Investors have wrung their hands over the last several weeks over the effect of lower oil prices on the broader S&P 500, but the relationship between the two is actually starting to break down.

Crude prices had dropped more than 10 percent in the trading week ended Dec. 12. That was largely responsible for a 3.5 percent drop in the S&P 500, as investors fled stocks over concerns about energy-sector bonds, corporate earnings, and expectations for world economic demand.

That seemed to change Thursday. The S&P 500 surged while oil fell, a potential change in sentiment among investors looking to focus on sectors that may benefit from an accelerating U.S. economy.

"The proof is that oil turned down and the market said, 'Oh, that was yesterday's news, today we're moving ahead,'" said Quincy Krosby, market strategist at Prudential Financial in Newark, New Jersey.

Bank of America Merrill Lynch credit strategist Hans Mikkelsen credited the decoupling partly to Fed Chair Janet Yellen's Wednesday news conference.

"She explained how declining oil prices are expected to be a net positive for the U.S. economy. Furthermore, she went out of her way to dismiss any downward pressure on inflation as transitory."

Investors may have already priced in the effect of cheaper oil on energy-sector earnings and are now starting to weigh the positives for other sectors.

In its 2015 global outlook, fund manager Pimco said the fall in energy costs, because it is largely supply-driven, should ultimately help growth in major economies, including the United States, Japan, and the euro zone.

Fourth-quarter energy-sector earnings are expected to decline 19.2 percent from a year ago; on October 1, growth of 6.6 percent was expected.

"You will see some pain in the short term because of fourth quarter earnings," said James Liu, global market strategist at JPMorgan Funds in Chicago. "So the broad S&P 500 will take a hit based on that, but over the next several quarters it is clearly going to be a good thing."

As recently as Tuesday, the 10-day correlation between the S&P 500 .SPX and Brent crude LCOc1 stood at 0.97, meaning each moved in almost perfect sync with the other. The correlation has been breaking down and last stood at 0.42, with Brent stumbling 3.1 percent, while the S&P 500 surged 2.4 percent, on Thursday.

According to data from S&P, energy .SPNY has fallen to a market share representation of 8.31 percent, from 9.7 percent at the end of the third quarter, as names such as Denbury Resources (DNR.N), Nabors Industries (NBR.N) and Halliburton (HAL.N) have each tumbled more than 35 percent.

With investors hoping oil prices have at least stabilized as Brent hovers around the $60 mark, selling pressure could resume on equities if the downward march for oil begins again, weighing on the broader S&P index and tightening the correlation.