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Lattice QCD study of a five-quark hadronic molecule



Lattice QCD study of a ?ve-quark hadronic molecule
M.S. Cook1, 2 and H.R. Fiebig1
2 1 Physics Department, FIU-University Park, Miami, Florida 33199 Department of Physics, Converse College, Spartanburg, South Carolina 29302 (Dated: February 1, 2008)

arXiv:0704.2385v2 [hep-lat] 26 Jul 2007

We compute the ground-state energies of a heavy-light K–Λ like system as a function of the relative distance r of the hadrons. The heavy quarks, one in each hadron, are treated as static. Then, the energies give rise to an adiabatic potential Va (r) which we use to study the structure of the ?ve-quark system. The simulation is based on an anisotropic and asymmetric lattice with Wilson fermions. Energies are extracted from spectral density functions obtained with the maximum entropy method. Our results are meant to give qualitative insight: Using the resulting adiabatic potential in a Schr¨dinger equation produces bound state wave functions which indicate that the o ground state of the ?ve-quark system resembles a hadronic molecule, whereas the ?rst excited state, having a very small rms radius, is probably better described as a ?ve-quark cluster, or a pentaquark. We hypothesize that an all light-quark pentaquark may not exist, but in the heavy-quark sector it might, albeit only as an excited state.
PACS numbers: 12.38.Gc, 12.39.Mk

INTRODUCTION

n
Lattice QCD studies of hadron-hadron interactions are the gateway to nuclear physics through ?rst principles [1]. From a lattice simulation point of view the nucleonnucleon interaction is unquestionably the most challenging case [2], and it might not be resolved in the foreseeable future. However, interactions in other two-hadron systems are worth investigating as well, because new insights into the structural features of already discovered [3] or yet unknown baryon resonances may emerge. In particular, one may ask if some of those may be understood as hadronic molecules, like the deuteron, or if more compact clusters, like a pentaquark [4], may also exist. We are here interested in pairs of hadrons containing one heavy quark each. In such systems a relative, residual, interaction is a well-de?ned concept. In the spirit of the Born-Oppenheimer approximation, the (slow) heavy quarks serve to de?ne the centers of the two hadrons while the (fast) light quarks and gluons provide the physics of the interaction. Exploratory studies along those lines have been done in the context of meson-meson and baryon-baryon systems [5, 6, 7]. Speci?cally, we here investigate a heavy-light mesonbaryon (?ve-quark) hadron with the quantum numbers of an S-wave K–Λ system. The heavy-quark propagator is treated in the static approximation. This allows us to compute the total energy as a function of the relative distance r between the heavy quarks, viz. hadrons. A production scheme is illustrated in Fig. 1. The resulting adiabatic (Born-Oppenheimer) potential Va (r) then may be used to address the possibility of molecule-like, or other, structures. In the static approximation to the heavy-quark propagator the potential Va (r) extracted from the lattice equally applies to the systems K–Λ, D–Λc , and B–Λb ,

d u d

d s s u d u s s u d

K0 r Λ K
0

γ

+

p

u u d

r Λ
0

γ

FIG. 1: Schematic view of K–Λ like molecule production and illustration of the current scheme for extracting a hadronic interaction at relative distance r. Thick lines indicate heavyquark propagators, and thin lines depict light quark propagators.

the s-?avored quarks being replaced with c and b ?avors, respectively. Within the framework of a Schr¨dinger o equation those systems can be studied using their respective physical reduced masses, provided of course, one is willing to accept the limitations of the adiabatic approximation and the non-relativistic nature of the framework. This is the main reason for classifying our results as qualitative.

OPERATOR CONSTRUCTION

The lattice simulation requires two-hadron interpolating ?elds. Those are constructed using as building blocks standard local operators for the K + and Λ0 particles [8]. They are placed at relative distance r and then projected to total momentum zero Oα (r; t) = N ?1/2
x y

δr, x ? y K + (xt)Λ0 (yt) . (1) α

2 Here the normalization N is the spatial lattice volume, the sums are over lattice sites, and α is a Dirac spinor index. Then, with
? ? O? (r; t) = Oα (r; t)γ4,α?

(2)

the time correlation function ? ? C(t, t0 ) = O? (r; t)O? (s; t0 ) ? O? (r; t) O? (s; t0 ) , (3) where s is the relative distance at the source, can be expressed in terms of fermion propagators. The ?avor assignment K + Λ0 ? su uds causes the separable term in ? (3) to vanish. Writing H(xt, yt0 ) and G(xt, yt0 ) for the heavy (s) and light (u,d) quark propagators, respectively, one obtains C(t, t0 ) = [H(yt, y + rt)H(r1 + st0 , r1 t0 ) ?H(yt, r1 t0 )H(r1 + st0 , y + rt)] ×
y

where U(x; t0 t) is the product of SU (3) link variables along a straight line from (xt0 ) to (xt) [8]. The factor (2κ)t?t0 gives rise to only a constant global energy shift ?m = ? ln(2κ), which we will ignore. The distance r = 0 is rather special [6] because a color singlet operator, as realized by (1), can also be achieved by a “color twisted” version of (1) where quarks across the hadrons K + and Λ0 , now at the same location, are combined into a color singlet. Because we do not consider color twisted operators in this work, we restrict ourselves to non-zero relative distance[26]. Thus, using H(yt, y + rt) ∝ δr,0 and H(r1 + st0 , r1 t0 ) ∝ δs,0 , the ?rst two terms in (4), and accordingly the top two diagrams in Fig. 2 vanish, and only the last two of those make a contribution to the correlation function for non-zero relative distance. By way of (5) those contributions are proportional to δy,r1 δr,s . Introducing r2 = r1 + s, to replace s, the correlation function (4) becomes (6) C(t, t0 ) = δr, r ? r H(r1 t, r1 t0 )H(r2 t0 , r2 t) 2 1 G(r1 t, r1 t0 )[?G(r1 t, r1 t0 )G(r2 t, r2 t0 ) +G(r1 t, r2 t0 )G(r2 t, r1 t0 )] for r = 0 . As a consequence of the static approximation, the site sum y has vanished from (4), and thus, is unfortunately no longer working to improve statistics. The ?nal correlator we use is extended from (6) to a K × K matrix by employing several levels k = 1 . . . K of operator smearing. The procedure amounts to re? placing in (1) all light-quark ?elds ψ, ψ with smeared {k} ?{k} ?elds ψ , ψ . We have used Wuppertal-style fermion smearing [9] along with APE-style gauge ?eld fuzzing [10]. The implementation speci?c to the asymmetric lattice used here is discussed below. No smearing, nor link variable fuzzing, was done for the heavy, static, quark ?elds in order to preserve spatial locality, i.e. the δ factor in (5). Thus, generically, replacing O → O{k} = ? O[ψ {k} , ψ {k} . . .] the correlator (6) becomes a K × K matrix ? {k C kk (t, t0 )(r; t, t0 ) = O{k} (r; t) O? } (r; t0 ) , ?
′ ′

G(yt, r1 t0 )[G(yt, r1 t0 )G(y + rt, r1 + st0 ) ?G(yt, r1 + st0 )G(y + rt, r1 t0 )] . (4) For clarity the rather involved color and spin index structure is not shown in (4). Also, translational invariance of the gauge ?eld average has been used to arrive at the above expression, and utilizing this freedom, an arbitrary space site r1 was introduced to ?x the source locations. A diagrammatic representation of (4) is shown in Fig. 2.
y t y+r t y y+r

t0 r1 y t r+s 1 y+r

t0 r1 y t r+s 1 y+r

(7)

t0 r1 r+s 1

t0 r1 r+s 1

FIG. 2: Diagrammatic representation of (4). Thick lines indicate heavy-quark propagators, and thin lines depict light (u and d) quark propagators.

The heavy-quark propagators are employed in the static approximation. For (unimproved) Wilson fermions with hopping parameter κ this means that the propagator is taken in the limit κ → 0 , resulting in 1 H(xt, yt0 ) = δx, y (2κ)t?t0 (1 + γ4 ) U ? (x; t0 t) , 2 (5)

with k, k ′ = 1 . . . K and a sum over ? is understood. ′ The expression for C kk (t, t0 ) in terms of quark propagators still has the form given by (6), however, lightquark propagator elements are replaced with smeared ′ ones, G → G{ kk } . Smearing and fuzzing prescriptions are the same at source and sink. Thus the correlation matrix (7) is hermitian by construction. We choose an asymmetric and anisotropic lattice with geometry L1 × L2 × L3 × L4 = 8 × 8 × 32 × 16. The bare lattice constants, in the respective directions, satisfy a1 = a2 = 2a3 = 2a4 . Subsequently we shall refer to a4 = a3 = a as the lattice constant a of the simulation, so the physical volume is 16a × 16a × 32a × 16a. The lattice has a ?ne resolution in 4-direction (time), and the

3 same ?ne resolution in the (spatial) 3-direction where the lattice size is twice as long as in the 1,2-directions. The idea here is to place the static quark sources along the 3-direction and thus achieve a ?ne spatial resolution of the relative distance as well as to provide enough space to allow for asymptotic separation of the hadrons. The spatial lattice asymmetry leads us to modify the procedure for operator smearing. A smearing iteration of k steps is understood to mean k steps in the 1,2-directions but 2k steps in the 3-direction, so the physical smearing extent is the same for all spatial directions (neglecting renormalization e?ects). Mutatis mutandis, the same is true for gauge ?eld fuzzing. An illustration is shown in Fig. 3. The implementation used here is described in more technical terms in reference [11].
Smearing direction i=2

proved lattice actions. We have therefore used the standard Wilson plaquette action with Wilson fermions in a quenched simulation [8]. The gauge ?eld couplings in the ?-ν planes and the hopping parameters in directions ? are given by, respectively, β?ν = β a1 a2 a3 a4 (a? aν )2 and κ? = κ
1 a? 4 4 1 ν=1 aν

.

(8)

The simulation was done at β = 6.2 with four values κ = 0.140, 0.136, 0.132, 0.128 of the hopping parameter employing a multiple mass solver [12]. A total of 343 gauge ?eld con?gurations were used.
MASS SPECTRUM ANALYSIS

k=1 2k i=3

i=2 Fuzzing direction i k=1 2k i=3

The time evolution of the eigenvalues of C(t, t0 ) determines the energy spectrum. Eigenvalues, behaving exponentially with t, may rapidly vanish into numerical noise. Conventional diagonalization methods do not work well under those circumstances. Singular value decomposition (SVD), on the other hand, is ideally suited to the task [13]. The SVD reads C(t, t0 ) = U (t, t0 ) Σ(t, t0 ) V ? (t, t0 ) , (9)

FIG. 3: Illustration of the prescription for quark ?eld smearing (top) and gauge ?eld fuzzing (bottom) on the asymmetric lattice. The geometry for k = 1 iteration is shown.

The positions of the static quark sources are chosen as x = (5, 5, n, 3) with n = 1, 2, 3, 4, 8, 11, 13, 17. Their spatial locations along the 3-direction, see Fig. 4, allow us to achieve any relative distance r = 1a . . . 16a by choosing an appropriate pair of source points. Some relative distances can be obtained more than once. In those cases appropriate averages over sets of source and sink pairs have been taken to compute the correlator matrix elements. Because periodic boundary conditions are applied in the spatial directions relative distances larger than r = 16a are redundant.
1 2 3 4 8 11 13 17

FIG. 4: Placement of static hadron sources along the 3direction of the lattice. Periodic boundary conditions are applied to the quark and gluon ?elds. All distances between r = 1a and r = 16a can be achieved, some multiple times.

where U (t, t0 ) and V (t, t0 ) are unitary in our case[27], and Σ(t, t0 ) = diag(σ1 (t, t0 ) . . . σK (t, t0 )) contains the singular values satisfying σk (t, t0 ) ≥ 0. If C(t, t0 ) is non-degenerate and positive semi-de?nite then the set of singular values {σk (t, t0 ) : k = 1 . . . K} and the set of eigenvalues are the same. A few more details can be found in [11]. For simplicity we will refer to σk (t, t0 ) as eigenvalues. To extract energy levels from the eigenvalues an often used procedure, sometimes starting from a generalized eigenvalue problem [14], is to construct e?ective mass functions from the eigenvalues and then try to identify plateaus in the asymptotic time region. Typically, only a narrow subset of the time slices is usable, which also is subject to some discretion. We shall not rely on e?ective mass functions here, but rather analyze correlator eigenvalues by means of the maximum entropy method [15] following the adaption described in [11], and in [16]. Among the advantages are that all available time slices may be used, if desired, and that excited states, if present, are easily revealed. The time dependence of the eigenvalues σk (t, t0 ) shall be ?t with the model


F (ρ|t) =
0

dω ρ(ω)e?ωt .

(10)

Given the approximations employed in this work, e.g. static heavy quarks, and a few more mentioned below, it would be of no advantage to employ sophisticated im-

for some set of discrete times slices t = t1 . . . t2 , where ρ(ω) is a spectral density function. The maximum entropy method (MEM) is based upon Bayesian statistics. In this context the numbers ρ(ω) are interpreted

4 as stochastic variables [15]. Their most likely values are obtained by simulated annealing as described in [11]. We refer the reader to this reference for technical details. At each of the 16 relative distances r we have used three (K = 3) operator smearing levels k = 1, 2, 3. A typical set of eigenvalues is displayed in Fig. 5 for distance r = 1a and the smallest (light) quark mass, κ = 0.140. Only two eigenvalue correlators are shown, the third one is indistinguishable from zero due to numerical noise. Note that eigenvalues are separated by about four orders of magnitude. Thus crossing ambiguities are nonexistent[28]. The eigenvalue sets for all other distances r exhibit the same features.

FIG. 5: Two sets of eigenvalues σk (t, t0 ), k = 1, 2, of the 3 × 3 time correlation matrix (9) for relative distance r = 1a and quark hopping parameter κ = 0.140. The third eigenvalue, k = 3, is not shown, it vanishes into numerical noise. Filled plot symbols belong to time slices used for the MEM ?ts employing the model (10), the curves are the corresponding results. Note that the source has been shifted to t0 = 0.

The eigenvalue correlators were ?t with the model (10) using, consistently in all cases, time slices t = 1 . . . 8 with the source t0 = 0 being omitted, and mass range of 0 ≤ ω ≤ 4 with discretization interval size ?ω = 0.05, all in units of a?1 . The maximum entropy method then yields spectral density functions ρ(ω). Representative examples are shown in Fig. 6 for r = 1, 4, 8, 16, in units of a, and κ = 0.140. The histogram lines in Fig. 6 represent ρ(ω), obtained from simulating annealing (cooling) following the very procedure put forward in [11]. Sixteen random annealing starts were used. The solid histogram lines represent the average of those 16 runs, the dashed histogram lines give the corresponding standard deviation. In most cases the spectra exhibit isolated peaks, say δn = {ω : ω ∈ peak #n}. Then one may compute, for each peak n, the volume Z, the energy E, and the

FIG. 6: Spectral density functions ρ, according to (10), for two-hadron operators with relative distance r = 1, 4, 8, 16, in units of a, thick histogram lines. Shown are eigenvalue spectra which contain ground state peaks. The dashed histogram lines indicate errors as explained in the text. The smooth dotted lines correspond to Gaussian ?ts to the discretized spectral densities.

5 width ?, according to Z =
δn

dωρ(ω) dωρ(ω)ω
δn

(11) (12)
2

E = Z ?1 ?2 = Z ?1
δn

dωρ(ω) (ω ? E) .

(13)

Typically the largest eigenvalues reveal the ground state. There are few exceptions. The spectra at r = 1a and r = 8a, shown in Fig. 6, are examples. Selecting the support δn of a peak can be slightly ambiguous, particularly if the peak sits on background noise, or if there is overlap with another one. We have therefore, as a matter of course, performed least-square ?ts to the average spectral functions ρ(ω) with one or two Gaussians, as required in each case. These ?ts then give values for Z, E, and ? as de?ned in (11)–(13) directly from the parameters of the Gaussians. In this manner we have obtained 16 sets, one for each r, of ?ve-quark hadron ground state energies. Each set contains four energies corresponding to hopping parameter values κ = 0.140, 0.136, 0.132, 0.128. The spectral density functions ρ(ω) typically exhibit some additional structure at the high end of the ω range, a broad peak in most cases. These secondary peaks appear because diagonalizing the correlator matrix separately on all time slices only ensures that its eigenvalues describe a single state from the physical spectrum at large values for t [14]. At small time slices, close to the source, the eigenvectors of C(t, t0 ) may still describe a mix of spectral states. In practice the large-t rule means that, in each eigen channel, one should only take the lowest peak into account. In physical terms the secondary peaks are separated from those by ≈3GeV, and larger, and thus should probably be considered lattice artifacts. In order to relate our results to the physical hadron spectrum we have also computed 3 × 3 correlation matrices with standard local [8] pseudoscalar meson (π), vector meson (ρ), and nucleon (N ) operators, using the same smearing and fuzzing prescription. The analysis was performed with the MEM in exactly the same way as described above. This now allows extrapolations of hadron masses to the physical pion mass region (amπ ≈ 0). We use the extrapolation model introduced in [11] y = c1 + c2 x + c3 ln(1 + x) , x = (amπ )2 , (14)

FIG. 7: Extrapolations of the vector meson and nucleon masses y = am based on the model (14) and four hopping parameter values, top panel. The bottom panel shows two typical examples of ?ve-quark hadron ground state energies y = aE, at distances r = 1a and r = 4a, and their respective extrapolation using the same procedure.

constant of a?1 = 2251MeV or a = 0.088fm. The corresponding vector meson and nucleon masses come out as 659MeV and 1196MeV, respectively, which deviate by 15% and 27% from their experimental values. We take these numbers as indicators of the systematic errors to be expected due to the extrapolation. Also in Fig. 7 two representative examples of extrapolations of ?ve-quark hadron ground state masses y = aE are shown, the relative distances are r = 1a and r = 4a. The error bars represent spectral peak widths ? according to (13). We utilize those to compute uncertainties on the extrapolated energies: At each of the four data points stochastically independent Gaussian random numbers with average and standard deviation given by the energies and peak widths of the data, respectively, were generated and again ?tted with (14). Repeating this a large number of times then gives rise to a standard deviation which we interpret as a MEM peak extrapolated width ?. The extrapolated energy spectrum obtained in this way is displayed in Fig. 8 as a function of the relative distance r.

and y = am. For a motivation of the same we refer the reader to said reference. The extrapolation prescription is, of course, a source of systematic error on our results. In Fig. 7 we show the vector meson and nucleon masses as a function of the squared pion mass and the extrapolations according to (14). The extrapolated values amρ = 0.293 and amN = 0.531, as x → 0, are used to set the reduced mass of the ρN system to its experimental value mρN = 424.7MeV. This results in a lattice

6
TABLE I: Comparison of the statistical errors S of the ground-state peak positions E, see (12), and the peak widths ?, see (13). Results are shown for the lightest quark mass, κ = 0.140, and for all relative distances r. r/a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 aE 1.193 1.434 1.514 1.526 1.512 1.378 1.206 1.120 1.190 1.223 1.348 1.390 1.465 1.338 1.315 1.297 a? 0.050 0.106 0.083 0.062 0.067 0.053 0.052 0.035 0.055 0.065 0.066 0.072 0.047 0.080 0.058 0.062 aS 0.038 0.031 0.037 0.059 0.076 0.032 0.024 0.029 0.030 0.020 0.034 0.030 0.024 0.030 0.040 0.009

FIG. 8: Extrapolated ground-state energy E of the heavylight K–Λ like system versus the relative distance r with MEM based uncertainties (plot symbols). The solid line represents a ?t with the model (16). The dashed line shows the same ?t, but the mirror term aV (L ? x) has been dropped.

The energies shown in Fig. 8 are also subject to statistical errors which originate from the gauge con?guration ensemble. To identify those a bootstrap error analysis [17] was performed. Using a resampling factor of eight on the 343 gauge con?gurations the MEM spectral analysis was repeated for each bootstrap sample[29]. In each case the same ?xed random number sequence was employed in order to eliminate ?uctuations in the annealing process. The statistical errors S thus obtained are compared to the peak widths ? in Tab. I. Results are for κ = 0.140 (lightest quark mass). The other κ values give very similar numbers because the underlying set of gauge con?gurations is the same. Clearly the statistical errors S are consistently smaller than the peak widths ?. On average over all relative distances r the ratio ?/S turns out to be slightly larger than two. Thus, judiciously keeping the larger errors, we have used the spectral peak widths as the principle input for computing uncertainties.
ADIABATIC POTENTIAL

Thus lattice sites with relative distances x and L3 ? x are equivalent. However, as the two-hadron operators for the K–Λ like system are essentially loop operators, see Fig. 2, those do not share the periodic behavior of the fundamental lattice ?elds. This e?ect should be most notable for distances around ≈ L3 /2 = 16. Second, interactions with hadron images in adjacent copies of the lattice are present. Their dominant e?ect is a constant interaction energy, as the relative distances of most of the mirror images remain constant with varying x. Thus motivated, empirically one ?nds that the lattice data are well represented by the model aVL (x) = α0 + aV (x) + aV (L ? x) , x = r/a , (16)

where α0 and L are additional parameters. The resulting ?t is shown in Fig. 8 as a solid line. The dashed line represents α0 + aV (x) only. Table II contains the ?t parameters. Note that L deviates somewhat from 32, as should
TABLE II: Results for the ?t parameters using the model (16). α0 α1 α2 α3 α4 L 1.213(93) 0.0409(37) ?0.89(10) 0.109(19) ?0.00328(52) 26.54(57)

The data points in Fig. 8 connect smoothly along a continuous curve. In order to ?nd an appropriate ?t model we note that the ground-state K–Λ like system should be in a relative S-wave. Our choice thus are Swave 3d harmonic oscillator functions [18]. The ?rst few terms are given by x = r/a , (15) with parameters α1 . . . α4 . This will not be su?cient though for two reasons. First, periodic boundary conditions have been imposed on the lattice in the 3-direction. aV (x) = exp(?α1 x2 )(α2 + α3 x2 + α4 x4 ) ,

be expected for the reason stated above. The χ2 per degree of freedom is 0.63, the uncertainties given stem from the covariance matrix of the Levenberg-Marquardt algorithm. We list those for completeness only, they will not be used in the subsequent analysis. Rather, because we

7 are mostly interested in uncertainties of possible bound state energies, as the case may be, we proceed as follows: Each data point in Fig. 8 is randomized by stochastically independent Gaussian random numbers with average and standard deviation given by the energies and peak widths of the data, respectively. Then the ?t with the model (16) is repeated, the corresponding set of parameters gives rise to a new potential V (x). The procedure is performed 32 times. We show the results in Fig. 9 along with the potential obtained directly from the original data points. Speci?cally, Fig. 9 displays adiabatic potentials
BOUND STATES AND WAVE FUNCTIONS

Va (r) = V (r/a)

(17)

on physical scales given by the lattice constant a, as determined above.

FIG. 9: Adiabatic potential Va (r) resulting from the lattice simulation in physical units, panel (a). Panel (b) shows potentials from 32 sets of randomizations of the ground state energies E as seen in Fig. 8.

Figure 9 represents the end result of the lattice simulation. Although we expect that systematic errors coming from extrapolations may be as large as 30%, and the uncertainties on the potential itself according to Fig. 9(b) are sizable, we believe that important physics can still be learned on a qualitative level. First, we observe that Va (r) of Fig. 9(a) has no repulsive regions. This is a reasonable, and con?dence inspiring, outcome because repulsion in a two-hadron system is typically Pauli repulsion, but of course the latter is absent in a K–Λ like system. Also, as mentioned earlier just before Eqn. (6), the relative distance r = 0 was excluded from the simulation because “color twisted” operators would then have to be included for all distances r, to be consistent. Without further work it remains an open question how such a family of operators might in?uence the form of Va (r). However, it seems reasonable to assume that those will not play a major role for large r because there they would create hadrons inconsistent with con?nement, whereas at r = 0 mixing with those states would lower the ground state energy according to general principles of quantum mechanics. Thus, we believe that the current simulation correctly captures the most important physics of the fourquark system. The salient features of Va (r) are the two distinct regions of attraction around r ? 0.5–1.0fm and r 0.2fm. An obvious question is whether the attraction is su?cient to generate bound states of the ?ve-quark molecule. To ?nd out we have, ?rst, computed scattering phase shifts δ? (p) employing the adiabatic potential Va (r) in a Schr¨dinger equation. For the reduced mass m we chose, o in turn, the three values of the physical K–Λ, D–Λc , and B–Λb systems [3]. These evaluate to am = 0.152, 0.415, and 1.172, respectively. Continuum boundary conditions were implemented by solving a Volterra integral equation within the Jost function formalism [19]. The resulting scattering phase shifts are shown in Fig. 10. According to Levinson’s theorem, see [19] for example, we have δ? (0) ? δ? (∞) = nπ where n is the number of bound states. In the S-wave n is one for the K–Λ system, two for the D–Λc system, and four for the B–Λb system. We need to caution though that relativistic e?ect are large in K–Λ, with increasing quark mass (s → c → b) this becomes less of a concern. The scattering phase shifts are quite featureless, but they give us an exact count of the number of bound states present in the simulation. In order to compute bound state energies and wave functions we solve the radial Schr¨dinger equation with o Va (r). Our conventions can be gleaned from ? ?(? + 1) d2 ua + [2mVa (r) + ]ua = 2mEua , 2 dr r2 (18)

8

FIG. 10: Scattering phase shifts (S-wave) for the K–Λ, D– Λc , and B–Λb systems (bottom to top, respectively) versus the relative momentum p.

FIG. 11: S-wave bound state wave function χ = ua (r) (dashed line) and its square χ = |ua (r)|2 (solid line) for the K–Λ system.

and


dr|ua (r)|2 = 1 .
0

(19)

The region between r = 0 and r = 30a was subdivided into 60 discretization intervals. A suitable cubic B-spline representation [20] for ua (r) was employed to enforce bound-state boundary conditions and provide smooth interpolation between the knots. This then leads to a generalized matrix eigenvalue problem. The scale neutral bound state energies are compiled in Tab. III for the three reduced masses of K–Λ, D–Λc , B–Λb , and three partial waves. The errors come from repeating the calculation with 32 randomizations of the potential, as seen in Fig. 9(b).
TABLE III: Scale neutral bound state energies aEn of the adiabatic potential Va (r) for three values am of the reduced mass m and partial waves ?. The reduced masses correspond to K–Λ, D–Λc , and B–Λb . ? 0 am 0.152 0.415 1.172 0.152 0.415 1.172 0.152 0.415 1.172 aE1 -0.22(12) -0.32(13) -0.39(13) -0.11(11) -0.28(13) -0.38(13) -0.20(13) -0.35(13) aE2 -0.04(13) -0.26(16) aE3 aE4

-0.15(12)

-0.001(88)

1

-0.14(12)

2

-0.11(12)

The S-wave bound state wave functions ua (r) and their squares |ua (r)|2 are displayed in Figs. 11,12 and 13. The wave function of the K–Λ system, Fig. 11, peaks at around r ? 0.6–0.9fm. This is a typical nuclear physics distance scale, and clearly points to the structure of a

hadronic molecule. With increasing (heavy) quark mass the number of bound states also increases. The ground state of the D–Λc system, Fig. 12, exhibits the same feature. However, the excited state wave function has sizable support at small distances r 0.2fm. The wave function starts to ‘feel’ the short-range attraction of the potential Va (r), see Fig. 9(a). This property is fully developed in B–Λb , see Fig. 13. The ground state wave function ‘lives’ in the large-distance trough of Va (r) and thus points to a hadronic molecule. On the other hand, the wave function of ?rst excited state ‘lives’ in the smalldistance regime of Va (r), which indicates a very tight spacial structure. There is nothing as distinct about the third state, and the forth one strongly resembles a continuum state. The wave functions of the ground and ?rst excited B– Λb states with aE1 = ?0.39 and aE2 = ?0.26, respectively, shown in Fig. 13 deserve further comment. Viewing the system as consisting of ?ve quarks, the ground state is clearly a pure two-hadron molecule, and the rmsradius evaluates to rrms ≈ 0.74fm. The excited state, on the other hand, is spatially compact in a dramatic way: Most of the probability density |ua (r)|2 is within a region of less than 0.3fm radius and peaks just above 0.1fm. Inevitably, this ?ve-quark state should be interpreted as a pentaquark. We should remark here that our original ?ve-quark operators (1) are indeed capable of describing such physics, because the set comprises operators where the heavy quark and anti-quark have very small distances, the smallest one being the lattice constant a = 0.088fm. Also, the B–Λb results are probably our most reliable ones, because relativistic e?ects are not very important. On the other hand, systematic errors, having to do with the adiabatic approximation, extrapolations to zero pion mass, and dismissal of relativistic e?ects, clearly render our interpretation a qualitative one. It is interesting to ask though if the reason for our inability to establish

9

FIG. 12: S-wave bound state wave functions χ = ua (r) (dashed lines) and their squares χ = |ua (r)|2 (solid lines) for the D–Λc system.

a clear experimental signal for a pentaquark [21, 22], and also to irrefutably establish a lattice QCD pentaquark [23, 24, 25], could be rooted in the assumption that the pentaquark shall be a ground state of bound light quarks. There is no a priori reason for this. The well-publicized quark model decouplet Θ particle [4] may well have a molecule-like structure. Thus we speculate that the pentaquark, if we understand it as a tight ?ve-quark bound state, may not exist for all light ?avored (u,d,s) quarks, but there is still hope in the heavy-?avor sector. However, a pentaquark may then reveal itself only as an excited state, the ground state being a hadronic molecule. Finally, as a matter of course, we show in Figs. 14 and 15 the P - and D-state wave functions of the calculation. And in Tab. IV we have compiled physical mass spectra of the considered ?ve-quark systems by subtracting the computed binding energies from the experimental single meson and baryon masses [3]. We give those numbers for completeness only, without claiming quantitative relevance.

FIG. 13: S-wave bound state wave functions χ = ua (r) (dashed lines) and their squares χ = |ua (r)|2 (solid lines) for the B–Λb system.

10

FIG. 15: D-wave bound state wave functions χ = ua (r) (dashed lines) and their squares χ = |ua (r)|2 (solid lines). The insets for am and aEn match the entries of Tab. III.

SUMMARY AND CONCLUSION

FIG. 14: P-wave bound state wave functions χ = ua (r) (dashed lines) and their squares χ = |ua (r)|2 (solid lines). The insets for am and aEn match the entries of Tab. III.

We have performed a lattice simulation of a ?ve-quark K–Λ like hadronic system on an anisotropic and asymmetric lattice. The heavy quark anti-quark pair was treated as static. Thus it was possible to compute the total energy of the system as a function of the relative distance r between the hadrons. The maximum entropy method was employed toward this end. An extrapolation into the physical pion mass region was also performed. The objective was to extract an adiabatic potential Va (r)

11
TABLE IV: Physical ?ve-quark hadron masses mn , in GeV, for three values of the reduced mass m corresponding to the systems K–Λ, D–Λc , and B–Λb . The layout matches that of Tab. III. ? 0 m 0.342 0.934 2.639 0.342 0.934 2.639 0.342 0.934 2.639 m1 1.11(28) 3.44(29) 10.02(30) 1.36(26) 3.54(29) 10.05(30) 3.70(29) 11.11(30) m2 4.07(29) 10.33(35) m3 m4

10.58(27)

10.90(20)

1

10.60(27)

2

10.66(26)

for the relative motion. The potential turns out to have two distinct attractive troughs, one at intermediate distances, r ≈ 0.7fm, and one at short range, r 0.2fm. To study the dynamics of the two-hadron system we have used Va (r) in a Schr¨dinger equation for three valo ues of the physical reduced mass, corresponding to K– Λ, D–Λc , and B–Λb . With increasing heavy-quark mass ms < mc < mb the number of S-wave bound states increases from one to four. We have examined the corresponding wave functions to study the nature of the ?vequark systems. Systematic errors predominantly originating from using the adiabatic approximation, extrapolation to zero pion mass, the non-relativistic framework, and possibly the quenched approximation, render the results of our study qualitative only. Nevertheless, we believe that the results for the B–Λb ground and excited state wave functions, respectively, bring to light a particularly interesting scenario with regard to ?ve-quark system physics: The ground state is best described as a hadronic molecule, with a relative hadron-hadron distance matching a nuclear physics scale, whereas the excited state exhibits a wave function with support on very short distances of r 0.2fm, or so. This leads us to interpret the excited state as a pentaquark. The results for the bquark system are less prone to relativistic corrections and therefore are our most reliable ones. The light-quark K– Λ exhibits one ground state which clearly is a hadronic molecule. In the light of our results it might be worthwhile to initiate future lattice QCD studies with this scenario in mind, but with a more realistic set of operators, lattices, actions, and lighter quark masses, etc. All things considered, there is ground for the hypothesis that an all light-quark pentaquark may not exist, but that the existence of a genuine pentaquark in the heavy-quark sector cannot be ruled out. However, if present, it may well be an excited state. This material is based upon work supported by the National Science Foundation under Grant No. 0300065.

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