We study the distribution P(\omega) of the random variable \omega = x_1/(x_1
+ x_2), where x_1 and x_2 are the wealths of two individuals selected at random
from the same tempered Paretian ensemble characterized by the distribution
\Psi(x) \sim \phi(x)/x^{1 + \alpha}, where \alpha > 0 is the Pareto index and
$\phi(x)$ is the cut-off function. We consider two forms of \phi(x): a bounded
function \phi(x) = 1 for L \leq x \leq H, and zero otherwise, and a smooth
exponential function \phi(x) = \exp(-L/x - x/H). In both cases \Psi(x) has
moments of arbitrary order.
  We show that, for \alpha > 1, P(\omega) always has a unimodal form and is
peaked at \omega = 1/2, so that most probably x_1 \approx x_2. For 0 < \alpha <
1 we observe a more complicated behavior which depends on the value of \delta =
L/H. In particular, for \delta < \delta_c - a certain threshold value -
P(\omega) has a three-modal (for a bounded \phi(x)) and a bimodal M-shape (for
an exponential \phi(x)) form which signifies that in such ensembles the wealths
x_1 and x_2 are disproportionately different.Comment: 9 pages, 8 figures, to appear in Physica 

AN Sosnovskii

AW Thomas

BR Holstein

CA Dominguez

CJ Christensen

DS Beder

G Bardin

G Jonkmans

H Haberzettl

HW Fearing

J Bijnens

JD Bjorken

JF Donoghue

K Kubodera

K Schreckenbach

ML Goldberger

NH Fuchs

P Ackerbauer

Particle Data Group

R Arndt

R Dashen

R Koch

RE Stuckey

RJ Blin-Stoyle

S Choi

S Theberge

SA Adler

T Yamaguchi

V Agrawal

V Bernard

V Roesch

WJ Cummings

English

arXiv

G. Oshanin

Yu. Holovatch

G. Schehr

Crossref

Physica A Statistical Mechanics and its Applications

Proportionate vs disproportionate distribution of wealth of two individuals in a tempered Paretian ensemble

MUCC (Crossref)

Current issues associated with nucleon axial matrix elements are studied, including the Goldberger-Treiman discrepancy, the induced pseudoscalar, and SU(3) chiral perturbation theory

Holstein, BR

ScholarWorks@UMass Amherst

University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics1998Nucleon Axial Matrix ElementsBR Holsteinholstein@physics.umass.eduFollow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubsPart of the Physical Sciences and Mathematics CommonsThis Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in PhysicsDepartment Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contactscholarworks@library.umass.edu.Recommended CitationHolstein, BR, "Nucleon Axial Matrix Elements" (1998). Physics Department Faculty Publication Series. 524.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/524arXiv:nucl-th/9806036v1  12 Jun 1998Nucleon Axial Matrix ElementsBarry R. HolsteinDepartment of Physics and AstronomyUniversity of MassachusettsAmherst, MA 01002 USAandInstitut fu¨r KernphysikForschungszentrum Ju¨lichD-52425 Ju¨lich, GermanyFebruary 9, 2008AbstractCurrent issues associated with nucleon axial matrix elements arestudied, including the Goldberger-Treiman discrepancy, the inducedpseudoscalar, and SU(3) chiral perturbation theory.01 IntroductionI have been given the task of speaking about the nucleon axial matrix ele-ments. In comparison with the many exciting things being discussed at thismeeting this may seem rather prosaic. However, I will try to convince youotherwise by discussing issues associated with the Goldberger-Treiman dis-crepancy, recent and future measurements of the induced pseudoscalar, andthe renormalizations of the axial couplings within SU(3) chiral perturbationtheory.2 The Goldberger-Treimen Discrepancy ORTime Dependence of Fundamental Con-stantsMany years ago Dirac noticed that the ratio of electrical to gravitationalforces between a pair of electrons was equal to α/(Gm2e) ∼ 1040. In askinghimself how such an enormous dimensionless number could arise he noticedthat 1040 is also the age of the universe measured in fundamental units oftime—i.e. the time it takes light to traverse an elementary particle—1017sec./ 10−23 sec.! He then asked if it were possible that the electrical togravitational ratio might change as the universe evolved. It turns out onfurther analysis that this is extremely unlikely—the consequences of evenrelatively small changes to either the fine structure or gravitational constantsturn out to be significant (the anthropic principle),[1] but I am preparedtoday to point out an arena where the changes in a fundamental couplinghave been major, and they have occured within a generation—-the nucleonaxial coupling. Below I give a list of values which I have gathered fromvarious sources, and it is clear that there has been a seven percent increasein gA within a decade!1959 : gA/gV = 1.17± 0.02[2] 1965 : gA/gV = 1.18± 0.02[3]1967 : gA/gV = 1.24± 0.01[4] 1969 : gA/gV = 1.26± 0.02[5] (1)In fact this number continues to increase—the latest published experimentsfrom Grenoble give gA/gV = 1.266±0.004,[6] which is the value I shall employin this note.1Now my facetious discussion in the previous paragraph is at one levelamusing, but at another has some important ramifications when consideredin terms of the Goldberger-Treiman relation[15]MNgA(0) = FpigpiNN(0) (2)which is required by chiral invariance and hence by QCD. Now I have in-dicated in Eq. 2 that the axial coupling gA and the pion-nucleon couplingconstant gpiNN are both to be evaluated at zero momentum transfer. Howeverwhile the former is the (”time-dependent”) number quoted above, the latteris not a physical quantity. What is directly measurable is the pion-nucleoncoupling evaluated at the pion mass-squared—gpiNN (m2pi), so it is useful toexamine the Goldberger-Treiman discrepancy∆pi = 1− MgA(0)FpigpiNN(m2pi)= 1− gpiNN(0)gpiNN(m2pi)(3)In the venerable text Bjorken and Drell this number is described as be-ing less than 0.1, but I want to argue that it must be much less. Indeed,while gpiNN(0) is not an observable, one can show that in reasonable mod-els such as the linear sigma model or via correlated π − ρ exchange oneshould expect ∆pi ≃ 0.02. However, there is another interesting approach viathe so-called Dashen-Weinstein theorem,[7] which uses the fact that whilethere does not exist a prediction for ∆pi in chiral SU(2), since it is given interms of an a priori unknown counterterm, in SU(3) this quantity is given interms of a sum of quark masses times an SU(3) octet operator. Thus a re-lation exists—the Dashen–Weinstein theorem—between corresponding kaonand pion quantities[8, 9]∆pi =√32FKFpimu +mdmu +ms(gΛKNgpiNN∆ΛK −1√6gΣKNgpiNN∆ΣK)(4)where∆K = 1− (MN +MΣ,Λ)gA(0)√2FKg(m2K)={0.32 Λ−0.05 Σ (5)where I have used the values[11, 12]g(m2K) gA(0)Λ −13.5 −0.72Σ 4.3 0.34(6)2Then using (mu +md)/(mu +ms) = m2pi/m2K we find ∆theopi ≃ 0.028. Besideschanges in the size of the axial coupling, however, the size of Fpi decreased by1% in 1990 when it was realized that previous evaluations had not includedthe running of the weak coupling constant,[10] and there has been continuousdebate about the size of gpiNN(m2pi), with current analyses favoring either theKarsruhe value 13.4[13] or the VPI number 13.05.[14] Thus we findgpiNN = 13.4 −→ ∆pi = 4.1%, or ms/mˆ ≈ 48gpiNN = 13.05 −→ ∆pi = 1.5%, or ms/mˆ ≈ 17 (7)so that if the low value of gpiNN is confirmed, the Goldberger-Treiman dis-crepancy would strongly favor the conventional χpt picture (ms/mˆ = 25)over its generalized version, which predicts ms/mˆ < 25.[9]3 The Induced PseudoscalarThe axial matrix element of the nucleon consists in general of two pieces, theusual axial coupling and the induced pseudoscalar1< p(p′)|Aµ|n(p) >= u¯(p′)(gA(q2)γµγ5 + gP (q2) qµ2Mγ5)u(p) (8)and chiral considerations require that this new piece is dominated by its pionpole contributiongP (q2) =4MFpim2pi − q2gpiNN(q2) ≃ 4MFpim2pi − q2gpiNN(m2pi)−2M23gA(0)r2A (9)where rA is the axial radius. This result is generally used in the combinationrP =mµ2MgA(0)gP (q2 = −0.9m2µ) = 6.7 (10)relevant for muon capture. This is the standard approach and is used be-cause the contraction of the four-vector qµ with the lepton tensor resultsin a factor of the lepton mass accompanied by the nucleon matrix element1In general one could also allow an axial tensor coupling, but this ”second class current”is disallowed by G-invariance.3of γ5, which brings in an additional suppression |~q|/2M , meaning that de-spite the extraordinary precision of modern nuclear beta decay experiments,any effects from gP arise only at O[rPm2e/(2Mmµ)] ∼ 10−5!. On the otherhand in muon capture this factor becomes rPmµ/2M , which means that thepseudosclar contributes at the same order as weak magnetism and becomesin principle measurable. The one problem here is that typically one hasonly a single number—the capture rate—to work with so that in order toextract the desired value of rP one must make three reasonable, but stillmodel-dependent, assumptions—i) the validity of CVC in order to extractfV (q2), fM(q2) from electron; scattering data; ii) the validity of the impulseapproximation to evaluate gA(q2); and iii) the assumption of G-invariance torule out the presence of second class currents. Using these assumptions onefinds the experimental valuesrP =6.5± 2.4 H [17]6.9± 0.2 3He[18]9.0± 1.7 12C[19](11)which are in agreement with the chiral expectations. It should be notedthat the extraordinary precision associated with the 3He number is allowedbecause of a spectacular new PSI experiment which measured the capturerate to 3%Γµ(3He) = 1496± 4sec.−1 (12)In order to eliminate some of this model dependence, there are additionalapproaches which have been and which are being pursuedi) Radiative muon capture on hydrogen: This is the approach which hasreceived the most recent attention, because the result[20]rP = 9.8± 0.7± 0.3 (13)is at variance with the chiral prediction at the 3σ level. Now thisTRIUMF measurement is extraordinarily difficult both because of thetiny 10−8 branching fraction compared to ordinary capture and be-cause of the presence of many possible experimental backgrounds.However, it has the advantage that at the maximum photon energykmax = 100MeV the momentum transfer is qmax = m2µ compared to thevalue q2µ = −0.9m2µ which obtains in the ordinary muon capture case.4Integrated over the photon spectrum this leads to an enhancement ofabout a factor of three for pseudoscalar effects in RMC over those inOMC.[21] Clearly this is an experiment that should be repeated.ii) Threshold pion photoproduction on hydrogen: This might seem astrange place to study nucleon axial matrix elements, but this is possi-ble because of the PCAC relation[22]< π+np′|V emµ |pp > q→0−→−i√2Fpi< np′|A−µ |pp > (14)The variation in q2 which is allowed by the use of electroproductionrather than photoproduction permits a check of the q2-variation of bothaxial matrix elements. A recent Saclay experiment produced in this waya measurement of the axial radius rA which was in good agreementwith parallel neutrino scattering measurements, when a small chiralsymmetry offset is included, but more importantly for our case for thefirst time a measurement was made of the shape of gP (q2) which wasin good agreement with the pion pole dominance assumption.[23]iii) Correlations in polarized muon capture on 3He: The final methodwhich is being pursued at present goes back to an old idea to mea-sure the correlation of the final neutrino direction with initial statepolarization in the case that the muon and target are polarized.[24] Ofcourse, the muon is almost completely longitudinally polarized at thetime of its capture, but unless the target too is polarized this polar-ization for the most part lost as the muon cascades down through thevarious atomic levels before finally reaching the ground state–1S–levelfrom which it is captured. In the general case, when one has muonpolarization P ′nˆ and (spin 1/2) target polarization P nˆ the decay dis-tribution is found to be of the formd2ΓµdΩkˆ= A− 2PP ′B − 12(P + P ′)Cnˆ · kˆ + 2PP ′D[(nˆ · kˆ)2 − 13](15)where the structure functions A,B,C,D are functions of the weak form-factors gv, gM , gA, gP whose specific forms can be found in the literature.The important feature for our case is that D has a strong dependence ongP , and thus measurement of the angular correlations allows one to pick5out the induced pseudoscalar. This measurement was attempted unsuc-cessfully many years ago at LAMPF, but only recently has the abilityto polarize 3He at high levels given hope that this experiment can ac-tually be carried out. Preliminary results at TRIUMF are encouraging,but the precision is not yet at a level where anything definitive can besaid.[25]4 Axial Couplings and SU(3)The existence of semileptonic hyperon decays such as Λ, n →pe−ν¯e,Σ−,Ξ− → Λe−ν¯e,Ξ− → Σ0e−ν¯e, etc. allows the probing of axialmatrix elements in SU(3). Indeed it has long been known that to leadingorder in chiral symmetry one can describe such decays in terms of simple f,dparameters, e.g.gpnV = fV gnpA = fA + dAgpΛV = −√32fV gpΛA = −√16(3fA + dA)gnΣ−V = −fV gnΣ−A = −fA + dA(16)This type of fit yields remarkably good results—χ2d.o.f ≈ 8.5 for ten degreesof freedom, when small ≤ 5% quark model symmetry breaking effects areadded.[26] One can try to do even better by including chiral loops usingheavy baryon chiral perturbation theory. At one loop one finds results[27]gijA = (fA, dA)ij +∑mβijmm2m lnm2mµ2(17)However, this inclusion of supposedly model-independent corrections bringsin modifications to the axial couplings at the level of 30-50% which resultsin a vastly increased χ2. Of course, one can restore experimental agreementby the addition of appropriately chosen higher order counterterms, but thenone worries about the convergence of the chiral expansion and is certainlyjustified in asking what is going on. Our answer is that this simple chiralpicture omits an important piece of physics, which is finite hadronic size.[28]The simple chiral expansion assumes (at lowest order) propagation of mesonsbetween point baryons, while in the real world any such propagation takesplace between objects about a fermi or so in size. That means that only6the long-distance component of the meson loop is really model-independentand to be trusted. One can eliminate such short distance components by useof a cutoff regularization with scale ∼300 MeV ≤ Λ ≤∼ 600 MeV of orderinverse baryon size rather than the usual dimensional regularization whichmixes both long and short distance effects.[29] The specific form of the cutofffunction is unimportant, so for calculational purposes it is useful to use asimple dipole. The result is that the heavy baryon integral responsible forloop corrections to axial couplingsIij(m2) =∫d4k(2π)4kikj(k0 − iǫ)2(k2 −m2 + iǫ) =−iδij16π2m2 lnm2µ2(18)is replaced byI˜ij =∫d4k(2π)4kikj(k0 − iǫ)2(k2 −m2 + iǫ)(Λ2Λ2 − k2)2=−iδij16π2J(m2) (19)whereJ(m2) =Λ4(Λ2 −m2)2m2 lnm2Λ2+Λ4Λ2 −m2 (20)We see then that unlike Eq. 18 which emphasizes heavy meson (short-distance) propagation over that of light mesons, in the large mass limitJ(m2)m2>>Λ2−→ Λ4m2lnm2Λ2→ 0 (21)On the other hand in the large cutoff limit we haveJ(m2)Λ2>>m2−→ Λ2 +m2 ln m2Λ2(22)which reproduces the usual dimensional regularization result plus a quadraticterm in Λ. That this latter piece does not destroy the chiral invariance canbe seen from the feature that it can be absorbed in a renormalization of thebasic couplings[30]drA = d(0)A −32dA(3d2A + 5f2A + 1)Λ216π2F 2pif rA = f(0)A −16fA(25d2A + 63f2A + 9)Λ216π2F 2pi(23)7dim. Λ=300 Λ=400 Λ=500 Λ=600gA(p¯n) 1.72 0.37 0.53 0.69 0.84gA(p¯Λ) -1.78 -0.34 -0.51 -0.67 -0.84gA(Λ¯Σ−) 1.17 0.23 0.34 0.45 0.56gA(n¯Σ−) 0.36 0.07 0.10 0.14 0.17gA(Λ¯Ξ−) 0.83 0.15 0.23 0.31 0.39gA(Σ¯0Ξ−) 2.46 0.45 0.68 0.91 1.15Table 1: Given are the nonanalytic contribtions to gA for various transitionsin dimensional regularization and for various values of the cutoff parameterΛ in MeV.However, in this procedure with reasonable values of the cutoff, the SU(3)chiral expansion is now under control, as can be seen in Table 1.This brings such results into agreement with typical chiral bag calcula-tions, such as the cloudy bag,[31] and there is no longer any need to appendlarge counterterm contributions in higher orders.5 ConclusionWe have above considered an old subject—that of nucleon axial matrixelements—from the point of view of modern experiments. I hope that I haveconvinced you that despite the age of the field, the new results in the areasof Goldberger-Treiman discrepancies, induced pseudoscalar measurements,and SU(3) chiral perturbative studies promise continued interest even as weapproach the millenium.AcknowlegementIt is a pleasure to acknowledge support from the Alexander von HumboldtFoundation and the hospitality of Forschungszentrum Ju¨lich. This work wasalso supported by the National Science Foundation.8References[1] For a recent slant and for further references see V. Agrawal et al., hep-ph/9707380.[2] A.N. Sosnovskii et al., Sov. Phys. JETP 8, 739 (1959).[3] S.A. Adler, Phys. Rev. Lett. 14, 1051 (1965).[4] C.J. Christensen et al., Phys. Lett. B26, 11 (1967); Phys. Rev. D5,1628 (1972).[5] R.J. Blin-Stoyle, Fundamental Interactions and the Nucleus,North-Holland, New York (1969).[6] K. Schreckenbach et al., Phys. Lett. B259, 353 (1991).[7] R. Dashen and M. Weinstein, Phys. Rev. 188, 2330 (1969).[8] C.A. Dominguez, Riv. del Nuovo Cimento 8,#6,1 (1985).[9] N.H. Fuchs, H. Sazdjian, and J. Stern, Phys. Lett. B238, 380 (1990).[10] B.R. Holstein, Phys. Lett. B244, 83 (1990).[11] Particle Data Group, Phys Rev. D50, 1173 (1995).[12] H. Haberzettl et al., nucl-th/9804051.[13] R. Koch and E. Pieterinin, Nucl. Phys. A336, 331 (1980).[14] R. Arndt et al., Phys. Rev. Lett. 65, 157 (1990).[15] M.L. Goldberger and S.B. Treiman, Phys. Rev. 110, 1478 (1958).[16] J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics,McGraw-Hill, New York (1964).[17] G. Bardin et al., Phys. Lett. B104, 320 (1981).[18] P. Ackerbauer et al., Phys. Lett. B417, 224 (1998).[19] V. Roesch et al., Phys. Rev. Lett. 46, 1507 (1981).9[20] G. Jonkmans et al., Phys. Rev. Lett. 77, 4512 (1996).[21] H.W. Fearing, Phys. Rev. C21, 1951 (1980); D.S. Beder and H.W. Fear-ing, Phys. Rev. D39, 3493 (1989).[22] A.I Vainshtein and V.I. Zakharov, Nucl. Phys. B36 (1972); V. Bernard,N. Kaiser, and U.-G. Meissner, Nucl. Phys, A607, 379 (1996).[23] S. Choi et al., Phys. Rev. Lett. 71, 3927 (1993).[24] See, e.g. B.R. Holstein, Phys. Rev. C3, 1964 (1972).[25] W.J. Cummings et al., Proc. WEIN ’95, ed. H. Ejiri, T. Kishimoto, andT. Sato, World Scientific, Singapore (1995), p. 381; G. Cates, privatecommunication.[26] J.F. Donoghue, B.R. Holstein, and S.W. Klimt, Phys. Rev. D35, 934(1987][27] J. Bijnens, H. Sonoda, and M.B. Wise, Nucl. Phys. B261, 185 (1985).[28] J.F. Donoghue, B.R. Holstein, and B. Borasoy, hep-ph/9804281.[29] J.F. Donoghue and B.R. Holstein, hep-ph/9803312.[30] M.A. Luty and M. White, Berkeley prepring LBL33993 (1993).[31] See, it e.g. T. Yamaguchi et al., Nucl. Phys. A500, 129 (1989); K.Kubodera et al., Nucl. Phys. A439, 695 (1985); S. Theberge et al.,Phys. Rev. D22, 2838 (1980); A.W. Thomas, J. Phys. G7, L283 (1981);R.E. Stuckey and M.C. Birse, J. Phys. G23, 29 (1997).10

Nucleon Axial Matrix Elements

This is the pre-published version received from arXiv.Current issues associated with nucleon axial matrix elements are studied, including the Goldberger-Treiman discrepancy, the induced pseudoscalar, and SU(3) chiral perturbation theory

Barry R. Holstein

Proportionate vs disproportionate distribution of wealth of two
  individuals in a tempered Paretian ensemble

Proportionate vs disproportionate distribution of wealth of two individuals in a tempered Paretian ensemble

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