study of h,h,a decays in the context of the mssm … · study of h;h;a! ˝ decays in the context of...
TRANSCRIPT
Study of h,H,A→ τµ decays in the
context of the MSSM within the Mass
Insertion Approximation
Roberto Morales
Universidad Nacional de La Plata - Instituto de Fısica La Plata - Argentina
IV Postgraduate Meeting on Theoretical Physics, IFT UAM-CSIC - November 18, 2015
Roberto Morales 1/28
Outline
• Motivation
• Experimental Bounds
• MSSM + general slepton mixing
• Mass Insertion Approximation (MIA)
• Lepton Flavor Violation Higgs Decays (LFVHD) processes
• Results
• Phenomenology
• Conclusions
Roberto Morales 2/28
References
Work based on:
• E.Arganda, M. J. Herrero, R. Morales and A. Szynkman,“Analysis of the h,H,A→ τµ decays induced from SUSY loops within the Mass InsertionAproximation”, arXiv:1510.04685[hep-ph].
In continuation to:
• M. Arana-Catania, E.Arganda and M. J. Herrero,“Non-decoupling SUSY in LFV Higgs decays: a window to new physics at the LHC”,JHEP 1309 (2013) 160 [JHEP 1510 (2015) 192] [arXiv:1304.3371 [hep-ph]].
Roberto Morales 3/28
Motivation
• Why Lepton Flavor Violation? LFV occurs in Nature.Neutrino oscillations −→ LFV
• LFV is a window to BSM physics: No LFV within SM.
• SUSY not seen yet at LHC (scale mSUSY at TeV range?)
• Higgs mediated processes are sensitive to SUSY via loops.
Main here:the MIA allows us to derive simple analytical expressions for decay rates. Useful for Pheno!
Roberto Morales 4/28
Experimental BoundsSome searched processes:
LFV process Present Upper Bound (90%CL) Future Sensitivity (?)
BR(µ→ eγ) 5.7× 10−13 (MEG 2013) 5× 10−14 (MEGup)
BR(τ → eγ) 3.3× 10−8 (BaBar 2010) 3× 10−9 (SuperB)
BR(τ → µγ) 4.4× 10−8 (BaBar 2010) 2.4× 10−8 (SuperB)
BR(µ→ eee) 1× 10−12 (SINDRUM 1988) 1× 10−16 (Mu3E-PSI)
BR(τ → eee) 2.7× 10−8 (Belle 2010) 1× 10−9,−10 (Belle2, SuperB)
BR(τ → µµµ) 2.1× 10−8 (Belle 2010) 1× 10−9,−10 (Belle2, SuperB)
BR(τ → µη) 2.3× 10−8 (Belle 2010) 1× 10−9,−10 (Belle2, SuperB)
CR(µ− e, Au) 7× 10−13 (SINDRUM2 2006)
CR(µ− e, Al) 3.1× 10−15 (COMET-I, J-PARC)
2.6× 10−17 (COMET-II, J-PARC)
2.5× 10−17 (Mu2E-FermiLab)
CR(µ− e, Ti) 4.3× 10−12 (SINDRUM2 2004) 1× 10−18 (PRISM, J-PARC)
New input from LHC: LFV Higgs DecaysBR(H → τµ) < 1.51× 10−2 (95%C.L.) [CMS 2015] (2.4σ excess?)
BR(H → τµ) < 1.85× 10−2 (95%C.L.) [ATLAS 2015]
Roberto Morales 5/28
The MSSM with general slepton mixingLow energy parametrization for general slepton mixing (model independent).
LLFVsoft = −L†m2
LL− R∗m2
RR +
(L†AlR∗H1 + h.c.
)
m2L
=
m2L1
δLL12 mL1mL2
δLL13 mL1mL3
δLL21 mL2mL1
m2L2
δLL23 mL2mL3
δLL31 mL3mL1
δLL32 mL3mL2
m2L3
∆
LLmk ≡ (m
2L
)mk = δLLmkmLmmLk
m2R
=
m2R1
δRR12 mR1mR2
δRR13 mR1mR3
δRR21 mR2mR1
m2R2
δRR23 mR2mR3
δRR31 mR3mR1
δRR32 mR3mR2
m2R3
∆
RRmk ≡ (m
2R
)mk = δRRmkmRmmRk
v1Al =
meAe δLR12 mL1
mR2δLR13 mL1
mR3
δLR21 mL2mR1
mµAµ δLR23 mL2mR3
δLR31 mL3mR1
δLR32 mL3mR2
mτAτ
∆LRmk ≡ (v1Al)mk = δ
LRmkv1
√mLmmRk
=⇒ δLRmk = δ
LRmk
v1√mLmmRk
∆RLmk ≡ (v1Al)km = δ
RLmkv1
√mRmmLk
=⇒ δRLmk = δ
RLmk
v1√mRmmLk
v1(2) = 〈H1(2)〉 and tanβ = tβ = v2/v1
⇒ LFV is originated from the off-diagonal real δABmk ’s via SUSY loops.
Roberto Morales 6/28
Full and MIA approach
Full Calculation
• Work with the mass basis −→ full diagonalization of mass matrix.
• Physical states: 6 sleptons lα and 3 sneutrinos να.
• Dependence on δABmk hidden in physical masses and rotation matrices.
• Numerical results for decay rates.
MIA Calculation
• Work with the EW interaction basis. Gauge states: lLi , lRi and νi where i denotes flavore, µ, τ .
• Non-diagonal elements of the mass matrix are considered as two-field interactions. Forexample, explicit dependence on ∆AB
mk appears in the lepton flavor violation insertions:
lAm lBk
−i∆ABmk
νm νk−i∆LL
mk
• Analytical expressions for decay rates as perturbative expansion of δABmk and other massratios.
Roberto Morales 7/28
LFVHD processes h,H,A→ lklm with m 6= k
In this context, LFV Higgs Decays occur at one-loop order:
Hx
lk
lm
px
pk
pm
∆mk
mSUSY
mEW
Our aim:to get the form factors F as an expansion valid for heavy SUSY
(mSUSY � mh,H,A,mlep,mW )
∆ABmkF (mHx ,mlep,mW ,mSUSY) ∼ ∆AB
mk
(F |mext=0 +O(M2
Hx/m2
SUSY) +O(M2W /m2
SUSY))
F |mext=0 ∼ O(M0Hx/m0
SUSY) −→ Non-Decoupling ND contributions (dominant terms).
O(M2Hx/m2
SUSY),O(M2W /m2
SUSY) −→ Decoupling D contributions (leading and subleading
corrections).
Roberto Morales 8/28
Results for LFVHD in the MIA
In the next slides,
• SUSY scenarios: all heavy masses are proportional to one scale mSUSY.
• All relevant one-loop diagrams for the full and MIA computation.
• Decay rate in terms of the form factors and ∆ABmk .
• Study of the cases ∆LLmk and ∆RR
mk : Non-Decoupling behavior.
• Study of the cases ∆LRmk and ∆RL
mk: Decoupling behavior.
• Perturbativity on δABmk ’s.
• Comparison between MIA/Full calculation and with Effective approach.
• Pheno implications: numerical study of maximum rates allowed by experimental boundsfrom τ → µγ and searches of neutral MSSM Higgs bosons.
Roberto Morales 9/28
SUSY scenarios
• Equal masses scenario: all the relevant parameters involved set to be equal
M1 = M2 = µ = mL = mR = Aµ = Aτ = mSUSY
• GUT approximation scenario: set an approximate GUT relation for the gaugino masses
M2 = 2M1 (GUT relation)
We also relate the soft parameters and the µ parameter to a common scale by choosing:
mL = mR = M2 = Aµ = Aτ = µ/a = mSUSY with a =3
4,
4
3
• Generic scenario: we set different values for all the mass parameters involved
M1 = 2.2mSUSY ,M2 = 2.4mSUSY , µ = 2.1mSUSY
mL1= 2mSUSY ,mL2
= 1.8mSUSY ,mL3= 1.6mSUSY
mR1= 1.4mSUSY ,mR2
= 1.2mSUSY ,mR3= mSUSY
Aµ = 0.6mSUSY , Aτ = 0.8mSUSY
Roberto Morales 10/28
One-loop diagrams for the Full results
Use mass basis for the internal SUSY particles: sleptons, sneutrinos, charginos andneutralinos.
[Arganda, Curiel, Herrero, Temes; 2005]
Hx
lk
lmχ−j
χ−i
να
(1)
Hx
lk
lm
χ−i
να
νβ(2)
Hx
lm
lk
χ−i
να
lm
(3)
Hx
lm
lk
χ−i
ναlk
(4)
Hx
lk
lmχ0j
χ0i
lα
(5)
Hx
lk
lm
χ0i
lα
lβ(6)
Hx
lm
lk
χ0i
lα
lm
(7)
Hx
lm
lk
χ0i
lαlk
(8)
Roberto Morales 11/28
Relevant one-loop diagrams for ∆LLmk
Use gauge basis for the internal SUSY particles: flavor sleptons, flavor sneutrinos, Bino,Winos, Higgsinos.
Hx
lk
lmW−, H−
H−, W−
νm
νk
(1a,1b)
Hx
lm
lk
H−
W−
νkνm
lm
(3a)
Hx
lm
lk
W−
νkνm
lm
(3b)
Hx
lm
lk
W−, H−H−, W−
νkνmlk
(4a,4b)
Hx
lm
lk
W−
νkνmlk
(4c)
Hx
lk
lmB, W3
H1
H2:1
lLm
lLk
(5a,5b:5c,5d)
Hx
lk
lm
B, W3
H1
H2:1lLm
lLk
(5e,5f:5g,5h)
Hx
lk
lm
B
lRk
lLk
lLm(6a)
Hx
lk
lm
B
lRm
lLm
lLk
(6b)
Roberto Morales 12/28
Relevant one-loop diagrams for ∆LLmk
Hx
lm
lk
H1
H2:1B, W3
lLklLm
lm
(7a,7b:7c,7d)
Hx
lm
lk
B, W3
lLklLm
lm
(7e,7f)
Hx
lm
lk
B
lRklLk
lLmlm
(7g)
Hx
lm
lk
B, W3 H1
H2:1
lLklLmlk
(8a,8b:8c,8d)
Hx
lm
lk
B, W3
lLklLmlk
(8e,8f)
Hx
lm
lk
B
lLk
lLm
lRmlk
(8g)
Hx
lm
lk
B, W3
H1
H2:1
lLklLmlk
(8h,8i:8j,8k)
Hx
lm
lk
B
lRk
lLk
lLmlk
(8l)
Other diagrams suppressed by extra factors of (MEW /mSUSY)n and/or (mlep/mSUSY)m.
Roberto Morales 13/28
Relevant one-loop diagrams for ∆LRmk and ∆RL
mk
• Case ∆LRmk
Hx
lk
lm
B
lRk
lLm(6c)
Hx
lm
lk
B
lRklLmlk
(8m)
• Case ∆RLmk
Hx
lk
lm
B
lLk
lRm(6d)
Hx
lm
lk
B
lLklRmlk
(8n)
Roberto Morales 14/28
Relevant one-loop diagrams for ∆RRmk
Hx
lk
lmB
H1
H2:1
lRm
lRk
(5i:5j)
Hx
lk
lm
B
H1
H2:1lRm
lRk
(5k:5l)
Hx
lk
lm
B
lLk
lRk
lRm(6e)
Hx
lk
lm
B
lLm
lRm
lRk
(6f)
Hx
lm
lk
H1
H2:1B
lRklRm
lm
(7h:7i)
Hx
lm
lk
B
lRklRm
lm
(7j)
Hx
lm
lk
B
lLklRk
lRmlm
(7k)
Hx
lm
lk
B
H1
H2:1
lRklRmlk
(8o:8p)
Hx
lm
lk
BH1
H2:1
lRklRmlk
(8q:8r)
Hx
lm
lk
B
lRklRmlk
(8s)
Hx
lm
lk
B
lLk
lRk
lRmlk
(8t)
Hx
lm
lk
B
lRk
lRm
lLmlk
(8u)
Roberto Morales 15/28
Analytical results in the MIA
The amplitude of Hx(p1)→ lk(−p2)lm(p3) for Hx = h,H,A is
iM = −igulk (−p2)(F(x)L PL + F
(x)R PR)vlm (p3)
The decay rate reads as follows:
Γ(Hx → lk lm) =g2
16πmHx
√√√√(1−(mlk +mlm
mHx
)2)(
1−(mlk −mlm
mHx
)2)
×(
(m2Hx−m2
lk−m2
lm)(|F (x)
L |2 + |F (x)R |2)− 4mlkmlmRe(F
(x)L F
(x)∗R )
)where the form factors are linear in ∆AB
mk
F(x)L,R = ∆LL
mkF(x)LLL,R + ∆LR
mkF(x)LRL,R + ∆RL
mkF(x)RLL,R + ∆RR
mkF(x)RRL,R
• All of them are UV convergent: they are expressed in terms of the loop functions C0, C2,D0, D0 of three and four points because there are 2 scalar and 1 fermionic propagators atleast.
• We perform a systematic expansion in powers of pext and keep: the leading contributionsO(p0
ext) and the next-to-leading O(p2ext).
Roberto Morales 16/28
Non-decoupling contributions for ∆LLmk and ∆RR
mk(∆LL
23 F(x)LLL
)ND
=
(g2
16π2
mτ
2MW
)[σ
(x)2 + σ
(x)∗
1 tβ
cβ
](δLL23 mL2
mL3
)
×[
3
2µM2D0(0, 0, 0,m
L2,m
L3, µ,M2)
− t2W
2µM1D0(0, 0, 0,m
L2,m
L3, µ,M1)
−t2WµM1D0(0, 0, 0,mL2,m
L3,m
R3,M1)
]
O(M0
Hx/m0
SUSY)
(∆RR
23 F(x)RRR
)ND
=
(g2t2W16π2
mτ
2MW
)[σ
(x)∗
2 + σ(x)1 tβ
cβ
](δRR23 m
R2mR3
)
×[µM1D0(0, 0, 0,m
R2,m
R3, µ,M1)
−µM1D0(0, 0, 0,mR2,m
R3,m
L3,M1)
]
O(M0
Hx/m0
SUSY)
whereHx = (h,H,A) , σ
(x)1 = (sα,−cα, isβ) , σ
(x)2 = (cα, sα,−icβ)
⇒ Simple analytical expressions: we factorized the contributions of the loops and the rest ofMSSM parameters (tβ and mA).
Roberto Morales 17/28
Non-decoupling contributions for ∆LLmk and ∆RR
mk
10-9
10-8
10-7
10-6
10-5
10-4
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
BR(φ
→ τ µ
)
mSUSY [TeV]
mA = 800 GeV, tan β = 40, δ23LL = 0.5
GUT approx. µ = 3/4 mSUSY h → τ µ FullA → τ µ Fullh → τ µ MIAA → τ µ MIA
10-12
10-11
10-10
10-9
10-8
10-7
10-6
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
BR(φ
→ τ µ
)
mSUSY [TeV]
mA = 800 GeV, tan β = 40, δ23RR = 0.5
Generic scenario
h → τ µ FullA → τ µ Fullh → τ µ MIAA → τ µ MIA
• Non-Decoupling behavior of BR(Hx → τµ) ∼ m0SUSY in contrast with the decoupling
behavior of BR(τ → µγ) ∼ m−4SUSY.
• First order MIA approximates the full calculation very well.
• Largest rates for LL and large tβ : BR(Hx → τµ) ∼ t2β .
• Similar results in other scenarios.
Roberto Morales 18/28
Decoupling contributions for ∆LRmk and ∆RL
mk(∆LR23 F
(x)LRL
)D
=
(g2t2W16π2
M1
2MW
)[σ(x)∗1
](δLR23 v
√mL2
mR3)
×[−C0(p2, p1,M1,mR3
,mL2) + C0(p3, 0,M1,mL2
,mR3)]O(M
2Hx/m
2SUSY)
⇒ Simple analytical expressions: we factorized the contributions of the loops and the rest ofMSSM parameters (tβ and mA).
10-17
10-16
10-15
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10-11
10-10
10-9
10-8
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
BR(φ
→ τ µ
)
mSUSY [TeV]
mA = 800 GeV, tan β = 40, δ~23LR = 0.5
GUT approx. µ = 4/3 mSUSY
h → τ µ FullA → τ µ Fullh → τ µ MIAA → τ µ MIA
10-18
10-17
10-16
10-15
10-14
10-13
10-12
10-11
10-10
10-9
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
BR(φ
→ τ µ
)mSUSY [TeV]
mA = 800 GeV, tan β = 40, δ~23LR = 0.5
Generic scenario
h → τ µ FullA → τ µ Fullh → τ µ MIAA → τ µ MIA
• Exact cancellations between Non-Decoupling terms.
• Decoupling behavior of BR(Hx → τµ) ∼ m−4SUSY like BR(τ → µγ).
• First order MIA for H and A approximates the full calculation better than the h.
• Very small rates for LR and RL.
• Similar results in other scenarios.
Roberto Morales 19/28
Limitations of the MIA results
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
-1 -0.5 0 0.5 1
BR(φ
→ τ µ
)
δ23LL
mA = 800 GeV, tan β = 40, mSUSY = 5 TeVGUT approx. µ = 4/3 mSUSY
h → τ µ FullA → τ µ Fullh → τ µ MIAA → τ µ MIA
10-14
10-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
-1 -0.5 0 0.5 1
BR(φ
→ τ µ
)
δ23RR
mA = 800 GeV, tan β = 40, mSUSY = 5 TeVGUT approx. µ = 4/3 mSUSY
h → τ µ FullA → τ µ Fullh → τ µ MIAA → τ µ MIA
10-21
10-20
10-19
10-18
10-17
10-16
10-15
10-14
10-13
10-12
-1 -0.5 0 0.5 1
BR(φ
→ τ µ
)
δ23LR
mA = 800 GeV, tan β = 40, mSUSY = 5 TeVGUT approx. µ = 4/3 mSUSY
~
h → τ µ FullA → τ µ Fullh → τ µ MIAA → τ µ MIA
10-21
10-20
10-19
10-18
10-17
10-16
10-15
10-14
10-13
10-12
-1 -0.5 0 0.5 1
BR(φ
→ τ µ
)
δ23RL
mA = 800 GeV, tan β = 40, mSUSY = 5 TeVGUT approx. µ = 4/3 mSUSY
~
h → τ µ FullA → τ µ Fullh → τ µ MIAA → τ µ MIA
Perturbativity works reasonably well within |δAB23 | ≤ O(1).
Roberto Morales 20/28
Equal masses scenario: form factors
Dominant contributions for each δ:
F(x)L,R = δLL23 F
(x)LLL,R + δLR23 F
(x)LRL,R + δRL23 F
(x)RLL,R + δRR23 F
(x)RRL,R
As in the generic scenario, for δLL23 we get
F(x)LLL =
g2
16π2
mτ
2MW
[σ
(x)2 + σ
(x)∗
1 tβ
cβ
]1− t2W
4
}ND
For δLR23 , δRL23 , δRR23 , strong cancellations produce
F(x)LRL =
gt2W16π2
1
24√
2
[σ
(x)∗
1
] m2Hx
m2S
F(x)RLR =
gt2W16π2
1
24√
2
[σ
(x)1
] m2Hx
m2S
F(x)RRR =− g2t2W
16π2
mτ
2MW
[2σ
(x)∗
2 + σ(x)1
120cβ
]m2Hx
m2S
D
Roberto Morales 21/28
Equal masses scenario: simplest effective vertices
The most relevant mixing for phenomenology is δLL23 .
For this case, we can write the dominant effective vertex −igV effHxτµPL
V effHxτµ
=g2
16π2
mτ
2MW
[σ
(x)2 + σ
(x)∗
1 tβ
cβ
](1− t2W
4
)δLL23
In the large tβ limit:
V effhτµ|tβ�1 = − g2
16π2
mτ
MW
M2Z
m2A
tβ
(1− t2W
4
)δLL23
V effHτµ|tβ�1 = −iV eff
Aτµ|tβ�1 = − g2
16π2
mτ
2MWt2β
(1− t2W
4
)δLL23
We deduce:
• For light Higgs: effective vertex suppressed by M2Z/m
2A (mA →∞, h is SM-like).
• For heavy Higgs: effective vertex enhanced by t2β (in agreement with [Brignole, Rossi;
2003])
Roberto Morales 22/28
Phenomenology
1.¥10-9
5.¥10-9
1.¥10-8
2.¥10-8
3.¥10-8
5.¥10-8
7.¥10-8
1 2 3 4 5
10
20
30
40
50
60
mSUSY HTeVL
tanb
BRHhÆtmLê»d23LL 2
Equal masses 1.¥10-6
5.¥10-6
0.00001
0.00002
0.000030.00005
0.00007
0.0001
1 2 3 4 5
10
20
30
40
50
60
mSUSY HTeVLtanb
BRHAÆtmLê»d23LL 2
Equal masses
• Shaded pink area: excluded by BR(τ → µγ) < 4.4× 10−8.• Shaded blue area: excluded by negative searches by ATLAS and CMS of neutral MSSM
Higgs bosons decaying to tau pairs.
⇒ Maximum allowed rates are ∼ 5× 10−5 for H and A (below LHC sensitivity).
Roberto Morales 23/28
Phenomenology
1.¥10-9
1.¥10-8
3.¥10-8
1.¥10-73.¥10-71.¥10-60.00001
200 400 600 800 1000
10
20
30
40
50
60
mA HGeVL
tanb
BRHhÆtmLê»d23LL 2
Equal masses 1.¥10-6
0.00001
0.00005
0.0001
0.00015
0.00020.00025
0.00030.00035
200 400 600 800 1000
10
20
30
40
50
60
mA HGeVLtanb
BRHAÆtmLê»d23LL 2
Equal masses
• We can set mSUSY = 4 TeV (ND terms) −→ no constraints by τ → µγ.• Shaded blue area: excluded by negative searches by ATLAS and CMS of neutral MSSM
Higgs bosons decaying to tau pairs.
⇒ Maximum allowed rates are ∼ 3.5× 10−4 for H and A.Taking in account both channels τµ and τµ, BR∼ 10−3 closer to LHC sensitivity!
Roberto Morales 24/28
h,H,A→ τµ at LHC
• Shaded gray area: excluded by τ → µγ.
• Shaded blue area: excluded by negative searches by ATLAS and CMS.
⇒ Most promising for H and A: in the region tβ ∼ 40− 60, mA ∼ 900− 1000 GeV and
mSUSY > 4 TeV, for√s = 14 TeV and L ∼100 fb−1 we predict O(1-10) events.
Roberto Morales 25/28
Conclusions
• We performed a diagramatic computation of LFVHD within the MIA. We found simpleanalytical expressions for the form factors. They are very useful for phenomenology andcomparison with data.
• We compared MIA vs Full results: good agreement in most cases, in particular for theNon-Decopling terms (phenomenological important ones).
• Also, we compared with the effective vertices approach: in accordance with previousresults in the literature.
• We understood the BR behavior with the MSSM parameters tβ and mA.
• We concluded that H,A→ τµ are promising channels at the LHC.
Roberto Morales 26/28
Back-up
Roberto Morales 27/28
Subleading corrections: O(M2W/m
2SUSY)
10-20
10-18
10-16
10-14
10-12
10-10
10-8
5 10 15 20 25 30 35 40
BR(h
→ τµ
)
tan β
Equal massesmA = 800 GeV, mSUSY = 5 TeV, δ23
XY = 0.5M1 = M2 = µ = mL = mR = Aµ,τ = mSUSY
LL FullLL MIA
RR FullRR MIA
LR FullLR MIA
10-18
10-16
10-14
10-12
10-10
10-8
10-6
10-4
5 10 15 20 25 30 35 40
BR(A
→ τµ
)
tan β
Equal massesmA = 800 GeV, mSUSY = 5 TeV, δ23
XY = 0.5M1 = M2 = µ = mL = mR = Aµ,τ = mSUSY
LL FullLL MIA
RR FullRR MIA
LR FullLR MIA
F(x)RRR =
g2t2W16π2
mτ
2MW cβ
M2W
m2S
t2β
1 + t2β
[(σ(x)1
60
(3t
2W + 13− 4t
2W tβ − 12tβ
)
−σ(x)∗1
5−
4σ(x)2
15−
2σ(x)∗2
15+σ(x)3
√1 + t2β
12tβ
(1 + t
2W
)
+
1 + t2W60tβ
(−8σ
(x)1 + 4σ
(x)∗1 + σ
(x)2 + σ
(x)∗2
)+σ(x)3
√1 + t2β
12t2β
(−1 + 5t
2W
)+
(1 + t2W30t2β
(−σ(x)
1 + σ(x)∗1 + σ
(x)2 − σ(x)∗
2
))]Roberto Morales 28/28