Phase structure of a generalized Nambu-Jona-Lasinio model with Wilson fermions in the mean-field or large-N expansion

A ailable online a www.sciencedi ec .com
Nuclea Physics B 875 (2013) 45–62
www.else ie .com/loca e/nuclphysb
Phase s uc u e o a gene alized Nambu–Jona-Lasinio
model wi h Wilson e mions in he mean- ield
o la ge-Nexpansion
V. Azcoi ia,G.DiCa lob, E. Follanaa, M. Gio danoc,A.Vaque od,∗
aDepa amen o de Física Teó ica, Facul ad de Ciencias, Uni e sidad de Za agoza, Cl. Ped o Ce buna 12,
E-50009 Za agoza, Spain
bINFN, Labo a o i Nazionali del G an Sasso, I-67100 Asse gi, L’Aquila, I aly
cIns i u e o Nuclea Resea ch o he Hunga ian Academy o Sciences (ATOMKI), Bem é 18/c,
H-4026 Deb ecen, Hunga y
dCompu a ion-based Science and Technology Resea ch Cen e (CaSToRC), The Cyp us Ins i u e,
20 Cons an inou Ka a i S ee , Nicosia 2121, Cyp us
Recei ed 10 Ap il 2013; accep ed 1 July 2013
A ailable online 9 July 2013
Abs ac
We analyze he acuum s uc u e o a gene alized la ice Nambu–Jona-Lasinio model wi h wo la o s
o Wilson e mions, such ha i s con inuum ac ion is he mos gene al ou - e mion ac ion wi h “ i ial”
colo in e ac ions, and ha ing a SU(2)V×SU(2)Asymme y in he chi al limi . The phase s uc u e o his
model in he space o he wo ou - e mion couplings shows, in addi ion o he s anda d Aoki phases, new
phases wi h ¯
ψγ5ψ=0, in close analogy o simila esul s ecen ly sugges ed by some o us o la ice
QCD wi h wo degene a e Wilson e mions. This esul shows how he phase s uc u e o an e ec i e model
o low-ene gy QCD canno be en i ely unde s ood om Wilson Chi al Pe u ba ion Theo y, based on he
s anda d QCD chi al e ec i e Lag angian app oach.
©2013 Else ie B.V. All igh s ese ed.
Keywo ds: Nambu–Jona-Lasinio; La ice QCD; Wilson e mions; Spon aneous symme y b eaking; Pa i y; Fla o
symme y
*Co esponding au ho .
E-mail add ess: a. aque [email protected] (A. Vaque o).
0550-3213/$ – see on ma e ©2013 Else ie B.V. All igh s ese ed.
h p://dx.doi.o g/10.1016/j.nuclphysb.2013.07.001
46 V. Azcoi i e al. / Nuclea Physics B 875 (2013) 45–62
1. In oduc ion
Since he i s nume ical in es iga ions o ou -dimensional non-abelian gauge heo ies wi h
dynamical Wilson e mions we e pe o med in he ea ly 80s [1,2], he unde s anding o he phase
and acuum s uc u e o la ice QCD wi h Wilson e mions a nonze o la ice spacing, and o he
way in which chi al symme y is eco e ed in he con inuum limi , has been a goal o la ice
ield heo is s. The complexi y o he phase s uc u e o his model has been known o a long
ime. The exis ence o a phase wi h pa i y and la o symme y b eaking was conjec u ed o his
model by Aoki in he middle 80s [3,4], and con i med la e on [5–27].
The s anda d wisdom on la ice QCD wi h Wilson e mions is ha e en i chi al symme y is
explici ly b oken a ini e la ice spacing aby he Wilson egula iza ion, his symme y will be
eco e ed and spon aneously b oken in he con inuum limi . Howe e , i is di icul o unde s and
why he e exis s a c i ical line a ini e la ice spacing along which he h ee pions a e massless.
Indeed, he pions canno be he h ee Golds one bosons associa ed wi h he spon aneous b eak-
ing o he SU(2)chi al symme y since, as p e iously s a ed, he Wilson egula iza ion b eaks
explici ly his symme y.
One o he main ea u es o Aoki’s pic u e was o cla i y his poin . In he Aoki phase, he
cha ged pions a e massless because hey a e he wo Golds one bosons associa ed wi h he spon-
aneous b eaking o he SU(2) la o symme ydown oU(1), wi h a non- anishing acuum
expec a ion alue o he i¯
ψγ5τ3ψcondensa e. The neu al pion, which is massi e in he Aoki
phase, becomes massless on he c i ical line because la o symme y is con inuously eco e ed
on his line which sepa a es he b oken (Aoki) phase om he unb oken (physical) phase. The
o he ele an ea u e o he Aoki scena io is ha i p o ides a coun e example o he Va a–Wi en
heo em on he impossibili y o spon aneously b eak pa i y in a ec o -like heo y wi h posi i e
de ini e in eg a ion measu e [28–32].
Aoki’s conjec u e has been suppo ed no only by nume ical simula ions o la ice QCD, bu
also by heo e ical s udies based on he Nambu–Jona-Lasinio model [5,33], on he linea sigma
model [6], and on applying Wilson Chi al Pe u ba ion Theo y (WχPT) o he con inuum e -
ec i e Lag angian [11]. The la e analysis p edic s, nea he con inuum limi , wo possible
scena ios, depending on he sign o an unknown low-ene gy coe icien . In he i s scena io,
la o and pa i y a e spon aneously b oken, and he e is an Aoki phase wi h a nonze o alue only
o he i¯
ψγ5τ3ψcondensa e, whe eas i¯
ψγ5ψ=0. In he o he one ( he “ i s -o de ” scena io)
he e is no spon aneous symme y b eaking.
This s anda d pic u e o he Aoki phase was ques ioned by h ee o us in [21], whe e we
conjec u ed on he appea ance o new acua in he Aoki phase, which can be cha ac e ized by a
non- anishing acuum expec a ion alue o he la o -single pseudoscala condensa e i¯
ψγ5ψ,
and which canno be connec ed o he Aoki acua by pa i y- la o symme y ans o ma ions.
Mo e ecen ly, we ha e ob ained esul s om nume ical simula ions o la ice QCD wi h wo
degene a e la o s o Wilson e mions, sugges ing ha ou conjec u e could be ealized [34].
Since hese esul s seem o ques ion he alidi y o he WχPT analysis [11], an app oach which
has been success ully applied in many con ex s, i is wo hwhile o analyze he possible o igins
o his disc epancy.
Fi s , one should no ice ha he chi al e ec i e Lag angian app oach is based on he con-
inuum e ec i e Lag angian w i en as a se ies o con ibu ions p opo ional o powe s o he
la ice spacing a, plus he cons uc ion o he co esponding chi al e ec i e Lag angian, keeping
only he e ms up o o de a2[11]. This means ha p edic ions based on his chi al e ec i e
Lag angian app oach should wo k close enough o he con inuum limi , whe e keeping e ms
V. Azcoi i e al. / Nuclea Physics B 875 (2013) 45–62 47
only up o o de a2can be jus i ied. Howe e , he da a epo ed in [34] we e ob ained a β=2.0,
and a e y ough es ima e gi es a la ice spacing o o de 3.0GeV
−1a his β. Hence, a possible
explana ion o he disc epancies ha we ound elies on he necessi y o including highe -o de
e ms in he chi al e ec i e Lag angian.
We wan o ecall he e ha he Aoki e ec i e po en ial was ob ained om a s ong cou-
pling expansion combined wi h a 1/N expansion [4], i.e., a away om he con inuum limi .
Fu he mo e, Aoki’s solu ion shows degene a e acua wi h i¯
ψγ5τ3ψ=0, i¯
ψγ5ψ=0 and
i¯
ψγ5τ3ψ=0, i¯
ψγ5ψ=0 espec i ely in he s ong coupling limi . The inclusion o highe -
o de con ibu ions o he s ong-coupling 1/N expansions b eaks he acuum degene acy by
selec ing he s anda d acuum wi h i¯
ψγ5τ3ψ=0, bu he no malized di e ence o acuum
ene gy densi ies o he wo acua is o o de 10−14 a β=2.0 and N=3, showing he ex emely
high ins abili y o he Aoki solu ion.
The second possible o igin o he disc epancies be ween he non-s anda d scena io o [21,34]
and he WχPT analysis o [11], he analysis o which will be he main subjec o his pape , lies
in he ollowing poin . The chi al e ec i e Lag angian app oach is based, as i is well known,
on he assump ion ha he ele an low-ene gy deg ees o eedom in QCD a e he h ee pions.
This assump ion can be eliable in he physical phase, up o he c i ical line, and also in he Aoki
phase, nea he c i ical line, bu i could b eak down as we go deep in he Aoki phase, whe e
he neu al pion is massi e. Indeed, QCD wi h wo degene a e la o s o Wilson e mions o
ba e mass m0=−4.0 in la ice uni s should also show degene a e acua wi h i¯
ψγ5τ3ψ=0,
i¯
ψγ5ψ=0 and i¯
ψγ5τ3ψ=0, i¯
ψγ5ψ=0 espec i ely, as discussed in [34].
Wi h he pu pose o es ablishing he ange o applicabili y o he s anda d QCD chi al e ec-
i e Lag angian app oach, we will analyze in his pape he acuum s uc u e o a gene alized
Nambu–Jona-Lasinio model (NJL) wi h Wilson e mions in he mean- ield o leading-o de 1/N
expansion. The model has been chosen o possess he mo e gene al SU(2)V×SU(2)Asymme-
y in he con inuum, in analogy o QCD. The elec ion o he NJL model o ou analysis was
mo i a ed by he ac ha ou -dimensional models wi hou gauge ields, and wi h ou - e mion
in e ac ions, a e conside ed as e ec i e models o desc ibe he low-ene gy physics o QCD1[35,
36].
The ou line o he pape is as ollows. In Sec ion 2we desc ibe he model in he con inuum and
i s la ice egula ized e sion wi h Wilson e mions, as well as he way in which he model can
be analy ically sol ed in he mean- ield 1/N expansion wi h he help o eigh auxilia y scala
and pseudoscala ields. The gap equa ions and he phase diag am o he mean- ield model in
he a ious physically ele an cases a e analyzed in Sec ion 3. In Sec ion 4we show how he
mean- ield equa ions o ou gene alized NJL model can be ob ained in he leading o de o he
1/N expansion o a ou - e mion model wi h non- i ial colo and la o in e ac ions, bu whe e
he ac ion is local and ee om he sign p oblem. Sec ion 5summa izes ou conclusions.
2. The model
The mos gene al ou - e mion con inuum Lag angian in Euclidean space wi h SU(2)V×
SU(2)Asymme y in he chi al limi and wi h i ial colo dependence can be w i en as ollows,
−L=−¯
ψ(/∂+m)ψ +G1(¯
ψψ)2+(i ¯
ψγ5τψ)2+G2(i ¯
ψγ5ψ)2+(¯
ψτψ)2,(1)
1Fo a e iew on he NJL model, see [37] and e e ences he ein.
48 V. Azcoi i e al. / Nuclea Physics B 875 (2013) 45–62
whe e ψis a e mion ield wi h ou Di ac and wo la o componen s, and τaa e he Pauli
ma ices ac ing in la o space. I is cus oma y, in o de o a oid he sign p oblem and/o o
pe o m a 1/N expansion, o add ano he (“colo ”) deg ee o eedom o he spino s, and o
s aigh o wa dly gene alize he in e ac ion by eplacing ¯
ψBψ →N
i=1¯
ψiBψi, whe e Bis any
o he ma ices appea ing in Eq. (1). Al hough he in e ac ion is no diagonal in colo space, i
will become so a e a Hubba d–S a ono ich ans o ma ion, and mo eo e i will be he same
o e e y colo : o his eason we will call i diagonal and i ial in colo , wi h a small abuse o
e minology. In he ollowing, we will e e o his s aigh o wa d gene aliza ion as he N-colo
model.
The NJL model gi en by ac ion (1) enjoys he same SU(2)V×SU(2)Asymme y o QCD and
i is an e ec i e model o desc ibe he low-ene gy physics o QCD [35]. This model, egula ized
on a hype cubic ou -dimensional la ice wi h Wilson e mions, was analyzed in he G2=0
limi and in he mean- ield o i s -o de 1/N expansion by Aoki e al. [5], who ound a phase,
o la ge alues o G1, in which bo h la o symme y and pa i y a e spon aneously b oken, in
close analogy o la ice QCD wi h Wilson e mions. The quali a i e esul s o Aoki e al. we e
also co obo a ed by Bi a and V anas in [33], whe e hey ound, using nume ical simula ions,
he exis ence o his pa i y- la o b oken phase in he wo-colo model.
The la ice ac ion o he N-colo model in he Wilson egula iza ion can be w i en as S=
S0+SI, wi h he ee pa o he ac ion being
S0=
x,y ¯
ψxxyψy,(2)
whe e now ψis a e mion ield wi h ou Di ac, wo la o and Ncolo componen s, and whe e
he Di ac–Wilson ope a o is gi en by
xy =1
2
4

μ=1(γμ− )δx+ˆμ,y −(γμ+ )δx−ˆμ,y+(4 +m0)δxy,(3)
wi h he Wilson pa ame e and m0 he ba e e mion mass. The in e ac ion pa is
−SI=
x
G1
N(¯
ψxψx)2+(¯
ψxiγ5τψx)2+G2
N(¯
ψxiγ5ψx)2+(¯
ψxτψx)2,(4)
we e we ha e con enien ly ede ined he coupling cons an s. As i is well known, he Wilson
e m b eaks explici ly he ull chi al symme y, and so only pa i y and ec o symme ies a e
kep in he la ice egula iza ion. The ou - e mion ac ion can be bilinea ized by pe o ming a
Hubba d–S a ono ich ans o ma ion, which implies he in oduc ion o eigh scala and pseu-
doscala auxilia y ields as ollows,
SB=N
xβ1σ2
x+π2
x+β2η2
x+ρ2
x+
x,y ¯
ψxMxyψy,(5)
whe e he e mion ma ix Mis
Mxy =xy +δxy(σx+iγ5τ·πx+iγ5ηx+τ·ρx), (6)
and mo eo e βi=1/(4Gi). He e we a e conside ing he case G1,G
2⩾0.
In he G2=0 case analyzed in [5,33] i is easy o see ha he e mion de e minan is eal2and
he e o e he heo y, wi h an e en numbe o colo s, is ee om he sign p oblem. This allows
2Reali y is eadily p o ed by no ing ha CMC†=M∗, wi h C=τ2γ1γ3and CC†=1.
V. Azcoi i e al. / Nuclea Physics B 875 (2013) 45–62 49
o consis en ly pe o m he 1/N expansion [5] and he nume ical simula ions in he wo-colo
model [33]. Un o una ely, in he gene al case (G1= 0, G2= 0) he e mion de e minan is
complex, and e en i he leading o de o he 1/N expansion is ee om he sign p oblem o
an e en numbe o colo s, he e y consis ency o his expansion is, a leas , doub ul. This is he
eason why we decided o s udy he in ini e ange model, o mean- ield app oxima ion, whe e
again one can easily show ha he sign p oblem is absen o an e en numbe o colo s. Howe e ,
in Sec ion 4we will show how he gap equa ions o he in ini e ange model a e jus he same
ob ained a leading o de in he 1/N expansion o a ou - e mion model wi h local in e ac ions,
he same symme ies, and ee om he sign p oblem.
3. The phase diag am o he mean- ield model
The in e ac ion pa S(MF)
Io he la ice ac ion S(MF)=S0+S(MF)
I o he in ini e- ange model
can be w i en as ollows,
−S(MF)
I=G1
N
1
V
x¯
ψxψx2
+
x¯
ψxiγ5τψx2
+G2
N
1
V
x¯
ψxiγ5ψx2
+
x¯
ψxτψx2,(7)
whe e Vis he numbe o la ice si es. Pe o ming again a Hubba d–S a ono ich ans o ma ion
we ge o he bilinea ized ac ion
S(MF)
B=VNβ1σ2+π2+β2η2+ρ2+
x,y ¯
ψxMxyψy,(8)
whe e he auxilia y ields a e now cons an ields, he e mion ma ix is
Mxy =xy +δxy(σ +iγ5τ·π+iγ5η+τ·ρ), (9)
and again βi=1/(4Gi), wi h G1,G
2⩾0. The in eg al o e he e mion ields can again be
done analy ically and in he limi o la ge olume V he model can be sol ed by w i ing down
and sol ing he saddle-poin equa ions.
In eg a ing ou he e mionic deg ees o eedom, he pa i ion unc ion o he mean- ield
model eads
Z=dσ d3πdηd
3ρDe Me
−NV[β1(σ 2+π2)+β2(η2+ρ2)]
≡dσ d3πdηd
3ρe
−2NVVe ,(10)
wi h Ve he e ec i e po en ial pe la o and colo . As we show in Appendix A, he e mionic
de e minan is eal in his case; since we a e aking an e en numbe o colo s, De Mis also
posi i e, so ha he e is no sign p oblem, and we can w i e De M=(De MM†)1
2.
In o de o compu e he de e minan i is con enien o go o e o momen um space. S a -
ing om a ini e la ice wi h pe iodic bounda y condi ions and hen aking he limi o in ini e
olume, one ob ains
De M=expV
2
B
d4p
(2π)4 log ˜
M(p) ˜
M(p)†+OV−1,(11)

50 V. Azcoi i e al. / Nuclea Physics B 875 (2013) 45–62
whe e
Mxy =
B
d4p
(2π)4e−ip·(x−y) ˜
M(p), ˜
M(p) =
x
eip·xMx0,(12)
and whe e Bis he i s B illouin zone pμ∈[0,2π],μ=1,...,4 (o equi alen ly pμ∈[−π,π]
due o pe iodici y), and s ands o he ace o e Di ac, la o and colo indices. A s aigh o -
wa d calcula ion shows ha
˜
M(p) =i
4

μ=1
γμsinpμ+ 4−
4

μ=1
cospμ+m0+σ+iγ5τ·π+iγ5η+τ·ρ. (13)
The e ec i e po en ial Ve can be compu ed explici ly, and eads
Ve =β1
2σ2+π2+β2
2η2+ρ2−
B
d4p
(2π)4logQ, (14)
wi h
Q=Σ(p)2+2Σ(p)w (p) +m0+σ2+π2+η2+ρ2
+w (p) +m0+σ2+π2−η2+ρ22
+4ηw (p) +m0+σ−ρ·π2,(15)
whe e we ha e se
Σ(p)=
4

μ=1
(sinpμ)2,w
(p) = 4−
4

μ=1
cospμ.(16)
No ice ha Q⩾0.
In he la ge olume limi , he pa i ion unc ion will be domina ed by he con ibu ion coming
om he minimum o he e ec i e po en ial, and so i can be compu ed h ough he saddle-poin
echnique. In o de o look o he minimum o he e ec i e po en ial, i is con enien o eo de
he e ms in Eq. (15). A li le algeb a allows o ew i e i as
Q=Σ(p)+w (p) +m0+σ2+Π2+η2+ρ22
−4ρw (p) +m0+σ+Πηcosθ2−4η2+ρ2Π2(sinθ)2,(17)
whe e we ha e se
Π=|π|,ρ=|ρ|,
π·ρ=Πρ cosθ. (18)
Se ing also
Q0=Σ(p)+w (p) +m0+σ2+Π2+η2+ρ22⩾0,
Q1=4ρw (p) +m0+σ+Πηcosθ2+4η2+ρ2Π2(sinθ)2⩾0,(19)
we ha e Q=Q0−Q1, and he e ec i e po en ial can be ew i en as
Ve =β1
2σ2+Π2+β2
2η2+ρ2−
B
d4p
(2π)4logQ0+log1−Q1
Q0.(20)
V. Azcoi i e al. / Nuclea Physics B 875 (2013) 45–62 51
I is con enien o ou pu poses o g oup he a ious e ms in wo di e en ways. The i s way
is
Ve =β1
2σ2+Π2+η2+ρ2−
B
d4p
(2π)4logQ0
+(β2−β1)
2η2+ρ2−
B
d4p
(2π)4log1−Q1
Q0,(21)
which as we will see is app op ia e o he case β1<β
2, and he second way is
Ve =β1
2σ2+β2
2Π2+η2+ρ2−
B
d4p
(2π)4logQ0
+(β1−β2)
2Π2−
B
d4p
(2π)4log1−Q1
Q0,(22)
which is app op ia e o he case β1>β
2. The key obse a ion is ha Q0depends only on σand
on he combina ion z2≡Π2+η2+ρ2, so ha he i s line in bo h equa ions depends only on
σand z. The e o e, he minimiza ion o he e ec i e po en ial a ixed σand zin ol es only he
second line o Eqs. (21) and (22).
To make hings mo e anspa en , le us in oduce he new se o a iables z,ω,ϕ,in e ms o
which one w i es
Π=zcosω,
η=zsinωcosϕ,
ρ=zsinωsinϕ. (23)
The ange o hese a iables is z⩾0, ω∈[0,π
2],ϕ∈[0,π], ha co esponds o he ange Π⩾0,
ρ⩾0, η∈Ro he o iginal a iables. In e ms o he new a iables, Eq. (21) eads
Ve (σ,z,ω,ϕ)=β1
2σ2+z2−
B
d4p
(2π)4logQ0(σ, z)
−β
2(z sinω)2−
B
d4p
(2π)4log1−Q1(σ,z,ω,ϕ)
Q0(σ, z) ,(24)
whe e β ≡β1−β2and we ha e made explici he dependence on he ele an a iables, and
Eq. (22) eads
Ve (σ,z,ω,ϕ)=β1
2σ2+β2
2z2−
B
d4p
(2π)4logQ0(σ, z)
+β
2(z cosω)2−
B
d4p
(2π)4log1−Q1(σ,z,ω,ϕ)
Q0(σ, z) .(25)
As we ha e al eady no ed, he i s line in Eqs. (24) and (25) depends only on σand z.Asa
consequence, in o de o minimize he e ec i e po en ial wi h espec o ωand ϕwe ha e o
52 V. Azcoi i e al. / Nuclea Physics B 875 (2013) 45–62
ocus on he second line only. Mo eo e , since Q⩾0, we ha e ha Q1⩽Q0, and so he las
e m in Eqs. (24) and (25) is posi i e o ze o,
V≡−
B
d4p
(2π)4log1−Q1(σ,z,ω,ϕ)
Q0(σ, z) ⩾0.(26)
The e o e, he second line in Eq. (24) is posi i e o ze o i β2⩾β1, and he second line in Eq. (25)
is posi i e o ze o i β1⩾β2; in pa icula , bo h e ms a e posi i e o ze o. I we can ind alues
o ωand ϕsuch ha hese lowe bounds a e sa u a ed, hen we ha e au oma ically minimized
he e ec i e po en ial wi h espec o ωand ϕ. In o de o do so, we need o make bo h e ms
anish, and in pa icula we need ha Q1 anishes iden ically as a unc ion o he momen um.3
In e ms o ou new a iables, Q1 eads
Q1=4(z sinω)2(z cosωsinθ)2+sinϕw (p) +m0+σ+zcosωcosϕcosθ2.
(27)
One sees immedia ely ha Q1 anishes iden ically i z=0, o i sinω=0, i.e., ω=0. I z=0,
ω=0, hen bo h e ms in b aces mus be ze o, and he second one mus be so independen ly
o p: his can happen only i sinϕ=0, i.e., ϕ=0, π, which in u n equi es ha cosω=0, i.e.,
ω=π
2.
Summa izing, Q1 anishes iden ically only i 4
1.ω=0;
2.ω=π
2,ϕ=0,π, (28)
independen ly o he alues o σand z. S a ed di e en ly, in e ms o he o iginal a iables, Q1
anishes iden ically only i
1.ρ=0,|η|=0;
2.ρ=0,Π=0.(29)
I is immedia e o check ha in case 1 he second line o Eq. (24) anishes, while in case 2 he
second line o Eq. (25) anishes, independen ly o β . Le us now discuss he a ious cases
sepa a ely.
Case β1<β
2.In his case, he minimum o he e ec i e po en ial lies on he cu e ω=0, and
is ob ained by minimizing he unc ional
V<=β1
2σ2+z2−
B
d4p
(2π)4logQ0(σ, z), (30)
wi h espec o σand z. Since ω=0 co esponds o η=ρ=0, in his case z=Π. Clea ly, he
minimum will be independen o β2. The gap equa ions ead he e o e
3In p inciple i is su icien ha Q1is nonze o only on a se o ze o measu e in he ou -dimensional momen um space,
bu i is easy o see ha ei he Q1 anishes iden ically o i anishes on a h ee-dimensional hype su ace.
4Roughly speaking, since a z=0 all alues o ωand ϕa e equi alen , hese wo cases include also he case z=0.
V. Azcoi i e al. / Nuclea Physics B 875 (2013) 45–62 53
Fig. 1. Bounda y o he phase wi h b oken pa i y and la o o β =β1−β2<0in he(β1,m
0)plane. When β =0,
i.e., β1=β2, wo degene a e acua wi h b oken pa i y exis , one wi h b oken and one wi h unb oken la o symme y.
He e we se =1.
0=β1
4σ−
B
d4p
(2π)4
w (p) +m0+σ
Σ(p)+(w (p) +m0+σ)2+Π2,
0=β1
4−
B
d4p
(2π)4
1
Σ(p)+(w (p) +m0+σ)2+Π2Π. (31)
The e a e wo solu ions o hese equa ions. The i s one has Π=0 and σde e mined by he
solu ion o he equa ion
β1
4σ=
B
d4p
(2π)4
w (p) +m0+σ
Σ(p)+(w (p) +m0+σ)2,(32)
which always exis s.5A second solu ion wi h nonze o Πis ob ained by sol ing
0=
B
d4p
(2π)4
w (p) +m0
Σ(p)+(w (p) +m0+σ)2+Π2,
β1
4=
B
d4p
(2π)4
1
Σ(p)+(w (p) +m0+σ)2+Π2.(33)
5To see his i is enough o show ha he igh -hand side is always ini e, and ha i anishes as 1/σ o la ge |σ|.The
second poin is i ial, while o p o e he i s one i is enough o bound he igh -hand side as ollows:

B
d4p
(2π)4
w (p) +m0+σ
Σ(p)+(w (p) +m0+σ)2⩽(8+m0+σ)
B
d4p
(2π)4
1
Σ(p) <∞.
60 V. Azcoi i e al. / Nuclea Physics B 875 (2013) 45–62
Appendix A. Absence o a sign p oblem in he in ini e- ange model
In his appendix, we p o e ha he e mion de e minan appea ing in Eq. (10) is eal and posi-
i e. To show his, one no ices i s ha γ4xyγ4=xPyP, wi h x=(x,x4)and xP=(−x,x4).
Since he de e minan s o Mand M(P ) a e equal, wi h M(P )
xy =MxPyP, one inds ha
De M=De M(P ) =De [γ4γ4+σ+iγ5τ·π+iγ5η+τ·ρ]
=De +(σ −iγ5τ·π−iγ5η+τ·ρ)
.(A.1)
Nex , one exploi s he uni a i y o he ma ix ˜
C=γ1γ3and he ac ha ˜
C ˜
C†=∗ o w i e
De M=De ∗+(σ −iγ5τ·π−iγ5η+τ·ρ).(A.2)
Finally, one pe o ms a o a ion Rin la o space in such a way ha he ec o s πR=Rπand
ρR=Rρlie in he (1,3) plane, i.e., so ha ρR∧πRis along di ec ion 2 in la o space. Since R
is implemen ed by a uni a y ope a o UR, and τ1and τ3a e eal, one has ha
De M=De ∗+(σ −iγ5τ·πR−iγ5η+τ·ρR)
=De +(σ +iγ5τ·πR+iγ5η+τ·ρR)∗
=De +(σ +iγ5τ·π+iγ5η+τ·ρ)∗=De M∗.(A.3)
Finally, since Mis diagonal and i ial in colo , M=¯
M1C, one has ha o an e en numbe o
colo s de M=(de ¯
M)N=(de ¯
Mde ¯
M†)N
2⩾0.
Appendix B. Quad a ic pa o he ac ion in he local model
In his appendix we epo a ew esul s ela ed o he local model discussed in Sec ion 4.We
use he ollowing no a ion:
I0(q,σ,z)=
B
d4p
(2π)4
1
g(p +q
2, z)g(p −q
2,z)4

μ=1
sinpμ+qμ
2sinpμ−qμ
2
+w p+q
2+m0+σw p−q
2+m0+σ+z2,
I1(q,σ,z)=
B
d4p
(2π)4[w (p +q
2)+m0+σ][w (p −q
2)+m0+σ]
g(p +q
2, z)g(p −q
2,z) ,
I2(q,σ,z)=
B
d4p
(2π)4
1
g(p +q
2, z)g(p −q
2,z),
I3(q,σ,z)=
B
d4p
(2π)4
w (p +q
2)+w (p −q
2)+2(m0+σ)
g(p +q
2, z)g(p −q
2,z) .(B.1)
The non- anishing en ies o he in e se p opaga o G−1
ij (q),Eq.(50), a e lis ed below o he
s anda d Aoki and non-s anda d Aoki cases. In he unb oken phase he in e se p opaga o is
diagonal, and can be eco e ed by simply se ing Π=0o η=0 in he equa ions below. He e
Π=|π|. S anda d Aoki phase:

V. Azcoi i e al. / Nuclea Physics B 875 (2013) 45–62 61
G−1
σσ(q) =8β1
4−I0(q,σ,Π)+2I1(q,σ,Π)
,
G−1
πaπb(q) =8β1
4−I0(q,σ,Π)
δab +2πaπbI2(q,σ,Π)
,
G−1
ηη (q) =8β2
4−I0(q,σ,Π)+2Π2I2(q,σ,Π)
,
G−1
ρaρb(q) =8β2
4−I0(q,σ,Π)+2I1(q,σ,Π)
δab +2I2(q,σ,Π)
Π2δab −πaπb,
G−1
σπa(q) =−8I3(q,σ,Π)π
a,
G−1
ηρa(q) =−8I3(q,σ,Π)π
a.
Non-s anda d Aoki phase:
G−1
σσ(q) =8β1
4−I0(q,σ,η)+2I1(q,σ,η)
,
G−1
πaπb(q) =8β1
4−I0(q,σ,η)+2η2I2(q,σ,η)
δab,
G−1
ηη (q) =8β2
4−I0(q,σ,η)+2η2I2(q,σ,η)
,
G−1
ρaρb(q) =8β2
4−I0(q,σ,η)+2I1(q,σ,η)
δab,
G−1
ση(q) =−8I3(q,σ,η)η,
G−1
πaρb(q) =−8I3(q,σ,η)ηδ
ab.(B.2)
The i s wo e ms in he small-qexpansion o I0and I2is gi en below.
I0(q,σ,z)=
B
d4p
(2π)4
1
g(p,z)
−q2
8
B
d4p
(2π)44
μ=1(cospμ)2+ 2(sinpμ)2
[g(p,z)]2+Oq4
=I0(0,σ,z)−q2I(2)
0(σ, z) +Oq4,
I2(q,σ,z)=
B
d4p
(2π)4
1
[g(p,z)]2
−q2
2
B
d4p
(2π)44
μ=1(sinpμ)2[cospμ+ (w (p) +m0+σ)]2
[g(p,z)]4+Oq4
=I2(0,σ,z)−q2I(2)
2(σ, z) +Oq4.(B.3)
No ice ha
I0(0,σ,z)=
B
d4p
(2π)4
1
g(p,z) =β1
4,in he s anda d Aoki phase,
β2
4,in he non-s anda d Aoki phase. (B.4)
62 V. Azcoi i e al. / Nuclea Physics B 875 (2013) 45–62
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