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Figure 13-9 Infinite momentum frame.
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with Qi equal to its charge (Fig. 13-10). Since its initial momentum is XiP, we have
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Therefore the above contribution can also be written
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giving an interpretation of the scaling variable x = W -1 as the fraction of total momentum carried by a constituent when scattering off the virtual photon. The model is completed by assuming the probability for such an occurrence to bef(x) with
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dxf(x) = 1
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For simplicity we have considered the case wheref(x) does not depend on the type i of the constituent.
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It then follows that the structure function W 2 , say, is given by
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- V W2 (v, q 2 ) = m F2 (x)
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(q2)
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I Qf xf(x)
(13-108)
\q ~, ~ ~
Figure 13-10 Parton contributions to the structure function.
QUANTUM FIELD THEORY
and automatically satisfies scaling. Similarly, mWI (v, q2) is equal to a function FI ( - q2/2v). The relation between FI and F2 depends on the constituent's spin, being
(13-109)
for spin i, which corresponds to R
(IXi = 1) we derive the sum rules
(JS/(JT
O. From the conservation of total energy momentum
dv - W (v, q2) 2 o v m
dv - -q2 - W2(v, q2) 2mov
fI fl
dx F (x) 2 x
I Qr
(13-110)
dx F2(x)
Q I ----'iN
The crucial hypothesis of this approach is the quasi free behavior of partons during the interaction with the external current and the neglect of transverse degrees of freedom.
An equivalent description amounts to substituting free constituent fields in (13-101) when estimating the contribution of the hadronic current in the light-cone region. The latter will read
J'"(x)
li/(x)yI'Q!/J(x)
(13-111)
where!/J is a fermion free field and Q the charge matrix. Using the anticommutator (3-170)
we find
When evaluated in a sum over polarizations of a diagonal element, only even terms in the J1 +-* V exchange contribute. We expand the product of fields in Taylor senes
and use
to obtain
"2 {[JI'(x), J v( - x)] + [Jv(x), JI'( = i
x)]}
nodd
XI'I
1 Xl'n f Oi3i1I'''i1n(gl'agvp n.
+ gl'pgva -
1 gl'vgaP) -8 aac;(xO)(j(x 2) n
(13-112)
ASYMPTOTIC BEHAVIOR
In the vicinity of x 2 = 0 the commutator has been expanded in an infinite series of regular local operators multiplying the same c-number distribution. To compute the tensor WIlV in the Bjorken limit we need the matrix elements
2" I <Pi
1 n OfJl'l'''I''ip) 2
an + 1(pPpIl1 pll,
+ trace terms)
(13-113)
The trace terms involving contractions of two indices will not contribute when multiplied by a string of coordinates as in (13-112). We have
We insert (13-113) into this expression and define a function f(x), where x will turn out to be the scaling variable (not to be confused with a configuration argument), in such a way that
1..., --,-an+l-~ nodd n. 21
~ (y- pt
ix(y
p)f(x) x
(13-114)
This means that the matrix elements an + 1 are the moments of the distribution f(x). Neglecting p2 = m 2 as compared to pO q, we find
rd y
e iq
y dx eix(y pJ(x)
x [Pll(q
+ xp)v + p.(q + xp)ll- gllvpoq] _t5(y2)
c;(yO)
Carrying out the integral over y we are led to the expression
Wllv=f~; (q;;V_gllv)+f~)(pll- ;2qll)(PV- ;2qV)
(13-115)
where of course x = W - 1 = - q2/2v. We recognize the results of the parton model with structure functions
F2(x) = mf(x)
mf(x) mW1 = Fl(X) = ~ F 2(x)
(13-116)
2xFt{x)
QUANTUM FIELD THEORY
Why is such a simple-minded approximation in good agreement with experiment and what are the expected corrections The answer to these questions is given by the existence of an asymptotically free theory. The detailed implications will be discussed in Sec. 13-5.
It is possible to measure the antisymmetric part of the current commutator in experiments with
polarized leptons and nucleolls. Measuring spin in the direction of the incident beam, show that
dO" it d<1 1i 4a 2 E' - - - - - = --2dO' dE' dO' dE'
- q E 4nm
+ E' cos 8)d(v, q2) -
(E - E' cos 8)(E
+ E')mg(v, q2)]
(13-117)
where the quantities d(v, q2) and g(v, q2) are defined through
and the nucleon polarization fourvector S satisfies S' p = S2
+ 1 = O.
13-4-3 Electron-Positron Annihilation
Electron-positron annihilation into hadrons at very high energies (several GeY) is a surprisingly fruitful domain of investigation. The discovery of narrow resonances such as the ljJ and ljJ' and the corresponding spectroscopy will not be discussed here because of lack of space. The cross-sections are of the order of a few tens of nanobarns (10- 33 cm 2 ) and are comparable to the rate of the electromagnetic annihilation e + e - ---+ 11 + 11-. This process is schematized on Fig. 13-11a. At very high energy where the square center of mass energy q2 is much larger than the masses it is given by d3 P+ d 3P e4 4 4 (Je+e--->I'+I'- = (2 n )6pO pO 0 0 (2 q 2)2 (2n) (j (p+ + p- - p+ - p-) + _p+p_v+_
4mx 2 86.9 3q2 = q2 (Gey2)
(13-118)
The hadronic annihilation involves matrix elements of the current Jw For unpolarized electrons and positrons, taking into account the hermiticity of the current, it assumes the form
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