HEAVY-QUARK FRAGMENTATION FUNCTIONS IN e e C · 2006. 4. 20. · HEAVY-QUARK FRAGMENTATION...

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H EAVY -Q UARK F RAGMENTATION F UNCTIONS IN ee C OLLISIONS Carlo Oleari Universit` a di Milano-Bicocca, Milan DIS2006, Tzukuba 21 April 2006 Theoretical introduction collinear logarithms soft logarithms Non-perturbative contributions Results for c (and b) fragmentation functions Conclusions
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Transcript of HEAVY-QUARK FRAGMENTATION FUNCTIONS IN e e C · 2006. 4. 20. · HEAVY-QUARK FRAGMENTATION...

  • HEAVY-QUARK FRAGMENTATION FUNCTIONSIN e

    e � COLLISIONS

    Carlo OleariUniversità di Milano-Bicocca, Milan

    DIS2006, Tzukuba

    21 April 2006

    � � Theoretical introduction

    – collinear logarithms

    – soft logarithms

    � � Non-perturbative contributions

    � � Results for c (and b) fragmentationfunctions

    � � Conclusions

  • What is a fragmentation function?

    pZ/γ

    q

    e-

    e+ Q

    X

    The differential cross section for the production of a massive quark Q in the process

    e

    e

    � � Z

    γ

    q

    � � Q

    p� � X

    is calculable in PQCD, because the mass acts as a cut-off for final-state collinear singularities

    dσdx

    x, q2 , m2

    � �

    ∑n � 0

    a

    n

    x, q2 , m2, µ2�

    ᾱns�

    µ2

    x �

    2 p � qq2

    E ��

    q2

    2

    with µ = renormalization scale and ᾱs

    αs

    � �

    .

    The heavy-quark fragmentation function (FF) in e

    e � annihilation is defined as

    x, q2 , m2

    � 1σtot

    dσdx

    x, q2 , m2

  • Differential cross-section at order αs

    pZ/γ

    q

    e-

    e+ Q

    Q

    pZ/γ

    q

    e-

    e+ Q

    Q

    x �

    EQ�

    EQ

    max

    1σ0

    dσdx

    � δ

    1 � x

    � � αs

    q2

    2πCF

    1 � lnq2

    m2

    1 � x2

    1 � x�

    ��

    ln

    1 � x

    1 � x

    ��

    1 � x2

    � 21 � x2

    1 � xlog x �

    12

    11 � x

    ��

    x2 � 6x � 2

    � ��

    23π2 �

    52

    δ

    1 � x

    ��

    � ��

    m2

    q2

    1

    0

    11 � x

    ��

    f

    x

    1

    0

    f

    x�

    � f�

    1

    1 � x

    � � σ � σ0

    1 �

    αs

    π

    � ��

    m2

    q2

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 2

  • Two main issues

    � � Is this a “well-behaved” perturbation expansion?

    � � We do NOT “see” free quarks, even if they are heavy, butbound states (mesons and baryons). How can we incorporatethese non-perturbative (hadronization) effects?

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 3

  • Collinear and soft logarithms

    1σ0

    dσdx

    ∑n � 0

    a

    n

    x, q2 , m2, µ2

    αs

    µ2

    � n

    � δ

    1 � x

    � � αs

    q2

    2πCF

    1 � lnq2

    m2

    1 � x2

    1 � x

    ��

    ln

    1 � x�

    1 � x�

    ��

    1 � x2

    � 21 � x2

    1 � xlog x �

    12

    11 � x

    ��

    x2 � 6x � 2

    � ��

    23π2 �

    52

    δ

    1 � x

    ��

    � ��

    m2

    q2

    � � In the limit q2 � m2,

    a

    n

    ��

    logq2

    m2

    � n

    and if αs logq2

    m2

    � 1 we cannot truncate the series at some fixed order, because eachterm in the series is of the same order as the first one � � we have to resum these largecontributions

    � � Same for soft logarithms, that arise when x � 1.

  • Quasi-collinear behavior

    M

    k

    pp+k

    � � M

    k

    pp/z

    S

    p � k

    � 2 � m2 � 2p � k

    � 2k0

    � �

    p

    2 � m2 �� �

    p

    cosθQg�

    t = (t0,0,tz) η = (1,0,-1)

    k⊥ = (0,k⊥,0)

    pν � ztν � ξ

    ην � kν �

    kν ��

    1 � z

    tν � ξ

    � �

    ην

    � kν �

    p2 � t2 � m2 k2 � 0 η2 � 0

    t � k � � 0 η � k � � 0 kν � k �ν � � k2 �

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 5

  • Factorization theorem

    � � factorization of the squared amplitude

    � � �

    p, k; . . .

    � � 2 � � � � p

    z; . . .

    � � 2 8παs2 p � k

    S �� � �

    p

    z; . . .

    � � 2 8παsz

    �1 � z

    k2 � ��

    1 � z

    2m2S

    S z,m2

    k2 �

    � CF

    1 � z2

    1 � z

    2z

    1 � z

    m2

    k2 � ��

    1 � z�

    2m2

    � � factorization of the phase space

    d

    n � 1

    � d

    n � 1d3 p

    32p0

    d3k

    32k0δ4

    . . . � p � k

    � � d�

    n � 1d3t

    32t0δ4

    . . . � t

    � �� �

    d

    n

    116π2

    dzz

    1 � z

    � dk2 �

    Then we can iterate to take into account multiple quasi-collinear emissions

    dσn � 1 �� � �

    p, k; . . .

    � � 2d

    n � 1

    � � � � t; . . .

    � � 2d

    nαs

    1

    0dz

    µ2f

    0dk2 �

    1k2 � �

    1 � z

    2m2S

    � dσnαs

    2πCF

    1

    0dz

    1 � z2

    1 � zlog

    1 �

    µ2f

    m21

    1 � z

    2

    �� . . .

  • Factorization theorem

    In the limit m2 � q2 we can neglect terms proportional to powers ofm2

    q2, and the single inclusive heavy-quark cross section can be writtenas (factorization theorem)

    dσdx

    x, q2 , m2

    � � ∑i

    1

    x

    dzz

    dσ̂idz

    z, q2 , µ2F

    D̂i

    � xz

    , µ2F , m2

    dσ̂idz

    MS-subtracted partonic cross section, for the production of the par-

    ton i (process dependent)

    D̂i MS fragmentation functions for the parton i to “fragment” into theheavy quark Q (process independent)

    µF factorization scale

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 7

  • Example

    dσdx

    x, q2 , m2

    � � ∑i

    1

    x

    dzz

    dσ̂idz

    z, q2 , µ2F

    D̂i

    � xz

    , µ2F , m2

    if i � g then

    Z / γ

    Q

    Q

    i=g

    � �� �

    m � 0 !

    process dependent

    dσ̂idz

    � D̂i

    i=g

    Q

    Q� �� �

    m

    � � 0

    process independent

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 8

  • Resummation of collinear logs

    dσdx

    x, q2 , m2

    � � ∑i

    1

    x

    dzz

    dσ̂idz

    z, q2 , µ2F

    D̂i

    � xz

    , µ2F , m2

    How to choose µF?

    If µ2F

    � q2 then no large logarithms of q2

    µ2F appear in dσ̂i

    dz and its perturbative expan-

    sion is reliable.The large logarithms are moved in the fragmentation functions that obey the (µF � µ)

    DGLAP evolution equations

    dD̂id log µ2

    x, µ2 , m2

    � � ∑j

    1

    x

    dzz

    Pi j� x

    z, ᾱs

    µ2

    D̂ j

    z, µ2 , m2

    where the splitting functions Pi j have the expansion

    Pi j

    x, ᾱs

    µ

    � � ᾱs

    µ

    P

    0

    i j

    �x

    � �ᾱ2s

    µ

    P

    1

    i j

    x

    � �

    ᾱ3s

    µ

    P

    2

    i j

    x

    � � ��

    α4s

    The DGLAP equations resum correctly all the large logarithms

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 9

  • Leading Log, Next-to-Leading Log. . .

    Introducing the shorthand notation L � log

    q2

    m2

    ✓ with an expansion for Pi j up to order αs, we resum all terms of the form

    dσdx

    x, q2 , m2

    ��������

    LL

    ∑n � 0

    β

    n

    x

    � �

    ᾱsL

    � n

    � �� �

    ᾱsL

    � n � � Leading Log

    ✓ with an expansion for Pi j up to order α2s , we resum all terms of the form

    dσdx

    x, q2 , m2

    ��������

    NLL

    ∑n � 0

    β

    n

    x

    � �

    ᾱsL� n �

    ∑n � 0

    γ

    n

    x

    ᾱs

    ᾱsL

    � n

    � �� �

    ᾱs

    ᾱsL

    � n � � Next-to-Leading Log

    ✓ . . . so on, so forth.

    The time-like splitting functions are known up to the third order [Mitov, Moch and Vogt, hep-ph/0604053]

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 10

  • Initial condition

    How can we compute the initial condition for D̂i

    x, µ2 , m2

    ?

    D̂i

    x, µ2 , m2

    � � d

    0

    i δ

    1 � x

    � �

    ᾱs

    µ2

    d

    1

    i

    x, µ2 , m2

    � � ��

    α2s

    .

    Use the factorization theorem

    dσdx

    x, q2 , m2

    � � ∑i

    1

    x

    dzz

    dσ̂idz

    z, q2 , µ2

    D̂i� x

    z, µ2 , m2

    and match the expansion up to order αs of the left- and right-hand side to obtain d

    0

    i and

    d

    1

    i [Mele and Nason, Nucl. Phys. B361 (91) 626]

    dσdx

    x, q2 , m2

    � � a

    0

    x, q2 , m2

    � � a

    1

    x, q2 , m2, µ2

    ᾱs

    µ2

    � � ��

    α2s

    dσ̂idx

    x, q2 , µ2

    � � â

    0

    i

    x

    � � â

    1

    i�

    x, q2 , µ2

    ᾱs

    µ2

    � � ��

    α2s

    The d

    2

    i terms are known too, and have been computed by [Melnikov and Mitov,

    �� �� � � �� �� �� �� ;

    Mitov, �� �� � �� � �� � � ], following a different strategy.

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 11

  • Final recipe for collinear-log resummation

    ✓ start with D̂i

    x, µ20 , m2 � , with µ20

    � m2, so that no large loga-rithms of the ratio µ20

    m2 appear in the initial conditions

    ✓ evolve D̂i

    x, µ20 , m2 � from the low to the high energy scale µ

    with the DGLAP equation to obtain D̂i�

    x, µ2 , m2

    ✓ use the factorization theorem to compute the resummedcross section.

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 12

  • Soft logarithms

    In the region of the phase space of multiple soft-gluon emission (x � 1), the differentialcross section contains enhanced terms proportional to

    a

    n

    � c

    n

    LLlog2n

    � 1 � 1 � x

    1 � x �� c

    n

    NLLlog2n

    � 2 � 1 � x

    �1 � x �

    � . . .

    These terms can be organized in towers of log N, where we introduce the Mellin transform

    f

    N

    � �

    1

    0dx xN

    � 1 f

    x

    � � �

    1

    0dx xN

    � 1 logk � 1 � x

    1 � x �� logk

    � 1 N

    The large-N contributions come from the regions where x � 1, associated to thebremsstrahlung spectrum of soft and collinear emission.Up to now, it is known how to resum all the Leading Log and Next-to-Leading Log [Dok-shitzer, Khoze and Troyan, �� �� � �

    �� � � � ; Cacciari and Catani, �� �� � �� � � � �� � �

    ]

    ∑n � 0

    c

    n

    LL αns log

    n � 1 N � log N gLL�

    αs log N

    ∑n � 0

    c

    n

    NLL αns log

    n N � gNLL

    αs log N

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 13

  • Non-perturbative effects

    ✗ The weak point of the factorization theorem comes from the initial condition for theevolution of the fragmentation function, which is computed as a power expansionin terms of αs

    m

    : irreducible, non-perturbative uncertainties of order

    QCD

    m arepresent.

    ✗ The soft-gluon resummation functions gLL and gNLL contain singularities at large Nwhich signal the eventual failure of perturbation theory and hence the onset of non-perturbative phenomena.

    � � in the initial condition, the region

    1 � x

    m ��

    (m

    N ��

    in moment space) issensitive to the decay of excited states of the heavy-flavoured hadrons, where

    isa typical hadronic scale of a few hundreds MeV.

    � � in the coefficient functions, when

    1 � x�

    q2 ��2, the mass of the recoil system

    approaches typical hadronic scales.

    The matching of perturbative results with non-perturbative physics is a delicate problem,which rests, first of all, on a proper definition of the perturbative series.

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 14

  • Non-perturbative fragmentation function

    We assume that all these effects are described by a non-perturbative fragmen-tation function DHNP, that takes into account all low-energy effects, includingthe process of the heavy quark turning into a heavy-flavoured hadron. Thefull resummed cross section, including non-perturbative corrections, is thenwritten as

    dσH

    dx

    x, q2

    � � ∑i

    dσ̂idx

    x, q2 , µ2F

    � D̂i

    x, µ2F , m2

    � DHNP

    x

    The non-perturbative part DHNP iswhat is missing to go from the par-tonic cross section to the hadronicone � � very sensitive to the per-turbative part

    It is expected to be universal � � exctract it from e

    e � data and use it inhadronic heavy-quark production.

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 15

  • Non-perturbative fragmentation function

    The Mellin transform of the full resummed cross section, including non-perturbative corrections, is

    σH

    N, q2

    � � σQ

    N, q2 , m2

    DNP

    N

    We [Cacciari, Nason and C.O., �� � � � � �� � � � � ] have fitted CLEO and BELLE D

    datausing the two-component form for DNP

    DNP

    x

    � � Norm. ��

    δ

    1 � x

    � � c

    1 � x

    � axb

    Na,b�

    , Na,b �

    1

    0

    1 � x

    � axb

    Simple phenomenological interpretation

    � � the hard term (the delta function) corresponds, in some sense, to the directexclusive production of the D

    � � the rest accounts for D

    ’s produced in the decay chain of higher reso-nances.

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 16

  • Non-perturbative fragmentation function

    DNP

    x

    � � Norm. ��

    δ

    1 � x

    � � c

    1 � x

    � axb

    Na,b

    , Na,b �

    1

    0

    1 � x� axb

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 17

  • CLEO and BELLE D

    fits

    ✓ D mesons data fits near the

    � �

    4S

    mass (10.6 GeV)

    ✓ Very good fit in the whole x range

    ✓ More fits in [Cacciari, Nason andC.O.], where D

    � � D X have beenmodeled from decay chains andbranching ratios

    ✓ B mesons data fits near the Z0

    mass (91.2 GeV)

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 18

  • ALEPH D

    fits at LEP

    ✗ very few useful points

    ✗ large error bars

    DNP

    x

    � � Norm. ��

    δ�

    1 � x

    � � c

    1 � x

    � axb

    Na,b

    ALEPH a � 2.4

    1.2 , b � 13.9

    5.7 c � 5.9

    1.7

    CLEO

    BELLE a � 1.8

    0.2 , b � 11.3

    0.6 c � 2.46

    0.07

    What about the supposed universality of the non-perturbative fragmentation function?

  • ALEPH with CLEO/BELLE parameters

    ✗ Discrepancy in the large-x (large-N)region

    ✗ The ratio data/theory well modeled by

    11 � 0.044

    N � 1�

  • Possible explanation

    We can only speculate about the possible origin of the discrepancy.If this were due to a non-perturbative correction to the coefficient function ofthe form

    � � 1 �

    C

    N � 1

    q2this would lead to the extra factor

    1 � C

    N � 1

    M2Z

    1 � C

    N � 1

    M2�

    � � � 11 � 0.044

    N � 1

    � if C � 5 GeV2 large value!!

    � � 1 �

    C

    N � 1

    Ewith E �

    q2

    2 then C � 0.52 GeV, a much more accept-

    able value.

    Demonstrating the absence (or the existence) of 1

    E corrections in fragmenta-tion functions would be a very interesting result, since it would help to vali-date or disprove renormalon-based predictions.

    Carlo Oleari Heavy-Quark Fragmentation Functions in e

    e

    Collisions 21

  • Conclusions

    σQ

    N, M2Z , m2 �

    σQ

    N, M2� , m2

    � �

    āq

    N, M2Z , µ2Z

    1 � αs

    µ2Z

    � �

    πE

    N, µ2Z , µ2�

    � 1 � αs

    µ2�� �

    π

    āq

    N, M2� , µ2��

    � � low-scale effects, both at the heavy-quarkmass scale and at the non-perturbativelevel, cancel completely

    � � its prediction is then entirely perturbative(the evolution function E is totally pertur-bative)

    � � scale-variation effects do not explain thediscrepancy

    � � mass effects in charm production onthe

    � �

    4S

    where computed at order α2s ,and found to be small [Nason and C.O.,

    �� �� � � �� � � � �� ]

    Unfortunately, the low precision of the available data does not allow, at the moment, to resolvethe issue of the absence (or the existence) of 1

    E corrections in fragmentation functions.