Compton mechanism of W and Z boson hadroproduction
It is argued that W/Z boson production in ultra-relativistic pp collisions in the fragmentation region, subject to a kinematic cut on the boson transverse momentum Q┴ > Q┴min, with 1 GeV/c << Q┴min, << MW/Z, must be dominated by the Compton mechanism qg → q'W/Z. We propose applic...
Gespeichert in:
| Datum: | 2012 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2012
|
| Schriftenreihe: | Вопросы атомной науки и техники |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/107006 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Compton mechanism of W and Z boson hadroproduction / M.V. Bondarenco // Вопросы атомной науки и техники. — 2012. — № 1. — С. 105-110. — Бібліогр.: 28 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-107006 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1070062016-10-11T03:02:32Z Compton mechanism of W and Z boson hadroproduction Bondarenco, M.V. Section B. QED Processes at High Energies It is argued that W/Z boson production in ultra-relativistic pp collisions in the fragmentation region, subject to a kinematic cut on the boson transverse momentum Q┴ > Q┴min, with 1 GeV/c << Q┴min, << MW/Z, must be dominated by the Compton mechanism qg → q'W/Z. We propose applications for boson hadroproduction in this kinematics, formulate the factorization theorem, and analyze the QCD enhancements. Отмечается, что рождение W и Z бозонов при столкновениях ультрарелятивистских протонов, регистрируемое во фрагментационной области и при наложении условия на поперечный импульс бозона Q┴ > Q┴min с 1 GeV/c << Q┴min << MW/Z, должно доминироваться комптоновским механизмом qg → q'W/Z. Мы предлагаем приложения для рождения электрослабых бозонов в данной кинематике, формулируем факторизационную теорему, и анализируем КХД-усиления. Відзначається, що народження W та Z бозонів при зіткненнях ультрарелятивістських протонів, за умови реєстрації у фрагментаційній області та за накладеної умови на поперечний імпульс бозона Q┴ > Q┴min з 1 GeV/c << Q┴min << MW/Z, повинно домінуватися комптонівським механізмом qg → q'W/Z. Ми пропонуємо застосування для народження електрослабких бозонів у даній кінематиці, формулюємо факторизаційну теорему та аналізуємо КХД-підсилення. 2012 Article Compton mechanism of W and Z boson hadroproduction / M.V. Bondarenco // Вопросы атомной науки и техники. — 2012. — № 1. — С. 105-110. — Бібліогр.: 28 назв. — англ. 1562-6016 PACS: 13.85.Qk, 13.60.Hb, 12.39.St, 12.40.Nn http://dspace.nbuv.gov.ua/handle/123456789/107006 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Section B. QED Processes at High Energies Section B. QED Processes at High Energies |
| spellingShingle |
Section B. QED Processes at High Energies Section B. QED Processes at High Energies Bondarenco, M.V. Compton mechanism of W and Z boson hadroproduction Вопросы атомной науки и техники |
| description |
It is argued that W/Z boson production in ultra-relativistic pp collisions in the fragmentation region, subject to a kinematic cut on the boson transverse momentum Q┴ > Q┴min, with 1 GeV/c << Q┴min, << MW/Z, must be dominated by the Compton mechanism qg → q'W/Z. We propose applications for boson hadroproduction in this kinematics, formulate the factorization theorem, and analyze the QCD enhancements. |
| format |
Article |
| author |
Bondarenco, M.V. |
| author_facet |
Bondarenco, M.V. |
| author_sort |
Bondarenco, M.V. |
| title |
Compton mechanism of W and Z boson hadroproduction |
| title_short |
Compton mechanism of W and Z boson hadroproduction |
| title_full |
Compton mechanism of W and Z boson hadroproduction |
| title_fullStr |
Compton mechanism of W and Z boson hadroproduction |
| title_full_unstemmed |
Compton mechanism of W and Z boson hadroproduction |
| title_sort |
compton mechanism of w and z boson hadroproduction |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2012 |
| topic_facet |
Section B. QED Processes at High Energies |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/107006 |
| citation_txt |
Compton mechanism of W and Z boson hadroproduction / M.V. Bondarenco // Вопросы атомной науки и техники. — 2012. — № 1. — С. 105-110. — Бібліогр.: 28 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT bondarencomv comptonmechanismofwandzbosonhadroproduction |
| first_indexed |
2025-07-07T19:21:43Z |
| last_indexed |
2025-07-07T19:21:43Z |
| _version_ |
1837017202216140800 |
| fulltext |
COMPTON MECHANISM OF W AND Z BOSON
HADROPRODUCTION
M.V. Bondarenco∗
National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received November 1, 2011)
It is argued that W/Z boson production in ultra-relativistic pp collisions in the fragmentation region, subject to
a kinematic cut on the boson transverse momentum Q⊥ > Q⊥min, with 1 GeV/c � Q⊥min � MW/Z , must be
dominated by the Compton mechanism qg → q′W/Z. We propose applications for boson hadroproduction in this
kinematics, formulate the factorization theorem, and analyze the QCD enhancements.
PACS: 13.85.Qk, 13.60.Hb, 12.39.St, 12.40.Nn
1. INTRODUCTION
Resonant production of W± and Z0 bosons at hh
colliders [1] has high probability and a clear exper-
imental signature when the boson decays to a lep-
ton pair, since both leptons have pT ∼ MV /2 ∼ 40
GeV (V = W or Z), thus being well separated from
the hadronic underlying event. That makes the elec-
troweak (EW) boson production process a convenient
quark-meter, well suited for determination of quark
momentum distributions in hadrons, complementary
to DIS [2], and also serve as a playground for new
physics searches.
Due to the W boson charge, measurement of its
asymmetry is convenient for probing valence quark
distributions [3, 4]. As for Z-bosons, they have nearly
the same differential distribution, but are easier to
reconstruct from detection of 2 charged leptons, and
serve as a benchmark. Albeit the formidable EW
boson mass does not afford probing the smallest x
frontier for the given proton beam energy (at LHC in
the central rapidity region x1,2 ∼ MV /
√
s ∼ 10−2),
but in the fragmentation region one of the x dimin-
ishes to ∼ 10−3, so in this kinematics the heavy bo-
son hadroproduction may serve as a probe of small-x
physics, as well.
Historically, it was suggested by Drell and Yan [5]
that the leading contribution to inclusive hadropro-
duction of a heavy gauge boson comes from qq̄ an-
nihilation, with q and q̄ carried by different hadrons
(Fig. 1, a). That approximation holds well at pp̄ colli-
sions and for not very high energies (
√
s ≤ MV
xval
∼ 0.5
TeV). However, in pp collisions and at modern col-
lider energies (particularly at LHC), antiquarks are
by far less abundant than gluons, and since an EW
boson can only be emitted from a quark line, the next
mechanism to be considered is qg → qV (known as
QCD Compton scattering, see Fig. 1, b). Next, since
at small x gluons tend to be more abundant than
even quarks, in the central rapidity region one also
has to take into account boson production trough
gg fusion. But as long as boson-gluon coupling is
not direct, proceeding through an auxiliary (virtual
or real) qq̄ pair creation (see Figs. 1, c and 1, d), gg
processes appear rather as a background for quark
physics studies, and it is desirable either to arrange
conditions in which their contribution is minor, or ap-
ply additional experimental selection criteria to elim-
inate them. Most obviously, to suppress gg fusion,
it should suffice to work in the fragmentation region,
where q(x) � αsg(x). Besides that the process at
Fig. 1, d gives 2 jets [6], while that of Fig. 1, c, as well
as Fig. 1, a a minimum bias event with only small
Q⊥ ∼ Λ ∼ 1 GeV. To compare with, the contri-
bution from the Compton mechanism is typically 1-
jet and broadly distributed in the boson’s transverse
momentum Q⊥, out to Q⊥ ∼ MV (which at inclu-
sive treatment of Q⊥ gives rise to ln MV
Λ ). So, the
criterion may be to select events with 1 jet balancing
the boson transverse momentum, in favor of 2-jet and
0-jet events.
Strictly speaking, the physical distinction be-
tween the Drell-Yan (DY) and Compton mechanisms
is not quite clear-cut, because one of the two Feyn-
man diagrams of the Compton process (Fig. 2,b) is
topologically similar to that of qq̄ pair production by
a qluon and subsequent annihilation of the q̄ with the
projectile quark. If in a proton all antiquarks stem
from gluon splitting g → qq̄, then one may expect the
DY mechanism to be contained in the Compton one.
However, before the annihilation, the antiquark may
interact with other constituents of the hadron and
get non-trivially entangled with them, while in the
Compton mechanism this possibility is not accounted
for. At evaluation of the Q⊥-integrated cross-section
of boson production, that problem is circumvented by
taking the approach similar to DIS: including the LO
gluon contribution into NLO antiquark contribution
∗E-mail address: bon@kipt.kharkov.ua
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 105-110.
105
at a higher factorization scale ∼ M2
V [7, 8].
At high pT , nonetheless, one may be pretty sure
that most of the (anti)quarks are due to direct gluon
splitting. Although even an intrinsic, low-pT anti-
quark can scatter on another constituent (of the same
hadron [9], or even of the opposite hadron [10]) and
acquire high pT , while staying non-trivially entangled
with the hadron constituents, but the probability of
hard scattering is ∼ α2
s(p2
T ), whereas that of direct
production of q̄ at high pT from gluon splitting is
∼ αs(p2
T ), i.e. of lower order in αs and thus greater.
A caveat is that there are several constituents in the
proton to hard-scatter from, but from DIS experi-
ence, that should not make the proton obscure with
respect to hard scattering, anyway. Besides that, the
event shapes in these cases contain different number
of minijets and can be rejected by additional experi-
mental criteria. Therefore, at high pT the distinction
between DY and Compton mechanisms is clearer.
a b
c d
Fig. 1. Contributions to W and Z boson production
in a high-energy proton-proton collision: qq̄ annihi-
lation (a); qg → qV (Compton) (b); gg fusion with
virtual qq̄ pair creation (c); gg fusion with real qq̄
pair creation (d)
The above discussion suggests a possibility to ex-
perimentally isolate the Compton contribution by im-
posing a kinematic cut Q⊥ > Q⊥min � Λ, which sup-
presses the DY contribution, and working in the frag-
mentation region, which suppresses the contribution
from gg-fusion. The contribution from the central
rapidity region may otherwise be suppressed by con-
structing a dσ(W+) − dσ(W−) difference. The stip-
ulated high Q⊥ will also alleviate the partonic sub-
process description, reducing the collinear gluon re-
absorption by the non-annihilated active quark, and
of the active quark non-factorizable rescatterings [10]
(they are entirely absorbable into g(xg)). Hence, it
may be feasible to use this mechanism as well in
global analysis of quark momentum distribution func-
tions, provided the factorization procedure is appro-
priately formulated. The latter, in fact, must be sim-
ilar to that in case of direct photon or jet production,
with the proviso that we do not actually need to be
involved in precise jet definition. That is rather nat-
ural since the final quark with high pT will emerge as
a jet. Incidentally, let us note that tagging the flavor
of the quark jet, e.g. its charm, in association with
a W boson, will give access to the sea strangeness
content of the nucleon (cf. [3]).
Comparing the procedures of pdf determination
from EW boson hadroproduction with a Q⊥ cut and
via inclusive Q⊥ treatment, we should note the fol-
lowing. At LO, the DY mechanism is very convenient
for pdf determination because from the reconstructed
V -boson momentum one determines longitudinal mo-
menta of both annihilating partons exactly. At NLO,
though, due to an additional unregistered gluon in
the final state it involves an x-convolution. For
Compton mechanism, the convolution arises already
at LO. However, since diagram 2b is similar to that of
DY, and moreover, it appears to dominate, the con-
volution kernel for Compton is also rather singular,
and perhaps even replaceable by a δ-function. This is
quite opposite, say, to the direct photon production
case, where the emitted photon is always the lighter
particle among the final products, and therefore tends
to carry away only a small fraction of energy. For
heavy boson production, the roles of the momentum-
conserving radiator and the soft radiation are re-
versed: the emitted boson assumes most of the mo-
mentum while the left-over quark is wee. The dynam-
ical reason is that in the dominating diagram 2b the
final quark and the virtual space-like quark tend to
be collinear with the parent gluon, and so the space-
like quark manifests itself more like an antiquark.
a b
Fig. 2. Feynman diagrams for the partonic sub-
process of Fig. 1b (Compton scattering)
Ultimately, the dominance of diagram 2b may
be utilized for probing the gluon distribution in a
close analogy with probing quark distribution at DIS.
Indeed, the virtual space-like (∼ −M2
V ) quark in
Fig. 2, b corresponds to the virtual photon in the DIS
LO diagram, and this virtual quark knocks out a
quasi-real gluon from the proton, converting it to a
quark. However, at small xg one must beware of mul-
tiple gluon exchanges between the probing quark and
the probed hadron, which can affect the universality
of the probabilistic gluon distribution. Those issues
will be discussed in the next section.
2. FACTORIZATION FOR THE
COMPTON MECHANISM
In this section we outline the factorization proce-
dure relating the fully differential cross-section of
EW boson hadroproduction with the corresponding
qg → q′V partonic cross-section. At small x val-
ues, it may be important to formulate the factoriza-
tion theorem non-perturbatively, beyond the notion
of gluon distribution probability. The irreducibility
to single gluon exchange amounts to quark scatter-
ing off an intense and coherent gluonic field. In
106
this respect the situation resembles that in QED,
where there is a familiar factorization theorem be-
yond the perturbative treatment of interaction with
the external field, first established for scattering in
a Coulomb field [11], and subsequently generalized
to high-energy scattering in compact field of an arbi-
trary shape [12] (see also [13]). The theorem presents
the process amplitude as a product of the electron
spin-independent (eikonal) scattering amplitude and
the electron spin-dependent perturbative amplitude
of real photon emission at absorption of a virtual
photon with the momentum equal to the total mo-
mentum transfer in scattering. In our case the ini-
tial gluon virtuality may be neglected compared with
the emitted boson mass. Therefore we may regard
the initial gluon as real, and apply the Weizsäcker-
Williams approximation. The latter, however, needs
to be generalized, factoring out not the gluon flow
but the full quark scattering amplitude.
Consider a non-diffractive high-energy pp collision
event containing a high-pT ll̄ pair. Suppose that by
reconstructing the total momentum of the ll̄ pair its
mass is identified to be at the W or Z boson res-
onance1, and the rapidity being > 1, say, positive.
The latter implies that this boson had most proba-
bly been emitted by one of the quarks of the hadron
moving in the positive (forward) direction.
Owing to the Lorentz-contraction of ultra-
relativistic hadrons, the interaction of the emit-
ter quark with the opposite hadron proceeds very
rapidly. Furthermore, owing to large value of MV
compared to Λ, the boson emission from the quarks
also passes very rapidly compared to the intra-hadron
timescale. Hence, sufficiently reliable must be the im-
pulse approximation, at which the emitting quark ini-
tial state is described by the (empirical) momentum
distribution function, while the rest of the partons
in that hadron are regarded as spectators. Thereby
we reduce the problem to that of V boson emission
by a relativistic quark scattering on a hadron. Since
due to the boson heavyness, the amplitudes of its
emission from different quarks within one (forward
moving) hadron do not interfere (the formfactor re-
duces to the number of quarks), the probability (dif-
ferential cross-section) of boson production in the pp
collision comes as an integral of the correspondent
quark-proton differential cross-section weighted with
the quark pdf f(x) in the first proton:
dΣ(P1, P2, Q) =
∫ 1
0
dxf(x)dσ(xP1 , P2, Q), (1)
Pμ
1 , Pμ
2 being the initial hadron 4-momenta, Q the fi-
nal boson momentum, and x the hadron momentum
fraction carried by the emitter quark. The factoriza-
tion scale for f(x) will be determined later on.
Relation with quark-hadron scattering dif-
ferential cross-section.
Applying the generalized Weizsäcker-Williams
procedure to the differential cross-section of boson
production in quark-hadron scattering, we obtain
dσ
dΓQ
= 16π
2E
E′ + p′z
dσ̂
dt
k2
⊥dσscat, (2)
where dΓQ = d3Q
(2π)32Q0
, and dσ̂ may be related with
QED or QCD virtual Compton cross-section:
dσ̂
dt
=
1
4παem
dσ(eγ → eV )
dt
=
2Nc
4παs
dσ(qg → qV )
dt
.
The value of the gluon longitudinal momentum, or
energy, ω is fixed by the 4-momentum conservation
law and on-shellness of the final undetected quark.
The quark-hadron quasi-elastic scattering dif-
ferential cross-section dσscat encodes all non-
perturbative aspects of fast quark-hadron interaction
in a model-independent way. Earlier, the differential
cross-section of quark-hadron scattering had already
been introduced in the context of high-energy pA [14]
and γA, γ∗A collisions of nucleons and nuclei.
Representation (2) also exhibits similarity with
kT -factorization [15], but there dσ̂ may go beyond
the WW approximation, while in place of dσscat one
has the BFKL kernel describing the growth of the
cross-section with the energy. In the next section we
shall discuss the latter issue as well, along with other
effects arising in QFT.
Relation with gluon distributions.
If we could rely on an approximation that the
high-energy small angle quark-hadron scattering pro-
ceeds only through a single t-channel gluon exchange
(presumably with a running αs), we might avoid de-
tailed description of the hadron creating the color
field. Then all we need to know is the equivalent
gluon flow. In the single gluon exchange approxima-
tion, the absorbed gluon may be treated on equal
footing with the initial quark, and so the description
of boson hadroproduction must become symmetric in
terms of initial quark and gluon distributions.
Implementing the single gluon exchange approxi-
mation into the factorization procedure and compar-
ing the final result with Eq. (2), we obtain a formula
for the unintegrated gluon density (DGLAP type)
xgg(xg, k
2
⊥, Q2
⊥) =
1
π
2Nc
4παs(Q2
⊥)
k2
⊥
dσscat
d2k⊥
. (3)
This function vanishes at k⊥ → 0 due to factor k2
⊥, as
well as at k⊥ → ∞ due to factor dσscat/dk2
⊥. Hence,
somewhere in between it must have a maximum, but
it is unobvious whether it belongs to the hard or soft
region, and whether the decrease immediately beyond
the maximum is exponential or ∼ 1/k2
⊥. The existing
parameterizations favor the hard scenario.
The corresponding conventional gluon density ob-
tained by k⊥-integration of Eq. (3) is
xgg(xg, Q
2
⊥) =
1
π
2Nc
4παs(Q⊥)
∫ Q2
⊥
0
dk2
⊥k2
⊥
dσscat
dk2
⊥
. (4)
This may be compared with DIS in the dipole picture
[23], and with the approach of [24].
1The resonance width Γ ∼ 2GeV � MV will be neglected in this article throughout, and so the boson is handled as a
quasi-free particle.
107
3. MODIFICATIONS ARISING IN QFT
In ordinary quantum mechanics, the differen-
tial cross-section of qh scattering appearing in
Eqs. (2), (3) would assume a finite value in the high-
energy limit. But it is now well-known that QFT
brings (fortunately, mild) modifications to the im-
pulse approximation and the parton model, for a
number of reasons. First of all, even a static field
created by an ensemble of point-like partons has
Coulomb singularities, resulting in a logarithmic de-
pendence of the transport or radiative cross-section
on some hard scale. Secondly, multiple emission of
soft quanta and particle pairs in the central rapid-
ity region generates various double logarithmic as-
ymptotics in the cross-section, which upon resumma-
tion to all orders may turn into power-law modifica-
tions [16]. We shall discuss these effects by turn as
applied to our specific problem.
Q⊥ as factorization scale for Compton.
At practice, transverse momenta of multiple final
hadrons produced within the underlying event are
usually not counted, and correspondingly, dσ/dΓQ
must be integrated over the unconstrained momen-
tum components of the initial gluon(s), i.e., over k⊥.
In so doing, it seems reasonable to neglect the k⊥-
dependence of the Compton subprocess cross-section
dσ̂/dt provided k2
⊥ � p · k. That leads to
dσ
dΓQ
= 16π
2E
E′ + p′z
dσ̂
dt
∫
d2k⊥k2
⊥
dσscat
d2k⊥
. (5)
However, at large k2
⊥ the scattering cross-section
has Rutherford asymptotics (cf., e.g., [17]):
dσscat
d2k⊥
∼
k⊥→∞
2α2
s(k2
⊥)
k4
⊥
[(
1− 1
N2
c
)
Nq + Nq̄
2
+ Ng
]
(6)
(with Nc = 3 the number of colors, and Nq, Nq̄, Ng
the mean numbers of quarks, antiquarks and gluons
in the proton), and therewith the k⊥-integral in (5)
appears to be logarithmically divergent at the up-
per limit. That means that at sufficiently large k⊥
one still needs to rely on the decrease of dσ̂/dt with
k⊥, providing the additional convergence factor. The
sensitivity of dσ̂/dt to k⊥ arises at
k⊥max ∼ min{MV , Q⊥}, (7)
which should be used as the upper limit in k⊥ integral
in Eq. (5) and serve as a natural factorization scale.
It is also to be used as a factorization scale for the
quark pdf in Eq. (1), if we wish at determination of
Q⊥ to be able to neglect the initial quark transverse
momentum. In what follows, we will be mostly con-
sidering the case Q⊥ < MV , whereby k⊥max ∼ Q⊥.
That differs from the case of DY mechanism, where
even at small Q⊥ the natural factorization scale is
MV [7, 18]2, and is in the spirit of factorization in
direct photon and jet production [20].
Distribution of the color sources.
With asymptotics (6) and upper limit (7), the k⊥-
integral in (5) with constant Nq, Ng would give ln Q⊥
Λ .
But in fact, Nq, Ng = const, because they express as
integrals from pdfs, which diverge at low x. If the as-
ymptotics of f(x′) is ∼ 1/x′, as is motivated by the
perturbation theory,
Nq =
∫ 1
k2
⊥/xs
dx′f(x′) ∼ ln
xs
k2
⊥
, (8)
where xs is the quark-hadron collision subenergy.
Substituting Eqs. (6), (8) to (5), we get:
∫
dk2
⊥k2
⊥
dσscat
d2k⊥
∼
∫ Q2
⊥
Λ2
d lnk2
⊥ ln
xs
k2
⊥
1
2
ln2 xs
k2
⊥
∣∣∣∣
Λ2
Q2
⊥
= ln
Q2
⊥
Λ2
ln
xs
Λ|Q⊥| . (9)
This equation is similar to the (Sudakov) double log-
arithms for the reggeized gluon, if Q2
⊥ stands for |t|.
But ln Q2
⊥
Λ2 may be absorbed into pdf definition.
In a more empirical approach, however, the diver-
gence proceeds as a power law [22]:
f(x′) ∼ x′−αP , αP > 1,
and the factorization scale must be taken � Q⊥ (cf.
CGC approach [21]). The x′-integration then gives
Nq ∼ (
xs/k2
⊥
)Δ
, Δ = αP − 1 > 0,
Therewith, the k⊥-integral in Eq. (5) converges on
the upper limit:
∫
d2k⊥k2
⊥
dσscat
dk2
⊥
∼ (xs)Δ
∫ ∞
Λ2
dk2
⊥
(k2
⊥)αP
∼
(xs
Λ2
)Δ
,
and the result is independent of the factorization
scale, provided Δ is. That must correspond to the
BFKL-regime [22].
Small-x behavior of the gluon distribution.
Since at present we can not reliably calculate the
gluon distribution function ab initio, it is to be in-
ferred on phenomenological basis. Although gluon is
not directly observable outside of hh collisions, in DIS
at small x its density must be proportional to that of
sea quark, which, in turn, is ∝ F2(x, μ2). At typical
scales of hadroproduction at LHC (Q2
⊥ ∼ 100 GeV2
and xg ∼ 10−3), the DIS data for the nucleon struc-
ture function are not available, but the data come
close, and seemingly admit safe extrapolation. One
must also duly incorporate the dependence on the
factorization scale μ2, since at x so small the scaling
is absent. To extrapolate both x and μ2 dependences,
one may utilize the observation [25] that at x < 0.01
the DIS γ∗p cross-section
σγ∗p = 4π2αemμ−2F2(x, μ2) = σγ∗p(τ)
obeys “geometrical scaling”, reducing to a function
of a single variable
τ =
μ2
μ2
0
(
xg
x0
)λ
, μ0 = 1 GeV, (10)
2For Q⊥-integrated distributions, the factorization scale is usually taken to be ∼ M2
V , as well see [19].
108
with the best-fit parameters [25]
λ ≈ 0.3, x0 ≈ 3 · 10−3.
Furthermore, in the domain τ � 1, to which our
parameters belong, the dependence on τ is a simple
power law in itself:
F2(x, μ2) =
μ2
4π2αem
40μb τ−Δ/λ (11a)
= 0.35
(
μ2
μ2
0
)1−Δ/λ (x0
x
)Δ
. (11b)
Next, we note that phenomenologically the expo-
nent in Eq. (11a) Δ/λ ≈ 0.75, and so in Eq. (11b)
1 − Δ/λ ≈ 0.25 ≈ Δ, i.e. exponents for μ2 and 1/x-
dependencies are approximately equal. Theoretically,
there might be some difference between them in con-
nection that integral
∫
d2k⊥ diverges and demands a
cutoff at ∼ μ2. But if for simplicity we assume the
equality of the exponents, and utilize the relation
μ2/x = W 2 + μ2 ≈ W 2,
the gluon pdf behavior is inferred to be
αs(μ2)xgg(xg, μ
2) ∝ F2(x, μ2) ≈ 0.07
(
W 2
μ2
0
)0.25
.
(12)
Recalling the relation with the differential cross-
section (3), equation (12) is quite natural from the
viewpoint of t-channel Reggeization. Then, we ob-
tain the same result in any treatment — through pdfs
or through qh scattering.
Reggeization in the Q⊥-dependence of the
boson production differential cross-section.
In general, the onset of energy-dependence of the
quark-hadron scattering amplitude, along with the
strong difference between the initial and final quark
energies in the hard subprocess may affect the bal-
ance of the Compton process Feynman diagrams, and
in principle violate the gauge invariance. Fortunately,
at Q⊥ � MV , only one of the two Feynman diagrams
dominates, wherein the final quark interacts with the
encountered proton. Thereat, the qh collision suben-
ergy is counted by the energy of the final quark. To
estimate it, note that
xg ∼ M2
V /xs,
p′ ·P2 =
p′ · k
xg
=
p′ · k
p · k xs =
p′+
p+
xs =
Q2
⊥
p · p′xs ∼ Q2
⊥
M2
V
xs.
(13)
The non-trivial property of subenergy (13) is the
Q2
⊥/M2
V = ρ2 factor. It means that as a result of
Reggeization, the differential cross-section multiplies
by ρ2Δ. That factor may alternatively be considered
as being due to the factorization scale Q⊥ depen-
dence of the gluon structure function. The rest of
the Q2
⊥-dependence comes from the partonic Comp-
ton differential cross-section, which in the perturba-
tive description (the sum of diagrams 2a and 2b) goes
as ∼ Q−2
⊥ for Λ � Q⊥ � MV and as ∼ Q−4
⊥ for
Q⊥ > MV . Hence, in the fragmentation region of ra-
pidities, and intermediate region of boson transverse
momenta Λ � Q⊥ � MV the boson hadroproduc-
tion differential cross-section should behave as
dσ
dQ⊥
∼ Q2Δ−1
⊥ ∼ Q
−1/2
⊥ (fragm. region). (14)
In the central rapidity region, Q⊥-dependence of
the quark pdf comes into play, and (14) modifies to
dσ
dQ⊥
∼ Q4Δ−1
⊥ ∼ Q0
⊥ (centr. region). (15)
However, one must keep in mind that in the central
region there are other mechanisms contributing be-
sides the Compton one.
Sudakov FFs in the Compton subprocess.
The above discussed power-law increase of sea and
gluon pdfs at small x physically owes to the open-
ing possibility of particle production in the central
rapidity region, with the phase space indefinitely in-
creasing with the collision energy. Theoretically it is
connected with gluon Reggeization and double loga-
rithmic asymptotics. But double log effects also arise
in the hard subprocess, since M2
V is a large scale,
while Q2
⊥ is a smaller subscale. Physically, at color
exchange the particle intensely emit soft and collinear
radiation quanta, but those do not essentially change
the emitting particle energy, only altering the trans-
verse momentum. For large-angle scattering, this is
unessential, but for small-angle it is. Resummation
of not completely compensating real and virtual con-
tributions leads to transverse Sudakov form-factors.
The Sudakov resummation for the Drell-Yan pro-
duction mechanism at Q⊥ � MV received a great
deal of attention (see, e.g., [26] for an overview). But
for the Compton mechanism the implementation of
the developed techniques is hampered by the pres-
ence of 3 eikonal lines instead of 2 in DY, though all
belonging to one plane.
The Sudakov formfactors lead to hardening of
the Q⊥-dependence, but in a way of redistribution,
whereas Regge effects are pure enhancements. So far,
Tevatron [27] and LHC [28] agreed well with the pre-
dictions of existing Monte-Carlo generators. Thus, in
order to discern Regge effects in the Q⊥-dependence
and disentangle them from Sudakov ones, more stud-
ies are required.
Acknowledgement
Thanks are to P.V. Sorokin for inspiring my interest
to the EW boson production problems at LHC.
References
1. R. Stroynowski // Phys. Rep. 1981, v. 71, p. 1;
J. Alitti et al. // Z. Phys. 1990, v. C47, p. 11;
T. Aaltonen et al. // Phys. Lett. 2010, v. B692,
p. 232; V.M. Abazov et al. // Phys. Lett. 2010,
v. B693, p. 522; V. Khachatryan et al. // JHEP.
2011, iss. 01, 080.
109
2. P.M. Nadolsky et al. // Phys. Rev. 2008, v. D78,
013004; A.D. Martin et al. // Eur. Phys.
J. 2009, v. C63, p. 189; M. Dittmar et al.
Parton Distributions // arXiv : 0901.2504 [hep-
ph]; M. Dittmar, F. Pauss, and D. Zurcher //
Phys. Rev. 1997, v. D56, p. 7284.
3. E.L. Berger et al. //Phys. Rev. 1989, v.D40, p. 83.
4. T. Aaltonen et al. // Phys. Rev. Lett. 2009,
v. 102, 181801; G. Aad et al. // Phys. Lett.
2011, v. B701, p. 31.
5. S.D. Drell and T.M. Yan // Phys. Rev. Lett.
1970, v. 25, p. 316 [Err.-ibid. 1970, v. 25, p. 902].
6. J. Campbell and R.K. Ellis // Phys. Rev. 2002,
v. D65, 113007.
7. H.D. Politzer // Nucl. Phys. 1977, v. B129,
p. 301; C.T. Sachrajda // Phys. Lett. 1978,
v. B73, p. 185.
8. G. Altarelli // Phys. Rep. 1982, v. 81, p. 1.
9. R.P. Feynman et al. // Phys. Rev. 1978, v. D18,
p. 3320; R.D. Field // Talk at 19th Int. Conf. on
High Energy Physics. Tokyo, 1978.
10. E.L. Berger and S. Brodsky // Phys. Rev. Lett.
1979, v. 42, p. 440.
11. H. Olsen et al. // Phys. Rev. 1957, v. 106, p. 27.
12. V.N. Baier and V.M. Katkov // Sov. Phys.
JETP. 1968, v. 26, p. 854; 1969, v. 28, p. 807.
13. M.V. Bondarenco // Phys. Rev. 2010, v. A82,
042723.
14. E. Levin // Phys. Lett. 1996, v. B380, p. 399;
R. Baier et al. // Nucl. Phys. 1997, v. B484,
p. 265.
15. S. Catani et al. // Phys. Lett. 1990, v. B242,
p. 97; Nucl. Phys. 1991, v. B366, p. 135.
16. V.G. Gorshkov // Sov. Phys. Usp. 1973, v. 16,
p. 322.
17. J.D. Bjorken // FERMILAB-PUB-82-059-THY.
1982.
18. J.M. Campbell, J.W. Huston, and W.J. Stirling
// Rep. Prog. Phys. 2007, v. 70, p. 89;
W. Greiner, S. Schramm, E. Stein. Quantum
chromodynamics. Berlin: “Springer”, 2002.
19. Ch. Anastasiou et al. // Phys. Rev. 2004, v. D69,
094008.
20. J.F. Owens //Rev.Mod. Phys. 1987, v. 59, p. 465.
21. F. Gelis et al. //Annu.Rev.Nucl. Part. Sci. 2010,
v. 60, p. 463.
22. V. Barone, E. Predazzi. High energy particle
diffraction. Berlin: “Springer”, 2002.
23. V. Barone et al. // Z. Phys. 1993, v. C58, p. 541;
Phys. Lett. 1994, v. B326, p. 161; N.N. Nikolaev
and B.G. Zakharov // Z. Phys. 1994, v. C64,
p. 631; A.H. Mueller // Nucl. Phys. 1994,
v. B415, p. 373.
24. M.A. Kimber, A.D. Martin, and M.G. Ryskin
// Eur. Phys. J. 2000, v. C12, p. 655; Phys. Rev.
2001, v. D63, 114027.
25. A.M. Stasto, K.J. Golec-Biernat, and J. Kwie-
cinski // Phys. Rev. Lett. 2001, v. 86, p. 596.
26. E. Fink // arXiv : 0105276 [hep-ph].
27. V.M. Abazov et al. // Phys. Lett. 2010, v. B693,
p. 522–530.
28. G. Aad et al. // arXiv : 1107.2381.
�����������
���
� �� ������ � W Z �������
���� �����
���
���������� ��
�
�
���� W � Z �
�
�
� ��� ��
���
������ �������������������� ��
�
�
� ���
����������
� �
�����������
��
�
������ � ��� ���
����� ���
��� �� �
�������� ������� �
�
�
�� Q⊥ > Q⊥min � � !"/c � Q⊥min � MW/Z
���
�����
������ �
���
�
����� ��������
�
qg → q′W/Z# $� ���
������ ����
�����
�� �
�
���� %�����
������ �
�
�
� �
���
� ����������
�
��������� ����
������
���& ��
���� � ����������� '()���������#
������������
���
���� �
�������� W �
Z �������
���� �����
���
*+
�����,���� -
���
����� W �� Z �
�
�+� ��� �+�������� �������������+������� ��
�
�+� ��
��
�� ��,�����+. � ����������+��+�
�����+ �� �� �����
��
. ��
�� �� �
�������� +������ �
�
��Q⊥ >
Q⊥min � � !"/c � Q⊥min � MW/Z �
����
�+�������� �
���
�+������ �����+��
� qg → q′W/Z#
$� ��
�
��,�
����
�������
�� ���
����� ������
������� �
�
�+� �
��+� �+�������+ �
����&,�
����
�����+��� ��
���� �� ����+��,�
'()��+
�������#
��/
|