1 Introduction

Exotic, hidden charm, mesons known as X, Y, Z resonances have been interpreted in [1, 2] as tetraquarks, namely states made by two diquark pairs \([cq][\bar{c}\bar{q}^\prime ]\) with \(q,q^\prime \) light quarks. Each pair is in color \(\varvec{3}\) or \(\bar{\varvec{3}}\) configuration, spin \(s,\bar{s}=1,0\) and relative orbital momentum \(L=0,1\). The scheme has met with some degree of success at explaining the rich phenomenology which has emerged from electron–positron and hadron collider experiments. More information is expected in the future data from LHCb, BES III, and Belle II.

The long-standing conviction, based on consideration of large-N QCD, that tetraquark states could only materialize in the form of hadronic resonances too broad to be experimentally resolved, has been recently proven incorrect in [3, 4]. Tetraquarks in large-N QCD have been further studied in [5–7]. The recent discovery of two pentaquark states of opposite parity [8] has reinforced the case for a new spectroscopic series of hadrons, in which diquarks (antidiquarks) replace antiquarks (quarks) in the classical scheme [9, 10].

Recently, a new paradigm for the spin–spin interactions in hidden-charm tetraquarks has been proposed, which assumes the dominance of spin–spin couplings inside the diquark or the antidiquark [11]. This simple ansatz reproduces the mass ordering of the three, well identified, spin \(1^+\) states, X(3872), Z(3900) and Z(4020) and the pattern of their observed decays. In addition in Ref. [11] the diquark spin assignments of \(L=1\) states is discussed, pointing out that Y(4260) has the same spin distribution as X(3872) namely

$$\begin{aligned} X= & {} |0_{cq},1_{\bar{c} \bar{q}};L=0\rangle +|1_{cq},0_{\bar{c} \bar{q}};L=0\rangle ,\nonumber \\ Y= & {} (|0_{c q},1_{\bar{c} \bar{q}};L=1\rangle +|1_{ c q},0_{\bar{c} \bar{q}};L=1\rangle )_{J=1}. \end{aligned}$$
(1)

States are in the basis \(|s,\bar{s}; L\rangle \) where s (\(\bar{s}\)) is the diquark (antidiquark) spin and L the relative orbital angular momentum.

A similar scheme has been extended to exotic, hidden beauty mesons [12, 13], and shown to give a consistent picture of the decays of \(\Upsilon (10890)\) into \(\Upsilon (nS)\pi ^+\pi ^-\) or \(h_b(nP)\pi ^+\pi ^-\), which occur via the intermediate \(Z_b, Z_b^\prime \) states [14].

The suppression of spin–spin interactions between a quark and an antiquark in different diquarks, underlined in [11], suggests that the overlap of the two constituents is very small, as if diquark and antidiquark were well separated entities inside the hadron. In the present paper we pursue this idea to the extreme consequences by considering the approximation where a diquark and an antidiquark can be described as pointlike. X, Y, Z would be, in this case, bound particle–antiparticle systems, that we call diquarkonia for brief. We shall see that this extremely simplified picture leads to a reasonable approximation to the mass spectrum of S and P wave tetraquarks.

The diquarkonium picture has been introduced by Ali et al. [15] to study the production and decay of the Y(10890) considered as the b-tetraquark analog to the Y(4008). The annihilation of a diquarkonium made by \(s, {\bar{s}}=0\) has been treated in [15] as the annihilation of a pair of spinless, pointlike particles. The extension to \(Y(4260)\rightarrow \mu ^+ \mu ^-\) deserves further consideration, given that the diquark and the antidiquark in the Y have not the same spin and the coupling to the photon is not simply determined by the charges.

We study the diquarkonia mass spectrum in the non-relativistic approximation, using the Cornell potential previously applied to charmonia [16–19] and then, equipped with the corresponding wave functions, we compute the predicted rate of the ED1 allowed transition with \(\Delta L=1,\Delta s=0\),

$$\begin{aligned} Y(4260) \rightarrow \gamma X(3872) ,\end{aligned}$$
(2)

which arises naturally from (1).

Using the masses of the identified X, Y, Z states, we find parameters of the potential rather similar to the Cornell parameters and confirm the identification of the Z(4430) as the first radial excitation of Z(3900).

We compute the rates of the radiative transition for isospin \(I=0,1\) of X(3872) and Y(4260). Assuming X(3872) to be an isospin singlet, we find

$$\begin{aligned}&\Gamma (Y(4260)\rightarrow \gamma X(3872))=\nonumber \\&= 496~\mathrm{keV}~ (I:~0\rightarrow 0),\end{aligned}$$
(3)
$$\begin{aligned}&= 179~\mathrm{keV}~(I:~1\rightarrow 0), \end{aligned}$$
(4)

and we compare this result to the available experimental information [20].

The rate of the radiative decay (2) has been computed in Ref. [21] in the molecular scheme, describing Y(4260) and X(3872) as \(DD_1\) and \(DD^\star \) bound states, respectively. The resulting rate turns out to be considerably smaller than the values indicated in (3) or (4).

2 Diquark masses

For S-wave states diquarkonia one writes the rest frame Hamiltonian

$$\begin{aligned} M(S\mathrm{-wave})=2M_{cq}+2\kappa _{cq}({\varvec{s}}_c \cdot {\varvec{s}}_q+{ \varvec{\bar{s}}}_c \cdot {\varvec{ \bar{s}}}_q) \end{aligned}$$
(5)

where \({\varvec{s}}\) (\( \varvec{\bar{s}}\)) denotes the quark (antiquark) spin and \(M_{cq}\) is the effective diquark mass. In the \(|s,\bar{s}\rangle _J\) basis, S-wave tetraquarks with \(J^P=1^+\) are described [11] by

$$\begin{aligned}&J^{P}=1^{+}\quad C=+,\nonumber \\&\quad X_1= \frac{1}{\sqrt{2}}\left( |1,0\rangle _1+|0,1\rangle _1\right) =X(3872), \end{aligned}$$
(6)
$$\begin{aligned}&J^{P}=1^{+},\quad G=+,\nonumber \\&\quad \left\{ \begin{array}{l}Z=\frac{1}{\sqrt{2}}\left( |1,0\rangle _1-|0,1\rangle _1\right) =Z(3900),\\ Z^{\prime }=|1,1\rangle _1=Z(4020) ;\end{array}\right. \end{aligned}$$
(7)

\(M_{cq}\) can be estimated from the X(3872) and Z(4020) masses, subtracting the spin–spin contributions

$$\begin{aligned}&M(X)=M(Z)=2M_{cq}-\kappa _{cq},\nonumber \\&M(Z^\prime )=2M_{cq}+\kappa _{cq},\nonumber \\&M_{cq}=\frac{1}{4}~\left( M[Z(3900)]+M[Z(4020)] \right) \approx 1980~\mathrm{MeV} . \end{aligned}$$
(8)

As a first approximation, we shall use \(M_{cq}\) as input mass in the Schrödinger equation that gives the diquarkonia wave functions and masses.

In the case of charmonium, the input charm quark mass in the Schrödinger equation is obtained from the leptonic width \(\Gamma (J/\Psi \rightarrow e^{+} e^{-})\); see [16–18]. In our case, the leptonic width of Y(4260) is not available yet and we shall be content to use the value (8) as input. We have verified that the various quantities are little sensitive (only to a few percent) to variations of the input diquark mass around this value.

3 Bound state masses

The simplest description of diquarkonia is in terms of a non-relativistic potential, V(r). For this first exploration we take the Cornell potential [16–19] with one chromo-Coulombic and one confining term

$$\begin{aligned} V=-A\,\frac{1}{r}+\nu \, r. \end{aligned}$$
(9)

For charmonia, one finds [19]

$$\begin{aligned} A=0.47,~\nu =0.19~\mathrm{GeV}^{2}~(\mathrm{charmonium~spectrum}). \end{aligned}$$
(10)

For diquarkonia, we leave the parameters as free variables to be determined by comparison of diquarkonia eigenvalues 1S, 2S and 2P, to the mass differences of the \(J=1\) states, X(3872) or Z(3900), Y(4260) and Z(4430), subtracted of spin dependent terms. The subtraction is straightforward for the S-wave states, but for P-waves it requires the determination of not well known spin-orbit couplings [11], which introduces a non-negligible uncertainty.

Let us assume, as in [1, 11], that we can write

$$\begin{aligned}&M(X)= M_0(1S)+ \mathrm{spin~ interaction~ terms}, \nonumber \\&M(Y)= M_0(2P)+ \mathrm{spin~ interaction~ terms},\nonumber \\&\mathrm{etc.} \end{aligned}$$
(11)

where in the r.h.s. we have introduced the eigenvalues of the Schrödinger equation, \(M_0(1S)\), etc.

Explicitly, spin interaction terms are obtained from a parametrization of the constituent quark Hamiltonian, which generalizes Eq. (5) to include orbital angular momentum excitation [11] Footnote 1

$$\begin{aligned} M=M_{00}+B_c\frac{\varvec{L}^2}{2} -2a~\varvec{L}\cdot \varvec{S} +2\kappa _{qc}~\left[ (\varvec{s}_q \cdot \varvec{s}_c)+(\varvec{s}_{\bar{q}} \cdot \varvec{s}_{\bar{c}})\right] . \end{aligned}$$
(12)

Obvious manipulations lead to

$$\begin{aligned} M&=M_{00}+B_c\frac{L(L+1)}{2} + a~\left[ L(L+1)+S(S+1) -2\right] \nonumber \\&\quad + \kappa _{qc}~\left[ s(s+1)+\bar{s} (\bar{s} +1)-3\right] , \end{aligned}$$
(13)

and we read

$$\begin{aligned}&M(X(3872))=M_{00} - \kappa _{qc},\nonumber \\&M(Y(4260))=M_{00} + B_c+ 2 a - \kappa _{qc},\nonumber \\&M(Z(4430))=M_{00}^\prime - \kappa _{qc}. \end{aligned}$$
(14)

\(M_{00}^\prime \) is the analog of \(M_{00}\) for the first radial excitation. Therefore

$$\begin{aligned}&M_0(1S)=M(X(3872))+\kappa _{qc},\nonumber \\&M_0(2P)=M(Y(4260))-2a+\kappa _{qc},\nonumber \\&M_0(2S)=M(Z(4430))+\kappa _{qc}, \end{aligned}$$
(15)

and

$$\begin{aligned}&M_0(2S)-M_0(1S)=M(Z(4430))-M(Z(3900)), \end{aligned}$$
(16)
$$\begin{aligned}&M_0(2P)-M_0(1S)=M(Y(4260))-M(X(3872))-2a.\nonumber \\ \end{aligned}$$
(17)
Table 1 Masses of the well identified X, Y, Z states used in the text [22]
Full size table

We use the mass values summarized in Table 1 [22] and take the value \(a=73\) MeV from the fit to the masses of Y states in [11]Footnote 2 to which we attribute a theoretical error estimated to be not less than 50 %. We find

$$\begin{aligned}&M_{2S}-M_{1S}= 0.60\pm 0.03~\mathrm{GeV},\nonumber \\&M_{2P}-M_{1S}=0.23\pm 0.07~\mathrm{GeV}. \end{aligned}$$
(18)

We solve numerically the Schrödinger equation [24] using the diquark mass in (8).

Results for the mass differences are reported in Fig. 1, in the plane of the eigenvalue differences \(2S-1S\) and \(2P-1S\). The result for the Cornell potential with charmonium parameters is given by the round dot, whereas the squared box with errors corresponds to the eigenvalue differences estimated in (18). Lines indicate the results computed with fixed A while varying \(\nu \). Approximate agreement with the mass formula point is obtained for

$$\begin{aligned} A\sim 0,~\nu =0.25~\mathrm{GeV}^{2}~(\mathrm{diquarkonium~spectrum}). \end{aligned}$$
(19)
Fig. 1
figure 1

Results for the mass differences in the plane \(M(2S)-M(1S)\) and \(M(2P)-M(1S)\) (in GeV). The squared box with errors corresponds to the eigenvalue differences estimated in (18). Lines indicate the results computed with fixed A and varying \(\nu \). The round dot represents the result for the Cornell potential with charmonium parameters given in Eq. (10)

Full size image

The difference \(2S-1S\) is well reproduced for both sets of parameters, Eqs. (10) and (19), reinforcing the case for Z(4430) to be the first radial excitation of Z(3900) [2, 25]. The difference between the parameters in (10) and (19) may be due to the inaccuracy of the mass formula or to the fact that the diquark is not as pointlike as the c quark and, therefore, less sensitive to the short distance effects embodied by the Coulomb term.

4 The ED1 transition

We consider the process

$$\begin{aligned} Y(4260)\rightarrow \gamma + X(3872) \end{aligned}$$
(20)

as the ED1 transition from a P-wave to a S-wave tetraquark with the same spin structure. Diquarks are taken as pointlike objects of electric charge Q

$$\begin{aligned} Q=\left\{ \begin{array}{c} +\frac{4}{3}~\mathrm{for}~[cu] \\ \\ +\frac{1}{3}~\mathrm{for}~[cd] .\end{array}\right. \end{aligned}$$
(21)

The Hamiltonian (radiation gauge) is

$$\begin{aligned} H=e Q~ {\varvec{v}}\cdot {\varvec{A}}({\varvec{x}}) \end{aligned}$$
(22)

where \(\varvec{A}\) is the vector potential, \(\varvec{x}\) the coordinate and \(\varvec{v}\) the relative velocity of the particles in the center of mass system, with the diquark reduced mass

$$\begin{aligned} \mu =\frac{1}{2} M_{2q} \end{aligned}$$
(23)

and \(M_{2q}\) given by (8). In the dipole approximation where we set \({\varvec{A}}({\varvec{x}})\approx {\varvec{A}}({\varvec{0}})\), the matrix element for the decay is

$$\begin{aligned} \mathcal{M}_{if}= & {} e~\frac{1}{\sqrt{2\omega }}~\langle X,m |Q\,{\varvec{v}}|Y,k\rangle ~\cdot {\varvec{\epsilon }}({\varvec{q}})=\end{aligned}$$
(24)
$$\begin{aligned}= & {} ie\omega ~\frac{1}{\sqrt{2\omega }}~\langle X,m |Q\,{\varvec{x}}|Y,k\rangle ~\cdot {\varvec{\epsilon }}({\varvec{q}}) \end{aligned}$$
(25)

where \(\varvec{\epsilon }\) and \(\varvec{q}\) are the polarization vector and momentum of the photon, \(\omega =E_f -E_i\) its energy and m and k label the spin states of X and Y, respectively.

The total rate is obtained by (25)

$$\begin{aligned} \Gamma= & {} ~e^2 \int \frac{\mathrm{d}^3q}{(2\pi )^3 2\omega }\,\omega ^2\,(2\pi )\,\delta (E_f-E_i -\omega )~\left( \delta _{ij}-n_i n_j\right) \nonumber \\&\times \, \frac{1}{3}\,\sum _{m,k}\langle Q\, x^i\rangle ~ \langle Q\, x^j\rangle ^\star \nonumber \\= & {} \frac{4\,\alpha \,\omega ^3}{9} \sum _{m,k,i}|\langle Q\,x^i \rangle |^2 \end{aligned}$$
(26)

where we used

$$\begin{aligned} \int \,\mathrm{d}\Omega \,(\delta _{ij}-n_in_j)=\frac{2}{3}(4\pi )\delta _{ij} \end{aligned}$$
(27)

with \(n^i=q^i/\omega \).

5 Diquarkonium wave functions and transition radius

Consider first diquarkonia with a given flavor composition, e.g. \(Y_u=[cu][\bar{c} \bar{u}]\) or \(Y_d=[cd][\bar{c} \bar{d}]\). In the non-relativistic approximation, state vectors corresponding to Y (P-wave) or X (S-wave) are written as

$$\begin{aligned}&N_Y \langle Y,k|=\langle 0|\,\int d^3x~R^{2P}(r)\frac{x^i}{r}\epsilon _{ijk} \nonumber \\&\quad \times \left[ d^j_a\left( \frac{{\varvec{x}}}{2}\right) (d_c)^a\left( -\frac{{\varvec{x}}}{2}\right) +d_a\left( \frac{{\varvec{x}}}{2}\right) (d_c)^{aj}\left( -\frac{{\varvec{x}}}{2}\right) \right] ,\end{aligned}$$
(28)
$$\begin{aligned}&N_X \langle X,m|=\langle 0|\,\int d^3x~R^{1S}(r)\nonumber \\&\quad \times \left[ d^m_a\left( \frac{{\varvec{x}}}{2}\right) (d_c)^a\left( -\frac{{\varvec{x}}}{2}\right) +d_a\left( \frac{{\varvec{x}}}{2}\right) (d_c)^{a,m}\left( -\frac{{\varvec{x}}}{2}\right) \right] \end{aligned}$$
(29)

where d and \(d_c\) (or \(d^m\) and \(d_c^m\)) are the destruction operators of diquark and antidiquark with spin \(S=0\) (\(S=1\)) and R(r) the radial wave functions. We have made explicit the color index \(a=1,2,3\). The normalization factors are obtained from (non-relativistic) identities of the form

$$\begin{aligned} \langle 0|d_a^j({\varvec{x}})[d_b^l({\varvec{y}})]^\dagger |0\rangle =\delta _a^b~\delta ^{jl}~\delta ^{(3)}({\varvec{x}}-{\varvec{y}}),~\mathrm{etc.} \end{aligned}$$
(30)

to wit

$$\begin{aligned}&N_Y^2= 2^6 (2N)\,\frac{2}{3}\,(4\pi ) ~\delta ^{(3)}(0),\end{aligned}$$
(31)
$$\begin{aligned}&N_X^2=2^6 (2N)\,(4\pi )~\delta ^{(3)}(0), \end{aligned}$$
(32)

where (27) has been used and the number of colors is \(N=3\).

The transition radius is then computed between normalized states to be

$$\begin{aligned}&\langle X,m|x^i|Y,k\rangle =\frac{1}{\sqrt{6}}~\epsilon _{mik}~\langle r \rangle ,\end{aligned}$$
(33)
$$\begin{aligned}&\langle r \rangle = \langle r \rangle _{2P\rightarrow 1S}= \frac{\int _0^{\infty } r \left[ y^{1S}(r)y^{2P}(r)\right] \mathrm{d}r}{\sqrt{\int _0^{\infty }\mathrm{d}r\, (y^{1S}(r))^2}\sqrt{\int _0^{\infty }\mathrm{d}r\, (y^{2P}(r))^2}}\nonumber \\ \end{aligned}$$
(34)

and we have introduced the reduced radial wave functions of the 1S and 2P wave functions \(y(r)=r R(r)\) computed numerically [24].

Table 2 Transition radius and corresponding decay widths for \(Y\rightarrow \gamma X\). In parentheses the branching ratio, assuming \(\Gamma _{Y(4260)}=120\) MeV [22]
Full size table

Finally, we consider the general isospin structure of Y(4260) and X(3872), defining

$$\begin{aligned}&X(3872)=\cos {\theta }\,X_u+\sin \theta \,X_d\nonumber \\&Y(4260)=\cos {\phi }\,Y_u+\sin \phi \,Y_d \end{aligned}$$
(35)

and obtain

$$\begin{aligned}&\langle X,m | Q\,x^i |Y, k\rangle =\frac{1}{\sqrt{6}}~\epsilon _{mik}~Q_\mathrm{eff}~\langle r \rangle \nonumber \\&Q_\mathrm{eff}=\left( \frac{4}{3}\cos {\theta }\cos {\phi } + \frac{1}{3}\sin {\theta }\sin {\phi } \right) . \end{aligned}$$
(36)

6 Diquarkonium rate

With (26) and (36), we obtain

$$\begin{aligned}&\Gamma (Y(4260)\rightarrow \gamma + X(3872))= \frac{4\,\alpha \, \omega ^3}{9}~Q_\mathrm{eff}^2 \,\langle r \rangle ^2 \nonumber \\&\quad = 154.2\times Q_\mathrm{eff}^2\left( \frac{\langle r \rangle }{\mathrm{GeV}^{-1}}\right) ^2~\mathrm{keV} . \end{aligned}$$
(37)

Note that \(0 \le Q_\mathrm{eff}^2\le (4/3)^2\), with zero attained when \(Y=Y_{u}\) and \(X=X_{d}\) or vice versa, and the maximum when Y and X have only u-flavor.

As indicated by data, we take X(3872) close to a pure \(I=0\) state. For the two sets of parameters of the potential, Eqs. (10) and (19), we summarize in Table 2 (i) the numerical values of the transition radius and (ii) the rate for Y(4260) with \(I=0,1\).

With the indicated numerical value of the radius, we are at the border of the dipole approximation, since \(\omega \langle r\rangle \simeq 0.8\), not so much smaller than one. The situation, however, is not so different from the radiative transition \(\chi _{c2}\rightarrow J/\Psi \gamma \), which has \(\omega \langle r\rangle = 0.86\), with estimated \(\sim \)10 % corrections; see [26].

The result found in Ref. [20] can be stated thus:

$$\begin{aligned} \frac{B(Y\rightarrow \gamma X)B(X\rightarrow J/\Psi ~\pi \pi )}{B(Y\rightarrow J/\Psi ~ \pi \pi )}=5 \times 10^{-3}, \end{aligned}$$
(38)

which, assuming [22]

$$\begin{aligned} B(X\rightarrow J/\Psi ~\pi \pi )\gtrsim 2.6 \times 10^{-2}, \end{aligned}$$
(39)

becomes

$$\begin{aligned} \frac{B(Y \rightarrow \gamma X)}{B(Y\rightarrow J/\Psi ~\pi \pi )}< 0.2. \end{aligned}$$
(40)

Using our result, we predict

$$\begin{aligned} B(Y\rightarrow J/\Psi ~\pi \pi )> \left\{ \begin{array}{c} 2.1\times 10^{-2}\quad (I: 0 \rightarrow 0) ,\\ 0.78\times 10^{-2}\quad (I: 1 \rightarrow 0).\end{array}\right. \end{aligned}$$
(41)

From the value of \(\Gamma (Y\rightarrow X\gamma )\) we can also estimate \(\Gamma (Y\rightarrow e^-e^+)\). We use the well-known formula for the peak cross section

$$\begin{aligned} \sigma (e^-e^+\rightarrow X\,\gamma )_{Y}=\frac{12\pi }{m_Y^2}\frac{\Gamma (Y\rightarrow X\gamma )\Gamma (Y\rightarrow e^-e^+)}{\Gamma (Y\rightarrow \mathrm {All})^2} \end{aligned}$$
(42)

with the experimental determination [27]

$$\begin{aligned} \sigma (e^-e^+\rightarrow Y(4260) \rightarrow X\,\gamma )=\frac{0.33~\mathrm {pb}}{\mathcal {B}(X\rightarrow \pi ^+\pi ^- J/\Psi )} \end{aligned}$$
(43)

and the input values in Tables 1 and 2 (diquarkonium potential)

$$\begin{aligned} \Gamma (Y\rightarrow e^-e^+)\lesssim \frac{226}{(\Gamma (Y\rightarrow X\gamma )/\mathrm {keV})}~\mathrm {keV}= \left\{ \begin{array}{c} 0.45 \\ 1.26 \end{array}\right. ~\mathrm{keV}\nonumber \\ \end{aligned}$$
(44)

and

$$\begin{aligned}&\sigma (e^+e^-\rightarrow \mu ^+\mu ^-)_Y \lesssim \frac{2871}{(\Gamma (Y\rightarrow X\gamma )/\mathrm {keV})^2}~\mathrm {pb}\nonumber \\&\quad = \left\{ \begin{array}{c} 0.01 \\ 0.09 \end{array}\right. ~\mathrm{pb.} \end{aligned}$$
(45)

7 Conclusions

We estimated the transition rates \(\Gamma (Y(4260)\rightarrow \gamma +X(3872))\) under both the assumptions that Y is an isospin singlet or a triplet bound state confined by a Cornell-like potential, a diquarkonium. We observe that the mass formula of the constituent quark model gives results for the mass differences between the radial excitations being closer to the results computed through an inter-diquark potential linearly rising with the distance and no chromo-Coulombic term. The results obtained, together with an upper bound estimate of the Y electronic width, can be confronted with future data, from electron–positron and hadron colliders.