Department of Physics, University of Milan and INFN, Sez. di Milano, via Celoria 16, I-20133 Milano, Italy
Received Date: December 12, 2007; Revised Date: March 28, 2008
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The implementation of the ’t Hooft ®-gauge in the symmetrically subtracted massive gauge theory based on the nonlinearly realized SU(2) gauge group is discussed. The gauge independence of the self-mass of the gauge bosons is proven by cohomological techniques.
A consistent subtraction scheme for massive non-Abelian gauge theories based on a nonlinearly realized gauge group has been recently proposed in . The symmetric subtraction algorithm was already successfully applied to the four-dimensional nonlinear sigma model in the flat connection formalism in –.
The Feynman rules of the nonlinearly realized massive gauge theory entail that already at one loop level there is an infinite number of divergent amplitudes involving the pseudo-Goldstone fields Φa . The latter amplitudes are uniquely fixed by implementing a defining local functional equation [5,7] which encodes the invariance of the path-integral Haar measure under local SU(2)L transformations2
is the element of the nonlinearly represented SU(2)L gauge group
vD is a D-dimensional mass scale and ta are the Pauli matrices. is the SU(2)L gauge connection. It is also convenient to introduce the SU(2)L flat connection
with the following SU(2)L transformation induced by the transformation of
The amplitudes not involving the pseudo-Goldstone fields are named ancestor amplitudes since they are at the top of the hierarchy induced by the local functional equation. At every loop order there is only a finite number of divergent ancestor amplitudes (weak power-counting theorem [5,10]). The requirement of physical unitarity is satisfied since the Slavnov-Taylor (ST) identity holds [5,9]. The ghost equation and the Landau gauge equation are also preserved by the symmetric subtraction . These symmetries, supplemented by global SU(2)R invariance and the weak power-counting, uniquely fix the tree-level vertex functional of the nonlinearly realized theory .
In  the Landau gauge was used for the sake of simplicity and conciseness. The aim of this note is to implement the ’t Hooft ®-gauge in a way compatible with all the symmetries required for the definition of the model (local functional equation, ST identity, ghost equation, B-equation for a general ®-gauge ) and the weak power-counting.
The validity of the local functional equation requires that the gauge-fixing functional transforms in the adjoint representation of SU(2)L. In the Landau gauge this was achieved by introducing an external vector source Vaμ and by making use of the gauge-fixing functional
where is the covariant derivative w.r.t. the vector source Vaμ. Ba
is the Nakanishi-Lautrup field . It transforms in the adjoint representation of SU(2)L. The local functional equation is preserved by the gauge-fixing (1.5).
It should be stressed that the local functional equation associated with the SU(2)L local invariance is not the standard background Ward identity -. The essential difference is that the local functional equation is bilinear in the vertex functional ¡, due to the presence of the nonlinear constraint Φ0 in Eq (1.2), which needs to be coupled to the scalar source K0 in the tree-level vertex functional.
The ’t Hooft ®-gauge is defined by the condition of the cancellation (once the Nakanishi- Lautrup field is eliminated via its equation of motion) of the mixed terms arising in the nonlinear theory from the mass invariant
The ’t Hooft (φ-dependent) gauge-fixing functional must transform in the adjoint representation of SU(2)L in order to preserve the local functional equation. For that purpose one needs to introduce an auxiliary matrix
with the same SU(2)L transformation as
has the correct transformation properties. Therefore one can consider the following gauge-fixing functional
where ® is the gauge parameter. With the choice in Eq (1.10) the local functional equation is respected and by integrating the Nakanishi-Lautrup field Bα the mixed terms arising from Eq (1.6) are canceled. The propagators obtained by using the gauge-fixing functional in Eq (1.10) have a UV behaviour compatible with the weak power-counting.
It is important to realize that b is not an element of SU(2). The reason is that the amplitudes involving and must be ancestors. Already at one loop level one cannot have a finite number of divergent amplitudes involving and is given by the SU(2) constraintOne must split in a linear way the constant component of by setting Since and are independent, by inspecting the Feynman rules one can then check that the weak power-counting is preserved. Moreover, since are independent variables, the BRST transformation can be extended to these sources by pairing them to external source ghosts as follows
Then form BRST doublets [15,16] and therefore they do not contribute to the cohomology H(s) of the BRST differential s. Hence they are not physical, as expected. The same technique can be used to prove that the vector source Vaμ does not modify the physical observables too .
The ghost-antighost part of the tree-level vertex functional is generated as usual by
with the standard BRST transformation of the antighost
Gauge dependence of the physical amplitudes
The extended ST identity is in fact sufficient to prove the independence of the physical quantities from α also in the nonlinear case. We sketch here the main lines of the proof. Dropping inessential terms involving the background ghosts the ST identity for the vertex functional ¡ is3
We define as usual the connected generating functional W by the Legendre transformation of ¡ w.r.t. the quantized fields (collectively denoted by
where is a short-hand notation for the sources of the quantized fields. Eq (2.2) yields stands for the source coupled to the field
By differentiating Eq (2.4) w.r.t and a set of sources coupled to physical BRSTinvariant local operators O1, . . . ,On one finds by going on-shell (all external sources set to zero)
i.e. the physical Green function is on-shell gauge-independent. is on-shell gauge-independent.
By differentiating Eq (2.2) w.r.t. and by setting all the fields and external sources to zero one gets
We decompose and into their transverse and longitudinal components as follows
By taking the transverse part of Eq (2.6) one finds (notice that the terms proportional to only contribute to the longitudinal part and thus drop out)
The self-mass is defined as the zero of
if no tadpoles are present as in the case under consideration. In passing it is worth noticing that the presence of tadpoles requires that the self-mass is defined as the pole of the transverse part of the connected two-point function (as it happens in the linearly realized theory).
By Eq (2.8) one finds
Moreover invertibility of is guaranteed in the loop expansion since
The above equation together with Eq (2.10) implies
i.e. the self-mass is gauge-independent. This behaviour is a typical property of the nonlinear theory. In the linear case the zero of the two-point 1-PI function is in general gauge-dependent. Gauge independence can only be recovered by taking into account the Higgs tadpole contributions. For a detailed comparison of the two-point 1-PI function in the linear and the nonlinear case see .
The formulation of the nonlinearly realized SU(2) massive gauge theory in the ’t Hooft gauge has been achieved in a way consistent with all the symmetries of the model and the weak powercounting. This requires the introduction of auxiliary external sources We have shown that this procedure does not alter the physical content of the model. Gauge independence of physical observables has been established by using cohomological methods. The self-mass, which can be computed in the nonlinearly realized theory as the zero of the transverse part of the 1-PI two-point function, has been proven to be gauge-independent.
This work was partially supported by INFN.
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