The distribution of gluon fields in hadrons is of fundamental interest in
QCD. Using lattice QCD we have observed the formation of gluon flux tubes
within 3 quark (baryon) and quark plus antiquark (meson) systems for a wide
variety of spatial distributions of the color sources. In particular we have
investigated three quark configurations where two of the quarks are close
together and the third quark is some distance away, which approximates a quark
plus diquark string. We find that the string tension of the quark-diquark
string is the same as that of the quark-antiquark string on the same lattice.
We also compare the longitudinal and transverse profiles of the gluon flux
tubes for both sets of strings, and find them to be of similar radii and to
have similar vacuum suppression.Comment: 6 pages, 9 figures and 1 tabl

A. I. Signal

D. B. Leinweber

F. Bissey

I. Montvay

K. Orginos

Z. Liu

Physical Review D

English

arXiv

Crossref

Comparison of gluon flux-tube distributions for quark-diquark and quark-antiquark hadrons

The distribution of gluon fields in hadrons is of fundamental interest in QCD. Using lattice QCD we have observed the formation of gluon flux tubes within three-quark (baryon) and quark plus antiquark (meson) systems for a wide variety of spatial distributions of the color sources. In particular we have investigated three-quark configurations where two of the quarks are close together and the third quark is some distance away, which approximates a quark plus diquark string. We find that the string tension of the quark-diquark string is the same as that of the quark-antiquark string on the same lattice. We also compare the longitudinal and transverse profiles of the gluon flux tubes for both sets of strings, and find them to be of similar radii and to have similar vacuum suppression.F. Bissey and A. I. Signal, D. B. Leinwebe

Bissey, F.

Signal, A.

Leinweber, D.

Adelaide Research &amp; Scholarship

PUBLISHED VERSION     Bissey, Francois; Signal, A. I.; Leinweber, Derek Bruce  Comparison of gluon flux-tube distributions for quark-diquark and quark-antiquark hadrons Physical Review D, 2009; 80(11):114506  ©2009 American Physical Society  http://link.aps.org/doi/10.1103/PhysRevD.80.114506            http://link.aps.org/doi/10.1103/PhysRevD.62.093023                         http://hdl.handle.net/2440/57606   PERMISSIONS http://publish.aps.org/authors/transfer-of-copyright-agreement   “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.”    3rd June 2013  Comparison of gluon flux-tube distributions for quark-diquark and quark-antiquark hadronsF. Bissey and A. I. SignalInstitute of Fundamental Sciences, Massey University, Private Bag 11 222, Palmerston North, New ZealandD.B. LeinweberCentre for the Subatomic Structure of Matter and School of Chemistry and Physics, University of Adelaide, SA 5005, Australia(Received 22 October 2009; published 28 December 2009)The distribution of gluon fields in hadrons is of fundamental interest in QCD. Using lattice QCD wehave observed the formation of gluon flux tubes within three-quark (baryon) and quark plus antiquark(meson) systems for a wide variety of spatial distributions of the color sources. In particular we haveinvestigated three-quark configurations where two of the quarks are close together and the third quark issome distance away, which approximates a quark plus diquark string. We find that the string tension of thequark-diquark string is the same as that of the quark-antiquark string on the same lattice. We also comparethe longitudinal and transverse profiles of the gluon flux tubes for both sets of strings, and find them to beof similar radii and to have similar vacuum suppression.DOI: 10.1103/PhysRevD.80.114506 PACS numbers: 12.38.Gc, 12.38.Aw, 14.70.DjI. INTRODUCTIONThere has recently been a renewal of interest in theproperties of diquarks in hadronic systems, as they mayplay an important role in the existence of exotic states, suchas theþ, or in explaining the scarcity of such exotics [1].While string-type models consisting of a quark plus adiquark have been studied using various analytic tech-niques (see [2] for a review), there have only been a fewstudies of this type of system on the lattice. These studieshave investigated the mass of diquarks [3–7], and, morerecently, the nature of diquark correlations [8]. Recentlywe investigated the formation of flux tubes in static baryonsystems on the lattice using a high statistics approachwhich enabled us to observe correlations between thevacuum action density and the quark positions in a gaugeindependent manner [9]. Such a study is similar to theinvestigations found in [10–14]. In that work the threequarks were positioned approximately equidistant, and aY-shaped flux tube was observed to form at large inter-quark distances. In this work we extend our study of three-quark systems to the case where two of the quarks are closetogether and the third is some distance away.In QCD, two quarks close together, a diquark, can trans-form either according to the conjugate representation ð3Þ orthe sextet (6) representation of SUð3Þ. The color hyperfineinteraction then leads to attraction in the spin singlet, scalardiquark channel, while the spin triplet, axial vector diquarkis disfavored. Hence low-lying diquarks should have posi-tive parity and belong to the color 3 representation, and sohave many properties similar to an antiquark. In latticeQCD this should lead to the formation of quark-diquarkflux tubes with similar physical characteristics to those ofquark-antiquark flux tubes. In particular we would expectthe long range linear part of the quark-diquark potential tohave the same slope as that of the quark-antiquark poten-tial, corresponding to the flux tubes having the same energydensity, and we would expect the flux tubes to have similartransverse size. In this work we investigate whether thesesimilarities do indeed hold.II. FLUX TUBES ON THE LATTICEIn order to study flux tubes on the lattice, we begin withthe standard approach of connecting static quark (andantiquark) propagators with spatial-link paths in a gauge-invariant manner. We use APE-smeared spatial-link pathsto propagate the quarks from a common origin to theirspatial positions as illustrated in Fig. 1. In earlier work wesaw that after approximately 30 APE smearing steps wehad obtained optimal overlap with the ground state. Thestatic quark propagators are constructed from time directedlink products at fixed spatial coordinate,QiUtð ~x; tiÞ, usingthe untouched ‘‘thin’’ links of the gauge configuration. Inprinciple, the ground state is isolated after sufficient timeevolution. Finally smeared-link spatial paths propagate thequarks back to the common spatial origin.The three-quark Wilson loop is thus defined asW3Q ¼ 13! "abc"a0b0c0Uaa01 Ubb02 Ucc03 ; (1)where Uj is a staple made of path-ordered link variablesUj  P expigZjdxAðxÞ; (2)and j is the path along a given staple as shown in Fig. 1. Incontrast the quark-antiquark Wilson loop is given by theproduct of two staplesWQ Q ¼ aba0b0Uaa01 ðUbb02 Þy: (3)The three-quark configurations we use to approximate aquark-diquark string are T shapes, with the origin at thePHYSICAL REVIEW D 80, 114506 (2009)1550-7998=2009=80(11)=114506(6) 114506-1  2009 The American Physical Societyjunction of the T. Two quarks are positioned one latticestep in opposite directions from the origin (approximatingthe diquark), and the third is placed from 1 to 12 latticesteps in an orthogonal direction, as shown in Fig. 2.In this work we have used 300 quenched QCD gauge-field configurations created with the Oða2Þ-mean-field im-proved Luscher-Weisz plaquette plus rectangle gauge ac-tion [15] on 163  32 lattices with the long dimensionbeing the x direction, making the spatial volume 162 32. Two hundred of these configurations were at  ¼ 4:60(as in our previous work) and 100 at  ¼ 4:80, to inves-tigate the use of a finer lattice. These values of  givelattice spacings a of 0.123 and 0.0945 fm, respectively.We use lattice symmetries to improve the signal to noiseratio of our measurements. These include translationalinvariance (any point on the lattice can be taken as theorigin), reflection in the x plane, and 90 rotational sym-metry about the x axis. The advantage of this approach isthat we do not have to perform any gauge fixing to find asignal in our flux distributions.We characterize the gluon-field fluctuations in our con-figurations using the gauge-invariant action density Sð ~y; tÞobserved at spatial coordinate ~y and Euclidean time trelative to the origin of the Wilson loop. We calculate theaction density using the highly-improvedOða4Þ three-loopimproved lattice field-strength tensor [16] on four-sweepAPE-smeared gauge links. While the use of this highly-improved action suppresses correlations close to the quarkpositions, it gives good resolution of the flux-tube correla-tions we are interested in.Defining the quark positions as ~ri relative to the origin ofthe Wilson loop, and denoting the Euclidean time extent ofthe loop by , we evaluate the following correlation func-tions:C3Qð ~y; ~r1; ~r2; ~r3; Þ ¼hW3Qð ~r1; ~r2; ~r3; ÞSð ~y; =2ÞihW3QðÞihSð ~y; =2Þi ; (4)CQ Qð ~y; ~r1; ~r2; Þ ¼hWQ Qð ~r1; ~r2; ÞSð ~y; =2ÞihWQ QðÞihSð ~y; =2Þi; (5)where h  i denotes averaging over configurations andlattice symmetries. These correlate the quark positions,via the Wilson loops, with the gauge-field action in agauge-invariant manner. For fixed quark positions andEuclidean time, the correlation functions are scalar fieldsin three dimensions. For values of ~y well away from thequark positions ~ri, there are no correlations and C! 1.Also the correlators are positive definite, eliminating anysign ambiguity on whether vacuum field fluctuations areenhanced or suppressed in the presence of static quarks.We find that C is generally less than 1, signaling thex OQQQ231yFIG. 2. Projection of the T-shape path on the x-y plane.FIG. 3 (color online). Expulsion of gluon-field fluctuationsfrom the region of static quark sources illustrated by the spheres.An isosurface of Cð ~yÞ is illustrated by the translucent surface. Asurface plot (or rubber sheet) describes the values of Cð ~yÞ for ~y inthe quark plane, ðy1; y2; 0Þ.FIG. 1. Gauge-link paths or ‘‘staples,’’ U1, U2, and U3, form-ing a three-quark Wilson loop with the quarks located at ~r1, ~r2,and ~r3. "abc and "a0b0c0 denote color antisymmetrization at thesource and sink, respectively, while  indicates evolution of thethree-quark system in Euclidean time.F. BISSEY, A. I. SIGNAL, AND D.B. LEINWEBER PHYSICAL REVIEW D 80, 114506 (2009)114506-2expulsion of vacuum fluctuations from the interior ofhadrons.In Figs. 3 and 4 we show examples of the expulsion ofvacuum fluctuations and the formation of flux tubes for ourquark-diquark and quark-antiquark configurations.III. EFFECTIVE POTENTIALSIn this section we extract the long range portion of thepotential for our quark-antiquark and quark-diquark fluxtubes. According to QCD, the effective potentials shouldhave the same slope, so long as the APE smearing in thespatial directions has smoothed the gluon fluctuations suf-ficiently to isolate the ground state and the propagation ofthe Wilson loops in the time direction is long enough forany excited states to decay. The effective potential isobtained from the Wilson loops in the standard manner:aVð~r; Þ ¼ lnWð ~r; ÞWð ~r; þ 1Þ: (6)As shown in Figs. 5 and 6, we obtain stable plateaus for thepotentials as a function of . The statistical uncertaintiesare estimated using the jackknife method [17].The quark-antiquark potential has the well-known formVQ QðrÞ ¼ V0 rþ Q Qr; (7)where  is the string tension. The three-quark potential is[18,19]V3QðrÞ ¼ 32V0 12Xj<kg2CF4rjkþ 3QLðrÞ; (8)whereCF ¼ 4=3 and LðrÞ is a length linking the quarks. Asshown in our earlier work [9], LðrÞ is given by the mini-mum length of string that connects the three quarks, or thesum of distances from the quarks to the Fermat (or Steiner)point. QCD suggests that the two string tensions Q Q and3Q are equal. In Fig. 7 we plot the extracted effectivepotentials for the quark-diquark and quark-antiquark fluxtubes at each of the values of  for our gauge configura-tions. The plots in Fermi show that the QCD prediction isconfirmed at both values of . Converting length measure-ments from lattice units to Fermi we obtain the quark-diquark string tension 3Q ¼ 0:97 0:01 GeV fm1,which is in excellent agreement with the quark-antiquarkstring tension Q Q ¼ 0:98 GeV fm1, defining the latticespacing.IV. FLUX-TUBE PROFILESWe can gain further insight into the properties of the fluxtubes by examining their profiles close to the quark. Westudy the values of the correlators C3Qð ~yÞ and CQ Qð ~yÞwhere ~y ¼ ðy1; y2; 0Þ is constrained to the plane of thecolor sources, and the origin is at the position of eitherFIG. 4 (color online). Formation of quark-antiquark flux tube.Details are as in Fig. 3.0 1 2 3 4 5 6 7 8T0.7511.251.51.752aV0 1 2 3 4 5 6 7 8T0.811.21.4aVFIG. 5 (color online). Effective static quark potential for 30-sweep smeared quark-diquark sources for  ¼ 4:6 (top) and ¼ 4:8 (bottom). From bottom up, the lines correspond toquark-diquark separation increasing from 1 to 12 lattice spac-ings.COMPARISON OF GLUON FLUX-TUBE . . . PHYSICAL REVIEW D 80, 114506 (2009)114506-3the antiquark or the join of the T. The quark is then at theposition ð; 0; 0Þ where  varies from 1 to 12 lattice steps.First we examine the longitudinal profiles of both quark-diquark and quark-antiquark flux tubes along the line ð ~yÞ ¼ðx; 0; 0Þ in Fig. 8. As expected, the vacuum expulsion closeto the diquark is stronger than in the vicinity of the anti-quark. However, near the quark the two flux tubes showvery similar profiles. Similar results are seen at  ¼ 4:6.Next we examine the transverse profiles along a lineorthogonal to the midpoint of the flux tube, i.e., alongð=2; y; 0Þ for  even, or along ððþ 1Þ=2; y; 0Þ for odd. In Fig. 9 we show profiles of both quark-diquarkand quark-antiquark flux tubes for  ¼ 12. We find thatas long as  is larger than one-third of the total length of thequark-antiquark system, the transverse profiles are close toidentical.The transverse profile of the flux tube is fitted well by aGaussian function CðyÞ ¼ 1 Aey2=r2 . The fit enables us0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6l (fm)46810121416V(fm-1 )QQQ(QQ)0 0.2 0.4 0.6 0.8 1 1.2l (fm)46810121416V(fm-1 )QQQ(QQ)FIG. 7 (color online). Comparison of quark-antiquark and quark-diquark effective potentials for  ¼ 4:6 (top) and  ¼ 4:8(bottom).0 1 2 3 4 5 6 7 8T0.250.50.7511.251.51.75aV0 1 2 3 4 5 6 7 8T0.40.60.81aVFIG. 6 (color online). Effective static quark potential for 30-sweep smeared quark-antiquark sources for  ¼ 4:6 (top) and ¼ 4:8 (bottom). From bottom up, the lines correspond toquark-diquark separation increasing from 1 to 12 lattice spac-ings.-4 0 4 8 12 16x0.920.940.960.981C(x)Q(QQ)QQFIG. 8 (color online). Comparison of longitudinal flux-tubeprofiles for  ¼ 4:8 at longitudinal separation of 12 lattice units.F. BISSEY, A. I. SIGNAL, AND D.B. LEINWEBER PHYSICAL REVIEW D 80, 114506 (2009)114506-4to estimate the radius ðrÞ and area ðAr ﬃﬃﬃﬃp Þ of the flux tubeas well as its depth. The fit parameters for our flux tubes aregiven in Table I. Again, the fits show that for long enoughflux tubes the transverse profiles of quark-diquark andquark-antiquark flux tubes are statistically identical.V. CONCLUSIONSWe have directly compared gluon flux tubes for quarkplus antiquark and three-quark systems. In the three-quarksystems we kept two quarks close together (two latticeunits separation), so that the system would approximate aquark-diquark string. We found that the string tension inthe quark-diquark string was the same as for the quark-antiquark string. In addition we compared the vacuumexpulsion in both sets of flux tubes. We found that, in thevicinity of the quark, there was no measurable differencebetween the transverse profiles of the quark-diquark fluxtubes and the quark-antiquark flux tubes. Also the longitu-dinal profiles of both sets of flux tubes were very similar.These findings confirm the expectation from QCD that adiquark has many properties in common with an antiquark.In particular the long range color interaction between adiquark and a quark is seen to be the same as that betweenan antiquark and a quark. This result is interesting in that itis obtained in the quenched approximation, where the colorhyperfine interaction should be small. This implies that theAPE smearing and propagation in Euclidean time we haveperformed has been sufficient for decuplet baryon states todecay. It would be interesting to repeat this work withdynamical quarks, where variation in the strength of thecolor hyperfine interaction could be investigated. This ispotentially of great importance to phenomenological mod-els of hadron structure.ACKNOWLEDGMENTSThis work has been done using the DoubleHelix com-puter at Massey University and supercomputing resourcesfrom the NCI National Facility and eResearch SA. The 3Drealizations have been rendered using OpenDX (http://www.opendx.org). The 2D plots and curve fitting havebeen done using Grace (http://plasma-gate.weizmann.a-c.il/Grace/).[1] R. L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91, 232003(2003); R. L. Jaffe, Phys. Rep. 409, 1 (2005).[2] M. Anselmino et al., Rev. Mod. Phys. 65, 1199 (1993).[3] M. Hess, F. Karsch, E. Laermann, and I. Wetzorke, Phys.Rev. D 58, 111502(R) (1998).[4] R. Babich et al., J. High Energy Phys. 01 (2006) 086.[5] K. Orginos, Proc. Sci., LAT2005 (2006) 054 [arXiv:hep-lat/0510082].[6] C. Alexandrou, Ph. de Forcrand, and B. Lucini, Phys. Rev.Lett. 97, 222002 (2006).[7] Z. Liu and T. DeGrand, Proc. Sci. LAT2006 (2006) 116[arXiv:hep-lat/0609038].[8] R. Babich et al., Phys. Rev. D 76, 074021 (2007).[9] F. Bissey et al., Nucl. Phys. B, Proc. Suppl. 141, 22(2005); Phys. Rev. D 76, 114512 (2007).[10] H. Ichie, V. Bornyakov, T. Streuer, and G. Schierholz,Nucl. Phys. B, Proc. Suppl. 119, 751 (2003).[11] V. G. Bornyakov et al. (DIK Collaboration), Phys. Rev. D70, 054506 (2004).[12] F. Okiharu and R.M. Woloshyn, Nucl. Phys. B, Proc.Suppl. 129, 745 (2004).[13] G. S. Bali, C. Schlichter, and K. Schilling, Phys. Rev. D51, 5165 (1995).[14] R.W. Haymaker, V. Singh, Y. C. Peng, and J. Wosiek,Phys. Rev. D 53, 389 (1996).[15] M. Luscher and P. Weisz, Commun. Math. Phys. 97, 59-8 -6 -4 -2 0 2 4 6y0.940.950.960.970.980.991C(y)β=4.6 - QQβ=4.6 - Q(QQ)β=4.8 - QQβ=4.8 - Q(QQ)FIG. 9 (color online). Transverse profiles for quark-diquark(light shade lines) and quark-antiquark (black lines) flux tubesat  ¼ 4:6 (solid lines) and  ¼ 4:8 (dashed lines).TABLE I. Values of the fit parameters of the function 1A expðy2=r2Þ to the transverse profiles of the flux tubes. r isreported in lattice units (LU) and fm. The last column indicatesthe area under Cð ~yÞ ¼ 1 for the fitted curve in units of relativedepth times LU.Flux Tube  A r (LU) r (fm) ArﬃﬃﬃﬃpQ Q 4.6 0.0511(1) 2.89(3) 0.355(3) 0.261(3)QQQ 4.6 0.0510(1) 2.91(2) 0.358(3) 0.263(3)Q Q 4.8 0.0243(3) 3.17(4) 0.299(4) 0.136(3)QQQ 4.8 0.0241(3) 3.21(3) 0.303(3) 0.137(3)COMPARISON OF GLUON FLUX-TUBE . . . PHYSICAL REVIEW D 80, 114506 (2009)114506-5(1985); 98, 433(E) (1985).[16] S. O. Bilson-Thompson, D. B. Leinweber, and A.G.Williams, Ann. Phys. (N.Y.) 304, 1 (2003).[17] I. Montvay and G. Mu¨nster, Quantum Fields on a Lattice(Cambridge University Press, Cambridge, UK, 1994),p. 389.[18] T. T. Takahashi, H. Matsufuru, Y. Nemoto, and H.Suganuma, Phys. Rev. Lett. 86, 18 (2001); T. T. Takahashi,H. Suganuma, Y. Nemoto, and H. Matsufuru, Phys. Rev. D65, 114509 (2002).[19] C. Alexandrou, P. de Forcrand, and A. Tsapalis, Phys. Rev.D 65, 054503 (2002); C. Alexandrou, P. de Forcrand, andO. Jahn, Nucl. Phys. B, Proc. Suppl. 119, 667 (2003).F. BISSEY, A. I. SIGNAL, AND D.B. LEINWEBER PHYSICAL REVIEW D 80, 114506 (2009)114506-6

Comparison of gluon flux-tube distributions for quark-diquark and
  quark-antiquark hadrons

Comparison of gluon flux-tube distributions for quark-diquark and quark-antiquark hadrons

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Comparison of gluon flux-tube distributions for quark-diquark and quark-antiquark hadrons

Abstract

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Full text

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Adelaide Research &amp; Scholarship

Adelaide Research & Scholarship