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In $\phi^4$ momentum conservation at each vertex comes from integration in the vertex. How can from momentum conservation at each vertex proof overall momentum conservation that is if $k_1,k_2 ,k_3,k_4$ are external lines how can we proof that the deltas function at each vertex turn into one delta function, namely $\delta(k_1+k_2 -k_3-k_4)$

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The number of 4-momentum conserving delta functions is one more than the number of (independent) internal momenta. After performing integrations over the internal momenta, you will be left with one remnant delta function. This is the total 4-momentum conserving delta function every amplitude must have. Can you see this for $\phi^4$ theory?

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  • $\begingroup$ Alternatively, given that OP already agrees that momentum is conserved in each vertex, just do a simple resummation to show that this holds for all diagrams (not just the ones with a single vertex). $\endgroup$ Nov 7, 2018 at 7:51

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