PUBLISHED VERSION Jonathan M. M. Hall and Derek B. Leinweber Flavor-singlet baryons in the graded symmetry approach to partially quenched QCD Physical Review D - Particles, 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 PERMISSIONS http://publish.aps.org/authors/transfer-of-copyright-agreement Permission 4.11.2015 “The author(s), and in the case of a Work Made For Hire, as defined in the U.S. Copyright Act, 17 U.S.C. §101, the employer named [below], shall have the following rights (the “Author Rights”): 3. The right to use all or part of the Article, including the APS-prepared version without revision or modification, on the author(s)’ web home page or employer’s website and to make copies of all or part of the Article, including the APS-prepared version without revision or modification, for the author(s)’ and/or the employer’s use for educational or research purposes.” 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 Matter (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 baryon excitations in lattice QCD presents new challenges in the determination of sea-quark 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 mesons, 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 meson- 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 hadrons 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 strange quark 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 kaon in a relative S and sink of the quark propagator at the current insertion wave about the nucleon. 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 gluon exchange, resolving a nontrivial contribution, which can be related to the proton and neutron 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 hadron 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 quarks 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 Particle 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 field 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 effective field theory. 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.