On the phase diagram of the Higgs SU (2) model C. Bonati, G. Cossu, A. D’Alessandro, M. D’Elia, A. Di Giacomo On the phase diagram of the Higgs SU (2) model – p. 1
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On the phase diagram ofthe Higgs SU(2) model
C. Bonati, G. Cossu, A. D’Alessandro,
M. D’Elia, A. Di Giacomo
On the phase diagram of the Higgs SU(2) model – p. 1
Summary
the model (notations)
On the phase diagram of the Higgs SU(2) model – p. 2
Summary
the model (notations)
features of the model
On the phase diagram of the Higgs SU(2) model – p. 2
Summary
the model (notations)
features of the model
previous results in literature
On the phase diagram of the Higgs SU(2) model – p. 2
Summary
the model (notations)
features of the model
previous results in literature
new results
On the phase diagram of the Higgs SU(2) model – p. 2
Summary
the model (notations)
features of the model
previous results in literature
new results
conclusionsOn the phase diagram of the Higgs SU(2) model – p. 2
The modelSU(2) gauge theory coupled with a Higgsdoublet in fundamental representation
S = SW [U ] −κ
2
∑
x,µ{Φ†(x)Uµ(x)Φ(x + µ̂) +
+Φ†(x + µ̂)U †µ(x)Φ(x)} + λ
∑
x[Φ†(x)Φ(x) − 1]2
On the phase diagram of the Higgs SU(2) model – p. 3
The modelSU(2) gauge theory coupled with a Higgsdoublet in fundamental representation
S = SW [U ] −κ
2
∑
x,µ{Φ†(x)Uµ(x)Φ(x + µ̂) +
+Φ†(x + µ̂)U †µ(x)Φ(x)} + λ
∑
x[Φ†(x)Φ(x) − 1]2
in this work λ = ∞
On the phase diagram of the Higgs SU(2) model – p. 3
The modelSU(2) gauge theory coupled with a Higgsdoublet in fundamental representation
S = SW [U ] −κ
2
∑
x,µ{Φ†(x)Uµ(x)Φ(x + µ̂) +
+Φ†(x + µ̂)U †µ(x)Φ(x)}
On the phase diagram of the Higgs SU(2) model – p. 4
The modelSU(2) gauge theory coupled with a Higgsdoublet in fundamental representation
S = SW [U ] −κ
2
∑
x,µ{Φ†(x)Uµ(x)Φ(x + µ̂) +
+Φ†(x + µ̂)U †µ(x)Φ(x)}
Φ̃(x) = iσ2Φ(x)∗ φ(x) =
Φ̃1(x) Φ1(x)
Φ̃2(x) Φ2(x)
ΦΛ = ΛΦ φΛ = Λφ
On the phase diagram of the Higgs SU(2) model – p. 4
The modelSU(2) gauge theory coupled with a higgsdoublet in fundamental representation
S = SW [U ] −κ
2
∑
x,µ{Φ†(x)Uµ(x)Φ(x + µ̂) +
+Φ†(x + µ̂)U †µ(x)Φ(x)}
S = β∑
x,µ<ν
{
1 −1
2ReTrPµν(x)
}
−
−κ
2
∑
x,µTr[φ†(x)Uµ(x + µ̂)φ(x + µ̂)]
On the phase diagram of the Higgs SU(2) model – p. 5
Features of themodel
Limiting cases:
κ = 0 SU(2) gauge theoryconfinement
On the phase diagram of the Higgs SU(2) model – p. 6
Features of themodel
Limiting cases:
κ = 0 SU(2) gauge theoryconfinement
β = ∞O(4) σ−modelspontaneous symme-try breaking
On the phase diagram of the Higgs SU(2) model – p. 6
Features of themodel
Limiting cases:
κ = 0 SU(2) gauge theoryconfinement
β = ∞O(4) σ−modelspontaneous symme-try breaking
β = 0 &unitarygauge indipendent Uµ(x)′s
On the phase diagram of the Higgs SU(2) model – p. 6
Features of themodel
Limiting cases:
κ = 0 SU(2) gauge theoryconfinement
β = ∞O(4) σ−modelspontaneous symme-try breaking
β = 0 &unitarygauge indipendent Uµ(x)′s
κ = ∞ &unitarygauge no dynamics
On the phase diagram of the Higgs SU(2) model – p. 6
Features of themodel
Fradkin-Shenker theorem:
β0 ∞
κ
∞Phys. Rev. D 19, 3682 (1979)
in the red regionlocal observables
are analytic
On the phase diagram of the Higgs SU(2) model – p. 7
Features of themodel
Fradkin-Shenker theorem:
β0 ∞
κ
∞Phys. Rev. D 19, 3682 (1979)
in the red regionlocal observables
are analytic
The theorem does not holdfor non-local observables!Grady, Phys. Lett. B 626, 161 (2005)
On the phase diagram of the Higgs SU(2) model – p. 7
Features of themodel
Supposed phase diagram at zero temperatureFradkin & Shenker Phys. Rev. D 19, 3682 (1979)
first order
Mean field
β0 ∞
κ
∞
On the phase diagram of the Higgs SU(2) model – p. 8
Features of themodel
Supposed phase diagram at zero temperatureFradkin & Shenker Phys. Rev. D 19, 3682 (1979)
first order
Mean field
β0 ∞
κ
∞
Confermed insimulations atλ = 0.5Bock et alt. Phys. Rev. D41, 2573 (1990)
On the phase diagram of the Higgs SU(2) model – p. 8
Results in literaturefor the λ = ∞ case
first numerical study on 44 lattice seem toconfirm theoretical predictionLang, Rebbi & Virasoro Phys. Lett. B 104, 294 (1981)
On the phase diagram of the Higgs SU(2) model – p. 9
Results in literaturefor the λ = ∞ case
first numerical study on 44 lattice seem toconfirm theoretical predictionLang, Rebbi & Virasoro Phys. Lett. B 104, 294 (1981)
claims of two-state signal on 124 lattice atβ = 2.3 (but with small statistics)Langguth & Montvay Phys. Lett. B 165, 135 (1985)
On the phase diagram of the Higgs SU(2) model – p. 9
Results in literaturefor the λ = ∞ case
first numerical study on 44 lattice seem toconfirm theoretical predictionLang, Rebbi & Virasoro Phys. Lett. B 104, 294 (1981)
claims of two-state signal on 124 lattice atβ = 2.3 (but with small statistics)Langguth & Montvay Phys. Lett. B 165, 135 (1985)
“the system exhibits a transient behavior upto L = 24 along which the order of thetransition cannot be discerned” (also in thiscase β = 2.3)Campos Nucl. Phys. B 514, 336 (1998)
On the phase diagram of the Higgs SU(2) model – p. 9
New resultsLocal observables analized
plaquette
Higgs-gauge interaction:1
2Tr[φ†(x)Uµ(x + µ̂)φ(x + µ̂)]
Z2 monopoles
On the phase diagram of the Higgs SU(2) model – p. 10
New resultsLocal observables analized
plaquette
Higgs-gauge interaction:1
2Tr[φ†(x)Uµ(x + µ̂)φ(x + µ̂)]
Z2 monopoles
Studied using
susceptibilities L4[〈O2〉 − 〈O〉2]
Binder fourth-order cumulant1 − 〈O4〉/[3〈O2〉2]
On the phase diagram of the Higgs SU(2) model – p. 10
New resultsLocal observables analized
plaquette
Higgs-gauge interaction:1
2Tr[φ†(x)Uµ(x + µ̂)φ(x + µ̂)]
Z2 monopoles
Studied using
susceptibilities L4[〈O2〉 − 〈O〉2]
Binder fourth-order cumulant1 − 〈O4〉/[3〈O2〉2]
resulted the mostsensitive one
On the phase diagram of the Higgs SU(2) model – p. 10
New results: β = 2.5
10 20 30 40L
0.5
1
Hig
gs-g
auge
inte
ract
ion
susc
eptib
ility
a+b x4
β=2.5
On the phase diagram of the Higgs SU(2) model – p. 11
New results: β = 2.5
0 5e-05 0.0001 0.00015 0.0002 0.00025
1/L4
0.664
0.6645
0.665
0.6655
0.666
0.6665
0.667
Hig
gs-g
auge
inte
ract
ion
Bin
der
cum
ulan
t
2.0/3.0
β=2.5
On the phase diagram of the Higgs SU(2) model – p. 12
New results: β = 2.5
0 5e-06 1e-05 1.5e-05
1/L4
0.6665
Hig
gs-g
auge
inte
ract
ion
Bin
der
cum
ulan
t 2.0/3.0
2.0/3.0 + b x + c x2
β=2.5
On the phase diagram of the Higgs SU(2) model – p. 13
New results: β = 3.5
10 15 20 25 30L
0.51
0.52
0.53
0.54
0.55
0.56
0.57
0.58
0.59
0.6
Hig
gs-g
auge
inte
ract
ion
susc
eptib
ility
β=3.5
On the phase diagram of the Higgs SU(2) model – p. 14
New results: β = 3.5
0 2e-05 4e-05
1/L4
0,666
0,6665
Hig
gs-g
auge
inte
ract
ion
Bin
der
cum
ulan
t 2.0/3.0
2.0/3.0 + a x + b x2
β=3.5
On the phase diagram of the Higgs SU(2) model – p. 15
New results: β = 30
10 15 20 25 30L
0.56
0.57
0.58
0.59
0.6
0.61
0.62
0.63
0.64
0.65
0.66
Hig
gs-g
auge
inte
ract
ion
susc
eptib
ility
β=30
On the phase diagram of the Higgs SU(2) model – p. 16
New results: β = 30
0 5e-05 0.0001
1/L4
0.6645
0.665
0.6655
0.666
0.6665
Hig
gs-g
auge
inte
ract
ion
Bin
der
cum
ulan
t 2.0/3.02.0/3.0 + a x
β=30
On the phase diagram of the Higgs SU(2) model – p. 17
New resultsNon-local observable analized
〈µ〉 magnetic monopole operator
On the phase diagram of the Higgs SU(2) model – p. 18
New resultsNon-local observable analized
〈µ〉 magnetic monopole operator
Studied using
ρ =∂
∂γlog〈µ〉
On the phase diagram of the Higgs SU(2) model – p. 18
New resultsNon-local observable analized
〈µ〉 magnetic monopole operator
Studied using
ρ =∂
∂γlog〈µ〉
Characteristic behavior
smooth crossover no singularity
at transition min ρ → −∞
On the phase diagram of the Higgs SU(2) model – p. 18
New results: β = 3.5
0.62 0.64 0.66 0.68 0.7 0.72 0.74γ
-200
-150
-100
-50
0
ρ
L = 16L = 24L = 32
β=3.5
On the phase diagram of the Higgs SU(2) model – p. 19
New results: β = 3.5
0.62 0.64 0.66 0.68 0.7 0.72 0.74γ
-200
-150
-100
-50
0
ρ
L = 16L = 24L = 32
β=3.5
Not yet firm results!
On the phase diagram of the Higgs SU(2) model – p. 19
Conclusionsfor all the β’s simulated only a smoothcrossover is seen
On the phase diagram of the Higgs SU(2) model – p. 20
Conclusionsfor all the β’s simulated only a smoothcrossover is seen
it seem natural to suppose that the first-orderline of transitions is not present in the λ = ∞case
On the phase diagram of the Higgs SU(2) model – p. 20
Conclusionsfor all the β’s simulated only a smoothcrossover is seen
it seem natural to suppose that the first-orderline of transitions is not present in the λ = ∞case
conservative point of view: we have shownthat, if it exists, the line of first ordertransitions ends for β much bigger than thevalue βc ≈ 2 previously thought as critical
On the phase diagram of the Higgs SU(2) model – p. 20
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