method of regions and its applications 2011.4.21 the interdisciplinary center for theoretical study,...

Method of Regionsand Its ApplicationsMethod of Regions

and Its Applications

2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC 1

Graduate University of the CAS

Deshan Yang

OutlineOutline


1. Introduction

2. Examples of Method of Regions

3. Connections to Effective Field Theory

4. Applications

5. Summary

Victor Frankenstein’s Idea of ScienceVictor 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 decayBeauty charmless decay


Many scales

Many couplings

Many hadrons

Difficulties： Strong

interactions

Way-out： Factorization

FactorizationFactorization


Questions to be answeredQuestions 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 regionsIntegration 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 integralExample 1: Two-masses dependent integral


Cut-off regularizationCut-off regularization


UV div.IR div.

Dimensional regularizationDimensional 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 ExpansionExample 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

4qp

qmy

2,, 21

2122

22

1

pppppqmpp

qkqk ~,~0

1

1 2 2 2 2

[ ] 4 1 ( )

( )( ) 2 1 2Eh dk

I ek k q k k q k q

ykqyk ~,/~0

2

)2/1(

))((

1)1(22212/

1

1

yq

ye

pkyk

kde

qI EE

d

dp

ykyk ~,~0

qykqyk /~,/~0

Adding upAdding up


11 2 1

1

2

1ˆ( ) (1/ 2,1 ,3 / 2; 1/ (4 ))

2

ˆ4 (4 ) ( 1/ 2)

8 2(1 2 )ˆ

E

E

I e y F y

ye

q y

/1 1 1 1

h p s usI I I I

Remarks on method of regionsRemarks on method of regions


Effective Field TheoryEffective Field Theory


Application 1: Effective weak HamiltonianApplication 1: Effective weak Hamiltonian


Effective operatorsEffective operators


First step factorization in B decaysFirst step factorization in B decays


Example of matching : Tree-levelExample of matching : Tree-level


One-loop level matching equationOne-loop level matching equation


(1)fulliM

...

1Q

1Q

...

One-loop matching equationOne-loop matching equation


1tr loopfulliM iM iM 1 1tr loop loop

hard IRiM iM iM

Hard partHard part


Putting togetherPutting together


RenormalizationRenormalization


Application 2: Heavy-to-light Form-factorsApplication 2: Heavy-to-light Form-factors


Factorization formulaFactorization 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 SCETFactorization formula in SCET


Matching procedureMatching procedure


method of regions and its applications 2011.4.21 the interdisciplinary center for theoretical study,...

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