sigma v for Majorana annihilation

Asked by samwell187


Using CalcHEP, I've computed cross-sections (sigma and sigma*v with v->0) for fermion DM annihilating to b quarks through scalar, vector, and axial vector couplings. I've also done these by hand. The CalcHEP numerical results agree with the by-hand results for all cases /except sigma*v with v->0/ for fermion DM annihilating through a vector with a purely axial coupling. Interestingly, sigma and sigma*v with v->0 for fermion DM annihilating through a spin-1 particle with vector couplings or with both vector and axial couplings agrees with the by-hand result, as does the result for sigma (not sigma*v) for the DM annihilating with only axial couplings.

Furthermore, if I extract the symbolic results (e.g. from symb1.m) and plug in the four-vectors that are required for the zero-velocity limit, then I /do/ agree with the by-hand calculation of sigma*v with v->0. So, seemingly, there is something that goes wrong when CalcHEP evaluates that symbolic result.

Any ideas what might be going wrong?


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Alexander Pukhov
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Alexander Pukhov (pukhov) said :

It should not be so.
Do you mean calculations in your toy model or in MSSM?
In which gauge?

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Sam McDermott (mcdermod) said :

In a toy model: I added a fermion DM candidate and a vector, where the vector has a new coupling to the b also. Since this is a Majorana particle, the couplings are only axial.

The result is the same (about 70% off) in Feynman or unitary gauge.

Thanks for your help!


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Alexander Pukhov (pukhov) said :

I don't see this problem.

I use model with

   EE |0.31343 | Electroweak coupling.
   Mb |5 | b-quark mass
  MNE |100 | neutralino mass
ooz |0.01 | neutralino Z coupling |
 MZ |91.188 | Z-boson mass
 MW |80.385 | W-boson mass

CW |MW/MZ % on-shell cos of th
SW |sqrt(1-CW^2) % sin of the Weinber

Z-boson |Z |Z |23 |2 |MZ |0 |1 |
b-quark |b |B |5 |1 |Mb |0 |3 |
neutralino |~o1 |~o1 |1000001 |1 |MNE |0 |1 |

Feynmam Rules:
B |b |Z | |-EE/(12*SW*CW) |+3*G(m3)*(1-G5)-4*(SW^ 2)*G(m3)
~o1 |~o1 |Z | |ooz |G5*G(m3)

Squared matrix element at rest in untary gauge is

sqme= 3* (ee*mb*mne*ooz/(sw*cw*mz**2))**2

Cross section :

where v is defined as relative velocity.

Factor 3.8937966E8 transforms GeV^{-2) cross section to [pb].

Using both CalcHEP v*sigma plot and analitic formula I get the same result 2.38E-4 pb

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samwell187 (samwell187) said :

The model that I used was just a modification of the Standard Model. So I had the SM particle content, plus two particles:

A A+ 2*spin color aux
J J 1 1 0
VA VA 2 1 0

with Feynman rules

J J VA gjjVA ( G(m3)*G5 )
b B VA gbbVA ( G(m3)*G5 )

This gave a squared matrix element at rest in Feynman gauge (and I believe also in unitary gauge) as

48 gbbVA^2 gjjVA^2 Mb^2 Mx^2 MVA^-4

with cross section times vrel of

sigma vrel = 3 gbbVA^2 gjjVA^2 Mb^2 Mx^2 Sqrt[1-Mb^2/Mx^2] / (2 Pi MVA^4)

This seems to agree with your expression, when I make the substitutions gbbVA -> e/4swcw and gjjVA->ooz. I reproduce your result using your values for the inputs (i.e. Mb=5 GeV, etc.)

When I evaluate my expression in the toy model using gbbVA = gjjVA = 0.1, Mb = 4.25GeV, Mx = 40 GeV, and MVA = 10 TeV I get 3.34E-11 pb ; when I extract the value from the CalcHEP sigma*v plot I get 1.92E-11 pb

Again, the answer that I get is correct when the DM is Dirac and has both axial and vector couplings. Not sure what could be going wrong...


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Best Alexander Pukhov (pukhov) said :

In general Feynman gauge vector boson has to be accompanied with scalar goldstone. Only in this case Feynmal gauge and Unitary gauge produce the same result. It seems you use Unitary gauge in CalcHEP and Feynman gauge in your calculation. Thus results are different. In case of vector interection coupling of goldstone with fermions is proportion to mass difference. So, it is zero and you have agreement with CalcHEP. But for axial current goldstone coupling is proportion to sum of fermion masses and you see a difference.

Just type 'g' in 'aux' column of your vector field. Then I guees CalcHEP will produce the same result as your calculation by hand.
Or use ubitary gauge in your calculations.

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samwell187 (samwell187) said :

Whether I use unitary or Feynman gauge or put the "G" or "g" in the aux column, I always get the same answer for the parameters listed previously -- CalcHEP always says 1.92 x 10^-11 pb, but by hand I always get 3.34 x 10^-11 pb

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Alexander Pukhov (pukhov) said :

I have implemented your mdel in calchep and have got 3.34E-11.

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samwell187 (samwell187) said :

Maybe I'm extracting that value incorrectly then. How do you get this number? I use the sigma*v plots option of n_calchep and plot Pcm from some arbitrarily low value, then use the plot indicator to read off the y-value. Is this the most sensitive way to get sigma*v as v -> 0?

(Thanks for all of your help!)

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Alexander Pukhov (pukhov) said :

I display plot for sigma*v in interval pcm=0.0001 - 0.01 . It is a constant. After that I click mouse to see value.
For pcm <=1.E-5 there is numerical instability.

Check value of parameters. May be they are restored form session.dat in a wrong way. Sometimes session.dat from previous session is not deleted.
Indeed for such huge vector boson mass both gauges lead to the same numerical results.
Calchep is used in micrOMEGAs to generate matrix elements for relic density calculation. So, I expect it should work correctly for v*sigma. From the other side correct calculation of vsigma is very important for us. By the way, your "Dark Mater" is a Dirac particle or Majorana one. In my caculations it is a Majorama particle.

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samwell187 (samwell187) said :

(The dark matter here is Majorana since it only has axial couplings. So in the model definition I make the particle and antiparticle have the same symbol.)

I do the exact same procedure to see sigma*v as v->0. The parameter values seem to be correct, but I still get the incorrect value of sigma*v

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Alexander Pukhov (pukhov) said :

My version of your model is here

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Sam McDermott (mcdermod) said :

Ok, thanks for your help. The issue is the b-quark mass. I did not realize the b mass was being RG evolved by default. I was just plugging in the value at the b pole. Since this process has no mediator pole and is helicity suppressed, it's more important for this process than the others I was checking.

Thanks again,