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Anomalous condensates and the equivalence theorem
Physics Letters B 323 (1994) 161-168 North-Holland
PHYSICS LETTERS B
Anomalous condensates and the equivalence theorem -AW i l l i a m B. K i l g o r e Theorettcal Phystcs Group, LawrenceBerkeley Laboratory, and Department of Physics, Umversltyof Cahforma, Berkeley, CA 94720, USA Received 9 December 1993 Editor H Georgl
A recently published report has called into question the validity of the eqmvalence theorem m dynamicallybroken gauge theories in which the fermlons making up the symmetry breaking condensate he m an anomalous representation of the broken gauge group Such a situation can occur if the gauge anomaly is cancelledby another sector of the theory Using the example of the one family Standard Model without scalar Hlggs structure, we analyze a low energy effective theory which preserves the symmetries of the fundamental theory and demonstrate the vah&ty of the eqmvalence theorem m this class of models
1. Introduction
Recently, Donoghue and T a n d e a n [ 1 ] have &scussed models of dynamical gauge symmetry brealong in which the fermions that participate in the symmetry breaking condensate lie in an anomalous representation of the broken symmetry To preserve the gauge symmetry in the quantlzed theory there must be no net gauge anomaly [ 2 ] Therefore there must also be additional fermlons which cancel the gauge anomaly but do not take part in the condensate. They conclude that the equivalence theorem [3,4], which relates the scattering amphtudes of high-energy (E>> M ) longitudinal gauge bosons to those of the unphystcal would-be Goldstone bosons, is invalid in such a theory Speofically, they find that there is a non-zero scattering amplitude for the production of would-be Goldstone bosons via the W e s s - Z u m l n o a n o m a l y interaction, while the production of gauge bosons, longitudinal or otherwise, through fermlon triangle diagrams is forbidden by the anomaly cancellation condition This result would be quite disturbing, since the equivalence theorem is proven [ 3 ] to follow from the BRS [ 5 ] identities of the theory al Here we resolve the apparent paradox by showing that the a n o m a l y cancellation condition does not imply that the production of gauge bosons via fermion trmngle diagrams must vanish A n o m aly cancellation guarantees gauge lnvariance a n d causes the mass-independent pieces of the triangle diagrams to cancel one another The mass d e p e n d e n t pieces are not constrained to cancel a n d in general do not cancel except when the fermlon masses are equal. W h e n the interactions of the would-be Goldstone bosons are properly formulated (1.e such that the symmetries of the full theory, such as gauge invarlance, are conserved), would-be Goldstone boson p r o d u c t m n is consistent with gauge boson production so that the equivalence theorem ~s mdeed valid in a n o m a l o u s condensate models
This work was supported by the Dwector, Office of Energy research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U S Department of Energyunder Contract DE-AC03-76SF00098 #1 The existing proofs of the equivalence theorem are all m the context of a Hlggs theory, but the proof can be carned over to a low energy effective field theory [6 ] 0370-2693/94/$07 00 © 1994 Elsevier Science B V All rights reserved SSD10370-2693 ( 93 )E 1609-2
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2. A toy model Let us carefully construct a simple theory that accomplishes spontaneous gauge symmetry breaking via an anomalous condensate We consider a theory with the gauge structure of the Standard Model, a single famdy of fermlons, and nothing else In particular, there is no scalar Hlggs sector to break the electroweak symmetry and give masses to the gauge bosons or the fermlons. Thus, the electroweak symmetry is broken dynamically, as in technicolor models, by the spontaneous chlral symmetry breakdown of QCD [ 7 ] The Lagrangian for our theory is ~ = - ~ G"U~G~,, - ~ W '~" W~,, - 1BU"Bu. + ~]Ld~qL+ ~]RII~qR+ fLd~/L + eRlI~eR,
(2 1 )
where G~,,. W ~,, and Bu,, are the QCD, SU (2)L, and hypercharge field strength tensors, respectively and
(R) l=(:)
q= d '
Du=Ou--lg3Ga.u1)~a--lg2~u I~7~L-IglBuY,
(22,
where 2 a acts only on the color triplet quarks, L is the left chwallty projection operator, L = ½(1-Ys), and Y represents the usual Standard Model hypercharge quantum number for each fermlon species Being massless, the quarks exhibit an exact SU (2) L® SU (2) R chlral symmetry At scale Az, a fermlon condensate forms with a non-zero vacuum expectation value, (au+dd)
(2 3)
SO,
which breaks the SU (2) L® SU (2) R symmetry down to SU (2) v Associated with this symmetry breakdown are three Goldstone bosons, the plons, one for each broken symmetry generator The plons transform under a nonhnear representaUon of SU (2) L® SU (2) R, forming a (hnear) triplet representation of the unbroken SU (2) v symmetry, but transforming nonhnearly under the broken axial SU (2) A All such nonlinear realizations are physically equivalent [ 8 ] and their interactions are descnbed by the so-called chlral Lagrangian, which may be written as ~GB= ~ F 2 T r ( D U X * D u ~ ) +
,
(2 4)
where
( O"1t I;=exp 1 ~ - ] ,
D u I ; = 0 u I J - l g 2 W u ½ a ' X + l g l B u X ½tr3 ,
(2 5)
the ~z' are the plon fields, the a' are the Pauh matrices, and F~, called the plon decay constant, is the strength of the coupling of the pmns to the axml SU(2 )A currents, ( 0 IJ~ InJ(q) )
=lfnquc~u
(2 6)
Under SU (2) L® SU (2) R transformations, IJ transforms as .~ ~. L.~R*
(2 7)
The elhpsls in eq (2 4) indicates terms that involve higher covariant derivatives and are assumed to be small at low energy In addition to plon kinetic terms and multl-pion interaction terms, we find that ~GB Includes vector boson mass terms and gauge boson-pion mixing terms which are exactly those found in the Standard Model with F~ taking the place of the Hxggs vacuum expectation value v and the plons serving as the would-be Goldstone bosons Thus the plons are not physical degrees of freedom, but are absorbed into the gauge bosons via the H~ggs mechanism Since the vector mass terms have changed in scale only ( v - , F . ) , the Wemberg angle, parametenz162
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ing the mixing o f W 3 a n d B into ~ a n d Z as well as the W t o Z mass ratio, is the same as in the S t a n d a r d Model
3. Chiral quarks In the following discussion we use the chlral constituent quark model [ 9 ] as a laboratory in which to study in detail the role o f the a n o m a l y in a n o m a l o u s condensate theories and to identify exactly where we disagree with ref. [ 1 ] The large mass limit o f the constituent quark model, in which the quarks are decoupled, is equivalent to the s t a n d a r d low energy theory without quarks As we will see, our basic result, the v a h d l t y o f the equivalence t h e o r e m in a n o m a l o u s condensate theories, is I n d e p e n d e n t o f the constituent quark mass and is thus also a valid result o f the s t a n d a r d low energy theory The quarks participate in the condensate and acquire mass through their coupling to it Below the chlral s y m m e t r y breaking scale, Az, the quark Lagrangian m a y be written as [ 9,10 ]
~q = #LI~)qL + #RII~qR -- mQ(#LZqR + 0RZtqL) +
(3 1 )
U p o n expanding I; in powers o f zt, one sees that the zeroth order term is the mass term for the quarks. This mass, m o, IS called the constituent mass o f the quarks and is expected to be o f the same order o f magnitude as the chlral s y m m e t r y breaking scale A z The Lagranglan in eq (3 1 ) gives us a perfectly good description o f the low energy quark interactions, but does not explicitly exhibit decoupllng o f the massive quarks from very low energy (s << rn~ ) processes F o r instance, the coupling o f two vector currents to the pseudoscalar plons via quark triangle diagrams (VVP triangles), does not vanish for s << m 3, even in the limit that m o approaches infinity [ 11 ] Gauge lnvariance forbids quark deeoupling, the leptons carry an a n o m a l o u s gauge d e p e n d e n c e which is exactly cancelled by the a n o m a l o u s gauge d e p e n d e n c e o f the quarks. If the quarks decoupled, there would be noting to cancel the gauge d e p e n d e n c e o f the leptons It is possible to write the Lagranglan in a form such that the quarks can be decoupled [ 12 ], but there are subtleties involved and we will examine the procedure carefully (We follow here the work o f M a n o h a r a n d M o o r e [ 10 ] ) The a n o m a l o u s gauge d e p e n d e n c e o f the quark Lagranglan in eq (3 1 ) is tied to the fact that the left- and right-handed quarks transform i n d e p e n d e n t l y under SU ( 2 ) L® SU ( 2 ) R rotations qL ~" LqL,
qR ~-~ RqR
(3 2)
However, it is possible to define a new basis for the quarks so that both left- and right-handed fields transform in the same way. To that end, we define the matrix ~ as ¢'-_-_r
(33)
U n d e r chlral rotations, ~ transforms as ¢ ~, L # U * ( x ) = U ( x ) ¢ R * ,
(34)
where U ( x ) is a u n i t a r y matrix l m p h c l t l y defined by the above equations Using ~ one m a y rotate the quarks to a new basis Q, defined by QL=~:tqL'
QR=~qR
(3 5)
whlch transform as QL,R ~ U(X)QL,R.
(3 6)
We define vector a n d axial vector fields as "/~= ½[¢(DuC*) + ¢ * ( D u ¢ ) ] ,
~, = ½1[¢(DuCt)-¢*(D.¢) l ,
(3 7)
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where
Du¢-Ou¢-lg2W~u ½a'¢-igtB u ~ ,
DuCt=OuCt-iglB.(l +½~3)¢t,
(3 8)
which transform as
~¢[,~ U:t[~U*+UOuUt , s ~
Uo~uU*
(3 9)
In this notation, the Goldstone boson Lagranglan is written as 5e6B=¼F~Tr(dUd.)+
,
(3 10)
and the quark Lagranglan as ~Q = (~(I~ Q - mQ) Q + Q~¢2)5Q +
,
where
~QQ= (~u-Ig3G~ ½2a+ "/~)Q
(3.11)
The Lagranglan in eq (3.1) and the above expression for 5°Q are not equivalent. Since QL and Qa transform in the same way, £PQis not anomalous under local chlral SU (2) c® SU (2)l~ rotations and since electroweak gauge rotations are simply a subset of these local chiral rotations, 5°Q has no anomalous gauge dependence Somehow, in changing quark bases, we seem to have lost the anomalous gauge dependence that we need so that the complete theory is gauge lnvariant This apparent paradox arises because the change of variables from q to Q changes the measure of the path integral, requiring a non-trivial Jacoblan factor This Jacobian factor carries the anomalous gauge dependence. A change in the measure of the path integral is difficult to incorporate in a perturbative expansion So, just as we move the gauge fixing condition and the Fadeev-Popov determinant out of the measure of the path integral and into the effective Lagrangian, here too we add a new term to the Lagranglan which exactly compensates for the change in measure induced by the quark rotation. This compensating term is the Wess-Zumino anomaly term [ 13 ], which for the case at hand is 1
L.ewz_ 16n2eu"p°{BuO.BpTr(½as[I~t(W~+Lo)I~-Wol)+OuB~Tr(WpLo)-IBuTr(L.LpL~)},
(3 12)
with Lu= (~uI;)~t, Wu= - I g z W~ ½a' and Bu= --lg~Bu Thus, our complete low-energy Lagrangian is
£e= _ l W'U"W~u.- ~BU"Bu.+ ~F 2~Tr(~Vu~cffu)+ ~gf-t- FEll])/L -t- eRlI~eR + Q(I~ Q- mQ)Q+ Os~y, Q +SPwz+
,
(3 13)
where 5~gfcontains the gauge fixing and Fadeev-Popov ghost terms necessary for gauge field quantlzatIon (We assume, of course, that electroweak gauge fixing is accomplished through a "~-gauge" condition [ 14], £~gf= ( 1/ 2~) (0uA u_ ~M;¢)2, in which we know how to prove the equivalence theorem ) The quarks now have derivative couplings to the pxons and thus the VVP mangles connecting pions to vector currents m the q basis are replaced by vector-vector-axial vector (VVA) triangles and gauge-Goldstone contact interactions In the Wess-Zumlno term The VVA triangles, and all other terms explicitly involving quarks, decouple as the quark mass is taken to infinity The Wess-Zumxno term does not explicitly involve the quarks and does not decouple It cannot decoupie because it is not gauge xnvariant In the q basis, the quarks cannot decouple because they must cancel the anomalous gauge dependence of the leptons In the Q basis, the Wess-ZumIno term balances the gauge dependence of the leptons, the Q quarks are gauge invanant and there is no obstacle to their decoupllng We can directly integrate out the quarks in the Q basis but not in the q basis.
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4. The equivalence theorem The simplest process involving the a n o m a l y which one might investigate is the well-known decay n°-~Ty, and its analog through the equivalence theorem Z L ~ y y This process, however, is not an appropriate test of the equivalence theorem which is exphcltly formulated to be valid for high energy processes ( E >> Mw,z), a n d relies upon the fact that the longitudinal polarization vector for fast moving gauge bosons is approximately proportional to the m o m e n t u m of the particle This approximation is clearly invalid in the rest frame Let us therefore examine a process where we can produce energetic gauge bosons such as g + v--~ZL+ ? This process provides a particularly clean example to study as the more realistic process e + + e - -~ ZL + y IS also more complicated because of Z - y interference The relevant diagrams for the q a n d Q bases are shown in figs 1 and 2 respectively (Implicitly included are the triangle diagrams with the fermlons propagating in the opposite direction around the loop ) In the q basis, there is no W e s s - Z u m i n o term, so the VVP triangle of fig 1c completely describes n°~, production, while in the Q basis noy p r o d u c t i o n results from the sum of the VVA triangle of fig 2c and the W e s s - Z u m l n o interaction of fig 2d The amplitude for n°7 production via the W e s s - Z u m l n o interaction is wz leg~ 1 - 4 s l n 2 0 w pT~(1-ys)U ,~p~pl,~p2~(p2 ) Jg,~ - 64nEF~ cosE0w s-Mz
(4
1)
where eY is the photon polarization vector The other amplitudes, computed in the limit that eZL(px)~P~u/
M z + 0 ( M z / E ) are 1
o wz( o#~=-~g~y 1-2
1--~c
dY ( v z _ y ) M 2 _ x y ( s _ M 2 ) + m 2 .
dx 0
0
,
(42)
"
z
z
PJ
z
z
Pl
Y
(a)
Y
Y
(b)
7
(a)
(b)
~o V
Y
(c) Fig 1 Feynman dmgramsfor ZL7and yTrproduction with quarks in the q basis The D1rac structure of the external ZL/Zrcouplings to the fermlon triangles is indicated The full vertex factors are (a) -T-0g2/2 cosOw)~(p)75~T-(1/F~)~75 for (]) quarks, (b) -~ (]/F~)~ys, and (c) + (rne/F~)ys
/
V
p~-cz Y
(el
V
~
/
~ 7
(d)
Fig 2 Feynman diagrams for ZLy and Tryproduction with quarks in the Q basis The Dlrac structure of the external ZL/ZCcouphngs to the fermlon triangles is indicated The full vertex factors are (a) T (lg2/2 cos Ow)~(p)ys~ +-(1/F~)I~ysfor (]) quarks, (b) (1/F~)I~ys,and (c) _+( 1/F~)t~y5 (d) represents ny production via the Wess-Zummo term 165
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a w -y z 4_ ~/Q ~gL _ -~-o -~-o ,r,
PHYSICS LETTERS B
jltzr = _ l~,WZ + O ( - - ~ )
d ¢ ~ y = - l~g~r + O(--~Z) =~//~r,
10 March 1994
(4.2 cont'd)
where x and y are Feynman parameters Up to a phase,
Jg,w + ~g,~ = J¢~ = ~g z~ + ~t{~ + 0 exactly as the equivalence theorem asserts There are several features worth noticing. First, by explicit calculation, J//nr-[-~'[[~, a wz IS exactly equal to o,#~ This IS as it must be since the q and Q formulations are equivalent Second, the validity of the equivalence theorem is independent of the consntuent quark mass m e. We see this most clearly in the Q basis where we find that gauge boson production via quark triangles is equivalent to would-be Goldstone production via quark triangles, and gauge boson production via lepton triangles is equivalent to would-be Goldstone production via the Wess-Zumlno interaction The quark triangles for gauge and would be Goldstone boson production have identical mass dependence while the lepton trmngles and the Wess-Zumlno interaction are independent of the quark mass Finally, for M 2 << s << rn~, the quark triangles, o,/[~ and ~/(~ are suppressed relative to the lepton triangles l and the Wess-Zumlno term J{,~ WZ respectively, providing nonvanlshlng production amphtudes for ZL7 and Jgz7 zr°y. For s >> m~, however, the quark triangles are not suppressed relative to the lepton triangles and the WessZumxno term and tend to cancel them so that the production amphtudes are quite small For example, at an intermediate energy M 2 << s << m ~, the cross section for ZL7 production is approximately s independent,
aZLy(M2z<
1-
6--~Q)
,
(4.4)
while at high energy, s >> m~, the cross section falls rapidly with s, a a 2 w ( 1 - 4 s l n Z 0 w ) ~z4m~ I tTZL~,(s >> m ~ ) ----
1 ( ) [ S" " 2~2
(45)
Let us now return to ref [ 1 ] Donoghue and Tandean compute n°y production exclusively through the WessZumino interaction, while finding that ZLy production vanishes because the quark triangles exactly cancel the lepton triangles. From eq. (4 2), we see that their computation of n°y production is a vahd low energy (heavy wz I ) while their result for ZLy production is a valid high quark) approximation, (s << m~ so that Io¢[~y[ << [~/t~ energy (light quark) approximation (s >> m~ so that ~g~y ~ --~/llzy ) Each is valid within a particular energy (quark mass) regime, but they are never simultaneously valid
5. Comments One complication that we omitted, though it must in general be allowed, is the renormahzatlon by strong QCD forces of the hadronlc part of the axial vector current in which case the Q~y5 Q term in eq (3 13 ) enters with a coefficient gA which could perhaps be determined experimentally The introduction of gA IS the only change that needs to be made to the Lagrangian in the Q basis Both ~//~ and ~,/[~ are multiplied by gA whale the lepton couphngs and the Wess-Zumlno interaction are unchanged so that the equivalence theorem remains intact More complicated modifications must be made in the q basis [ 10 ], but the conclusions are of course the same Another complication that we have not addressed involves enlarging the scalar sector This would happen if more than one electroweak doublet were to participate in the condensate. In this case, we would again find the equivalence theorem to be valid As discussed above, in the Q basis quark triangle production of high energy 166
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gauge bosons is manifestly equivalent to quark triangle p r o d u c t i o n o f would-be G o l d s t o n e bosons Since the strength o f the W e s s - Z u m l n o t e r m is d e t e r m i n e d by the strength o f the quark a n o m a l y [ 13 ] which, because o f a n o m a l y cancellation IS equal in m a g n i t u d e to the lepton anomaly, there is again equivalence between gauge boson p r o d u c t i o n via lepton triangles and would-be G o l d s t o n e production via the W e s s - Z u m i n o interaction Thus, the overall equivalence is m a i n t a i n e d
6. Conclusions The equivalence t h e o r e m is a direct consequence o f the BRS s y m m e t r y o f a spontaneously broken gauge theory Since BRS s y m m e t r y is the special form o f the classical gauge s y m m e t r y which survives the quantlzatlon procedure, the equivalence t h e o r e m is a necessary consequence o f gauge l n v a n a n c e In this letter, we have constructed an explicit example o f how the equivalence theorem is satisfied even in the presence o f an a n o m a l o u s s y m m e t r y breaking condensate. Gauge invarlance d e m a n d s that the a n o m a l y associated with the condensate be cancelled by some other part o f the theory Together, the two parts c o m b i n e to preserve gauge lnvarlance and in so doing guarantee the equivalence t h e o r e m In this analysis, we have used the constituent quark model to illustrate that one must c o m p u t e low energy processes consistently within the low energy theory If quarks o f a particular mass contribute to (or cancel) Zy production, the same quarks contribute to (or cancel) ny p r o d u c t i o n We have seen that the equivalence theorem is valid for any constituent quark mass that we might choose In particular, it is valid in the heavy quark limit and so is also valid in the conventional low energy theory without quarks In d e m o n s t r a t i n g the validity o f the equivalence theorem, we have seen that a n o m a l o u s condensate models p e r m i t gauge boson p r o d u c t i o n through fermlon triangle diagrams The expectation that fermlon triangle diagrams exactly cancel one another stems from the action o f gauge lnvarlance w~thln an unbroken chlral theory as in eq (2 1 ) However, triangle d i a g r a m cancellation is not a p r o p e r t y o f gauge m v a r l a n c e alone and need not occur in spontaneously b r o k e n chwal theories It m a y be that there are " a n o m a l o u s technicolor" models for which triangle d i a g r a m gauge boson p r o d u c t i o n has observable consequences [ 15 ]
Acknowledgement I would like to t h a n k M a r k u s Luty, Greg K e a t o n a n d Michael Chanowltz for m a n y helpful discussions This work was s u p p o r t e d by the Director, Office o f Energy research, Office o f High Energy and Nuclear Physics, Division o f High Energy Physics o f the US D e p a r t m e n t o f Energy under Contract DE-AC03-76SF00098
References [ l ] J F Donoghue and J Tandean, Phys Lett B 301 (1993) 372 [2]DJ Gross and R Jacklw, Phys Rev D 6 (1972)477, C Bouchlat, J Ilhopoulos and Ph Meyer, Phys Len B 38 (1972) 519 [ 3 ] M S Chanowltz and M K Galliard, Nucl Phys B 261 ( 1985 ) 379, G J Gounarls, R Kogerler and H Neufeld, Phys Rev D 34 (1986) 3257, J Bagger and C Schmldt, Phys Rev D 41 (1990) 264, H Veltman, Phys Rev D 41 (1990)2294, W Kalgore, Phys Lett B 294 (1992)257, H J He, Y P Kuang and X Y1, Phys Rev Lett 69 (1992) 2619, TUIMP preprlnt TUIMP-TH-92/51 167
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[4] J M Cornwall, D N Levln and G Taktopoulos, Phys Rev D 10 (1974) 1145, C E Vayonakxs, Lett Nuovo Omento 17 (1976) 383, BW Lee, C QmggandH Thacker, Phys Rev D 16 (1977) 1519, D SoperandZ Kunst, Nucl Phys B296 (1988)253 [5 ] C Becchl, A Rouet and R Stora, Commun Math Phys B 261 (1975)379 [ 6 ] W Kalgore, in preparatton [7] M Wemstem, Phys Rev D 7 (1973) 1854, D 8 (1973) 2511, S Wemberg, Phys Rev D 19 (1979) 1277, L Susskmd, Phys Rev D 20 (1979) 2619 [8 ] S Coleman, J Wess and B Zumlno, Phys Rev 177 (1969) 2239, C Callan, S Coleman, J Wess and B Zummo, Phys Rev 177 (1969)2247 [9] A Manohar and H Georgl, Nucl Phys B 234 (1984) 189 [ 10] A Manohar and G Moore, Nucl Phys B 243 (1984) 55 [ 11 ] J Stemberger, Phys Rev 76 (1949) 1180 [ 12 ] E D'Hoker and E Farhl, Nucl Phys B 248 (1984) 59, 77 [ 13] J Wess and B Zummo, Phys Lett B 37 ( 1971 ) 95, E Wltten, Nucl Plays B223 (1983) 422, WA Bardeen, Phys Rev 184 (1969)1848, K Fujlkawa, Phys Rev Lett 42 (1979) 1195, Phys Rev D 21 (1980) 2848, D 22 (1980) 1499, D 23 ( 1981 ) 2262, B Zummo, m Relativity, groups, and topology II, eds B S DeW~tt and R Stora (North-Holland, Amsterdam, 1984) p 1291, B Zummo, Y -S wu and A Zee, Nucl Phys B 239 (1984) 477, W A Bardeen and B Zumlno, Nucl Phys B 244 (1984) 421, L Alvarez-Gaum6 and P Gmsparg, Ann Phys 161 (1985) 423, N K Pak and P Rossl, Nucl Phys B 250 (1985) 279 [ 14] K Fujlkawa, B W Lee and A I Sanda, Phys Rev D 6 (1972) 2923 [ 15 ] W Kalgore, in preparation
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Anomalous condensates and the equivalence theorem -AW i l l i a m B. K i l g o r e Theorettcal Phystcs Group, LawrenceBerkeley Laboratory, and Department of Physics, Umversltyof Cahforma, Berkeley, CA 94720, USA Received 9 December 1993 Editor H Georgl
A recently published report has called into question the validity of the eqmvalence theorem m dynamicallybroken gauge theories in which the fermlons making up the symmetry breaking condensate he m an anomalous representation of the broken gauge group Such a situation can occur if the gauge anomaly is cancelledby another sector of the theory Using the example of the one family Standard Model without scalar Hlggs structure, we analyze a low energy effective theory which preserves the symmetries of the fundamental theory and demonstrate the vah&ty of the eqmvalence theorem m this class of models
1. Introduction
Recently, Donoghue and T a n d e a n [ 1 ] have &scussed models of dynamical gauge symmetry brealong in which the fermions that participate in the symmetry breaking condensate lie in an anomalous representation of the broken symmetry To preserve the gauge symmetry in the quantlzed theory there must be no net gauge anomaly [ 2 ] Therefore there must also be additional fermlons which cancel the gauge anomaly but do not take part in the condensate. They conclude that the equivalence theorem [3,4], which relates the scattering amphtudes of high-energy (E>> M ) longitudinal gauge bosons to those of the unphystcal would-be Goldstone bosons, is invalid in such a theory Speofically, they find that there is a non-zero scattering amplitude for the production of would-be Goldstone bosons via the W e s s - Z u m l n o a n o m a l y interaction, while the production of gauge bosons, longitudinal or otherwise, through fermlon triangle diagrams is forbidden by the anomaly cancellation condition This result would be quite disturbing, since the equivalence theorem is proven [ 3 ] to follow from the BRS [ 5 ] identities of the theory al Here we resolve the apparent paradox by showing that the a n o m a l y cancellation condition does not imply that the production of gauge bosons via fermion trmngle diagrams must vanish A n o m aly cancellation guarantees gauge lnvariance a n d causes the mass-independent pieces of the triangle diagrams to cancel one another The mass d e p e n d e n t pieces are not constrained to cancel a n d in general do not cancel except when the fermlon masses are equal. W h e n the interactions of the would-be Goldstone bosons are properly formulated (1.e such that the symmetries of the full theory, such as gauge invarlance, are conserved), would-be Goldstone boson p r o d u c t m n is consistent with gauge boson production so that the equivalence theorem ~s mdeed valid in a n o m a l o u s condensate models
This work was supported by the Dwector, Office of Energy research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U S Department of Energyunder Contract DE-AC03-76SF00098 #1 The existing proofs of the equivalence theorem are all m the context of a Hlggs theory, but the proof can be carned over to a low energy effective field theory [6 ] 0370-2693/94/$07 00 © 1994 Elsevier Science B V All rights reserved SSD10370-2693 ( 93 )E 1609-2
161
Volume 323, number 2
PHYSICS LETTERS B
10 March 1994
2. A toy model Let us carefully construct a simple theory that accomplishes spontaneous gauge symmetry breaking via an anomalous condensate We consider a theory with the gauge structure of the Standard Model, a single famdy of fermlons, and nothing else In particular, there is no scalar Hlggs sector to break the electroweak symmetry and give masses to the gauge bosons or the fermlons. Thus, the electroweak symmetry is broken dynamically, as in technicolor models, by the spontaneous chlral symmetry breakdown of QCD [ 7 ] The Lagrangian for our theory is ~ = - ~ G"U~G~,, - ~ W '~" W~,, - 1BU"Bu. + ~]Ld~qL+ ~]RII~qR+ fLd~/L + eRlI~eR,
(2 1 )
where G~,,. W ~,, and Bu,, are the QCD, SU (2)L, and hypercharge field strength tensors, respectively and
(R) l=(:)
q= d '
Du=Ou--lg3Ga.u1)~a--lg2~u I~7~L-IglBuY,
(22,
where 2 a acts only on the color triplet quarks, L is the left chwallty projection operator, L = ½(1-Ys), and Y represents the usual Standard Model hypercharge quantum number for each fermlon species Being massless, the quarks exhibit an exact SU (2) L® SU (2) R chlral symmetry At scale Az, a fermlon condensate forms with a non-zero vacuum expectation value, (au+dd)
(2 3)
SO,
which breaks the SU (2) L® SU (2) R symmetry down to SU (2) v Associated with this symmetry breakdown are three Goldstone bosons, the plons, one for each broken symmetry generator The plons transform under a nonhnear representaUon of SU (2) L® SU (2) R, forming a (hnear) triplet representation of the unbroken SU (2) v symmetry, but transforming nonhnearly under the broken axial SU (2) A All such nonlinear realizations are physically equivalent [ 8 ] and their interactions are descnbed by the so-called chlral Lagrangian, which may be written as ~GB= ~ F 2 T r ( D U X * D u ~ ) +
,
(2 4)
where
( O"1t I;=exp 1 ~ - ] ,
D u I ; = 0 u I J - l g 2 W u ½ a ' X + l g l B u X ½tr3 ,
(2 5)
the ~z' are the plon fields, the a' are the Pauh matrices, and F~, called the plon decay constant, is the strength of the coupling of the pmns to the axml SU(2 )A currents, ( 0 IJ~ InJ(q) )
=lfnquc~u
(2 6)
Under SU (2) L® SU (2) R transformations, IJ transforms as .~ ~. L.~R*
(2 7)
The elhpsls in eq (2 4) indicates terms that involve higher covariant derivatives and are assumed to be small at low energy In addition to plon kinetic terms and multl-pion interaction terms, we find that ~GB Includes vector boson mass terms and gauge boson-pion mixing terms which are exactly those found in the Standard Model with F~ taking the place of the Hxggs vacuum expectation value v and the plons serving as the would-be Goldstone bosons Thus the plons are not physical degrees of freedom, but are absorbed into the gauge bosons via the H~ggs mechanism Since the vector mass terms have changed in scale only ( v - , F . ) , the Wemberg angle, parametenz162
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ing the mixing o f W 3 a n d B into ~ a n d Z as well as the W t o Z mass ratio, is the same as in the S t a n d a r d Model
3. Chiral quarks In the following discussion we use the chlral constituent quark model [ 9 ] as a laboratory in which to study in detail the role o f the a n o m a l y in a n o m a l o u s condensate theories and to identify exactly where we disagree with ref. [ 1 ] The large mass limit o f the constituent quark model, in which the quarks are decoupled, is equivalent to the s t a n d a r d low energy theory without quarks As we will see, our basic result, the v a h d l t y o f the equivalence t h e o r e m in a n o m a l o u s condensate theories, is I n d e p e n d e n t o f the constituent quark mass and is thus also a valid result o f the s t a n d a r d low energy theory The quarks participate in the condensate and acquire mass through their coupling to it Below the chlral s y m m e t r y breaking scale, Az, the quark Lagrangian m a y be written as [ 9,10 ]
~q = #LI~)qL + #RII~qR -- mQ(#LZqR + 0RZtqL) +
(3 1 )
U p o n expanding I; in powers o f zt, one sees that the zeroth order term is the mass term for the quarks. This mass, m o, IS called the constituent mass o f the quarks and is expected to be o f the same order o f magnitude as the chlral s y m m e t r y breaking scale A z The Lagranglan in eq (3 1 ) gives us a perfectly good description o f the low energy quark interactions, but does not explicitly exhibit decoupllng o f the massive quarks from very low energy (s << rn~ ) processes F o r instance, the coupling o f two vector currents to the pseudoscalar plons via quark triangle diagrams (VVP triangles), does not vanish for s << m 3, even in the limit that m o approaches infinity [ 11 ] Gauge lnvariance forbids quark deeoupling, the leptons carry an a n o m a l o u s gauge d e p e n d e n c e which is exactly cancelled by the a n o m a l o u s gauge d e p e n d e n c e o f the quarks. If the quarks decoupled, there would be noting to cancel the gauge d e p e n d e n c e o f the leptons It is possible to write the Lagranglan in a form such that the quarks can be decoupled [ 12 ], but there are subtleties involved and we will examine the procedure carefully (We follow here the work o f M a n o h a r a n d M o o r e [ 10 ] ) The a n o m a l o u s gauge d e p e n d e n c e o f the quark Lagranglan in eq (3 1 ) is tied to the fact that the left- and right-handed quarks transform i n d e p e n d e n t l y under SU ( 2 ) L® SU ( 2 ) R rotations qL ~" LqL,
qR ~-~ RqR
(3 2)
However, it is possible to define a new basis for the quarks so that both left- and right-handed fields transform in the same way. To that end, we define the matrix ~ as ¢'-_-_r
(33)
U n d e r chlral rotations, ~ transforms as ¢ ~, L # U * ( x ) = U ( x ) ¢ R * ,
(34)
where U ( x ) is a u n i t a r y matrix l m p h c l t l y defined by the above equations Using ~ one m a y rotate the quarks to a new basis Q, defined by QL=~:tqL'
QR=~qR
(3 5)
whlch transform as QL,R ~ U(X)QL,R.
(3 6)
We define vector a n d axial vector fields as "/~= ½[¢(DuC*) + ¢ * ( D u ¢ ) ] ,
~, = ½1[¢(DuCt)-¢*(D.¢) l ,
(3 7)
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where
Du¢-Ou¢-lg2W~u ½a'¢-igtB u ~ ,
DuCt=OuCt-iglB.(l +½~3)¢t,
(3 8)
which transform as
~¢[,~ U:t[~U*+UOuUt , s ~
Uo~uU*
(3 9)
In this notation, the Goldstone boson Lagranglan is written as 5e6B=¼F~Tr(dUd.)+
,
(3 10)
and the quark Lagranglan as ~Q = (~(I~ Q - mQ) Q + Q~¢2)5Q +
,
where
~QQ= (~u-Ig3G~ ½2a+ "/~)Q
(3.11)
The Lagranglan in eq (3.1) and the above expression for 5°Q are not equivalent. Since QL and Qa transform in the same way, £PQis not anomalous under local chlral SU (2) c® SU (2)l~ rotations and since electroweak gauge rotations are simply a subset of these local chiral rotations, 5°Q has no anomalous gauge dependence Somehow, in changing quark bases, we seem to have lost the anomalous gauge dependence that we need so that the complete theory is gauge lnvariant This apparent paradox arises because the change of variables from q to Q changes the measure of the path integral, requiring a non-trivial Jacoblan factor This Jacobian factor carries the anomalous gauge dependence. A change in the measure of the path integral is difficult to incorporate in a perturbative expansion So, just as we move the gauge fixing condition and the Fadeev-Popov determinant out of the measure of the path integral and into the effective Lagrangian, here too we add a new term to the Lagranglan which exactly compensates for the change in measure induced by the quark rotation. This compensating term is the Wess-Zumino anomaly term [ 13 ], which for the case at hand is 1
L.ewz_ 16n2eu"p°{BuO.BpTr(½as[I~t(W~+Lo)I~-Wol)+OuB~Tr(WpLo)-IBuTr(L.LpL~)},
(3 12)
with Lu= (~uI;)~t, Wu= - I g z W~ ½a' and Bu= --lg~Bu Thus, our complete low-energy Lagrangian is
£e= _ l W'U"W~u.- ~BU"Bu.+ ~F 2~Tr(~Vu~cffu)+ ~gf-t- FEll])/L -t- eRlI~eR + Q(I~ Q- mQ)Q+ Os~y, Q +SPwz+
,
(3 13)
where 5~gfcontains the gauge fixing and Fadeev-Popov ghost terms necessary for gauge field quantlzatIon (We assume, of course, that electroweak gauge fixing is accomplished through a "~-gauge" condition [ 14], £~gf= ( 1/ 2~) (0uA u_ ~M;¢)2, in which we know how to prove the equivalence theorem ) The quarks now have derivative couplings to the pxons and thus the VVP mangles connecting pions to vector currents m the q basis are replaced by vector-vector-axial vector (VVA) triangles and gauge-Goldstone contact interactions In the Wess-Zumlno term The VVA triangles, and all other terms explicitly involving quarks, decouple as the quark mass is taken to infinity The Wess-Zumxno term does not explicitly involve the quarks and does not decouple It cannot decoupie because it is not gauge xnvariant In the q basis, the quarks cannot decouple because they must cancel the anomalous gauge dependence of the leptons In the Q basis, the Wess-ZumIno term balances the gauge dependence of the leptons, the Q quarks are gauge invanant and there is no obstacle to their decoupllng We can directly integrate out the quarks in the Q basis but not in the q basis.
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4. The equivalence theorem The simplest process involving the a n o m a l y which one might investigate is the well-known decay n°-~Ty, and its analog through the equivalence theorem Z L ~ y y This process, however, is not an appropriate test of the equivalence theorem which is exphcltly formulated to be valid for high energy processes ( E >> Mw,z), a n d relies upon the fact that the longitudinal polarization vector for fast moving gauge bosons is approximately proportional to the m o m e n t u m of the particle This approximation is clearly invalid in the rest frame Let us therefore examine a process where we can produce energetic gauge bosons such as g + v--~ZL+ ? This process provides a particularly clean example to study as the more realistic process e + + e - -~ ZL + y IS also more complicated because of Z - y interference The relevant diagrams for the q a n d Q bases are shown in figs 1 and 2 respectively (Implicitly included are the triangle diagrams with the fermlons propagating in the opposite direction around the loop ) In the q basis, there is no W e s s - Z u m i n o term, so the VVP triangle of fig 1c completely describes n°~, production, while in the Q basis noy p r o d u c t i o n results from the sum of the VVA triangle of fig 2c and the W e s s - Z u m l n o interaction of fig 2d The amplitude for n°7 production via the W e s s - Z u m l n o interaction is wz leg~ 1 - 4 s l n 2 0 w pT~(1-ys)U ,~p~pl,~p2~(p2 ) Jg,~ - 64nEF~ cosE0w s-Mz
(4
1)
where eY is the photon polarization vector The other amplitudes, computed in the limit that eZL(px)~P~u/
M z + 0 ( M z / E ) are 1
o wz( o#~=-~g~y 1-2
1--~c
dY ( v z _ y ) M 2 _ x y ( s _ M 2 ) + m 2 .
dx 0
0
,
(42)
"
z
z
PJ
z
z
Pl
Y
(a)
Y
Y
(b)
7
(a)
(b)
~o V
Y
(c) Fig 1 Feynman dmgramsfor ZL7and yTrproduction with quarks in the q basis The D1rac structure of the external ZL/Zrcouplings to the fermlon triangles is indicated The full vertex factors are (a) -T-0g2/2 cosOw)~(p)75~T-(1/F~)~75 for (]) quarks, (b) -~ (]/F~)~ys, and (c) + (rne/F~)ys
/
V
p~-cz Y
(el
V
~
/
~ 7
(d)
Fig 2 Feynman diagrams for ZLy and Tryproduction with quarks in the Q basis The Dlrac structure of the external ZL/ZCcouphngs to the fermlon triangles is indicated The full vertex factors are (a) T (lg2/2 cos Ow)~(p)ys~ +-(1/F~)I~ysfor (]) quarks, (b) (1/F~)I~ys,and (c) _+( 1/F~)t~y5 (d) represents ny production via the Wess-Zummo term 165
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a w -y z 4_ ~/Q ~gL _ -~-o -~-o ,r,
PHYSICS LETTERS B
jltzr = _ l~,WZ + O ( - - ~ )
d ¢ ~ y = - l~g~r + O(--~Z) =~//~r,
10 March 1994
(4.2 cont'd)
where x and y are Feynman parameters Up to a phase,
Jg,w + ~g,~ = J¢~ = ~g z~ + ~t{~ + 0 exactly as the equivalence theorem asserts There are several features worth noticing. First, by explicit calculation, J//nr-[-~'[[~, a wz IS exactly equal to o,#~ This IS as it must be since the q and Q formulations are equivalent Second, the validity of the equivalence theorem is independent of the consntuent quark mass m e. We see this most clearly in the Q basis where we find that gauge boson production via quark triangles is equivalent to would-be Goldstone production via quark triangles, and gauge boson production via lepton triangles is equivalent to would-be Goldstone production via the Wess-Zumlno interaction The quark triangles for gauge and would be Goldstone boson production have identical mass dependence while the lepton trmngles and the Wess-Zumlno interaction are independent of the quark mass Finally, for M 2 << s << rn~, the quark triangles, o,/[~ and ~/(~ are suppressed relative to the lepton triangles l and the Wess-Zumlno term J{,~ WZ respectively, providing nonvanlshlng production amphtudes for ZL7 and Jgz7 zr°y. For s >> m~, however, the quark triangles are not suppressed relative to the lepton triangles and the WessZumxno term and tend to cancel them so that the production amphtudes are quite small For example, at an intermediate energy M 2 << s << m ~, the cross section for ZL7 production is approximately s independent,
aZLy(M2z<
1-
6--~Q)
,
(4.4)
while at high energy, s >> m~, the cross section falls rapidly with s, a a 2 w ( 1 - 4 s l n Z 0 w ) ~z4m~ I tTZL~,(s >> m ~ ) ----
1 ( ) [ S" " 2~2
(45)
Let us now return to ref [ 1 ] Donoghue and Tandean compute n°y production exclusively through the WessZumino interaction, while finding that ZLy production vanishes because the quark triangles exactly cancel the lepton triangles. From eq. (4 2), we see that their computation of n°y production is a vahd low energy (heavy wz I ) while their result for ZLy production is a valid high quark) approximation, (s << m~ so that Io¢[~y[ << [~/t~ energy (light quark) approximation (s >> m~ so that ~g~y ~ --~/llzy ) Each is valid within a particular energy (quark mass) regime, but they are never simultaneously valid
5. Comments One complication that we omitted, though it must in general be allowed, is the renormahzatlon by strong QCD forces of the hadronlc part of the axial vector current in which case the Q~y5 Q term in eq (3 13 ) enters with a coefficient gA which could perhaps be determined experimentally The introduction of gA IS the only change that needs to be made to the Lagrangian in the Q basis Both ~//~ and ~,/[~ are multiplied by gA whale the lepton couphngs and the Wess-Zumlno interaction are unchanged so that the equivalence theorem remains intact More complicated modifications must be made in the q basis [ 10 ], but the conclusions are of course the same Another complication that we have not addressed involves enlarging the scalar sector This would happen if more than one electroweak doublet were to participate in the condensate. In this case, we would again find the equivalence theorem to be valid As discussed above, in the Q basis quark triangle production of high energy 166
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gauge bosons is manifestly equivalent to quark triangle p r o d u c t i o n o f would-be G o l d s t o n e bosons Since the strength o f the W e s s - Z u m l n o t e r m is d e t e r m i n e d by the strength o f the quark a n o m a l y [ 13 ] which, because o f a n o m a l y cancellation IS equal in m a g n i t u d e to the lepton anomaly, there is again equivalence between gauge boson p r o d u c t i o n via lepton triangles and would-be G o l d s t o n e production via the W e s s - Z u m i n o interaction Thus, the overall equivalence is m a i n t a i n e d
6. Conclusions The equivalence t h e o r e m is a direct consequence o f the BRS s y m m e t r y o f a spontaneously broken gauge theory Since BRS s y m m e t r y is the special form o f the classical gauge s y m m e t r y which survives the quantlzatlon procedure, the equivalence t h e o r e m is a necessary consequence o f gauge l n v a n a n c e In this letter, we have constructed an explicit example o f how the equivalence theorem is satisfied even in the presence o f an a n o m a l o u s s y m m e t r y breaking condensate. Gauge invarlance d e m a n d s that the a n o m a l y associated with the condensate be cancelled by some other part o f the theory Together, the two parts c o m b i n e to preserve gauge lnvarlance and in so doing guarantee the equivalence t h e o r e m In this analysis, we have used the constituent quark model to illustrate that one must c o m p u t e low energy processes consistently within the low energy theory If quarks o f a particular mass contribute to (or cancel) Zy production, the same quarks contribute to (or cancel) ny p r o d u c t i o n We have seen that the equivalence theorem is valid for any constituent quark mass that we might choose In particular, it is valid in the heavy quark limit and so is also valid in the conventional low energy theory without quarks In d e m o n s t r a t i n g the validity o f the equivalence theorem, we have seen that a n o m a l o u s condensate models p e r m i t gauge boson p r o d u c t i o n through fermlon triangle diagrams The expectation that fermlon triangle diagrams exactly cancel one another stems from the action o f gauge lnvarlance w~thln an unbroken chlral theory as in eq (2 1 ) However, triangle d i a g r a m cancellation is not a p r o p e r t y o f gauge m v a r l a n c e alone and need not occur in spontaneously b r o k e n chwal theories It m a y be that there are " a n o m a l o u s technicolor" models for which triangle d i a g r a m gauge boson p r o d u c t i o n has observable consequences [ 15 ]
Acknowledgement I would like to t h a n k M a r k u s Luty, Greg K e a t o n a n d Michael Chanowltz for m a n y helpful discussions This work was s u p p o r t e d by the Director, Office o f Energy research, Office o f High Energy and Nuclear Physics, Division o f High Energy Physics o f the US D e p a r t m e n t o f Energy under Contract DE-AC03-76SF00098
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[4] J M Cornwall, D N Levln and G Taktopoulos, Phys Rev D 10 (1974) 1145, C E Vayonakxs, Lett Nuovo Omento 17 (1976) 383, BW Lee, C QmggandH Thacker, Phys Rev D 16 (1977) 1519, D SoperandZ Kunst, Nucl Phys B296 (1988)253 [5 ] C Becchl, A Rouet and R Stora, Commun Math Phys B 261 (1975)379 [ 6 ] W Kalgore, in preparatton [7] M Wemstem, Phys Rev D 7 (1973) 1854, D 8 (1973) 2511, S Wemberg, Phys Rev D 19 (1979) 1277, L Susskmd, Phys Rev D 20 (1979) 2619 [8 ] S Coleman, J Wess and B Zumlno, Phys Rev 177 (1969) 2239, C Callan, S Coleman, J Wess and B Zummo, Phys Rev 177 (1969)2247 [9] A Manohar and H Georgl, Nucl Phys B 234 (1984) 189 [ 10] A Manohar and G Moore, Nucl Phys B 243 (1984) 55 [ 11 ] J Stemberger, Phys Rev 76 (1949) 1180 [ 12 ] E D'Hoker and E Farhl, Nucl Phys B 248 (1984) 59, 77 [ 13] J Wess and B Zummo, Phys Lett B 37 ( 1971 ) 95, E Wltten, Nucl Plays B223 (1983) 422, WA Bardeen, Phys Rev 184 (1969)1848, K Fujlkawa, Phys Rev Lett 42 (1979) 1195, Phys Rev D 21 (1980) 2848, D 22 (1980) 1499, D 23 ( 1981 ) 2262, B Zummo, m Relativity, groups, and topology II, eds B S DeW~tt and R Stora (North-Holland, Amsterdam, 1984) p 1291, B Zummo, Y -S wu and A Zee, Nucl Phys B 239 (1984) 477, W A Bardeen and B Zumlno, Nucl Phys B 244 (1984) 421, L Alvarez-Gaum6 and P Gmsparg, Ann Phys 161 (1985) 423, N K Pak and P Rossl, Nucl Phys B 250 (1985) 279 [ 14] K Fujlkawa, B W Lee and A I Sanda, Phys Rev D 6 (1972) 2923 [ 15 ] W Kalgore, in preparation
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