kinematicHopfAlgebra

This program generates the BCJ numerator in HEFT,YM, (arxiv:2111.15649, arxiv:2208.05519). YMS+ $\phi^3$ (arxiv:2208.05886), QCD with fermions, YM+ $F^3+F^4$arxiv:2208.05519, $DF^2$+YM 2403.04614. It can be used in constructing EYM and GR+HEFT amplitude via double copy. Examples are included. Local BCJ numerator for YM is also included.

Install KiHA

1. Copy "KiHA.wl" to your computer with "mathematica >10" installed
2. Load the package use Needs["KiHA","your local directory/KiHA.wl"]`
3. enjoy!

Heavy-mass effective theory (HEFT) and Yang-Mills (YM)

Related paper for KiHA in HEFT are arxiv:2111.15649, arxiv:2208.05519 . In HEFT, kinematic algebra is taken as current algebra. The building blocks are - vector current $T^{(i)}{(i)}$: correspond to the vector currents and map to a vector current(just the velocity $v$) product with a polarisation vector $\varepsiloni$ . - tensor current $T^{(\alpha)}{(\tau1),(\tau2),\cdots, (\taur)}$: Tensor currents and map to all multiplicity universal tensor product with multi polarisation vectors - fusion product $\star$: The fusion rules from lower order rank tensor currents to higher order tensor currents. This fusion product is bilinear and associative. - convolution map $\langle \bullet \rangle$: This is a linear map from abstract algebra general to physics kinematic expression, which is, in general, non-local and manifestly gauge invariant.

The major output of this program is

$$\widehat {\mathcal{N}} (12\ldots n{-}2)={T{(1)}^{(1)}}\star T{(2)}^{(2)}\star \ldots \star T_{(n{-}2)}^{(n{-}2)}$$

which is known as an algebraic pre-numerator. Another major function is the convolution map, which maps the abstract generator to the physical function of kinematic information. In HEFT, the algebraic pre-numerator is from

$$\widehat{\mathcal N}(123)= T{(1)}^{(1)} \star T{(2)}^{(2)} \star T_{(3)}^{(3)}.$$

num = \[FivePointedStar] @@ (T /@ List /@ Range[1, 3]); We get

$$T{\text{(1)},\text{(2)},\text{(3)}}+T{\text{(1)},\text{(3)},\text{(2)}}+T{\text{(2)},\text{(1)},\text{(3)}}+T{\text{(2)},\text{(3)},\text{(1)}}+T{\text{(3)},\text{(1)},\text{(2)}}+T{\text{(3)},\text{(2)},\text{(1)}}$$ $$-T{\text{(1)},\text{(23)}}-T{\text{(23)},\text{(1)}}-T{\text{(12)},\text{(3)}}-T{\text{(3)},\text{(12)}}-T{\text{(13)},\text{(2)}}-T{\text{(2)},\text{(13)}}+T_{\text{(123)}}$$

After taking the convolution map, we have

$${\mathcal{N}}(123,v)=\langle\widehat {\mathcal{N}}(123)\rangle= \langle T{(1)}^{(1)} \star T{(2)}^{(2)} \star T_{(3)}^{(3)}\rangle.$$

preNumerator = num /. T -> Tp /. rmzero, where the Tp is the convolution map. Then we get the output of the pre-numerator

$${\mathcal{N}}(123,v)=-\frac{v\cdot F1\cdot F2\cdot v p{1,2}\cdot F3\cdot v}{3v\cdot p1 v\cdot p{1,2}}-\frac{v\cdot F1\cdot F3\cdot v p1\cdot F2\cdot v}{3v\cdot p1 v\cdot p{1,3}}+\frac{v\cdot F1\cdot F2\cdot F3\cdot v}{3v\cdot p1}$$

All other BCJ numerators are obtained directly from the BCJ numerator by crossing symmetry. For the $n$ point YM amplitude, you only need to replace the velocity by the polarisation vector of the last line $\varepsilon_n$

Yang-Mills-scalar theory

The related paper for KiHA in Yang-Mills-scalar arxiv:2208.05886. The kinematic algebra is taken as field algebra in the full theory of Yang-Mills-scalar+ $\phi^3$. The building blocks are * vector field ${\mathsf Ki}=T{(i)}^{(i)}$ * scalar field ${\mathsf Kj}=T^{(j)}$ * tensor field $T^{(\alpha)}{(\tau1),(\tau2),\cdots, (\tau_r)}$: fields for multi-particle states lie on the interline, which is all multiplicity university mapping to the gauge invariant functions. * fusion product $\star$: The fusion rules from a fewer-particle field to a more-particle field. This fusion product is bilinear and associative. * convolution map $\langle \bullet \rangle$: This is a linear map from abstract algebra general to physics kinematic expression. This is the inner product between multi-particle states and single outgoing particle states. For each algebraic generator, the mapping value is in general non-local and manifestly gauge invariant.

The major output of this program is

$$\widehat {\mathcal N}(1,2,\ldots, n{-}1)= {\mathsf K1}\star {\mathsf K2} \star \ldots \star {\mathsf K{n-1}}$$ which is known as an algebraic pre-numerator. Another major function is the convolution map, which maps the abstract generator to the physical function of kinematic information. For the amplitude with two scalars, ``` preNumerator = [FivePointedStar][[ScriptCapitalK][1, 1], [ScriptCapitalK][2, 1], [ScriptCapitalK][3, 0]] /. ET[f] :> ET2F2s[ET[f]] /. CenterDot[f_] :> tr[f, t^a[n]] /. rmzero //. niceF ``` one can get

$${\mathcal N}(1,2,\overline 3,\overline 4)=\langle {\mathsf K1} \star {\mathsf K2} \star {\mathsf K3} \rangle = \langle T{(1)}^{(1)}\star T{(2)}^{(2)}\star T^{(3)} \rangle=-\frac{p3\cdot F1\cdot F2\cdot p3 \text{tr}\left(t^{a3},t^{a4}\right)}{p{3,1}\cdot p_{3,1}}$$

For the amplitude with more than three scalars, preNumerator = \[FivePointedStar][\[ScriptCapitalK][1, 0], \[ScriptCapitalK][3, 1], \[ScriptCapitalK][2, 0]] /. ET[f__] :> ET2F[ET[f]] /. CenterDot[f__] :> tr[f, t^a[n]] //. niceF you get

$${\mathcal N}(\overline 1,\overline 2,3,\overline 4)=\langle {\mathsf K1} \star {\mathsf K2} \star {\mathsf K3} \rangle= \langle T^{(1)}\star T^{(2)}\star T{(3)}^{(3)} \rangle=\frac{2 p1\cdot F3\cdot p2 \text{tr}\left(t^{a1},t^{a2},t^{a4}\right)}{p{1,2}\cdot p{1,2}}.$$

BCJ numerator for YM+two massive scalar amplitude

For the Yang-Mills amplitude with two massive scalar, we can use the following more efficient code n = 3; Timing[numpreP = \[FivePointedStar] @@ (T /@ List /@ Range[1, n]);] bpeff2 = BinaryProduct[Range[n]] nod = numpreP /. T -> TScalar /. rmzero See more details in the Notebook "YMSAlgebraNew.nb"

higher-derivative gauge field theory

The related paper for KiHA in higher-derivative gauge field theory is 2310.11943 . We consider the gauge field theory with higher order contraction of the strengthen tensor.

$$ \int \mathrm{d}^D x \text{Tr}{\frac{1}{4} F{\mu \nu} F^{\mu \nu}+\frac{2 \alpha^{\prime}}{3} F\mu^\nu F\nu^\lambda F\lambda^\mu+\frac{\alpha^{\prime 2}}{4}[F{\mu \nu}, F{\lambda \rho}][F^{\mu \nu}, F^{\lambda \rho}] } $$

num = \[FivePointedStar][T[{1}], T[{2}], T[{3}]] /. rmzeroT /. T[f__] :> T2FF3F4[T[f]]; num=num /. W -> WFun /. F[i__] :> Sequence @@ (F /@ {i}); using the nice function ClearAll[nicesp, niceFT] nicesp = Join[{\[Epsilon][p[i_]] :> \[Epsilon][i], p[f__] :> Subscript[p, StringJoin[ToString /@ {f}]], F[f__] :> Subscript[F, StringJoin[ToString /@ {f}]], CenterDot[Subscript[p, f_], Subscript[p, f_]] :> \!\(\*SubsuperscriptBox[\(p\), \(f\), \(2\)]\)}, nice, {v[i_] :> v}]; niceFT = Join[ niceT, {List[f_List, i_Integer] :> StringJoin["(", ToString /@ f, ")"]^ToString[i], FT[f__] :> Subscript[J, f]}];

you can see readable form of the pre-numerator num//.nicesp

BCJ numerator in the DF^2+YM theory

This part is to generate the BCJ numerator in the DF2+YM theory. This theory contains massless gluon, massive gluon and tachyon, see 1803.05452 for reference.

generate the HEFT BCJ numerator n=7; num = \[FivePointedStar][T[{1}], T[{2}], T[{3}],T[{4}],T[{5}]] /. rmzeroT /. T[f__] :> T2FF3F4[T[f]]; num = num /. W -> WFun0; num = num /. W[od__] :> WFunDF2[od] /. repNormalOrder /. W[f__] :> W2basis[W[f]] //. dotRules // Expand; ng=n-2; Do[ Print[Length[wfun]]; Monitor[ num = Sum[ num[[jjj]] /. W -> WFunDF2 /. repNormalOrder /. W[f__] :> W2basis[W[f]] /. W0[f__] :> W02basis[W0[f]] //. dotRules // Expand, {jjj, Length@num}];, jjj]; , {id, ng - 2}]

generate the W prime function ng=5; wfun = (-1)^(ng - 1) (WFunDF2 @@ Range[ng]) /. repNormalOrder /. W[f__] :> W2basis[W[f]] //. dotRules // Expand; Do[ Print[Length[wfun]]; Monitor[ wfun = Sum[wfun[[jjj]] /. W -> WFunDF2 /. repNormalOrder /. W[f__] :> W2basis[W[f]] /. W0[f__] :> W02basis[W0[f]] //. dotRules // Expand, {jjj, Length@wfun}];, jjj]; , {id, ng - 2}] More details can be found in the file "DF2YMExample.nb".

QCD BCJ numerator

The amplitude with two fermion line and multi gluon lines are also of color-kinematic duality. The kinematic algebra is also quasi-shuffle hopf algebra. The pre-numerator is obtained as n = 5 numJ = \[FivePointedStar] @@ Table[FT[{{i}, 1}], {i, 1, n - 2}] /. rmzeroT /. FT[f__] :> FT2F[FT[f]]; numJ = % /. sp[f__] :> spAB[p[n], f, p[n - 1]] // Expand; numJ = numJ /. F[i__] :> Sequence @@ (F /@ {i});

Another version of the bcj numerator can be obtained recursively. numRecQCD[5] // contractSp

Local BCJ numerator for Yang-Mills

For the local BCJ numerator in Yang-Mills theory, the kinematic algebra is also quasi-shuffle hopf algebra. The BCJ numerator is obtained as ng = 8; rmzeroDF2 = {T[{i1_, ils__}, g__] :> 0 /; i1 != 1}; Timing[num2 = (\[FivePointedStar] @@ (Table[ T[{ii}], {ii, ng - 1}])) /. T[{i1__}, g___] :> T[{i1, ng}, g] /. rmzeroDF2; num2 = num2 /. T[f__] :> T2YMLocal[T[f]];]

Another version of the bcj numerator can be obtained recursively. We first set the replace rules for the factors in the W' function repdotnum2 = {W0[1, f___, ng] :> -1/\[Alpha] dot[\[Epsilon][1], Sequence @@ F /@ {f}, \[Epsilon][ng]] /. dot[i1_, F[], i2_] :> dot[i1, i2]}; repdotden2 = {(1 - \[Alpha] dot[p[f__], p[f__]]) :> 2 \[Alpha] Length[Complement[Range[ng], {f}]] x}; repdotFnum2 = {dot[f__, F[i_], p[gs___, ng]] :> -dot[f, \[Epsilon][i]] x, dot[f__, F[rs__, i_], p[gs___, ng]] :> -dot[f, F[rs], \[Epsilon][i]] x}; Then we use the recursive rules that contribute to the leading order of $\alpha'$ and in local numerator limit, after applying the above replacement rules, we can get the local BCJ numerator as "wfunYM" Timing[wfun = (-1)^(ng - 1) (WFun2YM @@ Range[ng]); wfun = Sum[ wfun[[ii]]*(1 - \[Alpha] dot[p @@ Range[ng]]) // Simplify, {ii, Length@wfun}]; wfun = wfun //. W -> WFun2YM // Expand; wfunYM = wfun /. repdotnum2 /. repdotden2 /. repdotFnum2; ]