Divergence Free QED Lagrangian in (2 + 1)-Dimensional Space-Time with Three Different Regularization Prescriptions

Quantum field theory can be understood through gauge theories. It is already established that the gauge theories can be studied either perturbatively or non-perturbatively. Perturbative means using Feynman diagrams and non-perturbative means using Path-integral method. Operator regularization (OR) is one of the exceptional methods to study gauge theories because of its two-fold prescriptions. That means in OR two types of prescriptions have been introduced, which gives us the opportunity to check the result in self consistent way. In an earlier paper, we have evaluated basic QED loop diagrams in (3 + 1) dimensions using the both methods of OR and Dimensional regularization (DR). Then all three results have been compared. It is seen that the finite part of the result is almost same. In this paper, we are interested to evaluate the same basic loop diagrams in (2 + 1) space-time dimensions, because of two reasons: the main reason in (2 + 1) space-time dimensions, these loops diagrams are finite, on other hand, there are divergences in (3 + 1) space-time dimensions and the other reason is to see validity of using OR to evaluate Feynman loop diagrams in all dimensions. Here we have used both prescriptions of OR and DR to evaluate the basic loop diagrams and compared the results. Interestingly the results are almost same in all cases.

We have clearly explained how this OR can be applied for evaluating Feynman diagrams in ref. [18]. For self consistence let us write a few main steps of this prescription which has to be used in evaluating the one-loop Feynman diagrams in (2 + 1)-dimensions.
In gauge theories we mainly deal with the generating functions. Then after some simplification we end up with some types operator and inverse operators.
Then how one can take care of these operators has been explained in this OR.
If we have an operator Ω then according to OR we can write In facing no divergences we can always choose n to be greater than or equal to the number of "loop momentum integrals" or in other words order in  .
Hence,   d  1  det  exp lim  d  e  e  d e  e  d  2   d  d  e  e  3   u  t  t  t  u t  I  I  I   u  t  u v  t  uv t  I  I  I   t  tt tr  t  u   t  uu where, 0 I Ω = Ω + Ω with 0 Ω is independent of the background field i f and I Ω is at least linear in i f .
Then following the steps described in ref. [15] we can find the result of the problems in consideration.
On the other hand if we want to use perturbatuve method then we have to take n = 1 for one -loop, take n = 2 for two-loops in Equations (2.2) and (2.3) and so on.
From Equation (2.3b) we can write the general prescription of Operator regularization for the Feynman diagrams following [18]: where the n α s are arbitrary. For one-loop diagrams it is enough to use n = 1. When m = 2 and n = 1, then Equation (2.7) taken the form ( ) In one loop calculations we can use (2.8) for operators. In the following sections we will use this prescription for evaluating the three basic one loop diagrams.

One Loop Fermions Correction in
Using the Feynman identity for combining the denominators, we can write The term linear in l′ integrates to zero because of symmetric integration, so Which is taken as the common starting point for both Dimensional and Operator regularization.
Using Feynman identity and then γ-algebra, the above result becomes, Thus according to dimensional regularization, we see that there is no divergent part in (2 + 1)-dimensional space-time, because the integrals are finite in 3-dimensions.
2) Operator Regularization Method: The same one-loop correction to fermion can be evaluated using OR, following the rule cited in Equations (2.5) and (2.6) in ref. [8]. The amplitude of the self-energy diagram as This is the same form as like as obtained by dimensional regularization approach.

One Loop Photon Correction in (2 + 1) Dim. Using Dimensional and Operator Regularizations
Let us consider the Feynman diagram for the one loop correction to the photon line shown in Figure 2 which is represented by The QED one loop correction to the photon line in (2 + 1)-dimensions is   Combining the denominator using the Feynman identity and simplifying, we Which is again taken as the common starting point for both Dimensional and Operator regularization for one loop correction to the photon lines.
Using Feynman identity-II, the above result becomes, Journal of Applied Mathematics and Physics Thus according to dimensional regularization, we see that there is no divergent part in (2 + 1)-dimensional space-time, because the integrals are finite in 3-dimensions.
Now proceeding with operator regularization and again following the same route, we get, This is the same form as we obtained by dimensional regularization approach.

One Loop Vertex Correction in (2 + 1) Dim. Using Dimensional and Operator Regularizations
Let us now consider the Feynman diagram for the one loop correction to the vertex shown in Figure 3 which is represented by The QED one loop correction to the vertex in (2 + 1)-dimensions is Applying the 3-parameter Feynman formula for combining the denominator Journal of Applied Mathematics and Physics and shifting the variable of integration l l px qy → + + and simplify the denominator and numerator, we obtain,

∫ ∫ ∫
Now performing the momentum integral -I from above, then we get, Now using the Equation (2.1.4) and γ-algebra, then above equation reduces to, where, This is the same form as like as obtained by dimensional regularization approach.

Path Integral Form of Operator Regularization for One Loop Generating Functional in QED
The path integral form of OR for one-loop case is described in ref. [15]. That is if we consider the QED Lagrangian as, Here we see that 1 Z is the ratio of determinant of operators. Each of the de-

One-Loop Generating Functional and Loop Corrections for External Boson Lines
To find the loop corrections or to write the generating functional for external boson lines one has to make a close look at the numerator of Equation ( To one-loop order this series plays the same role as Feynman rules in the usual perturbation theory. Here we want to evaluate the one-loop correction to the two-point function for external photon in QED; we restrict our attention to the term bilinear in V µ on the right-hand side of Equation (