Techniques for automated lattice Feynman diagram calculations

1
Techniques for automated lattice Feynman diagram
calculations

• Antonio Rago
• Università di Milano
• Trento, September 2005

Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
2
Outline

• 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

How to calculate a two loop Feynman diagram?
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
3
Motivations

• 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.B445429-450,1995
• 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

Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
4
Caracciolo, Menotti and Pelissetto
Nucl.Phys.B375195-242,1992
One loop Feynman Diagrams
(The bosonic case)
Every bosonic one-loop lattice integral with zero
external momentum can be written as a linear
combination of terms of the form
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
5
Recursion relations
First observation each integral can be
analytically reduced to a sum of integrals of the
same type with
by using recursion relations like


Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
6
Recursion relations
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
Each one loop bosonic integral can be expressed
on a basis of 3 lattice integral plus a
polinomial in
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
7
Burgio, Caracciolo, Pelissetto Nucl.Phys.B478687-
722,1996
One loop Feynman Diagrams
(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
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
8
Recursion relations
... it is again possible to write a set of
recursion relations
which allows us to write every one-loop
bosonic-fermionic integral as a linear
combination of
12 finite constants (lattice integrals)
a logarithmic term
but they are more involved than the bosonic case
... and for your good luck I will not show them
now!
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
9
Numerical evaluation of the basis
How do we get a very precise numerical
determination of the integrals of the basis?
We use again the recursion rules!
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
10
Numerical evaluation of the basis
We need eight linearly independent equations.
... few minutes of cputime more...
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
11
A side remark
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...
... even if the convergence for the numerical
determination of the basis is faster ...
...the procedure is not applicable to the
fermionic case.
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
12
The Coordinate space method for the two-loop
Feynman integrals
Having shown how to deal with the one-loop
integrals...
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
13
The Coordinate space method
An example to start
Consider, to fix the ideas, a two-loop integral
like
The integral is finite and can be easily
numerically evaluated
Capitani, Caracciolo, Pelissetto and Rossi,
Nucl.Phys.Proc.Suppl.63802-804,1998
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
14
An example to fix the ideas




Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
15
An example to fix the ideas
how to evaluate the integral in the Coordinate
approach?
A generic two-loop integral can be written as
In this notation our previous example corresponds
to
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
16
An example to fix the ideas
... but in the evaluation of these sums we make
use of the following advantages

• 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

Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
17
An example to fix the ideas
... moreover many symmetries can be used to
reduce the number of integrals that we must
evalute

• 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

Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
18
How to compute the lattice sums?
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.
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
19
... computing lattice sums...
Thus we can estimate
Notice that it depends on the power of the
asymptotic expansion
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
20
... computing lattice sums...
Following this last observation we can also
define an improved estimate for by
... and notice that the larger is the best is
the estimate
So if we consider a subtracted lattice function
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
21
An example to fix the ideas
Coming back to our example
by subtracting an increasing number of terms of
the asymptotic expansion, we get
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
22
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
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
23
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
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
24
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!
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
25
Subtraction of the divergences
A bosonic example
We want to write our integral as
where
with
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
26
Some results!
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
S. Caracciolo, A. Pelissetto and A.R.
Phys.Rev.D64094506,2001
S. Caracciolo, A. Pelissetto and A.R.
Nucl.Phys.Proc.Suppl.106835-837,2002
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
27
Some results!
At one-loop order
Follana, Panagopoulos .Phys.Rev.D63017501,2001
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
28
Some results!
At two-loop order
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
29
Some results!
At two-loop order
Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano
30
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

Techniques for automated lattice Feynman diagram
calculations
Antonio Rago
Università di Milano