from Inclusive b c ‘ Decays: An Alternative...

Vcb from Inclusive b → c`ν Decays:

An Alternative Method

Matteo Fael

25 Mar 2019 – Moriond QCD 2019

in collaboration with T. Mannel and K. Vos

JHEP 02 (2019) 177

Problem: How to measure Vcb?

M. Fael Moriond 19 Mar. 25, 2019 1

From Inclusive Decays

• B̄ → Xc`ν̄

with Xc = D,D∗,Dπ,DKK , . . .

|Vcb| = (42.2± 0.8)× 10−3

Gambino et al, PRL 114 (2015) 061802

From Exclusive Decays

• B̄ → D `ν̄

• B̄ → D∗`ν̄

|Vcb| = (39.2± 0.7)× 10−3(LQCD, CLN)

HFLAV ’17, EPJ C 77, 895

1.0 1.1 1.2 1.3 1.4 1.5

0.4

0.6

0.8

1.0

1.2

1.4

1.6

w

10

3η

EW

2

Vc

b

2ℱ

2

CLN + LCSR

BGL + LCSR

1.0 1.1 1.2 1.3 1.4 1.5

0.4

0.6

0.8

1.0

1.2

1.4

1.6

w

10

3η

EW

2

Vc

b

2ℱ

2

CLN + LCSR

BGL + LCSR

Bigi, Gambino, Schacht, PLB 769 441 (2017).

New analyses:

B̄ → D∗`ν̄` + LQCD + BGL

|Vcb| = (41.9+2.0−1.9)× 10−3

Belle, arXiv:1702.01521 [hep-ex];

Grinstein, Kobach, PLB 771 359 (2017);

Bernlochner, Ligeti, Robinson, hep-ph/1902.09553.


Inclusive Decays

• Optical Theorem

B

f

2ℓ

ν̄

2 ImBB

ℓ

ν̄

f

∑f

|〈f |Heff(0) |B〉|2 = 2 Im

∫d4x e−iq·x 〈B|T{H†eff(x),Heff(0)} |B〉


Inclusive Decays

• Optical Theorem

• Operator Product Expansion (OPE)

B

f

2ℓ

ν̄

2 ImBB

ℓ

ν̄

f

=∑i

Ci(µ, αs) 〈B | Oi |B〉µ

µ : matching scale.

Ci(µ, αs) : short distance (perturbative) effects.

〈B| Oi |B〉µ : large distance (non-perturbative) effects.


Inclusive Decays

• Optical Theorem


• Heavy Quark Expansion (HQE)

b

light quark cloud

v

• B meson:

pB = mBv with v 2 = 1

• b quark:

pb = mbv + k with k � mb


Inclusive Decays

• Optical Theorem



B

f

2ℓ

ν̄

2 ImBB

ℓ

ν̄

f

=∑i ,j

Cij(µ, αs)

mib

〈B | Od=3+ij |B〉µ

How many 〈B| Od=3+ij |B〉 are there?


HQE Parameters

• 1/m2b

Kinetic energy: 2mBµ2π = −〈B| b̄v (iD)2bv |B〉

Chromomagnetic moment: 2mBµ2G = 〈B| b̄v (iDµ)(iDν)(−iσµν)bv |B〉

• 1/m3b

Darwin term: 2mBρ3D = 〈B| b̄v (iDµ)(ivD)(iDµ)bv |B〉

Spin-orbit: 2mBρ3LS = 〈B| b̄v (iDµ)(ivD)(iDν)(−iσµν)bv |B〉

• 1/m4b: 9 parameters (tree level);

• 1/m5b: 18 parameters (tree level).

Dassinger, Mannel, Turczyk, JHEP 0703 (2007) 087;

Mannel, Turczyk, Uraltsev, JHEP 1011 (2010) 109;

Kobach, Pal, hep-ph/1810.02356.


Inclusive Decays

• Optical Theorem



Observables can be written as:

dΓ = dΓ0 + dΓµπµ2π

m2b

+ dΓµGµ2G

m2b

+ dΓρDρ3D

m3b

+ dΓρLSρ3LS

m3b

+ . . .

Reviews:

Benson, Bigi, Mannel, Uraltsev, Nucl.Phys. B665 (2003) 367;

Dingfelder, Mannel, Rev.Mod.Phys. 88 (2016) 035008.


Moments of the spectrum

Charged lepton energy

〈E n〉cut =

∫E`>Ecut

dE` En`

dΓdE`∫

E`>EcutdE`

dΓdE`

Experiment n Ecut [GeV]

BABAR 3 0.6, . . . , 1.5

Belle 4 0.4, . . . , 2.0 (GeV/c)e

*BE

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

En

trie

s p

er

0.1

GeV

/c

0

200

400

600

800

1000

Belle

Fig: Belle, PRD 75 (2007) 032001;

BABAR, PRD 69 (2004) 111104; BABAR, PRD 81 (2010) 032003.



Hadronic invariant mass

⟨(M2

X )n⟩

cut=

∫E`>Ecut

dM2X (M2

X )n dΓdM2

X∫E`>Ecut

dM2X

dΓdM2

X

Experiment n Ecut [GeV]

BABAR 3 0.8, . . . , 1.9

Belle 1,2,4 0.7, . . . , 1.9

]2

[GeV/cX

m0 1 2 3 4

2en

trie

s /

80

MeV

/c

0

400

800

1200

1600

2000

]2

[GeV/cX

m0 1 2 3 4

2en

trie

s /

80

MeV

/c

0

400

800

1200

1600

2000

]2

[GeV/cXm0 1 2 3 4

2en

trie

s /

80

MeV

/c0

50

100

150

200

250

300

350

]2

[GeV/cXm0 1 2 3 4

2en

trie

s /

80

MeV

/c0

50

100

150

200

250

300

350

Fig: BABAR, PRD 81 (2010) 032003

Belle, PRD 75 (2007) 032005.



Fractional branching ratio

R∗(Ecut) =

∫E`>Ecut

dE`dΓdE`∫

0dE`

dΓdE`

∆Br(Ecut) =

∫E`>Ecut

dE`dΓdE`

ΓB

Experiment Ecut [GeV]

BABAR 0.6 . . . 1.5

Belle 0.4, . . . , 2.0

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Ge

V)

-3 (

10

1M

1400

1500

1600

1700

1800

1900

2000

2100

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

)2

Ge

V-3

(1

02

M

0

20

40

60

80

100

120

140

160

180

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4)

3 G

eV

-3 (

10

3M

-25

-20

-15

-10

-5

0

5

Belle

cut (GeV)*BeE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

-2

10

× B

R

∆

0

2

4

6

8

10 Belle

Fig: Belle, PRD 75 (2007) 032005

BABAR, PRD 81 (2010) 032003


R∗(Ecut) 〈E n〉cut 〈(M2X )n〉cut

µπ, µG , ρD , ρLS ,mb, (mc)

Br(B̄ → Xc`ν̄) ∝ |Vcb|2τB

[Γ0 + Γµπ

µ2π

m2b

+ ΓµG

µ2G

m2b

+ ΓρDρ3D

m3b

]

Vcb = (42.21± 0.78)× 10−3

see: Gambino, Schwanda, PRD 89 (2014) 014022;

Alberti, Gambino et al, PRL 114 (2015) 061802


Vcb at 1%?

tree αs α2s α3

s

1 3 3 3 !Jezabek, Kuhn, NPB 314 (1989) 1; Gambino et al., NPB 719 (2005) 77;

Melnikov, PLB 666 (2008) 336; Pak, Czarnecki, PRD 78 (2008) 114015.

µπ 3 3 ! Becher, Boos, Lunghi, JHEP 0712 (2007) 062.

µG 3 3 !Alberti, Gambino, Nandi, JHEP 1401 (2014) 147;

Mannel, Pivovarov, Rosenthal, PRD 92 (2015) 054025.

ρD 3 Mannel, Pivovarov, SI-HEP-2018-36.

ρLS 3 !

1/m4b 3 Dassinger, Mannel, Turczyk, JHEP 0703 (2007) 087

1/m5b 3 Mannel, Turczyk, Uraltsev, JHEP 1011 (2010) 109

mkinb 3 3 3 !

Bigi, Shifman, Uraltsev, Vainshtein, PRD 56 (1997) 4017;

Czarnecki, Melnikov, Uraltsev, PRL 80 (1998) 3189.


• Problem: number of HQE parameters at higher orders

• 4 up to 1/m3b

• 13 up to 1/m4b

• 31 up to 1/m5b

• Lowest State Saturation Approximation (LSSA)

〈B| b̄v (iD)2(iD)2bv |B〉 ∼ 〈B| b̄v (iD)2 |B〉〈B| (iD)2bv |B〉

Mannel, Turczyk, Uraltsev, JHEP 1011 (2010) 109; Heinonen, Mannel, NPB 889 (2014) 46.

• Higher order fit: LSSA estimates as priors

(60% Gaussian uncertainty).

• Fit is unchanged, slightly smaller theoretical errors.

|Vcb| shifts by −0.25%.

Healey, Turczyk, Gambino, PLB 763 (2016) 60.


Can we reduce the number

of HQE parameters?


Reparametrization Invariance

• b quark momentum


• this decomposition is NOT unique:{v → v + δv/mb

k → k − δv• The RP transformations:

• δRPvµ = δvµ with v · δv = 0

• δRPiDµ = −mbδvµ

• δRPbv (x) = imb(x · δv)bv (x), in particular δRPbv (0) = 0

Luke, Manohar, PLB 286, 348 (1992);

Manohar, PRD 82, 014009 (2010);

Heinonen, Hill, Solon, PRD 86 094020 (2012).


B

f

2ℓ

ν̄

2 ImBB

ℓ

ν̄

f

=∑n

C(n)µ1...µn

(v) b̄v(iDµ1 . . . iDµn)bv

• δRPΓtot = 0

• The RPI relation:

δRPC(n)µ1···µn(S) = mb δv

α

[C

(n+1)αµ1···µn(S)+C

(n+1)µ1αµ2···µn(S)+· · ·+C

(n+1)µ1···µnα(S)

]

Mannel, Vos, JHEP 1806 (2018) 115


Reduced Set

• 1

• 2mBµ3 = 〈b̄vbv 〉 = 2mB

(1− µ2

π − µ2G

2mb

)• 1/m2

b

• 2mBµ2G = 〈b̄v (iDα)(iDβ)(−iσαβ)bv 〉

• 1/m3b

• 2mB ρ̃3D =

1

2〈b̄v

[(iDµ) ,

[(ivD +

1

2m(iD)2

), (iDµ)

]]bv 〉

• 1/m4b

• 2mB r4G = 〈b̄v [(iDµ) , (iDν)] [(iDµ) , (iDν)] bv 〉

• 2mB r4E = 〈b̄v [(ivD) , (iDµ)] [(ivD) , (iDµ)] bv 〉

• 2mBs4B = 〈b̄v [(iDµ) , (iDα)] [(iDµ) , (iDβ)] (−iσαβ)bv 〉

• 2mBs4E = 〈b̄v [(ivD) , (iDα)] [(ivD) , (iDβ)] (−iσαβ)bv 〉

• 2mBs4qB = 〈b̄v [iDµ , [iDµ , [iDα , iDβ ]]] (−iσαβ)bv 〉

• ρLS and four HQE parameters of O(1/m4b) do not appear.

Mannel, Vos, JHEP 1806 (2018) 115


Which Observables are RPI?

dΓ =

∫w(v , pe , pν)〈ImT (S)〉L(pe , pν)dΦ3

B

f

2ℓ

ν̄

• The observable dΓ is RPI if δRPw(v , pe , pν) = 0

dΓ w(v , pe , pν) RPI

Total Rate 1 3

Moments charged lepton energy (v · pe)n 7

Moments hadronic invariant mass (MBv − q)2n 7

Moments leptonic invariant mass (q2)n 3

MF, Mannel, Vos, JHEP 02 (2019) 177


RPI Observables

• Ratio between the rate with and without a cut

R∗(q2cut) =

∫q2>q2

cut

dq2 dΓ

dq2

/∫0

dq2 dΓ

dq2

• q2 moments

⟨(q2)n

⟩cut

=

∫q2>q2

cut

dq2 (q2)ndΓ

dq2

/∫q2>q2

cut

dq2 dΓ

dq2


R∗(q2cut) 〈(q2)n〉cut

µ3, µG , ρ̃D ,rE , rG , sE , sB , sqB ,mb,mc

Br(B̄ → Xc`ν̄) ∝ |Vcb|2τB

[Γµ3µ3 + ΓµG

µ2G

m2b

+ Γρ̃Dρ̃3D

m3b

+ΓrE

r4E

m4b

+ ΓrG

r4G

m4b

+ ΓsB

s4B

m4b

+ ΓsE

s4E

m4b

+ ΓsqB

s4qB

m4b

]

Vcb = ?

MF, Mannel, Vos, hep-ph/1812.07472


Experimental Perspectives

• Need to reconstruct neutrino momentum

• BABAR

• Belle, Belle-II

Dingfelder, Mannel, Rev. Mod. Phy. 88 035008

• BABAR: 433 fb−1 of data at Υ(4S)

Note: 〈E ne 〉 & 〈Mn

X 〉 obtained with 210 fb−1

BABAR, PRD 69 (2004) 111104; PRD 81 (2010) 032003.

• Belle: 711 fb−1 of data at Υ(4S)

Note: 〈E ne 〉 & 〈Mn

X 〉 obtained with 140 fb−1

Belle, PRD 75 (2007) 032005; PRD 75 (2007) 032001

• Belle-II = Belle × 50


Conclusions

• Current fits consider HQE up to 1m3

b(6 non-pert. param.)

or 1/m4b + 1/m5

b+LSSA priors (9+18 extra param.).

• Reparametrization invariance links HQE at different orders.

• Total rate and q2 moments are RPI:

8 param. instead of 13 up to 1/m4b .

• New method: Vcb from Γtot, ∆Br(q2cut) and 〈(q2)n〉cut up to 1/m4

b,

completely data driven.

• Neutrino momentum reconstruction well established at BABAR and

Belle, based on Btag algorithm.

• Belle and BABAR data are still there, Belle-II just started . . .


Backup


1.0 1.1 1.2 1.3 1.4 1.5

0.4

0.6

0.8

1.0

1.2

1.4

1.6

w

10

3η

EW

2

Vc

b

2ℱ

2

CLN + LCSR

BGL + LCSR

Bigi, Gambino, Schacht, PLB 769 (2017) 441

Belle, arXiv:1702.01521 [hep-ex]


q2 > 3.6 GeV2

0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

2.0

2.5

q2 > 8.4 GeV2

0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

2.0

2.5

MF, Mannel, Vos, JHEP 02 (2019) 177


OPE for Semileptonic Decays

• The Effective Hamiltonian

Heff =4GFVcb√

2jµq Lµ

• The decay rate: Γ(B → Xc`ν̄) ∝ 2 ImTµνLµν

T µν = i

∫dx4e−iqx 〈B |T

{b̄(x)γµPLc(x), c̄(0)γνPLb(0)

}|B〉


Heavy Quark Expansion

• B meson:

pB = mBv with v 2 = 1

• b quark:


“Rephase” the field b(x) :

b(x) = exp(−imbv · x)bv (x)

iDµb(x) = e−imbv ·x (mbvµ + iDµ) bv (x)

b

light quark cloud

v


T µν = i

∫dx4e i(mbv−q)·xT

{b̄v (x)γµPLc(x), c̄(0)γνPLbv (0)

}

ν µ

bv bv

c

ր pb

տ q

→Q+ k

with Q = mbv − q


• Take matrix elements with quarks and gluons

ν µ

bv bv

c

ր pb

տ q

→Q+ k

= b̄vγµPL

[i

/Q + /k −mc

]γνPLbv

• Now expand . . .

S =i

/Q + /k −mc

=i

/Q −mc+

i

/Q −mc(−/k)

i

/Q −mc+

i

/Q −mc(−/k)

i

/Q −mc(−/k)

i

/Q −mc+ . . .


Technical Ingredients

We need to calculate:

b̄vΓ†[

i

/Q −mc+

i

/Q −mc(−i /D)

i

/Q −mc+

i

/Q −mc(−i /D)

i

/Q −mc(−i /D)

i

/Q −mc+ . . .

]Γbv

• Reduce all possible matrix elements to scalar operators

(also redundant ones):

b̄v (iDµ1 )(iDµ2 ) . . . (iDµn)Γbv

with Γ = 1, γ5, γα, γαγ5,−iσµν

• Use equation of motion:

• /vbv = bv − i /Dmb

bv

• (iv · D)bv = − 12mb

(i /D)(i /D)bv

• Example:

〈b̄v (iDµ)(iDν)bv 〉 = 2mB

[1

3µπ(vµvν − gµν) +

1

3σG1(4vµvν − gµν)

]M. Fael Moriond 19 Mar. 25, 2019 29

Phase space integration

• Im

(1

(p2 −m2 + iε)n

)= −π (−1)n

n!δ(n)(p2 −m2)

• Integrate:∫dq2dv ·qdEeθ(q2)θ(4E 2

e +4Eev ·q−q2)f (q2, v ·q,E`)δ(n)(q2+m2b−2v ·q−m2

c)

• q2-specturm:

dΓ

dq̂2= freg(q̂2) + δ(z(q̂2))f1(q̂2) + δ′(z(q̂2))f2(q̂2) + . . .

with z(q̂2) = 1− 2√

q̂2 + q̂2 − ρ


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