We impose the requirement that the isovector axial vector current for the
soliton sector of the chiral quark model transforms correctly under particle
conjugation. This forces us to choose an otherwise arbitrary ordering of
collective space operators in such a way that the next--to--leading $1/N_C$
correction to $g_A$ vanishes.Comment: 6 pages LaTeX, SU--4240--588, UNITU--THEP--23/199

Schechter, J.

Weigel, H.

English

arXiv

J. SCHECHTER

H. WEIGEL

Crossref

PARTICLE CONJUGATION AND THE 1/N
                    c

We impose the requirement that the isovector axial vector current for the soliton sector of the chiral quark model transforms correctly under particle conjugation. This forces us to choose an otherwise arbitrary ordering of collective space operators in such a way that the next--to--leading 1/N_C correction to g_A vanishes

Schechter, Joseph

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Physics College of Arts and Sciences 10-19-1994 Particle Conjugation and the 1/N_C Corrections to g_A Joseph Schechter Department of Physics, Syracuse University, Syracuse, NY H. Weigel Syracuse University and Tubingen University Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation Schechter, Joseph and Weigel, H., "Particle Conjugation and the 1/N_C Corrections to g_A" (1994). Physics. 313. https://surface.syr.edu/phy/313 This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Physics by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:hep-ph/9410320v1  19 Oct 1994SU–4240–588UNITU–THEP–23/1994October 1994hep-ph/9410320Particle Conjugation and the 1/NC Corrections to gAJ. Schechtera and H. Weigela,baDepartment of Physics, Syracuse UniversitySyracuse, NY 13244–1130, USAbInstitute for Theoretical Physics, Tu¨bingen UniversityD-72076 Tu¨bingen, GermanyABSTRACTWe impose the requirement that the isovector axial vector current for the soliton sectorof the chiral quark model transforms correctly under particle conjugation. This forces us tochoose an otherwise arbitrary ordering of collective space operators in such a way that thenext–to–leading 1/NC correction to gA vanishes.1Recently, the very interesting observation has been made [1]–[3] that the too small valueof the neutron beta decay constant gA predicted in many chiral soliton models might be dra-matically improved by including subleading 1/NC corrections. These calculations, however,require one to make a certain choice about the ordering of two collective space operatorswhich appear in the expression for the current. An ambiguity arises when one replaces clas-sical objects by quantum operators while doing the collective quantization. In the presentnote we show that particle conjugation invariance can provide useful operator ordering in-formation. It implies that the next–to–leading 1/NC correction to gA should vanish in thecontext of collectively quantizing the static soliton configuration. We will use the chiralquark model [4] but the calculation is essentially identical to those in other models of thistype [1]–[3]. More detailed aspects as well as a similar study for the isovector part of themagnetic moment of the nucleon will be presented elsewhere [5].In chiral models it is useful to employ the non–linear parametrization U = exp (iτ · pi/fpi)of the pion field, pi. The static soliton configuration is described in terms of the hedgehogansatzU0(r) = exp[iτ · rˆF (r)], (1)where the radial function F (r) is obtained by minimizing the static energy functional whilemaintaining definite boundary conditions. The projection of the soliton onto physical statesis accomplished by introducing [6] time dependent collective coordinates A(t) ∈ SU(2)parametrizing the isospin (and/or spin) orientation of the hedgehogU(r, t) = A(t)U0(r)A†(t). (2)Due to the hedgehog structure of U0 a rotation in space is equivalent to a right transformationof A(t), i.e. the spin operator represents the right generator in the collective space[J , A] = Aτ2. (3)The angular velocity Ω, defined byi2τ ·Ω = A†∂∂tA, (4)2measures the right transformation at the classical level. When quantizing A(t) canonically,the spin is identified as the momentum conjugate to ΩJ =∂L∂Ω= α2Ω, (5)where L = L(A,Ω) denotes the collective Lagrangian. The moment of inertia, α2, is afunctional of the chiral angle, F (r) and is of order NC .In the chiral quark model [4] one considers a Dirac spinor field in the background field ULq = Ψ(i∂/−mUγ5)Ψ, (6)where m parametrizes the quark–pion coupling. The contribution of the quarks to the staticenergy functional is given byEclq = NCsgn(B)ǫval, (7)where ǫval denotes the lowest (in magnitude) eigenvalue of the Dirac Hamiltonianh(F ) = α · p+ βm[cosF (r) + iγ5τ · rˆ sinF (r)]. (8)In (7) the sign of the baryon number carried by the soliton is included as an additional factorto accommodate the hole interpretation of the Dirac theory. Next the collective coordinatesare taken to be time dependent. Transforming to the flavor rotating frame Ψ → AΨ addsthe Coriolis term τ · Ω/2 as a perturbation to (8). Then ǫval is changed at second orderyielding the quark part of the collective LagrangianLq = −Eclq +12sgn(B)α2qΩ2 , α2q =NC2∑µ6=val|〈µ|τ3|val〉|2ǫval − ǫµ, (9)while the wave–function acquires a so–called cranking correction to become [7]Ψcrank = A(t)Ψval + 12∑µ6=valΨµ〈µ|τ ·Ω|val〉ǫval − ǫµ . (10)Now, the contribution of the quarks to the isovector axial vector current Aai is obtainedby substituting (10) into sgn(B)NCΨ¯γiγ5(τa/2)Ψ. The factor sgn(B) has to be carried alongbecause this current is a Noether current of the functional (9). Furthermore Ω is replaced3by J/α2 according to (5) yielding the quark part of the axial current operator in the spaceof the collective coordinatesA(q)ai = sgn(B)NCDabΨ†valσiτ b2Ψval (11)+sgn(B)NC2α2[(1− ξ)JjDab + ξDabJj ]∑µ6=val〈val|τj |µ〉ǫval − ǫµΨ†µσiτ b2Ψval.+sgn(B)NC2α2[(1− η)DabJj + ηJjDab]∑µ6=val〈µ|τj|val〉ǫval − ǫµΨ†valσiτ b2Ψµ.Here ξ and η denote a priori undetermined parameters labeling possible orderings betweenDab =12tr(τaAτbA†)and J . Their presence reflects the ambiguity occurring when goingfrom the classical to the quantum description. As α2 is of the order NC the ambiguous termsare subleading.We note that the eigenvalues ǫµ are degenerate with respect to the grand spin (i.e. totalspin plus isospin) projection, Mµ. The associated sum introduces a kind of reduced matrixelement Tµ∑Mµ〈val|σ3τb|µ,Mµ〉〈µ,Mµ|τj|val〉 = iTµǫ3bj . (12)An explicit calculation shows that Tµ is real [5]. The current (11) then yields the quarkcontribution to gAgqA = 〈2A(q)33 〉N = sgn(B)NC{〈D3b〉N〈val|σ3τb|val〉 (13)+1− ξ − η2α2∑µ6=valiTµǫ3bjǫval − ǫµ[〈D3bJj〉N−〈JjD3b〉N] },where the subscript, N , indicates a matrix element between nucleon states. Here it should bestressed that the expression in square brackets vanishes classically; however, in the quantumdescription J does not commute with the rotation matrices[Ji, Dab] = iǫibcDac. (14)Then one finds for (13)gqA = sgn(B)NC〈D3b〉N〈val|σ3τb|val〉 −1− ξ − ηα2〈D33〉N∑µ6=valTµǫval − ǫµ . (15)4The 1/NC correction depends on the parameters ξ and η and is hence ambiguous. In variousquark soliton models it has been shown that the prescription ξ = η = 0, which is suggestedby the form of the cranked wave–function (10), results in sizable corrections to gA improvingthe agreement with the experimental data [1]–[2]. However, we will now argue that requiringthe proper behavior of the axial current under particle conjugation symmetry requires usto choose an ordering prescription so that the matrix elements of the 1/NC terms actuallyvanish.The particle conjugation symmetry is most conveniently implemented as G–parity in-variance [8]. This transformation represents charge conjugation followed by an isorotationof angle π around the y–axis. Under G–parity the pion field changes sign, i.e.U(r, t)G−→ U †(r, t). (16)Noting (2), we see that G–parity reflection for the rotating soliton corresponds toF (r)G−→ − F (r) (17)while keeping the collective coordinates, A(t) unchanged. At the soliton level the G–parityreflection agrees with the particle conjugation because the baryon number density reversesits sign under the transformation (17), i.e. B → −B. This property is a consequence of thefact that the isoscalar vector current has negative G–parity. The effect of the transformation(17) on the Dirac Hamiltonian may be simplified by introducing the self–adjoint unitarytransformation J = iβγ5 and noting thath(−F ) = −J †h(F )J . (18)From this one may read off the particle conjugation properties of the eigenvalues and eigen-states of the Dirac Hamiltonian (8):ǫµF→−F−→ − ǫµ and |µ〉F→−F−→ J |µ〉, (19)where µ labels the particular eigenstate. Since J does not affect isopsin it is obvious thatthe Lagrangian (9) is invariant under particle conjugation, i.e. the mass of the anti–baryonis identical to that of the baryon. Furthermore, as J has negative parity, the parity of the5anti–baryon is opposite to that of the baryon, as it should be. Since the isovector axial vectorcurrent Aaµ, is directly related to the pion field via PCAC we must haveAaµG−→ − Aaµ, (20)under G–parity reflection.Now we have collected all necessary ingredients and may consider the effect of particleconjugation on gA. The reduced matrix element Tµ in (12) is invariant under this transfor-mation because J σi = σiJ . Thus we findagqAF→−F−→ g¯qA = sgn(B¯)NC〈D3b〉N¯〈val|σ3τb|val〉+1− ξ − ηα2〈D33〉N¯∑µ6=valTµǫval − ǫµ (21)since the energies ǫµ reverse their signs. The leading term changes sign under particleconjugation since sgn(B¯) = −sgn(B). This agrees with the required transformation propertyof the axial current under G–parity (20). However, the 1/NC correction does not change signunder (19). Stated otherwise, the result for gA violates the G–parity reflection symmetryunless1− ξ − η ≡ 0. (22)Thus G–parity invariance implies an operator ordering which rules out the 1/NC correctionsto gA. We stress that this conclusion has been obtained without specifying the explicit wave–functions of the baryon or the anti–baryon. The constraint (22) is merely a consequence of thefact that the leading and next–to–leading order in 1/NC terms in the approximate expression(13) for gA transform oppositely under particle conjugation. It is interesting to note thatthe hermitean ordering ξ = η = 1/2 satisfies the constraint (22).Here we have been studying only the contributions of the quarks to gA in the chiral quarkmodel. The leading order term of the mesonic axial current contains only odd powers ofF (r) and no 1/NC corrections appear. Thus the transformation property (20) is triviallysatisfied for the total current Aaµ.Although we have only considered the chiral quark model as the simplest containingsoliton solutions, our studies apply to more complicated ones as well. In the literature theaFor taking matrix elements we may note 〈p ↑ |D3b|p ↑〉 = 〈p ↑ |D33|p ↑〉δ3b = 〈n¯ ↑ |D33|n¯ ↑〉δ3b =−1/3δ3b.6chiral bag [2] and Nambu–Jona–Lasinio (NJL) [1] models have been considered in connectionwith the 1/NC corrections to gA. In both models matrix elements of the structure (11,12)are involved in the subleading terms for gA. Since the moment of inertia carries massdimension one, the function multiplying the dimensionless reduced matrix element must beodd in the energy eigenvalues ǫµ. Thus the corrections do not transform properly underparticle conjugation and an operator ordering which makes these corrections vanish has tobe adopted.Similar ambiguities appear in the next–to–leading order in the 1/NC expansion of theisovector part of the magnetic moment of the nucleon, µV . We have also considered thebehavior of these corrections under particle conjugation [5]. For this observable too, theproper behavior requires an ordering so that the 1/NC corrections are zero.To summarize, we have seen that particle conjugation symmetry provides useful con-straints on operator ordering ambiguities which may occur when quantizing the chiral soliton.We have seen explicitly that these constraints require an ordering so that the next–to–leading1/NC correction to the axial charge of the nucleon vanishes in the chiral quark model.AcknowledgementsWe would like to thank R. Alkofer for helpful discussions. One of us (HW) acknowledgessupport by a Habilitanden–scholarship of the Deutsche Forschungsgemeinschaft (DFG). Thiswork was supported in part by the US DOE contract number DE-FG-02-85ER 40231.References[1] M. Wakamatsu and T. Watabe, Phys. Lett. B312 (1993) 184;A. Blotz, M. Prasa lowicz and K. Goeke, Phys. Lett. B317 (1993) 195;C. V. Christov, K. Goeke, P. Pobilista, V. Petrov, M. Wakamatsu and T. Watabe, Phys. Lett.B325 (1994) 467;A. Blotz, M. Prasa lowicz and K. Goeke, Axial Properties of the Nucleon with 1/NC Correctionsin the Solitonic SU(3) NJL Model, Bochum University preprint RUB–TPII-41/93, March 1994.[2] A. Hosaka and H. Toki, Phys. Lett. B322 (1993) 1.[3] M. Wakamatsu, Phys. Lett. B331 (1994) 425.7[4] S. Kahana, G. Ripka and V. Soni, Nucl. Phys. A415 (1984) 351;M. C. Birse and M. K. Banerjee, Phys. Rev. D31 (1985) 118;P. Jain, R. Johnson and J. Schechter, Phys. Rev. D38 (1988) 1571.[5] J. Schechter and H. Weigel, Resolving Ordering Ambiguities in the Collective Quantization byParticle Conjugation Constraints, Syracuse University preprint SU–4240–587, UNITU–THEP–22/1994, October 1994.[6] G. S. Adkins, C. R. Nappi and E. Witten, Nucl. Phys. B228 (1993) 552.[7] D. R. Inglis, Phys. Rev. 96 (1954) 1059; 103 (1956) 1786;T. D. Cohen and W. Broniowski, Phys. Rev. D34 (1986) 3472.[8] K. Nishijima, Prog. Theor. Phys. 6 (1951) 614, 1027;A. Pais and R. Jost, Phys. Rev. 87 (1952) 871;L. Michel, Nuovo Cim. 10 (1953) 319;D. Amati and B. Vitale, Nuovo Cim. 2 (1955) 719;C. Goebel, Phys. Rev. 103 (1956) 258;T. D. Lee and C. N. Yang, Nuovo Cim. 10 (1956) 749.8

Particle Conjugation and the 1/N_C Corrections to g_A

https://surface.syr.edu/cgi/viewcontent.cgi?article=1312&amp;context=phy

Particle Conjugation and the $1/N_C$ Corrections to $g_A$

Abstract

Similar works

Full text

Available Versions

Crossref

Syracuse University Research Facility and Collaborative Environment

Particle Conjugation and the 1/NC1/N_C1/NC​ Corrections to gAg_AgA​

Abstract

Similar works

Full text

Available Versions

Crossref

Syracuse University Research Facility and Collaborative Environment

Particle Conjugation and the $1/N_C$ Corrections to $g_A$