FeynCalc manual (development version)

CompleteSquare

CompleteSquarep[exp, x] completes the square of a second order polynomial in the momentum x.

See also

Overview, ExpandScalarProduct.

Examples

CompleteSquare[4 SP[p] + SP[b, p] + c, p]

4(b8+p)2b216+c4 (\frac{\overline{b}}{8}+\overline{p})^2-\frac{\overline{b}^2}{16}+c

CompleteSquare[4 SP[p] + SP[b, p] + c, p, q]

{b216+4q2+c,qb8+p}\left\{-\frac{\overline{b}^2}{16}+4 \overline{q}^2+c,\overline{q}\to \frac{\overline{b}}{8}+\overline{p}\right\}

ex1 = 5 SP[2 p + 3 r, p + r]

5((p+r)(2p+3r))5 \left((\overline{p}+\overline{r})\cdot (2 \overline{p}+3 \overline{r})\right)

ex2 = CompleteSquare[ex1, p]

10(p+5r4)25r2810 (\overline{p}+\frac{5 \overline{r}}{4})^2-\frac{5 \overline{r}^2}{8}

ex1 - ex2 // ScalarProductExpand // Expand

00

CompleteSquare[5 SP[2 p + 3 r, p + r], p, q]

{10q25r28,qp+5r4}\left\{10 \overline{q}^2-\frac{5 \overline{r}^2}{8},\overline{q}\to \overline{p}+\frac{5 \overline{r}}{4}\right\}

SPD[a] + 2 SPD[a, b] 
 
ex = CompleteSquare[%, a]

2(ab)+a22 (a\cdot b)+a^2

(a+b)2b2(a+b)^2-b^2

ex // StandardForm

(*-Pair[Momentum[b, D], Momentum[b, D]] + Pair[Momentum[a, D] + Momentum[b, D], Momentum[a, D] + Momentum[b, D]]*)