1. K  πνν, K  π l + l - on the Lattice A feasibility study Paolo Turchetti LNF Spring School Frascati, 19 th May

2. Rare decays: the present procedure Rare decays: K +  π + νν, K +/-  π +/- l + l –, K S  π 0 l + l – Wilson coefficients Operator matrix elements Top, W, beauty, charm integrated out Experiments χPT Perturbation Theory + renormalization group Phenomenological Predictions

3. Problems with this procedure The problems affecting this procedure are connected with the importance of long distance physics in the description of this decays. For example in the K  πνν family there is only the weak contribution. In this case a hard GIM mechanism is active The errors affecting the theoretical estimates of charm contribution (the long distance one) are relevant O(20%-30%). By courtesy of G. Isidori Operators with dimension greater than 6 Truncated perturbative expression for Wilson Coefficients (NNLO terms are not included) Is perturbation theory valid at a such low energy scale (< 1 GeV)? Error sources

4. Rare decays: K +  π + νν, K +/-  π +/- l + l –, K S  π 0 l + l – Wilson coefficients Operator matrix elements Top, W, beauty, integrated out Experiments + LQCD Perturbation Theory + renormalization group Now charm is a dynamical degree of freedom Phenomenological Predictions Lattice QCD and Rare decays

5. Impact of LQCD computations By means of LQCD we could improve our phenomenological predictions lowering the theoretical errors affecting our estimates of total decay amplitude by an order of magnitude for both K +  π + νν and K  π l + l - decays Owing to the great sensibility of these decays to V td and to physics beyond the Standard Model, an improved theoretical comprehension of these transitions through LQCD computation could have an important impact on our understanding of the dynamics of quark-flavour mixing.

6. LQCD approach If we consider the charm as a dynamical degree of freedom (dof), we can restrict ourselves to compute, through LQCD numerical simulations at a scale greater than the charm mass, the physical amplitude in which are involved the operators In this way we can take into account all the long distance contributions exactly we don’t need to evolve Wilson coefficients down to a too low scale, in such a way that NNLO contributions are small LQCD computations are based only on first principles What do we need to compute?

7. What do we really compute? The physical quantities of our interest are encoded in Where J is the electro-magnetic or weak current. The main issue one has to face is the possible presence of power divergences in the expression of the T-product. These divergences introduce ambiguities that make the extraction of physical informations through numerical simulations impossible. The task of our work is to make an analysis as complete as possible of these power divergences.

8. Classifications of power divergences The power divergences potentially present in our T-product originate from two diverse singularity sources. We can face the first one renormalizing the operators involved in the T-product. In this way one can get rid of the divergences associated with the diagrams +

9. In the second class are included all the divergences due to the singularities appearing in when i.e. those divergences associated with contact terms and with the diagrams This last kind of divergences can’t be removed by simply renormalizing the relevant operators.

10. The Bubble (1) At this point is useful to emphasize an important issue regarding the differences existing on the lattice between electromagnetic and weak current… These topologies are connected by Fierz transformations so that we are allowed to restrict our attention only on one of them

11. The Bubble (2) The Z 0 boson is too heavy (M Z ~ 80 GeV) and can’t be considered as a dynamical dof on the lattice because the lattice cutoff at our disposal are only O(3/4 GeV). It must be integrated out and the weak interactions mediated by a Z boson have to be considered as local ones. Of course this is not the case for photon. So we can associate with the photon a non-local interaction. This is a fundamental observation because it brings with it some very important consequences about the way one handles this two cases on the lattice.

12. Bubbles on a lattice In fact, on the lattice we can associate with the photon a gauge-current that is implemented by a splitted (non local) current. With weak current we have to associate a local current that, on the lattice is not a gauge current. This difference is clear if we consider the Feynman rules associated with the two currents Gauge current Local current

13. Photon case In this –lucky– case the constraints imposed by gauge symmetry cause the algebraic cancellation of power divergences. The residual divergences are only logarithmic. We have worked out explicit computations in different QCD regularizations on euclidean lattice (Wilson, clover, twisted mass) and found that

14. Where In all lattice regularizations the power divergences are absent! The only difference between different regularizations concerns the finite term L. are constants andCandL

15. Z boson case As outlined earlier in this case we don’t have a gauge current and the gauge symmetry doesn’t work anymore. So we don’t have any constraints implying the algebraic cancellation of power divergences… As a result we HAVE quadratic divergences left if we are working with Wilson regularization of lattice QCD!!!!! But…

16. Twisted mass fermions GIM mechanism Z 0 case (quadratically divergent) Z 0 case (logarithmically divergent) In all this work one has to cope with the perturbative lattice computations. In particular the twisted mass case is the most involved. Appling the recent theory with twisted mass fermions (Frezzotti, Rossi hep- lat/0306014) we have explicitly demonstrated, through the first computation ever done, to our knowledge, in perturbation theory with this regularization in the “physical basis”, that

17. First conclusion We don’t have any power divergences associated with contact terms both in photon case and in Z boson case! We can handle the residual logarithmically divergent terms by usual perturbative methods. The perturbative procedure is very laborious, but doesn’t imply any conceptual problems. G. Isidori, G. Martinelli and P. Turchetti to be published.

18. The renormalization ambiguity Let’s now face the problem of effective hamiltonian renormalization. The Wilson term, necessary to solve the fermion doubling problem of naive discretization, breaks chiral symmetry explicitly even if we are working with massless fermions. This implies that under the renormalization procedure a generic operator mixes with other operators of equal or lower dimension (in energy) having, in general, different chiral properties. In our case we have that Where are the renormalized operators are the bare ones is the field stenght tensor and

19. Symmetries By dimensional counting we see that But, if we consider the constraints introduced by GIM mechanism and CPS symmetry with We can infer that A and B are at most logarithmically divergent and So we need non perturbative methods to subtract these power divergences.

20. Renormalization condition Just limit ourselves to the photon case. For parity reasons the pseudoscalar density doesn’t appear in this case and the renormalization condition we need to impose takes the form Where C is a constant, J is the electromagnetic current and π and K are the operators interpolating the pion and the Kaon Who can fix the value on r.h.s? The question is:

21. Perturbation theory would be the ideal tool to impose the correct renormalization condition, but the presence of power divergences prevents us to use it. The arbitrariness in the renormalization condition introduces the outlined ambiguities and thus it makes the extraction of reliable physical informations through numerical simulations impossible. In correspondence of every different value of finite term we are given a different estimate of physical quantities: this is the ambiguity. Can this ambiguity really affect the physical quantities?

22. The answer to this question we found is very encouraging. The physical quantities we are looking for doesn’t need any subtraction. This implies that it isn’t affected by any ambiguities! By means of Ward identities… Can this ambiguity really affect the physical quantities?

23. Ward identities … we are given this relation, derivable by means of usual methods, Where S(x) is the scalar density. From this expression we deduce that the insertion of this quantities in our correlator, needed to subtract the power divergences present in the insertion of Q +/-, has a singularity structure completely different from that associated with the physical quantity we are interested in.

24. Pole structure (1) As can be derived by the previous Ward identity the power divergences are proportional to this pole structure i.e. they are proportional to the sum of two correlators characterized each by a single double pole. MinkowskiEuclidean

25. Pole structure (2) so that any other quantity having another singularity structure doesn’t interfere with it. The physical quantities we are interested in are characterized by the product of two simple poles, one of which is associated with the mass of the Kaon and the other one with the mass of the pion Minkowski Euclidean

26. Second Result As a matter of fact the power divergences, due to their pole structure, can’t affect the physical quantities we want to extract. We can estimate the matrix elements of Q +/- operators without any ambiguities. G. Isidori, G. Martinelli and P. Turchetti to be published.

27. Summary In conclusion we can summarize our results: We have demonstrated, through explicit computations, the algebraic cancellation of power divergences due to the contact terms present in the relevant T-product We have demonstrated the absence of any ambiguities in the extraction of physical quantities due to renormalization conditions.

28. Fermion doubling problem Naive discretization

29. Physical content 16 solutions 16 lattice degrees of freedom One of the main consequences is that this theory is anomaly-free

30. Wilson Action Wilson term. This term breaks the chiral symmetry explicitly even if we are considering massless fermions

31. Twisted mass action Where the quarks are organized in mass-degenerate doublet so that And where Twisted Wilson Term

32. LSZ reduction formulas assert that we can extract S-matrix elements for a particular transition taking into account the Fourier transforms of T-product of appropriate operators and than going on-shell. For example if we consider a four particles transition we’ll have Where m i are the masses of the particles involved in this process and O i are the operators interpolating the particles. LSZ reduction formula