Incorrect prefactor in vector boson contribution to 1-loop fermion self-energy

[AlexanderVoigt]

Hi Florian,

Benjamin Summ and I think we have found a small bug in the 1-loop self-energy of fermions: The R and L contribution coming from vector bosons misses a global factor 2.

In the file Package/loopCorrections.m line 690ff it is written

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Switch[corrections[[4]],
	...
	FFV,
		Sigma1LoopS= sum[gI1,1,getGen[corrections[[1]]],sum[gI2,1,getGen[corrections[[2]]],
		         - 4 m11 coupL1*coupR2*(B0[p^2,m12,m22] -1/2 rMS)corrections[[6]]*corrections[[5]]]];
		Sigma1LoopR = sum[gI1,1,getGen[corrections[[1]]],sum[gI2,1,getGen[corrections[[2]]],
		         - coupL1*coupL2*(B1[p^2,m12,m22]+1/2 rMS)corrections[[6]]*corrections[[5]]]];
		Sigma1LoopL=sum[gI1,1,getGen[corrections[[1]]],sum[gI2,1,getGen[corrections[[2]]],
		         - coupR1*coupR2*(B1[p^2,m12,m22]+1/2 rMS) corrections[[6]]*corrections[[5]]]];
	];

The prefactor 4 in the scalar contribution is correct and comes from the contraction of two gamma matrices. However, in the R and L contribution we would expect a global factor 2 from the contraction of three gamma matrices (a p-slash sandwiched by two gamma matrices, which are contracted via their Lorentz index). As far as we understand, this factor 2 is neither hidden in any of the vertices, nor in the additional symmetry factors. So, we believe there is a bug here.

We think the correct expression would be

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Switch[corrections[[4]],
	...
	FFV,
		Sigma1LoopS= sum[gI1,1,getGen[corrections[[1]]],sum[gI2,1,getGen[corrections[[2]]],
		         - 4 m11 coupL1*coupR2*(B0[p^2,m12,m22] -1/2 rMS)corrections[[6]]*corrections[[5]]]];
		Sigma1LoopR = sum[gI1,1,getGen[corrections[[1]]],sum[gI2,1,getGen[corrections[[2]]],
		         - 2 coupL1*coupL2*(B1[p^2,m12,m22]+1/2 rMS)corrections[[6]]*corrections[[5]]]];
		Sigma1LoopL=sum[gI1,1,getGen[corrections[[1]]],sum[gI2,1,getGen[corrections[[2]]],
		         - 2 coupR1*coupR2*(B1[p^2,m12,m22]+1/2 rMS) corrections[[6]]*corrections[[5]]]];
	];

Best regards,
Alex

[FStaub]

Hi Alex,

looking at the Fortran output (SPhenoLoopMasses.m)

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WriteString[sphenoLoop, 
 "B1m2 = -4._dp*(" <> "Real(SA_" <> pre <> "B1(p2," <> m12 <> "," <>   m22 <> "),dp) + 0.5_dp*" <> pre <> "rMS) \n"];
 WriteString[sphenoLoop,  "B0m2 = -8._dp*" <> m1 <> "*(" <> "Real(SA_" <> pre <> "B0(p2," <>   m12 <> "," <> m22 <> "),dp) - 0.5_dp*" <> pre <> rMS) \n"];
...
 

it seems correct that only a relative factor of 2 between the B0 and B1 contributions should be there. I just wonder why this hasn't been observed in the comparisons before?!

Cheers,
Florian

[AlexanderVoigt]

Hi Florian,

I agree, there should be a factor 2 difference between the scalar and the R/L contribution.

I'm not sure why this has not been discovered before. One reason may be that in FlexibleSUSY the gluon contribution is removed from the SARAH-generated top self energy and added back "by hand" later in a correct form. However, the photon, W and Z contributions should have been discovered before, strange... Also Pierce et.al. seem to have a factor 4 different between the S and R/L contributions, see Eq.(D.27)ff for example. But maybe I'm missing something?

BTW, why is there a factor 4 and 8 in the SPheno expressions (not 2 and 4 as in loopCorrections.m)?

Best regards,
Alex

[FStaub]

Hi,

sorry, I should have looked at the entire part of the code which reads:

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WriteString[sphenoLoop, 
 "B1m2 = -4._dp*(" <> "Real(SA_" <> pre <> "B1(p2," <> m12 <> "," <>   m22 <> "),dp) + 0.5_dp*" <> pre <> "rMS) \n"];
WriteString[sphenoLoop,  "B0m2 = -8._dp*" <> m1 <> "*(" <> "Real(SA_" <> pre <> "B0(p2," <>   m12 <> "," <> m22 <> "),dp) - 0.5_dp*" <> pre <> "rMS) \n"];
...
WriteString[sphenoLoop, "SumSR(gO1,gO2) = coupL2*coupR1*B0m2 \n"];
WriteString[sphenoLoop, "SumSL(gO1,gO2) = coupR2*coupL1*B0m2 \n"];
WriteString[sphenoLoop, "sumL(gO1,gO2) = coupR1*coupR2*B1m2 \n"];
WriteString[sphenoLoop, "sumR(gO1,gO2) = coupL1*coupL2*B1m2 \n"];

In SPheno, I don't calculate anymore Sigma_S but Sigma_SL and Sigma_SR which are needed for the one-loop decays. The mass corrections are proportional to Sigma_SL+Sigma_SR, i.e. there is indeed a factor of 4.
The overall factor is a matter of convention and absorbed in the definition of the loop corrected masses. This reads in SARAH:

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WriteString[sphenoLoop,   "mat1 = mat1a - 0.5_dp*(Conjg(SigSL) + SigSR + & \n"];
WriteString[sphenoLoop,   "      & 0.5_dp*(MatMul(Transpose(SigL),mat1a) + MatMul(SigR,mat1a) \+ & \n"];
WriteString[sphenoLoop,   "      & MatMul(mat1a,Transpose(SigR)) + MatMul(mat1a,SigL))) \n \n\"];

Maybe, also there the factor 2 is buried which you are looking for?

Cheers,
Florian

[FStaub]

Hi again,

I was checking another reference which writes

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\delta m_f = 1/2 m_f (\Sigma^L+Sigma^R+2 Sigma^S)

for a fermion appearing in one generation. This differs by a relative factor of 2 compared to the Pierce et al. conventions, correct?

Cheers,
Florian

[AlexanderVoigt]

Hi Florian,

you are right, it is matter of convention. We checked that if you use the S,L,R components of the SARAH-generated self energies and combine them as written in http://stauby.de/sarah_wiki/index.php?title=Loop_Masses then the resulting mass is correct. If I understand correctly, the SARAH convention is to decompose the fermion self energy as

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  Sigma = Sigma_S + 2 pslash ( PL Sigma_L + PR Sigma_R )

With this convention you obtain the same expressions as in the BPMZ paper. In this convention there is a factor 4 difference between the vector boson contribution of S and R/L. But this is correct.

Benjamin and I were assuming that SARAH uses

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  Sigma = Sigma_S + pslash ( PL Sigma_L + PR Sigma_R )

which is obviously not the case. In this convetion there would only be a factor 2 different.

Anyway, SARAH is correct.
Sorry for the confusion.

Best regards,
Alex