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  • 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

    1 0

    3 η

    E W

    2 

    V c

    b 

    2 ℱ

    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

    1 0

    3 η

    E W

    2 

    V c

    b 

    2 ℱ

    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

    1 0

    3 η

    E W

    2 

    V c

    b 

    2 ℱ

    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

    1 0

    3 η

    E W

    2 

    V c

    b 

    2 ℱ

    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

    2 ℓ

    ν̄

    2 Im BB

    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

    2 ℓ

    ν̄

    2 Im BB

    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

    2 ℓ

    ν̄

    2 Im BB

    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

    2 ℓ

    ν̄

    2 Im BB

    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: 2mBµ 2 π = −〈B| b̄v (iD)2bv |B〉

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

    • 1/m3b

    Darwin term: 2mBρ 3 D = 〈B| b̄v (iDµ)(ivD)(iDµ)bv |B〉

    Spin-orbit: 2mBρ 3 LS = 〈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 + dΓµπ µ2π m2b

    + dΓµG µ2G m2b

    + dΓρD ρ3D m3b

    + dΓρLS ρ3LS 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` E n `

    dΓ dE`∫

    E`>Ecut dE`

    dΓ dE`

    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

    E n

    tr ie

    s p

    e r

    0 .1

    G e V

    /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

    〈 (M2X )

    n 〉

    cut =

    ∫ E`>Ecut

    dM2X (M 2 X )

    n dΓ dM2X∫

    E`>Ecut dM2X

    dΓ dM2X

    Experiment n Ecut [GeV]

    BABAR 3 0.8, . . . , 1.9

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

    ] 2

    [GeV/c X

    m 0 1 2 3 4

    2 e n

    tr ie

    s /

    8 0

    M e V

    /c

    0

    400

    800

    1200

    1600

    2000

    ] 2

    [GeV/c X

    m 0 1 2 3 4

    2 e n

    tr ie

    s /

    8 0

    M e V

    /c

    0

    400

    800

    1200

    1600

    2000

    ] 2

    [GeV/cXm 0 1 2 3 4

    2 e n

    tr ie

    s /

    8 0

    M e V

    /c 0

    50

    100

    150

    200

    250

    300

    350

    ] 2

    [GeV/cXm 0 1 2 3 4

    2 e n

    tr ie

    s /

    8 0

    M e V

    /c 0

    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` dΓ dE`∫

    0 dE`

    dΓ dE`

    ∆Br(Ecut) =

    ∫ E`>Ecut

    dE` dΓ dE`

    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

    G e

    V )

    -3 (

    1 0

    1 M

    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

    G e

    V -3

    ( 1

    0 2

    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

    e V

    -3 (

    1 0

    3 M

    -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

    1 0

    × 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|2 τB

    [ Γ0 + Γµπ

    µ2π m2b

    + ΓµG µ2G m2b

    + ΓρD ρ3D 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 α 2 s α

    3 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 07