To create a temporal profile that models purchase to payment processes in a retail business setting, we need to consider various activities involved such as ordering goods ('Order'), waiting for approval ('Approval'), processing financial details ('Processing'), and then completing the transaction ('Pay').

The time spent on each task can vary based on factors like availability of resources within the system/database, approval process standards set by internal policy documents (quickly approved orders vs those requiring extensive review), or external dependencies such as bank account verification times.

Let's define our hypothetical temporal profile where 'A' represents an order action, 'B' represents Approval Action, 'C' Processing Action ('D') Payment Action:

Here we can make assumptions based on the time taken commonly observed:

1. **'Order to Approval (A-B)'**: This might typically take anywhere between minutes to hours depending on how quickly orders are usually reviewed and approved in practice.

2. **'Approval to Processing (B-C)'**: Assuming that the review process is quick compared to the rest of operations involved.

3. **'Processing to Payment (C-D)'**: Time taken depends heavily on the backend processing systems utilized by retailers but considering it may vary depending on when and how transactions are handled electronically.

Assuming each of these transitions has a standard deviation to account for any irregularities:

Temporal Profile = { ('A', 'B'): (.avgTime, stdDev), 
                    ('B', 'C'): (.avgTime, stdDev),
                    ('C', 'D'): (.avgTime, stdDev) }

For instance,

{'Order': ('A', ('86400', 'PT24H')),  # Assuming average days to approve order is roughly equivalent to one day (~86400 seconds)
         'Approval': ('B', ('3600', 'PT1H')),   # Approval process takes approximately hourly (~3600 seconds),
         'Processing': ('C', ('PT1H', '1H')),    # Processing the order might take around one hour,
         'Payment': ('D', ('PT6H', 'PH6H'))}    # Time to complete payment process might average roughly half day or so (~6 hours) with some variation. 

In this hypothetical case:
- 'A' -> 'B': AVG of approximately a day (86400 seconds), STDEV indicating acceptable fluctuation around one day limit.
- 'B' -> 'C': AVG typically hourly (~3600 seconds), STDEV for variability in the approval process time.
- 'C' -> 'D': AVG roughly half-day (~PT1H) and ~PH6H for the entire process encompassing order processing to payment completion.

Remember that these are approximations based on common practices but specifics may vary greatly depending on actual policies, regulations governing transactions, systems used etc., as well as the specific business operation details.

A real-world model would require detailed data about activities involved in purchase-to-pay processes including their execution time and how they inter-relate through an entire workflow for multiple samples over different periods of observation.