Second-Order Fermions and the Standard Model

0.24 MB·English

by  Johnny Espin

Second-Order Fermions and the Standard Model
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2 Johnny Espin∗†

n SchoolofMathematicalSciences
a UniversityofNottingham
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E-mail: [email protected]
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] We present a construction of a non-hermitian fermionic Lagrangian which has a second-order
h
kinetic term. Despite the non-hermicity of the latter, the theory is unitary and the perturbation
t
-
p theory that can be derived is equivalent to the usual one derived from a first-order formalism.
e
Havingthisinmind,theconstructionofasecond-orderStandardModelallowsamorecompact
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[ descriptionofthetheory. Thisworkisbasedon[1,2]andcitationstherein.

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ProceedingsoftheCorfuSummerInstitute2014"SchoolandWorkshopsonElementaryParticlePhysics
andGravity",
3-21September2014
Corfu,Greece
∗Speaker.
†WorkdoneincollaborationwithKirillKrasnov.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Second-OrderFermionsandtheStandardModel JohnnyEspin
1. Introduction
Since the emergence of Quantum Mechanics (QM), much effort was put into finding rela-
tivistic wave equations that would govern the dynamics of mechanical systems. Schrödinger and
then Klein and Gordon formulated a second-order wave equation, but at that time it seemed that
the nature of the latter violated some fundamental properties of mechanical systems: the Klein-
Gordonwaveadmittedapositiveandanegativeenergysolution. BritishphysicistPaulA.M.Dirac
believed that the issue relied on the second-order nature of the differential equation. He therefore
soughtafirst-orderdifferentialequationthatrespectedtherelativityprinciple. Histheorywasfor-
mulatedin1928andtheDiracequationwaslatershowntodescriberelativisticspin1/2particles:
fermions. This was followed by the development of Quantum Electrodynamics (QED), the rela-
tivisticquantumtheoryoflightandmatterinteractionswhichwasthengeneralisedintoYang-Mills
(YM)theory,thetheoryofnon-abelianSU(n)gaugefields. Thissummarisesthesuccessofparti-
cle physics in the last century, success that culminated with the edification of the Standard Model
(SM)ofparticlephysics. Yetsomethingcanbeseenaspuzzling. Indeed,oneofDirac’sreasonsto
seek a first-order differential equation was the misinterpretation of the negative energy solutions.
Howeverimmediatelyafterthediscoveryofhisequation,itbecameclearthatNatureadmittedpar-
ticles and antiparticles (positive and negative energy solutions). Nonetheless, fermions remained
the only dynamical system that only admitted a first-order description. Indeed, both General Rel-
ativity and Yang-Mills theory admit a first- and second-order formulation [3, 4]. As for fermions,
because their equation is derived from the relativity principle, it also satisfies the Klein-Gordon
equation;itsLagrangianis,however,firstorderinderivatives. Hence,itisnaturaltoaskwhethera
(fully)second-orderformulationoffermionsthatcontainsalltheinformationoftheoriginaloneis
possible.
Letusmakethelastargumentmoreprecise. ADiracspinorcanbewrittenasthesumoftwounitary
infinitedimensionalrepresentationsoftheLorentzgroupSO(3,1)(oritsdoublecoverSL(2,C)):
Ψ ∈(1/2,0)⊕(0,1/2) (1.1)
D
which we call left-(unprimed) and right-handed (primed) respectively. The Dirac equation is then
derivedfromtheDiracLagrangian,herein3+1dimensionswiththemetricη =(−,+,+,+):
µν
L =Ψ¯ (cid:0)−i∂/−m(cid:1)Ψ (1.2)
D D D
with∂/=γµ∂ andwehavethealgebraof(Dirac)gammamatrices:
µ
{γµ,γν}=−2ηµν, (γµ)†=γ0γµγ0, γ =iγ0γ1γ2γ3, (γ )†=γ (1.3)
5 5 5
ThisLagrangiangeneralisesinastraightforwardwaysoastoincludetheinteractionoffermions
andphotons. WeseethattheDiracequation
(cid:0) (cid:1)
−i∂/−m Ψ =0 (1.4)
D
relatesspinorsofonechiralitytotheotherthroughtheoff-diagonalentriesoftheDiracmatri-
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Second-OrderFermionsandtheStandardModel JohnnyEspin
ces1. ThisisheuristicallywhyasecondorderLagrangianofthetype
L =? Ψ¯ (cid:0)−(cid:3)+m2(cid:1)Ψ (1.5)
D D D
does not work since the Klein-Gordon operator is diagonal and hence we lose information con-
tainedintheDiracequation.
A lot of insight can be gained on the issue when one expresses all the quantities in terms of
two-componentspinors(seeforexample[5]). Inthispaper,wewillconstructthesimplestsecond-
orderspinorfieldtheories;thoseofa(Weyl-)MajoranaandofaDiracfermion. Then,wewillapply
thesameconstructiontotheStandardModel. Finally,wewillshortlydiscussPerturbationTheory
andUnitarityinthisframework.
2. Second-order(Weyl-)MajoranaandDiracfermions
We start by considering the much simpler setup of a single (Weyl-)Majorana spinor and then
repeattheprocedureforaDiracspinor.
2.1 First-orderLagrangian
TheLagrangianforasingleMajoranafermionreads:
√
L =−i 2λ†∂λ−(m/2)λλ−(m/2)λ†λ† (2.1)
Maj
Hereλ isatwo-componentspinor,λ† isitsHermitianconjugateandθA(cid:48)A isthesolderingform
A A µ
θA θ A(cid:48) =η (2.2)
µA(cid:48) νA µν
whereη =diag(−1,1,1,1). Wehavealsointroducedthenotation∂AA(cid:48) :=θµAA(cid:48)∂ andwritten
µν µ
theLagrangianinanindex-freeway. Ourindex-freeconventionisthattheunprimedfermionsare
always contracted as λχ ≡λAχ , while the primed fermions are contracted in the opposite way
A
λ†χ†≡(λ†) (χ†)A(cid:48).
A(cid:48)
2.2 Equationsofmotionandsecond-ordertheory
The(first-order)equationsofmotionfortheunprimedspinorare:
√
−i 2∂λ−mλ†=0 (2.3)
The second-order formulation can then be obtained by carrying out the Berezin path integral
overtheprimedspinors(whicharetreatedasindependentvariables),whicheffectivelyamountsto
substitutingtheaboveequationofmotionintheLagrangian. Thenwehave:
1
L =− ∂λ∂λ−(m/2)λλ (2.4)
Maj
m
1InthecaseofMajoranafermionsthespinorislinkedtoitshermitianconjugatethroughtheDiracequation.
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Second-OrderFermionsandtheStandardModel JohnnyEspin
whichisequivalent,afteranappropriaterescalingofthefieldsto:
L =−∂λ∂λ−(m2/2)λλ (2.5)
Maj
Thesecond-orderWeyltheoryisthensimplyobtainedfromthemasslesslimit(afterthepath
integraloftheprimedspinorshasbeensolved):
L =−∂λ∂λ (2.6)
Weyl
2.3 Diracfermions
In order to describe Dirac fermions, we consider two massive Majorana fermions of equal
mass. The system is then invariant under and internal SO(2) symmetry. Since SO(2) ∼ U(1),
we can introduce a complex linear combination of the spinors and make the second symmetry
explicit. It can then be made local by introducing a U(1) gauge field and by minimally coupling
thefermions. Thus,wedefine
Dξ =(∂−ieA)ξ, Dχ =(∂+ieA)χ, (2.7)
where AAA(cid:48) =θµAA(cid:48)A is the electromagnetic potential and e is the electron charge. The gauge
µ
transformation rules are: for the fermions ξ →eiαξ,χ →e−iαχ, and for the electromagnetic po-
tentialA →A −(1/e)∂ α. TheLagrangianbecomes
µ µ µ
√ √
L =−i 2ξ†Dξ−i 2χ†Dχ−mχξ−mξ†χ†, (2.8)
Dirac
where as before D:=θµD . In order to obtain the second-order Lagrangian, the same procedure
µ
isappliedtobothprimedspinors,andweobtain
√ √
i 2 i 2
L =−2DχDξ−m2χξ, ξ†=− Dχ, χ†=− Dξ. (2.9)
Dirac
m m
3. ShortreviewoftheStandard-Model
Forsimplicity,weonlyconsiderthequarksectoroftheSM,forthecompletepicture,werefer
thereaderto[1].
3.1 StandardModelquarks
TheSMquarkscanbeputtogetherinthefollowingtable
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Second-OrderFermionsandtheStandardModel JohnnyEspin
(1/2,0)reps SU(3) SU(2) Y T Q=T +Y
3 3
(cid:32) (cid:33)
u triplet 1/6 1/2 2/3
i
Q = doublet
i
d triplet 1/6 −1/2 −1/3
i
u¯ triplet singlet −2/3 0 −2/3
i
d¯ triplet singlet 1/3 0 1/3
i
All quarks here are unprimed two-component spinors. Therefore, the Hermitian conjugate
of u is denoted by u†, and u¯ is another independent unprimed spinor. Notice that each SU(n)
i i i
multiplet corresponds to a collection of unprimed spinors. Thus, e.g. u, which is a color triplet,
i
has two types of indices suppressed: the usual spinor index, as well as the strong SU(3) index.
With the index structure made explicit, this field is denoted by uα, where A = 1,2 is the usual
iA
spinorindex,andα =1,2,3istheindexonwhichSU(3)acts. Onlytheflavourindexi=1,2,3is
leftexplicitinwhatfollows.
3.2 Higgsfield
This is the field that plays the central role in the the Standard Model. It is a complex scalar
fieldofU(1)hyperchargechargeY =1/2. ItisalsoaweakSU(2)doublet,i.e.itcanbewrittenas
acolumn
Higgs SU(3) SU(2) Y T Q=T +Y
3 3
(cid:32) (cid:33)
φ+ 1/2 1/2 1
φ = singlet doublet
φ0 1/2 −1/2 0
Note that being an SU(2) doublet, it is really a collection of 2 complex scalar fields φ+ and
φ0 (withQ=0). WeshalldenotetheweakSU(2)indexbya,b,...=1,2. Thuswecanwritethe
Higgsfieldasφ ,withφ =φ+ andφ =φ0.
a 1 2
3.3 QuarksectoroftheStandardModel
Usingindex-freenotations,theLagrangianforthequarksectoroftheStandardModelreads:
√ √ √
L = −i 2Q†iDQ −i 2u¯†iDu¯ −i 2d¯†iDd¯
quarks i i i
(3.1)
+YijφTεQu¯ −Yijφ†Qd¯ −(Y†)iju¯†Q†εφ∗−(Y†)ijd¯†Q†φ
u i j d i j u i j d i j
Here as before DAA(cid:48) ≡θµAA(cid:48)D , where D is the covariant derivative taht acts on each field
µ µ
according to its representation. The quantitiesYij are arbitrary complex 3×3 Yukawa matrices.
Thequantityε ≡ε b isthematrix
a
(cid:32) (cid:33)
0 1
ε b= . (3.2)
a
−1 0
which plays the role of a SL(2,C) metric. Then the object φTεQ≡(φT)aε bQ is invariant
a a
under the action of the latter via Q→gQ,φ →gφ since gTεg=ε. In particular, φTεQ is SU(2)
invariant. Itisthenclearthatalltheinteractiontermsin(3.1)areSU(2)invariant. Thehypercharge
invarianceiseasilycheckedusingthechargestables.
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4. SecondorderformulationoftheStandardModel
Wenowformalisetheprocedureofintegratingouttheprimedtwo-componentspinorsfromthe
significantlymoreinvolvedquarksLagrangian. Theequationsofmotionfortheunprimedspinors
are:
√
Q†: i 2DQi= −(εφ∗)u¯†(Y†)ji−φ d¯†(Y†)ji
i j u j d
√
u¯†: i 2Du¯i= −(Y†)ijQ†(εφ∗) (4.1)
i u j
√
d¯†: i 2Dd¯i= −(Y†)ijQ† φ
i d j
Noticethatsomesymmetrystructureismakingitselfexplicitintheequationsofmotion. Thus,
letuscombinethecomponentsoftheHiggsfieldintothefollowing2×2matrix:
(cid:32) (cid:33)
(φ0)∗ φ+
ρΦ†:=(εφ∗,φ)≡ . (4.2)
−φ− φ0
UndertheweakSU(2)thematrixֆ transformsas:
Φ† (cid:55)→ ΩΦ†, (4.3)
while the field ρ remains invariant. It is clear that ρ2 ≡|φ|2 is just the modulus squared of the
Higgsfield.
Inordertofurthersimplifytheequationsofmotion,aseriesoffieldredefinitionsareneeded,
[1]. Thisleadsto:
√
Q†: i 2DQ = −ρ Φ†(cid:0)Q¯†Λ(cid:1)
i i i
(4.4)
√
Q¯†: i 2DQ¯ = −ρ Q†Φ†
i i i
Here, the doublet Q transforms under the weak SU(2), and so does the Higgs matrix Φ†,
i
while Q¯ does not transform. This suggests that we define new composite SU(2)-invariant quark
i
variablesΦQ
i
ΦQ :=Qinv. (4.5)
i i
ThiscorrespondstoaHiggs-fielddependentSU(2)gaugetransformationoftheoriginalquarkdou-
blet. Assuch,itcanbepulledthroughthe(gaugedependent)covariantderivativeifonetransforms
the SU(2) gauge fields accordingly. The new vector field so defined will be an SU(2)-invariant
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Second-OrderFermionsandtheStandardModel JohnnyEspin
object. Notice that the SU(2) invariance does not disappear from the theory, it is simply “frozen”
bythenewsetofvariablesthatwechoseasaconvenientbasistoworkin. Thischangeofvariables
for the gauge field is equivalent to working in the unitary gauge, but it should be stated that the
second order formulation exists independently of the choice of variables. Keeping in mind this
changeinthederivativeoperatorwecanwritethefieldequationsas:
√
Q†: i 2DQ = −ρ (cid:0)Q¯†Λ(cid:1)
i i i
(4.6)
√
Q¯†: i 2DQ¯ = −ρ Q†
i i i
wherefromnowonwedropthesuperscriptinvfromtheQ asitisclearthatwewillbeworking
i
withthissetofvariablesexclusively. Weseethattheequationsbecomemuchsimplerthaninterms
oftheoriginalfields.
We now substitute the primed spinors obtained from the above field equations into the La-
grangian(3.1)andobtainthefollowingsecond-orderLagrangian:
L =−2DQ¯iDQ −ρ(cid:0)ΛQ¯(cid:1)iQ, (4.7)
quarks i i
ρ
where it should be remembered that the covariant derivative acting on Q in (4.7) takes into
i
accountthefieldredefinition(4.5).
Notice that this new second-order Lagrangian, even though it contains fewer terms that its
first-order counterpart, is clearly non-polynomical in the (physical) Higgs scalar field ρ, because
of the presence of 1/ρ in the kinetic term. In the case of Dirac theory (2.9) a constant rescaling
of the spinors was all that we needed to obtain a canonical kinetic term. The same procedure can
be applied to (4.7). However, ρ is now a dynamical field and thus, absorbing it into the fermion
fields (again) changes the covariant derivative operators acting on both Q¯ ,Q. Denoting the new
i i
covariantderivativebyD,wefinallywrite:
L =−2DQ¯iDQ −ρ2(cid:0)ΛQ¯(cid:1)iQ (4.8)
quarks i i
√
where 1/ ρ was absorbed into each spinor field. The new covariant derivative D contains non-
polynomialHiggs-quarksinteractionsaswellasthephysicalSU(2)-frozengaugefieldswhenact-
ingontheunbarreddoublet. ItisclearthatinteractionverticeswiththeHiggscanbeofarbitrarily
highvalency(duetonon-polynomialityinρ).
Thefieldequations(4.6)forthenewfermionicfieldsofmassdimensiononeread
√ √
i 2DQ =−ρ (cid:0)Q¯†Λ(cid:1) , i 2DQ¯ =−ρ Q†. (4.9)
i i i i
and are now interpreted as the reality conditions, whose linearised versions are to be imposed on
theexternallines.
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Second-OrderFermionsandtheStandardModel JohnnyEspin
5. PerturbationtheoryandperturbativeUnitarity
We now discuss, in the framework of Dirac theory, how pertubation theory is modified but
leadstothesameresultsasinthefirst-orderformalism,andalsohowUnitarityholdseventhough
theLagrangianisnon-Hermitian. Moredetailscanbefoundin[2].
5.1 Feynmanrules
Thepropagatorbecomesverysimple:
−i
(cid:104)0|T{χ (p)ξ (−p)}|0(cid:105)≡D(p) = ε (5.1)
A B AB p2+m2 AB
It is nothing but a scalar-type propagator, the espilon tensor in the numerator being the identity
over the space of unprimed spinors. The complexity of the Dirac propagator is now shifted to the
vertices. WehavetwointeractionverticeswithFeynmanrules(incomingmomenta):
(cid:104) (cid:105)
2ie k A(cid:48)ε +p A(cid:48)ε , −2ie2εA(cid:48)B(cid:48)ε ε (5.2)
C BA B CA AB CD
NoticethatthequarticvertexissimplytheidentityoverbothMinkowskispacetimeandthespace
ofunprimedspinors.
AA(cid:48) χ
A C
χ
B
AA(cid:48)
A
ξ
C
AB(cid:48) ξ
B D
Figure1: Interactionvertices
5.2 Spinaveragedprobabilities
When we sum (or average) over photon polarisation states, one can make use of the Ward
identitiestoobtain:
∑ε ε∗ → η (5.3)
µ ν µν
pol.
Inourcase,thiswillbecome:
∑ε ε∗ → −ε ε (5.4)
AA(cid:48) BB(cid:48) AB A(cid:48)B(cid:48)
pol.
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Second-OrderFermionsandtheStandardModel JohnnyEspin
Asforthesecond-orderfermions,wehave:
(cid:114)
2
ε+ε∗+(k)+ε−ε∗−(k)= k (5.5)
A A(cid:48) A A(cid:48) m2 AA(cid:48)
5.3 e−µ−→e−µ− scattering
Consider the simplest QED process: electron-muon scattering at tree level in the limit m (cid:28)
e
m ,Fig.2.
µ
Figure2: e−µ−→e−µ−
Let us first compute the amputated amplitude M for an incoming electron with momen-
ABCD
tumk scatteredoffanincomingmuonwithmomentum p . Wehave:
1 1
(ie)2(−i)(cid:16) (cid:17) (cid:16) (cid:17)
M =4 k E(cid:48)ε −k E(cid:48)ε εEFε p F(cid:48)ε −p F(cid:48)ε
ABCD q2 1A DE 2D AE E(cid:48)F(cid:48) 1B CF 2C BF
(5.6)
4ie2(cid:104) (cid:105)
=− (k ·p ) ε −(k ·p ) ε −(k ·p ) ε +(k ·p ) ε
q2 1 1 AB CD 2 1 DB AC 1 2 AC BD 2 2 CD AB
wherewedefined:
(k·p) :=k C(cid:48)p (5.7)
AB A BC(cid:48)
and q2 =(k −k )2 =(p −p )2 =t. The complex conjugate amplitude is simply given by re-
1 2 1 2
placing every unprimed spinor by a primed one and every primed spinor by an unprimed one, so
that:
4ie2(cid:104) (cid:105)
M∗ =− (k ·p ) ε −(k ·p ) ε −(k ·p ) ε +(k ·p ) ε
A(cid:48)B(cid:48)C(cid:48)D(cid:48) q2 1 1 A(cid:48)B(cid:48) C(cid:48)D(cid:48) 2 1 D(cid:48)B(cid:48) A(cid:48)C(cid:48) 1 2 A(cid:48)C(cid:48) B(cid:48)D(cid:48) 2 2 C(cid:48)D(cid:48) A(cid:48)B(cid:48)
(5.8)
andwedefined:
(k·p) :=k p C (5.9)
A(cid:48)B(cid:48) A(cid:48)C B
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Second-OrderFermionsandtheStandardModel JohnnyEspin
Theunpolarisedcross-sectionisthenobtainedfrom:
1
|M|2= ∑M M∗ εAε∗A(cid:48)(k )εBε∗B(cid:48)(p )εCε∗C(cid:48)(p )εDε∗D(cid:48)(k )
4 ABCD A(cid:48)B(cid:48)C(cid:48)D(cid:48) 1 1 2 2
pol.
(5.10)
1 2 2
= M M∗ kAA(cid:48)pBB(cid:48)pCC(cid:48)kDD(cid:48)
4 ABCD A(cid:48)B(cid:48)C(cid:48)D(cid:48) 1 1 2 2 m2m2
e µ
Withabitofalgebra,itcanbeshownthatthefollowingequalityholds:
2 2
M∗ kAA(cid:48)pBB(cid:48)pCC(cid:48)kDD(cid:48) =−MABCD (5.11)
A(cid:48)B(cid:48)C(cid:48)D(cid:48) 1 1 2 2 m2m2
e µ
Sothat:
1
|M|2=− M MABCD (5.12)
ABCD
4
In the above formula, we only need to compute three different expressions. Consider four
momentak, p, q, l describingmassiveparticles. Wehave:

(k·p)ABεCD =m2m2
 k p

(k·p)ABεCD (k·q)ACεBD =−12m2k(p·q) (5.13)


(q·l)CDεAB =(k·p)(q·l)
where
k·p=kA pA(cid:48) =k pµ (5.14)
A(cid:48) A µ
Usingthisandneglectingtermsproportionaltotheelectronmass,weobtain:
8e4(cid:104) (cid:105)
|M|2= (k ·p )(k ·p )+(k ·p )(k ·p )+m2(k ·k ) (5.15)
q4 1 1 2 2 1 2 2 1 µ 1 2
whichisthewellknowsquaredamplitudefortheunpolarisedprocess.
5.4 One-loopchargerenormalisation
Letusnowlookatasimpleone-loopexample. Althoughtherearehalfthenumberoffermions
inourtheory,fermionsloopsareequivalentinbothformalisms. Indeed,letuslookattheamplitude
forthechargerenormalisationinthesecond-orderformalism:
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