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Gluon - W-meson scattering via different renormalization schemes

The one loop gluon - W-meson amplitude is calculated by means of the gauge-invariant generalized Pauli-Villars regularization and with the help of dimensional regularization. It is shown that in the former case the amplitude satisfies Generalized Ward Identities, whereas in the latter case the amplitude differs from the first one by the constant.


Introduction
An important part of the calculations of radiative corrections in the Standard model is a gauge invariant renormalization procedure. The dimensional regularization which is mainly used for practical calculations does not preserve chiral gauge invariance as there is no self-consistent definition of γ 5 matrix in arbitrary dimension. Of course one can use for calculations noninvariant regularization as well and restore gauge invariance of renormalized theory by adding noninvariant counterterms. However in this case to fix renormalization freedom one has to use Generalized Ward Identities (GWI). In the case of the Standard model it complicates calculations considerably. A good illustration of this problem is a calculation of the one loop diagram describing the scattering of two gauge particles. This diagram is superficially divergent but gauge invariance fixes the renormalization freedom completely and imposing on the amplitude GWI one gets unambiguous finite result. This program was realized long ago for scattering of light by light in QED [1,2].
In principle analogous procedure may be applied to the QCD process, for example the gluon-W-meson amplitudes. However in this case one has to use GWI related to the different gauge groups (vectorial SU(3) and chiral SU(2)), and contrary to the QED case the amplitude in question is not symmetric with respect to the interchange of all arguments. It makes the procedure quite cumbersome.
One may avoid those complications by using a manifestly gauge invariant procedure. In this case GWI are fulfilled automatically and one gets an unambiguous result for this amplitude.
A gauge invariant regularization procedure for the Standard model (generalized Pauli-Villars regularization or GPV) was proposed in [3]. In the present paper we apply this regularization to the one loop amplitude describing the transition of two gluons to two W-mesons. We demonstrate that GPV regularization indeed produces an unambiguous finite result which is consistent with GWI. The formal using of the dimensional regularization with γ 5 -matrix defined as γ 5 = iγ 0 γ 1 γ 2 γ 3 gives the amplitude which differs from the former one by a constant.
The following calculations were carried out for the Standard model. To simplify the calculations we embed the Standard model gauge group SU(3) × SU(2) × U(1) into the the SO(10) group.
One may note that the term, proportional to Γ 11 vanishes, and I (r,+) = I (r,−) . The expression for the I (0) in (4) is the particular case of the I (r,+) (when r = 0). For explicit calculations Feynman parametrization is used: α + β + γ + δ = 1, and the integration over the Feynman parameters is defined as follows: For simplicity integration variable in the Π(k (1) , k (2) , k (3) , k (4) ) is shifted: Thus I (r) µνλσ (1, 2, 3, 4) can be rewritten in the form: (16) After this shift, the terms in the numerator proportional to p and p 3 may be omitted. The denominator can be represented as: Regularized expression for the Feynman diagram shown at Fig.1 which can be rewritten as derivatives of Since after the Wick rotation p 2 + F becomes negative, one sees that integrand of Π is proportional to: π M |p 2 + F |sinh(π |p 2 + F|/M) .
This expression decreases rapidly providing the regularization. But eq. (21) is too complicated for further calculations. Equation (21) arises after summation of contributions of an infinie series of PV fields. The necessity to introduce an infinite system of PV fields is due to the fact that these fields belong to the 32-dimensional representation of SO(10) and they contribute to the amplitude under consideration both I (r,+) and I (r,−) whereas the original fields ψ contribute only I (0,+) . For this reason as was discussed in ref. [3], a finite number of PV fields does not allow to satisfy the PV conditions providing the convergence of regularized amplitude. To construct a Lagrangian gauge invariant regularization for the Standard model with odd number of generations an infinite set of PV fields is needed. However being interested in the calculation of a particular four-point amplitude we can omit the terms I (r,−) without spoiling the gauge invariance. (As for this amplitude I (r,−) = I (r,+) it corresponds to replacing the determinant which arise after integration over the PV fields by the square root of the determinant.) Then to regularize the model it is sufficient to take a finite number of PV fields (r=1).
So instead of (18) the following formula for Π may be used: To illustrate this statement let us note that the integrand of the difference between (18) and (22) is proportional to the following: thus the difference between (18) and (22) tends to zero when Pauli-Villars mass M → ∞, and the gauge invariance is preserved. The complete four-point function G is the sum of Π's corresponding to the diagrams at Fig. 1, . . . , 6, which differ by permutations of W + , W − and g: Below we demonstrate that this amplitude satisfies generalized Ward Identities and in the limit M → ∞ leads to a finite gauge invariant expression.
These identities differ from QED only by the absence of symmetry with respect to the interchange of the momenta. Moreover, one can show that the fourth order fermion loop diagram have to satisfy the eq. (26) also in the model with spontaneously broken symmetry, when the fermions acquire nonzero masses. In principle GWI in this case will include a contribution of the Higgs fields, but for the diagram under consideration this contribution vanishes. For this reason in the following we consider a general case when the fermions are massive. The physical fermion propagator looks as follows: and the interaction vertices include the chiral projector. One can note that ). Thus terms proportional to tr(γ 5 . . .) in G cancel in pairs identically. For example, such contributions into I µσλν (1,4,3,2) and I µνλσ (1, 2, 3, 4) cancel each other: Thus the expression for this four-point function G is equal to the expression for the four-point function in non-chiral theory multiplied by 1/2. Therefore one expects that the fourth order scattering amplitude calculated with the use of GPV regularization is gauge invariant.
As it was expected the scattering amplitude calculated with the help of GPV method satisfy automatically generalized Ward Identities. In what follows this regularization method will be compared with other methods.

The renormalization with the help of the Generalized Ward Identities (GWI)
The only term in the numerator of the integrand of I (0) , which can lead to divergency is: (In what follows we use the notation I (0) ≡ I.) The part of I, which contains only this term in the numerator will be called as the 'potentially divergent part'. One may note that for some µ, ν, λ, σ the potentially divergent part of I vanishes. With the help of the identity: this potentially divergent part of I µνλσ can be represented as follows: Taking trace over spinoral indices one gets the following result: If δ µν δ λσ +δ µσ δ νλ −2δ µλ δ νσ = 0, then I µνλσ is finite and does not need renormalization. The set of indices {µ, ν, λ, σ} for which the expression in (30) is zero (i.e. I µνλσ -finite) is large enough for finding I with arbitrary indices with the help of the GWI (26).

Discussion
The comparison of different regularizations methods for the four-point function is carried out. The contribution of the spinor loop to the four-point function for W + W − → gg transition in the Standard model is examined by means of the Generalized Pauli-Villars regularization, dimensional regularization and with the help of Generalized Ward Identities.
It is found that GPV regularization and renormalization with the help of the GWI give correct result, whereas inconsistency in the definition of γ 5 -matrix in the framework of the dimensional regularization leads to the result, which differs from the correct one by a constant.
For the diagram we considered, all three methods require a comparable amount of work. However for more complicated diagrams the manifestly gauge invariant GPV regularization seems to be preferable.