# Baryons and Low-Density Baryonic Matter in 1+1 Dimensional Large QCD with Heavy Quarks

###### Abstract

This paper studies baryons and baryonic matter in the combined large and heavy quark mass limits of QCD in 1+1 dimension. This regime allows for a simple physical picture, which is computationally tractable. In this non-relativistic limit, baryons are composed of quarks that interact, at leading order in , through a color Coulomb potential. Using variational techniques, single baryon masses and interaction energies of low-density baryon crystal are calculated. The single baryon calculations are very accurate and can be used as a cross-check to the general numerical approach based on Hamiltonian methods for baryons and baryonic matter in 1+1 dimension for arbitrary quark mass recently proposed by Bringoltz, which is based on a lattice in a finite box. It is noteworthy that the Bringoltz method differs from a previous approach developed by Salcedo, et al. in its treatment of an effect due to the finite box size with periodic boundary conditions – namely, the existence of gauge configurations which wind around the box. As these are finite volume effects, one might expect them to be small for boxes that are large enough so that the baryon density approaches zero to high accuracy at the edges of the box. However, the effects of these windings appear to be quite large even in such boxes. The large mass infinite volume calculations performed here are consistent with the results of numerical calculations using the Bringoltz approach and confirm the importance of gauge fields that wrap around for finite systems. The calculation of the baryon crystal interaction energy requires the assumption that at low-densities the ground state is composed of individual baryons, each in color-singlet states, with wave functions orthogonal to one another. While plausible, this assumption is somewhat ad hoc in that one can construct configurations in which the entire state is color-singlet but cannot be broken into individual color-singlet baryons. However, the interaction energy of low-density baryon crystals calculated with this assumption is consistent with numerical results based on Bringoltz’s approach. This suggests that the assumptions regarding the color structure is justified in 1+1 dimension. This is useful, as it supports a similar assumption that was made in 3+1 dimensions, where there is no alternative means of calculation.

###### pacs:

11.15.Pg, 12.39.Hg, 14.20.-c, 21.65.Mn## I Introduction

The problem of understanding cold nuclear matter from QCD first principles has remained quite challenging due to the intrinsic non-perturbative nature of the problem borici ; politzer . While many non-perturbative problems such as the finite temperature problem can be studied using lattice QCD techniques, the problem of finite density QCD is marred by the presence of the fermion sign problem politzer ; hands ; barbour ; alford ; only low densities can be studied and that too only at large temperatures stepanov as all available methods are only valid for small values of . As such it is useful to find certain limiting cases for the study of cold nuclear matter, in which the problem of finite density QCD becomes somewhat more tractable.

Large QCD is a limit where the physics simplifies considerably tHooft1 . Only planar diagrams contribute at leading order in while quark loops effects are suppressed. Despite these simplifications, the problem of large QCD has remained unsolved. However, the problem in 1+1 dimension is more tractable tHooft ; bars ; here the physics simplifies considerably because gluons, due to the absence of magnetic fields in one spatial dimension, do not propagate. Additionally, with appropriate choice of gauge (either axial or Coulomb), gluon-gluon interactions are absent and hence the theory looks very much like QED, in which color charges (i.e. quarks) interact via a Coulomb potential. The self-energy corrections to the quark propagator come from an infinite number of ladder diagrams, which remarkably can be summed up exactly in spite of the need to solve a bootstrap equation. Furthermore, the meson spectrum can be solved – albeit only numerically via a Bethe-Salpeter equation; a linear “Regge trajectory” (with no continuum) is observed.

Unlike the meson problem in 1+1 large QCD, the baryon problem is highly non-trivial. Many attempts have been made to study the problem: Ref. thies studies the problem in the chiral limit, where baryons assume the form of a Sine-Gordon soliton. Ref. narayanan invokes Eguchi-Kawai reduction EK to argue controversially barak that 1+1 large QCD is independent of baryon chemical potential. Ref. negele formulates the large baryon problem for arbitrary quark masses at infinite volume. Numerical solutions of the problem require a lattice treatment in a finite-sized box. The most exhaustive work was done recently by Bringoltz in Ref. barak . It is based on the axial gauge representation of QCD axialQCD and the coherent states formalism yaffe . Ref. barak solves both the single baryon and baryon crystal problems in a finite sized box for arbitrary quark masses. The principal difference between the approaches of refs. barak and negele is the treatment of the effects of finite box sizes. In Ref. negele the infinite volume Hamiltonian is derived and simply used on the finite box by imposing boundary conditions. In contrast, the idea underlying Ref. barak is that the quarks making up a baryon in a finite box interact via a color-Coulomb potential of the form , where is the physical separation of the color charges, Q is an integer and L is the size of the box. These additional windings are due to the inability to gauge away the gluon degrees of freedom in the problem barak ; axialQCD . In effect, the treatment of Ref. negele corresponds to including only the sector.

The study of these windings in Ref. barak was motivated largely for theoretical reasons associated with QCD volume reduction. Clearly in small boxes the two treatments might be expected to be quite different. A priori, however, one might expect that the difference between calculations based on Refs. barak and negele for a baryon mass would be quite small in cases where the box sizes are large enough to comfortably contain a baryon since the difference between the two is a finite volume effect. However, as will be discussed in section III, the differences can be quite large, even in such circumstances. Indeed, the differences are large enough that one might worry the differences suggest a possible problem with the formulation of Ref. barak . Unfortunately, It is difficult to verify numerically that this is the case by going to exceptionally large box sizes.

Given this situation, it is quite useful to have a regime in which one can independently compute the mass of a baryon to verify that the winding effects are as large as they appear to be for practical-sized boxes. The regime of heavy quark masses and large serves this purpose extremely well. In this paper, we perform a highly accurate variational calculation of single baryons using Witten’s mean-field argument witten , which is similar to an analogous 3+1 dimensional calculation that was done recently in Ref. 3+1 . It was argued in Ref. witten that baryons are composed of quarks in a color singlet state, where each quark moves in an average potential due to the remaining quarks. In the non-relativistic limit with quark masses taken to infinity, the average potential has a simple form because the only contributions to the interaction come from single gluon exchange. Note that the principal difference between the heavy quark system using Witten’s approach and the general result of Ref. barak is that the vacuum is trivial in the former case.

We find that the single baryon results using Witten’s variational formulation agree with the results based on Ref. barak that incle the nontrivial effects of the winding with remarkable accuracy. These results support the approach taken in Ref. barak that include a large effect of the windings, albeit only in the restricted regime of large quark masses. Our result is also quite useful in providing a simple and physically intuitive picture for at least one regime of QCD in 1+1 dimension and, as such, nicely complements the Bringoltz method, which obtains results largely through numerical means.

The study of the heavy quark regime in 1+1 dimension is useful for another purpose: the study of baryonic matter at low density. Studies of low-density baryonic matter in the heavy quark regime of large QCD were previously done in 3+1 dimensions 3+1 . The principal attraction of this regime is that it appears to be tractable due to Witten’s mean-field arguments and the non-relativistic nature of heavy quark dynamics. However, as discussed in Ref. 3+1 for the case of baryonic matter (as opposed to a single baryon) a plausible but ad hoc assumption was needed: that at low densities the ground state is composed of individual baryons each in a color-singlet state with the wave function of quarks for each baryon orthogonal to the wave function of quarks in any other baryon. This structure appears quite plausible, but it is ad hoc in that one can construct mean-field configurations in which the entire state is color-singlet but cannot be broken into individual color singlet baryons. In 3+1 dimensions, it was not easy to verify whether this assumption was justified. However, in 1+1 dimension, one can use the methods of Ref. barak to numerically compute the interaction energy and compare it with the result in the mean-field approach using the assumption of no hidden-color states. We find good agreement between them, which suggests that this assumption is indeed valid. This agreement in 1+1 dimension greatly increases our confidence that the assumption made for the 3+1 dimensional case is valid.

The paper is organized as follows. In the following section, we calculate the single baryon energy and wave function in the limit of large quark masses using Witten’s mean-field argument. We compare this variational solution to the numerical results based on the recent Hamiltonian approach proposed of Ref. barak and with a previous Hamiltonian in Ref. negele that did not include the windings. In the next section, we calculate the interaction energy per unit cell of a low density baryon crystal, again using Witten’s mean-field argument. We compare the interaction energies with numerical results for the many baryon sector based on Ref. barak . The paper concludes with a brief discussion of the results.

## Ii Baryons in the Combined Large and Large Limits

### ii.1 The Energy Functional in the Mean Field Approximation

The nonrelativistic mean-field energy functional for baryons in 3+1 dimensional large QCD for heavy quarks has been calculated in Ref. 3+1 based on Witten’s classic analysis in Ref. witten . QCD in 1+1 dimension at large is generally simpler than in 3+1 dimensions. In 1+1 dimension, quarks are spinless, nonrelativistic fermions and gluon-gluon couplings can be eliminated by appropriate choice of gauge (e.g. the axial gauge as in refs. tHooft and negele ). Thus the leading order contribution to the energy comes from a coherent one-gluon exchange between pairs of quarks. Interactions involving multiple gluons are suppressed at least at a relative order of . Furthermore, quarks loops are suppressed at relative order of at least as a result of the large limit. However, for the case of heavy quarks, these simplifications do not help. For heavy quarks, the interactions are pairwise color-Coulomb in any event so the lack of gluon-gluon interactions adds no new simplifications. Similarly, spin effects are suppressed for heavy quarks. Thus, the leading-order energy functional in the dual expansion, is derived in an essentially identical manner to the 3+1 dimensional case 3+1 . However, the color-Coulomb potential in the context of one spatial dimension is linear in distance as opposed to dropping off as . The strength of the coupling is chosen to be , adopting the conventions in refs. tHooft , barak and negele .

The energy of the baryon per color at leading order in and is

(1) | |||

where is the quark mass, is the normalized baryon wave function and . Minimizing the energy functional with respect to the wave function while imposing the normalization leads to a Schrődinger equation for the ground state baryon.

(2) |

where is the Lagrange multiplier that ensures the normalization of the wave function.

### ii.2 The Variational Calculation

We choose to calculate the ground state baryon wave function using a variational approach. From here on, will be taken to be the ground state wave function . To proceed, we define a scaling variable and a scaled wave function where is the dimensionless variable defined as . Then, the baryon wave function is:

(3) |

The normalization of the wave function is independent of the scaling variable as long as is normalized, i.e. .

Next we define functionals for the kinetic and potential energies, and :

(4) |

Rewriting the energy functional in terms of the new variables , and , we get:

(5) |

For simplicity, we choose a normalized functional form for to be a polynomial times a Gaussian:

(6) |

where indicates the number of terms kept. Only even powers of appear in the wave function because the ground state is expected to be symmetric. We found that truncation at was sufficient to obtain the energy with a very high accuracy. We now have an energy functional that is a function of and . The next step is to minimize the functional with respect to each of the variables, i.e. for . Minimizing with respect to gives

(7) |

which yields:

(8) |

To proceed we minimize with respect to each for with chosen to be . Including , there are a total of 10 parameters in the variational calculation.

The binding energy of the baryon is the second term in Eq. (10) and is characterized by the quantity , the number of parameters, , in the trial wave function. Minimizing this quantity yields . It converges rapidly with increasing

(9) |

which fixes the wave function.

It is sensible to present the results in units of i.e. , which is the only scale present in large QCD. Hence masses will be written in the units of and lengths in the units of .

The ground state energy per color in units of is:

(10) |

The variational calculation converged well and the interaction energy is estimated to be accurate to five significant figures. The parameter is given by:

(11) |

the dimensionless Lagrange multiplier is

(12) |

### ii.3 Asymptotic Behavior of the wave function

The nature of variational calculations is such that they determine energies more accurately than the wave functions. The Gaussian form of the trial wave function we chose was very convenient and provided for an extremely accurate calculation for the energy. Moreover, with a sufficient number of terms, it provides a rather accurate form for the wave functions over the bulk of the region that contributes substantially to the energy. However, it does not capture the correct behavior of the wave function at asymptotically large distances. At large , the drop off of the true solution of Eq. (2) is slower than a Gaussian 3+1 . Since one principal goal of this paper is to compute the interaction energy of low density baryonic matter – which is highly sensitive to the long range part of the wave function – it is important to determine the correct behavior at asymptotic distances.

To proceed, we study Eq. (2) in the limit where . In doing so, we obtain a Schrődinger equation that captures the asymptotic behavior of the baryon wave function:

(13) |

The general solution is a linear combination of the Airy functions. Taking only the normalizable solution, in units where and , we find

(14) |

where is a constant.

The asymptotic form is thus fixed entirely by the determination of the constant . This can be done numerically provided that there is a region for which is large enough so that the asymptotic form of the wave function of Eq. (14) is accurate while simultaneously being small enough that the variational wave function is accurate. If so, one can match to the two forms in the region of simultaneous validity of the two forms. It is easy to verify that such a region exists: a plot of the ratio of the two forms of the wave function must have a plateau with a value near unity over an extended region. Note that the size of such a region of stability depends on the accuracy of the variational wave function. It was largely for this reason, that we used as many terms as we did in the variational wave function. Such a stable region was found in the vicinity of and the asymptotic wave function takes the form:

(15) |

where the asymptotic form of the Airy function has been used.

## Iii Comparison of single baryon with more general approaches

In this section, we compare our single baryon energy results with numerical results based on the Hamiltonian approach of Refs. barak and negele . These approaches are more general than ours as they are valid for arbitrary quark masses. Our approach complements these; while the regime is far more limited, the approach is more transparent and far less reliant on numerical work. Moreover, there is real value in having the ability to independently check the analysis of Ref. barak in a regime where the results of infinite box size are clearly under control.

Ref. barak analytically finds the most general Hamiltonian for large finite density QCD in the baryon sector for arbitrary quark masses in a finite box. The Hamiltonian for a box of size is given by barak :

(16) |

where the the trace is only over Dirac indices. The Hamiltonian is given in terms of density matrices but can also be represented in terms of the conjugate space density matrix as

(17) |

The index represents integers and is defined as:

(18) |

In Eq. (18), represents the average over all possible gauge field configurations of some gauge-invariant quantity

(19) |

is a path-ordered (denoted ) spatial gauge field operator

(20) |

and represents appropriately normalized coherent states

(21) |

Note that has implicit dependence on gauge fields (denoted ) via , the gauge-invariant generators of the fermionic coherent group barak

(22) |

are the corresponding weights to and is a color singlet reference state that satisfies the condition =0, where is a member of a set of single particle annihilation operators chosen to ensure that the reference state has the baryon number of interest in the box.

To recap, the diagonal projection onto coherent states defined in Eq. (18) is for a particular gauge configuration; however, reflects averaging over all possible gauge configurations. In this averaging, the only non-trivial structure that survives are the windings around the periodic box.

Note that has the following properties:

(23) |

Here is the density matrix associated with the baryon Dirac sea and gives a formally divergent contribution to the integral in the absence of regularization. is the baryon number in the box, which can be written in terms of , the baryon density as .

The baryon Hamiltonian of Eq. (16) can only be solved numerically. To make the problem numerically tractable, we truncated the theory such that in Eq. (16) is kept finite. This truncation introduces corrections to observables (including the baryon mass) that scale as . To obtain the observables in the full theory, the observables are calculated for different values of and the result in the limit is obtained by extrapolation.

To formulate the problem numerically, the Hamiltonian is also discretized in space and the continuum results are obtained by extrapolation to the continuum limit. A latticized Hamiltonian using staggered fermions is discussed in detail in Ref. barak . Note that the act of latticizing the theory introduces an ultraviolet cutoff thereby regularizing the theory and rendering the vacuum density matrix, , finite. Here, we only present the form of the density matrix latticized using staggered fermions. These matrices satisfy the properties in Eq. (23) can be written in terms of a finite dimensional basis of orthogonal wave functions for each with the indices , where is the number of lattice sites. The discretized density matrix is

(24) |

with the following orthonormality constraints on the wave functions:

(25) |

The sum from in Eq. (24) is the lattice version of the Dirac trace of vacuum density matrix in Eq. (23). The integral becomes regularized on the lattice and gives a finite contribution.

The Hamiltonian of Eq. (16) was derived in the axial gauge representation of QCD axialQCD . It is well-known that in the Hamiltonian approach to either QCD or QED, Gauss’ law appears as a constraint and not as an equation of motion. Ref. axialQCD explicitly constructed a QCD Hamiltonian, with the Gauss’ law constraint built in a manner that only the physical states of the Hilbert space play a role.

Ref. barak , uses the constrained Hamiltonian of Ref. axialQCD and the coherent states approach of Ref. yaffe to solve the Hamiltonian in the baryonic sector, which is the leading order physics in a expansion. This follows from the fact that the coherent states form an overcomplete set so that any physical state can be expressed as an integral over them and that the overlap between coherent states becomes sharply peaked as .

The Hamiltonian has a number of interesting features. First, for translationally invariant states, the classical Hamiltonian for the baryons exhibits volume independence and partial volume independence barak ; unsal . Second, color charges interact via a linear Coulomb potential of the form , where is the size of the box and refers to the number of windings around the box. The sign of determines the direction of the windings. The presence of these windings is a result of the inability to gauge away all the spatial gauge degrees of freedom in a finite box. Specifically, there are spatial gluon windings that remain in the axial gauge after the imposition of the Gauss’ law constraint.

Ref. negele , on the other hand, formulates the baryon Hamiltonian in infinite volume, where the windings represented by are absent; the form of the Hamiltonian is exactly the same as that of Eq. (16) except that . However, in constructing the lattice regularized version of the resulting Hartree-Fock equations, Ref. negele assumes that the correct approach is to use a box large enough such that finite volume effects on the baryon structure is small and with lattice spacing small enough that the baryon density can be easily resolved. For this purpose, Ref. negele constructs a lattice potential, which has the form in the continuum and infinite volume limits, while continuing to ignore the windings. Thus, the resulting latticized theory of Ref. negele is equivalent to the truncation of the Hamiltonian in Eq. (16) and, for small boxes, may have significant errors. One expects these errors to go to zero as the box size goes to infinity.

### iii.1 Comparison of Single Baryon Mass and Density Profile

We find that in the limit of large quark masses, there is excellent agreement between the variational calculation of section II and a numerical calculation based on the latticized baryon Hamiltonian of Ref. barak . In Figs. 1 and 2, we first plot the baryon mass minus the quark mass contribution (i.e. the interaction energy) as an extrapolation (in ) for boxes of size for and for . Note that the points plotted are the baryon masses in the continuum limit but for different finite values of . We have points for .

The baryon masses (including numerical uncertainties) are for quark masses of and for quark masses of . The uncertainties in the baryon masses are associated with taking the continuum and limits and also the systematic errors due to the numerical nature of the Hartree-Fock calculation. We find that the extrapolated baryon masses are in agreement with the masses from the variational calculation presented in Eq. (10), up to the numerical uncertainties of the calculation; the baryon masses from the variational calculation are and for quark masses of and respectively.

Next, in Figs. 3 and 4, we plot baryon density profiles for the two calculations with the line representing the result of the variational calculation and the dots representing the result of the numerical calculation of the general finite baryon density Hamiltonian. The densities are plotted for and with the number of lattice points, . The two results agree quite well in spite of the errors associated with discretization of space and truncation in the number of windings.

It is useful to see the extent to which including the windings (i.e. ) is crucial for getting interaction energies accurately for system with relatively large-sized boxes. From the plots in Figs. 3 and 4 it is clear that the baryons fit well within the boxes in the sense that the baryon density has dropped extraordinarily close to zero well before the edge of the box. Thus, one might be tempted to conclude that these boxes are big enough so that the effects of winding should be quite small. However, this is not the case. Comparing the interaction energy of the baryon mass (which we label by ) with the interaction energy after extrapolation (labelled ) using the quantity baryon masses. We also quote the error in the baryon mass, which comes from systematic errors associated with the numerical nature of the Hartree-Fock calculation and the continuum extrapolation. For a quark mass of , the baryon mass is and for quark masses of , the baryon mass is . Thus, we find that the relative difference in interaction energy between the result is approximately 7 per cent, which is quite significant given the degree to which the box fully contains the baryon. , which measures the relative difference in the interaction pieces of the extrapolated and the

The large difference between the and results show the practical importance of the windings in realistic calculations even for very large boxes. Of course, the effect of the winding is expected to be large for small boxes and was introduced largely to explain the behavior of the system in small boxes including large volume independence barak . To some extent, the large effect of the windings on the boxes computed here is rather surprising. Indeed, they are large enough that one might question whether there is something wrong with the formalism in barak . However, the fact that the results extrapolated to going to infinity agree so well with the results derived with the formalism of this paper, which are known to be valid for heavy quark masses, shows that this is not the case.

## Iv Baryonic matter

### iv.1 Introduction to the Multi-baryon Problem

Now that we have established that Witten’s heavy-quark formulation for single baryons is consistent with results based on the Hamiltonian approach of Ref. barak , we turn our attention to baryonic matter. It is believed that baryonic matter forms a crystal at large : the baryons are heavy and have a tendency lie in equilibrium at the potential wells created by the other baryons.

We consider infinite nuclear matter at low enough densities such that only interbaryon interactions between nearest neighbors are relevant.

Here, we assume that the many-baryon sector consists of distinct color-singlet clusters of quarks with identical wave functions, as in Witten’s discussion of baryon scattering witten . An analogous assumption was made in Ref. 3+1 for heavy quark, large QCD in 3+1 dimensions. This assumption is highly plausible but as noted in Ref. 3+1 , it has not been rigorously established. A possible concern is that hidden color states could contribute at leading order. Thus, it is useful to test this assumption in a 1+1 dimensional setting. We can do that by comparing the results based on this assumption using our heavy quark formalism with numerical results using the methods of Ref. barak , which are known to be valid at large generally and do not involve any additional assumptions.

Let us begin by deriving the interaction energy per baryon in a 1+1 dimensional crystal at low density using the heavy quark non-relativistic formalism. The energy per unit-cell of the low-density baryon crystal in terms of a reference baryon is:

(26) |

where is the wave function of the reference baryon, are the wave functions of the nearest-neighbor baryons, the first term is the contribution to the intrabaryon kinetic energy, the second term is the contribution to the intrabaryon potential energy, and the last two terms are the contributions to the interbaryon energy. In this expression both and have been scaled out of the energy density using the following change of variables:

(27) |

By assumption, the density is sufficiently small so that the effect of interactions with next-to-nearest neighbors is negligible.

The wave function for the baryon crystal is constrained by the Pauli exclusion principle, which precludes two fermions from simultaneously occupying the same position. This implies that the wave function of a reference baryon state must be completely orthogonal to the the wave function of the neighboring baryon states and .

(28) |

where is some normalized wave function. Here, we have assumed that densities are low enough that the reference baryon wave function is dominated by the single baryon wave function except for corrections centered around . The choice of and is determined by the orthogonality constraint of the neighboring baryons and by normalization of the baryon wave function. We find and in terms of , which characterizes the overlap at the same spatial points of the neighboring baryon wave functions and the quantity , which characterizes the orthogonality of the neighboring wave functions:

(29) |

We have disregarded effects because becomes exponentially small with decreasing density and in the low density regime, the higher order effects are parametrically small. Effectively, we are writing the interaction energy in the form (where is the characteristic width of a baryon and is the separation of neighboring baryons) by developing an appropriate power expansion for .

To proceed, we first assume that , which appears to be the most efficient way to orthogonalize neighboring crystal wave functions. We will show a posteriori that this assumption is correct in the sense that it will minimize the total energy up to corrections which are parametrically small at low density. The orthogonality constraint in doing so gives

(30) |

As can be seen from the explicit form of , which we calculate in the following subsection, for parametrically low densities . Hence, it is plausible that in the low density regime, where only nearest-neighbor interactions are relevant, the choice of is indeed the correct one. In the section IV.4, we introduce corrections such that , where is a normalized wavefunction and argue that contributions to energy at do not contribute at the order to which we work.

### iv.2 Calculation of overlap function

Before proceeding with the calculation of energy, it is useful to calculate the overlap function , which is defined in Eq. (29). Most of the contribution to comes from the localized baryon density at . In the low density regime, the contributions to , localized at and are exponentially suppressed. The contribution to the center comes from the tails of neighboring baryons, which have the form of Airy functions. Using the asymptotic form of Airy functions, we get

(31) |

where is the strength of the asymptotic wave function of Eq. (15).

The function in the exponent has a sharp minima at and hence the integrand contributing to has a maxima at . We use the standard steepest descent method to calculate the integral, which involves series expanding the exponent around the turning point, evaluating the denominator of at and performing the integral. This gives

(32) |

with a leading order correction term that is proportional to . The correction is suppressed for parametrically large distances, which is the relevant regime for this paper.

### iv.3 Energy at low density

Here, we calculate the interaction energy per unit cell of a low density baryon crystal up to and leading order in using units of Eq. (27). Note that is the characteristic width of an isolated baryon and is the interbaryon separation. We divide the energy into two parts: the first line of Eq. (26), which label the intrabaryon contributions, and the second line, which label the interbaryon contribution. The intrabaryon contributions contain a contribution from the kinetic energy and another from a potential energy. Note that the intrabaryon energy is not the same as in the free space baryon since the wave function has been modified due to the interactions with the other baryons.

#### iv.3.1 Interbaryon Energy

The contribution to the interbaryon energy under the assumption that only nearest neighbor interactions are relevant is presented in Eq. (33).

(33) |

(34) |

The two terms in the equation correspond to the interbaryon energies in a unit cell associated with two nearest neighbors of a reference baryon state in the crystal. Note that all terms in the interbaryon energy in Eq. (33) are of the form , which is of the same structure as the Coloumb energy associated with a one-dimensional charge density . Thus, for example, the first term in Eq. (33) has an effective charge density . Moreover, note that in one spatial dimension, unlike in three spatial dimensions, the Coloumb energy of a charge distribution grows with the size of a system.

In Eq. (34), we present the effective charge density that determines the interbaryon energy in terms of the single baryon wavefunction , and . Equation (29) completely determines the two parameters. Since for a low-density baryon crystal, we have and . Therefore, and .

To proceed, we note that the effective charge density relevant for the interbaryon contribution is localized. In Fig. 5, we show a schematic effective charge density plot characterizing the nature of in Eq. (34) upto . Note that the densities are sharply peaked at . These peaks in Fig. 5 correspond to the first three terms of Eq. (34). The smaller peaks at and correspond to the second and the third terms of Eq. (34) respectively. Since the isolated baryon wave funtion, , is normalized, the terms each have a total baryon charge of . Furthermore, the width (in units of Eq. (27)) of the peak at , which is proportional to the charge density, , is of and does not scale with as can be seen from the results of Eqs. (3), (6) and (11). The same properties are true for the charge density centered at . However, the effective charge density centered around behaves differently. For large interbaryon separation , the width (in units of Eq. (27)) scales as , which can be easily calculated from the standard steepest descent calculation in the integrand of Eq. (29). Also note that the total effective charge in the center peak is for parametrically small baryon densities. Although the width of the central peak depends on (i.e. the baryon density), it is much smaller than the interparticle spacing at low density. Thus in the low-density regime, these three regions of charge are well separated from each other with a spacing much larger then the width of each lump.

To calculate the leading contribution interbaryon energy per unit cell of a baryon crystal in the low-density regime, the effective charge densities can be approximated as delta functions with the appropriate baryon charges. The leading correction is the fact that the spatial extent of the charge density centered around grows with . The size of this correction can be calculated using the approximation to the asymptotic form of the Airy function that was used to calculate the overlap integral in Eq. (33). This integral can be calculated easily using the method of steepest descent.

For parametrically low baryon densities, the total interbaryon energy becomes:

(35) |

where is the correction to the interbaryon energy associated with the fact that the center peak scales with :

(36) |

The interbaryon energy above has the form – charge squared times a width – which is as expected. Also, note that the energy contributions due to the finite extent of the two regions of effective charge at the the ends is and hence is parametrically smaller at large .

#### iv.3.2 Intrabaryon Energy: Potential Contribution

In this subsection and the next, we calculate intrabaryon energy for the low-density baryon crystal at . There are two contributions to this, the potential and the kinetic.

The potential contributions to the intrabaryon energy of the baryon crystal comes from Eq. (37). The effective charge density is presented in Eq. (38). Additionally, we present in Fig. 5 a schematic plot of , the shift in the effective baryon charge density from the free case, , minus the isolated baryon density, , to emphasize the various charge enhancements. This quantity is even around and hence we only plot the region .

(37) |

(38) |

To proceed, we calculate the effective charge density, , upto to obtain the intrabaryon energy at the same order. Note that in Eq. (38), the first term, , of the effective charge density is centered around , the second term proportional to is centered around , and the third term, which is proportional to , is centered around .

Furthermore, at the intrabaryon energy contributions for the baryon crystal come from a cross-term between the effective charge from the unperturbed baryon, , and the effective charge due to the perturbation induced by the near-neighbor baryons. To the extent that is large, the original charge is well-localized at the origin, it creates an effective electric field which is constant for (or a potential linear in ). With a linear field dependence, the potential energy is sensitive to the center of charge the effective charge due to the perturbation. Thus, we can approximate the charge densities in Eq. (38) as localized delta functions with the appropriate strengths, which are associated with the total baryon charges of each of the terms. Now for the case of the first term, which has a strength of and is localized at the origin, the delta function approximation is problematic, since it is not in the region where the field from the unperturbed effective charge is linear. However, the potential energy from this term is of , while the leading behavior scales as . Thus, this term can be neglected as being parametrically small. The contributions from the other two terms yield,

(39) |

As noted above, there are corrections to this of relative order from the perturbation in the effective charge due to . There are other corrections due to the fact that unperturbed effective charge has a spread and hence the effective field is not precisely constant. These corrections are exponentially small in .

#### iv.3.3 Intrabaryon Energy: Kinetic Contribution

The kinetic energy contribution to the intrabaryon energy at leading order in is presented in Eq. (40). Note that the first three terms are independent of the interbaryon spacing , whereas the remaining terms are not; in the low-density regime, the contribution from these terms independent of are parametrically small. Furthermore, we ignore the term because it is the contribution to the single baryon kinetic energy and not the crystal interaction energy. We also ignore the last term because it is not nearest neighbor.