Download Initial State Parton Evolution beyond the Leading Logarithmic Order

963
Progress of Theoretical Physics, Vol. 110, No. 5, November 2003
Initial State Parton Evolution beyond the Leading Logarithmic
Order of QCD
Hidekazu Tanaka
Department of Physics, Rikkyo University, Tokyo 171-0021, Japan
(Received July 29, 2003)
A Monte-Carlo model based on the evolution of momentum distributions proposed in
a previous work is extended to the next-to-leading logarithmic (NLL) order of quantum
chromodynamics (QCD). Scaling violation of initial state parton distributions are generated
using parton showers to accuracy of the NLL order by using the information from only
splitting functions and initial parton distributions at some fixed low energy.
Interference contributions due to the NLL order terms are taken into account for the
generation of the transverse momenta in initial state parton radiations.
§1.
Introduction
Lepton-hadron and hadron-hadron collisions are significantly affected by initial
state parton radiations. However, the leading-logarithmic (LL) order of QCD is insufficient to evaluate these effects. Therefore, the next-to-leading logarithmic (NLL)
order contributions should be taken into account. However, the inclusion of the
NLL order terms into Monte-Carlo simulations of the initial state radiations is not
a straightforward task.
The scaling violation of parton distributions inside hadrons can be understood
as the sequential evolution of partons in the initial state. Conventionally, the scaling
violation of the parton distributions is calculated by solving the renormalization
group equations in moments. Then these solutions are numerically inverted to yield
momentum fractions of partons.1), 2) Alternatively, one can use parton shower models
in order to evaluate the scaling violation of the parton distributions.3), 4)
In a previous work,4) two types of algorithms that can reproduce the scaling
violation for flavor singlet partons were compared to the LL order of QCD. One
algorithm is a model based on the evolution of the particle number distributions
(Model I), which is conventionally used in various event generators.5) The other
algorithm is a model based on the evolution of momentum distributions (Model II).
In both models, the scaling violation of the parton distributions are generated by
using only information from splitting functions of the parton branching vertices and
input distributions at a given energy Q0 . It has been found that both methods
reproduce the scaling violation of the flavor singlet parton distributions up to their
normalizations.
For Model I, a non-trivial weight factor must be introduced in order to reproduce
the energy scale dependence of the total number of partons. The results thereby
obtained depend on a cutoff parameter for small momentum fraction, even in the
LL order approximation.
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H. Tanaka
By contrast, in Model II, conservation of the total momentum of the initial state
partons is guaranteed. Furthermore, for this model, it is not necessary to introduce
non-trivial weight factors in order to reproduce the energy scale dependence of the
momentum distribution of partons in the initial state. These are advantages of this
type of model for the extension of the parton shower model to the NLL order of
QCD.
The purpose of this paper is to extend Model II studied in the previous paper to
the NLL order of QCD. In §2, the algorithm proposed in Ref. 4) is briefly explained,
including the NLL order terms. An algorithm for the generation of transverse momenta of the initial state partons is presented in §3, where interference contributions
due to the NLL order terms are taken into account. In §4, some numerical results
for the distribution functions obtained using the proposed model are presented, and
these results are compared with distribution functions obtained using the renormalization group equations. The transverse momentum distributions for the generated
partons are also presented. Section 4 contains a summary and some comments.
§2.
Generation of momentum distributions
In this section, the parton shower model based on the evolution of the momentum
distributions is briefly explained.4) In this model, the parton evolutions develop
according to non-branching probabilities determined by real emission terms (without
regularizing terms for the soft gluon radiations) of the splitting functions multiplied
by z [i.e., zPij (αs , z)], where z and αs represent the momentum fraction of the
daughter parton and the running coupling constant of QCD, including the NLL
order terms. Here i and j indicate the types of partons (q for a flavor singlet quark
and g for a gluon).
Including the NLL order terms, Pij (αs , z) is written
α 2
αs (0)
s
(1)
Pij (z),
(2.1)
Pij (αs , z) =
Pij (z) +
2π
2π
(0)
(1)
where Pij (z) and Pij (z) are the splitting functions of LL order and of NLL order
of QCD, respectively.6)
For the evolution of the momentum distributions, the non-branching probability
is defined by
2
K2
dK 2 1−ε
(i)
2
2
dzzPi (αs , z) ,
(2.2)
ΠN B (K2 , K1 ) = exp −
2
K12 K
0
for i = q, g. Here, the partons inside an initial state hadron have space-like virtualities, ki2 ≡ −Ki2 < 0, and ε denotes the resolution of the momentum fraction of the
final state partons. The quantity Pi (αs , z) is a combination of the splitting functions
Pij (αs , z) and is defined by
Pq (αs , z) = Pqq (αs , z) + Pgq (αs , z),
Pg (αs , z) = 2Nf Pqg (αs , z) + Pgg (αs , z),
(2.3)
Initial State Parton Evolution beyond the Leading Logarithmic Order
965
where Nf denotes the number of flavors.
Because the second moment of the function Pi (αs , z) with infrared regularizing
terms at z = 1, denoted by [ ]+ , vanishes due to the momentum conservation of the
parton momentum fractions,6) i.e.,
1
dzz[Pi (αs , z)]+ = 0,
(2.4)
0
the momentum distributions are reproduced up to their normalizations without the
introduction of any non-trivial weight factor. In this model, the total momentum of
the initial state hadron,
1
dxx[Σ(x, K 2 ) + G(x, K 2 )] = 1,
(2.5)
0
K2
is conserved, where x is the momentum fraction of a parton inside the
for any
hadron. Here Σ(x, K 2 ) and G(x, K 2 ) are the particle number distribution function
of the flavor singlet quarks and that of the gluons inside the hadron, respectively.
Therefore, the model guarantees the total normalization of the momentum distributions for the flavor singlet partons.
In actual computation for event generation, a cutoff parameter δ for small z is
introduced in order to avoid the situation in which the generated random numbers
take values close to z = 0, because the NLL order terms of the splitting functions
behave as zP (1) (z) ∼ (log(z))2 at small z. However, the integrated contributions
below δ ( 1) are at most O(δlog2 δ). Therefore, the cutoff dependence is negligible
for small δ.
The particle number distributions can be obtained from the momentum distributions by multiplying by a factor of 1/xF for each event, where xF is the momentum
fraction of a parton that finished the evolution.
Using the algorithm presented above, one can easily obtain the momentum distribution for each flavor quark by selecting the parton flavor in each branching step.
The actual steps involved in the generation of the momentum fraction xF using the
Monte-Carlo method are similar to those for the LL order case presented in Ref. 4).
§3.
Generation of transverse momenta
The algorithm for the generation of transverse momenta is similar to that for
the Monte-Carlo model presented in Ref. 4) to the LL order of QCD, except that
the effects due to the three-body decay functions are taken into account for each
branching step. Various processes contribute to the initial state parton radiations.7)
The important contributions are q(S) → q(S) + X and g(S) → g(S) + X, where
S represents a parton with space-like virtuality. Particularly, two-gluon radiation,
such as
a(p) → g(k1 ) + g(k2 ) + a(k3 ),
(3.1)
becomes large in the soft gluon region. Here, a represents a quark (a = q) or a gluon
(a = g) with space-like virtuality. The momenta of these partons are denoted by p,
k1 , k2 and k3 , respectively.
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H. Tanaka
In the calculation of the three-body decay functions, the parton momenta are
set as p2 = k12 = k22 = 0 and k32 = s < 0, because the collinear contributions for
−p2 , k12 , k22 −s are extracted in the jet calculus.7)
The three-body decay functions are coefficients of the 1/(−s) contribution
(collinear contribution) for the branching vertex, which is given by
V (3) =
α 2
s
2π
δ(1 − x1 − x2 − x3 )dx1 dx2 dx3
D
J [j]
j=A
d(−s)
.
−s
(3.2)
The momentum fraction is defined by
xi =
ki n
,
pn
(3.3)
where n is a light-like vector that specifies the light-cone gauge. The quantity J [j]
in Eq. (3·2) is written
J
[j]
=
(−s)
M02
[j]
L log
dKj2
Kj2
+ L[j] logW [j] + N [j]
(3.4)
for j = A, B1, B2 and
J [j] = L[j] logW [j] + N [j]
(3.5)
for j = C1, C2, D. Here j = A − D indicate the types of squared matrix elements
2 = s , K 2 = −s
defined by the structures of the propagators.7) Furthermore, KA
12
23
B1
2
2
and KB2 = −s13 , respectively, with sij = (ki + kj ) .
In Eq. (3·4), M0 is the minimum mass scale of the phase space integration. In
the Monte-Carlo calculation, M0 corresponds to the absolute value of the virtuality
for the parent parton. The explicit expressions of L, N and W in the light-cone
gauge are presented in Ref. 7). As shown there, the functions L[j] for j = A, B1, B2
are the convolutions of the splitting functions of the two-body branching vertices at
the LL order of QCD. The first term of Eq. (3·4) is regarded as the O(αs2 ) term of
the LL order contribution, which should be subtracted form V (3) in Eq. (3·2). The
interference terms (types [C] and [D]) are free from the mass singularity for fixed s.
Therefore a log(−s/M02 ) term does not appear in Eq. (3·5).
As a result of the subtraction of the LL order terms from V (3) , there is some freedom in defining the NLL order terms. The three-body decay functions are modified
as
(−s)fA
(−s)fB1
(−s)fB2
2
2
2
[A] dKA
[B1] dKB1
[B2] dKB2
.
L
−
L
−
L
J M = J0 −
2
2
2 , (3 6)
KA
KB1
KB2
M02
M02
M02
D
[j]
where J0 =
j=A J . Here fB1 , fB2 and fA are the functions that depend on
xi . The subtracted NLL contributions are included in the kinematical constraints
2 < (−s)f , K 2 < (−s)f
2
of the branching vertices as KA
A
B1 and KB2 < (−s)fB2 ,
B1
respectively.
Initial State Parton Evolution beyond the Leading Logarithmic Order
967
A simple choice for the NLL order terms is that in which the virtualities of
2 , K 2 < (−s), and therefore
the initial state partons are strongly ordered, as KB1
B2
fB1 = fB2 = 1. Furthermore, one can stipulate that all the NLL contributions are
included in the phase space restriction for the radiated virtual gluon with momentum
2 (= s )
k1 + k2 in a process of type [A] (JM = 0). The kinematic boundary for KA
12
[A]
˜
is given by fA = exp(J0 /L ), with
J˜0 = J0 − (L[A] + L[B1] + L[B2] )ln(−s/M02 ),
(3.7)
where J˜0 is independent of −s and M02 . The asymptotic form for small x1 (soft gluon
radiation) with the above choice is fA x1 for x1 x2 , x3 , which corresponds to
the angular ordering condition θk1 k2 < θpk2 7), 8) for the small angle approximation,
that is, the case θk1 k2 , θpk2 1.
In the case considered above, the two-body branching process is allowed only
in the branching steps of the initial state parton radiations, as is usually the case
in LL order parton shower models, except that all of the contributions due to the
NLL order terms are included in the kinematic boundary for the two-body decays.
However, the upper limit of the virtuality for the radiated parton is not properly
determined at the LL order.
In the numerical results presented in the next section, the transverse momenta
of partons were generated using the two-body branching restricted by the boundary
condition due to JM = 0.
§4.
Numerical results
In this section, some results obtained using the algorithm explained in the previous sections are presented. In Fig. 1, the momentum distributions for the gluons and
for the singlet quarks obtained with the Monte-Carlo simulations are compared with
those obtained using GRV(98).2) The distribution functions obtained with GRV(98)
contain Q2 evolution calculated using the renormalization group equations, and these
effects are fully parameterized in the NLL order approximation of QCD. In the
Monte-Carlo calculation, the parton showers start from the momentum distributions at Q20 = 5 GeV2 , denoted by the dashed curve for the gluons and by the dotted
curve for the singlet quarks with fixed resolution ε = 10−6 . These may correspond to
the solutions of the renormalization group equations integrated over the final state
parton momenta. Here, the results obtained from the Monte-Carlo calculation are
plotted as dN/dyF , with yF = −lnxF , where dN denotes the averaged number of
events generated within the range yF ± dyF /2. The quantity dN divided by dyF
corresponds to the momentum distribution multiplied by xF [i.e., xF (xF f (xF , Q2 ))].
The results for the momentum distributions (multiplied by xF ) at Q2 = 103 GeV2
are represented by ‘+’ for the gluons and by ‘×’ for the flavor singlet quarks, where
the contribution
from a b quark loop is taken into account in the QCD coupling constant above Q2 = mb (= 4.5 GeV).2) In the figure, the solid curves are the results
obtained with GRV(98) for the corresponding values of Q2 .
In Figs. 2 and 3, the momentum distribution for each quark and anti-quark,
respectively, are presented. In these figures, the input momentum distributions at
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H. Tanaka
10
2
GRV98 GL(5GeV )
2
GRV98 SQ(5GeV )
3
2
GRV98(103GeV2)
MC GL(10 GeV )
3
2
MC SQ(10 GeV )
1
dN/dyF
0.1
0.01
0.001
0.0001
0.001
0.01
0.1
1
xF
Fig. 1. The momentum distribution functions (multiplied by xF ) for gluons (GL) and singlet quarks
(SQ) obtained using the algorithm explained in §2. The results obtained from the Monte-Carlo
simulation are denoted by ‘+’ for the gluons and ‘×’ for the singlet quarks, respectively at
Q2 = 103 GeV2 . The solid curves are the corresponding results obtained using GRV(98). The
input distributions at Q20 = 5 GeV2 are represented by the dashed curve for the gluons and by
the dotted curve for the singlet quarks. The parton showers were generated with ε = 10−6 . Here,
xF denotes the momentum fraction of a parton that finished the evolution, and yF = −lnxF .
Q20 = 5 GeV2 are represented by dashed curves.
The transverse momenta for the partons are restricted by the NLL order terms,
as explained in §3. At the LL order of QCD, the upper limit of the virtuality for
the generated partons is restricted only by a large energy scale, which contributes
to the branching vertex. In actual Monte-Carlo generation, the allowed kinematic
boundary for the two-body branching is determined by the condition that the squared
transverse momenta of the generated partons be positive.
The transverse momentum of the parton with momentum k3 for the branching
a(p) → g(k1 + k2 ) + a(k3 ) is given by
2
−s KA
2
2
−
,
(4.1)
pT = x3 y3 p +
x3
y3
2 = (k + k )2 , s = k 2 and p2 = 2 with y = 1 − x . Here, K 2 is restricted
k3T
where KA
1
2
3
3
3
T
A
2 ≤ f (−s) at the NLL order, instead of K 2 ≤ y /x (−s), by
by the condition KA
3
3
A
A
the kinematic boundary of the two-body branching due to the fact that p2T ≥ 0 for
2.
−p2 −s, KA
2
As shown in Figs. 4 and 5, for small x3 , the upper limits of the virtuality KA
are far below the allowed kinematic boundary for the two-body branching vertex,
due to the NLL contributions. This suppression changes the transverse momentum
distribution of the partons in the initial state.
2 , the quantity p2 is restricted by 0 ≤ p2 ≤ y (−s) by the twoFor −p2 −s, KA
3
T
T
Initial State Parton Evolution beyond the Leading Logarithmic Order
969
2
GRV98( 5 3GeV2)
GRV98(10 GeV )
MC U
MC D
MC S
1
dN/dyF
0.1
0.01
0.001
0.0001
0.001
0.01
0.1
1
xF
Fig. 2. The momentum distribution functions (multiplied by xF ) for each flavor of quark obtained
using the algorithm explained in §2. The results of the Monte-Carlo simulation are represented
by ‘+’ for u-quarks (U), ‘×’ for d-quarks (D), and ‘2’ for s-quarks (S), with ε = 10−6 at
Q2 = 103 GeV2 . The solid curves are the corresponding results obtained with GRV(98). The
input distributions at Q20 = 5 GeV2 are denoted by the dashed curves. The remaining notation
is the same as in Fig. 1.
0.1
GRV98( 5 3GeV22)
GRV98(10 GeV )
MC UB
MC DB
MC SB
dN/dyF
0.01
0.001
0.0001
0.001
0.01
0.1
1
xF
Fig. 3. The momentum distribution functions (multiplied by xF ) for each flavor of anti-quark obtained with the algorithm explained in §2. The results obtained from the Monte-Carlo simulation
are denoted by ‘+’ for anti-u-quarks (UB), ‘×’ for anti-d-quarks (DB), and ‘2’ for anti-s-quarks
(SB), with ε = 10−6 at Q2 = 103 GeV2 . The solid curves are the corresponding results obtained
with GRV(98). The input distributions at Q20 = 5 GeV2 are denoted by the dashed curves. The
remaining notation is the same as in Fig. 1.
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H. Tanaka
1e+006
2
pT =0
x1=0.5
-1
x1=10-2
x1=10
100000
3
-s=10 GeV
g -> ggg
fA(-s)
10000
2
1000
100
10
1
0.001
0.01
0.1
1
x3
2
Fig. 4. The x3 dependence of the upper limits of KA
for the g(S) → g(T )g(T )g(S) process with
−1
−2
x1 = 0.5, 10 , 10 . The solid curve denotes the upper limit given by p2T = 0 in the two-body
branching.
1e+006
pT2=0
x1=0.5
x1=10-1
x1=10-2
100000
-s=103 GeV2
q -> ggq
fA(-s)
10000
1000
100
10
1
0.001
0.01
0.1
1
x3
2
Fig. 5. The x3 dependence of the upper limits of KA
for the q(S) → g(T )g(T )q(S) process with
−1
−2
x1 = 0.5, 10 , 10 . The solid curve denotes the upper limit given by p2T = 0 in the two-body
branching.
body decay kinematics. At the NLL order, the transverse momenta for the initial
state partons are restricted by (y3 − x3 fA )(−s) ≤ p2T ≤ y3 (−s). For on-shell gluon
2 = 0) at the LL order, the transverse momentum of the parton with
radiation (KA
momentum k3 is determined by p2T = y3 (−s).
Finally, the transverse momentum distributions of the initial state partons are
Initial State Parton Evolution beyond the Leading Logarithmic Order
971
LLA (case 1)
LLA (case 2)
NLL (JM = 0)
1
Gluons
2
3
2
Q =10 GeV
dN/dpT
0.1
0.01
0.001
0
5
10
15
pT(GeV)
20
25
30
Fig. 6. The transverse momentum distributions for the gluons that finished the evolution for 0.09 ≤
xF ≤ 0.11 at Q2 = 103 GeV2 . The results obtained from the Monte-Carlo simulation are
2
2
represented by ‘+’ for KA
= l02 (case 1), ‘×’ for l02 ≤ KA
≤ y3 /x3 (−s) (case 2), and ‘2’ for
2
2
2
2
l0 ≤ KA ≤ fA (−s) (JM = 0), with l0 = 1 GeV .
LLA (case 1)
LLA (case 2)
NLL (JM = 0)
1
Singet Quarks
Q2=103 GeV2
dN/dpT
0.1
0.01
0.001
0
5
10
15
pT(GeV)
20
25
30
Fig. 7. The transverse momentum distributions for the singlet quarks that finished the evolution
with 0.09 ≤ xF ≤ 0.11. The notations are the same as those in Fig. 6.
represented by ‘2’ in Figs. 5 and 6 for 0.09 ≤ xF ≤ 0.11 at Q2 = 103 GeV2 . For
2 = l2 (case 1, represented
comparison, the case of on-shell parton radiation with KA
0
2
2
by ‘+’) and that with l0 ≤ KA ≤ y3 /x3 (−s) (case 2, represented by ‘×’) at the LL
order are also presented.
2
As seen in the figures, due to the strong suppression of the allowed region of KA
972
H. Tanaka
at the NLL order, the transverse momentum distribution for the initial state parton
that finished the evolution becomes harder than that for case 2 used in the LL order
Monte-Carlo algorithm.
§5.
Summary and comments
In this paper, a parton shower model based on the evolution of the momentum
distributions was extended to the NLL order of QCD. In this type of model, the total
momentum of the initial state partons is conserved. Therefore, it is not necessary
to introduce non-trivial weight factors into this model in order to reproduce scaling
violation of the flavor singlet parton distributions up to their normalization. The
results obtained from the algorithm presented in this paper have been compared with
those obtained using GRV(98), in which the Q2 evolutions are calculated using the
renormalization group equations in the NLL order approximation of QCD. These
two methods give consistent results for the scaling violation of the flavor singlet
parton distributions. It may be important to examine the scaling violation for the
parton distribution functions, which are widely used to evaluate the cross sections,
by comparing them with the results obtained using different algorithms.
In the generation of transverse momenta for the initial state partons with spacelike virtuality, the NLL order terms are included in the kinematic conditions for
2 , for
the two-body branching vertices. The kinematic boundary of the virtuality, KA
the radiated virtual gluon, which subsequently decays into two gluons, is strongly
suppressed by the NLL order contribution determined by the three-body decay functions, particularly for soft gluon radiation and for a small momentum fraction of the
initial state partons. Due to the strong suppression of the phase space for the twobody branching, the transverse momentum distribution for the initial state parton
2 ≤ y /x (−s),
that finished the evolution becomes harder than that in the case KA
3
3
used in the LL order Monte-Carlo algorithm.
The results presented in this paper suggest that the NLL order terms contribute
not only to the evolution of the longitudinal momentum distributions, which is usually taken into account in the evaluation of the scattering cross sections, but also
to the transverse momentum distribution for the initial state partons that couple to
hard scattering processes. Therefore, the transverse structure of the hard scattering
is affected by the phase space restriction due to the NLL order contributions in the
initial state parton radiations.
In order to allow for the application of the method presented in this paper to
realistic processes, the matching problem should be solved, that is double counting
between hard scattering and parton showers must be avoided. Recently, this problem
has been studied9) in hadron-hadron collisions using the LL order parton shower
model proposed in Ref. 4). It may be useful to combine the algorithm presented in
this paper with the method presented in Ref. 9) for the purpose of constructing a
more accurate Monte-Carlo generator.
Initial State Parton Evolution beyond the Leading Logarithmic Order
973
Acknowledgements
This work was supported in part by a Rikkyo University Grant for the Promotion of Research and RCMAS (the Research Center for Measurement in Advanced
Science) of Rikkyo University.
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