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Source Theory Analysis of Electron-Positron Annihilation

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Source Theory Analysis of Electron-Positron Annihilation Proc. Nat. Acad. Sci. USA Vol. 72, No. 12, pp. 4725-4728, December 1975 Physics Source theory analysis of electron-positron annihilation experiments (high energy/electrodynamics/pion production/spectral forms/differential cross sections) JULIAN SCHWINGER University of California, Los Angeles, Calif. 90024 Contributed by Julian Schwinger, September 23, 1975 ABSTRACT The phenomenological viewint already sional discussion on the oversimplified picture of the form applied to deep inelastic scattering is extended to the discus- factor that is conveyed by the reaction (,y - po) + ir+ g sion of electron-positron annihilation experiments. Some heuristic arguments lead to simple forms for the pion differ- W+: ential cross section that are in reasonable accord with the published experimental data in the energy interval 3-4.8 F(q2) 1 + (q2/mp2)' [2] Gev. where mp = 0.77 Gev, as compared with mrn = 0.14 Gev. Two articles previously published in these Proceedings (1, 2) Then, in contrast with [I 33], we have gave a source theoretic discussion of deep inelastic scattering of electrons (or muons) on nucleons. The deflection of the <« 1: h1'(f) - const, h,'( ) H- [3] electron creates an electromagnetic field, and inelastic scat- tering on protons (p+), for example, can be viewed as the re- The comparison of [1] with the double spectral form, sim- action y + p+ > p+ + any(thing), where y here indicates a ilar to [I 31], supplies such information as space-like virtual photon. Now, there is a mathematical sub- stitution, "crossing," whereby a charged particle entering a hl,2= 1, [4] reaction is replaced by an oppositely charged particle that emerges from the reaction. Accordingly, inelastic scattering where we recall that these are the coefficients of the product on protons is related to the reaction y - p+ + p- + any, of two elastic scattering delta functions in the spectral where the virtual photon, now necessarily time-like, can be weight functions hl,2(M+2, M-2). If we identify an average realized in electron-positron annihilation experiments: e+ + value of Ik (M_2 -mr2) with mp2, the analogue of mo2 in e - p+ + p- + any. Such experiments have been per- the nucleon discussion, we get formed in the past few years with results, most dramatically in the last months, that have sent shock waves through the M_ = (2mpg + mr2)1/2 = .1 Gev, [5] theoretical physics community. It is my intention to focus the light of nonspeculative source theory on this area of in- remarkably close to the mass of the 1+ particle Al. This par- vestigation. ticle, beloved of the theorists, but still under a cloud experi- Let it be quickly pointed out, however, that proton-anti- mentally, is the first resonance available in inelastic scatter- ing, according to the reaction p0 + r+ A1+. In the nu- proton production is actually relatively unimportant at the cleon situation the first resonance is A, at the mass 1.24 Gev, energy levels attained in current experiments. The dominant while the average excitation mass turned out to be 1.5 Gev. process is pion production, as in e+ + e- r+ + ir- + Perhaps the pion result is attributable to the special family any, which is analogously related to the inelastic scattering y relationships among the bosons 7r, p, and Al, or, equivalent- + 7r+ -7r+ + any. Hence, we begin our discussion with the ly, to the comparatively low density of such resonances. latter reaction, although, unlike the nucleon situation, no di- Before continuing with the application to electron-posi- rect experimental information is available. The general for- tron annihilation experiments, two changes in the formalism malism of I*, which refers to unpolarized nucleons, seems to will be introduced. First, we look again at elastic scattering be applicable to the spinless or meson (but, see later). The on pions and conclude from the difference in high q behav- first explicit statement about the nature of the particle oc- ior of Im H1 and Im H2 (Eq. [3]), as compared with the sit- curs in the discussion of elastic scattering. Instead of the two uation in nucleon scattering (Eq. [I 33]), that the basis ten- form factors characteristic of spin ½, there is now only one, sors used in the latter discussion are inappropriate to the F(q2). The analogue of Eq. [I 27] is found to be pion considerations. In response, we redefine Tl.,L (Eq. [I 4]) m+2 Mi2: m2q2 ImH,,2 to read Ti = (m2 + 1q2)(q q q- g ) [6] _2M 1 + 4q2JF, 2, [1] where m is, naturally, the pion mass (a subscript will be sup- which amends Eq. [1] to plied only when needed to avoid confusion). Since no reli- able experimental information is at hand, we base a provi- M+2 ,Mi2: m2q21 ImH, = m2q 2ImH2 * Ref. 1 is cited as I in the text. = _____ - 1]F2, [7] 4725 Downloaded by guest on October 3, 2021 4726 Physics: Schwinger Proc. Nat. Acad. Sci. USA 72 (1975) and, correspondingly, [3] now becomes where p2 = E2-m2 is the squared pion momentum and E its energy, while 0 is the pion emission angle relative to the col- v << 1: h1'(O) = h2'(t) - v- [a] lision axis. The differential cross section inferred from the persistence amplitude [I 3] is then Then, with the coming switch from positive to negative values of q2, which, incidentally, invalidates the real expo- nential representation of [I 21], attention will shift to the du = dwp 3Q2 [-(Q2 - 4m2)ImH, counterpart of the variable v, since q2 is fixed in each annihi- lation experiment at a prescribed total energy. Accordingly, + p2(1 + CsO2O)ImH2], [17] we now write in which 1 1 1 q2 + 1(M+2 + M_2)-m2 2mv + M 2 M2) dwp = (1/4 r2)pjdE2dcosO.2) [18] =Re i exp -iM 2{2m + - M+24] The variable w, originally defined for inelastic scattering JoM+2 [ + by Eq.[I 38], continues to be useful. (Despite this connection [9] it is often designated in the literature by the symbol x.) After making the crossing substitution p -- -p that converts an where we recall that M+2 = M2 + 2mv - q2. This replaces incoming into an outgoing particle, it appears as [I 22,23] with 2qp -2QE 2E [1] ImH1,2 - (M+2)2Re fid exp[-i(2mv/M+2)] q2TeQ2Qsin Then, since X hjf2iv, m'2/ M+2), hj,2(it, m'2/M+2)= M+2 = m2 + Q2(1 - c) > m2, [20] dM_2 M_2_ M+2 of w interval Xf M 2 exp[-' 2M 2l hl,2(M+212M2) [10] the domain the variable is in the 2m/Q < w < 1. [21] where, since m = mr is a quite small mass on the hadronic scale, a reference mass m' has been introduced; it is reason- Inasmuch as our primary concern is with inelastic processes ably anticipated to be of the order of the p mass. In the ap- (w < 1) and magnitudes of Q2 >10 Gev2, the squared pion plication to elastic scattering, where M+2 = m2 and 2mv = mass, mr,2 0.02 Gev2, is generally negligible in M+2 and q2, the previous results apply, with it substituted for A. we therefore write the first entry of Eq.[10] as (Re is under- The annihilation of unpolarized electrons and positrons, of stood) respective momenta qk, q = q+ + q-, produces an electro- magnetic field represented by ImH1,_ Q4(1 ()2f) idt exp[1 _ i AT(-q)Av(q) = [e2/(q2)icdh+dw_ x Qh2 1- [22] X [qMq-" + qj"q v - q+q g"M], [11] For this initial encounter with the insufficiently reported where the lepton mass has been neglected and and not overly accurate data on electron-positron annihila- tion, we make several additional simplifications. One of q2 = -2q+q_ = Q2 > 0. [12] these is based on the statement (3) that the pion angular dis- tribution is roughly isotropic for Q values in the interval be- In the center of mass frame that is realized in this experi- tween 3 and 5 Gev. Accordingly, we omit the Im H2 term in ment, only spatial components in [11] are nonvanishing, and [17] (in view of the imprecise and limited nature of the ob- (kl 1,2,3) servations, this does not require that Im H2 be much smaller than Im H1) and present the differential in c cross section as Ak(-q)AI(q) = -(e2/Q2)dwo+d.(6kl - nknI) [13] 2 dch/dw = a2[w2 - (2m /Q2]1/2(-Q2)ImH,, [23] in which n is the unit vector along the collision axis. The in- or variant flux for this collision is Q2dojdw = a2[A2 - (2m /Q)Y]12 (1 A d F = dw+dwA-41q q - = dw+dw2Q2 [14] and thus exp~lii]h~t 2l_() [24] Ak(-q)Al(q)/F = (lralQ4)(6kl - nknl). I[15] The so-called scaling limit of this differential cross section The combination of bkl-nkil with the two basis tensors is will be reached if, for Q >> m', the function hi becomes de- pendent only on iA: 1 (6k1 - nnTll-= -.VQ m) - < 1: << l: h iA, _ h(-iD), (bkl(-Qnkn,)T2k, =Q2p2(l + +WS26)o)16]I ml'/Q2 Q2(1- [25] Downloaded by guest on October 3, 2021 Physics: Schwinger Proc. Nat. Acad. Sci. USA 72 (1975) 4727 leading to (-i*A) Q2 dwa = a 2(1 - W)2 dA exp[ 1 - Adh(])) [26] a function of the single variable w = 2E/Q, when E >> ma. To illustrate this formula, consider the simple one-parameter class of functions h(v) = rA¢°, [27] which yields a Q2days = r2AF(a+ (1 - Wral b fl2 = r 2 '( 1 a [28] N d scale(A 10 or, as a function of E under the circumstances c = 2E/Q << 1, d= + a dE 27ra2Ar(a 1)(E .
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