A framework for the chiral extrapolation of the charmed baryon ground-state masses

We consider the chiral Lagrangian for charmed baryon fields with $J^P =\frac{1}{2}^+$ or $J^P =\frac{3}{2}^+$ quantum numbers. A chiral expansion framework for the baryon ground state masses is worked out to N$^3$LO as to compute their dependence on the up, down and strange quark masses for finite box QCD lattice simulations. It is formulated in terms of on-shell meson and baryon masses. The convergence of such a scheme is illustrated with physical masses as taken from the PDG. The counter terms relevant at N$^3$LO are correlated systematically by large-$N_c$ sum rules to leading and subleading order in a manner that keeps the renormalization scale invariance of the approach.

QCD lattice simulations offer the opportunity to determine low-energy parameters of the chiral Lagrangian. Since the simulations are performed also at quark masses distinct to those needed to reproduce the physical hadron masses new information is generated that may help to determine so far unknown low-energy constants.
Such programs have already been successfully set up for the masses of baryons and mesons in their ground states with J P = 1 2 + , 3 2 + and J P = 0 − , 1 − quantum numbers [1][2][3].
Corresponding sets of low-energy parameters to be used in flavor SU(3) chiral Lagrangians were established from the available lattice data on such hadron masses [1,2].
The purpose of the present work is to establish a corresponding framework for the masses of charmed baryons, which can then eventually be applied to the current QCD lattice data.
Given the rather scarce data set that is provided so far on the charmed baryon masses [4][5][6][7][8][9] it is important to derive additional constraints from QCD that will guide a fit of the lowenergy constants to such data. An important first step in this direction are the recent works [10,11], in which all counter terms that turn relevant in a chiral expansion of the charmed baryon masses are constructed and correlated by the heavy-quark spin symmetry and sum rules derived from large-N c QCD. Here we complement these results by deriving explicit expressions for the various contributions to the baryon masses that arise at next-to-next-tonext-to leading order (N 3 LO).
As was argued in our previous works [1][2][3] a chiral expansion around the flavor SU(3) limit of QCD in terms of bare meson and baryon masses is not convergent for the physical up, down and strange quark masses. Any attempt to apply such a conventional expansion strategy to the QCD lattice data set is futile and should be abandoned [12]. Instead, it was demonstrated that a reformulated expansion scheme that uses the on-shell meson and baryon masses appears to have a significantly larger convergence domain, that is applicable to the physical masses for up, down and strange quarks [1,2]. In this work the details of such an approach for the charmed baryon masses are presented. In particular its convergence properties are illustrated at the hand of a chiral decomposition of the bubble loop contributions at physical meson and baryon masses. This is supplemented by the derivation of additional sum rules for the counter terms that arise from the condition of renormalization scale invariance. Given the results of this work an application to the data set from lattice QCD group is feasible.
This should eventually lead to a faithful set of low-energy constants.
The work is organized as follows. In section II the relevant parts of the chiral Lagrangian are collected. All expressions required at N 3 LO are detailed in section III. In section IV the various large-N c sum rules are studied at the one-loop level. The paper continues with a convergence study in section V and VI of the one-loop bubble contributions as decomposed into their chiral moments. A short summary given with section VII. The appendix provides a glossary for our notations and conventions.

II. CHIRAL LAGRANGIAN WITH CHARMED BARYON FIELDS
The chiral Lagrangian is a reliable tool, once it is combined with appropriate counting rules leading to a systematic approximation strategy. In the following we recall the leading order (LO) terms [10,13,14]. It is convenient to decompose the fields into their isospin multiplets with Φ = τ · π(140) + α † · K(494) + K † (494) · α + η(547) λ 8 , where the matrices λ i are the standard Gell-Mann generators of the SU(3) algebra. The numbers in the brackets recall the approximate masses of the particles in units of MeV. Note that we do not consider the η ′ meson as an active degree of freedom in our current study [15,16] It should be noted that Ξ c and Ξ ′ c have the same quantum numbers and therefore a mixing of the two fields needs to be considered [17]. We introduce a Ξ c − Ξ ′ c mixing angle ǫ by where the physical fields are denoted byΞ c andΞ ′ c . The off-diagonal self energy Σ Ξc Ξ ′ c = 0 reflects the fact that the fields Ξ c and Ξ ′ c are unphysical. Only for the physical fieldsΞ c and Ξ ′ c we expect their corresponding off-diagonal self energy ΣΞ cΞ ′ c → 0 to vanish for on-shell conditions.
and 6 structures which parameterize the three-point interactions of the Goldstone bosons with the charmed baryon fields [13,14]. From the kinetic terms one can read off the Weinberg-Tomozawa interaction terms on which the coupled-channel computation of [18] rests. It follows upon an expansion of the kinetic terms in powers of the Goldstone boson fields. At leading order in a chiral expansion, the bare masses M 1/2 [6] , M 3/2 [6] and M 1/2 [3] may be identified with the flavor average of the sextet and anti-triplet baryon masses. Note the classical vector and axial-vector source functions v µ and a µ of QCD in (3) were instrumental in the derivation of our large-N c sum rules [10,11].
We proceed with the terms at next-to-leading order (NLO) where there are symmetry conserving and symmetry breaking terms [10,11]. We recall the 7 symmetry breaking counter terms with χ 0 = 2 B 0 diag(m, m, m s ) proportional to the quark-mass matrix. We do not consider isospin violating effects in this work. The low-energy constants of (4) imply a linear quarkmass dependence for the charmed baryon masses with Ωc − M A complete list of chiral symmetry conserving Q 2 counter terms, relevant for the calculation of the charm baryon masses at N 3 LO, was given in [10,11]. In these works the Q 2 counter terms are grouped according to their Dirac structure. Here we display the scalar and vector terms relevant for our study only with D,[33] tr B [3] i γ α U α , U β (D β B [3] ) + h.c.
where further possible terms that are redundant owing to the on-shell conditions of spin-3 2 fields with γ µ B µ [6] = 0 and ∂ µ B µ [6] = 0 are eliminated systematically. The counter terms recalled in (7) contribute to the baryon masses at the one-loop level.
They imply renormalization scale dependent contributions that need to be balanced by a set of symmetry breaking counter terms in L (4) χ . We close this section with a partial collection of terms contributing to L (4) χ that are relevant in a chiral extrapolation of the baryon masses at N 3 LO. There are 16 such symmetry breaking counter terms [3] B [3] tr χ 2 + + c 2,[33] tr B [3] B [3] tr χ + 2 + c 3,[33] tr B [3] χ + B [3] tr χ + + c 4,[33] tr B [3] Altogether we count 54 low-energy constants in this section that have to be determined by some data set. Clearly, any additional constraints from heavy-quark spin symmetry or large-N c QCD are desperately needed to arrive at any significant result. Such constraints were derived in [10,11,19] to subleading order in the 1/N c expansion and are summarized in Appendix A for the readers' convenience.

III. CHIRAL EXPANSION OF THE CHARMED BARYON MASSES
We turn to the computation of the baryon masses. The baryon self energy, Σ B (p /), may be considered to be a function of p µ γ µ only, with the 4-momentum p µ of the baryon B. This is obvious for the spin-one-half baryons, but less immediate for the spin-three-half baryons.
We refer to [20] for technical details. To order Q 4 the self energy receives contributions of symmetry breaking counter terms, the tadpole and the one-loop bubble diagram where the index B stands for the members of the flavor multiplets with J P = 1 The on-shell mass of the baryon M B is determined by the condition M J=1/2 [6] ≡ M [6] for B ∈ [6] M J=3/2 [6] ≡ M [4] for where M [3] , M [4] and M [6] are the renormalized and scale-independent masses of the baryon multiplets in the flavor SU(3) limit.
The separation of the baryon self energies into a loop and a tree-level contribution is not unique depending on the renormalization scheme. In this work we apply a recent approach developed for the chiral extrapolation of the baryon octet and decuplet masses [1,2,21].
It is based on the χMS scheme [20] and can directly be adapted to the charmed baryons, the focus of the current work. A matching with alternative renormalization schemes is most economically performed by a direct comparison with the explicit expressions of our study.
Given the renormalization scheme [1] the low-energy constants b n or d n do specify the linear quark mass dependence of the baryon masses as already detailed in (5). The particular subtraction scheme for the loop contributions as introduced in [1] was constructed to ensure  Let us begin with the tadpole contributions, which in a finite volume take the following where the various Clebsch coefficients G (χ) BQ are summarized in Tab. I for the J P = 1 2 + states 1 . Note that like in our previous works [1][2][3] we use the letter Q in a context specific manner. It may either denote a chiral order, or as in (11) or Tab. I if used as an index it runs over the eight Goldstone bosons properly grouped into their isospin multiplets.
The set of finite-box scalar tadpole integralsĪ (n) Q were introduced in [21]. Here we recall their infinite volume limit only, with the renormalization scale µ of dimensional regularization. Explicit expressions forĪ Q appropriate for their finite volume generalization are given in equations (2) and (19) of [21].
It remains to detail the Clebsch coefficients G

3
,g such that the desired Clebsch coefficients can be read off from Tab. I by replacing g Consider now the terms quadratic in the quark masses, which we denote with Σ They supplement the terms linear in the quark masses, which were already considered in (5). Together, both classes of terms are combined with Σ tree−level B in (9). We follow here our 1 For the tadpole contribution to the Ξ c Ξ ′ c mixing it follows M (0)    (8) is recovered in application of the Gell-Mann-Oakes-Renner relations, e.g.
previous works [1,2,21] in which we keep the on-shell meson masses in the tadpole contri- in terms of which our unambiguous results are simplified significantly. The self energies for the flavor anti-triplet states are detailed in the first part of Tab. II with the coupling constantsc n . As can be seen from Tab. II only the particular termc 1 keeps the original structure being a product of two quark masses. We identify a single parameter combinatioñ c 3 that probes the product of a quark mass with the second power of some meson mass.
The remaining parametersc 2 andc 4 select the terms involving the fourth power of a meson mass.
Like in the previously studied cases [1,2] there is a subtle issue as how to treat the flavor singlet structures proportional to g (S) 0,[33] and b 1, [33] . While the first term stems from a chiral symmetric interaction, the second one from a structure that breaks the chiral symmetry explicitly. Nevertheless, the two terms end up with identical tadpole type contributions if the Gell-Mann-Oakes-Renner relations are used. The request of renormalization scale invariance implies that the two contributions in (11) have to be dealt with identically, i.e.
we take the replacement in G as used in the lower parts of Tab. II. We point at the one-to-one correspondence of the coefficients in (14) and (16)    part in (14). Note that here the replacements BQ respectively. We turn to the bubble loop contributions properly derived in the subtraction scheme [1,2]. The generic form of the loop contributions can be taken over from our previous works [1,2,20,21]. Consider first the contributions to the masses of the J P = where we encounter the subtraction terms α QR (see (21)). We first recall the scalar tadpole and bubble integrals with where all finite volume effects are collected into ∆Ī Q and ∆Ī QR . For explicit expressions for the latter the reader is referred to [21]. The sums in (17)  We note that all terms proportional to m 2 n QĪ Q with n ≥ 1 are dropped in (17) as either higher order or as terms that can be absorbed into our tadpole terms. This requires to use renormalized low-energy parametersḡ (S,V ) in (11) of the following form    (17). It is recalled that the subtraction makes sure that the scalar bubbleĪ QR (M B ) will vanish in the chiral limit with m Q → 0 strictly. This protects the tree level slope parameters b n, [ab] in (5)  QR is required to protect a chiral theorem. There are non-analytic terms proportional to m 3 Q that arise from the bubble loop contributions. Only in the presence of the subtraction terms α (B) QR they take their proper form. From [1,2] it is recalled where we note that the last expression for α 1 in (21) was not needed in [1] since there only one flavor multiplet of baryon states with J P = 1 2 + occurs.
We close this section with the bubble loop contribution for the J P = 3 2 + states. Again the form for the loop contributions can be inferred from our previous work [1,20]. We find For a more in depth discussion of the various arguments in favour of the applied renormalization scheme we refer to our previous works [1,2].

IV. LARGE-N c SUM RULES AT THE ONE-LOOP LEVEL
In our previous work [11] we derived sum rules for our low-energy constants as they arise in QCD with a large number of colors (N c ). The analysis was performed for treelevel expressions derived from the chiral Lagrangian. As was pointed out already for the analogous case of a study for the baryon octet and decuplet masses [1], such relations need to be supplemented by constraints that are implied by the renormalization scale invariance condition.
This is readily understood if one considers the scale dependence of the symmetry breaking counter terms proportional to c i and e i of (8). The request that their contributions to the baryon masses are renormalization scale invariant is readily derived with where all Γ c i, [ab] and Γ e i, [ab] are detailed in Appendix B. They depend on the symmetry conserving two-body terms g and h in (7), but also on the symmetry breaking parameters b and d in (4). In turn if we insist on the leading order sum rules for the c i and e i of Appendix where we used the three large-N c sum rules forḡ in terms of which all low-energy constants can be expressed most conveniently. Superficially the expressions (30) appear singular at either M [6] → M [4] or M [6] → M [3] . However this is not the case since in the later limits there are additional relations that ensure that all lowenergy constants remain finite in those limits. We remind the reader of the sum rules that arise in large-N c QCD at leading order. Here it follows M [6] = M [3] but also thath While we can conclude from (31) that the low-energy parameters in (29) remain finite at large-N c this does not yet follow in the heavy-quark mass limit. From the previous work [10] we recall that there is no immediate reason thath  27)). Note a subtle issue concerning the order at which the two limits M c → ∞ and N c → ∞ have to be applied. Consistent results follow only if the heavy-quark mass limit with M [6] → M [4] at M [3] = M [4] is applied first.
At subleading order altogether we have 21 = 16 + 5 sum rules. Thus from the 54 lowenergy constants, there remain only 24 = 33 -8 -1 parameters that we have to adjust to the QCD lattice data set on the charmed baryon masses. Even at subleading order we deem this to be a significant result which paves the way towards a quantitative and controlled approach to chiral dynamics of charmed baryons.

V. A CONVERGENCE STUDY FOR THE BUBBLE LOOP
The purpose of the following section is to decompose the loop functionΣ bubble B into power and illustrate the convergence properties of such an expansion at hand of the physical meson and baryon masses. It is emphasized that any conventional chiral expansion in terms of bare meson and baryon masses appears futile at physical up, down and strange quark masses, at least for the baryon masses with zero charm content. From our previous study of the chiral expansion for the charm meson masses [2] we already learned that such a conventional strategy appears ill defined even for charmed systems. Though, a conventional expansion for the charm baryon masses may not be as disastrous as it is for the baryons with zero charm content, we anticipate that our expansion in terms of on-shell masses generates much more useful and convincing results.
Given our framework and notations the required expressions can be readily deduced from our previous work [1], where however a slight adaptation is necessary. Any of the moments in (32) receives three types of contributions Σ bubble−n

B∈[a]
=Σ bubble−n B∈[a], [3] +Σ bubble−n B∈[a], [6] +Σ bubble−n B∈[a], [4] , which are classified according to the flavor or spin multiplicity of the intermediate charmed baryon states. We will exemplify such results for the leading order term in the expansion.
With the construction of the third order terms (34, 35) it is straightforward to correctly identify the corresponding fourth and fifth order terms from [1]. Note that the higher order terms involve additional coefficients α n ,α n and γ n ,γ n and β n ,β n and δ n ,δ n that are detailed at the beginnings of Appendix A and B of our previous work [1]. All such coefficients are dimension less and depend on the ratio ∆/M only. It should be noted that if the ratio ∆/M turns out to be significantly smaller than 1/3 a further expansion of our results in powers of such a ratio may be justified. However, this can be decided only after a full analysis of the lattice data set has been performed.

VI. SOME NUMERICAL RESULTS
We now generate some numerical results illustrating the convergence properties of the chiral expansion. Since the relevant set of low-energy parameters is basically unknown we focus on the chiral decomposition of the one-loop bubble functions as detailed in the previous chapter.
Any numerical estimate requires the values of the on-shell baryon and meson masses involved. Those we take from the PDG [22]. While for any of the hadron masses m Q or M B , M R we apply an isospin average to the values of the PDG [22], for the chiral limit masses M [3] , M [6] and M [4] we take the flavor SU(3) average of the corresponding multiplet masses from the PDG. The latter assumption is ad-hoc and constitutes a zeroth order estimate for such values only.
It is left to set the axial-vector coupling constants F and C. In [10]    MeV. Thus such contributions will play a decisive role in any chiral extrapolation study of the charmed baryon masses.
We turn to our second scenario in Tab  Like for the contributions in Tab. V we observe a stunning convergence behaviour.
The self energyΣ bubble B truncated at the fifth order is reproduced with an uncertainty of about 0.5%. Note that in this scenario the flavor anti-triplet baryons do not receive any contributions. This is so since at leading order in the large-N c expansion it holds F [33] = 0.
We conclude that a chiral decomposition of the one-loop contributions formulated in terms of on-shell meson and baryon masses appears well converging also for the charmed baryon masses. Thus, in any realistic application to QCD lattice data, which should be minimally at N 3 LO, it is not required to work with loop expressions truncated to some order. Since the fifth order terms are about 10 MeV on average it is well justified to apply the loop functions as they are specified in Chapter III for the finite volume case. This is the strategy followed also in our previous works on the chiral extrapolation of other hadron masses in [1,2]. Note that the size of the systemtic error in the charmed baryon masses from current QCD lattice ensembles is at least of that size. As we repeatedly emphasized, any significant results from a fit to the lattice data can be expected only if for a given lattice ensemble the set of eight coupled and non-linear equations is solved that determines the charmed baryon masses.

VII. SUMMARY
We considered the self energies for the charmed baryon masses from the chiral Lagrangian    conventions used for the various low-enery constants.
The large number of unknown low-energy constants is reduced by sets of sum rules that follow from a systematic 1/N c expansion [10,11] where we correct the results for C [66] and F [33] of [10] and apply the notation Note that the parameterĝ (V ) 1,[33] cannot be determined in an analysis of the charm baryon masses. We close this appendix with a short summary of the implications from the heavyquark spin symmetry that arises in the limit of an infinitely heavy charm quark mass [13,14,19,23]. The mass parameters M 1/2 [3] , M 1/2 [6] and M 3/2 [6] may be expanded in inverse powers of the charm quark mass M c . A matching with QCD's properties [13,14,19,23] leads to the scaling properties which implies that the two sextet masses are degenerate in this limit. We recall that the leading order large-N c sum rules are supplemented by one additional relationh (S) 5,[66] = 0 from [10,11] if the heavy-quark mass limit is applied. Note that at subleading order in the 1/N c expansion heavy-spin symmetry breaking terms occur. For instance b 1,[36] or c n, [36] need no longer to vanish.