arX

iv:1

503.

0212

5v3

[hep

-ph]

5 S

ep 2

015

Photoproduction of the charged charmoniumlike Z+c (4200)

Xiao-Yun Wang1,2,3∗ and Xu-Rong Chen1,31Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China

2University of Chinese Academy of Sciences, Beijing 100049, China3Research Center for Hadron and CSR Physics, Institute of Modern Physics of CAS and Lanzhou University, Lanzhou 730000, China

Alexey Guskov†1Joint Institute for Nuclear Research, Dubna 141980, Russia

In this work, inspired by the observation of charmoniumlikeZ+c (4200), we study the photoproduction ofcharged charmoniumlikeZ+c (4200) with an effective Lagrangian approach and the Regge trajectories model.The numerical results indicate that the Reggeized treatment can lead to a lower total cross section of theZ+c (4200) photoproduction and the peak position of cross section was moved to the higher energy point whenthe Reggeized treatment was added. Moreover, using the datafrom the COMPASS experiment and presentedtheoretical predictions, an upper limit of the decay width of Zc(4200)→ J/ψπ is estimated. The relevant resultsnot only shed light on the further experiment of searching for the charmoniumlikeZc(4200) state via mesonphotoproduction, but also provide valuable informations for having a better comprehension of the nature ofcharmoniumlikeZc(4200) state.

PACS numbers: 13.60.Le, 11.10.Ef, 11.55.Jy, 12.40.Vv

I. INTRODUCTION

As of now, most of hadrons can be well described by theclassical constituent quark model in the picture ofqq formesons andqqq for baryons. However, according to the quan-tum chromodynamics, the exotic states (such as the multi-quark states, molecule states etc.) are also allowed to existin our Universe. Therefore, searching and explaining theseexotic states arouse great interest among researchers.

In the experiments, a series of charmoniumlike and bot-tomoniumlike states referred toXYZ have been observed [1–16]. Especially, those chargedZ states are even more exoticsince they have a minimal quark content of

∣

∣

∣ccud⟩

(Z+c ) or∣

∣

∣bbud⟩

(Z+b ) [17–21]. Later, some neutralZ states (including

Z0c (3900),Z0

b (10610) andZ0c (4020)) were reported by experi-

ments [22–24], which provide important informations of con-firming and understanding the exoticZ states.

On theoretical aspects, these exotic states are interpreted asa hadronic molecule, a tetraquark, hadrocharmonium or justacusp effect. [17–21, 25–43],et al. Moreover, several hiddencharm baryons composed by|ccqqq〉 have been predicted andinvestigated [44–47]. These studies enriched the picture ofexotic states.

Recently, Belle Collaboration claimed that a new chargedcharmoniumlikeZ+c (4200) was observed in the invariant massspectrum ofJ/ψπ+ with a significance of 6.2σ [13]. Its massand width areMZc(4200) = 4196+31+17

−29−13 MeV/c2 andΓZc(4200) =

370+70+70−70−132 MeV [13], respectively. Meanwhile, the quantum

number ofZ+c (4200) was determined to beJP = 1+ since otherhypotheses withJP ∈ {0−, 1−, 2−, 2+} were excluded [13]. InRef. [40], the calculations show thatZc(4200) is a strong can-

∗xywang@impcas.ac.cn†avg@jinr.ru

didate of the lowest axial-vector tetraquark state within theframework of the color-magnetic interaction. In Refs. [41–43], using the QCD sum rule approach, the relevant resultsalso support the tetraquark interpretation ofZc(4200). Be-sides, theZc(4200) was described as a molecule-like state in[48]. The above informations indicate that theZc(4200) is anideal candidate for investigating the nature of exotic charmo-niumlike states.

As of now, the charmoniumlikeXYZ states are only ob-served in four ways [17], i.e., thee+e− annihilation (e+e− →XYZ or e+e− → J/ψ+ XYZ), γγ fusion process (γγ→ XYZ),B meson decay (B → K + XYZ) and hidden-charm dipiondecays of higher charmonia or charmoniumlike states. There-fore, searching for the charmoniumlike states through otherproduction process is an important topic, which will be use-ful in confirming and understanding these exoticXYZ states.For example, Keet al. suggested to search for the chargedZ±c (4430) by the nucleon-antinucleon scattering [49], whilethe production of neutralZ0

c (4430) andZ0c (4200) states in ¯pp

reaction were investigated in Refs. [50, 51]. Moreover, inRefs. [52–55], the meson photoproduction process were pro-posed to be an effective way to search for the charmonium-like states. Soon after, according to the theoretical predic-tions obtained in Ref. [54], an experiment of searching fortheZ±c (3900) throughγN → Z±c (3900)N → J/ψπ±N was car-ried out by the COMPASS Collaboration [56]. Unfortunately,no signal of exclusive photoproduction of theZ±c (3900) stateand its decay intoJ/ψπ± was found. Thus it is important todiscuss whether there are other charmoniumlikes that have adiscovery potential throughγN → J/ψπ±N channel. Besides,a more accuracy theoretical prediction is necessary.

Usually, for the meson photoproduction process, themesonic Reggeized treatment will play important role at highphoto energies. The exchange of dominant meson Regge tra-jectories already used to successfully describe the meson pho-toproduction in Refs. [57–59]. Since a high photon beam en-ergy is required for the production of charmoniumlike states

2

through meson photoproduction process, the Reggeized treat-ment will be necessary to ensure the result accuracy. In thiswork, within the frame of an effective Lagrangian approachand the Regge trajectories model, we systematically study theproduction of chargedZc(4200) by meson photoproductionprocess in order to provide a reliable theoretical results andshed light on our understanding of the properties and produc-tion mechanism of chargedZc(4200) state.

This paper is organized as follows. After an introduction,we present the investigate method and formalism. The numer-ical result and discussion are given in Sec. III. In Sec. IV, wediscuss the upper limit of decay width ofZc(4200)→ J/ψπ.Finally, this paper ends with a brief conclusion.

II. FORMALISM AND INGREDIENTS

Since theZc(4200) have a strong coupling withJ/ψπ[13, 40, 43], the photoproductionprocessγp→ Z+c (4200)n→J/ψπ+n may be an ideal reaction channel of searching andstudying production of the chargedZ+c (4200). Moreover, con-sidering the signal ofZ+c (4200) are mainly from the contri-butions ofπ exchange, while the contributions fromρ anda0

exchange can be negligible1, the process as depicted in Fig.1 are regard as the source of signal ofZ+c (4200). Besides,the reactionγp → J/ψπ+n via Pomeron exchange (as shownin Fig.2) are also calculated, which is considered to be thebackground for theZ+c (4200) photoproduction. To investigate

γ

p

Z+(4200)

n

π+

p1

p2

qz

p3

J/ψ

(a)

qπ

γ

p

Z+(4200)

n

π+p1

p2

qz

p3

J/ψ

(b)

qπ

p4

p5

J/ψ

π+

FIG. 1: (Color online) The Feynman diagram forγp → Z+c (4200)nreaction (a) andγp→ J/ψπ+n reaction (b) throughπ exchange.

1 In Refs. [60–62], the results indicate that the pion exchange plays a majorrole in theγp → Xn process by analyzing the HERA data. Besides, InRefs. [63, 64], it is found that the contributions ofρ anda0 exchange intheγ∗p → Xn reaction are very small. Thus, in the present work we onlyconsider the contribution from the one pion exchange. Here,theγ∗ standfor the virtual photon.

γ

p

Pomeron

p1

p2

(a)

qP

J/ψ

n

p4

qsp5

p3

π+

γ

p

Pomeron

p1

p2

(b)

qP

J/ψ

π+

p4

qup3

p5

n

FIG. 2: (Color online) The Feynman diagram ofγp → J/ψπ+n pro-cess through the Pomeron exchange.

Z+c (4200) production, an effective Lagrangian approach andthe Regge trajectories model in terms of hadrons will be usedin the follows.

A. Feynman diagrams and effective interaction Lagrangiandensities

Fig. 1 show the basic tree level Feynman diagram for theproduction ofZ+c (4200) inγp → Z+c (4200)n → J/ψπ+n re-action via pion exchange. To gauge the contributions of thesediagrams, we need to know the effective Lagrangian densitiesfor each interaction vertex.

For the interaction vertex ofπNN, we use the effectivepseudoscalar coupling2 [68–70],

LπNN = −igπNN Nγ5~τ · ~πN (1)

whereN andπ stand for the fields of nucleon and pion meson,while~τ is the Pauli matrix. The coupling constant of theπNNinteraction was given in many theoretical works, and we takeg2πNN/4π = 14.4 [71].As mentioned above, the spin-parity ofZ+c (4200) has been

determined by Belle Collaboration to beJP = 1+ [13]. Thusthe relevant effective Lagrangian for the vertex3 of Zψπ read

2 It should be noted that some works [65, 66] have pointed out that the sim-ple pseudoscalar coupling between nucleons and pions is incomplete andinconsistent with chiral symmetry. Thus the pseudovector coupling is sug-gested in Refs. [65, 66]. However, since the new pseudovector formalismmay not yet be ready for phenomenological use [67], the pseudoscalar cou-pling is adopted in the present work.

3 For the sake of simplicity, we useZ andψ denoteZc(4200) andJ/ψ, re-spectively.

3

as [52],

LZψπ =gZψπ

MZ(∂µψν∂µπZν − ∂µψν∂νπZµ), (2)

whereZ, andψ denote the fields ofZ(4200) andJ/ψ meson,respectively. With the effective Lagrangians above, the cou-pling constantgZψπ can be determined by the partial decaywidthsΓZc(4200)→J/ψπ,

ΓZ(4200)→J/ψπ =

(

gZψπ

MZ

)2 |~p c.m.π |

24πM2Z

×

(M2Z − m2

ψ − m2π)

2

2+ m2

ψE2π

, (3)

with

|~p c.m.π | =

λ1/2(M2Z ,m

2ψ,m

2π)

2MZ, (4)

Eπ =

√

|~p c.m.π |2 + m2

π, (5)

whereλ is the Kallen function withλ(x, y, z) = (x−y−z)2−4yz.As of now, no relevant experiment datas aboutΓZ(4200)→J/ψπ

can be available [72]. However, in Ref. [43], the authors ob-tained the partial decay widthΓZc(4200)→J/ψπ = 87.3 ± 47.1MeV with the QCD sum rule approach, which allow us to esti-mate the lower (upper) limit of the decay width ofZ(4200)→J/ψπ, ΓZ(4200)→J/ψπ = 40.2(134.4) MeV. With MZ = 4196MeV/c2 and ΓZ = 370 MeV [72], we getgZψπ/Mz =

1.174, 1.731, 2.147 MeV, which correspond to three typicalpartial decay widthΓZ(4200)→J/ψπ = 40.2, 87.3, 134.4 MeV, re-spectively.

For the interaction vertex ofZγπ, we need to derive it by thevector meson dominance (VMD) mechanism [73–75]. In theVMD mechanism for photoproduction, a real photon can fluc-tuate into a virtual vector meson, which subsequently scattersoff the target proton. Thus within the frame of VMD mecha-nism, we get the Lagrangian of depicting the coupling of theintermediate vector mesonJ/ψ with a photon as follows,

LJ/ψγ = −em2

ψ

fψVµAµ, (6)

wherem2ψ and fψ are the mass and the decay constant ofJ/ψ

meson, respectively. With the above equation, one gets theexpression for theJ/ψ→ e+e− decay,

ΓJ/ψ→e+e− =

(

efψ

)2 8α∣

∣

∣~p c.m.e

∣

∣

∣

3

3m2ψ

, (7)

where~p c.m.e indicate the three-momentum of an electron in the

rest frame of theJ/ψ meson, whileα = e2/4~c = 1/137 is theelectromagnetic fine structure constant. Thus, in the lightofthe partial decay width ofJ/ψ→ e+e− [72]

ΓJ/ψ→e+e− ≃ 5.547 keV, (8)

we get the constante/ fψ ≃ 0.027.In Fig. 2, we present the Feynman diagram for the

γp → J/ψπ+n process through Pomeron exchange, whichis considered as the main background contributions toγp →Z+c (4200)n → J/ψπ+n process. To depict the Pomeron ex-change process, the relevant formulas which were used inRefs. [52, 76, 77] are adopted in this work. The Pomeron-nucleon coupling is described as follow,

Fµ(t) =3β0(4m2

N − 2.8t)

(4m2N − t)(1− t/0.7)2

γµ = F(t)γµ, (9)

wheret = q2P is the exchanged Pomeron momentum squared.

β20 = 4 GeV2 stands for the coupling constant between a single

Pomeron and a light constituent quark.For the vertex ofγψP, with an on-shell approximation for

keeping the gauge invariance, we have

VγψP =2βc × 4µ2

0

(m2ψ − t)(2µ2

0 + m2ψ − t)

Tµρνǫνψǫ

µγPρ, (10)

with

T µρν = (p1 + p4)ρgµν − 2pν1gρµ

+2{

pµ1gρν +pν4p2

4

(p1 · p4gρµ − pρ1pµ4 − pµ1pρ4)

−p2

1pµ4p2

4p1 · p4(p2

4gρν − pρ4pν4)}

+ (p1 − p4)ρgµν,(11)

whereβ2c = 0.8 GeV2 is the effective coupling constant be-

tween a Pomeron and a charm quark withinJ/ψ meson, whileµ0 = 1.2 GeV2 denotes a cutoff parameter in the form factorof Pomeron.

B. Cross sections for the γp→ Z+c (4200)n reaction

After the above preparations, the invariant scattering ampli-tudeA for theγ(p1)p(p2) → Z+c (4200)(qz)n(p3) reaction byexchanging aπ meson read as,

A = (√

2gπNNgZψπ

MZ

efψ

)u(p3)γ5u(p2)ǫ∗µZ

×ǫνγ[

p1 · (qz − p1)gµν − p1µ(qz − p1)ν]

× 1q2π − m2

π

FπNN (q2π)FZψπ(q

2π) (12)

whereFπNN (q2π) andFZψπ(q2

π) are the form factors for the ver-tices ofπNN andZψπ, respectively. We have the followingdefinitions for both form factors,

FπNN (q2π) =

Λ2π − m2

π

Λ2π − q2

π

, (13)

and

FZψπ(q2π) =

m2ψ − m2

π

m2ψ − q2

π

, (14)

4

whereΛπ is the cutoff parameter for theπNN vertex. In thenext calculations, we take the typical value ofΛπ = 0.7 GeVas used in Refs. [52, 54, 58, 78].

As mentioned above, a higher photon beam energy is re-quired for the production of charmoniumlike states throughmeson photoproduction process. Thus, to better describe thephotoproduction ofZ+c (4200) at high photon energies we in-troduce a pion Reggeized treatment by replacing the Feynmanpropagator 1

q2π−m2

πwith the Regge propagator as follows [57–

59, 79],

1q2π − m2

π

→ Rπ = (s

sscale)απ(t)

πα′πΓ[1 + απ(t)]

e−iπαπ(t)

sin[παπ(t)], (15)

whereα′π is the slope of the trajectory and the scale factorsscale

is fixed at 1 GeV2, while s = (p1 + p2)2 andt = (p2 + p3)2

are the Mandelstam variables. In addition, the pionic Reggetrajectoryαπ(t) read as [58, 59, 79]

απ(t) = 0.7(t − m2π). (16)

The unpolarized differential cross section for theZ+c (4200)photoproduction shown in Fig. 1(a) then reads

dσd cosθ

=1

32πs

∣

∣

∣~q c.m.z

∣

∣

∣

∣

∣

∣~p c.m.1

∣

∣

∣

14

∑

spins

|A|2

, (17)

where~p c.m.1 and~q c.m.

z are the three-momentum of initial pho-ton and finalZ+c (4200) state, whileθ denotes the angle of theoutgoingZ+c (4200) state relative to the photon beam directionin the c.m. frame. The total cross section can be easily ob-tained by integrating the above equation.

In Fig. 3, the total cross sectionσ(γp → Z+c n) throughπ meson or pionic Regge trajectory exchange are presented

(b)

W (GeV)

meson exchange Pionic Regge trajectory

FIG. 3: (Color online) The total cross section forγp → Z+c (4200)nprocess throughπ meson or pionic Regge trajectory exchange. Here,the numerical results (the blue solid line and the red dashedline)correspond to the partial decay widthΓZc(4200)→J/ψπ = 87.3 MeV,while the bands stand for the uncertainties with the variation ofΓZc(4200)→J/ψπ from 40.2 to 131.4 MeV.

ddc

os(

) (b/

sr)

cos ( )

meson exchange Pionic Regge trajectory

7.5 GeV

9.0 GeV

ddc

os(

) (b/

sr)

cos ( )

meson exchange Pionic Regge trajectory

FIG. 4: (Color online) The differential cross section forγp →Z+c (4200)n process throughπ meson or pionic Regge trajectory ex-change. The notation of the lines and bands as in Fig. 3.

with Λπ = 0.7 GeV. Since the total cross section is propor-tional to the partial decay widthΓZ(4200)→J/ψπ, thus we notethat the cross section changes by a factor of 3 to 4 when thepartial widthΓZ(4200)→J/ψπ is varied from 40.2 to 134.4 MeV.Besides, it is found that the total cross section through theReggeized treatment is about five times smaller than that of re-sult through aπ exchange, which indicate that the Reggeizedtreatment can lead to a lower cross section of theZ+c (4200)photoproduction at high photon energies. Moreover, we notethat the peak position of total cross section was moved to thehigher energy point when the Reggeized treatment is used inthe calculations.

Fig. 4 are the differential cross section for theγp → Z+c nprocess by exchanging theπ meson or pionic Regge trajec-tory at different energies, respectively. From Fig. 4 one cansee that, relative to the results related to theπ exchange, thedifferential cross section by exchanging the pionic Regge tra-jectory are very sensitive to theθ angle and gives a consider-able contribution at forward angles.

5

C. Cross sections for the γp→ J/ψπ+n reaction

With the Feynman rules and above Lagrangian densities,we obtain the invariant scattering amplitudeMsignal

Z for theγp → J/ψπ+n process throughπ exchange (as depicted inFig. 1(b)) as follows,

MsignalZ = (

√2gπNN

gZψπ

MZ

efψ

)u(p3)γ5u(p2)

×(p1 · qπgµα − pα1qµπ)(p4 · p5gρν − pρ4pν5)

× 1q2π − m2

π

GραZ (qz)ǫγµǫ∗ψν

(

Λ2π − m2

π

Λ2π − q2

π

)

×

m2ψ − m2

π

m2ψ − q2

π

m2ψ − M2

Z

m2ψ − q2

z

, (18)

where GµαZ are the propagators of theZ(4200), taking the

Breit-Wigner form [80],

GραZ (q) =

−gρα + qzρqzα/M2Z

q2z − M2

Z + iMZΓZ. (19)

Just as above practice, by replacing the Feynman propagator1

q2π−m2

πwith the Regge propagatorRπ, we can get the scattering

amplitude for theγp → J/ψπ+n process through the pionicRegge trajectory exchange.

Since the Pomeron can mediate the long-range interactionbetween a confined quark and a nucleon, thusγp → J/ψπ+nvia the Pomeron exchange (as described in Fig. 2) are themainly background contribution to theγp → Z+c (4200)n →J/ψπ+n reaction. The invariant scattering amplitudesMs

P andMu

P for Figs. 2(a) and 2(b) can be written, respectively, as

MsP = 8

√2βcµ

20gπNN FN(q2

s)F(t)GP(s, t)

(m2ψ − t)(2µ2

0 + m2ψ − t)

×T µρνǫ∗ψνǫγµu(p3)γ5/qs + mN

q2s − m2

N

γρu(p2), (20)

MuP = 8

√2βcµ

20gπNN FN(q2

u)F(t)GP(s, t)

(m2ψ − t)(2µ2

0 + m2ψ − t)

×T µρνǫ∗ψνǫγµu(p3)γρ/qu + mN

q2u − m2

N

γ5u(p2) (21)

with

GP(s, t) = −i(η′s)η(t)−1 (22)

whereη(t) = 1 + ǫ + η′t is the Pomeron trajectory. Here, theconcrete valuesǫ = 0.08 andη′ = 0.25 GeV−2 are adopted.

Considering the size of the hadrons, the monopole form fac-tor for the off-shell intermediate nucleon is introduced as inthe Bonn potential model [81]:

FN(q2i ) =

Λ2N − m2

N

Λ2N − q2

i

, i = s, u (23)

whereΛN andqi(qs = p3+ p5, qu = p2− p5) are the cut-off pa-rameter and four-momentum of the intermediate nucleon, re-spectively. For the value ofΛN , we will discuss it in the nextsection. It is worth mentioning that the form factor is phe-nomenological and has a great uncertainty. Thus the dipoleform factor is deserved to be discussed and compared withthe monopole form.

Combining the signal terms and background amplitudes,we get the total invariant amplitude

M =MsignalZ +Ms

P +MuP. (24)

Thus the total cross section of theγp → J/ψπ+n reactioncould be obtained by integrating the invariant amplitudes inthe three body phase space,

dσ(γp → J/ψπ+n) =m2

N

|p1 · p2|

14

∑

spins

|M|2

×(2π)4dΦ3(p1 + p2; p3, p4, p5), (25)

where the three-body phase space is defined as [72]

dΦ3(p1+ p2; p3, p4, p5) = δ4

p1 + p2 −5

∑

i=3

pi

5∏

i=3

d3pi

(2π)32Ei.

(26)

III. NUMERICAL RESULTS AND DISCUSSION

With the FOWL code in the CERN program library, thetotal cross section including both signal and background con-tributions can be calculated. In these calculations, the cutoffparametersΛN related to the Pomeron term is a free param-eter. Thus we first need to give a constraint on the value ofΛN . Fig. 5 (a) and Fig. 5 (b) present the variation of crosssection from the background contributions forγp → J/ψπ+nwith monopole and dipole form factor, respectively. It is obvi-ous that the Pomeron exchange contributions with dipole formfactor are more sensitive to the values of the cutoff ΛN thanthat of monopole form factor. Thus monopole form factor isadopted in the following calculation.

At present, no experiment data is available for theγp →J/ψπ+n process. However, we notice that the similar reac-tion γp → J/ψp and pp → J/ψπ0 have been measured bysome experiment [82–85], where the measured cross sectionsof these two process are about 1 nb and 10 nb, respectively.Here, we naively think that the cross section ofγp→ J/ψπ+nmay be equal to or less than that ofγp → J/ψp, while itgreater than that of ¯pp → J/ψπ0. Thus we constrain the cut-off to beΛN = 0.96 GeV as used in Ref. [54, 55], which willbe used in our calculations.

To better understand that the effects of Reggeized treatmenton the final results, we calculate the total cross section of theγp → J/ψπ+n reaction without or with Reggeized treatmentas presented in Fig. 6 and Fig. 7, respectively.

Fig. 6 show the total cross sections forγp→ J/ψπ+n reac-tion including bothπ exchange and Pomeron exchange con-tributions by takingΛZ = 0.7 GeV andΛN = 0.96 GeV. We

6

5 10 15 20 2510-6

10-5

10-4

10-3

10-2

10-1(b)

W (GeV)

N=0.95 GeV N=0.96 GeV N=0.97 GeV N=1.0 GeV

5 10 15 20 2510-7

10-6

10-5

10-4

10-3

10-2 (b)

(b)

W (GeV)

FIG. 5: (Color online) (a): The cross section of background from thePomeron exchange forγp → J/ψπ+n process with monopole formfactor at the different values of the cutoff parameterΛN . (b) is sameas the (a), but for the case of dipole form factor.

notice that the line shape of total cross section goes up veryrapidly and has a peak aroundW ≃ 7.5 GeV. In this energyregion, the cross section of signal are larger than that of back-ground when taking the partial width valuesΓZc(4200)→J/ψπ =

87.3 or 134.4 MeV.In contrast, Fig. 7 present the total cross sections for

γp → J/ψπ+n reaction including both pionic Regge trajec-tory exchange and Pomeron exchange contributions by takingΛZ = 0.7 GeV andΛN = 0.96 GeV. It is found that the totalcross section shows a peak at center of mass energyW ≃ 9GeV, and the contributions from signal are driven down whenusing the Reggeized treatment. We note that the cross sectionof signal just a little bit higher than that of background at cen-ter of mass energyW ≃ 9 GeV even a larger partial decaywidth value (ΓZc(4200)→J/ψπ = 134.4 MeV) are adopted.

To demonstrate the feasibility of searching for the chargedcharmoniumlikeZ+c (4200) through theγp → J/ψπ+n reac-tion, we further give the Dalitz plot and invariant mass spec-trum for theγp→ J/ψπ+n process.

Fig. 8 present the Dalitz plot andJ/ψπ+ invariant mass

spectrum for theγp → J/ψπ+n process with the Reggeizedtreatment at different center of mass energy, where the nu-merical results are obtained by taking the partial decay widthΓZc(4200)→J/ψπ = 134.4 MeV. From Dalitz plot we notice thatthere exist a vertical band and a horizontal band, which arefrom the signal and background contributions, respectively.Moreover, one notice that the signal ofZ+c (4200) withW = 9.0is more explicit than that withW = 7.5 or 12 GeV, which isconsistent with the result in Fig. 7. Thus we can conclude thattheW = 9.0 GeV is the best energy window for searching forthe chargedZ+c (4200) via theγp → J/ψπ+n process. By ana-lyzing theJ/ψπ+ invariant mass spectrum in Fig. 8, one findsthat the number of events ofJ/ψπ+ can reach up to 500/2GeV2 at W = 9.0 GeV when taking 50 million collisions ofγp.

Moreover, we take the center of mass energyW = 9.0 GeVas one of the inputs to calculate the Dalitz plot andJ/ψπ+

invariant mass spectrum related to the smaller partial decaywidth, which are presented in Fig. 9. From Dalitz plot in Fig.9 we notice that there exist an clear vertical band which re-lated to theZ+c (4200) signal when taking partial decay widthΓZc(4200)→J/ψπ = 87.3 MeV. Since the signal and backgroundcontribution do not interfere with each other as shown inDalitz plot, the signal ofZ+c (4200) can also be distinguishedfrom the background. Thus we can expect about 375/2 GeV2

events for the production ofJ/ψπ+ in 50 million collisions ofγp atW = 9.0 GeV if takingΓZc(4200)→J/ψπ = 87.3 MeV, whichis enough to meet the requirements of the experiment. How-ever, we also see that the signal ofZ+c (4200) are submerged inthe background and will be difficult to distinguish it from thebackground if takingΓZc(4200)→J/ψπ = 40.2 MeV.

For comparison, we calculate the Dalitz plot andJ/ψπ+ in-variant mass spectrum for theγp → J/ψπ+n process with-

5 10 15 20 250.00

0.02

0.04

0.06

0.08

(b)

W (GeV)

Total ( i=134.4 MeV) ( i=134.4 MeV) Total ( i=87.3 MeV) ( i=87.3 MeV) Total ( i=40.2 MeV) ( i=40.2 MeV) Pomeron

FIG. 6: (Color online) The energy dependence of the total crosssections for theγp → J/ψπ+n reaction. Here,σPomeron and σπ

denote the results via the Pomeron andπ exchange, respectively,while σT otal is the total cross section ofγp → J/ψπ+n. The vari-ation ofσπ andσT otal to W with several typical partial width valuesΓZ(4200)→J/ψπ = 40.2,87.3, 134.4 MeV are also presented.

7

5 10 15 20 250.00

0.01

0.02

0.03(b)

W (GeV)

Total ( i=134.4 MeV) Regge ( i=134.4 MeV) Total ( i=87.3 MeV) Regge ( i=87.3 MeV) Total ( i=40.2 MeV) Regge ( i=40.2 MeV) Pomeron

FIG. 7: (Color online) The energy dependence of the total cross sec-tions for theγp → J/ψπ+n reaction. Here,σPomeron andσRegge de-note the results via the Pomeron exchange and pionic Regge trajec-tory exchange, respectively, whileσT otal is the total cross section ofγp → J/ψπ+n. The variation ofσRegge andσT otal to W with severaltypical partial width valuesΓZ(4200)→J/ψπ = 40.2, 87.3, 134.4 MeV arealso presented.

out the Reggeized treatment atW = 7.5 GeV, as presented inFig. 10. One finds that a vertical band related to the signalof Z+c (4200) appears in Dalitz plot even if the lowest partialdecay width (ΓZc(4200)→J/ψπ = 40.2 MeV) is adopted, which isobvious different from that with Reggeized treatment.

IV. UPPER LIMIT OF THE DECAY WIDTH ΓZc(4200)→J/ψπ

The J/ψπ± mass spectrum presented by the COMPASScollaboration in [56], which studied exclusive photoproduc-tion of a J/ψπ± state at a nuclear target in the range from 7GeV to 19 GeV in the centre-of-mass energy of the photon-nucleon system, does not exhibit any statistically significantstructure at about 4.2 GeV. Nevertheless it can be used forestimation of an upper limit for the valueBR(Zc(4200) →J/ψπ) × σγN→Zc(4200)N .

A sum of two exponential functions for a continuum anda Breit-Wigner curve for a possible contribution ofZ±c (4200)photoproduction was fitted to the mass spectrum in the rangefrom 3.4 GeV to 6.0 GeV. The massMZc(4200) = 4196 MeVand the widthΓZc(4200) = 370 MeV were used as the fixedparameters. Doing this we ignore possible contribution ofany other resonances likeZc(3900) and their interference withZc(4200). TheJ/ψπ± mass spectrum with the fitting curve isshown in Fig. 11. The obtained from the fit possible numberof Zc(4200) events isNZc(4200)= 58±31. It can be converted tothe upper limitNUL

Zc(4200) < 98 events corresponding to a con-fidence level of CL= 90%. According to the normalizationused in [56] this limit corresponds to the result

BR(Zc(4200)→ J/ψπ) × σγN→Zc(4200)N < 340 pb. (27)

This result can be used for estimation of an upper limit forthe partial widthΓJ/ψπ of the decayZc(4200)→ J/ψπ basedon the Reggeized treatment. The production cross section,averaged over theW-range covered by COMPASS, is aboutΓJ/ψπ × 91 pb/MeV. So

ΓJ/ψπ

Γtot× σγN→Z±c (4200) N =

Γ2J/ψπ × 90 pb/MeV

Γtot< 340 pb.

(28)AssumingΓtot = 370 MeV, we obtain an upper limitΓJ/ψπ <37 MeV.

Photoproduction of theZ+c (4200) state could also be testedusing the data on the HERMES experiment. It covers therange 2 GeV< W < 6.3 GeV [86] where the differencebetween production cross sections calculated through pionicRegge trajectory exchange and virtual pion exchange is max-imal.

V. SUMMARY

In this work, we study the chargedZc(4200) production inγp→ J/ψπ+n reaction with an effective Lagrangian approachand the Regge trajectories model. Since the charmoniumlikeZc(4200) was only observed inB meson decay process, it is aninteresting and important topic to study the charmoniumlikeZc(4200) by different processes.

Through of analysis and comparison, our numerical resultsindicate:

(I) The effect of introducing the Reggeized treatment hasbeen to significantly reduce the magnitude of crosssection for theZc(4200) photoproduction. The to-tal cross section for theγp → Z+c (4200)n via pionicRegge trajectory exchange is smaller than that of with-out Reggeized treatment and the predictions in Refs.[52–55].

(II) We finds that the differential cross section for theγp→Z+c (4200)n by exchanging the pionic Regge trajectoryare very sensitive to theθ angle and gives a considerablecontribution at forward angles, which can be checkedby further experiment and may be an effective way toexamine the validity of the Reggeized treatment.

(III) The total cross section for theγp → J/ψπ+n processwith Reggeized treatment is lower than that of with-out Reggeized treatment. The calculations indicate thatthe partial decay widthΓZc(4200)→J/ψπ is a key parame-ter in studying the production ofZc(4200) viaγp col-lision. Adopting the partial decay width predicted inRef. [43] by assuming that theZc(4200) is a tetraquarkstate, we find that the signal ofZ+c (4200) can also bedistinguished from the background atW = 9.0 GeVif taking ΓZc(4200)→J/ψπ = 87.3 MeV, but not for thecase of takingΓZc(4200)→J/ψπ = 40.2 MeV. In Ref. [48],by assuming theZc(4200) as an axial-vector molecule-like state, the partial decay widthΓZc(4200)→J/ψπ = 24.6MeV was obtained with QCD sum rule. If the predicted

8

FIG. 8: (Color online) The Dalitz plot (top) and theJ/ψπ+ invariant mass spectrum (bottom) for theγp→ J/ψπ+n reaction with the Reggeizedtreatment at different center of mass energyW = 7.5,9, 12 GeV. Here, the numerical result corresponds to the partial decay widthΓZ(4200)→J/ψπ =

134.4 MeV.

ΓZc(4200)→J/ψπ = 24.6 MeV in Ref. [48] is reliable,then the signal ofZ+c (4200) produced inγp collisionwill be difficult to be distinguished from background.Thus the experiment of the meson photoproduction ofZc(4200) may provide a useful adjunctive informationfor the confirmation of the inner structure ofZc(4200).

(IV) The peak position of total cross section for theγp →Z+c (4200)n→ J/ψπ+n process was moved to the higherenergy point when adding the Reggeized treatment,which means that a higher beam energy is necessaryfor the meson photoproduction ofZc(4200). The resultsshow thatW ≃ 9.0 GeV is the best energy window forsearching for theZc(4200) viaγp collision. All thesecalculations can be checked in the future experiment.

(V) Using data on exclusive photoproduction of aJ/ψπ±

state from COMPASS we estimated the upper limit forthe value ofZc(4200) production cross section multi-plied by the branching ratio of theZc(4200)→ J/ψπdecay to be above 340 pb (CL=90 %). According to theReggeized treatment it corresponds to the upper limitof ΓZc(4200)→J/ψπ of about 37 MeV, which is coincidencewith the prediction ofΓZc(4200)→J/ψπ = 24.6 MeV by as-suming theZc(4200) as a molecule-like state in [48].

Since the Reggeized treatment used in this work has beenproven to be more precise than the general effective La-grangian approach in the pion and kaon photoproduction [57–59], our theoretical results may provide a valuable informa-tion, both for searching for theZc(4200) viaγp collision orexplaining the lack of observation ofZc(4200) in experiment.Therefore, the more experiment about the photoproduction ofZc(4200) are suggested, which will be important to improveour knowledge of the nature ofZc(4200) and the Regge the-ory.

VI. ACKNOWLEDGMENTS

The authors would like to acknowledge the COMPASS col-laboration for allowing us to use the data ofJ/ψπ± mass spec-trum. Meanwhile, X. Y. W. is grateful Dr. Qing-Yong Linfor the valuable discussions and help. This work is partlysupported by the National Basic Research Program (973 Pro-gram Grant No. 2014CB845406), the National Natural Sci-ence Foundation of China (Grant No. 11175220) and the OneHundred Person Project of the Chinese Academy of Science(Grant No. Y101020BR0).

9

FIG. 9: (Color online) The Dalitz plot (top) and theJ/ψπ+ invariant mass spectrum (bottom) for theγp→ J/ψπ+n reaction with the Reggeizedtreatment at center of mass energyW = 9 GeV. Here, the numerical results correspond to the partialdecay width valuesΓZ(4200)→J/ψπ =

87.3, 40.2 MeV.

[1] S. K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 91,262001 (2003).

[2] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 110,222001 (2013).

[3] B. Aubertet al. (BARBAR Collaboration), Phys. Rev. Lett. 95,142001 (2005).

[4] C. Z. Yuan et al. (Belle Collaboration), Phys. Rev. Lett. 99,182004 (2007).

[5] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 110,252001 (2013).

[6] Z. Q. Liu et al. (Belle Collaboration), Phys. Rev. Lett. 110,252002 (2013).

[7] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 112,132001 (2014).

[8] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 111,242001 (2013).

[9] R. Mizuk et al. (Belle Collaboration), Phys. Rev. D 78, 072004

(2008).[10] S. K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 100,

142001 (2008).[11] K. Chilikin et al. (Belle Collaboration), Phys. Rev. D 88,

074026 (2013).[12] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 112,

222002 (2014).[13] K. Chilikin et al. (Belle Collaboration), Phys. Rev. D 90,

112009 (2014).[14] A. Bondar et al. (Belle Collaboration), Phys. Rev. Lett. 108,

122001 (2012).[15] I. Adachi et al. (Belle Collaboration), Phys. Rev. Lett. 108,

032001 (2012).[16] I. Adachiet al. (Belle Collaboration), arXiv:1105.4583.[17] X. Liu, Chin. Sci. Bull. 59, 3815 (2014).[18] A. Esposito, A. L. Guerrieri, F. Piccinini, A. Pilloni and A. D.

Polosa, Int. J. Mod. Phys. A 30, 1530002 (2014).

10

FIG. 10: (Color online) The Dalitz plot (top) and theJ/ψπ+ invariant mass spectrum (bottom) for theγp → J/ψπ+n reaction withoutthe Reggeized treatment at center of mass energyW = 7.5 GeV. Here, the numerical results correspond to the partialdecay width valuesΓZ(4200)→J/ψπ = 134.4, 87.3, 40.2 MeV.

[19] S. L. Olsen, Hyperfine Interact. 229, 7 (2014).[20] M. Nielsen, F. S. Navarra and S. H. Lee, Phys. Rept. 497, 41

(2010).[21] M. Nielsen and F. S. Navarra, Mod. Phys. Lett. A 29, 1430005

(2014).[22] T. Xiao et al., Phys. Lett. B 727, 366 (2013).[23] P. Krokovny et al. (Belle Collaboration), Phys. Rev. D 88,

052016 (2013).[24] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 113,

212002 (2014).[25] S. L. Zhu, Phys. Lett. B 625, 212 (2005).[26] Y.-R. Liu et al., Phys. Rev. D 82, 014011 (2010).[27] N. Li and S.-L. Zhu, Phys. Rev. D 86, 074022 (2012).[28] H. Hogaasen, J. M. Richard, and P. Sorba, Phys. Rev. D 73,

054013 (2006).[29] D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Lett. B 634,

214 (2006).[30] N. Barnea, J. Vijande, and A. Valcarce, Phys. Rev. D 73,

054004 (2006).[31] Q. Wang, C. Hanhart, and Q. Zhao, Phys. Rev. Lett. 111,

132003 (2013).[32] E. Braaten, Phys. Rev. Lett. 111, 162003 (2013).[33] D. Y. Chen, X. Liu, and T. Matsuki, Phys. Rev. D 88, 036008

(2013).[34] C. F. Qiao and L. Tang, Eur. Phys. J. C 74, 2810 (2014).[35] F. Aceti, M. Bayar, and E. Oset, Eur. Phys. J. A 50, 103 (2014).

[36] I. V. Danilkin, V. D. Orlovsky and Yu. A. Simonov, Phys. Rev.D 85, 034012 (2012).

[37] D. Bugg, Europhys. Lett. 96, 11002 (2011).[38] R. D. Matheus,et al., Phys. Rev. D 75, 014005 (2007).[39] L. Maiani et al., Phys. Rev. D 89, 114010 (2014).[40] L. Zhao, W. Z. Deng and S. L. Zhu, Phys. Rev. D 90, 094031

(2014).[41] W. Chen and S. L. Zhu, Phys. Rev. D 83, 034010 (2011).[42] W. Chen and S. L. Zhu, EPJ Web Conf. 20, 01003 (2012).[43] W. Chenet al., arXiv:1501.03863 [hep-ph].[44] J. J. Wu, R. Molina, E. Oset and B. S. Zou, Phys. Rev. Lett.105,

232001 (2010).[45] J. J. Wu, R. Molina, E. Oset and B. S. Zou, Phys. Rev. C 84,

015202 (2011).[46] X. Y. Wang and X. R. Chen, Europhys. Lett. 109, 41001 (2015).[47] X. Y. Wang and X. R. Chen, Eur. Phys. J. A 51, 85 (2015).[48] Z. G. Wang, arXiv:1502.01459.[49] H. W. Ke and X. Liu, Eur. Phys. J. C 58, 217 (2008).[50] X. Y. Wang, J. J. Xie and X. R. Chen, Phys. Rev. D 91, 014032

(2015).[51] X. Y. Wang, X. R. Chen, Adv. High Energy Phys. 2015, 918231

(2015).[52] X. H. Liu, Q. Zhao, and F. E. Close, Phys. Rev. D 77, 0944005

(2008).[53] J. He and X. Liu, Phys. Rev. D 80, 114007 (2009).[54] Q. Y. Lin et al., Phys. Rev. D 88, 114009 (2013).

11

, GeVπψJ/M3 3.5 4 4.5 5 5.5 60

5

10

15

20

25

30

35

40

45

COMPASS data

FIG. 11: (Color online) Mass spectrum of theJ/ψπ state obtainedby COMPASS [56]. The fitted function is shown as a red solid line.Dashed blue line corresponds to the continuum description.

[55] Q. Y. Lin et al., Phys. Rev. D 89, 034016 (2014).[56] C. Adolphet al. (COMPASS Collaboration), Phys. Lett. B 742,

330 (2015).[57] M. Guidal, J. M. Laget, and M. Vanderhaeghen, Nucl. Phys. A

627, 645 (1997).[58] G. Galata, Phys. Rev. C 83, 065203 (2011).[59] J. He, Phys. Rev. C 89, 055204 (2014).[60] A. B. Kaidalovet al., Eur. Phys. J. C 47, 385 (2006).[61] V. A. Khoze, A. D. Martin and M. G. Ryskin, Eur. Phys. J. C

48, 797 (2006).[62] S. Chekanov,et al. (ZEUS Collaboration), Nucl. Phys. B 637,

3 (2002).[63] B. Z. Kopeliovich, B. Povh and I. Potashnikova, Z. Phys.C 73,

125 (1996).[64] H. Holtmann,et al., Phys. Lett. B 338, 393 (1995).[65] M. Burkardtet al., Phys. Rev. D 87, 056009 (2013).[66] Y. Salamuet al., Phys. Rev. Lett. 114, 122001 (2015).[67] F. Carvalhoet al., arXiv:1507.07758.[68] K. Tsushima, S. W. Huang, and A. Faessler, Phys. Lett. B 337,

245 (1994).[69] K. Tsushima,et al., Phys. Lett. B 411, 9 (1997), Erratum-ibid.

Phys. Lett. B 421, 413 (1998).[70] K. Tsushima,et al., Phys. Rev. C 59, 369 (1999), Erratum-ibid.

Phys. Rev. C 61, 029903 (2000).[71] Z. Lin, C. M. Ko, and B. Zhang, Phys. Rev. C 61, 024904

(2000).[72] K. A. Olive et al. (Particle Data Group), Chin. Phys. C, 38,

090001 (2014).[73] T. Bauer and D. R. Yennie, Phys. Lett. 60B, 165 (1976).[74] T. Bauer and D. R. Yennie, Phys. Lett. 60B, 169 (1976).[75] T. H. Bauer,et al., Rev. Mod. Phys. 50, 261 (1978); 51, 407(E)

(1979).[76] A. Donnachie and P. V. Landshoff, Phys. Lett. B 185, 403

(1987).[77] M. A. Pichowsky and T. S. H. Lee, Phys. Lett. B 379, 1 (1996).[78] V. P. Goncalves and M. L. L. da Silva, Phys. Rev. D 89, 114005

(2014).[79] R. J. Eden, Rep. Prog. Phys. 34, 995 (1971).[80] W. H. Liang, P. N. Shen, J. X. Wang and B. S. Zou, J. Phys. G

28, 333 (2002).[81] R. Machleidt, K. Holinde, and Ch. Elster, Phys. Rep. 149, 1

(1987).[82] https://www.jlab.org/Hall-C/talks/08 21 06/Chudakov.pdf.[83] A. Levy, arXiv:0711.0737.[84] T. A. Armstronget al., Phys. Rev. Lett. 69, 2337 (1992).[85] M. Andreotti et al., Phys. Rev. D 72, 032001 (2005); D. Joffe,

arXiv:hep-ex/0505007.[86] A. Movsisyan, EPJ Web Conf. 73 (2014) 02017 (2014)