Position-space renormalization schemes for four-quark operators in HQET
Joshua Lin William Detmold Stefan Meinel

[
    [
    25 April 2024
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§ INTRODUCTION



Lattice Quantum Chromodynamics (LQCD) calculations are crucial to nonperturbative determinations of Standard-Model parameters and searches for beyond-the-Standard-Model physics in the flavor sector <cit.>. 
These calculations (see FLAG <cit.> for a review) typically require renormalization of composite operators, as the hadronic matrix-elements are renormalization scale-dependent, and must be combined with Wilson coefficients to determine renormalization-scale-independent physical quantities. 
The most commonly used renormalization scheme in phenomenological applications of QCD is the modified minimal-subtraction () scheme because of its perturbative simplicity. However, LQCD has no direct access to the  scheme which is only defined in dimensionally-regulated perturbation theory. 
Lattice perturbation theory <cit.> can be used to convert bare matrix-elements computed in LQCD to -renormalized matrix-elements, but tends to suffer from poor convergence properties <cit.>. 
Nonperturbative renormalization schemes bypass these problems by renormalizing composite lattice operators in an intermediate, regulator-independent scheme, before perturbatively matching between this intermediate scheme and the  scheme using dimensional regularization. 
A common choice of an intermediate scheme is to impose a momentum-space renormalization condition, as is done in the Regularization-Independent Momentum Subtraction (RI-(S)MOM) methods <cit.>. A drawback of RI-(S)MOM schemes is that gauge-fixing is required due to the use of gauge-noninvariant quark and/or gluon states, giving rise to additional mixing with gauge-noninvariant operators <cit.>. Furthermore, Gribov copies (the discrete set of intersections between a gauge orbit and a gauge-fixing condition) introduce a systematic error to numerically computed renormalization constants, although numerical studies often suggest that this is practically negligible compared to statistical noise <cit.>. 

To circumvent these problems, one can instead impose renormalization conditions based on position-space correlation functions of gauge-invariant operators. In the literature this has been called the X-space scheme <cit.>, or the gauge-invariant-renormalization-scheme (GIRS) <cit.>. 
For multiplicatively renormalizable operators, the X-space renormalized operator 𝒪^(X) can be related to the bare operator O^(0) by O^(X) = Z_𝒪^(X)𝒪^(0), where Z_𝒪^(X) is an (often divergent) renormalization constant. 
One possible  X-space renormalization scheme can be defined by the requirement that the renormalized two-point correlation function built from 𝒪^(X) is equal to its non-interacting (NI) value when the operators are separated by a fixed spacetime distance x:

    ⟨𝒪^† (X)(0) 𝒪^(X)(x) ⟩ = ⟨𝒪^†(0)(0) 𝒪^(0)(x) ⟩|_NI ,

where √(x^2) is the scale at which the operator is renormalized, and the non-interacting correlation function is defined as the α_S → 0 value of the correlation function[The non-interacting correlation function in X-space schemes is sometimes referred to as `tree-level', though this is a misnomer due to the fact that the noninteracting calculation of position-space correlation functions of composite operators involves loops. Note that the superscript labels `(0)' on the operators in the RHS of <ref> are redundant as the non-interacting value for the bare and renormalised operators is the same.]. Although the scheme is gauge invariant by definition, it requires more-complicated perturbative calculations in order to match to other schemes in the continuum. Matching calculations from X-space schemes to  have been performed for light-quark bilinears <cit.>, heavy-light quark bilinears <cit.>, dimension-5 operators appearing in the energy-momentum tensor <cit.>, and supercurrent operators in supersymmetric theories <cit.>. 

When implementing RI-(S)MOM renormalization conditions at momentum p with a lattice discretisation, there is a `window problem' where Λ_QCD≪ p ≪ a^-1 is required to keep all systematic uncertainties under control. 
Indeed, the RI-SMOM scheme <cit.> was introduced to remove infrared convergence issues in applications of the original RI-MOM scheme where some momenta were not in the desired range. 
Here, a is the lattice discretisation-scale that regulates the ultraviolet (UV) behaviour of the theory and Λ_QCD is the typical QCD scale that emerges through dimensional transmutation. 
The same window problem affects any position-space scheme, where √(x^2)≪Λ_QCD^-1 is required to control perturbation-theory errors appearing in the perturbative matching to , and a ≪√(x^2) is required to control discretisation artifacts. In practice, this window problem must be investigated on a case-by-case basis, and various investigations have been performed in X-space schemes for the local light-quark bilinear operators using Wilson fermions <cit.>, twisted-mass fermions <cit.>, as well as domain-wall fermions <cit.>. 
Furthermore, numerical studies of the feasibility of X-space renormalization conditions for renormalizing the QCD energy-momentum tensor <cit.>, heavy-light quark bilinear operators <cit.>, and operators in supersymmetric field theories <cit.> have been undertaken. There have also been numerical investigations of the possibility of using position-space schemes to match between three and four-flavor QCD <cit.>. 

The X-space scheme is particularly suited to renormalizing Heavy Quark Effective Theory (HQET) operators. Choosing a reference frame in which the heavy-quark velocity has spatial components that vanish, v = (1,0,0,0)^T, the bare Euclidean propagator for a heavy quark Q in the static limit is naturally written in position space <cit.> as

    ⟨ Q^(0)(0) Q^(0)(x_E) ⟩_F = δ_x⃗_E,0⃗ θ(-t_E)  W^(0)(0,x_E) 1 + γ_0/2,

where ⟨·⟩_F indicates the path integral is performed over all the fermionic degrees of freedom but not the gauge degrees of freedom, and W^(0)(a,b) is the bare straight Wilson line from a to b. A complication in lattice regularizations of the static theory is that the static-quark self-energy has a power divergence <cit.>, which is caused by mixing between the kinetic term Q D_0 Q and a mass-like term m_statQ Q, where m_stat∼ O(α_S)/a is radiatively generated.  
The X-space scheme proposed in this work (<Ref>) utilizes ratios of three-point to two-point position-space correlation functions to nonperturbatively cancel this power divergence. Nonperturbatively-renormalized matrix-elements of HQET operators can thus be extracted without needing to determine m_stat explicitly, which would otherwise constitute another source of uncertainty. 

In this work, a set of X-space schemes for renormalizing four-quark HQET operators is proposed, and the O(α_S) matching to  is calculated in the static limit, extending the X-space approach used in Refs. <cit.>. 
The first set of operators that are considered are isospin-nonsinglet, four-quark operators τ_f f' (QΓ q_f)(q_f'Γ' Q) where Γ,Γ' are spin-colour tensors, q_f ∈{q_u,q_d} are light-quark fields, and τ_f f' is a Pauli matrix in the light-quark flavor space. In the Heavy Quark Expansion (HQE) formalism, matrix elements of these Δ Q = 0 (heavy-quark flavor preserving) operators are known as `spectator contributions', and are the dominant O(1/m_Q^3)-corrections to the inclusive lifetimes of heavy (charm or bottom) hadrons <cit.>. 
The second set of operators that are considered are Δ B = 2 four-quark operators, relevant for determinations of B -B-mixing <cit.>. 
Precise determinations of hadronic matrix-elements of these quantities will allow for better constraints on fundamental parameters of the Standard Model such as the CKM matrix-elements, and will also further constrain extensions of the Standard Model. 
Existing LQCD studies of these four-quark operators have utilized lattice perturbation-theory to perform matching to  <cit.>, and the nonperturbative renormalization-conditions proposed in this work will allow for more precise LQCD determinations of the renormalized matrix-elements <cit.>. 

The four-quark operators studied in this work do not renormalize multiplicatively but rather mix within multiplets of operators with the same quantum numbers, and therefore <ref> must be generalized to define a scheme that determines the entire mixing-matrix. 
In order to provide a sufficient number of renormalization conditions to determine the full mixing-matrix, the X-space scheme proposed here utilize three-point correlation functions involving the four-quark operator with multiple choices of source and sink operators. In particular, the source and sink operators used are heavy-light mesonic operators QΓ q and heavy-light-light baryonic operators ϵ^abc [q^aTΓ_1 q^b] Γ_2 Q^c for various choices of the Dirac matrices Γ,Γ_1,Γ_2. 
Perturbative calculations of two-point correlation functions constructed from these operators have been performed in the literature, and these can be used to determine matching coefficients between X-space schemes and the  scheme, which are presented to O(α_S) in <Ref>. 
The four-quark operators also mix into evanescent operators
(operators that explicitly vanish in d = 4) in dimensional regularization. 
To be able to utilize the X-space scheme as a regulator-independent scheme for conversion of lattice matrix-elements, the scheme is defined in terms of evanescent-subtracted operators. 
The X-space schemes, and O(α_S)-matching to the  scheme for the isospin non-singlet Δ B = 0 operators and the Δ B = 2 operators are presented in <Ref>, respectively. An outlook is presented in <Ref>.
An overview of the conventions and integration techniques used in this paper is given in <Ref>. 




§ MULTIPLICATIVELY RENORMALIZABLE OPERATORS



 §.§ X-space schemes for heavy-light bilinear and heavy-light-light trilinear operators



In this section, the O(α_S) matching-factors between X-space-renormalized operators and -renormalized operators for heavy-light mesonic, and heavy-light-light baryonic operators in HQET are presented. Choosing the frame in which the static heavy quark propagates purely in the Euclidean time direction, the Euclidean HQET Lagrangian used in the following calculations is given by

    ℒ = 1/4 F_μν F_μν + ∑_f = {u,d}q_f γ_μ D_μ q_f +  Q D_0 Q,

with two massless light-quarks q_u and q_d, and one static heavy-quark Q that satisfies 1 + γ_0/2 Q = Q. 
To regulate the continuum theory, dimensional regularization (DR) is used, where the dimension of spacetime is analytically continued to d = 4 - ϵ, and γ_5 =  γ_0 γ_1 γ_2 γ_3 is treated in the 't Hooft-Veltman (HV) scheme <cit.> (see <Ref> for a review of the conventions used). 
Due to the heavy-quark term in the action not containing any Dirac matrices, in d = 4 there is an SU(2)_h heavy-quark spin symmetry

    Q ↦ e^-iθ_j γ_5  γ_j Q,   Q↦Q e^iθ_j γ_5  γ_j

that leaves the action invariant, where γ_j ∈{γ_1,γ_2,γ_3}. 
The resulting effect of this symmetry is that local heavy-light mesonic operators are related by heavy-quark symmetry in the following SU(2)_h doublets:

    SU(2)_h  H_f^+ (0^+) : q_f Q 
    H_f,i^*+ (1^+) : q_f γ_5 γ_i Q
    ,  
    SU(2)_h  H_f^- (0^-) : q_f  γ_5 Q 
    H_f,i^*- (1^-) : q_f  γ_i Q
    ,

where the J^P quantum numbers of the state are listed along with the corresponding local heavy-light operator. Note that the ± superscripts refer to the parity of the operator, rather than the electromagnetic charge of the state. In <Ref> and what follows, γ_i indexes the (d-1)-dimensional spatial Dirac matrices, such that γ_i γ_i = (d-1) 1.[In d = 4, where spatial Dirac matrices γ_i take values in γ_i ∈{γ_1,γ_2,γ_3}, the heavy-quark spin symmetry is indeed SU(2)_h. In dimensional regularization the spatial index i now varies over the (d-1)-dimensions that are not timelike, and formally the symmetry group is no longer SU(2)_h. This distinction, however, does not have any implications for the calculations presented in this section. ] 
Since the antiparticle is integrated out in HQET, the operators shown in <ref> form a basis for the heavy-light bilinear operators with no derivatives in d = 4. 

The heavy-light-light baryonic operators of the form ϵ^abc [q^aTτΓ_1 q^b] Γ_2 Q^c for varying Dirac matrices Γ_1,Γ_2 and isospin matrices τ are also multiplicatively renormalizable. In the following, C is the charge-conjugation matrix satisfying C γ_μ C^-1 = -γ_μ^T, with an explicit construction in dimensional regularization given in <Ref>. The isospin matrices are written in terms of the antisymmetric τ^A or symmetric τ^S_α matrices

    τ^A := [  0  1; -1  0 ],   τ^S_1 := [ 1 0; 0 0 ],   τ^S_0 := 1/√(2)[ 0 1; 1 0 ],   τ^S_-1 := [ 0 0; 0 1 ].

Operators coupling to Λ_Q baryons (isospin singlets) are given by


    Λ_1 (12^+)   : ϵ^abc [q^aTτ^A C  γ_5 q^b] Q^c , 
    Λ_2 (12^+ )   : ϵ^abc [q^aTτ^A C  γ_5 γ_0 q^b] Q^c , 
    Λ_1^- ( 12^-)    : ϵ^abc [q^aTτ^A C q^b] Q^c ,



    SU(2)_h  Λ_2^- (12^-): ϵ^abc [q^aTτ^A C γ_5 γ_i q^b] γ_i γ_5 Q^c ,
    Λ^*-_i ( 32^- ) : ϵ^abc [q^aTτ^A C γ_5 γ_i q^b] Q^c  - 13ϵ^abc [q^aTτ^A C γ_5 γ_j q^b] γ_i γ_j Q^c .

The operators are labelled by their angular momentum representation and parity, J^P. 
The SU(2)_h-doublet {Λ_2^-, Λ_i^*-} arises from decomposing the tensor product of a spin-1 light-quark doublet with the spin-1/2 heavy quark into irreducible spin representations. In particular, spin-3/2 operators such as Λ_i^*- satisfy the condition γ_i Λ_i^*- = 0 in d = 4. 
Operators coupling to Σ_Q-baryons which transforms in the isospin-triplet representation, are given by




    SU(2)_h Σ_1,α(12^+): ϵ^abc [q^aTτ^S_α C γ_i q^b] γ_i γ_5 Q^c ,
    Σ_1,α,i^* ( 32^+ ) : ϵ^abc[q^aTτ^S_α C γ_i q^b] Q^c  - 13ϵ^abc [q^aTτ_A C γ_j q^b] γ_i γ_j Q^c ,




    SU(2)_h Σ_2,α(12^+): ϵ^abc [q^aTτ^S_α C γ_0 γ_i q^b] γ_i γ_5 Q^c ,
    Σ_2,α,i^* ( 32^+ ) : ϵ^abc [q^aTτ^S_α C γ_0 γ_i q^b] Q^c  - 13ϵ^abc [q^aTτ^S_α C γ_0 γ_j q^b] γ_i γ_j Q^c ,




    SU(2)_h Σ_1,α^- (12^-): ϵ^abc [q^aTτ^S_α C γ_5 γ_0 γ_i q^b] γ_i γ_5 Q^c, 
    Σ_1,α,i^*-( 32^- ) : ϵ^abc[q^aTτ^S_α C γ_5 γ_0 γ_i q^b] Q^c  - 13ϵ^abc[q^aTτ^S_α C γ_5 γ_0 γ_j q^b] γ_i γ_j Q^c ,


    Σ_2,α^- (12^-): ϵ^abc [q^aTτ^S_α C  γ_0 q^b] Q^c ,

where α is the isospin index and i is the vector index of the spin-3/2 field. The operators in <ref> form a complete basis in d = 4 for local HQET heavy-light-light currents with no derivatives, of which the positive-parity local operators were previously classified in Refs <cit.>. 

The X-space scheme for multiplicatively-renormalizable currents is specified at a renormalization scale t^-1 by the following condition:

    ⟨𝒪^† (R,X)(-t,0⃗) 𝒪^(R,X)(t,0⃗) ⟩ = ⟨𝒪^†(0)(-t,0⃗) 𝒪^(0)(t,0⃗) ⟩|_NI,

where R is a regulator (either DR or the lattice regulator), and X specifies the operators are renormalized in the X-space scheme. The subscript NI on the right-hand side of <Ref> refers to the non-interacting value of the correlator. If the operator has additional indices (such as isospin triplet indices α in the case of the Σ-interpolating operators in <ref>, or spatial gamma-matrix indices i) these additional indices are summed over on both sides of the renormalization condition. 
For example, in the 1^- mesonic channel (the B^* mesons if Q is a bottom quark), the renormalization condition is

    ∑_i ⟨ (qγ_i Q)^† (R,X)(-t,0⃗) (qγ_i Q)^(R,X)(t,0⃗) ⟩ = ∑_i ⟨ (qγ_i Q)^(0)†(-t,0⃗) (qγ_i Q)^(0)(t,0⃗) ⟩|_NI.

Furthermore, the open Dirac indices of the static heavy-quark and heavy anti-quark in the baryonic operators are traced over. 



The position-space diagrams that contribute to the O(α_S) determination of the two-point correlation function for the heavy-light currents are shown in Fig. <ref>. 
Although chiral symmetry is broken by the HV-prescription for γ_5, the massless nature of the light quark still has consequences for the symmetries present in the renormalization factors. 
In particular, the specific Γ-matrix appearing in the interpolating operator 𝒪 = qΓ Q only affects the two-point correlation function by a constant factor, as the correlation function (to any order in α_S) has the form

    ⟨( Q^(DR,0)Γ q^(DR,0))(-t,0⃗)  (q^(DR,0)Γ Q^(DR,0))(t,0⃗) ⟩   = A  Tr(1 + γ_0/2Γγ_0 Γ) 
       = A P  Tr(1 + γ_0/2ΓΓ),

where A is a constant that does not depend on Γ, P ∈{-1,+1} is the parity of the state (γ_0Γ = P Γγ_0), and Γ = γ_0 Γ^†γ_0. This is because of the fact that regardless of the number of gluons attached to the light-quark line in the two-point correlation function, there are always an odd number of γ-matrices inserted (one for each massless propagator and one for each vertex). For example, shown below is the case with three gluons attached to the light quark:



    < g r a p h i c s >



 The only four-vector available for contraction is the purely timelike heavy-quark velocity v; thus, after performing integration over loop momenta and Fourier transforming to position space, the light-quark line is proportional to v = γ_0. 
The result of <Ref> is that all heavy-light operators qΓ Q renormalize multiplicatively with the same factor to all orders of α_S in the X-space scheme, regardless of the choice of Γ. 

The diagrams in Fig. <ref> can be computed by first calculating the corresponding two-loop momentum-space diagrams (also known as p-type integrals <cit.>) and taking a Fourier transform; further details of this computation are given in <Ref>. Writing (qΓ Q)^(DR,S) = Z_(qΓ Q)^(DR,S) (q^(0)Γ Q^(0)) for the heavy-light currents (where S is a choice of renormalization scheme), the X-space renormalization factors are given by[O(α_S^2) calculations of the two-point position-space correlation function of heavy-light currents can be found in Ref <cit.>.]

    Z_(qΓ Q)^(DR,X) (t,μ) =  1 - α_S(μ)/πϵ - α_S(μ)/π( 4/3 + 2 π^2/9 + 1/2log(4 π e^γ_Eμ^2 t^2) )

in d = 4 - ϵ dimensions. Here, μ is the scale obtained after writing the renormalized coupling-constant as a dimensionless object g^(DR,) = 1/Z_gμ^-ϵ/2g^(0), where α_S := (g^(DR,))^2/4 π, and γ_E is the Euler-Mascheroni constant. It is possible to define an X-space scheme strong coupling α_S^(DR,X) (for example by fixing the two-point correlation function for the gluon correlator), but this work focuses on the matching of composite operators between X-space schemes and , and hence the α_S that is used is always renormalized in  at scale μ. The  renormalization factor 

    Z_(qΓ Q)^(DR,)(μ)= 1 - α_S(μ)/πϵ - α_S(μ)/2πlog (4 π e^-γ_E),

can be read off from <ref> as the 1/ϵ piece and the corresponding  logarithms. Comparing this to <Ref>, note that the -counterterm contains factors of +α_S γ_E/π, whereas the X-space counterterm contains corresponding factors of -α_Sγ_E/π. The difference in sign arises from the d-dimensional Fourier transform of the logarithmic structure <cit.>. This causes the conversion factor 𝒪^(DR,) = C^(DR,;DR,X)(t,μ) 𝒪^(DR,X)(t) between X-space and  to contain γ_E factors: 

    C^(DR,;DR,X)_(qΓ Q)(t,μ) = Z_(qΓ Q)^(DR,)(μ)/Z^(DR,X)_(qΓ Q)(t,μ) = 1 + α_S(μ)/π( 4/3 + 2 π^2/9 + 1/2log(e^2 γ_Eμ^2 t^2 ) ).

In numerical studies utilizing X-space schemes, μ should be varied to provide an indication of the size of the error caused by truncating the perturbative series. A natural candidate for a central value of μ is given by the condition μ^2 = e^-2 γ_E t^-2 which would cancel the logarithm that appears in the matching factor. 



For the heavy-light-light baryons, the O(α_S) conversion factors can be parametrized similarly as

    C^(DR,; DR,X)_𝒪(t,μ) = Z_𝒪^(DR,)(μ)/Z^(DR,X)_𝒪(t,μ) = 1 + α_S/π( C_1,𝒪 +  2π^2/9 + C_2,𝒪/2log ( e^2 γ_Eμ^2 t^2) ),

where the coefficients C_1,𝒪,C_2,𝒪 are given in <Ref> for 𝒪 taken from <Ref>.[Note that Z^(DR,MS)_𝒪 (which contains only the 1/ϵ piece) can be read off as Z^(DR,MS)_𝒪 = -α_S/πϵ C_2,𝒪 as the γ_E factors are directly proportional to the 1/ϵ pole.] 
The calculations of the conversion factors proceed similarly to those for the heavy-light mesonic operators, by evaluating the baryonic analogues to the diagrams appearing in Fig. <ref>. 



 §.§ Ratios of decay constants

The renormalization conditions presented in this section are not directly applicable to two-point correlation functions computed with the static heavy-quark action in Lattice-HQET as 𝒪^(DR,X)(t) ≠𝒪^(lat,X)(t). The reason for this disagreement is that the lattice regulator introduces a power-divergent mixing between the static kinetic operator Q D_0 Q and a radiatively generated mass-term m_statQ Q where m_stat∼ O(α_S)/a. The relationship between matrix elements of the operators renormalized in the two different regulators is 

    ⟨⋯ | 𝒪^(lat,X)(t) | ⋯⟩ = e^m_stat t⟨⋯ | 𝒪^(DR,X)(t) | ⋯⟩,

where | ⋯⟩ represents an arbitrary state. Without an additional renormalization condition that can be used to extract m_stat, it is not possible to match matrix elements of 𝒪^(lat,X) to matrix elements in continuum renormalization schemes. 

Since the self-energy power divergence affects all the two-point correlation functions discussed in this section in the same fashion, it is, however, possible to take ratios to cancel this self-energy divergence. 
As an example application, consider the following QCD matrix elements:

    ⟨ 0 | (q_f γ_μγ_5 b)^(DR,)  | B_f(p) ⟩   = i p_μ f_B_f^, 
    ⟨ 0 |  (ϵ^abc [q^aTτ_A C  γ_5 q^b] 1 + γ_02 b^c_α)^(DR,) | Λ_b(p,s) ⟩   = i m_Λ_b N_α(p,s) f_Λ_b,1^,

where b is a (relativistic) bottom quark field, p_μ is a Euclidean four-momentum, s is the spin of the Λ_b-baryon, and N_α is the baryon Dirac spinor satisfying N(p,s) N(p,s') = 2 m_Λ_bδ_s s'. Here, f_B_f is the mesonic decay-constant, and f_Λ_b,1 is the normalization constant for one of the distribution amplitudes of the Λ_b baryon <cit.>. The states in <Ref> have standard relativistic normalization

    ⟨B_f(p') | B_f(p) ⟩   = 2 E_B_f(p) (2 π)^3 δ^3(p⃗ - p⃗'), 
    ⟨Λ_b(p',s') | Λ_b(p,s) ⟩   = 2 δ_ss' E_Λ_b(p,s) (2 π)^3 δ^3(p⃗ - p⃗').

Note that, given the conventions of <ref>, the mass dimension of f_B_f is 1, while the mass dimension of f_Λ_b,1 is 2 (using the normalization for distribution amplitudes from Ref <cit.>). The decay constants f_B_f, f_Λ_b,1 as defined above do not have well-defined limits as m_b →∞, as the relativistic normalization of states does not behave well in this limit. The m_b →∞ limit can be studied by switching to a non-relativistic normalization for the HQET states:

    _NR⟨B_f(v,k') | B_f(v,k) ⟩_NR   = (2 π)^3 δ^3(k⃗ - k⃗'), 
    _NR⟨Λ_b(v,k',s') | Λ_b(v,k,s) ⟩_NR   = δ_ss' (2 π)^3 δ^3(k⃗ - k⃗'),

where p_μ = m_X v_μ + k_μ. The relativistically normalised and non-relativistically normalised states additionally differ by O(1/m_b) corrections, which are irrelevant in a static limit analysis. Furthermore, matching the QCD operators to HQET operators with  matching factors D(μ), and dropping the O(1/m_b) contributions gives <cit.>

    H_f^(DR,)   = (q_f γ_5 Q)^(DR,) 
    = D_H_f(μ) (q_f γ_0 γ_5 b)^(DR,),
    Λ_1,α^(DR,)   = ([q^aTτ_A C  γ_5 q^b] Q^c_αϵ_abc)^(DR,) 
    = D_Λ_1(μ) ([q^aTτ_A C  γ_5 q^b] 1 + γ_02 b^c_αϵ_abc)^(DR,).

Combining the matching of the operators and states from QCD to HQET with <Ref> gives the relationships

    i m_B_f f_B_f^stat,   = D_H_f(μ)^-1√(2 m_B_f)⟨ 0 | H_f^(DR,) | H_f(v,k) ⟩_NR, 
     
    i m_Λ_b N_α(p,s) f_Λ_b,1^stat,   = D_Λ_1(μ)^-1√(2 m_Λ_b,1)⟨ 0 | Λ_1,α^(DR,) | Λ_b(v,k,s) ⟩_NR,

where the superscript `stat' has been prepended to the label of the decay constants to emphasize that O(1/m_b) corrections have been dropped in the derivation. The matrix elements on the right-hand sides of <ref> are defined completely in the static HQET limit, and hence have no heavy-quark mass dependence. Therefore, up to logarithmic corrections due to the matching factors, the combinations f_B_f^stat,√(m_B_f) and f_Λ_b,1^stat, m_Λ_b are constant in the m_b →∞ limit. 

To extract the ratio of the static decay constants using the X-space scheme as an intermediate nonperturbative renormalization scheme, the bare two-point correlation functions are first calculated in Lattice HQET:

    T_H_f(t)    := ⟨ H_f^(lat,0)(t,0⃗) H_f^(lat,0)† (0,0⃗) ⟩ = |Z_H_f|^2 e^-E_H_f t + excited states, 
    
    T_Λ_1(t)    := ∑_α⟨Λ_1,α^(lat,0)(t,0⃗) Λ_1,α^(lat,0)† (0,0⃗)  ⟩ = |Z_Λ_1|^2 e^-E_Λ_Q,1 t + excited states,

and fitted at large Euclidean time-separations, t, to extract Z_H_f and Z_Λ_1. Note that the energies E_H_f,E_Λ_1 are the binding energies of the respective hadrons shifted by the static quark mass m_stat. By renormalizing the operators in the X-space scheme at reference scale t_0^-1 and then matching to the  scheme, an expression for the ratio of the decay constants in the  scheme can be derived as

    f_B_f^stat,/f_Λ_b,1^stat,(μ) 
    =
    ( T_H_f(t_0)/T_H_f^NI(t_0)T_Λ_1^NI(t_0)/T_Λ_1(t_0))^1/2   ×( C^DR,; DR,X_QΓ q (t_0,μ)/C^DR,; DR,X_Λ_1 (t_0,μ)) ×( D_Λ_1(t_0)/D_H_f(t_0)) 
    ×Z_H_f√(2) m_Λ_b/Z_Λ_1√(m_B_f)

where the first factor renormalizes the bare operators in the X-space scheme, the second factor converts to the  scheme in the dimensionally-regulated continuum, and the third factor matches the renormalized HQET operators to QCD operators in the  scheme. All the renormalization and matching in this expression is performed at the scale t_0^-1 for simplicity, but hybrid schemes where the renormalized operators are run before performing matching are also possible. For instance, running the operators in  from the scale t_0^-1 to the scale m_b before matching the HQET operators to QCD is usually desirable, as this cancels the large logarithms appearing if the t_0^-1 scale is very different from m_b. 




§ FOUR-QUARK OPERATORS




 §.§ Flavor non-singlet Δ Q = 0 four-quark operators



By performing an Operator Product Expansion (OPE) for the product of two weak currents and matching to HQET (a procedure known in the literature as the Heavy Quark Expansion <cit.>), the lifetimes of hadrons containing a heavy quark Q (Q = c,b) can be expressed as a sum over matrix elements of operators of increasing dimension. 
Of the various O(1/m_Q^3) corrections, the “spectator effects"  
arising for four-quark operators where a light spectator quark q_f in the hadron participates in the decay along with the heavy-quark are phase-space enhanced in the OPE by a factor of 16 π^2 <cit.>. In HQET, the operators of interest are conventionally written in the basis

    O^f_1    :=  (Qγ_μ P_L q_f)(q_f γ_μ P_L Q),  
       O^f_2    :=  (Q  P_L q_f)(q_f  P_R Q) ,
    
    O^f_3    :=  (Qγ_μ P_L T^A q_f)(q_f γ_μ P_L T^A Q),  
       O^f_4    :=  (Q  P_L T^A q_f)(q_f  P_R T^A Q),

where f ∈{u,d}, P_L = 1 - γ_5/2, P_R = 1 + γ_5/2 are the left/right projectors, and the T^A are color matrices satisfying Tr(T^A T^B) = 1/2δ^AB. Note that the flavor-singlet combinations of the operators such as O^u + O^d will mix with the lower-dimensional Q Q operator in a power-divergent way. When acting on B-hadrons, QQ is the identity operator, leading to an a^-3 additive mixing to the operators in lattice-HQET computations. This section focuses on the renormalization of the isospin-nonsinglet contributions such as O^u - O^d where the mixing with the Q Q operators cancel, and the four-quark operators are protected from power-divergent mixing. The f-label on the four-quark operators is omitted in what follows, as it should be understood that all operators refer to the isospin-nonsinglet versions. 

In dimensional regularization, four-quark operators such as those listed in <Ref> mix with evanescent operators, which are operators that formally vanish in d = 4 due to their Dirac structure. Different choices of basis for the evanescent operators lead to finite shifts in the  renormalized matrix elements <cit.>. Working in the HV scheme, the basis of evanescent operators appearing at O(α_S) chosen here is written as




    E_1    := (Qγ_μ P_L γ_αγ_β q) (qγ_βγ_αγ_μ P_L Q) - 
    4 O_1,  
    
    E_2    := (Q P_L γ_αγ_β q) (qγ_βγ_α P_R Q) - 4 O_2,  
    
    E_3    := (Qγ_μ P_L γ_αγ_β T^A q) (qγ_βγ_αγ_μ P_L T^A Q) - 4 O_3,  
    
    E_4    := (Q P_L γ_αγ_β T^A q) (qγ_βγ_α P_R T^A Q) - 4 O_4.

The Dirac structures present in these evanescent operators occur in the one-loop diagram with a gluon attached to the two light quarks, as shown in Fig. <ref>. In order for the proposed X-space scheme to be regulator-independent, evanescent contributions to operators must be subtracted for all regulators, and the renormalization conditions must be formulated for the subtracted operators <cit.>. In general, all operators in {𝒪_1,𝒪_2,𝒪_3,𝒪_4} will mix with each other, but to O(α_S) in  the operators mix in sub-bases, with {𝒪_1,𝒪_3,E_1,E_3} having the same mixing pattern as {𝒪_2,𝒪_4,E_2,E_4}. The MS-renormalization conditions (not containing the logarithmic factors included in ) for i ∈{1,2} are given by


    [   O^(0)_i; O^(0)_i+2 ] =  
    [ 1 + 2 α_Sπϵ              -3 α_S2 πϵ                       0                 α_S8 πϵ;     -α_S3πϵ             1 + α_S4 πϵ               α_S 36 πϵ              7 α_S48 πϵ ][   O^(MS)_i; O^(MS)_i+2;   E^(MS)_i; E^(MS)_i+2 ].

The first generation of bare evanescent operators {E_1,⋯,E_4} themselves mix at O(α_S) with a second generation of bare evanescent operators containing even more complicated Dirac structures (such as (Q P_L γ_α_1γ_α_2γ_α_3γ_α_4 q)(qγ_α_4γ_α_3γ_α_2γ_α_1 P_R Q)) <cit.>. Such higher-order evanescent operators are omitted in <ref> as the matching conditions presented later between  and X-space schemes are not sensitive to them at O(α_S). 

Subtracting the 1/ϵ evanescent contributions to the physical operators gives evanescent-subtracted operators 𝒪̃_i that can be used in regulator-independent schemes. By reading off the coefficients from <ref> they are defined to be, for i ∈{1,2},

    Õ_i^(0) = O_i^(0) - α_S/8 πϵ E_i+2^(0),  Õ_i+2^(0) = O_i+2^(0) - α_S/36 πϵ E_i^(0) - 7 α_S/48 πϵ E_i+2^(0).


As the four-quark operators being considered are Δ Q = 0, and the static quark can only travel in the timelike direction, an X-space scheme utilising two-point correlation functions of 𝒪̃_i (similar to that proposed in <ref> for the bilinear and trilinear operators) is not possible to define. The reason is that the corresponding two-point correlation functions are zero (due to the θ(-t_E) portion of the static heavy quark propagator shown in <Ref>). A possible way to rectify this is to compactify the time direction (for instance, in a thermal calculation), but this would likely be significantly more complicated due to the Matsubara sums required in the computation <cit.>. 

Instead, three-point correlation functions combining 𝒪̃_i with different choices of source and sink operators can be used to define an X-space renormalization scheme. Such an approach was also considered, for example, in the X-space renormalization of the QCD stress-energy tensor <cit.>. Taking ratios of three-point correlation functions to appropriate two-point correlation functions cancels the renormalization factors of the source and sink operators, provided they are multiplicatively renormalizable. In HQET, this has the added benefit of cancelling the static-quark self-energy divergence.  
Writing the renormalized Δ Q = 0 operators as 𝒪̃^(R,X)_i(t) = Z^(R,X)_ij(t) 𝒪̃^(0)_j,  
the renormalization condition is defined here by a choice of four source-operator/sink-operator combinations (labelled as J_n,K_n respectively, n ∈{1,2,3,4}) and is given by

    ⟨ J_n^†(-t,0⃗) 𝒪̃_i^(R,X)(0,0⃗) K^_n(t,0⃗)⟩/√(|⟨ J_n^†(-t,0⃗) J_n^(t,0⃗) ⟩⟨ K_n^†(-t,0⃗) K_n^(t,0⃗) ⟩|) 
    = 
    ⟨ J_n^†(-t,0⃗) 𝒪̃^(0)_i(0,0⃗) K_n(t,0⃗)⟩/√(| ⟨ J_n^†(-t,0⃗) J_n(t,0⃗)  ⟩⟨ K_n^†(-t,0⃗) K_n(t,0⃗)  ⟩|)|_NI,

(for all n; no sum over n implied) at a fixed t. In the same way as for the two-point X-space condition presented in <Ref>, additional indices on the source and sink operators should be summed over on both sides of the condition, and the open spinor index is traced over if the source and sink are baryonic. Since the source and sink operators are multiplicatively renormalizable, this causes the Z-factors of the source and sink to cancel in these ratios, so they are not labelled as bare or renormalized. Furthermore, the static-quark self-energy cancels in these ratios of correlation functions with the same physical length of the Wilson line, allowing for nonperturbative renormalization of the operators without determination of m_stat.
Defining

    T_i,n(t) := ⟨ J_n^†(-t,0⃗) 𝒪̃_i^(0)(0,0⃗) K^_n(t,0⃗)⟩/√(| ⟨  J_n^†(-t,0⃗) J_n^(t,0⃗) 
     ⟩⟨  K_n^†(-t,0⃗) K_n^(t,0⃗) 
     ⟩|) = T_i,n^(0)(t) + α_S T_i,n^(1)(t) + O(α_S^2),

where T^(0)_i,n(t) is the noninteracting value, the X-space renormalization conditions can then be solved as

    Z_ij^(X) = ∑_n T_i,n^(0)(t) T^-1_n,j(t) = 1_i,j - α_S  ∑_n T^(1)_i,n(t) (T^(0)(t))^-1_n,j + O(α_S^2),

which is well-defined as long as the four source/sink pairs, n ∈{1,2,3,4}, are chosen so that the noninteracting matrix T^(0)_i,n(t) is invertible in d = 4.  



The natural candidates for the source and sink operators are the mesonic and baryonic currents discussed in <ref>. 
The requirement that T^(0)_i,n(t) is invertible means that it is not possible to use four mesonic source/sink pairs, as, for any mesonic source/sink pair (J_M,K_M), the matrix element ⟨ J_M^† (-t,0⃗) 𝒪̃^(0)_i K_M(t,0⃗) ⟩|_NI vanishes for i ∈{3,4} due to the color trace. 
Chiral symmetry, heavy quark symmetry, and spin representations cause many source/sink choices to give vanishing matrix elements with all the operators, further restricting the number of distinct choices. 
For the remaining nonzero ratios of correlation functions, the noninteracting ratios are parametrized as

    T_i,n^(0)(t) = T_i,n^(0,0)1/π^2 t^3-ϵ( π/4 e^γ_E)^ϵ/2 + T_i,n^(0,1)ϵ/π^2 t^3-ϵ  + O(ϵ^2).

In dimensional regularization, after removing factors of δ^d-1 corresponding to the δ-function in position space from the static quark propagators, T_i,n(t) has dimension 3-ϵ, accounted for by the factor of t^-(3-ϵ) in <Ref>. The specific source/sink pairs that are studied in this section are the negative-parity heavy-light mesonic operators and the positive-parity heavy-light-light baryonic operators discussed in <Ref>. The values of the decomposition for the noninteracting ratio for these source/sink operators (J,K) are tabulated in <Ref>. Every choice of four linearly independent source/sink operators from this list (there are 28 different choices in total) defines a different X-space scheme. In a similar way to the parametrization of the noninteracting contribution to the ratio of correlation functions in <Ref>, the O(α_S)-contribution to the ratios are parametrized by

    T_i,n^(1) =  T_i,n^(1,0)1/ϵπ^3 t^3-2ϵ(π/4 e^γ_Eμ)^ϵ + T_i,n^(1,1)1 /π^3 t^3-2ϵ + T_i,n^(1,2)1/π t^3-2ϵ + O (ϵ).

For the various source/sink pairs, the O(α_S) matrix elements for the ratio have been calculated and are tabulated in <Ref>. 



Choosing the specific source/sink pairs n ∈{1,2,3,4} from the list of source/sink pairs presented in <Ref>, the X-space renormalization matrix can be calculated using <Ref> as

    Z^(X)_ij,n ∈{1,2,3,4} = 1 - 
           α_S(μ) (π/4 e^γ_E t^2 μ^2)^ϵ/2·[        2/πϵ + 24 + 4 π^2/9 π                            0   -3/2 πϵ + -9 + 24 π^2/24 π                       -5/4 π;                            0        2/πϵ + 24 + 4 π^2/9 π                      -1/16 π  -3/2 πϵ + -63 + 72 π^2/72 π;   -1/3 πϵ + -15 + 4 π^2/54 π                       -1/9 π   1/4 πϵ + 90 + 56 π^2/144 π                      -1/12 π;                      -1/36 π   -1/3 πϵ + -15 + 4 π^2/54 π                      -3/16 π 1/4 πϵ + 414 + 168 π^2/432 π ],

where 1 is the 4 × 4 identity matrix. As a check, it can be verified that the 1/ϵ divergent pieces match up with the MS counterterms presented in <Ref>. The corresponding conversion factor from the X-space scheme to  is given by

    C^(;X)_ij,n ∈{1,2,3,4} := ∑_k Z^()_ik (Z^(X))^-1_kj,n ∈{1,2,3,4} = 1 + 
           α_S(μ)(
    [      log (β )/π+4 π/9+8/3 π                           0    -3 log (β )/4 π+π -3/8 π                      -5/4 π;                           0      log (β )/π+4 π/9+8/3 π                     -1/16 π    -3 log (β )/4 π+π -7/8 π; -log (β )/6 π+2 π/27-5/18 π                      -1/9 π   log (β )/8 π+7 π/18+5/8 π                     -1/12 π;                     -1/36 π -log (β )/6 π+2 π/27-5/18 π                     -3/16 π log (β )/8 π+7 π/18+23/24 π;                             ]),

where β := e^2 γ_Eμ^2 t^2/16. A natural choice for μ is μ^2 = 16e^-2γ_E/t^2 which would cancel the factors of log(β) appearing in the matching coefficient. In principle, when converting matrix elements computed with lattice HQET to -renormalized matrix elements, varying over the different choices of source/sink pairs, as well as varying over the scale t that the X-space scheme is defined at before perturbatively running to a common scale, will give an indication on the error due to O(α_S^2) terms that have been neglected in this study. 




 §.§ Δ Q = 2 four-quark operators


In the Standard Model, neutral B-mesons (B^0, B^0_s) mix with their own antiparticles. The lowest-order diagram contributing to this in the Standard Model is a box diagram with the exchange of two weak bosons, which after integrating out physics at and above the weak scale leads to Δ B = 2 four-quark operators. When matching these operators to HQET, the Lagrangian is expanded to include a static antiquark Q_- that travels in the opposite direction to the static quark Q_+ (compare to the Q_+-propagator <Ref>):

    ⟨ Q_-^(0)(0) Q_-^(0)(x_E) ⟩_F = δ_x⃗_E,0⃗ θ(t_E)  W^(0)(0,x_E) 1 - γ_0/2,

where 1 ±γ_0/2 Q_± = Q_±. A basis of the relevant Δ Q = 2 operators in HQET is

    O_1 := (Q_+  P_L q) (Q_-  P_L q), 
       O_2 := (Q_+ P_L T^A q) (Q_- P_L T^A q), 
       O_3 := (Q_+ P_L q)(Q_- P_R q),
       O_4 := (Q_+ P_L T^A q) (Q_- P_R T^A q).

The operators O_1,O_2 contribute to neutral B-meson mixing in the Standard Model, and O_3,O_4 are contributions that could arise from possible new physics <cit.>. The full basis of Δ Q = 2 operators also contains {O_1,O_2} with left-handed projectors swapped with right-handed projectors (P_L ↔ P_R), but these are related by parity and so renormalize with the same factors as {O_1,O_2}. There are also additional Δ Q = 2 operators when Q is relativistic, but these are related to the operators in <Ref> in the static quark limit <cit.>:

    (Q_+ γ_μ P_L q) (Q_- γ_μ P_L q) = 8/3 O_1 + 4 O_2     in   d = 4.

The evanescent operators are defined as

    E_1    := (Q_+ P_L γ_αγ_β q)(Q_- P_L γ_αγ_β q) - 32/3 O_1 - 16 O_2, 
      
    E_2    := (Q_+ P_L T^A γ_αγ_β) (Q_- P_L T^A γ_αγ_β q) - 32/9 O_1 - 16/3 O_2 
    
    E_3    := (Q_+ P_L γ_αγ_β q)(Q_- P_R γ_αγ_β q) - 4 O_3, 
     
    E_4    := (Q_+ P_L T^A γ_αγ_β) (Q_- P_R T^A γ_αγ_β q) - 4 O_4,

where the γ-matrix structure is governed by the O(α_S) diagram with a single gluon attached to the two light quarks. The mixing pattern is given by the following MS renormalization:

    [ O_1^(0); O_2^(0) ] = 
    [ 1 + 14 α_S9 πϵ      4 α_S3 πϵ              0       -α_S8 πϵ;     8 α_S27 πϵ 1 + 10 α_S9 πϵ      -α_S36 πϵ       α_S24 πϵ ][ O_1^MS; O_2^MS; E_1^MS; E_2^MS ],


    [ O_3^(0); O_4^(0) ] = 
    [   1 + 2 α_Sπϵ     3 α_S2 πϵ             0      -α_S8 πϵ;       α_S3 πϵ 1 + 3 α_S2 πϵ     -α_S36 πϵ      α_S24 πϵ ][ O_3^MS; O_4^MS; E_3^MS; E_4^MS ].

Correspondingly, the evanescent-subtracted operators are defined for i ∈{1,3} as

    Õ_i^(0) = O_i^(0) + α_S/8 πϵ E_i+1^(0),   Õ_i+1^(0) = Õ_i+1^(0) + α_S/36 πϵ E_i^(0) - α_S/24 πϵ E_i+1^(0)


Baryonic heavy-light-light currents cannot be used as source/sink pairs for the Δ B = 2 four-quark operators, as the corresponding three-point functions all vanish. Fortunately, enough constraints can be derived with the mesonic heavy-light currents as source/sink pairs to constitute a valid X-space scheme. Although chiral symmetry is formally broken by the HV γ_5 scheme, the massless nature of the light quarks causes the operators to mix in the 2 × 2 subblocks presented in <Ref>. Thus, only two source/sink pairs are needed in the X-space scheme. Using the choice (Qγ_5 q, qγ_5 Q) and (Qγ_i γ_5 q, qγ_i γ_5 Q), the O(α_S) matching matrices are found to be

    C^(,X)_{O_1,O_2} = 1 + α_S/π(
    [   7 log (β)9+4 π ^29+239                                2 log(β)3-π ^23+43; 4 log (β)27-2 π ^227+827                          5 log (β) 9 +5 π ^29+199;                          ]),



    C^(,X)_{O_3,O_4} = 1 + α_S/π(
    [   log(β) +4 π ^29+259                           3log(β)4-π ^23+76;  log(β)6-2 π ^227+727                       3log(β)4+5 π ^29+4318;                       ]),

where β := e^2 γ_Eμ^2 t^2/16. 



§ CONCLUSION


In this work, a set of X-space renormalization schemes for isospin-nonsinglet Δ Q = 0 and Δ Q = 2 four-quark HQET operators have been proposed, and the O(α_S) matching coefficients between these schemes and  in the dimensionally regulated continuum have been calculated. This allows for a gauge-invariant, nonperturbative renormalization matrix elements calculated in lattice HQET, without the need to extract the power-divergent self-energy contribution m_stat. 
Precise computations of these matrix elements with lattice HQET will reduce theory uncertainties on lifetimes of heavy hadrons, and help constrain physics beyond the Standard Model. 
Note that, when implementing the X-space scheme for these four-quark operators in lattice HQET, it is convenient to use Ginsparg-Wilson discretizations of the light quarks (e.g., with the domain-wall fermion action) to avoid additional mixing between the four-quark operators of interest and operators in other chiral representations.   

Next-to-next to leading order calculations of the matching coefficients presented in <Ref> at O(α_S^2) are possible, but the computation is complicated by the fact that, unlike p-type integrals that only have one external scale, the perturbative calculations shown in <Ref> have two external scales x_src,x_snk corresponding to the source and sink locations of the three-point renormalization scheme proposed. Corrections due to finite light-quark masses are more easily calculable (though they are likely smaller than the O(α_S^2) corrections on typical lattice-QCD ensembles), and require computation of the three-loop integrals in <Ref> either analytically in the light-quark mass, or by expanding in powers of the light quark mass. O(1/m_Q) corrections are in principle also calculable, but require considering mixing of the four-quark operators with dimension-7 operators that contain an additional covariant derivative, as well as considering the O(1/m_b) corrections to the static HQET lagrangian. 

The calculations and techniques used in this work can be readily applied to X-space schemes for other classes of operators. For instance, using the auxiliary-field formalism, nonlocal operators such as q(x) W(x,y) q(y), where W(x,y) is a Wilson line, are transformed into products of local operators qQ_y - x(x) Q_y-x q(y), which can be renormalized by the techniques presented in <Ref>. The calculations of the three-loop diagrams involving gluons attached to the light-quark propagators presented in <Ref> can also be applied to renormalize massless four-quark operators such as the Δ S = 1 four-quark operators relevant for kaon decays. 



The authors gratefully acknowledge useful discussions with Artur Avkhadiev and Aditya Pathak. S.M. is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics under Award Number DE-SC0009913.
W.D. and J.L. are supported in part by the
U.S. Department of Energy, Office of Science under grant Contract Number DE-SC0011090 and by the SciDAC5 award DE-SC0023116 and are additionally supported by the National Science Foundation under Cooperative Agreement PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, http://iaifi.org/). 







§ INTEGRALS AND CONVENTIONS




 §.§ Dimensional regularization and charge conjugation


The calculations in this work use dimensionally regulated integrals ∫d^4 k/(2 π)^4→μ^4-d∫d^d k/(2 π)^d where d = 4 - ϵ. 
γ_5 is treated as in the 't Hooft-Veltman scheme <cit.> (γ_5 := i γ_0 γ_1 γ_2 γ_3 in Minkowski signature). 
Charge-conjugation matrices C are used in the construction of the baryonic operators such as ϵ^abc [q^aT C Γ q^b] Q^c (where Γ is a Dirac matrix), and it is natural to assume that the defining relation C γ_μ C^-1 = -γ_μ^T holds in dimensional regularization.
However, to our knowledge, an explicit charge-conjugation matrix satisfying the defining relations for an explicit basis of infinite-dimensional γ-matrices has not been constructed previously in the literature for dimensional regularization (a construction was presented for dimensional reduction in Ref <cit.>). 
The explicit construction shows that enforcing C γ_μ C^-1 = -γ_μ^T does not lead to inconsistencies in Dirac traces, unlike how naively enforcing the anticommuting relation {γ_5,γ_μ} = 0 leads to inconsistencies in certain Dirac traces. 

In what follows, a construction of the d-dimensional gamma matrices as well as an explicit charge-conjugation matrix C satisfying C γ_μ C^-1 = γ_μ^T are presented.
The Minkowski-signature gamma matrices are defined inductively following the algorithm of Ref <cit.> (up to a trivial reordering that enforces (γ^μ)^T = (-1)^μγ^μ):




  1) Set

    γ^0_(1) := [  1  0;  0 -1 ],   γ^1_(1) := [  0  1; -1  0 ].


  2) For ω∈ℤ_≥ 1, define

    γ̂_(ω) := i^ω - 1γ^0_(ω)⋯γ^2 ω - 1_(ω),


    γ^μ_(ω + 1) := [ γ^μ_(ω)       0;       0 γ^μ_(ω) ]  for   0 ≤μ < 2 ω,


    γ^2 ω_(ω + 1) := [        0 i γ̂_(ω); i γ̂_(ω)        0 ] ,   γ^2 ω + 1_(ω + 1) := [        0     γ̂_ω; - γ̂_(ω)        0 ].


  3) The infinite-dimensional γ-matrices are defined by block-diagonal copies of the finite-dimensional construction, so that for μ∈ℤ_≥ 0, choosing any ω≥⌊μ/2⌋ + 1,

    γ^μ := [ γ^μ_(ω)                ;         γ^μ_(ω)        ;                       ⋱ ].


The γ matrices defined here satisfy

    {γ_μ , γ_ν} = 2 g_μν,   γ_μ^† = γ_0 γ_μγ_0,

where the metric is written in the mostly-negative convention, g_μν = diag(+1,-1,-1,⋯). 
With this construction, no finite product of γ matrices will satisfy the charge-conjugation matrix condition C γ_μ C^-1 = -γ_μ^T. Modifying the basis of γ-matrices by eliminating γ_4,γ_6,γ_8… from the basis, such that the new basis γ_μ is given by the relabelling

    {γ_0,γ_1,γ_2,γ_3,γ_4,γ_5,γ_6,⋯} = {γ_0,γ_1,γ_2,γ_3,γ_5,γ_7,γ_9,⋯},

a charge-conjugation matrix can be defined. These γ matrices still satisfy <ref>, but now we can define C = i γ_0 γ_2, which satisfies

    C γ_μ^T C^-1 = -γ_μ^T,    C γ_5 C^-1 = γ_5,    C^-1 = C^T = C^† = -C.

The trace is normalized such that Tr(1) = 4. Euclidean γ-matrices are obtained by defining
γ_0^E := γ_0, γ_i^E := -i γ_i such that {γ_μ^E, γ_ν^E} = 2 δ_μν. The `E' labels are dropped from all Euclidean γ-matrices in the main text as all calculations are presented in Euclidean space. 




 §.§ Integrals






Calculations were performed in Mathematica, with the aid of the Tracer package <cit.> for Dirac traces in the `t Hooft-Veltman scheme, and HypExp <cit.> for expansions of hypergeometric functions. Because three-quark baryonic sources were used in defining the X-space schemes (which only exist in N_c = 3, and introduce an ϵ^abc tensor), color traces are explicitly evaluated without attempting to write in terms of N_c. Color matrices are normalised so that Tr(T^A T^B) = 1/2δ^AB. 

The two-point correlation functions required for renormalizing the heavy-light and heavy-light-light operators from <Ref> can be computed first in momentum space and then by taking a Fourier transform, as shown in Fig. <ref>. As such, these can be evaluated by using `p-type' integrals <cit.>. In the chiral limit for the light quark, there is only a single dimensionful scale p in the momentum-space integral corresponding to the momenta running through the diagram as seen in Fig. <ref>. Hence, integration-by-parts relations based on the identity ∫d^d k/(2 π)^d∂/∂ k_μ f(k) = 0 are relatively easy to derive. 

For three-point position-space functions, at O(α_S), the diagrams can all be split up into a number of component pieces. For example, one diagram topology that appears in the O(α_S) contribution to the three-point mesonic function ⟨QΓ^† q (-t_0,0⃗) ·  (QΓ_L q)(qΓ_R Q) (0,0⃗) ·qΓ Q (t_0,0⃗) ⟩ can be written as




    < g r a p h i c s >




Here, α_i, β_i, ρ_i, δ_i are Dirac-color indices. As well as the position-space propagators and self-energy diagrams, there are three O(α_S) diagrams to compute, corresponding to a gluon attaching to two heavy-quark propagators, a gluon attaching on one end to a heavy-quark propagator and on the other end to a light-quark propagator, and a gluon attaching to two light-quark propagators. In the case of the Δ Q = 0 four-quark operators, the diagram with a gluon attaching to two heavy-quark propagators can be directly calculated in Minkowski space as follows:


    < g r a p h i c s >


    = [ (i g v_μ T_A) ]_α_1 β_1[ (ig v_μ T_A) ]_ρ_1 δ_1·∫_-t_0^0 ds_L ∫_0^t_0 ds_R ∫d^d k/(2 π)^d-i e^i s_L k - i s_R k/k^2
        = [v T_A ]_α_1 β_1 [v T_A ]_ρ_1 δ_1( -α_S/πϵ - α_S/2 π(2 + log(- 1/4 e^γ_Eπμ^2 t_0^2) ) ),

where the static nature of the heavy quark has been utilized to integrate the vertex insertions at s_L and s_R along the line connecting the three operators. For the two diagrams involviong gluons attaching to light-quarks, additional master integrals are required. Relabelled from the basis from Appendix A of <cit.>, and in Minkowski space, there is a Tripod diagram T and a Wedge diagram W:




    < g r a p h i c s >





    T(x_L, x_R; n_1,n_2,n_3) := ∫d^d p_L d^d p_R/(2 π)^2de^i p_L x_L e^- i p_R x_R/(-p_L^2)^n_1 (-(p_L - p_R)^2)^n_2 (-p_R^2)^n_3,


    W(x_L,x_R; n_1,n_2,n_3,n_4,n_5,n_6) := 
         ∫d^d p_L d^d p_R d^d k/(2 π)^3de^i p_L x_L e^- i p_R x_R/(-p_L^2)^n_1 (-(p_L - k)^2)^n_2 (-(p_R - k)^2)^n_3 (-p_R^2)^n_4 (-k^2)^n_5(-(p_L-p_R)^2)^n_6.

The W master integral can be reduced to a few base cases by use of integration-by-parts relations (derived from inserting ∂_k · k in front of the integrand of <ref>):

    W(x_L,x_R; n⃗) = n_2 2^+ (5^- - 1^-) + n_3 3^+ (5^- - 4^-)/d - n_2 - n_3 - 2n_5 W(x_L,x_R; n⃗),

where n⃗ = (n_1,⋯,n_6), and m^± W(x_L,x_R,n⃗) = W(x_L,x_R,n⃗') with n⃗' = n⃗ for all components except the m-th component, n'_m = n_m ± 1 (this is the notation used in Ref <cit.>). <Ref> reduces the W master integral to base cases where either n_2,n_3 or n_5 equals zero in the argument of W. In these cases, the integral reduces to a p-type integral and the T master integral, which can be performed explicitly using Schwinger parameters:

    T(x_L,x_R;n_1,n_2,n_3) = -Γ(d/2-n_1) Γ(d - n_1 - n_2 - n_3)/Γ(n_2) Γ(n_3) Γ(d/2) 4^n_1 + n_2 + n_3π^d (-x_R^2)^-d + n_1 + n_2 + n_3
       ∫_0^1 dx (1-x_1)^-d/2 + n_1 + n_2 - 1 x_1^-d/2 + n_1 + n_3 - 1_2F_1(d/2 - n_1, d - n_1 - n_2 - n_3, d/2, - (x_L - x_1 x_R)^2/x_1 (1 - x_1) x_R^2).

Explicit evaluations of T can be performed at relevant values of n_1,n_2 and n_3. Diagrams with a gluon attaching to a heavy-quark propagator on one end and to a light-quark propagator on the other end can be calculated in Minkowski space, for instance:


    < g r a p h i c s >


    = ∫d^d p_L d^d p_R/(2π)^2d∫_-t_0^0 ds_L [i/p_R (i g γ_μ T_A) i/p_R - p_L]_α_1 β_1[1 + v/2(i g v_μ T^A) ]_ρ_1 δ_1-i e^i p_L^0 s_L - i p_R^0 t_0/p_L^2
       = -i g^2 ∫_-t_0^0 ds_L  [ γ_αvγ_β T^A ]_α_1 β_1[ 1 + v/2 T^A]_ρ_1 δ_1∂/∂ x_R^α(∂/∂ x_R^β + ∂/∂ x_L^β) T(x_L,x_R; 1,1,1) |_x_L → (s_L,0⃗) 
     x_R → (t_0, 0⃗)

Finally, the diagram with a gluon attaching to two light lines can be reduced to the W master integral



    [valign=c]figs/fig9/tikz9.pdf = -i g^2 μ^4-d (γ^αγ^μγ^β T^a)_αβ (γ^ργ^μγ^δ T^a)_ρδ
                 ×∫d^d p_L d^d p_R d^d k/(2 π)^3de^i p_L x_L - i p_R x_R p_R^α (p_R - k)^β (p_L - k)^ρ p_L^δ/(-p_L^2)(-(p_L-k)^2)(-(p_R-k)^2)(-p_R^2)(-k^2),

where the factors of p_L, p_R in the numerator can be handled by differentiating with respect to x_L, x_R. Calculating <ref> at the relevant values of x_L and x_R is the main computation involved in calculating the O(α_S) contribution to the ratios of three-point correlation functions to two-point correlation functions presented in <ref>. 





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