Zoltan Ligeti - web.mit. csuggs/www/scet_talks/Ligeti-InclusiveV_ub.pdf  Outline Introduction Complete

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Text of Zoltan Ligeti - web.mit. csuggs/www/scet_talks/Ligeti-InclusiveV_ub.pdf  Outline Introduction...

  • Inclusive |Vub| unlimitedZoltan Ligeti

    SCET Workshop, MIT, March 2426, 2009

  • Outline

    Introduction

    Complete description of B Xs [ZL, Stewart, Tackmann, arXiv:0807.1926]

    Complete description of B Xu` [ZL, Stewart, Tackmann, to appear]

    A glimpse at SIMBA [Bernlochner, Lacker, ZL, Stewart, Tackmann, Tackmann, to appear]

    Conclusions

    p.1

  • The importance of |Vub|

    |Vub| determined by tree-level decaysCrucial for comparing tree-dominatedand loop-mediated processes

    |Vub|`LQCD = (3.5 0.5) 103|Vub|inclBLNP = (4.32 0.35) 103

    |Vub| = (5.2 0.5 0.4fB) 103

    SM CKM fit, sin 21, favors small value

    Fluctuation, bad theory, new physics?

    The level of agreement between themeasurements often misinterpreted

    3

    3

    2

    2

    dmK

    K

    sm & dm

    SLubV

    ubV

    1sin 2

    (excl. at CL > 0.95)

    < 01

    sol. w/ cos 2

    excluded at CL > 0.95

    2

    1

    3

    1.0 0.5 0.0 0.5 1.0 1.5 2.0

    1.5

    1.0

    0.5

    0.0

    0.5

    1.0

    1.5excluded area has CL > 0.95

    Moriond 09

    CKMf i t t e r

    1020% non-SM contributions to most loop-mediated transitions are still possible

    p.2

  • |Vub| matters. . .

    Including B , the SM is disfavored at >2

    dh0.0 0.2 0.4 0.6 0.8 1.0

    d

    0.0

    0.5

    1.0

    1.5

    2.0

    2.5

    3.0

    0.0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1.01-CL

    Moriond 09

    CKMf i t t e r

    w/o B

    dh0.0 0.2 0.4 0.6 0.8 1.0

    d0.0

    0.5

    1.0

    1.5

    2.0

    2.5

    3.0

    0.0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1.01-CL

    Moriond 09

    CKMf i t t e r

    Parameterize NP in B0B0 mixing: M12 = MSM12 (1 + hd e2id)

    1020% non-SM contributions to most loop-mediated transitions are still possiblep.3

  • The challenge of inclusive |Vub| measurements

    Total rate known with 4% accuracy, similar to B(B Xc`) [Hoang, ZL, Manohar]

    To remove the huge charm background(|Vcb/Vub|2 100), need phase space cuts

    Phase space cuts can enhance perturbativeand nonperturbative corrections drastically

    d(bc)/dEe

    10 d(bu)/dEe

    Ee (GeV)

    d/d

    Ee

    E

    0

    20

    40

    60

    80

    0 1 2

    Nonperturbative effects shift endpoint 12 mb 12 mB and determine shape

    Endpoint region determined by b quark PDF in B; want to extract it from data to

    make predictions at lowest order B Xs photon spectrum[Bigi, Shifman, Uraltsev, Vainshtein; Neubert]

    p.4

  • |Vub|: lepton endpoint vs. B Xs

    b quark decayspectrum

    with a model forb quark PDF

    0 0.5 1 1.5 2 2.5El

    ddEl

    p.5

  • |Vub|: lepton endpoint vs. B Xs

    b quark decayspectrum

    with a model forb quark PDF

    0 0.5 1 1.5 2 2.5El

    ddEl

    ddEl

    p.5

  • |Vub|: lepton endpoint vs. B Xs

    b quark decayspectrum

    with a model forb quark PDF

    0 0.5 1 1.5 2 2.5El

    ddEl

    ddEl

    difference:

    2 3 4

    p.5

  • |Vub|: lepton endpoint vs. B Xs

    b quark decayspectrum

    with a model forb quark PDF

    0 0.5 1 1.5 2 2.5El

    ddEl

    ddEl

    difference:

    2 3 4

    1850801-007

    40

    0

    DataSpectator Model

    Wei

    ghts

    / 10

    0 M

    eV

    1.5 2.5 3.5 4.5E (GeV)

    [CLEO, 2001]

    p.5

  • |Vub|: lepton endpoint vs. B Xs

    b quark decayspectrum

    with a model forb quark PDF

    0 0.5 1 1.5 2 2.5El

    ddEl

    ddEl

    difference:

    2 3 4

    1850801-007

    40

    0

    DataSpectator Model

    Wei

    ghts

    / 10

    0 M

    eV

    1.5 2.5 3.5 4.5E (GeV)

    [CLEO, 2001]

    Both of these spectra are determined at lowest order by the b quark PDF in the B Lots of work toward extending beyond leading order; many open issues...

    p.5

  • Past efforts: BLNP (best so far)

    Treated factorization & resummationin shape function region correctly

    Use fixed functional forms to modelshape function (similar to PDFs)

    Analysis tied to shape functionscheme for mb, 1(One scheme for each approach)

    Need for tail gluing to get correctperturbative tail (other approachesdont even do this [DGE, GGOU, ADFR])

    Hard to assess uncertainties

    [Bosch, Lange, Neubert, Paz]

    p.5

  • Start with B Xs

    [ZL, Stewart, Tackmann, PRD 78 (2008) 114014, arXiv:0807.1926]

  • Regions of phase space

    B Xs gives one of the strongest bounds on NP p+X mB 2E

  • The shape function (b quark PDF in B)

    The shape function S(, ) contains nonperturbative physics and obeys a RGEIf S(, ) has exponentially small tail, any smallrunning gives a long tail and divergent moments

    [Balzereit, Mannel, Kilian]

    S(, i) =

    ZdUS( , i, )S(, )

    Constraint: moments (OPE) + B Xs shapeConstraint: How to combine these?

    Consistent setup at any order, in any scheme

    Stable results for varying (SF modeling scale, must be part of uncert.)

    Similar to how all matrix elements are defined e.g., BK() = bBK [s()]2/9(1 + . . .)

    Derive: [ZL, Stewart, Tackmann, 0807.1926]

    S(, ) =

    Zdk C0(k, )F (k)

    0

    0

    0

    0

    1

    1

    1

    1 22

    0.5

    0.5

    0.5

    0.5

    1.5

    1.5

    1.5

    1.5 2.52.5

    0.50.5

    [GeV] [GeV]

    S(

    ,2.5

    GeV

    )[G

    eV

    1]

    S(

    ,2.5

    GeV

    )[G

    eV

    1] = 2.5 GeV

    = 1.8 GeV = 1.3 GeV = 1.0 GeV

    nonpert. perturbative

    ModelS (dash)F (solid)

    run to 2.5GeV

    Can consistently impose moment constraints on F (k), but not on S(, )

    p.7

  • The shape function (b quark PDF in B)

    The shape function S(, ) contains nonperturbative physics and obeys a RGEIf S(, ) has exponentially small tail, any smallrunning gives a long tail and divergent moments

    [Balzereit, Mannel, Kilian]

    S(, i) =

    ZdUS( , i, )S(, )

    Constraint: moments (OPE) + B Xs shapeConstraint: How to combine these?

    Consistent setup at any order, in any scheme

    Stable results for varying (SF modeling scale, must be part of uncert.)

    Similar to how all matrix elements are defined e.g., BK() = bBK [s()]2/9(1 + . . .)

    Derive: [ZL, Stewart, Tackmann, 0807.1926]

    S(, ) =

    Zdk C0(k, )F (k)

    0

    0

    0

    0

    1

    1

    1

    1 22

    0.5

    0.5

    0.5

    0.5

    1.5

    1.5

    1.5

    1.5 2.52.5

    0.50.5

    [GeV] [GeV]

    S(

    ,2.5

    GeV

    )[G

    eV

    1]

    S(

    ,2.5

    GeV

    )[G

    eV

    1] = 2.5 GeV

    = 1.8 GeV = 1.3 GeV = 1.0 GeV

    nonpert. perturbative

    ModelS (dash)F (solid)

    run to 2.5GeV

    Consistent to impose moment constraints on F (k), but not on S(, ) w/o cutoffp.7

  • Derivation of the magic formula (1)

    The shape function is the matrix element of a nonlocal operator:S(, ) = B| bv (iD+ + ) bv| {z }

    O0(,)

    |B, = mB mb

    Integrated over a large enough region, 0 , one can expand O0 as

    O0(, ) =X

    Cn(, ) bv (iD+ )n bv| {z }Qn

    + . . . =X

    Cn( , ) bv (iD+)n bv| {z }eQn + . . .The Cn are the same for Qn and Qn (since O0 only depends on )

    Matching: bv|O0(+, )|bv = XCn(, ) bv| eQn|bv = C0(, ), bv| eQn|bv = 0nbv(k+)|O0( + , )|bv(k+) = C0( + k+, ) =

    X kn+n!

    dnC0(, )

    dn

    bv(k+)|O0( + , )|bv(k+) =X

    Cn(, )bv| eQn|bv = XCn(, ) kn+ Comparing last two lines: Cn(, ) = 1

    n!dnC0(, )

    dn[Bauer & Manohar]

    p.8

  • Derivation of the magic formula (2)

    Define the nonperturbative function F (k) by: [ZL, Stewart, Tackmann; Lee, ZL, Stewart, Tackmann]S(, ) =

    dk C0( k, )F (k), C0(, ) = bv|O0( + , )|bv

    uniquely defines F (k): F (y) = S(y, )/C0(y, )

    Expand in k: S(, ) = n

    1n!

    dnC0(, )dn

    dk (k)nF (k)

    Compare with previous page

    dk knF (k) = (1)n B|Qn|B

    B|Q0|B = 1 , B|Q1|B = , B|Q2|B = 1

    3+

    2

    More complicated situation for higher moments, so stop here

    This treatment is fully consistent with the OPEp.9

  • Changing schemes: mb

    Converting results to a short distance mass scheme removes dip at small : Want to define short distance

    (hatted) quantities such that:

    S() =

    Zdk C0( k)F (k)

    S() =

    Zdk bC0( k) bF (k)

    Switch from pole to short dis-tance scheme:

    mb = bmb + mb1 = b1 + 1

    0

    0

    0

    0

    1

    1

    1

    1 22

    0.5

    0.5

    0.5

    0.5

    1.5

    1.5

    1.5

    1.5 2.52.50.50.5

    [GeV] [GeV]

    S(

    ,2.5

    GeV

    )[G

    eV

    1]

    S(

    ,2.5

    GeV

    )[G

    eV

    1] mpoleb

    m1Sbmkinb

    dashed:

    solid:

    NLL

    NNLL

    ( mB 2E)

    bC0() = C0( + mb) 16

    d2

    d2C0() =

    1 + mb

    d

    d+

    (mb)

    2

    21

    6

    d2

    d2

    C0()

    Can use any short distance mass scheme (1S, kinetic, PS, shape function, . . . )p.10

  • Changing schemes: 1

    We find that kinetic scheme, 2 kin1 , oversubtracts; similar to mb(mb) issues

    Introduce invisible scheme: i1 = 10sR2 2s()

    2CFCA

    4

    2

    31

    (R = 1 GeV)

    Yet to be seen if shape func-tion scheme for 1 gives goodbehavior

    0

    0

    0

    0

    1

    1

    1

    1 22