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and X$_{1}$ ( x ) is the QCD correction. The expression for X$_{1}$ ( x ) is given in [56]. The function X can also be written as 

Here η$_{X}$ represents the NLO corrections. 

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The function X l NL is the function corresponding to X ( x$_{t}$ ) in the charm sector. It results from the NLO calculations and its explicit form is given in [60, 69]. 

The branching fraction of K + → π $^{+}$ν ¯ ν can be written as follows 

where 

and r$_{K}$$_{+}$ = 0 . 901 summarizes the isospin breaking corrections in relating the K + → π $^{+}$ν ¯ ν to the well measured leading decay K + → π $^{0}$e$^{+}$ν . 

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