Date: 11/21/16-12:41:33 AM Z

Dear Jongping,

sorry for the late reply, I had to finish something very important.

> Thank you very much for your answers. But the problem is not resolved.The reason is as follows:
>
> Your conventions of the Feynman diagrams etc. are slightly different from mine.But this is not important because they lead to the same (incorrect) result after using FeynCalc to carry out OneLoop with the.nb file you sent me in the email.

I still don't think that there any issues with the FeynCalc result.

> am interested only in the logarithmically divergent terms,i.e. the coefficient of the function B0(p^2,0,0).

You have to take into account that the C0 function that appears in the
result is also log-divergent and hence cannot be simply neglected. The
tricky thing here is that C0 is not UV but IR divergent. Nevertheless,
as in DR we usually regularize both divergences with the same regulator,
this IR divergence manifests itself as another 1/eps pole.

One can see it either through a direct computation or by noticing that
C0 is proportional to d^D q/[q^4 (q-p)^2] and hence can be reduced to B0
via Integration-By-Parts (IBP) identities.

> My question is: Why it does not produce the expected (Lorentz) tensor structure? My hand-calculation and the calculation of Politzer
> did produce the correct structure, as expected. Could you please use FeynCalc to check the consistce?

So if you are interested only in the 1/eps piece of the result, you can
do either

Simplify[AA //. {FCI[
C0[0, SP[p, p], SP[p, p], 0, 0,
0]] -> -(( (-3 + D) B0[FCI[SP[p, p]], 0, 0])/SP[p, p]),
FCI[B0[FCI[SP[p, p]], 0, 0]] -> 1/(16*Epsilon*Pi^4)}];
Series[FCI[%] /. D -> 4 - 2 Epsilon, {Epsilon, 0, 0}] // Normal //
Collect2[#, Epsilon] &
SelectNotFree[%, 1/Epsilon]

or

ints = FCI[{FCI[B0[SP[p, p], 0, 0]] -> 1/(16*Epsilon*Pi^4),
FCI[C0[0, SP[p, p], SP[p, p], 0, 0, 0]] -> -1/(16*Epsilon*Pi^4*SP[p,
p])}]
Series[AA /. ints /. D -> 4 - 2 Epsilon, {Epsilon, 0, 0}] // Normal
//Collect2[#, Epsilon] &
SelectNotFree[%, 1/Epsilon]

This way you obtain precisely the tensor structure that you mentioned.

p^nu p^si) when finite parts of the result are also taken into account.
This is also mentioned in Pascual and Tarrach. The structure
that multiplies 1/eps pole is of course the expected one.

> Five or six years ago I and a student used FeynCalc to calculate the graviton self-energy in a new theory called 'quantum Yang-Mills gravity' that I formulated. I do not remember such inconsistence in the loop calculations.

In self-energy calculations you usually don't have C0 functions (unless
you keep the full gauge dependence), only A0's and B0's. In usual DR
those are always only UV divergent, so one does not have to care about
1/eps poles from IR divergences.

> Is it possible that now Mathematica has evolved to version 11 which may not be completely consistent with FeynCalc?

I'm using Mathematica 11 on a daily basis, so there are no known
incompatibilities with FeynCalc. However, I believe that the first part
of this e-mail already explains why the given result is consistent.

Hope this helps.

Cheers,

Am 17.11.2016 um 22:46 schrieb Jongping Hsu:
> Please see the attached letter and .nb file. JP
>
> HSU Jongping,
> Chancellor Professor
> Department of Physics
> Univ. of Massachusetts Dartmouth,
> North Dartmouth, MA 02747. FAX (508)999-9115
> http://www.umassd.edu/engineering/phy/people/facultyandstaff/jong-pinghsu/
> recent monograph: Space-Time Symmetry and Quantum Yangâ€“Mills Gravity
>
> ------------------------------------------------------------------------
> *To: *feyncalc@feyncalc.org
> *Sent: *Monday, November 14, 2016 4:48:03 PM
> *Subject: *Re: ?
>
> Good evening (according to Munich time),
>
> Rolf "delegated" the development of FeynCalc to me since some time,
> so he is now more acting as a supervisor, while I do the coding.
>
> First of all let me remark, that the amplitude for the first diagram
> does not look correct to me. I can only guess, what routing of momenta
> was meant here, but assuming that it is
> mu
> /|\ /|~~~~~~~~ -> p
> q | / |
> | / | /|\
> ~~~~~~~~~~/ | | p-q
> | \la | |
> | \ |
> -q \|/ \ |
> \|~~~~~~~~ -> -p
> nu
>
> the amplitude should be written as
>
> g1 = GluonGhostVertex[{0, la, a}, {q, la1, a1}, {-q, la2, a2}] //
> Explicit;
> g2 = GluonGhostVertex[{p, mu, b}, {-q, mu1, b1}, {q - p, mu2, b2}] //
> Explicit;
> g3 = GluonGhostVertex[{-p, nu, c}, {p - q, nu1, c1}, {q, nu2, c2}] //
> Explicit
> gp = g1*g2*g3 // Contract // Explicit // Simplify
> g12 = GhostPropagator[q, a1, b2] // Explicit
> g23 = GhostPropagator[p - q, b1, c2] // Explicit;
> g31 = GhostPropagator[q, c1, a2] // Explicit;
> gpro = g12*g23*g31 // FeynAmpDenominatorCombine;
> tp = gpro*gp // Contract // SUNSimplify // Simplify
>
> i.e. in my view g2 and g3t in the attached notebook need to be corrected.
>
> One can check this explicitly by comparing to the FeynArts output (
> on a new kernel):
>
> << FeynCalc`
>
> \$FAVerbose = False;
>
> top = CreateTopologies[1, 1 -> 2,
> ExcludeTopologies -> {WFCorrections, SelfEnergies, Tadpoles}];
> diags = InsertFields[top, {V[5]} -> {V[5], V[5]}, Model -> "SMQCD",
> InsertionLevel -> {Particles},
> ExcludeParticles -> {F[__], S[_], V[_], U[1 | 2 | 3 | 4]}];
> Paint[diags, ColumnsXRows -> {2, 1}, SheetHeader -> False,
> SheetHeader -> None, Numbering -> None, ImageSize -> {512, 256}];
>
> amps = FCFAConvert[
> CreateFeynAmp[diags, Truncated -> True, GaugeRules -> {},
> PreFactor -> 1], IncomingMomenta -> {p1},
> OutgoingMomenta -> {p2, p3}, LoopMomenta -> {q},
> DropSumOver -> True, UndoChiralSplittings -> True,
> ChangeDimension -> D, List -> True, SMP -> True,
> LorentzIndexNames -> {\[Lambda], \[Mu], \[Nu]}] // Contract //
> SUNSimplify
>
> amps[[2]] /. p3 -> -p2 /. p2 -> p
>
> Notice that here the prefactor 1/(2Pi)^D is understood but not written
> down explicitly.
>
> My second remark is that the replacement D->4 should not be done when
> the result contains Passarino-Veltman function multiplied by polynomials
> in D. As PaVe funtions have poles in 1/Eps, taking the limit naively by just
> putting D=4 leads to the wrong finite part in the final result.
>
> As to the correctness of the result for the 3-gluon vertex, I currently
> do not have a source the values of for both pieces separately at hand.
> However, I believe that the result returned by FeynCalc is correct,
> including the overall prefactor. A simple way to check it, is to look at
> the UV-part of the ghost triangle, which can be found e.g. in Pascual
> and Tarrach, QCD: Renormalization for Practitioner, Eq III.45
>
> The 3-gluon vertex function (with all momenta ingoing) can be
> parametrized as
>
> "diagram" = i g T^{mu nu si}_{a b c}
>
> with
>
> T^{mu nu si}_{a b c} = - i f_abc (g^{mu nu} (p-q)^si + g^{nu si} (q-k)^mu +
> g^{nu si} (q-k)^mu (k-p)^nu ) T1(p^2,q^2,k^2)
>
> According to Pascual and Tarrach the UV part of T1(p^2,q^2,k^2) is given by
>
> g^2/(4 Pi)^2 CA/8 1/(3 Eps)
>
> By making the momenta p2 and p3 of the FeynArts amplitude ingoing
>
> amps2 = (amps /. {p2 -> -p2, p3 -> -p3});
>
> doing the tensor decomposition
>
> ampsPT = (TID[#, q, ToPaVe -> True, UsePaVeBasis -> True] & /@ amps2);
>
> and extracting the UV parts of the PaVe functions (c.f.
> arXiv:hep-ph/0609282 for tabulated results)
>
>
> uvParts = {PaVe[
> 1, {SPD[p2, p2], SPD[p2, p2] + 2 SPD[p2, p3] + SPD[p3, p3],
> SPD[p3, p3]}, {0, 0, 0}, PaVeAutoOrder -> True,
> PaVeAutoReduce -> True] -> 0,
> PaVe[2, {SPD[p2, p2], SPD[p2, p2] + 2 SPD[p2, p3] + SPD[p3, p3],
> SPD[p3, p3]}, {0, 0, 0}, PaVeAutoOrder -> True,
> PaVeAutoReduce -> True] -> 0,
> PaVe[0, 0, {SPD[p2, p2], SPD[p2, p2] + 2 SPD[p2, p3] + SPD[p3, p3],
> SPD[p3, p3]}, {0, 0, 0}, PaVeAutoOrder -> True,
> PaVeAutoReduce -> True] -> 1/(64 EpsilonUV \[Pi]^4),
> PaVe[1, 1, {SPD[p2, p2], SPD[p2, p2] + 2 SPD[p2, p3] + SPD[p3, p3],
> SPD[p3, p3]}, {0, 0, 0}, PaVeAutoOrder -> True,
> PaVeAutoReduce -> True] -> 0,
> PaVe[1, 2, {SPD[p2, p2], SPD[p2, p2] + 2 SPD[p2, p3] + SPD[p3, p3],
> SPD[p3, p3]}, {0, 0, 0}, PaVeAutoOrder -> True,
> PaVeAutoReduce -> True] -> 0,
> PaVe[2, 2, {SPD[p2, p2], SPD[p2, p2] + 2 SPD[p2, p3] + SPD[p3, p3],
> SPD[p3, p3]}, {0, 0, 0}, PaVeAutoOrder -> True,
> PaVeAutoReduce -> True] -> 0,
> PaVe[0, 0,
> 1, {SPD[p2, p2], SPD[p2, p2] + 2 SPD[p2, p3] + SPD[p3, p3],
> SPD[p3, p3]}, {0, 0, 0}, PaVeAutoOrder -> True,
> PaVeAutoReduce -> True] -> -(1/(192 EpsilonUV \[Pi]^4)),
> PaVe[0, 0,
> 2, {SPD[p2, p2], SPD[p2, p2] + 2 SPD[p2, p3] + SPD[p3, p3],
> SPD[p3, p3]}, {0, 0, 0}, PaVeAutoOrder -> True,
> PaVeAutoReduce -> True] -> -(1/(192 EpsilonUV \[Pi]^4)),
> PaVe[1, 1,
> 1, {SPD[p2, p2], SPD[p2, p2] + 2 SPD[p2, p3] + SPD[p3, p3],
> SPD[p3, p3]}, {0, 0, 0}, PaVeAutoOrder -> True,
> PaVeAutoReduce -> True] -> 0,
> PaVe[1, 1,
> 2, {SPD[p2, p2], SPD[p2, p2] + 2 SPD[p2, p3] + SPD[p3, p3],
> SPD[p3, p3]}, {0, 0, 0}, PaVeAutoOrder -> True,
> PaVeAutoReduce -> True] -> 0,
> PaVe[1, 2,
> 2, {SPD[p2, p2], SPD[p2, p2] + 2 SPD[p2, p3] + SPD[p3, p3],
> SPD[p3, p3]}, {0, 0, 0}, PaVeAutoOrder -> True,
> PaVeAutoReduce -> True] -> 0,
> PaVe[2, 2,
> 2, {SPD[p2, p2], SPD[p2, p2] + 2 SPD[p2, p3] + SPD[p3, p3],
> SPD[p3, p3]}, {0, 0, 0}, PaVeAutoOrder -> True,
> PaVeAutoReduce -> True] -> 0};
>
> ampsPT2 = ampsPT /. FCI[uvParts]
>
> ampsPTUVpart1 = Total[ampsPT2] // FCE // Collect2[#, {Epsilon, MTD}] &
>
> we obtain
>
> -((CA (2 FVD[p2, \[Nu]] + FVD[p3, \[Nu]]) MTD[\[Lambda], \[Mu]] SMP[
> "g_s"]^3 SUNF[a, b, c])/(384 EpsilonUV \[Pi]^2)) + (
> CA (FVD[p2, \[Mu]] + 2 FVD[p3, \[Mu]]) MTD[\[Lambda], \[Nu]] SMP[
> "g_s"]^3 SUNF[a, b, c])/(384 EpsilonUV \[Pi]^2) + (
> CA (FVD[p2, \[Lambda]] - FVD[p3, \[Lambda]]) MTD[\[Mu], \[Nu]] SMP[
> "g_s"]^3 SUNF[a, b, c])/(384 EpsilonUV \[Pi]^2)
>
> this can be further simplified by exploiting the kinematics p1+p2+p3 = 0
>
> ampsPTUVpart2 =
> ampsPTUVpart1 /. {FVD[p2, i_] + 2 FVD[p3, i_] :>
> FVD[p3, i] - FVD[p1, i],
> FVD[p3, i_] + 2 FVD[p2, i_] :> FVD[p2, i] - FVD[p1, i]}
>
> The Lorentz structure of the vertex function from Pascual and Tarrach is
> given by
>
> PTVertexFuLorentzStruct[{p_, q_, k_}, {mu_, nu_, si_}, {a_, b_,
> c_}] :=
> -I SUNF[a, b,
> c] (MTD[mu, nu] FVD[p - q, si] + MTD[nu, si] FVD[q - k, mu] +
> MTD[si, mu] FVD[k - p, nu])
>
> so the UV part of T1(p^2,q^2,k^2) reads
>
> Cancel[FCI[ExpandScalarProduct[ampsPTUVpart2]]/
> ExpandScalarProduct[
> FCI[PTVertexFuLorentzStruct[{p1, p2,
> p3}, {\[Lambda], \[Mu], \[Nu]}, {a, b, c}]]]] // Simplify
> res = %/(I SMP["g_s"])
>
> comparing this to the result from the book we see the full agreement
>
> PTGhostTriangleResult = SMP["g_s"]^2/(4 Pi)^2 CA/8 1/(3 EpsilonUV)
>
> res - PTGhostTriangleResult
>
> (* gives 0*)
>
> So everything should be correct. I hope this helps to clarify the
> discrepancy.
>
> Cheers,
>
>
> Am 14.11.2016 um 03:20 schrieb Jongping Hsu:
>> Dear Dr. Mertig:
>> FeynCalc calculation leads to incorrect results (in comparison with my
>> hand-calculations and those in the literature (e.g.,Politzer's) for the
>> 3rd. order Feynman diagram involving 3 ghost propagators and 3 external
>> gluon lines [with one external gluon momentum set to be zero for
>> simplicity] in the calculation of the asymptotic freedom)? Thank you
>> very much for your help.
>>
>> I used Mathematica 11 for calculations(with SU(2) in mind). My short
>> program is attached for your reference.
>>
>>
>> 1. The vertices g1*g2*g3 contains 2 terms. For clarity, I first
>> calculate the first term by using the vertex g1*g2*g3t. The OneLoop[...]
>> gives the result A1, which is only half of the correct result A10 (see
>> In[2182]).
>> 2. I calculate the second term by using g1*g2*g3tt, which lead to the
>> oneloop result A2. Only when it is multiplied by(3/2), one gets the
>> correct result. See In[2198].
>> 3. In[2198]: By observation, A1*(4/2)+A2*(3/2) give the correct result.
>> 4. For the crossed diagram (of external gluons with non-zero momenta), a
>> similar corrections for the one-loop results (T1 and T) are done for the
>> 2 terms: B1=2*T1, B2=(3/2)*T.
>> 5. The corrected result, {A1*2 + A2*(3/2)}+{2*T1 + (3/2)*T}, leads to
>> the correct structure and magnitude of the diagram involving 3 ghost
>> internal lines, as shown in the coefficient of B0 in Out[2231] and in
>> Out[2232].
>>
>> HSU Jongping,
>> Chancellor Professor
>> Department of Physics
>> Univ. of Massachusetts Dartmouth,
>> North Dartmouth, MA 02747. FAX (508)999-9115
>> http://www.umassd.edu/engineering/phy/people/facultyandstaff/jong-pinghsu/
>> recent monograph: Space-Time Symmetry and Quantum Yangâ€“Mills Gravity