Magnetic dipole moment of the $Z_{cs}(3985)$ state: diquark–antidiquark and molecular pictures

Abstract

We calculate the magnetic dipole moment of the newly observed charged hidden-charmed open strange $Z_{cs}(3985)^-$ state, recently observed by BESIII Collaboration. Based on the information provided by the experiment and theoretical studies followed the observation, we assign the quantum numbers $J^{P} = 1^{+}$ and the quark composition $c {{\bar{c}}} s{{\bar{u}}}$ to this state and estimate the magnetic dipole moment of this resonance in both the compact diquark–antidiquark and molecular pictures. We apply the light cone QCD formalism and use the distribution amplitudes of the on-shell photon with different twists. The obtained results in both pictures are consistent with each other within the errors. The magnitude of the magnetic dipole moment shows that it is accessible in the experiment.

Appendix: Photon distribution amplitudes and wave functions

In this Appendix, we present the matrix elements $\langle \gamma (q)\left| {\bar{q}}(x) \Gamma _i q(0) \right| 0\rangle$ and $\langle \gamma (q)\left| {\bar{q}}(x) \Gamma _i G_{\mu \nu }q(0) \right| 0\rangle$ in terms of the photon DAs and wave functions of different twists [67],

$$\begin{aligned}&\langle \gamma (q) \vert {{\bar{q}}}(x) \gamma _\mu q(0) \vert 0 \rangle = e_q f_{3 \gamma } \left( \varepsilon _\mu - q_\mu \frac{\varepsilon x}{q x} \right) \int _0^1 du e^{i {{\bar{u}}} q x} \psi ^v(u) \\&\quad \langle \gamma (q) \vert {{\bar{q}}}(x) \gamma _\mu \gamma _5 q(0) \vert 0 \rangle = - \frac{1}{4} e_q f_{3 \gamma } \epsilon _{\mu \nu \alpha \beta } \varepsilon ^\nu q^\alpha x^\beta \int _0^1 du e^{i {{\bar{u}}} q x} \psi ^a(u) \\&\quad \langle \gamma (q) \vert {{\bar{q}}}(x) \sigma _{\mu \nu } q(0) \vert 0 \rangle = -i e_q \langle {{\bar{q}}} q \rangle (\varepsilon _\mu q_\nu - \varepsilon _\nu q_\mu ) \int _0^1 du e^{i {{\bar{u}}} qx} \left( \chi \varphi _\gamma (u) + \frac{x^2}{16} {\mathbb {A}} (u) \right) \\&\qquad -\frac{i}{2(qx)} e_q {{\bar{q}}}q \left[ x_\nu \left( \varepsilon _\mu - q_\mu \frac{\varepsilon x}{qx}\right) - x_\mu \left( \varepsilon _\nu - q_\nu \frac{\varepsilon x}{q x}\right) \right] \int _0^1 du e^{i \bar{u} q x} h_\gamma (u) \\&\quad \langle \gamma (q) | {{\bar{q}}}(x) g_s G_{\mu \nu } (v x) q(0) \vert 0 \rangle = -i e_q \langle {{\bar{q}}} q \rangle \left( \varepsilon _\mu q_\nu - \varepsilon _\nu q_\mu \right) \int {{{\mathcal {D}}}}\alpha _i e^{i (\alpha _{{{\bar{q}}}} + v \alpha _g) q x} {{{\mathcal {S}}}}(\alpha _i) \\&\quad \langle \gamma (q) | {{\bar{q}}}(x) g_s {{\tilde{G}}}_{\mu \nu }(v x) i \gamma _5 q(0) \vert 0 \rangle = -i e_q \langle {{\bar{q}}} q \rangle \left( \varepsilon _\mu q_\nu - \varepsilon _\nu q_\mu \right) \int {{{\mathcal {D}}}}\alpha _i e^{i (\alpha _{{{\bar{q}}}} + v \alpha _g) q x} \tilde{{{\mathcal {S}}}}(\alpha _i) \\&\quad \langle \gamma (q) \vert {{\bar{q}}}(x) g_s {{\tilde{G}}}_{\mu \nu }(v x) \gamma _\alpha \gamma _5 q(0) \vert 0 \rangle = e_q f_{3 \gamma } q_\alpha (\varepsilon _\mu q_\nu - \varepsilon _\nu q_\mu ) \int \mathcal{D}\alpha _i e^{i (\alpha _{{{\bar{q}}}} + v \alpha _g) q x} \mathcal{A}(\alpha _i) \\&\quad \langle \gamma (q) \vert {{\bar{q}}}(x) g_s G_{\mu \nu }(v x) i \gamma _\alpha q(0) \vert 0 \rangle = e_q f_{3 \gamma } q_\alpha (\varepsilon _\mu q_\nu - \varepsilon _\nu q_\mu ) \int \mathcal{D}\alpha _i e^{i (\alpha _{{{\bar{q}}}} + v \alpha _g) q x} \mathcal{V}(\alpha _i) \\&\quad \langle \gamma (q) \vert {{\bar{q}}}(x) \sigma _{\alpha \beta } g_s G_{\mu \nu }(v x) q(0) \vert 0 \rangle \\&\qquad = e_q \langle {{\bar{q}}} q \rangle \left\{ \left[ \left( \varepsilon _\mu - q_\mu \frac{\varepsilon x}{q x}\right) \left( g_{\alpha \nu } - \frac{1}{qx} (q_\alpha x_\nu + q_\nu x_\alpha )\right) \right. \right. q_\beta \\&\qquad - \left( \varepsilon _\mu - q_\mu \frac{\varepsilon x}{q x}\right) \left( g_{\beta \nu } - \frac{1}{qx} (q_\beta x_\nu + q_\nu x_\beta )\right) q_\alpha - \left( \varepsilon _\nu - q_\nu \frac{\varepsilon x}{q x}\right) \left( g_{\alpha \mu }\right. \\&\qquad \left. - \frac{1}{qx} (q_\alpha x_\mu + q_\mu x_\alpha )\right) q_\beta \\&\qquad + \left. \left( \varepsilon _\nu - q_\nu \frac{\varepsilon x}{q.x}\right) \left( g_{\beta \mu } - \frac{1}{qx} (q_\beta x_\mu + q_\mu x_\beta )\right) q_\alpha \right] \int {{{\mathcal {D}}}}\alpha _i e^{i (\alpha _{{{\bar{q}}}} + v \alpha _g) qx} {{{\mathcal {T}}}}_1(\alpha _i) \\&\qquad + \left[ \left( \varepsilon _\alpha - q_\alpha \frac{\varepsilon x}{qx}\right) \left( g_{\mu \beta } - \frac{1}{qx}(q_\mu x_\beta + q_\beta x_\mu )\right) \right. q_\nu \\&\qquad - \left( \varepsilon _\alpha - q_\alpha \frac{\varepsilon x}{qx}\right) \left( g_{\nu \beta } - \frac{1}{qx}(q_\nu x_\beta + q_\beta x_\nu )\right) q_\mu \\&\qquad - \left( \varepsilon _\beta - q_\beta \frac{\varepsilon x}{qx}\right) \left( g_{\mu \alpha } - \frac{1}{qx}(q_\mu x_\alpha + q_\alpha x_\mu )\right) q_\nu \\&\qquad + \left. \left( \varepsilon _\beta - q_\beta \frac{\varepsilon x}{qx}\right) \left( g_{\nu \alpha } - \frac{1}{qx}(q_\nu x_\alpha + q_\alpha x_\nu ) \right) q_\mu \right] \int {{{\mathcal {D}}}} \alpha _i e^{i (\alpha _{{{\bar{q}}}} + v \alpha _g) qx} {{{\mathcal {T}}}}_2(\alpha _i) \\&\qquad +\frac{1}{qx} (q_\mu x_\nu - q_\nu x_\mu ) (\varepsilon _\alpha q_\beta - \varepsilon _\beta q_\alpha ) \int {{{\mathcal {D}}}} \alpha _i e^{i (\alpha _{{{\bar{q}}}} + v \alpha _g) qx} {{{\mathcal {T}}}}_3(\alpha _i) \\&\qquad + \left. \frac{1}{qx} (q_\alpha x_\beta - q_\beta x_\alpha ) (\varepsilon _\mu q_\nu - \varepsilon _\nu q_\mu ) \int {{{\mathcal {D}}}} \alpha _i e^{i (\alpha _{{{\bar{q}}}} + v \alpha _g) qx} {{{\mathcal {T}}}}_4(\alpha _i) \right\} ~, \end{aligned}$$

where $\varphi _\gamma (u)$ is the distribution amplitude of leading twist-2, $\psi ^v(u)$, $\psi ^a(u)$, ${{{\mathcal {A}}}}(\alpha _i)$ and ${{{\mathcal {V}}}}(\alpha _i)$, are the twist-3 amplitudes, and $h_\gamma (u)$, ${\mathbb {A}}(u)$, ${{{\mathcal {S}}}}(\alpha _i)$, $\mathcal{{{{\tilde{S}}}}}(\alpha _i)$, ${{{\mathcal {T}}}}_1(\alpha _i)$, ${{{\mathcal {T}}}}_2(\alpha _i)$, ${{{\mathcal {T}}}}_3(\alpha _i)$ and ${{{\mathcal {T}}}}_4(\alpha _i)$ are the twist-4 photon DAs. The measure ${{{\mathcal {D}}}} \alpha _i$ is defined as

$$\begin{aligned} \int {{{\mathcal {D}}}} \alpha _i = \int _0^1 d \alpha _{{{\bar{q}}}} \int _0^1 d \alpha _q \int _0^1 d \alpha _g \delta (1-\alpha _{\bar{q}}-\alpha _q-\alpha _g)~. \end{aligned}$$

The expressions of the DAs entering into the above matrix elements are defined as:

$$\begin{aligned} \varphi _\gamma (u)= & {} 6 u {{\bar{u}}} \left( 1 + \varphi _2(\mu ) C_2^{\frac{3}{2}}(u - {{\bar{u}}}) \right) , \\ \psi ^v(u)= & {} 3 \left( 3 (2 u - 1)^2 -1 \right) +\frac{3}{64} \left( 15 w^V_\gamma - 5 w^A_\gamma \right) \left( 3 - 30 (2 u - 1)^2 + 35 (2 u -1)^4 \right) , \\ \psi ^a(u)= & {} \left( 1- (2 u -1)^2\right) \left( 5 (2 u -1)^2 -1\right) \frac{5}{2} \left( 1 + \frac{9}{16} w^V_\gamma - \frac{3}{16} w^A_\gamma \right) , \\ h_\gamma (u)= & {} - 10 \left( 1 + 2 \kappa ^+\right) C_2^{\frac{1}{2}}(u - {{\bar{u}}}), \\ {\mathbb {A}}(u)= & {} 40 u^2 {{\bar{u}}}^2 \left( 3 \kappa - \kappa ^+ +1\right) + 8 (\zeta _2^+ - 3 \zeta _2) \left[ u {{\bar{u}}} (2 + 13 u {{\bar{u}}}) \right. \\&+ \left. 2 u^3 (10 -15 u + 6 u^2) \ln (u) + 2 {{\bar{u}}}^3 (10 - 15 {{\bar{u}}} + 6 {{\bar{u}}}^2) \ln ({{\bar{u}}}) \right] , \\ {{{\mathcal {A}}}}(\alpha _i)= & {} 360 \alpha _q \alpha _{{{\bar{q}}}} \alpha _g^2 \left( 1 + w^A_\gamma \frac{1}{2} (7 \alpha _g - 3)\right) , \\ {{{\mathcal {V}}}}(\alpha _i)= & {} 540 w^V_\gamma (\alpha _q - \alpha _{{{\bar{q}}}}) \alpha _q \alpha _{{{\bar{q}}}} \alpha _g^2, \\ {{{\mathcal {T}}}}_1(\alpha _i)= & {} -120 (3 \zeta _2 + \zeta _2^+)(\alpha _{\bar{q}} - \alpha _q) \alpha _{{{\bar{q}}}} \alpha _q \alpha _g, \\ {{{\mathcal {T}}}}_2(\alpha _i)= & {} 30 \alpha _g^2 (\alpha _{{{\bar{q}}}} - \alpha _q) \left( (\kappa - \kappa ^+) + (\zeta _1 - \zeta _1^+)(1 - 2\alpha _g) + \zeta _2 (3 - 4 \alpha _g)\right) , \\ {{{\mathcal {T}}}}_3(\alpha _i)= & {} - 120 (3 \zeta _2 - \zeta _2^+)(\alpha _{\bar{q}} -\alpha _q) \alpha _{{{\bar{q}}}} \alpha _q \alpha _g, \\ {{{\mathcal {T}}}}_4(\alpha _i)= & {} 30 \alpha _g^2 (\alpha _{{{\bar{q}}}} - \alpha _q) \left( (\kappa + \kappa ^+) + (\zeta _1 + \zeta _1^+)(1 - 2\alpha _g) + \zeta _2 (3 - 4 \alpha _g)\right) , \\ {{{\mathcal {S}}}}(\alpha _i)= & {} 30\alpha _g^2\{(\kappa + \kappa ^+)(1-\alpha _g)+(\zeta _1 + \zeta _1^+)(1 - \alpha _g)(1 - 2\alpha _g) \\&+\zeta _2[3 (\alpha _{{{\bar{q}}}} - \alpha _q)^2-\alpha _g(1 - \alpha _g)]\}, \\ \tilde{{{\mathcal {S}}}}(\alpha _i)= & {} -30\alpha _g^2\{(\kappa -\kappa ^+)(1-\alpha _g)+(\zeta _1 - \zeta _1^+)(1 - \alpha _g)(1 - 2\alpha _g) \\&+\zeta _2 [3 (\alpha _{{{\bar{q}}}} -\alpha _q)^2-\alpha _g(1 - \alpha _g)]\}. \end{aligned}$$

Numerical values of parameters used in distribution amplitudes are: $\varphi _2(1~GeV) = 0$, $w^V_\gamma = 3.8 \pm 1.8$, $w^A_\gamma = -2.1 \pm 1.0$, $\kappa = 0.2$, $\kappa ^+ = 0$, $\zeta _1 = 0.4$, $\zeta _2 = 0.3$.