While more specifics would be helpful, I think that I understand what you are asking and can give you at least a partial reply.
You have an LLC that is engaged in two lines of business. For clarity of exposition I will call those lines of business: Widget selling (the original line of business), and the "new service" that the LLC intends to get in the business of selling under a different trade name.
It appears that since you call it "my LLC" that the LLC is currently a single member LLC owned by you which is disregarded for income tax purposes and treated as a sole proprietorship whose income from Widget selling you report on Schedule C or C-EZ to your personal form 1040.
I assume that this is a limited liability company in the United States since this is the only jurisdiction that I am aware of where the name LLC is commonly used for a form of business entity.
General Legal Background Regarding LLCs
A limited liability company in the United States is governed for tax purposes by check the box regulations.
The default rule for an LLC formed under U.S. law is that a single member LLC is disregarded for tax purposes (and is treated as a sole proprietorship reporting income on the owner's Schedule C or C-EZ to form 1040 when the owner is a natural person) and that multi-member LLCs are taxed under Subchapter K of the Internal Revenue Code as partnerships (which file a form 1065 each issue and issue a form K-1 to each partner in connection with doing so which the partner reports on their form 1040 were indicated by the form K-1).
An LLC can also elect by "checking a box" to be taxed instead as a C corporation, but if you are going to do that, outside rare instances where state law prohibits businesses from being organized as corporations (e.g. farms in certain Great Plains states) or there is already an established identity as an LLC and you just want to change the tax treatment without informing the public of that fact, it is usually easier to form a corporation rather than an LLC in the first place, rather than checking the box.
In a limited liability company in the United States, equity owners are called "members" rather than partners or shareholders. Anyone who is an owner of the business or a part of the business, is a member of the LLC for state law purposes, by definition, and is a partner in the company for federal income tax purposes.
Special Allocations in LLCs
An LLC with multiple members, unlike an S corporation (which is required to divide all profits and losses pro-rata), is permitted under federal income tax laws to make what is called a "special allocation" of profits and losses from the entity. Some of the most common reasons this is done are:
(1) to treat distributions returning originally invested capital differently than distributions once all invested capital has been returned to the original investors,
(2) for the similar purpose of giving a member (often an active manager with little personal wealth) an interest in profits but not invested capital, or
(3) to allow members to have different interests in different lines of business (as you appear to be proposing).
Special Allocations In This Case
In the case you present, you could draft an operating agreement for the LLC which contained a special allocation that allocated all profits and losses from the Widget selling line of business to you, and allocated all profits and losses from the new service line of business 60% to you and 40% to your new partner. You would have to figure out to handle situations when one line of business had a loss or was unable to pay its debts from assets associated with its line of business, and the other had a profit or was at least not insolvent.
Your partner in the new service line of business would become a member of the LLC and the LLC would now have to file a form 1065 each year.
You would also have to decide under the operating agreement what matters could be decided by you unilaterally and what matters would require the approval of both of you.
For example, your new partner would probably need the assurance that his percentage interest in profits from the new service line of business would require mutual consent and could not be decided unilaterally by you, but day to day management decisions in the business might be decided by you unilaterally.
Why This Is A Horrible Idea
This said, while you could do that, the structure of the transaction you are contemplating is horribly ill advised.
Higher Transaction Costs
It is very expensive and very complex to draft an operating agreement that accomplishes what you are contemplating. But, it is very cheap and very easy, to form a new LLC for the new service line of business owned 60-40 by you and your partner, and to leave the existing LLC in the Widget line of business undisturbed.
I would probably charge something like $6,000-$7,000 to draft the combined operating agreement, but would probably charge something more like $1,500-$2,000 to set up a new LLC plus the nominal filing fee for a new entity.
The tax returns for the two separate businesses each year would also be profoundly less complicated if you had two separate businesses than if they were a single entity that had to report its income each year on a single form 1065. You might even be able to do your own tax returns if they were two separate entities, while there is no humanly possible way that a non-tax professional without specialized experience in partnership taxation could do the tax returns correctly for the combined entity.
Usually, the only people who are concerned about setting up new LLCs are people in Delaware and California that have fairly significant minimum annual taxes for LLCs, with California's franchise tax board being particularly ruthless in this regard. (Most states charge no minimum fee or tax for merely having an LLC or only a nominal amount like $10-$50 to file an update of contact information report each year.) But, the dollar amount of the annual minimum taxes even in those two jurisdictions would be totally overwhelmed by the additional CPA charges you would need to incur to file returns for them each year and the additional legal fees you would incur over the lifetime of the entity.
Liability and Governance In Combined v. Separate LLCs
If you had two separate LLCs, the assets and income flow of the Widget line of business would not be subject to liability incurred in the new service line of business and visa versa.
Also, your partner in the new service line of business would not have a right to information about your Widget line of business as your partner would if the two were combined in one LLC, and would not have any legal rights to allege that you were mismanaging the Widget line of business as your partner would if the two were combined in one LLC.
Risk Of Forming An Unlimited Liability General Partnership
If you didn't treat the new partner as a member of the existing LLC in the operating agreement and otherwise, you would also have made a grave error.
This is because then you would have formed a general partnership without limited liability protection between your LLC and your new partner, and you would have to comply with special tax rules governing cases in which one partner has unlimited liability for partnership debts (the new partner) and the other (you via the LLC) has limited recourse for partnership debts. The tax laws that apply in this hybrid situation are insanely complicated.
Also, you are needlessly exposing your partner to unnecessary liability for all debts of the company if the business fails in this situation (and your partner may sue you for failing to disclose this liability if you indeed do fail to disclose this risk).
Your desire to avoid forming a new LLC is penny wise and pound foolish. It makes far more sense to achieve your objectives by forming a new LLC for the new service line of business and to leave your existing LLC as it is.