Dispersion relation for hadronic light-by-light scattering: two-pion contributions
Gilberto Colangelo, Martin Hoferichter, Massimiliano Procura, Peter Stoffer
IINT-PUB-17-009CERN-TH-2017-041NSF-KITP-17-036
Dispersion relation for hadronic light-by-light scattering:two-pion contributions
Gilberto Colangelo a , Martin Hoferichter b,c , Massimiliano Procura d † , Peter Stoffer e,fa Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics,University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland b Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA c Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA d Theoretical Physics Department, CERN, Geneva, Switzerland e Helmholtz-Institut für Strahlen- und Kernphysik (Theory) and Bethe Center for Theoretical Physics,University of Bonn, 53115 Bonn, Germany f Department of Physics, University of California at San Diego, La Jolla, CA 92093, USA
Abstract
In this third paper of a series dedicated to a dispersive treatment of the hadronic light-by-light (HLbL)tensor, we derive a partial-wave formulation for two-pion intermediate states in the HLbL contribution to theanomalous magnetic moment of the muon ( g − µ , including a detailed discussion of the unitarity relation forarbitrary partial waves. We show that obtaining a final expression free from unphysical helicity partial wavesis a subtle issue, which we thoroughly clarify. As a by-product, we obtain a set of sum rules that could beused to constrain future calculations of γ ∗ γ ∗ → ππ . We validate the formalism extensively using the pion-boxcontribution, defined by two-pion intermediate states with a pion-pole left-hand cut, and demonstrate how thefull known result is reproduced when resumming the partial waves. Using dispersive fits to high-statistics datafor the pion vector form factor, we provide an evaluation of the full pion box, a π -box µ = − . × − . Asan application of the partial-wave formalism, we present a first calculation of ππ -rescattering effects in HLbLscattering, with γ ∗ γ ∗ → ππ helicity partial waves constructed dispersively using ππ phase shifts derived fromthe inverse-amplitude method. In this way, the isospin- part of our calculation can be interpreted as thecontribution of the f (500) to HLbL scattering in ( g − µ . We argue that the contribution due to charged-pionrescattering implements corrections related to the corresponding pion polarizability and show that these aremoderate. Our final result for the sum of pion-box contribution and its S -wave rescattering corrections reads a π -box µ + a ππ,π -pole LHC µ,J =0 = − × − . † On leave from the University of Vienna. a r X i v : . [ h e p - ph ] M a y ontents ( g − µ . . . . . . . . . . . . . . . . . . 52.1.1 BTT decomposition of the HLbL tensor . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Master formula for the HLbL contribution to ( g − µ . . . . . . . . . . . . . . 82.2 Dispersion relations for the HLbL tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Mandelstam representation for HLbL . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Two-pion contributions beyond the pion box . . . . . . . . . . . . . . . . . . . . 122.2.3 Single-variable dispersion relation for two-pion contributions . . . . . . . . . . . 162.3 Sum rules for the BTT scalar functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Relation to observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.1 Construction of the singly-on-shell basis . . . . . . . . . . . . . . . . . . . . . . 212.4.2 Scalar functions for the two-pion dispersion relations . . . . . . . . . . . . . . . 232.4.3 Physical sum rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 Helicity amplitudes and partial-wave expansion . . . . . . . . . . . . . . . . . . . . . . 262.5.1 Unitarity relation in the partial-wave picture . . . . . . . . . . . . . . . . . . . 262.5.2 Approximate partial-wave sum rules . . . . . . . . . . . . . . . . . . . . . . . . 292.5.3 Result for arbitrary partial waves . . . . . . . . . . . . . . . . . . . . . . . . . . 302.6 Summary of the formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 γ ∗ γ ∗ → ππ helicity partial waves from the inverse-amplitude method . . . . . . . . . . 404.2 A first numerical estimate of the ππ -rescattering contribution to ( g − µ . . . . . . . . 434.3 Role of the pion polarizabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ( g − µ A.1 Tensor structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48A.2 Scalar functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
B New kernel functions for the master formula 49C Feynman-parameter representation of the pion box 51D Scalar functions for the two-pion dispersion relations 52E Basis change and sum rules 53
E.1 Unphysical polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53E.2 Comparison to forward-scattering sum rules . . . . . . . . . . . . . . . . . . . . . . . . 542
Basis change to helicity amplitudes 57
F.1 Calculation of tensor phase-space integrals . . . . . . . . . . . . . . . . . . . . . . . . . 57F.2 Direct matrix inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
G Partial-wave expansion of the γ ∗ γ ∗ → ππ pion-pole contribution 61H Pion polarizability and γγ → ππ in ChPT 63References 64 The long-standing discrepancy between the standard-model determination and the experimental mea-surement [1] (updated to the latest muon–proton magnetic moment ratio [2]) a exp µ = 116 592 089(63) × − (1.1)of the anomalous magnetic moment of the muon ( g − µ has triggered substantial interest in the subjecton both the theoretical and the experimental side. The ongoing E989 experiment at Fermilab [3] aswell as complementary efforts by J-PARC E34 [4] aim at improving the precision by a factor of , see [5]for a detailed account of the experimental strategies in both cases. On the theory side, the uncertaintyis dominated by hadronic effects [6–8], while QED [9] and electroweak [10] contributions are undercontrol at the level of at least × − . Currently, the dominant source of hadronic uncertainties ishadronic vacuum polarization (HVP) at O ( α ) in the fine-structure constant, closely followed by the O ( α ) hadronic light-by-light (HLbL) contribution, depicted in Fig. 1, and with higher-order insertionsof the same hadronic amplitudes already under sufficient control [11–14]. In view of improved datainput for the dispersion relation for HVP [15], it is likely that the stumbling block will eventuallybecome the sub-leading HLbL contribution.Current estimates for HLbL scattering in ( g − µ are largely based on hadronic models [16–27],which despite implementing different limits of QCD, such as large- N c , chiral symmetry, or constraintsfrom perturbative QCD, all involve a certain amount of uncontrollable uncertainties without offeringa systematic path forward. In order to improve the determination of the HLbL contribution, weproposed a dispersive framework [28], based on the fundamental principles of analyticity, unitarity,gauge invariance, and crossing symmetry, which opens up a path towards a data-driven evaluation [29].As the next step [30, 31], we presented a comprehensive solution to the task of constructing a basisfor the HLbL tensor devoid of kinematic singularities, defining scalar functions that are amenable toa dispersive treatment. In particular, we derived a Lorentz decomposition of the HLbL tensor thatmanifestly implements crossing symmetry and gauge invariance, with scalar coefficient functions freeof kinematic singularities and zeros that fulfill the Mandelstam double-spectral representation. In thisframework, we worked out how to define unambiguously and in a model-independent way both thepion-pole and the pion-box contribution. With pion- as well as η -, η (cid:48) -pole contributions determined by their doubly-virtual transition formfactors, which by themselves are strongly constrained by unitarity, analyticity, and perturbative QCDin combination with experimental data [38–46], we here apply our framework to extend the partial-waveformulation of two-pion rescattering effects for S -waves [28] to arbitrary partial waves. To this end, weidentify a special set of (unambiguously defined) scalar functions that fulfill unsubtracted dispersionrelations and can be expressed as linear combinations of helicity amplitudes. Their imaginary part, theinput required in the dispersion relations, is provided in terms of helicity partial waves for γ ∗ γ ∗ → ππ For a dispersive approach not for the HLbL tensor, but for the Pauli form factor instead see [32]. Complementary tothe dispersive approach, a model-independent determination of the HLbL contribution could be achieved using latticeQCD, see [33–37] for recent progress in this direction. igure 1: HLbL contribution to the anomalous magnetic moment of the muon ( g − µ . by means of unitarity. Working out explicitly the basis change to the helicity amplitudes, we generalizethe unitarity relation derived in [28] up to D -waves only to arbitrary partial waves. We demonstratethat indeed the summation of the partial waves reproduces the known full result for the pion box, towhich the ππ -rescattering contribution is expected to produce the dominant correction. We providethe details of a first numerical analysis [47] of these rescattering effects based on helicity partial wavesfor γ ∗ γ ∗ → ππ that we construct dispersively from a pion-pole left-hand cut (LHC) and ππ phase shiftsfrom the inverse-amplitude method, an approach that isolates pure ππ contributions and thus, in theisospin- channel, provides an estimate for the impact of the f (500) resonance on HLbL scattering.In the same way, our γ ∗ γ ∗ → ππ amplitudes reproduce the phenomenological value for the charged-pion polarizability, thereby clarifying the role of the associated corrections in ( g − µ [48–50]. Theseresults lay the groundwork for a future global analysis of two-meson intermediate states in the HLbLcontribution.The outline is as follows: Sect. 2 is devoted to a thorough derivation of partial-wave dispersionrelations for the HLbL tensor, with tensor decomposition, dispersion relations, sum rules, and partial-wave expansion addressed in Sects. 2.1–2.5. A short summary of the strategy is provided at thebeginning of Sect. 2, complemented by a summary of the most important results in Sect. 2.6. InSect. 3, a numerical evaluation of the pion box is provided based on fits of the pion vector form factorto high-statistics time-like and space-like data. The pion box is further used to explicitly verify thegeneral results derived in Sect. 2, in particular to demonstrate the convergence of the partial-waveexpansion for its contribution to ( g − µ . Rescattering corrections to the pion box are discussedin Sect. 4, including a numerical analysis of the S -wave contribution, before we conclude in Sect. 5.Further details of the formalism are provided in the Appendices. In this section, we derive the formalism for the evaluation of the HLbL two-pion contribution to ( g − µ . The goal of our treatment is to relate this contribution to helicity partial waves for the sub-process γ ∗ γ ∗ → ππ , which in principle are measurable input quantities or at least can be reconstructeddispersively.The outline of this derivation is illustrated as a flowchart in Fig. 2. The first step is the decom-position of the HLbL tensor into Lorentz structures and scalar functions that are free of kinematicsingularities and zeros. We have solved this problem in [31] and recapitulate the results in Sect. 2.1.This representation, referred to as BTT tensor decomposition [51, 52] in Fig. 2, allows us to write theHLbL contribution to ( g − µ in full generality as a master formula that involves only three inte-grals. This master formula (2.25) applies to any conceivable HLbL tensor, as long as it is consistentwith general properties that should be fulfilled by any admissible HLbL amplitude: gauge invariance,crossing symmetry, and the principle of maximal analyticity [53], i.e. the principle that the scatteringamplitude can be represented by a complex function that exhibits no further singularities except forthose required by unitarity and crossing symmetry. Any such singularities are of dynamical origin, and4hus have to be contained within the scalar functions ¯Π i in the master formula. Phrased differently, if agiven amplitude for the HLbL tensor cannot be expressed in the BTT basis, e.g. due to the appearanceof kinematic singularities, this automatically implies that this amplitude is at odds with said generalproperties.The dynamics of HLbL scattering is thus contained in the scalar functions, which are the objectsthat we describe dispersively. In [31], we have used the Mandelstam representation for the scalarfunctions to study the pion-box contribution. In Sect. 2.2, we extend the dispersive treatment andderive from the Mandelstam representation single-variable dispersion relations for general two-pioncontributions. Combining these single-variable dispersion relations with unitarity constraints requiresa basis change to helicity amplitudes, since the partial-wave unitarity relation becomes diagonal onlyfor definite helicity amplitudes. However, this basis change is complicated by the appearance of redun-dancies in the representation which, together with the requirement that longitudinal polarizations foron-shell photons not contribute in the final HLbL representation, necessitates a more careful study ofthe BTT scalar functions and their relation to helicity amplitudes. The solution to this problem is theexplicit derivation of a basis that removes all redundancies and apparent contributions from unphysicalpolarizations, which is presented in Sect. 2.4. As a by-product we obtain a set of physical sum rulesto be fulfilled by the scalar functions and thereby the helicity amplitudes.After the basis change to helicity amplitudes, we can then employ the unitarity relation to determinethe imaginary parts in the dispersion integrals in terms of helicity amplitudes for γ ∗ γ ∗ → ππ . Inparticular, we perform a partial-wave expansion of the helicity amplitudes and generalize the S -waveresult of [28] to arbitrary partial waves, which is the main result of Sect. 2.5. In performing thisanalysis the partial waves for γ ∗ γ ∗ → ππ are treated as known, given quantities, which unfortunatelythey are not. The lack of experimental information can be partly compensated by theory constraints,in particular by dispersion relations in the form of Roy–Steiner equations [54–57]. A simplified, S -wavevariant of these will be solved in Sect. 4.A summary of the main results is provided in Sect. 2.6, including a glossary of the notation forthe scalar functions. The subtleties in the various basis changes unfortunately require the introductionof different sets of scalar functions, whose dimension, defining equation, and main properties aresummarized in Table 1. ( g − µ In this subsection, we recapitulate the decomposition of the HLbL tensor into a sum of gauge-invariantLorentz structures times scalar functions that are free of kinematic singularities. We slightly modifyand improve the master formula presented in [30, 31] in such a way that crossing symmetry betweenall three off-shell photons remains manifest. The dynamical input in the master formula is encoded inonly six different scalar functions and their crossed versions.
The HLbL tensor is defined as the hadronic Green’s function of four electromagnetic currents in pureQCD: Π µνλσ ( q , q , q ) = − i (cid:90) d x d y d z e − i ( q · x + q · y + q · z ) (cid:104) | T { j µ em ( x ) j ν em ( y ) j λ em ( z ) j σ em (0) }| (cid:105) , (2.1)where the electromagnetic current includes the three lightest quarks: j µ em := ¯ qQγ µ q, q = ( u, d, s ) T , Q = diag (cid:18) , − , − (cid:19) . (2.2)5TT tensor decompositionSects. 2.1 and 2.4 master formula (2.25): a HLbL µ = (cid:90) d ˜Σ dr dφ ( . . . ) (cid:88) i =1 T i ¯Π i Mandelstam representationSect. 2.2 ππ dispersion relation (2.36), (2.64): ˇΠ i ( s ) = (cid:90) ds (cid:48) Im ˇΠ i ( s (cid:48) ) s (cid:48) − s unitarity relationSect. 2.5 imaginary parts of scalar functions (2.87):Im ˇΠ i ( s ) ∝ (cid:88) j ˇ c ij (cid:88) J ( h J,λ λ h ∗ J,λ λ ) j Roy–Steiner equations for γ ∗ γ ∗ → ππ Sect. 4 helicity partial waves for γ ∗ γ ∗ → ππ : h J,λ λ ( s, q , q ) Figure 2:
Outline of the formalism for the HLbL two-pion contribution to ( g − µ . The dashed lines denotea derivation or calculation, the double lines indicate the insertion of results. The hadronic contribution to the helicity amplitudes for (off-shell) photon–photon scattering isgiven by the contraction of the HLbL tensor with polarization vectors: H λ λ ,λ λ = (cid:15) λ µ ( q ) (cid:15) λ ν ( q ) (cid:15) λ λ ∗ ( − q ) (cid:15) λ σ ∗ ( q )Π µνλσ ( q , q , q ) , (2.3)where q = q + q + q .The usual Mandelstam variables s := ( q + q ) , t := ( q + q ) , u := ( q + q ) (2.4)fulfill the linear relation s + t + u = (cid:88) i =1 q i =: Σ . (2.5)Gauge invariance requires the HLbL tensor to satisfy the Ward–Takahashi identities { q µ , q ν , q λ , q σ } Π µνλσ ( q , q , q ) = 0 . (2.6)Based on a recipe by Bardeen, Tung [51], and Tarrach [52] (BTT), we have derived in [30, 31] adecomposition of the HLbL tensor Π µνλσ = (cid:88) i =1 T µνλσi Π i , (2.7)6ith tensor structures reproduced here for completeness (all remaining ones follow from crossing sym-metry [31]) T µνλσ = (cid:15) µναβ (cid:15) λσγδ q α q β q γ q δ ,T µνλσ = (cid:16) q µ q ν − q · q g µν (cid:17)(cid:16) q λ q σ − q · q g λσ (cid:17) ,T µνλσ = (cid:16) q µ q ν − q · q g µν (cid:17)(cid:16) q · q (cid:16) q λ q σ − q · q g λσ (cid:17) + q λ q σ q · q − q λ q σ q · q (cid:17) ,T µνλσ = (cid:16) q µ q ν − q · q g µν (cid:17)(cid:16) q · q (cid:16) q λ q σ − q · q g λσ (cid:17) + q λ q σ q · q − q λ q σ q · q (cid:17) ,T µνλσ = (cid:16) q µ q ν − q · q g µν (cid:17)(cid:16) q λ q · q − q λ q · q (cid:17)(cid:16) q σ q · q − q σ q · q (cid:17) ,T µνλσ = (cid:16) q µ q · q − q µ q · q (cid:17) (cid:16) q ν q λ q σ − q ν q λ q σ + g λσ ( q ν q · q − q ν q · q )+ g νσ (cid:16) q λ q · q − q λ q · q (cid:17) + g λν ( q σ q · q − q σ q · q ) (cid:17) ,T µνλσ = q σ (cid:16) q · q q · q q µ g λν − q · q q · q q ν g λµ + q µ q ν (cid:16) q λ q · q − q λ q · q (cid:17) + q · q q µ q ν q λ − q · q q µ q ν q λ + q · q q · q (cid:16) q ν g λµ − q µ g λν (cid:17) (cid:17) − q λ (cid:16) q · q q · q q µ g νσ − q · q q · q q ν g µσ + q µ q ν ( q σ q · q − q σ q · q )+ q · q q µ q ν q σ − q · q q µ q ν q σ + q · q q · q ( q ν g µσ − q µ g νσ ) (cid:17) (2.8) + q · q (cid:16) (cid:16) q λ q µ − q · q g λµ (cid:17) ( q ν q σ − q · q g νσ ) − (cid:16) q λ q ν − q · q g λν (cid:17) ( q µ q σ − q · q g µσ ) (cid:17) . The BTT decomposition has the following properties: • all the Lorentz structures fulfill the Ward–Takahashi identities, i.e. { q µ , q ν , q λ , q σ } T iµνλσ ( q , q , q ) = 0 , ∀ i ∈ { , . . . , } , (2.9) • there are only seven distinct Lorentz structures, the remaining 47 ones are crossed versionsthereof, • the scalar functions Π i are free of kinematic singularities and zeros.The first two properties make gauge invariance and crossing symmetry manifest, while the third prop-erty provides the foundation for writing dispersion relations: in a dispersive treatment, we exploit theanalytic structure of the scalar functions dictated by unitarity and we have to make sure that thesingularity structure due to the hadronic dynamics is not entangled with kinematic singularities.Since the number of helicity amplitudes for fully off-shell photon–photon scattering is 41, the setof 54 structures { T µνλσi } does not form a basis, but exhibits a 13-fold redundancy, as we discussedin detail in [31]. While 11 linear relations hold in general, two additional ones are present in fourspace-time dimensions [58]. Away from four space-time dimensions, a subset of 43 Lorentz structuresforms a basis: Π µνλσ = (cid:88) i =1 B µνλσi ˜Π i , (2.10)where the basis-coefficient functions ˜Π i are no longer free of kinematic singularities. However, theexplicit structure of their kinematic singularities follows from the projection of the BTT decompositiononto this “basis.” 7 .1.2 Master formula for the HLbL contribution to ( g − µ Based on a projection technique in Dirac space, one can extract the HLbL contribution to a µ :=( g − µ / from the following expression: a HLbL µ = − e m µ (cid:90) d q (2 π ) d q (2 π ) q q ( q + q ) p + q ) − m µ p − q ) − m µ × Tr (cid:16) ( /p + m µ )[ γ ρ , γ σ ]( /p + m µ ) γ µ ( /p + /q + m µ ) γ λ ( /p − /q + m µ ) γ ν (cid:17) × (cid:88) i =1 (cid:18) ∂∂q ρ T iµνλσ ( q , q , q − q − q ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) q =0 Π i ( q , q , − q − q ) . (2.11)There are only 19 independent linear combinations of the structures T µνλσi that contribute to ( g − µ ,hence we can make a basis change in the 54 structures Π µνλσ = (cid:88) i =1 T µνλσi Π i = (cid:88) i =1 ˆ T µνλσi ˆΠ i , (2.12)in such a way that in the limit q → the derivative of 35 structures ˆ T µνλσi vanishes. Since the loopintegral and the propagators are symmetric under q ↔ − q , in [31] we made sure to preserve crossingsymmetry under exchange of q and q , but did not yet exploit the fact that it is even possible to preservecrossing symmetry between all three off-shell photons—the limit q → singles out one of the photons,but the remaining three are completely equivalent. For the sake of simplifying further calculations,we present here new structures ˆ T µνλσi and the corresponding scalar functions ˆΠ i , superseding the onesgiven in [31].The 19 structures ˆ T µνλσi contributing to ( g − µ can be chosen as follows: ˆ T µνλσi = T µνλσi , i = 1 , . . . , , , , , , , , , ˆ T µνλσ = 13 (cid:16) T µνλσ + T µνλσ + T µνλσ (cid:17) . (2.13)The 35 structures (cid:8) ˆ T µνλσi (cid:12)(cid:12) i = 12 , , , . . . , , , . . . , , , (cid:9) (2.14)do not contribute to ( g − µ and are given in App. A.The set of 19 linear combinations of scalar functions that give a contribution to ( g − µ is definedby (replacing Eq. (D.1) in [31]) ˆΠ = Π + q · q Π , ˆΠ = Π − q · q (Π − Π ) − q · q (Π − Π ) + q · q q · q Π , ˆΠ = Π − Π + q · q Π , ˆΠ = Π + Π + Π − Π , ˆΠ = Π + Π + Π , ˆΠ = Π − Π + Π , (2.15)together with the crossed versions thereof ˆΠ = C (cid:2) ˆΠ (cid:3) , ˆΠ = C (cid:2) ˆΠ (cid:3) , ˆΠ = C (cid:2) ˆΠ (cid:3) , ˆΠ = C (cid:2) ˆΠ (cid:3) , ˆΠ = C (cid:2) ˆΠ (cid:3) , ˆΠ = C (cid:2) C (cid:2) ˆΠ (cid:3)(cid:3) , ˆΠ = C (cid:2) ˆΠ (cid:3) , ˆΠ = C (cid:2) ˆΠ (cid:3) , ˆΠ = C (cid:2) C (cid:2) ˆΠ (cid:3)(cid:3) , ˆΠ = C (cid:2) ˆΠ (cid:3) , ˆΠ = C (cid:2) ˆΠ (cid:3) , ˆΠ = −C (cid:2) ˆΠ (cid:3) , ˆΠ = C (cid:2) ˆΠ (cid:3) , (2.16)8here the crossing operators C ij exchange momenta and Lorentz indices of the photons i and j , e.g. C [ f ] := f ( µ ↔ ν, q ↔ q ) , C [ f ] := f ( µ ↔ σ, q ↔ − q ) . (2.17)The following intrinsic crossing symmetries are preserved (we do not list the symmetries involving thefourth photon): ˆΠ = C (cid:2) ˆΠ (cid:3) , ˆΠ = C (cid:2) ˆΠ (cid:3) , ˆΠ = C (cid:2) ˆΠ (cid:3) , ˆΠ = C (cid:2) ˆΠ (cid:3) = C (cid:2) ˆΠ (cid:3) = . . . , ˆΠ = −C (cid:2) ˆΠ (cid:3) , (2.18)where the dots denote three more crossing relations that follow from the given ones. Hence, thescalar functions ˆΠ i contributing to ( g − µ fall into only six distinct classes that are closed undercrossing symmetry of the off-shell photons 1, 2, and 3. Apart from ˆΠ , which is fully symmetric, therepresentatives in (2.15) are picked because they share a common property: their s -channel is specialas follows from the observation that the corresponding Lorentz structures ˆ T µνλσi are (anti-)symmetricunder either C or C (or both). This is reflected in the intrinsic crossing symmetries (2.18). The HLbL contribution to ( g − µ can now be written as a HLbL µ = − e (cid:90) d q (2 π ) d q (2 π ) q q ( q + q ) p + q ) − m µ p − q ) − m µ × (cid:88) i ∈ G ˆ T i ( q , q ; p ) ˆΠ i ( q , q , − q − q ) , (2.19)where G := { , . . . , , , , , , , , , } and ˆ T i ( q , q ; p ) := 148 m µ Tr (cid:16) ( /p + m µ )[ γ ρ , γ σ ]( /p + m µ ) γ µ ( /p + /q + m µ ) γ λ ( /p − /q + m µ ) γ ν (cid:17) × (cid:18) ∂∂q ρ ˆ T iµνλσ ( q , q , q − q − q ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) q =0 . (2.20)As in [31], we perform a Wick rotation, average the result over the direction of the Euclidean four-momentum of the muon, and use the Gegenbauer polynomial technique [59] to perform five of the eightintegrals in full generality, i.e. without prior knowledge of the functions ˆΠ i . The symmetry propertiesof the loop integral and the kernels ˆ T i under q ↔ − q allow us to write the master formula for theHLbL contribution to ( g − µ containing a sum of only 12 terms: a HLbL µ = 2 α π (cid:90) ∞ dQ (cid:90) ∞ dQ (cid:90) − dτ (cid:112) − τ Q Q
32 12 (cid:88) i =1 T i ( Q , Q , τ ) ¯Π i ( Q , Q , τ ) , (2.21)where Q := | Q | and Q := | Q | denote the norm of the Euclidean four-vectors. The 12 scalarfunctions ¯Π i are a subset of the functions ˆΠ i : ¯Π = ˆΠ , ¯Π = ˆΠ , ¯Π = ˆΠ , ¯Π = ˆΠ , ¯Π = ˆΠ , ¯Π = ˆΠ , ¯Π = ˆΠ , ¯Π = ˆΠ , ¯Π = ˆΠ , ¯Π = ˆΠ , ¯Π = ˆΠ , ¯Π = ˆΠ . (2.22)They have to be evaluated for the reduced ( g − µ kinematics s = q = − Q = − Q − Q Q τ − Q , t = q = − Q , u = q = − Q , q = 0 . (2.23)Due to the basis change, the kernel functions T i differ slightly from the ones given in [31]. Weprovide the explicit expressions in App. B. The composition of two crossing operators is understood to act e.g. in the following way: C [ C [ f ( q , q , q , q )]] = C [ f ( q , q , q , q )] = f ( q , q , q , q ) . ˆ T µνλσ is symmetric under C , but not under C . One could split the six elements in the crossing class of ˆΠ intotwo classes, one with an additional even, one with an odd intrinsic crossing symmetry. = 0 Q = Q = r = 0 r = 1 φ = πφ = π φ = π τ = 1 τ = − Figure 3:
Integration region for ( g − µ . The border of the integration region is at r = 1 and correspondsto τ = − for π/ < φ < π/ (solid gray line) and τ = 1 otherwise (dashed gray line). The angles φ = π/ , φ = π , and φ = 5 π/ correspond to Q = Q , Q = Q , and Q = Q , respectively. The three points where oneof the Q i is zero are singularities of the integration kernels. The height of the equilateral triangle is given by ˜Σ . In [60] a different parametrization of the ( g − µ integration region has been proposed, which provedadvantageous for the numerical implementation. We perform the following variable transformation inthe master formula (note that ˜Σ = − Σ is the sum of the squared Euclidean virtualities, whereas Σ denotes the sum of the squared Minkowskian virtualities): Q = ˜Σ3 (cid:16) − r φ − r √ φ (cid:17) ,Q = ˜Σ3 (cid:16) − r φ + r √ φ (cid:17) ,Q = Q + 2 Q Q τ + Q = ˜Σ3 (1 + r cos φ ) . (2.24)The range of integration is then ˜Σ ∈ [0 , ∞ ) , r ∈ [0 , , and φ ∈ [0 , π ] . The integration region inthe Mandelstam plane and the meaning of the variables is illustrated in Fig. 3. After the variabletransformation, the master formula becomes a HLbL µ = α π (cid:90) ∞ d ˜Σ ˜Σ (cid:90) dr r (cid:112) − r (cid:90) π dφ (cid:88) i =1 T i ( Q , Q , τ ) ¯Π i ( Q , Q , τ ) , (2.25)where Q , Q , and τ are understood as functions of ˜Σ , r , and φ .The master formula for the HLbL contribution to ( g − µ is exact and completely general: givenany representation of the HLbL tensor, one can project out the six scalar functions ˆΠ i in (2.15). Usingthese and their crossed versions, one can construct the 12 scalar functions ¯Π i in (2.22), which encodethe entire dynamical content of HLbL scattering relevant for ( g − µ . After their insertion into themaster formula (2.25), only a three-dimensional integral has to be carried out.In a next step, we aim at reconstructing the scalar functions ¯Π i using dispersive methods, whichwill be the content of the remainder of this section.10 + + . . . Figure 4:
Intermediate states in the direct channel: pion pole and two-pion cut.
In this subsection, we discuss the dispersive framework that we employ for the reconstruction of thescalar functions. The starting point is the Mandelstam representation, which is a double-dispersionrelation. Unitarity allows us to write the HLbL tensor as a sum of contributions from different in-termediate states. After reviewing in Sect. 2.2.1 the most important properties of the pion-pole andpion-box contributions, we continue by considering general two-pion intermediate states in Sect. 2.2.2.In order to calculate the two-pion contributions beyond the pion box, input on the sub-process γ ∗ γ ∗ → ππ is needed. This input will be in the form of helicity partial waves which, in principle,could be measured or, in the absence of data on the doubly-virtual process, have to be reconstructeddispersively [54–57]. The partial-wave expansion turns, however, the amplitude into a polynomial inthe crossed-channel Mandelstam variables, i.e. the cut structure in the crossed channel due to heavier(e.g. multi-pion) intermediate states gets lost. Therefore, with γ ∗ γ ∗ → ππ helicity partial waves asinput, one has to use a single-variable dispersion relation. We derive in Sect. 2.2.3 a suitable form forsuch a dispersion relation that follows from the Mandelstam representation. In [31], we have used Mandelstam’s double-spectral representation [61] for the BTT scalar functions Π i in order to split the HLbL contribution to ( g − µ into the following sum: a HLbL µ = a π -pole µ + a π -box µ + a ππµ + . . . (2.26)This sum directly reflects the sum over intermediate states in the unitarity relation in which, bydefinition, all intermediate states enter on-shell. While unitarity alone defines the imaginary parts, thereal parts are obtained from the dispersion integrals. In short, this amounts to the following procedure: • Write down the unitarity relation for the HLbL tensor. • In the sum over intermediate (on-shell) states, the one-pion state contributes as a δ -function tothe imaginary part, which offsets the dispersion integral and defines the π -pole contribution. • The next-heavier intermediate state in the unitarity relation is a two-pion state. So far, weconcentrate on one- and two-pion intermediate states, shown in Fig. 4. • In the two-pion contribution, write down the crossed-channel unitarity relation for the sub-process γ ∗ γ ∗ → ππ . The one-pion contribution in this unitarity relation defines the π -pole contributionto γ ∗ γ ∗ → ππ . Separating this pole contribution corresponds to further splitting the two-pioncontribution to HLbL into different box-type topologies, shown in Fig. 5. • The two-pion phase-space integral in the HLbL unitarity relation can be converted into a second(crossed-channel) dispersion integral. This nontrivial but essential technical step is described indetail in App. D of [30]. 11 + + + + . . .
Figure 5:
Two-pion contributions to HLbL. Further crossed diagrams are not shown explicitly. • Finally, the symmetrization over the different channels produces the Mandelstam representation.The double-spectral representation for the pion box has the following form: Π π -box i ( s, t, u ; { q j } ) = 1 π (cid:90) ∞ M π ds (cid:48) (cid:90) ∞ t + ( s (cid:48) ; { q j } ) dt (cid:48) ρ π -box i ; st ( s (cid:48) , t (cid:48) ; { q j } )( s (cid:48) − s )( t (cid:48) − t )+ 1 π (cid:90) ∞ M π ds (cid:48) (cid:90) ∞ u + ( s (cid:48) ; { q j } ) du (cid:48) ρ π -box i ; su ( s (cid:48) , u (cid:48) ; { q j } )( s (cid:48) − s )( u (cid:48) − u )+ 1 π (cid:90) ∞ M π dt (cid:48) (cid:90) ∞ u + ( t (cid:48) ; { q j } ) du (cid:48) ρ π -box i ; tu ( t (cid:48) , u (cid:48) ; { q j } )( t (cid:48) − t )( u (cid:48) − u ) , (2.27)where the functions ρ π -box i denote the double-spectral densities, which have been derived (though notgiven explicitly) in [31]. The borders of the double-spectral regions t + and u + are defined in App. G.3of [31].In [31], we have explicitly shown that the Mandelstam representation for the pion box is math-ematically equivalent to a scalar QED (sQED) one-loop calculation, multiplied by appropriate pionvector form factors for the off-shell photons. First, the form factors only depend on the virtualities { q i } and can be pulled out of the double-dispersion integral. Second, triangle and bulb diagrams appearin the sQED calculation only in order to ensure gauge invariance: indeed when projected onto ourgauge-invariant tensor structures, the analytic structure of sQED is the one of pure box topologies.In order to calculate the pion-box contribution numerically, it is convenient to rather use a Feynmanparametrization instead of the dispersive representation. It turns out that in the limit of ( g − µ kinematics, the Feynman parametrization of the scalar functions ˆΠ i defined in (2.15) is very compact.Due to the limit q → , only two-dimensional Feynman parameter integrals appear: ˆΠ π -box i ( q , q , q ) = F Vπ ( q ) F Vπ ( q ) F Vπ ( q ) 116 π (cid:90) dx (cid:90) − x dyI i ( x, y ) , (2.28)where F Vπ is the electromagnetic pion vector form factor and the integrands I i can be found in App. C,written in a way that shows explicitly the absence of kinematic singularities.The main goal of the present article is to describe two-pion contributions beyond the pion box, i.e.the topologies that involve a crossed-channel intermediate state heavier than one pion in one or bothsub-processes. Let us examine in more detail the form of the Mandelstam representation as sketched in the previoussubsection. The starting point is a fixed- t dispersion relation with a discontinuity given by the two-pion12ontribution to the unitarity relation for the HLbL tensor:Im ππs Π µνλσ = 132 π σ π ( s )2 (cid:90) d Ω (cid:48)(cid:48) s (cid:18) W µν + − ( p , p , q ) W λσ + −∗ ( p , p , − q )+ 12 W µν ( p , p , q ) W λσ ∗ ( p , p , − q ) (cid:19) , (2.29)where W µν are the matrix elements for γ ∗ γ ∗ → ππ . The subscripts { + − , } denote the charges and p , the momenta of the intermediate pions. The phase-space factor is σ π ( s ) := (cid:114) − M π s . (2.30)In order to analytically continue the unitarity relation, these matrix elements have to be expressed interms of fixed- s dispersion relations for the scalar functions in a proper tensor decomposition, see [31]: W µν + − = (cid:88) i =1 T µνi (cid:32) ρ s ;+ − i ; t ( s ) t − M π + ρ s ;+ − i ; u ( s ) u − M π + 1 π (cid:90) ∞ M π dt D s ;+ − i ; t ( t ; s ) t − t + 1 π (cid:90) ∞ M π du D s ;+ − i ; u ( u ; s ) u − u (cid:33) ,W µν = (cid:88) i =1 T µνi (cid:32) π (cid:90) ∞ M π dt D s ;00 i ; t ( t ; s ) t − t + 1 π (cid:90) ∞ M π du D s ;00 i ; u ( u ; s ) u − u (cid:33) . (2.31) W µν does not contain any pole terms because the photon does not couple to two neutral pions due toangular momentum conservation and Bose symmetry.If we pick the contribution of the pole terms on both sides of the cut, we single out box topologies: Im ππs Π µνλσ (cid:12)(cid:12)(cid:12) box = 132 π σ π ( s )2 (cid:90) d Ω (cid:48)(cid:48) s (cid:88) i,j =1 , T µνi T λσj (cid:32) ρ s ;+ − i ; t ( s ) t (cid:48) − M π + ρ s ;+ − i ; u ( s ) u (cid:48) − M π (cid:33)(cid:32) ρ s ;+ − j ; t ( s ) t (cid:48)(cid:48) − M π + ρ s ;+ − j ; u ( s ) u (cid:48)(cid:48) − M π (cid:33) ∗ , (2.32)where the primed variables belong to the sub-process on the left-hand side and the double-primedvariables to the sub-process on the right-hand side of the cut. This contribution was the subject ofstudy in [31]. We consider now the contributions with discontinuities either in one or both of thesub-processes: Im ππs Π µνλσ (cid:12)(cid:12)(cid:12) = 132 π σ π ( s )2 (cid:90) d Ω (cid:48)(cid:48) s (cid:88) i,j =1 T µνi T λσj × (cid:34)(cid:18) ρ s ;+ − i ; t ( s ) t (cid:48) − M π + ρ s ;+ − i ; u ( s ) u (cid:48) − M π (cid:19)(cid:18) π (cid:90) ∞ M π dt D s ;+ − j ; t ( t ; s ) t − t (cid:48)(cid:48) + 1 π (cid:90) ∞ M π du D s ;+ − j ; u ( u ; s ) u − u (cid:48)(cid:48) (cid:19) ∗ + (cid:18) π (cid:90) ∞ M π dt D s ;+ − i ; t ( t ; s ) t − t (cid:48) + 1 π (cid:90) ∞ M π du D s ;+ − i ; u ( u ; s ) u − u (cid:48) (cid:19)(cid:18) ρ s ;+ − j ; t ( s ) t (cid:48)(cid:48) − M π + ρ s ;+ − j ; u ( s ) u (cid:48)(cid:48) − M π (cid:19) ∗ (cid:35) , Im ππs Π µνλσ (cid:12)(cid:12)(cid:12) = 132 π σ π ( s )2 (cid:90) d Ω (cid:48)(cid:48) s (cid:88) i,j =1 T µνi T λσj × (cid:34) (cid:18) π (cid:90) ∞ M π dt D s ;+ − i ; t ( t ; s ) t − t (cid:48) + 1 π (cid:90) ∞ M π du D s ;+ − i ; u ( u ; s ) u − u (cid:48) (cid:19) × (cid:18) π (cid:90) ∞ M π dt D s ;+ − j ; t ( t ; s ) t − t (cid:48)(cid:48) + 1 π (cid:90) ∞ M π du D s ;+ − j ; u ( u ; s ) u − u (cid:48)(cid:48) (cid:19) ∗ + 12 (cid:18) π (cid:90) ∞ M π dt D s ;00 i ; t ( t ; s ) t − t (cid:48) + 1 π (cid:90) ∞ M π du D s ;00 i ; u ( u ; s ) u − u (cid:48) (cid:19) × (cid:18) π (cid:90) ∞ M π dt D s ;00 j ; t ( t ; s ) t − t (cid:48)(cid:48) + 1 π (cid:90) ∞ M π du D s ;00 j ; u ( u ; s ) u − u (cid:48)(cid:48) (cid:19) ∗ (cid:35) . (2.33)13 a) ρ st (b) ρ su (c) ρ us (d) ρ ut Figure 6:
Unitarity diagrams representing the “2disc”-box contributions that are (partially) accessible througha fixed- t dispersion relation. If the order of phase-space and dispersive integrals are exchanged, the phase-space integrals can beperformed by applying a tensor reduction to the quantities (cid:90) d Ω (cid:48)(cid:48) s (cid:88) i,j =1 T µνi T λσj t − t (cid:48) t − t (cid:48)(cid:48) . (2.34)The reduced scalar phase-space integrals can then be transformed into another dispersive integral.Together with the dispersion integral ds (cid:48) of the primary cut, this produces a double-dispersion relation.The case of the simplest scalar phase-space integral is explained in [30]. Here, we do not try to calculateexplicitly the tensor phase-space integrals, because we are interested just in the analytic structure ofthe “1disc” and “2disc” contributions, i.e. the boxes with heavier intermediate states in one or both ofthe sub-processes.In order to obtain the full double-spectral representation, one has to consider not only a fixed- t dispersion relation as a starting point but also the crossed versions, i.e. fixed- s and fixed- u dispersionrelations. The symmetrization leads to the Mandelstam representation. For a more detailed discussionin the case of the pion box, see again [31]. We consider now the “1disc” and “2disc” contributions, wherethe pole in one or both of the sub-processes is replaced by a discontinuity. As the symmetrizationprocedure is identical in both cases, we only discuss the case of a discontinuity in both sub-processes.Fig. 6 shows the unitarity diagrams corresponding to the double-spectral representations that aregenerated if we start in our derivation from the fixed- t dispersion relation: the diagrams 6a and 6bgenerate a cut for s > M π , which is the right-hand cut in the fixed- t dispersion relation. Thediagrams 6c and 6d are responsible for the left-hand cut for u > M π . In all cases the first cut isalways the one through the two-pion intermediate state.As discussed in [30, 31], an ( st ) -box diagram can be represented either by a fixed- s , fixed- t , or fixed- u dispersion relation: in the case of a fixed- t representation, there appears only one dispersion integralalong the right-hand s -channel cut. Likewise, in a fixed- s representation, only one dispersion integralalong the t -channel cut is present. In the case of a fixed- u representation, however, an ( st ) -box generatestwo integrals along both the s - and the t -channel cut. This particularity translates directly into thedouble-spectral representation: the ( st ) -box can be written as only one double-dispersion integral ifone starts from a fixed- s or fixed- t representation. If one starts from the fixed- u representation, oneobtains a sum of two double-dispersion integrals, see App. G.3 of [31].Consider now the Mandelstam diagram in Fig. 7, which shows the double-spectral regions that wegenerate if we start from a fixed- t dispersion relation. Because we consider in the primary cut onlytwo-pion intermediate states, not all the contributions from the displayed double-spectral regions aregenerated. We understand from the above discussion of the ( st ) -box that ρ st and ρ ut are complete, butthat the contributions from ρ us and ρ su are not, because only one double-spectral integral for each ofthese contributions is obtained. However, two double-spectral integrals would be needed to generatethe full contribution of these regions: one of the two integrals has a primary cut at the higher threshold M π and is neglected in the fixed- t representation. Of course, two more double-spectral regions ρ ts and ρ tu , which correspond to crossed boxes, are completely missing in the fixed- t representation.14 = 0 s = 4 M π s = 16 M π t = t = M π t = M π u = u = M π u = M π s -channel t -channel u -channel ρ st ρ su ρ us ρ ut Figure 7:
Mandelstam diagram for HLbL scattering for the case q = q = q = − M π , q = 0 . Only thosedouble-spectral regions for “2disc”-box topologies are shown that are reconstructed from the fixed- t dispersionrelation. The dashed line marks a line of fixed t with its s - and u -channel cuts highlighted in gray. The complete set of double-spectral regions, which is obtained after symmetrization, is shown inFig. 8. In the symmetric version, the double-spectral integrals over ρ st and ρ ut are taken from thefixed- t representation, ρ ts and ρ us come from the fixed- s representation, and finally ρ su and ρ tu stemfrom the fixed- u dispersion relation.In summary, we can write the contribution of higher intermediate states in the secondary channelas a double-spectral representation (we suppress the explicit dependence on the virtualities): Π ππi ( s, t, u ) = 1 π (cid:90) ∞ M π ds (cid:48) (cid:90) ∞ t + ( s (cid:48) ) dt (cid:48) ρ ππi ; st ( s (cid:48) , t (cid:48) )( s (cid:48) − s )( t (cid:48) − t ) + 1 π (cid:90) ∞ M π ds (cid:48) (cid:90) ∞ u + ( s (cid:48) ) du (cid:48) ρ ππi ; su ( s (cid:48) , u (cid:48) )( s (cid:48) − s )( u (cid:48) − u )+ 1 π (cid:90) ∞ M π dt (cid:48) (cid:90) ∞ s + ( t (cid:48) ) ds (cid:48) ρ ππi ; ts ( t (cid:48) , s (cid:48) )( t (cid:48) − t )( s (cid:48) − s ) + 1 π (cid:90) ∞ M π dt (cid:48) (cid:90) ∞ u + ( t (cid:48) ) du (cid:48) ρ ππi ; tu ( t (cid:48) , u (cid:48) )( t (cid:48) − t )( u (cid:48) − u )+ 1 π (cid:90) ∞ M π du (cid:48) (cid:90) ∞ s + ( u (cid:48) ) ds (cid:48) ρ ππi ; us ( u (cid:48) , s (cid:48) )( u (cid:48) − u )( s (cid:48) − s ) + 1 π (cid:90) ∞ M π du (cid:48) (cid:90) ∞ t + ( u (cid:48) ) dt (cid:48) ρ ππi ; ut ( u (cid:48) , t (cid:48) )( u (cid:48) − u )( t (cid:48) − t ) . (2.35)The border functions of the double-spectral regions approach asymptotically t + ( s ) s →∞ −→ M π for the“1disc” contribution or M π for the “2disc” contribution.15 = 0 s = 4 M π s = 16 M π t = t = M π t = M π u = u = M π u = M π s -channel t -channel u -channel ρ st ρ ts ρ su ρ us ρ ut ρ tu Figure 8:
Mandelstam diagram for HLbL scattering for the case q = q = q = − M π , q = 0 with all thedouble-spectral regions for “2disc”-box topologies. When we expand the sub-process γ ∗ γ ∗ → ππ into partial waves, we obtain a polynomial in the crossed-channel Mandelstam variables. This means that we neglect the crossed channel cut of the “1disc” or“2disc” boxes, reducing them effectively to triangle (in the case of “1disc” boxes) and bulb topologies (inthe case of “2disc” boxes), as illustrated in Fig. 9. After having applied the approximation, there is noway to distinguish e.g. in Fig. 9g between contributions coming originally from ρ st or ρ su . Therefore,we discuss in the following what kind of single-variable dispersion relation is appropriate in the case ofa partial-wave expanded input for the sub-process.16 a) ρ st (b) ρ su (c) ρ ts (d) ρ tu (e) ρ us (f ) ρ ut (cid:124) (cid:123)(cid:122) (cid:125) ≈ (cid:124) (cid:123)(cid:122) (cid:125) ≈ (cid:124) (cid:123)(cid:122) (cid:125) ≈ (g) (h) (i)Figure 9: (a)–(f ) Unitarity diagrams representing the complete set of “2disc”-box contributions. (g)–(i)
Partial-wave approximation: the sub-process becomes a polynomial in the crossed variable.
Consider again the situation for a fixed- t dispersion relation with the corresponding Mandelstamdiagram in Fig. 7. When constructing the Mandelstam representation, we selected from this rep-resentation only the contributions from ρ st and ρ ut . After the partial-wave expansion, however, weare no longer able to drop the incomplete contributions from ρ us and ρ su . Instead, let us assumethat the neglected contributions from these two double-spectral regions are small: they are only dueto the higher thresholds M π (in the case of “1disc”) or M π (in the case of “2disc”). Furthermore,their discontinuities, being generated by multi-particle intermediate states, are phase-space suppressed.Instead of combining the completely reconstructed double-spectral regions from fixed- s , fixed- t , andfixed- u representations, we can simply sum all contributions from all three fixed- ( s, t, u ) representa-tions. Apart from the neglected higher cuts, each double-spectral contribution appears twice in thissum. The appropriate representation is therefore one half the sum of fixed- ( s, t, u ) representations: Π ππi ( s, t, u ) ≈ (cid:18) π (cid:90) ∞ M π dt (cid:48) Im Π ππi ( s, t (cid:48) , u (cid:48) ) t (cid:48) − t + 1 π (cid:90) ∞ M π du (cid:48) Im Π ππi ( s, t (cid:48) , u (cid:48) ) u (cid:48) − u + 1 π (cid:90) ∞ M π ds (cid:48) Im Π ππi ( s (cid:48) , t, u (cid:48) ) s (cid:48) − s + 1 π (cid:90) ∞ M π du (cid:48) Im Π ππi ( s (cid:48) , t, u (cid:48) ) u (cid:48) − u + 1 π (cid:90) ∞ M π ds (cid:48) Im Π ππi ( s (cid:48) , t (cid:48) , u ) s (cid:48) − s + 1 π (cid:90) ∞ M π dt (cid:48) Im Π ππi ( s (cid:48) , t (cid:48) , u ) t (cid:48) − t (cid:19) . (2.36)In the limit of infinitely heavy intermediate states in the crossed channel this relation is exact. Inparticular, the dominant ππ -rescattering contributions that we consider in this paper can be understoodas a unitarization of the pion pole in the crossed channel on a partial-wave basis. In this case, thedispersion relation (2.36) provides a model-independent representation of the contribution of resonanteffects in the ππ spectrum. 17 q = 0 q = 4 M π q = q = M π q = q = M π (a) A point p inside the ( g − µ integration region is selected and defines the external kinematics. pρ st ρ su ρ tu s = 0 s = 4 M π fixed s = q fi x e d t = q t = t = M π fi x e d u = q u = u = M π (b) Mandelstam diagram for the selected kinematics of point p . The double-spectral regions for the pion boxare shown. Lines of fixed s , t , and u running through the point p with ( s, t, u ) = ( q , q , q ) are shown. Theydo not intersect any double-spectral region. Figure 10:
For ( g − µ kinematics, the paths of the single-variable dispersion integrals never enter anydouble-spectral region, which enables a partial-wave expansion.
18n the case of ( g − µ kinematics, we are interested only in space-like momenta of the virtualphotons. The lines of fixed- ( s, t, u ) therefore never enter the double-spectral regions, see Fig. 10. Thisimplies that a partial-wave expansion is valid without restrictions. This is true even in the case of thepion box, which provides the opportunity to check the partial-wave formalism in a case where we knowthe full result. However, one has to bear in mind that the double-spectral representation for the pionbox differs from the “1disc” and “2disc” boxes: in the case of the pion box, there are only two-pionintermediate states, hence only three box topologies exist and there are only three double-spectralregions. Each fixed- ( s, t, u ) representation reconstructs already all three double-spectral contributions,so that the full result can be obtained from a fixed- ( s, t, u ) dispersion relation separately. Hence,in a symmetrized version for the pion-box one has to take one third of the sum of fixed- ( s, t, u ) representations: Π π -box i ( s, t, u ) = 13 (cid:18) π (cid:90) ∞ M π dt (cid:48) Im Π π -box i ( s, t (cid:48) , u (cid:48) ) t (cid:48) − t + 1 π (cid:90) ∞ M π du (cid:48) Im Π π -box i ( s, t (cid:48) , u (cid:48) ) u (cid:48) − u + 1 π (cid:90) ∞ M π ds (cid:48) Im Π π -box i ( s (cid:48) , t, u (cid:48) ) s (cid:48) − s + 1 π (cid:90) ∞ M π du (cid:48) Im Π π -box i ( s (cid:48) , t, u (cid:48) ) u (cid:48) − u + 1 π (cid:90) ∞ M π ds (cid:48) Im Π π -box i ( s (cid:48) , t (cid:48) , u ) s (cid:48) − s + 1 π (cid:90) ∞ M π dt (cid:48) Im Π π -box i ( s (cid:48) , t (cid:48) , u ) t (cid:48) − t (cid:19) , (2.37)and the relation is exact. The Lorentz decomposition of the HLbL tensor is only unique up to transformations that do notintroduce kinematic singularities, hence there is a fair amount of freedom in choosing a particularrepresentation. One important aspect of such transformations concerns the fact that the differentmass dimensions of the Lorentz structures imply different mass dimensions of the scalar functions Π i ,which must be reflected in a different asymptotic behavior. Indeed if we assume, as it is natural, auniform asymptotic behavior of the whole HLbL tensor, i.e. in all Mandelstam variables and for alltensor components, this implies that functions multiplying Lorentz structures of higher mass dimensionshould fall down even faster for asymptotic values of the Mandelstam variables. In order to have apredictive framework, we require, that all BTT scalar functions satisfy unsubtracted (i.e. parameter-free) dispersion relations, and in particular that those multiplying the Lorentz structures with lowestmass dimensions fall down like the inverse of the Mandelstam variables at infinity. This hypothesis,which will be tested later on, implies that the HLbL tensor behaves asymptotically as Π µνλσ (cid:16) s, t, u, (2.38)and that the BTT scalar functions behave (up to logarithmic corrections) according to: Π , Π (cid:16) s , t , u , Π , Π , Π , Π (cid:16) s , t , u , Π (cid:16) s , t , u , (2.39)with analogous asymptotics for the functions related by crossing symmetry. Under this assumption,the functions Π , . . . , Π fulfill an unsubtracted dispersion relation. However, as they fall down to zeroeven faster, the functions Π , . . . fulfill not only unsubtracted dispersion relations, but even a set ofsum rules. These sum rules ensure that the result for the HLbL tensor is independent of the choice ofthe tensor decomposition: the difference between the Mandelstam representations for one set of scalar19oefficient functions and a second, equally valid set of functions will vanish as a consequence of thesum rules (also known as “superconvergence relations” [62]).Consider for example Π for fixed t = t b = q + q . At this kinematic point, the Tarrach singularityis absent and Π = ˜Π is unambiguously defined (up to the redundancy in 4 space-time dimensions),see [31]. It fulfills an unsubtracted fixed- t dispersion relation: Π (cid:12)(cid:12) t = t b = 1 π (cid:90) ∞ s ds (cid:48) Im Π ( s (cid:48) , t b , Σ − t b − s (cid:48) ) s (cid:48) − s + 1 π (cid:90) ∞ u du (cid:48) Im Π (Σ − t b − u (cid:48) , t b , u (cid:48) ) u (cid:48) − u , (2.40)where s and u denote the threshold in the respective channel. Due to the asymptotic behavior, s Π fulfills an unsubtracted dispersion relation as well: s Π (cid:12)(cid:12) t = t b = 1 π (cid:90) ∞ s ds (cid:48) s (cid:48) Im Π ( s (cid:48) , t b , Σ − t b − s (cid:48) ) s (cid:48) − s + 1 π (cid:90) ∞ u du (cid:48) (Σ − t b − u (cid:48) ) Im Π (Σ − t b − u (cid:48) , t b , u (cid:48) ) u (cid:48) − u . (2.41)We subtract this equation once using s (cid:48) − s = 1 s (cid:48) + ss (cid:48) ( s (cid:48) − s ) (2.42)as well as s (cid:48) − s = 1Σ − t b − u (cid:48) − (Σ − t b − u ) = − u (cid:48) − u (2.43)in the u -channel integral to obtain: Π (cid:12)(cid:12) t = t b = 1 s (cid:18) π (cid:90) ∞ s ds (cid:48) Im Π ( s (cid:48) , t b , Σ − t b − s (cid:48) ) − π (cid:90) ∞ u du (cid:48) Im Π (Σ − t b − u (cid:48) , t b , u (cid:48) ) (cid:19) + 1 π (cid:90) ∞ s ds (cid:48) Im Π ( s (cid:48) , t b , Σ − t b − s (cid:48) ) s (cid:48) − s + 1 π (cid:90) ∞ u du (cid:48) Im Π (Σ − t b − u (cid:48) , t b , u (cid:48) ) u (cid:48) − u . (2.44)The comparison with (2.40) gives the following sum rule: π (cid:90) ∞ s ds (cid:48) Im Π ( s (cid:48) , t b , Σ − t b − s (cid:48) ) − π (cid:90) ∞ u du (cid:48) Im Π (Σ − t b − u (cid:48) , t b , u (cid:48) ) = 0 . (2.45)In the case of Π , an even higher-degree sum rule is fulfilled. Starting from the unsubtractedfixed- s dispersion relation t Π (cid:12)(cid:12) s = s b = 1 π (cid:90) ∞ t dt (cid:48) t (cid:48) Im Π ( s b , t (cid:48) , Σ − s b − t (cid:48) ) t (cid:48) − t + 1 π (cid:90) ∞ u du (cid:48) (Σ − s b − u (cid:48) ) Im Π ( s b , Σ − s b − u (cid:48) , u (cid:48) ) u (cid:48) − u , (2.46)with s b = q + q , two subtractions lead to Π (cid:12)(cid:12) s = s b = 1 t (cid:18) π (cid:90) ∞ t dt (cid:48) t (cid:48) Im Π ( s b , t (cid:48) , Σ − s b − t (cid:48) ) − π (cid:90) ∞ u du (cid:48) (Σ − s b − u (cid:48) ) Im Π ( s b , Σ − s b − u (cid:48) , u (cid:48) ) (cid:19) + 1 t (cid:18) π (cid:90) ∞ t dt (cid:48) Im Π ( s b , t (cid:48) , Σ − s b − t (cid:48) ) − π (cid:90) ∞ u du (cid:48) Im Π ( s b , Σ − s b − u (cid:48) , u (cid:48) ) (cid:19) + 1 π (cid:90) ∞ t dt (cid:48) Im Π ( s b , t (cid:48) , Σ − s b − t (cid:48) ) t (cid:48) − t + 1 π (cid:90) ∞ u du (cid:48) Im Π ( s b , Σ − s b − u (cid:48) , u (cid:48) ) u (cid:48) − u . (2.47)Both large brackets have to vanish, producing two independent sum rules for Π . We have verifiedthese sum rules explicitly in the case of sQED, see Sect. 3.2. All imaginary parts are understood to be evaluated on the upper rim of the cut in the respective channel. .4 Relation to observables In Sect. 2.2, we have derived the form of the dispersion relation for general two-pion contributions to ( g − µ , writing the results (2.36) and (2.37) for a generic BTT function Π i . In a next step, we wantto use this dispersion relation for the actual input in the ( g − µ master formula (2.25). Our goal is toestablish via unitarity a relation between the two-pion contribution to ( g − µ and helicity amplitudesfor the sub-process γ ∗ γ ( ∗ ) → ππ .While the BTT decomposition solves the problem of kinematic singularities, the 54 scalar functions Π i have the disadvantage to form a redundant set: there are 11 Tarrach redundancies [31] and twofurther ambiguities in four space-time dimensions [58], as the number of helicity amplitudes for HLbLscattering is 41. Furthermore, in the on-shell limit of the external photon contributions from itslongitudinal polarization must not survive. This reduces the number of helicity amplitudes to × =27 . In the limit of ( g − µ kinematics, the number of independent amplitudes is further reduced to 19,see Sect. 2.1.2. Note, however, that this limit applies to the outer kinematics in the master formula andnot to the imaginary parts inside the dispersion integrals, where one Mandelstam variable is integratedover and thus not fixed to ( g − µ kinematics (2.23).While working with a redundant set of functions may, at first sight, seem as a minor nuisancewhich should in the final result take care of itself and lead to a unique and correct answer, this is notthe case in our context. The origin of the problem is that (i) establishing the relation between thephysical observables (i.e. the helicity amplitudes) and the BTT functions, (ii) projecting on partialwaves, and (iii) writing down dispersion relations, are not necessarily commuting operations. In [31],we constructed single-variable dispersion relations that are free of the Tarrach redundancies. However,for most of the scalar functions, we only found a dispersion relation in one of the three channels,which was sufficient to obtain the dispersive reconstruction of the pion box. For general two-pioncontributions (2.36) we need all three fixed- ( s, t, u ) dispersion relations. Furthermore, the fact thatlongitudinal polarizations of the external photon do not contribute is not immediately manifest indispersion relations for the BTT functions Π i . In order to solve all these problems we must constructanother basis that is appropriate for the kinematics in the imaginary parts of the dispersion integrals.The scalar functions of this basis, which we will call ˇΠ i , are in one-to-one correspondence with the 27singly-on-shell helicity amplitudes.In Sect. 2.4.1, we explain how to derive these singly-on-shell basis functions ˇΠ i . In the construction,we make use of the sum rules for the BTT scalar functions Π i , derived in Sect. 2.3. Readers who arenot interested in the technical details of the derivation may skip the following subsection and jumpdirectly to Sect. 2.4.2, where we present the solution for the ˇΠ i functions. As a by-product in thederivation of the singly-on-shell basis, we find a set of 15 sum rules for fixed- t kinematics, presented inSect. 2.4.3. These physical sum rules are of relevance for the construction of the input on γ ∗ γ ( ∗ ) → ππ and can be considered a generalization of certain sum rules for forward HLbL scattering from [63]. The most efficient way to obtain a representation for the two-pion HLbL contribution to ( g − µ involving only physical helicity amplitudes is the construction of a basis { ˇΠ i } for singly-on-shell kine-matics that can be used together with unsubtracted single-variable dispersion relations. In such a basis,contributions from longitudinal polarizations of the external photon are manifestly absent. As we willsee, this construction is possible due to the presence of the sum rules for the BTT scalar functionsderived in Sect. 2.3. The rather surprising fact that contributions from unphysical polarizations arenot trivially absent in a representation involving redundancies is explained in App. E.1.Let us define the transformation from the BTT functions to ˆΠ i as the × matrix t : ˆΠ i = (cid:88) j =1 t ij Π j . (2.48)21n terms of the Lorentz structures ˆ T µνλσi , both the 11 Tarrach redundancies ( r ) and the two ambiguitiesin four space-time dimensions ( a ) can be written as linear relations: (cid:88) i =1 ˆ T µνλσi r ij = 0 , j = 1 , . . . , , (cid:88) i =1 ˆ T µνλσi a ij = 0 , j = 1 , . (2.49)Next, we study the unphysical polarizations: they multiply structures that are proportional to q or q σ . Hence, we determine all linear dependencies of the tensor structures in the limit q , q σ → ,which leads to a matrix u of rank 25: (cid:88) i =1 (cid:18) lim q ,q σ → ˆ T µνλσi (cid:19) u ij = 0 , j = 1 , . . . , , rank( u ) = 25 . (2.50)If we join u with the two 4d ambiguities, the rank is 27: rank( u, a ) = rank( u, a, r ) = 27 . (2.51)Since −
27 = 27 , this is consistent with the fact that in the singly-on-shell limit there are 27independent helicity amplitudes. In this limit, the 11 Tarrach redundancies r are linearly dependenton a and u . Moreover, in the singly-on-shell limit the transformations u and a can be interpreted asan ambiguity in the scalar functions: ˆΠ i (cid:55)→ ˆΠ i + (cid:88) j =1 ¯ u ij ∆ j , (2.52)where we denote by ¯ u the × matrix ( u, a ) .We consider now the limit q → and t → q , which is relevant for the fixed- t dispersion relation.For a suitable choice of u and a , we still have rank(¯ u ) = 27 in this kinematic limit. The goal is now tofind all linear combinations of scalar functions ˆΠ i that are invariant under the transformation (2.52)and satisfy an unsubtracted dispersion relation. Hence, we have to determine the matrix ˆ p , such that ˆ p ki ˆΠ i + (cid:88) j =1 ¯ u ij ∆ j = ˆ p ki ˆΠ i , (2.53)for arbitrary ∆ j , which corresponds to the null-space of ¯ u : (cid:88) i =1 ˆ p ki ¯ u ij = 0 . (2.54)However, the requirement that ˆ p ki ˆΠ i satisfy an unsubtracted dispersion relation does not allow arbi-trary ˆ p . Here, the sum rules for the BTT scalar functions Π i , derived in Sect. 2.3, are employed in anessential way: the linear combinations (cid:88) i =1 ˆ p ki ˆΠ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q =0 t = q = (cid:88) i,j =1 ˆ p ki t ij Π j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q =0 t = q =: (cid:88) j =1 p kj Π j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q =0 t = q (2.55)must only involve coefficients p kj that depend linearly on s for j ≥ or at most quadratically for j ∈ { , . . . , } , because the scalar functions Π j satisfy the linear sum rule for j ≥ and the quadraticsum rule for j ∈ { , . . . , } . Meanwhile, the coefficients p kj can have an arbitrary dependence on thevirtualities q i , which in the dispersion relation are fixed external quantities. Hence, we write p kj ( s ) = (cid:88) l =0 p kjl s l (2.56)22nd bear in mind the mentioned restrictions for p kj and p kj . Solving this linear algebra exercise isthe major problem of the calculation. With the help of computer algebra, we obtain a × matrix ( p kj , p kj , p kj ) , whose contraction p kj ( s ) has again rank 27 and is in one-to-one correspondence withthe 27 singly-on-shell helicity amplitudes.In a last step, we consider the limit s → q (which is now equivalent to q → ) and search forlinear relations ˆΠ i (cid:12)(cid:12)(cid:12) q =0 = (cid:88) k =1 54 (cid:88) j =1 b ik p kj ( q )Π j (cid:12)(cid:12)(cid:12) q =0 , i = 1 , . . . , , , , , , , , , , (2.57)for all the functions contributing to ( g − µ , where the coefficients b ik are functions of the virtualities q , q , and q . The solution of this system is not unique: p kj is a × matrix of rank 27, hencethere exist 15 null relations (cid:88) k =1 n ik p kj ( q ) , i = 1 , . . . , , (2.58)again with coefficients n ik depending only on q , q , and q .With the constructed solution for p kj , we can build a singly-on-shell basis by selecting a convenientset of 27 independent linear combinations. We choose the basis functions ˇΠ i in such a way that onlythe first 19 contribute to ( g − µ : ˇΠ i ( s ; q i ) := (cid:88) k =1 54 (cid:88) j =1 b g i k p kj ( s )Π j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q =0 t = q , i = 1 , . . . , , (2.59)where { g i } := G = { , . . . , , , , , , , , , } . In Sect. 2.4.1, we have detailed the derivation of the 27 singly-on-shell basis functions ˇΠ i . Thesefunctions have the following four important properties.1. They are linear combinations of the BTT functions Π i for fixed- t with coefficients depending on s only in such a way that the sum rules in Sect. 2.3 allow an unsubtracted dispersion relation forthe ˇΠ i : ˇΠ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q =0 t = q = 1 π (cid:90) ∞ M π ds (cid:48) Im ˇΠ i ( s (cid:48) , q , u (cid:48) ) s (cid:48) − s + 1 π (cid:90) ∞ M π du (cid:48) Im ˇΠ i ( s (cid:48) , q , u (cid:48) ) u (cid:48) − u . (2.60)2. In the limit q → , a subset of 19 functions reproduces the input for the master formula (2.25)for ( g − µ : ˇΠ i (cid:12)(cid:12)(cid:12) s = q = ˆΠ g i (cid:12)(cid:12)(cid:12) q =0 , i = 1 , . . . , . (2.61)3. They are free from Tarrach redundancies [31] and the ambiguity in four space-time dimen-sions [58].4. A basis change relates them to the 27 singly-on-shell helicity amplitudes, hence the imaginaryparts in the dispersion integrals (2.60) can be expressed in terms of physical helicity amplitudesfor γ ∗ γ ∗ → ππ and γ ∗ γ → ππ . 23he first point reflects the need to obtain a parameter-free prediction for the two-pion contributionto ( g − µ . The second point implies that we can construct a dispersive representation for ˆΠ i ofthe form (2.36) (or (2.37) for the pion box), by summing fixed- ( s, t, u ) representations. The fixed- t representation is given directly by (2.60), while fixed- s and fixed- u representations follow from thecrossing relations (2.16). The last two properties mean that we can relate the two-pion contributionto ( g − µ to observable quantities. In particular, longitudinal polarizations for the external photonmust drop out in the limit q → .The 19 functions contributing to ( g − µ can be written as (for q = 0 and t = q ) ˇΠ i = ˆΠ g i + ( s − q ) ¯∆ i + ( s − q ) ¯¯∆ i , (2.62)where { g i } = G = { , . . . , , , , , , , , , } and where ¯∆ i = (cid:88) j =7 ¯ d ij Π j , ¯¯∆ i = (cid:88) j =31 ¯¯ d ij Π j (2.63)are given explicitly in App. D. The coefficients ¯ d ij and ¯¯ d ij depend only on q , q , and q . To verifythat the functions ˇΠ i fulfill unsubtracted fixed- t dispersion relations, we observe π (cid:90) ds (cid:48) Im ˇΠ i ( s (cid:48) ) s (cid:48) − s = 1 π (cid:90) ds (cid:48) Im ˆΠ g i ( s (cid:48) ) s (cid:48) − s + 1 π (cid:90) ds (cid:48) ( s (cid:48) − q ) Im ¯∆ i ( s (cid:48) ) s (cid:48) − s + 1 π (cid:90) ds (cid:48) ( s (cid:48) − q ) Im ¯¯∆ i ( s (cid:48) ) s (cid:48) − s = 1 π (cid:90) ds (cid:48) Im ˆΠ g i ( s (cid:48) ) s (cid:48) − s + ( s − q ) 1 π (cid:90) ds (cid:48) Im ¯∆ i ( s (cid:48) ) s (cid:48) − s + ( s − q ) π (cid:90) ds (cid:48) Im ¯¯∆ i ( s (cid:48) ) s (cid:48) − s + 1 π (cid:90) ds (cid:48) Im ¯∆ i ( s (cid:48) ) + 1 π (cid:90) ds (cid:48) ( s + s (cid:48) − q ) Im ¯¯∆ i ( s (cid:48) )= ˆΠ g i ( s ) + ( s − q ) ¯∆ i ( s ) + ( s − q ) ¯¯∆ i ( s )= ˇΠ i ( s ) , (2.64)where we have used the sum rules for the BTT functions: (cid:90) ds (cid:48) Im Π i ( s (cid:48) ) = 0 , i ∈ { , . . . , } , (cid:90) ds (cid:48) s (cid:48) Im Π i ( s (cid:48) ) = 0 , i ∈ { , . . . , } , (2.65)and written both channels schematically as one integral. This proves that the dispersion relation for ˇΠ i is indeed fulfilled. In particular, the limit s → q provides a fixed- t representation for ˆΠ g i , the inputfor the ( g − µ master formula. The solutions for fixed- s and fixed- u follow immediately from thecrossing relations (2.16) and (2.18).Unfortunately, it turns out that it is not possible to find a representation for the functions ˇΠ i withcoefficients ¯ d ij and ¯¯ d ij in (2.63) free of all kinematic singularities. This is a final relic of the redundancyin the tensor decomposition which is, however, not a real problem at all. Indeed the contribution of ¯∆ i and ¯¯∆ i in the dispersion relation for ˆΠ g i vanishes due to the sum rules, and the same is true forthe residue of any kind of kinematic singularity in the coefficients ¯ d ij and ¯¯ d ij . The residue is definedin terms of physical quantities only and can thus be subtracted explicitly, to obtain a representationthat is manifestly free of kinematic singularities.Using the above sum rules, we can optimize the representation to a certain degree. We have chosenour preferred representation in App. D according to the following criteria:24 We have avoided for scalar functions Π i that receive S -wave contributions to mix into otherfunctions in (2.62). • We have made the singularity structure of the coefficients ¯ d ij and ¯¯ d ij as simple as possible. • We have optimized the convergence of the partial-wave representation of ( g − µ for the pionbox.The minimal singularity structure for the coefficients ¯ d ij and ¯¯ d ij consists of simple poles in / ( q + q ) and singularities of the type /λ ( q , q , q ) , where λ ( a, b, c ) := a + b + c − ab + bc + ca ) is theKällén triangle function. The first singularity lies on a straight line outside the ( g − µ integrationregion, see Fig. 3. Writing λ := λ ( q , q , q ) = −
13 ˜Σ (1 − r ) , (2.66)we see that the second type of singularity lies on the border of the ( g − µ integration region. In the ( g − µ master formula (2.25), we subtract the residue of this singularity at r = 1 , which vanishesdue to the sum rules, to obtain a representation without any kinematic singularities in the ( g − µ integration region. In the derivation of the singly-on-shell basis functions ˇΠ i , we have encountered the 15 null rela-tions (2.58), which lead to sum rules involving only physical (singly-on-shell) quantities. We buildthe 15 functions N i ( s ; q i ) := (cid:88) k =1 54 (cid:88) j =1 n ik p kj ( s )Π j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q =0 t = q , i = 1 , . . . , . (2.67)By using the null relations (2.58), we subtract zero on the right-hand side and obtain N i ( s ; q i ) = (cid:88) k =1 54 (cid:88) j =1 n ik (cid:0) p kj ( s ) − p kj ( q ) (cid:1) Π j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q =0 t = q = (cid:88) k =1 54 (cid:88) j =7 n ik (cid:0) p kj ( s ) − p kj ( q ) (cid:1) Π j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q =0 t = q , (2.68)where the second equality follows from the fact that p kj ( s ) = p kj ( q ) is constant for j < . For j ≥ , p kj ( s ) is linear in s or quadratic for j ∈ { , . . . , } . Hence, we can write p kj ( s ) − p kj ( q ) = ( s − q )˜ p kj ( s ) , j ≥ , (2.69)where ˜ p kj is either constant or linear in s for j ∈ { , . . . , } . Inserting N i into a dispersion integralleads to 15 linear combinations of the sum rules for the scalar functions, discussed in Sect. 2.3: π (cid:90) ds (cid:48) Im N i ( s (cid:48) ) s (cid:48) − q = (cid:88) k =1 n ik (cid:88) j =7 π (cid:90) ds (cid:48) ˜ p kj ( s (cid:48) ) Im Π j ( s (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q =0 t = q = 0 . (2.70)These 15 sum rules are special: they are free of any ambiguity and only involve physical helicityamplitudes, i.e. amplitudes with a transversely polarized external photon. They can be used to modifythe fixed- t representations (2.62) of the 19 ˇΠ i functions contributing to ( g − µ . The 15 sum rules can25e written in very compact form in terms of the singly-on-shell basis functions ˇΠ i , defined in App. D: (cid:90) ds (cid:48) Im ˇΠ i ( s (cid:48) ) , i = 7 , , , , , , , , , , , , (cid:90) ds (cid:48) Im (cid:16) ˇΠ ( s (cid:48) ) + ˇΠ ( s (cid:48) ) − ˇΠ ( s (cid:48) ) (cid:17) , (cid:90) ds (cid:48) Im (cid:16) ˇΠ ( s (cid:48) ) − ˇΠ ( s (cid:48) ) + ˇΠ ( s (cid:48) ) (cid:17) , (cid:90) ds (cid:48) Im (cid:16) ˇΠ ( s (cid:48) ) − ˇΠ ( s (cid:48) ) + ˇΠ ( s (cid:48) ) (cid:17) , (2.71)where fixed- t kinematics is implicit. These sum rules are related to certain sum rules for forwardHLbL scattering derived in [63], although we have derived them for a different kinematic situation(non-forward scattering but q = 0 ). A detailed comparison is provided in App. E.2. In order to determine the two-pion contribution to the scalar functions in the master formula (2.25),we write fixed- ( s, t, u ) dispersion relations of the form (2.36), where we take only the contribution ofthe two-pion intermediate state to the imaginary parts into account. The scalar functions that fulfillsingle-variable dispersion relations and reproduce the scalar functions in the master formula are givenin (2.62). The last missing piece in the formalism for two-pion contributions to ( g − µ is thus thelink with helicity amplitudes and partial waves for γ ∗ γ ( ∗ ) → ππ .Unitarity determines the imaginary part of the scalar functions, which is the input in the dispersionrelations, and is most conveniently expressed in the basis of helicity amplitudes, expanded into partialwaves: for helicity partial waves the unitarity relation is diagonal. Furthermore, the input on γ ∗ γ ( ∗ ) → ππ is available in the form of helicity partial waves: these are in principle observable quantities, eventhough given the absence of double-virtual data they will have to be reconstructed dispersively bymeans of the solution of a system of Roy–Steiner equations [28, 31, 55]. In Sect. 4, we will providea first estimate of the two-pion rescattering contribution by solving the Roy–Steiner equations for S -waves, using a pion-pole LHC and ππ phase shifts based on the inverse-amplitude method [64–69].The step from the singly-on-shell basis to the basis of helicity amplitudes for HLbL is again rathertedious. The helicity amplitudes can be easily expressed in terms of BTT scalar functions or thesingly-on-shell basis by contracting the HLbL with appropriate polarization vectors, but expressingthe scalar functions in terms of helicity amplitudes requires the analytic inversion of a × matrix,which is a formidable task. Here, we present the solution to this problem and discuss the subtletiesof the partial-wave expansion in connection with ( g − µ . In Sect. 2.5.1, we recall the definitions forthe helicity amplitudes from [31]. In Sect. 2.5.2, we comment on the implication of the sum rules forthe partial waves. In Sect. 2.5.3, we discuss the result for the dispersion relation in terms of helicitypartial waves, generalizing the S -wave result of [28] to arbitrary partial waves. Some technical partsof the calculation are relegated to App. F. Although the direct inversion of the × matrix is feasible, see App. F.2 for a summary of how toachieve this task, there is a more elegant way to derive the partial-wave unitarity relation without theneed for a full inversion. We checked that both derivations lead to identical results, but pursue thelatter, more physical approach in the main part of the paper.The strategy that avoids the inversion of the matrix describing the basis change relies on thefollowing idea: by expanding the sub-process γ ∗ γ ∗ → ππ into helicity partial waves, we can explicitlycalculate the phase-space integral in the unitarity relation and determine the imaginary part as a sumof products of helicity partial waves. The phase-space integrals become more and more complicated26or higher partial waves, but due to the fact that unitarity is diagonal for helicity partial waves, thecontribution of arbitrary partial waves is determined as soon as the S -, D -, and G -wave discontinuitiesare calculated.In phenomenological applications, we expect the contribution of partial waves beyond D -waves tobe negligible. However, the calculation of higher partial waves allows us to check the convergence ofthe partial-wave series to the full result in the test case of the pion box and provides a very strong testof the formalism for the single-variable partial-wave dispersion relations. The numerical checks of theconvergence will be discussed in Sect. 3.3.In the following, we define the helicity amplitudes for HLbL and the sub-process γ ∗ γ ( ∗ ) → ππ . Thedefinitions of angles and polarization vectors can be found in [31].The helicity amplitudes of γ ∗ γ ∗ → ππ are defined as H λ λ = e i ( λ − λ ) φ (cid:15) λ µ ( q ) (cid:15) λ ν ( q ) W µν ( p , p , q ) . (2.72)For two off-shell photons, there are in principle = 9 helicity combinations. However, due to parityconservation and with our convention for the polarization vectors, we have the relation H − λ − λ = ( − λ + λ H λ λ , (2.73)which implies that only − + 1 = 5 amplitudes are independent: H ++ = H −− , H + − = H − + , H +0 = − H − , H = − H − , H . (2.74)Similarly, for the HLbL helicity amplitudes, defined by H λ λ ,λ λ = (cid:15) λ µ ( q ) (cid:15) λ ν ( q ) (cid:15) λ λ ∗ ( − q ) (cid:15) λ σ ∗ ( q )Π µνλσ ( q , q , q ) , (2.75)there are helicity amplitudes, but only − + 1 = 41 independent ones.We introduce rescaled helicity amplitudes that remain finite in the limit q i → : H λ λ =: κ λ κ λ ¯ H λ λ , H λ λ ,λ λ =: κ λ κ λ κ λ κ λ ¯ H λ λ ,λ λ , (2.76)where κ i ± = 1 , κ i = q i ξ i , (2.77)and ξ i refers to the normalization of the longitudinal polarization vectors. Since only the ¯ H amplitudesappear in the final results, this procedure avoids any confusion that might originate from a particularchoice of normalization.We define the helicity partial-wave expansion for γ ∗ γ ∗ → ππ by ¯ H λ λ ( s, t, u ) = (cid:88) J (2 J + 1) d Jm ( z ) h J,λ λ ( s ) , (2.78)where m = | λ − λ | , z is the cosine of the scattering angle, and d Jm m ( z ) denotes the Wigner d -function.For HLbL, we expand the helicity amplitudes as follows into partial waves: ¯ H λ λ ,λ λ ( s, t, u ) = (cid:88) J (2 J + 1) d Jm m ( z ) h Jλ λ ,λ λ ( s ) , (2.79)where m = λ − λ , m = λ − λ .Unitarity is diagonal for helicity partial waves, i.e.Im ππs h Jλ λ ,λ λ ( s ) = η i η f σ π ( s )16 πS h J,λ λ ( s ) h ∗ J,λ λ ( s ) , (2.80)27here S is the symmetry factor of the two pions and η i = (cid:26) − if λ − λ = − , otherwise , η f = (cid:26) − if λ − λ = − , otherwise (2.81)account for the sign convention in (2.78). We find the relationIm ππs h Jλ λ , − λ − λ ( s ) = Im ππs h Jλ λ ,λ λ ( s ) , (2.82)where the ratio of η f factors compensates the sign ( − λ + λ from (2.73).The HLbL tensor is written in terms of the redundant BTT Lorentz decomposition as Π µνλσ = (cid:88) i =1 T µνλσi Π i = (cid:88) i =1 B µνλσi ˜Π i . (2.83)For fixed t = q and q = 0 , we have defined the singly-on-shell basis functions ˇΠ i . The helicityamplitudes form a basis of the HLbL tensor, hence Π i = (cid:88) j =1 c ij ¯ H j , ˜Π i = (cid:88) j =1 ˜ c ij ¯ H j , ˇΠ i = (cid:88) j =1 ˇ c ij ¯ H j , j = { λ , λ , λ , λ } . (2.84)The coefficients c ij contain 13 redundancies, the ˜ c ij still two (in four space-time dimensions). In therelation for ˇΠ i , fixed- t kinematics is implicit and the coefficients ˇ c ij are free from redundancies.We define the following “canonical” ordering of j : j ∈ { { ++ , ++ } , { ++ , +0 } , { ++ , + −} , { ++ , } , { ++ , } , { ++ , −} , { ++ , − + } , { ++ , − } , { ++ , −−} ,
10 = { +0 , ++ } ,
11 = { +0 , +0 } ,
12 = { +0 , + −} ,
13 = { +0 , } ,
14 = { +0 , } ,
15 = { +0 , −} ,
16 = { +0 , − + } ,
17 = { +0 , − } ,
18 = { +0 , −−} ,
19 = { + − , ++ } ,
20 = { + − , +0 } ,
21 = { + − , + −} ,
22 = { + − , } ,
23 = { + − , } ,
24 = { + − , −} ,
25 = { + − , − + } ,
26 = { + − , − } ,
27 = { + − , −−} ,
28 = { , ++ } ,
29 = { , +0 } ,
30 = { , + −} ,
31 = { , } ,
32 = { , } ,
33 = { , −} ,
34 = { , − + } ,
35 = { , − } ,
36 = { , −−} ,
37 = { , ++ } ,
38 = { , +0 } ,
39 = { , + −} ,
40 = { , } ,
41 = { , }} , (2.85)and the subsets { l j } j := { , , , , , , , , } , { k j } j := { , , , , , , } , { ¯ k j } j := { , , , , , , } , { n j } j := { , , , , , , , , } , { ¯ n j } j := { , , , , , , , , } . (2.86)The meaning of these subsets is the following: the subset { l j } j corresponds to helicity amplitudes with ¯ H ¯ j = ± ¯ H j , where ¯ j := { λ , λ , − λ , − λ } . For the subset { k j } j , the Wigner d -functions for j and ¯ j are identical up to a sign, while for the subset { n j } j this is not the case.28he imaginary parts of the scalar functions are given byIm ππs ˇΠ i = (cid:88) j =1 (cid:88) J ˇ c ij (2 J + 1) d Jm j m j ( z ) Im ππs h Jj ( s )= (cid:88) J (cid:20) (cid:88) j =1 ˇ c il j (2 J + 1) d Jl j ( z ) Im ππs h Jl j ( s )+ (cid:88) j =1 (cid:16) ˇ c ik j + ζ j ˇ c i ¯ k j (cid:17) (2 J + 1) d Jk j ( z ) Im ππs h Jk j ( s )+ (cid:88) j =1 (cid:16) ˇ c in j d Jn j ( z ) + ˇ c i ¯ n j d J ¯ n j ( z ) (cid:17) (2 J + 1) Im ππs h Jn j ( s ) (cid:21) , (2.87)where the signs { ζ j } j = { + , − , + , − , + , + , + } (2.88)come from the relation d J − m − m ( z ) = ( − m − m d Jm m ( z ) = d Jm m ( z ) . (2.89)The explicit Wigner d -functions are { d Jl j } j = { d J , d J , d J , − d J , d J , − d J , d J , d J , d J } , { d Jk j } j = { d J , − d J , d J , d J , d J , d J , − d J } , { d Jn j } j = { d J , − d J , d J − , d J , d J , d J − , d J − , − d J − , d J } , { d J ¯ n j } j = { d J − , d J − , d J , d J − , d J − , d J , d J , d J , d J − } , (2.90)where the signs are due to the use of relation (2.89).In order to identify the coefficients ˇ c il j and (ˇ c ik j + ζ j ˇ c i ¯ k j ) , it is sufficient to know the contributionto the unitarity relation from the lowest partial waves h Jl j and h Jk j (which are either S - or D -waves).However, as the Wigner d -functions d Jn j are different from d J ¯ n j , we need to know the contribution fromthe two lowest partial waves h Jn j in order to identify the coefficients ˇ c in j and ˇ c i ¯ n j separately. Therefore,the generalization to arbitrary partial waves is possible as soon as the contributions from S -, D -, and G -waves are determined.The explicit calculation of the partial-wave unitarity relation involves rather complicated phase-space integrals, see App. F.1. By calculating the fully-off-shell unitarity relation, projecting onto BTT,and working out the imaginary parts of the functions ˇΠ i , we have verified explicitly that the coefficients ˇ c ij for the longitudinal polarization λ = 0 vanish. Therefore, ˇ c ij is effectively an invertible × matrix. As mentioned above, we have also computed the matrix ˇ c ij by direct inversion of the basischange from helicity amplitudes to the scalar functions, see App. F.2. The fact that the result agreeswith the one from the phase-space calculation provides a very strong cross check, and in addition thefull inversion allows one to separate the ˇ c ik j and ˇ c i ¯ k j coefficients. Before returning to the final result, we comment on the role of the sum rules in the context of apartial-wave expansion. In Sect. 2.4.3, we have derived a set of 15 sum rules for the ˇΠ i functions,which, after a basis change, can be written in terms of the 27 singly-on-shell helicity amplitudes forHLbL scattering. By construction, these sum rules only hold true for the full helicity amplitudes.29n particular, when expanding the imaginary part of the helicity amplitudes into partial waves andtruncating the partial-wave series, there is no reason why the sum rules should still be satisfied exactly:sum-rule violations of a size consistent with higher partial waves are expected, so that the sum rulesare fulfilled only approximately. This has some important consequences.Due to the presence of the sum rules, the formal relation between the master formula input ˆΠ i at q = 0 and the singly-on-shell basis functions ˇΠ i is not unique, but can be modified by linear combina-tions of the sum rules. If the sum rules hold exactly, all these representations are equivalent. Violatingthe sum rules by a truncation of the partial-wave series implies that a dependence on the precise repre-sentation of the ˇΠ i functions is introduced. Our preferred representation of the ˇΠ i functions, discussedin Sect. 2.4.2 and App. D, leads to a fast convergence of the partial-wave expansion in the test caseof the pion box, see Sect. 3, but we also checked other variants and convinced ourselves in each casethat indeed the sum-rule violations are consistent with a meaningful partial-wave expansion and onlyslight losses in the rate of convergence.The dependence on the representation of the ˇΠ i functions or the violation of the sum rules concernsonly the truncated higher partial waves. Hence, we can reverse the argument: with the assumptionthat sufficiently high partial waves are negligible, the included partial waves have to fulfill the sumrules, which also removes the dependence on the representation. This can be used as a check of oreven a constraint on the input for the γ ∗ γ ∗ → ππ helicity partial waves, in a similar way as sum rulesfor forward HLbL scattering have been used to derive constraints on transition form factors of higherresonances [63, 70, 71].Out of the 15 sum rules, only a single one involves helicity amplitudes starting with S -waves. If wetruncate the partial-wave expansion after S -waves, this sum rule reads (cid:90) ∞ M π ds (cid:48) s (cid:48) − q λ ( s (cid:48) ) (cid:16) Im h , ++ ( s (cid:48) ) − ( s (cid:48) − q − q ) Im h , ++ ( s (cid:48) ) (cid:17) + higher waves , (2.91)where λ ( s ) := λ ( s, q , q ) denotes the Källén triangle function. Verifying that the correspondingsum rule is approximately fulfilled for the γ ∗ γ ∗ → ππ amplitudes constructed in Sect. 4 provides animportant check on the calculation. In fact, it is precisely this sum rule that proves that the S -waveresult derived here based on the BTT formalism and the one from [28] are equivalent. We note that inthe limit of forward kinematics the sum rule (2.91) reduces to the S -wave approximation of the sumrule (27b) in [63]. The calculations of the previous sections allow one to reconstruct the full result for the dispersionrelation for HLbL two-pion contributions to ( g − µ . The imaginary part of the functions ˇΠ i , whichhave to be inserted into the dispersion integrals, are provided by (2.87). Evaluated at s = q , thedispersion relations give the s -channel contribution for the fixed- t representation of all 19 ˆΠ i functionsthat contribute to ( g − µ . Using the crossing relations (2.16) and (2.18), we obtain the five othercontributions: the u -channel contribution for fixed- t as well as both channels in the fixed- s and fixed- u representations. Hence, all six integrals in a dispersion relation for the functions ˆΠ i of the form (2.36)or (2.37) can be calculated.The crucial ingredient in this calculation is the basis change ˇ c ij from scalar functions to helicityamplitudes, which enables the generalization of the S -wave result of [28] to arbitrary partial waves.The matrix ˇ c ij contains two types of ostensible kinematic singularities:1. The kinematic singularities of the singly-on-shell basis ˇΠ i are present, as explained in Sect. 2.4.2.In the dispersion relation, their residues vanish due to the sum rules, hence they can be subtractedexplicitly in the master formula for ( g − µ .30. Additional kinematic singularities ( − q ) − n/ , n = 1 , . . . , , show up in the coefficients ˇ c ij . Theyare introduced by the basis change to helicity amplitudes, i.e. they cancel against kinematic zerosin the helicity amplitudes, present in (2.87) in the Wigner- d functions for fixed- t kinematics.Unfortunately, the matrix ˇ c ij is too lengthy to be shown here in full, but is provided as supplementalmaterial in the form of a Mathematica notebook. In contrast, the explicit results for the two-pion dispersion relation in the S -wave approximationare very compact: ˆΠ J =04 = 1 π (cid:90) ∞ M π ds (cid:48) − λ ( s (cid:48) )( s (cid:48) − q ) (cid:16) s (cid:48) Im h , ++ ( s (cid:48) ) − ( s (cid:48) + q − q )( s (cid:48) − q + q ) Im h , ++ ( s (cid:48) ) (cid:17) , ˆΠ J =05 = 1 π (cid:90) ∞ M π dt (cid:48) − λ ( t (cid:48) )( t (cid:48) − q ) (cid:16) t (cid:48) Im h , ++ ( t (cid:48) ) − ( t (cid:48) + q − q )( t (cid:48) − q + q ) Im h , ++ ( t (cid:48) ) (cid:17) , ˆΠ J =06 = 1 π (cid:90) ∞ M π du (cid:48) − λ ( u (cid:48) )( u (cid:48) − q ) (cid:16) u (cid:48) Im h , ++ ( u (cid:48) ) − ( u (cid:48) + q − q )( u (cid:48) − q + q ) Im h , ++ ( u (cid:48) ) (cid:17) , ˆΠ J =011 = 1 π (cid:90) ∞ M π du (cid:48) λ ( u (cid:48) )( u (cid:48) − q ) (cid:16) Im h , ++ ( u (cid:48) ) − ( u (cid:48) − q − q ) Im h , ++ ( u (cid:48) ) (cid:17) , ˆΠ J =016 = 1 π (cid:90) ∞ M π dt (cid:48) λ ( t (cid:48) )( t (cid:48) − q ) (cid:16) Im h , ++ ( t (cid:48) ) − ( t (cid:48) − q − q ) Im h , ++ ( t (cid:48) ) (cid:17) , ˆΠ J =017 = 1 π (cid:90) ∞ M π ds (cid:48) λ ( s (cid:48) )( s (cid:48) − q ) (cid:16) Im h , ++ ( s (cid:48) ) − ( s (cid:48) − q − q ) Im h , ++ ( s (cid:48) ) (cid:17) , (2.92)where the dependence of the helicity amplitudes on the virtualities is not written explicitly. This resultagrees with [30]. It slightly differs from the S -wave result presented in [28], but, as explained in theprevious section, this difference is precisely of the form of the sum rule (2.91) and thus simply relatedto a different choice of basis.The above result is given in a form that corresponds to the dispersion relation (2.36). In orderto apply it to the pion box, one has to use (2.37), hence the dispersion integrals in (2.92) need to bemultiplied by a factor / . For the proper evaluation of the ππ -rescattering corrections, the contributionof the pion box to the partial waves has to be subtracted: we define the operator S , which takes careof the symmetry factor and the subtraction of the pole × pole term [28]. The imaginary part for the ππ -rescattering contribution is then given byIm ππs h Jλ λ ,λ λ ( s ) = η i η f σ π ( s )16 π S (cid:104) h J,λ λ ( s ) h ∗ J,λ λ ( s ) (cid:105) , (2.93)where S (cid:104) h c J,λ λ ( s ) h c ∗ J,λ λ ( s ) (cid:105) := h c J,λ λ ( s ) h c ∗ J,λ λ ( s ) − N J,λ λ ( s ) N ∗ J,λ λ ( s ) , S (cid:104) h n J,λ λ ( s ) h n ∗ J,λ λ ( s ) (cid:105) := 12 h n J,λ λ ( s ) h n ∗ J,λ λ ( s ) . (2.94)The superscripts refer to charged (c) and neutral (n) pions, respectively, and N J,λ i λ j denotes thepartial-wave projection of the pure pion-pole term, explicitly given in App. G. Arguably the most important result of this paper, especially in view of future applications and gener-alizations, concerns the derivation of the ˇΠ i functions, which allows us to establish a direct correspon-dence between singly-on-shell helicity amplitudes and the scalar functions ¯Π i that enter the master In this notebook, we make use of
FeynCalc [72, 73]. Π i
54 (2.7) 4 off-shell BTT scalarfunctions redundant set; free of kinematic singulari-ties and zeros; full crossing symmetry ˜Π i
43 (2.10) 4 off-shell basis true off-shell basis away from 4 space-timedimensions; no Tarrach redundancies, buttwo ambiguities in 4 space-time dimen-sions; kinematic singularities, see [31] ˆΠ i
54 (2.12) 4 off-shell “basis” changefor ( g − µ redundant set; free of kinematic singulari-ties and zeros; crossing symmetry for pho-tons 1, 2, and 3 ˆΠ g i
19 (2.15) q = 0 contributing to ( g − µ subset of 19 functions ˆΠ i that contributeto ( g − µ : { g i } = { , . . . , , , , , , , , , } ¯Π i
12 (2.22) q = 0 scalar functionsin masterformula correspond to the 19 functions ˆΠ i con-tributing to ( g − µ modulo crossing sym-metry q ↔ − q ˇΠ i
27 (2.62) fixed t = q , q = 0 singly-on-shellbasis fulfill unsubtracted dispersion relations;kinematic singularities depending on q , q , and q only; contain in the limit q → as a subset the 19 functions ˆΠ i contribut-ing to ( g − µ ¯ H j
41 (2.76) 4 off-shell helicity ampli-tudes off-shell HLbL helicity amplitudes; com-plicated kinematic singularities; simpleunitarity relation ¯ H j (cid:12)(cid:12)(cid:12) λ (cid:54) =0
27 3 off-shell singly-on-shellhelicity ampli-tudes helicity amplitudes for the case of an ex-ternal on-shell photon
Table 1:
Scalar functions appearing in the formalism for the two-pion HLbL contribution to ( g − µ . formula (2.25) for the HLbL contribution to ( g − µ . The key quantities in this construction arethe various scalar amplitudes, a glossary of which is provided in Table 1, including a reference to theequation where they are defined and a short definition and explanation. They can be roughly dividedinto four classes: first, the Π i and ˜Π i are related to the general BTT decomposition of the HLbLtensor, irrespective of any application to ( g − µ or dispersion relations. Second, the ˆΠ i and ¯Π i isolatethe functions actually relevant for ( g − µ , by forming suitable subsets and taking the appropriatekinematic limit, but are otherwise still completely general. Third and fourth, the ˇΠ i are constructed asthe crucial intermediate step in the derivation of single-variable partial-wave dispersion relations, byeliminating redundancies in the representation and thereby allowing a well-defined transition to helicityamplitudes ¯ H j . In combination with partial-wave unitarity, this last step completes the derivation ofthe dispersion relation for two-pion intermediate states in the HLbL contribution to ( g − µ . The interest in the pion box is twofold. On the one hand, it gives a unique meaning to the notion ofa pion loop, by virtue of its dispersive definition as two-pion intermediates with a pion-pole LHC, and32s expected to provide the most important contribution to HLbL scattering beyond the pseudoscalarpoles. Phenomenologically, the pion box is fully determined by the pion vector form factor, whichallows us to pin down its numerical value to very high precision, as we will show in Sect. 3.1 includingan error analysis for the form factor input.On the other hand, the pion box constitutes an invaluable test case for the partial-wave formalismthat we have developed in Sect. 2. Given a certain representation of the pion vector form factor, thefull pion box is known exactly, see App. C. Since the partial-wave expansion and the single-variabledispersion relations are valid not only for the rescattering contribution but also for the pion box,provided the correct prefactor in (2.37) according to the counting of double-spectral regions is takeninto account, we can use the pion box to check whether the partial-wave representation converges tothe full result upon resummation of the partial waves, and we can study the details of the convergencebehavior numerically.In a similar way, the pion box provides a test case for the sum rules for the HLbL scalar functions.In Sect. 3.2 we demonstrate that they are indeed fulfilled, which is a prerequisite for the unsubtractedsingle-variable dispersion relations derived in Sect. 2. In Sect. 3.3, we investigate the convergencebehavior of its partial-wave representation and discuss the implications for applications beyond thepion box, such as the ππ -rescattering contribution discussed in Sect. 4. For the numerical evaluation of the pion box, the representation in terms of Feynman-parameterintegrals given in App. C proves most efficient. This representation is based on the equivalence of thepion box with the FsQED amplitude [28], which we proved in [31]. It requires the numerical evaluationof two-dimensional Feynman integrals with the pion vector form factor as the only input. For a reliableevaluation of the pion-box contribution to ( g − µ , we therefore need a precise representation of thepion vector form factor in the space-like region.Since about of the final pion-box ( g − µ integral originate from virtualities below GeV, itis most critical that the low-energy properties be correctly reproduced. Experimentally, the availableconstraints derive from e + e − → π + π − data, which determine the time-like form factor [74–79], andspace-like measurements by scattering pions off an electron target [80, 81]. We have also checked thatour representation is consistent with extractions of the space-like form factor from e − p → e − π + n data [82–85], although due to the remaining model dependence of extrapolating to the pion pole wedo not use these data in our fits. To obtain a representation that allows us to simultaneously fit space-and time-like data, and thereby profit from the high-statistics form factor measurements motivatedmainly by the two-pion contribution to HVP, we adopt the formalism suggested in [86, 87] (similarrepresentations have been used in [88–93]), whose essential ingredients will be briefly reviewed in thefollowing.The form factor is decomposed according to F Vπ ( s ) = Ω ( s ) G ρω ( s ) G inel ( s ) . (3.1)The Omnès factor Ω ( s ) = exp (cid:40) sπ (cid:90) ∞ M π ds (cid:48) δ ( s (cid:48) ) s (cid:48) ( s (cid:48) − s ) (cid:41) (3.2)would provide the exact answer if only the elastic ππ channel contributed to the unitarity relation ofthe form factor. It is fully determined by the P -wave phase shift δ . Next, G ρω describes the isospin-violating coupling to the π system, which becomes relevant in the vicinity of the ρ peak as reflectedby ρ – ω mixing. In practice, a one-parameter ansatz G ρω ( s ) = 1 + (cid:15) ρω ss ω − s , s ω = (cid:18) M ω − i Γ ω (cid:19) , (3.3)33 s (cid:2) GeV (cid:3) | F V π | NA7JLab s (cid:2) GeV (cid:3) || Figure 11:
Left: space-like pion form factor from our dispersive fit in comparison to data from NA7 [81] andJLab [83–85] (the latter are not included in the fit). The error band represents the variation observed betweendifferent time-like data sets. Right: pion form factor in the time-like region from the combined fit to NA7and [77], chosen here for illustrative purposes only. Fits to the other time-like data sets look very similar andlead to the same numerical results within the accuracy quoted in (3.5). proves indistinguishable from a dispersively improved version that eliminates the imaginary part belowthe π threshold [86, 87]. Finally, G inel parameterizes the effect of higher inelastic channels. We use aconformal mapping G inel ( s ) = 1 + p (cid:88) i =1 c i (cid:0) z ( s ) i − z (0) i (cid:1) , z ( s ) = √ s πω − s − √ s πω − s √ s πω − s + √ s πω − s , (3.4)where s ωπ = ( M ω + M π ) denotes the threshold where phenomenologically π inelasticities first start toset in and the second parameter is fixed at s = − GeV . The ππ phase shift is taken from the extendedRoy-equation analysis of [94], which determines δ up to s m = (1 . GeV ) in terms of its values at s m and s A = (0 . GeV ) . Our representation thus involves p free parameters: the ρ – ω mixingparameter (cid:15) ρω , the two values of the phase shift at s m and s A , and p parameters from the conformalexpansion of G inel . This representation ensures that the form factor behaves as /s asymptoticallyas long as the phase shift approaches π , up to logarithms in agreement with the expectation fromperturbative QCD [95–99]. We impose this asymptotic behavior by smoothly extrapolating δ to π from the boundary s m of the applicability of the Roy solution, but checked that introducing effectsfrom ρ (cid:48) , ρ (cid:48)(cid:48) excitations as suggested in [40] does not impact the space-like form factor. The form of G inel can be further constrained by requiring that the imaginary part exhibit the expected P -wavebehavior and respect the Eidelman–Łukaszuk bound [100], but again the impact on the space-like formfactor proves to be small.We fit this representation simultaneously to the space-like data from [81] as well as one of the time-like data sets [74–79] (restricted to data points below GeV). Moreover, we varied s , p = 1 , , andconstructed an error band for the uncertainties in δ apart from the phase shifts at s m and s A . We findthat the results for the space-like form factor are extremely stable to all these variations, the largesteffect being produced by the differences between the time-like data sets. For the accuracy requiredin HLbL scattering we can therefore simply take the largest variation among them as an uncertaintyestimate, without having to perform a careful investigation of the statistical and systematic errors thatare crucial when combining the different data sets for HVP. The result for the space-like form factor34s shown in Fig. 11, leading to a numerical evaluation for the pion box of a π -box µ = − . × − . (3.5) In Sect. 2.3, we have presented sum rules for the BTT scalar functions that follow from a uniformasymptotic behavior of the HLbL tensor and ensure the independence from the choice of the tensorbasis. These sum rules prove essential for the derivation of single-variable dispersion relations that canbe used with input on the γ ∗ γ ∗ → ππ helicity partial waves. Furthermore, an important consequenceof the BTT sum rules are the physical sum rules in Sect. 2.4.3, which can be expressed in terms ofhelicity amplitudes.An important test case for our partial-wave formalism is the pion box: the fact that we knowthe full result allows us to test the convergence behavior of the partial-wave approximation. Beforeturning to the tests of the full formalism in Sect. 3.3, here we check that the sum rules as a necessaryprerequisite for the single-variable dispersion relations are indeed fulfilled in the case of the pion box.Due to the equivalence of the pion box with the FsQED amplitude [28, 31], these tests can be directlyperformed with sQED.Although we have formulated the sum rules in terms of the BTT functions Π i , an explicit calculationmust avoid the Tarrach ambiguities present in this set. In Sect. 2.3, we have derived the sum rules at acertain kinematic point where the ambiguity vanishes. The most convenient and complete method tocheck the sum rules uses the basis coefficient functions ˜Π i , see [31]. In this set, the Tarrach redundancyis traded for kinematic singularities. We remove these singularities by multiplying the ˜Π i functionswith the denominators of the Tarrach poles, i.e. we consider q · q ˜Π = q · q Π − q · q q · q Π ,q · q ˜Π = q · q Π + q · q Π ,q · q ˜Π = q · q Π + q · q q · q Π ,q · q q · q ˜Π = q · q q · q Π − q · q q · q Π ,q · q ˜Π = q · q Π + q · q Π . (3.6)The functions ˜Π , . . . , ˜Π are not involved in sum rules, while the functions ˜Π , . . . , ˜Π vanish insQED. All the remaining functions are related to the ones above by crossing. Apart from q · q q · q ˜Π ,the combinations in (3.6) have a mass dimension that suggests an asymptotic behavior (cid:16) s − , t − , u − .The BTT sum rules can therefore be formulated as the requirement that the functions in (3.6) fulfillan unsubtracted Mandelstam representation. In contrast, in [31] we only verified that subtractedMandelstam representations which follow from unsubtracted ones for the BTT functions Π i are actuallyfulfilled.In analogy to [31], we extract the sQED double-spectral densities of these functions from the explicitexpression of the loop calculation in terms of Passarino–Veltman amplitudes [102, 103]: in such adecomposition into scalar loop functions the double-spectral densities are given by the coefficients ofthe D functions times the D spectral densities. By inserting the double-spectral densities into anunsubtracted Mandelstam representation of the form π (cid:90) ds (cid:48) dt (cid:48) ρ st ( s (cid:48) , t (cid:48) )( s (cid:48) − s )( t (cid:48) − t ) + 1 π (cid:90) ds (cid:48) du (cid:48) ρ su ( s (cid:48) , u (cid:48) )( s (cid:48) − s )( u (cid:48) − u ) + 1 π (cid:90) dt (cid:48) du (cid:48) ρ tu ( t (cid:48) , u (cid:48) )( t (cid:48) − t )( u (cid:48) − u ) , (3.7)we have verified numerically that the functions (3.6) are reproduced. Surprisingly, even q · q q · q ˜Π fulfills an unsubtracted Mandelstam representation, which is not expected from the mass dimension The multidimensional integrals required for the numerical evaluation of (2.25) are performed using the
CUBA library [101]. ( g − µ single-variable dispersionintegrals, and thus allow one to impose constraints on the γ ∗ γ ∗ → ππ helicity amplitudes used as inputfor a numerical evaluation. We have verified that these sum rules are fulfilled in the case of the pionbox, by extracting the imaginary parts of the ˇΠ i functions from the sQED calculation and calculatingthe integrals numerically. These sQED tests thereby allow one to establish the validity of nearly allsum rules—except for the last one involving ˇΠ − ˇΠ + ˇΠ = ˜Π − ˜Π , which vanishes identically insQED. It should be stressed that the underlying assumptions follow solely from demanding a uniformasymptotic behavior of the HLbL tensor, but as the discussion in App. E.2 shows, similar conclusionscan be drawn from Regge models as well. Together with the explicit checks in the case of sQED thereis therefore compelling evidence for our assumptions regarding the asymptotic behavior of the HLbLtensor. In the following, we perform tests of the helicity partial-wave dispersion relations developed in Sect. 2by applying the formalism to the pion box. In this case, a dispersion relation of the form (2.37) has tobe used in order to account for the fact that only three different double-spectral regions are present.We emphasize that in this test case each single-variable dispersion relation reconstructs the full pionbox. Therefore, we can test the three channels separately—each must converge to the full result uponresummation of the partial-wave series.The input for the γ ∗ γ ∗ → ππ helicity partial waves in the case of the pion box is given by thepartial-wave projection of the pure pion-pole terms, see App. G. In order to simplify the convergencechecks, we use a simple vector-meson dominance representation for the pion vector form factor: F Vπ,
VMD ( q ) = M ρ M ρ − q . (3.8)Such a form factor leads to a π -box, VMD µ = − . × − , which is very close to the full result obtainedwith the dispersive representation of the form factor discussed in Sect. 3.1. The convergence behaviorof the partial-wave expansion is not affected by the details of the form factor implementation.Since our formalism for single-variable dispersion relations is valid for arbitrary partial waves, wecan extend these tests in principle to an arbitrary angular momentum J . In practice, our numericalimplementation becomes less reliable for large values of J , so that we performed the numerical testsup to J = 20 and estimated the truncation error by extrapolation.The HLbL contribution to ( g − µ is given as a sum of 12 terms in the master formula (2.25),which, in principle, are completely independent. However, in the case of the pion box it turns out thatespecially for the lower partial waves a numerical cancellation occurs that leads to a faster convergenceof a µ than for the individual terms. Therefore, we define the following vector in the 12-dimensionalspace of the contributions to the master formula: a HLbL µ := (cid:8) a HLbL µ,i (cid:9) i ,a HLbL µ,i := α π (cid:90) ∞ d ˜Σ ˜Σ (cid:90) dr r (cid:112) − r (cid:90) π dφ T i ( Q , Q , τ ) ¯Π i ( Q , Q , τ ) , (3.9)36xed- s fixed- t fixed- u average J max δ J max ∆ J max δ J max ∆ J max δ J max ∆ J max δ J max ∆ J max .
0% 100 . − .
2% 35 . − .
2% 35 .
4% 29 .
2% 55 . .
1% 50 . − .
3% 5 .
6% 7 .
3% 8 .
0% 10 .
4% 20 . .
8% 28 . − .
5% 2 .
1% 3 .
6% 3 .
9% 4 .
3% 11 . .
7% 16 . − .
7% 1 .
1% 2 .
1% 2 .
2% 2 .
4% 6 . .
5% 9 . − .
4% 0 .
6% 1 .
3% 1 .
4% 1 .
5% 3 . .
3% 5 . − .
2% 0 .
4% 0 .
9% 1 .
0% 1 .
0% 2 . .
7% 3 . − .
1% 0 .
3% 0 .
7% 0 .
7% 0 .
7% 1 . .
3% 2 . − .
1% 0 .
2% 0 .
5% 0 .
5% 0 .
6% 1 . .
0% 1 . − .
0% 0 .
2% 0 .
4% 0 .
4% 0 .
4% 0 . .
8% 1 . − .
0% 0 .
1% 0 .
3% 0 .
3% 0 .
4% 0 . .
7% 0 . − .
0% 0 .
1% 0 .
3% 0 .
3% 0 .
3% 0 . Table 2:
Convergence of the partial-wave expansion in the case of the pion box: the three single-variabledispersion relations and their average are compared. See main text for the definition of the relative deviations. so that a HLbL µ = (cid:88) i =1 a HLbL µ,i . (3.10)In order to quantify the convergence behavior, we define the following two quantities: the relativedeviation between the full pion-box contribution to ( g − µ and its partial-wave approximation δ J max := 1 − a π -box, PW µ,J max a π -box µ , (3.11)as well as the analogous quantity in the 12-dimensional space of the contributions to the master formula ∆ J max := (cid:12)(cid:12)(cid:12) a π -box, PW µ,J max − a π -box µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a π -box µ (cid:12)(cid:12) , (3.12)where | · | denotes the 12-dimensional Euclidean norm. Due to cancellations between the 12 terms inthe master formula, δ J max will indicate a faster convergence than ∆ J max , which is more robust againstcancellations.Table 2 shows the results of a detailed study of the convergence behavior of the partial-waverepresentation for the test case of the pion box. Both measures δ J max and ∆ J max for the deviation fromthe full pion box result are displayed for fixed- s , fixed- t , and fixed- u dispersion relations, as well as forthe average of the three single-variable dispersion relations (2.37). In this context we use the notion offixed- ( s, t, u ) as follows: it defines the dispersion relation for each of the six representatives in (2.15),while the remaining scalar functions are obtained via the crossing relation (2.16). In particular, thisimplies that in the so-called fixed- s evaluation, we do use a fixed- s dispersion relation for ˆΠ , but afixed- t dispersion relation for ˆΠ and a fixed- u dispersion relation for ˆΠ .Next, we comment on the following two observations:1. The S -wave approximation shows a particular pattern: the fixed- s representation vanishes, whilefixed- t and - u agree. 37. The fixed- s representation exhibits a slower convergence than the other two dispersion relations.In order to understand the first point, consider the explicit S -wave representation for the ˆΠ i func-tions, (2.92). We note that S -waves contribute only to the s -channel discontinuities in ˆΠ and ˆΠ ,while the t - and u -channel discontinuities for these two functions start with D -waves (the situation forthe other functions follows from crossing symmetry). A discontinuity in the s -channel contributes to afixed- t and fixed- u dispersion relation, while in a fixed- s dispersion relation the integral runs only over t - and u -channel discontinuities. This means that in the fixed- s representation in Table 2, no S -wavediscontinuity is encountered at all, hence in this representation the first non-vanishing contributionis obtained from D -waves. Furthermore, because the S -wave s -channel discontinuity has no angulardependence, it contributes identically to a fixed- t and fixed- u dispersion relation, which makes thefixed- t and fixed- u results in Table 2 agree at J max = 0 .The second point can be understood as follows. For each of the six representatives in (2.15) the s -channel is special with respect to the other two. This is due to the fact that the associated Lorentzstructure exhibits an s -channel symmetry, either C , C , or both, the only special case being ˆΠ ,which is totally crossing symmetric in all three channels. For instance, the Lorentz structure ˆ T µνλσ isthe one that belongs to the s -channel (pseudo-scalar) π -pole contribution to HLbL scattering, while ˆ T µνλσ can be related to an s -channel scalar amplitude, which manifests itself as the S -wave s -channel ππ contribution. It is therefore not surprising that even in the case of the pion box, the s -channeldiscontinuity for the functions (2.15) is more important than the other two discontinuities. Since thisis the discontinuity that evades the fixed- s dispersion relation, we observe a slower convergence patternin this case.We have performed these convergence tests not only with our preferred representation for the ˇΠ i functions, but also with different versions that are modified by terms that vanish due to the sumrules (2.71). While the exact numbers do differ—as expected given the fact that the sum rules onlyhold for the full amplitudes but not the individual partial waves—the sum rule violations in the caseof the pion box due to the partial-wave approximation are reasonably small and the overall pictureremains the same.To fully understand the partial-wave convergence of the pion box we also studied the remainingdeviation from the full result at J = 20 . Empirically, we observe that the size of the individual termsfor given J is well described by a fit function a π -box, PW µ,J ∼ cJ x , (3.13)which in a double-log plot produces the straight lines in Fig. 12. The fact that the first few terms donot fall on this line indicates that the form (3.13) is only asymptotic, and might also be related to theabovementioned cancellations for low J (the fit therefore excludes the points for J ≤ ). The figureshows that the rate of convergence is actually similar for fixed- s and fixed- u , both of which yield anexponent x ≈ − , while the fixed- t representation converges with x ≈ − . The slower convergence ofthe fixed- s results seen in Table 2 is therefore a remnant of the missed S -wave contribution that leadsto larger deviations for small J , not the overall rate of convergence. The resummation of the termswith J > based on the fit function then removes all remaining discrepancies, providing a strongcheck of the partial-wave formalism developed in Sect. 2.Finally, we discuss the consequences for the application of the formalism to the case of two-pioncontributions beyond the pion box, most importantly the unitarity (or rescattering) correction. Themost important difference is related to the fact that for these applications, instead of (2.37), thedispersion relation (2.36) applies, where due to the different double-spectral regions an overall factor / instead of / is required. However, this means that for the rescattering contribution the slowerconvergence of the fixed- s dispersion relation is of no significance: let us assume that an importantresonant contribution shows up in a partial wave in the s -channel. This resonance will be captured We thank Martin J. Savage for suggesting this ansatz. log J l og (cid:12)(cid:12) a π - b o x µ × (cid:12)(cid:12) fixed- s fixed- t fixed- u average Figure 12:
Extrapolation of the partial-wave results for the fixed- ( s, t, u ) representations as well as theiraverage, see main text for details. by the fixed- t and fixed- u dispersion relation (though not by the fixed- s dispersion relation). Sincethe full result is given by the sum of the three dispersion relations weighted by / , this behavior isactually expected and the absence of the resonance in the fixed- s representation does not impact theaverage in the symmetric representation (2.36) (in contrast to the pion box, where the missed S -wavecontribution needs to be recovered by the higher partial waves). Therefore, the average in the caseof the pion box in Table 2 should rather be regarded as a worst-case scenario—for the convergencebehavior in the case of rescattering contributions the fixed- t and fixed- u dispersion relations are morerepresentative.The second important difference concerns the presence of resonances in the rescattering contribu-tion, a feature that does not occur in the pion box. We expect the rescattering contribution to bedominated by resonant effects, whereas the convergence behavior established for the pion box can beunderstood as a weighting of the partial waves. In the truncated partial-wave series, the resonances inthe included partial waves are fully reproduced. The approximate fulfillment of the sum rules indicatesthen whether neglected higher partial waves still play an important role, to the effect that the size ofthe sum-rule violations allows one to estimate the accuracy of the calculation. The natural application of the partial-wave formalism developed in the main part of this paper concerns ππ rescattering effects, which can be considered a unitarization of the pure pion-pole LHC that definesthe pion box. To isolate this contribution, it suffices to subtract the pure pion-pole piece in the partial-wave unitarity relation, and insert for the remainder phenomenological input for the γ ∗ γ ∗ → ππ partialwaves. The construction of such input is by itself challenging, given that direct experimental results,at least for the doubly-virtual case, are not expected in the near future.In the on-shell case, available data on γγ → ππ [104–109] (in combination with γγ → ¯ KK [110–116]) are now sufficient to perform a partial-wave analysis [117], but such an approach appears unre-alistic to control the dependence on the photon virtualities. However, approaches that exploit morecomprehensively the analytic properties of the amplitude, see [54, 55, 118] for on-shell photons, canbe extended towards the off-shell case with limited data input required to determine parameters, asdemonstrated for the singly-virtual process in [56]. The essential features of the generalization towards39he doubly-virtual case, i.e. the appearance of anomalous thresholds for time-like kinematics [57] andthe modifications to tensor basis and kernel functions [28, 31], have already been laid out in previouswork, but the practical implementation involves a number of challenges: due to the strong couplingbetween the ππ/ ¯ KK channels in the isospin- S -wave a single-channel analysis is limited to rather lowenergies [54, 56, 117, 118], assumptions for the LHC and number of subtractions need to be carefullystudied to reliably assess the sensitivity to the high-energy input in the dispersive integrals [55], a fullanalysis of the generalized Roy–Steiner equations [28, 31, 55] involves solving coupled S - and D -wavesystems of various helicity projections, and last but not least constraints on the γ ∗ γ ∗ → ππ amplitudesfrom asymptotic behavior and the sum rules derived in Sect. 2.4.3 need to be incorporated. A fullanalysis along these lines will be left for future work.To obtain a first estimate of rescattering effects, we concentrate on S -waves and consider a furthersimplified system: first, we use ππ phase shifts from the inverse-amplitude method, which reproducesthe phenomenological phase shifts as well as the f (500) properties at low energies, and in additionallows one to separate the ππ rescattering from the ¯ KK channel in a well-defined manner, all whileproviding a reasonable extrapolation for high energies. In addition, we restrict ourselves to a pion-pole LHC in the solution of the Roy–Steiner equations, which has the advantage that the off-shellbehavior is still described by the pion vector form factor. In the following, we lay out the details ofthis calculation, and discuss the consequences for rescattering effects in ( g − µ . γ ∗ γ ∗ → ππ helicity partial waves from the inverse-amplitude method Unitarization within the inverse-amplitude method (IAM) [64–69] is based on the observation thatelastic unitarity Im t ( s ) = σ π ( s ) | t ( s ) | (4.1)for a ππ partial-wave amplitude t ( s ) impliesIm t ( s ) = − σ π ( s ) , (4.2)which together with the chiral expansion t ( s ) = t ( s ) + t ( s ) + O ( p ) and perturbative unitarityIm t ( s ) = 0 , Im t ( s ) = σ π ( s ) | t ( s ) | , (4.3)already concludes the naive derivation of the IAM prescription t IAM ( s ) = 1 Re t ( s ) − iσ π ( s ) = (cid:0) t ( s ) (cid:1) t ( s ) − t ( s ) . (4.4)However, in the single-channel case the IAM approach can be justified much more rigorously basedon dispersion relations, where the only approximation involves replacing the LHC by its chiral expan-sion [119]. In this way, one can also remedy the fact that the standard IAM fails to correctly reproducethe Adler zero [120, 121], and is thus not fully consistent with chiral symmetry. The modified form ofthe IAM (mIAM) becomes [119] t mIAM ( s ) = (cid:0) t ( s ) (cid:1) t ( s ) − t ( s ) + A mIAM ( s ) , (4.5)where the additional term A mIAM ( s ) = (cid:18) t ( s ) t (cid:48) ( s ) (cid:19) (cid:20) t ( s )( s − s ) − s − s A ( s − s )( s − s A ) (cid:18) t (cid:48) ( s ) − t (cid:48) ( s ) + t ( s ) t (cid:48)(cid:48) ( s ) t (cid:48) ( s ) (cid:19)(cid:21) (4.6) For ππ scattering the expression simplifies because t (cid:48)(cid:48) ( s ) = 0 and t ( s ) /t (cid:48) ( s ) = s − s . .4 0.6 0.8 1 1.2012345 √ s [GeV] δ [ r a d ] IAMBernMadrid/Krakow √ s [GeV] δ [ r a d ] IAMBernMadrid/Krakow
Figure 13: I = 0 (left) and I = 2 (right) ππ S -wave phase shifts from the IAM (black solid line), in comparisonto the Bern (red dashed line) [94, 124] and Madrid/Krakow (blue dot-dashed line) [125] Roy-equation analyses. ensures that the Adler zero s A = s + s + O ( p ) occurs at its O ( p ) position, i.e. t ( s ) = 0 , t ( s + s ) + t ( s + s ) = 0 . (4.7)This form of the IAM thus correctly describes the low-energy phase shifts as well as resonance prop-erties, and has indeed been used in recent years to determine the quark-mass dependence of σ and ρ resonances [122, 123]. For our purposes, the single-channel IAM for ππ scattering conveniently sep-arates the ππ channel from its mixing to ¯ KK in the vicinity of the f (980) and defines a reasonablecontinuation to high energies, without compromising the low-energy physics.We use the -loop IAM with low-energy constants as specified in [123], which produces the phaseshifts shown in Fig. 13. As expected, there is good agreement throughout, apart from the fact thatthe IAM I = 0 phase shift avoids the rise related to the f (980) and the coupling to the ¯ KK channel.We also checked that the σ properties [126] are reproduced: for the pole position we find √ s σ =(0 .
443 + i . GeV, to be compared to √ s σ = (0 .
441 + i . GeV [127] and similar numbers fromother recent dispersive extractions [118, 128]. Accordingly, the width comes out a bit too low, as doesthe residue at the pole g σππ . This deviation is consistent with earlier IAM analyses, see e.g. [122] forthe analogous calculation including the mIAM correction, and can certainly be tolerated to obtain anestimate for the HLbL rescattering contribution, which, after all, only requires the amplitude on thereal axis, not the analytic continuation into the complex plane where the slight discrepancy in thewidth would matter most. Similarly, one can check the coupling to two photons | g σγγ /g σππ | ∼ . ,well in line with | g σγγ /g σππ | = 0 . and . from [55] and [118], respectively.With the input for the ππ phase shifts specified, the γ ∗ γ ∗ → ππ amplitudes follow by solving thegeneralized Roy–Steiner equations derived in [28, 31] for doubly-virtual kinematics. For the S -waves,these dispersion relations take the form (isospin indices are suppressed for the time being) h , ++ ( s ) = ∆ , ++ ( s ) + 1 π (cid:90) ∞ M π ds (cid:48) (cid:20)(cid:18) s (cid:48) − s − s (cid:48) − q − q λ ( s (cid:48) ) (cid:19) Im h , ++ ( s (cid:48) ) + 2 q q λ ( s (cid:48) ) Im h , ( s (cid:48) ) (cid:21) ,h , ( s ) = ∆ , ( s ) + 1 π (cid:90) ∞ M π ds (cid:48) (cid:20)(cid:18) s (cid:48) − s − s (cid:48) − q − q λ ( s (cid:48) ) (cid:19) Im h , ( s (cid:48) ) + 2 λ ( s (cid:48) ) Im h , ++ ( s (cid:48) ) (cid:21) , (4.8)with LHC singularities represented by the inhomogeneities ∆ , ++ ( s ) and ∆ , ( s ) . These equationscan be rewritten as h , ++ ( s ) ± (cid:113) q q h , ( s ) = ∆ , ++ ( s ) ± (cid:113) q q ∆ , ( s )+ s − (cid:0)(cid:112) q ∓ (cid:112) q (cid:1) π (cid:90) ∞ M π ds (cid:48) Im (cid:2) h , ++ ( s (cid:48) ) ± (cid:112) q q h , ( s (cid:48) ) (cid:3) ( s (cid:48) − s ) (cid:0) s (cid:48) − ( (cid:112) q ∓ (cid:112) q ) (cid:1) . (4.9)41he new combinations still fulfill Watson’s theorem [129]Im (cid:104) h , ++ ( s ) ± (cid:113) q q h , ( s ) (cid:105) = sin δ ( s ) e − iδ ( s ) (cid:104) h , ++ ( s ) ± (cid:113) q q h , ( s ) (cid:105) θ (cid:0) s − M π (cid:1) , (4.10)so that the dispersion relation reduces to a standard Muskhelishvili–Omnès (MO) problem [130, 131],whose solution reads h , ++ ( s ) ± (cid:113) q q h , ( s ) = ∆ , ++ ( s ) ± (cid:113) q q ∆ , ( s ) (4.11) + Ω ( s ) (cid:16) s − (cid:0)(cid:112) q ∓ (cid:112) q (cid:1) (cid:17) π (cid:90) ∞ M π ds (cid:48) (cid:2) ∆ , ++ ( s (cid:48) ) ± (cid:112) q q ∆ , ( s (cid:48) ) (cid:3) sin δ ( s (cid:48) )( s (cid:48) − s ) (cid:0) s (cid:48) − ( (cid:112) q ∓ (cid:112) q ) (cid:1) | Ω ( s (cid:48) ) | , with the Omnès function Ω ( s ) = exp (cid:40) sπ (cid:90) ∞ M π ds (cid:48) δ ( s (cid:48) ) s (cid:48) ( s (cid:48) − s ) (cid:41) . (4.12)For convenience, we finally rewrite the result in terms of the original helicity amplitudes according to h , ++ ( s ) = ∆ , ++ ( s )+ Ω ( s ) π (cid:90) ∞ M π ds (cid:48) sin δ ( s (cid:48) ) | Ω ( s (cid:48) ) | (cid:20)(cid:18) s (cid:48) − s − s (cid:48) − q − q λ ( s (cid:48) ) (cid:19) ∆ , ++ ( s (cid:48) ) + 2 q q λ ( s (cid:48) ) ∆ , ( s (cid:48) ) (cid:21) ,h , ( s ) = ∆ , ( s )+ Ω ( s ) π (cid:90) ∞ M π ds (cid:48) sin δ ( s (cid:48) ) | Ω ( s (cid:48) ) | (cid:20)(cid:18) s (cid:48) − s − s (cid:48) − q − q λ ( s (cid:48) ) (cid:19) ∆ , ( s (cid:48) ) + 2 λ ( s (cid:48) ) ∆ , ++ ( s (cid:48) ) (cid:21) . (4.13)For a pion-pole LHC ∆ , ++ ( s ) and ∆ , ( s ) simply correspond to the partial-wave projection of theBorn terms, given in App. G, which shows that the dependence on the virtualities, apart from themodified kernel functions in the MO solution, is still governed by the pion vector form factor. Inparticular, the corresponding factor F Vπ ( q ) F Vπ ( q ) can be moved out of the integrals in (4.13), so thatone can simply calculate a reduced amplitude, with the dependence on the pion form factors fullyfactorized. Further, in the solution of Roy–Steiner equations, a MO representation similar to (4.13) isoften required for the low-energy region only, in order to match to some known high-energy input, andto this end a finite matching point is introduced [55, 132–135]. In case the amplitudes are assumedto vanish above the matching point, it effectively acts as a cutoff both in (4.13) and in the Omnèsfunction. We will use this variant of the MO solution to estimate the sensitivity to the high-energyextrapolation of the phase shifts, referring for more details of its implementation to [55, 132].Finally, the justification why an unsubtracted representation such as (4.13) is still expected toprovide a decent description is two-fold: first, by removing the ¯ KK intermediate states the Omnèsfunctions are smoothened considerably around the nominal f (980) position, which eliminates mostof the need for subtractions necessary otherwise in a single-channel description to suppress the corre-sponding peak in the Omnès function. Second, while in general a precision description does requiresubtractions [54, 55], we observe in the on-shell case that the results particularly for the chargedchannel are reasonably close to the twice-subtracted variants studied in [55], see Fig. 14 for a cutoff Λ = 1
GeV. The upper panel shows the modulus | h I , ++ | for isospin I = 0 and I = 2 , which for theunsubtracted IAM emerges remarkably close to the twice-subtracted variant in both cases. However,this agreement is largely driven by the projection of the Born term, while a more realistic picture canbe obtained by considering the rotated amplitudes | h , ++ | c = 1 √ | h , ++ | + 1 √ | h , ++ | , | h , ++ | n = 1 √ | h , ++ | − (cid:114) | h , ++ | , (4.14)42 .4 0.6 0.8-10123 √ s [GeV] | h , ++ | √ s [GeV] | h , ++ | √ s [GeV] | h , ++ | c − N , ++ √ s [GeV] | h , ++ | n Figure 14:
Comparison of the γγ → ππ S -waves from this work (black solid line) to the different subtractionschemes from [55] as indicated. Upper/lower panel: left/right corresponds to I = 0 / and charged/neutralchannel, respectively, as explained in the main text. and subtracting the Born term in the charged channel. In this way, we find that the agreement is stillvery good for the charged combination, while the neutral channel is less well reproduced based on thepion-pole LHC alone, see lower panel in Fig. 14. To improve the quantitative agreement, the intro-duction of subtraction constants becomes unavoidable. These subtraction constants can be identifiedwith pion polarizabilities and were taken from -loop ChPT [136, 137] in [55]. The agreement in thecharged channel implies that the corresponding sum rules for the subtraction constants, just based onthe pion-pole LHC, are reasonably well fulfilled, while significant corrections are expected in the neu-tral channel. This interplay with the pion polarizabilities will be discussed in more detail in Sect. 4.3.For the moment, the fact that the dominant rescattering correction is generated by the charged-pionintermediate states, with neutral pions first entering at three-loop order in the chiral expansion, en-sures that the Roy–Steiner solution (4.13) captures the phenomenology of unitarity corrections to thepion-pole LHC, i.e. the rescattering effects required to unitarize the pion-box contribution. ππ -rescattering contribution to ( g − µ Based on the amplitudes calculated from (4.13) we are now in the position to present a first numericalevaluation for the S -wave ππ rescattering effects. For simplicity, we use a VMD pion form factor,which proves to be very close to a full phenomenological determination extrapolated from the time-likeregion [138], see Sect. 3.1. Restoring isospin indices, symmetry factors, virtualities, and subtractingthe corresponding isospin projection of the pion-pole terms N J,λ λ , the relevant imaginary parts in43utoff GeV . GeV GeV ∞ I = 0 − . − . − . − . I = 2 2 . . . . sum − . − . − . − . Table 3:
Results for the S -wave rescattering contribution to ( g − µ in units of − . The cutoff refers tothe finite-matching-point analog of (4.13). cutoff GeV . GeV GeV ∞ ++ , ++ 6 . . . . I = 0 00 , ++ − . − . − . − . sum − . − . − . − . , ++ − . − . − . − . I = 2 00 , ++ 1 . . . . sum . . . . Table 4:
Contribution to the sum rule (2.91) from h , ++ and h , ++ as well as their sum once integratedover momenta and virtualities in the ( g − µ master formula as explained in the main text, in units of − . the HLbL integral becomeIm h ,I ++ , ++ (cid:0) s ; q , q , q , (cid:1) = σ π ( s )32 π (cid:16) h I , ++ ( s ; q , q ) h I , ++ ( s ; q , − c I N , ++ ( s ; q , q ) N , ++ ( s ; q , (cid:17) , Im h ,I , ++ (cid:0) s ; q , q , q , (cid:1) = σ π ( s )32 π (cid:16) h I , ( s ; q , q ) h I , ++ ( s ; q , − c I N , ( s ; q , q ) N , ++ ( s ; q , (cid:17) , (4.15)with isospin factors c = 4 / , c = 2 / .The numerical results for the S -wave contribution then follow from (2.36) together with the dis-persive representation for the scalar functions derived in Sect. 2. Since the full integration becomesnumerically costly—with the dispersion integral in (4.13), the ( g − µ dispersion integral, and threeintegrals in the master formula (2.25) this would amount to a delicate -dimensional integral, whereinin addition the Omnès factor requires the numerical evaluation of yet another integral—we calculatethe γ ∗ γ ∗ → ππ amplitudes on a three-dimensional grid in ( s, q , q ) and then interpolate in the re-maining -dimensional ( g − µ integration. Using up to grid points in each variable the resultsbecome insensitive to the interpolation uncertainty, and we obtain the values listed in Table 3. Asexpected based on the size of the phase shifts, the I = 2 contribution is much smaller than its I = 0 counterpart, while in both cases the variation with respect to the cutoff amounts to about one unit.Accordingly, this estimate can be interpreted as evidence for a rescattering contribution correspondingto f (500) degrees of freedom of about − × − in the HLbL contribution to ( g − µ .Another check on our input for γ ∗ γ ∗ → ππ follows from the sum rule (2.91). In fact, it is preciselythis sum rule that ensures that the S -wave rescattering contribution as formulated in [28] and the onefrom Sect. 2.5 are strictly equivalent. Furthermore, this observation immediately suggests a way howto condense the full sum rule into a single number: the difference between the two representationsamounts to a shift in ˆΠ of the size ∆ ˆΠ = 2 π (cid:90) ∞ M π ds (cid:48) s (cid:48) − q ) λ ( s (cid:48) ) (cid:16) Im h , ++ ( s (cid:48) ) − (cid:0) s (cid:48) − q − q (cid:1) Im h , ++ ( s (cid:48) ) (cid:17) , (4.16)44 GeV . GeV GeV ∞ ChPT ( α − β ) π ± (cid:2) − fm (cid:3) . . . . . . α − β ) π (cid:2) − fm (cid:3) . . . . − . α − β ) π ± (cid:2) − fm (cid:3) . . . . . . α − β ) π (cid:2) − fm (cid:3) . . . . . . Table 5:
Pion polarizabilities from the sum rules (4.19) for a pion-pole LHC and different values of the cutoff Λ ,in comparison to the chiral two-loop prediction from [136, 137]. The two numbers in the case of the charged-pionquadrupole polarizability refer to two different sets of low-energy constants. and accordingly in ˆΠ and ˆΠ from crossing, so that the convolution in the ( g − µ integral should bedone with the corresponding kernel function. Still subtracting the pion-pole terms since the validityof the sum rule in sQED is already known, we find the results for the separate contribution from h , ++ and h , ++ as listed in Table 4. The expected cancellation already works at the level of with S -waves only, and even better for the larger values of the cutoff. Such a error on the actualrescattering contributions from Table 3 would yield a very similar uncertainty estimate as the variationobserved from the cutoff dependence before. In total, these results lead us to quote a ππ,π -pole LHC µ,J =0 = − × − (4.17)for the S -wave rescattering corrections to the pion-pole LHC. The low-energy behavior of the on-shell γγ → ππ amplitudes is strongly constrained by the pionpolarizabilities, which therefore encode valuable information on the two-pion rescattering contributionsto HLbL. The precise relation can be expressed in terms of the expansion αM π s ˆ h , ++ ( s ) = α − β + s
12 ( α − β ) + O ( s ) (4.18)for the Born-term-subtracted on-shell amplitudes ˆ h , ++ = h , ++ − N , ++ . Here, α − β and α − β referto dipole and quadrupole polarizabilities, respectively. The soft-photon zero required as a consequenceof Low’s theorem [139] ensures that ˆ h , ++ indeed vanishes for s → .Accordingly, the representation (4.13) implies the following sum rules for the pion polarizabilities M π α ( α − β ) = (cid:20) ∆ , ++ ( s ) − N , ++ ( s ) s (cid:21) s =0 + 1 π (cid:90) ∞ M π ds (cid:48) sin δ ( s (cid:48) )∆ , ++ ( s (cid:48) ) | Ω ( s (cid:48) ) | s (cid:48) , (4.19) M π α ( α − β ) = (cid:20) ∂∂s ∆ , ++ ( s ) − N , ++ ( s ) s (cid:21) s =0 + 1 π (cid:90) ∞ M π ds (cid:48) sin δ ( s (cid:48) )∆ , ++ ( s (cid:48) ) | Ω ( s (cid:48) ) | s (cid:48) (cid:18) ˙Ω (0) + 1 s (cid:48) (cid:19) , where ˙Ω (0) denotes the derivative of the Omnès factor at s = 0 and the first term in each linedisappears for a pion-pole LHC.The numerical evaluation for ∆ , ++ = N , ++ , see Table 5, confirms the observation from Sect. 4.1that the charged-pion amplitude is better reproduced than its neutral-pion analog. In fact, the charged-pion dipole polarizability comes out in perfect agreement with ChPT [137], as well as with the recentmeasurement by COMPASS ( α − β ) π ± = 4 . . stat (1 . syst × − fm [140]. The quadrupolepolarizability is more sensitive to poorly-determined low-energy constants, but the sum-rule value lieswithin the range quoted in [137] and is also close to ( α − β ) π ± = 15 . . × − fm obtained in [55]45y combining the more stable chiral prediction for the neutral-pion quadrupole polarizability with afinite-matching-point sum rule for I = 2 .In contrast, both neutral-pion polarizabilities differ by about units each from the full result,a deficiency that signals the impact of higher contributions to the LHC, as we will demonstrate inthe following. The next such contribution is generated by the exchange of vector-meson resonances V = ρ, ω , whose impact can be roughly estimated within a narrow-width approximation. Startingfrom a vector–pion–photon coupling of the form L V πγ = eC V (cid:15) µνλσ F µν ∂ λ πV σ , (4.20)with coupling constant related to the partial width according to Γ V → πγ = αC V ( M V − M π ) M V , (4.21)we obtain [54, 141] ∆ V , ++ ( s ) = 2 C V (cid:20) − M V σ π ( s ) log x V ( s ) + 1 x V ( s ) − s (cid:21) , x V ( s ) = s + 2( M V − M π ) sσ π ( s ) . (4.22)Unfortunately, the polynomial piece ∝ s is ambiguous and would even appear with a different sign inan antisymmetric-tensor description of the vector-meson fields [54, 142]. It is for this reason that ina full Roy–Steiner approach only the imaginary parts are employed, while the low-energy parametersenter via subtraction constants. However, in order to predict the numerical values of the polarizabilitiesin terms of the lowest contributions to the LHC in γγ → ππ we do need the full amplitude in (4.22).Parameterizing the ambiguity according to s → ξ V s , we find M π α ( α − β ) V = 2 C V (cid:20) ξ V − M V M V − M π (cid:21) , M π α ( α − β ) V = C V M V (3 M V − M π )3( M V − M π ) . (4.23)Adding ρ, ω contributions using masses and partial widths from [143], the quadrupole polarizabilitiesare shifted by ( α − β ) π ± V = 0 . × − fm and ( α − β ) π V = 10 . × − fm , which explains howvector-meson contributions can restore agreement with ChPT for the neutral pion without spoilingthe charged channel. In fact, the hierarchy can be attributed almost exclusively to the large ω → π γ branching fraction Γ ω BR [ ω → π γ ] + Γ ρ BR [ ρ → π γ ]Γ ρ BR [ ρ ± → π ± γ ] ∼ , (4.24)which ensures that the same mechanism applies for the dipole polarizability as well.In any case, such corrections are not contained in our estimate (4.17), but at least at the on-shellpoint the impact is expected to be moderate due to the fact that the charged-pion intermediate statesare most important. In particular, the physics related to the low-energy constants ¯ l − ¯ l , which appearat two-loop level in the chiral expansion for the HLbL tensor [48], only contribute to the charged-pionpolarizability (a more detailed comparison to ChPT is provided in App. H). Our calculation thereforedemonstrates in a model-independent way that such next-to-leading-order corrections are moderatein size, in agreement with [50], but in contradiction to the large corrections suggested in [49]. Thisconclusively settles the role of the charged-pion dipole polarizability in the HLbL contribution to ( g − µ . 46 Conclusions
In this paper we presented an in-depth derivation of the general formalism required for the analysis oftwo-pion-intermediate-state contributions to HLbL scattering in ( g − µ . As a first step we gained adetailed understanding of the properties of the HLbL tensor, including its decomposition into scalarfunctions, projection onto helicity amplitudes, and the relation between the different sets we neededto introduce in the course of our derivation, see Table 1. Some of the more subtle issues that arose inthis derivation are related to the fact that, in order to write down dispersion relations for the HLbLtensor, we had to start with a redundant set of functions. At first sight, the relation between the latterand the physically observable helicity amplitudes seems to suffer from ambiguities. To show that thisarbitrariness is only apparent we invoked a set of sum rules, which follow from a simple assumptionon the asymptotic behavior of the HLbL tensor. These sum rules allowed us to construct a basisfor kinematics with one single on-shell photon (singly-on-shell) that satisfies unsubtracted dispersionrelations. In addition they lead to physically relevant sum rules that constrain the helicity amplitudesfor γ ∗ γ ∗ → ππ . After working out the basis change from the singly-on-shell basis to helicity amplitudes,we combined this general formalism with a partial-wave expansion to address two-pion-rescatteringcontributions.In a second step we thoroughly tested our formalism using the example of the pion box, whosefull result is known thanks to an exact relation to the scalar QED pion loop we established earlier. Inparticular, we demonstrated that the sum rules that follow from our assumptions on the asymptoticbehavior of the HLbL tensor are fulfilled. Moreover we studied whether the partial-wave expansionof the pion box converges to the full answer after resummation, and demonstrated that it does sosufficiently quickly. Given that the pion-box contribution can be expressed exactly in terms of thepion vector form factor—much as the HVP contribution of two pion intermediate states is completelydetermined by this form factor—we showed that by fitting a dispersive representation of the pion vectorform factor to a combination of space- and time-like data, the space-like form factor required for theHLbL application can be constrained to a very high precision, leading to a π -box µ = − . × − for the pion-box contribution.The main motivation for developing a partial-wave framework is to be able to calculate rescatteringcorrections, since only in a partial-wave basis for helicity amplitudes do unitarity relations become di-agonal. Accordingly, as a first application of the formalism developed here we studied the unitarizationof the pion box, a correction whose evaluation requires the use of partial-wave amplitudes. Concentrat-ing on S -wave ππ -rescattering effects, we presented a first numerical estimate, which, together withthe pion-box evaluation, combines to a π -box µ + a ππ,π -pole LHC µ,J =0 = − × − (5.1)for the leading two-pion contributions to ( g − µ . The improvement in accuracy with respect toprevious model-dependent analyses is striking. It derives: (i) from our model-independent approachbased on dispersion relations that allows us to express this contribution, in a rigorous way, in terms ofhadronic observables, and (ii) from the fact that all quantities needed in this calculation (the pion vectorform factor and the ππ S -wave phase shifts) are very well known. Remaining two-pion contributionsthat have not been addressed yet are likely to lead to larger uncertainties, but given that the errorquoted in (5.1) lies an order of magnitude below the experimental accuracy goal, we are confidentthat the final estimate for the total HLbL contribution should be sufficiently accurate to make thesemeasurements of ( g − µ a sensitive test of the Standard Model.Many of the technical advances described here are not specific to the two-pion intermediate statebut completely general and thus lay the groundwork for a full phenomenological analysis of HLbLscattering. Armed with these, we are now poised to study other contributions and apply furtherrefinements to the numerical analysis of the two-pion channel and beyond.47 cknowledgements We thank B. Kubis, A. Manohar, M. J. Ramsey-Musolf, and M. J. Savage for useful discussions.Financial support by the DFG (SFB/TR 16, “Subnuclear Structure of Matter,” SFB/TR 110, “Sym-metries and the Emergence of Structure in QCD”), the DOE (Grant No. DE-FG02-00ER41132 andDE-SC0009919), the National Science Foundation (Grant No. NSF PHY-1125915), and the SwissNational Science Foundation is gratefully acknowledged. M.P. is supported by a Marie Curie Intra-European Fellowship of the European Community’s 7th Framework Programme under contract num-ber PIEF-GA-2013-622527 and P.S. by a grant of the Swiss National Science Foundation (Project No.P300P2_167751).
A Transformed tensor decomposition for the contribution to ( g − µ For the calculation of ( g − µ , we make a linear transformation of the BTT tensor decomposition: Π µνλσ = (cid:88) i =1 T µνλσi Π i = (cid:88) i =1 ˆ T µνλσi ˆΠ i . (A.1)Only 19 of the new structures ˆ T µνλσi contribute to ( g − µ , which is the minimal number of independentcontributions in the ( g − µ kinematic limit. The symmetry under q ↔ − q reduces this to 12 termsin the master formula. A.1 Tensor structures
Here, we give the tensor structures ˆ T µνλσi explicitly in terms of the BTT structures [31]. The 19structures contributing to ( g − µ are defined in (2.13). The remaining 35 structures, which do notcontribute to ( g − µ , are defined by ˆ T µνλσi = T µνλσi , i = 12 , , , , , , , , , , , , , , , , , , , , , ˆ T µνλσ = q · q T µνλσ + T µνλσ + T µνλσ , ˆ T µνλσ = − q · q q · q T µνλσ − q · q T µνλσ − q · q T µνλσ + T µνλσ , ˆ T µνλσ = T µνλσ − T µνλσ , ˆ T µνλσ = − q · q ( T µνλσ + T µνλσ + T µνλσ ) − T µνλσ + T µνλσ − T µνλσ + T µνλσ − T µνλσ − T µνλσ , (A.2)together with the crossed structures ˆ T µνλσ = C [ ˆ T µνλσ ] , ˆ T µνλσ = C [ ˆ T µνλσ ] , ˆ T µνλσ = C [ C [ ˆ T µνλσ ]] , ˆ T µνλσ = C [ C [ ˆ T µνλσ ]] , ˆ T µνλσ = C [ ˆ T µνλσ ] , ˆ T µνλσ = C [ ˆ T µνλσ ] , ˆ T µνλσ = C [ ˆ T µνλσ ] , ˆ T µνλσ = C [ ˆ T µνλσ ] , ˆ T µνλσ = C [ ˆ T µνλσ ] , ˆ T µνλσ = C [ ˆ T µνλσ ] . (A.3)48 .2 Scalar functions In terms of the BTT functions Π i , the transformed scalar functions ˆΠ i that contribute to ( g − µ aredefined in (2.15) and (2.16). The ones that do not contribute to ( g − µ are given by: ˆΠ i = Π i , i = 12 , , , . . . , , , . . . , , , , , , , ˆΠ = 13 ( − Π + 2Π − Π ) , ˆΠ = C [ ˆΠ ] . (A.4) B New kernel functions for the master formula
Compared to [31], we choose a different basis for the Lorentz structures contributing to ( g − µ inorder to preserve crossing symmetry between all three off-shell photons. This modifies slightly thekernel functions in the master formula (2.21).The kernel functions T , . . . , T are identical to the ones in [31], while T = T [31] . For complete-ness, here we provide the full set of the new kernels, superseding Sect. E.2 in [31]: T = Q τ (cid:0) σ E − (cid:1) (cid:0) σ E + 5 (cid:1) + Q τ (cid:0) σ E − (cid:1) (cid:0) σ E + 5 (cid:1) + 4 Q Q (cid:0) σ E + σ E − (cid:1) − τ m µ Q Q Q m µ + X (cid:32) (cid:0) τ − (cid:1) Q − m µ (cid:33) ,T = Q (cid:0) σ E − (cid:1) (cid:0) Q τ (cid:0) σ E + 1 (cid:1) + 4 Q (cid:0) τ − (cid:1)(cid:1) − τ m µ Q Q Q m µ + X (cid:0) τ − (cid:1) (cid:0) m µ − Q (cid:1) Q m µ ,T = 1 Q (cid:18) − (cid:0) σ E + σ E − (cid:1) m µ − Q τ (cid:0) σ E − (cid:1) (cid:0) σ E + 7 (cid:1) Q m µ + 8 τQ Q − Q τ (cid:0) σ E − (cid:1) (cid:0) σ E + 7 (cid:1) Q m µ + Q (cid:0) − σ E (cid:1) Q m µ + Q (cid:0) − σ E (cid:1) Q m µ + 2 Q + 2 Q (cid:19) + X (cid:18) m µ − τQ Q (cid:19) ,T = 1 Q (cid:32) (cid:0) τ (cid:0) σ E − (cid:1) + σ E − (cid:1) m µ − Q τ (cid:0) σ E − (cid:1) (cid:0) σ E − (cid:1) Q m µ + 4 τQ Q − Q τ (cid:0) σ E − (cid:1) (cid:0) σ E − (cid:1) Q m µ + 2 Q (cid:0) σ E − (cid:1) Q m µ − Q + X (cid:18) − Q τ m µ − Q Q τm µ − Q m µ + 16 Q τQ + 16 (cid:19) (cid:33) ,T = 1 Q (cid:32) Q (cid:32) τ (cid:0) σ E − (cid:1) (cid:0) σ E + 3 (cid:1) + 4 (cid:0) σ E + σ E − (cid:1) m µ − Q (cid:33) − Q τ (cid:0) σ E − (cid:1) (cid:0) σ E − (cid:1) m µ + Q τ (cid:0) σ E − (cid:1) (cid:0) σ E + 5 (cid:1) Q m µ + Q (cid:32) Q τ (cid:0) σ E + 5 σ E − (cid:1) m µ − τQ (cid:33) + 2 Q (cid:0) σ E − (cid:1) Q m µ − τ + X (cid:32) Q (cid:18) Q (cid:0) τ + τ (cid:1) − Q τm µ (cid:19) + Q (cid:18) τ − Q (cid:0) τ + 1 (cid:1) m µ (cid:19) + Q (cid:18) τQ − Q τm µ (cid:19) − Q m µ (cid:33)(cid:33) , = 1 Q (cid:32) Q (cid:0) τ (cid:0)(cid:0) σ E − (cid:1) σ E − σ E + 29 (cid:1) + 2 (cid:0) − σ E + σ E + 4 (cid:1)(cid:1) m µ + Q Q τ (cid:16) τ (cid:16)(cid:0) σ E − (cid:1) − σ E (cid:17) − σ E + σ E (cid:0) σ E − (cid:1) + 37 (cid:17) m µ − τQ + Q (cid:0) τ (cid:0) − σ E + σ E (cid:0) σ E − (cid:1) + 29 (cid:1) − (cid:0) σ E + 2 σ E − (cid:1)(cid:1) m µ + Q τ (cid:0) σ E − (cid:1) (cid:0) σ E − (cid:1) Q m µ + Q τ (cid:0) σ E − (cid:1) (cid:0) σ E − (cid:1) Q m µ + 8 Q τQ + 2 Q (cid:0) − σ E (cid:1) Q m µ + 4 Q Q + X (cid:32) Q Q (cid:0) τ + 22 τ (cid:1) m µ + Q (cid:0) τ − (cid:1) m µ + Q (cid:32) Q (cid:0) τ + 18 (cid:1) m µ − (cid:0) τ + 1 (cid:1)(cid:33) + Q (cid:0) τ + 4 (cid:1) m µ + Q (cid:32) Q (cid:0) τ + 34 τ (cid:1) m µ − Q τ (cid:0) τ + 5 (cid:1)(cid:33) − Q (cid:0) τ + 1 (cid:1) − Q τQ (cid:33)(cid:33) ,T = 1 Q (cid:32) Q (cid:0) (cid:0) σ E + σ E − (cid:1) − τ (cid:0)(cid:0) σ E + 10 (cid:1) σ E + 8 σ E − (cid:1)(cid:1) m µ + Q (cid:32) Q τ (cid:0) τ (cid:0) σ E − (cid:1) (cid:0) σ E − (cid:1) − σ E + σ E (cid:0) σ E + 4 (cid:1) − (cid:1) m µ − τQ (cid:33) + Q τ (cid:0) σ E − (cid:1) (cid:0) σ E − (cid:1) m µ + Q τ (cid:0) σ E − (cid:1) (cid:0) σ E − (cid:1) Q m µ + 4 τ + X (cid:32) Q Q (cid:0) τ + 6 τ (cid:1) m µ + Q (cid:18) Q τm µ − Q (cid:0) τ + τ (cid:1)(cid:19) + Q (cid:0) τ − (cid:1) m µ + Q (cid:32) Q (cid:0) τ − (cid:1) m µ − (cid:0) τ + 1 (cid:1)(cid:33) (cid:33)(cid:33) ,T = 1 Q (cid:32) Q (cid:32) Q − (cid:0) τ + 1 (cid:1) (cid:0) σ E + σ E − (cid:1) m µ (cid:33) + Q (cid:32) τQ − Q τ (cid:0) τ + 1 (cid:1) (cid:0) σ E − (cid:1) m µ (cid:33) − Q τ (cid:0) σ E − (cid:1) Q m µ + Q (cid:0) − σ E (cid:1) Q m µ + X (cid:32) Q (cid:0) τ + 4 (cid:1) m µ + Q (cid:32) Q τ (cid:0) τ + 2 (cid:1) m µ − τQ (cid:33) + Q (cid:32) Q (cid:0) τ + 4 (cid:1) m µ − τ (cid:33)(cid:33) (cid:33) ,T = Q (cid:32) σ E − Q m µ + σ E − Q m µ − Q Q (cid:33) + X (cid:18) − Q m µ + 8 Q τQ + 8 Q τQ + 8 (cid:0) τ + 1 (cid:1)(cid:19) , = 12 Q (cid:32) − Q (cid:0) τ (cid:0) σ E − (cid:1) (cid:0) σ E + 3 (cid:1) + 2 (cid:0) σ E + σ E − (cid:1)(cid:1) m µ − Q τ (cid:0) σ E − (cid:1) (cid:0) σ E + 3 (cid:1) Q m µ − Q (cid:0) τ (cid:0) σ E − (cid:1) (cid:0) σ E + 3 (cid:1) + 2 (cid:0) σ E + σ E − (cid:1)(cid:1) m µ − Q τ (cid:0) σ E − (cid:1) (cid:0) σ E + 3 (cid:1) Q m µ + Q (cid:32) τQ − Q τ (cid:0)(cid:0) σ E + 4 (cid:1) σ E + σ E (cid:0) σ E + 4 (cid:1) − (cid:1) m µ (cid:33) + 8 Q τQ + 8 τ + X (cid:0) − Q (cid:0) τ − (cid:1) − Q Q τ (cid:0) τ − (cid:1) − Q (cid:0) τ − (cid:1)(cid:1) (cid:33) + X (cid:18) Q Q τm µ + 4 Q m µ + 4 Q m µ (cid:19) ,T = 12 m µ Q Q Q (cid:32) Q τ (cid:16) − σ E + σ E + 5 (cid:17) + 8 Q (cid:0) − σ E + 2 Q (cid:0) τ + 1 (cid:1) X + 1 (cid:1) + 4 Q Q τ (cid:0) − σ E + 2 Q (cid:0) τ + 9 (cid:1) X + 7 (cid:1) + 4 Q Q (cid:0) τ (cid:0) − σ E − σ E + 8 Q X + 4 (cid:1) − (cid:0) σ E + σ E − (cid:1) + 5 Q X (cid:1) + Q Q τ (cid:16) τ (cid:0) − σ E − σ E + 2 Q X + 2 (cid:1) − σ E − σ E − σ E + 16 Q X + 35 (cid:17) + 2 Q Q (cid:16) τ (cid:16) − σ E + σ E + 9 (cid:17) − σ E − σ E + 2 Q X + 4 (cid:17) − m µ (cid:16) − Q τ + 2 Q (cid:0) Q (cid:0) τ X + X (cid:1) − (cid:1) + Q Q τ (cid:0) Q (cid:0) τ + 3 (cid:1) X − (cid:1) + Q Q (cid:0) τ (cid:0) Q X − (cid:1) + 2 Q X − (cid:1) + 8 Q Q τ X (cid:17)(cid:33) ,T = 14 m µ Q Q Q (cid:32) Q τ (cid:16) − Q σ E + Q (cid:0) σ E − (cid:1) − m µ (cid:17) − Q Q (cid:0) τ (cid:0) σ E + 8 Xm µ − (cid:1) − σ E + σ E + 8 Xm µ + 2 (cid:1) + Q τ (cid:16) − Q (cid:0) τ − (cid:1) (cid:0) σ E − σ E (cid:1) + Q σ E + 8 m µ + 8 Q (cid:0) τ − (cid:1) X (cid:17) + 2 Q Q (cid:0) τ (cid:0) σ E + 8 Xm µ − (cid:1) + σ E − σ E + 8 Xm µ − Q X + 2 (cid:1) + Q τ (cid:0) − σ E − Q (cid:0) τ − (cid:1) X + 5 (cid:1) + 4 Q Q X (cid:33) , (B.1)where X = 1 Q Q x atan (cid:18) zx − zτ (cid:19) , x = (cid:112) − τ ,z = Q Q m µ (1 − σ E )(1 − σ E ) , σ Ei = (cid:115) m µ Q i ,Q = Q + 2 Q Q τ + Q . (B.2) C Feynman-parameter representation of the pion box
In the limit q → , the pion-box contribution to the scalar functions that appear in the master formulacan be written as a two-dimensional Feynman parameter integral: ˆΠ π -box i ( q , q , q ) = F Vπ ( q ) F Vπ ( q ) F Vπ ( q ) 116 π (cid:90) dx (cid:90) − x dy I i ( x, y ) , (C.1)51here I ( x, y ) = 8 xy (1 − x )(1 − y )∆ ∆ ,I ( x, y ) = 4(1 − x − y )(1 − x − y )∆ ∆ (cid:18) (1 − x − y ) ∆ − − x (3 − x ) − y (3 − y )∆ (cid:19) + 16 xy (1 − x )(1 − y )∆ ∆ ,I ( x, y ) = − xy (1 − x − y )(1 − x ) (1 − y )∆ ,I ( x, y ) = 16 xy (1 − x )(1 − y )∆ ∆ (cid:18) − x − y ∆ + 1 − y ∆ (cid:19) ,I ( x, y ) = 8 xy (1 − x − y )(1 − x )(1 − y )(1 − x − y )∆ ,I ( x, y ) = − xy (1 − x − y )(1 − x )(1 − y )( x − y )∆ ∆ (cid:18) + 1∆ (cid:19) , (C.2)and ∆ ijk = M π − xyq i − x (1 − x − y ) q j − y (1 − x − y ) q k , ∆ ij = M π − x (1 − x ) q i − y (1 − y ) q j . (C.3)The remaining functions entering the master formula can be obtained with the crossing relations (2.16). D Scalar functions for the two-pion dispersion relations
Here, we give the explicit solution for the scalar functions ˇΠ i , which fulfill unsubtracted single-variabledispersion relations and only depend on physical helicity amplitudes. First, we define the followinglinear combinations of BTT functions: Π A := Π + Π − Π , Π B := Π − Π + Π , Π C := Π + Π − Π + Π , Π D := Π − Π − + 2Π − , Π E := Π − Π , Π F := Π − Π − Π , Π G := Π + Π − Π − Π − Π + Π + Π − Π + Π , Π H := Π + Π + Π , Π I := 4Π + Π − − Π + Π + Π + Π + 2Π + Π − Π + Π + Π + 3Π − + 6Π + 3Π , Π J := Π + Π − Π − Π − Π + 2Π − Π − Π − Π − Π + Π − Π + Π , Π K := Π − Π + Π , (D.1)as well as Π ci := C (cid:2) Π i (cid:3) .The 19 functions that contribute to ( g − µ can be written in the form (for q = 0 and t = q ) ˇΠ i = ˆΠ g i + ( s − q ) ¯∆ i + ( s − q ) ¯¯∆ i , (2.62)52here { g i } = { , . . . , , , , , , , , , } , ¯∆ = −
12 Π A + q q λ Π B + q λ Π C − q q λ Π cC , ¯∆ = − Σ q ( q − q )2 λ ( q + q ) Π B − Σ q (2 q + q )4 λ ( q + q ) Π C + Σ q (2 q + q )4 λ ( q + q ) Π cC − q q + q ) Π D − Σ4( q + q ) Π E + Σ q ( q + 2 q )4 λ ( q + q ) Π K , ¯∆ = −C (cid:2) ¯∆ (cid:3) , ¯∆ = 12 Π cA + q ( q − q ) λ Π B − q ( q − q ) λ Π C + 2 q ( q − q ) λ Π cC −
12 Π F − q q λ Π K , ¯∆ = −
12 Π A + 12 Π cA + 2 q ( q − q ) λ Π B + (cid:18)
12 + q ( q + 2 q ) λ (cid:19) Π C − (cid:18)
12 + q ( q + 2 q ) λ (cid:19) Π cC + q (2 q + q )Σ4 λ ( q + q ) Π K , ¯∆ = −C (cid:2) ¯∆ (cid:3) , ¯∆ = − ¯∆ = − ¯∆ = − q λ Π B − q λ Π C − q λ Π cC , ¯∆ = − Σ( q − q ) λ ( q + q ) Π B − Σ( q + 2 q )2 λ ( q + q ) Π C + Σ( q + 2 q )2 λ ( q + q ) Π cC − q + q ) (Π D − Π E ) , ¯∆ = q λ Π B + 2 q λ Π C − q λ Π cC , ¯∆ = −C (cid:2) ¯∆ (cid:3) , (D.2)and ¯¯∆ = q q + q ) Π , ¯¯∆ = − q + q ) Π . (D.3)All other ¯∆ i and ¯¯∆ i are zero. We use the abbreviations q ijk := q i + q j − q k , Σ = q + q + q , and λ := λ ( q , q , q ) for the Källén function.We define five additional scalar functions ˇΠ i that appear in sum rules: ˇΠ := q Π G − q Π H + q ( s − q − q )2 Π + q q , ˇΠ := C (cid:2) ˇΠ (cid:3) , ˇΠ := λ Π − q q Π B − q Π C + 2 q q Π cC + λ Π cF , ˇΠ := C (cid:2) ˇΠ (cid:3) , ˇΠ := q Π I + q Π J + q Π cI − q − q )( s − q )Π . (D.4)The singly-on-shell basis consists of 27 elements. The three functions ˇΠ , ˇΠ , and ˇΠ are not givenexplicitly as they have no significance in the connection with ( g − µ . E Basis change and sum rules
E.1 Unphysical polarizations
In the following, we explain why unphysical polarizations are not trivially absent in any representa-tion. In short, although unphysical polarizations cannot contribute to any observable, the absence of53uch unphysical contributions is manifest only if the basis is well chosen. Otherwise, their apparentcontribution vanishes only due to the presence of sum rules for the scalar functions.Suppose we have a decomposition of the HLbL tensor into a “physical” and an “unphysical” piece, Π µνλσ = Π µνλσ phys + Π µνλσ unph = (cid:88) i T µνλσi, phys Π phys i + (cid:88) i T µνλσi, unph Π unph i , (E.1)where the scalar functions Π phys i are linear combinations of helicity amplitudes with only transversepolarizations of the external photon. The scalar functions Π unph i contain also contributions fromthe longitudinal polarization. Because these scalar functions cannot contribute to an observable, theunphysical tensor structures have to fulfill T µνλσi, unph ∝ q σ , q . (E.2)Such structures do not contribute to ( g − µ , because the derivative with respect to q ρ either vanishesfor q → or is symmetric in ρ ↔ σ .Next, we apply the following transformation, which mixes the physical and unphysical part: T µνλσa, phys Π phys a + T µνλσb, unph Π unph b = T µνλσa, phys (cid:16) Π phys a + α Π unph b (cid:17) + (cid:16) T µνλσb, unph − αT µνλσa, phys (cid:17) Π unph b . (E.3)Because not all tensor structures have the same mass dimension, the coefficient α can be dimensionful,e.g. α = q · q if the mass dimension of T µνλσb, unph is larger by two units than the one of T µνλσa, phys , allwhile avoiding kinematic singularities. The new structure (cid:16) T µνλσb, unph − αT µνλσa, phys (cid:17) still cannot contributeto ( g − µ if α ∝ q . However, we have introduced a new combination of unphysical and physicalhelicity amplitudes into the scalar coefficient functions of T µνλσa, phys . If we make such a transformation inthe discontinuity appearing in an s -channel dispersion integral, the factor α = q · q becomes in the ( g − µ limit q · q → −
12 ( s (cid:48) − q ) , (E.4)where we have replaced the Mandelstam variable s by the integration variable of the dispersion in-tegral s (cid:48) . This factor cancels with the Cauchy kernel / ( s (cid:48) − q ) , producing an apparent polynomialcontribution that depends on both physical and unphysical helicity amplitudes. As shown in Sect. 2.3this polynomial contribution actually vanishes due to sum rules, but in practice it can be tediousto identify the combination of physical and unphysical helicity amplitudes that corresponds to thisvanishing polynomial, and, worse, in a partial-wave expansion these sum rules are only fulfilled afterresumming all partial waves. Since the above example implies that setting by hand only the unphysicalpolarizations to zero leads to a wrong result, a practical implementation requires a basis where thiscontribution is manifestly absent from the beginning. The construction of this basis is performed inSect. 2.4.1. E.2 Comparison to forward-scattering sum rules
In [63], sum rules have been derived for the case of forward HLbL scattering. In the following, wecompare them to our fixed- t sum rules derived in Sect. 2.4.3. To this end, we consider the case ofgeneral forward kinematics, i.e. q = − q , q = q , (E.5)which implies for the Lorentz invariants t = 0 , u = 2 q + 2 q − s, q = q , q = q . (E.6)54he common limit of forward and singly-on-shell fixed- t kinematics is obtained for q → .It is convenient to define the variable [144] ν := q · q = 14 ( s − u ) . (E.7)In the case of forward scattering, only eight independent helicity amplitudes exist [144]. Consis-tently, starting with the BTT decomposition (2.7) and taking the limit of forward kinematics, only eightindependent Lorentz structures survive. Interestingly, the two ambiguities in four space-time dimen-sions [58] disappear, but even for forward kinematics one redundancy of Tarrach’s type remains [52].Therefore, the forward HLbL tensor can be written as Π µνλσ FW = (cid:88) i =1 T µνλσi, FW Π FW i , (E.8)where the tensor structures are given by T µνλσ , FW = 12 (cid:16) T µνλσ + T µνλσ (cid:17) , T µνλσ , FW = T µνλσ ,T µνλσ , FW = 12 (cid:16) T µνλσ + T µνλσ (cid:17) , T µνλσ , FW = 12 (cid:16) T µνλσ + T µνλσ (cid:17) ,T µνλσ , FW = 12 (cid:16) T µνλσ + T µνλσ (cid:17) , T µνλσ , FW = 14 (cid:16) T µνλσ + T µνλσ + T µνλσ + T µνλσ (cid:17) ,T µνλσ , FW = 12 (cid:16) T µνλσ − T µνλσ (cid:17) , T µνλσ , FW = 12 (cid:16) T µνλσ − T µνλσ − T µνλσ + T µνλσ (cid:17) ,T µνλσ , FW = 14 (cid:16) T µνλσ − T µνλσ − T µνλσ + T µνλσ (cid:17) , (E.9)with BTT structures on the right-hand side of the equations evaluated in the limit (E.5). The redun-dancy reads ν T µνλσ , FW + q q T µνλσ , FW = 0 . (E.10)In terms of the BTT functions, the forward scalar functions are given by Π FW = Π + Π − ν (cid:0) Π − Π − Π + Π (cid:1) , Π FW = Π − ν (cid:0) Π − Π − Π + Π (cid:1) , Π FW = Π + Π + q (cid:0) Π + Π + Π + Π (cid:1) + q (cid:0) Π + Π + Π + Π (cid:1) − q q (cid:0) Π + Π + Π + Π (cid:1) + ν (cid:0) Π − Π − Π + Π − Π + Π + Π − Π (cid:1) , Π FW = Π + Π − Π − Π , Π FW = Π + Π − Π − Π , Π FW = − (cid:0) Π + Π + Π + Π (cid:1) − ν (cid:0) Π + Π − Π − Π (cid:1) − Π − Π − Π − Π − Π − Π − Π − Π + Π + Π + Π + Π , Π FW = Π − Π + Π − Π + ν (cid:0) Π + Π + Π + Π (cid:1) + q (cid:0) Π − Π − Π + Π − (cid:1) + q (cid:0) Π − Π − Π + Π − (cid:1) + q q (cid:0) − Π − Π + Π + Π (cid:1) , Π FW = Π − Π − ν (cid:0) Π + Π + Π + Π − (cid:0) Π + Π + Π + Π + Π + Π + Π + Π (cid:1)(cid:1) − ν (cid:0) Π + Π − Π − Π (cid:1) , Π FW = Π − Π − Π + Π + 2 (cid:0) Π − Π − Π + Π (cid:1) + ν (cid:0) Π + Π + Π + Π (cid:1) . (E.11)55he functions Π FW i are even in ν for i = 1 , . . . , and odd for i = 7 , , , which corresponds tothe crossing symmetries C or C . We further have C (cid:2) C (cid:2) Π FW (cid:3)(cid:3) = Π FW , while the other sevenfunctions are invariant under this transformation. According to our assumption for the asymptoticbehavior (2.39), all the functions Π FW i fulfill an unsubtracted dispersion relation. Note, however, thatdue to the redundancy (E.10) Π FW , enter in observables only in the linear combination q q Π FW − ν Π FW , (E.12)which requires a once-subtracted dispersion relation. The subtraction constant vanishes in the quasi-real limit of one of the photons.With our assumption for the asymptotic behavior, we find three physical sum rules: (cid:90) dν Im Π FW i ( ν ) = 0 , i = 4 , , . (E.13)Due to the symmetry in ν , the first two are trivially fulfilled: the integrals over the left- and right-handcuts cancel. This leaves a single sum rule involving Π FW .Next, we consider the basis change to helicity amplitudes. The eight forward-scattering amplitudesare given by [63, 144] H FW := H ++ , ++ + H + − , + − , H FW := H ++ , −− , H FW := H , ,H FW := H +0 , +0 , H FW := H , , H FW := H ++ , + H +0 , − ,H FW := H ++ , ++ − H + − , + − , H FW := H ++ , − H +0 , − , (E.14)where the first six are even, the last two are odd in ν . With our conventions for the polarizationvectors, they are related to the scalar functions (E.11) by H FW = − ( ν − q q )Π FW − q q Π FW − ν Π FW − ν q Π FW − ν q Π FW − νq q Π FW ,H FW = ( ν − q q )Π FW − ν Π FW − νq q Π FW ,H FW = − Π FW − Π FW − q Π FW − q Π FW − ν Π FW ,H FW = − q Π FW − ( q ) Π FW − ν Π FW ,H FW = − q Π FW − ν Π FW − ( q ) Π FW ,H FW = q q Π FW − ν Π FW + ν Π FW ,H FW = ν ( q q Π FW − ν Π FW ) + q q Π FW ,H FW = − ν Π FW −
12 ( ν + q q )Π FW . (E.15)In terms of the helicity amplitudes the sum rule reads (cid:90) ∞ ν dν ν − q q ) (cid:18) ν Im (cid:104) H FW ( ν ) + H FW ( ν ) + 2 q q H FW ( ν ) − q H FW ( ν ) − q H FW ( ν ) (cid:105) − ν + q q ) Im H FW ( ν ) (cid:19) = 0 , (E.16)where ν denotes the threshold in ν . Taking the quasi-real limit q → of this equation and accountingfor the different conventions for the polarization vectors, we reproduce the sum rule (27b) of [63]. Inaddition, two more sum rules (superconvergence relations) were derived in [63]. They originate indifferent assumptions about the asymptotic behavior based on the Regge model of [144]. In Table 6,we compare the assumptions on the asymptotic behavior of the helicity amplitudes: concerning thenumber of subtractions needed in a dispersion relation for the helicity amplitudes, this leads in most56his work Ref. [63] H FW (cid:16) ν (cid:16) ν α P (0) H FW (cid:16) ν (cid:16) ν α π (0) H FW (cid:16) ν − (cid:16) ν α P (0) H FW (cid:16) ν (cid:16) ν α P (0) H FW (cid:16) ν (cid:16) ν α P (0) H FW (cid:16) ν (cid:16) ν α π (0) − H FW (cid:16) ν (cid:16) ν α π (0) H FW (cid:16) ν (cid:16) ν α π (0) − Table 6:
Comparison of the assumptions about the asymptotic behavior of the helicity amplitudes. In [63], α P (0) ≈ . and α π (0) ≈ − . was assumed. cases to identical results. For H FW , our assumption is more restrictive. In fact, a similar behaviorwas used in [63] to derive an additional sum rule for a low-energy constant in the effective photonLagrangian, stressing that this sum rule cannot be justified based on the Regge model of [144]. In ourapproach this sum rule emerges naturally by demanding a uniform asymptotic behavior of the HLbLtensor, which in turn determines the asymptotics of the BTT functions and thereby of the helicityamplitudes. For H FW , , the assumption in [63] is more restrictive and leads to two additional sum rules,Eqs. (27a) and (27c) in [63].We note that the constraints from gauge invariance that were determined in [63] based on aneffective photon Lagrangian are all implemented in the BTT decomposition of the HLbL tensor and canbe read off directly from the relations between the helicity amplitudes and the BTT scalar functions.Finally, with the above description of forward scattering in terms of BTT functions, we can easilyestablish the link to our sum rules derived for singly-on-shell fixed- t kinematics. By setting q = q and taking the limit q → in both situations, we reach the common kinematic configuration, i.e. thecase of singly-on-shell forward scattering. We can then easily find the embedding of the forward sumrule into the sum rules for the ˇΠ i functions: lim q → Π FW = − q → ,q = q (cid:0) ˇΠ + ˇΠ − ˇΠ − ˇΠ + 2 ˇΠ − (cid:1) , (E.17)where the right-hand side is a combination of functions fulfilling the sum rules (2.71). We also notethat in the S -wave approximation, the sum rule (27b) of [63] reduces to the forward limit of (2.91). F Basis change to helicity amplitudes
F.1 Calculation of tensor phase-space integrals
If we consider only S -waves in γ ∗ γ ∗ → ππ , the phase-space integral in the ππ unitarity relationfor HLbL is trivial and the unitarity relation factorizes. We have calculated the D -wave unitarityrelation in [28] for an external on-shell photon and in [30] for the fully off-shell case by using a tensordecomposition. In this approach, the unitarity relation requires the calculation of tensor integralswith additional factors of the γ ∗ γ ∗ → ππ scattering angles, which are replaced by scalar products ofexternal and internal (loop) momenta. Then the unitarity relation can be written as contractions of Note that even (odd) subtractions vanish for a function that is odd (even) in ν . This implies that for H FW , , , , thesubtraction schemes are identical although the exact assumptions for the asymptotic behavior slightly differ. D -waves extremely inefficient. In order to avoid the excessive amount of contrac-tions in the calculation of the phase-space integral, here we present an alternative way to calculate thetensor integrals directly by taking derivatives of scalar integrals. This allows us to calculate even the G -wave unitarity relation, as required for the present application in Sect. 2.5. In the case of D -waves,we have checked that both methods give the same result.We first consider the scalar integrals with additional Legendre polynomials of the scattering angles: I nm := (cid:90) d p (2 π ) p d p (2 π ) p (2 π ) δ (4) (cid:0) Q − p − p (cid:1) P n ( z (cid:48) ) P m ( z (cid:48)(cid:48) ) , (F.1)where Q := q + q = q − q and z (cid:48) , z (cid:48)(cid:48) denote the scattering angles z (cid:48) = q − q − q − q ) · p σ π ( s ) λ / ( s ) , z (cid:48)(cid:48) = q − q + 2( q + q ) · p σ π ( s ) λ / ( s ) (F.2)with λ ( s ) = λ ( s, q , q ) , λ ( s ) = λ ( s, q , q ) . The HLbL scattering angle is defined as z = ( q − q )( q − q ) + s ( t − u ) λ / ( s ) λ / ( s ) . (F.3)The angles fulfill cos θ (cid:48)(cid:48) = cos θ (cid:48) cos θ + sin θ (cid:48) sin θ cos φ (cid:48) , (F.4)where z = cos θ , z (cid:48) = cos θ (cid:48) , z (cid:48)(cid:48) = cos θ (cid:48)(cid:48) , and φ (cid:48) is the azimuthal angle of (cid:126)p in the centre-of-massframe. The phase-space integral can be understood as an integral over the variables θ (cid:48) and φ (cid:48) .As a first step, direct calculation leads to I nm = 116 π (cid:90) ∞ dp p M π + p δ ( Q − (cid:112) M π + p ) (cid:90) d Ω P n ( z (cid:48) ) P m ( z (cid:48)(cid:48) )= 18 π σ π ( s ) δ nm P n ( z )2 n + 1 , (F.5)where we have used the addition theorem for the Legendre polynomials. Next, we define P := q − q and R := q + q and write the angles as z = Q ( P · R ) − ( P · Q )( R · Q )(( P · Q ) − P Q ) / (( R · Q ) − R Q ) / ,z (cid:48) = P · Q − P · p σ π ( Q )(( P · Q ) − P Q ) / , z (cid:48)(cid:48) = 2 R · p − R · Qσ π ( Q )(( R · Q ) − R Q ) / . (F.6)58aking the derivatives of the angles with respect to P µ and R µ gives ∂z∂P µ = Q λ / ( Q ) λ / ( Q ) (cid:18) Q µ (cid:0) P ( Q · R ) − ( P · Q )( P · R ) (cid:1) + P µ (cid:0) Q ( P · R ) − ( Q · P )( Q · R ) (cid:1) + R µ (cid:0) ( P · Q ) − P Q (cid:1)(cid:19) ,∂z∂R µ = Q λ / ( Q ) λ / ( Q ) (cid:18) Q µ (cid:0) R ( P · Q ) − ( P · R )( Q · R ) (cid:1) + P µ (cid:0) ( R · Q ) − R Q (cid:1) + R µ (cid:0) Q ( P · R ) − ( Q · P )( Q · R ) (cid:1)(cid:19) ,∂z (cid:48) ∂P µ = Q µ − p µ σ π ( Q ) λ / ( Q ) + z (cid:48) Q P µ − ( P · Q ) Q µ λ ( Q ) ,∂z (cid:48)(cid:48) ∂R µ = 2 p µ − Q µ σ π ( Q ) λ / ( Q ) + z (cid:48)(cid:48) Q R µ − ( R · Q ) Q µ λ ( Q ) ,∂z (cid:48) ∂R µ = ∂z (cid:48)(cid:48) ∂P µ = 0 . (F.7)Observing that a loop momentum with an open Lorentz index, p µ , can be written in terms of thederivative of a γ ∗ γ ∗ → ππ angle with respect to P µ or R µ and functions of angles and external momentaonly, we can write all tensor integrals in terms of derivatives of scalar integrals, since the phase-spaceintegral does not depend on P or R . With this method no additional contractions of Lorentz indices arenecessary and the complexity of the calculation is reduced significantly. This enables the calculationof the G -wave unitarity relation.Explicitly, we define tensor integrals involving factors of the scattering angles according to: I µ ...µ i i,nm := (cid:90) d p (2 π ) p d p (2 π ) p (2 π ) δ (4) (cid:0) Q − p − p (cid:1) p µ · · · p µ i z (cid:48) n z (cid:48)(cid:48) m . (F.8)For the G -wave unitarity relation, we need to know the integrals I µ ...µ i i,nm with i + n + m ≤ and i ≤ .The scalar integrals with i = 0 can be calculated easily using (F.5): I , = I ,I , = I , = 13 I , I , = z I ,I , = I , = 15 I , I , = I , = z I , I , = 1 + 2 z I ,I , = I , = 17 I , I , = I , = z I , I , = I , = 1 + 4 z I , I , = z (3 + 2 z )35 I ,I , = I , = 19 I , I , = I , = z I , I , = I , = 1 + 6 z I , I , = I , = z (3 + 4 z )63 I ,I , = 3 + 24 z + 8 z I , (F.9)where I = 18 π σ π ( s ) , (F.10)while all I ,nm with n + m odd vanish. 59ext, we calculate the remaining integrals with i = 1 , . . . , successively using the derivative trick.The integrals with n = m = 0 are pure tensor integrals and can be cross-checked with the results fromthe tensor decomposition method.In order to compute the integrals I µ ,nm , we consider the following derivative: ∂∂P µ (cid:16) z (cid:48) n +1 z (cid:48)(cid:48) m (cid:17) = ( n + 1) z (cid:48) n z (cid:48)(cid:48) m ∂z (cid:48) ∂P µ = ( n + 1) z (cid:48) n z (cid:48)(cid:48) m (cid:32) Q µ − p µ σ π ( Q ) λ / ( Q ) + z (cid:48) Q P µ − ( P · Q ) Q µ λ ( Q ) (cid:33) . (F.11)Since the phase-space integral does not depend on P or R , we can commute it with the derivative andfind I µ ,nm = 12 Q µ I ,nm + σ π ( s ) λ / ( s )2 (cid:20) sP µ − ( q − q ) Q µ λ ( s ) I ,n +1 m − n + 1 ∂∂P µ I ,n +1 m (cid:21) . (F.12)Similarly, the tensor integrals I µν ,nm can be calculated by considering the double derivative ∂ ∂P µ ∂P ν (cid:16) z (cid:48) n +2 z (cid:48)(cid:48) m (cid:17) . (F.13)Finally, by taking multiple derivatives the tensor integrals I µνλ ,nm and I µνλσ ,nm can be calculated. F.2 Direct matrix inversion
The expressions for the helicity amplitudes in terms of the scalar coefficient functions in the tensordecomposition are easily obtained by contracting the HLbL tensor with the polarization vectors. Ex-pressing the scalar functions in terms of the helicity amplitudes requires the inversion of these relations.If we consider the singly-on-shell case, this amounts to the inversion of a × matrix. The directanalytic inversion of a general matrix of this size is not possible, but in this case it can be reconstructedalong the following lines.Let us define the basis change from the singly-on-shell helicity amplitudes to scalar functions as ¯ H j (cid:12)(cid:12)(cid:12) λ (cid:54) =0 = (cid:88) i =1 η ji ˇΠ i , (F.14)where η is a × matrix. Its inverse is effectively the matrix ˇ c in (2.84) (restricted to λ (cid:54) = 0 ) thatwe need to determine in order to obtain the imaginary parts of the scalar functions through unitarity.First, we note that the basis of helicity amplitudes suffers from the presence of kinematic singulari-ties, which makes the expressions for η more involved. These singularities can be removed by applyingthe general recipe of [62] for the construction of amplitudes free of kinematic singularities: first, thesingularities at the boundary of the physical region can be removed by ˆ H j := (cid:16) z (cid:17) − | m + m | (cid:16) − z (cid:17) − | m − m | ¯ H j , (F.15)where m = λ − λ , m = λ − λ , and z is the cosine of the scattering angle. In our case of fixed- t singly-on-shell kinematics, we have z = s − q + q λ / ( s ) . Next, the parity-conserving amplitudes ˆ H j ± ˆ H ¯ j (F.16)are formed (see Sect. 2.5.1 for the notation). Finally, the remaining singularities can be removedby multiplying with the appropriate powers of √ s , λ / ( s ) , and λ / ( s ) , see [62]. We note that the60artin–Spearman amplitudes constructed in this way are free of kinematic singularities, but have anasymptotic behavior that is much worse than the one of the BTT scalar functions.Since all square-root singularities have been removed, the basis change from the scalar functions ˇΠ i to the Martin–Spearman amplitudes is now meromorphic in s , q , q , and q . We determine allmatrix entries with partly numerical methods as follows.Numerically, the inversion of the × matrix is straightforward. The denominators of themeromorphic matrix entries can be guessed from the pole structure of the numerical inversion: theyare products of simple polynomials such as λ , λ ( s ) , ( q − q + q ) etc. We calculate numericallythe matrix inversion as a function of each of the Lorentz invariants in turn, keeping the other threeinvariants fixed. A plot of the matrix entries as a function of the varying variable reveals the polesand therefore the exact form of the denominators. This simple but tedious task has to be performedfor all × entries. The remaining numerators are then polynomials of the form (cid:88) i + j + k + l = n a ijkl s i ( q ) j ( q ) k ( q ) l , (F.17)where the mass dimension of the numerator is n and known beforehand. In most cases, n is a smallnumber, although for very few entries we encounter a maximal value of n = 9 , which results in apolynomial with terms. We perform the numerical inversion on a grid consisting of points inthe four-dimensional space of s , q , q , and q and determine the integer coefficients a ijkl for each of thenumerators of the × matrix entries by a fit. In contrast to the determination of the denominatorsby hand, this fit of the numerators can be easily automatized.Combining the results with the (simple) basis change from helicity to Martin–Spearman amplitudesthen leads to the full analytic expression for the inverted basis change ˇ c . In particular, it is straightfor-ward to check analytically that the matrix ˇ c determined partly with numerical methods is indeed theexact inverse of η . The result is provided as supplementary material in the form of a Mathematica notebook.
G Partial-wave expansion of the γ ∗ γ ∗ → ππ pion-pole contribution In order to test the partial-wave formalism, we expand the pion-pole contribution to γ ∗ γ ∗ → ππ intopartial waves. The scalar functions are given by [31] (with isospin conventions from [28]): A π = F Vπ ( q ) F Vπ ( q ) (cid:18) t − M π + 1 u − M π (cid:19) ,A π = F Vπ ( q ) F Vπ ( q ) 2 s − q − q (cid:18) t − M π + 1 u − M π (cid:19) ,A π = A π = A π = 0 . (G.1)The helicity amplitudes become: ¯ H π ++ = ¯ H π −− = F Vπ ( q ) F Vπ ( q ) (cid:18) t − M π + 1 u − M π (cid:19) (cid:32) −
12 ( s − q − q )+ 14 ( s − M π ) (cid:18) ( s − q − q ) + (cid:18) ( q − q ) s − ( q + q ) (cid:19) z (cid:19) s − q − q (cid:33) , ¯ H π + − = ¯ H π − + = − F Vπ ( q ) F Vπ ( q ) 12 ( s − M π )(1 − z ) (cid:18) t − M π + 1 u − M π (cid:19) , ¯ H π +0 = − ¯ H π − = − F Vπ ( q ) F Vπ ( q ) 12 (cid:114) s ( s − M π ) z (cid:112) − z s + q − q s − q − q (cid:18) t − M π + 1 u − M π (cid:19) , H π = − ¯ H π − = − F Vπ ( q ) F Vπ ( q ) 12 (cid:114) s ( s − M π ) z (cid:112) − z s − q + q s − q − q (cid:18) t − M π + 1 u − M π (cid:19) , ¯ H π = − F Vπ ( q ) F Vπ ( q ) (cid:18) − s − M π ) z s − q − q (cid:19) (cid:18) t − M π + 1 u − M π (cid:19) . (G.2)We calculate the partial-wave expansion thereof: N J,λ λ ( s ) := 12 (cid:90) − dz d Jm ( z ) ¯ H πλ λ ( s, t ( s, z ) , u ( s, z )) , (G.3)where m = | λ − λ | . With the relation t − M π = − σ π ( s ) λ / ( s ) 1 x − z , u − M π = − σ π ( s ) λ / ( s ) 1 x + z , (G.4)where x = s − q − q σ π ( s ) λ / ( s ) , (G.5)we can calculate the pion-pole contribution to the helicity partial waves in terms of the Legendrefunctions of the second kind, defined by Q J ( x ) = 12 (cid:90) − P J ( z ) x − z dz. (G.6)They satisfy the relations [145] Q J ( x ) P J − ( x ) = P J ( x ) Q J − ( x ) − J − J ( J − x, ( J + 1) Q J +1 ( x ) = (2 J + 1) xQ J ( x ) − J Q J − ( x ) , (G.7)which, together with the recursion relation for the Legendre polynomials ( J + 1) P J +1 ( x ) = (2 J + 1) xP J ( x ) − J P J − ( x ) (G.8)leads to the following expressions for the pion-pole helicity partial waves: N J, ++ ( s ) = F Vπ ( q ) F Vπ ( q ) (cid:40) σ π ( s ) λ / ( s ) (cid:18) sq q λ ( s ) + M π (cid:19) Q J ( x ) + 2 δ J ( q − q ) − s ( q + q ) λ ( s ) (cid:41) ,N J, + − ( s ) = F Vπ ( q ) F Vπ ( q ) 2 sσ π ( s ) λ / ( s ) J (cid:115) ( J − J + 2)! (cid:110) xQ J − ( x ) − (cid:0) ( J + 1) − x ( J − (cid:1) Q J ( x ) (cid:111) ,N J, +0 ( s ) = F Vπ ( q ) F Vπ ( q ) 2 √ sσ π ( s ) λ / ( s ) s + q − q s − q − q (cid:114) JJ + 1 x (cid:110) xQ J ( x ) − Q J − ( x ) (cid:111) ,N J, ( s ) = F Vπ ( q ) F Vπ ( q ) 2 √ sσ π ( s ) λ / ( s ) s − q + q s − q − q (cid:114) JJ + 1 x (cid:110) xQ J ( x ) − Q J − ( x ) (cid:111) ,N J, ( s ) = F Vπ ( q ) F Vπ ( q ) 4 λ ( s ) (cid:40) ( q − q ) − s σ π ( s ) λ / ( s ) Q J ( x ) + 2 s δ J (cid:41) . (G.9) We use a different convention than in [31] and do not (anti-)symmetrize the partial waves with respect to q ↔ q . Pion polarizability and γγ → ππ in ChPT The one-loop amplitude for γγ → ππ takes the form [146, 147] h c , ++ ( s ) (cid:12)(cid:12) ChPT = N , ++ ( s ) + ¯ l − ¯ l π F π s − s π F π (cid:0) M π C ( s ) (cid:1) ,h n , ++ ( s ) (cid:12)(cid:12) ChPT = − s − M π π F π (cid:0) M π C ( s ) (cid:1) , (H.1)where we have suppressed the arguments for the virtualities, ¯ l − ¯ l refers to a combination of SU (2) low-energy constants [148], and the loop function is given by C ( s ) = (cid:90) dxsx log (cid:20) − x (1 − x ) sM π (cid:21) . (H.2)Unitarity is only fulfilled perturbatively, so that at the one-loop levelIm h c , ++ ( s ) (cid:12)(cid:12) ChPT = σ π ( s )3 N , ++ ( s ) (cid:0) t ( s ) + t ( s ) (cid:1) = M π πF π log 1 + σ π ( s )1 − σ π ( s ) , Im h n , ++ ( s ) (cid:12)(cid:12) ChPT = 2 σ π ( s )3 N , ++ ( s ) (cid:0) t ( s ) − t ( s ) (cid:1) = s − M π πF π log 1 + σ π ( s )1 − σ π ( s ) , (H.3)with tree-level ππ partial waves t IJ ( s ) . Due to the pathological high-energy behavior of these imaginaryparts, the chiral amplitudes do not fulfill an unsubtracted dispersion relation, but only a subtractedvariant of the form h c , ++ ( s ) (cid:12)(cid:12) ChPT = N , ++ ( s ) + ¯ l − ¯ l π F π s + s π (cid:90) ∞ M π ds (cid:48) Im h c , ++ ( s (cid:48) ) (cid:12)(cid:12) ChPT s (cid:48) ( s (cid:48) − s ) ,h n , ++ ( s ) (cid:12)(cid:12) ChPT = − s π F π + s M π π (cid:90) ∞ M π ds (cid:48) Im h n , ++ ( s (cid:48) ) (cid:12)(cid:12) ChPT s (cid:48) ( s (cid:48) − s ) , (H.4)to be contrasted with h , ++ ( s ) = ∆ , ++ ( s ) + sπ (cid:90) ∞ M π ds (cid:48) Im h , ++ ( s (cid:48) ) s (cid:48) ( s (cid:48) − s ) (H.5)for the full amplitudes provided that the imaginary parts fall off sufficiently fast. If the MO inho-mogeneity is approximated by the Born term that is indeed the case, which, by comparison to thechiral amplitudes, allows one to predict the derivatives at s = 0 and thereby the pion polarizabilitieswithin this approximation. At the one-loop level this implies a sum rule for ¯ l − ¯ l , whose numericalevaluation ¯ l − ¯ l = 2 . . . . . for the same range of cutoffs as in Sect. 4.3 indeed comes out very closeto the phenomenological value ¯ l − ¯ l = 3 . . [137, 149, 150]. As discussed in Sect. 4.3, only thecharged-pion polarizability is reproduced in this way, indicating that higher contributions to the LHCare required in the case of the neutral pion.In ChPT the value of ¯ l − ¯ l can be empirically understood in terms of resonance saturation,explicitly one has [142, 151, 152] ¯ l − ¯ l (cid:12)(cid:12) sat = 48 π F A M A ∼ π F π M ρ = 3 . , (H.6)where F A and M A refer to decay constant and mass of axial resonances to be related to pion decayconstant and vector masses by short-distance constraints, see [151]. The fact that resonance saturationindeed reproduces the empirical value of ¯ l − ¯ l rather accurately has motivated the construction ofmodels based on explicit a resonances to incorporate the corresponding effects related to the charged-pion polarizability into HLbL scattering [49, 50]. 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