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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) [2–4] 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) [1,7–10]; 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 [11–14] expanded around a non-trivial background, the Heavy Quark Eﬀective

Theory (HQET) [13,15–17], see also the reviews [18–22].

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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 [26–28] 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 [32–34], theories of non-relativistic bound states such as nrQCD [35–37],

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 [2–4], 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. [7–10,40] 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 ⊃1−∆R(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) αs≡R(1)

Q−R(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,γµ→γµγ5, and vµ→vµγ5

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

2∆R(1) +V(1)

HL,1−V(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

µ−4

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 ηV→ηAand

γµ→γµγ5. The results can be parameterized as (compare with eqs. (3.98) and (3.101) in

ref. [22])

ηV= 1 + 1

2∆R(1)

1+ ∆R(1)

2αs+ ∆V(1)

HH αs,(1.8a)

ηA=ηV−2

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

m1−m2m1ln m2

µ−m2ln m1

µ,(1.9)

where m1,2corresponds to the mass of Q1,2. The matching coeﬃcients then take the form

ηV= 1 + αs

π−2 + m1+m2

m1−m2

ln m1

m2,(1.10a)

ηA= 1 + αs

π−8

3+m1+m2

m1−m2

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

hv−cmag(µ)gs¯

hv

σµν Gµν

4mQ

hv.(1.11)

The Renormalization Group Equations (RGEs) are [44–46]

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/

D−mQQ , (2.1)

where Qis our heavy quark, and the covariant derivative only includes the QCD interac-

tions, Dµ=∂µ−igsGa

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 vµ−δkµ

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) = e−imQv·xhhv(x) + Hv(x)i.(2.5)

Plugging eq. (2.5) into eq. (2.1) yields

L ⊃ ¯

hviv ·Dhv−¯

Hviv ·D+ 2mQHv+¯

Hvi/

D⊥hv+¯

hvi/

D⊥Hv,(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

⊥≡Dµ−vµ(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/

D⊥hv,(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

2mQ−1

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 i factors 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±i 1

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+i 1

v·(q+p)+2mQ−i .(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)

Q−R(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)2−m2

Qi−c.t.

=−i(A−Act)mQ+ (B−Bct )/

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

Qx−p2x(1 −x)−,(3.2a)

Bp2=−αs

3π4π µ2Γ() 2 (1 −)Z1

0

dx(1 −x)m2

Qx−p2x(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(A−Act) = 4

3

αs

π 3

2−m2

Q

p2ln m2

Q

µ2+m2

Q−p2

p2ln m2

Q−p2

µ2!,(3.3a)

lim

→0(B−Bct) = −αs

3π 1−m4

Q

p4ln m2

Q

µ2+m4

Q−p4

p4ln m2

Q−p2

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=dΣQCD(/

p)

d/

p/

p=mQ

= 2 m2

Qd(A−Act)

dp2+d(B−Bct)

dp2p2=m2

Q

+ (B−Bct)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)1−2Γ(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 −)Γ1

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·k→0 to derive eq. (3.7b). If instead we took

v·k→0 ﬁrst, we would eﬀectively be using dim. reg. to regulate the IR divergence as well.

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JHEP06(2020)164

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 lim→0ΣHQET 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