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Question data given below:

Let us consider the 𝐴 βˆ’ 𝐡 βˆ’ πœ™ model that describes 3 real scalar fields 𝐴, 𝐡 and πœ™ governed by the following action $𝑆[𝐴, 𝐡,πœ™] = ∫ 𝑑^ 4 π‘₯ (β„’_𝐴 + β„’_𝐡 + β„’_πœ™ + β„’_{𝑖𝑛𝑑})$

$ β„’_𝐴 [𝐴] = \frac{1} {2} πœ•_ πœ‡π΄ πœ•^πœ‡π΄ βˆ’ \frac{1} {2} π‘š_𝐴^ 2 𝐴^ 2$

$β„’_𝐡[𝐡] = \frac{1} {2} πœ•_ πœ‡π΅ πœ•^πœ‡π΅ βˆ’ \frac{1} {2} π‘š_B^ 2 𝐡^ 2$

$β„’πœ™[πœ™] = \frac{1} {2} πœ•_ πœ‡πœ™ πœ•^πœ‡πœ™$

$β„’_{𝑖𝑛𝑑}[𝐴, 𝐡,πœ™] = βˆ’ \frac{1} {2} (𝑔_𝐴 𝐴^ 2 + 𝑔_𝐡 𝐡^ 2 ) πœ™$

In this exercise, we shall determine the 4-point function

$β„³ = ⟨0| 𝑇(𝐡(π‘₯_4 ) 𝐴(π‘₯_3 ) 𝐡(π‘₯_2 ) 𝐴(π‘₯_1 )) |0⟩$ in terms of Feynman diagrams through path integral expressions. It is known that $β„³ = \frac{𝑍_β„³} {𝑍_0}$

where $𝑍_β„³$, $𝑍_0$ are given by the following path integrals

$𝑍_β„³ = ∫ π’ŸΞ¦ 𝐡(π‘₯_4 ) 𝐴(π‘₯_3 ) 𝐡(π‘₯_2 ) 𝐴(π‘₯_1 ) 𝑒^ {𝑖𝑆}$

$𝑍_0 = ∫ π’ŸΞ¦ 𝑒^ {𝑖𝑆}$ and where $π’ŸΞ¦ ≔ π’Ÿπ΄ π’Ÿπ΅ π’Ÿπœ™$

denotes integration over classical field configurations of all three fields.

The question is how to show through diagrams or algebric path integral in the position space that two (actually three) vaccum diagrams that recieved by expanding $Z_β„³$ and $Z_0$ up to order 2 in $g_A$ and $g_B$, are vanished when we expanding $β„³$ to second order? (Vaccum diagrams are diagrams that including connection of the propogator to itself and not to an external legs, such as "2-tadpoles")

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