FeynCalc manual (development version)

AnomalousDimension

AnomalousDimension[name] is a database of anomalous dimensions of twist 2 operators.

AnomalousDimension["gnsqg0"] yields the non-singlet one-loop contribution to the anomalous dimension γS,qg(0),m\gamma_{S,qg}^{(0),m} in the MS-bar scheme etc.

See also

Overview, SplittingFunction, SumS, SumT.

Examples

Polarized case:

SetOptions[AnomalousDimension, Polarization -> 1]

{Polarization1,Simplify  FullSimplify}\{\text{Polarization}\to 1,\text{Simplify}\to \;\text{FullSimplify}\}

γNS,qq(0)\gamma _{NS,qq }^{(0) } polarized:

AnomalousDimension[gnsqq0]

CF(8S1(m1)+4m+4m+16)C_F \left(8 S_1(m-1)+\frac{4}{m}+\frac{4}{m+1}-6\right)

γS,qg(0)\gamma _{S,qg }^{(0)} polarized:

AnomalousDimension[gsqg0]

(8m16m+1)Tf\left(\frac{8}{m}-\frac{16}{m+1}\right) T_f

γS,gq(0)\gamma _{S,gq }^{(0)}polarized:

AnomalousDimension[gsgq0]

(4m+18m)CF\left(\frac{4}{m+1}-\frac{8}{m}\right) C_F

γS,gg(0)\gamma _{S,gg}^{(0)} polarized:

AnomalousDimension[gsgg0]

CA(8S1(m1)8m+16m+1223)+8Tf3C_A \left(8 S_1(m-1)-\frac{8}{m}+\frac{16}{m+1}-\frac{22}{3}\right)+\frac{8 T_f}{3}

γPS,qq(0)\gamma _{PS,qq}^{(0)} polarized:

AnomalousDimension[gpsqq1]

16(2m31m2+1m+1+3(m+1)2+2(m+1)31m)CFTf16 \left(\frac{2}{m^3}-\frac{1}{m^2}+\frac{1}{m+1}+\frac{3}{(m+1)^2}+\frac{2}{(m+1)^3}-\frac{1}{m}\right) C_F T_f

γNS,qq(1)\gamma _{NS,qq }^{(1)} polarized:

AnomalousDimension[gnsqq1]

CACF(16S~2(m1)m16S~2(m1)m+1+16S~3(m1)32S~12(m1)+443m25369S1(m1)+883S2(m1)16S3(m1)+2129m7489(m+1)43(m+1)216(m+1)3+173)(CF2(32S~2(m1)m+32S~2(m1)m+132S~3(m1)+64S~12(m1)+8m3+16S1(m1)m2+16S1(m1)(m+1)2+16S2(m1)m+16S2(m1)m+124S2(m1)+32S12(m1)+32S21(m1)40m+40m+1+16(m+1)2+40(m+1)3+3))CFNf(83m2+809S1(m1)163S2(m1)89m+889(m+1)83(m+1)223)-C_A C_F \left(-\frac{16 \tilde{S}_2(m-1)}{m}-\frac{16 \tilde{S}_2(m-1)}{m+1}+16 \tilde{S}_3(m-1)-32 \tilde{S}_{12}(m-1)+\frac{44}{3 m^2}-\frac{536}{9} S_1(m-1)+\frac{88}{3} S_2(m-1)-16 S_3(m-1)+\frac{212}{9 m}-\frac{748}{9 (m+1)}-\frac{4}{3 (m+1)^2}-\frac{16}{(m+1)^3}+\frac{17}{3}\right)-\left(C_F^2 \left(\frac{32 \tilde{S}_2(m-1)}{m}+\frac{32 \tilde{S}_2(m-1)}{m+1}-32 \tilde{S}_3(m-1)+64 \tilde{S}_{12}(m-1)+\frac{8}{m^3}+\frac{16 S_1(m-1)}{m^2}+\frac{16 S_1(m-1)}{(m+1)^2}+\frac{16 S_2(m-1)}{m}+\frac{16 S_2(m-1)}{m+1}-24 S_2(m-1)+32 S_{12}(m-1)+32 S_{21}(m-1)-\frac{40}{m}+\frac{40}{m+1}+\frac{16}{(m+1)^2}+\frac{40}{(m+1)^3}+3\right)\right)-C_F N_f \left(-\frac{8}{3 m^2}+\frac{80}{9} S_1(m-1)-\frac{16}{3} S_2(m-1)-\frac{8}{9 m}+\frac{88}{9 (m+1)}-\frac{8}{3 (m+1)^2}-\frac{2}{3}\right)

γS,qg(1)\gamma _{S,qg }^{(1)} polarized:

AnomalousDimension[gsqg1]

16CATf(2S~2(m1)m+4S~2(m1)m+1+2m32S1(m1)m23m2S12(m1)m+2S12(m1)m+1+4S1(m1)(m+1)2S2(m1)m+2S2(m1)m+14m+3m+1+8(m+1)2+12(m+1)3)+8CFTf(2m31m2+2S12(m1)m4S12(m1)m+12S2(m1)m+4S2(m1)m+1+14m19m+18(m+1)2+4(m+1)3)16 C_A T_f \left(-\frac{2 \tilde{S}_2(m-1)}{m}+\frac{4 \tilde{S}_2(m-1)}{m+1}+\frac{2}{m^3}-\frac{2 S_1(m-1)}{m^2}-\frac{3}{m^2}-\frac{S_1^2(m-1)}{m}+\frac{2 S_1^2(m-1)}{m+1}+\frac{4 S_1(m-1)}{(m+1)^2}-\frac{S_2(m-1)}{m}+\frac{2 S_2(m-1)}{m+1}-\frac{4}{m}+\frac{3}{m+1}+\frac{8}{(m+1)^2}+\frac{12}{(m+1)^3}\right)+8 C_F T_f \left(-\frac{2}{m^3}-\frac{1}{m^2}+\frac{2 S_1^2(m-1)}{m}-\frac{4 S_1^2(m-1)}{m+1}-\frac{2 S_2(m-1)}{m}+\frac{4 S_2(m-1)}{m+1}+\frac{14}{m}-\frac{19}{m+1}-\frac{8}{(m+1)^2}+\frac{4}{(m+1)^3}\right)

γS,gq(1)\gamma _{S,gq }^{(1)} polarized:

AnomalousDimension[gsgq1]

8CACF(4S~2(m1)m2S~2(m1)m+14m3+283m22S12(m1)m+S12(m1)m+1+16S1(m1)3m5S1(m1)3(m+1)+2S2(m1)mS2(m1)m+1569m209(m+1)383(m+1)26(m+1)3)+32CFTf(23m22S1(m1)3m+S1(m1)3(m+1)+79m29(m+1)+13(m+1)2)+4CF2(4m3+8S1(m1)m212m2+4S12(m1)m2S12(m1)m+18S1(m1)m+2S1(m1)m+14S1(m1)(m+1)2+4S2(m1)m2S2(m1)m+1+15m6m+1+3(m+1)22(m+1)3)8 C_A C_F \left(\frac{4 \tilde{S}_2(m-1)}{m}-\frac{2 \tilde{S}_2(m-1)}{m+1}-\frac{4}{m^3}+\frac{28}{3 m^2}-\frac{2 S_1^2(m-1)}{m}+\frac{S_1^2(m-1)}{m+1}+\frac{16 S_1(m-1)}{3 m}-\frac{5 S_1(m-1)}{3 (m+1)}+\frac{2 S_2(m-1)}{m}-\frac{S_2(m-1)}{m+1}-\frac{56}{9 m}-\frac{20}{9 (m+1)}-\frac{38}{3 (m+1)^2}-\frac{6}{(m+1)^3}\right)+32 C_F T_f \left(-\frac{2}{3 m^2}-\frac{2 S_1(m-1)}{3 m}+\frac{S_1(m-1)}{3 (m+1)}+\frac{7}{9 m}-\frac{2}{9 (m+1)}+\frac{1}{3 (m+1)^2}\right)+4 C_F^2 \left(\frac{4}{m^3}+\frac{8 S_1(m-1)}{m^2}-\frac{12}{m^2}+\frac{4 S_1^2(m-1)}{m}-\frac{2 S_1^2(m-1)}{m+1}-\frac{8 S_1(m-1)}{m}+\frac{2 S_1(m-1)}{m+1}-\frac{4 S_1(m-1)}{(m+1)^2}+\frac{4 S_2(m-1)}{m}-\frac{2 S_2(m-1)}{m+1}+\frac{15}{m}-\frac{6}{m+1}+\frac{3}{(m+1)^2}-\frac{2}{(m+1)^3}\right)

γS,gg(1)\gamma _{S,gg }^{(1)} polarized:

v1 = AnomalousDimension[gsgg1]

4CA2(8S~2(m1)m16S~2(m1)m+1+4S~3(m1)8S~12(m1)8m3+8S1(m1)m2+583m216S1(m1)(m+1)2+1349S1(m1)+8S2(m1)m16S2(m1)m+1+4S3(m1)8S12(m1)8S21(m1)1079m+2419(m+1)863(m+1)248(m+1)3163)+32CATf(13m259S1(m1)+149m199(m+1)13(m+1)2+13)+8(4m310m210m+1+2(m+1)2+4(m+1)3+10m+1)CFTf4 C_A^2 \left(\frac{8 \tilde{S}_2(m-1)}{m}-\frac{16 \tilde{S}_2(m-1)}{m+1}+4 \tilde{S}_3(m-1)-8 \tilde{S}_{12}(m-1)-\frac{8}{m^3}+\frac{8 S_1(m-1)}{m^2}+\frac{58}{3 m^2}-\frac{16 S_1(m-1)}{(m+1)^2}+\frac{134}{9} S_1(m-1)+\frac{8 S_2(m-1)}{m}-\frac{16 S_2(m-1)}{m+1}+4 S_3(m-1)-8 S_{12}(m-1)-8 S_{21}(m-1)-\frac{107}{9 m}+\frac{241}{9 (m+1)}-\frac{86}{3 (m+1)^2}-\frac{48}{(m+1)^3}-\frac{16}{3}\right)+32 C_A T_f \left(-\frac{1}{3 m^2}-\frac{5}{9} S_1(m-1)+\frac{14}{9 m}-\frac{19}{9 (m+1)}-\frac{1}{3 (m+1)^2}+\frac{1}{3}\right)+8 \left(\frac{4}{m^3}-\frac{10}{m^2}-\frac{10}{m+1}+\frac{2}{(m+1)^2}+\frac{4}{(m+1)^3}+\frac{10}{m}+1\right) C_F T_f

γS,gg(1)\gamma _{S,gg }^{(1)} polarized (different representation):

v2 = AnomalousDimension[GSGG1];

Check that all odd moments give the same for the two representations of γS,gg(1)\gamma _{S,gg }^{(1)}:

Table[v1 - v2 /. OPEm -> ij, {ij, 1, 17, 2}] // Simplify

{0,0,0,0,0,0,0,0,0}\{0,0,0,0,0,0,0,0,0\}