown recently that using this method one can actually disentangle the possible dominant production modes of S coupling to different initial state patterns okay so the so I guess I only have like 5 minutes left or so no you'll have more time okay so then otherwise I would have skipped this so let me just briefly go through this argument namely a possible size on total width is a challenge for perturbative weakly coupled models and this can be most easily shown again for the thermal singlet scalar case now in this case if S has a sizable width of like 0.6 then this puts lower bounds of the effective scales suppressing the effective operators coupling S to gluons and photons and the the most probably at the moment the most popular UV completion of such an effective theory is in terms of some heavy vector like fermions with masses comparable or larger than those that of S and then one can write the dominant directions of these fermions to S in terms of effective Yuccava couplings and these will at one loop level generate these effective dimension 5 operators coupling S to gluons or photons now one can do a matching of the EFT to this perturbative models and one sees that one then sees that these perturbative realizations come with severe suppression factors coming from the loop expansion or perturbative coupling expansion which turns out to be the leading contribution turned to be at one loop and so since this suppression factor is already bigger than the, so it's larger than the inferred suppression that is required to feed the data this leads us to the theory, these models or theories to live in the regime of large Hoft coupling namely one either needs large couplings or large multiplicity of states and so to test the validity or to constrain the regime of validity of such perturbative descriptions one should of course examine the RGs of the models and a particular observable which is very sensitive to any large couplings or multiplicities in a theory is of course the quartic of the scalar because it's sensitive to all interactions coupling to S. In particular one can show that the beta function of the S quartic receives large corrections which are both due to the quartic itself and due to the possible Eucala interactions of additional heavy fermions and this can then be used to put a bound on the on the Eucala couplings multiplicities and charges of hypothetical new heavy fermions generating the effective couplings of S to photons and gluons and a conservative bound that what can impose is that the beta function of the of the quartic should not exceed 16 pi. So because the argument goes that if the beta function exceeds is very large and this means that basically the theory will run into a landopole or an instability within a decade of running. So basically the validity of theory is restricted to within less than order of magnitude of where it's defined. So it basically becomes meaningless to talk about a perturbed description of such theory. And we'll see that this actually allows one to put constraints on both the the required couplings so Eucala couplings of these extra fermions in particular one is forced to live in a regime of large, either large charges or large multiplicities. So one can, of course more generally, one can also look at not only the running of the quartic of S, but also of the other parameters, the models in particular of the Eucalas themselves and also of the gauge couplings and one can combine all of these different RGE constraints to constrain the parameter space of such interpretive models. And we see that even though one is there are, this is again for a particular example although there are regions of parameter space where these models are valid at least in a limited regime of energies this regime is actually very narrow. So typically within a few TV these theories run into into some non-perturbative phenomena. So there is a, so there are also of course similar constraints where can you both coming from both vacuous stability and also unitarity of processes involving S and these heavy fermions and all of them point to the generalization that generically perturbative models of S, if they try to accommodate a sizable width, have a very limited range of validity, in particular typically below a few TV. There is one caveat or one loophole that one can exploit, in particular this conclusion can be avoided if these new interactions which generate the cutlings of S2 bonds and photons are reside near an infrared fixed point they exhibit an infrared fixed point then a large coupling regime can be valid even to arbitrary high scales. But it's very theoretically restricted of course to actually have such an infrared fixed point. And this, what is interesting also is that this regime of large couplings can actually be again, can be tested experimentally, in this case the relevant observable is S-spare production, which would be anomalously enhanced if heavy fermions coupling to S have very large couplings. Okay, so let me use the remaining couple of minutes to now focus on the other possibility, namely that S is more effectively described in terms of some strong dynamics so being interpreted as a composite state. And to motivate if you look at just the the composite theory we all know namely QCD and look at the neutral resonances appearing in QCD we see that their widths are generically of the right sides so they're generically broad which one expects in composite or strongly coupled theories. Now there are a plethora of candidates which appear in strongly coupled models of new physics which one might consider as candidates for the bifoton resonance in particular in models of composite Higgs which address the electric hierarchy problem one typically encounters scalar resonances but also additional pseudogolston bosons in addition to the Higgs boson and generically in theories exhibiting a high-scale conformal symmetry there is also a dilatum similarly in strongly coupled theories which are not directly necessarily directly related to the electric weak scale which go under the turn vector like confinement one generically encounters again pseudogolston bosons which can be thought of as some technique interaction form of the eta meson or one can think of more massive coonium like so QQ bar like states. Of course these strongly interactive theories can often be reinterpreted and discussed also in terms of models with extra warp dimensions. In this case for example one can identify models of the dilaton in terms of a radian although there is a special example where these interpretations are especially useful these extra dimension repetitions is the case of spin 2. So these extra dimension models are rare exceptions where one can actually discuss on a firm footing the possibility that S has a spin 2 which is otherwise very difficult in a purely four-dimensional setting. In this case of course S would be interpreted as a cakey gravity and there is already some literature on it and I will not have time to discuss this in detail and so do I still have a minute or I should skip the last two slides No, you can say it you can continue. Okay, so just because it's something that I've recently worked on and because I think it's something that has not been discussed so much in the literature especially not before the diphthora resonance appears. Let me discuss briefly the case of coonium because it's also not so well known. So these QQ bar states appear in models of actual confinement. I also think of them as like technical or like theories but where the electric symmetry is not broken by technical dynamics. So these states appear where the fermions charge under these technical confining interactions have vector like masses which are comparable or above the confinement scale of the strong interactions. Then in this case the lightest scalars the lightest states in the theory can be bound QQ bar states and of course technical globals. And a clear prediction in this case is that for these heavy fermions residing in a given standalone representation R one can expect resonances appearing in the product representations of R times R bar. In particular one generically expects colored resonances of spin both 0 and 1. A typical spectrum of these resonances is shown on the right hand side here where the lightest one is typically expected to have to be a scalar of odd parity so in particular a pseudo scalar but one also expects an almost degenerate spin one vector with otherwise the same electric quantum numbers while and the other higher excited states are excited to be heavier. So all of these states should come both as QCD singlets and also octets and this is a very interesting prediction of this construction in particular all the most relevant phenomenology namely the decay rate and prompt production of the lowest line resonances is controlled by very few parameters. In particular specifying the standalone representations of these vector like fermions and the basic parameters of these confining interactions one can determine or parametrize all the leading decay modes and production modes of both spin 0 and spin 1 resonances. The important parameters coming from the strongly interacting sector is of course the binding energy of these resonances and the size of the bound state which can be also parameterized in terms of the wave function the q-cubar wave function at the origin for these bound states. One can ask some form of estimate of these quantities can be also obtained in the Coulomb limit where one can just compute these bound states of perturbation. So in this case the digit resonance signal which is already constraining in case of gluon fusion dominated production of S is now dominated by color octet partners of S and this can be used to constrain the possible quantum numbers of these heavy fermions in particular current bounds already constrain the charge or the hyper charge in case of SU2 singlets of these heavy fermions to be above 0.6. And another interesting phenomenon that appears here is that besides prompt production one expects to have this S and it's colored and higher spin partners to be also produced from fragmentation of q-cubar production at high pt and one can actually compare these production modes and the relevant terminology in terms of just two basic parameters and the stress energy and the size of the bound states and superimposing all of these constraints in terms of these two parameters one can first see that a good fit to data is possible even close to the Coulomb limit so the Coulomb limit on this plot is this purple dashed line here and superimposing this are the perturbative production for heavy colored fermions and normalized to the number of technicolors in picobar and in blue is the ratio of the prompt production normalized to this perturbative q-cubar production and we can see that the prompt production can easily dominate perturbative production so estimating the prompt production via this prompt production can be a valid exercise in a region of parameter space and furthermore on the same plot I'm showing three examples of models where these heavy fermions have particular standard model representations referring to u so basically this is an SU2 singlet which charge through thirds and also two different possibilities either an SU2 singlet with charge four thirds or an SU2 doublet with hyper charge five-sixths and what we also just for illustration also showing on the same plot the values of the binding energy and the wave function which would correspond to the known q-cubar bound states the standard model quarks namely the eta-b and eta-c so the lowest line botomonium and germanium and what we also see from this plot is actually there is room for additional decay modes of this coronium which would push if one adds additional decay modes one pushes these preferred regions for this benchmark models upwards to measure values of the wave function so this allows one to consider the case for example of this s to hidden sector states in particular for example to technical globals or technical pions if they exist in the theory another interesting feature of this coronium state is that higher n states higher radial excitations of these bound states are typically split by just a few percent and this can in fact be used to fake a broad aspic in the diphoton data so on the right-hand side I'm showing the current spectrum of atlas with its current binning which is of the order of 20 gv and in this crude binning that is currently available one cannot discern several closely spaced resonances however this should be discernible with more data in the photomass spectrum because the environment mass resolution is much better than what is currently being publicly presented and in particular just for illustration I'm showing on the upper plot I'm showing the corresponding spectrum what we'll see assuming an experimental resolution of 1% so this spectrum above once one averages or sums over the atlas binns results in a perfect fit to the current atlas data but in fact on close inspection one would actually expect to see several resonances okay so let me conclude with this example, simple example I think that the current experimental hits hints of a diphoton excess of 750 gv are tantalizing yet unfortunately inconclusive what we know already is that they should be at some point accompanied by resonances at 750 gv decaying to ZZWWN or ZGAMF states at least two of those now at the moment fitting current data to new physics model is rather easy as you can infer from observing the more than 200 papers that have been published so far on the success at the same point, at the same time the larger parent width preferred by atlas would point towards strong interaction models or possible multiple states and also the absence of signals in other decay models beside the electrovec decays might motivate connections with dark matter in the future it will be of course paramount to probe alternative and additional production and decay models to test the preferred quantum numbers of S both CP, spin and electrovec quantum numbers and these alternative decay or production models can also be used as probes of the total width and can test possible uv realizations of S so at this point I can maybe venture a robust prediction that in July with the 2016 data set of atlas in CMS we can already expect a mass model extinction event to occur thank you very much thank you very much and I guess I was very interested in the talk in fact it was worth to follow it so before to start with the question now I'm just going to tell to the people that is following the streaming of this webinar that you can make questions if you are already in the google plus event of this webinar you can just have to click in the upper right part of the screen and there is a Q&A button so if you want to make any question you have just to write it there and we are going to address it to your name so maybe if we can pass to the question from the people that is here in the hangout session if you have any questions please unmute yourself ok ok sorry I have a question it is evident that we need to see events with two zeds or one zed and a photo so that clearly will affect the width so regarding your initial plots how does the picture change if you consider the fact that it will be producing zeds and zed gammas you are referring to which initial plot right initially you just considered that you had couplings to 2 photons or 2 blue yeah let me show you ok so you see on this plot I already considered the possibility of having additional decay modes and actually gauge mozons and you see there is a ww in red you don't see the zed zed and zed gamma at all because they are already so much constrained that they cannot appreciably contribute to the total decay ok because they can be at most a factor of a few more frequent than the gamma gamma and we know that the gamma gamma cannot be the dominant component in the width because if it was then the 8 and 13 TV data compatibility would suffer ok Anderson ok so just a second question would be so what would be what should we expect then in July so what channels should we on the look out for should we expect say for leptoman analysis what kind of data do you I don't know if you are aware the ones which would most eagerly await would be the zed gamma zed gamma the zed zed and then the digents because it's very difficult to have this thing produced and not appearing in digents right nevertheless those channels would work mostly to confirm this excess but would they be of any use to exclude other models yes so maybe let me go to here right so you see in this plot you see that the right now the constraints are shown now imagine you push the zed gamma the zed gamma bound to be below the diphoton bound signal ok so for rz gamma to be below 1 if you do that then you see that you have a definite prediction that zz and ww should both be of order one or larger so at the point where all these three modes are pushed below the gamma gamma so if the zed gamma zed zed and ww turn out to be all constrained to be below 5 m2 bar then the whole thing becomes very very difficult to explain because then you cannot use eft because eft is at that point eft is violated ok fantastic thank you very much so is there another question maybe for the other participants otherwise we can at least we can start with one of the questions from the followers of the streaming the first time I'm going to talk about the one from Abelino Vicente that first of all he acknowledged you for a very nice review and he's asking if you could please comment on how to use the vector Poisson fusion production to test the scenarios with invisible STKs ok yeah so basically what happens is that once you determine your dominant production mode for any given dominant production mode you predict the corresponding vbf product vbf signal and if this prediction only depends then at this point only depends on the total width of s and so if you account for all the modes of s into the standard model states and you see that the you constrain the vbf production below a certain value you know you have to have contributions to the width which go beyond the standard model states at this point you know you have other decay modes of s which you are not seeing which you have not constrained so this is in the same paper this one that discusses this SU2 is discussed in detail unfortunately I don't think I have any plots on this but they are in the paper so as I said so for any given production mode there is an upper bound on the vbf production so this is basically production associated with two jets where you put the typical vbf cuts on the two jets and so there is always a lower expected value of the vbf for any given production mode if the total width is saturated with the standard model final states if you validate this then you have other decay modes besides the standard model ok and now I just have a little reply thanks for the answer so meanwhile I can start with one of the other questions this is a question which is the future strategy to break the model of the regeneracy in the case of composite states since there could be many solutions for the same problem maybe some of these models have different signatures in different channels I don't know so for example typically in models where this s is a part of a goldstone sector involving the Higgs then one typically expects s to decay to ptbar or dihiggs so expect a resonance in dihiggs a resonance in ptbar on the other hand this is not expected for example in this Qonion picture in this Qonion picture you expect on the other hand colored resonances which are degenerate with s also you expect spin 1 resonances which are almost degenerate with s and on the other hand if I again in this goldstone picture this s can actually be separated a little bit in mass with the rest of the spectrum so in those modes you don't expect additional spin 1 resonances being very very close to the mass of s so this is one example where so at the end it will be probably several different observables which will be needed to pinpoint what is going on I see so another question that is about the in the case of these perturbative models in which you have loops inside the loops different new fermions especially do you have further constraints on this kind of exotic fermions that number of fermions or masses or can they be light or they can be very heavy so we know that in order to induce a couple of photons they need to be charged so having charged new massive charged fermions one can use existing direct search bounds on this and these typically lie in the ballpark of 100 to 300 gv so one can for sure not have new charged fermions lighter than about 100 gv because these are subject to lap constraints they are most robust but if even at the LHC unless one makes the decays quite intricate then one will be subject to more severe constraints coming from the LHC and then these will be closer to like 300 gv now the optimal range to get the maximal effect in the loop of these fermions inducing an s to gamma gamma is in fact when their masses are roughly half the mass of s meaning that they are about 400 gv in mass so this is at the moment this is perfectly consistent in general this is perfectly consistent with direct searches and would be the maximal possible effect one can get if these fermions are lighter than half the mass then their effect in the loop starts to decouple so in the color limit they decouple and also in the heavy limits if their masses are much heavier then as that again they decouple I hope this answers the other question yeah in fact this is one of the the ideas that I mean what I was trying to see if there are constraints or not like you were saying so we have another question this time it's from Jan Manbrini and his question is follow-up is very long let's see when talking about the violation of weak isospin invariance one should add that the ratio set to gamma gamma can you sorry because he just removed the question if he can put it back okay I found it okay when talking about the when talking about violation of weak isospin invariance one should add that the ratio gamma gamma set set not conform with the weak relation sorry that doesn't meet the violation of isospin but for instance intermediate states with four collimated photons I don't know if you can so in case one would the question is what the experiments are seeing are not actually the case to die photons but the case to to some intermediate states which then decays to collimated photons in that case one would expect also z gamma z z and so forth well in fact I don't I don't think the signature would look exactly like this so all the conclusions that I had were assuming s is actually decaying to die photons if s is decaying in a more complicated way then of course these implications can be different yes maybe there is another question yes I have one too okay can I ask one yeah sure hi Erne how you doing thanks for the talk so people have tried to explain the Higgs for example have tried to fix all the Higgs data with a light dilaton and one of the concerns is that to get a light dilaton even a light as 125 GB requires a lot of fine tuning and large explicit breaking of conformal invariance wouldn't these be worse in the case of a 750 GB scalar if it is to be the dilaton yeah so it's true that the dilaton is not is not a very elegant solution to this precisely for the reason that you're saying one needs to have a large breaking one large departures from the limit of conformal invariance to make it work this I actually have it in the in the backup slide let me see the case of the dilaton yes so the reason is the following in the case of the dilaton since the couplings of the dilaton to to fermions and to the Higgs are fixed by conformal invariance and then in this case you get a severe constraint coming from the dilaton actually wanting to decay to TT bar and diluting all the other rates and so in the dilaton of course the dominant contributions to radiative decays are coming from from the from the loops of whatever matter is coupling to it but the problem is that of course whatever you put there will be diluted severely if the dilaton couples to top and one needs to somehow break this in order to try to fit the data so at the end so it turns out that I think it's at the moment it's already clear that it's inco-system to try to interpret the 750 as a dilaton and having the Higgs as a pseudo-gallon cannot have balls at the same time okay and also in the dilaton case it is possible that the dilaton can mix with the Higgs in some way right through some some kinetic mixing or something like that and that's even worse if you try to make the top a composite of the conformal cycle okay so there there was a proposal which was phrased in terms of an RS model an actual dimensional model where they interpret the radion as the 750 so which would be equivalent to a dilaton in a 4D setting and they showed that one actually it's the mixing with the Higgs and they tuned the mixing with the Higgs in a way to explain the data thank you I'll check that reference thanks your name okay I think I don't know if there are other questions from the people here or in the Q&A let me just check otherwise I think it's okay for today I mean it was very interesting in fact I don't have any questions but later on the people we're looking in archive and so on are going to discover what is happening with this 750 GB resonance so I guess first of all I have to acknowledge the journey for this very nice thought and explanation and everything and I guess we can start to say bye for the next webinar and just let me talk to the people that is going to be next week is going to be Ricardo Sturani and we're going to have a talk about all the observational gravitational waves in this case the in the case of the recent discovery of these such a ways so I guess for my side is everything that I have to say and I think we can say goodbye and see it for the next time for the next webinar Bye


Thank you for the awesome introduction, since there are some non-dutch speaking folks in the room I'll be doing the talk in English and I'll be talking about debugging workers today. Who am I? I'm the guy on the screen here, I've been working at Combell, it's a Belgian hosting company for approximately 10 years now, I work as a support engineer, so on day-to-day basis I deal with customers calling or mailing and saying, hey my site is broken, it's down, this isn't working, technically every kind of issue, both with WordPress and other CMSs, with custom applications, websites and things like that. I've been a WordPress user since version 1.5 back all the way in 2005, I have a passion for security and performance and I'm on Twitter at Altebrechtreikarts.com, now who knows this whenever launching a new website, or even auditing plugins or things, or even when migrating your website, just by show of hands, who has seen this before, pretty much everybody, and things like this I assume as well, doesn't it make you go like banging the keyboards against your desk and things like that. Now before we can dive into debugging we need to understand what kind of errors there are and basically let's say 95% of all cases you'll be seeing two kinds of error codes, you've got the 400 error codes and the 500 error codes, now any of you know the difference between a 400 and a 500, anyone, yeah, Sherwin? Yeah I do, but the mission related was 500 more code errors. It's actually far more simple to explain it, 400 you stuffed up, 500 your server stuffed up, basically. Just to give you an example, 400, I'll just go back, here you go, since it needs to be politically correct, I said stuffed up but you all know which word I mean, now in the 400 series there's like the things you will see most are like 404, 403, things like that, 405 sometimes, 401 is also one that sometimes happens and well yeah, the explanations I think they speak for themselves, so there are pages with the RFCs with the entire status codes which are pretty fun, one of them is called I'm a Teapot, just think it up, it's really funny, and then the 500 series of codes are exactly what they are, server related, so in terms of server error, server's unavailable, the one you'll probably see most is the 503, 503 or 504, both are pretty common, so anybody has an idea of how to debug this, anything, just shout. Check the logs, yeah, check the logs, yeah, exactly, there's actually three things I use, server logs, very important, WP debug, which often gets forgotten, sadly enough, and WP-CLI which is actually just a command line interface to manage WordPress, but it can be really, really helpful in case of errors, so first up, server logs, okay, so there are a couple of types of errors you can see in logs, you've got the fatal error, you've got warnings and notices, and you've got several limit related errors, just to give you an example on how a server log might look, this is an example of some excerpts of an error log of one of my own sites, which I just changed to some website.com, just to keep it a bit anonymous, because throughout the talk you'll be seeing actual examples of tickets I've worked on for the last month and a half, and errors that were triggered on sites, and to keep some anonymity for our customers, I just changed them all to some website.com. So, this one here, I'll just go over some of them in detail and highlighting, so client denied by server configuration is one that we often get, usually in the case that something is pretty darn wrong with your HTX file, so if you get a client denied by server configuration, it's probably a good idea to start by looking at your HTX, and there's things like this, a timeout, which can point in the direction of several things, but usually it's, for example, an external element that needs to be checked or a server that needs to respond, and it doesn't happen, or just keeps on waiting, and eventually the server will kill the process and will get this kind of error thrown, and then there's of course things like this one, everybody hates this one, because, well, it's obvious, it's obvious fatal error means PHP will stop processing the code and just kill it all, and throw no output or limited output or whatever, it just won't render, it won't get the requested result, like for, in this example, you'll see the fatal error, so this will give you an indication of this is a breaking point for my loading process, for my process I'm trying to do, whether it's loading aside or doing some action in the back end or whatever, and then it helps to just follow the line and just actually just read what kind of component is triggering this, so in this case if we look into this log excerpt, we'll see here that it's concerned to the, or it's related to the WP, to the WooCommerce multilingual plugin, specifically this one, now if for example you just updated your WooCommerce multilingual plugin, you knew because of this that this broke, so the easiest way would be to revert the update and just get in touch with the developers or look further into what's actually blocking this, and you'll also always get further information on what the exact issue is here, so this is why the logs are really, really important and certainly, well, based on the amount of calls or mails I get, I'd be happy to say if 25% of our customers look at the logs, I'm afraid that's not the case, so they just call it, it's broken, then there's an amazing tool within WordPress called WPDbook, it's always there in our WPD config file and so often forgotten, even with my colleagues at the support team, so they just go into the server logs and if they don't find it there, oh I can't find it, Brecht, will you look at it? And I always say have you tried WPDbook? So in WPD config we have one simple line, which is WPDbook false, and just to enable the debug function, you can just switch it to true and it will output extra information on the errors, but of course that's not very, how should I put it, it's public, the errors are public for anyone to see, just if you do that, it's blatantly public. And there's extra statements you can give, define, so this is actually a line you just can add beneath or above the WPDbook true line, and this will actually create a log file in the WPD content folder, so all the extra information that you're probably missing in the server error logs will be added there. And then this is a very good one, especially when you need to do this on a production environment of a customer, is just setting WPDbook display on false, so you have the ideal setup here, which enables WPDbook, it'll enable writing the output to a log file, which is not public technically, or at least it's not showing on the rendered version of your website, and this disables the rendering of the errors on your website, so for the customer, or client, or whatever, or your room project stairs, for the visitor, there's nothing to see, but you're logging every single instance of a failure within WordPress, so this is a very, very good setup if you want to go run a debug. And what file did you chase this? This is actually all entries in the WP config file, so by default you have this first little line, and those two are ones I add just to keep the output hidden and logged into a file, which makes it easier to just in VI or whatever your preferred editor is on the command line just to analyze logs and search for specific entries. Then we have WPCLi, which has recently become an official WordPress project, which is a very good thing. These are some of the commands I use. Now, for example, a customer calls us and just WP this plugin and on my website isn't loading anymore. Okay, no problem. First thing we can try is just see if WP plugin list gives us an output. This provides just a simple list of all plugins and status if they're active or not and if there's an upgrade available. But the fun thing is if that works, you can easily say, okay, WP plugin deactivate, and for example, Yoast-SEO, and it will deactivate that plugin even if you don't have WP Admin access anymore. So this can be a live seeker. The same thing with teams. You have WP team list, activate and deactivate. And then, especially in certain issues, when the log files don't tell you everything, WP debug doesn't show any specific indication, the server logs don't give you any good instance, it might be a good thing to test the WordPress website and the plugins by using WP checksum core or WP checksum plugins. Now, what this does is it will run a test and verify the files of your installation against the one in the repository. And in, I'd say, 5 to 10% of all cases, this results in finding or hacking related files, which become an issue in the rendering process or who intertwine with the running, which causes a conflict between plugins or things like that. So this can be a true live seeker, and it's actually rather fast, so please use it. Another cool thing is, and I'm pretty sure it's not an intended use of WP CLI, but sometimes whenever I run a command, for example in this case, this is a recent one from a couple of days ago, I just ran the command WP team list to see what teams were running, and I got this output. So this isn't the normal output for WP CLI, but the fun thing is, this was code, this was an error which wasn't in the error logs prior to running this specific command, nor was it in the WP debug. So, I don't know if anyone can tell me what this would indicate. So it's a resupply expert, an expert experiment to do one to be erased, and then it says Fevicon underscore 316779.ico evaltcode evaltcode evaltcode. Basically, this site was hacked, and the entire error, which was a blank page, in a certain screen in the back end, was triggered by an infection of the XMET plugin. And this was found through using WP CLI, just listing the team, because else I wouldn't have found it, or except after a long search probably, but using WP CLI in this case made it much easier for me to just trace it back to where it came from. Now, there are also several, I call them typical errors, and easy fixes or quick fixes or things like that. How many of you have seen the library, the media library in this state? Presumably after a migration, I would say. Some people, I suppose, okay? So usually this is the culprit. So in the WP options table, there's a specific field called upload path, and now when you're moving a site from one server to another, that path can be different between the two servers. Now the problem is if the path is different, and you don't change it, either your website will load endlessly, try to load endlessly, or your media library will not show. So if this is the case, just empty the field, clean it, and usually it'll work. Anyone has seen this before? Just a CSS less render. It's also one thing I often see when clients are migrating from one server to another, especially when it's from the non-combal server to a combal server. Directly structures are different, but this is not something which has to do with that. In this case, I just broke it deliberately just to show you, but in this case, usually the site URL has something to see. I often see this when customers change their domain name, and they make it a type of graphical error. You can look pretty long and hard for it, but usually just checking out the site URL and the home URL are a pretty good place to start. It won't fix everything, but to my experience, in 90% of cases, this was the cause. Then this is also one, which you'll find often in the server box. Specifically, you cannot modify header information when the header is already sent. I don't know just by show of hands but most of you are active PHP developers. All of you, okay? I'm doing the better part, even, which is great. So most of you, in this case, will know that a PHP script is bound to just give header output once. Now, in WordPress, I don't want to point fingers, but some teams or some plugins don't always uphold the best coding standards, which can cause the situation where header output was already sent and there's more header output being sent and sometimes triple or more. Well, PHP can't handle that very well. So there's an easy fix for this, which is in our PHP settings. This just enable output buffering. It's just a simple change in the PHP.NE or in your control panel for whatever reason. And that'll fix it for you. So this is actually one of the errors I get most often these days. Mostly when customers are using one kind of page builder, which I won't name, which sounds with composer. I'm not a fan, sorry. So this is one, yeah, sure. Why isn't that set by default there? Well, that would depend actually, it would be logical to set it by default. I can't actually figure out a use case at this point why you wouldn't want to set it. Any questions concerning? Because it's, yeah, this is pretty much it. Whoa, that's it. Could Thais go next time? I could ask, sure. Okay, thanks. Okay. What kind of question was that? For a question like that, you have to make it a serious question as well. Could you elaborate on the housebook, for a little more? Thank you. Yeah, sure. It depends on what you are looking for. Are you looking for a more technical response on how the function works? Or are you more into the causes? The causes. Okay. The basic explication is that if there are several sources of output, whether it be the team, several plugins, they all are sending some amount of header information. When the request is being processed, it expects only one set of headers. When there's multiple, it goes wrong if you don't have output buffering active. And what output buffering would do is just, it'll hold open the processing of the script or the requests until all header output was received. It'll be keeping it in some kind of buffer, which is obvious, if you see the name, output buffering. It'll then all solidify them in one major header and then go over to the rendering process and then return the results. Yeah, right. Yeah. Hope that covers it a bit. It sounds like misbehaving plugins. In most cases it is. That's why I actually said it's, it's why not every single plugin or team developer upholds the best coding practices, because that's usually the case. It's often a space before or at the end of the, pre-opening tag or at the end of the, before closing tag. Yeah, true. Simple typing error. Yeah. That's what I mean. Any more questions? Maybe city, but are there any plugins that could prevent bugs? What would you recommend? I do, I use a draft and we're fans, but is there any more which I get used to? There are plugins, but not necessarily for WordPress. In that case I would, this is actually more a bit of the responsibility of the developer, of the plugin or team or even core developers. There's a plugin called PHP Code Sniffer, for example, which you can use in either via command line or for example it can integrate perfectly in PHP Store, which will just scan your code and see if it upholds to the best coding practices. They also have specific, a specific set of libraries to check WordPress related tags and hooks and things like that. But you can also verify your code against PSR1 or 2, things like that. And those will help tremendously to avoid blatant errors, for example, a space before or after the tag. That might help, but it's not something we can usually prevent on the WordPress site. However, there is some logic built into WordPress. Let's say you're trying to activate a plugin which you wrote yourself, it has a code, it'll just deactivate the activation process and just tell you, okay, there's a fatal error here, we couldn't activate. You'll at least have an indication, but when it goes wrong on an update, it's not always covered by WordPress. And since WordPress at that point will break, there's no plugin in WordPress that will help you out, because it's going to get executed anymore as well. Anymore? Well, to summarize, I think very good attention, I hope. 400 errors is the one that I messed up, 500 errors is the one you messed up as a server. If something breaks, plugs, the LAPCLI, and then if nothing else works, blame the hosting company. I would say at least call the hosting company and get them to help you. Thank you very much.