如何使用AJAX执行PHP电子邮件脚本？[当前代码不工作]

<form id="emailForm">
  <input type="email" name="email" placeholder="请输入您的电子邮件" required>
  <textarea name="message" placeholder="请输入消息内容" required></textarea>
  <button type="submit">发送邮件</button>
</form>
<div id="response"></div>

document.getElementById("emailForm").addEventListener("submit", function(event) {
  event.preventDefault(); // 阻止表单默认提交行为

  var form = document.getElementById("emailForm");
  var formData = new FormData(form);

  var xhr = new XMLHttpRequest();
  xhr.open("POST", "email_script.php", true);
  xhr.onreadystatechange = function() {
    if (xhr.readyState === XMLHttpRequest.DONE && xhr.status === 200) {
      document.getElementById("response").innerHTML = xhr.responseText;
    }
  };
  xhr.send(formData);
});

<?php
if ($_SERVER["REQUEST_METHOD"] === "POST") {
  $email = $_POST["email"];
  $message = $_POST["message"];

  // 执行电子邮件发送逻辑，这里只是一个示例
  $to = "recipient@example.com";
  $subject = "新邮件";
  $body = "发件人：" . $email . "\n\n" . $message;
  $headers = "From: sender@example.com";

  if (mail($to, $subject, $body, $headers)) {
    echo "邮件发送成功！";
  } else {
    echo "邮件发送失败！";
  }
}
?>