Hello - you have not responded to the information request and I must go offline, so I will have to go with the information I do have. Feel free to reply to continue the correspondence if needed.
If there is no custody between the parents formally via the Court the parent wants to file to establish custody. This is a lawsuit. See HERE.
The Court decides on custody based on the rule of thumb of "best interest of the child." This includes, but is not limited to, general stability of the parent, financial stability, indoctrination of the child in the current school and environment, household condition and living condition of the child, other persons living in the house, etc. The courts generally do not like to split the custody 50/50 since this is hard on the child unless both parents agree to this.
One parent usually becomes the custodian and the other parent becomes the "visiting" parent which is generally one day a week, every other weekend, and alternating holidays. The nuanced points of the custody can either be decided by the parties or the Court, if the parties cannot come to an agreement.
Even if a parent does not get managing custody, they are almost guaranteed visitation unless they have a drug problem, alcohol dependency, or an unsafe home environment. Abuse and or neglect of the child or previous children are an almost automatic bar for even visitation, although supervised visitation may be granted by the Court.
Of course, that is the standard order of possession. That order can be modified if both parties agree or if the Court finds that it is an extraordinary situation.
Now of course, if the other parent cannot be found, the Court may agree simply not to award them any visitation at all.
One that note, if they cannot be found, one would have to file stating that they do not know where they are and ask for alternative service. See HERE and scroll down to "Substituted Service."
Now the important part. As part of the suit, one would ask the Court for permission to be the only parent required to sign the passport. This is what the US Dept. wants to see. If the court agrees, the decree serves as this authorization. NOTE that since this may take a few weeks or even months, one can ask for temporary orders in the meantime to get permission as the only parent required for the passport, while the suit itself is pending.
If there is already custody between the parents but the father simply has disappeared, or, is not signing, respond and let me know, as my answer will then very.
Please note: If I tell you simply what you wish to hear, this would be unfair to you. I want to be honest with you and sometimes this means providing information that is not optimal. Negative ratings are reserved for experts who are rude or for erroneous information. Please rate me on the quality of my information; do not punish me for my honesty.
I hope this helps and clarifies. Gentle Reminder: Use the SEND or REPLY button to keep chatting, or please RATE when finished. You may always ask follow ups at no charge after rating. Kindly rate my answer as one of TOP THREE FACES/STARS and then SUBMIT, as this is how I get credit for my time with you. Rating my answer the bottom two faces/stars or failing to submit the rating does not give me credit and reflects poorly on me, even if my answer is correct. I work very hard to formulate an informative and honest answer for you; please reciprocate my good faith.