Effective multi-quark interactions with explicit breaking of chiral symmetry

# Effective multi-quark interactions with explicit breaking of chiral symmetry

A. A. Osipov111Email address: osipov@nu.jinr.ru, B. Hiller222Email address: brigitte@teor.fis.uc.pt and A. H. Blin333Email address: alex@teor.fis.uc.pt Centro de Física Computacional, Departamento de Física da Universidade de Coimbra, 3004-516 Coimbra, Portugal
###### Abstract

In a long distance Lagrangian approach to the low lying meson phenomenology we present and discuss the most general spin zero multi-quark interaction vertices of non-derivative type which include a set of effective interactions proportional to the current quark masses, breaking explicitely the chiral and symmetries. These vertices are of the same order in counting as the ’t Hooft flavor determinant interaction and the eight quark interactions which extend the original leading in four quark interaction Lagrangian of Nambu and Jona-Lasinio. The assignements match the counting rules based on arguments set by the scale of spontaneous chiral symmetry breaking. With path integral bosonization techniques which take appropriately into account the quark mass differences we derive the mesonic Lagrangian up to three-point mesonic vertices. We demonstrate that explicit symmetry breaking effects in interactions are essential to obtain the correct empirical ordering and magnitude of the splitting of certain states such as for the pseudoscalars and in the scalar sector, and achieve total agreement with the empirical low lying meson mass spectra. With all parameters of the model fixed by the spectra we analyze further a bulk of two body decays at tree level of the bosonic Lagrangian: the strong decays of the scalars , , , , as well as the two photon decays of , and mesons and the anomalous decays of the pseudoscalars , and . Our results for the strong decays are within the current expectations and the pseudoscalar radiative decays are in very good agreement with data. The radiative decays of the scalars are smaller than the observed values for the and the , but reasonable for the . A detailed discussion accompanies all the results.

###### pacs:
11.30.Rd, 11.30.Qc, 12.39.Fe, 12.40.Yx, 14.40.Aq, 14.65.Bt

## I Introduction

A long history of applying the Nambu – Jona-Lasinio (NJL) model in hadron physics shows the importance of the concept of effective multi-quark interactions for modelling QCD at low energies. Originally formulated in terms of the gauge invariant nonlinear four-fermion coupling Nambu:1961 (); Vaks (), the model has been extended to the realistic three flavor and color case with breaking six-quark ’t Hooft interactions Hooft:1976 (); Hooft:1978 (); Bernard:1988 (); Bernard:1988a (); Reinhardt:1988 (); Weise:1990 (); Vogl:1990 (); Weise:1991 (); Takizawa:1990 (); Klevansky:1992 (); Hatsuda:1994 (); Bernard:1993 (); Dmitrasinovic:1990 (); Birse:1996 (); Naito:2003 () and an appropriate set of eight-quark interactions Osipov:2005b (). The last ones complete the number of vertices which are important in four dimensions for dynamical chiral symmetry breaking Andrianov:1993a (); Andrianov:1993b ().

The explicit breaking of chiral symmetry in the NJL model is described by introducing the standard light quark mass term of the QCD Lagrangian (light means consisting of and quarks), e.g. Ebert:1986 (); Bijnens:1993 (). The current quark mass dependence is of importance for several reasons, in particular for the phenomenological description of meson spectra and meson-meson interactions, and for the critical point search in hot and dense hadronic matter, where it has a strong impact on the phase diagram Kunihiro:2010 (). The values of the current quark masses are determined in the Higgs sector of the Standard Model. In this regard they are foreign to QCD and, at an effective description, can be included through the external sources, interacting with the originally massless quark fields. This is why the explicit chiral symmetry breaking (ChSB) by the standard mass term of the free Lagrangian is only a part of the more complicated picture arising in effective models beyond leading order Gasser:1982 (). Chiral perturbation theory Weinberg:1979 (); Pagels:1975 (); Gasser:1984 (); Gasser:1985 () gives a well-known example of a self consistent accounting of the mass terms, order by order, in an expansion in the masses themselves. In fact, extended NJL-type models should not be an exception from this rule. If one considers multi-quark effective vertices, to the extent that suppressed ’t Hooft and eight-quark terms are included in the Lagrangian, certain mass dependent multi-quark interactions must be also taken into account.

The aim of the present work is precisely to analyze these higher order terms in the quark mass expansion. Our consideration proceeds along the following steps. We start from the three-flavor NJL-type model with self-interacting massless quarks. The chiral symmetry of the Lagrangian is known to be dynamically broken to its subgroup at some scale , with being one of the model parameters. There is also explicit symmetry breaking due to the bare quark masses , which are taken to transform as under . Since the Lagrangian contains, in general, an unlimited number of non-renormalizable multi-quark and -quark interactions (scaled by some powers of ), we formulate the power counting rules to classify these vertices in accordance with their importance for dynamical symmetry breaking. Then we bosonize the theory by using the path-integral method. The functional integrals are calculated in the stationary phase approximation and by using the heat kernel technique. As a result one obtains the low-energy meson Lagrangian. At last we fix the parameters of the model by confronting it to the experimental data. In particular, we show the ability of the model to describe the spectrum of the pseudo Goldstone bosons, including the fine tuning of the splitting, and the spectrum of the light scalar mesons: or , , , and .

The coupling constants of multi-quark vertices, fixed from mass-spectra, enter the expressions for meson decay amplitudes and lead to a bulk of model predictions. It is interesting to note that certain multi-quark vertices of the model encode implicitly in the couplings of the tree level bosonized Lagrangian the signature of and more complex quark structures which are elsewhere obtained by considering explicitly meson loop corrections, tetraquark configurations and so on Jaffe:1977 (); Black:1999 (); Wong:1980 (); Narrison:1986 (); Beveren:1986 (); Latorre:1985 (); Alford:1998 (); Achasov:1984 (); Isgur:1990 (); Schechter:2008 (); Schechter:2009 (); Close:2002 (); Klempt:2007 (). It seems appropriate, therefore, to examine the possible physics opportunities connected with the discovery and study of such multi-quark structures in hadrons. For instance, by calculating the mass spectra and the strong decays of the scalars, one can realize which multi-quark interactions are most relevant at the scale of spontaneous ChSB. On the other hand, by analyzing the two photon radiative decays, where a different scale, associated with the electromagnetic interaction, comes into play, one can study the possible recombinations of quarks inside the hadron. We will show, for example, that the meson couples with a large strength of the multi-quark components to the two kaon channel in its strong decay to two pions, but evidences a dominant component in its radiative decay. As opposed to this, the and mesons do not display an enhanced component neither in their two photon decays nor in the strong decays.

There are several direct motivations for this work. In the first place, the quark masses are the only parameters of the QCD Lagrangian which are responsible for the explicit ChSB, and it is important for the effective theory to trace this dependence in full detail. In this paper it will be argued that it is from the point of view of the expansion that the new quark mass dependent interactions must be included in the NJL-type Lagrangian already when the breaking ’t Hooft determinantal interaction is considered. This important point is somehow completely ignored in the current literature.

A second reason is that nowadays it is getting clear that the eight-quark interactions, which are almost inessential for the mesonic spectra in the vacuum, can be important for the quark matter in a strong magnetic background Hiller:2007 (); Gatto:2010 (); Gatto:2011 (); Frasca:2011 (); Gatto:2012 (). The simplest next possibility is to add to that picture a set of new effective quark-mass-dependent interactions, discussed in this work. Such feature of the quark matter has not been studied yet, but probably contains interesting physics.

Further motivation comes from the hadronic matter studies in a hot and dense environment. It is known that lattice QCD at finite density suffers from the numerical sign problem. This is why the phase diagram is notoriously difficult to compute “ab initio”, except for the extremely high density regime where perturbative QCD methods are applicable. In such circumstances effective models designed to shed light on the phase structure of QCD are valuable, especially if such models are known to be successful in the description of the hadronic matter at zero temperature and density. Reasonable modifications of the NJL model are of special interest in this context and our work aims also at future applications in that area.

The paper is organized as follows. In section II the effective Lagrangian in terms of quark degrees of freedom and bosonic sources with specific quantum numbers is derived using a classification scheme which selects all possible non-derivative vertices according to the symmetries of the strong interaction and which are relevant at the scale of spontaneous chiral symmetry breaking. It is then shown that this scheme can be equally organized in terms of the large counting rules, which in turn allow to attribute to the couplings of the interactions encoded signatures of and more complex structures involving four fermions. We obtain in this section also that a set of interactions lead to the Lagrangian specific Kaplan-Manohar ambiguity associated with the current quark masses.

In section III we proceed to bosonize the multi-quark Lagrangian in two steps. First, we introduce in section III-A a set of auxiliary scalar fields. By these new variables the multi-quark interactions can be brought to the Yukawa form that is quadratic in Fermi fields. Consequently one obtains a Gauss-type integral over quarks, and a set of integrals over auxiliary fields. The latter are evaluated by the stationary phase method. We obtain here the vertices up to the cubic power in the meson fields, needed for the study of the meson spectra and of the two-body decays. Then, in section III-B, we integrate over quark fields. The arising quark determinant of the Dirac operator is a complicated non-local functional of the collective meson fields. We calculate it in the low-energy regime by using the Schwinger-DeWitt technique, based on the heat kernel expansion. In this approximation one can adequately incorporate the effect of different quark masses contained in the modulus of the one-loop quark determinant. We derive the kinetic terms of the collective meson fields, as well as the heat kernel part of contributions to meson masses and interactions. In the end of this section we present the complete bosonized Lagrangian, give the mixing angle conventions used, and the expressions for the strong decay widths. In section III-C we obtain the expressions for the radiative widths of the pseudoscalars and scalars.

In section IV we present the numerical results and discussion, in IV-A for the meson mass spectra and weak decay constants, in IV-B for the strong decays and in IV-C for the radiative decays.

We conclude in section V with a summary of the main results.

## Ii Effective multi-quark interactions

The chiral quark Lagrangian has predictive power for the energy range which is of order  GeV Georgi:1984 (). characterizes the spontaneous chiral symmetry breaking scale. Consequently, the effective multi-quark interactions, responsible for this dynamical effect, are suppressed by , which provides a natural expansion parameter in the chiral effective Lagrangian. The scale above which these interactions disappear and QCD becomes perturbative enters the NJL model as an ultraviolet cut-off for the quark loops. Thus, to build the NJL type Lagrangian we have only three elements: the quark fields , the scale , and the external sources , which generate explicit symmetry breaking effects – resulting in mass terms and mass-dependent interactions.

The color quark fields possess definite transformation properties with respect to the chiral flavor global symmetry of the QCD Lagrangian with three massless quarks (in the large limit). It is convenient to introduce the Lie-algebra valued field , where , , and , , being the standard Gell-Mann matrices for . Under chiral transformations: , where , and . Hence, , and . The transformation property of the source is supposed to be .

Any term of the effective multi-quark Lagrangian without derivatives can be written as a certain combination of fields which is invariant under chiral transformations and conserves and discrete symmetries. These terms have the general form

 Li∼¯giΛγχαΣβ, (1)

where are dimensionless coupling constants (starting from eq. (III.1) the dimensional couplings will be also considered). Using dimensional arguments we find (in four dimensions) , with integer values for and .

We obtain a second restriction by considering only the vertices which make essential contributions to the gap equations in the regime of dynamical chiral symmetry breaking, i.e. we collect only the terms whose contributions to the effective potential survive at . We get this information by contracting quark lines in , finding that this term contributes to the power counting of in the effective potential as , i.e. we obtain that (we used the fact that in four dimensions each quark loop contributes as ).

Combining both restrictions we come to the conclusion that only vertices with

 α+β≤4 (2)

must be taken into account in the approximation considered. On the basis of this inequality one can conclude that (i) there are only four classes of vertices which contribute at ; those are four, six and eight-quark interactions, corresponding to and respectively; the class is forbidden by chiral symmetry requirements; (ii) there are only six classes of vertices depending on external sources , they are: ; ; and .

Let us consider now the structure of multi-quark vertices in detail Osipov:2013 (). The Lagrangian corresponding to the case (i) is well known

 Lint = ¯GΛ2tr(Σ†Σ)+¯κΛ5(detΣ+detΣ†) (3) + ¯g1Λ8(trΣ†Σ)2+¯g2Λ8tr(Σ†ΣΣ†Σ).

It contains four dimensionful couplings .

The second group (ii) contains eleven terms

 Lχ=10∑i=0Li, (4)

where

 L0 = −tr(Σ†χ+χ†Σ) L1 = −¯κ1ΛeijkemnlΣimχjnχkl+h.c. L2 = ¯κ2Λ3eijkemnlχimΣjnΣkl+h.c. L3 = ¯g3Λ6tr(Σ†ΣΣ†χ)+h.c. L4 = ¯g4Λ6tr(Σ†Σ)tr(Σ†χ)+h.c. L5 = ¯g5Λ4tr(Σ†χΣ†χ)+h.c. L6 = ¯g6Λ4tr(ΣΣ†χχ†+Σ†Σχ†χ) L7 = ¯g7Λ4(trΣ†χ+h.c.)2 L8 = ¯g8Λ4(trΣ†χ−h.c.)2 L9 = −¯g9Λ2tr(Σ†χχ†χ)+h.c. L10 = −¯g10Λ2tr(χ†χ)tr(χ†Σ)+h.c. (5)

Each term in the Lagrangian is hermitian by itself, but because of the parity symmetry of strong interactions, which transforms one of them into the other, they have a common coupling .

Some useful insight into the Lagrangian above can be obtained by considering it from the point of view of the expansion. Indeed, the number of color components of the quark field is , hence summing over color indices in gives a factor of , i.e. one counts .

The cut-off that gives the right dimensionality to the multi-quark vertices scales as , as a direct consequence of the gap equations (see eq. (37) below), which imply ; on the other hand, since the leading quark contribution to the vacuum energy is known to be of order , the first term in (3) is estimated as , and we conclude that .

Furthermore, the anomaly contribution (the second term in (3)) is suppressed by one power of , it yields .

The last two terms in (3) have the same counting as the ’t Hooft term. They are of order . Indeed, Zweig’s rule violating effects are always of order with respect to the leading order contribution . This reasoning helps us to find . The term with is also suppressed. It represents the next to the leading order contribution with one internal quark loop in counting. Such vertex contains the admixture of the four-quark component to the leading quark-antiquark structure at .

Next, all terms in eq. (II), except , are of order 1. The argument is just the same as before: this part of the Lagrangian is obtained by succesive insertions of the -field ( counts as ) in place of fields in the already known suppressed vertices. It means that , , and .

There are two important conclusions here. The first is that at leading order in only two terms contribute: the first term of eq. (3), and the first term of eq. (II). This corresponds exactly to the standard NJL model picture, where mesons are pure states with constituents which have a non-zero bare mass. At the next to leading order we have thirteen terms additionally. They trace the Zweig’s rule violating effects , and an admixture of the four-quark component to the one (, ). Only the phenomenology of the last three terms from eq. (3) has been studied until now. We must still understand the role of the other ten terms to be consistent with the generic expansion of QCD.

The second conclusion is that the counting justifies the classification of the vertices made above on the basis of the inequality (2). This is seen as follows: the equivalent inequality is obtained by restricting the multi-quark Lagrangian to terms that do not vanish at (it follows from (1) that by noting that , where is the nearest integer greater than or equal to ).

The total Lagrangian is the sum

 L=¯qiγμ∂μq+Lint+Lχ. (6)

In this symmetric chiral Lagrangian we neglect terms with derivatives in the multi-quark interactions, as usually assumed in the NJL model. We follow this approximation, because the specific questions for which these terms might be important, e.g. the radial meson excitations, or the existence of some inhomogeneous phases, characterized by a spatially varying order parameter, are not the goal of this work.

Finally, having all the building blocks conform with the symmetry pattern of the model, one is now free to choose the external source . Putting , where

 M=diag(μu,μd,μs),

we obtain a consistent set of explicitly breaking chiral symmetry terms. This leads to the following mass dependent part of the NJL Lagrangian

 Lχ→Lμ=−¯qmq+8∑i=2L′i (7)

where the current quark mass matrix is equal to

 m = (8) + ¯g104Λ2(trM2)M,

and

 L′2=¯κ22Λ3eijkemnlMimΣjnΣkl+h.c.L′3=¯g32Λ6tr(Σ†ΣΣ†M)+h.c.L′4=¯g42Λ6tr(Σ†Σ)tr(Σ†M)+h.c.L′5=¯g54Λ4tr(Σ†MΣ†M)+h.c.L′6=¯g64Λ4tr[M2(ΣΣ†+Σ†Σ)]L′7=¯g74Λ4(trΣ†M+h.c.)2L′8=¯g84Λ4(trΣ†M−h.c.)2 (9)

Let us note that there is a definite freedom in the definition of the external source . In fact, the sources

 χ(ci) = χ+c1Λ(detχ†)χ(χ†χ)−1+c2Λ2χχ†χ (10) + c3Λ2tr(χ†χ)χ

with three independent constants have the same symmetry transformation property as . Therefore, we could have used everywhere that we used . As a result, we would come to the same Lagrangian with the following redefinitions of couplings

 ¯κ1→¯κ′1=¯κ1+c12,¯g5→¯g′5=¯g5−¯κ2c1, ¯g7→¯g′7=¯g7+¯κ22c1,¯g8→¯g′8=¯g8+¯κ22c1, ¯g9→¯g′9=¯g9+c2−2¯κ1c1, ¯g10→¯g′10=¯g10+c3+2¯κ1c1. (11)

Since are arbitrary parameters, this corresponds to a continuous family of equivalent Lagrangians. This family reflects the known Kaplan – Manohar ambiguity Manohar:1986 (); Leutwyler:1990 (); Donoghue:1992 (); Leutwyler:1996 () in the definition of the quark mass, and means that several different parameter sets (II) may be used to represent the data. In particular, without loss of generality we can use the reparametrization freedom to obtain the set with .

The effective multi-quark Lagrangian can be written now as

 L=¯q(iγμ∂μ−m)q+Lint+8∑i=2L′i. (12)

It contains eighteen parameters: the scale , three parameters which are responsible for explicit chiral symmetry breaking , and fourteen interaction couplings , . Three of them, , contribute to the current quark masses . Seven more describe the strength of multi-quark interactions with explicit symmetry breaking effects. These vertices contain new details of the quark dynamics which have not been studied yet in any NJL-type models. We shall now see how important they are.

## Iii Bosonization: meson masses and decays

### iii.1 Stationary phase contribution

The model can be solved by path integral bosonization of the quark Lagrangian (12). Indeed, following Reinhardt:1988 () we may equivalently introduce auxiliary fields , and physical scalar and pseudoscalar fields . In these variables the Lagrangian is a bilinear form in quark fields (once the replacement has been done the quarks can be integrated out giving us the kinetic terms for the physical fields and )

 L = ¯q(iγμ∂μ−σ−iγ5ϕ)q+Laux, Laux = saσa+paϕa−sama+Lint(s,p) (13) + 8∑i=2L′i(s,p,μ).

It is clear, that after the elimination of the fields by means of their classical equations of motion, one can rewrite this Lagrangian in its original form (12). The term bilinear in the quark fields in (III.1) will be integrated out using the heat kernel technique in the next subsection. The remaining higher order quark interactions collected in will be integrated in the stationary phase approximation (SPA). In terms of auxiliary bosonic variables one has

 Lint(s,p) = L4q+L6q+L(1)8q+L(2)8q, L4q(s,p) = ¯G2Λ2(s2a+p2a), L6q(s,p) = ¯κ4Λ5Aabcsa(sbsc−3pbpc), (14) L(1)8q(s,p) = ¯g14Λ8(s2a+p2a)2, L(2)8q(s,p) = ¯g28Λ8[dabedcde(sasb+papb)(scsd+pcpd) + 4fabefcdesascpbpd],

and the quark mass dependent part is as follows

 L′2 = 3¯κ22Λ3Aabcμa(sbsc−pbpc), L′3 = ¯g34Λ6μa[dabedcdesb(scsd+pcpd)−2fabefcdepbpcsd], L′4 = ¯g42Λ6μbsb(s2a+p2a), L′5 = ¯g54Λ4μbμd(dabedcde−fabefcde)(sasc−papc), L′6 = L′7 = ¯g7Λ4(μasa)2, L′8 = −¯g8Λ4(μapa)2, (15)

where

 Aabc=13!eijkemnl(λa)im(λb)jn(λc)kl, (16)

and the antisymmetric and symmetric constants are standard.

Our final goal is to clarify the phenomenological role of the mass-dependent terms described by the Lagrangian densites of eq. (III.1). We can gain some understanding of this by considering the low-energy meson dynamics which follows from our Lagrangian. For that we must exclude quark degrees of freedom in (III.1), e.g., by integrating them out from the corresponding generating functional. The standard Gaussian path integral leads us to the fermion determinant, which we expand by using a heat-kernel technique Osipov:2006a (); Osipov:2001 (); Osipov:2001a (); Osipov:2001b (). The remaining part of the Lagrangian, , depends on auxiliary fields which do not have kinetic terms. The equations of motion of such a static system are the extremum conditions

 ∂L∂sa=0,∂L∂pa=0, (17)

which must be fulfilled in the neighbourhood of the uniform vacuum state of the theory. To take this into account one should shift the scalar field . The new -field has a vanishing vacuum expectation value , describing small amplitude fluctuations about the vacuum, with being the mass of constituent quarks. We seek solutions of eq. (17) in the form:

 ssta = ha+h(1)abσb+h(1)abcσbσc+h(2)abcϕbϕc+… psta = h(2)abϕb+h(3)abcϕbσc+… (18)

Eqs. (17) determine all coefficients of this expansion giving rise to a system of cubic equations to obtain , and the full set of recurrence relations to find higher order coefficients in (III.1). We can gain some insight into the physical meaning of these parameters if we calculate the Lagrangian density on the stationary trajectory. In fact, using the recurrence relations, we are led to the result

 Laux =haσa+12h(1)abσaσb+12h(2)abϕaϕb +13σa[h(1)abcσbσc+(h(2)abc+h(3)bca)ϕbϕc]+…

Indicated are all the terms which are necessary to analyze the mass spectra and two particle decays. Here define the quark condensates, , contribute to the masses of scalar and pseudoscalar states, and higher order ’s are the couplings that measure the strength of the meson-meson interactions. The transition from the Lagrangian in (III.1) to its form in (III.1) can be viewed as a Legendre transformation.

We proceed now to explain the details of determining . We address first the coefficients , , and . In particular, eq. (17) states that , if , while (), after the convenient redefinition to the flavor indices

 hα=eαihi,eαi=12√3⎛⎜⎝√2√2√2√3−√3011−2⎞⎟⎠, (20)

satisfy the following system of cubic equations

 +μi4[3g3h2i+g4h2+2(g5+g6)μihi+4g7μh] +κ2tijkμjhk=0. (21)

Here ; is a totally symmetric quantity, whose nonzero components are ; there is no summation over the open index but we sum over the dummy indices, e.g. .

In particular, eq. (8) reads in this basis

 mi=μi(1+g94μ2i+g104μ2)+κ12tijkμjμk. (22)

For the set the current quark mass coincides precisely with the explicit symmetry breaking parameter .

Note that the factor multiplying in the third term of eq. (III.1) is the same for each flavor. This quantity also appears in all meson mass expressions, and there is no further dependence on the couplings involved for meson states with . Thus there is a freedom of choice which allows to vary these couplings, condensates and quark masses , without altering this part of the meson mass spectrum.

To obtain the coefficients , in the Lagrangian (III.1), it is sufficient to collect in the stationary phase equations (17) only the terms linear in the fields, as can be seen from the structure of the solutions (III.1). Moreover, for any coefficient multiplying a certain number of fields in it is required to consider terms only up to order in fields in the expansion (III.1). For instance, the inverse matrices to and are

 −2(h(1)ab)−1=(2G+g1h2+g4μh)δab+4g1hahb +3Aabc(κhc+2κ2μc)+g2hrhc(dabedcre+2dacedbre) +g3μrhc(dabedcre+dacedbre+daredbce) +2g4(μahb+μbha)+g5μrμc(daredbce−farefbce) +g6μrμcdabedcre+4g7μaμb. (23)
 −2(h(2)ab)−1=(2G+g1h2+g4μh)δab −3Aabc(κhc+2κ2μc)+g2hrhc(dabedcre+2farefbce) +g3μrhc(dabedcre+farefbce+facefbre) −g5μrμc(daredbce−farefbce) +g6μrμcdabedcre−4g8μaμb. (24)

These coefficients are totally defined in terms of and the parameters of the model. Eqs. (III.1)-(III.1) can be easily converted into explicit formulae for , .

Finally, to obtain the , , of the interactions involving three fields in , one equates the factors of , , in (17) independently to zero. After some algebra, this results into the following expressions

 h(1)abc = [3κ4A¯a¯b¯c+g1(h¯aδ¯b¯c+2h¯cδ¯a¯b) (25) + g2h¯r(d¯a¯b¯ρd¯r¯c¯ρ+12d¯a¯r¯ρd¯b¯c¯ρ) + g34m¯r(2d¯a¯c¯ρd¯b¯r¯ρ+d¯b¯c¯ρd¯a¯r¯ρ−f¯b¯c¯ρf¯a¯r¯ρ) + g42(m¯aδ¯b¯c+2m¯cδ¯a¯b)]h(1)a¯ah(1)b¯bh(1)c¯c
 h(2)abc = [−3κ4A¯a¯b¯c+g1h¯aδ¯b¯c (26) + g2h¯r(f¯a¯b¯ρf¯c¯r¯ρ+12d¯a¯r¯ρd¯b¯c¯ρ) − g34m¯r(2f¯a¯c¯ρf¯b¯r¯ρ+f¯b¯c¯ρf¯a¯r¯ρ−d¯b¯c¯ρd¯a¯r¯ρ) + g42m¯aδ¯b¯c]h(1)a¯ah(2)b¯bh(2)c¯c
 h(3)abc = [−3κ2A¯a¯b¯c+2g1h¯cδ¯b¯a (27) + g2h¯r(d¯a¯b¯ρd¯c¯r¯ρ+f¯r¯a¯ρf¯c¯b¯ρ+f¯r¯b¯ρf¯c¯a¯ρ) + g32m¯r(d¯a¯b¯ρd¯c¯r¯ρ+f¯b¯c¯ρf¯a¯r¯ρ+f¯a¯c¯ρf¯b¯r¯ρ) + g4m¯cδ¯b¯a]h(2)a¯ah(2)b¯bh(1)c¯c.

Contracting with in eq. (III.1), one sees that the term going with is simply half the one going with , and simplifies to

 Laux =haσa+12h(1)abσaσb+12h(2)abϕaϕb (28) +σa(13h(1)abcσbσc+h(2)abcϕbϕ