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By Csaba Csaki, Scott Dodelson
This quantity provides a collection of pedagogical lectures that introduce particle physics past the normal version and particle cosmology to complicated graduate scholars.
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Extra info for Physics of the Large and the Small: Tasi 2009, Proceedings of the 2009 Theoretical Advanced Study Institute in Elementary Particle Physics
Example text
Symmetry 6 factor, while there are choices of the external lines in Fig. 6(a). 3 The term with the propagator on the right-hand side of Eq. (50) together with diagram (b) in Fig. 6 give the modification of the ϕ4 interaction in Eq. (48). Diagram (b) is associated with the 3! · 3 factor, where 3! comes from the permutations of lines without the derivative and 3 comes from placing the derivative on either of the 3 external lines. Combined with 3! in Eq. (50), we get 3! · 3 + 3! = 4! to reproduce the coefficient of the ϕ4 term in Eq.
4! 3! (∂µ ϕ′ )2 m2 ′2 η ηc2 ′6 = − ϕ − ( + c2 m2 )ϕ′4 − (c1 + )ϕ + . . , (47) 2 2 4! 3! where we omitted terms quadratic in the coefficients c1,2 . This field redefinition removed the ϕ3 ∂ 2 ϕ term and converted it into the ϕ6 term. Field redefinitions are equivalent to using the lowest oder equations of motions to find redundancies among higher dimensional operators. The equation of motion following from the η 3 Lagrangian in Eq. (46) is ∂ 2 ϕ = −m2 ϕ − 3! ϕ . Substituting the derivative part Lϕ→ December 22, 2010 9:24 WSPC - Proceedings Trim Size: 9in x 6in tasi2009 Effective field theory and precision electroweak measurements 29 of the ϕ3 ∂ 2 ϕ operator with the equation of motion gives LD>4 = −c1 ϕ6 + c2 ϕ3 ∂ 2 ϕ → −c1 ϕ6 + c2 ϕ3 (−m2 ϕ − η 3 ϕ ) 3!
A3µ T A3ν = −Nc T · = 2 igµǫ 2 dd k i2 tr[γ µ PL (/ k + MT )γ ν PL (/ k + MT )] 2 d 2 2 (2π) (k − MT ) iNc g 2 MT2 g µν 2 (4π)2 1 µ2 + ln( 2 ) , ǫ MT (63) 5 where PL = 1−γ 2 . The diagram with the B quark in the loop gives the same answer, except for the MT → MB replacement. The two diagrams with external A1µ bosons are identical and can be evaluated as A1µ T A1ν = −Nc B · = = 2 igµǫ 2 dd k i2 tr[γ µ PL (/ k + MT )γ ν PL (/ k + MB )] 2 d 2 2 (2π) (k − MT )(k − MB2 ) iNc g 2 g µν 1 dx(xMT2 + (1 − x)MB2 ) 2(4π)2 0 µ2 1 + ln( ) · 2 ǫ xMT + (1 − x)MB2 iNc g 2 g µν 2(4π)2 MT2 + MB2 2ǫ 2 2 + µ µ 4 MT4 ln( M 2 ) − MB ln( M 2 ) + T B 4 MT4 −MB 2 2(MT2 − MB2 ) .