Resolving Charged Hadrons in QED -- Gauge Invariant Interpolating Operators

Standard interpolating operators for charged mesons, e.g. $J_{B} = \bar b i \gamma_5 u$ for $B^-$, are not gauge invariant in QED and therefore problematic for perturbative methods. We propose a gauge invariant interpolating operator by adding an auxiliary charged scalar $\Phi_B$, ${\cal J}_{B}^{(0)} = J_B \, \Phi_B$, which reproduces all the universal soft and collinear logs. The modified LSZ-factor is shown to be infrared finite which is a necessary condition for validating the approach. At ${\cal O}(\alpha)$, this is equivalent to a specific Dirac dressing of charged operators. A generalisation thereof, using iterated integrals, establishes the equivalence to all orders and provides a transparent alternative viewpoint. The method is discussed by the example of the leptonic decay $B^- \to \ell^- \bar \nu$ for which a numerical study is to follow. The formalism itself is valid for any spin, flavour and set of final states (e.g. $B^- \to \pi^0 \ell^- \bar \nu$).


Introduction
Scalar QED or the point-like approximation is a consistent framework which works numerically well in many cases, but precision in CKM matrix elements and testing of lepton flavour universality [1,2] are calling for stucture-dependent computations. 1 Approaches include chiral perturbation theory (ChPT) [5][6][7], soft collinear effective theory (SCET) [8,9] and lattice Monte Carlo simulations (with a range of methods to contain the massless photon in finite volume [10][11][12][13] and applications thereof in [14][15][16][17][18]). ChPT applies to low energy physics and might be viewed as a successful expansion around the point-like approximation where the challenge relies in determining the finite counterterms. In SCET, mesons are described by light-cone distribution amplitudes which necessitate the introduction of process dependent Wilson lines [19,20]. In lattice Monte Carlo simulations, hadrons are described by either gauge variant or gauge invariant interpolating operators (on which we comment at the end of Sec. 4.1.1). No method exists for perturbative approaches with interpolating operators for charged mesons. 2 It is the aim of this paper to fill this gap.
This work is organised as follows. In Sec. 2, some preliminary materials such as the problem of gauge variance and the universal infrared (IR) logs are discussed. In Sec. 3, we introduce the idea of the long distance B-meson as part of the modified gauge invariant interpolating operator. In Sec. 3.2, it is shown and argued that its LSZ-factor is IRfinite. In Sec. 4, we establish the connection with the Dirac dressing at O(α), and a generalisation using iterated integrals establishes the connection to all orders. The paper ends with conclusions and discussions in Sec. 5. In App. A, we comment on the necessity of gauge invariant interpolating operators, in connection with the hard photon approach to B − → −ν γ. In App. B, we briefly review the QCD sum rules approximation to the LSZ approach. 1 Scalar QED is sufficient in reducing the RK uncertainty to O(1%) as there are no enhanced O(α) ln m m B logarithms (logs) beyond scalar QED [3] and charmonium resonances are under control for q 2 < 6 GeV 2 [4]. 2 The approach we have in mind is QCD sum rules (cf. App. B) where the Lehmann Symanzik Zimmerman (LSZ) approach (e.g. [21]) is implemented via semi-global quark-hadron duality [22,23] 2 Preliminaries

Gauge variance of the QCD interpolating operator
Let us first discuss the problem in a pedestrian way through the example of a leptonic decay of the type B − → −ν . In QCD, mesons such as the B − are interpolated by 3 , (2.1) where Z B plays the rôle of the LSZ-factor, This means that the matrix element can be extracted from the correlation function 3) since the latter satisfies a dispersion relation as indicated. In (2.3), disc s Π(s) = Π(s + i0) − Π(s − i0) is the discontinuity across the real line, the dots stand for higher states in the spectrum and Z * B = Z B has been assumed. The shorthands x = d 4 x and xp = x · p, used above, are occasionally assumed hereafter.
When one considers QED, the picture is fundamentally disturbed in perturbation theory since the operator J B (or Z B ) ceases to be gauge invariant 4 here and below), as is well-known, f B ceases to be an observable once QED corrections are considered (e.g. [24] for a discussion for f π ). Note that (2.5) implies the D = ∂ +iA convention with the electric charge absorbed into the photon field A such that n photon fields correspond to O(e n ).

Universal infrared sensitive logs in QED
One of the main features of QED is the appearance of IR sensitive logs which can overcome the small coupling constant α = e 2 /4π ≈ 1/137 and a fair amount is known about them. In particular, their physics is dictated and controlled by gauge invariance and thus complications are to be expected when gauge invariance is not manifest. One may distinguish two classes of logs: first, the so-called (hard) collinear logs, which arise from (charged) particles − with small mass, allowing for collinearity with the photon up to O(m 2 ) and resulting in 3 In QCD, it is advantageous to include (m b + mu) as a prefactor in JB as it becomes a renormalisation group invariant quantity then. However, in QED, this will be of no use since this property is spoiled by the fact that the mq renormalises as e 2 Q 2 q and thus we omit the masses in JB. 4 c.f. App. A.2 for comments on the non-perturbative case.
sizeable O(α) lnm -terms (hatted quantities are divided by m B hereafter). Second, there are soft and soft-collinear logs of the form O(α) lnm γ and O(α) lnm γ lnm which are true IR divergences but turn into O(α) lnδ and O(α) lnδ lnm when the photon emission, of E γ < δ, is added. Aspects of resummation of these logs are known [25,26] but it is rather their degree of independence of the structure (or universality) that is of interest to us. Soft and soft-collinear logs are universal since the soft photons cannot resolve the structure of the mesons. For hard-collinear logs, the situation is not as transparent as photon energies can be as large as the kinematics allow for. It turns out that gauge invariance and the KLN-theorem (cancellation of all real and virtual IR-logs in the m → 0 limit, in the photon inclusive case) are sufficient premises to show that there are no further hard-collinear logs of the form O(α) lnm [3]. In other words, structure-dependent lnm terms are subleading, either in O(α)m 2 lnm or O(α 2 ) lnm , both of which are negligible. 5 However, when the leading order (LO) process is O(m 2 )-suppressed, the theorem does not apply as then real and virtual contributions are not linked by the KLN-theorem [27]. This is precisely the case for B − → −ν enabled by V-A interactions, as in the Standard Model (SM). These non-universal structure-dependent logs complicate the investigation of the validity of the approach. However, we can easily bypass this issue by resorting to an S-P interaction L (S−P ) which is not O(m 2 )-suppressed. Hence, the hard-collinear logs are universal and, in particular, reproducible from the splitting function since the KLN-theorem applies [27]. We stress that the reasoning for choosing an S-P interaction that it allows us to test whether or not our method is capable of reproducing universal collinear logs. In summary, quoting almost verbatim from Sec. 3.2.1 of that reference, one has where the soft factor is 8) and the (hard) collinear part, reproducible from the splitting function, reads This has been backed up by an explicit computation [28]. We note that in the photon inclusive limit, 2δ → 1 −m 2 , which implies that F coll (δ) → O(m 2 ) as required by the KLN-theorem since these logs are not suppressed by a factor of m 2 in the rate (2.7). The essential starting point of this paper is the observation that for the gauge variant interpolating current J B (2.1), these universal logs are not reproduced in a perturbative computation. We shall see how to remedy it and how to interpret it in due course.

The Long Distance B-meson as a Scalar Field Φ B
It is well-known that off-shell correlation functions are not gauge invariant, be it in QED or QCD, and this is at the heart of the issue raised in the introduction; namely, that the universal IR-logs are not reproduced when computing with the interpolating operator J B (2.1). On the other hand, in scalar QED (point-like approximation), no such problems occur since the external particles can all be put on-shell. It is thus tempting to cure both, gauge invariance and the universality problem, by introducing a long distance (on-shell) B-meson in terms of a scalar field Φ B of mass m B as follows The matrix element Z where both terms, the LSZ-factor |Z (0) B | 2 and the integrand, are computed separately, and δ dΦ γ is the integral over the photon phase space with E γ < δ (cf. [3] for more detail). The amplitude squared is given by where δ(Φ γ ) is a delta function in the photon variables as appropriate for the virtual contribution. This is the famous Bloch-Nordsieck mechanism at work which bypasses the QED IR-problem of charged particles. As previously mentioned, the Φ B − -particle can be considered as the long distance version of the B-meson which splits into its partons at the J

The main process
In this section, we describe how the main process, by which we mean everything in (3.3) but the LSZ-factor, is computed. We depart from the following correlation function (with L W , a shorthand for the weak Lagrangian, see (2.6)) 6 Since we invoke the Bloch-Nordsieck mechanism, we may ignore that the virtual and the real emission part of the B-meson is not a well-isolated state.
the radiative or real emission part). The last diagram is specific to the Φ B -particle.
where r ≡ p Φ B − p B is introduced in order to distinguish the p Φ B -and the p B -momenta, even though both are to be set on-shell (to m 2 B ) in the end. 7 For p Φ B , this is straightforward (3.6) since it plays the rôle of an elementary particle. The formal definition of the matrix element (times the LSZ-factor) is then obtained from (3.5) as We stress that (3.7) serves only as a formal definition of the matrix element since in practice, as is well-known, it is impossible to extract a bound state pole with perturbative methods, since bound states are non-perturbative. The real emission diagrams are depicted in Fig. 1 and the computation of their discontinuities is straightforward. In practice, the main challenge is to compute the discontinuity of the virtual diagrams of the type shown in Fig. 2. We have performed this task by using Cutkosky rules as the virtual diagrams involve two loops and have a considerable number of scales. The results, with more details to be reported in [28], are the following: 1. All universal collinear logs (2.7) are reproduced separately for the virtual and the real rates. They originate from the sum of all bū-cuts as the sum of all būγ-cuts is free from collinear logs (in the S-P case).
2. The universal soft (and soft-collinear) logs, given in (2.8), are equally reproduced and emerge as lnδ and lnδ lnm terms respectively. 8 In order to reproduce the soft logs the auxiliary Φ B is crucial. Its on-shellness gives rise to the correct soft-structure in the integrand.
the non-radiative part, hence no (γ) superscript). Top line is the LO diagram and the third line and the last diagram are specific to the Φ B -particle. 3. Unphysical IR divergences in ln m u and ln m γ , of the collinear and soft type, cancel for the sum of all cuts. We note that this must be the case since the corresponding momentum p B is off-shell and IR-finiteness follows from the Kinoshita-Poggio-Quinn theorem [29].
Let us remark to this end that taking all the cuts is what is usually done in the case of virtual QCD computations. However, in the case of hard photon emission, B − → −ν γ, where the photon is energetic (or non-soft), the procedure has been a different one in the literature for many decades. Why this is a valid procedure is explained in App. A.1.

The LSZ-factor is infrared finite
The LSZ-factor (3.1) can be extracted from the following diagonal correlation function where, as before, the dots stand for higher states and this time we do not show the LSZ procedure for the Φ B -particle explicitly, as it is straightforward. The quantity of interest is then determined from (3.8) The computation proceeds in the same way as for the main process and the diagrams are shown in Fig. 3. One may be concerned as to whether the correlation function (or its discontinuity) are IR-finite as the Φ B -particle is on-shell. Fortunately, |Z B | 2 turns out to be IR-finite and this follows from a physical argument. We may interpret |Z (0) B | 2 , via the optical theorem, as an inclusive decay rate of Φ B (p B ) →bqX 0 (r) induced by the hypothetical effective Lagrangian "L eff = J (0) B X 0 " where X 0 is a neutral particle (of momenta r, which decouples when r → 0). Hence, by virtue of the KLN-theorem, which is based on unitarity, we know that its discontinuity must be IR-finite (also in the m u → 0 limit). We have checked that this is true by an explicit computation. Once more, it is important that one takes the sum of all cuts as individual cuts are IR divergent.

Summary of the basic interpolating operator approach
In summary, since the numerator reproduces all universal IR-sensitive logs and the denominator is IR-finite, this strongly suggests that the proposed procedure is correct. In particular, the IR-finiteness of Z (0) B means that the expression (3.3) has "forgotten" about its interpolating operator, as required, since it is an auxiliary in the LSZ formalism. The incorporation for several particles is straightforward from the viewpoint of the interpolating operators; one can add as many as one desires to. The same applies to non-scalar particles; for a proton one adds a scalar Φ P and not a spin 1 2 -particle. The O(E 0 γ ) term in Low's theorem (A.1), related to spin, is reproduced from the spinor in the form factor decomposition and is entirely kinematical.
, where three diagrams with the photon coupling to the u-quarks are omitted as they are completely analogous to the b-quark ones.

Relation to Non-local Operators
There are gauge invariant formulations of QED, which are functionals of the photon field. These date back to the work of Dirac in 1955 [30] 9 and Mandelstam in 1962 [32] where the photon field is integrated over spacetime and a spacelike path respectively. We first review the Dirac dressing in Sec. 4.1, including the concept of dual gauges in Sec. 4.1.1. In Sec. 4.1.2 we show how to embed our approach at O(α) into it. In Sec. 4.2, the all order equivalence is established, going beyond the Dirac dressing, using iterated integrals.

Dirac Dressing
One may introduce a gauge (compensating) factor U (P ) which is a functional of the photon field A and a current I. The latter has no direct relation to the electromagnetic current but it is required to satisfy the differential equation above and vanishing boundary condition at infinity. There are many solutions to this equation and that will be the point of discussion soon. First, let us observe that is gauge invariant since the gauge transformation (2.5) of the gauge factor and the fermion (or any other field) act to compensate each other. In effect, the gauge flux is transported by U (ψ) I to infinity where it is assumed not to matter (implicit by the imposed boundary condition).

Dual gauges
It is now immediate to define a gauge invariant version of J B (2.1) is now a functional of I. This raises the question of whether specific choices are more convenient than others and or even more legitimate. Let us first set aside the latter point. It turns out that for a given I, one can often choose a gauge for which the gauge factor becomes trivial i.e. U I = 1. We may think of this in terms of the following equivalence or duality where "gauge" is now an index for a specific current I and the subscript after the vertical bar on the right hand side indicates that the computation is to be performed in the specific gauge. This calls for examples, for which we will choose the Coulomb and Lorenz gauges (for the further example of the axial gauge, we refer the reader to [12] where these aspects are nicely discussed).
• The Coulomb gauge: here, the current I, satisfying the differential equation (4.1), is x i y i . The gauge factor may be integrated by parts to and trivialises in the Coulomb gauge ∂ · A = 0.
• The Lorenz gauge: in this case, the current I, satisfying the differential equation, is Integrating by parts, the gauge factor reads and trivialises in the Lorenz/Landau gauge, ∂ · A = 0. 10 We wish to stress that the choice of gauge here is nothing but a computational trick or a matter of convenience. The element of complexity in the gauge factor U Igauge is moved into computing with the gauge variant operator J B in a specific gauge. These two effects of gauge dependence act to compensate each other. Let us now return to the question, alluded to before, of whether all choices of I are equally valid. We would think that the answer to this question ought to be yes in approaches with an exact LSZ formula. However, if the LSZ formula is approached in the sense of duality as in QCD sum rules (cf. App. B), this is not the case as it turns out that neither the Coulomb nor the Lorenz gauge current reproduce the universal IR-logs in (2.7). This was explicitly verified using the gauge factor U I .

The Φ B -particle in Dirac dressing
Hence, the natural question is whether our approach which is gauge invariant can be captured in this formalism with a specific current I. The following expression achieves this task where p ≡ p Φ B for brevity and on-shell momentum (p 2 = m 2 B ). As a solution to the differential equation (4.1) with the appropriate boundary condition, the Feynman propagator ϕ(x) = i∆ F (x, m 2 B ) is chosen. The gauge factor integrates by part to a most familiar form. Namely, the exponent becomes the Feynman rule for scalar QED with the scalar being our Φ B -particle! In fact, a hint of this possibility was given by the Lorenz gauge case (4.8) which, however, corresponds to the massless propagator with zero momentum insertion (p µ = 0). Here, we have in effect extended this mechanism to the massive propagator with a non-zero momentum. For clarity, let us quote the corresponding interpolating operator J (D) where the superscript (D) stands for Dirac. A natural question, in view the discussion in Sec. 4.1.1, is whether there exists a dual gauge (that trivialises the gauge factor (4.11))?
The answer is yes, which is a peculiar axial gauge for which the photon propagator in momentum space assumes the form with k {µ n ν} = k µ n ν + k ν n µ , and n µ ∆ µν = 0 as required.

Iterated integral approach
It is clear that the form in (4.11) is not suitable for higher order computations, or already the Φ B self-energy correction. Matters can be improved by writing an expression with iterated integrals. For that purpose, let us define the following kernel suppressing the p and A dependence in K. Then, the improved and final version reads where consists of the iterated kernels K. In essence, this formula is the Dyson series for the Φ Bparticle where the propagators have been contracted already and this is the reason why the 1/n!-factor from the exponential has disappeared! To establish the gauge transformation, , by direct computation is not a simple matter. However, it is clear that it must hold since the Φ B -formalism is gauge invariant. In addition, we have verified the transformation law explicitly up to fourth order, witnessing intricate cancellations. The J B interpolating operator is the most transparent and most generally valid version obtained in this paper as it clarifies a number of questions. For example, does the Φ Bparticle need to be included into the running of the fine structure constant α? The answer is negative since it just "lives" inside the factor V I and does not figure in the Lagrangian of the theory. This means that there is no coupling to charged fermions other than through the interpolating operator itself. As such it serves as a justification for the rules applied previously. In summary we thus have the following relation formulated in terms of the respective LSZ-factors. 11

Conclusions and Discussions
In this work, a method was proposed for incorporating charged hadrons via gauge invariant interpolating operators for perturbative methods such as QCD sum rules. Technically, this consists of adding the long distance field Φ B to the gauge variant operator J B (2.1), that is J Intuitively, Φ B takes on the rôle of the long distance B-meson splitting into its two valence partonsū and b, thereby resolving the dilemma that bound states (e.g. hadrons) are beyond perturbation theory, but essential for infrared-sensitive physics. Formally, Φ B solves two linked problems at once: J B → J B , which can be found in Eqs. (4.12) and (4.11). An improved version J B , valid to all orders, generalising the Dirac dressing by using iterated integrals has been given in Eqs. (4.16) and (4.17). Reassuringly, this generalisation makes it clear that the Φ B -particle does not contribute to the running of the fine structure constant α since it does not appear in the Lagrangian of the theory. The dual gauge, trivialising the gauge factor, has been identified as a peculiar axial gauge, cf. Eqs. (4.13) and (4.14). 11 The JB-formulation is related to the coherent state framework in the sense that the soft logs (not the hard-collinear log) are reproduced. This is achieved by taking the coherent state function to be the eikonal factor ωµ ∝ pµ/(p · k) which defines the coherent state |ω ∝ exp( dΦγω µ a † µ )|0 with a † µ being the photon creation operator [33] (and [27] for a more complete set of references). Again, this has to be the case since soft resummation is equivalent to the coherent state approach at the leading log level. 12 For inclusive enough quantities, which are free of IR-logs, one may not need to use gauge invariant interpolating operators since the on-shell cuts will render it gauge invariant. This happens for ∆m = m B + − m B 0 type quantities [34].
Note that the method generalises to any number of particles and any types of spins with remarks at the end of Sec. 3.3. One can add for each charged particle an operator of the form (4.16). However, one can use the trick of the trivialising gauge only once. Explicit results of the computation for leptonic decays, which necessitate one interpolating operator only, including numerics, are to follow in a forthcoming publication [28].

Acknowledgments
We would to thank Martin Beneke, Luigi Del Debbio, Matteo Di Carlo, Giulio Falcioni, Einan Gardi, Max Hansen, Anton Ilderton Agostino Patella, Antonin Portelli and further participants of the "QED in Weak Decays" workshop in Edinburgh for useful discussions and feedback. In particular, we acknowledge the input of Matt Rowe to this work in terms of proofreading, discussions and computations RZ is supported by an STFC Consolidated Grant, ST/P0000630/1. SN is supported by the GLUODYNAMICS project funded by the "P2IO LabEx (ANR-10-LABX-0038)" in the framework "Investissements d'Avenir" (ANR-11-IDEX-0003-01) managed by the Agence Nationale de la Recherche (ANR), France.

A On the Necessity of Gauge Invariant Interpolating Operators
In this appendix, we comment under which circumstances interpolating operators for charged mesons are necessary or not. This seems mandatory since processes with charged hadrons have been considered in the literature using gauge variant interpolating operators. This includes i) B − → −ν γ where γ is a hard photon, to be discussed in App. A.1, and ii) leptonic decays in lattice Monte Carlo simulations, to be discussed in App. A.2.

A.1 Relation to computations of B − → −ν γ with a hard photon
There is some good tradition in using QED gauge dependent interpolating operators J B (2.1) for B − → −ν γ in QCD sum rule approaches; e.g at O(α 0 s ) [35,36] and O(α s ) [37] or for the g B − B * − γ coupling at O(α 0 s ) [36] and O(α s ) [38]. This raises the obvious question of how the issues raised, at the beginning of the paper, were avoided.
First and foremost, at O(e), that is, for a single photon emission, these observables are formally analogous to B − → −ν ρ 0 (k) and g B − B * − ρ 0 respectively. The main point is that the ρ 0 or the hard photon (say E γ ≥ Λ QCD ) are considered as separate particles and formally, this implies that {q 2 , p 2 B , k 2 } are the independent kinematic variables, referred to as the hard photon approach. 13 The situation is illustrated in Fig. 4 with further comments in the caption.
Computing O(α s ) corrections to B − → −ν γ and g B − B * − γ is demanding, but straightforward, because J B is QCD gauge invariant. However, complications arise if we were to compute O(α) radiative corrections as J B is QED gauge variant. In that case, the introduction of the gauge invariant interpolating operator becomes, in our opinion, a necessity in perturbation theory. One should regard B − → −ν γ with the hard photon as a LO process, and it is only its radiative corrections that necessitate the introduction of the soft photon (to complement the virtual corrections). This is in line with the picture of coherent states (e.g. [27] and relevant references therein).
soft photon hard photon Figure 4. Comparison of soft and hard photon approaches. In the soft photon approach (top diagrams), q 2 → p 2 B − 2p B · k (since k 2 = 0) and p B · k becomes an independent variable. Top left (right) diagram correspond to the Φ B (quarks)-term in (A.5). Note that the top left diagram reproduces the Low term and the top right diagram is finite as p B · k → 0. In the hard photon approach (bottom diagrams), q 2 is an independent variable. The Low term is produced by the bottom left diagram (cutting in p 2 B ), while the bottom right diagram probes the structure. In particular, the blob is the photon distribution amplitude (DA) and is generated by a sum over intermediate mesons ρ 0 , ω, φ with quantum numbers of the photon J P C = 1 −− . For an elaborate discussion of this viewpoint, see App. A of [39]. The mappings of the diagrams between the two approaches are as follows: top left ⊂ bottom left, bottom right ⊂ top right. The first relation follows from the fact that the point-like interaction on the top-left does not "know" anything about the non-perturbative ρ − γ conversion that the photon DA represents and this also implies the second relation.
It is still an interesting question as to how B − → −ν γ at O(e) with gauge invariant interpolating operator J B reduces to the case of the gauge variant operator J B ; in particular, of how the Low terms emerge, giving rise to the universal IR-logs for which we had to introduce the Φ B -particle. Low's theorem [40], quoted with the same convention as in [27], states that adding a real photon to a transition α → β the two first terms in an E γ -expansion are universal and given by p j is the orbital angular momentum operator (and square brackets denote anti-symmetrisation). Hatted quantities are plus(minus) its value for outgoing(incoming) particles.
To further simplify matters, we consider the S-P interaction (2.6), in which case there are no B → γ form factors by helicity conservation and it all reduces to the Low-term corresponding to the emission from the B-meson (and the charged lepton of course). More concretely, at LO, the amplitude factorises where g S−P is an irrelevant constant for our purposes and the dots stand for photon emission from the lepton. We focus on the first term which, by Low's theorem, gives , is the photon's polarisation vector, and by parity, only the γ 5 -part contributes. Two remarks are in order: i) as this is an on-shell matrix element, hence p 2 B = m 2 B ii) the expression (A.4) is exact, as stated before. Now, we would like to see how this works out in our approach (with gauge invariant interpolating operator). The correlation function (3.5) evaluates to (A.5) where the dots once more stand for the photon emission from the lepton (and (p B −p Φ B )· *terms which have to be dropped as they are unphysical). The first term in the p B · *structure corresponds to emission from the Φ B -particle, the second term to emission from the quarks and Π P (s) is the following 2-point function which is further scrutinised in Sec. B. Note that the extra factor of J B arises from the quark part of the weak Lagrangian. The quark emission term in (A.5) can be established without computation as being the unique solution to the QED Ward identity. This goes hand in hand with the statement that (A.4) is exact. Now, we have all the information in order to investigate the differences between the two approaches.
• Hard photon approach {q 2 , p 2 B , k 2 }: in this case q 2 is considered an independent variable and the discontinuity in p 2 B is given by which reproduces the  to the extent that p 2 B → m 2 B , which holds in QCD and approximately for QCD sum rules at the level of quark hadron duality as noticed in [37] (and cf. App. B). The latter is a sufficiently good approximation as long as q 2 is not too close to m 2 B which is the premise of the hard photon QCD sum rule approach.
• Soft photon approach {p B ·k, p 2 B , k 2 }: it is characterised by E γ Λ QCD and it makes more sense to expand around E γ (or better its covariant form p B · k). Eq. (A.5), omitting its arguments, then becomes and formally we traded q 2 for p B · k. Note that, at the end, p B · k is set equal to p Φ B · k = E γ m B The first thing to notice is that in the k → 0 limit, the second term becomes Π P (p 2 B ), the derivative of the 2-point function, and contributes to c 0 , but does not reproduce the c −1 -term in (A.1). 14 This rôle is reserved the Φ B -term! This highlights the necessity of introducing the Φ B -particle in this approach. Most importantly, the two terms in Π P (q 2 ) exactly cancel each other and the Low term appears in its exact form (A.4) (after the LSZ formula in p 2 B is applied).
The example of the S-P interaction almost appears a bit too simple to illustrate the point but in fact it is not. In the V-A case, there would simply be many other terms contributing to the structures in (A.1) other than the Low-term (e.g. [37]).
An interesting question that one could raise is the following: how can taking cuts in p 2 B in the hard photon approach be equivalent to taking cuts in p 2 B itself and an extra cut in q 2 = p 2 B − 2p B · k in the soft photon approach? The answer is that for where s 0 is the (effective) continuum threshold, this extra cut does not actually contribute to the sum rule (cf. (B.6)). This translates to q 2 < 14 GeV 2 for typical values of s 0 = 35 GeV 2 and m b = 4.6 GeV (pole or kinetic scheme m b -mass). This is what is usually assumed in the light-cone expansion indeed (e.g. [37]). Hence for q 2 > 14 GeV 2 , or well-above, the hard photon approach gradually breaks down and the soft photon variables become appropriate.

A.2 Exact LSZ formula and gauge variant interpolating operators
Let us comment on the necessity of gauge invariant interpolating operators. If one aims only at O(α) in a decay process and one has an exact LSZ formula, then it would seem possible to work with gauge variant interpolating operators for hadrons. This is because the hadrons are the QCD hadrons (not corrected by QED) and those states are well-isolated in the spectrum. The Euclidean correlation function then assumes the form where the dots stand for exponentially suppressed terms (cf. (B.9)) and Z B corresponds to the gauge variant LSZ-factor of the B-interpolating operator in use. The exact LSZ formula emerges in the limit of infinite Euclidean time separation of the meson source. In this case, the exponentially suppressed terms disappear as t E → ∞ and the sole gauge dependence is in Z B and can be cancelled by computing it from an appropriate correlation function. This is the idea underlying the QED L [10] and the QED mγ [11] lattice approaches. When aiming for corrections beyond O(α), matters are more delicate since charged states are not isolated anymore (QED IR-problem, see [27] for references) and an explicit gauge invariant formulation seems more appropriate. This is the idea behind the C *boundary method [12]. So far, this method has only been applied to hadronic masses and the specifics for (leptonic) decay rates have not been proposed to date.

B Brief review of the LSZ formalism and QCD sum rule approach
In this appendix, we briefly review the LSZ formalism (e.g. [21]) itself and how it is handled in QCD sum rules [22]. We do so by considering the 2-point function in Eq. (A.6) relevant to B − → −ν γ for S-P interactions. In QCD and perturbative QCD (pQCD), the most important terms for our considerations are where N c are the number of colours. The dots stand for higher states in the spectrum in (B.1) and non-logarithmic terms, condensate terms and O(α s )-contributions in (B.2) (for pQCD, cf. [38] for explicit results). In the LSZ approach, repeating the steps in (2.2), one would extract the amplitude A LO , referred to below (A.4), by where "cut" marks the start of the discontinuity and s 0 is just below the onset of the first excited states which are of the order of (m B + 2m π ) 2 to (m B + m ρ ) 2 . The difference between QCD and QCD sum rules is now most clear as it reduces to which density is in use. The approximation made can be quantified by 15 and is sometimes referred to as semi-global quark-hadron duality [23]. In practice, it may be expected to hold to within 30% and if the pole term dominates by 60%, this leads to an uncertainty of roughly 10% [22]. In practice, most sum rules are ratios of sum rules in fact, such as (3.3), and this effect cancels to a considerable extent. Eq. (B.8) can be expected to work well when the higher spectrum is broad, that is, if there are no further narrow resonances, which is most often the case. At last, it is worthwhile to sketch the analogue of the LSZ formula in Euclidean field theory in which the positive frequency correlation function in the time-momentum representation is considered Above the dots stand for exponentially suppressed terms and Z B has been defined below (B.3). The physical matrix element emerges in the t E → ∞ such that the suppressed terms disappear. 15 It is tempting to take the limit M 2 → 0 as then the higher states would decouple. However, the problem with this is that then the operator products expansion does not converge in that case. M 2 → 0 is in some sense the analogue of infinite Euclidean time separation in lattice QCD.