flavour physics (of quarks)€¦ · m. kenzie 3 / 56. recap last time we looked at i the ckm matrix...

Flavour Physics (of quarks)

Part 4: Flavour Changing Neutral Currents

Matthew Kenzie

Warwick Week Graduate Lectures

July 2020

Checkpoint Reached

1. Recap

M. Kenzie 2 / 56

Overview

Lecture 1: Flavour in the SM

I Flavour in the SM

I Quark Model History

I The CKM matrix

Lecture 2: Mixing and CP violation

I Neutral Meson Mixing (no CPV)

I B-meson production and experiments

I CP violation

Lecture 3: Measuring the CKM parameters

I Measuring CKM elements and phases

I Global CKM fits

I CPT and T -reversal

I Dipole moments

Lecture 4: Flavour Changing Neutral Currents (Today)

I Effective Theories

I New Physics in B mixing

I New Physics in rare b→ s processes

I Lepton Flavour Violation

M. Kenzie 3 / 56

Recap

Last time we looked at

I The CKM matrix

M. Kenzie 4 / 56

FCNC processes

I FCNC processes can probe incredibly high massscales (well beyond those directly accesible at theLHC)I If there are new particles at the TeV-scale, why

don’t they manifest themselves in FCNC processes

(the so-called “flavour problem”)

I There are two types of FCNC process:I ∆F = 2: meson anti-meson mixingI ∆F = 1: “rare decays” e.g.B0

s →µ+ µm or B0

→K∗0µ+ µm

I In the SM these processes are heavily suppressedI They are loop processes that are CKM suppressed

and (depending on the process) can also be GIM

suppressed and/or helicity suppressed

T. Blake

FCNC processes• Two types of FCNC process:

∆F = 2, meson anti-meson mixing.

∆F = 1, e.g. Bs→!+!- . (commonly described as rare decays).

!

• In the SM these processes are suppressed: Loop processes that are CKM suppressed

and can (depending on the process) be highly GIM suppressed.

B0

s B0s

B0

K0

µ+

µ

µ+

µ

B0

s

4

t

t

t

W W

t

W

, Z0

Z0W

M. Kenzie 5 / 56

Checkpoint Reached

2. Effective Theories

M. Kenzie 6 / 56

Effective Theories

I In meson/baryon decays there is a clear separation of scales which we can “decouple”I b quark states have m ∼ 5 GeV while particles in loops (W±,t) have m ∼ 100 GeV

mw mb > ΛQCD (1)

I We want to study the physics of the mixing/decay at or below a scale, Λ, in a theory

which has contributions from particles at a scale below and above Λ

I We can replace the full theory with an effective theory (which is renormalisable) valid

at Λ

L(φL, φH)→ L(φL) + Leff = L(φL) +∑i

CiOi(φL)

︸︷︷︸operator product expansion

(2)

I In other words for interactions originating at a high scale (i.e. SM+NP) we get an

effective matrix element

〈f |Heff |i〉 =∑k

1

Λk

∑i

Ck,ishort distancecontribution

(physics> Λ)

〈f |Ok|i〉|Λ“Local operators”

long distancecontribution

(physics> Λ)

(3)

I The so-called “Wilson coefficients” are independent of Λ

M. Kenzie 7 / 56

Effective Theories

Non-leptonic b decay

e.g. b→ ucs

𝑏

𝑢!

#𝑢"

𝑑!

Heff(b→ u1u2d1) =

GF√2Vu1bV

∗u2d1 [C1Ou1u2d1

1 + C2Ou1u2d12 ]

Ou1u2d11 = (uα1 γµ(1− γ5)bβ)(dβ1γ

µ(1− γ5)uα2 )

Ou1u2d12 = (uα1 γµ(1− γ5)bα)(dβ1γ

µ(1− γ5)uβ2 )

EW penguin

e.g. b→ s`+`−

𝑏

𝑠

𝑙!

𝑙"

Heff(b→ s`+`−) =

2GF√2VtsV

∗tb

∑i=9,10

CiOi

O9 = (sLγµbL)(¯γµ`)

O10 = (sLγµbL)(¯γµγ5`)

C = CSM + CNP and is complex

M. Kenzie 8 / 56

Fermi’s theory

I In the Fermi model of the weak interaction, the full electroweak Lagrangian (which

was unknown at the time) is replaced by a low-energy theory (QED) plus a single

operator with an effective coupling constant

T. Blake

Fermi’s theory• In the Fermi model of the weak interaction, the full electroweak

Lagrangian (which was unknown at the time) is replaced by the low-energy theory (QED) plus a single operator with an effective coupling constant.

7

d u d u

e e

W

LEW ! LQED +GFp

2(ud)(e)

limq2!0

g2

m2W q2

=

g2

m2W

At low energies:

i.e. the full theory can be replaced by a 4-fermion operator and a coupling constant, GF.

Simplifies Lagrangian

I At low energies the full theory can be replaced by a 4-fermion operator and a single

coupling constant, GF , as

limq2→0

(g2

m2W − q2

)=

g2

m2W

(4)

I The Lagrangian simplifies to

LEW → LQED +GF√

2(ud)(eν) (5)

M. Kenzie 9 / 56

FCNC constraints

T. Blake

FCNC constraints

9

NP

/2NP

2/2NP

direct searches !(in reality this won’t be completely independent of !, you need some coupling to SM particles to produce particles in pp collisions).

mixing,

ΔF=1 processes,

mass scale of !new particles

coupling

0.0

slide from Tom Blake

I In reality the direct searches do have some dependence on κ as you need a coupling

to SM particles in order to produce the new particles in pp collisions

M. Kenzie 10 / 56

Checkpoint Reached

3. ∆F = 2 processes (NP in B mixing)

M. Kenzie 11 / 56

GIM mechanism

I Take neutral B mixing diagram as an example

B0 B0

u, c, t

W± W±

u, c, t

d b

b d

𝑞 𝑏

𝑞𝑞

𝑉!!" #𝑞 = ,

𝑉!"#∗𝑉!""

𝑉!!#∗

I Have an amplitude (summed over up-type quarks in the loop, u1, u2)

A(B0q → B

0q) =

∑u1,u2

(V ∗u1bVu1q)(V∗u2bVu2q)Au1u2 where Au1u2 ∝ mu1mu2/m

2W

(6)

I Inserting the known CKM constraint V ∗ubVud + V ∗cbVcd + V ∗tbVtd = 0 gives

A(B0q → B

0q) =

∑u1

(V ∗u1bVu1d[V∗tbVtd(Atu1 −Auu1) + V ∗cbVcd(Acu1 −Auu1)] (7)

so for the B system the top totally dominates as Atu1 Acu1 Auu1

M. Kenzie 12 / 56

New physics in B mixing

I Introducing new physics at some higher scale, ΛNP, with coupling, κNP

Leff = LSM +∑i

κ2NP

Λd−4NP

O(d)i (8)

I With the SM contribution from the box diagram

(V ∗tbVtd)2 g4m2

t

16π2m4W

I and a NP contribution (at dimension 6)

κ2NP

Λ2NP

M. Kenzie 13 / 56

New physics in B mixing

I Quantify the NP contribution to B mixing with a multiplicative factor such that

M12 = M12,SM ·∆q (9)

I Constraints provided by CKM fitter show that the result is consistent with the SM

(i.e. Re(∆) = 1 and Im(∆) = 0)

)s

(BSL

) & ad

(BSL

& aSLA

expαsm∆ & dm∆

)d

β+2d ∆φsin(

SM point

d∆Re

2 1 0 1 2 3

d∆

Im

2

1

0

1

2

excluded area has CL > 0.68

Summer14

CKMf i t t e r

SL mixing with Ad NP in B

)s

(BSL

) & ad

(BSL

& aSLA

)0

fψ(J/sτ) &

K+(Ksτ & FSsτ & sΓ ∆

sm∆ & dm∆

sβ2s

∆φ

SM point

s∆Re

2 1 0 1 2 3

s∆

Im

2

1

0

1

2

excluded area has CL > 0.68

Summer14

CKMf i t t e r

SL mixing with As NP in B

M. Kenzie 14 / 56

New physics constraints from neutral mixing

I So far everything shows consistency with the SM

I We can use this to set limits on the size of the NP scale (Λ) or coupling to SM (κ)

[arXiv:1002.0900]

Operator Bounds on ! in TeV (cij = 1) Bounds on cij (! = 1 TeV) Observables

Re Im Re Im

(sL!µdL)2 9.8! 102 1.6! 104 9.0! 10!7 3.4! 10!9 "mK ; "K

(sR dL)(sLdR) 1.8! 104 3.2! 105 6.9! 10!9 2.6! 10!11 "mK ; "K

(cL!µuL)2 1.2! 103 2.9! 103 5.6! 10!7 1.0! 10!7 "mD; |q/p|,#D

(cR uL)(cLuR) 6.2! 103 1.5! 104 5.7! 10!8 1.1! 10!8 "mD; |q/p|,#D

(bL!µdL)2 5.1! 102 9.3! 102 3.3! 10!6 1.0! 10!6 "mBd

; S!KS

(bR dL)(bLdR) 1.9! 103 3.6! 103 5.6! 10!7 1.7! 10!7 "mBd; S!KS

(bL!µsL)2 1.1! 102 7.6! 10!5 "mBs

(bR sL)(bLsR) 3.7! 102 1.3! 10!5 "mBs

TABLE I: Bounds on representative dimension-six "F = 2 operators. Bounds on ! are quoted assuming an

e#ective coupling 1/!2, or, alternatively, the bounds on the respective cij ’s assuming ! = 1 TeV. Observables

related to CPV are separated from the CP conserving ones with semicolons. In the Bs system we only quote

a bound on the modulo of the NP amplitude derived from "mBs (see text). For the definition of the CPV

observables in the D system see Ref. [15].

central value for the CP-violating phase, contrary to the SM expectation. The errors are, however,

still large and the disagreement with the SM is at about the 2$ level. If the disagreement persists,

becoming statistically significant, this would not only signal the presence of physics beyond the

SM, but would also rule out a whole subclass of MFV models (see Sect. IV).

(iv) In D " D mixing we cannot estimate the SM contribution from first principles; however,

to a good accuracy this is CP conserving. As a result, strong bounds on possible non-standard

CP-violating contributions can still be set. The resulting constraints are only second to those from

"K , and unlike in the case of "K are controlled by experimental statistics and could possibly be

significantly improved in the near future.

A more detailed list of the bounds derived from "F = 2 observables is shown in Table I,

where we quote the bounds for two representative sets of dimension-six operators: the left-left

operators (present also in the SM) and operators with a di#erent chirality, which arise in specific

SM extensions. The bounds on the latter are stronger, especially in the kaon case, because of the

larger hadronic matrix elements. The constraints related to CPV correspond to maximal phases,

and are subject to the requirement that the NP contributions are smaller than 30% (60%) of the

total contributions [9] in the Bd (K) system. Since the experimental status of CP violation in the

Bs system is not yet settled we simply require that the new physics contributions are smaller than

9

M. Kenzie 15 / 56

http://arxiv.org/abs/1002.0900

New physics constraints from neutral mixing

I So far everything shows consistency with the SM

I We can use this to set limits on the size of the NP scale (Λ) or coupling to SM (κ)

[arXiv:1002.0900]

New Physics from Flavour sector

I Direct NP discovery by ATLAS/CMS is still possibleI The flavour sector provides another window of opportunity

I High Energy scales, I Low coupling scales, z

Q(6)AB zij [qi

Aqj ] [qiBqj ]

Figure 2. Current constraints of neutral meson oscillation measurements on new F = 2dimension six operator contributions, given in terms of the eective operator scale (for genericflavour structures on the left and in the MFV limit on the right) or Wilson coecients’ size (inthe centre). Bounds on the CP conserving and CP violating contributions are shown in blueand red, respectively (see text for details).

operators involving only SM fields [4] via the matching procedure

LBSM ! LSM +

d>4

Q(d)i

d4, (3)

where d is the canonical operator dimension. Below the EW breaking scale, these newcontributions can lead to (a) shifts in the Wilson coecients corresponding to Qi present inLe

weak already within the SM; (b) the appearance of new eective local operators. In both cases,the resulting eects on the measured flavour observables can be computed systematically. Giventhe overall good agreement of SM predictions with current experimental measurements, suchprocedure typically results in severe bounds on the underlying NP flavour breaking sources inLBSM.

Let us consider the canonical example of NP in F = 2 processes associated with oscillationsof neutral mesons (for recent extended discussion see [5]). The leading (d = 6) NP operators

are of the form Q(6)AB zij [qi

Aqj ] [qiBqj ], where qi denote the SM quark fields, while A,B

denote the Cliord algebra generators. Assuming zij to be generic O(1) complex numbers,z exp(iNP), the reach of current constraints in terms the probed NP scales are shownin Fig. 2 (left). It is important to stress that most of these constraints are currently limitedby theory (i.e. lattice QCD inputs [6]) and parametric uncertainties. Consequently, significantfuture improvements will require a corroborative eort of mostly lattice QCD methods on thetheory side, as well as improved experimental determinations of SM CKM parameters by flavourexperiments, most notably LHCb and Belle II. Among the few F = 2 observables whichremain largely free from theoretical uncertainties are those related to CP violation in D0 andBs oscillations. These are expected to remain eective experimental null-tests of the SM in theforeseeable future.

The current severe flavour bounds could be interpreted as a requirement on beyond SM(BSM) degrees of freedom to exhibit a large mass gap with respect to the EW scale (if theNP flavour and CP breaking sources are of order one and not aligned with Yu,d). Conversely,TeV scale NP (c.f. Fig. 2 (centre)) can only be reconciled with current experimental results,provided it exhibits sucient flavour symmetry or structure, such that |zij | 1 (the extremecase being minimal flavour violation (MFV) [7], where one requires Yu,d to be the only sourcesof flavour breaking even BSM) . However, even in this most minimalistic scenario, the suggestivepattern of masses and mixing observed in both the quark and lepton (neutrino) sectors remainslargely unexplained. It thus remains as one of the ultimate goals of flavour physics to determinewhether the observed hierarchies and structures of flavour parameters are purely accidental, or

3

Figure 2. Current constraints of neutral meson oscillation measurements on new F = 2dimension six operator contributions, given in terms of the eective operator scale (for genericflavour structures on the left and in the MFV limit on the right) or Wilson coecients’ size (inthe centre). Bounds on the CP conserving and CP violating contributions are shown in blueand red, respectively (see text for details).

operators involving only SM fields [4] via the matching procedure

LBSM ! LSM +

d>4

Q(d)i

d4, (3)

where d is the canonical operator dimension. Below the EW breaking scale, these newcontributions can lead to (a) shifts in the Wilson coecients corresponding to Qi present inLe

weak already within the SM; (b) the appearance of new eective local operators. In both cases,the resulting eects on the measured flavour observables can be computed systematically. Giventhe overall good agreement of SM predictions with current experimental measurements, suchprocedure typically results in severe bounds on the underlying NP flavour breaking sources inLBSM.

Let us consider the canonical example of NP in F = 2 processes associated with oscillationsof neutral mesons (for recent extended discussion see [5]). The leading (d = 6) NP operators

are of the form Q(6)AB zij [qi

Aqj ] [qiBqj ], where qi denote the SM quark fields, while A,B

denote the Cliord algebra generators. Assuming zij to be generic O(1) complex numbers,z exp(iNP), the reach of current constraints in terms the probed NP scales are shownin Fig. 2 (left). It is important to stress that most of these constraints are currently limitedby theory (i.e. lattice QCD inputs [6]) and parametric uncertainties. Consequently, significantfuture improvements will require a corroborative eort of mostly lattice QCD methods on thetheory side, as well as improved experimental determinations of SM CKM parameters by flavourexperiments, most notably LHCb and Belle II. Among the few F = 2 observables whichremain largely free from theoretical uncertainties are those related to CP violation in D0 andBs oscillations. These are expected to remain eective experimental null-tests of the SM in theforeseeable future.

The current severe flavour bounds could be interpreted as a requirement on beyond SM(BSM) degrees of freedom to exhibit a large mass gap with respect to the EW scale (if theNP flavour and CP breaking sources are of order one and not aligned with Yu,d). Conversely,TeV scale NP (c.f. Fig. 2 (centre)) can only be reconciled with current experimental results,provided it exhibits sucient flavour symmetry or structure, such that |zij | 1 (the extremecase being minimal flavour violation (MFV) [7], where one requires Yu,d to be the only sourcesof flavour breaking even BSM) . However, even in this most minimalistic scenario, the suggestivepattern of masses and mixing observed in both the quark and lepton (neutrino) sectors remainslargely unexplained. It thus remains as one of the ultimate goals of flavour physics to determinewhether the observed hierarchies and structures of flavour parameters are purely accidental, or

3

Isidori, Nir and Perez, [Ann. Rev. Nucl. Part. Sci (2010) 60:355]15/41

Scale of NP if κ = 1 Size of κ if Λ = 1 TeV

M. Kenzie 15 / 56


Small couplings?

I New flavour violating sources (if there are any) must be highly tunedI Either come with a very small coupling constantI Or must have a very large mass

I For an O(1) effect:I generic tree-level

κNP ∼ 1 −→ ΛNP & 104 TeV

I generic loop-order

κNP ∼ 1(4π)2

−→ ΛNP & 103 TeV

I tree-level with “alignment”

κNP ∼ (ytV ∗tiVtj)2 −→ ΛNP & 5 TeV

I loop-order with “alignment”

κNP ∼ (ytV∗tiVtj)

2

(4π)2−→ ΛNP & 0.5 TeV

M. Kenzie 16 / 56

Minimal Flavour Violation

I One way of achieving small couplings is to build models that have a flavour structurewhich is “aligned” with the CKM matrixI Require that the Yukawa couplings are also the unique source of flavour breaking beyond

the SM

I This is referred to as minimal flavour violation (MFV)

I The couplings to new particles are naturally supressed by the Hierarchy of CKM

elements

I Clearly this massively degrades the sensitivty to finding it

M. Kenzie 17 / 56

Checkpoint Reached

4. ∆F = 1 processes (Rare B decays)

M. Kenzie 18 / 56

∆F = 1 FCNC decays

I FCNC transitions only occur at loop order (and beyond) in the SM

I The SM diagrams involve the charged current interaction (W±)

T. Blake

ΔF = 1 FCNC decays• Flavour changing neutral current transitions only occur at loop order

(and beyond) in the SM.

!

!

!• New particles can also contribute:

!

!

!

Enhancing/suppressing decay rates, introducing new sources of CP violation or modifying the angular distribution of the final-state particles.

18

b s

µ+

µ!!

W! W+

tb s

µ+

µ!

t

!, Z0

W!

b s

µ+

µ!!

H! H+

tb s

µ+

µ!

di

!, Z0

"0b s

µ+

µ!

di

H0

g b s

µ+

µ!Z "

SM diagrams involve the charged current interaction.

I New particles can also contribute (at either tree or loop level depending on the NP

characteristics)

T. Blake

ΔF = 1 FCNC decays• Flavour changing neutral current transitions only occur at loop order

(and beyond) in the SM.

!

!

!• New particles can also contribute:

!

!

!

Enhancing/suppressing decay rates, introducing new sources of CP violation or modifying the angular distribution of the final-state particles.

18

b s

µ+

µ!!

W! W+

tb s

µ+

µ!

t

!, Z0

W!

b s

µ+

µ!!

H! H+

tb s

µ+

µ!

di

!, Z0

"0b s

µ+

µ!

di

H0

g b s

µ+

µ!Z "

SM diagrams involve the charged current interaction.

I The effect of the NP amplitudes can be to enhance (or suppress) decays, introduce

new sources of CP violation or modify angular distributions of final-state particles (as

their spin structure and coupling will be different to the SM)

M. Kenzie 19 / 56

Properties of ∆F = 1 processes

I There are a large number of other observables that can be considered

I In the SM, photons from b →sγ decays are predominantly left-handed

(C7/C′7 ∼ mb/ms) due to the charged current interaction

T. Blake

Properties of ΔF = 1 processes

• In the SM, photons from b→s! decays are predominantly left-handed (C7/C’7 ∼ mb/ms) due to the charged- current interaction.

• Flavour structure of SM implies that the rate of b→d processes is suppressed by compared to b→s processes.

• In the SM, the rate due to the universal coupling of the gauge bosons (except the Higgs) to the different lepton flavours. Any differences in the rate are due to phase-space.

• Lepton flavour violation is unobservable in the SM at any conceivable experiment due to the small size of the neutrino mass.

19

bR(L) sL(R)W!

!L(R)

t

Vtb Vts

Large number of other observables that can be considered

|Vtd/Vts|2

[B ! Mµ+µ] [B ! Me+e]

I The flavour structure of the SM implies that the rate of b →s processes is suppressed

by |Vtd/Vts|2 relative to b →s processes

I In the SM

Γ(B →Mµ+µ−) ≈ Γ(B →Me+e−)

due to the universal couplings of the gauge bosons (except the Higgs) to the different

lepton flavours (known as lepton universality). The only differences in the rate come

down to phase-space considerations

I Direct lepton flavour violation is unobservable in the SM at any conceivable

experiment due to the small size of the neutrino mass

M. Kenzie 20 / 56

The effective theory for rare b→ s decays

I Can write an effective theory Hamiltonian as

Heff = −4GF√2VtbV

∗tsαe4π

∑i

Ci(µ)Oi(µ) (10)

Weak decay, (1/mW )2

CKM supprresion

Loop suppression, (1/4π)2

Wilson coefficient (integrating out scales above µ)

Local operator with different Lorentz structure (vector, axial vector etc.)

I Then introduce new particles that give rise to corrections

∆Heff =κrmNP

Λ2NP

ONP (11)

NP scale

local operator

I The constant κ can share some, all or none of the suppression of the SM process

M. Kenzie 21 / 56

Leptonic decay operators

I Have already seen some of the non-lepontic operators (and the b→ s`+`− operators

O9 and O10)

O7 =mb

esσµνPRbFµν

EW penguin

O8 = gsmb

e2sσµνPRT

αbGαµν

gluonic penguin

O9 = sγµPLb¯γµ`

vector current

O10 = sγµPLb¯γµγ5`

axial-vector current

O′7 =mb

esσµνPLbFµν

O′8 = gsmb

e2sσµνPLT

αbGαµν

O′9 = sγµPRb¯γµ`

O′10 = sγµPRb¯γµγ5`

right handed currents(suppressed in the SM)

M. Kenzie 22 / 56

NP operators

I Scalar and pseudo-scalar operators (e.g. from Higgs penguins)

OS = sPRb ¯ , O′S = sPLb ¯

OP = sPRb¯γ5`, O′P = sPLb¯γ5`

I Tensor operators

OT = sσµνb¯σµν`, O′T 5 = sσµνb¯σ

µν`

I All of these are vanishingly small in the SM

I In principle one could also introduce LFV versions of every operator

M. Kenzie 23 / 56

Generic ∆F = 1 process

I In the effective theory we then have

A(B → f) = V ∗tbVtq∑i

Ci(MW )U(µ,mW )〈f |Oi(µ)|B〉had. mat. elem,

(12)

I For inclusive processes the sum over exclusive states is related to the quark level

decays

B(B → Xsγ) = B(b→ sγ) +O(Λ2QCD/m

2B) (13)

I For exclusive processes we need to compute form-factors / decay constants

I In leptonic decays the matrix element can be factorised into a leptonic current and a

B meson decay consant, fBq

〈`+`−|j`jq|Bq〉 = 〈`+`−|j`|0〉〈0|jq|Bq〉 ≈ 〈`+`−|j`|0〉 · fBq (14)

I In semi-leptonic decays the matrix element can be factorised into a leptonic current

times a form-factor

〈`+`−M |j`jq|Bq〉 = 〈`+`−|j`|0〉〈M |jq|Bq〉 ≈ 〈`+`−|j`|0〉·F (q2) +O(ΛQCD/mB) (15)

although, due to hadronic contributions, this factorisation is not exact

M. Kenzie 24 / 56

Form-factors

I Alas, we never have free quarks to we need to compute hadronic matrix elements

(form-factors and decay constant) which relate us back to a real life mesonic or

baryonic decay system

I This is the non-perturbative regime of QCD i.e. very difficult (and very nasty) to

estimate

I Fortunately there have been considerable recent developments (last 10-20 years) which

do provide us the tools to make some calculations in different kinematic regimes

What we can do

T. Blake

Form factors• Unfortunately, we don’t just have free quarks and we need to

compute hadronic matrix elements (form-factors and decay constants).

Non-perturbative regime of QCD, i.e. difficult to estimate.

27

fBFortunately we have tools to help us in different kinematic regimes.

e.g how to deal with this:

Real life

T. Blake




27



M. Kenzie 25 / 56

Theoretical QCD calculation tools

I Lattice QCDI Non-perturbative approach to QCD using a discretised system of points in space and

timeI As the lattice becomes infinitely large and the spacing infinitely small the continuum of

QCD is reached

I Light-Cone-Sum-Rules (LCSR)I Exploit parton-hadron duality to compute form-factors and decay constants

I Operator product expansions (OPE)I Match physics at relevant scales

I Heavy quark expansionI Exploit the heaviness of the b quark, mb ΛQCD

I QCD factorisationI Light quark has large energy in the meson decay frameI e.g. in B → π decays, quarks in the π have high energy in the B rest frame

I Soft Collinear Effective TheoryI Model the system as highly energetic quarks interacting with soft collinear gluons

I Chrial perturbation theory

M. Kenzie 26 / 56

Lattice QCD

I Theory developments have been rapid over the past decade

I Takes huge scale collaborations (for theorists anyway)

I Lattice QCD is a numerical approach to non-perturbative calculationsI Recall the QCD Lagrangian has massless gluons and nearly massless quarksI There is a strong coupling =⇒ non-perturbative

I Perform the Feynman path integral inEuclidean space on the “lattice”(space-time grid) using Markov ChainMC (MCMC)I Correlation lengths → massesI Amplitudes → matrix elements

T. Blake

Lattice QCD• QCD lagrangian has massless gluon

fields sand almost massless quarks.

• Strong coupling → non-perturbative.

• Lattice QCD is a numerical approach to non-perturbative calculations.

• Perform path integral in Euclidean space on the lattice (space-time grid) using MCMC.

• Correlation lengths → masses.

• Amplitudes → matrix elements.

29

M. Kenzie 27 / 56

Checkpoint Reached

5. FCNC Experimental Results

M. Kenzie 28 / 56

FCNC Experimental Results

I Will mainly focus on recent measurements of B decay processes, predominantly

involving b→ s transitions

I These are some of the less well tested (only recently had sufficient samples of B

decays for many of these measurements

I FCNC decays of charm and strange can also be studied however the GIM mechanismis much more effective (i.e. there is a larger natural cancellation) for themI For the charm mesons the masses and mass differences are small (i.e. mc −ms)I For strange the top contribution is considerably suppressed relative to the B decays

because Vts Vtb

I These are some of the arguments that make B physics so compelling (at least to

some)

M. Kenzie 29 / 56

The B0(s) → µ+µ− decay

I B0s →µ+µ− is the golden channel for study of FCNC decays

I It is highly suppressed in the SM

1. Loop suppressed

2. CKM suppressed

3. Helicity suppressed (pseudo-scalar B to two spin- 12

muons)

SM process

with neutral current (axial-vector)

There is also a contribution from W± box

diagrams

T. Blake




27



T. Blake

Bs→!+!-

• Golden channel to study FCNC decays.

• Highly suppressed in SM.

1. Loop suppressed.

2. CKM suppressed (at least one off diagonal element)

3. Helicity suppressed (pseudo-scalar B to two spin-½ muons)

B0

s

µ+

µ

Z0

t

W

b

s

33

neutral current (axial-vector)

also receives contributions from W box diagrams

Possible NP process

with scalar operators

No helicity suppression e.g. SUSY at high

tan(β)

T. Blake

Bs→!+!-

• Golden channel to study FCNC decays.

• Highly suppressed in SM.

1. Loop suppressed.

2. CKM suppressed (at least one off diagonal element)

3. Helicity suppressed (pseudo-scalar B to two spin-½ muons)

B0

s

µ+

µb

s

34

Interesting probe of models with new or enhanced scalar operators (no helicity suppression), e.g. SUSY at high tan β.

b

h0

M. Kenzie 30 / 56

B0s → µ+µ− in the SM

I Nice and clean because only one operator contributes in the SM

O10 = (sγµb)(µγµγ5µ) (16)

I The branching fraction in the SM is

B(B0s → µ+µ−) = |V ∗tbVts|2

CKM factors

G2Fα

2e

16π3ΓH

Decay constant〈0|sγµγ5b|B〉=ifBpµ

MBM2µ f2

B

√1− 4M2

µ

M2B

|C10(mb)|2(M2µ

M2B

)helicity

suppression

I Beyond the SM

B(B0s → µ+µ−)NP

B(B0s → µ+µ−)SM

=1

|CSM|2[(

1− 4m2µ

m2B

) ∣∣∣∣ mB

2mµ(CS − C′S)

∣∣∣∣2+

∣∣∣∣ mB

2mµ(CP − C′P ) + (C10 − C′10)

∣∣∣∣2 ]

M. Kenzie 31 / 56

B0s →µ+µ− experimental results

I Observation is the end of a long road of searches

I B0s → µ+µ− (B0 → µ+µ−) now observed at > 7σ (∼ 3σ). Both are consistent with

the SM predictions

I No sign of NP here (unfortunately) but this does set some very strong constraints on

many models

]2c [MeV/−µ+µm5000 5200 5400 5600 5800 6000

)2C

andi

date

s / (

50

MeV

/c

0

5

10

15

20

25

30

35 Total−µ+µ → s

0B−µ+µ → 0B

Combinatorial−

h'+ h→ (s)0B

µν+µ)−(K−π → (s)0B

−µ+µ0(+)π → 0(+)B

µν−µ p→ b0Λ

µν+µψ J/→ +cB

LHCb

BDT > 0.5

LHCb latest

M. Kenzie 32 / 56

Photon polarisation

I In radiative B decays allows both

bL → sRγR (17)

bR → sLγL (18)

I However the charged current interaction only couples to left-handed quarks

I Need a helicity flip (boost into suitable frame) to either the b or s quark

I The right-handed contribution is therefore suppressed by

A(bL → sRγR)

A(bR → sLγL)∼ ms

mb

T. Blake

Photon polarisation• In radiative B decays, angular momentum

conservation allows

!

!

• However, the charged current interaction only couples to left handed quarks. Need to helicity flip the b- or s-quark.

• The right-handed contribution is therefore suppressed by

41

bR(L) sL(R)W!

!L(R)

t

Vtb VtsbL ! sRR

bR ! sLL

A(bL ! sRR)

A(bR ! sLL) ms

mb

M. Kenzie 33 / 56

Radiative decays

I Constraints on right-handed currents in b→ sγ decays

I Results are consistent with the LH polarisation expectated in the SM

Inclusive branching

fraction

T. Blake

Radiative decays• Constraints on right-handed currents in b→s! decays:

time dependent CP violation in B ! [K0

S0]

inclusive branching fraction.

angular distribution of B ! Ke+e

42

Results are consistent with LH polarisation expected in SM

Time-dependent CP

violation in B → [K0Sπ

0]γ

T. Blake



S0]



42

Results are consistent with LH polarisation expected in SM

Angular distribution of

B → K∗e+e−

T. Blake



S0]



42

Results are consistent with LH polarisation expected in SMM. Kenzie 34 / 56

Is the photon polarised?

I Yes, in B+ →K+π−π+γ decays the photon has a preferred direction with respect to

the K+π−π+ decay plane

I This can only happen if the photon is polarised

[arXiv:1402.6852]

]2c) [MeV/ππM(K1200 1400 1600 1800

udA

-0.1

-0.05

0

0.05

0.1

LHCb

M. Kenzie 35 / 56


b→ s`+`−

I A very important class of decays for FCNC limits are b→ s`+`− transitions

I Understanding distributions with respect to the invariant mass of the di-lepton

spectrum (q2) is vital

T. Blake

b→s!+!−decay spectrum

44

!"#$%&$%$"'$(

J/(1S)

(2S)C()7

C()7 C

()9

C()9 C

()10

4 [m(µ)]2 q2

ddq2

)"*(

+,"-(*!.#)"'$(

',"#%!/01,".(&%,2(

)/,3$(,4$"('5)%2(

#5%$.5,6*((

cc

Photon pole enhancement (no pole for B→P!! decays)

Dimuon mass squared

Spectrum dominated by narrow charmonium resonances. (vetoed in data)

Form-factors from Lattice QCD

Form-factors from LCSR calculations parameterisation

slide from Tom BlakeM. Kenzie 36 / 56

Branching fractions in b→ sµ+µ−

I The LHCb (and CMS) Run 1 datasets already have precise measurements of

differential branching fractions with at least comparable precision to the SM theory

expectations

[arXiv:1403.8044] [arXiv:1606.04731], [arXiv:1507.08126]

T. Blake

]4c/2 [GeV2q0 5 10 15 20

]4 c-2

GeV

-710

× [2 qdB

R/d

0

0.5

1

1.5

µµ *0K →0B

• We already have precise measurements of branching fractions in the run1 datasets with at least comparable precision to SM expectations:

!

!

!

!

!

!!

• SM predictions have large theoretical uncertainties from hadronic form factors (3 for B→K and 7 for B→K* decays). For details see [Bobeth et al JHEP 01 (2012) 107] [Bouchard et al. PRL111 (2013) 162002] ![Altmannshofer & Straub, EPJC (2015) 75 382].

]4c/2 [GeV2q0 5 10 15 20

]2/G

eV4 c ×

-8 [1

02 q

/dBd 0

1

2

3

4

5

LCSR Lattice Data

LHCb−µ+µ+ K→+B

Branching fraction measurements

45

[LHCb, JHEP 1406 (2014) 133]

LHCb [JHEP 11 (2016) 047]!CMS [PLB 753 (2016) 424]3fb-1

3 fb-1

20.5 fb-1

−+

I SM predictions have large theory uncertainties from the hadronic form-factors (of

which there are 3 for B± → K± and 7 for B → K∗)

I Details of theory predictions in [arXiv:1111.2558], [arXiv:1306.0434] and [arXiv:1411.3161]

M. Kenzie 37 / 56







The B0 → K∗0µ+µ− angular basis

I We have a four-body final state (as K∗0 → K+π−)I The angular distribution provides many

observables that are sensitive to new

physicsI The branching fraction might not be

affected (or affected at a very small level)

however angular distributions can be

affected by different spin structure of NP

particlesI For example, at low q2, the angle between

the two decays planes, φ, is sensitive to

the photon polarisation

I The four-body system is described by threedecay angles (defined in the helicity basis)and the dimuon invariant mass squared, q2

I φ angle between the two decay planes in

the B rest-frameI θ`, θK angle between the B momentum in

the B frame and the Kπ or `+`−

momentum in their decay frameT. Blake

B0→K*0!+!− angular basis• Four-body final state.

Angular distribution provides many observables that are sensitive to NP.

e.g. at low q2 the angle between the decay planes, " , is sensitive to the photon polarisation.

• System described by three angles and the dimuon invariant mass squared, q2.

Use helicity basis for the angles.

46

K+

K0 K

µ+

µ

B0

`

(a) K and ` definitions for the B0 decay

µ

µ+

K+

B0

K0

K+

nK

pK

µ

µ+

nµ+µ

(b) definition for the B0 decay

+

KK0

µ

µ+

B0

K +

nK

pK

µ

µ+

nµµ+

(c) definition for the B0 decay

M. Kenzie 38 / 56

The B0 → K∗0µ+µ− angular distribution

I A rather complex angular distribution with many observables (which depend on

form-factors for the B → K∗ transition plus the Wilson coefficients

I The CP -averaged angular decay rate (where Ω = (θK , θ`, φ)) is

1

d(Γ + Γ)/dq2

d3(Γ + Γ)

d~Ω

∣∣∣∣P

=9

32π

[34(1− FL ) sin2 θK + FL cos2 θK

+ 14(1− FL ) sin2 θK cos 2θ`

− FL cos2 θKcos2θ` + S3 sin2 θK sin2 θ` cos 2θ

+ S4 sin 2θK sin 2θ` cosφ+ S5 sin 2θK sin θ` cosφ

+ 43AFB sin2 θK cos θ` + S7 sin 2θK sin θ` sinφ

+ S8 sin 2θK sin 2θ` sinφ+ S9 sin2 θK sin2 θ` sin 2φ

]FL fractional longitudinal polarisation of the K∗0

AFB forward-backward asymmetry of the dilepton system

S5 particularly sensitive to C9M. Kenzie 39 / 56

Form-factor “free” observables

I Several experiments have produced such an angular analysis (LHCb is the most

sensitive)

I In QCD factorisation / SCET there are only two form factorsI One is associated with A0 and the other with A‖ and A⊥

I Can then construct ratios of observables which are independent of the form-factors

(at least to leading order) e.g.

P ′5 = S5/√FL(1− FL)

I Historically there has been quite a bit of

tension between predictions and

measurement of P ′5. In the latest LHCb

measurement ([arXiv:]) this specific tension

is a bit reduced but there remains an overall

considerable tension with the SM (arising

from discrepancies in P ′5 and AFB and FL)

[arXiv:2003.04831]

0 5 10 15]4c/2 [GeV2q

1−

0.5−

0

0.5

1

5'P

(1S)

ψ/J

(2S)

ψ

LHCb Run 1 + 2016SM from DHMV

M. Kenzie 40 / 56

http://arxiv.org/abs/


Global fits

I These measurements then lead to some very nice interpretations in terms of the

Wilson coefficients with global fits to b→ s data

I Note a general pattern of consistency between experiments/measurements and data

seems to favour a modified vector coupling (CNP9 6= 0) at ∼ 4− 5σ (if you entirely

trust the theory assumptions

[arXiv:1903.09578] (updated 2020)

M. Kenzie 41 / 56


Interpretation of global fits

Pessimist’s view point

T. Blake


54

Optimist’s view point Pessimist’s view point

Vector-like contribution could come from new tree level contribution from a Z’ with a mass of a few TeV (the Z’ will also contribute to mixing, a challenge for model builders)

Vector-like contribution could point to a problem with our understanding of QCD, e.g. are we correctly estimating the contribution for charm loops that produce dimuon pairs via a virtual photon.

More work needed from experiment/theory to disentangle the two

I A vector-like contribution could point

to a problem with our understanding

of QCD

I e.g. are we correctly estimating the

contribution from charm loops that

produce dimuon pairs via a virtual

photon?

Optimist’s view point

T. Blake


54

Optimist’s view point Pessimist’s view point

Vector-like contribution could come from new tree level contribution from a Z’ with a mass of a few TeV (the Z’ will also contribute to mixing, a challenge for model builders)

Vector-like contribution could point to a problem with our understanding of QCD, e.g. are we correctly estimating the contribution for charm loops that produce dimuon pairs via a virtual photon.

More work needed from experiment/theory to disentangle the two

I Vector-like contribution could come

from a new tree-level contribution (e.g.

Z′ with m ∼ O(1) TeV)

I A Z′ should also give effects elsewhere

(e.g. particularly in mixing, which it

doesn’t) so a challenge for model

builders who need to suppress this

Which one are you?

Further work is needed from both experiment and theory to establish what is going on here

M. Kenzie 42 / 56

Lepton Universality

I In the SM ratios like

RK =

∫dΓ(B+ → K+µ+µ−)/dq2 · dq2∫dΓ(B+ → K+e+e−)/dq2 · dq2

(19)

should only differ from unity by phase space

I The dominant SM processes couple equally to the different lepton flavour (apart from

the Higgs)

I Incredibly theoretically clean since hadronic uncertainties cancel in the ratio (they

have the same hadronic matrix element). The only consideration is from small

electroweak corrections as q2 approaches 0

I Experimentally these are much more challeneging, primarily due to differences inmuon/electron reconstructionI In particular Bremsstrahlung radiation from the electronsI LHCb does not have a high resolution ECALI Electron efficiency is much poorer than muon efficiency at LHCb (trigger and

reconstruction)

M. Kenzie 43 / 56

B+ → K+`+`− candidates

I Have to correct electrons for energy loss due to Bremmstrahlung (look for ECAL

clusters (i.e. photons) associated with the electron track

I This is successful to some extent but even after Bremmstrahlung recovery there are

significant differences in mass resolution between the dielectron and dimuon final

states

5.0 5.2 5.4 5.6 5.8 6.0]2c [GeV/)-µ+µ+m(K

0

5

10

15

20

25]4 c/2 [G

eV2 q

1

10

210

310

410

510LHCb

4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0]2c [GeV/)-e+e+m(K

0

5

10

15

20

25]4 c/2 [G

eV2 q

1

10

210

310

410

510LHCb

𝐵! → 𝐽/𝜓𝐾! and 𝐵! → 𝜓 2𝑆 𝐾!resonances

Resonances are much broader in the 𝑒!𝑒" mode (note the wider scale also)

𝐵! → 𝜇!𝜇"𝐾!visible by eye

𝐵! → 𝑒!𝑒"𝐾! not visible by eye (in this plot)

M. Kenzie 44 / 56

Lepton universality results

RK from B± → K±`+`−

[arXiv:1903.09252]

]4c/2 [GeV2q0 5 10 15 20

KR

0.0

0.5

1.0

1.5

2.0

BaBarBelleLHCb Run 1LHCb Run 1 + 2015 + 2016

LHCb

RK from B± → K±`+`−

[arXiv:1705.05802]

0 5 10 15 20

q2 [GeV2/c4]

0.0

0.5

1.0

1.5

2.0

RK∗0

LHCb

LHCb

BaBar

Belle

M. Kenzie 45 / 56



Rare kaon decays

I Two new rare kaon decay experimentsI KOTO at J-PARC, searching for K0

L → π0ννI NA62 at CERN, searching for K+ → π+νν

I The advantage (theoretically) of final states with neutrinos is that there is no

contribution from quark loops involving light quarks (which can annihilate to produce

charged leptons e.g. charm loops)

I The challenege experimentally is these are incredibly rare (and contain just one

charged track in the final state)

T. Blake

Rare kaon decays• Two new rare kaon decay experiments:

KOTO at J-PARC, searching for NA62 at CERN, searching for

• The main advantage of final states with neutrinos is that there is no contribution from quark loops involving light quarks (which can annihilate to produce charged leptons).

59

K+ ! +

K0L ! 0

KOTO NA62

M. Kenzie 46 / 56

NA62

I Aim to collect a dataset of ∼ 100 K+ → π+νν decays

I Currently have ∼ 3 events in analysed data (2016+2017) giving

B(K+ → π+νν) = (4.7+7.2−4.7)× 10−11

i.e. consistent with zero

I Also search for lepton number violating K± → π∓`±`± decays

M. Kenzie 47 / 56

Checkpoint Reached

6. Lepton Flavour Violation

M. Kenzie 48 / 56

Lepton Flavour Violation

I Essentially forbidden in the SM by the smallness of the neutrino mass

B(µ→ eγ) ∝ m4ν

m4W

∼ 10−54 (20)

I Very powerful null test of the SM

I Any visible signal is a clear sign of New Physics

T. Blake

Lepton flavour violation• Essentially forbidden in SM by smallness of

the neutrino mass.

Powerful null test of the SM.

• Any visible signal would be an indication of BSM physics.

62

x µ

e

B(µ ! e) / m4

m4W

1054

I Different signatures include

1. µ→ eγ at rest (MEG at PSI, Mu2E at PSI)

2. µ→ 3e (Mu3e at PSI)

3. µ conversion in field of Au nucleus (SINDRUM II as PSI)

M. Kenzie 49 / 56

Lepton Flavour Violation in τs

I A large number of experimental signatures

I Global summary (averages) provided by HFLAV

HFLAV2018

10−8

10−6

e− γ

µ− γ

e− π

0

µ− π

0

e− K S

0

µ− K S

0

e− η

µ− η

e− η

′ (958

)µ

− η′ (9

58)

e− ρ

0

µ− ρ

0

e− ω

µ− ω

e− K

∗ (892

)µ

− K∗ (8

92)

e− K

∗ (892

)µ

− K∗ (8

92)

e− φ

µ− φ

e− f 0(

980)

µ− f 0(

980)

e− e

+ e−

e− µ

+ µ−

µ− e

+ µ−

µ− e

+ e−

e− µ

+ e−

µ− µ

+ µ−

e− π

+ π−

e+ π

− π−

µ− π

+ π−

µ+ π

− π−

e− π

+ K−

e− K

+ π−

e+ π

− K−

e− K S

0 K S0

e− K

+ K−

e+ K

− K−

µ− π

+ K−

µ− K

+ π−

µ+ π

− K−

µ− K S

0 K S0

µ− K

+ K−

µ+ K

− K−

π− Λ

π− Λ

pµ− µ

−

pµ+ µ

−

ATLAS BaBar Belle CLEO LHCb

90% CL upper limits on τ LFV decays

M. Kenzie 50 / 56

Charged LFV future

I Data taking has begun at MEG-II

(aiming for O(10−14))

I New µ→ 3e experiment (Mu3e) at

PSI

I Two new conversion experiments

(Mu2e) at PSI and (COMET) at

J-PARC

I Expect improvements for LFV τ

decays from Belle 2

M. Kenzie 51 / 56

Checkpoint Reached

7. Recap

M. Kenzie 52 / 56

New Physics?

I We have seen in these lectures the incredible success of the CKM matrix as a

predictive tool for properties of flavour decays

I Our various measurements which constrain the CKM picture are all consistent with

the SM predictions

I However, there are some very tantalising hints that could suggest New Physics

I Tension in Vub (and to a lesser extent Vcb)I Enhancement / tension in B → D(∗)τντI Anomalies in B → K(∗)`+`− decaysI Muon g-2

all at & 3σ

I These should all be resolved in the next 5-10 years

I It’s an exciting time to be a flavour physicist!

M. Kenzie 53 / 56

Recap

In this lecture we have covered

I Effective theories

I Flavour Changing Neutral Current processes

I Experimental constraints on new particles in ∆F = 1 and ∆F = 2 FCNCs

I Minimal Flavour Violation

I Lepton Flavour Violation

I Future Flavour Violation Experiments

M. Kenzie 54 / 56

Checkpoint Reached

End of Lecture 4

M. Kenzie 55 / 56

GAME OVERThanks for playing (listening)!

M. Kenzie 56 / 56

flavour physics (of quarks)€¦ · m. kenzie 3 / 56. recap last time we looked at i the ckm matrix...

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