Effective Field Theory approaches to Flavor Physics beyond ... · ×U(1) Y local symmetry Global flavor symmetry The Standard Model and its flavor structure Particle physics is described

Effective Field Theory approachesto Flavor Physics beyond the SM

Gino Isidori[ University of Zürich ]

G. Isidori – EFT tools for Flavor Physics JOINT FGZ-PH Summer School – Garching, July 2017

Lecture I [The SM as an EFT]Lecture II [EFT tools for B Physics]Lecture III [Flavor Physics beyond the SM]

Effective Field Theory approachesto Flavor Physics beyond the SM


Gino Isidori[ University of Zürich ]

Lecture I [The SM as an EFT]The Standard Model and its flavor structureProperties of the CKM matrix and status of CKM fitsThe SM as an Effective Field Theory

Lecture II [EFT tools for B Physics] Lecture III [Flavor Physics beyond the SM]

The Standard Model and its flavor structure


Particle physics is described with good accuracy by a simple and economical theory:

ℒSM = ℒgauge (Aa, ψi) + ℒHiggs(ϕ, Aa, ψi )




Natural

Experimentally tested with high accuracy

Stable with respect to quantum corrections

Highly symmetric:

Σa - (Fμνa)2 + Σψ Σi ψi iD ψi

1

4ga2ℒgauge =

SU(3)c×SU(2)L×U(1)Y local symmetry

Global flavor symmetry




Natural



Highly symmetric

Ad hoc

Necessary to describe data [clear indication of a non-invariant vacuum] weakly tested in its dynamical form

Not stable with respect to quantum corrections

Origin of the flavor structure of the model [and of all the problems of the model...]





Natural



Highly symmetric

Ad hoc

Necessary to describe data [clear indication of a non-invariant vacuum] weakly tested in its dynamical form

Not stable with respect to quantum corrections

Origin of the flavor structure of the model [and of all the problems of the model...]



Elegant & stable, but also a bit boring...

Ugly & unstable, but is what makes nature interesting...!



3 identical replica of the basic fermion family [ψ = QL , uR, dR, LL, eR ] ⇒ huge flavor-degeneracy

The gauge Lagrangian is invariant under 5 independent U(3) global rotations for each of the 5 independent fermion fields

E.g.: QLi → Uij

QLj

U(1) flavor-independent phase +

SU(3) flavor-dependent mixing matrix


Σ ψ = QL , uR ,dR ,LL ,eR Σi=1..3 ψi iD ψi

QL = uL

dL

LL = νL

eL

, uR , dR , , eR


3 identical replica of the basic fermion family [ψ = QL , uR, dR, LL, eR ] ⇒ huge flavor-degeneracy: U(3)5 global symmetry


U(1)L × U(1)B × U(1)Y × SU(3)Q × SU(3)U × SU(3)D ×...

Flavor mixingLepton numberBarion number

Hypercharge




Within the SM the flavor-degeneracy is broken only by the Yukawa interaction:

QLi YD

ik dRk ϕ + h.c. → dL

i MDik dR

k + ...

QLi YU

ik uRk ϕc + h.c. → uL

i MUik uR

k + ...

in the quarksector:

_

_

_

_





QLi YD


i MDik dR

k + ...

QLi YU


i MUik uR

k + ...

in the quarksector:

_

_

_

_

The Y are not hermitian → diagonalized by bi-unitary transformations:

VD+ YD

UD = diag(yb , ys , yd)

VU+ YU

UU = diag(yt , yc , yu)yi = ≈

2½ mqi

〈 ϕ〉

mqi

174 GeV





QLi YD


i MDik dR

k + ...

QLi YU


i MUik uR

k + ...

in the quarksector:

_

_

_

_

YD = diag(yd ,ys ,yb)

YU = V+ × diag(yu ,yc ,yt)

The residual flavor symmetry let us to choose a (gauge-invariant) flavor basis where only one of the two Yukawas is diagonal:

unitary matrix

YD = V × diag(yd ,ys ,yb)

YU = diag(yu ,yc ,yt)

or


To diagonalize also the second mass matrix we need to rotate separately uL & dL (non gauge-invariant basis) ⇒ V appears in charged-current gauge interactions:

Jwμ = uL

γ μ dL → uL

V γμ dL

Cabibbo-Kobayashi-Maskawa(CKM) mixing matrix

_ _

MD = diag(md ,ms ,mb)

MU = V+ × diag(mu ,mc ,mt)

_

_

_

_

...however, it must be clear that this non-trivial mixing originates only from the Higgs sector: Vij → δij if we switch-off Yukawa interactions !

QLi YD

ik dRk ϕ → dL

i MDik dR

k + ...

QLi YU

ik uRk ϕc → uL

i MUik uR

k + ...



Jwμ = uL

γ μ dL → uL

V γμ dL


_ _



_

_

_

_QL

i YDik dR

k ϕ → dL

i MDik dR

k + ...

QLi YU

ik uRk ϕc → uL

i MUik uR

k + ...

V CKM = [V ud V us V ub

V cd V cs V cb

V td V ts V tb]

3 real parameters (rotational angles)

+

1 complex phase (source of CP violation)

Several equivalent parameterizations[unobservable quark phases] in terms of

V V+ = I



Jwμ = uL

γ μ dL → uL

V γμ dL


_ _



_

_

_

_QL

i YDik dR

k ϕ → dL

i MDik dR

k + ...

QLi YU

ik uRk ϕc → uL

i MUik uR

k + ...


V cd V cs V cb

V td V ts V tb]


+

1 complex phase (source of CP violation)

The SM quark flavor sector is described by 10 observable parameters:

6 quark masses3+1 CKM parameters

Note that:

The rotation of the right-handed sector is not observableNeutral currents remain flavor diagonal


Properties of the CKM matrix & status of CKM fits


the area of these triangles is:always the samephase-convention independentzero in absence of CP violation

6 triangular relations:

Va1 (V+)1b

+Va2(V+)2b+Va3 (V

+)3b = 0

Some properties of the CKM matrix


+1 complex phase (source of CP violation)



V cd V cs V cb

V td V ts V tb] V CKM V CKM

= I


A, |ρ+iη| = O(1)

Wolfenstein, '83

λ = 0.22

1-λ2/2 λ Aλ3(ρ-iη)

- λ 1-λ2/2 Aλ2

Aλ3(1-ρ-iη) -Aλ2 1

≈

Experimental indication of a strongly hierarchical structure:

The b → d UT triangle:

Vub*Vud

+ Vcb

*Vcd + Vtb

*Vtd = 0

-Aλ3

Aλ3(ρ+iη)

only the 3-1 triangles have all sizes of the same order in λ

Aλ3(1-ρ-iη)



V cd V cs V cb


= I


V cd V cs V cb


= I


1-λ2/2 λ Aλ3(ρ-iη)

- λ 1-λ2/2 Aλ2

Aλ3(1-ρ-iη) -Aλ2 1

≈

Experimental indication of a strongly hierarchical structure:

The b → d UT triangle:

Vub*Vud

+ Vcb

*Vcd + Vtb

*Vtd = 0

0 1

γ β

α

(ρ, η)η

ρ

Note: often you'll find experimental results shown as constraints in the ρ, η plane. These new parameters are defined by ρ = ρ (1-λ2/2)-½ (same for η) to keep into account higher-order terms in the expansion in powers of λ.

_ __


VCKM

= [V udV

usV

ub

Vcd

Vcs

Vcb

Vtd

Vts

Vtb]

Once we assume unitarity,the CKM matrix can be completely determinedusing only exp. info from processes mediated by tree-level c.c. amplitudes

l

di uk

ν W

Nuclear β decay K→ πlν

Excellent determination (error ~ 0.1%)Very good determination (error ~ 0.5%) Good determination (error ~ 2 %)Sizable error (5-15 %)Not competitive with unitarity constraints

|Vik |

1-λ2/2 λ Aλ3(ρ-iη)

- λ 1-λ2/2 Aλ2

Aλ3(1-ρ-iη) -Aλ2 1

N.B.: Also the phase γ = arg(Vub) can be obtained by (quasi-) tree-level processes



V cd V cs V cb

V td V ts V tb]

Once we assume unitarity,the CKM matrix can be completely determinedusing only exp. info from processes mediated by tree-level c.c. amplitudes

l

di uk

ν W

Nuclear β decay K→ πlν

v + d c + l

Excellent determination (error ~ 0.1%)Very good determination (error ~ 0.5%) Good determination (error ~ 2 %)Sizable error (5-15 %)Not competitive with unitarity constraints

|Vik |

b → ulν

b → clν

t → blν

c → slν

1-λ2/2 λ Aλ3(ρ-iη)

- λ 1-λ2/2 Aλ2

Aλ3(1-ρ-iη) -Aλ2 1

N.B.: Also the phase γ = arg(Vub) can be obtained by (quasi-) tree-level processes


b (s) s (d)W

Z, γ

q

u = u,c,t

b (s)

ΔF =1 FCNC ΔF=2 (neutral-meson mixing)

GIM mechanism [ large top-quark contribution: A ~ mt

2 Vtq* Vtb ]

Rare B decays [ B → Xs γ, B → K(*)

l+l−, Bs,d → l

+l−, ...]

Rare K decays [ K → πνν, ...]

s (d)

b (s)s (d)

q

u = u,c,t

The only CKM elements we cannot access via tree-level processes are Vts & Vtd

Loop-induced amplitudes:

Bd(s)- Bd(s) mixing

K0- K0 mixing

_

_


Present status of CKM fits

At present all the measurements of quark flavor-violating observables show a remarkable success of the CKM picture: we have a redundant and consistent determination of various CKM elements.


Present status of CKM fits

At present all the measurements of quark flavor-violating observables show a remarkable success of the CKM picture: we have a redundant and consistent determination of various CKM elements.

The agreement between data and SM expectations is even more striking if we consider other observables, not appearing in CKM fits, such as BR(B → Xs γ) or the Bs mixing phase

Are there interesting measurements still to be done, within this field ?


The SM as an Effective Theory


The SM as an effective theory

Several theoretical arguments [inclusion of gravity, instability of the Higgs potential, neutrino masses, ...] and cosmological evidences [dark matter, inflation, cosmological constant, ... ] point toward the existence of physics beyond the SM.




Even the observed hierarchical structure of quark and lepton masses, and their mixing pattern, seems to points toward some new dynamics:

VCKM ~

It does not look “accidental”...




Even the observed hierarchical structure of quark and lepton masses, and their mixing pattern, seems to points toward some new dynamics.

If this new dynamics is not too far from the electroweak scale, we can expect modifications of the SM predictions for a few low-energy observables in the sector of flavor physics

still a lot of work to be done in this perspective


+ “heavy fields”

ℒSM = renormalizable part of ℒeff

All possible operators with d ≤ 4, compatible with the gauge symmetry, depending only on the “light fields” of the system

The modern point of view on the SM Lagrangian is that is only the low-energy limit of a more complete theory, or an effective theory.

New degrees of freedom are expected at a scale Λ above the electroweak scale.

ℒeff = ℒgauge (Aa, ψi) + ℒHiggs(ϕ, Aa, ψi )



+ “heavy fields”




The concept of effective QFT allows us to give a more general meaning to the concept (request) of “renormalizability” in QFT

No fundamental distinction between renormalizable & non-renormalizable QFTs (once we consider them both as effective QFTs)



d=4 g Aμ ψ γμψ, λ ϕ4, ...

(ψ γμψ)2, ϕ6, ... 1

Λ21

Λ2d > 4

Contributions to scattering amplitudes:

mϕ2

E2

g(E), λ(E)

mψ

E dominant in the IR

potentially relevant at all energies

E2

Λ2irrelevant at low energies, but bad UV behavior

mϕ2 ϕ2, mψ ψ ψ, ... d < 4

S=∫ d 4 x ℒ

The concept of effective QFT allows us to give a more general meaning to the concept (request) of “renormalizability” in QFT:

Some general remarks on effective QFT


To define an effective QFT we need to define three basic ingredients:

I. The nature of the “light fields”, or the nature of the degrees of freedom we can directly excite in the validity regime of the theory [e.g. the SM fields]

II. The symmetries of the theory [e.g. the SU(3)×SU(2)×U(1) gauge symmetry]

III. The validity regime of the theory [typically an UV cut-off]

In general, the most general Lagrangian compatible with I & II contains and infinite set of operators; however, there is a finite number of operators with a given canonical dimension → at low-energies we can focus only on a finite sets of operators given terms of high dimension given small contributions




To define an effective QFT we need to define three basic ingredients:

I. The nature of the “light fields”, or the nature of the degrees of freedom we can directly excite in the validity regime of the theory [e.g. the SM fields]

II. The symmetries of the theory [e.g. the SU(3)×SU(2)×U(1) gauge symmetry]

III. The validity regime of the theory [typically an UV cut-off]

In general, the most general Lagrangian compatible with I & II contains and infinite set of operators; however, there is a finite number of operators with a given canonical dimension → at low-energies we can focus only on a finite sets of operators given terms of high dimension given small contributions


“renormalizable” EFTs [e.g. SM]No clear indication about the validity

range of the theory

“non-renormalizable” EFTs [e.g. Fermi theory, CHPT, ...]

Clear indication about the maximal validity range of the theory



+ Σ On(d) (ϕ, Aa, ψi)

cn

Λd-4




Operators of d ≥ 5 containing SM fields only (compatible with the SM gauge symmetry)

heavy dynamics

SM field

SM field SM field

SM field

This is the most general parameterization of the new (heavy) degrees of freedom, as long as we do not have enough energy to directly produce them.

ℒSM = renormalizable part of ℒeff




cn

Λd-4 ℒeff = ℒgauge (Aa, ψi) + ℒHiggs(ϕ, Aa, ψi )

(LLi σ2 ϕ)( LL

j σ2 ϕ

T )gν

ij

ΛLN

gνij v2

ΛLN

(mν)ij =

v =〈 ϕ〉

T

A clear evidence of a non-vanishing term in the series of higher-dim. ops. (so far the only one...) comes from neutrino masses:

Neutrino masses are well described by the only d=5 term allowed by the SM gauge symmetry.


The SM as an effective theory & neutrino masses

(→ Exercise: check in detail)


cn


(LLi σ2 ϕ)( LL

j σ2 ϕ

T )gν

ij

ΛLN

gνij v2

ΛLN

(mν)ij =

v =〈 ϕ〉

Possible dynamical origin of this d=5 ops.:

TLL LL

νR

ϕ ϕ


In this case (see-saw mechanism):

gν

Λ

YνTYν

MRN=




cn


(LLi σ2 ϕ)( LL

j σ2 ϕ

T )gν

ij

ΛLN

gνij v2

ΛLN

(mν)ij =

v =〈 ϕ〉

T


The smallness of mν seems to point toward a veryhigh value of Λ (that fits well with the idea of an underlying GUT at very high energies):

O(1) ×(246 GeV)2

1015 TeVLN

~ 0.06 eV

0.3 eV < mνmax < 0.04 eV ~ ~

(cosmological bounds)

(atmosphericoscillations)




cn


(LLi σ2 ϕ)( LL

j σ2 ϕ

T )gν

ij

ΛLN

gνij v2

ΛLN

(mν)ij =

v =〈 ϕ〉

T


However, this d=5 ops. violates lepton number, which is a global symmetry of the SM → the high value of Λ may be related to the breaking of LN.

We can expect lower values of Λ for effective operators that do not violate any of the symmetries of the SM Lagrangian

The smallness of mν seems to point toward a veryhigh value of Λ




cn


Indeed an indication of a much lower value of Λ comes from the only d=2 term in this Lagrangian, namely from the Higgs sector:

If the effective theory is “natural” we should expect some new degrees of freedom (respecting SM symmetries and coupled to the Higgs sector) not far from the TeV scale to stabilize the Higgs mass term.

μ2 ϕ

+ ϕ Λ2 ϕ

+ ϕ quantum

corrections

The“naturalness” problem


μ2(tree)

= ½ mh2


N.B.: the quadratic sensitivity of the Higgs mass from the cut-off is not a pure “technical” issue: it implies a quadratic sensitivity to the the new degrees of freedom (if we assume there is something else beyond the SM, such e.g. RH neutrinos):

ΔmH2 ~ Σb cbMb

2 – Σf cf Mf2

boson

fermion


μ2 ϕ

+ ϕ Λ2 ϕ

+ ϕ quantum

corrections

If Λ >> mh, either mh is “fine-tuned” or the NP does not couple to the Higgs sector



N.B.: the quadratic sensitivity of the Higgs mass from the cut-off is not a pure “technical” issue: it implies a quadratic sensitivity to the the new degrees of freedom


μ2 ϕ

+ ϕ Λ2 ϕ

+ ϕ quantum

corrections

N.B. (II): a “warning” that the naturalness argument should be taken with some care comes from the cosmological constant

V0 + μ2 ϕ

+ ϕ Λ4 + Λ2 ϕ

+ ϕ quantum

corrections

“exp”

~ (100 MeV)2~ (0.002 eV)4






cn


Which is the energy scale of New Physics (or the value of Λ)

Which is the symmetry structure of the new degrees of freedom (or the structure of the cn)

Two key questions of particle physics today:

High-energy searches [the high-energy frontier]

High-precision measurements [the high-intensity frontier]

The key questions of particle physics


More precisely, on both “frontiers” we have two independent sets of questions, a “difficult one” and a “more pragmatic one”:

What determines the Higgs mass term (or the Fermi scale)?

Is there anything else beyond the SM Higgs at the TeV scale?

What determines the observed pattern of masses and mixing angles of quarks and leptons?

Which are the sources of flavor symmetry breaking accessible at low energies? [Is there anything else beside SM Yukawa couplings & neutrino mass matrix?]

High-energy searches[the high-energy frontier]













Lecture I [The SM as an EFT] Lecture II [EFT tools for B Physics]

PremiseEFT tools for B PhysicsΔF=2 amplitudes and the gauge-less limitRare B decays

Lecture III [Flavor Physics beyond the SM]









Premise


3 identical replica of the basic fermion family ⇒ huge flavor-degeneracy [U(3)5 symmetry]

Flavor-degeneracy broken only by the Yukawa interaction

several new sources of flavor symmetry breaking are, in principle, allowed

What we have only started to investigate is the flavor structure of the new degrees of freedom

which hopefully will show up above the electroweak scale


cn



Premise

Probing the flavor structure of physics beyond the SM requires the following three main steps:

Identify processes where the SM is calculable with good accuracy using the tree-level inputs, or sufficiently suppressed for null tests.

Determine the CKM elements from theoretically clean and non-suppressed tree-level processes, where the SM is likely to be largely dominant.

Measure with good accuracy these rare processes and determine the allowed room

for new physics.

ΔF=2 Neutral meson mixing [ K, Bd, Bs]

Rare decays

Exclusive and inclusive semi-leptonic b→u decays ( |Vub| )

Selected non-leptonic B decays sensitive to γ

1

Λ2 A = A0 cSM + cNP

1

MW2


EFT tools for B Physics


b s

newd.o.f.

γ, l+l-, νν

B Xs

q

No SM tree-level contribution

Strong suppression within the SM because of CKM hierarchy

Low-energy tools: general considerations

ΔF=2 meson-antimeson mixing amplitudes & rare decays mediated by Flavour Changing Neutral Current (FCNC) amplitudes are, in principle, the best probes for precision tests of flavor dynamics beyond the SM

newd.o.f.

b q

bq

B B_

b s

newd.o.f.

γ, l+l-, νν

B Xs

q

In both cases the key point to be addresses for a good TH control is a proper “scale separation” (from the e.w. scale down to the hadronic scale) by means of suitable EFT approaches


In general, this involves a 3-step procedure:

1st step: Construction of a local (partonic) eff. Hamiltonian at the electroweak scale “integrating out” all the heavy fields around mW (including the heavy SM fields)

Heff = Σi Ci(MW) Qi

b s

c c


“Matching procedure”(Perturb. Theory,

within the SM)





b s

c c


within the SM)


2nd step: Evolution of Heff down to low scales using RGE

Heff = Σi Ci(μ) Qi

RGE evolution dictated by light SM

d.o.f. only (QCD, QED)[ ΛQCD << μ < MW ]





b s

c c



Heff = Σi Ci(μ) Qi

RGE evolution dictated by light SM

d.o.f. only (QCD, QED)[ ΛQCD << μ < MW ]

3rd step: Evaluation of the hadronic matrix elements

A(i → f ) = Σi Ci(μ) ⟨ f | Qi

|i ⟩ (μ)non-perturbative effects...


within the SM)


ΔF=2 amplitudes and the gauge-less limit


b d

bd Vqd Vq'b*

Vqb* Vq'd

Highly suppressed amplitude potentially very sensitive

to New PhysicsB B

_

q'

q

No SM tree-level contributionStrong suppression within the SM because of CKM hierarchy Calculable with good accuracy since dominated by short-distance dynamics [power-like GIM mechanism → top-quark dominance]Measurable with good accuracy from the time evolution of the neutral meson system

ΔF=2 mixing


b d

bd Vqd Vq'b*

Vqb* Vq'd

q'

power-like GIM mechanism:

AΔF=2 = Σq,q'=u,c,t (Vqb*Vqd) (Vq'b

*Vq'd) Aq'q q

Aqq' ~

mq mq'

mW2

g4

16π2mW2

Const. + + ... × (dL γμ bL)2


_

Only d=6 operator generated within the SM


b d

bd Vqd Vq'b*

Vqb* Vq'd

q'



*Vq'd) Aq'q q

AΔF=2 = Σq=u,c,t (Vqb*Vqd) [ Vtb

*Vtd (Atq-Auq) + Vcb*Vcd (Acq-Auq) ]

Vub*Vud = - Vtb

*Vtd - Vcb*Vcd [CKM unitarity]

Aqq' ~

mq mq'

mW2

g4

16π2mW2

Const. + + ... × (dL γμ bL)2


_

Only d=6 operator generated within the SM


b d

bd Vqd Vq'b*

Vqb* Vq'd

q'



*Vq'd) Aq'q q

AΔF=2 = Σq=u,c,t (Vqb*Vqd) [ Vtb

*Vtd (Atq-Auq) + Vcb*Vcd (Acq-Auq) ]

Vub*Vud = - Vtb

*Vtd - Vcb*Vcd [CKM unitarity]

Aqq' ~

mq mq'

mW2

g4

16π2mW2

Const. + + ... × (dL γμ bL)2


_

HΔF=2 ~ (Vtb*Vtd)2

g4mt2

16π2mW4

[expansion of the loop amplitude for small

(internal) quark masses]

(dL γμ bL)2(d=6) _


bL dL

bLdL Yqd Yq'b*

Yqb* Yq'd

q'R

qRThe structure of this result can be better

understood “switching-off” gauge interactionsϕ+

dLi YU

ik uRk ℒYukawa →

_ϕ- + h.c.

YU = V+ × diag(yu, yc, yt)

The “gauge-less” limit:

HΔF=2 ~ (Vtb*Vtd)2

g4mt2

16π2mW4

(dL γμ bL)2(d=6) _


bL dL

bLdL Yqd Yq'b*

Yqb* Yq'd

tR

tR ϕ+

dLi YU

ik uRk ℒYukawa →

_ϕ- + h.c.

YU = V+ × diag(yu, yc, yt)

≈ V+ × diag(0, 0, yt)

ADF=2 ~ (Vtb*Vtd)2 ~ (Vtb

*Vtd)2(yt)

4

16π2mt2

This way we obtain the exact result of the amplitude in the limit mt ≫mW :

gaugeless g4 mt2

16π2mW4

mt = yt v / √2

mW = g v / 2

ADF=2 = ADF=2 × [ 1 + O(g2) ]full gauge-less

The “gauge-less” limit:

The structure of this result can be better understood “switching-off” gauge interactions

HΔF=2 ~ (Vtb*Vtd)2

g4mt2

16π2mW4

(dL γμ bL)2(d=6) _

(→ Exercise: check in detail)



HΔF=2 (MW) =

(Vtb*Vtd)2

16π2mW2

Q0 = (dL γμ bL)2(d=6)_

C0(MW) Q0

C0(MW) = g4 mt

2

mW2 × [ 1 + O(g2, gs

2) ]


In this case the RGE evolution is particularly simple since there is only one effective operator (even beyond the 1-loop level) at d=6 → “multiplicative” renormalization of C0 (dictated by the anomalous dimension of Q0)



HΔF=2 (MW) =

(Vtb*Vtd)2

16π2mW2

Q0 = (dL γμ bL)2(d=6)_

C0(MW) Q0

C0(MW) = g4 mt

2

mW2 × [ 1 + O(g2, gs

2) ]


3rd step: Evaluation of the hadronic matrix element

⟨ B | Qi | B ⟩

In this case the RGE evolution is particularly simple since there is only one effective operator (even beyond the 1-loop level) at d=6 → “multiplicative” renormalization of C0 (dictated by the anomalous dimension of Q0)

_All non-perturbative effects are encoded into a single hadronic parameter,

→ reliable estimates (~20%) by means of Lattice QCD

N.B.: the “weak phase of the amplitude” has negligible hadronic uncertainties


Rare B decays


Similarly to ΔF=2 mixings, rare decays mediated by Flavour Changing Neutral Current (FCNC) amplitudes are useful probes for precision tests of flavor dynamics beyond the SM

b s (d)

newd.o.f.

γ, l+l-, νν

Rare B decays



Predicted with high precision within the SM at the partonic level: NNLO pert. calculations available for all the main B modes (mb ≫ ΛQCD)


The key point is the relation between patonic & hadronic worlds

Fully inclusive decays usually good precision thanks

to heavy-quark symmetry

Γ(b → sγ) Γ(B → Xs γ) mb ∞

Exclusive decays generally more difficult than inclusive,

with some notable exceptions:

B → (0, K, K*) + μ-μ+

Similarly to ΔF=2 mixings, rare decays mediated by Flavour Changing Neutral Current (FCNC) amplitudes are useful probes for precision tests of flavor dynamics beyond the SM

b s (d)

newd.o.f.

γ, l+l-, νν

Rare B decays



Predicted with high precision within the SM at the partonic level: NNLO pert. calculations available for all the main B modes (mb ≫ ΛQCD)



The interesting short-distance info is encoded in the Ci(MW) (initial conditions) of the FCNC operators (NP effects expected to be small in the four-quark ops.)

Q9V

=Qf(b s)

V−A( f f )

V

Q10A

=Qf(b s)

V−A( f f )

A

FCNC operators: Four-quark (tree-level) ops.:

Q1=(b s)

V−A(c c)

V−A

Q2=(bc)

V−A(c s)

V−A

b sb s

c c

⋮Q

6=(b s)

V −A(q q)

V

⋮

[Gluon penguins]

[E.W. penguins]

1st step: Construction of a local eff. Hamiltonian at the electroweak scale



The interesting short-distance info is encoded in the Ci(MW) (initial conditions) of the FCNC operators (NP effects expected to be small in the four-quark ops.)

N.B.: Only one of the FCNC ops. (Q10A) exhibits the power-like GIM behavior seen in the ΔF=2 case (→ Exercise: try to understand why. Help: gauge-less limit)

Q9V

=Qf(b s)

V−A( f f )

V

Q10A

=Qf(b s)

V−A( f f )

A

FCNC operators: Four-quark (tree-level) ops.:

Q1=(b s)

V−A(c c)

V−A

Q2=(bc)

V−A(c s)

V−A

b sb s

c c

⋮Q

6=(b s)

V −A(q q)

V

⋮

[Gluon penguins]

[E.W. penguins]

1st step: Construction of a local eff. Hamiltonian at the electroweak scale


Sources of long-distance effects: [dilution of the interesting short-distance info]:

Mixing of the four-quark Qi into the FCNC Qi [perturbative long-distance contribution]

g

Q2 c, u

p ~ μ

b

s

e.g.:


Penguin operators: Four-quark (tree-level) ops.:

Q9V =Q f b sV−A f f VQ10A=Q f b sV−A f f A

Q1=b s V−Ac c V−A

Q2=b cV−Ac sV−A⋮Q6 =b sV−Aq qV

⋮

Small in the case of electroweak penguins (Q10A) because of the power-like GIM mechanism [mixing parametrically suppressed by O(mc

2/ mt

2)]

Sizable for photon penguinsHuge for gluon penguins


Heff = Σi Ci(μ ~ mb) Qi



sensitivity to long-distances (cc threshold, mc dependence,...) distinction between inclusive (OPE + 1mb,c expansion) and exclusive modes (hadronic form factors) irreducible large theory errors in the case of exclusive non-leptonic final states

A(B → f ) = Σi Ci(μ) ⟨ f | Qi |B ⟩ (μ) [ μ ~ mb ]



A(B → f ) = Σi Ci(μ) ⟨ f | Qi

|B ⟩ (μ)

sensitivity to long-distances (cc threshold, mc dependence,...) distinction between inclusive (OPE + 1mb,c expansion) and exclusive modes (hadronic form factors) irreducible large theory errors in the case of exclusive non-leptonic final states

NNLO SM th. estimate: To be compared with: B(B → Xs γ) = (3.57±0.24)×10

-4

Putting all the ingredients together in the case of B → Xs γ[best inclusive mode so far ]:

A great success for the SM... ...and a great challenge for many of its extensions !

[Misiak et al. '07] [present exp. W.A.]

B(B → Xs γ) = (3.15±0.23)×10-4

[ μ ~ mb ]


The accuracy on exclusive FCNC B decays of the type B H+(γ, l+l−) depends on the th. control of B H hadronic form factors :

Several progress in the last few years [Light-cone sum rules, Heavy-quark expansion, Lattice] but typical errors still > 30%

A B f = ∑iC i ⟨ f ∣Qi∣B ⟩ μ ~ mb

Exclusive rare B decays:

~


The most difficult exclusive observables are the total branching ratios; however, f.f. uncertainties can be considerably reduced in appropriate ratios (or differential distributions) or considering very peculiar final states.

The accuracy on exclusive FCNC B decays of the type B H+(γ, l+l−) depends on the th. control of B H hadronic form factors :

Several progress in the last few years [Light-cone sum rules, Heavy-quark expansion, Lattice] but typical errors still > 30%

I. BR(Bs,d → μ+μ−) (→ Exercise: try to evaluate it in the gauge-less limit)

II. LFU ratios in B → H l+l−

A B f = ∑iC i ⟨ f ∣Qi∣B ⟩ μ ~ mb

Exclusive rare B decays:

Notable examples:

~


∫ dΓ(B → H μμ)

∫ dΓ(B → H ee)RH =

H = K, K*

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