Towards a Next-to-Next-to-Leading Order analysis of matching in $B^0$-$\bar{B}^0$ mixing

TL;DR

This paper calculates advanced two-loop perturbative corrections to the matching coefficients between QCD and HQET for the $B^0-ar{B}^0$ mixing, improving precision in theoretical predictions.

Contribution

It provides the first analytical two-loop correction results for the matching coefficients relevant to $B^0-ar{B}^0$ mixing in the QCD-HQET framework.

Findings

Analytical two-loop corrections at order α_s^2 obtained.

Clarified operator basis choice in HQET.

Enhanced precision in $B^0-ar{B}^0$ mixing calculations.

Abstract

We compute perturbative corrections to matching coefficients of QCD to HQET for the matrix element of the $\Delta B=2$ operator that determines the mass difference in $B^0-\bar{B}^0$ system of states. This involves the technical point of choosing the operator basis in HQET, separating physical operators from evanescent ones. We obtain an analytical result for some of the two-loop corrections at order $\alpha_s^2$.

Equations

$$\mathcal{H}=C(M_W,m_t,\mu ,{\alpha }_s(\mu ))Q(\mu )$$

$$Q(\mu )=({\bar{q}}_L{\gamma }_{\alpha }b)({\bar{q}}_L{\gamma }_{\alpha }b)(\mu )$$

$$\mathcal{H}=C(M_W,m_t,\mu ,{\alpha }_s(\mu ))Q(\mu )+\sum _i{C}_{E_i}{E}_i(\mu ).$$

$$⟨E_i(\mu )⟩=0.$$

$$\displaystyle\left(\begin{array}[]{c}Q(\mu^{\prime})\\ E(\mu^{\prime})\end{array}\right)=\left(\begin{array}[]{cc}Z_{Q}(\mu^{\prime},\mu)&Z_{QE}(\mu^{\prime},\mu)\\ Z_{EQ}(\mu^{\prime},\mu)&Z_{EE}(\mu^{\prime},\mu)\end{array}\right)\left(\begin{array}[]{c}Q(\mu)\\ E(\mu)\end{array}\right)$$

$$Q({\mu }^{\prime })=Z_Q({\mu }^{\prime },\mu )Q(\mu )+Z_{QE}({\mu }^{\prime },\mu )E(\mu ).$$

$$\left(\begin{array}[]{c}\mathrm{Q}\\ \mathrm{E}\end{array}\right)=Z\left(\begin{array}[]{c}Q(\mu)\\ E(\mu)\end{array}\right)\,.$$

$$Z=1+\frac{\alpha_{s}}{4\pi\varepsilon}\left(\begin{array}[]{cc}z_{QQ}&z_{QE}\\ \epsilon z_{EQ}&z_{EE}\end{array}\right)\,,$$

$$\Gamma \otimes \Gamma \equiv {\bar{q}}_L\Gamma b{\bar{q}}_L\Gamma b.$$

$${\gamma }_{\mu }{\gamma }_{\nu }{\gamma }_{\alpha }=g_{\mu \nu }{\gamma }_{\alpha }+g_{\nu \alpha }{\gamma }_{\mu }-g_{\mu \alpha }{\gamma }_{\nu }+i{ϵ}_{\mu \nu \alpha \beta }{\gamma }_{\beta }{\gamma }_5$$

$${\gamma }_{\mu }{\gamma }_{\nu }{\gamma }_{\alpha }\otimes {\gamma }_{\mu }{\gamma }_{\nu }{\gamma }_{\alpha }\to 16{\gamma }_{\alpha }\otimes {\gamma }_{\alpha },$$

$${\gamma }_{\mu }{\gamma }_{\nu }{\gamma }_{\alpha }\otimes {\gamma }_{\mu }{\gamma }_{\nu }{\gamma }_{\alpha }\to 16{\gamma }_{\alpha }\otimes {\gamma }_{\alpha }+E^{\mathrm{evan}}$$

$${\gamma }_{\mu }{\gamma }_{\nu }{\gamma }_{\alpha }\otimes {\gamma }_{\mu }{\gamma }_{\nu }{\gamma }_{\alpha }\to (16-4ϵ){\gamma }_{\alpha }\otimes {\gamma }_{\alpha }+E^{QCD}$$

$$E^{QCD}={\gamma }_{\mu }{\gamma }_{\nu }{\gamma }_{\alpha }\otimes {\gamma }_{\mu }{\gamma }_{\nu }{\gamma }_{\alpha }-(16-4ϵ){\gamma }_{\alpha }\otimes {\gamma }_{\alpha }.$$

$$({\bar{q}}_L{t}^a{\Gamma }^{[3]}b)({\bar{q}}_L{t}^a{\Gamma }^{[3]}b)$$

$$E^{QCD}$$

$$Q^{\prime }(\mu )=\left[1-a{z}_{QE}\frac{{\alpha }_s(\mu )}{4\pi }\right]Q(\mu ).$$

$$Q(\mu )=2\sum _{i=l,s}{C}_i(\mu )O_i(\mu )+\mathcal{O}\left(\frac{{\Lambda }_{\text{QCD}}}{m_b}\right)$$

$$O_l=({\bar{q}}_L{\gamma }_{\mu }{h}_{+})({\bar{q}}_L{\gamma }_{\mu }{h}_{-}),O_s=({\bar{q}}_L{h}_{+})({\bar{q}}_L{h}_{-}).$$

$$O_n=(\bar{q}{\gamma }_{\perp }^{[n]}{h}_{+})(\bar{q}{\gamma }_{\perp }^{[n]}{h}_{-}),O_n^{\prime }=({\bar{q}}_i{\gamma }_{\perp }^{[n]}{h}_{+}^j)({\bar{q}}_j{\gamma }_{\perp }^{[n]}{h}_{-}^i),$$

$${\gamma }_{\perp }^{\mu }={\gamma }^{\mu }-v^{\mu }/v.$$

$${\Gamma }^{[n]}\otimes {\Gamma }^{[n]}$$

$$O_2$$

$$O_0^{\prime }$$

$$O_2^{\prime }$$

$$e_1$$

$$e_3$$

$$e_5$$

$$O_l={\Gamma }^{[1]}\otimes {\Gamma }^{[1]}=O_1-O_0,O_s={\Gamma }^{[0]}\otimes {\Gamma }^{[0]}=1\otimes 1=O_0.$$

$${\gamma }_{\mu }{\gamma }_{\nu }{\gamma }_{\alpha }\otimes {\gamma }_{\mu }{\gamma }_{\nu }{\gamma }_{\alpha }$$

$$=$$

$$O_2$$

$$O_3$$

Full text

SI-HEP-2017-13

QFET-2017-10

**Towards a Next-to-Next-to-Leading Order analysis

of matching in $B^{0}$–$\bar{B}^{0}$ mixing **

Andrey G. Grozin

Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russia and

Novosibirsk State University, Novosibirsk, 630090, Russia

Thomas Mannel and Alexei A. Pivovarov

Theoretische Elementarteilchenphysik, Naturwiss.–techn. Fakultät,

Universität Siegen, 57068 Siegen, Germany

We compute perturbative corrections to matching coefficients of QCD to heavy quark effective theory (HQET) for the matrix element of the $\Delta B=2$ operator that determines the mass difference in $B^{0}-\bar{B}^{0}$ system of states. This involves the technical point of choosing the operator basis in HQET, separating physical operators from evanescent ones. We obtain an analytical result for some of the two-loop corrections at order $\alpha_{s}^{2}$.

PACS: 12.38.Bx, 12.38.Lg, 12.39.Hg, 14.40.Nd

1 Introduction

Theoretical predictions within the Standard Model (SM) of particle physics for the oscillation frequency and the lifetime differences in the systems of neutral bottom mesons have become much more accurate during the last decade [1, 2, 3]. This progress is pretty much related to the numerical computation of the relevant hadronic matrix elements using lattice simulations of QCD (e.g. [4, 5]). $B^{0}$-$\bar{B}^{0}$ oscillations are flavor-changing neutral current processes and hence are sensitive to possible effects beyond the SM, so our ability to perform precise calculations is of prime importance [6, 7]. In fact, the high precision of experimental data provides us with good opportunities for searches of physics beyond the SM [8], in particular in the mixing of states in the systems of neutral flavored mesons. The mixing in the neutral kaon and the charmed-meson systems is strongly long-distance sensitive and thus depends on nonperturbative QCD effects. This is in contrast to the oscillation frequency in the neutral $B$ mesons which is dominated by the top-quark contribution and hence it is short distance dominated even in the SM, thereby making a clean SM prediction possible.

Highly precise predictions for observables in the $B$-meson system are based on the method of effective field theory (EFT), which allows us to separate vastly different mass scales and to perform an expansion in the small ratios of these scales [9, 10, 11]. This involves, on the one hand, the use of EFT for weak interactions as an expansion in $m_{b}/M_{W/Z}$ and $m_{b}/m_{t}$, and, on the other hand, the use of heavy quark effective theory (HQET) as an expansion in $\Lambda_{\rm QCD}/m_{b}$. However, as soon as the scales become so low that a perturbative evaluation is impossible, nonperturbative input becomes necessary in form of hadronic matrix elements. To this end, observables in the $B$-meson system can be generically written as perturbatively computable Wilson coefficients and low-energy hadronic matrix elements [12, 13]. While the Wilson coefficients have rather accurate theoretical predictions within perturbation theory [14] the recent progress in lattice produces the comparable accuracy of computation for matrix elements.

However, the data to be expected in the near future will require to further improve the accuracy. This eventually requires even more precise values of Wilson coefficients as well as improved lattice calculations. Clearly, one of the most crucial points of such an improvement are the perturbative corrections at the lowest scales involved in the analysis. As the first step of this program one has to compute the next-to-next-to leading order (NNLO) corrections to the matching of QCD to heavy quark effective theory (HQET) that is applicable at the scales of order few $\Lambda_{\text{QCD}}$. In the present paper we discuss the choice of the operator basis within dimensional regularization and give explicitly some of the NNLO contributions.

The paper is organized as follows. We give a short necessary description of the procedure of computation of the effective Hamiltonian at the bottom quark mass scale $m_{b}$ and definition of the operator basis for four-quark operators with introduction of evanescent operators in QCD. Then we define the operator basis in HQET and discuss the correspondence of the two bases. With two bases at hand we consider a simple computation of matching coefficients with fermion bubbles to set the stage for the full computation at NNLO. In Sect. 4 we consider the contribution of evanescent operators to the correlator for the sum rules analysis in HQET. In Sect. 5 we discuss the bag parameter for $B_{s}$ meson within unitary symmetry with expansion in the strange quark mass. Sect. 6 contains our conclusions.

2 Wilson coefficients and operator

basis

In the SM the amplitude for $B^{0}$-$\bar{B}^{0}$ transitions vanishes at tree level. The leading contribution are the box diagrams appearing at the one-loop level. The dominant part is induced by the loop diagram involving the top quark and the $W$ boson. On the other hand, the relevant scale for this process is much lower, of the order of the $b$-quark mass. Switching to an EFT for weak interactions allows us to integrate out the top quark and the $W$ boson, leaving us with local four-quark operators.

The most important corrections to the leading term are the contributions of strong interactions. They can be computed within QCD perturbatively as the relevant scale is of the order of the meson mass $m_{B}$ and is much larger than QCD infrared scale $\Lambda_{\text{QCD}}$. Finally, the matrix element of the four-quark operator still contains the scale $m_{b}$, which can be treated in QCD perturbation theory. By switching from QCD to HQET we may extract also these (perturbative) contributions, leaving us with matrix elements of static HQET fields.

2.1 QCD: From $M_{W}$ to $m_{b}$

The scales between $M_{W}$, $m_{t}$ and $m_{b}$ are in a perturbative domain for strong interactions. After integrating the large scales $M_{W}$, $m_{t}$ out perturbatively, the process of $B^{0}$–$\bar{B}^{0}$ is described by an effective Hamiltonian of the form

[TABLE]

with

[TABLE]

being a local operator with $\Delta B=2$ renormalized at the scale $\mu$. The amplitude for $B^{0}$-$\bar{B}^{0}$ mixing is obtained by computing a matrix element of (2.1) between $B^{0}$ and $\bar{B}^{0}$ states. Choosing the scale $\mu\sim m_{b}$ as the renormalization scale will avoid the appearance of the large scales $M_{W}$ and $m_{t}$ in the matrix elements; these dependencies reappear in the Wilson coefficient $C(m_{W},m_{t},\mu,\alpha_{s}(\mu))$ that accounts for the effects of the evolution between $m_{t}$, $M_{W}$ and $\mu\sim m_{b}$, or just resums those large logarithms of the scales ratio, $\ln(M_{W}/m_{b})$. It is a standard tool to use the renormalization group (RG) to evolve the scale $\mu$ from $m_{t}$, $M_{W}$ to $m_{b}$, thereby resumming large logarithms of the scales ratio, $\ln(M_{W}/m_{b})$. The RG technique requires the calculation of the $\beta$ function governing the running of $\alpha_{s}$ and the anomalous dimensions $\gamma$, which can be done in QCD perturbation theory. Using the one-loop results for $\beta$ and $\gamma$ in combination with tree-level matching at $\mu=M_{W}$ yields the leading order result, while the input of $\beta$ and $\gamma$ at two loops together with the one-loop matching yields the result at next-to-leading order (NLO).

At present the coefficient $C(M_{W},m_{t},\mu,\alpha_{s}(\mu))$ in (2.1) is known at the NLO of the expansion in the strong coupling constant that gives the accuracy of a few percent [12, 13, 14]. The RG function $\beta$ and $\gamma$ are related to divergences in Feynman diagrams, which need to be regularized. Technically, the best way of regularization in perturbative multiloop calculations is dimensional regularization (DimReg) where the Feynman diagrams are computed in $d$ dimensions. However, the mixing process in the SM contains left-handed quark fields and DimReg has no unique or even simple satisfactory way of treating the Dirac matrix $\gamma_{5}$. This problem is vital for supersymmetric theories and some solutions have been suggested. Dimensional reduction allows for the computation of corrections to $\Delta F=1$ Hamiltonian [15]. There are also techniques by ’t Hooft and Veltman (HV) [16] and a naive dimensional regularization prescription (NDR) [17]. For four-quark operators one can use techniques of refs. [18, 19, 20] that are a practical way out through an extension of the operator basis by using so-called evanescent operators. Note that the two-loop anomalous dimensions of baryon operators, or three-quark operators, have been computed earlier within a similar approach [21, 22]. A clear presentation of the techniques is given in [23].

In an extended operator basis with evanescent operators $E_{i}(\mu)$ the effective Hamiltonian is written as

[TABLE]

The number of evanescent operators and their structures depend on the order of perturbation theory to which the Hamiltonian is computed. The appearance of evanescent operators is an artifact of using dimensional regularization and these operators play an auxiliary role. The generic feature of evanescent operators is that they are, in a sense, equivalent to zero after going to four-dimensional space-time, $d\to 4$. More precisely, these operators are defined such that their Green functions with fundamental quark-gluon fields vanish in perturbation theory for $d=4$

[TABLE]

Here the brackets mean to compute any Green function of the operators $E_{i}(\mu)$ and renormalized quark and gluon fields within perturbation theory (PT) in an infrared safe kinematical point. In practice, one computes an amputated connected Green’s function of the operators $E_{i}(\mu)$ with four fermionic fields that is sufficient for extracting the UV properties of the operator basis in eq. (2.3). However, the coefficient $C(M_{W},m_{t},\mu,\alpha_{s}(\mu))$ of the physical operator $Q(\mu)$ and also the very definition of $Q(\mu)$ itself depends on the choice of the whole basis $\{Q(\mu),E_{i}(\mu)\}$, and, therefore, the concrete set of evanescent operators $E_{i}(\mu)$. Physical operator(s) $Q(\mu)$ mix with the evanescent operators $E_{i}(\mu)$ under renormalization. For finite renormalization it means that with the change of the renormalization parameter one has

[TABLE]

and for the physical sector

[TABLE]

Here both $Q$ and $E$ can be columns of operators and any $Z$ is a matrix. In terms of bare operators $(\mathrm{Q},\mathrm{E})$ the renormalization reads

[TABLE]

At one loop the renormalization constant $Z$ has the form

[TABLE]

where $z_{EQ}$ is obtained from the requirement that corresponding Green’s functions of the renormalized evanescent operator $E(\mu)$, as in eq. (2.4), vanish.

In the following we shall use, as a short-hand notation, a symbol of direct product $\Gamma\otimes\Gamma$ for the Dirac structure of four-quark operators. Thus, one has

[TABLE]

The color structure of the operator can be also indicated in this manner. It can be either $1\otimes 1$ or $t^{a}\otimes t^{a}$, or some mixture of both (see below). If not explicitly said, the short-hand notation tacitly implies that a relation between operators is independent of color arrangement of quark fields.

The evanescent operators of dimension six with four-quark structure naturally emerge for DimReg treatment of flavor mixing processes since the Dirac algebra of gamma matrices becomes formally infinite dimensional in $d=4-2\epsilon$ dimensional space-time for arbitrary $\epsilon$. Therefore, the basis in Dirac algebra contains the products of gamma matrices with any number of those. Higher products of gamma matrices are not reducible to lower ones anymore contrary to four-dimensional case. The first example emerges already with the product of three gamma matrices. In four dimensions, the relation

[TABLE]

allows to reduce the number of gamma matrices in the product. This is not possible in $d$ dimensions anymore. However, for four-quark operators with left quarks that are eigenstates of chirality one can simply parametrize the difference between a higher product of gamma matrices and its four-dimensional limit with a new object that is called an evanescent operator. Thus, while in four dimensions there exists a reduction of the form

[TABLE]

in $d$ dimensions it becomes a relation

[TABLE]

that defines an evanescent operator $E^{\rm evan}$. Note also that the renormalization of four-quark operators would require an introduction of evanescent operators even in pure vectorlike theories if DimReg is used.

The choice of the evanescent operator $E^{\rm evan}$ is not unique (see, e.g. [20]). A concrete recipe of treating evanescent operators in QCD for flavor changing processes within naive dimensional regularization has been formulated in ref. [18] where Wilson coefficients at NLO have been analysed in detail. At a computational level the particular recipe of ref. [18] reduces to a substitution

[TABLE]

with $d=4-2\epsilon$. Thus, the evanescent operator in QCD is defined in the original publication [18] as

[TABLE]

An infrared safe Green’s function of (renormalized) $E^{QCD}(\mu)$ being computed over the quark states vanishes in $d=4$ in perturbation theory up to NLO. In higher orders of PT new evanescent operators will appear and the coefficient of the physical structures will have higher orders of $\epsilon$ expansion. The choice of $E^{QCD}$ is not unique. The actual choice of the basis in ref. [18] is a nonminimal choice of the basis in a sense that there is an addition of explicit order $\epsilon$ of a physical operator to the reduction relation (2.16). This choice is motivated by the requirement of validity of Fierz transformation in the physical sector and discussed in detail in [18, 20].

Note that the one-loop counterterm to the operator $Q$ has the form [24]

[TABLE]

where $\Gamma^{[3]}$ is a totally antisymmetric product of three Dirac gamma matrices, and $t^{a}$ are $SU(3)$ color generators of QCD. The totally antisymmetric products of $n$ Dirac gamma matrices $\Gamma^{[n]}$ form also a convenient basis in the Dirac algebra in $d$-dimensions. In terms of this basis the above evanescent operator in QCD reads

[TABLE]

with $\Gamma^{[1]}\otimes\Gamma^{[1]}\equiv\gamma_{\nu}\otimes\gamma_{\nu}$. The anomalous dimension of the operator $Q(\mu)$ and the Wilson coefficient $C(\mu)$ at NLO in QCD are given in the literature mainly for this particular choice of the basis (2.17).

The renormalized operator $Q(\mu)$ (in contrast to the bare one $\mathrm{Q}$) depends on the choice of the evanescent one [20, 23, 25, 26]. If we choose $\mathrm{E}^{\prime}=\mathrm{E}+a\epsilon\mathrm{Q}$ instead of $\mathrm{E}$, we obtain a different renormalized operator $Q^{\prime}(\mu)$. At the one-loop level the relation between the two operators goes as

[TABLE]

2.2 HQET: Below $m_{b}$

The second step in the effective theory analysis of the $B^{0}$–$\bar{B}^{0}$ mixing is to remove an explicit dependence on $m_{b}$ from the matrix element or the mixing amplitude at low energy since this scale is still QCD perturbative, $m_{b}\gg\Lambda_{\text{QCD}}$. The removal of $m_{b}$ scale is achieved by using HQET [27, 28, 29]. One requires matching QCD to HQET at the scales around $m_{b}$. At scales $\mu$ below the $b$ quark mass the QCD operators are expanded into a series of local HQET operators. This is called heavy quark expansion (HQE) and symbolically denoted as expansion in $\Lambda_{\text{QCD}}/m_{b}$.

In particular, the HQE of the operator $Q(\mu)$ goes

[TABLE]

where the HQET operators ${O}_{l,s}(\mu)$ are defined as

[TABLE]

The bare field $h_{+}$ annihilates the heavy quark in HQET (moving with the four velocity $v$), and $h_{-}$ creates the heavy antiquark (again moving with the four velocity $v$), which is a completely separate particle in HQET framework. Note that a single physical operator of QCD, $Q(\mu)$, is expanded over two independent operators of HQET, ${O}_{l,s}(\mu)$.

In general one has to define in HQET its own set of physical and evanescent operators. It is convenient to choose four-quark operators in analogy with QCD. We introduce

[TABLE]

and $q$ is always a left-handed fermion $q\equiv q_{L}$. It is usually a light-flavored one, $d$ or $s$. A natural choice for a basis in HQET is an antisymmetrized product of transverse gammas,

[TABLE]

Then $\gamma_{\perp}^{[n]}$ is an antisymmetrized product of transverse gamma matrices with a rank (number of matrices) $n$. We use further notation: the capital $\Gamma^{[n]}$ is full antisymmetric as in QCD, small $\gamma^{[n]}_{\perp}$ is transverse. Then

[TABLE]

in a sense of four-quark operators in HQET of the form eq. (2.23). Up to NLO computation (one loop) one can meet in the calculations a product of not more than three gamma matrices only (and four in HQET).

The simplest basis of physical and evanescent operators emerges after a direct reduction of four-quark operators in four dimensions with the use of the Fierz rearrangement. With $O_{0},O_{1}$ from (2.23) being chosen as a physical pair the four-dimensional reduction of the operators of the form given in eq. (2.23) reads

[TABLE]

Thus a set of corresponding evanescent operators reads

[TABLE]

and $O_{0},O_{1}$ form a physical pair. Other convenient choices of the bases are discussed in Appendix A in some detail. Here we only mention that one possible physical basis is also

[TABLE]

This basis is such that the operator $O_{s}$ appears only at NLO in matching. It has been used in original papers on matching QCD onto HQET [30, 31, 32] and is indicated in eq. (2.22).

The basis (2.2) in HQET is fully appropriate for further use in HQET but it does not match a QCD evanescent definition of ref. [18] in a sense that the evanescent operator $E^{\rm QCD}$ does not match onto evanescent operators of an HQET basis (2.2). While it is not a necessary requirement such a property would be convenient in practical computation.

Let us work out the basis which has such a property. An expansion of antisymmetric product of gammas relevant for QCD over the HQET basis of transverse products reads

[TABLE]

where $\gamma_{\perp}^{[0]}\otimes\gamma_{\perp}^{[0]}\equiv 1\otimes 1$. Any basis is “legal” to use in the calculation. In some bases, however, the evanescents give a contribution to a physical sector of HQET due to radiative corrections and have to be explicitly matched. One can choose a basis when QCD evanescent matches to HQET evanescent. Of course, this happens with some accuracy in $\epsilon$ expansion only. In our case one sees that it suffices to shift the definition of the highest rank operator ($O_{3}$) with both color structures in accordance with the prescription of ref. [18]. Then the canonical QCD evanescent operator matches to HQET evanescent.

Finally, our working basis in HQET is $\{O_{0},O_{1},e_{i}\}$ with

[TABLE]

With such a choice the QCD canonical evanescent operator $E^{QCD}$ eq. (2.17) matches to pure HQET evanescent operators with necessary accuracy in $\epsilon$-expansion.

3 Matching computation

After fixing the choice of the basis we compute the matching coefficients $C_{i}$ in eq. (2.21). The basis in terms of transverse antisymmetric gammas allows for projecting diagrams and completely automatic handling of the whole computation. We use the symbolic system REDUCE for these calculations.

The computation has been done in leading logs in [33, 34] where LO anomalous dimension has been found. It happens to be equal to twice the anomalous dimension of a bilinear current. In higher orders it is not so. The standard result at NLO is reproduced [30, 31, 32]. Note that the use of the minimal basis of eq. (2.2) gives a different answer for the NLO matching coefficient and therefore requires an explicit matching of the QCD evanescent operator $E^{QCD}$ for reproducing the result given in the literature. The adjusted basis of eq. (2.2) reproduces the correct NLO matching coefficient automatically. This is a clear difference with computation of anomalous dimensions. While the presence of evanescent operators starts to influence the computation of anomalous dimensions at two-loop level, the matching coefficients are sensitive to the particular basis of evanescent operators already at one-loop level.

In this paper, as a first step of NNLO calculation, we compute the leading order in $n_{l}$ only. In this approximation the operators $O_{l}$ and $O_{s}$ do not mix in matching coefficients. We need, in addition to the computation of the two-loop diagrams, the heavy quark on-shell renormalization constant $Z_{2}$ up to NLO, and the corresponding contributions to the anomalous dimension.

A single bare operator $\mathrm{Q}$ is expanded over the basis of renormalized operators in the form

[TABLE]

therefore,

[TABLE]

Bare evanescent operators are expanded through

[TABLE]

Finally $E^{QCD}(\mu)=0$ and the quantity $Z_{EQ}$ is designed so that $\langle\mathrm{E^{QCD}}\rangle=Z_{EQ}\langle Q(\mu)\rangle$. At LO the quantity $Z_{EQ}$ has no poles in $\epsilon$ that makes the renormalization nonminimal.

Arranging things so that $E^{\rm QCD}\to e^{\rm HQET}$ one need not consider NNLO evanescent but NLO evanescent should be matched at one-loop order. At matching, one has to make sure that the one-loop matching of $E^{\rm QCD}$ is safe and the tree-level of new evanescent is at least of $O(\epsilon^{2})$ in the physical sector.

We define the expansion of the coefficients as

[TABLE]

At NNLO we compute explicitly only the contribution of light fermion loops that is gauge invariant.

The leading order result goes at $\mu=m_{b}$ [30, 31, 32]

[TABLE]

At NNLO the result for the scalar operator is

[TABLE]

Here $T_{F}=\frac{1}{2}$, $n_{l}$ is a number of light (massless) quarks. Numerically the contribution gives a rather reasonable shift of order 20% for $n_{l}=4$.

The result for the vector operator $O_{l}$ reads

[TABLE]

and for $N_{c}=3$

[TABLE]

Note that we disagree with the entry $n_{l}$ in the two-loop anomalous dimension of ref. [35]. The matching coefficient which relates the renormalized operators in QCD and HQET must be finite at $\epsilon\to 0$. This requirement is satisfied when we use the anomalous dimension of $O_{l}$ derived in Appendix C. It is not satisfied if we use the anomalous dimension from [35].

The shift in the coefficient $C_{l}$ for $n_{l}=4$ is

[TABLE]

that amounts to 20% of the NLO result and should be taken into account in precision analysis. Thus, NNLO corrections can be large and shift the results of NLO computation by at least 10–20%. This is an argument that the NNLO corrections require full computation.

4 Operator product expansion in HQET for the sum rules analysis

In this section we discuss the role of evanescent operators in the sum rules (SR) analysis for the matrix element of the four-quark operator $O_{l}$ within HQET. This analysis is important for our computation of bag parameters with three-loop correlator in [36]. In the previous section we established that $E^{\rm QCD}$ matches to evanescent operators in HQET provided the basis is chosen as in (2.2) which implies that only physical operators $O_{l},O_{s}$ should be considered in SR analysis. However, for the choice of the basis as in (2.2) it is not the case. Here we explicitly demonstrate that the operator $E^{QCD}$ after matching to HQET gives no contribution to SR if the basis (2.2) is used for the computation of the matching coefficients $C_{l},C_{s}$. Thus, this section gives an explicit demonstration of usefulness of introducing correlated evanescent operators in QCD and HQET. One has to consider the Buras evanescent operator $E^{QCD}$ that is used for computing the Wilson coefficient $C(\mu)$ at NLO in QCD. It suffices to compute the LO contribution of the evanescent operator to the three-point correlator. It requires a two-loop calculation of the operator product expansion (OPE) but the integral factorizes.

To evaluate the matrix element of the mixing we use a vertex (three-point) correlation function [37]. This correlator reveals the factorizable structure of the matrix element more clearly than the two-point function [38] but is significantly more difficult to compute at NLO compared to the calculation of the two-point function. For the present analysis we however set up a three-point sum rule in HQET where the computational difficulties have been solved [39]. We consider a correlator [36]

[TABLE]

of the operator $O_{l}$. Here we compute in dimensional regularization with $d=4-2\varepsilon$ and with anticommuting $\gamma_{5}$. The currents

[TABLE]

interpolate the ground state of a $B$ meson in a static approximation.

The bare QCD operator is

[TABLE]

and $E$ is an evanescent operator that contains $\Gamma^{[3]}$ and has $t^{a}\otimes t^{a}$ color structure. It suffices to match $E$ to HQET at LO. Then one has to compute a correlator at LO as well that in our case is given by a two-loop integral but of a factorized form (just a product of two one-loop integrals).

Thus, one has to compute the three-point correlator at LO and make sure that there is no contribution of the evanescent operator. Due to the color structure of $E$ only $\infty$-type topology or one-trace part of the correlator survives. It goes symbolically as

[TABLE]

Here $S_{L}(k)=\not{k}(1+\gamma_{5})/k^{2}$ is a light quark propagator, $H_{\pm}(l,\omega,v)=(1\pm\not{v})/(-2lv+\omega)$ is a heavy quark propagator, $\Gamma$ is either $\Gamma^{[3]}$ or $\Gamma^{[1]}$ for the evanescent operator. Note that one can also use the basis of full gamma matrices for the evanescent operators that results only in reshuffling the basis.

One finds for the contribution of a four-quark operator of a general gamma structure

[TABLE]

where $\mathcal{K}_{0}$ is a scalar structure of the integral

[TABLE]

The $d$-depending factor of $\Gamma^{[3]}$ operator is $-12+8\epsilon$.

We remind one that the evanescent operator is

[TABLE]

and it matches onto HQET operators at LO that retain its Dirac gamma structure and color structure. Thus, one finds the contribution of the evanescent operator to the correlator to be

[TABLE]

There is a pole in $\epsilon$ from the renormalization constant and finite factors $\rho(\omega_{1})\rho(\omega_{2})$ after taking the imaginary parts in $\omega_{1,2}$. All together this gives a total factor of $\epsilon$ multiplying the finite result for the correlator and the contribution of the evanescent operator vanishes as it should be.

Thus, the calculation in ref. [36] is not affected by the presence of evanescent operators as soon as they are defined according to the standard rules used for the computation of the Wilson coefficients. This fact has not been explicitly mentioned in ref. [36].

In general, an HQET basis should respect the relation

[TABLE]

for the $t^{a}\otimes t^{a}$ color structure where only one trace is possible for the three-point correlator. Our choice of the basis agrees with this requirement.

5 A comment on $B_{s}$ mixing

Mixing effects occur in the $B_{d}$ as well as in the $B_{s}$ system, and in both systems the mixing parameters have been measured quite precisely. In both cases, the mixing frequency is dominated by the top quark and thus can be computed in terms of the local four-quark matrix elements discussed above. Aside from the different Cabibbo–Kobayashi–Maskawa (CKM) factors [$(V_{td}V_{tb}^{*})^{2}$ compared to $(V_{ts}V_{tb}^{*})^{2}$] the hadronic matrix element involves an $s$ quark instead of a $d$ quark, and the states need to be replaced accordingly. It has become customary to define the matrix element

[TABLE]

in terms of the corresponding decays constant $f_{B_{q}}$ multiplied by a bag factor $B_{B_{q}}$ which is unity in naive factorization. Phenomenological CKM fits make use of lattice calculations of the relevant hadronic matrix elements. It turns out that the ratio

[TABLE]

can be computed quite precisely on the lattice, since many systematic uncertainties cancel in the ratio. The quantity $\xi$ in combination with the perturbative calculation of the Wilson coefficient is the basis for the extraction of $V_{td}{V_{tb}}^{*}$ which fixes one of the sides of the unitarity triangle.

In a recent paper [36] we have discussed the matrix element (5.1) for a $B_{d}$ meson, using a QCD sum rule within HQET. While in a lattice calculation the decomposition of the matrix element into decay constant and bag parameter is irrelevant, it is important in the sum-rule calculation, since the contributing Feynman diagrams can be uniquely attributed to either $f_{B_{q}}$ or $B_{B_{q}}$. Furthermore, the sum rule allows us to estimate the deviation of $B_{B_{q}}$ from unity, which eventually leads to a quite precise result for the bag factor $B_{B_{d}}$.

A similar estimate for the bag parameter $B_{B_{s}}$ requires one to take into account $SU(3)$ breaking effects, which are induced by the strange-quark mass $m_{s}$. In a sum rule, this parameter appears, on the one hand, explicitly in the perturbative calculation; on the other hand, it will also induce the difference between the strange- and the light-quark condensates.

It is well known that the $SU(3)$ breaking is in general not small as one can see for light mesons $K$ and $\pi$. For example, the dependence on $m_{s}$ is well seen in the sum-rule calculation for the leptonic decay constants $f_{K}$ and $f_{\pi}$ [40]. It is not small as it emerges at the tree level and is of the order $\frac{m_{s}}{\omega_{c}}\approx 0.15$ with $m_{s}=150\leavevmode\nobreak\ \mathrm{MeV}$ and ${\omega_{c}}=1\leavevmode\nobreak\ \mathrm{GeV}$ is a typical scale of sum-rules computation. Indeed, experimentally we have $(f_{K}-f_{\pi})/f_{\pi}=0.19$, which is parametrically close to the above estimate.

However, if we look at bag parameters, we see that the leading term of the sum-rules computation is completely factorized and predicts $B=1$ for both ${B_{s}}$ and ${B_{d}}$ mesons. This means in turn that the ratio of the two bag parameters emerges only at NLO level and is of the order

[TABLE]

Thus our best estimate is

[TABLE]

This may be compared to the prediction from the lattice which can be derived by using the lattice predictions for the decay constants to be [7]

[TABLE]

which is compatible with our observation.

We conclude that the deviation from unity of the quantity $\xi$ is almost completely driven by the $SU(3)$ in the leptonic decays constants. There are also recent sum-rule estimates for these matrix elements given in [41]. In particular, we obtain for the ratio [41]

[TABLE]

The contributions of power corrections to the sum rules for mixing analysis are pretty small [40, 42, 43, 44]. Making use of (5.3) we obtain

[TABLE]

to be compared with the lattice result [7]

[TABLE]

We see that our result (5.6) agrees with the lattice value (5.7) within our uncertainties but it is less precise. The main uncertainty comes from the ratio of decay constants where the sum-rule estimate remains less precise as the current lattice evaluations.

On the other hand, the QCD sum-rule estimate reveals the relative size of the different contributions to the four-quark matrix elements as well as to the parameter $\xi$. The key observation is that naive factorization of the four-quark matrix elements is in fact a quite good approximation; QCD sum rules indicate that the correction to this assumption is small. This is even more true for the ratio $\xi$, which is (up to a small correction of the order of 2%) driven by the $SU(3)$ breaking in the leptonic decay constants. Indeed, by taking the world average results for the decay constants [45]

[TABLE]

we find

[TABLE]

which is comparable with the lattice result (5.7).

6 Summary

We discuss the problem of higher order corrections in HQET for the analysis of mixing in sum rules. We fix the basis of HQET operators with a set of evanescent operators necessary in higher orders within computation in naive dimensional regularization. We have considered matching of QCD to HQET at NNLO where the precise definition of evanescent operators is a must. We have computed the contribution due to light fermion loops and discovered that our result disagrees with the entry of the anomalous dimension used before. We have computed leading $n_{l}$ contributions. They happen to be large at the level of 10%. If there is no cancellation in the full result our numbers show that the precision of matrix elements (ME) at the level of a few percent requires the matching coefficients at NNLO. Note in passing that with the NNLO accuracy of the leading term one may need to account also for nonleading terms of HQE [46].

We have revisited the calculation of the bag parameter in sum rules at three-loop level [36]. We stress that the form of the Hamiltonian in the physical sector is not sufficient: for any computation of ME (or Green functions) one needs to know the set of related evanescent operators. This makes the computation rather cumbersome as the renormalized physical operators in such a scheme have no meaning without an explicit form of the evanescent ones. We show that our choice of evanescent operators coordinated with the standard ones in QCD does not affect our calculation of the three-loop correction to the bag parameter in HQET [36].

We also discuss the flavor dependence of the bag parameters for strange bottom meson. Within the SR approach, in contrast to lattice, one can see the anatomy of contributions and guarantee that the flavor shift is very small. This makes the prediction of the ratio $\xi$ dependable only on the ratio of leptonic couplings that is rather precisely known.

7 Acknowledgment

We thank U.Nierste for interest in the work and kind indication on papers [20, 25]. A.G. is grateful to Siegen University for hospitality; his work has been partially supported by the Russian Ministry of Education and Science. This work is supported by the DFG Research Unit FOR 1873 ”Quark Flavour Physics and Effective Theories”.

Appendix A HQET bases for evanescent operators

The physical pair can be chosen differently than in the main text. Let’s take $O_{0},O^{\prime}_{0}$ that look rather symmetric and differ by color arrangement only. Then the reduction of operators up to rank three reads

[TABLE]

and this determines a new set of evanescent operators.

Yet another physical basis is $O_{+}=O_{0}+O^{\prime}_{0}$, $O_{-}=O_{0}-O^{\prime}_{0}$. These operators are Fierz eigenstates by construction with $\pm 1$ parity at tree level in four dimensions. The reduction in this basis is

[TABLE]

Note again that this is a pure four dimensional reduction that leads to evanescent operators of a minimal choice.

Clearly, the freedom of the definition of evanescents is not only the choice of a physical pair but deviation from minimality. By adding a physical operator to an evanescent with a coefficient vanishing in four-dimensional space gives a nonminimal basis. For example, the shift of the evanescent $e$ with a physical operator $O$ with a coefficient of order $\epsilon$

[TABLE]

changes the basis and, therefore, anomalous dimensions and matching coefficients for physical operators in higher orders.

Our working basis is

[TABLE]

It means that one first does such a shift and then applies the standard minimal reduction. Or, in addition to this,

[TABLE]

is applied.

Combining two expansions (A.2) and (A) one obtains the basis as

[TABLE]

and we have redefined $e_{6}$ compared to the minimal basis. With such a substitution the QCD canonical evanescent operator matches to pure HQET evanescent with necessary order in $\epsilon$-expansion.

Appendix B Renormalization

Here we fix some notation. The $\overline{\text{MS}}$ renormalized coupling constant is

[TABLE]

A renormalized operator $O(\mu)$ is related to the bare one $\mathrm{O}$ by $\mathrm{O}=Z(\alpha_{s}(\mu))O(\mu)$. Its anomalous dimension is

[TABLE]

and the renormalization constant must have the form

[TABLE]

The operator $Q$ (2.2) has

[TABLE]

For the matching calculation we need the on-shell renormalization constant $Z_{2}^{OS}$ [47]:

[TABLE]

where only the $n_{l}$ contribution to the $g_{0}^{4}$ term is written, and $m_{b}^{\text{os}}$ is the mass in the on-shell scheme.

Appendix C The $n_{l}\alpha_{s}^{2}$ term

in the anomalous dimension of $O_{l}$

An easy way to calculate this anomalous dimension is to use infrared rearrangement. We nullify all external momenta (including HQET residual ones) and introduce a gluon mass as an IR regulator. The gluon propagator with a light-quark loop insertion is transverse; it is convenient to keep this property also for the propagator without insertions, i. e., to use Landau gauge. The free gluon propagator becomes

[TABLE]

The quark-loop insertion is $i\Pi(k^{2})\left(k^{2}g_{\mu\nu}-k_{\mu}k_{\nu}\right)$, $\Pi(k^{2})=\beta_{0}A_{0}e^{-\gamma\epsilon}D(\epsilon)(-k^{2})^{-\epsilon}$, where $A_{0}=g_{0}^{2}/((4\pi)^{d/2}\epsilon)$ and $D(\epsilon)=1+\frac{5}{3}\epsilon+\cdots$ (we will keep only the $-\frac{4}{3}T_{F}n_{l}$ term in $\beta_{0}$). Therefore, the gluon propagator with up to 1 quark-loop insertion is (C.1) times

[TABLE]

Let’s first discuss the bilinear current $\tilde{\jmath}^{\alpha}=\bar{q}\gamma^{\alpha}h_{+}$. Its matrix element is $(Z_{h}Z_{q})^{1/2}V\gamma^{\alpha}$, where the vertex is

[TABLE]

The vector integral with $k^{\mu}$ in the numerator can be directed only along $v^{\mu}$, and we may substitute $k^{\mu}\to k\cdot v\,v^{\mu}$. Loop corrections vanish, and we get

[TABLE]

where $\gamma_{h,q}$ are in Landau gauge. In this way we easily reproduce the $\alpha_{s}$ and $n_{l}\alpha_{s}^{2}$ terms in $\tilde{\gamma}_{j}$ [48],

[TABLE]

Now we turn to the operator $O_{l}$. Its matrix element is $Z_{h}Z_{q}V$, where the vertex is

[TABLE]

where $O=\gamma^{\alpha}\otimes\gamma_{\alpha}$. The color structure of the tree diagram is $T_{1}=\delta_{a_{1}}^{b_{1}}\delta_{a_{2}}^{b_{2}}$ where $a_{1,2}$ are the color indices of the heavy external legs, $b_{1,2}$ are those of the light legs, and 1, 2 number the fermion lines. The first loop diagram in (C.6) is the same as the one for the current $\tilde{\jmath}$, and hence it vanishes. The second diagram differs from the first one only by the color factor: instead of $C_{F}T_{1}$, it is now

[TABLE]

it also vanishes.

The heavy–heavy diagram in (C.6) is

[TABLE]

Averaging over $k$ directions, we may substitute $1/(k\cdot v)^{2}\to-(d-2)/k^{2}$ [49] (this has been rigorously proved in [50]). This gives

[TABLE]

where

[TABLE]

Re-expressing this result via the renormalized

[TABLE]

we finally obtain

[TABLE]

The light–light diagram in (C.6) is

[TABLE]

We may average over $k$ directions: $k^{\mu}k^{\nu}\to(k^{2}/d)g^{\mu\nu}$, and obtain

[TABLE]

where $O_{3}=\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}\otimes\gamma_{\mu}\gamma_{\nu}\gamma_{\alpha}$. Neglecting the evanescent operator (which does not contribute to the $n_{l}\alpha_{s}^{2}$ term in the anomalous dimension) we may replace $O_{3}\to(16-4\epsilon)O$ and get

[TABLE]

or finally

[TABLE]

The renormalized operators $O_{l}^{\prime}=O_{l}$, so that we may replace $T\to T_{F}(1-1/N_{c})T_{1}$. Combining (C.8) with (C.9) we see that the vertex (C.6) is $T_{1}OV$ with

[TABLE]

We have $\log V=\log Z_{V}+\log V_{r}$, where negative powers of $\epsilon$ go to $\log Z_{V}$, while non-negative ones to $\log V_{r}$. Taking logarithm of $V$ (C.10) we note that the square of the 1-loop term does not contribute to the $n_{l}\alpha_{s}^{2}$ structure, and $\log Z_{V}=\frac{3}{2}T\beta_{0}A^{2}\epsilon$ (we keep only the $n_{l}$ term in $\beta_{0}$). The anomalous dimension of the operator $O_{l}$ is

[TABLE]

Finally, we arrive at

[TABLE]

The result of [35] (where $T_{F}=\frac{1}{2}$) leads to

[TABLE]

Our result (C.11) differs from it by the absence of the term $N_{c}/3$ in the bracket. This color structure cannot appear in the diagrams which contribute to the $n_{l}\alpha_{s}^{2}$ term in $\Delta\gamma$. Probably, other structures, without $n_{l}$, which appear in the 2-loop $\Delta\gamma$, also need rechecking.