is the advantage of this method we will find that the gauge invariance of A\u03bc is

Is the advantage of this method we will find that the

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is the advantage of this method; we will find that the gauge invariance of¯Aμis retainedthroughout the calculation.We add to our action the gauge-fixing termSgf=1g2Zd4xtr (¯DμδAμ)2(2.62)The choice of overall coefficient of the gauge fixing term is arbitrary. But nice thingshappen if we make the choice above. To see why, let’s focus on the¯DμδA¯DδAμtermin (2.60) . Integrating by parts, we haveZd4xtr¯DμδA¯DδAμ=-Zd4xtrδA¯Dμ¯DδAμ=-Zd4xtrδA[¯Dμ,¯D] +¯D¯DμδAμ=Zd4xtrh(¯DμδAμ)2+iδA[¯Fμ,δAμ]iThe first of these terms is then cancelled by the gauge fixing term (2.62), leaving uswithSY M+Sgf=1g2Zd4xtr12¯Fμ¯Fμ+ 2¯Fμ¯DμδA+¯DμδA¯DμδA-2i¯Fμ[δAμ,δA]-2i¯DμδA[δAμ,δA]-12[δAμ,δA][δAμ,δA]and we’re left with just two terms that are quadratic inδA.We’ll return to theseshortly.The next step of the Faddeev-Popov procedure is to implement the gauge fixingcondition (2.61) as a delta-function constraint in the path integral.We denote thegauge transformed fields as¯A!μ=¯AμandδA!μ=δAμ+¯D!-i[δAμ,!]. We then usethe identityZD! δ(G(¯A!,δA!)) det@G(¯A!,δA!)@!= 1– 68 –
The determinant can be rewritten through the introduction of adjoint-valued ghostfieldsc. For the gauge fixing condition (2.61), we havedet@G(¯A,δA!)@!=ZDcDcexp-1g2Zd4xtrh-c¯D2c+ic[¯DμδAμ, c]iwhere we’ve chosen to include an overall factor of 1/g2in the ghost action purely as aconvenience; it doesn’t eect subsequent calculations. The usual Faddeev-Popov storytells us that the integrationRD!now decouples, resulting in a unimportant overallconstant. We’re left with an action that includes both the fluctuating gauge fieldδAμand the ghost fieldc,S=SY M+Sgf+Sghost,S=1g2Zd4xtr12¯Fμ¯Fμ+ 2¯Fμ¯DμδA+¯DμδA¯DμδA-2i¯Fμ[δAμ,δA] +¯Dμc¯Dμc-2i¯DμδA[δAμ,δA]-12[δAμ,δA][δAμ,δA] +ic[¯DμδAμ, c]As previously, we have arranged the terms so that the middle line is quartic in fluctu-ating fields, while the final line is cubic and higher.One-Loop DeterminantsOur strategy now is to integrate out the fluctuating fields,δAμandc, to determinetheir eect on the dynamics of the background field¯Aμ.e-Se[¯A]=ZDδADcDce-S[¯A,δA,c]Things are simplest if we take our background field to obey the classical equations ofmotion,¯Dμ¯Fμ, which ensures that the term linear inδAμin the action disappears.Furthermore, at one loop it will suffice to ignore the terms cubic and quadratic influctuating fields that sit on the final line of the action above. We’re then left just withGaussian integrations, and these are easy to do,e-Se[¯A]= det-1/2Δgaugedet+1Δghoste-12g2Rd4xtr¯Fμ¯Fμwhere the quadratic fluctuation operators can be read ofrom the action and are givenbyΔμgauge=-¯D2δμ+ 2i[¯Fμ·]andΔghost=-¯D2where the¯Fμ