Two-particle Threshold Research Articles

Tcc in the diabatic diquark model: Effects of D*D isospin

Tcc+ is an isoscalar 4-quark state with mass lying barely below the D*+D0 threshold, and several times further below the D*0D+ threshold. It allows both dimeson molecular and elementary diquark-antidiquark (cc)(u¯d¯) substructures. The diabatic generalization of the adiabatic approximation within the Born-Oppenheimer formalism rigorously incorporates the mixing of such elementary eigenstates with states corresponding to two-particle thresholds. We examine the separate influence of the two D*D isospin channels and find that the influence of D*+D0 is larger than that of D*0D+ but not overwhelmingly so, and that Tcc+ contains an O(10%) (cc)(u¯d¯) component. We then explore the variation of these results if the isospin breaking between the dimeson thresholds is varied, and also the sensitivity of our results to variation of the mixing-potential parameters. Published by the American Physical Society 2024

Finite-volume scattering on the left-hand cut

The two-particle finite-volume scattering formalism derived by Lüscher and generalized in many subsequent works does not hold for energies far enough below the two-particle threshold to reach the nearest left-hand cut. The breakdown of the formalism is signaled by the fact that a real scattering amplitude is predicted in a regime where it should be complex. In this work, we address this limitation by deriving an extended formalism that includes the nearest branch cut, arising from single particle exchange. We focus on two-nucleon (NN → NN) scattering, for which the cut arises from pion exchange, but give expressions for any system with a single channel of identical particles. The new result takes the form of a modified quantization condition that can be used to constrain an intermediate K-matrix in which the cut is removed. In a second step, integral equations, also derived in this work, must be used to convert the K-matrix to the physical scattering amplitude. We also show how the new formalism reduces to the standard approach when the N → Nπ coupling is set to zero.

Scattering amplitudes from dispersive iterations of unitarity

We present and numerically implement a computational method to construct relativistic scattering amplitudes that obey analyticity, crossing, elastic and inelastic unitarity in three and four spacetime dimensions. The algorithm is based on the Mandelstam representation of the amplitude and iterations of unitarity. The input for the iterative procedure is given by the multi-particle double spectral density, the S-wave inelasticity, and the value of the amplitude at the crossing-symmetric point. The output, obtained at the fixed point of the iteration of unitarity, is a nonperturbative scattering amplitude. The amplitudes we obtain exhibit interesting features, such as non-zero particle production, intricate high-energy and near the two-particle threshold behavior. Scattering amplitudes obtained by initializing the iteration process with zero (or small) multi-particle input end up close to saturating the S-matrix bounds derived by other methods. There is a version of the iterative algorithm that is directly related to Feynman diagrams: it effectively re-sums infinitely many two-particle reducible planar Feynman graphs in the ϕ4 theory, which remarkably produces a unitary nonperturbative scattering amplitude function. Finally, we discuss how the algorithm can be further refined by including multi-particle unitarity.

Role of Left-Hand Cut Contributions on Pole Extractions from Lattice Data: Case Study for T_{cc}(3875)^{+}.

We discuss recent lattice data for the T_{cc}(3875)^{+} state to stress, for the first time, a potentially strong impact of left-hand cuts from the one-pion exchange on the pole extraction for near-threshold exotic states. In particular, if the left-hand cut is located close to the two-particle threshold, which happens naturally in the DD^{*} system for the pion mass exceeding its physical value, the effective-range expansion is valid only in a very limited energy range up to the cut and as such is of little use to reliably extract the poles. Then, an accurate extraction of the pole locations requires the one-pion exchange to be implemented explicitly into the scattering amplitudes. Our findings are general and potentially relevant for a wide class of hadronic near-threshold states.

Lattice QCD study of the H dibaryon using hexaquark and two-baryon interpolators

We present a lattice QCD spectroscopy study in the isospin singlet, strangeness $-2$ sectors relevant for the conjectured $H$ dibaryon. We employ both local and bilocal interpolating operators to isolate the ground state in the rest frame and in moving frames. Calculations are performed using two flavors of O($a$)-improved Wilson fermions and a quenched strange quark. Our initial point-source method for constructing correlators does not allow for bilocal operators at the source; nevertheless, results from using these operators at the sink indicate that they provide an improved overlap onto the ground state in comparison with the local operators. We also present results, in the rest frame, using a second method based on distillation to compute a hermitian matrix of correlators with bilocal operators at both the source and the sink. This method yields a much more precise and reliable determination of the ground-state energy. In the flavor-SU(3) symmetric case, we apply L\"uscher's finite-volume quantization condition to the rest-frame and moving-frame energy levels to determine the $S$-wave scattering phase shift, near and below the two-particle threshold. For a pion mass of 960 MeV, we find that there exists a bound $H$ dibaryon with binding energy ${\Delta}E=(19\pm10)$ MeV. In the 27-plet (dineutron) sector, the finite-volume analysis suggests that the existence of a bound state is unlikely.

Overlap singularity and time evolution in integrable quantum field theory

We study homogeneous quenches in integrable quantum field theory where the initial state contains zero-momentum particles. We demonstrate that the two-particle pair amplitude necessarily has a singularity at the two-particle threshold. Albeit the explicit discussion is carried out for special (integrable) initial states, we argue that the singularity is inevitably present and is a generic feature of homogeneous quenches involving the creation of zero momentum particles. We also identify the singularity in quenches in the Ising model across the quantum critical point, and compute it perturbatively in phase quenches in the quantum sine-Gordon model which are potentially relevant to experiments. We then construct the explicit time dependence of one-point functions using a linked cluster expansion regulated by a finite volume parameter. We find that the secular contribution normally linear in time is modified by a t ln t term. We additionally encounter a novel type of secular contribution which is shown to be related to parametric resonance. It is an interesting open question to resum the new contributions and to establish their consequences directly observable in experiments or numerical simulations.

Lattice QCD investigation of the structure of the a0(980) meson

We investigate the quark content of the scalar meson $a_0(980)$ using lattice QCD. To this end we consider correlation functions of six different two- and four-quark interpolating fields. We evaluate all diagrams, including diagrams, where quarks propagate within a timeslice, e.g. with closed quark loops. We demonstrate that diagrams containing such closed quark loops have a drastic effect on the final results and, thus, may not be neglected. Our analysis shows that in addition to the expected spectrum of two-meson scattering states there is an additional energy level around the two-particle thresholds of $K + \bar{K}$ and $\eta + \pi$. This additional state, which is a candidate for the $a_0(980)$ meson, couples to a quark-antiquark as well as to a diquark-antidiquark interpolating field, indicating that it is a superposition of an ordinary $\bar{q} q$ and a tetraquark structure. The analysis is performed using AMIAS, a novel statistical method based on the sampling of all possible spectral decompositions of the considered correlation functions, as well as solving standard generalized eigenvalue problems.

Erratum: Threshold expansion of the three-particle quantization condition [Phys. Rev. D 93 , 096006 (2016)

We recently derived a quantization condition for the energy of three relativistic particles in a cubic box. Here we use this condition to study the energy level closest to the three-particle threshold when the total three-momentum vanishes. We expand this energy in powers of $1/L$, where $L$ is the linear extent of the finite volume. The expansion begins at ${\cal O}(1/L^3)$, and we determine the coefficients of the terms through ${\cal O}(1/L^6)$. As is also the case for the two-particle threshold energy, the $1/L^3$, $1/L^4$ and $1/L^5$ coefficients depend only on the two-particle scattering length $a$. These can be compared to previous results obtained using nonrelativistic quantum mechanics and we find complete agreement. The $1/L^6$ coefficients depend additionally on the two-particle effective range $r$ (just as in the two-particle case) and on a suitably defined threshold three-particle scattering amplitude (a new feature for three particles). A second new feature in the three-particle case is that logarithmic dependence on $L$ appears at $\mathcal O(1/L^6)$. Relativistic effects enter at this order, and the only comparison possible with the nonrelativistic result is for the coefficient of the logarithm, where we again find agreement. For a more thorough check of the $1/L^6$ result, and thus of the quantization condition, we also compare to a perturbative calculation of the threshold energy in relativistic $\lambda \phi^4$ theory, which we have recently presented. Here all terms can be compared and we find full agreement.

Threshold expansion of the three-particle quantization condition

We recently derived a quantization condition for the energy of three relativistic particles in a cubic box. Here we use this condition to study the energy level closest to the three-particle threshold when the total three-momentum vanishes. We expand this energy in powers of $1/L$, where $L$ is the linear extent of the finite volume. The expansion begins at ${\cal O}(1/L^3)$, and we determine the coefficients of the terms through ${\cal O}(1/L^6)$. As is also the case for the two-particle threshold energy, the $1/L^3$, $1/L^4$ and $1/L^5$ coefficients depend only on the two-particle scattering length $a$. These can be compared to previous results obtained using nonrelativistic quantum mechanics and we find complete agreement. The $1/L^6$ coefficients depend additionally on the two-particle effective range $r$ (just as in the two-particle case) and on a suitably defined threshold three-particle scattering amplitude (a new feature for three particles). A second new feature in the three-particle case is that logarithmic dependence on $L$ appears at $\mathcal O(1/L^6)$. Relativistic effects enter at this order, and the only comparison possible with the nonrelativistic result is for the coefficient of the logarithm, where we again find agreement. For a more thorough check of the $1/L^6$ result, and thus of the quantization condition, we also compare to a perturbative calculation of the threshold energy in relativistic $\lambda \phi^4$ theory, which we have recently presented. Here all terms can be compared and we find full agreement.

Solving 2D QCD with an adjoint fermion analytically

We present an analytic approach to solving 1+1 dimensional QCD with an adjoint Majorana fermion. In the UV this theory is described by a trivial CFT containing free fermions. The quasi-primary operators of this CFT lead to a discrete basis of states which is useful for diagonalizing the Hamiltonian of the full strongly interacting theory. Working at large-$N$, we find that the decoupling of high scaling-dimension quasi-primary operators from the low-energy spectrum occurs exponentially fast in their scaling-dimension. This suggests a scheme, whereby, truncating the basis to operators of dimension below $\Delta_{max}$, one can calculate the low-energy spectrum, parametrically to an accuracy of $e^{-\Delta_{max}}$ (although the precise accuracy depends on the state). Choosing $\Delta_{max} =9.5$ we find very good agreement with the known spectrum obtained earlier by numerical DLCQ methods. Specifically, below the first three-particle threshold, we are able to identify all six single-particle bound-states, as well as several two-particle thresholds.

Analytic Perturbation Theory for Bound States in the Transfer Matrix Spectrum of Weakly Correlated Lattice Ferromagnetic Spin Systems

We consider general d-dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region (β≪1). Each model is characterized by a single site apriori spin distribution taken to be even. We also take the parameter α=〈s4〉−3〈s2〉2>0, i.e. in the region which we call Gaussian subjugation, where 〈sk〉 denotes the kth moment of the apriori distribution. Associated with the model is a lattice quantum field theory known to contain a particle of asymptotic mass −lnβ and a bound state below the two-particle threshold. We develop a β analytic perturbation theory for the binding energy of this bound state. As a key ingredient in obtaining our result we show that the Fourier transform of the two-point function is a meromorphic function, with a simple pole, in a suitable complex spectral parameter and the coefficients of its Laurent expansion are analytic in β.

Couplings of light I=0 scalar mesons to simple operators in the complex plane

The flavour and glue structure of the light scalar mesons in QCD are probed by studying the couplings of the I=0 mesons $\sigma(600)$ and $f_0(980)$ to the operators $\bar{q}q$, $\alpha_s G^2$ and to two photons. The Roy dispersive representation for the $\pi\pi$ amplitude $t_0^0(s)$ is used to determine the pole positions as well as the residues in the complex plane. On the real axis, $t_0^0$ is constrained to solve the Roy equation together with elastic unitarity up to the $K\Kbar$ threshold leading to an improved description of the $f_0(980)$. The problem of using a two-particle threshold as a matching point is discussed. A simple relation is established between the coupling of a scalar meson to an operator $j_S$ and the value of the related pion form-factor computed at the resonance pole. Pion scalar form-factors as well as two-photon partial-wave amplitudes are expressed as coupled-channel Omn\`es dispersive representations. Subtraction constants are constrained by chiral symmetry and experimental data. Comparison of our results for the $\bar{q}q$ couplings with earlier determinations of the analogous couplings of the lightest I=1 and $I=1/2$ scalar mesons are compatible with an assignment of the $\sigma$, $\kappa$, $a_0(980)$, $f_0(980)$ into a nonet. Concerning the gluonic operator $\alpha_s G^2$ we find a significant coupling to both the $\sigma$ and the $f_0(980)$.

Thermodynamics of SU(N) Yang-Mills theories in 2 + 1 dimensions I — The confining phase

We compute the equation of state in the confining phase of SU(N) Yang-Mills theories with N=2, 3, 4, 5 and 6 colors in 2+1 dimensions, via lattice simulations. At low enough temperatures, the results are accurately described by a gas of glueballs, including all known states below the two-particle threshold. Close to the deconfinement temperature, however, this prediction underestimates the numerical results, and the contribution from heavier glueballs has to be included. We show that the spectral density of the latter can be accurately described using a simple bosonic string model.

High-precision thermodynamics and Hagedorn density of states

We compute the entropy density of the confined phase of QCD without quarks on the lattice to very high accuracy. The results are compared to the entropy density of free glueballs, where we include all the known glueball states below the two-particle threshold. We find that an excellent, parameter-free description of the entropy density between 0.7Tc and Tc is obtained by extending the spectrum with the exponential spectrum of the closed bosonic string.

Dispersion representations and anomalous singularities of the triangle diagram

We discuss dispersion representations for the triangle diagram F(p{sub 1}{sup 2},p{sub 2}{sup 2},q{sup 2}), the single dispersion representation in q{sup 2} and the double dispersion representation in p{sub 1}{sup 2} and p{sub 2}{sup 2}, with special emphasis on the appearance of the anomalous singularities and the anomalous cuts in these representations. For the double dispersion representation in p{sub 1}{sup 2} and p{sub 2}{sup 2}, the appearance of the anomalous cut in the region q{sup 2}>0 is demonstrated, and a new derivation of the anomalous double spectral density is given. We point out that the double spectral representation is particularly suitable for applications in the region of p{sub 1}{sup 2} and/or p{sub 2}{sup 2} above the two-particle thresholds. The dispersion representations for the triangle diagram in the nonrelativistic limit are studied and compared with the triangle diagram of the nonrelativistic field theory.

Meson-meson bound state in a2+1lattice QCD model with two flavors and strong coupling

We consider the existence of bound states of two mesons in an imaginary-time formulation of lattice QCD. We analyze an SU(3) theory with two flavors in $2+1$ dimensions and two-dimensional spin matrices. For a small hopping parameter and a sufficiently large glueball mass, as a preliminary, we show the existence of isoscalar and isovector mesonlike particles that have isolated dispersion curves (upper gap up to near the two-particle threshold $\ensuremath{\sim}\ensuremath{-}4\mathrm{ln}\ensuremath{\kappa}$). The corresponding meson masses are equal up to and including $\mathcal{O}({\ensuremath{\kappa}}^{3})$ and are asymptotically of order $\ensuremath{-}2\mathrm{ln}\ensuremath{\kappa}\ensuremath{-}{\ensuremath{\kappa}}^{2}$. Considering the zero total isospin sector, we show that there is a meson-meson bound state solution to the Bethe-Salpeter equation in a ladder approximation, below the two-meson threshold, and with binding energy of order $\overline{b}{\ensuremath{\kappa}}^{2}\ensuremath{\simeq}0.02359{\ensuremath{\kappa}}^{2}$. In the context of the strong coupling expansion in $\ensuremath{\kappa}$, we show that there are two sources of meson-meson attraction. One comes from a quark-antiquark exchange. This is not a meson exchange, as the spin indices are not those of the meson particle, and we refer to this as a quasimeson exchange. The other arises from gauge field correlations of four overlapping bonds, two positively oriented and two of opposite orientation. Although the exchange part gives rise to a space range-one attractive potential, the main mechanism for the formation of the bound state comes from the gauge contribution. In our lattice Bethe-Salpeter equation approach, this mechanism is manifested by an attractive distance-zero energy-dependent potential. We recall that no bound state appeared in the one-flavor case, where the repulsive effect of Pauli exclusion is stronger.

On the relaxation properties of stochastic Ginzburg–Landau type models

Considering the project of studying the basic properties of commonly used stochastic nonlinear dynamical systems, we investigate and present new properties of the relaxation to equilibrium of stochastic Ginzburg–Landau type models with polynomial interaction with nonlocal nonlinear terms. In our approach, we map the initial stochastic model on a imaginary-time quantum field theory. We show that (with a suitable nonlocality) a two-particle bound state appears below the two-particle threshold even if all the coefficients in the polynomial interaction are positive. The existence of such a bound state changes the relaxation rate to equilibrium, and so, it is an experimentally observable effect of physical interest. Moreover, we show that the bound state masses are sensitive to changes in the rate of nonlocality, which may favor other phenomena related to the low-lying spectrum of the dynamics generator. Our results involve a perturbative analysis (supported by previous rigorous results): in the computation of the bound state mass we use a Bethe–Salpeter equation in the ladder approximation.

Effective action approach to the Leggett's mode in two-band superconductors

We investigate a collective excitation (Leggett's mode) corresponding to small fluctuations of the relative phase of two condensates in two-band superconductor using the effective ``phase only'' action. We consider the possibility of observing Leggett's mode in MgB$_2$ superconductor and conclude that for the known at present values of the two-band model parameters for MgB$_2$ Leggett's mode arises above the two-particle threshold.

Low-energy gluon to the vacuum polarization of heavy quarks

We calculate a correction to the electromagnetic current induced by a heavy quark loop. The contribution of this correction to the vacuum polarization function appears at the O(alpha_s^3) order of perturbation theory and has a qualitatively new feature -- its absorptive part starts at zero energy in contrast to other contributions where the absorptive parts start at the two-particle threshold. Our result imposes a constraint on the order n of the moments used in the heavy-quark sum rules, n<4.

Noise strength effects on the relaxation properties of weakly coupled Ginzburg-Landau models.

We search for new effects and extend previous investigations on the relaxation properties of weakly coupled stochastic Ginzburg-Landau models under noise with different intensity. We calculate the one-particle mass up to second order in the Ginzburg-Landau potential coupling constant and show that, as previously predicted, it is in fact less sensitive to the noise strength than the two-particle bound state. For d=1 and 2 and a negative quartic term in the potential, we show that the difference between these masses becomes smaller as we increase the noise strength, which indicates the possibility of a crossover for large noise (i.e., the two-particle bound state mass may become smaller than the one-particle mass) or of a phase transition (the masses are going to zero). For d=1 and in the ladder approximation (first order in the coupling constant), we show the absence of resonances in the spectrum close to the two-particle threshold and the existence of one antibound state for a positive quartic term in the potential, which indicates the possibility of some bound state due to changes in the system (e.g., in the noise intensity).