FeynCalc manual (development version)

FCFeynmanRegularizeDivergence

FCFeynmanRegularizeDivergence[exp, div] regularizes the divergence div in the Feynman parametric integral exp. Provided that all divergences have been regularized in this fashion, upon expanding the integrand around ε=0\varepsilon = 0 one can safely integrate in the Feynman parameters.

Notice that div can be also a list made of divergences found by FCFeynmanFindDivergences.

This function uses the method of analytic regularization introduced by Erik Panzer in 1403.3385, 1401.4361 and 1506.07243.

Its current implementation is very much based on the code of the dimregPartial routine from the Maple package HyperInt by Erik Panzer.

Here div must be of the form {{x[i], x[j], ...}, {x[k], x[l], ...}, sdd}, where {x[i],x[j], ...} need to approach zero, while {x[k], x[l], ...} must tend towards infinity to generate the superficial degree of divergence sdd.

See also

Overview, FCFeynmanParametrize, FCFeynmanProjectivize, FCFeynmanFindDivergences.

Examples

int = SFAD[l, k + l, {{k, -2 k . q}}]
fpar = FCFeynmanParametrize[int, {k, l}, Names -> x, FCReplaceD -> {D -> 4 - 2 Epsilon}]

1(l2+iη).((k+l)2+iη).(k2−2(k⋅q)+iη)\frac{1}{(l^2+i \eta ).((k+l)^2+i \eta ).(k^2-2 (k\cdot q)+i \eta )}

{(x(1)x(2)+x(3)x(2)+x(1)x(3))3ε−3(q2x(1)2(x(2)+x(3)))1−2ε,−Γ(2ε−1),{x(1),x(2),x(3)}}\left\{(x(1) x(2)+x(3) x(2)+x(1) x(3))^{3 \varepsilon -3} \left(q^2 x(1)^2 (x(2)+x(3))\right)^{1-2 \varepsilon },-\Gamma (2 \varepsilon -1),\{x(1),x(2),x(3)\}\right\}

This Feynman parametric integral integrand contains logarithmic divergences for x1→∞x_1 \to \infty and x2,3→0x_{2,3} \to 0

divs = FCFeynmanFindDivergences[fpar[[1]], x]

({{},{x(1)}}ε{{x(2),x(3)},{}}ε)\left( \begin{array}{cc} \{\{\},\{x(1)\}\} & \varepsilon \\ \{\{x(2),x(3)\},\{\}\} & \varepsilon \\ \end{array} \right)

Regularizing the first divergence we obtain

intReg = FCFeynmanRegularizeDivergence[fpar[[1]], divs[[1]]]

−3(ε−1)q2x(1)2x(2)x(3)(x(2)+x(3))(x(1)x(2)+x(3)x(2)+x(1)x(3))3ε−4(q2x(1)2(x(2)+x(3)))−2εε-\frac{3 (\varepsilon -1) q^2 x(1)^2 x(2) x(3) (x(2)+x(3)) (x(1) x(2)+x(3) x(2)+x(1) x(3))^{3 \varepsilon -4} \left(q^2 x(1)^2 (x(2)+x(3))\right)^{-2 \varepsilon }}{\varepsilon }

It turns out that there are no further divergences left

FCFeynmanFindDivergences[intReg, x]

{}\{\}

Now one can expand the integrand in Epsilon and perform the integration in Feynman parameters order by order in Epsilon

Series[intReg, {Epsilon, 0, 0}] // Normal

3q2x(1)2x(2)x(3)(x(2)+x(3))ε(x(1)x(2)+x(3)x(2)+x(1)x(3))4−3q2x(1)2x(2)x(3)(x(2)+x(3))(2log⁡(q2x(1)2(x(2)+x(3)))−3log⁡(x(1)x(2)+x(3)x(2)+x(1)x(3))+1)(x(1)x(2)+x(3)x(2)+x(1)x(3))4\frac{3 q^2 x(1)^2 x(2) x(3) (x(2)+x(3))}{\varepsilon (x(1) x(2)+x(3) x(2)+x(1) x(3))^4}-\frac{3 q^2 x(1)^2 x(2) x(3) (x(2)+x(3)) \left(2 \log \left(q^2 x(1)^2 (x(2)+x(3))\right)-3 \log (x(1) x(2)+x(3) x(2)+x(1) x(3))+1\right)}{(x(1) x(2)+x(3) x(2)+x(1) x(3))^4}

Here is an example of regularizing two divergences at a time

FCFeynmanRegularizeDivergence[(y[1]*(y[1] + y[2] + y[3])^(2*ep)*(y[1]^2 - 4*y[2]*y[3])^(-2 - 
        ep))/(x[1] + x[2])^2, {{{{y[2]}, {y[3]}}, -2*ep}, {{{y[3]}, {y[2]}}, -2*ep}}]

12  ep(x(1)+x(2))2y(1)(y(1)+y(2)+y(3))2(ep−1)(y(1)2−4y(2)y(3))−ep−2(2  epy(1)2+4  epy(2)y(1)+4  epy(3)y(1)+8  epy(2)y(3)−y(2)y(1)−y(3)y(1)−4y(2)y(3))\frac{1}{2 \;\text{ep} (x(1)+x(2))^2}y(1) (y(1)+y(2)+y(3))^{2 (\text{ep}-1)} \left(y(1)^2-4 y(2) y(3)\right)^{-\text{ep}-2} \left(2 \;\text{ep} y(1)^2+4 \;\text{ep} y(2) y(1)+4 \;\text{ep} y(3) y(1)+8 \;\text{ep} y(2) y(3)-y(2) y(1)-y(3) y(1)-4 y(2) y(3)\right)