University of LjubljanaFaculty of Mathematics and Physics

Department of Physics

Seminar

Flavour-Changing NeutralCurrents

Marko Petric

Advisor: Prof. Dr. Peter Krizan

25th May 2008

Abstract

In this paper we will present the theory of Flavour-Changing NeutralCurrents and the possible implications for the discovery of new particles.We also present a search for such currents in decays of B0

s → µ+µ− andB0 → µ+µ− , which has direct implications for the constraints of newtheories, especially Supersymmetry.

Contents

1 Introduction 1

2 Theory 12.1 Electroweak interaction . . . . . . . . . . . . . . . . . . . . . . . 1

2.1.1 Charged currents . . . . . . . . . . . . . . . . . . . . . . . 22.1.2 Neutral currents . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 The quark mixing matrix . . . . . . . . . . . . . . . . . . . . . . 32.3 The Cabibbo-GIM scheme . . . . . . . . . . . . . . . . . . . . . . 32.4 Why study FCNC? . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 New physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Experiment 73.1 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Conclusion 12

1 Introduction

The Standard Model (SM) is a gauge theory that describes three out of fourfundamental interactions which make up all of the matter known to us. It unifieselectroweak theory and quantum chromodynamics into a structure denoted bySU(3)C ⊗ SU(2)L ⊗ U(1)Y . With the help of quantum field theory, whichunifies quantum mechanics and special relativity, it proved itself to be a greatinstrument for describing current measurements that are in the reach of today’senergies [1]. In the face of this quality of the SM, it is palpable that this cannotbe the final solution, because it does not take into account certain observedphenomena like massive neutrinos, gravity, . . . , and because it has eighteennumerical parameters that must be put into the theory “by hand”. Thus themajority of the particle physics community is in the search for physics beyondthe SM.

Till now we were unable to produce energies required (on the TeV scale)to directly study the processes beyond the SM. Henceforth we choose to studyrare processes and try to compare them with SM predictions. The idea behindsuch an approach is that rare processes should be very sensitive to new physicsand such deviations could help us to constrain parameters on new theories. Theusual field of study of rare decays are heavy mesons such as B0, D0, . . . or heavybaryons. Usually the field of study are mesons, because the experiments andthe calculations can be done more easily.

This paper will give an overview of the Glashow-Iliopoulos-Maiani (GIM)and Cabibbo-Kobayashi-Maskawa (CKM) mechanisms. Specifically, it will fo-cus on flavour-changing neutral currents (FCNC) – such decays are highly sup-pressed in the SM and are a good starting point for searches of new physics.

1

2 Theory

In this paper we shall not cover quantum chromodynamics and will focus onthe electroweak sector.

2.1 Electroweak interaction

Low-energy experiments provide us with a large amount of valuable informationabout the dynamics of flavour-changing processes. Most notably, the analysisof the angular distribution of β decays such as µ− → e−νe νµ or n→ p e−νeestablished that only the left-handed (right-handed) fermion (antifermion) chi-ralities take part in these weak transitions. It was also established that thestrength of the interaction appears to be universal. This was further corrob-orated through the study of processes like π− → e−νe or π− → µ−νµ ,which show that neutrinos have left-handed chiralities while anti-neutrinos areright-handed.

Data from neutrino scattering allowed physicists to deduce not only the ex-istence of different neutrino types (νe 6= νµ), but also that there are separatelyconserved lepton quantum numbers which distinguish neutrinos from antineu-trinos. Thus we observe the transitions νe p→ e+n , νe n→ e−p , νµ p→ µ+nor νµ n → µ−p , but we do not see processes like νe p 6→ e+n , νe n 6→ e−p ,νµ p 6→ e+n or νµ n 6→ e−p .

Taking into account considerations related to unitarity, for a proper high-energy behaviour and the absence of flavour-changing neutral-current transitionsat tree level (µ− → e−e−e+ , Γi/Γ < 10−12 ), this low-energy information wassufficient to formulate the electroweak theory.

2.1.1 Charged currents

W±µ!

!µ

!e

e!

W±

e!

!µ

!e

µ!

Figure 1: Tree-level Feynman diagrams for µ− → e−νe νµ and νµ e− → µ−νe.

We can divide electroweak interaction basically into two groups. The firstgroup is characterised by interactions of quarks and leptons with the W± bosons

2

(Fig. 1). In such interactions only left-handed fermions and right-handed an-tifermions couple to the W± bosons. As a consequence we observe a 100%breaking of parity P (left ↔ right) and charge conjugation C (particle ↔ an-tiparticle). However, the combined transformation CP is still a good symmetry.Furthermore, the W± bosons couple to the fermionic doublets. All fermion dou-blets couple to the W± bosons with the same strength.

2.1.2 Neutral currents

!, Z0

e!

e!

µ!

µ!

Z0

e!

e!

!e

!e

Figure 2: Tree-level Feynman diagrams for e+e− → µ+µ− and e+e− → νe νe.

The second group of electroweak interactions is the interaction of Z0 bosonswith quarks and leptons. Couplings of this interaction (Fig. 2) have interactingvertices that conserve flavour. Both the γ and the Z couple to a fermion andits own antifermion, i.e., γ f f and Z0 f f . Transitions of the type µ 6→ eγ orZ 6→ e±µ∓ have never been observed [3].

The interactions depend on the fermion electric charge Qf . Fermions withthe same Qf have the same universal couplings. Neutrinos do not have elec-tromagnetic interactions (Qν = 0), but they have a non-zero coupling to theZ0 boson. Photons have the same interaction for both fermion chiralities, butthe Z0 couplings are different for left-handed and right-handed fermions. Theneutrino coupling to the Z0 involves only left-handed chiralities.

2.2 The quark mixing matrix

Leptons and quarks participate in weak interactions through charged currentsconstructed from pairs of left-handed fermion states: (νe, e−) , (νµ, µ−) and(u, d). All these charged currents couple with a universal coupling constantG. If we now want to extend this to the doublet (c, s) and (t, b) with the sameuniversal coupling we are struck by a problem. Since we know that decay modeslike K+ → µ+νµ exist (K+ is made of u and s), there has to be a weak currentbetween quarks up and anti-charm, which is contrary to our beginning assump-tion. Here physicists came to the conclusion that the weak interaction couplesto the linear superposition of mass eigenstates.

(d′, s′, b′) = V(d, s, b) . (1)

3

This introduces unitary matrix V, which is the Cabibbo – Kobayashi – Maskawa(CKM) mixing matrix [4]. Instead of defining the current as Jµq = uuγ

µ 12

(1− γ5

)ud

we define the current as Jµ =(u, c, t

)γµ 1

2

(1− γ5

)V (d, s, b)T . The corre-

sponding matrix element can now be written as M = 4G√2JµJ†µ.

W+

d

u

Vud

W+

s

u

Vus

Figure 3: Feynman diagrams with CKM weights on weak interaction vertices

We can now also easily construct amplitudes for semileptonic decays fromthe product of quark and lepton currents.

A reader might ask why there is no mixing matrix in the leptonic sector. Theanswer is simple: because the neutrino has zero mass (in the SM), it follows thatthere is no difference between weak and mass eingenstates, thus making mixingunobservable.

2.3 The Cabibbo-GIM scheme

In 1970 Glashow, Iliopoulos and Maiani proposed a mechanism [5] that wouldassure that there are no transitions that would change flavour but not charge.All the experimental evidence at the time and today points to this conclusion,as we do not observe processes like K0 → µ−µ+.

To see where this condition come from we rewrite (1) as

d′i =∑j

Vijdj , (2)

denoting d1 ≡ dL, d2 ≡ sL, etc.; here L denotes left-handed quark states. Thematrix V should be unitary if we want to follow the universal weak couplinghypothesis. Thus we can write∑

i

d′id′i =

∑i, j, k

djV†jiVikdk =

∑i

didi . (3)

We see that d → d and s → s transitions are allowed, but s → d transitionswhich alter the flavour are forbidden.

4

Z0

s

d

Z0

µ!

e!

Figure 4: Feynman diagrams for forbidden flavour changing neutral currents inthe leptonic and fermionic sector

The matrix V is unitary with dimensions 3× 3 and is characterised by fourreal parameters. We can omit one overall phase which is not measurable. Ifwe compare this a to an orthogonal 3 × 3 matrix, which has just three realparameters, we see that with rendering the quark phases we will not be able toobtain a real matrix. Therefore we must also include a phase of the form eiδ.

Another good explanation for the absence of FCNC is the fact that the SMLagrangian corresponding to the Z0 boson current is proportional to ZµuLγµuL.The Dirac matrix does not affect the flavour structure, so if we would applyunitary transformations, the matrices would simply cancel out – hence thereare no transitions between quarks of different flavours.

In higher order terms we can also encounter FCNC. With the help of aW± boson in the loop we can create an overall flavour-changing neutral process(Fig. 5). One endpoint of the W± gives a factor of Vbi while the other gives afactor of Vib, depending upon which u quark it couples to.

b

!

ui

W±

d

W±

"!

!

Vbi

Vid

Figure 5: FCNC with higher order electroweak terms

The amplitudes of such diagrams are proportional to∑iV†biVid. We know

that the CKM matrix is unitary and thus the product is a unit matrix. How-

5

ever, the amplitudes are proportional to the off diagonal elements, which wouldbe zero in this case. In our summation through all diagrams we did not in-clude the masses of the particles, so their propagators are different. The properapproximation of the amplitude would be

g4 14π2

1M2W±

(∑i

V†biVidmi

)2

, (4)

where we have used Taylor expansion for mi � MW± . FCNC at one loopare suppressed by the smallness of the quark mass degeneracies. The up-typequarks masses form a matrix in flavour space, which is not proportional to theunit matrix. This has as the result that when we sum over i the result will notbe proportional to unity. We can write the flavour structure of non-degeneratequarks in the form ∑

i, j

V†biMijVjd 6= δbd , (5)

where M is used to denote the general flavour structure from the up-type quarkpropagator. We have established that the mass dependence of the up-quarkpropagator in the loop allows us a way to permit loop-level flavour-changingneutral currents.

2.4 Why study FCNC?

As already mentioned, such processes are very rare and if there would be newmassive particles, they could drastically alter the amplitudes. Furthermore it isimportant to mention that the most popular candidates for beyond the standardmodel are Supersymmetry(SUSY), Technicolour,. . . have problems with FCNC.

In SUSY for every type of boson there exists a corresponding type of fermion,and vice-versa. Since superpartners of the particles of the Standard Model havenot been observed, supersymmetry, if it exists, must be a broken symmetryallowing the sparticles to be heavy. If supersymmetry exists close to the TeVenergy scale, it allows the solution of two major puzzles in particle physics. Oneis the hierarchy problem and the other is the unification of the weak interactions,the strong interactions and electromagnetism.

SUSY is often touted as being superior in that there is no FCNC prob-lem. This is only partially correct. To avoid FCNCs, it must be true that thesquark mass matrices are approximately diagonal in the same basis that thecorresponding quark mass matrices are diagonal. In addition, since the domi-nant contributions to the squark masses arise from soft-SUSY breaking terms,one finds that the squarks must be roughly degenerate in mass [9]. All thisconditions are signs of fine tuning, which a serious physicist wants to avoid.

We see that FCNCs do not only offer a possibility for the search of newparticles, they are also important for the benchmark of new theories.

6

µ!

e!

S

Figure 6: A diagram showing a possible FCNC in Supersymmetry involving asquark

2.5 New physics

In this section we will give a short account of how to measure possible newparticles. Instead of direct production of these particles, we are looking forquantum effect, i.e. we will measure their influence in virtual loops.

The only problem with such an approach is that for large mass differences,this is suppressed by the momentum conservation. The solution to this problemis that new particles radiate away the extra momenta, so that processes canoccur on-shell and we can measure the effects of new physics on the branchingratio. Such a process is depicted in (Fig. 7).

W±

b s

t t

!, f !, f

new

b s

!, f !, f

Figure 7: Penguin diagram for quantum fluctuations involving observed particles[Left] and hypothetical particles [Right]

3 Experiment

We will present the FCNC decays of B0s (B0) → µ+µ−, which occur in the

SM through higher order diagrams (Fig. 8). The SM expectations for these

7

branching fractions are B(B0s → µ+µ−) = (3.42 ± 0.54) × 10−9 and B(B0 →

µ+µ−) = (1.00±0.14)×10−10 [12, 13], which are one order of magnitude smallerthan current experimental sensitivity.

Many new-physics models contain mechanisms to increate B0s (B0)→ µ+µ−.

In SUSY models, contributions from diagrams including supersymmetric parti-cles (Fig. 9) can increase B(B0

s (B0)→ µ+µ−) by several orders of magnitude atlarge tanβ, the ratio of vacuum expectation values of the Higgs doublets [14, 15].In the minimal supersymmetric standard model (MSSM), the enhancement isproportional to tan6β.

Global analyses including all existing experimental constraints suggest thatthe large tanβ region is of interest [16, 17, 18, 19]. In contrast, SUSY R-parityviolating models [16] and non-minimal flavour violating models [20] can bothenhance B0

s → µ+µ− and B0 → µ+µ− separately even at low tanβ.In the absence of an observation, limits on B(B0

s → µ+µ−) are comple-mentary to those provided by other experimental measurements, and togetherwould significantly constrain the allowed supersymmetric parameter space. Forexample, if the lightest neutralino in SUSY models is a cold dark matter (CDM)particle, B(B0

s → µ+µ−) and constraints on the amount of CDM in the uni-verse from cosmic microwave anisotropy measurements can be exploited in thisway [16, 17, 19]. Then, for instance, in minimal supergravity (mSUGRA) mod-els limits on B(B0

s → µ+µ−) will correspond to bounds on superpartner particlemasses that are beyond the sensitivity of the corresponding direct searches forthose particles in colliding-beam experiments [16]. In general, the search forthese rare decays is central to exploring a large class of new-physics models.

s µ

b µ

sµ

b

Z0

µ

sµ

b

h

µ

Figure 8: Feynman diagrams that contribute to B0s → µ+µ−

s µ

d

!0

b

!0

"

µ

sµ

g

d

b

dZ0

µ

sµ

A0

d

b

dA0

µ

Figure 9: Feynman diagrams that contribute to B0s → µ+µ−that are specific

for MSSM

8

3.1 Detector

At the moment, the only detector performing measurements of this kind isthe CDF II detector 10. The detector [21, 22] has a silicon vertex detector(SVX II) [23], located immediately outside the beam pipe, provides precisethree-dimensional track reconstruction and is used to identify displaced verticesassociated with b and c hadron decays. The momentum of charged particles ismeasured precisely in the central outer tracker (COT) [24], a multi-wire driftchamber that sits inside a 1.4 T superconducting solenoidal magnet. Outside theCOT are electromagnetic and hadronic calorimeters arranged in a projective-tower geometry, covering the pseudo-rapidity region |η| < 3.5. Drift chambersand scintillator counters in the region |η| < 1.5 provide muon identificationoutside the calorimeters. Pseudorapidity is a spatial coordinate describing theangle of a particle relative to the beam axis. It is defined as η = − ln

[tan

(θ2

)].

Figure 10: The CDF II detector [22]

The SVX II consists of double-sided micro-strip sensors arranged in fiveconcentric cylindrical shells. The detector is divided into 3 contiguous five-layersections along the beam direction for a total z coverage of 90 cm. Each barrelis divided into twelve azimuthal wedges of 30◦ each. Each of the five layersin a wedge is further divided into two electrically independent modules calledladders. There are a total of 360 ladders in the SVX II detector.

The COT is the main tracking chamber in CDF. It is a cylindrical driftchamber segmented into eight concentric superlayers filled with a mixture of50% Argon and 50% Ethane. The active volume covers |z| < 155 cm and 40 to140 cm in radius. Each superlayer is sectioned in φ into separate cells. A cellis defined as one sense plane with two adjacent grounded field sheets. In themiddle of the sense planes, a mechanical spacer made of polyester/fiber glassrod is epoxied to each wire to limit the stepping of wires out of the plane dueto electrostatic forces.

The CDF central muon detector (CMU) [25] is located outside of the centralhadron calorimeter at a radius of 347 cm. The calorimeter is formed from 48

9

wedges, 24 on the east (positive z), and 24 on the west (negative z), each wedgecovering 15◦ in φ. The calorimeter thickness is about 5.5 interaction lengthsfor hadron attenuation. The muon drift cells with seven wires parallel to thebeamline are 226 cm long and cover 12.6◦ in φ. The pseudorapidity coveragerelative to the center of the beam-beam interaction volume is 0.03 < |η| <0.63. Each wedge is further segmented azimuthally into three modules. Eachmodule consists of four layers of four rectangular drift cells. The sense wiresin alternating layers are offset by 2 mm for ambiguity resolution. The smallestunit in the CMU, called a stack, covers about 1.2◦ and includes four drift cells,one from each layer.

A second set of muon drift chambers is located behind an additional 60 cmof steel. The chambers are arranged axially to form a box around the centraldetector. This system is called the CMP, and muons which register a tracksegment in both the CMU and the CMP are called CMUP muons.

Luminosity is measured using low-mass gaseous Cherenkov luminosity coun-ters (CLC) [26, 27]. There are two CLC modules in the CDF detector installedat small angles in the proton and antiproton directions. Each module consistsof 48 long, thin conical counters filled with isobutane gas and arranged in threeconcentric layers around the beam pipe.

3.2 Analysis

Events are recorded for subsequent analysis if they have either of two require-ments:

• events where both muon candidates are triggered using the central muondetectors,

• events where one of the muons is triggered in the central muon detectorand one in the higher pseudorapidity region.

In the further analysis we have a series of requirements. We select twooppositely charged muon candidates within a dimuon invariant mass window of4.669 < mµµ < 5.969 GeV/c2 around the B0

s and B0 masses. Backgrounds fromhadrons misidentified as muons are suppressed by selecting muon candidatesusing a likelihood function. In addition, backgrounds from kaons that penetratethrough the calorimeter to the muon system or decay in flight outside the driftchamber are further suppressed by a loose selection based on dE/dx [28].

To reduce combinatorial backgrounds, the muon candidates are required tohave transverse momentum relative to the beam direction pT > 2.0(2.2) GeV/cfor CMU(CMX), and |~p µµT | > 4 GeV/c, where ~p µµT is the transverse componentof the sum of the muon momentum vectors.

The remaining pairs of muon tracks are fit under the constraint that theycome from the same three-dimensional (3D) space point. To enhance signaland background separation a neural network(NN) discriminant, νN , was con-structed, based on all the discriminating variables except mµµ, which is used todefine signal and sideband background regions.

10

51015202530

< 0.95N!0.8 < sBdB

2Ca

ndid

ates

/24

MeV

/c

2

4

6

8

10

< 0.995N!0.95 < sBdB

)2Candidate Mass (GeV/c4.8 5 5.2 5.4 5.6 5.8

2468

10

< 1.0N!0.995 < sBdB

Figure 11: The µ+µ− invariant mass distribution for events satisfying all selec-tion criteria for the final three ranges of νN .

For measuring efficiencies, estimating backgrounds, and optimizing the anal-ysis, samples of B0

s (B0)→ µ+µ−, B+ → J/ψK+, and B → h+h− are generatedwith the pythia simulation program [29] and a CDF II detector simulation.Muon reconstruction efficiencies are estimated as a function of muon pT usingobserved event samples of inclusive J/ψ → µ+µ− decays. Systematic uncertain-ties in the efficiency ratio largely cancel with the exception of the kaon efficiencyfrom the B+ decay. The uncertainty is dominated by kinematic differences be-tween inclusive J/ψ → µ+µ− and B0

s (B0)→ µ+µ− decays. The efficiency, εN ,is estimated from the simulation. A relative systematic uncertainty is assignedon εN of 6% based on comparisons of NN performance in simulated and ob-served B+ → J/ψK+ event samples and the statistical uncertainty on studiesof the B0

s pT and I distributions from observed B0s → J/ψφ event samples.

The expected background is obtained by summing contributions from thecombinatorial continuum and from B → h+h− decays. The contribution fromother heavy-flavour decays is negligible. The B → h+h− contributions are abouta factor of ten smaller than the combinatorial background and are estimatedusing efficiencies taken from simulations. The two-body invariant mass distribu-tion of the simulated B → h+h− candidates is calculated from the momentumof the hadrons assuming the muon mass hypothesis. The background estimatesare cross-checked using three independent control samples: µ±µ± events anda misidentified muon-enhanced µ+µ− sample by requiring one muon candidate

11

to fail the muon quality requirements. Then a comparison is done between thepredicted and observed number of events in these samples for a wide range ofνN requirements and no significant discrepancies were observed.

The signal region is divided into five equal mass bins of 24 MeV/c2 andthree νN bins delineated at 0.8, 0.95, 0.995 and 1.0. Summing over the massbins in each slice of νN , the corresponding εN s are estimated to be 12%, 23%,and 44% and the expected SM yields of B0

s → µ+µ− events are 0.08 ± 0.03,0.15 ± 0.05, and 0.30 ± 0.10, respectively. The expected yield of B0 → µ+µ−

events is ten times smaller. Using these optimized selection criteria, a expectedlimit of B(B0

s → µ+µ−) < 4.9× 10−8 at 95% C.L. was computed.The uncertainty on the background estimate is dominated by the statistical

uncertainty of the sideband sample. The µ+µ− invariant mass distributions forthe three different νN ranges are shown in Fig. 11. The observed event ratesare consistent with SM background expectations. At 95% (90%) C.L. limits onecan extract:

Channel Bound (95% CL) SM Prediction

B0 → µ+µ− < 5.8× 10−8 (1.00± 0.14)× 10−10

B0s → µ+µ− < 1.8× 10−8 (3.42± 0.54)× 10−9

4 Conclusion

In this paper we have presented the theory of FCNC and the possible implica-tions for the discovery of new particles. We also presented a search for FCNCin B0

s and B0 mesons, which has direct implications for the constraints of newtheories [14, 15, 16, 17, 18, 19, 20] . Sadly, measurements have not yieldedresults which would give any conclusive results.

However, we anticipate that in the near future there will be enough datagathered and new detectors built, so that we will know which path high energyphysics will go.

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