We report on variability and correlation studies using multiwavelength observations of the blazar Mrk 421 during the month of February, 2010 when an extraordinary flare reaching a level of $sim$27~Crab Units above 1~TeV was measured in very-high-energy (VHE) $gamma$-rays with the VERITAS observatory. This is the highest flux state for Mrk 421 ever observed in VHE $gamma$-rays. Data are analyzed from a coordinated campaign across multiple instruments including VHE $gamma$-ray (VERITAS, MAGIC), high-energy (HE) $gamma$-ray (Fermi-LAT), X-ray (Swift}, RXTE, MAXI), optical (including the GASP-WEBT collaboration and polarization data) and radio (Metsahovi, OVRO, UMRAO). Light curves are produced spanning multiple days before and after the peak of the VHE flare, including over several flare `decline epochs. The main flare statistics allow 2-minute time bins to be constructed in both the VHE and optical bands enabling a cross-correlation analysis that shows evidence for an optical lag of $sim$25-55 minutes, the first time-lagged correlation between these bands reported on such short timescales. Limits on the Doppler factor ($delta gtrsim 33$) and the size of the emission region ($ delta^{-1}R_B lesssim 3.8times 10^{13},,mbox{cm}$) are obtained from the fast variability observed by VERITAS during the main flare. Analysis of 10-minute-binned VHE and X-ray data over the decline epochs shows an extraordinary range of behavior in the flux-flux relationship: from linear to quadratic to lack of correlation to anti-correlation. Taken together, these detailed observations of an unprecedented flare seen in Mrk 421 are difficult to explain by the classic single-zone synchrotron self-Compton model.
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