(2018) Journal of High Energy Physics. 2018, 1, 8. Abstract
In the search for QFT’s that admit boundstates, we reinvestigate the two dimensional Gross-Neveu model, but with massive fermions. By computing the self-energy for the auxiliary boundstate field and the effective potential, we show that there are no bound states around the lowest minimum, but there is a meta-stable bound state around the other minimum, a local one. The latter decays by tunneling. We determine the dependence of its lifetime on the fermion mass and coupling constant.
(2014) Journal of High Energy Physics. 12, Abstract
In a previous paper [ 1], we analyzed the theory of massive fermions in the fundamental representation coupled to a U(N) Chern-Simons gauge theory in three dimensions at level K. It was done in the large N, large K limits where lambda = N/K was kept fixed. Among other results, we showed there that there are no high mass "quark anti-quark" bound states. Here we show that there are no bound states at all.
(2013) Journal of High Energy Physics. 12, Abstract
We analyze the theory of massive fermions in the fundamental representation coupled to a U(N) Chern-Simons gauge theory at level K. It is done in the large N, large K limits where lambda = N/K is kept fixed. Following  we obtain the solution of a Schwinger-Dyson equation for the two point function, the exact expression for the fermion propagator and the partition function at finite temperature. We prove that in the large K limit there exists an infinite set of classically conserved high spin currents also when a mass is introduced, breaking the conformal invariance. In analogy to the seminal work of 't Hooft on two dimensional QCD, we write down a Bethe-Salpeter equation for the wave function of a "quark anti-quark" bound state. We show that unlike the two dimensional QCD case, the three dimensional Chern-Simons theory does not admit a confining spectrum.
(2013) Journal of High Energy Physics. 2013, 8, 96. Abstract
Recent observations by Belle and BESIII of charged quarkonium-like resonances give new stimulus for theoretical investigation of exotic hadrons in general and heavy tetraquarks in particular. We use QED(2), a confining theory, as a model for the masses and decays of tetraquarks. Here we discuss the states (Q (Q) over barq (q) over bar) and (QQ (q) over bar(q) over bar) (and its anti-particle), where Q and q are two fermion flavors with masses M and in, so that M > m. We then discuss decay modes of these states into (Q (Q) over bar), (q (q) over bar), (Q (q) over bar), ((Q) over barq). It turns out that (Q (Q) over barq (q) over bar) is stable, while (QQ (q) over bar(q) over bar) is not.
(2006) Nuclear Physics B-Proceedings Supplements. 161, p. 2-7 Abstract
We consider a "one-current" state obtained by application of a colour current on the "adjoint" vacuum in QCD(2) with quarks in the fundamental representation. The quarks are taken to be massless. The theory on the light-front can be "currentised" namely formulated in terms of currents only. The adjoint vacuum is obtained by applying a current derivative at zero momentum on the singlet vacuum. In general, the "one-current" states are not eigenstates of M-2 = 2P(+)P(-), except in the large N-f limit. Problems with infrared regularisations are pointed out. Connection to fermionic structure is made. This talk is based on the paper of Ref.  by the same authors.
Exotic baryons in two-dimensional QCD(2005) Journal of High Energy Physics. 8, Abstract
Two-dimensional QCD has often been used as a laboratory for studying the full four-dimensional theory, providing, for example, an explicit realization of baryons as solitons. We review aspects of conventional baryons in two-dimensional QCD, including the classical and quantum contributions to their masses. We then discuss the spectrum of exotic baryons in two-dimensional QCD, commenting on the solitonic radius inferred from the excitation spectrum as well as the two-dimensional version of the Goldberger-Treiman relation relating meson couplings to current matrix elements. Our treatment of two-dimensional QCD, via the semi-classical quantization of collective coordinates, is consistent with chiral-soliton approaches to normal and exotic baryons in four dimensions, but is not able to resolve all the issues arising in four-dimensional models.
(2004) Nuclear Physics B. 703, 2-Jan, p. 320-332 Abstract
We consider a "one current" state, which is obtained by the application of a color current on the "adjoint" vacuum. This is done in QCD(2), with the underlying quarks in the fundamental representation. The quarks are taken to be massless, in which case the theory on the light-front can be "currentized", namely, formulated in terms of currents only. The adjoint vacuum is shown to be the application of a current derivative, at zero momentum, on the singlet vacuum. We apply the operator M-2 = 2P(+)P(-) on these states and find that in general they are not eigenstates of M-2 apart from the large N-f limit. Problems with infra-red regularizations are pointed out. We discuss the fermionic structure of these states. (C) 2004 Elsevier B.V. All rights reserved.
(2003) Physics Letters B. 566, 4-Mar, p. 201-206 Abstract
Extending earlier work on strong-coupling meson-baryon scattering in QCD(2), we derive the effective meson-baryon action for any value of the coupling constant, in the large-N-C limit. Colour degrees of freedom play an important role, and we show that meson-baryon scattering can be formulated as a relativistic potential problem. We distinguish two cases that are non-trivial for unequal quark masses, and present the resulting equations for meson-baryon scattering amplitudes. (C) 2003 Published by Elsevier B.V.
(2003) Ichep 2002, Proceedings. p. 391-393 Abstract
Extending previous works on the spectrum of QCD(2), we now investigate the 2D analogue of meson-baryon scattering. We use semi-classical methods, perturbing around classical soliton solutions. In the case of one flavor, we find that the effective potential is reflectionless. In the case of several flavors, the method yields a potential which depends on the momentum of the incoming particle. In this case there is both transmission and reflection. In both cases no resonances appear.
(2002) Physics Letters B. 541, 4-Mar, p. 273-280 Abstract
Extending previous works on the spectrum of QCD(2), we now investigate the 2D analogue of meson-baryon scattering. We use semi-classical methods, perturbing around classical soliton solutions. We start with the Abelian case, corresponding to one flavor, and find that in this case the effective potential is reflectionless. We obtain an explicit expression for the forward phase shift. In the non-Abelian case of several flavors, the method yields a potential which depends on the momentum of the incoming particle. In this case there is both transmission and reflection. In both cases no resonances appear. As a byproduct, we derive the general conditions for a 2D scalar quantum field theoretical action to yield a reflectionless effective potential when one expands in small fluctuations about the classical solution. (C) 2002 Elsevier Science B.V. All rights reserved.
(2001) Nuclear Physics B. 596, 2-Jan, p. 459-470 Abstract
We discuss the spectra of multi-flavor massless QCD(2). An approximation in which the Hilbert space is truncated to two currents states is used. We write down a 't Hooft like equation for the wave function of the two currents states. We solve this equation for the lowest massive state and find an excellent agreement with the DLCQ results. In addition, the 't Hooft model and the large NS limit spectra are re-derived by using a description in terms of currents. (C) 2001 Elsevier Science B.V. All rights reserved.
(1999) International Journal of Modern Physics A. 14, 16, p. 2475-2493 Abstract
We review and elaborate on properties of the string tension in two-dimensional gauge theories. The first model we consider is massive QED in the m
(1999) Physics Letters B. 449, 2-Jan, p. 76-80 Abstract
We show that the string tension in N=1 two-dimensional super Yang-Mills theory vanishes independently of the representation of the quark anti-quark external source. We argue that this result persists in SQCD(2) and in two-dimensional gauge theories with extended supersymmetry or in chiral invariant models with at least one massless dynamical fermion. We also compute the string tension for the massive Schwinger model, as a demonstration of the method of the calculation. (C) 1999 Elsevier Science B.V. AU rights reserved.
(1999) Nuclear Physics B. 537, 3-Jan, p. 503-515 Abstract
Massless QCD(2) is dominated by classical configurations in the large-N-f limit. We use this observation to study the theory by finding solutions to equations of motion, which are the non-abelian generalization of the Schwinger equation. We find that the spectrum consists of massive mesons with M-2 = e(2)N(f)/2 pi, which correspond to abelian solutions. We generalize previously discovered non-abelian solutions and discuss their interpretation. We prove a no-go theorem ruling out the existence of soliton solutions. Thus the semi-classical approximation shows no baryons in the case of massless quarks, a result derived before in the strong-coupling limit only. (C) 1999 Elsevier Science B.V.
(1998) Physical Review Letters. 80, 3, p. 430-433 Abstract
We compute the string tension in massive QCD(2). It is shown that the string tension vanishes when the mass of the dynamical quark is zero, with no dependence on the representations of the dynamical or of the external charges. When a smalt mass (m much less than e) is added, a tension appears and we calculate its value as a function of the representations.
(1997) Nuclear Physics B. 496, 2-Jan, p. 285-301 Abstract
We analyze the question of screening versus confinement in bosonized massless QCD in two dimensions. We deduce the screening behavior of massless SU(N-c) QCD with flavored fundamental fermions and fermions in the adjoint representation. This is done by computing the potential between external quarks as well, as by bosonizing also the external sources and analyzing the states of the combined system, We write down novel ''non-Abelian Schwinger like'' solutions of the equations of motion, compute their masses and argue that an exchange of massive modes of this type is associated with the screening mechanism. (C) 1997 Elsevier Science B.V.
(1997) Physical review D. 55, 7, p. 3994-4000 Abstract
We calculate in two-dimensional QCD (QCD,) the ratios of baryonic matrix elements of Delta I=2 and Delta I=0 four-fermion operators, with a view to understanding better the mechanism of Delta I=1/2 enhancement in QCD(4). We find relatively small suppressions of both the scalar-scalar and vector-vector Delta I=2 four-fermion operators. We discuss the possible implications of these results, in view of a suggestion that gluon condensation may be an important contributing factor in the Delta I=1/2 enhancement seen in QCD(4). At the technical level, our calculation of the vector-vector operator matrix element requires a treatment of the lime dependence of the QCD(2) soliton which had not been developed in previous phenomenological calculations within this model.
(1994) Nuclear Physics B. 429, 1, p. 75-89 Abstract
It is shown that in a formulation of Yang-Mills theory in two dimensions in terms of A = if(-1)partial derivative f, (A) over bar = if partial derivative (f) over bar(-1) with f(z, (z) over bar>, (f) over bar(z, (z) over bar) is an element of [SU(N-C)](c) the complexification of SU(N-C), reveals certain subtleties. ''Physical'' massive color singlet states seem to exist. When coupled to N-F quarks the coupling constant is renormalized in such a way that it vanishes for the pure Yang-Mills case. This renders the above states massless and unphysical. In the abelian case, on the other hand, the known results of the Schwinger model are reproduced with no need of such a renormalization. The massless QCD(2) theory is analyzed in similar terms and peculiar massive states appear, with a mass of e(c) root N-F/2 pi.
(1994) Nuclear Physics B. 424, 1, p. 3-17 Abstract
We derive an effective low-energy action for QCD in 4 dimensions. The low energy dynamics is described by chiral fields transforming nontrivially under both color and flavor. We use the method of anomaly integration from the QCD action. The solitons of the theory have the quantum numbers of quarks. They are expected to be the constituent quarks of hadrons. In two dimensions our result is exact, namely the bosonic gauged action of WZW.
(1993) Physics Reports-Review Section Of Physics Letters. 223, 6, p. 309-348 Abstract
Quantum Chromodynamics (QCD) is believed to describe the strong interactions. In the asymptotic domain of large momenta, improved perturbation theory describes phenomena by use of point-like quarks and gluons. But the spectrum and wave functions are in the nonperturbative domain, for which not much can be done analytically in four dimensions. In order to develop analytical methods physicists turned to simpler models, like QCD2, the theory in one space and one time dimensions. This review is devoted to the application of bosonization techniques to two-dimensional QCD. We start with a description of the ''abelian bosonization''. The methods of the abelian bosonization are applied to several examples like the Thirring model, the Schwinger model and QCD2. The failure of this scheme to handle flavored fermions is explained. Witten's non-abelian bosonization rules are summarized including the generalization to the case of fermions with color and flavor degrees of freedom. We discuss in detail the bosonic version of the mass bilinear of colored-flavored fermions in various schemes. The color group is gauged and the full bosonized version of massive multiflavor QCD2 is written down. The strong coupling limit is taken in the ''product scheme'' and then in the U(N(F) x N(C)) scheme. Once the multiflavor QCD2 action in the interesting region of the low energies is written down, we extract the semiclassical low-lying baryonic spectrum. First, classical soliton solutions of the bosonic action are derived. Quantizing the flavor space around those classical solutions produces the masses as well as the flavor properties of the two-dimensional baryons. In addition, low-lying multibaryonic solutions are presented, as well as wave functions and matrix elements of interest, like qqBAR content.
(1993) Journal Of Physics A-Mathematical And General. 26, 2, p. 301-312 Abstract
Field-theoretic models for fields taking values in quantum groups are investigated. First we consider SU(q)(2) sigma model (q real) expressed in terms of basic notions of non-commutative differential geometry. We discuss the case in which the sigma models fields are represented as products of conventional sigma fields and of the coordinate-independent algebra. An explicit example is provided by the U(q)(2) sigma model with q(N) = 1, in which case quantum matrices U(q)(2) are realised as 2N x 2N unitary matrices. Open problems are pointed out.
(1992) Nuclear Physics B. 382, 2, p. 189-212 Abstract
We exhibit static solutions of multi-flavour QCD in two dimensions that have the quantum numbers of baryons and mesons, constructed out of quark and anti-quark solitons. In isolation the latter solitons have infinite energy, corresponding to the presence of a string carrying the non-singlet colour flux off to spatial infinity. When N(c) solitons of this type are combined, a static, finite-energy, colour singlet solution is formed, corresponding to a baryon. Similarly, static meson solutions are formed out of a soliton and an anti-soliton of different flavours. The stability of the mesons against annihilation is ensured by flavour conservation. The static solutions exist only when the fundamental fields of the bosonized lagrangian belong to U(N(c) X N(f)) rather than to SU(N(c))xU(N(f)). Discussion of flavour-symmetry breaking requires a careful treatment of the normal-ordering ambiguity. Our results can be viewed as a derivation of the constituent quark model in QCD2, allowing a detailed study of constituent mass generation and of the heavy-quark symmetry.
(1991) Physics Letters B. 272, 4-Mar, p. 333-338 Abstract
We exhibit soliton solutions of QCD in two dimensions that have the quantum numbers of quarks. They exist only for quarks heavier than the dimensional gauge coupling, and have infinite energy, corresponding to the presence of a string carrying the non-singlet color flux off to spatial infinity. The quark solitons also disappear at finite temperature, as the temperature-dependent effective quark mass is reduced in the approach to the quark/hadron phase transition.
(1980) Physics letters. B. 91, 2, p. 239-244 Abstract
The operator product expansion, of appropriate products of quark fields, is used to find the anomalous dimensions which control the short distance behavior of hadronic wave functions. This behavior in turn controls the high-Q2 limit of hadronic form factors. In particular, we relate each anomalous dimension of the nonsinglet structure functions to a corresponding logarithmic correction factor to the nominal αS(Q2)/Q2 fall off of meson form factors. Unlike the case of deep inelastic lepton-hadron scattering, the operator product necessary here involves extra terms which do not contribute to forward matrix elements.
(1980) Nuclear physics. B. 165, 1, p. 185-188 Abstract
The general condition for the validity of the usual perturbation expansion, in gauges with copies, is given. This includes abelian and non-abelian cases. The condition is that the potentials of copies be “far” from each other.
(1979) Nuclear physics. B. 148, 1-2, p. 74-92 Abstract
Non-Abelian gauge theory of quarks and gluons with SU(n) symmetry, in one-space and one-time dimensions, is investigated in the large-n limit. Special attention is given to the infrared regularization. The fermion propagator turns out to be cut-off independent. It is shown that in the n→∞ approximation, tachyons appear in the two-point function of the scalar density for mass m=0 of the quarks. This is not the case when m≠0 with m→0 is taken. Thus m=0 is singular. It is also shown that the singlet vector current has a free two-point function for m=0, which entails a consistency condition on bound state wave functions.
(1979) Physical review D. 19, 2, p. 540-548 Abstract
The large-order behavior of perturbation theory of φ4 theory in four dimensions in the presence of mass terms is estimated. A new scale comes into the coefficient functions. For mass m and order K this scale is m√K.
(1977) Physical review. D, Particles and fields. 15, 8, p. 2275-2281 Abstract
It is shown that the standard infrared-cutoff procedures are inconsistent with the general axial gauge in 't Hooft's two-dimensional model of confinement.
(1975) Physical review. D, Particles, fields, gravitation and cosmology. 11, 10, p. 2781-2802 Abstract
Four-fermion interactions of the current-current type with U(n) symmetry, in one space and one time dimension, are investigated. It is shown that the equations of motion yield scale-invariant solutions only for two values of the coupling gv of the SU(n) currents, namely gv=0 and gv=4π(n+1). This holds for any value of the coupling gB of the U(1) currents. For the above two values of gv and any gB the theory is solved completely. Operator products of spinor fields are shown to be equal to c-number functions singular on the light cone times analytic bilocal operators expressed in terms of currents and free spinor fields. The currents are free for the above two values of gv. The connection with the coupling as defined through four-point functions is discussed, and it turns out that the combination corresponding to SU(n) coupling is zero for both solutions. However, the solution for gv=4π(n+1) exhibits nontrivial four-point functions also for gB=0. It is shown, in an expansion around gv=0, that there is only one Callan-Symanzik function β which depends only on gv and that gv=0 is relevant to the ultraviolet limit of the gv>0 theories. When mass terms are introduced, this still holds in an infinite interval for gB, which is bounded below by a certain negative value and in which the mass term is soft.
(1973) Physics letters. B. 46, 3, p. 439-442 Abstract
Generalizations of the Thirring model to Fermi fields with U(n) symmetry are treated. When interactions quadratic in the SU(n) currents are introduced, scale invariance (with anomalous dimensions) is maintained only for values of the coupling gv=0 or gv=4φ(n+1).
(1973) Physical review letters. 31, 2, p. 135-138 Abstract
It is argued that light-cone dominance ideas are compatible with canonical scaling in the process e-e+ → hadrons + anything and a logarithmic increase in hadronic multiplicity.
(1972) Physical review. D, Particles and fields. 6, 4, p. 988-1007 Abstract
Exact expansions of operator products, in terms of c-number functions singular on the light cone and regular operators, are exhibited explicitly in the Thirring model. For the products ψ1(x)ψ†1(x′) and ψ2(x) ψ†2(x′) of fermion fields the expansion reduces to one term only, with the c-number function having a singularity on the light cone which depends on the coupling constant, and the regular operator depending only on the currents, which are free. The resulting formula allows one to calculate all Wightman functions in terms of current matrix elements and thereby provides a simple and complete solution to the Thirring model. The different charge sectors are realized as inequivalent irreducible representation spaces of the canonical current commutation relations, on which the charged field ψ acts as an intertwining operator.
(1971) Annals of Physics. 66, 1, p. 373-389 Abstract
The recently proposed expansion of products of local operators when their space-time distance approaches the light cone is motivated and discussed. Attempts to prove the expansion from Wilson's short distance expansion are analyzed. Applications to processes involving currents in certain asymptotic domains are reviewed and difficulties are pointed out.
(1971) Physical review letters. 26, 13, p. 798-801 Abstract
It is shown that light-cone dominance ideas apply to exclusive processes as well. These processes must include a photon of large virtual mass, as for example deep pion electroproduction. The method provides for a new general parametrization for amplitudes in the Bjorken limit. If correct, it will lend strong support to the ideas of light-cone expansions of operator products and provide a way to study the structure, at short distances, of the commutators of electromagnetic currents and strong sources.
(1970) Physical review letters. 25, 14, p. 966-969 Abstract
Consequences of the assumption that current commutators have power-type singularities near the light cone are derived. An extension of Wilson's short-distance expansion of commutators is suggested for almost lightlike distances. Consequences for power-type singularities are derived for processes involving currents. Especially, generalized scaling laws are obtained for inelastic electron-nucleon scattering. The connection with Regge behavior is discussed. An application to electron-positron annihilation into hadrons is also made.
(1969) Physical Review. 180, 5, p. 1556-1571 Abstract
Free fields of massless particles transforming covariantly under the Poincaré group are constructed. The allowed infinite- and finite-dimensional representations of the Lorentz group are obtained. The wave functions are calculated in these representations in various bases. The commutation rules are computed, and turn out to be nonlocal for any infinite-dimensional fields. The transformation law of a certain irreducible infinite-dimensional representation is shown to coincide, for its lowest-spin component, with the usual, radiation-gauge, vector-potential transformation law, as already discovered by Bender.
(1969) Physical review letters. 22, 11, p. 572-576 Abstract
It is shown that a class of theories, of which Sugawara's theory of currents and certain Yang-Mills type theories are examples, possess a particular symmetry which is not, even approximately, present in the observed strong interactions. The symmetry would imply that all mesons and all massive fermions have partners with opposite parity and opposite charge conjugation.
(1968) Physical Review. 171, 5, p. 1442-1452 Abstract
We discuss the possibility of constructing, out of particle creation and destruction operators, local quantum fields that transform as representations of the homogeneous Lorentz group. Our immediate goal is to write down a consistent local quantum field theory which can simultaneously describe many particles with different masses and spins. In the case that the field is a finite-dimensional irreducible Lorentz tensor, we are able to carry through our program with no restrictions on the masses considered as functions of the spin, provided the usual connection between spin and statistics is satisfied. However, when the field transforms as a unitary irreducible representation of the homogeneous Lorentz group (an infinite-dimensional representation), the requirement of locality, along with the physical assumption that the masses are bounded below, m(j)≥m0>0, leads to the restriction that the masses are independent of the spin. This property is shown to hold when the transformation law of the field is taken to be an irreducible finite-dimensional representation ⊗ a unitary irreducible representation. The physical consequences of this result and possible methods for evading it are discussed. Finally, an Appendix is included, where the related problem of orthogonality properties of timelike solutions to infinite-component wave equations is examined. In particular, we show that when the solutions of such wave equations transform as unitary irreducible representations of the homogeneous Lorentz group, only the Majorana representations support a scalar product, which is orthogonal for different spins.
(1968) Physical Review. 170, 5, p. 1353-1359 Abstract
We show that Sugawara's theory of currents can be obtained as a formal limit of the massive Yang-Mills theory. In this limit, g0→0 and m0→0 in such a way that m02g02=C, the constant appearing in Sugawara's theory. The limiting procedure is used to incorporate electromagnetism, the hypothesis of partially conserved axial-vector current, and SU(3) breaking into the theory, retaining canonical structure for electromagnetism.
(1968) Physical Review. 165, 5, p. 1555-1556 Abstract
Equations for the divergence of the vector and axial-vector currents follow from the assumptions of Lorentz invariance, locality, the chiral SU(3)×SU(3) commutators between charges and charge densities, and the usual electromagnetic and weak Hamiltonians. The divergence conditions lead to derivations of the low-energy meson theorems which do not involve Schwinger terms.
(1967) Physical Review. 159, 5, p. 1428-1430 Abstract
It is shown, directly from proper Lorentz invariance and a positive Hilbert-space metric, that the vacuum expectation value ⟨0∣∣[j0(x),jk†(y)]∣∣0⟩ cannot vanish unless jμ(x)∣∣0⟩=jμ†(x)∣∣0⟩≡0. Neither locality nor Källén-Lehmann—type representations are needed. The same is demonstrated for ⟨0∣∣[Skl(x),S0m†(y)]∣∣0⟩, for any antisymmetric tensor Sμν. The explicit dependence of jμ and Sμν on the fields with which they interact is an immediate consequence in our approach. Similarly, it is easy to show that χ(x), χ(x)=∂μjμ(x), does not commute with j0†(y) for y0=x0, unless χ(x)∣∣0⟩=χ†(x)∣∣⟩≡0.
(1967) Physical review. 156, 5, p. 1623-1624 Abstract
The dependence of spontaneous symmetry breaking on the representation is discussed. In case the usual symmetry-breaking condition is not realized, a modified condition may be found, which replaces the former in proofs of the Goldstone theorem.
(1966) Physical Review. 143, 4, p. 1135-1138 Abstract
Closed formulas are given for the relativistic helicity eigenstates of a wave function for any spin. Helicity amplitudes for the production of higher spin particles are then computed and brought to a simple form for pion and vector-meson exchange.
Average charge distributions in systems of nucleons and pions(1966) Proceedings of the 9th International Conference on Cosmic Rays, London, September 1965. Bombay: . Vol. 2. p. 946-949 Abstract
Closed expressions are derived for the average charge distributions in systems of nucleons and pions with given total isotopic spin and charge. It is also shown that in nucleon-nucleus collisions with multiple pion production, on the average one third of the pions produced are neutral and the difference between the numbers of positive and negative pions tends to +1/2 or 1/ for proton- or neutronnucleus collisions, respectively, as the total number of produced pions tends to infinity
(1965) Journal of Mathematical Physics. 6, 12, p. 1855-1856 Abstract
A simple connection between the radial Schrödinger equation for the bound states of a hydrogen atom with angular momentum l and that of an isotropic harmonic oscillator of even dimension from 2 to 4l + 4 is noted. The case of highest dimensionality 4l + 4 is shown to lead in a very simple way to the energy levels and degeneracies of the hydrogen atom, once the energy levels of a one‐dimensional harmonic oscillator are known.
(1965) Physical Review. 140, 4B, p. B1151-B1156 Abstract
A general method is given for calculating the exact form of the helicity eigenstates of a wave function for any spin. Explicit expressions of the wave functions for spin S=1, 32, 2, and 52 are listed. Examples of helicity amplitudes for the production of S=32+ and S=52+ isobars in proton-proton scattering are computed.
Derivation of the muon charge ratio using a simple pion-production model(1964) Proceedings of the International Conference on Cosmic Rays, Jaipur, December 2-14, 1963 / edited by R.R. Daniel ... [et al.] ; under the auspices of the International Union of Pure and Applied Physics, and the Department of Atomic Energy, Government of In. Bombay: . Vol. 6. p. 45-50 Abstract
The observed vertical meson (ut)/() ratio at sea level was previously explained by several collision models, but these models were unsuccessful in some respects. Another modification of the simple mesons(7) only model is considered. The values so obtained agree very well with observed charge ratio values. The experimental and calculated charge ratios are tabulated and compared.