1

16. Heavy-Quark and Soft-Collinear Effective Theory

16. Heavy-Quark and Soft-Collinear Effective Theory
Revised August 2023 by C.W. Bauer (LBNL) and M. Neubert (PRISMA, Mainz Inst. for Theor.
Physics, JG U.).

16.1 Effective Field Theories
Quantum field theories provide the most precise computational tools for describing physics at
the highest energies. One of their characteristic features is that they almost inevitably involve
multiple length scales. When trying to determine the value of an observable, quantum field theory
demands that all possible virtual states and hence all particles be included in the calculation.
Since these particles have widely different masses, the final prediction is sensitive to many scales.
This fact represents a formidable challenge from a practical point of view. No realistic quantum
field theories can be solved exactly, so that one needs to resort to approximation schemes; these,
however, are typically most straightforward when only a single scale is involved at a time.
Effective field theories (EFTs) provide a general theoretical framework to deal with the multiscale problems of realistic quantum field theories. This framework aims at reducing such problems
to a combination of separate and simpler single-scale problems; simultaneously, however, it provides
an organization scheme whereby the other scales are not omitted but allowed to play their role ina separate step of the computation. The philosophy and basic principles of this approach are very
generic, and correspondingly EFTs represent a widely used method in many different areas of highenergy physics, from the low-energy scales of atomic and nuclear physics to the high-energy scalesof (partly yet unknown) elementary-particle physics, see [1–3] for some early references. EFTs canplay a role both within analytic perturbative computations and in the context of non-perturbative
numerical simulations. One of the simplest applications of EFTs to particle physics concerns thedescription of an underlying theory that is only probed at energy scales E < Λ. Any particle
with mass m > Λ cannot be produced as a real state and therefore only leads to short-distancevirtual effects. Thus, one can construct an effective theory in which the quantum fluctuations ofsuch heavy particles are “integrated out” from the generating functional for Green functions. This
results in a simpler theory containing only those degrees of freedom that are relevant to the energy
scales under consideration. In fact, the standard model of particle physics itself is widely viewed
as an EFT of some yet unknown, more fundamental theory [4, 5].
The development of any effective theory starts by identifying the degrees of freedom that are
relevant to describe the physics at a given energy (or length) scale and constructing the Lagrangian
describing the interactions among these fields. Short-distance quantum fluctuations associatedwith much smaller length scales are absorbed into the coefficients of the various operators in theeffective theory. These coefficients are determined in a matching procedure, by requiring that the
EFT reproduces the matrix elements of the full theory up to power corrections. In many cases theeffective Lagrangian exhibits enhanced symmetries compared with the fundamental theory, giving
non-trivial relations between different observables.

16.2 Heavy-Quark Effective Theory
Heavy-quark systems provide prime examples for applications of the EFT technology, because
the hierarchy mQ  ΛQCD (with Q = b, c) provides a natural separation of scales. Physics at thescale mQ is of a short-distance nature and can be treated perturbatively, while for heavy-quark
systems there is always also some hadronic physics governed by the confinement scale ΛQCD of thestrong interaction. Being able to separate the short-distance and long-distance effects associatedwith these two scales is crucial for any quantitative description. For instance, if the long-distancehadronic matrix elements are obtained from lattice QCD, then it is necessary to analytically comS. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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16. Heavy-Quark and Soft-Collinear Effective Theory

pute the effects of short-wavelength modes that do not fit on the lattice. In many other instances,the long-distance physics can be encoded in a small number of hadronic parameters.
16.2.1 General idea and derivation of the effective Lagrangian
The simplest effective theory for heavy-quark systems is the heavy-quark effective theory (HQET)[6–9] (see [10, 11] for detailed discussions). It provides a simplified description of the soft interactions of a single heavy quark with light partons. This includes the interactions that bind the heavy
quark with other light partons inside heavy mesons and baryons.
A softly interacting heavy quark is nearly on-shell. Its momentum may be decomposed as
pQ = mQ v + k, where v is the 4-velocity of the hadron containing the heavy quark. The “residual
momentum” k results from the soft interactions of the heavy quark with its environment andsatisfies v · k ∼ ΛQCD and k 2 ∼ Λ2QCD , which in the rest frame of the heavy hadron reduces tok µ ∼ ΛQCD . In the limit mQ  ΛQCD , the soft interactions do not change the 4-velocity of theheavy quark, which is therefore a conserved quantum number that is often used as a label on theeffective heavy-quark fields.
A nearly on-shell Dirac spinor has two large and two small components. We define
Q(x) = e−imQ v·x [hv (x) + Hv (x)] ,(16.1)

where

1 + /v
1 − /v
Q(x) ,Hv (x) = eimQ v·xQ(x)(16.2)
2
2
are the large (“upper”) and small (“lower”) components of the spinor field, respectively. The extraction of the phase factor in (16.1) implies that the fields hv and Hv carry the residual momentum
k. The field Hv is 1/mQ suppressed relative to hv and describes quantum fluctuations far off themass shell. Integrating it out using its equations of motion yields the HQET Lagrangianhv (x) = eimQ v·x1gh̄v (iDs )2 hv + Cmag (µ) h̄v σµν Gµν
s hv + . . . .
2mQ2

LHQET = h̄v iv · Ds hv +

(16.3)

The covariant derivative iDsµ = i∂ µ + gAµs and the field strength Gµν
s contain only the soft gluon
field. Hard gluons have been integrated out, and their effects are contained in the Wilson coefficientsof the operators in the effective Lagrangian. From the leading operator one derives the Feynman
rules of HQET. The new operators entering at subleading order are referred to as the “kinetic
energy” and “chromo-magnetic interaction”. The kinetic-energy operator corresponds to the firstcorrection term in the Taylor expansion of the relativistic energy E = mQ + p~ 2 /2mQ + . . . . Lorentz
invariance, which is encoded as a reparametrization invariance of the effective Lagrangian [12],ensures that its Wilson coefficient is not renormalized (Ckin ≡ 1). The coefficient Cmag of thechromo-magnetic operator receives corrections starting at one-loop order [6].
16.2.2 Spin-flavor symmetry
The leading term in the HQET Lagrangian exhibits a global spin-flavor symmetry. Its physicalmeaning is that, in the infinite mass limit, the properties of hadronic systems containing a single
heavy quark are insensitive to the spin and flavor of the heavy quark [13, 14]. The spin symmetry
results from the fact that there are no Dirac matrices in the leading term of the effective Lagrangianin (16.3), implying that the interactions of the heavy quark with soft gluons leave its spin unchanged.
The flavor symmetry arises since the mass of the heavy quark does not appear at leading order.
For nQ heavy quarks moving at the same velocity, one can simply extend (16.3) by summing overnQ identical terms for heavy-quark fields hiv . The result is invariant under rotations in flavor space.
When combined with the spin symmetry, the symmetry group becomes promoted to SU (2nQ ).
These symmetries are broken by the operators at subleading power in the 1/mQ expansion.
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16. Heavy-Quark and Soft-Collinear Effective Theory

The spin-flavor symmetry leads to many interesting relations between the properties of hadrons
containing a heavy quark. The most direct consequences concern the spectroscopy of such states[15]. In the heavy-quark limit, the spin of the heavy quark and the total angular momentum jof the light degrees of freedom are separately conserved by the strong interactions. Because ofheavy-quark symmetry, the dynamics is independent of the spin and mass of the heavy quark.
Hadronic states can thus be classified by the quantum numbers (flavor, spin, parity, etc.) of thelight degrees of freedom. The spin symmetry predicts that, for fixed j 6= 0, there is a doublet ofdegenerate states with total spin J = j ± 21 . The flavor symmetry relates the properties of stateswith different heavy-quark flavor.
16.2.3 Weak decay form factors
Of particular interest are the relations between the weak decay form factors of heavy mesons,which parametrize hadronic matrix elements of currents between two mesons containing a heavy
quark. These relations have been derived by Isgur and Wise [14], generalizing ideas developed by
Nussinov and Wetzel [16] and Voloshin and Shifman [17]. For the purpose of this discussion, it isconvenient to work with a mass-independent normalization of meson states and use velocity rather
than momentum variables.
Consider the elastic scattering of a pseudoscalar meson, P (v) → P (v 0 ), induced by an external
vector current coupled to the heavy quark contained in P , which acts as a color source moving withthe meson’s velocity v. The action of the current is to replace instantaneously the color source by
one moving at velocity v 0 . Soft gluons need to be exchanged in order to rearrange the light degrees
of freedom and build up the final state meson moving at velocity v 0 . This rearrangement leads
to a form-factor suppression. The important observation is that, in the mQ → ∞ limit, the form
factor can only depend on the Lorentz boost γ = v · v 0 connecting the rest frames of the initialand final-state mesons (as long as γ = O(1)). In the effective theory the hadronic matrix element
describing the scattering process can therefore be written ashP (v 0 )| h̄v0 γ µ hv |P (v)i = ξ(v · v 0 )(v + v 0 )µ ,(16.4)

with a form factor ξ(v · v 0 ) that is real and independent of mQ . By flavor symmetry, the form factor
remains identical when one replaces the heavy quark Q in one of the meson states by a heavy
quark Q0 of a different flavor, thereby turning P into another pseudoscalar meson P 0 . At the sametime, the current becomes a flavor-changing vector current. This universal form factor is called the
Isgur-Wise function [14]. For equal velocities the vector current J µ = h̄v γ µ hv is conserved in theeffective theory, irrespective of the flavor of the heavy quarks. The corresponding conserved charges
are the generators of the flavor symmetry. It follows that the Isgur-Wise function is normalized
at the point of equal velocities: ξ(1) = 1. Since the recoil energy of the daughter meson P 0 in
the rest frame of the parent meson P is Erecoil = mP 0 (v · v 0 − 1), the point v · v 0 = 1 is referred
to as the zero-recoil limit. The heavy-quark spin symmetry leads to additional relations among
weak decay form factors. It can be used to relate matrix elements involving vector mesons tothose involving pseudoscalar mesons, which once again can be described completely in terms of theuniversal Isgur-Wise function.
The form factor relations imposed by heavy-quark symmetry describe the semileptonic decay
processes B̄ → D ` ν̄ and B̄ → D∗ ` ν̄ in the limit of infinite heavy-quark masses. They are modelindependent consequences of QCD. The known normalization of the Isgur-Wise function at zero
recoil can be used to obtain a model-independent measurement of the element |Vcb | of the CabibboKobayashi-Maskawa (CKM) matrix. The semileptonic decay B̄ → D∗ ` ν̄ is particularly well suitedfor this purpose [18]. Experimentally this is a very clean mode, since the reconstruction of theD∗ meson mass provides a powerful rejection against background. From the theoretical point of
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16. Heavy-Quark and Soft-Collinear Effective Theory

view, it is ideal since the decay rate at zero recoil is protected by Luke’s theorem against first-order
power corrections in 1/mQ [19]. This is described in more detail in Section 12. Corrections to
the heavy-quark symmetry relations for the B̄ → D(∗) form factors near zero recoil can also beconstrained using sum rules derived in the small-velocity limit [20, 21].
16.2.4 Decoupling transformation
At leading order in 1/mQ , the couplings of soft gluons to heavy quarks in the effective Lagrangian(0)(16.3) can be removed by the field redefinition hv (x) = Yv (x) hv (x), where Yv (x) is a soft Wilsonline along the direction of v, extending from minus infinity to the point x. In terms of the new fields(0)(0)
the leading-order HQET Lagrangian becomes LHQET = h̄v iv · ∂ hv . It describes a free theory as
far as the strong interactions of heavy quarks are concerned. However, the theory is neverthelessnon-trivial in the presence of external sources. Consider, e.g., the case of a weak-interaction heavyquark current(0)
h̄v0 γ µ (1 − γ5 )hv = h̄v0 γ µ (1 − γ5 ) Yv†0 Yv h(0)(16.5)v ,where v and v 0 are the velocities of the heavy mesons containing the heavy quarks. Unless thetwo velocities are equal, corresponding to the zero-recoil limit discussed above, the object Yv†0 Yv isnon-trivial, and hence the soft gluons do not decouple from the heavy quarks inside the current
operator. One may interpret Yv†0 Yv as a Wilson loop with a cusp at the point x, where the twopaths parallel to the different velocity vectors intersect. The presence of the cusp leads to non-trivial
ultra-violet behavior (for v 6= v 0 ), which is described by a cusp anomalous dimension Γcusp (v · v 0 )that was calculated at two-loop order in [22]. It coincides with the velocity-dependent anomalous
dimension of heavy-quark currents, which was introduced in the context of HQET in [23]. Theinterpretation of heavy quarks as Wilson lines is a useful tool, which was put forward in one of thevery first papers on the subject [6]. This technology will be useful in the study of the interactionsof heavy quarks with collinear degrees of freedom discussed later in this review.
16.2.5 Heavy-quark expansion for inclusive decays
The theoretical description of inclusive decays of hadrons containing a heavy quark exploits
two observations [24–28]: bound-state effects related to the initial state can be calculated using
the heavy-quark expansion, and the fact that the final state consists of a sum over many hadronic
channels eliminates the sensitivity to the properties of individual final-state hadrons. The second
feature rests on the hypothesis of quark-hadron duality, i.e. the assumption that decay rates arecalculable in QCD after a smearing procedure has been applied [29]. In semileptonic decays, theintegration over the lepton spectrum provides a smearing over the invariant hadronic mass of the
final state (global duality). For nonleptonic decays, where the total hadronic mass is fixed, thesummation over many hadronic final states provides an averaging (local duality). Since global
duality is a much weaker assumption, the theoretical control of inclusive semileptonic decays is on
firmer footing.
Using the optical theorem, the inclusive decay width of a hadron Hb containing a b quark canbe written in the form
1
Im hHb | iΓ (Hb ) =MH b

Z

d4 x T {Heff (x), Heff (0)} |Hb i .

(16.6)The effective weak Hamiltonian for b-quark decays consists of dimension-6 four-fermion operatorsand dipole operators [30]. Because of the large mass of the b quark, it follows that the separationof fields in the time-ordered product in (16.6) is small, of order x ∼ 1/mb . It is thus possible
to construct an operator-product expansion (OPE) for the time-ordered product, in which it isrepresented as a series of local operators in HQET. The leading operator h̄v hv has a trivial matrix
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16. Heavy-Quark and Soft-Collinear Effective Theory

element. The next contributions arise at O(1/m2b ) and give rise to two parameters µ2π (Hb ) andµ2G (Hb ), which are defined as the matrix elements of the heavy-quark kinetic energy and chromomagnetic interaction inside the hadron Hb , respectively [31]. For the ground-state heavy mesons
and baryons, one has µ2G (B) = 3(m2B ∗ − m2B )/4 ' 0.36 GeV2 and µ2G (Λb ) = 0. Thus, the total
inclusive decay rate of a hadron Hb can be written as [25, 26]"

G2 m5 |Vcb |2µ2 (Hb )µ2 (Hb )1
Γ (Hb ) = F b 3
c1 + c2 π 2 + c3 G 2 + O
192π
2mb
2mb
m3b


#



+ ... ,

(16.7)

where the prefactor arises from the loop integrations and is proportional to the fifth power of the bquark mass. The coefficient functions ci are calculable order by order in perturbation theory. Whilec1 corresponds to the decay rate of a free heavy quark, the higher-order coefficients systematicallyaccount for bound-state effects. The coefficients of the subleading operators and of the leading
operator at third order in 1/mb have been calculated at NLO [32–36]. At tree level the heavyquark expansion has been performed up to fifth order in 1/mb [37].
From the fully inclusive width in (16.7) one can obtain the lifetime of a heavy hadron via
τ (Hb ) = 1/Γ (Hb ). Due to the universality of the leading term in the heavy-quark expansion,
lifetime ratios such as τ (B − )/τ (B̄ 0 ), τ (B̄s0 )/τ (B̄ 0 ) and τ (Λb )/τ (B̄ 0 ) are particularly sensitive tothe hadronic parameters determining the power corrections in the expansion. In order to understandthese ratios theoretically, it is necessary to include phase-space enhanced power corrections of order(ΛQCD /mb )3 [38, 39] as well as short-distance perturbative effects [40] in the calculation (see [41]for a discussion of the corresponding calculations).
A formula analogous to (16.7) can be derived for differential distributions in specific inclusive
decay processes, assuming that these distributions are integrated over a sufficiently large regionof phase space to ensure quark-hadron duality. Important examples are the distributions in the
lepton energy and the lepton invariant mass, as well as moments of the invariant hadronic mass
distribution in the semileptonic processes B̄ → Xu ` ν̄ and B̄ → Xc ` ν̄. A global fit of semileptonic
decay distributions can be used to determine the CKM matrix elements |Vub | and |Vcb | alongwith heavy-quark parameters such as the masses mb , mc and the hadronic parameters µ2π (B),µ2G (B). These determinations provide some of the most accurate values for these parameters (seee.g. [42–45]).
16.2.6 Shape functions and non-local power corrections
In certain regions of phase space, in which the hadronic final state in an inclusive heavyhadron decay is made up of light energetic partons, the local OPE for inclusive decays must be
replaced by a more complicated expansion involving hadronic matrix elements of non-local light-ray
operators [46, 47]. Prominent examples are the radiative decay B̄ → Xs γ for large photon energy
Eγ near mB /2, and the semileptonic decay B̄ → Xu ` ν̄ at large lepton energy or small hadronic
invariant mass. In these cases, the differential decay rates at leading order in the heavy-quark
expansion can be written in the factorized form dΓ = H J ⊗ S [48], where the hard function H
and the jet function J are calculable in perturbation theory. The characteristic scales for thesefunctions are set by mb and (mb ΛQCD )1/2 , respectively. The soft function
Z

S(ω) =

dt −iωtehB̄(v)| h̄v (tn)Yn (tn)Yn† (0)hv (0)|B̄(v)i
4π(16.8)

is a non-perturbative object called the shape function [46, 47]. Here Yn are soft Wilson lines along
a light-like direction n aligned with the momentum of the hadronic final-state jet. The jet function
and the shape function share a common variable ω ∼ ΛQCD , and the symbol ⊗ denotes a convolution
in this variable.
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16. Heavy-Quark and Soft-Collinear Effective Theory

While the hard functions are different for the decays B̄ → Xs γ and B̄ → Xu `ν̄, the jet and soft
functions are identical at leading order in ΛQCD /mQ . This is particularly important for the shapefunction, which introduces non-perturbative physics into the theoretical predictions for the decay
rates in the regions of experimental interest. The fact that both processes depend on the same
non-perturbative function makes it possible to use the measured shape of the B̄ → Xs γ photon
spectrum to reduce the theoretical uncertainties in the determination of the CKM element |Vub |from semileptonic decays. In higher orders of the heavy-quark expansion, an increasing number ofsubleading jet and soft functions are required to describe the decay distributions [49]. These havebeen analyzed in detail at order 1/mb [50–52]. In the case of B̄ → Xs γ (and also in the related¯ some of these non-local effects survive in the total decay rate and give rise tocase of B̄ → Xs ``),
irreducible hadronic uncertainties [53]. The technology for deriving the corresponding factorization
theorems relies on the soft-collinear effective theory, to which we now turn.

16.3 Soft-Collinear Effective Theory
As discussed in the previous section, soft gluons that bind a heavy quark inside a heavy meson
cannot change the virtuality of that heavy quark by a significant amount. The ratio ΛQCD /mQ
provides the expansion parameter in HQET, which is a small parameter since mQ  ΛQCD . Thisobviously does not work when considering light quarks. However, if the energy Q (not to be
confused with the flavor of a heavy quark) of the quarks is large, the ratio ΛQCD /Q provides asmall parameter, which can be used to construct an effective theory. One major difference to
HQET is that light energetic quarks cannot only emit soft gluons, but they can also emit collinear
gluons (an energetic gluon in the same direction as the original quark), without parametricallychanging their virtuality. Thus, to fully reproduce the long-distance physics of energetic quarks
requires that one includes their interactions with both soft and collinear particles. The resulting
effective theory is therefore called soft-collinear effective theory (SCET) [54–57] (see [58] for areview).
A single energetic particle can always be boosted to a frame where all momentum components
have similar size, in which case there is no small expansion parameter. Thus, the presence ofenergetic particles must refer to a reference frame defined by external kinematics. SCET has awide range of applications; some examples are the production of energetic, light states in the decayof a heavy particle in its rest frame, the production of energetic jets in collider environments, andthe scattering of energetic particles off a target at rest. In this brief review we will outline the mainfeatures of this effective theory and mention a few selected applications.
16.3.1 General idea of the expansion
Consider a quark with virtuality much less than its energy Q, moving along the direction ~n. It isconvenient to parameterize the momentum pn of this particle in terms of its light-cone components,+ ⊥⊥µdefined by (p−n) and n̄µ = (1, −~n) are light-liken , pn , pn ) = (n̄ · pn , n · pn , pn ), where n = (1, ~⊥⊥reference vectors, and n · pn = n̄ · pn = 0. The subscript n on the momentum indicates the
direction of the collinear particle. In terms of these light-cone components, the virtuality satisfies−⊥2
p2n = p+n pn + pn . The individual components of the momentum obey
+ ⊥
2
(p−
n , pn , pn ) ∼ Q(1, λ , λ) ,(16.9)

where λ2 = p2 /Q2 is the expansion parameter of SCET. The virtuality of such an energetic particle
remains parametrically unchanged if it interacts with energetic particles in the same direction n,or with soft particles with momentum scaling as
+ ⊥
2 2 2
(p−
s , ps , ps ) ∼ Q(λ , λ , λ ) .
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16. Heavy-Quark and Soft-Collinear Effective Theory

SCET is constructed in such a way as to reproduce the long-distance dynamics arising from theinteractions of collinear and soft degrees of freedom.
In the above power counting the transverse momenta of soft degrees of freedom scale as p⊥s ∼2
Qλ , which is much smaller than the transverse momenta p⊥∼
Qλofcollinearfields.
This
theoryiscusually called SCETI . If the external kinematics require that the transverse momenta of both soft⊥and collinear fields are of the same size, p⊥c ∼ ps , then the appropriate degrees of freedom have the2scaling pc ∼ Q(1, λ , λ) and ps ∼ Q(λ, λ, λ). This theory is usually called SCETII and is required,e.g., for exclusive hadronic decays such as B̄ → Dπ, where the virtuality of both collinear and soft
degrees of freedom are set by ΛQCD , or for the description of transverse-momentum distributions
at colliders. SCETI power counting is assumed in the following sections, while SCETII is discussed
in more detail in 16.3.6.
16.3.2 Leading-order Lagrangian
The derivation of the SCET Lagrangian follows similar steps as described for HQET in Section 16.2.1. One begins by deriving the Lagrangian for a theory containing only a single collinear sector. Similar to HQET, one separates the full QCD field into two components, qn (x) = ψn (x)+ηn (x),where (with n · n̄ = 2)
n
/n
/̄
n/
/̄ n
ψn (x) =qn (x) ,ηn (x) =qn (x) .
(16.11)
4
4
The degrees of freedom described by the field ηn are far off shell and can therefore be eliminatedusing its equation of motion. This gives
1n/̄Ln = ψ̄n (x) in · D + iD/iD/⊥
ψn (x) .
in̄ · D2
⊥



(16.12)

As a next step, one separates the large and small momentum components by decomposing thecollinear momentum into a “label” and a residual momentum, pµ = P µ + k µ with n · P = 0.
One then performs a phase redefinition on the collinear fields, such that ψn (x) = e−iP ·x ξn (x).
Derivatives acting on the fields ξn (x) now only pick out the residual momentum. Since unlike in
HQET the label momentum in SCET is not conserved, one defines a label operator P µ acting as
P µ ξn (x) = P µ ξn (x) [56], as well as a corresponding covariant label operator iDnµ = P µ + gAµn (x).
Note that at leading order in power counting iDnµ does not contain the soft gluon field. This leads
to the final SCET Lagrangian [56, 57, 59, 60]Ln = ξ¯n (x) in · Dn + gn · As + iD
/ n⊥

1
n
/̄
iD
/ n⊥
ξn (x) + . . . ,in̄ · Dn2


(16.13)where we have split in · D into a collinear piece in · Dn = in · ∂ + gn · An and a soft piece gn · As . Thislatter term gives rise to the only interaction between a collinear quark and soft gluons at leading
power in λ. The ellipses represent higher-order interactions between soft and collinear particles.
The Lagrangian describing collinear fields in different light-like directions is simply given by theP
sum of the Lagrangians for each direction n, i.e. L = n Ln . The soft gluons are the same in eachindividual Lagrangian. An alternative way to understand the separation between large and small
momentum components is to derive the Lagrangian of SCET in position space [60]. In this case nolabel operators are required, but the collinear field operators are separated by a light-like distance.
An important difference between SCET and HQET is that the SCET Lagrangian is not correctedby short distance fluctuations. The physical reason is that in the construction described above
no high-momentum modes have been integrated out [60]. Such hard modes arise when different
collinear sectors are coupled via some external current (e.g. in jet production at e+ e− or hadroncolliders), or when collinear particles are produced in the rest frame of a decaying heavy object
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16. Heavy-Quark and Soft-Collinear Effective Theory(such as in B decays). Short-distance effects are then incorporated in the Wilson coefficients of theexternal source operators.
16.3.3 Collinear gauge invariance and Wilson lines
An important aspect of SCET concerns the implementation of local gauge invariance. Because the effective field operators describe modes with certain momentum scalings, the effective
Lagrangian respects only residual gauge symmetries. One of them satisfies the collinear scaling(n̄ · ∂, n · ∂, ∂ ⊥ ) Un (x) ∼ Q(1, λ2 , λ) Un (x) ,(16.14)(n̄ · ∂, n · ∂, ∂ ⊥ ) Us (x) ∼ Q(λ2 , λ2 , λ2 ) Us (x) .

(16.15)and one the soft scaling

While the soft gauge transformation is common for all fields, collinear fields in different directions
each transform under their own collinear gauge transformation. In other words, each collinear
sector has to be separately gauge invariant under its collinear gauge transformation. This requires
the introduction of collinear Wilson lines [56]Wn (x) = P exp − ig

Z 0
−∞



ds n̄ · An (sn̄ + x) ,(16.16)

which transform under collinear gauge transformations according to Wn → Un Wn . Thus, thecombination χn ≡ Wn† ψn is gauge invariant. In a similar manner, one can define the gauge-invariant
gluon field Bnµ = g −1 Wn† iDnµ Wn [61, 62]. Collinear operators in SCET are typically constructedfrom such collinearly gauge-invariant building blocks.
16.3.4 Derivation of factorization theorems
One of the important applications of SCET is to understand how to factorize cross sections or
decay rates involving energetic particles moving in different directions into simpler quantities thatcan either be calculated perturbatively or determined from data. For the purpose of this discussion,
we focus on factorization in the context of collider physics. Note that factorization theorems havebeen around for much longer than SCET (see [63] for a review). However, the effective theory
allows for a conceptually simpler understanding of certain classes of factorization theorems [61],since most simplifications happen already at the level of the Lagrangian. The discussion in thissection is valid to leading order in the power counting of the effective theory.
As discussed in the previous section, the Lagrangian of SCET does not involve any direct
couplings between collinear particles moving in different directions. A process involving highly energetic initial- and/or final-state particles moving along different directions ni is therefore describedin SCET by operators involving gauge-invariant collinear building blocks for each direction, which
only interact amongst each other via soft gluon exchange. At leading power, soft gluons couple tocollinear quarks only through the term ξ¯n gn · As (/̄n/2) ξn in the effective Lagrangian in (16.13),
which is similar to the coupling of soft gluons to heavy quarks in HQET, see Section 16.2.4. This
coupling, and the corresponding coupling to collinear gluons can be removed by means of the field
redefinitions [57]ψn (x) = Yn (x) ψn(0) (x) ,Aan (x) = Ynab (x) Ab(0)(16.17)n (x) ,
where the soft Wilson lines Yn and Ynab live in the fundamental and adjoint representations of SU (3),respectively. Each collinear building block gives rise to a soft Wilson line. After this decoupling, thedifferent collinear sectors no longer interact with each other and the soft interactions are entirely
encoded in the product of soft Wilson lines. This is the basis of factorization proofs in SCET.
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916. Heavy-Quark and Soft-Collinear Effective Theory

The simplest applications of this formalism concern inclusive processes, such as the Drell-Yan
production of electroweak bosons in the threshold region, pp → Z/H + Xs , where Xs is a final
state containing only soft particles, and event shape distributions in e+ e− collisions in the two-jet
limit. For Drell-Yan the factorization formula is given by [61, 64]σ(pp → ZXs ) =

X





Hq (µ) fq/p (µ) fq̄/p (µ) ⊗ SF (µ) ,(16.18)qσ(pp → HXs ) = Hg (µ) fg/p (µ) fg/p (µ) ⊗ SA (µ) ,where the collinear matrix elements give rise to the parton distributions fa/p (µ), the soft function
are vacuum matrix element of the soft Wilson lines in the fundamental or adjoint representation,and the hard functions are the squares of the Wilson coefficients of the relevant SCET operators.
The convolution symbol indicates a convolution over the longitudinal momentum fractions of thecolliding partons. The objects in the factorization theorem depend on the renormalization scaleµ, whereas the cross sections are scale independent. Solving the renormalization group equations(RGE) allows to resum large logarithms in the cross section (see below).
For event shapes such as thrust T , no initial-state partons are present, and the factorization
formula takes the form (with τ = 1 − T ) [65, 66]dσ= H(µ) Jn (µ)Jn̄ (µ) ⊗ Snn̄ (µ) ,dτ

(16.19)where the the jet functions are vacuum matrix elements of the outgoing collinear quark fields, which
are convoluted with the soft function Snn̄ defined as another vacuum matrix element of the soft
Wilson lines.
For a more general high-energy scattering process such as pp → N jets, where the incoming protons move along directions na,b and the jets define directions n1 , . . . , nN , the factorization theorem
for the differential cross section can be written as
dσ ∼

X









Hab...N (µ) Ba/p (µ) Bb/p (µ) ⊗ J1 (µ) . . . JN (µ) ⊗ Sab...N (µ) .

(16.20)abwhere the beam functions are generalized parton distributions [67] and the jet functions dependon the algorithm used to define the jets. It should be mentioned that the most difficult part of
traditional factorization proofs involves showing that so-called Glauber gluons do not spoil theabove factorization theorem [68]. The description of Glauber effects within SCET [69] has beendeveloped in [70], where a closed form for the effective Lagrangian describing these interactions
was derived. The formalism has been extended to Glauber quarks in [71]. In this context, a proofof factorization requires demonstrating that this Lagrangian has no impact on a particular crosssection. If an observable is non-global, in the sense that energetic radiation is restricted to certain
regions in phase space, the above factorization theorem needs to be generalized [72–74].
16.3.5 Resummation of large logarithms
Many processes of interest depend on multiple scales which can be widely separated. The
perturbative series of such processes depends logarithmically on large scale ratios. For high-energy
processes involving soft and collinear particles, there can be up to two powers of large logarithms foreach power of the coupling constant. These double logarithms are referred to as Sudakov logarithms
and can spoil the perturbative expansion.
SCET can be used to sum large logarithms L using renormalization-group equations. This
reorganizes the perturbative series in such a way that the product αs L is treated as an O(1)31st May, 2024

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16. Heavy-Quark and Soft-Collinear Effective Theory

parameter. In many cases, one can write a perturbative series for the logarithm of a cross section,such thatσ ∼ exp [L g0 (αs L) + g1 (αs L) + αs g2 (αs L) + . . . ] ,(16.21)
with functions gn (x) that need to be determined. Note that each term in this series in the exponent
is one power of αs suppressed relative to the previous term.
The important ingredient in achieving this resummation is the fact that SCET factorizes agiven cross section into simpler pieces, each of which depends on a single physical scale. The only
dependence on that scale can arise through logarithms of its ratio with the renormalization scaleµ. Thus, for each of the components in the factorization theorem one can choose a renormalization
scale µ for which the large logarithmic terms are absent. Of course, the factorization formularequires a common renormalization scale µ in all its components, and one therefore has to use therenormalization group (RG) to evolve the various component functions from their preferred scale
to the common scale µ. A novel feature of RG equations in SCET, as opposed to other EFTs, isthat the anomalous dimensions entering the evolution equations of the hard, beam, jet and soft
functions in a factorization formula such as (16.20) contain a single power of the logarithm of therelevant energy scale. For example, the anomalous dimension γH of the hard function has the formγH (µ) = cH Γcusp (αs ) ln

Q2
+ γ(αs ) ,µ2(16.22)where cH is a process-dependent coefficient and Γcusp denotes the so-called cusp anomalous dimension [22, 75]. Collinear and soft functions have similar anomalous dimensions, which also involvea cusp and a non-cusp part. The non-cusp part γ of the anomalous dimensions is process (andobservable) dependent. The presence of a logarithm in the anomalous dimension is characteristicof Sudakov problems and arises since the perturbative series contains double logarithms of scaleratios.
Solving the RG equations one can systematically resum all large logarithms of scale ratios in
the factorized cross section and express the functions gn (αs L) introduced above in terms of ratiosof running coupling constants. In order to compute the first two terms L g0 (αs L) + g1 (αs L) in
ln σ, corresponding to the next-to-leading logarithmic (NLL) approximation, one needs two-loop
expressions for the cusp anomalous dimension and β function, one-loop expressions for the non-cusp
pieces in the anomalous dimensions, and tree-level matching conditions for all component functions
at their characteristic scales. To calculate the next term αs g2 (αs L) in the expansion, corresponding
to NNLL order, one needs to go one order higher in the loop expansion, and so on.
16.3.6 Factorization and resummation in SCETIIThe effective theory SCETII contains collinear and soft particles with momenta scaling as
+ ⊥
2
− + ⊥
(p−
,
n pn , pn ) ∼ Q(1, λ , λ) and (ps , ps , ps ) ∼ Q(λ, λ, λ). They have the same small virtuality(p2n ∼ p2s ∼ Q2 λ2 ) but differ in their rapidities. An important class of observables, for which thisscaling is relevant, contains cross sections for processes in which the transverse momenta of particles
are constrained by external kinematics. The prime example are the transverse-momentum distributions of electroweak gauge bosons or Higgs bosons produced at hadron colliders. The partontransverse momenta are constrained by the fact that their vector sum must be equal and opposite
to the transverse momentum qT of the boson. Standard RG evolution in the effective theory controls the logarithms arising from the fact that the virtualities of the collinear and soft modes aremuch smaller than the hard scale Q in the process (the boson mass). However, additional large
logarithms arise since the rapidities of collinear and soft modes are parametrically different, suchthat e|yc −ys | ∼ 1/λ. These logarithms can be traced to a new source of divergences and an unusual
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16. Heavy-Quark and Soft-Collinear Effective Theory

failure of dimensional regularization. They need to be factorized in the cross section and resummedby other means.
Two approaches exist for how to deal with the additional rapidity logarithms in SCET. In theapproach of [76], they are interpreted as a consequence of a “collinear anomaly” of the effective
theory SCETII , resulting from the fact that a classical rescaling symmetry of the effective Lagrangian is broken by quantum effects. The extra large logarithms can be resummed by means ofsimple differential equations, which typically state that, to all orders in perturbation theory (andin an appropriate space), the logarithm of the cross section contains only a single extra logarithmof λ ∼ qT /Q not contained in the hard function. Another approach to resum the rapidity logarithms uses the “rapidity renormalization group” [77], in which the relevant differential equations
are obtained by considering a new type of scale variation in a parameter ν, which separates thephase space for collinear and soft particles along a hyperbola in the (p− , p+ ) plane. In contrast
to the standard RG, there is no running coupling involved in the ν evolution, since the different
contributions live at the same virtuality.
As an important application, consider the Higgs-boson production cross section in gluon fusion
at the LHC, defined with a jet veto stating that no jet in the final state has transverse momentum
above a threshold pveto
T , which can be factorized in the form [78, 79]σ(pvetoT ) = H(mH , µ)
Z 1



νB
νS

−2Fgg (R,pveto ,µ)
T

dz
νB
×
Bg/P z, R, pveto
T , µ,mHτ z


Sgg (R, pveto
T , µ,





Bg/P

νS
)
pveto
T

τ
νB
, R, pveto
T , µ,
zmH(16.23),

where τ = m2H /s, and µ ∼ pveto
is a common factorization scale. The beam functions Bg/P ,T
the soft function Sgg and the exponent Fgg all depend on the jet radius R as well as the jetclustering algorithm. The scale dependence of the hard function H is controlled by standard RG
evolution in SCET. The beam functions can be factorized further into calculable collinear kernels
convoluted with parton distribution functions. In addition to the renormalization scale µ, thebeam and soft functions depend on two rapidity scales νB ∼ mH and νS ∼ pveto
T , respectively.
veto
In [78] the default values νB = mH and νS = pT are used for these scales, and the soft function
Sgg is absorbed into the beam functions. In [79] the exponent Fgg is called −γνg /2. The second
factor on the right-hand side of the factorization formula (16.23), which resums large rapidity
logarithms, implies that the logarithm of the jet-veto cross section contains a single large logarithm
veto
ln σ = −2Fgg (R, pveto
T , µ) ln(mH /pT ) + . . . not contained in the hard function. Its coefficient canbe calculated in fixed-order perturbation theory.
SCETII also plays an important role in the study of factorization for a variety of exclusive B
meson decays, such as B̄ → π`ν̄, B̄ → K ∗ γ and B̄ → ππ, for which the virtualities of energetic(collinear) final-state particles are of order ΛQCD , which is also the scale for the soft light degrees
of freedom contained in the initial-state B meson.
16.3.7 Applications
Most of the applications of SCET are either in flavor physics, where the decay of a heavy B
meson can give rise to energetic light partons, or in collider physics, where the presence of jets
naturally leads to collimated sets of energetic particles. For some of these applications alternative
approaches existed before the invention of SCET, but the effective theory has opened up alternative
ways to understand the physics of these processes. For many examples, however, SCET has allowed
new insights and new applications.
The investigation of heavy-to-light form factors has been instrumental for understanding factorization in exclusive semileptonic B decays [80]. SCET has also provided a field-theoretic basis for
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16. Heavy-Quark and Soft-Collinear Effective Theory

the QCD factorization approach to exclusive, non-leptonic decays of B mesons [81]. Using SCET
methods, proofs of factorization were derived for the color-allowed decay B̄ 0 → D+ π − [82], thecolor-suppressed decay B̄ 0 → D0 π 0 [83], and the radiative decay B̄ → K ∗ γ [84]. In recent years,
the factorization approach has been extended to include QED effects [85,86]. Further examples arefactorization theorems and the resummation of endpoint logarithms for quarkonia production [87],the resummation of large logarithmic terms for the thrust [88] and jet broadening [89] distributions
in e+ e− annihilation beyond NLL order, the development of new factorizable observables to vetoextra jets [67, 90], all-order factorization theorems for processes containing electroweak Sudakov
logarithms [91], and the resummation of threshold (soft gluon) logarithms in momentum space forseveral important processes at hadron colliders [64, 92, 93]. There has also been a lot of activity
describing pT -based resummation at hadron colliders. Prominent examples are the transversemomentum distributions of electroweak bosons [76, 77, 94], which give access to the pT -dependent
parton distribution functions. Higher-order resummations of event shape distributions [88], combined with two-loop fixed-order expressions [95,96] and including nonperturbative effects by meansof a shape function, have been used for an accurate determination of the strong coupling constantαs [97,98]. A method based on N -jettiness (TN ) slicing [99,100] has provided a new way of performing the IR subtractions needed for precise fixed-order calculations, using the fact that the singulardependence on TN can be calculated using SCET at NNLO. This has been used to compute various
processes with final states containing up to one hard, colored particle [101–105]. Calculations of
the leading power corrections in T0 /Q [106, 107] have helped to improved the numerical stability ofthe method. The N -jettiness slicing method has also been used in the combination of higher-orderresummation with parton showers [108–110]. Finally, SCET has given new insights into the jet
substructure methods (see [111] for a review).
A novel type of SCET factorization theorem is needed to describe non-global observables, inwhich energetic radiation is limited to certain regions of phase space, e.g. inside jets. Jet events
play a prominent role in collider physics, because they closely mirror the underlying short-distancedynamics. In the theoretical description of such observables one encounters so-called “non-global”
logarithms (NGLs) [112, 113], whose resummation cannot be performed using standard techniques.
Using SCET, this problem has been reduced to the solution of a RG evolution equation derived
from a novel type of factorization theorem, in which the hard and soft functions are operators incolor space and in the (infinite) space of parton multiplicities [72, 74] (see also [73, 114]). In e+ e−
collisions, solving the RGE in the large-Nc limit using parton-shower techniques [115] gives a resultthat agrees with the solution of the BSM integral equation for the NGLs [116]. The first applicationof this formalism was to the light jet mass distribution [117], and the extension of the resummation
to NLL accuracy has been accomplished in [118,119]. Non-global observables at hadron colliders areyet more intricate, since Glauber-gluon exchange between the colliding partons lead to a new typeof double logarithmic corrections in high orders of perturbation theory, the so-called superleading
logarithms (SLLs) [120,121]. Based on the corresponding SCET factorization theorem, the all-orderresummation of SLLs has been accomplished for arbitrary pp → n jet processes [122].
Recent years have seen a strong effort to push the applications of SCET factorization and resummation techniques to next-to-leading power (NLP) in the expansion in λ. The subleading SCET
Lagrangian [60, 123, 124] and current operators arising in B-meson decays and their anomalous
dimensions [59, 60, 125–127] have been studied a long time ago. For important collider processes,
such as Drell-Yan or Higgs production, the general sets of operators have been identified [128–130],and several of their anomalous dimensions have been calculated [130, 131]. Resummed results at
subleading power have been presented for event shapes [132] and the Drell-Yan process [133]. Moreapplications of SCET can be found in Section 6.4 of the review article [134].
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16. Heavy-Quark and Soft-Collinear Effective Theory

16.4 Open issues and perspectives
HQET has successfully passed many experimental tests, and there are not many open questions
that still need to be addressed. One concept that has not been derived from first principles isthe notion of quark-hadron duality, which underlies the application of HQET to the descriptionof inclusive decays of B mesons. The validity of global duality (at energies even lower than those
relevant in B decays) has been tested experimentally using high-precision data on semileptonic B
decays and on hadronic τ decays. However, assigning a theoretical uncertainty due to possible
duality violations remains a difficult task. Another known issue is that the measured values of
the CKM elements |Vub | and |Vcb | extracted from exclusive or inclusive decays of B mesons differ
from each other by several standard deviations (see Section 76). This measurement relies on
the heavy-quark limit, and the uncertainty quoted includes a theoretical estimate of the effect ofpower corrections arising from the finite b-quark mass. It remains an open question whether thediscrepancy is due to underestimated theoretical or experimental uncertainties.
SCET, on the other hand, is still an active field of research, and new results are being obtained
regularly. In particular, it has been realized that at NLP SCET factorization theorems generically
suffer from endpoint-divergent convolution integrals [135–140]. To remove these divergences, one
uses exact D-dimensional refactorization conditions, which relate different terms in the NLP factorization theorem in the singular limits, to rearrange these terms in such a way that the endpoint
divergences are systematically removed. Successful applications of this technique include the NLP
factorization theorem for the b-quark induced contribution to the diphoton decay amplitude of the
Higgs boson [139,141] and the gg → H production amplitude [142], the “off-diagonal gluon thrust”
in e+ e− collisions [143], and the NLP factorization theorems for some exclusive [144, 145] and inclusive [146] B-meson decays. On the other hand, for some other processes, such as electron-muon
backward scattering [147], the factorization of endpoint singularities is still a subject of ongoing
research.
Another active field concerns the further study and applications of Glauber gluons and quarks
in SCET and their relation to the BFKL equation familiar from small-x physics [114, 148]. The
theoretical formalism developed in [70, 71] sets the basis for a solid understanding of the impact ofGlauber exchanges on factorization theorems. Moreover, using the rapidity renormalization groupone can derive the BFKL equation in the presence of a single Glauber exchange describing forward
scattering [70, 148] and for small x resummation in deep inelastic scattering [149]. Glauber gluons
play an important role in SCET-based analysis of jet propagation in dense QCD media [150–152],
which gives rise to the jet-quenching phenomenon in heavy-ion collisions. They are also a crucial
ingredient in studies of violations of collinear factorization at hadron colliders [69, 153–157].
We close this short review by mentioning a particularly nice application combining the methodsof heavy-particle EFTs such as HQET and non-relativistic QCD with SCET in the context of describing the interactions of heavy dark matter (with mass M  v) with SM particles. In [158] it was
realized that the interactions of heavy, weakly interacting massive particles (WIMPs) with nuclear
targets can be described in a model-independent way using heavy-particle EFTs. The WIMPs arecharged under SU (2)L and can interact with electroweak gauge bosons and the Higgs boson. The
WIMP EFT was later extended by describing the produced, highly energetic electroweak gaugebosons in terms of soft or collinear fields in SCET [159–161]. This allows one to systematically separate all relevant mass scales, resum electroweak Sudakov logarithms and disentangle the so-called
Sommerfeld enhancement from the short-distance hard annihilation process.
References[1] E. Witten, Nucl. Phys. B122, 109 (1977).
[2] S. Weinberg, Phys. Lett. 91B, 51 (1980).
31st May, 2024

14

16. Heavy-Quark and Soft-Collinear Effective Theory[3] L. J. Hall, Nucl. Phys. B178, 75 (1981).
[4] W. Buchmuller and D. Wyler, Nucl. Phys. B 268, 621 (1986).
[5] B. Grzadkowski et al., JHEP 10, 085 (2010), [arXiv:1008.4884].
[6] E. Eichten and B. R. Hill, Phys. Lett. B 234, 511 (1990).
[7] H. Georgi, Phys. Lett. B240, 447 (1990).
[8] B. Grinstein, Nucl. Phys. B339, 253 (1990).
[9] T. Mannel, W. Roberts and Z. Ryzak, Nucl. Phys. B368, 204 (1992).
[10] M. Neubert, Phys. Rept. 245, 259 (1994), [hep-ph/9306320].
[11] A. V. Manohar and M. B. Wise, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 10, 1(2000).
[12] M. E. Luke and A. V. Manohar, Phys. Lett. B286, 348 (1992), [hep-ph/9205228].
[13] E. V. Shuryak, Phys. Lett. 93B, 134 (1980).
[14] N. Isgur and M. B. Wise, Phys. Lett. B232, 113 (1989).
[15] N. Isgur and M. B. Wise, Phys. Rev. Lett. 66, 1130 (1991).
[16] S. Nussinov and W. Wetzel, Phys. Rev. D36, 130 (1987).
[17] M. A. Shifman and M. B. Voloshin, Sov. J. Nucl. Phys. 45, 292 (1987), [Yad.
Fiz.45,463(1987)].
[18] M. Neubert, Phys. Lett. B264, 455 (1991).
[19] M. E. Luke, Phys. Lett. B252, 447 (1990).
[20] I. I. Y. Bigi et al., Phys. Rev. D52, 196 (1995), [hep-ph/9405410].
[21] N. Uraltsev, Phys. Lett. B501, 86 (2001), [,195(2000)], [hep-ph/0011124].
[22] G. P. Korchemsky and A. V. Radyushkin, Nucl. Phys. B283, 342 (1987).
[23] A. F. Falk et al., Nucl. Phys. B343, 1 (1990).
[24] J. Chay, H. Georgi and B. Grinstein, Phys. Lett. B247, 399 (1990).
[25] I. I. Y. Bigi, N. G. Uraltsev and A. I. Vainshtein, Phys. Lett. B293, 430 (1992), [Erratum:
Phys. Lett.B297,477(1992)], [hep-ph/9207214].
[26] A. V. Manohar and M. B. Wise, Phys. Rev. D49, 1310 (1994), [hep-ph/9308246].
[27] T. Mannel, Nucl. Phys. B413, 396 (1994), [hep-ph/9308262].
[28] A. F. Falk, M. E. Luke and M. J. Savage, Phys. Rev. D49, 3367 (1994), [hep-ph/9308288].
[29] E. C. Poggio, H. R. Quinn and S. Weinberg, Phys. Rev. D13, 1958 (1976).
[30] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996), [hepph/9512380].
[31] A. F. Falk and M. Neubert, Phys. Rev. D47, 2965 (1993), [hep-ph/9209268].
[32] T. Becher, H. Boos and E. Lunghi, JHEP 12, 062 (2007), [arXiv:0708.0855].
[33] A. Alberti, P. Gambino and S. Nandi, JHEP 01, 147 (2014), [arXiv:1311.7381].
[34] T. Mannel, A. A. Pivovarov and D. Rosenthal, Phys. Lett. B741, 290 (2015),[arXiv:1405.5072].
[35] T. Mannel, A. A. Pivovarov and D. Rosenthal, Phys. Rev. D92, 5, 054025 (2015),[arXiv:1506.08167].
[36] T. Mannel and A. A. Pivovarov, Phys. Rev. D100, 9, 093001 (2019), [arXiv:1907.09187].
31st May, 2024

15

16. Heavy-Quark and Soft-Collinear Effective Theory[37] T. Mannel, S. Turczyk and N. Uraltsev, JHEP 11, 109 (2010), [arXiv:1009.4622].
[38] M. Neubert and C. T. Sachrajda, Nucl. Phys. B483, 339 (1997), [hep-ph/9603202].
[39] M. Beneke, G. Buchalla and I. Dunietz, Phys. Rev. D54, 4419 (1996), [Erratum: Phys.
Rev.D83,119902(2011)], [hep-ph/9605259].
[40] M. Beneke et al., Phys. Lett. B459, 631 (1999), [hep-ph/9808385].
[41] M. Kirk, A. Lenz and T. Rauh, JHEP 12, 068 (2017), [arXiv:1711.02100].
[42] C. W. Bauer et al., Phys. Rev. D 70, 094017 (2004), [hep-ph/0408002].
[43] P. Gambino and C. Schwanda, Phys. Rev. D89, 1, 014022 (2014), [arXiv:1307.4551].
[44] F. U. Bernlochner et al. (SIMBA), Phys. Rev. Lett. 127,
[arXiv:2007.04320].

10,102001 (2021),[45] Y. S. Amhis et al. (HFLAV), Phys. Rev. D 107, 052008 (2023), [arXiv:2206.07501].
[46] M. Neubert, Phys. Rev. D49, 3392 (1994), [hep-ph/9311325].
[47] I. I. Y. Bigi et al., Int. J. Mod. Phys. A9, 2467 (1994), [hep-ph/9312359].
[48] G. P. Korchemsky and G. F. Sterman, Phys. Lett. B340, 96 (1994), [hep-ph/9407344].
[49] C. W. Bauer, M. E. Luke and T. Mannel, Phys. Rev. D68, 094001 (2003), [hep-ph/0102089].
[50] K. S. M. Lee and I. W. Stewart, Nucl. Phys. B721, 325 (2005), [hep-ph/0409045].
[51] S. W. Bosch, M. Neubert and G. Paz, JHEP 11, 073 (2004), [hep-ph/0409115].
[52] M. Beneke et al., JHEP 06, 071 (2005), [hep-ph/0411395].
[53] M. Benzke et al., JHEP 08, 099 (2010), [arXiv:1003.5012].
[54] C. W. Bauer, S. Fleming and M. E. Luke, Phys. Rev. D63, 014006 (2000), [hep-ph/0005275].
[55] C. W. Bauer et al., Phys. Rev. D63, 114020 (2001), [hep-ph/0011336].
[56] C. W. Bauer and I. W. Stewart, Phys. Lett. B516, 134 (2001), [hep-ph/0107001].
[57] C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D65, 054022 (2002), [hep-ph/0109045].
[58] T. Becher, A. Broggio and A. Ferroglia, Lect. Notes Phys. 896, pp.1 (2015), [arXiv:1410.1892].
[59] J. Chay and C. Kim, Phys. Rev. D65, 114016 (2002), [hep-ph/0201197].
[60] M. Beneke et al., Nucl. Phys. B643, 431 (2002), [hep-ph/0206152].
[61] C. W. Bauer et al., Phys. Rev. D66, 014017 (2002), [hep-ph/0202088].
[62] R. J. Hill and M. Neubert, Nucl. Phys. B657, 229 (2003), [hep-ph/0211018].
[63] J. C. Collins, D. E. Soper and G. F. Sterman, Adv. Ser. Direct. High Energy Phys. 5, 1
(1989), [hep-ph/0409313].
[64] T. Becher, M. Neubert and G. Xu, JHEP 07, 030 (2008), [arXiv:0710.0680].
[65] S. Fleming et al., Phys. Rev. D 77, 074010 (2008), [hep-ph/0703207].
[66] M. D. Schwartz, Phys. Rev. D 77, 014026 (2008), [arXiv:0709.2709].
[67] I. W. Stewart, F. J. Tackmann and W. J. Waalewijn, Phys. Rev. D 81, 094035 (2010),[arXiv:0910.0467].
[68] J. C. Collins, D. E. Soper and G. F. Sterman, Nucl. Phys. B261, 104 (1985).
[69] C. W. Bauer, B. O. Lange and G. Ovanesyan, JHEP 07, 077 (2011), [arXiv:1010.1027].
[70] I. Z. Rothstein and I. W. Stewart, JHEP 08, 025 (2016), [arXiv:1601.04695].
[71] I. Moult et al., JHEP 02, 134 (2018), [arXiv:1709.09174].
[72] T. Becher et al., Phys. Rev. Lett. 116, 19, 192001 (2016), [arXiv:1508.06645].
31st May, 2024

16

16. Heavy-Quark and Soft-Collinear Effective Theory[73] A. J. Larkoski, I. Moult and D. Neill, JHEP 09, 143 (2015), [arXiv:1501.04596].
[74] T. Becher et al., JHEP 11, 019 (2016), [Erratum: JHEP05,154(2017)], [arXiv:1605.02737].
[75] I. A. Korchemskaya and G. P. Korchemsky, Phys. Lett. B287, 169 (1992).
[76] T. Becher and M. Neubert, Eur. Phys. J. C71, 1665 (2011), [arXiv:1007.4005].
[77] J.-Y. Chiu et al., JHEP 05, 084 (2012), [arXiv:1202.0814].
[78] T. Becher and M. Neubert, JHEP 07, 108 (2012), [arXiv:1205.3806].
[79] I. W. Stewart et al., Phys. Rev. D89, 5, 054001 (2014), [arXiv:1307.1808].
[80] M. Beneke and T. Feldmann, Nucl. Phys. B685, 249 (2004), [hep-ph/0311335].
[81] M. Beneke et al., Phys. Rev. Lett. 83, 1914 (1999), [hep-ph/9905312].
[82] C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. Lett. 87, 201806 (2001), [hepph/0107002].
[83] S. Mantry, D. Pirjol and I. W. Stewart, Phys. Rev. D68, 114009 (2003), [hep-ph/0306254].
[84] T. Becher, R. J. Hill and M. Neubert, Phys. Rev. D72, 094017 (2005), [hep-ph/0503263].
[85] M. Beneke, C. Bobeth and R. Szafron, JHEP 10, 232 (2019), [Erratum: JHEP 11, 099(2022)], [arXiv:1908.07011].
[86] M. Beneke et al., JHEP 10, 223 (2021), [arXiv:2107.03819].
[87] S. Fleming, A. K. Leibovich and T. Mehen, Phys. Rev. D68, 094011 (2003), [hep-ph/0306139].
[88] T. Becher and M. D. Schwartz, JHEP 07, 034 (2008), [arXiv:0803.0342].
[89] T. Becher and G. Bell, JHEP 11, 126 (2012), [arXiv:1210.0580].
[90] I. W. Stewart, F. J. Tackmann and W. J. Waalewijn, Phys. Rev. Lett. 105, 092002 (2010),[arXiv:1004.2489].
[91] J.-y. Chiu, R. Kelley and A. V. Manohar, Phys. Rev. D78, 073006 (2008), [arXiv:0806.1240].
[92] V. Ahrens et al., Eur. Phys. J. C62, 333 (2009), [arXiv:0809.4283].
[93] X. Liu, S. Mantry and F. Petriello, Phys. Rev. D86, 074004 (2012), [arXiv:1205.4465].
[94] M. G. Echevarria, A. Idilbi and I. Scimemi, JHEP 07, 002 (2012), [arXiv:1111.4996].
[95] A. Gehrmann-De Ridder et al., Phys. Rev. Lett. 99, 132002 (2007), [arXiv:0707.1285].
[96] S. Weinzierl, Phys. Rev. Lett. 101, 162001 (2008), [arXiv:0807.3241].
[97] R. Abbate et al., Phys. Rev. D83, 074021 (2011), [arXiv:1006.3080].
[98] A. H. Hoang et al., Phys. Rev. D91, 9, 094018 (2015), [arXiv:1501.04111].
[99] R. Boughezal, X. Liu and F. Petriello, Phys. Rev. D91, 9, 094035 (2015), [arXiv:1504.02540].
[100] J. Gaunt et al., JHEP 09, 058 (2015), [arXiv:1505.04794].
[101] R. Boughezal et al., Phys. Lett. B748, 5 (2015), [arXiv:1505.03893].
[102] R. Boughezal et al., Phys. Rev. Lett. 116, 15, 152001 (2016), [arXiv:1512.01291].
[103] J. M. Campbell, R. K. Ellis and C. Williams, JHEP 06, 179 (2016), [arXiv:1601.00658].
[104] J. M. Campbell et al., JHEP 07, 148 (2016), [arXiv:1603.02663].
[105] G. Heinrich et al., JHEP 03, 142 (2018), [arXiv:1710.06294].
[106] I. Moult et al., Phys. Rev. D95, 7, 074023 (2017), [arXiv:1612.00450].
[107] R. Boughezal, X. Liu and F. Petriello, JHEP 03, 160 (2017), [arXiv:1612.02911].
[108] S. Alioli et al., JHEP 09, 120 (2013), [arXiv:1211.7049].
[109] S. Alioli et al., Phys. Rev. D92, 9, 094020 (2015), [arXiv:1508.01475].
31st May, 2024

17

16. Heavy-Quark and Soft-Collinear Effective Theory[110] S. Alioli et al., Phys. Rev. D 104, 9, 094020 (2021), [arXiv:2102.08390].
[111] A. J. Larkoski, I. Moult and B. Nachman (2017), [arXiv:1709.04464].
[112] M. Dasgupta and G. P. Salam, Phys. Lett. B512, 323 (2001), [hep-ph/0104277].
[113] R. B. Appleby and M. H. Seymour, JHEP 12, 063 (2002), [hep-ph/0211426].
[114] S. Caron-Huot, JHEP 03, 036 (2018), [arXiv:1501.03754].
[115] M. Balsiger, T. Becher and D. Y. Shao, JHEP 08, 104 (2018), [arXiv:1803.07045].
[116] A. Banfi, G. Marchesini and G. Smye, JHEP 08, 006 (2002), [hep-ph/0206076].
[117] T. Becher, B. D. Pecjak and D. Y. Shao, JHEP 12, 018 (2016), [arXiv:1610.01608].
[118] M. Balsiger, T. Becher and D. Y. Shao, JHEP 04, 020 (2019), [arXiv:1901.09038].
[119] T. Becher, T. Rauh and X. Xu, JHEP 08, 134 (2022), [arXiv:2112.02108].
[120] J. R. Forshaw, A. Kyrieleis and M. H. Seymour, JHEP 08, 059 (2006), [hep-ph/0604094].
[121] J. R. Forshaw, A. Kyrieleis and M. H. Seymour, JHEP 09, 128 (2008), [arXiv:0808.1269].
[122] T. Becher, M. Neubert and D. Y. Shao, Phys. Rev. Lett. 127, 21, 212002 (2021),[arXiv:2107.01212].
[123] M. Beneke and T. Feldmann, Phys. Lett. B 553, 267 (2003), [hep-ph/0211358].
[124] C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D68, 034021 (2003), [hep-ph/0303156].
[125] D. Pirjol and I. W. Stewart, Phys. Rev. D67, 094005 (2003), [Erratum:
Rev.D69,019903(2004)], [hep-ph/0211251].

Phys.

[126] R. J. Hill et al., JHEP 07, 081 (2004), [hep-ph/0404217].
[127] M. Beneke and D. Yang, Nucl. Phys. B736, 34 (2006), [hep-ph/0508250].
[128] I. Moult, I. W. Stewart and G. Vita, JHEP 07, 067 (2017), [arXiv:1703.03408].
[129] I. Feige et al., JHEP 11, 142 (2017), [arXiv:1703.03411].
[130] M. Beneke et al., JHEP 03, 001 (2018), [arXiv:1712.04416].
[131] S. Alte, M. König and M. Neubert, JHEP 08, 095 (2018), [arXiv:1806.01278].
[132] I. Moult et al., JHEP 08, 013 (2018), [arXiv:1804.04665].
[133] M. Beneke et al., JHEP 03, 043 (2019), [arXiv:1809.10631].
[134] F. Gross et al. (2022), [arXiv:2212.11107].
[135] I. Moult, I. W. Stewart and G. Vita, JHEP 11, 153 (2019), [arXiv:1905.07411].
[136] M. Beneke et al., JHEP 09, 101 (2019), [arXiv:1907.05463].
[137] I. Moult et al., JHEP 05, 089 (2020), [arXiv:1910.14038].
[138] M. Beneke et al., JHEP 07, 078 (2020), [arXiv:1912.01585].
[139] Z. L. Liu and M. Neubert, JHEP 04, 033 (2020), [arXiv:1912.08818].
[140] M. Beneke et al., JHEP 10, 196 (2020), [arXiv:2008.04943].
[141] Z. L. Liu et al., JHEP 01, 077 (2021), [arXiv:2009.06779].
[142] Z. L. Liu et al., JHEP 06, 183 (2023), [arXiv:2212.10447].
[143] M. Beneke et al., JHEP 07, 144 (2022), [arXiv:2205.04479].
[144] T. Feldmann et al., Phys. Rev. D 107, 1, 013007 (2023), [arXiv:2211.04209].
[145] C. Cornella, M. König and M. Neubert, Phys. Rev. D 108, 3, L031502 (2023),[arXiv:2212.14430].
[146] T. Hurth and R. Szafron, Nucl. Phys. B 991, 116200 (2023), [arXiv:2301.01739].
31st May, 2024

18

16. Heavy-Quark and Soft-Collinear Effective Theory[147] G. Bell, P. Böer and T. Feldmann, JHEP 09, 183 (2022), [arXiv:2205.06021].
[148] S. Fleming, Phys. Lett. B735, 266 (2014), [arXiv:1404.5672].
[149] D. Neill, A. Pathak and I. W. Stewart, JHEP 09, 089 (2023), [arXiv:2303.13710].
[150] A. Idilbi and A. Majumder, Phys. Rev. D80, 054022 (2009), [arXiv:0808.1087].
[151] G. Ovanesyan and I. Vitev, JHEP 06, 080 (2011), [arXiv:1103.1074].
[152] V. Vaidya, JHEP 11, 064 (2021), [arXiv:2010.00028].
[153] S. Catani, D. de Florian and G. Rodrigo, JHEP 07, 026 (2012), [arXiv:1112.4405].
[154] J. R. Forshaw, M. H. Seymour and A. Siodmok, JHEP 11, 066 (2012), [arXiv:1206.6363].
[155] J. R. Gaunt, JHEP 07, 110 (2014), [arXiv:1405.2080].
[156] M. Zeng, JHEP 10, 189 (2015), [arXiv:1507.01652].
[157] M. D. Schwartz, K. Yan and H. X. Zhu, Phys. Rev. D 96, 5, 056005 (2017), [arXiv:1703.08572].
[158] R. J. Hill and M. P. Solon, Phys. Lett. B707, 539 (2012), [arXiv:1111.0016].
[159] M. Bauer et al., JHEP 01, 099 (2015), [arXiv:1409.7392].
[160] M. Baumgart, I. Z. Rothstein and V. Vaidya, Phys. Rev. Lett. 114, 211301 (2015),[arXiv:1409.4415].
[161] G. Ovanesyan, T. R. Slatyer and I. W. Stewart, Phys. Rev. Lett. 114, 21, 211302 (2015),[arXiv:1409.8294].

31st May, 2024