Flavor physics and lattice quantum chromodynamics

2.4 MB·English

by  Laurent Lellouch

Flavor physics and lattice quantum
chromodynamics
2
1
0
2
n Laurent Lellouch
a
J
Centre de Physique Th´eorique 1
6
CNRS UMR 7332
2
Aix-Marseille U. and U. Sud Toulon-Var

F-13288 Marseille Cedex 9
]
t France
a
l
-
p
e
h
[

2
v
4
8
4
5
.
4
0
1
1
:
v
i
X
r
a
Summer school on “Modern perspectives in lattice QCD”
´
Ecole de Physique des Houches, August 3–28, 2009
1CPTisresearchunitUMR7332oftheCNRS,ofAix-MarseilleU.andofU.SudToulon-
Var; it is also affiliated with the CNRS’ research federation FRUMAM (FR 2291).

To Annemarie, Benjamin and Niels
Acknowledgements
I am indebted to my fellow organizers for an enjoyable collaboration in preparing
this Summer School and for unanimously designating me to give the traditional pub-
lic lecture! A school is only as good as the students who attend it are, and I would
like to thank them for their strong motivation, unrelenting questioning and ability to
put together great parties seven nights a week. I am also grateful to the other teach-
ers for preparing excellent lectures which were profitable, not only for the students.
Moreover, I wish to thank Leticia Cugliandolo for her masterful direction of the E´cole
des Houches, and to thank Brigitte Rousset and Murielle Gardette for their seamless
running of the program. Finally, the help of Antonin Portelli and Alberto Ramos in
preparing many of the Feynman diagrams in these notes, as well as the careful read-
ing of the manuscript by J´erome Charles, Marc Knecht, Thorsten Kurth, Eduardo de
Rafael and Alberto Ramos, are gratefully acknowledged. This work is supported in
partbyEUgrantMRTN-CT-2006-035482(FLAVIAnet),CNRSgrantGDR2921and
a CNRS “formation permanente” grant.
Preface
Quark flavor physics and lattice quantum chromodynamics (QCD) met many years
ago, and together have given rise to a vast number of very fruitful studies. All of
these studies certainly cannot be reviewed within the course of these lectures. Instead
of attempting to do so, I discuss in some detail the fascinating theoretical and phe-
nomenologicalcontextandbackgroundbehindthem,andusetherichphenomenology
of nonleptonic weak kaon decays as a template to present some key techniques and to
show how lattice QCD can effectively help shed light on these important phenomena.
Even though the lattice study of K ππ decays originated in the mid-eighties, it
→
is still highly relevant. In particular, testing the consistency of the Standard Model
with the beautiful experimental measurements of direct CP violation in these decays
remains an important goal for the upcoming generation of lattice QCD practitioners.
The course begins with an introduction to the Standard Model, viewed as an
effectivefieldtheory.Experimentalandtheoreticallimitsontheenergyscalesatwhich
New Physics can appear, as well as current constraints on quark flavor parameters,
are reviewed. The role of lattice QCD in obtaining these constraints is described. A
second section is devoted to explaining the Cabibbo-Kobayashi-Maskawa mechanism
for quark flavor mixing and CP violation, and to detailing its most salient features.
ThethirdsectionisdedicatedtothestudyofK ππ decays.Itcomprisesdiscussions
of indirect CP violation through K0-K¯0 mixing→, of the ∆I = 1/2 rule and of direct
CP violation. It presents some of the lattice QCD tools required to describe these
phenomena ab initio.

Contents
1 Introduction and motivation 1
1.1 The Standard Model as a low-energy effective field theory 1
1.2 Flavor physics phenomenology 6
1.3 Flavor physics and lattice QCD 8
1.4 Low-energy effective field theories of the Standard Model 10
2 Standard Model and quark flavor mixing 13
2.1 On the origin of quark flavor mixing in the Standard Model 13
2.2 Properties of the CKM matrix 14
2.3 CP violation and rephasing invariants 17
3 A lattice case study: K ππ, CP violation and ∆I =1/2 rule 21
→
3.1 K ππ phenomenology 22
3.2 K0→-K¯0 mixing in the Standard Model 26
3.3 The theory of K0-K¯0 mixing 29
3.4 Computation of bare B 33
K
3.5 Renormalization of the Standard Model ∆S =2 operator 36
3.6 Final words on K0-K¯0 mixing | | 40
3.7 Phenomenology of the ∆I =1/2 rule 41
3.8 The ∆I =1/2 rule in the Standard Model 42
3.9 Euclidean correlation functions and the Maiani-Testa theorem 45
3.10 Two-pion states in finite volume 46
3.11 K ππ in finite volume 50
→
3.12 K ππ in finite volume: a simple relativistic quantum field
→
theory example 54
4 Appendix: integral representation for Z (1;q2) 64
00
References 67
1
Introduction and motivation
1.1 The Standard Model as a low-energy effective field theory
If elementary particles were massless, their fundamental interactions would be well
described by the most general perturbatively renormalizable 1 relativistic quantum
field theory based on:
the gauge group
•
SU(3) SU(2) U(1) , (1.1)
c L Y
× ×
wherethesubscriptcstandsfor“color”,Lforleft-handedweakisospinandY for
hypercharge;
three families of quarks and leptons
•
u d e− ν
e
c s µ− ν , (1.2)
 µ
t b τ− ν
 τ
withprescribedcouplingstothegaugefields(i.e.inspecificrepresentationsofthe
gauge groups);
and the absence of anomalies.
•
InthepresenceofmassesfortheweakgaugebosonsW±andZ0,forthequarksandfor
the leptons, the most economical way known to keep this construction perturbatively
renormalizableistoimplementtheHiggsmechanism(EnglertandBrout,1964;Higgs,
1964), as done in the Standard Model (SM). However, this results in adding a yet
unobserved degree of freedom to the model, the Higgs boson.
Bycallingatheoryrenormalizablewemeanthatitcanbeusedtomakepredictions
of arbitrarily high accuracy over a very large interval of energies, ranging from zero
to possibly infinite energy, with only a finite number of coupling constants. 2 These
couplingsareassociatedwithoperatorsofmassdimensionlessorequaltofourin3+1
dimensions.
1Beyond fixed-order perturbation theory, the U(1)Y of hypercharge is trivial: the renormalized
couplingconstantvanisheswhenthecutoffoftheregularizedtheoryistakentoinfinity,anotionfirst
suggestedin(WilsonandKogut,1974).
2If one sticks to perturbation theory, the precision reached is actually limited by the fact that
perturbativeexpansionsinfieldtheoryaretypicallyasymptoticexpansions.Moreover,thetriviality
of the Higgs and U(1)Y sectors means that the cutoff, which we generically call Λ here, has to be
keptfinite.Thislimitstheaccuracyofpredictionsthroughthepresenceofregularizationdependent
corrections which are proportional to powers of E/Λ, where E is an energy typical of the process
studied. Inthat sense,only asymptoticallyfree theoriescan be fundamentalsincethey arethe only
onesthatcanbeusedtodescribephenomenauptoarbitrarilyhighenergies.
2 Introduction and motivation
Renormalizable field theories are remarkable in many ways. Consider an arbitrary
high-energy theory described by a Lagrangian (e.g. a GUT, a string theory,
UV
L
...) with given low-energy spectrum and symmetries. At sufficiently low energies this
theory is described by the unique renormalizable theory with the given spectrum and
symmetries, whose Lagrangian we will denote . Moreover, the deviations between
ren
L
the predictions of the two theories can be parametrized through a local low-energy
effective field theory (EFT)
C
= + d,i O(d) , (1.3)
LUV Lren Λd−4 i
d≥4 i i
(cid:88)(cid:88)
wheretheO(d) areoperatorsofmassdimensiond 4builtupfromfieldsof .The
i ≥ Lren
Λ are mass scales which are much larger than the masses in the spectrum of –
i ren
L
theremaybeoneormanyofthemdependingonthenumberofdistinctscalesin .
UV
L
The C are dimensionless coefficients whose sizes depend on how the corresponding
d,i
operators are generated in the UV theory, e.g. at tree or loop level.
Thus, very generally, we can write down the Lagrangian of particle physics as a
low-energy EFT with the gauge group of Eq. (1.1), the matter content of Eq. (1.2)
and a Higgs mechanism:
1 C
eff = + O(5) + d,i O(d) , (1.4)
LSM LSM M Maj Λd−4 i
d≥6 i i
(cid:88)(cid:88)
where the left-handed neutrino Majorana mass term, O(5) , and the O(d) must be
Maj i
invariantundertheStandardModelgaugegroup(1.1).InEq.(1.4), istherenor-
SM
L
malizable Standard Model Lagrangian
= + + + . (1.5)
SM g+f flavor EWSB ν
L L L L L
where contains the gauge and fermion kinetic and coupling terms, , the
g+f flavor
L L
Higgs-Yukawa terms, , the Higgs terms and , the possible renormalizable
EWSB ν
L L
neutrino mass and right-handed neutrino kinetic terms. In that sense, the renormal-
izable Standard Model is a low-energy approximation of a more complete high-energy
theory involving scales of New Physics much larger than M .
W
Schematically, the gauge and fermion Lagrangian reads
1
= Fa Fµν +ψ¯D/ψ . (1.6)
Lg+f 4 µν a
It has 3 parameters, the gauge couplings (g ,g ,g ), and is very well tested through
1 2 3
experiments conducted at LEP, SLC, the Tevatron, etc. Its parameters are known to
better than per mil accuracy.
The Higgs-Yukawa terms are given by
= ψ¯(−1/2)Y φ†ψ ψ¯(1/2)Y φ˜†ψ +h.c. , (1.7)
Lflavor − R (−1/2) L− R (1/2) L
with ψ corresponding to the left-handed SU(2) doublets and ψ(±1/2) the right-
L L R
handed SU(2) singlets, associated with the I = 1 component of the doublets.
L 3 ±2
The Standard Model as a low-energy effective field theory 3
In this equation, φ is the Higgs field and φ˜ its conjugate, (φ0, φ+∗). The flavor
−
componentoftheStandardModelLagrangianhasmanymorecouplings,13infact.It
gives rise to the 3 charged lepton masses, 6 quark masses and the quark flavor mixing
matrix which has 3 mixing angles and 1 phase. 3 The understanding of this quark
mixing and its associated CP violation will be the main focus of the present course.
There is also the electroweak symmetry breaking (EWSB) contribution
=(D φ)†(Dµφ) µ2φ†φ λ(φ†φ)2 . (1.8)
EWSB µ
L − −
It has only 2 couplings, the Higgs mass and self-coupling (µ,λ), and is very poorly
tested so far, a situation which will change radically with the LHC.
As for the neutrino Lagrangian, little is known from experiment about its form.
There are theoretically two possible, nonexclusive scenarios:
1. There are no right-handed neutrinos in sight. Thus, we give our left-handed neu-
trinos a mass without introducing a right-handed partner. In that case, = 0
ν
L
in Eq. (1.5) and we have a Majorana mass term for the left-handed neutrinos in
Eq. (1.4), with
1
O(5) = νTCφ˜TALφ˜ν +h.c. , (1.9)
Maj −2 L ν L
where C is the charge conjugation matrix (see Eq. (2.22)). That is, after EWSB
the neutrino acquire a Majorana mass through the introduction of a nonrenor-
malizable dimension-5 operator. This implies that the Standard Model is an
EFT and that we already have a signal for a new mass scale. Indeed, with
m 0.1 eV (a plausible value), eigenvalues of the coupling matrix AL of or-
ν ∼ ν
der 1 and φ 246GeV, one finds for the mass scale M of Eq. (1.4)
(cid:104) (cid:105)∼
φ 2
M (cid:104) (cid:105) 1015GeV (1.10)
∼ m ∼
ν
which is tantalizingly close to a possible unification scale.
2. Wechoosetoallowright-handedneutrinos,N .Theseneutrinosmustbesinglets
R
under the Standard Model group. Thus, they themselves may have a Majorana
mass, but this time a renormalizable one, in addition to allowing the presence of
a Dirac mass term:
1
=N i∂/N (L¯ Y†φ˜N + NTCMRN +h.c.) (1.11)
Lν R R− L ν R 2 R ν R
1 0 Y†φ0 ν
=N i∂/N (νT,NcT)C ν L + , (1.12)
R R− 2 L R Y∗φ0 MR Nc ···
(cid:18) ν ν (cid:19)(cid:18) R(cid:19)
where L stands for the left-handed lepton doublets, and Nc for the charge
L R
conjugate of N (see Eq. (2.22)).
R
There are here three more possibilities:
a) MR =0
ν
In that case, the three neutrinos have Dirac masses and lepton number is
conserved.
3Rememberthatwehaveseparatedoutinto ν possiblerenormalizableneutrinomassterms.
L