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.

M. Fael Moriond 19 Mar. 25, 2019 2

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.

M. Fael Moriond 19 Mar. 25, 2019 2

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〉

M. Fael Moriond 19 Mar. 25, 2019 3

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.

M. Fael Moriond 19 Mar. 25, 2019 4

Inclusive Decays

• Optical Theorem

• Operator Product Expansion (OPE)

• 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

M. Fael Moriond 19 Mar. 25, 2019 5

Inclusive Decays

• Optical Theorem

• Operator Product Expansion (OPE)

• Heavy Quark Expansion (HQE)

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?

M. Fael Moriond 19 Mar. 25, 2019 6

Inclusive Decays

• Optical Theorem

• Operator Product Expansion (OPE)

• Heavy Quark Expansion (HQE)

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?

M. Fael Moriond 19 Mar. 25, 2019 6

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.

M. Fael Moriond 19 Mar. 25, 2019 7

Inclusive Decays

• Optical Theorem

• Operator Product Expansion (OPE)

• Heavy Quark Expansion (HQE)

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.

M. Fael Moriond 19 Mar. 25, 2019 8

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.

M. Fael Moriond 19 Mar. 25, 2019 9

Moments of the spectrum

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.

M. Fael Moriond 19 Mar. 25, 2019 10

Moments of the spectrum

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

M. Fael Moriond 19 Mar. 25, 2019 11

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

M. Fael Moriond 19 Mar. 25, 2019 12

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.

M. Fael Moriond 19 Mar. 25, 2019 13

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.

M. Fael Moriond 19 Mar. 25, 2019 13

• 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.

M. Fael Moriond 19 Mar. 25, 2019 14

Can we reduce the number

of HQE parameters?

M. Fael Moriond 19 Mar. 25, 2019 14

Reparametrization Invariance

• b quark momentum

pb = mbv + k with k � mb

• 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).

M. Fael Moriond 19 Mar. 25, 2019 15

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

M. Fael Moriond 19 Mar. 25, 2019 16

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

M. Fael Moriond 19 Mar. 25, 2019 17

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

M. Fael Moriond 19 Mar. 25, 2019 18

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

M. Fael Moriond 19 Mar. 25, 2019 19

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

M. Fael Moriond 19 Mar. 25, 2019 20

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

M. Fael Moriond 19 Mar. 25, 2019 21

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 . . .

M. Fael Moriond 19 Mar. 25, 2019 22

Backup

M. Fael Moriond 19 Mar. 25, 2019 22

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]

M. Fael Moriond 19 Mar. 25, 2019 23

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

M. Fael Moriond 19 Mar. 25, 2019 24

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〉

M. Fael Moriond 19 Mar. 25, 2019 25

Heavy Quark Expansion

• B meson:

pB = mBv with v 2 = 1

• b quark:

pb = mbv + k with k � mb

“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

M. Fael Moriond 19 Mar. 25, 2019 26

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

M. Fael Moriond 19 Mar. 25, 2019 27

• 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+ . . .

M. Fael Moriond 19 Mar. 25, 2019 28

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

M. Fael Moriond 19 Mar. 25, 2019 30