Diquarks may play an important role in hadron spectroscopy, baryon decays and
color superconductivity. We investigate the existence of diquark correlations
in lattice QCD by considering systematically all the lowest energy diquark
channels in a color gauge-invariant setup. We measure mass differences between
the various channels and show that the positive parity scalar diquark is the
lightest. Quark-quark correlations inside the diquark are clearly seen in this
channel, and yield a diquark size of order 1 fm.Comment: Version as published in Phys. Rev. Lett.97, 222002,2006; 4 pages, 5
  figure

B. Lucini

C. Alexandrou

K. Orginos

Ph. de Forcrand

S. Kabana

English

arXiv

Diquarks may play an important role in hadron spectroscopy, baryon decays and color superconductivity. We investigate the existence of diquark correlations in lattice QCD by considering systematically all the lowest energy diquark channels in a color gauge-invariant setup. We measure mass differences between the various channels and show that the positive parity scalar diquark is the lightest. Quark-quark correlations inside the diquark are clearly seen in this channel, and yield a diquark size of order 1 fm.Diquarks may play an important role in hadron spectroscopy, baryon decays and color superconductivity. We investigate the existence of diquark correlations in lattice QCD by considering systematically all the lowest energy diquark channels in a color gauge-invariant setup. We measure mass differences between the various channels and show that the positive parity scalar diquark is the lightest. Quark-quark correlations inside the diquark are clearly seen in this channel, and yield a diquark size of order 1 fm

Alexandrou, C.

de Forcrand, Ph.

Lucini, B.

CERN Document Server

arXiv:hep-lat/0609004v2   11 Dec 2006Evidence for diquarks in lattice QCDC. Alexandrou a, Ph. de Forcrand b,c and B. Lucini b,da Department of Physics, University of Cyprus, CY-1678 Nicosia, Cyprusb Institute for Theoretical Physics, ETH Zurich, CH-8093 Zurich, Switzerlandc CERN, Physics Department, TH Unit, CH-1211 Geneva 23, Switzerlandd Department of Physics, University of Wales Swansea, SA2 8PP Swansea, UKDiquarks may play an important role in hadron spectroscopy, baryon decays and color super-conductivity. We investigate the existence of diquark correlations in lattice QCD by consideringsystematically all the lowest energy diquark channels in a color gauge-invariant setup. We measuremass differences between the various channels and show that the positive parity scalar diquark isthe lightest. Quark-quark correlations inside the diquark are clearly seen in this channel, and yielda diquark size of O(1) fm.PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.Aw, 12.38.-t, 14.70.DjDiquarks were originally proposed several decadesago [1] as a natural way to explain the absence of a largenumber of exotics otherwise predicted by QCD. Recently,it has been realized that they provide a natural explana-tion for an exotic baryon antidecuplet, the Θ+ [2], thatcan not be accomodated in the quark model. The firstevidence for the Θ+ was reported by the LEPS Collabo-ration [3], but subsequent experiments cast doubts on itsexistence [4]. Independently of whether a pentaquark ex-ists or not, diquark correlations may play an importantrole in the description of quark distributions and frag-mentation functions and in explaining the systematicsof non-leptonic weak decays of light quark hadrons [8].In addition diquarks are the central ingredient of cold,dense matter where they condense to form a color super-conductor. Despite their potential role in explaining sucha variety of phenomena in hadronic physics, quantitativeanalyses that can directly assess whether diquark corre-lations are present in QCD are lacking. Lattice QCD isthe method of choice for studying hadronic states. How-ever, diquarks are colored objects and need an appropri-ate formalism to study them using lattice simulations.One way is to fix the gauge as done in an early studyon the mass of the diquarks [5]. In this work we presenta gauge invariant formalism, where we create color sin-glet states by considering diquarks in the background ofa static quark [6, 7]. This enables us to extract massdifferences between baryons containing diquarks in thevarious channels. These mass differences, unlike diquarkmasses themselves, are well-defined and gauge invariant.One gluon exchange indicates quark-quark attractionin the color antitriplet 3¯c, flavor antisymmetric 3¯f andspin singlet positive parity channel. Diquarks in thischannel are referred to as “good” diquarks [8]. The aimof this work is to check whether QCD dynamics supportsattraction in this channel and compare it with other di-quark channels. Possible diquark configurations are cre-ated by two quark operators and insertions of the co-variant derivative. In this work we consider only diquarkconfigurations with no derivatives since these are lower inenergy. We therefore consider all 16 diquark multipletsthat can be created by operators bilinear in the quarkfields of the form qTCΓq with C = iγ0γ2 the charge con-jugation operator, and Γ = 1, γµ, γ5, γ5γµ, σµν . Thepositive parity channels are qTCγ5q and qTCγ5γ0q withspin zero and qTCγiq and qTCσ0iq with spin one. Thenegative parity channels are qTCq and qTCγ0q with spinzero and qTCγ5γiq and qTCσijq with spin one. Theycreate states that vanish in the non-relativistic limit andare excluded in quark models. We expect them to behigher in energy.In addition to mass differences, we probe correlationsdirectly by examining the spatial distribution of the twoquarks forming the diquark in the background of thestatic quark [6]. Correlations which persist as the staticquark is removed are intrinsic to the diquark. We studythese correlations using gauge invariant two-density cor-relators, used before [9] to probe hadron structure:CΓ(ru, rd, t) = 〈0|JΓ(0, 2t)Ju0 (ru, t)Jd0 (rd, t)J†Γ(0, 0)|0〉 (1)where Jf0 (r, t) =: f¯(r, t)γ0f(r, t) :, f = u, d andJΓ(x) = ǫabc[uT a(x)C Γdb(x) ± dT a(x)C Γub(x)]sc(x)(2)where the + (-) sign corresponds to the flavor symmetric(antisymmetric) combination, and sc denotes the staticquark. Latin indices denote color. We use two degener-ate flavors of Wilson fermions on quenched configurationsgenerated at β = 5.8, 6.0 and 6.2 corresponding to latticespacing a = 0.136, 0.093 and 0.068 fm as determined fromthe static quark force [10]. At each lattice spacing, weperform measurements at three values of the pion massin the range 570 − 910 MeV. By comparing results ob-tained at different lattice spacings but at the same pionmass we can assess discretization effects. In most casesthe quenched approximation is good when the mass ofthe pion is higher than 600 MeV, but is expected to failin the chiral limit. In order to check the validity of ourresults in this range of pion masses, we repeat our masssplitting measurements on a set of configurations withtwo degenerate flavors of dynamical Wilson fermions atβ = 5.6 [11]. In Table I we collect the parameters of ourcalculation.2TABLE I: Summary of our simulations, including the valueof κ, which determines the bare quark mass, the pion mass,mpi, and nucleon mass, MN , in lattice and, for the latter, inphysical units and the number of gauge configurations.κ ampi aMN [MN GeV] number of confsQuenched 163 × 32 β = 5.8, a−1 = 1.47 GeV0.1560 0.619(2) 1.139(8) [2.04(1)] 3560.1575 0.549(2) 1.052(9) [1.55(1)] 1600.1590 0.473(2) 0.961(9) [1.41(1)] 200Quenched 163 × 32 β = 6.0, a−1 = 2.15 GeV0.153 0.423(1) 0.783(8)[1.68(2)] 3640.154 0.366(2) 0.716(7)[1.54(2)] 5030.155 0.300(2) 0.634(8)[1.36(2)] 287Quenched 203 × 40 β = 6.2, a−1 = 2.94 GeV0.1510 0.286(3) 0.563(8) [1.66(2)] 1160.1520 0.214(3) 0.487(9) [1.43(3)] 1660.1523 0.188(3) 0.460(11)[1.35(3)] 157Unquenched Wilson 243 × 40 β = 5.6, a−1 = 2.42 GeV0.1575 0.270(3) 0.580(7)[1.40(2)] 185The mass of our static-light-light baryon can beobtained from the large time limit of the correlatorGΓ(t) = 〈JΓ(~x, t)J†Γ(~x, 0)〉. The effective mass, meff(t) ≡− log [GΓ(t)/GΓ(t− 1)], becomes time independent forlarge t when the lightest state dominates (plateau re-gion), yielding its mass. To reduce the statistical errors,we need to keep t small by isolating the ground stateas fast as possible. This is accomplished by suppressinghigher excitations using Wuppertal smearing [12] on thesource and sink with hypercubic (HYP) smeared spatiallinks [13] for the Wuppertal smearing function [14]. Inaddition we use HYP smearing on the temporal gaugelinks U4(x) that enter in the construction of the staticpropagator given bySstat.(x2, t2,x1, t1) = δ3(x2 − x1)„1 + γ02«"t2−aYt=t1U4(x1, t)#†(3)for t2 > t1. The exponential prefactor exp(−mq(t2− t1))has been dropped, since this introduces a constant shiftin all energies by the bare heavy quark mass mq, whichcancels in the mass differences that we measure betweenbaryons with diquarks in different channels.We show in Fig. 1 the effective mass differences, ∆meff ,between the “good” (scalar) diquark created by takingΓ = γ5, and the other channels. As expected the massdifference is zero for the other scalar diquark with posi-tive parity created by taking Γ = γ0γ5. The other pos-itive parity channels, i.e. the “bad” (vector) diquarks,are degenerate and are clearly higher in energy than the“good” diquarks as pointed out in Ref. [8]. The nega-tive parity states, as expected, have even larger energies.Their effective masses are much noisier making it difficultto identify a clear plateau void of contamination from yethigher states. The general behavior is however similar tothe positive parity diquarks: the vector channels are alsodegenerate and tend to have a larger mass than the scalarFIG. 1: Effective mass difference, ∆meff , in the various di-quark channels, at β = 6.0 for our lightest quarks. The ar-rows (from left to right) indicate the time slice of the source,of the density insertions and of the sink.channel created by taking Γ = I. To verify that the massdifference seen between the two scalar channels is signif-icant one would need to improve the quality of the massplateaus.Effective color-spin Hamiltonian arguments [8] predictthat for heavy constituent quarks the mass difference∆m ≡ mγi − mγ5 between the “good” and “bad” di-quarks scales like 1/mq1mq2 , where mq1,2 are the massesof the constituent quarks. On the other hand in the lightquark regime we havem2pi ∝ mq. Therefore for intermedi-ate values ofmq where both relations hold approximatelywe make the Ansatz ∆m ∝ 1/m4pi. In the opposite limitmq → 0, ∆m is expected to approach a constant value.These two behaviors can be connected by the Ansatz∆m =c11 + (mpi/c2)4, (4)with c1 and c2 to be determined from the lattice data.We show in Fig. 2 results for ∆m as a function of m4pifor our three β values. All lattice data should fall ona universal line if we are close to the continuum limit.We observe that this is indeed the case on our two finestlattices whereas for the coarsest, scaling violations areapparent. In the same figure we also show ∆m ob-tained using unquenched, dynamical Wilson configura-tions. It nicely falls on the same curve as the fine-lattice quenched results. This corroborates that scal-ing has set in and that quenching effects at these quarkmasses are small. Therefore a quenched study of diquarkproperties in the quark mass range used here is a verygood approximation and will be adopted in the rest ofthe discussion. The Ansatz given in Eq. (4) providesa good fit to the lattice data. Using the lattice datawith good scaling behavior at β = 6.0 and 6.2, we findc2 = 0.78± 0.15 GeV, and c1 = 0.138± 0.01 GeV, whichis the mass difference at the chiral limit [15]. This masssplitting can be compared to the ∆-nucleon mass split-ting, δm∆N . Using our quenched data at β = 6.0 we findfor the ratio ∆m/δm∆N = 0.67(7), 0.73(8) and 0.67(8)3at κ = 0.153, 0.154 and 0.155 respectively. Therefore ourestimate of ∆m is consistent with the predicted 2/3 ofthe ∆-nucleon mass difference given in Ref. [8].FIG. 2: Mass difference between “good” and “bad” diquarksas a function of m4pi at β = 5.8 (crosses), β = 6.0 (opentriangles), β = 6.2 (filled circles) and unquenched result (filledsquare). The dashed lines are fits to Eq. (4).FIG. 3: CΓ(r/a = 7.35, θ) versus cos(θ) for β = 6.2 andκ = 0.1520 for the “good” diquark ( asterisks) and the otherdiquark channels using the same notation is as in Fig. 1. Cor-relators for the “bad” diquark have been multiplied by twoand for all negative parity channels by ten.Having determined the diquark spectrum, we now turnto the analysis of their structure by studying the density-density correlators defined in Eq. (1). The time t wherethe density operators are inserted is shown in Fig. 1 bythe arrow (at t/a = 8), and is within the plateau rangeof the effective mass. We have verified that the correla-tors remain, within errors, unchanged when we vary thesource-sink separation or equivalently t. Our aim is tolook for spatial correlations between the two light quarksin the various diquark channels. We take the location ofthe static quark as the reference coordinate from whichthe distances ru and rd of the two quarks of different fla-vors are measured. We are interested in intrinsic diquarkcorrelations, which persist as the static quark is movedaway from the diquark. We therefore consider sphericalFIG. 4: Left: CΓ(r/a = 5.1, θ)/Cγ5 (r/a = 5.1, 0) versuscos(θ). Right: CΓ(r = 0.5 fm, rud)/Cγ5(r = 0.5 fm, 0) ver-sus rud, for the “good” (asterisks) and “bad” (filled triangles)diquarks at the lightest pion for our three lattice spacings.shells |ru| = |rd| = r of increasing radius r. Since thesystem is spherically symmetric the correlator dependsonly on r and the angle θ = arccos(rˆu.rˆd) [6]. In the ab-sence of any correlation, the distribution of CΓ(r, θ) willbe uniform as a function of cos(θ). Attraction will showup as an enhancement at small angles, near cos(θ) = 1.Our cubic lattice breaks rotational symmetry and dis-torts the uniform spherical distribution, particularly atsmall angle θ. To remove such lattice artifacts, we nor-malize our distributions by a uniform lattice distribution.We show the resulting density correlators in Figs. 3 and4 for various shell radii r. The physically relevant corre-lations are those that survive when r is large. We indeedobserve that the qualitative behavior of the distributionsdoes not depend on r, confirming that the color field gen-erated by the static quark does not affect the physics ofdiquarks once r is large enough.In Fig. 3 we show CΓ(r, θ) for all the different channelsas a function of cos(θ) when the shell radius is fixed tor = 0.5 fm (r/a = 7.35). We clearly observe that the“good” diquark shows enhanced correlations at small θ,indicating attraction between the quarks. There is alsoa gradual increase in the distribution for the “bad” di-quark indicating a weaker attraction in this channel. Thisbehavior persists as we increase or decrease the quarkmasses. For the negative parity channels, correlations areabsent or very weak, except for the vector channel withΓ = γ5γi. However those disappear for lighter quark4masses. The negative-parity correlators are noisier be-cause the diquarks are heavier, so that a higher statisticsanalysis will be needed to fully determine the charac-teristics of their distributions. For this reason, we nowrestrict our analysis to the positive parity diquarks.            δL = (2/√3− 1/2√3)ǫ=√3ǫ/2ǫ/2increases by ǫ/√3decreases by ǫ/2√3ǫ/2FIG. 5: String formation for three quarks, where one quarkhas been taken to the distant left. As a function of the sepa-ration between the other two quarks, the static potential riseswith an effective string tension σeff = σ√32.In Fig. 4 we show the density correlators for the “good”(scalar) and “bad” (vector) diquarks for all three latticespacings at our lightest quark mass. The left hand setshows the correlator as a function of cos(θ) normalized to1 at θ = 0 for r/a = 5.1, which corresponds to r = 0.69,0.47 and 0.35 fm at β = 5.8, 6.0 and 6.2 respectively.As can be seen, irrespectively of the distance the “good”diquark shows stronger spatial correlations. Within thisframework we can also extract the diquark size: At fixedr we look at the dependence of the correlator on the rel-ative u − d separation, rud = 2r sin (θ/2). In the righthand of Fig. 4, we show the correlator (normalized to 1 atthe origin) as a function of rud for a fixed physical shellradius r = 0.5 fm, at our lightest quark mass. An ex-ponential dependence of Cγ5(r, rud) ∝ exp(−rud/r0(r))provides a gauge invariant definition of the diquark sizer0(r) for a given value of r. The curves in Fig. 4, ob-tained from fits to an exponential dependence, describewell Cγ5(r, rud) at all β values. Moreover the physicalsize r0(r) is the same on the two finer lattices, confirm-ing continuum-like behaviour as for the diquark masses.We can then evaluate r0 as a function of the shell ra-dius r. Our measurements show a mild increase of r0with r, from ∼ 0.9 fm to ∼ 1.3 fm as r increases from0.3 to 0.75 fm. For large r & 0.5 fm, r0 is consistentwith having reached a plateau. This suggests that thestatic source has a small influence on the “good” diquarkgiving a characteristic size of about 1.1 ± 0.2 fm. Forcomparison, at the same quark mass and using the samedefinition of size, we find for the ρ a size of 0.7 fm [9]. Thelarge diquark size can be understood from the followingqualitative argument: As the static quark is moved awayfrom the diquark a q-q string tension develops. Consid-ering an increase ǫ in the separation between the quarksin the diquark as illustrated in Fig. 5 one finds that theeffective q-q string tension is(√3/2)σ where σ is the q-q¯string tension. Since in the perturbative regime, the q-qattraction is also weaker than the q-q¯ attraction, this timeby a factor 1/2, the conclusion is that for all distancesone expects the q-q attraction to be weaker. Thus a q-q diquark should be somewhat larger than a q-q¯ meson,which is what we find. Size measurements for the “bad”diquark, on the other hand, show neither scaling nor con-vergence to a plateau value. The large values obtained,often similar to our box size, corroborate the weaknessof spatial correlations in this channel.In conclusion, we have evaluated the mass splittingsand density correlators of the complete set of diquarkchannels created by local diquark fields. Both observ-ables confirm the phenomenological expectation thatQCD dynamics favors the formation of “good” diquarks,i.e. in the scalar positive parity channel. The character-istic size of this diquark, O(1) fm, is large but consistentwith the scale O(200) MeV of the attraction. Even a“good” diquark is a large object, which may limit its rel-evance to hadron structure. The positive parity vectorchannel is higher in energy by about 2/3 the ∆-nucleonmass splitting, and forms an even larger object. All thenegative parity channels have much higher energies.Acknowledgments: We thank B. Orth, Th. Lippertand K. Schilling [11] for providing the unquenched con-figurations. This work is supported in part by the EU In-tegrated Infrastructure Initiative Hadron Physics (I3HP)under contract RII3-CT-2004-506078. The work of B.L.has been partially supported by the Royal Society.[1] M. Ida and R. Kobayashi, Prog. Theor. Phys. 36, 846(1966). See M. Anselmino et al., Rev. Mod. Phys. 65,1199(1993) for a review.[2] R. L.Jaffe and F. Wilczek, Phys. Rev. Lett. 91, 232003(2003).[3] T. Nakano et al., [LEPS Collaboration], Phys. Rev. Lett.91, 012002 (2003).[4] S. Kabana, AIP Conf. Proc.7 56 195 (2005).[5] M. Hess, F. Karsch, E. Laermann and I. Wetzorke, Phys.Rev. D 58 111502 (1998).[6] C. Alexandrou, Ph. de Forcrand and B. Lucini, PoSLAT2005, 053 (2006).[7] K. Orginos, PoS LAT2005, 054 (2006).[8] R. L. Jaffe, Phys. Rept. 409, 1 (2005).[9] C. Alexandrou, Ph. de Forcrand and A. Tsapalis, Phys.Rev. D 66, 094503 (2002).[10] S. Necco and R. Sommer, Nucl Phys. B622 328 (2002).[11] B. Orth, Th. Lippert and K. Schilling, Phys. Rev. D 72,014503 (2005).[12] C. Alexandrou et al., Nucl. Phys. B414 815 (1994).[13] A. Hasenfratz and F. Knechtli, Phys. Rev. D 64, 034504(2001).[14] C. Alexandrou, G. Koutsou, J. W. Negele and A.Tsapalis, Phys. Rev. D 74, 034508 (2006).[15] Diquark mass splittings, measured in Ref. [7] for lighterquarks tend to be larger than ours, with large uncertain-ties.

Evidence for diquarks in lattice QCD

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Evidence for Diquarks in Lattice QCD

Diquarks may play an important role in hadron spectroscopy, baryon decays and color superconductivity. We investigate the existence of diquark correlations in lattice QCD by considering systematically all the lowest energy diquark channels in a color gauge-invariant setup. We measure mass differences between the various channels and show that the positive parity scalar diquark is the lightest. Quark-quark correlations inside the diquark are clearly seen in this channel, and yield a diquark size of order 1 fm

Biagio Lucini

Cronfa at Swansea University

Evidence for diquarks in lattice QCD

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Cronfa at Swansea University