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- 1. Vector meson condensation within NJL model Marco Frasca Quantum Chromodynamics in strong magnetic ﬁelds, Catania, September 23, 2013
- 2. Plan of the Talk 2 / 36 Plan of the talk Low-energy QCD Scalar ﬁeld theory Yang-Mills theory Yang-Mills Green function QCD in the infrared limit ρ meson condensation Emerging of ρ condensation NJL and magnetic ﬁeld Mass of the ρ in NJL model Quantum phase transition in QCD Conclusions
- 3. Low-energy QCD Scalar ﬁeld 3 / 36 Scalar ﬁeld A classical ﬁeld theory for a massless scalar ﬁeld is given by φ+λφ3=j The homogeneous equation can be exactly solved by [M. Frasca, J. Nonlin. Math. Phys. 18, 291 (2011)] Exact solution φ=µ( 2 λ ) 1 4 sn(p·x+θ,i) p2=µ2 λ 2 being sn an elliptic Jacobi function and µ and θ two constants. This solution holds provided the given dispersion relation holds and represents a free massive solution notwithstanding we started from a massless theory. Mass arises from the nonlinearities when λ is taken to be ﬁnite.
- 4. Low-energy QCD Scalar ﬁeld 4 / 36 Scalar ﬁeld When there is a current we ask for a solution in the limit λ→∞ as our aim is to understand a strong coupling limit. We consider φ = φ[j] and put φ[j]=φ0(x)+ d4x δφ[j] δj(x ) j=0 j(x )+ d4x d4x δ2φ[j] δj(x )δj(x ) j=0 j(x )j(x )+... This is a current expansion where ∆(x−x )= δφ[j] δj(x ) j=0 is the Green function One can prove that this is indeed so provided Green function equation ∆(x)+3λ[φ0(x)]2∆(x)=δ4(x). This is an exactly solvable linear equation.
- 5. Low-energy QCD Scalar ﬁeld 5 / 36 Scalar ﬁeld The exact solution of the equation ∆(x)+λ[φ0(x)]2∆(x)=δ4(x) it is straightforwardly obtained and its Fourier transformed counterpart is [M. Frasca, J. Nonlin. Math. Phys. 18, 291 (2011)] ∆(p)= ∞ n=0(2n+1)2 π3 4K3(i) e −(n+ 1 2 )π 1+e−(2n+1)π . 1 p2−m2 n+i provided the phases of the exact solutions are ﬁxed to θm=(4m+1)K(i) being K(i)=1.311028777.... The poles are determined by mn=(2n+1) π 2K(i) (λ 2 ) 1 4 µ.
- 6. Low-energy QCD Scalar ﬁeld 6 / 36 Scalar ﬁeld We can formulate a quantum ﬁeld theory for the scalar ﬁeld starting from the generating functional Z[j]= [dφ] exp[i d4x(1 2 (∂φ)2−λ 4 φ4+jφ)]. We can rescale the space-time variable as x → √ λx and rewrite the functional as Z[j]= [dφ] exp[i 1 λ d4x(1 2 (∂φ)2−1 4 φ4+ 1 λ jφ)]. Then we can seek for a solution series as φ= ∞ n=0 λ−nφn and rescale the current j → j/λ being this arbitrary. We work on-shell given the equation for the exact solution to hold φ0+φ3 0=0 This is completely consistent with our preceding formulation [M. Frasca, Phys. Rev. D 73, 027701 (2006)] but now all is fully covariant.
- 7. Low-energy QCD Scalar ﬁeld 7 / 36 Scalar ﬁeld Using the current expansion that holds at strong coupling λ→∞ one has φ[j]=φ0+ d4x ∆(x−x )j(x )+... it is not diﬃcult to write the generating functional at the leading order in a Gaussian form Z0[j]=exp[ i 2 d4x d4x j(x )∆(x −x )j(x )]. This conclusion is really important: It says that there exists an inﬁnite set of scalar ﬁeld theories in d=3+1 that are trivial in the infrared limit and that obtain a mass gap even if are massless. This functional describes a set of free particles with a mass spectrum mn=(2n+1) π 2K(i) (λ 2 ) 1 4 µ that are the poles of the propagator, the same as in classical theory.
- 8. Low-energy QCD Yang-Mills ﬁeld 8 / 36 Yang-Mills ﬁeld A classical ﬁeld theory for the Yang-Mills ﬁeld is given by ∂µ∂µAa ν −(1− 1 ξ )∂ν (∂µAa µ)+gf abc Abµ(∂µAc ν −∂ν Ac µ)+gf abc ∂µ(Ab µAc ν )+g2f abc f cde AbµAd µAe ν =−ja ν . For the homogeneous equations, we want to study it in the formal limit g → ∞. We note that a class of exact solutions exists if we take the potential Aa µ just depending on time, after a proper selection of the components [see Smilga (2001)]. These solutions are the same of the scalar ﬁeld when spatial coordinates are set to zero (rest frame). Diﬀerently from the scalar ﬁeld, we cannot just boost away these solutions to get a general solution to Yang-Mills equations due to gauge symmetry. But we can try to ﬁnd a set of similar solutions with the proviso of a gauge choice. This kind of solutions will permit us to prove that a set of them exists supporting a trivial infrared ﬁxed point to build up a quantum ﬁeld theory.
- 9. Low-energy QCD Yang-Mills ﬁeld 9 / 36 Yang-Mills ﬁeld Exactly as in the case of the scalar ﬁeld we assume the following solution to our ﬁeld equations Aa µ[j]=Aa µ[0]+ d4x Dab µν (x−x )jbν (x )+... Also in this case this boils down to an expansion in powers of the currents as already guessed in the ’80 [R. T. Cahill and C. D. Roberts, Phys. Rev. D 32, 2419 (1985)]. This implies that the corresponding quantum theory, in a very strong coupling limit, takes a Gaussian form and is trivial. The crucial point, as already pointed out in the eighties [T. Goldman and R. W. Haymaker, Phys. Rev. D 24, 724 (1981), T. Cahill and C. D. Roberts (1985)], is the determination of the gluon propagator in the low-energy limit.
- 10. Low-energy QCD Yang-Mills ﬁeld 10 / 36 Yang-Mills ﬁeld The question to ask is: Does a set of classical solutions exist for Yang-Mills equations supporting a trivial infrared ﬁxed point for the corresponding quantum theory? The answer is yes! These solutions are instantons in the form Aa µ = ηa µφ with ηa µ a set of constants and φ a scalar ﬁeld. By direct substitution into Yang-Mills equations one recovers the equation for φ that is ∂µ∂µφ− 1 N2−1 1− 1 ξ (ηa·∂)2φ+Ng2φ3=−jφ being jφ=ηa µjµa. In the Landau gauge (Lorenz gauge classically) this equation is exactly that of the scalar ﬁeld given above and we can solve it accordingly. So, a set of solutions of the Yang-Mills equations exists supporting a trivial infrared ﬁxed point. Our aim is to study the theory in this case.
- 11. Low-energy QCD Yang-Mills ﬁeld 11 / 36 Yang-Mills-Green function The instanton solutions given above permit us to write down immediately the propagator for the Yang-Mills equations in the Landau gauge for SU(N) being exactly the same given for the scalar ﬁeld: Gluon propagator in the infrared limit ∆ab µν (p)=δab ηµν − pµpν p2 ∞ n=0 Bn p2−m2 n+i +O 1√ Ng being Bn=(2n+1)2 π3 4K3(i) e −(n+ 1 2 )π 1+e−(2n+1)π and mn=(2n+1) π 2K(i) Ng2 2 1 4 Λ This is the propagator of a massive free ﬁeld theory due to the K¨allen-Lehman representation.
- 12. Low-energy QCD Yang-Mills ﬁeld 12 / 36 Yang-Mills ﬁeld We can now take the form of the propagator given above, e.g. in the Landau gauge, as Dab µν (p)=δab ηµν − pµpν p2 ∞ n=0 Bn p2−m2 n+i +O 1√ Ng to do a formulation of Yang-Mills theory in the infrared limit. Then, the next step is to use the approximation that holds in a strong coupling limit Aa µ[j]=Aa µ[0]+ d4x Dab µν (x−x )jbν (x )+O(j2) and we note that, in this approximation, the ghost ﬁeld just decouples and becomes free and one ﬁnally has at the leading order Z0[j]=N exp[ i 2 d4x d4x jaµ(x )Dab µν (x −x )jbν (x )]. Yang-Mills theories exist that have an infrared trivial ﬁxed point in the limit of the coupling going to inﬁnity and we expect the running coupling to go to zero lowering energies. So, the leading order propagator cannot conﬁne.
- 13. Low-energy QCD Yang-Mills ﬁeld 13 / 36 Yang-Mills ﬁeld A direct comparison of our results with lattice data gives the following ﬁgure: that is strikingly good below 1 GeV (ref. A. Cucchieri, T. Mendes, PoS LAT2007, 297 (2007)).
- 14. Low-energy QCD QCD at infrared limit 14 / 36 QCD at infrared limit When use is made of the trivial infrared ﬁxed point, QCD action can be written down quite easily. Indeed, we will have for the gluon ﬁeld Sgf =1 2 d4x d4x jµa(x )Dab µν (x −x )jνb(x )+O 1√ Ng +O(j2 ) and for the quark ﬁelds Sq= q d4x¯q(x) i /∂−mq−gγµ λa 2 d4x Dab µν (x−x )jνb(x ) −g2γµ λa 2 d4x Dab µν (x−x ) q ¯q (x ) λb 2 γν q (x )+O 1√ Ng +O(j2 ) q(x) We recognize here an explicit Yukawa interaction and a Nambu-Jona-Lasinio non-local term. Already at this stage we are able to recognize that NJL is the proper low-energy limit for QCD.
- 15. Low-energy QCD QCD at infrared limit 15 / 36 QCD in the infrared limit Now we operate the Smilga’s choice ηa µηb ν = δab(ηµν − pµpν/p2) for the Landau gauge. We are left with the infrared limit for QCD using conservation of currents Sgf =1 2 d4x d4x ja µ(x )∆(x −x )jµa(x )+O 1√ Ng +O(j2 ) and for the quark ﬁelds Sq= q d4x¯q(x) i /∂−mq−gγµ λa 2 d4x ∆(x−x )ja µ(x ) −g2γµ λa 2 d4x ∆(x−x ) q ¯q (x ) λa 2 γµq (x )+O 1√ Ng +O(j2 ) q(x) We want to give explicitly the contributions from gluon resonances. In order to do this, we introduce the bosonic currents ja µ(x) = ηa µj(x) with the current j(x) that of the gluonic excitations.
- 16. Low-energy QCD QCD at infrared limit 16 / 36 QCD in the infrared limit Using the relation ηa µηµa=3(N2 c −1) we get in the end Sgf =3 2 (N2 c −1) d4x d4x j(x )∆(x −x )j(x )+O 1√ Ng +O(j2 ) and for the quark ﬁelds Sq= q d4x¯q(x) i /∂−mq−gηa µγµ λa 2 d4x ∆(x−x )j(x ) −g2γµ λa 2 d4x ∆(x−x ) q ¯q (x ) λa 2 γµq (x )+O 1√ Ng +O(j2 ) q(x) Now, we recognize that the gluon propagator is just a sum of Yukawa propagators weighted by exponential damping terms. So, we introduce the σ ﬁeld and truncate at the lowest excitation. This means the we can write the bosonic currents contribution as coming from a single bosonic ﬁeld written down as σ(x)= √ 3(N2 c −1)/B0 d4x ∆(x−x )j(x ).
- 17. Low-energy QCD QCD at infrared limit 17 / 36 QCD at infrared limit So, low-energy QCD yields a non-local NJL model as given in [M. Frasca, Phys. Rev. C 84, 055208 (2011)] Sσ= d4x[1 2 (∂σ)2−1 2 m2 0σ2 ] and for the quark ﬁelds Sq= q d4x¯q(x) i /∂−mq−g B0 3(N2 c −1) ηa µγµ λa 2 σ(x) −g2γµ λa 2 d4x ∆(x−x ) q ¯q (x ) λa 2 γµq (x )+O 1√ Ng +O(j3 ) q(x) Now, we obtain directly from QCD (2G(0) = G is the standard NJL coupling) G(p)=−1 2 g2 ∞ n=0 Bn p2−(2n+1)2(π/2K(i))2σ+i =G 2 C(p) with C(0) = 1 ﬁxing in this way the value of G using the gluon propagator. This yields an almost perfect agreement with the case of an instanton liquid (see ref. in this page).
- 18. Low-energy QCD QCD at infrared limit 18 / 36 QCD at infrared limit Nambu-Jona-Lasinio model from QCD at low-energies SNJL = q d4 x¯q(x) i /∂ − mq q(x) + d4 x G(x − x ) q q ¯q(x) λa 2 γµ ¯q (x ) λa 2 γµq (x )q(x) + d4 x 1 2 (∂σ)2 − 1 2 m2 0σ2 . We need to: 1 Turn on the electromagnetic interaction. 2 Fierz transform it. 3 Bosonize it.
- 19. ρ meson condensation Emerging of ρ condensation 19 / 36 Emerging of ρ condensation A rather elementary argument [Chernodub, Phys. Rev. D 82, 085011 (2010)] starts by simply observing that a relativistic particle in a magnetic ﬁeld has ε2 n,sz (pz) = p2 z + (2n − 2sz + 1)eBext + m2 . n is a non-negative integer and sz the spin projection. For the π (spin 0 mass about 139 MeV) and ρ meson (spin 1 and mass about 770 MeV) happens that m2 π± (Bext) = m2 π± + eBext , m2 ρ± (Bext) = m2 ρ± − eBext . the π becomes heavier while the ρ becomes lighter possibly hindering the decay ρ → ππ. But, more important, from this simple argument a critical magnetic ﬁeld seems to exist that, once overcame, the ρ change into a tachyonic mode: A condensate forms Bc = m2 ρ e ≈ 1016 Tesla .
- 20. ρ meson condensation Emerging of ρ condensation 20 / 36 Emerging of ρ condensation A more substantial argument [Chernodub, ibid.] is based on the vector meson Lagrangian [Djukanovic et al. Phys. Rev. Lett. 95, 012001 (2005)] L = − 1 4 FµνFµν − 1 2 (Dµρν − Dνρµ)† (Dµ ρν − Dν ρµ ) + m2 ρ ρ† µρµ − 1 4 ρ(0) µν ρ(0)µν + m2 ρ 2 ρ(0) µ ρ(0)µ + e 2gs Fµν ρ(0) µν . For a ρ meson the gyromagnetic ratio is anomalously large being 2. This Lagrangian describes a non-minimal coupling between vector mesons and electromagnetic ﬁeld with gs ≈ 5.88. High gs and so high g is the reason for condensation. If we consider the homogeneous case for a moment we get 0(ρ)=1 2 B2 ext +2(m2 ρ−eBext ) |ρ|2+2g2 s |ρ|4 . being ρ− ≡ √ 2ρ, ρ+ = 0 and this permits to draw an identical conclusion as in the aforementioned simplest argument.
- 21. ρ meson condensation Emerging of ρ condensation 21 / 36 Emerging of ρ condensation Condensation of charged particles implies a superconductor state. One needs to show that NJL model should also conﬁrm ρ condensation. A ﬁrst computation was given in Chernodub, Phys. Rev. Lett. 106, 142003 (2011). A ﬁrst computation on lattice conﬁrmed ρ condensation [Braguta et al., Phys. Lett. B718 (2012) 667]. The idea of spin-1 particles that condensate apparently clashes with Vafa-Witten and Elitzur theorems [Hidaka and Yamamoto, Phys. Rev. D 87, 094502 (2013)]. Hidaka and Yamamoto also performed lattice computations disclaiming ρ condensation. Chernodub responses are Phys.Rev.D 86, 107703 (2012), arXiv:1309.4071 [hep-ph]. NJL model evades Vafa-Witten theorem and condensation could be seen there. But we just proved that a non-local NJL model is the low-energy limit of QCD so, condensation in NJL is condensation in QCD.
- 22. ρ meson condensation NJL and magnetic ﬁeld 22 / 36 Turning on the electromagnetic ﬁeld Nambu-Jona-Lasinio model in an electromagnetic ﬁeld Introduce the gauge covariant derivative Dµ = ∂µ − ieqAµ and then, SNJL = q d4 x¯q(x) i /∂ + eq /A − mq q(x) + d4 x G(x − x ) q q ¯q(x) λa 2 γµ ¯q (x ) λa 2 γµq (x )q(x) + d4 x 1 2 (∂σ)2 − 1 2 m2 0σ2
- 23. ρ meson condensation NJL and magnetic ﬁeld 23 / 36 Fierz transforming Nambu-Jona-Lasinio model after Fierz transformation We take the local and two-ﬂavor approximations that are enough for our aims. So, SNJL = q=u,d d4 x¯q(x) i /∂ + eq /A − mq q(x)+ d4 x 1 2 (∂σ)2 − 1 2 m2 0σ2 + G 2 q=u,d d4 x (¯q(x)q(x))2 + (¯q(x)iγ5 τq(x))2 − G 4 q=u,d d4 x (¯q(x)γµ τaq(x))2 + (¯q(x)γ5 γµ τaq(x))2 from which we get GS = G and GV = GS /2 = G/2.
- 24. ρ meson condensation NJL and magnetic ﬁeld 24 / 36 Bosonization Nambu-Jona-Lasinio model after bosonization Introducing bosonic degrees of freedom SB = d4 x 1 2 (∂σ)2 − 1 2 m2 0σ2 + q=u,d ¯q i ˆD − mq q − 1 2G σ2 + π · π + 1 G (Va · Va + Aa · Aa) being i ˆD = i /D − mq + /ˆV + γ5 /ˆA − (σ + iγ5π · τ). After integration of the fermionic degrees of freedom yields SB = −iTrln(i ˆD − mq) + d4x[− 1 2G (σ2+π·π)+ 1 G (Va·Va+Aa·Aa)]. The fermionic determinant is singular and needs regularization.
- 25. ρ meson condensation NJL and magnetic ﬁeld 25 / 36 One-loop potential One-loop potential is easily obtained by V1L = −iTrlnS−1 q (x, x ) The fermionic propagator is obtained through Ritus-Leung-Wang technique Sq(x,y)= ∞ k=0 dpt dpy dpz (2π)4 EP (x)Λk i /P−mf ¯EP (y). This yields (βk = 2 − δk0 degeneracy of Landau levels) V1L = −NcT q=u,d |eqB| 2π ∞ k=0 βk ∞ n=−∞ dpz 2π × ln((2n + 1)2 π2 T2 + p2 z + 2k|eqB| + v2 ).
- 26. ρ meson condensation NJL and magnetic ﬁeld 26 / 36 Moving to ﬁnite temperature and magnetic ﬁeld Moving rules From the fermionic determinant and the one-loop potential we learn the rules to move from a standard NJL model to ﬁnite temperature and magnetic ﬁeld: 1 Change euclidean integrals to d4pE (2π)4 → T q |eqB| 2π k βk ∞ n=−∞ dpz 2π 2 Change p0 → (2n + 1)πT 3 Change the third component of momenta to p2 z → p2 z + 2k|eqB| 4 Finally, regularize.
- 27. ρ meson condensation Mass of the ρ in NJL model 27 / 36 ρ mass and NJL model ρ mass in NJL model comes out as the pole of the corresponding propagator [Ebert, Reinhardt and Volkov, Prog. Part. Nucl. Phys. 33, 1 (1994); Bernard, Meissner and Osipov, Phys. Lett. B 324, 201 (1994)]. This result is well-known in literature and is m2 ρ = 3 8GV NcJ2(0) being in our case GV = G/2. J2(0) is the following integral J2(0) = −iNf d4p (2π)4 1 (p2 − v2 + i )2 in need of regularization being divergent. Various techniques of regularization (e.g. Pauli-Villars) provide the value J2(0) = Nf 16π2 ln Λ2 v2 .
- 28. ρ meson condensation Mass of the ρ in NJL model 28 / 36 ρ mass and NJL model One can yield a numerical evaluation of the ρ mass from the above formula starting from the values in [Frasca, AIP Conf. Proc. 1492, 177 (2012); Nucl. Phys. Proc. Suppl. 234, 329 (2013)]. Λ si NJL cut-oﬀ while v is the eﬀective quark mass. At this level of approximation one gets 670 MeV that is rather good. Further corrections can be added [Klevansky and Lemmer, hep-ph/9707206 (unpublished)]. Our aim is to generalize the equation for the mass to ﬁnite temperature and magnetic ﬁeld. This will be accomplished through the moving rules introduced above.
- 29. ρ meson condensation Mass of the ρ in NJL model 29 / 36 ρ mass at ﬁnite T and B We need to move the integral J2(0) to the case of ﬁnite temperature and magnetic ﬁeld. This is immediately done through the moving rules introduced in the NJL model yielding NJL integral for the ρ mass J2(B, T) = T q |eqB| 2π k βk ∞ n=−∞ dpz 2π × 1 [(2n + 1)2π2T2 + p2 z + 2k|eqB| + v2]2 We are interested in the limit of decreasing T (T → 0).
- 30. ρ meson condensation Quantum phase transition in QCD 30 / 36 Strategy A quantum phase transition occurs when some other parameter is varied rather than the temperature (taken to be 0). In our case this parameter is the magnetic ﬁeld. We have to show that the integral J2 changes sign crossing 0, at T=0, for a critical magnetic ﬁeld. We observe that such a zero in J2 is just signaling that non-local eﬀects come into play at the critical magnetic ﬁeld.
- 31. ρ meson condensation Quantum phase transition in QCD 31 / 36 Computation Firstly, we perform the Matsubara sum that gives ∞ n=−∞ 1 [(2n+1)2π2T2+p2 z +2k|eqB|+v2]2 = 1 4T cosh ωk (pz ) 2T sinh ωk (pz ) 2T − ωk (pz ) 2T ω3 k (pz ) cosh2 ωk (pz ) 2T . At small T yields ∞ n=−∞ 1 [(2n+1)2π2T2+p2 z +2k|eqB|+v2]2 ≈ 1 4T 1− 2ωk (pz ) T e − ωk (pz ) T ω3 k (pz ) . being ωk (pz )= √ p2 z +2k|eqB|+v2.
- 32. ρ meson condensation Quantum phase transition in QCD 32 / 36 Computation Integrating on pz one gets J2(B, T) ≈ q |eqB| 16π2 k βk 1 2k|eqB| + v2 − 2 T F √ 2k|eqB|+v2 T 2k|eqB| + v2 being F(a) = π − 2a K0(a) + π 2 (K0(a)L1(a) + K1(a)L0(a)) where Kn are Macdonald functions and Ln modiﬁed Struve functions. We are in need to regularize the harmonic series and this is done in a standard way as ∞ k=0 βk 1 2k|eqB| + v2 = − 1 v2 − 1 |eqB| ψ v2 2|eqB| being ψ(a) the polygamma function.
- 33. ρ meson condensation Quantum phase transition in QCD 33 / 36 Recovering the ρ mass ρ mass at B = 0 can ber ecovered if we put an ultraviolet cut-oﬀ on Landau levels, N, and expand the polygamma function [Ruggieri, Tachibana and Greco,arXiv:1305.0137 [hep-ph]]. The harmonic sum takes the form N k=0 βk 1 2k|eqB| + v2 ≈ 1 |eqB| ln Λ2 v2 + 1 3 |eqB| v4 + . . . . Notice that the corrections due to the temperature goes to 0 exponentially for T → 0. So, we have recovered the J2 integral for B ≈ 0 J2(B, 0) B→0 = 1 8π2 ln Λ2 v2 + q=u,d 1 3 |eqB|2 v4
- 34. ρ meson condensation Quantum phase transition in QCD 34 / 36 High magnetic ﬁeld limit We can collect all the results given above to get J2(B,T)≈ 1 16π2 q − |eqB| v2 −ψ v2 2|eqB| − 2|eqB| T ∞ k=0 F √ 2k|eqB|+v2 T √ 2k|eqB|+v2 . In the limit T → 0, the contributions due to the temperature are exponentially damped and one should ask if the remaining function can change sign and for what value of the magnetic ﬁeld this happens. Indeed, this is so yielding the conclusion Critical magnetic ﬁeld |eBc| v2 = 0.9787896 . . . This shows that in the NJL at increasing magnetic ﬁeld, with the temperature going to zero, there is a quantum phase transition.
- 35. Conclusions 35 / 36 Conclusions We derived the Nambu-Jona-Lasinio model from QCD with all the parameters properly ﬁxed. The NJL model has been extended to the case of interaction with a magnetic ﬁeld. Computations have been performed providing the dependence of the ρ meson mass from the magnetic ﬁeld and the temperature. It has been shown that a quantum phase transition happens increasing the magnetic ﬁeld to a critical value for the ρ particle. This yields strong support to the hypothesis by Chernodub of a superconductive state for such mesons. Enlightening discussions with Marco Ruggieri are gratefully ac- knowledged.
- 36. Conclusions 36 / 36 Thanks a lot for your attention!

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