veff = 0 ) + hy(1) + h2 0eff) + ... eff eff in .NET framework

Printing PDF-417 2d barcode in .NET framework veff = 0 ) + hy(1) + h2 0eff) + ... eff eff

2 veff = 0 ) + hy(1) + h2 0eff) + ... eff eff
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(9-117)
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Using (9-114) we have first
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00) = eff 2
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+ V(q
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(9-118)
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In the determinant occurring in (9-112) q> is now a constant and the propagator [0 + m 2 + VI/(q>)]-l is thus diagonal in momentum space:
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(9-119)
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_ 12 (2)4 In [1 V 1/ (q p-m+lI: ] 2 n
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This expression is, of course, meaningless before ultraviolet subtractions. To be concrete let us pick the potential VI/(q
Aq>2 2
To comply with the prescription of normal ordering we should also have added a term in q>2. Otherwise we have to include tadpole diagrams, corresponding to a contraction of two fields at the same vertex. This is precisely the only divergent one-loop diagram of the two-point function r(2) (Fig. 9-5a), and it contributes a quadratically divergent term proportional to q>2 in the expansion of V;;~:
Aq>2
4 d p i (2n)4 p2 - m 2 + is
To this order we also have the logarithmic divergence of the four-point function r(4) associated with the diagram of Fig. 9-5b: 4 d p 1 i (Aq>2)2 4 2 (2n)4 (p2 - m 2 + is)2
FUNCTIONAL METHODS
Figure 9-5 Divergent diagrams to the one-loop order.
Higher powers of q> lead to convergent integrals. Renormalization introduces counterterms needed to insure that
d[,(2) dp2
(m~hYs) = 1
(9-120)
and a constraint on the four-point function giving the meaning of the renormalized coupling constant, for instance, its value at the on-shell symmetric point S:
pr = m~hYS
+ Pj =
%m~hYs
i-=lj
(9-121)
P4)(Pi) Is =
AphYs
These conditions are useful when dealing with some actual application to a scattering problem. They are, however, cumbersome to compute quantities in the effective action expanded at zero external momenta. Up to a finite renormalization, we can replace (9-120) and (9-121) by
[,(2)(0)
= _m 2
dp 2 )
~(O)
['(4)(0)
= -A
(9-122)
To emphasize the distinction we have introduced the label "physical" to the mass and coupling occurring in the previous normalizations. The virtue of Eqs. (9-122) is that they can be translated as conditions on the effective renormalized potential and function Zeff(q as
dq>2
Ye~O) = m
dq>4 Yeff(O) = A
(9-123)
Zeff(O)
These requirements are obviously fulfilled to order 11 as shown by (9-118). At this level we also have
=1 (9-124) eff The counterterms in q>2 and q>4 cancel the corresponding terms in the higher orders of Yeff and Zeff' Consequently, the correct one-loop contribution to the
Z(O)
QUANTUM FIELD THEORY
(9-125) It is useful at this stage to perform the Wick rotation Po -+ ipo. We denote by k the corresponding euclidean four-momentum so that 4 1 d k [( Acp2/2 ) Acp2/2 1 ( Acp2/2 ~p =:2 (2n)4 In 1 + k2 + m2 - k 2 + m2 +:2 k2 + m 2 (9-126)
Performing the integral and adding this expression to the zeroth-order term given by (9-118) yields 2 m cp2 Acp4 [(Acp2 Ac(2) Acp2 Acp2 Yeff=-2-+--::t:!+(8n T+m2 In 1+2m2 - T :2T+m2 + ...
)2 (
(9-127)
We observe that the large-cp behavior of V(cp) is modified by the quantum corrections. To exhibit it more transparently, it is useful to introduce yet another set of normalization conventions to be able to set m = o. We may define a new coupling constant AM such that
(9-128)
From (9-127) it follows that
AM = A +
(8~2 [6 In ~~: + 16 + o (;:)J
We use this definition in (9-127) and consider the limiting massless theory for which cp4 AL-cp4 ( cp2 25) (9-129) Yeff(CP) = AM 4T + (16n In M2 -""6 + ... a result due to Coleman and Weinberg. It is not possible to directly set m = 0 in Eq. (9-127). This arises from the structure of the ultraviolet subtractions in Eq. (9-126), the second of which, designed to enforce the condition (d 4Yeff/dcp4)(0) = 0, introduces an infrared divergence in the limit m = O. We have to choose an arbitrary but nonvanishing subtraction point cp = M to define the massless theory. Stated differently, the r functions at zero momentum generated by the expansion of (9-127) are singular as 111 goes to zero. The behavior when cp -+ CIJ is clear from the fact that the dimension of V is four so that dominant terms must be proportional to cp4 up to logarithms in cp/M. Note that the arbitrariness in the point M implies that a related change in M and AM must leave Yeff invariant. We encounter here a manifestation of the renormalization group to be discussed at length later.
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