Step 1: Decide how sick the patient is:
The key facts: atrial fibrillation, pain out of proportion to the abdominal findings, and a stable circulation. There is no peritonitis in the stem.
So we have a high-risk patient with probable embolic acute mesenteric ischemia, who is not yet in the stage of dead bowel or shock.
Step 2: Choose the path for a stable patient:
There are two paths in this condition. One is fast surgery, and the other is a quick, accurate scan first.
Fast surgery is for patients with peritonitis, shock or other signs that bowel has already died. A stable patient has time for a scan, and the scan guides the exact treatment.
The scan used is a CT angiogram of the abdomen, done in the arterial and venous phase. It shows the clot in the artery, bowel wall changes and the extent of disease.
Step 3: Rule out the remaining choices:
Doppler: results depend on skill and on bowel gas, and it only shows the start of the SMA well. It cannot decide the plan.
Plain X-ray: usually normal in early disease. Waiting for changes on it only delays care.
Laparotomy: a major operation, which is not the first step when the patient is stable and peritonitis is absent.
Step 4: Link the answer to what follows:
Once CT angiography confirms the embolus, start heparin and plan revascularisation by embolectomy or endovascular means, with a second look at the bowel when needed.
So the correct next step is urgent CT angiography, option (2).
Final Answer:
For a stable patient with suspected embolic mesenteric ischemia, urgent CT angiography is the next step. This is option (2).
\[ \boxed{\text{Option (2)}} \]