string.l

%{
  #include ... // your includes

  // ensure that string_builder is initialized within the scanner, otherwise reading string tokens will fail
  static string_builder_t *sb = sb_init();
%}

...the rest of your lexer stuff

%%
<STRING>{

\" {
  BEGIN(INITIAL);
  if (!sb_is_empty(sb)) {
    yytext = sb_build_str(sb);
    sb_reset(sb);
  }

  return yy::Parser::token::T_STR;
}

{string_char} {
  if (YYText()[0] != '\0') {
    sb_append(sb, YYText());
  }
}

\\t  { sb_append(sb, yytext); }
\\n  { sb_append(sb, yytext); }
\\r  { sb_append(sb, yytext); }
\\b  { sb_append(sb, yytext); }
\\f  { sb_append(sb, yytext); }
\\'  { sb_append(sb, yytext); }
\\\" { sb_append(sb, yytext); }
\\   { sb_append(sb, yytext); }

<<EOF>> {
  // in case if string is unterminated and there's EOF
  fprintf(stderr, "unterminated string on line %d\n", yylineno);
  BEGIN(INITIAL);
  warning_count++;
}
  
{newline} {
  fprintf(stderr, "unterminated string on line %d\n", yylineno);
  BEGIN(INITIAL);
  warning_count++;
}


}