diff --git a/planner/core/rule_eliminate_projection.go b/planner/core/rule_eliminate_projection.go
index 83936e7cd9e94..1459896f30b01 100644
--- a/planner/core/rule_eliminate_projection.go
+++ b/planner/core/rule_eliminate_projection.go
@@ -144,16 +144,7 @@ func eliminatePhysicalProjection(p PhysicalPlan) PhysicalPlan {
 		}
 	})
 
-	oldSchema := p.Schema()
 	newRoot := doPhysicalProjectionElimination(p)
-	newCols := newRoot.Schema().Columns
-	for i, oldCol := range oldSchema.Columns {
-		oldCol.Index = newCols[i].Index
-		oldCol.ID = newCols[i].ID
-		oldCol.UniqueID = newCols[i].UniqueID
-		oldCol.VirtualExpr = newCols[i].VirtualExpr
-		newRoot.Schema().Columns[i] = oldCol
-	}
 	return newRoot
 }
 
diff --git a/planner/core/stats.go b/planner/core/stats.go
index 81b09494aad40..be1c66d5528c4 100644
--- a/planner/core/stats.go
+++ b/planner/core/stats.go
@@ -1497,7 +1497,7 @@ func (p *LogicalCTE) DeriveStats(_ []*property.StatsInfo, selfSchema *expression
 				return nil, err
 			}
 		}
-		recurStat := p.cte.recursivePartPhysicalPlan.Stats()
+		recurStat := p.cte.recursivePartLogicalPlan.statsInfo()
 		for i, col := range selfSchema.Columns {
 			p.stats.ColNDVs[col.UniqueID] += recurStat.ColNDVs[p.cte.recursivePartLogicalPlan.Schema().Columns[i].UniqueID]
 		}
diff --git a/planner/core/testdata/plan_suite_out.json b/planner/core/testdata/plan_suite_out.json
index 3a7507f4a3b35..50a54ccba0596 100644
--- a/planner/core/testdata/plan_suite_out.json
+++ b/planner/core/testdata/plan_suite_out.json
@@ -2225,8 +2225,8 @@
       {
         "SQL": "with recursive cte(a) as (select 1 union select a from cte) select * from cte; -- recursive cte cannot be inlined",
         "Plan": [
-          "CTEFullScan 1.00 root CTE:cte data:CTE_0",
-          "CTE_0 1.00 root  Recursive CTE",
+          "CTEFullScan 2.00 root CTE:cte data:CTE_0",
+          "CTE_0 2.00 root  Recursive CTE",
           "├─Projection(Seed Part) 1.00 root  1->Column#2",
           "│ └─TableDual 1.00 root  rows:1",
           "└─CTETable(Recursive Part) 1.00 root  Scan on CTE_0"