Search

We study the problem of online convex optimization with memory and predictions over a horizon $T$. At each time step, a decision maker is given some limited predictions of the cost functions from a finite window of future time steps, i.e., values of the cost function at certain decision points in the future. The decision maker then chooses an action and incurs a cost given by a convex function that depends on the actions chosen in the past. We propose an algorithm to solve this problem and show that the dynamic regret of the algorithm decays exponentially with the prediction window length. Our algorithm contains two general subroutines that work for wider classes of problems. The first subroutine can solve general online convex optimization with memory and bandit feedback with $\sqrt{T}$-dynamic regret with respect to $T$. The second subroutine is a zeroth-order method that can be used to solve general convex optimization problems with a linear convergence rate that matches the best achievable rate of first-order methods for convex optimization. The key to our algorithm design and analysis is the use of truncated Gaussian smoothing when querying the decision points for obtaining the predictions. We complement our theoretical results using numerical experiments., Comment: 28 pages, 2 figures

An integer-valued polynomial $P(x,y,z)$ is said to be universal (over $\mathbb Z$) if each nonnegative integer can be written as $P(x,y,z)$ with $x,y,z\in\mathbb Z$. In this paper, we mainly introduce a new technique to determine the universality of some sums in the form $x(a_1x+a_2)/2+y(b_1y+b_2)/2+z(c_1z+c_2)/2$ (with $a_1-a_2,b_1-b_2,c_1-c_2$ all even) conjectured by Sun, using various identities of Ramanujan's theta functions. For example, we prove that $x(3x+1)+y(3y+2)+2z(3z+2)$ and $x(4x+r)+y(3y+1)/2+z(7z+1)/2\ (r=1,3)$ are universal., Comment: 22 pages, refined version with some remarks added

We investigate the supersymmetry breaking soft terms for all the viable models in the complete landscape of three-family supersymmetric Pati-Salam models arising from intersecting D6-branes on a $\mathbb{T}^6/(\mathbb{Z}_2\times \mathbb{Z}_2)$ orientifold in type IIA string theory. The calculations are performed in the general scenario of $u$-moduli dominance with the $s$-moduli turned on, where the soft terms remain independent of the Yukawa couplings and the Wilson lines. The results for the trilinear coupling, gaugino-masses, squared-mass parameters of squarks, sleptons and Higgs depend on the brane wrapping numbers and the susy breaking parameters. We find that unlike the Yukawa couplings which remain unchanged for the models dual under the exchange of two SU(2) sectors, the corresponding soft term parameters only match for the trilinear coupling and the mass of the gluino. This can be explained by the internal geometry where the Yukawa interactions depend only on the triangular areas of the worldsheet instantons while the soft terms have an additional dependence on the orientation-angles of D6-branes in the three two-tori. In the special limit of parameter space we find universal masses for the Higgs and the gauginos., Comment: 61 pages + appendix, 18 figures. arXiv admin note: substantial text overlap with arXiv:2409.09110

We examine the vacuum stability of gauge symmetry breaking in five dimensions, compactified on the $S_1/(\mathbb{Z}_2 \times \mathbb{Z}'_2)$ orbifold. We consider $SU(N)$, $Sp(N)$, $SO(2N)$ and $SO(2N+1)$ theories in the bulk, and provide an exhaustive classification of possible parity assignments that lead to stable orbifolds and of the corresponding symmetry breaking patterns. We use these results in the search for viable asymptotic grand unification theories (aGUT), testing the stability criteria on models based on $SU(6)$ and $SU(8)$. As a result, we identify two viable aGUTs: a unique $SU(6)$ pathway down to the Standard Model, and one $SU(8)$ model leading to an intermediate Pati-Salam partial unification., Comment: 44 pages, 4 figures

Recently, the complete landscape of three-family supersymmetric Pati-Salam models from intersecting D6-branes on a type IIA $\mathbb{T}^6/(\mathbb{Z}_2\times \mathbb{Z}_2)$ orientifold has been enumerated consisting of 33 independent models with distinct gauge coupling relations at the string scale. Here, we study the phenomenology of all such models by providing the detailed particle spectra and the analysis of the possible 3-point and the 4-point Yukawa interactions in order to accommodate all standard-model fermion masses and mixings. We find that only 17 models contain viable Yukawa textures to explain quarks masses, charged-leptons' masses, neutrino-masses, quarks' mixings and leptons' mixings. These viable models split into four classes, viz. a single model with 3 Higgs fields from the bulk and sixteen models with either 6, 9 or 12 Higgs from the $\mathcal{N}=2$ sector. The models perform successively better with the increasing number of Higgs pairs. Remarkably, the class of models with 12 Higgs naturally predicts the Dirac-type neutrino masses in normal ordering consistent with both the experimental constraints as well as the bounds from the swampland program., Comment: 89 pages + appendix, 21 figures, accepted in JHEP

Let $p=2n+1$ be an odd prime. In this paper, we mainly evaluate determinants involving $(\frac {j+k}p)\pm(\frac{j-k}p)$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. When $p\equiv1\pmod4$, we determine the characteristic polynomials of the matrices $$\left[\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)\right]_{1\le j,k\le n}\ \ \text{and}\ \ \left[\left(\frac{j+k}p\right)-\left(\frac{j-k}p\right)\right]_{1\le j,k\le n},$$ and also prove that \begin{align*} &\ \left|x+\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)+\left(\frac jp\right)y+\left(\frac kp\right)z+\left(\frac{jk}p\right)w\right|_{1\le j,k\le n} \\=&\ (-p)^{(p-5)/4}\left(\left(\frac{p-1}2\right)^2wx-\left(\frac{p-1}2y-1\right)\left(\frac{p-1}2z-1\right)\right), \end{align*} which was previously conjectured by the second author., Comment: 16 pages. Add Theorem 1.3 and its proof

A recent analysis of the LHCb data [Phys. Rev. D 105 (2022) L031503] obtained a sizable negative effective range for the $X(3872)$. This has attracted intensive discussions on whether $X(3872)$ can be deemed as a $D\bar{D}^*$ molecular state. This work explicitly demonstrates that the negative effective range of the $X(3872)$ does not contradict the molecular picture, adopting an effective field theory formulation of the $D\bar{D}^*$ interaction that can simultaneously reproduce the binding energy and effective range of the $X(3872)$. We elaborate on the implications of the large negative effective range of $X(3872)$ and the small binding energy on the underlying $D\bar{D}^*$ interaction. Such results are relevant for a better understanding of hadronic molecules and their binding mechanism., Comment: 6 pages, 4 figures; Comments and suggestions are welcome

Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. In this paper, we study the determinant $$\det\left[\left(\frac{j^2-k^2}p\right)+\left(\frac{jk}p\right)w\right]_{\delta\le j,k\le (p-1)/2}$$ with $\delta\in\{0,1\}$. For example, we prove that the determinant does not depend on $w$ if $p\equiv3\pmod4$ and $\delta=0$., Comment: 7 pages

We propose a novel class of Dark Matter (DM) candidates in the form of a heavy composite Axion-Like Particle (ALP) with highly suppressed electromagnetic interactions, being stable even for masses exceeding the GeV scale. We argue that such a composite ALP emerges as a bound state -- the dark glueball -- due to confinement in a pure Yang-Mills dark sector. In a minimal ultraviolet complete QCD-like model, cosmological production of dark gluons as well as photons occurs via heavy fermion annihilation which effectively reheats both the dark and visible sectors setting up their temperature scales. Furthermore, effective interactions between glueballs and photons, resembling those of standard ALPs, are radiatively generated by heavy fermion loops. Consequently, DM glueballs interacting with photons are dubbed `Glueball ALPs' (GALPs). We uncover novel phenomenology of GALPs focusing on their unique astrophysical and cosmological signatures., Comment: 7 pages, 1 figure

In this paper, we prove a conjecture of the second author by evaluating the determinant $$\det\left[x+\left(\frac{i-j}p\right)+\left(\frac ip\right)y+\left(\frac jp\right)z+\left(\frac{ij}p\right)w\right]_{0\le i,j\le(p-3)/2}$$ for any odd prime $p$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. In particular, the determinant is equal to $x$ when $p\equiv 3\pmod4$., Comment: 16 pages. Make the main result more general

The DarkSHINE experiment aims to search for dark photons by measuring the energy loss of the electrons recoiled from fixed-target. Its electromagnetic calorimeter is primarily responsible for accurately reconstructing the energy of the recoil electrons and bremsstrahlung photons. The performance of the electromagnetic calorimeter is crucial, as its energy measurement precision directly determines the sensitivity to the search for dark photons. The DarkSHINE electromagnetic calorimeter uses LYSO crystals to form a fully absorptive electromagnetic calorimeter. It utilizes SiPMs to detect scintillation light in the crystals, and its readout electronics system deduces the deposited energy in the crystals by measuring the number of photoelectric signals generated by the SiPMs. The DarkSHINE electromagnetic calorimeter aims to operate at an event rate of 1-10 MHz, detecting energies ranging from 1 MeV to 1 GeV. To meet the requirements of high energy measurement precision, high event rate, and large dynamic range, we have researched and designed a readout electronics system based on dual-channel high-speed ADCs and a customized DAQ. The front-end amplification part of this system uses low-noise trans-impedance amplifiers to achieve high-precision waveform amplification. It successfully achieves a dynamic range up to a thousandfold through a double-gain readout scheme. The digital part uses 1 GSPS high-speed ADCs to achieve non-dead-time, high-precision waveform digitization. The DAQ part uses JESD204B high-speed serial protocol to read out the signal from ADC, and transmit it to PC software for processing and storage. Test results show a signal-to-noise ratio greater than 66 dBFS and an ENOB greater than 10.6 bits. Energy spectra measurements have been conducted using LYSO crystals and SiPMs, and an energy resolution of 5.96% at the 2.6 MeV gamma peak of Th-232 has been achieved.

The ground-breaking discovery of the first fully charmed tetraquark state $X(6900)$ in the $J/\psi J/\psi$ invariant mass distribution by the LHCb collaboration has inspired intensive theoretical studies. Various interpretations, such as molecular states, compact tetraquark states, and coupled-channel effects, have been proposed for these states. Of particular interest is the ongoing search for the triple-$J/\psi$ state--a fully-charmed hexaquark state. To deepen our understanding of the triple-$J/\psi$ state and to guide future experimental searches, we study the triple-$J/\psi$ system in this work employing the Gaussian expansion method and the $J/\psi J/\psi$ potential parameterized to yield a shallow bound state, as suggested in several theoretical works. Our results support a triple-$J/\psi$ bound state, even in cases where the attractive interaction between the two $J/\psi$ mesons is very weak. Moreover, our analysis implies the Efimov effect in the triple-$J/\psi$ system. In addition, we extend our investigation to the triple-$\Upsilon (1S)$ system and obtain results similar to those for the triple-$J/\psi$ system., Comment: 8 pages, 3 figures, 4 tables; comments welcome

Analyzing the supersymmetric Pati-Salam landscape on a $\mathbb{T}^6/(\mathbb{Z}_2 \times \mathbb{Z}_2)$ orientifold in IIA string theory, we have found only two models that accurately account for all standard model fermion masses and mixings. The models are dual to each other under the exchange of two SU(2) sectors and feature 12 adjoint scalars, the maximum number allowed in the landscape, whose linear combination yields the two light Higgs eigenstates. Dirac neutrino-masses in normal ordering $(50.4,~10.5,~6.1)~\mathrm{meV}$ satisfying the swampland constraints are predicted, a testable prospect for string phenomenology., Comment: 5 pages

Calibrating the optical properties within the detection medium of a neutrino telescope is crucial for determining its angular resolution and energy scale. For the next generation of neutrino telescopes planned to be constructed in deep water, such as the TRopIcal DEep-sea Neutrino Telescope (TRIDENT), there are additional challenges due to the dynamic nature and potential non-uniformity of the water medium. This necessitates a real-time optical calibration system distributed throughout the large detector array. This study introduces a custom-designed CMOS camera system equipped with rapid image processing algorithms, providing a real-time optical calibration method for TRIDENT and other similar projects worldwide. In September 2021, the TRIDENT Pathfinder experiment (TRIDENT Explorer, T-REX for short) successfully deployed this camera system in the West Pacific Ocean at a depth of 3420 meters. Within 30 minutes, about 3000 images of the T-REX light source were captured, allowing for the in-situ measurement of seawater attenuation and absorption lengths under three wavelengths. This deep-sea experiment for the first time showcased a technical demonstration of a functioning camera calibration system in a dynamic neutrino telescope site, solidifying a substantial part of the calibration strategies for the future TRIDENT project.

In this paper, we study arithmetic properties of certain determinants involving powers of $i^2+cij+dj^2$, where $c$ and $d$ are integers. For example, for any odd integer $n>1$ with $(\frac dn)=-1$ we prove that $\det [ (\frac{i^2+cij+dj^2}{n})]_{0\le i,j\le n-1}$ is divisible by $\varphi(n)^2$, where $(\frac{\cdot}{n})$ is the Jacobi symbol and $\varphi$ is Euler's totient function. This confirms a previous conjecture of the second author., Comment: 13 pages

Giant resonances (GRs) provide crucial insights into nuclear physics and astrophysics. Exciting GRs using particles like electrons is effective, yet the angular momentum (AM) transfer of electrons, including both intrinsic spin and orbital degrees of freedom in inelastic scattering, has never been studied. Here, we investigate AM transfer in GRs excited by plane-wave and vortex electrons, developing a comprehensive AM-resolved inelastic electron scattering theory. We find that even plane-wave electrons can model-independently extract transition strengths of higher multipolarity by selecting specific AM states of scattered electrons. Additionally, relativistic vortex electrons with orbital angular momentum (OAM) $\pm1$ can be efficiently generated. Vortex electrons can also be used to extract GR transition strength as in the plane-wave case, regardless of the position of nucleus relative to the beam axis. Furthermore, relativistic vortex electrons with larger OAM can be generated for on-axis nuclei due to AM conservation. Our method offers new perspectives for nuclear structure research and paves the way for generating vortex particles.

In this paper we investigate determinants whose entries are linear combinations of Legendre symbols. We deduce some new results in this direction; for example, we prove that for any prime $p\equiv3\pmod4$ we have $$\det\left[x+\left(\frac{j-k}p\right)+\left(\frac jp\right)-\left(\frac kp\right)\right]_{0\le j,k\le(p-1)/2}=4,$$ where $(\frac{\cdot}p)$ is the Legendre symbol. We also pose many conjectures for further research. For example, for any prime $p>3$ we conjecture that \begin{align*}&\ \det\left[\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)+\left(\frac{jk}p\right)\right]_{1\le j,k\le(p-1)/2} \\=&\ \begin{cases}(\frac 2p)p^{(p-5)/4}&\text{if}\ p\equiv1\pmod4, \\(-1)^{(h(-p)-1)/2}(1-(2-(\frac 2p))h(-p))p^{(p-3)/4}&\text{if}\ p\equiv3\pmod4, \end{cases}\end{align*} where $h(-p)$ is the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$., Comment: 24 pages. Refined version with typos corrected

There have been extended and heated discussions on the nature of the two exotic states, $Z_c(3900)$ and $Z_{cs}(3985)$, particularly whether they are resonances or virtual states. We demonstrate for the first time that the femtoscopic technique can be used to unambiguously distinguish between such two scenarios. More concretely, we show that the $D^0D^{*-}$/$D^0D_s^{*-}$ correlation functions are significantly different in the high-momentum region, especially in small collision systems of the order of 1 fm, as produced in pp collisions at the LHC, which can unambiguously tell whether $Z_c(3900)$/$Z_{cs}(3985)$ is a resonant or a virtual state. For the $Z_{cs}(3985)$, the $D^0D_s^{*-}$ results at zero momentum are significantly different in the two scenarios. We hope all these discoveries can stimulate further experimental studies and help clarify the nature of the many exotic states discovered., Comment: 4 pages, 2 figures. Comments are welcome

We give a direct proof of the following known result: the Grothendieck group of a triangulated category with a silting subcategory is isomorphic to the split Grothendieck group of the silting subcategory. Moreover, we obtain its cluster-tilting analogue., Comment: added a new section to discuss the cluster-tilting analogue

Let $p>3$ be a prime, and let $d\in\mathbb Z$ with $p\nmid d$. For the determinants $$S_m(d,p)=\det\left[(i^2+dj^2)^{m}\right]_{1\leqslant i,j \leqslant (p-1)/2}\ \ \left(\frac{p-1}2\leqslant m\leqslant p-1\right),$$ Sun recently determined $S_m(d,p)$ modulo $p$ when $m\in\{p-2,p-3\}$ and $(\frac {-d}p)=-1$. In this paper, we obtain $S_{p-2}(d,p)$ modulo $p$ in the remaining case $(\frac{-d}p)=1$, and determine the Legendre symbols $(\frac{S_{p-3}\,(d,p)}p)$ and $(\frac{S_{p-4}\,(d,p)}p)$ in some special cases., Comment: 14 pages

Quantum vacuum fluctuations of the electromagnetic field result in two signatures on a harmonically trapped charged particle: a shift from the natural trap frequency and generation of quantum coherences. We assess the role of the long-wavelength and rotating-wave approximations in estimating this frequency shift. We estimate the magnitude of the frequency shift using parameters from a single-electron cyclotron experiment and also demonstrate how the dependence of the frequency shift on the magnetic field of the cyclotron is tied to the rotating-wave approximation. We expect the frequency shift to be observable in future experiments. We also suggest a possible route to detecting vacuum-generated quantum coherences. These experiments should settle the debate on the choice of approximations and gauge in capturing the effect of the quantum vacuum fluctuations., Comment: 6 pages, 1 figure

In the past two decades, a plethora of hadronic states beyond the conventional quark model of $q\bar{q}$ mesons and $qqq$ baryons have been observed experimentally, which motivated extensive studies to understand their nature and the non-perturbative strong interaction. Since most of these exotic states are near the mass thresholds of a pair of conventional hadrons, the prevailing picture is that they are primarily hadronic molecules. In principle, one can verify the molecular nature of these states by thoroughly comparing their masses, decay widths, and production rates in a particular picture with experimental data. However, this is difficult or impossible. First, quantum mechanics allows for the mixing of configurations allowed by symmetries and quantum numbers. Second, data are relatively scarce because of their small production rates and the many difficulties in the experimental measurements. As a result, other alternatives need to be explored. This review summarizes three such approaches that can help disentangle the nature of the many exotic hadrons discovered. In the first approach, based on the molecular interpretations for some exotic states, we study the likely existence of multiplets of hadronic molecules related by various symmetries, such as isospin symmetry, SU(3)-flavor symmetry, heavy quark spin/flavor symmetry, and heavy antiquark diquark symmetry. In the second approach, starting from some hadronic molecular candidates, one can derive the underlying hadron-hadron interactions. With these interactions, one can study related three-body systems and check whether three-body bound states/resonances exist. In the third approach, one can turn to the femtoscopy technique to derive the hadron-hadron interactions, hence inaccessible. This technique provided an unprecedented opportunity to understand the interactions between unstable hadrons., Comment: 140 pages, 40 figures; any comments are appreciated!

TRopIcal DEep-sea Neutrino Telescope (TRIDENT) plans to incorporate silicon photomultipliers (SiPMs) with superior time resolution in addition to photomultiplier tubes (PMTs) into its detection units, namely hybrid Digital Optical Modules (hDOMs), to improve its angular resolution. However, the time resolution significantly degrades for large-area SiPMs due to the large detector capacitance, posing significant challenges for the readout electronics of SiPMs in hDOM. We analyzed the influences of series and parallel connections when constructing a large-area SiPM array and designed a series-parallel connection SiPM array with differential output. We also designed a high-speed pre-amplifier based on transformers (MABA-007159) and radio frequency amplifiers (BGA2803), and an analog multi-channel summing circuit based on operational amplifiers (LMH6629). We measured the single photon time resolution (SPTR) of a $4\times4$ SiPM (Hamamatsu S13360-3050PE) array ($12\times12~\mathrm{mm}^2$) of approximately 300 ps FWHM. This front-end readout design enables the large-area SiPM array to achieve high-precision single photon time measurement in one readout channel., Comment: Revised version. 12 pages, 10 figures

Arrays of neutral atoms have emerged as promising platforms for quantum computing. Realization of high-fidelity two-qubit gates with robustness is currently a significant important task for large-scale operations. In this paper, we present a convenient approach for implementing a two-qubit controlled-phase gate using Rydberg blockade. We achieve the noncyclic geometric control with a single modulated pulse. As compared with the control scheme by cyclic evolution that determined by dynamical parameters, the robustness of the proposal against systematic errors will be remarkably improved due to the geometric characteristic. Importantly, the noncyclic geometric control reduces the gate time for small rotation angles and will be more insensitive to the decoherence effect. We accelerate the adiabatic control with the aid of shortcuts to adiabaticity to further shorten the operation time. We apply our protocol to the algorithm of quantum Fourier transformation to show the actual acceleration. Therefore, the proposed scheme will provide an analytical waveforms for arbitrary two-qubit gates and may have important use in the experiments of atomic arrays., Comment: 8 pages, 5 figures

In this paper we evaluate several determinants involving quadratic residues modulo primes. For example, for any prime $p>3$ with $p\equiv3\pmod4$ and $a,b\in\mathbb Z$ with $p\nmid ab$, we prove that $$\det\left[ 1+\tan\pi\frac{aj^2+bk^2}p \right]_{1\le j,k \le \frac{p-1}2} = \begin{cases}-2^{(p-1)/2}p^{(p-3)/4}&\text{if}\ (\frac{ab}p)=1, \\p^{(p-3)/4}&\text{if}\ (\frac{ab}p)=-1, \end{cases}$$ where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also pose some conjectures for further research., Comment: 25 pages. Mainly add Theorem 1.1(i) and its proof

Using cyclotomic multiple zeta values of level $8$, we confirm and generalize several conjectural identities on infinite series with summands involving $\binom{2k}k8^k/(\binom{3k}k\binom{6k}{3k})$. For example, we prove that \[\sum_{k=0}^\infty\frac{(350k-17)\binom{2k}k8^k} {\binom{3k}k\binom{6k}{3k}}=15\sqrt2\,\pi+27\] and \[\sum_{k=1}^\infty\frac{\left\{(5k-1)\left[16\mathsf H_{2k-1}^{(2)}-3\mathsf H_{k-1}^{(2)}\right]-\frac{12(6k-1)}{(2k-1)^2}\right\}\binom{2k}k8^k} {k(2k-1)\binom{3k}k\binom{6k}{3k}}=\frac{\pi^3}{12\sqrt2},\] where $\mathsf H^{(2)}_m$ denotes the second-order harmonic number $\sum_{0

We perform polylogarithmic reductions for several classes of infinite sums motivated by Z.-W. Sun's related works in 2022--2023. For certain choices of parameters, these series can be expressed by cyclotomic multiple zeta values of levels $4$, $5$, $6$, $7$, $8$, $9$, $10$, and $12$. In particular, we obtain closed forms of the series $$\sum_{k=0}^\infty\frac{x_0^k}{(k+1)\binom{3k}k} \ \ \text{and}\ \ \sum_{k=1}^\infty\frac{x_0^k}{k^2\binom{3k}k}$$ for any $x_0\in(-27/4,27/4)$., Comment: 23 pages

The braiding operations of quantum states have attracted substantial attention due to their great potential for realizing topological quantum computations. In this paper, we show that a three-fold degenerate eigen subspace can be obtained in a four-level Hamiltonian which is the minimal physical system. Braiding operations are proposed to apply to dressed states in the subspace. The topology of the braiding diagram can be characterized through physical methods once that the sequential braiding pulses are adopted. We establish an equivalent relationship function between the permutation group and the output states where different output states correspond to different values of the function. The topological transition of the braiding happens when two operations overlap, which is detectable through the measurement of the function. Combined with the phase variation method, we can analyze the wringing pattern of the braiding. Therefore, the experimentally-feasible system provides a platform to investigate braiding dynamics, the SU(3) physics and the qutrit gates., Comment: 10 pages, 6 figures

We determine anomalous dimensions of a family of fixed hypercharge operators in the Standard Model featuring the general Cabibbo-Kobayashi-Maskawa structure. The results are obtained at infinite orders in the couplings and to leading and subleading orders in the charge. The computed anomalous dimensions are shown to agree with the maximum known order in perturbation theory. We further show that the large hypercharge sector of the Standard Model is characterised by a non-Abelian vector condensation phase., Comment: 27 pages in LaTeX

We have conducted a model independent analysis of the $K^+ \bar{K}^0$ pair correlation function obtained from ultra high energy $pp$ collisions, with the aim of extracting the information encoded in it related to the $K\bar{K}$ interaction and the coupled channel $\pi^+ \eta$. With the present large errors at small relative $K^+\bar{K}^0$ momenta, we find that the information obtained about the scattering matrix suffers from large uncertainties. Even then, we are able to show that the data imply the existence of the $a_0$ resonance, $a_0(980)$, showing as a strong cusp close to the $K\bar{K}$ threshold. We also mention that the measurement of the $\pi^+ \eta$ correlation function will be essential in order to constrain more the information on $K\bar{K}$ dynamics that can be obtained from correlation functions., Comment: 7 pages, 9 figures