FORMULATION OF THE TRANSVERSE LATTICE. Bob F. Klindworth, Jr. A Dissertation submitted to the Graduate School

Transcription

1 THE Q Q POTENTIAL IN THE COLOR-DIELECTRIC FORMULATION OF THE TRANSVERSE LATTICE BY Bob F. Klindworth, Jr. A Dissertation submitted to the Graduate School in partial fulllment of the requirements for the Degree Doctor of Philosophy Subject: Physics New Mexico State University Las Cruces, New Mexico May 1999

2 \The Q Q Potential in the Color-Dielectric Formulation of the Transverse Lattice" a dissertation prepared by Bob F. Klindworth, Jr. in partial fulllment of the requirements for the degree, Doctor of Philosophy, has been approved and accepted by the following: Timothy J. Pettibone Dean of the Graduate School Matthias Burkardt Chair of the Examining Committee Date Committee in charge: Dr. Matthias Burkardt, Chair Dr. Srinivas Aluru Dr. Thomas M. Hearn Dr. Stephen F. Pate Dr. Vladimir M. Shalaev ii

3 VITA May 6, 1971{Born in Norfolk, Nebraska 1993{B.A., Gustavus Adolphus College PROFESSIONAL AND HONORARY SOCIETIES American Association of Physics Teachers PUBLICATIONS Laith J. Abu-Raddad, Bob Klindworth, and Longzhe Zhang Two Theories of Baryons To appear in the Proceedings of HUGS 1996 M. Burkardt and Bob Klindworth Calculating the Q Q Potential in 1 dimensional Light-Front QCD Phys. Rev. D55, 1001 (1997) Bob Klindworth and Matthias Burkardt The Q Q Potential in the Color Dielectric Formulation of the Transverse Lattice To appear in Proceedings of \Quark Connement and the Hadron Spectrum III" hep-ph/ FIELD OF STUDY Major Field: Physics iii

4 ABSTRACT THE Q Q POTENTIAL IN THE COLOR-DIELECTRIC FORMULATION OF THE TRANSVERSE LATTICE BY Bob F. Klindworth, Jr. Doctor of Philosophy in Physics New Mexico State University Las Cruces, New Mexico, 1999 Dr. Matthias Burkardt, Chair The Color-Dielectric formulation of the Transverse Lattice seems to be a promising tool to describe Quantum Chromodynamics. The Transverse Lattice blends the advantages of Light Front eld theories, with their intuitive interpretation of physical observables, and that of Euclidean Lattice Monte Carlo theories with their powerful numerical capabilities. The Color-Dielectric formalism regards the Transverse Lattice elds as being smeared variables much like the electric eld is regarded as being an average eld in condensed matter physics. The advantage of this approach is that long distance physics (eg. masses) can be represented using fewer, simpler degrees of freedom. The main drawback is that, because of iv

5 the diculty in determining the exact V eff, one is forced into making an ansatz for V eff which has free parameters. One can determine these by either tting these parameters to physical observables or determining them from rst principles. One physical observable that is used to t the parameters is the static QQ potential. The QQ potential is an ideal starting point as it probes a symmetry that is not manifest on the Light Front, namely rotational invariance. In addition, the heavy quark potential is well known hence there is a dearth of data, both from theoretical calculations and from experimental measurements. I begin my study of the Color-Dielectric formulation of the Transverse Lattice by investigating the QQ potential in 1 dimensions. I then expand my work to 31 dimensional Transverse Lattice QCD. Finally, I work on the Euclidean Lattice to directly calculate the parameters of the Color-Dielectric theory from a set of Schwinger-Dyson equations. v

6 TABLE OF CONTENTS LIST OF TABLES viii LIST OF FIGURES ix 1 INTRODUCTION LIGHT FRONT COORDINATES LATTICE GAUGE THEORIES BLENDING TWO METHODS: THE TRANSVERSE LATTICE CALCULATING THE EFFECTIVE POTENTIAL THE QQ POTENTIAL ON THE TRANSVERSE LATTICE THE HAMILTONIAN SOLUTIONS IN 1 DIMENSIONS SOLUTIONS IN 31 DIMENSIONS CALCULATING THE EFFECTIVE POTENTIAL DETERMINING THE EFFECTIVE EUCLIDEAN ACTION FINDING THE TRANSVERSE LATTICE V eff CONCLUSIONS AND FUTURE WORKS A DETERMINING THE TRANSVERSE LATTICE HAMILTONIAN. 56 B CALCULATING THE QQ POTENTIAL ON THE LIGHT FRONT 60 C EQUATIONS RELATING m AND? TO OBSERVABLES D DERIVATION OF THE EQUATIONS FOR THE COUPLINGS vi

7 BIBLIOGRAPHY vii

8 LIST OF TABLES 3.1 Results of the t of the parameters in S eff Results of transforming the fourth order couplings back to the Transverse Lattice viii

9 LIST OF FIGURES 1.1 Space-time cartoon of Deep Inelastic Scattering A set of elementary plaquettes Space-time view of a Transverse Lattice Contour plot of the Q Q Potential in the strong coupling limit The 1 dimensional Q Q Potential Examples of eld congurations that mix with one another via the plaquette interaction. a) (1,1) displacement. b) (,1) displacement The 31 dimensional Q Q potential Schematic view of the terms in the blocking transformation Schematic picture of the terms in S eff C.1 Schematic view of the \window" and \hourglass" diagrams ix

10 1. INTRODUCTION Quantum Chromodynamics has emerged as the theory of the strong nuclear interaction. It is a gauge theory, meaning that its form is motivated by the familiar gauge principle which brought about the very successful quantum theory of electrodynamics: QED. The fundamental dierence between QCD and QED is that the gauge group of QED is the abelian group U(1), while the gauge group of QCD is the nonabelian group SU(3). This leads to one of the fundamental complications of QCD: the gauge elds, the gluons, interact with one another. Yet another complication of the theory is that the strong coupling constant, s, is not a small quantity. It is on the order of one. This means that one can not expect to get meaningful results by performing perturbation theory in this coupling constant as the most complicated diagrams would be the ones that contribute most. Because of this complication it seems that if one wants to study strong interaction physics one must develop methods which are nonperturbative in nature. Two promising tools for helping one understand the nonperturbative nature of QCD are Light Front eld theories and Lattice Gauge Theories. 1.1 LIGHT FRONT COORDINATES Light Front eld theories are quantum eld theories where one transforms to the Light Front variables, x = 1 p x 0 x 3 (1.1) 1

11 Because the Minkowski metric changes the sign of the spatial coordinates but leaves the time coordinate the same one nds, x = 1 p (x 0 x 3 ) ) x = x (1.) In the equal time formalism one chooses x 0 = 0 as one's initial surface of quantization. On the Light Front, one chooses x = 0. Thus x is regarded as the Light Front time coordinate. The operator conjugate to this variable is P = P?, and this is then regarded as the Hamiltonian of the system. The invariant mass squared is found from, m = p p p? p?? ~p? = p? p? ~p? (1.3) The use of these coordinates is tantamount to boosting one's system to the innite momentum frame. For this reason it is not surprising that these coordinates seem particularly useful for describing observables which are extracted from deep inelastic scattering experiments such as nucleon structure functions. Indeed, the parton distribution functions only have the simple interpretation of momentum densities if one is in the innite momentum frame. In other frames, the picture that they paint is much more dicult to interpret. Another way to see that Light Front coordinates seem to be a good way to describe deep inelastic scattering experiments is to consider what happens when the high energy electron interacts with a quark inside a nucleon. The struck quark

12 x - x t e - q q q Figure 1.1: Space-time cartoon of Deep Inelastic Scattering z will be moving at very near the speed of light, ie very near to the light cone. If one wishes to calculate a response function one will need to evaluate the wavefunction at, for example, the two points indicated on the struck quark line in Figure 1.1. In the real time formalism these two points correspond to two dierent times. Hence, in order to calculate the response function one is required to solve for the eigenmodes of the Hamiltonian so that one could evolve the initial state into the nal state. If, instead, one uses Light Front coordinates then there is no need to evolve the state because the wavefunctions used in calculating the response function are evaluated at equal Light Front time. 3

13 1. LATTICE GAUGE THEORIES Lattice gauge theories are also very useful in describing the nonperturbative aspects of QCD. In such theories 31 dimensional space-time is discretized with the quarks existing on the lattice sites, while the gluons are replaced with gauge elds which exist only on the links between the sites. The necessity of the link elds can be seen by considering the quantity, y (x dx )(x) (1.4) where in this case is the quark eld, x represents the position of the eld on the lattice, and dx represents a displacement of one lattice spacing in the direction from x. Such a quantity arises when one discretizes the derivative term in the Lagrangian, turning it into a nite dierence. The problem with this quantity is that it is not invariant under a local gauge transformation, G(x), (x)! G(x)(x) (1.5) ) y (x dx )(x)! y (x dx )G y (x dx )G(x)(x) (1.6) That this term is not gauge invariant comes as no surprise. In the continuum theory the unmodied derivative term also spoils gauge invariance. This problem in the continuum is alleviated by introducing the covariant derivative which cancels the phase dependent piece introduced by the unmodied derivative. The crux of the problem on the lattice is that by considering quantities which have contributions from dierent lattice sites one introduces an arbitrary 4

14 phase. This phase comes from the underlying continuum gauge eld, A. By introducing a link eld between the two sites in question one could cancel this phase, y (x dx )U (x)(x) (1.7) where, U (x) e igr dx A (1.8) x determines where the link eld originates, and indicates its orientation and is therefore not summed over in the integral. With the addition of the link eld (1.7) is found to be invariant under local gauge transformations. In the next step one introduces a discrete version of the continuum QCD action to be used in the calculation of the Feynman path integral. For pure gauge QCD the continuum action is, S =? 1 4 Z d 4 xf a F a (1.9) The discrete version will involve nite dierences in the eld strength tensors instead of derivatives and will make use of the link elds to keep it gauge invariant. The details can be found in [14]. The resulting action is called the Wilson action and has the form, S W = 1? 1 T r U (1.10) where represents an elementary square (or plaquette) of the lattice with side 5

15 U3 U4 U U1 Figure 1.: A set of elementary plaquettes. length equal to the lattice spacing, U is the matrix generated by multiplying all the link elds around the elementary plaquette, and is Nc. N g c is the number of colors. As an example in Figure 1., U = U 4 U 3 U U 1 (1.11) One can now use this action in the path integral, Z DUe?S W (1.1) The technique for calculating observables is to start with some initial conguration of the link elds and to vary these elds using Monte Carlo techniques. One then calculates expectation values of certain operators using the congurations generated by this Monte Carlo approach. The problem with this method is 6

16 mainly technological, further advances being limited by the available computing power. With the advent of improved actions, which have much in common with the techniques developed in this work, these limitations are being overcome. However, even if lattice gauge theory had sucient computing power to solve QCD, such a solution would be dicult to interpret. Thus, phenomenological models and eective theories would still have a place as they provide intuition into the subtleties of QCD. 1.3 BLENDING TWO METHODS: THE TRANSVERSE LATTICE It seems clear that a blend of Light Front eld theories and Lattice Gauge theories could lead to a theory whose results can be readily interpreted by our intuition yet still be numerically compelling. Such a theory was introduced by Bardeen and Pearson in 1976 []. It was dubbed the Transverse Lattice. The original motivation for this theory was the observation that only two of the gauge elds in QCD are dynamical. Using Light Front coordinates and Light Front gauge, one can therefore eliminate these gauge elds from the theory. Doing so, however, induces an instantaneous linearly conning potential in the 11 dimensional plane spanned by x (The details of this calculation can be found in Appendix A). This leads one to discretize two of the spatial dimensions while leaving the Light Front coordinates, x, continuous. As in lattice gauge theories, gauge invariance is only preserved in the discrete direction if one introduces link elds between the sheets. Ideally, these 7

17 6? space (discrete) P P long. space (continuous) P P Pq 1 time (continuous) Figure 1.3: Space-time view of a Transverse Lattice. link elds would be members of the gauge group of QCD, SU(N c ). Unfortunately, it is very dicult to enforce such a complicated, nonlinear constraint on the Light Front. In the canonical formalism one normally just expands the elds around the origin. The reason this is not possible on the Light Front is that in the continuum the action for the gauge elds resembles the \Mexican hat" potential of the nonlinear sigma model, therefore one would be expanding the elds around a maximum. One might be tempted to use the same trick as in the nonlinear sigma model, namely to simply shift the elds to the minimum away from the origin. In Light Front theories, this is not possible since there is no absolute minimum for the elds, only a narrow valley with degenerate values. To arbitrarily choose a point in this valley would break gauge invariance. For this reason the link elds of this theory are taken to be unconstrained complex matrices, M, instead of the local matrices, U. One then enforces the SU(N c ) constraint by introducing an eective potential which is peaked for M SU(N c ). These complex matrices can 8

18 be regarded as blocked or smeared variables which represent some average over the local gluonic degrees of freedom. Thus our inability to rigidly enforce the SU(N c ) constraint can be turned into an advantage. If one wants to calculate long distance physics using the Transverse Lattice it is clear that using microscopic degrees of freedom is a poor choice. Such a choice would necessitate the use of many microscopic variables. It would correspond to attempting to model the large scale electromagnetic properties of a solid using a distance scale on the order of the atomic size. In such a case, one would in principle need to know the position of all the atoms in the solid. This analogy with electromagnetism motivates the name of this formalism: the Color-Dielectric formalism. Within this paradigm, one hopes to determine long range physics using relatively few, well-chosen degrees of freedom. There is a large dierence between the way one does calculations on the Transverse Lattice and the way calculations are done using the Euclidean Lattice. On the Euclidean Lattice one uses Monte Carlo techniques to vary the link eld conguration, calculating expectation values of observables using these congurations. On the Transverse Lattice, one expands in a Fock Space basis and calculates Hamiltonian matrix elements. Determining the eigenvalues and eigenvectors of this matrix will give both the masses and the momentum space wavefunctions. Thus while these two theories are fundamentally similar, the methods by which they are solved are very dierent. 9

19 Naturally, there is some disadvantage to the Transverse Lattice formalism. Formally one can dene an exact expression for the action of the blocked theory. e?s block [M ] = Z DU (M? U)e?S W [U ] (1.13) where S W [U] is the canonical Wilson action in terms of the standard, SU(N c ) link elds, M is the blocked eld, U is the averaged eld, and Sblock [M] is the exact action of the blocked theory. Unfortunately while the path integral on the right hand side of (1.13) is well-dened it is extremely dicult to calculate in practice. In principle S block contains an innite number of terms. Thus, one must resort to an eective action. This eective action in the blocked elds will have some free parameters. In essence, in return for a more physical starting point one gives up precise knowledge of the Hamiltonian. In order to determine the values of these free parameters, one could choose to t to data. The Q Q potential is one such piece of data. There is one very important reason that the QQ potential is a natural observable to study on the Transverse Lattice: the Transverse Lattice does not exhibit manifest rotational invariance. This means that, although we expect angular momentum to be conserved, the angular momentum is not a kinematic variable as it is in the more familiar equal time formalism. Rotational invariance is thus conserved only by the dynamics of the system. Therefore, the rotational invariance of the QQ potential is dependent on the dynamics which are included in the Hamiltonian. Hence rotational invariance can be used as a criteria for limiting the freedom of the parameters in the eective potential. 10

20 It bears noting that two distinct meanings of the word \potential" will be used throughout this work. The rst refers to the eective potential. This is simply a set of operators which are added to the Light Front Hamiltonian to describe the new distinct dynamics of the eective degrees of freedom. This particular use will always be found in the combination \eective potential." The other meaning for potential will be in the context of the Q Q potential. This is quite distinct from the above meaning, and is the observable obtained by measuring the energy of the link eld string at a xed quark separation. Work on the Q Q potential has gained some popularity of late. The Euclidean Lattice has produced a wealth of calculations of this important observable.[10] In addition, there have been calculations of the many excited states of the potential. One can interpret these various states as adiabatic potential energy surfaces on which the quarks move. The physics of the lowest energy state has a simple interpretation as the potential experienced by a qq pair in a normal meson such as the pion. If one wished to do heavy quark meson calculations, one could use this potential as the potential energy in a Schrodinger equation for the heavy quarks where a nonrelativistic approximation would be valid. The physics of the higher potential energy states is a little more complicated. To understand this physics one could use the paradigm of the Flux Tube model. Consider a qq pair separated by a distance r. The color electric eld lines spread out from the q source and end on the q sink. This is in analogy to the 11

21 electric eld lines between an e e? pair in QED. The main dierence between QED and QCD is that the gauge elds of QED (photons) do not interact with one another while the gauge elds of QCD (gluons) do. This result causes the eld lines between the q q pair to behave very dierently from the analogous QED picture. In QED, the electric eld lines spread out and cover a wide area. In QCD, the color eld lines are pulled together by their self-interactions. The result is a narrow tube of color electric ux which connects the qq pair. In the Flux Tube model one assumes that this narrow tube of color ux oscillates quantum mechanically, much like a quantum violin string. The ground state, where the string simply executes its zero point motion, corresponds to a normal meson. The excited states of oscillation are the hybrid meson modes. In a valence picture of hadrons, the normal mesons are those with only a valence qq pair which is color neutral. It is also possible to couple qqg to color singlet (neutral). This type of color neutral object is distinct from the normal mesons and is called a hybrid meson. As of now, such hybrid mesons have not been conclusively observed[1]. The reason for this is that, in the region where these states are expected to exist, there are many other non-hybrid states. Extracting a conclusive signal from this plethora of states is very challenging experimentally. However, calculating the precise energies of these excited modes and determining their quantum numbers should help experimentalists nd these elusive states. The discovery of such a hybrid state would put QCD on an even rmer footing as the 1

22 theory of strong interactions since these states are a prediction of SU(3) gauge theory. Calculations on the Transverse Lattice make use of the fact that the link elds are aggregate degrees of freedom. If the transformation from the local elds to these new elds produces degrees of freedom which mimic the string of the Flux Tube model then one should be able to compare the features of the Flux Tube predictions to that of the Transverse Lattice. The Flux Tube model is a phenomenological tool for describing hybrids. The Transverse Lattice is an eective theory based on Light Front QCD and Lattice QCD. The Transverse Lattice could then provide a bridge between phenomenology and more exact calculations. One is able to limit the freedom of the parameters in the eective potential by demanding that the Transverse Lattice reproduce some of the physical characteristics of QCD. Examples of this are rotational invariance and the distinguishing behavior of the hybrid modes. Clearly, one would much rather be predicting these properties than using them to determine the eective potential. To calculate these parameters from rst principles would be ideal. 1.4 CALCULATING THE EFFECTIVE POTENTIAL Another method for determining the parameters in the eective potential was used by Pirner and Wroldsen[1]. In this work, one begins with a Euclidean Lattice and performs a blocking transformation. One then directly relates the couplings in the eective potential to observables using an overdetermined set of 13

23 Schwinger-Dyson equations. This program has two great advantages. First, it is self-consistent. There is no need to t the eective potential to data. One simply solves a set of linear equations in the couplings. Second, since it is done on the Euclidean Lattice there is no need to truncate the Fock Space on which the Hamiltonian acts. Once the eective potential is determined, it is possible to calculate various observables within the Color-Dielectric formalism. Using the Transverse Lattice one could then, in principle, calculate hadron masses. More importantly, one would then be able to extract useful information about the structure of hadrons from the easily interpreted light front wavefunctions. A hindrance to this comes from the fact that one is interested in matrix elements of operators involving the unblocked degrees of freedom. Clearly it is necessary to relate the smeared variables to the local variables. This is discussed in more detail in Section 3.. Yet it remains clear that determining the eective potential is the rst stage in the quest for this holy grail. 14

24 . THE Q Q POTENTIAL ON THE TRANSVERSE LATTICE The QQ potential is simply the energy of an innitely massive QQ pair at a xed separation as measured in their rest frame. As was discussed in the previous chapter this observable is familiar to physicists, for the concept of a potential permeates physics. The only concept that may seem daunting is the method of interaction, namely QCD. While QCD is certainly a complicated theory, the means by which one thinks of the potential should be familiar. Another conceptual diculty could arise when one thinks of measuring this observable using the Light Front frame. It must be stressed that the observable that is calculated in this work is not the heavy quark potential in the Light Front frame. The Light Front frame is both a means of simplifying the calculation and a tool for extracting more information from this important observable. The QQ potential which is extracted is the potential one would measure in the rest frame of the quarks. The details of calculating the Q Q potential using the Light Front frame are discussed in Appendix A..1 THE HAMILTONIAN In order to calculate the Q Q potential on the Transverse Lattice one must rst determine the Light Front Hamiltonian. The details of this are in Appendix B. In 1 dimensions the result is, 15

25 P? = c g n Z dx? Z h i dy? : T r J n (x? ) j x?? y? j J n (y? ) : V eff (M) (.1) where, J n = M y n M n? M y n1 M n1 (.) and M n is the link eld matrix connecting site n and n1. As was discussed in the previous chapter, one would ideally wish to demand M n SU(N c ) but one nds that such a constraint is extremely complicated to enforce in practice. It is also possible that one may not want to consider these link elds as members of SU(N c ). To do this would mean that these elds would correspond to microscopic degrees of freedom. If one wants to study the long distance behavior of this theory, one could conceive of a program to block or smear these local elds. The result would be aggregate degrees of freedom which would not necessarily satisfy the same dynamics as the original gauge group. If one properly chose a blocking procedure and a corresponding blocked action one should be able to describe fundamental physics even though the results may have been calculated at some nite lattice spacing (Such a procedure will be shown explicitly in the next chapter). This is the assumption of the Color-Dielectric Formalism and also is the prime motivation for much of the work on improved and perfect actions in the Euclidean Lattice community [13]. From now on, it is assumed that the link elds are unconstrained 16

26 matrix elds which result from some smearing procedure which preserves gauge invariance. Unfortunately, making such an assumption introduces the need for new contributions to the Transverse Lattice action since the blocked theory is not the same as the unblocked theory. One therefore introduces an eective potential which has the following properties, It should be peaked for unitary matrices in the continuum, meaning it should enforce the constraint M SU(N c ) It should be invariant under the local gauge transformation, M n! G n (x)m n G y n1(x) The astute reader might notice that making the link elds unconstrained causes them to have a norm (ie. determinant) ranging from 0 to 1. If one now considers the Euclidean Lattice action for these elds, one nds, S = 1? 1 T rm (.3) where M is an elementary plaquette with four M's in it. It is thus fourth order in the elds. With no constraint on the norm of the M's, this action is clearly minimized when all the M's in the lattice have norm 1, its value at that point being?1. Denitely not a very physical conguration. It is the duty of the eective potential to regulate this problem, but it is rather obvious that terms second order in M (ie. a mass term) will not suce as the canonical term is 17

27 already fourth order. For consistency, it seems clear that one must include terms of at least order four to obtain sensible results. This motivates one to make the following ansatz for the eective potential, V eff = T r(m y M) T r(m y M y MM)? T r(m y M y? M?M) T r(m) : : : (.4) In principle the eective potential could contain terms of all orders in M. Also I choose to work in the limit of N c! 1. In the large-n c limit only terms which contain an even number of M 0 s contribute to the eective potential. Possible improvements to this work include investigating the eects of sixth order terms on the results. The number of possible sixth order operators makes this a daunting task. The couplings of this eective potential enter the theory as free parameters. Ideally one wishes to calculate these parameters from rst principles. This is extremely dicult on the Light-Front. I therefore postpone the discussion of directly calculating these couplings until the next chapter where I use Euclidean Lattice Monte Carlo techniques to determine these couplings. An alternative to calculating the couplings from rst principles is to adjust the parameters by trial and error in order to t to some physical observables. As the title of this dissertation suggests, the observable of choice for this work is the static QQ potential. The discussion of the previous chapter made clear the fact that the Q Q potential is an ideal choice to study since rotational invariance is not a manifest symmetry on the Light-Front. If one calculates the QQ potential 18

28 and nds that it is not rotationally invariant one can be certain that one has not included the essential dynamics in the eective Hamiltonian.. SOLUTIONS IN 1 DIMENSIONS Now that one has determined the form of the Hamiltonian one must employ a method of solving for the eigenvalues. These eigenvalues will be easily related to the various potential energy surfaces which are experienced by mesons. Since the problem is very complicated one resorts to numerical methods of solution. For this purpose one imposes the conditions of Discrete Light Cone Quantization (DLCQ) [11]. In DLCQ one expands the link elds as a superposition of plane waves and imposes periodic boundary conditions. Thus one takes, M x; = 1 p Z 1 i 0 dk h p k; e?ikx y eikx k; k Y k; e?ikx Y y k; eikxi (.5) Now dene, A k; 1 p ( k; iy k; ) B k; 1 p ( k;? iy k; ) (.6) Then substituting these expressions, one nds, M x; = p 1 Z 1 dk p A k; e?ikx B y eikx k; (.7) k 0 19

29 One can now express (.1) in terms of these creation and annihilation operators. As a simple example of the procedure, consider the mass term in V eff. P? m = x = x Z h i dx 0 T r M y x M x Z " Z 1 dx 0 T r 0 A i k e?ik0x B y x k eik0 dk p k Z 1 0 dk p 0 A y k 0 k eikx B k e?ikx = x Z dk Z 1 0 dk k 0 A y k A k B y k B k (.8) The Fock Space on which the creation and annihilation operators act is, in principle, only limited by the Gauss' Law constraint. This means that all states which have a string of link elds which begin and end on a quark are allowed. As an example consider a conguration in which a heavy Q Q pair is separated by one link. Since the quarks are at xed separation and are innitely massive, there will not be momentum conservation when the link eld interacts with one of the quarks. Thus, such a state can be written as, j i = Z 1 0 dp (p)b y pj0i (.9) where (p) is the momentum space wavefunction and j0i is the Fock Vacuum. B y p has the normal interpretation as the operator which creates a link eld with momentum p. One then requires that the quantity, h jp? j i h j i (.10) 0

30 is stationary under variation of the wavefunction. In general, this results in an integral equation. For one link, the resulting integral equation is, P? (p) = p v! p (p) Z h 1 G dq (q p) (p)? (q)e i(p?q)x? = 0 p pq(q? p) Z h 1 G dq (q p) (p)? (q)e i(q?p)x? = 0 p (.11) pq(q? p) The rst term is the \recoil term" which was derived in Appendix A. The two interaction terms result from the link eld interacting with the innitely massive quark or anti-quark. A more general case results from considering a separation of more than one link. In such a case the two link elds interact with one another in addition to interacting with the quarks. The resulting equation is, P? (p 1 ; p ) = p 1 p v! (p 1 ; p ) p 1 p 1 Z h 1 G dq (q p 1) (p 1 ; p )? (q; p )e i(p1?q)x? = 0 p p 1 q(q? p 1 ) Z h 1 G dq (q p ) (p 1 ; p )? (p 1 ; q)e i(q?p)x? = 0 p p q(q? p ) G Z p1 p 0 dq (q p 1)(p 1 p? q) 4 q p 1 p q(p 1 p? q) i i [ (p 1 ; p )? (q; p )] (q? p 1 ) ) G 4 p p 1 p (p 1 ; p ) (.1) Note that in both the integral equations above that use have been made of the \Coulomb subtraction" trick where one adds and subtracts a term in the interaction. The resulting integrand vanishes for constant wavefunctions. Since the 1 i i

31 resulting wavefunctions are nearly constant, this speeds convergence. Explicitly what was done is, Z 1 0 = " dy (x y) p (x? y) xy Z 1 0 # (y)? (x) ( dy (x y) p [ (y)? (x)] (x? y) xy ) (x) x (.13) Obviously, the only real change when using this trick for the one link case is the redenition of the mass due to the last term. A similar trick is used for the link-link interaction, Z 1 0 = dy (x? y) Z q dy (x? y) (x y)(? x? y) xy(1? x)(1? y) 8 < : q x(1? x) (y)? 4 (x) 3 5 (x y)(? x? y) [ (y)? (x)] qxy(1? x)(1? y) (x) 4 x(1? x) 9 = ; (x) (.14) The method for calculating the Q Q potential is the following. One rst chooses a separation of the quarks, both x? and x L. Next, the momentum cuto,, and v are chosen. There is a delicate interplay between these two variables. The reason for this is that the peak of the wavefunction in momentum space occurs for momentum k v G. This can be seen from the fact that there is only one energy scale for the system, G. Boosting to this new frame means that the momentum gets boosted to this mass scale, G, times the 4-velocity. Since the momenta of the link elds are discrete, v essentially provides an infrared cut o for the theory. By making v G large the peak of the wavefunction

32 is shifted away from the low-lying momentum states, and one is less sensitive to the discretization. It is also clear from the dependence of the peak of the wavefunction on v that must be a great deal larger than v G. If this were not true then the momentum cut-o could in principle be close to the peak of the wavefunction, and this would seriously aect the results of the calculation. Choosing the momentum cut-o to be large increases the size of the matrix that one must diagonalize. Thus, the main limitation was the amount of memory one has available on the computer. For these reasons, one chooses the maximal value of v where the values of the potential are relatively well-converged in. In the next step the Fock Space is constructed on which the Hamiltonian acts. Due to limitations in computer memory, one is forced to introduce an ad hoc approximation which reduces the size of the Fock Space. Among many choices, one could choose to only consider states with a minimal number of link quanta, ie the link eld string is not allowed to wander throughout the lattice. The fact that I choose to work in the large-n c limit also reduces the size of the Fock Space since quark pair creation is prohibited. Using these approximations, all states with momentum up to and including the momentum cut o are constructed. One then calculates the Hamiltonian matrix for this particular conguration using these states as a basis. The matrix can grow very large, but many of the elements are zero. Since the matrix is very sparse it makes sense to employ Lanczos Tridiagonalization to determine the eigenvalues and eigenvectors of the Hamiltonian. The 3

33 eigenvalues are then extrapolated to both and v large. The details of this extrapolation scheme are in Appendix E. The next step in the procedure is to set the scale of the calculation. This is done by demanding that the string tension in the longitudinal direction matches the string tension in the transverse direction. The longitudinal string tension is obtained by assuming that the quark is only displaced in the longitudinal direction, x?. For this simple case, the Hamiltonian reduces to, P? = G jx? j (.15) This is a result of the Fock Space truncation. In general states which have a link eld going away from the site and directly back could also contribute. These are neglected due to the Fock Space truncation, therefore one is left with only the instantaneous Coulomb interaction. As is shown in Appendix A, this longitudinal separation of the quarks is related to the separation of the quarks in their rest frame through, x L = x? v (.16) Similarly, V (x L ; x? ) = P? v (.17) Thus, V (x L ; 0) = G x L (.18) ) L = G (.19) 4

34 There are two alternatives to calculate the transverse string tension. The most obvious one is to calculate the long distance Q Q potential for purely transverse displacements of the quarks. Measuring the slope of this line in lattice units would then yield the transverse string tension. The other method is to consider a toroidal geometry where one imposes periodic boundary conditions. One could then ignore quarks and construct Fock States where the string of link elds wraps once around this periodic lattice. By calculating the energy of such congurations for increasing size of the lattice one would be able to extract the transverse string tension. The main advantage of such a technique is that, for a given momentum cut o, the number of Fock States is much smaller than the case where the link elds interact with the xed color sources due to momentum conservation. Due to the quality of the extrapolation scheme (Appendix E), it was found that the string tension measured in either scheme was essentially the same. To extract the transverse string tension from this data one simply ts a line and determines the slope. One nds, M(n) G = n?a n(:84 0:01) (.0) n is the number of links, a is the lattice spacing, and? is the transverse string tension. The transverse lattice spacing is determined by demanding the equality of our two string tensions. Using equation (.19) one nds, a = 1 (0:905 0:003) (.1) G 5

35 Figure.1: Contour plot of the Q Q Potential in the strong coupling limit. This is not enough to guarantee the rotational invariance of the system. It will basically demand that the points that correspond to purely transverse displacements will lie on the same line as the points which correspond to purely longitudinal displacements. Points which correspond to simultaneous displacements in both the longitudinal and transverse directions can still lie o of this line. Such an anisotropy would indicate the violation of rotational invariance. One example in which rotational invariance is violated in this way is in the strong coupling limit of Hamiltonian Lattice Gauge theory. In this limit the QQ potential has the form, V (x L ; x? ) = (jx L j jx? j) (.) 6

36 Figure.: The 1 dimensional Q Q Potential. Clearly such a potential is not rotationally invariant. A contour plot of this limit is shown in Figure.1. While this work has many things in common with the strong coupling limit, one is clearly not in the strong coupling regime since one obtains a rotationally invariant QQ potential. The lowest eigenvalue of the Hamiltonian matrix is the rest frame Q Q potential for the given conguration of the quarks. By solving for the eigenvalues for several congurations one can plot the potential as a function of the separation of the quarks and search for anisotropies. The resulting potential is in Figure.. It is clear that rotational invariance is actually quite good; for the most part all points in the ground state lie on the same line. The reason for the lack of points below rg 1 is that there is a lower bound on the transverse separation of the 7

37 quarks: the lattice spacing. In principle one could ll this region with points which have purely longitudinal displacements, but this would be redundant since they lie on a perfect straight line due to the conning potential of (.19). Also of interest is the rst excited state. The blocking transformation causes the standard gauge elds of the theory (gluons) to be replaced by new eective degrees of freedom which are supposed to represent, in an aggregate way, the average properties of the underlying theory. One can thus arrive at a string picture for the link elds (\String" in the sense of the old string theory. This picture has no direct insight from SUSY or Super String Theory). The ground state of such a theory represents the mode where the string executes only its zero point motion. The rst excited state corresponds to the situation where there is an excitated mode of oscillation for the gauge eld string. In an adiabatic approximation to the quark model this rst excited state corresponds to the potential experienced by quarks in a hybrid meson. The results in 1 dimensions are quite remarkable. Even, with a rather simple eective potential and very limiting restrictions on the Fock Space, rotational invariance still seems to be a good symmetry of the Transverse Lattice Hamiltonian. This result gives one condence in the dynamics which has been included; enough condence, in fact, that one could contemplate expanding the work to include the full 31 dimensional space-time. 8

38 .3 SOLUTIONS IN 31 DIMENSIONS The method of solution in 31 dimensions is identical to that used in the previous section. The main dierence in the dynamics is the addition of a few new terms in the eective potential. First consider the plaquette interaction. Clearly such a term could not contribute in 1 dimensions since there was only one discrete direction and, hence, no plaquettes. This term is necessary here since without it there is no dierence in energy between those states where the quarks are separated by two links and those states where the quarks are separated by one link in one direction and one link in another direction. If the potential is rotationally invariant the former should give twice the energy of the state where the quarks are separated by a single link, while the latter should give a factor of p. Another term which is new in 31 dimensions is a dierent nonlocal coupling. In 1 dimensions the nonlocal coupling could act only on link elds that met on a straight line. In 31 dimensions, there is a distinct nonlocal coupling that acts on link elds that meet at a corner. Unfortunately, adding the second discrete direction will increase the number of states in the Fock Space. The reason for this is that specifying the total number of links separating the quarks is not sucient to completely specify the state of the link eld string. There are, in principle, many paths that the link eld string can follow and still remain within the minimal string approximation. 9

39 a) b) Figure.3: Examples of eld congurations that mix with one another via the plaquette interaction. a) (1,1) displacement. b) (,1) displacement. This multiplicity of paths is the factor by which the number of Fock states is increased for a given momentum cut o. The practical eect on the results is that the momentum cuto must be reduced to allow for all the specic orientations of the link eld string to be included. This means that one is more dependent on the extrapolation scheme since the raw data will be further away from the continuum limit as discussed in the previous section. The procedure for determining the new parameters in the eective potential is to consider the conguration where the quarks are displaced by one lattice unit in both transverse directions with no longitudinal displacement. One then adjusts the new couplings such that this conguration lies on the line that was found in the 1 dimensional case. The reason that one can simply use the 1 30

40 dimensional points is because of the minimal string approximation. A general Fock basis, with link elds allowed to wander around the lattice, would cause there to be contributions to the Hamiltonian from the plaquette interaction even when the displacement is purely in one of the transverse directions. In the minimal string approximation there will not be any contribution for such a displacement and thus the 1 dimensional potential can be used for such displacements. Once these couplings are xed, one considers other types of 31 dimensional congurations. In general, these need not lie on the same 1 dimensional line unless the eective potential is properly chosen. Clearly with only one observable one could not hope to exactly determine all of the parameters in the eective potential. In essence one can only determine a trajectory in parameter space which gives a rotationally invariant QQ potential. For (1,1) displacements (see, for example, Figure.3a) this trajectory is?? = constant. One can motivate this result using a matrix model where one ignores longitudinal momentum factors and considers only the action of the eective potential on certain congurations which have the same longitudinal momentum. To start, one assumes that in the absence of the nonlocal couplings each link of separation between the quarks yields an equal contribution to the energy, m. Thus, for a displacement of two links this would be m, while a displacement of three links would be 3m. If one now considers the Hamiltonian of the two states 31

41 in Figure.3a one nds,! H = m? m? where? is the nonlocal coupling for states which turn a corner, and is the plaquette coupling. The ground state eigenvalue of this Hamiltonian is, E = m?? (.3) To construct the Hamiltonian of the three states in Figure.3b it is simpler to consider the states, j~1i = j1i j3i p j~i = ji j~3i = j1i? j3i p (.4) Then the matrix is, H = 0 3m? p 0 p 3m? m? where is the \straight" nonlocal coupling. This is the only nonlocal coupling 1 C A which entered in the 1 dimensional case. The eigenvalues of this case are simplied if one chooses =?. Then the lowest eigenvalue is, E 3 = 3m?? p (.5) The two relations (.3) and (.5) are very useful for tuning the parameters. If, for instance, one has tuned the couplings so that the (1,1) displacement (eg. 3

42 Figure.3a) is exactly on the 1 dimensional line, one may still not have the parameters correct to force the (,1) displacement (eg. Figure.3b) to be on this same line. Equation (.3) shows that as long as one varies? and such that their dierence remains constant, the (1,1) displacement will remain the same. However, varying them in this way will change the value of the (,1) displacement since? and have dierent weights in equation (.5). Another way of further restricting the parameters is to use the excited state modes, the hybrids. If one thinks of the hybrid mode as an excitation of the link eld string then there are two distinct kinds of excitations: those that are transversely polarized and those that are longitudinally polarized (In 1 dimensions it is obvious that the only possible polarization is the longitudinal polarization). In a rotationally invariant theory, these should be degenerate. Unfortunately, this was found to be impossible for the above choice of =?. Making these parameters distinct complicates the eigenvalues for the (,1) displacement, but it is clear that? and will enter the equation with dierent weights. One can also see that the excited state eigenvalues will enter, not with a dierence between? and, but with a sum. Thus it is possible to tune these parameters according to these equations to make the transversely polarized hybrid mode degenerate with the longitudinally polarized hybrid. It was also veried that the rotational invariance of the dierent states is insured as long as one varies the parameters according to Equations (.3) and (.5). 33

43 Figure.4: The 31 dimensional Q Q potential. The results of this relatively simple model were tested by calculating the Q Q potential for values of the couplings which were tuned using this procedure. They were shown to exhibit rotational invariance. The 31 dimensional Q Q potential for this set of parameters is shown in Figure.4. Again, rotational invariance is shown to be restored by the specic choice of parameters in the eective potential. As before, the rst excited state corresponds to the rst hybrid mode. Since this is a fully 31 dimensional calculation it is possible to compare these results to that found by Euclidean Lattice Monte Carlo calculations of the various potentials. In [10], the static heavy quark potential and a few of the lowest hybrid modes were calculated in SU() gauge theory. One can not hope to accurately re- 34

44 produce the short distance behavior of QCD with our rather large lattice spacings. One can, however, examine the long distance behavior of this theory and compare to the results on the Euclidean Lattice. These results on the Euclidean Lattice are qualitatively similar to the results obtained in this work. As an example of the agreement with the Euclidean Lattice, consider the excitation energy of the lowest hybrid mode above the ground state at the separation where the hybrid potential reaches its minimum. This excitation energy was noted to be approximately the same as that found by the Euclidean Lattice. Another notable feature of the hybrid mode is its asymptotic approach to the ground state curve for large separations. As in the Euclidean Lattice calculation, this mode approaches the ground state curve much more slowly than the prediction of the Flux Tube model, where the asymptotic splitting of the two states is. Thus, while one can use the Flux Tube model to interpret the rst r excited mode, the detailed physics is distinct from that predicted by the Flux Tube model. As was hoped, these aggregate degrees of freedom provide a bridge between the phenomenology of the Flux Tube model and the complicated but more precise calculations of the Euclidean Lattice. Though the results on the 31 dimensional Transverse Lattice are quite good, they are less satisfying than one could hope. The main weakness of this theory is its number of free parameters. Only the rotational invariance of the Q Q potential was tested, and thus one is only able to specify a trajectory in 35

45 parameter space instead of a unique solution. In principle one could continue the analysis of the eective potential by studying other observables like glueball masses [7]. Continuing such a program would require one to calculate many dierent observables in order to t the entire eective potential. Expanding to higher orders in the eective potential is also problematic. A better way to determine the eective potential is to calculate it from rst principles. This is the topic of the next chapter. 36