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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

(2)

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 . . .

(3)

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

(4)

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

¯

p2n2, 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

(5)

Momenta k ∼ g

2

T 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)

(6)

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

(7)

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.

(8)

3d effective theory

4D SM action: Physics at T = 0

perturbatively

S(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

(9)

Standard Model

(10)

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]

(11)

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!

(12)

3d effective Lagrangian:

L =1

4FijaFija+1 4BijBij+

(Diφ)Diφ + m23φφ + λ3φ)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

(13)

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

(14)

Higgs susceptibility & pseudocritical temperature

Define pseudocritical T : maximum location of thehφφi susceptibility χφφ:

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

(15)

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

(16)

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!

(17)

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

(18)

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

dVFµνµν

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

(19)

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 use

I 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

(20)

Evolution of N

CS

Symmetric 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.

(21)

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 Tfrom

Γ(T) T3

= 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]

(22)

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

(23)

MSSM

[Laine, Nardini, K.R.]

(24)

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.]

(25)

3d effective Lagrangian:

L3d = 1

2Tr Gij2+ (DisU)(DisU) + mU2(T )UU +λU(UU)2 + γ1UUH1H12UUH2H2+h

γ12UUH1H2+ H.c.i

+ 1

2Tr Fij2+ (DiwH1)(DiwH1) + (DiwH2)(DiwH2) + m12(T )H1H1+ m22(T )H2H2+h

m122(T )H1H2+ H.c.i + λ1(H1H1)22(H2H2)23H1H1H2H24H1H2H2H1

+ h

λ5(H1H2)26H1H1H1H27H2H2H1H2+ 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; m122ji12 depend on the 4d physical parameters (incl.

temperature T ).

(26)

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

(27)

Results: Higgs field expectation value hH

2

H

2

i

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

(28)

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

(29)

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

(30)

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

(31)

2HDM

[Kainulainen, Keus, Niemi, K.R., Tenkanen, Vaskonen]

(32)

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

(33)

Strong transition

−0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6

2

φ

2

i

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)

(34)

Results

Largish variation in results

Note: Veff in 3d relies on the same 3d effective theory than 3d lattice Transition stronger on the lattice

(35)

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

References

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