On the absorption and production cross sections of $K$ and $K^*$

We have computed the isospin and spin averaged cross sections of the processes $\pi K^*\to \rho K$ and $\rho K^*\to \pi K$, which are crucial in the determination of the abundances of $K^*$ and $K$ in heavy ion collisions. Improving previous calculations, we have considered several mechanisms which were missing, such as the exchange of axial and vector resonances ($K_1(1270)$, $K^*_2(1430)$, $h_1(1170)$, etc...) and also other processes such as $\pi K^*\to \omega K, \phi K$ and $\omega K^*,\,\phi K^*\to \pi K$. We find that some of these mechanisms give important contributions to the cross section. Our results also suggest that, in a hadron gas, $K^*$ production might be more important than its absorption.


I. INTRODUCTION
The study of nucleus-nucleus collisions at high energies [1][2][3][4], such as Au+Au at center of mass energies of 200 GeV or Pb-Pb at center of mass energies of 2.76 TeV, hints towards the existence of a phase transition from nuclear matter to a locally thermalized state of deconfined quarks and gluons, the quark-gluon plasma (QGP) [5]. After a hot initial stage, the QGP cools and hadronizes forming a hadron gas, where the produced mesons and baryons interact inelastically and the relative abundances are changed. After further cooling, the system reaches chemical equilibrium, where only elastic collisions take place. This is also called "chemical freeze-out" and at this point the abundances are frozen. Finally, at the "kinetic freeze-out", the density becomes small, the interactions no longer occur and the particles stream freely to the detectors [6,7]. After hadronization and before the kinetic freeze-out the hadrons can interact and different production and absorption reactions (including the formation and decay of resonances) will change the hadron abundances. These changes will be different for different hadron species and they depend on the details of hadron dynamics, especially on possible resonance formation.
Particularly interesting is the case of the K * (892) meson. The lifetime of this meson is around 4 fm/c, which is smaller than that of the QGP formed in heavy-ion collisions (∼ 10 fm/c [7]). This means that, from hadronization up to the kinetic freeze-out, a K * meson present in the QGP has enough time to decay into K and π. It can also be absorbed, as well as produced, by other mesons present in the medium. All these reactions can change the abundance of the K * at the kinetic freeze-out.
In Refs. [1][2][3][4], K * production was investigated considering data from Au+Au at center of mass energies of 200 GeV, from Cu+Cu at 62. 4  b 1 (1235) for I = 1, G = +1. The nature of these resonances has been tested in Refs. [14][15][16] where their decay widths in several channels were calculated and a good description of the experimental data was found. The inclusion of these resonances can contribute to the cross section of ρK * → πK.

and 200 GeV and from Pb+Pb collisions
The main purpose of the present work is to include the exchange of all these resonances in the study of the processes K * π → Kρ and K * ρ → Kπ. Besides resonance exchange, some other mechanisms are missing in the determination of the cross sections of K * π → Kρ and K * ρ → Kπ performed in Ref. [9]. For example, the exchange of a vector meson in the t-channel and a pseudoscalar in the s-channel were taken into account to study the reaction K * π → Kρ, but other mechanisms like u-channel exchange or s-channel exchange of vectors were not. Some of such missing diagrams involve anomalous vertices [17,18] (i.e., the natural parity is not conserved in the vertex, which is described by a Lagrangian containing the Levi-Civita pseudotensor). In Refs. [19,20] it was shown that interaction terms with anomalous parity couplings have a strong impact on the corresponding cross sections, and the relevance of such anomalous terms in the determination of the abundance of X(3872) in heavy ion collisions was computed in Ref. [21]. Such processes, involving the anomalous vertices, were missed in the earlier work of Ref. [22]. Similar is the case of the reaction K * ρ → πK: in Ref. [9] Feynman diagrams related to the exchange of a pseudoscalar meson in the t-channel and a vector meson in the s-channel were considered.
However, other contributions, as u-channel exchange diagrams and exchange of other mesons in the t-and s-channels were not taken into account. In this work we are going to evaluate the contribution from all such mechanisms and calculate the cross sections of the reactions K * π → Kρ, Kω, Kφ and K * ρ, K * ω, K * φ → Kπ for the absorption of the K * meson and the corresponding cross section for its production.

II. FORMALISM
In the model of Ref. [9], the effect of absorption and production of K * and K mesons in a hadron gas appears in the thermal average cross sections of such processes. These cross sections affect the time evolution of the abundance of K * and K. As concluded in Ref. [9], the most important absorption and production processes of K * and K correspond to πK * → ρK, ρK * → πK, K * → πK and the inverse reactions.
In the present work, we calculate these cross sections including the following reactions πK * → ρK, ωK, φK and ρK * , ωK * , φK * → πK. The cross sections associated with the corresponding inverse reactions can be obtained using the principle of detailed balance. Note that in Ref. [9], the cross sections related to processes involving ω and φ in the initial or final states were not evaluated in spite of their mass similarity with ρ as well as similar dynamics involved in the corresponding reactions.
We will calculate the cross section of the process a + b → c + d. For a specific reaction mechanism r, we can write σ r in the center of mass frame as [9,20,22] where s and t are the Mandelstam variables for the reaction r, m a,r and m b,r represent the masses of the two particles in the initial state of the reaction r, λ(a, b, c) is the Källén function and M r is the reduced matrix element for the process r.
The symbol S,I in Eq. (1) represents the sum over the spins (S) and isospins (I) projections of the particles in the initial and final states, weighted by the isospin and spin degeneracy factors of the two particles forming the initial state for the reaction r, i.e., S,I |M r | 2 → 1 (2I a,r + 1)(2I b,r + 1) 1 (2s a,r + 1)(2s b,r + 1) where, In Eq. (3), i and j represent the initial (a + b) and final (c + d) channels in the reaction r for a particular charge In Figs. 1 and 2 we show the different diagrams contributing to the processes πK * → ρK, ωK, φK and ρK * , ωK * , φK * → πK (without specifying the charge of the reaction).
Each of the amplitudes M ij of Eq. (3) can be written as where T ij , U ij and S ij are the contributions related to the t-, u-and s-channel diagrams shown in Figs. 1 and 2 for the process i → j for a particular total charge of the reaction r.
The L V V P Lagrangian written above contains the Levi-Civita pseudotensor since it describes an anomalous vertex, which involves a violation of the natural parity in the vertex [17,18].
In Eq. (5), P and V µ are matrices containing the octet of pseudoscalars and vectors mesons and the singlet of SU(3), respectively, which in the physical basis and considering ideal mixing for η and η as well as for ω and φ read as [10,11,28]: The couplings appearing in Eq. (5) are given by [29][30][31] with m V being the mass of the vector meson, which we take as the mass of the ρ meson, Refs. [10,11], and we list them in Tables I and II of the Appendix A for the convenience of the reader.
After defining all the ingredients needed for the evaluation of the contribution of the diagrams in Figs. 1 and 2, we can start writing the contributions explicitly. The t-channel, T ij , u-channel, U ij , and s-channel, S ij , amplitudes for the diagrams shown in Figs. 1a, 1b, and 1c, respectively, for a reaction r of the type i (a + b) → j (c + d) are given by where T ij k , T ij k , U ij k , S ij K and S ij K * are coefficients which depend on the initial i and final j channels, as well as the exchanged particle k, and they are given in Tables III-VII of the Appendix A. In Eqs. (9), (10) and (11), p, k are, respectively, the four-momentum of the π and K * in the initial state, and p and k correspond, respectively, to the four-momentum of the vector meson (ρ, ω or φ) and the K in the final state, µναβ is the Levi-Civita tensor and µ (q) is the polarization vector associated with the particle exchanged, with four momentum q. To arrive to these expressions we have made use of the Lorenz gauge, in which (p) · p = 0, and the fact that the contraction of an antisymmetric tensor, like the Levi-Civita tensor, with a symmetric one gives 0. The m P k in Eq. (9) corresponds to the mass of the exchanged pseudoscalar in Fig. 1a and m V k and Γ V k are the mass and width, respectively, of the exchanged vector. We have considered [13]: Γ φ = 4.3 MeV, Γ ω = 8. In Eqs. (10) and (11), g αβ is the Minkowski metric tensor. The M K 1 ,l , Γ K 1 ,l and g K 1 ,l present in Eq. (11) are the mass, width and coupling of the pole l (to the initial i and final j channels) associated with the axial state K 1 (1270). These values can be found in Table I of the Appendix A. In case of the t-channel amplitude of Eq. (9), we have considered the exchange of pseudoscalars as well as vector mesons. A note here is in order. When exchanging a pion in the t-channel in the reaction πK * → ρK, the energy-momentum conservation in the vertex π → πρ of Fig. 1a is such that the exchanged pion can become on-shell. Because of this, in some regions of the phase-space, the pion propagator develops a pole originating a singular cross section [32,33]. This latter singularity in the cross section can be removed by the so-called Peierls method [32], where the basic idea is to introduce a complex four-momentum for the unstable particle in the vertex by considering its decay width. As a consequence, the four-momentum of the exchanged particle gets an imaginary part through the energy-momentum conservation, which leads to [32,33] 1 where E π and E ρ are the energies for the external ρ and π in the center of mass frame. For In the case of the t-, u-and s− channel diagrams in Figs. 2a−2c, respectively, we find the following contributions where T ij , T ij , U ij K , U ij K * and S ij are coefficients which are given in Tables VIII−XII of the Appendix A. In Eqs. (13), (14), (15), and all diagrams depicted in Fig. 2, p, k, are, respectively, the four momenta of the external vector meson without strangeness (ρ, ω, or φ) and of the external K * , while p and k are the four momenta of the external π and K, A and g (2) A in Eq. (13) represent, respectively, the mass, width and coupling constants to the two vertices shown in Fig. 2a for the pole associated with the exchanged axial resonance R A (see Table II of the Appendix A for the numerical values). In Eq. (14), K 1 ,l and g (2) K 1 ,l correspond to the mass, width and coupling constants to the two vertices shown in Fig. 2b for the pole l related to the K 1 (1270) state and their numerical values are listed in Table I  (1)-(6) for the determination of the cross section.
As can be seen in Fig. 3, the contribution from the t-channel exchange of a pseudoscalar meson (not considered in Ref. [9] ) gives rise to the largest cross section and the other mechanisms considered produce small corrections to it. Note that due to a reordering of the particles in the vertices, the t-channel (u-channel) exchange in Ref. [9] corresponds to the u-channel (t-channel) exchange in the present work to which we refer throughout the text. It is also interesting to notice that the u-channel exchange of a vector meson (considered in Ref. [9]) leads to a larger cross section than that associated with the t-channel exchange of a vector meson (not evaluated in Ref. [9]) and the s-channel exchange of a pseudoscalar. The process in which a vector meson is exchanged in the s-channel (not taken into account in Ref. [9]) gives a larger contribution to the cross section when compared with the one arising from the exchange of a pseudoscalar in the s-channel (considered in Ref. [9]). It should be mentioned that the contribution of the K 1 (1270) exchange in the s-channel to the cross section is negligible (compare the solid and dotted lines of Fig. 3).
In Fig. 4, we show the results for the reactions πK * → ωK (left panel) and πK * → φK (right panel), reactions which were not considered in Ref. [9]. As can be seen, the final cross section for both reactions (solid lines) have similar magnitude and both are smaller than the one for the process πK * → ρK (solid line in Fig. 3) by around one order of magnitude. This finding indicates that the absorption mechanism of a K * by a pion, producing a K together with an ω or a φ may probably not be relevant in the determination of the time evolution of the abundances found in Ref. [9] for K * and K. Note, however, that without the contribution to the cross section of πK * → ρK from a diagram involving ρππ and K * πK vertices (not   Fig. 4) gives larger contribution to the cross section than the exchange of a vector or pseudoscalar mesons in the s-channel or a vector meson in the t-channel, with the latter mechanism being more important in case of the process πK * → ωK.
In Fig. 5 we show the cross section calculated with Eq. (1) for the process ρK * → πK  Table II in the t-channel and the exchange of K 1 (1270) in the u-channel produces a small modification in the total cross section. The contribution to the cross section from the exchange of a pseudoscalar meson in the t-channel is larger than that related to the exchange of a vector meson in the t-or u-channel (both missing in Ref. [9]) and that of a pseudoscalar meson in the u-channel (considered in Ref. [9]).
Since the s-channel exchange of a vector meson (taken into account in Ref. [9]) turns out to give a very small contribution to the cross section, the other mechanisms considered here become relevant.
Similar to the case ρK * → πK, the resonance exchange in the t-and u-channels for the reactions ωK * → πK and φK * → πK produces a weak modification in the cross section (compare the solid and dotted lines in both panels of Fig. 6). Interestingly, the final cross sections for ρK * → πK, ωK * → πK and φK * → πK have comparable magnitude. C. Resonance exchange in ρK * , ωK * , φK * → πK through triangular loops.
In addition to the mechanisms discussed so far to determine the cross sections of the reactions ρK * , ωK * , φK * → πK (see Fig. 2), one could also consider the possibility of exchanging a resonance in the s-channel, as in case of the K 1 (1270) exchange in πK * collisions (see Fig. 1). Indeed, in Ref. [34] the interaction of K * with ρ, ω and φ in s-wave (orbital angular momentum 0) was investigated and several K * resonances with I = 1/2 and different spin were found as a consequence of the dynamics involved: a J P = 0 + resonance with mass 1643 MeV and width of 48 MeV, which is a prediction of the theory; a 1 + resonance with mass 1737 MeV and width of 164 MeV which is associated with the state K * 1 (1650) listed by the PDG [13]; a J P = 2 + state with mass 1431 MeV and 56 MeV of width which is identified with the K * 2 (1430) listed by the PDG. Thus, exchange of these K * S states (with S indicating the spin) in the s-channel, as shown in Fig. 7, can be important while calculating the cross section for ρK * → πK. As can be seen in Fig. 7, one of the vertices involved in the process is the K * S πK vertex. From Ref. [34], we have information on the pole positions of these K * S states and their couplings to the channels ρK * , ωK * and φK * (which we list in Table XIII of the Appendix B), but the couplings to two pseudoscalars are not available.
However, one can still consider K * S exchange in the s-channel through an effective vertex, represented by a filled box in Fig. 7, by describing it through triangular loops (see Fig. 8). The details related to the determination of the amplitude for the process depicted in Fig. 7 can be found in Appendix B. Since the interaction of the initial vector-vector system in the diagram of Fig. 7 would generate these K * S states, the quantum numbers for the external vectors system can be J P = 0 + , 1 + or 2 + . The final state in Fig. 7 consists of two pseudoscalars (total spin 0), thus, the only way of getting J = 1 is with one unit of orbital angular momentum which leads the two pseudoscalar system to have J P = 1 − instead of the initial 1 + . This means that in the diagram of Fig. 7 we can not have a transition in the s-channel through the exchange of the K * 1 resonance found in Ref. [34]. Similarly, we can not have interference between the diagrams in Fig. 2c, which involves the exchange of a pseudoscalar or vector meson (thus, a initial state having negative total parity), and the diagram in Fig. 7. In Fig. 9 we show the cross section for the process ρK * → πK considering s-channel exchange of the K * S resonances. As can be seen by comparing with the results shown in Fig. 5, the contribution to the cross section of the mechanism depicted in Fig. 7 is very relevant. These results suggest that the inclusion of the process shown in Fig. 7 must strongly affect the production of K * and K in heavy ion collisions.
In Fig. 10 we show the results found for the cross section related to the s-channel exchange of K * S in case of the reactions ωK * → πK (left panel) and φK * → πK (right panel). By comparing with the results found in Fig. 6, this mechanism also produces changes in the cross section obtained without the s-channel K * S exchange, although milder than in case of the reaction ρK * → πK.  Table XIII.

D. Cross sections for the inverse reactions
We can obtain the cross section for the production of K * from the reactions ρK, ωK, φK → πK * and πK → ρK * , ωK * , φK * using the principle of detailed balance: if σ ab→cd is the cross section for the process a + b → c + d, calculated via Eq. (1), we can determine the cross section for the inverse reaction, c + d → a + b, as σ cd→ab = (2s a + 1)(2s b + 1)(2I a + 1)(2I b + 1) (2s c + 1)(2s d + 1)(2I c + 1)(2I d + 1) In Fig. 11 we show the results obtained for the K * production cross sections using the As can be seen in Fig. 11 (left panel), the absorption cross sections of K * by π are smaller than the corresponding ones for the production processes through collisions of K with ρ, ω or φ. The trend is the same in case of the absorption of K * by ρ for excitation energies above ∼ 90 MeV (right panel), while the absorption cross sections of K * by ω or φ are larger than those related to its production from collisions of π and K. However, for excitation energies bigger than ∼ 90 MeV the cross section for the πK → ρK * process dominates above all.
Very recently, K and K * formation in relativistic heavy-ion collisions has been investigated in the context of the Parton-Hadron-String dynamics (PHSD) transport approach [39,40], which considers the in-medium effects in the K andK * states through the modification of their spectral properties during the propagation through the medium. The authors conclude that final state interactions (in the hadron gas) contribute to reduce the ratio K * /K, corroborating the findings of [9].
Our main results summarized in Fig. 11. should be useful in the determination of the abundance ratio of K * and K from heavy ion collisions with more accuracy. Based on these results, we may anticipate that, in contrast to the previous expectations [9,39,40], the interactions in a pos-QGP hadronic medium may lead to an enhancement of the K * yield, not a suppression. A detailed analysis based on the study of the rate equations is in progress and will be published soon.

IV. CONCLUSIONS
We have determined the cross sections related to the processes πK * → ρK, ωK φK and ρK * , ωK * φK * → πK considering the exchange of pseudoscalars, vectors and several resonances. The reactions πK * → ρK and ρK * → πK, together with K * → Kπ, Kπ → K * , were found in Ref. [9] to be the reactions contributing dominantly to the abundance ratio of K * and K in heavy ion collisions. However, several mechanisms which could contribute to the cross sections of πK * → ρK and ρK * → πK were missing in Ref. [9]. With the purpose of obtaining information on such processes, we consider a more complete formalism, which takes into account more mechanisms and calculate cross sections. We find that some of these new contributions turn out to be especially important, as the pseudoscalar exchange in the t-channel for the processes πK * → ρK and ρK * → πK, exchange of resonances in the s-channel, like K * 2 (1430), for ρK * → πK, etc. We have also determined the cross sections for the inverse processes using the principle of detailed balance. The comparison between direct and inverse processes, shown in Fig. 11, suggests that the production of K * in a hadron gas is more important that its absorption. Our results should be useful in obtaining a more accurate time evolution for the abundance ratio of K * and K in heavy ion collisions.

V. ACKNOWLEDGEMENTS
being the total isospin and I 3 its third projection) In this way, for example, from Table II, we have that the coupling of the isospin 0, Gparity positive, state f 1 (1285) to 1 √ 2 [K * K + K * K ] (which corresponds to a positive G-parity combination) is g = 7230+i 0. This means that the state f 1 (1285) couples to the combination 1 √ 2 |K * K, I = 0, I 3 = 0 + |K * K , I = 0, from which we get  [10,11]. A two pole structure is found for K 1 (1270) in Refs. [10,11] and the values shown in this table have been taken from Ref. [11]. In Tables III-XII we TABLE III. Coefficients T ij k present in Eq. (9) for the reactions r = πK * → ρK, ωK, φK for total charge −1, 0, 1 and 2. The index i represents the initial state πK * for a particular charge configuration and the index j corresponds to the final state for the same total charge. The index k corresponds to the exchanged pseudoscalar, which we indicate in brackets next to the coefficient.
The absence of the coefficient for some k means that the coefficient is 0 for that exchanged particle.
If no exchanged particle is indicated next to the coefficient, the coefficient is 0 independently of the exchanged particle.   Table III for the notation used here.
Coefficients U ij k present in Eq. (10) for the reactions r = πK * → ρK, ωK, φK for total charge −1, 0, 1 and 2. See the caption of Table IV for the notation used here.
TABLE VI. Coefficients S ij K present in Eq. (11) for the reactions r = πK * → ρK, ωK, φK for total charge −1, 0, 1 and 2. In this case, a K 0 is exchanged for those processes whose total charge is 0 and a K + for total charge +1. For total charge −1 and 2, no particle can be exchanged in the s-channel.
TABLE VII. Coefficients S ij K * present in Eq. (11) for the reactions r = πK * → ρK, ωK, φK for total charge −1, 0, 1 and 2. In this case, a K * 0 is exchanged for those processes whose total charge is 0 and a K * + for total charge +1. In case of total charge −1 and 2, no particle can be exchanged in the s-channel.  (13) for the reactions r = ρK * , ωK * , φK * → πK for total charge −1, 0, 1 and 2. In this case, a vector meson is exchanged and we write the exchanged particle next to the coefficient. If the coefficient is 0 the process can not proceed via vector meson exchange.
Appendix B: Evaluation of the s-channel exchange of resonances in the reactions ρK * , ωK * , φK * → πK In this appendix, we determine the amplitude related to the process depicted in Fig. 7 in which the K * S states (where the subscript S indicates spin) found in Ref. [34] are exchanged in the s-channel through triangular loops (see Fig. 8). We have summarized the properties found in Ref. [34] for these K * S in Table XIII. We have the following expression for the amplitude of the process depicted in Fig. 7 S S S ij where the coefficients S (k 1 k 2 k 3 ) are given in Table XIV. The symbols M K * S and Γ K * S in Eq. (B1) are the mass and width of the poles related to the exchanged K * S state, while g are, respectively, the coupling constants of those poles to the initial state and to the vector mesons present in the triangular loops shown in Fig. 8. The numerical values for these quantities can be found in Table XIII.
To get Eq. (B1), we have used the following amplitude for the coupling of the K * S states to the vector mesons which, when involved in the triangular loop, give a nonzero coefficient.
where P S is a spin projector, which is given for the case of spin S = 0, 2 by [38] In Eq. (B3), q and p + k − q represent, respectively, the four momenta of the vector meson without strangeness and the K * meson present in the triangular loop of Fig. 8 and which are coupled to K * S . Since K * S can be considered as molecular state of ρK * and coupled channels [34] with its hadron components being in s-wave, the vector mesons present in the triangular loops and which couple to K * S , although being off-shell, should not be very far from being onshell (i.e., their respective modulus of the three-momenta are negligible as compared to their energies). Within such an interpretation of K * S , the temporal part of the polarization vectors (∼ modulus of momentum divided by mass) of the mesons at the resonance-mesonmeson vertex should be negligible as compared to the spatial components. This means that for the external as well as the internal vector mesons coupled to K * S we can use the approximation [34,38] polarizations with i and j being spatial indices. However, it would be more appropriate to maintain the covariant formalism instead of working with mixed indices (some spatial and other temporalspatial). This can be achieved by writting for the vector mesons coupled to K * S present in the triangular loop of Fig. 8. This approximation implies the inclusion, in the result, of a very small contribution arising from the temporal part of the polarization vector of these vector mesons. We have made use of this approximation to get Eq. (B1). When summing over the polarizations of the external vector which will produce negligible values for the temporal and temporal-spatial components. This is so because, as mentioned above, the external vectors, when interacting in s-wave and for energies not far away from the threshold (as in our case), generate the K * S (following the interpretation of Ref. [34]). Thus, the modulus of their momenta is much smaller than their energies, so polarizations 0 (k) 0 (k) = −g 00 + k 0 k 0 m 2 and same is the case for (p). Then, the use of Eqs. (B5) and (B6) is in line with the approximation in Eq. (B4).
The summation over the polarizations of the vector mesons in the triangular loop coupled to K * S gives rise to the P µν S present in Eq. (B1), which is a spin projector for the external vector mesons coupled to K * S . Within the approximation of Eqs. (B5) and (B6), we have for spin S = 0, 2 P µν 0 = 1 3 (p) · (k)g µν , These expressions can be compared with the spin projectors found in Ref. [38] for the case of spatial indices and neglecting the temporal part of the polarization vector, In this case, Eq. (B4) is used to sum over the polarizations.
In Eq. (B1), I (k 1 k 2 k 3 ) ν corresponds to the following integral with m V k 1 , m V k 2 = m K * being the masses of the two vector mesons which couple to K * S in the triangular loop of Fig. 8 and m P k 3 is the mass of the exchanged pseudoscalar. Using Lorentz covariance, the integral of Eqs. (B10) can be written as and we need to determine the coefficients a (k 1 k 2 k 3 ) , b (k 1 k 2 k 3 ) , · · · appearing in this expression.
The momentum and mass assignations for the particles involved in the triangular loop diagrams is shown in Fig. 12.
The determination of the four coefficients of Eq. (B11) can be done by making use of the Feynman parametrization and writing where α ≡ q 2 − m 2 In this way, [α + (β − α)x + (γ − β)y] = q 2 + r (k 1 k 2 k 3 ) ,  Table XIII. The mass m P k 3 is related to the pseudoescalars (π, η, η ) which can be exchanged. .