watchers for form input validation in Vue.js don’t work

My page is displaying user data, which he can edit.
I’m working with views, on a small project. I want to do form validation with regex, but nothing happens. Exemple, When I write an email that does not respect the syntax of a regex no message is displayed.
When the regex is valid the validation message also does not appear.

<script>
import UsersDataService from "../Service/UsersDataService";
import VueJwtDecode from "vue-jwt-decode";
export default {
  name: "ProfilConnect",
  data() {
    return {
      user: {},
      firstname: "",
      lastname: "",
      email: "",
    };
  },

  watch: {
    email(email) {
      this.email = email;
      this.validateEmail(email);
    },
  },

  methods: {
    validateEmail(email) {
      //**** Regex pour l'email***/
      if (
        /^[a-zA-Z0-9.!#$%&’*+/=?^_`{|}~-]+@[a-zA-Z0-9-]+(?:.[a-zA-Z0-9-]+)*$/.test(
          email
        )
      ) {
        //******Message d'alerte******/
        this.msg["email"] = "Valid";
      } else {
        this.msg["email"] = "Email no valid";
      }
    },
    updateProfil() {
      let data = {
        firstname: this.user.firstname,
        lastname: this.user.lastname,
        email: this.user.email,
      };
      UsersDataService.putUser(data);
      this.$store
        .dispatch("logout")
        .then((response) => {
          this.user.email = this.email;
          this.$router.push("/login");
          console.log("Data: ", response.data);
        })
        .catch((error) => {
          console.error("Something went wrong!", error);
        });
    },
  
  },

  mounted() {
    this.getProfilUser();
    this.email = "";
  },
};
</script>
                    
            
  <form class="background-style">
                      <div class="form-group">
                        <label class="form-label">Prénom</label>
                        <input
                          type="text"
                          class="form-control"
                          :value="user.firstname"
                        />
                      </div>
                      <div class="form-group">
                        <label class="form-label">Nom</label>
                        <input
                          type="text"
                          class="form-control"
                          :value="user.lastname"
                        />
                      </div>

                      <div class="form-group">
                        <label for="email" class="form-label">Email</label>

                        <input
                          type="email"
                          class="form-control"
                          :value="user.email"
                        />
                      </div>

                      <div class="btn rounded p-1">
                       

                        <button
                          class="rounded p-2"
                          @click.prevent="updateProfil"
                        >
                          Update
                        </button>
                      </div>
                    </form>