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Basis States for Hamiltonian QCD

arXiv:hep-lat/9210006v1 2 Oct 1992BUHEP-92-33October, 1992Basis States for Hamiltonian QCDwith Dynamical QuarksTimothy E. VaughanDepartment of PhysicsBoston University590 Commonwealth Ave.Boston, MA 02215email: tvaughan@buphyk.bu.eduAbstractWe discuss the construction of basis states for Hamiltonian QCD onthe lattice, in particular states with dynamical quark pairs. We cal-culate the matrix elements of the operators in the QCD Hamiltonianbetween these states.

Along with the “harmonic oscillator” states in-troduced in previous pure SU(3) work, these states form a workingbasis for calculations in full QCD.

1IntroductionAs detailed in papers with Bronzan [2, 3], we have been studying latticeQCD using an operator and states approach rather than the usual MonteCarlo simulation of the Feynman path integral. As we have emphasized, thisrequires the use of a suitable set of single degree-of-freedom (DOF) stateson the SU(3) manifold.

Here “suitable” means that matrix elements of theQCD Hamiltonian must be calculable in closed form using these states, andthat the states have tunable parameters so they can model the QCD wavefunctions at all values of the coupling constant.We have introduced a “harmonic oscillator” basis of states as one whichsatisfies the above criteria and is therefore useful in making calculations inHamiltonian QCD. Since these states describe only gauge degrees of freedom,they are actually designed for studying only a pure SU(3) gauge theory.In this paper we will introduce a similarly suitable set of dynamical-quark basis states to complement the harmonic oscillator states.

We willdiscuss how we construct a quark ground state at arbritrary values of thecoupling constant, create quark excitations, and calculate matrix elementsof the QCD Hamiltonian between these states. Together with the harmonicoscillator states, then, we have a complete basis for the study of full QCD.The layout of this paper is as follows.In Section 2 we briefly reviewour work on the construction of harmonic oscillator states.

In Section 3 wedescribe the construction of the quark vacuum as well as the states whichinclude virtual quark pairs. In Section 4 we discuss the combination of theharmonic oscillator states with the quark states to form a basis for simulatingfull QCD.

In Section 5 we derive the matrix elements of the operators of theQCD Hamiltonian between these states. We conclude in Section 6 with adiscussion of the usefulness of these states.2Review of pure gauge statesEach gauge degree of freedom (link variable on the lattice) can be describedby a wave function which is a “harmonic oscillator” state on the SU(3) man-ifold.

The harmonic oscillator states are derived from a “Gaussian” state on1

the manifold. This Gaussian can be writtenψα1(α, t) =Xp,qd(p, q)e−tλ(p,q)χ(p,q),(2.1)where d(p, q) = (p + 1)(q + 1)(p + q + 2)/2 is the dimension of the (p, q)representation of SU(3), λ(p, q) = (p + q)2/4 + (p −q)2/12 + (p + q) is thequadratic Casimir eigenvalue, and χ(p,q) is the character.

α ≡(α1, α2, . .

. , α8)is a parameterization of the adjoint representation of SU(3).

The parametert controls the width of the Gaussian, which is centered at α1.To make connection with a more familiar object, note that the corre-sponding wave function on the flat, three-dimensional manifold R3 wouldbeψx1(x, t) = e−(x−x1)2/4t(4πt)4,(2.2)where x and x1 are ordinary vectors. On this flat manifold, harmonic os-cillator states can be generated by applying a polynomial in the operators−i∂∂x1a to the wave function, then setting x1 to 0.

On the SU(3) manifold, thecorresponding operators are JLa(α1), the generators of SU(3) in differentialform.1The harmonic oscillator states that we have formed on the manifold canbe labeled by the number of SU(3) indices on the state. Thus, the zero-,one-, and two-color states areφ = φ(α, t)=ψα1(α, t)|α1=0,φa = φa(α, t)=JLa(α1)ψα1(α, t)|α1=0,φab = φab(α, t)={JLa(α1), JLb(α1)}ψα1(α, t)|α1=0.

(2.3)For comparison, the analogous states on a flat manifold areφ(x, t)=e−x2/4t,φa(x, t)=xae−x2/4t,φab(x, t)=xaxbe−x2/4t. (2.4)1When applied to a representation of SU(3), JLa(α1) has the effect of left-multiplyingthe representation by the ath generator.

There are also operators JRa(α1) which right-multiply the representation and could be used to generate a slightly different set of har-monic oscillator states.2

In the papers we have cited we described the orthogonalization of these statesas well as the calculation of matrix elements of the QCD Hamiltonian betweenthem. We will not need those details here, however.These single-DOF wave functions must now be pieced together to formmulti-DOF states suitable for computations in Hamiltonian QCD.

Each multi-DOF state will be a product of single-DOF states. The most basic state isone with no excitations.

Specifically, each degree of freedom is a zero-colorstate:|Ψ>= |φφφφ · · ·> . (2.5)The next simplest states are those with just one excitation:|Ψ>= |φaφφφ · · ·> .

(2.6)However, these states are not SU(3) singlets, as evidenced by the color indexwhich is left hanging. In order to maintain global SU(3) invariance, we mustsum over all indices.

Therefore, the next possible states are|Ψ>= δab|φaφbφφ · · ·>,(2.7)where δab is a Kronecker δ-function. There are of course similar states wherethe excitations lie on different degrees of freedom.

We call these one-pairstates. In analogous fashion, we can form two-pair states, three-pair states,and so on.Arbitrarily intricate states can be constructed by using highly colored,single-DOF states along with complicated coupling schemes.

For instance,the following is a valid (that is, gauge-invariant) state:|Ψ>= δabTr(λcλdλe)|φacφbdφφe · · ·>,(2.8)where the λi’s are Gell-Mann matrices.The states described above provide the basis for a pure SU(3) gaugetheory calculation, and they will likewise provide the basis for the gaugesector of our full QCD calculation. In the next sections we will describe howto form similar states that include dynamical quarks, to complete our basis.3Quark sector statesTo formulate our quark states, we use staggered fermions [4].

(We followclosely the notation of Banks et al. [1]).

Specifically, the quarks are created3

and destroyed by applications of single-component fields χ†i(s) and χi(s),where the index i signifies color in the fundamental representation of SU(3),and the argument s = (x, y, z) specifies a site on the lattice. The operatorsact on a simple Fock space, and any quark configuration can be specified bygiving the location of the quarks on the lattice.3.1Strong-coupling quark vacuumJust as we did for the gauge states, we would like to use quark states which areflexible enough to treat both the strong- and weak-coupling regimes.

Towardthis end, we will begin with a description of the strong-coupling vacuum,then discuss how to obtain the weak-coupling vacuum from it. We will thendescribe the excitations of the vacuum.At very strong coupling the vacuum state is the “checkerboard” state,in which alternating sites are fully occupied [1].

Two of these states canbe formed, depending on which sublattice is filled (the “red” squares or the“black” squares of the checkerboard). The two states are|ψeven >=Yeven sites16ǫijkχ†i(s)χ†j(s)χ†k(s)|E >,|ψodd >=Yodd sites16ǫijkχ†i(s)χ†j(s)χ†k(s)|E >,(3.1)where |E > is the empty lattice, and even/odd means that x + y + z = ±1.2These two states transform into one another under a chirality transforma-tion.

We can form two degenerate vacua, the symmetric and antisymmetriccombinations of these two states, which are eigenstates of the chirality oper-ator. We have shown numerically that as we decrease the coupling constant,the true vacuum is the symmetric combination, so we use|0>strong= |ψeven > + |ψodd>(3.2)as the vacuum state at strong coupling.2One must be careful to maintain a fixed ordering for the product over the sites, lestminus sign errors creep in.

Although we will not display it explicitly, we assume that thesites have been ordered in some appropriate manner.4

3.2Weak-coupling quark vacuumWe derive the weak-coupling vacuum from the strong-coupling vacuum bymeans of a projection operator. We write|0>weak= limα→∞e−αHw|0>strong .

(3.3)Hw is the weak-coupling limit of the quark Hamiltonian,Hw = 12aXs,s′χ†(s)Mss′χ(s′),(3.4)whereMss′ =Xˆn[δs,s′+ˆn + δs,s′−ˆn]η(ˆn),(3.5)η(x) = (−1)z, η(y) = (−1)x, η(z) = (−1)y, and a is the lattice spacing. (There is a suppressed color index in this expression.) Note that in weakcoupling the quarks decouple from the gauge degrees of freedom, and Hw isequal to the free-field quark Hamiltonian.The projection works as follows.α is an adjustable parameter whichgoverns the strength of the projection.

When α = 0, there is no projection,and we still have the strong-coupling vacuum.As α becomes large, theprojection operator damps out the pieces of |0>strong which have high energyin the weak-coupling limit, leaving intact the lowest energy weak-couplingstate. So, for large α, this operator does exactly what we want: it projectsout the weak-coupling vacuum.3 In practice, α = 1 will give nearly completeprojection.As we vary α from 0 to 1, we obtain a vacuum state whichinterpolates smoothly between the g →∞and g = 0 vacua.

We thereforehope to obtain a good approximate quark vacuum for arbitrary values of thecoupling constant by using α as a variational parameter in our simulations.We will designate this approximate vacuum |0q >.3.3Quark excitationsHaving established a quark-sector vacuum, we must now understand how tocreate excited quark states. This is much easier and more familiar than it3Of course, we are assuming here that the strong- and weak-coupling vacua are notorthogonal to each other.

This can be demonstrated a posteriori.5

was for the gluon states, where we were dealing with the complicated SU(3)manifold. Here, we have a simple Fock space, and we merely need to decidewhich states from that Fock space to include in our basis.We can exclude a large number of states by observing that the “checker-board” vacuum state is half filled and that the Hamiltonian preserves quarknumber.

We are interested in studying the glueball spectrum, which has thesame quantum numbers as the vacuum, so there is no reason to include anyquark state that isn’t half-filled. If we wish to study the spectrum of particleswith other quantum numbers, we likewise limit ourselves to those sectors.We can obtain a new half-filled state directly from the vacuum state byapplying one annihilation operator and one creation operator:|Ψ>= χ†i(s)χi(s′)|0q > .

(3.6)Recall that the index i indicates color, in the fundamental representationof SU(3), reflecting the correct group properties of the quarks. Notice thatwe are summing over i, thereby maintaining global gauge invariance in thequark sector in an analogous way to the gauge sector.The state shown above corresponds to exciting one virtual quark pairout of the vacuum.

Clearly, we can form states with an arbitrary number ofvirtual pairs by applying the appropriate number of creation and annihilationoperators. One convenient feature of this scheme is that we can control howmany such pairs we wish to have in the simulation simply by choosing whichbasis states to include.

We will comment on this in the final section.4Combining the gauge and quark statesSections 2 and 3 described our construction of states for the gauge and quarkdegrees of freedom in our simulations. The simplest way to form a combinedstate suitable for a full QCD simulation is to make a direct product of onestate from the gauge sector with one from the quark sector.

A typical state,then could be written|Ψ>= δabδcd|φaφbφcφdφ · · ·> ⊗χ†i(s1)χi(s′1)χ†j(s2)χj(s′2)|0q > . (4.1)Let us emphasize again that all of the SU(3) indices are contracted, ensuringthat global gauge invariance is maintained.

Note in particular that the indices6

are summed separately within the two sectors, indicating that the gauge andquark sectors are separately invariant under the global SU(3) transformation.There is a second way to form combined states, in which the two sec-tors are not separately invariant. To do this, we need an object which canintertwine the SU(3) adjoint representation of the gauge sector with the fun-damental representation of the quark sector.

The simplest way to do this iswith the Gell-Mann matrices, which carry both three- and eight-dimensionalindices. A simple state of this form is|Ψ>= |φaφφφφ> ⊗χ†i(s)λaijχj(s′)|0q > .

(4.2)This completes the description of our basis states for full QCD calcula-tions on the lattice. In the next section we will show how to compute thematrix elements necessary for our simulations.5Matrix elementsThe matrix elements of our combined states are sums of products of gaugematrix elements with quark matrix elements.

The results of our gauge sectorcalculations have already appeared in [3], and we will not repeat them here.The most basic quark matrix element that we must calculate is the overlapof our projected vacuum, <0q|0q >. This, in turn, can be broken down intosums of matrix elements of the projected checkerboard states:<0q|0q >=< ψeven|e−2αHw|ψeven > + < ψeven|e−2αHw|ψodd >+< ψodd|e−2αHw|ψeven > + < ψodd|e−2αHw|ψodd > .

(5.1)We will explicitly calculate only the first of these four nearly identical ex-pressions. Recall that|ψeven >=Yeven sites16ǫijkχ†i(s)χ†j(s)χ†k(s)|E >=[χ†1(s1)χ†2(s1)χ†3(s1)] · · · [χ†1(sM)χ†2(sM)χ†3(sM)]|E >, (5.2)where M =N32is the number of even sites on our N × N × N lattice,4and s1, .

. .

, sM specify the locations of those sites. First, we segregate the4We assume N is an even number.7

creation operators by color. This involves an even number of permutations,so no minus sign is picked up.|ψeven >= [χ†1(s1) · · ·χ†1(sM)][χ†2(s1) · · ·χ†2(sM)][χ†3(s1) · · · χ†3(sM)]|E > .

(5.3)Likewise, we can segregate Hw into a product of single-color factors:Hw≡H1w + H2w + H3w=12aXs,s′[χ†1(s)Mss′χ1(s′) + χ†2(s)Mss′χ2(s′) + χ†3(s)Mss′χ3(s′)]. (5.4)Since the separate color factors commute, we can writee−2αHw = e−2αH1we−2αH2we−2αH3w(5.5)and thereby write our matrix element in color-factorized form:< ψeven|e−2αHw|ψeven >= .

(5.6)We now want to commute the exponential factors all the way to the left,where we will be able to take advantage of the fact that

We prove thisrelation in the Appendix.8

Commuting each of the exponential factors to the left gives< ψeven|e−2αHw|ψeven >= . (5.10)The M sites s′1, .

. .

, s′M can be chosen in any way from the even sites, butunless they are a permutation of s1, . .

. , sM, the contribution to the matrixelement is zero.

Thus, we get< ψeven|e−2αHw|ψeven >==[Xperms PKs1,sP (1) · · · KsM ,sP (M)]3. (5.11)Introduce the M × M matrix LEE(α), whose (m, n)th element is given byLEEmn(α) = Ksm,sn(2α).

(5.12)The EE refers to the fact that we are calculating the even-even matrix ele-ment. We can then write our even-even matrix element in its final, decep-tively compact form:< ψeven|e−2αHw|ψeven >= [det LEE(α)]3.

(5.13)The even-odd, odd-even, and odd-odd pieces of < 0q|0q > can be calculatedin analogous fashion, the only difference being the composition of the corre-sponding L matrix. We therefore have completed the calculation of the normof the quark vacuum:<0q|0q >= [det LEE(α)]3 + [det LEO(α)]3 + [det LOE(α)]3 + [det LOO(α)]3.

(5.14)9

Now that the machinery is set up, the remaining matrix elements requiremuch less effort. The next simplest matrix element to calculate is<0q|χi(s0)χ†j(s′0)|0q > .

(5.15)We need not consider operators χχ or χ†χ†, which change quark number andtherefore have trivially vanishing matrix elements. (As we have mentionedalready, the QCD Hamiltonian preserves quark number, and therefore doesnot include operators of this type.

)Similarly, if i ̸= j, then the matrixelement vanishes because there would be a mismatch in the number of quarksof a given color.The calculation of this matrix element proceeds exactly as the overlapcalculation, except that one of the colors (suppose it’s color 1) will have theextra operators χ1(s0) and χ†1(s′0) to commute through the exponential. Thisgives color 1 a contribution ofXs0,s′0Ks0s0(−α)Ks′0s′0(−α) .

(5.16)If we define the extended (M + 1) × (M + 1) matrixLEE(s0, s′0) =δs0,s′0Ks0,s1(α)· · ·Ks0,sM(α)Ks1,s′0(α)Ks1,s1(2α)· · ·Ks1,sM(2α)............KsM,s′0(α)KsM ,s1(2α)· · ·KsM,sM(2α),(5.17)then we can again write a compact expression for our matrix element,<0q|χi(s0)χ†j(s′0)|0q >=δij det LEE(s0, s′0)[det LEE(α)]2+EO + OE + OO contributions. (5.18)The factors of det LEE(α) are the contribution of the two unmodified colors.Matrix elements involving more χχ† pairs can be calculated in analogywith the above derivation.

Each additional pair requires the introduction ofa further extended matrix, for the case in which all the χ’s act on the samecolor. For instance, the next extension of the matrix comes from<0q|χ1(s0)χ1(s00)χ†1(s′0)χ†1(s′00)|0q >=det LEE(s0, s′0; s00, s′00)[det LEE(α)]2+EO + OE + OO contributions,(5.19)10

whereLEE(s0, s′0; s00, s′00) =δs00,s′00δs00,s′0Ks00,s1(α)· · ·Ks00,sM(α)δs00,s′0δs0,s′0Ks0,s1(α)· · ·Ks0,sM(α)Ks1,s′00(α)Ks1,s′0(α)Ks1,s1(2α)· · ·Ks1,sM(2α)...............KsM,s′00(α)KsM,s′0(α)KsM ,s1(2α)· · ·KsM,sM(2α). (5.20)We have calculated matrix elements of this type for up to four χχ† pairs,which is the requirement for simulations with two dynamical quark pairs.56SummaryWe have introduced a set of basis states that is suitable for making calcula-tions in an “operator and states” approach to the study of lattice QCD.

Theprevious section detailed the computation of matrix elements of the quark-sector states. In previous work we calculated matrix elements for pure gaugesimulations.

Taken together, these matrix elements provide us the tools toconstruct a Hamiltonian matrix on a basis of states in full QCD. The eigen-values of this matrix are then estimates of masses in the QCD spectrum.We would like to emphasize the particular power that this method haswith respect to testing the quenched approximation.

Unlike Monte Carlosimulations of the Feynman path integral, dynamical quarks pose no partic-ular problems to our method. The states which include quark pairs are “justanother configuration.” Moreover, by choosing which states to include, wecan limit the simulation to having as many virtual quark pairs as we wish.By comparing runs with and without quark pairs, we hope to be able toprovide some insight into the quenched approximation.We would like to acknowledge the contributions made to this work byJ.

B. Bronzan.5If one fixes the gauge by our current method[5], the QCD Hamiltonian can contributeup to two pairs of operators, rather than just one. In this case we would need two more“extensions” of the L matrix to do the equivalent simulations.11

AAppendixTo prove the relationχ(s)e−σHw =Xs′Ks,s′(σ)e−σHwχ(s′),(A.1)we first need to diagonalize Hw. Recall that Hw is bilinear in the quarkcreation and annihilation operators:Hw = 12aXs,s′χ†(s)Mss′χ(s′),(A.2)whereMss′ =Xˆn[δs,s′+ˆn + δs,s′−ˆn]η(ˆn).

(A.3)Diagonalizing Mss′ amounts to solving the Dirac equation on the lattice. Weintroduce quark mode operators, which annihilate “plane-wave” states:χ(k, i) =Xsχ(s)φ(s; k, i).

(A.4)The φ(s; k, i) are the eigenfunctions of Mss′:Xs′Mss′φ(s′; k, i) = λ(k, i)φ(s; k, i),(A.5)where k is the lattice momentum, i enumerates the modes, and λ(k, i) is theeigenvalue of the mode.6 On a lattice with N×N×N sites, 1 ≤kx, ky, kz ≤N2and 1 ≤i ≤N3. The derivation of the eigenfunctions is straightforward butrather tedious.

We have carried it out explicitly only for a 2 × 2 × 2 lattice.The weak-coupling quark Hamiltonian is (by design) diagonal using thesenew operators:Hw=12aXs,s′χ†(s)Mss′χ(s′)=12aXs,s′Xk,k′Xi,i′χ†(k, i)φ(s; k, i)Mss′φ(s′; k′, i′)χ(k′, i′)=12aXk,iλ(k, i)χ†(k, i)χ(k, i). (A.6)6The plane-wave operators also have a color index, which we suppress here.12

Next, we define the expressionS(σ) = eσHwχ(k, i)e−σHw. (A.7)ThendS(σ)dσ= eσHw[Hw, χ(k, i)]e−σHw.

(A.8)Substituting the diagonal form of Hw we just derived, we have[Hw, χ(k, i)] = −λ(k, i)χ(k, i). (A.9)Therefore,dS(σ)dσ= −λ(k, i)S(σ),(A.10)which implies thatS(σ) = constant × e−σλ(k,i).

(A.11)Noting that S(0) = χ(k, i), we haveS(σ) = e−σλ(k,i)χ(k, i),(A.12)giving us the relationeσHwχ(k, i)e−σHw = e−σλ(k,i)χ(k, i). (A.13)If we now multiply both sides of this by φ(s; k, i), sum over k and i, andchange back to configuration space variables χ(s), we obtaineσHwχ(s)e−σHw =Xs′Xk,iφ(s; k, i)e−σλ(k,i)φ(s′; k, i)χ(s′).

(A.14)Substituting the variable K, and left-multiplying by e−σHw, we get the rela-tion we set out to prove,χ(s)e−σHw =Xs′Ks,s′(σ)e−σHwχ(s′). (A.15)13

References[1] T. Banks et al. Phys.

Rev., D15:1111, 1977. [2] J.

B. Bronzan and T. E. Vaughan. Phys.

Rev., D43:3499, 1991. [3] J.

B. Bronzan and T. E. Vaughan. Phys.

Rev., D44:3264, 1991. [4] L. Susskind.

Phys. Rev., D16:3031, 1977.

[5] T. E. Vaughan. A Hamiltonian Study of Lattice Quantum Chromody-namics with Dynamical Quarks.

PhD thesis, Rutgers University, 1992.14

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