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Jonathan M. M. Hall and Derek B. Leinweber Flavor-singlet in the graded approach to partially quenched QCD Physical Review D - , Fields, Gravitation and Cosmology, 2016; 94(9):094004-1- 094004-10

© 2016 American Physical Society.

Originally published by American Physical Society at: http://dx.doi.org/10.1103/PhysRevD.94.094004

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29 November 2016

http://hdl.handle.net/2440/102852

PHYSICAL REVIEW D 94, 094004 (2016) Flavor-singlet baryons in the graded symmetry approach to partially quenched QCD

Jonathan M. M. Hall and Derek B. Leinweber Special Research Centre for the Subatomic Structure of (CSSM), Department of Physics, University of Adelaide, Adelaide, South Australia 5005, Australia (Received 29 September 2015; published 4 November 2016) Progress in the calculation of the electromagnetic properties of excitations in lattice QCD presents new challenges in the determination of sea- loop contributions to matrix elements. A reliable estimation of the sea-quark loop contributions represents a pressing issue in the accurate comparison of lattice QCD results with experiment. In this article, an extension of the graded symmetry approach to partially quenched QCD is presented, which builds on previous theory by explicitly including flavor-singlet baryons in its construction. The formalism takes into account the interactions among both octet and singlet baryons, octet , and their ghost counterparts; the latter enables the isolation of the quark-flow disconnected sea-quark loop contributions. The introduction of flavor-singlet states enables systematic studies of the internal structure of Λ-baryon excitations in lattice QCD, including the topical Λð1405Þ.

DOI: 10.1103/PhysRevD.94.094004

I. INTRODUCTION successful chiral unitary approaches [30–42], which have elucidated the two-pole structure [17]. In describing the A. Quark loop contributions to matrix elements Λð1405Þ resonance at quark masses larger than the physical The calculation of hadronic matrix elements in non- masses, the introduction of a bare state dressed by - perturbative QCD typically encounters nontrivial contri- baryon interactions has also been explored [28]. In all butions from quark-flow disconnected diagrams in which cases, the disconnected loops of Fig. 1 will make important the current interacts with the of interest, first contributions to the matrix elements of these states. through a virtual sea-quark loop. An example process Recent lattice QCD calculations of the electromagnetic for a baryon undergoing a virtual meson-baryon transition, form factors of the Λð1405Þ have reported evidence of an B → BΦ , involving a disconnected sea-quark loop, is important KN molecular component in the Λð1405Þ [13]. depicted in Fig. 1. These disconnected-loop diagrams are On the lattice, the contribution to the difficult to calculate in numerical simulations of QCD due magnetic form factor of the Λð1405Þ approaches zero, to the space-time coordinate dependence of both the source signaling the formation of a spin-zero in a relative S and sink of the quark at the current insertion wave about the . However, the light u and d point. As the loop is correlated with other quark degrees of valence-quark sector of the Λð1405Þ makes a nontrivial freedom only via exchange, resolving a nontrivial contribution, which can be related to the and signal requires high statistics through innovative methods. electromagnetic form factors. While there has been recent success in isolating dis- To understand the physics behind this contribution connected-loop contributions in ground-state baryon and enable a comparison with experiment, one needs to matrix elements, even at nonzero momenta [1,2], the challenging nature of these calculations suggests that resolving similar contributions for hadronic excitations in lattice QCD [3–16] will be elusive. However, it is here that the meson-baryon dressings of excitations are particularly important. Consider, for example, the Λð1405Þ resonance which is of particular contemporary interest. It is widely agreed that there is a two-pole structure for this resonance [17–28], associated FIG. 1. The quark-flow disconnected diagram for the process with attractive interactions in the πΣ and KN channels. The B → BΦ. The cross on the disconnected quark loop line indicates πΣ two poles lie near the and KN thresholds and both the insertion of the matrix-element current. For an incoming channels can contribute to the flavor-singlet components of baryon B, each of the three valence entering on the left- the resonance, studied in detail herein. This two-pole hand side may contribute to the internal meson state Φ. In the structure, as first analyzed in Ref. [17], has now been graded symmetry approach, the conventional u, d and s sea-quark entered into the Data Group tables [29]. This loops are complemented by their commuting ghost counterparts achievement reflects decades of research using the u~, d~ and s~, respectively.

2470-0010=2016=94(9)=094004(10) 094004-1 © 2016 American Physical Society JONATHAN M. M. HALL and DEREK B. LEINWEBER PHYSICAL REVIEW D 94, 094004 (2016) understand the role of disconnected sea-quark loop valence content of the baryon comprises three unique quark contributions to the matrix elements of the Λð1405Þ. flavors (uds) [50]. These contributions have not been included in present- Once the coupling strength of disconnected sea-quark day lattice QCD calculations and present a formidable loop contributions to hadron two-point functions is under- challenge to the lattice QCD community. stood, one can use this information to estimate the size of When disconnected sea-quark loop contributions are sea-quark loop contributions to matrix elements [51–57]. unknown, one often seeks observables insensitive to this The formalism presented herein is based on the original contribution. For example, the isovector baryon combina- work of Labrenz and Sharpe [43] and builds upon the tion is insensitive to a sea-quark contribution as it is similar constructions of that work in a consistent manner. We for both isospin variants and cancels. For baryon states that create a new, completely symmetric baryon in the are not in an isovector formation, a reliable estimation of graded symmetry Lagrangian and couple this field to octet the sea-quark loop contributions remains a pressing issue in mesons and octet baryons. Through the admission of the the accurate comparison of lattice results with experiment. commuting “ghost” counterparts, introduced to isolate the disconnected loop component of each coupling, the par- B. Flavor-singlet symmetry ticles collectively fall into representations of the vector subgroup of the graded symmetry group SUð3j3Þ ,asin Λ 1405 V In lattice QCD calculations, the ð Þ is excited from Ref. [43]. The anticommuting behavior of the standard the vacuum using interpolating fields dominated by a quark degrees of freedom enables our symmetric baryon flavor-singlet operator. Thus, a complete understanding field in the graded symmetry Lagrangian to represent the Λ 1405 of the ð Þ necessarily involves understanding the role flavor-singlet baryon. of disconnected sea-quark loop contributions in the flavor- Our aim here is to disentangle quark-flow connected and singlet sector of . Away from the disconnected couplings between singlet baryons and octet 3 SUð Þ-symmetric limit, flavor-singlet and flavor-octet mesons and baryons in modern 2 þ 1-flavor lattice QCD Λ 1405 interpolators mix in isolating the ð Þ [4] and therefore simulations. As such, there are no issues with the artifacts this new information in the flavor-singlet sector will of quenched QCD, such as a light double-pole η0 meson. complement the flavor-octet sector. Moreover, we will consider quark masses similar to those Flavor symmetry is important in effective field theory as of nature, with mu ≃ md ≤ ms. it provides guidance on the meson-baryon couplings to be used in coupled-channel analyses. It also has the ability to D. Outline describe the relative strength between quark-flow con- nected and disconnected contributions to observables. The structure of the paper is as follows. Section II This separation is well understood in the flavor-octet sector expands upon the graded symmetry approach to include the of effective field theory but is unknown in the flavor-singlet couplings of flavor-singlet baryons to octet mesons and sector. This gap in understanding is addressed in this baryons, and the Lagrangian is established. In Sec. III, article. particle labels for baryons containing a single ghost quark It is perhaps important to note that the physical Λð1405Þ are explicitly written. This partly follows the method is a mix of flavor symmetries due to the explicit breaking of described in Ref. [46], with a few differences in the SUð3Þ symmetry by the quark masses. Our task is not to notation, which are introduced for convenience in identi- determine this mixing but rather to disclose the separation fying the quantum numbers and symmetry properties of the of connected and disconnected contributions in the flavor- baryons considered. As a relevant test case, the flavor- singlet sector for the first time. Thus, in referring to flavor- singlet component of the Λð1405Þ is described in Sec. IV. singlet baryons in the following, it is to be understood that we are referring to the flavor-singlet component of a II. GRADED SYMMETRY FORMALISM baryon. In this manner, our results are applicable to Λ baryons in general. In this section, the details of the graded symmetry approach are briefly reviewed. In order to derive couplings to flavor-singlet baryons, the standard graded symmetry C. Graded symmetry approach approach requires a complementary augmentation in order The separation of connected and disconnected contribu- to include interactions between octet and singlet baryons, tions to the baryon couplings can be handled using the graded octet mesons, and their ghost counterparts. symmetry approach [43–49] or the diagrammatic approach For 2 þ 1-flavor dynamical- lattice QCD calcu- 6 3 [50,51]. Although it has been demonstrated that both lations, the relevant graded group is SUð j ÞV, composed methods are consistent with each other [50,51], in this work, of three valence-quark flavors, three sea-quark flavors and the graded symmetry approach is used because of its ability three ghost-quark flavors [46]. While the sea-quark masses to be easily generalized to include flavor-singlet baryons. are free to differ from the valence sector, the ghost sector The diagrammatic approach becomes cumbersome when the masses are matched to the valence sector such that their

094004-2 FLAVOR-SINGLET BARYONS IN THE GRADED SYMMETRY … PHYSICAL REVIEW D 94, 094004 (2016) 0 1 1 0 1 fermion-determinant contributions exactly cancel the pffiffi π þ pffiffi ηπþ Kþ fermion-determinant contributions of the valence sector, B 2 6 C 1 B − 1 0 1 0 C thus ensuring the valence sector is truly valence only. π ¼ pffiffiffi B π − pffiffi π þ pffiffi η K C 2 @ 2 6 A However, a common approach used in practice, when − 0 2 K K − pffiffi η separating connected and disconnected contributions, is to 6 3 3 work within the simpler graded group SUð j ÞV of η0 þ pffiffiffi I3: ð3Þ quenched QCD. Here the standard quark sector is com- 6 plemented by a ghost quark sector with matching quark masses to render the quark sector to a valence quark sector. With the inclusion of ghost quarks, the flavor of the ghost Identification of the ghost sector contributions to quark- quark or antiquark may specified through either the usual flow diagrams is then sufficient to identify the sea-quark quark mixing, to form π~ 0, η~ and η~0 particles, or through loop contribution of the standard quark sector that is being neutral meson labels η~u, η~d and η~s. Both representations removed. This approach is used herein. have their merit. When the full flavor symmetry of the We note, however, that there are important differences meson dressing is present, the standard representation with between the two graded groups, particularly for the flavor- π~ 0, η~ and η~0 is of particular utility as the mass of the η0 3 3 η0 singlet meson. In SUð j ÞV of quenched QCD, the intermediate meson is manifest. Therefore, we consider the appears only as a double-hairpin diagram, whereas in χ† 6 3 η0 following form for : SUð j ÞV the appears in a partially quenched generali- 0 0 1 zation of the standard η . In our case of interest, it is 1 0 1 ~ þ pffiffi π~ þ pffiffi η~ π~ þ K sufficient to hold the masses of each flavor in the valence, B 2 6 C 1 B − 1 0 1 0 C sea and ghost sectors equal, such that all mesons and † π~ − pffiffi π~ pffiffi η~ ~ χ ¼ pffiffiffi B 2 þ 6 K C baryons participating in the diagrams take their standard 2 @ A − 0 2 ~ ~ − pffiffi η~ properties. K K 6 In deriving a Lagrangian that couples flavor-singlet η~0 baryons to octet baryons and mesons under the graded þ pffiffiffi I3: ð4Þ 3 3 6 symmetry subgroup SUð j ÞV, the constructions intro- duced in Ref. [43] will be used as much as possible, so that the extension to singlet couplings can draw on the On the other hand, the flavor symmetry can be incomplete. In this case, it is more transparent to work with the neutral previously established formalism. η~ η~ η~ In the graded symmetry approach, the quark fields are u, d and s mesons along the diagonal of Eq. (4). This is extended to include bosonic ghost counterparts for each useful when considering the ghost sector of a flavor-octet flavor. The new field Q ¼ðu; d; s; u;~ d;~ s~ÞT transforms baryon transition to a flavor-singlet baryon and an under the vector subgroup of the graded symmetry group octet meson. 3 3 The group transformation properties of the meson field SUð j ÞV as a fundamental representation, analogous to three-flavor chiral perturbation theory (χPT): can be made compatible with the baryon fields via the methods of χPT, where the exponential of the mesons is incorporated into an axial-vector field → → † Qi UijQj; Qi QjUji; ð1Þ ξðxÞ¼expðiΦðxÞ=fπÞ; ð5Þ where the indices i, j run from 1 to 6. i ξ∂ ξ† − ξ†∂ ξ Aμ ¼ 2 ð μ μ Þ: ð6Þ A. Mesons The elements of Aμ transform under the graded symmetry The meson field is expanded to a 6 × 6 matrix, with four subgroup SUð3j3Þ as blocks of mesons corresponding to the ordinary mesons V πðqqÞ, the entirely ghost mesons π~ðq~ q~Þ, and the fermionic μ → μ † Aij Uii0 A 0 0 U 0 : ð7Þ composite mesons χðq~ qÞ and χ†ðqq~Þ in the off-diagonal i j j j blocks B. Octet and singlet baryons πχ† In the graded symmetry approach, the baryon field is Φ ¼ : ð2Þ χ π~ generalized to a rank-3 tensor, allowing the three constitu- ent flavors of each baryon to be specified separately by each flavor index. This is convenient, as it readily allows The elements of the ordinary meson block follow the one to identify the valence quark content of a baryon simply standard convention by inspection. The introduction of graded symmetry into

094004-3 JONATHAN M. M. HALL and DEREK B. LEINWEBER PHYSICAL REVIEW D 94, 094004 (2016) 0 1 1 0 1 the transformation matrices, U, in Eq. (1) requires the pffiffi Σ þ pffiffi Λ Σþ p introduction of a grading factor, defined as B 2 6 C B − 1 0 1 C B Σ − pffiffi Σ þ pffiffi Λ n C  B ¼ @ 2 6 A 1 if i ¼ 1–3; − 0 2 Ξ Ξ − pffiffi Λ ηi ¼ ð8Þ 0 if i ¼ 4–6; 6 Λ0 þ pffiffiffi I3: ð15Þ which takes into account the bosonic nature of the ghost 2 3 quarks—swapping two quark fields incurs a minus sign only if both are anticommuting (Dirac-spinor field) quarks. Λ0 denotes a positive- flavor-singlet baryon. The transformation properties of the octet baryon field Similarly, a matrix B can be defined with the same form, can be derived from the standard nucleon interpolating field representing the lowest-lying negative-parity baryons. The negative-parity flavor-singlet baryon is then denoted by Λ0. Bγ ∼ α;a β;b γ;c − α;a γ;c β;b ϵ γ ijk ½Qi Qj Qk Qi Qj Qk abcðC 5Þαβ; ð9Þ The baryon construction of Eq. (14) contains both octet and singlet fields. However, it is known that interactions where a–c are color indices, α–γ are Dirac indices, and C is that include a singlet baryon have couplings that are not the standard charge conjugation matrix. This construction related via flavor symmetry to the interaction couplings ensures the antisymmetry of the latter two Dirac indices between octet baryons only. It is therefore important (β, γ) and thus eliminates the flavor-singlet baryon [43]. to separate the baryon field into “octet-only” and The pure flavor-singlet baryon field, denoted BS, on the “singlet-only” pieces, so that the terms in the Lagrangian other hand, is totally symmetric in quark flavor Q. This can can be assigned independent coefficients. be encoded into the form of the baryon tensor by permuting To extract the purely octet component of Bijk, Labrenz and the color and Dirac indices while maintaining the flavor- Sharpe antisymmetrized over the j and k indices of Q in label ordering (ijk) as in Eq. (9), such that the established Eq. (9), so that the symmetric component (containing the baryon transformation rules are maintained: singlet) is removed. In terms of the standard baryon field, with indices i, j,andk restricted to 1–3 and fermion BSγ ∼ α;a β;b γ;c β;b γ;c α;a γ;c α;a β;b anticommutation taken into account, this takes the following ijk ½Qi Qj Qk þ Qi Qj Qk þ Qi Qj Qk β α γ γ β α α γ β form: þ Q ;bQ ;aQ ;c þ Q ;cQ ;bQ ;a þ Q ;aQ ;cQ ;b i j k i j k i j k 1 ϵ γ B ¼ pffiffiffi ðϵ 0 B 0 þ ϵ 0 B 0 Þ: ð16Þ × abcðC 5Þαβ: ð10Þ ijk 6 ijk kk ikk jk pffiffiffi By transforming the individual quark operators (Q) The factor of 1= 6 is chosen so that the baryon wave within the baryon fields of Eqs. (9) and (10) and collecting functions have the correct normalization. This becomes the transformation matrices (U) together using the com- apparent when assigning conventional baryon labels to the mutation rule [47] elements of the B tensor (discussed in Sec. III). For example, extracting the proton states involves selecting the elements η η η pffiffiffi pffiffiffi −1 ið jþ kÞ QiUjk ¼ð Þ UjkQi; ð11Þ B112, B121 and B211, with weights of 1= 6, 1= 6 and pffiffiffi −2= 6, respectively, whose squares sum to 1. the resultant transformation rules for the baryon and Similarly, the baryon tensor containing the flavor-singlet antibaryons (suppressing the Dirac indices) are baryon follows from Eq. (10) for BS: η η η η η η η i0 ð jþ j0 Þþð i0 þ j0 Þð kþ k0 Þ 1 Bijk → ð−1Þ Uii0 Ujj0 Ukk0 Bi0j0k0 ; ð12Þ S B ¼ pffiffiffi ðþϵ 0 B 0 þ ϵ 0 B 0 þ ϵ 0 B 0 ijk 3 2 ijk kk jkk ik kik jk

η 0 ðη þη 0 Þþðη 0 þη 0 Þðη þη 0 Þ † † † B → −1 i j j i j k k B 0 0 0 − ϵ 0 B 0 − ϵ 0 B 0 − ϵ 0 B 0 Þ: ð17Þ kji ð Þ k j i Uk0kUj0jUi0i; ð13Þ jik kk kjk ik ikk jk pffiffiffi 3 2 −1 for both B and BS. Again, the normalization factor of ð Þ ensures the squares of the coefficients sum to 1. For the Λ0, one In matching the elements of the generalized baryon field S S S S S considers the six terms B123; B132; B213; B231; B312 and to those of the standard baryon matrix Bkk0 , the indices are BS , each of which provides a contribution with magnitude restricted from 1 to 3, and the elements are assigned so that 321pffiffiffi the result is antisymmetric under i ↔ j: Λ0= 6. Similarly pffiffiffi B ∝ ϵ − 1 − 3 ϵ BS 6Λ0 − 1–3 ijk ijk0 Bkk0 ði k ¼ Þ; ð14Þ ijk ijk ¼ ði k ¼ Þ: ð18Þ where the convention used for the standard baryon field of The symmetry relations that describe the relationships χPT is among the elements of the baryon tensors are important

094004-4 FLAVOR-SINGLET BARYONS IN THE GRADED SYMMETRY … PHYSICAL REVIEW D 94, 094004 (2016) pffiffiffi ðη þη Þðη þη 0 Þ S when considering their general form in the graded L ¼ 2 2g ð−1Þ i j k k B A 0 B 0 ; ð21Þ – S s kji kk ijk symmetry group, i.e. where the indices i k run from 1 ffiffiffi B p η η η η to 6. For the octet field , the following two symmetry ð iþ jÞð kþ k0 Þ S S L ¼ 6g ð−1Þ B A 0 B 0 : ð22Þ relations are sufficient to specify the two group invariants SS ss kji kk ijk of the symmetry, α and β [43], which are related to the We note these are special cases of the general irreducible standard D and F couplings: representations of graded Lie groups classified by Balantekin and Bars [58,59]. The leading coefficient in ηjηkþ1 Bikj ¼ð−1Þ Bijk; ð19aÞ LS is selected to define the coupling associated with the 0 η η η η η η η η process Λ → π0Σ0 as gs. B −1 i j B −1 i jþ j kþ k i B ijk ¼ð Þ jik þð Þ kji: ð19bÞ Our investigation of the separation of quark-flow con- nected and disconnected contributions to standard baryons Similarly, the following symmetry relations exist for the constrains the indices i–k to 1–3, whereas k0 1–6 BS ¼ field : incorporates the ghost meson-baryon contributions, which η η reveal the disconnected contributions. BS −1 i j BS ; jik ¼ð Þ ijk ð20aÞ In summary, the first-order interaction Lagrangian for η η low-lying octet and singlet mesons, octet baryons and low- BS −1 j k BS ikj ¼ð Þ ijk; ð20bÞ lying negative-parity singlet baryons, including the Dirac structure, takes the form η η η η BS −1 i kþ i j BS jki ¼ð Þ ijk; ð20cÞ Lð1Þ L L L L η η η η BA ¼ α þ β þð S þ H:c:Þþ SS BS −1 i kþ j k BS kij ¼ð Þ ijk; ð20dÞ η η η η μ 2α −1 ð iþ jÞð kþ k0 ÞB γ γ B ¼ ð Þ kji μ 5Akk0 ijk0 S ηiηjþηiηkþηjηk S μ B ¼ð−1Þ B : ð20eÞ 2βB γ γ B 0 kji ijk þ kji μ 5Aik0 k jk pffiffiffi ðηiþηjÞðηkþη 0 Þ þ 2 2gsð−1Þ k C. The Lagrangian BS γ μ B B γ μ BS × f μA 0 ijk0 þ k0ji μA 0 g The singlet Lagrangian will couple the singlet-baryon ffiffiffikji kk k k ijk p η η η η μ 6 −1 ð iþ jÞð kþ k0 ÞBS γ γ BS field to a combination of octet-meson and octet-baryon þ gssð Þ kji μ 5Akk0 ijk0 ; ð23Þ fields, A · B. Given the maximal symmetry for BS in BS Eq. (10), there are three different ways of contracting the where ijk denotes the odd-parity generalized field asso- 0 octet-baryon flavor indices. In keeping with the baryon ciated with B and thus Λ via Eq. (17). The loss of γ5 in construction of Labrenz and Sharpe [43],weconsidera the terms coupling octet- and singlet-flavor baryons reflects Lagrangian term similar to their term with coefficient α, the odd parity of the low-lying Λ0. BS B B kjiAkk0 ijk0 , where the first two indices of are contracted with BS; a term similar to their term with coefficient β, III. GHOST BARYON STATE IDENTIFICATION S B A 0 B 0 , where the latter two indices of B are contracted kji ik k jk In this section we review briefly the process of relating S BS B with B ; and a third candidate kjiAkk0 ik0j,wheretheouter the elements of B and BS to baryons having one ghost indices of B are contracted with BS. Noting that the octet quark and two ordinary quarks. This is relevant to our field of Eq. (9) has a symmetry constraint under the investigation of the ordinary Λ0 baryon, where one ghost exchange of the latter two indices as in Eq. (19a), this third quark can be introduced through the disconnected candidate is related to the first candidate and may be sea-quark loop. The ordinary baryons (those without ghost discarded. Similarly, the relation of Eq. (20a) for BS is quark content) adopt the labeling convention assigned in opposite to that of Eq. (19a) for B and therefore the second Eqs. (15)–(17). The case for one ghost quark in B was first candidate vanishes. Thus, there is only one group invariant present by Chen and Savage [46]. We will extend the combination. formalism to include BS. In the general case where all indices run from 1 to 6, the In identifying the intermediate states of the meson- commutation or anticommutation behavior of the elements baryon channels dressing a baryon, we find it is useful of the transformation matrices U depends on their location to assign state labels that provide information about the in the block matrix. As a result, grading factors must be charge, quark composition and flavor symmetry. Thus, in included when swapping the order of the indices. Noting defining the baryon labels, there are some differences in our BS that Eq. (20e) indicates that ijk is antisymmetric under notation with respect to Ref. [46]. Here, the choice in exchange of the outer indices in both the normal quark and baryon labels is to manifest the symmetry and flavor single ghost-quark sectors, the interaction Lagrangians take properties of baryons with a single ghost quark. If one the following forms: of the quarks is a (commuting) ghost particle, it may be

094004-5 JONATHAN M. M. HALL and DEREK B. LEINWEBER PHYSICAL REVIEW D 94, 094004 (2016) combined with the two remaining ordinary quarks in a second and third index positions of B are combined with Σ-like symmetric or a Λ-like antisymmetric configuration the antisymmetric baryons of Eq. (25) in of ordinary quarks. For example, for a proton with a ghost ~ þ ~ þ 1 1 , one defines new particles Σ ~ , and Λ ~ , which B ϵ ffiffiffi p;u p;u jik ¼ 2 tip pjk þ p sijk; ð27aÞ correspond to symmetric and antisymmetric combinations 6 of the two remaining quark flavors, respectively. B −B In addition, unphysical states such as doubly charged jki ¼ jik; ð27bÞ “ ” Σ~ þþ ~ “ ” p;u~ ðu; u; uÞ and negatively charged 4–6 – 1–3 1–3 ~ − ~ where i ¼ , j k ¼ and p ¼ . These two Σ ~ ðd; d; dÞ are members of the single ghost-quark bary- n;d equations map the symmetric baryon states to two elements pffiffiffi pffiffiffiffiffiffiffiffi ons. These states occur in both the graded symmetry of B with weight 1= 6 such that the weighting of 2=3 in approach and the diagrammatic approach by considering Eq. (26) provides a sum of squares equal to one. Similarly, all possible quark-flow topologies. Note that these states do Eqs. (27) distribute the antisymmetric ghost baryon states not contribute in full QCD, where the cancellation of the of Eqs. (25) over four elements of B with weight 1=2 such connected and disconnected quark-flow diagrams associ- that a sum of squares again equals one. ated with them enforces the Pauli exclusion principle. The complete flavor antisymmetry of BS for ordinary Following Ref. [46], the flavor-symmetric variants are quarks restricts its first ghost sector to baryon states having encoded in a rank-3 tensor sijk, which is symmetric in the antisymmetry in the ordinary quarks, i.e. the baryons of last two indices reserved for ordinary quarks: Eq. (25):

~ þþ ~ þ ~ þ s411 ¼ Σ ~ ;s511 ¼ Σ ~ ;s611 ¼ ΣΣ ~; 1 p;u p;d ;s BS ¼ pffiffiffi t0 ϵ ; ð28aÞ pffiffiffi pffiffiffi pffiffiffi ijk 6 ip pjk Σ~ þ 2 Σ~ 0 2 Σ~ 0 2 s412 ¼ p;u~ = ;s512 ¼ n;d~ = ;s612 ¼ Σ;s~= ; Σ~ 0 Σ~ − Σ~ − BS ¼þBS ; ð28bÞ s422 ¼ n;u~ ;s522 ¼ ~ ;s622 ¼ Σ;s~; jik ijk ffiffiffi n;d ffiffiffi ffiffiffi p 0 p 0 p Σ~ þ 2 Σ~ 2 Σ~ 2 S S s413 ¼ Σ;u~ = ;s513 ¼ Σ ~ = ;s613 ¼ Ξ;s~= ; B ¼þB : ð28cÞ ffiffiffi ;d ffiffiffi ffiffiffi jki ijk ~ 0 p ~ − p ~ − p s423 ¼ ΣΣ ~ = 2;s523 ¼ Σ ~ = 2;s623 ¼ ΣΞ ~= 2; ;u Σ;d ;s where i ¼ 4–6 and j–k ¼ 1–3. Equations (28), now with a Σ~ 0 Σ~ − Σ~ − prime to denote the association with the flavor-singlet s433 ¼ Ξ;u~ ;s533 ¼ Ξ;d~ ;s633 ¼ Ω;s~: baryon sector, map each baryon of Eqs. (25) over six ð24Þ elements of BS. Analogous results are obtained for the odd-parity BS using t0 . The baryon subscripts reflect the combined number of ijk ip In studies of the Λ0, the low-lying ghost states reside in strange quarks and strange ghost quarks and the super- the octet meson and octet baryon sectors. The odd-parity scripts record the total electric charge. The flavor of the BS residing ghost quark is explicit in the subscript. We note ghost states of ijk are of relevance to other states transiting Λ0 that although Ω has been used to denote the ðs;~ s; sÞ to and a meson. As an example, consider an octet combination, this baryon is actually associated with octet baryon transition to a singlet baryon and an octet meson. as opposed to decuplet baryons. Such physics is relevant to understanding the disconnected Using a similar notation, the Λ-like particles are encoded loop contributions in Σ → πΛ0 or Ξ → KΛ0. These become particularly important in lattice QCD, where the in an antisymmetric tensor, denoted tij: initial state can be a finite-volume excitation associated Λ~ 0 Λ~ − Λ~ − with a resonance. t41 ¼ Σ;u~ ;t51 ¼ Σ;d~ ;t61 ¼ Ξ;s~; This defines the method by which baryon labels are Λ~ þ Λ~ 0 Λ~ 0 t42 ¼ Σ;u~ ;t52 ¼ Σ;d~ ;t62 ¼ Ξ;s~; assigned to different elements of the three-index baryon S ~ þ ~ 0 ~ 0 tensors B and B and their odd-parity analogues. By t43 ¼ Λ ~ ;t53 ¼ Λ ~ ;t63 ¼ ΛΣ ~: ð25Þ p;u n;d ;s writing out the full Lagrangian in Eq. (23), one can simply These baryons are distributed over the elements of B in pick out the relevant contributions to each baryon in each the following manner. The manifestly symmetric part of B interaction channel available for the symmetry sub- 3 3 has the simple relation group SUð j ÞV. rffiffiffi 2 B 4 − 6 − 1 − 3 IV. FLAVOR-SINGLET Λ-BARYON COUPLINGS ijk ¼ 3sijk ði ¼ ;j k ¼ Þ; ð26Þ In this section we examine the coupling strengths for which places the ghost quark in the first index position of transitions involving the flavor-singlet component Λ0.As B. The symmetric states with ghost quarks mapped to the described at the beginning of Sec. II, the couplings of

094004-6 FLAVOR-SINGLET BARYONS IN THE GRADED SYMMETRY … PHYSICAL REVIEW D 94, 094004 (2016) 3 3 ~ the ghost sector in SUð j ÞV provide the couplings of quarks in an antisymmetric formation. The Σ baryons do quark-flow disconnected sea-quark loop contributions in not contribute. 6 3 SUð j ÞV that we are seeking. In writing out the relevant In addition, flavor-singlet symmetry is manifest. Each terms of the Lagrangian, we report the meson-baryon → meson charge state participating in the quark-flow dis- baryon transition terms, noting that the baryon → meson- connected loop contributes in a uniform manner with a 2 2 baryon terms are given by the Hermitian conjugate. coupling strength of 3 gs. The ghost-quark subscript on the baryon field enables A. Flavor-octet baryon dressings of the singlet baryon the identification of the flavor of the quark participating in the loop. Thus, matrix elements can be separated into their Here we consider the specific case relevant to the Λ 1405 quark-flow connected and disconnected parts for each low-lying odd-parity ð Þ baryon, known to have an quark flavor. Similarly, the baryon subscripts facilitate important flavor-singlet component [4]. We commence by the incorporation of SU 3 -flavor breaking effects through Λ0 → BΦ → Λ0 ð Þ considering the process via the variation of the hadron masses. Lagrangian of Eq. (23). The essential feature to be drawn from this analysis is that In this case, the u, d and s valence quarks that comprise the relative contribution of quark-flow connected and dis- Λ0 Φ the state may each contribute to the meson loop, connected diagrams is now known. Consider the process shown in Fig. 1. The ghost quark sector may enter through − 2 Λ0 → K p → Λ0 proportional to g . This process proceeds the disconnected loop, and therefore there are three choices s through a combination of quark-flow connected diagrams for the antiquark plus three choices for the ghost antiquark. and the quark-flow disconnected diagram of Fig. 1. See, for Noting that the flavor-singlet η0 meson cannot be combined example, Ref. [51]. The ghost Lagrangian reports a term with an octet baryon to form a flavor singlet; there are 3 6 − 1 17 Λ0ΦB rffiffiffi ð × Þ ¼ terms in the Lagrangian for the 2 coupling. At one loop, the baryon B contains at most one − Λ0 ~ −Λ~ þ gs 3 K p;u~ ; ð31Þ ghost quark for which the meson contains the correspond- ing antighost quark. indicating that this process proceeds through the quark-flow The SUð3j3Þ -flavor composition of the interaction V disconnected diagram with strength 2 g2. That is, two-thirds Lagrangian of Eq. (23) contains the regular full-QCD 3 s Λ0 → − → Λ0 meson-baryon couplings to the singlet Λ0: of the full-QCD process K p proceeds by the quark-flow disconnected diagram with a u quark flavor QCD 0 − participating in the loop. The fact that two-thirds of the L ¼ g Λ0fK n þ K p S;Λ0 s strength lies in the loop is perhaps surprising. 0 þ πþΣ− þ π0Σ0 þ π−Σþ The full-QCD processes involving the neutral π or η mesons are only slightly more complicated, as one needs to þΞ− 0Ξ0 ηΛ þ K þ K þ gð29Þ sum over more than one quark flavor participating in the sea-quark loop. After summing over the participating and the ghost meson to ghost baryon couplings: flavors, each full-QCD process proceeds with one-third rffiffiffi of the strength in quark-flow connected diagrams and 2 0 LGhost − Λ0 ~ −Λ~ þ ~ Λ~ 0 two-thirds of the strength in quark-flow disconnected 0 ¼ gs ðK p;u~ þ K ~ S;Λ 3 n;d diagrams. π~ þΛ~ − π~ −Λ~ þ þ Σ;d~ þ Σ;u~ ~ þ ~ − ~ 0 ~ 0 B. Flavor-singlet baryon dressings of octet baryons þ K ΛΞ;s~ þ K ΛΞ;s~Þ 1 It is also of interest to examine processes where the ffiffiffi π~ Λ~ 0 − π~ Λ~ 0 flavor-singlet baryon participates as an intermediate state þ p ð 0 Σ;u~ 0 Σ;d~ Þ 3 dressing of an octet baryon. The full Lagrangian describing S 1 ~ 0 ~ 0 ~ 0 the processes B → ΦB → B, where B is an octet baryon, η~ΛΣ ~ η~Λ ~ − 2η~ΛΣ ~ ; 30 þ 3 ð ;u þ Σ;d ;sÞg ð Þ can likewise be decomposed into full-QCD and ghost meson to ghost baryon components, suppressing the Dirac where the Dirac structure has been suppressed. In the full- structure: QCD case, the relative coupling strengths of the πΣ, KN, KΞ and ηΛ channels are consistent with early work [60] as LQCD 0Λ0 þΛ0 S;Λ0 ¼ gsfnðK ÞþpðK Þ expected. Σ− π−Λ0 Σ0 π0Λ0 Σþ πþΛ0 The ghost terms, however, present novel features. Only þ ð Þþ ð Þþ ð Þ ~ antisymmetric Λ ghost baryons contribute. Thus, the þ Ξ−ðK−Λ0ÞþΞ0ðK0Λ0Þ coupling of a baryon with a single ghost quark to a Λ ηΛ0 flavor-singlet baryon is constrained to have its two ordinary þ ð Þg; ð32Þ

094004-7 JONATHAN M. M. HALL and DEREK B. LEINWEBER PHYSICAL REVIEW D 94, 094004 (2016)

LGhost π~ −Λ~ þ0 η~ Λ~ 00 ~ 0Λ~ 00 η~ Λ~ þ0 π~ þΛ~ 00 ~ þΛ~ 00 S;Λ0 ¼ gsfnð p;u~ þ d n;d~ þ K Σ;s~Þþpð u p;u~ þ n;d~ þ K Σ;s~Þ Σ− π~ −Λ~ 00 η~ Λ~ −0 ~ 0Λ~ −0 Σþ η~ Λ~ þ0 π~ þΛ~ 00 ~ þΛ~ 00 þ ð Σ;u~ þ d Σ;d~ þ K Ξ;s~ Þþ ð u Σ;u~ þ Σ;d~ þ K Ξ;s~Þ 1 0 − ~ þ0 ~ 00 ~ 00 þ ~ −0 ~ 0 ~ 00 ~ þ ~ −0 þ pffiffiffi Σ ð−π~ ΛΣ ~ þ η~ ΛΣ ~ − η~ Λ ~ þ π~ Λ ~ − K ΛΞ ~ þ K ΛΞ ~ Þ 2 ;u u ;u d Σ;d Σ;d ;s ;s Ξ− ~ −Λ~ 00 ~ 0Λ~ −0 η~ Λ~ −0 Ξ0 ~ −Λ~ þ0 ~ 0Λ~ 00 η~ Λ~ 00 þ ðK Σ;u~ þ K Σ;d~ þ s Ξ;s~ Þþ ðK Σ;u~ þ K Σ;d~ þ s Ξ;s~Þ 1 0 − ~ þ0 ~ 00 ~ 00 ~ 00 þ ~ −0 ~ − ~ þ0 ~ ~ 00 ~ 0 ~ 00 ~ þ ~ −0 þ pffiffiffi Λðπ~ ΛΣ ~ þ η~ ΛΣ ~ þ η~ Λ ~ − 2η~ ΛΣ ~ þ π~ Λ ~ − 2K Λ ~ − 2K Λ ~ þ K ΛΞ ~ þ K ΛΞ ~ Þg: ð33Þ 6 ;u u ;u d Σ;d s ;s Σ;d p;u n;d ;s ;s

In each channel of Eq. (33), there are terms which there is no ss contribution in any channel apart from the octet represent the disconnected loop contributions of the normal Λ. In the Σ, where both light flavor ηq mesons appear, the full-QCD interactions of Eq. (32), and there are additional flavor symmetry of the is apparent. Similarly, in the Λ terms that serve to cancel the unphysical quark-flow channel, the flavor symmetry of the η contributions iden- connected diagrams associated with each channel. For q tifies the octet-η participating to cancel the full-QCD process example, in the case of the neutron channel n, the relevant Λ ηΛ0 η η ~ 0 ~ 00 ð Þ. In all other cases, the light-quark u and d mesons ghost term K ΛΣ ~ comes with the same magnitude as the ;s propagate with the mass of the pion. normal process, gs, indicating that there is no completely connected contribution for this process. The physical process proceeds through a quark-flow disconnected dia- C. Flavor-singlet baryon dressings of the singlet baryon π~ −Λ~ þ0 gram. The other two terms in this channel, p;u~ and Finally, we also consider the quark-flow disconnected ~ 00 loop contributions to the process Λ0 → ΦBS → Λ0. η~dΛ ~ , occur to cancel the aforementioned unphysical n;d We find connected diagrams. These contributions are also important and must be included when evaluating the quark-flow LQCD Λ0η0Λ0 disconnected contributions to baryon matrix elements. SS;Λ0 ¼ gss ð34Þ Note that, unlike the flavor-singlet Λ-baryon couplings of Eq. (30), where the structure of the π~ 0 and η~ is manifest, here and

rffiffiffi g 2 0 0 − Ghost ssffiffiffi 0 ~ −0 ~ þ0 ~ 0 ~ 0 þ0 ~ 0 −0 ~ þ0 L 0 ¼ p Λ ðK Λ ~ þ K Λ ~ þ π~ Λ ~ þ π~ ΛΣ ~ SS;Λ 2 3 p;u n;d Σ;d ;u 1 ~ þ0 ~ −0 ~ 00 ~ 00 00 ~ 00 00 ~ 00 þ K ΛΞ ~ þ K ΛΞ ~Þþpffiffiffi ðπ~ ΛΣ ~ − π~ Λ ~ Þ ;s ;s 3 ;u Σ;d  1 0 ~ 00 0 ~ 00 0 ~ 00 η~ ΛΣ ~ η~ Λ ~ − 2η~ ΛΣ ~ : 35 þ 3 ð ;u þ Σ;d ;sÞ ð Þ

Here we have added a prime to the meson labels to remind important and must be included when evaluating the the reader that in full QCD these contributions enter with quark-flow disconnected contributions to baryon matrix the meson propagating with the mass of the flavor-singlet elements. η0. The symbols serve to indicate the manner in which specific quark-flow disconnected loops contribute to the V. CONCLUSION full-QCD process. Again, we observe the physical process proceeding An extension of the graded symmetry approach to through a quark-flow disconnected diagram. Here, the partially quenched QCD has been presented. The extension final term, indicating the ghost-η0 contributions, pro- builds on previous theory by explicitly including flavor- vides a total contribution with strength gss, precisely singlet baryons in its construction. The aim is to determine removing the full-QCD contribution of Eq. (34).The the strength of both the quark-flow connected and quark- remaining ghost terms enter to ensure unphysical flow disconnected diagrams encountered in full-QCD contributions from quark-flow connected diagrams do calculations of hadronic matrix elements containing fla- not contribute in full QCD. Their contributions are vor-singlet baryon components.

094004-8 FLAVOR-SINGLET BARYONS IN THE GRADED SYMMETRY … PHYSICAL REVIEW D 94, 094004 (2016) This is particularly important in light of recent progress flavors participating in the loops and their relative con- in the calculation of the electromagnetic properties of the tributions are provided in Eq. (30) of Sec. IVA. Λð1405Þ in lattice QCD [13], where flavor-singlet sym- We have also considered the separation of quark-flow metry is known to play an important role [4]. There only the connected and disconnected contributions for the flavor- quark-flow connected contributions are calculated. singlet baryon dressings of octet baryons in Sec. IV B and Because a determination of the quark-flow disconnected flavor-singlet baryon dressings of the flavor-singlet baryon in contributions to baryon excited states presents formidable Sec. IV C. challenges to the lattice QCD community, it is essential to When combined with previous results for flavor-octet have an understanding of the physics missing in these baryons, this information enables a complete understanding contemporary lattice QCD simulations. An understanding of the contributions of disconnected sea-quark loops in the of the couplings of the quark-flow disconnected contribu- matrix elements of baryons and their excitations, including tions to baryon matrix elements is vital to understanding the Λð1405Þ. This allows an estimation of quark-flow dis- QCD and ultimately comparing with experiment. connected corrections to lattice QCD matrix elements or an We have discovered that the full-QCD processes adjustment of other observables prior to making a comparison with the quark-flow connected results of lattice QCD. Λ0 → K−p → Λ0; Λ0 → K0n → Λ0; ACKNOWLEDGMENTS Λ0 → πþΣ− → Λ0; Λ0 → π0Σ0 → Λ0; We thank Tony Thomas for inspiring this study, Phiala Λ0 → π−Σþ → Λ0; Λ0 → KþΞ− → Λ0; Shanahan for helpful discussions and Stephen Sharpe for Λ0 → K0Ξ0 → Λ0; and Λ0 → ηΛ → Λ0 insightful comments in the preparation of this manuscript. This research is supported by the Australian Research proceed with two-thirds of the full-QCD strength in the Council through Grants No. DP120104627, No. DP1401 quark-flow disconnected diagram. Details of the quark 03067 and No. DP150103164 (D. B. L.).

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