It is shown that in the simple context of the elementary one-meson exchanges, the use of the "improper" delta "functions" could lead to the physically correct results. The same results were obtained by involving limiting values of the integrals over "proper" functions, thus providing the examples of the sequences connected with delta "functions". That formulation of sequences of integrals emerged automatically from the physical considerations concerning hypernuclear processes

Horvat, Dubravko

Narančić, Zoran

Tadić, Dubravko

University of Zagreb Repository

ISSN1330–0016CODEN FIZBE7MESON EXCHANGE FORMALISM AND THE DEFINITION OF DELTAFUNCTIONSDUBRAVKO HORVATa,1, ZORAN NARANCˇIC´a AND DUBRAVKO TADIC´b,2aDepartment of Physics, Faculty of Electrical Engineering, University of Zagreb, 10 000Zagreb, CroatiabDepartment of Physics, University of Zagreb, 10 000 Zagreb, Croatia1E-mail address: dubravko.horvat@fer.hrReceived 1 August 2003; Accepted 8 December 2003Online 19 April 2004It is shown that in the simple context of the elementary one-meson exchanges, theuse of the “improper” delta “functions” could lead to the physically correct results.The same results were obtained by involving limiting values of the integrals over“proper” functions, thus providing the examples of the sequences connected withdelta “functions”. That formulation of sequences of integrals emerged automati-cally from the physical considerations concerning hypernuclear processes.PACS numbers: 21.30.-x, 13.75.Cs UDC 539.141Keywords: delta function, one-meson exchange, integral over proper function, hypernu-clear processes1. IntroductionThe elementary calculation of an amplitude corresponding to a simple mesonexchange process can illustrate and confirm a general use of “delta function” [1].Such functions have appeared in theoretical physics in the 19th century [2]. Use-ful, compact and self consistent formalism has been developed by Dirac who said:“Strictly, of course, δ(x) is not a proper function of x, but can be regarded only asa limit of a certain sequence of functions. All the same one can see δ(x) as thoughit were a proper function for practically all the purposes of quantum mechanicswithout getting incorrect results” [3]. This improper function has been rigorouslydefined as sequence which in the limit defines the distribution delta [2,4,5]. Thecalculation of the meson exchanged amplitudes is easily and simply carried out in2DeceasedFIZIKA B 12 (2003) 4, 267–274 267horvat et al.: meson exchange formalism and the definition . . .the complex momentum transfer plane leading to the well known Yukawa-functionradial dependence [6–8]. If the baryon masses are not equal, one immediately ob-tains a shifted Yukawa function [9], as encountered in the study of the hypernucleardecay. When the baryon masses are equal, the distributions appear in the directcalculation and no integration in the complex plane is needed. The unequal-masscase leads in the equal-mass-case limit to the same result “not involving improperfunctions” [11], i.e. when one deals with the strong nucleon–pion interaction. In theequal-mass case, one can also find the sequence, whose form is based on physicalconsiderations and intuition. In the limit, that sequence leads to the same result.2. One-meson exchangeA transition amplitude AfΛ corresponding to the one-pion (or any meson) ex-change has a standard textbook generic form [6–10]. This form corresponds toeither strong or weak meson exchange [9]. In the most general case displayed belowAfΛ =12∫d4xd4y∫d4ke−ik(x−y)k2 − µ2 + iǫ〈 f |T(S(x)W (y))|Λ 〉, (1a)withk2 = k20 −~k2,kx = k0x0 − ~k · ~x . (1b)Here S(x) and W (x) are some baryon densities (scalar or pseudoscalar, see Eq. (5a)below) which are sources of the meson (pion) field, with mass µ. The states | f 〉and |Λ 〉 are some baryonic bound states. If made out of nucleons, like state | f 〉,they symbolize atomic nuclei. Detailed form of those quantities need not concernus here. If all baryons are nucleons N , then the calculation can be found, forexample, in Ref. [8], p. 213. In a more general case, the density W (x) can containa strange baryon, for example Λ hyperon [9], and one of the sources will be weak,meaning that neither parity nor strangeness need not be conserved. The detailsof their spatial properties and their precise particle content are immaterial forour arguments. The integration over dx0, dy0 and d4k can be carried out withoutreferring to these properties of the baryon densities. The results of such integrationsdo reveal the functional (distributional) properties mentioned in Introduction.The time-ordered product in Eq. (1a) can be written as〈 f |T(S(x)W (y))|Λ 〉 = θ(x0 − y0)∑n〈 f |S(x) |n 〉〈n |W (y) |Λ 〉+θ(y0 − x0)∑s〈 f |W (y) | s 〉〈 s |S(x) |Λ 〉 . (2a)268 FIZIKA B 12 (2003) 4, 267–274horvat et al.: meson exchange formalism and the definition . . .Here the complete set of states∑i | i 〉〈 i | = 1 was introduced [7]. If the weakinteractions are involved [9], some of the nuclear states will describe hypernuclei.The density S (i.e. strong) conserves the strangeness while the density W (weak)changes the strangeness. In each matrix element in Eq. (2), as for example in〈 f |S(x) |n 〉 , (2b)〈 f |W (y) | s 〉 , (2c)the strangeness is conserved [8–10]. Thus the intermediate states|n 〉; S = 0 (3)have no strangeness, while the states| s 〉; S = −1 (4)must contain a strange baryon.When both source densities correspond to strong interactions, i.e. if they are ofthe well known form [8]S(x) ∼ ψN (x) γ5 ψ(x) , (5a)then all states are nucleon states (i.e. nuclei) with no strangeness. In that case, onecan replace |Λ 〉 with a nucleon state | i 〉. The general expression (1) can be easilyspecialized for the strong interaction case (see Eq. (16) below). In the expression(1), one can first carry out the integration over times (x0, y0) using the relation (2).A useful identity is [8,10]〈 f |S(x) |n 〉 = ei(Ef−En)x0〈 f |S(~x) |n 〉. (5b)An analogous expression holds for other matrix elements. Alsodx0 dy0 = 2dξ dη,x0 = ξ + η,y0 = −ξ + η. (6)The integration dη can be immediately carried out which results inAfΛ =∫d3xd3y dξ∫d4ke−2iξk0+i~k·(~x−~y)k2 − µ2 + i ǫ2πδ(Ef − EΛ)×[θ(ξ)∑nei∆nξαn(~x, ~y) + θ(−ξ)∑se−i∆sξβs(~y, ~x)]. (7a)FIZIKA B 12 (2003) 4, 267–274 269horvat et al.: meson exchange formalism and the definition . . .Here∆i = Ef + EΛ − 2Ei,αn = 〈 f |S(~x) |n 〉〈n |W (~y) |Λ 〉, (7b)βs = 〈 f |W (~y) | s 〉〈 s |S(~x) |Λ 〉,and Ek are relativistic energies.Integrating over dξ, one obtainsAfΛ = 2πi δ(Ef − EΛ)∫d3xd3y∫d4kei~k·(~x−~y)k2 − µ2 + i ǫ×[∑nαn1∆n − 2k0 + i ǫ+∑sβs1∆s + 2k0 + i ǫ]. (8)Integration in the complex k0 plane over the contours shown in Fig. 1 leads toAfΛ = 2π2δ(Ef − EΛ)∫d3xd3y∫d3k ei~k·(~x−~y)×[∑nαn(~x, ~y)ω(∆n − 2ω)+∑sβs(~y, ~x)ω(∆s − 2ω)], (9)ω =√~k2+ µ2 .Fig. 1. The contours in the k0plane. Here ω2 = ~k2 + µ2.ω−Σ(∆  +     )εΣε−ω+ ε(− ∆ −      )εIn the nonrelativistic limit, the energy differences ∆i can be approximated bythe corresponding baryon mass differences. Schematically one can use the followingbaryonic contents:| f 〉 K nucleons,|Λ 〉 1Λ + (K − 1) nucleons,| s 〉 1Λ + (K − 1) nucleons,|n 〉 K nucleons.270 FIZIKA B 12 (2003) 4, 267–274horvat et al.: meson exchange formalism and the definition . . .ThusEf → K ·mN ,EΛ → (K − 1) ·mN +mΛ,Es → (K − 1) ·mN +mΛ,En → K ·mN ,∆s = Ef + EΛ − 2Es → −2δ,∆n = Ef + EΛ − 2En → 2δ,δ = (mΛ −mN )/2.(10)In such an approximation, the factors depending on ω and ∆i can be taken out ofthe summations in Eq. (9). In the nonrelativistic treatment [9], one finds∑n αn =∑n βn = 〈 f |O(x, y) |Λ 〉. Thus one obtainsAf Λ = 2π2 δ(Ef − EΛ)∫d3xd3y∫d3k ei~k·(~x−~y)×12[−1ω(δ + ω)+1ω(δ − ω)]〈 f |O(~x, ~y) |Λ 〉, (11)O(~x, ~y) = S(~x)W (~y).The integration over ~k gives the shifted Yukawa function [5]∫d3k ei~k·(~x−~y) 12[−1ω(δ + ω)+1ω(δ − ω)]= −∫d3kei~k·(~x−~y)ω2 − δ2= −2π2e−κrr,r = |~x− ~y|, κ =√µ2 − δ2 =√µ2 −14(mΛ −mN )2. (12)Eventually, one finds a generic formAfΛ = −4π4Nδ(Ef − EΛ)∫d3xd3ye−κrr〈 f |O(~x, ~y) |Λ 〉. (13)When one deals with the strong interactions among the nucleons, then all states(10) contain nucleons only. One hasEf , EΛ ≡ Ei, Es, En → KmN ,∆s = ∆n = 0, (14a)δ = 0,consequently κ→ µ (14b)and one finds the well known form [8]VY (r) =e−µrr. (15)FIZIKA B 12 (2003) 4, 267–274 271horvat et al.: meson exchange formalism and the definition . . .More precisely, one should start from the formula (9) and write it in the limit(10b) and (14a) asAfi = 2π2 δ(Ef − Ei)∫d3xd3y d3k ei~k·(~x−~y)×[∑nαˆn(−12ω2)+∑sβˆs(−12ω2)](16a)withαˆ = 〈 f |S(~x) |n 〉〈n |S(~y) | i 〉 andβˆ = 〈 f |S(~y) | s 〉〈 s |S(~x) | i 〉. (16b)Integration over d3k givesAfi = −2π4 δ(Ef − Ei)∫d3xd3ye−µrr[∑nαˆn +∑sβˆs]. (16c)If one was interested in the strong pion exchange [8] only, one would start directlywith formula (7) neglecting k0 in the first denominator and using Eqs. (14a) and(16b). Only the first factor in thus redefined (7) depends on k0. One finds∫dk0 e−2iξk0∫d3kei~k·(~x−~y)− (~k2 + µ2) + i ǫ= −2π3δ(ξ)e−µrr(17)and ends withAfi → −4π4δ(Ef − Ei)∫d3xd3y dξe−µrr×δ(ξ)[∑nθ(ξ)αˆn +∑sθ(−ξ)βˆs]. (18)The formula (16c) can be equal to (18) only if1∫δ(ξ)θ(ξ) dξ =∫δ(−ξ)θ(ξ) dξ =12. (19)This corresponds to the identity which can be read from Ref. [1]. In the equal-masslimit (14), (17), |Λ 〉 = | i 〉 the previous result (16) can be considered as leading tothe identity (19), which is a consequence of the generalized delta function definition[1].1The second term in (19) follows from the first by a trivial coordinate change ξ → −ξ.272 FIZIKA B 12 (2003) 4, 267–274horvat et al.: meson exchange formalism and the definition . . .A very interesting result follows if terms which were neglected in (7) in order toobtain Eq. (18) are marked by the parameter λ. For that purpose, one writes Eq.(7a) asAfi =∫d3xd3y dξ∫d3k dk0e−2iξk0+i~k·(~x−~y)λ2k20 − ω2 + i ǫ2πδ(Ef − Ei)×[θ(ξ)∑neiλ∆nξαˆn(~x, ~y) + θ(−ξ)∑se−iλ∆sξβˆs(~x, ~y)]. (20)The parameter λ determines the sequence which for λ → 0 can be understood asand equivalent to the integration over distributions appearing in Eq. (18). Keepingλ finite, the integrations over ξ and k0 in the formula (20) can be carried outexplicitly. Instead of Eq. (9), one findsAfi(λ) = 2π2δ(Ef − Ei)∫d3xd3y∫d3k ei~k·(~x−~y)×[∑nαˆn(~x, ~y)ω(λ2∆n − 2ω)+∑sβˆs(~x, ~y)ω(λ2∆s − 2ω)]. (21a)This posseses the well defined limit for λ→ 0 which is shown in Eq. (16), i.e.limλ→0Afi(λ) = Afi (Eq. (16)). (21b)3. SummaryIn all above derivations and comparisons, the discussion of either the weak or thestrong interactions was immaterial. Only very general physical properties, commonto any physical process, the time dependence and the energy conservation, areessential. That led to the physical expressions which modelled the mathematicaldescription of “improper” functions [2,3,11] as distributions [12]. All what wasneeded was the introduction of a different mass baryon. That can be considered asa purely mathematical device which allows to have a good sequence of functionswhich in the limit leads to a delta function identity. It is gratifying for a physicistthat such a selection corresponds to a well known (i.e. hypernuclei decay) case[9]. Formally that can be expressed through the parameter λ dependent expression(21), i.e. mΛ in (10b). The limit λ → 0 in Eq. (21) exists and that expression canbe understood as a defining a sequence of functions which lead to the relation (17).Furthermore, the omission of the terms in Eq. (21) marked by λ can be justifiedby physical considerations. When one starts directly from the modified formula (7)(i.e., k2−µ2 → −(~k 2 +µ2)), the integration over k0 simply defines the Dirac deltafunction (17). No complex plane k0 integration (Fig. 1) is ever needed.Presented results illustrate the general Dirac’s statement about the usefulnessof delta-functions [2]: “Therefore it should be possible to rewrite the theory in aFIZIKA B 12 (2003) 4, 267–274 273horvat et al.: meson exchange formalism and the definition . . .form in which the improper functions appear all through only in integrands. . .Theuse of improper functions thus does not involve any lack of rigour in the theory butis merely a convenient notation, enabling us to express in a concise form certainrelations which we could, if necessary, rewrite in a form not involving improperfunctions, but only in a cumbersome way which tend to obscure the argument.” Itis interesting to note how the mathematical contributions of Hevyside and Dirac[2], inspired by physical processes, were perceived by a famous mathematician [13].He wrote “. . . I heard of the Dirac function for the fist time in my second year atENS. . . . which absolutely disgusted us, but it is true that those formulas were socrazy from the mathematical point of view that there was simply no question ofaccepting them. . . . nine years later, I discovered distributions. . . .This at least canbe deduced from the whole story: it’s good thing that theoretical physicists do notwait for mathematical justification before going ahead with their theories.”References[1] D. Zhang, Y. Ding and T. Ma, Am. J. Phys. 57 (1989) 281.[2] M. Jammer, The Conceptual Development of Quantum Mechanics, McGraw Hill, NewYork (1966).[3] P. A. M. Dirac, Proc. Roy. Soc. (A) 113 (1926) 621.[4] E. Butkov, Mathematical Physics, Addison-Wesley, Reading (1968).[5] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Academic Press,New York (1995).[6] H. Yukawa, Proc. Phys. Math. Soc. Japan 17 (1935) 48.[7] J. J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, Redwood City (1985).[8] J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, NewYork (1964) p. 211.[9] J. F. Dubach, G. B. Feldman, B. R. Holstein and L. de la Torre, Ann. Phys. (N.Y.)249 (1996) 146; A. Parren˜o, A. Ramos and C. Bennhold, Phys. Rev. C 56 (1997) 339.[10] S. S. Schweber, An Introduction to Relativistic Quantum Field Theory , Row, Petersonand Co., Evanston (1961); E. D. Commins and P. H. Bucksbaum, Weak Interactionsof Leptons and Quarks, Cambridge University Press, Cambridge (1983).[11] P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford at the Claredon Press,Oxford (1976).[12] L. Schwartz, The`orie des Distributions, Hermann et Cie, Paris (1950); I. M. Gelfandand G. E. Shilov, Generalized Functions, Academic Press, New York (1964).[13] L. Schwartz, Mathematician Grappling with this Century , Birkha¨user Verlag, Basel(2001).FORMALIZAM IZMJENE MEZONA I DEFINICIJA DELTA FUNKCIJEPokazujemo da u okviru elementarnih jedno-mezonskih izmjena, upotreba “nepravih”delta “funkcija” mozˇe davati fizicˇki ispravne rezultate. Postigli smo iste rezultate uzprimjenu granicˇnih vrijednosti integrala preko “pravih” funkcija, i tako pripremiliprimjere nizova povezanih s delta “funkcijama”. Ta formulacija nizova integralasama proizlazi iz fizicˇkih razmatranja hipernuklearnih procesa.274 FIZIKA B 12 (2003) 4, 267–274

Meson exchange formalism and the definition of delta functions

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Meson exchange formalism and the definition of delta functions

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