Serving notice period is mandatory as the appointment letter is a contract between the employer and employee. All the terms and conditions must have been accepted by you at the time of joining, by signing on the duplicate appointment letter, which will be in your personnel file maintained by HR. My suggestion is to either convince your HR guys in relieving you by waiving off the notice period or you may waive the notice period by paying the one month salary. It is also not good to abandon the employment as it will affect all the future background checks that may be carried out by your prospective employers, which you may take up in future.
Immediately talk to your Old HR Department and try to buy back your notice period. Ask them to deduct the salary for the one month period or pay a draft/cheque for the remaining notice period. Do the proper handing over to your successor who is going to take over your responsibility. Go friendly and explain about your position to your line manager and to HR manager. Surely they will understand and help you out.
All the best