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\noindent PHY--396~L.\quad
Problem set \#15.\quad
Due February 17, 2005.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
First, a calculational exercise.
In class, I described the higher-derivative UV regularization scheme
in which propagators behave at high Euclidean momenta as
$$
\pspicture[0.25](0,-.18)(3,+.18)
\psline(0,0)(3,0)
\endpspicture\quad
=\ {1\over p^2\,+\,m^2}\ -\ {1\over p^2\,+\,\Lambda^2}\
\approx\ {1\over \Lambda^{-2}p^4\,+\,p^2\,+m^2}\,.
\eqn\HDC
$$
Use this UV cutoff to calculate the scalar loop
$$
\pspicture[](-4.33,0)(+4.33,2.5)
\psarc(0,-2.5){5}{30}{150}
\psarc(0,+5){5}{210}{330}
\pscircle*(-3.3,1.25){0.1}
\pscircle*(+3.3,1.25){0.1}
\endpspicture
\eqn\Loop
$$
Your result should have the same form as the loop with a hard-edge cutoff,
provided the respective cutoff parameters are related according to
$$
\Lambda^2_{\rm higher-derivative}\
=\ \Lambda^2_{\rm hard-edge}\times\rm a\ numerical\ constant.
\eqno\eq
$$

\bigskip
\point
And now a reading assignment,
\S7.3 of the {\it Peskin \& Schroeder} textbook.
Read about the {\sl Optical Theorem} in QFT and Cutkoski's cutting rules.

\bye
