Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 1...

Antonio Rago

Università di MilanoTechniques for automated lattice Feynman diagram calculations 1

Antonio Rago

Università di MilanoTechniques for automated lattice Feynman diagram calculations

Techniques for Techniques for automated lattice automated lattice Feynman diagram Feynman diagram

calculationscalculationsAntonio RagoAntonio Rago

Università di MilanoUniversità di Milano

Trento, September 2005Trento, September 2005

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• 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 Rago


Outline

How to calculate a two loop Feynman diagram?

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RagoUniversità di MilanoTechniques for automated lattice

Feynman diagram calculations

MotivatiMotivationsonsWe 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

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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 One loop Feynman DiagramsDiagrams

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First observation:each integral can be analytically reduced to a sumof integrals of the same type with

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by using recursion relations like:

Recursion Recursion relationsrelations

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Each one loop bosonic integral can be expressed on a basis of 3 lattice integral plus a

polinomial in

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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:


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(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 One loop Feynman DiagramsDiagrams

Burgio, Caracciolo, Pelissetto Nucl.Phys.B478:687-722,1996

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... 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:


12 finite constants (lattice integrals)a logarithmic terma polynomial in

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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 basisNumerical 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

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Using and and setting all appearing there to zero we get our determination

of the basic constants.

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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 basisNumerical evaluation of the basisWe need eight linearly independent

equations.

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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 remarkA side remark

... even if the convergence for the numerical determination of the basis is faster ...

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The Coordinate space method The Coordinate space method for the two-loop Feynman for the two-loop Feynman

integralsintegrals

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Having shown how to deal with the one-loop integrals...Having shown how to deal with the one-loop integrals...

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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 The Coordinate space methodmethod

Capitani, Caracciolo, Pelissetto and Rossi, Nucl.Phys.Proc.Suppl.63:802-804,1998

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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 An example to fix the ideasthe ideas

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how to evaluate the integral in the Coordinate approach?

Let:

In this notation our previous example corresponds to


A generic two-loop integral can be written as:

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• 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

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... but in the evaluation of these sums we make use of the following

advantages


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• 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:

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An example to fix An example to fix the ideasthe ideas... moreover many symmetries can be

used to reduce the number of integrals that we must evalute:

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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 How to compute the lattice sums?lattice sums?

on the lattice Let

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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 ... computing lattice sums...sums...

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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

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So if we consider a “subtracted” lattice function

... computing lattice ... computing lattice sums...sums...

this term behaves like for large

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Coming back to our example

by subtracting an increasing number of terms of the asymptotic expansion, we get


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Subtraction of the Subtraction of the divergencesdivergences

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

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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 integralsThis term is a two-loop finite integral

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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!

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Subtraction of the Subtraction of the divergencesdivergencesA bosonic

example:

We want to write our integral as

where:

with:

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Some Some results!results!

S. Caracciolo, A. Pelissetto and A.R. Phys.Rev.D64:094506,2001S. 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.

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Some Some results!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

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The -th diagram gives a contribution of the form

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Some Some results!results!At two-loop

order:

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Some Some results!results!At two-loop

order:

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at the end... at the end... if you if you

are still awakeare still awake

Conclusions and PerspectiveConclusions 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

Antonio RagoUniversità di Milano Techniques for automated lattice Feynman diagram calculations 1...

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