AAntikaon-induced K (1270) + meson production on nucleinear threshold 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 axial-vector meson K (1270) + production in K − A reactionsat near-threshold laboratory incident antikaon momenta within a nuclear spectral functionapproach, which describes incoherent direct K (1270) + meson production in K − meson–proton K − p → K (1270) + Ξ − production processes and accounts for three different options for its in-medium mass shift (or for its effective scalar potential) at central density ρ . We calculate theabsolute differential and total cross sections for the production of K (1270) + mesons on C and
W target nuclei at laboratory angles of 0 ◦ –45 ◦ by K − mesons with momenta of 2.5, 2.8 and3.5 GeV/c, which are close to the threshold momentum ( ≈ K (1270) + mesonproduction off the free target proton at rest. The intrinsic properties of carbon and tungstentarget nuclei have been described in terms of their spectral functions, which take into accountthe momenta of target protons and the energies of their separation from the considered nuclei.We show that the differential and total (absolute and relative) K (1270) + antikaon-inducedproduction cross sections at initial momenta not far from threshold – at momenta ∼ K (1270) + mass, studied in the paper, both in the K (1270) + meson low-momentum (0.1–1.0 GeV/c) and in its full-momentum ranges. Thiswould permit evaluating this shift. Experimental data necessary for this aim can be obtainedin a dedicated experiment at the J-PARC Hadron Experimental Facility. a r X i v : . [ nu c l - t h ] F e b Introduction
The study of the in-medium properties (masses and widths) of hadrons at finite density hasreceived considerable interest in the last two decades due to the hope to extract valuable informationon the partial restoration of chiral symmetry in a nuclear medium (see, for example, [1–6]). Anotherprominent case of medium modification of hadrons is that 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 ), which are chiral partners and whose in-medium mass difference, as is expected [7–10], issensitive to the chiral order parameter and, hence, will give the possibility to identify unambiguouslythe effect of chiral symmetry breaking in nuclear medium. Their isospins, spins-parities quantumnumbers are I ( J P ) = (1 − ) for the K ∗ (892) and I ( J P ) = (1 + ) for the K (1270). They have arelatively small vacuum decay widths of 50 and 90 MeV, respectively.At present, in literature there is a number of publications devoted to the study of the in-mediumproperties of hadronic resonances K ∗ (892) and K (1270). The properties of ¯ K ∗ (892) and K ∗ (892)mesons in cold nuclear matter have been studied in Refs. [11–13] and [13, 14], respectively, basingon chirally motivated model of the meson selfenergies. Thus, for instance, a moderately attractiveand repulsive real low-energy nuclear potentials (or their in-medium mass shifts) of about -50 and+40 MeV at density ρ have been predicted for the ¯ K ∗ (892) and K ∗ (892) mesons, respectively.These are similar to those for light strange mesons ¯ K and K . On the other hand, a negative massshift of about -20 MeV has been predicted for the K ∗ (892) + meson, having the same quark content u ¯ s as the K + one, at rest at saturation density ρ within the quark-meson coupling model [15].Other recent calculations [8] based on QCD sum rule predict that the upper limits of the mass shiftsof K − and K +1 mesons in nuclear matter are -249 and -35 MeV, respectively.The influence of the K ∗ (892) + meson in-medium mass shift on its yield from π − A reactions atnear-threshold laboratory initial pion momenta of 1.4–2.0 GeV/c has been recently analyzed withinthe collision model [16] using an eikonal approximation. It has been shown that both differentialand total K ∗ (892) + production cross sections possess a high sensitivity to changes in this shift in thelow-momentum region of 0.1–0.6 GeV/c. In addition to a feasibility study performed in Ref. [16]in order to investigate the impact of the mass shift of K ∗ (892) + mesons in nuclear matter on theiryield in near-threshold π − A reactions, we give here the predictions for the absolute differential andtotal cross sections for near-threshold production of K (1270) + mesons in K − C → K (1270) + X and K − W → K (1270) + X reactions at laboratory angles of 0 ◦ –45 ◦ by incident antikaons withmomenta of 2.5–3.5 GeV/c as well as for their relative yields from these reactions within threedifferent scenarios for the K (1270) + in-medium mass shift. These nuclear targets were also adoptedin calculations [16] of K ∗ (892) + meson production in π − A reactions and, therefore, can be usedin studying the K − A → K (1270) + X interactions here. The present calculations are based on afirst-collision model, developed in Ref. [16] for the description of the inclusive K ∗ (892) + mesonproduction and extended to account for different scenarios for the K (1270) + in-medium mass shiftand width change. The medium modification of the K (1270) ± (and K ∗ (892) ± ) mesons could beprobed at the J-PARC Hadron Experimental Facility through the K − reaction on various nucleartargets [8] . Comparison the results of such measurement for K (1270) + and those from K ∗ (892) + It is appropriate to note that the measuring of K (1270) − and K ∗ (892) − mesons would require less energyto produce compared to that for K (1270) + and K ∗ (892) + using the K − beam due to more lower their lowestproduction thresholds in elementary reactions K − p → K (1270) − p and K − p → K ∗ (892) − p ( ≈ K (1270) + and K ∗ (892) + production in the relevant elementary channels K − p → K (1270) + Ξ − and K − p → K ∗ (892) + Ξ − ( ≈ K (1270) − and K ∗ (892) − productionon nuclei in near-threshold K − A interactions to get a deeper insight into the possibility of observing here of theirmodification in nuclear medium in these interactions. π − A reactions with our calculations – previous [16] and present – will shed lighton the partial restoration of chiral symmetry in nuclear matter [8]. K (1270) + meson productionin nuclei Direct production of K (1270) + mesons in K − A (A= C and
W) interactions at near-threshold incident antikaon beam momenta below 3.5 GeV/c may occur in the following K − p elementary process, which has the lowest free production threshold momentum ( ≈ K − + p → K (1270) + + Ξ − . (1)We can ignore in the momentum domain of interest the contribution to the K (1270) + yield fromthe processes K − p → K (1270) + Ξ − π , K − p → K (1270) + Ξ π − , K − n → K (1270) + Ξ − π − and K − p → K (1270) + Ξ(1530) − due to larger their production thresholds ( ≈ K − N collisions.Following [16], we simplify the ensuing calculations via accounting for the in-medium modifica-tion of the final K (1270) + meson, involved in the production process (1), in terms of its averagein-medium mass < m ∗ K > instead of its local effective mass m ∗ K ( | r | ) in the in-medium cross sectionof this process, with < m ∗ K > defined as: < m ∗ K > = m K + V < ρ N >ρ , (2)where m K is the K (1270) + free space mass, V is the K (1270) + effective scalar nuclear potential(or its in-medium mass shift) at normal nuclear matter density ρ , and < ρ N > is the averagenucleon density. In the present work, for nuclei C and
W of interest, the ratio < ρ N > /ρ ,was chosen as 0.55 and 0.76, respectively. No experimental data exist presently on the K (1270) + N interaction. Recent calculations performed in Ref. [8] within the QCD sum rule approach show, aswas noted above, that the upper limit of the mass shift of K (1270) + mesons in nuclear matter is-35 MeV at zero momentum relative to the surrounding nuclear matter. Therefore, we will adoptfor the quantity V in this work three following options: i) V = −
35 MeV, ii) V = −
20 MeV andiii) V = 0 MeV. Since there is no information about momentum dependence of this mass shift,we will use the above options for it at finite momenta, accessible in calculation of the K (1270) + production in K − A reactions at beam momenta of interest, with allowance for the Fermi motion ofintranuclear protons (at momenta ∼ − N interaction is poor currently. The study of Ξ − hypernuclei provides a valuable informationon this interaction at low energies [6, 17]. At present, there are only a few sets of experimentaldata on the Ξ − hyperon properties in the nuclear medium. Following phenomenological informationdeduced in Refs. [18, 19] from old emulsion data , from missing-mass measurements [22–25] inthe inclusive ( K − , K + ) reaction on nuclear targets at incident momenta of 1.6–1.8 GeV/c in theΞ − bound and quasi-free regions with insufficiently good (10 MeV or worse) [22–24] and moderate(5.4 MeV) [25] energy resolutions, from the analysis the results of these measurements yet in Refs.[26, 27] using Green’s function method of the DWIA, the theoretical predictions [28–31] that theΞ − cascade hyperon feels only a weak attractive potential in nuclei, the depth of which is not solarge, ∼ -(10–20) MeV at central nuclear density and at rest as well as the predictions [32, 33] It is worth noting that the new recent emulsion KEK-PS E373 [20] and J-PARC E07 [21] experiments, in whichthe remarkable events named, respectively, ”KISO” and ”IBUKI” were observed, reported the evidence of a likelythe Coulomb-assisted nuclear 1 p single-particle Ξ − state of the bound Ξ − – N(g.s.) system with binding energy ofabout 1 MeV, which also testifies in favor of shallow Ξ − –nucleus potential. − single-particle potential in symmetric nuclear matterbecomes almost zero at high Ξ − momenta around 1.3 GeV/c corresponding to the K (1270) + momentum range of ∼ K (1270) + production cross sections are the greatest,we will ignore the modification of the free space mass m Ξ − of the Ξ − hyperons, produced togetherwith the K (1270) + mesons in the process (1), in the nuclear medium. To improve essentiallyour understanding of the Ξ − N interaction the high-quality data on the doubly strange S = − C( K − , K + ) reaction around the Ξ − production threshold and the X-ray data on the level shifts and width broadening of the Ξ − atomicstates in the Ξ − atoms will become available from the planned dedicated E70 and E03 experimentsat the J-PARC Hadron Experimental Facility [34]. Accounting for that the in-medium thresholdenergy √ s ∗ th = < m ∗ K > + m Ξ − of the process (1) looks like that for the final charged particles,influenced also by the respective Coulomb potentials, due to the cancelation of these potentials, wewill neglect here their impact on these particles as well.The total energy E (cid:48) K of the K (1270) + meson in nuclear matter is expressed via its averageeffective mass < m ∗ K > and its in-medium momentum p (cid:48) K by the expression [16]: 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 (1270) + momentum p K as follows [16]: 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 (1270) + total energy in vacuum.As the K (1270) + –nucleon elastic cross section is expected to be small similar to the K ∗ (892) + N elastic cross section [35], we will ignore quasielastic K (1270) + N rescatterings in the present work.Then, taking into account the attenuation of the incident antikaon and the final K (1270) + mesonin the nuclear matter in terms, respectively, of the K − N total cross section σ tot K − N and the totalwidth Γ K ( | r | ) in the K (1270) + rest frame, taken at the point r inside the nucleus, as well as usingthe results given in [16], we represent the inclusive differential cross section for the production of K (1270) + mesons with vacuum momentum p K on nuclei in the direct process (1) as follows: dσ (prim) K − A → K (1270) + X ( p K − , p K ) d p K = I V [ A, θ K ] (cid:18) ZA (cid:19) (cid:42) dσ K − p → K (1270) + Ξ − ( p K − , 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 K − 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) Determining mainly the strength of the K (1270) + production cross sections in near-threshold antikaon–nucleuscollisions. tot K − N = ZA σ tot K − p + NA σ tot K − n , (9) λ K ( | r | ) = p (cid:48) K m ∗ K ( | r | )Γ K ( | r | ) , m ∗ K ( | r | ) = m K + V ρ N ( | r | ) ρ (10)and (cid:42) dσ K − p → K (1270) + Ξ − ( p K − , 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 (11) × (cid:40) dσ K − p → K (1270) + Ξ − [ √ s ∗ , < m ∗ K >, m Ξ − , p (cid:48) K ] d p (cid:48) K (cid:41) ,s ∗ = ( E K − + E t ) − ( p K − + p t ) , (12) E t = M A − (cid:113) ( − p t ) + ( M A − m N + E ) . (13)Here, dσ K − p → K (1270) + Ξ − [ √ s ∗ , < m ∗ K >, m Ξ − , p (cid:48) K ] /d p (cid:48) K is the off-shell inclusive differential crosssection for the production of K (1270) + meson and Ξ − hyperon with modified mass < m ∗ K > and free mass m Ξ − , respectively. The K (1270) + meson is produced with in-medium momentum p (cid:48) K in process (1) at the K − p center-of-mass energy √ s ∗ . E K − and p K − are the total energyand momentum of the incident antikaon ( E K − = (cid:113) m K + p K − , m K is the free space kaon mass); ρ ( r ) and P A ( p t , E ) are the local nucleon density and the spectral function of the target nucleusA normalized to unity (the concrete information about these quantities, used in the subsequentcalculations, is given in Refs. [36–39]); p t and E are the internal momentum and removal energyof the target proton involved in the collision process (1); Z and N are the numbers of protons andneutrons in the target nucleus ( A = Z + N ), M A and R are its mass and radius; m N is the freespace nucleon mass; θ K is the polar angle of vacuum momentum p K in the laboratory system withz-axis directed along the momentum p K − of the incident antikaon beam; σ tot K − p and σ tot K − n are thetotal cross sections of the free K − p and K − n interactions at beam momenta belonging to the rangeof 2.5–3.5 GeV/c . In this range σ tot K − p ≈
28 mb and σ tot K − n ≈
22 mb [41]. With these, the quantity σ tot K − N , entering into Eqs. (6) and (9), amounts approximately to 25 mb for both considered targetnuclei C and
W. We will use this value in our present calculations.According to [16], we suppose that the off-shell differential cross section dσ K − p → K (1270) + Ξ − [ √ s ∗ ,
594 GeV is replaced by the in-medium one √ s ∗ th and the free center-of-mass energysquared s , presented by the formula s = ( E K − + m N ) − p K − , (19)is replaced by the in-medium one s ∗ , defined by the expression (12).For the free total cross section σ K − p → K (1270) + Ξ − ( √ s, √ s th ) we have adopted the following es-timates, based on the available only two experimental data points (6.2 ± µ b and (1.7 ± µ b for the vacuum total cross sections of the processes K − p → K (1270) + Ξ − → Kρ Ξ − and K − p → K (1270) + Ξ − → K ∗ (892) π Ξ − at 0.4 GeV excess energy √ s − √ s th (or at the beammomentum of 4.15 GeV/c) [42] as well as on the relatively large amount of existing experimentaldata for the total cross section of the K − p → K + Ξ − reaction [41]. Accounting for the branchingratios of (42 ± ± K (1270) + → Kρ and K (1270) + → K ∗ (892) π and these data points, the values of (14.8 ± µ b and (10.6 ± µ b for the free cross section σ K − p → K (1270) + Ξ − ( √ s, √ s th ) are obtained at √ s − √ s th =0.4 GeV. They are shown in Fig. 1 by fulltriangles. Here, the vacuum total cross section of the reaction K − p → K + Ξ − is presented as well asa function of the respective excess energy √ ˜ s − √ ˜ s th available in this reaction with threshold energy √ ˜ s th = m K + m Ξ − . Full squares are experimental data taken from the compilation of Flaminio etal. [41]. Solid curve is their parametrization by the formula σ K − p → K + Ξ − ( √ ˜ s, √ ˜ s th ) = 235 . (cid:16) − √ ˜ s th / √ ˜ s (cid:17) . (cid:16) √ ˜ s th / √ ˜ s (cid:17) . [mb] , (20)suggested in Ref. [43]. Visual inspection of Fig. 1 yields the following ratio of K (1270) + Ξ − to K + Ξ − production cross sections at the same excess energies √ s − √ s th and √ ˜ s − √ ˜ s th above the K (1270) + Ξ − and K + Ξ − thresholds equal to 0.4 GeV: σ K − p → K (1270) + Ξ − ( √ s, √ s th ) /σ K − p → K + Ξ − ( √ ˜ s, √ ˜ s th ) ≈ . . (21)Since no data on the total cross section σ K − p → K (1270) + Ξ − ( √ s, √ s th ) exist for other excess energies,it is natural to estimate it at them in the threshold region assuming that the ratio (21) is alsovalid at the same excess energies √ s − √ s th and √ ˜ s − √ ˜ s th differing from 0.4 GeV. Under thisassumption, the center-of-mass energies √ ˜ s and √ s , at which the cross sections entering into theratio (21) are calculated, are linked by the relation √ ˜ s − √ ˜ s th = √ s − √ s th . (22)Thus, we have √ ˜ s = √ s − √ s th + √ ˜ s th = √ s − m K + m K . (23)6 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 002 04 06 08 01 0 01 2 01 4 01 6 01 8 02 0 02 2 02 4 02 6 02 8 03 0 03 2 0 s [ m b] e x c e s s e n e r g i e s [ G e V ] K - p - > K + X - , d a t a K - p - > K + X - , f i t K - p - > K ( 1 2 7 0 ) + X - K - p - > K ( 1 2 7 0 ) + X - , d a t a K - p - > K ( 1 2 7 0 ) + X - , f i t Figure 1: (Color online) Total cross sections for the reactions K − p → K + Ξ − and K − p → K (1270) + Ξ − as functions of the available excess energies √ ˜ s − √ ˜ s th and √ s − √ s th above theirthresholds √ ˜ s th and √ s th , respectively. For notation see the text.When reactions K − p → K + Ξ − and K − p → K (1270) + Ξ − go in medium on an off-shell targetprotons, instead of the quantity √ ˜ s th , appearing in Eqs. (20)–(23), we should use the in-mediumthreshold energy √ ˜ s ∗ th = < m ∗ K > + m Ξ − , where < m ∗ K > is the K + meson average in-mediummass defined by the expression analogous to that given by Eq. (2) and in which V = +22 MeV[38]. And instead of the quantities √ s and √ s th , entering into Eqs. (21)–(23), one needs to adopt,respectively, the in-medium expression (12) and in-medium threshold energy √ s ∗ th determined above.The foregoing leads, as is easy to see, to the following substitutions m K → < m ∗ K > , m K → < m ∗ K > in the second relation of Eq. (23).At incident antikaon momenta of interest p K − ≤ √ s ≤ .
785 GeV and √ ˜ s ≤ . s and thetotal energy ˜ E K − and momentum ˜ p K − of the K − meson inducing the reaction K − p → K + Ξ − issimilar to that given above by Eq. (19), we easily get that the latter corresponds to its laboratorymomenta ˜ p K − ≤ σ K − p → K + Ξ − ( √ ˜ s, √ ˜ s th ) at these momenta we willemploy in our calculations the expression (20). Using it and the relations (21)–(23), we calculatedthe vacuum total cross section σ K − p → K (1270) + Ξ − ( √ s, √ s th ) of the process (1). It is plotted with adashed curve in Fig. 1. As can be seen from this figure, the on-shell cross section σ K − p → K (1270) + Ξ − amounts approximately to 16 µ b for the initial antikaon momentum of 3.5 GeV/c, corresponding to7 width change, total width [MeV] m a s s s h i f t V [ M e V ] w i d t h c h a n g e t o t a l w i d t hK ( 1 2 7 0 ) + r N = r Figure 2: (Color online) In-medium change of the K (1270) + meson width and its total in-mediumwidth in its rest frame as functions of the K (1270) + in-medium mass shift V at normal nuclearmatter density.the excess energy √ s −√ s th of about 0.2 GeV. This offers the possibility of measuring the K (1270) + yield in K − A collisions both at the above-threshold and below-threshold beam momenta at the J-PARC Hadron Experimental Facility with sizable strength. Here, the K (1270) + mesons could beregistered via the hadronic decays K (1270) + → Kρ and K (1270) + → K ∗ (892) π .In Eqs. (6)–(8) we suppose that the direction of the K (1270) + meson three-momentum is notchanged during its propagation inside the nucleus in the relatively weak nuclear field, consideredin the present work, from the production point here to the vacuum. As a result, the quantities (cid:68) dσ K − p → K (1270) + Ξ − ( p K − , 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 expressed inthe following simple forms (cid:68) dσ K − p → K (1270) + Ξ − ( p K − , 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 (1270) + momentum p K in thelaboratory system. Taking into account the fact that in the considered incident momentum region K (1270) + mesons are mainly ejected, due to the kinematics, in a narrow cone along the beamline , we will calculate the K (1270) + momentum differential and total production cross sectionson C and
W targets for laboratory solid angle ∆ Ω K =0 ◦ ≤ θ K ≤ ◦ , and 0 ≤ ϕ K ≤ π .Integrating the full inclusive differential cross section (5) over this angular domain, we can represent Thus, for instance, at a beam momentum of 3.5 GeV/c the K (1270) + laboratory production polar angles inreaction (1) proceeding on the target proton being at rest are ≤ ◦ . . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 01 E - 61 E - 51 E - 41 E - 30 . 0 10 . 111 0 p K - = 3 . 5 G e V / c q K 1 = 0 - 4 5 K - + C - > K ( 1 2 7 0 ) + + X d s /dpK1 [ m b/(GeV/c)] p K 1 [ G e V / c ] K - + W - > K ( 1 2 7 0 ) + + X V = 0 M e V V = - 2 0 M e V V = - 3 5 M e V d s /dpK1 [ m b/(GeV/c)] p K 1 [ G e V / c ]
Figure 3: (Color online) Momentum differential cross sections for the production of K (1270) + mesons from the direct K − p → K (1270) + Ξ − channel in the laboratory polar angular range of0 ◦ –45 ◦ in the interaction of K − mesons of momentum of 3.5 GeV/c with C (left) and
W(right) nuclei, calculated for different values of the K (1270) + meson effective scalar potential V atdensity ρ indicated in the inset. The arrows indicate the boundary between the low-momentumand high-momentum regions of the K (1270) + spectra.the differential cross section for K (1270) + meson production in K − A reactions from the directprocess (1), corresponding to this angle, in the following form: dσ (prim) K − A → K (1270) + X ( p K − , p K ) dp K = (cid:90) ∆ Ω K d Ω K dσ (prim) K − A → K (1270) + X ( p K − , p K ) d p K p K (24)= 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 K − p → K (1270) + Ξ − ( p K − , p (cid:48) K , θ K ) dp (cid:48) K d Ω K (cid:43) A . We define now the K (1270) + meson total in-medium width Γ K ( | r | ) appearing in Eq. (10) andadopted in our calculations of K (1270) + production in K − A interactions. According to Ref. [8],we can represent it in the following form:Γ K ( | r | ) = Γ K + Γ ρ N ( | r | ) ρ , (25)9 . 0 0 . 5 1 . 0 1 . 5 2 . 01 E - 61 E - 51 E - 41 E - 30 . 0 10 . 11 p K - = 2 . 8 G e V / c q K 1 = 0 - 4 5 K - + C - > K ( 1 2 7 0 ) + + X d s /dpK1 [ m b/(GeV/c)] p K 1 [ G e V / c ] K - + W - > K ( 1 2 7 0 ) + + X V = 0 M e V V = - 2 0 M e V V = - 3 5 M e V d s /dpK1 [ m b/(GeV/c)] p K 1 [ G e V / c ]
Figure 4: (Color online) The same as in Fig. 3, but for the incident antikaon beam momentum of2.8 GeV/c.where Γ K = 90 MeV is the vacuum total K (1270) + decay width in its rest frame and Γ isthe in-medium change of the K (1270) + width at saturation density ρ . A work [8] finds a linearcorrelation between the width change Γ and the in-medium mass shift V , shown by dashed curvein Fig. 2. Analytically, a relation between the quantities Γ and V can be easily expressed asfollows: Γ = 38 MeV + (38 MeV /
35 MeV) · V , −
35 MeV ≤ V ≤ . (26)One can see that the maximum change of the width is +38 MeV when the K (1270) + meson massremains the same in the medium, while for the upper limit of its mass shift of -35 MeV the width isnot changed here at all. In these cases, the resulting total in-medium width (25) of the K (1270) + meson, depicted by the solid curve in Fig. 2, reaches the values of 128 and 90 MeV, respectively. So,once the change of the K (1270) + meson mass is allowed, its total in-medium width gets smaller.This will lead to a more larger K (1270) + decay mean ”free” path λ K , which in turn will causemore weaker absorption of K (1270) + mesons in the nuclear matter. Thus, for example, Eq. (10)shows that for typical values p (cid:48) K ≈ m ∗ K and total in-medium K (1270) + meson decay width in itsrest frame of 128 MeV, corresponding to no in-medium effects on its mass, this mean ”free” pathis equal to 1.5 fm. Whereas in the case of maximum change of K (1270) + mass in the medium,when the above total in-medium width equals to the vacuum width Γ K = 90 MeV, the K (1270) + decay mean ”free” path λ K is larger and is equal to 2.2 fm. These values are comparable with theradius of C of 3 fm and they are much less than that of
W of 7.4 fm. The above means that10 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 81 E - 71 E - 61 E - 51 E - 41 E - 30 . 0 10 . 1 p K - = 2 . 5 G e V / c q K 1 = 0 - 4 5 K - + C - > K ( 1 2 7 0 ) + + X d s /dpK1 [ m b/(GeV/c)] p K 1 [ G e V / c ] K - + W - > K ( 1 2 7 0 ) + + X V = 0 M e V V = - 2 0 M e V V = - 3 5 M e V d s /dpK1 [ m b/(GeV/c)] p K 1 [ G e V / c ]
Figure 5: (Color online) The same as in Fig. 3, but for the incident antikaon beam momentum of2.5 GeV/c.the dropping K (1270) + mass scenario will lead to an enhancement (see below) of the yield of the K (1270) + mesons in K − A reactions at near-threshold incident beam momenta of interest both dueto in-medium shift of the elementary production threshold to lower energy and due to their weakerabsorption in the nuclear matter. The model described above makes it possible to calculate the absolute K (1270) + momentumdifferential cross sections from the direct K (1270) + production process in K − C and K − Wcollisions. These cross sections were calculated according to Eq. (24) for three adopted values ofthe K (1270) + in-medium mass shift V at density ρ at laboratory angles of 0 ◦ –45 ◦ and for initialantikaon momenta of 3.5, 2.8 and 2.5 GeV/c. They are presented, respectively, in Figs. 3, 4 and5. One can see from these figures that the K (1270) + meson differential cross sections are notablysensitive to its in-medium mass shift, mostly in the low-momentum region of 0.1–1.0 GeV/c, for bothtarget nuclei and for all antikaon momenta considered. Here, the differences between all calculationscorresponding to different options for the K (1270) + in-medium mass shift are well separated andexperimentally distinguishable. They are practically similar to each other for each target nucleus atinitial antikaon momenta considered. Thus, for example, for incident K − and outgoing K (1270) + . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 001234567891 0 p K - = 3 . 5 G e V / c q K 1 = 0 - 4 5 K - + C - > K ( 1 2 7 0 ) + + X ratio p K 1 [ G e V / c ] K - + W - > K ( 1 2 7 0 ) + + X V = 0 M e V V = - 2 0 M e V V = - 3 5 M e V ratio p K 1 [ G e V / c ]
Figure 6: (Color online) Ratio between the differential cross sections for K (1270) + production on C (left) and
W (right) target nuclei in the angular range of 0 ◦ –45 ◦ by 3.5 GeV/c K − mesons,calculated with and without the K (1270) + meson in-medium mass shift V at density ρ indicatedin the inset, as a function of K (1270) + momentum. The arrows indicate the boundary betweenthe low-momentum and high-momentum regions of the K (1270) + spectra.meson momenta of 3.5 and 0.5 GeV/c, respectively, and in the case of C nucleus the K (1270) + yield is enhanced at mass shift V = −
20 MeV by about a factor of 2.2 as compared to that obtainedfor the shift V = 0 MeV. When going from V = −
20 MeV to V = −
35 MeV the enhancementfactor is about 1.8. In the case of
W target nucleus these enhancement factors are about 3.3and 2.0. At initial antikaon momentum of 2.8 GeV/c and the same outgoing kaon momentum of0.5 GeV/c the corresponding enhancement factors are similar and are about 1.9 and 1.7 as well as1.7 and 1.4 in the cases of C as well as
W target nuclei, respectively. And for incident beammomentum of 2.5 GeV/c and final kaon momentum of 0.5 GeV/c these enhancement factors areabout 1.9 and 1.5 as well as 1.7 and 1.5 for C as well as
W nuclei, correspondingly. However,the K (1270) + low-momentum (and high-momentum) production differential cross sections at beammomentum of 2.5 GeV/c are less than those at antikaon momenta of 2.8 and 3.5 GeV/c by aboutof one to two orders of magnitude. Therefore, the K (1270) + meson differential cross sectionsmeasurements in the near-threshold incident K − momentum region (at 2.8–3.5 GeV/c) will openan opportunity to shed light on its possible mass shift in cold nuclear matter. Such measurementscould be performed in the future at the J-PARC Hadron Experimental Facility using the high-intensity and high-momentum (up to 10 GeV/c) separated secondary K − beams in the designed12 . 0 0 . 5 1 . 0 1 . 5 2 . 001234567891 0 p K - = 2 . 8 G e V / c q K 1 = 0 - 4 5 K - + C - > K ( 1 2 7 0 ) + + X ratio p K 1 [ G e V / c ] K - + W - > K ( 1 2 7 0 ) + + X V = 0 M e V V = - 2 0 M e V V = - 3 5 M e V ratio p K 1 [ G e V / c ]
Figure 7: (Color online) The same as in Fig. 6, but for the incident antikaon beam momentum of2.8 GeV/c.now the K10 beamline [34, 44].More detailed information about the sensitivity of the differential cross sections, presented inFigs. 3, 4 and 5, to the K (1270) + mass shift is contained in Figs. 6, 7 and 8, where the momentumdependence of the ratios of these cross sections calculated for the K (1270) + mass shift V at thecentral density ρ to the analogous cross section as determined at V = 0 MeV is shown on a linearscale for carbon and tungsten nuclei at K − momenta of 3.5, 2.8 and 2.5 GeV/c, respectively . It should be pointed out that such relative observables are more favorable compared to thosebased on the absolute cross sections for the aim of obtaining the information on particle in-mediummodification, since the theoretical uncertainties associated with the particle elementary productioncross section essentially cancel out in them. It can be seen from these figures that at the K (1270) + meson momenta less than 1.0 GeV/c there are indeed substantial differences between the resultsobtained by using considered options for its in-medium mass shift for both target nuclei and alladopted initial momenta. Moreover, Figs. 7 and 8 clearly show that the relative K (1270) + yieldexperiences rather noticeable variation for the mass shift range of V = 0 MeV to -35 MeV also inthe high-momentum region (at kaon momenta ≥ K − momenta of 2.8and 2.5 GeV/c. Thus, for incident antikaon momentum of 2.8 GeV/c and in the case of a carbonnucleus, the distinctions between the ratios corresponding to zero mass shift and shift values of We recall that a comparison of a similar model cross section ratios with data on η (cid:48) meson photoproduction offcarbon and niobium nuclei was employed in [45, 46] to extract its in-medium mass shift at a central nuclear density. . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 801234567891 0 p K - = 2 . 5 G e V / c q K 1 = 0 - 4 5 K - + C - > K ( 1 2 7 0 ) + + X ratio p K 1 [ G e V / c ] K - + W - > K ( 1 2 7 0 ) + + X V = 0 M e V V = - 2 0 M e V V = - 3 5 M e V ratio p K 1 [ G e V / c ]
Figure 8: (Color online) The same as in Fig. 6, but for the incident antikaon beam momentum of2.5 GeV/c.-20 and -35 MeV are of the order of 30% and 60% in the K (1270) + momentum range ∼ C as well as
Wnuclei, respectively. At the same time, as the antikaon beam momentum increases to 3.5 GeV/c,the sensitivity of the high-momentum parts of the ratios (and absolute momentum distributions)considered to variations in the K (1270) + in-medium mass shift becomes somewhat lower. Thus,at this incident momentum and the same final kaon momenta as above, the respective distinctions,as follows from Fig. 6, are smaller (but yet are measurable) than at beam momenta of 2.8 and2.5 GeV/c: they are about 15% and 30% as well as 20% and 40% for C as well as
W nuclei,correspondingly. This means, accounting for the above considerations, that the measurements ofthe K (1270) + meson momentum distributions (absolute and relative) might permit to shed lightalso on the momentum dependence of its in-medium mass shift (or of its effective scalar potential V ) at saturation density ρ . It should be noticed that the ratios of differential cross sections for K (1270) + meson production on W nucleus by 3.5 and 2.8 GeV/c antikaons, shown in Figs. 6and 7, respectively, exhibit dips at momenta ∼ K − p collision and14 K - + C - > K ( 1 2 7 0 ) + + X s [ m b] p K 1 = 0 . 1 - 1 . 0 G e V / c - 4 0 - 3 5 - 3 0 - 2 5 - 2 0 - 1 5 - 1 0 - 5 0 50 . 0 10 . 11 p K 1 = 0 . 1 - 1 . 0 G e V / c K - + W - > K ( 1 2 7 0 ) + + X - 4 0 - 3 5 - 3 0 - 2 5 - 2 0 - 1 5 - 1 0 - 5 0 51 E - 30 . 0 10 . 111 0 q K 1 = 0 - 4 5 K - + C - > K ( 1 2 7 0 ) + + X s [ m b] m a s s s h i f t V [ M e V ]a l l a l l o w e d m o m e n t a - 4 0 - 3 5 - 3 0 - 2 5 - 2 0 - 1 5 - 1 0 - 5 0 50 . 0 10 . 111 0 a l l a l l o w e d m o m e n t a K - + W - > K ( 1 2 7 0 ) + + X p K - = 3 . 5 G e V / c p K - = 2 . 8 G e V / c p K - = 2 . 5 G e V / cm a s s s h i f t V [ M e V ] Figure 9: (Color online) Total cross sections for the production of K (1270) + mesons from thedirect K − p → K (1270) + Ξ − channel on C and W target nuclei with momenta of 0.1–1.0 GeV/c(upper two panels) and with all allowed momenta ≥ ◦ –45 ◦ by 2.5, 2.8 and 3.5 GeV/c K − mesonsas functions of the K (1270) + in-medium mass shift V at normal nuclear density. The lines arevisual guides.the role played by the nucleus-related effects such as the target proton binding and Fermi motion,encoded in the nuclear spectral function P A ( p t , E ). The spectral functions for C and
W, usedin our calculations, are different [37–39].The sensitivity of the K (1270) + meson production differential cross sections to its in-mediummass shift V , shown in Figs. 3, 4, 5 and 6, 7, 8, can also be studied from the measurements ofthe total cross sections for K (1270) + production in K − C and K − W collisions at the thresholdincident K − momenta. Such cross sections, calculated by integrating Eq. (24) over the K (1270) + momentum p K at 2.5, 2.8 and 3.5 GeV/c antikaon momenta in the low-momentum region (0.1–1.0GeV/c) and in the full-momentum region allowed for given beam momentum are shown in Fig. 9as functions of the mass shift V . It is seen from this figure that again the low-momentum regionof 0.1–1.0 GeV/c shows the highest sensitivity to this shift. Thus, for example, the ratios betweenthe total cross sections for K (1270) + production by 2.5, 2.8, 3.5 GeV/c K − mesons on C and
W nuclei in this momentum region, calculated with the shift V = −
35 MeV, and the same crosssections as those obtained with V = 0 MeV, are about 2.1, 1.9, 1.6 and 2.3, 2.3, 1.8, respectively.Whereas the same ratios in the full-momentum regions are only somewhat smaller: they are about15 p K 1 = 0 . 1 - 1 . 0 G e V / c K - + C - > K ( 1 2 7 0 ) + + X ratio - 4 0 - 3 5 - 3 0 - 2 5 - 2 0 - 1 5 - 1 0 - 5 0 50 . 51 . 01 . 52 . 02 . 5 p K 1 = 0 . 1 - 1 . 0 G e V / c K - + W - > K ( 1 2 7 0 ) + + X - 4 0 - 3 5 - 3 0 - 2 5 - 2 0 - 1 5 - 1 0 - 5 0 50 . 51 . 01 . 52 . 02 . 5 q K 1 = 0 - 4 5 K - + C - > K ( 1 2 7 0 ) + + Xa l l a l l o w e d m o m e n t a ratio m a s s s h i f t V [ M e V ] - 4 0 - 3 5 - 3 0 - 2 5 - 2 0 - 1 5 - 1 0 - 5 0 50 . 51 . 01 . 52 . 02 . 5 a l l a l l o w e d m o m e n t a K - + W - > K ( 1 2 7 0 ) + + X p K - = 3 . 5 G e V / c p K - = 2 . 8 G e V / c p K - = 2 . 5 G e V / cm a s s s h i f t V [ M e V ] Figure 10: (Color online) Ratio between the total cross sections for the production of K (1270) + mesons from the direct K − p → K (1270) + Ξ − channel on C and
W target nuclei at laboratoryangles of 0 ◦ –45 ◦ with momenta of 0.1–1.0 GeV/c (upper two panels) and with all allowed momenta ≥ K − mesons,calculated with and without the K (1270) + in-medium mass shift V at normal nuclear density, asfunction of this shift. The lines are visual guides.2.0, 1.6, 1.2 for C and 2.2, 2.0, 1.4 for
W, correspondingly. The highest sensitivity of the totalcross sections for K (1270) + production in the low-momentum and full-momentum ranges underconsideration to the mass shift V is observed, as one would expect, at beam momentum of 2.5GeV/c. However, the cross sections at this momentum are small and they are less than those at K − momenta of 2.8 and 3.5 GeV/c by about of one and two orders of magnitude, respectively. Sincethe latter ones have a measurable strength ∼ µ b, the total cross section measurements of K (1270) + meson production on nuclei both in the low-momentum (0.1–1.0 GeV/c) and in the fullmomentum regions for incident antikaon momenta not far from threshold (for momenta ∼ K (1270) + in-medium mass shift V at sat-uration density ρ for incident K − momenta of interest are nicely supported also by the resultspresented in Fig. 10. Here, the ratios of the K (1270) + meson production total cross sections calcu-lated for its mass shift V and given in Fig. 9 to the analogous cross sections determined at V = 016eV are shown as functions of this mass shift. It should be mentioned that an analysis of theseratios, as in the case of those shown in Figs. 6–8, has the advantage that the theoretical uncertain-ties associated with the elementary cross section for particle production substantially cancel out inthem. It is clearly seen from this figure that the highest sensitivity of the ratios in both consideredkinematic ranges to the mass shift V is indeed observed at far subthreshold antikaon momentum of2.5 Gev/c, corresponding to the free space excess energy of -164 MeV. For instance, at this momen-tum and for these ranges the cross section ratios for V = −
20 MeV are about 1.5 for both targetnuclei. At this mass shift they are similar to those at momentum of 2.8 GeV/c. The latter onesare about 1.4 and 1.5 for both considered kinematic ranges for C and
W, respectively. As the K − momentum increases to 3.5 GeV/c, the sensitivity of the total cross section ratios to changesin the K (1270) + in-medium mass shift V decreases. Thus, in the case where K (1270) + mesons ofmomenta of 0.1–1.0 GeV/c are produced by 3.5 GeV/c antikaons incident on C and
W targets,the ratios being considered for V = −
20 MeV take smaller but yet measurable values of about1.3 and 1.4, respectively. The analogous ratios for the production of the K (1270) + mesons in thefull-momentum ranges by 3.5 GeV/c K − mesons in C and
W nuclei are somewhat yet smaller:they are about 1.1 and 1.2, respectively.Therefore, one can conclude that a comparison of the low-momentum and full-momentum ”in-tegral” results shown in Figs. 9, 10 with the respective precise experimental data, which couldbe also obtained in future experiments at J-PARC [34, 44] employing near-threshold K − beams,should allow to distinguish between possible weak ( V ∼ -20 MeV) and relatively weak ( V ∼ -35MeV) K (1270) + meson mass shifts in nuclear matter, considered in the present work.Finally, accounting for the above considerations, we can conclude that the K (1270) + differentialand total cross section measurements in K − A reactions at initial momenta not far from threshold(at momenta ∼ K (1270) + meson low-momentum (0.1–1.0 GeV/c) andin its full-momentum ranges will allow us to shed light on the possible K (1270) + in-medium massshift. Our present study was aimed at studying the possibility of extracting information about thepredicted in-medium change in the K (1270) + meson mass. For this, we calculated the absolutedifferential and total cross sections for the production of K (1270) + mesons on C and
W tar-get nuclei at laboratory angles of 0 ◦ –45 ◦ by K − mesons with momenta of 2.5, 2.8 and 3.5 GeV/c,which are close to the threshold momentum (2.95 GeV/c) for K (1270) + meson production offthe free target proton at rest. These calculations have been performed within a nuclear spectralfunction approach, which describes incoherent direct K (1270) + meson production in K − meson–proton K − p → K (1270) + Ξ − production processes and accounts for three different options for itsin-medium mass shift (or for its effective scalar potential) at central density ρ . We show that thedifferential and total (absolute and relative) K (1270) + antikaon-induced production cross sectionsat initial momenta not far from threshold – at momenta ∼ K (1270) + mass, studied in the paper, both in the K (1270) + meson low-momentum (0.1–1.0GeV/c) and in its full-momentum ranges. This would permit evaluating this shift. Experimentaldata necessary for this aim can be obtained in a dedicated experiment at the J-PARC HadronExperimental Facility. 17 eferences [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].[5] V. Metag, M. Nanova, and E. Ya. Paryev, Prog. Part. Nucl. Phys. , 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] S. H. Lee and S. Cho, Int. J. Mod. Phys. E , 1330008 (2013);arXiv:1302.0642 [nucl-th].[8] T. Song, T. Hatsuda, and S. H. Lee, Phys. Lett. B , 160 (2019);arXiv:1808.05372 [nucl-th].[9] S. H. Lee, arXiv:1904.09064 [nucl-th].[10] S. H. Lee, Nucl. Part. Phys. Proc. , 111 (2020).[11] L. Tolos, R. Molina, E. Oset, and A. Ramos, Phys. Rev. C , 045210 (2010);arXiv:1006.3454 [nucl-th].[12] E. Oset et al. , Int. J. Mod. Phys. E , 1230011 (2012);arXiv:1210.3738 [nucl-th].[13] A. Ilner, D. Cabrera, P. Srisawad, and E. Bratkovskaya, Nucl. Phys. A , 249 (2014);arXiv:1312.5215 [hep-ph].[14] D. Cabrera et al. , Journal of Physics: Conf. Series , 012017 (2014);arXiv:1312.4343 [hep-ph].L. Tolos, EPJ Web of Conf. , 09003 (2018).[15] K. Tsushima, A. Sibirtsev, and A. W. Thomas, Phys. Rev. C , 064904 (2000);arXiv:nucl-th/0004011.[16] E. Ya. Paryev, Chinese Physics C, Vol. , No. (11), 114106 (2020);arXiv:2007.10192 [nucl-th].[17] E. Friedman and A. Gal, Phys. Rep. , 89 (2007);arXiv:0705.3965 [nucl-th].[18] C. B. Dover and A. Gal, Annals of Phys. , 309 (1983).[19] S. Aoki et al. , Phys. Lett. B , 45 (1995).1820] K. Nakazawa et al. , Prog. Theor. Exp. Phys. , 033D02 (2015).[21] S. H. Hayakawa et al. (J-PARC E07 Collaboration), arXiv:2010.14317 [nucl-ex].[22] T. Fukuda et al. , Phys. Rev. C , 1306 (1998).[23] P. Khaustov et al. , Phys. Rev. C , 054603 (2000);arXiv:nucl-ex/9912007.[24] T. Iijima et al. , Nucl. Phys. A , 588 (1992).[25] T. Nagae et al. (J-PARC E05 Collaboration), PoS INPC , 038 (2017);AIP Conf. Proc. , 020015 (2019).[26] H. Maekawa et al. , arXiv:0704.3929 [nucl-th].[27] H. Maekawa, K. Tsubakihara and A. Ohnishi, Eur. Phys. J. A , 269 (2007);arXiv:nucl-th/0701066.[28] J. Hu and H. Shen, Phys. Rev. C , 054304 (2017);arXiv:1710.08613 [nucl-th].[29] E. Hiyama et al. , Phys. Rev. C , 054316 (2008);arXiv:0811.3156 [nucl-th].[30] Y. Jin, X.-R. Zhou, Yi-Yu. Cheng and H.-J. Schulze, arXiv:1910.05884 [nucl-th].[31] T. Miyatsu and K. Saito, Prog. Theor. Phys. , 1035 (2009);arXiv:0903.1893 [nucl-th].[32] M. Kohno and Y. Fujiwara, Phys. Rev. C , 054318 (2009);arXiv:0904.0517 [nucl-th].[33] M. Kohno, Phys. Rev. C , 014003 (2010);arXiv:0912.4330 [nucl-th].[34] H. Ohnishi, F. Sakuma, and T. Takahashi, arXiv:1912.02380 [nucl-ex].[35] K. P. Khemchandani et al. , Phys. Rev. D , 094008 (2015);arXiv:1406.7203 [nucl-th].[36] E. Ya. Paryev, Chinese Physics C, Vol. , No. (8), 084101 (2018);arXiv:1806.00303 [nucl-th].[37] S. V. Efremov and E. Ya. Paryev, Eur. Phys. J. A , 99 (1998).[38] E. Ya. Paryev, Eur. Phys. J. A , 521 (2000).[39] E. Ya. Paryev, Eur. Phys. J. A , 127 (2000).[40] T. Harada and Y. Hirabayashi, Phys. Rev. C , 024618 (2020);arXiv:2006.15627 [nucl-th].[41] V. Flaminio et al. , Compilation of Cross Sections.II: K + and K − Induced Reactions. CERN-HERA , (1983).[42] Ph. Gavillet et al. , Phys. Lett. B , 517 (1978).1943] F. Li, L.-W. Chen, C. M. Ko and S. H. Lee, Phys. Rev. C , 064902 (2012);arXiv:1204.1327 [nucl-th].[44] W. J. Briscoe et al. , Eur. Phys. J. A , 129 (2015);arXiv:1503.07763 [hep-ph].[45] M. Nanova et al. (CBELSA/TAPS Collaboration), Phys. Lett. B , 417 (2013);arXiv:1311.0122 [nucl-ex].[46] M. Nanova et al. (CBELSA/TAPS Collaboration), Phys. Rev. C94