It is 'branchy' and lies low evading the lawn mower. Do you know what it is?
It looks like creeping buttercup (Ranunculus repens) to me. If you get any yellow flowers, that would confirm the identification.
See https://www.lawnweeds.co.uk/weed-identification/creeping-buttercup for some pictures.
It spreads rapidly by putting out runners, so getting rid of it "without using chemicals" is not easy. A broad-leaf lawn weedkiller should work but you may need more than one application.