• output

The leading-order contribution:

((i/2) (c_2 (-m_{\rm K}^2 + q^2) (-m_{\rm K}^2 + 4 m_{\rm \pi}^2 - 3 s + q^2) + 2 c_5 (m_{\rm K}^2 \
- m_{\rm \pi}^2) (-m_{\rm K}^2 + s + q^2)))/(f^2 (-m_{\rm K}^2 + q^2))

The logs:

((-i/384) (-4 m_{\rm K}^2 + m_{\rm \pi}^2) (t^2 + u^2) (6 c_5 (-m_{\rm K}^2 + m_{\rm \pi}^2) + c_2 \
(m_{\rm K}^2 - q^2)) \log(m_{\rm \eta}^2/\mu^2))/(f^4 (m_{\rm K}^2 - m_{\rm \pi}^2) \pi^2 (-m_{\rm \
K}^2 + q^2)) - ((i/96) m_{\rm K}^2 (t^2 + u^2) (3 c_5 (m_{\rm K}^2 - m_{\rm \pi}^2) + 2 c_2 \
(-m_{\rm K}^2 + q^2)) \log(m_{\rm K}^2/\mu^2))/ (f^4 (m_{\rm K}^2 - m_{\rm \pi}^2) \pi^2 (-m_{\rm \
K}^2 + q^2)) + ((i/384) m_{\rm \pi}^2 (t^2 + u^2) (30 c_5 (m_{\rm K}^2 - m_{\rm \pi}^2) + 11 c_2 \
(-m_{\rm K}^2 + q^2)) \log(m_{\rm \pi}^2/\mu^2))/(f^4 (m_{\rm K}^2 - m_{\rm \pi}^2) \pi^2 (-m_{\rm \
K}^2 + q^2))

((i/2304) (((2 c_5 (-m_{\rm K} + m_{\rm \pi}) (m_{\rm K} + m_{\rm \pi}) (36 m_{\rm \pi}^6 - 36 \
m_{\rm K}^2 (m_{\rm K}^2 + s - q^2)^2 + m_{\rm \pi}^4 (-96 m_{\rm K}^2 - 84 s + 44 q^2) + m_{\rm \
\pi}^2 (-75 m_{\rm K}^4 + 2 m_{\rm K}^2 (123 s - 103 q^2) + 9 (s - q^2)^2)) + c_2 (-196 m_{\rm K}^8 \
- 4 m_{\rm \pi}^6 (s - 6 q^2) + 12 m_{\rm \pi}^4 q^2 (s + q^2) + m_{\rm K}^6 (-163 m_{\rm \pi}^2 + \
376 s + 28 q^2) + 3 m_{\rm \pi}^2 q^2 (5 s^2 - 6 s q^2 + q^4) + m_{\rm K}^4 (140 m_{\rm \pi}^4 - \
234 m_{\rm \pi}^2 s + 60 s^2 + (289 m_{\rm \pi}^2 - 384 s) q^2 + 180 q^4) - m_{\rm K}^2 (24 m_{\rm \
\pi}^6 + 8 m_{\rm \pi}^4 (-3 s + 19 q^2) + 12 q^2 (5 s^2 - 6 s q^2 + q^4) + 3 m_{\rm \pi}^2 (5 s^2 \
- 52 s q^2 + 43 q^4)))) \log(m_{\rm \eta}^2/\mu^2))/((m_{\rm K}^2 - m_{\rm \pi}^2) \pi^2) + (6 (c_2 \
(-21 m_{\rm K}^8 - 3 m_{\rm \pi}^2 s q^2 (-3 s + q^2) + m_{\rm K}^6 (166 m_{\rm \pi}^2 - 141 s + 59 \
q^2) + m_{\rm K}^2 (-9 m_{\rm \pi}^2 s^2 - 2 (-74 m_{\rm \pi}^4 + 72 m_{\rm \pi}^2 s + s^2) q^2 - 9 \
(-6 m_{\rm \pi}^2 + s) q^4 + 5 q^6) + m_{\rm K}^4 (-148 m_{\rm \pi}^4 + 2 s^2 + 162 s q^2 - 43 q^4 \
+ 5 m_{\rm \pi}^2 (27 s - 44 q^2))) - 6 c_5 (-m_{\rm K} + m_{\rm \pi}) (m_{\rm K} + m_{\rm \pi}) \
(m_{\rm K}^2 (11 m_{\rm K}^4 - 15 m_{\rm K}^2 s - 4 s^2 - 3 s q^2 + q^4) + m_{\rm \pi}^2 (2 m_{\rm \
K}^4 + 3 s (s + q^2) + m_{\rm K}^2 (7 s + 2 q^2)))) \log(m_{\rm K}^2/\mu^2))/((m_{\rm K}^2 - m_{\rm \
\pi}^2) \pi^2) - (3 (c_2 (-3 m_{\rm K}^6 (m_{\rm \pi}^2 - 8 s) + 12 m_{\rm \pi}^4 q^2 (-11 s + q^2) \
- 4 m_{\rm \pi}^6 (9 s + 4 q^2) - m_{\rm K}^4 (100 m_{\rm \pi}^4 + 114 m_{\rm \pi}^2 s - 72 s^2 + \
(35 m_{\rm \pi}^2 + 48 s) q^2) + m_{\rm \pi}^2 q^2 (71 s^2 - 30 s q^2 + 7 q^4) + m_{\rm K}^2 (16 \
m_{\rm \pi}^6 + 168 m_{\rm \pi}^4 s - 71 m_{\rm \pi}^2 s^2 + (88 m_{\rm \pi}^4 + 144 m_{\rm \pi}^2 \
s - 72 s^2) q^2 + (31 m_{\rm \pi}^2 + 24 s) q^4)) - 6 c_5 (-m_{\rm K} + m_{\rm \pi}) (m_{\rm K} + \
m_{\rm \pi}) (20 m_{\rm \pi}^6 + 4 m_{\rm \pi}^4 (6 m_{\rm K}^2 - 7 s + q^2) + 8 m_{\rm K}^2 s \
(-m_{\rm K}^2 + s + q^2) + m_{\rm \pi}^2 (m_{\rm K}^4 - 7 s^2 - 14 s q^2 + 5 q^4 + 10 m_{\rm K}^2 \
(s + q^2)))) \log(m_{\rm \pi}^2/\mu^2))/((m_{\rm K}^2 - m_{\rm \pi}^2) \pi^2) + 96 (-m_{\rm K}^2 + \
m_{\rm \pi}^2) (-m_{\rm \pi}^2 + q^2) (6 c_5 (-m_{\rm K}^2 + m_{\rm \pi}^2) + 5 c_2 (-m_{\rm K}^2 + \
q^2)) k(m_{\rm K}^2, m_{\rm \eta}^2) + 96 (-m_{\rm K}^2 + m_{\rm \pi}^2) (-m_{\rm \pi}^2 + q^2) (6 \
c_5 (-m_{\rm K}^2 + m_{\rm \pi}^2) + c_2 (-m_{\rm K}^2 + q^2)) k(m_{\rm \pi}^2, m_{\rm K}^2)))/(f^4 \
(-m_{\rm K}^2 + q^2))

The loop-function parts of the loop contribution:

((-i/144) c_5 (m_{\rm K}^2 - m_{\rm \pi}^2) (12 (2 m_{\rm K}^4 - m_{\rm K}^2 (m_{\rm \pi}^2 + u - 5 \
q^2) + m_{\rm \pi}^2 (-7 m_{\rm \pi}^2 + u + q^2)) K(t, m_{\rm K}^2, m_{\rm \eta}^2) + 36 (2 m_{\rm \
K}^4 - m_{\rm K}^2 (m_{\rm \pi}^2 + u - q^2) + m_{\rm \pi}^2 (-3 m_{\rm \pi}^2 + u + q^2)) K(t, \
m_{\rm \pi}^2, m_{\rm K}^2) + 12 (2 m_{\rm K}^4 - m_{\rm K}^2 (m_{\rm \pi}^2 + t - 5 q^2) + m_{\rm \
\pi}^2 (-7 m_{\rm \pi}^2 + t + q^2)) K(u, m_{\rm K}^2, m_{\rm \eta}^2) + 36 (2 m_{\rm K}^4 - m_{\rm \
K}^2 (m_{\rm \pi}^2 + t - q^2) + m_{\rm \pi}^2 (-3 m_{\rm \pi}^2 + t + q^2)) K(u, m_{\rm \pi}^2, \
m_{\rm K}^2) - 8 m_{\rm \pi}^2 (-9 m_{\rm K}^2 + 9 s + q^2) \overline{J}(s, m_{\rm \eta}^2) - 108 s \
(-m_{\rm K}^2 + s + q^2) \overline{J}(s, m_{\rm K}^2) - 72 (-m_{\rm \pi}^2 + 2 s) (-m_{\rm K}^2 + s \
+ q^2) \overline{J}(s, m_{\rm \pi}^2) - (-46 m_{\rm K}^4 + 5 m_{\rm \pi}^4 + 18 t (t - u) + m_{\rm \
K}^2 (-31 m_{\rm \pi}^2 + 24 t + 84 u - 45 q^2) - 3 m_{\rm \pi}^2 (8 t + 4 u - 7 q^2)) \
\overline{J}(t, m_{\rm K}^2, m_{\rm \eta}^2) - 9 (-3 m_{\rm \pi}^4 - 2 m_{\rm K}^2 (3 m_{\rm K}^2 + \
8 t - 2 u) + 2 t (9 t - u) + m_{\rm \pi}^2 (25 m_{\rm K}^2 - 24 t + 4 u) + (3 m_{\rm K}^2 + 13 \
m_{\rm \pi}^2 - 8 t) q^2) \overline{J}(t, m_{\rm \pi}^2, m_{\rm K}^2) + (46 m_{\rm K}^4 - 5 m_{\rm \
\pi}^4 + 18 (t - u) u + m_{\rm \pi}^2 (31 m_{\rm K}^2 + 12 t + 24 u - 21 q^2) - 3 m_{\rm K}^2 (28 t \
+ 8 u - 15 q^2)) \overline{J}(u, m_{\rm K}^2, m_{\rm \eta}^2) + 9 (6 m_{\rm K}^4 + 3 m_{\rm \pi}^4 \
- 4 m_{\rm K}^2 (t - 4 u) + 2 (t - 9 u) u + m_{\rm \pi}^2 (-25 m_{\rm K}^2 - 4 t + 24 u) + (-3 \
m_{\rm K}^2 - 13 m_{\rm \pi}^2 + 8 u) q^2) \overline{J}(u, m_{\rm \pi}^2, m_{\rm K}^2) + 108 \
(-m_{\rm K}^2 + m_{\rm \pi}^2) (-m_{\rm \pi}^2 + q^2) Mr(t, m_{\rm K}^2, m_{\rm \eta}^2) + 108 \
(-m_{\rm K}^2 + m_{\rm \pi}^2) (-m_{\rm \pi}^2 + q^2) Mr(t, m_{\rm \pi}^2, m_{\rm K}^2) + 108 \
(-m_{\rm K}^2 + m_{\rm \pi}^2) (-m_{\rm \pi}^2 + q^2) Mr(u, m_{\rm K}^2, m_{\rm \eta}^2) + 108 \
(-m_{\rm K}^2 + m_{\rm \pi}^2) (-m_{\rm \pi}^2 + q^2) Mr(u, m_{\rm \pi}^2, m_{\rm K}^2)))/(f^4 \
(-m_{\rm K}^2 + q^2)) + ((i/864) c_2 (12 (-35 m_{\rm \pi}^4 + m_{\rm K}^2 (-26 m_{\rm K}^2 - 5 u + \
13 q^2) + m_{\rm \pi}^2 (31 m_{\rm K}^2 + 5 u + 17 q^2)) K(t, m_{\rm K}^2, m_{\rm \eta}^2) + 36 (17 \
m_{\rm \pi}^4 + m_{\rm \pi}^2 (23 m_{\rm K}^2 + u - 11 q^2) - m_{\rm K}^2 (10 m_{\rm K}^2 + u + 19 \
q^2)) K(t, m_{\rm \pi}^2, m_{\rm K}^2) + 12 (-35 m_{\rm \pi}^4 + m_{\rm K}^2 (-26 m_{\rm K}^2 - 5 t \
+ 13 q^2) + m_{\rm \pi}^2 (31 m_{\rm K}^2 + 5 t + 17 q^2)) K(u, m_{\rm K}^2, m_{\rm \eta}^2) + 36 \
(17 m_{\rm \pi}^4 + m_{\rm \pi}^2 (23 m_{\rm K}^2 + t - 11 q^2) - m_{\rm K}^2 (10 m_{\rm K}^2 + t + \
19 q^2)) K(u, m_{\rm \pi}^2, m_{\rm K}^2) + 24 m_{\rm \pi}^2 (-13 m_{\rm K}^2 + 4 m_{\rm \pi}^2 + 9 \
s - 3 q^2) \overline{J}(s, m_{\rm \eta}^2) - 108 s (3 m_{\rm K}^2 - 3 s + q^2) \overline{J}(s, \
m_{\rm K}^2) - 216 (-m_{\rm \pi}^2 + 2 s) (m_{\rm K}^2 - 4 m_{\rm \pi}^2 + 3 s - q^2) \
\overline{J}(s, m_{\rm \pi}^2) + (302 m_{\rm K}^4 - 97 m_{\rm \pi}^4 + 18 t (7 t + 5 u) - 60 m_{\rm \
K}^2 (8 t + 7 u) + m_{\rm \pi}^2 (443 m_{\rm K}^2 - 24 t + 60 u) - 3 (-91 m_{\rm K}^2 + 19 m_{\rm \
\pi}^2 + 24 t) q^2) \overline{J}(t, m_{\rm K}^2, m_{\rm \eta}^2) + 9 (-42 m_{\rm K}^4 - 13 m_{\rm \
\pi}^4 + m_{\rm \pi}^2 (103 m_{\rm K}^2 - 64 t - 4 u) - 4 m_{\rm K}^2 (6 t + u) + 2 t (23 t + u) + \
(-19 m_{\rm K}^2 + 35 m_{\rm \pi}^2 - 16 t) q^2) \overline{J}(t, m_{\rm \pi}^2, m_{\rm K}^2) + (302 \
m_{\rm K}^4 - 97 m_{\rm \pi}^4 + m_{\rm \pi}^2 (443 m_{\rm K}^2 + 60 t - 24 u) + 18 u (5 t + 7 u) - \
60 m_{\rm K}^2 (7 t + 8 u) - 3 (-91 m_{\rm K}^2 + 19 m_{\rm \pi}^2 + 24 u) q^2) \overline{J}(u, \
m_{\rm K}^2, m_{\rm \eta}^2) + 9 (-42 m_{\rm K}^4 - 13 m_{\rm \pi}^4 + m_{\rm \pi}^2 (103 m_{\rm \
K}^2 - 4 t - 64 u) - 4 m_{\rm K}^2 (t + 6 u) + 2 u (t + 23 u) + (-19 m_{\rm K}^2 + 35 m_{\rm \pi}^2 \
- 16 u) q^2) \overline{J}(u, m_{\rm \pi}^2, m_{\rm K}^2) + 540 (-m_{\rm K}^2 + m_{\rm \pi}^2) \
(-m_{\rm \pi}^2 + q^2) Mr(t, m_{\rm K}^2, m_{\rm \eta}^2) + 108 (-m_{\rm K}^2 + m_{\rm \pi}^2) \
(-m_{\rm \pi}^2 + q^2) Mr(t, m_{\rm \pi}^2, m_{\rm K}^2) + 540 (-m_{\rm K}^2 + m_{\rm \pi}^2) \
(-m_{\rm \pi}^2 + q^2) Mr(u, m_{\rm K}^2, m_{\rm \eta}^2) + 108 (-m_{\rm K}^2 + m_{\rm \pi}^2) \
(-m_{\rm \pi}^2 + q^2) Mr(u, m_{\rm \pi}^2, m_{\rm K}^2)))/f^4

The polynomial parts of the loop contribution:

((i/1152) (c_2 (-m_{\rm K}^2 + q^2) (-16 m_{\rm K}^4 + 78 m_{\rm \pi}^4 + 111 m_{\rm K}^2 s + 87 \
s^2 - m_{\rm \pi}^2 (17 m_{\rm K}^2 + 237 s) - 3 t^2 - 3 u^2 + (-13 m_{\rm K}^2 + 37 m_{\rm \pi}^2 \
- 36 s) q^2 + 3 q^4) + 2 c_5 (-m_{\rm K}^2 + m_{\rm \pi}^2) (3 (-4 m_{\rm K}^4 + 2 m_{\rm \pi}^4 - \
9 m_{\rm K}^2 s + 23 s^2 - m_{\rm \pi}^2 (7 m_{\rm K}^2 + 3 s) - t^2 - u^2) + (-9 m_{\rm K}^2 - 7 \
m_{\rm \pi}^2 + 54 s) q^2 + 3 q^4)))/(f^4 \pi^2 (-m_{\rm K}^2 + q^2))

Weak counterterms contributing also to K->2π:

((-i/3) c_2 (2 m_{\rm K}^4 (-8 (n_{10} + n_{11}) + 3 n_{5} + 3 n_{8}) - 4 m_{\rm \pi}^4 (-6 n_{10} \
- 14 n_{11} - 6 n_{12} + 6 n_{5} + 12 n_{7} + 3 n_{8}) - (2 (m_{\rm K}^2 - m_{\rm \pi}^2) (m_{\rm \
\pi}^2 n_{11} + 2 m_{\rm K}^2 (n_{10} + n_{11})) (m_{\rm K}^2 + 3 s - q^2))/(-m_{\rm K}^2 + q^2) + \
m_{\rm K}^2 (m_{\rm \pi}^2 (-8 (n_{10} + 5 n_{11} + 3 n_{12}) + 48 n_{7} - 21 n_{8} - 18 n_{9}) + 6 \
(n_{5} - 4 n_{7} + 3 n_{8} + 2 n_{9}) s - 6 (n_{5} + n_{8}) q^2) + 3 m_{\rm \pi}^2 ((4 n_{5} + 8 \
n_{7} + 3 n_{8} + 2 n_{9}) s - (n_{8} + 2 n_{9}) q^2)))/f^4

Weak conterterms not contributing to K->2π:

((i/6) c_2 (-3 m_{\rm K}^4 (2 n_{21} + n_{22} + 2 n_{23}) - 3 n_{20} (-m_{\rm K}^4 - 6 m_{\rm \
\pi}^4 + 2 m_{\rm \pi}^2 s + m_{\rm K}^2 (-2 m_{\rm \pi}^2 + s) + t^2 + 4 t u + u^2 + (-2 m_{\rm \
K}^2 - 10 m_{\rm \pi}^2 + 7 s) q^2 - 3 q^4) + 3 n_{19} (-m_{\rm K}^4 - 6 m_{\rm \pi}^4 + 2 m_{\rm \
\pi}^2 s + m_{\rm K}^2 (-2 m_{\rm \pi}^2 + s) + t^2 + 4 t u + u^2 + (-4 m_{\rm K}^2 - 2 m_{\rm \
\pi}^2 + s) q^2 - q^4) - 4 m_{\rm \pi}^2 (3 (n_{22} + 2 n_{23}) s + n_{21} (6 s - 4 q^2)) + m_{\rm \
K}^2 (2 n_{21} (12 m_{\rm \pi}^2 + 3 s - 5 q^2) + 3 (n_{22} + 2 n_{23}) (4 m_{\rm \pi}^2 + s + \
q^2))))/f^4

Strong counterterms:

((-4 i) c_5 (m_{\rm K}^2 - m_{\rm \pi}^2) (2 (2 L_{4} + L_{5} - 4 L_{6} - 2 L_{8}) m_{\rm K}^4 + 10 \
L_{3} m_{\rm \pi}^4 - 6 L_{3} m_{\rm \pi}^2 s - 10 L_{4} m_{\rm \pi}^2 s - 4 L_{5} m_{\rm \pi}^2 s \
+ 4 L_{6} m_{\rm \pi}^2 s + 4 L_{8} m_{\rm \pi}^2 s - 6 L_{3} m_{\rm \pi}^2 t + 3 L_{3} s t - 6 \
L_{3} m_{\rm \pi}^2 u + 3 L_{3} s u + 2 L_{3} t u + 6 L_{3} m_{\rm \pi}^2 q^2 + 6 L_{4} m_{\rm \
\pi}^2 q^2 + 4 L_{6} m_{\rm \pi}^2 q^2 + 4 L_{8} m_{\rm \pi}^2 q^2 - 3 L_{3} s q^2 - 3 L_{3} t q^2 \
- 3 L_{3} u q^2 + 3 L_{3} q^4 + 8 L_{1} (-2 m_{\rm \pi}^2 + s) (m_{\rm K}^2 - s + q^2) + 4 L_{2} \
(-2 m_{\rm \pi}^4 + 2 m_{\rm \pi}^2 s - s^2 + m_{\rm K}^2 (-2 m_{\rm \pi}^2 + s) + 2 t u + (-2 \
m_{\rm K}^2 - 2 m_{\rm \pi}^2 + s) q^2) + m_{\rm K}^2 (4 (2 L_{5} - 9 L_{6} - 5 L_{8}) m_{\rm \
\pi}^2 + L_{3} q^2 - 2 (L_{5} - 4 L_{6} - 2 L_{8}) (s + q^2) - 2 L_{4} (-13 m_{\rm \pi}^2 + 6 s + 2 \
q^2))))/ (f^4 (-m_{\rm K}^2 + q^2))

Converted by Mathematica  (July 10, 2003)