Some Aspects of Three-Quark Potentials
We analytically evaluate the expectation value of a baryonic Wilson loop in a holographic model of an pure gauge theory. We then discuss three aspects of a static three-quark potential: an aspect of universality which concerns properties independent of a geometric configuration of quarks; a heavy diquark structure; and a relation between the three and two-quark potentials.
pacs:12.39.Pn, 12.90.+b, 12.38.Lg
Since many years there has been a great interest in triply heavy baryons bj (); richard (). The challenge here is to reach the level of knowledge similar to that of charmonium and bottomonium. On the theoretical side, the challenge is to explain the structure and properties of such baryons. It is expected that the potential models would be useful in doing so. Furthermore, in the process one would expect to gain important insights into understanding how baryons are put together from quarks.
The three-quark potential is one of the most important inputs of the potential models and also a key to understanding the quark confinement mechanism in baryons. However, so far, there is no reliable formula to describe the three-quark potential. The best known phenomenological models are ansätze which are a kind of the Cornell model for a three-quark system. The -law states that the static three-quark potential is a sum of two-quark potentials delta-law ().111For more details, see the Appendix D. In practice, it is taken to be the sum of Cornell-type potentials
where is the position vector of the -th quark, , and are parameters, and is the string tension. The relation with the quark-antiquark potential implies that . The -law predicts that the energy grows linearly with the perimeter of the triangle formed by quarks. Alternatively, the -law states that the energy grows linearly with the minimal length of a string network which has a junction at the Fermat point of the triangle artu (); isgur (); cornwall (). In this case, a slight modification of (1.1) is of common use in the potential models. It is simply
Unfortunately, there have not yet been any experimental results concerning triply heavy baryons. Thus, the predictions222For a recent development, see, e.g., pheno (). based on either the -law or the -law can not be compared to the real world, and the last word has not yet been said on this matter. In such a situation lattice gauge theory is the premier method for obtaining quantitative and nonperturbative predictions from strongly interacting gauge theories. The three-quark potentials are being studied on the lattice review (). Although the accuracy of numerical simulations has been improved, the situation is still not completely clear. There is no problem with the fit by the -law at long distances suganuma3q (); jahn3q (); suganuma3q-new (). At shorter distances the conclusion is mixed in the sense that some results favor the -law pdf3q (); jahn3q (), and others favor the -law suganuma3q (); suganuma3q-new (); koma ().
The situation is even worse with the hybrid three-quark potentials. So far, there are no phenomenological predictions and very little is known about those from lattice studies.333See, however, suganuma-hybrid (). These potentials remain almost unknown and, therefore, merit more attention, as in the case of the hybrid quark-antiquark potentials review (); hybrid-review ().
One of the implications of the AdS/CFT correspondence malda () is that it opened a new window for studying strongly coupled gauge theories and, as a result, resumed interest in finding a string description (string dual) of QCD. It is worth noting that it is not an old fashioned four-dimensional string theory in Minkowski space, but a five (ten)-dimensional one in a curved space.
In this paper we continue a series of studies a-bar (); a-Nq () devoted to the three-quark potentials within a five (ten)-dimensional effective string theory. The model we are developing represents a kind of soft wall model of son (), where the violation of conformal symmetry is manifest in the background metric q2 (). It would be unwise to pretend that such a model is dual to QCD or that it can be deduced starting from the AdS/CFT correspondence. Our reasons for now pursuing this model are:
(1) Because there still is no string theory which is dual to QCD. It would seem very good to gain what experience we can by solving any problems that can be solved with the effective string model already at our disposal. The AdS/CFT correspondence is a good starting point, because we already know a lot about Wilson loops in supersymmetric Yang-Mills theory ads ().444There is an obvious question. If , why is it a good starting point? We have no answer and take this as an assumption, but (2) partially resolves the question.
(2) Because the results provided by this model are consistent with the lattice calculations and QCD phenomenology. In some cases the quantitative agreement is so good that one could not even expect such a success from a simple model. In az1 (), we have computed the quark-antiquark potential. Subsequent work made it clear that the model should be taken seriously, particular in the context of consistency with the lattice white () and quarkonium spectrum carlucci (). Another instance of perfect consistency between the model a-Nq () and the lattice calculations pdf3q (); suganuma3q (); suganuma3q-new () is the three-quark potential obtained on an equilateral triangle. In addition, the model also reproduces one of the hybrid potentials of kuti (). It is the potential hybrids ().
(3) Because analytic formulas are obtained by solving the model. This allows one to easily compare the results with the lattice and QCD phenomenology.
(4) Because the model is able to explore QCD properties in the transition region between confinement and asymptotic freedom. As seen from Figure 3, it can operate at length scales down to . This is a big advantage of the model over any type of an old fashioned four-dimensional string model in Minkowski space.
(5) Because the aim of our work is to make predictions which may then be tested by means of other methods, e.g., numerical simulations.
Of course, it is worth keeping in mind that this model, as any other model, has its own limitations. In particular, it breaks down at very small length scales that makes impossible to compare the results with those of perturbative QCD.
The paper is organized as follows. For orientation, we begin by setting the framework and recalling some preliminary results. Then, we consider the three-quark potential of collinear quarks. This allows us to compare the results with those obtained on the equilateral geometry and make predictions on what is expected to be universal (independent of a geometrical configuration of quarks) at short and long distances. We go on in Section III to discuss the quark-diquark and quark-quark potentials as the limiting cases of that obtained for the collinear geometry. Our goal here is to determine the leading terms in the interquark potential. In Section IV, we give an example of the three-quark hybrid potential. Here we also consider a relation between two and three hybrid quark potentials and make a prediction of universality for a gap between the potential and its hybrid at long distances. We conclude in Section V with a discussion of some open problems and possibilities for further study. Some technical details are given in the Appendices.
Ii Three-Quark Potential via Gauge/String Duality
In this section we will derive the three-quark potential for two different geometries. We start with an equilateral triangle geometry and then consider a collinear geometry. Although we mainly concentrate on the collinear geometry, as our basic example, the approach is equally applicable for any geometry. The Appendices A and B provide the necessary toolkit for doing so.
The static three-quark potential can be determined from the expectation value of a baryonic Wilson loop review (). The Wilson loop in question is defined in a gauge-invariant manner as , with the path-ordered exponents along the lines shown in Figure 1.
In the limit the expectation value of is given by
Here is called the three-quark potential (ground state energy) if the corresponding contribution dominates the sum as approaches infinity. In other words, it requires for any and any quark configuration. If so, then the remaining ’s are called hybrid three-quark potentials (excited state energies).
First, we take the ansatz for the background metric
Thus, the geometry in question is a one-parameter deformation, parameterized by , of a product of the Euclidean space of radius and a 5-dimensional compact space (sphere) whose coordinates are . In (2.2), there are the two free parameters and . Note that the first combines with from the Nambu-Goto action (A.1) such that then is to be fitted. Since we deal with an effective string theory based on the Nambu-Goto formulation 555This has some limitations. The reason is that the Green-Schwarz formulation similar to that for is still missing. So, we consider the model as an effective theory rather than an ultimate solution for QCD., knowing the background geometry is sufficient for our purposes. There are good motivations for taking this ansatz. First, such a deformation of leads to linear Regge-like spectra for mesons son (); q2 () and the quark-antiquark potential az1 () which is very satisfactory in the light of lattice gauge theory and phenomenology white (); carlucci (). Second, the deformation of is motivated by thermodynamics a-pis ().
Next we consider the baryon vertex. In static gauge we take the following ansatz for its action
where and are parameters. Note that the parameter is the same as in (2.2) so that only is a new one.
In what follows we will assume that quarks are placed at the same point in the internal space.666It is worth noting that this assumption makes the problem effectively five-dimensional and hence more tractable. From the five-dimensional point of view, the vertex looks like a point-like object (particle). Therefore the detailed structure of is not important, except the exponential warp factor depending on the radial direction. The motivation for such a form of the warp factor is drawn from the AdS/CFT construction, where the baryon vertex is a five-brane witten (). Taking a term from the world-volume action of the brane results in . This is, of course, a heuristic argument but it leads to the very satisfactory result that the lattice data of pdf3q (); suganuma3q (); suganuma3q-new () obtained on an equilateral triangle can be described by a single parameter a-Nq ().
One of the important differences between QCD and AdS/CFT is that QCD has got much further than AdS/CFT in addressing the issue of hybrid potentials hybrid-review (). The common wisdom is that those potentials correspond to excited strings isgur (). The structure of string excitations is quite complicated and is made of many different kinds of elementary excitations like vibrational modes, loops, knots etc. Adopting the view point that some excitations imply a formation of cusps, one can model them by inserting local objects (defects) on a string.777A similar idea was used in four-dimensional string models, but with a different goal, such as a description of linear baryons. For more discussion and references, see nesterenko (). Thus, what we need is an action for such an object. In static gauge we take the action to be of the form
where and are parameters. Again, is the same as in (2.2).
The form like (2.4) seems natural if one thinks of the defect as a tiny loop formed by a pair of baryon-antibaryon vertices connected with fundamental strings. In such a scenario, the expression (2.4) is based on an assumption that the strings do not change the radial dependence in (2.3). This is also a heuristic argument but it can be confronted with the lattice data of kuti (). For , the result is in good agreement hybrids (). Notice that (2.4) contains only one free parameter that makes it attractive from the phenomenological point of view.
Finally, we place heavy quarks at the boundary points of the five-dimensional space ( but at the same point in the internal space . We consider configurations in which each quark is the endpoint of the Nambu-Goto string, with the strings join at the baryon vertex in the interior as shown in Figure 2. For excited strings, we also place defects on them as shown in Figure 8. The total action of a system has, in addition to the standard Nambu-Goto actions, also contributions arising from the baryon vertex and defect. The expectation value of the Wilson loop is then
where is the minimal action of the system. Combining it with gives the the three-quark potential.888Like in AdS/CFT, this formula is oversimplified for various reasons, but it seems acceptable for the purposes of the effective string theory based on the Nambu-Goto formulation.
ii.2 Equilateral triangle geometry
As a warmup, let us give a derivation of the three-quark potential in the case when the quarks are at the vertices of an equilateral triangle of length a-Nq (). To this end, we consider the configuration shown in Figure 2. Here, gravity pulls the baryon vertex toward the boundary that allows the result to be consistent with the lattice data pdf3q (); suganuma3q (); jahn3q (); suganuma3q-new ().
The symmetry of the problem immediately implies that the projection of onto the plane is a center of the triangle and all the strings have an identical profile. In this case, a radius of the circumscribed circle is given by (A.16) and, as a consequence, the triangle’s side length is999We abbreviate to when this is not ambiguous.
where and . Note that and , with shown in Figure 10.
It is straightforward to compute the total energy of the configuration . The energy of a single string is given by (A.20). Then, using this expression and the expression (2.3) for the gravitational energy of the baryon vertex, we find
At this point, we can complete the parametric description of the three-quark potential for the equilateral triangle geometry. To this end, the gluing conditions at the vertex should be involved. When the triangle is equilateral, we impose the condition (B.10)
which is nothing else but the balance of force in the radial direction. Here we have abbreviated to . Note that a negative value of , as sketched in Figure 2, implies that . In this case, the gravitational force is directed in the downward vertical direction.
Here , with a solution to .
Thus the potential is given in parametric form by and . The parameter takes values on the interval . This is the main result of this section. It is a special case of the result announced in a-Nq ().
Now we wish to compare this result to the lattice data and see some of the surprising features of our model. But first let us look at the behavior of at short and long distances.
A simple analysis shows that is a monotonically increasing function on the interval , and that goes to zero as and goes to infinity as . Thus, to analyze the short distance behavior of , we need to find the asymptotic behavior of and near .101010For details, see the Appendix C. In this case, we restrict ourselves to the two leading terms that allows us to easily obtain the energy as a function of the triangle’s side length. Thus, at short distances the three-quark potential is given by
Here is the physical string tension. It remains universal and unaltered in all the cases: the quark-antiquark az1 (), hybrid hybrids (), three-quark potentials a-bar (), and also in the examples we consider below. It is notable that the constant terms at short and long distances are different. Because of scheme ambiguities, each of those has no physical meaning, but the difference
Having found the asymptotic behaviors at short and long distances, we can compare our model of the three-quark potential with the results of numerical simulations. We proceed along the lines of a-Nq (). First, we set and , i.e., to the same values as those of hybrids () used for modeling the quark-antiquark potentials of kuti (). Then the remaining parameter is fitted to be using the data of pdf3q () from numerical simulations of the baryonic Wilson loops. The result is plotted in Figure 3.
We see that the model reproduces the lattice data remarkably well with just one free parameter.
where is the quark-antiquark potential and denotes the distance between the vertices and . With our choice of normalization, the normalization constant for the energy of a single baryon configuration is equal to , while that of the quark-antiquark pair is (see (A.23)). This is consistent with the relation (2.16). Actually, it is the -law suggested in delta-law (). The analysis of pdf3q (); jahn3q (); a-Nq () shows that it is a good approximation to the lattice at distances shorter than .
The underlying physical picture is not necessarily very accurate for baryons. We suggest that the -law can be treated in a way that helps clarify the physics of strong interactions and at the same time complies with the lattice. The picture that emerges from this point of view is that at short distances the three-quark potential is described by a sum of two-body potentials. At length scales where a diquark can be treated as a point-like object, it is simply
|with a quark-diquark potential. This is plausible because in this case coincides with the quark-antiquark potential such that (2.17a) reduces to (2.16). Alternatively, it may be written as a sum over all quark-quark pairs|
Actually, in connection with the above analysis, some additional questions should be asked. To reach the conclusion that in the interval the three-quark potential is well approximated by a sum of two-body potentials we analyzed the short distance behavior of the expressions (2.6) and (2.7). It turns out that the first terms of the series (2.10) provides a good approximation in this interval, as can be seen from Figure 3. This is a model artifact. Does it imply that (2.10) is also reliable at shorter distances? This is unclear. Certainly, no resummation of perturbative series of a pure gauge theory is known. But it might occur that resummation will lead to the form of (2.10). In addition, the relation (2.15) between and is similar to that of the tree level result. What may be the reason? In the model we are considering is a function of the parameter , while is a constant. So, there may be any relation between those coefficients. The peculiar one is the result of fitting to the lattice. Again, to really explain why it is so, resummation of perturbative series is needed.
What happens at longer distances? We can reasonably expect that (2.12) is a proper approximation for . This is indeed the case a-Nq (). It is interesting that the Lüscher correction to (2.12) turns out to be negligible. In fact, such a behavior is also provided by the Y-law at long distances artu (); isgur ().
Thus, the physical picture we have is that our model incorporates two-body interactions, the -law, at short distances and a genuine three-body interaction, the -law, at longer distances. In string context, the two-body interaction is described by a single string stretched between a quark and a diquark, while the three-body interaction is the standard one. It includes three strings meeting at a common junction. Mathematically, is a complicated function of whose asymptotic behavior is described by the and -laws.
If one includes the Lüscher correction, like in the Y-law,111111This issue was raised by H. Suganuma. Here we replace by to better fit the data.
then the whole picture does not change and remains the same as before, except the transition between the two behaviors occurs at a slightly larger scale, of order as seen from Figure 3.
So far, we have tacitly assumed that gravity pulls the vertex toward the boundary. This is an important difference between our model and those in the literature devoted to duals of large (supersymmetric) gauge theories. There are obvious questions one can ask about what took place. Is the expansion really a good approximation? What is the origin for being negative? Is it a model artifact? Unfortunately, no real resolution of this problem will be proposed here. Our criterion is to mimic QCD and provide the basis for further calculations.
ii.3 Symmetric collinear geometry
Given the set of parameters that we have just fitted, it is straightforward to determine the three-quark potential for other geometries and make some predictions in the cases when there are no lattice data available. To get some intuition, we first consider the symmetric collinear geometry.
As before, we place the quarks at the boundary points of the five-dimensional space and consider a configuration in which each of the quarks is the endpoint of a fundamental string. The strings join at a baryon vertex in the interior as shown in Figure 4, on the left.
For convenience, the quarks are on the -axis such that , , and .
Since the quark configuration is symmetric under reflection through , the side strings have an identical profile and the middle string is stretched in the radial direction. Given this, we can use the general formula (A.16) to write
where and , with shown in Figure 10.
As before, the normalization constant is given by .
A complete description has to include the gluing conditions at the vertex. Such conditions are given by (B.12) with and . So we have
The analysis of in the two limiting cases, long and short distances, is formally similar to that of the previous section. The first step is to learn that is an increasing function on the interval . Moreover, it goes from zero to infinity. Having learned this, we can obtain the short distance behavior by expanding and near and then reducing the two equations to a single one
denotes the imaginary error function. As before, the constant terms and c turn out to be different.
Now we wish to make some predictions. Otherwise, this will remain as an academic exercise in gauge/string duality. So far there are no lattice data available for the symmetric collinear geometry, therefore we compare the results with the phenomenological ansätze and those from the previous section.
In Figure 5,
we display the three-quark potential obtained from equations (2.19) and (2.20). Because the phenomenological and -laws coincide for the collinear configuration, one could expect that the potential is well described by121212We set to fit on the interval from to .
However, as seen from Figure 5, this is not the case. The ansatz (2.27) seems very good for short and long distances but not for intermediate ones. The picture is similar to that of Figure 3, on the right. In other words, is a more complicated function of than the function given by equation (2.27).
Let us make some estimates. In units of and , we get
The last ratio shows that there is a deviation from the -law defined by (1.1).
Looking again at (2.28) and back to (2.15), we see that the coefficients in the short distance expansion are not the same. It is a bit surprising but true that for the triangle geometry the coefficients are larger131313We use the subscripts and to denote the following geometries: collinear and triangle.
Thus, these expansion coefficients are geometry-dependent.
The long distance expansion is also puzzling. Although the physical string tension is universal (geometry-independent), the constant terms are not. A simple estimate yields
Here we have assumed that the normalization constants are equal in both cases.
This is one of the important aspects of the model. The first that comes to mind when thinking about a possible explanation to what we have found is that it has a stringy origin. The point may be that a string junction affects the shape of the three-quark potential not only at long but even at short distances. As a result, the coefficients in the short distance expansion become dependent of the geometry in question. This is another objection to the -law where for any geometry. Of course, it would be very interesting to see what this means for heavy baryon spectroscopy because neither the nor -law takes this effect into account.
ii.4 Generic collinear geometry
Consider the symmetric configuration sketched in Figure 4 on the left. To get further, we need to modify it by moving the first quark to a new position, say at . Without loss of generality, we may move it closer to the origin such that . The resulting configuration will be a deformation of the initial one, as shown in Figure 4 on the right.
First, we will describe the gluing conditions that is the easier thing to do. From (B.12), we immediately obtain
Here we have abbreviated to .
Next we should mention an important subtlety that arises when one tries to move closer to . This subtlety is related to a flip of sign in , the slope of the first string at the vertex. As a result, the shape of the first string changes from that of Figure 10 on the right to that on the left. In the meantime, the other strings keep their shapes.
ii.4.1 Configuration with
We will now make the discussion more concrete. Consider a small deformation of the symmetric configuration. In that case, we still have and , but with , as shown in Figure 4 on the right.
If we consider the third string, then, in virtue of (A.6), can be written as an integral. We will thus have
Notice that is negative. The reason for this is a force balance at the vertex.
The energies of the first and second strings can be read from (A.20), while that of the third from (A.9). Combining those with the expression (2.3) for the gravitational energy of the vertex, we find the total energy
First, let us see what happens if . In this case, vanishes and becomes equal to . In addition, the expression (2.35) reduces to that of equation (2.20). So, the above formulas are consistent with those of subsection C.
Now, let us try to understand how the potential can be written parametrically as , , and . It is easy to see from (A.17) that
with . After a substitution into the first equation of (2.31), we find
Unfortunately, we do not know how to explicitly express one parameter as a function of two others. In practice it is convenient to choose the ’s as independent parameters and then solve equation (2.38) for numerically.
To summarize, the three-quark potential is given in parametrical form by , , and . The parameters take values on the interval and obey the inequality .
ii.4.2 Configuration with
There is one important situation in which a transition between the two types of profile occurs. This is the case . Although the calculations are simple, it is useful for the purposes of the present paper to have explicit formulas.
A helpful observation which makes the analysis easy is the following. If , then , which follows from (2.36). Hence there is only one independent parameter.
In the meantime, the expression for the total energy of the configuration becomes
For completeness, we include the corresponding equations for and in our analysis. A short calculation shows that (2.37) becomes
and (2.38) becomes
Even for this simplified form, we do not know how to explicitly express as a function of , but of course it can be done numerically.
Thus, in the case the potential can be parametrically written as , , and . The parameter takes values on the interval . But one important fact about such a configuration is that it occurs only at specific values of and , as follows from the last two equations.
ii.4.3 Configuration with
Finally, to complete the picture, we need to consider the configurations with . Such configurations exist for smaller . This can be done by slightly extending what we have described so far. The novelty is that the first string has a shape similar to that of Figure 10, on the left.
Plugging this into the second equation of (2.31) gives
This equation can be used to solve, at least numerically, for under the given values of and .
At this point it is worth noting that at all the above formulas reduce to those of the previous section. This fact can be used as a self-consistency check of the results.
To summarize, the three-quark potential is given in parametric form by , , and . The parameters and take values on the intervals and , where is a solution to equation (2.43) at .
ii.4.4 What we have learned
In Figure 6, we display our result for the potential obtained from the three different types of configurations. For simplicity, we restrict to the
case . Since the function is symmetric under the exchange of and , one can easily obtain for the opposite case by a reflection in the line .
One lesson to learn from this example is that in the limit , along the curve given by , the value of goes to infinity, while that of stays bounded. Indeed, it follows from (2.39) that is an increasing function of whose limiting value at is given by
The same conclusion is true for the configuration with . The separation is bounded from above by , while is not. This can also be seen from Figure 6.
Another lesson is that at large separations between the quarks, when both and go to infinity, the potential behaves like
with c given by (2.26). To deduce this, we consider the configuration with and use the fact that this limiting case corresponds to a small region near the point in the parameter space . As in subsection C, we expand the right hand sides of (2.33), (2.34), and (2.35) near and then reduce the three equations to the single equation (2.49).
Besides looking for possible applications to phenomenology, it would be very interesting to confront our predictions with future numerical simulations.
Iii More on Limiting Cases
In Section II, we took the collinear geometry as the basic example to illustrate the use of the string theory technique in practice. There is more to say if we consider the situation when two quarks get close to each other. In this case, a simplified treatment is to consider a system of two particles, the quark and diquark. The analysis of Section II allows us to shed some light on that situation.
iii.1 Quark-diquark potential
It is known that in the limit of large quark masses QCD has a symmetry which relates hadrons with two heavy quarks to analogous states with one heavy antiquark wise (). This implies that we should be able to reproduce the heavy quark-antiquark potential from the results of Section II.
To show this let us consider a collinear configuration with is fixed and .141414One can think of as a diquark size. In this case, we should pick the configuration with . A short analysis shows that in terms of the parameters what we need is to take the limit with fixed.
These formulas mean that in this limit the angle becomes a right angle.
Notice that is a continuously increasing function of . It vanishes at and develops a logarithmic singularity at .
Taking the limit in (2.45) requires some care. Because of divergences at , we need a regulator that renders finite. The right procedure, which is consistent with what we did before, is to first replace by its lower bound. It is given by as it follows from the lower bound on . Then the regularized expression for is given by
A little calculation reveals that the expansion in powers of takes the form
where is given by (C.10). Notice that the singular term contains divergences coming from an infinitely large quark mass as well as self-energies of the vertex and strings. It is important that the coefficient does not depend on and, as a consequence, on . This allows one to deal with the divergence in a fashion similar to the standard treatment of power divergences in Wilson loops.
Subtracting the term and letting , we get a finite result
Here is a normalization constant which is scheme dependent.
iii.2 Quark-quark potential
We will describe a static potential which represents a part of the inter-quark interaction inside heavy diquarks of small size. As above, we pick the configuration with . However, we now consider the case when is small but non-zero, and , as sketched in Figure 7.
This means that one quark is placed far away from the others such that its impact on the diquark can be easily assessed. From (2.34) and (2.44) it follows that in terms of the parameters and we need to consider a small region near the point in the parameter space . The difference with what we have done in subsection A is that in the expansions in powers of we should keep track of small terms which give rise to a term linear in in the expression for the total energy.
With the help of the expansions (C.8), one readily sees that for small and large the three-quark potential behaves like
The ’s and ’s are given by equations (C.9) and (C.10), respectively. Notice that in (3.7) the difference between c and is written as an integral which is equal to that obtained in the quark-antiquark potential at long distances hybrids (). This is consistent with the symmetry wise ().
There is an important subtlety that arises when one tries to extract the quark-quark potential from the above expression. This subtlety is related to the fifth dimension. The point is that in the limit we are considering the -coordinate of the vertex turns out to be . In other words, the vertex is located between the and quarks, as shown in Figure 7. In contrast, from the four-dimensional perspective it should be located at because in the case of the collinear geometry the vertex (the Fermat point) coincides with .151515Note that in the model we are considering a projection of on the boundary coincides with the Fermat point only in the infrared limit, when the quarks are far away from each other a-bar (). Thus, we have to subtract from the term proportional to and a constant term. Such a constant term has to include a contribution from the integral on the right-hand side of the last equation in (3.7). These two terms represent the quark-diquark binding energy at large distances. As a result, we get
with a normalization constant. This is the main result of this section.
Having derived the static quark-quark potential, we can make some estimates of and . Using the fitted value of , we get
Here we have used from the quark-antiquark potential. Thus, our estimates suggest that the ratio is close to , and that the effective string tension inside a heavy diquark is approximately 20 percent less than the physical tension.
In phenomenology161616See, e.g., Ebert () and references therein., the quark-quark potential is usually related to the quark-antiquark potential by Lipkin’s rule. This is an ansatz which says that Lipkin (). Our estimate of is relatively close to it, whereas that of is somewhat larger.
It is also worth noting that in four dimensions a simple estimate based on a string model yields a larger value for the ratio . It is pdf-diquark (). The caution here is that the reason of decreasing the string tension is geometric and nothing similar happens once three quarks are placed on a straight line.
Iv An Example of Hybrid Three-Quark Potential
Here we will describe a special type of hybrid three-quark potentials, namely, the potentials whose gluonic excitations are similar to those in the potential for the quark-antiquark. For simplicity, we illustrate only the case when one string is excited. The question we want to address is this: what is the impact of gluonic excitations on the physical picture we found for the ground state?
Consider the configuration of Figure 2. We are going to modify it by exciting one string via gluonic excitations so that the quarks remain at the same positions, i.e., at the vertices of the equilateral triangle. Unfortunately, we know of no efficient way to describe all possible excitations kuti (); bali-pineda () within our model. The only exception is the so called excitations which have zero angular momentum. From the five dimensional point of view, such excitations may be modeled by cusps on the Nambu-Goto strings hybrids (). Importantly, cusps are allowed only in the radial direction that has no impact on smoothness of a four-dimensional picture.
One way to implement this is to insert a local object, called the defect, on a string hybrids (). Technically, it is quite similar to what we did before for the baryon vertex, with the only difference that two strings join at a defect in the interior as shown in Figure 11. By inserting the defect on the third string of Figure 2, we construct a new configuration shown in Figure 8. It is an example of a string picture
(five-dimensional) of hybrid baryons. Notice that the configuration has a reflection symmetry with respect to the -axis.
Let us begin our analysis with the gluing conditions. As follows from the discussion in the Appendix B, the only condition to be satisfied at the defect is that
with . One can obtain it from equation (B.7) by simply replacing with and with . The conditions at the vertex are a bit more involved since there are two equations for force equilibrium, in the and -directions. From (B.14) it follows that
In Section II, we observed that a string shape can change as a quark attached to its endpoint approaches another one. In the example we are analyzing, there is a similar story. A heuristic understanding of this can be obtained without resorting to explicit calculations, as follows. For large , the configuration looks pretty much like that of Figure 2 because a local defect has a little impact on very long strings. This implies that is negative. For small , the strings become short, and we might expect the opposite to happen. It does happen when the defect has the dominant effect. In this case, shown in Figure 8, the third string is stretched along the -axis from the boundary at to a position of the defect at , whereas the vertex is located at such that . It is obvious that is now positive. This means that a flip of sign in occurs at some value of .
iv.0.1 Configuration with
We begin with the case that the non-excited strings have negative slopes at the vertex. Equation (A.16) tells how to write the distance between the points and via an integral. It is
Note that in order for the integrals to remain well-defined and finite, the condition must hold that , where is a solution of equation . In particular, in the limit one has . In this limit, also goes to infinity with going to which is defined by (2.9). For