In the first chapter, I introduce Wilson's brilliantly insightful concept of the 'smooth effective Lagrangian' [9], and extensively discuss the renormalization theorems derived by Polchinski in this picture [10]. The ideal way to approach this problem is to study the effective Lagrangian flow.
The Effective Lagrangian Flow Equations for ,\¢} Theory
By expanding the interaction S in the powers of the field cjJ, we can rewrite the flow in the form of dimensionless coefficient functions:. The equation of motion for the W vertices is identical to that for the V vertices, so I won't rewrite it.).
Description of the Flow in Perturbation
The importance of this result is that we have fixed the two ends of the flow. Values of the vertex functions for momenta outside this range will then be unphysical, and therefore ignorable.
The Renormalisation Theorem
In the proof of the A vertices, the integrated equation (1.48) now contains the initial values (1.62). Suppose (p~)1 are the initial relevant parameters that flow down to the solid surface defined by 1 using form 1 of the flow equations.
The Renormalisation of Composite Operators
The difference between fermions and bosons only lies in the signs of different terms in AfJS j f) A and they vanish while taking boundaries. size ~ d. You might think that we have in fact already proven this simply by replacing S with STAtal in the previous paragraphs, but in reality we haven't, because.
Summary
In general, an operator will mix with all those of the same symmetry with the same or lower canonical dimension. This actually results in a breakdown of the renormalization theorems in Chapter I, because the estimates (1.36), which were crucial to the argument, do not hold. If we consider that all the poles in B have a positive mass squared, we find that B is essentially the sum of the form terms. and so the sum of the residuals is zero.
In an obviously Lorentz-covariant formalism, we need only use representations of the Lorentz group - but a 4-vector has one more degree of freedom than we require, and due to the signature of the spacetime it appears as a ghost or a tachyon. We really need to preserve unity, locality and the existence of the vacuum (the absence of tachyons). The only known acceptable solutions to the dilemma are gauge theories, for which one proves that the S-matrix is indeed unitary when restricted to a 'physical' subspace of the full Hilbert space.
This 'decoupling' of the physical and non-physical parts depends crucially on the congregation's identities, and it is to these that I now devote this chapter. But any such prospective argument is bound to be inductive in the order of the couplings (or n), and so it is disastrous if the input, the regularized Ward identity, requires an infinite expansion in n before we can begin. Previous work [11] has failed right here, due to a misunderstanding of the fact that the inclusion of higher covariant derivatives in the action only softens the structure of the n 2 2-loop divergences instead of dismantling it.
The Method of Higher Covariant Derivatives
E = 1 is negligible, since the tadpole graphs disappear with symmetric integration, which is allowed by the presence of the pre-regulating cutoff of momentum r.). Alternatively, regulators of the same type as the measuring field could be included, in which case there would also be single-loop deviations from graphs with 2 or 4 external regulators. 11 showing the contributions of the same topology, from the gauge field and the Pauli–Villars field to the two-point gauge function.
It can sufficiently preserve the one-loop divergence structure, while also reducing the type of regulating fields. Now to estimate the freedom in the parameters, consider for the moment a real scalar regulator of type 2r in representation R of the gauge group. In fact, there is no dependence, since an insertion of the longitudinal part of the gauge propagator automatically makes any one-loop graph f-finite.
I define the variation of the BRS fields of the regulator as only the variation of the gauge, 8j = i(Ta)jkTJa£. The Jacobian of the transformation is indeed unity (see Appendix 2), so we have, according to (2.8), The regularized Ward identity holds regardless of the nature of the preregulator.
The Renormalised Ward Identity
Green's functions from (2.37) are Mo-finite for a certain choice of bare couplings p~ and generically we have p~ and p~ of order MJ PN(logMo/ MR) and. This is because the longitudinal and transverse parts of the gauge propagator (2.35b) in particular have different boundary coefficients. All machines are now established for an inductive proof of the renormalized Ward's identity, where the induction is of order n.
By an appropriate choice of the relevant parameters p~ on the low scale, the renormalized interaction S(Mo, N) can be written as the sum of Sreg, cf. e.g. In what follows, I change the normalization of the Green's functions by a factor 1i1-E, so that L ~ P. Consider the longitudinal part of the gauge 2-point function calculated from the shifted interaction at MR, namely S(MR, oo ). (Remember, we can calculate the Green's functions using the interaction at any scale we want.) The mo divergences at first order in 1i come solely from 8 pf and 8 p~, since changes in the other parameters only contribute to the Green's work by forming loops and therefore they only come into higher order.
We have to show that the four-dimensional part of the regulator interaction is also of gauge invariant form at 0(1i1. This is proved in exactly the same way mentioned in the next section as for a non-chiral substance coupled to a gauge field. Using reasoning we can derive that for each n, 2.48). This completes the proof that we can construct an order by order in 1i renormalized interaction of the form (2.33), which gives Green's functions finite in the limit Mo, N ---t oo, which satisfy the renormalized Ward's identity (2.34) .
Non-chiral Matter Couplings
Unitarity
I argue that by choosing the interpolation gauge (2.51) and with M in the transverse region, we still have a renormalized interaction of the form (2.33), despite the Lorentz invariance breaking. This is because the embedding of the nonlinear field function does not have full singularity.). They are derived from actions with the same scale setting, but the values of the renormalization constants Z1,.
Mo-divergences can be hidden in the choice of bare parameters, and the only difference between the renormalizable and non-renormalizable cases is whether the number of parameters is finite or infinite. The presence of trivial operators only affects the relations between bare and renormalized couplings and does not affect the (physical) relations between the renormalized Green's functions themselves. In the above context, gauge theories are special in that we essentially require that some relevant operators do not appear in S(Ao).
However, the generic results of renormalization theory give us a bare mass term of order A5 times a polynomial of log (Ao/Ephys), so clearly we need to characterize the nature of the effective Lagrangian flow more finely. It is this that provides the necessary constraints on the form of the 'bare interaction', and it enables us to obtain the renormalized Ward Identity. The final step of the argument is to link the renormalized Ward identity to unity.
Appendix 1. 'Construction of the HCD regulator'
In (A1.2), the values given for divergences and 6 must all be multiplied by T(R), which is the normalization constant for the group generators, i.e. the values of the real scalar contributions can be obtained from (A1.2) by setting Ar = 1,7rr = 0, replacing En'TJr,Kr by ar, f3n In and dividing by 8. Now we see that the divergences 2,3, 4 and 6 can be taken care of by using our freedom in the parameters ar , f3r, .
In fact, to be sure of their cancellation, we need to calculate the gauge and ghost contributions, calling them d1 and ds respectively. Now the Lagrangian for the gauge and ghost fields is Eq. It is clear that divergence 5 is not a problem, and d1 can be canceled by real scalars in the adjoint representation, which is satisfactory. A1.2) provides us with no less than nine independent real parameters and one can trivially find combinations showing cancellation.
Appendix 2. 'The Jacobian of the BRS transformation'
Supersymmetry
Moreover, a consideration of the 'D-algebra' shows that the purely chiral operators J d2fJ¢l are indeed Ao-finite [5], so that the only A0 divergence comes in the wave function renormalization, z~¢>·. It is worth reminding ourselves at this point of the difficulties that standard renormalization theory faces in dealing with N = 1 Super Yang-Mills. The discussion of the r-divergences and r-finite parts and the existence of a compensating operator then follows exactly that of Appendix 4.
The proof of the renormalized Ward identities from (3.28) follows the arguments presented above for chiral gauge theories. The mass and self couplings of the matter fields, ffiij and 9ijkl are not renormalized, as in a pure matter theory, and neither is the gauge coupling g. This showed that the matrix S computed on G is the same as that computed on any of the transverse gauges, and hence that it is renormalizable (as well as unique).
Instead, we require the absence of some of the relevant actors to ensure the renormalized Ward identities and thus unitarity. To see the non-trivial effect of the presence of gravity, we take the action (3.9) and make it 'generally covariant'. Again the possible obstacles G are fully classified at the level of one loop, and must have the form, c.f.
The expressions in 8Lpre due to the pulse-off pre-regulator can be analyzed exactly as in Appendix 3, so that all that remains are operator insertions due to the intrinsic non-invariance of the gauge couplings of the regulator. However, it is difficult to justify them, since strictly using the d = 4 algebra is inconsistent [20].
