2282 relevant interactions Lets look at \u03c6 3 theory first then the mass terms We

2282 relevant interactions lets look at φ 3 theory

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22.8.2 relevant interactions Let’s look at φ 3 theory first, then the mass terms. We are interested in radiative corrections, so let’s couple φ to something else, say a fermion ψ . We will set the mass of φ to zero for simplicity. The Lagrangian is L = 1 2 φ square φ + g 3! φ 3 + λφψ ¯ ψ + ψ ¯ ( i∂ M ) ψ (22.59) 22.8 Super-renormalizable field theories 259
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Now consider the renormalization of g . We can compute this by evaluating the 3-point function ( 0 | T { φ ( p ) φ ( q 1 ) φ ( q 2 ) }| 0 ) = g + (22.60) There is a radiative correction from the loop of φ . This is UV finite, proportional to g 3 and not particu- larly interesting. A more important radiative correction comes from the loop of the fermion. M 2 = p q 2 k q 1 k q 1 k + q 2 = λ 3 integraldisplay d 4 k (2 π ) 4 Tr [ 1 k M 1 k q 1 M 1 k + q 2 M ] (22.61) = λ 3 8 π 2 M integraldisplay 0 1 dz integraldisplay 0 1 z dy bracketleftBigg M 2 + p 2 ( y + z 3 yz 1 2 ) M 2 p 2 yz + 3 log Λ 2 M 2 p 2 yz bracketrightBigg (22.62) Taking p 2 = 0 , this gives ( 0 | T { φ ( p ) φ ( q 1 ) φ ( q 2 ) }| 0 ) = g + λ 3 16 π 2 M parenleftbigg 1 + 3 log Λ 2 M 2 parenrightbigg (22.63) So we find a (divergent) shift in g proportional to the mass of the fermion M . This is fine if M g , but parametrically it’s really weird. As M gets larger, the correction grows. That means that the theory is sensitive to scales much larger than the scale associated with g . So, Super-renormalizable theories are sensitive to UV physics Is this a problem? Not in the technical sense, because the correction can be absorbed in a renormalization of g . We just add a counterterm by writing g = g R λ 3 16 π 2 M parenleftbigg 1 + 3 log Λ 2 M 2 parenrightbigg (22.64) This will cancel the radiative correction exactly (at p = 0 ). Again, this is totally reasonable if g R M , and theoretically and experimentally consistent for all g , but if g R M it looks strange . Will there be problems due to the quantum corrections? We can look at the Green’s function at a dif- ferent value of p 2 . This would contribute to something like the scalar Coulomb potential. We find that of, course, the divergence cancels, and the result at small p is something like ( 0 | T { φ ( p ) φ ( q 1 ) φ ( q 2 ) }| 0 ) ≈ g R + λ 3 16 π 2 bracketleftbigg p 2 M + 3 M log parenleftbigg 1 p 2 M 2 parenrightbigg + bracketrightbigg (22.65) g R + λ 3 16 π 2 p 2 M (22.66) So for p M these corrections are tiny and do in fact decouple as M → ∞ . As for the non-renormalizable case, we enjoy speculating that the ultimate theory of nature is finite. Then Λ is physical. If we suppose that there is some bare g which is fixed, then we need a very special value of Λ to get the measured coupling to be g R M . Of course, there is some solution for Λ , for any g 0 and g R . But if we change Λ by just a little bit, then we get g R g R + λ 3 16 π 2 M log Λ old 2 Λ new 2 (22.67) which is a huge correction if M g R .
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