the dirac operator spectrum from a perturbative approach

The Dirac operator spectrum from a perturbative approachThe Dirac operator spectrum from a perturbative approachFrancesco Di RenzoFrancesco Di Renzo

Seul - 3-5 August 2009Seul - 3-5 August 2009xQCD 2009xQCD 2009

The Dirac operator spectrumThe Dirac operator spectrumfrom a perturbative approachfrom a perturbative approach

F. Di RenzoF. Di Renzo

M. Brambilla, M. Dall’arnoM. Brambilla, M. Dall’arno

Università di Parma Università di Parma andand INFN, Parma, Italy INFN, Parma, Italy

DisclaimerDisclaimer: : this is work in progress …this is work in progress …

In what follows I collect mainly ideas and very preliminary results: this is really work in progress. Still, I think there’s already some flavour of what we aim at. Let’s have a very first glance

QuickTime™ and a decompressor

are needed to see this picture.

My own expertise has been for quite a long time in a (non diagrammatic) way of doing Lattice Perturbation Theory. I have to warn you that this is still another application of NSPT!

OutlineOutline

Preludio: the Preludio: the spectrum spectrum of theof the Dirac operator Dirac operator as aas a probe probe forfor chiral chiral (and(and deconfinement deconfinement ?!)?!) transition transition..

Polyakov loopPolyakov loop,, Z(3) Z(3),, differentdifferent boundary conditions boundary conditions and all that...and all that...

A skecth of the technique by which computations were made (A skecth of the technique by which computations were made (NSPTNSPT):):

from Stochastic Quantization to Stochastic Perturbation Theoryfrom Stochastic Quantization to Stochastic Perturbation Theory from SPT to Numerical SPTfrom SPT to Numerical SPT The case of Lattice Gauge Theories and The case of Lattice Gauge Theories and different vacuadifferent vacua..

The computation of the spectrum:The computation of the spectrum:a a degeneratedegenerate eigenvalue problemeigenvalue problem in Perturbation Theory in Perturbation Theory

Very preliminary Very preliminary resultsresults: where do the Dirac eigenvalues accumulating near : where do the Dirac eigenvalues accumulating near zero come from?zero come from?

Our spectra are highly degenerate: do we need a regulator?Our spectra are highly degenerate: do we need a regulator?

OutlookOutlook

The Dirac spectrum: whyThe Dirac spectrum: whyThe transition associated to chiral symmetry breaking has a natural order parameter The transition associated to chiral symmetry breaking has a natural order parameter and this is connected to the Dirac operator spectrum (and this is connected to the Dirac operator spectrum (Banks and CasherBanks and Casher,, 1980 1980). This ). This made the Dirac eigenvalue density (which is not a natural observable in Field made the Dirac eigenvalue density (which is not a natural observable in Field Theory) a natural quantity to be interested in.Theory) a natural quantity to be interested in.

the relation we are looking for was put forward by the relation we are looking for was put forward by Banks and CasherBanks and Casher. The . The chiral chiral condensatecondensate (the order parameter of the transition at hand) can be expressed as (the order parameter of the transition at hand) can be expressed as

we are led to look for an we are led to look for an accumulation of Dirac eigenvalues near zeroaccumulation of Dirac eigenvalues near zero (otherwise a small (otherwise a small quark mass would be dominated by much larger eigenvalues).quark mass would be dominated by much larger eigenvalues).Defining the Defining the density of eigenvaluesdensity of eigenvalues

What does spontaneous What does spontaneous chiral symmetry breakingchiral symmetry breaking actually mean? ( actually mean? (VerbaarschotVerbaarschot))A small quark mass leads to a macroscopic reallignement of the QCD vacuumA small quark mass leads to a macroscopic reallignement of the QCD vacuum. Since. Since

The The small eigenvaluessmall eigenvalues we are looking for are we are looking for are not there in the free casenot there in the free case (deep perturbative, (deep perturbative, chirally simmetric regime). So, they must be chirally simmetric regime). So, they must be due to the interactiondue to the interaction mediated by gauge fields. mediated by gauge fields.

As a matter of fact, any As a matter of fact, any interaction in quantun mechanics produces a repulsion among interaction in quantun mechanics produces a repulsion among eigenvalueseigenvalues and this is a natural mechanism to account for the rearrangement we are after. and this is a natural mechanism to account for the rearrangement we are after.

This is actually one natural candidate for the Physics of Banks Casher: eigenvalues sitting This is actually one natural candidate for the Physics of Banks Casher: eigenvalues sitting near zero are near zero are coming from the bulkcoming from the bulk. .

(Not the only candidate: istantons contributions?)(Not the only candidate: istantons contributions?)

With this respect Perturbation Theory is in a tantalizing situationWith this respect Perturbation Theory is in a tantalizing situation

On one side, On one side, it sits (deep!) in the chirally symmetric phaseit sits (deep!) in the chirally symmetric phase, while we are after an effect , while we are after an effect (non zero eigenvalue density in the low end of the spectrum) which lives at its boundary!(non zero eigenvalue density in the low end of the spectrum) which lives at its boundary!

On the other side, On the other side, repulsion among eigenvaluesrepulsion among eigenvalues is a phenomenon is a phenomenon we can typicall inspect we can typicall inspect in PTin PT! (canonical example is level splitting in non-relativistic QM)! (canonical example is level splitting in non-relativistic QM)

We will compute the We will compute the spectrum of Dspectrum of D††D in PTD in PT (plain (plain Wilson fermionsWilson fermions). ).

We start our journey with some no-go barriers standing on our way:We start our journey with some no-go barriers standing on our way:

• We sit (deep!) in the We sit (deep!) in the chirally restored regimechirally restored regime while we look for a way to get to its frontier… while we look for a way to get to its frontier…

• We are aware of the We are aware of the asymptotic natureasymptotic nature of any perturbative expansion… of any perturbative expansion…

• We deal with Wilson fermions (ok… well… chiral properties not so brilliant…)We deal with Wilson fermions (ok… well… chiral properties not so brilliant…)

We can nevertheless hope to turn (at least part) of the difficulties into opportunities:We can nevertheless hope to turn (at least part) of the difficulties into opportunities:

• It is interesting enough to understand It is interesting enough to understand how far PT can lead ushow far PT can lead us towards chiral symmetry towards chiral symmetry breaking (coming from the other side!)breaking (coming from the other side!)

• We can get some piece of information from the (at least apparent) We can get some piece of information from the (at least apparent) convergence propertiesconvergence properties of an asymptotic perturbative expansion.of an asymptotic perturbative expansion.

We can hope to follow the eigenvalues in their way to zero!We can hope to follow the eigenvalues in their way to zero!

Take care of Take care of spectral observables renormalization propertiesspectral observables renormalization properties! (! (M. LuscherM. Luscher,, L. Giusti L. Giusti). ).

Dealing with Wilson fermions, in our case we also have to care of critical mass counterterms. Dealing with Wilson fermions, in our case we also have to care of critical mass counterterms.

As a matter of fact spectral observables are not that natural as objects (observables?) in As a matter of fact spectral observables are not that natural as objects (observables?) in Quantum Field Theory.Quantum Field Theory.

Define the Define the average number of eigenvaluesaverage number of eigenvalues of of DDmm††DDmm within a given thresholdwithin a given threshold

Then consider the Then consider the spectral sumsspectral sums defined as (it is sufficient to understand their renormalization defined as (it is sufficient to understand their renormalization properties because one can invert to get mode number)properties because one can invert to get mode number)

It turns out that these spectral sums can be It turns out that these spectral sums can be mapped to composite operators in Twisted Mass mapped to composite operators in Twisted Mass QCDQCD, in terms of which they have a natural renormalization prescription (renormalize all the , in terms of which they have a natural renormalization prescription (renormalize all the masses which are around with masses which are around with ZZPP…)…)

Polyakov loop, Z(3) and all that…Polyakov loop, Z(3) and all that…Are spontaneous chiral symmetry breaking and confinement in the end related?Are spontaneous chiral symmetry breaking and confinement in the end related?

The relation beween The relation beween chiral symmetry breakingchiral symmetry breaking and and confinementconfinement is one the long standing is one the long standing problem of our field. As the problem of our field. As the chiral condensatechiral condensate is an order parameter for the first transition, is an order parameter for the first transition, Polyakov loopPolyakov loop is for sure related to the second. is for sure related to the second.

The Polyakov loop is an order parameter for the The Polyakov loop is an order parameter for the quenched casequenched case, in which , in which Z(3) symmetryZ(3) symmetry and and its breaking are in place. Due to its its breaking are in place. Due to its relation to static quark-antiquark potentialrelation to static quark-antiquark potential, it is a good , it is a good indicator for confinement also for the complete theory.indicator for confinement also for the complete theory.

A few years ago A few years ago GattringerGattringer put forward a relation between Polyakov loop and Dirac spectrum put forward a relation between Polyakov loop and Dirac spectrum in the background of different Z(3) vacua.in the background of different Z(3) vacua.

In his more recent words, “In his more recent words, “the response of Dirac spectra to different temporal boundary the response of Dirac spectra to different temporal boundary conditions contains information about confinementconditions contains information about confinement”.”.

Our idea: Our idea: compute the Dirac spectrum in the background of different Z(3) vacuacompute the Dirac spectrum in the background of different Z(3) vacua..

• Is the repulsion of eigenvalues much the same?Is the repulsion of eigenvalues much the same?• How do convergence properties of the series compare?How do convergence properties of the series compare?• How does the reconstruction of Polyakov loop work in perturbation theory?How does the reconstruction of Polyakov loop work in perturbation theory?

From From Stochastic QuantizationStochastic Quantization to to NSPTNSPTNSPT NSPT comes almost for free from the framework of comes almost for free from the framework of Stochastic QuantizationStochastic Quantization ( (Parisi and Parisi and WuWu,, 1980 1980). From the latter originally both a non-perturbative alternative to ). From the latter originally both a non-perturbative alternative to standard Monte Carlo and a new version of Perturbation Theory were developed. NSPT standard Monte Carlo and a new version of Perturbation Theory were developed. NSPT in a sense interpolates between the two. in a sense interpolates between the two.

Now, the Now, the main assertionmain assertion is very simply stated: is very simply stated: asymptoticallyasymptotically

Stochastic Quantization Stochastic Quantization andand Stochastic Stochastic Perturbation TheoryPerturbation Theory

In the previous formula, In the previous formula, is a is a gaussian noisegaussian noise, from which the stochastic nature of the , from which the stochastic nature of the equation originates.equation originates.

Given a field theory, Stochastic Quantization basically amounts to giving to the field an extra Given a field theory, Stochastic Quantization basically amounts to giving to the field an extra degree of freedom, to be thought of as a degree of freedom, to be thought of as a stochastic timestochastic time in which an evolution takes place in which an evolution takes place according to the according to the Langevin equationLangevin equation

To understand, take the standard example: To understand, take the standard example: 44 theory ... theory ...

The free case is easy to solve in term of a propagator ...The free case is easy to solve in term of a propagator ...

... and for the interacting case you can always trade the differential equation for an integral one ... ... and for the interacting case you can always trade the differential equation for an integral one ...

Without entering into details: solve by iteration …Without entering into details: solve by iteration … Diagrammatically ... Diagrammatically ...

+ + λλ + + λλ22 ( ( + + ... ) + O(... ) + O(λλ33 ))

+ 3 + 3 λλ ( ( ++ ) + O() + O(λλ22))... and this is a propagator ...... and this is a propagator ...

And hereAnd here Stochastic Perturbation TheoryStochastic Perturbation Theory comes comes

If you insert the previous expansion in the Langevin equation, the latter gets translated into a If you insert the previous expansion in the Langevin equation, the latter gets translated into a hierarchy of equationshierarchy of equations, each for each order, each dependent on lower orders., each for each order, each dependent on lower orders.

Numerical Stochastic Perturbation TheoryNumerical Stochastic Perturbation TheorySince the solution of Langevin equation will depend on the coupling constant of the theory, look for Since the solution of Langevin equation will depend on the coupling constant of the theory, look for the solution as a the solution as a power expansionpower expansion

Observation: we can get power expansions from Stochastic Quantization’s main assertion, e.g.Observation: we can get power expansions from Stochastic Quantization’s main assertion, e.g.

Now, also Now, also observablesobservables are expanded are expanded

NSPTNSPT ( (Di Renzo, Marchesini, Onofri 94Di Renzo, Marchesini, Onofri 94) simply amounts to the ) simply amounts to the numerical integrationnumerical integration of SPT of SPT equations on a computer! Of courseequations on a computer! Of course this time we are dealing with a this time we are dealing with a LATTICELATTICE regularization in x- regularization in x-space and the time evolution has of course to be discretized ...space and the time evolution has of course to be discretized ...

Langevin equation for LGT Langevin equation for LGT goes back to the 80’s (goes back to the 80’s (Cornell Group 84Cornell Group 84): the main point is to formulate ): the main point is to formulate a a stochastic process in the group manifoldstochastic process in the group manifold..

NSPT for Lattice Gauge Theories NSPT for Lattice Gauge Theories ((JHEP0410:073JHEP0410:073))

Then one has to implement a finite difference integration scheme (i.e. Euler) Then one has to implement a finite difference integration scheme (i.e. Euler)

1 1 is not the only trivial order for our expansion! Other is not the only trivial order for our expansion! Other vacua vacua are viable choices as well!are viable choices as well!

NSPT around non trivial vacuaNSPT around non trivial vacua

Since dynamics is dictated by the equations of motion, any Since dynamics is dictated by the equations of motion, any classical solution classical solution is good enough!is good enough!

UUxx(t;(t;

Fermionic observables Fermionic observables are then constructed by are then constructed by invertinginverting (maybe several times) the (maybe several times) the Dirac matrixDirac matrix on on convenient sources. The Dirac matrix in turn is a function of the gluonic field, and because of that is convenient sources. The Dirac matrix in turn is a function of the gluonic field, and because of that is expressed as a expressed as a seriesseries as well as well

The good point is that The good point is that free partfree part is diagonal in is diagonal in p-spacep-space, while , while interactionsinteractions are diagonal in are diagonal in x-spacex-space: go : go back and forth via back and forth via FFTFFT! This is also crucial in taking into account fermions in the evolution.! This is also crucial in taking into account fermions in the evolution.

Degenerate eigenvalues PTDegenerate eigenvalues PTA textbook computation … still many textbooks mess up with it!A textbook computation … still many textbooks mess up with it!

We express the solution in terms of projectors inside and outside the degeneration space of We express the solution in terms of projectors inside and outside the degeneration space of the starting (unperturbed, i.e. free field) eigenvalue. Notice that inside the free field the starting (unperturbed, i.e. free field) eigenvalue. Notice that inside the free field eigenspace there is the component selected by diagonalizing the perturbation and a eigenspace there is the component selected by diagonalizing the perturbation and a component perpendicular to it.component perpendicular to it.

On the (perturbative) configurations we produce via NSPT dynamics we want to compute On the (perturbative) configurations we produce via NSPT dynamics we want to compute something like (a very general notation)something like (a very general notation)

Our goal is to get the perturbative solutionsOur goal is to get the perturbative solutions

Notice that one recognizes the standard features of perturbative eigenvalues:Notice that one recognizes the standard features of perturbative eigenvalues:

• Construction is Construction is iterativeiterative (by the way, our iterative inversion procedure is in place). (by the way, our iterative inversion procedure is in place).• Eigenvalues repel each otherEigenvalues repel each other!!

Beware! This works if degeneracy is lifted at first order. Otherwise one has to go for a Beware! This works if degeneracy is lifted at first order. Otherwise one has to go for a different formalism (i.e. there is a third projector to be taken into account)different formalism (i.e. there is a third projector to be taken into account)

We then rewrite our equationWe then rewrite our equation

Results now follow by applying the projectors to the eigenvalue equationResults now follow by applying the projectors to the eigenvalue equation

Some resultsSome resultsPlots are always instructive to look at when you deal with distributionsPlots are always instructive to look at when you deal with distributions

The first (trivial) order (g) of the correction to the second eigenvalue (ordering refers to The first (trivial) order (g) of the correction to the second eigenvalue (ordering refers to unperturbed spectrum, i.e. free field)unperturbed spectrum, i.e. free field)

The first non trivial order (gThe first non trivial order (g22) of the correction to the second eigenvalue (ordering refers to ) of the correction to the second eigenvalue (ordering refers to unperturbed spectrum, i.e. free field)unperturbed spectrum, i.e. free field)

Now it is good to go back to our movie: this is Now it is good to go back to our movie: this is density of eigenvaluesdensity of eigenvalues up to up to one loopone loop, lowering , lowering from 40 to 6 from 40 to 6 (on a (on a 6644 lattice) lattice)

Where do eigenvalues moving to zero come from?Where do eigenvalues moving to zero come from?Plots are always instructive to look at when you deal with distributionsPlots are always instructive to look at when you deal with distributions

We plot the plain value of the eigenvalues (ordered at tree level). Red is tree level, black is We plot the plain value of the eigenvalues (ordered at tree level). Red is tree level, black is correction at one loop (4correction at one loop (444))

Is it an artifact?Is it an artifact?Always look for finite volume effects …Always look for finite volume effects …

Again. Red is tree level, black is correction at one loop (6Again. Red is tree level, black is correction at one loop (644))

Once again it is good to go back to a movie: let’s have a look at the repulsionOnce again it is good to go back to a movie: let’s have a look at the repulsion

Why only one loop?Why only one loop?NSPT is there to go higher than that …NSPT is there to go higher than that …

• Distributions for Distributions for higher loopshigher loops display display hugehuge tailstails!!

Is there anything going really wrong?Is there anything going really wrong?Always check results, if you can …Always check results, if you can …

It was ok!It was ok!Notice that there is a potential source of numerical instability which is easy to spot and we are Notice that there is a potential source of numerical instability which is easy to spot and we are now trying to investigate:now trying to investigate:The dimension of degenerate eigenspaces at tree level can be large; if by numerical artifacts The dimension of degenerate eigenspaces at tree level can be large; if by numerical artifacts one misses a residual degeneracy, the effect is going to be huge!one misses a residual degeneracy, the effect is going to be huge!

• A consistency check was performed by computing both directly and via the eigenvalues A consistency check was performed by computing both directly and via the eigenvalues quantitites likequantitites like

Remember:Remember:

IDEAIDEA: use it as a regulator when first order corrections almost coincide…: use it as a regulator when first order corrections almost coincide…

Now, Now, if degeneracy is still there at first orderif degeneracy is still there at first order, one has to look for , one has to look for

ConclusionsConclusions

• I only discussed preliminary results.I only discussed preliminary results.

• There is something already valuable: one can really see that eigenvalues There is something already valuable: one can really see that eigenvalues repulsion is indeed there (and they come from the bulk to zero).repulsion is indeed there (and they come from the bulk to zero).

• We now should try to understand what is going on at higher loops.We now should try to understand what is going on at higher loops.

• Computations in the background of different Z(3) vacua are under their way.Computations in the background of different Z(3) vacua are under their way.

• Having the whole spectrum at disposal opens the way to a variety of Having the whole spectrum at disposal opens the way to a variety of computations…computations…

the dirac operator spectrum from a perturbative approach

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