Entanglement suppression and low-energy scattering of heavy mesons

Recently entanglement suppression was proposed to be one possible origin of emergent symmetries. Here we test this conjecture in the context of heavy meson scatterings. The low-energy interactions of $D^{(*)}\bar D^{(*)}$ and $D^{(*)} D^{(*)}$ are closely related to the hadronic molecular candidates $X(3872)$ and $T_{cc}(3875)^+$, respectively, and can be described by a nonrelativistic effective Lagrangian manifesting heavy-quark spin symmetry, which includes only constant contact potentials at leading order. We explore entanglement suppression in a tensor-product framework to treat both the isospin and spin degrees of freedom. Using the $X(3872)$ and $T_{cc}(3875)^+$ as inputs, we find that entanglement suppression indeed leads to an emergent symmetry, namely, a light-quark spin symmetry, and as such the $D^{(*)}\bar D^{(*)}$ or $D^{(*)} D^{(*)}$ interaction strengths for a given total isospin do not depend on the total angular momentum of light (anti)quarks. The $X(3872)$ and $T_{cc}(3875)^+$ are predicted to have five and one isoscalar partner, respectively, while the corresponding partner numbers derived solely from heavy-quark spin symmetry are three and one, respectively. The predictions need to be confronted with experimental data and lattice quantum chromodynamics results to further test the entanglement suppression conjecture.


I. INTRODUCTION
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 SU(2) isospin × SU(2) spin symmetry of nucleon-nucleon scattering enlarges to Wigner's SU(4) symmetry [11][12][13] when the entanglement vanishes, leading to the conjecture that entanglement suppression could be the origin of emergent symmetries.
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 ( * ) 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 Λ QCD .Therefore, when studying physical processes involving momentum scales of O(Λ QCD ), we can treat Λ 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 → ∞), 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 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 ( * ) 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 A. Then in Sec.III B, we demonstrate how to relate the parameterization of the S-matrix to the amplitudes.In Sec.III C, we present the effective Lagrangians and compute the amplitudes for D ( * ) D ( * ) and D ( * ) 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 climit analysis for the heavy meson scatterings in Sec.VI.Finally, a concise summary is provided in Sec.VII.

II. ENTANGLEMENT POWER
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 |ψ⟩, the commonly employed linear entropy is defined as (see, e.g., Refs.[3,4]) where ρ = |ψ⟩ ⟨ψ| is the density matrix, and ρ 1 = Tr 2 (ρ) is the reduced density matrix obtained after tracing over subsystem 2. E(|ψ⟩) serves as a semi-positive definite measure of entanglement which vanishes only on tensor-product states |ψ⟩ = |ψ 1 ⟩ ⊗ |ψ 2 ⟩, as shown in Appendix A.
Entanglement measure quantifies the entanglement in a quantum state |ψ⟩, 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]: 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 |ψ⟩ remains a tensorproduct state for any 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 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 ⊗ 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 CP 1 manifold, also known as the 2-sphere S 2 or the Bloch sphere [3,43].It can be parameterized as with θ ∈ [0, π] and ϕ ∈ [0, 2π).Therefore, the incoming state of two isospin-1/2 particles is mapped to a point on the product manifold, CP 1 × CP 1 , while the entanglement power E(S) of the S-matrix is defined as where we have defined ρ = |ψ out ⟩⟨ψ out | and In the spin-1 case, we have three spin states (a qutrit) which involve three complex parameters, similar to the isospin space of the ππ scattering discussed in Ref. [3].Four real parameters are left after considering normalization and removing the overall phase, and they parameterize the CP 2 manifold.Thus, an arbitrary qutrit can be written as where α, β ∈ [0, π/2] and µ, ν ∈ [0, 2π).Similarly to Eq. ( 4), the entanglement power E(S) of the S-matrix can be defined as with dω = (2/π 2 ) cos α sin 3 αdα cos β sin βdβdµdν the normalized measure that describes the geometry of CP 2 [43,44].

III. HEAVY MESON SCATTERING A. S-matrix
In this paper, we primarily study the scattering of heavy mesons in the near-threshold region, which is dominated by the S-wave interaction.In general, the S-matrix can be expressed as where we define J J ⊗ I I the projection operators onto subspaces of definite isospin I and total spin J, and δ IJ the corresponding phase shift.
Let us start with D ( * ) D ( * ) scattering.The construction of the S-matrix proceeds straightforwardly [3,4]: with the isospin space projectors and the spin space projectors where (t a 1,2 ) bc = −iϵ abc , τ are Pauli matrices in the flavor space, and the D * D * scattering phase shifts are denoted as δ IJ * , to be distinguished from the D ( * ) D scattering phase shifts δ IJ .The S-matrices for DD and D * D scattering are exclusively parameterized in the isospin space.This is because in these two processes, the total spin has only one specific value for each.Additionally, the Bose-Einstein statistics dictates that: (i) δ 00 = 0 for DD scattering; (ii) δ 00 * = δ 02 * = δ 11 * = 0 for D * D * scattering, i.e., the total isospin I = 0 projects into spin-triplet 3 S 1 while I = 1 projects into both spin-singlet 1 S 0 and quintuplet 5 S 2 [26].
For D ( * ) D( * ) scattering, there are two additional intricacies: (i) electrically neutral D ( * ) D( * ) combinations should have definite C-parities; (ii) there is no Bose-Einstein statis-tics.This implies that the S-matrix for D ( * ) D( * ) scattering should be written as The phase shifts of D D and D * D * scatterings are denoted as δIJ and δIJ * , respectively.
The J P C combinations that a pair of D ( * ) and D( * ) can form are as follows [31,45]: Thus, it is seen in Eq. ( 14) that there are two independent S-matrices in D D * channel, with C = ± and thus the corresponding subindex " ± ".Here the phase convention for the charge conjugation is chosen as Ĉ |D⟩ = D and Ĉ |D * ⟩ = − D * .

B. Effective range expansion
In this subsection, we will first derive the effective range expansion that will be utilized later, and then discuss how to relate phase shifts to amplitudes in different physical cases.
We start by considering the effective Lagrangian for two nonrelativistic spinless bosons ϕ i with only the LO contact interaction in a derivative (nonrelativistic) expansion: It is easy to write down the amplitude for the process ϕ 1 ϕ 2 → ϕ 1 ϕ 2 shown in Fig. 1: FIG. 1.The first few diagrams contributing to the S-wave amplitude for the process The solid black dot represents the −iC 0 vertex.
where we define the two-point loop function with m 1 and m 2 the boson masses, µ the reduced mass, p = √ 2µE the magnitude of the center-of-mass momentum, and Λ the cut-off introduced to regularize the loop integral.
Based on the above discussion, we can deduce the contact potential C 0 for certain phase shift values.In particular, the noninteracting and unitary limit cases are of utmost importance at LO: In fact, at LO only these two limits are momentum-independent since the phase shift δ(p) is in general a function of p, so if the entanglement suppression constraint (see Sec. IV) is enforced at one value of momentum it will generically not hold at other values of momentum.Additionally, both the free theory (δ = 0) and the theory at the unitarity limit (δ = π/2) are invariant under the Schrödinger symmetry [2,49], which is the nonrelativistic conformal group and the largest symmetry group preserving the Schrödinger equation.
They correspond to the two fixed points of renormalization group running of nonrelativistic two-body scattering by a short-range potential [50].

C. LO effective field theory for heavy meson scattering
At very low energies, the LO D ( * ) D ( * ) interaction in the nonrelativistic effective field theory follows from the effective Lagrangian which contains only constant contact potentials [26,51], Moreover, the LO Lagrangian for the low-energy S-wave interaction between a pair of heavy and anti-heavy mesons containing only constant contact terms reads [45,52,53] 1 1 One can check that the double-trace form can also be rewritten in the single-trace form like Eq. ( 25) using the completeness relation for the Pauli matrices, 2δ l i δ j k = δ j i δ l k + σ j i • σ l k , as shown in [41].
Again, there are four light-flavor-independent LECs F (τ ) A,B .
In the above Lagrangians, the heavy and anti-heavy mesons are grouped into superfields as [54,55] with P a and P * a annihilating the ground-state pseudoscalar and vector charmed mesons, respectively, and P a and P * a annihilating the anti-charmed mesons.
The contact potentials for the different isospin/spin-parity D ( * ) D ( * ) S-wave channels derived from the Lagrangian of Eq. ( 25) read [26] T All other potentials vanish, where we have used the fact that the isoscalar (isovector) wave function for two identical particles with isospin I = 1/2 are antisymmetric (symmetric), respectively.
When it comes to the D ( * ) D( * ) case, the four LECs that appear in Eq. ( 26) are often rewritten for convenience into C 0a , C 0b and C 1a , C 1b [56], which stand for the LECs in the isospin I = 0 and I = 1 channels, respectively.The relations read Then the contact potentials can be expressed utilizing these four LECs [53]: where the lower indices " ± " represent (D D * ∓ D * D)/ √ 2 with different C-parities in Eq. ( 16).

IV. RESULTS
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 * 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 I I ′ ≡ I I ′ ⊗ 1 to S = I,J I I ⊗ J J e 2iδ IJ * gives So the vanishing of entanglement implies that the entanglement vanishes in this spin subspace with isospin I ′ .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 ( * ) 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 µ = 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): The Bose-Einstein forbidden cases of DD and D * D * scatterings are formally included by requiring δ 00 = 0 and δ 00 * = δ 02 * = δ 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 which vanishes only when The solutions to equation |δ 0J − δ 1J | = π/2, namely δ 0J = 0 and δ 1J = π/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: Again, Bose-Einstein statistics requires δ 00 * = δ 02 * = δ 11 * = 0 for D * D * scattering, while it does not constrain anything for D * D * scattering, as mentioned above.The entanglement power can be calculated using Eq. ( 6) and reads which has only two non-entangling solutions: 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.

V. CONSEQUENCES ON HEAVY-MESON HADRONIC MOLECULES
It is already known that the near-threshold interaction between a pair of groundstate 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 D * hadronic molecule with [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 π + and D + D 0 π 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 D * 0 and D * + D 0 thresholds, respectively, 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 DDπ three-body effects in Ref. [26].
The existence of the isoscalar X(3872) and T cc (3875) + states so close to the D 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 δ 01 = π/2.Then one obtains two solutions: or In both scenarios, an additional D * D * zero-energy bound molecular state in the isoscalar , can be predicted based on Eqs. ( 49) and ( 50), i.e., δ 01 * = π/2.However, it is not a result of entanglement suppression but stems from HQSS [26,27], as can be seen from T IJ=01 (D * D) = T IJ=01 (D * D * ) 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 TABLE I. Partners of the T cc (3875) + predicted by HQSS or the two solutions of entanglement suppression given in Eqs.(49) and (50).The symbol "⊙" denotes the input T cc (3875) + state, "⊗" represents its predicted partners, "⊘" indicates that no near-threshold state is allowed, " " is forbidden by Bose-Einstein statistics, and "−" signifies that no prediction can be made without further inputs.

HQSS
Eq. ( 49) predictions Eq. ( 50) predictions near the D ( * ) D ( * ) threshold.The results are shown in Table I.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)×SU(2) to SU(4).
It is also instructive to explicitly write out the solution of the LECs for the first case (49): which yields the Lagrangian as 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, For D ( * ) D( * ) scattering, the X(3872) implies δ01+ = π/2, where we use δ to denote D ( * ) D( * ) scattering phase shifts and the subscript " + " denotes the positive C-parity com-  53) and ( 54).The symbol "⊙" denotes the input X(3872), "⊗" represents its predicted partners, "⊘" indicates no near-threshold state is allowed, and "−" signifies that no prediction can be made without further inputs.Moreover, "⊕" means that the corresponding meson pair needs to be mixed with another one to get a spin partner of X(3872), see Eqs. ( 57) and (58).
In both scenarios, we conclude that X(3872) should have five spin partner states, all of them being isoscalar states, like the X(3872) itself, so there are totally six weakly bound states in the isospin-0 channels of D ( * ) D( * ) scattering.Also, it is noted that HQSS predicts only three isoscalar spin partners in the strict heavy-quark limit [59,60], one D * D * state with J P C = 2 ++ (40) and two mixing states with J P C = 0 ++ and J P C = 1 +− : Therefore, as in the double-charm case, entanglement suppression again enlarges the symmetry for the spin degree of freedom of the light quarks from SU(2)×SU(2) to SU(4), see Eqs. ( 53) and ( 54), predicting more states than HQSS.
Furthermore, if Nature chooses the solution in Eq. ( 54), there would be six isovector hadronic molecules in addition, as listed in Table II.Two of these isovector states have quantum numbers J P C = 1 +− , and thus are in line with the existence of Z c (3900) [61,62] and Z c (4020) [63,64]  amplitude is This amplitude is 1/N c more suppressed than the isoscalar D One sees that the 1/N c , together with HQSS and the X(3872) input, would lead to the same scenario as in Eq. ( 53).However, the existence of Z c (3900), Z c (4020) and Z b (10610), Z b (10650) implies subleading O(N −2 c ) contributions in the 1/N c expansion might be important for the scattering of a heavy-antiheavy meson pair. 2 As such, the solution in Eq. ( 54) is beyond consequences of the large-N limit.Moreover, HQSS constrains that the LO D ( * ) D ( * ) or D ( * ) D( * ) interaction strengths for each isospin depends on two LECs (for instance, C 0a , C 0b for I = 0 and C 1a , C 1b for I = 1 in Eq. ( 34)).Without introducing more detailed dynamics, the large-N c limit does not provide a connection between these two LECs. 3On the contrary, entanglement suppression predicts that the interaction strengths for each isospin are the same.

VII. SUMMARY AND DISCUSSION
In this study, inspired by the findings of previous works [1-4], we studied consequences of entanglement suppression in the low-energy D ( * ) D ( * ) and D ( * ) D( * ) scatterings.These processes are currently of high interest due to the discoveries of the X(3872) and T cc (3875) + , which are proposed to be hadronic molecules of D D * and D * D, respectively.Using the X(3872) and T cc (3875) + as inputs, we found that entanglement suppression results in the same interaction strengths for all D ( * ) D ( * ) pairs with the same isospin, unless the interaction is forbidden due to Bose-Einstein statistics; the same also holds for D ( * ) D( * ) .The isoscalar channels are at the unitary limit, thanks to the inputs of X(3872) and T cc (3875) + , and molecular states are predicted as listed in Tables I and II.The isovector channels would be uncertain, being either noninteracting or at the unitary limit, corresponding to the two possible fixed points of two-body nonrelativistic scattering by a short-range potential.However, for the D ( * ) D( * ) pairs, the existence of a single Z c (3900) or Z c (4020) (or Z b (10610) or Z b (10650) in the bottomonium sector) state with J P C = 1 +− allows one to select the solution with all isovector channels at the unitary limit.In this case, entanglement suppression together with HQSS predicts five more isovector states around the D ( * ) D( * ) thresholds, see Table II.
where σ denotes the Pauli matrices in the SU(2) spinor space, D 00,01,10,11 are light-flavorindependent low-energy constants (LECs), Tr[•] takes trace in the spinor space, and τ • τ sums over all Pauli matrices in the flavor space.The above Lagrangian respects both HQSS and isospin symmetry.

FIG. 2 .
FIG. 2. One of the LO diagrams for D ( * )+ D( * )0 scattering (left), and the corresponding diagram with the double-line representation of gluons (right).

2 c 4 = 4 =
( * ) D( * ) amplitude, in line with the Okubo-Zweig-Iizuka rule [70-72].The isoscalar scattering can proceed through annihilating and creating light quark-antiquark pairs, and thus the LO color line diagrams without gluonic vertices consist of only one closed color loop.Correspondingly, one has T I=0 (D ( * ) D( * ) ) = O N c ÷ N 1/the D ( * ) D ( * ) sector is different, since the two charm mesons can always interact through exchange of light antiquarks no matter whether the total isospin is 0 or 1, see Fig. 3. Therefore, the 1/N c scaling of the scattering amplitude is similar to that in Eq. (60), T I=0,1 (D ( * ) D ( * ) ) = O N c ÷ N 1/2 c O(N −1 c ).

TABLE II .
Partners of the X(3872) predicted by HQSS or the two solutions of entanglement suppression given in Eqs. (