The role of the Delta isobar in chiral perturbation theory
델타 이소바르는 핵심적인 역할을 하는데, 치환 이론에서는 대피온 상태(Δ-π states)가 포함되어 있지 않지만 헤지호그 모델에서는 대피온이 핵심 요소로 취급된다.
해당 논문에서 저자들은 델타 이소바르의 역할에 대한 연구를 통해, 표준 치환 이론과 헤지호그 모델 간의 일치를 설명하고, 이러한 일치가 두 모델 모두의 정답을 구하기 위한 보완적인 단서를 제공한다고 주장한다.
한글 요약 끝.
영어 요약 시작:
Chiral perturbation theory (χPT) and hedgehog soliton models both predict the leading nonanalytic behavior of certain observables, but their results differ by an overall factor which depends on the quantum numbers of the operator. This discrepancy is due to the role of the Δ-isobar in χPT calculations.
In standard treatments of χPT, the N-Δ splitting is assumed to be much larger than mπ and contributions from Δ-π states are suppressed. In contrast, hedgehog models include these states as degenerate with the nucleon.
Here, we study the relationship between χPT and hedgehog soliton models, focusing on the leading nonanalytic behavior in the chiral limit (m2π → 0). We find that the hedgehog model results for scalar-isoscalar quantities are a factor of three larger than one-loop χPT predictions, while vector-isovector quantities differ by a factor of 3/2.
We argue that this discrepancy is due to the role of the Δ-isobar in chiral loops, which is not included in conventional χPT. We provide an explicit calculation of the Δ-π loop contribution and find that it gives rise to large corrections to both hedgehog models and χPT.
Our results suggest how hedgehog model results should be corrected to account for effects due to finite N-Δ splitting. This correction is necessary for a consistent description of nucleon properties in both theories.
영어 요약 끝.
The role of the Delta isobar in chiral perturbation theory
arXiv:hep-ph/9208253v1 26 Aug 1992DOE/ER/40322-154U. of MD PP #92-191The role of the Delta isobar in chiral perturbation theoryand hedgehog soliton modelsThomas D. Cohen and Wojciech Broniowski 1Department of Physics and AstronomyUniversity of Maryland, College Park, Maryland 20742-4111AbstractHedgehog model predictions for the leading nonanalytic behavior (in m2π) of certainobservables are shown to agree with the predictions of chiral perturbation theory up to anoverall factor which depends on the operator.
This factor can be understood in terms ofcontributions of the ∆isobar in chiral loops. These physically motivated contributions areanalyzed in an expansion in which both mπ and M∆−MN are taken as small parameters,and are shown to yield large corrections to both hedgehog models and chiral perturbationtheory.1On leave of absence from H. Niewodnicza´nski Institute of Nuclear Physics, ul.
Radzikowskiego 152,31-342 Cracow, POLAND
There is a wide spread consensus that the pion cloud plays an important role in thestructure of the nucleon — the pion is very light and therefore very long ranged. Moreover,the pion is rather well understood in terms of spontaneous chiral symmetry breaking ofthe underlying theory, QCD.
Accordingly, the pion is emphasized in a large number ofhedgehog soliton models of the nucleon, including various incarnations of the Skyrmemodel [1], the chiral or hybrid bag model [2], the chiral quark-meson model [3], the chiralcolor-dielectric model [4], or the Nambu–Jona-Lasinio model in the solitonic treatment[5].Apart from these models, there is another approach to the structure of hadrons whichemphasizes the role of pions, namely chiral perturbation theory (χPT). Its basic premiseis that there is a separation of scales between the pion mass and all other mass scales in theproblem (this separation becomes increasingly good as one approaches the chiral limit).Low momentum observables are studied via systematic expansion in m2π (or equivalentlythe quark mass).In this note we study the relationship between χPT and hedgehog soliton models Ourcentral point is that for a certain class of observables (those whose long range physicsis dominated by pion bilinears and which which are scalar-isoscalar or vector-isovector)the hedgehog models agree with the predictions of (χPT) for the leading nonanalyticbehavior (in m2π) up to an overall factor which depends on the quantum numbers of theoperator.
This factor, equal to 3 for scalar-isoscalar operators, and 3/2 for vector-isovectoroperators, can be traced to the noncommutativity of the large-Nc (number of colors) limit1
and the chiral limit. The essential physics behind this noncommutativity is the role ofthe ∆resonance.
The ∆makes important contributions in the hedgehog models, whereit is treated as degenerate with the nucleon. On the other hand, its contributions tothe leading nonanalytic behavior are not included in conventional χPT, since the N-∆splitting is assumed to be much larger than mπ.
We analyze the role of the ∆in chiralloops in the spirit of Ref. [6], taking physical values for the N-∆splitting and mπ, and findlarge corrections to both hedgehog models and chiral perturbation theory.
This suggestshow hedgehog results and χPT results should be corrected to account for effects finiteN-∆splitting.The hedgehog soliton models are designed to be used at the mean-field level, whichcan be justified in the large-Nc limit of QCD [7]. Stable mean-field configurations arehedgehog solitons in which the internal isospin index is correlated with the spatial index.For pions, the hedgehog configuration is πa = f(r)ˆra, where a is the isospin index, f is aspherically symmetric profile function and ˆra is a spatial unit vector pointing out from thecenter of the soliton.
Appropriate forms can be written for other fields. As is well known,these hedgehogs break both the rotational and isorotational symmetries of the modellagrangian, while preserving the “grand rotational symmetry” generated by K = I + J.As a result, the hedgehog does not have quantum numbers of physical baryons.
Instead,it represents a deformed intrinsic state which corresponds to a band of states. Informationabout physical states is obtained with a semiclassical projection method [8, 9], which givesthe matrix elements of arbitrary operators in a form which is manifestly correct to leading2
order in the 1/Nc expansion.In χPT, beyond the lowest order, one typically has to include both tree diagrams andloops (which are suitably cut offat the separation scale) [10, 11]. Infrared divergencesin the pion loops (at m2π = 0) lead to effects which are nonanalytic in m2π.
Recently thesystematic treatment of χPT has been extended to the nucleon sector [12]. It is believedthat the leading nonanalytic behavior is given by a single pion-nucleon loop calculation.The hedgehog models to leading order in 1/Nc are essentially classical so it is by nomeans obvious that the physics of quantum pion loops should be present.
Somewhat sur-prisingly, this is precisely what happens for a class of observables which have nonanalyticbehavior, in particular for observables which diverge as m−1π , as will be demonstrated inexamples below. Generally, the connection can be seen as follows: We start from an ex-pression for the pion-nucleon loop, and perform the integration over the time-componentof the momentum flowing around the loop.
The leading divergence picks up contributionsfrom poles in the pion propagator(s) only — the nucleon can be treated non-relativistically,and its recoil enters at a subleading level. The resulting expression, after rewriting it ina Fourier transformed manner, involves a single spatial integral of a quadratic expressionin Hankel functions (or derivatives thereof).
Explicitly, the pion tail in a soliton has theform φasymp.a= (3gA)/(8πFπ)caibxi(mπ + 1/r)exp(−mπr)/r, and involves the same Hankelfunction (the collective variables cai = Tr[τaBτiB†] are discussed in Refs. [8, 9]).
Theabove outline shows there is nothing mysterious about hedgehog models reproducing someof the physics of the chiral loops. The noncommutativity of the large-Nc and chiral limits3
leads, however, to a mismatch by a constant, which is the key point discussed below 2.Perhaps the most straightforward way to see this is to study some explicit cases.Here, we will compare the leading singular behavior as mπ →0 of several quantitiescalculated using standard leading order in Nc hedgehog model techniques with the samequantities calculated in χPT at one loop. In the case of the hedgehog models the chirallysingular behavior comes from the long-range tail of the pion field, φasymp.a, whose amplitudedepends on gπNN (or gA).
We consider the isovector mean squared magnetic radius (whichis expressed in terms of the matrix element of a vector-isovector operator), and twoscalar-isoscalar quantities: d2MN/d(m2π)2 = d(σπN/m2π)/d(m2π), and the isoscalar electricpolarizability αN = (αp + αn)/2.All these quantities diverge as 1/mπ near the chirallimit. The hedgehog model expression for d2MN/d(m2π)2 can be obtained easily from theidentity dMN/d(m2π) = 12 < N |R d3x(φasymp.
)2 | N >, the form for < r2 >I=1mis given in2For observables not considered in this paper (vector-isoscalar or scalar isovector) hedgehog models donot predict correct chiral singularities (e.g., for the electric isovector mean squared radius hedgehogs givem−1πrather than log(mπ)). Evaluation of these observables explicitly involves the dynamics of cranking(results depend on the moment of inertia).
In the semiclassical projection one finds rotating solutionsto the time-dependent classical equations of motion to leading order in 1/Nc (slow rotations); centrifugalstretching and other order 1/Nc effects are ignored. However, centrifugal effects increase with distancefrom the center of the soliton.
The longest distance part of the configuration is the region for whichthe 1/Nc approximation does the worst job in cranking, i.e. the difference between the leading order1/Nc solution of the classical equations of motion and the exact solution increases with distance.
In thiscase the issue of nonocommutativity of the large-Nc and chiral limits is much more complicated than forobservables considered in this paper.4
Refs. [8, 9], while the electric polarizability is given in Ref.
[13]. The χPT one-loopexpressions for the same expressions can be extracted from Refs.
[14, 12, 15].The hedgehog model expressions and the χPT predictions are compared in Table I. Wesee, as advertised, that for the scalar-isoscalar quantities the hedgehog model results area factor of three larger than the one loop χPT predictions while for the vector-isovectorquantity the factor is 32. The reason for this is associated with the noncommutativityof the large-Nc and chiral limits.In chiral perturbation it is assumed that the pionis very light compared to all other scales in the problem, and, consequently, dominantcontributions in chiral loops come from N-π states, which are the lightest excited stateswith the appropriate quantum numbers.
These states become increasingly dominant asthe chiral limit is approached (at least for process which are infrared divergent) and theylead to nonanalytic behavior in m2π.In the large-Nc limit the nucleon is essentially degenerate with the ∆isobar (the masssplitting goes as 1/Nc). Consequently, the pion is not much lighter than all other scales inthe problem and the nucleon-pion states are no longer the only light intermediate states inthe problem — ∆-π states are also light.
In one-pion-loop calculations these ∆-π statesshould also be included. We now see explicitly the issue of ordering of the limits.
Instandard treatments of hedgehog models one implicitly takes the large-Nc limit beforegoing to the chiral limit. Thus ∆-π states are degenerate with the N-π states and henceare not suppressed due to the mass difference.
In contrast, conventional χPT correspondsto the opposite ordering of limits (first chiral, than large-Nc), in which a finite energy5
denominator prevents the ∆-π states from giving rise to chiral singularities. The relativesize of ∆-π contributions in one loop calculations is determined by the relative strengthof the π-N-∆coupling to the the π-N-N coupling, and by the N-∆mass difference.In hedgehog models at large-Nc, one has MN = M∆, and gπN∆= 3/2 gπNN (with thenormalization of Ref.
[8]). Then, it is straightforward to see that the contribution of the ∆-π loop to a nucleon matrix element of some operator O is just a numerical coefficient timesthe nucleon contribution.This coefficient is determined simply from Clebsch-Gordanalgebra, and depends on the quantum numbers of O.
For our vector-isovector and scalar-isoscalar cases we find< N|OI=J|N >1−loop∆< N|OI=J|N >1−loopN=CO,CO = 2 for I = J = 0,CO = 1/2 for I = J = 1. (1)It is easy to understand Eq.
(1) from the point of view of the hedgehog modelsthemselves. Consider a generic operator constructed from two pion fields including anynumber of derivatives.
Such an operator can always be written as O = habXaYb + h.c.,where X and Y are operators composed of one pion field and any number of spatialderivatives and a and b are isospin indices. To leading order in Nc , X and Y in thehedgehog model are given by [8, 9]:Xa = caiXhhi , Ya = caiY hhi,(2)where Xhh and Y hh are the mean-field hedgehog expressions for the fields, and cai is thecollective isorotation operator described in Refs.
[8, 9]. We wish to consider nucleon6
matrix elements, thus O must be either isovector or isoscalar and hab is either ǫabc or δab.Using the properties of the collective matrix elements and properties of the SU(2) groupit is straightforward to demonstrate that⟨N | OI=0 | N⟩= ⟨N | XiYi + h.c. | N⟩= 2Xhhi Y hhi(3)and spatial integrals of this operator will only be nonvanishing if OI=0 is scalar. One canexplicitly evaluate the XY products in terms of collective intermediate states, and isolatecontributions from intermediate collective N and ∆states:XN′⟨N | Xa | N′⟩⟨N′ | Ya | N⟩+ h.c.=2/3 Xhhi Y hhi,X∆⟨N | Xa | ∆⟩⟨∆| Ya | N⟩+ h.c.=4/3 Xhhi Y hhi.
(4)Thus, in agreement with Eq. (1), the nucleon intermediate state accounts for 1/3 of thetotal in this scalar-isoscalar channel and the ∆for 2/3 of the total.
Similarly, one findsthat for isovector operators the nucleon intermediate states give 2/3 of the total, and the∆for 1/3.Next, let us consider how large are the ∆contributions for physical values of mπ andN-∆mass splitting. Let us introduced = M∆−MNmπ.
(5)χPT implicitly assumes d →∞, while the large-Nc limit used in hedgehog models assumesd →0. In nature, neither of these extremes is true: d ≃2.1, which raises troubling7
questions about the validity of both approaches. It seems plausible that a more usefulway to organize the problem is to include both the nucleon and ∆explicitly and toexpand by assuming that both mπ and M∆−MN are much smaller than other scalesin the problem but with no prejudice as to their relative size.
This is in fact the spiritof the works of Jenkins and Manohar [6]. We note that leading singularities in this newexpansion should also come from one-pion-loop diagrams with both nucleons or deltasare included in the intermediate states.
Moreover, to determine the contribution to theleading nonanalytic behavior it is legitimate to ignore the recoil of the baryon and usenonrelativistic baryon propagators.It is useful to compare the contribution to some quantity O of diagrams with the ∆intermediate state, O∆, to ON, the contribution with the nucleon in the intermediatestate. One can write this in the form:O∆ON= gπN∆gπNNghhπNNghhπN∆!2COSO(d),(6)where the first factor is corrects for the fact that in nature the ratio of the π coupling tothe ∆need not be what is in the hedgehog models to leading order in the 1/Nc expansion(although in practice this ratio is within a few percent of unity), CO is a factor whichonly depends on the quantum numbers of O, and SO(d) is a “∆mass suppression factor”which is normalized to be unity at d = 0.
The spin-isospin factor CO is defined in Eq. (1).Somewhat surprisingly, all three quantities considered in this note, the isovector magneticradius, the electric polarizability and d2MN/d(m2π)2 all have the same ∆mass suppression8
factor:S(d) = 4πArctanq1−d1+d/√1 −d2for d ≤1Arctanhqd−11+d/√d2 −1for d > 1(7)This function is plotted in Fig. I.
In the case of conventional χPT we have S(d →∞) = 0,while for the large-Nc approximation S(d = 0) = 1. We note that for the physical value,d ≃2 (the blob in Fig.I), we find S ≃0.5.This means that for scalar-isoscalarquantities (with CO = 2) the ∆-π intermediate states contribute as strongly as the N-π state, and hedgehogs overestimate the total (N + ∆) contribution by a factor of ∼3/2, while conventional χPT underestimates it by a factor of ∼1/2.
Since for vector-isovector quantities the value of CO is four times smaller, the effect is reduced: hedgehogsoverestimate by a factor of ∼1.2, while χPT underestimates by a factor of ∼0.8.Figure I illustrates how far we are from the chiral limit and how slowly it is approached.While it is formally true that as d →∞, S →0, the falloffis very slow, S(d) ∼log(d)/d.Even when d ∼10, the ∆-π contribution to scalar-isoscalar quantities is still ∼40% ofthe nucleon contribution!In summary, we have shown that the leading nonanalytic behavior for certain observ-ables in large-Nc hedgehog models agrees with leading order χPT (in its conventionalversion) up to an overall factor, whose origin can be traced to the role of the ∆. These∆effects are large.
Neither approach treats them properly, which suggests the need forsignificant corrections in both. Our study shows how the magnitude of these correctionscan be estimated.
In hedgehog models one can make a “quick and dirty” fix. One simplyisolates the nonanalytic part of an observable, and corrects it according to Eq.
(6). Also,9
our analysis supports the inclusion of an explicit ∆degree of freedom in a modified χPT,along the lines of Refs. [6].Support of the the National Science Foundation (Presidential Young Investigatorgrant), and of the U.S. Department of Energy is gratefully acknowledged.We thankManoj Banerjee for many useful discussions.One of us (WB) acknowledges a partialsupport of the Polish State Committee for Scientific Research (grants 2.0204.91.01 and2.0091.91.01).10
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QuantityI=JHedgehogχPTd2MNd(m2π)2 = d(σπN /m2π)d(m2π)0−1mπ27128πg2AF 2π−1mπ9128πg2AF 2παN0e24π1mπ532πg2AF 2πe24π1mπ596πg2AF 2π(κp −κn)⟨r2⟩I=1m1MNmπ316πg2AF 2πMNmπ18πg2AF 2πTable I: Comparison of hedgehog model predictions with chiral perturbation theoryfor the leading nonanalitic term of selected observables.13
Figure captionFigure I: The ∆mass suppression factor, S(d), where d = (M∆−MN)/mπ. The blobindicates the physical point.
S(d) determines the relative contribution of ∆-π to N-πstates in chiral loops, up to an overall spin-isospin factor. See Eqs.
(1,6).14
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