for Heavy Quarks - the School on Flavour Physics 2010

<strong>Heavy</strong> Quark Expansion 

Soft Collinear Effective Theory 

Effective Field Theories 

<strong>for</strong> <strong>Heavy</strong> <strong>Quarks</strong> 

Part III: <strong>Heavy</strong> Quark Expansion and 


Thomas Mannel 

Siegen University 

FlaviaNet <strong>School</strong> 2010, Uni. Bern 

Thomas Mannel, Uni. Siegen 

Part III: HQE and SCET



What is still missing? 

Discussion of inclusive decays: 

B → X c l¯ν l , Lifetimes Mixing ... 


Decays heavy hadrons into light quarks: 

Very Energetic light degrees of freedom: E ∼ m b 






Literature <strong>for</strong> Part III 

A Manohar, M. Wise, 

<strong>Heavy</strong> Quark Physics, Cambridge UP 

T. Mannel, Springer Tracts 2005 

SCET Paper series by Bauer, Pirjol, Stewart 

SCET Paper series by Beneke, Feldmann 





Contents Part III 

1 <strong>Heavy</strong> Quark Expansion 

Set up 

A short look at lifetimes 

Spectra of inclusive decays 

2 Soft Collinear Effective Theory 

The Endpoint Region 

A look at SCET 

The SCET Lagrangian 





Set up 



<strong>Heavy</strong> Quark Expansion <strong>for</strong> Inclusive Decays: Set up 

<strong>Heavy</strong> Quark Expansion = Operator Product Expansion 

(Chay, Georgi, Bigi, Shifman, Uraltsev, Vainstain, Manohar. Wise, Neubert, M,...) 

Γ ∝ ∑ (2π) 4 δ 4 (P B − P X )|〈X|H eff |B(v)〉| 2 

X 

∫ 

= d 4 x 〈B(v)|H eff (x)H † eff (0)|B(v)〉 

∫ 

= 2 Im d 4 x 〈B(v)|T {H eff (x)H † eff (0)}|B(v)〉 

∫ 

= 2 Im d 4 x e −imbv·x 〈B(v)|T { ˜H eff (x) ˜H † eff (0)}|B(v)〉 

Last step: p b = m b v + k, 

Expansion in <strong>the</strong> residual momentum k 





Set up 



Per<strong>for</strong>m an OPE: m b is much larger than any scale 

appearing in <strong>the</strong> matrix element 

∫ 

d 4 xe −im bvx T { ˜H eff (x) ˜H † eff (0)} 

= 

∞∑ 

n=0 

( 1 

2m Q 

) n 

C n+3 (µ)O n+3 

→ The rate <strong>for</strong> B → X c l¯ν l can be written as 

Γ = Γ 0 + 1 Γ 1 + 1 Γ 

m Q mQ 

2 2 + 1 Γ 

mQ 

3 3 + · · · 

The Γ i are power series in α s (m Q ): 

→ Perturbation <strong>the</strong>ory! 





Set up 



Γ 0 is <strong>the</strong> decay of a free quark (“Parton Model”) 

Γ 1 vanishes due to <strong>Heavy</strong> Quark Symmetries 

Γ 2 is expressed in terms of two parameters 

2M H µ 2 π = −〈H(v)| ¯Q v (iD) 2 Q v |H(v)〉 

2M H µ 2 G = 〈H(v)| ¯Q v σ µν (iD µ )(iD ν )Q v |H(v)〉 

µ π : Kinetic energy and µ G : Chromomagnetic moment 

Γ 3 two more parameters 

2M H ρ 3 D = −〈H(v)| ¯Q v (iD µ )(ivD)(iD µ )Q v |H(v)〉 

2M H ρ 3 LS = 〈H(v)| ¯Q v σ µν (iD µ )(ivD)(iD ν )Q v |H(v)〉 

ρ D : Darwin Term and ρ LS : Chromomagnetic moment 





Set up 



Γ 4 : Nine more Parameters At O(1/m 4 ) 

Dimension 7 matrix elements = four derivatives 

In terms of physical quantities: 

Spin-independent 

2M B m 1 = 〈 ( (⃗p) 2) 2 

〉 

2M B m 2 = g 2 〈 E ⃗ 2 〉 

2M B m 3 = g 2 〈 B ⃗ 2 〉 

2M B m 4 = g〈⃗p · rot B〉 ⃗ 





Set up 



Spin-dependent 

2M B m 5 = g 2 〈 S ⃗ · ( E ⃗ × E)〉 ⃗ 

2M B m 6 = g 2 〈 S ⃗ · ( B ⃗ × B)〉 ⃗ 

2M B m 7 = g〈( S ⃗ · ⃗p)(⃗p · ⃗B)〉 

2M B m 8 = g〈( S ⃗ · ⃗B)(⃗p) 2 〉 

2M B m 9 = g〈∆(⃗σ · ⃗B)〉 

Beyond this <strong>the</strong>re is a real proliferation of parameters! 

This becomes useless unless one has a way to 

compute <strong>the</strong> matrix elemets 





Set up 



Remarks: 

All <strong>the</strong>se parameters are of <strong>the</strong> order (Λ QCD /m b ) n with 

appropriate n. 

For Λ QCD ∼ 500MeV and m b = 4.6 GeV: 

Λ QCD /m b = 0.11 

The leading nonperturbative Corrections are of order 

(Λ QCD /m b ) 2 = 1% 

Nonperturbative corrections are expected to be small! 

This is an embarrasment: 

Significant lifetime difference between D 0 and D + 

Even larger lifetime differences between D 0 and Λ c 

... 






Set up 



Calculate <strong>the</strong> Coefficients ... 

Up to Isospin breaking µ π (B + ) = µ π (B 0 ) 

→ Meson lifetime differences are order (Λ QCD /m b ) 3 

Expectation: 

τ(M − ) 

τ(M 0 ) 

τ(M s ) 

τ(M 0 ) 

τ(Λ Q ) 

τ(M 0 ) 

= 1 + O(1/m 3 Q ) , 

= 1 + O(1/m 3 Q ) , 

= 1 + O(1/m 2 Q) 

This does not look good <strong>for</strong> charm, however ... 





Set up 



Look at <strong>the</strong> diagrams: 

These diagrams have one less loop compared to <strong>the</strong> 

leading terms 

... yields a relative factor 16π 2 ∼ 158 





Set up 



Hence 

τ(B − ) 

τ(B d ) = 1+ 1 m 3 b 

[ 

a 0 + 1 [ 

a 1 + · · · 

]+ 16π2 b 

m b mb 

3 0 + 1 ] 

b 1 + · · · 

m b 

, 

Relative enhancement of <strong>the</strong> 1/m 3 Q terms 

Satisfactory explanation of <strong>the</strong> lifetme patterns 





Spectra of Inclusive Decays 

Set up 



Calculation proceeds in <strong>the</strong> same way 

Denominator of <strong>the</strong> charm propagator: 

(m b v + k − q) 2 − m 2 c = m 2 b − m 2 c + 2m b (vq) + q 2 + · · · 

Sbould be large against Λ QCD 

... this cannot be in all phase space 





Lepton Energy Epectrum 

Set up 



2 

1.5 

1_ dΓ __ 

Γ dy 

b 

1 

0.5 

0.2 0.4 0.6 0.8 1 

y 

Endpoint region: ρ = mc/m 2 b 2, y = 2E l/m b 

[ 

( ) 2 { 

dΓ 

dy ∼ Θ(1−y−ρ) λ 1 ρ 

2 + 

3 − 4 ρ } ] 

(m b (1 − y)) 2 1 − y 1 − y 

Break-down of <strong>the</strong> HQE in <strong>the</strong> endpoint region 

... but reliable calculation <strong>for</strong> moments of specta 





Moments of Spectra 

Set up 



Charged lepton energy spectrum 

Hadronic invariant mass spectrum 

〈MX〉 n = 1 ∫ ∫ 

dM X MX 

n d 2 Γ 

dE l 

Γ 

E cut 

dM x dE l 

〈El n 〉 = 1 ∫ 

dM X dE l E 

Γ 

∫E n d 2 Γ 

l 

cut 

dM x dE l 

Moment are sensitive to higher orders of HQE: 

[( ) n ] 1 

〈MX〉 n ∼ O 

m b 

May be used to extract <strong>the</strong> HQE parameters! 





Set up 



Hadronic Invariant Mass Moments (Buchmüller, Flächer) 





Set up 



Lepton Energy Moments I (Buchmüller, Flächer) 





Set up 



Lepton Energy Moments II (Buchmüller, Flächer) 





Set up 



V cb,incl = (41.54 ± 0.44 ± 0.59 HQE ) × 10 −3 

(PDG 2010) 





Set up 



<strong>Heavy</strong> to light inclusive: B → X u l¯ν 

Formulae also apply <strong>for</strong> b → u, taking <strong>the</strong> limit 

m c → 0 and V cb → V ub 

Useless <strong>for</strong> <strong>the</strong> extraction of V ub ! 

Due to <strong>the</strong> b → c background. 

Phase space cuts ruin <strong>the</strong> stadard OPE 

(at last <strong>for</strong> most of <strong>the</strong> proposed methods) 

Partial Resumation: Nonperturbative input will be a 

“shape functon” 

Simple way to explain this: BtoX s γ 








Photon Spectrum in B → X s γ 

The photon spectrum becomes (with x = 2E γ /m b ) 

dΓ 

dx = G2 F αm5 b 

32π |V tsV 4 tb ∗| 2 |C 7 | 2 

( 

δ(1 − x) − λ 1 + 3λ 2 

δ ′ (1 − x) + λ ) 

1 

δ ′′ (1 − x) + · · · 

2mb 

2 6mb 

2 

General Structure at tree level (no real gluons) 

[ 

dΓ ∑ 

( ) ] 

i 1 

dx = Γ 0 a i δ (i) (1 − x) + O((1/m b ) i+1 δ (i) (1 − x) 

m b 

i 








The same happens with <strong>the</strong> o<strong>the</strong>r spectra ! 

Interpretation: The expansion parameter is not 1/m, 

ra<strong>the</strong>r it is 1/[m(1 − x)]: Expansion breaks down in 

<strong>the</strong> Endpoint x ∼ 1. 

Comparison with experiment: Moments: 

M n = 

∫ 1 

M 0 = Γ M 1 = − λ 1 + 3λ 2 

0 

2m 2 b 

dx (1 − x) n dΓ 

dx 

M 2 = λ 1 

6m 2 b 

Power counting <strong>for</strong> <strong>the</strong> moments: M n = O(1/m n ) 

· · · 








Resummation and shape function 

Leading terms can be resummed into a shape 

function: 

dΓ 

dx = G2 F αm5 b 

32π 4 |V tsV tb ∗| 2 |C 7 | 2 f (1 − x) 

where 2M B f (ω) = 〈B| ¯Q v δ(ω + n · (iD))Q v |B〉 

n · (iD): light-cone component of residual momentum. 

Fourier trans<strong>for</strong>m of a (light-like) Wilson line: 

∫ 

( ∫ dt 

t 

) 

f (ω) = 

2π 〈B(v)| ¯Q v (0)P exp −i ds n · A(n · s) Q v (n·t)|B(v)〉 

0 








Shape Function: Kinematics 

Where in phase space is <strong>the</strong> shape function relevant? 

p B = M B v = q + p with q 2 = 0 

Light cone vectors n and ¯n with n · ¯n = 2 

One of which is q = 1 (n · q)¯n 

2 

The second is from v = 1(n + ¯n) 

2 

m b v − q = 1 2 [m b n + (m b − n · q)¯n] 

Shape function region: 

Final state invariant mass (p B − q) 2 ∼ O(Λ QCD m b ) 

Hadronic energy M B − v · q ∼ O(m b ) 



Examples 






B → X s γ 

q 2 γ = 0 and q = E γ ¯n 

E γ = M B 

2 

− O(Λ QCD ) 

→ Endpoint of <strong>the</strong> Photon Energy Spectrum (2E γ → M B ) 

B → X u l¯ν l 

m l = 0 and q = E l¯n 

E l = M B 

2 

− O(Λ QCD ) 

→ Endpoint of <strong>the</strong> Lepton Energy Spectrum (2E l → M B ) 

The same non-perturbative input ! 








Simple (tree level) derivation 

We start from: 

∫ 

d 4 xe −i(m bv−q)·x 〈B(v)|T {b v (x)Γq(x)¯q(0)Γb v (0)}|B(v)〉 

and contract <strong>the</strong> light quark 

〈B(v)|T {b v (x)Γq(x)¯q(0)Γb v (0)}|B(v)〉 

= 〈B(v)|h v (x)Γ 〈0|T {q(x)¯q(0)}|0〉 Γb v (0)|B(v)〉 

expand <strong>the</strong> light (massless) Propagator (in external field) 

〈0|T {q(x)¯q(0)}|0〉 F = 

.T . 

m b /v − /q + i /D 

(m b − q) 2 + 2(m b − q) · (iD) + (i /D) 2 + iɛ 








Numerator: m b /v − /q + i /D = m b 

2 /n + O(Λ QCD) 

Denominator: 

(m b − q) 2 + 2(m b − q) · (iD) + (i /D) 2 

= m b [m b − n · q + n · (iD) + iɛ] + O(Λ 2 QCD) 

Collect everything: 

∫ 

d 4 xe −i(mbv−q)·x 〈B(v)|T {b v (x)Γq(x)¯q(0)Γb v (0)}|B(v)〉 

= 〈B(v)|h v Γ 1 ( 

) 

2 /nΓ 1 

h v |B(v)〉 

m b − n · q + n · (iD) + iɛ 

+ · · · 

→ Leads to a convolution with <strong>the</strong> shape function 





Why and where SCET? 




Power Counting: (p fin : light final state) 

p fin = 1 2 (n · p fin)¯n and v = 1 2 (n + ¯n) 

Momentum of a light quark in such a meson: 

p light = 1 2 [(n · p light)¯n + (¯n · p light )n] + p ⊥ light 

The light quark invariant mass (or virtuality) is 

assumed to be 

p 2 light = (n · p light )(¯n · p light ) + (p ⊥ light) 2 ∼ λ 2 m 2 b 

Quark momentum has to scale as 

(n · p light ) ∼ m b (¯n · p light ) ∼ λ 2 m b p ⊥ light ∼ λm b 





The Lagrangian of SCET 




Analogous to HQET: 

Pick <strong>the</strong> O(λ) and O(λ 2 ) of <strong>the</strong> quark momentum: 

Q = 1 2 (n · p light)¯n + p ⊥ light 

Rephase <strong>the</strong> quark fields: ψ(x) = ∑ Q 

e −iQ·x ψ n,Q (x) 

Project out <strong>the</strong> “large” and “small” components using 

ξ n,Q (x) = Pψ n,Q (x) with P = 1 4 /n/¯n 

and 

η n,Q (x) = Qψ n,Q (x) with Q = 1 4 /¯n/n 








The Lagrangian becomes 

L = ∑ [ 

e −ix(p−p′ ) /n 

¯ξ n,p ′ 

2 (in · D)ξ /n 

n,p + ¯η n,p ′ 

2 [¯n · p + (i ¯n · D)]η n,p 

p,p ′ ] 

+ ¯ξ n,p ′ (/p ⊥ + i /D ⊥ ) η n,p + ¯η n,p ′ (/p ⊥ + i /D ⊥ ) ξ n,p 

Integrate out <strong>the</strong> “heavy”degree of freedom η 

→ use <strong>the</strong> equations of motion <strong>for</strong> η 

L = ∑ [ 

e −ix(p−p′ ) /n 

¯ξ n,p ′ 

2 (in · D)ξ n,p + 

p,p ′ 

¯ξ n,p ′ (/p ⊥ + i /D ⊥ ) 

1 

¯n · p + (i ¯n · D) η n,p (/p ⊥ + i /D ⊥ ) /n ] 

2 ξ n,p 

Similarly <strong>for</strong> Gluons ... 








Matching of <strong>Heavy</strong>-Light Currents 

¯qΓQ −→ ¯ξ n,p ′W(0)Γh v : W: Resummation of 

collinear gluons 

p 

p 

q m 

+ perms ! 

q 2 

q 

p b 

q 1 

q 2 

SCET contains non-localities, expressed in terms of 

Wilson-lines 

( ∫ 0 

) 

W(x) = P exp −i ds ¯n · A c (x + s¯n) 

−∞ 








Radiative corrections in heavy-to-light inclusive decays 

Matching proceeds in two steps: 

(a) Matching QCD → SCET 

T [ ¯Q(x)Γq(x)¯q(0)¯ΓQ(0) ] −→ 

T [¯hv (x)ΓW † (x)ξ n,p ′(x)¯ξ n,p ′W(0)¯Γh v (0) ] 

(b) Matching SCET → Nonlocal operators 

T [¯hv (x)ΓW † (x)ξ n,p ′(x)¯ξ n,p ′W(0)¯Γh v (0) ] −→ 

∫ 

dω C(ω)h v δ(ω + in · D)h v 








Leads to a convolution of <strong>the</strong> structure 

dΓ = H ∗ J ∗ f 

H: Hard Contibution 

J: Jet Function: 

collinear modes (contains <strong>the</strong> Sudakov Logs.) 

f : Shape Function: 

soft contributions, nonperturbative 








This has numerous applications: 

Semileptonic Decays: Precision determination of V ub 

Precise predictions <strong>for</strong> B → X s γ 

Applies also <strong>for</strong> exclusive (non-leptonic) decays: 

QCD Factorization, SCET ... 

ongoing debate with <strong>the</strong> PQCD group 

seems to have problems to explain some of <strong>the</strong> data 





Summary of Part III 




Inclusive decays can be described also in a HQE 

Make use of <strong>the</strong> optical <strong>the</strong>orem 

Standard OPE has a certain set of non-perturbative 

parameters: µ π , µ G , , ρ D , ρ LS ... 

In <strong>the</strong> standard approach: Light degrees of freedom 

have to be “slow”: (vp) 2 ∼ p 2 ∼ Λ 2 QCD 

O<strong>the</strong>r extreme can also be handled: 

vp ∼ m b but p 2 ∼ Λ 2 QCD 

energetic light degrees of freedom: SCET 





Summary of “Everything” 




After approx 20 Years of 1/m Q expansion 

... we are able to perfome QCD-based appraoches 

... many models have been superseeded 

... and <strong>the</strong> model dependence has been pushed back 

into subleading terms 

... we entered an era of precision flavour physics, 

... at least <strong>for</strong> many quantities 

Fruitful overlapp with o<strong>the</strong>r QCD based methods 

QCD Sum Rules 

Lattice QCD 








Of course, <strong>the</strong>re are problems ... 

Exclusive Non-Leptonic Decays 

Important <strong>for</strong> CP violation and searches <strong>for</strong> “New 

Physics” 

Tensions between exclusive and inclusive 

determinations of CKM parameters 

Proliferation of parameters in higher orders 1/m b 

Limits our ability to compute 

Higher orders in α s are a real technical challenge 

More data may clarify a few of <strong>the</strong>se things: 

LHCb and Superflavour ...

for Heavy Quarks - the School on Flavour Physics 2010

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