Parity Doubling as a Tool for Right-handed Current Searches

The V-A structure of the weak interactions leads to definite amplitude hierarchies in exclusive heavy-to-light decays mediated by $b \to (d,s)\gamma$ and $b \to (d,s) \ell \bar{\ell}$. However, the extraction of right-handed currents beyond the Standard Model is contaminated by V-A long-distance contributions leaking into right-handed amplitudes. We propose that these quantum-number changing long-distance contributions can be controlled by considering the almost parity-degenerate vector meson final states by exploiting the opposite relative sign of left- versus right-handed amplitudes. For example, measuring the time-dependent rates of a pair of vector $V(J^P=1^-)$ and axial $A(1^+)$ mesons in $B \to (V,A) \gamma$, up to an order of magnitude is gained on the theory uncertainty prediction, controlled by long-distance ratios to the right-handed amplitude. This renders these decays clean probes to null tests, from the theory side.


Introduction
It is well-known that the V-A structure of the weak interaction leaves characteristic traces in the polarisation of weak decays e.g. [1][2][3]. This is particularly attractive in heavy-to-light flavour-changing neutral currents (FCNCs), such as b → Dγ or b → D ¯ with D = d, s and = e, µ, τ . These transitions therefore make excellent probes to search for new physics with V+A structure, referred to as right-handed currents (RHC) below. Schematically, the effective Hamiltonian for such decays reads where 2d L(R) = (1 ∓ γ 5 )d, and O ( ) r stands for the (r)emaining part of the operator, to be made more precise in section 2. The b → sγ case is recovered by d → s, with according changes in the CKM factors. In the Standard Model (SM), (C /C) SM ∼ m D /m b , whereas in generic New Physics (NP) models (C /C) NP (C /C) SM [4][5][6][7], with current constraints on the electroweak penguin operator around (C 7 /C 7 ) NP ∼ 1/4 [8][9][10]. In the terminology of the Minimal Flavour Violation (MFV) effective field theory framework, the O-and O -type operators, with the above mentioned hierarchies, are referred to as MFV and non-MFV type [11].
Non-perturbative matrix elements, connecting H eff to amplitudes, can dilute the cleanliness of the signal. ForB → V γ, where V is a vector meson (e.g. ρ, K * , . . . ), the form factors, referred to as short-distance (SD) contributions hereafter, obey exact algebraic relations, leading to accidental control in the SD part. However, sizeable tree-level four-quark operators with charm and up quarks, H eff ∼d L γ µ UŪ L γ µ b (U = u, c), induce genuine long-distance (LD) effects, which are more difficult to control. It was argued, based on studying the inclusivē B → X s γ decay, that such contaminations could be rather significant [12], whereas actual computations show smaller effects in exclusive channels [13][14][15][16][17].
In this article, we show that these LD effects can be controlled by a symmetry which in turn also explains the smallness found in the concrete computation [13] quoted just above. The symmetry in question is the chiral restoration limit. The crucial point is that decays of opposite parity, such asB → ρ(1 −− )γ versusB → a 1 (1 ++ )γ, are opposite in sign in the righthanded amplitude between the exact SD and LD contributions (originating from the sizeable V-A part). 1 While decays of axial mesons have received some attention as complementary probes for RHC (e.g. [21][22][23]), we advocate that the combination of the two decay channels allows for a cleaner extraction of the relevant observables controlled by ratios of vector to axial LD amplitudes. In light-cone approaches, this necessitates axial vector meson distribution amplitudes (DAs) [24], whose symmetry relations with vector meson DAs can be studied rather systematically [25].
The paper is organised as follows. In section 2 it is shown, using the path integral, that the fraction of LD-over SD-RHC flips sign for parity doublers. In section 3, the parity doublers are listed (section 3.1), followed by a discussion of the sources of correction to the symmetry limit in section 3.2. Applications to the time-dependent rates ofB → (V, A)γ, a detailed breakdown ofB s → φ(f 1 )γ, and remarks on B → (V, A) ¯ are presented in sections 4.1, 4.2 and 4.3 respectively. The paper ends with conclusions in section 5, including comments on the experimental feasibility of the measurement. A discussion on the chiral order parameter, illustrating some technical aspects of the paper, is given in appendix A.

The use of parity doubling for right-handed current searches
After briefly discussing the structure of B → V γ amplitudes in section 2.1, we demonstrate in section 2.2 how the left-and right-handed amplitudes of opposite parity states come with a relative minus sign.

Chirality hierarchy of amplitudes in the Standard Model
The B → V γ amplitude can be expressed in terms of the two photon polarisations as 2 where Levi-Civita sign convention 0123 = 1, contractions of vectors are understood, and ε and η are the polarisation vectors of the photon 1 We choose to use the ρ and a1 as the prime examples for general discussions on historical grounds, in connection with the Weinberg sum rules [18,19]. Other parity doubling pairs, with considerably smaller widths, are tabulated in section 3.1. For an exhaustive review on the physics and history of parity doubling, we refer the reader to reference [20]. Some more discussion can found in appendix B. 2 The extension to the notation of B → V ¯ is as follows: A L(R) ∼ H∓, with photon polarisation vectors ε(±) = (0, ±1, i, 0)/ √ 2, and the amplitudes are frequently written in terms of √ 2A ⊥( ) = H+ ∓ H− [26,27]. Away from q 2 = 0, one also needs the amplitude A0, corresponding to the longitudinal polarisation of the vector meson or the off-shell photon. and the meson respectively. The bar refers toB-(b → Dγ), where D = (d, s), as opposed to the B-transition (b →Dγ). Each chirality amplitude can then be decomposed into contributions from O and O operators (1):ĀB dropping the superscript for brevity. The V-A interactions imply |Ā χ | |Ā χ |, which is, for example, encoded in C 7 /C 7 = m D /m b in the SM, where the C 7 and C 7 are Wilson coefficients of the effective Hamiltonian (U = u, c and λ (D) with more detailed definitions in appendix C. Normalising to the dominant SD contribution, the amplitudes (3) read with summation over i = u, c implied, −2T 1 (0)S L(R) = V |s L(R) σ · F b|B , and i V,χ includes the ratio of Wilson coefficients and the QCD matrix element but not the CKM contributioñ where by convention, ∆ R ≥ 0, and φ ∆ R is the weak (CP-odd) phase relative to the SM phase originating from λ t . For further discussion, it is convenient to break down the relative parts into the following tablē AB →V γ χĀ SD,χĀLD,χĀ SD,χĀ LD,χ The two zero entries in (7) are due to the algebraic relation σ αβ γ 5 = − i 2 αβγδ σ γδ , which descends to the form-factor relation T 1 (0) = T 2 (0).
The relative importance of the LD contributions in b → Dγ depends on the CKM hierarchy (43). More specifically, theĀ LD,χ are not of major importance, as only the C 8 O 8 -operator contributes, and it was shown in [28] that, at leading twist,Ā LD,L = 0, whileĀ LD,R is at the percent level in the normalisation above. The i V,L,R are the, potentially sizeable, LD contributions. Throughout this presentation we assume i V,L,R 1, which is a circumstance that can be checked experimentally for the i V,R contribution (cf. section 4.2). Crucially, the breakdown (7) reveals that, in a vector-meson final state of definite parity, i V,R cannot be distinguished from the RHC ∆ R e iφ ∆ R . It is the aim of this work to show, however, that ∆ R e iφ ∆ R can be unambiguously identified when two parity-doubler vector meson final states, to be listed in section 3.1, are taken into account. In order to gain some insight, we first discuss the procedure in the chiral symmetry restoration limit, before returning to QCD in section 3.

The chiral symmetry restoration limit
We consider the effect of the chiral symmetry restoration limit on the breakdown (7), using B → ρ versus B → a 1 as a template. In this limit, suppressing the Baryon number U(1) V , the global flavour symmetry of N F fermions is restored: where SU(N F ) V × SU(N F ) A SU(N F ) L × SU(N F ) R , and the dots stand for other SU(N F ) A × U(1) A -violating condensates such as qσ·Gq (see appendix A for further discussion). Let us mention in passing that such a situation can be simulated on the lattice at temperatures above the chiral phase transition, cf. footnote 13 in appendix B. Figure 1: Diagram representing the procedure outlined in the text, using the relation (13), which necessitates both the limits m q → 0 and qq → 0 in (8). The argument only requires that the weak vertex h eff be a local operator, and thus applies to both SD (form-factor) and LD (charm-loop) contributions. The schematic correlation functions on the left and right are exactly equal, from where the information on the matrix elements can be assessed via the LSZ formalism or dispersion relations, when taking into account finite-width effects. Corrections to this exact equality, beyond the chiral symmetry limit, are discussed in section 3.

Path integral representation of matrix elements
In establishing our main result, the path-integral formalism, with quarks propagating in a background gluon field, will prove powerful. 3 The ρ-and a 1 -meson can be represented by the interpolating currents, with isospin I, while the B-meson J P C = 0 −+ can be interpolated by J B =bγ 5 q. The correlation function 4 provides all the necessary information to understand the properties of the matrix element V |H eff |B , e.g. by using the LSZ formalism or dispersion representations in case finite-width effects are to be studied. Above, V I µ stands for either of the interpolating currents defined in (17), and is a schematic substitute for H eff (4), where (v, a) = (1, ±1) correspond to O-and O -like operators. SD and LD charm contributions correspond to O r ∼ 1,cΓ r c for example. Lorentz/colour contractions over Γ and Γ r are suppressed, as these have no impact on our argument. Integrating out the quarks in the path integral, the matrix element (10) assumes the form where is the path integral measure, S(G) the Yang-Mills action, and M f denotes the mass matrix, which comprises of all flavours. The quark propagator in the gluon background field is S in the restoration limit (8). We stress that (13), upon which the argument is based, necessitates both the vanishing SU(2) A × U(1) A -violating condensates and the limit m q → 0, with some more detail on related matters deferred to appendix A.

Relating matrix elements of parity doublers
Now comes the main step, where we replace γ µ → γ µ (γ 5 ) 2 and use (13) to arrive at which can also be written symbolically as This relation translates into the amplitudes (2), with CO-and C O -contributions in (4): Moreover, in terms of the breakdown (7), the relations (14) and (15) lead tō AB →ρ(a 1 )γ χĀ SD,χĀLD,χĀ SD,χĀ LD,χ We wish to emphasise that the procedure in this section leading to (14) works for any local operator of the form in (11), and so, in particular, applies to both the LD and SD contributions, which is reflected in the second amplitude breakdown (16). This argument establishes the main point of this work, and we now turn to the discussion of how this can be applied beyond the symmetry limit.

Beyond the symmetry limit
We briefly discuss the question of how to isolate the relevant phenomenology using parity doubling. 5

Parity doubling for phenomenologically relevant vector mesons
where q = u, d and Lorentz indices have been suppressed on the left-hand side. Vector and axial mesons couple to these currents as where square brackets denote antisymmetrisation in indices. The canonical parity doublers are listed on the horizontal lines. However, by SU(N F ) V flavour symmetry, it will become clear that the measurement of a single parity doubler can reveal important information on primed Wilson coefficients C 7,8 (and C 9,10 ) by disentangling the LD contributions.

Sources of correction to the symmetry limit
In the real world, SU(N F ) A × U(1) A is broken, e.g. by qq µ=1GeV (−0.24(1) GeV) 3 , raising the question of the corrections to (14) and the resultant breakdown (16). We find it advantageous to distinguish two sources: corrections to the γ 5 -trick (13), and those arising from the hadronic parameters of the vector mesons. Corrections to (14), for instance, can be understood by considering the light-cone Operator Product Expansion (OPE), with the interpolating current replaced by a B-meson light-cone DA, in place of the complete path integral (12). In this case, (13) results in both m q -and qq -corrections, along with other condensates. Whereas the former are parametrically small, the latter are suppressed in effect by | qq /M 3 | (1/4) 3 , where M 1 GeV is the Borel mass scale. This suppression can be verified in explicit computations. The remaining corrections to the symmetric breakdown (16) arise from differences in the hadronic parameters, namely the meson mass and the DA parameters. The latter are either (partly) known or are expressible in terms of m q , qq , and higher-dimensional condensates [25]. Moreover, these can be assessed experimentally, cf. section 4. Table 1: Mass (m V ) and width (Γ V ) data for the neutral mesons, from the latest PDG data [29].
Uncertainties in these parameters are also indicated in brackets alongside the central values.
The I G and J P C quantum numbers have also been indicated. The K particles are separated, as they do not have definite G-parity states. The interpolating operators under the O V -column are described in the main text in (17) and and superscripts s andss denote replacements of the light quarks q → s. In addition, there are "exotic" 1 −+ states, with interpolating The π 1 (1400) [29] (I G = 1 − ) is a candidate particle for carrying these quantum numbers. Such states are rather broad, e.g. Γ π1 /m π1 ∼ 1/4, and are not well-studied.

Applications to experimental searches 4.1. Right-handed currents from time-dependent rates
A relevant question is how to test the chirality hierarchy (7). The decay rate does not lend itself to such tests, as the right-handed amplitude is dominated by the left-handed one. The situation is, however, favourable in the case where the B-meson is neutral and undergoes mixing [2] (and/or decays to at least 3 hadrons and a photon [21,[30][31][32]). The mixing is driven by particle and antiparticle having a common final state, e.g.B → V γ L ← B. In this case, one of the amplitudes is chirally suppressed, giving rise to a direct linear behaviour in righthanded amplitudes, Γ mix ∼Ā R , compared to the unfavourable behaviour of the t = 0-rate Γ ∼ |Ā L | 2 + |Ā R | 2 .
The time-dependent rate of a B D meson, produced at t = 0, assuming CPT-invariance and |p/q| = 1, 6 takes the form where ∆Γ D ≡ Γ D the mass difference, of the heavy (H) and light (L) mass eigenstates. The quantities S and C are related to indirect and direct CP violation respectively. In the Particle Data Group (PDG) notation [29], H ≡ A ∆Γ . In terms of the decomposition (2), dropping the superscripts for brevity, these quantities read 6 The quantities p and q describe the transition matrix from mass to flavour eigenstates. In BD-BD mixing they are indeed compatible with the assumption |p/q| = 1, up to negligible corrections [33].
In the SM, there are three weak phases orginating from λ u,c,t , one of which can be eliminated by the unitarity relation λ u + λ c + λ t = 0. Hence, one may writē where the result on the right follows by CP conjugation, and ξ V is the CP eigenvalue of V . Assuming |C 7 | |C 7 | and i V (A)L 1, as previously discussed, then S and H are wellapproximated by where the sines and cosines refer to S and H, and the signs ± follow from the breakdown (16). It is instructive to expand (22) for specific modes. There are four classes of B → V γ decays, due to the choice of initial state meson, B d or B s , while the transition itself can be either b → d or b → s. Since ∆Γ d is too small to have an observable effect, this makes the H B d parameter unobservable in practice. Below, we present the observables (22) for three of these classes, postponing details on B s →K * (b → s) to [13] as the decay is experimentally less attractive. The formulae in (22) take the form −1, and we have used ξ V,A = 1. For S B d →K * (K 1 )γ , either (near) CP eigenstate (K S,L π 0 ) can be observed in the subsequent decay, so we indicate the explicit CP eigenvalue ξ K * (K 1 ) in this case. 7 The vanishing of S Bs→φ(f 1 )γ 0 in the SM comes from the cancelation of all weak phases involved, and this quantity is therefore a null test for weak phases of RHC.
The expressions (23) are for the J P C = 1 ++ parity doublers. If, instead, one were to use doublers with the J P C = 1 +− quantum numbers, then ξ (1 −+ ) = −1, and the corresponding observables S and H pick up an additional minus sign. (23), it can be seen that both observables S and H in (22) are of interest, and can be observed experimentally, in the decay B s → φ(f 1 /h 1 )γ, where f 1 ≡ f 1 (1420) and h 1 ≡ h 1 (1380).

The approach exemplified in
Inserting (26) into (23) and using ξ h 1 = −1, we find and where the initial state in the subscript is dropped for brevity, and (26) Equations (24,25) contain the essential idea of this work. In particular, (24) shows that it is possible to measure the sum of the LD (charm) contributions without any compromise from the symmetry breaking, thanks to the exact form factor relation T 1 (0) = T 2 (0).
The RHC (25) can be extracted modulo the symmetry breaking quantity R i V,A , which we expect to be close to unity, and will be addressed in a future computation [13]. Let us address the last point in some more detail. Assuming c φ,R c f 1 (h 1 ),R , but keeping as a free value δR i V,A , the extraction of ∆ R e iφ ∆ R from the last line (23) is then improved by (25) by a factor (δR c V,A ) −1 . Assuming, for example, that R i V,A 1.2 or 1.5, this leads to an improvement of factors 11 and 5 respectively. The quantity in (24), appearing in (25), can either be taken from a theory computation or from experimental measurements in a data-driven approach.

B → V ¯ and other decay channels
Decays such as B → K * (→ Kπ)µ + µ − are an important probe of NP in the flavour sector [34][35][36][37], as their angular distributions allow the assessment of a wealth of observables, twelve for the dimension-six H eff [38]; higher-dimensional operators are also accessible by extracting higher moments (modulo QED effects) [27].
The parity-doubling approach can be extended to B → (V, A) ¯ rather straightforwardly by considering the angular observables. In the notation of [27], the moments, or angular coefficients, are G l k ,l m , where l k, denote the partial wave of the Kπ and µµ-pair respectively, and m is the relative helicity difference. An interference of left-and right-handed polarisation corresponds to the helicity difference m = 2, which leads to the real and imaginary parts of G l K ,l m → G 2,2 2 being the observables of interest. These have indeed been identified, with the acronyms P 1 = A (2) T ∼ Re[G 2,2 2 ], P 3 ∼ Im[G 2,2 2 ], some time ago (e.g. [39,40]) as giving access to RHC at low q 2 . 8 A measurement of the right-handed LD contribution at q 2 = 0, or at low q 2 , could also provide invaluable information on the angular anomalies (most notably, 1 ]) observed in B → K * ¯ at LHCb, in connection with approaches using analyticity [41,42]. In this respect, B → K * e + e − is an even more promising channel, studied at the LHCb experiment [43], and with good prospects at Belle II. Exploring the potential of time-dependent angular distributions would also seem to be an interesting possibility [44].
Thus, B → (V, A) ¯ allows the assessment of all the right-handed operators O 7,8,9,10 , or their respective Wilson coefficients, without resorting to time-dependent amplitudes, meaning that decay modes for charged mesons can also be assessed in this manner.
We wish to emphasise that the ideas in this paper, whilst they have been discussed primarily in the context of B → (V, A)γ and B → (V, A) ¯ decays, can be extended to other decays of interest, as long as such systems admit parity-doubling partner decays. 9 Examples include the D → V γ( ¯ ) sector, e.g. [45,46], as well as higher-spin states e.g. B → K 2 γ( ¯ ), since parity doubling occurs in those modes [20]. 10 Applications to symmetry-based amplitude parametrisations in B → P V (A), with P being a pseudoscalar meson, are another possibility, although one would expect stronger breaking effects in final-state interactions than in the LD amplitude of radiative decays.

Discussion and conclusions
In this paper, we have advocated that the contamination of right-handed currents in B → V γ( ¯ ) decays due to long-distance effects can be controlled by considering in addition the corresponding decay B → Aγ( ¯ ). It was shown, in the chiral symmetry restoration limit (8), that the leaking of the V-A contributions into the right-handed amplitude for the parity doublers comes with exactly the opposite sign, as compared to the leading short-distance contributions (16).
The case beyond the symmetry limit is briefly discussed in the previous section for B → V (A) ¯ and for B → V (A)γ the summary is as follows. The sum of the vector and axial longdistance contribution can still be extracted from experiment (e.g. (24)), because of the exact form factor relation T 1 (0) = T 2 (0). This then allows to test theory predictions of exclusive long-distance contributions [13,14,16,48] and contrast them with the, somewhat larger, indirect estimation from the inclusive b → X s γ channel [12]. More concretely, the charm contamination was computed to be 0.6% for S B d →K * γ in [16], whereas [12] estimated the contaminations to be up to 6%. In addition, the smallness of the theoretical result can be understood by invoking the parity-doubling limit [13].
In terms of extracting the right-handed currents, parametrised by ∆ R e iφ ∆ R (6), the parity doubling improves the situation by the factor (δR c V,A ) −1 (25,26) , which can be expected to be around 5 to 10 [13]. This implies a theory error for S B d →K * γ just below 0.1%, which is considerably improved and lower than the experimental precision that can be attained in the near future. For Belle II, an uncertainty of 3% is anticipated in this observable with a data set of 50 ab −1 [49]. This estimate does not yet take into account the gain in photon efficiency in differences and sums of rates, relevant to the parity-doubling approach cf. (24,25).
On the experimental side, axial vector meson final states are more challenging, because of their decay chains. However, the measurement of a single channel can still provide invaluable information on the long-distance contributions (24), as the ratios R V,A (26) can be expected to be approximately universal, owing to SU(N F ) V flavour symmetry. Promising states are the K 1 (1270) and the f 1 (1285) or f 1 (1420). Belle has reported a time-dependent measurement of B 0 → ρ 0 K s γ [50] and the rate B(B + → K + 1 (1270)γ) 4.3(9)(9) · 10 −5 [51]. At the LHCb experiment, the K 1 states were seen in B → K + π + π − at the LHCb [52], and the first observation of B (B s,d → f 1 (1285)J/Ψ) 7 · 10 −5 , 8 · 10 −6 was reported in [53].
A. Chiral restoration limit and γ 5 S (q) The pion decay constant, F π 92 MeV in QCD, is defined by π b (p)|J A,a µ |0 = δ ab p µ F π , with J A,a µ =qT a γ µ γ 5 q, and q are N F light quarks. F π is the order parameter of spontaneous chiral symmetry breaking. At the heuristic level, F π = 0 implies the non-invariance of the vacuum with respect to the axial flavour charge Q A,a = V d 3 xJ A,a 0 . In this appendix, we aim to show that our formula (13), depends, as expected, on the restoration limit. This result serves to illustrate (13) in some more detail than discussed in the main text. We first proceed to show that γ 5 S (q) For this purpose, consider the correlation function, used to derive the Weinberg sum rules [18], where J a,L(R) µ = 2qT a γ µ (γ 5 )q L(R) , and (a, b) are SU(N F ) (L,R) flavour indices. One may integrate out the fermion in the path integral formulation to obtain where the hat denotes the Fourier transform, and the path integral measure, already defined below Eq. (12), is Dµ G = DG µ det( / D + iM f )e iS(G) . Using γ 5 S (q) G (w, z)γ 5 (13), one concludes that (Π a,b LR ) µν (x) = 0 by commuting the γ 5 through the fermion propagator and the γ-matrix in (29).
are point-by-point identical. The spectral densities are given by where the dots stand for higher states, and the fact that we used the narrow width approximation for the ρ and a 1 meson is immaterial. The only crucial point is that (30) necessarily implies F π = 0, since there is no massless particle in the vector channel.
In the second part, we show that m q , qq = qσ·Gq = · · · = 0 ⇔ γ 5 S (q) G (w, z)γ 5 , which is what was used in the main text in section 2.2.1. In practical computations, using the OPE one uses the formula [54] q m (x 1 )q n (x 2 ) = N c 1 12 where x 12 ≡ (x 1 − x 2 ), m and n are Dirac indices, V a q V a f = qγ µ t a q ff γ µ t a f , and the dots stand for higher dimensional condensates. We only consider the terms which do not vanish in the m q → 0 limit. It is readily seen that qq and qσ·Gq are obstructions to (13), which is expected since they are not invariant under SU(N F ) A (and U(1) A ). The contrary applies to V a q V a f . The statement about qq can be made slightly more rigorous. Following the argument of [55], the fermion propagator in the external gluon field may be written as where φ n are the Dirac operator eigenmodes i / Dφ n = λ n φ n . Noting that γ 5 φ n is an eigenmode of eigenvalue −λ n , and that the zero eigenmodes are suppressed by V −1/2 in the infinite-volume limit, one finds where the function ρ(λ) is the Dirac eigenmode density. In the limit m q → 0, one obtains the celebrated Banks-Casher relation qq = −sign(m) πρ(0) [55]. Therefore, one concludes that qq = 0 (ρ(0) = 0) is a necessary condition for γ 5 S (q) G (w, z)γ 5 (13) to hold. Similar arguments would apply to further terms in (33), but are more difficult to render rigorous. Since F π = 0 ⇒ m q , qq = · · · = 0, this finally results in (27).

B. Parity doubling
In this section, we provide some minimal background on parity doubling which has a long history in particle physics [20]. Parity doubling achieved its modern paradigm shift with the advent of the Weinberg sum rules [18], partly described in the previous section, and has recently been investigated on the lattice [56][57][58]. 12,13 The basic idea is that a global symmetry, generated by a charge Q, induces degeneracies in the spectrum, as it commutes with the Hamiltonian. Examples include supersymmetry, with degeneracies between bosons and fermions, or simply the global SU(N f ) V flavour symmetry, leading to isospin multiplets. In the restoration limit (8), which leads to the enhanced flavour symmetry SU(N f ) V → SU(N f ) V × SU(N f ) A × U(1) A , the same types of degeneracy can be expected. There is, however, another important point that accompanies this effect: namely, that the additional global symmetry gives rise to new quantum numbers.
For the sake of concreteness, let us choose N f = 2 below. In the case at hand, SU(2) V × SU(2) A SU(2) L ×SU(2) R , this leads to both a left and right-handed isospin quantum number (I L , I R ) instead of just the isospin I V itself. The classification of the representations is discussed in [63]. More precisely, the particles are classified according to the parity-chiral group SU(2) × SU(2) × C i , where C i is the space reflection. The lowest irreducible representations (I L , I R ), of dimension (2I L + 1)(2I R + 1), are listed in figure 2. The splitting from left to right can be understood as the branching rule of (I L , I R )| SU V (2) ; e.g.
The two interpolating currents discussed as templates in the text (17) correspond to the (1, 0)⊕(0, 1) multiplet. The two (1/2, 1/2) representations denoted by superscripts a and b are distinct by the parity operation. The use of and ⊥ as superscripts is non-standard, and inspired by the notation of the corresponding decay constants. As a last generic remark, let us add that, in the real world, the ρ and the ρ = ρ(1450) are admixtures of the ρ and ρ ⊥ -states. In the following subsection, we give an example where one can explicitly see how the U(1) A × SU(2) A -violating condensates control differences in the hadronic data between the ρ and a 1 .

B.1. The Weinberg sum rules as an example
From group-theoretic considerations, one can argue that the first correction to the chirality correlation function (28) is given by where q 2 ≡ −Q 2 (e.g. [66]), 4O χSB =qT 3 λ i q Lq T 3 λ i q R , and λ i are the SU(3) colour matrices. One can then power-expand the denominator and arrive at the first two Weinberg sum rules ∞ 0 s n (ρ V (s) − ρ A (s)) = 0 , n = 0, 1 .
The leading correction, or the third sum rule, is given by So far, everything is exact. Since the correlation functions are well-described at high q 2 by perturbation theory, which is equal for the V -and the A-channel, ρ V (s) ρ A (s) will hold for some s > s 0 . Weinberg [18] assumed that s 0 is just above the a 1 resonance, and restricted himself to the parametrisations found in (32), from which he deduced along with enhanced symmetries, has been found in lattice simulations with truncated eigenmodes of the Dirac operator [56,57]. The motivation for this truncation is that the lowest eigenvalues are related to the breaking of chiral symmetry by the Banks-Casher relation [55]. The restoration of the flavour symmetry, and some of the symmetry enhancements, have been confirmed by the above-mentioned finite-temperature simulations [58]. Whereas more is to be learnt from this interesting topic in the future, the precise outcomes are not important for our practical purposes. However, if one were able to perturb in qq and mq, then the exact limit would be of interest. Figure 2: Lowest-lying particles coupling to vector and tensor currents of isosinglet-and isotriplettype, making a total of 16 vector mesons. For simple comparison, we use the same graphic presentation with similar notation as used in [56]. The interpolating operators are defined in (17) where the coupling to the states is specified further below. In the restoration limit (8), one would expect a 4-, 4-and 6-plet degeneracy by the restoration of the SU(2) A (red arrows). In the case where U(1) A (blue arrows) symmetry is also restored, this leads to a 8and 6-plet degeneracy. The actual degeneracy is found to be larger [56], compatible with an emergent SU(4) [64] (or even SU(4) × SU(4) [65]) symmetry.

SU(2)
These sum rules are rather well-satisfied at the empirical level. The last equation nicely illustrates, in a concrete setting, how the restoration limit is controlled by the SU(2) A × U(1) A -violating condensate O χSB . We note that, in the vacuum factorisation approximation,

C. Definition of effective Hamiltonian
Here we describe in more detail the effective Hamiltonian for b → (d, s)γ and b → (d, s) ¯ decays, clarifying the notation in (4). In the basis of [67], the operators contributing to (4) are: the four-quark tree-level operators the loop-induced four-quark operators, O 3,..., 6 : and the electromagnetic and QCD dipoles (with D µ = ∂ µ − ig s G µ − ieA µ convention) The C i are scale-dependent Wilson coefficients, which can be calculated perturbatively using renormalisation group methods (e.g. [67,68]). The most relevant C i for the discussion in the paper are (C 2 , C 1 , C 7 )(m b ) (1, −0.13, −0.37).