Entanglement suppression and low-energy scattering of heavy mesons

Symmetries play a crucial role in physics, serving as fundamental principles for understanding Nature and revealing the properties of elementary particles and interactions. Particularly at low energies, the behavior of many systems can be attributed to the influence of their symmetries. Symmetries at low energies often manifest as local approximations of high-energy theories, and additional symmetries, known as “emergent symmetries” which are not in the action of the theory, may arise. In recent years, the concept of quantum entanglement has been introduced to the study and description of emergent symmetries, providing a new perspective for uncovering novel physical phenomena and understanding the behavior of low-energy systems [1, 2, 3, 4, 5, 6, 7].

Entanglement measures the degree to which a system is entangled and quantifies the deviation from tensor-product structure for a given state [8, 9] (and if the tensor product structure is considered quasi-classical, then entanglement signifies the deviation from the classical structure). Given the ability to quantify the entanglement of a state, it is natural to extend this concept to quantify the entanglement of an operator [10]. This can be achieved by averaging the entanglement measure of the states produced by applying the operator to all tensor-product states. It is evident that the entanglement of an operator measures its capacity to generate entanglement, termed as “entanglement power” in the literature.

Since the $S$-matrix is also an operator, it is conceivable to assign an entanglement power to it. We know that the $S$-matrix carries all the information of a scattering process, hence its entanglement power measures the deviation of a specific scattering process from classical structure. It is natural to consider extreme cases, such as when the entanglement of the $S$-matrix reaches its maximum or minimum value (which can always be zero). Ref. [1] examines the latter, revealing some intriguing clues: the inherent $\text{SU}(2)_{\text{isospin}}\times\text{SU}(2)_{\text{spin}}$ symmetry of nucleon-nucleon scattering enlarges to Wigner’s $\text{SU}(4)$ symmetry [11, 12, 13] when the entanglement vanishes, leading to the conjecture that entanglement suppression could be the origin of emergent symmetries.

If the hypothesis holds, one may speculate that minimal entanglement would constrain the parameter space of low-energy hadron reactions, and thus would determine the emergence of new structures in hadronic interactions in the low-energy region. Consequences of entanglement suppression has been examined for nucleon-nucleon interactions [1, 2], the pionic scattering [3], the scattering between light octet baryons [4, 6], and the relativistic scattering of Higgs doublets [5]. In this paper, we will extend the study of entanglement suppression to the scattering of heavy mesons, where there are numerous intriguing near-threshold structures under intensive investigations. Two of the prominent examples in this regard are the $X(3872)$ [14], also known as the $\chi_{c1}(3872)$ [15], and the $T_{cc}(3875)^{+}$ [16, 17], which have been proposed to be potential $D\bar{D}^{*}$ [18, 19, 20, 21, 22, 23] and $D^{*}D$ [24, 25, 26, 27] hadronic molecular states, respectively (for reviews, see Refs. [28, 29, 30, 31, 32, 33, 34, 35, 36, 37]). We will investigate to what consequences their existence together with entanglement suppression can lead to.

The interaction between a pair of ground-state heavy mesons near threshold is closely related to formation of hadronic molecular states [31]. Therefore, we will analyze the near-threshold scattering processes of a pair of heavy mesons (namely, the $D^{(*)}D^{(*)}$ and $D^{(*)}\bar{D}^{(*)}$ scattering), where particles can be treated using a nonrelativistic approximation and the interaction is dominated by the lowest partial wave, i.e., the $S$-wave. Moreover, the mass of charm quark, $m_{c}$, is much larger than the nonperturbative energy scale of quantum chromodynamics (QCD), denoted by $\Lambda_{\text{QCD}}$. Therefore, when studying physical processes involving momentum scales of $\mathcal{O}(\Lambda_{\text{QCD}})$, we can treat $\Lambda_{\text{QCD}}/m_{c}$ as a small parameter and expand it in a power series to construct an effective field theory. The leading order (LO) is given by the heavy-quark limit ($m_{c}\rightarrow\infty$), where the heavy-quark spin symmetry (HQSS) [38] exists. HQSS has been used to predict heavy-quark spin partners of hadronic molecules containing heavy quark(s) [39, 40, 41, 42]. This paper will investigate whether entanglement suppression will enlarge HQSS, then obtaining an emergent symmetry, which can predict more potential siblings of $X(3872)$ and $T_{cc}(3875)^{+}$ than HQSS does.

Furthermore, enlarged symmetries in the low-energy region can also emerge in the large-$N_{c}$ limit, with $N_{c}$ the number of colors. For the Wigner’s $\text{SU}(4)$ symmetry, the large-$N_{c}$ limit makes the same prediction, but for some other cases [1, 4], the results obtained from the entanglement suppression and the large-$N_{c}$ limit differ, making it also meaningful to examine the differences between the two in the $D^{(*)}D^{(*)}$ and $D^{(*)}\bar{D}^{(*)}$ systems.

The outline of this paper is as follows: In Sec. II, the entanglement power is considered in detail. The $S$-matrix is formulated in a basis convenient for calculation in Sec. III.1. Then in Sec. III.2, we demonstrate how to relate the parameterization of the $S$-matrix to the amplitudes. In Sec. III.3, we present the effective Lagrangians and compute the amplitudes for $D^{(*)}D^{(*)}$ and $D^{(*)}\bar{D}^{(*)}$ scatterings. In Sec. IV, we derive the constraints imposed by entanglement suppression on the $S$-matrix, and in Sec. V, these results are connected to hadronic molecules, and we will show that with the $X(3872)$ and $T_{cc}(3875)^{+}$ as inputs, there emerges a light-quark spin symmetry. Subsequently, we conduct a brief large-$N_{c}$-limit analysis for the heavy meson scatterings in Sec. VI. Finally, a concise summary is provided in Sec. VII.

The degree to which a system is entangled, or its deviation from a tensor-product structure, provides a measure of how “non-classical” it is [1, 2]. An entanglement measure is a way to quantify the degree of entanglement of any given state. For a bipartite system $\left|\psi\right>$, the commonly employed linear entropy is defined as (see, e.g., Refs. [3, 4])

$$E(\left|\psi\right>)=1-\text{Tr}_{1}[\rho_{1}^{2}],$$

(1)

where $\rho=\left|\psi\right>\left<\psi\right|$ is the density matrix, and $\rho_{1}=\operatorname{Tr}_{2}(\rho)$ is the reduced density matrix obtained after tracing over subsystem 2. $E(\left|\psi\right>)$ serves as a semi-positive definite measure of entanglement which vanishes only on tensor-product states $\left|\psi\right>=\left|\psi_{1}\right>\otimes\left|\psi_{2}\right>$, as shown in Appendix A.

Entanglement measure quantifies the entanglement in a quantum state $\left|\psi\right>$, while entanglement power measures the ability of a quantum-mechanical operator $U$ to generate entanglement by averaging over all states obtained by acting it on tensor-product states [10]:

$$E(U)=\overline{E\left(U\left|\psi\right>\right)},\quad\left|\psi\right>=\left|\psi_{1}\right>\otimes\left|\psi_{2}\right>.$$

(2)

By describing the average action of $U$ transiting a tensor-product state to an entangled state, entanglement power expresses a state-independent entanglement measure that is also semi-positive definite and vanishes, i.e., is minimized, only when $U\left|\psi\right>$ remains a tensor-product state for any $\left|\psi\right>=\left|\psi_{1}\right>\otimes\left|\psi_{2}\right>$.

In general, a low-energy scattering event can entangle position, spin, and other quantum numbers, and it is therefore natural to assign an entanglement power to the $S$-matrix for such a scattering process. Moreover, the small mass splitting between $u$ and $d$ quarks leads to the approximate $\text{SU}(2)$ isospin symmetry, so it is instructive to take into account the isospin invariance, which introduces interesting interplay between flavor and spin quantum numbers [4]. Based on the above discussion, we choose to define the entanglement power of the $S$-matrix in the initial two-particle isospin $\otimes$ spin space. Since the $S$-wave heavy mesons are isospin-1/2 and spin-0 ($D$) or spin-1 ($D^{*}$), it is expedient to focus on the following two cases.

The first one is where there are two isospin states (a qubit) for each particle, the isospin-1/2 case. This is just like the discussion of the spin states for the nucleon-nucleon case in Ref. [3]. The most general initial isospin-1/2 state can be parameterized using the two complex parameters or four real parameters. Among them, one parameter can be removed by normalization and one gives an overall irrelevant phase. Finally only two real parameters are left, which parameterize a $\mathbb{CP}^{1}$ manifold, also known as the 2-sphere $S^{2}$ or the Bloch sphere [3, 43]. It can be parameterized as

$\displaystyle\left|\psi\right>=\left(\cos\frac{\theta}{2},e^{i\phi}\sin\frac{\theta}{2}\right)^{T},$

(3)

with $\theta\in[0,\pi]$ and $\phi\in[0,2\pi)$. Therefore, the incoming state of two isospin-1/2 particles is mapped to a point on the product manifold, $\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ , while the entanglement power $E(S)$ of the $S$-matrix is defined as

$$E(S)=1-\int\frac{\mathrm{d}\Omega_{1}}{4\pi}\frac{\mathrm{d}\Omega_{2}}{4\pi}\mathrm{Tr}_{1}[\rho_{1}^{2}],$$

(4)

where we have defined $\rho=|\psi_{\mathrm{out}}\rangle\langle\psi_{\mathrm{out}}|$ and $|\psi_{\mathrm{out}}\rangle=S|\psi_{\mathrm{in}}\rangle$,  $|\psi_{\mathrm{in}}\rangle=|\psi_{1}\rangle\otimes|\psi_{2}\rangle$.

In the spin-1 case, we have three spin states (a qutrit) which involve three complex parameters, similar to the isospin space of the $\pi\pi$ scattering discussed in Ref. [3]. Four real parameters are left after considering normalization and removing the overall phase, and they parameterize the $\mathbb{CP}^{2}$ manifold. Thus, an arbitrary qutrit can be written as

$$\left|\psi\right>=(\cos\beta\sin\alpha,e^{i\mu}\sin\beta\sin\alpha,e^{i\nu}\cos\alpha)^{T},$$

(5)

where $\alpha,\beta\in[0,\pi/2]$ and $\mu,\nu\in[0,2\pi)$. Similarly to Eq. (4), the entanglement power $E(S)$ of the $S$-matrix can be defined as

$$E(S)=1-\int\mathrm{d}\omega_{1}\mathrm{d}\omega_{2}\mathrm{Tr}_{1}[\rho_{1}^{2}],$$

(6)

with $\mathrm{d}\omega=(2/\pi^{2})\cos\alpha\sin^{3}\alpha\mathrm{d}\alpha\cos\beta\sin\beta\mathrm{d}\beta\mathrm{d}\mu\mathrm{d}\nu$ the normalized measure that describes the geometry of $\mathbb{CP}^{2}$ [44, 43].

Having set up the theoretical framework, we can now calculate the entanglement power and require it to vanish, which gives constraints on phase shifts. In this way, we can check the consequences of the constraints on amplitudes, which lead to relations among the LECs in the Lagrangian. It is not difficult for scatterings involving pseudoscalar mesons, as entanglement occurs solely in the isospin space, while for $D^{*}D^{*}$ and $D^{*}\bar{D}^{*}$ scatterings, it is useful to express what minimal entanglement means in a tensor-product space. Clearly, entanglement being zero in a large space is equivalent to it being zero in all of its subspaces. For instance, applying $\mathcal{I}_{I^{\prime}}\equiv\mathcal{I}_{I^{\prime}}\otimes 1$ to $S=\sum_{I,J}\mathcal{I}_{I}\otimes\mathcal{J}_{J}\,e^{2i\delta_{IJ*}}$ gives

$$\mathcal{I}_{I^{\prime}}S=\sum_{I,J}{\mathcal{I}_{I^{\prime}}\mathcal{I}_{I}\otimes\mathcal{J}_{J}}e^{2i\delta_{IJ*}}=\mathcal{I}_{I^{\prime}}\otimes\sum_{J}{\mathcal{J}_{J}}e^{2i\delta_{IJ^{\prime}*}}\equiv\mathcal{I}_{I^{\prime}}\otimes S_{I^{\prime}}.$$

(41)

So the vanishing of entanglement implies that the entanglement vanishes in this spin subspace with isospin $I^{\prime}$. Similarly, for any isospin subspace with a specific total spin, the entanglement should also be zero. Nucleon-nucleon scattering is also a process entangled in both spin and isospin spaces. In Ref. [1], the parameterization was carried out only in spin space. This is because Fermi-Dirac statistics results in entanglement effectively occurring in only one space. Similarly, we find that in $D^{(*)}D^{(*)}$ scattering, spin entanglement and isospin entanglement also yield completely consistent results, so it is actually sufficient to compute entanglement in only one space. However, for $D^{(*)}\bar{D}^{(*)}$ scattering, there is no Bose-Einstein statistics, so this tensor-product structure is necessary, and such a formalism can be extended to cases entangling more quantum numbers.

Based on the above discussion, we only need to compute the entanglement power in two scenarios using Eqs. (4) and (6). At LO of the heavy quark expansion, the $D$ and $D^{*}$ masses are the same. We also consider the isospin symmetric limit such that charged and neutral mesons in the same isospin multiplet are degenerate. Thus we will take $\mu=M/2$ with $M$ denoting the charmed meson mass in the following.

We start with the $S$-matrix in the isospin subspace with a specific total spin $J$ (both particles are isospin-1/2 states):

$$S_{J}=\mathcal{I}_{0}e^{2i\delta_{0J}}+\mathcal{I}_{1}e^{2i\delta_{1J}}.$$

(42)

The Bose-Einstein forbidden cases of $DD$ and $D^{*}D^{*}$ scatterings are formally included by requiring $\delta_{00}=0$ and $\delta_{00*}=\delta_{02*}=\delta_{11*}=0$ (recall that in Eq. (10) we have introduced the “_∗” subindex for vector-vector scattering phase shifts), respectively. Evaluating the entanglement power using Eq. (4) yields

$$E(S_{J})=\frac{1}{6}\sin^{2}[2(\delta_{0J}-\delta_{1J})],$$

(43)

which vanishes only when

$\displaystyle\left|\delta_{0J}-\delta_{1J}\right|=0~{}~{}\text{or}~{}~{}\frac{\pi}{2}.$

(44)

The solutions to equation $\left|\delta_{0J}-\delta_{1J}\right|=\pi/2$, namely $\delta_{0J}=0$ and $\delta_{1J}=\pi/2$ (or vice versa), correspond to the cases of no interaction and the unitarity limit, respectively. The latter is equivalent to taking the limit of exactly infinite scattering length.

For vector meson scatterings, one also needs to consider the $S$-matrix in the spin subspace with a specific isospin $I$:

$$S_{I}=\mathcal{J}_{0}e^{2i\delta_{I0*}}+\mathcal{J}_{1}e^{2i\delta_{I1*}}+\mathcal{J}_{2}e^{2i\delta_{I2*}}.$$

(45)

Again, Bose-Einstein statistics requires $\delta_{00*}=\delta_{02*}=\delta_{11*}=0$ for $D^{*}D^{*}$ scattering, while it does not constrain anything for $D^{*}\bar{D}^{*}$ scattering, as mentioned above. The entanglement power can be calculated using Eq. (6) and reads

	$\displaystyle E(S_{I})=$	$\displaystyle\,\frac{1}{648}\Big{\{}156-6\cos[4(\delta_{I0*}-\delta_{I1*})]-65\cos[2(\delta_{I0*}-\delta_{I2*})]$		(46)
		$\displaystyle-10\cos[4(\delta_{I0*}-\delta_{I2*})]-60\cos[4(\delta_{I2*}-\delta_{I1*})]-15\cos[2(\delta_{I0*}+\delta_{I2*}-2\delta_{I1*})]\Big{\}},$

which has only two non-entangling solutions:

$$\left|\delta_{I0*}-\delta_{I1*}\right|=\left|\delta_{I2*}-\delta_{I1*}\right|=0~{}~{}\text{or}~{}~{}\frac{\pi}{2}.$$

(47)

The subsequent step involves relating these solutions to the amplitudes using Eq. (22), which can be then be confronted to experimental or lattice QCD results, or use such empirical results as further input to select solutions and explore their implications.

It is already known that the near-threshold interaction between a pair of ground-state heavy mesons is closely related to formation of hadronic molecular states [31]. The $X(3872)$ [14] has been proposed as a candidate of an isoscalar $D\bar{D}^{*}$ hadronic molecule with $J^{PC}=1^{++}$ quantum numbers [57] for a long while  [18, 19, 20, 21, 22, 23]. Moreover, in 2021 the LHCb Collaboration announced the discovery of $T_{cc}(3875)^{+}$ with preferred quantum numbers $I(J^{P})=0(1^{+})$ [16, 17], a double-charm $D^{*}D$ molecular candidate [24, 25, 26, 27], which reveals itself as a high-significance peaking structure in the $D^{0}D^{0}\pi^{+}$ and $D^{+}D^{0}\pi^{0}$ invariant mass distributions just below the nominal $D^{*+}D^{0}$ threshold. The masses of these two particles are extremely close to the $D^{0}\bar{D}^{*0}$ and $D^{*+}D^{0}$ thresholds, respectively,

$\displaystyle M_{X}-M_{D^{0}}-M_{\bar{D}^{*0}}=0.00_{-0.15}^{+0.09}\,\text{MeV},~{}M_{T_{cc}^{+}}-M_{D^{*+}}-M_{D^{0}}=(-0.36\pm 0.04)\,\text{MeV},$

(48)

where we have used the charmed meson masses from Ref. [15], the $X(3872)$ mass from the Flatté analysis in Ref. [58], and the $T_{cc}(3875)^{+}$ mass from the coupled-channel analysis with full $DD\pi$ three-body effects in Ref. [26].

The existence of the isoscalar $X(3872)$ and $T_{cc}(3875)^{+}$ states so close to the $D\bar{D}^{*}$ and $D^{*}D$ thresholds, respectively, implies that the near-threshold $S$-wave interactions in both channels approach the unitary limit, with the corresponding $S$-wave scattering lengths being infinitely large. By taking these conditions as input, the entanglement suppression solutions can be further pinned down. Consequently, partners of the $X(3872)$ and $T_{cc}(3875)^{+}$ states can be predicted. If some of these partners are not predicted by the intrinsic HQSS, one can assert that they arise from an emergent symmetry dictated by entanglement suppression.

For $D^{(*)}D^{(*)}$ scattering, the $T_{cc}(3875)^{+}$ implies $\delta_{01}=\pi/2$. Then one obtains two solutions:

$$\delta_{01}=\delta_{01*}=\frac{\pi}{2},\quad\delta_{10}=\delta_{11}=\delta_{10*}=\delta_{12*}=0,$$

(49)

or

$$\delta_{01}=\delta_{01*}=\delta_{10}=\delta_{11}=\delta_{10*}=\delta_{12*}=\frac{\pi}{2}.$$

(50)

In both scenarios, an additional $D^{*}D^{*}$ zero-energy bound molecular state in the isoscalar $J^{P}=1^{+}$ sector, $T_{cc}^{*+}$, can be predicted based on Eqs. (49) and (50), i.e., $\delta_{01*}=\pi/2$. However, it is not a result of entanglement suppression but stems from HQSS [27, 26], as can be seen from $T^{IJ=01}\left(D^{*}D\right)=T^{IJ=01}\left(D^{*}D^{*}\right)$ in Eqs. (29) and (31). The additional consequences of entanglement suppression is that the interaction strengths of the isovector channels are all the same, either noninteracting as in Eq. (49) or at the unitary limit as in Eq. (50). In the latter instance, we would also anticipate four extra weakly bound states near the $D^{(*)}D^{(*)}$ threshold. The results are shown in Table 1. In both cases, it means that the symmetry for the spin degree of freedom of the light quarks in the heavy meson pair is enlarged from SU(2)$\times$SU(2) to SU(4).

It is also instructive to explicitly write out the solution of the LECs for the first case (49):

$$D_{00}+D_{10}=0,\quad D_{01}=\frac{\pi}{4\mu\Lambda},\quad D_{11}=-\frac{\pi}{4\mu\Lambda},$$

(51)

which yields the Lagrangian as

	$\displaystyle\mathcal{L}_{HH}=$	$\displaystyle-\frac{D_{00}}{8}\mathrm{Tr}\left[H^{a\dagger}H_{b}H^{b\dagger}H_{a}\right]-\frac{\pi}{32\mu\Lambda}\mathrm{Tr}\left[H^{a\dagger}H_{b}\sigma^{m}H^{b\dagger}H_{a}\sigma^{m}\right]$		(52)
		$\displaystyle+\frac{D_{00}}{8}\mathrm{Tr}\left[H^{a\dagger}H_{b}H^{c\dagger}H_{d}\right]\boldsymbol{\tau}_{a}^{d}\cdot\boldsymbol{\tau}_{c}^{b}+\frac{\pi}{32\mu\Lambda}\mathrm{Tr}\left[H^{a\dagger}H_{b}\sigma^{m}H^{c\dagger}H_{d}\sigma^{m}\right]\boldsymbol{\tau}_{a}^{d}\cdot\boldsymbol{\tau}_{c}^{b}.$

It is seen in Eq. (49) that all amplitudes for isovector channels vanish, in agreement with the Lagrangian (52) proportional to the projector onto isoscalar subspace, $\mathcal{I}_{0}=(1-\boldsymbol{\tau}_{1}\cdot\boldsymbol{\tau}_{2})/4$.

For $D^{(*)}\bar{D}^{(*)}$ scattering, the $X(3872)$ implies $\bar{\delta}_{01+}=\pi/2$, where we use $\bar{\delta}$ to denote $D^{(*)}\bar{D}^{(*)}$ scattering phase shifts and the subscript “₊” denotes the positive $C$-parity combination of $D\bar{D}^{*}$ in Eq. (16). One obtains

$$\bar{\delta}_{00}=\bar{\delta}_{01\pm}=\bar{\delta}_{0J*}=\frac{\pi}{2},\quad\bar{\delta}_{10}=\bar{\delta}_{11\pm}=\bar{\delta}_{1J*}=0,$$

(53)

or

$$\bar{\delta}_{00}=\bar{\delta}_{01\pm}=\bar{\delta}_{0J*}=\bar{\delta}_{10}=\bar{\delta}_{11\pm}=\bar{\delta}_{1J*}=\frac{\pi}{2},$$

(54)

where $J=0,1,2$. Then the solutions of the LECs read:

$$C_{0a}=-\frac{2\pi}{\mu\bar{\Lambda}},\quad C_{0b}=C_{1a}=C_{1b}=0,$$

(55)

or

$$C_{0a}=C_{1a}=-\frac{2\pi}{\mu\bar{\Lambda}},\quad C_{0b}=C_{1b}=0,$$

(56)

where $\bar{\Lambda}$ denotes the cut-off for the $D^{(*)}\bar{D}^{(*)}$ system.