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Consider the Lagrangian 1 1 M2 M2 #- c=_a 2 _a 2__12__x2__x 2 2( p.431) +2{ ax) 2 (51 2 X 24: where girl and X are real scalar elds.

Please answer the following question. It is from the book of Schwartz(QFT and the Standard Model) Prob 7.4

Lagrangian.jpegQuestion.jpeg

Lagrangian.jpeg

Consider the Lagrangian 1 1 M2 M2 #-
c=_a 2 _a 2__12__x2__x 2
2( p.431) +2{ ax) 2 (£51 2 X 24:51): where girl and X are real scalar fields.

Question.jpeg

Typically the hilinear terms [i.e. terms containing exactly two fields) in the Lagrangian
are referred to as the kinetic terms, since they determine how the field propagates through
space rather than how it interacts with other fields. One can, however, think of the mass terms, e.g. —%M12q5¥, as interaction terms rather than
kinetic ones. Note that this question closely follows problem 7.4 in Schwartz. a) What is the Feynman rule for the qblgai; interaction? 1)) Draw the [somewhat degenerate looking} Feynman diagrams that contribute to the 2-
point function {0|T{¢1(m)qa1(y)}|fl), using only the interaction term £111; 2 —%M12¢-%,
up to order M {5, and write down the amplitudes for each diagram. Associate external
lines with propagators. c] Sum the series to all orders in M12 and show you reproduce the same propagator as
if you were to treat the mass term as part of the free Lagrangian, i.e.
M? 2 1
£0 = §(a_u.¢tl2 — Y 1-

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