Method of Regions and Its Applications The Interdisciplinary Center for Theoretical Study, USTC 1 Graduate University of the CAS Deshan Yang

Outline The Interdisciplinary Center for Theoretical Study, USTC 2 1.Introduction 2.Examples of Method of Regions 3.Connections to Effective Field Theory 4.Applications 5.Summary

Victor Frankenstein’s Idea of Science The Interdisciplinary Center for Theoretical Study, USTC 3  Modern Physics Understand the nature of the Universe qualitatively and quantitatively.  What can we do? Anatomy--approaching to the truth gradually Cut the body into pieces and study each part Stitch them together and hope for the best Scientist: Frankenstein To create the Frankenstein’s monster or an angel?

Beauty charmless decay The Interdisciplinary Center for Theoretical Study, USTC 4 Many scales Many couplings Many hadrons Difficulties ： Strong interactions Way-out ： Factorization

The Interdisciplinary Center for Theoretical Study, USTC 5

Questions to be answered  How to separate the contributions from the different scales?  How to establish the RGEs to resum the large logarithms?  How to estimate or compensate the loss due to the power corrections? The Interdisciplinary Center for Theoretical Study, USTC 6 Method of regions can help!

Integration by regions For a Feynman integral containing small parameters (multiple- scale problem) in dimensional regularization  Divide the space of the loop momenta into various regions and, in each region, expand the integrand into a Taylor series with respect to the parameters that are considered small there;  Integrate the integrand, expanded in the appropriate way in every region, over the whole integration domain of the loop momenta;  Add up all the expanded integrals in all regions, we reproduce the Taylor series of the original Feynman integral with respect to the small parameters exactly.  Finally, a multiple-scale problem is divided into single (less) scale problems The Interdisciplinary Center for Theoretical Study, USTC 7

Example 1: Two-masses dependent integral The Interdisciplinary Center for Theoretical Study, USTC 8

Cut-off regularization The Interdisciplinary Center for Theoretical Study, USTC 9 UV div. IR div.

Dimensional regularization The Interdisciplinary Center for Theoretical Study, USTC 10  The expansion is valid up to any order of a;  The integral in each region is the function of only one scale and simpler than the original integral;  The factious divergence in each region is cancelled after adding up the contributions from large scale region and small scale region. UV div.IR div.

Example 2: Threshold Expansion Beneke & Smirnov, NPB1998  Small parameter:  Hard region:  Potential region:  Soft/Ultra-soft region: or Tadpole diagrams: 0 in DR The Interdisciplinary Center for Theoretical Study, USTC 11

Adding up The Interdisciplinary Center for Theoretical Study, USTC 12

Remarks on method of regions The Interdisciplinary Center for Theoretical Study, USTC 13

Effective Field Theory The Interdisciplinary Center for Theoretical Study, USTC 14

Application 1: Effective weak Hamiltonian The Interdisciplinary Center for Theoretical Study, USTC 15

Effective operators The Interdisciplinary Center for Theoretical Study, USTC 16

First step factorization in B decays The Interdisciplinary Center for Theoretical Study, USTC 17

Example of matching : Tree-level The Interdisciplinary Center for Theoretical Study, USTC 18

One-loop level matching equation The Interdisciplinary Center for Theoretical Study, USTC 19

One-loop matching equation The Interdisciplinary Center for Theoretical Study, USTC 20

Hard part The Interdisciplinary Center for Theoretical Study, USTC 21

Putting together The Interdisciplinary Center for Theoretical Study, USTC 22

Renormalization The Interdisciplinary Center for Theoretical Study, USTC 23

Application 2: Heavy-to-light Form-factors The Interdisciplinary Center for Theoretical Study, USTC 24

Factorization formula The Interdisciplinary Center for Theoretical Study, USTC 25 There’s another factorization formula in which the transverse momenta of the patrons are invoked to avoid the endpoint singularity. Kurimoto, Li, Sanda 2002

Factorization formula in SCET The Interdisciplinary Center for Theoretical Study, USTC 26

Matching procedure The Interdisciplinary Center for Theoretical Study, USTC 27

More on matching The Interdisciplinary Center for Theoretical Study, USTC 28

“Hard” contribution The Interdisciplinary Center for Theoretical Study, USTC 29

Wilson coefficients The Interdisciplinary Center for Theoretical Study, USTC 30

Wilson coefficients The Interdisciplinary Center for Theoretical Study, USTC 31

RGEs The Interdisciplinary Center for Theoretical Study, USTC 32

Jet functions The Interdisciplinary Center for Theoretical Study, USTC 33

Application 3: B two-body charmless decay The Interdisciplinary Center for Theoretical Study, USTC 34

Matching onto SCETII The Interdisciplinary Center for Theoretical Study, USTC 35

Factorization formula The Interdisciplinary Center for Theoretical Study, USTC 36

Hard-spectator interaction The Interdisciplinary Center for Theoretical Study, USTC 37

NNLO vertex corrections The Interdisciplinary Center for Theoretical Study, USTC 38 Complete NNLO: G.Bell, 2009; Beneke,Li,Huber 2009

Application 4: Exclusive single quarkonium production The Interdisciplinary Center for Theoretical Study, USTC 39

NRQCD factorization For single quarkonium production  : NRQCD operator with definite velocity power counting  multi-scale problem: Q>>m  stability of the perturbation: large log(Q/m) may need the resummation The Interdisciplinary Center for Theoretical Study, USTC 40

Refactorization The Interdisciplinary Center for Theoretical Study, USTC 41 At the leading power of velocity,  The hard kernel is the same as the similar process in which the quarkonium is replaced by a flavor singlet light meson.  Since, the LCDA of bounded heavy quark and anti-quark can be calculated perturbatively. Ma and Si, PRD 2006; Bell and Feldmann, JHEP 2007;

Example: The Interdisciplinary Center for Theoretical Study, USTC 42 NRQCD factorization up to leading power of velocity: The short-distance contribution is parameterized as The equivalent computation is to calculate the on-shell heavy quark anti-quark pair with equal momentum and the same quantum number as the quarkonium. At the tree level,

One-loop level The Interdisciplinary Center for Theoretical Study, USTC 43 Sang, Chen, arXiv: ; Li, He, Chao arXiv:

Leading regions The Interdisciplinary Center for Theoretical Study, USTC 44  Hard Region:  Collinear region:  Anti-collinear region:  Potential region:  Soft region:  Ultra-soft region: NRQCD regions Non-perturbative

Form factor The Interdisciplinary Center for Theoretical Study, USTC 45 NRQCD: Collinear factorization:  Hard-kernel: at tree level  Light-cone distribution amplitude Ma and Si, PRD 2006; Bell and Feldmann, JHEP 2007;

RGE for LCDA The Interdisciplinary Center for Theoretical Study, USTC 46 Brodsky-Lepage kernel: Resum the leading logarithms where

NLO results (preliminary) The Interdisciplinary Center for Theoretical Study, USTC 47 Braaten, PRD 1981; Ma and Si, PRD 2006; Bell and Feldmann, JHEP 2007;  Hard Part  Collinear Part  Total Results Sang, Chen, arXiv: ; Li, He, Chao arXiv:

Summary  Method of regions: Not mathematically proved, but no counter-examples so far.  Intimately connected to the calculation of the matching coefficients in EFT.  Advantages: Multiple scale problems simplified to single scale problems;  Disadvantages: How to find the relevant regions? (No general procedure!) The Interdisciplinary Center for Theoretical Study, USTC 48

谢谢！ The Interdisciplinary Center for Theoretical Study, USTC 49