Twisted Mass F ermions

The modiation of Wilson type fermions by a hirally twisted mass term has been

introdued about a deade ago [81, 82℄. Some years later Frezzotti and Rossi realised

that thetheoryis automatially

O (a)

^{-improved iftheoriginal untwisted massisset to}

its ritialvalue[83 ℄. Ourpresentation oftwisted masslattiefermions isbasedon the

reviewbyShindler[84 ℄.

3.3.1. Twisted Mass Formulation

The entral keyto the introdution of twisted massfermions isthetransformation

ψ =

^ei

^{ωγ 5 τ 3 /2} χ

^and

ψ = χ

^ei

^{ωγ 5 τ 3 /2}

^(3.20)

whihisasymmetryoftheontinuumationifthemassterm isreinterpreted

appropri-ately.

ψ

^{,inthe so-alledphysialbasis,and}

χ

^{,inthetwistedbasis,areavour doublets}

and thethirdPauli matrix

τ ³

^{ats intheaording avour spae. It isthenpossibleto}

reformulate the QCDation as

S _F = Z

d

4 x χ(x) γ _µ D _µ + m ₀ +

ⁱ

µ ₀ γ ₅ τ ³

χ(x) .

^(3.21)

The barequarkmassisrelated to the untwisted andtwisted omponents,

µ ₀ = m _q sin ω ,

^(3.22a)

m ₀ = m _q cos ω ,

^(3.22b)

m q = q

m ² ₀ + µ ² ₀ ,

^(3.22)

withthe orrespondingtwistangle

ω

^,

tan ω = µ ₀

m ₀ .

^(3.23)

Thetwist angle obviously hanges underrenormalisation astheuntwisted and twisted

massomponentsdonotsharethesamerenormalisationfator. Therenormalisedquark

massis determined as

m _R = q

Z _m ² (m ₀ − m _c ) ² + Z _µ ² µ ² ₀ .

^(3.24)

Only if the untwisted quark mass is set to its ritial value, the angle,

ω = π/2

^{, is}

independent ofrenormalisation. This aseisalledmaximal twist.

Applyingthe Wilson disretisation to theontinuumation withtwisted massterm,

weobtain

S _F [U, ψ, ψ] = X

x

χ(x) 1 − κD _W [U] + 2

ⁱ

κaµ ₀ γ ₅ τ ³

χ(x) .

^(3.25)

Apartiular feature ofthetwisted massfermionmatrix,

M

^tm

= 1 − κD W [U] + 2

ⁱ

κaµ 0 γ 5 τ ³ ,

^(3.26)

thatwasimportant fortheoriginalmotivationtointroduethis modiationofWilson

fermionsis that the twisted massparameter ats as aregulator proteting simulations

fromzeromodes. Thesezeromodeshavebeenasoure ofonernintheuntwisted

the-ory[82 ℄. Thispropertyan beseenwhen evaluatingtheavour determinant expliitly,

Det

avour

M

tm

= M _W M _W ^† + (aµ ₀ ) ² ,

^(3.27)

where

M _W

^{denotes the standard Wilson fermionmatrix for one avour.}

Inversionof equation(3.26) inmomentumspae for afreeeldleads to the

determi-nationof the propagator,

S

^tm

(p) = a −

ⁱ

P

ν γ ν p _ν + ¹ ₂ p ˆ ² + am 0 −

ⁱ

aµ 0 γ 5 τ ³ p ² + ¹ ₂ p ˆ ² + am ₀ 2

+ (aµ ₀ ) ² ,

^(3.28)

whih isimportant for the later analytial alulations in thefree and weakly oupled

limits.

3.3.2. Automati Improvement

Sinethe twistrotations(3.20) arenomoreasymmetryofthelattieationdue tothe

Wilsonterm, thefermiondisretisation providesdierent regularisations dependingon

thetwistangle. Thatmeansinpartiularthattheutoeetsarefuntionsofthetwist.

Ithasbeen shownthattheleading

O (a)

^{eetsvanishinmostrelevantquantities ifthe}

ationis tunedto maximaltwist. Morepreisely,this statement appliesto parityeven

quantities as an be derived from the symmetries of the Symanzikexpansion [84, 85℄.

Thisfeature is known as automati

O (a)

-improvement and is the ruial property for urrent appliationsof twisted massfermions.

We now briey sketh the main idea for the proof of automati

O (a)

improvement following the reviewbyShindler[84℄. ThestartingpointistheSymanzikexpansionfor

thelattie ation,

S

e

= S ₀ + aS ₁ + . . . ,

^(3.29)

where

S ₀

^{isthe ontinuum ation. Thehigher order} ontributions,

S n = Z

d

4 y L ⁿ ,

^(3.30)

are onstruted from all possible ounterterms. Espeially, we have innext-to-leading

order

L 1 =

5

X

k=1

c _k O k .

^(3.31)

We do not give the expliit form of the operators

O k

^{here but refer to [84 ℄ where the}

omplete set is given. The oeients

c _k

^{have then to be tuned suh that ontinuum}

alulationswith

S

^{ereprodue thelattietheoryuptothehosenorderof}

a

^{. Asimilar}

expansion holds foranyeldobservable,

φ

^e

= φ ₀ + aφ ₁ + . . . .

^(3.32)

With theabove expansions, one an identify the possible disretisation eetsto

O (a)

for an arbitraryexpetation value,

h φ(x ₁ ) . . . φ(x _n ) i = 1 Z

Z

D [ψ, ψ, A _µ ] φ(x ₁ ) . . . φ(x _n )

^e

^{−S 0} (1 − aS ₁ + . . .)

= h φ ₀ (x ₁ ) . . . φ ₀ (x _n ) i 0

− a Z

d

4 y h φ ₀ (x ₁ ) . . . φ ₀ (x _n ) L 1 (y) i ₀

^(3.33)

+ a

n

X

k=1

h φ ₀ (x ₁ ) . . . φ ₁ (x _k ) . . . φ ₀ (x _n ) i 0 + O (a ² ) ,

where ontat termsan be absorbed into a redenition of theelds. For theproof of

automati improvement one anutilise thefollowing symmetryof thelattieation,

R ₅ ¹ × D × [µ → − µ] ,

^(3.34)

where

D

^{multiplies all terms by}

( − 1) ^{d m}

^{aording to their mass dimension}

d _m

^.

R ¹ ₅

transforms theelds as

χ →

ⁱ

γ 5 τ ¹ χ ,

^(3.35a)

χ → χ

ⁱ

γ ₅ τ ¹ .

^(3.35b)

This symmetryholds for the lattie ationeven inthe innitevolume limit. Therefore

all termson the right handside of equation(3.33) have to respetthe symmetry. This

neessitates the absene of

O (a)

^{termsfor anyeld}

φ

^thatrespets

R ¹ ₅

^{sine thenboth}

ontributions to thatorder in equation(3.33) areodd undertheabove symmetry. For

the rst part,

φ ₀ (x ₁ ) . . . φ ₀ (x _n ) L 1 (y)

^{, this is true beause of the operator insertions in}

L 1

^{. Theseond part,}

φ ₀ (x ₁ ) . . . φ ₁ (x _k ) . . . φ ₀ (x _n )

^{is odd beause of themassdimension}

of

φ 1

^{thathas to be dierent by 1 fromthat of}

φ 0

^{so thatall terms inequation (3.32)}

possess the same symmetries. Introduing an untwisted quark massto the lattie

a-tion expliitly breaksthe ruial symmetry (3.34) and therefore there is no automati

improvement exept for maximal twist. Note that aording to the above arguments

alsoquantities withvanishingexpetationvaluean have disretisationeets in

O (a)

^.

Apossible aveatisposedbypionpoles,i.e.byutoeetsthatareof

O (a ^2k /m ^2h _π )

with

2k ≥ h ≥ 1

^{. Thesepolesan beavoidedbya suitabledenitionof maximaltwist.}

For this reason ETMC have adopted the ondition of a vanishing PCAC (partially

onserved axialurrent)quarkmass[86℄,

m

PCAC

= P

x h ∂ ₄ A ^a ₄ (x, t)P ^a (0) i 2 P

x h P ^a (x, t)P ^a (0) i ,

^(3.36)

where

P ^a = ψγ ₅ τ ^a /2ψ

^,

A ^a _µ = ψγ _µ γ ₅ τ ^a /2ψ

^{. From thenumerial sidea quenhed test of}

a ²

^{-saling withthis onditionhasbeen presented in[87 ,88, 89℄.}

On the level of free lattie fermions the improvement an be observed easily, for

instane, for the dispersion relation that is obtained from the pole of the propagator,

S F (p, p 4 =

ⁱ

E)

^,

E(p) = q

p ² + m ² _q − a m ³ _q 2

cos(ω) q

p ² + m ² _q

+ O (a ² ) .

^(3.37)

The improvement for maximal twist,

ω = π/2

^{is apparent. Note that for a theory}

withoutspontaneousbreakingofhiral symmetry,like inthefreelimit,thelinearuto

eets have to be proportional to the quark mass. This is not neessarily the ase in

thepresene ofspontaneous hiral symmetrybreaking [85℄.

Automati improvement also holds at nite temperature. Thisis expeted sine the

relevant symmetries do not mixspatial and temporaldiretions sothat thenite time

extent whihharaterises simulations atnite temperature doesnotinterferewiththe

improvement. A report of observed

a ²

^{-saling at nite} temperature in the quenhed ase has been given by our tmfT ollaborators [90℄. Our perturbative analysis of the

pressure,disussed insetion 4.2,nds improvement at maximaltwist aswell.

Another possible drawbak of twisted mass fermions is thebreaking of avour

sym-metry. Flavour singlet and doublet quantities suh as

m ⁰ _π

^and

m ^± _π

^{may show large}

splittingsfornitelattiespaing. Indeedinaseofthepiontheselargesplittingshave

been observed [91 ℄. A theoretial analysisintheSymanzikexpansion shemeallows to

identify the leading orders ofuto eets[92 ℄,

m ⁰ _π 2

= m ² _π + a ² ζ _π + O (a ² m _π , a ⁴ ) ,

^(3.38a)

m ^± _π 2

= m ² _π + O (a ² m ² _π , a ⁴ ) .

^(3.38b)

Theleading order of thesplitting,

ζ _π

^{,originates solely in theneutral setorand turns}

out to be large. Important to note is that the large size of splitting uto eets is

restritedto thepionmassand relatedquantities.

3.3.3. Phase Diagram

Asfor all Wilson type fermions, twisted mass fermions an exhibit unphysial phases

in their bare parameter spae that is spanned by the hopping parameter, the

lat-tie oupling and the twisted mass. Its vauum struture has been studied

exten-sively [93 , 94 ,95 ,96 , 97 ,98 ℄. From ordinary Wilsonfermions it isknownthat there is

Figure 3.3.:Phasediagramin

(κ, β, µ ⁰ )

^{parameterspae, from[96℄. The strongoupling}

re-gion, i.e. theregion of small lattieoupling

β

^{, exhibits the Aoki phasein the}

planeofvanishingtwistedmassparameter

µ ⁰

^{. Forweakerouplingsasurfaeof}

bulktransitionsisenounteredthat ontainsthelineofritialhopping

parame-ter,

κ c (β)

^andextentsto

µ ⁰ 6 = 0

^{. Thewidthof thissurfaevanishestowardsthe}

ontinuumlimit.

the so-alled Aoki phase of broken parity-avour symmetry in the strong oupling

re-gion[99 ,100℄. Fortwistedmassfermions,thatadditionallyhavethe

µ ₀

^{-axisintheirbare}

parametersphasediagram,anotherunphysialphaseappearsforintermediateouplings

that shrinks towards the ontinuum limit. This phaseis a surfae of rst order

transi-tions thatinorporatesthe lineofritialhoppingparameters

κ _c (β)

^{. Thesendingsare}

summarised inthe skethingure 3.3.

Thetheoretialexplanation forthesetwo phasesanbeworked outintheframework

ofhiral perturbationtheory[101 ,102℄whihallowstoinvestigatethevauumstruture

of its eetive Lagrangian by means of minimising the orresponding potential. For

this purpose, the ontinuum Lagrangian given in equation (2.30) has, of ourse, to be

supplementedbythe leading utoeets,

L χ = f ² 4

^Tr

∂ _µ Σ∂ _µ Σ ^†

− f ² 4

^Tr

χΣ ^† + Σχ ^†

− f ² 4

^Tr

ρΣ ^† + Σρ ^†

,

^(3.39)

with

χ = 2B ₀ (m1 −

ⁱ

µτ ³ )

^and

ρ = 2aW ₀ 1

^{, slightly adjusting thenotation in [102 ℄.}

ρ

parametrisestheutoeets. NotethatitenterstheLagrangianinthesamewayasthe

quarkmasssothatthe leadinguto eetsessentiallyleadtoashiftinthebarequark

mass,f.alsothedisussionin[101 ℄. Thepotentialthatanbeextratedfromtheabove

Lagrangian anbeexpressed interms ofa unitfour vetor,

u = (u ₀ , u ₁ , u ₂ , u ₃ )

^{[102 ℄,}

V = − c ₁ u ₀ + c ₂ u ² ₀ + c ₃ u ₃ .

^(3.40)

This vetorparametrises

Σ

^,

Σ = u ₀ 1 +

ⁱ

u _a τ ^a ,

^(3.41)

sothat itsomponents determine the vauum struture,inpartiular,

u 0 ∼ ψψ

,

^(3.42a)

u ₃ ∼

ψγ ₅ τ ³ ψ

.

^(3.42b)

m ₀ µ

B

m ₀ µ

µ _c

Figure3.4.:Possiblephasetransition senariosfor Wilson fermionswith twistedmass term.

Left: Aokiphase. Right: Normalsenario,alsoalledSharpe-Singletonsenario.

(from[102℄)

The oeients

c 1 = 2f ² (B 0 m q + W 0 a)

^and

c 3 = 2f ² B 0 µ 0

^{are diretly related to the}

quark mass and uto eets. However, the ruial observation is that the sign of

c ₂

^{is not determined from the} Lagrangian. Therefore, the vauum struture an be qualitatively dierent depending on that sign. The two possible senarios have been

mapped out in [102 ℄. The rst possibility,

c ₂ > 0

^{, leads to thepreviously known Aoki}

phase of broken avour symmetry,

ψγ ₅ τ ³ ψ

6

= 0

^{, if the twisted mass vanishes. The}

seondsenario,

c ₂ < 0

^{, orrespondsto the} Sharpe-Singleton plane of bulk transitions that has been found for weaker ouplings numerially. This plane inorporates the

ritial untwisted quark mass and extends to non-vanishing twisted mass. It ends for

some ritial

µ _c ∼ a ²

^{and thus vanishes in the ontinuum limit. A sketh of the two}

senariosis showningure3.4.

Im Dokument OPUS 4 | The thermal transition of quantum chromodynamics with Twisted Mass Fermions (Seite 30-35)