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

2011.4.21 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 ScienceVictor Frankenstein’s Idea of Science

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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

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Many scales

Many couplings

Many hadrons

Difficulties： Strong

interactions

Way-out： Factorization

FactorizationFactorization

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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?

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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.

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Example 1: Two-masses dependent integralExample 1: Two-masses dependent integral

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Cut-off regularizationCut-off regularization

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UV div.IR div.

Dimensional regularizationDimensional regularization

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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

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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

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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

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Effective Field TheoryEffective Field Theory

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Application 1: Effective weak HamiltonianApplication 1: Effective weak Hamiltonian

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Effective operatorsEffective operators

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First step factorization in B decaysFirst step factorization in B decays

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

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One-loop level matching equationOne-loop level matching equation

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(1)fulliM

...

1Q

1Q

...

One-loop matching equationOne-loop matching equation

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1tr loopfulliM iM iM 1 1tr loop loop

hard IRiM iM iM

Hard partHard part

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Putting togetherPutting together

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RenormalizationRenormalization

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

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Factorization formulaFactorization formula

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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

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Matching procedureMatching procedure

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More on matchingMore on matching

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“Hard” contribution“Hard” contribution

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Wilson coefficientsWilson coefficients

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Wilson coefficients Wilson coefficients

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RGEsRGEs

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Jet functionsJet functions

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Application 3: B two-body charmless decayApplication 3: B two-body charmless decay

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Matching onto SCETIIMatching onto SCETII

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Factorization formulaFactorization formula

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Hard-spectator interactionHard-spectator interaction

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NNLO vertex correctionsNNLO vertex corrections

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Complete NNLO: G.Bell, 2009; Beneke,Li,Huber 2009

Application 4: Exclusive single quarkonium productionApplication 4: Exclusive single quarkonium production

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NRQCD factorizationNRQCD factorizationFor 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.

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RefactorizationRefactorization

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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:Example:

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Qe e

NRQCD factorization up to leading power of velocity:

The short-distance contribution is parameterized as

The equivalent computation is to calculate the on-shell heavyquark anti-quark pair with equal momentum and the samequantum number as the quarkonium. At the tree level,

One-loop levelOne-loop level

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 Sang, Chen, arXiv:0910.4071; Li, He, Chao arXiv:0910.4155

Leading regionsLeading regions

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Hard Region:

Collinear region:

Anti-collinear region:

Potential region:

Soft region:

Ultra-soft region:

2( , ) ~ (1, , ),n k k n k s ~ / ,Qm s 2 2~ Qk m

2( , ) ~ ( , ,1),n k k n k s

~ ,k s 2 ~k s

2 2~ Qk m

NRQCD regionsNon-perturbative

Form factorForm factor

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NRQCD:

Collinear factorization:

Hard-kernel:

at tree level

Light-cone distribution amplitude

Ma and Si, PRD 2006; Bell and Feldmann, JHEP 2007;

RGE for LCDARGE for LCDA

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Brodsky-Lepage kernel:

Resum the leading logarithms

where

NLO results (preliminary)NLO results (preliminary)

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Braaten, PRD 1981;

Ma and Si, PRD 2006; Bell and Feldmann, JHEP 2007;

Hard Part

Collinear Part

Total Results

2(1) 2 ln

( ) (3 2ln ) ln ln 9 ( 1 )4 1S FC x x

T x x x x xx s i x

Sang, Chen, arXiv:0910.4071; Li, He, Chao arXiv:0910.4155

2(1) (0) (1) 2 2(1/ 2) ((9 6ln 2) ln 9ln 2 3ln 2 27 )

mT T

s

SummarySummary

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!)

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谢谢！谢谢！2011.4.21 The Interdisciplinary Center for Theoretical Study, USTC49