I often see precision measurements of the tt¯Htt¯H process at the (HL-)LHC justified by the statement that "it's the only cross-section that's directly proportional to the Higgs-top Yukawa coupling
Yes, the process is g g --> t t~ h, there are at least 2 QCD couplings and the Yukawa coupling h-t, so the amplitude is proportional to the Yukawa coupling. What is the question?
Imagine two gluons which are producing a t-tbar pair each. This process creates two top quarks and two top anti quarks in total. Now imagine that one top quark and one top antiquark fuse together to form a neutral Higgs scalar. Then, in this way, the final state is: a top quark, a top anti quark, and a neutral Higgs.
A top quark, a top antiquark and a neutral Higgs couples via the top Yukawa coupling. Therefore the whole process is proportional to the top Yukawa coupling, which is large compared to other couplings in the Yukawa sector.
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