Chiral Gauge Theories and Fermion-Higgs Systems

17 Pages·0.27 MB·English

by  Donald N. Petcher

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Chiral Gauge Theories and Fermion–Higgs Systems
Donald N. Petchera
aDepartment of Physics, Washington University, Saint Louis, Missouri 63130 USA
Thestatusofseveralproposalsfordefiningatheoryofchiralfermions onthelatticeisreviewedandsomenew
3
estimates for theupperbound on theHiggs mass are presented.
9
9
1
1. Introduction proach
n
a
The amount of work on lattice theory relat- • the Rome approach and its relation to the
J
ing to chiral gauge theory has dramatically in- above
2
creased this year with some forty papers in the
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subject, represented by about twenty contribu- • the staggered fermion approach

1 tionstothisconference. Incontrast,onlyahand-
• mirror fermions
v ful of papers have been written in the broader
5
fieldoffermion–Higgssystemswithoutaneyeto- • Zaragoza“replica” fermions
1
ward chiral gauge theory. Therefore, with one
0
1 exception, I have decided to limit my review to • the topological method
0 thetopicofchiralgaugetheories,andwithinthat
3 I will primarily emphasize the conceptual devel- • contributionofdoublerstotheS–parameter
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opments rather than numerical simulations. The
/ • the Banks “U(1) problem”
t exception is some work in the pure Higgs sec-
a
l tor concerning the regularization dependence of Suggestions that were not represented at the
-
p the upper boundforthe Higgsmass[1]towhichI meeting are:
e would like to draw your attention.
h LastMarch,averypleasantworkshopwasheld • the Bodwin-Kovacs proposal
:
v in Rome, on the topic of non-perturbative ap-
• reflection positive fermions
i proaches to chiral gauge theories. In that work-
X
shop, virtually all proposals around at the time which I would also like to mention, and finally I
r
a were presented, following which some quite in- would like to spend some time to discuss a pro-
tense discussions took place in the proverbial posalmade after the conference which is perhaps
smoke-filled back room1 in an attempt to under- themostinterestingproposalpresentlyontheta-
stand the problems of some of the proposals and ble:
the connections between them. Some valuable
• domain wall fermions
insights were gained in these discussions and in
my talk I will attempt to portray the underlying
Aside from these, a rather pessimistic contribu-
theme that we arrived at during that meeting.
tion concerning the use of the random lattice as
The topics discussed in the Rome workshop are
a solutionto the chiralfermionproblem was pre-
the following:
sented at this conference[2], although I am not
• the failure of the Wilson–Yukawa approach prepared to discuss this work.
Naturally with the large amount of material
• consequences for the Eichten–Preskill ap- represented above, I have been forced to make
1Although due to federal regulations the smoke was not some subjective choices about what to cover and
allowed what to omit. So keeping in mind my attempt
2
to focus on the conceptual development of the
field, I have decided to omit discussion of mirror
fermions[3] for the most part with a twofold jus-
tification. First, that theory is a vector theory in
principle and not an attempt at producing a chi-
ral theory. The chiral theory is modeled at low
energy,with the chiralpartners ofthese fermions
appearing close to the electroweakscale. Second,
and perhaps more to the point, the theory has
already been well represented in plenary talks of
the last two lattice conferences[4] with little con-
ceptual development since then. I would like to
comment though that Montvay gave us a nice
talk in Rome indicating that the theory of mir-
rorfermionsisnotincontradictiontoanyknown
experimental phenomena to date, which should
force us not to foreclose on the possibility that
mirror fermions actually do exist in nature, and
the world is not chiral after all. Work goes on
in the numerical arena, and as this is the first
year that the effects of fermions on the upper
Figure 1. Graphs containing numerical data and
bound of the Higgs mass have been reported, I
results from large N calculations for hypercubic
will mention some results obtained fromthe mir-
(HC), F and Symanzik improved lattice regu-
4
ror fermion method when I come to this topic.
larization schemes, and with Pauli–Villars (PV)
I will also say little about the Banks problem
regularization,includingresultsfromactionswith
other than to point out what it is and how it
dimension 6 operators.
can be resolved in various of the proposals, and
I will not have time to discuss the contribution
of doubler fermions to the S–parameter[5]. All
other topics mentioned above will find their way have been substantially broadened, with investi-
into mytalk,althoughinseveralcasesI willhave gations of theories with actions including higher
time only to present the barest essentials. derivativeterms,usingthe1/N approximationas
well Monte Carlo calculations[1].
A typical graph of the results of a calculation
2. HIGGS MASS UPPER BOUND
of the Higgs mass in units of f versus the Higgs
π
Before a discussion of chiral fermions I would mass in cutoff units is given in figure 1. These
like to briefly describe some progress on a topic graphs represent calculations on an F lattice, a
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that falls under my jurisdiction: recent results hypercubic lattice with standardaction,a hyper-
onregularizationdependence ofthe upper bound cubic lattice with a Symanzik improved action,
to the Higgs mass. In past years, calculations of and a calculation with Pauli–Villars regulariza-
the upper bound to the Higgs mass have concen- tion, as well as results from actions including
tratedprimarilyonatheorywithstandardaction higherdimensionoperators. Theresultsfromthe
and hypercubic lattice[6,7], with the only devia- 1/N expansion (extrapolatedto N =4) with the
tion being studies on an F lattice[8]. Fixing f standard action and an action including higher
4 π
at its experimental value of 246 GeV (within the dimension operators are given in each case by
σ-model context), a typical value for the upper the dotted and solid lines respectively. The main
bound ofthe Higgsmassinthese studies was640 point I would like to make concerning this graph
GeV. This year, for the first time the horizons is that both adding ‘irrelevant’ terms to the ac-
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tionwhichrepresentcutoffeffects,andlookingat 4. WILSON–YUKAWA,
a broader class of regularization schemes, allows EICHTEN–PRESKILL AND ROMA
theupperboundontheHiggsmasstoberelaxed.
There is very strong evidence by now that the
Withtheusualrequirementthatthecutoffeffects
Wilson–Yukawa approach fails to result in a the-
are only a few per cent on pion scattering, from
ory of lattice chiral fermions, and that unfortu-
this analysis a conservative estimate for the up-
nately its failure takes downthe Eichten–Preskill
per bound is about 750 GeV ± 50 GeV which is
approach with it. In this section I review why
somewhat higher than the old bound. In addi-
this is so, and indicate how the findings are re-
tion, the decay width is found to be about 290
lated to the Rome approach and why the latter
GeV which is somewhat broader than the tree
may escape the problems that arise in the first
level value of 210 GeV, indicating that in the re-
two.
gion in question the Higgs particle may be more
strongly coupled than previously believed. I will
4.1. The Wilson–Yukawa approach
come back to this topic shortly when I consider
To understandthe problems involved,it is suf-
the effects of fermions on these numbers.
ficient to consider a simple case of the Wilson–
Yukawa model containing only a single species of
fermion, and in which only the left-handed chi-
3. THE BANKS “U(1) PROBLEM” ralsymmetryisgauged,leavingtheright–handed
chiral symmetry rigid. The model is defined by
Up to a little over a year ago, most models
the fermion action[10,11,12]
whichwereunder investigationas theoriesofchi-
ralfermionspossessedarigidU(1)symmetrycor- S = 1ψ γ (D +D˜ )ψ
WY 2 x µ µ µ x
responding to fermion number which commutes
Xx,µ
with the gauge symmetry. Banks pointed out
+ y [ψ V ψ +ψ V†ψ ] (1)
that for a realistic theory which hopes to repro- Lx x Rx Rx x Lx
duce the standard model either directly or as a Xx
w
low energy approximation to some other chiral − [ψ V 2ψ +ψ 2(V†ψ )]
2 Lx x Rx Rx x Lx
theoryobtainedfromthe lattice,this presenteda Xx
problem from the point of view of anomalies[9].
along with the usual actions for the gauge and
Forinthestandardmodel,baryonnumberB and
scalarfields,whereψ andψ areright-andleft-
R L
leptonnumberLareeachindependently violated handedchiralfermionfields 1(1−γ )ψ and 1(1+
2 5 2
by instanton processes, and only the difference γ )ψrespectively,D andD˜ aretheforwardand
5 µ µ
B−L is conserved, whereas the rigid U(1) sym-
backward lattice derivatives respectively, gauged
metry in the lattice theories would insure that
with respect to the gauge field for the left chiral
bothB andLwouldbe conservedindependently. symmetry UL, 2isthe lattice laplacian,andV is
Although this problem had been overlooked for µ
a scalar field with frozen radius taking its value
the most part2 in the face of other more severe
in a groupG which for our purposes is takento
L
problems, the comments by Banks did force the
be U(1) or SU(2).
rest of us to ‘come clean’ and solutions to the
Aside from the term ‘Wilson–Yukawa’ term
problem for the various models formed a part of
containing w, this action is just that which pro-
our discussion in the Rome workshop. There are
duces a naive (doubled) theory of gauged chiral
several ways around the problem that are either
fermions coupled to a scalar field via a Yukawa
built in explicitly or realized dynamically by the
interaction. The only differences between this
various models which I will point out as we go
and the usual standard model are the presence
along.
ofthedoublers,andanextraneutrinowithright-
handed chiral symmetry. In the broken phase at
2It should be pointed out that Eichten and Preskill did treeleveltheWilson–Yukawatermtakestheform
addressthisproblemdirectlyintheirapproach. wvψ2ψ where v =hVi, so that naively,this term
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hVi=v, including the masses of the doublers:

m =v(y+2nw) n=0,1,2,3,4. (3)
f
FM
Therefore, in the PMW phase (region I of figure
2), all 16 fermions are massless, and in the bro-
FM
kenphase(regionII),thedegeneracyislifteddue
to a non-vanishing value of v. However, as the
II III
I IV continuum limit is approached, v scales to zero,
andalldoublersappearinthephysicalspectrum.
PMW PMS
Thustheweakcouplingregionisclearlynotare-
gioninwhichtolookforasolutiontothedoubler
w
problem. This has also been checked via Monte
Carlo calculations[14].
PMS To study the strong w region, it is convenient
tochangevariablestofermionfields,neutralwith
respect to g-gauge symmetry:
y
ψ(n) =V†ψ , (4)
x x x
in terms of this field, the fermion part of the ac-
tion becomes
Figure 2. the phase diagram for the Smit–Swift
S′ = [1ψ(n)γ (D(Wµ)+D˜(Wµ))ψ(n)
model at small gauge coupling WY x,µ 2 Lx µ µ µ Lx
P+ 1ψ(n)γ (∂ +∂˜ )ψ(n)] (5)
2 Rx µ µ µ Rx
looks like a Wilson mass term, which could serve + ψx(n)(y− w22)ψx(n),
as a remedy for the doubling problem. Xx
The action possesses a local chiral gauge sym- whereD(Wµ)isacovariantderivativewithrespect
µ
metry
to the composite gauge field W =V†U V :
µ x µx x+µˆ
ψ → g ψ
VLx → gxVLx (2) Dµ(Wµ)ψL(nx) =WµxψL(nx)+µˆ−ψL(nx). (6)
x x x
UL → g UL g† Weshouldnotethatthisactionlookslikethatofa
µx x µx x+µˆ
chiral gauge theory invariant under the following
which we will refer to as g-symmetry. symmetry [15,16], (which we will here denote as
The phase diagram for this model for small h-symmetry):
gauge coupling is well known by now [4] and is
representedinfigure2. ThepureHiggsmodelex- ψ(n) → h ψ(n), (7)
Lx x Lx
hibitsaphasetransitionforsomeκc,belowwhich W → h W h† ,
liesasymmetricphaseandaboveabrokenphase. µx x µx x+µˆ
Taking y →0 insures the presence of atleast one along with a symmetry breaking Wilson mass
light fermion[13], so we will limit our discussion term. This form of the lagrangian appears as
to that case. In the weak w region, the critical the starting point for the Rome proposal with
point extends into the w − κ plane, separating therelationbeingmademanifestinunitarygauge
a symmetric (PMW) phase from a broken (FM) (V =1),inwhichψandψ(n) areequivalentasare
phase. Perturbationtheoryappliesinthisregion, U and W . We will return to this point later.
µ µ
andjustasinusualstandardmodelperturbation (See also [17,18].)
theory,allfermionmassesareproportionaltothe At strong w, the Schwinger–Dyson equations
vacuum expectation value of the Higgs particle for the right-handed fermion, which happen to
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correspond to the Ward identities for a rigid strated in the broken phase taking into account
fermionshift symmetry, leadto someconclusions the contributions of the scalar field exactly[16].
about the effective action of the model[19]. In The interpretation is that once the scalar fields
particular, ψ(n) does not appear in any interac- formboundstateswiththefermionssoastoform
R
tions, and the fermion mass spectrum is Dirac fermions which are neutral with respect to
the g-symmetry, they then screen these fermions
m(n) = Z (y+2nw), n=0,1,2,3,4, (8) from any interaction with the gauge fields.
f 2
p A quick way to look at this is to notice that
where Z is the wave function renormalization
2 in the interaction lagrangian, W couples to the
µ
whichcanbedeterminedbyexpansiontechniques fermions through the operator V†D V − h.c.
x µ x
or by Monte Carlo calculations. The important
which is a dimension three operator in contrast
thing is that it is finite and non-vanishing. Thus
to the usual gauge field. Thus in the continuum
in the broken (FM) phase all fermions are mas-
limit the interactiontermis dimension6andwill
sive, and a single massless fermion can be ar-
need two extra powers of the lattice spacing to
ranged by tuning y to zero. In the symmetric
compensate. Inthebrokenphasethismeansthat
phase the spectrum is exactly the same. theinteractionmustbeproportionaltov2 ≡v2a2
In addition, an examination of the propa-
(where v is dimension 1), and in the symmetric
gator for the composite field ψx(c) = Vxψx(n) phase a factor of Λ2a2 will emerge, where Λ is
which has non-zero charge with respect to g-
the scale associated with the gauge fields (e.g.
symmetry reveals that in the absence of gauge
“Λ ”). In both cases, the interaction vanishes
QCD
fields, apparently no fermion states with non- in the continuum limit.
singlet g-quantum numbers appear in the sym-
Now we can summarized what we learned so
metric phase[20,21,22,23]. One can understand
far. First,fermionspairupasvectorseverywhere
this by realizing that the large coupling w has in the phase diagram. Second, There are no g-
caused the original left-handed charged fermion
charged fermions in the PMS phase. And third,
ψ to form a bound state with V† and this in
L in the strong w region, the g-neutral (h-charged)
turn pairs up with ψ to form a Dirac fermion,
R fermion is a free particle. It is possible to define
screening the charge of the original state. In the
atheorywithg-chargedstatesthatappearinthe
broken phase, ψ(c) mixes with ψ(n) and does not
physical spectrum, by changing the form of the
represent an indepedent state.
Wilson–Yukawa term, and these do couple with
With only the neutral fermions as physical
the gauge field, but they do so in a vector-like
states, we next must examine the couplings to manner[19]. Hence we have learned several valu-
thesefermionstoseeiftheyinteract. Wehaveal-
able lessons from the study of this model which
ready noted that in the effective action, all bare
may serve as warnings for future investigations:
interactions of the right-handed fermion vanish.
For the left-handed fermion, it is a simple mat- • If there is a strong coupling in the game,
ter to see that its Yukawa interaction with the andchirallyoppositeboundstatescanform
Higgs field is proportional to v which vanishes topairupasDiracfermions,theylikelywill.
in the PMS phase and scales to zero in the FM
phase near the phase transition (region III). The • In this model, the scalar field V is the cul-
situation is similar for higher order couplings be- prit. V screensthechiralgaugeinteractions
tween this fermion field and the scalar. Finally, anddecouplingV bylettingitsmassdiverge
for small gauge coupling and hopping parameter (as deep in the PMS phase) doesn’t work.
α = 1/(4w + y), one can show that the gauge
• Itisnotaprioriclearwhichstatesarephys-
coupling to the neutral field also vanishes3 in
ical by looking at the lagrangian.
the scaling region[19,25], and this can be demon-
3One author is not in agreement[24], but I believe the • Anomalies play no role (see also next sec-
argumentsreferredtoheretobecorrect. tion)
6
One final comment: note the dimension depen- explicit symmetry breaking terms to the action.
dence of the conclusions regarding the gauge in- The action for the SO(10) theory can be written
teraction term. In two dimensions, the operator as follows.
V†D V −h.c.isdimensiononeandthequickar-
guxmeµntxabove would not require the interaction SEP = 21[ψL†xσˆµψLx+µˆ−h.c.] (9)
to vanish. Indeed there is evidence that in a two Xxµ
dimensional theory one can constructa theory of − λ [(ψTσ Tψ )2+h.c.]
chiral fermions using the Smit–Swift model[26]. 24 L 2 L
Xx
This case also gives us the first scenario for es- r
− [∆(ψTσ Tψ )2+h.c.],
caping the Banks “U(1) problem”. In two di- 48 L 2 L
mensions, apparently the fermion forms a bound Xx
state with the scalar to form a vector multiplet, where ψi is a Weyl spinor in the 16 representa-
L
but the left-handed component nevertheless in- tionofSO(10),Taij isanSO(10)invarianttensor
teracts with the composite gauge field in a chiral
transforming as 16×16×10, σˆ =(1,i~σ), where
µ
fashion. Thus h-symmetry emerges as the sym-
~σ are the Pauli matrices, and
metry associated with this coupling. This turns
outtobe arealizationofthe Dugan-Manoharso- ∆(ψ ψ ψ ψ) = −1 [ψ ψ ψ ψ (10)
i j k l 2 ix+µˆ jx kx xl
lution to this problem[27], namely that the cur- X±µ
rent associated with the relevant gauge symme-
+ ···−4ψ ψ ψ ψ ].
ix jx kx xl
try is not what one would expect a priori. In-
deed the U(1) rigid symmetry commutes with g- This latter construction allows the term in the
symmetry, but the states are classified according actionproportionaltor toplaytheroleofaWil-
to h-symmetry. The currentfor this symmetry is son mass term, breaking the degeneracy in the
related to that for g-symmetry by a local coun- mass spectrum between the fermion and its dou-
terterm, and so the Noether current for the rigid blers. Note that the quantity ψTσ Taψ explic-
L 2 L
U(1) symmetry is not the h-gauge invariant cur- itly breaks fermion number and so by construc-
rent. In other words, the rigid U(1) symmetry tiontheEichten–Preskillmodeldoesnothavethe
and the h-symmetry do not commute. Although Banks problem.
the Dugan–Manohar scenario is not realized in Studying the model for r = 0, Eichten and
four dimensions due to the screening of the in- Preskill found a symmetric phase in the strong λ
teraction, it still teaches us that the way to the region with massive fermions, and another sym-
continuum can be more subtle than first imag- metricphaseintheweakcouplingregionbutwith
ined. massless fermions. They then hoped that at the
phase transition between them, the degeneracy
4.2. The Eichten–Preskill Model ofthefermionanditsdoublerswouldbeliftedby
Eichten and Preskill assumed from the start turningonthecouplingr,andthataregionwould
that when constructing a lattice theory with exist where the one fermion can be tuned to be-
any hope of defining a theory of chiral fermions, comemassless,whereasthe doublershavemasses
one must pay careful attention to anomalies[28]. of the order of the cutoff. They also pointed out
Therefore they proposedto look at a theory that that such a scenario would not be realized if the
respects alldesiredsymmetries ofthe targetcon- twosymmetricphaseswereseparatedbyabroken
tinuum theory,andexplicitly breaksany symme- phase.
try that should be broken. To satisfy this crite- The model above is rather like the Nambu–
rion, they chose to study a lattice version of a Jona-Lasinio model in appearance, and recent
chiral SU(5) grand unified theory. work showing the equivalence of a model of this
For simplicity and with no loss I will discuss type with the standardmodel[29] suggeststhat a
an SO(10) theory with the gauge fields turned similarassociationcanbemadewiththeEichten–
off. TheSU(5)theorycanbeobtainedbyadding Preskillmodel. Indeedthisisthecase. Anaction
7
with the same symmetries as the model in ques- the Smit–Swift model applies here, and again
tion is no theory of chiral fermions is realized. Finally
we should emphasize that even though Eichten
S′ = 1[ψ† σˆ ψ −h.c.] (11)
EP 2 Lx µ Lx+µˆ and Preskill were careful to pay attention to the
Xxµ
anomalystructureofthetheory,thisplaysnorole
+ 1 φ φ −κ φ φ in its failure.
2 x x x x+µˆ
Xx Xxµ
4.3. The Rome proposal
+ 1y ψTσ Tψ φ+h.c.
2 L 2 L An understanding of the preceding problems,
Xx and in particular that the culprit for the failure
1
− w ψTσ T2ψ φ+h.c. of the Smit–Swift model to produce a theory of
4 L 2 L
Xx chiral fermions is the scalar field forming bound
stateswiththefermionstocreateDiracpartners,
where we have introduced a scalar φa in the
suggests a direction for proceeding. The scalar
10 representation of SO(10). This model with
field represents just those gauge degrees of free-
Wilson–Yukawa term obviously has a similar
domwhichareunphysical,andwouldberemoved
structureastheSmit–Swiftmodel. Tomakecon-
bygaugefixing. Thisiseasilyseeninatransverse
tactwiththe Eichten–Preskillmodel, onecanin-
gaugeforwhichthescalarfieldsrepresentthelon-
ventanadditional‘flavor’andperforma largeN
gitudinal modes. Since we have learned that de-
expansion in the number of flavors. To do this,
coupling ofthese modes by keeping their mass at
westudied the Weylfermionψ (explicit in both
L
thecutoffdoesnotpreventtheboundstatesfrom
models),itsright-handedpartnerψ (formedasa
R
being formed and destroying the chiral structure
bound state in both models), and the scalar field
of the theory, perhaps enforcing gauge fixing on
(a bound state in the Eichten–Preskill model),
the lattice would do the trick. This is precisely
whose composite forms are
the proposal of the Rome group[31], although
field Eichten–Preskill Wilson–Yukawa historically this proposal was quite independent
from the reasoning I have presented here. Orig-
ψ ψ ψ
L L L
inally the motivation of the Rome group was to
ψ σ T†aψ∗(ψ†σ T†aψ∗) φaσ T†aψ∗ (12)
R 2 L L 2 L 2 L define a theory motivated by perturbative gauge
φa ReψLTσ2TaψL φa fixing as a prescription for obtaining a full non-
perturbativeasymptoticallyfreechiralgaugethe-
where a sum over a is implied when repeated.
ory.
Not surprisingly, the phase diagram of the above
The action for the Rome group’s proposal
Wilson–Yukawa model is similar to that of the
starts with that given in eq. (6), but without
Smit–Swiftmodel(figure2),withtheparameters
the scalar field (as it appears in unitary gauge).
(y,w) playing the role of (λ,r) in the Eichten–
Thentheyaddgaugefixingandghosttermsofthe
Preskill model. One can find matching condi-
form required for a particular gauge choice such
tions between the couplings of the two models
as Landau gauge. Thus they start with a theory
sothatallGreenfunctionsinvolvingtheparticles
thatviolatesgaugeinvariance(herewearetalking
abovecoincideinthe largeN expansion[30]. The
abouth-gaugeinvariance,notinvolvingthescalar
physics of the Eichten–Preskill model is realized
fields),andthefinalingredientistoaddallcoun-
intheκ=0plane,andinparticular,thetwosym-
terterms necessary to allow tuning to impose the
metric phases are separated by a broken phase.
satisfying of the BRST identities associated with
In the strong λ region, the right-handed fermion
h-symmetry:
forms as a bound state and pairs with the orig-
inal fermion to give a theory of vector fermions S =S′ (V =1)+S +S +S .(13)
Roma WY g.f. ghosts c.t.
interactingwithgaugefields,andintheweakcou-
plingregionthe doublersarepresentinthe phys- With this procedure the longitudinal modes are
ical spectrum. Thus all that we learned about decoupled not by putting their mass at the cut-
8
offscale,butbydecouplingtheirinteractionwith reduces the number of countertermsneeded from
other particles of the theory. sevendowntotwo,whichisasubstantialsavings,
Can his decoupling be accomplished? It ap- and similarly in the non-abelian theory. There is
pears that the prescription should work in prin- also a proposal on the market for calculating the
ciple to all orders in perturbation theory, and in- phase[35] using topological methods following a
deedthere arenowsomeexplicittwoloopresults definitionbyAlvarez-Gaum´eet. al.[36],soacom-
indicatingthatthingsareworkingatleasttothat binationofallthreeoftheseideasmayeventually
order[32]. The question arises how well it can prove fruitful, if not elegant.
workasanon-perturbativeprescription,inwhich Finally,IwouldliketopointouthowtheRome
thetuningmust(atleastinpart)bedonenumer- approach gets around the Banks problem. Actu-
ically. Toaddressthisquestion,letustakealook allysofartwowayshavebeenproposed[15]. One
atthe pathintegralofthe theory. The pathinte- wayarounditistowriteeachchiralmultipletasa
gral for the Rome proposal is given by charged chiral fermion and a neutral ‘spectator’
fermion, in which the neutral spectator fermion
[dU ][dψ][dψ]e−SRoma. (14) obeys shift symmetry. Then using the decou-
µ
Z
pling theorem[13] the spectator fermions should
One procedure to approach this path integral is all decouple, and the S-matrix should factorize
to multiply by the trivialintegral [dV]=1,and into charged particles and spectators. Kikukawa
thenmakeatransformationofvarRiablestorotate hasalreadyshownthatthecombinationofaright
V back into the action[17,18], so that the first and left charged fermion along with the respec-
term looks like S as given in eq. (1). As Smit tive spectators can produce the correct anomaly
WY
has pointed out (see e.g. [33]) the longitudinal responsible for baryon number violation[37]. So
modes are actually present in the Haar measure despite the fact that the full S-matrix including
forthe gaugefields,andthe abovetrickismerely spectators would be U(1) invariant, due to the
making explicit what already is there. Thus in decoupling of the spectators, the baryon number
so far as the Rome proposal would work, it is would be carried off by the spectators. The sec-
obviousthatitdependsonhowwelltheyareable ond method is to use Wilson-Yukawa terms that
to decouple the scalars. If the decoupling of the havetheformofMajoranamassterms[38]. These
scalars is very sensitive to tuning then values of explicitly violate the U(1) symmetry, and Pryor
the coupling that are a little off may not prevent has shown (in the context of the Smit-Swift type
the scalars from forming bound states with the models) that such a term reproduces the correct
fermions andwewouldbe backtothe scenarioof anomaly[39].
the Smit–Swift model. Because it is expensive to
tune couplings numerically, this is a potentially
5. Staggered fermion approach
serious problem. As to whether the problem is
realized,“theproofisinthepudding”sotospeak, With the demise of the Wilson–Yukawa ap-
andwewillhavetowaitforarealisticattemptat proach, this past year has also seen renewed
doing the non-perturbative problem. interest in the staggered fermion[40,41,42] ap-
Bodwin and Kovacs[34] have made an obser- proach toward constructing a lattice standard
vation that could make the problem more man- model[18,33,43]. For details of the method, see
ageable,providedthe technicaldifficulties canbe [33,44]. First, the basic idea of the staggered
worked out. They observe that the magnitude fermion theory is to spread out the spin and fla-
of the chiral fermion determinant is equal to the vorcomponentsofafermiononthelattice,result-
positive squarerootof a vectordeterminant, and ing in the decoupling of the original 16 doubled
providedamethodisfoundtodealwiththephase fermion components into 4 independent 4-plets.
of the chiral determinant, the calculation could Then only one of these 4-plets is kept to formu-
be done with a vector theory plus an extra cal- late the theory, thus reducing the 16 original fla-
culation for the phase. In the abelian case this vors to 4. Because of this spreading out of the
9
spin components, the hypercubic symmetry and doublet, and the three colors of u and d quarks
flavorsymmetry are now mixed, and thus for ex- can be represented by three fields χ with the
a
ample,ashiftonthelatticeofonelatticespacing color index a = 1,2,3. (In this formulation, the
mixesflavors,whereasittakestwoshiftstogener- U (1)symmetrymayleadtofermionnumbercon-
ǫ
ate a spatial translation. Also γ corresponds to servationinthe scaling limit andpresumably the
5
afour link operator. Now the beauty ofthe stag- Banks problem arises. A somewhat less elegant
geredfermionapproachisthatratherthantrying embedding of standard model quantum numbers
to get rid of the remaining doublers, they are to intostaggeredfermiondoubletscanbemadehow-
be used as physical flavors in the theory. The ever that explicitly breaks U (1) and avoids the
ǫ
important consequence of this is that if the four problem[33].)
flavorsare to correspond to flavors in the contin- To gauge the model, a more useful organiza-
uumtheory,sincetheircomponentsaresittingon tion of the fermion components is to define the
differentsitesofthelattice,theglobalchiralsym- Grassmann matrices
metry is broken and therefore cannot be gauged
1
inaninvariantway. (QCDescapesthisbyadding Ψx = 8 Γ(x,b)12(1−ǫx+b)χx+b, (16)
anindependentcolorindextoeachflavor,andthe Xb
new symmetry is gauged.) This is reminiscent of Ψx = 1 1Γ(x,b)(1+ǫ )χ , (17)
the Rome proposal in which chiral gauge invari- 8 2 x+b x+b
Xb
anceisbrokenexplicitlyontheleveloftheaction
Γ(x,b) = γx1+b1γx2+b2γx3+b3γx4+b4 (18)
andmust be restoredby tuning, andso it is with 1 2 3 4
staggeredfermions. wherebissummedoverallcornersofaunitlattice
It is useful to keep track of two remnant U(1) cell. Intermsofthesefieldsonecandefineachiral
symmetriesthatarenotbroken,onetransforming modelinanotationreminiscentofthatofthecon-
each fermion component by the same phase, re- tinuum theory. For example,andSU(2)×SU(2)
latingtofermionnumberandthesecondrotating modelinwhichonly SU(2)×U(1)is gauged,can
alternate components by opposite phases: be defined through the action
exp(iωǫx)∈Uǫ(1), ǫx =(−1)x1+x2+x3+x4. (15) S = − 1tr[Ψ γ Ψ U†
2 x µ x+µˆ µx
Theseareofcoursepartofarichlatticesymmetry Xxµ
group including many other discrete symmetries − Ψ γ Ψ U ]
x+µˆ µ x µx
which I will not enumerate here. The group goes
+ m ρ tr[Ψ Ψ γ ], (19)
underthenameSF orstaggeredfermionsymme- µ x x x µ
Xxµ
try group[44].
If χ is the component of a fermion at a partic- where the gauge fields are the appropriate ones
ular site, one can cut the number of components for gauging the groups desired and the last term
in half by letting χ = χ on each site (equivalent is present since no symmetry excludes it. ρ is
x
to letting χ live only on even sites and χ on odd a spacetime dependent amplitude factor. Keep
sites). These arereferredtoasreducedstaggered in mind that because the components of Ψ lie on
fermions. Each reduced fermion field represents different sites, this action is not gauge invariant.
a doublet of fermion flavors, and because of the However, if one sets m to zero, the action does
µ
‘Majorana’constraintthatdefinesthem,theU(1) lead to the proper gaugeinvariantcontinuum ac-
relating to fermion number is broken and only tion in the classical continuum limit.
U (1) is left. Now recall that in the Smit-Swift model one
ǫ
Itisstraightforwardnow tobuild alattice the- could freely transform between the g-charged
ory with particle content equivalent to the stan- fields ψ(c) and the g-neutral fields ψ(n), and rep-
dard model. For example[33], for the first gen- resent the path integral in terms of either. Such
eration of the standard model, the electron and a transformation is not possible here because
neutrino together can be represented by one χ of the lack of local gauge invariance. One can
10
however expose the longitudinal degrees of free- volume[45]. To define the model, the action for
dom by making the transformation of variables the scalar field
U → V U V† [18], then one arrives at the
acµtxion (mxµ =µx0)x+µˆ Sscalar =κ12 tr(Φ†xΦx+µˆ+Φ†x+µˆΦx) (24)
Xxµ
S = − xµ 12tr[ΨxγµΨx+µˆVx+µˆUµ†xVx† (20) −1 tr[Φ† Φ +λ(Φ† Φ )2], (25)
2 x x x x
P −Ψ γ Ψ V U V† ] (21) Xx
x+µˆ µ x x µx x+µˆ
is added to eq. (23). The case of 2 reduced stag-
which appears similar in form to the version of
gereddoubletsor4flavorsandthecaseof2multi-
the Smit–Swift model written in terms of neu-
plets ofnaivefermioncorrespondingto 32flavors
tral fields in eq. (6). The obvious question based
were studied.
on our previous experience is whether the corre-
First of all, the phase diagram has been
sponding desired ‘charged’ states appear in the
mapped out and no surprises were found. For
physical spectrum, or equivalently, whether the
weak values of y and large κ a ferromagnetic
propagator
phase occurs. As one reduces κ a second or-
hV†Ψ Ψ V i (22) der transition appears to a paramagnetic phase
x x y y
whichcontinuesonintothe negativeκregion,af-
has a pole structure (when the gauge fields are
ter which for large enough negative κ an antifer-
turned off). This is a crucial test as to whether
romagnetic phase occurs. For much larger values
a gauge theory emerges which includes charged
of y the diagram moves directly from the ferro-
states in the usual sense.
magnetic to a ferrimagneticphase for some value
Alternatively, one can define another fermion–
of κ below zero. As a by-product in the simula-
Higgs model through an action which has the
tion it was discoveredthat the staggeredfermion
form of eq. (1)
method appears to offer a very efficient way to
simulate such models.
S = − 1tr[Ψ γ Ψ −Ψ γ Ψ ]
2 x µ x+µˆ x+µˆ µ x Another question addressed is O(4) symmetry
Xxµ
restoration. Because of the explicit breaking of
+ y trΨ Ψ Φ +S , (23) the symmetry, two counterterms O(1) and O(2):
x x x scalar
Xx
O(1) = Φ4 , O(2) = (Φ −Φ )2,(26)
where Φ = Φ γ . Because there is no for- µx µx+µˆ µx
µ µx µ Xxµ Xxµ
mal transforPmation between the models defined
by eq. (21) and by eq. (23) it is a non-trivial are necessary to restore it as the continuum is
question whether the two quantum theories are approached, which in principle would mean that
equivalent. Hence it makes sense to first study twocouplingsneedtobe tuned. Whatwasfound
the two fermion–Higgssystems alone, for if these however, is that in the region studied, the O(4)
are seen to be equivalent, one might expect the symmetry is already present to a good approx-
gauged theories to be also. Then the more press- imation, and that the breaking effects are only
ingquestioncanbeaddressed: whetherthey(one on the order of a few per cent. Two other facets
ortheotherorboth)giverisetothedesiredgauge of the calculation deserve mention before going
invariant continuum theory (assuming the neces- on. The authors have included one loop fermion
sary tuning is performed). effects infitting the scalarparameters,andinor-
As a first serious investigation of these ques- der to control systematic errors due to finite size
tions, recently a group has begun to study an effects, the authors have performed calculations
SU(2) × SU(2) invariant version of the latter on lattices of different physical sizes, in order to
modelnumerically[43]. Iwillsummarizethehigh- make an extrapolation to large volume. Among
lights here as befit the flow of my theme; a more results obtained from these models are estimates
complete report can be found elsewhere in this of the effects of fermions on the upper bound