11institutetext: Institute of Physics, Academia Sinica, Taipei, Taiwan 115,
Republic of China

# Quantum Chromodynamics

Hsiang-nan Li

###### Abstract

I review the basics of perturbative QCD, including infrared divergences and
safety, collinear and $k_{T}$ factorization theorems, and various evolution
equations and resummation techniques for single- and double-logarithmic
corrections. I then elaborate its applications to studies of jet substructures
and hadronic two-body heavy-quark decays.

##  0.1 Introduction

One of the important missions of the Large Hadron Collider (LHC) is to search
for new physics beyond the standard model. The identification of new physics
signals usually requires precise understanding of standard-model background,
whose contributions mainly arise from quantum chromodynamics (QCD). Many
theoretical approaches have been developed based on QCD, which are appropriate
for studies of processes in different kinematic regions and involving
different hadronic systems. The theoretical framework for high-energy hadron
collisions is known as the perturbative QCD (pQCD). I will focus on pQCD
below, introducing its fundamental ingredients and applications to LHC
physics. Supplementary material can be found in [1].

The simple QCD Lagrangian reveals rich dynamics. It exhibits the confinement
at low energy, which accounts for the existence of various hadronic bound
states, such as pions, protons, $B$ mesons, and etc.. This nonperturbative
dynamics is manifested by infrared divergences in perturbative calculations of
bound-state properties like parton distribution functions and fragmentation
functions. On the other hand, the asymptotic freedom at high energy leads to a
small coupling constant, that allows formulation of pQCD. Therefore, it is
possible to test QCD in high-energy scattering, which is, however, nontrivial
due to bound-state properties of involved hadrons. That is, high-energy QCD
processes still involve both perturbative and nonperturbative dynamics. A
sophisticated theoretical framework needs to be established in order to
realize the goal of pQCD: it is the factorization theorem [2], in which
infrared divergences are factorized out of a process, and the remaining piece
goes to a hard kernel. The point is to prove the universality of the infrared
divergences, namely, the independence of processes the same hadron
participates in. Then the infrared divergences are absorbed into a parton
distribution function (PDF) for the hadron, which just needs to be determined
once, either from experimental data or by nonperturbative methods. The
universality of a PDF guarantees the infrared finiteness of hard kernels for
all processes involving the same hadron. Convoluting these hard kernels with
the determined PDF, one can make predictions. In other words, the universality
of a PDF warrants the predictive power of the factorization theorem.

Though infrared divergences are factorized into a PDF, the associated
logarithmic terms may appear in a process, that is not fully inclusive. To
improve perturbative expansion, these logarithmic corrections should be
organized by evolution equations or resummation techniques. For the summation
of different single logarithms, the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi
(DGLAP) equation [3] and the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [4]
have been proposed. For different double logarithms, the threshold resummation
[5, 6, 7] and the $k_{T}$ resummation [8, 9] have been developed. Besides, an
attempt has been made to combine the DGLAP and BFKL equations, leading to the
Ciafaloni-Catani-Fiorani-Marchesini (CCFM) equation [10]. Similarly, the
threshold and $k_{T}$ resummations has been unified under the joint
resummation [11, 12], which is applicable to processes in a wider kinematic
range. A simple framework for understanding all the above evolution equations
and resummation techniques will be provided.

After being equipped with the pQCD formalism, we are ready to learn its
applications to various processes, for which I will introduce jet
substructures and hadronic two-body heavy-quark decays. It will be
demonstrated that jet substructures, information which is crucial for particle
identification at the LHC and usually acquired from event generators [13], are
actually calculable using the resummation technique. Among jet substructures
investigated in the literature, the distribution in jet invariant mass and the
energy profile within a jet cone will be elaborated. For the latter, it will
be shown that the factorization theorem goes beyond the conventional naive
factorization assumption [14], and provides valuable predictions for branching
ratios and CP asymmetries of hadronic two-body heavy-quark decays, that can be
confronted by LHCb data. Specifically, I will concentrate on three major
approaches, the QCD-improved factorization [15], the perturbative QCD [16, 17,
18, 19], and the soft-collinear-effective theory [20, 21, 22, 23]. Some long-
standing puzzles in $B$ meson decays and their plausible resolutions are
reviewed. For more details on this subject, refer to [24].

##  0.2 Factorization Theorem

The QCD lagrangian is written as

$\displaystyle{\cal
L}_{QCD}=\bar{\psi}(i\not\\!D_{a}T_{a}-m)\psi-\frac{1}{4}F^{\mu\nu}_{a}F_{\mu\nu
a},$ (1)

with the quark field $\psi$, the quark mass $m$, and the covariant derivative
and the gauge field tensor

$\displaystyle D^{\mu}_{a}$ $\displaystyle=$
$\displaystyle\partial^{\mu}+igA^{\mu}_{a},$ $\displaystyle F^{\mu\nu}_{a}$
$\displaystyle=$
$\displaystyle\partial^{\mu}A^{\nu}_{a}-\partial^{\nu}A^{\mu}_{a}-gf_{abc}A^{\mu}_{b}A^{\nu}_{c},$
(2)

respectively. The color matrices $T_{a}$ and the structure constants $f_{abc}$
obey

$\displaystyle[T^{(F)}_{a},T^{(F)}_{b}]=if_{abc}T^{(F)}_{c},\;\;\;\;(T_{a}^{(A)})_{bc}=-if_{abc},$
(3)

where $F$ $(A)$ denotes the fundamental (adjoint) representation. Adding the
gauge-fixing term in the path-integral quantization to remove spurious degrees
of freedom, Eq. (1) becomes

$\displaystyle{\cal L}_{QCD}$ $\displaystyle=$
$\displaystyle\bar{\psi}(i\not\\!D_{a}T_{a}-m)\psi-\frac{1}{4}F^{\mu\nu}_{a}F_{\mu\nu
a}-\frac{1}{2}\lambda(\partial_{\mu}A^{\mu}_{a})^{2}+\partial_{\mu}\eta^{\dagger}_{a}(\partial^{\mu}+gf_{abc}A^{\mu}_{c})\eta_{b},$
(4)

with the gauge parameter $\lambda$, and the ghost field $\eta$. The last term
in the above expression comes from the Jacobian for the variable change, as
fixing the gauge.

The Feynman rules for QCD can be derived from Eq. (4) following the standard
procedures [25]. The quark and gluon propagators with the momentum $p$ are
given by $i\not\\!p/(p^{2}+i\epsilon)$ and $-ig^{\mu\nu}/p^{2}$ in the Feynman
gauge, respectively. The quark-gluon-quark vertex and the ghost-gluon-ghost
vertex are written as $-ig\gamma_{\mu}T_{a}$ and $gf_{abc}p^{\prime}_{\mu}$,
respectively, where the subscripts $\mu$ and $a$ are associated with the
gluon, $p^{\prime}$ is the momentum of the outgoing ghost, and $b$ ($c$) is
associated with the outgoing (incoming) ghost. The three-gluon vertex and the
four-gluon vertex are given by

$\displaystyle\Gamma_{3g}$ $\displaystyle=$ $\displaystyle-
gf_{a_{1}a_{2}a_{3}}[g^{\nu_{1}\nu_{2}}(p_{1}-p_{2})^{\nu_{3}}+g^{\nu_{2}\nu_{3}}(p_{2}-p_{3})^{\nu_{1}}+g^{\nu_{3}\nu_{1}}(p_{3}-p_{1})^{\nu_{2}},$
$\displaystyle\Gamma_{4g}$ $\displaystyle=$ $\displaystyle-
ig^{2}[f_{ea_{1}a_{2}}f_{ea_{3}a_{4}}(g^{\nu_{1}\nu_{3}}g^{\nu_{2}\nu_{4}}-g^{\nu_{1}\nu_{4}}g^{\nu_{2}\nu_{3}})+f_{ea_{1}a_{3}}f_{ea_{4}a_{2}}(g^{\nu_{1}\nu_{4}}g^{\nu_{3}\nu_{2}}-g^{\nu_{1}\nu_{2}}g^{\nu_{3}\nu_{4}})$
(5)
$\displaystyle\;\;\;\;+f_{ea_{1}a_{4}}f_{ea_{2}a_{3}}(g^{\nu_{1}\nu_{2}}g^{\nu_{4}\nu_{3}}-g^{\nu_{1}\nu_{3}}g^{\nu_{4}\nu_{2}})],$

respectively, where the subscripts $a_{1}$, $a_{2}$, $\cdots$ and $\nu_{1}$,
$\nu_{2}$, $\cdots$ are assigned to gluons counterclockwise. The particle
momenta flow into the vertices in all the above Feynman rules.

###  0.2.1 Infrared Divergences and Safety

The first step to establish the factorization theorem is to identify infrared
divergences in Feynman diagrams for a QCD process at quark-gluon level. We
start with the vertex correction to the amplitude $\gamma^{*}(q)\to
q(p_{1})\bar{q}(p_{2})$, in which a virtual photon of momentum $q=p_{1}+p_{1}$
splits into a quark of momentum $p_{1}$ and an anti-quark of momentum $p_{2}$.
Given the Feynman rules, one has the loop integral

$\displaystyle\int\frac{d^{4}l}{(2\pi)^{4}}(-ig\gamma^{\nu}T_{a})\frac{i(\not\\!p_{1}-\not\\!l)}{(p_{1}-l)^{2}+i\epsilon}(-ie\gamma_{\mu})\frac{-i(\not\\!p_{2}-\not\\!l)}{(p_{2}-l)^{2}+i\epsilon}(-ig\gamma_{\nu}T_{a})\frac{-i}{l^{2}+i\epsilon},$
(6)

where $l$ is the loop momentum carried by the gluon, and the inclusion of the
corresponding counterterm for the regularization of a ultraviolet divergence
is understood. The appearance of infrared divergences becomes more
transparent, as performing the contour integration in the light-cone frame, in
which the coordinates $l^{\mu}=(l^{+},l^{-},{\bf l}_{T})$ are defined by

$\displaystyle l^{\pm}=\frac{l^{0}\pm l^{z}}{\sqrt{2}},\;\;\;\;{\bf
l}_{T}=(l^{x},l^{y}).$ (7)

When an on-shell particle moves along the light cone, only one component of
its momentum is large in this frame. For example, the above quark momenta can
be chosen as $p_{1}^{\mu}=(p_{1}^{+},0,{\bf 0}_{T})$ and
$p_{2}^{\mu}=(0,p_{2}^{-},{\bf 0}_{T})$.

In terms of the light-cone coordinates, Eq. (6) is reexpressed as

$\displaystyle\int\frac{dl^{+}dl^{-}d^{2}l_{T}}{(2\pi)^{4}}\frac{1}{2(l^{+}-p_{1}^{+})l^{-}-l_{T}^{2}+i\epsilon}\frac{1}{2l^{+}(l^{-}-p_{2}^{-})-l_{T}^{2}+i\epsilon}\frac{1}{2l^{+}l^{-}-l_{T}^{2}+i\epsilon},$
(8)

where only the denominators are shown, since infrared divergences are mainly
determined by pole structures. The poles of $l^{-}$ are located, for
$0<l^{+}<p_{1}^{+}$, at

$\displaystyle
l^{-}=\frac{l_{T}^{2}}{2(l^{+}-p_{1}^{+})}+i\epsilon,\;\;l^{-}=p_{2}^{-}+\frac{l_{T}^{2}}{2l^{+}}-i\epsilon,\;\;l^{-}=\frac{l_{T}^{2}}{2l^{+}}-i\epsilon.$
(9)

As $l^{+}\sim O(p_{1}^{+})$, the contour of $l^{-}$ is pinched at $l^{-}\sim
O(l_{T}^{2}/p_{1}^{+})$ by the first and third poles, defining the collinear
region. As $l^{+}\sim O(l_{T})$, the contour of $l^{-}$ is pinched at
$l^{-}\sim O(l_{T})$, defining the soft region. That is, the collinear (soft)
region corresponds to the configuration of
$l^{\mu}\sim(E,\Lambda^{2}/E,\Lambda)$
($l^{\mu}\sim(\Lambda,\Lambda,\Lambda)$), where $E$ and $\Lambda$ denote a
large scale and a small scale, respectively. Another leading configuration
arises from the hard region characterized by $l^{\mu}\sim(E,E,E)$. A simple
power counting implies that all the above three regions give logarithmic
divergences. Picking up the first pole in Eq. (9), Eq. (8) becomes

$\displaystyle\frac{-i}{2p_{1}^{+}}\int\frac{dl^{+}d^{2}l_{T}}{(2\pi)^{3}}\frac{p_{1}^{+}-l^{+}}{2p_{2}^{-}l^{+}(p_{1}^{+}-l^{+})+p_{1}^{+}l_{T}^{2}}\frac{1}{l_{T}^{2}}\approx\frac{-i}{4p_{1}\cdot
p_{2}}\frac{1}{(2\pi)^{3}}\int\frac{dl^{+}}{l^{+}}\int\frac{d^{2}l_{T}}{l_{T}^{2}},$
(10)

which produces the double logarithm from the overlap of the collinear (the
integration over $l^{+}$) and soft (the integration over $l_{T}$)
enhancements.

The existence of infrared divergences is a general feature of QCD corrections.
An amplitude is not a physical quantity, but a cross section is. To examine
whether the infrared divergences really call for attention, we extend the
calculation to the cross section of the process $e^{-}e^{+}\to X$, the
$e^{-}e^{+}$ annihilation into hadrons. A cross section is computed as the
square of an amplitude, whose Feynman diagrams are composed of those for the
amplitude connected by their complex conjugate with a final-state cut between
them. The cross section at the Born level $e^{-}e^{+}\to\gamma^{*}\to
q\bar{q}$ is written as

$\displaystyle\sigma^{(0)}=N_{c}\frac{4\pi\alpha^{2}}{3Q^{2}}\sum_{f}Q_{f}^{2},$
(11)

where $N_{c}=3$ is the number of colors, $\alpha$ is the electromagnetic
coupling constant, $Q^{2}$ is the $e^{-}e^{+}$ invariant mass, and $Q_{f}$ is
the quark charge in units of the electron charge. The virtual one-loop
corrections, including those to the gluon vertex in Eq. (6) and to the quark
self-energy, give in the dimensional regularization [25]

$\displaystyle\sigma^{(1)V}=-2N_{c}C_{F}\sum_{f}Q_{f}^{2}\frac{\alpha\alpha_{s}}{\pi}Q^{2}\left(\frac{4\pi\mu^{2}}{Q^{2}}\right)^{2\epsilon}\frac{1-\epsilon}{\Gamma(2-2\epsilon)}\left[\frac{1}{\epsilon^{2}}+\frac{3}{2}\frac{1}{\epsilon}-\frac{\pi^{2}}{2}+4+O(\epsilon)\right],$
(12)

with the color factor $C_{F}=4/3$, the strong coupling constant $\alpha_{s}$,
the renormalization scale $\mu$, and the Gamma function $\Gamma$. The double
pole $1/\epsilon^{2}$ is a consequence of the overlap of the collinear and
soft divergences. The one-loop corrections from real gluons lead to [25]

$\displaystyle\sigma^{(1)R}=2N_{c}C_{F}\sum_{f}Q_{f}^{2}\frac{\alpha\alpha_{s}}{\pi}Q^{2}\left(\frac{4\pi\mu^{2}}{Q^{2}}\right)^{2\epsilon}\frac{1-\epsilon}{\Gamma(2-2\epsilon)}\left[\frac{1}{\epsilon^{2}}+\frac{3}{2}\frac{1}{\epsilon}-\frac{\pi^{2}}{2}+\frac{19}{4}+O(\epsilon)\right].$
(13)

Figure 1: Final-state cut on self-energy corrections to a virtual photon
propagator.

It is a crucial observation that the infrared divergences cancel in the
summation over the virtual and real corrections in Eqs. (12) and (13),
respectively: the double and single poles have a minus sign in the former, but
a plus sign in the latter. It is easy to understand the infrared cancellation
by means of self-energy corrections to the propagator of a virtual photon.
Since a virtual photon does not involve a low characteristic scale, the loop
corrections must be infrared finite. As taking the final-state cut shown in
Fig. 1, the imaginary piece of a particle propagator is picked up, ${\rm
Im}(1/(p^{2}+i\epsilon))\propto\delta(p^{2})$, which corresponds to the
Feynman rule for an on-shell particle. Because the self-energy corrections are
infrared finite, their imaginary part, i.e., the $e^{-}e^{+}\to X$ cross
section, is certainly infrared finite. The above observation has been
formulated into the Kinoshita-Lee-Nauenberg (KLN) theorem [26], which states
that a cross section is infrared safe, as integrating over all phase spaces of
final states. Combining Eqs. (11), (12), and (13), one derives the
$e^{-}e^{+}\to X$ cross section up to next-to-leading order (NLO)

$\displaystyle\sigma=N_{c}\frac{4\pi\alpha^{2}}{3Q^{2}}\sum_{f}Q_{f}^{2}\left[1+\frac{3}{4}\frac{\alpha_{s}(Q)}{\pi}C_{F}\right],$
(14)

that has been used to determine the strong coupling constant $\alpha_{s}(Q)$
at the scale $Q$.

###  0.2.2 DIS and Collinear Factorization

Though a naive perturbation theory applies to the $e^{-}e^{+}$ annihilation,
it fails for more complicated ones, such as the deeply inelastic scattering
(DIS) of a nucleon by a lepton, $\ell(k)N(p)\to\ell(k^{\prime})+X$. Even as
the momentum transfer squared $-q^{2}=(k-k^{\prime})^{2}\equiv Q^{2}$ is
large, the quark-level cross section for the DIS suffers infrared divergences
at high orders, which reflect the nonperturbative dynamics in the nucleon. A
special treatment of the infrared divergences is then required. It will be
demonstrated that they can be factorized out of the scattering process, and
absorbed into a nucleon PDF.

Consider the two structure functions $F_{1,2}(x,Q^{2})$ involved in the DIS,
where the Bjorken variable is defined as $x\equiv-q^{2}/(2p\cdot
q)=Q^{2}/(2p\cdot q)$, and take $F_{2}$ as an example. We shall not repeat
loop integrations, but quote the NLO corrections to the quark-level diagrams
[25]:

$\displaystyle F_{2}^{q}(x,Q^{2})$ $\displaystyle=$ $\displaystyle
x\left\\{\delta(1-x)+\frac{\alpha_{s}}{2\pi}C_{F}\left[\frac{1+x^{2}}{1-x}\left(\ln\frac{1-x}{x}-\frac{3}{4}\right)+\frac{1}{4}\left(9+5x\right)\right]_{+}\right.$
(15)
$\displaystyle\left.+\frac{\alpha_{s}}{2\pi}C_{F}\left(\frac{1+x^{2}}{1-x}\right)_{+}\left(4\pi\mu
e^{-\gamma_{E}}\right)^{\epsilon}\int_{0}^{Q^{2}}\frac{dk_{T}^{2}}{k_{T}^{2+2\epsilon}}+\cdots\right\\},$

where the superscript $q$ denotes the initial-state quark, $\gamma_{E}$ is the
Euler constant, and the first term comes from the leading-order (LO)
contribution. The subscript $+$ represents the plus function, which is
understood as a distribution function via

$\displaystyle\int_{0}^{1}dx\frac{f(x)}{(1-x)_{+}}\equiv\int_{0}^{1}dx\frac{f(x)-f(1)}{1-x}.$
(16)

The integration over $k_{T}^{2}$ generates an infrared divergence, that is
regularized in the dimensional regularization with $\epsilon<0$,

$\displaystyle\int_{0}^{Q^{2}}\frac{dk_{T}^{2}}{k_{T}^{2+2\epsilon}}=\frac{1}{-\epsilon}\left(Q^{2}\right)^{-\epsilon}.$
(17)

Hence, the infrared divergence does exist in the perturbative evaluation of
the DIS structure function, even after summing over the virtual and real
corrections. This divergence arises from the collinear region with the loop
momentum being parallel to the nucleon momentum, since it can also be
regularized by introducing a mass to the initial-state quark. It is related to
the confinement mechanism, and corresponds to a long-distance phenomenon
associated with a group of collimated on-shell particles. The other terms in
Eq. (15) represent the hard NLO contribution to the structure function.
Comparing the results for the DIS and for the $e^{-}e^{+}$ annihilation, the
former involves the integration over final-state kinematics, but not over
initial-state kinematics. This is the reason why the KLN theorem does not
apply to the infrared divergences associated with the initial-state nucleon,
and the above collinear divergence exists. Note that the soft divergences
cancel between virtual and real diagrams due to the fact that a nucleon is
color-singlet: a soft gluon with a huge space-time distribution cannot resolve
the color structure of a nucleon, so it does not interact with it.

Besides, the collinear gluon emissions modify a quark momentum, such that the
initial-state quark can carry various momenta, as it participates in hard
scattering. It is then natural to absorb the collinear divergences into a PDF
for the nucleon, $\phi_{q/N}$, which describes the probability for quark $q$
to carry certain amount of the nucleon momentum. In other words, the quark-
level collinear divergences are subtracted by those in the PDF in perturbation
theory, and the remaining infrared finite piece contributes to the hard kernel
$H$. We write the quark-level structure function as the following expansion in
the strong coupling constant,

$\displaystyle
F_{2}^{q}(x,Q^{2})=H^{(0)}\otimes\phi_{f/N}^{(0)}+\frac{\alpha_{s}}{2\pi}H^{(1)}\otimes\phi_{q/N}^{(0)}+\frac{\alpha_{s}}{2\pi}H^{(0)}\otimes\phi_{q/N}^{(1)}+\cdots,$
(18)

where $H^{(i)}$ ($\phi_{q/N}^{(i)}$) is the hard kernel (PDF) of the $i$-th
order. The symbol $\otimes$ represents a convolution in the parton momentum
fraction $\xi$:

$\displaystyle
H\otimes\phi_{q/N}\equiv\int_{x}^{1}\frac{d\xi}{\xi}H(x/\xi,Q,\mu)\phi_{q/N}(\xi,\mu).$
(19)

We are ready to assign each term in Eq. (15) into either $H^{(i)}$ or
$\phi_{q/N}^{(i)}$. The first term $\delta(1-x)$ goes to
$H^{(0)}\otimes\phi_{q/N}^{(0)}$ with the definitions

$\displaystyle
H^{(0)}(x/\xi,Q,\mu)=\delta(1-x/\xi),\;\;\;\;\phi_{q/N}^{(0)}(\xi,\mu)=\delta(1-\xi),$
(20)

which confirm $H^{(0)}\otimes\phi_{q/N}^{(0)}=\delta(1-x)$. The second term in
Eq. (15) is assigned to $H^{(1)}\otimes\phi_{q/N}^{(0)}$ and the third term to
$H^{(0)}\otimes\phi_{q/N}^{(1)}$ with

$\displaystyle H^{(1)}(x,Q,\mu)$ $\displaystyle=$ $\displaystyle
P_{qq}^{(1)}(x)\ln\frac{Q^{2}}{\mu^{2}}+\cdots,$
$\displaystyle\phi_{q/N}^{(1)}(\xi,\mu)$ $\displaystyle=$
$\displaystyle\left(4\pi\mu
e^{-\gamma}\right)^{\epsilon}P_{qq}^{(1)}(\xi)\int_{0}^{\mu^{2}}\frac{dk_{T}^{2}}{k_{T}^{2+2\epsilon}},$
(21)

and the quark splitting function

$\displaystyle P_{qq}^{(1)}(x)=C_{F}\left(\frac{1+x^{2}}{1-x}\right)_{+}.$
(22)

The definition of the PDF in terms of a hadronic matrix element is given by

$\displaystyle\phi_{q/N}(\xi,\mu)$ $\displaystyle=$
$\displaystyle\int\frac{dy^{-}}{2\pi}\exp(-i\xi p^{+}y^{-})$ (23)
$\displaystyle\times\frac{1}{2}\sum_{\sigma}\langle
N(p,\sigma)|\bar{q}(0,y^{-},0_{T})\frac{1}{2}\gamma^{+}W(y^{-},0)q(0,0,0_{T})|N(p,\sigma)\rangle,$

where $|N(p,\sigma)\rangle$ denotes the bound state of the nucleon with
momentum $p$ and spin $\sigma$, $y^{-}$ is the minus component of the
coordinate of the quark field after the final-state cut, the first factor
$1/2$ is attributed to the average over the nucleon spin, and the matrix
$\gamma^{+}/2$ is the spin projector for the nucleon. Here $\mu$ is called the
factorization scale, which is similar to a renormalization scale, but
introduced in perturbative computations for an effective theory. The Wilson
lines are defined by $W(y^{-},0)=W(0)W^{\dagger}(y^{-})$ with

$\displaystyle W(y^{-})={\cal P}\exp\left[-ig\int_{0}^{\infty}dzn_{-}\cdot
A(y+zn_{-})\right],$ (24)

where $\cal P$ represents a path-ordered exponential. The Wilson line behaves
like a scalar particle carrying a color source. The two quark fields in Eq.
(23) are separated by a distance, so the above Wilson links are demanded by
the gauge invariance of the nonlocal matrix element. Since Eq. (23) depends
only on the property of the nucleon, but not on the hard processes it
participates in, a PDF is universal (process-independent). This is the most
important observation, that warrants the predictive power of the factorization
theorem.

The Wilson line appears as a consequence of the eikonalization of the final-
state quark, to which the collinear gluons attach. The eikonalization is
illustrated below by considering the loop correction to the virtual photon
vertex. Assuming the initial-state quark momentum $p=(p^{+},0,\bf 0_{T})$ and
the final-state quark momentum $p^{\prime}=(0,p^{\prime-},\bf 0_{T})$, we have
the partial integrand

$\displaystyle\not\\!p^{\prime}\gamma^{\nu}\frac{\not\\!p^{\prime}+\not
l}{(p^{\prime}+l)^{2}}\gamma^{\mu}\frac{\not\\!p+\not
l}{(p+l)^{2}}\gamma_{\nu}\approx\not\\!p^{\prime}\gamma^{-}\frac{\not\\!p^{\prime}+\not
l}{(p^{\prime}+l)^{2}}\gamma^{\mu}\frac{\not\\!p+\not
l}{(p+l)^{2}}\gamma^{+}\approx\not\\!p^{\prime}\gamma^{-}\frac{\not\\!p^{\prime}}{2p^{\prime}\cdot
l}\gamma^{\mu}\frac{\not\\!p+\not l}{(p+l)^{2}}\gamma^{+},$ (25)

as the loop momentum $l$ is collinear to $p$, where $\not\\!p^{\prime}$ comes
from the Feynman rule for the final-state quark, $\gamma^{\mu}$ is the photon
vertex, and the subleading contribution from the transverse components of
$\gamma^{\nu}$ has been neglected. Applying the identity
$\gamma^{-}\not\\!p^{\prime}=2p^{\prime-}-\not\\!p^{\prime}\gamma^{-}$ and
$\not\\!p^{\prime}\not\\!p^{\prime}=p^{\prime 2}=0$ leads the above expression
to

$\displaystyle\not\\!p^{\prime}\gamma^{\mu}\frac{\not\\!p+\not
l}{(p+l)^{2}}\gamma^{+}\frac{p^{\prime-}}{p^{\prime}\cdot
l}\approx\not\\!p^{\prime}\gamma^{\mu}\frac{\not\\!p+\not
l}{(p+l)^{2}}\gamma^{+}\frac{n_{-}^{-}}{n_{-}\cdot
l}\approx\not\\!p^{\prime}\gamma^{\mu}\frac{\not\\!p+\not
l}{(p+l)^{2}}\gamma_{\nu}\frac{n_{-}^{\nu}}{n_{-}\cdot l},$ (26)

where the dimensionless vector $n_{-}=(0,1,\bf 0_{T})$ is parallel to
$p^{\prime}$, and the subleading contribution from $\nu=T$ has been restored.
The factor $n_{-}^{\nu}$ and $1/n_{-}\cdot l$ are called the eikonal vertex
and the eikonal propagator, respectively.

It is then shown that the Feynman rule $n_{-}^{\nu}/n_{-}\cdot l$ for the
eikonalized final-state quark is derived from the Wilson line in Eq. (24).
Consider the expansion of the path-order exponential in $W(0)$ up to order of
$\alpha_{s}$, and Fourier transform the gauge field into the momentum space,

$\displaystyle-ig\int_{0}^{\infty}dzn_{-}\cdot\int d^{4}l\exp[iz(n_{-}\cdot
l+i\epsilon)]\tilde{A}(l)$ (27) $\displaystyle=$ $\displaystyle-ig\int
d^{4}l\frac{\exp[iz(n_{-}\cdot l+i\epsilon)]}{i(n_{-}\cdot
l+i\epsilon)}\bigg{|}_{z=0}^{z=\infty}n_{-}\cdot\tilde{A}(l)=\int
d^{4}l\frac{gn_{-}^{\nu}}{n_{-}\cdot l+i\epsilon}\tilde{A}_{\nu}(l),$

where the term $i\epsilon$ has been introduced to suppress the contribution
from $z=\infty$. The field $\tilde{A}(l)$ is contracted with the gauge field
from the initial-state quark with interaction to form the gluon propagator
$-i/(l^{2}+i\epsilon)$. The expansion of the second piece $W(y^{-})$ gives the
Feynman rules for the eikonal propagator appearing after the final-state cut.
In this case the additional exponential factor $\exp(il\cdot y)$ is combined
with $\exp(-i\xi p^{+}y^{-})$, implying that the valence quark
$q(0,y^{-},0_{T})$ after the final-state cut carries the momentum $\xi p-l$.
In summary, the first (second) piece of Wilson lines corresponds to the
configuration without (with) the loop momentum flowing through the hard
kernel. The above discussion verifies the Wilson lines in the PDF definition.

After detaching the collinear gluons from the final-state quark, the fermion
flow still connects the PDF and the hard kernel. To achieve the factorization
in the fermion flow, we insert the Fierz identity,

$\displaystyle I_{ij}I_{lk}$ $\displaystyle=$
$\displaystyle\frac{1}{4}I_{ik}I_{lj}+\frac{1}{4}(\gamma_{\alpha})_{ik}(\gamma^{\alpha})_{lj}+\frac{1}{4}(\gamma^{5}\gamma_{\alpha})_{ik}(\gamma^{\alpha}\gamma^{5})_{lj}$
(28)
$\displaystyle+\frac{1}{4}(\gamma^{5})_{ik}(\gamma^{5})_{lj}+\frac{1}{8}(\gamma^{5}\sigma_{\alpha\beta})_{ik}(\sigma^{\alpha\beta}\gamma^{5})_{lj},$

with $I$ being the identity matrix and $\sigma_{\alpha\beta}\equiv
i[\gamma_{\alpha},\gamma_{\beta}]/2$. At leading power, only the term
$(\gamma_{\alpha})_{ik}(\gamma^{\alpha})_{lj}/4$ contributes, in which the
structure $(\gamma^{\alpha})_{lj}/2\approx(\gamma^{+})_{lj}/2$ goes to the
definition of the PDF in Eq. (23), and
$(\gamma_{\alpha})_{ik}/2\approx(\gamma^{-})_{ik}/2$ goes into the evaluation
of the hard kernel. The other terms in Eq. (28) contribute at higher powers.
Similarly, we have to factorize the color flow between the PDF and the hard
kernel by inserting the identity

$\displaystyle
I_{ij}I_{lk}=\frac{1}{N_{c}}I_{ik}I_{lj}+2(T^{c})_{ik}(T^{c})_{lj},$ (29)

where $I$ denotes the $3\times 3$ identity matrix, and $T^{c}$ is a color
matrix. The first term in the above expression contributes to the present
configuration, in which the valence quarks before and after the final-state
cut are in the color-singlet state. The structure $I_{lj}/N_{c}$ goes into the
definition of the PDF, and $I_{ik}$ goes into the evaluation of the hard
kernel. The second term in Eq. (29) contributes to the color-octet state of
the valence quarks, together with which an additional gluonic parton comes out
of the nucleon and participates in the hard scattering.

The factorization formula for the nucleon DIS structure function is written as

$\displaystyle
F_{2}(x,Q^{2})=\sum_{f}\int_{x}^{1}\frac{d\xi}{\xi}H_{f}(x/\xi,Q,\mu)\phi_{f/N}(\xi,\mu),$
(30)

with the subscript $f$ labeling the parton flavor, such as a valence quark, a
gluon, or a sea quark. The hard kernel $H_{f}$ is obtained following the
subtraction procedure for the collinear divergences, and its LO and NLO
expressions have been presented in Eqs. (20) and (21), respectively. The
universal PDF $\phi_{f/N}$, describing the probability for parton $f$ to carry
the momentum fraction $\xi$ in the nucleon, takes a smooth model function. It
must be derived by nonperturbative methods, or extracted from data.

###  0.2.3 Predictive Power

The factorization theorem derived above is consistent with the well-known
parton model. The nucleon travels a long space-time, before it is hit by the
virtual photon. As $Q^{2}\gg 1$, the hard scattering occurs at point space-
time. Relatively speaking, the quark in the nucleon behaves like a free
particle before the hard scattering, and decouples from the rest of the
nucleon. Therefore, the cross section for the nucleon DIS reduces to an
incoherent sum over parton flavors under the collinear factorization. That is,
the approximation

$\displaystyle\left|\sum_{i}{\cal
M}_{i/N}\right|^{2}\approx\sum_{i}\left|{\cal M}_{f}\right|^{2}\phi_{f/N},$
(31)

holds, where ${\cal M}_{i/N}$ represents the scattering amplitude for partonic
state $i$ of the nucleon $N$ (it could be a multi-parton state), and ${\cal
M}_{f}$ represents the infrared finite scattering amplitude for parton $f$.

Comparing the factorization theorem with the operator product expansion (OPE),
the latter involves an expansion in short distance $y^{\mu}\sim 0$. A typical
example is the infrared safe $e^{-}e^{+}\to X$, whose cross section can be
expressed as a series $\sigma\approx\sum_{i}C_{i}(y)O_{i}(0)$. The Wilson
coefficients $C_{i}$ and the local effective operators $O_{i}$ appear in a
product in the OPE. A factorization formula involves an expansion on the light
cone with small $y^{2}\sim 0$, instead of $y^{\mu}\sim 0$. A typical example
is the DIS structure function, in which the existence of the collinear
divergences implies that a parton travels a finite longitudinal distance
$y^{-}$. It is also the reason why the hard kernel $H_{f}$ and the PDF
$\phi_{f/N}$ appear in a convolution in the momentum fraction.

Figure 2: CT10 NNLO (solid color) and NLO (dashed) parton distribution
functions.

The factorization procedure introduces the factorization scale $\mu$ into the
hard kernel $H_{f}$ and the PDF $\phi_{f/N}$, as indicated in Eq. (30).
Higher-order corrections produce the logarithms $\ln(Q/\mu)$ in $H_{f}$ and
$\ln(\mu/Q_{0})$ in $\phi_{f/N}$, which come from the splitting of
$\ln(Q/Q_{0})$ in the structure function $F_{2}$, $Q_{0}$ being a low scale
characterizing $\phi_{f/N}$. One usually sets $\mu=Q$ to eliminate the
logarithm in $H_{f}$, such that the input $\phi_{f/N}(\xi,Q)$ for arbitrary
$Q$ is needed. The factorization scale does not exist in QCD diagrams, but is
introduced when a physical quantity like the structure function is factorized.
The independence of the factorization scale, $\mu dF_{2}/d\mu=0$, leads to a
set of renormalization-group (RG) equations

$\displaystyle\mu\frac{d}{d\mu}\phi_{f/N}(\xi,\mu)=\gamma_{f}\phi_{f/N}(\xi,\mu),$
$\displaystyle\mu\frac{d}{d\mu}H_{f}(x/\xi,Q,\mu)=-\gamma_{f}H_{f}(x/\xi,Q,\mu),$
(32)

where $\gamma_{f}$ denotes the anomalous dimension of the PDF. A solution of
the RG equations describes the evolution of the PDF in $Q$

$\displaystyle\phi_{f/N}(\xi,Q)=\phi_{f/N}(\xi,Q_{0})\exp\left[\int_{Q_{0}}^{Q}\frac{d\mu}{\mu}\gamma_{f}(\alpha_{s}(\mu))\right],$
(33)

as a result of the all-order summation of $\ln(Q/Q_{0})$. Hence, one just
extracts the initial condition $\phi(\xi,Q_{0})$ defined at the initial scale
$Q_{0}$ from data. The PDF at other higher scales $Q$ is known through the
evolution. That is, the inclusion of the RG evolution increases the predictive
power of the factorization theorem.

Fitting the factorization formulas for those processes, whose dynamics is
believed to be clear, such as Eq. (30) for DIS, one has determined the PDFs
for various partons in the proton. The CTEQ-TEA CT10 models at the accuracy of
NLO and next-to-next-to-leading order (NNLO) for hard kernels are displayed in
Figs. 2 [27, 28]. The increase of the gluon and sea-quark PDFs with the
decrease of the momentum fraction $\xi$ is a consequence of more radiations in
that region in order to reach a lower $\xi$. The comparison of the PDFs at
$Q=2$ GeV and $Q=85$ GeV indicates that the valence $u$-quark and $d$-quark
PDFs become broader with $Q$, while the gluon and sea-quark PDFs increase with
$Q$.

Note that a choice of an infrared regulator is, like an ultraviolet regulator,
arbitrary; namely, we can associate an arbitrary finite piece with the
infrared pole $1/(-\epsilon)$ in $\phi^{(1)}_{f/N}$. Shifts of different
finite pieces between $\phi_{f/N}$ and $H_{f}$ correspond to different
factorization schemes. Hence, the extraction of a PDF depends not only on
powers and orders, at which QCD diagrams are computed, but on factorization
schemes. Since perturbative calculations are performed up to finite powers and
orders, a factorization scheme dependence is unavoidable. Nevertheless, the
scheme dependence of pQCD predictions would be minimized, if one sticks to the
same factorization scheme. Before adopting models for PDFs, it should be
checked at which power and order, at which initial scale, and in what scheme
they are determined.

Figure 3: Comparison of ATLAS data for inclusive jet $p_{T}$ distribution with
a theoretical prediction using CT10 NNLO.

At last, I explain how to apply the factorization theorem to make predictions
for QCD processes. A nucleon PDF $\phi_{f/N}$ is infrared divergent, if
evaluated in perturbation theory due to the confinement mechanism. The QCD
diagram for a DIS structure function involving quarks and gluons as the
external particles are also infrared divergent. It has been demonstrated that
the infrared divergences cancel between the QCD diagrams and the effective
diagrams for $\phi_{f/N}$, as taking their difference, which defines the hard
kernel $H^{\rm DIS}$. One then derives the factorization formula for other
processes, such as the Drell-Yan (DY) process
$N(p_{1})N(p_{2})\to\ell^{+}\ell^{-}(q)+X$, and computes the corresponding
hard kernel $H^{\rm DY}$. The point is to verify that the infrared divergences
in the QCD diagrams for DY and in the effective diagrams for the nucleon PDF
cancel, and $H^{\rm DY}$ is infrared finite. If it is the case, the
universality of the nucleon PDF holds, and the factorization theorem is
applicable. If not, the factorization theorem fails. After verifying the
factorization theorem, one makes predictions for the DY cross section using
the formula $\sigma^{\rm DY}=\phi_{f_{1}/N}\otimes H^{\rm
DY}\otimes\phi_{f_{2}/N}$. As an example, the predictions for the inclusive
jet $p_{T}$ distribution derived from the factorization theorem [28] are
presented in Fig. 3. The consistency between the predictions and the ATLAS
data is obvious.

###  0.2.4 $k_{T}$ Factorization

The collinear factorization theorem introduced above has been intensively
investigated and widely applied to many QCD processes up to higher powers and
orders. The evolution of PDFs from low to high factorization scales is
governed by the DGLAP equation. The databases for PDFs have been constructed,
such as the CTEQ models. Other nonperturbative inputs like soft functions, jet
functions, and fragmentation functions have been all explored to some extent.
However, another more complicated framework, the $k_{T}$ factorization theorem
[29, 30, 31], may be more appropriate in some kinematic regions or in semi-
inclusive processes. The collinear factorization applies, when the DIS is
measured at a finite Bjorken variable $x$. The cross section is written as the
convolution of a hard kernel with a PDF in a parton momentum fraction $\xi$.
As $x\to 0$, $\xi\geq x$ can reach a small value, at which the parton
transverse momentum $k_{T}$ is of the same order of magnitude as the
longitudinal momentum $\xi p$, and not negligible. Once $k_{T}$ is kept in a
hard kernel, a transverse-momentum-dependent (TMD) function
$\Phi(\xi,k_{T},\mu)$ is needed to describe the parton distribution not only
in the momentum fraction $\xi$, but also in the transverse momentum $k_{T}$.
The DIS cross section is then written, in the $k_{T}$ factorization theorem,
as the convolution

$\displaystyle F_{2}(x,Q^{2})=\sum_{f}\int_{x}^{1}\frac{d\xi}{\xi}\int
d^{2}k_{T}H_{f}(x/\xi,k_{T},Q,\mu)\Phi_{f/N}(\xi,k_{T},\mu).$ (34)

The $k_{T}$ factorization theorem is also applicable to the analysis of low
$p_{T}$ spectra of final states, like direct photon and jet productions, for
which $k_{T}\sim p_{T}$ is not negligible.

A collinear gluon emission, modifying a parton longitudinal momentum,
generates a parton transverse momentum $k_{T}$ at the same time. The
factorization of a TMD from the DIS is similar to that of a PDF, which relies
on the eikonal approximation in the collinear region. This procedure results
in the eikonal propagator $n_{-}^{\nu}/n_{-}\cdot l$, represented by the
Wilson lines similar to that defined in Eq. (24). A naive TMD definition as an
extension of the PDF in Eq. (23) is given by

$\displaystyle\Phi_{q/N}(\xi,k_{T},\mu)$ $\displaystyle=$
$\displaystyle\int\frac{dy^{-}}{2\pi}\int\frac{d^{2}y_{T}}{(2\pi)^{2}}e^{-i\xi
p^{+}y^{-}+i{\bf k}_{T}\cdot{\bf y}_{T}}$ (35)
$\displaystyle\times\frac{1}{2}\langle
N(p,\sigma)|\bar{q}(0,y^{-},y_{T})\frac{1}{2}\gamma^{+}W(y^{-},y_{T},0,0_{T})q(0,0,0_{T})|N(p,\sigma)\rangle,$

with the Wilson links
$W(y^{-},y_{T},0,0_{T})=W(0,0_{T})I_{0,y_{T}}W^{\dagger}(y^{-},y_{T})$.
Because the valence quark fields before and after the final-state cut are
separated by a transverse distance in this case, the vertical links
$I_{0,y_{T}}$ located at $y^{-}=\infty$ are demanded by the gauge invariance
of a TMD [32]. More investigations on the vertical Wilson links can be found
in [33].

Though we do need the $k_{T}$ factorization theorem, many of its aspects have
not yet been completely understood. For example, the naive definition in Eq.
(35) is actually ill-defined, due to the existence of the light-cone
singularity, that arises from a loop momentum parallel to the Wilson line
direction $n_{-}$. A plausible modification is to rotate the Wilson line away
from the light cone, namely, to replace $n_{-}$ by a vector $n$ with
$n^{2}\not=0$. This rotation is allowed, since the collinear divergences are
insensitive to the direction $n$ as illustrated in Eq. (26) [34]: even when
$n_{-}$ is rotated to $n$, only the minus component $n^{-}$ is relevant for
the evaluation of the collinear divergences. A detailed discussion on this
subtle issue can be found in [35]. Besides, a parton is off-shell by
$-k_{T}^{2}$, once $k_{T}$ is retained. Then whether a hard kernel obtained in
the $k_{T}$ factorization theorem is gauge invariant becomes a concern [36].
Dropping the $k_{T}$ dependence of the hard kernel in Eq. (34), the
integration of the TMD over $k_{T}$, $\int d^{2}k_{T}\Phi_{f/N}(\xi,k_{T})$,
can be worked out. How this integral is related to the PDF $\phi_{f/N}(\xi)$
in Eq. (23) is worth of a thorough study.

##  0.3 Evolution and resummation

As stated in the previous section, radiative corrections in pQCD produce large
logarithms at each order of the coupling constant. Double logarithms appear in
processes involving two scales, such as $\ln^{2}(p^{+}b)$ with $p^{+}$ being
the large longitudinal momentum of a parton and $1/b$ being the small inverse
impact parameter, where $b$ is conjugate to the parton transverse momentum
$k_{T}$. In the region with large Bjorken variable $x$, there exists
$\ln^{2}(1/N)$ from the Mellin transformation of $\ln(1-x)/(1-x)_{+}$, for
which the two scales are the large $p^{+}$ and the small infrared cutoff
$(1-x)p^{+}$ for gluon emissions from a parton. Single logarithms are
generated in processes involving one scale, such as $\ln p^{+}$ and
$\ln(1/x)$, for which the relevant scales are the large $p^{+}$ and the small
$xp^{+}$, respectively. Various methods have been developed to organize these
logarithmic corrections to a PDF or a TMD: the $k_{T}$ resummation for
$\ln^{2}(p^{+}b)$ [8, 9], the threshold resummation for $\ln^{2}(1/N)$ [5, 6,
7], the joint resummation [11, 12] that unifies the above two formalisms, the
DGLAP equation for $\ln p^{+}$ [3], the BFKL equation for $\ln(1/x)$ [4], and
the CCFM equation [10] that combines the above two evolution equations. I will
explain the basic ideas of all the single- and double-logarithmic summations
in the Collins-Soper-Sterman (CSS) resummation formalism [8, 9].

###  0.3.1 Resummation Formalism

Collinear and soft divergences may overlap to form double logarithms in
extreme kinematic regions, such as low $p_{T}$ and large $x$. The former
includes low $p_{T}$ jet, photon, and $W$ boson productions, which all require
real gluon emissions with small $p_{T}$. The latter includes top pair
production, DIS, DY production, and heavy meson decays $B\to X_{u}l\nu$ and
$B\to X_{s}\gamma$ [16, 37, 38] at the end points, for which parton momenta
remain large, and radiations are constrained in the soft region. Because of
the limited phase space for real gluon corrections, the infrared cancellation
is not complete. The double logarithms, appearing in products with the
coupling constant $\alpha_{s}$, such as $\alpha_{s}\ln^{2}(E/p_{T})$ with the
beam energy $E$ and $\alpha_{s}\ln(1-x)/(1-x)_{+}$, deteriorate perturbative
expansion. Double logarithms also occur in exclusive processes, such as
Landshoff scattering [39], hadron form factors [40], Compton scattering [41]
and heavy-to-light transitions $B\to\pi(\rho)$ [42] and $B\to D^{(*)}$ [43] at
maximal recoil. In order to have a reliable pQCD analysis of these processes,
the important logarithms must be summed to all orders.

(a) (b) (c)

Figure 4: (a) Jet subprocess defined in Eq. (36). (b) and (c) LO diagrams of
(a).

The resummation of large logarithms will be demonstrated in the covariant
gauge $\partial\cdot A=0$ [38], in which the role of the Wilson line direction
$n$ and the key technique can be explained straightforwardly. Take as an
example a jet subprocess defined by the matrix element

$J(p,n)u(p)=\langle 0|{\cal P}\exp\left[-ig\int_{0}^{\infty}dzn\cdot
A(nz)\right]q(0)|p\rangle\;,$ (36)

where $q$ is a light quark field with momentum $p$, and $u(p)$ is a spinor.
The abelian case of this subprocess has been discussed in [44]. The path-
ordered exponential in Eq. (36) is the consequence of the factorization of
collinear gluons with momenta parallel to $p$ from a full process, as
explained in the previous section. For convenience, it is assumed that $p$ has
a large light-cone component $p^{+}$, and all its other components vanish. A
general diagram of the jet function $J$ is shown in Fig. 4(a), where the path-
ordered exponential is represented by a double line along the vector $n$. As
explained before, varying the direction $n$ does not change the collinear
divergences collected by the Wilson line.

It is easy to see that $J$ contains double logarithms from the overlap of
collinear and soft divergences by calculating the LO diagrams in Fig. 4(b),
the self-energy correction, and in Fig. 4(c), the vertex correction. In the
covariant gauge both Figs. 4(b) and 4(c) produce double logarithms. In the
axial gauge $n\cdot A=0$ the path-ordered exponential reduces to an identity,
and Fig. 4(c) does not exist. The essential step in the resummation technique
is to derive a differential equation $p^{+}dJ/dp^{+}=CJ$ [38, 16, 42], where
the coefficient function $C$ contains only single logarithms, and can be
treated by RG methods. Since the path-ordered exponential is scale-invariant
in $n$, $J$ must depend on $p$ and $n$ through the ratio $(p\cdot
n)^{2}/n^{2}$. The differential operator $d/dp^{+}$ can then be replaced by
$d/dn$ using a chain rule

$p^{+}\frac{d}{dp^{+}}J=-\frac{n^{2}}{v\cdot
n}v_{\alpha}\frac{d}{dn_{\alpha}}J,$ (37)

with the vector $v=(1,0,{\bf 0}_{T})$ being defined via $p=p^{+}v$.

Figure 5: Derivative $p^{+}dJ/dp^{+}$ in the covariant gauge.

Equation (37) simplifies the analysis tremendously, because $n$ appears only
in the Feynman rules for the Wilson line, while $p$ may flow through the whole
diagram in Fig. 4(a). The differentiation of each eikonal vertex and of the
associated eikonal propagator with respect to $n_{\alpha}$,

$\displaystyle-\frac{n^{2}}{v\cdot
n}v_{\alpha}\frac{d}{dn_{\alpha}}\frac{n_{\mu}}{n\cdot l}=\frac{n^{2}}{v\cdot
n}\left(\frac{v\cdot l}{n\cdot l}n_{\mu}-v_{\mu}\right)\frac{1}{n\cdot
l}\equiv\frac{{\hat{n}}_{\mu}}{n\cdot l},$ (38)

leads to the special vertex ${\hat{n}}_{\mu}$. The derivative $p^{+}dJ/dp^{+}$
is thus expressed as a summation over different attachments of
${\hat{n}}_{\mu}$, labeled by the symbol $+$ in Fig. 5. If the loop momentum
$l$ is parallel to $p$, the factor $v\cdot l$ vanishes, and ${\hat{n}}_{\mu}$
is proportional to $v_{\mu}$. When this ${\hat{n}}_{\mu}$ is contracted with a
vertex in $J$, in which all momenta are mainly parallel to $p$, the
contribution to $p^{+}dJ/dp^{+}$ is suppressed. Therefore, the leading regions
of $l$ are soft and hard.

Figure 6: (a) $O(\alpha_{s}^{2})$ examples for the differentiated $J$. (b)
Factorization of $K$ at $O(\alpha_{s})$. (c) Factorization of $K$ at
$O(\alpha_{s}^{2})$. (d) Factorization of $G$ at $O(\alpha_{s})$.

According to this observation, we investigate some two-loop examples exhibited
in Fig. 6(a). If the loop momentum flowing through the special vertex is soft
but another is not, only the first diagram is important, giving a large single
logarithm. In this soft region the subdiagram containing the special vertex
can be factorized using the eikonal approximation as shown in Fig. 6(b), where
the symbol $\otimes$ represents a convoluting relation. The subdiagram is
absorbed into a soft kernel $K$, and the remainder is identified as the
original jet function $J$, both being $O(\alpha_{s})$ contributions. If both
the loop momenta are soft, the four diagrams in Fig. 6(a) are equally
important. The subdiagrams, factorized according to Fig. 6(c), contribute to
$K$ at $O(\alpha_{s}^{2})$, and the remainder is the LO diagram of $J$. If the
loop momentum flowing through the special vertex is hard and another is not,
the second diagram in Fig. 6(a) dominates. In this region the subdiagram
containing the special vertex is factorized as shown in Fig. 6(d). The right-
hand side of the dashed line is absorbed into a hard kernel $G$ as an
$O(\alpha_{s})$ contribution, and the left-hand side is identified as the
$O(\alpha_{s})$ diagram of $J$. If both the loop momenta are hard, all the
diagrams in Fig. 6(a) are absorbed into $G$, giving the $O(\alpha_{s}^{2})$
contributions.

Extending the above reasoning to all orders, one derives the differential
equation

$p^{+}\frac{d}{dp^{+}}J=\left[K(m/\mu,\alpha_{s}(\mu))+G(p^{+}\nu/\mu,\alpha_{s}(\mu))\right]J,$
(39)

where the coefficient function $C$ has been written as the sum of the soft
kernel $K$ and the hard kernel $G$. In the above expression $\mu$ is a
factorization scale, the gauge factor in $G$ is defined as $\nu=\sqrt{(v\cdot
n)^{2}/|n^{2}|}$, and a gluon mass $m$ has been introduced to regularize the
infrared divergence in $K$. It has been made explicit that $K$ and $G$ depend
on a single infrared scale $m$ and a single ultraviolet scale $p^{+}$,
respectively.

The $O(\alpha_{s})$ contribution to $K$ from Fig. 6(b) is written as

$\displaystyle K$ $\displaystyle=$ $\displaystyle-
ig^{2}C_{F}\mu^{\epsilon}\int\frac{d^{4-\epsilon}l}{(2\pi)^{4-\epsilon}}\frac{{\hat{n}}_{\mu}}{n\cdot
l}\frac{g^{\mu\nu}}{l^{2}-m^{2}}\frac{v_{\nu}}{v\cdot l}-\delta K,$ (40)

$\delta K$ being an additive counterterm. The $O(\alpha_{s})$ contribution to
$G$ from Fig. 6(d) is given by

$\displaystyle
G=-ig^{2}C_{F}\mu^{\epsilon}\int\frac{d^{4-\epsilon}l}{(2\pi)^{4-\epsilon}}\frac{{\hat{n}}_{\mu}}{n\cdot
l}\frac{g^{\mu\nu}}{l^{2}}\left(\frac{\not p+\not
l}{(p+l)^{2}}\gamma_{\nu}-\frac{v_{\nu}}{v\cdot l}\right)-\delta G,$ (41)

where the second term in the parentheses acts as a soft subtraction to avoid
double counting, and $\delta G$ is an additive counterterm. A straightforward
evaluation shows that Eqs. (40) and (41) contain only the single logarithms
$\ln(m/\mu)$ and $\ln(p^{+}\nu/\mu)$, respectively, as claimed before.
Organizing these single logarithms using RG methods, and then solving Eq.
(39), one resums the double logarithms $\ln^{2}(p^{+}/m)$ in $J$.

To explain all the known resummations and evolution equations, we first
construct a master equation for the TMD $\Phi(x,k_{T})$, which is a
differential equation in the hadron momentum $p^{+}$. The dependence on the
factorization scale $\mu$ is implicit. If the parton is a quark, $\Phi$ is
defined by Eq. (35). If the parton is a gluon, the nonlocal operator in the
hadronic matrix element of Eq. (35) is replaced by
$F^{+}_{\mu}(y^{-},y_{T})F^{\mu+}(0)$. Similarly, $n$ is varied arbitrarily
away from the light cone with $n^{2}\not=0$. Then $\Phi$ depends on $p^{+}$
via the ratio $(p\cdot n)^{2}/n^{2}$, so the chain rule in Eq. (37) relating
the derivative $d\Phi/dp^{+}$ to $d\Phi/dn_{\alpha}$ applies. Following the
derivation in the previous subsection, one obtains the master equation

$\displaystyle p^{+}\frac{d}{dp^{+}}\Phi(x,k_{T})=2{\bar{\Phi}}(x,k_{T}),$
(42)

where $\bar{\Phi}$ contains the special vertex, and the coefficient 2 is
attributed to the equality of $\bar{\Phi}$ with the special vertex on either
side of the final-state cut.

The function $\bar{\Phi}$ is factorized into the convolution of the soft and
hard kernels with $\Phi$:

$\displaystyle{\bar{\Phi}}(x,k_{T})={\bar{\Phi}}_{s}(x,k_{T})+{\bar{\Phi}}_{h}(x,k_{T}),$
(43)

with the soft contribution

$\displaystyle{\bar{\Phi}}_{s}$ $\displaystyle=$
$\displaystyle\left[-ig^{2}C_{F}\mu^{\epsilon}\int\frac{d^{4-\epsilon}l}{(2\pi)^{4-\epsilon}}\frac{{\hat{n}}\cdot
v}{n\cdot ll^{2}v\cdot l}-\delta K\right]\Phi(x,k_{T})$ (44) $\displaystyle-
ig^{2}C_{F}\mu^{\epsilon}\int\frac{d^{4-\epsilon}l}{(2\pi)^{4-\epsilon}}\frac{{\hat{n}}\cdot
v}{n\cdot lv\cdot l}2\pi i\delta(l^{2})\Phi(x+l^{+}/p^{+},|{\bf k}_{T}+{\bf
l}_{T}|),$

where the first term is the same as in Eq. (40), and the second term
proportional to $\delta(l^{2})$ arises from the real soft gluon emission. The
hard contribution is given by
${\bar{\Phi}}_{h}(x,k_{T})=G(xp^{+}\nu/\mu,\alpha_{s}(\mu))\Phi(x,k_{T})$, in
which the hard kernel $G$ is the same as in Eq. (41).

###  0.3.2 $k_{T}$ Resummation and BFKL Equation

The TMD definition in Eq. (35) contains three scales: $(1-x)p^{+}$, $xp^{+}$,
and $k_{T}$. We first consider the soft approximation corresponding to the
rapidity ordering of real gluon emissions in a ladder diagram. Assume that a
parton carries the longitudinal momentum $xp^{+}+l_{2}^{+}+l_{1}^{+}$, which
becomes $xp^{+}+l_{1}^{+}$ after emitting a gluon of longitudinal momentum
$l_{2}^{+}$ and transverse momentum $l_{2T}$, and then becomes $xp^{+}$ after
emitting a gluon of longitudinal momentum $l_{1}^{+}$ and transverse momentum
$l_{1T}$. In the kinematic configuration with $l_{2}^{+}\gg l_{1}^{+}$ and
$l_{2T}\sim l_{1T}$, the original parton momentum is approximated by
$xp^{+}+l_{2}^{+}+l_{1}^{+}\approx xp^{+}+l_{2}^{+}$. The loop integral
associated with the first gluon emission is then independent of $l_{1}^{+}$,
and can be worked out straightforwardly, giving a logarithm. The loop integral
associated with the second gluon emission, involving only $l_{1}^{+}$, also
gives a logarithm. Therefore, a ladder diagram with $N$ rung gluons generates
the logarithmic correction $(\alpha_{s}L)^{N}$ under the above rapidity
ordering, where $L$ denotes the large logarithm. Following the rapidity
ordering, we adopt the approximation for the real gluon emission in Eq. (44)

$\Phi(x+l^{+}/p^{+},|{\bf k_{T}+\bf l_{T}}|)\approx\Phi(x,|{\bf k_{T}+\bf
l_{T}}|),$ (45)

where the $l^{+}$ dependence has been neglected. The transverse momenta
$l_{T}$, being of the same order as $k_{T}$ in this kinematic configuration,
is kept. The variable $l^{+}$ in $K$ is then integrated up to infinity, such
that the scale $(1-x)p^{+}$ disappears.

Equation (44) is Fourier transformed into the impact parameter $b$ space to
decouple the $l_{T}$ integration. Hence, in the intermediate $x$ region $\Phi$
involves two scales, the large $xp^{+}$ that characterizes the hard kernel $G$
and the small $1/b$ that characterizes the soft kernel $K$. The master
equation (42) becomes

$\displaystyle
p^{+}\frac{d}{dp^{+}}\Phi(x,b)=2\left[K(1/(b\mu),\alpha_{s}(\mu))+G(xp^{+}\nu/\mu,\alpha_{s}(\mu))\right]\Phi(x,b),$
(46)

whose solution with $\nu=1$ leads to the $k_{T}$ resummation

$\displaystyle\Phi(x,b)=\Delta_{k}(x,b)\Phi_{i}(x),$ (47)

with the Sudakov exponential

$\displaystyle\Delta_{k}(x,b)=\exp\left[-2\int_{1/b}^{xp^{+}}\frac{dp}{p}\int_{1/b}^{p}\frac{d\mu}{\mu}\gamma_{K}(\alpha_{s}(\mu))\right],$
(48)

and the initial condition $\Phi_{i}$ of the Sudakov evolution. The anomalous
dimension of $K$, $\lambda_{K}=\mu d\delta K/d\mu$, is given, up to two loops,
by [45]

$\displaystyle\gamma_{K}=\frac{\alpha_{s}}{\pi}C_{F}+\left(\frac{\alpha_{s}}{\pi}\right)^{2}C_{F}\left[{C}_{A}\left(\frac{67}{36}-\frac{\pi^{2}}{12}\right)-\frac{5}{18}n_{f}\right]\;,$
(49)

with $n_{f}$ being the number of quark flavors and $C_{A}=3$ being a color
factor.

Figure 7: Scattering amplitude for direct photon production.

The $k_{T}$ resummation effect on the low $p_{T}$ spectra of the direct photon
production depicted in Fig. (7)has been analyzed [46]. The initial-state and
final-state radiations are constrained in the low $p_{T}$ region, where the
$k_{T}$ resummation is necessary for improving the perturbation theory. Figure
8 shows the deviation (Data -Theory)/Theory of the NLO pQCD predictions,
obtained using the CTEQ4M PDFs [47], from the experimental data as a function
of $x_{t}=2p_{T}/\sqrt{s}$, $\sqrt{s}$ being the center-of-mass energy. The
deviation is huge as expected, especially at low $x_{t}$ of each set of the
data. After including the $k_{T}$ resummation effect [46], it is clear that a
significant improvement on the agreement between theoretical predictions and
the data is achieved. As to the intermediate- and high-$p_{T}$ regions of the
direct photon production, NLO pQCD works reasonably well in accommodating the
data as indicated in Fig. 9. The threshold resummation effect, which will be
introduced in the next subsection, is more relevant in these regions: it
slightly improves the consistency between predictions and the data [48].

Figure 8: Low $p_{T}$ direct photon spectra before (upper) and after (lower)
including the $k_{T}$ resummation. Figure 9: High $p_{T}$ direct photon
spectrum under the threshold resummation.

In the small $x$ region with $xp^{+}\sim k_{T}$, or $xp^{+}\sim 1/b$ in the
$b$ space, the two-scale case reduces to the single-scale one. In this region
contributions from gluonic partons dominate, so $\Phi$ represents the gluon
TMD below. The source of double logarithms, i.e., the integral containing the
anomalous dimension $\gamma_{K}$, is less important. Because only the soft
scale exists, one drops the hard kernel $G$, and keeps the soft kernel with an
ultraviolet cutoff. The right-hand side of Eq. (42) becomes

$\displaystyle{\bar{\Phi}}(x,k_{T})$ $\displaystyle=$ $\displaystyle-
ig^{2}N_{c}\int\frac{d^{4}l}{(2\pi)^{4}}\frac{{\hat{n}}\cdot v}{n\cdot lv\cdot
l}\left[\frac{\theta(k_{T}^{2}-l_{T}^{2})}{l^{2}}\Phi(x,k_{T})\right.$ (50)
$\displaystyle\left.+2\pi i\delta(l^{2})\phi(x,|{\bf k}_{T}+{\bf
l}_{T}|)\right],$

where the color factor $C_{F}$ has been replaced by $N_{c}$ for the gluon TMD.
The $\theta$ function introduces the ultraviolet cutoff on $l_{T}$ mentioned
above. To make variation in $x$ via variation in $p^{+}$, a fixed parton
momentum is assumed. Under this assumption, the momentum fraction $x$ is
proportional to $1/p^{+}$, and one has $p^{+}d\Phi/dp^{+}=-xd\Phi/dx\Phi$
[49]. Performing the integrations over $l^{+}$ and $l^{-}$ in Eq. (50), the
master equation (42) reduces to the BFKL equation [50],

$\displaystyle\frac{d\phi(x,k_{T})}{d\ln(1/x)}={\bar{\alpha}}_{s}\int\frac{d^{2}l_{T}}{\pi
l_{T}^{2}}\left[\phi(x,|{\bf k}_{T}+{\bf
l}_{T}|)-\theta(k_{T}^{2}-l_{T}^{2})\phi(x,k_{T})\right],$ (51)

with the coupling constant ${\bar{\alpha}}_{s}=N_{c}\alpha_{s}/\pi$.

Figure 10: Effects from LO and NLO BFKL equations.

A remarkable prediction of the above LO BFKL equation is that a high-energy
cross section increases with the center-of-mass energy,

$\displaystyle\sigma\approx\frac{1}{t}\left(\frac{s}{t}\right)^{\omega_{P}-1},$
(52)

with the momentum transfer squared $t$. It turns out that Eq. (52), with the
Pomeron intercept $\omega_{P}-1=4{\bar{\alpha}}_{s}\ln 2$, violates the
Froissart (unitarity) bound $\sigma<{\rm const.}\times\ln^{2}$ [51]. The
unsatisfactory prediction of the LO BFKL equation called for the NLO
corrections [52], which were, however, found to be dramatic as indicated by
the $x$ dependence of the derivative of the structure function $dF_{L}/d\ln
Q^{2}$ in Fig. 10 [53]: the NLO effect is nearly as large as the LO result for
$x\sim 0.001$, and becomes dominant at lower $x$. It even turns $dF_{L}/d\ln
Q^{2}$ negative below $x\sim 0.0001$ in the upper of Fig. 10. That is, the
perturbative solution is not at all stable. Choosing a running coupling
constant [53], the NLO effect is not overwhelming, but still significant as
exhibited in the lower of Fig. 10.

###  0.3.3 Threshold Resummation and DGLAP Equation

We then consider the soft approximation corresponding to the $k_{T}$ ordering
of real gluon emissions in a ladder diagram. Assume that a parton without the
transverse momentum, carries $-{\bf l}_{1T}$ after emitting a gluon of
longitudinal momentum $l_{1}^{+}$ and transverse momentum ${\bf l}_{1T}$, and
then carries $-{\bf l}_{1T}-{\bf l}_{2T}$ after emitting a gluon of
longitudinal momentum $l_{2}^{+}$ and transverse momentum ${\bf l}_{2T}$. In
the kinematic configuration with $l_{2T}\gg l_{1T}$ and $l_{2}^{+}\sim
l_{1}^{+}$, the final parton momentum can be approximated by $-{\bf
l}_{2T}-{\bf l}_{1T}\approx-{\bf l}_{2T}$, such that the loop integral
associated with the first gluon emission involves only $l_{1T}$, and can be
worked out straightforwardly, giving a logarithm. The loop integral associated
with the second gluon emission involves only $l_{2T}$, and also gives a
logarithm. Hence, a ladder diagram with $N$ rung gluons generates the
logarithmic correction $(\alpha_{s}L)^{N}$ under the above $k_{T}$ ordering.
In this case $\Phi$ is independent of $l_{T}$, and we have the approximation
for the real gluon emission in Eq. (44)

$\Phi(x+l^{+}/p^{+},|{\bf k_{T}+\bf l_{T}}|)\approx\Phi(x+l^{+}/p^{+},k_{T}),$
(53)

in which $x$ and $l^{+}/p^{+}$ are of the same order. The dependence on
$k_{T}$ can then be integrated out from both sides of the master equation
(42), and the TMD $\Phi$ reduces to the PDF $\phi$. The scale $k_{T}$
disappears, and the scale $(1-x)p^{+}$ is retained.

The Mellin transformation is employed to bring ${\bar{\phi}}_{s}$ from the
momentum fraction $x$ space to the moment $N$ space,

$\displaystyle{\bar{\phi}}_{s}(N)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dxx^{N-1}{\bar{\phi}}_{s}(x),$ (54)

under which the $l^{+}$ integration decouples. In the large $x$ region $\phi$
involves two scales, the large $xp^{+}\sim p^{+}$ from the hard kernel $G$ and
the small $(1-x)p^{+}\sim p^{+}/N$ from the soft kernel $K$. To sum
$\ln(1/N)$, we rewrite the derivative $p^{+}d\phi/dp^{+}$ as

$p^{+}\frac{d\phi}{dp^{+}}=\frac{p^{+}}{N}\frac{d\phi}{d(p^{+}/N)}.$ (55)

The solution of the master equation (42) then gives the threshold resummation,

$\displaystyle\phi(N)=\Delta_{t}(N)\phi_{i}$ (56)

with the exponential

$\displaystyle\Delta_{t}(N)=\exp\left[-2\int_{p^{+}/N}^{p^{+}}\frac{dp}{p}\int_{p^{+}}^{p}\frac{d\mu}{\mu}\gamma_{K}(\alpha_{s}(\mu))\right],$
(57)

or its equivalent expression

$\displaystyle\Delta_{t}(N)=\exp\left[\int_{0}^{1}dz\frac{1-z^{N-1}}{1-z}\int_{(1-z)^{2}}^{1}\frac{d\lambda}{\lambda}\gamma_{K}(\alpha_{s}(\sqrt{\lambda}p^{+}))\right].$
(58)

Figure 11: Dependence of the total cross section for the top-pair production
on the top mass at the LHC with $\sqrt{s}=$ 7 TeV.

An application of the threshold resummation is found in the analysis of the
top-quark pair production, which was performed at the next-to-next-to-leading-
logarithmic (NNLL) accuracy [54]. It has been observed that the threshold
resummation effect enhances the NLO total cross section by few percents as
shown in Fig. 11, where the bands sandwiched by the thinner lines denote the
theory uncertainty. The above formalism can be used to determine of the top
quark mass as indicated in Fig. 12, where the solid lines represent the
central values, and the total uncertainties of the theoretical and
experimental results [55] are given by the external dashed lines.

Figure 12: Mass dependence of the theoretical cross section with the threshold
resummation effect (red) and of the measured cross section (black).

In the intermediate $x$ region the two-scale case reduces to the single-scale
one because of $xp^{+}\sim(1-x)p^{+}$, and the source of double logarithms is
less important. Without the Mellin transformation, the sum in Eq. (43), with
the approximation in Eq. (53) being inserted, leads to the DGLAP equation
[49],

$\displaystyle
p^{+}\frac{d}{dp^{+}}\phi(x)=\int_{x}^{1}\frac{d\xi}{\xi}P(x/\xi)\phi(\xi)\;,$
(59)

with the kernel

$\displaystyle P(z)=\frac{\alpha_{s}(p^{+})}{\pi}C_{F}\frac{2}{(1-z)_{+}}\;,$
(60)

where the variable change $\xi=x+l^{+}/p^{+}$ has been made. The argument of
$\alpha_{s}$, i.e, the factorization scale $\mu$, has been set to the scale
$xp^{+}\sim(1-x)p^{+}\sim O(p^{+})$. Note that the kernel $P$ differs from the
splitting function $P_{qq}$ in Eq. (22) by the term $(z^{2}-1)/(1-z)_{+}$,
which is finite in the $z\to 1$ limit. The reason is that the real gluon
emission was evaluated under the soft approximation as deriving $P$, while it
was calculated exactly as deriving $P_{qq}$.

Figure 13: Diagrams for the DGLAP splitting functions. Figure 14: $Q^{2}$
evolutions of the valence quark PDFs for some parameter values in the DGLAP
solutions (solid and dashed lines).

Gluon emissions in Fig. 13 cause the mixing between the quark and gluon PDFs,
giving the complete set of DGLAP equations with four splitting functions

$\frac{\partial}{\partial\ln Q^{2}}\left(\begin{array}[]{c}\phi_{q}\\\
\phi_{g}\end{array}\right)=\left(\begin{array}[]{cc}P_{qq}&P_{qg}\\\
P_{gq}&P_{gg}\end{array}\right)\otimes\left(\begin{array}[]{c}\phi_{q}\\\
\phi_{g}\end{array}\right).$ (61)

The evolution of the $u$-quark and $d$-quark PDFs in $Q^{2}$ predicted by the
LO DGLAP equation [56] is shown in Fig. 14, where the inputs at the initial
scale $Q_{0}=1$ GeV were taken from MRST2001 [57]. It is observed that the
valence quark PDFs increase with $Q^{2}$ at small $x$, namely, they become
broader with $Q^{2}$, a feature consistent with what was stated in the
previous section. The predictions for the deuteron structure function derived
from the LO, NLO, and NNLO DGLAP equations are displayed in Fig. 15 [58],
which agree with the NMC data [59].

Figure 15: Predictions from the DGLAP equation and the NMC data for the
deuteron structure function. For clarity, data are scaled up by +0.2i (in
Fig.(a)) and +i (in Fig.(b)) (with i = 0,1,2,3) starting from the bottom of
the graphs.

###  0.3.4 Joint Resummation and CCFM Equation

At last, a unified resummation formalism for large and intermediate $x$ and a
unified evolution equation for intermediate and small $x$ can be derived by
retaining the $l^{+}$ and $l_{T}$ dependencies of $\Phi$ in Eq. (44), which
corresponds to the so-called angular ordering. In this case both the Fourier
and Mellin transformations are applied to Eq. (44), leading to

$\displaystyle{\bar{\Phi}}_{s}(N,b)=K(p^{+}/(N\mu),1/(b\mu),\alpha_{s}(\mu))\Phi(N,b)\;,$
(62)

with the soft kernel [11]

$\displaystyle K$ $\displaystyle=$ $\displaystyle-
ig^{2}C_{F}\mu^{\epsilon}\int_{0}^{1}dz\int\frac{d^{4-\epsilon}l}{(2\pi)^{4-\epsilon}}\frac{{\hat{n}}\cdot
v}{n\cdot lv\cdot l}\left[\frac{\delta(1-z)}{l^{2}}\right.$ (63)
$\displaystyle\left.+2\pi
i\delta(l^{2})\delta\left(1-z-\frac{l^{+}}{p^{+}}\right)z^{N-1}e^{i{\bf
l}_{T}\cdot{\bf b}}\right]-\delta K,$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}(\mu)}{\pi}C_{F}\left[\ln\frac{1}{b\mu}-K_{0}\left(\frac{2\nu
p^{+}b}{N}\right)\right],$

$K_{0}$ being the modified Bessel function. As $p^{+}b\gg N$, we have
$K_{0}\to 0$, and the soft scale inferred by the above expression approaches
$1/b$ for the $k_{T}$ resummation. As $N\gg p^{+}b$, we have
$K_{0}\approx-\ln(\nu p^{+}b/N)$, and the soft scale approaches $p^{+}/N$ for
the threshold resummation.

Following the procedures similar to Eqs. (46)-(48), we derive the joint
resummation

$\Phi(N,b)=\Delta_{u}(N,b)\Phi_{i},$ (64)

with the exponential

$\Delta_{u}(N,b)=\exp\left[-2\int_{p^{+}\chi^{-1}(N,b)}^{p^{+}}\frac{dp}{p}\int_{p^{+}\chi^{-1}(1,b)}^{p}\frac{d\mu}{\mu}\gamma_{K}(\alpha_{s}(\mu))\right].$
(65)

The dimensionless function [12]

$\displaystyle\chi(N,b)=\left(N+\frac{p^{+}b}{2}\right)e^{\gamma_{E}},$ (66)

is motivated by the limits discussed above. It is apparent that Eq. (65)
reduces to Eq. (48) and Eq. (57) in the $b\to\infty$ and $N\to\infty$ limits,
respectively. The effect from the joint resummation on the $q_{T}$ spectra of
selectron pairs produced at the LHC with $\sqrt{S}=14$ TeV has been
investigated in [60]. It is seen in Fig. 16 that the joint and $k_{T}$
resumations exhibit a similar behavior in the small-$q_{T}$ region as
expected, but the jointly-resummed cross section is about 5%-10% lower than
the $k_{T}$-resummed cross section in the range 50 GeV $<q_{T}<$ 100 GeV.

Figure 16: Transverse-momentum distribution of selectron pairs at the LHC in
the framework of joint (full) and $k_{T}$ (dotted) resummations.

In the intermediate and small $x$ regions, it is not necessary to resum the
double logarithms $\ln^{2}(1/N)$. After extracting the $k_{T}$ resummation,
the remaining single-logarithmic summation corresponds to a unification of the
DGLAP and BFKL equations, since both the $l^{+}$ and $l_{T}$ dependencies have
been retained. The function $\Phi(x+l^{+}/p^{+},b)$ in Eq. (44) is
reexpressed, after the Fourier transformation, as

$\displaystyle\Phi(x+l^{+}/p^{+},b)=\theta((1-x)p^{+}-l^{+})\Phi(x,b)$
$\displaystyle\hskip
42.67912pt+[\Phi(x+l^{+}/p^{+},b)-\theta((1-x)p^{+}-l^{+})\Phi(x,b)].$ (67)

The contribution from the first term is combined with the first term in Eq.
(44), giving the soft kernel $K$ for the $k_{T}$ resummation. The second term
in Eq. (67) contributes

$\displaystyle-iN_{c}g^{2}\int\frac{d^{4}l}{(2\pi)^{4}}\frac{{\hat{n}}\cdot
v}{n\cdot lv\cdot l}2\pi i\delta(l^{2})e^{i{\bf l}_{T}\cdot{\bf
b}}[\Phi(x+l^{+}/p^{+},b)-\theta((1-x)p^{+}-l^{+})\Phi(x,b)],$ (68)

which will generate the splitting function below. The color factor has been
replaced by $N_{c}$, since the gluon TMD is considered here.

The master equation (42) then becomes

$\displaystyle p^{+}\frac{d}{dp^{+}}\Phi(x,b)$ $\displaystyle=$
$\displaystyle-2\left[\int_{1/b}^{xp^{+}}\frac{d\mu}{\mu}\gamma_{K}(\alpha_{s}(\mu))-{\bar{\alpha}}_{s}(xp^{+})\ln(p^{+}b)\right]\Phi(x,b)$
(69)
$\displaystyle+2{\bar{\alpha}}_{s}(xp^{+})\int_{x}^{1}dzP_{gg}(z)\Phi(x/z,b),$

with the splitting function

$P_{gg}=\left[\frac{1}{(1-z)_{+}}+\frac{1}{z}-2+z(1-z)\right],$ (70)

obtained from Eq. (68). The term $-2+z(1-z)$ finite as $z\to 0$ and $z\to 1$
has been added. The exponential $\Delta$ is extracted from the $k_{T}$
resummation,

$\displaystyle\Delta(x,b,Q_{0})=\exp\left(-2\int_{xQ_{0}}^{xp^{+}}\frac{dp}{p}\left[\int_{1/b}^{p}\frac{d\mu}{\mu}\gamma_{K}(\alpha_{s}(\mu))-{\bar{\alpha}}_{s}(p)\ln\frac{pb}{x}\right]\right),$
(71)

$Q_{0}$ being an arbitrary low energy scale. It is trivial to justify by
substitution that the solution is given by

$\displaystyle\Phi(x,b)$ $\displaystyle=$
$\displaystyle\Delta(x,b,Q_{0})\Phi_{i}$ (72)
$\displaystyle+2\int_{x}^{1}dz\int_{Q_{0}}^{p^{+}}\frac{d\mu}{\mu}{\bar{\alpha}}_{s}(x\mu)\Delta_{k}(x,b)P_{gg}(z)\Phi(x/z,b),$

which can be regarded as a modified version of the CCFM equation [10].

##  0.4 PQCD for jet physics

Jets, abundantly produced at colliders [61], carry information of hard
scattering and parent particles, which is crucial for particle identification
and new physics search. Study of jet physics is usually done using event
generators, which, however, suffer ambiguity from parameter tuning. Hence, we
are motivated to establish an alternative approach free of the ambiguity. I
will demonstrate that jet dynamics can be explored and jet properties can be
predicted in the pQCD resummation formalism.

Figure 17: Dijet final state in $e^{-}e^{+}$ annihilation.

We start from the dijet production in the $e^{-}e^{+}$ annihilation, which is
part of its total cross section. The physical dijet final state, described in
Fig. 17, contains two jet cones of half angle $\delta$ and isotropic soft
gluons within the energy resolution $\epsilon Q$, $Q$ being the $e^{-}e^{+}$
invariant mass. The Born cross section is the same as the total one in Eq.
(11). With the constrained phase space for real gluons, the infrared
cancellation is not complete, and logarithmic enhancement appears. The
explicit NLO calculations imply that the isotropic soft gluons give a
contribution proportional to $2\ln^{2}(2\epsilon Q/\mu)-\pi^{2}/6$, the
collinear gluons in the cones with energy higher than the resolution give
$-3\ln(Q\delta/\mu)-2\ln^{2}(2\epsilon)-4\ln(Q\delta/\mu)\ln(2\epsilon)+17/4-\pi^{2}/3$,
and the virtual corrections contribute
$-2\ln^{2}(Q/\mu)+3\ln(Q/\mu)-7/4+\pi^{2}/6$. The total NLO corrections
indicate that the dijet cross section is infrared finite, but logarithmically
enhanced:

$\displaystyle
3\ln\delta+4\ln\delta\ln(2\epsilon)+\frac{\pi^{2}}{3}-\frac{5}{2},$ (73)

where the double logarithm $\ln\delta\ln(2\epsilon)$ is attributed to the
overlap of the collinear and soft logarithms.

###  0.4.1 Jet in Experiments

To describe the kinematics for jets, we define the pseudorapidity
$\eta=\ln[\cot(\theta/2)]$, which is related to the polar angle $\theta$ with
respect to the beam direction, and the azimuthal angle $\phi$. That is,
$\theta=0$, $90^{\circ}$, and $180^{\circ}$ correspond to $\eta=+\infty$, 0
and $-\infty$, respectively. Comparison of theoretical and experimental
descriptions for jet observables is nontrivial. One needs jet algorithms that
map experimental measurements with theoretical calculations as close as
possible. The infrared safety [61] is an important guideline for setting up a
jet algorithm. There are two major classes of jet algorithms in the
literature: cone algorithms and sequential algorithms. The former is a
geometrical method, which stamps out jets on the $\eta$-$\phi$ plane as with a
cookie cutter. The latter combines particle four-momenta one by one following
given kinematic criteria.

I take the seeded cone algorithm as an example to explain the operation in the
first class of jet algorithms, which aims at finding stable cones via an
iterative procedure. Start from a seed particle $i$, and consider a set of
particles $j$ with separations smaller than jet cone of radius $R$,

$\displaystyle\Delta
R_{ij}^{2}\equiv(\eta_{i}-\eta_{j})^{2}+(\phi_{i}-\phi_{j})^{2}<R^{2}.$ (74)

Calculate the new cone center $J$ by summing all particle four-momenta in the
cone. A stable cone is composed of a set of particles $i$ satisfying $\Delta
R_{iJ}<R$. If the cone is stable, the procedure stops. Otherwise, take $J$ as
a new seed, and repeat the above procedure.

However, the seeded cone algorithm suffers the problem of infrared
divergences. Such a geometrical algorithm does not differentiate infrared
gluons from energetic gluons, so final outcomes depend on soft radiation and
collinear splitting. This problem can be illustrated by considering a system
of two particles 1 and 2, separated by $R_{12}$ with $R<R_{12}<2R$. Each of
particles 1 and 2, taken as a seed, forms a stable jet. One then adds a soft
gluon to this system. It is obvious that a virtual soft gluon exchanged
between jets 1 and 2 does not change the outcome; namely, a virtual soft gluon
contributes to the dijet cross section. On the contrary, adding a real soft
seed between jets 1 and 2 will merge the two jets because of $R<R_{12}<2R$.
Therefore, a real soft gluon contributes to the single jet cross section. As a
result, the soft divergences do not cancel between the virtual and real
corrections. One may speculate that starting from the hardest particle may
avoid the difficulty caused by the soft seed. It turns out that the collinear
splitting would change the outcome. Including a more energetic particle into
the above system, which is emitted between particles 1 and 2. Taking this
central particle as the seed, one constructs a single stable jet formed by the
three particles. A self-energy correction to the central particle does not
change this final state, and contributes to the single jet cross section.
However, the splitting of the central particle may produce two particles,
which are less energetic than particles 1 and 2. Then one has to take particle
1 or 2 as the seed, and ends up with two stable jets. That is, the collinear
splitting contributes to the dijet cross section, and there is no cancellation
between virtual and real corrections. It is concluded that a seeded cone
algorithm is not infrared safe.

Next I introduce sequential algorithms by taking the $k_{T}$ algorithm as an
example. For any pair of particles $i$ and $j$, find the minimum of the
following three distances

$\displaystyle d_{ij}=\min(k_{Ti}^{2},k_{Tj}^{2})\frac{\Delta
R_{ij}^{2}}{R^{2}},\;\;\;\;d_{iB}=k_{Ti}^{2},\;\;\;\;d_{jB}=k_{Tj}^{2},$ (75)

with $k_{T}$ being is a jet transverse momentum. If the minimum is $d_{iB}$ or
$d_{jB}$, $i$ or $j$ is a jet, and removed from the list of particles.
Otherwise, $i$ and $j$ are merged into a new jet. Repeat the above procedure
until no particles are left. The other sequential algorithms include the
Cambridge/Aachen and anti-$k_{T}$ ones with the definitions of the distances

$\displaystyle d_{ij}=\frac{\Delta
R_{ij}^{2}}{R^{2}},\;\;\;\;d_{iB}=1,\;\;\;\;d_{jB}=1,$ $\displaystyle
d_{ij}=\min(k_{Ti}^{-2},k_{Tj}^{-2})\frac{\Delta
R_{ij}^{2}}{R^{2}},\;\;\;\;d_{iB}=k_{Ti}^{-2},\;\;\;\;d_{jB}=k_{Tj}^{-2},$
(76)

respectively. The grouping starts from soft (energetic) particles and usually
leads to an irregular (round) jet shape in the $k_{T}$ (anti-$k_{T}$)
algorithm. Note that a sequential algorithm differentiates infrared gluons
from energetic ones: adding a soft real gluon does not modify a cone center,
so it does not change the outcome.

###  0.4.2 Jets in Theory

As outlined in the Introduction, we intend to establish a theoretical
framework for jet study, following the idea of the factorization theorem for
the DIS in Sec. 2. At NLO, a jet is produced in DIS, as the gluon emitted by
the initial-state or final-state quark is collimated to the final-state quark.
The restricted phase space of the final-state quark and the gluon in a small
angular separation renders an incomplete cancellation between the virtual and
real corrections. Hence, jet production is expected to be enhanced by
collinear dynamics. Similarly, the initial-state quark propagator can be
eikonalized in this collinear region, such that collinear gluons are detached
from the initial-state quark and absorbed into a jet function. To all orders,
the collinear gluons are collected by the Wilson link with the path-ordered
exponential

$\displaystyle W={\cal P}\exp\left[-ig\int_{0}^{\infty}dzn\cdot A(zn)\right],$
(77)

with an arbitrary vector $n$. The collinear gluon emitted by the final-state
quark can be factorized into the jet function straightforwardly by applying
the Fierz transformation. A more sophisticated factorization formula for the
jet production in the DIS is then written as a convolution of a hard kernel
$H$ with a PDF and a jet function $J$. $H$ denotes the contribution with the
collinear pieces for the initial and final states being subtracted. This
factorization formalism is the basis for the application of pQCD to jet
physics.

The light-quark and gluon jet functions are defined by [62]

$\displaystyle J_{q}(M_{J}^{2},P_{T},\nu^{2},R,\mu^{2})$ $\displaystyle=$
$\displaystyle\frac{(2\pi)^{3}}{2\sqrt{2}(P_{J}^{0})^{2}N_{c}}\sum_{N_{J}}Tr\left\\{\not\xi\langle
0|q(0)W^{(\bar{q})\dagger}|N_{J}\rangle\langle
N_{J}|W^{(\bar{q})}\bar{q}(0)|0\rangle\right\\}$
$\displaystyle\times\delta(M_{J}^{2}-\hat{M}_{J}^{2}(N_{J},R))\delta^{(2)}(\hat{e}-\hat{e}(N_{J}))\delta(P_{J}^{0}-\omega(N_{J})),$
$\displaystyle J_{g}(M_{J}^{2},P_{T},\nu^{2},R,\mu^{2})$ $\displaystyle=$
$\displaystyle\frac{(2\pi)^{3}}{2(P_{J}^{0})^{3}N_{c}}\sum_{N_{J}}\langle
0|\xi_{\sigma}F^{\sigma\nu}(0)W^{(g)\dagger}|N_{J}\rangle\langle
N_{J}|W^{(g)}F_{\nu}^{\rho}(0)\xi_{\rho}|0\rangle$ (78)
$\displaystyle\times\delta(M_{J}^{2}-\hat{M}_{J}^{2}(N_{J},R))\delta^{(2)}(\hat{e}-\hat{e}(N_{J}))\delta(P_{J}^{0}-\omega(N_{J})),$

where $|N_{J}\rangle$ denotes the final state with $N_{J}$ particles within
the cone of size $R$ centered in the direction of the unit vector $\hat{e}$,
$\hat{M}_{J}(N_{J},R)$ ($\omega(N_{J})$) is the invariant mass (total energy)
of all $N_{J}$ particles, and $\mu$ is the factorization scale. The above jet
functions absorb the collinear divergences from all-order radiations
associated with the energetic light jet of momentum
$P_{J}^{\mu}=P_{J}^{0}v^{\mu}$, in which $P_{J}^{0}$ is the jet energy, and
the vector $v$ is given by $v^{\mu}=(1,\beta,0,0)$ with
$\beta=\sqrt{1-(M_{J}/P_{J}^{0})^{2}}$. $\xi^{\mu}=(1,-1,0,0)$ is a vector on
the light cone. The coefficients in Eq. (78) have been chosen, such that the
LO jet functions are equal to $\delta(M_{J}^{2})$ in a perturbative expansion.

Underlying events include everything but hard scattering, such as initial-
state radiation, final-state radiation, and multiple parton interaction (MPI).
The Wilson lines in Eq. (78) have collected gluons radiated from both initial
states and other final states in a scattering process, and collimated to the
light-particle jets. Gluon exchanges between the quark fields $q$ (or the
gluon fields $F^{\sigma\nu}$ and $F_{\nu}^{\rho}$) correspond to the final-
state radiations. Both the initial-state and final-state radiations are
leading-power effects in the factorization theorem, and have been included in
the jet function definition. A chance of involving more partons in hard
scattering is low, so the contribution from MPI is regarded as being
subleading-power. This contribution should be excluded from data, but it is
certainly difficult to achieve in experiments. Nevertheless, it still makes
sense to compare predictions for jet observables based on Eq. (78) at the
current leading-power accuracy with experimental data. At last, pile-up events
must be removed in experiments [63], since they cannot be handled
theoretically so far.

Figure 18: Some NLO real corrections to the quark jet function. Figure 19:
Some NLO real corrections to the gluon jet function, where the dashed line
represents a ghost field. Figure 20: Jet mass distribution at NLO.

The NLO diagrams for the light-quark and gluon jet functions are displayed in
Figs. 18 and 19, respectively. Evaluating the jet functions up to NLO, a
divergence is observed at small jet invariant mass $M_{J}$ as shown in Fig.
20, that implies the nonperturbtive nature of the jet functions. The total NLO
corrections in Mellin space indicate the existence of double logarithms, which
hint the implementation of the resummation technique. Both the angular and
energy resolutions are related to the jet mass: when $M_{J}$ is not zero,
particles in a jet cannot be completely collimated, and the jet must have
finite minimal energy. This accounts for the source of the double logarithms.
Recall that low $p_{T}$ spectra of direct photons, dominated by soft and
collinear radiations, are treated by the $k_{T}$ resummation. The jet
invariant mass is attributed to soft and collinear radiations, so the mass
distribution can also be derived in the resummation formalism.

Varying the Wilson line direction $n$, we derive the differential equation for
the light-quark jet function [64]

$\displaystyle-\frac{n^{2}}{v\cdot
n}v_{\alpha}\frac{d}{dn_{\alpha}}{J}_{q}(M_{J}^{2},P_{T},\nu^{2},R,\mu^{2})=2(K+G)\otimes{J}_{q}(M_{J}^{2},P_{T},\nu^{2},R,\mu^{2}).$
(79)

The above equation implies that the soft gluons in $K$ are associated with the
jet function $J$, a feature consistent with the anti-$k_{T}$ algorithm. The
solution to Eq. (79) resums the double logarithms in the jet function. One
then convolutes the light-quark and gluon jet functions with the constituent
cross sections of LO partonic dijet processes at the Tevatron and the PDF
CTEQ6L [65]. The resummation predictions for the jet mass distributions at
$R=0.4$ and $R=0.7$ are compared to the Tevatron CDF data [66] in Fig. 21 [67]
with the kinematic cuts $P_{T}>400$ GeV and the rapidity interval
$0.1<|Y|<0.7$. The abbreviation NLL refers to the accuracy of the resummation,
and NLO to the accuracy of the initial condition of the jet function solved
from Eq. (79). The consistency of the resummation results with the CDF data is
satisfactory.

Figure 21: Comparison of resummation predictions for the jet mass
distributions to Tevatron CDF data with the kinematic cuts $P_{T}>400$ GeV and
$0.1<|Y|<0.7$ at $R=0.4$ and $R=0.7$. The inset shows the detailed comparison
in large jet mass region.

###  0.4.3 Jet Substructure

It is known that a top quark produced almost at rest at the Tevatron can be
identified by measuring isolated jets from its decay. However, this strategy
does not work for identifying a highly-boosted top quark produced at the LHC.
It has been observed that an ordinary high-energy QCD jet [68, 69] can have an
invariant mass close to the top quark mass. A highly-boosted top quark,
producing only a single jet, is then difficult to be distinguished from a QCD
jet. This difficulty also appears in the identification of a highly-boosted
new-physics resonance decaying into standard-model particles, or Higgs boson
decaying into a bottom-quark pair. Hence, additional information needs to be
extracted from jet internal structures in order to improve the jet
identification at the LHC. The quantity, called planar flow [70], has been
proposed for this purpose, which utilizes the geometrical shape of a jet: a
QCD jet with large invariant mass mainly involves one-to-two splitting, so it
leaves a linear energy deposition in a detector. A top-quark jet, proceeding
with a weak decay, mainly involves one-to-three splitting, so it leaves a
planar energy deposition. Measuring this additional information, it has been
shown with event generators that the top-quark identification can be improved
to some extent. Investigations on various observables associated with jet
substructures are usually done using event generators. For a review on recent
theoretical progress and the latest experimental results in jet substructures,
see [71].

Here I focus on a jet substructure, called the energy profile, and explain how
to calculate it in the resummation formalism [64]. This quantity describes the
energy fraction accumulated in the cone of size $r$ within a jet cone $R$,
i.e., $r<R$. Its explicit definition is given by [72]

$\Psi(r)=\frac{1}{N_{J}}\sum_{J}\frac{\sum_{r_{i}<r,i\in{J}}P_{Ti}}{\sum_{r_{i}<R,i\in{J}}P_{Ti}},$
(80)

with the normalization $\Psi(R)=1$, where $P_{Ti}$ is the transverse momentum
carried by particle $i$ in the jet $J$, and $r_{i}<r$ $(r_{i}<R)$ means the
flow of particle $i$ into the jet cone $r$ $(R)$. Different types of jets are
expected to exhibit different energy profiles. For example, a light-quark jet
is narrower than a gluon jet; that is, energy is accumulated faster with $r$
in a light-quark jet than in a gluon jet. A heavy-particle jet certainly has a
distinct energy profile, which can be used for its identification. The
importance of higher-order corrections and their resummation for studying a
jet energy profile have been first emphasized in [73]. Another approach based
on the soft-collinear effective theory and its application to jet production
at an electron-positron collider can be found in Refs. [74, 75, 76].

We first define the jet energy functions
$J^{E}_{f}(M_{J}^{2},P_{T},\nu^{2},R,r)$ with $f=q(g)$ denoting the light-
quark (gluon), which describe the energy accumulation within the cone of size
$r<R$. The definition is chosen, such that
$J^{E(0)}_{f}=P_{T}\delta(M_{J}^{2})$ at LO. The Feynman rules for $J^{E}_{f}$
are similar to those for the jet functions $J_{f}$ at each order of
$\alpha_{s}$, except that a sum of the step functions
$\sum_{i}k_{i}^{0}\Theta(r-\theta_{i})$ is inserted, where $k_{i}^{0}$
($\theta_{i}$) is the energy (the angle with respect to the jet axis) of
particle $i$. For example, the jet energy functions $J^{E}_{f}$ are expressed,
at NLO, as

$\displaystyle J_{q}^{E(1)}(M_{J}^{2},P_{T},\nu^{2},R,r,\mu^{2})$
$\displaystyle=$
$\displaystyle\frac{(2\pi)^{3}}{2\sqrt{2}(P_{J}^{0})^{2}N_{c}}\sum_{\sigma,\lambda}\int\frac{d^{3}p}{(2\pi)^{3}2p^{0}}\frac{d^{3}k}{(2\pi)^{3}2k^{0}}$
$\displaystyle\times[p^{0}\Theta(r-\theta_{p})+k^{0}\Theta(r-\theta_{k})]$
$\displaystyle\times{\rm Tr}\left\\{\not\xi\langle
0|q(0)W^{(\bar{q})\dagger}|p,\sigma;k,\lambda\rangle\langle
k,\lambda;p,\sigma|W^{(\bar{q})}\bar{q}(0)|0\rangle\right\\}$
$\displaystyle\times\delta(M_{J}^{2}-(p+k)^{2})\delta^{(2)}(\hat{e}-\hat{e}_{\bf{p}+\bf{k}})\delta(P_{J}^{0}-p^{0}-k^{0}),$
$\displaystyle J_{g}^{E(1)}(M_{J}^{2},P_{T},\nu^{2},R,r,\mu^{2})$
$\displaystyle=$
$\displaystyle\frac{(2\pi)^{3}}{2(P_{J}^{0})^{3}N_{c}}\sum_{\sigma,\lambda}\int\frac{d^{3}p}{(2\pi)^{3}2p^{0}}\frac{d^{3}k}{(2\pi)^{3}2k^{0}}$
(81)
$\displaystyle\times[p^{0}\Theta(r-\theta_{p})+k^{0}\Theta(r-\theta_{k})]$
$\displaystyle\times\langle
0|\xi_{\sigma}F^{\sigma\nu}(0)W^{(g)\dagger}|p,\sigma;k,\lambda\rangle\langle
k,\lambda;p,\sigma|W^{(g)}F_{\nu}^{\rho}(0)\xi_{\rho}|0\rangle$
$\displaystyle\times\delta(M_{J}^{2}-(p+k)^{2})\delta^{(2)}(\hat{e}-\hat{e}_{\bf{p}+\bf{k}})\delta(P_{J}^{0}-p^{0}-k^{0}),$

where the expansion of the Wilson links in $\alpha_{s}$ is understood. The
factorization scale is set to $\mu=P_{T}$ to remove the associated logarithms,
so its dependence will be suppressed below.

The Mellin-transformed jet energy function ${\bar{J}}_{q}^{E}$ obeys a similar
differential equation [64]

$\displaystyle-\frac{n^{2}}{v\cdot
n}v_{\alpha}\frac{d}{dn_{\alpha}}{\bar{J}}_{q}^{E}(N=1,P_{T},\nu^{2},R,r)=2({\bar{K}}+G){\bar{J}}_{q}^{E}(N=1,P_{T},\nu^{2},R,r),$
(82)

which can be solved simply. Inserting the solutions to Eq. (82) into Eq. (80),
the jet energy profile is derived. Note that a jet energy profile with $N=1$
is not sensitive to the nonperturbative contribution, so the predictions are
free of the nonperturbative parameter dependence, in contrast to the case of
the jet invariant mass distribution. It has been found that the light-quark
jet has a narrower energy profile than the gluon jet, as exhibited in Fig. 22
for $\sqrt{s}=7$ TeV and the interval $80$ GeV $<P_{T}<$ $100$ GeV of the jet
transverse momentum. The broader distribution of the gluon jet results from
stronger radiations caused by the larger color factor $C_{A}=3$, compared to
$C_{F}=4/3$ for a light-quark jet.

Figure 22: Resummation predictions for the energy profiles of the light-quark
(solid curve) and gluon (dotted curve) jets with $\sqrt{s}=7$ TeV and 80 GeV
$<P_{T}<$ 100 GeV.

Figure 23: Comparison of resummation predictions for the jet energy profiles
with $R=0.7$ to Tevatron CDF data in various $P_{T}$ intervals. The NLO
predictions denoted by the dotted curves are also displayed.

One then convolutes the light-quark and gluon jet energy functions with the
constituent cross sections of the LO partonic subprocess and CTEQ6L PDFs [65]
at certain collider energy. The predictions are directly compared with the
Tevatron CDF data [72] as shown in Fig. 23. It is evident that the resummation
predictions agree well with the data in all $P_{T}$ intervals. The NLO
predictions derived from $\bar{J}_{f}^{E(1)}(1,P_{T},\nu_{\rm fi}^{2},R,r)$
are also displayed for comparison, which obviously overshoot the data. The
resummation predictions for the jet energy profiles are compared with the LHC
CMS data at 7 TeV [77] from the anti-$k_{T}$ jet algorithm [78] in Fig. 24,
which are also consistent with the data in various $P_{T}$ intervals. Since
one can separate the contributions from the light-quark jet and the gluon jet,
the comparison with the CDF and CMS data implies that high-energy (low-energy)
jets are mainly composed of the light-quark (gluon) jets. Hence, a precise
measurement of the jet energy profile as a function of jet transverse momentum
can be used to experimentally discriminate the production mechanism of jets in
association with other particles, such as electroweak gauge bosons, top quarks
and Higgs bosons.

Figure 24: Resummation predictions for the jet energy profiles with $R=0.7$
compared to LHC CMS data in various $P_{T}$ intervals. The NLO predictions
denoted by the dotted curves are also displayed.

##  0.5 Hadronic heavy-quark decays

Hadronic decays of heavy-quark bound states, such as $B$, $B_{s}$, and
$\Lambda_{b}$, are one of the focuses of LHCb physics, whose precision
measurement may reveal new physics in the flavor sector. They are difficult to
analyze theoretically because of complicated QCD dynamics and multiple
characteristic scales they involve: the $W$ boson mass $m_{W}$, the $b$ quark
mass $m_{b}$, and the QCD scale $\Lambda_{\rm QCD}$. The standard procedure is
first to integrate out the scale $m_{W}$, such that QCD dynamics is organized
into an effective weak Hamiltonian [79]. For the $B\to D\pi$ decays, the
effective Hamiltonian is written as

$\displaystyle{\cal H}_{\rm
eff}={G_{F}\over\sqrt{2}}\,V_{cb}V_{ud}^{*}\Big{[}C_{1}(\mu)O_{1}(\mu)+C_{2}(\mu)O_{2}(\mu)\Big{]},$
(83)

where $G_{F}$ is the Fermi coupling constant, $V_{cb}V_{ud}^{*}$ is the
product of the Cabibbo-Kobayashi-Maskawa matrix elements, $\mu$ is the
renormalization scale, $C_{1,2}$ are the Wilson coefficients, and the four-
fermion operators are defined by

$\displaystyle O_{1}=(\bar{d}b)_{V-A}(\bar{c}u)_{V-A}\;,\qquad\qquad
O_{2}=(\bar{c}b)_{V-A}(\bar{d}u)_{V-A}.$ (84)

For exclusive processes, such as hadron form factors, the collinear
factorization was developed in [80, 81, 82, 83]. The range of a parton
momentum fraction $x$, contrary to that in the inclusive case, is not
experimentally controllable, and must be integrated over between 0 and 1.
Hence, the end-point region with a small $x$ is not avoidable. If there is no
end-point singularity developed in a hard kernel, the collinear factorization
works. If such a singularity occurs, indicating the breakdown of the collinear
factorization, the $k_{T}$ factorization should be employed, because the
parton transverse momentum $k_{T}$ is not negligible. To derive $B\to D\pi$
decay amplitudes, one evaluates the hadronic matrix elements $\langle
D\pi|O_{i}(\mu)|B\rangle$. Different theoretical approaches have been
developed for this purpose, which include the factorization assumption, the
QCD-improved factorization, the perturbative QCD, the soft-collinear effective
theory, the light-cone QCD sum rules, and the quark-diagram parametrization.
In this section I briefly introduce the basic ideas of the first three
approaches [24].

###  0.5.1 Factorization Assumption

Intuitively, decay products from a heavy $b$ quark move fast without further
interaction between them. This naive picture is supported by the color-
transparency argument [84]: the Lorentz contraction renders energetic final
states emitted from the weak vertex have small longitudinal color dipoles,
which cannot be resolved by soft gluons. Therefore, the hadronic matrix
element $\langle O(\mu)\rangle$ is factorized into a product of two matrix
elements of single currents, governed by decay constants and form factors,
without soft gluon exchanges between them. This factorization assumption (FA)
[14] was first proved in the framework of large energy effective theory [85],
and justified in the large $N_{c}$ limit [86]. For the $B\to D\pi$ decays, the
color-allowed (color-suppressed) amplitude, involving the $B\to D$ ($B\to\pi$)
transition form factor, is proportional to the Wilson coefficient
$a_{1}=C_{2}+C_{1}/N_{c}$ ($a_{2}=C_{1}+C_{2}/N_{c}$).

In spite of its simplicity, the FA encounters three principal difficulties.
First, a hadronic matrix element under the FA is independent of the
renormalization scale $\mu$, as the vector or axial-vector current is
partially conserved. Consequently, the amplitude $C(\mu)\langle O\rangle_{\rm
fact}$ is not truly physical as the scale dependence of the Wilson coefficient
does not get compensation from the matrix element. This problem may not be
serious for color-allowed modes, because the parameter $a_{1}$ is roughly
independent of $\mu$. It is then not a surprise that the simple FA gives
predictions in relatively good agreement with data of these modes. However,
the parameter $a_{2}$ depends strongly on the renormalization scale and on the
renormalization scheme, because of the similar magnitude and different sign of
the $C_{1}(\mu)$ and $C_{2}(\mu)/N_{c}$ terms (calculated in the NDR scheme
and for $\Lambda_{\overline{MS}}^{(5)}=225$ GeV, the Wilson coefficients have
the values $C_{1}(m_{B})=-0.185$ and $C_{2}(m_{B})=1.082$ [79], $m_{B}$ being
the $B$ meson mass). This may be the reason why the FA fails to accommodate
data of color-suppressed modes. It also means that $a_{2}$ is more sensitive
to subleading contributions.

The second difficulty is related to the first one: nonfactorizable effects
have been neglected in the FA. This neglect may be justified for color-allowed
modes due to the large and roughly $\mu$-independent value of $a_{1}$, but not
for color-suppressed modes, such as $B\to J/\psi K^{(*)}$. The $J/\psi$ meson
emitted from the weak vertex is not energetic, and the color-transparency
argument does not apply. To circumvent this difficulty, nonfactorizable
contributions were parameterized into the parameters $\chi_{i}$ [87, 88],

$\displaystyle a_{1}^{\rm eff}$ $\displaystyle=$ $\displaystyle
C_{2}(\mu)+C_{1}(\mu)\left[{1\over N_{c}}+\chi_{1}(\mu)\right],$
$\displaystyle a_{2}^{\rm eff}$ $\displaystyle=$ $\displaystyle
C_{1}(\mu)+C_{2}(\mu)\left[{1\over N_{c}}+\chi_{2}(\mu)\right].$ (85)

The $\mu$ dependence of the Wilson coefficients is assumed to be exactly
compensated by that of $\chi_{i}(\mu)$ [89]. It is obvious that the
introduction of $\chi_{i}$ does not really resolve the scale problem in the
FA.

Third, strong phases are essential for predicting CP asymmetries in exclusive
$B$ meson decays. These phases, arising from the Bander-Silverman-Soni (BSS)
mechanism [90], are ambiguous in the FA: the charm quark loop contributes an
imaginary piece proportional to

$\int duu(1-u)\theta(q^{2}u(1-u)-m_{c}^{2}),$ (86)

where $q^{2}$ is the invariant mass of the gluon attaching to the charm loop.
Since $q^{2}$ is not precisely defined in the FA, one cannot obtain definite
information of strong phases from Eq. (86). Moreover, it is legitimate to
question whether the BSS mechanism is an important source of strong phases in
$B$ meson decays. Viewing the above difficulties, the FA is not a complete
model, and it is necessary to go beyond the FA by developing reliable and
systematic theoretical approaches.

###  0.5.2 QCD-improved Factorization

The color-transparency argument allows the addition of hard gluons between the
energetic mesons emitted from the weak vertex and the $B$ meson transition
form factors. These hard gluon exchanges lead to higher-order corrections in
the coupling constant $\alpha_{s}$ to the FA. By means of Feynman diagrams,
they appear as the vertex corrections in the first two rows of Fig. 25 [15].
It has been shown that soft divergences cancel among them, when computed in
the collinear factorization theorem. These $O(\alpha_{s})$ corrections weaken
the $\mu$ dependence in the Wilson coefficients, and generate strong phases.
Besides, hard gluons can also be added to form the spectator diagrams in the
last row of Fig. 25. Feynman rules of these two diagrams differ by a minus
sign in the soft region resulting from the involved quark and anti-quark
propagators. Including the above nonfactorizable corrections to the FA leads
to the QCD-improved factorization (QCDF) approach [15]. The gluon invariant
mass $q^{2}$ in the BSS mechanism can be unambiguously defined and related to
parton momentum fractions in QCDF. Hence, the theoretical difficulties in the
FA are resolved. This is a breakthrough towards a rigorous framework for two-
body hadronic $B$ meson decays in the heavy quark limit.

Figure 25: $O(\alpha_{s})$ corrections to the FA in the QCDF approach.

Corrections in higher powers of $1/m_{b}$ to the FA can also be included into
QCDF, such as those from the annihilation topology in Fig. 26, and from
twist-3 contributions to the spectator amplitudes. However, it has been found
that endpoint singularities exist in these high-power contributions, which
arise from the divergent integral $\int_{0}^{1}dx/x$, $x$ being a momentum
fraction. These singularities have the same origin as those in the collinear
collinear factorization formulas for $B$ meson transition form factors [91].
Because of the endpoint singularities, the annihilation and twist-3 spectator
contributions must be parameterized as [15]

$\displaystyle\ln\frac{m_{B}}{\Lambda_{h}}\left(1+\rho_{A}e^{i\delta_{A}}\right)\;,\;\;\;\;\ln\frac{m_{B}}{\Lambda_{h}}\left(1+\rho_{H}e^{i\delta_{H}}\right)\;,$
(87)

respectively, with the hadronic scale $\Lambda_{h}$. A QCDF formula then
contains the arbitrary parameters $\rho_{A,H}$ and $\delta_{A,H}$. Setting
these parameters to zero, one obtains predictions in the “default” scenario,
and the variation of the arbitrary parameters gives theoretical uncertainties.
If tuning these parameters to fit data, one obtains results in the scenarios
“S”, “S2”,…[92].

Figure 26: Annihilation contributions.

###  0.5.3 Perturbative QCD

The endpoint singularities signal the breakdown of the collinear factorization
for two-body hadronic $B$ meson decays. Motivated by the removal of these
singularities, the perturbative QCD (PQCD) approach based on the $k_{T}$
factorization theorem was developed [16, 17, 18, 19]. A parton transverse
momentum $k_{T}$ can be generated by gluon radiations, before hard scattering
occurs. The endpoint singularities from the small $x$ region simply indicate
that $k_{T}$ is not negligible. Taking into account $k_{T}$, a particle
propagator does not diverge as $x\to 0$. The $B$ meson transition form
factors, and the spectator and annihilation contributions are then all
calculable in the framework of the $k_{T}$ factorization theorem. It has been
shown that a $B\to M_{1}M_{2}$ decay amplitude is factorized into the
convolution of the six-quark hard kernel, the jet function and the Sudakov
factor with the bound-state wave functions as shown in Fig. 27,

$\displaystyle A(B\to M_{1}M_{2})=\phi_{B}\otimes H\otimes J\otimes
S\otimes\phi_{M_{1}}\otimes\phi_{M_{2}}.$ (88)

The jet function $J$ comes from the threshold resummation, which exhibits
suppression in the small $x$ region [93]. The Sudakov factor $S$ comes from
the $k_{T}$ resummation, which exhibits suppression in the small $k_{T}$
region [39, 40]. These resummation effects guarantee the removal of the
endpoint singularities. $J$ ($S$), organizing double logarithms in the hard
kernel (meson wave functions), is hidden in $H$ (the three meson states) in
Fig. 27. The arbitrary parameters introduced in QCDF are not necessary, and
PQCD involves only universal and controllable inputs.

Figure 27: Perturbative QCD factorization.

The theoretical difficulties in the FA are also resolved in PQCD but in a
different manner. The FA limit of the PQCD approach at large $m_{b}$, which is
not as obvious as in QCDF, has been examined [93]. It was found that the
factorizable emission amplitude decreases like $m_{b}^{-3/2}$, if the $B$
meson decay constant $f_{B}$ scales like $f_{B}\propto m_{b}^{-1/2}$. This
power-law behavior is consistent with that obtained in [15, 94]. The higher-
order corrections to the FA have been included in PQCD, which moderate the
dependence on the renormalization scale $\mu$. The ratio of the spectator
contribution over the factorizable emission contribution decreases with
$m_{b}$ in PQCD, showing a behavior close to that in QCDF. The gluon invariant
mass $q^{2}$ in the BSS mechanism is unambiguously defined and related to
parton momentum fractions. The penguin annihilation amplitude is almost
imaginary in PQCD [18], whose mechanism is similar to the BSS one [90]: in the
annihilation topology, the loop is formed by the internal particles in the LO
hard kernel and by infinitely many Sudakov gluons exchanged between two
partons in a light meson. A sizable strong phase is generated, when the
internal particles go on mass shell. In terms of the principal-value
prescription for the internal particle propagator, the strong phase is given
by [18]

$\displaystyle\frac{1}{xm_{B}^{2}-k_{T}^{2}+i\epsilon}=\frac{P}{xm_{B}^{2}-k_{T}^{2}}-i\pi\delta(xm_{B}^{2}-k_{T}^{2}).$
(89)

###  0.5.4 Soft-Collinear Effective Theory

The soft-collinear effective theory (SCET) based on the collinear
factorization is formulated in the framework of OPE [20, 21, 22, 23]. The
matching at different scales involved in $B$ meson decays has been carefully
handled in SCET. Take the simple $B\to\pi$ transition form factor in Fig. 28
as an example. The soft spectator in the $B$ meson carries the momentum $r\sim
O(\Lambda_{\rm QCD})$, because it is dominated by soft dynamics. If the
spectator in the energetic pion carries the momentum $p_{2}\sim O(m_{b})$, the
virtual gluon in Fig. 28 is off-shell by $p_{g}^{2}=(p_{2}-r)^{2}=-2p_{2}\cdot
r\sim O(m_{b}\Lambda_{\rm QCD})$. Then the virtual quark in Figs. 28(a) is
off-shell by $(m_{b}v+k+p_{g})^{2}-m_{b}^{2}\sim O(m_{b}^{2})$, where $v$ is
the $b$ quark velocity and $k\sim O(\Lambda_{\rm QCD})$ denotes the Fermi
motion of the $b$ quark. Hence, $B$ meson decays contain three scales below
$m_{W}$: $m_{b}$, $\sqrt{m_{b}\Lambda_{\rm QCD}}$, and $\Lambda_{\rm QCD}$.

Figure 28: Diagrams for the $B\to\pi$ form factor in QCD.

The separate matching at the two scales $m_{b}$ and $\sqrt{m_{b}\Lambda_{\rm
QCD}}$ is briefly explained below [95]. The first step is to integrate out the
lines off-shell by $m_{b}^{2}$ in QCD, and the resultant effective theory is
called SCETI. One then derives the zeroth-order effective current $J^{(0)}$
from the $b\to u$ weak vertex, and the first-order effective current $J^{(1)}$
by shrinking the virtual $b$ quark line in Fig. 28(a). The next step is to
integrate out the lines off-shell by $m_{b}\Lambda_{\rm QCD}$ in SCETI,
arriving at SCETII. The relevant diagrams to start with are displayed in Fig.
29. Shrinking all the lines off-shell by $m_{b}\Lambda_{\rm QCD}$, one derives
the corresponding Wilson coefficients, i.e., the jet functions, and the
effective four-fermion operators. Sandwiching these four-fermion operators by
the initial $B$ meson state and the final pion state leads to the $B$ meson
and pion distribution amplitudes. The $B\to\pi$ transition form factor is then
factorized as depicted in Fig. 30. The factorization of two-body hadronic $B$
meson decays is constructed in a similar way, and the result is also shown in
Fig. 30.

Figure 29: Diagrams for the $B\to\pi$ form factor in SCETI.

Figure 30: Factorization of the $B\to\pi$ form factor and of the $B\to
M_{1}M_{2}$ decay in SCET.

At leading power in $1/m_{b}$, there is no large source of strong phases in
SCET (the annihilation contribution is parametrically power-suppressed). To
acquire strong phases, it has been argued that $c\bar{c}$ (charming) penguins
could give long-distance effects at leading power [96]. This contribution is
nonperturbative, so it must be parameterized as an arbitrary amplitude
$A^{c\bar{c}}$. Including the charming penguin, SCET has been applied as an
QCD-improved parametrization, and $A^{c\bar{c}}$ is determined together with
other hadronic inputs from data. It should be mentioned that the long-distance
charming-penguin contribution is power-suppressed according to QCDF, PQCD and
light-cone sum rules [97].

###  0.5.5 Puzzles in $B$ Physics

Before concluding, I review the long-standing puzzles in hadronic two-body $B$
meson decays, which have not yet been fully resolved so far. According to a
naive estimate of the color-suppressed tree amplitude, the hierarchy of the
branching ratios $B(B^{0}\to\pi^{0}\pi^{0})\sim
O(\lambda^{2})B(B^{0}\to\pi^{\mp}\pi^{\pm})$ with the CKM parameter
$\lambda\approx 0.2$ is expected. However, the data [98]

$\displaystyle B(B^{0}\to\pi^{\mp}\pi^{\pm})$ $\displaystyle=$
$\displaystyle(5.10\pm 0.19)\times 10^{-6}\;,$ $\displaystyle
B(B^{0}\to\pi^{0}\pi^{0})$ $\displaystyle=$
$\displaystyle(1.91^{+0.22}_{-0.23})\times 10^{-6}\;,$ (90)

imply $B(B^{0}\to\pi^{0}\pi^{0})\sim O(\lambda)B(B^{0}\to\pi^{\mp}\pi^{\pm})$,
giving rise to the $B\to\pi\pi$ puzzle. As observed in [99], the NLO
corrections, despite of increasing the color-suppressed tree amplitude
significantly, are not enough to enhance the $B^{0}\to\pi^{0}\pi^{0}$
branching ratio to the measured value. A much larger color-suppressed tree
amplitude, about the same order as the color-allowed tree amplitude, must be
obtained in order to resolve the puzzle [100, 101]. To make sure that the
above NLO effects are reasonable, the PQCD formalism has been applied to the
$B\to\rho\rho$ decays [99], which also receive the color-suppressed tree
contribution. It was observed that the NLO PQCD predictions are in agreement
with the data $B(B^{0}\to\rho^{0}\rho^{0})=(0.73^{+0.27}_{-0.28})\times
10^{-6}$ [98]. One concludes that it is unlikely to accommodate the measured
$B^{0}\to\pi^{0}\pi^{0}$ and $\rho^{0}\rho^{0}$ branching ratios
simultaneously in PQCD, and that the $B\to\pi\pi$ puzzle remains.

It has been claimed that the $B\to\pi\pi$ puzzle has been resolved in the QCDF
approach [15] with an input from SCET [102, 103, 104]: the inclusion of the
NLO jet function, the hard coefficient of SCETII, into the QCDF formula for
the color-suppressed tree amplitude gives sufficient enhancement of the
$B^{0}\to\pi^{0}\pi^{0}$ branching ratio, if adopting the parameter scenario
”S4" [105]. It is necessary to investigate whether the proposed new mechanism
deteriorates the consistency of theoretical results with other data. The
formalism in [102] has been extended to the $B\to\rho\rho$ decays as a check
[99]. It was found that the NLO jet function overshoots the observed
$B^{0}\to\rho^{0}\rho^{0}$ branching ratio very much as adopting ”S4". That
is, it is also unlikely to accommodate the $B\to\pi\pi$ and $\rho\rho$ data
simultaneously in QCDF.

Table 1: Polarization fractions in the penguin-dominated $B\to VV$ decays.  Mode | BABAR | Belle
---|---|---
$\phi K^{*+}$ | $0.49\pm 0.05\pm 0.03$ | $0.52\pm 0.08\pm 0.03$
$K^{*+}\rho^{0}$ | $0.78\pm 0.12\pm 0.03$ |
$K^{*0}\rho^{+}$ | $0.52\pm 0.10\pm 0.04$ | $0.43\pm 0.11^{+0.05}_{-0.02}$
$K^{*+}K^{*0}$ | $0.75^{+0.16}_{-0.26}\pm 0.03$ |

For penguin-dominated $B\to VV$ decays, such as those listed in Table 1 [98],
the polarization fractions deviate from the naive counting rules based on
kinematics [106]. This is the so-called the $B\to\phi K^{*}$ puzzle. Many
attempts to resolve the $B\to\phi K^{*}$ polarizations have been made [107],
which include new physics [108, 109, 110, 111, 112], the annihilation
contribution [113, 114] in the QCDF approach, the charming penguin in SCET
[115], the rescattering effect [116, 117, 118], and the $b\to sg$ (the
magnetic penguin) [119] and $b\to s\gamma$ [120] transitions. The annihilation
contribution from the scalar penguin operators improves the consistency with
the data, because it is of the same order for all the three final helicity
states, and could enhance the transverse polarization fractions [106].
However, the PQCD analysis of the scalar penguin annihilation amplitudes
indicates that the $B\to\phi K^{*}$ puzzle cannot be resolved completely
[107]. A reduction of the $B\to K^{*}$ form factor $A_{0}$, which is
associated with the longitudinal polarization, further helps accommodating the
data [121].

The penguin-dominated $B\to K^{*}\rho$ decays are expected to exhibit similar
polarization fractions. This is the reason why the longitudinal polarization
fraction in the $B^{+}\to K^{*0}\rho^{+}$ decay, which contains only the
penguin contribution, is close to $f_{L}(\phi K^{*})\sim 0.5$ as listed in
Table 1. Another mode $B^{+}\to K^{*+}\rho^{0}$, nevertheless, exhibits a
large longitudinal polarization fraction around 0.8. This mode involves tree
amplitudes, which are subdominant, and should not cause a significant
deviation from $f_{L}\sim 0.5$. Though the data of $f_{L}(K^{*0}\rho^{+})$
from BABAR still suffer a large error, the different longitudinal polarization
fractions, $f_{L}(K^{*+}\rho^{0})\not=f_{L}(K^{*0}\rho^{+})$, call for a
deeper understanding. The $B^{+}\to K^{*+}K^{*0}$ decay shows a longitudinal
polarization fraction smaller than unity, but larger than 0.5. A more thorough
study of the $B\to K^{*}K^{*}$ decays can help discriminating the various
resolutions for the $B\to\phi K^{*}$ puzzle [121, 122].

The $B^{0}\to K^{\pm}\pi^{\mp}$ decays depend on the tree amplitude $T$ and
the QCD penguin amplitude $P$. The data of the direct CP asymmetry
$A_{CP}(B^{0}\to K^{\pm}\pi^{\mp})\approx-10\%$ then imply a sizable relative
strong phase between $T$ and $P$, which verifies the LO PQCD prediction made
years ago [18]: the scalar penguin annihilation provides an important source
of strong phases. The PQCD predictions for significant penguin annihilation
have been confirmed by the recent measurement of the pure annihilation mode,
$B(B_{s}\to\pi^{+}\pi^{-})=(0.73\pm 0.14)\times 10^{-6}$, which is consistent
with $0.57\times 10^{-6}$ obtained in the LO PQCD approach [123]. The
$B^{\pm}\to K^{\pm}\pi^{0}$ decays contain the additional color-suppressed
tree amplitude $C$ and electroweak penguin amplitude $P_{ew}$. Since both $C$
and $P_{ew}$ are subdominant, the approximate equality for the direct CP
asymmetries $A_{CP}(B^{\pm}\to K^{\pm}\pi^{0})\approx A_{CP}(B^{0}\to
K^{\pm}\pi^{\mp})$ is expected. However, this naive expectation is in conflict
with the data [98],

$\displaystyle A_{CP}(B^{0}\to K^{\pm}\pi^{\mp})$ $\displaystyle=$
$\displaystyle-0.086\pm 0.007$ $\displaystyle A_{CP}(B^{\pm}\to
K^{\pm}\pi^{0})$ $\displaystyle=$ $\displaystyle 0.040\pm 0.021,$ (91)

making the $B\to K\pi$ puzzle.

While LO PQCD gives a negligible $C$ [18, 19], it is possible that this
supposedly tiny amplitude receives a significant subleading correction. Note
that the small $C$ is attributed to the accidental cancellation between the
Wilson coefficients $C_{1}$ and $C_{2}/N_{c}$ at the scale of $m_{b}$. In
[124] the important NLO contributions to the $B\to K\pi$ decays from the
vertex corrections, the quark loops, and the magnetic penguins were
calculated. It was observed that the vertex corrections increase $C$ by a
factor of 3, and induce a large phase about $-80^{o}$ relative to $T$. The
large and imaginary $C$ renders the total tree amplitude $T+C$ more or less
parallel to the total penguin amplitude $P+P_{ew}$ in the $B^{\pm}\to
K^{\pm}\pi^{0}$ decays, leading to nearly vanishing $A_{CP}(B^{\pm}\to
K^{\pm}\pi^{0})=(-1^{+3}_{-6})\%$ at NLO (it is about -8% at LO). One
concludes that the $B\to K\pi$ puzzle has been alleviated, but not yet gone
away completely. Whether new physics effects [125, 126] are needed will become
clear when the data get precise. More detailed discussion on this subject can
be found in [127].

##  0.6 Summary

Despite of nonperturbative nature of QCD, theoretical frameworks with
predictive power can be developed. They are based on the factorization
theorems, in which nonperturbative dynamics is absorbed into PDFs, and the
remaining infrared finite contributions go to hard kernels. A PDF is universal
(process-independent) and can be extracted from data, while a hard kernel is
calculable in perturbation theory. Both the collinear and $k_{T}$
factorization theorems are the fundamental tools of pQCD. The collinear
factorization theorem is a simpler version, and has been intensively studied
and widely applied. The $k_{T}$ factorization theorem is more complicated, and
many of its aspects have not been completely explored.

Sophisticated evolution equations and resummation techniques have been
developed in pQCD, which enhance predictive power, and increase theoretical
precision. All the known single- and double-logarithm summations, including
their unifications, have been explained in the CSS resummation formalism. The
point is the treatment of real gluon emissions under different kinematic
orderings, and the resultant logarithmic summations are summarized in Table
LABEL:T2. The $k_{T}$ and threshold resummations, and the DGLAP and BFKL
equations have been applied to various QCD processes.

Table 2: Single- and double-logarithmic summations under different kinematic orderings. | $|$ | small $x$ | $|$ | intermediate $x$ | $|$ | large $x$
---|---|---|---|---|---|---
rapidity ordering | $|$ | BFKL equation | $|$ |  $k_{T}$ resummation | $|$ |
$k_{T}$ ordering | $|$ |  | $|$ | DGLAP equation | $|$ | threshold resummation
angular ordering | $|$ | CCFM |  | equation; joint |  | resummation

Experimental and theoretical studies of jet physics have been reviewed.
Especially, it was pointed out that jet substructures could be calculated in
pQCD: starting with the jet function definition, applying the factorization
theorem and the resummation technique, one can predict observables, which are
consistent with data. Because fixed-order calculations are not reliable at
small jet invariant mass, and event generators have ambiguities, pQCD provides
an alternative approach, that resolves the above difficulties. The pQCD
formalism will improve the jet identification and new particle search at the
LHC.

We have been able to go beyond the factorization assumption fr hadronic two-
body heavy-quark decays by including QCD corrections. Different approaches
have been discussed and commented: in QCDF the high-power corrections must be
parameterized due to the existence of the endpoint singularities. There are no
endpoint singularities in PQCD, which is based on the $k_{T}$ factorization
theorem, and in SCET, which employs the zero-bin subtraction [128]. A major
difference arises from the treatment of the annihilation contribution, which
is parameterized in QCDF and neglected in SCET, but is the main source of
strong phases in PQCD.

Many subtle subjects on pQCD deserve more exploration, including the
legitimate definition of TMDs, the gauge invariance of the $k_{T}$
factorization, resummations of other types of logarithms, such as rapidity
logarithms, non-global logarithms, and etc., jet substructures of boosted
heavy particles, and the long-standing puzzles in $B$ physics. pQCD remains as
one of the most challenging research fields in high-energy physics.

## Acknowledgment

This work was supported in part by the National Science Council of R.O.C.
under Grant No. NSC-101-2112-M-001-006-MY3, and by the National Center for
Theoretical Sciences of R.O.C.. The author acknowledges the hospitality of the
organizers during the First Asia-Europe-Pacific School of High-energy Physics
at Fukuoka, Japan in Oct., 2012.

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