QUANTUM FIELD THEORY in .NET framework

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QUANTUM FIELD THEORY
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Yi(g) = A
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a~ In Zi(~' go)
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(13-93)
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If goo is an ultraviolet fixed point, the effective dimension of the operator Oi will differ from the canonical one di according to
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(13-94) Here we only deal with. the case di < 4. In general, mass corrections will remain negligible as long as Wilson's criterion is satisfied: (13-95) This will be automatically verified in an asymptotically free theory where Yi(goo) vanishes. We recall, however, that logarithmic corrections still affect the canonical behavior. A closely related procedure has been proposed by Weinberg. It relies on the construction of counterterms independent of the renormalized mass up to trivial dimensional factors. In other words, we do not fix the latter explicitly. A mean to achieve this result is to use dimensional regularization and renormalization. Therefore we shall no longer have to neglect the insertion of the mass term in the renormalization group equation. In cp4 theory the solution to the corresponding equation will have the form (13-96) where
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Show that to lowest order the contribution of the one-loop diagram of Fig. 13-7 leads to
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g y" = (4n)2
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As A --> 00, the condition m 2 (A) --> 0 is YLl(goo) < 2, and is therefore equivalent to Wilson's criterion (13-95). Only in this case is a massless scale-invariant theory meaningful, even though this condition does not appear perturbatively. We should not conclude from the previous analysis that a soft mass insertion can be treated perturbatively. Indeed, derivatives of Green functions with respect to the mass of a sufficient high order are singular in the massless limit. This is due to the fact that after a certain point increasing the number of insertions does not improve the ultraviolet behavior (see the discussion of Weinberg's
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Figure 13-7 Lowest-order mass insertion in the
theory.
ASYMPTOTIC BEHAVIOR
theorem in Chap. 8). An intuitive argument is the following. The second (third) derivative with respect to m of a bosonic (fermionic) propagator induces in general when m --> 0 a logarithmic infrared singularity in a Feynman diagram. We therefore expect singularities of the type m~(1n mB)a[mJ(ln mF)a] for small masses.
13-4 DEEP INELASTIC LEPTON-HADRON SCATTERING AND ELECTRON-POSITRON ANNIHILATION INTO HADRONS
We consider here in more detail a subject already introduced in Chap. 11. Our aim is to discuss in the next section the implications of a field theoretic model for the description of high transfer phenomena induced by leptons.
13-4-1 Electro production
We begin with a discussion of the electromagnetic scattering of a charged lepton (electron or muon) off a nucleon. The process has been depicted in Fig. 11-8. The initial and final lepton momenta are I and l' and at high energy we neglect the lepton mass. The initial nucleon of momentum p (mass m) breaks into a final state X which is not observed; hence the name inclusive process. Up to radiative corrections the electromagnetic interaction acts to lowest order. A photon carries the space-like momentum transfer q = I - l' from the lepton to the hadron vertex. In practice the measured quantities are the laboratory initial and final lepton energies E and E' and the scattering angle B. We shall not discuss here polarization effects. The kinematical invariants are
- q2
= 4EE'
sin 2
(13-97)
E') - 4EE' sin 2
v = p. q = m(E - E')
M2 = (p
+ q =
+ 2m(E -
The nucleon being the lightest state with baryonic number one, the stability condition M2 ;::::: m 2 implies that the Bjorken variable
x=w -1
satisfies the inequalities
(13-98)
(13-99)
The upper limit corresponds to elastic scattering. Both notations x and ware found in the literature.
QUANTUM FIELD THEORY
Let J p be the hadronic component of the electromagnetic current. The scattering amplitude reads
(13-100)
For unpolarized leptons and nucleons the inclusive cross section is therefore
1 d 1'" ( )4 ~4( EE' (2n)3 'T 2n u Px
+ [' -
[)(eZ)Z 1 tvf') qZ "2 tr y "2 y "2
<piJp(O)iPx) <PxiJv(O)ip)
The notation implies a sum over nucleon polarizations. Explicit computation yields - tr (yP IfI) 8
Wl'V + NP -
gPV[-l')
(2n)4b 4(px - p - q)
<pjJp(O)iPx)<pxiJv(O)ip)
~ I fd
eiQ,x<piJp(x)Jv(O)ip)
In the kinematical region of interest, Wpv may also be expressed as the Fourier transform of the current commutator
(13-101)
Relativistic in variance, current conservation, and parity in the electromagnetic case enable us to express W pv in terms of two structure functions Wi and Wz which generalize the elastic form factors [compare with Eq. (3-203)J
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