Saturation of gluon density and soft p-p collisions at LHC

0.25 MB·English

by  G. I. Lykasov

Saturation of gluon density and soft pp collisions at LHC
G. I. Lykasov, A. A.Grinyuk and V. A. Bednyakov
Joint Institute for Nuclear Research - Dubna 141980, Moscow region, Russia
Abstract
3
1 Wecalculatetheunintegratedgluondistributionatlowintrinsictransversemomenta
0
2 and its parameters are found from the best description of the SPS and LHC data on
the pp collision in the soft kinematical region. It allows us to study the saturation
n
a of the gluon density at low Q2 more carefully and find the saturation scale.
J

2
2

] 1. Introduction
h
p
-
As is well known, hard processes involving incoming protons, such as deep-inelastic
p
e lepton-proton scattering (DIS), are described using the scale-dependent parton den-
h
[ sity functions. Usually, these quantities are calculated as a function of the Bjorken

variable x and the square of the four-momentum transfer q2 = Q2 within the
1
−
v framework of popular collinear QCD factorization based on the DGLAP evolution
6
equations [1]. However, for semi-inclusive processes (such as inclusive jet production
5
1 in DIS, electroweak boson production [2, 3], etc.) at high energies it is more appro-
5 priate to use the parton distributions unintegrated over the transverse momentum
.
1 k in the framework of k -factorization QCD approach [4], see, for example, reviews
t t
0
[5, 6] for more information. The k -factorization formalism is based on the BFKL
3 t
1 [7] or CCFM [8] evolution equations and provides solid theoretical grounds for the
:
v effects of initial gluon radiation and intrinsic parton transverse momentum k . The
t
i
X theoreticalanalysisoftheunintegratedquarkq(x,k )distribution(u.q.d.) andgluon
t
r g(x,k ) distribution (u.g.d.) can be found, for example, in [9]-[13]. In this paper we
t
a
estimate the u.g.d. at low intrinsic transverse momenta k 1.5 1.6 GeV/c and its
t
≤ −
parameters extracted from the best description of the LHC data at low transverse
momenta p of the produced hadrons. We also show that our u.g.d. similar to the
t
u.g.d. obtained in [9, 10] at large k and different from it at low k . The u.g.d. is
t t
directly related to the dipole-nucleon cross section within the model proposed in [9],
see also [13]-[18], that is saturated at low Q or large transverse distances r 1/Q
∼
between quark q and antiquark q¯ in the qq¯ dipole created from the splitting of the
∗
virtual photon γ in the ep DIS. So, we find also a new parameterization for this
dipole-nucleon cross section, as a function of r, from the modified u.g.d. and analyze
the saturation effect for the gluon density.
1
2. Inclusive spectra of hadrons in pp collisions
2..1 Unintegrated gluon distributions
As was mentioned above, the unintegrated gluon density in a proton are a subject of
intensive studies, and various approaches to investigate these quantities have been
proposed [7]. At asymptotically large energies (or very small x) the theoretically
correct description is given by the BFKL evolution equation [7] where the leading
ln(1/x) contributions are taken into account in all orders. Another approach, valid
for both small and large x, is given by the CCFM gluon evolution equation [8].
It introduces angular ordering of emissions to correctly treat the gluon coherence
effects. In the limit of asymptotic high energies, it almost equivalent to BFKL [7],
but also similar to the DGLAPevolution for largex 1. The resulting unintegrated
∼
gluon distribution depends on two scales, the additional scale q¯is a variable related
to the maximum angle allowed in the emission and plays the role of the evolution
scale µ in the collinear parton densities. Also it is possible to obtain the two-scale
involvedunintegratedquarkandgluondensitiesfromtheconventional onesusingthe
Kimber-Martin-Ryskin (KMR) prescription [5, 6]. In this way the k dependence
T
in the unintegrated parton distributions enters only in last step of the evolution,
and usual DGLAP evolution equations can be used up to this step. Such procedure
is expected to include the main part of the collinear higher-order QCD corrections.
Finally, a simple parameterization of the unintegrated gluon density was obtained
within the color-dipole approach in [9] on the assumption of a saturation of the
gluon density at low Q2 which successfully described both inclusive and diffracting
EPA scattering. This gluon density xg(x,k2,Q2) is given by [9, 10]
t 0
3σ 1 x λ/2
xg(x,k ,Q ) = 0 R2k2exp R2(x)k2 ; R = , (1)
t 0 4π2α (Q ) 0 t − 0 t 0 Q x
s 0 0 (cid:18) 0(cid:19)
(cid:16) (cid:17)
where σ = 29.12 mb, α = 0.2, Q = 1 GeV, λ = 0.277 and x = 4.1 10−5. This
0 s 0 0
·
simple expression corresponds to the Gaussian form for the effective dipole cross
section σˆ(x,r) as a function of x and the relative transverse separation r of the
qq¯ pair [9]. In fact, this form could be more complicated. In this paper we study
this point and try to find the parameterization for xg(x,k ,Q ), which is related to
t 0
σˆ(x,r), fromthebest description oftheinclusive spectra ofchargehadronsproduced
in pp collisions at LHC energies and mid-rapidity region.
2..2 Quark-gluon string model (QGSM) including gluons
As is well known, the soft hadron production in pp collisions at not large trans-
fer can be analyzed within the soft QCD models, namely, the quark-gluon string
model (QGSM) [19]-[22] or the dual parton model (DPM) [23]. The cut n-pomeron
graphs calculated within these models result in a reasonable contribution at small
but nonzero rapidities. However, it has been shown recently [24, 25] that there are
2
some difficulties in using the QGSM to analyze the inclusive spectra in pp collisions
in the mid-rapidity region and at the initial energies above the ISR one. However,
it is due to the according to Abramovsky-Gribov-Kancheli cutting rules (AGK) [26]
at mid-rapidity (y 0), when only one-pomeron Mueller-Kancheli diagrams con-
≃
tribute to the inclusive spectrum ρ (y 0,p ). To overcome these difficulties it
h t
≃
was assumed in [24] that there are soft gluons or the so called intrinsic gluons in
the proton [27], which are split into qq¯pairs and should vanish at the zero intrinsic
transverse momentum (k 0). The total spectrum ρ (y 0,p ) was split into
t h t
∼ ≃
two parts, the quark contribution ρ (y 0,p ) and the gluon one and their energy
q t
≃
dependence was calculated [24, 25]
ρ(p ) = ρ (x = 0,p )+ρ (x = 0,p ) = g(s/s )∆φ¯ (0,p )+ g(s/s∆ σ φ¯ (0,p ) . (2)
t q t g t 0 q t 0 − nd g t
(cid:16) (cid:17)
Here
ρ (x = 0,p ) = g(s/s )∆φ¯ (0,p ) (3)
q t 0 q t
and
ρ (x = 0,p ) = g(s/s∆ σ φ¯ (0,p ) (4)
g t 0 − nd g t
(cid:16) (cid:17)
˜ ˜
The following parameterization for φ (0,p ) and φ (0,p ) was found [24]:
q t g t
˜
φ (0,p ) = A exp( b p )
q t q q t
−
˜
φ (0,p ) = A √p exp( b p ), (5)
g t g t g t
−
where s = 1GeV2,g = 21mb,∆ = 0.12. The parameters are fixed from the fit to
0
the data on the p distribution of charged particles at y = 0 [24]: A = 4.78 0.16
t q
(GeV/c)−2, b = 7.24 0.11 (Gev/c)−1 and A = 1.42 0.05 (GeV/c)−2;±b =
q g g
3.46 0.02 (GeV/c)−1.±Figure 1 illustrates the best fit of±the inclusive spectrum of
±
charged hadrons produced in pp collisions at √s =7 TeV and the central rapidity
region at the hadron transverse momenta p 1.6 Gev/c; the solid line corresponds
t
≤
to the quark contribution ρ , the dashed line is the gluon contribution ρ , and the
q g
dotted curve is the sum of these contributions ρ given by Eq.(2).
h
In Figs. (2,3) the inclusive spectra of the charged hadrons produced in pp colli-
sions at the mid-rapidity region and √s =2.36 GeV, 7 Tev, 500 GeV, 900 GeV are
presented with the inclusion of the calculations within the perturbative LO QCD,
see details in [25]. One can see from Figs. (1-3) that the calculations within the soft
QCD including both the quark contribution (Eq.(3)) and the gluon one (Eq.(4))
and the contribution, which corresponds to the LO QCD calculation, results in the
satisfactorily description of these spectra in the wight region of the initial energies.
¯
It stimulates us to study the form of the gluon contribution φ (x 0,p ) related to
g ht
ρ (x 0,p ) in detail and find the information on the distribution of the soft gluons
g t
at small transverse momenta.
3
Figure 1: The inclusive spectrum of the charged hadrons as a function of p (GeV/c)
t
in the central rapidity region (y = 0) at √s =7 TeV at p 1.6 GeV/c compared
t
≤
with the CMS [28] which are very close to the ATLAS data [29]. The solid line is
the quark contribution ρ (x = 0,p ) (Eq.(3), the long dashed curve corresponds to
q t
the gluon one ρ (x = 0,p ) (Eq.(4), the dotted line is the sum of the quark and
g t
gluon contributions (Eq.(2).
2..3 Modified unintegrated gluon distributions
As can be seen in Figs. (1-3) the contribution to the inclusive spectrum at y 0 due
≃
to the intrinsic gluons is sizable at low p < 2 GeV/c, e.g., in the soft kinematical re-
t
gion. Therefore, we can estimate this contribution within the nonperturbative QCD
˜
model, similar to the QGSM [19]. We calculate the gluon contribution φ (x 0,p )
g t
≃
entering into Eq.(5) as the cut graph (Fig. 4, right) of the one-pomeron exchange
in the gluon-gluon interaction (Fig. 4, left) using the splitting of the gluons into the
qq¯ pair. The right diagram of Fig. 4 corresponds to the creation of two colorless
strings between the quark/antiquark (q/q¯) and antiquark/quark (q¯/q). Then, after
their brake qq¯are produced and fragmented to the hadron h. Actually, the calcula-
tion can be made in a way similar to the calculation of the sea quark contribution
˜
to the inclusive spectrum within the QGSM [19], e.g., the contribution φ (0,p ) is
g t
presented as the sum of the product of two convolution functions
˜
φg(x,pt) = Fq(x+,pht)Fq¯(x−,pht)+Fq¯(x+,pht)Fq(x−,pht) , (6)
4
s1/2 = 2.36 TeV s1/2 = 7 TeV
100 100
soft QCD (quarks) soft QCD (quarks)
soft QCD (gluons) soft QCD (gluons)
10 Perturbative QCD 10 Perturbative QCD
2c] SQCD (quarks + gluons) 2c] SQCD (quarks + gluons)
-2V* 1 SEQxpC. Dda +ta P (QCCMDS) -2V* 1 SEQxpC. Dda +ta P (QCCMDS, ATLAS)
e e
G G
0 [ 0.1 0 [ 0.1
= =
y y
2, 0.01 2, 0.01
Pt Pt
d d
dy/ 0.001 dy/ 0.001
N/ N/
d d
]ev 0.0001 ]ev 0.0001
N N
1/ 1/
[ 1e-05 [ 1e-05
1e-06 1e-06
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
p, [GeV/c] p, [GeV/c]
t t
Figure 2: The inclusive spectrum of charged hadron as a function of p (GeV/c)
t
in the central rapidity region (y = 0) at √s =2.36 TeV (left) and √s =7 TeV
(right) compared with the CMS [28] and ATLAS [29] data. The solid line (soft
QCD (quarks)) is the quark contribution ρ (x = 0,p ) (Eq.(3)), the dotted curve
q t
(soft QCD (gluons)) corresponds to the gluon one ρ (x = 0,p ) (Eq.(4), the long
g t
dashedline(SQCD(quarks+gluons)) isthesumofthequarkandgluoncontributions
(Eq.(2), theshort dashed curve (Perturbative QCD) corresponds tothe perturbative
LOQCD[25]andthedash-dottedline(SQCD+PQCD)isthesumofthecalculations
within the soft QCD including the gluon contribution (Eq.(2)) and the perturbative
LO QCD.
where the function F (x ,p ) corresponds to the production of final hadrons from
q(q¯) + ht
decay of qq¯string. It is calculated as the following convolution:
1 x±
Fq(q¯)(x±,pt;pht) = dx1 d2k1tfq(q¯)(x1,k1t)Gq(q¯)→h ,pht kt) , (7)
Zx± Z (cid:18)x1 − (cid:19)
˜ ˜ ˜
Here G (z,k ) = zD (z,k ), D (z,k ) is the fragmentation function
q(q¯)→h t q(q¯)→h t q(q¯)→h t
˜
(FF) of the quark (antiquark) to a hadron h, z = x±/x1,kt = pht kt, x± =
−
0.5( x2 +x2 x),x = 2 (m2 +p2)/s. The distribution of sea quarks (antiquark)
t ± t h t
fq(q¯)qis related to the splitqting function g→qq¯ of gluons to qq¯by
P
1 z dz
1
fq(q¯)(z,kt) = g(z1,kt,Q0) g→qq¯( ) , (8)
P z z
Zz 1 1
where g(z1,k1t,Q0) is the u.g.d.. The gluon splitting function g→qq¯ was calculated
P
within the Born approximation. In Eq.(8) we assumed the collinear splitting of the
intrinsic gluon to the qq¯pair because values of k are not zero but small.
t
Calculating the diagram of Fig. (4) (right) by the use of Eqs.(4-8) for the gluon
contribution ρ we took the FF to charged hadrons, pions, kaons, and pp¯ pairs
g
5
s1/2 = 540 GeV s1/2 = 900 GeV
100 100
soft QCD (quarks) soft QCD (quarks)
soft QCD (gluons) soft QCD (gluons)
10 Perturbative QCD 10 Perturbative QCD
2c] SQCD (quarks + gluons) 2c] SQCD (quarks + gluons)
-2V* 1 SEQxpC. Dda +ta P (QUCAD1) -2V* 1 SEQxpC. Dda +ta P (QCCMDS)
e e
G G
0 [ 0.1 0 [ 0.1
= =
y y
2, 0.01 2, 0.01
Pt Pt
d d
dy/ 0.001 dy/ 0.001
N/ N/
d d
]ev 0.0001 ]ev 0.0001
N N
1/ 1/
[ 1e-05 [ 1e-05
1e-06 1e-06
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
p, [GeV/c] p, [GeV/c]
t t
Figure 3: The inclusive spectrum of charged hadron as a function of p (GeV/c)
t
in the central rapidity region (y = 0) at √s =540 GeV (left) and √s =900 GeV
(right) compared with the UA1 [30] and ATLAS [29] data. The solid line (soft QCD
(quarks)) isthequarkcontribution ρ (x = 0,p )(Eq.(3), thedottedcurve (softQCD
q t
(gluons)) corresponds to the gluon one ρ (x = 0,p ) (Eq.(4), the long dashed line
g t
(SQCD(quarks+gluons)) is the sum of the quark and gluon contributions (Eq.(2),
the short dashed curve (Perturbative QCD) corresponds to the perturbative LO
QCD [25] and the dash-dotted line (SQCD+PQCD) is the sum of the calculations
within the soft QCD including the gluon contribution (Eq.(2)) and the perturbative
LO QCD.
Figure 4: The one-pomeron exchange graph between two gluons in the elastic pp
scattering (left) and the cut one-pomeron due to the creation of two colorless strings
between quarks /antiquarks (right) [19].
obtained within the QGSM [31]. From the best description of ρ (x 0,p , see its
g ht
≃
parameterization given by Eq.(5), we found the form for the xg(x,k ,Q ) which was
t 0
fitted in the following form:
3σ
xg(x,k ,Q ) = 0 C (1 x)bg
t 0 4π2α (Q ) 1 − ×
s 0
6
R2(x)k2 +C (R (x)k )a exp R (x)k d(R (x)k )3 , (9)
0 t 2 0 t − 0 t − 0 t
(cid:16) (cid:17) (cid:16) (cid:17)
The coefficient C was found from the following normalization:
1
Q20
g(x,Q2) = dk2g(x,k2,Q2) , (10)
0 t t 0
Z0
and the parameters
a = 0.7;C 2.3;λ = 0.22;b = 12;d = 0.2;C = 0.3295
2 g 3
≃
were found from the best fit of the LHC and SPS data on the inclusive spectrum of
charged hadrons produced in pp collisions and in the mid-rapidity region at p 1.6
t
≤
GeV/c, see the dashed lines (SQCD (quark+gluons)) in Figs. (1-3) and Eq.(5).
(cid:1)
M(cid:21)(cid:22)(cid:23)(cid:24)(cid:23)(cid:16)(cid:22)(cid:25)(cid:0)(cid:26)(cid:0)(cid:22)(cid:0)
O(cid:27)(cid:23)(cid:26)(cid:23)(cid:28)(cid:29)(cid:30)(cid:15)(cid:31)!(cid:25)(cid:0)(cid:26)(cid:0)(cid:22)(cid:0)
0
x (cid:0)(cid:1)
(cid:11)
x
(cid:7)0
)(cid:9)(cid:10)0
(cid:7),t
(cid:7)(cid:8)
(cid:6)x
x(cid:5) (cid:0) (cid:1)
(cid:0) (cid:1)
(cid:0)(cid:2) (cid:1) (cid:1)(cid:0)(cid:2) (cid:3) (cid:3)(cid:0)(cid:2) (cid:4)
k(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)
Figure 5: The unintegrated gluon distribution xg(x,k ,Q )/C as a function of k
t 0 0 t
at x = x and Q = 1.GeV/c. The dashed curve corresponds to the original GBW
0 0
[9, 10], Eq.(1), and the solid line is the modified u.g.d. given by Eq.(9).
Figure 5 presents the modified u.g.d. obtained by calculating the cut one-
pomeron graph of Fig. 4 and the original GBW u.g.d. [9, 10] as a function of
the transverse gluon momentum k . Here C = 3σ /(4π2α (Q )). One can see that
t 0 0 s 0
the modified u.g.d. (the solid line in Fig. 5) is different from the original GBW
u.g.d. [9, 10] at k < 1.5 GeV/c and coincides with it at larger k . This is due to
t t
the sizable contribution of ρ (Eqs.(4,5)) to the inclusive spectrum ρ(p ) of charged
g t
hadrons produced in pp collisions at the LHC and SPS energies in the mid-rapidity
region, see the dashed lines (soft QCD(gluons)) in Figs. (1-3).
Let us also note that, as is shown recently in [32], the modified GBW given
by Eq.(9) does not contradict the HERA data on the longitudinal structure func-
tion F (Q2) at low x, the charm structure function F (x,Q2) and the bottom one
l 2c
F (x,Q2).
2b
7
3. Saturation dynamics
According to[9], the u.g.d. canberelated tothecross section σˆ(x,r) oftheqq¯dipole
with the nucleon. This dipole is created from the split of the virtual exchanged
∗
photon γ to qq¯pair in the e p deep inelastic scattering (DIS).This relation at the
−
fixed Q2 is the following: [9]:
0
4πα (Q2) d2k
σˆ(x,r) = s 0 t 1 J (rk ) xg(x,k ) (11)
3 k2 { − 0 t } t
Z t
Inserting the simple form for xg(x,k ) given by Eq.(1) to Eq.(11) on can get the
t
following form for the dipole cross section:
r2
σˆ (x,r) = σ 1 exp (12)
GBW 0( − −4R02(x)!)
However, the modified u.g.d. given by Eq.(9), inserted to Eq.(11) results inthe more
complicated form for σˆ(x,r):
b r b r2
1 2
σˆ (x,r) = σ 1 exp , (13)
Modif 0( − −R0(x) − R02(x)!)
where b = 0.045,b = 0.3.
1 2
There are a few forms for the dipole cross sections suggested in [13]-[18]. The
dipole cross section can be presented in the general form: [9]:
σˆ(x,r) = σ g(rˆ2) , (14)
0
where rˆ= r/(2R (x)). The function g(rˆ2) was presented in the form [13]
0
1
g(rˆ2) = rˆ2ln 1+ (15)
rˆ2
(cid:18) (cid:19)
or in the form [17]:
1
g(rˆ2) = 1 exp rˆ2ln +e (16)
− − Λr
(cid:26) (cid:18) (cid:19)(cid:27)
which both of them are saturated when r grows. The function g(rˆ2) was also pre-
sented in the form of type [14, 15]:
g(rˆ2) = ln(1+rˆ2) (17)
thatis notsaturatedwhen r increases. Figure(6)illustrates thedipolecross sections
σˆ/σ at x = x which are saturated at r > 0.6 fm, see [13, 16, 17]. They are
0 0
compared with our calculations (solid line, Modified σ) given by Eq.(13). The solid
line in Fig. 6 corresponds to the modified u.g.d. given by Eq.(9) the application
8
Modified σ
1.2 GBW σ
Nikolaev, Zakharov σ
Albacete, Marquet σ
1
0.8
0
σ
)/0
x
σ(r, 0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
r, [fm]
Figure 6: The dipole cross section σˆ/σ at x = x as a function of r. The green
0 0
line “GBW σ” is calculation of [9], the red line “Modified σ” corresponds to our
calculation; the curve ”Nikolaev,Zakharov σ” is the calculation of [13]; the line
”Albacete,Marquet σ” is the calculation of [17].
of that allowed us to describe satisfactorily the LHC data on inclusive spectra of
hadrons produced in the mid-rapidity region of pp collision at low p . Therefore, the
t
form of the dipole-nucleon cross sections presented in Fig. 6 can be verified by the
last LHC data on hadron spectra in soft kinematical region.
Comparing the solid line (Modified σ) and dashed curve (GBW σ) in Fig. 6 one
can see that σˆ (x,r) given by Eq.(13) saturates faster than σˆ (x,r) given by
Modif GBW
Eq.(12) when the qq¯dipole distance r increases. If R = 1/GeV(x/x )λ/2, according
0 0
to [9], then for saturation scale has the form Q 1/R = Q (x /x)λ/2, where
s 0 s0 0
Q = 1 GeV=0.2 fm−1. The saturation in the dipo∼le cross section (Eq.(12) sets in
s0
when r 2R or Q (Q /2)(x /x)λ/2). Comparing the saturation properties of
0 s s0 0
∼ ∼
the Modified σ and GBW σ presented in Fig. 6 one can get slightly larger value for
Q in comparison to Q = 1 GeV/c.
s0 s0
4. Conclusion
Assumingcreationofsoftintrinsicgluonsintheprotonatlowtransversemomentak
t
and calculating the cut one-pomeron graph between two gluons in colliding protons
we found the form for the unintegrated gluon distribution (modified u.g.d) as a
function of x and k at the fixed value of Q2. The parameters of this u.g.d. were
t 0
found from the best description of the SPS and LHC data on the inclusive spectra
9
of the charged hadrons produced in the mid-rapidity pp collisions at low p . It was
t
shown that the modified u.g.d. is different from the original GBW u.g.d. obtained
in [9, 10] at k 1.6 GeV/c and it coincides with the GBW u.g.d. at k > 1.6GeV/c.
t t
≤
Usingthemodifiedu.g.d. wecalculatedtheqq¯dipole-nucleoncrosssectionσˆ
Modif
as a function of the transverse distance r between q and q¯ in the dipole and found
that it saturates faster than the σˆ obtained within the GBW dipole model [9].
GBW
It allowed us to find the saturation scale Q for the gluon density that is larger than
s
the one obtained in [9]. Moreover, we showed that the satisfactory description of
the LHC data using the modified u.g.d. can verify the form of the dipole-nucleon
cross section and the property of the saturation of the gluon density at low Q2.
Acknowledgments
The authors are grateful to S.P. Baranov, B.I. Ermolaev, H. Jung, A.V.Kotikov,
A.V.Lipatov, M.G. Ryskin, Yu.M. Shabelskiy and N.P.Zotov for useful discussions
and comments. This research was supported by the RFBR grant 11-02-01538-a.
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