Sigh

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Middle English (as a verb): probably a back-formation from sighte, past tense of siche, sike, from Old English sīcan .

wiktionary

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From Middle English sighen, syghen, syȝȝen, (also syken, sychen, syghten, etc.), from Old English sīcan(“to sigh, moan, groan”), from Proto-West Germanic *sīkan, perhaps ultimately from Proto-Indo-European *seykʷ-(“to pour out”). The Middle English infinitive forms in ȝ/gh are backformations of the past tense forms sighte, siȝhte, isiȝt, etc. Doublet of sike.

etymonline

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sigh (v.)

mid-13c., probably a Middle English back-formation from sighte, past tense of Old English sican "to sigh," perhaps echoic of the sound of sighing. Related: Sighed; sighing.

sigh (n.)

early 14c., from sigh (v.).