1. CKM, Dec. 2006 Aneesh Manohar & I.S. hep-ph/0605001 Arnesen, Ligeti, Rothstein, & I.S. hep-ph/0607001; Lange, Manohar, & I.S. hep-ph/0? ? ?001 Iain Stewart MIT Iain Stewart MIT Dissecting Penguins and Annihilation with SCET: rapidity factorization and the zero-bin Dissecting Penguins and Annihilation with SCET: rapidity factorization and the zero-bin hep-ph/0611356

2. Form Factors Nonleptonic universality at Bauer, Pirjol, Rothstein, I.S. Bauer, Pirjol, Rothstein, I.S. Factorization with SCET Formalism,,,,,...

3. with Latest Data Numbers Isospin + bare minimum from expansion small strong phase between two “tree” amplitudes small strong phase between two “tree” amplitudes (incl. expt. and theory errors) Bauer, Rothstein, I.S.,, fit still gives hard-scattering & soft form-factor similar in size fit still gives hard-scattering & soft form-factor similar in size expt. only

4. Beneke, Buchalla, Neubert, Sachrajda Beneke, Buchalla, Neubert, Sachrajda “BBNS approach” “SCET approach” perturbative universal functions Bauer, Pirjol, Rothstein, I.S. Bauer, Pirjol, Rothstein, I.S. left as a form factor,, Singularity: SCET II

5. Keum, Li, Sanda “pQCD approach” instead of form factor get: however, definition of these functions is generically still singular ‘s

6. Consider quark m.elt. in light-cone gauge, or with lines in Feyn. gauge: tree singular one-loop 0 0 y y Collins (hep-ph/0304122) Details on “still singular”: eg.

7. Keum, Li, Sanda “pQCD approach” instead of form factor get: however, definition of these functions is still singular ‘s Rapidity Factorization (0-bin) Manohar & I.S. rapidity parameter rapidity parameter

8. Penguin ology

9. data+isospin: Non.Pert. Charm Penguin Ciuchini et al, Colangelo et al Ciuchini et al, Colangelo et al Chiral Enh. terms Chiral Enh. terms BBNS LO terms Beneke, Jager hep-ph/0610322; Jain, Rothstein, I.S. (unpub) Beneke, Jager hep-ph/0610322; Jain, Rothstein, I.S. (unpub) Arnesen et al. singular Annihilation terms Annihilation terms Keum, Li, Sanda Keum, Li, Sanda Penguin Phenomenology theory: Using factorization for Trees and  fit Using factorization for Trees and  fit phase relative to phase relative to

10. theory: Big ? Big ?? Ciuchini et al; Bauer et al;... Keum, Li, Sanda ; BBNS For all groups: The terms used to bring theory in agreement with data depend on model parameters For all groups: The terms used to bring theory in agreement with data depend on model parameters data+isospin: Penguin Phenomenology Using factorization for Trees and  fit Using factorization for Trees and  fit Large Im. part Large Im. part

11. The Zero-Bin SCET modes have restrictions to avoid double counting SCET modes have restrictions to avoid double counting collinear label momentum removes support of collinear integrand in ultrasoft region removes support of collinear integrand in ultrasoft region Needed for anom. dimensions. Needed for real radiation calculations. Formulates SCET with arbitrary regulators. Needed for anom. dimensions. Needed for real radiation calculations. Formulates SCET with arbitrary regulators.

12. Cross-Check: Three singularities in non-relativistic field theory (NRQCD, NRQED) 1) Static potential in perturbative QCD is IR divergent Static potential in perturbative QCD is IR divergent 2) Lamb shift in positronium 3) pinch singularity with two heavy static (soft) particles pinch singularity with two heavy static (soft) particles 3) 1,2) These singularities come from taking a double limit: 3) soft overlaps potential region 1,2) soft overlaps ultrasoft region Zero-bin avoids double counting and avoids these singularities, so that proper momentum mode can describe the region Zero-bin avoids double counting and avoids these singularities, so that proper momentum mode can describe the region

13. For the endpoint divergence, the singularity also comes from taking a double limit: collinear SCET II which encounters the soft region Rapidity Fact. in SCET distinguishes the collinear and soft d.o.f. Rapidity Fact. in SCET distinguishes the collinear and soft d.o.f. In a scheme with no hard cutoff, the zero-bin subtractions remove endpt. divergences & rapidity regulator is needed for the UV. In a scheme with no hard cutoff, the zero-bin subtractions remove endpt. divergences & rapidity regulator is needed for the UV. get Works with both cutoff and dim.reg. type regulators Works with both cutoff and dim.reg. type regulators well defined and REAL Rapidity parameters also give well-defined kT parton distribution functions (see Collins). Rapidity parameters also give well-defined kT parton distribution functions (see Collins). nonpert. functions

14. What about the messenger modes? In the case with endpoint singularities we effectively absorb the messenger into a combination of and “messenger” scales show up in perturbation theory Becher, Hill, Neubert Beneke, Feldmann; Bauer, Dorsten, Salem Must consider effect of confinement. but only for special choices of the IR regulators but only for special choices of the IR regulators

15. Apply rapidity factorization: lowest order annihilation is REAL lowest order annihilation is REAL Annihilation in B-Decays Arnesen, Ligeti, Rothstein, & I.S.

16. Annihilation is real at lowest order Suffers from endpoint divergences. But they do not introduce a phase. Suffers from endpoint divergences. But they do not introduce a phase. Leading order, its same size as a). No endpoint divergences here. No imaginary part here either. Leading order, its same size as a). No endpoint divergences here. No imaginary part here either. A soft rescattering annihilation contribution DOES have a strong phase, but is one higher order in A soft rescattering annihilation contribution DOES have a strong phase, but is one higher order in Might be large if intermediate scale expansion did not converge here hep-ph/0611356 However: Arnesen et al.

17. Conclusion Differential formulation of The zero-bin. continuum EFT. Resolves singularities, but formalism is in its infancy. Still theory (eg. RGE) to work out. Resolves singularities, but formalism is in its infancy. Still theory (eg. RGE) to work out. Annihilation Amplitudes are real at lowest order with a factorization theorem in rapidity space (unless pert. theory at int. scale breaks down, but most factorization analyses assume that it does not) Annihilation Amplitudes are real at lowest order with a factorization theorem in rapidity space (unless pert. theory at int. scale breaks down, but most factorization analyses assume that it does not) (in frag. function literature this RGE is known as the Collins-Soper Eqtn.)

18. THE END