A chiral covariant approach to \(\varvec{\rho \rho }\) scattering

The European Physical Journal C, Jul 2017

We analyze vector meson–vector meson scattering in a unitarized chiral theory based on a chiral covariant framework restricted to \(\rho \rho \) intermediate states. We show that a pole assigned to the scalar meson \(f_0(1370)\) can be dynamically generated from the \(\rho \rho \) interaction, while this is not the case for the tensor meson \(f_2(1270)\) as found in earlier work. We show that the generation of the tensor state is untenable due to the extreme non-relativistic kinematics used before. We further consider the effects arising from the coupling of channels with different orbital angular momenta which are also important. We suggest to use the formalism outlined here to obtain more reliable results for the dynamical generation of resonances in the vector–vector interaction.

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A chiral covariant approach to \(\varvec{\rho \rho }\) scattering

Eur. Phys. J. C A chiral covariant approach to ρρ scattering D. Gülmez 2 U.-G. Meißner 1 2 J. A. Oller 0 0 Departamento de Física, Universidad de Murcia , 30071 Murcia , Spain 1 Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics, Forschungszentrum Jülich , 52425 Jülich , Germany 2 Helmholtz-Institut für Strahlenund Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn , 53115 Bonn , Germany We analyze vector meson-vector meson scattering in a unitarized chiral theory based on a chiral covariant framework restricted to ρρ intermediate states. We show that a pole assigned to the scalar meson f0(1370) can be dynamically generated from the ρρ interaction, while this is not the case for the tensor meson f2(1270) as found in earlier work. We show that the generation of the tensor state is untenable due to the extreme non-relativistic kinematics used before. We further consider the effects arising from the coupling of channels with different orbital angular momenta which are also important. We suggest to use the formalism outlined here to obtain more reliable results for the dynamical generation of resonances in the vector-vector interaction. 1 Introduction It is now commonly accepted that some hadron resonances are generated by strong non-perturbative hadron–hadron interactions. Arguably the most famous example is the (1405), which arises from the coupled-channel dynamics of the strangeness S = −1 ground -state octet meson–baryon channels in the vicinity of the π and K − p thresholds [1]. This resonance also has the outstanding feature of being actually the combination of two near poles, the so-called two-pole nature of the (1405). In a field-theoretic sense, one should consider this state as two particles. This fact was predicted theoretically [2,3] and later unveiled experimentally [4] (see also the discussion in Ref. [5]). Another example is the scalar meson f0(980) close to the K¯ K threshold, which is often considered to arise due to the strong S-wave interactions in the π π –K¯ K system with isospin zero [6–8]. A new twist was given to this field in Ref. [9] where the S-wave vector–vector (ρρ) interactions were investigated and it was found that due to the strong binding in certain channels, the f2(1270) and the f0(1370) mesons could be explained as ρρ bound states. This approach offered also an explanation why the tensor state f2 is lighter than the scalar one f0, as the leading order attraction in the corresponding ρρ channel is stronger. This work was followed up by extensions to SU( 3 ) [10], to account for radiative decays [11,12], and much other work; see e.g. the short review in Ref. [13]. These findings concerning the f2(1270) are certainly surprising and at odds with well-known features of the strong interactions. It is a text-book result that the f2(1270) fits very well within a nearly ideally mixed P-wave qq¯ nonet comprising as well the a2(1320), f2(1525) and Ks∗(1430) resonances [14–17]. Values for this mixing angle can be obtained from either the linear or quadratic mass relations as in Ref. [17]. Non-relativistic quark model calculations [18], as well as with relativistic corrections [19], predict that the coupling of the tensor mesons to γ γ should be predominantly through helicity two by an E 1 transition. This simple qq¯ picture for the tensor f2(1270) resonance has been recently validated by the analyses performed in Ref. [20] of the high-statistics Belle data [21–25] on γ γ → π π in both the neutral and the charged pion channels. Another point of importance in support of the qq¯ nature of the f2(1270) is Regge theory, since this resonance lies in a parallel linear exchange-degenerate Regge trajectory with a “universal” slope parameter of around 1 GeV [26–30]. Masses and widths of the first resonances with increasing spin lying on this Regge trajectory (ρ, f2, ρ3, f4) are nicely predicted [31] by the dual-hadronic model of Lovelace–Shapiro–Veneziano [32–34]. One should stress that the results of Ref. [9] were obtained based on extreme non-relativistic kinematics, pi2/m2ρ 0, with p the rho-meson three-momentum and mρ the vectormeson mass. This approximation, however, leads to some severe simplifications: • Due to the assumed threshold kinematics, the full ρ propagator was reduced to its scalar form, thus enabling the use of techniques already familiar from the pion–pion interaction [8]. This was applied when considering the iteration of the interactions in the Bethe–Salpeter equation. • Based on the same argument, the algebra involving the spin and the isospin projectors of the two vector-meson states could considerably be simplified. However, as √sth = 2mρ = 1540 MeV, the lighter of the bound states obtained in Ref. [9] is already quite far away from the 2ρ threshold, with a nominal three-momentum of modulus larger than a 60% of the ρ mass. It is therefore legitimate to question the assumptions there involved. In this work, we will reanalyze the same reactions using a fully covariant approach. This is technically much more involved than the formalism of the earlier work. However, as our aim is to scrutinize the approximations made there, we stay as much as possible close to their choice of parameters. Besides, we also consider ρρ higher partial waves since the authors of Ref. [9] only considered scattering in S-wave because of the same type of near-threshold arguments. The inclusion of these higher partial waves is also important when moving away from threshold. As it will be shown, the near-threshold approximation is only reliable very close to threshold. There is one further issue not considered in Ref. [9] concerning the multiple coupled-channel nature in the energy region where these resonances (for the f0(1370) and f2(1272)) appear. Regarding this point, a set of 13 coupled channels were taken into account in the study of Ref. [35] of the J PC = 0++ meson scattering. Of special significance in their analysis among the channels considered are the twopseudoscalar ones of π π , K K¯ , ηη, ηη and the two-resonance modes (mimicking multi-meson states) of σ σ , ρρ and ωω. A main conclusion of this reference is that the coupled-channel dynamics is crucial to fully understand the meson spectrum in the energy region around 1.5 GeV, which spans the ρρ threshold region, as also concluded in Ref. [36]. In particular Ref. [35] found that the f0(1370) was mostly a bare state stemming from the SU ( 3 ) octet of bare 0++ scalar resonances with mass around 1.4 GeV. This conclusions was reached by the almost coincidence of the couplings of this resonance to the two-pseudoscalar channels calculated both from the tree-level Lagrangian and from the residue of the meson–meson amplitudes at the resonance pole. Furthermore, the mass from the pole position of the f0(1370) is also very close to the bare mass of this octet of scalar resonances. Reference [35] derived the couplings of the vector–vector states to the pseudoscalar–pseudoscalar states and to σ σ by invoking minimal coupling [37], although the direct transitions between the vector–vector channels were not included. We fill partially this gap here by considering the direct scattering between the ρρ states but at the same time we keep the strong assumption of Ref. [9] of neglecting any other channels. In this sense it would be very valuable to improve the work of Ref. [35] by incorporating this new formalism to treat vector–vector scattering. This, however, goes beyond the scope of this research, as our main emphasis here lies on extending and critically reviewing the model of Ref. [9] by taking into account the relativistic corrections and higher partial waves. Our work is organized as follows: in Sect. 2 we outline the formalism to analyze ρρ scattering in a covariant fashion. In particular, we retain the full propagator structure of the ρ, which leads to a very different analytic structure of the scattering amplitude compared to the extreme non-relativistic framework. We also perform a partial-wave projection technique, which allows us to perform the unitarization of the tree-level scattering amplitudes using methods well established in the literature. An elaborate presentation of our results is given in Sect. 3, where we also give a detailed comparison to the earlier work based on the non-relativistic framework. Next, we consider the effect of the coupling between channels with different orbital angular momentum. We also improve the unitarization procedure by considering the first-iterated solution of the N /D method in Sect. 4, reinforcing our results obtained with the simpler unitarization method. We conclude with a summary and a discussion in Sect. 5. A detailed account of the underlying projection formalism is given in Appendix A. 2 Formalism The inclusion of vector mesons in a chiral effective Lagrangian can be done in a variety of different ways, such as treating them as heavy gauge bosons, using a tensor field formulation or generating them as hidden gauge particles of the non-linear σ -model. All these approaches are equivalent, as shown e.g. in the review [37]. While in principle the tensor field formulation is preferable in the construction of chiral-invariant building blocks, we stick here to the hidden symmetry approach as this was also used in Ref. [9]. To be specific, the Lagrangian for the interactions among vector mesons is taken from the pure gauge-boson part of the non-linear chiral Lagrangian with hidden local symmetry [38,39], 1 L = − 4 Fμν F μν . Here, the symbol . . . denotes the trace in SU( 2 ) flavor space and the field strength tensor Fμν is Fμν = ∂μVν − ∂ν Vμ − i g[Vμ, Vν ], ( 1 ) ( 2 ) ρ+ ρ− ρ+ ρ− ρ+ ρ− ρ+ ρ0 ρ0 with the coupling constant g = MV /2 fπ and fπ ≈ 92 MeV [5] the weak pion decay constant. The vector field Vμ is Vμ = √1 ρ0 2 ρ− ρ+ − √12 ρ0 . From the Lagrangian in Eq. ( 1 ) one can straightforwardly derive the interaction between three and four vector mesons, with the corresponding Lagrangians denoted as L3 and L4, respectively. The former one gives rise to ρρ interactions through the exchange of a ρ meson and the latter corresponds to purely contact interactions. We did not include the ω resonance in Eq. ( 3 ), since it does not contribute to the interaction part (in the isospin limit). Consider first the contact vertices for the 4ρ interaction. These can be derived from Eq. ( 2 ) by keeping the terms proportional to g2, leading to g2 L4 = 2 VμVν V μV ν − VμV μVν V ν . The three different isospin (I ) amplitudes for ρρ scattering (I = 0, 1 and 2) can be worked out from the knowledge of the transitions ρ+( p1)ρ−( p2) → ρ+( p3)ρ−( p4) and ρ+( p1)ρ−( p2) → ρ0( p3)ρ0( p4) by invoking crossing as well. We have indicated the different four-momenta by pi , i = 1, . . . , 4. The scattering amplitude for the former transition is denoted by A( p1, p2, p3, p4) and the latter one by B( p1, p2, p3, p4), which are shown in Figs. 1 and 2, respectively. ( 3 ) ( 4 ) ρ0 ρ+ ρ− ρ+ ρ− ρ0 ρ+ ρ− The contributions to those amplitudes from L4, cf. Eq. ( 4 ), are indicated by the subscript c and are given by Ac(k1, k2, k3, k4) = −2g2(2 ( 1 )μ ( 2 )ν ( 3 )ν ( 4 )μ Bc(k1, k2, k3, k4) = 2g2(2 ( 1 )μ ( 2 )μ ( 3 )ν ( 4 )ν In this equation, the (i )μ corresponds to the polarization vector of the i th ρ. Each polarization vector is characterized by its three-momentum pi and third component of the spin σi in its rest frame, so that (i )μ ≡ (pi , σi )μ. Explicit expressions of these polarization vectors are given in Eqs. (A.9) and (A.10) of Appendix A. In the following, so as to simplify the presentation, the tree-level scattering amplitudes are written for real polarization vectors. The same expressions are valid for complex ones by taking the complex conjugate of the polarization vectors attached to the final particles.1 Considering the one-vector exchange terms, we need the three-vector interaction Lagrangian L3. It reads L3 =i g (∂μVν − ∂ν Vμ)V μV ν . The basic vertex is depicted in Fig. 3 which after a simple calculation can be written as √ V3 = − 2g (qμ ( 1 )ν − qν ( 1 )μ) ( 3 )μ ( 2 )ν − (kμ ( 2 )ν − kν ( 2 )μ) ( 1 )μ ( 3 )ν − ( pμ ( 3 )ν − pν ( 3 )μ) ( 2 )μ ( 1 )ν . In terms of this vertex, one can straightforwardly calculate the vector exchange diagrams in Figs. 1 and 2. The expression for the t -channel ρ-exchange amplitude, the middle diagram 1 The polarization vectors (p, σ ) in Appendix A are complex, so that the polarization vectors associated with the final-state ρρ should be complex conjugated in this case. ( 5 ) ( 6 ) ( 7 ) As ( p1, p2, p3, p4; 1, 2, 3, 4) = At ( p1, − p3, − p2, p4; 1, 3, 2, 4). The total amplitudes for ρ+ρ− → ρ+ρ− and ρ+ρ− → ρ0ρ0 are with the usual arguments ( p1, p2, p3, p4; 1, 2, 3, 4). By crossing we also obtain the amplitude for ρ+ρ+ → ρ+ρ+ [that we denote C ( p1, p2, p3, p4; 1, 2, 3, 4)] from the one for ρ+ρ− → ρ+ρ− by exchanging p2 ↔ − p4 and 2 ↔ 4, that is, C ( p1, p2, p3, p4; 1, 2, 3, 4) = A( p1, − p4, p3, − p2; 1, 4, 3, 2). The amplitude C is purely I = 2, which we denote T ( 2 ). The amplitude B is an admixture of the I = 0, T (0), and I = 2 amplitudes, (k, σ2) A = Ac + At + As , B = Bc + At + Au , ( 10 ) ( 11 ) ( 12 ) ( 13 ) ( 14 ) ( 15 ) ( 16 ) (p, σ3) (q, σ1) Fig. 3 Three-ρ vertex from L3 in Fig. 1, and denoted by At ( p1, p2, p3, p4; 1, 2, 3, 4), is At ( p1, p2, p3, p4; 1, 2, 3, 4) 2g2 = ( p1 − p3)2 − m2ρ + i 0+ × ( p1( p2 + p4) + p3( p2 + p4)) 1 · 3 2 · 4 + 4( 1 · k3 4 · k2 2 · 3 + 1 · k3 2 · k4 3 · 4 + 3 · k1 4 · k2 1 · 2 + 2 · k4 3 · k1 1 · 4) where for brevity, we have rewritten (i ) → i , and the scalar products involving polarization vectors are indicated with a dot. The u-channel ρ-exchange amplitude Au ( p1, p2, p3, p4; 1, 2, 3, 4) can be obtained from the expression of At by exchanging p3 ↔ p4 and 3 ↔ 4. In the exchange for the polarization vectors they always refer to the same arguments of three-momentum and spin, that is, (p3, σ3) ↔ (p4, σ4). In this way, Au ( p1, p2, p3, p4; 1, 2, 3, 4) = At ( p1, p2, p4, p3; 1, 2, 4, 3). Notice that the second diagram in Fig. 2 is a sum of the t -channel and u-channel ρ-exchange diagrams. The s-channel exchange amplitude (the last diagram in Fig. 1) can also be obtained from At by performing the exchange p2 ↔ − p3 and 2 ↔ 3, with the same remark as above for the exchange of polarization vectors. We then have from which we find that T (0) = 3B + C. To isolate the I = 1 amplitude, T ( 1 ), we take the ρ+ρ− elastic amplitude A, which obeys the following isospin decomposition: 1 T ( 2 ) A = 6 1 T ( 1 ) + 2 1 T (0). + 3 Taking into account Eqs. ( 14 ) we conclude that T ( 1 ) = 2 A − 2B − C. In terms of these amplitudes with well-defined isospin, the expression in Eq. (A.48) for calculating the partial-wave amplitudes in the S J I basis (states with well-defined total angular momentum J , total spin S, orbital angular momentum and isospin I ), denoted as T (SJ;I¯)S¯ (s) for the transition ( ¯S¯ J I ) → ( S J I ), simplifies to Y 0¯(zˆ) T (SJ;I¯)S¯ (s) = 2(2 J + 1) σ1,σ2,σ¯1 σ¯2,m d pˆ Y m (p )∗(σ1σ2 M|s1s2 S) × (m M M¯ | S J )(σ¯1σ¯2 M¯ |s¯1s¯2 S¯)(0M¯ M¯ | ¯S¯ J ) × T (I )( p1, p2, p3, p4; 1, 2, 3, 4), ( 17 ) with s the usual Mandelstam variable, p1 = |p|zˆ, p2 = −|p|zˆ, p3 = p and p4 = −p , M = σ1 + σ2 and M¯ = σ¯1 + σ¯2 The Mandelstam variables t and u for ρρ scattering in the isospin limit are given by t = −2p2(1 − cos θ ) and u = −2p2(1 + cos θ ), with θ the polar angle of the final momentum. The denominator in At due to the ρ propagator, cf. Eq. ( 8 ), vanishes for t = m2ρ and similarly the denominator in Au for u = m2 . When performing the angular proρ jection in Eq. ( 17 ) these poles give rise to a left-hand cut starting at the branch point s = 3m2 . This can easily be ρ seen by considering the integration on cos θ of the fraction 1/(t − m2ρ + i ε), which gives the same result both for the t and the u channel exchange, Here, G(s) is a diagonal matrix made up by the two-point loop function g(s) with ρρ as intermediate states, g(s) → i d4q 1 (2π )4 (q2 − m2ρ )(( P − q)2 − m2 ) , ρ where P2 = s and within our normalization, cf. Eq. (A.53), Im g(s) = −|p|/8π √s. The loop function is logarithmically divergent and it can be calculated once its value at a given reference point is subtracted. In this way, one can write down a once-subtracted dispersion relation for g(s) whose result is3 1 g(s) = (4π )2 m2ρ a(μ) + log μ2 + σ log(σ + 1) − log(σ − 1) , , with ε → 0+. The argument of the log becomes negative for 4p2 < −m2ρ , which is equivalent to s < 3m2ρ . Because of the factor p2ε the imaginary part of the argument of the log below the threshold is negative which implies that the proper value of the partial-wave amplitude on the physical axis below the branch point at s = 3m2ρ is reached in the limit of vanishing negative imaginary part of s. The presence of this branch point and left-hand cut was not noticed in Ref. [9], where only the extreme non-relativistic reduction was considered, so that the ρ propagators in the ρ-exchange amplitudes collapsed to just a constant. Once we have calculated the partial-wave projected treelevel amplitude we proceed to its unitarization making use of standard techniques within unitary chiral perturbation theory [2,8,40]. This is a resummation technique that restores unitarity and also allows one to study the resonance region. It has been applied to many systems and resonances by now, e.g. in meson–meson, meson–baryon, nucleon–nucleon and W W systems. Among many others we list some pioneering work for these systems [2,3,8,41–56]. In the last years this approach has been applied also to systems containing mesons and baryons made from heavy quarks, some references on this topic are [57–61]. The basic equation to obtain the final unitarized T matrix in the subspace of coupled channels S J I , with the same J I , is 2 T (J I )(s) = I − V (J I )(s) · G(s) · V (J I )(s). −1 ( 19 ) ( 20 ) ( 21 ) ( 22 ) ( 23 ) ( 24 ) 2 In order to easier the comparison with Ref. [9] we take the same sign convention for matrices V (s) and T (s) as in that reference. 3 It is the same result as calculating g(s) in dimensional regularization, d = 4 + 2 , and replacing the 1/ divergence by a constant; cf. [62]. and μ is a renormalization scale typically taken around mρ , such the sum a(μ) + log m2ρ /μ2 is independent of μ. The subtraction constant in Eq. ( 21 ) could depend on the quantum numbers , S and J , but not on I due to the isospin symmetry [3]. To compare with the results of Ref. [9], we also evaluate the function g(s) introducing a three-momentum cutoff qmax, the resulting g(s) function is denoted by gc(s), 0 qmax dq q2 w(s − 4w2 + i ε) q2 + m2ρ . This integral can be done algebraically 1 gc(s) = 2π 2 with w = [62] 1 ⎛ ⎡ ⎛ m2ρ ⎞ gc(s) = (4π )2 ⎝ σ ⎣ log ⎝ σ 1 + qm2ax + 1⎠ ⎛ m2 − log ⎝ σ 1 + qm2ρax − 1⎠ ⎦ ⎞ ⎤ ⎧ ⎛ +2 log ⎨ qmmρax ⎝ 1 + ⎩ m2 ⎞ ⎫ ⎞ ρ ⎬ 1 + qm2ax ⎠ ⎭ ⎠ . Typical values of the cutoff are around 1 GeV. The unitarity loop function g(s) has a branch point at the ρρ threshold (s = 4m2ρ ) and a unitarity cut above it (s > 4m2ρ ). The physical values of the T -matrix T (J I )(s), with s > 4m2ρ , are reached in the limit of vanishing positive imaginary part of s. Notice that the left-hand cut present in V (J I )(s) for s < 3m2ρ does not overlap with the unitarity cut, so that V (J I )(s) is analytic in the complex s-plane around the physical s-axis for physical energies. In this way, the sign of the vanishing imaginary part of s for V (J I )(s) is of no relevance in the prescription stated above for reaching its value on the real axis with s < 3m2ρ according to the Feynman rules. We can also get a natural value [2] for the subtraction constant a in Eq. ( 21 ) by matching g(s) and gc(s) at threshold where σ = 0. For μ = mρ , a usual choice, the final expression simplifies to qmax a = −2 log mρ ⎝ 1 + By varying qmax in this equation in the range of typical values around 1 GeV (avoiding the explicit resolution of hadrons into its quark–gluon constituents for shorter wave lengths) one studies the generation of states within the asymptotic hadronic degrees of freedom. In this way one circumvents the fine tuning of subtraction constants, a process that is associated with pre-existing (quark–gluon) states [63]. It is also worth noticing that Eq. ( 19 ) gives rise to a T matrix T (I J )(s) that is gauge invariant in the hidden local symmetry theory because this equation just stems from the partial-wave projection of a complete on-shell tree-level calculation within that theory, which certainly is gauge invariant. 3 Results One of our aims is to check the stability of the results of Ref. [9] under relativistic corrections, particularly regarding the generation of the poles that could be associated with the f0(1370) and f2(1270) resonances as obtained in that paper. The main source of difference between our calculated V (J I )(s) and those in Ref. [9] arises from the different treatment of the ρ-meson propagator. The point is that the authors of Ref. [9] take the non-relativistic limit of this propagator so that from the expression 1/(t − m2ρ ), cf. Eq. ( 8 ), or 1/(u − m2ρ ), only −1/m2ρ is kept. This is the reason that the tree-level amplitudes calculated in Ref. [9] do not have the branch-point singularity at s = 3m2ρ nor the corresponding left-hand cut for s < 3m2ρ . It turns out that, for the isoscalar tensor case, the resonance f2(1270) is below this branch point, so that its influence cannot be neglected when considering the generation of this pole within this approach. 3.1 Uncoupled S-wave scattering The issue on the relevance of this branch-point singularity in the ρ-exchange amplitudes was not addressed in Ref. [9] and it is indeed very important. This is illustrated in Fig. 4 where we plot the potentials V (J I )(s) in S-wave ( = 0) (only S-wave scattering is considered in Ref. [9]).4 From top to bottom and left to right we show in the figure the potentials for the quantum numbers ( J, I ) equal to (0, 0), ( 2, 0 ), ( 0, 2 ), ( 2, 2 ) and ( 1, 1 ). The red solid and black dotted lines correspond to the real and imaginary parts of our full covariant calculation of the V (J I )(s), respectively, while the blue dashed ones are the results of Ref. [9]. The imaginary part in our results for V (J I )(s) appears below s < 3m2ρ due to the left-hand cut that arises from the t - and u-channel ρexchanges. It can be seen that our results and those of Ref. [9] are typically similar near threshold (s = 4m2 ) but for lower ρ values of s they typically depart quickly due to the onset of the branch-point singularity at s = 3m2 . The strength of ρ this singularity depends on the channel, being particularly noticeable in the ( J, I ) = ( 2, 0 ) channel, while for the (0, 0) channel it is comparatively weaker. The strongest attractive potentials in the near-threshold region occur for ( J, I ) = (0, 0) and ( 2, 0 ) and in every of these channels Ref. [9] found a bound-state pole that the authors associated with the f0(1370) and f2(1272) resonances, respectively. For the (0, 0) quantum numbers the pole position is relatively close to the ρρ threshold, while for ( 2, 0 ) it is much further away. Two typical values of the cutoff qmax were used in Ref. [9], qmax = 875 and 1000 MeV. We employ these values here, too, together with qmax = mρ (so that we consider three values of qmax separated by around 100 MeV), and study the pole positions for our T (J I )(s) amplitudes in S wave. We only find a bound state for the isoscalar scalar case, while for the tensor case no bound state is found. In Table 1 we give the values of the pole positions for our full calculation for qmax = mρ (first), 875 (second) and 1000 MeV (third row). For comparison we also give in round brackets the bound-state masses obtained in Ref. [9], when appropriate. As indicated above, the strong differences for V ( 20 )(s) between our full covariant calculation and the one in Ref. [9] in the extreme non-relativistic limit, cf. Fig. 4b, imply the final disappearance of the deep bound state for the isoscalar tensor case. The nominal threemomentum of a ρ around the mass of the f2(1270) has a modulus of about 0.6mρ 460 MeV and for such high values of three-momentum relativistic corrections are of importance, as explicitly calculated here. On the contrary, the (0, 0) pole is located closer to the ρρ threshold and the results are 4 Partial waves with = 0 are considered in Sect. 3.2. more stable against relativistic corrections, though one still finds differences of around 20 MeV in the bound-state mass. In addition we also show in the third column of Table 1 the residue of T (00)(s) at the pole position sP . For a generic partial wave T (J I ) (s), its residue at a pole is denoted by S; ¯S γ (SJ I )γ (J I ) and is defined as ¯S¯ γ (SJ I )γ ¯(S¯J I ) = − sl→imsP (s − sP )T (SJ;I¯)S¯ (s). In terms of these couplings one can also calculate the compositeness X (JS I ) associated with this bound state [64–66], sP , which in our case determines the ρρ component in such bound state. Notice that the derivative of g(s) from Eq. ( 21 ) (which is negative below threshold) does not depend on the subtraction constant; the dependence on the latter enters implicitly by the actual value of the pole position sP . Of course, if one uses a three-momentum cutoff then gc(s) must be employed in the evaluation of X (JS I ). The compositeness obtained for the pole positions in Table 1 is given in the fourth column of the same table. As expected the ρρ component is dominant, with X 0(000) > 0.5, and increases as the pole moves closer to threshold, so that it is 73% for qmax = mρ and √sP = 1516 MeV. We can also determine the pole positions when g(s) is calculated with exact analytical properties, Eq. ( 21 ), and taking for a the values from Eq. ( 25 ) as a function of qmax. The results are given Table 2, where we also give the residue at the pole position and the calculated compositeness, in the same order as in Table. 1. The results obtained are quite close to those in this table so that we refrain of further commenting on them. Nonetheless, we should stress again that we do not find any pole for the isoscalar tensor case. We could try to enforce the generation of an isoscalar tensor pole by varying qmax, when using gc(s), or by varying a, if Eq. ( 21 ) is used. In the former case a much lower value of qmax is required than the chiral expansion scale around 1 GeV (qmax 400 MeV), while for the latter a qualitatively similar situation arises when taking into account the relationship between a and qmax of Eq. ( 25 ). Even more serious are two facts that happen in relation with this isoscalar tensor pole. First, one should stress that such pole appears associated to the evolution with qmax or a of a pole in the first Riemann sheet, which violates analyticity. This is shown in Fig. 5 where we exhibit the evolution of this pole as a function of qmax. We start the series at a low value of qmax = 300 MeV, where we have two poles on the real axis, and increase the cutoff in steps of δqmax = 50 MeV. These two poles get closer and merge for qmax = 403.1 MeV. For larger values of the cutoff the resulting pole moves deeper into the complex plane of the physical or first Riemann sheet. Second, we find that X 0(220) is larger than 1. For example, for qmax = 400 MeV, there are two poles at 1422.4 and 1463.4 MeV with X 0(220) = 2.7 and 3.8, in order, which of course makes no sense as compositeness factors have to be less or equal to one. Next, we take into account the finite width of the ρ meson in the evaluation of the unitarity two-point loop function g(s). As a result the peak in the modulus squared of the isoscalar scalar amplitude now acquires some width due to the width itself of the ρ meson. To take that into account this effect we convolute the g(s) function with a Lorentzian mass squared distribution for each of the two ρ mesons in the intermediate state [9,45,46]. The resulting unitarity loop function is denoted by g(s) and is given by 1 g(s) = N 2 with (m) the width of the ρ meson with mass m. Due to the P-wave nature of this decay to π π , we take into account its strong cubic dependence on the decaying pion three-momentum and use the approximation with mπ the pion mass and ρ =∼ 148 MeV [5]. The function g(s, m2, m22) is the two-point loop function with different 1 masses, while in Eq. ( 21 ) we give its expression for the equal mass case. When evaluated in terms of a dispersion relation it reads 1 g(s, m12, m22) = 16π 2 a(μ) + log to the case without convolution. The resulting peak positions are given in the last three rows of Tables 1 and 2. The effects of the non-zero ρ width are clearly seen in Fig. 6, where we plot |T (00)(s)|2 for the different values of qmax shown in Table 1. The shape of the peaks follows quite closely a Breit–Wigner form, though it is slightly wider to the right side of the peak. We find that the width decreases with the increasing value of qmax, being around 45, 65 and 95 MeV for qmax = 1000, 875 and 775 MeV, respectively, of similar size as those found in Ref. [9]. When using a subtraction constant instead of qmax, relating them through Eq. ( 25 ), the picture is quite similar. The peak positions are given in the last three columns of Table 2 while the widths obtained are around 105, 70 and 50 MeV for a = −1.70, −1.94 and −2.14, in order. These widths are significantly smaller than the PDG values assigned to the f0(1370) resonance of 200–500 MeV [5]. Due to the coupling of the ρρ and π π , this pole could develop a larger width. This is approximated in Ref. [9] by considering the imaginary part of the π π box diagram, with a ρ → π π vertex at each of the vertices of the box. These vertices are also worked out from the non-linear chiral Lagrangian with hidden gauge symmetry [38,39]. We refer to Ref. [9] for details on the calculation of this contribution. According to this reference one has to add to V0(00;00)0 and to V0(22;00)2 the contribution V2(πJ I ), given by with λ1/2(s) = s2 + m41 + m24 − 2sm21 − 2sm22 − 2m12m22. The algebraic expression of this function when calculated with a three-momentum cutoff for different masses can be found in Ref. [62], to which we refer the interested reader. When using the convoluted g(s) function we find similar masses for the peak of |T (00)|2 in the (0, 0) channel compared In the calculation of the function V!ππ , Ref. [9] introduces a monopole form factor F (q) for each of the four ρ → π π vertices in the pion box calculation, F (q) = 2 − m2π 2 − (k − q)2 log λ1/2(s) + s − m22 + m21 , ( 31 ) V2(π00) = 20i ImV!ππ , V2(π20) = 8i ImV!ππ . ( 32 ) ( 33 ) with k0 = √s/2, k = 0, q0 = √s/2 and q the integration variables. This introduces a sizable dependence of the results on the value of . Nonetheless, in order to compare with Ref. [9] we follow the very same scheme of calculation and take the same values for , that is, 1200, 1300 and 1400 MeV.5 The inclusion of the π π box diagram, on top of the convolution with the ρ mass squared distribution for calculating the g(s) function, does not alter the previous conclusion on the absence of a pole in the isoscalar tensor channel. However, the isoscalar scalar pole develops a larger width around 200–300 MeV, which increases with , as can be inferred from Fig. 7, where we plot |T (00)(s)|2. On the other hand, the position of the peak barely changes compared to the one given in the last two rows of Table 1. Tentatively this pole could be associated to the f0(1370) resonance, which according to Refs. [35,67] decays mostly to π π with a width around 200 MeV. In the PDG [5] the total width of the f0(1370) is given with a large uncertainty, within the range 200–500 MeV and the π π decay mode is qualified as dominant. The nearby f0(1500) resonance has a much smaller width, around 100 MeV, and its coupling and decay to π π is suppressed. These properties of the f0(1500) are discussed in detail in Ref. [35]. 3.2 Coupled-channel scattering We now consider the impact on our results when allowing for the coupling between ρρ channels with different orbital angular momenta, an issue not considered in Ref. [9]. In Table 3 we show the different channels that couple for given J I quantum numbers and pay special attention to the ( J, I ) = (0, 0) 5 Another more complete scheme is two work explicitly with coupledchannel scattering as done in Ref. [35], where ρρ and π π channels, among many others, were explicitly included. In this way resonances develop decay widths in a full non-perturbative fashion because of the coupling between channels. Table 3 Coupled channels with different orbital angular momentum (J, I ) (0, 0) ( 2, 0 ) ( , S) channels (0, 0), ( 2, 2 ) (0, 2), ( 2, 0 ), ( 2, 2 ) and (2, 0) channels. Apart from the conservation of J and I , one also has to impose invariance under parity, which avoids the mixing between odd and even ’s. When including coupled-channel effects, one finds two poles in the channels with ( J, I ) = (0, 0), which are reported in Table 4 for various values of qmax (shown in the first column). We give from left to right the pole mass (second column), the residues (third and fourth ones) and compositeness coefficients (fifth and sixth ones) of the different channels, ( , S) = (0, 0) and ( 2, 2 ), respectively. One of the poles is heavier and closer to the ρρ threshold with similar properties as the pole in the uncoupled case, compare with Table 1, particularly for qmax = 775 MeV. Nonetheless, as qmax increases the difference of the properties of this pole between the coupled and uncoupled cases is more pronounced. In particular let us remark that now X 0(000) is always 0.7 and for qmax = 1 GeV the residue to the channel ( , S) = (0, 0) is much larger than in the uncoupled case. Additionally, we find now a lighter pole which lies above the branch-point singularity at √3 mρ 1343 MeV. For lower values of the cutoff qmax this pole couples more strongly to the ( , S) = ( 2, 2 ) channel, but as the cutoff increases its residue for the channel ( , S) = (0, 0) also increases in absolute value and it is the largest for qmax = 1 GeV. It is then clear that both channels ( , S) = (0, 0) and ( 2, 2 ) are relevant for the origin of this pole. Note that the residues for this lighter pole are negative, which is at odds with the standard interpretation of the residue (γ (S00))2 of a bound state as the coupling squared. This implies that the compositeness coefficients X (0S0) are all negative, which is at odds with a probabilistic interpretation as suggested in Refs. [64–66] for bound states. The moduli of 775 875 1000 Table 4 Bound-state poles in the partial-wave amplitudes of quantum numbers ( J, I ) = (0, 0) with varying cutoff qmax, which is indicated in the first column. The masses (second column), the residues to ( , S) = (0, 0) and ( 2, 2 ) (third and fourth columns) and the compositeness coefficients X0000) and X2020) (sixth and seventh columns) are also given. For the lighter poles the compositeness coefficients are small and negative, so that they cannot be interpreted as physical states contrary to common wisdom [64–66] 57.2 Table 5 Bound-state poles in the partial-wave amplitudes of quantum numbers ( J, I ) = ( 2, 0 ) with different cutoffs qmax, indicated in the first column. The masses (second column) and the residues to ( , S) = ( 0, 2 ), ( 2, 0 ) and ( 2, 2 ) (third, fourth and fifth columns) are shown. Here, “−0.0” denotes a small but negative number the | X (0S0)| are all small because this lighter pole lies quite far from the ρρ threshold. The fact that its mass is not far from the strong branch-point singularity at √3 mρ implies that this pole is very much affected by the left-hand cut discontinuity. In this respect, it might well be that the presence of this pole with anomalous properties is just an artifact of the unitarization formula of Eq. (19), which treats the left-hand cut discontinuity of the potential perturbatively. One can answer this question by solving exactly the N / D method [68–72], so that the left-hand cut discontinuity of the potential is properly treated and the resulting amplitude has the right analytical properties. Let us recall that Eq. ( 19 ) is an approximate algebraic solution of the N / D method by treating perturbatively the left-hand cut discontinuities of the coupled partial waves [40, 49, 52, 53]. For the uncoupled scattering such effects are further studied in detail in the next section. For the ( J, I ) = ( 2, 0 ) partial waves we have three coupled channels, ( , S) = ( 0, 2 ), ( 2, 0 ) and ( 2, 2 ) and, contrary to the uncoupled case, we now find a pole that lies above the branch-point singularity. We give its mass and residues for different qmax in Table 5, with the same notation as in Table 4. Notice that these pole properties are very stable under the variation of qmax. This pole couples by far much more strongly to the channel with ( , S) = ( 2, 2 ) than to any other channel. This indicates that it is mainly due to the dynamics associated with the ( , S) = ( 2, 2 ) channel. But the same comments are in order here as given above for the lighter isoscalar scalar pole, because its residues shown in Table 5 are negative and so are the corresponding compositeness coefficients. Hence, the lighter pole for ( J, I ) = (0, 0) and the one found for ( 2, 2 ) cannot be considered as robust results of our analysis. This has to be contrasted to the case of the heavier isoscalar scalar pole that is stable under relativistic corrections, coupled-channel effects and has quite standard properties regarding its couplings and compositeness coefficients. 4 First-iterated solution of the N/ D method In this section for definiteness we only consider uncoupled scattering. We have in mind the ( J, I ) = (0, 0) and ( J, I ) = ( 2, 0 ) quantum numbers to which special attention has been paid in the literature concerning the generation of poles that could be associated with the f0(1370) and f2(1270) resonances, as discussed above. Further applications of the improved unitarization formalism presented in this section are left for future work. According to the N / D method [73] a partial-wave amplitude can be written as T = N (s) D(s) , D(si ) = 0. where the function D(s) has only the unitarity or right-hand cut (RHC) while N (s) only has the left-hand cut (LHC). The secular equation for obtaining resonances and bound states corresponds to look for the zeros of D(s), ( 34 ) ( 35 ) Below threshold along the real s axis this equation is purely real because D(s) has a non-vanishing imaginary part only for s > sth. However, with our unitarization procedure from leadingorder unitary chiral perturbation theory (UChPT)6 we have obtained the approximation V (J I )(s) T (J I )(s) = 1 − V (J I )(s)gc(s) , and the resulting equation to look for the bound states is DU (s) = 1 − V (s)gc(s) = 0. Notice that Eq. ( 36 ), contrary to the general Eq. ( 35 ), has an imaginary part below the branch-point singularity at s = 3m2. Nonetheless, if the LHC were perturbative this imaginary part would not have prevented having a strong peak in the amplitude if there were already a bound state below the branch-point singularity when the LHC is neglected. However, this is by far not the case for the pole associated with the f2(1270) in Ref. [9], which clearly indicates the instability of this approach under the inclusion of relativistic corrections. We can go beyond this situation by considering the firstiterated solution to the N /D method. This is indeed similar to Eq. ( 36 ) but improving upon it because it allows us to go beyond the on-shell factorization employed in this equation. In the first-iterated N /D solution one identifies the numerator function N (s) to the tree-level calculation V (J I )(s) and employs the exact dispersive expression for D(s). Namely, it reads7 N (s) = V (J I )(s), D(s) = γ0 + γ1(s − sth ) + 21 γ2(s − ssth )2 + (s − sth)s2 π ∞ sth ds ρ(s )V (J I )(s ) (s − sth)(s − s)(s )2 , ( 36 ) ( 37 ) ( 38 ) N (s) TN D(s) = D(s) , with the phase space factor ρ(s) given by ρ(s) = σ (s)/16π . We have taken three subtractions in the dispersion relation for D(s) because V (J I )(s) diverges as s2 for s → ∞. From our present study we have concluded that T (J I )(s) is stable in the threshold region under relativistic corrections as 6 In spite of the fact that the input for the unitarization procedure comes from the hidden local symmetry Lagrangians and not from chiral perturbation theory, we keep the standard notation of UCHPT to refer to the unitarization procedure employed and its degree of sophistication [2,40,49]. 7 A comprehensive introduction to the N /D method is given in Refs. [40,70–72,74–76]. 1.5 1 0.5 ()s 0 D -0.5 -1 -1.51 well as under the addition of coupled ρρ higher partial waves. Because of the stability of the results in this region under relativistic corrections and by visual inspection of the potentials in Fig. 4 one concludes that the near-threshold region is quite safe of the problem related to the branch-point singularity of the LHC associated with one-ρ crossed-channel exchanges. We then determine the subtractions constants, γ0, γ1 and γ2 in D(s) by matching TN D(s), Eq. ( 38 ), and T (J I )(s), Eq. ( 36 ), around threshold (sth). At the practical level it is more convenient to match 1/ T (s), so that in the threshold region up to O(s3) one has γ0 + γ1(s − sth ) + 21 γ2(s − ssth )2 = 1 − V (s)gc(s) − (s − sth)s2 π ρ(s )V (s ) (s − sth)(s − s)(s )2 × sth ≡ ω(s). ∞ ds In this way, γ0 = 1 − V (sth)gc(sth), γ1 = ω (sth), γ2 = ω (sth). ( 39 ) ( 40 ) The dependence of our present results on the cutoff used in T (J I )(s) stems from the matching conditions of Eq. ( 40 ). However, let us stress that the analytical properties of D(s) and N (s) are correct, they have the RHC and LHC with the appropriate extent and branch-point singularities, respectively, and the resulting amplitude is unitarized. (J,I)=(0,0) D(s) qmax=0.7 GeV qmax=1 GeV qmax=1.3 GeV 0 1.4 s (GeV) 1.1 1.2 1.3 1.5 1.6 1.7 1.8 Fig. 8 D(s) function, Eq. ( 38 ), for (J, I ) = (0, 0). Above threshold only the real part is shown (J,I)=(0,0) ReD(s) ReD, qmax=1 GeV ReDU, qmax=1 GeV 0 1.1 1.2 1.3 1.5 1 ))0.5 s ( D ( e R 0 We plot D(s) for ( J, I ) = (0, 0) in Fig. 8 for qmax = 0.7, 1 and 1.3 GeV by the red solid, green dashed and blue dashdotted lines, in order. The crossing with the zero line (dotted one) indicates the mass of the bound state. This mass decreases with increasing qmax, being around 1.4 GeV for the largest cutoff and very close to threshold for the smallest. In Fig. 9 we compare the real (left) and imaginary parts (right panel) of D(s) and DU (s) = 1 − V (s)gc(s) for a cutoff of 1 GeV. We do not show more values of the cutoff because the same behavior results. The function D(s) and DU (s) match up to around the branch-point singularity at √s = 3m2ρ . Below it DU (s) becomes imaginary, cf. Eq. ( 37 ), while D(s) remains real and has this right property by construction, cf. Eq. ( 38 ). Above threshold the imaginary parts of both functions coincide as demanded by unitarity. We see that for these quantum numbers our new improved unitarization formalism and the one used to derive Eq. ( 19 ) agree very well. The bound-state mass remains the same as given in Table 1 because the functions D(s) and DU (s) match perfectly well in the region where these poles occur, as is clear from Fig. 9. This should be expected because for ( J, I ) = (0, 0) the branch-point singularity is much weaker than for other cases, e.g. ( J, I ) = ( 2, 0 ), as discussed above. 4.2 Results J = 2, I = 0 We plot D(s) for ( J, I ) = ( 2, 0 ) in Fig. 10 for qmax = 0.7, 1 and 1.3 GeV by the red solid, green dashed and blue dashdotted lines, respectively. One can see that in this region the function D(s) is large and negative so that there is by far no pole in the f2(1270) region. In order to show the curves more clearly we use only qmax = 1 GeV in Fig. 11, for other values (J,I)=(0,0) ImD(s) ImD, qmax=1 GeV ImDU, qmax=1 GeV 0 s (GeV) (J,I)=( 2,0 ) D(s) 1.4 s (GeV) Fig. 10 D(s) function, Eq. ( 38 ), for (J, I ) = ( 2, 0 ). Above threshold only the real part is shown of qmax the behavior is the same. In the left panel we compare the real parts of TN D(s) (black solid) and T ( 20 )(s) (red dashed) while in the right panel we proceed similarly for the real parts of D(s) and DU (s) = 1 − V ( 20 )(s)gc(s), with the same type of lines in order. All the functions match near the threshold region and above it, but they strongly depart once we approach the LHC branch-point singularity at s = 3m2 ρ and beyond (for smaller values of s). Notice that D(s), which has no such branch-point singularity, follows then the smooth decreasing trend already originating for 3m2 < s < 4m2. For ( J, I ) = (0, 0) the corresponding smooth trend is that of a decreasing function; cf. Fig. 8. The branch-point singularity is clearly seen in T ( 20 )(s) because it is proportional to V ( 20 )(s). In summary, the conclusions obtained in Sect. 3.1 regarding the generation of the pole that could be associated with 1.4 s (GeV) 1.4 s (GeV) 1.1 1.2 1.3 1.5 1.6 1.7 1.8 1.1 1.2 1.3 1.5 1.6 1.7 1.8 (J,I)=( 2,0 ) ReD(s) D, Q=1 GeV DU, Q=1 GeV 2 0 -2 -4 )) -6 s (D -8 ( eR-10 -12 -14 -16 600 the f0(1370) and the absence of that associated with the f2(1270) as claimed in Ref. [9] fully hold. As a matter of fact, they get reinforced after considering the more elaborated unitarization process that is obtained here by taking the first-iterated N /D solution. 5 Summary and conclusions In this paper, we have revisited the issue of resonance generation in unitarized ρρ scattering using a chiral covariant formalism. The main results of our study can be summarized as follows: (i) We have developed a partial-wave projection formalism that is applicable to the covariant treatment of ρρ scattering. In particular, we point out that accounting for the full ρ-meson propagator leads to a branch point in the partial-wave projected amplitudes at s = 3m2ρ 1.8 GeV2, about 208 MeV below the 2ρ threshold. This branch point does not appear in the extreme nonrelativistic treatment of the propagator. (ii) Evaluating the T -matrix using the standard form, see Eq. ( 19 ), which treats the left-cut perturbatively, we find a pole in the scalar isoscalar channel close to the ρρ threshold that can be associated with the f0(1370) resonance, in agreement with the findings of Ref. [9], though there are relatively small quantitative differences. (iii) In contrast to Ref. [9], we do not find a tensor state below the scalar one. This can be traced back to the influence of the aforementioned branch point. We therefore conclude that the state that is identified in Ref. [9] with the f2(1270) is an artifact of the non-relativistic approximation and its generation does not hold from the arguments given in that reference. (iv) We have also worked out the effects of the coupling between ρρ channels with higher orbital angular momenta, which soon become strong when one departs from threshold and lead to additional states. These, however, have negative composite coefficients and are thus not amenable to a simple bound-state interpretation. As these states are close to the branch point at s = 3m2ρ , the perturbative treatment of the left-hand cut, as employed in Sect. 3.2, is certainly not sufficient to decide as regards their relevance. (v) We have improved the treatment of the left-hand cut by employing the first-iterated N /D method, in particular this method avoids the factorization approach of leading order UChPT. We worked the solutions that follow for uncoupled scattering in the ( J, I ) = (0, 0) and ( 2, 0 ) channels. The outcome fully agrees with the conclusions already obtained from UChPT and, notably, the absence of a pole that could be associated with the f2(1270) is firmly reinforced. A lesson from points (iii) and (v) is clear. A strongly attractive interaction around the threshold of a given channel is by far not sufficient condition to generate a multi-hadron bound state. This argument, as used in Ref. [9], is in general terms too naive because it does not take into account the possible rise of a singularity in the true potential between the range of validity of the approximation used and the predicted boundstate mass from the latter. It could be rephrased as trying to deduce the values of the function 1/(1 + x ) for x < −1 by knowing its values for x around 0. We conclude that the approach presented here should be used to investigate the possible generation of meson resonances from the interaction of vector mesons. In the next steps, we will investigate how the relativistic effects affect the conclusions of the SU( 3 ) calculation of Ref. [10] and will further sharpen the framework along the lines mentioned, in particular by solving exactly the N /D equations [70–72]. On top of this, a realistic treatment of the meson spectroscopy in the region above 1 GeV definitely requires the inclusion of more channels than just ρρ (or more generally vector– vector ones). However, the formalism developed here could be very valuable to accomplish this aim phenomenologically if implemented in multi-channel studies, e.g. by extending the work of Ref. [35]. Acknowledgements We thank Maxim Mai for useful discussions and for his contribution during the early stages of this investigation. We would also like to thank E. Oset for some criticism which led us to add some additional material to the manuscript. This work is supported in part by the DFG (SFB/TR 110, “Symmetries and the Emergence of Structure in QCD”). JAO would like to acknowledge partial financial support from the MINECO (Spain) and ERDF (European Commission) grants FPA2013-40483-P and FPA2016-77313-P, as well as from the Spanish Excellence Network on Hadronic Physics FIS201457026-REDT. The work of UGM was supported in part by The Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative (PIFI) Grant No. 2015VMA076. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3. A Partial-wave projection formalism In this appendix we detail the projection formalism used in this work to calculate the different ρρ partial waves. First, we give the expression for the polarization vectors for a massive spin-one particle with a three-momentum p and third component of spin σ in the z axis of its rest frame, which we denote by ε(p, σ ). In the rest frame they are given by Notice that one could also include any arbitrary rotation around the zˆ axis to the right end of Eq. (A.5). Of course, this does not have any affect on either Eqs. (A.4) and (A.6) (for the latter one let us note that Bz (|p|) commutes with a rotation around the zˆ axis). The action of U (p) on (0, σi ) gives us the polarization vectors with definite three-momentum p, whose expressions are that takes zˆ to pˆ , R( pˆ)zˆ = pˆ. with E p = $m2 + p2. We also introduce the rotation R( pˆ) In terms of the polar (θ ) and azimuthal (φ) angles of pˆ, this rotation is defined as R( pˆ) = Rz (φ)Ry (θ ), with the subscripts z and y indicating the axis of rotation. For later convenience we write the Lorentz transformation U (p) as U (p) = R( pˆ)Bz (|p|)R( pˆ)−1, −β and β = |p|/E p. Namely, where Bz (|p|) is a boost along the zˆ axis with velocity v = Bz (|p|) = ⎜⎜ ⎝ and γ = 1 $1 − β2 ε(0, σ ) = with " 0 # εσ , ⎛ 0 ⎞ ε0 = ⎝ 0 1 ⎛ 1 ⎞ ∓1 ⎠ , ε±1 = √2 ⎝ ±i ⎠ . 0 (A.1) (A.2) Next, we take a Lorentz transformation U (p) along the vector p that takes the particle four-momentum at rest to its final value, U (p) " m # 0 = " E p # p , ε(p, σ ) = γ β pˆ · εσ εσ + pˆ (γ − 1) pˆ · εσ # with the corresponding similar expressions for the ρ±(x ) fields. Here a(p, σ ) and a(p, σ )† refer to the annihilation and creation operators, with the canonical commutation relation [a(p , σ ), a(p, σ )†] = δσ σ (2π )32E pδ(p − p ). (A.12) In order to check the time-reversal and parity-invariance properties of the vector–vector scattering amplitudes worked out from the chiral Lagrangians in Eq. ( 1 ) we notice that the polarization vectors in Eq. (A.9) satisfy the following transformation properties: ε(−p, σ )∗ = (−1)σ (−ε(p, σ )0, ε(p, σ )), ε(−p, σ ) = (−ε(p, σ )0, ε(p, σ )). A one-particle state |p, σ is obtained by the action of the creation operators on the vacuum state, |p, σ =a(p, σ )†|0, σ . From Eq. (A.12) it follows the following normalization for such states: p , σ |p, σ = δσ σ (2π )32E pδ(p − p). 1 4π Next, we consider a two-body state characterized by the CM three-momentum p and the third components of spin σ1 and σ2 in their respective rest frames. This state is denoted by |p, σ1σ2 . Associated to this, we can define the two-body state with orbital angular momentum with its third component of orbital angular momentum m, denoted by | m, σ1σ2 as | m, σ1σ2 = √ dpˆ Y m ( pˆ)|p, σ1σ2 . (A.16) Let us show first that this definition is meaningful because the state | m, σ1σ2 transforms under the rotation group as the direct product of the irreducible representations associated to the orbital angular momentum and the spins s1 and s2 of the two particles. Every single-particle state |p, σ under the action of a rotation R transforms as × D(s2)(R)σ2σ2 | m , σ1σ2 . (A.24) 8 For a general Lorentz transformation these manipulations give rise to the Wigner rotation [76]. (A.10) R|p, σ = RU (p)|0, σ = U (p )U (p )−1 RU (p)|0, σ , (A.17) and p = Rp.8 It is straightforward to show that R = U (p )−1 RU (p). For that we explicitly write the Lorentz transformations U (p ) and U (p) as in Eq. (A.6) so that U (p )−1 RU (p) = R( pˆ )Bz (|p|)−1 R( pˆ )−1 R R( pˆ) × Bz (|p|)R( pˆ)−1. Next, the product of rotations R( pˆ )−1 R R( pˆ) is a rotation around the z axis, Rz (γ ), since it leaves invariant zˆ. Thus, R( pˆ )−1 R R( pˆ) = Rz (γ ), or, in other terms, R( pˆ ) = R R( pˆ)Rz (γ )−1, (A.11) (A.13) (A.14) (A.15) (A.18) (A.19) (A.20) (A.21) (A.23) Taking into account Eqs. (A.20) and (A.21) in Eq. (A.19) it follows the result in Eq. (A.18) because Bz (|p|) and Rz (γ ) commute. Then Eq. (A.17) implies that R|p, σ = U (p )R|0, σ = D(s)(R)σ σ |p , σ , (A.22) σ with D(s)(R) the rotation matrix in the irreducible representation of the rotation group with spin s. Now, we can use this result to find the action of the rotation R on the state |p, σ1σ2 , which is the direct product of the states |p, σ1 and | − p, σ2 (once the trivial CM movement is factorized out [76]). In this way, R|p, σ1σ2 = D(s1)(R)σ1σ1 D(s2)(R)σ2σ2 |p , σ1σ2 . We are now ready to derive the action R on | m, σ1σ2 , R| m, σ1σ2 = D(s1)(R)σ1σ1 D(s2)(R)σ2σ2 dpˆ Y m (R−1 pˆ )|p , σ1σ2 D( )(R)m m D(s1)(R)σ1σ1 σ1,σ2 = σ1,σ2 1 × √4π σ1,σ2,m In this equation we have made use of the property of the spherical harmonics Y m (R−1 pˆ ) = D( )(R)m m Y m ( pˆ ). (A.25) Equation (A.24) shows that under rotation the states defined in Eq. (A.16) have the right transformation under the action of a rotation R, and our proposition above is shown to hold. Now, because of the transformation in Eq. (A.24), corresponding to the direct product of spins s1, s2 and , we can combine these angular momentum indices and end with the L S J basis. In the latter every state is labeled by the total angular momentum J , the third component of the total angular momentum μ, orbital angular momentum and total spin S (resulting from the composition of spins s1 and s2). Namely, we use the notation | J μ, S for these states which are then given by |J μ, S = (σ1σ2 M|s1s2 S)(m Mμ| S J )| m, σ1σ2 , σ1,σ2,m,M (A.26) where we have introduced the standard Clebsch–Gordan coefficients for the composition of two angular momenta.9 Next we introduce the isospin indices α1 and α2 corresponding to the third components of the isospins τ1 and τ2. This does not modify any of our previous considerations since isospin does not transform under the action of spatial rotations. Within the isospin formalism the ρρ states obey Bose– Einstein statistics and these symmetric states are defined by |p, σ1σ2, α1α2 S 1 = √2 (|p, σ1σ2, α1α2 + | − p, σ2σ1, α2α1 ) , with the subscript S indicating the symmetrized nature of the state under the exchange of the two particles. One can invert Eq. (A.16) and give the momentum-defined states in terms of those with well-defined orbital angular momentum, |p, σ1σ2, α1α2 = Y m ( pˆ)∗| m, σ1σ2, α1α2 = √4π √4π ,m J,μ, ,m S,M,I,t3 Y m ( pˆ)∗ × (σ1σ2 M |s1s2 S)(m M μ| S J ) × (α1α2t3|τ1τ2 I )| J μ, S, I t3 , (A.28) 9 The Clebsch–Gordan coefficient (m1m2m3| j1 j2 j3) is the composition for j1 + j2 = j3, with mi referring to the third components of the spins. with I the total isospin of the particle pair and t3 is the third component. Taking into account this result we can write the symmetrized states as |p, σ1σ2, α1α2 S = √4π J,μ, ,m S,M,I,t3 We can also express the state | J μ, S, I t3 in terms of the states |p, σ1σ2, α1α2 without symmetrization by inverting Eq. (A.28). We would obtain the same expression as Eq. (A.31), but with a factor 1/√4π instead of 1/√8π , namely | J μ, S, I t3 = √ dpˆ Y m ( pˆ)(σ1σ2 M |s1s2 S) × (m M μ| S J )(α1α2t3|τ1τ2 I )|p, σ1σ2, α1α2 . (A.32) The extra factor of 1/√2 in Eq. (A.31) is a symmetrization factor because of the Bose–Einstein symmetry properties of the two-particle state in the symmetrized states |p, σ1σ2, α1α2 S, which disappears when employing the nonsymmetrized states. In order to obtain the normalization of the states | J μ, S, I t3 it is indeed simpler to use Eq. (A.32) 1 8π σ1,σ2 M,m α1,α2 1 4π σ1,σ2 M,m α1,α2 (A.27) × (m M μ| S J )(α1α2t3|τ1τ2 I )|p, σ1σ2, α1α2 S. (A.31) though, of course, the same result is obtained if starting from Eq. (A.31). The two-body particle states with definite threemomentum satisfy the normalization invariant under the transformation Tˆ → R( pˆ)Tˆ R( pˆ)†, which implies at the level of the matrix elements that S p , σ1σ2, α1α2|Tˆ |p, σ¯1σ¯2, α¯ 1α¯ 2 S The total energy conservation guarantees that the modulus of the final and initial three-momentum in Eq. (A.34) is the same, which we denote by |p|. In terms of this result and Eq. (A.32) it follows straightforwardly by taking into account the orthogonal properties of Clebsch–Gordan coefficients and spherical harmonics that We are interested in the partial-wave amplitude corresponding to the transition between states with quantum numbers J ¯S¯ I to states J S I , which corresponds to the matrix element T (J I ) S; ¯S¯ = J μ, S, I t3|Tˆ | J μ, ¯S¯, I t3 , with Tˆ the T -matrix scattering operator. Here we take the convention that the quantum numbers referring to the initial state are barred. Of course, the matrix element in Eq. (A.35) is independent of μ and t3 because of invariance under rotations in ordinary and isospin spaces, respectively. We can calculate this scattering matrix element in terms of those in the basis with definite three-momentum by replacing in Eq. (A.35) the states in the J S basis as given in Eq. (A.31). We then obtain in a first step Here we have not shown the explicit indices over which the sum is done in order not to overload the notation.10 We use next the rotation invariance of the T -matrix operator Tˆ to simplify the previous integral so that, at the end, we have just the integration over the final three-momentum angular solid. There are several steps involved that we give in quite detail. The referred rotational invariance of Tˆ implies that it remains 10 They correspond to those indicated under the summation symbol in Eq. (A.31) both for the initial and final states. (A.37) Under the action of the rotation R( pˆ)† (R( pˆ)† pˆ = zˆ and R( pˆ)†pˆ = pˆ ) the final and initial states transform as, cf. Eq. (A.23), R( pˆ)†|p, σ¯1σ¯2, α¯ 1α¯ 2 S R( pˆ)†|p , σ1σ2, α1α2 S = = with the convention that R inside the argument of the rotation matrices refers to R( pˆ). We insert Eqs. (A.37) and (A.38) into Eq. (A.36), and next transform pˆ → pˆ as integrations variables, take into account the invariance of the solid angle measure under such rotation and use Eq. (A.25) for Y ¯m¯ ( pˆ) = Y m¯ (R( pˆ)zˆ) = ¯ Dm(¯¯)m¯ (R†)Y ¯m¯ (zˆ), Y m ( pˆ ) = Y m (R( pˆ) pˆ ) = Dm( )m (R†)Y m ( pˆ ). (A.39) m ¯ m Then Eq. (A.36) for T (J I ) can be rewritten as S; ¯S¯ dpˆ dpˆ (σ1σ2 M |s1s2 S)(m M μ| S J ) × (α1α2t3|τ1τ2 I )Dσ(s11σ)1 (R†)∗ D(s2) (R†)∗ σ2σ2 × Dm( )m (R†)∗Y m ( pˆ )∗(σ¯1σ¯2 M¯ |s¯1s¯2 S¯) × (m¯ M¯ μ| ¯S¯ J )(α¯ 1α¯ 2t3|τ¯1τ¯2 I ) × D(s¯1) (R†)Dσ(¯s¯22σ¯)2 (R†)Dm(¯¯)m¯ (R†)Y ¯m¯ (zˆ) σ¯1σ¯1 × S p , σ1σ2, α1α2|Tˆ | |p|zˆ, σ¯1σ¯2, α¯ 1α¯ 2 S. (A.40) Let us recall that from the composition of two rotation matrices one has [76,77] m1,m2 = M (m1m2 M | 1 2 L)Dm(11m)1 (R)Dm(22m)2 (R) (m1m2 M | 1 2 L)D(ML)M (R). (A.41) We apply this result first to two combinations in Eq. (A.40): σ1,σ2 The same relation in Eq. (A.41) is applied once more to the following combinations in Eq. (A.43): (m M μ| S J )D(MS)M (R†)Dm( )m (R†) (m M μ | S J )Dμ(J μ)(R†), (m¯ M¯ μ| ¯S¯ J )D(S¯) (R†)Dm(¯¯)m¯ (R†) M¯ M¯ (m¯ M¯ μ¯ | ¯S¯ J )Dμ(¯J μ)(R†). We take Eq. (A.44) into Eq. (A.43) which now reads Now, the partial-wave amplitude T (SJ;I¯)S¯ is independent of μ so that we have μ=−J (A.46) (A.42) (A.44) Sˆ = I − i Tˆ The same result in Eq. (A.45) is obtained with the product Dμ(J μ)(R†)∗ Dμ(¯J μ)(R†) replaced by 1 2 J + 1 J μ=−J Dμ(J μ)(R†)∗ Dμ(¯J μ)(R†) = 2δJμ¯+μ 1 , (A.47) as follows from the unitarity character of the rotation matrices. As a consequence any dependence in pˆ present in the integrand of Eq. (A.45) disappears in the average of Eq. (A.46), the integration in the solid angle pˆ is trivial and it gives a factor 4π . Taking into account the Kronecker delta from Eq. (A.47) in the third component of the total angular momentum and a new one that arises because Y m¯ (zˆ) is ¯ enxoptrzeesrsoioonnfloyr fTor(J mI¯) := 0, we then end with the following S; ¯S¯ dpˆ Y m ( pˆ )∗(σ1σ2 M |s1s2 S) T (J I ) Y ¯0(zˆ) S; ¯S¯ = 2(2 J + 1) σ1,σ2,σ¯1 σ¯2,α1,α2 α¯1,α¯2,m where we have removed the primes on top of the spin and orbital angular momentum third-component symbols and in the previous sum M = σ1 + σ2 and M¯ = σ¯1 + σ¯2. Next, we derive the unitarity relation corresponding to our normalization for the partial-wave projected amplitudes T (SJ;I¯)S¯ . We write the Sˆ matrix as which satisfies the standard unitarity relation Sˆ · Sˆ† = I, with I the identity matrix. In terms of the T -matrix, cf. (A.49), this implies that Tˆ − Tˆ † = −i Tˆ Tˆ †. Expressed with the matrix elements in the basis S J this relation becomes 2ImTˆ (J I ) S; ¯S¯ = − J μ, S, T t3|Tˆ Tˆ †| J μ, ¯S¯, T t3 . In deriving the left-hand side of this equation we have taken into account that because of time-reversal symmetry T (J I ) S; ¯S¯ = T (J I ) . On the right-hand side we introduce now ¯S¯; S a two-body resolution of the identity of states | J μ, S, I t3 (A.49) (A.50) (A.51) (A.52) (we have restricted our vector space to the one generated by these states) such that, taking into account their normalization in Eq. (A.34), one ends with ImT (J I ) S; ¯S¯ = − ,S |p | T (J I ) 8π √s ,S; ,S T (J I )∗ ,S ; ¯S¯ . The phase space factor is included in the diagonal matrix such that now the diagonal matrix elements of the identity operator I and S are just 1 and ηi e2iδi , in order, where ηi is the inelasticity for channel i and δi its phase shift. Finally, let us mention that a partial-wave expansion of the helicity amplitudes (not S J partial waves as here) for vector–vector scattering was developed in Ref. [78]. 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D. Gülmez, U.-G. Meißner, J. A. Oller. A chiral covariant approach to \(\varvec{\rho \rho }\) scattering, The European Physical Journal C, 2017, 1-21, DOI: 10.1140/epjc/s10052-017-5018-z