HEAVY-QUARK FRAGMENTATION FUNCTIONS

IN e

�

e � COLLISIONS

Carlo OleariUniversita 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 �αn

s�

µ2 �

x � 2 p � qq2 E �

�

q2

2

with µ = renormalization scale and αs

αs

� �

2π

�

.

The heavy-quark fragmentation function (FF) in e

�

e � annihilation is defined as

D

�

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 � 1

2

�

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

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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

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Collisions 3

Collinear and soft logarithms

1σ0

dσdx

�

�

∑n � 0

a

n

�

x, q2 , m2, µ2 �

�

αs

�

µ2 �

2π

� 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 � 1

2

�

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 log q2

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

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Collisions 5

Factorization theorem

� � factorization of the squared amplitude

� � �

p, k; . . .

� � 2 � � � �

p

�

z; . . .

� � 2 8παs

2 p � kS � � � �

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

�

2π

�

32p0

d3k

�

2π

�

32k0δ4 �

. . . � p � k

� � d�

n � 1d3t

�

2π

�

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

2π

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

dσidz

�

z, q2 , µ2F

Di

� xz

, µ2F , m2

dσidz

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

ton i (process dependent)

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

µF factorization scale

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Collisions 7

Example

dσdx

�

x, q2 , m2 � � ∑i

1

x

dzz

dσidz

�

z, q2 , µ2F

Di

� xz

, µ2F , m2

if i � g then

Z / γ

Q

Q

i=g

� �� �

m � 0 !

process dependent

dσidz

� Di

i=g

Q

Q� �� �

m

� � 0

process independent

Carlo Oleari Heavy-Quark Fragmentation Functions in e

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Collisions 8

Resummation of collinear logs

dσdx

�

x, q2 , m2 � � ∑i

1

x

dzz

dσidz

�

z, q2 , µ2F

Di

� xz

, µ2F , m2

How to choose µF?

If µ2F

� q2 then no large logarithms of q2 �

µ2F appear in dσi

�

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

dDi

d log µ2

�

x, µ2 , m2 � � ∑j

1

x

dzz

Pi j� x

z, αs

�

µ2 �

D j

�

z, µ2 , m2 �

where the splitting functions Pi j have the expansion

Pi j

�

x, αs

�

µ

�

� αs

�

µ

�

P

0

i j

�x

� �

α2s

�

µ

�

P

1

i j

�

x

� �

α3s

�

µ

�

P

2

i j

�

x

� � � �

α4s

The DGLAP equations resum correctly all the large logarithms

Carlo Oleari Heavy-Quark Fragmentation Functions in e

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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

� �

αsL

� n

� �� �

�

αsL

� 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

� �

αsL� n �

�

∑n � 0

γ

n

�

x

�

αs

�

αsL

� n

� �� �

αs

�

αsL

� 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

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Collisions 10

Initial condition

How can we compute the initial condition for Di

�

x, µ2 , m2 �

?

Di

�

x, µ2 , m2 � � d

0

i δ

�

1 � x

� �

αs

�

µ2 �

d

1

i

�

x, µ2 , m2 � � � �

α2s

.

Use the factorization theorem

dσdx

�

x, q2 , m2 � � ∑i

1

x

dzz

dσidz

�

z, q2 , µ2

Di� 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 �

αs

�

µ2 � � � �

α2s

dσidx

�

x, q2 , µ2 � � a

0

i

�

x

� � a

1

i�

x, q2 , µ2 �

αs

�

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

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Collisions 11

Final recipe for collinear-log resummation

✓ start with Di

�

x, µ20 , m2 �

, with µ20

� m2, so that no large loga-rithms of the ratio µ2

0

�

m2 appear in the initial conditions

✓ evolve Di

�

x, µ20 , m2 �

from the low to the high energy scale µ

with the DGLAP equation to obtain Di�

x, µ2 , m2 �

✓ use the factorization theorem to compute the resummedcross section.

Carlo Oleari Heavy-Quark Fragmentation Functions in e

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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 logn �

1 N � log N gLL�

αs log N

�

�

∑n � 0

c

n

NLL αns logn N � gNLL

�

αs log N

�

Carlo Oleari Heavy-Quark Fragmentation Functions in e

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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 systemapproaches 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

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Collisions 14

Non-perturbative fragmentation function

We assume that all these effects are described by a non-perturbative fragmen-tation function DH

NP, 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

dσidx

�

x, q2 , µ2F

� Di

�

x, µ2F , m2

� DHNP

�

x

�

The non-perturbative part DHNP is

what 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

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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

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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

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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 and

C.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

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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

�

q2 this 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

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Collisions 21

Conclusions

σQ

�

N, M2Z , m2 �

σQ

�

N, M2� , m2

� � aq

�

N, M2Z , µ2

Z

�

1 �

αs

�

µ2Z

� �

πE

�

N, µ2Z , µ2�

� 1 �

αs

�

µ2�� �

π

aq

�

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.