$Y(4626)$ as a $P$-wave $[cs][\bar{c}\bar{s}]$ tetraquark state

Motivated by the Belle Collaboration's new observation of $Y(4626)$, we investigate the possibility of its configuration as a $P$-wave $cs$-scalar-diquark $\bar{c}\bar{s}$-scalar-antidiquark state from QCD sum rules. Eventually, the extracted mass $4.60^{+0.14}_{-0.20}~\mbox{GeV}$ agrees well with the experimental data of $Y(4626)$, which could support its interpretation as a $P$-wave $[cs][\bar{c}\bar{s}]$ tetraquark state.

The new observation of Y (4626) by Belle immediately aroused one's great interest [26][27][28][29][30][31][32]. With an eye to the multiquark viewpoint, an assignment of Y (4626) was proposed as a D * sDs1 (2536) molecular state in a quasipotential Bethe-Salpeter equation approach with the one-boson-exchange model [27]. Later, the mass spectrum of a D * sDs1 (2536) system was calculated within the framework of Bethe-Salpeter equations [29], and in the end the authors may not think Y (4626) to be a D * sD s1 (2536) bound state, but something else. Otherwise, some authors employed a multiquark color flux-tube model with a multibody confinement potential and one-glue-exchange interaction to make an exhaustive investigation on the diquark-antidiquark state [31], and they concluded that Y (4626) can be well interpreted as a P -wave [cs][cs] state.
Under the circumstance, it is interesting and of significant to study that whether Y (4626) could be a candidate of P -wave [cs][cs] tetraquark state by different means. It is known that one has to face the complicated nonperturbative problem in QCD while handling a hadronic state. Established on the QCD basic theory, the QCD sum rule [33] acts as one authentic way for evaluating nonperturbative effects, which has been successfully applied to plenty of hadronic systems (for reviews see [34][35][36][37] and references therein). Therefore, in this work we devote to investigate that whether Y (4626) could be a P -wave [cs][cs] tetraquark state with the QCD sum rule method. This paper is organized as follows. The QCD sum rule for the P -wave tetraquark state is derived in Sec. II, followed by the numerical analysis in Sec. III. The last part is a brief summary. scalar-diquarkcs-scalar-antidiquark configuration and a derivative could be included to generate L = 1, Here the index T means matrix transposition, C denotes the charge conjugation matrix, D µ is the covariant derivative, and d, e, f , d ′ , and e ′ are color indices. In general, the two-point correlator can be parameterized as Furthermore, the part Π (1) (q 2 ) of the correlator proportional to g µν is employed to attain the sum rule, which can be evaluated in two different ways: at the hadronic level and at the quark level. Phenomenologically, Π (1) (q 2 ) can be written as where M H denotes the hadron's mass. In the OPE side, Π (1) (q 2 ) can be expressed as for which the spectral density ρ(s) = 1 π ImΠ (1) (s). To derive ρ(s), one works at leading order in α s . The s quark is treated as a light one and the diagrams are considered up to the order m s . Keeping the heavy-quark mass finite, one uses the heavy-quark propagator in momentum space [39]. The correlator's light-quark part is calculated in the coordinate space and Fourier-transformed to the momentum space in D dimension, which is combined with the heavy-quark part and then dimensionally regularized at D = 4 [37,40,41]. Lastly, the spectral density is concretely given by ρ(s) = ρ pert + ρ ss + ρ g 2 G 2 + ρ gsσ·Gs + ρ ss 2 + ρ g 3 G 3 , with which is coincident with our previous work [22]. It is defined as r = (α + β)m 2 c − αβs and κ = 1 + α − 2α 2 + β + 2αβ − 2β 2 . The integration limits are α min = (1 − 1 − 4m 2 c /s)/2, α max = (1 + 1 − 4m 2 c /s)/2, and β min = αm 2 c /(sα − m 2 c ). For the four-quark condensate ss 2 , a general factorization ssss = ̺ ss 2 [35,42] has been used, where ̺ is a constant, which may be equal to 1 or 2.
After equating the two expressions (4) and (5) of the correlator, assuming quark-hadron duality, and making a Borel transform, the sum rule can be given by Eliminating the hadronic coupling constant λ, one could yield

III. NUMERICAL ANALYSIS
Performing the numerical analysis of sum rule (7), the s-quark mass and the running charm quark mass are chosen as updated values [43] Fig. 3 as a function of M 2 from sum rule (7). In the work windows, the mass value is computed to be 4.60 ± 0.11 GeV, for which the numerical error reflects the uncertainty due to variation of s 0 and M 2 . By this time, the input QCD parameters have been kept at the central values.
Next, varying the quark masses and condensates and one could arrive at 4.60 ± 0.11 +0.03 −0.04 GeV (the first error resulted from the uncertainty due to variation of s 0 and M 2 , and the second error reflects the variation of QCD parameters) or concisely 4.60 +0.14 −0.15 GeV. At last, taking into account the variation of factorization factor ̺ in four-quark condensate ss 2 from 1 to 2, one could extract the final mass value 4.60 +0.14 −0.20 GeV for the P -wave [cs][cs] tetraquark state, which is in good agreement with the experimental data for Y (4626) and could support its P -wave [cs][cs] explanation. The OPE convergence is shown by comparing the relative contributions of perturbative, two-quark condensate ss , two-gluon condensate g 2 G 2 , mixed condensate gsσ · Gs , four-quark condensate ss 2 , and three-gluon condensate g 3 G 3 from sum rule (6) for √ s0 = 5.2 GeV.   For the future, one can expect that further experimental observations and continually theoretical studies may shed more light on the nature of Y (4626).