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PHYSICAL REVIEW D 101, 074037 (2020)

Properties of Zc(3900) tetraquark in a cold nuclear

K. Azizi 1,2 and N. Er3 1Department of Physics, University of Tehran, North Karegar Avenue, Tehran 14395-547, Iran 2Department of Physics, Dogus University, Acibadem-Kadikoy, 34722 Istanbul, Turkey 3Department of Physics, Abant İzzet Baysal University, Gölköy Kampüsü, 14980 Bolu, Turkey

(Received 25 January 2020; accepted 13 April 2020; published 27 April 2020)

The study of medium effects on properties of embedded in nuclear matter is of great importance for understanding the nature and internal - organization as well as exact determination of the quantum numbers, especially of the exotic states. In this context, we study the physical properties of one of the famous charmoniumlike states, Zcð3900Þ, in a cold dense matter. We investigate the possible shifts in the mass and current- coupling of the Zcð3900Þ state due to the dense medium at saturation density ρsat by means of the in-medium sum rules. We also estimate the vector self-energy of this state at saturation nuclear matter density. We discuss the behavior of the spectroscopic parameters of this state with respect to the density up to a high density corresponding to the core of stars, ρ ≈ 5ρsat. Both the mass and current coupling of this state show nonlinear behavior and decrease with respect to the density of the medium: the mass reaches roughly 30% of its vacuum value at ρ ¼ 5ρsat, while the current coupling approaches zero at ρ ≈ 2.1ρsat, when the central values of the auxiliary and other input parameters are used.

DOI: 10.1103/PhysRevD.101.074037

I. INTRODUCTION In the first trials, the new charmoniumlike resonances, The standard are divided into qq¯ and qqq=q¯ q¯ q¯ discovered in the above experiments, were evaluated as the ¯ systems. Neither the nor the QCD excludes excited states of ordinary cc charmonium states. However, the existence of the structures out of these configurations. the obtained data showed that some resonances do not Hence, search for exotic states is inevitable: now we have conform to standard spectroscopy, and thus some new many exotic states observed in the experiment. We have nonconventional models have been developed. The new also made good progress in the determination of different models differ from each other in terms of their components aspects of these states in theory. Most of the discovered and mechanisms. There are plenty of exotic states are tetraquarks of the XYZ family. The term studies for the physical picture interpretation of the XYZ comes from the generalization of states Xð3872Þ, charmoniumlike states using the new models: for instance, Yð4260Þ, and Z ð3900Þ. The XYZ states are the charmo- QCD tetraquarks [6,7], weakly bound hadronic c – – – niumlike or bottomoniumlike resonances, which, because [8 14], charmonium hybrids [15 19], threshold cups [20 of their mass, cannot be placed in the charmonium or 23], and hadro-charmonium [24,25]. Understanding the bottomonium picture: These resonances, mainly with nonperturbative behavior of QCD and the strong interaction ¯ ¯ dynamics that cause the production and structure of these QQqq quark content, have different properties than the ’ standard excited states. In the last decade, nonconventional states is very important for today s exper- many XYZ states have been observed by the Belle, BESIII, imental and theoretical studies. 3900 BABAR, LHCb, CMS, D0, CDF, and CLEO-c collabora- As candidates of tetraquark states, Zc ð Þ were tions [1], and their masses, widths and quantum numbers reported simultaneously by the BESIII Collaboration þ − JPC have been predicted. Detailed analysis of the exper- [26] using e e annihilation at the vector resonance Yð4260Þ and by the Belle Collaboration [27] using the imental status of these states and various theoretical models ϒ 1 2 … 5 can be found in numerous new review articles [2–5]. same process but at or near the ðnSÞ (n ¼ ; ; ; ) resonances. For the full history, see [28–30]. They were confirmed by the CLEO-c Collaboration using 586 pb−1 of eþe− annihilation data taken at the CESR collider at pffiffiffi s ¼ 4170 MeV, the peak of the charmonium resonance Published by the American Physical Society under the terms of 0 ψð4160Þ. They also reported evidence for Zcð3900Þ, which the Creative Commons Attribution 4.0 International license. is the neutral member of this isospin triplet [31].In Further distribution of this work must maintain attribution to pffiffiffi þ − → π ¯ ∓ 4 26 the author(s) and the published article’s title, journal citation, Ref. [32] e e ðDD Þ at s ¼ . GeV using and DOI. Funded by SCOAP3. 525 pb−1 of data collected at the BEPCII storage ring, the

2470-0010=2020=101(7)=074037(12) 074037-1 Published by the American Physical Society K. AZIZI and N. ER PHYS. REV. D 101, 074037 (2020)

3883 9 1 5 3900 determined pole mass Mpole ¼ð . . ðstatÞ medium on the parameters of the Zcð Þ state and look 4 2 2 Γ 24 8 . ðsystÞÞ MeV=c , and pole width pole ¼ð . for the behavior of the mass, current-meson coupling, and 3.3ðstatÞ11.0ðsystÞÞ MeV were reported with a signifi- vector self-energy of this charmoniumlike state, consider- cance of 2σ and 1σ, respectively. It was referred to as ing it as a compact tetraquark state. The study is organized as follows. In Sec. II, we derive Zcð3885Þ. BESIII also observed a new neutral state 0 þ − 0 0 the spectral densities associated with the state Zcð3900Þ by Zcð3900Þ in a process e e → π π J=ψ with a significance of 10.4σ. The measured mass and width were ð3894.8 applying the two-point sum rule technique, and we obtain 2 the QCD sum rules for the mass, current-meson coupling, 2.3ðstatÞ3.2ðsystÞÞ MeV=c and Γ ¼ð29.6 8.2ðstatÞ and vector self-energy using the obtained spectral densities. 8.2ðsystÞÞ MeV, respectively [33].InRef.[34], after the full In Sec. III, after fixing the auxiliary parameters using the construction of the pair and the bachelor π in the standard prescriptions of the method, the numerical analy- final state, the existence of the charged structure Zc ð3885Þ ¯ ∓ sis of the physical observables, both in vacuum and cold was confirmed in the ðDD Þ system, and its pole mass and nuclear matter, is performed. Section IV is devoted to a 3881 7 1 6 width were measured as Mpole ¼ð . . ðstatÞ discussion and concluding remarks. We collect some 1 6 2 Γ 26 6 2 0 . ðsystÞÞ MeV=c and pole ¼ð . . ðstatÞ lengthy expressions obtained from the calculations in the 2.1ðsystÞÞ MeV, respectively. In the processes eþe− → Appendix. pffiffiffi DþD−π0 þ c:c: and eþe− → D0D¯ 0π0 þ c:c: at s ¼ 0 4.226 and 4.257 GeV, the neutral structure Zcð3885Þ II. IN-MEDIUM MASS AND CURRENT 4 3 3885 7þ . COUPLINGS OF Zc(3900) was observed with the pole mass ð . −5.7 ðstatÞ 8 4 2 35þ11 . ðsystÞÞ MeV=c and pole width ð −12 ðstatÞ Hadrons are formed as a result of some nonperturbative 15ðsystÞÞ MeV [35]. In a very recent study by the D0 effects at low energies very far from the asymptotic region Collaboration, the authors presented evidence for the of QCD. Hence, to investigate their properties, some Zc ð3900Þ state decaying to J=ψπ in semi-inclusive weak nonperturbative methods are needed. QCD sum rules decays of b-flavored hadrons [36]. appear as a reliable, powerful, and predictive approach On the theoretical side, for investigation of the Zcð3900Þ in this respect. In this approach, the hadrons are represented resonance, many different models and approaches are used; by interpolating currents, written by considering the quark some of them are mentioned here as examples. In a recent content and all the quantum numbers of the hadrons. From study [37], using the three-channel Ross-Shaw theory, the the theoretical studies in vacuum and the experimental data, authors have obtained constraint conditions that need to be the quantum numbers JPC ¼ 1þ− have been assigned to satisfied by various parameters of the theory in order to Zcð3900Þ. The comparison of the theoretical predictions on have a narrow resonance close to the threshold of the third some parameters of this state with the experimental data channel; this is relevant to the structure. Using the QCD leads us to consider a compact tetraquark of a and sum rule method, the same state was considered as a an antidiquark structure for this state [39,40]. Thus, the compact tetraquark state of a diquark-antidiquark configu- interpolating current representing the ccu¯ d¯ quark content ration in [38–40] and a hadronic in [41,42].In and JPC ¼ 1þ− quantum numbers can be written as these studies, many parameters related to the Z ð3900Þ c ϵ ϵ state were calculated. Its mass was already calculated i abc dec T ¯ T JμðxÞ¼ pffiffiffi f½u ðxÞCγ5c ðxÞ½d ðxÞγμCc¯ ðxÞ within the framework of a nonrelativistic quark model in 2 a b d e [43] as well. T ¯ T − ½ua ðxÞCγμcbðxÞ½ddðxÞγ5Cc¯e ðxÞg; ð1Þ Despite a lot of theoretical and experimental efforts, unfortunately, the nature and internal structures of most of where ϵabc and ϵdec are antisymmetric Levi-Civita symbols the exotic states including Zcð3900Þ are not exactly clear. in three dimensions with color indices a; b; c; d, and e. The Hence, investigation of their properties at a dense or hot letter T represents a transpose in Dirac space, γ5 and γμ are medium can play an important role. Experiments like Dirac matrices, and C is the charge conjugation operator. PANDA will provide a possibility to explore these states In the framework of QCD sum rules, we aim to obtain in a dense medium. These studies will help us better the sum rules for the mass, current coupling, and vector understand the quark-gluon organization of the exotic states self-energy of the exotic state Zc in cold nuclear matter. In as well as their interactions with the particles existing in the the generalization of the vacuum QCD sum rules to finite medium. In our previous study, Ref. [44], we investigated density, we start with the same correlation function of the Xð3872Þ state by applying a diquark-antidiquark-type interpolating currents as the vacuum with the difference current in the framework of in-medium QCD sum rules. We that, in this case, the time ordering product of the calculated the mass, current-meson coupling, and also the interpolating currents is sandwiched between the ground vector self-energy of this state and found that these states of a finite density medium instead of vacuum [45]. parameters strongly depend on the density of the medium. So, we start with the following in-medium two-point In the present study, we investigate the effects of a dense correlation function as the building block of the method:

074037-2 PROPERTIES OF ZCð3900Þ TETRAQUARK IN A COLD … PHYS. REV. D 101, 074037 (2020) Z 4 ip·x † which further simplifies Eq. (3). By summing over the ΠμνðpÞ¼i d xe hψ 0jT ½JμðxÞJνð0Þjψ 0i; ð2Þ polarization vectors, we can recast the phenomenological side of the correlation function into the form T where p is the four-momentum of the Zc state, is the time   2 2 ordering operator, and jψ 0i is the parity and time-reversal m f pμpν Phe Zc Zc Πμν ðpÞ¼− −gμν þ þ: ð5Þ symmetric ground state of nuclear matter. p2 − m2 m2 In cold nuclear matter, we formulate the sum rules for Zc Zc the modified mass, m , and current coupling constant, Zc To proceed, we introduce two self-energies: the scalar self- fZ , of the ground state Zc. For this purpose and in energy Σ ¼ m − m and the vector self-energy Συ, c s Zc Zc accordance with the philosophy of the method, the which appears in the expression of the in-medium momen- correlation function is calculated in two different perspec- tum, pμ ¼ pμ − Συuμ [46], where uμ is the four-velocity tives: the phenomenological (Phe) and operator product vector of the cold nuclear medium. We work in the rest expansion (OPE) windows. The results of these two frame of the medium, i.e., uμ ¼ð1; 0Þ. As a result, one can representations are then matched under some conditions write like the quark- duality assumption to relate the hadronic parameters to fundamental QCD parameters. f2 Phe Zc 2 We focus on the ground state; hence, we should apply Πμν ðpÞ¼− ½−gμνm þ pμpν 2 − μ2 Zc some transformations like Borel and continuum subtrac- p 2 tion to enhance the ground-state contribution and suppress − Συpμuν − Συpνuμ þ Συuμuνþ; ð6Þ the contributions coming from the higher states and 2 2 2 continuum. where μ ¼ m − Συ þ 2p0Συ, with p0 ¼ p:u being the Zc energy of the . The Borel transformed form A. Phenomenological window (Borel with respect to p2) of the phenomenological The phenomenological side of the correlation function is representation reads determined in a few steps: First, it is saturated by a full set Phe 2 −μ2=M2 2 Πμν ðpÞ¼f e ½−gμνm of hadronic states carrying the same numbers as the Zc Zc interpolating current, and then the contribution of the 2 þ pμpν − Συpμuν − Συpνuμ þ Συuμuν ground state is isolated. As a result, we get þ; ð7Þ † ψ ν ψ Phe h 0jJμjZcðpÞihZcðpÞjJ j 0i 2 Πμν ðpÞ¼− þ; ð3Þ where M is the Borel mass parameter to be fixed later p2 − m2 Zc using the standard recipe of the method. where p is the in-medium momentum and dots denote B. OPE or QCD window contributions arising from higher resonances and con- The QCD side of the calculations can be obtained by tinuum states. The decay constant or current-meson cou- inserting the interpolating current JμðxÞ into the correlation ε pling is expressed in terms of the polarization vector μ of function and performing all possible contractions of the Zc as quark pairs. The resultant equation is a expression in terms ij ij of the in-medium light quark (Sq ) and heavy quark (S ) Q hψ 0jJμjZ ðpÞi ¼ f m εμ; ð4Þ c Zc Zc propagators:

Z QCD i 4 ipx ˜ aa0 bb0 ˜ e0e d0d ˜ e0e Πμν p − ε ε 0 0 0 ε ε 0 0 0 d xe γ5S x γ5S x γμS −x γνS −x − γμS −x ð Þ¼ 2 abc a b c dec d e c fTr½ u ð Þ c ð ÞTr½ c ð Þ d ð Þ Tr½½ c ð Þ γ d0d − γ ˜ aa0 γ bb0 − γ ˜ aa0 γ bb0 γ ˜ e0e − γ d0d − × 5Sd ð xÞTr½ νSu ðxÞ 5Sc ðxÞ Tr½ 5Su ðxÞ μSc ðxÞTr½ 5Sc ð xÞ νSd ð xÞ γ ˜ aa0 γ bb0 γ ˜ e0e − γ d0d − þ Tr½ νSu ðxÞ μSc ðxÞTr½ 5Sc ð xÞ 5Sd ð xÞgjψ 0i; ð8Þ

˜ T where SqðcÞ ¼ CSqðcÞC. The in-medium light and heavy quark propagators in coordinate space in the fixed point gauge and mq → 0 limit are given as

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1 ij i ij i j purpose of the QCD side of the calculations is to calculate Sq ðxÞ¼ δ =x þ χ ðxÞχ¯qð0Þ 2π2 ðx2Þ2 q these spectral densities. To this end and after insertion of the quark propagators into the correlation function, we use igs ij 1 μν μν − Fμνð0Þ ½=xσ þ σ =xþ; ð9Þ the relation 32π2 x2 Z 1 dDk and −ik·x −1 mþ12D−2mπD=2 2 ¼ e ið Þ Z  ðx Þm ð2πÞD i δ   ij 4 −ik·x ij Γ D=2 − m 1 D=2−m Sc ðxÞ¼ 4 d ke ½ − ð2πÞ =k − m × 2 ; ð14Þ c Γ½m k ij μν 0 σμν =k m =k m σμν − gsF ð Þ ð þ cÞþð þ cÞ 4 ðk2 − m2Þ2 to bring x to the exponential. In this way, the resultant   c  expressions contain four four-integrals: one over four-k π2 α 2 sGG δ k þ mc=k coming from the above relation, one initially existing in the þ 3 π ijmc 2 2 4 þ : ð10Þ ðk − mcÞ correlation function (integral over four-x), and the last two over four-k1 and four-k2 coming from the heavy quark χi χ¯j In the above equations, q and q are the Grassmann propagators. The next step is to perform the four-integral background quark fields and over x, which gives a Dirac delta function. We use this

ij function to perform an integral over k1. The remaining two μν A ij;A 1 2 … 8 F ¼ Fμνt ;A¼ ; ; ; ; ð11Þ integrals over four-k2 and four-k are performed using the A Feynman parametrization and the formula where Fμν are classical background gluon fields, and ij;A Z ij;A λ λij;A 2 m 2 m−n t ¼ 2 , with being the standard Gell-Mann matri- ðl Þ iπ ð−1Þ Γ½m þ 2Γ½n − m − 2 d4l ¼ : ces. The next step is to use the expressions of the quark ðl2 þ ΔÞn Γ½2Γ½nð−ΔÞn−m−2 propagators in Eq. (8). This leads to two main contribu- tions: perturbative and nonperturbative. The perturbative ð15Þ contributions, which represent the short distance effects, are calculated directly by applying the Fourier, Borel, and Finally, by applying the relation continuum subtraction procedures. The nonperturbative or    D 1 D=2−n ð−1Þn−1 long distance contributions are expressed in terms of the in- Γ − − −Δ n−2 −Δ 2 n Δ ¼ − 2 ! ð Þ ln½ ; ð16Þ medium quark, gluon, and mixed condensates. For the ðn Þ explicit expressions of the condensates and the details of −Δ their expansions in terms of different operators, see, for and using the expansion of ln½ , we get the imaginary ϒQCD instance, Ref. [44]. After insertion of the in-medium parts of the i functions. condensates in terms of various operators, the Fourier We apply the Borel transformation with respect to the 2 and Borel transformations as well as continuum subtraction variable p and perform the subtraction procedure accord- are also applied to the nonperturbative part of the ing to the standard prescriptions of the method. As a result, QCD side. we get Z The QCD side of the correlation function can be s0 − s QCD 2 QCD 2 decomposed over the selected Lorentz structures as ϒ ðM ;s0Þ¼ dsρ ðsÞe M ; ð17Þ i 2 i 4mc QCD QCD 2 QCD 2 Πμν −ϒ ϒ ðpÞ¼ 1 ðp Þgμν þ 2 ðp Þpμpν where s0 is the in-medium continuum threshold parameter − ϒQCD 2 − ϒQCD 2 3 ðp Þpμuν 4 ðp Þpνuμ separating the contributions of the ground state Zc and QCD 2 higher resonances and continuum. It will be fixed later. The þ ϒ ðp Þuμuν; ð12Þ 5 spectral density related to each structure is the sum of the spectral densities of the perturbative (pert), two-quark (qq), ϒQCD 1 … 5 where the invariant functions i (i ¼ ; ; )in two-gluon (gg), and mixed quark-gluon (qgq) parts: Eq. (12) are given in terms of the following dispersion integrals: ρQCD ρpert ρqq ρgg ρqgq i ðsÞ¼ i ðsÞþ i ðsÞþ i ðsÞþ i ðsÞ: ð18Þ Z ∞ ρQCD s ϒQCD 2 i ð Þ As an example, for the gμν structure, these spectral densities ðp Þ¼ 2 ds; ð13Þ i 2 − 4mc s p are collected in the Appendix. To obtain the QCD sum rules for the mass, current ρQCD where i ðsÞ are the two-point spectral densities related coupling constant, and vector self-energy of the Zc state, ΠPhe to the imaginary parts of the selected coefficients. The main the coefficients of the same structures from both the i

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ΠQCD TABLE I. Working windows for M2 and s. and i functions are equated. We get the following sum 0 rules for the physical quantities under consideration: 2 2 M ½3–5 GeV 2 2 s0 ½17.6–19.4 GeV − μ 2 2 2 QCD 2 −m f e M ¼ ϒ ðM ;s Þ; Zc Zc 1 0 2 − μ 2 2 QCD 2 f e M ¼ ϒ ðM ;s Þ; show that the OPE series of sum rules converge very nicely Zc 2 0 2 within the working windows of the auxiliary parameters. − μ 2 2 QCD 2 −Συf e M ¼ ϒ ðM ;s Þ; Zc 3 0 Prior to the investigation of the behavior of the in- 2 − μ medium physical quantities, we extract the vacuum mass 2 2 QCD 2 −Συf e M ¼ ϒ ðM ;s Þ; Zc 4 0 value for the Zc state. The in-medium sum rules in the limit 2 ρ → 0 − μ lead to the average value of vacuum mass as 2 2 2 QCD 2 103 Συf e M ¼ ϒ5 ðM ;s0Þ: ð19Þ þ Zc ð3932−84 Þ MeV. We compare this value with the exper- imental data and other theoretical predictions in vacuum in III. NUMERICAL RESULTS Table II. As is seen, our prediction for the vacuum mass of Zc obtained via in-medium sum rules in the ρ → 0 limit is In this section, the QCD sum rules presented in Eq. (19) are consistent with other presented results within the errors used to investigate the behavior of the mass, current-meson arising mainly from the choice of the auxiliary parameters coupling, and vector self-energy of the Zc stateincoldnuclear M2 and s. Note that for Ref. [52], we presented only one of matter. To this end, we need the values of some input 0 the results obtained via different scenarios. This result is the parameters and in-medium condensates entering the expres- closest to the average experimental value. sions of the obtained sum rules. The in-medium expectation Now, we proceed to display the behavior of physical values of different operators, together with some other input quantities under consideration, with respect to the Borel parameters used in the calculations, are as follows: ρsat ¼ 3 3 † 3 mass and continuum threshold parameters at saturation 0.11 GeV , p0 ¼ 3887.2 2.3 MeV [1], hq qiρ ¼ ρ, 3 3 2 nuclear matter density, ρsat ¼ 0.11 GeV . In this respect, 3 3 σπ hqq¯ i0 ¼ð−0.241Þ GeV [47], hqq¯ iρ ¼hqq¯ i0 þ N ρ [48], 2mq we plot the ratio of the in-medium mass to vacuum mass, α muþmd 0 00345 s 2 0 33 mq ¼ 2 ¼ . GeV [1], h π G i0 ¼ð. mZ =mZc , and the ratio of the in-medium current meson α α c 0 04 4 4 s 2 s 2 − 0 65 0 15 ρ coupling to its vacuum value, f =f , in Fig. 2.Asitis . Þ GeV , h π G iρ ¼hπ G i0 ð . . Þ GeV , Zc Zc † † q iD0q 0.18 GeV ρ, qiD¯ 0q 0, q iD0iD0q clearly seen, the physical quantities show elegant stability h iρ ¼ h iρ ¼ h iρN ¼ 2 1 † 2 2 0.031GeV ρ − q g σGq , qg¯ σGq m qq¯ against the changes in the parameters M and s0 in their N 12h s iρN h s i0 ¼ 0h i0 2 2 working regions. At the saturation nuclear matter density, [45], m0 ¼ 0.8 GeV [47], hqg¯ sσGqiρ ¼hqg¯ sσGqi0þ 2 † 2 † the in-medium mass of the Zc state decreases to approx- 3 GeV ρ, hq g σGqiρ ¼ −0.33 GeV ρ, hq iD0iD0qiρ ¼ s imately 77% of its vacuum mass, and the shift in the 0 031 2 ρ − 1 † σ . GeV 12 hq gs Gqiρ [45,48]. The - average current coupling value of the state due to nuclear σ 0 045 sigma term πN ¼ . GeV [49] is used. The quark masses medium is about 15%. In Fig. 3, the quantities m =m 0 49 0 48 Zc Zc m 2 16þ . m 4 67þ . 2 are taken as u ¼ . −0.26 MeV, d ¼ . −0.17 MeV, and Σν=m with respect to M at the average value of the 1 27 0 02 Zc and mc ¼ . . GeV [1]. continuum threshold and saturation nuclear matter density Besides these inputs, the sum rules in Eqs. (19) require that we fix two auxiliary parameters: the Borel mass parameter M2 and the in-medium continuum threshold s0. To proceed, we need to determine their working regions, such that the in-medium mass, current coupling, and the vector self-energy show mild variations with respect to the changes in these parameters. The upper and lower limits for M2 are determined via imposing some conditions accord- ing to the standard prescriptions of the method (for details in the case of doubly heavy , see, for instance, Refs. [50,51]). The working window for s0 is such that the maximum possible pole contribution is obtained, and the physical observables show weak dependence on the aux- iliary parameters. These requirements lead to the working windows for auxiliary parameters displayed in Table I.As seen in Fig. 1, the average pole contribution changes in the FIG. 1. The pole contribution in the Z channel as a function of 2 2 c interval [57%–23%], corresponding to M ¼½3–5 GeV , M2 at saturation nuclear matter density and different fixed values which is reasonable in the case of tetraquarks. Our analyses of the in-medium continuum threshold.

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TABLE II. Experimental result and theoretical predictions of different methods for the vacuum mass of the Zc state. PS means present study, AAD means amplitude analysis of the data, and IMQCDSR refers to the in-medium QCD sum rules.

Method mZc ρ → 0 þ103 PS IMQCDSR ( ) ð3932−84 Þ MeV [1] Experiment ð3887.2 2.3Þ MeV [13] QCDSR ð3.88 0.17Þ GeV 3 91þ0.11 [53] QCDSR ð . −0.09 Þ GeV 3 89þ0.09 [54] QCDSR ð . −0.09 Þ GeV 3893 2þ5.5 [52] AAD ð . −7.7 Þ MeV þ125 [40] QCDSR ð3901−148 Þ MeV 3 86 0 27 2 [19] QCDSR ð . . Þ GeV FIG. 3. The ratios m =m and Σν=m as functions of M at Zc Zc Zc average values of the continuum threshold and at saturation nuclear matter density. are shown. As already mentioned, the average negative shift in the modified mass of the Z state due to cold nuclear c average values of the continuum threshold and Borel matter (scalar self-energy) is about 23% of the vacuum parameter. The saturation nuclear matter mass density mass, whereas the vector self-energy Σν is obtained as is ρsat 2.7 × 1014 g=cm3, which is equivalent to approximately 32% of the vacuum value. ¼ ρsat 0 16 −3 The main objective in the present study is to investigate ¼ . fm . However, we need to know the behavior the variations of the physical quantities with respect to the of hadrons at higher densities, which will be accessible in density of the nuclear medium. To this end, we plot heavy ion collision experiments. Neutron stars as natural sat laboratories are very compact and dense and may produce in Fig. 4 m =mZ and f =fZ as functions of ρ=ρ at Zc c Zc c and even heavy baryons and exotic states based on the processes that may occur inside them. For a neutron

FIG. 2. The ratios m =m and f =f as functions of M2 at Zc Zc Zc Zc the saturated nuclear matter density ρsat ¼ 0.113 GeV3 and at FIG. 4. The ratios m =m and f =f as functions of ρ=ρsat Zc Zc Zc Zc fixed values of the continuum threshold. at mean values of the continuum threshold and Borel parameter.

074037-6 PROPERTIES OF ZCð3900Þ TETRAQUARK IN A COLD … PHYS. REV. D 101, 074037 (2020) star with mass ∼1.5M⊙, the relevant core density is sat sat approximately (2ρ − 3ρ ), and for the mass ∼2M⊙ the same density is about 5ρsat [55]. Therefore, we discuss the behavior of the mass and coupling constants up to densities comparable with the densities of the neutron stars. However, as is clear from Fig. 4, the in-medium sum rules give reliable results up to ρ=ρsat ¼ 1 and ρ=ρsat ¼ 1.1 for m =m and f =f , respectively. Hence, we need to Zc Zc Zc Zc extrapolate the results to include the higher densities. Our analyses show that the following fit functions well describe the ratios under consideration when the central values of the auxiliary and other input parameters are used:

FIG. 6. Comparison between polynomial and exponential fit −0 252 m =m ¼ e . x ð20Þ functions for f =f changes with respect to ρ=ρsat. Zc Zc Zc Zc

m 1 and Zc ¼ 2 ; ð22Þ m 1 x p Zc ½ þ pa f =f ¼ −0.251x2 þ 0.054x þ 1.028; ð21Þ Zc Zc where p ¼ 2.035 and a ¼ 3.056, and the exponential fit for sat f =f : where x ¼ ρ=ρ . From Fig. 4, we see that the fit results Zc Zc coincide with the sum rule predictions at lower densities. The results show that the mass exponentially decreases −x=0.879 f =fZ ¼ 1.169 − 0.111e : ð23Þ with respect to x and reaches roughly 30% of the vacuum Zc c mass at a density 5ρsat. The coupling constant, however, rapidly changes with respect to x and goes to zero at x ¼ 2.1. This point may be considered as a pseudocritical density, at which hadrons are melted. It is necessary to check the results at higher densities using different fit functions that may end up in different predictions. Among various fit functions, the best ones are as follows: exponential fit for the mass ratio and poly- nomial fit for the coupling constant ratio. Our analyses show that another set of fit functions, although not as good as the above selected functions, describe the behaviors of the quantities under study with respect to density as well. These are the p-pole fit for m =m : Zc Zc

FIG. 7. The ratios m =m and f =f as functions of ρ=ρsat Zc Zc Zc Zc FIG. 5. Comparison between exponential and p-pole fit func- when the uncertainties of the auxiliary and other input parameters tions for m =m changes with respect to ρ=ρsat. are considered. Zc Zc

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In Figs. 5 and 6, we compare the predictions of these fit the saturation nuclear matter density. The processes that functions with the predictions of the previous fit functions. occur at high densities may produce different kinds of In the case of the mass ratio, although both fitting results hadrons, even exotic states. Experiments that take place at show a similar behavior, the p-pole fitting leads to a result high densities will serve as a good opportunity to study the which is 6.4% larger than the exponential fitting result at exotic states like Zc. Hence, to provide some phenomeno- the density 5ρsat. In the case of the coupling constant, logical predictions, we investigated the behavior of the state f =f becomes zero at ρ=ρsat ¼ 2.13 and ρ=ρsat ¼ 2.07, under consideration with respect to the density of the Zc Zc for the polynomial and exponential fittings, respectively. medium at higher densities. The in-medium sum rules are The small differences in the predictions of different fit truncated at around ρ=ρsat ¼ð1 − 1.1Þ. Thus, we used functions remain inside the uncertainties allowed by the some fit functions in order to extrapolate the results to a method used. density around 5ρsat. Our analyses show that the mass and Now, we add the uncertainties in the values of the current coupling constant exhibit nonlinear behavior and auxiliary parameters and errors of other inputs to the decrease with respect to density. The mass reaches roughly predictions at higher densities. For the case of best fits 30% of the vacuum mass at a density 5ρsat if the central for the mass and coupling ratios, i.e., Eqs. (20) and (21), the values of the auxiliary and other input parameters, as well swept areas presented in Fig. 7 are obtained. As we can see, as the best fit, are considered. When the uncertainties of the the values of the ratio m =m vary in the region [0.26, parameters are taken into account, this value lies in the Zc Zc sat 0.39] at ρ=ρ ¼ 5. The band of the coupling ratio f =fZ interval [26%, 39%]. The current coupling, however, Zc c ≈2 1ρsat becomes zero and crosses the ρ=ρsat axis in the interval rapidly decreases and goes to zero at a density . [1.9, 2.8]. if the central values of all parameters are considered, which may be considered as a pseudocritical density for melting IV. DISCUSSION AND CONCLUSION the exotic charmoniumlike states. When the uncertainties in the values of the auxiliary and input parameters are taken Despite much experimental and theoretical effort, the into account, the band of f =f becomes zero and crosses Zc Zc structure, quark-gluon organization, and nature of most the ρ=ρsat axis in the region [1.9, 2.8]. 3900 exotic states remain unclear. The tetraquark state Zcð Þ Investigation of properties of hadrons at finite temper- is among the charmoniumlike resonances that deserve ature and density constitutes one of the main directions of further investigation in vacuum and a dense or hot medium research in high energy and . Understanding in order to fix its nature. We considered it as a compact the hadronic behavior under extreme conditions can help us tetraquark and assigned a diquark-antidiquark structure to it not only understand the internal structure and nature of PC 1þ− with quantum numbers J ¼ . We constructed in- hadrons and gain knowledge of the QCD as the theory of medium sum rules to calculate its mass, current coupling, the strong interaction, but also analyze the results of future and vector self-energy in cold nuclear matter. The result experiments as well as understand different possible phases ρ → 0 obtained for its mass in the limit is compatible with of matter. the experimental data and vacuum theory predictions. At saturation density, the mass and current coupling receive ACKNOWLEDGMENTS negative shifts with respect to the vacuum values. These shifts amount to 23% and 15% for the mass and current The authors thank TUBITAK for the partial support coupling, respectively. This state receives a repulsive vector provided under Grant No. 119F094. self-energy with an amount of 32% of its vacuum mass value at saturation nuclear matter density. APPENDIX: SPECTRAL DENSITIES The in-medium experiments aim to reach higher den- sities. The neutron stars as compact and dense objects are We present the explicit forms of the spectral densities for natural laboratories with densities 2–5 times greater than the gμν structure used in the calculations:

Z Z 1 1 1−z 1 ρpert − − 1 − 2 3 2 2 − 1 1 ðsÞ¼ 6 dz dw 8 ½wz½ðswzðw þ z Þ mc½ðw þ w ð z Þ 3072π 0 0 ξϑ 2 2 4 3 2 2 2 þ 2wðz − 1Þz þðz − 1Þz ½3mc½w þ w ð2z − 1Þþ2wðz − 1Þz þðz − 1Þz 2 4 3 2 2 2 2 2 − 26mcswz½w þ w ð3z − 2Þþw ð4z − 5z þ 1Þþwzð3z − 5z þ 2Þþðz − 1Þ z þ 35s2w2z2ðw þ z − 1Þ2Θ½Lðs; z; wÞ; ðA1Þ

074037-8 PROPERTIES OF ZCð3900Þ TETRAQUARK IN A COLD … PHYS. REV. D 101, 074037 (2020) Z Z  1 1 1−z ρqq mcw 3 4 3 2 2 − 1 2 − 1 − 1 2 2 1 ðsÞ¼ 4 dz dw 5 ½ mc½w þ w ð z Þþ wðz Þz þðz Þz 4π 0 0 4ϑ 2 4 3 2 2 2 2 2 − 10mcswz½w þ w ð3z − 2Þþw ð4z − 5z þ 1Þþwzð3z − 5z þ 2Þþðz − 1Þ z 2 m m p0w zξ 7s2w2z2 w z − 1 2 uu¯ q c 5m2 w3 w2 2z − 1 þ ð þ Þ h iρ þ ϑ5 ½ cð þ ð Þ m z 2w z − 1 z z − 1 z2 − 7swzξ u†u c 3m4 w3 w2 2z − 1 þ ð Þ þð Þ Þ h iρ þ 4ϑ5 ½ c½ þ ð Þ 2 2 2 4 3 2 2 þ 2wðz − 1Þz þðz − 1Þz − 10mcswz½w þ w ð3z − 2Þþw ð4z − 5z þ 1Þ 2 2 2 2 2 2 2 ¯ þ wzð3z − 5z þ 2Þþðz − 1Þ z þ7s w z ðw þ z − 1Þ hddiρ 2 m m p0wz ξ q c 5m2 w3 w2 2z − 1 2w z − 1 z z − 1 z2 − 7swzξ þ ϑ5 ½ cð þ ð Þþ ð Þ þð Þ Þ  † × hd diρ Θ½Lðs; z; wÞ; ðA2Þ Z Z  1 1 1−z m m ρgg − q c 2 4 6 5 24 2 − 8 1 1 ðsÞ¼ 4 dz dw i 4 ½mcð w z þ w ð z z þ Þ 96π 0 0 ϑ þ w4ð52z3 − 44z2 þ 6z − 1Þþw3zð60z3 − 80z2 þ 55z − 2Þþw2z2ð40z3 − 68z2 3 3 2 4 2 3 þ 95z − 34Þþ6wz ð2z − 4z þ 13z − 11Þþ33ðz − 1Þz Þþwzξð4p0ð4w z þ w2ð16z2 − 4z þ 1Þþ12wðz − 1Þz2 þ z2Þ − 3sð4w3z þ w2ð16z2 − 4z þ 1Þ  2 2 † 1 4 2 þ 12wðz − 1Þz þ 17z ÞÞ hu iD0uiρ þ ½m ðw − 1Þðw 1536ðw − 1Þξ2ϑ6 c þ wðz − 1Þþðz − 1ÞzÞ2ð24w8z þ 12w7ð11z2 − 8z þ 3Þþ3w6ð112z3 − 156z2 þ 45z − 36Þþ2w5ð258z4 − 516z3 þ 332z2 − 201z þ 54Þþw4ð516z5 − 1320z4 þ 1574z3 − 1198z2 þ 471z − 36Þþw3zð336z5 − 1032z4 þ 1707z3 − 1769z2 þ 858z − 132Þ þ w2z2ð132z5 − 468z4 þ 1122z3 − 1253z2 þ 807z − 180Þþ4wz3ð6z5 − 24z4 þ 152z3 2 4 3 2 2 10 − 155z þ 96z − 27Þþ8z ð23z − 29z þ 9z − 3ÞÞ − mqsðw − 1Þwzð24w z þ 12w9ð13z2 − 12z þ 3Þþw8ð492z3 − 852z2 − 89z − 180Þþw7ð984z4 − 2424z3 þ 283z2 þ 672z þ 360Þþw6ð1368z5 − 4296z4 þ 2149z3 þ 1121z2 − 270z − 360Þþw5ð1368z6 − 5160z5 þ 4057z4 − 420z3 þ 275z2 − 676z þ 180Þþw4ð984z7 − 4296z6 þ 4151z5 − 809z4 þ 1804z3 − 1901z2 þ 615z − 36Þþw3zð492z7 − 2424z6 þ 2485z5 þ 834z4 þ 43z3 − 2396z2 þ 1098z − 132Þþw2z2ð156z7 − 852z6 þ 898z5 þ 1887z4 − 2785z3 − 27z2 þ 903z − 180Þþ12wðz − 1Þ2z3ð2z5 − 8z4 þ 18z3 þ 89z2 − 12z − 9Þþ8ðz − 1Þ3z4ð23z2 þ 30z þ 3ÞÞ − 12swz2ðw þ z − 1Þ3ðw5ðð17s þ 4Þz − 4Þ þ w4ð5ð3s þ 2Þz2 − 2ð17s þ 4Þz þ 2Þþw3zðsð32z2 − 13z þ 17Þþ4ð3z2 − 4z þ 1ÞÞ þ 2w2z2ðsð16z2 − 32z − 1Þþ7z2 − 10z þ 3Þ − 8wðz − 1Þz3ð4s − z þ 1Þþ2w6  α 4 − 1 2 4 s 2 Θ þ ðz Þ z Þh π G iρ ½Lðs; z; wÞ; ðA3Þ and

Z Z 1 1−z ρqgq ρqgq ρqgq ρqgq ρqgq ρqgq Θ 1 ðsÞ¼ dz dw½ 1;1 ðsÞþ 1;2 ðsÞþ 1;3 ðsÞþ 1;4 ðsÞþ 1;5 ðsÞ ½Lðs; z; wÞ; ðA4Þ 0 0

074037-9 K. AZIZI and N. ER PHYS. REV. D 101, 074037 (2020) where  1 m ρqgq s − c m2 4w6z w5 24z2 − 8z 1 w4 52z3 − 44z2 − 2z − 1 w3z 1;1 ð Þ¼96π4 κ6 ½ cð þ ð þ Þþ ð Þþ × ð60z3 − 80z2 þ 31z þ 6Þþw2z2ð40z3 − 68z2 þ 71z − 18Þþ2wz3ð6z3 − 12z2 4 2 3 2 9 þ 35z − 29Þþ33ðz − 1Þz Þðw þ wðz − 1Þþðz − 1ÞzÞ þ wzðw þ z − 1Þð4p0ð4w z þ w8ð28z2 − 16z þ 1Þþ3w7ð28z3 − 36z2 þ 25z − 1Þþw6ð160z4 − 300z3 þ 371z2 − 169z þ 3Þþw5ð208z5 − 508z4 þ 838z3 − 694z2 þ 157z − 1Þþw4zð192z5 − 564z4 þ 1116z3 − 1276z2 þ 567z − 51Þþ2w3z2ð62z5 − 210z4 þ 463z3 − 642z2 þ 409z − 82Þ þ w2ðz − 1Þ2z3ð52z3 − 92z2 þ 239z − 164Þþ3wðz − 1Þ3z4ð4z2 − 4z þ 17Þ þðz − 1Þ3z5Þ − sð12w9z þ w8ð84z2 − 48z þ 3Þþw7ð252z3 − 324z2 þ 121z − 9Þ þ w6ð480z4 − 900z3 þ 713z2 − 195z þ 9Þþw5ð624z5 − 1524z4 þ 1730z3 − 986z2 þ 159z − 3Þþw4zð576z5 − 1692z4 þ 2500z3 − 2060z2 þ 709z − 49Þþ2w3z2ð186z5 − 630z4 þ 1125z3 − 1218z2 þ 635z − 98Þþw2ðz − 1Þ2z3ð156z3 − 276z2 þ 589z − 292Þ  3 4 2 3 5 þ wðz − 1Þ z ð36z − 36z þ 193Þþ51ðz − 1Þ z ÞÞ huiD¯ 0iD0uiρ; ðA5Þ  1 m wz2ξ ρqgq s c m2 w z w2 w z − 1 z − 1 z 3 − wzξ 1;2 ð Þ¼12π4 ϑ7 ½ cð þ Þð þ ð Þþð Þ Þ 2 4 3 2 2 2 2 × ð8p0ð3w þ w ð7z − 6Þþw ð7z − 6Þþw ð10z − 14z þ 3Þþwzð6z − 13z þ 7Þ þ 3ðz − 1Þ2z2Þ − sð5w4 þ 2w3ð6z − 5Þþw2ð17z2 − 24z þ 5Þþ2wzð5z2 − 11z  2 2 ¯ þ 6Þþ5ðz − 1Þ z ÞÞ hdiD0iD0diρ; ðA6Þ  1 m ρqgq s c wz w z − 1 2p2 4w9z w8 28z2 − 16z 1 1;3 ð Þ¼96π4 4ϑ7 ½ ð þ Þð 0ð þ ð þ Þ þ 3w7ð28z3 − 36z2 þ 25z − 1Þþw6ð160z4 − 300z3 þ 371z2 5 5 4 3 2 4 − 169mcz þ 3Þþw ð208z − 508z þ 838z − 694z þ 157z − 1Þþw z × ð192z5 − 564z4 þ 1116z3 − 1276z2 þ 567z − 51Þþ2w3z2ð62z5 − 210z4 þ 463z3 − 642z2 þ 409z − 82Þþw2ðz − 1Þ2z3ð52z3 − 92z2 þ 239z − 164Þþ3wðz − 1Þ3z4ð4z2 − 4z þ 17Þþðz − 1Þ3z5Þþsð4w9z þ w8ð28z2 − 16z − 59Þþw7ð84z3 − 108z2 − 401z þ 177Þþw6ð160z4 − 300z3 − 1197z2 þ 1259z − 177Þþw5ð208z5 − 508z4 − 2022z3 þ 3534z2 − 1271z þ 59Þ þ w4zð192z5 − 564z4 − 2204z3 þ 5368z2 − 3185z þ 425Þþ2w3z2ð62z5 − 210z4 − 751z3 þ 2380z2 − 1945z þ 464Þþw2ðz − 1Þ2z3ð52z3 − 92z2 − 861z þ 760Þ 3 4 2 3 5 2 2 þ wðz − 1Þ z ð12z − 12z − 173Þþ25ðz − 1Þ z ÞÞ − mcðw þ wðz − 1Þ þðz − 1ÞzÞ3ð4w6z þ w5ð24z2 − 8z − 35Þþw4ð52z3 − 44z2 − 170z þ 35Þþw3zð60z3 − 80z2 − 353z þ 174Þþw2z2ð40z3 − 68z2 − 301z þ 234Þþ2wz3ð6z3 − 12z2 − 37z  4 þ 43Þþ9ðz − 1Þz Þ hug¯ sσGuiρ; ðA7Þ

074037-10 PROPERTIES OF ZCð3900Þ TETRAQUARK IN A COLD … PHYS. REV. D 101, 074037 (2020)  1 m wz2ξ ρqgq s c 13m2 w z w2 w z − 1 z − 1 z 3 wzξ 1;4 ð Þ¼48π4 ϑ7 ½ cð þ Þð þ ð Þþð Þ Þ þ 2 4 3 2 2 2 × ð4p0ð3w þ w ð7z − 6Þþw ð10z − 14z þ 3Þþwzð6z − 13z þ 7Þ þ 3ðz − 1Þ2z2Þ − sð31w4 þ w3ð66z − 62Þþw2ð97z2 − 132z þ 31Þþ2wzð31z2 − 64z  2 2 ¯ þ 33Þþ31ðz − 1Þ z ÞÞ hdgsσGdiρ; ðA8Þ and  1 p0z ρqgq s m2 16w7z w6 48z2 − 48z 31 w5 80z3 − 128z2 137z − 62 1;5 ð Þ¼192π4 ϑ4 ½ cð þ ð þ Þþ ð þ Þ þ w4ð80z4 − 176z3 þ 237z2 − 164z þ 31Þþw3zð48z þ 4 − 128z3 þ 223z2 − 190z þ 59Þ þ w2z2ð16z4 − 48z3 þ 109z2 − 110z þ 33Þþwz3ð25z2 − 38z þ 13Þ  2 4 † þ 8ðz − 1Þ z Þ hu gsσGuiρ: ðA9Þ

The shorthand notations in the above equations are given as

ξ ¼ðw þ z − 1Þ; ϑ ¼ðw2 þ wðz − 1Þþðz − 1ÞzÞ; ðð−1 þ wÞð−ðswzð−1 þ w þ zÞÞ þ m2ðw3 þ 2wð−1 þ zÞz þð−1 þ zÞz2 þ w2ð−1 þ 2zÞÞÞÞ L½s; w; z¼ c : ðA10Þ ðw2 þ wð−1 þ zÞþð−1 þ zÞzÞ2

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