Low Energy Spectrum of SU(2) Lattice Gauge Theory: An Alternate Proposal via Loop Formulation

We show that, prepotential formulation of gauge theories on honeycomb lattice yields local loop states, which are free from any spurious loop degrees of freedom and hence exact and orthonormal. We also illustrate that, the dynamics of orthonormal loop states are exactly same in both the square and honeycomb lattices. We further extend this construction to arbitrary dimensions. Utilizing this result, we make a mean field ansatz for loop configurations for SU(2) lattice gauge theory in $2+1$ dimension contributing to the low energy sector of the spectrum. Using variational analysis, we show that, this type of mean loop configurations has two distinct phases in the strong and weak coupling regime and shows a first order transition at $g=1$. We then propose a reduced Hamiltonian to describe the dynamics of the theory within the mean field ansatz. We further work with the mean loop configuration obtained at the weak coupling regime and analytically calculate the spectrum of the reduced Hamiltonian. The spectrum matches with that of the existing literature in this regime, establishing our ansatz to be a valid alternate one which is far more easier to handle for computation.


Introduction
Understanding the low energy behaviour of gauge theories is one of the most important problem of particle physics. Formulation of gauge theories on discrete lattice [1] has shown many major investigations in this direction over past few decades. Using Monte Carlo method, many important physical quantities can be computed numerically [2]. However, understanding the vacuum as well as excited states in this sector is still open for investigation. Hence, there should always be attempts to make analytic approximations, specifically in the weak coupling regime of lattice gauge theory. This present work proposes such an approximation for SU (2) lattice gauge theories in 2+1 dimensions, which can as well be generalized to higher dimensions and higher gauge groups.
We work within the Hamiltonian formulation of lattice gauge theory [3] and use prepotential [4] framework, which gives an useful reformulation in terms of gauge invariant loops. Loop formulation of gauge theories is always desired by theorists [5] as one can get rid of spurious gauge degrees of freedom. However, working with loop does not guarantee to work with only physical degrees of freedom as the loop space itself is highly over-complete [6]. Working in terms of gauge invariant loops in the weak coupling regime is again particularly difficult, as all possible loops of all shapes and sizes do contribute to the low energy spectrum of weakly coupled gauge field theories. In this scenario, the prepotential formulation [4] gives a great advantage over the standard Wilson loop approach as it is possible to extract only physically relevant loop degrees of freedom and study the dynamics of those.
In prepotenntial formulation of lattice gauge theory [4], one constructs gauge invariant loop variables, locally at each site. This particular feature makes it extremely useful for different directions of exploring the subject starting from analytic calculations [7,8,9] to the recent approach of quantum simulating lattice gauge theory [10]. However, the original motivation for formulating prepotential approach was to make better understanding at the weak coupling regime of the theory. Towards this direction, a very important step is to construct the exact and orthonormal loop Hilbert space, containing only physical degrees of freedom. In this work, (also in another parallel recent work by Anishetty et al. [9]), we have proposed a general technique of constructing explicit orthonormal loop states for SU (2) lattice gauge theory in any arbitrary dimension in an extremely easy way even without going into the complicated Clebsch-Gordon coeffiecients. The prepotential formulation on square lattice reveals that the physically relevant loops are the non-intersecting ones [7]. We propose that, if one virtually splits each site of the lattice into two virtual ones following figure 8, the resulting lattice is an hexagonal one in two dimension. It has also been demonstrated that the dynamics of orthogonal loop states on square lattice is exactly equivalent to that of all possible loops on hexagonal one. The prepotential formulation on hexagonal lattice keeps all the important features of this particular formulation intact, such as local loop description by constructing intertwiners and Abelian weaving of those intertwiners leads to standard Wilson loops of the theory [4]. Moreover, the extra advantage that is obtained on hexagonal lattice, is that the local loop space constructed at each site is exact and orthonormal. This is a tremendous advantage for the purpose of any computation be it analytic or numerical as one needs to work within a really small Hilbert space without bothering about complicated Mandelstam constraint. In [9], it has also been shown that there exists, more than one way to split each point, and the resulting lattices turn out to be of different types (i.e hexagon,octagon, square etc) as well as of different translational symmetries. It has also been argued in [9] that, dynamics on these virtual lattices are exactly equivalent to that of the original lattice by gauge fixing on the virtual links connecting splitted lattice sites. One can choose any splitting scheme as per the calculational convenience. For the purpose of present work we fix the splitting scheme given in figure 8 at each site, and get a hexagonal lattice to work with.
We establish its equivalence with the original lattice by computing the dynamics explicitly for random loop configurations generated on both of these.
In this work, we start from prepotential formulation of pure SU (2) gauge theory on square lattice. Then using the virtual point splitting technique of [9], we move to the virtual hexagonal lattice. On this particular lattice, we make a mean field ansatz that only an average loop configuration contribute to the low energy spectrum of SU(2) lattice gauge theory. We also show that this average loop configuration has two distinct phases at the weak and strong coupling regime of the theory and shows a first order phase transition at g = 1. We also construct a reduced Hamiltonian, starting from the Kogut-Susskind Hamiltonian, which keeps the dynamics of the theory within the mean field ansatz. We further perform a variational calculation, to fix the mean field configuration at different values of coupling. This analysis also shows that the average value of the fluxes flowing across each site shows a distinct jump as one moves from strong to weak coupling regime at g = 1. We are however interested in the the loop configuration at small values of g . This analysis reveals that large fluxes contribute to the low energy spectrum at weak coupling regime as opposed to the zero flux at strong coupling regime. For the original Kogut-Susskind Hamiltonian, largest contribution should come from large loops carrying large fluxes. Working with prepotentials makes us free from considering large loops at all, as all loops are now local [4,7,9]. Hence, our ansatz of the weak coupling vacuum consists of only large fluxes flowing across the sites. We finally compute the lower lying spectrum of that reduced Hamiltonian at weak coupling regime and show that we get reasonably acceptable results within this approximation.
The organization of the paper is as follows: in section 2 we briefly discuss the loop formulation of SU(2) gauge theory on hexagonal lattice and compare its dynamics with that of the square lattice. In section 3, we illustrate the the origin of hexagonal lattice from square lattice by virtual point splitting in arbitrary dimension. In section 4, we discuss the average loop configuration for prepotential formulation of SU(2) theory in 2 + 1 dimentions on virtual hexagonal lattice. In section 5, we propose a reduced Hamiltonian and discuss its dynamics within the mean field ansatz and finally compute low energy spectrum within this ansatz.
Finally we summarize and discuss future aspects of this study in section 6. (2) (7) Let us consider SU(2) pure gauge theory, formulated on 2 dimensional spatial lattice consisting of hexagonal plaquettes as shown in figure 1. As in the Kogut-Susskind Hamiltonian formulation [3], each of the links (n, i) originating from site n along direction i 1 carries a link variables U (n, i) and there are SU(2) generators E a (n, i), a = 1, 2, 3 present at each end of the links. The Hamiltonian, in terms of these canonically conjugate variables reads as:

Loop formulation of SU
Within prepotential framework [4], we attach e set of prepotential doublet (as we are working with SU (2)) a † α (L)&a † α (R) at left and right end of each link (n, i) with α = 1, 2 and i = 1, 2, 3. In terms of preotentials [4], the electric field is given by, satisfying the SU(2) algebra at each end. The link variable at site n along the direction i takes the form given by: 1 On hexagonal lattice, i=1,2,3 as denoted in figure 1, although the physical dimension of lattice is only two.
where, N i = a † (L) · a(L) = a † (R) · a(R) counts the number of prepotentials along direction i.
From (3), we find that, the link operator is a sum of a creation part, which increases the flux by one unit and another annihilation part, where flux is decreased by same. Hence, pictorially we can think of the link as shown in figure 2. For hexagonal plaquette this decomposition Figure 2: Link operators consisting of two parts in prepotential, one of which increases flux along that link by one unit and another one decreases. yields 2 6 plaquette terms (as opposed to 2 4 terms in the case of square plaquette). Likewise on square lattice, the prepotential formulation on hexagonal lattice, yields a local loop description of the theory which we will show to be free from any loop redundancy. Each of the plaquette operators basically consists of six local loop operators glued together following Abelian Gauss law. Let us first concentrate on the local loop operators and loop states constructed on at each site of the hexagonal plaquette.
At each site of the hexagonal lattice, links emerge in three directions, labelled as 1, 2, 3.
Each link is associated with a prepotential creation and annihilation operators {a α (n, i), a † α (n, i)} for i = 1, 2, 3 and α = 1, 2. The gauge invariant operator constructed out of these are: where, i, j are the direction indices with i = j. These local loop operators on the hexagonal lattice are represented pictorially in figure 3. It is clear from the above set of equations that, acting on strong coupling vacuum only the first operator, i.e the one given in (4) will give non-zero contribution and will build up the local loop Hilbert space. For two dimensional hexagonal lattice, we characterize the local loop space by three independent linking numbers l ij denoting the flux flowing along three (ij) directions, namely (12), (23)  Hence, the most general loop state at each site is given by: where, N is the normalization factor. Note that, on hexagonal lattice, only three linking numbers are present at each characterizing a complete basis. The local and orthonormal loop states at each site is explicitly obtained as, The |l ij basis on hexagonal lattice is exactly equivalent to the number operator basis, where three number operators n 1 , n 2 , n 3 counts the number of prepotentials along each direction.
These two basis are related as: n 1 = l 12 + l 31 , n 2 = l 12 + l 23 , n 3 = l 23 + l 31 (9) or equivalently, From the above set of equations, note that, the l ij variables are truly independent positive integers ranging from zero to infinity, whereas the transformed n i variables not truly independent but derived from l ij basis following (9). Unlike l ij variables, the ranges of n i 's are restricted to satisfy the triangle inequalities given in (10) keeping the l ij 's positive semi-definite. In this regard, working with the l ij basis is more convenient as it involve no further constraint unlike the n i basis, however one can always move from one to another following (9) or (10). Next is to find out the action of the local loop operators (4,5,6) on orthonormal local loop states (8).
These actions are far more simple than that in case of square lattice, derived in [7]. The first and simplest operator (4) involves no annihilation operator, and hence simply gives: O ++ ij |l ij = (l ij + 1)(l 12 + l 23 + l 31 + 2)|l ij + 1 (11) Note that, the coefficient is obtained to create another normalized state from the one on which it acts. This loop action is pictorially represented in figure 5(a) showing that the resultant flux along ij direction increases by one unit.
The next loop operator in (5) involves only annihilation operator and its action on a general loop state is: The right hand side of (12) is obtained by shifting both the annihilation operators of O −− ij to the right using commutation relations and adjusting the normalization factor as given in (8).
Another loop operator (6) involves one creation as well as one annihilation operator and in the very same way its action on arbitrary loop state is obtained as: Note that, as there are only three links emerging from each site, for a given operator O +− ij , the direction k, on which there will be change in flux configuration, but no change in net flux is always fixed, and hence there is no summation over index k, in the right hand side of (13). The last two loop actions (12,13) are also represented pictorially in figure 5 (b) and (C) respectively.
Let us now concentrate to the complete plaquette, around which the Hamiltonian dynamics evolves. As stated earlier, plaquettes are the smallest Wilson loops and the basic gauge invariant variables to appear in the Kogut-Susskind Hamiltonian. But in terms of prepotentials, it is not the fundamental one but consists of six vertices as shown in figure 1 each consisting of the full set of loop operators and states. All possible loop operators at each vertex which are given in figure 2, which are of type listed in (4)(5)(6). Their action on loop state as derived above is listed in detail in table 1 for convenience. One important thing to note here is that, as per our convention of defining loop operators Sl no. At vertices Action on |l 12 , l 23 , l 31 Coefficient 1. c and f O ++ 12 |l 12 , l 23 , l 31 = C 1 |l 12 + 1 C 1 = ( l 12 +1)(l 12 +l 23 +l 31 +1) (l 12 +l 31 +2)(l 12 +l 23 +1) 2. c and f O −+ 12 |l 12 , l 23 , l 31 = −C 2 |l 23 + 1, l 31 − 1 C 2 = l 31 (l 23 +1) (l 12 +l 31 +2)(l 12 +l 23 +1) 3.

Dynamics on hexagonal lattice vs. dynamics on square lattice
At this point we compare the Hamiltonian dynamics on hexagonal lattice with that on the square lattice numerically. For this purpose, we generate an arbitrary but valid loop configuration around one particular hexagonal plaquette of the lattice. This is done by specifying a set of three positive semi-definite integers denoting n 1 , n 2 , n 3 at each of the alternate site (say at site a, c and e of the plaquette in figure 1). We denote these alternate sites as the even sites of the lattice. The set of three integers when satisfy triangle inequalities are accepted as a valid loop configuration. For the neighbouring odd sites, i.e for sites b,d and f, two numbers are fixed by even sites a, c, e, and the third one is generated randomly satisfying triangle inequalities. The prefixed ones are: and randomly generate n 2 (b), n 3 (d) & n 1 (f ) satisfying triangle inequalities at b,d and f sites as well. 2 Having generated a loop configuration denoted by n 1 , n 2 , n 3 at each of the six sites around the plaquette, we readily compute the corresponding l 12 , l 23 , l 31 at each of them following (10). All possible loop operators residing at each of the six vertices around the hexagonal plaquette, changes the loop configurations following a coefficient listed in table 2.
The dynamics of an arbitrary hexagonal plaquette in compact form is given below: where, C ± i for i = 1, 2, ..., 6 's are some algebraic coefficients which are functions of number operators. In the full plaquette operator, they always come in pairs as in the six vertices of the plaquette. Product of two such C ± i s are the vertex coefficient C j for j = 1, 2, .., 12 listed in table 2. We compute those coefficeints for our configuration to find out the matrix element (M h ) of magnetic Hamiltonian between a randomly selected initial and final states, few of which are listed in table 2.
Having calculated numerical values of the matrix elements of the magnetic Hamiltonian for hexagonal plaquette, we will now compare those matrix elements with that for a square plaquette (Matrix elements of magnetic Hamiltonian on a square lattice is given in Appendix A).
An important point to note at this point is, even for a square plaquette, the loop configuration actually changes over a virtual hexagonal plaquette consisting of four real fluxes flowing around a plaquette together with two internal fluxes (coupled angular momenta, see Appendix A for detail). We provide the dictionary of identifying square plaquette to hexagonal plaquette has not been written explicitly as they remain same in the particular plaquette interaction, as evident in (53) and (16) respectively.
The dictionary for shifting between square (given in appendix A) and hexagonal plaquette (given in section 2) is as follows: • Each site (s ≡ a, b, c, d) on the square lattice has the following orthonormal angular momentum to have net angular momentum zero at site x: • Identify the flux around a square plaquette abcd as: • At each site (s ≡ a, b, c, d, e, f ) of the hexagonal plaquette, the orthonormal states are characterized by l x 12 , l x 23 , l x 31 or equivalently by • Identify the flux (marked with 'h') around the hexagonal plaquette 'abcdef' as: (17) • Now, the same flux around the hexagonal plaquette, can be identified as the dynamic flux around square plaquette 'abcd' in the following way: • Moreover, the external links of plaquettes 'abcd' and 'abcdef' can also be identified, but we are not writing them explicitly as we find them to remain unchanged in this particular plaquette dynamics.
Having established the connection between the particular square and hexagonal plaquette of interest we can now compute the matrix elements for magnetic Hamiltonian for both the cases.
We now compare the above calculated dynamics around an hexagonal plaquette with that of the square plaquette given in (53). For this purpose we identify the fluxes around the hexagonal plaquette with those around a square plaquette as given in figure 7. To compare the dynamics on square lattice and that on the hexagonal lattice, we simulate a random loop configuration on hexagonal lattice and compute the matrix element of the magnetic Hamiltonian as discussed above. Next we identify the same loop configuration on square lattice 3 Note that, the hexagonal lattice contains alternate odd (b,d,f) and even (a,c,e) sites. The links emerge emerge in the direction 1, 2, 3 from even sites and in1,2,3 from odd sites. However, for most of the purposes, we will not differentiate even and odd sites in general and will consider links to emerge from all sites in direction 1, 2, 3
following prescription listed above, and compute the matrix element of Magnetic Hamiltonian on square lattice for this state following (53). In this comparison it is easy to observe that the nontrivial delta functions in (16)

Point Splitting and virtual hexagonal lattice
In the last section, we have established the equivalence in dynamics of orthonormal loop configurations in a square and hexagonal lattice. In this section, we prescribe a virtual point splitting technique, which translates any square lattice to its hexagonal counter part. As a result of this transition, we gain a theory formulated in terms of only explicitly orthonormal loop degrees of freedom at each site, and pay the price of an extra Abelian Gauss law constraint. This price is actually negligible as the square lattice already had the Abelian constraints to solve and the extra one in hexagonal lattice is on very same footing as those.
The two dimensional hexagonal lattice, that we demonstrated here, is obtained by virtually splitting of each lattice site of a two dimensional square lattice. Elaborating a bit, consider a lattice site 'x' as shown in figure 8. From this site on a 2d lattice, clearly 4 links emerge where, S is the splitting index, which denotes each site has been splitted into S virtual sites;  (2) lattice gauge theory in any arbitrary dimensions without getting involved with complicated Clebsch Gordon coefficients.

The Hamiltonian and Average loop configurations
In this work, we consider pure SU(2) gauge theory defined on a square lattice. The system is described by the Kogut-Susskind Hamiltonian, The virtual point splitting, discussed in last section now enables us to define the theory on hexagonal lattice. In prepotential formulation, this electric part of Hamiltonian H E counts the fluxes which are actually related to number of prepotentials sitting at each end of the links of the original lattice [4].
The magnetic part of the Hamiltonian H mag is anyway more complicated to analyse.
Clearly, at weak coupling regime, this part contributes most, and hence it is essential to simplify it as much as possible to make analytic calculations feasible. The part however is altered from the square lattice, as it must contain trace of the product of link operators along the full hexagonal plaquette to make the smallest closed loop.
The magnetic Hamiltonian on hexagonal lattice contains 2 6 different gauge invariant plaquette terms for prepotential formulation of SU(2) gauge theory on hexagonal lattice. Pictorially, these plaquette terms contain n d number of dotted link and 6 − n d number of solid links around a plaquette, for n d = 1, 2, .., 6. These solid and dashed line comes in different combination yielding all of the 64 plaquette terms as illustrated in figure 11.
Let us now, concentrate on the loop configurations of the system and proceed towards a mean field ansatz (i.e average loop configuration) for low energy limit of the theory.
We have already characterized the local loop states at a particular site x of a hexagonal lattice in terms of linking numbers as |l 12 , l 23 , l 31 x in (7) and also discussed how these linking numbers are related to the occupational number basis in (9). The original Wilson loops, which are non-local are obtained by weaving these local loops at neighbouring sites along the three directions, using an extra Abelian Gauss law [4] constraint at each link direction as given below:

Mean Field Ansatz
In this sub-section, we make an ansatz for the vacuum loop configuration of the SU (2) (7) at a site x. The electric part of the Hamiltonian (20) in Prepotential formulation reads as:

The reduced Hamiltonian and its spectrum
In the last section we have made a mean field ansatz, in which an arbitrary loop state defined at each site of the virtual hexagonal lattice is given by the average value for the linking number variables is given by: In this section, we will reduce the full Kogut-Susskind Hamiltonian to a sub Hamiltonian, which describes the dynamics within this mean field ansatz. With this reduced system, one can get reasonable physical results with minimal calculational effort and hence establishes this as a valid toy system to understand weak coupling regime of the gauge theory analytically, numerically as well as by quantum simulating the system.

The Sub-Hamiltonian
As we have already explained in the last section, the electric part of the Kogut-Susskind Hamiltonian in the mean field approximation reads as: Next we concentrate on the magnetic part of the Hamiltonian within this mean field approximation. The magnetic Hamiltonian contains 64 plaquette operators, each of which is a combination of six local vertex operators listed in table 4.1 within the mean field ansatz.
Let us now analyze each and every plaquette term illustrated in figure 11 : Each of these two types of plaquette terms has 6 rotationally symmetric contribution.
Hence this particular type of rotational symmetric Hermitian operator contains total of 12 individual plaquette operators.
Combining the above three, we find that the magnetic part of the reduced Hamiltonian contains 2 + 6 + 12 = 20 plaquette operators and is completely rotationally symmetric and Hermitian.
Let us now construct a particular basis consisting of the state |L, M and the twenty plaquette operators listed above, acting on it. Hence we have a 21 dimensional basis vectors listed as below: where, the braces in the last three states denote the sets consisting of rotationally symmetric and Hermitian conjugate states. Each of these states consists of one of those three types of vertices at each site. We first calculate the local action of E 2 on each vertex, depending on whether it is a junction of two solid line (pp-vertex ≡ |pp ), two dashed line (mm-vertex ≡ |mm ) and of one solid and one dashed line (pm vertex ≡ |pp ). Using the 12 vertices described in table 4.1, the action of E 2 within mean field approach is given by: However, our basis now consists of 21 states given in (33), each having six of the above vertices.
The action of E 2 on these states are given as below: At this point, we import another notation for the basis. Let the basis be denoted as |i , f ori = 0, .., 20, with |0 = |L, M , and |i = H i |0 for i = 1, 2, .., 20, where H i denotes the 20 plaquette operators described before (given as H pmpmpm , H pppmmm , H ppmmpm ).
Hence, the reduced Hamiltonian takes the form: The electric part of the Hamiltonian acting on a state of this basis gives: 1, .., 21 (38) and the magnetic part gives: In this particular basis, the Hamiltonian matrix for each plaquette takes the form given below: Note that, the diagonals have contribution from the electric term as well from the constant term in magnetic Hamiltonian. Other than the diagonal entries, other non zero matrix elements of the Hamiltonian lies along the first row and first column of the matrix, and those are basically given by the coefficients sitting in front of the 20 plaquette terms discussed above.
We label these coefficients as C i , for i = 1, .., 20. The advantage of working with such a basis is that the Hamiltonian matrix takes a very special form, namely form of an arrowhead matrix which in turn enables us to solve for the eigenvalues analytically.
Let us assume the Hamiltonian matrix to satisfy the following eigenvalue equation: This matrix equation is equivalent to the set of eigenvalue equations mentioned below: Now, from (43), Putting (44) back in (42), we get, Solving (45) would yield theλ, which in tern gives all the eigenvalues of the 21×21 Hamiltonian matrix. To simplify (45), we further make the following observation. Note that, the diagonal elements g 2 E i + 1/g 2 are highly degenerate, as one finds in (36) that, E i 's can take only three possible values, namely 6v, (2v + 2v p + 2v m ) and (4v + v p + v m ). Hence, the sum in (45) reduces to the following sum of three terms: where,C 1 ,C 2 ,C 3 are combinations of the coefficients C i taking into account of the degeneracy.
(46) is a transcendental equation and can be solved graphically, however approaching a rough solution is quite easy. We plot the right hand side of this equation and get divergences at three poles precisely atλ = 0, g 4 , 2g 4 respectively. The plot of left hand side, i.e straight line y = x cuts the right hand side curve very close to the position of the poles. The exact value ofλ yields the value of the eigenvalues as: Note that, the smallest eigenvalue λ, actually corresponds to the highestλ from (47). To get the exact numerical value of the solution one needs to consider the coefficientsC i 's appearing in right hand side of (46). Note that, the coefficientsC i 's are functions of the mean fields L & M . Hence, it is necassary to calculate the exact value of these mean fields. For this purpose, we take the following approach.
• For some fixed g, we numerically calculate (using Mathematica) the value of L and M , for which lowest eigenvalue of the Hamiltonian matrix reaches a minima. In the strong  we have not used any assumption for weak coupling limit, except taking a mean field ansatz. We now fix the mean field configuration in a way, such that we are in the weak coupling regime of the theory. From the variational study discussed above, we see, that the ground state energy shows a perfect first order phase transition at coupling g = 1, above which the vacuum is the strong coupling vacuum, which in the mean field ansatz gives |L = 0, M = 0 . However, as g → 0, the lowest energy mean field state turns out to be a state comprising of large average flux at each site, implying the weak coupling vacuum to be consisted of loops carrying large fluxes, throughout the lattice. As g → 0, L >> M >> 0. The coefficients, listed in table 3 also changes to particular limiting values as g → 0.
• We now consider a particlular value of g, in the weak coupling regime, and the corresponding mean field configuration. For each configuration, we exactly diagonalize the Hamiltonian matrix, and calculate the eigenvalues. We list the spacings between the lowest one and first excited one as ∆λ 1 and similarly between the first and second excited one as ∆λ 2 . We list these two gaps and their ratios at different values of the coupling constant g → 0 in  lattices, as all the computation was done locally at each site.
The scaling of mass-gaps ∼ g 2 is consistent with both the weak coupling perturbation expansion in [11] 4 , variational analysis [12,13] 5 , using cluster algorithm [15] 6 for SU (2)  On a further note, the analytically calculated value of the ratio of energy differences for the reduced Hamiltonian within the mean field ansatz, matches with previous calculations at weak coupling limit for finite lattices. For example, the mass spectrum of 0 ++ sector for SU(2) lattice gauge theory in 2 + 1 dimensions is obtained in [12] as follows: m 1 a = (2.0560.00l)g 2 , m 2 a = (3.640.03)g 2 , m 3 a = (5.150.10)g 2 This result is obtained with a lattice consisting of 25 plaquettes and in the weak coupling region 1.8 < 1/g 2 < 3.6. Note that, the results obtained in [12] are consistent with the results obtained with Monte Carlo calculations within Euclidean formalism as well.
(48) establishes the validity of our much simplified system to extract out weak coupling results for practical purposes, which can be exploited to address various problems in future researches.

Summary and future directions
In this work we have proposed and justified an effective mean field description for the low energy spectrum of SU(2) lattice gauge theory in 2 + 1 dimension and have calculated the 4 ∆E = 0.2637g 2 5 ∆E ≈ 2g 2 6 ∆E ≈ 2.2g 2 7 ∆E ≈ 2.1g 2 spectrum at the weak coupling limit analytically. Starting from prepotential formulation on square lattice, we perform virtual splitting of each lattice site into two and end up with a virtual hexagonal lattice. On this hexagonal lattice, all of the local loop states in prepotential formulations constitutes an exact and orthonormal loop basis, with no further Mandelstam constraints. We have proposed a mean value ansatz for the loop configurations throughout the lattice contributing to the low energy spectrum of the theory. We have shown that such average loop configurations have two distinct phases at the strong and weak coupling regime.
Next, we have chosen a reduced Hamiltonian, from the full Kogut-Susskind Hamiltonian, which keeps the dynamics of the loops confined into our ansatz. Variational study shows that this reduced system with mean value-loop configuration shows a clear jump between the weak and strong coupling vacuum. As we are interested to explore weak coupling regime of the theory, we choose the relevant average loop configuration in that regime and calculate the spectrum for the reduced Hamiltonian we choose. In this spectrum we find ∆E ∼ g 2 which is the expected weak coupling behaviour for mass gap of the theory. The spacings of the spectrum obtained in this work is as well consistent with the available literature at weak coupling regime of 2 + 1 dimensional SU(2) lattice gauge theory. We have discussed in detail, how the point splitting lattices are constructed in higher dimensions, which can as well be exploited to extend this work beyond 2 + 1 dimensions in a straight forward way.
In a recent and parallel work [9], the point splitting lattice is constructed and utilized to analytically study the weak coupling limit of SU(2) lattice gauge theory in 2 + 1 dimension as well. In that work, they have used the path integral representation of the phase space to analytically compute the dispersion relation at the lowest order in weak coupling perturbation expansion.
However, the particular study demonstrated in this paper shows that the physical results at the weak coupling regime of SU(2) gauge theory can be extracted from a much simpler mean field approximation made within prepotential formulation of the theory. Being completely gauge invariant, and formulated only in terms of relevant physical degrees of freedom, this technique is suited for both analytic calculations and numerical simulations. From analytic perspective, this study gives a clear notion of the weak coupling vacuum for pure gauge theory and its dynamics. From numerical perspective, this particular formulation is most suited for quantum Monte-Carlo simulation of Hamiltonian lattice gauge theory using a complete gauge invariant basis characterized by only integers. Till date, this aspect has not been studied extensively, but worth investigating in near future. That study will lead to explore some of the very important physics such as calculation of the entanglement entropy of lattice gauge theory. Last but not the least, there is a tremendous progress going on, in the recently developed research interests for quantum simulating gauge theories using ultracold atoms in optical lattices as well as using different other techniques. The prepotential formulation has already been explored to propose quantum simulator for gauge theories [10]. However, most of such proposals till date have addressed the abelian gauge theories and the strong coupling regime. This present work, shows the way to construct quantum simuator to simulate the loop dynamics of non-Abelian lattice gauge theory at weak coupling regime using ultracold atoms in optical lattices. The work in this direction is in progress and will be reported shortly.

A Orthonormal loop basis and its dynamics on square lattice
Within prepotential framework of the theory one is equipped with a set of orthonormal basis states, defined at each lattice site [4,7]. For SU(2) lattice gauge theory on 2 dimensional spatial lattice, such an orthonormal basis can be easily obtained in terms of angular momentum fluxes.
In this section, we briefly quote the result of plaquette (the smallest closed loop) dynamics from [4] which we have used to compare with the dynamics of hexagonal plaquette.