Pos(QNP2012)112 £ Richard.Williams@Uni-Graz.At Reinhard.Alkofer@Uni-Graz.At Helios.Sanchis-Alepuz@Uni-Graz.At Speaker

Home , Delta baryon

PoS(QNP2012)112 http://pos.sissa.it/ £ [email protected] [email protected] [email protected] Speaker. A calculation of the massesgether and with electromagnetic their evolution properties with of the current theSalpeter quark Delta approach mass and is with presented. Omega the Hereby baryon interaction a to- The generalized truncated model Bethe- to dependence is a explored dressed by one-gluon investigating two exchange forms is for employed. the dressed gluon exchange. £ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

Sixth International Conference on Quarks andApril Nuclear 16-20, Physics 2012 Ecole Polytechnique, Palaiseau, Paris Reinhard Alkofer University of Graz, Austria E-mail: Richard Williams University of Graz, Austria E-mail: Helios Sanchis-Alepuz University of Graz, Austria E-mail: Baryon properties from the covariant Faddeev equation PoS(QNP2012)112 2 ], Z ]. 16 (2.1) (2.2) (2.3) 13 , 12 , 11 , its bare counter- 10 multiplied by the ) : p 2 ]. For more details, ) ( q Helios Sanchis-Alepuz p 1 = 18

0 ν

S q k µ : ( q ν the quark mass function. γ µν

: ) ) D 2 2 ν such that ) p µν q γ 2 p ( ( ) δ q ; p

eff M k ; ( µ k α ν ) = ( γ ) q ν ) Γ and q ( q ) Γ ( . ) ( k ] at the level of the Rainbow-Ladder (RL) 2 µν ( 9 µν p T µν S , T ( T µ 8 with 2 2 A 2-body γ ) , Z = 2 4 7 K µν ) q k , 2 ( 4 2 π 6 D q ) = d 2 eff p ( is decomposed into an irreducible three-quark contri- α ; k 2 2 Z ( f Z ν 1 follows from Slavnov-Taylor identities. Consideration of Γ π Z 3-body ) 4 2 2 2 K q g is the inverse quark propagator, with Z ( = ¡ + µν ) ] with a model for the quark-gluon vertex [ . We combine 1 2 2 D

p 0 q 17 ( S 2 2-body π = 2 g ) 4 K M 2 Z f q 1 + ( Z Z ] and references therein. ) = is just the transverse projector ip 9 p

], whereas today calculations have reached parity with meson studies in the ( ]. This dominance is well-earned since it performs well phenomenologically. ) 5 1 µν 2 ,

D p 15 4 S ( , , A 3 14 is an effective interaction subsuming non-perturbative features of the gluon propaga- , 2 ) ) = 2 , p q 1 ( ( are renormalisation constants of the quark propagator and quark-gluon vertex respectively. 1 ]. eff

f 1 S α We start with the DSE for the quark propagator, With the truncation and effective interactions thus chosen, we shall proceed to calculate the A covariant description of relativistic two- and three-body bound states is provided by the In the case of baryons, only one RL interaction has typically been tested, known as the Maris- In this paper we compare two interactions. The ﬁrst is the Maris-Tandy (MT) model men- 19 Z gluon dressing function combining the gluon dressing [ see [ In Landau gauge, tor and the quark-gluon vertex,chiral and symmetry leads to the two-body kernel in RL approximation, 2. Framework Here and Here For the baryon, the three-body kernel form of a more intricate three-body description [ appropriate meson and baryon masses, together withof a the calculation Delta of the baryon electromagnetic in properties Rainbow-Ladderfor approximation. details Here we we refer provide to a [ summary of the results; approximation. In the meantime, meson studies have made progress beyond RL [ However, it does not draw ontions the and plethora so of we information choose we here now to know investigate about in QCD addition Green’s another func- effective interaction. bution and the sum of permuted two-body kernels (generalised) Bethe-Salpeter equations. Until recently quark-diquarkstandard calculations [ of baryons were Tandy model [ Baryon properties from the covariant Faddeev equation 1. Introduction part. The quark wave-function renormalisation is 1 tioned above. The second, which we will refer to as (AFW), has been proposed in reference [ PoS(QNP2012)112 ]. 22 ] and pNRQCD [ Helios Sanchis-Alepuz 21 , 20 . 1 4.4 4.9 4.7 4.9(0.25) MT AFW exp. MT AFW lattice pNRQCD 0.94 0.971.26 1.221.72 0.94 1.80 1.23 1.67 13.7 13.8 14.4 14.5(0.25) ccc bbb ∆ N Ω 10%. This implies that we have a qualitative Ω Ω 3 for MT and AFW models. We compare to lattice data; 2 π m , we see that the charm quarks exhibit similar trends 1 1 GeV, relevant for DCSB. As the quark masses evolve to . This is not surprising as they both exhibit similar features 1 0.138 0.496 2.980 9.391 y y y y masses with ∆ 0.139 0.497 2.980 9.390 and N y y y y , ρ MT AFW exp. MT AFW exp. 0.9421.075 0.9613.163 0.892 1.1149.466 1.020 3.302 3.097 9.621 9.460 0.743 0.710 0.775 0.140 0.496 2.979 9.388 + )

) ) ψ ) 0 1 ? ) ) ) ) ) c b = Evolution of ϒ ρ K φ J π K η η = = ( ( ( ( ( ( ( ( ( Meson and baryon masses (GeV) for both interactions (ﬁtted values marked †), compared to exper- ] for references. s s 8 b n s c n c b PC PC s , c c n n J J n b n b For the heavy quarks, see also Table Both models reproduce the pattern of dynamical chiral symmetry breaking (DCSB) and related 7 Table 1: iment. Heavy-Omega baryons are not yet observed thus we compare to lattice [ model independence within RL. We collect results in Table 3. Hadron Masses observables. This is also seen fordescribe light the states data such well, as as the seen rho, in nucleon Fig. and delta where both models at the important momentum region of heavier values, a greater deviation iswith seen lattice between the and/or two experiment models is although always qualitative agreement within Figure 1: see [ Baryon properties from the covariant Faddeev equation PoS(QNP2012)112 , 0 E G (4.1) gauging 50 72 : : 4 4 + ; Exp. , 691 MeV (DW3) ) 4 i 3.54 baryon. For both ]. These presently F = P ( + π 26 β Helios Sanchis-Alepuz ... 0 ∆ m 1 . We use the β 3 F P M 0 G β . We compare our two model ]. Improvement of this is in 2 ) 0 is less so. One may speculate 9 Q 0 0 β M 0 ( α 1 4 α bbb M Q δ Ω G µ µ M M P P 2 4 F F and for 509 MeV (DW2) and

E ]. 0 = µ µ are the initial and ﬁnal baryon total momenta, 2 E γ γ 29 r π f i i , D ) ) P m 2 4 4 28 and magnetic octupole F F ∆ 2 terms, as discussed in [ and + + E i 6 1 3 DW1 DW2 DW3 P G F F Q form factors. Note that owing to isospin symmetry, for 2.35 (16) 2.68 (13) 2.589 (78) ( ( 0.373 (21) 0.353 (12) 0.279 (6) = + ∆ + ) f and 1 P ( 4 magnetic dipoles with small statistics. Thus we compare with a 0 ] to couple our baryon to an external EM ﬁeld such that gauge 384 MeV (DW1), Q + = αα = particle is characterised by four form factors ∆ 24 P . These form factors can be related to the usual electric monopole 2 , π i / mass is well reproduced however the 3 P m 23 and ϒ

) = f good agreement with the lattice where the quark core is probed. At smaller 0.672.22 0.60 2.33 Q F-MT F-AFW P ++ ; , electric quadrupole Q P 1 their form-factors differ only by a factor corresponding to their charge. Due to ∆ ( = M )

2 Q G αβ ∆ ) ; we show the computed results for the charge radius of the 0 µ we show results for the ( 2 J (fm technique, [ 1 and

2 M 0 0 2 E G r ∆ Comparison of results for the charge radius is the Rarita-Schwinger projector,

, P ]. Where available we also show experiment [ ]. ++ We see at large In Table In Fig. The current for a spin- 9 27 ∆ , , 25 26 respectively, and magnetic dipole calculations to the lattice at values the pion cloud isin not this captured region, by the our electricto model quadrupole cancellations and enhanced and so by magnetic deviations 1 octupole are feature expected. numerical Furthermore inaccuracies due where of equations suffer from large errors, especially foron the the electric magnetic quadrupole octupole. form factor and give no information progress. Table 2: [ symmetry is preserved. Definitions[ and conventions for this and further details can be found in models, our results appear considerably higher than those of the lattice. A possible explanation the the short lifetime of the Deltaare resonance it restricted is to difficult the to study it experimentally, and EM properties Baryon properties from the covariant Faddeev equation with AFW giving heavier masses thancoincide. MT. Unexpectedly We see for that bottom the quarks thehere calculated as masses to the relevancemodels of to three-body the interactions. heavy quark To make sectorevolution precise where to corrections statements light beyond one quarks. RL should are fit Weeffects suppressed, remark at the and that u/d then quark the study masses majority the in of the RL parameterisation models of the already interaction. capture beyond-RL 4. Electromagnetic Form Factors of the lattice calculation using dynamical Wilson fermions at different pion masses [ PoS(QNP2012)112 509 MeV (red) and = Helios Sanchis-Alepuz decay channel opens π π m N ! ∆ ]. Since in our calculation we have no 384 MeV (green), 30 = π m 5 for the MT and AFW models. The results are compared + ∆ PT shows than when the χ ] for dynamical Wilson fermions at 26 (color online) EM form factors for the 691 MeV (blue). We have shown results for the electromagnetic form factors of the Delta baryon using a three- We thank G. Eichmann and C. Fischer for useful discussions. This work was supported by = π Figure 2: body covariant Bethe-Salpeter equation and tworesults different show models good for the agreement effective with interaction.a lattice Our qualitative data, model especially independence. at Calculations large ofas photon the well electromagnetic momentum, as form and refinements factors in feature for the the numerical Omega techniques are in progress. Acknowledgments the Austrian Science Fund FWFDoctoral Program under W1203 Projects (Doctoral No. Program “Hadrons in P20592-N16 vacuum, and nuclei No. and stars”). M1333-N16 and the m 5. Summary and Outlook Baryon properties from the covariant Faddeev equation is the pion-mass dependence ofphysical the value charge from radius, above. which grows Moreover as the pion massmechanism for approaches the the Delta to decay, our larger result is not unreasonable. to lattice data [ the charge radius changes abruptly to a lower value [ PoS(QNP2012)112 (1999) 60 (2012) 034013. (2011) 096003. (2009) 106. 85 84 (2011) 041. 324 (2010) 201601. Helios Sanchis-Alepuz 104 (2009) 054028. -TNT-II (1997) 679. 80 (1998) 2459. (2012) 2019. (2004) 014014. 627 58 72 70 (2008) 53. (2010) 114017. (2010) 114017. 38 82 82 (2010) 075021. (1999) 044003. (1999) 044004. 37 60 60 (2009) 122001. (2009) 081601. (2005) 471. 103 (2008) 074006. 6 103 755 (1997) 3369. 78 (1999) 055214. 56 (2009) 115. 60 825 (2011) 581. (2011) 014014. 84 1374 (2002) 272001. (2009) 092. 89 09 , Phys. Rev. Lett. 062201. M. Vanderhaeghen, Nucl. Phys. A A. Tsapalis, PoS CD et al. [5] D. Nicmorus, G. Eichmann and R. Alkofer, Phys. Rev. D [4] D. Nicmorus, G. Eichmann, A. Krassnigg and R. Alkofer, Phys. Rev. D [1] G. Hellstern, R. Alkofer, M. Oettel and H. Reinhardt, Nucl. Phys. A [2] M. Oettel, G. Hellstern, R. Alkofer[3] and H.J. Reinhardt, C. Phys. R. Rev. Bloch, C C. D. Roberts, S. M. Schmidt, A. Bender and M. R. Frank, Phys. Rev. C [6] G. Eichmann, R. Alkofer, A. Krassnigg[7] and D.G. Nicmorus, Eichmann, Phys. Phys. Rev. Lett. Rev. D [8] H. Sanchis-Alepuz, G. Eichmann, S. Villalba-Chavez[9] and R.H. Alkofer, Phys. Sanchis-Alepuz, Rev. PhD D Thesis, University of Graz (2012), arXiv:1206.5190 [hep-ph]. Baryon properties from the covariant Faddeev equation References [10] C. S. Fischer and R. Williams, Phys. Rev. D [11] C. S. Fischer and R. Williams, Phys. Rev. Lett. [13] R. Williams, EPJ Web of Conferences[14] 3 (2010)P. 03005. Maris and C. D. Roberts,[15] Phys. Rev. C P. Maris and P. C. Tandy, Phys. Rev. C [12] L. Chang and C. D. Roberts, Phys. Rev. Lett. [21] R. Lewis, AIP Conf. Proc. [16] R. Alkofer, C. S. Fischer and[17] R. Williams,R. Eur. Phys. Alkofer, W. J. Detmold, A C. S.[18] Fischer and P.R. Maris, Alkofer, Phys. C. Rev. S. D Fischer, F.[19] J. Llanes-Estrada andH. K. Sanchis-Alepuz, Schwenzer, R. Annals Alkofer, Phys. G. Eichmann[20] and R.T. Williams, -W. PoS Chiu QCD and T. -H. Hsieh, Nucl. Phys. A [22] F. J. Llanes-Estrada, O. I. Pavlova and R. Williams, Eur. Phys. J. C [23] A. N. Kvinikhidze and B. Blankleider, Phys. Rev. C [24] A. N. Kvinikhidze and B. Blankleider, Phys. Rev. C [25] D. Nicmorus, G. Eichmann and R. Alkofer, Phys. Rev. D [26] C. Alexandrou, T. Korzec, G. Koutsou, C. Lorce, J. W. Negele, V. Pascalutsa, A. Tsapalis and [27] C. Alexandrou, T. Korzec, G. Koutsou, C. Lorce, V. Pascalutsa, M. Vanderhaeghen, J.[28] W. NegeleK. and Nakamura et al. (Particle Data Group), J. Phys. G [29] M. Kotulla, J. Ahrens, J. R. M. Annand, R. Beck, G. Caselotti, L. S. Fog, D. Hornidge and S. Janssen [30] T. Ledwig, J. Martin-Camalich, V. Pascalutsa and M. Vanderhaeghen, Phys. Rev. D