Symmetry factors for tadpoles

[tueda]

I think it doesn't matter for physical calculations, but if I compare the symmetry factors generated by FeynGraph 0.1.0b4 and those by QGRAF 3.6.10 (and also FORM with form-dev/form#744) for 4 diagrams in 1-loop quark self-energy $q \to q$ in QCD without any filtering, they are different for 2 diagrams with tadpoles (quark-loop and ghost-loop).

FeynGraph	-1/2	+1/2	-1/2	1
QGRAF	-1	+1/2	-1	1

Input files

feyngraph.toml:

template = "json.jinja"
[process]
in = ["qua"]
out = ["qua"]
loops = 1
model = "qcd.mdl"
momenta = ["p1", "q1", "k1"]

qgraf.dat:

output = 'qgraf.out';
style = 'form.sty';
model = 'qcd.mdl';
in = qua;
out = qua;
loops = 1;
loop_momentum = k;
options = ;

form.sty:

<prologue>
*
* file generated by <program>
*
<command_loop><command_line_loop>* <command_data><end><end>*
*
<diagram>
*--#[[ d<diagram_index>:
     <sign><symmetry_factor>
<vertex_loop>    *vx(<ray_loop><field>(<propagator_index>),<end><back>)
<end><back>
*--#]] d<diagram_index>: 
<epilogue>
*
* end
*
<exit>

qcd.mdl:

[qua,QUA,-]
[glu,glu,+]
[gho,GHO,-]
[QUA,qua,glu]
[glu,glu,glu]
[glu,glu,glu,glu]
[GHO,gho,glu]

Output

diagrams.out:

{
  "0": {
    "symmetry_factor": 2,
    "sign": -1,
    "legs": {
      "0": {"particle": "qua", "vertex": 0, "ray_index": 0},
      "1": {"particle": "qua", "vertex": 0, "ray_index": 1}
    },
    "propagators": {
      "2": {"particle": "glu", "vertices": [0, 1], "ray_indices": [2, 0], "momentum": ""},
      "3": {"particle": "qua", "vertices": [1, 1], "ray_indices": [1, 2], "momentum": "k1"}
    },
    "vertices": {
      "0": {"interaction": "V_1", "propagators": [1, 0, 2]},
      "1": {"interaction": "V_1", "propagators": [3, 3, 2]}
    }
  },
  "1": {
    "symmetry_factor": 2,
    "sign": 1,
    "legs": {
      "0": {"particle": "qua", "vertex": 0, "ray_index": 0},
      "1": {"particle": "qua", "vertex": 0, "ray_index": 1}
    },
    "propagators": {
      "2": {"particle": "glu", "vertices": [0, 1], "ray_indices": [2, 0], "momentum": ""},
      "3": {"particle": "glu", "vertices": [1, 1], "ray_indices": [1, 2], "momentum": "k1"}
    },
    "vertices": {
      "0": {"interaction": "V_1", "propagators": [1, 0, 2]},
      "1": {"interaction": "V_2", "propagators": [2, 3, 3]}
    }
  },
  "2": {
    "symmetry_factor": 2,
    "sign": -1,
    "legs": {
      "0": {"particle": "qua", "vertex": 0, "ray_index": 0},
      "1": {"particle": "qua", "vertex": 0, "ray_index": 1}
    },
    "propagators": {
      "2": {"particle": "glu", "vertices": [0, 1], "ray_indices": [2, 0], "momentum": ""},
      "3": {"particle": "gho", "vertices": [1, 1], "ray_indices": [1, 2], "momentum": "k1"}
    },
    "vertices": {
      "0": {"interaction": "V_1", "propagators": [1, 0, 2]},
      "1": {"interaction": "V_4", "propagators": [3, 3, 2]}
    }
  },
  "3": {
    "symmetry_factor": 1,
    "sign": 1,
    "legs": {
      "0": {"particle": "qua", "vertex": 0, "ray_index": 0},
      "1": {"particle": "qua", "vertex": 1, "ray_index": 0}
    },
    "propagators": {
      "2": {"particle": "glu", "vertices": [0, 1], "ray_indices": [1, 1], "momentum": "p1+k1"},
      "3": {"particle": "qua", "vertices": [0, 1], "ray_indices": [2, 2], "momentum": "-k1"}
    },
    "vertices": {
      "0": {"interaction": "V_1", "propagators": [3, 0, 2]},
      "1": {"interaction": "V_1", "propagators": [1, 3, 2]}
    }
  }
}

qgraf.out:

*
* file generated by qgraf-3.6.10
*
* output = 'qgraf.out';
* style = 'form.sty';
* model = 'qcd.mdl';
* in = qua;
* out = qua;
* loops = 1;
* loop_momentum = k;
* options = ;
*
*
*--#[ d1:
     -1
    *vx(QUA(-2),qua(-1),glu(1))
    *vx(QUA(2),qua(2),glu(1))
*--#] d1:
*--#[ d2:
     +1/2
    *vx(QUA(-2),qua(-1),glu(1))
    *vx(glu(1),glu(2),glu(2))
*--#] d2:
*--#[ d3:
     -1
    *vx(QUA(-2),qua(-1),glu(1))
    *vx(GHO(2),gho(2),glu(1))
*--#] d3:
*--#[ d4:
     +1
    *vx(QUA(1),qua(-1),glu(2))
    *vx(QUA(-2),qua(1),glu(2))
*--#] d4:
*
* end
*

[Jens-Braun]

This was indeed a wrong symmetry factor in FeynGraph caused by a small oversight for self-loops. The symmetry factor due to self-loop propagators was previously calculated as $2^n n!$, where $n$ is the number of identical self-loop propagators of a vertex. This is only correct for self-conjugate propagators however, otherwise the correct expression is just $n!$. With the previous commit, the symmetry factors should now be correct.
Thank you very much for the report!