Thermal phase transition at the 2-loop level beyond temperature - - PowerPoint PPT Presentation
Thermal phase transition at the 2-loop level beyond temperature expansions Eibun Senaha (Nagoya U, E-ken) Feb. 14, 2014 @U of Toyama in collaboration with Koichi Funakubo (Saga U) Ref. PRD87, 054003 (2013) Outline Motivation A tractable
in collaboration with Koichi Funakubo (Saga U)
❒ 1st-order electroweak phase transition (EWPT) is necessary for successful electroweak baryogenesis (EWBG). ❒ Sunset diagram can change strength of 1st-
❒ Thermal resummation:
∼ T 2
dominant at high T. ❒ Noncovariant:
Dµν(p) = −1 p2 − m2
L(T)Lµν(p) +
−1 p2 − m2
T (T)Tµν(p),
T00 = T0i = Ti0 = 0, Tij = gij − pipj −p2 , Lµν = Pµν − Tµν, Pµν = gµν − pµpν p2 ,
Lμν, Tμν are projections: (in thermal rest frame) scalar scalar vector e.g.
pµTµν = pµLµν = 0
(4dim. transverse)
❒ Approximation breaks down around m/T=O(1)
10 20 30 40 0.001 0.01 0.1 1 10 a
KHTE(a) = −π2 3 (ln a2 + 3.48871)
a=m/T HTE formula:
[R.R.Parwani, PRD45, 4695 (1992)]
❒ HTE has been exclusively used in the EWBG calculation. Our aim: to devise a tractable calculation scheme beyond HTE. ❒ standard calculation methods:
❒ This raise a question about the reliability of the EWPT analysis in the MSSM.
❒ Approximation breaks down around m/T=O(1)
10 20 30 40 0.001 0.01 0.1 1 10 a
KHTE(a) = −π2 3 (ln a2 + 3.48871)
a=m/T HTE formula:
[R.R.Parwani, PRD45, 4695 (1992)]
❒ HTE has been exclusively used in the EWBG calculation. Our aim: to devise a tractable calculation scheme beyond HTE. ❒ standard calculation methods:
region relevant to EWBG ❒ This raise a question about the reliability of the EWPT analysis in the MSSM.
L = −1 4FµνF µν + |DµΦ|2 − V0(|Φ|2)
Fµν = ∂µAν − ∂νAµ, DµΦ = (∂µ − ieAµ)Φ
V0(|Φ|2) = −ν2|Φ|2 + λ 4 |Φ|4 Φ(x) = 1 √ 2
h = −ν2 + 3λ
4 v2, m2
a = −ν2 + λ
4 v2, m2
A = e2v2.
where ❒ As an illustration, we consider Abelian-Higgs model.
LB = LR + LCT → LR − ∆m2
ΦΦ†Φ + 1
2Aµ ∆m2
LLµν(i∂) + ∆m2 T Tµν(i∂)
+ LCT + ∆m2
ΦΦ†Φ − 1
2Aµ ∆m2
LLµν(i∂) + ∆m2 T Tµν(i∂)
LB(R): bare (renormalized) Lagrangian, LCT: counterterms ∆m2
Φ = 3e2 + λ
12 T 2, ∆m2
T = 0,
∆m2
L = e2
3 T 2
Explicitly,
[Buchmuller et al, NPB407 (’93) 387 .]
❒ We adopt a resummation method in which thermal masses are added and subtracted in the Lagrangian. ❒ Dominant thermal corrections are taken into account.
LB = LR + LCT → LR − ∆m2
ΦΦ†Φ + 1
2Aµ ∆m2
LLµν(i∂) + ∆m2 T Tµν(i∂)
+ LCT + ∆m2
ΦΦ†Φ − 1
2Aµ ∆m2
LLµν(i∂) + ∆m2 T Tµν(i∂)
LB(R): bare (renormalized) Lagrangian, LCT: counterterms ∆m2
Φ = 3e2 + λ
12 T 2, ∆m2
T = 0,
∆m2
L = e2
3 T 2
Explicitly,
[Buchmuller et al, NPB407 (’93) 387 .]
❒ We adopt a resummation method in which thermal masses are added and subtracted in the Lagrangian. ❒ Dominant thermal corrections are taken into account.
LB = LR + LCT → LR − ∆m2
ΦΦ†Φ + 1
2Aµ ∆m2
LLµν(i∂) + ∆m2 T Tµν(i∂)
+ LCT + ∆m2
ΦΦ†Φ − 1
2Aµ ∆m2
LLµν(i∂) + ∆m2 T Tµν(i∂)
new counterterm
LB(R): bare (renormalized) Lagrangian, LCT: counterterms ∆m2
Φ = 3e2 + λ
12 T 2, ∆m2
T = 0,
∆m2
L = e2
3 T 2
Explicitly,
[Buchmuller et al, NPB407 (’93) 387 .]
❒ We adopt a resummation method in which thermal masses are added and subtracted in the Lagrangian. ❒ Dominant thermal corrections are taken into account.
Resummed propagators
m2
L,T (T) = m2 A + ∆m2 L,T
∆h(p) = 1 p2 − m2
h(T),
∆a(p) = 1 p2 − m2
a(T),
Dµν(p) = −1 p2 − m2
L(T)Lµν(p) +
−1 p2 − m2
T (T)Tµν(p),
m2
h,a(T) = m2 h,a + ∆m2 Φ
With these, we compute 1- and 2-loop effective potentials. 1-loop resummed potential At 1-loop level, there is no complication to obtain the resummed effective potential.
V (1)
R (v; T) = 1
2
ln
m2
h(T)
2
ln
m2
a(T)
2
ln
m2
L(T)
2
ln
m2
T (T)
2π2
¯ m2
h(T)
T 2
¯ m2
a(T)
T 2
¯ m2
L(T)
T 2
¯ m2
T (T)
T 2
Resummed SSV and SVV type diagrams noncovariant due to
Tµν(k), Lµν(k)
It is not easy to evaluate the divergence and finite parts.
Dµν
Dµν
∆h ∆h ∆a
DR
SV V (m; m1L, m1T ; m2L,2T ) = −4
Dµν(k; m1)Dµν(q; m2)∆(k + q)
DR
SSV (m1, m2; mL, mT ) = −4
kµkνDµν(q)∆1(k)∆2(k + q)
Dµν(p) = (1 − r)Dr=0
µν (p) + rDr=1 µν (p)
= −(1 − r) p2 − m2
L
+ −r p2 − m2
T
p2 − m2
T
− −1 p2 − m2
L
Tµν(p) − rPµν(p)
Dµν(p) = −1 p2 − m2
L
Pµν(p) +
p2 − m2
T
− −1 p2 − m2
L
µν (p),
Dµν(p) = −1 p2 − m2
T
Pµν(p) +
p2 − m2
L
− −1 p2 − m2
T
µν (p).
In general, Resummed gauge boson propagator is devised as follows. covariant noncovariant
(r ∈ R)
Using Pµν(p) = gµν − pµpν
p2
= Tµν(p) + Lµν(p), Lµν(p) or Tµν(p) can be elimi- nated.
The loop calculation is simplified if r is determined in the following way.
gµν Tµν(p) − rPµν(p)
r = d − 2 d − 1
Dcov
µν (p) =
p2 − m2
L
+ −(d − 2) p2 − m2
T
Pµν(p) d − 1 , δDµν(p) =
p2 − m2
T
− −1 p2 − m2
L
Tµν(p) − d − 2 d − 1Pµν(p)
Dµν(p) = Dcov
µν (p) + δDµν(p),
pµDcov
µν (p) = pµδDµν(p) = 0 and gµνδDµν(p) = 0
+1 2
Π(q) − ∆m2
LL(q) − ∆m2 T T(q)
µν δDµν(q)
Dµν Dcov
µν
covariant part Diagrams involving the gauge boson can be decomposed in the following way.
+1 2
Π(q) − ∆m2
LL(q) − ∆m2 T T(q)
µν δDµν(q)
Dµν Dcov
µν
covariant part Diagrams involving the gauge boson can be decomposed in the following way.
from thermal CT
In general, self-energy can be written as
where uT
µ = uµ − qµ(u · q)/q2 with uµ = (1, 0).
Πµν(q) = ΠL(q)Lµν(q) + ΠT (q)Tµν(q) + ΠG(q)qµqν q2 + ΠS(q)qµuT
ν + qνuT µ
Finally, the noncovariant part is reduced to
1 2
Π(q) − ∆m2
LL(q) − ∆m2 T T(q)
µ Dµ(q) → m2
L − m2 T
482
T (T) − () L (T)
T≠0: T=0: To leading order no v-dependence. -> no effect on phase transition.
ΠT =0
µ (q) →
L,T +
2()
L,T (T)
(q0 = 0, q → 0)
ΠT =0
µν (q)δDµν(q) = 0
ΠT =0
µν
Π0(q)
q2
v-dependent masses: ❒ Landau gauge ξ=0 is taken. ❒ We study the thermal phase transition in the MSSM-like toy model. stop and gluon-like particles Note: gluon-like particle is a U(1) gauge boson
LAbelian−Higgs = −1 4FµνF µν + (DµΦ)∗DµΦ − V (|Φ|2),
DµΦ = (∂µ − ieAµ)Φ, V (|Φ|2) = −ν2|Φ|2 + λ 4 |Φ|4, Φ = 1 √ 2 (v + h + ia). ∆L = 1 4(∂µGν − ∂νGµ)2 + |(∂µ − ig3Gµ)˜ t|2 − m2
0|˜
t|2 − ˜ λ 4 |˜ t|4 − y2
t |Φ|2|˜
t|2.
¯ m2
h = −ν2 + 3λµ
4 v2, ¯ m2
a = −ν2 + λµ
4 v2, ¯ m2
A = e2µv2,
¯ m2
˜ t = m2 0 + y2µ
2 v2, m2
G = 0.
L = LAbelian−Higgs + ∆L
T=0 T≠0
S
S S S S S V
V V S
S S V S
S V
V (1)(v; T) =
ni
m2
i ) + T 4
2π2 IB ¯ m2
i
T 2
t = 2, nA = 3
F0( ¯ m2
i ) =
¯ m4
i
64π2
m2
i
¯ µ2 − Ci
IB(a2) = ∞ dx x2 ln
√ x2+a2
.
MS : Ch = Ca = C˜
t = 3/2, CA = 5/6 V
T=0 T≠0
S
S S S S S V
V V S
S S V S
S V
V (1)(v; T) =
ni
m2
i ) + T 4
2π2 IB ¯ m2
i
T 2
t = 2, nA = 3
F0( ¯ m2
i ) =
¯ m4
i
64π2
m2
i
¯ µ2 − Ci
IB(a2) = ∞ dx x2 ln
√ x2+a2
.
MS : Ch = Ca = C˜
t = 3/2, CA = 5/6 V
where
divergent part finite part
Focus on the parts which have 2 thermal distributions (nB(Ei) nB(Ej))
˜ H(m1, m2, m3) K−−(a1, a2, a3) + K−−(a2, a3, a1) + K−−(a3, a1, a2), ai mi/T H(m1, m2, m3) =
1 (P 2 + m2
1)(Q2 + m2 2)[(P + Q)2 + m2 3]
= Hdiv(m1, m2, m3) + ˜ H(m1, m2, m3)
≡ µT
(2π)D−1 , P 2 = (2mπT)2 + P 2
nB(E) = 1 eE/T − 1
K−−(a1, a2, a3) = 1 ds ea1 − s 1 dt ea2 − t ln
Y (0)
+ (s, t; a1, a2, a3)
¯ Y (0)
− (s, t; a1, a2, a3)
¯ Y (0)
± (s, t; a1, a2, a3) = 16
4
3
3
1 ln t(ln t − 2a2) + a2 2 ln s(ln s − 2a1)
± (a2
1 + a2 2 − a2 3)
2 .
S S S
finite part
finite part
Therefore, we focus on the scalar sunset diagram H ❒ Finite parts can be written in terms of the scalar sunset diagrams etc.
DSSV (m1, m2, M) =
4Q2 − 4(P · Q)2/P 2 (P 2 + M 2)(Q2 + m2
1)((P + Q)2 + m2 2)
= Ddiv
SSV (m1, m2, M) + ˜
DSSV (m1, m2, M), DSV V (m, M1, M2) =
4(D − 2) + 4(P · Q)2/(P 2Q2) (P 2 + M 2
1 )(Q2 + M 2 2 )((P + Q)2 + m2)
= Ddiv
SV V (m, M1, M2) + ˜
DSV V (m, M1, M2)
divergent part divergent part
˜ DSSV (m1, m2, M) ˜ H(m1, m2, M), ˜ DSV V (m, M1, M2) ˜ H(m, M1, M2)
S S V
V V S
DR
SSV (m1, m2; ML, MT )
1 D − 1 h DSSV (m1, m2, ML) + (D − 2)DSSV (m1, m2, MT ) i DR
SV V (m; M1L, M1T ; M2L, M2T )
= 1 (D − 1)2 DSV V (m; M1L, M2L) + (D − 2) n DSV V (m; M1L, M2T ) + DSV V (m; M1T , M2L)
❒ In our scheme, the noncovariant parts are dropped. ❒ Resummed sunset diagrams are written in terms of the unresummed
m2 → m2(T) = m2 + ∆m2(T)
V (2)
R (v; T)
= −λ2v2 16
H(mh) + ˜ H(mh, ma, ma)
2 ˜ DR
SSV (mh, ma; mL, mT )
− e4v2 4 ˜ DR
SV V (mh; mL, mT ; mL, mT ) − g2 3
2 ˜ DR
SSV (m˜ t, m˜ t; mGL, mGT ) − y4
2 v2 ˜ H(mh, m˜
t, m˜ t)
+ λ 16
I2
−(mh) + 3¯
I2
−(ma) + 2¯
I−(mh)¯ I−(ma)
2
I−(mh) + ¯ I−(ma) ¯ I−(mL) + 2¯ I−(mT ) + m2
T
8π2
e2 16π2
h + m2 a − m2 T
3 − 3e2v2
I−(mT ) + ˜ λ 2 ¯ I2
−(m˜ t)
+
˜ t + y2
2
I−(mh) + ¯ I−(ma)
3
I−(mGL) + 2¯ I−(mGT ) + m2
GT
8π2
I−(m˜
t)
+ g2
3
16π2
˜ t − m2 GT
3
I−(mGT ) − 1 2
h ¯
I−(mh) + ∆T m2
a ¯
I−(ma) + ∆T m2
L ¯
I−(mL) + 2∆T m2
T
I−(mT ) + m2
T
16π2
We study the thermal PT with/without the high-T expansion. After renormalization, the 2-loop resummed Veff is given by
■ order parameter
[From K. Funakubo’ s slide] ■ EWBG requires
■ order parameter
[From K. Funakubo’ s slide] ■ A negative
■ EWBG requires
■ order parameter
[From K. Funakubo’ s slide] ■ A negative
■ EWBG requires
e.g.
V (boson)
1
|const| · |m(v)|3T
˜ H(m) ≡ ˜ H(m, m, m) = − 3m2 (4π)4
¯ µ2 − 3 ln m2 ¯ µ2 + 7 2 + π2 12 + 2 3f2
64π2
¯ µ2 + ln 2 − 2
π2 j−(0) − cH − 2 ln a2
If −m2 ˜ H(m), vC/TC gets smaller
where f2 = −1.7579, j(0) = ζ(2)(1 − γ5) + ζ(2) and cH = 5.3025.
a=m/T
[P. Arnold, O. Espinosa, PRD47, (’93) 3546, J.R. Espinosa, NPB475, (’96) 273 etc]
DSSV 3 m2 ˜ H(m)
˜ H(m) ≡ ˜ H(m, m, m) = − 3m2 (4π)4
¯ µ2 − 3 ln m2 ¯ µ2 + 7 2 + π2 12 + 2 3f2
64π2
¯ µ2 + ln 2 − 2
π2 j−(0) − cH − 2 ln a2
If −m2 ˜ H(m), vC/TC gets smaller
where f2 = −1.7579, j(0) = ζ(2)(1 − γ5) + ζ(2) and cH = 5.3025.
a=m/T
[P. Arnold, O. Espinosa, PRD47, (’93) 3546, J.R. Espinosa, NPB475, (’96) 273 etc]
DSSV 3 m2 ˜ H(m)
5x105 106 100 150 200 250 300 1-loop 2-loop 2-loop HTE 50
v = 246 GeV, mh = 35 GeV, m2
0 = 0,
y = 1.0, e = 0.5, ˜ λ, g3 = 1.2
❒ vC/TC is enhanced due to the 2-loop contributions. ❒ If HTE is used, each vC and TC can be enhanced by 10%, and by 50% for barrier hight. ❒ Errors can be reduced in vC/TC.
K(a) ≡ K−−(a, a, a) → KHTE(a) = −π2 3 (ln a2 + 3.48871)
50 100 150 200 250 35 40 45 50 55 60 2-loop 2-loop HTE 1-loop 1.5 2 2.5 3 1 1.2 1.4 1.6 1.8 2 1-loop 2-loop 2-loop HTE
❒ We have proposed a scheme that can calculate the 2-loop resummed effective potential without using the HTE.
and noncovariant parts, simplifying the 2-loop calculation.
and the barrier hight by about 50%. ❒ Generalization to nonAbelian gauge theories.
❒ Thermal phase transition is studied based on the new scheme.
Strong 1st-order EWPT is driven by a right-handed stop with a mass below 120 GeV . (called light stop (LSS) scenario) difficult to see it directly
˜ t1 → c˜ χ0
1, |m˜ t1 − m˜ χ0
1| < 35 GeV
Recently, LSS has been declared dead at least twice in the literature.
Light red bands: theory errors (higher order corrections and χ±, χ0 corrections.)
Solid red bands: range of predictions for m˜
tR ∈ (80, 115) GeV.
mA = 300 GeV [D. Curtin, P. Jaiswall, P. Meade., arXiv:1203.2932]
For mH ≃125 GeV, MSSM EWBG is ruled out at greater than 98% CL (mA>1 TeV), at least 90% CL for light value of mA (~300 GeV)
10 20 30 40 50 60 70 80 90 1 2 3 4
qq,ll,VV (ggF) qq,ll,VV (VBF) γγ (ggF) γγ (VBF)
(σ × BR) Point G
M2=200 GeV µ=200 GeV
mχ0
1
[GeV]
σ×BR (σ×BR)SM
[M. Carena, G. Nardini, M. Quiros, CEM. Wagner, arXiv:1207 .6330]
Higgs invisible mode is open ➔ σ(gg -> H -> VV) is recuded ➔ tension is relaxed.
χ0
1 <
But, stop main decay mode changes as
˜ t1 → bW + ˜ χ0
1, b˜
χ0
1 ¯
ff
“Very Light Scalar Top Quarks at the LHC, K. Krizka, A. Kumar, D. Morrissey, arXiv:1212.4856” claims that
“Ours results suggest that such a state is ruled out by existing LHC analyses, at least if it decays promptly in the FV , 4B or 3B modes. ”