My wife recently started becoming very distant, and over Christmas went by herself to visit her brother in a different country than where I live. I took this as betrayal to our home and child,
Why in the world would you view your W going to visit family as "betrayal"? Is it your opinion that she is not "allowed" any time for herself? That's a HUGE problem. In healthy marriages both spouses have their independent lives as well as their married life together. Are you denying your wife her independence?
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because she left on the 26th of December and came back on the 3rd of January, leaving me and our child alone to spend the New Year's.
You make it sound like spending a week alone with your child is a prison sentence. Do you feel the child is totally your wife's responsibility?
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After she came back I confronted her
Confronted her about what? Taking a week off to visit family?
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I am willing to set things straight and do whatever it takes to win her back.
Have you read DR? That's step 1. Also read Sandi2's 37 Rules daily, and live those rules.
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and I was entirely and completely unaware that I was causing her so much pain.
Please describe what you did in the M that caused her pain.
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I need help. What can I do?
Read DR and other R books. Determine what you did wrong in the M. Do 180's on those things. Give your W time and space. Change, and be consistent with your changes. Show her the spouse (and father) only a fool would leave.