Unintegrated CCFM parton distributions and pion production in proton-proton collisions at high energies

1 MB·English

by  M. Czech

Unintegrated CCFM
parton distributions
and pion production
in proton-proton collisions
at high energies
5
0
0 M. Czech 1,2 and A. Szczurek 1,3
2
n 1 Institute of Nuclear Physics PAN
a PL-31-342 Cracow, Poland
J
2 Institute of Physics, Jagiellonian University
1
2 PL-30-059 Cracow, Poland
3 University of Rzesz´ow
2
v PL-35-959 Rzesz´ow, Poland
6
2 Abstract
0
1
Inclusive cross sections for pion production in proton-proton col-
0
5 lisions are calculated for the first time based on unintegrated parton
0 (gluon, quark, antiquark) distributions (uPDF). We use Kwiecin´ski
/
h uPDF’s and phenomenological fragmentation functions from the lit-
t
- erature. In addition to the gg g diagram used recently for appli-
l →
c cations at RHIC we include also gq q and qg q diagrams. We
u → →
find that the new contributions are comparable to the purely gluonic
n
: one at midrapidities and dominate in the fragmentation region. The
v
new mechanisms are responsible for π+ π asymmetry. We discuss
i −
X −
how the asymmetry depends on x and p . Inclusive distributions in
F t
r x (or rapidity) and transverse momentum for partons and pions are
a F
shown for illustration. In contrast to standard collinear approach in
ourapproach therangeof applicability can beextendedtowards much
lower transverse momenta.
PACS: 12.38.Bx, 13.85.Hd, 13.85.Ni
1 Introduction
The distributionsofmesons atlargetransverse momenta inpporpp¯collisions
are usually calculated in the framework of perturbative QCD using collinear
1
factorization (see e.g. [1, 2, 3, 4]). While the shape at transverse momenta
larger than 2-4 GeV can be relatively well explained, the difference between
the data and the lowest-order computation is quantified in terms of a so-
called K-factor, independent quantity for each energy [4]. The K-factor is
found to systematically decrease with growing energy [5]. In order to extend
the calculation towards lower values of meson transverse momenta it was
suggested to add an extra Gaussian distribution in transverse momentum
[6, 7, 8] 1. In this approach the standard collinear integrals are replaced as:
dx p(x,µ2) dxd2k g(k )p(x,µ2) , (1)
t t
→
where the extra distribution function in transverse momentum is normalized
to unity
d2k g(k ) = 1. (2)
t t
Z
It is customary to use Gaussian distributions for g(k ). It becomes clear that
t
this procedure is effective in the following sense. The transverse momentum
originates either from the nonperturbative “really internal” momentum dis-
tributions of partons in nucleons (of the order of a fraction of GeV) and/or
is generate dynamically as the inital state radiation process (of the order
of GeV). In principle, the second component may depend on the values of
longitudinal momentum fractions x and/or x . The formalism used by us
1 2
in the present paper will include both these effects separately and explicitly.
The recent results from RHIC (see e.g. [13]) have attracted a renewed
interest inbetterunderstandingthedynamicsofparticleproduction,notonly
in nuclear collisions. Quite different approaches have been used to describe
the particle spectra from the nuclear collisions [14]. The model in Ref.[10]
withaneducatedguessforUGDdescribessurprisinglywellthewholecharged
particle rapidity distribution by means of gluonic mechanisms only. Such
a gluonic mechanism would lead to identical production of positively and
negatively charged hadrons. The recent results of the BRAHMS experiment
concerning heavy ion collisions [15] show that the π /π+ and K /K+ ratios
− −
differ fromunity. Thisput intoquestionthesuccessful description ofRef.[10].
In the light of this experiment, it becomes obvious that the large rapidity
regions have more complicated flavour structure. At lower energies these
ratios are known to differ from unity drastically [28].
InRef.[11]oneofushascalculatedinclusive pionspectrainproton-proton
collisions based on different models of unintegrated gluon distributions taken
from the literature. In the present paper in addition to the gg g mecha-
→
1
A similar procedure was used e.g. for prompt photon production [9]
2
nism we include also q g q and gq q mechanisms and similar ones
f f f f
→ →
for antiquarks in order to obtain a fully consistent description.
Manyunintegratedgluondistributionsintheliteratureareadhocparametriza-
tions of different sets of experimental data rather than derived from QCD.
An example of a more systematic approach, making use of familiar collinear
distributions can be found in Ref.[17]. Recently Kwiecin´ski and collabora-
tors [18, 19, 20] have shown how to solve the so-called CCFM equations
by introducing unintegrated parton distributions in the space conjugated to
the transverse momenta [18]. We present first results for pion production
based on unintegrated parton (gluon, quark, antiquark) distributions ob-
tained by solving a set of coupled equations developed by Kwiecin´ski and
collaborators. Recently these parton distributions were tested for inclusive
gauge boson production in proton-antiproton collisions [21] and for charm-
anticharm correlations in photoproduction [22]. While in the first process
one tests mainly quark and antiquark distributions at scales µ2 M2 ,M2,
∼ W Z
in the second reaction one tests mainly gluon distributions at scales µ2 m2.
∼ c
In comparison to those reactions in the present application one tests both
gluon as well as quark and antiquark distributions in a “more nonperturba-
tive” region of smaller scales of the order down to µ p (parton) 0.5 -
t
∼ ∼
1.0 GeV, which corresponds to pion transverse momenta p (pion) 0.25 -
t
∼
0.5 GeV. This is a region where perturbative and nonperturbative effects are
believed to mix up and the application of the pQCD is doubtful. On the
other hand, this is an interesting region of phase space responsible for the
bulk of hadronic production. We shall discuss how far down to small pion
transverse momenta one can apply the present approach.
Some preliminary results of the present study were presented at a confer-
ence [12].
2 Kwiecin´ski unintegrated parton distributions
Kwiecin´ski has shown that the evolution equations for unintegrated parton
distributions takes a particularly simple form in the variable conjugated to
the parton transverse momentum. The two possible representations are in-
terrelated via Fourier-Bessel transform
f (x,κ2,µ2) = ∞db bJ (κ b)f˜(x,b,µ2) ,
k t 0 t k
Z0 (3)
f˜(x,b,µ2) = ∞dκ κ J (κ b)f (x,κ2,µ2) .
k t t 0 t k t
Z0
3
The index k above numerates either gluons (k=0), quarks (k> 0) or anti-
quarks (k< 0). In the impact-parameter space the Kwiecin´ski equation takes
the following simple form
∂f˜ (x,b,µ2) α (µ2) 1 x
NS = s dzP (z) Θ(z x)J ((1 z)µb)f˜ ,b,µ2
∂µ2 2πµ2 qq − 0 − NS z
Z0 (cid:20) (cid:16) (cid:17)
f˜ (x,b,µ2) ,
NS
−
(cid:21)
∂f˜ (x,b,µ2) α (µ2) 1 x
S = s dz Θ(z x)J ((1 z)µb) P (z)f˜ ,b,µ2
∂µ2 2πµ2 − 0 − qq S z
Z0 (cid:26) (cid:20) (cid:16) (cid:17)
x
+P (z)f˜ ,b,µ2 [zP (z)+zP (z)]f˜ (x,b,µ2) ,
qg G qq gq S
z −
(cid:21) (cid:27)
(cid:16) (cid:17)
∂f˜ (x,b,µ2) α (µ2) 1 x
G = s dz Θ(z x)J ((1 z)µb) P (z)f˜ ,b,µ2
∂µ2 2πµ2 − 0 − gq S z
Z0 (cid:26) (cid:20) (cid:16) (cid:17)
x
+P (z)f˜ ,b,µ2 [zP (z)+zP (z)]f˜ (x,b,µ2) .
gg G gg qg G
z −
(cid:21) (cid:27)
(cid:16) (cid:17)
(4)
We have introduced here the short-hand notation
˜ ˜ ˜ ˜ ˜
f = f f , f f ,
NS u − u¯ d − d¯ (5)
˜ ˜ ˜ ˜ ˜ ˜ ˜
f = f +f +f +f +f +f .
S u u¯ d d¯ s s¯
The unintegrated parton distributions in the impact factor representation
are related to the familiar collinear distributions as follows
x
f˜(x,b = 0,µ2) = p (x,µ2) . (6)
k k
2
On the other hand, the transverse momentum uPDF’s are related to the
integrated distributions as
xp (x,µ2) = ∞dκ2 f (x,κ2,µ2) . (7)
k t k t
Z0
While physically f (x,κ2,µ2) should be positive, there is no obvious reason
k t
for such a limitation for f˜(x,b,µ2).
k
In the following we use leading-order parton distributions from ref.[29]
as the initial condition for QCD evolution. The set of integro-differential
equations in b-space was solved by the method based on the discretisation
made with the help of the Chebyshev polynomials (see e.g. [20]). Then the
unintegrated parton distributions were put on a grid in x, b and µ2 and the
grid was used in practical applications for Chebyshev interpolation (see next
section).
4
3 Inclusive cross sections for partons
The approach proposed by Kwiecin´ski is very convenient to introduce the
nonperturbative effectslike internal(nonperturbative) transverse momentum
distributions of partons in nucleons. It seems reasonable, at least in the first
approximation, to include the nonperturbative effects in the factorizable way
f˜(x,b,µ2) = f˜CCFM(x,b,µ2) Fnp(b) . (8)
q q · q
The form factor responsible for the nonperturbative effects must be normal-
ized such that
FNP(b = 0) = 1 (9)
in order not to spoil the relation (6). In the following, for simplicity, we use
a flavour and x-independent form factor
b2
Fnp(b) = Fnp(b) = exp (10)
q −4b2
(cid:18) 0(cid:19)
which describes the nonperturbative effects. The Gaussian form factor in b
means also a Gaussian initial momentum distribution exp( k2b2) (Fourier
∝ − t 0
transform of a Gaussian function is a Gaussian function). Gaussian form
factor is often used to correct collinear pQCD calculations for the so-called
internal momenta. Other functional forms in b are also possible.
The gg g mechanism considered in the literature is not the only one
→
possible. In Fig.1 we show two other leading order diagrams. They are
potentially important in the so-called fragmentation region. The formulae
for inclusive quark/antiquark distributions are similar to the formula for
gg g [16]. The formulae for all the processes mentioned are listed below:
→
for diagram A (gg g):
→
dσA 16N 1
c
=
dyd2p N2 1p2
t c − t
α (Ω2) f (x ,κ2,µ2) f (x ,κ2,µ2)
s g/1 1 1 g/2 2 2
Z
δ(2)(~κ +~κ ~p ) d2κ d2κ , (11)
1 2 t 1 2
−
for diagram B (q g q ):
1 f f
→
dσB1 16N 4 1
c
=
dyd2p N2 1 9 p2
t c − (cid:18) (cid:19) t
α (Ω2) f (x ,κ2,µ2) f (x ,κ2,µ2)
s qf/1 1 1 g/2 2 2
f Z
X
δ(2)(~κ +~κ p~ ) d2κ d2κ , (12)
1 2 t 1 2
−
5
for diagram B (g q q ):
2 f f
→
dσB2 16N 4 1
c
=
dyd2p N2 1 9 p2
t c − (cid:18) (cid:19) t
α (Ω2) f (x ,κ2,µ2) f (x ,κ2,µ2)
s g/1 1 1 qf/2 2 2
f Z
X
δ(2)(~κ +~κ p~ ) d2κ d2κ . (13)
1 2 t 1 2
−
These seemingly 4-dimensional integrals can be written as 2-dimensional in-
tegrals after a siutable change of variables [11]
d2q
... δ(2)(~κ +~κ p~ ) d2κ d2κ = ... t . (14)
1 2 t 1 2
− 4
Z Z
The integrands of these “reduced” 2-dimensional integrals in ~q = κ~ κ~ are
t 1 2
−
generally smooth functions of q and corresponding azimuthal angle φ . In
t qt
the following we use two different prescriptions for the factorization scale µ2:
µ2 = p2 with freezing for p2 < µ2,
• t t
µ2 = p2 +µ2.
• t 0
In Eqs.(11), (12) and (13) the longitudinal momentum fractions
p2 +m2
x = t x exp( y) , (15)
1/2
√s ±
p
where m is the effective mass of the parton.
x
The sums in (12) and (13) run over both quarks and antiquarks. The
argument of the running coupling constant Ω2 above was not specified ex-
plicitly yet. In principle, it can be p2 or a combination of p2, κ2 and κ2. In
t t 1 2
the standard transverse momentum representation it is reasonable to assume
Ω2 = min(p2,κ2,κ2) (see e.g. [11]). In the region of very small p usually
t 1 2 t
p2 < κ2,κ2 and Ω = p2 is a good approximation.
t 1 2 2 t
Assuming for simplicity that Ω2 = Ω2(p2) or p2 (function of transverse
t t
momentum squared of the “produced” parton, or simply transverse momen-
tum squared) and taking the following representation of the δ function
1
δ(2)(κ~ +κ~ ~p ) = d2b exp (κ~ +κ~ ~p )~b , (16)
1 2 − t (2π)2 1 2 − t
Z
h i
theformulae(11), (12)and(13)canbewrittenintheequivalent way interms
of parton distributions in the space conjugated to the transverse momentum.
The corresponding formulae read:
6
for diagram A:
dσA 16N 1
= c α (p2)
dyd2p N2 1p2 s t
t c − t
f˜ (x ,b,µ2) f˜ (x ,b,µ2)J (p b) 2πbdb , (17)
g/1 1 g/2 2 0 t
Z
for diagram B :
1
dσB1 16N 4 1
= c α (p2)
dyd2p N2 1 9 p2 s t
t c − (cid:18) (cid:19) t
f˜ (x ,b,µ2) f˜ (x ,b,µ2)J (p b) 2πbdb , (18)
qf/1 1 g/2 2 0 t
f Z
X
for diagram B :
2
dσB2 16N 4 1
= c α (p2)
dyd2p N2 1 9 p2 s t
t c − (cid:18) (cid:19) t
f˜ (x ,b,µ2) f˜ (x ,b,µ2)J (p b) 2πbdb . (19)
g/1 1 qf/2 2 0 t
f Z
X
These are 1-dimensional integrals. The price one has to pay is that now the
integrands are strongly oscillating functions of the impact factor, especially
for large p . The formulae (17), (18) and (19) are very convenient to di-
t
rectly use the solutions of the Kwiecin´ski equations discussed in the previous
section.
When extending running α to the region of small scales we use a param-
s
eter free analytic model from ref.[23].
4 From partons to hadrons
In Ref.[10] it was assumed, based on the concept of local parton-hadron
duality, that the rapidity distribution of particles is identical to the rapidity
distribution of gluons. In the present approachwe followa different approach
which makes use of phenomenological fragmentation functions (FF’s). In the
following we assume θ = θ . This is equivalent to η = η = y , where η
h g h g g h
and η are hadron and gluon pseudorapitity, respectively. Then
g
m
t,h
y = arsinh sinhy , (20)
g h
p
(cid:18) t,h (cid:19)
7
where the transverse mass m = m2 +p2 . In order to introduce phe-
t,h h t,h
nomenological FF’s one has to definqe a new kinematical variable. In accord
with e+e and ep collisions we define a quantity z by the equation E = zE .
− h g
This leads to the relation
p
t,h
p = J(m ,y ) , (21)
t,g t,h h
z
where the jacobian J(m ,y ) reads
t,h h
1/2
m2 −
J(m ,y ) = 1 h . (22)
t,h h − m2t,hcosh2yh!
Now we can write a given-type parton contribution to the single particle
distribution in terms of a parton (gluon, quark, antiquark) distribution as
follows
dσp(η ,p )
h t,h = dy d2p dz D (z,µ2 )
dηhd2pt,h p t,p p→h D
Z Z
zp~ dσ(y ,p )
δ(y η ) δ2 p~ t,p p t,p . (23)
p − h t,h − J · dy d2p
(cid:18) (cid:19) p t,p
Please note that this is not an invariant cross section. The invariant cross
section can be obtained via suitable variable transformation
dσp(yh,pt,h) ∂(yh,pt,h) −1 dσp(yh,pt,h)
= , (24)
dy d2p ∂(η ,p ) dη d2p
h t,h (cid:18) h t,h (cid:19) h t,h
where
m2h+p2t,h +sinh2η +sinhη
1 p2 h h
t,h
y = logr  . (25)
h
2 m2h+p2t,h +sinh2η sinhη
 p2 h − h
r t,h 
 
Making use of the δ function in (23) the inclusive distributions of hadrons
(pions, kaons, etc.) are obtained through a convolution of inclusive distribu-
8
tions of partons and flavour-dependent fragmentation functions
dσ(η ,p ) zmax J2
h t,h
= dz
dη d2p z2
h t,h Zzmin
dσA (y ,p )
Dg→h(z,µ2D) gdg→yggd2pgt,gt,g (cid:12)(cid:12) yg=ηh
(cid:12)pt,g=Jpt,h/z
+ 3 D (z,µ2 )dσqBf1g→qf(yqf,pt,qf)(cid:12)(cid:12)
fX=−3 qf→h D dyqfd2pt,q (cid:12)(cid:12)(cid:12)pt,qy=qJ=pηth,h/z
+ 3 D (z,µ2 )dσgBq2f→qf(yqf,pt,qf) (cid:12)(cid:12) . (26)
fX=−3 qf→h D dyqfd2pt,q (cid:12)(cid:12)(cid:12)pt,qy=qJ=pηth,h/z
(cid:12)
(cid:12)
One dimensional distributions of hadrons can be obtained through the inte-
gration over the other variable. For example the pseudorapidity distribution
is
dσ(η ) dσ(η ,p )
h = d2p h t,h . (27)
dη t,h dη d2p
h h t,h
Z
There are a few sets of fragmentation functions available in the literature
(see e.g. [24], [25]).
5 Results
Asanillustrationoftheformalism, inthepresent paper,weshallshowresults
forenergies adequateforCERNSPSi.e. fortheenergies atwhich themissing
mechanisms should play an important role.
Beforewe adressthedistributions ofpionswewishtodiscuss theinclusive
spectra of “produced” partons.
5.1 Parton distributions
In the familiar collinear approach the contributions of diagrams involving
quarks and antiquarks are not negligible even at large energies. A nice, quan-
titative discussion of this issue can be found in [3]. In this section we shall
make a similar analysis in our approach based on CCFM uPDF’s. In Fig.2
we display the contributions of partons (gluons, quarks, and antiquarks) for
all diagramsof Fig.1for the center-of-mass energy W = 17.3GeV, i.e. energy
of recent experiments of the NA49 collaboration at CERN as a function of
9
jet (minijet) x . The parton rapidity is related to the parton x as follows
F F
m2 +x2 s +p2 +x √s
1 x F4 t F 2
y = log , (28)
2 qm2 +x2 s +p2 x √s 
x F4 t − F 2
q 
where the parton mass here is the same as in Eq.(15). The corresponding
cross section is obtained by integration over parton transverse momenta in
the interval 0.2 GeV < p < 4 GeV. The gg g contribution, claimed
t
→
to be the dominant contribution at RHIC [10], is somewhat smaller than
the contribution of diagrams B and B . The contribution of diagram B
1 2 1
(dashed line) dominates at negative Feynman-x , while the contribution of
F
diagram B (dotted line) at positive Feynman-x . By symmetry require-
2 F
ments dσB2/dx (x ) = dσB1/dx ( x ).
F F F F
−
In order to understand the intriguing asymmetry in x of contributions
F
of diagrams B and B , in Fig.3 we present a further decomposition of the B
1 2 1
contributionintosea-glue,valence-gluesubcontributionsandcorrespondingly
of the B contribution into glue-sea, glue-valence subcontributions. This de-
2
composition shows that the sea-glue and glue-sea contributions are of similar
size as the valence-glue and glue-valence, respectively. It is interesting to
note a different x -asymmetry of the contributions involving sea and valence
F
quarks. This decomposition explains the shift of maxima of contributions
corresponding to diagram B and B seen in Fig.2.
1 2
For completeness in Fig.4 we show transverse momentum distribution of
“produced” gluons corresponding to diagramAand of quarks and antiquarks
corresponding to diagrams B and B . In this calculation we have integrated
1 2
over parton x (or parton rapidity). For the two contributions one obtains a
F
rather similar functional behaviour. It is worth noticing that in contrast to
the standard collinear case in our approach the partonic cross section is fully
integrable. This seems promissing in extending the region of applicability of
the “perturbative” 2 QCD towards smaller transverse momenta of hadrons.
The rise of the cross section above p 0.5 GeV is due to a rapid increase
t
≈
ofgluonicradiationabovep = µ ,whereµ2 istheminimalfactorizationscale
t 0 0
for the GRV parton distributions. On the other hand, the rise of the cross
section towards p < µ is an artifact of freezing factorization scale below µ2
t 0 0
in the conjuction of the singular behaviour of the denominator in the parton
cross section formulae. To elucidate this issue somewhat better in Fig.5 we
present also the results with the second prescription for the factorization
2
Our approach is not completely perturbative. Some nonperturbative effects are con-
tained inthe nonperturbativeformfactor,wayof freezingαs or the scale ofpartondistri-
butions.
10