Towers of positive parity excited baryons and their mixing in the large $N_c$ limit
Cintia Willemyns, Carlos Schat
TL;DR
This paper analyzes the structure and mixing of positive parity excited baryons in the large N_c limit, revealing a simplified pattern of state mixing and degeneracy into six towers labeled by K, with implications for quark model state interactions.
Contribution
It demonstrates that in the large N_c limit, excited baryon states organize into six degenerate towers and mixing is simplified to only involve states with the same K label, parameterized by two constants.
Findings
States form six degenerate towers labeled by K.
Mixing occurs only among states with the same K.
Configuration mixing can be described by two parameters.
Abstract
We consider configuration mixing for the nonstrange positive parity excited baryons in the and quark model multiplets contained in the band. Starting from the effective mass operator for these states we show by an explicit calculation that in the large limit they fall into six towers of degenerate states labeled by . We find that the mixing of the quark model states is much simpler than what is naively expected. To leading order in only states carrying the same label can mix, which implies that for the spin-flavor states we started with configuration mixing can be parameterized by just two constants, and .
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Towers of positive parity excited baryons and their mixing in the large limit
Cintia Willemyns
Departamento de Física, FCEyN, Universidad de Buenos Aires and IFIBA,
CONICET, Ciudad Universitaria, Pabellón 1, (1428) Buenos Aires, Argentina
Carlos Schat
Ökumenisches Gymnasium zu Bremen,
Oberneulander Landstr.143a, 28355 Bremen, Germany
Abstract
We consider configuration mixing for the nonstrange positive parity excited baryons in the and quark model multiplets contained in the band. Starting from the effective mass operator for these states we show by an explicit calculation that in the large limit they fall into six towers of degenerate states labeled by . We find that the mixing of the quark model states is much simpler than what is naively expected. To leading order in only states carrying the same label can mix, which implies that for the spin-flavor states we started with configuration mixing can be parameterized by just two constants, and .
I Introduction
The quark model in its different versions has been a useful tool in exploring the spectrum and properties of excited baryons, as well as for testing different model assumptions on the effective interactions between quarks Capstick:2000qj ; Klempt:2009pi ; Crede:2013kia . Although the model interactions between constituent quarks are QCD inspired, a clear connection to the fundamental theory of the strong interactions remains elusive to this date. Recent successes of lattice QCD calculations Edwards:2011jj ; Edwards:2012fx seem to validate the classification scheme of baryon states in multiplets, as put forward by early quark model studies. These numerical lattice calculations have the advantage of being based on the fundamental theory of the strong interactions, but lack the simplicity of an analytic approach, that provides a physical picture in terms of effective degrees of freedom, effective interactions and symmetries. Such an analytic scheme to study the phenomenology of baryons and their excited states, that also makes contact with QCD, can be obtained starting from the large number of colors () limit of QCD 'tHooft:1973jz Witten:1979kh . For a review on the relevance of the large limit for gauge theories, see also Ref. Lucini:2012gg .
In the large limit it has been shown that the spin-flavor symmetry for ground state baryons can be justified from the contracted symmetry derived from consistency relations for pion-nucleon scattering Gervais:1983wq Gervais:1984rc Bardakci:1983ev Dashen:1993as Jenkins:1993zu . The predictions of this symmetry for the masses and the couplings explain some of the successes of the non-relativistic quark model Dashen:1993jt Carone:1993dz Luty:1993fu Dashen:1994qi . Even more important than that, the breaking of the spin-flavor symmetry can be studied systematically in a expansion using quark operators, establishing a close connection between QCD and the quark model Carone:1993dz Luty:1993fu Dashen:1994qi , see Refs. Manohar:1998xv ; Jenkins:1998wy ; Lebed:1998st for a pedagogical introduction.
The operator construction of the expansion was later extended to study the masses of the negative parity excited baryons Goity:1996hk Pirjol:1997bp Pirjol:1997sr Carlson:1998vx Schat:2001xr Goity:2002pu with great success. The strong and electromagnetic decays of these states belonging to the multiplet, as well as the masses and decays of baryon resonances in other spin-flavor multiplets were also studied in the expansion (see Matagne:2014lla for a recent review, and references therein), establishing a comprehensive framework to study the phenomenology of excited baryons at the physical value . Lattice studies of excited baryons offer further tests of the expansion by providing predictions for all states in a spin-flavor multiplet and probing the dependence of observables on the quark masses Jenkins:2009wv Fernando:2014dna . They are even starting to explore the mass spectrum for values larger than three Cordon:2014sda DeGrand:2016pur .
It is important to note that the classification scheme for baryon resonances based on irreducible representations (irreps) of , grouped into excitation bands , is based on the quark model and cannot be justified from QCD. Physical states appear as admixtures of different irreps, something known as configuration mixing, which so far has been neglected in phenomenological studies that use the expansion.
In the large limit a different symmetry structure is present at leading order and states are classified according to irreducible representations of the contracted symmetry , dubbed as towers and labeled by , leading to degeneracies in the mass spectrum of excited baryons Pirjol:1997bp ; Pirjol:1997sr . These predicted degeneracies in the large limit have been verified by explicit calculations, first in Pirjol:2003ye ; Cohen:2003tb for the nonstrange states that constitute the band and later also for the nonstrange states of the band Matagne:2011sn , but up to now the effect of configuration mixing, which is not suppressed Goity:2004pw ; Goity:2005gs , has been neglected in an explicit calculation of the spectrum using the effective quark operators.
Here we consider the entire space of states spanned by the states and , with . We restrict ourselves to the nonstrange states and show by an explicit calculation that configuration mixing preserves the tower structure of the mass spectrum in the large limit. We also study the mixing pattern, which in the basis of states turns out to be much simpler than what would have been expected starting from the spin-flavor basis of the quark model.
In order to perform the calculation including the leading order effects of configuration mixing we need to extend the construction of the leading order mass operator. To allow for transitions, we will introduce a generic spatial operator in place of the angular momentum operator used in previous studies Carlson:1998vx . This is similar to the construction of the transition operators for the decay processes Goity:2004ss , as we need the most general operators that mix the multiplets to leading order.
In this work we present the most general form of the leading order mass operator that incorporates configuration mixing and the explicit expressions obtained for its matrix elements between states of the band. Our results provide important consistency checks on the contracted symmetry predictions for the masses and mixings in the large limit, and the correctness of the usual construction of the effective mass operators in terms of core and excited quark operators Carlson:1998vx .
The paper is organized as follows: In Sec. II we present the states, in Sec. III we present the effective mass operator, in Sec. IV we give the explicit form of the mass matrices for and states. In Sec. V we discuss the spectrum and the mixing pattern we obtain. We finally conclude in Sec. VI. In App. A we give the general expressions for calculating the matrix elements of the leading mass operators and in App. B we give the explicit expressions of the matrix elements for arbitrary .
II The states
The positive parity, orbitally excited modes of an three quark system differ from each other in their orbital angular momentum and their behaviour under permutations. The spin-flavor symmetry is a useful classification scheme for these states. Each type of permutational symmetry of three objects, symmetric , mixed symmetric and antisymmetric corresponds to an multiplet, , and , respectively. For the nonstrange members of these multiplets, the spin-flavor symmetry is only broken by the spin-dependent interactions. Here and in what follows we will concentrate on the nonstrange members of each multiplet. In the case of harmonic forces between quarks the five multiplets (), () and () are degenerate and constitute the harmonic oscillator band. We do not consider the antisymmetric states in this work, as their relevance for the observed physical states at is not clear yet. In the generalization of the lowest-energy multiplets to arbitrary the additional quarks are taken in a completely symmetric spin-flavor combination.
In order to identify the nonstrange and physical states ( and ) with the large states, it is useful to keep in mind the usual decomposition that labels the states
[TABLE]
where , with the quark spin of the state. It should always be clear if stands for the spin, or if it denotes the symmetric irrep of the permutation group. To count the nonstrange and states contained in the , irreducible representations of the permutation group for arbitrary it is helpful to recall that for the symmetric representation the spin and isospin of the states are related by , while for the mixed symmetric irrep the spin is obtained from the vector sum , so that
[TABLE]
We see that in there appear states with higher spin that are absent in . These are ghost states that decouple from the physical states in the limit, as has been noticed in Pirjol:2003ye Cohen:2006en and we will also see explicitly here when discussing the matrix elements of App. B. After coupling with the orbital angular momentum we obtain the states
[TABLE]
where we indicate , the total spin of the nonstrange states , as given by the vector sum . The primes indicate the different values of quark spin , e.g. correspond to with , respectively, while correspond to with , respectively.
In Eq. (3) we show all the states that we will consider to compute the mass spectrum. There are 30 states distributed as 11 -states and 19 -states, where 11 of the -states are ghost states. That reduces the number of physical states at to 19 states.
The representations reduce in the limit to the ,, and quark model multiplets. We use the prime on to distinguish it from the ground state baryons , usually labeled as in the limit.
In a quark model calculation all states with same mix, giving rise to large mixing matrices and a complex pattern, see e.g. Ref. Isgur:1978wd . In the large limit the mixing pattern is much simpler, something that only becomes clear after classifying the states in a different way, according to the irreps of the contracted spin-flavor symmetry . These irreps are labeled by , which relates and as , so that in the symmetric irreps and in the mixed symmetric ones Pirjol:1997bp Pirjol:1997sr . In contrast to the spin-flavor states for arbitrary of Eq. (3), the 30 large states carry an additional label and can be grouped as “tower states”
[TABLE]
where each tower state is in general in an admixture of the spin-flavor states shown in Eq. (3). The nonstrange states in belong to a tower, the ones in appear in the decomposition of states, the ones in contribute to states and finally the states appear in the decomposition of states. As we will show by an explicit calculation in Sec. V, in the basis only states with the same label mix. This implies that in the large limit the tower structure that was first found by performing explicit calculations within a single spin-flavor irrep Pirjol:2003ye Cohen:2003tb Matagne:2011sn is also preserved when including the effect of configuration mixing.
In the next Section we will present the simple leading order in mass operator from where this mixing pattern follows.
III The mass operator
The leading order mass operator needed for our calculation is obtained by slightly generalizing the construction of Ref. Carlson:1998vx as follows. As explained in detail in Ref. Carlson:1998vx , the large states can be constructed as product states of a symmetric core of quarks and an -th quark in a proper linear combination, so that the and irreps with the desired permutation properties are obtained. The operators contributing to the mass operator at different orders in can then be constructed from the generators acting on the symmetric core and on the quark that was singled out, and the orbital angular momentum operator . Here we will replace the orbital angular momentum operator by a generic spatial operator to allow for configuration mixing of spin-flavor representations with different . This is similar to the construction of the effective operators for decay processes Carone:1994tu Goity:2004ss , where we also needed to describe transitions between states with different .
The leading order Hamiltonian including operators up to order that will correctly describe all possible mixings in our configuration space, spanned by the states given in Eq. (3), has then the following form
[TABLE]
where and , stand for the irreps . The coefficients are order and encode the details of the spatial wave function, as has been shown explicitly within a single spin-flavor representation by different matching calculations Collins:1998ny Pirjol:2007ed Galeta:2009pn Pirjol:2010th Willemyns:2015hgy . They take different values on different spin-flavor irreps . In our case we also have off-diagonal matrix elements, so that the coefficients depend on both the and irreps that are mixing. For the diagonal matrix elements we have and we use the notation . The unit operator only contributes to diagonal matrix elements.
The general expressions for the matrix elements of the operators in Eq. (5) are given in App. A following closely the notation of Ref. Carlson:1998vx .
IV The mass matrices
In this Section we present the explicit form of the mass matrices we obtain by computing the matrix elements of Eq. (5), after expanding in and taking the large limit. The matrix elements for arbitrary are given in App. B. We also give the expressions for the corresponding eigenvalues and eigenstates. The reader can skip through this Section and continue reading Sec. V, where we work out an explicit example that better illustrates the relevant points of the discussion.
It should be noted that in the coefficients we absorb common group theoretical numerical factors and the reduced matrix elements of the operator, as appearing in Eqs. (152, 159) or more explicitly, as in Tables 1, 2, 3 and 4 of App. B. We do not distinguish them from the original appearing in Eq. (5) to keep the notation simple.
IV.1 The states
The states with the same can mix among each other. We have three , four , three and one states. Their mass matrices are given below. The 11 mass eigenvalues for the large , states will be labeled as and their degeneracies will be discussed in the next Section. Table 1 in App. B shows all the matrix elements for the nucleons at finite .
In the large limit the mass matrix for the states in the {, , } basis is
[TABLE]
The mass matrix and all mass matrices that follow are symmetric. We only show the upper right half of the matrix to keep the expressions more readable. The eigenvalues are and . Their explicit dependence on the coefficients is given in the next Section. The corresponding eigenstates are
[TABLE]
where each row vector on the right hand side indicates the composition of the eigenstate, e.g. for the second eigenstate we have
[TABLE]
where can be expressed in terms of the coefficients. We will use this matrix notation to show the composition of the eigenstates throughout the rest of the paper. This is very convenient to make the mixing pattern manifest. In the limit of no mixing () the eigenstates are normalized. In the general case there is still a normalization factor of that has to be taken into account.
The mass matrix for states in the large limit is given in the {, , , } basis by
[TABLE]
We denote the eigenvalues as and . The eigenstates of this matrix are
[TABLE]
For the states in the {, , } basis we obtain
[TABLE]
with eigenvalues and and eigenstates
[TABLE]
Finally, the matrix element for the large state in is
[TABLE]
IV.2 The states
The states with the same can mix among each other. We have four , six , five , three and one states. Their mass matrices are given below. The 19 mass eigenvalues for the large , states will be labeled as and their degeneracies will be discussed in the next Section.
Table 2 in App. B shows the matrix elements for at finite . In the large limit the mass matrix for the states in the {, , , } basis is
[TABLE]
with eigenvalues and . The corresponding eigenstates are
[TABLE]
Table 3 shows the matrix elements for at finite . For the states in the { , , , , , } basis we obtain
[TABLE]
with eigenvalues , and and eigenstates
[TABLE]
Table 4 in App. B shows the matrix elements for , and at finite . For the states in the { , , , , } basis we have
[TABLE]
with eigenvalues , and and eigenstates
[TABLE]
The matrix for the states in the large limit in the { , , } basis is
[TABLE]
with eigenvalues , and , and with eigenstates
[TABLE]
Finally, the matrix element for the large state in is
[TABLE]
V The large spectrum
The diagonalization of the and mass matrices presented in the previous Section leads to 30 mass eigenvalues. In the large limit we find as a result of our explicit calculation the remarkable result that the masses assume only six different values, leading to a highly degenerate spectrum. The 11 and 19 masses are grouped in six energy levels as follows
[TABLE]
The “tower masses” correspond to the tower states listed in Eq. (4) and the degeneracy in the spectrum reflects the symmetry present in the large limit. Notice that there are two towers with labels and , their masses are unrelated by the symmetry. The explicit expressions we obtain for the tower masses in terms of the coefficients and result from the diagonalization of the mass matrices given in Sec. IV and can be written in compact form as
[TABLE]
where
[TABLE]
Given the complexity of the mass matrices from where we started, these expressions are surprisingly simple. It is possible to understand this by looking at the general structure of the corresponding eigenstates shown in the previous Section: only some subset of spin-flavor states are mixed among each other in the large limit, namely those with same assignment. As we will show next by working out an explicit example, all our results can be understood as a two level mixing. To make this manifest it is useful to write the tower masses we obtained in terms of the mass eigenvalues that we would have in the absence of configuration mixing
[TABLE]
We see that for and for the states we have
[TABLE]
where and are the matrix elements that mix the two states. For , i.e. and , the mixing is strong and the energy levels get corrections due to configuration mixing.
This mixing pattern, which is not obvious at all when starting from the mass matrices written in the spin-flavor basis, can be made manifest by a change of basis. The states constitute a good example to see this, as they have the largest mass matrix, of dimension six and given by Eq. (LABEL:example32), where all states appear as eigenstates. To find the change of basis we need, we first compute the eigenstates in the absence of mixing by setting in Eq. (LABEL:example32). In our matrix notation they are given by the row vectors of the following expression
[TABLE]
This provides us the change of basis matrix . We obtain the large mass matrix in the basis { , , , , , } as follows
[TABLE]
where we only show matrix elements that are non-zero.
The eigenstates in this basis are given now by the rows of the matrix below and take the simple form
[TABLE]
Finally, from we recover as the rows of the eigenstates in the spin-flavor basis as given by Eq. (78).
[TABLE]
The explicit expressions for that relate them to the mixing matrix elements are
[TABLE]
As it is clear from Eqs. (130,137), only the two and towers mix through the mixing matrix elements and , respectively. This explains why in the large limit all the mixing that can occur among the spin-flavor states of Eq. (3) can be expressed in terms of only two parameters, and it confirms that the effective mass operator given by Eq. (5) correctly accounts for the symmetry structure expected in large QCD.
VI Conclusions
We have extended the large analysis of excited baryons to include configuration mixing. In particular, in this paper we studied the configuration mixing of the symmetric and mixed symmetric spin-flavor irreps that belong to the band. Rather than the spin-flavor symmetry of the quark model, in the large limit we have a contracted symmetry , which gives rise to numerous mass degeneracies and also fixes the mixing pattern among the states. Degenerate states fill irreps (“towers”), which are labeled by . They contain an infinite number of states with increasing spin and isospin. Here we restricted ourselves to two flavors and to the low spin and isospin states that are identified with the physical states at .
We found by an explicit calculation that, in contrast to the complex mixing pattern of spin-flavor irreps in the quark model due to hyperfine interactions (see e.g. Ref. Isgur:1978wd ), in the large limit the mixing pattern is governed by the symmetry: only states carrying the same label mix. For the states considered, this mixing can be described by just two parameters related to the mixing of the mixed symmetric states with and ( and ) and the mixing of the symmetric and mixed symmetric states with ( and ).
We performed the calculation of the mass spectrum by slightly extending the construction of Ref. Carlson:1998vx and using a common mass operator for all states, Eq. (5), expressed in the quark operator basis. We verified explicitly that this simple leading order effective mass operator correctly describes the configuration mixing pattern expected from the symmetry present in the large limit. We also checked explicitly that the inclusion of configuration mixing preserves the large tower structure in the spectrum of positive parity excited baryons: The 11 -states and 19 -states are the lowest isospin members of six degenerate towers of large states labeled by . The matrix elements presented in App. B show that ghost states that only exist for decouple from the physical states. This decoupling is a general feature of large calculations Pirjol:2003ye , that was also pointed out in the meson-baryon scattering picture Cohen:2006en . Another important point to note is that only the presence of core operators makes the mixing of symmetric and mixed-symmetric states possible. A mass operator constructed solely in terms of symmetric generators would not mix spin-flavor states in different irreps of the permutation group.
All these are non-trivial checks of the correctness of the core-excited quark picture and the operator construction in terms of generators that started the program of applying the large expansion to the study of excited baryons Goity:1996hk Carlson:1998vx . Most importantly, the leading order calculation presented here provides the first step towards a systematic inclusion of configuration mixing effects at subleading order in the large analysis of the phenomenology of excited baryons. The predicted large spectrum and the configuration mixing pattern could also be checked in the future by lattice calculations, providing a useful guide for the ongoing explorations of the baryon spectrum at arbitrary values DeGrand:2016pur .
Acknowledgements
C.S. and C.W. thank the kind hospitality of the Institut für Theoretische Physik II, University of Bochum, were this work was initiated. We both thank Rodolfo Sassot for his support and encouragement.
Appendix A Calculation of matrix elements
After replacing the orbital angular momentum operator by a generic spatial operator , we obtain the generalization of equation (A7) of Ref. Carlson:1998vx :
[TABLE]
where denote a symmetric () or mixed symmetric () irrep and the coefficients with and are given by
[TABLE]
In all other Sections we use to label the symmetric irrep . Here, for the sake of clarity, we use the label to distinguish it from the we also use to denote the total spin of the quarks. The rank two tensor operator in the effective mass operator accounts for the mixing of and states. It generalizes Eq. (A9) in Ref. Carlson:1998vx to
[TABLE]
The reduced matrix elements and are unknown and can be absorbed in the operator coefficients of the expansion.
Appendix B Explicit matrix elements for arbitrary
We list in this Appendix all the explicit matrix elements for the operators , and , for finite . We defined , and , which contain the reduced matrix elements of the generic operator. Note that the ghost states decouple from the physical states through factors, see also Cohen:2006en for a related discussion.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1(1) S. Capstick and W. Roberts, Prog. Part. Nucl. Phys. 45 , S 241 (2000) [nucl-th/0008028].
- 2(2) E. Klempt and J. M. Richard, Rev. Mod. Phys. 82 , 1095 (2010) [ar Xiv:0901.2055 [hep-ph]].
- 3(3) V. Crede and W. Roberts, Rept. Prog. Phys. 76 , 076301 (2013) [ar Xiv:1302.7299 [nucl-ex]].
- 4(4) R. G. Edwards, J. J. Dudek, D. G. Richards and S. J. Wallace, Phys. Rev. D 84 , 074508 (2011) [ar Xiv:1104.5152 [hep-ph]].
- 5(5) R. G. Edwards et al. [Hadron Spectrum Collaboration], Phys. Rev. D 87 , no. 5, 054506 (2013) [ar Xiv:1212.5236 [hep-ph]].
- 6(6) G. ’t Hooft, Nucl. Phys. B 72 , 461 (1974).
- 7(7) E. Witten, Nucl. Phys. B 160 , 57 (1979).
- 8(8) B. Lucini and M. Panero, Phys. Rept. 526 , 93 (2013) [ar Xiv:1210.4997 [hep-th]].