With all due respect to Greaves and I don't wish to diminish his achievements but he scored 1 goal in major tournaments and the rest were mostly against very poor opposition. Not saying Greaves was poor but how is that any better than Rooney's goals record? His hatricks were against Northern Ireland, Peru, Scotland, Norway and Luxembourg. Those sides (excluding Luxembourg) were a great deal poorer then than they are now.
Rooney - scored in 38 games
Macedonia 1
Liechtenstein 1
Denmark 2
Iceland 3
Switzerland 4
Croatia 4
Argentina 1
Netherlands 1
Estonia 2
Russia 1
Kazakhstan 3
Andorra 2
Bulgaria 2
Belarus 2
Slovakia 2
Ukraine 1
San Marino 5
Poland 2
Brazil 2
Montenegro 2
Equador 1
Uruguay 1
Norway 1
Lithuania 1
Slovenia 2
Scotland 2
France 1
Australia 1
Greaves - scored in 27 games
Peru 4
Wales 5
Yugoslavia 2
Northern Ireland 10
Luxembourg 3
Spain 2
Scotland 4
Italy 1
Austria 1
Argentina 1
Czechoslovakia 2
Rest of the World 1
Republic of Ireland 1
Brazil 1
Netherlands 1
Hungary 1
Norway 4