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Functional methods for Heavy Quark Effective Theory

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Abstract

We use functional methods to compute one-loop effects in Heavy Quark Effective Theory. The covariant derivative expansion technique facilitates the efficient extraction of matching coefficients and renormalization group evolution equations. This paper pro- vides the first demonstration that such calculations can be performed through the algebraic evaluation of the path integral for the class of effective field theories that are (i) constructed using a non-trivial one-to-many mode decomposition of the UV theory, and (ii) valid for non-relativistic kinematics. We discuss the interplay between operators that appear at intermediate steps and the constraints imposed by the residual Lorentz symmetry that is encoded as reparameterization invariance within the effective description. The tools presented here provide a systematic approach for computing corrections to higher order in the heavy mass expansion; precision applications include predictions for experimental data and connections to theoretical tests via lattice QCD. A set of pedagogical appendices comprehensively reviews modern approaches to performing functional calculations algebraically, and derives contributions from a term with open covariant derivatives for the first time.A preprint version of the article is available at ArXiv.
JHEP06(2020)164
Published for SISSA by Springer
Received:January 27, 2020
Accepted:June 7, 2020
Published:June 26, 2020
Functional methods for Heavy Quark Effective Theory
Timothy Cohen,aMarat Freytsisband Xiaochuan Lua
aInstitute for Fundamental Science, Department of Physics, University of Oregon,
Eugene, OR 97403, U.S.A.
bNHETC, Department of Physics and Astronomy, Rutgers University,
Piscataway, NJ 08854, U.S.A.
E-mail: tcohen@uoregon.edu,marat.freytsis@rutgers.edu,
xlu@uoregon.edu
Abstract: We use functional methods to compute one-loop effects in Heavy Quark Effec-
tive Theory. The covariant derivative expansion technique facilitates the efficient extraction
of matching coefficients and renormalization group evolution equations. This paper pro-
vides the first demonstration that such calculations can be performed through the algebraic
evaluation of the path integral for the class of effective field theories that are (i) constructed
using a non-trivial one-to-many mode decomposition of the UV theory, and (ii) valid for
non-relativistic kinematics. We discuss the interplay between operators that appear at
intermediate steps and the constraints imposed by the residual Lorentz symmetry that is
encoded as reparameterization invariance within the effective description. The tools pre-
sented here provide a systematic approach for computing corrections to higher order in the
heavy mass expansion; precision applications include predictions for experimental data and
connections to theoretical tests via lattice QCD. A set of pedagogical appendices compre-
hensively reviews modern approaches to performing functional calculations algebraically,
and derives contributions from a term with open covariant derivatives for the first time.
Keywords: Effective Field Theories, Heavy Quark Physics
ArXiv ePrint: 1912.08814
Open Access,c
The Authors.
Article funded by SCOAP3.https://doi.org/10.1007/JHEP06(2020)164
JHEP06(2020)164
Contents
1 Introduction 1
1.1 Summary of results 4
2 Heavy Quark Effective Theory 7
2.1 Decoupling the heavy quark 9
3 Residues using Feynman diagrams 10
3.1 Residue in QCD 11
3.2 Residue in HQET 12
3.3 Residue difference from the method of regions 13
4 Matching and running using functional methods 15
4.1 1PI effective action from a functional determinant 15
4.2 Matching from a functional determinant 16
4.3 RGEs from the 1PI effective action 18
5 HQET matching from a functional determinant 18
6 Residue difference from a functional determinant 20
6.1 When does RPI become manifest? 23
7 More matching and running calculations 26
7.1 Heavy-light current matching 26
7.2 Heavy-heavy current matching 29
7.3 HQET RGEs 32
8 Conclusions 36
A Loop integrals 37
A.1 Combining denominators 37
A.2 Single scale relativistic integrals 37
A.3 Two scale relativistic integrals 38
A.4 Integrals with relativistic and linear propagators 40
B Covariant derivative expansion 42
B.1 Tools and tricks 42
B.1.1 Baker-Campbell-Hausdorff formula 43
B.1.2 Gauge coupling 43
B.1.3 Functional traces 43
B.1.4 Product rule for covariant derivative 44
B.1.5 Territory of covariant derivative 45
B.1.6 Integration by parts for covariant derivative 47
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JHEP06(2020)164
B.2 Algebraic evaluation of functional traces 47
B.2.1 Partial Derivative Expansion 48
B.2.2 Simplified CDE 52
B.2.3 Original CDE 55
B.3 Functional variations for gauge bosons 59
B.4 Applications 60
B.4.1 Expansions up to mass dimension six 60
B.4.2 Functional traces and determinants 63
C More on RGEs with CDE 67
C.1 RGEs for gauge couplings 67
C.2 Example RGE calculations 69
C.2.1 Scalar φ4theory 69
C.2.2 Heavy-light scalar theory 69
C.2.3 Gauge theory 71
D Heavy-heavy current matching at finite recoil 75
1 Introduction
The modern perspective on Effective Field Theories (EFTs) takes them to be well-defined
quantum field theories in their own right, with all the attendant structure that implies. In
particular, one can define an EFT at the quantum level using a path integral. However,
the standard approach to extracting observables is to sidestep this intimidating object
by organizing perturbation theory through the use of Feynman diagrams. The recent
reintroduction [1] of the covariant derivative expansion (CDE) [24] in a form tailored
to tackling a broader set of EFTs has sparked a renaissance in the extraction of physics
directly from the path integral using functional techniques. A noteworthy example of this
progress has been a general matching formula at one-loop [5,6] that captures the far off-
shell fluctuations of heavy fields. This result can be used for the extraction of the effective
operators and their Wilson coefficients that characterize the effects of the heavy physics,
encoded in the so-called Universal One-Loop Effective Action (UOLEA) [1,710]; one of its
features is that it elegantly packages together what would be multiple independent Feynman
diagram calculations. This has seen ready application to beyond the Standard Model (SM)
scenarios by facilitating one-loop matching onto the so-called SMEFT. However, there exist
a large number of EFTs whose relationship to UV physics cannot be captured by simply
integrating out an entire heavy field. Here, we present the first application of the modern
functional approach to such EFTs. Specifically, we work in a particular low-energy limit
of the SM [1114] expanded around a non-trivial background, the Heavy Quark Effective
Theory (HQET) [13,1517], see also the reviews [1822].
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JHEP06(2020)164
There are many novel features of HQET when compared to the SMEFT. Perhaps the
most striking is that — as a model of the long distance fluctuations of a heavy particle —
HQET is a non-relativistic EFT. Obviously, the path integral is a valid description of a
non-relativistic theory (after all, it was invented as an alternative description of quantum
mechanics [23,24]), but the concrete demonstration that the functional approach is useful
for precision non-relativistic field theory computations has been lacking until now.1An-
other intriguing aspect of HQET is that it invokes the concept of a mode expansion. A
single heavy quark field is decomposed into a pair of fields, which model short and long
distance fluctuations. Then, assuming a particular kinematic configuration, only the long
distance modes can be accessed as external states. The short distance fluctuations can
therefore be integrated out, which generates a tower of local EFT operators. It is this
description that is called HQET. This type of one-to-many correspondence plays a role
in many modern formulations of EFTs, and for the first time here we put such models on
even firmer theoretical footing by computing observables directly from the path integral.
Furthermore, a theoretically appealing aspect of this work is that the covariant derivative
expansion of the functional integral manifests the symmetries of the theory in a transparent
way. This is obviously true for gauge invariance, but we will also be able to approach the
residual Lorentz symmetry known as Reparameterization Invariance (RPI) from a novel
vantage point. In particular, we will identify an intermediate stage in our calculations where
RPI becomes manifest. This provides a nice contrast to the Feynman diagram approach,
where this invariance only holds once one sums the full set of relevant diagrams.
Through our application of the functional approach to HQET, we will expose some
important features of the general formalism. Matching a UV theory onto an EFT, first
done for HQET in ref. [29], can be performed diagrammatically by equating matrix ele-
ments computed with the two descriptions of the theory at a common kinematic point.
This requires knowing the EFT operator expansion and identifying the operators that are
relevant to the EFT matrix element calculation. By contract, using functional methods the
generation of operators and matching of Wilson coefficients occurs in a single step. There
is no need to specify the structure of the EFT before performing a matching calculation.
Working through the example of HQET will show that the problem for how to marry the
mode decomposition with functional methods has a nearly universal solution. Specifically,
given an implementation of the mode decomposition using operator-valued projectors, one
can derive a functional equation of motion for the short distance modes. This can be
used to integrate out these modes, yielding a non-local EFT description that encodes the
complete dynamics of the full theory in the relevant kinematic limit. This justifies the
construction of the EFT as a full path integral over a well defined field.
Deriving concrete predictions using conventional techniques typically involves several
steps and many subtleties, but a functional approach makes the procedure more algorith-
mic. The complicated multi-mode matching calculation has a simple form whose structure
is elucidated by analyzing the resulting integrals using the method of regions [30,31]. As
1To our knowledge, the only time the path integral has been discussed in the context of HQET was
in [25] where the tree-level formulation of the theory was first derived, and in [2628] where RPI was briefly
discussed in the context of the path integral.
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JHEP06(2020)164
is well known, matching in HQET only receives support from diagrams that have loops
with both a short distance mode and a mode that propagates in the EFT. Understanding
how to access this exact class of diagrams using functional methods has been the subject
of some confusion. Showing that we can derive matching and running in HQET with these
methods is a conclusive demonstration that functional methods provide a complete frame-
work at one-loop. These results will be summarized below as a simple master formula that
encodes the matching of QCD onto HQET at one-loop and to any order in the heavy mass
expansion, see eq. (5.4).
There are important practical implications of this work. One of the primary purposes
of HQET is to provide efficient methods for precision calculations within the SM. There are
a number of EFTs widely used to facilitation precise predictions including Soft Collinear
Effective Theory [3234], theories of non-relativistic bound states such as nrQCD [3537],
and others. Improving on the precision of a calculation often requires the application of
novel theoretical approaches, with a goal to provide computational benefits over a naive
perturbative expansion evaluated using Feynman diagrams. Specifically, functional tech-
nology has seen little use in this context, despite the fact that some of the simplifications
it provides are arguably most relevant to the questions these kinematic EFTs are designed
to answer. By showing we can reproduce non-trivial matching and running results for
HQET here, we open the door to understanding how to apply functional techniques in
these other contexts. We additionally lay the foundation for performing new calculations
within HQET itself. In particular, one can now perform matching calculations to higher
order in the heavy mass expansion, which would be relevant to high precision measure-
ments made at experiments such as LHCb and Belle II, and for connecting lattice gauge
theory calculations performed in the heavy quark limit to their continuum limits [38,39].
The rest of the paper is organized as follows. In the rest of section 1, we provide a
summary of the known results that we reproduce in a novel way throughout this paper.
Next, section 2provides a condensed introduction to HQET. Section 3provides a summary
of the traditional method of calculating the simplest piece of the one-loop HQET match-
ing, the residue matching. (Readers familiar with HQET could skip these two sections.)
In section 4, we review how to extract matching and running from a functional determi-
nant. (Readers familiar with functional methods could skip this section.) In section 5,
we introduce the use of functional integration to construct kinematic EFTs, and clarify
how the method of regions simplifies the derivation of the resulting master matching for-
mula. Section 6is then dedicated to an explanation of how such matching is performed
to one-loop order. This is followed by section 7, which strengthens the case for functional
methods by providing additional matching calculations, along with an example that shows
how operator running can be derived in the formalism as well. Section 8then concludes.
An extensive set of pedagogical appendices provide an introduction to many of the
relevant technical details. The “covariant derivative expansion” technique used here was
originally invented in 1980s [24], we refer to this as “original CDE” in appendix B. This
was reintroduced in the context of modern EFT calculations in ref. [1], and has been applied
by refs. [710,40] to develop one-loop universal effective actions. A closely related variant
of the original CDE, which we call “simplified CDE” in appendix B, was proposed in ref. [5].
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JHEP06(2020)164
This more rudimentary version of the CDE turns out to be significantly more convenient for
extracting operators that do not involve a gauge field strength. Most of the results in the
main text of this paper were derived using this simplified CDE. In appendix B, we clarify
the relations between these two versions of the CDE. For completeness, in appendix B
we also provide some simple universal results (tabulated in appendix B.4.2) for functional
traces derived using the CDE. These include the famous elliptic operator, see eq. (B.84a),
which is the central object of study in the development of the UOLEA. Additionally, we
provide the first computation of the contributions to the UOLEA from a term with an open
covariant derivative (truncated at dimension four). This result is provided in eq. (B.84b),
and was previously unknown as emphasized by refs. [10,41,42]. A variety of RGE example
calculations are given in appendix C, and a generalization of the Heavy-Heavy matching
calculation is provided in appendix D.
1.1 Summary of results
In what follows, we will present our methods by way of a few canonical matching and run-
ning calculations. First, we derive the high scale HQET Wilson coefficients by matching
QCD onto the EFT for the heavy-light currents ¯q γµ(γ5)Qand the heavy-heavy currents
¯
Q1γµ(γ5)Q2using purely functional methods. We also present a functional derivation of
running effects, using the first subleading operators in the HQET Lagrangian as a concrete
example.
In order to fix our notation and make comparison to standard results straightforward,
we provide a brief compendium of the results we will reproduce in this paper. The conven-
tional derivation is presented in much more detail in standard references, e.g., refs. [19,22].2
This list provides a useful context for our goals here.
Propagator residue. When performing a matching calculation in any off-shell scheme,
such as MS, it is critical to track the difference in propagator residues (necessary for obtain-
ing the desired S-matrix elements using the LSZ reduction procedure) when moving from
the full theory to the EFT. Taking the functional point of view, the resulting effect shows
up as a rescaling of the kinetic terms for the EFT fields. This can then be moved to its
canonical position in the Wilson coefficients by a field redefinition. Thus, the first step per-
formed in what follows is to derive the one-loop corrections to the kinetic terms from QCD:
LHQET 1R(1) αs¯
hv(iv ·D)hv,(1.1)
where hvmodels the long distance fluctuations of the heavy quark field, vµis the refer-
ence vector defined in eq. (2.2) below, Dµis the covariant derivative, αsis the strong fine
2Notation. We have mostly chosen to follow the notation of ref. [22], but have made a number of
minor changes, which is partially why we provide this summary here. For dimensional regularization
(dim. reg.), our convention is to work in d= 4 2dimensions. We have also chosen to use a different
standard notation of the EFT heavy quark fields. Additionally, we have reserved the superscript numeral in
parenthesis notation to denote loop order, R(0) is tree-level, R(1) is the one-loop correction, and so on. This
is different from the standard notation in the HQET community, e.g. in ref. [22] the one-loop correction
to the residue is denoted as R1. Additionally, we note that we will only denote the loop order of the
renormalized terms, i.e., counterterms are implicit.
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JHEP06(2020)164
structure constant, and ∆R(1) is the matching correction for the propagator residue. In a
traditional matching calculation, the correction comes from the difference of the residues
between the full and EFT descriptions:
R(1) αsR(1)
QR(1)
hαs=1
3
αs
π 3 ln µ2
m2
Q
+ 4!.(1.2)
While this result is sensitive to the choice of matching scale µdue to presence of a UV
divergence; the cancellation of terms that depend on the IR regulator serves as a check
that the IR behavior of the two theories is identical.
Heavy-light current. At leading order in the heavy-mass expansion, two HQET oper-
ators have the same quantum numbers as the heavy-light vector current of QCD, implying
that they can appear in the matching:
¯q γµQ=CV ,1mQ
µ, αs(µ)¯q γµhv+CV ,2mQ
µ, αs(µ)¯q vµhv,(1.3)
where CV,i is the to-be-calculated matching coefficient for the vector current, which is a
function of the heavy quark mass mQand the strong coupling; there is an analogous ex-
pression for the axial current derived by replacing CV,i CA,i,γµγµγ5, and vµvµγ5
in eq. (1.3). The answer up to one-loop (see eq. (3.48) in ref. [22]) can be written as
CV,1= 1 + "1
2R(1) +V(1)
HL,1V(1)
eff #αs+··· ,(1.4a)
CV,2=V(1)
HL,2αs+··· ,(1.4b)
where ∆R(1) is defined in eq. (1.2), and the other terms (see eqs. (3.66) and (3.73) in
ref. [22], respectively) are
V(1)
HL,1αs=1
3
αs
π1
γE+ ln 4 π+ 2,(1.5a)
V(1)
HL,2αs= +2
3
αs
π,(1.5b)
V(1)
eff αs=1
3
αs
π1
γE+ ln 4 π,(1.5c)
where 1
γE+ ln 4 πis the standard factor that is subtracted when using the MS scheme,
with γEdenoting the Euler-Mascheroni constant. Then the axial matching coefficients are
given by CA,1=CV,1and CA,2=CV,2. The one-loop matching coefficients are thus
(compare with eq. (3.74) in ref. [22])
CV,1= 1 + αs
πln mQ
µ4
3+Oα2
s,(1.6a)
CV,2=2
3
αs
π+Oα2
s.(1.6b)
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JHEP06(2020)164
Heavy-heavy current. In case of the heavy-heavy currents, for simplicity we will take
the special kinematic choice of zero recoil, corresponding to v1=v2in the EFT (the
generalization to v16=v2, first done in ref. [43], is discussed in appendix D). In this limit,
all possible HQET operators at leading order in the mass expansion are equal by the
equations of motion, and the matching between QCD and HQET is simply3
¯
Q1γµQ2=ηV¯
hv1γµhv2,(1.7)
with an analogous expression for the axial current given by the replacement ηVηAand
γµγµγ5. The results can be parameterized as (compare with eqs. (3.98) and (3.101) in
ref. [22])
ηV= 1 + 1
2R(1)
1+ ∆R(1)
2αs+ ∆V(1)
HH αs,(1.8a)
ηA=ηV2
3
αs
π,(1.8b)
where ∆R(1)
1and ∆R(1)
2are eq. (1.2) for the heavy quarks Q1and Q2respectively, and
V(1)
HH =2
3
αs
π1 + 3
m1m2m1ln m2
µm2ln m1
µ,(1.9)
where m1,2corresponds to the mass of Q1,2. The matching coefficients then take the form
ηV= 1 + αs
π2 + m1+m2
m1m2
ln m1
m2,(1.10a)
ηA= 1 + αs
π8
3+m1+m2
m1m2
ln m1
m2,(1.10b)
in agreement with eqs. (3.97), (3.99), and (3.101) of ref. [22].
β-functions. Finally, we reproduce the expressions for the running of the HQET match-
ing coefficients at one-loop and at O(1/mQ) (compare with eq. (4.8) in ref. [22]):
L1=ckin(µ)¯
hv
D2
2mQ
hvcmag(µ)gs¯
hv
σµν Gµν
4mQ
hv.(1.11)
The Renormalization Group Equations (RGEs) are [4446]
µd
dµckin = 0 ,(1.12a)
µd
dµcmag =αs
4π2CAcmag ,(1.12b)
where CAdenotes the Casimir factor for the adjoint representation
In what follows, we will show how to reproduce all of these results using functional
methods equipped with the CDE technique.
3The choice of notation for the HQET fields here is made for of ease of legibility in later sections,
but deserves comment here so as to not mislead. The labels v1,2differ only to keep track of finite-mass
corrections. They do not indicate different velocities, and the relation between the QCD and HQET
operators is not valid except in the zero-recoil limit. (Compare with eq. (3.89) in ref. [22].)
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JHEP06(2020)164
2 Heavy Quark Effective Theory
One of our main goals here is to initiate the study of functional methods for higher-order
calculations in EFTs that are derived by performing multi-modal decomposition of full
theory fields. Such EFTs are quite common; they occur when the kinematics of the process
being studied selects a preferred reference frame due to, e.g. a conservation law preventing
the decay of a heavy particle, or a measurement function that forces the external states
into a particular region of phase space. These restrictions imply that there are full theory
modes which cannot be put on-shell within the EFT regime of validity, and so it is sensible
to integrate them out. This procedure breaks the full theory space-time symmetries to
some subgroup, while potentially also introducing new internal ones. A theory of this
type is the Heavy Quark Effective Theory (HQET), which describes the fluctuations of a
heavy quark (mQΛQCD) in the presence of light QCD charged degrees of freedom. The
simplifications and universal behavior of QCD in the heavy-mass limit were first appreciated
by refs. [11,12] and especially refs. [13,14]. A non-covariant EFT making this behavior
manifest was later developed and shown to be well-behaved in perturbation theory [15,17],
and finally given a covariant formulation [16]. We provide a review of HQET here, with
a particular emphasis on the equation of motion due to the critical role it plays in what
follows. The reader familiar with HQET can skip ahead to section 4, while more details
can be found in, e.g. section 4.1 of ref. [22].
The full theory (QCD) Lagrangian for a heavy quark includes
LQCD ¯
Qi/
DmQQ , (2.1)
where Qis our heavy quark, and the covariant derivative only includes the QCD interac-
tions, Dµ=µigsGa
µTa. In the following, additional interactions of the heavy quark
will be modeled through the introduction of current operators as needed.
Naively, it does not seem that the Lagrangian of eq. (2.1) has a good expansion about
the mQ→ ∞ limit. The key insight that allows one to circumventing this issue lies in an
appropriately chosen phase redefinition of the field, whose purpose is to cancel the mass
term for certain components. The full quark field can then be separated into so-called short
distance and long distance fields, where the latter become approximately mass-independent.
Physically, this is motivated by the realization that the heavy quark cannot be pushed
very far off-shell by degrees of freedom for which |q|.ΛQCD. To make this manifest, we
decompose a heavy quark’s momentum as
pµ=mQvµ+kµ,(2.2)
where vµis a unit time-like vector, and kµis the residual heavy quark momentum which
models small fluctuations about its mass shell. For kinematic configurations such that all
Lorentz invariants depending on kµare small compared to mQ, a truncation at finite order
in the |k|/mQexpansion is justified.4Since the theory is expanded around the mQ→ ∞
4When we expand assuming |k|  mQ, we take each element of the kµvector to be much smaller
than mQ.
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JHEP06(2020)164
limit, the structure of HQET does not know about the mass of the heavy quark except
through various non-dynamical quantities. In particular, no sensitivity to mQappears in
any of the calculations beyond what is encoded in the structure of the matching coefficients.
As a brief aside, we note that the decomposition in eq. (2.2) is not unique. In particular,
a simultaneous transformation of kµand vµby a fixed vector:
kµ
RPI kµ+δkµand vµ
RPI vµδkµ
mQ
,(2.3)
leaves pµunchanged. Enforcing that vµremains a unit vector implies that δkµmust
satisfy v·δk =δk2/(2mQ). Reparameterization invariance (RPI) is then the statement
that physical observables cannot depend on δkµ, thereby enforcing the residual Lorentz
invariance of the underlying full theory. We will explore the interplay between RPI and
the functional approach in section 6.1 below.
In order to make use of eq. (2.2), we decompose the heavy quark field into two fields
by first extracting the rapidly-varying phase that remains unchanged by low-energy in-
teractions, and then using vµ-dependent projectors to split the remaining field into short
distance and long distance components:
hv(x) = eimQv·x1 + /
v
2Q(x),(2.4a)
Hv(x) = eimQv·x1/
v
2Q(x),(2.4b)
or equivalently
Q(x) = eimQv·xhhv(x) + Hv(x)i.(2.5)
Plugging eq. (2.5) into eq. (2.1) yields
L ⊃ ¯
hviv ·Dhv¯
Hviv ·D+ 2mQHv+¯
Hvi/
Dhv+¯
hvi/
DHv,(2.6)
since the projectors enforce /
v hv=hvand /
v Hv=Hv(and hence ¯
Hv/
v hv=¯
hv/
v Hv= 0),
and we have defined
Dµ
Dµvµ(v·D).(2.7)
The Lagrangian in eq. (2.6) makes the interpretation of Hvas the heavy mode manifest —
this field has an effective mass of 2mQ, permitting a description at lower energies in terms
of hvalone.
To formally integrate out Hvat tree-level, we solve for its equation of motion
Hv=1
iv ·D+ 2mQ
i/
Dhv,(2.8)
and plug it into eq. (2.6), which yields
Lnon-local
HQET ¯
hv iv ·D+i/
D
1
iv ·D+ 2mQ
i/
D!hv.(2.9)
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JHEP06(2020)164
Provided that the momentum of the field hvsatisfies |k|  mQ, eq. (2.9) can be expressed
into a convergent series of local terms
1
iv ·D+ 2mQ
=1
2mQ1
2mQ2(iv ·D) + 1
2mQ3(iv ·D)2+··· .(2.10)
Since the leading term is then insensitive to the Dirac structure of the spinor components
of the field, the theory has gained an approximate SU(2) heavy-spin symmetry, which can
be expanded to SU(2nh) in the presence of nhheavy flavors. This is an example of an
aforementioned emergent symmetry of the EFT.
To understand how this procedure for deriving the EFT Lagrangian can be interpreted
at the path integral level, we note that the projection operators in eq. (2.4) imply that
hvand Hvcan be treated as two orthogonal projections of Q. As such, the path-integral
measure factorizes: ZDQ=ZDhvZDHv.(2.11)
Since the resulting Lagrangian is quadratic in Hv, the Gaussian integral over the short-
distance field immediately yields eq. (2.9). Therefore, the procedure described here allows
us to literally integrate out the short distance modes of our original quark. This elegant
derivation of the tree-level HQET Lagrangian by way of the path-integral measure was
first presented in ref. [25]. Furthermore, it also justifies the use of the functional approach
to matching and running developed in ref. [5]. Note that there would be no significant ob-
structions even if our projectors were quantum operators instead of simply being functions
of the kinematics as above, since all objects in the path integral are treated as operators
acting on fields. As long as the mode decomposition that is used to define an EFT can be
written in terms of operator-valued projectors, we expected the methods developed here
to be universally applicable.
2.1 Decoupling the heavy quark
The decomposition of the full QCD Lagrangian given in eq. (2.6) makes it clear that Hvcan
be identified as a short distance mode which one can integrate out in the limit that |k| 
mQfor all fields in a given process. This procedure yields the non-local Lagrangian given
in eq. (2.9), which makes predictions that are equivalent to QCD for processes involving
only hvmodes in the external states. Expanding the non-local Lagrangian using eq. (2.10)
and truncating the series at some order in 1/mQyields an EFT which is valid for momenta
|k|  mQ. In this subsection, we will briefly review the connection between this procedure
and the fact that the heavy quark should not contribute to the running of the QCD gauge
coupling at scales below mQ. This both has a conceptual benefit, and will also be of
practical importance since similar arguments will be used below when we derive our master
formula for one-loop matching given in eq. (5.4).
For this argument, we will work directly with the hvand Hvfields, and we will use the
terms that are diagonal in these fields to derive propagators, while the mixed terms will be
treated as interactions. Two features are of critical importance, the fact that the kinetic
terms are linear in the momentum of the state, and the relative minus sign between the
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hvand the Hvkinetic terms. The later fact implies that the ifactors in the propagators
will take the opposite sign such that the same Wick rotation can be used to Euclideanize
diagrams involving both hvand Hvpropagators.
The QCD Lagrangian written as eq. (2.6) has three types of couplings between the
heavy quark modes and the gluon: diagonal couplings hvhvGµ,HvHvGµ, and an off-
diagonal coupling HvhvGµ. The integral that results when computing the contribution
to the vacuum polarization for the gluon from the two diagonal loops schematically take
the form
Idiag Zddq
(2π)d1
v·q+m±i1
v·(q+p) + m±i,(2.12)
where mis an IR regulator for the hvloop and is equal to 2mQfor the Hvloop. Due to
the linear nature of the kinetic terms and the sign on the factor of i, when integrating
over q0, the poles reside on only one side of the real axis, and one can deform the contour
away from all them yielding zero contribution. However, the situation is different for the
mixed loop, where the integral takes the form
Ioff-diag Zddq
(2π)d1
v·q+m+i1
v·(q+p)+2mQi .(2.13)
Now we see that the opposite sign on the i terms implies that there is a pole in both the
positive and negative Im q0half-planes. The contour will enclose a pole for any possible
deformation, yielding a non-zero contribution to the β-function.
The argument above makes it clear why in HQET the heavy quark does not contribute
to the RGEs of the gauge coupling. Once we construct HQET by integrating out Hvand
expanding, the heavy field Hvis non-propagating. Therefore, there are no diagrams that
yield contributions of the type in eq. (2.13). The mode hvonly yields potentially relevant
integrals of the type in eq. (2.12), which vanish as we have argued. Similar reasoning will
be critical to the derivation of the master formula for matching in eq. (5.4) below.
3 Residues using Feynman diagrams
In this section, we will review the diagrammatic approach to calculating matching coeffi-
cients, using the simple example of the residue of the on-shell propagator for concreteness.
Along the way, we will encounter an order of limits issue, which is a manifestation of the
IR divergence structure of QCD. We will then revisit the calculation by relying on the
so-called method of regions [30,31] that will avoid the need to deal with this subtlety di-
rectly. This has the additional benefit of providing a familiar setting to review the method
of regions, which will be a critical tool in the derivation of our master formula for HQET
matching coefficients below.
When performing matching calculations, one typically equates matrix elements as cal-
culated in the full theory and the EFT. Particular care must be taken to include the
appropriate residue factors for the external states to ensure that the LSZ reduction is cor-
rectly implemented. One way to extract the residue for the heavy quark Qis to take the
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derivative of the 1PI corrections to the propagator iΣ(/
p), and evaluate it on the mass
shell. We will compute this factor for a quark in QCD RQ= 1 + R(1)
Qαs+···, and for a
quark in HQET Rh= 1 + R(1)
hαs+···, from which we get the quantity that appears in
matching calculations ∆R(1) =R(1)
QR(1)
h, see eq. (1.2).
3.1 Residue in QCD
The one-loop QCD residue R(1)
Qcan be obtained by computing the two-point 1PI function
iΣQCD(/
p) = 4
3g2
sµ2Zddq
(2π)d
(2 d)(/
q+/
p) + d mQ
q2h(q+p)2m2
Qic.t.
=i(AAct)mQ+ (BBct )/
p.(3.1)
Here as well as throughout this paper, “c.t.” denotes the counter term contributions, and
we take the Feynman gauge ξ= 1 for gauge boson propagators. Performing standard
manipulations, we derive5
Ap2=αs
3π4π µ2Γ() 4 1
2Z1
0
dxm2
Qxp2x(1 x),(3.2a)
Bp2=αs
3π4π µ2Γ() 2 (1 )Z1
0
dx(1 x)m2
Qxp2x(1 x),(3.2b)
Act =αs
3π41
γE+ ln 4 π,(3.2c)
Bct =αs
3π1
γE+ ln 4 π,(3.2d)
where the MS counter terms Act and Bct are derived by taking the expansion of Ap2
and Bp2. This yields
lim
0(AAct) = 4
3
αs
π 3
2m2
Q
p2ln m2
Q
µ2+m2
Qp2
p2ln m2
Qp2
µ2!,(3.3a)
lim
0(BBct) = αs
3π 1m4
Q
p4ln m2
Q
µ2+m4
Qp4
p4ln m2
Qp2
µ2+m2
Q
p2!.(3.3b)
These results are finite but non-analytic at p2=m2
Q. In particular, their derivatives with
respect to p2are divergent when evaluated at p2=m2
Q. These are a manifestation of IR
divergences that appear when taking on-shell kinematics.
5See eqs. (3.53), (3.55), and (3.57) in ref. [22], noting again that we use d= 4 2, while ref. [22]
uses d= 4 .
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One way to side step this issue, thereby allowing us to extract the residue, is to keep
6= 0 until after taking the derivative. It is then straightforward to derive
R(1)
Qαs=QCD(/
p)
d/
p/
p=mQ
= 2 m2
Qd(AAct)
dp2+d(BBct)
dp2p2=m2
Q
+ (BBct)p2=m2
Q
=αs
3π"21
γE+ ln 4 π+ 3 ln µ2
m2
Q
+ 4#,(3.4)
where is specifically regulating the IR divergence. Next, we will derive the residue in
HQET, where we will see that the same IR divergent terms appear. Therefore, the object
of interest ∆R(1) is IR finite. This is to be expected, since one of the standard tests that
one has correctly implemented the matching procedure, namely that one is working with
the correct low energy description, is to check that the IR of the full theory and EFT have
the same divergence structure. We will provide a procedure for directly extracting ∆R(1)
that avoids this IR subtlety utilizing the method of regions, see section 3.3 below.
3.2 Residue in HQET
Next, we perform the two-point function calculation in HQET. Diagrammatically, the
structure is identical to the QCD calculation where the relativistic quark propagator is
replaced by the HQET propagator, yielding
iΣHQET(v·k) = 4
3g2
sµ2Zddq
(2π)d
1
q2v·(q+k)c.t.
=ihC(v·k)Cct(v·k)i,(3.5)
which evaluates to6
C(v·k) = 2
3
αs
π4π µ2Γ()(v·k)12Γ(1 ) Γ1
2+
(1 2) Γ1
2,(3.6a)
Cct(v·k) = 2
3
αs
πv·k1
γE+ ln 4 π,(3.6b)
where the counter term is again determined in the MS scheme.7Noting that to zeroth
order in 1/mQthe on-shell condition for hvis v·k= 0, we evaluate
dC(v·k)
d(v·k)v·k=0
=2
3
αs
π4π µ2Γ() (v·k)2Γ(1 1
2+
Γ(1
2)v·k=0
= 0 ,(3.7a)
dCct(v·k)
d(v·k)v·k=0
=2
3
αs
π1
γE+ ln 4 π,(3.7b)
6See eqs. (3.67) and (3.69) in ref. [22].
7Note that when computing the counter term in dim. reg., one must be careful to isolate the UV
divergence. This is done here by keeping v·k6= 0 as an IR regulator at intermediate steps. This is why
we must take a derivative of eq. (3.6b) before sending v·k0 to derive eq. (3.7b). If instead we took
v·k0 first, we would effectively be using dim. reg. to regulate the IR divergence as well.
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which yields
R(1)
hαs=dC(v·k)dCct(v·k)
d(v·k)v·k=0
=2
3
αs
π1
γE+ ln 4 π.(3.8)
Similar to above, the lim0ΣHQET is not analytic at v·k= 0 due to an IR divergence,
and so we had to defer taking the 0 limit until after taking the derivative with respect
to v·k.
3.3 Residue difference from the method of regions
The IR divergences in eqs. (3.4) and (3.8) are the same, as they must be if the EFT
correctly captures the dynamics of the full theory below a certain scale. This implies that
the residue difference is IR finite:
R