The residual mass in lattice Heavy Quark Effective Theory to the third order

2. Francesco Di Renzo(1)&Luigi Scorzato(2) (1) Università di Parma and INFN, Parma, Italy (2) Humboldt-Universität, Berlin, Germany The residual mass in lattice Heavy Quark Effective Theory to the third order

3. Motivation Why one should deal with the residual mass (self-energy)δm: theb-quarkmass in a HQET approach (Martinelli et al) Computational Setup Numerical Stochastic Perturbation Theory(Parma group after Parisi & Wu) The computation itself The perturbative evaluation of the static inter-quark potential from (big) Wilson Loops δm Perspectives The impact of the computation Outline

4. >> HQET in a Non-Perturbative framework >> HQET + Perturbation Theory >> NRQCD V. Gimenez et al, JHEP 0003:018, 2000 S. Collins, Quark Confin. and the Hadron Spec., World Scientific (2002), 325 ALPHA Collaboration, JHEP 0402:022, 2004 >> Step-scaling method G. De Divitiis et al, Nucl.Phys.B675 (2003), 309 The b-quark mass from the lattice Despite the fact that you can not accommodate the b-quark on the lattice, its mass can be determined to a very good accuracy from the lattice.

5. It is quite difficult to figure out something easier to write down than a lattice version of HQET ... A very important one emerges from the very fundamental relation one would like to exploit in order to connect the mass of a physical hadron to the HQET expansion mass parameter and the (linearly divergent!) binding energy HQET and its LATTICE counterpart ... but as a matter of fact HQET theory itself appears conceptually simple, but is full of subtleties, many of which have to do with a proper definition of the heavy quark mass itself.

6. How to get a measure ... V. Gimenez et al, JHEP 0003:018, 2000 By matching the QCD propagator to its lattice HQET counterpart one gets a relation involving the pole mass (which has been matched to the MS mass) ... ... where a new character enters the stage: a linearly divergent additive mass counterterm (residual mass), which we can compute in Perturbation Theory. As already said, the pole mass and the MS one are related in PT So that one can put everything together

7. Many things are taking place in the fundamental relation, of which the perturbative expansion of the residual mass is in charge A few comments are in order! The expansion of δm is in charge of canceling the linear divergence of the binding energy It is in charge of canceling a renormalon ambiguity as well (the latter comes from the pole mass) Everything takes place at a fixed order in Perturbation Theory! Since D0,D1,D2 are known (Chetyrkin et al, Melnikov et al), one needs X0,X1,X2.

8. Computational Setup (NSPT) 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 Stochastic Quantization (Parisi & Wu): 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 Langevin equation The main assertion is (remember: η is gaussian noise) We now simply implement on a computer the expansion which is the starting point of Stochastic Perturbation Theory Both the Langevin equation and the main assertion get translated in a tower of relations ...

9. For a generic Wilson Loop (also for a P-line) where in the first factor we can see the linear divergence we are interested in. In the second factor there are a couple of log divergences: one is the usual divergence that one can absorb in the redefinition of the coupling, the other has to do with corners (Dotsenko & Vergeles). If we compute the (approximants for the) potential via Creutz’s ratios the corner divergences disappear. As for the coupling, there is a standard way to renormalize it: Computational strategy F. Direnzo, L. Scorzato, JHEP 0102:020, 2001

10. 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 Schröder, Christou et al): For the quenched and unquenched (Nf = 2) case we get (first two terms were already known; third term for quenched case computed four years ago and also computed by Trottier et al) so that we only have to fit the residual mass! (but remember: we work on 324) We fit our data to the expected form by varying T and the R-interval, with R>3 and T>2.5*mean(R). We then choose in terms of χ2: errors refer to the interval embraced by letting χ2 vary within a given interval. On top of that we also inspect the impact of lattice artifacts (an handle is the value of known parameters entering the matching relations).

13. The impact of the computation V. Gimenez, private communication Already four years ago one could inspect the impact of the quenched computation Now one can look at what happens for the unquenched (Nf = 2) case > Results differ, even if they are compatible within errors. > In both cases, the effect of the third order in the expansion of the residual mass on the last errors (the ones connected to indeterminations in the series) is quite important (errors roughly halved). > The dependence on the lattice spacing is not dramatic and gets decreased by the new terms. > The control over renormalon ambiguities seems quite firm.