Resonance studies at STAR

Preliminary results from measurements of resonances (K ^*0(892), $\overline {K*^0 } (892)$ , Φ(1020), and ρ(770)) and weakly decaying particles (Λ(1116), $\bar \Lambda (1116)$ , and K _S ⁰ (498)) are presented. The measurements are performed at mid-rapidity by the STAR detector in $\sqrt {s_{NN} } = 130$ GeV Au?Au collisions at RHIC. The ratios K ^*0/h?, $\overline {K*^0 } /K$ , and $\bar \Lambda /\Lambda $ are compared to measurements at different energies and colliding systems. Estimates of thermal parameters, such as temperature and baryon chemical potential, are also presented.     

K S 0 , Λ and ≡ production at intermediate to high pT from Au+Au collisions at \sqrt {s_{NN} } = 39, 11.5 and 7.7 GeV

We report on the p _T dependence of nuclear modification factors (R _CP) for K _S ⁰ , ??, ?? and the $\bar NK_S^0 $ ratios at mid-rapidity from Au+Au collisions at $\sqrt {s_{NN} } $ = 39, 11.5 and 7.7 GeV. At $\sqrt {s_{NN} } $ = 39 GeV, the R _CP data show a baryon/meson separation at intermediate p _T and a suppression for K _S ⁰ for p _T up to 4.5 GeV/c; the $\bar \Lambda K_S^0 $ shows baryon enhancement in the most central collisions. However, at $\sqrt {s_{NN} } $ = 11.5 and 7.7 GeV, R _CP shows less baryon/meson separation and $\bar NK_S^0 $ shows almost no baryon enhancement. These observations indicate that the matter created in Au+Au collisions at $\sqrt {s_{NN} } $ = 11.5 or 7.7 GeV might be distinct from that created at $\sqrt {s_{NN} } $ = 39 GeV.     

Search for Θ+(1540) in exclusive proton-induced reaction p + C(N) \to \Theta ^ + \bar \kappa ^0 + C(N) at the energy of 70 GeV

A search for narrow Θ⁺(1540), a candidate for a pentaquark baryon with positive strangeness, has been performed in an exclusive proton-induced reaction $p + C(N) \to \Theta ^ + \bar \kappa ^0 + C(N)$ on carbon nuclei or quasifree nucleons at $E_{beam} = 70GeV(\sqrt s = 11.5GeV)$ studying nK ⁺, pK _S ⁰ , and pK _L ⁰ decay channels of Θ⁺(1540) in four different final states of the $\Theta ^ + \bar K^0 $ system. In order to assess the quality of the identification of the final states with neutron or K _L ⁰ , we reconstructed Λ(1520) → nK _S ⁰ and ? → K _L ⁰ K _S ⁰ decays in the calibration reactions p + C(N) → Λ (1520)K ⁺+C(N) and p+C(N) → p?+C(N). We found no evidence for a narrow pentaquark peak in any of the studied final states and decay channels. Assuming that the production characteristics of the $\Theta ^ + \bar K^0 $ system are not drastically different from those of the Λ(1520)K ⁺ and p? systems, we established upper limits on the cross-section ratios $\sigma (\Theta ^ + \bar K^0 )/\sigma (\Lambda (1520)K^ + ) < 0.02$ and $\sigma (\Theta ^ + \bar K^0 )/\sigma (p\phi ) < 0.15$ at 90% C.L. and a preliminary upper limit for the forward-hemisphere cross section $\sigma (\Theta ^ + \bar K^0 )$ nb/nucleon.     

Strange and non-strange (Anti-) baryon production at 200 A GeV/c

Rapidity distributions of net hyperons $\left( {\Lambda - \bar \Lambda } \right)$ are compared to distributions of participant protons $\left( {p - \bar p} \right)$ . Strangeness production (mean multiplicities of produced Λ/Σ⁰ hyperons and $\left\langle {K + \bar K} \right\rangle $ in central nucleusnucleus collisions is shown for different collision systems at different energies. An enhanced production of $\bar \Lambda $ compared to $\bar p$ is observed at 200 GeV per nucleon.     

Quark mixing angles in the left-ringt symmetric model with lightW R fromK 0-K 0 mixing

K _L?K _S mass difference and the CP violation parameter, ?, of theK ⁰ ? \(\overline {K^0 } \) system are used to set bounds on the right-handed Cabibbo-like angle and the CP violating phase angle in the left-right symmetric electroweak model of four quarks. The corresponding mixing and phase angles in typical left-right asymmetric models (g _L≠g _R) are also determined.     

Measurement of the masses and lifetimes of the charmed mesonsD 0,D + andD s +

We present the final results on the measurement of the masses and lifetimes of the mesonsD ⁰,D ⁺ andD _s ⁺ in the NA32 experiment at the CERN SPS, using silicon microstrip detectors and charge-coupled devices for vertex reconstruction. We measure the following lifetimes: \(\tau _{D^0 } = 3.88 \pm _{0.21}^{0.23} \cdot 10^{ - 13} s\) using a sample of 479D°→K ^?π⁺π^?π⁺ and 162D°→K ^?π⁺ decays; \(\tau _{D^ + } = 10.5 \pm _{0.72}^{0.77} \cdot 10^{ - 13} s\) with a sample of 317D ⁺→K ^?π⁺π⁺ decays; \(\tau _{D_s^ + } = 4.69 \pm _{0.86}^{1.02} \cdot 10^{ - 13} s\) with a sample of 54D _s ⁺ →K ⁺ K ^?π⁺ decays. We measure the following masses:m _D ⁰=1864.6±0.3±1.0 MeV,m _D ⁺=1870.0±0.5±1.0 MeV and \(m_{D_s^ + } \) =1967.0±1.0±1.0 MeV.     

Localization in the ground state of the ising model with a random transverse field

We study the zero-temperature behavior of the Ising model in the presence of a random transverse field. The Hamiltonian is given by $$H = - J\sum\limits_{\left\langle {x,y} \right\rangle } {\sigma _3 (x)\sigma _3 (y) - \sum\limits_x {h(x)\sigma _1 (x)} } $$ whereJ>0,x,y∈Z ^d, σ₁, σ₃ are the usual Pauli spin 1/2 matrices, andh={h(x),x∈Z ^d} are independent identically distributed random variables. We consider the ground state correlation function 〈σ₃(x)σ₃(y)〉 and prove:

1. Letd be arbitrary. For anym>0 andJ sufficiently small we have, for almost every choice of the random transverse fieldh and everyx∈Z ^d, that $$\left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle \leqq C_{x,h} e^{ - m\left| {x - y} \right|} $$ for ally∈Z ^d withC _{x h} <∞.
2. Letd≧2. IfJ is sufficiently large, then, for almost every choice of the random transverse fieldh, the model exhibits long range order, i.e., $$\mathop {\overline {\lim } }\limits_{\left| y \right| \to \infty } \left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle > 0$$ for anyx∈Z ^d.

     

Bound kaon approach for the ppK − system in the Skyrme model

The lightest nuclear $\overline K$ bound state, ppK ^?, is investigated in the Skyrme model. We describe the ppK ^? as two-Skyrmion around which a kaon field fluctuates. The two-Skyrmion is projected onto the (pp) _S=0 state using the collective coordinate quantization method. We find that the energy of K ^? can be considerably small, and that ppK ^? is a molecular state. The binding energy of the ppK ^? and the mean pp distance are estimated to be B.E. = 104–126?MeV and $\sqrt{\langle r_{pp}^{2}\rangle}$ = 1.6–1.8?fm, respectively.     

The predissociation of the 2σ u −1 (c 4Σ u − , v states of the oxygen molecular ion

The dynamics of predissociation of the 2σ _u ^?1 (c ⁴Σ _u ^? ), v vibrational states of the O ₂ ⁺ ion was studied theoretically using the method of coupled differential equations. The main equations describing the vibrational motions of nuclei in the adiabatic and diabatic approximations are given. The applicability scope of approximate methods for solving these equations was studied. The predissociation widths for the v = 0 and 1 vibrational levels were found to be Γ₀ = 0.054 meV and Γ₁ = 9.71 meV. This substantiated the results of recent observations of neutral fragments formed after the dissociation of the O₂ molecule. About 99% of the O ₂ ⁺ ions in the 2σ _u ^?1 (c ⁴Σ _u ^? ), v states were found to decompose to the O(¹ D) + O⁺(⁴ S) dissociation products.     

Experimental determination of penetration parameter forM1 conversion,and theE2 admixtures for 77 and 191 keV transitions in197Au

TheL-subshell conversion for 77 keV transition andK,L ₁,L ₂-shell conversion for 191 keV transition in¹⁹⁷Au, as well asK-shell conversion transition of 158 keV in¹⁹⁹Hg were measured by means of Π√2-iron free electron spectrometer. Relative gamma-ray intensities have been determined by Ge(Li) spectrometer. From these measurements the α _K conversion coefficient value has been deduced for 191 keV transition as α_K(191 keV)=0.86±0.03. This value shows that the spin of the level at 268 keV in¹⁹⁷Au is 3/2⁺. For the penetration parameter (λ) and intensity ratio \(\left( {\delta ^2 = \frac{{\left\langle {E2} \right\rangle ^2 }}{{\left\langle {MI} \right\rangle ^2 }}} \right)\) the following values are obtained: $$\begin{gathered} \lambda = 3.4 \pm _{1.5}^{1.9} \delta ^2 = 0.11 \pm 0.03for 77 keV transition \hfill \\ \lambda = 3.2 \pm _{0.6}^{0.8} \delta ^2 = 0.17 \pm 0.04for 191 keV transition. \hfill \\ \end{gathered} $$ The agreement of these results with the predictions of De Shalit model is discussed.     

Production of π0 mesons and charged hadrons in\bar v neon and ν neon charged current interactions

The average multiplicities of charged hadrons and of π⁺, π^? and π⁰ mesons, produced in \(\bar v\) Ne and νNe charged current interactions in the forward and backward hemispheres of theW ^±-nucleon center of mass system, are studied with data from BEBC. The dependence of the multiplicities on the hadronic mass (W) and on the laboratory rapidity (y _Lab) and the energy fraction (z) of the pion is also investigated. Special care is taken to determine the π⁰ multiplicity accurately. The ratio of average π multiplicities \(\frac{{2\left\langle {n_{\pi ^O } } \right\rangle }}{{[\left\langle {n_{\pi ^ + } } \right\rangle + \left\langle {n_{\pi ^ - } } \right\rangle ]}}\) is consistent with 1. In the backward hemisphere \(\left\langle {n_{\pi ^O } } \right\rangle \) is positively correlated with the charged multiplicity. This correlation, as well as differences in multiplicities between \(\mathop v\limits^{( - )} \) and \(\mathop v\limits^{( - )} \) , \(\mathop v\limits^{( - )} \) scattering, is attributed to reinteractions inside the neon nucleus of the hadrons produced in the initial \(\mathop v\limits^{( - )} \) interaction.     

Further study of theE/f 1 (1420) meson in central production

We have studied the reactions \(({{\pi ^ + } \mathord{\left/ {\vphantom {{\pi ^ + } p}} \right. \kern-0em} p})p \to ({{\pi ^ + } \mathord{\left/ {\vphantom {{\pi ^ + } p}} \right. \kern-0em} p})(K\bar K\pi )p\) where the \(K\bar K\pi \) system is centrally produced, at 85 GeV/c and 300 GeV/c using the CERN Omega spectrometer. A spin-parity analysis of theK _S ⁰ K ^± π ^? system shows the presence of a strongJ ^PC=1⁺⁺ signal which we identify as theE/f ₁ (1420) meson. We also find evidence for the decayE/f ₁(1420)→K _S ⁰ K _S ⁰ π ⁰ which determines theC-parity of this state to be positive. Alternative explanations of the data have been tested and ruled out. Hence we obtain the quantum numbers of theE/f ₁ (1420) to beI ^G(J^PC)=0⁺(1⁺).     

CP violation with beautiful baryons

CP violation can be studied in modes of charmed or bottom baryons when a decay process is compared with its charge-conjugated partner. It can show up as a rate asymmetry and in a study of other decay parameters. Neither tagging nor time-dependences are required to observeCP violation with modes of baryons, in contrast to the conventionalB ⁰ modes. Numerous modes of bottom baryons have the potential to show largeCP-violating effects within the Standard Model. Those effects can be substantial for modes with aD ⁰, which is seen in a final state that can also be fed from a \(\bar D^0 \) . For instance, a comparison of theΛ _b→Λ _CP ⁰ with the \(\bar \Lambda _b \to \bar \Lambda D_{CP}^0 \) process can show sizeableCP violation. HereD _CP ^o denotesCP eigenstates ofD ⁰, which occur at a few percent. Six related processes, such asΛ _b→ΛD ⁰, \(\Lambda _b \to \Lambda \bar D^0 \) ,Λ _b→Λ _CP ⁰ , and their charge-conjugated counterparts, can extract ?, which is the most problematic angle of the unitarity triangle and which is conventionally probed with theB _s→ρ⁰ K _S asymmetry. HereD ⁰ andD ^?0 are identified by their charged kaon or lepton. We predictB(Λ _b→ΛD ⁰)～10^?5, thusB(Λ _b→Λ _CP ⁰ )～10^?7. Under favourable circumstances,CP violation can occur at the few tens of percent level. Thus 10²–10³ Λ _b→Λ _CP ⁰ decays start probing ?. Tables list many additional modes with typical branching ratios at the 10^?5–10^?6 level, with large detection efficiencies (in contrast to theD _CP ⁰ ), and with potentially largeCP-violating effects, such as Ξ _b ⁰ →ΛΨ, Λ?, ΛK^*0; Ξ _b ^? →ΛK^(*)?, Ξ^?K_s, Ξ^?K^*0, Ω _b ^? →Ξ^?φ, Ξ^?ρ⁰, ΛK^(*)?, ΩK_s, Ω^?K^*0.     

Uniform boundedness of conditional gauge and Schrödinger equations

We prove that for a bounded domainD ?R ⁿ withC ² boundary and \(q \in K_n^{loc} (n \geqq 3) if E^x \exp \int\limits_0^{\tau _D } {q(x_t )dt} \mathop \ddag \limits_--- \infty \) inD, then $$\mathop {\sup }\limits_{\mathop {x \in D}\limits_{z \in \partial D} } E_z^x \exp \int\limits_0^{\tau _D } {q(x_t )dt}< + \infty $$ ({x _t : Brownian motion}) The important corollary of this result is that if the Schrödinger equation Δ/2u+qu=0 has a strictly positive solution onD, then for anyD ₀ ? ?D, there exists a constantC=C(n,q,D,D ₀) such that for anyf εL ¹(?D, σ), (σ: area measure on ?D) we have $$\mathop {\sup |}\limits_{x \in D_0 } u_f (x)| \mathop< \limits_ = C\int\limits_{\partial D} {|f(y)|\sigma (dy)} $$ whereu _f is the solution of the Schrödinger equation corresponding to the boundary valuef. To prove the main result we set up the following estimate inequalities on the Poisson kernelK(x,z) corresponding to the Laplace operator: $$C_1 \frac{{d(x,\partial D)}}{{|x - z|^n }}\mathop< \limits_ = K(x,z)\mathop< \limits_ = C_2 \frac{{d(x,\partial D)}}{{|x - z|^n }},x \in D,z \in \partial D$$ whereC ₁ andC ₂ are constants depending onn andD.     

Evidence for a newK * \bar K state at a mass of 1,530 MeV withIJ PC=01++ observed inK − p interactions at 4.2 GeV/c

Evidence is presented for a newK ^* \(\bar K\) +c.c. resonance with a mass of (1,526±6) MeV, a width of (107±15) MeV and quantum numbersIJ ^PC=01⁺⁺. We call itD′ meson. Initially it is observed as aK ^* \(\bar K\) +c.c. enhancement in the reactionsK ^? p→(K _s ⁰ K ^±π^?)Λ at 4.2 GeV/c. The isospin assignmentI=0 comes from its further observation in the reactionsK ^? p→(K _s ⁰ K ^±π^?)Σ ⁰ andK ^? p→(K _s ⁰ K ^±π^?)Σ(1,385)⁰ but not inK ^? p→(K ⁺ K ^?π^?)Σ⁺ orK ^? p→(K _s ⁰ K ^±π^?)Σ(1,385)⁺. A maximum likelihood analysis of the (K \(\bar K\) π) decay Dalitz plots in the reactionsK ^? p→(K _s ⁰ K ^±π^?) determines theJ ^PC of theD′ meson to be 1⁺⁺. A satisfactorySU(3) fit is obtained to a 1⁺⁺ nonet composed of theI-1A ₁, theI=1/2Q _A with theD(1,285) and theD′(1,526) as theI=0 members having a mixing angle close to the magic one.     

A coupled-channel treatment of Cabibbo-angle favoured (D,D s + )→VP decays

We have performed a two-channel calculation of Cabibbo-angle favoured decays,D _s ⁺ →VP. We find a satisfactory fit toS _s ⁺ →φπ ⁺,ρ ⁰ π ⁺ andK ⁺ \(\bar K^{ * 0} \) data from ARGUS and E-691. We have also studied Cabbibo-angle favouredD→VP decays in a coupled channel formalism. We coupleD→K ^*π,K _? and \(\bar K^0 \phi \) channels inI=1/2 state, andK ^*π andK _? channels inI=3/2 state. We leave the two channels, \(\bar K^0 \omega \) and \(\bar K^{ * 0} \eta \) out of our unitarization scheme. Particular attention is paid to the role of the weak annihilation term in these decays.     

Determination of the CP violating phase γ by a sum over common decay modes toB s and\bar B_s

To help the difficult determination of the angle γ of the unitarity triangle, Aleksan, Dunietz and Kayser have proposed the modes of the typeK ^? D _s ⁺ , common toB _s and $\bar B_s $ . We point out that it is possible to gain in statistics by a sum over all modes with ground state mesons in the final state, i.e.K ^? D _s ⁺ ,K ^*? D _s ⁺ ,K ^? D _s ^*+ ,K ^* D _s ^* . The delicate point is the relative phase of these different contributions to the dilution factorD of the time dependent asymmetry. Each contribution toD is proportional to a product $F^{cb} F^{ub} f_{D_s } f_K $ whereF denotes form factors andf decay constants. Within a definite phase convention (i.e. for example the one defined by the heavy quark symmetry in the limit of heavy quarks), lattice calculations do not show any change in sign when extrapolating to light quarks the form factors and decay constants. Then, we can show that all modes contribute constructively to the dilution factor, except theP-waveK ^* D _s ^*+ , which is small. Quark model arguments based on wave function overlaps also confirm this stability in sign. By summing over all these models we find a gain of a factor 6 in statistics relatively toK ^? D _s ⁺ . The dilution factor for the sumD _tot is remarkably stable for theoretical schemes that are not in very strong conflict with data onB→ψK(K ^*) or extrapolated from semileptonic charm form factors, givingD _tot≥0.6, always close toD(K ^? D _s ⁺ ).     

On the free energy of the hopfield model

The general theory of inhomogeneous mean-field systems of Raggio and Werner provides a variational expression for the (almost sure) limiting free energy density of the Hopfield model $$H_{N,p}^{\{ \xi \} } (S) = - \frac{1}{{2N}}\sum\limits_{i,j = 1}^N {\sum\limits_{\mu = 1}^N {\xi _i^\mu \xi _j^\mu S_i S_j } } $$ for Ising spinsS _i andp random patterns ξ^μ=(ξ ₁ ^μ ,ξ ₂ ^μ ,...,ξ _N ^μ ) under the assumption that $$\mathop {\lim }\limits_{N \to \gamma } N^{ - 1} \sum\limits_{i = 1}^N {\delta _{\xi _i } = \lambda ,} \xi _i = (\xi _i^1 ,\xi _i^2 ,...,\xi _i^p )$$ exists (almost surely) in the space of probability measures overp copies of {?1, 1}. Including an “external field” term ?ξ _μ ^p h^μμξ _i=1 ^N ξ _i ^μ S_i, we give a number of general properties of the free-energy density and compute it for (a)p=2 in general and (b)p arbitrary when λ is uniform and at most the two componentsh ^μ1 andh ^μ2 are nonzero, obtaining the (almost sure) formula $$f(\beta ,h) = \tfrac{1}{2}f^{ew} (\beta ,h^{\mu _1 } + h^{\mu _2 } ) + \tfrac{1}{2}f^{ew} (\beta ,h^{\mu _1 } - h^{\mu _2 } )$$ for the free energy, wheref ^cw denotes the limiting free energy density of the Curie-Weiss model with unit interaction constant. In both cases, we obtain explicit formulas for the limiting (almost sure) values of the so-called overlap parameters $$m_N^\mu (\beta ,h) = N^{ - 1} \sum\limits_{i = 1}^N {\xi _i^\mu \left\langle {S_i } \right\rangle } $$ in terms of the Curie-Weiss magnetizations. For the general i.i.d. case with Prob {ξ _i ^μ =±1}=(1/2)±?, we obtain the lower bound 1+4?²(p?1) for the temperatureT _c separating the trivial free regime where the overlap vector is zero from the nontrivial regime where it is nonzero. This lower bound is exact forp=2, or ε=0, or ε=±1/2. Forp=2 we identify an intermediate temperature region between T_*=1?4?² and T_c=1+4?² where the overlap vector is homogeneous (i.e., all its components are equal) and nonzero.T _* marks the transition to the nonhomogeneous regime where the components of the overlap vector are distinct. We conjecture that the homogeneous nonzero regime exists forp≥3 and that T_*=max{1?4?²(p?1),0}.