m ethod of r egions and i ts a pplications

Method of Regionsand Its Applications

2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC 1

Graduate University of the CAS

Deshan Yang

Outline


1. Introduction

2. Examples of Method of Regions

3. Connections to Effective Field Theory

4. Applications

5. Summary

Victor Frankenstein’s Idea of Science


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: FrankensteinTo create the Frankenstein’s monster or an angel?

Beauty charmless decay


Many scales

Many couplings

Many hadrons

Difficulties： Strong interactions

Way-out： Factorization

Factorization


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?


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.


Example 1: Two-masses dependent integral


Cut-off regularization


UV div. IR div.

Dimensional regularization

2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC10

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


2221 ))()((

][pkykqkykqk

dkI

222

2

4qpqmy

2,, 21

2122

22

1pppppqmpp

qkqk ~,~0

1

1 2 2 2 2

[ ] 4 1 ( )( )( ) 2 1 2

Eh dkI ek k q k k q k q

ykqyk ~,/~0

2)2/1(

))((1)1(

22212/

1

1

yq

yepkyk

kdeq

I EEd

dp

ykyk ~,~0

qykqyk /~,/~0

Adding up


11 2 1

1

2

1 ˆ( ) (1/ 2,1 ,3 / 2; 1/ (4 ))2

ˆ4 (4 ) ( 1/ 2)8 2(1 2 )ˆ

E

E

I e y F y

yeq y

/1 1 1 1

h p s usI I I I

Remarks on method of regions


Effective Field Theory


Application 1: Effective weak Hamiltonian


Effective operators


First step factorization in B decays


Example of matching : Tree-level


One-loop level matching equation


(1)fulliM

...

1Q

1Q

...

One-loop matching equation


1tr loopfulliM iM iM 1 1tr loop loop

hard IRiM iM iM

Hard part


Putting together


Renormalization


Application 2: Heavy-to-light Form-factors


Factorization formula


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


Matching procedure


m ethod of r egions and i ts a pplications

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