Near-threshold K ∗ (892 ) + meson production in the interaction of π − mesons with nuclei
NNear-threshold K ∗ (892) + meson production in theinteraction of π − mesons with nuclei E. Ya. Paryev , Institute for Nuclear Research, Russian Academy of Sciences,Moscow 117312, Russia Institute for Theoretical and Experimental Physics,Moscow 117218, Russia
Abstract
We study the inclusive strange vector meson K ∗ (892) + production in π − A reactions atnear-threshold laboratory incident pion momenta of 1.4–2.0 GeV/c within a nuclear spec-tral function approach. The approach accounts for incoherent primary π − meson–proton π − p → K ∗ (892) + Σ − production processes as well as the influence of the scalar K ∗ (892) + –nucleus potential (or the K ∗ (892) + in-medium mass shift) on these processes. We calculatethe absolute differential and total cross sections for the production of K ∗ (892) + mesons offcarbon and tungsten nuclei at laboratory angles of 0 ◦ –45 ◦ and at these momenta within fivescenarios for the above shift. We show that the K ∗ (892) + momentum distributions and theirexcitation functions (absolute and relative) possess a high sensitivity to changes in the in-medium K ∗ (892) + mass shift in the low-momentum region of 0.1–0.6 GeV/c. Therefore, themeasurement of such observables in a dedicated experiment at the GSI pion beam facility inthe near-threshold momentum domain will allow to get valuable information on the K ∗ (892) + in-medium properties. a r X i v : . [ nu c l - t h ] J u l Introduction
The study of the modification of the hadronic properties (masses and widths) of light non-strange vector mesons ρ , ω , φ , light strange pseudoscalar mesons K and ¯ K , pseudoscalar mesons η , η (cid:48) as well as mesons with open and hidden charm D and J/ψ in a strongly interacting environmenthas received considerable interest in recent years owing to the expectation to observe a partialrestoration of chiral symmetry in a nuclear medium (see, for example, [1–13]). The in-mediumproperties of hyperons at finite density have also been matter of intense theoretical investigationsin the last two decades [14–24]. Another interesting case of medium renormalization of hadrons isthat of the strange vector K ∗ (892) and axial-vector K (1270) mesons with the same charge states(or with the same quark structure q ¯ s or ¯ qs with q = u, d ), whose in-medium mass difference , asis expected [25–27], is sensitive to the chiral order parameter and, hence, will give the possibilityto identify unambiguously the effect of chiral symmetry breaking in nuclear medium. The K ∗ (892)and the K (1270) mesons in the quark model are a kaonic excitations with angular momenta oneand with opposite parities. Namely, their isospins, spins-parities are I ( J P ) = (1 − ) for the K ∗ (892)and I ( J P ) = (1 + ) for the K (1270). They are chiral partners and have relatively large vacuumdecay widths of 50 and 90 MeV, respectively, corresponding to a mean live-times of 4 and 2.2 fm/c.On the theoretical side, in literature there are a lot of publications devoted to the study of the in-medium properties of hadronic resonances K ∗ (892) and K (1270). Thus, the properties of ¯ K ∗ (892)and K ∗ (892) mesons in cold nuclear matter have been investigated in Refs. [28–30] and [30, 31],respectively, on the basis of chirally motivated model of the meson selfenergies. In particular, it wasshown that the ¯ K ∗ (892) in-medium width is enlarged beyond 200 MeV at normal nuclear matterdensity ρ , whereas that of K ∗ (892) is barely influenced by nuclear matter. The model predictsalso for the ¯ K ∗ (892) and K ∗ (892) mesons, respectively, a moderately attractive and repulsive reallow-energy nuclear potentials (or their in-medium mass shifts) of about -50 and +40 MeV at density ρ . These are similar to those for light strange mesons ¯ K and K . On the other hand in contrast,a negative mass shift of about -20 MeV has been predicted for the K ∗ (892) + meson, having thesame quark composition u ¯ s as the K + one, at rest at saturation density ρ within the quark-mesoncoupling model [32]. Mass shifts of the K ∗ (892) and K (1270) mesons of about -40 and -150 MeV atdensity ρ were obtained in a more recent calculations in the framework of the three-flavor extendedlinear sigma model [33]. Other recent calculations performed in Ref. [26] using QCD sum rule showthat the upper limits of the mass shifts of K − and K +1 mesons in nuclear matter are -249 and -35MeV, respectively.What concerns the experimental situation, up to now only a scarce data on K ∗ production inheavy-ion and proton–proton collisions have been collected in the experiments performed in theSPS [34], RHIC [35] and LHC [36] energy domains. At SIS energies, the subthreshold and deepsubthreshold production of K ∗ (892) mesons in Al+Al and Ar+KCl collisions at a beam kineticenergies of 1.9 A GeV and 1.76 A GeV, respectively, has been reported by the FOPI [37] and theHADES [38] Collaborations. While the K ∗ (892) / K yield ratio deduced in the FOPI experimentis found in good agreement with the corresponding prediction of the UrQMD transport model, thisratio extracted in the HADES experiment is overestimated by the model by factor of about two.Probably, less discrepancy might appear here if in-medium modifications of kaon properties will beimplemented into this transport model. The medium modification of the K (1270) meson couldbe probed at J-PARC through the K − reaction on various nuclear targets [26]. Such measurementtogether with that of K ∗ (892) will shed light on the partial restoration of chiral symmetry in nuclearmatter [26].As a guidance for such future dedicated experiments and as a first step in the implementationof this programme, in the present study we give the predictions for the absolute differential andtotal cross sections for near-threshold production of K ∗ (892) + mesons in π − C → K ∗ (892) + X and π − W → K ∗ (892) + X reactions at laboratory angles of 0 ◦ –45 ◦ by incident pions with momenta2elow 2.0 GeV/c as well as for their relative yields from these reactions within different scenarios forthe K ∗ (892) + in-medium mass shift. These nuclear targets were employed in recent measurements[39] of φ meson production in π − A reactions at the GSI pion beam facility using the HADES spec-trometer and, therefore, can be adopted in studying the π − A → K ∗ (892) + X interactions here. Thecalculations are based on a first-collision model using an eikonal approximation, developed in Refs.[9, 10, 21] for the description of the inclusive φ and η (cid:48) meson as well as Λ(1520) hyperon productionand extended to account for different scenarios for the K ∗ (892) + in-medium mass shift. This modelis based on the quasiparticle picture and, therefore, it is more appropriate for consideration of the K ∗ meson production in nuclei than for the study of the ¯ K ∗ creation here since, contrary to the ¯ K ∗ ,the K ∗ meson behaves in the medium as a quasiparticle with a single-peak spectral function anda modified effective mass [30, 31]. Our calculations can be used as an important tool for possibleextracting of the valuable information on the K ∗ (892) + in-medium mass shift from the data whichcould be taken in a dedicated experiment at the GSI pion beam facility. K ∗ (892) + meson production onnuclei Since we are interested in near-threshold incident pion beam momenta below 2.0 GeV/c, we haveaccounted for the following direct elementary K ∗ (892) + production process which has the lowestfree production threshold momentum (1.84 GeV/c) : π − + p → K ∗ (892) + + Σ − . (1)For numerical simplicity, in our calculations we will account for the medium modification of the final K ∗ (892) + meson, participating in the production process (1), by adopting its average in-mediummass < m ∗ K ∗ > instead of its local effective mass m ∗ K ∗ ( | r | ) in the in-medium cross section of thisprocess, with < m ∗ K ∗ > defined according to Refs. [9, 10] as: < m ∗ K ∗ > = m K ∗ + V < ρ N >ρ . (2)Here, m K ∗ is the K ∗ (892) + free space mass, V is the K ∗ (892) + effective scalar nuclear potential(or its in-medium mass shift) at normal nuclear matter density ρ , and < ρ N > is the averagenucleon density. For target nuclei C and
W, the ratio < ρ N > /ρ , was chosen as 0.55 and0.76, respectively, in the present work. With regards to the quantity V , we will adopt for it in linewith above-mentioned the five following options: i) V = −
40 MeV, ii) V = −
20 MeV, iii) V = 0MeV, iv) V = +20 MeV, and v) V = +40 MeV throughout the study. Following the predictionsof the chiral effective field theory approach [18, 40], SU(6) quark model [41, 42] for the fate ofhyperons in nuclear matter and phenomenological information deduced from hypernuclear data [6,43] that the Σ hyperon experiences only a moderately repulsive nuclear potential of about 10–40MeV at central nuclear densities and finite momenta as well as a weakly attractive potential at thesurface of the nucleus, we will ignore the modification of the mass of the Σ − hyperons, producedtogether with the K ∗ (892) + mesons in the process (1), in the nuclear medium. Accounting for thatthe in-medium threshold energy √ s ∗ th = < m ∗ K ∗ > + m Σ − of the process (1) looks like that for the We can ignore in the momentum domain of interest the contribution to the K ∗ (892) + yield from the processes π − p → K ∗ (892) + Λ π − and π − N → K ∗ (892) + Σ π due to larger their production thresholds ( ≈ π − p and π − N collisions. Moreover, taking into consideration the results of the study [9] of pion-induced φ meson production on C and
W nuclei at beam momentum of 1.7 GeV/c, we neglect in this domainby analogy with [9] the secondary pion–nucleon πN → K ∗ (892) + Λ and πN → K ∗ (892) + Σ production processes. Determining mainly the strength of the K ∗ (892) + production cross sections in near-threshold pion–nucleuscollisions. E (cid:48) K ∗ of the K ∗ (892) + meson in nuclear matter is expressed via its averageeffective mass < m ∗ K ∗ > and its in-medium momentum p (cid:48) K ∗ by the expression [9, 10]: E (cid:48) K ∗ = (cid:113) ( p (cid:48) K ∗ ) + ( < m ∗ K ∗ > ) . (3)The momentum p (cid:48) K ∗ is related to the vacuum K ∗ (892) + momentum p K ∗ as follows [9, 10]: E (cid:48) K ∗ = (cid:113) ( p (cid:48) K ∗ ) + ( < m ∗ K ∗ > ) = (cid:113) p K ∗ + m K ∗ = E K ∗ , (4)where E K ∗ is the K ∗ (892) + total energy in vacuum.Since the K ∗ (892) + –nucleon total cross section is expected to be small [44], we will neglect bothinelastic and quasielastic K ∗ (892) + N interactions in the present study. Then, accounting for thedistortion of the incident pion in nuclear matter and the attenuation of the flux of the K ∗ (892) + mesons in the nucleus due to their decays here as well as using the results given in [9, 10, 21],we represent the inclusive differential cross section for the production of K ∗ (892) + mesons withvacuum momentum p K ∗ on nuclei in the direct process (1) as follows: dσ (prim) π − A → K ∗ (892) + X ( p π − , p K ∗ ) d p K ∗ = I V [ A, θ K ∗ ] (cid:18) ZA (cid:19) (cid:42) dσ π − p → K ∗ (892) + Σ − ( p π − , p (cid:48) K ∗ ) d p (cid:48) K ∗ (cid:43) A d p (cid:48) K ∗ d p K ∗ , (5)where I V [ A, θ K ∗ ] = A R (cid:90) r ⊥ dr ⊥ √ R − r ⊥ (cid:90) − √ R − r ⊥ dzρ ( (cid:113) r ⊥ + z ) exp − σ tot π − N A z (cid:90) − √ R − r ⊥ ρ ( (cid:113) r ⊥ + x ) dx (6) × π (cid:90) dϕ exp − l ( θ K ∗ ,ϕ ) (cid:90) dxλ K ∗ ( (cid:113) x + 2 a ( θ K ∗ , ϕ ) x + b + R ) ,a ( θ K ∗ , ϕ ) = z cos θ K ∗ + r ⊥ sin θ K ∗ cos ϕ, b = r ⊥ + z − R , (7) l ( θ K ∗ , ϕ ) = (cid:113) a ( θ K ∗ , ϕ ) − b − a ( θ K ∗ , ϕ ) , (8) λ K ∗ ( | r | ) = p (cid:48) K ∗ m ∗ K ∗ ( | r | )Γ K ∗ , m ∗ K ∗ ( | r | ) = m K ∗ + V ρ N ( | r | ) ρ (9)and (cid:42) dσ π − p → K ∗ (892) + Σ − ( p π − , p (cid:48) K ∗ ) d p (cid:48) K ∗ (cid:43) A = (cid:90) (cid:90) P A ( p t , E ) d p t dE (10) × (cid:40) dσ π − p → K ∗ (892) + Σ − [ √ s, < m ∗ K ∗ >, m Σ − , p (cid:48) K ∗ ] d p (cid:48) K ∗ (cid:41) ,s = ( E π − + E t ) − ( p π − + p t ) , (11) E t = M A − (cid:113) ( − p t ) + ( M A − m N + E ) . (12)Here, dσ π − p → K ∗ (892) + Σ − [ √ s, < m ∗ K ∗ >, m Σ − , p (cid:48) K ∗ ] /d p (cid:48) K ∗ is the off-shell inclusive differential crosssection for the production of K ∗ (892) + meson and Σ − hyperon with modified mass < m ∗ K ∗ > and Eq. (9) shows that for typical values p (cid:48) K ∗ ≈ m ∗ K ∗ and vacuum total K ∗ (892) + decay width in its rest frameΓ K ∗ = 50 MeV the K ∗ (892) + decay mean free path λ K ∗ is equal to 4 fm. This value is comparable with the radiusof C of 3 fm and it is much less than that of
W of 7.4 fm. m Σ − , respectively. The K ∗ (892) + meson is produced with in-medium momentum p (cid:48) K ∗ inprocess (1) at the π − p center-of-mass energy √ s . E π − and p π − are the total energy and momentumof the incident pion ( E π − = (cid:113) m π + p π − , m π is the free space pion mass); ρ ( r ) and P A ( p t , E ) arethe local nucleon density and the spectral function of the target nucleus A normalized to unity (theconcrete information about these quantities, used in the subsequent calculations, is given in Refs.[9, 45–47]); p t and E are the internal momentum and removal energy of the struck target protoninvolved in the collision process (1); σ tot π − N is the total cross section of the free π − N interaction (weuse in our calculations the value of σ tot π − N = 35 mb for initial pion momenta of interest); Z and A are the numbers of protons and nucleons in the target nucleus, and M A and R are its mass andradius; m N is the free space nucleon mass; and θ K ∗ is the polar angle of vacuum momentum p K ∗ in the laboratory system with z-axis directed along the momentum p π − of the incident pion beam.In line with [9], we assume that the off-shell differential cross section dσ π − p → K ∗ (892) + Σ − [ √ s,
089 GeV is replaced by the in-medium one √ s ∗ th and the free collision energy s = ( E π − + m N ) − p π − – by the in-medium expression (11). For the free total cross section σ π − p → K ∗ (892) + Σ − ( √ s, √ s th ) we have adopted the following parametrization of the available scarceexperimental data [48]: σ π − p → K ∗ (892) + Σ − ( √ s, √ s th ) = . (cid:16) √ s − √ s th (cid:17) . [ µ b] for 0 < √ s − √ s th ≤ .
355 GeV , . / (cid:16) √ s − √ s th (cid:17) . [ µ b] for √ s − √ s th > .
355 GeV . (18)As can be seen from Fig. 1, the parametrization (18) (solid line) fits reasonably well the data [48] p p - = 1 . 4 G e V / cp t = 0 . 5 G e V / c s [ m b] s - s t h 1 / 2 [ G e V ] p - p - > K * ( 8 9 2 ) + S - p a r a m e t r i z a t i o np p - = 1 . 7 G e V / cp t = 0 . 2 5 G e V / c Figure 1: (color online) Total cross section for the reaction π − p → K ∗ (892) + Σ − as a function of theexcess energy √ s − √ s th . The left and right arrows indicate the excess energies √ s − √ s th =46 MeVand √ s − √ s th =99 MeV corresponding to the incident pion momenta of 1.4 and 1.7 GeV/c and atarget proton bound in C by 16 MeV and having momenta of 500 and 250 MeV/c, respectively.The latter ones are directed opposite to the incoming pion beam. The middle arrow indicates theexcess energy √ s − √ s th =70 MeV corresponding to the initial pion momentum of 2.0 GeV/c anda free target proton at rest. For the rest of notation see text.(full circles) for the π − p → K ∗ (892) + Σ − reaction. One can also see that the on-shell cross section σ π − p → K ∗ (892) + Σ − amounts approximately to 31 µ b for the initial pion momentum of 2.0 GeV/c anda free target proton being at rest. The off-shell cross section σ π − p → K ∗ (892) + Σ − , calculated in linewith Eqs. (11), (12), (18) for a pion momenta of 1.4 and 1.7 GeV/c and a target proton bound in C by 16 MeV and having relevant internal momenta of 500 and 250 MeV/c, is about 28 and 35 µ b, respectively . This opens the possibility of measuring the K ∗ (892) + yield in π − A reactions It should be noted that these data correspond to the initial laboratory π − momenta belonging to the range1.97 GeV/c ≤ p π − ≤ It is interesting to note that the excess energy is equal to -67 MeV for a pion momentum of 1.4 GeV/c anda target proton bound in C by 16 MeV and having internal momentum of 250 MeV/c directed opposite to theinitial pion beam. This means that the main contribution to the deep subthreshold K ∗ (892) + production on nucleicomes from the dynamically formed compact nucleonic configurations – in particular, from pairs of correlated pn , pp clusters. K ∗ (892) + three- p p - = 2 . 0 G e V / c q K * = 0 - 4 5 p - + C - > K * ( 8 9 2 ) + + X d s /dpK* [ m b/(GeV/c)] p K * [ G e V / c ] p - + W - > K * ( 8 9 2 ) + + X V = 4 0 M e V V = 2 0 M e V V = 0 M e V V = - 2 0 M e V V = - 4 0 M e V d s /dpK* [ m b/(GeV/c)] p K * [ G e V / c ] Figure 2: (color online) Momentum differential cross sections for the production of K ∗ (892) + mesonsfrom the primary π − p → K ∗ (892) + Σ − channel in the laboratory polar angular range of 0 ◦ –45 ◦ inthe interaction of π − mesons of momentum of 2.0 GeV/c with C (left) and
W (right) nuclei,calculated for different values of the K ∗ (892 + ) meson effective scalar potential V at density ρ indicated in the inset. The arrows indicate the boundary between the low-momentum and high-momentum regions of the K ∗ (892) + spectra.momentum is not changed during the propagation from its production point inside the nucleus in therelatively weak nuclear field, considered in the work, to the vacuum far away from the nucleus. As aconsequence, the quantities (cid:68) dσ π − p → K ∗ (892) + Σ − ( p π − , p (cid:48) K ∗ ) /d p (cid:48) K ∗ (cid:69) A and d p (cid:48) K ∗ /d p K ∗ , entering into Eq.(5), can be put in the simple forms (cid:68) dσ π − p → K ∗ (892) + Σ − ( p π − , p (cid:48) K ∗ , θ K ∗ ) /p (cid:48) K ∗ dp (cid:48) K ∗ d Ω K ∗ (cid:69) A and p (cid:48) K ∗ /p K ∗ ,where Ω K ∗ ( θ K ∗ , ϕ K ∗ ) = p K ∗ /p K ∗ . Here, ϕ K ∗ is the azimuthal angle of the K ∗ (892) + momentum p K ∗ in the laboratory system. Accounting for the HADES spectrometer acceptance as well as the factthat in the considered energy region K ∗ (892) + mesons are mainly emitted, due to the kinematics, inforward directions , we will calculate the K ∗ (892) + momentum differential and total productioncross sections on C and
W target nuclei for laboratory solid angle ∆ Ω K ∗ =0 ◦ ≤ θ K ∗ ≤ ◦ , and0 ≤ ϕ K ∗ ≤ π . Integrating the full inclusive differential cross section (5) over this angular domain,we can represent the differential cross section for K ∗ (892) + meson production in π − A collisionsfrom the direct process (1), corresponding to the HADES acceptance window , in the following Thus, for instance, at a beam momentum of 2.0 GeV/c the K ∗ (892) + laboratory production polar angles inreaction (1) proceeding on the target proton being at rest are ≤ ◦ . At HADES the K ∗ (892) + mesons could be identified via the hadronic decays K ∗ (892) + → K π + with abranching ratio of 2/3 or via their radiative decays K ∗ (892) + → K + γ with sizable branching ratio of 10 − [49]. . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 21 E - 51 E - 41 E - 30 . 0 10 . 111 0 d s /dpK* [ m b/(GeV/c)] p K * [ G e V / c ] p - + C - > K * ( 8 9 2 ) + + Xp p - = 1 . 7 G e V / c q K * = 0 - 4 5 p - + W - > K * ( 8 9 2 ) + + X V = 4 0 M e V V = 2 0 M e V V = 0 M e V V = - 2 0 M e V V = - 4 0 M e V d s /dpK* [ m b/(GeV/c)] p K * [ G e V / c ] Figure 3: (color online) The same as in Fig.2, but for the incident pion beam momentum of 1.7GeV/c.form: dσ (prim) π − A → K ∗ (892) + X ( p π − , p K ∗ ) dp K ∗ = (cid:90) ∆ Ω K ∗ d Ω K ∗ dσ (prim) π − A → K ∗ (892) + X ( p π − , p K ∗ ) d p K ∗ p K ∗ (19)= 2 π (cid:18) ZA (cid:19) (cid:32) p K ∗ p (cid:48) K ∗ (cid:33) (cid:90) cos 45 ◦ d cos θ K ∗ I V [ A, θ K ∗ ] (cid:42) dσ π − p → K ∗ (892) + Σ − ( p π − , p (cid:48) K ∗ , θ K ∗ ) dp (cid:48) K ∗ d Ω K ∗ (cid:43) A . In the beginning, we consider the absolute K ∗ (892) + momentum differential cross sections fromthe direct K ∗ (892) + production mechanism in π − C and π − W collisions. These cross sectionswere calculated according to Eq. (19) in five considered scenarios for the K ∗ (892) + in-medium massshift at density ρ at laboratory angles of 0 ◦ –45 ◦ and for incident pion momenta of 2.0, 1.7 and 1.4GeV/c. They are presented, respectively, in Figs. 2, 3 and 4. It is seen that the K ∗ (892) + mesonmomentum distributions are appreciable sensitive to its in-medium mass shift mainly in the low-momentum region of 0.1–0.6 GeV/c for both target nuclei and for all considered beam momenta.Here there are a sizeable and experimentally accessible differences between the results obtained byemploying different K ∗ (892) + in-medium mass shifts under consideration, which for each target8 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 71 E - 71 E - 61 E - 51 E - 41 E - 30 . 0 1 p p - = 1 . 4 G e V / c q K * = 0 - 4 5 p - + C - > K * ( 8 9 2 ) + + X d s /dpK* [ m b/(GeV/c)] p K * [ G e V / c ] p - + W - > K * ( 8 9 2 ) + + X V = 4 0 M e V V = 2 0 M e V V = 0 M e V V = - 2 0 M e V V = - 4 0 M e V d s /dpK* [ m b/(GeV/c)] p K * [ G e V / c ] Figure 4: (color online) The same as in Fig.2, but for the incident pion beam momentum of 1.4GeV/c.nucleus are practically similar to each other at these initial pion momenta. Thus, for example, forincident pion and outgoing K ∗ (892) + meson momenta of 2.0 and 0.3 GeV/c, respectively, and inthe case of C nucleus the K ∗ (892) + yield is enhanced at mass shift V = +20 MeV by about afactor of 4.1 as compared to that obtained for the shift V = +40 MeV. When going from V = +20MeV to V = 0 MeV, from V = 0 MeV to V = −
20 MeV and from V = −
20 MeV to V = − W target nucleus theseenhancement factors are about 14.0, 3.5, 2.1 and 1.7. At initial beam momentum of 1.4 GeV/cand the same outgoing kaon momentum of 0.3 GeV/c the corresponding enhancement factors aresimilar and are about 5.0, 3.3, 2.6 and 2.5 as well as 6.0, 3.3, 2.4 and 2.0 in the cases of C as wellas
W target nuclei, respectively. However, the K ∗ (892) + low-momentum production differentialcross sections at beam momentum of 1.4 GeV/c are very small (in the range of ∼ µ b/(GeV/c)) and they are less than those at pion momenta of 1.7 and 2.0 GeV/c by about oftwo–three orders of magnitude. Therefore, the measurements of the K ∗ (892) + differential crosssections in π − A reactions in the near-threshold incident pion momentum region (at 1.7–2.0 GeV/c)with the aim of distinguishing between considered options for the K ∗ (892) + mass shift in nuclearmatter look promising.The sensitivity of the low-momentum parts of the K ∗ (892) + meson production differential crosssections to its in-medium mass shift V , shown in Figs. 2, 3, 4, can be also studied from suchintegral measurements as the measurements of the total cross sections for K ∗ (892) + production in9 p - + C - > K * ( 8 9 2 ) + + X s [ m b] p K * = 0 . 1 - 0 . 6 G e V / c - 4 0 - 2 0 0 2 0 4 01 E - 30 . 0 10 . 111 0 p K * = 0 . 1 - 0 . 6 G e V / c p - + W - > K * ( 8 9 2 ) + + X - 4 0 - 2 0 0 2 0 4 01 E - 51 E - 41 E - 30 . 0 10 . 111 0 p - + C - > K * ( 8 9 2 ) + + X s [ m b] p o t e n t i a l V [ M e V ]a l l a l l o w e d m o m e n t a - 4 0 - 2 0 0 2 0 4 01 E - 30 . 0 10 . 111 01 0 0 q K * = 0 - 4 5 p - + W - > K * ( 8 9 2 ) + + X p p - = 2 . 0 G e V / c p p - = 1 . 7 G e V / c p p - = 1 . 4 G e V / cp o t e n t i a l V [ M e V ]a l l a l l o w e d m o m e n t a Figure 5: (color online) The total cross sections for the production of K ∗ (892) + mesons from theprimary π − p → K ∗ (892) + Σ − channel on C and W target nuclei with momenta of 0.1–0.6 GeV/c(upper two panels) and with all allowed momenta ≥ ◦ –45 ◦ by 1.4, 1.7 and 2.0 GeV/c π − mesonsas functions of the effective scalar K ∗ (892) + potential V at normal nuclear density. The lines areto guide the eye. π − C and π − W reactions by 1.4, 1.7 and 2.0 GeV/c pions at laboratory angles of 0 ◦ –45 ◦ inthe low-momentum (0.1–0.6 GeV/c) and in the allowed for given beam momentum full-momentumregions. These cross sections, calculated by integrating Eq. (19) over the K ∗ (892) + momentum p K ∗ in these regions, are shown in Fig. 5 as functions of the mass shift (or effective scalar potential) V . It can be seen from this figure that again the low-momentum range of 0.1–0.6 GeV/c shows thehighest sensitivity to this potential. Thus, for instance, the ratios between the total cross sections of K ∗ (892) + production by 1.4, 1.7, 2.0 GeV/c pions on C and
W target nuclei in this momentumrange, calculated with the potential V = −
40 MeV, and the same cross sections, obtained in thescenario V = +40 MeV, respectively, are about 21.0, 5.0, 3.0 and 14.0, 10.0, 4.0. While the sameratios in the full-momentum regions are about 21.0, 3.0, 1.4 for C and 14.0, 4.0, 1.7 for
W. Inthe low-momentum region of interest the highest sensitivity of the K ∗ (892) + production total crosssections to the potential V is observed, as is expected, at initial pion momentum of 1.4 GeV/c.However, these cross sections are small and they are less than those at beam momenta of 1.7 and 2.0GeV/c by several orders of magnitude. Since the latter ones have a measurable strengh ∼ µ b,the low-momentum total cross section measurements of K ∗ (892) + meson production on nuclei in10 p K * = 0 . 1 - 0 . 6 G e V / c p - + C - > K * ( 8 9 2 ) + + X p p - = 2 . 0 G e V / c p p - = 1 . 7 G e V / c p p - = 1 . 4 G e V / c R - 4 0 - 2 0 0 2 0 4 0012345 p K * = 0 . 1 - 0 . 6 G e V / c p - + W - > K * ( 8 9 2 ) + + X - 4 0 - 2 0 0 2 0 4 0012345 a l l a l l o w e d m o m e n t a p - + C - > K * ( 8 9 2 ) + + X R p o t e n t i a l V [ M e V ] - 4 0 - 2 0 0 2 0 4 0012345 q K * = 0 - 4 5 a l l a l l o w e d m o m e n t a p - + W - > K * ( 8 9 2 ) + + Xp o t e n t i a l V [ M e V ] Figure 6: (color online) Ratio between the total cross sections for the production of K ∗ (892) + mesons from the primary π − p → K ∗ (892) + Σ − channel on C and
W target nuclei at laboratoryangles of 0 ◦ –45 ◦ with momenta of 0.1–0.6 GeV/c (upper two panels) and with all allowed momenta ≥ π − mesons,calculated with and without the K ∗ (892) + in-medium mass shift V at normal nuclear density, asfunction of this shift. The lines are to guide the eye.the near-threshold incident pion momentum region ∼ K ∗ (892) + in-medium mass shift V at the central density ρ is clearly supported also by the resultsgiven in Fig. 6. Here, the ratios R of the K ∗ (892) + meson production total cross sections calculatedfor its mass shift V and presented in Fig. 5 to the analogous cross sections determined at V = 0MeV are shown as functions of this mass shift. It is worth noting that an analysis of these ratios hasthe advantage that they do not depend on the absolute normalization of calculated and measuredcross sections. As is seen from this figure, the highest sensitivity of the ratios in both consideredkinematic ranges to the quantity V is indeed observed at pion momentum of 1.4 GeV/c. Forexample, at this momentum and for these ranges the cross section ratios R for V = −
40 MeVare about 4.5 and 3.2 for C and
W, respectively. As the pion-beam momentum increases to1.7 and 2.0 GeV/c, the sensitivity of the cross section ratios to variations in the mass shift V . 4 1 . 5 1 . 6 1 . 7 1 . 8 1 . 9 2 . 01 E - 51 E - 41 E - 30 . 0 10 . 111 01 0 0 p K * = 0 . 1 - 0 . 6 G e V / c p - + C - > K * ( 8 9 2 ) + + X s [ m b] p K * = 0 . 1 - 0 . 6 G e V / c p - + W - > K * ( 8 9 2 ) + + X a l l a l l o w e d m o m e n t a p - + C - > K * ( 8 9 2 ) + + X s [ m b] p i o n m o m e n t u m [ G e V / c ] a l l a l l o w e d m o m e n t a p - + W - > K * ( 8 9 2 ) + + X V = - 4 0 M e V V = - 2 0 M e V V = 0 M e V V = 2 0 M e V V = 4 0 M e V p i o n m o m e n t u m [ G e V / c ] q K * = = 0 - 4 5 Figure 7: (color online) The total cross sections for the production of K ∗ (892) + mesons from theprimary π − p → K ∗ (892) + Σ − channel on C and
W target nuclei at laboratory angles of 0 ◦ –45 ◦ with momenta of 0.1–0.6 GeV/c (upper two panels) and with all allowed momenta ≥ K ∗ (892) + in-medium mass shift V at normal nuclear density depicted in the inset, as functions of the incident pion momentum.becomes somewhat lower. Thus, in the case where K ∗ (892) + mesons of momenta of 0.1–0.6 GeV/care produced by 1.7 and 2.0 GeV/c pions incident to a C as well as
W targets, the ratiosbeing considered take for V = −
40 MeV smaller but yet a sizeable values of 2.1 and 1.6 as well as2.6 and 1.8, respectively. The analogous ratios for the production of the K ∗ (892) + mesons in thefull-momentum regions by 1.7 and 2.0 GeV/c pions on C as well as
W nuclei are somewhat yetsmaller. Namely, they are about 1.6 and 1.2 as well as 1.9 and 1.3, respectively.Therefore, we come to the conclusion that a comparison of the low-momentum ”integral” resultsshown in Figs. 5, 6 with the respective near-threshold experimental data, which could be taken infuture experiments using π − beams at the GSI pion beam facility or at J-PARC [50], will also allowto study the in-medium properties of the K ∗ (892) + mesons.These properties can be also investigated [5] from such another integral meaurements as themeasurements of the excitation functions for K ∗ (892) + production in π − C and π − W reactionsat laboratory angles of 0 ◦ –45 ◦ in the low-momentum (0.1–0.6 GeV/c) and in the full-momentumregions. They were calculated for five adopted scenarios for the K ∗ (892) + in-medium mass shift12nd are given in Fig. 7. One can see that the absolute values of the excitation functions show awider variation for the mass shift range of V = −
40 to +40 MeV in the low-momentum region forall considered beam momenta. In this momentum region and at beam momenta not far below thethreshold (at p π − ∼ ∼ C and ∼ W) between all calculations corresponding todifferent options for the K ∗ (892) + in-medium mass shift. Here, the total K ∗ (892) + production crosssections have a measurable strength ∼ K ∗ (892) + meson mass shift on its yield becomes somewhat lower. Here,the respective differences are ∼ C and ∼ W, but one might expect tomeasure their as well in future experiments at the GSI pion beam facility .Taking into account the above consideration, one can conclude that the near-threshold K ∗ (892) + differential and total cross section measurements at K ∗ (892) + momenta of 0.1–0.6 GeV/c in π − A interactions will allow to shed light on the possible K ∗ (892) + in-medium mass shift at these mo-menta. In this paper we study the inclusive strange vector meson K ∗ (892) + production in π − A re-actions at near-threshold laboratory incident pion momenta of 1.4–2.0 GeV/c within a nuclearspectral function approach. The approach accounts for incoherent primary π − meson–proton π − p → K ∗ (892) + Σ − production processes as well as the influence of the scalar K ∗ (892) + –nucleuspotential (or the K ∗ (892) + in-medium mass shift) on these processes. We calculate the absolutedifferential and total cross sections for the production of K ∗ (892) + mesons on carbon and tungstentarget nuclei at laboratory angles of 0 ◦ –45 ◦ and at these initial pion momenta within five scenar-ios for the above shift. We show that the K ∗ (892) + momentum distributions and their excitationfunctions (absolute and relative) possess a high sensitivity to changes in the in-medium K ∗ (892) + mass shift in the low-momentum region of 0.1–0.6 GeV/c. Therefore, the measurement of such ob-servables in a dedicated experiment at the GSI pion beam facility in the near-threshold momentumdomain will allow to get valuable information on the K ∗ (892) + in-medium properties. References [1] R. Rapp and J. Wambach, Adv. Nucl. Phys. , 1 (2000);arXiv:hep-ph/9909229.[2] R. S. Hayano and T. Hatsuda, Rev. Mod. Phys. , 2949 (2010);arXiv:0812.1702 [nucl-ex].[3] S. Leupold, V. Metag, and U. Mosel, Int. J. Mod. Phys. E , 147 (2010);arXiv:0907.2388 [nucl-th].[4] G. Krein, A. W. Thomas, and K. Tsushima, Prog. Part. Nucl. Phys. , 161 (2018);arXiv:1706.02688 [hep-ph]. Since, as one may hope, the precision of these experiments can reach the same value ∼
15% as was achieved inrecent measurements here [39] of π − meson-induced K + meson production in π − C → K + X and π − W → K + X reactions at 1.7 GeV/c beam momentum. , 199 (2017);arXiv:1706.09654 [nucl-ex].[6] A. Gal, E. V. Hungerford and D. J. Millener, Rev. Mod. Phys. , 035004 (2016);arXiv:1605.00557 [nucl-th].[7] K. Tsushima et al. , Phys. Rev. C , 065208 (2011) [arXiv:1103.5516 [nucl-th]];G. Krein, A. W. Thomas, and K. Tsushima, Phys. Lett. B , 136 (2011) [arXiv:1007.2220[nucl-th]].[8] E. Ya. Paryev, Yu. T. Kiselev, and Yu. M. Zaitsev, Nucl. Phys. A , 1 (2017);E. Ya. Paryev and Yu. T. Kiselev, Nucl. Phys. A , 201 (2018) [arXiv:1810.01715 [nucl-th]];E. Ya. Paryev and Yu. T. Kiselev, Phys. Atom. Nucl. Vol. , No.1, 67 (2017);E. Ya. Paryev and Yu. T. Kiselev, Phys. Atom. Nucl. Vol. , No.5, 566 (2018);E. Ya. Paryev, Nucl. Phys. A , 121711 (2020) [arXiv:2003.00788 [nucl-th]].[9] E. Ya. Paryev, Chinese Physics C, Vol. , No. (8), 084101 (2018);arXiv:1806.00303 [nucl-th].[10] E. Ya. Paryev, Nucl. Phys. A , 24 (2019);arXiv:1906.02185 [nucl-th].[11] S. D. Bass and P. Moskal, arXiv:1810.12290 [hep-ph].[12] M. Nanova et al. , Eur. Phys. J. A : 182 (2018);arXiv:1810.01288 [nucl-ex].[13] N. Tomida et al. , Phys. Rev. Lett. , 202501 (2020);arXiv:2005.03449 [nucl-ex].[14] M. M. Kaskulov and E. Oset, Phys. Rev. C , 045213 (2006);arXiv:nucl-th/0509088.[15] M. M. Kaskulov and E. Oset, AIP Conf. Proc. , 483–5 (2006).[16] M. F. M. Lutz, C. L. Copra and M. Moeller, Nucl. Phys. A , 124 (2008);arXiv:0707.1283 [nucl-th].[17] D. Cabrera et al. , Phys. Rev. C , 055207 (2014);arXiv:1406.2570 [hep-ph].[18] S. Petschauer et al. , Eur. Phys. J. A , 15 (2016);arXiv:1507.08808 [nucl-th].[19] E. Ya. Paryev, M. Hartmann, and Yu. T. Kiselev, Chinese Physics C,Vol. , No. (12), 124108 (2017);arXiv:1612.02767 [nucl-th].[20] Z. Q. Feng, W. J. Xie, and G. M. Jin, Phys. Rev. C , 064604 (2014).[21] E. Ya. Paryev and Yu. T. Kiselev, Nucl. Phys. A , 121622 (2019);arXiv:1910.02755 [nucl-th].[22] M. Kaskulov, L. Roca and E. Oset, Eur. Phys. J. A , 139 (2006);arXiv:nucl-th/0601074. 1423] E. Ya. Paryev, Phys. Atom. Nucl. Vol. , No.12, 1523 (2012).[24] E. Ya. Paryev, J. Phys. G: Nucl. Part. Phys. , 105101 (2010);arXiv:1010.0111 [nucl-th].[25] S. H. Lee and S. Cho, Int. J. Mod. Phys. E , 1330008 (2013);arXiv:1302.0642 [nucl-th].[26] T. Song, T. Hatsuda, and S. H. Lee, Phys. Lett. B , 160 (2019);arXiv:1808.05372 [nucl-th].[27] S. H. Lee, arXiv:1904.09064 [nucl-th].[28] L. Tolos, R. Molina, E. Oset, and A. Ramos, Phys. Rev. C , 045210 (2010);arXiv:1006.3454 [nucl-th].[29] E. Oset et al. , Int. J. Mod. Phys. E , 1230011 (2012);arXiv:1210.3738 [nucl-th].[30] A. Ilner, D. Cabrera, P. Srisawad, and E. Bratkovskaya, Nucl. Phys. A , 249 (2014);arXiv:1312.5215 [hep-ph].[31] D. Cabrera et al. , Journal of Physics: Conf. Series , 012017 (2014);arXiv:1312.4343 [hep-ph].L. Tolos, EPJ Web of Conf. , 09003 (2018).[32] K. Tsushima, A. Sibirtsev, and A. W. Thomas, Phys. Rev. C , 064904 (2000);arXiv:nucl-th/0004011.[33] D. Suenaga and P. Lakaschus, Phys. Rev. C , 035209 (2020);arXiv:1908.10509 [nucl-th].[34] T. Anticic et al. (NA 49 Collaboration), Phys. Rev. C , 064909 (2011);arXiv:1105.3109 [nucl-ex].[35] J. Adams et al. (STAR Collaboration), Phys. Rev. C , 064902 (2005) [arXiv:nucl-ex/0412019];M. M. Aggarwal et al. (STAR Collaboration), Phys. Rev. C , 034909 (2011) [arXiv:1006.1961[nucl-ex]].[36] B. Abelev et al. (ALICE Collaboration), Eur. Phys. J. C , 2183 (2012).[37] X. Lopez et al. (FOPI Collaboration), Phys. Rev. C , 061902 (2010);arXiv:1006.1905 [nucl-ex].[38] G. Agakishiev et al. (HADES Collaboration), Eur. Phys. J. A : 34 (2013).[39] J. Adamczewski-Musch et al. (HADES Collaboration), Phys. Rev. Lett. , 022002 (2019);arXiv:1812.03728 [nucl-ex].[40] J. Haidenbauer and Ulf-G Meissner, Nucl. Phys. A , 29 (2015);arXiv:1411.3114 [nucl-th].[41] M. Kohno and Y. Fujiwara, Phys. Rev. C , 054318 (2009);arXiv:0904.0517 [nucl-th]. 1542] M. Kohno, Phys. Rev. C , 014003 (2010);arXiv:0912.4330 [nucl-th].[43] E. Friedman and A. Gal, Phys. Rep. , 89 (2007);arXiv:0705.3965 [nucl-th].[44] K. P. Khemchandani et al. , Phys. Rev. D , 094008 (2015);arXiv:1406.7203 [nucl-th].[45] S. V. Efremov and E. Ya. Paryev, Eur. Phys. J. A , 99 (1998).[46] E. Ya. Paryev, Eur. Phys. J. A , 521 (2000).[47] E. Ya. Paryev, Eur. Phys. J. A , 127 (2000).[48] V. Flaminio et al. , Compilation of Cross Sections.I: π + and π − Induced Reactions. CERN-HERA , (1983).[49] T. Hatsuda, arXiv:nucl-th/9702002.[50] M. Moritsu et al. (J-PARC E19 Collaboration), Phys. Rev. C90