After giving their general expressions valid at all x, double differential
1-particle inclusive distribution inside a quark and a gluon jet produced in a
hard process, together with the inclusive kt distribution, are given at small x
in the Modified Leading Logarithmic Approximation (MLLA), as functions of the
transverse momentum kt of the outgoing hadron.Comment: 4 LaTeX page, 8 figures. To appear in proceedings of Rencontres de
  Moriond 2006, "QCD and hadronic interactions

Ramos, Redamy Perez

English

arXiv

4 LaTeX page, 8 figures. To appear in proceedings of Rencontres de Moriond 2006, "QCD and hadronic interactions"International audienceAfter giving their general expressions valid at all x, double differential 1-particle inclusive distribution inside a quark and a gluon jet produced in a hard process, together with the inclusive kt distribution, are given at small x in the Modified Leading Logarithmic Approximation (MLLA), as functions of the transverse momentum kt of the outgoing hadron

Perez Ramos, Redamy

Hal-Diderot

Inclusive hadronic distributions at small x
Redamy Perez Ramos
To cite this version:
Redamy Perez Ramos. Inclusive hadronic distributions at small x. QCD and Hadronic In-
teractions, La Thuile, Italy, 18-25 Mar 2006., Mar 2006, La Thuile, Val d’Aosta, Italy. 2006.
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May 2006
Inclusive hadronic distributions at small x inside jets
Redamy Perez-Ramos 1
Laboratoire de Physique The´orique et Hautes Energies 2
Unite´ Mixte de Recherche UMR 7589
Universite´ Pierre et Marie Curie-Paris6; CNRS; Universite´ Denis Diderot-Paris7
Abstract: After giving their general expressions valid at all x, double differential 1-particle inclusive
distribution inside a quark and a gluon jet produced in a hard process, together with the inclusive k⊥
distribution, are given at small x in the Modified Leading Logarithmic Approximation (MLLA), as func-
tions of the transverse momentum k⊥ of the outgoing hadron.
Introduction
This work concerns the production of two hadrons inside a high energy jet (quark or gluon); they
hadronize out of two partons at the end of a cascading process that we calculate in pQCD[1]. Con-
sidering this transition as a “soft” process is the essence of the “Local Parton Hadron Duality” (LPHD)
hypothesis[2]. We give indeed, in the MLLA scheme of resummation, the double differential inclusive
1-particle distribution and the inclusive k⊥ distribution as functions of the transverse momentum of the
emitted hadrons in the limit that proved successful when describing energy spectra of particles in jets,
that is Q0 ≈ ΛQCD, the so called “limiting spectrum”[2].
The process under consideration
In a hard collision (pp, pp¯ collisions, e+e− annihilation) we consider a jet of half opening angle Θ0
initiated by a parton A0, which can be a quark or a gluon, see Fig.1. A0, by a succession of partonic
emissions (quarks, gluons), produces a jet of half opening angle Θ0, which, in particular, contains the
parton A; A splits into B and C , which hadronize respectively into the two hadrons h1 and h2. Θ is
the angle between B and C . Because the virtualities of B and C are much smaller than that of A, Θ
can be considered to be close to the angle between h1 and h2; angular ordering (AO) is also a necessary
condition for this property to hold. A0 carries the energy E and gives rise to the (virtual) parton A, which
carries the fraction u of the energy E. A→ B +C occurs with probability ∝ Φ, Φ is the corresponding
DGLAP splitting function, B and C carry respectively the fractions uz and u(1− z) of E; h1 carries the
fraction x1 of E; h2 carries the fraction x2 of E. One sets Θ ≤ Θ0.
Θ0
Θ
ΦD
h1
x1E
B
C
D
D
E
A 0 A h2
x2EuE
uzE
u(1−z)E
Fig. 1: process under consideration: two hadrons h1 and h2 inside one jet.
Since we are concerned with observables depending on the transverse momentum of outgoing hadrons,
1E-mail: perez@lpthe.jussieu.fr
2LPTHE, tour 24-25, 5e`me e´tage, Universite´ P. et M. Curie, BP 126, 4 place Jussieu, F-75252 Paris Cedex 05 (France)
1
k⊥ should be defined with respect to the jet axis which is identified with the direction of the energy
flow[1].
Double differential 1-particle inclusive distribution at all values of x1
The energy conservation sum rule, considered together with DGLAP evolution equations and AO lead
the general expression for the double differential 1-particle inclusive distribution
d2N
dx1 d lnΘ
=
d
d ln Θ
∑
A
∫ 1
x1
duDAA0 (u,EΘ0, uEΘ)D
h1
A
(x1
u
, uEΘ, Q0
)
(1)
which is valid at all x1. d
2N
dx1 d lnΘ
in (1) is described indeed by the convolution of two fragmentation
functions, DAA0 and D
h1
A , over the energy fraction u of A. Namely, DAA0 is purely partonic and describes
the probability to emit parton A with energy uE off parton A0, moreover, it takes into account the
evolution of the jet between Θ0 and Θ; Dh1A describes the probability to produce hadron h1 of energy
fraction x1/u at angle Θ with respect to the direction of the energy flow inside the subjet A. On the
other hand, the transverse momentum of h1 should be bigger than the collinear cut-off parameter Q0
(uEΘ ≥ Q0). Consequently, a lower bound for Θ is obtained: Θ ≥ Θmin ≈ Q0/x1E.
Perturbative expansion parameter and kinematics
We conveniently define the variables ℓ1= ln 1x1 , y1= ln
x1EΘ
Q0
. In what follows, we use the anomalous
dimension γ0 defined through the running coupling constant αs as
γ20(k
2
⊥)=
2Ncαs(k
2
⊥
)
π
=
2
β ln
k2
⊥
Λ2QCD
≡γ20(ℓ1+y1)=
1
β(ℓ1+y1+λ)
=
1
β(YΘ+λ)
, λ=ln
Q0
ΛQCD
. (2)
It determines the rate of multiplicity growth with energy. Nc=3 for SU(3), β= 14Nc (
11
3
Nc− 43TR)
nf=3
=
0.75 with TR= 12nf where nf=3 is the number of light quarks, ΛQCD is the QCD scale, YΘ=ℓ1+y1 =
ln EΘ
Q0
and YΘ0=ln EΘ0Q0 . For instance, in the LHC environment we can take the typical value YΘ0=7.5
⇒ γ0≃0.4 by setting λ=0 (Q0=ΛQCD), Θ=Θ0 and Q0=250 MeV in (2). γ0 can be therefore treated
as the small parameter of the perturbative expansion at MLLA.
Double differential 1-particle inclusive distribution at small x1, x1 ≪ 1
The convolution integral (1) is dominated by u ≈ 1. In order to obtain an analytical expression of (1),
since x1/u ≪ 1, we perform a perturbative expansion in γ0 such that (1) gets factorized and derived in
the region of soft multi-particle production[1]
d2N
dℓ1 d ln k⊥
=
< C >q,g
Nc
d
dy1
D˜g(ℓ1, y1) +
1
Nc
D˜g(ℓ1, y1)
d
dy1
< C >q,g, (3)
d
dy1
D˜g(ℓ1, y1) = O(γ0) = O(
√
αs),
d
dy1
< C >q,g= O(γ20) = O(αs). (4)
The first term in (3) is the main contribution to the double differential 1-particle inclusive distribution
while the second one constitutes its MLLA correction of relative order O(γ0) (4). Dh1A in (1) has been
replaced by the distribution Dg(ℓ1, y1) in (3) that describes the MLLA “hump-backed plateau” in the
limit where Q0 can be taken down to ΛQCD (“limiting spectrum”)[2]. < C >q,g is the total colour
current that describes the evolution of the jet between Θ0 and Θ. It decomposes into its leading term
< C >0q,g and the MLLA correction δ < C >q,g= O(γ0)[1]. In Fig.2 we represent < C >0q,g (straight
line) and the full expression < C >q,g=< C >0q,g +δ < C >q,g (curve) for two values of ℓ1; ℓ1 =
2
2.5(left), 3.5(right) in function of y1. Two types of MLLA corrections are displayed in this figure.
Namely, δ < C >q,g< 0 is given by the vertical difference between the straight and curved lines. The
other correction is given by the slope of the curve, that is d
dy1
< C >q,g in (4), this one is large and
positive for y1 ≥ 1.5. For ℓ1 = 2.5 and y1 ≈ 1.5, δ < C >q,g represents 50% of < C >q,g while
for ℓ1 = 3.5 > 2.5 it gets under control. We are thus allowed to set the range of applicability of
our soft approximation to ℓ1 ≥ ℓ1,min ≈ 2.5 (x1 . 0.08) ⇒ y1 ≤ y1,max = YΘ0 − ℓ1,min = 5.0.
 1
 1.5
 2
 2.5
 3
 3.5
 4
 0  1  2  3  4  5
y
YΘ0 = 7.5
<C>0G, l = 2.5
<C>0G+δ<C>G, l = 2.5
<C>0Q, l = 2.5
<C>0Q+δ<C>Q, l = 2.5
 1
 1.5
 2
 2.5
 3
 3.5
 4
 0  0.5  1  1.5  2  2.5  3  3.5  4
y
YΘ0 = 7.5
<C>0G, l = 3.5
<C>0G+δ<C>G, l = 3.5
<C>0Q, l = 3.5
<C>0Q+δ<C>Q, l = 3.5
Fig. 2: < C >0A0 and < C >
0
A0
+δ< C >A0 for quark and gluon jets, as functions of y,
for YΘ0 = 7.5, ℓ = 2.5 on the left and ℓ = 3.5 on the right.
Furthermore, one should stay in the perturbative regime, which needs y1 ≥ 1 (k⊥ > 2.72ΛQCD ≈
0.7 GeV). We finally get an estimate of the range of applicability of the MLLA scheme of resummation
to be 1.0≤ y1 ≤ 5.0 in the LHC environment. In Fig.3 below, we represent the double differential 1-
particle inclusive distribution (3) for ℓ1 = 3.5 in function of y1. We compare our results with a naive
DLA-inspired case where one does not take into account the evolution of the jet between Θ0 and Θ but
sets < C >q= CF (quark jet) and < C >g= Nc (gluon jet).
 0
 0.5
 1
 1.5
 2
 2.5
 3
 3.5
 4
 0  0.5  1  1.5  2  2.5  3  3.5  4
y
YΘ0 = 7.5
naive case, l = 3.5
cone Θ,  l = 3.5
 0
 0.5
 1
 1.5
 2
 2.5
 3
 0  0.5  1  1.5  2  2.5  3  3.5  4
y
YΘ0 = 7.5
naive case, l = 3.5
cone Θ,  l = 3.5
Fig. 3: d2N
dℓ1 d lnk⊥
for a gluon jet (left) and for a quark jet (right) at fixed ℓ1 = 3.5,
MLLA and naive approach.
For both, the quark and gluon jets, new MLLA corrections arising from (4) push up the distribution when
y1 increases as compared with the naive case. On the other hand, it is enhanced when y1 → 0 by the
running of αs(k⊥) at k⊥ → ΛQCD.
Transverse momentum inclusive k⊥ distribution
Integrating (3) over the whole phase space in the logarithmic sense (ℓ1 = ln(1/x1)) we obtain the
inclusive k⊥ distribution inside a quark and a gluon jet[1](
dN
d ln k⊥
)
q,g
=
∫ YΘ0−y1
0
dℓ1
(
d2N
dℓ1 d ln k⊥
)
q,g
. (5)
We give results for YΘ0 = 7.5 in Fig.4 below and compare with the naive case.
3
 0
 5
 10
 15
 20
 25
 1  2  3  4  5  6
y
YΘ0 = 7.5
(dN/dy)G
(dN/dy)naiveG
 0
 2
 4
 6
 8
 10
 12
 14
 1  2  3  4  5  6
y
YΘ0 = 7.5
(dN/dy)Q
(dN/dy)naiveQ
Fig. 4: dN
d lnk⊥
for a gluon jet, MLLA and naive approach with enlargement
We observe in particular that the positivity of the distribution when y1 increases is restored as compared
with the naive case. This stems from the global role of MLLA corrections in the range.
 0
 20
 40
 60
 80
 100
 0  2  4  6  8  10  12
y
YΘ0 = 15
(dN/dy)MLLAG
(dN/dy)MLLAQ
Fig. 5: dN
d ln k⊥
for a gluon jet (blue) and for a quark jet (green) at YΘ0 = 15.0
In Fig.5 above, we represent the evidence of two competing effects at the unrealistic value YΘ0 = 15.0.
One observes that as y1 → 0 the distribution is depleted by QCD coherence effects. Indeed, in this
region of the phase space (k⊥ → Q0 ≈ ΛQCD) gluons are pushed at larger angles, they are thus emitted
independently from the rest of the partonic ensemble. At the opposite, when the value k⊥ ≈ Q0 ≈ ΛQCD
is reached, the distribution is enhanced by the running of αs.
Conclusion
Results for the double 1-particle inclusive distribution and the inclusive k⊥ distribution at small x have
been discussed and displayed. Sizable differences with the naive approach in which one forgets the evo-
lution of the jet between its half opening angle Θ0 and the emission angle Θ have been found. The global
role of new MLLA corrections is emphasized to recover the positivity of the inclusive k⊥ distribution.
On the other hand, at realistic energy scales (LHC, Tevatron, LEP), QCD coherence effects are screened
by the running of αs (see Fig.4) which, furthermore, forbids extending the confidence domain for y1 ≤ 1.
The range of applicability of the soft approximation has been discussed from the analysis of corrections
not to exceed ymax = YΘ0 − ℓmin = 5.0; it is indeed smaller at LEP and Tevatron energies (smaller
YΘ0). Our results will be compared with forthcoming data from CDF.
References
[1] R. Perez-Ramos & B. Machet: “MLLA inclusive hadronic distributions inside one jet at high energy
colliders”, hep-ph/0512236, JHEP 04 (2006) 043.
[2] Yu.L. Dokshitzer, V.A. Khoze, S.I. Troyan and A.H. Mueller: Rev. Mod. Phys. 60 (1988) 373-388.
4


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