At a loss

detriment, disadvantage, or deprivation from failure to keep, have, or get:
to bear the loss of a robbery.
something that is lost:
The painting was the greatest loss from the robbery.
an amount or number lost:
The loss of life increased each day.
the state of being deprived of or of being without something that one has had:
the loss of old friends.
death, or the fact of being dead:
to mourn the loss of a grandparent.
the accidental or inadvertent losing of something dropped, misplaced, stolen, etc.:
to discover the loss of a document.
a losing by defeat; failure to win:
the loss of a bet.
failure to make good use of something, as time; waste.
failure to preserve or maintain:
loss of engine speed at high altitudes.
destruction or ruin:
the loss of a ship by fire.
a thing or a number of related things that are lost or destroyed to some extent:
Most buildings in the burned district were a total loss.
Military.

the of soldiers by death, capture, etc.
Often, losses. the number of soldiers so .

Insurance. occurrence of an event, as death or damage of property, for which the insurer makes indemnity under the terms of a policy.
Electricity. a measure of the power lost in a system, as by conversion to heat, expressed as a relation between power input and power output, as the ratio of or difference between the two quantities.
at a loss,

at less than cost; at a financial loss.
in a state of bewilderment or uncertainty; puzzled; perplexed:
We are completely at a loss for an answer to the problem.

noun
the act or an instance of losing
the disadvantage or deprivation resulting from losing: a loss of reputation
the person, thing, or amount lost: a large loss
(pl) military personnel lost by death or capture
(sometimes pl) the amount by which the costs of a business transaction or operation exceed its revenue
a measure of the power lost in an electrical system expressed as the ratio of or difference between the input power and the output power
(insurance)

an occurrence of something that has been insured against, thus giving rise to a claim by a policyholder
the amount of the resulting claim

at a loss

uncertain what to do; bewildered
rendered helpless (for lack of something): at a loss for words
at less than the cost of buying, producing, or maintaining (something): the business ran at a loss for several years

n.

Old English los “loss, destruction,” from Proto-Germanic *lausa- (see lose). The modern word, however, probably evolved 14c. with a weaker sense, from lost, the original past participle of lose. Phrase at a loss (1590s) originally refers to hounds losing the scent. To cut (one’s) losses is from 1885, originally in finance.
Below cost, as in The store was doing so badly that it was selling merchandise at a loss.
Puzzled, perplexed, in a state of uncertainty, as in When his letters were returned unopened, John was at a loss as to what to do next. This usage was originally applied to hounds who had lost the scent or track of their prey. [ Mid-1600s ]
at a loss for words. Unable or uncertain as to what to say. For example, Father’s tirade left us all at a loss for words. [ Late 1600s ]
see:

at a loss
cut one’s losses
dead loss