Scattering of composite particles in a gauge theory with confinement

Nuclear Physics B (Proc. Supp) .) 17 (1990) 744-748 North-Holland

744

SCATTERING OF COMPOSITE PARTICLES IN A GAUGE THEORY WITH CONFINEMENT D. Bérubé l , J.F . Brière', H. Kr®ger', K .J .M . Moriartya .3,4 and J . Potvin 4 - 5 Département de Physique, Université Laval, Québec, Prov . Qué ., GlK 7P4, Canada

z Institute for Advanced Study, Princeton, NJ 08540 3 John von Neumann National Super Computer Center, Princeton, NJ 08540 4 Department of Mathematics, Statistics and Computer Science, Dalhousie University, Halifax, Nova 5

Scotia, Canada Department of Physics, Boston University, Boston, MA 02215

Firstly, we discuss a new method based on the Hamiltonian formulation and on the Monte Carlo projector technique in order to compute the Minkowski time evolution and S-matrix . We present results for NN scattering in a non-relativistic potential model . Secondly, we present numerical results of a lattice calculation on scattering of two composite particles in QED,, .,,, which confines single fermions . The composite particles are taken as neutral, fermion-antifermion, lowest mass eigen states of the Hamiltonian . We use the light-cone momentum representation on a lattice and compute the time evolution and S-matrix by diagonalizing the lattice Hamiltonian . 1 . S-MATRIX FROM MONTE CARLO PROJECTOR TECHNIQUE The computation of the Minkowski time evolution, S-matrix or decay amplitudes is a numerical difficult problem if one starts from the

Euclidean lattice formulation with a

nonvanishing imaginary action . The Monte-Carlo projector method [1] based on a Hamiltonian formulation has been applied to compute ground state properties of glueballs [2] . Here we

e .g . such that it has less degrees of freedom, but with the constraint to give the same physics for energies in the relevant energy interval . H + H(block) can be considered as a renormalization group transformation . We suggest to'construct H(block) as follows : One called broad states 10(broad)> . Then one

group ideas and the Monte-Carlo

projector method to compute the Minkowski S-

constructs so-called zoomed states

matrix I T(zoom)> = §Pin1S1 0 init> - limt- < 0Pin 1 exp[iH°®t] exp [ -i2Ht]exp[iH ° -t]10,,_,,>,

(1)

where H" denotes an asymptotic Hamiltonian and the asymptotic states are assumed to have a wave packet distribution in energy in the interval [E,,_W,E,,,] . In field theory, the Hamiltonian has infinitely many degress of freedom . In the

scattering process described by the above matrix element, only the the degrees of freedom corresponding

to energies in [E,._W,E P] play a

0920-5632/90/$3 .50 © E!sevier Science Publishers B.V . North-Holland

replace the

constructs a finite dimensional basis of so-

suggest to use a Hamiltonian formulation, use renormalization

role . Thus we have the freedom to

Hamiltonian H by a new Hamiltonian H(block),

f

E p dE exp [ -(H-E)2 %,_ ._] 1 1(block)>

E 3. oW

In the limit i ZOO,

(2) , the contributions

of the broad state corresponding to energies E outside of [E L_W ,E  p ] are filtered out . The block Hamiltonian is constructed by taking matrix elements of the original Hamiltonian between the zoomed states . The projection operator is positive (z,, parameter) with a rapid

,*, is a real exponential fall-off

behavior for contributions corrosponding to an

D. Bérubé et al ./ Scattering of' composite particles

745

energy outside of [E loW ,E p ] . This is suitable for application

of the Monte-Carlo projector

method [1] . The S-matrix is replaced by a block Smatrix, by substituting H --> H(block) in eq .(1), and replacing the time-limes by a finite scattering time . The numerical computation of the block S-matrix is straightforward after algebraic diagonalization of the block Hamiltonian . We have tested this prescription for NN scattering in a non-relativistic potential model (Yamaguchi potential [3]) . We consider s-wave scattering . The reference S-matrix is computed on a fine grid momentum lattice (40 nodes) . Numerical results from the block method are presented in Figs .[1,2] . A criterion for the usefulness of the block method is the

FIGURE 1 NN scattering dependence of S-matrix on statistical error of Hamiltonian (Eq . 3) . Solid line correspond to reference solution .

propagation of statistical errors from the block Hamiltonian to the block S-matrix . We have simulated this by introducing a statistical error H(block) (3) i , 3 4 H(block) i , j ( 1+ e *P(x)),

S

where P has a Gaussian distribution . The results displayed in Fig .[1] correspond to block dimension = 3 (small block size) and tizoom = 0 .1 . The important result is the linear behavior of the error of the S-matrix as a function of the error of the Hamiltonian in the region of small errors . For large errors, there is some saturation, probably due to the conservation of the unitarity of the block S-matrix . The results for the S-matrix using the MonteCarlo projector method to compute the block Hamiltonian are presented in Fig .[2] . The Smatrix is displayed as a function of the coupling constant à . The original Yamaguchi model (describing NN physics up to 100 MeV) corresponds to

. A = 0 .0278811 fm-2

FIGURE 2 NN scattering dependence of S-matrix on coupling J1 . Solid line correspond to reference solution .

D. Bér06 et al. /Scattering of composite particles

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2 . COMPOSITE PARTICLE SCATTERING IN QED,, .,. Because the aim to compute scattering in QCD on the lattice still seems

to be very far away,

it is reasonable to study models which are similar, but simpler . QED, ., is such a model, because it displays confinement of single fermions [4] .

Here we want to present results

element is ~~BB fi n ISl~a~i nit> = lim t .el ",,, .Iexp[iE-t] exp[-i2Ht]exp[iE-t] I~$Hinit> " In order to compute these quantities

on the light-cone momentum lattice, one replaces the Hamiltonian by the light-cone Hamiltonian P- (L) and the time by the lightcone time . P- (L) corresponds to finite

of a lattice calculation of scattering of

dimensional Hermitian matrix, which can be

composite particles in QED,,, .,,, in order to

diagonalized numerically and yields a spectrum

simulate positronium-positronium scattering in

of energies and wave functions .

QED or meson-meson scattering in QCD_ As tools

numerical computation one diagonalizes the

we have used a Hamiltonian formulation, using

light-cone Hamiltonian P"(L), and

light-cone momentum variables on the lattice and

the exponential using the corresponding eigen

a time-dependent method for the computation of

representation . Replacing the continuous

the S-matrix . Eller, Pauli and Brodsky [5] have

Hamiltonian by

computed the mass spectrum for this model using

requires also to replace the time limes x+

the light-cone momentum representation on the

by a finite value of x + . For more details see

lattice . The light-cone coordinates are x' = x°

Ref . [6] . In order to obtain a physical S-

± x 1 . In order to quantize the corresponding

matrix one has to carry out a mass

Hamiltonian, one chooses the light-cone gauge

coupling renormalization (For details see

A+ = 0 . Because there are no transverse photons

Refs .[5,6]) . In the numerical calculations we

in one space dimension, the Hamiltonian can be

have taken as bare model parameters :

expressed in terms

MeV, g/m = 0 .5 . We have used the following

of fermion fields only . One

ä

In the computes

discretized Hamiltonian

4

and

m = 3 .14

constructs the light-cone momentum operator P + ,

lattice parameters : L=15 MeV -1 , A = 18 . The

the light-cone Hamiltonian P - and the charge

bare coupling g has been chosen from a region

operator Q,

which turned out in calculations of the mass

which all commute . By a Fourier

transformation one goes over to the variables

spectrum to be a region of weak to medium

k`, k - , i .e . light-cone momentum and energy .

coupling . We found for the invariant mass of

Then a light-cone momentum lattice is introduced

the composite particle state M = 6 .0?. MeV . Thus

via k}  = 2 n n/L, n=1, . . . ., A .

the difference with the bare mass of the

We have taken as composite particle a fermion anti-fermion state, which

forms a boson . It is

constituents is 6m = 0 .26 MeV . For comparison, one obtains from Eller et al .

[5] the value M =

a simultaneous eigenstate of the light-cone

5 .99 MeV .

Hamiltonian, momentum, invariant mass and charge

coupling, in particular for g/m = 0 .5,

operator . We have taken the lowest mass M

results of Ref [5] agree well with those by

(lowest

Crewther and Hamer [7] from a standard space

eigenvalue of the mass spectrum) and

charge Q=O . This state IB(k+)> is then given by its light-cone momentum k+ (or light-cone energy k - , related by the mass shell condition k'k =M 2 ) . In order to compute scattering of two of those

bosons we have taken as asymptotic state

as a wave packet superposition of two bosens, to a given total momentum k',_, . The S-matrix

Also in the region of weak to medium

time lattice . In Fig .

the

[3], we have displayed

the results for composite particle-composite particle (B-B) elastic scattering, using a reduced Fock space of 2 fermion - 2 antifermion states . We display the imaginary part of the Smatrix and

A« which is a measure for the

violation of energy in the scattering process

D. Bérubé et al./Scattering of composite particles

747

on the lattice, as a function of the light-cone

ACKNOWLEDGEMENT

time . One observes a region of stability of the

H . Krdger is grateful for computer time given at CRAY XMP 48 by HLRZ (Jülich), CRAY 1 S

S-matrix in the neighborhood of the light-cone scaii.aring time 0 .48 MeV -1 , i .e . at the location of the minimum of the violation of energy conservation .

by CRAY Research (Montreal) and CRAY XMP 22 by CLSC (Toronto), and for support by NSERC of Canada . K .J .M . Moriarty and J . Potvin would like to thank John Buchanan, Terence R .B . Donahoe and Don Cameron of the Goverment of Nova Scotia

for their continued interest,

support and encouragement and grant support, James R . Berrett, Samuel W . Adams and Paul W . Crum for access to the NEC SX-2 in the Houston Area Research Center in The Woodlands, Texas, the National Allocation Committee for the John von Neumann National Supercomputer Center for access to the two CDC CYBER 205's and the two ETA-10's at the JVNC [Grant Nos . 110128, 171805, E171805, 171812, E171812, 171813, 551701-551706] where part of the calculations were performed .We acknowledge the Natural Sciences and Engineering Research Council of Canada [Grant Nos . NSERC A8420 and NSERC A9030] for financial support and the Canada/Nova FIGURE 3 Boson-boson scattering in QED, + , . Dependence of the imaginary part of the S-matrix and the violation of energy A,,, as a function of light cone time .

Scotia Technology Transfer and Industrial Innovation Agreement [Grant Nos . 871101, 88TTII01 and 89TTII01]for further financial support . REFERENCES

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[3]

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[4]

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[7]

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compute the (Minkowski) time evolution .