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The process to create a new column in a Pandas dataframe by utilizing two columns concurrently from a distinct dataframe is as follows:

1. First, ensure that both dataframes have at least one column in common.
2. Merge the two dataframes using the common column as the merge key. This can be done using the merge() function from Pandas.
3. Once merged, you can access the columns from both dataframes using the merged dataframe. You can create a new column in the merged dataframe by applying a function to the two columns from both dataframes.
4. To create a new column in the original dataframe, you can use the join() function to add the column from the merged dataframe to the original dataframe.

Here is an example:

import pandas as pd

# Create the first dataframe
df1 = pd.DataFrame({'A': [1, 2, 3, 4], 'B': [5, 6, 7, 8]})

# Create the second dataframe
df2 = pd.DataFrame({'A': [1, 2, 3, 4], 'C': [9, 10, 11, 12]})

# Merge the two dataframes using the common column 'A'
merged_df = pd.merge(df1, df2, on='A')

# Create a new column in the merged dataframe by applying a function to columns 'B' and 'C'
merged_df['D'] = merged_df['B'] + merged_df['C']

# Add the new column 'D' from the merged dataframe to the original dataframe 'df1'
df1 = df1.join(merged_df['D'])

# Print the result
print(df1)

This will output:

   A  B   D
0  1  5  14
1  2  6  16
2  3  7  18
3  4  8  20