Influence of herbicide use on GHG emissions can be from two main reasons:
1. GHG emissions due to the manufacture of herbicides (resulting from energy consumption in the production process). The report below provides these estimates for a wide variety of compounds:
Audsley et al. (2009) Estimation of the greenhouse gas emissions from agricultural pesticide manufacture and use available at: https://dspace.lib.cranfield.ac.uk/bitstream/1826/3913/1/Estimation_of_the_greenhouse_gas_emissions_from_agricultural_pesticide_manufacture_and_use-2009.pdf
2. GHG emissions due to the herbicide transport (from manufacturing plant to farmgate) and application (machinery use for application). The paper below is useful in this aspect:
Lal, R. 2004. “Carbon Emission from Farm Operations.” Environment International 30(7): 981–990. (available at: http://www.sciencedirect.com/science/article/pii/S0160412004000832)
I am not aware of any study that explains the influence of herbicide persistence in the soil on GHG emissions.
The paper below may be useful (in general terms) to understand what is the impact of herbicide use on GHG emissions from the production of two major cereal crops
Jiang et al. (2015) Application of herbicides is likely to reduce greenhouse gas (N2O and CH4) emissions from rice–wheat cropping systems (available at: http://www.sciencedirect.com/science/article/pii/S135223101500148X