the residual mass in lattice heavy quark effective theory to the third order

The XXII International Symposium on Lattice Field TheoryThe XXII International Symposium on Lattice Field TheoryLattice 2004Lattice 2004

June 25, 2004June 25, 2004Fermi National Accelerator Laboratory Fermi National Accelerator Laboratory



The residual mass in lattice The residual mass in lattice Heavy Quark Effective TheoryHeavy Quark Effective Theory

to the third orderto the third order

Francesco Di RenzoFrancesco Di Renzo ((11)) && Luigi ScorzatoLuigi Scorzato ((22))

(1) (1) Università di Parma Università di Parma andand INFN, INFN, Parma, ItalyParma, Italy

(2) (2) Humboldt-UniversitHumboldt-Universitäät,t,Berlin, GermanyBerlin, Germany



OutlineOutline

MotivationMotivationWhy one should deal with the residual mass (self-Why one should deal with the residual mass (self-energy)energy) δmδm: : thethe bb-quark-quark mass mass in a HQET approach in a HQET approach ((Martinelli et alMartinelli et al))

Computational SetupComputational SetupNumerical Stochastic Perturbation TheoryNumerical Stochastic Perturbation Theory ((Parma Parma group after group after Parisi & WuParisi & Wu))

PerspectivesPerspectives The impact of the computationThe impact of the computation

The computation itselfThe computation itselfThe perturbative evaluation of the static inter-quark The perturbative evaluation of the static inter-quark potential from (big) Wilson Loops potential from (big) Wilson Loops δmδm



The The bb-quark -quark massmass from the from the latticelattice

Despite the fact that you can not accommodate the b-quark on Despite the fact that you can not accommodate the b-quark on the lattice, its mass can be determined to a very good accuracy the lattice, its mass can be determined to a very good accuracy from the lattice.from the lattice.

V. Gimenez et al, JHEP 0003:018, 2000 >>>> HQET + Perturbation Theory HQET + Perturbation Theory

S. Collins, Quark Confin. and the Hadron Spec., World Scientific (2002), 325>>>> NRQCD NRQCD

ALPHA Collaboration, JHEP 0402:022, 2004 >>>> HQET in a Non-Perturbative framework HQET in a Non-Perturbative framework

G. De Divitiis et al, Nucl.Phys.B675 (2003), 309 >> Step-scaling method >> Step-scaling method



HQETHQET and its and its LATTICELATTICE counterpart counterpart

... but as a matter of fact HQET theory itself appears ... but as a matter of fact HQET theory itself appears conceptually simple, but is full of subtleties, many of which conceptually simple, but is full of subtleties, many of which have to do with a proper definition of the heavy quark mass have to do with a proper definition of the heavy quark mass itself. itself.

It is quite difficult to figure out something easier to write down It is quite difficult to figure out something easier to write down than a lattice version of HQET ...than a lattice version of HQET ...

A very important one emerges from the very fundamental A very important one emerges from the very fundamental relation one would like to exploit in order to connect the relation one would like to exploit in order to connect the mass mass of a physical hadronof a physical hadron to the HQET expansion to the HQET expansion mass parametermass parameter and the (and the (linearly divergent!linearly divergent!) ) binding energybinding energy



How to get a measure ...How to get a measure ...V. Gimenez et al, JHEP 0003:018, 2000

By matching the QCD propagator to its lattice HQET counterpart one By matching the QCD propagator to its lattice HQET counterpart one gets a relation involving the gets a relation involving the pole masspole mass (which has been matched to (which has been matched to the the MS massMS mass) ...) ...

... where a new character enters the stage: a linearly divergent ... where a new character enters the stage: a linearly divergent additive mass counterterm (additive mass counterterm (residual massresidual mass), which we can compute in ), which we can compute in Perturbation Theory. Perturbation Theory.

As already said, the pole mass and the MS one are related in PTAs already said, the pole mass and the MS one are related in PT

So that one can put everything togetherSo that one can put everything together



A few comments are in order!A few comments are in order!

The expansion of The expansion of δmδm is in charge of is in charge of canceling the linear divergencecanceling the linear divergence of the binding energyof the binding energy

It is in charge of It is in charge of canceling a renormalon ambiguitycanceling a renormalon ambiguity as well (the latter as well (the latter comes from the pole mass)comes from the pole mass)

Everything takes place at a Everything takes place at a fixed orderfixed order in Perturbation Theory! in Perturbation Theory!

Many things are taking place in the fundamental relation, of which the Many things are taking place in the fundamental relation, of which the perturbative expansion of the residual mass is in chargeperturbative expansion of the residual mass is in charge

Since DSince D00,D,D11,D,D22 are known ( are known (Chetyrkin et alChetyrkin et al, , Melnikov et alMelnikov et al), one needs ), one needs XX00,X,X11,,XX22..



Computational Setup (Computational Setup (NSPTNSPT))F. Di Renzo, G. Marchesini, E. Onofri, Nucl.Phys. B457 (1995), 202

F. Direnzo, L. Scorzato, JHEP 0102:020, 2001

NSPT comes as an application of NSPT comes as an application of Stochastic QuantizationStochastic Quantization ( (Parisi & WuParisi & Wu): ): the field is given an extra degree of freedom, to be thought of as a the field is given an extra degree of freedom, to be thought of as a stochastic time, in which an evolution takes place according to the stochastic time, in which an evolution takes place according to the LangevinLangevin equation equation

The main assertion is (remember: The main assertion is (remember: η is gaussian noise)η is gaussian noise)

Both the Langevin equation and the main assertion get translated in Both the Langevin equation and the main assertion get translated in a tower of relations ...a tower of relations ...

We now simply implement on a computer the We now simply implement on a computer the expansionexpansion which is the which is the starting point of Stochastic Perturbation Theorystarting point of Stochastic Perturbation Theory



Computational strategyComputational strategyF. Direnzo, L. Scorzato, JHEP 0102:020, 2001

where in the first factor we can see the where in the first factor we can see the linear divergencelinear divergence we are we are interested in. In the second factor there are a couple of interested in. In the second factor there are a couple of loglog divergences: one is the usual divergence that one can absorb in the divergences: one is the usual divergence that one can absorb in the redefinition of the redefinition of the couplingcoupling, the other has to do with , the other has to do with corners corners ((Dotsenko & VergelesDotsenko & Vergeles). ).

For a generic Wilson Loop (also for a P-line)For a generic Wilson Loop (also for a P-line)

If we compute the (approximants for the) potential via Creutz’s ratiosIf we compute the (approximants for the) potential via Creutz’s ratios

the corner divergences disappear. As for the coupling, there is a the corner divergences disappear. As for the coupling, there is a standard way to renormalize it:standard way to renormalize it:



We fit our data to the expected form by varying We fit our data to the expected form by varying TT and the and the RR-interval, -interval, with with R>3R>3 and and T>2.5*mean(R)T>2.5*mean(R). We then choose in terms of . We then choose in terms of χχ22: errors : errors refer to the interval embraced by letting χrefer to the interval embraced by letting χ2 2 vary within a given vary within a given interval. On top of that we also inspect the impact of lattice artifacts interval. On top of that we also inspect the impact of lattice artifacts (an handle is the value of known parameters entering the matching (an handle is the value of known parameters entering the matching relations).relations).

so that we only have to fit the residual mass! (but remember: we so that we only have to fit the residual mass! (but remember: we work on work on 323244))

What we have done this way is defining the coupling in the potential What we have done this way is defining the coupling in the potential scheme. We know its matching to the lattice scheme (due to work of scheme. We know its matching to the lattice scheme (due to work of SchrSchröderöder, , Christou et alChristou et al):):

For the quenched and unquenched (NFor the quenched and unquenched (Nff = 2) case we get (first two = 2) case we get (first two terms were already known; third term for quenched case computed terms were already known; third term for quenched case computed four years ago and also computed by four years ago and also computed by Trottier et alTrottier et al))



1 2 3 4 5 6 7 8 9 102.5

3

3.5

4

4.5

R

VT(1)(R)

1 2 3 4 5 6 7 8 9 1010

15

20

25

R

VT(2)(R)



1 2 3 4 5 6 7 8 9 1060

70

80

90

100

110

120

130

140

R

VT(3)(R)



The impact of the computationThe impact of the computationV. Gimenez, private communication

>> Results differ, even if they are compatible within errors. Results differ, even if they are compatible within errors.

>> In both cases, the effect of the third order in the expansion of the In both cases, the effect of the third order in the expansion of the residual mass on the last errors (the ones connected to residual mass on the last errors (the ones connected to indeterminations in the series) is quite important (indeterminations in the series) is quite important (errors roughly errors roughly halvedhalved).).>> The dependence on the lattice spacing is not dramatic and gets The dependence on the lattice spacing is not dramatic and gets decreased by the new terms.decreased by the new terms.

>> The control over renormalon ambiguities seems quite firm. The control over renormalon ambiguities seems quite firm.

Already four years ago one could inspect the impact of the quenched Already four years ago one could inspect the impact of the quenched computationcomputation

Now one can look at what happens for the unquenched (NNow one can look at what happens for the unquenched (Nff = 2) case = 2) case

the residual mass in lattice heavy quark effective theory to the third order

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