Decay properties of $P_c$ states through the Fierz rearrangement

We systematically study hidden-charm pentaquark currents with the quark configurations $[\bar c u][u d c]$, $[\bar c d][u u c]$, and $[\bar c c][u u d]$. Some of their relations are derived using the Fierz rearrangement of the Dirac and color indices, and the obtained results are used to study strong decay properties of $P_c$ states as $\bar D^{(*)} \Sigma_c$ hadronic molecules. We calculate their relative branching ratios for the $J/\psi p$, $\eta_c p$, $\chi_{c0} p$, $\chi_{c1} p$, $\bar D^{(*)0} \Lambda_c^+$, $\bar D^{0} \Sigma_c^{+}$, and $\bar D^{-} \Sigma_c^{++}$ decay channels. We propose to search for the $P_c(4312)$ in the $\eta_c p$ channel and the $P_c(4440)/P_c(4457)$ in the $\bar D^{0} \Lambda_c^+$ channel.


I. INTRODUCTION
Since the discovery of the X(3872) in 2003 by Belle [1], many charmonium-like XY Z states were discovered in the past twenty years [2]. Besides, the LHCb Collaboration observed three enhancements in the J/ψp invariant mass spectrum of the Λ b → J/ψpK decays [3,4]: These structures contain at least five quarks,ccuud, so they are perfect candidates of hidden-charm pentaquark states. Together with the charmonium-like XY Z states, their studies are significantly improving our understanding of the non-perturbative behaviors of the strong interaction at the low energy region [5][6][7][8][9][10][11][12][13][14].
In this paper we shall apply the Fierz rearrangement of the Dirac and color indices to study strong decay properties of P c states asD ( * ) Σ c hadronic molecules, which method has been used in Ref. [56] to study strong decay properties of the Z c (3900) and X(3872). The present approach follows the idea of the QCD factorization method [57][58][59], which has been widely and successfully used to study weak decay properties of heavy hadrons. A similar arrangement of the spin and color indices in the nonrelativistic case was used to study decay properties of XY Z and P c states in Refs. [48,55,[60][61][62][63][64].
In this paper shall use thec, c, u, u, and d (q = u/d) quarks to construct hidden-charm pentaquark currents with the three configurations: [cu][udc], [cd] [uuc], and [cc] [uud]. In Refs. [65][66][67] we have found that these three configurations can be related as a whole, while in the present study we shall further find that two of them are already enough to be related to each other, just with the color-octet-color-octet meson-baryon terms included. Using these relations, we shall study strong decay properties of P c states asD ( * ) Σ c molecular states.
The above two equations can be easily used to calculate the relative branching ratio of the P c decay into η c p to its decay into J/ψp [71]. Detailed discussions on this will be given below. This paper is organized as follows. In Sec. II we systematically study hidden-charm pentaquark currents with the quark contentccuud. We consider three different configurations, [cu][udc], [cd] [uuc], and [cc] [uud], whose relations are derived in Sec. III using the Fierz rearrangement of the Dirac and color indices. In Sec. IV we extract some strong decay properties ofD ( * )0 Σ + c andD ( * )− Σ ++ c molecular states, which are combined in Sec. V to further study strong decay properties ofD ( * ) Σ c molecular states with I = 1/2. The results are summarized in Sec. VI.

II. HIDDEN-CHARM PENTAQUARK CURRENTS
We can usec, c, u, u, and d (q = u/d) quarks to construct many types of hidden-charm pentaquark currents. In the present study we need the following three, as illustrated in Fig. 1: where Γ η/ξ/θ 1/2/3 are Dirac matrices, the subscripts a · · · e are color indices, and the sum over repeated indices (both superscripts and subscripts) is taken.
All the independent hidden-charm tetraquark currents of J P C = 1 +± have been constructed in Refs. [56,[72][73][74][75]. However, in this case there are hundreds of hiddencharm pentaquark currents, and it is difficult to find out all the independent ones (see Refs. [65,66] for relevant discussions). Hence, in this paper we shall not construct all the currents, but just investigate those that are needed to study decay properties of the P c (4312), P c (4440), and P c (4457). We shall separately investigate their color and Lorentz structures in the following subsections.
to describe the former, while there are three color-octetcolor-octet meson-baryon terms for the latter: Only two of them are independent due to which is consistent with the group theory that there are two and only two octets in Similar argument applies to ξ(x, y) and θ(x, y). In Refs. [65,66] we use the color rearrangement together with the Fierz rearrangement to derive the relations among all the three types of currents, e.g., we can transform an η current into the combination of many ξ and θ currents: In the present study we further derive another color rearrangement: Note that the other color-octet-color-octet meson-baryon term λ ae n ǫ cdf λ f b n can also be included, but the first coefficient 1/3 always remains the same. This is reasonable because the probability of the relevant fall-apart decay is just 33% if only considering the color degree of freedom, as shown in Figs. 2(a) and 3(a).
Using the above color rearrangement in the color space, together with the Fierz rearrangement in the Lorentz space to interchange the u b and c e quark fields, we can transform an η current into the combination of many θ currents (both color-singlet-color-singlet and color-octetcolor-octet ones). Similar arguments can be applied to relate whose explicit formulae will be given in Sec. III.  Operators

y) and heavy baryon fields
In this subsection we construct the η(x, y) and ξ(x, y) currents. To do this, we need charmed meson operators as well as their couplings to charmed meson states, which can be found in Table I (see Ref. [56] and references therein for detailed discussions). We also need "ground-state" charmed baryon fields, which have been systematically constructed and studied in Refs. [81][82][83] using the method of QCD sum rules [84,85] within the heavy quark effective theory [86][87][88]. We briefly summarize the results here.
The interpolating fields coupling to the J P = 1/2 + ground-state charmed baryons Λ c and Σ c are Their couplings are defined as where u B is the Dirac spinor of the charmed baryon B, and the decay constants f B have been calculated in Refs. [81][82][83] to be f Λc = 0.015 GeV 3 , (17) f Σc = 0.036 GeV 3 .
The above results are evaluated within the heavy quark effective theory, but for light baryon fields we shall use full QCD decay constants (see Sec. II C). This causes some, but not large, theoretical uncertainties.
Actually, there are several other charmed baryon fields, such as: • the "ground-state" field of pure J P = 3/2 + J µ which couples to the J P = 3/2 + ground-state charmed baryons Σ * + c , with P µα 3/2 the J = 3/2 projection operator • the "excited" charmed baryon field which contains the excited diquark field ǫ abc u T a Cd b of J P = 0 − . For completeness, we list all of them in Appendix B, and refer to Ref. [89] for detailed discussions. The major advantage of using the heavy quark effective theory is that within this framework all these charmed baryon fields do not couple to the J P = 1/2 + ground-state charmed baryons Λ c and Σ c [90]. However, some of them, both "ground-state" and "excited" fields, can couple to the J P = 3/2 + ground-state charmed baryon Σ * c . Hence, we do/can not study decays of P c states into theDΣ * c final state in the present study.
Combing charmed meson operators and ground-state charmed baryon fields, we can construct the η(x, y) and ξ(x, y) currents. In the molecular picture the P c (4312), P c (4440), and P c (4457) can be interpreted as theDΣ c hadronic molecular state of J P = 1/2 − , theD * Σ c one of J P = 1/2 − , and theD * Σ c one of J P = 3/2 − [19][20][21]: where and In the above expressions we have written J B as B for simplicity.

C. θ(x, y) and light baryon fields
In this subsection we construct the θ(x, y) currents, which can be constructed by combing charmonium operators and light baryon fields. Hence, we need charmonium operators as well as their couplings to charmonium states, which can be found in Table I (see Ref. [56] and references therein for detailed discussions). We also need light baryon fields, which have been systematically studied in Refs. [68][69][70][91][92][93][94][95]. We briefly summarize the results here.
According to the results of Ref. [91], we can use u, u, and d (q = u/d) quarks to construct five independent baryon fields: where the projection operator P µναβ All the other light baryon fields including 1,2,3,4,5 , as shown in Appendix B. Among the five fields defined in Eqs. (31), the former two N 1,2 have pure spin J = 1/2, and the latter three N µ(ν) 3,4,5 have pure spin J = 3/2. In the present study we shall study decays of P c states into charmonia and protons, but not study their decays into charmonia and ∆/N * , since the couplings of N µ(ν) 3,4,5 to ∆/N * have not been (well) investigated in the literature. Therefore, we only keep N 1,2 but omit N µ(ν) 3,4,5 . Moreover, we shall find that all the terms in our calculations do not depend on N 1 + N 2 , so we only need to consider the Ioffe's light baryon field This field has been well studied in Refs. [68][69][70] and suggested to couple to the proton through with the decay constant evaluated in Ref. [96] to be

III. FIERZ REARRANGEMENT
In this section we study the Fierz rearrangement of the η(x, y) and ξ(x, y) currents, which will be used to investigate fall-apart decays of P c states in Sec. IV. Taking η(x, y) as an example, when thec a (x) and c e (y) quarks meet each other and the u b (x), u c (y), and d d (y) quarks meet together at the same time, aD ( * )0 Σ + c molecular state can decay into one charmonium meson and one light baryon. This is the decay process depicted in Fig. 2 The first step is a dynamical process, during which we assume that all the color, flavor, spin and orbital structures remain unchanged, so the relevant current also remains the same. The second and third steps can be described by applying the Fierz rearrangement to interchange both the color and Dirac indices of the u b (y ′ ) and c e (x ′ ) quark fields. Still taking η(x, y) as an example: when thec a (x) and u c (y) quarks meet each other and the u b (x), d d (y), and c e (y) quarks meet together at the same time, aD ( * )0 Σ + c molecular state can decay into one charmed meson and one charmed baryon, as depicted in Fig. 2(b); when thē c a (x) and d d (y) quarks meet each other and the u b (x), u c (y), and c e (y) quarks meet together at the same time, ā D ( * )0 Σ + c molecular state can also decay into one charmed meson and one charmed baryon, as depicted in Fig. 2(c). Similarly, decays ofD ( * )− Σ ++ c molecular states can be investigated through the ξ(x, y) currents, as depicted in Fig. 3(a,b,c).
In the following subsections we shall study the above fall-apart decay processes, by applying the Fierz rearrangement [97] of the Dirac and color indices to relate the η, ξ, and θ currents. This method has been used to systematically study light baryon and tetraquark operators/currents in Refs. [72-74, 91-95, 98-102]. We note that the Fierz rearrangement in the Lorentz space is actually a matrix identity, which is valid if each quark field in the initial and final operators is at the same location, e.g., we can apply the Fierz rearrangement to transform a non-local η current with the quark fields into the combination of many non-local θ currents with the quark fields at Hence, this rearrangement exactly describes the third step of Eq. (36).
A. η → θ and ξ → θ Using Eq. (13), together with the Fierz rearrangement to interchange the u b and c e quark fields, we can transform an η(x, y) current into the combination of many θ currents: In the above transformations we have changed the coordinates according to the first step of Eq. (36), which are not shown explicitly here for simplicity. Besides, we have omitted in · · · that: a) the color-octet-color-octet mesonbaryon terms, and b) terms depending on the J = 3/2 light baryon fields N µ(ν) 3,4,5 . Hence, we have only kept, but kept all, the color-singlet-color-singlet meson-baryon terms depending on the J = 1/2 fields N 1 and N 2 . This  is not an easy task because we need to use many identities given in Eqs. (B22) and (B23) of Appendix B in order to safely omit N µ(ν) 3,4,5 . Moreover, we can find in the above expressions that all terms contain the Ioffe's light baryon field N ≡ N 1 − N 2 , and there are no terms depending on The above transformations can be used to describe the fall-apart decay process depicted in Fig. 2(a) forD ( * )0 Σ + c molecular states. Similarly, we can investigate the fallapart decay process depicted in Fig. 3(a) forD ( * )− Σ ++ c molecular states. To do this, we need to use Eq. (13), together with the Fierz rearrangement to interchange the d b and c e quark fields, to transform a ξ(x, y) current into the combination of many θ currents: B. η → η and η → ξ First we derive a color rearrangement similar to Eq. (13): Using this identity, together with the Fierz rearrangement to interchange the u b and u c quark fields, we can transform an η(x, y) current into the combination of many η currents. Besides, we can derive another similar color rearrangement: Using this identity, together with the Fierz rearrangement to interchange the u b and d d quark fields, we can transform an η(x, y) current into the combination of many ξ currents. The above two transformations describe the fall-apart decay processes depicted in Fig. 2(b,c) forD ( * )0 Σ + c molecular states. Altogether, we obtain: In the above transformations we have only kept, but kept all, the color-singlet-color-singlet meson-baryon terms depending on the J P = 1/2 + "ground-state" charmed baryon fields given in Eqs. (15). Again, this is not an easy task because we need to carefully omit the terms depending on the other charmed baryon fields, ,µ , and B U 6,µν , whose definitions can be found in Appendix B.
Following the procedures used in the previous subsection, we can transform a ξ(x, y) current into the combination of many η currents (without ξ currents): The above transformations describe the fall-apart decay processes depicted in Fig. 3
The obtained results will be combined in Sec. V to further study decay properties ofD ( * ) Σ c molecular states with definite isospins.
In this subsection we study strong decay properties of |D 0 Σ + c ; 1/2 − through the η 1 (x, y) current. First we use the Fierz rearrangement given in Eq. (37) to study the decay process depicted in Fig. 2(a), i.e., decays of |D 0 Σ + c ; 1/2 − into one charmonium meson and one light baryon. Together with Table I, we extract the following decay channels that are kinematically allowed: where u and u p are the Dirac spinors of the P c state with J P = 1/2 − and the proton, respectively; a 1 is an overall factor, related to the coupling of η 1 (x, y) to |D 0 Σ + c ; 1/2 − as well as the dynamical process of Fig. 2(a); the two coupling constants A ηcp and A ′ ηcp are defined for the two different effective Lagrangians 2. The decay of |D 0 Σ + c ; 1/2 − into J/ψp is contributed by [c a γ µ c a ] γ µ γ 5 N : Then we use the Fierz rearrangement given in Eq. (45) to study the decay processes depicted in Fig. 2(b,c), i.e., decays of |D 0 Σ + c ; 1/2 − into one charmed meson and one charmed baryon. Together with Table I, we extract only one decay channel that is kinematically allowed: where u Λc is the Dirac spinor of the Λ + c ; a 2 is an overall factor, related to the coupling of η 1 (x, y) to |D 0 Σ + c ; 1/2 − as well as the dynamical processes of Fig. 2(b,c); the coupling constant AD * Λc is defined for LD * Λc = AD * ΛcPc γ µ γ 5 Λ + cD * ,µ .
In the molecular picture the P c (4312) is usually interpreted as theDΣ c hadronic molecular state of J P = 1/2 − . Accordingly, we assume the mass of |D 0 Σ + c ; 1/2 − to be 4311.9 MeV (more parameters can be found in Appendix A), and summarize the above decay amplitudes to obtain the following (relative) decay widths: There are two different effective Lagrangians for the |D 0 Σ + c ; 1/2 − decays into the η c p final state, as given in Eqs. (52) and (53). It is interesting to see their individual contributions: Hence, the former is about four times larger than the latter. We note that their interference can be important, but the phase angle between them, i.e., the phase angle between the two coupling constants A ηcp and A ′ ηcp , can not be well determined in the present study. We shall investigate its relevant uncertainty in Appendix C.
In this subsection we follow the procedures used in the previous subsection to study decay properties of |D − Σ ++ c ; 1/2 − , through the ξ 1 (x, y) current and the Fierz rearrangements given in Eqs. (40) and (48). Again, we assume its mass to be 4311.9 MeV, and obtain the following (relative) decay widths: Here b 1 and b 2 are two overall factors, which we simply assume to be b 1 = a 1 and b 2 = a 2 in the following analyses.
In this subsection we follow the procedures used in Sec. IV A to study decay properties of |D * 0 Σ + c ; 1/2 − through the η 2 (x, y) current. First we use the Fierz rearrangement given in Eq. (38) to study the decay process depicted in Fig. 2(a): where c 1 is an overall factor.
2. The decay of |D * 0 Σ + c ; 1/2 − into J/ψp is contributed by both [c a γ µ c a ] γ µ γ 5 N and [c a σ µν c a ] σ µν γ 5 N : where the two coupling constants C ψp and C ′ ψp are defined for 3. The decay of |D * 0 Σ + c ; 1/2 − into χ c0 (1P )p is contributed by [c a c a ] γ 5 N : where C χc0p is defined for 4. The decay of |D * 0 Σ + c ; 1/2 − into χ c1 (1P )p is contributed by [c a γ µ γ 5 c a ] γ µ N : This decay channel may be kinematically allowed, depending on whether the P c (4457) is interpreted as |D * 0 Σ + c ; 1/2 − or not. Then we use the Fierz rearrangement given in Eq. (46) to study the decay processes depicted in Fig. 2(b,c): 5. The decay of |D * 0 Σ + c ; 1/2 − intoD 0 Λ + c is contributed by [c a γ 5 u a ] Λ + c : where c 2 is an overall factor, and the coupling constant CD Λc is defined for 6. The decay of |D * 0 Σ + c ; 1/2 − intoD * 0 Λ + c is contributed by [c a σ µν u a ] σ µν γ 5 Λ + c : where C ′D * Λc is defined for 7. Decays of |D * 0 Σ + c ; 1/2 − into theD 0 Σ + c and D − Σ ++ c final states are: In the molecular picture the P c (4440) is sometimes interpreted as theD * Σ c hadronic molecular state of J P = 1/2 − . Accordingly, we assume the mass of |D * 0 Σ + c ; 1/2 − to be 4440.3 MeV, and summarize the above decay amplitudes to obtain the following (relative) decay widths: Besides, |D * 0 Σ + c ; 1/2 − can also couple to χ c1 p, but this channel is kinematically forbidden under the assumption M |D * 0 Σ + c ;1/2 − = 4440.3 MeV. There are two different effective Lagrangians for the |D * 0 Σ + c ; 1/2 − decays into the J/ψp final state, as given in Eqs. (63) and (64). It is interesting to see their individual contributions: Hence, the former is about four times smaller than the latter. Again, the phase angle between them can be important, whose relevant uncertainty will be investigated in Appendix C.
In this subsection we follow the procedures used in the previous subsection to study decay properties of |D * − Σ ++ c ; 1/2 − , through the ξ 2 (x, y) current and the Fierz rearrangements given in Eqs. (41) and (49). Again, we assume its mass to be 4440.3 MeV, and obtain the following (relative) decay widths: Here d 1 and d 2 are two overall factors, which we simply assume to be d 1 = c 1 and d 2 = c 2 in the following analyses.
In this subsection we follow the procedures used in Sec. IV A and Sec. IV C to study decay properties of |D * 0 Σ + c ; 3/2 − through the η α 3 (x, y) current. First we use the Fierz rearrangement given in Eq. (39) to study the decay process depicted in Fig. 2(a): where u α is the spinor of the P c state with J P = 3/2 − , and e 1 is an overall factor.

The decay of |D
This decay channel may be kinematically allowed, depending on whether the P c (4457) is interpreted as |D * 0 Σ + c ; 3/2 − or not.
Then we use the Fierz rearrangement given in Eq. (47) to study the decay processes depicted in Fig. 2(b,c): where e 2 is an overall factor.
In this subsection we follow the procedures used in the previous subsection to study decay properties of |D * − Σ ++ c ; 3/2 − , through the ξ α 3 (x, y) current and the Fierz rearrangements given in Eqs. (42) and (50). Again, we assume its mass to be 4457.3 MeV, and obtain the following (relative) decay widths: Here f 1 and f 2 are two overall factors, which we simply assume to be f 1 = e 1 and f 2 = e 2 in the following analyses.

V. ISOSPIN OFD ( * ) Σc MOLECULAR STATES
In this section we collect the results calculated in the previous section to further study decay properties of D ( * ) Σ c molecular states with definite isospins.
TheD ( * ) Σ c molecular states with I = 1/2 can be obtained by using Eqs.
Combining the results of Sec. IV E and Sec. IV F, we obtain: Comparing the above values with those given in Eqs. (58), (76), and (85), we find that the decay widths of the threeD ( * ) Σ c molecular states with I = 1/2 into the η c p, J/ψp, χ c0 p, χ c1 p,D 0 Λ + c , andD * 0 Λ + c final states also with I = 1/2 are increased by three times, and their decay widths into theD 0 Σ + c andD − Σ ++ c final states are decreased by three times. We shall further discuss these results in Sec. VI.
For completeness, we also list here the results for the threeD ( * ) Σ c molecular states with I = 3/2 (as if they existed), which can be obtained by using Eqs. (21), (22), and (23) with θ i = 35 o : Naively assuming their masses to be 4311.9 MeV, 4440.3 MeV, and 4457.3 MeV, respectively, we obtain the following non-zero (relative) decay widths: Comparing them with Eqs. (58), (76), and (85), we find that the threeD ( * ) Σ c molecular states with I = 3/2 can not fall-apart decay into the η c p, J/ψp, χ c0 p, χ c1 p, D 0 Λ + c , andD * 0 Λ + c final states with I = 1/2, their widths into theD 0 Σ + c final state are increased by a factor of 8/3, and their widths into theD − Σ ++ c final state are reduced to two third. We summarize these results in Appendix C, which we shall not discuss any more.

VI. SUMMARY AND CONCLUSIONS
In this paper we systematically study hidden-charm pentaquark currents with the quark contentccuud. We In the molecular picture the P c (4312) is usually interpreted as theDΣ c hadronic molecular state of J P = 1/2 − , and the P c (4440) and P c (4457) are sometimes interpreted as theD * Σ c hadronic molecular states of J P = 1/2 − and 3/2 − respectively (sometimes interpreted as states of J P = 3/2 − and 1/2 − respectively) [19][20][21]. Using their masses measured in the LHCb experiment [4] as inputs, we calculate some of their relative decay widths. The obtained results have been summarized in Eqs. (88), (89), and (90), from which we further obtain: • We obtain the following relative branching ratios for the |DΣ c ; 1/2 − decays: • We obtain the following relative branching ratios for the |D * Σ c ; 1/2 − decays: is the parameter measuring which processes happen more easily, the processes depicted in Figs. 2&3(a) or the processes depicted in Figs. 2&3(b,c). Generally speaking, the exchange of one light quark with another light quark seems to be easier than the exchange of one light quark with another heavy quark [103], so it can be the case that t ≥ 1. There are two phase angles, which have not been taken into account in the above expressions yet. We investigate their relevant uncertainties in Appendix C, where we also give the relative branching ratios for theD ( * ) Σ c hadronic molecular states of I = 3/2, and separately for theD ( * )0 Σ + c andD ( * )− Σ ++ c hadronic molecular states. To extract these results: • We have only considered the leading-order fallapart decays described by color-singlet-colorsinglet meson-baryon currents, but neglected the O(α s ) corrections described by color-octet-coloroctet meson-baryon currents, so there can be other possible decay channels.
• We have omitted all the charmed baryon fields of J = 3/2, so we can not study decays of P c states into theDΣ * c final state. However, we have kept all the charmed baryon fields that can couple to the J P = 1/2 + ground-state charmed baryons Λ c and Σ c , i.e., fields given in Eqs. (15), so decays of P c states into theD ( * ) Λ c andDΣ c final states have been well investigated in the present study.
• We have omitted all the light baryon fields of J = 3/2, so we can not study decays of P c states into charmonia and ∆/N * . However, we have kept all the light baryon fields of J P = 1/2 + , i.e., terms depending on N 1 and N 2 , so decays of P c states into charmonia and protons have been well investigated in the present study.
Our conclusions are: • Firstly, we compare the η c p and J/ψp channels: These ratios are quite similar to those obtained using the heavy quark spin symmetry [48]. Since the width of the |DΣ c ; 1/2 − decay into the η c p final state is comparable to its decay width into J/ψp, we propose to confirm the existence of the P c (4312) in the η c p channel.
• Secondly, we compare theD ( * ) Λ c and J/ψp channels: and Accordingly, we propose to observe the P c (4312), P c (4440), and P c (4457) in theD * 0 Λ + c channel. Moreover, theD 0 Λ + c channel can be an ideal channel to extract the spin-parity quantum numbers of the P c (4440) and P c (4457).
• Thirdly, we compare theDΣ c and J/ψp channels: Accordingly, we propose to observe the P c (4440) and P c (4457) in theD − Σ ++ c channel, which is another possible channel to extract their spin-parity quantum numbers.
We list masses of charmonium mesons and charmed mesons used in the present study, taken from PDG [2] and partly averaged over isospin: In this paper we only investigate two-body decays, and their widths can be easily calculated. In the calculations we use the following formula for baryon fields of spin 1/2 and 3/2:

Appendix B: Heavy and light baryon fields
First we construct charmed baryon interpolating fields. We refer to Ref. [89] for detailed discussions. There are altogether nine independent charmed baryon fields: where In the above expressions, a, b, c are color indices and the sum over repeated indices is taken; A, B, G, U are SU (3) flavor indices, so that q A = {u, d, s}; ǫ ABG is the totally antisymmetric matrix with G = 1, 2, 3, so that B Ḡ 3,i belong to the SU (3) flavor3 F representation; S U AB are the totally symmetric matrices with U = 1 · · · 6, so that B U 6,i belong to the SU (3) flavor 6 F representation; c c is the charm quark field with the color index c; C is the chargeconjugation matrix; P 3/2 µν and P 3/2 µναβ are two J = 3/2 projection operators.
Among the nine fields given in Eqs. (B1-B9) ,µ , and B U 6,µν , all of which do not couple to the J P = 1/2 + ground-state charmed baryons Λ c and Σ c within the framework of heavy quark effective theory [90].
Appendix C: Uncertainties due to phase angles There are two different effective Lagrangians for the |D 0 Σ + c ; 1/2 − (and |D − Σ ++ c ; 1/2 − ) decay into the η c p final state, as given in Eqs. (52) and (53): There are also two different effective Lagrangians for the |D * 0 Σ + c ; 1/2 − (and |D * − Σ ++ c ; 1/2 − ) decay into the J/ψp final state, as given in Eqs. (63) and (64): There can be a phase angle θ between g ηcp and g ′ ηcp and another phase angle θ ′ between g ψp and g ′ ψp , both of which can not be determined in the present study. In this appendix we rotate θ/θ ′ and redo all the calculations.
• We obtain the following relative branching ratios for theD ( * ) Σ c hadronic molecular states of I = 1/2: