The electromagnetic form factors of the transition from the spin-3/2 Sigma to the Lambda hyperon
Olov Junker, Stefan Leupold, Elisabetta Perotti, Timea Vitos
TThe electromagnetic form factors of the transition from the spin-3/2 Sigma to theLambda hyperon
Olov Junker, Stefan Leupold, Elisabetta Perotti, and Timea Vitos
Institutionen f¨or fysik och astronomi, Uppsala Universitet, Box 516, S-75120 Uppsala, Sweden (Dated: October 17, 2019)The three electromagnetic form factors for the transition from a 3 / + Σ ∗ hyperon to the ground-state Λ hyperon are studied. At low energies, combinations of the transition form factors can bededuced from Dalitz decays of the Σ ∗ hyperon to Λ plus an electron-positron pair. It is pointedout how more information can be obtained with the help of the self-analyzing weak decay of the Λ.In particular it is shown that these transition form factors are complex quantities already in thiskinematical region. Such measurements are feasible at hyperon factories as for instance the Facilityfor Antiproton and Ion Research (FAIR). At higher energies, the transition form factors can bemeasured in electron-positron collisions. The pertinent relations between the transition form factorsand the decay distributions and differential cross sections are presented. Using dispersion theory, thelow-energy electromagnetic form factors for the Σ ∗ -to-Λ transition are related to the pion vector formfactor. The additionally required input, i.e. the two-pion–Σ ∗ –Λ amplitudes are determined fromrelativistic next-to-leading-order (NLO) baryon chiral perturbation theory including the baryonsfrom the octet and the decuplet. A poorly known NLO parameter is fixed to the experimental valueof the Σ ∗ → Λ γ decay width. Pion rescattering is taken into account by dispersion theory solvinga Muskhelishvili-Omn`es equation. Subtracted and unsubtracted dispersion relations are discussed.However, in view of the fact that the transition form factors are complex quantities, the currentdata situation does not allow for a full determination of the subtraction constants. To reduce thenumber of free parameters, unsubtracted dispersion relations are used to make predictions for thetransition form factors in the low-energy space- and timelike regions. I. INTRODUCTION AND SUMMARY
Electromagnetic form factors have become an impor-tant tool to study the structure of strongly interactingobjects, see e.g. [1–12] and references therein. Dependingon the invariant mass of the virtual photon, one achievesdifferent resolutions and different degrees of freedom be-come relevant. At very large energies, one “sees” theminimal quark content of the probed object [6, 13, 14].Asymptotic freedom causes a suppression of the influenceof any non-minimal quark or gluon content of the probedstate. At low energies, the dynamics of pions has an im-portant influence on the shape of form factors. Dynam-ical chiral symmetry breaking causes the appearance ofGoldstone bosons [15], the pions. Because they are muchlighter than any other hadron, pions can be excited withenergies so low that all other degrees of freedom are stillfrozen. Both aspects, dominance of minimal quark con-tent at high energies and universal pion dynamics at lowenergies, are model-independent consequences of Quan-tum Chromodynamics (QCD). A complete description ofa form factor must include both of these aspects.The purpose of [11] and of the present and future workis to provide the low-energy input for such a complete de-scription of form factors in the hyperon sector. Here weextend previous work of the Uppsala group [11, 16] wheredispersion theory is used to relate in a model-independentway isovector form factors of baryons to pion-baryonscattering amplitudes. In a second step, these scatteringamplitudes are approximated by relativistic chiral per-turbation theory ( χ PT) including the baryon octet anddecuplet as active degrees of freedom. Conceptually this is close in spirit to [12, 17–19]. On a more technicallevel, the rescattering of pions is treated differently in[12] and in [11]. In [12] an N/D method is used; in [11] aMuskhelishvili-Omn`es (MO) equation is solved. As hasbeen demonstrated in [16] for the nucleon case, solving anMO equation with input from χ PT up to next-to-leadingorder leads to better results when compared to a fullydispersive calculation [20, 21]. For the case of hyperons,the use of dispersively reconstructed pion-baryon ampli-tudes is not an option because there are no direct data onpion-hyperon scattering. Therefore, we rely also in thepresent work on input from χ PT [7, 22–25] and solve anMO equation. A combined use of dispersion theory and χ PT has been pioneered in [26]; see also [27] for a briefreview.A general motivation for studying hyperon form fac-tors has been provided in [11] in great detail. With thepresent work, we extend the approach of [11] to electro-magnetic form factors of hyperons with spin 3/2. Ourframework is suited for the determination of isovectorform factors. Therefore we focus in the present work onthe only electromagnetic form factors of hyperons thatare purely isovector (and involve a spin 3/2 state). Theseare the form factors for the transition of the lowest lyingspin 3/2 decuplet state Σ ∗ to the spin 1/2 ground stateΛ.In the timelike region, these transition form factors(TFFs) can be measured at low energies via the Dalitzdecay Σ ∗ → Λ e + e − . It can be expected that theseDalitz decays will be addressed in the near future by thecollaborations HADES [28] and PANDA [29] at the Fa-cility for Antiproton and Ion Research (FAIR). Therefore a r X i v : . [ h e p - ph ] O c t we regard our present work as very timely.Concerning the Σ ∗ -Λ transition, two distinct qualita-tive aspects are noteworthy; one is more case specific,one is universal. We start with the latter. Whenever pi-ons are excited, they rescatter. In the isovector channel,the p-wave pion phase shift shows a relatively broad, es-sentially elastic resonance, the ρ meson [30, 31]. Thus,the universal pion dynamics gives rise to the coupling ofthe virtual photon to the ρ -meson. Phenomenologically,this is covered by the concept of vector meson dominance[32]. In the dispersive framework this is covered by thepion phase shift and the pion vector form factor. Wewill explore the quantitative importance of these effectsin the present paper.There is a second aspect, however, which is also cov-ered by our dispersive framework, but is typically miss-ing in a vector meson dominance approach. This is theaspect that we called “more case specific”. Being reso-nances, the Σ ∗ hyperons are unstable. In particular, theΣ ∗ can decay to Σ ± π ∓ . This pair can rescatter into aΛ and a real or virtual photon. Therefore, the TFFs arecomplex quantities in all kinematically allowed regimes:in the spacelike scattering region of Σ ∗ e − ↔ Λ e − ; atthe photon point Σ ∗ → Λ γ ; in the low-energy timelikeDalitz decay region of Σ ∗ → Λ e + e − ; and in the high-energy production region of e + e − → Σ ∗ ¯Λ. This is incontrast to TFFs for hadrons that are stable with re-spect to the strong interaction. For stable hadrons, theTFFs are essentially real in all regimes except for theproduction region.Complex form factors allow for non-trivial interfer-ence patterns between them. Those can be measured,e.g., with the help of the self-analyzing weak decays ofthe “stable” hyperons. In practice, this means that inthe succession of the two decays Σ ∗ → Λ e + e − andΛ → p π − , the angular distribution of the second de-cay contains interesting information about the interfer-ence of the TFFs. This information is accessible withoutinvolving the production process or the spin orientationof the Σ ∗ and without determining the spin orientationof the proton [33, 34]. On the other hand, in a strictvector meson dominance scenario, the Σ ∗ couples justvia a pointlike interaction to ρ Λ. There, a form factorcan only become complex where the ρ becomes unstable.This happens essentially only above the two-pion thresh-old. But in the Dalitz decay region of Σ ∗ → Λ e + e − , themaximally possible dielectron invariant mass (“ ρ -mesoninvariant mass”) is m Σ ∗ − m Λ < m π [35]. Thus in real-ity, the TFFs are complex but in a simple vector mesondominance approach they are real in the Dalitz decay re-gion. We will also explore the quantitative importance ofthese effects. One peculiarity we observe is that even ifthe imaginary part of a TFF at the photon point is verysmall, it gets larger for the transition radius.Ideally we would like to use subtracted dispersion rela-tions, but the available experimental input is too scarceto allow for it. For the case at hand there are threeTFFs and therefore three complex valued subtraction constants. For the time being we choose to use unsub-tracted dispersion relations; being aware of the large un-certainties they carry, we still expect to obtain results ofthe correct order of magnitude.In the first part of the paper we define the Σ ∗ -Λ TFFsand relate them to several observables, accessible in dif-ferent kinematical regions. Directly after we enter thecore of the theoretical work: we derive the appropriatedispersion relations for pion-hyperon scattering ampli-tudes and TFFs. Finally the results are presented. Thereare several appendices with various purposes. AppendixA clarifies how the individual contributions of meson andbaryon dynamics influence the final results. The otherscomplement the main text with technical details. II. TRANSITION FORM FACTORS ANDOBSERVABLES
Following essentially [5] we define three TFFs via (cid:104) | j µ | Σ ∗ ¯Λ (cid:105) = e ¯ v Λ ( p Λ , λ ) Γ µν ( p Σ ∗ , p Λ ) u ν Σ ∗ ( p Σ ∗ , σ ) (1)withΓ µν ( p Σ ∗ , p Λ ) := − ( γ µ q ν − (cid:54) q g µν ) m Σ ∗ γ F ( q )+ ( p µ Σ ∗ q ν − p Σ ∗ · q g µν ) γ F ( q )+ ( q µ q ν − q g µν ) γ F ( q ) (2)and q := p Σ ∗ + p Λ . Conventions for the spin-3/2 spinor u µ are provided in Appendix B. The neutral spin-3/2 Sigmahyperon is denoted by Σ ∗ . The helicities (not spins!) ofΣ ∗ and ¯Λ are called σ and λ , respectively.The TFFs defined via (1) are appropriate for a disper-sive representation where we study formally the reactionΣ ∗ ¯Λ → π + π − → γ ∗ . Physically, however, we study thereactions e + e − → γ ∗ → ¯Σ ∗ Λ and Σ ∗ → Λ γ ∗ → Λ e + e − .In addition, if one wants to compare the results of theelectromagnetic form factors for the transition Σ ∗ → Λwith the ones for ∆ → N it is convenient to adapt to theconventions used in the ∆ sector where mostly electro-production is studied [6, 7] and not Dalitz decays. Thusone should also look at the reaction e − Λ → e − Σ ∗ or moreformally Λ γ ∗ → Σ ∗ . Therefore we present the transitionform factors also for other kinematical regimes.In principle, the reactions Σ ∗ ¯Λ → γ ∗ , γ ∗ → ¯Σ ∗ Λ andΣ ∗ → Λ γ ∗ are related by crossing symmetry. For Λ γ ∗ → Σ ∗ one might involve charge conjugation and then againcrossing symmetry.For the amplitude relevant for the Dalitz decay, Σ ∗ → Λ γ ∗ , one finds (cid:104) Λ | j µ | Σ ∗ (cid:105) = e ¯ u Λ ( p Λ , λ ) Γ µν ( p Σ ∗ , − p Λ ) u ν Σ ∗ ( p Σ ∗ , σ ) . (3)In practice this leads to the very same expression as onthe right-hand side of (2) but with q := p Σ ∗ − p Λ .For the production amplitude γ ∗ → ¯Σ ∗ Λ one has tospecify the meaning of the two-fermion bra state: (cid:104) ¯Σ ∗ Λ | := | ¯Σ ∗ Λ (cid:105) † . (4)The structure corresponding to (2) would beΓ µν ( − p Σ ∗ , − p Λ ), but it is not convenient to define q as − p Σ ∗ − p Λ . Therefore we rather provide a fullyexplicit version of the TFFs adapted to the productionprocess: (cid:104) ¯Σ ∗ Λ | j µ | (cid:105) = − e ¯ u Λ ( p Λ , λ ) ˜Γ µν ( p Σ ∗ , p Λ ) v ν Σ ∗ ( p Σ ∗ , σ ) , (5)˜Γ µν ( p Σ ∗ , p Λ ) := ( γ µ q ν − (cid:54) q g µν ) m Σ ∗ γ F ( q )+ ( p µ Σ ∗ q ν − p Σ ∗ · q g µν ) γ F ( q )+ ( q µ q ν − q g µν ) γ F ( q ) (6)with q := p Σ ∗ + p Λ .Finally we obtain for the excitation process: (cid:104) Σ ∗ | j µ | Λ (cid:105) = − e ¯ u ν Σ ∗ ( p Σ ∗ , σ ) ˜Γ µν ( p Σ ∗ , − p Λ ) u Λ ( p Λ , λ ) . (7)Here the pertinent expression for ˜Γ µν agrees with theright-hand side of (6) provided one defines q := p Σ ∗ − p Λ .Next we introduce linear combinations of F , F and F , which correspond to TFFs with fixed helicity combi-nations. We denote them by G m ( m = σ − λ = 0 , ± G − ( q ) := ( − m Λ ( m Λ + m Σ ∗ ) + q ) F ( q )+ 12 ( m ∗ − m + q ) F ( q ) + q F ( q )for σ = − , λ = + 12 , (8) G ( q ) := m ∗ F ( q ) + m ∗ F ( q )+ 12 ( m ∗ − m + q ) F ( q )for σ = + 12 , λ = + 12 , (9)and G +1 ( q ) := m Σ ∗ ( m Λ + m Σ ∗ ) F ( q )+ 12 ( m ∗ − m + q ) F ( q ) + q F ( q )for σ = + 32 , λ = + 12 . (10)In the following we adopt the reference frame from [11]where the virtual photon is at rest, i.e. the Σ ∗ -¯Λ center-of-mass system, and where the Σ ∗ is moving in the z -direction. In this frame the three-momentum of the Σ ∗ is given by (cid:126)p Σ ∗ = p z (cid:126)e z with p z = (cid:112) λ ( q , m ∗ , m )2 (cid:112) q (11)where we have introduced the K¨all´en function λ ( a, b, c ) := a + b + c − ab + bc + ac ) . (12)We find ¯ v Λ ( − p z , /
2) Γ ν u ν Σ ∗ ( p z , +1 / v Λ γ u ∗ q m ∗ − m + q G ( q ) , (13) ¯ v Λ ( − p z , /
2) Γ ν u ν Σ ∗ ( p z , − /
2) = ¯ v Λ γ u ∗ G − ( q ) , (14)¯ v Λ ( − p z , /
2) Γ ν u ν Σ ∗ ( p z , +3 /
2) = ¯ v Λ γ u ∗ G +1 ( q ) . (15)The spinors on the right-hand side are evaluated with thesame arguments as on the respective left-hand side. Notethat in these relations the explicit “photon” indices 3 and1 are covariant, not contravariant as it is the case for thecorresponding relations in [11]. This will lead to a signchange in (44) below as compared to the conventions of[11].To make further contact with the existing literature,we relate our TFFs to the ones introduced in [6]. Therein,the transition from nucleon to ∆ is considered. We re-place ∆ → Σ ∗ and N → Λ to obtain our case at hand.The conventions for this process are provided in (7). Itis convenient to define Q := − q . Since one studies nowreactions with Q >
0, it is meaningful to introduce also Q := (cid:112) Q . The TFFs of Carlson [6] (in the followinglabeled with “Ca”) are related to our TFFs by G Ca − = Q − m Λ G +1 ,G Ca+ = Q − √ m Λ G − ,G Ca0 = Q Q − √ m Λ m Σ ∗ G (16)with Q − := (cid:112) Q + ( m Λ − m Σ ∗ ) .In [7] various conventions for the TFFs are related toeach other, including the ones from [6]. With the help of(16) and [7] our TFFs can be easily related to any otherTFF combinations and conventions.At large spacelike momenta, i.e. for large Q , onefinds the following asymptotic behavior from perturba-tive QCD [6]: G − ( − Q ) ∼ Q , G , +1 ( − Q ) ∼ Q ,F ( − Q ) ∼ Q , F , ( − Q ) ∼ Q . (17)Since we will provide only a low-energy representation forthe various TFFs, one cannot expect to reproduce thisasymptotic behavior without involving physics beyondthe low-energy region. In general, this requires too much There is a mismatch between the conventions used in [6] and here.This is essentially based on the fact that we introduce our TFFsvia the coupling of a virtual timelike photon to a spin-3/2 baryonand a spin-1/2 antibaryon where the latter has helicity +1 /
2; see(13)-(15). In [6] the TFFs are introduced via the coupling of avirtual spacelike photon to an incoming spin-1/2 baryon and anoutgoing spin-3/2 baryon. The former has helicity +1 /
2. If onetranslates our case to the one in [6] our antibaryon turns to abaryon with helicity − / /
2. This sign change relatesour TFF G m to Carlson’s TFF G Ca − m for all m = 0 , ± modeling. Nonetheless, it might be reasonable to aim fora representation where the TFFs fall off with 1 /Q atleast. We will come back to this point below.Pion-loop contributions to the TFFs can be most easilyaddressed for fixed helicity combinations. This favors theuse of the TFFs (8), (9), (10). However, these combina-tions are subject to kinematical constraints, i.e. there is akinematical point where these TFFs are not independentfrom each other. This happens at q = ( m Λ + m Σ ∗ ) where G +1 = G − = G ( m Λ + m Σ ∗ ) /m Σ ∗ . Disper-sion relations should be formulated for constraint-freequantities [36, 37], otherwise one might have to involveadditional subtractions. The construction procedure of[36, 37] leads to the TFFs of (2). Therefore it can beuseful to invert the relations (8), (9), (10), which yields F ( q ) = G +1 ( q ) − G − ( q )( m Σ ∗ + m Λ ) − q ,F ( q ) = 2 λ ( m ∗ , m , q ) × (cid:2) − q G ( q )+ ( m Σ ∗ m Λ − m + q ) G +1 ( q )+ ( m ∗ − m Σ ∗ m Λ ) G − ( q ) (cid:3) ,F ( q ) = 2 λ ( m ∗ , m , q ) × (cid:2) ( m ∗ − m + q ) G ( q ) − m ∗ (cid:0) G +1 ( q ) + G − ( q ) (cid:1)(cid:3) (18)with the K¨all´en function given in (12).Let us turn now to observable production and decayprocesses. In terms of the TFFs the decay width of Σ ∗ → Λ γ is given byΓ = e ( m ∗ − m )96 πm ∗ ( m Σ ∗ − m Λ ) × (cid:0) | G +1 (0) | + | G − (0) | (cid:1) . (19)For the differential cross section of the reaction e + e − → ¯Σ ∗ Λ (see also [5]) we obtain in the center-of-mass frameand neglecting the electron mass: (cid:18) d σ dΩ (cid:19) CM ( q , θ ) = e π q p z (cid:112) q q − ( m Σ ∗ − m Λ ) ) × (cid:104) (1 + cos θ ) (cid:0) | G +1 ( q ) | + | G − ( q ) | (cid:1) + 4 q m ∗ sin θ | G ( q ) | (cid:105) (20)with the center-of-mass momentum p z given in (11).For the Dalitz decay distribution of Σ ∗ → Λ e + e − weprovide one version keeping the electron mass and onewhere only the kinematical velocity factor is kept. Weintroduce the electron velocity by β e := (cid:115) − m e q (21) with the electron mass m e . The doubly-differential decayrate is given bydΓd q d cos θ = e (2 π ) m ∗ q p z (cid:112) q β e (cid:0) ( m Σ ∗ − m Λ ) − q (cid:1) × (cid:20)(cid:18) θ + 4 m e q sin θ (cid:19) × (cid:0) | G +1 ( q ) | + | G − ( q ) | (cid:1) + 4 (cid:18) sin θ + 4 m e q cos θ (cid:19) q m ∗ | G ( q ) | (cid:21) ≈ e (2 π ) m ∗ q p z (cid:112) q β e (cid:0) ( m Σ ∗ − m Λ ) − q (cid:1) × (cid:104) (cid:0) θ (cid:1) (cid:0) | G +1 ( q ) | + | G − ( q ) | (cid:1) + 4 q m ∗ sin θ | G ( q ) | (cid:105) . (22)Here θ denotes the angle between electron and Λ in therest frame of the electron-positron pair. If one calculatesthe integrated decay rate, the integration in θ ranges from π to 0 such that the cos θ integration ranges from − q of the photon is limited in the kinematical region4 m e ≤ q ≤ ( m Σ ∗ − m Λ ) (23)and so the factor ( m Σ ∗ − m Λ ) − q will always be non-negative. If one blindly neglected the electron mass, onewould obtain a divergent integrated decay rate. Thephase-space factor β e and the proper integration range(23) ensure a physical, finite result.For later use we also introduce a QED version of (22),which is supposed to describe the situation where thestructure of hyperons is not resolved. In practice we re-place the TFF combinations by their q = 0 expressionsand make in this way also contact with the real photoncase (19):dΓ QED d q d cos θ := e (2 π ) m ∗ q p z (cid:112) q β e (cid:0) ( m Σ ∗ − m Λ ) − q (cid:1) × (cid:18) θ + 4 m e q sin θ (cid:19) × (cid:0) | G +1 (0) | + | G − (0) | (cid:1) . (24)Conceptually, small momenta go along with small q and with treating the mass difference m Σ ∗ − m Λ as small.By inspecting (22), we see that at small momenta thedecay rate is dominated by the combination 3 | G +1 | + | G − | . In turn, (8) and (10) show that for low momentathe dominant contribution to G +1 and G − originatesfrom F . At high momenta, G − is dominant, as can beread off from (17); see also [6]. We deduce from (8) and(17) that it is again F that dominates G − . Thus inboth limiting cases, low and high momenta, the TFF F plays the dominant role.More detailed access to the TFFs can be obtained bydetermining the angular distribution of the subsequentweak decay of the Λ; see also [34]. To this end considerthe decay Λ → pπ − governed by the amplitude [35] M weak = G F m π ¯ u p ( p ) ( A − Bγ ) u Λ ( p ) . (25)It is useful to introduce the asymmetry parameter α Λ := 2Re( T ∗ s T p ) | T s | + | T p | (26)with the s-wave amplitude T s := A , the p-wave ampli-tude T p := p f B/ ( E p + m p ) and mass m p , energy E p andmomentum p f of the proton in the rest frame of the de-caying Λ hyperon, i.e. E p = m + m p − m π m Λ (27)and p f = λ / ( m , m p , m π )2 m Λ (28)with the K¨all´en function (12).For the differential decay width of the whole four-bodydecay Σ ∗ → Λ e + e − → pπ − e + e − one finds (neglectingagain the electron mass where meaningful)dΓd q d cos θ dΩ p ≈ (29) e (2 π ) m ∗ q p z (cid:112) q β e (cid:0) ( m Σ ∗ − m Λ ) − q (cid:1) Br Λ → pπ − × (cid:104) (1 + cos θ ) (cid:0) | G +1 ( q ) | + | G − ( q ) | (cid:1) + 4 q m ∗ sin θ | G ( q ) | + 4 (cid:112) q m Σ ∗ α Λ Im (cid:0) G ( q ) G ∗− ( q ) (cid:1) cos θ sin θ sin θ p sin φ p (cid:105) . Here Br Λ → pπ − denotes the branching ratio while θ p and φ p are the angles of the proton three-momentum mea-sured in the rest frame of Λ. The coordinate system inthis frame is defined by (cid:126)q pointing in the negative z -direction (i.e. in the rest frame of the virtual photon theΣ ∗ and Λ direction defines the positive z -axis) and theelectron moves in the x - z plane with positive momentumprojection on the x -axis. In this frame, θ p is the angleof the proton momentum relative to the z -axis and φ p isthe angle between the x -axis and the projection of theproton momentum on the x - y plane, i.e. (cid:126)p p = p f (sin θ p cos φ p , sin θ p sin φ p , cos θ p ) ,(cid:126)q = | (cid:126)q | (0 , , − ,(cid:126)p e − · (cid:126)e y = 0 , (cid:126)p e − · (cid:126)e x > , (cid:126)e y = (cid:126)p e − × (cid:126)q | (cid:126)p e − | | (cid:126)q | . (30) Note the subtlety that θ is measured in the rest frameof the virtual photon while Ω p denotes angles in the restframe of the Λ-hyperon. In terms of Lorentz invariantquantities the angles are related to p Λ · k e = − λ / ( m ∗ , m , q ) cos θ ,(cid:15) µναβ k µe p ν Λ p αp q β = − (cid:112) q λ / ( m ∗ , m , q ) × p f sin θ sin θ p sin φ p (31)with k e := p e − − p e + , q = p e − + p e + = p Σ ∗ − p Λ and theconvention [38] for the Levi-Civita symbol: (cid:15) = − . (32)A peculiar feature of (29) is the presence of the com-bination Im( G G ∗− ), which is non-zero even below thetwo-pion threshold. This is a consequence of the Σ ∗ be-ing unstable with respect to the strong interaction. Thisproperty plays a crucial role throughout the developmentof this paper, and constitutes the main difference fromthe analogous Σ-Λ case [11]. III. DISPERSIVE MACHINERY
Essentially this goes along the same lines as describedin [11, 16]. In particular we use the same Omn`es function,Ω( s ) = exp s ∞ (cid:90) m π d s (cid:48) π δ ( s (cid:48) ) s (cid:48) ( s (cid:48) − s − i(cid:15) ) (33)where δ denotes the pion p-wave phase shift [30, 31]. Thepion vector form factor F Vπ is taken from [16] (see also[21, 39, 40]): F Vπ ( s ) = (1 + α V s ) Ω( s ) . (34)For the pion phase shift from [31], a value of α V = 0 .
12 GeV − (35)yields an excellent description of the data on the pionvector form factor from tau decays [41] for energies below1 GeV; see [16]. A. Dispersion relations
Based on the asymptotic behavior (17), the three TFFsintroduced in (2) satisfy unsubtracted dispersion rela-tions F j ( q ) = (cid:90) d s πi disc F j ( s ) s − q (36)for j = 1 , ,
3. Here “disc” denotes the discontinuity ofthe function F j .How does this translate to the TFFs G m introducedin (8), (9), (10)? We can discuss this rather generally: Ifone defines two new TFFs, A and B , via A ( q ) := F ( q ) + F ( q ) ,B ( q ) := F ( q ) + q s F ( q ) , (37)one sees that they are subject to the kinematical con-straint A ( s ) = B ( s ) . (38)The dispersion relation for A can be formulated withoutproblems. For B one obtains B ( q ) = (cid:90) d s πi s − q (cid:18) disc F ( s ) + q s disc F ( s ) (cid:19) = (cid:90) d s πi s − q (cid:18) disc B ( s ) + q − ss disc F ( s ) (cid:19) = (cid:90) d s πi disc B ( s ) s − q − s (cid:90) d s πi disc F ( s ) . (39)This shows that in general one has to deal with an ad-ditional constant in a dispersive calculation of B . It isthis constant that ensures that (38) holds. In addition,we have implicitly assumed that the dispersive integralover disc B actually converges.For the TFFs F j that show the high-energy behavior(17), the situation is actually simpler. This high-energybehavior provides conditions for the integrals over disc F j .In particular, the conditionlim Q →∞ Q F j ( − Q ) = 0 (40)leads to (cid:90) d s πi disc F j ( s ) = 0 . (41)Thus, the additional constant in (39) vanishes. One ob-tains standard unsubtracted dispersion relations for A and for B . In view of the relations (8), (9), (10) one cantherefore conclude that also all the G m ’s satisfy unsub-tracted dispersion relations: G m ( q ) = (cid:90) d s πi disc G m ( s ) s − q (42)for m = 0 , ± B. General considerations about the analyticstructure
At low energies, it can be expected that the q behaviorof the TFFs is determined by the lowest-energy statesthat can be excited. For the isovector TFFs that we study here, the lowest energetic states are pion pairs.Therefore we can write in complete analogy to [11]: G m ( q ) = 112 π ∞ (cid:90) m π d sπ T m ( s ) p . m . ( s ) F V ∗ π ( s ) s / ( s − q − i(cid:15) )+ G anom m ( q ) + . . . (43)where the ellipsis denotes other intermediate states asfor instance four-pion or baryon-antibaryon states. The“anomalous” piece will be determined later. It is relatedto anomalous thresholds.The pion-hyperon scattering amplitudes T m are ob-tained in a two-step procedure: In line with (13), (14),(15) we define first the reduced amplitudes K ± ( s ) := − π (cid:90) d θ sin θ × M ( s, θ, / ± , / v Λ ( − p z , / γ u ∗ ( p z , / ± p c . m . ,K ( s ) := − m ∗ − m + s s π (cid:90) d θ sin θ cos θ × M ( s, θ, / , / v Λ ( − p z , +1 / γ u ∗ ( p z , +1 / p c . m . . (44)Here p c . m . denotes the modulus of the momenta of thepions in the center-of-mass frame. We have introduced M ( s, θ, σ, λ ) as the approximation to the Feynman am-plitude for the reaction Σ ∗ ¯Λ → π + π − . In practice, M ( s, θ, σ, λ ) does not include the rescattering effect ofthe pions. This will be taken care of in the second step.In addition, we want to distinguish conceptually betweenprocesses with left-hand cut structures and purely poly-nomial terms. In practice, the reduced amplitudes K originate from the left-hand cut structures only, whilewe denote the polynomial terms by P . All the formulaepresented explicitly for K apply also to P .Pion rescattering is taken into account by solving aMuskhelishvili-Omn`es equation [42, 43]. The result is T m ( s ) = K m ( s ) + Ω( s ) P m + T anom m ( s )+ Ω( s ) s ∞ (cid:90) m π d s (cid:48) π K m ( s (cid:48) ) sin δ ( s (cid:48) ) | Ω( s (cid:48) ) | ( s (cid:48) − s − i(cid:15) ) s (cid:48) . (45)As already spelled out, K m takes care of the left-handcut structures. P m is a constant (per channel) that canbe obtained ideally from a fit to data or estimated from χ PT. We have used here a once-subtracted dispersionrelation. In principle, one could use more subtractions,which brings in a polynomial instead of a constant. Butthis would worsen the high-energy behavior. In the fol-lowing, we will occasionally suppress the index m whenpresenting generic formulae.If there is an anomalous threshold, there might bean extra piece T anom ( s ) that is added to the amplitude.Such a situation can occur if the mass m exch of the ex-changed state in the t/u-channel is “too light”. For ourreaction the condition to have an anomalous threshold is[44] m < (cid:0) m ∗ + m − m π (cid:1) . (46)For the formal reaction Σ ∗ ¯Λ → π one has to deal withthe exchange of states carrying strangeness. In practicewe will take into account the exchange of Σ and Σ ∗ hy-perons. The condition (46) does not hold for the Σ ∗ ex-change, but is satisfied for the Σ exchange. In the lattercase the logarithm obtained from the partial-wave projec-tion (44) requires a proper analytic continuation. If onetakes the partial-wave projection literally (straight-lineintegral) as given in (44), then the obtained logarithmhas a cut in the complex s plane that intersects with theright-hand cut (unitarity cut), i.e. part of this cut lieson the physical Riemann sheet. To disentangle the cuts,one can define the cut of the logarithm such that it con-nects the branch point to the start of the unitarity cutby a straight line. The additional contribution T anom ( s )takes care of the extra cut. A general discussion is pro-vided in Appendix C.To be more concrete, we note that the p-wave partial-wave projection of type (44) for a t- or u-channel ex-change process produces a term K ( s ) = g ( s ) − f ( s ) Y ( s ) κ ( s )+ f ( s ) 1 κ ( s ) log Y ( s ) + κ ( s ) Y ( s ) − κ ( s ) (47)with the functions Y , κ and σ defined in Appendix C for m → m Σ ∗ , m → m Λ . In addition, we have introduced f ( s ), g ( s ) as functions without cuts. These functionsmight have poles at kinematical thresholds, but they con-spire such that no poles show up for K as given in (47).If one expands the log function in powers of κ/Y one seesthat there are no poles for κ →
0. Concrete formulae aregiven in Section V.If one considers the standard logarithm with a cutalong the real negative axis, then (47) is ill-defined for Y ( s ) = 0. This point lies on the unitarity cut if (46) issatisfied. To disentangle the two cuts one starts with aproper analytic continuation of the logarithm along theunitarity cut. To this end we introduce the following fourpoints: • At s := ( m Σ ∗ + m Λ ) we have κ = 0. Abovethis point, i.e. for s real and larger than s , there It does not hold for any exchange of a many-particle state thatcontains a hyperon. The lightest such state would be a Λ andone pion. Using that the Σ ∗ is lighter than a Λ and two pions,it is easy to check that the condition (46) is not satisfied for m exch ≥ m Λ + m π . is the true scattering region. There, κ is real and Y is positive and larger than κ . The logarithmin (47) can be defined as the real-valued standardlogarithm of positive numbers. • At s := m ∗ + m + 2 m π − m we have Y = 0.For s real and between s and s the function κ ispurely imaginary and Y is still positive. • At s := 4 m π we have κ = 0. For s real and be-tween s and s the function κ is purely imaginaryand Y is negative. • At s := ( m Σ ∗ − m Λ ) we have κ = 0. For s realand between s and s the function κ is real (and Y is negative).For the case of a Σ exchange we have 0 < s < s s , K ( s ) := g ( s ) − f ( s ) Y ( s ) κ ( s ) + 2 f ( s ) κ ( s ) | κ ( s ) | arctan | κ ( s ) | Y ( s )(49)for s < s < s , and K ( s ) := g ( s ) − f ( s ) Y ( s ) κ ( s )+ 2 f ( s ) κ ( s ) | κ ( s ) | (cid:18) arctan | κ ( s ) | Y ( s ) + π (cid:19) (50)for s < s < s . Here the standard logarithm for pos-itive real numbers is used and the arctan function withvalues between − π/ π/
2. Note that at the two-pion threshold s = s the quantity K ( s ) of (50) di-verges ∼ πf ( s ) / ( κ ( s ) | κ ( s ) | ) ∼ /σ ( s ), but the prod-uct K ( s ) sin δ ( s ) in (45) remains finite due to sin δ ( s ) ∼ σ ( s ) for the p-wave pion phase shift [30, 31]. K ( s ) alsoappears in the combination K ( s ) p . m . ( s ) in (43) whichremains also finite.The second issue is the definition of T anom ; see alsothe discussion in Appendix C. The branch points of thelogarithm in (47) are defined by Y ( s ) = κ ( s ). Theyare located at s ± = − m + 12 (cid:0) m ∗ + m + 2 m π (cid:1) − m ∗ m − m π ( m ∗ + m ) + m π m ∓ λ / ( m ∗ , m , m π ) λ / ( m , m , m π )2 m . (51)We take s + as the solution that has a positive imagi-nary part for small values of m ∗ . If one replaces m ∗ by m ∗ + i(cid:15) and follows the motion of s + for increas-ing values of m ∗ , then s + moves towards the real axisand intersects with the unitarity cut where (46) turnsto an equality. For larger values of m ∗ one finds s + inthe lower half plane of the first Riemann sheet. This isthe situation for the physical value of m ∗ for the case m = m . Thus we have s + = − m + 12 (cid:0) m ∗ + m + 2 m π (cid:1) − m ∗ m − m π ( m ∗ + m ) + m π m − i λ / ( m ∗ , m , m π ) (cid:0) − λ ( m , m , m π ) (cid:1) / m (52)with positive square roots.The anomalous contribution that enters (45) is thengiven by T anom ( s ) = Ω( s ) s (cid:90) d x d s (cid:48) ( x )d x s (cid:48) ( x ) − s × f ( s (cid:48) ( x ))( − λ ( s (cid:48) ( x ) , m ∗ , m )) / κ ( s (cid:48) ( x )) × t ( s (cid:48) ( x ))Ω( s (cid:48) ( x )) s (cid:48) ( x ) (53)with the straight-line path s (cid:48) ( x ) := (1 − x ) s + + x m π (54) that connects the branch point (52) of the logarithm of(47) and the branch point 4 m π of the unitarity cut.One also needs the scattering amplitude t ( s ) in thecomplex plane. Following [45], one could use an analyticcontinuation of the amplitude as constructed from χ PTand unitarized by the inverse amplitude method. Thisrepresentation does not show a decent high-energy be-havior. Therefore we will use it only for the anomalouspart. There the whole integration region is rather closeto the two-pion threshold. Therefore an expression from χ PT or a unitarized version thereof should be sufficientlyclose to the true scattering amplitude. We take from[45] the following expressions (extended to the complexplane). The approximation from χ PT is given by t χ PT ( s ) ≈ t ( s ) + t ( s ) (55)and its unitarized version is t IAM ( s ) = t ( s ) t ( s ) − t ( s ) (56)with t ( s ) = sσ πF , (57) t ( s ) = t ( s )48 π F (cid:20) s (cid:18) ¯ l + 13 (cid:19) − m π − m π s (cid:16) − L σ (cid:0) − σ (cid:1) + 3 L σ (cid:0) − σ + 3 σ (cid:1)(cid:17)(cid:21) − ˆ σ ( s ) t ( s ) , (58) L σ = 1 σ (cid:18) σ log 1 + σ − σ − (cid:19) . (59)The functions σ ( s ) and ˆ σ ( s ) are defined in (C6) and (C9),respectively. Note that there is no square root ambiguityin the definition of σ since all expressions are even in σ → − σ . The square root appearing in the definition ofthe function ˆ σ has its cut on the negative real axis. Thenthe function ˆ σ has the unitarity cut (and also a cut alongthe negative real axis).The value for the pion decay constant in the chirallimit F is taken from the ratio F π /F = 1 . F π = 92 . l = 5 . ρ -meson resonanceon the second Riemann sheet. In this work instead weuse ¯ l = 6 .
47 which is obtained by requiring agreementbetween the pion p-wave phase shifts from (56) and from[31] around the two-pion threshold.Finally, we provide the anomalous piece of the TFFs.As described in Appendix C one can relate the anoma-lous piece of the TFF to the anomalous piece of T − K . Therefore we obtain G anom m ( q ) = 112 π (cid:90) d x d s (cid:48) ( x )d x s (cid:48) ( x ) − q × f m ( s (cid:48) ( x )) s (cid:48) ( x ) F Vπ ( s (cid:48) ( x )) − − λ ( s (cid:48) ( x ) , m ∗ , m )) / . (60)Note that the Omn`es function (33) that enters the pionvector form factor (34) is defined everywhere on the firstRiemann sheet via the pion phase shift along the right-hand cut. Therefore, (60) can be calculated withoutproblems.Note that without any anomalous piece the TFF inte-gral in (43) would be real below the two-pion threshold.However, the TFF should be complex because the Σ ∗ is unstable. The imaginary part emerges from the fol-lowing process: Irrespective of the invariant mass of thephoton, the Σ ∗ can decay to a pion and a Σ. These statescan rescatter into a Λ and a real or virtual photon. Theanomalous pieces take care of this physical process. C. Subtracted dispersion relations
Though the intermediate states not explicitly consid-ered in (43) might have a minor influence on the shape of the TFFs at low energies, it is likely that they havean impact on the overall size; see e.g. the discussion in[11, 16, 21]. A way to enhance the importance of thelow-energy region in a dispersive integral is the use ofa subtracted dispersion relation. The most conservativeapproach that does not make use of any high-energy in-put is to start again from the unconstrained TFFs F i . Asubtracted dispersion relation reads F i ( q ) = F i (0)+ q π Λ (cid:90) m π d sπ T i ( s ) p . m . ( s ) F V ∗ π ( s ) s / ( s − q − i(cid:15) ) + F anom i ( q ) (61)for i = 1 , ,
3. The last, “anomalous” piece will be speci-fied below.In principle, the scattering amplitudes T i are againgiven by (45) but now we need the amplitudes K i , i = 1 , , K +1 , , − inthe same way as the TFFs F i are obtained from G +1 , , − ,i.e. K ( s ) = K +1 ( s ) − K − ( s )( m Σ ∗ + m Λ ) − s , (62) K ( s ) = 2 λ ( m ∗ , m , s ) [ − s K ( s )+ ( m Σ ∗ m Λ − m + s ) K +1 ( s )+ ( m ∗ − m Σ ∗ m Λ ) K − ( s ) (cid:3) ,K ( s ) = 2 λ ( m ∗ , m , s ) (cid:2) ( m ∗ − m + s ) K ( s ) − m ∗ ( K +1 ( s ) + K − ( s )) (cid:3) . We have introduced a cutoff Λ in (61). Since wehave only the low-energy part under control where thetwo-pion state dominates, it is not reasonable to extendthe integral into the uncontrolled high-energy region.In practice, the two-pion state dominates the isovectorchannel up to about 1 GeV. To explore the uncertaintiesof our low-energy approximation we will vary the cutoffbetween 1 and 2 GeV.Finally we come back to the anomalous piece in (61): F anom i ( q ) = q π (cid:90) d x ds (cid:48) ( x ) dx s (cid:48) ( x ) − q × f i ( s (cid:48) ( x )) F Vπ ( s (cid:48) ( x )) − − λ ( s (cid:48) ( x ) , m ∗ , m )) / . (63)The drawback of (61) is that one needs experimentalinput for the three complex-valued(!) subtraction con-stants F i (0). This is on top of the constants P m in (45),which are ideally also fitted to experimental data. At themoment such an amount of experimental information is not available. Therefore we will explore an alternative inthe next subsection. D. Unsubtracted dispersion relations
At large energies, the TFFs F i determined via (61) ap-proach a constant, in sharp contrast to the correct scalingbehavior (17). The TFFs G m obtained from (8), (9), (10)even diverge. All this is not a fundamental problem sinceby construction the representation (61) is designed to beaccurate at low energies only. Nonetheless, the represen-tation (61) requires the knowledge of several subtractionconstants, all of them in principle complex, because theΣ ∗ is unstable. Thus, it might be of advantage to explorethe predictive power of an unsubtracted dispersion rela-tion. As shown, e.g., in [11, 16, 21], one cannot expect toobtain completely correct values for the subtraction con-stants from the unsubtracted dispersion relations, if oneuses only the two-pion intermediate states. However, itmight be reasonable to use a simple effective pole to ap-proximate the impact of all the other, higher-lying inter-mediate states on the low-energy quantities [21, 46, 47].The pole position might be varied in a reasonable rangeof masses of excited vector mesons [35] while the residuecan be chosen such that a more reasonable high-energybehavior is achieved.Enforcing a more realistic high-energy behavior pro-vides an additional advantage. As already pointed out,one can then formulate simple dispersion relations alsofor the TFFs G m , m = 0 , ±
1. In practice we write G m ( q ) = 112 π Λ (cid:90) m π d sπ T m ( s ) p . m . ( s ) F V ∗ π ( s ) s / ( s − q − i(cid:15) )+ G anom m ( q ) + c m M V M V − q , (64)which is only valid for q (cid:28) M V . The anomalous part isgiven in (60). The dimensionless constant c m is adjustedsuch that lim Q →∞ Q G m ( − Q ) = 0 . (65)This leads to c m M V = − π Λ (cid:90) m π d sπ T m ( s ) p . m . ( s ) F V ∗ π ( s ) s / − π (cid:90) d x d s (cid:48) ( x )d x f m ( s (cid:48) ( x )) s (cid:48) ( x ) F Vπ ( s (cid:48) ( x )) − − λ ( s (cid:48) ( x ) , m ∗ , m )) / . (66)To explore the uncertainties of this approach one mightvary the effective pole between the masses of the excitedvector mesons [35], 1 . < M V < . ∗ → Λ γ and Σ ∗ → Λ e + e − must show if (64), (65)0is a reasonable approach or if one has to resort to thesubtracted dispersion relations (61). So far there are noDalitz decay data available. In Section V we present nu-merical results utilizing (64), (66). IV. INPUT FROM CHIRAL PERTURBATIONTHEORY
The leading-order (LO) chiral Lagrangian includingthe decuplet states is given by [7, 22, 24, 25] L (1)baryon = tr (cid:0) ¯ B ( i /D − m (8) ) B (cid:1) + ¯ T µabc ( iγ µνα D α − γ µν m (10) ) ( T ν ) abc + D B γ µ γ { u µ , B } ) + F B γ µ γ [ u µ , B ])+ h A √ (cid:0) (cid:15) ade ¯ T µabc ( u µ ) bd B ce + (cid:15) ade ¯ B ec ( u µ ) db T abcµ (cid:1) − H A m R (cid:15) µναβ (cid:0) ¯ T µabc ( D ν T α ) abd ( u β ) cd + ( D ν ¯ T α ) abd ( T µ ) abc ( u β ) dc (cid:1) (67)with tr denoting a flavor trace.We have introduced the totally antisymmetrized prod-ucts of two and three gamma matrices [38], γ µν := 12 [ γ µ , γ ν ] (68)and γ µνα := 16 ( γ µ γ ν γ α + γ ν γ α γ µ + γ α γ µ γ ν − γ µ γ α γ ν − γ α γ ν γ µ − γ ν γ µ γ α )= 12 { γ µν , γ α } = + i(cid:15) µναβ γ β γ , (69)respectively. Our conventions are: γ := iγ γ γ γ and(32), the latter in agreement with [38] but opposite to[7, 48]. If a formal manipulation program is used to calcu-late spinor traces and Lorentz contractions a good checkfor the convention for the Levi-Civita symbol is the lastrelation in (69).The octet baryons are collected in ( B ab is the entry inthe a th row, b th column) B = √ Σ + √ Λ Σ + p Σ − − √ Σ + √ Λ n Ξ − Ξ − √ Λ . (70) Throughout this work, when using the phrase “gamma matrices”we have the four gamma matrices γ µ , µ = 0 , , ,
3, in mind, not γ . The decuplet is expressed by a totally symmetric flavortensor T abc with T = ∆ ++ , T = 1 √ + ,T = 1 √ , T = ∆ − ,T = 1 √ ∗ + , T = 1 √ ∗ , T = 1 √ ∗− ,T = 1 √ ∗ , T = 1 √ ∗− , T = Ω . (71)The Goldstone bosons are encoded inΦ = π + √ η √ π + √ K + √ π − − π + √ η √ K √ K − √ K − √ η ,u := U := exp( i Φ /F π ) , u µ := i u † ( ∇ µ U ) u † = u † µ . (72)The fields have the following transformation propertieswith respect to chiral transformations [15, 22]: U → L U R † , u → L u h † = h u R † ,u µ → h u µ h † , B → h B h † , (73) T abcµ → h ad h be h cf T defµ , ¯ T µabc → ( h † ) da ( h † ) eb ( h † ) fc ¯ T µdef . In particular, the choice of upper and lower flavor indicesis used to indicate that upper indices transform with h under flavor transformations while the lower componentstransform with h † .For a (baryon) octet the chirally covariant derivativesare defined by D µ B := ∂ µ B + [Γ µ , B ] , (74)for a decuplet T by( D µ T ) abc := ∂ µ T abc + (Γ µ ) aa (cid:48) T a (cid:48) bc + (Γ µ ) bb (cid:48) T ab (cid:48) c + (Γ µ ) cc (cid:48) T abc (cid:48) , (75)for an anti-decuplet by( D µ ¯ T ) abc := ∂ µ ¯ T abc − (Γ µ ) a (cid:48) a ¯ T a (cid:48) bc − (Γ µ ) b (cid:48) b ¯ T ab (cid:48) c − (Γ µ ) c (cid:48) c ¯ T abc (cid:48) , (76)and for the Goldstone boson fields by ∇ µ U := ∂ µ U − i ( v µ + a µ ) U + iU ( v µ − a µ ) (77)with Γ µ := 12 (cid:0) u † ( ∂ µ − i ( v µ + a µ )) u + u ( ∂ µ − i ( v µ − a µ )) u † (cid:1) , (78)where v and a denote external sources.In (67) m (8) ( m (10) ) denotes the mass of the baryonoctet (decuplet) in the chiral limit. For the next-to-leading-order (NLO) calculation that we perform in thepresent work we use the physical masses [35] of all states.Indeed, for the octet and decuplet the flavor breaking1terms that appear at NLO, cf. (81), (82) below, are ca-pable of splitting up the baryon masses such that theyare sufficiently close to the physical masses; see, e.g. thecorresponding discussion in [9, 25].For the coupling constants we use D = 0 . F =0 .
46 which implies for the pion-nucleon coupling constant g A = F + D = 1 .
26. The value for h A can be determinedfrom the partial decay width Σ ∗ → π Λ or Σ ∗ → π Σyielding h A = 2 . ± . N c , one obtains the following relations for two or threeflavors, respectively: h A = 3 g A / √ ≈ .
67 according to[7, 48, 49] or h A = 2 √ D ≈ .
26 according to [23, 50].Finally one has to specify H A . In absence of a simpledirect observable to pin it down we take estimates fromlarge- N c considerations: H A = g A ≈ .
27 [7, 48] or H A = 9 F − D ≈ .
74 [23, 50]. Numerically we explorethe range H A = 2 . ± .
3. We have checked explicitlythat the sign of H A is in agreement with [7, 48] and alsowith [50]. For quark-model estimates of these couplingconstants see [51, 52]. For our purposes the interactionterm proportional to H A effectively reduces to+ H A m R F π (cid:15) µναβ ¯ T µabc ∂ ν ( T α ) abd ∂ β Φ cd . (79)Working with relativistic spin-3/2 Rarita-Schwingerfields is plagued by ambiguities how to deal with thespurious spin-1/2 components. In the present contextthe interaction term ∼ h A causes not only the proper ex-change of spin-3/2 resonances, but induces an additionalcontact interaction. This unphysical contribution can beavoided by constructing interaction terms according tothe Pascalutsa description [7, 48, 49, 53]. It boils downto the replacement T µ → − m R (cid:15) νµαβ γ γ ν ∂ α T β (80)where m R denotes the resonance mass. Strictly speakingthis procedure induces an explicit flavor breaking, butsuch effects are anyway beyond leading order. In prac-tice, we take the mass of the Σ ∗ resonance. The H A termof (79) is already constructed such that only the spin-3/2components contribute.We will explore both the standard interaction term ∼ h A from (67) and the corresponding one obtained by(80). We will show explicitly that differences can beaccounted for by contact interactions of the chiral La-grangian at NLO and beyond. Quantitatively, it is inter-esting to see how much the contact terms P m in (45) arechanged when switching from the standard to the Pasca-lutsa interaction. This provides an uncertainty estimateif P m is not determined from a fit to form factor data. Inprinciple we could do the same for the H A term and startinstead with a simpler Lagrangian ∼ ¯ T µabc [ /u ] cd γ T abdµ . Butwe refrain from this exercise. Now we turn to the Lagrangian of second order inthe chiral counting. A complete and minimal NLO La-grangian has been presented in [25]. For our present pur-pose we need terms that lift the mass degeneracies thathold at LO and we need terms that provide interactionsfor Σ ∗ π → Λ π (or formally Σ ∗ ¯Λ → π ) with the twopions in a p-wave.The relevant part of the NLO Lagrangian for thebaryon octet sector reads [25, 54, 55] L (2)8 = b χ,D tr( ¯ B { χ + , B } ) + b χ,F tr( ¯ B [ χ + , B ]) (81)with χ ± = u † χu † ± uχ † u and χ = 2 B ( s + ip ) obtainedfrom the scalar source s and the pseudoscalar source p .The low-energy constant B is essentially the ratio of thelight-quark condensate and the square of the pion-decayconstant; see, e.g. [15, 56–58]. While at LO all baryonoctet states are degenerate in mass, the NLO terms of(81) lift this degeneracy and essentially move all massesto their respective physical values. Technically this isachieved if one replaces the scalar source s by the quarkmass matrix. Numerical results for the octet mass m (8) in (67) and the splitting parameters b χ,D/χ,F in (81) aregiven, for instance, in [9]. In practice we use the physicalmasses. Therefore we do not specify these parametershere.The relevant part of the NLO Lagrangian for thebaryon decuplet sector reads [25] L (2)10 = − d χ, (8) ¯ T µabc ( χ + ) cd γ µν ( T ν ) abd . (82)It provides a mass splitting for the decuplet baryons suchthat m Ω − m Ξ ∗ = m Ξ ∗ − m Σ ∗ = m Σ ∗ − m ∆ , in good agree-ment with phenomenology [35]. In the present work weonly deal with the Σ ∗ . In practice we use the physicalmass of the neutral Σ ∗ . In that way the physical thresh-olds are exactly reproduced.More concretely we use the following masses (in GeV): m π = 0 . m Λ = 1 . m Σ = 1 .
193 and m Σ ∗ =1 . ∗ ¯Λ → π the relevant part ofthe NLO Lagrangian [25] is given by L (2)8 − → c F F π ¯Λ γ µ γ Σ ∗ ν (cid:0) ∂ µ π + ∂ ν π − − ( µ ↔ ν ) (cid:1) . (83)A vector-meson-dominance estimate for c F is provided inAppendix D. V. RESULTSA. Matrix elements
The first step is the calculation of the pion-hyperontree-level amplitudes, i.e. χ PT amplitudes up to (includ-ing) NLO. In practice, the extraction of the reduced am-plitudes is simplified and systemized by a projector for-malism presented in Appendix B.2The Feynman matrix element for the reaction Σ ∗ ¯Λ → π + π − up to (including) NLO is given by − Dh A √ F π t − m + i(cid:15) p µπ + g µα ¯ v Λ /p π − γ ( /p Σ ∗ − /p π + + m Σ ) u α Σ ∗ + Dh A √ F π u − m + i(cid:15) p µπ − g µα ¯ v Λ /p π + γ ( /p Σ ∗ − /p π − + m Σ ) u α Σ ∗ + h A H A √ m Σ ∗ F π i(cid:15) λναβ p ν Σ ∗ p βπ + p µπ − ¯ v Λ S µλ ( p Σ ∗ − p π + ) u α Σ ∗ − h A H A √ m Σ ∗ F π i(cid:15) λναβ p ν Σ ∗ p βπ − p µπ + ¯ v Λ S µλ ( p Σ ∗ − p π − ) u α Σ ∗ + c F F π ( p µπ + p απ − − p απ + p µπ − ) g αβ ¯ v Λ γ µ γ u β Σ ∗ . (84)Here S µν denotes the spin-3/2 propagator given in (B7).The Σ and Σ ∗ exchange diagrams yield the following amplitudes: K +1 = Dh A √ F π ( C +1 + D +1 R oct .s ) + h A H A √ F π ( E +1 + F +1 R dec .s ) ,K − = Dh A √ F π ( C − + D − R oct .s ) + h A H A √ F π ( E − + F − R dec .s ) ,K = Dh A √ F π ( C + D R oct .d ) + h A H A √ F π ( E + F R dec .d ) (85)with R oct .s = − Y Σ κ (cid:18) − (cid:18) − Y κ (cid:19) | κ | Y Σ (cid:18) arctan (cid:18) | κ | Y Σ (cid:19) + π Θ( s − s ) (cid:19)(cid:19) ,R oct .d = 4 κ (cid:18) − Y Σ | κ | (cid:18) arctan (cid:18) | κ | Y Σ (cid:19) + π Θ( s − s ) (cid:19)(cid:19) ,R dec .s = − Y Σ ∗ κ (cid:18) − (cid:18) − Y ∗ κ (cid:19) | κ | Y Σ ∗ arctan (cid:18) | κ | Y Σ ∗ (cid:19)(cid:19) ,R dec .d = 4 κ (cid:18) − Y Σ ∗ | κ | arctan (cid:18) | κ | Y Σ ∗ (cid:19)(cid:19) (86)and Y Σ = 2 m − m ∗ − m − m π + s , (87) Y Σ ∗ = m ∗ − m − m π + s , (88) κ = 1 s ( s − m π ) λ ( s, m ∗ , m ) , (89) s = m ∗ + m + 2 m π − m . (90)Note that κ is negative in the range 4 m π < s < ( m Σ ∗ + m Λ ) , i.e. | κ | = √− κ . Only for negative κ the expressions (86) are correct. For positive κ one haslog’s instead of arctan’s. Finally the coefficient functions in (85) are given by C +1 = − m Σ ∗ − m Λ ) ( m Λ + m Σ ) s − ( m Σ ∗ − m Λ ) , (91) C − = − m Σ ∗ − m Λ ) ( m Λ + m Σ ) s − ( m Σ ∗ − m Λ ) , (92) C = ( m Σ ∗ + m Λ ) ( m Σ ∗ + m Σ ) s − m Σ ∗ ( m Λ + m Σ ) s − ( m Σ ∗ − m Λ ) , (93)3 D +1 = 3 m Σ ( m Λ + m Σ ) + 3 ( m Σ ∗ − m Λ ) ( m Λ + m Σ ) ( m π + m Σ ∗ m Λ − m ) s − ( m Σ ∗ − m Λ ) , (94) D − = 3 m Σ ∗ ( m Λ + m Σ ) ( m π − m ∗ + m Σ ∗ m Σ − m ) + 9 ( m Σ ∗ − m Λ ) ( m Λ + m Σ ) ( m π + m Σ ∗ m Λ − m ) s − ( m Σ ∗ − m Λ ) , (95) D = 3 m Σ ( m Λ + m Σ )( m ∗ − m Σ ∗ m Σ − m π + m ) − m Σ ∗ ( m Λ + m Σ )( m Σ ∗ m Λ + m π − m ) s − ( m Σ ∗ − m Λ ) + 3( m Σ ∗ + m Λ )( m Σ + m Λ ) s (cid:16) m ∗ m Λ − m Σ ( m Σ ∗ − m Λ )( m ∗ + m π ) + 2 m ∗ m π − m (cid:0) m Σ ∗ ( m Σ ∗ + m Λ ) + 2 m π (cid:1) + 2 m Σ ∗ m Λ m π − m ( m Λ − m Σ ∗ ) + m π + m (cid:17) , (96) E +1 = ( m Σ ∗ − m Λ ) (cid:0) ( m Σ ∗ + m Λ ) − m π (cid:1) m Σ ∗ ( s − ( m Σ ∗ − m Λ ) ) , (97) E − = ( m Σ ∗ − m Λ ) (cid:0) ( m Σ ∗ + m Λ ) − m π (cid:1) m Σ ∗ ( s − ( m Σ ∗ − m Λ ) ) , (98) E = − ( m Σ ∗ + m Λ )(2 m ∗ + 2 m Σ ∗ m Λ − m π )6 m Σ ∗ s + ( m Σ ∗ + m Λ ) − m π s − ( m Σ ∗ − m Λ ) ) , (99) F +1 = − s − m π (2 m Σ ∗ + 3 m Λ )2 m Σ ∗ + 5( m Σ ∗ + m Λ ) m Σ ∗ − m Λ )(( m Σ ∗ + m Λ ) − m π )( m ∗ − m Σ ∗ m Λ − m π )2 m Σ ∗ ( s − ( m Σ ∗ − m Λ ) ) , (100) F − = 3 s m π ( m ∗ + m Σ ∗ m Λ − m ) + m π m ∗ − m Σ ∗ + m Λ )
2+ 3( m Σ ∗ − m Λ )(( m Σ ∗ + m Λ ) − m π )( m ∗ − m Σ ∗ m Λ − m π )2 m Σ ∗ ( s − ( m Σ ∗ − m Λ ) ) , (101) F = 3 m ∗ s − m π (7 m ∗ − m Σ ∗ m Λ + 2 m ) + m ∗ ( m Σ ∗ + m Λ ) m π + 4 m ∗ m π ( m Σ ∗ − m Λ )( m Σ ∗ + m Λ ) − m π (2 m ∗ + m ∗ m Λ + m ) + m π ( m Σ ∗ + m Λ )2 m Σ ∗ s + 3(( m Σ ∗ + m Λ ) − m π )( m Σ ∗ ( m Λ − m Σ ∗ ) + m π ) s − ( m Σ ∗ − m Λ ) ) . (102)The explicit expressions for the polynomial terms are P +1 = Dh A √ F π h A H A √ F π m Σ ∗ + m Λ )6 m Σ ∗ ,P − = Dh A √ F π m Σ ∗ − m Λ − m Σ ) m Σ ∗ + h A H A √ F π s − m π − ( m Σ ∗ + m Λ )(6 m Σ ∗ − m Λ )6 m ∗ ≈ Dh A √ F π m Σ ∗ − m Λ − m Σ ) m Σ ∗ + h A H A √ F π − ( m Σ ∗ + m Λ )(6 m Σ ∗ − m Λ )6 m ∗ ,P = Dh A √ F π + h A H A √ F π m Σ ∗ − m Λ m Σ ∗ . (103) For P − we dropped terms which are suppressed by twoorders in the chiral counting.The Σ ∗ Λ π + π − contact diagram produces the followingpolynomials: P NLO χ PT+1 = c F m Σ ∗ + m Λ F π ,P NLO χ PT0 = c F m Σ ∗ F π ,P NLO χ PT − = c F s − m Λ ( m Σ ∗ + m Λ )2 F π m Σ ∗ ≈ − c F m Λ ( m Σ ∗ + m Λ )2 F π m Σ ∗ . The amplitudes (84) become slightly different when thePascalutsa prescription is used: new contact terms pop4up but the pole terms and therefore (85) are not affected.In particular we have: P P +1 = P +1 + h A H A √ F π m ∗ (( m Λ + m Σ ∗ )(2 m Σ ∗ + 3 m Λ ) − s ) ,P P = P − h A H A √ F π ,P P − = P − − h A H A √ F π m ∗ (( m Λ + m Σ ∗ )(3 m Σ ∗ + 2 m Λ ) − s ) . (104)As expected the Σ exchange diagrams do not get anycontribution since the external Σ ∗ hyperon is on-shell.It is illuminating to translate these contact interactionsto the i = 1 , , P NLO χ PT1 = c F m Σ ∗ F π , P NLO χ PT2 , = 0 (105)and P P = P + 5 h A H A √ F π m ∗ ,P P = P − h A H A √ F π m ∗ ,P P = P . (106)Thus the NLO contact term can be used to compensatefor the difference between naive and Pascalutsa interac-tion concerning the i = 1 amplitude structure, but not for i = 2. In fact, there is a one-to-one correspondence be-tween the contact terms of the pion-hyperon scatteringamplitudes and the constraint-free TFFs introduced atthe very beginning in (2). Chiral power counting showsthat in χ PT, the TFF F i receives tree-level contributionsstarting at chiral order i +1. At an NLO accuracy, one hasonly full access to F . Correspondingly, the NLO contactinteraction for the pion-hyperon amplitudes contributesonly to P as shown explicitly in (105). To compensatethe difference between naive and Pascalutsa interactionconcerning the i = 2 amplitude structure, one needs acontact term from the next-to-next-to-leading-order La-grangian. This is beyond the scope of the present work. B. Numerical results
The results below have been obtained using Pascalutsaamplitudes. They consist in unsubtracted dispersion re- lations for the TFFs G m (64), evaluated at the photonpoint ( q = 0), followed by the corresponding radii: (cid:104) r m (cid:105) := 6 G m (0) d G m ( q )d q (cid:12)(cid:12)(cid:12)(cid:12) q =0 . (107)Other quantities of interest are the integrated decay ratefor Σ ∗ → Λ e + e − and the decay width for Σ ∗ → Λ γ .We start by fixing the input parameters h A , H A and M V to the respective central value. We will explorelater the impact of their uncertainties on the final results,while we will not vary D nor the pion phase shift sincethey are better constrained. We also want to investigatethe dependence on the cutoff Λ, which we will take equalto 1 and 2 GeV, respectively. Furthermore recall that inorder to account for the anomalous contribution (53) tothe scattering amplitudes, one needs to know the pionscattering amplitude t ( s ) in the complex plane. We willexplore two options: an approximation from χ PT (55),denoted by t χ PT , and its unitarized version (56), t IAM .Our strategy is to adjust the dimensionless constants c m ’s according to (66) and fix the NLO low-energy con-stant c F to the experimental value of the decay widthΣ ∗ → Λ γ which is 0.452 MeV [35]. In doing so one hastwo possible values ( c F = − .
33 GeV − and c F = 2 . − ) to choose from. We pick the first, being closerto the VMD estimate (D4). The chosen value of c F iskept unchanged throughout the whole analysis, while theconstants c m ’s are adjusted by (66) each time any otherparameter is varied. For completeness we report the c m values obtained when choosing Λ = 2 GeV, t IAM , centralvalues for h A , H A and M V : c − = − . − . i, c =1 . − . i, c +1 = 0 . − . i . This scenario gives riseto the results of Table I, right column.From Table I we get the encouraging message thatvarying the cutoff has a rather small impact. In TableII we compare the choice of using t χ PT versus t IAM . Asexpected both approaches lead essentially to the sameresults. Finally we study the changes of the G m (0)’s, theradii, the partial widths Γ Σ ∗ → Λ e + e − and Γ Σ ∗ → Λ γ whenvarying h A , H A and M V , one at a time, as shown in Ta-ble III. The uncertainties related to h A , H A and M V turnout to be moderate and comparable. It is satisfying toobserve that the G m (0) values are not subject to largechanges and the radii are even less sensitive to these vari-ations. In fact the dispersive machinery is supposed towork better for the radii since by definition they receivea suppressed contribution from the high-energy region.As previously stated we stick to Pascalutsa amplitudeshere, but the very same analysis can be carried out usingthe naive couplings instead. Note that it would then benecessary to refit c F since the meaning of the contactinteraction changes based on which three-point couplingis used. The results corresponding to the other choice for c F reflect thefact that the two possible values have opposite signs. Otherwisethe results are qualitatively similar. Again, the final results show similar qualitative behavior as inthe Pascalutsa case. quantity Λ = 1 GeV Λ = 2 GeV G (0) − . − . i − . − . i (cid:104) r (cid:105) [GeV − ] 21 . . i . . iG +1 (0) − . − . i − . − . i (cid:104) r (cid:105) [GeV − ] 16 . . i . . iG − (0) 3 . − . i . − . i (cid:104) r − (cid:105) [GeV − ] 16 . − . i . − . i Γ Σ ∗ → Λ e + e − [keV] 3 . . Σ ∗ → Λ γ [MeV] 0 .
39 0 . TABLE I: Comparison of the results for various observables using t IAM , c F = − .
33 GeV − , central values for h A , H A and M V and varying the cutoff Λ. quantity t χ PT t IAM G (0) − . − . i − . − . i (cid:104) r (cid:105) [GeV − ] 20 . . i . . iG +1 (0) − . − . i − . − . i (cid:104) r (cid:105) [GeV − ] 16 . . i . . iG − (0) 3 . − . i . − . i (cid:104) r − (cid:105) [GeV − ] 16 . − . i . − . i Γ Σ ∗ → Λ e + e − [keV] 3 . . Σ ∗ → Λ γ [MeV] 0 .
45 0 . TABLE II: Same as Table I for the comparison between t χ PT and t IAM using Λ = 2 GeV, c F = − .
33 GeV − , centralvalues for h A , H A and M V . quantity h A = 2 . h A = 2 . H A = 1 . H A = 2 . M V = 1 . M V = 1 . G (0) − . − . i − . − . i − . − . i − . − . i − . − . i − . . i (cid:104) r (cid:105) [GeV − ] 20 . . i . . i . . i . . i . . i . . iG +1 (0) − . − . i − . − . i − . − . i − . − . i − . − . i − . . i (cid:104) r (cid:105) [GeV − ] 16 . . i . . i . . i . . i . . i . . iG − (0) 3 . − . i . − . i . − . i . − . i . − . i . . i (cid:104) r − (cid:105) [GeV − ] 16 . − . i . − . i . − . i . − . i . − . i . − . i Γ Σ ∗ → Λ e + e − [keV] 3 . . . . . . Σ ∗ → Λ γ [MeV] 0 .
47 0 .
43 0 .
51 0 .
40 0 .
41 0 . TABLE III: Same as Table I using t IAM , Λ = 2 GeV, c F = − .
33 GeV − and varying h A , H A and M V one at a time.Still using t IAM , central values for h A , H A and M V and cutoff Λ = 2 GeV, we plot the real and imaginarypart of the TFFs G m ( q ) (64) in the space- and timelikeregions, up to q = ( m Σ ∗ − m Λ ) . As shown in Figs. 1, 2,3, all three functions are complex, already below the two-pion threshold. Technically this is a consequence of theadditional anomalous cut located on the first Riemannsheet.We plot the single differential decay width dΓ / d q forthe Dalitz decay Σ ∗ → Λ e + e − , i.e. the angular integralof (22), in the Dalitz region 4 m e < q < ( m Σ ∗ − m Λ ) . In particular in Fig. 4 we show a comparison with the corre-sponding QED case (24), for which the q -dependence ofthe TFFs is not resolved. The two curves show a slightoff-set in the central region, but essentially coincide overthe whole range. This implies that high resolution isneeded from the experimental side in order to appreci-ate this discrepancy and gain new insight on the internalstructure of hyperons.6 q [GeV ] G Re[ G ]Im[ G ] FIG. 1: Real and imaginary part of G ( q ). q [GeV ] G + Re[ G +1 ]Im[ G +1 ] FIG. 2: Real and imaginary part of G +1 ( q ). q [GeV ] G − Re[ G − ]Im[ G − ] FIG. 3: Real and imaginary part of G − ( q ). q [10 − GeV ] d Γ Σ ∗ Λ e + e − / d q [ − G e V − ] FFs ( q ) QED
FIG. 4: Single-differential decay width for the Σ ∗ → Λ e + e − Dalitz decay. The curve labeled “FFs ( q )” is theangular integral of (22). The other curve is the QEDanalogue, given by (24).In the Dalitz region it is also meaningful to plot thethree combinations of TFFs that appear in front of thetrigonometric functions in the four-body decay expres-sion (29), in order to compare their magnitude. Fig. 5shows that one of them, the linear combination of | G +1 | and | G − | is dominant, making it very challenging toextract information on the individual TFFs. Yet withsufficient statistics and angular resolution for the four-body decay Σ ∗ → Λ e + e − → pπ − e + e − one might getaccess to the smaller form factor combinations. q [10 − GeV ] | G +1 | + | G − | q /m Σ ∗ | G | q q /m Σ ∗ Im[ G G ∗− ] FIG. 5: A comparison of the three combinations of TFFsin front of the trigonometric functions in (29) for theΣ ∗ → pπ − e + e − decay.The situation might be compared to the history ofthe experimental determination of the pion-to-photonTFF and the corresponding radius from Dalitz decays7 π → γe + e − as documented in the citations of [35]. Alsothere one had to establish first the mere existence of thedecay, then the approximate agreement with the QEDcase and finally with much higher experimental effortsthe existence of a non-trivial form factor. We are lookingforward to this future endeavor for the hyperon sector. ACKNOWLEDGMENTS
SL thanks Martin Hoferichter and Bastian Kubis forvery valuable discussions on anomalous thresholds.
Appendix A: Meson vs baryon dynamics
The purpose of this appendix is to discuss the differentphysical aspects that are contained in a dispersive deter-mination of the low-energy TFFs. As an integral part ofthe main text it might distract the reader too much fromthe presentation of conceptual developments and results.Therefore we dedicate this appendix to this discussion. Σ ∗ γ ∗ Λ π πY Σ ∗ γ ∗ Λ V ππ Σ ∗ γ ∗ Λ ππ FIG. 6: One-loop diagrams contained in our approach.The shaded blob denotes the pion vector form factor.The first diagram leads to the amplitude K in (45).To understand the physical content of our approach, it might be illuminating to study a form factor on the one-loop level. This is displayed in Fig. 6. Before discussingthese diagrams, we stress that the dispersive approachcontains more than these one-loop diagrams by includ-ing in (45) the complete rescattering of pions via theirmeasured pion phase shift. The first diagram in Fig.6 displays the exchange of a hyperon Y in the crossedchannels. The second diagram shows the formation of avector meson V . The third diagram contains a contactinteraction between the hyperons and the pions. A con-tact interaction is without structure. It contributes witha polynomial to the hyperon-pion scattering amplitude.Thus the contact interaction provides a contribution tothe polynomial P in (45). For the following discussionwe call this contribution P c .Next we want to specify the relevant exchange hadrons Y and V . If such a hadron is very heavy, its pole andcut structures caused by its propagator are not resolved.It contributes effectively like a contact interaction. Thuswhat is not covered (at the one-loop level) by the third di-agram of Fig. 6 are exchanges of light hadrons. Concern-ing the baryon exchange diagrams, we have included ex-plicitly the relevant lightest baryon states from the octetand decuplet. We call the impact of these processes onthe form factors the “aspect of baryon dynamics”. Be-low we will show a calculation that focuses only on thisaspect. The second and third diagram of Fig. 6 couplethe external baryons directly to mesons. Therefore wecall the impact of these processes on the form factors the“aspect of meson dynamics”. This part might be linkedto the notion of vector meson dominance [32]. Belowwe will also show a calculation that focuses only on thisaspect. Σ ∗ γ ∗ Λ ππ ππ FIG. 7: Diagrammatic representation of all processesthat do not contain the cross-channel exchange of lightbaryons. The shaded blob with four pion legs representsthe S-matrix of pion scattering. The black dot containsthe contact interaction of the third diagram of Fig. 6 andthe strength mediated by the vector meson of the seconddiagram of Fig. 6. This black dot leads to the polynomial P in (45).Finally, let us look at the second diagram of Fig. 6 inmore detail. The dynamics of the lightest vector meson,8the ρ -meson, is automatically contained in the measuredpion phase shift because the ρ -meson couples essentiallywith 100% to a two-pion state. Diagrammatically thesecond and the third diagram of Fig. 6 are covered bythe diagram of Fig. 7.What is not automatically covered is the initial cou-pling strength with which the vector meson V couplesthe pions to the hyperons. Schematically ig BV ¯ Bγ µ γ T ν V µν + iG V [ u µ , u ν ] V µν → g BV G V M V ¯ Bγ µ γ T ν [ u µ , u ν ] , (A1)which leads to P = P c + P V with P V ∼ g BV G V M V . (A2)In Appendix D and in [59], respectively, the flavor struc-ture of (A1) is specified, which is, however, of no con-cern for our qualitative discussion. We will show belowin more detail how the dynamics contained in the seconddiagram of Fig. 6 emerges from the dispersive frameworkby translating and simplifying this framework to the vec-tor meson dominance language.The result of the present discussion is that our dis-persive framework contains all processes of Fig. 6 if thecontact interaction strength ∼ P of Fig. 7 is determinedby a fit to experiment. Without further theory input,this needs to be done separately for each form factor. Ifwe need to estimate the size of P on the theory side, wemust include the influence of vector mesons as shown in(A2) and carried out in Appendix D. In this context, wenote that a pion-hyperon contact term of a given orderin χ PT leads to a contribution of the same order for theform factor. To be concrete, the TFFs F i of (2) start atsecond, third and fourth chiral order for i = 1 , ,
3, re-spectively. Correspondingly, to fully account for the con-tribution of the ρ -meson to the TFF F i requires a pion-hyperon contact interaction from the chiral Lagrangianof ( i + 1)th order. With our present NLO input, we havea full coverage of F only. In turn, F constitutes theleading contribution to the TFFs G ± in (8), (10). Inaddition, our formalism contains the pertinent contribu-tions to all TFFs from the baryon dynamics induced bythe first diagram of Fig. 6.Baryon form factors are influenced by meson dynamicsand by baryon dynamics. Therefore it might be illumi-nating to disentangle the meson and the baryon dynamicsby switching off one of the two aspects. This will be dis-cussed in the following two subsections. Yet we wouldlike to stress that both cases miss part of the physics.
1. Pure meson dynamics
To focus on the impact of pion rescattering, we switchoff the aspect of baryon dynamics, i.e. we put K → T in (45). Consequently, we put f → c m →
0. Thus we obtain finally G pure meson ( q ) = P π ∞ (cid:90) m π d sπ Ω( s ) p . m . ( s ) F V ∗ π ( s ) s / ( s − q − i(cid:15) ) . (A3)Since the subtraction constant = contact interactionstrength P is just a number that does not influence the s dependence of the integrand, we have not specified theTFF by any label. We show its generic form in Fig. 8a,in the unphysical region between the two-pion thresholdand 1 GeV . Obviously, the TFF displays the influenceof the ρ -meson, i.e. the mesonic aspects are very wellcovered. An unphysical aspect emerges from the factthat the imaginary part of the TFF vanishes below thetwo-pion threshold. In reality, the TFF is complex ev-erywhere, since the Σ ∗ is unstable. This is, however,hardly visible in the full results for G − in Fig. 8b, sincethe imaginary part at q ≈ m π becomes extremely tiny.From the comparison of Figs. 8a, 8b, one sees that if oneadjusted the ρ -peak of the imaginary parts to the samesize, then the peak of the real part of the full calculationwill be somewhat smaller than the one of the pure-mesoncalculation. Moreover at low energies the curvature inthe real part of the full G − is milder with respect to thepure-meson calculation.A relation to strict vector meson dominance can bededuced from (A3). Suppose that the width of the vec-tor meson is small. Essentially this means that thepion phase shift is zero below the vector meson massand π above. This leads to Ω( s ) = m ρ / ( m ρ − s ).With a slight refinement, m ρ − s → m ρ − s − im ρ Γ ρ ,one obtains Ω( s ) F V ∗ π ( s ) ∼ δ ( s − m ρ ) / Γ ρ and therefore G pure meson ( q ) ∼ P/ Γ ρ · / ( m ρ − q ). The appearanceof the ratio P/ Γ ρ has a natural interpretation: In a vec-tor meson dominance picture the contact term ∼ P forthe hyperon-pion scattering amplitude emerges from in-tegrating out the vector meson, see (A1), (A2). Thisleads to P ∼ g Bρ G V . On the other hand, in strict vectormeson dominance, the coupling of the vector meson tothe pions must be adjusted such that the correct elec-tric charge of the pion emerges that is independent ofstrong-interaction coupling constants, i.e. G V ∼ /F V where F V denotes the coupling strength with which thephoton couples to the vector meson. Thus one finds P/ Γ ρ ∼ P/G V ∼ g Bρ /G V ∼ g Bρ F V . This is exactlywhat one expects as the coefficient of a form factor ob-tained in the vecor meson dominance framework, i.e. infull analogy to (A1) one finds ig BV ¯ Bγ µ γ T ν V µν + F V f µν + V µν → i g BV F V M V ¯ Bγ µ γ T ν f µν + . (A4)9 q [GeV ] G pu r e m e s o n Re[ G ]Im[ G ] (a) q [GeV ] G − Re[ G − ]Im[ G − ] (b) FIG. 8: Pure mesonic contribution to the TFFs (a) in arbitrary units, compared with the full G − ( q ) described bymeson and baryon dynamics together (b), in the unphysical timelike region 4 m π < q < .Thus the dispersive framework reproduces and refinesthe vector meson dominance aspects. The only input oneneeds is the initial strength (contact interaction) withwhich the pions couple to the baryons. In the vector me-son dominance setup this is obtained by integrating outthe vector meson. One can also rephrase it in the fol-lowing way: If one integrates out the vector meson forthe interaction terms between the vector meson and thebaryons and between the vector meson and the photon,one obtains a photon-baryon coupling right away. Thedispersive framework produces the same with the pionsas intermediate agents. Of course, the dispersive frame-work based on data for the pion phase shift and the pionvector form factor is more acurate than the schematicand model-dependent vector meson dominance scenariobut it covers qualitatively the same physics. In addition,the dispersive framework presented in this paper containsalso the aspects of baryon dynamics that is completelymissing in the vector meson dominance approach.
2. Pure baryon dynamics
To focus on the impact of the processes where baryonsare exchanged in the cross channel, we switch off the con-tact interaction and the pion rescattering. For simplicitywe use an unsubtracted dispersion relation. Thus we put F Vπ → c m →
0. For the calculation of the scatter-ing amplitude T we put P → δ → T m → K m in (64). This leads to G pure baryon m ( q ) = 112 π Λ (cid:90) m π d sπ K m ( s ) p . m . ( s ) s / ( s − q − i(cid:15) )+ 112 π (cid:90) d x ds (cid:48) ( x ) dx s (cid:48) ( x ) − q × f m ( s (cid:48) ( x )) s (cid:48) ( x ) − − λ ( s (cid:48) ( x ) , m ∗ , m )) / . (A5)In Fig. 9a this contribution is plotted for the TFF G − ( q ), in the range − < q [GeV ] <
1. As expectedthe form factor has an imaginary part for all values of q , even if very tiny for q <
0. The baryon exchangediagrams contain the physical aspect that the Σ ∗ is un-stable. What is missing, of course, is the influence ofthe ρ -meson, i.e. the mesonic aspects. For comparisonwe show in Fig. 9b also the complete result for G − ( q ),taking into account the contributions of both meson andbaryon dynamics, again across the space- and timelike re-gions. Note that the same quantity has been previouslyplotted (Fig. 8b) but in a different range. This time weinclude the negative q physical region and focus on theregion around q = 0. There we notice in Fig. 9a a steeprise in the real part, which is mitigated in Fig. 9b by the ρ -meson tail. In summary, even if the ρ -meson dictatesin general the shape and size of the form factor, the low-energy behavior is significantly influenced by the baryondynamics.0 q [GeV ] G − , pu r e b a r y o n Re[ G − ]Im[ G − ] (a) q [GeV ] G − Re[ G − ]Im[ G − ] (b) FIG. 9: Pure baryonic contribution to G − ( q ) (a) as compared to the full result for G − ( q ) described by meson andbaryon dynamics together (b), in the range − < q [GeV ] < Appendix B: Projector formalism for helicityamplitudes
Spin-3/2 objects can be obtained from the coupling ofspin-1/2 and spin-1 states. Thus we construct a spin-3/2vector-spinor [60] by u µ ( p, σ ) = (cid:88) ρ,λ (cid:18) , σ (cid:12)(cid:12)(cid:12)(cid:12) , ρ ; 12 , λ (cid:19) u ( p, λ ) ε µ ( p, ρ ) (B1)with a spin-1/2 spinor u , a spin-1 polarization vector ε µ and a Clebsch-Gordan coefficient ( J, M | j , m ; j , m ).Here in slight contrast to the rest of this work the spinprojections on a given quantization axis (and not the he-licities) are denoted by σ , λ and ρ , respectively. Yet ifone chooses the quantization axis along the flight direc-tion (as we will do in a moment) then helicity and spinprojection coincide.It is useful to provide (B1) in an explicit form: u µ ( p, ± /
2) = u ( p, ± / ε µ ( p, ± ,u µ ( p, ± /
2) = 1 √ u ( p, ∓ / ε µ ( p, ± √ √ u ( p, ± / ε µ ( p, . (B2)For the spin-1/2 spinors we use the conventions of [38].For the spin-1 polarization vectors for massive states weprovide here only their explicit form for the case wherethe z -direction constitutes both the spin quantization axis and the direction of motion of the particle [61, 62]: ε µ ( p z , ±
1) = ∓ √ , , ± i, ,ε µ ( p z ,
0) = ( p z /m, , , E/m ) (B3)where m denotes the mass of the particle and E its en-ergy. Note that the coefficient ∓ (cid:88) σ u µ ( p, σ ) ¯ u ν ( p, σ ) = − ( /p + m ) P / µν ( p ) (B4)where p = (cid:112) m + (cid:126)p denotes the energy of the parti-cle described by the vector-spinor and m its mass. Theprojector on spin 3/2 is defined by P / µν ( p ) := g µν − γ µ γ ν − p ( /p γ µ p ν + p µ γ ν /p ) . (B5)Note that for (B4) the scalar product p appearing in(B5) can be replaced by m .For our Lagrangian (67) a spin-3/2 (vector-spinor) field ψ µ ( x ) has the following propagator [63, 64] (cid:104) | T ψ µ ( x ) ¯ ψ ν ( y ) | (cid:105) = (cid:90) d p (2 π ) i S µν ( p ) e − ip ( x − y ) (B6)with S µν ( p ) := − /p + mp − m + i(cid:15) P / µν ( p ) + 23 m ( /p + m ) p µ p ν p − m p µ p α γ αν + γ µα p α p ν p . (B7)1Note that for the propagator of (B6), (B7) the scalarproduct p appearing in (B5) cannot be replaced by m .The propagator (B7) describes propaging modes of spin3/2 and frozen modes of spin 1/2 [63].The scattering amplitudes for the reaction Σ ∗ ¯Λ → π + π − have the following structure:¯ v Λ ( p Λ , λ ) M µ ( p Σ ∗ , p Λ , k ) u µ Σ ∗ ( p Σ ∗ , σ ) (B8)with k := p π + − p π − . Feynman rules can provide a quitelengthy expression for the spinor 4 × M µ . There-fore we aim at a projector formalism where (B8) is relatedto scalar quantities a i and a pre-defined set of spinor ob-jects such that only the scalar quantities depend on theexplicit form of M µ , i.e.¯ v Λ M µ u µ Σ ∗ = (cid:88) i a i ¯ v Λ M µi g µν u ν Σ ∗ . (B9)The tasks are to construct a complete set of linearly in-dependent structures M µi and to find a convenient wayto determine a i from an arbitrary M µ . Such an endeavoris similar in spirit to [62].Due to parity symmetry we can focus on the casewhere the ¯Λ has positive helicity, λ = +1 /
2. Thenwe need four pre-defined spinor objects correspondingto the possible values for the helicity of the Σ + baryon, σ = +3 / , +1 / , − / , − /
2. It is convenient to intro-duce the following four-vectors: q := p Σ ∗ + p Λ , ¯ k := p Σ ∗ − p Λ ,r := ¯ k − ¯ k · qq q , k ⊥ := k − k · rr r . (B10)In the center-of-mass frame with the three-momentum ofthe Σ ∗ pointing in the z -direction and the reaction takingplace in the x - z plane one finds that q has only a zerothcomponent, r has only a third ( z ) component and k ⊥ hasonly a first ( x ) component.We are looking now for four independent structures oftype M µ in (B8). In general, M µ contains products of γ matrices and exactly one γ . All γ matrices that arecontracted with p Λ , p Σ ∗ or the spin-3/2 spinor u Σ ∗ canbe moved towards ¯ v Λ or u Σ ∗ and eliminated by equationsof motion. This results in structures with less many γ matrices. If two γ matrices are contracted with eachother or with the very same four-momentum, then onecan also simplify the expression.This whole procedure leaves us with four independentstructures of M µ type: γ k µ ⊥ , γ p µ Λ , /k ⊥ γ p µ Λ , /k ⊥ γ k µ ⊥ . (B11) Note that alternatively to a γ one might involve a Levi-Civitasymbol. However, this can be related to one γ and a product of γ matrices. It is simpler, however, to use the following linear combi-nations: M µ := (cid:0) q − ( m Σ ∗ + m Λ ) (cid:1) γ k µ ⊥ − m Σ ∗ /k ⊥ γ p µ Λ ,M µ := γ p µ Λ ,M µ := /k ⊥ γ p µ Λ ,M µ := (cid:0) q − ( m Σ ∗ − m Λ ) (cid:1) /k ⊥ γ k µ ⊥ − m Σ ∗ k ⊥ γ p µ Λ . (B12)They are constructed such that in the center-of-massframe they satisfy¯ v Λ ( p Λ , +1 / M µi g µν u ν Σ ∗ ( p Σ ∗ , σ ) ∼ δ i i Σ ∗ (B13)with i Σ ∗ := 5 / − σ . In other words, each M µi contributesonly to one helicity amplitude. Thus the sum in (B9)reduces to only one term.The remaining task is to find the scalar quantity a i for a given M µ . What makes the task non-trivial is thefact that different M µ lead to the same a i because of theequations of motion for ¯ v Λ and u ν Σ ∗ . Therefore we con-struct on- and off-shell projectors to decompose a com-pletely general M µ . Since M µ is a 4 × µ ranging from 0 to 3 we need in general a basisof 64 Lorentz-spinor structures. Due to parity symmetrywe can restrict ourselves to 32 structures. For the first 4terms we use T iµ := P Λon M νi P Σ ∗ on P / νµ (B14)introducing the projectors [63] P Λon := 12 m Λ ( m Λ − /p Λ ) , P Λoff := 12 m Λ ( m Λ + /p Λ ) ,P Σ ∗ on := 12 m Σ ∗ ( m Σ ∗ + /p Σ ∗ ) ,P Σ ∗ off := 12 m Σ ∗ ( m Σ ∗ − /p Σ ∗ ) ,P / µν := 13 γ µ γ ν + 13 p ∗ ( /p Σ ∗ γ µ g να + g µα γ ν /p Σ ∗ ) p α Σ ∗ ,P / µν := g µν − P / µν . (B15)The other 28 structures are obtained from (B14) by ex-changing P Λon by P Λoff and/or P Σ ∗ on by P Σ ∗ off and/or P / νµ by P / νµ . We do not specify how we enumerate thesestructures from i = 5 to i = 32 because we will not needthem in the end. We also introduce the Dirac adjointstructures¯ T iµ := γ ( T iµ ) † γ , for i = 1 , . . . , . (B16)Provided that the T iµ form 32 linearly independent struc-tures we can decompose any M µ as M µ = (cid:88) i =1 a i T iµ (B17)with a i = (cid:88) j =1 ( C − ) ij Tr( ¯ T jµ M µ ) (B18)2and the 32 ×
32 matrix C with elements C ij := Tr( ¯ T iµ T jν ) g µν . (B19)Here Tr denotes the spinor trace. We have checked ex-plicitly that det C (cid:54) = 0 which shows that the 32 structures T iµ are linearly independent, i.e. form a basis to constructthe most general M µ .Inserting (B17) in (B8) and using the equations of mo-tion for the spinors shows¯ v Λ M µ u µ Σ ∗ = (cid:88) i =1 a i ¯ v Λ M µi g µν u ν Σ ∗ . (B20)Thus we only need to determine the four scalar quantities a i with i = 1 , , , T iµ T jν ) g µν = 0 for i > , j = 1 , , , . (B21)In addition, we have checked by an explicit calculation C ij ∼ δ ij for i, j = 1 , , , . (B22)A result that one could have anticipated already from(B13). Finally (B18) simplifies to a i = Tr( ¯ T iµ M µ ) C i (B23)with C i := Tr( ¯ T iµ T iν ) g µν . (B24)Explicit expressions are given by C := k ⊥ m Σ ∗ m Λ (cid:0) ( m Σ ∗ + m Λ ) − q (cid:1) λ ( q , m ∗ , m ) ,C := − m ∗ m Λ (cid:0) ( m Σ ∗ − m Λ ) − q (cid:1) λ ( q , m ∗ , m ) ,C := k ⊥ m ∗ m Λ (cid:0) ( m Σ ∗ + m Λ ) − q (cid:1) λ ( q , m ∗ , m ) ,C := − ( k ⊥ ) m Σ ∗ m Λ (cid:0) ( m Σ ∗ − m Λ ) − q (cid:1) λ ( q , m ∗ , m )(B25)with the K¨all´en function defined in (12). In the center-of-mass frame one finds k ⊥ = − p . m . sin θ (B26)with the center-of-mass momentum of the pions p c . m . := (cid:112) q − m π / θ denoting the angle between thethree-momenta of Σ ∗ and π + .To summarize, for a given amplitude structure M µ anda given helicity σ we find in the center-of-mass frame¯ v Λ ( p Λ , +1 / M µ u µ Σ ∗ ( p Σ ∗ , σ )= Tr( ¯ T iα M α ) C i ¯ v Λ ( p Λ , +1 / M µi g µν u ν Σ ∗ ( p Σ ∗ , σ ) (B27) with i = 5 / − σ . Note in particular that in (B27) thereis no implicit summation over i , it is fixed by the choiceof σ , the helicity of the Σ ∗ .In the main text we have introduced reduced am-plitudes (44) for the dispersive representation of theTFFs. To make contact with these reduced amplitudeswe present here the ratios¯ v Λ ( − p z , +1 / M µ g µν u ν Σ ∗ ( p z , +3 / v Λ ( − p z , +1 / γ u ∗ ( p z , +3 / p c . m . = − θ (cid:0) q − ( m Σ ∗ + m Λ ) (cid:1) , ¯ v Λ ( − p z , +1 / M µ g µν u ν Σ ∗ ( p z , +1 / v Λ ( − p z , +1 / γ u ∗ ( p z , +1 / p c . m . = 2 q m ∗ − m + q p z p c . m . , ¯ v Λ ( − p z , +1 / M µ g µν u ν Σ ∗ ( p z , − / v Λ ( − p z , +1 / γ u ∗ ( p z , − / p c . m . = − θ q − ( m Σ ∗ + m Λ ) m Σ ∗ (B28)with p z := λ / ( q , m ∗ , m ) / (2 (cid:112) q ) denoting thecenter-of-mass momentum of Σ ∗ and ¯Λ. Note that for the M case (non-flip amplitude) there will always be a factor p z p c . m . from the partial-wave projection of Tr( ¯ T α M α ).Together with the last ratio on the right-hand side of thecorresponding equation in (B28) this leads to an expres-sion for the reduced amplitude without any square roots.In practice the whole task of dealing with a Feynmanscattering amplitude for given helicities is reduced to thecalculation of one spinor trace Tr( ¯ T iα M α ). Appendix C: Dispersive representations, cuts andanomalous thresholds
This appendix has two purposes. First, we providea detailed discussion of the analytic structure of a scalartriangle diagram. This resembles the first diagram shownin Fig. 6, except that one deals with p-waves there andwith s-waves in the scalar case. But the appearance ofanomalous thresholds has the very same pattern. There-fore we use the scalar triangle as a test case to check thatwe include all bits and pieces in the correct way for ourTFF calculations. The second purpose is the derivationof (53), (60), and (63).3Consider the result of a triangle loop diagram [44, 65–67], C ( s ) = 1 iπ (cid:90) d l l − m ) (( l + p ) − m π ) (( l − p ) − m π ) , (C1)which can be calculated directly when rewritten as C ( s ) = (cid:90) d x dx dx δ (1 − x − x − x ) (cid:2) x x m + x x m + x x s − x m π − x m π − x m (cid:3) − (C2)with s := ( p + p ) , m := p and m := p . We considerfirst the case that m is large enough and m and m are small enough. A quantitative specification will followlater. In this case, the imaginary part of C (for real valuesof s ) is just given by cutting [68] the two pion lines of theFeynman diagram. The result isIm C ( s ) = − π σ ( s ) κ ( s ) log Y ( s ) + κ ( s ) Y ( s ) − κ ( s ) Θ( s − m π ) (C3)where Y ( s ) := s + 2 m − m − m − m π , (C4) κ ( s ) := λ / ( s, m , m ) σ ( s ) , (C5)and σ ( s ) := (cid:114) − m π s . (C6)We use the log and the square root function both with acut on the real negative axis.The triangle function C can be represented by a disper-sive integral in the variable s ranging from the two-pionthreshold to infinity (unitarity cut): C ( z ) = ∞ (cid:90) m π ds (cid:48) π Im C ( s (cid:48) ) s (cid:48) − z = ∞ (cid:90) m π ds (cid:48) π σ ( s (cid:48) ) l ( s (cid:48) ) s (cid:48) − z (C7)with l ( s ) := − πκ ( s ) log Y ( s ) + κ ( s ) Y ( s ) − κ ( s ) . (C8)Here z is an arbitrary complex number that does not lieon the unitarity cut, i.e. z (cid:54)∈ [4 m π , + ∞ [.It should be possible to find a dispersive representationof the function C for any values of the masses. But it isnecessary to study the cut structure of the logarithm in(C8). If this cut intersects with the unitarity cut, oneneeds a proper analytic continuation of the logarithmalong the unitarity cut and one picks up an anomalouscontribution.To understand these statements, we consider first thecase where (C7) works. In this case, l ( s ) from (C8) is asmooth function along and in the vicinity of the unitaritycut. Concerning the function σ ( s ), it has a cut for s ∈ [0 , m π ]. It is convenient to define a function that has acut along the unitarity cut [69]:ˆ σ ( z ) := (cid:114) m π z − . (C9)For s ∈ [4 m π , + ∞ [ it satisfiesˆ σ ( s ± i(cid:15) ) = ∓ iσ ( s ) . (C10)By construction, the function C ( z ) is defined via (C7)in the whole complex plane except for the unitarity cut.This cut defines a second Riemann sheet. We construct afunction C II ( z ) that constitutes an analytic continuationof C through the cut. For s ∈ [4 m π , + ∞ [ we demand C II ( s + i(cid:15) ) ! = C ( s − i(cid:15) ) = C ( s + i(cid:15) ) − iσ ( s ) l ( s )= C ( s + i(cid:15) ) + 2ˆ σ ( s + i(cid:15) ) l ( s + i(cid:15) ) . (C11)In the last step we have used (C10) and the assumptionthat l is a smooth function around the unitarity cut. Thisassumption will be critically reviewed below.For the case at hand, we can use (C11) to define ananalytic continuation of C on the second Riemann sheet: C II ( z ) := C ( z ) + 2ˆ σ ( z ) l ( z ) . (C12)The cut structure of C II originates from the unitarity cut,from the additional cut of ˆ σ along the negative real axisand from the cut of the logarithm in the expression (C8)for the function l . We note that the square root functionsthat define κ in (C5) and therefore enter (C8) do not cause an additional cut because l is an even function in κ . Let us first focus on the unitarity cut. For s ∈ [4 m π , + ∞ [, we find C II ( s − i(cid:15) ) = C ( s − i(cid:15) ) + 2ˆ σ ( s − i(cid:15) ) l ( s )= C ( s + i(cid:15) ) . (C13)Thus the unitarity cut connects just the two Riemannsheets.Next we focus on the log function. The branch pointsof the logarithm in (C8) are given by Y ( s ) = κ ( s ).4They are located at s ± = − m + 12 (cid:0) m + m + 2 m π (cid:1) − m m − m π ( m + m ) + m π m ∓ λ / ( m , m , m π ) λ / ( m , m , m π )2 m = 14 m (cid:104) ( m − m ) − (cid:16) λ / ( m , m , m π ) ± λ / ( m , m , m π ) (cid:17) (cid:105) . (C14)The problem is that as a function of the masses, the val-ues of s ± move through the complex plane that consti-tutes the second Riemann sheet. If any of the two branchpoints hits the unitarity cut then this branch point moveson the physical (=first) Riemann sheet. To be specific,we take s + as the solution that has a positive imaginarypart for small values of m . If one replaces m by m + i(cid:15) and follows the motion of s + for increasing values of m ,then s + moves towards the real axis and could intersectwith the unitarity cut. Fig. 10 shows the trajectory of s + in the complex plane obtained by varying m , havingfixed m = m and m = m . Note the intersectionwith the unitarity cut, which implies that an additionalcut must be located on the first Riemann sheet. The reddot indicates the actual position of s + for the physicalchoice m = m ∗ . Re[ s + ][GeV ] I m [ s + ][ G e V ] s + trajectoryunitarity cut FIG. 10: Real and imaginary part of s + obtained byvarying m . The red dot corresponds to m = m ∗ ,which is our case of interest.Indeed, for m + m − m π − m = 0 (cross point) (C15) For completeness we note that s − does not intersect with the uni-tarity cut and therefore does not enter the first Riemann sheet. the two K¨all´en functions in (C14) become identical andone finds at this point s + | cross point = 4 m π , ∂s + ∂m (cid:12)(cid:12)(cid:12)(cid:12) cross point = 0 ,∂ s + ∂ ( m ) (cid:12)(cid:12)(cid:12)(cid:12) cross point = 2 m π λ ( m , m , m π ) . (C16)Therefore we obtain s + ( m + i(cid:15), . . . ) (cid:12)(cid:12) cross point ≈ (cid:20) s + ( m , . . . ) + i(cid:15) ∂s + ( m , . . . ) ∂m − (cid:15) ∂ s + ( m , . . . ) ∂ ( m ) (cid:21) cross point = 4 m π − (cid:15) m π λ ( m , m , m π ) . (C17)In other words, the motion of s + just turns around (van-ishing derivative) at the two-pion threshold. s + intersectswith the unitarity cut if λ ( m , m , m π ) < . (C18)One can already see in the original expression (C8)that something goes wrong if m becomes so large that(C15) is satisfied. On the real axis, the log in (C8) isill-defined for Y ( s ) = 0. From (C4) we see that this zeroof Y is small as long as m and m are small and m is large. But the zero of Y reaches the unitarity cut, i.e.the branch point at the two-pion threshold for (C15). Foreven larger values of m , i.e. for m + m − m π − m > , (C19)one needs a smooth analytic continuation of the loga-rithm along the unitarity cut. Otherwise, the dispersiverepresentation (C7) does not make sense. In addition,(C7) is incomplete, because one has to circumvent alsothe branch point s + , which is now on the physical Rie-mann sheet. It is convenient to choose the branch thatstarts at s + such that it intersects with the unitarity cutjust at its own branch point at the two-pion threshold[67].The two conditions for s + being located on the firstRiemann sheet are (C18) and (C19). The dispersive rep-resentation of the triangle function (C1) is then givenby C ( s ) = 12 πi (cid:90) d s (cid:48) disc C ( s (cid:48) ) s (cid:48) − s = 12 πi ∞ (cid:90) m π d s (cid:48) disc unit C ( s (cid:48) ) s (cid:48) − s + 12 πi (cid:90) d x d z ( x )d x disc anom C ( z ( x )) z ( x ) − s (C20)5with the straight-line path connecting s + and the two-pion threshold, z ( x ) := (1 − x ) s + + x m π , (C21)the functiondisc anom C ( z ) = − π i ( − λ ( z, m , m )) / , (C22)and a piecewise defined function given by (cf. (C3))disc unit C ( s )2 i = (C23) − π σ ( s ) κ ( s ) (cid:20) log Y ( s ) + κ ( s ) Y ( s ) − κ ( s ) + 2 πi Θ (cid:0) ( m − m ) − s (cid:1)(cid:21) for λ ( s, m , m ) > unit C ( s )2 i = − π σ ( s )˜ κ ( s ) (cid:20) arctan ˜ κ ( s ) Y ( s ) + π Θ( − Y ( s )) (cid:21) (C24) for λ ( s, m , m ) <
0. This function is continuous alongthe unitarity cut except if s = ( m − m ) lies on thecut; there one has an integrable divergence. We haveintroduced ˜ κ ( s ) := ( − λ ( s, m , m )) / σ ( s ) . (C25)In Fig. 11a the real and imaginary part of the trianglefunction (C1) are plotted using m = m Σ ∗ , m exch = m Σ and m = m Λ . We have checked that the dispersiverepresentation (C20) for s + i(cid:15) with arbitrary real s fullyagrees with the direct calculation (C2). We want to stressthat ignoring the integration along the anomalous cutproduces a very incomplete result, shown in Fig. 11b.Having established the correct analytic structure, weleave the case of the scalar triangle behind and turn toour TFFs, which have a different partial-wave structureand include the full pion rescattering. s [GeV ] ReIm (a) s [GeV ] ReIm (b)
FIG. 11: Comparison between (a) the triangle function (C1), obtained by either (C2) or (C20), and (b) its incompletedispersive representation (C7), where only the unitarity cut has been taken into account, neglecting the presence ofthe anomalous cut. The masses involved here are m = m Σ ∗ , m exch = m Σ and m = m Λ .For triangle diagrams with full two-pion rescatteringwe extend the usual formulae to allow for the presenceof the anomalous cuts. We introduce the values of afunction A to the left ( A + ) and to the right ( A − ) of a(directed) cut line. The discontinuity of A is then definedby disc A := A + − A − . (C26)For a cut along the real axis this yields the well-known relations disc A ( s ) = A ( s + i(cid:15) ) − A ( s − i(cid:15) )= A ( s + i(cid:15) ) − A ∗ ( s + i(cid:15) )= 2 i Im A ( s + i(cid:15) ) . (C27)The optical theorem that leads to (61) and (45) gen-eralizes to disc F = 2 i π T + σp F Vπ − (C28)6and disc( T − K ) = 2 iT + σt − (C29)with the p-wave pion scattering amplitude t . Along theunitarity cut, the amplitude t is given by t = sin δ e iδ /σ .We recall how (C29) is solved [43]. The Omn`es func-tion is introduced as a solution ofdiscΩ = 2 i Ω + σt − . (C30)This allows to calculatedisc T − K Ω = ( T − K ) + Ω − − ( T − K ) − Ω + Ω + Ω − = ( T − K ) + Ω − − ( T − K ) + Ω + Ω + Ω − + ( T − K ) + Ω + − ( T − K ) − Ω + Ω + Ω − = 2 iK + σt − Ω − . (C31)The product Kσ is essentially proportional to the discon-tinuity of the triangle function C . The proportionalityfactor h is a rational function of s , i.e. has no cuts. Withthe previous construction of disc C we have achieved thatthe two cuts (unitarity cut and anomalous cut) do notintersect. Therefore we can write for the discontinuity(C31) along the unitarity cutdisc T − K Ω = 2 iKσt − Ω − = 2 iK sin δ | Ω | (C32)because here K is by construction a continuous functionand Ω has the same phase as t . This leads to the standarddispersive part for T − K explicitly given in (45). Alongthe anomalous cut we havedisc T − K Ω = 2 i h disc anom
C t − Ω − , (C33)which leads to (53).Finally we have to solve (C28). For the unitarity cutwe can just integrate the right-hand side of (C28). Forthe anomalous cut we use (C29) and finddisc anom F = 124 π disc anom ( T − K ) p F Vπ − t − . (C34)Since we have a dispersive representation for T − K in(45), (53) we just need to read off the discontinuity alongthe anomalous cut. This leads to (63).If one compares the expressions (53) and (63), one no-tices that (53) looks more complicated with Ω appearingoutside and inside of the integral. Isn’t it possible towrite (53) in a simpler way? After all, Ω is continuousalong the anomalous cut. From (C33) one sees that thediscontinuity of T − K along the anomalous cut is in-deed just 2 i h disc anom C t . The same can be obtainedfrom (53). But the expression (53) inherits from Ω also a discontinuity along the unitarity cut. Therefore a directdispersive representation of T − K instead of the ratio( T − K ) / Ω leads to an integral where in the integrandthe integral of (53) appears. For the form factor we havethis situation of a double integral anyway in (61) wherethe integral expressions for T enter in the integrand. Butfor T itself one can avoid the double integral representa-tion if one lives with Ω appearing outside and inside ofthe integrals. Appendix D: Estimate for the NLO four-pointpion-baryon coupling constant
Ideally the low-energy constant c F from (83) shouldbe determined from experiment. To have a rough es-timate for its size we apply a vector-meson-dominance(resonance-saturation) assumption [32, 59, 70, 71]. Toget a feeling for its accuracy we will make the same es-timate for the octet sector. To this end, we consider thefollowing part of the NLO Lagrangian of [25]: L (2) V := i c M ( O µν ) bd ( f µν + ) db + h . c . + 14 c F ( O µν ) bd ([ u µ , u ν ]) db + h . c . + b M,D tr( ¯ B { f µν + , σ µν B } )+ i b , tr( ¯ B { [ u µ , u ν ] , σ µν B } ) (D1)with ( O µν ) bd := (cid:15) ade ¯ B ec γ µ γ T abcν . (D2)Estimates for c M and b M,D have been provided in [25],based on fits to experimental data on radiative decaysand magnetic moments, respectively: | c M | ≈ . − and b M,D ≈ .
32 GeV − .Vector-meson dominance [32] implies that the couplingstrengths of hadrons to two pions (in a p-wave) andto photons are correlated. In the χ PT framework thismight be rephrased as the statement that the two build-ing blocks [ u µ , u ν ] and f µν + appear in a fixed combination,i.e. as the chiral field strength [59, 70]Γ µν := 14 [ u µ , u ν ] − i f µν + . (D3)Under this assumption, we obtain the following esti-mates: c F ≈ − c M and b , ≈ b M,D . In [71], based ona resonance-saturation approach, the vector-meson con-tribution to the parameter b , (denoted by b therein)has been estimated to ≈ . − , i.e. about 50% largerthan our value for b M,D . Therefore, we use as an esti-mate: | c F | = (4 . ± .
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