Baryon Masses and Axial Couplings in the Combined and Chiral Expansions
The effective theory for baryons with combined and chiral expansions is analyzed for non-strange baryons. Results for baryon masses and axial couplings are obtained in the small scale expansion, to be coined as the -expansion, in which the and the low energy power countings are linked according to . Masses and axial couplings are analyzed to and respectively, which correspond to next-to-next to leading order evaluations, and require one-loop contributions in the effective theory. The spin-flavor approximate symmetry, consequence of the large limit in baryons, plays a very important role in the real world with as shown by the analysis of its breaking in the masses and the axial couplings. Applications to the recent lattice QCD results on baryon masses and the nucleon’s axial coupling are presented. It is shown that those results are naturally described within the effective theory at the order considered in the -expansion.
pacs:11.15-Pg, 11.30-Rd, 12.39-Fe, 14.20-Dh
- I Introduction
- II Framework for the combined expansion and Baryon Chiral Perturbation Theory
- III Baryon masses
- IV Axial couplings
- V Analysis of lattice QCD results for baryon masses and the nucleon’s axial coupling
- VI Discussion and conclusions
- A Spin-flavor Algebra
- B Non-linear realization of chiral symmetry and spin-flavor transformations
- C Tools for building effective Lagrangians
- D Matrix elements of spin-flavor operators in the symmetric representations of
The low energy effective theory for baryons is a topic that has evolved over time through several approaches and improvements. The early version of baryon Chiral Perturbation Theory (ChPT) Pagels (1975) evolved into the various effective field theories based on effective chiral Lagrangians Weinberg (1968); Coleman et al. (1969); Callan et al. (1969), starting with the relativistic version Gasser et al. (1988); Bernard et al. (1995) or Baryon ChPT (BChPT), followed by the non-relativistic version based in an expansion in the inverse baryon mass Jenkins and Manohar (1991a); Bernard et al. (1995) or Heavy Baryon ChPT (HBChPT), and by manifestly Lorentz covariant versions based on the IR regularization scheme Ellis and Tang (1998); Becher and Leutwyler (1999); Fuchs et al. (2003). In all these versions of the baryon effective theory a consistent low energy expansion can be implemented. The most important issue, which became apparent quite early, was the convergence of the low energy expansion. Being an expansion that progresses in steps of in contrast to the expansion in the pure Goldstone Boson sector where the steps are , it is natural to expect a slower rate of convergence. However, a key factor with the convergence has to do with the important effects due to the closeness in mass of the spin 3/2 baryons. It was realized Jenkins:1991es, that the inclusion of those degrees of freedom play an important role in improving the convergence of the one-loop contributions to certain observables such as the - scattering amplitude and the axial currents and magnetic moments. There have been since then numerous works including spin 3/2 baryons Hemmert et al. (1997, 1998, 2003); Fettes and Meissner (2001); Procura et al. (2006); Hacker et al. (2005); Bernard et al. (2003, 2005); Procura et al. (2007). The key enlightenment resulted from the study of baryons in the large limit of QCD ’t Hooft (1974). It was realized that in that limit baryons behave very differently than mesons Witten (1979), in particular because their masses scale like and the -baryon couplings are . These properties were shown to require for consistency, that at large baryons must respect a dynamical contracted spin-flavor symmetry , being the number of light flavors Gervais and Sakita (1984a, b); Dashen and Manohar (1993a, b), broken by effects ordered in powers of and in the quark mass differences. The inclusion of the consistency requirements of the large limit into the effective theory came naturally through a combination of the expansion and HBChPT Jenkins (1996), which is the framework followed in the present work. The study of one-loop corrections in that framework was first carried out in Refs. Jenkins (1996); Flores-Mendieta et al. (1998); Flores-Mendieta and Hofmann (2006). In the combined theory one has to deal with the fact that the and Chiral expansions do not commute Cohen and Broniowski (1992). The reason is due to the presence of the baryon mass splitting scale of ( mass difference), for which it becomes necessary to specify its order in the low energy expansion. Thus the and Chiral expansions must be linked. Particular emphasis will be given to the specific linking in which the baryon mass splitting is taken to be in the Chiral expansion, and which will be called the -expansion. Following references Jenkins (1996); Flores-Mendieta et al. (1998); Flores-Mendieta and Hofmann (2006), the theoretical framework is presented here in detail, in particular the power countings, the renormalization, and the linked and low energy expansions, along with observations that further clarify the significance of the framework.
The very significant contemporary progress in the calculations of baryon observables in lattice QCD (LQCD) Hagler (2010); Fodor and Hoelbling (2012); Alexandrou (2012) opens new opportunities for further understanding the low energy effective theory of baryons. The determination of the quark mass dependence of the various low energy observables, such as masses, axial couplings, magnetic moments, electromagnetic polarizabilities, etc., are of key importance as a significant test of the effective theory, in particular its range of validity in quark masses, as well as for the determination of its low energy constants (LECs). Lattice results for the and masses Durr et al. (2008); Walker-Loud et al. (2009); Aoki et al. (2009); Lin et al. (2009); Alexandrou et al. (2009); Aoki et al. (2010, 2011); Bietenholz et al. (2011) and the axial coupling of the nucleon Edwards et al. (2006); Bratt et al. (2010); Alexandrou et al. (2011); Yamazaki et al. (2008, 2009); Lin et al. (2008) at varying quark masses are analyzed with the purpose of testing the effective theory presented here. This in turn can give insights on LQCD results, in particular an understanding on the role and relevance of including the spin 3/2 baryons consistently with large requirements.
This work is organized as follows. In Section II the framework for the combined and HBChPT expansions is presented. Section III presents the evaluation of the baryon masses and Section IV the one for axial couplings at the one-loop level. Section V is devoted to applying those results in the -expansion to LQCD results. Finally, Section VI is devoted to observations and conclusions . Several appendices present useful material used in the calculations, namely, Appendix A on spin-flavor algebra, Appendix B on symmetries, Appendix C on the construction of effective Lagrangians, and Appendix D on useful matrix elements of spin-flavor operators.
Ii Framework for the combined expansion and Baryon Chiral Perturbation Theory
In this section the framework for the combined and chiral expansions in baryons is presented in some detail along similar lines as in the original works Jenkins (1996); Flores-Mendieta et al. (1998); Flores-Mendieta and Hofmann (2006). The symmetries that the effective Lagrangian must respect in the chiral and large limits are chiral and contracted dynamical spin-flavor symmetry Gervais and Sakita (1984a, b); Dashen and Manohar (1993b, a) 111See also Appendix B.. is the number of light flavors, and in this work . In the limit the spin-flavor symmetry requires baryons to belong into degenerate multiplets of . In particular, the ground state (GS) baryons belong into a symmetric multiplet, which consists of states with , where the baryon spin and its isospin. At finite the spin-flavor symmetry is broken by effects suppressed by powers of , and the baryon mass splittings in the GS multiplet are proportional to . The effects of finite are then implemented as an expansion in at the level of the effective Lagrangian. Because baryon masses scale as proportional to , it becomes natural to use the framework of HBChPT Jenkins and Manohar (1991a, b), where the expansion in inverse powers of the baryon mass becomes part of the expansion. The framework presented next follows that of Refs. Jenkins (1996); Flores-Mendieta et al. (1998).
The non-relativistic baryon field, denoted by , consists of the symmetric spin-flavor multiplet with states , ( odd). Chiral symmetry is realized in the usual non-linear way on , namely Weinberg (1968); Coleman et al. (1969); Callan et al. (1969):
where is a transformation, is given in terms of the pion fields by , where the isospin generators are normalized by the commutation relations , MeV, and is an isospin transformation which in any representation of Isospin satisfies . The chiral covariant derivative is given by:
where and are gauge sources. Another necessary building block of the effective chiral Lagrangian is the axial Maurer-Cartan one-form:
For later use, the following notation will be used: for flavor traces, and the definition , where is in the fundamental representation, which implies that in an arbitrary isospin representation (since in the fundamental representation, ). The definition is used.
Since , , and contain different orders in the expansion in powers of . The contracted transformations (see Appendix A) are generated by , where are semiclassical at large , i.e., commute with each other. The ordering in of the matrix elements of the spin-flavor generators in states with are as follows: , , and . While infinitesimal transformations generated by correspond to the usual isospin transformations when acting on pions, the ones generated by affect only the baryons (one can define these generators to not affect the pion field as shown in Appendix B). The effective Lagrangian can be systematically written as a power series in the low energy expansion or Chiral expansion, and simultaneously in . It is most convenient to write the Lagrangian to be manifestly chiral invariant as is usually done. The low energy constants (LECs) will themselves admit an expansion in powers of . For the HBChPT expansion the large mass of the expansion is taken to be the spin-flavor singlet component of the baryon masses, ( can be considered here to be a LEC defined in the chiral limit and which will have itself an expansion in ). To baryon masses will read Dashen and Manohar (1993b, a):
In the following we will define
which will be useful in the implementation of the expansion discussed later. The baryon mass splittings due to the hyperfine term, second term in Eq. (4), must be considered to be a small energy scale. It becomes necessary to establish of what order that term is in the low energy expansion, as it naturally appears in combinations with powers of when loop diagrams are calculated. This fact implies that the low energy and expansions do not commute Cohen and Broniowski (1992); Cohen (1996), and the natural way to proceed is therefore to link the two expansions. For the purpose of organizing the effective Lagrangian it is convenient to establish the link between the two expansions. In the real world with the mass splitting is about 300 MeV, and therefore it is reasonable to count that quantity as in the low energy expansion: the expansion where will be adopted in what follows, and it will be called -expansion. This power counting corresponds to the so called small scale expansion (SSE) Hemmert et al. (1998), now consistently implemented in the context of the expansion. Whenever appropriate, it will be indicated which aspects of the analysis are general and which are only valid in that expansion. Up to the baryon effective Lagrangian reads Jenkins (1996):
where is the axial coupling in the chiral and large limits (it has to be rescaled by a factor 5/6 to coincide with the usual axial coupling as defined for the nucleon), is the source containing the quark masses: specifically (see Appendix C ). Here one notes an important point which will be present in other instances as well: the baryon mass dependence on the current quark mass behaves at ( is of zeroth order in ), and this indicates that in a strict large limit the expansion in the quark masses of certain quantities such as the baryon masses cannot be defined due to divergent coefficients of .
The Lagrangian is manifestly invariant under chiral transformations, translations and rotations (the latter also involving obviously the action of the generators of ). Under an infinitesimal transformation generated by the spin-flavor generators , the Lagrangian (6) is transformed according to:
According to this, and using the commutation relations in Appendix A, the kinetic term changes by terms , the term proportional to , which contains the interaction and the leading order terms of the axial currents, changes by terms which are a factor smaller than the original term, and the term proportional to , which gives the leading order (LO) -term in the baryon masses, is a spin-flavor singlet and thus invariant under spin-flavor transformations. Finally, the hyperfine term proportional to is the one providing the dominant spin-flavor symmetry breaking effects, because it is modified by terms , which is the same order as the hyperfine term itself (this is so because ). The construction of higher order Lagrangians can be accomplished using the tools provided in Appendix C.
The operators appearing in the effective Lagrangian are normalized in such a way that all the LECs are of zeroth order in . Therefore, the power of a Lagrangian term with pion fields is given by Dashen et al. (1995):
where the spin-flavor operator is -body ( is the number of factors of generators appearing in the operator), and is basically the number of factors of the generators remaining after reducing the operator using commutators. The last term, , stems from the factor carried by any term with pion fields. It is opportune to point out that commutators of spin-flavor operators will always reduce the -bodyness of the product of operators: e.g., let be any generator of , and consider the commutator . In principle this looks like a three-body operator, but because is a 1-body operator, is actually a 2-body operator.
ii.1 Consistency of the expansion
The consistency of the expansion in QCD gives rise to the dynamical spin-flavor contracted symmetry in baryons at large . At the baryon level that symmetry can be deduced as the result of consistency or correct power counting of observables in which pion-baryon couplings are involved. This is because the pion-baryon coupling is from Witten’s counting rules Witten (1979). In particular the consistency of pion-baryon scattering is a direct way of deriving the existence of the dynamical spin-flavor symmetry Dashen and Manohar (1993b, a). In general, for any quantity there must be cancellations between the terms with the “wrong”power counting stemming from different Feynman diagrams. For instance, baryon masses are , and therefore pion loop contributions cannot give contributions which scale with a higher power of . On the other hand, the baryon mass splittings are , and loop contributions must respect that scaling. Similarly, in the axial currents, whose matrix elements are such cancellations occur when loop corrections are calculated. All this will be illustrated in the application to baryon masses and axial couplings discussed later. Although certain key cancellations must be exact in the large limit, the analysis of LQCD results will show that they are very significant in the physical world where .
ii.2 power counting
The terms in the effective Lagrangian are constrained in their dependence by the requirement of the consistency of QCD at large . This constraint is in the form of a lower bound in the power in for each term one could write down in the Lagrangian. This leads to constraints on the dependencies of the ultra-violet (UV) divergencies, which have to be subtracted by the corresponding counter-terms in the Lagrangian. One very important point to mention is that the UV divergencies are necessarily polynomials in low momenta (derivatives), in and in (modulo factors of due to factors in terms where pions are attached). Therefore, the structure of counter-terms is independent of any linking between the and chiral expansions. For this reason, one can simply take the large and low energy limits independently in order to determine the UV divergencies. For a connected diagram with external baryon legs, external pion legs, vertices of type which has baryon legs and pion legs, and loops, the following topological relations hold Weinberg (Cambridge University Press, New York (1995, 1991):
where is the number of pion propagators and the number of baryon propagators.
The chiral or low energy order of a diagram, where is the chiral power of the vertex of type , is then given by Weinberg (1991):
Note that is equal to 0 or 2 in the single baryon sector.
On the other hand, the power of a connected diagram is determined by looking only at the vertices: the order in of a vertex of type is given according to Eq. (8) by: , where is the order of the spin-flavor operator. Thus, the power of a diagram, upon use of the third Eq. (9), is given by:
where is the number of external pions, and the order of the spin-flavor operator of the vertex of type . Since can be negative (due to factors of in vertices), one can think of individual diagrams with negative and violating large consistency, requiring cancellation with other diagrams. Such a sum will have to respect the mentioned lower bound on the power corresponding to the sum of such diagrams. The explicit example of such cancellation in the axial currents at one-loop is given in Section IV.
One can determine now the nominal counting of the one-loop contributions to the baryon masses and axial currents. The LO baryon masses are , Eq. (4).The one loop correction shown in Fig. 1 has: giving as it is well known, and . Since there is only one possible diagram, this must be consistent by contributing to the spin-flavor singlet component of the masses, which is the case as shown in the next section. For the axial currents one has the diagrams in Fig. 2. The current at tree level is , and the sum of the diagrams cannot scale like a higher power of . Performing the counting for the individual diagrams one obtains: for , and , and . Thus a cancellation must occur of the terms when the contributions to the axial currents by diagrams 1, 2 and 3 are added. Since the acceptable bound is that the sum be , one concludes that the axial current has, at one-loop, corrections or higher.
One can consider the case of two-loop diagrams, in particular diagrams where the same pion-baryon vertex Eq.(6) appears four times. For the masses one has , and individual diagrams give . A cancellation must occur to restore the bound on the counting for the masses, i.e., . Thus, at two-loops the UV divergencies of the masses must be or higher. For the axial currents a similar discussion requires that counter-terms to the axial currents must be or higher.
The -power counting of the UV divergencies is obvious from the earlier discussion. At one-loop one finds that the masses have and counter-terms, while the axial currents will have and counter-terms. To two loops one expects and , and and counter-terms for masses and axial currents respectively. The non-commutativity of limits is manifested in the finite terms where and or momenta and appear combined in non-analytic terms, and are therefore sensitive to the linking of the two expansions.
Iii Baryon masses
In this section baryon masses are analyzed to order , or next-to-next to leading order (NNLO), in the limit of exact isospin symmetry. To that order the mass of the baryon of spin reads:
where involves contributions from the one-loop diagram in Fig. 1, and CT denotes counter-terms. From both types of contributions, there are and terms, and the calculation is exact at the latter order, as can be deduced from the previous discussion on power counting. Notice that is equal to the LO term in in the real world .
The leading 1-loop correction to the baryon self energy, diagram in Fig. 1, can be calculated through the matrix element , with:
where indicates the possible intermediate baryon spin-isospin states in the loop, are the corresponding spin-flavor projection operators, , and the loop integral is calculated in dimensional regularization with the result,
where , , and is the renormalization scale which will be taken later to be of the order of . For the specific evaluation of for a given baryon state denoted by , , where is a residual energy (when evaluated on an on-shell baryon it is the kinetic energy which is ). The non-commutativity of the and Chiral expansions of course resides in the non-analytic terms of the loop integral through their dependence on the ratio . Notice that when the one loop integrals are written in terms of the residual momentum , they do not depend on the spin-flavor singlet piece of , namely the -term in Eq.(5).
Appendix D provides all the necessary elements for the evaluation of the spin-flavor matrix elements in Eq. (14) as well as in the calculation of the one-loop corrections to the axial currents below. The explicit final expressions for the self energy are not given here because they are too lengthy, but with those elements the reader can easily obtain them.
The one-loop contribution to the wave function renormalization constant is given by:
The explicit evaluation of the ultraviolet divergent pieces of the self energy gives:
The UV divergent pieces start at . Note that the UV divergencies in the mass (term independent of ) is produced by the contribution of the partner baryon and is proportional to the mass splitting. As is well known, they are absent in HBChPT without explicit . The UV divergence is spin-flavor singlet and proportional to , while the contributions to mass splittings are . Notice that the leading UV divergence of is : this is necessary as shown later for rendering the one-loop calculation of the axial currents consistent in the large limit. Since the calculation is accurate to , additional terms in the effective Lagrangian up to that order are necessary for renormalization. The terms necessary for renormalizing the self energy are therefore the following:
where the residual energy has been identified with the operator . All LECs are here of the form . Writing , one renormalizes the self energy to . The coefficients are determined from given above. While the counter-terms are defined such that is , it is possible that is of higher order in . Notice that among the higher order terms there are terms which can be simply absorbed into corrections to the LECs of the lowest order Lagrangian, and into .
Finally, the baryon masses are given by:
Note that the correction to the wave function renormalization factor enters in the expression for the mass corrections: this is because starts with terms and starts at , therefore the terms of the mass correction involve these lower order terms of the wave function renormalization.
The one-loop corrections and corresponding counter-terms contribute to the masses at and , while in a strict large limit the following ordering is found:
Obviously the term stems from the expansion of non-analytic terms and shows the non-commutativity of limits.
The one loop correction with the vertex proportional to in Eq.(6) gives contributions to the masses, and is therefore beyond the accuracy considered here.
The -terms for and , defined by (), are with the one-loop corrections contributing up to . The difference is and at that order it receives only finite contributions from the loop. This implies that the slopes of the and masses as functions of are the same up to deviations. This seems to be closely followed by the lattice QCD results analyzed later. In the large limit, obviously . The terms of that order are necessarily spin-flavor singlet, and taking the limit at fixed one finds , a result similar to the one in the -expansion.
Iv Axial couplings
In this section the evaluation of the axial couplings including corrections is presented. At that order the one-loop corrections must be calculated.
The matrix elements of interest for the axial currents are evaluated at vanishing external 3-momentum. The axial couplings are then defined by:
The axial couplings defined here are . The of the matrix elements of the axial currents is due to the operator . The factor mentioned earlier is included so that at exactly corresponds to the usual nucleon , which has the value Beringer et al. (2012). This definition of the axial couplings is convenient in the context of the expansion, as the differences between the different axial couplings are .
The determination of the axial couplings to require the calculation of the 1-loop corrections to the axial current. Only the contributions with no pion pole are necessary, and they are given by the diagrams in Fig. 2. The resulting 1-loop contribution to the axial currents reads:
where is given by a factor 1/2 times the no-baryon-pole contributions of diagrams (2+3). The different contributions read as follows, where one needs to take the limits :
Obviously, and do not commute in general. The pion tadpole integral in the last term is given by:
Notice that the contribution by diagram (4) is actually , and thus beyond the degree of accuracy of the present calculation. It can serve however as a measure of the size of the NNNLO corrections.
The corrections to the axial currents must scale as with . While diagram (4) is and therefore consistent in itself, diagrams (1) and (2+3) above are . As shown in Ref. Flores-Mendieta et al. (1998), the offending terms cancel upon adding the diagrams. To test the cancellation it is sufficient to take the large limit at fixed . A straightforward evaluation leads to:
where indicate further terms which are consistent with the power counting. The suppression of the terms is direct consequence of the appearance of the commutator of two generators , which is , when the diagrams are added up. In consequence the displayed terms are .
The UV divergent contributions of the individual diagrams read:
One notices that only the terms proportional to in diagrams (1) and (2+3) diverge as proportional to , while the terms proportional to are . Thus, only the terms proportional to need to be cancelled to give consistency. One can easily check that such a cancellation indeed occurs, leaving only terms . An explicit evaluation of these UV divergent terms using the results from Appendix D finally gives:
The terms in the Lagrangian needed to renormalize the axial currents are then the following:
These are all the terms which can contribute to the axial currents up to , which will determine the axial couplings up to , i.e., NNLO, which is what is needed for our purpose. There are several very important observations concerning the -power counting. The corrections to the axial couplings start at , and the individual contributions of the different baryons in the loop diagrams are also . Even the difference of different axial couplings starts at . These latter differences are UV finite. The large cancellations do not seem manifest. However, at , where the -expansion is used, cancellations do occur numerically as shown by Fig. 4 in Section V. Thus, the smallness of terms in the axial couplings is a result of the incipient manifestation of the cancellations in the large limit. If one would consider the strict large limit, the one loop corrections and counter-terms considered give the following power counting:
where, as expected, the latter differences are UV finite as in the expansion.
The explicit expression for at is give here for completeness: