The first question, which we will only touch upon, is whether wide shoulder pieces are necessary to connect the two sides (front and back) of the tzitzit. The earliest stringent source on the matter is the Maharil, quoted by the Magen Avraham.[2] The simple reading of this source indicates that if the neck hole is bigger than either one of the shoulder pieces, the tallit katan is considered two separate small garments with two sets of tzitzit each, and it is therefore invalid. The Chazon Ish[3] understands the Maharil this way and suggests that one should try to follow his opinion. The Machatzit HaShekel[4] argues that it suffices that the shoulder pieces be wide enough to be deemed real parts of the garment and not just thin connectors; they do not have to be wider than the hole.
There are also questions of what counts toward the minimum size of a garment. Do the front and back parts combine, or must each one separately be the minimum size? What impact does the hole for the head have? The Mishna Berura[5] posits that the hole for the head does not count, and he seems to assume that the requisite size is necessary in both the front and back. If the hole is relatively small in comparison to the shoulder pieces, it is easier to contemplate counting that section or at least combining the front and the back sections. In any event, we will assume the worst case scenario – that the remaining width of the shoulder piece that is not ripped would not be enough to create a kosher tallit katan garment.
The question that arises when repairing a tallit katan is whether the principle of ta’aseh v’lo min he’asuy applies. This rule means that something that the Torah says to make, such as tzitzit, must be turned into a halachic entity by a direct action; it cannot come about indirectly through a situation that emerged incidentally. A classic application is when one attaches three tzitzit fringes to a three-cornered garment (which is not required to have tzitzit) and only afterward forms a fourth corner. In that case, we disqualify the three existing tzitzit fringes because they were not made into halachic tzitzit, but rather became significant indirectly when the part of the garment to which they were attached later required tzitzit.[6]
Let us consider a tallit that was severed into two, with each side retaining two tzitzit fringes and with each side including enough fabric to be considered a valid tallit that requires tzitzit. The Shulchan Aruch[7] writes that ta’aseh v’lo min he’asuy is not a problem here. The logic is that all the tzitzit were made properly, as they were required at the time they were tied. The Taz[8] says that if one reattaches the garment, the tzitzit of only one side are considered valid, whereas the part that is considered reattached needs to have its tzitzit removed and redone. According to the Magen Avraham,[9] however, nothing has to be redone when the garment is reconnected, as the situation is just a return to the garment as it used to be with properly attached tzitzit. The Acharonim infer that if, when severed in two, a tallit katan turned into two pieces in a manner that neither constituted a halachic tallit katan garment (which is usually the case according to some opinions), all the tzitzit must be redone.[10]
Our case is more lenient. First, only one side is affected, and there is a machloket among Acharonim when a tallit katan is not severed into two separate garments but instead remains connected on one of the two sides.[11] Furthermore, there is a machloket regarding whether to view a garment that is ripped a majority of the way but is still somewhat connected as being severed or not. The Lechem Mishneh and Mishneh LaMelech write that even a small amount of connection is sufficient, whereas the Artzot HaChayim,[12] who cites them, says a majority connection is required. In your case, the rip is not significant and is unlikely to be a majority of the fabric. On the other hand, when one adds the rip to the significant opening for the head, it may not leave enough connection.
Nevertheless, there is a widely accepted leniency that applies regarding this point.[13] The Chazon Ish[14] says that if the shoulder pieces of a tallit katan are clearly intact but are somewhat ripped, the ripped part counts toward the amount of garment needed to connect the sides, despite the hole for the head. Thus, the tallit katan can halachically be used as is. Consequently, sewing it (which is a good idea if for no other reason than to prevent further ripping) introduces no problem of ta’aseh v’lo min he’asuy.