Symmetric multivariate polynomials as a basis for three-boson light-front wave functions
aa r X i v : . [ phy s i c s . c o m p - ph ] J u l Symmetric multivariate polynomialsas a basis for three-boson light-front wave functions
Sophia S. Chabysheva, Blair Elliott, and John R. Hiller
Department of PhysicsUniversity of Minnesota-DuluthDuluth, Minnesota 55812 (Dated: August 20, 2018)
Abstract
We develop a polynomial basis to be used in numerical calculations of light-front Fock-spacewave functions. Such wave functions typically depend on longitudinal momentum fractions thatsum to unity. For three particles, this constraint limits the two remaining independent momentumfractions to a triangle, for which the three momentum fractions act as barycentric coordinates. Forthree identical bosons, the wave function must be symmetric with respect to all three momentumfractions. Therefore, as a basis, we construct polynomials in two variables on a triangle thatare symmetric with respect to the interchange of any two barycentric coordinates. We find that,through the fifth order, the polynomial is unique at each order, and, in general, these polynomialscan be constructed from products of powers of the second and third-order polynomials. The useof such a basis is illustrated in a calculation of a light-front wave function in two-dimensional φ theory; the polynomial basis performs much better than the plane-wave basis used in discretelight-cone quantization. PACS numbers: 11.15.Tk, 11.10.Ef, 02.60.Nm . INTRODUCTION Light-front quantization [1, 2] is a natural choice for the nonperturbative solution ofa quantum field theory. The eigenstates are built as expansions in terms of Fock states,states of definite particle number and definite momentum, where the coefficients are boost-invariant wave functions. The vacuum state is simply the Fock vacuum, thereby giving thewave functions a standard, quantum mechanical interpretation.The light-front time coordinate is chosen to be x + ≡ t + z/c , and the corresponding light-front spatial coordinate is x − ≡ t − z/c ; the other spatial coordinates are unchanged. Theconjugate light-front energy is p − = E − cp z , and the light-front longitudinal momentum is p + = E/c + p z . A boost-invariant momentum fraction x i = p + i /P + is defined for the ithparticle with momentum p + i in a system with total momentum P + . Because the light-frontlongitudinal momentum is always positive, these momentum fractions are between zero andone. Also, momentum conservation dictates that they sum to one.In the three-particle case, the three momentum fractions correspond to the barycentriccoordinates of a triangle. Any two can be treated as the independent variables. For a wavefunction that describes three identical bosons, there must be symmetry under the interchangeof any two of the three coordinates, not just symmetry under the interchange of the twochosen as independent. Any set of basis functions to be used in numerical approximationsof such a wave function should share this symmetry. However, the usual treatment of two-variable polynomials on a triangle is limited to consideration of symmetry with respect toonly the two independent variables [3, 4]. Here we consider the full-symmetry constraint.We find that full symmetry among all three barycentric coordinates dramatically reducesthe number of polynomials at any given order. For the lowest orders, there is only one; atthe sixth order, there are two. In general, for polynomials of order N , the number of linearlyindependent polynomials is the number of combinations of two nonnegative integers n and m such that N = 2 n + 3 m . These polynomials can be chosen to be products of n factorsof the second-order polynomial and m factors of the third-order polynomial. They are notorthonormal, but given such a set of polynomials one can, of course, systematically generatean orthonormal set.As a test of the utility of these polynomials, we consider a problem in two-dimensional φ theory where the mass of the eigenstate is shifted by coupling between the one-boson sectorand the three-boson sector. The results obtained are quite encouraging. For comparisonwe also consider discrete light-cone quantization (DLCQ) [2, 5] which uses a periodic plane-wave basis and therefore quadratures in momentum space that use equally spaced points.The DLCQ results would require extrapolation to obtain an accurate answer, whereas thesymmetric-polynomial basis immediately converges.The content of the remainder of the paper is as follows. In Sec. II, we specify theconstruction of the fully symmetric polynomials. The first subsection describes the lowestorder cases, where a first-order polynomial is found to be absent and the second and third-order polynomials are found to be unique. The second subsection gives the analysis at anyfinite order, with details of a proof left to an Appendix. The illustration of the use ofthese polynomials, as a basis for the three-boson wave function in φ theory, is presented inSec. III. A brief summary is given in Sec. IV.2 I. FULLY SYMMETRIC POLYNOMIALSA. Lowest orders
We consider polynomials in Cartesian coordinates x and y , on the triangle defined by0 ≤ x ≤ , ≤ y ≤ , ≤ − x − y ≤ , (2.1)that are fully symmetric with respect to interchange of the coordinates x , y , and z = 1 − x − y .These can be viewed as the restriction of three-variable polynomials on the unit cube to theplane x + y + z = 1. The construction of the fully symmetric three-variable polynomials onthe cube is trivial; at order N , the possible polynomials are linear combinations of the form x i y j z k + x j y k z i + x k y i z j + x j y i z k + x i y k z j + x k y j z i , (2.2)with i , j , and k nonnegative integers such that N = i + j + k . The linearly independentpolynomials would correspond to some particular ordering of these indices, such as i ≤ j ≤ k .For N = 0 or 1 there is only one polynomial, but for N ≥ x + y + z = 1 is, however, a severe constraint.As we will see, the fully symmetric two-variable polynomials are unique up through N = 5.For N = 1, the constraint eliminates the only candidate; the restriction from the cube tothe plane makes x + y + z just a constant. For N = 2, we have two candidates x + y + z and xy + xz + yz. (2.3)Substitution of z = 1 − x − y quickly shows that they are equivalent up to terms of orderless than two. Similarly, for N = 3, the three candidates x + y + z , x y + x z + xy + xz + y z + yz , and xyz (2.4)reduce to equivalent polynomials, up to terms of order less than three, upon substitution of z = 1 − x − y . Equivalence does not exclude the possibility that the polynomials will differby fully symmetric polynomials of lower order. The terms of order three are the same, andthe polynomials differ by at most symmetric polynomials of lower order.To proceed in this fashion to higher orders is, of course, possible but tedious. Insteadwe develop a direct analysis of the possible two-variable polynomials and the symmetryconstraints, as described in the next subsection. B. General analysis
In order to avoid complications due to lower-order contributions, we first change variablesfrom x, y, z to u, v, w defined by u = x − / , v = y − / , w = z − / − ( u + v ) . (2.5)Any polynomial P on the triangle, for which each term is of order N , can be written in theform P ( u, v ) = N X n =0 c n u n v N − n , (2.6)3nd, unlike replacement of x or y by z = 1 − x − y , powers of w = − ( u + v ) that appear inreplacements of u or v do not introduce lower-order contributions.Symmetry with respect to just u and v restricts the coefficients to be such that c n = c N − n .If symmetry with respect to v → w = − ( u + v ) is imposed, the coefficients must satisfy theconstraint N X n =0 c n u N − n v n = X n =0 ,N c n u N − n ( − n ( u + v ) n = N X n =0 c n u N − n ( − n n X m =0 (cid:18) nm (cid:19) u n − m v m . (2.7)These are sufficient to guarantee that the resulting polynomial has all the desired symmetries.The symmetry conditions can be reduced to a linear system for the coefficients. With achange in the order of the sums on the right of (2.7) and an interchange of the summationindices m and n , we find N X n =0 c n u N − n v n = N X n =0 N X m = n ( − m (cid:18) mn (cid:19) c m u N − n v n . (2.8)Therefore, the coefficients must satisfy the linear system c n = c N − n , N X m = n ( − m (cid:18) mn (cid:19) c m = c n . (2.9)This system may at first seem to be overdetermined, but instead it is typically underdeter-mined. A solution exists for any N other than N = 1. For N = 2 , ,
4, and 5, there is onelinearly independent solution; and, for N ≥
6, there can be two or more linearly independentsolutions.For example, with N = 6 the system can be expressed in matrix form as − − − − − − − −
10 150 0 0 − −
10 200 0 0 0 0 − − c c c c c c c = . (2.10)The determinant is obviously zero, as is the case for any N , allowing nontrivial solutions.The system reduces to two equations3 c − c = 0 , c − c + c = 0 (2.11)for the four unknowns, leaving two linearly independent solutions, such as u + 3 u v + 5 u v + 3 uv + v and u v + 2 u v + u v . (2.12)For any value of N , one finds that the number of independent solutions is always thenumber of ways that N can be written as 2 n + 3 m for nonnegative integers n and m . Aproof of this conjecture for arbitrary N is given in the Appendix. Thus, in each of these cases,4 fully symmetric polynomial can be chosen to be the product of n copies of the second-order polynomial and m copies of the third-order polynomial, or a linear combination ofsuch polynomials. Returning to the original Cartesian coordinates, we take these two basepolynomials to be C ( x, y ) = x + y + (1 − x − y ) and C ( x, y ) = xy (1 − x − y ) . (2.13)We then have that all fully symmetric polynomials can be constructed from linear combi-nations of the products C nm ( x, y ) = C n ( x, y ) C m ( x, y ) . (2.14)These do not form an orthonormal set. To construct such a set, we apply the Gramm–Schmidt process, relative to the inner product Z dx Z − x dyP ( i ) n ( x, y ) P ( j ) m ( x, y ) = δ nm δ ij , (2.15)where P ( i ) n is the ith polynomial of order n . The first few polynomials are P = √ , (2.16) P = 0 ,P = √ (cid:2) x + 4 yx − x + 4 y − y + 1 (cid:3) ,P = √ (cid:2) − yx + 20 x − y x + 160 yx − x + 20 y − y + 8 / (cid:3) ,P = √ (cid:2) x + 120 yx − x + 180 y x − yx + 80 x + 120 y x − y x + 100 yx − x + 60 y − y + 80 y − y + 5 / (cid:3) ,P = √ (cid:2) − yx + 210 x − y x + 5040 yx − x − y x + 7560 y x − yx + 280 x − y x + 5040 y x − y x + 1120 yx − x + 210 y − y + 280 y − y + 4 (cid:3) ,P (1)6 = r (cid:2) x + 720720 yx − x + 1441440 y x − yx + 826980 x + 1681680 y x − y x + 1709400 yx − x + 1441440 y x − y x + 2203740 y x − yx + 120204 x + 720720 y x − y x + 1709400 y x − y x + 146664 yx − x + 240240 y − y + 826980 y − y + 120204 y − y + 581] ,P (2)6 = r (cid:2) − x − yx + 49308 x + 399630 y x − yx − x + 881440 y x − y x + 202440 yx + 18200 x + 399630 y x − y x + 826560 y x − yx − x − y x − y x + 202440 y x − y x + 31080 yx − x − y + 49308 y − y + 18200 y − y − y + 28 (cid:3) . If there is only one polynomial at a particular order, the i index is dropped.5 II. ILLUSTRATION
As a sample application, we consider the integral equation for the three-boson wavefunction in two-dimensional φ theory. This equation is obtained from the fundamentalHamiltonian eigenvalue problem on the light front [2], P − | ψ ( P + ) i = M P + | ψ ( P + ) i and P + | ψ ( P + ) i = P + | ψ ( P + ) i . (3.1)The second equation is automatically satisfied by expanding the eigenstate in Fock states | p + i ; P + , n i of n bosons with momentum p + i such that P i p + i = P + : | ψ ( P + ) i = X n ( P + ) ( n − / Z n − Y i =1 dx i ! ψ n ( x , ..., x n ) | x i P + ; P + , n i . (3.2)Here ψ n is the n -boson wave function, and the factor ( P + ) ( n − / is explicit in order that ψ n be independent of P + .The light-front Hamiltonian for φ theory is P − = Z dp + µ p + a † ( p + ) a ( p + ) (3.3)+ λ Z dp +1 dp +2 dp +3 π p p +1 p +2 p +3 ( p +1 + p +2 + p +3 ) × (cid:2) a † ( p +1 + p +2 + p +3 ) a ( p +1 ) a ( p +2 ) a ( p +3 )+ a † ( p +1 ) a † ( p +2 ) a † ( p +3 ) a ( p +1 + p +2 + p +3 ) (cid:3) + λ Z dp +1 dp +2 π p p +1 p +2 Z dp ′ +1 dp ′ +2 p p ′ +1 p ′ +2 δ ( p +1 + p +2 − p ′ +1 − p ′ +2 ) × a † ( p +1 ) a † ( p +2 ) a ( p ′ +1 ) a ( p ′ +2 ) . The mass of the constituent bosons is µ , and λ is the coupling constant. The operator a † ( p + )creates a boson with momentum p + ; it obeys the commutation relation[ a ( p + ) , a † ( p ′ + ] = δ ( p + − p ′ + ) (3.4)and builds the Fock states from the Fock vacuum | i in the form | x i P + ; P + , n i = 1 √ n ! n Y i =1 a † ( x i P + ) | i . (3.5)The terms of the light-front Hamiltonian are such that P − changes particle number not atall or by two; therefore, the number of constituents in a contribution to the eigenstate isalways either odd or even.We consider the odd case, and, to have a finite eigenvalue problem, we truncate the Fock-state expansion at three bosons. We also simplify to a problem with an exact solution bydropping from the Hamiltonian the two-body scattering term, the last term in (3.3). The6ction of the light-front Hamiltonian then yields the following coupled system of integralequations: M ψ = µ ψ + λ √ Z dx dx π √ x x x ψ ( x , x , x ) , (3.6) M ψ = µ (cid:18) x + 1 x + 1 x (cid:19) ψ + λ √ ψ π √ x x x . (3.7)It is understood that x = 1 − x − x .To create a single integral equation for ψ , we use the first equation to eliminate ψ fromthe second, leaving M ψ = µ (cid:18) x + 1 x + 1 x (cid:19) ψ − λ π ) µ − M √ x x x Z dx ′ dx ′ p x ′ x ′ x ′ ψ ( x ′ , x ′ , x ′ ) . (3.8)This is no longer a simple eigenvalue problem for M , but it can be rearranged into aneigenvalue problem for the reciprocal of a dimensionless coupling, defined as ξ = 6(1 − M /µ ) (cid:18) πµ λ (cid:19) . (3.9)The rearrangement yields ξψ = (cid:20) x + 1 x + 1 x − M µ (cid:21) − √ x x x Z dx ′ dx ′ p x ′ x ′ x ′ ψ ( x ′ , x ′ , x ′ ) . (3.10)To symmetrize the kernel of this equation, we replace ψ by ψ ( x , x , x ) = (cid:20) x + 1 x + 1 x − M µ (cid:21) − / f ( x , x , x ) (3.11)and obtain ξf = 1 √ x x x (cid:20) x + 1 x + 1 x − M µ (cid:21) − / (3.12) × Z dx ′ dx ′ p x ′ x ′ x ′ (cid:20) x ′ + 1 x ′ + 1 x ′ − M µ (cid:21) − / f ( x ′ , x ′ , x ′ ) . This rearrangement also accomplishes an important step toward the use of a polynomialexpansion. The leading small- x i behavior of ψ is √ x i , and, as can be seen from the structureof the pre-factor in (3.11), the leading behavior of f is just a constant.Because the kernel factorizes, the equation can be solved analytically. The function f must be of the form f ( x , x , x ) = A √ x x x (cid:20) x + 1 x + 1 x − M µ (cid:21) − / , (3.13)with a normalization A . Substitution of this form into the equation for f yields the conditionfor the eigenvalue: ξ = Z dx dx x x x (cid:20) x + 1 x + 1 x − M µ (cid:21) − . (3.14)7 value can be computed when the ratio M/µ is specified.To solve the equation for f with the symmetric polynomial basis, we substitute thetruncated expansion f = N X n,i a ni P ( i ) n (3.15)and obtain a matrix eigenvalue problem for the coefficients N X m,j b ni b mj a mj = ξa ni , (3.16)with b ni ≡ Z dx dx √ x x x P ( i ) n ( x , x , x ) p /x + 1 /x + 1 /x − M /µ . (3.17)The eigenvalue is then approximated by ξ ≃ N X n,i b ni . (3.18)A set of values for different N is given in Table I for M = 0 . µ . The convergence to theexact value is quite rapid. Similar behavior occurs for other values of M . TABLE I. Sequence of eigenvalue approximations obtained with use of the fully symmetric poly-nomials P ( i ) N up to the eighth order for M = µ . Orders six and eight appear twice, becausethere are two polynomials in each case; however, the result changes little with the addition of thesecond polynomial. These results are to be compared with the exact value of ξ = 2 . N ξ By way of comparison, we also consider the DLCQ approach. In the present circumstance,DLCQ yields a trapezoidal approximation to the integral in Eq. (3.14), with the step sizesin x and x taken as 1 /N for an integer resolution N . Points on the edge of the triangle,which correspond to zero-momentum modes, are usually ignored. The DLCQ approximationis then ξ ≃ N N − X i =1 N − i − X j =1 N ij ( N − i − j ) (cid:20) Ni + Nj + NN − i − j − M µ (cid:21) − . (3.19)Results for the two approximations are presented in Fig. 1. The symmetric polynomialapproximation converges much faster. The primary difficulty for the DLCQ approximationis the integrable singularity at each corner of the triangle. To be fair, we should point out that DLCQ is used primarily for many-body problems, where basis functionexpansions are difficult to implement, and can be combined with an extrapolation procedure to obtainconverged results. /N x FIG. 1. Comparison of convergence rates for the fully symmetric polynomial basis (filled circles)and DLCQ (filled squares). The dimensionless eigenvalue ξ is plotted versus 1 /N , the reciprocal ofthe basis order and of the DLCQ resolution, for the case where M = 0 . µ . The horizontal lineis at the exact value, ξ = 2 . IV. SUMMARY
We have constructed an orthonormal set of fully symmetric polynomials on a trianglethat can be used as a basis for three-boson longitudinal wave functions in field theoriesquantized on the light front [1, 2]. At each order, the number of polynomials is quite small,the limitation to symmetry under the interchange of all three barycentric coordinates beinga much stronger constraint than just symmetry under interchange of the two independentvariables. A list of the first six polynomials is given in Eq. (2.16). In general, the polynomialsare formed by first constructing a non-orthonormal set according to Eq. (2.14), and thenapplying an orthogonalizing procedure, such as the Gramm–Schmidt process.As a sample application, we have considered a light-front Hamiltonian eigenvalue prob-lem in φ theory, limited to the coupling of one-boson and three-boson Fock states. Thepolynomial expansion for the wave function yields rapidly converging results, particularlyin comparison with a DLCQ approximation, as can be seen in Table I and Fig. 1.The original motivation for these developments was to find an expansion applicable to thenonlinear equations of the light-front coupled-cluster (LFCC) method [6]. In this method,there is no truncation of Fock space, but approximations for the wave functions for higherFock states are determined from the wave functions of the lowest states by functions thatsatisfy nonlinear integral equations. In bosonic theories, these functions must have the fullsymmetry, and any basis used should have this symmetry. The sample application herecan be interpreted as a linearization of the φ LFCC equations. Thus, we expect the newpolynomial basis to be of considerable utility.9
CKNOWLEDGMENTS
This work was supported in part by the Department of Energy through Contract No.DE-FG02-98ER41087.
Appendix A: Proof of the conjecture
Here we give a proof that any fully symmetric polynomial on a triangle can be expressedas a linear combination of products of powers of two fundamental polynomials of order twoand three. We work in terms of the translated variables u , v , and w defined in (2.5), sothat the constraint of being on the triangle is u + v + w = 0. The structure of the proof isfirst to characterize unconstrained polynomials on the unit cube and then to restrict thesepolynomials to the triangle.Any symmetric polynomial built from mononials of order N is a linear combination ofpolynomials ˜ P ijk ( u, v, w ) defined by˜ P ijk ( u, v, w ) = u i v j w k + permutations , (A1)with i + j + k = N and i ≤ j ≤ k . Thus, the ˜ P ijk form a basis for symmetric three-variablepolynomials with each term of order N . The size of this basis is S N ≡ [ N/ X i =0 [( N − i ) / X j = i , (A2)where [ x ] means the integer part of x . The limits on the sums guarantee the order i ≤ j ≤ k ,with k = N − i − j .We can also build symmetric polynomials from linear combinations of˜ C lnm ( u, v, w ) = ˜ C l ( u, v, w ) ˜ C n ( u, v, w ) ˜ C m ( u, v, w ) , (A3)where ˜ C = u + v + w, ˜ C = uv + uw + vw, ˜ C = uvw, (A4)and N = l + 2 n + 3 m . However, is this sufficient to generate all such polynomials? Thenumber of polynomials ˜ C lmn is Ξ N ≡ [ N/ X m =0 [( N − m ) / X n =0 , (A5)which counts the number of ways that the integers l , n , and m can be assigned, with l = N − n − m . The substitutions m = i and n = j − i yieldΞ N = [ N/ X i =0 [( N − i ) / − i ]+ i X j = i [ N/ X i =0 [( N − i ) / X j = i . (A6)Therefore, Ξ N is equal to S N , and the ˜ C lnm do form an equivalent basis on the unit cube.10he projection onto the triangle u + v + w = 0 eliminates ˜ C and any basis polynomial˜ C lnm with l >
0. Thus, the basis polynomials on the triangle can be chosen as products ofpowers of second and third-order polynomials. The powers n and m , respectively, includeall possible integers that satisfy N = 2 n + 3 m . In terms of the Cartesian variables x and y ,we then have the basis set specified by (2.13) and (2.14). [1] P.A.M. Dirac, Rev. Mod. Phys. , 392 (1949).[2] For reviews of light-cone quantization, see M. Burkardt, Adv. Nucl. Phys. , 1 (2002); S.J.Brodsky, H.-C. Pauli, and S.S. Pinsky, Phys. Rep. , 299 (1998).[3] See, for example, G.M.-K. Hui and H. Swann, Contemporary Mathematics , 438 (1998).[4] For general discussion of multivariate polynomials, see C.F. Dunkl and Y. Xu, Orthogonal Poly-nomials of Several Variables , (Cambridge, New York, 2001); P.K. Suetin,
Orthogonal Polyno-mials in Two Variables , (Gordon and Breach, Amsterdam, 1999).[5] H.-C. Pauli and S.J. Brodsky, Phys. Rev. D , 1993 (1985); , 2001 (1985).[6] S.S. Chabysheva and J.R. Hiller, Phys. Lett. B , 417 (2012)., 417 (2012).