Charge distribution of the gauge-mediation type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Q</mml:mi> </mml:math> ball

We numerically study the formation of the gauge-mediation type <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>Q</a:mi> </a:math> balls in the logarithmic square potential on three-dimensional lattices. We obtain the broad charge distribution of the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>Q</c:mi> </c:math> ball of this type for the first time, to the best of our knowledge. The charge of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>Q</e:mi> </e:math> ball at the peak of the distribution is smaller than what we estimated as the average of the largest tens of the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>Q</g:mi> </g:math> balls in the logarithmic potential for the same initial amplitude of the field at the onset of its oscillation. We also discuss some impacts of the broad distribution on cosmology and astrophysics. In the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>B</i:mi> </i:math> ball ( <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>Q</k:mi> </k:math> being the baryon number) case, the broad distribution would lead to the coexistence of both stable and unstable <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>B</m:mi> </m:math> balls. We find that stable <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>B</o:mi> </o:math> balls can account for the dark matter of the Universe without affecting successful big bang nucleosynthesis by the decay of the unstable <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>B</q:mi> </q:math> balls, but the baryon number of the Universe cannot be explained by them. On the other hand, the large <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>L</s:mi> </s:math> balls ( <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>Q</u:mi> </u:math> being the lepton number) would be the dark matter as well while avoiding the constraints on the x- and/or <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mi>γ</w:mi> </w:math> -rays from the decay of the smaller <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi>L</y:mi> </y:math> balls.