answered on 15/05/2018
B likely cannot sue A.
However, C may be able to sue A depending on the contents of the email.
First, C may claim against A under the tort of defamation (specifically libel) provided the elements of defamation can be proven: generally, the statement must harm the reputation of the victim, the statement must refer to the victim, and must have been communicated to a third party (B in this case). There are defences available such as fair comment and justification. Additionally, defamation is a criminal offence under section 499 of the Penal Code.
Second, C may also claim against A for malicious falsehood if A has maliciously made false representations with a view to injuring A's goodwill or economic reputation. C would have to prove that he/she has suffered loss as a result, such as loss of sales.
Third, C may be able to sue A under the economic tort of inducing breach of contract if B has breached a contractual obligation owed to A as a result of the email. Specifically, it must be proven that A knows of an existing contractual relationship between B and C, and that A has intended the breach of the obligation. The email must be a direct and effective cause of B's breach, and damage must have resulted from the breach of contract.