FeynCalc manual (development version)

ExpandScalarProduct

ExpandScalarProduct[expr] expands scalar products of sums of momenta in expr.

ExpandScalarProduct does not use Expand on expr.

See also

Overview, Calc, MomentumExpand, MomentumCombine, FCVariable

Examples

SP[p1 + p2 + p3, p4 + p5 + p6] 
 
% // ExpandScalarProduct

$$(\overline{\text{p1}}+\overline{\text{p2}}+\overline{\text{p3}})\cdot (\overline{\text{p4}}+\overline{\text{p5}}+\overline{\text{p6}})$$

$$\overline{\text{p1}}\cdot \overline{\text{p4}}+\overline{\text{p1}}\cdot \overline{\text{p5}}+\overline{\text{p1}}\cdot \overline{\text{p6}}+\overline{\text{p2}}\cdot \overline{\text{p4}}+\overline{\text{p2}}\cdot \overline{\text{p5}}+\overline{\text{p2}}\cdot \overline{\text{p6}}+\overline{\text{p3}}\cdot \overline{\text{p4}}+\overline{\text{p3}}\cdot \overline{\text{p5}}+\overline{\text{p3}}\cdot \overline{\text{p6}}$$

SP[p, p - q] 
 
ExpandScalarProduct[%]

$$\overline{p}\cdot (\overline{p}-\overline{q})$$

$$\overline{p}^2-\overline{p}\cdot \overline{q}$$

FV[p - q, \[Mu]] 
 
ExpandScalarProduct[%]

$$\left(\overline{p}-\overline{q}\right)^{\mu }$$

$$\overline{p}^{\mu }-\overline{q}^{\mu }$$

SPD[p - q, q - r] 
 
ExpandScalarProduct[%]

$$(p-q)\cdot (q-r)$$

$$p\cdot q-p\cdot r+q\cdot r-q^2$$

Using the option Momentum one can limit the expansion to particular momenta

SP[p1 + p2 + p3, p4 + p5 + p6] 
 
ExpandScalarProduct[%, Momentum -> {p1}]

$$(\overline{\text{p1}}+\overline{\text{p2}}+\overline{\text{p3}})\cdot (\overline{\text{p4}}+\overline{\text{p5}}+\overline{\text{p6}})$$

$$\overline{\text{p1}}\cdot (\overline{\text{p4}}+\overline{\text{p5}}+\overline{\text{p6}})+(\overline{\text{p2}}+\overline{\text{p3}})\cdot (\overline{\text{p4}}+\overline{\text{p5}}+\overline{\text{p6}})$$

By default ExpandScalarProduct does not apply linearity to Levi-Civita tensors

LC[\[Mu]][p1 + p2, p3 + p4, p5 + p6] 
 
ExpandScalarProduct[%]

$$\bar{\epsilon }^{\mu \overline{\text{p1}}+\overline{\text{p2}}\;\overline{\text{p3}}+\overline{\text{p4}}\;\overline{\text{p5}}+\overline{\text{p6}}}$$

$$\bar{\epsilon }^{\mu \overline{\text{p1}}+\overline{\text{p2}}\;\overline{\text{p3}}+\overline{\text{p4}}\;\overline{\text{p5}}+\overline{\text{p6}}}$$

Using the option EpsEvaluate takes care of that

LC[\[Mu]][p1 + p2, p3 + p4, p5 + p6] 
 
ExpandScalarProduct[%, EpsEvaluate -> True]

$$\bar{\epsilon }^{\mu \overline{\text{p1}}+\overline{\text{p2}}\;\overline{\text{p3}}+\overline{\text{p4}}\;\overline{\text{p5}}+\overline{\text{p6}}}$$

$$\bar{\epsilon }^{\mu \overline{\text{p1}}\;\overline{\text{p3}}\;\overline{\text{p5}}}+\bar{\epsilon }^{\mu \overline{\text{p1}}\;\overline{\text{p3}}\;\overline{\text{p6}}}+\bar{\epsilon }^{\mu \overline{\text{p1}}\;\overline{\text{p4}}\;\overline{\text{p5}}}+\bar{\epsilon }^{\mu \overline{\text{p1}}\;\overline{\text{p4}}\;\overline{\text{p6}}}+\bar{\epsilon }^{\mu \overline{\text{p2}}\;\overline{\text{p3}}\;\overline{\text{p5}}}+\bar{\epsilon }^{\mu \overline{\text{p2}}\;\overline{\text{p3}}\;\overline{\text{p6}}}+\bar{\epsilon }^{\mu \overline{\text{p2}}\;\overline{\text{p4}}\;\overline{\text{p5}}}+\bar{\epsilon }^{\mu \overline{\text{p2}}\;\overline{\text{p4}}\;\overline{\text{p6}}}$$

One can use the options EpsEvaluate and Momentum together

LC[\[Mu]][p1 + p2, p3 + p4, p5 + p6] 
 
ExpandScalarProduct[%, EpsEvaluate -> True, Momentum -> {p1}]

$$\bar{\epsilon }^{\mu \overline{\text{p1}}+\overline{\text{p2}}\;\overline{\text{p3}}+\overline{\text{p4}}\;\overline{\text{p5}}+\overline{\text{p6}}}$$

$$\bar{\epsilon }^{\mu \overline{\text{p1}}\;\overline{\text{p3}}+\overline{\text{p4}}\;\overline{\text{p5}}+\overline{\text{p6}}}+\bar{\epsilon }^{\mu \overline{\text{p2}}\;\overline{\text{p3}}+\overline{\text{p4}}\;\overline{\text{p5}}+\overline{\text{p6}}}$$

Of course, the function is also applicable to Cartesian quantities

CSP[p1 + p2, p3 + p4] 
 
ExpandScalarProduct[%]

$$(\overline{\text{p1}}+\overline{\text{p2}})\cdot (\overline{\text{p3}}+\overline{\text{p4}})$$

$$\overline{\text{p1}}\cdot \overline{\text{p3}}+\overline{\text{p1}}\cdot \overline{\text{p4}}+\overline{\text{p2}}\cdot \overline{\text{p3}}+\overline{\text{p2}}\cdot \overline{\text{p4}}$$

CLC[][p1 + p2, p3 + p4, p5 + p6] 
 
ExpandScalarProduct[%, EpsEvaluate -> True]

$$\bar{\epsilon }^{\overline{\text{p1}}+\overline{\text{p2}}\;\overline{\text{p3}}+\overline{\text{p4}}\;\overline{\text{p5}}+\overline{\text{p6}}}$$

$$\bar{\epsilon }^{\overline{\text{p1}}\;\overline{\text{p3}}\;\overline{\text{p5}}}+\bar{\epsilon }^{\overline{\text{p1}}\;\overline{\text{p3}}\;\overline{\text{p6}}}+\bar{\epsilon }^{\overline{\text{p1}}\;\overline{\text{p4}}\;\overline{\text{p5}}}+\bar{\epsilon }^{\overline{\text{p1}}\;\overline{\text{p4}}\;\overline{\text{p6}}}+\bar{\epsilon }^{\overline{\text{p2}}\;\overline{\text{p3}}\;\overline{\text{p5}}}+\bar{\epsilon }^{\overline{\text{p2}}\;\overline{\text{p3}}\;\overline{\text{p6}}}+\bar{\epsilon }^{\overline{\text{p2}}\;\overline{\text{p4}}\;\overline{\text{p5}}}+\bar{\epsilon }^{\overline{\text{p2}}\;\overline{\text{p4}}\;\overline{\text{p6}}}$$

Sometimes one would like to have external momenta multiplied by symbolic parameters in the propagators. In this case one should first declare the corresponding variables to be of FCVariable type

DataType[a, FCVariable] = True;
DataType[b, FCVariable] = True;

ExpandScalarProduct[SP[P, Q] /. P -> a P1 + b P2] 
 
StandardForm[%]

$$a \left(\overline{\text{P1}}\cdot \overline{Q}\right)+b \left(\overline{\text{P2}}\cdot \overline{Q}\right)$$

(*a Pair[Momentum[P1], Momentum[Q]] + b Pair[Momentum[P2], Momentum[Q]]*)