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(Je+e--->hadrons
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(g/lVp+ 'p- - p'tp"-- - p+p~)
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(13-119)
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Figure 13-11 (a) Born diagram for e+ e- ---+ p+ p-; (b) the process e+ e- ---+ hadrons; (e) the cross section ~hadrons in terms of the hadronic contribution to the vacuum polarization.
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Since q is time-like positive, we can also write
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d4 x eiq x <O[JI'(x)JJO) [0)
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d4 x eiq . x <Or [J I'(x), J v(O)J [0)
(13-120)
Annihilation at high energy allows therefore to test the vacuum matrix element of a current commutator at short time-like distance. We are led once again to investigate the asymptotic domain. However, changing from space-like to timelike short distances may be more than an innocent modification. The Fourier transform of the commutator in (13-120) is related to the hadronic contribution to the forward amplitude through
d x eiq . x <Or [JI'(x) , Jv(O)J [0)
[i J
d4 x eiq . x <Or T JI'(x)Jv(O) [0)] (13-121)
(Fig. 13-11c). This is also the hadronic part of the vacuum polarization tensor, up to a factor e2 :
d4 x eiq . x <0 [ T JI'(x)Jv(O) [0) = (gl' vq 2 - ql'qv)6i(q2)
(13-122)
Thus (13-123)
QUANTUM FIELD THEORY
Note that (Je+e-~ + _ has the same structure with 1m (7-11) !' !'
replaced by the muon-loop contribution
(13-124) Therefore
in agreement with (13-118).
It is also traditional to call R the ratio of hadronic to electromagnetic annihilation cross sections:
R(q2) =
O"e+e- -->hadrons
(J e+ e- -"J1+ 11-
12n 1m (ijh(q2)
(13-125)
An unsophisticated application of the parton model yields for the limiting value of this ratio !im R(q2) = Reo =
qJ--+OO
Ii Qr
(13-126)
by simply adding the contributions of the lowest-order interactions of the charged spin i elementary constituents. Models for the internal symmetries of hadronic states yield different values for -Reo according to the number, type, and charges of the constituents. For the octet model of quarks with charges ~, -j., and -j. and three color indices, the predicted value is Reo = 2. If additional quantum numbers are present this ratio increases. For instance, a charmed quark of charge ~ would contribute an extra
J/if;
It~~ hllii/
NtIJllllilill1i
E(GeV)
Figure 13-12 The ratio R
E= measured at SLAC. The data were compiled by R. F. Schwitters and K. Strauch, Ann. Rev. Nucl. Sci., vol. 26, p. 89, 1976.
= (Je+e-~hadrons!(Je+e-~!,+!,-
as a function of the total center of mass energy
ASYMPTOTIC BEHAVIOR
quantity j, leading to Ko = 1,j-. A massive lepton around 2 GeV produced in pairs and not distinguished from the hadronic states increases again Ko by one unit. In any case the experimental results as of 1976 are shown in Fig. 13-12, and support the idea that R(q2) might stabilize around a value compatible with these predictions. A new generation of experiments might provide surprises. The asymptotic property (13-126) may equivalently be stated by assuming a free-field behavior for the vacuum matrix element close to x = 0:
(13-127)
We can guess the kind of corrections to Ko which follows from an asymptotically free-field theoretic model. From the fact that conserved currents are not renormalized (see below), we expect that for large q2 the function R is approximately given by
g2 (A) R(A2 q2 ,g) ~ R [ q2 ,g(A)J ~ Reo [ 316n2 Tf 1+
+ ... ]
(13-128)
where g(A) is the running strong coupling constant. The value indicated in (13-128) uses the two-loop calculation of the vacuum polarization [Eq. (13-13)J to estimate the term in g2(A) ofIm Ii}. The internal fermion quantum numbers are responsible for the trace factor Tf and Reo is given by (13-126). For large A, g2(A) is given by
2 3(4n)2 g (A) ~ (llC _ 4T ) In ..1,2
(13-129)
according to Eqs. (13-81) and (13-90). Such a correction predicts that Reo is slowly approached from above [as (In q2) - 1J. In an ordinary perturbative expansion the dominant contribution would be obtained by expanding (13-124) in powers of m 2/q2, with the result
(13-130)
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