product_info$total_qty <- aggregate(col~row,which(outer(product_info$productId,transact_data$productId,`==`)&outer(product_info$beg_date,transact_data$date,`<=`)&outer(product_info$end_date,transact_data$date,`>=`),arr.ind=T),function(x) sum(transact_data$qty[x]))$col;
product_info;
##   productId old_price new_price   beg_date   end_date total_qty
## 1         A      0.50      0.70 2014-05-01 2014-05-31        25
## 2         B      0.10      0.11 2014-06-01 2014-06-31        20
## 3         A      0.11      0.12 2014-05-01 2014-05-31        25
## 4         C      0.12      0.11 2014-06-01 2014-06-31         9
## 5         C      0.30      0.20 2014-05-01 2014-05-31         2
## 6         B      0.40      0.30 2014-06-01 2014-06-31        20
Explanation
First, a logical matrix is constructed for each of the three match criteria, using outer() to compare every record in product_info with every record in transact_data. These three logical matrices are logical-ANDed together to form a final logical matrix representing which combinations of records match.
outer(product_info$productId,transact_data$productId,`==`)
&outer(product_info$beg_date,transact_data$date,`<=`)
&outer(product_info$end_date,transact_data$date,`>=`)
##       [,1]  [,2]  [,3]  [,4]  [,5]  [,6]  [,7]  [,8]  [,9] [,10] [,11] [,12]
## [1,]  TRUE FALSE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [2,] FALSE  TRUE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE
## [3,]  TRUE FALSE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [4,] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE FALSE
## [5,] FALSE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE
## [6,] FALSE  TRUE FALSE FALSE FALSE FALSE FALSE  TRUE FALSE FALSE FALSE FALSE
Then, the row and column indexes with TRUE are ascertained via a call to which() with arr.ind=T. Row indexes represent the matching records from product_info (since it was on the left of the outer() calls), and column indexes represent the matching records from transact_data.
which(...,arr.ind=T)
##       row col
##  [1,]   1   1
##  [2,]   3   1
##  [3,]   2   2
##  [4,]   6   2
##  [5,]   1   5
##  [6,]   3   5
##  [7,]   5   7
##  [8,]   2   8
##  [9,]   6   8
## [10,]   4  10
Since we want to sum qty values from transact_data for each record in product_info, we can aggregate() the col indexes grouping by row by writing a custom aggregation function to index transact_data$qty with the col indexes and sum() them to return a single value for each row.
aggregate(col~row,...,function(x) sum(transact_data$qty[x]))
##   row col
## 1   1  25
## 2   2  20
## 3   3  25
## 4   4   9
## 5   5   2
## 6   6  20
Finally, we can assign the result directly to product_info$total_qty to complete the solution.
product_info$total_qty <- ...$col;
I'm not entirely sure if it is a guarantee that aggregate() will always return its result ordered by the grouping column(s). I just asked this at Does aggregate() guarantee that the result will be ordered by the grouping columns?.
Also, I just realized that direct assignment will fail if not all records in product_info had at least one matching record in transact_data.
If either of those assumptions are violated, the solution can be fixed as follows:
product_info$total_qty <- with(aggregate(col~row,which(outer(product_info$productId,transact_data$productId,`==`)&outer(product_info$beg_date,transact_data$date,`<=`)&outer(product_info$end_date,transact_data$date,`>=`),arr.ind=T),function(x) sum(transact_data$qty[x])),col[match(1:nrow(product_info),row)]);
product_info;
##   productId old_price new_price   beg_date   end_date total_qty
## 1         A      0.50      0.70 2014-05-01 2014-05-31        25
## 2         B      0.10      0.11 2014-06-01 2014-06-31        20
## 3         A      0.11      0.12 2014-05-01 2014-05-31        25
## 4         C      0.12      0.11 2014-06-01 2014-06-31         9
## 5         C      0.30      0.20 2014-05-01 2014-05-31         2
## 6         B      0.40      0.30 2014-06-01 2014-06-31        20
Now, instead of the final step of dereferencing $col, we must construct a complete vector of length equal to the number of rows in product_info, and match() the qty sums (which are inside col) to their corresponding indexes (inside row), with a little help from with().
product_info$total_qty <- with(...,col[match(1:nrow(product_info),row)]);