Phase transitions and the lattice
Kari Rummukainen
University of Helsinki and Helsinki Institute of Physics
Farakos, D’Onofrio, Kainulainen, Kajantie, Keus, Laine, Nardini, Niemi, Sannino, Shaposhnikov, Tenkanen, Tranberg, Tsypin, Tuominen, Vaskonen
Nordita 2019
Phase transitions at Electroweak scale
Interesting physics:
1st order transition?
→ Gravitational waves?
→ Baryon number generation?
→ Magnetic fields?
Baryon number violation stops
No phase transitions in the Standard Model (atµ = 0)
I QCD and EW “phase transitions” are cross-overs
Many BSM models may have a first order EW phase transition7→
I MSSM
I 2HDM
I Composite Higgs, Technicolor . . .
In this talk:
Why non-perturbative?
3d effective theory
Precision results of the Standard Model:
I Equation of state - “softness”, width of the cross-over
I Sphaleron rate
I Why study SM? Background e.g. to leptogenesis, precision physics MSSM
2HDM
Loops at T 6= 0:
Finite T ensemble: euclidean with imaginary time extent 1/T , with (anti)periodic b.c for bosons (fermions)⇒
1
p2 −→ 1
¯
p2+ωn2, ωn=
2nπT n∈ Z Bosons
(2n + 1)πT n∈ Z Fermions Z d4p
(2π)4 −→ TX
n
Z d3~p (2π)3
Thus, all n6= 0 Bosonic modes and all Fermionic modes acquire a “mass” ∼ πT . Clearly, onlybosonic n = 0 modes are infrared sensitive.
1/T
Momenta k ∼ g
2T non-perturbative:
Let us consider vacuum diagram (pressure) at finite T , where we add a (fictitious) mass term m to keep track of the scale: [Linde 80]
N loops →
(N− 1) 4-vertices (2N− 2) propagators
T Z
d3p]N(g2)N−1
1
q2+ m2
2N−2
∝g6T4 g2T m
N−4
Ifm = g2T (“magnetic” scale), all loops contribute to pressure at g6! Perturbatively m = 0 for magnetic gauge modes⇒
Loop expansion fails when k<∼g2T (3d confinement).
⇒ Any quantity is perturbatively computable only up to some fixed order!
Note: Loop expansion OK when k ∼ πT (hard scales, n 6= 0) k ∼ gT (Debye, electric scales)
Slow convergence
Perturbative effective potential converges slowly: big differences between 1 and 2 loops: (MSSM)
0.0 0.2 0.4 0.6 0.8 1.0
v/T 0.0000
0.0002 0.0004 0.0006
V(v)/T4
mQ=1 TeV, m~U=65 GeV, mH=105 GeV
Tc1−loop=93.9 GeV Tc2−loop=88.5 GeV
[Laine, K.R. 2002]
Perturbatively weak 1st order transitions may vanish altogether (this happens
Method: 3d effective theory
Tool for perturbative and lattice computations
Modes p> g2T are perturbative (at weak coupling): can be integrated out in stages:
1. p>∼T : fermions, non-zero Matsubara frequencies
→ 3d theory (dimensional reduction)
1/ T
2. Electric modes p ∼ gT
Obtain a “magnetic theory” for modes p<∼g2T . Contains fully the non-perturbative thermal physics.
3d effective theory
4D SM action: Physics at T = 0
perturbatively
(αS(MW), GF, MH, MW, MZ, Mtop) Fix physics -
Integration over scales T and gT (at 2–loop level)
3D continuum effective theory
3D lattice theory B
B B N 2–loop counterterms in lattice perturbation theory
Standard Model
Phase diagram of the SM
After lots of activity on and off the lattice:
→ No phase transition at all, smooth “cross-over” for mHiggs>
∼72 GeV
50 60 70 80 90
mH/GeV 80
90 100 110 120 130
Tc/GeV
symmetric phase
broken Higgs phase 1st order transition
2nd order endpoint
[Kajantie,Laine,K.R.,Shaposhnikov,Tsypin 95–98]
see also
[Csikor,Fodor, Heitger]
[G¨urtler,Ilgenfritz,Schiller,Strecha]
Overall EOS
[Laine, Schr¨oder 2006]
101 102 103 104 105 106 T / MeV
0 2 4 6 8 10 12
p / T4
mH = 150 GeV mH = 200 GeV
101 102 103 104 105 106 T / MeV
0.1 0.2 0.3 0.4
w cs 2
1/3
Perturbation theory + Lattice QCD + Hadron RG Here EW transition “featureless” – can do better!
3d effective Lagrangian:
L =1
4FijaFija+1 4BijBij+
(Diφ)†Diφ + m23φ†φ + λ3(φ†φ)2
SU(2) + U(1) gauge + Higgs Parameters:
g32∼ gW2T x ≡ λ3/g32 y ≡ m32/g34 z ≡ g302/g32
0.3925 0.393 0.3935 0.394
g3 2 /T
-0.6 -0.3 0 0.3
y
0.285 0.29 0.295
x
140 150 160 170
T/GeV 0.3095
0.31 0.3105
z
Precise mapping between T and 3d parameters
Higgs field expectation value
Red dots: 3D lattice (continuum limit) Green bands:
I Broken phase: 2-loop Coleman-Weinberg[Kajantie et al 95]
I Symmetric phase: 3-loop
[Laine and Meyer 2015]
Agreement remarkably good away from the cross-over P.T. does not converge near the cross-over
140 145 150 155 160 165 170
T/GeV 0
0.1 0.2 0.3 0.4 0.5 0.6
<φ+φ>/T
Higgs susceptibility & pseudocritical temperature
Define pseudocritical T : maximum location of thehφ†φi susceptibility χφ†φ:
Tc = 159.6± 0.1 ± 1.5 GeV 1st error: lattice errors 2nd error: estimate of the
systematic uncertainty in 3d action
Right: continuum extrapolation ofχ
140 145 150 155 160 165 170
T/GeV 0
2 4 6 8 10
χφ+φ
β=6 β=9 β=16
156 157 158 159 160 161 162
T/GeV 2
4 6 8 10
χφ+φ
β=6 β=9 β=16
Pressure and energy density
140 145 150 155 160 165 170
T/GeV 33.6
33.8 34 34.2 34.4 34.6
e/T4, 3p/T4
e/T4
3p/T4
1%
Pressure:
p(T )
T4 −p(T0) T04 =
Z T T0
dT0∆(T0) T0 Use reference temperature T0= 140 GeV,
pert. pressure p(T0)/T04= 11.173
[Laine, Schr¨oder 2006]
Energy density e = ∆ + 3p p(T0) has∼ 1% uncertainty → uncertainty for e and p
More thermodynamics
140 145 150 155 160 165 170
T/GeV 137
138 139 140 141 142
CV/T3
140 145 150 155 160 165 170
T/GeV 0.324
0.326 0.328 0.33 0.332 0.334
c s 2
w 1/3
Heat capacity CV = e0(T ) Speed of sound: cs2= p0/e0 EOS parameter w = p/e
Cross-over well defined, but very soft!
Masses
Higgs and W3 screening masses
140 145 150 155 160 165 170
T/GeV 0
0,2 0,4 0,6 0,8
m/T
mHβG=9 mHβG=16 mW3βG=9 mW3βG=16
Effective cos2θWeinberg
140 145 150 155 160 165 170
T/GeV 0.7
0.8 0.9 1 1.1
Aγ
cos2θW
Baryon number violation: sphaleron rate
Anomaly: baryon number B and gauge topology are connected:
∆B = ∆L = 3∆NCS= 3 32π2
Z t 0
dt Z
dVFµνF˜µν
Baryogenesis
Rate in thermal equilibrium:
Γ = lim
V ,t→∞
h(∆NCS(t))2i Vt In the symmetric phase, Γ∝ α5WT4[Arnold, Son, Yaffe 97]
or rather Γ∝ α5Wlog(1/αW)T4[B¨odeker 98]
In the broken phase the rate is exponentially suppressed
Turning off of the rate is important for some baryogenesis scenarios
Calculation of the sphaleron rate
Non-perturbative⇒ real-time lattice simulations Several methods:
Classical EOM[Ambjorn, Krasnitz 95; + many]
I UV modes (HTL) make result lattice spacing depednent[Arnold 97]
Classical + HTL effective theories[Moore, Hu, Muller 97; Bodeker, Moore, K.R 99]
I More control over UV modes, no continuum limit
I Used also in studies of plasma instabilities in HIC B¨odeker method: [B¨odeker 98], heat bath version[Moore 98]
I Fully dissipative evolution of soft (g2T ) modes
X
Exact to leading log order log(1/g2), ∃ continuum limit, very simple to useI Same “action” and cont. limit as in 3D thermo simulation
∃ lot of lattice results in pure gauge, few in broken EW phase.
Here: physical Higgs mass
Evolution of N
CSSymmetric T = 152GeV, broken T = 145GeV, deeply broken T = 140GeV (with mH= 113GeV)[D’Onofrio et al 12]
0.0 5.0×104 1.0×105 1.5×105 2.0×105 Measurements
-20 0 20 40 60
Chern-Simons number NCS
0 1×105 2×105 3×105 4×105 5×105 Measurements
-2 0 2 4 6 8 10 12
Chern-Simons number NCS
0 1×105 2×105 3×105 4×105 5×105 Measurements
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Chern-Simons number NCS
Rate strongly suppressed in the broken phase. Use special “multicanonical”
methods to overcome this.
Result: sphaleron rate
Symmetric phase:
Γ
T4 = (8.0 ± 1.3) × 10−7≈ (18 ± 3)α5W
Broken phase: parametrized as log Γ
T4 = (0.83 ± 0.01) T
GeV− (147.7 ± 1.9) Errors dominated by systematics
Cosmology: Hubble cooling ˙T = −HT Freeze-out temperature T∗from
Γ(T∗) T∗3
= αH(T∗) with α ≈ 0.1015[Burnier et al 05]
Baryon number freeze-out T∗= 131.7 ± 2.3 GeV
130 140 150 160 170
T / GeV -45
-40 -35 -30 -25 -20 -15 -10
log Γ/Τ
4
standard multicanonical fit perturbative
pure gauge
log[αH(T)/T]
Overview of the Standard Model
Standard Model equation of state solved to 1% level Pseudocritical temperatureTc= 159.6± 0.1 ± 1.5 GeV Cross-over weak but well defined
Width of the cross-over region 2-3 GeV
Baryon number freeze-outT∗= 131.7± 2.3 GeV Symmetric phase sphaleron rateΓ/T4= (18± 3)α5W
Broken phase rate can be parametrized as log Γ/T4= (0.83± 0.01)T /GeV − (147.7 ± 1.9) . . . can be fed in to e.g. some leptogenesis scenarios
MSSM
[Laine, Nardini, K.R.]
Parameters:
80 100 120 140
mh / GeV 140
150 160 170
mtR~
v / T = 0.9 triple point mQ = 7 TeV, µ = M2 = mA = 150 GeV
tanβ 2
15
A
t / m
Q
0.02
0.35
Strong transition when right-handed stop U is light: mt˜R < mtop
(U: Higgs for SU(3) color!) Choose MSSM parameters
˜
mU= 70.4 GeV mQ = 7TeV
µ = M2= mA= 150GeV tanβ = 15
At/mQ = 0.02 These correspond to
Higgs mass mh= 126 GeV and the right-handed stop mass m˜tR ≈ 155 GeV
[Laine, Nardini, K.R.]
3d effective Lagrangian:
L3d = 1
2Tr Gij2+ (DisU)†(DisU) + mU2(T )U†U +λU(U†U)2 + γ1U†UH1†H1+γ2U†UH2†H2+h
γ12U†UH1†H2+ H.c.i
+ 1
2Tr Fij2+ (DiwH1)†(DiwH1) + (DiwH2)†(DiwH2) + m12(T )H1†H1+ m22(T )H2†H2+h
m122(T )H1†H2+ H.c.i + λ1(H1†H1)2+λ2(H2†H2)2+λ3H1†H1H2†H2+λ4H1†H2H2†H1
+ h
λ5(H1†H2)2+λ6H1†H1H1†H2+λ7H2†H2H1†H2+ H.c.i . Gij: SU(3) gauge
U: right-handed stop Fij: SU(2) gauge
H1, H2: two SU(2) scalars (Higgses)
Parameters gW2 ; gS2; mi2; m122;λj;γi;γ12 depend on the 4d physical parameters (incl.
temperature T ).
Simulation volumes
βw = 4/(gW2 Ta) volumes 8 123, 163
10 163
12 163, 203, 323, 122× 36, 202× 40 14 243, 142× 42, 242× 48
16 243, 162× 48, 202× 60, 242× 72 20 323, 202× 60, 262× 72, 322× 64 24 243, 323, 483, 242× 78, 302× 72
30 483
Results: Higgs field expectation value hH
2†H
2i
72 76 80 84 88
T* / GeV 0.0
0.5 1.0 1.5 2.0
<H2 + H2> / T*2 , <U+ U> / T*2
<H
2 +H
2> / T*2
<U+U> / T*2
Results extrapolated to continuum Strong transition
Stop field U participates
Histograms
0.0 0.2 0.4 0.6 0.8 1.0
H2+H2 / Tc*2 10-7
10-6 10-5 10-4 10-3 10-2 10-1 100
P(H2
+ H2)
162x48 202x60 242x72
0.0 0.2 0.4 0.6
H2 +H2/T*2 -0.15
-0.10 -0.05 0.00 0.05 0.10
U+U/T*2
Results: continuum limits
0.00 0.03 0.06 0.09 0.12
1 / βω 76
77 78 79 80
Tc* / GeV
Tc = 79.17± 0.10 GeV 2-loop pert. theory∼ 84.4 GeV
0.00 0.03 0.06 0.09 0.12
1 / βω 0.0
0.5 1.0 1.5 2.0
v(Tc*) / Tc *
v/Tc = 1.117± 0.005 pert. theory 0.9
Results compared with 2-loop perturbation theory
lattice perturbative (Landau gauge) Transition temperature Tc/GeV 79.17(10) 84.4
Higgs discontinuity v/Tc 1.117(5) 0.9 Latent heat L/Tc4 0.443(4) 0.26 Surface tension σ/Tc3 0.035(5) 0.025
The transition is clearly stronger on the lattice.
75 80 85 90 95
* 0.0
0.5 1.0 1.5 2.0
v(T*) / T*
mQ* = 7 TeV, m~U* = 70.5 GeV, mh*--~ 126 GeV
1-loop 2-loop
lattice
2HDM
[Kainulainen, Keus, Niemi, K.R., Tenkanen, Vaskonen]
Benchmark points
2HDM is not so strongly constrained than MSSM by collider phenomenology.
Strong phase transition→ large scalar couplings λi
→ problems in perturbation theory; accuracy of eff. 3d description?
→ Landau pole is close
BM1: “Inert doublet model”, studied perturbatively by[Laine, Meyer, Nardini 2017]
BM2: Approaches[Dorsch et al.] point but more restrictedλi
Strong transition
−0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6
hφ
†2φ
2i
3d/T
−6
−5
−4
−3
−2
−1 0
log
10(P )
βG=10 βG=12 βG=16 βG=20 βG=24 βG=32
Heavy modes at Tc⇒ need to go to very small lattice spacing (large βG)
Results
Largish variation in results
Note: Veff in 3d relies on the same 3d effective theory than 3d lattice Transition stronger on the lattice
Conclusions:
the Standard Model cross-over is resolved
If couplings are small, 3d effective theory simulations give very accurate results
Strong transitions seen in MSSM, 2HDM
Effective theory method applicable to many ”Higgs-like” models Bubble nucleation rate can be measured