standard model with gravity couplings
TRANSCRIPT
,
elysgugeat itson,
PHYSICAL REVIEW D 15 MAY 1996VOLUME 53, NUMBER 10
0556-28
Standard model with gravity couplings
Lay Nam Chang*Department of Physics, Virginia Tech, Blacksburg, Virginia 24061-0435
Chopin Soo†
Center for Gravitational Physics and Geometry, Department of Physics, Pennsylvania State UniversityUniversity Park, Pennsylvania 16802-6300
~Received 21 September 1995!
In this paper we examine the coupling of matter fields to gravity within the framework of the standard modof particle physics. The coupling is described in terms of Weyl fermions of a definite chirality, and emploonly ~anti-!self-dual or left-handed spin connection fields. We review the general framework for introducinthe coupling using these fields, and show that conditions ensuring the cancellation of perturbative chiral gaanomalies are not disturbed. We also explore a global anomaly associated with the theory, and argue thremoval requires that the number of fundamental fermions in the theory must be multiples of 16. In additiwe investigate the behavior of the theory under discrete transformationsP, C, andT, and discuss possibleviolations of these discrete symmetries, includingCPT, in the presence of instantons and the Adler-Bell-Jackiw anomaly.
PACS number~s!: 04.62.1v, 11.30.Er, 12.10.Dm
elav-r-alni-
a-.
n-bet.
nds
rellsthda-.he,-cef
red
I. INTRODUCTION
Ashtekar@1# has introduced a set of variables to descrigravity, which makes essential use of the chiral decompotion of the connection one-forms of the local Lorentz grouWhat has been shown is that, at least for Einstein manifowithout matter, the full set of field equations of general retivity can be recovered by use of only one of the two chirprojections of these connection forms. We may either useself-dual or the anti-self-dual connections and their resptive conjugate variables. The constraints and reality contions of the theory then define what the other set has to
Chiral projections are used routinely in particle physicindeed, the fermion fields that define the standard modelall chiral Weyl spinor fields. Within the Ashtekar contexleft-handed spinor fields are coupled to one of these conntion forms, sayA2, while right-handed ones are coupledthe other. Since only one of these is all one needs to degeneral relativity, the actual Lagrangian must be expresentirely in terms of either left-handed or right-handed Wespinor fields.
That the standard model can be so described is of coualready known. For example, in SO~10! grand unificationschemes@2#, one employs a single 16-dimensional lefhanded Weyl field to describe one generation of fermioWhat is less clear is how the coupling of Ashtekar fielaffects the resultant physics.
In what follows, we describe some facets of these conquences. Since Ashtekar gravity makes use of only onethe chirally projected Lorentz connections, there arisesquestion of whether anomalies which are normally presensuch theories are under control. We show that the usual c
*Electronic address: [email protected]†Present address: Department of Physics, Virginia Tech, Bla
burg, VA 24061-0435. Electronic address: [email protected]
5321/96/53~10!/5682~10!/$10.00
besi-p.ldsla-altheec-di-be.s;aret,ec-tofinesedyl
rse
t-ns.ds
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ditions for anomaly cancellations for the standard modgauge groups remain true in the presence of Ashtekar grity, and that the new fields do not introduce any further peturbative anomalies. However, they do introduce globanomalies. Cancellation of these obstructions for grand ufied theories~GUT’s! in the most general context results inthe rather strong constraint that the total number of fundmental fermions in the theory must be a multiple of 16Grand unification schemes based upon groups such as SU~5!are therefore inconsistent when coupled to gravity. As a cosequence, there is every likelihood that neutrinos mustmassive when consistency with gravity is taken into accounIn particular, the SO~10! GUT with 16 fundamental Weylfermions per generation is singled out as the preeminent asimplest choice free from the global anomaly, if one allowfor generalized spin structures and arbitrary topologies.
We also discuss how the usual discrete symmetries aimplemented in the presence of Ashtekar gravity. We shashow that it is possible to posit discrete transformation lawfor the fermion and Ashtekar fields which are consistent bowith our general notions of what parity, time-reversal, ancharge conjugation transformations are, and with the fundmental canonical commutation relations of all of the fieldsDiscrete symmetries for bispinors can be implemented in tclassical limit modulo certain reality conditions. Howeverwhat we shall show is that, inevitably, the underlying quantum theory is not invariant under parity due to the occurrenaxial anomaly in quantum field theory. The question oCPT invariance will also be briefly discussed.
II. THE SAMUEL-JACOBSON-SMOLIN ACTIONAND THE ASHTEKAR VARIABLES
We first consider spacetimes of Lorentzian signatu(2,1,1,1) and start with the gravitational action proposeby Samuel, and Jacobson and Smolin@3#
ck-du
5682 © 1996 The American Physical Society
l
kartheits
se
o-
53 5683STANDARD MODEL WITH GRAVITY COUPLINGS
S SJS7 5E
MLG
7
51
8pGEMS7a`Fa76 i
l
3~16pG!EM
S7a`Sa7.
~1!
The ~anti-!self-dual two-forms S7 which obey*S7a57 iS7a are defined as
S7a[S 2e0`ea6i
2eabce
b`ecD . ~2!
F7 are the curvature two-forms of the SO(3,C) Ashtekarconnections, i.e.,
Fa75dAa
711
2ea
bcAb7`Ac
7 , ~3!
and eA ,A50, . . . ,3 denote the vierbein one-forms in foudimensions,eabc[e0abc, while l is the cosmological con-stant. Latin indices label flat Lorentz indices while spacetimindices will be denoted by Greek indices. Lower case Laindices run from 1 to 3 while upper case Latin indices ranfrom 0 to 3.
The Ashtekar variables@1# are sometimes referred to a~anti-!self-dual variables because the equations of motionthe Samuel-Jacobson-Smolin action with respect toA7,
r
etinge
sof
D7S7a50, ~4!
imply that A2 and A1 are the anti-self-dual and self-duapart of the spin connectionv, respectively: i.e.,
Aa756 ivoa2
1
2ea
bcvbc . ~5!
It is easy to see that the action reproduces the Ashtevariables and constraints. For convenience, we work inspatial gauge in which the components of the vierbein andinverse can be written in the form
eAm5F N 0
Njea j eaiG , Em
A5F N21 0
2~Ni /N! s iaG . ~6!
The form assumed in~6! is compatible with the Arnowitt-Deser-Misner~ADM ! @4# decomposition of the metric
ds25eAmeA
ndxmdxn
52N2~dx0!21gi j ~dxi1Nidx0!~dxj1Njdx0! ~7!
with the spatial metricgi j5eaiea j . Thus we see that thechoice ~6! in no way compromises the values of the lapand shift functions,N andNi , which have geometrical inter-pretations in hypersurface deformations. With this decompsition, it is straightforward to rewrite
S SJS7 5
1
16pGE d4x$62i s iaAia762iA0a
7 Di sia62iN j s iaFi ja
7 %
21
16pGE d4xHN> S eabcsias jbFi j
7c1l
3eabce> i jk s
ias jbskcD J 1boundary terms, ~8!
anda-of
i-
eby
onsarkarve
with s andN> defined as
s ia[1
2e i jkeabcejbekc ,
N> [det~eai!21N. ~9!
The tildes above and below the variables indicate that thare tensor densities of weight 1 and21, respectively. There-fore,6(2i s ia/16pG) are readily identified as the conjugatvariables toAia
7 , and we have the commutation relations
@s ia~xW ,t !,Ajb7 ~yW ,t !#56~8pG!d j
idbad3~xW2yW !. ~10!
The variablesA0a7 , Ni , andN> are clearly Lagrange mul-
tipliers for the Ashtekar constraints, which can be identifias Gauss’ law generating SO~3! gauge invariance,
Ga[2iD i sia'0, ~11!
and the supermomentum and super-Hamiltonian constrai
ey
e
ed
nts
Hi[2i s jaFi ja'0,~12!
H[eabcsias jbS Fi j
c 1l
3e> i jk s
kcD'0,
respectively. Ashtekar showed that these constraintstheir algebra, despite their remarkable simplicity, are equivlent in content to the constraints and constraint algebrageneral relativity@1#. Note that both the self-dual and antself-dual versions,S SJS
6 , describepuregravity equally well.
III. COUPLING TO MATTER FIELDS
The coupling of matter fields to gravity described by th~anti-!self-dual Ashtekar variables have been consideredothers before@5–7#, and will be examined closely in thiswork. It should be emphasized that besides self-interactiin pure gravity, only fermions couple directly to the Ashtekconnections. They are hence direct sources for the Ashteconnection. Conventional scalars and Yang-Mills fields hadirect couplings only to the metric~hence, tos) rather than
-
,
n
-
5684 53LAY NAM CHANG AND CHOPIN SOO
to the Ashtekar-Sen connection. Thus when one substituthe conventional gravitational action with the SamueJacobson-Smolin action, complications can come fromfermionic sector, primarily because one has to make a chobetween the actions described byA1 andA2 and once thechoice is made, it is permissible to couple only right or lehanded fermions~but not both! to the theory.1
We first look at the conventional Dirac action, with couplings to ordinary spin connections. We then describe cplings of Ashtekar fields to fermions, and explore the phycal implications in the following sections.
Consider the conventional Dirac action for an electrona single quark of a particular color in the presence of only tconventional gravitational field. The action can be written
S D52i
2EM ~*1 !CgAEAbDvC1H. c., ~13!
where the covariant derivative with respect to the spin conection,v, is defined by
DvC5dxm~]m1 12 vmBCS
BC!C ~14!
with the generatorS AB5 14 @gA,gB#. We adopt the conven-
tion
$gA,gB%52hAB ~15!
with hAB5diag(21,11,11,11).We use the chiral representation henceforth for con
nience and clarity. In the chiral representation
tesl-theice
ft-
-ou-si-
orheas
n-
ve-
gA5S 0 i tA
i tA 0 D , ~16!
where ta52 ta (a51,2,3! are Pauli matrices, andt05 t052I 2 . We also have
g55 ig0g1g2g3
5S I 2 0
0 2I 2D ~17!
and the Dirac bispinor is expressed in terms of twocomponent left and right-handed Weyl spinorsfL,R as
C5S fR
fLD . ~18!
The contractions in~13! are defined by
EAbDC[EmADmC. ~19!
Note that the vierbein vector fields and one-formsEA5Em
A]m andeA5eAmdxm, satisfyEAbeB5dA
B .In the chiral representation, the covariant derivative ca
be written as
DC[dxmH ]mI 42 iS Ama1
ta
20
0 Ama2
ta
2
D J S fR
fLD ,
~20!
and in conventional coupling of fermions to spin connections,A7 are, precisely,
Aa756 iv0a2
12 ea
bcvbc . ~21!
Written in terms of the Weyl spinors, the Dirac action is
inors. So
~22!
where
1We shall choose the2 action. There is an ambiguity in the conventions of the Dirac matrices,gA, which allows one to couple eitherA1 or A2 to left-handed Weyl spinors. We have adopted a choice which couples anti-self-dual spin connections andA2 to left-handedspinors. To the extent that there are no right-handed neutrinos in nature, we should describe nature with only left-handed Weyl sponce the initial choice of couplingA2 to the neutrino is made, the theory cannot be described byA1 and right-handedWeyl spinors.
sds
on
ir
r
t
erre-
a-r-s,
a
ns
53 5685STANDARD MODEL WITH GRAVITY COUPLINGS
D2fL[S d2 iAa2
ta
2 DfL , D1fR[S d2 iAa1
ta
2 DfR ,
~D2fL!†[dfL†1 ifL
†Aa1
ta
2, ~23!
~D1fR!†[dfR†1 ifR
†Aa2
ta
2.
Notice that in ~22!, terms ~1! and ~4! containA2 but notA1, while ~2! and~3! involve A1 but notA2. Furthermore,it is well known that right-handed spinors can be writtenterms of left-handed ones through the relation
fR52 i t2xL* ~24!
and vice versa, so the term~4! which involvesA2 can bewritten in terms of totallyleft-handed~anti-self-dual! spinconnections and Weyl spinors as
EM
~*1 !S 2i
2DxL†tAEAbD
2xL . ~25!
Similar remarks apply to the term~2! with regard to right-handed spinors and the fieldA1.
In order to couple spinors to Ashtekar connections of onone chirality, we can now define modified ‘‘chiral’’ Diracactions
S D2[2$~1!1~4!%
5EM
~*1 !~2 i !~fL†tAEAbD
2fL1xL†tAEAbD
2xL!,
~26!
S D1[2$~2!1~3!%
5EM
~*1 !~2 i !@fR† t AEAbD
1fR1EAb~D2fL!†tAfL#.
~27!
Next, we make the identification that the quantitiesA7, asanticipated by the notation, are precisely the connectionstroduced by Ashtekar in his simplification of the constrainof general relativity, and the very same variables in thSamuel-Jacobson-Smolin action of Eq.~1!. Instead of theconventional sum of the Einstein-Hilbert-Palatini action anthe full Dirac action,S D , the total action is now taken to be(S D
21S SJS2 ). Remarkably, it is possible to show@for in-
stance, by using Eq.~37!# that this total action reproduces thecorrect classical equations of motion for general relativiwith spinors@5,8#.
IV. DISCRETE TRANSFORMATIONS AND SPINORS
The usual discrete transformations for the second qutized spinor fields are
in
ly
in-tse
d
ty
an-
P:fL~x!↔2 i t2xL* „P21~x!…, i.e., @fL~x!↔fR„P
21~x!…#,
C:fL~x!↔xL~x!, i.e., ~Cc5CCT!,
T:fL~x!°2 i t2fL„T21~x!…. ~28!
A set of discrete transformations for the gravity variableconsistent with the ones described above for fermion fielcan then be defined as follows:
P:„s ia~xW ,t !,Aia7~xW ,t !…°~ s ia
„P21~xW ,t !…,Aia6„P21~xW ,t !…!,
C:„s ia~xW ,t !,Aia7~xW ,t !…°„s ia~xW ,t !,Aia
7~xW ,t !…, ~29!
T:„s ia~xW ,t !,Aia7~xW ,t !…°~ s ia
„T21~xW ,t !…,Aia7„T21~xW ,t !…!.
These transformations are consistent with the commutatirelations~10!. In dealing with curved spacetimes , it is moreconvenient to rewrite these transformations in terms of theaction on the one-forms (eA,Aa
7). These give
P:~e0,ea;Aa7!°~e0,2ea;Aa
6!,
C:~eA;Aa7!°~eA;Aa
7!, ~30!
T:~e0,ea;Aa7!°~2e0,ea;Aa
7!.
P and T transformations for the Ashtekar variables fopure gravity have been discussed previously@9#. However,that discussion did not coverC and CPT, nor take intoaccount the effect of coupling to fermions. Note also thaP andT are orientation-reversing transformations. WhilePandC are to be implemented by unitary transformations,T isto be implemented anti-unitarily, so that underT, c numbersare complex conjugated.
We emphasize that the Ashtekar variables are howevnot necessarily orientation-reversal invariant. In fact theare four-manifolds with no orientation reversing diffeomorphisms. The signature invariant,t, of a four-manifold is oddunder orientation reversal. Therefore a manifold with nonvnishingt cannot possess an orientation reversing diffeomophism. One can show that for the Ashtekar variable*MFa
2`F2a2* MFa1`F1a}t @10#. Thus for manifolds
with nonvanishingt ’s, A1 can neither be diffeomorphic norSO(3,C) gauge equivalent toA2.
To discuss the effect of the discrete transformations inconcise manner, we first defineAAB52ABA such that
A0a[1
2i~Aa
22Aa1!,
~31!
Abc[21
2eabc~Aa
21Aa1!,
and keep in mind the effect of the discrete transformatiodisplayed in~30!.
The curvature
FAB[dAAB1AAC`ACB ~32!
then has components
nayedv-irt
n-
enorewein
e
theasd,
e
5686 53LAY NAM CHANG AND CHOPIN SOO
F0a51
2i~Fa
22Fa1!, Fbc52
1
2eabc~Fa
21Fa1!, ~33!
and the ‘‘torsion’’ TA(A7), which depends onA7, is de-
fined asTA[deA1AA
B`eB. ~34!
With all this, it can be shown that the ‘‘chiral’’ gravita-tional and Dirac actions are related to the conventional oby
S SJS7 5
1
~16pG!EMeA`eB`* FAB
7i
~16pG!EM
@2d~eA`TA!1TA`TA#
22l
16pGEMe0`e1`e2`e3 ~35!
and
S D75S D1E
MF6
i
4cAeABCDe
C`eD`TB
7i
2~3! !d~cAeABCDe
B`eC`eD!G , ~36!
wherecA[fL†tAfL1xL
†tAxL .The combined fermionic and gravitational total action
therefore,
S total7 5S D
71S SJS7
5S D11
~16pG!EMeA`eB`* FAB
7i
2 EMdH 2
1
~8pG!eA`TA
11
3!~eABCDc
AeB`eC`eD!J7
i
~16pG!EM
QA`QA22l
16pGEMe0`e1`e2`e3,
~37!
whereQA[TA1~2pG!eABCDc
BeC`eD. ~38!
Observe that the sum of the conventional Einstein-HilbePalatini and Dirac actions is given by the first two termsthe second line of~37!.
We then find that underP ~and alsoCP andCPT), thechange is
nS D25PS D
2P212S D2
5 i EM
1
3!d~eABCDc
AeB`eC`eD!
21
2eABCDc
AeC`eD`TB ~39!
nes
is,
rt-in
and the change of the total action is then
nS total2 5nS SJS
2 1nS D2
5 i EMdH 2
1
~8pG!eA`TA
11
3!~eABCDc
AeB`eC`eD!J1
i
~8pG!EM
QA`QA. ~40!
It should be noted that for spacetimes with Lorentziasignature, the Ashtekar variables are not real but rather mbe required to satisfy reality conditions which are supposto be enforced by a suitable inner product for quantum graity @1#. Moreover, the actions are not explicitly real and thehermiticity could be tied to the inner product for the yeunavailable quantum theory of gravity.2
However, if we are interested in examining second quatized matter in a gravitational background~for that matter, inflat spacetime!, we may enforce the reality conditions onA2 by hand. We assume for our present purposes, that evthough we do not at the moment have an inner product fquantum gravity that will enforce these reality conditions, wcan pass over to the second order formulation wherebyeliminate the Ashtekar connections in terms of the vierbeand spinors through the equations of motion forA2 and en-force the reality conditions by inspection.
So, varying the total first order action,S D21S SJS
2 , withrespect toA2 yields
DS2a5~4pG!F13 ebcdcaeb`ec`ed1
1
2eabcc
0eb`ec`e0
1 icbea`eb`e0G ~41!
which implies that the Ashtekar connection is
Aa25 iv0a2
1
2ea
bcvbc2~2pG!eabcF12 ebcABcAeB2 icbecG .
~42!
Upon imposing the usual Hermiticity requirements on thspinors ~which implies that cA is Hermitian!,(A2)†5A1and the requirement that the spin connectionv isreal, one finds thatQ50, which for the case of pure gravity,reduces to the torsionless condition. Thus passing over tosecond order formulation where the Ashtekar connection hbeen eliminated and reality conditions have been imposewe find that the change in the action underP ~and alsoCPandCPT) is
2Recall that pure imaginary local Lorentz invariant pieces of thaction areCPT odd.
-
be
to
p-
le
ts
e
ac-o
rd
is
edf-
a-he
53 5687STANDARD MODEL WITH GRAVITY COUPLINGS
nS total2 52
i
12EMd$eABCDcAeB`eC`eD%
52i
2EM]m@det~e! j 5m#d4x
52i
2EM ~¹m j 5m!det~e!d4x
52i
2nQ5 , ~43!
where
j 5m5CEA
mgAg5C
52EAm~fL
†tAfL2fR† tAfR!
52EAm~fL
†tAfL1xL†tAxL! ~44!
is precisely the chiral current, andnQ5 is the change in thechiral ~axial! charge. Therefore, it would appear that if thchiral current is not conserved, the action fails to beHermit-ian and invariant underP, CP, andCPT. In this respect, theAdler-Bell-Jackiw~ABJ! anomaly or the axial anomaly@11#induced by global instanton effects may lead to violationsCPT through chirality changing transitions. Chiralitychanging transitions due to QCD instantons have been invtigated by others before@12#, including ’t Hooft @13# in hisresolution of the U(1)A problem. We shall postpone the discussion on the physical implications and bounds on theparent non-Hermiticity of the action and the implied violations of discrete symmetries to a separate report. A reladiscussion on the effects of discrete transformations ontions which utilize self-dual variables in the descriptionfour-dimensional gravity can be found in Ref.@14#.
Although the expressions and discussions above are fsingle quark or electron, it is easy to incorporate more mafields into the theory simply by replacingcA with
cA[( FL†tAFL , ~45!
where the sum is over all Weyl fermions, including righhanded fermions which are to be written as left-handed onConventional standard model Yukawa mass couplings aHiggs fields can be introduced without modifications.
V. THE STANDARD MODEL WITH ASHTEKAR FIELDS
We next examine how the quark and lepton fields of tstandard model are to be coupled to the Ashtekar fieSince we are allowing couplings toA2 rather than the wholespin connection, only left-handed fermions can be coupledthe theory. By writing all right-handed fields in terms oleft-handed ones, it is possible to couple all the standmodel quark and lepton fields to Ashtekar gravity in a cosistent manner. We shall show in the next section that tcoupling will not disturb the cancellations of all perturbativanomalies, given the multiplets of the standard model. Tquestion of how to deal with the global anomaly that may
e
of-es-
-ap--tedac-of
or atter
t-es.nd
helds.
tofardn-hisehebe
present in the theory will be examined in the following section.
We shall label collectively byWiTi the gauge connection
one-forms of the standard model. TheTi ’s denote generatorsof the standard model gauge group, and there should notany confusion between this indexi and spatial indices. Thecovariant derivative inS D
2 in the action is then replaced bythe total covariant derivative
D2fLI5F S d2 iAa
2ta
2 D d IJ2 iWi~Ti ! IJGfLJ
, ~46!
where the indexI associated withfLIdenotes internal ‘‘fla-
vor’’ and/or ‘‘color.’’ In this modified model, the right-handed Weyl fields of the conventional standard model arebe written as left-handed Weyl fermions via the relation
fRI52 i t2xLI
* . ~47!
So an electron or a single quark of a particular color is reresented as a pair of left-handed Weyl fermions (fL andxL), and a left-handed neutrino is represented by a singleft-handed Weyl fermion.
The total action is still to beS SJS2 1S D
2 with the totalcovariant derivative of~46! but summed over all left-handedmultiplets, including ‘‘right-handed’’ multiplets which arewritten as left-handed ones. Using Eq.~47!, right-handedcurrents can be written in terms of left-handed currenthrough
JRm i5fRI
† tm~TRi ! IJfRJ
5xLI† tm~TL8
i ! IJxLJ, ~48!
with
TL8i52~TR
i !T. ~49!
Notice however that terms containing the ordinary gaugconnectionsW are Hermitian, so the factor 2 multiplyingterms denoted by~1! and ~4! in ~22! takes care of the con-tributions from terms withW in ~2! and ~3! which wouldhave been present had we used the conventional Diraction. Similarly, this holds also for the left-handed neutrinalthough it is now described by only the term~1! ~summedover the species! instead of~1! and~2!. The net result of allthis is that the difference between the ‘‘modified standamodel’’ with gravity described byS D
21S SJS2 with couplings
only to left-handed fermions and the conventional actionstill as described by~37!.
VI. CANCELLATION OF PERTURBATIVE GAUGEANOMALIES
We shall first demonstrate that the standard model definentirely in terms of left-handed fermions, and the anti-seldual Ashtekar fieldsA2, is free of perturbative chiral gaugeanomalies.
Recall that the anomalies can be determined via Fujikwa’s Euclidean path integral method. We can expand tleft-handed multiplet,CLI
5(fLI
0 ), andCLI, in terms of the
rd
a-
eoup
fer-ye
e
ly.nge,
p-
fo--n-alw-arer-
fto
ly
5688 53LAY NAM CHANG AND CHOPIN SOO
complete orthonormal set$XnL ,Xn
R% ~see, for instance, Ref.@15#! with
XnL~x!5S 12g5
A2 DXn~x!, ln.0,
5S 12g5
2 DX0~x!, ln50,
~50!
XnR~x!5S 11g5
A2 DXn~x!, ln.0,
5S 11g5
2 DX0~x!, ln50.
There can be more than one chiral zero modes with thleft-right asymmetry governed by the Atiyah-Patodi-Singindex theorem. In our present instance, the curved spDirac operator contains the full set ofA6 connections be-sidesW, and remains Hermitian providedTA50; andXn’sare the eigenvectors with eigenvaluesln of the full Euclid-ean Dirac operator. The chiral projections in~50! and ~51!serve the purposes of defining a chiral fermion determinafor the effective action as well as selecting only theA2 Ash-tekar fields in the action*(*1) CLig
mDmCL . This isachieved by using the Euclidean expansions@16#
CL~x!5 (n:ln>0
anXnL~x!,
~51!
CL~x!5 (n:ln>0
bnXnR†~x!.
The diffeomorphism invariant Euclidean measure for ealeft-handed multiplet,
)xD@e1/2~x!CL~x!#D@e1/2~x!CL~x!#, ~52!
wheree[det(e), is ~up to a phase! equivalent to~see Ref.@15#!
dm5)n
dan)n
dbn . ~53!
Under a gauge transformation with generatorTi ,
CL~x!→e2 ia~x!TiCL~x! ~54!
the measure transforms as
dm→dmexpH 2 i E a~x!Ai~x!dxJ . ~55!
The anomaly can be computed as in Ref.@15#. This gives
eirerace
nt
ch
Ai~x!5 (all n
Xn†~x!g5TiXn~x!det~e!
521
16p2TrHTi 12 emnabGabGmnJ , ~56!
and is proportional to Tr(Ti$Tj ,Tk%). Here,Gmni is the field
strength associated withWmi . So, the condition for cancella-
tion of perturbative gauge anomalies is that Tr(Ti$Tj ,Tk%)when summed over all fields coupled toWiT
i vanishes. Butas we have seen, if ‘‘right-handed’’ spinors in the standamodel coupled toWiTR
i are written as left-handed spinorscoupled to WiTL8
i such that the representationsTL852(TR)
T, then
Tr~T8Li $T8L
j ,T8Lk%!52Tr~TR
i $TRj ,TR
k %!, ~57!
and, the condition for the perturbative chiral gauge anomlies of left and ‘‘right’’-handed fermions to cancel is
Tr~TLi $TL
j ,TLk%!1Tr~T8L
i $T8Lj ,T8L
k%!50, ~58!
which is precisely equivalent to the well-known condition@17#
Tr~TLi $TL
j ,TLk%!5Tr~TR
i $TRj ,TR
k %!. ~59!
The Ashtekar fields do not give rise to perturbativanomalies because the generators of Ashtekar gauge grbelong to su~2!, and SU~2! is a ‘‘safe group’’ where Tr(Ti$Tj ,Tk%)50 for any representation. The introduction othe Ashtekar fields therefore does not disturb the usual pturbative anomaly cancellation conditions. But the theorcan still be afflicted with global anomalies, which is the issuwe shall address next.
VII. GLOBAL ANOMALY AND FERMION CONTENT
Witten @18# showed that in four dimensions, theories withan odd number of Weyl fermion doublets coupled to gaugfields of the SU~2! group are inconsistent. Without gravity,the standard model has four SU(2)weakdoublets in each gen-eration, and so it is not troubled by such a global anomaBut what about the Ashtekar gauge fields? Strictly speakifor manifolds of Lorentzian rather than Euclidean signaturthe group is complexified SU~2! and isomorphic to SL(2,C). But both these groups have the homotopy grouP4(G)5Z2 , and the nontrivial transformations of the complexified SU~2! group inP4(G) are associated with the ro-tation group. As Witten@18# has argued, the presence onontrivial elements of this homotopy group can produce glbal SU~2! anomalies. So it would appear that in four dimensions there could be further constraints on the particle cotent in order to ensure that the theory be free of the globanomaly associated with the Ashtekar gauge group. Hoever, unlike the pure gauge case, gravitational instantonsstrongly correlated with the topology of spacetime, and aguments based onP4(G) cannot be carried over straightfor-wardly. In what follows, we present a unified treatment oboth gravitational and pure gauge instanton contributionsthe global anomaly for Weyl fermions.
We review the essential points of the global anoma
n
theiralfor
acbe
53 5689STANDARD MODEL WITH GRAVITY COUPLINGS
@18,19#. To begin with, consider a suitable Wick rotation othe background spacetime into a Riemannian manifold. Nall manifolds with Lorentzian signature have straightforwacontinuations to Euclidean signature and vice versa. A mgeneral setting for this section is the scenario of matcoupled to gravity in the context of path integral Euclideaquantum gravity@20#. The Dirac operator is then Hermitianwith respect to the Euclidean inner product^XuY&5*d4x det(e)X†(x)Y(x) @15#. The fermions can then be expanded in terms of the complete set of the eigenfunctionsthe Dirac operator. The expansions will be the same@16# asin Eq. ~51! with the eigenfunctions normalized so that
EMd4x det~e!Xm
† ~x!Xn~x!5dmn . ~60!
fotrdoretern
-of
Consider next a chiral transformation byp which mapseach two-component left-handed Weyl fermioCL(x)°exp(ipg5)CL(x)5exp(2ip)CL(x)52CL(x). Obvi-ously such a map is a symmetry of the action. However,measure is not necessarily invariant under such a chtransformation because of the ABJ anomaly. Instead,each left-handed fermion, the measure transforms as
dm°dm expS 2 ipEMd4x det~e!(
nXn†~x!g5Xn~x! D . ~61!
The expression*Md4x(n det(e)Xn
†(x)g5Xn(x) is formallyequal to (n12n2), wheren6 are the number of normaliz-able positive and negative chirality zero modes of the Diroperator. Upon regularization, the expression works out to
(n
det~e!Xn†~x!g5Xn~x
dof
dedn
b-rene
rneont-the-
ly,d
x
s
t
-
to
dofa-eit,a-el-
-al
ald
5690 53LAY NAM CHANG AND CHOPIN SOO
Now, in a scenario where all of the fermions are couplto one another via gauge fields, the index theorem shouldapplied to the trace current involving the sum over all tfermion fields. The resultant restriction onN is the conditionthat the index for the total Dirac operator withSpinG(4) connections, N12N2 , is even. Otherwise,Green’s functions which contain (N12N2) more CL thanCL variables in the odd index sector will be inconsistenThe relevant index of the total Dirac operator in the presenof an additional grand unification gauge field other than tspin connection is
N12N252Nt
82
1
8p2EMTr~F`F !, ~63!
whereF5FiT i is the curvature of the internal grand unification gauge connection. The global anomaly consistecondition given above then implies that for arbitraryt ’sthere can only be 16k fermions in total in the theory, wherek is an integer; and the gauge group should be selected sthat the instanton number,(1/8p2) *MTr(F`F), is alwayseven for nonsingular configurations.
If one counts the number of left-handed Weyl fermionthat are coupled to the Ashtekar connection for the standmodel, one finds that the number is 15 per generation, giva total of 45 for 3 generations. This comes about becaeach bispinor is coupled twice to the Ashtekar connectwhile each Weyl spinor is coupled once~e.g., for the firstgeneration, the number is 2 for each electron and each udown quark of a particular color, and 1 for each left-handneutrino.! Thus even if one restricts tot58k, the globalanomaly with respect to the Ashtekar gauge group seemimply that there should be additional particle~s!. For ex-ample, there could be a partner for each neutrino, makN to be 16 per generation, or a partner for just thet neutrino,or even four generations. As a result, even witht58k, grandunification schemes based upon groups such as SU~5! andodd number of Weyl fermions would be inconsistent whcoupled to gravity.
The net conclusion is therefore that inclusion of manifolwith arbitrary t ’s in the integration range is possible, provided we allow for generalized spin structures, with a totalN516k fermions, in order that the global anomaly is abseIn the event that the structures are defined by simple Ggroups, the simplest choice would be SO~10!. It is easy tocheck that the 16 left-handed Weyl fermions in the SO~10!5Spin(10)/Z2 GUT indeed belong to the 16-dimensionachiral representation of Spin(10) and satisfy the generalizspin structure equivalence relation for$Spin(4)3Spin(10)%/Z2 .
It is worth emphasizing that this generalized spin structuimplies an additional ‘‘isospin-spin’’ relation in that fermions must belong to Spin(10) representations, while bosmust belong to SO~10! representations of the GUT. This haimplications for spontaneous symmetry breaking via fundmental bosonic Higgs fields, which cannot belong to tspinorial representations of SO~10! if one allows for arbitraryt ’s. More generally, the presence of the extra particle~s! im-plied by global anomaly considerations can generate mas
edbe
he
t.cehe
-ncy
uch
sardinguseion
p ored
s to
ing
en
ds-ofnt.UT
led
re-onssa-he
ses
for neutrinos, thereby giving rise to neutrino oscillations, anmay also play a significant role in the cosmological issue‘‘dark matter.’’
A more mathematical treatment of the perturbative anglobal anomaly cancellation conditions we have discusscan be given. It is based on the fact that the chiral fermiodeterminant is actually asection of the determinant linebundle@24–27#. In order for the functional integral over theremaining fields to make sense, there must not be any ostruction to the existence of a global section. It is therefonecessary for the first Chern class of the determinant libundle to vanish@24,25#. This gives the result for perturba-tive anomaly cancellation. However, a vanishing first Checlass does not necessarily imply a trivial connection for thdeterminant line bundle. There can be a further obstructito a consistent definition of the chiral fermion determinandue to nontrivial parallel transport around a closed loop. Indeed, it can be shown that under suitable assumptions,holonomy of the Quillen-Bismut-Freed connection of the determinant line bundle is given by (21)index@Dirac(M ,A)#
exp(22ipj). Herej5 1/2 (h1h), with h andh being thespectral asymmetry and number of zero modes, respectiveof the Dirac operator on the five-dimensional manifolM3S1 endowed with a suitably scaled metric onS1 @26,27#.In order to evaluatej, one can viewM3S1 as the boundaryof a six-dimensional manifold,M6 , with a product metric onthe boundary, and apply the Atiyah-Patodi-Singer indetheorem to obtain
index~Dirac!M652E
M6
A`ch~F !2j~M3S1!. ~64!
It is instructive to work out the terms and their implicationexplicitly. For the case relevant to us,M is four-dimensionaland we pick out the six-forms in the integrand in the firsterm on the right-hand side. These are of the form Tr(F`F`F) andRAB`RAB`Tr(F). The former is the Abeliananomaly inD12 dimensions~hereD54) which gives riseto the perturbative non-Abelian gauge anomaly inD dimen-sions@24,25,28#. It vanishes when the perturbative anomalyfree condition Tr(T i$T j ,T k%)50 is satisfied. The latter,which is the gravitational-gauge cross term, is requiredvanish@Tr(F)50# for the absence of mixed Lorentz-gaugeanomalies when there are couplings to Abelian fields@29#.These terms occur in the holonomy theorem of Bismut anFreed because the holonomy measures the nontrivialitythe connection of the determinant line bundle, and pertubtive anomalies which correspond to the nontriviality of thfirst Chern class must necessarily contribute. In this spirdiscussions of global anomalies make sense only if perturbtive anomalies cancel. With the perturbative anomaly canclation conditions, we therefore conclude thatj is an integersince index~Dirac!M6
is an integer. Thus the holomony theorem tells us that the remaining obstruction, i.e., the globanomaly is due to (21)index@Dirac(M ,A)#, and the consistencycondition is again the requirement that the index of the totchiral Dirac operator in four dimensions with generalizespin structure is even.
the5-nder-
terenter
53 5691STANDARD MODEL WITH GRAVITY COUPLINGS
We have presented our arguments of the global anomin terms of the Ashtekar variables for definiteness. We wolike to end by emphasizing that our results would also obtin the setting of Weyl fermions coupled to conventiongravity and ordinary spin connections instead of the Astekar connections@30#. This is essentially because thanomaly can be understood as coming from the inconsischange in the fermionic measure or the chiral fermion detminant, and for both schemes, the changeexp@2ip(N12N2)# 5(21)index (Dirac).
alyuldainalh-etenter-is
ACKNOWLEDGMENTS
The research for this work has been supported byDepartment of Energy under Grant No. DE-FG092ER40709, the NSF under Grant No. PHY-9396246, aresearch funds provided by the Pennsylvania State Univsity. One of us~C.S.! would like to thank Abhay Ashtekar,Lee Smolin, Jose Mourao, and other members of the Cenfor Gravitational Physics and Geometry for encouragemand helpful discussions. L.N.C. would like to thank PetHaskell for a helpful discussion.
l
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