The holographic model of QCD we work with is constructed from an intersecting D4/D8-brane system [2,3]. The D4-branes wrap around a circle on which a SUSY-breaking boundary condition is imposed, and yield gluons of gauge group on the worldsheet at low energy compared with the circle radius . D8- and -branes are placed at the anti-podal points of the SUSY-breaking circle. Quantization of D4–D8 and D4– strings gives left- and right-handed quarks in the fundamental representation of , respectively. This system has a manifest chiral symmetry.

The holographic dual of this model is formulated by replacing the D4-branes with a solution of type IIA supergravity with a nontrivial dilaton [20]: (2.1) (2.2)

Here, denote the indices of the four-dimensional Minkowski spacetime where QCD is defined, is the metric of a unit , and .³ is the coordinate of the SUSY-breaking circle. In addition, there exist units RR four-form flux over the : (2.3)

It is useful to define

The metric in Eq. (2.1) is defined in the decoupling limit, where the dependence on , the string length, factorizes as a prefactor. As a consequence, the string theory on this background is independent of . This allows one to set (2.4) in units of so that . (See Ref. [4] for more details on this point.) It follows that the stringy excitation modes have mass of and may be neglected at low energies for .

Assuming ,⁴ the D8-branes can be regarded as probes with no backreaction to the metric in Eq. (2.1) taken into account. It has been shown [2] that the D8- and -brane pairs interpolate with each other smoothly at , and the resultant D8-brane worldvolume is specified by the embedding equation . In this setup, the mesons are identified with the open strings attached on the D8-branes that can move along the direction.

In order to incorporate baryon degrees of freedom into the model, we introduce a baryon vertex [15], which is given by a single D4-brane wrapping around at . We refer to this D4-brane as a D4 in order to distinguish it from color D4-branes. The RR flux in Eq. (2.3) forces open strings to extend between the D8-branes and the baryon vertex. This configuration is identified with a single baryon. It is argued in Ref. [2] that this brane system is realized as an instanton solution on the D8-brane worldvolume theory. By analyzing the moduli space quantum mechanics corresponding to this instanton solution, Refs. [7,8] showed that aspects of the baryon dynamics are reproduced from this model both qualitatively and quantitatively. One of the limitations in this analysis, however, is that describing a baryon vertex as a classical solution of the gauge theory on the D8-branes is valid only for low-lying baryons. This is because the gauge theory is an effective theory of the D8-branes with only the massless degrees of freedom taken into account. In addition, the moduli space approximation only keeps light degrees of freedom in the fluctuations around the soliton solution. In fact, these are the main reasons why the analysis in Ref. [7] leads to only baryons with the spin and isospin equal to each other for the case. For the purpose of obtaining more general baryons, we thus have to consider stringy effects in the baryon vertex.

It is highly difficult to make a full quantization of a string that propagates in the curved background of Eq. (2.1) in the presence of the RR flux in Eq. (2.3). In order to circumvent this problem, we follow Ref. [4]. We first take the large- limit, where the curved background can be approximated with a ten-dimensional flat spacetime. Then, the baryon configuration reduces to a system with D8-branes and a D4-brane with open strings stretched between them in the flat background. For a technical reason, it is useful to formally T-dualize the system in the direction. The D8/D4-brane system gets mapped to the D9/D5-brane configuration shown in Table 1. The - and -directions are labeled by indices and , respectively. The -directions span , which results from the that is decompactified for . Quantization of a 9-5 and 5-5 string is performed most easily by using a light-cone quantization, where the light-cone coordinate is taken to be . The manifest spacetime symmetry of the brane system is .

We first study the light-cone quantization of a 9-5 string. The equations of motion (EOM) of the worldsheet boson in the 6789-directions is solved in terms of Fourier expansions with an integer modding, while that in the 123-directions in terms of those with a half-integer modding, because of the boundary conditions imposed on them. For the worldsheet fermions in the Neveu–Schwarz (Ramond) sector, the solutions of the EOM in the 6789-directions are written in terms of Fourier expansions with a half-integer (integer) modding, while those in the 123-directions are written in terms of those with an integer (half-integer) modding. It follows that the Neveu–Schwarz (NS) ground state is degenerate due to the fermion zero modes, belonging to a spinor representation of . The Ramond (R) ground state is degenerate too, and belongs to a spinor representation of . We label an irreducible representation of by , where and are the spin of and , respectively. The (integer spin) representation of is labeled by Young tableaux as , , , , etc., where the subscripts denote the dimensions. Then, the low-lying 9-5 string states in the NS sector with the Gliozzi–Scherk–Olive (GSO) projection imposed are summarized in Table 2. Although it is not manifest in the light-cone quantization, the six-dimensional Lorentz symmetry on the -brane worldvolume allows one to summarize the massive excitations into the irreducible representations of the little group , which contains as a subgroup. Table 3 gives a list of the low-lying 9-5 string states in the NS sector in terms of .

We next study the mass spectrum of a 5-5 string using the light-cone quantization. The worldsheet bosons can be Fourier expanded with an integer modding for both - and -directions. The worldsheet fermions in the NS (R) sector can be Fourier expanded with a half-integer (integer) modding for the - and -directions. The physical ground state in the NS sector is massless and given by and . Here, is tachyonic, being GSO-projected out. The first excited 5-5 string states in the NS sector that survive the GSO projection are given by acting on with a set of the creation operators with total excitation number equal to . These have the mass squared and are listed in Table 4.

As in the 9-5 string states, any massive state of the 5-5 string is summarized into an irreducible representation of . It is found that the first excited states with in Table 4 are rearranged as where the Young tableaux are those of . In fact, these states are obtained as the decomposition of of , which is the same as the first excited 9-9 string states considered in Ref. [4].

Reference [4] discusses that the D4/D8-brane system has discrete symmetries that are identified with those in massless QCD. The parity and charge conjugation are given by (2.6) respectively, where is spacetime involution along the directions, is a worldsheet parity, and is a spacetime fermion number in the left-moving sector of a string worldsheet. A -brane placed at ⁵ is invariant under , while it is mapped to a -brane under . To see the latter, note that when the action generated by is gauged, a background has an O6-plane at , and it is known that the -brane has to be paired with a -brane in the presence of the O6-plane [21]. This is consistent with the fact that the baryon is invariant under the parity, up to sign of the wavefunction, while it is mapped to an anti-baryon under the charge conjugation.

In order to see how acts on the NS ground state of the 9-5 string considered in Sect. 2.2, it is useful to write the parity operator in a bosonized form. We note that the worldsheet fermions of a 9-5 string can be expressed using free worldsheet complex scalars and as

Parity acts on the worldsheet fermions as which in turn induces the transformation of as (2.7) with a choice of . The vertex operator corresponding to the NS ground state of a 9-5 string is given by (2.8) up to a ghost sector that is invariant under , with for and for . Therefore, the parity transformation in Eq. (2.7) acts as the chirality operator on the spinor representation of up to a sign ambiguity. We choose and in Eq. (2.7) such that and are parity even and odd, respectively. With this convention, the parity of the proton and the neutron turn out to be even. This is consistent with the conventional choice of the parity in QCD, in which the parity of quarks are chosen to be even. For a -brane, which represents an anti-baryon, since the GSO projection is opposite, the parity of the the NS ground state is odd. This is again consistent with the fact that the anti-quarks have odd parity.

Then, the parity of the excited states can be computed by using the transformation laws of the creation operators that act on the ground state. Namely, and with are parity-odd and and with are parity-even operators.

In addition to these symmetries, the D4/D8-system admits a discrete symmetry that has no counterpart in QCD. This is called -parity⁶ and defined as (2.9)

As discussed in Ref. [4], both the quarks that originate from 4-8 and 4- strings in the open string picture and the gluons that originate from the 4-4 strings are even under -parity. This implies that all the states that can be interpreted as the genuine color singlet states of QCD have to be -parity even as well. There are -parity-odd states in the spectrum of the bound states in our model. However, such states are artifacts of the model which do not have counterparts in QCD, and we will not consider them in the following.

Assuming that the -brane is placed at , one can show that the -brane is invariant under the -parity . To see this, we note that maps the D4 to a and maps it back to a D4.

For the purpose of reading off the -parity of an open string state, it is useful to work in the T-dualized description used in Sect. 2.2. When the -direction is T-dualized, is mapped to (2.10) where is the T-dualized coordinate of . This is simply a 180 rotation in the 9- plane and it is easy to find the action of from the representation of listed in Table 3 and Eq. (2.5).

In addition to the -parity discussed above, we can also use the isometry of in the background to single out the open string states that could be used to construct a baryon in QCD. It is easy to see that both quarks and gluons are invariant under this , and hence the baryons in QCD have to be an singlet. In the flat spacetime limit, the requirement of the invariance amounts to demanding that the states be -singlet and carry no momentum along the 6789-directions. In the T-dualized picture, we should also impose the condition that the momentum along is zero, since the original direction is not compactified and there is no winding mode along . Therefore, among the open string states obtained in Sect. 2.2, we only consider the states that are invariant under and the -parity , and carry no momentum along the directions.

We first derive the 9-5 string states that meet the conditions discussed in the last subsection. The requirement of invariance implies that the R sector must be removed because all the states in the R sector are -nonsinglet. It follows from the -parity condition that among the -singlet NS states, only those with an even number of the spacetime index are allowed. The NS ground state satisfies these conditions. For the first excited states (those with ) listed in Table 3, only the state with is allowed. From the second excited states with , we pick up

Finally, we set the momenta along the direction to zero, which is equivalent to omitting the dependence of the corresponding wavefunctions on and . These results are summarized in Table 5, where we also list the representation (spin) of , which is related to the subgroup of by . Note that symmetry appears only in the flat spacetime limit and it is broken to due to the -dependence of the background. The masses of these states in the flat spacetime limit are proportional to the excitation number as (2.11) where we have used the relation in Eq. (2.4).

The quantum field corresponding to the 9-5 massless state is denoted by , which reduces to a function of time only as discussed above. Here, is the spin index for and is the index for the flavor symmetry.

Next, we discuss the 5-5 string states. As in the 9-5 string case, all the R states are non-singlet under and thus ruled out. The NS massless states that satisfy all the conditions are given by () only. The corresponding fields are denoted as . Again, these fields reduce to functions of . Among the first excited states with listed in Eq. (2.5), the following states satisfy all the conditions: (2.12)

Note here that there are two states, and one of them comes from in Eq. (2.5) with two indices. The masses are given by (2.13)

The results for the 5-5 strings are summarized in Table 6.

The action for the open string states attached on the baryon vertex -brane is written as (3.1) where is the Lagrangian for the ground states while is the part that involves the excited states. is derived in Ref. [14] as (3.2) where is a complex matrix variable with a spin () index and a flavor () index , () is a real four-component variable, and the gauge field on the -brane; and correspond to the ground state for 8-4 strings and 4-4 strings, respectively. The value of represents the position of the -brane in the four-dimensional space parametrized by . The dot denotes the time derivative as and (3.3) is the covariant derivative. The potential terms are given by (3.4) (3.5)

Here, is the Pauli matrix and we have used the notation for a complex matrix . , , , , and are constants; , , and are related to the number of colors and the ’t Hooft coupling as⁸ (3.6)

The potential in Eq. (3.4) is obtained by integrating out the auxiliary fields in Ref. [14]. The condition is equivalent to the ADHM constraints for the ADHM construction of the self-dual instanton solution. The first term of in Eq. (3.5) represents the fact that the -brane is attracted to the origin in the -direction due to the curved background. The second and third terms in Eq. (3.5) are added rather phenomenologically. is chosen to be in Ref. [14] so that the second term in Eq. (3.5) recovers the corresponding term in the soliton approach [7]. The third term in Eq. (3.5) was not present in Ref. [14], but one could add it to have more flexibility. We treat and as unspecified parameters for the moment.⁹

is the Lagrangian with the excited states obtained in Sect. 2. It can be written as (3.7) where and denote the fields corresponding to the excited states created by 8-4 strings and 4-4 strings, respectively. We call these “massive fields” in the following. The indices and label all the excited states, and and are the mass squared of these states given in Eqs. (2.11) and (2.13), which are of order . The are complex fields that couple with the gauge field with the unit charge, while the are real fields, which are neutral under the gauge symmetry. gives the interaction terms for the massive fields that may also contain massless fields. We put the overall factor by convention so that all the fields have the dimension of length. Since the evaluation of the interaction terms including the massive states is beyond the scope of this paper, we assume that the contribution from is small as far as the qualitative features of the baryon spectrum are concerned. In Sect. 3.7 we argue that though most of the possible terms in are suppressed in the large- limit, there are some terms that could survive even in the large- limit.

To quantize our system, we follow the approach developed recently in Ref. [13]. We take the gauge and impose the EOM for (Gauss law constraint) as a physical state condition on the Hilbert space. The Gauss law constraint can be written as (3.8) where (3.9)

These and correspond to the charge associated with the phase rotation symmetries and , respectively, which are approximate symmetries that exist when the interaction term is neglected. The Gauss law constraint in Eq. (3.8) represents the fact that open strings have to be attached on the -brane, and is interpreted as the number of excited open strings associated with .¹⁰

It is interesting to note that the Gauss law constraint in Eq. (3.8) implies that the spin of the baryon state is half-integer or integer for odd or even , respectively.¹¹ Indeed, the wavefunction for the baryon state satisfying the Gauss law constraint in Eq. (3.8) is of the form¹² (3.10)

Here, is a -invariant wavefunction that is written only through invariants. Because 8-4 strings ( and ) and 4-4 strings ( and ) carry half-integer and integer spin, respectively, can only have an integer spin and the spin of the state in Eq. (3.10) is .

Omitting , the Hamiltonian in the gauge is given by (3.11) with (3.12) (3.13) where , , , and are the momenta conjugate to , , , and , respectively. in Eq. (3.13) is simply a collection of harmonic oscillators associated with the excited open string states obtained in Sect. 2. The quantum mechanics for in Eq. (3.12) has been studied in Refs. [13,14], though the part with is treated in a different way in the following.

We are particularly interested in the cases with , in which is a complex matrix and can be parametrized as (3.14) where is the unit matrix. transforms as the (complex) four-dimensional vector representation of , where and with corresponds to the spin and isospin groups, respectively. The kinetic term for in Eq. (3.12) is written as (3.15) where is the Laplacian n .

Using the relations (3.16) where and , the ADHM potential can be written as (3.17)

The minimum of this potential is parametrized by (3.18)

Note that together with correspond to the collective coordinates of the one-instanton configuration considered in Ref. [7]. More explicitly, (3.19) corresponds to the size and the orientation of the instanton solution, respectively.¹³ One way to include the components that are orthogonal to the directions along Eq. (3.18) is to parametrize as¹⁴ (3.20) with (3.21) where () are the generators of acting on , which are chosen to be pure imaginary anti-symmetric matrices. See Appendix A for the explicit forms. One can easily show that (3.22) where and the ADHM potential becomes (3.23)

Note that the parametrization in Eq. (3.20) has a redundancy induced by the transformation (3.24)

When the wavefunction is written in terms of , , and instead of , we should impose the invariance of the wavefunction under this transformation.

In this paper we consider the cases that takes small values so that does not generate an additional mass term for . One important observation is that the kinetic term of the Hamiltonian in Eq. (3.12) contains a term as (3.25) for [see Eq. (3.29)]. Since is the generator of the phase rotation of , we have the relation (3.26) in the quantum mechanics. When we consider the cases with , has to be of because of the Gauss law constraint in Eq. (3.8). In such cases, the term in Eq. (3.25) gives a potential of the form (3.27) up to a numerical factor in the large- limit, which has the effect of pushing to have a larger value. Let be the value of that minimizes the effective potential given by adding this term to in Eq. (3.5). Assuming that the third term in Eq. (3.5) is either negligible or of the same order as Eq. (3.27), i.e. , we find , which is consistent with the results in Refs. [7,9,10]. We will shortly obtain an explicit expression for in the large- limit [see Eq. (3.32)], and show that it has the effect of generating a large mass term for in the next subsection.

Now, let us figure out which terms in are important in the large- limit. First, we decompose as , and regard , , and to be order 1 variables,¹⁵ which means that (3.28)

Then, the leading () and subleading () terms in the Laplacian turn out to be (3.29)

Keeping these terms, the Hamiltonian for and becomes (3.30) where (3.31)

Here we have imposed the condition that minimizes the potential for , which reads (3.32)

The Hamiltonian in Eq. (3.30) is a sum of the harmonic oscillators for and .

A few comments are in order. First, coincides with in Eq. (3.6) for used in Ref. [14], which is consistent with Ref. [7]. Second, the value of in Eq. (3.32) agrees with that in Ref. [14] when and . However, as pointed out in Ref. [14], it is larger than the value in Refs. [7,9,10] by a factor of . One can adjust the value of as to match with the value in Refs. [7,9,10]. Third, on the right-hand side of in Eq. (3.31), the first term is of order , while the other terms are of order 1. Recall that the masses of the excited open string states are . This means that although arises as the ground states (the open string states with ), it acquires a large mass comparable to the massive excited states due to the ADHM potential in Eq. (3.4) together with the Gauss law constraint in Eq. (3.8).

As argued in Sect. 3.3, the Hamiltonian is reduced to a collection of harmonic oscillators in the large- limit, which can be easily solved. Then, the masses of the baryons are obtained as (3.33) where , , , , , and are non-negative integers corresponding to the excitation levels of the harmonic oscillators associated with , , , , , and , respectively; , , and are given in Eqs. (3.6) and (3.31); and are the masses for the corresponding open string states given in Eqs. (2.11) and (2.13), respectively; and is a (-dependent) constant whose classical value is (3.34) where the first term comes from the tension of the -brane placed at and the second term is the first term in Eq. (3.30). It also contains the contributions from the zero-point energies of all the fields in the system, including those neglected in Sect. 2. Since there are infinitely many fields involved, it is not easy to evaluate it explicitly.¹⁶ For this reason, we leave as an unknown parameter and focus on the mass differences.

Note that the mass in Eq. (3.33) implicitly depends on the value of through the parameters and . Because the Gauss law constraint in Eq. (3.8) implies that is related to and by (3.35) these parameters are state dependent.

As a consistency check, one can show that the formula in Eq. (3.33) agrees with the leading-order terms in the baryon mass formula obtained in Ref. [7] when and . In fact, the baryon mass formula in Ref. [7] can be written as (3.36) (3.37) where is related to the spin and isospin as . The dependence appears because the Laplacian in the -space, (3.38) contains the Laplacian on parametrized by , denoted by , whose eigenvalue is . In Eq. (3.29) we have neglected this contribution, though it also appears in if we keep the term.

In Ref. [7] was chosen to be odd (or even) for odd (or even) by hand, so that the spin of the baryon obtained in the soliton approach is consistent with that in the quark model, as it is also the case for the Skyrme model with . In our case, this condition is replaced with , which automatically follows from the fact that the eigenfunction of is given by (3.39) where is a traceless symmetric tensor of rank , and appears in the wavefunction as an overall factor . As explained around Eq. (3.24), the wavefunction has to be invariant under the transformation in Eq. (3.24), which implies .

As discussed above, the Hamiltonian of the one-baryon quantum mechanics is a collection of infinitely many harmonic oscillators in the large- limit. The eigenfunction can be written as a product of a function of , , , and and a function of , , and as (3.40)

We call and wavefunctions for the massless and massive sectors, respectively.¹⁷

The massless sector wavefunction can be written as (3.41) where is the wavefunction for the plane wave with momentum , is defined in Eq. (3.39), and , , and are the eigenfunctions of the harmonic oscillators for , , and with the excitation numbers , , and , respectively. We set in the following for simplicity. We also use the bra–ket notation as (3.42)

Here, is included in the notation to remember that the massless sector wavefunction also depends on .

If is trivial, agrees with the large- limit of the wavefunction obtained in Ref. [7]. As shown in Ref. [7], has a degeneracy of that corresponds to the states in the representation of . The mass formula in Eq. (3.33) appears to be independent of , because the dependence is a subleading effect in the large- limit. Upon taking finite- effects into account, we expect that the energy is an increasing function of , as in Ref. [7].¹⁸

Note that since is parity odd, has parity . As mentioned in Sect. 3.4, coincides with for and hence the states with and are degenerate. This implies a degeneracy between parity-even and -odd states for those with . This could be a hint toward an understanding of the parity-doubling phenomenon in the excited baryons.¹⁹

is a wavefunction for a three-dimensional harmonic oscillator with respect to (). The energy contribution in the mass formula in Eq. (3.33) for this part is , with (3.43)

The degeneracy is (3.44) and the eigenspace for a given can be decomposed into a direct sum over the states with isospin or for even or odd , respectively. For example, for the state with and , the massless wavefunction has spin and isospin .

The wavefunction for the massive sector is given by the eigenfunctions of the harmonic oscillators associated with , , and , which is written in the bra–ket notation as (3.45)

In order to classify these states, we introduce the notation (3.46) which we call the level of a baryon, with (3.47) where and are the excitation numbers for and given in Tables 5 and 6, respectively.²⁰ It will become increasingly complicated to extract the spin and isospin for the states with larger . We will give some explicit examples of the baryon states in Sect. 4.

Here, we make some comments on in Eq. (3.7). First, we classify depending on the order of the massive fields multiplied and assume that each term contains at least two massive fields so that the trivial configuration is a solution of the EOM for the massive fields. Note that the overall factor in the Lagrangian in Eq. (3.7) is proportional to , which reflects the fact that the leading terms of the open string action are given by the string worldsheet of disk topology. As always, we neglect the loop corrections of string theory which are suppressed by . Then, is order 1 in the expansion with fixed . If one writes down the Lagrangian using canonically normalized massive fields (3.48) one finds that all the terms with more than two massive fields are suppressed in the large- limit. Therefore, the terms in that survive in the large- limit are quadratic with respect to the massive fields. For the same reason, it should not contain , , or . Then, the possible terms consistent with the gauge symmetry are schematically written as (3.49) with properly contracted indices and their complex conjugates. As we have seen in Sects. 3.3 and 3.4, is treated as an order 1 variable and these terms may appear even in the large- limit.

One might think that these terms are perhaps suppressed for large . Unfortunately, however, the answer is no. Consider, for example, a term proportional to for . As observed in Sect. 3.4, the leading term in is . Recall that all the fields have the dimension of length in our convention. To have the correct dimensions, there should be an appropriate number of or in the coefficient of Eq. (3.49) to saturate the correct dimension of . A possible term is of the form (3.50) which shifts the mass for in the same order as the original mass term. This is the same mechanism as the mass generation of discussed in Sect. 3.4. may also induce mixing terms as well, and the diagonalization of the mass matrix may become very complicated. Because we do not know the explicit form of we are not able to evaluate it explicitly, and leave the detailed analysis including for future research.

Here, we focus on the baryons listed in Table 7, which are the lightest baryons with and () found in the experiments.

These baryons have been considered to be described by an excited (rotating) open string with a quark and diquark pair attached on the two end points [26–29].²¹ An analogous object in our model is a -brane with 8-4 strings in the ground state and only one 8-4 string being excited as increases. The aim of this subsection is to discuss whether our model gives us plausible predictions assuming that this is the correct interpretation. More explicitly, the lightest one in Table 7, which is the nucleon (proton or neutron), is identified with , , and .²² The excited nucleons with spin in Table 7 are interpreted as the highest spin state among those with , , , and for some . These states are most likely to be the lightest state among the highest spin states with isospin 1/2 for each level. Let us discuss if the quantum numbers and the masses of these states are consistent with the experimental data with this interpretation.

The states we consider are labeled uniquely by the level introduced in Eq. (3.46). Let denote the baryon mass for a given . The nucleon corresponds to the case , which has and mass given by (4.1)

Here, is considered to be a function of and , as argued in Sect. 3.5. As it is technically hard to compute the quantum , we regard it as an unknown parameter.

For , because , the massless sector has vanishing spin and isospin. Then, the total spin of the excited baryons with is fixed by the massive sector. Let the excitation number of the excited 8-4 string be , which is to be identified with the level for the excited nucleons as seen before. For each , the highest spin states are contained in the states of the form (4.2) which belongs to the spin representation of . Here, we have included the flavor index to show that it is an isospin state for . Decomposing this under the vector-like subgroup , one finds that the highest spin is given by . The parities of these excited nucleons are given by , because the state in Eq. (4.2) has parity and the massless sector is parity even for . Therefore, the spin, isospin, and parity for the excited nucleon states constructed above are consistent with those in Table 7.

The baryon mass formula in Eq. (3.33) implies that the masses for these excited nucleons with states are (4.3) where . This formula can be recast as a formula for spin as a function of mass : (4.4)

It has been observed that, when the spin is plotted as a function of the mass squared , the excited nucleon states listed in Table 7 lie on a linear trajectory that satisfies (4.5) with and . Our formula in Eq. (4.4) is a nonlinear function with respect to , and one would think it disagrees with the observation. However, choosing (4.6) we get the plot shown in Fig. 1, which shows that it can fit the data reasonably well. Due to the nonlinear term in Eq. (4.4), the trajectory in Figure 1 is curved toward the left and the value of mass squared for becomes significantly smaller compared to that of the nucleons (proton or neutron). This is, however, not a problem of the formula in Eq. (4.4) as it is derived for the states with . Our expression for the nucleon mass is given in Eq. (4.1). Though we are not able to predict its value, this observation suggests that the difference between Eq. (4.1) and , (4.7) is positive, as expected.²³

We emphasize that the values in Eq. (4.6) should not be considered to be an accurate estimate, because we have neglected all the and corrections, as well as the possible contributions from the interaction term in Eq. (3.50) for the massive fields. Nevertheless, let us make a few comments here on the value of . In Refs. [2,3], the parameters and were chosen to be (4.8) to fit the experimental values of the -meson mass and the pion decay constant. If we use these values and the relation in Eq. (2.4), we obtain , which is a bit small compared with the value in Eq. (4.6). On the other hand, the value of evaluated from the Regge slope of the -meson trajectory is . In Ref. [8], the -meson Regge behavior is analyzed theoretically using the same holographic model of QCD as in the present paper. It was argued there that the -meson trajectory has some nonlinear corrections similar to that in Eq. (4.4), and the value of that fits well with the experimental data turned out to be around . The value of in Eq. (4.6) is close to neither of these values, though it is not too far from them. It is important to resolve this discrepancy by making a more accurate estimate of .

Note that the slope of the linear Regge trajectory in Eq. (4.5) for the excited nucleons is very close to that of the -mesons. This is one of the motivations for conjecturing that both of them are described by open strings with some particles attached on the end points, as investigated in Refs. [26–29]. Our description is similar to these models in that only one of the strings attached on the baryon vertex gets excited while the rest remain in the ground state. This system may be approximated with a single open string by regarding the effect of the baryon vertex as a massive end point. However, a clear distinction from the models in Refs. [26–29] is that the mass of the end point in the present model is of and considered to be much heavier than the energy scale determined by the string tension. In fact, it is not difficult to verify that a rotating open string with a massive end point of mass has a classical energy that reduces in the heavy end point limit to (4.9) which agrees with Eq. (4.4) up to an additive constant and the contributions from the zero-point energy in .²⁴ We note that the difference between the mass formula in Eq. (4.4) and Eq. (4.9) is due to quantum corrections.

In this subsection we show some examples of low-lying excited baryons that are obtained in a manner explained in the previous sections. For simplicity, we set and . The states in the massless and massive sectors are denoted by and , respectively, where only nonvanishing quantum numbers are indicated explicitly for notational simplicity.

We start from the sector . This sector is constructed only from the excitation of a single 8-4 string with , because any 4-4 excited state has .²⁵ The corresponding field belongs to under with the subscript denoting parity, and yields a harmonic oscillator with angular frequency given by . The condition is satisfied when or . The excited states and have the same energy eigenvalue of , with spin given by . We consider only the former, because this leads to so that Eq. (3.34) shows that the corresponding massless sector has less energy compared to that with , which corresponds to . As is even, the massless sector is allowed to have . We first consider the case , which yields the lightest state in the massless sector , which belongs to the trivial representation and has even parity because . Hence, the tensor product state of with has spin given by (4.10) where the superscripts represent the isospin. The tensor product state of with decomposes under as

It is straightforward to decompose all these states in terms of ; the results are summarized in Table 8. Note that appearing in the first row is identified with N(1520) in the previous section.

We next turn to discussing the states. This is possible only when or . The first condition is further divided into two cases: (i) and (ii) for . Note that the 8-4 massive states with are given by the three states with listed in Table 5. Again, we focus on the lightest states in each case, implying that we pick up only and with . The second condition is solved by with , with the rest of the excitation numbers set to zero. Note that the 4-4 string states labeled by are given by the excited states with shown in Table 6. As no excitation is made by any 8-4 string mode, this case gives .

Let us now work out the baryon states for the above three cases. For the first case, the state in the massive sector is given by , which transforms under as (4.11)

The massless sector for this case is characterized by . We are thus allowed to set as the lightest state, whose spin is given by . By taking the tensor product of this state with , we find the following baryon states: (4.12)

For the second case, we take the massive sector state to be with . This corresponds to . We can take . The state has a trivial spin so that the tensor product of this state with has the same spin as . The massless sector with has spin given by . The tensor product of this state with is easy to evaluate for each .

Finally, the massive sector for the third case is characterized by the four states with . As noted before, this corresponds to so that odd is allowed. We pick up , which is expected to give the lightest state among those with odd , and take its tensor product with . Note that any 4-4 string state has a vanishing isospin. The same computation is easy to perform for the next-lightest state with .

All the results are summarized in Table 9. Decomposing these states in terms of is straightforward.

Now we discuss possible identifications of the states listed in Tables 8 and 9 with the baryons found in experiments. Because we have not been able to derive the dependence in the baryon mass formula of Eq. (3.33), we have to rely on some qualitative arguments. Our guiding principles are as follows. First, we expect that the states with the same , , and are nearly degenerate. Second, for a given , the states with are heavier than those with . Third, for a given , the mass is an increasing function of both and except for the state with listed in the first row of Table 9, which is expected to be heavier than the others according to the baryon mass formula in Eq. (3.33).²⁶

The predictions for the low-lying excited baryons with are summarized in Table 10, whose data are taken from Tables 8 and 9. We will not attempt to relate the states with in this table to those in the baryon summary table in Ref. [24] here, because these might be regarded as excited states with nonvanishing , , and without excitations in the massive sector.²⁷ Note that the states with and in the first and third rows of Table 10 are identified with N(1520) and N(1680), respectively, in Sect. 4.1. The state at is expected to be the lightest state with this quantum number and hence it may be identified with N(1675), which is the lightest baryon with the same quantum number listed in the baryon summary table. Then, the states at the second row are expected to have mass nearly equal to N(1675). A natural candidate for one of them is N(1700).²⁸

As for the states, we find that the states in the third row of Table 10 are expected to have mass nearly equal to N(1680). A natural candidate for one of them is N(1720).²⁹ Since the fourth row has larger values of and compared with the third row, the state in the fourth row is expected to be heavier than N(1680) and N(1720). A natural candidate for it is N(1860), though this state has not been established in experiments. If this is the case, the states in the fourth row are expected to be nearly degenerate with N(1860). These states could be identified with N(1900) and N(1875). The baryon states in the fifth row contain a state with . The only baryon with this quantum number listed in the baryon summary table is N(1990), though this is not considered to be established. Then, the and states in the fifth row of Table 10 could be identified with N(2000) and N(2040), respectively, which are again poorly established in experiments.

Unfortunately, the identification we have made is not a clear one-to-one correspondence. There is more than one candidate state in the model for many of the baryons listed in the baryon summary table. In particular, the degeneracy of the states in Table 10 does not match the experimental data perfectly. Furthermore, as mentioned in the footnotes, some of the baryons may be identified with the states that are not listed in Table 10. This lack of a one-to-one correspondence could in part be because all the excited baryons we consider are unstable resonances (for finite ), and many of them, in particular the heavier ones, are probably not easy to identify in experiments. Furthermore, some of the states in Tables 10 and 11 could be artifacts of the model. Although, as discussed in Sect. 2, we have imposed invariance with respect to the symmetry and -parity to get rid of artifacts, we are not able to show that this is sufficient to exclude all of them. It is expected that incorporation of full corrections into the baryon mass formula makes the artifacts of the model infinitely heavy in the () limit with kept fixed. However, the extrapolation to the small- regime is a notoriously difficult problem in the holographic description, because we have to deal with all the stringy corrections in a highly curved spacetime. A similar observation was also made in Ref. [4]. We leave as an open problem the study of a dictionary between the theoretical predictions and the experimental data in more detail.

We also examine the mass spectrum of baryons with isospin . The theoretical predictions for this case are summarized in Table 11, whose data is taken from Tables 8 and 9. It is natural to identify the and states in the first and second rows of Table 11 with the lightest baryons having the same quantum numbers listed in the baryon summary table, which are and , respectively. This suggests that the states in the second row of Table 9 are nearly degenerate with . A good candidate to be identified with one of these states is . However, this identification is problematic: although our formula in Eq. (3.33) suggests that the states are significantly heavier than the states, and are nearly degenerate and is even lighter than .