Higgs mediated
Flavour Changing Neutral Currents
Leonardo Pedro,
October 16th 2014
1 Introduction
1.1 Particle Physics
About the interactions between the particles
(what are the EM waves? ⇒Maxwell: aether)
Not about what the elementary particles are
Experimental Physics [www.elab.ist.eu, H + at
]
Theoretical Physics [Higgs mediated FCNC]
Math./Comp. Physics [Real Poincare Reps, C++ software]
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1.2 Standard Model and its limits
the simplest to break spontaneously:
SU (3)C × SU (2)L × U (1)Y → SU (3)C × U (1)em
⇒ One Higgs Doublet
Ellis (2013)
QCD, ElectroWeak, Flavour ⇒ Experiments
ν masses and mixing, baryon asymmetry, dark matter, CMB fluctuations
⇒ New Physics
gravity;cosmological const.(dark energy);hierarchy;strong CP;
arbitrariness;meta-stability; accidental suppression of FCNCs,EDMs,p+ decay
⇒ Problems
Beyond:Inflaton,SUSY,Seesaw,GUTs,Strings,(discrete) symmetries, axion, vector-like quark ...
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1.3 Beyond the Standard Model: a modular approach
“If we want things to stay as they are, things will have to change.” — Il Gattopardo (1958)
Physical constraints on the BGL models
A contribution for the search for FCNCs (...)
⇒ Higgs mediated Flavour Violation
Botella, Branco, Carmona, Nebot, Pedro, and Rebelo (2014)
If time is enough...
Real representations of the Poincare group & Position operator
Position measurement as a Fock-space projection-valued measure (...)
⇒A localizable Spin-0 Poincare Rep is not a complex scalar
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2 Higgs mediated Flavour Violation
2.1 Yukawa couplings in 2HDM
LY = −Q0L ∆1 Φ̃1 + ∆2 Φ̃2 u0R − Q0L Γ1 Φ1 + Γ2 Φ2 d0R
0
− L0L Σ1 Φ̃1 + Σ2 Φ̃2 νR
− L0L Π1 Φ1 + Π2 Φ2 `0R + h.c.
Spontaneous symmetry breaking
1
0
hΦ1 i = √
2 v1 eiα1
q
!
,
1
0
hΦ2 i = √
2 v2 eiα2
v12 + v22 = v ' 246 GeV ,
v2
≡ tan β ,
v1
!
(α1 − α2 ) ≡ θ
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• Quark Yukawa couplings + Mass terms
LY ⊃ −u0L
− d0L
where
and
1
Mu (v + H 0 ) + Nu0 R + iNu0 I u0R
v
1
Md (v + H 0 ) + Nd0 R + iNd0 I d0R
v
√
2 0 0 0
†
uL Nd dR − u0R Nu0 d0L H + + h.c.
−
v
1
Mu = √ v1 ∆1 +v2 eiθ ∆2 ,
2
1
Nu0 = √ v2 ∆1 −v1 eiθ ∆2 ,
2
1
Md = √ v1 Γ1 +v2 eiθ Γ2
2
1
Nd0 = √ v2 Γ1 −v1 eiθ Γ2
2
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†
• Mixing matrix (CKM), V = UuL
UdL
• The “drama”:
†
UuL
Mu UuR = Du ≡ diag (mu , mc , mt )
†
UdL
Md UdR = Dd ≡ diag (md , ms , mb )
is fine, but
†
UuL
Nu0 UuR ≡ Nu = ?
†
UdL
Nd0 UdR ≡ Nd = ?
gives flavour changing couplings with R and I,
the non-SM neutral scalars!
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2.2 Top-down and Bottom-up approaches
Buras and Girrbach (2014)
models (e.g. BGL) vs. effective field theory (e.g. MFV)
For BGL:
correlations between observables
less sensitive to free parameters
patterns of flavour violation
may differ from the SM and MFV
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2.3 Generalized Minimal Flavour Violation
Two Higgs Doublets
LY (quark, Higgs) =
√ + 2H
H0 ¯ dd
−
ū V Nd γR − Nu† V γL d −
ūDu u + dD
v
2v
i
h
i
I
Rh
¯ d γR d + i ūNu γR u − dN
¯ d γR d + h.c.
ūNu γR u + dN
−
v
v
Botella, Branco, and Rebelo (2010)
flavour changing couplings dependent on VCKM : necessary & sufficient
Nd = λ1 Dd + λ2i Pi Dd + λ3i V † Pi V Dd + ...
Nu = τ1 Du + τ2i Pi Du + τ3i V Pi V † Du + ...
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2.4 Minimal Flavour Violation
only 2 spurions break the global
flavor symmetry SU (3)Q × SU (3)U × SU (3)D
Dery, Efrati, Hiller, Hochberg, and Nir (2013)
D2
2
Nd = λ1 Dd + λ2 v2d Dd + λ3 V † Dv2u V Dd + ...
2
Nu = τ1 Du + τ2 Dv2u Du + τ3 V
Dd2 †
V DU
v2
+ ...
flavour changing couplings dependent on VCKM : necessary
but not sufficient
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2.5 Models
• Discrete symmetries & Natural Flavour Conservation
[Glashow & Weinberg, PRD 15 (1977), . . . ]
– Type I: Φ2 couples to uR , dR , eR
– Type II: Φ2 couples to uR , Φ1 couples to dR , eR
– Lepton specific: Φ2 couples to uR , dR , Φ1 couples to eR
– Flipped: Φ2 couples to uR , eR , Φ1 couples to dR
• Aligned 2HDM: ∆2 ∝ ∆1 , Γ2 ∝ Γ1
[Pich & Tuzón, PRD 80 (2009), . . . ]
– Effective alignment
[Serôdio, PLB 700 (2011), Medeiros-Varzielas, PLB 701 (2011)]
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• suppression factors from symmetries breaking
[Joshipura & Rindani, PLB 260 (1991)]
[Antaramian, Hall & Rasin, PRL 69 (1992)]
[Hall & Weinberg, PRD 48 (1993)]
...
• symmetry imposes small FCNC
[Lavoura, Int.J.Mod.Phys. A9 (1994)]
[Branco, Grimus & Lavoura (BGL), PLB 380 (1996)]
[Botella, Branco & Rebelo, PLB 687 (2010)]
[Botella, Branco, Nebot & Rebelo, JHEP 1110 (2011)]
...
[Bhattacharyya, Das & Kundu, PRD 89 (2014)]
– In the BGL case:
FCNC proportional to fermion masses & mixings!
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3 Physical constraints on the BGL models
3.1 Relating Nd with the CKM
• Rewrite
†
UdL
Nd0 UdR
v2
v2
Dd − √
=
v1
2
v2 v1
† iθ
+
UdL
e Γ2 UdR
v1 v2
† iθ
• Problem with UdL
e Γ2 UdR
• Necessary and Sufficient conditions for BGL:
∆†1 ∆2 = 0; ∆1 ∆†2 = 0; Γ†1 ∆2 = 0; ∆†1 Γ2 = 0
“They [Branco, Grimus, and Lavoura (1996)] have developed the only possible
implementation of a relation between FCNSI and the CKM matrix which
uses abelian symmetries and is consistent with the sufficient conditions
above.” —Ferreira and Silva (2011)
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3.2 The abelian symmetry
Q0Lj 7→ eiτ Q0Lj ,
u0Rj 7→ ei2τ u0Rj ,
Φ2 7→ eiτ Φ2
and j = 3, leading to FCNC in the DOWN sector
Nu = −
v1
v2
diag(0, 0, mt ) + diag(mu , mc , 0)
v2
v1


md 0
0
v2 

Nd =
 0 ms 0 
v1
0
0 mb
md |Vtd |2 ms Vtd∗ Vts mb Vtd∗ Vtb


md Vts∗ Vtd ms |Vts |2 mb Vts∗ Vtb 
md Vtb∗ Vtd ms Vtb∗ Vts mb |Vtb |2

−
v1 v2
+
v2 v1
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but we can as well choose j = 1 (or j = 2),
then leading to FCNC in the DOWN sector
. . . or start with this symmetry
Q0Lj 7→ eiτ Q0Lj ,
d0Rj 7→ ei2τ d0Rj ,
Φ2 7→ eiτ Φ2
which would lead to FCNC in the UP sector
Models: 3 up quark + 3 down quark × 3 ` + 3 ν = 36 (!)
When V ≡ 1 the couplings still break the global flavour symmetry
SU (3)Q × SU (3)U × SU (3)D ⇒ BGL is not MFV
...phenomenology Botella, Branco, Carmona, Nebot, Pedro, and Rebelo (2014)
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|gµ /ge |2
Br(B + → e+ ν)
Br(B + → τ + ν)
Br(D+ → e+ ν)
Γ(π + →e+ ν)
Γ(π + →µ+ ν)
−
− − +
Br(τ → e e e )
Br(τ − → µ− µ− e+ )
Re(∆d )
Br(KL → µ± e∓ )
Br(D0 → e− e+ )
Br(D0 → µ− µ+ )
Br(Bs0 → µ± e∓ )
Br(µ → eγ)
Br(τ → µγ)
S
|gRR,τ
µ|
+
Br(Ds → e+ ν)
Br(Ds+ → τ + ν)
Br(D+ → µ+ ν)
Γ(τ − →π − ν)
Γ(π + →µ+ ν)
−
− −
Br(τ → µ µ µ+ )
Br(τ − → µ− e− µ+ )
Re(∆s )
Br(π 0 → µ± e∓ )
Br(B 0 → e+ e− )
Br(B 0 → µ− µ+ )
Br(B 0 → τ + τ − )
Br(B → Xs γ)NNLO
SM
FZbb̄
S
|gRR,τ
e|
+
Br(B → µ+ ν)
Br(B → Dτ ν)
Br(D+ → τ + ν)
Γ(K + →e+ ν)
Γ(K + →µ+ ν)
−
− − +
Br(τ → e e µ )
Br(µ− → e− e− e+ )
Im(∆d )
Br(KL → e− e+ )
Br(D0 → µ± e∓ )
Br(Bs0 → e+ e− )
Br(Bs0 → µ− µ+ )
Br(τ → eγ)
∆T
S
|
|gRR,µe
+
Br(Ds → µ+ ν)
Br(B → D∗ τ ν)
log C (K → π`ν)
Γ(τ − →K − ν)
Γ(K + →µ+ ν)
−
− − +
Br(τ → e µ e )
|K |N P ∆mK
Im(∆s )
Br(KL → µ− µ+ )
Br(B 0 → τ ± e∓ )
Br(B 0 → τ ± µ∓ )
K
D
2|M12
| , 2|M12
|
Br(B → Xs γ)
∆S
Table 1: Constraints on processes mediated at tree level by H ± ; tree level by R, I;
loop level by R, I, H ±
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simplified scenario: H 0 , R (& I) are the mass eigenstates
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428 < MH + /GeV
493 < MR0 /GeV
493 < MI 0 /GeV
428 < MH + /GeV
488 < MR0 /GeV
488 < MI 0 /GeV
290 < MH + /GeV
305 < MR0 /GeV
314 < MI 0 /GeV
57 < MH + /GeV
97 < MR0 /GeV
97 < MI 0 /GeV
676 < MH + /GeV
641 < MR0 /GeV
641 < MI 0 /GeV
646 < MH + /GeV
597 < MR0 /GeV
597 < MI 0 /GeV
32 < MH + /GeV
97 < MR0 /GeV
111 < MI 0 /GeV
161 < MH + /GeV
245 < MR0 /GeV
265 < MI 0 /GeV
339 < MH + /GeV
359 < MR0 /GeV
369 < MI 0 /GeV
32 < MH + /GeV
57 < MR0 /GeV
57 < MI 0 /GeV
646 < MH + /GeV
621 < MR0 /GeV
621 < MI 0 /GeV
651 < MH + /GeV
602 < MR0 /GeV
602 < MI 0 /GeV
389 < MH + /GeV
443 < MR0 /GeV
448 < MI 0 /GeV
389 < MH + /GeV
443 < MR0 /GeV
448 < MI 0 /GeV
260 < MH + /GeV
265 < MR0 /GeV
280 < MI 0 /GeV
107 < MH + /GeV
126 < MR0 /GeV
126 < MI 0 /GeV
661 < MH + /GeV
606 < MR0 /GeV
611 < MI 0 /GeV
646 < MH + /GeV
582 < MR0 /GeV
577 < MI 0 /GeV
161 < MH + /GeV
166 < MR0 /GeV
191 < MI 0 /GeV
354 < MH + /GeV
384 < MR0 /GeV
394 < MI 0 /GeV
52 < MH + /GeV
116 < MR0 /GeV
121 < MI 0 /GeV
57 < MH + /GeV
67 < MR0 /GeV
67 < MI 0 /GeV
636 < MH + /GeV
606 < MR0 /GeV
611 < MI 0 /GeV
641 < MH + /GeV
587 < MR0 /GeV
587 < MI 0 /GeV
240 < MH + /GeV
265 < MR0 /GeV
285 < MI 0 /GeV
275 < MH + /GeV
295 < MR0 /GeV
314 < MI 0 /GeV
250 < MH + /GeV
245 < MR0 /GeV
265 < MI 0 /GeV
57 < MH + /GeV
107 < MR0 /GeV
107 < MI 0 /GeV
666 < MH + /GeV
611 < MR0 /GeV
616 < MI 0 /GeV
542 < MH + /GeV
483 < MR0 /GeV
473 < MI 0 /GeV
32 < MH + /GeV
97 < MR0 /GeV
111 < MI 0 /GeV
161 < MH + /GeV
240 < MR0 /GeV
265 < MI 0 /GeV
339 < MH + /GeV
359 < MR0 /GeV
369 < MI 0 /GeV
42 < MH + /GeV
102 < MR0 /GeV
102 < MI 0 /GeV
661 < MH + /GeV
611 < MR0 /GeV
616 < MI 0 /GeV
597 < MH + /GeV
542 < MR0 /GeV
542 < MI 0 /GeV
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4 A contribution for the search for FCNCs
4.1 Next steps
the analysis on flavour physics suggest that the CKM’s hierarchy makes
some models competitive against NFC models (e.g. MSSM)
No a priori reason to expect a worse performance then NFC against
LHC data. It must be checked.
Study the correlations among the observables, find interesting patterns
(on progress...)
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4.2 ...and more
Porod, Staub, and Vicente (2014) Flavour Kit⇒ about a model:
expressions for all involved masses and couplings
renormalization group equations to get the running parameters at the
considered scale
expressions to calculate the EFT operators
formulae to obtain the observables from the EFT operators
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4.3 Software
Giac and Ginac (C++ algebra systems) available (also Flavour Kit,
etc.);
Moreover, using LLVM the code generation of GiNaC can be improved;
Library containing many known formulas for decays important for
FCNC;(contribution)
Library making global fits, from models, from formulas, from
experimental data (contribution)
Library containing the experimental data distributions.
(CERN-based ROOSTATS-like package for FCNCs?)
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4.4 A systematic treatment of the systematic uncertainties
Systematic uncertainties are not good guesses
the experimental results depend from each other and from the
theoretical results.
mean ± errors is ok to draw conclusions such as:
was a Higgs boson detected in the CMS experiment?
is a BGL pattern present in the data from many experiments? Not ok
Output the data distributions as a function of the parameters which do
not depend on the experiment.
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Example: Charged Higgs searches at the LHC
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5 On the real representations of the Poincare
group
5.1 Interesting facts
The Dirac equation for a free particle is:
(iγ µ ∂µ − m)Ψ = 0
In a Majorana basis:
+1
0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 +1
iγ 1 =
iγ 0 =
0
0
−1
0
0
0
0
−1
+1
0
0
0
0
+1
0
0
iγ 2 =
iγ 5 =
0
0
+1
0
0
0
0
+1
+1
0
0
0
0
+1
0
0
0
+1
0
0
−1
0
0
0
0
0
0
−1
0
0
+1
0
iγ 3 =
0
+1
0
0
+1
0
0
0
0
0
0
−1
0
0
−1
0
≡ −γ 0 γ 1 γ 2 γ 3
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the 5 anti-commuting 4 × 4 real matrices are unique (up to isomorphisms)
⇒ η = diag(−1, 1, 1, 1) , but no euclidean metric η = diag(1, 1, 1, 1)
The Fourier-Majorana transformation ψ(~
p) of a Majorana spinor Ψ(~x):
ψ(~
p) ≡
Z
p
0
/γ 0 + m
p
Ψ(~x)
d3 ~x e−iγ p~ · ~x p
Ep + m 2Ep
Where m ≥ 0 p0 = Ep =
p
p~2 + m2 .
ψ(~
p) ⊗ C is in a Newton-Wigner representation, obtained with a
Fourier followed by a Foldy-Wouthuysen transformation
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5.2 Map from complex to real representations
QM formulation on a real Hilbert space can be made equivalent to the
QM formulation on a complex Hilbert space, using the density matrix.
There is a map from the complex to the real unitary representations of
a group.
A reference for the energy positivity of a Poincare representation can
be another Poincare representation: V ⊗ C → V1 ⊗ V2 , such that,
(iγ 0 )(i) = −1 → (iγ 0 )1 (iγ 0 )2 = −1.
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5.3 Localization
A localizable unitary representation of the Poincare group verifies:
1) it’s a direct sum of irreducible representations which are massive or
massless with discrete helicity; (related with the result Yngvason
(1970))
2) it respects causality;
3) if it is complex it contains necessarily both positive and negative
energy subrepresentations;
4) it is an irreducible representation of the Poincare group (including
parity) if and only if it is: a)real and b)massive with spin 1/2 or
massless with helicity 1/2. Coordinates match the Dirac equation
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6 Position measurement as a Fock-space
projection-valued measure
The (free) Dirac equation was deduced from the QM principles:
unitary reps & projection-valued measure
The previous results lead to a “solution” to the QM position operator
Not a new solution: QM needs many particles
We have the tools to build a relativistic Quantum Mechanics theory.
(modest improvement of our knowledge about QFT, Consistent Histories play a role)
Important for the deduction of quantum models from first principles
(instead of classical model→ quantum model through quantization)
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7 Conclusion
Experimental results do not necessarily kill models with natural FCNCs
at tree level, such as the BGL models, CKM hierarchy is enough.
FCNCs may be the next candle towards New Physics,
but a systematic/collaborative approach is needed
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The complex Hilbert space do not generalizes a real Hilbert space, the
localizable represntations of the Poincare group force us to consider
real Hilbert spaces
A relativistic Quantum Mechanics formulation using Fock-space
projection-valued measures is being developed, important for the
future deduction of quantum models from first principles
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References
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Energy Physics, 7:78, July 2014, 1401.6147.
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