## Weinberg on the anomaly

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Does anybody have more details about the argument developed in\nWeinberg, v II p 381-382 (around formula 22.3.39) where it is argued\nthat "the anomalies in a given set of symmetries therefore are\nunaffected by the possible presence of fermions with a mass allowed by\nthese symmetries"?\n\nWeinberg does not give bibliography for this part of the topic; any\npointer?\n\nIf I understand it well, it means that if for instance the top mass is\nallowed by the fundamental symmetry group, then the top could not\ncontribute to anomalies in the fundamental theory and, by \'t hooft\nprinciple, it could not contribute neither to anomalies in effective\ntheories built upon the fundamental.\n\nI wonder if I could then to conclude that mesons of the third family\nare unaffected by the chiral anomaly.\n\nYours,\n\nAlejandro\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Does anybody have more details about the argument developed in
Weinberg, v II $p 381-382$ (around formula 22.3.39) where it is argued
that "the anomalies in a given set of symmetries therefore are
unaffected by the possible presence of fermions with a mass allowed by
these symmetries"?

Weinberg does not give bibliography for this part of the topic; any
pointer?

If I understand it well, it means that if for instance the top mass is
allowed by the fundamental symmetry group, then the top could not
contribute to anomalies in the fundamental theory and, by 't hooft
principle, it could not contribute neither to anomalies in effective
theories built upon the fundamental.

I wonder if I could then to conclude that mesons of the third family
are unaffected by the chiral anomaly.

Yours,

Alejandro



Al.Rivero@gmail.com wrote: > I wonder if I could then to conclude that mesons of the third family > are unaffected by the chiral anomaly. The generations are the degeneracy of the charge spectrum. Therefore, each one is a clone in charge space and exactly the same cancellations applies to each. The only essential restriction requires is that a complete copy of the first generation appear in the 2nd and 3rd. So, for instance, if there's a bottom, there'd have to be a top. The constraint to remove the anomaly corresponds to $Tr(\gamma^5 Y_a{Y_b,Y_c}) = 0,$ where the Y's are the gauge generators. The SU(2) and SU(3) generators automatically fall through correctly; as does anything else that is parity-symmetric. The only culprit is the hypercharge Y. In fact, if you start out with just SU(2) and SU(3) and ask what other c(generation-independent) charges can fit with these; you get one of two results: (1) If neutrinos are represented as Weyl particles, with a missing right-neutrino; left-antineutrino sector; then you get the hypercharge (2) If neutrinos are represented as Dirac particles, including an (as-of-yet unseen) right-neutrino; left-antineutrino; then you get the hypercharge AND the (baryon - lepton) quantum number. Calling $G = 1/2$ (baryon - lepton), and Y = hypercharge (scaled to -1 for the right-electron), you find the culprit behind the result: $Y - G$ behaves as the right-handed analogue of isospin and particles naturally arrange into 4 classes given by $(Y - G) = +/- 1/2; I_3 = +/- 1/2$. So, the anomaly removal has a tendency to drive SU(2) into a "protective envelope" algebra U(2) which contains the extra generator Y $- G;$ or into the larger parity-symmetric algebra $SU(2)_L x SU(2)_R,$ with $Y - G = I3_R$. In other words, the anomaly is basically forcing a parity symmetric algebra to re-emerge. It's of interest to note that the charge generator Q (electric charge) which remains after symmetry breaking will then take on the revealing form $Q = I3 + Y = I3 + I3_R + G;$ i.e., a parity symmetric residue of a possible $SU(2)_R x SU(2)_L$.



markwh04@yahoo.com ha escrito: > Al.Rivero@gmail.com wrote: > > I wonder if I could then to conclude that mesons of the third family > > are unaffected by the chiral anomaly. > > The generations are the degeneracy of the charge spectrum. Therefore, > each one is a clone in charge space and exactly the same cancellations > applies to each. The only essential restriction requires is that a > complete copy of the first generation appear in the 2nd and 3rd. So, > for instance, if there's a bottom, there'd have to be a top. Yep, and on another side, the mass of the top is very differen from the other two generations, and I wonder if it indicates a different underlying symmetry and thus a different anomaly. > The constraint to remove the anomaly corresponds to $Tr(\gamma^5 Y_a$ > ${Y_b,Y_c}) = 0,$ where the Y's are the gauge generators. > > The SU(2) and SU(3) generators automatically fall through correctly; as > does anything else that is parity-symmetric. The only culprit is the > hypercharge Y. > > (...) > > In other words, the anomaly is basically forcing a parity symmetric > algebra to re-emerge. ... > It's of interest to note that the charge generator Q (electric charge) > which remains after symmetry breaking will then take on the revealing > form > $Q = I3 + Y = I3 + I3_R + G;$ > i.e., a parity symmetric residue of a possible $SU(2)_R x SU(2)_L$. ... But, can your arguments be used to show that the chiral anomaly reponsible of pion decay emerges from the gauge anomaly of the stardard model?