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Intrinsic Parity of the (j, 0) ⊕(0, j) Mesons∗

arXiv:hep-ph/9211245v1 13 Nov 1992LA-UR-92-3726Intrinsic Parity of the (j, 0) ⊕(0, j) Mesons∗D. V. AhluwaliaMedium Energy Physics Division, LAMPF, MS H-844, Los Alamos National LaboratoryLos Alamos, New Mexico 87544, USA; E-mail: av@lampf.lanl.govContrary to the usual belief, by carefully examining the operation of parity trans-formation on the (1, 0)⊕(0, 1) mesons in the generalized canonical representation, weestablish that the (j, 0) ⊕(0, j) meson-antimeson pair have opposite intrinsic parity.This opens up the possibility that while the particles without an internal structuremay utilize one representation of the Lorentz group, phenomenologies of compositeparticles may exploit a different representation.

As such (perhaps, only some of) themeson structures beyond the standard q Q may exploit the (j, 0) ⊕(0, j) generalizedcanonical representation of the Lorentz group – this would result in a meson and theassociated antimeson to manifest themselves in different partial waves.Typeset Using REVTEX∗This work was done under the auspices of the U. S. Department of Energy.1

In this letter we argue that the relative intrinsic-parity2of the particle-antiparticlepair emerges as a (Lorentz group) representation-dependent kinematical object, in apparentcontradiction to the usual3 belief. Our motivation for this investigation is to continue our abinitio study, based on Refs.

[1,3], to establish a relativistic phenomenology [4] for high spinhadronic resonances. These resonances will become increasingly more accessible at CEBAF,NIKHEF, RHIC, a possible upgrade of LAMPF, and other new medium energy nuclearphysics facilities.For concreteness we look at the relativistic phenomenology of the j = 1 mesons.

Fol-lowing conventions defined by Ryder [5] the (1/2, 1/2) representation of the Lorentz groupcorresponds to the description of the j = 1 matter fields in terms of the vector field Aµ(x).The vector field Aµ(x) satisfies the Proca equation.Within this framework the relativeintrinsic parity of the meson-antimeson pair is same. By considering the (1, 0) ⊕(0, 1) rep-resentation in detail we will show that the relative intrinsic parity for the meson-antimeson,described by the generalized canonical [4a,b] representation (1, 0) ⊕(0, 1) matter field, isopposite.The generalization of this result to the mesons with j > 1 will be seen as es-sentially obvious.

As noted in the abstract, this result opens up the possibility that whilestructureless fundamental particles may utilize one representation of the Lorentz group, say(1/2, 1/2), the composite particles may find their phenomenological description in terms ofother representations, such as (1, 0) ⊕(0, 1).We begin with the classical considerations similar to the ones found for the (1/2, 0) ⊕(0, 1/2) Dirac field in the standard texts, such as Nachtmann’s treatment in Ref. [6, Sec.4.5].

The (1, 0) ⊕(0, 1) wave function satisfies the spin one Weinberg [1, 4h] equationγµν ∂µ ∂ν + m2ψ(t, ⃗x ) = 0. (1)2There is a slight ambiguity even in the definition of relative intrinsic parity.

For this ambiguitythe curious reader is referred to footnote [11] on p. 570 of Ref. [7].3 For example, see Ref.

[1, footnote 13] and Ref. [2].2

The 6 × 6 ten γµν matrices in the generalized canonical representation [4h] are given byγ◦◦=I00−I,γℓ◦= γ◦ℓ=0−JℓJℓ0,γℓ= γℓ=I00−Igℓ+{Jℓ, J}00−{Jℓ, J};(2)where ⃗J are the 3 × 3 spin one matrices with Jz diagonal; I are the 3 × 3 identity matrices;gµν is the flat spacetime metric with diag (1, −1, −1 −1); ℓ, run over the spacial indices1, 2, 3; and {Jl, J} is the anticommutator of Jℓand J. We seek the parity-transformedwave function4ψ′(t′, ⃗x ′) = S(ΛP) ψ(t, ⃗x),(3)such that Eq.

(1) holds true for ψ(t′, ⃗x ′)γµν ∂′µ ∂′ν + m2ψ(t′, ⃗x ′ ) = 0. (4)It is a straight forward exercise to find that S(ΛP) must simultaneously satisfy the followingrequirementsS−1(ΛP) γ◦◦S(ΛP) = γ◦◦,S−1(ΛP) γ◦S(ΛP) = −γ◦,S−1(ΛP) γ◦S(ΛP) = −γ◦,S−1(ΛP) γℓS(ΛP) = γℓ.

(5)Referring to Eqs. (2), we now note that while γ◦◦commutes with γℓit anticommutes withγ◦[γ◦◦, γℓ] = [γ◦◦, γℓ] = 0,{γ◦◦, γ◦} = {γ◦◦, γ◦} = 0.

(6)As a result, confining to the norm preserving transformations (and ignoring a possible globalphase factor), we identify S(ΛP) with γ◦◦, yielding4The operation of parity is defined via x′µ = Λµν xν, with ΛP ≡[Λµν] = diag(1, −1, −1, −1).All conventions, unless indicated otherwise, are that of Ref. [5].3

ψ′(t′, ⃗x ′) = γ◦◦ψ(t, ⃗x)⇐⇒ψ′(t′, ⃗x ′) = γ◦◦ψ(t′, −⃗x ′),(7)This prepares us to proceed to the field theoretic considerations.The (1, 0) ⊕(0, 1)matter field operator may be defined as follows [4a]Ψ(x) =Xσ = 0, ±1Zd3p(2π)312 ω⃗p×huσ(p) aσ(⃗p ) exp(−ip · x) + vσ(p ) b†σ(⃗p ) exp(+ip · x)i,(8)with ω⃗p =qm2 + ⃗p 2. The explicit general-canonical-representation expressions [4h,b,f] forthe (1, 0) ⊕(0, 1) spinors uσ(⃗p ) and vσ(⃗p ), which appear in Eq.

(8), areu+1(p) =m +(2p2z + p+p−)/2(E + m)pzp+/√2(E + m)p2+/2(E + m)pzp+/√20, u0(p) =pzp−/√2(E + m)m +p+p−/(E + m)−pzp+/√2(E + m)p−/√20p+/√2,4

u−1(p) =p2−/2(E + m)−pzp−/√2(E + m)m +(2p2z + p+p−)/2(E + m)0p−/√2−pz,vσ(p) =0II0uσ(p). (9)In the above expression we have defined p± = px ± i py.

The transformation propertiesof the “particle” [“antiparticle”] creation operators a†σ(⃗p )[b†σ(⃗p )] are obtained from theconditionU(ΛP) Ψ(t′, ⃗x ′ ) U−1(ΛP) = γ◦◦Ψ(t′, −⃗x ′ ),(10)where U(ΛP) represents a unitary operator which governs the operation of parity in theHilbert space of the single particle-antiparticle states. Using the definition of γ◦◦, Eqs.

(2),and the explicit expressions for the (1, 0) ⊕(0, 1) spinors uσ(⃗p ) and vσ(⃗p ) given by Eqs. (9),we findγ◦◦uσ(p′) = + uσ(p),γ◦◦vσ(p′) = −vσ(p),(11)with p′ the parity-transformed p — i.e.

for pµ = (p◦, ⃗p ), p′µ = (p◦, −⃗p ). The observa-tion (11) when coupled with the requirement (10) immediately yields the transformationproperties of the particle-antiparticle creation operatorsU(ΛP) a†σ(⃗p ) U−1(ΛP) = + a†σ(−⃗p )U(ΛP) b†σ(⃗p ) U−1(ΛP) = −b†σ(−⃗p ).

(12)5

Under the assumption that the vacuum is invariant under the parity transformation,U(ΛP) |⟩= |⟩, we arrive at the result that the “particles” (described by the u-spinors)and “antiparticles” (described by the v-spinors) have opposite relative intrinsic paritiesU(ΛP) |⃗p, σ⟩u = + | −⃗p, σ⟩u,U(ΛP) |⃗p, σ⟩v = −| −⃗p, σ⟩v. (13)The results (12) and (13) are precisely what we set out to prove.

While the particle-antiparticle pairs in the (1/2, 1/2) representation of the Lorentz group have same relativeintrinsic-parity, the particle-antiparticle pairs in the (1, 0) ⊕(0, 1) representation have op-posite intrinsic parity. Because of the general structure of the Weinberg’s equations andthe (j, 0) ⊕(0, j) spinors, we assert5 that this result is true for all spins.

Whether (at leastsome of the) mesons beyond the standard q Q structures exploit the (j, 0)⊕(0, j) representa-tion of the Lorentz group remains an open experimental question. To sum up, the analysisof this work establishes that the relative intrinsic parity of a particle-antiparticle pair is aLorentz-representation-dependent kinematical object.

Which of the various representations,for a given spin, is actually realized in nature, such as in constructing the phenomenolo-gies of composite particles, can be (contrary to the usual belief — for example, see Ref. [2])experimentally determined.ACKNOWLEDGMENTSAn anonymous referee (of a related work) is to be thanked for bringing to my attentionfootnote 13 of Ref.

[1]. Mikkel Johnson and Mikolaj Sawicki are to be thanked for beinginsistent that we understand intrinsic-parity at a deeper level within the context of our othercollaborative efforts — in addition they kindly read the rough draft of this work and provided5The reader may wish to note that all (j, 0) ⊕(0, j) spinors satisfy relations very similar to Eq.

(11); etc.6

comments and suggestions. It is also my pleasure to extend thanks to Dick Arnowitt, TerryGoldman and Barry Holstein for conversations on the subject matter of this work.

Finally,I thankfully acknowledge financial support via a postdoctoral fellowship by the Los AlamosNational Laboratory.7

REFERENCES[1] S. Weinberg, Phys. Rev.

B 133, 1318 (1964). [2] S. Weinberg, Nucl.

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[3] E. Wigner, Ann. Math.

40, 149 (1939). [4] (a) D. V. Ahluwalia and D. J. Ernst, Phys.

Rev. C 45, 3010 (1992); (b) Int.

J. Mod. Phys.E (in press); (c) Mod.

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A 7, 1967 (1992); (d) Phys. Lett.

B 287, 18 (1992);(e) in “Classical and Quantum Systems – Foundations and Symmetries,” Proceedings ofthe II International Wigner Symposium, 1991, Goslar, Germany,” (1992) (in press); (f)in “Proceedings of the Computational Quantum Physics Conference: AIP Conf. Proc.Vol 260, Nashville, 1991,” edited by A. S. Umar, V. E. Oberacker, M. R. Strayer and C.Bottcher (Vanderbilt University, TN, USA, 1992); (g) D. V. Ahluwalia and M. Sawicki,“Natural Hadronic Degrees of Freedom for an Effective QCD Action in the Front Form,”LA-UR-92-3133 and ISU-NP-92-11, (submitted for publication); (h) D. V. Ahluwalia,Ph.

D. thesis (Texas A&M University), UMI-92-06454-fiche (Aug. 1991).

[5] L. H. Ryder, “Quantum Field Theory,” Cambridge University Press, 1987. [6] O. Nachtmann, “Elementary Particle Physics,” Springer-Verlag, 1990.

[7] K. Gottfried and V. F. Weisskopf, “Concepts of Particle Physics,” Vol. II, Oxford Uni-versity Press, 1986.8

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