UV-IR Mixing in Non-Commutative Plane

Poincar\'e-invariant quantum field theories can be formulated on non-commutative planes if the coproduct on the Poincar\'e group is suitably deformed \cite{Dimitrijevic:2004rf, Chaichian:2004za}.(See also especially Oeckl \cite{Oeckl:1999jun},\cite{Oeckl:2000mar} and Grosse et al.\cite{Grosse:2001mar}) As shown in \cite{Balachandran:2005eb}, this important result of these authors implies modification of free field commutation and anti-commutation relations and striking phenomenological consequences such as violations of Pauli principle \cite{Balachandran:2005eb,Bal3}. In this paper we prove that with these modifications, UV-IR mixing disappears to all orders in perturbation theory from the $S$-Matrix. This result is in agreement with the previous results of Oeckl \cite{Oeckl:2000mar}.


Introduction
The non-commutative Groenwold-Moyal plane is the algebra A θ (R d+1 ) of functions on R d+1 with the * -product as the multiplication law. The latter is defined as follows.
Here x 0 is the time coordinate, and the rest are spatial coordinates. Henceforth, we will write α * θ β as α * β.
The appearance of constants θ µν would at first sight suggest that the diffeomorphism group Diff (R d+1 ) of R d+1 , and in particular its Poincaré subgroup is not an automorphism of A θ (R d+1 ). But the work of [1] and [2] (and the earlier work of [4] and [5]) have shown that this appearance is false. Thus there exists a deformed coproduct on Diff (R d+1 ) which depends on θ µν . With this deformation, Diff (R d+1 ) does act as the automorphism group of A θ (R d+1 ).
In [6] (and the earlier work of [4] and [5]), it was shown that the standard commutation relations are not compatible with the deformed action of Poincaré group. Rather they too have to be deformed. If a(p) is the annihilation operator of a free field for momentum p, then for example, a(p) a(q) = η e ipµθ µν qν a(q) a(p), where η is a Lorentz-invariant function of p and q. The choices η = ±1 correspond, for θ = 0, to bosons and fermions. There are similar relations involving a(p) † 's as well. All of them follow from the relations where c(p) and c(p) † are the standard oscillators a(p) | θ=0 , a(p) † | θ=0 for θ = 0, and P µ is the translation generator: dµ(p) here is the Poincaré-invariant measure. For a spin 0 field of mass m, There are striking consequences of the deformed commutation relation [6] such as the existence of Pauli-forbidden levels and attendant phenomenology [7]. In this note, we show another striking result: Non-planar graphs and UV-IR mixing completely disappear from the S-matrix S θ because of the deformed statistics. S θ is in fact independent of θ µν so that S θ = S 0 . This does not mean that scattering amplitudes are independent of θ, as the inand out-state vectors are different, being subject to deformed statistics. Our treatment here covers both time-space and space-space noncommutativity. In the former case, although there were initial claims of loss of unitarity, the work of Doplicher et al. [8] showed how to construct unitary theories. These ideas were subsequently applied to construct unitary quantum mechanics as well [9,10]. So there is no good theoretical reason to set θ 0i = 0. The work we present here is quite general as regards the choice of θ µν , allowing also the choice θ 0i = 0.
We present the calculations for a real scalar field with the interaction The generality of the results will be evident from this example.
There is considerable overlap of the results of this work with those of Oeckl [4]. He too uses nontrivial twisted statistics, but does not use Poincaré symmetry implemented with a twisted coproduct [1,2]. In contrast, our previous work [6] deduced twisted statistics from Poincaré invariance. Oeckl then deduces an expression for the n-point function in agreement with ours. His derivation is based on braided quantum field theory developed by him [3]. Its relation to our approach awaits clarification. But we point out that once the appropriately twisted spacetime algebra and statistics are accepted as axioms, both Oeckl and us get the same final answer without ever invoking Poincaré invariance or any other spacetime symmetry except translations.

The Model
The free scalar field φ of mass m in the Moyal plane has the Fourier expansion The interaction Hamiltonian, in the interaction representation, is taken to be where : : denote normal ordering of a(p)'s and a(p) † 's.
The operator H I (x 0 ) is self-adjoint for any choice of θ µν , even with time-space noncommutativity. Hence the S-matrix is unitary. We will now show that S θ is independent of θ. That means in particular that there is no UV-IR mixing. Let e p be the plane wave of momentum p: e p = e ip.x . The * -product of plane waves is simple : e p * e q = e − i 2 pµθ µν qν e p+q .
Let us introduce the notation a(p) † = a(−p) where p 0 is also reversed by the dagger. Then 3 The Proof i) n = 2 First consider n = 2, just as an example. Then the O(λ) term of S θ is A typical term in φ * φ is 1 a(p)a(q) e p * e q = a(p)a(q) e (14) 1 Here we have used e p * e q = e i 2 pµθ µν qν e p+q , which requires replacing θ µν by −θ µν in (1). The reason for this change is explained in [6] after Eq.(2.33).
Note how the phases e ∓ i 2 pµθ µν qν cancel. Using ∂ µ e p+q = i(p + q) µ e p+q , we can write this as c(p)c(q) e p+q e Expanding the exponential, integrating and discarding the surface terms, we find that is independent of θ µν . The only delicate issue here concerns the surface term.
Here and in what follows, we will assume that such surface terms vanish. In the absence of long range forces, the assumption should be correct.
Next consider the O(λ 2 ) term where the differentials act only on e p 1 +q 1 , e p 2 +q 2 and phases involving just p µ and q µ cancelling out as before.
Next note that since we can in fact allow ∂x 20 to act on the θ-function as well. But then all terms involving θ µν in the power series expansion of the exponential are total differentials and vanish upon integrating over d d+1 Similar calculations show that S θ is independent of θ µν exactly, to all orders in θ µν .