1. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 1 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Techniques for automated lattice Feynman diagram calculations Antonio Rago Università di Milano Trento, September 2005

2. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 2 Motivations The Coordinate space representation One loop Feynman diagrams Recursion relations Numerical evaluation of the basis The Coordinate space method An example to fix the ideas Evaluation of the lattice sums Asymptotic expansion Estimate of the errors Subtraction of the infrared divergences Some Results Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Outline How to calculate a two loop Feynman diagram?

3. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 3 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Motivatio ns We want to apply the coordinate-space by Lüscher and Weisz to the computation of two-loop diagrams in full QCD with Wilson fermions on the lattice Lüscher and Weisz Nucl.Phys.B445:429-450,1995 The essential ingredient is the high-precision determination of mixed fermionic-bosonic propagators The first step is to show how to calculate one-loop integrals

4. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 4 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Caracciolo, Menotti and Pelissetto Nucl.Phys.B375:195-242,1992 Every bosonic one-loop lattice integral with zero external momentum can be written as a linear combination of terms of the form: where: (The bosonic case) One loop Feynman Diagrams

5. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 5 First observation: each integral can be analytically reduced to a sum of integrals of the same type with Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations by using recursion relations like: Recursion relations

6. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 6 Each one loop bosonic integral can be expressed on a basis of 3 lattice integral plus a polinomial in Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Second observation: the left integrals can be expressed in terms of a finite number of them, by using again a recursion rule:... by applying the reduction relations just shown we obtain: Recursion relations

7. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 7 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations (The bosonic-fermionic case) Every mixed one-loop lattice integral with zero external momentum can be written as a linear combination of terms of the form: where in the following One loop Feynman Diagrams Burgio, Caracciolo, Pelissetto Nucl.Phys.B478:687-722,1996

8. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 8 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations... it is again possible to write a set of recursion relations but they are more involved than the bosonic case... and for your good luck I will not show them now! which allows us to write every one-loop bosonic-fermionic integral as a linear combination of: Recursion relations 12 finite constants (lattice integrals) a logarithmic term a polynomial in

9. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 9 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations How do we get a very precise numerical determination of the integrals of the basis? We use again the recursion rules! Numerical evaluation of the basis A determination of the basic constants is obtained by applying the reduction procedure to four values of two nearby values of. (for instance we could use and )... and we add another ingredient: for fixed

10. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 10 Using and and setting all appearing there to zero we get our determination of the basic constants. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Using and and setting all appearing there to zero we get our determination of the basic constants.... few minutes of cputime more... Numerical evaluation of the basis We need eight linearly independent equations.

11. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 11 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations A similar procedure for the bosonic case, but based on a different sets of recursion relations and on a different basis, was proposed by Vohwinkel but......the procedure is not applicable to the fermionic case. A side remark... even if the convergence for the numerical determination of the basis is faster...

12. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 12 The Coordinate space method for the two-loop Feynman integrals Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Having shown how to deal with the one-loop integrals...

13. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 13 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations An example to start: Consider, to fix the ideas, a two-loop integral like: The integral is finite and can be easily numerically evaluated The Coordinate space method Capitani, Caracciolo, Pelissetto and Rossi, Nucl.Phys.Proc.Suppl.63:802-804,1998

14. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 14 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations A possibility is rewriting the integral as: with Then, using an extrapolation of the form obtain the infinite volume estimate: evaluate the sums for increasing values of ex An example to fix the ideas

15. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 15 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations how to evaluate the integral in the Coordinate approach? Let: In this notation our previous example corresponds to An example to fix the ideas A generic two-loop integral can be written as:

16. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 16 only four infinite lattice sums must be computed can be determined with the desired precision, for a sufficiently large domain of values of, by using our algebraic algorithm the asymptotic expansion for large values of of the can be analytically computed Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations... but in the evaluation of these sums we make use of the following advantages An example to fix the ideas

17. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 17 Because of translation invariance, for every on the lattice: We can again use the definition of the bosonic propagator: By integration by parts of terms of the form: Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations An example to fix the ideas... moreover many symmetries can be used to reduce the number of integrals that we must evalute:

18. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 18 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Of course we will not be able to sum over the whole lattice. We will perform a sum over a finite domain of the type: The problem is to give an estimate of the error. How to compute the lattice sums? on the latticeLet

19. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 19 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Thus we can estimate If decreases for large as we expect the sum restricted to to behave as Notice that it depends on the power of the asymptotic expansion... computing lattice sums...

20. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 20 This sum can be computed directly on the infinite lattice by using harmonic polinomial and -functions Following this last observation we can also define an improved estimate for by... and notice that the larger is the best is the estimate Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations So if we consider a “subtracted” lattice function... computing lattice sums... this term behaves like for large

21. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 21 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Coming back to our example by subtracting an increasing number of terms of the asymptotic expansion, we get An example to fix the ideas

22. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 22 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Subtraction of the divergences Having shown how to deal with the two-loop finite integrals. What is left to do is to show how to deal with the infinite integrals! In Coordinate space representation (as for the momentum representation), we can classify two different types of divergences

23. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 23 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Subtraction of the divergences First case: Singular divergence Just one or more of the one-loop propagators is singular We just subtract the singular part of the propagator This is the product of two one-loop integrals This term is a two-loop finite integral

24. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 24 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Subtraction of the divergences Second case: Global divergence (logarithmic) The sum over the lattice is divergent We need again to subtract the divergent part of the sum, and express the subtracted part as product of one-loop integrals The leading order of the subtraction term can be computed on the continuum, and it is all we need!

25. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 25 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Subtraction of the divergences A bosonic example: We want to write our integral as where: with:

26. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 26 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Some results! S. Caracciolo, A. Pelissetto and A.R. Phys.Rev.D64:094506,2001 S. Caracciolo, A. Pelissetto and A.R. Nucl.Phys.Proc.Suppl.106:835-837,2002 The dressed inver fermion propagator has the form: The additive mass renormalization is obtained by requiring This equation can be solved in perturbation theory by expanding We have computed and for, gauge group and fermionic flavour species.

27. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 27 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Some results! At one-loop order: We report in the first line the result of Follana et al., and in the second line our result, obtained by means of the coordinate-space method. Follana, Panagopoulos.Phys.Rev.D63:017501,2001

28. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 28 The -th diagram gives a contribution of the form Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Some results! At two-loop order:

29. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 29 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations Some results! At two-loop order:

30. Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 30 Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations at the end... if you are still awake Conclusions and Perspective The coordinate space method has been can be used in two-loop full QCD It allow us to express analytically the divergences It can achieve arbitrary high precision in the determination of the numerical values We are now working on the determination of the renormalization constants of all the fermionic bilinear