Yet another way would be to set up a function(ref) which you just call for every line.  Initially, the called function checks for /alpha/, but once alpha is encountered, it replaces itself with the routine do_beta that is then called directly for the remaining beta lines. That way, the condition is no longer tested from there on.

my $f;

$f = sub {
    if (/alpha/) {
        do_alpha();
        $f = \&do_beta;
    } else {
        do_beta();
    }
};

while (<DATA>) {
    $f->();
}


sub do_alpha {
    print "do_alpha(): \$f = $f, \$_ = $_";
}
sub do_beta {
    print "do_beta():  \$f = $f, \$_ = $_";
}


__DATA__
beta
beta
beta
beta
alpha
beta
beta
beta
beta
beta
beta
beta
beta
[download]

Output:

do_beta():  $f = CODE(0x796e88), $_ = beta
do_beta():  $f = CODE(0x796e88), $_ = beta
do_beta():  $f = CODE(0x796e88), $_ = beta
do_beta():  $f = CODE(0x796e88), $_ = beta
do_alpha(): $f = CODE(0x796e88), $_ = alpha
do_beta():  $f = CODE(0x7c6ee8), $_ = beta
do_beta():  $f = CODE(0x7c6ee8), $_ = beta
do_beta():  $f = CODE(0x7c6ee8), $_ = beta
do_beta():  $f = CODE(0x7c6ee8), $_ = beta
do_beta():  $f = CODE(0x7c6ee8), $_ = beta
do_beta():  $f = CODE(0x7c6ee8), $_ = beta
do_beta():  $f = CODE(0x7c6ee8), $_ = beta
do_beta():  $f = CODE(0x7c6ee8), $_ = beta
[download]

As you can see, the coderef changes after alpha has been encountered.

(upd: as it seems, BrowserUk posted essentially the same idea while I wrote up my reply...)