Can the up quark still be massless? It used to be commonly discussed that the bare mass of the up quark can be $0$. This was because we can't observe its effect directly. To my knowledge the up quark can only be measured by its effect on the pion mass due to chiral perturbation theory, but relating the mass to the up quark mass results in some large uncertainties (see for example here for a discussion on this). However, I can't find any recent reference still discussing this possibility. Is this still allowed experimentally?
 A: This admittedly, isn't much of an answer, as I'm merely repeating information from the Particle Data Group page about the up-quark, which I consider up-to-date. Their current combination is that $m_u = 2.3^{+0.7}_{-0.5}\,\text{MeV}$, but they warn that

The $u$-, $d$-, and $s$-quark masses are estimates of so-called "current-quark masses," in a mass-independent subtraction scheme such as MS-bar. The ratios $m_u/m_d$ and $m_s/m_d$ are extracted from pion and kaon masses using chiral symmetry. The estimates of $d$ and $u$ masses are not without controversy and remain under active investigation. Within the literature there are even suggestions that the $u$-quark could be essentially massless

Naively, then, the up-quark isn't massless (for a reasonable level of agreement with data), but there are indeed complications and caveats, which mean that the idea that it might be massless is alive.
A: My own impression is that the ball got stuck in someone' roof, but I am partial because some of my numerology did coincide with the relationship $(m_u,m_d,m_s) \propto (0, 2 - \sqrt 3, 2+\sqrt 3)$ proposed by Harari Haut Weyers (1978) (presented by Harari here) when trying to find some first-principled calculation of Cabibbo angle.
My understanding is that for most calculations it is possible to work with an initially massless up quark and then to propose that it gets a initial correction $$\delta m_u \propto {m_d m_s \over \Lambda_\chi} $$ with $\Lambda_\chi$ about the chiral scale or pion mass, but that the concrete mechanism is unknown, some instanton effect of any non perturbative thing. Of course the same correction for the permuted quarks is almost zero, because the up quark enters the product. 
The instanton argument appears in Georgi, Ian McArthur 
HUTP-81/A011 (http://inspirehep.net/record/164546?ln=en) and is also more detailed in Phys.Rev.Lett.61:794,1988 and Nucl.Phys.B383:58-72,1992. After that, nothing. I can not see if the modern lattice calculations really address the question of which part of the mass comes from the instanton and which part comes from the bare lagrangian; most of them refer to JHEP 9911:027,1999. And some of them leave the issue open.
A recent revisit to the topic is Dine et al, http://arxiv.org/abs/1410.8505 (pointed to me by Ohwilleke in physicsforums). 
