Hybrid Seiberg-Witten map, itsθ-exact expansion, and the antifield formalism

Abstract

We deduce an evolution equation for an arbitrary hybrid Seiberg-Witten map for compact gauge groups by using the antifield formalism. We show how this evolution equation can be used to obtain the hybrid Seiberg-Witten map as an expansion, which is θ-exact, in the number of ordinary fields. We compute explicitly this expansion up to order three in the number of ordinary gauge fields and then particularize it to case of the Higgs of the noncommutative standard model.

Quantum duality under the θ-exact Seiberg-Witten map

“…A (n) µ and C (n) are totally symmetric under the permutations with respect to the set of the parameter-triples {(a i , µ i , p i )|i = 1, ..., n}, which have the property -of key importance in our later discussion-that only the momenta which are not contracted with θ µν build up polynomials which never occur in the denominator [25,26]. Now, let us introduce the ordinary background-field splitting…”

“…The θ-exact Seiberg-Witten map for scalar or fermion fields in the adjoint reads -see [26]: µ 1 , p 1 ), ......, (a n , µ n , p n ); (a, p); θ]…”

“…Now, taking into account the recursive equations -see section III of ref. [26]-which yieldΦ [A µ , Φ, θ] (x) have similar θ and momentum structure to the one for µ [A µ , θ], it is easy to see that F (n) [(a 1 , µ 1 , p 1 ), ......, (a n , µ n , p n ); (a, p); θ] is also a linear combination of functions of the type in (2.24). Hence, one also concludes that the Jacobian determinant for the change of variables Φ a →Φ a (x) = tr(Φ(x)T a ) = tr(T aΦ [A µ , Φ, θ] (x)) in the path integral over the fields Φ a is one, i.e.…”

Photon polarization tensor on deformed spacetime: A four-photon-tadpole contribution

“…Despite significant progresses made in [23] the result is incomplete because there also exists four photon self-coupling induced by the second order θ -exact Seiberg-Witten map expansion, which is, historically, largely an untouched subject due to mathematical sophistication. Following the recent results on higher order θ -exact Seiberg-Witten map [17,32,18], we present here the full θ -exact four photon self-interaction in the SW mapped deformed pure U(1) gauge theory on Moyal space and the evaluation of the resulting four-photon tadpole diagram, thus complete the one loop corrections to the photon polarization tensor.…”

“…One of most renowned issues is the quadratic infrared (IR) divergence in the one-loop 1-PI two point function of the gauge fields [3,4,5]. Actually its existence in a very large and important subcategory of the deformed gauge theories on Moyal space, namely those defined via Seiberg-Witten (SW) map [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21], was confirmed not long ago [22,23]. Unlike the quadratic UV/IR mixing in the scalar field theory on Moyal space [24], this quadratic IR divergence does not have a quadratic UV divergent counterpart, and appears to be difficult to control [26,27,28,29,2].…”