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He Song

Publications and source records attributed to He Song.

15 recordsLinked to original sources
arXiv2026

Regge trajectories for the doubly heavy triquarks $((Qq)\bar{Q}')$

We attempt to apply the Regge trajectory approach to the doubly heavy triquarks $((Qq)\bar{Q}^{\prime})$ $(Q,\,Q'=b,\,c; q=u,\,d,\,s)$. We propose the Regge trajectory relations for the doubly heavy triquarks, and then employ them to crudely estimate the spectra of the triquarks $((cu)\bar{c})$, $((cu)\bar{b})$, $((cs)\bar{c})$, $((cs)\bar{b})$, $((bu)\bar{c})$, $((bu)\bar{b})$, $((bs)\bar{c})$, and $((bs)\bar{b})$. The $\lambda$-trajectories and the $\rho$-trajectories are investigated. The triquark Regge trajectory becomes a new and very simple approach for estimating the spectra of triquarks. It also provides a simple method to investigate the $\rho$-mode and $\sigma$-mode excitations of pentaquarks and hexaquarks in the triquark picutre. Moreover, the spin-averaged masses of the ground states of pentaquarks $(\bar{c}(cu))(cu)$, $(\bar{b}(bu))(bu)$ and $(\bar{c}(cu))(bu)$ are estimated, which are consistent with other theoretical predictions.

hep-ph↗
arXiv2025

$\lambda$ and $\rho$ Regge trajectories for bottom-charm tetraquarks $(bq)(\bar{c}\bar{q}')$ and $(cq)(\bar{b}\bar{q}')$

Using the newly proposed tetraquark Regge trajectory relations, we investigate three series of Regge trajectories for bottom-charm tetraquarks $(bq)(\bar{c}\bar{q}')$ and $(cq)(\bar{b}\bar{q}')$ with $q,q'=u,d,s$: the $\rho_1$-, $\rho_2$-, and $\lambda$-trajectories. We provide rough estimates for the masses of the $\rho_1$-, $\rho_2$-, and $\lambda$-excited states. Except for the $\lambda$-trajectories, the complete forms of the other two series of Regge trajectories for bottom-charm tetraquarks are lengthy and cumbersome. We show that the $\rho_1$- and $\rho_2$-trajectories cannot be obtained by simply imitating meson Regge trajectories, because mesons have no substructures. To derive these trajectories, the tetraquarks' structure and substructure must be taken into consideration. Otherwise, the $\rho_1$- and $\rho_2$-trajectories would have to rely solely on fitting existing theoretical results or future experimental data. Consequently, the fundamental relationship between the slopes of the obtained trajectories and string tension would become unobvious, and the predictive power of the Regge trajectories would be compromised. Moreover, we show that the lengthy complete forms of the $\rho_1$- and $\rho_2$-trajectories can be well approximated by simple fitted formulas. For the bottom-charm tetraquarks $(bq)(\bar{c}\bar{q}')$ and $(cq)(\bar{b}\bar{q}')$, $\rho_1$- and $\rho_2$-trajectories exhibit a behavior of $M{\sim}x^{1/2}$ $(x=n_{r_1},n_{r_2},l_1,l_2)$, whereas $\lambda$-trajectories exhibit a behavior of $M{\sim}x^{2/3}$ $(x=N_{r},L)$. All three series of trajectories display concave downward behavior in the $(M^2,\,x)$ plane when the confining potential is linear. This conclusion holds irrespective of whether light-quark masses are included, owing to the large masses of the heavy quarks.

hep-ph↗
arXiv2025

$\lambda$ and $\rho$ Regge trajectories for the pentaquark $P_{cc\bar{c}bb}$ in the diquark-triquark picture

We propose the Regge trajectory relations for the fully heavy pentaquark $P_{cc\bar{c}bb}$ utilizing both diquark and triquark Regge trajectory relations. Using these new relations, we discuss four series of Regge trajectories: the $\rho_1$-, $\rho_2$-, $\lambda_1$-, and $\lambda_2$-trajectories. We provide rough estimates for the masses of the $\rho_1$-, $\rho_2$-, $\lambda_1$-, and $\lambda_2$-excited states. Except for the $\lambda_1$-trajectories, the complete forms of the other three series of Regge trajectories for the pentaquark $P_{cc\bar{c}bb}$ are lengthy and cumbersome. We show that the $\rho_1$-, $\rho_2$-, and $\lambda_2$-trajectories can not be obtained by simply imitating the meson Regge trajectories because mesons have no substructures. To derive these trajectories, pentaquark's structure and substructure should be taken into consideration. Otherwise, the $\rho_1$-, $\rho_2$-, and $\lambda_2$-trajectories must rely solely on fitting existing theoretical or future experimental data. Consequently, the fundamental relationship between the slopes of the obtained trajectories and constituents' masses and string tension will become unobvious, and the predictive power of the Regge trajectories would be compromised. Moreover, we show that the lengthy complete forms of the $\rho_1$-, $\rho_2$-, and $\lambda_2$-trajectories can be well approximated by the simple fitted formulas. Four series of Regge trajectories for the pentaquark $P_{cc\bar{c}bb}$ all exhibit a behavior of $M{\sim}x^{2/3}$, where $x=n_{r_1},n_{r_2},l_1,l_2,N_{r_1},N_{r_2},L_1,L_2$. All four series of trajectories exhibit concave downward behavior in the $(M^2,\,x)$ plane.

hep-ph↗
arXiv2025

$\lambda$ and $\rho$ trajectories for the doubly heavy baryons in the diquark picture

We present the explicit form of the Regge trajectory relations for the doubly heavy baryons $\Xi_{QQ'}$ and $\Omega_{QQ'}$ $(Q,Q'=b,c)$ in the diquark picture. Using the derived Regge trajectory relations, we estimate the masses of the $\lambda$-excited states and the $\rho$-excited states, which are consistent with other theoretical predictions. Both the $\lambda$-trajectories and $\rho$-trajectories are discussed. We show that the $\rho$-trajectories behave differently from the $\lambda$-trajectories. Specifically, the $\rho$-trajectories behave as $M{\sim}x_{\rho}^{2/3}$ $(x_{\rho}=n_r,l)$, whereas the $\lambda$-trajectories follow $M{\sim}x_{\lambda}^{1/2}$ $(x_{\lambda}=N_r,L)$. By using the obtained relations, the baryon Regge trajectory provides a straightforward and easy method for estimating the spectra of both the $\lambda$-excited states and $\rho$-excited states.

hep-ph↗
arXiv2024

Regge trajectories for the triply heavy triquarks

We attempt to apply the Regge trajectory approach to the triply heavy triquarks $((QQ')\bar{Q}^{\prime\prime})$ $(Q,\,Q',\,Q^{\prime\prime}=b,\,c)$. We present the triquark Regge trajectory relations, and then employ them to crudely estimate the spectra of the triquarks $((cc)\bar{c})$, $((cc)\bar{b})$, $((bc)\bar{c})$, $((bc)\bar{b})$, $((bb)\bar{c})$, and $((bb)\bar{b})$. The $\lambda$-trajectories and the $\rho$-trajectories are discussed. The triquark Regge trajectory becomes a new and very simple approach for estimating the spectra of triquarks. Moreover, the spin-averaged masses of the ground states of pentaquarks $(\bar{c}(cc))(cc)$, $(\bar{b}(cc))(cc)$ and $(\bar{c}(bb))(cc)$ are estimated, which are consistent with other theoretical predictions.

hep-ph↗
arXiv2024

Regge trajectories for the triply heavy bottom-charm baryons in the diquark picture

We present the explicit form of the Regge trajectory relations for the triply heavy bottom-charm baryons, which can be applied to investigate both the $\lambda$-mode excited states and the $\rho$-mode excited states. We estimate the masses of the $\lambda$-excited states and the $\rho$-excited states. The results are in agreement with other theoretical predictions. Both the $\lambda$-trajectories and the $\rho$-trajectories are discussed. Moreover, the behaviors of the $\lambda$- and $\rho$-trajectories for various baryons are discussed. It is shown that both the $\lambda$-trajectories and the $\rho$-trajectories for baryons are concave downwards in the $(M^2,\,x)$ plane. The Regge trajectories for the light baryons are approximately linear and become concave as the masses of the light constituents are considered.

hep-ph↗
arXiv2024

$\lambda$ and $\rho$ Regge trajectories for hidden bottom and charm tetraquarks $(Qq)(\bar{Q}\bar{q}')$

We propose the Regge trajectory relations for the heavy tetraquarks $(Qq)(\bar{Q}\bar{q}')$ $(Q=b,\,c;\,q,\,q'=u,\,d,\,s)$ with hidden bottom and charm. By employing the new relations, both the $\lambda$-trajectories and the $\rho$-trajectories for the tetraquarks $(Qq)(\bar{Q}\bar{q}')$ can be discussed. The masses of the $\lambda$-mode excited states and the $\rho$-mode excited states are estimated, and they agree with other theoretical predictions. We show that the behaviors of the $\rho$-trajectories are different from those of the $\lambda$-trajectories. The $\rho$-trajectories behave as $M{\sim}x_{\rho}^{1/2}$ $(x_{\rho}=n_r,\,l)$ while the $\lambda$-trajectories behave as $M{\sim}x_{\lambda}^{2/3}$ $(x_{\lambda}=N_r,\,L)$. Moreover, the Regge trajectory behaviors for other types of tetraquarks are investigated based on the spinless Salpeter equation. We show that both the $\lambda$-trajectories and the $\rho$-trajectories are concave downward in the $(M^2,\,x)$ plane. The Regge trajectories for the tetraquarks containing the light diquark and/or the light antidiquark also are concave in the $(M^2,\,x)$ plane when the masses of the light constituents are included and the confining potential is linear.

hep-ph↗
arXiv2023

Regge trajectories for the light diquarks

We attempt to present an unified description of the light meson spectra and the light diquark spectra by applying the Regge trajectory approach. However, we find that the direct application of the linear Regge trajectory formula for the light mesons and baryons fails. To address this issue, we fit the experimental data of light meson spectra and the light diquark spectra obtained by other theoretical approaches. By considering the light quark mass and the parameter $C$ in the Cornell potential, we provide a provisional Regge trajectory formula. We also crudely estimate the masses of the light diquarks $(ud)$, $(us)$, and $(ss)$, and find that they agree with other theoretical results. The diquark Regge trajectory not only becomes a new and very simple approach for estimating the spectra of the light diquarks, but also can explicitly show the behavior of the masses with respect to $l$ or $n_r$. Moreover, it is expected that the diquark Regge trajectory can provide a simple method for investigating the $\rho$-mode excitations of baryons, tetraquarks and pentaquarks containing diquarks.

hep-ph↗
arXiv2018

Uniform spanning forests associated with biased random walks on Euclidean lattices

The uniform spanning forest measure ($\mathsf{USF}$) on a locally finite, infinite connected graph $G$ with conductance $c$ is defined as a weak limit of uniform spanning tree measure on finite subgraphs. Depending on the underlying graph and conductances, the corresponding $\mathsf{USF}$ is not necessarily concentrated on the set of spanning trees. Pemantle~\cite{PR1991} showed that on $\mathbb{Z}^d$, equipped with the unit conductance $ c=1$, $\mathsf{USF}$ is concentrated on spanning trees if and only if $d \leq 4$. In this work we study the $\mathsf{USF}$ associated with conductances induced by $\lambda$--biased random walk on $\mathbb{Z}^d$, $d \geq 2$, $0 < \lambda < 1$, i.e. conductances are set to be $c(e) = \lambda^{-|e|}$, where $|e|$ is the graph distance of the edge $e$ from the origin. Our main result states that in this case $\mathsf{USF}$ consists of finitely many trees if and only if $d = 2$ or $3$. More precisely, we prove that the uniform spanning forest has $2^d$ trees if $d = 2$ or $3$, and infinitely many trees if $d \geq 4$. Our method relies on the analysis of the spectral radius and the speed of the $\lambda$--biased random walk on $\mathbb{Z}^d$.

math.PR↗
arXiv2017

Cutpoints for Random Walks on Quasi-Transitive Graphs

We prove that a simple random walk on quasi-transitive graphs with the volume growth being faster than any polynomial of degree 4 has a.s. infinitely many cut times, and hence infinitely many cutpoints. This confirms a conjecture raised by I. Benjamini, O. Gurel-Gurevich and O. Schramm [2011, Cutpoints and resistance of random walk paths, {\it Ann. Probab.} {\bf 39(3)}, 1122-1136] that PATH of simple random walk on any transient vertex-transitive graph has a.s. infinitely many cutpoints in the corresponding case.

math.PR↗
arXiv2016

Rainbow connection of bridgeless outerplanar graphs with small diameters

In this paper, we investigate rainbow connection number $rc(G)$ of bridgeless outerplanar graphs $G$ with diameter 2 or 3. We proved the following results: If $G$ has diameter $2,$ then $rc(G)=3$ for fan graphs $F_{n}$ with $n\geq 7$ or $C_5,$ otherwise $rc(G)=2;$ if $G$ has diameter $3,$ then $rc(G)\leq 4$ and the bound is sharp.

math.CO↗
arXiv2014

Locality of percolation critical probabilities: uniformly nonamenable case

Let $\{G_n\}_{n=1}^{\infty}$ be a sequence of transitive infinite connected graphs with $\sup\limits_{n\geq 1} p_c(G_n) < 1,$ where each $p_c(G_n)$ is bond percolation critical probability on $G_n.$ Schramm (2008) conjectured that if $G_n$ converges locally to a transitive infinite connected graph $G,$ then $p_c(G_n) \rightarrow p_c(G)$ as $n\rightarrow\infty.$ We prove the conjecture when $G$ satisfies two rough uniformities, and $\{G_n\}_{n=1}^{\infty}$ is uniformly nonamenable.

math.PR↗
arXiv2014

Connective Constants on Cayley Graphs

For a transitive infinite connected graph $G$, let $\mu(G)$ be its connective constant. Denote by $\mathbf{\cal G}$ the set of Cayley graphs for finitely generated infinite groups with an infinite-order generator which is independent of other generators. Assume $G\in\mathbf{\cal G}$ is a Cayley graph of a finitely presented group, and Cayley graph sequence $\{G_n\}_{n=1}^{\infty}\subset \mathbf{\cal G}$ converges locally to $G.$ Then $\mu(G_n)$ converges to $\mu(G)$ as $n\rightarrow\infty.$ This confirms partially a conjecture raised by Benjamini [2013. {\it Coarse geometry and randomness.} Lect. Notes Math. {\bf 2100}. Springer.] that connective constant is continuous with respect to local convergence of infinite transitive connected graphs.

math.PR↗