Download Concepts and Applications of Effective Field Theories: Flavor

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Concepts and Applications of Effective Field
Methodische Vielfalt
Theories: Flavor Physics and Beyond
Breiter Bereich der physikalischen Skalen;
„von den
niedrigsten zu den höchsten Energien“
Matthias
Neubert
Mainz Institute for Theoretical Physics (MITP)
Johannes
Gutenberg
University,
Mainz, Germany
TRIGA
Ionenfallen
MAMI
LHC
IceCube
!
BCVSPIN-MSPF-MITCHELL Joint School 2014
0
4
14
-18
-13
10
10
10
10
10
Las Hadas Golf Resort & Marina ultrakalte Colima,
HadronenHiggs, SUSY,
Antiprotonen
Manzanillo,
Mexico,
8-13 December
2014 Neutrinos
Neutronen
01.02.2010
struktur
Extra-Dimensionen
Energie [GeV]
Effective Field Theory
Effective field theories are a very powerful tool in quantum field
theories:
• systematic formalism for the analysis of multi-scale problems
• simplifies practical calculations, often makes them feasible
(“Taylor expansion of Feynman graphs”)
• particularly important in QCD, where short-distance effects are
calculable perturbatively, but long-distance effects are not
• provides new perspective on renormalization
• basis of factorization (i.e. scale separation) and resummation
of large logarithmic terms
Effective Field Theory
Useful reviews:
• E. Witten, Nucl. Phys. B 122 (1977) 109
• S. Weinberg, Phys. Lett. B 91 (1980) 51
• L. Hall, Nucl. Phys. B 178 (1981) 75
• J. Polchinsky, hep-th/9210046
• A. Buras, hep-ph/9806471
• M. Neubert, hep-ph/0512222
Effective Field Theory
Lecture I:
• General concepts of EFTs
• Scale separation, integrating out high-energy modes,
low-energy effective Lagrangian
• Modern view of QFTs and general principles
Lecture II:
• Applications • The Standard Model as an effective field theory
• Interesting insights
Lecture I: Concepts of Effective Field Theory
uch calculations feasible. As we will discuss, EFT also provides a new
enormalization”.
Derivation of the effective Lagrangian
idea of EFT is simply stated: Consider a quantum field theory with
cale M. This could be the mass of a heavy particle, or some large (E
Consider a QFT with a characteristic (fundamental) high-energy
ansfer. Suppose we are interested in physics at energies E (or momenta
scale M
M. How can we expand scattering or decay amplitudes in powers of E/
We areproceeds
interested
performing
question
in in
several
steps:experiments at energies
E ⌧ M
< Mfields
Step Λ
1: <
Choose
a cutoff
and divide
quantum
a cutoff
M and
divide⇤ the
of thealltheory
intofields
low-frequ
into high- and low-frequency components ( ! > ⇤ and ! < ⇤ ):
uency modes,
φ = φL + φH ,
!
M
⇤
Recall:
the Fourier modes with frequency ω < Λ, while φH con
L contains
Z
⇣ We can think of⌘ the cutoff as a “
g modes with frequency
dω3 k> Λ.
† ik·x
ik·x
(x) =that we 3may pretend
ak e
+ know
ak e nothing about the t
ance” in the sense
to
(2⇡) 2Ek
bove Λ (which is indeed often the case). By construction, low-energ
E
bed in terms of the φL fields. Everything we ever wish to know a
Feynman diagrams, scattering amplitudes, cross sections, decay rates,
ed from vacuum correlation functions of these fields. These correlato
uch calculations feasible. As we will discuss, EFT also provides a new
fundamental
scale
This could
be orthe
mass
of a heavy
particle,
or The
som
maller
than M. How
canM.
we expand
scattering
decay
amplitudes
in powers
of E/M?
enormalization”.
nswer
this question
proceeds
in several
momentum
transfer.
Suppose
we
are
interested
in physics
at energies
E(
often to
makes
such
calculations
feasible.
Assteps:
we will
discuss, EFT
also provides
a new, modern
Derivation of the effective Lagrangian
idea
of
EFT
is
simply
stated:
Consider
a
quantum
field
theory
with
meaning
to
“renormalization”.
smaller
than
M.
How
can
we
expand
scattering
or
decay
amplitudes
p
1. Choose a cutoff Λ < M and divide the fields of the theory into low-frequency in
and
The
ideacould
of EFTbe
is simply
stated:of Consider
a quantum
a large
cale
M.main
This
the
mass
a heavy
particle,fieldortheory
somewith
large
(E
high-frequency
modes,
answer
to
this
question
proceeds
in
several
steps:
fundamental
scalea M.
This
could
be the mass of a heavy
particle, or some
large (Euclidean)
Consider
QFT
with
a characteristic
(fundamental)
high-energy
φ =inφLphysics
+ φH , at energies E (or momenta
(2)
ansfer.
Suppose
we
are
interested
momentum
transfer.
Suppose we are interested in physics at energies E (or momenta p) much
scale
M
1. Choose
a expand
cutoff
Λexpand
< modes
M
and
ofpowers
theory
where
φL M.
contains
thewe
Fourier
with
frequency
ω fields
< Λ, in
while
φpowers
contains
the
M.
How
can
we
scattering
ordivide
decay
amplitudes
inthe
ofinto
E/
smaller
than
How
can
scattering
or
decaythe
amplitudes
E/M?
The
H of
remaining
modes
with
frequency
ω > steps:
Λ.experiments
We can think at
of energies
the cutoff as
a⌧
“threshold
answer
to
this
question
proceeds
in
several
high-frequency
modes,
E
M
We
are
interested
in
performing
question proceeds in several steps:
of ignorance” in the sense that we may pretend to
know
nothing
about
the
theory
for
φof=theφLtheory
+ φHinto
, low-frequency and
1.
Choose
a
cutoff
Λ
<
M
and
divide
the
fields
scales
above
Λ (whichaiscutoff
indeed ⇤
often
the and
case).divide
By construction,
low-energy
physics
<
M
Step
1:
Choose
all
quantum
fields
a cutoff
Λ <in M
and
divide
the fields
of the
theory
into
low-frequ
modes,
is high-frequency
described
terms
of
the
φ
fields.
Everything
we
ever
wish
to
know
about
the
L
where
φ
contains
the
Fourier
modes
with
frequency
ω
<
Λ,
wh
!
>
⇤
!
<
⇤
L
into
highand
low-frequency
components
(
and
):
φ
=
φ
+
φ
,
(2
L
H cross sections, decay rates, etc.) can
theory
(Feynman diagrams, scattering amplitudes,
uency
modes,
remaining
modescorrelation
with frequency
ωof >
Λ.fields.
We These
can think
of the
cu
be where
derived
from
vacuum
functions
these
correlators
can
be
φ
contains
the
Fourier
modes
with
frequency
ω
<
Λ,
while
φ
contains
the
L
H
φ
=
φ
+
φ
,
!
L we H
of ignorance” in the sense that
may pretend to know nothing
obtained
using
remaining
modes with frequency ω > Λ. We can think of the cutoff as a “threshold
!
"
!E
"while
ofscales
ignorance”
in
the
that is
weindeed
may
pretend
to the
know
nothing
about
the
theory
fo
above
Λ sense
(which
often
case).
By
construction
#
the
Fourier
modes
with
frequency
ω
<
Λ,
φ
con
⌧
⇤
Physics
(any
Green
function)
at
low
energies
is
entirely
L contains
H
δ
δ
1often the case).
#
scales
above
Λ
(which
is
indeed
By
construction,
low-energy
physics
−i
.
.
.
−i
Z[J
]
,wish
(3)
⟨0|
T
{φ
(x
)
.
.
.
φ
(x
)}
0⟩
=
#
L
L
1
L
n
is
described
in
terms
of
the
φ
fields.
Everything
we
ever
described
in
terms
of
the
fields
;
Green
functions
of
these
g modes
with
frequency
ω
>
Λ.
We
can
think
of
the
cutoff
as
a
“
L
Z[0]
δJ
(x
)
δJ
(x
)
J
=0
L
L
1
L
n
is described in terms of the φL fields. Everything we ever wish to know about the
fields
can
be derived
from
thepretend
generating
functional:
theory
(Feynman
diagrams,
scattering
amplitudes,
cross
sections,
dt
ance”
in
the
sense
that
we
may
to
know
nothing
about
the
theory
(Feynman
diagrams,
scattering
amplitudes,
cross
sections,
decay
rates,
etc.)
can
where
$
% D fields.of
bebe
derived
from
vacuum
correlation
functions
offunctions
these
These
correlators
can
be
derived
vacuum
correlation
these
fields.
Thes
bove Λ
(which
is from
indeed
often
the
case).
By
low-energ
iS(φL ,φH )+i
d construction,
x JL (x) φL (x)
Z[JL ] = DφL DφH e
(4)
obtained
using
bed inobtained
terms ofusing
the φL fields. !Everything
we
ever
wish
to
know
a
%
" !
"
# action, D
D
is
the
generating
functional
of
the
theory.
Here
S(φ
,
φ
)
=
d
x
L(x)
is
the
δ
δ
1
L
H
!cross
" Z[J
! ]## rates,
Feynman
diagrams,
scattering
amplitudes,
sections,
decay
−i
.
.
.
−i
(3
⟨0|
T
{φ
(x
)
.
.
.
φ
(x
)}
0⟩
=
L light ,fields,
L 1 of space-time,
L n
δ
δ
1
is the dimension
andZ[0]
we have δJ
only
included
sources
J
for
the
δJL (xnL)
JL =0
L (x1 )
−i
. . . −i
⟨0| T {φtoLcompute
(x
φL (x
)} 0⟩ =functions
ed from
correlation
functions
of these
correlato
1 ) . . . the
n
as this vacuum
suffices
correlation
in (3).fields. These
%
to this question proceeds in several steps:
iS(φL ,φH )+i dD x JL (x) φL (x)
Z[JL ] =$
here
DφL DφH e
%
D
al ofDerivation
the
theory.
Here
S(φ the
, Lagrangian
φ fields
) =of the
d theory
x L(x)
the
of
effective
=
Choose
a cutoff
ΛZ[J
<L ]the
M
and
divide
intoislow-freq
is the generating functional of
is the action
%
igh-frequency
modes,
time,
and
we
have
only
included
sources
for
the
lig
is
the
dimension
of
space-time,
and
we
have
only
included
sources
for
the
light
fie
L
D JJ
L
the generating functional of the theory. Here S(φL , φH ) = d x L(x) is the action
φ =functions
φL + φHin,do
Step
2: Since
the high-frequency
fields
as this
suffices
to compute
the correlation
(3).not appear in the
%
dD x JL (x) φL (x)
% D
H
DφL DφH L
e
the theory. Here S(φL , φH ) = d x L(x)
iS(φL ,φH )+i
dimension
of space-time,
and we have in
only(3).
included sources JL for the light fiel
ethethe
correlation
functions
generating functional, we can “integrate them out” in the path
sIn
this
to compute
thethe
correlation
functions
in (3).
where
φnext
the Fourier
modes
with
ω < Λ, fields.
whileThis
φH yic
thesuffices
step,
we perform
path
integral
overfrequency
the
high-frequency
L contains
integral:
modes
with
frequency
ω >
Λ.over
We
can
think of thefields.
cutoff
asyie
a
$ integral
%the
nemaining
the the
next step,
we perform
the path
high-frequency
This
rm
path
integral
over
the
high-frequency
fields.
T
iSΛ (φL )+i dD x JL (x) φL (x)
Z[JL ]that
≡$ we
Dφmay
!
f ignorance”
in the sense
pretend to know ,nothing about the
Le
% D
$
iSΛ (φ
d x JL (x)
φL (x)
L )+icase).
cales above Λ (which
isL ]indeed
often
the
By
construction,
low-ene
Z[J
≡
Dφ
e
,
%
L
where
!
$D x J (x) φ (x)
iS
(φ
)+i
d
s[Jdescribed
in
terms
of
φ
fields.
Everything
we
ever
wish
to
know
Λthe L
L iS(φLL
L
iS
(φ
)
,φ
)
Λ
L
H
]
≡
Dφ
e
,
e
=
Dφ
e
L
L
H
here where
$ amplitudes, cross sections, decay rate
heory
(Feynman diagrams, scattering
iSΛ (φL )
iS(φL ,φH )
e
=
Dφ
e
eis derived
vacuum
correlation
fields. this
These
correla
called thefrom
“Wilsonian
effective
action”.functions
NoteHthat,ofbythese
construction,
action
depe
!
on the choice
of the cutoff
Λ used to define the split between low- and high-freque
$
btained
using
called the “Wilsonian effective action”.
µ Note that, by construction, this action depen
modes.
S
is
non-local
on
scales
∆x
∼
1/Λ,
because
high-frequency
fluctuations
h
Λ (φ )
iS
iS(φ
,φ
)
iscutoff
called
the Wilsonian
action
and high-frequen
!
"
!
"
Λ of L
Ltheeffective
H
nbeen
theand
choice
the
Λ
used
to
define
split
between
low#
e
DφThe
e
removed
from=
the theory.
process
of
removing
these
modes
is
often
referred
H µ1
δ
δ
#
modes.
S
is
non-local
on
scales
∆x
∼
1/Λ,
because
high-frequency
fluctuations
h
Λ L (x1 ) out”
−i in the functional
. . . −i
Z[JL ]#
T {φ
. . . φLthe
(xnhigh-frequency
)} 0⟩ =
as⟨0|
“integrating
fields
integral.
Z[0]
δJ
δJ
(x
1 ) condition
L the
n ) referred
een removed
from the
The process
of removing
these modes
often
⇤ enters
Dependence
ontheory.
the cutoff
viaL (x
the
onis
s “integrating
out”of
thethe
high-frequency
fields
inconstruction,
the functional integral.
ffective
action”.
Note
by
this
action
frequencies
fields that,
where
$
%
ff Λ used to define
the
split
between
iS(φ ,φ )+i dlowx J (x)and
φ (x) high-f
Z[JL ] = DφL DφH e
L
µ
H
D
L
L
Derivation of the effective Lagrangian
Step 3: Effective action is non-local on the scale t ⇠ 1/! ,
corresponding to the propagation of high-energy modes that
have been removed from the Lagrangian the final step, we expand the non-local action functional in terms of local opera
the final
step,
we
expand the
non-local
action functional
in
terms
of local opera
!
<
⇤
Since
the
remaining
fields
have
energies
,
the
non-local
mposed of light fields. This process is called the (Wilsonian) operator-product ex
mposed
of lightaction
fields. can
Thisbe
process
is called
the
(Wilsonian)
operator-product
ex
effective
expanded
in
an
infinite
series
of
local
n (OPE). This expansion is possible because E ≪ Λ by assumption. The result
n (OPE).
This expansion is possible because E ≪ Λ by assumption. The result
operators:
cast in the form
!
cast in the form
! D eff
SΛ (φL ) = d Dx LΛeff (x) ,
!
SΛ (φL ) = d x LΛ (x) ,
ere where:
here
"
" g Q (φ (x)) .
Leff
(x)
=
i i
L
LΛeff
(x)
=
g
Q
(φ
i i
L (x)) .
Λ
i
i
is object is called the “effective Lagrangian”. It is an infinite sum over local opera
his object is called the “effective Lagrangian”. It is an infinite sum over local opera
multiplied
by
coupling
constants
g
i , which are referred to as Wilson coefficients
multiplied by coupling constants gi, which are referred to as Wilson coefficients
eral, all
by
symmetries
the theory
theory
are
generated in
in
constants
localof
built out of
neral,
all operators
operators allowed
allowedcoupling
by the
the
symmetries
ofoperators
the
are
generated
(Wilson coefficients)
fields in
their
derivatives
L and
struction
of
the
effective
Lagrangian
and
appear
this
sum.
nstruction of the effective Lagrangian and appear in this sum.
Dimensional analysis
Does a Lagrangian consisting of an infinite number of
interactions and hence an infinite number of (renormalized)
coupling constants give us any predictive power?
• Not if one adopt an old-fashioned view about renormalization
and renormalizable QFTs
• But not all is lost...
Can use naive dimensional analysis to estimate the size of
individual terms in the infinite sum to a given matrix element
here of the effective Lagrangian and
uction
"appear in this sum.
eff
LΛ (x) =
gi Qi (φL (x)) .
analysis
i
ere isDimensional
always an infinite
number of such
operators, the question arises: Ho
low-energy
theorythebe“effective
predictive?
This is where
the
simple,
powerful
his
object is called
Lagrangian”.
It is an
infinite
sumbut
over
local opet
by coupling
constants
gAs
are referred
to asin
Wilson
coefficien
nsional
comes
to play.
is physics,
common
practice
high-energy
p
As isanalysis”
common
practice
in particle
we
adopt
units
i multiplied
i , which
−1 generated
eneral,
allwhere
operators
by
of =
the[x−1
theory
~ =h̄c = c1allowed
where
,=such
thatthe
in units
1. Then
[m]symmetries
= [E] = [p]
] = [tare
] are
are all
all me
onstruction
the
and
appear
in thisofsum.
measured
ineffective
the
(mass
units)
units.
We of
denote
by same
[gi ] Lagrangian
= units
−γi the
mass
dimension
the effective coupli
at
e there is always an infinite number of such
operators, the question arises: Ho
−γ
i
Denote by [gi ] =
mass
dimension
of
the
coupling
i the
g
=
C
M
i
i This is where the simple, but powerful t
ctive low-energy theory be predictive?
constants in the effective Lagrangian
dimensional
analysis”Ccomes
to play.
As is common
practice
ina high-energy
ph
sionless
coefficients
.
Since
by
assumption
there
is
only
single
funda
i
−1
−1
work
in
units
where
h̄
=
c
=
1.
Then
[m]
=
[E]
=
[p]
=
[x
]
=
[t
]asarethe
allhyp
mea
the theory,
we
expect
that
C
=
O(1).
This
assertion
is
known
Since by assumption thei theory has only a single fundamental
ame units. We denote by [gi ] = −γi the mass dimension of the effective couplin
ess”.scale
Unless
is a specific
that could explain the smallness
M, there
it follows
that: mechanism
ws that
ss numbers Ci , we should assume
those
numbers
to
be
of
O(1).
The
prese
−γ
i
g = Ci M
6
−6i
rge (e.g. 10 ) or small (e.g. 10 ) numbers in a theory would appear “unna
mensionless
coefficients
Ci . Since
by assumption
further
explanation.
where
by
naturalness
we expect
that Ci = there
O(1) is only a single fundam
in the(E
theory,
thatcontribution
Ci = O(1). This
is knownQas in
thethe
hypo
nergy
≪ Λwe<expect
M), the
of aassertion
given operator
eff
i
uralness”. Unless there is a specific mechanism that could explain the smallness
to an observable (which for simplicity we assume to be dimensionless) is ex
onless numbers Ci , we should assume those numbers to be of O(1). The prese
−6
ly large (e.g. 106 ) or small (e.g. 10⎧
) numbers in a theory would appear “unna
⎪
#
$
O(1)
;
if
γ
i = 0,
γ
⎨
i
l for further explanation.E
with dimensional
dimensionless coefficients
Ci .comes
Since by
is only a single
fundament
ive
analysis”
toassumption
play. Asthere
is common
practice
in
scale M in the theory, we expect that Ci = O(1). This assertion is known as the−1
hypothes
Dimensional
analysis
us
work
in
units
where
h̄ a=specific
c = 1.
Then [m]
= [E]explain
= [p]the=smallness
[x ] =of [tt
of “naturalness”. Unless there is
mechanism
that could
he
same units.
WeCidenote
byassume
[gi ] =those
−γinumbers
the mass
the e
dimensionless
numbers
, we should
to bedimension
of O(1). Theof
presence
6
−6
unusually
large
(e.g.
10
)
or
small
(e.g.
10
numbers in a theory
ollows
thatenergy, it follows that the) contribution
At low
of a would
givenappear
term “unnatura
and call for further explanation.
−γi
g
=
C
M
gi low
Qi to
an observable
(which
for simplicity
i
At
energy
(E ≪ Λ < M),
the contribution
of ia we
givenassume
operator to
Q be
in the effecti
i
dimensionless)
scales
like:
Lagrangian
to an observable
(which
for simplicity we assume to be dimensionless) is expect
hto dimensionless
coefficients
C
.
Since
by
assumption
there
is
only
a
i
scale as
⎧
le M in the theory, we expect
Ci =
⎪
# $γthat
O(1)
; O(1).
if γi = 0,This assertion is know
⎨
i
E
Ci is a specific
= ≪ 1 ; mechanism
(1
if γi > 0,
“naturalness”.
Unless there
that
could
explain
!
⎪
M
⎩
≫ 1 ; if γthose
i < 0. numbers to be of O(
mensionless numbers Ci , we should assume
!
It followslarge
that only
operators
whose
couplings
0 are important
low energy.
Th
usually
(e.g.
106 ) or
small
(e.g. have
10−6γ)i ≤numbers
in a at
theory
would
fact
what only
makes
the
OPE
a
useful
tool.
Depending
on
the
precision
goal,
one
m
d very
callTherefore,
forisfurther
explanation.

0
E
⌧
M
operators with i
are important for
truncate the series in (8) at a given order in E/M. Once this is done, only a finite (oft
At
low
energy
(E ≪QΛ
<
M),
the
contribution
of
a
given
operato
small)
number
of
operators
and
couplings
g
need
to
be
retained.
i
i
This is what makes the
effective Lagrangian
useful! grangian
tothrough
an observable
(which for
assume
to be Assumin
dimen
Let us go
the above arguments
oncesimplicity
again, being we
slightly
more careful.
1
weak
coupling,
we on
can the
use the
free action
to assign
scaling
behaviorthe
withinfinite
E to all fields an
scaleDepending
as
precision
goal,
oneacan
truncate
⎧
1
Interactions
canterms
change the
γ , as we
will
see
this
#dimensions
$operators
O(1)
; later.
if γFor
0,reason,
sum over
bynaive
onlyscaling
retaining
whose
value
is the γi a
i =
γi i ⎪
⎨
referred to as “anomalous dimensions”.
E
smaller than a certain value
Ci
= ≪ 1 ; if γi > 0,
⎪
M
⎩
≫ 1 ; if γi < 0.
Dimensional analysis
Since the Lagrangian has mass dimension D = dimensionality of
spacetime (the action is dimensionless), it follows that !
i
= [Qi ] = D +
i
Hence
we1:can
summarize:
Table
Classification
of operators and couplings in the effective Lagrangian
!
!
!
!
Dimension
Importance for E → 0
δi < D, γi < 0
grows
δi = D, γi = 0
constant
δi > D, γi > 0
falls
Terminology
relevant operators
(super-renormalizable)
marginal operators
(renormalizable)
irrelevant operators
(non-renormalizable)
Only a finite number of relevant and marginal operators exist!
couplings in the low-energy effective theory. Consider scalar φ4 theory as an example. The
action is
"
#
!
2
1
m 2 λ 4
D
µ
Dimensional analysis
Table 1: Classification of operators and couplings in the effective Lagrangian
Comments:
!
Dimension
Importance for E → 0
δi < D, γi < 0
grows
!
δi = D, γi = 0
constant
δi > D, γi > 0
falls
!
!
Terminology
relevant operators
(super-renormalizable)
marginal operators
(renormalizable)
irrelevant operators
(non-renormalizable)
• “relevant” operators are usually unimportant, since they are
4
couplings
in
the
low-energy
effective
theory.
Consider
scalar
φ
theory
an example.
forbidden by some symmetry (otherwise they
giveasrise
to a The
action is
#
2
hierarchy problem)
! D " 1
m 2 λ 4
µ
S=
d x
∂µ φ ∂ φ −
φ −
φ
.
2
2
4!
• “marginal” operators are
all there is in renormalizable QFTs
(11)
Using that x ∼ E −1 and ∂ µ ∼ E, and requiring that the action scale like O(1) (in units of
D
•
“irrelevant”
operators
are the
interesting
since
by δi most
the mass
dimension ofones,
an operator
Qi , then
h̄), we see that φ ∼ E 2 −1 . If we denote
about
the fundamental
scale M
γi =they
δi − D.tell
Forus
thesomething
operators in the
Lagrangian
(11) we find:
δi
γi
Coupling
Example:
- theory at weak
δi > D, γi >40
falls
(renormalizable)
irrelevant operators
coupling
(non-renormalizable)
Use the free Lagrangian to derive the mass dimension of all
4
fields
and
couplings,
assuming
the theory
uplings
in the
low-energy
effective
theory. Consider
scalarisφweakly
theory coupled:
as an example. The
tion is
!
S=
!
D
d x
"
#
2
1
m 2 λ 4
µ
∂µ φ ∂ φ −
φ − φ .
2
2
4!
(11)
sing that x ∼ E −1 and ∂ µ ∼ E, and requiring that the action scale like O(1) (in units of
D
In
D
dimensions,
If follows
we denotethat:
by δi the mass dimension of an operator Qi , then
, we see that φ ∼ E 2 −1 . it
= δi − D. For the operators D
in the Lagrangian (11) we find:
!
Hence:
• The mass
[ ]=
2
1,
[m] = 1 ,
[ ]=4
D
δi
γi
Coupling
∂µ φ ∂ µ φ
D
0
1
2D − 4 D − 4 λ ∼ Λ4−D
φ4
term is 2a relevant operator
2
φ
D−2
−2
m ∼ Λ2
• The interaction term is marginal in D=4 (relevant in D<4)
ore generally, an operator with n1 scalar fields and n2 derivatives has
"
#
"
#
D
D
δi = n1
− 1 + n2 ,
γi = (n1 − 2)
− 1 + (n2 − 2) .
2
2
(12)
δi >
>0 0
δi >
D,D,
γi γ>
i4
Example:
(renormalizable)
(renormalizable)
irrelevant
irrelevantoperators
operators
(non-renormalizable)
(non-renormalizable)
falls
falls
- theory at weak coupling
Use the free Lagrangian to derive the mass dimension
of
all
couplings in the low-energy effective theory. Consider scalar φ4 4theory as an example. The
fields
and
couplings,
assuming
the theory
uplings
in the
low-energy
effective
theory. Consider
scalarisφweakly
theory coupled:
as an example. The
action is
tion is
!
"
#
2
" 1
m2 2 λ 4 #
µ
S= D
d x 1 ∂µ φ ∂µ φ − m φ 2− λφ 4 .
S = d x 2∂µ φ ∂ φ − 2 φ −4! φ .
2
2
4!
µ
!
D
(11)
(11)
!
Using that x ∼ E −1 and ∂ ∼ E, and requiring that the action scale like O(1) (in units of
−1
D µ
sing
that
x
∼
E
and
requiring
that
thedimension
action scale
O(1) (in
we and
denote
by δi the
mass
of anlike
operator
Qi , units
then of
h̄), we see that φ ∼ D
E 2∂−1 . ∼If E,
In
D
dimensions,
−1 it follows that:
2
. If we
by δi the
dimension of an operator Qi , then
, γwe
see
that
φ
∼
E
=
δ
−
D.
For
the
operators
in denote
the Lagrangian
(11)mass
we find:
i
i
= δi − D. For the operators D
in the Lagrangian (11) we find:
!
[ ]=
δi [m] =γi1 , Coupling
[ ]=4
2
δD
γ0i
Coupling
∂µ φ ∂ µ φ
1
i
∂µ φ ∂φµ4 φ 2DD− 4 D −
0 4 λ ∼ Λ14−D
2
2
4−D
42
φ
D
−
2
−2
m
∼
Λ
2D − 4 D − 4 λ ∼ Λ
φ
Hence:
1,
D
• An operator containing
n1 fields and 2n2 derivatives
has
2
2
φ n1 scalar
D − 2fields −2
m ∼ Λ has
More generally,
an
operator
with
and
n
derivatives
2
dimension:
"
#
"
#
D with n1 scalar fields and n2 derivatives
D
ore generally, an operator
has
δi = n1
− 1 + n2 ,
γi = (n1 − 2)
− 1 + (n2 − 2) .
(12)
" 2 #
"2
#
!
D
D
δi =
nD
1 few
+ noperators
γihave
= (nγ1 −
2)
− 1 + (n2 − 2) .
(12)
1 > 2−
2,
It follows
that
for
only
≤
0.
i
• For D>2, adding
fields or derivatives
increases
the dimension!
2
2
A summary of these considerations is presented in Table 1. The common terminology
Comments
Examples of effective field theories: !
!
!
!
High-energy theory Fundamental scale
Standard Model
MW ∼ 80 GeV
MGUT ∼ 1016 GeV
GUT
String theory
MS ∼ 1018 GeV
11-dim. M theory
...
...
...
Low-energy theory
Fermi theory
Standard Model
QFT
String theory
...
5 GeV
NRQCD
! arguments justQCD
The
presented providemab ~new
perspectiveHQET,
on renormalization.
Instead o
a paradigm of renormalizable theories M
based
on the concept of systematic
“cancellations o
ChPT
ChSM ~ 1 GeV
!
nfinities”,
we should adopt the following, more physical point of view:
• SM andphysics
GUTs depends
are perturbative
QFTs
structure of the fundamental theor
• Low-energy
on the short-distance
via
relevant
and marginal
couplings,
and possibly
through some
irrelevant coupling
• Fermi
theory
contains
only irrelevant
operators
(4 fermions)
provided measurements are sufficiently precise.
• String/M theory: fundamental theory is non-local and even
• “Non-renormalizable”
interactions
not forbidden;
on the contrary, irrelevant opera
spacetime breaks
down atare
short
distances
tors always contribute at some level of precision. Their effects are simply numericall
Comments
Examples of effective field theories: !
!
!
!
High-energy theory Fundamental scale
Standard Model
MW ∼ 80 GeV
MGUT ∼ 1016 GeV
GUT
String theory
MS ∼ 1018 GeV
11-dim. M theory
...
...
...
Low-energy theory
Fermi theory
Standard Model
QFT
String theory
...
5 GeV
NRQCD
! arguments justQCD
The
presented providemab ~new
perspectiveHQET,
on renormalization.
Instead o
a paradigm of renormalizable theories M
based
on the concept of systematic
“cancellations o
ChPT
ChSM ~ 1 GeV
!
nfinities”,
we should adopt the following, more physical point of view:
• QCD at physics
low energy:
with strong
coupling,
the theor
• Low-energy
dependsexample
on the short-distance
structure
of thewhere
fundamental
degrees
of freedom
energy
(hadrons)
are
viarelevant
relevant and
marginal
couplings, at
andlow
possibly
through
some irrelevant
coupling
provided
measurements
sufficiently
precise.
different
from the are
degrees
of freedom
of QCD
• “Non-renormalizable”
interactions
are not
forbidden;yet
on ChPT
the contrary,
irrelevant opera
• Low-energy theory
is strongly
coupled,
is useful
tors always contribute at some level of precision. Their effects are simply numericall
Running couplings / Wilson coefficients
Often the fields H correspond to heavy particles, whose effects
become unimportant at low energies
But the frequency decomposition implies that high-energy
excitations of massless particles (such as gauge bosons) are
also integrated out from the low-energy effective theory
M
⇤
Consider now the situation where we lower the cutoff ⇤
without crossing the threshold for a heavy particle that
could be integrated out:
• the structure of the operators Qi in the effective
Lagrangian remains the same
• hence, the effect of lowering the cutoff must be entirely
absorbed into the values of the coupling constants gi
Follows that gi = gi (⇤) are running, ⇤-dependent parameters!
E
Modern quantum field theory
“Theorem of modesty”: • no QFT ever is complete on all length and energy scales
• all QFTs are low-energy effective theories valid in some energy
range, up to some cutoff
⇤
Giving up renormalizability as a construction criterion for
“decent” QFTs: • at low energy, any effective theory will automatically reduce to
a “renormalizable” QFT, meaning that “non-renormalizable”
interactions give rise to small contributions ~(E/M)n
• this does not make renormalization irrelevant, but it provides a
different point of view (Wilsonian picture of the RG)
Modern quantum field theory
We should forget the folklore about “cancellations of infinities”
Adopt the more physical viewpoint that: • low-energy physics depends on the short-distance
dynamics of the fundamental theory only through a small
number of relevant and marginal couplings, and possibly
through some irrelevant couplings if our measurements are
sufficiently precise
• this finite number of couplings can be renormalized (i.e.,
infinities can be removed consistently) using a finite number
of experimental data
• the criterion of “renormalizability” is automatically fulfilled
(approximately) by any effective field theory
Modern quantum field theory
We should forget the folklore about “cancellations of infinities”
Adopt the more physical viewpoint that: • contrary to the old paradigm of strictly forbidding irrelevant
interactions, we always expect them to be present and give
rise to small effects, which may or may not be observable at a
given level of accuracy
• this provides an “indirect way” to search for hints of physics
beyond the (current) Standard Model:
!
!
low-energy, high-precision measurements
• e.g.: flavor physics, neutrino physics, (g-2)μ, EDMs, darkphoton searches, …
Modern quantum field theory
Instead, relevant (“super-renormalizable”) interactions cause
problems!
2 2
Consider, e.g., the mass term m
in scalar field theory
2
2
2
m
⇠
M
⇠
⇤
Dimensional analysis suggests that UV
But then a light scalar particle should not be present in the lowenergy effective theory!
!
Hierarchy problem!
The same argument applies for all mass terms
in any QFT!
And likewise for the cosmological constant!
Victor Weisskopf
Modern quantum field theory
New paradigm: EFTs must be natural in the sense that all mass
terms should be forbidden by (exact or broken) symmetries!
Indeed: • gauge invariance: forbids mass terms for gauge fields
(photons and gluons in the Standard Model)
• chiral symmetry: forbids mass terms for fermions (all matter
fields in the Standard Model)
Explains why the SM is a (broken) chiral gauge theory!
• But the Higgs boson exists and causes a naturalness problem!
• Supersymmetry: would link the masses of scalars and
fermions and hence, in combination with chiral symmetry,
forbid mass terms for scalar fields (solves hierarchy problem)
Is nature supersymmetric?
• The Higgs boson exists and causes a naturalness problem!
• Supersymmetry: would link the masses of scalars and
fermions and hence, in combination with chiral symmetry,
forbid mass terms for scalar fields (solves hierarchy problem)
gs&und&das&Weltbild&der&Physik&
nde&Idee:&Supersymmetrie&
et&jedem&SMVTeilchen&einen&
r&unentdeckten&Partner&zu&
Die&fieberhage&Suche&nach&diesen&
vermutlich&sehr&schweren&Teilchen&
am&LHC&war&bisher&leider&erfolglos&&
Is nature supersymmetric?
Lecture II: Some applications of EFTs
Standard Model as an effective field theory
Some interesting insights can be gained by considering the
Standard Model (SM) as a low-energy effective theory of some
more fundamental theory (supersymmetry, extra dimensions,
new strongly coupled physics, GUT, ...)
We will denote the scale of New Physics by M; this could be
as large as 1016 GeV for some applications, but as small as
103 GeV (= 1 TeV) for others
The SM Lagrangian should then be extended to an effective
Lagrangian, which besides the SM terms contains additional,
irrelevant operators
These operators must respect the symmetries of the SM
(gauge invariance, Lorentz symmetry, CPT) but are otherwise
unrestricted
QϕW B
ϕ τ ϕ Wµν B QϕW BQdW ϕ τ(q̄ϕ
drB
)τ ϕ WµνQdW Qϕd(q̄p σ (ϕd
p σWµν
light particles exist, like axions or sterile neutrinos.
† I $I
µν
µν
† I(q̄ $
I d )ϕ
µν B
µνϕ
QϕW
ϕ
τ
ϕ
W
B
Q
σ
Q
i(
!
Q
ϕ
τ
ϕ
W
B
Q
(q̄
σ
"B
dB
p
r
µν
ϕud
"
dB
p
µν
µν
ϕW B
In most of beyond-SM theories that have been considered to date, reduction
low energies proceeds via
decoupling
of heavyoperators
particles
with than
masses
order Λoto
Table
2: Dimension-six
the of
four-fermion
Table 2: other
Dimension-six
operators
a decoupling at the perturbative level is described by the Appelquist-Carazzon
effective
Lagrangian
up to operator
dimension operators
D=6 reads:
This inevitably
leads to appearance
of higher-dimensional
in the SM La
3
The
complete
set
of
dimension-five
and
-si
3
The
complete
set
of
dimen
are suppressed by powers of Λ
" #
!
!
This Section
to
presenting
our
final
results
in Secs.
Section
is(6)
devoted
to presenting
final 5,
res6
1
1(derived our
1is devoted
(5) This
(5)
(6)
(4)
(5)
(6)
(6)
Ck Qof
+
Ck. operators
Qk +independence
OQ(5) 3and, Qmeans
LSM
LSM +
k Qindependent
2
of =
independent
operators
and
Q
Their
thatind
n
n
n
nΛ
n . Their
Λ k
Λ k
of them and their Hermitian
conjugates
is Hermitian
EOM-vanishing
up to is
total
der
of them
and their
conjugates
EOM(5)
(4) Imposing the SM gauge Imposing
symmetry
constraints
on
Q
out
just a
SM SM
gauge
symmetry
constraints
n leaves
where LSM is the usual “renormalizable” partthe
of the
Lagrangian.
It
contains
(n)
up to Hermitian conjugation
flavour assignments.
It reads
up remaining
toand
Hermitian
conjugation
and
flavour
assignmo
and -four operators only.1 In the
terms,
Q
denote
dimension-n
unique operator (neutrinokmasses):
(n)
j m dimensionless
k T
n
†coupling
T j m† constants
Ck stand forQthe
corresponding
coeffi
=
ε
ε
ϕ
ϕ
(l
)
Cl
≡
(
ϕ
!
l
)
!(llrpk),
Q
=
ε
ε
ϕϕ
)T Clrn ≡(Wilson
(ϕ
!† lp )T C(
ϕ
!†
νν
jk mn
p ϕC(
jk mn
p
rνν
(n)
the underlying high-energy theory is specified, all the coefficients Ck can be d
2
2
Weinberg
(1979)
where
C
is
the
charge
conjugation
matrix.
Q
violates
the
lepton
nu
where
C
is
the
charge
conjugation
matrix.
Q
νν
ν
integrating out the heavy fields.
electroweak
symmetry
breaking,
it generates
neutrino
masses
and of
mixing
electroweak
symmetry
breaking,
it generates
neu
Our goal
in this paper
is to find
a complete
set of independent
operators
dim
can
doare
the
job. Thus,
consistency
ofjob.
thesymmetr
SM (as
dimension-six
terms
can
Thus,
co
that are the
builtdimension-six
out of the SMterms
fieldsthe
and
consistent
with
the do
SMthe
gauge
andoriginal
Tab. 1)analysis
with observations
crucially
on this
dimension-five
and operators
Tab.
1) with
observations
crucially
rely on the
of such
bydepends
Buchmüller
and
Wylerdepends
[3] buttero
All the independent
operators
that
allowed
by
the St
All the
the outset.
independent
operators
the full classification
once againdimension-six
from
One ofdimension-six
theare
reasons
for
repeatin
arethat
listed
in Tabs.
3.
in the
left
of each
sho
areTheir
listednames
in
2 and
3.column
Their
names
inblock
the left
is the fact
many
linear2 and
combinations
of Tabs.
operators
listed
in Ref.
[3] vanish
by
with
generation
indices
of
theare
fermion
fields
whenever
necessary,
e.g.,whe
Q
with
generation
indices
the give
fermion
fields
Of Motion
(EOMs).
Such
operators
redundant,
i.e.ofthey
no contributi
indices are always contracted
within
the brackets,
andwithin
not displayed.
The
indices
are
always
contracted
the
brackets
matrix elements, both in perturbation theory (to all orders) and beyond [4–9].
2been
presence of2 several
EOM-vanishing
combinations
in Ref.and
[3]
has
already
In the Dirac
representation2 CIn
=the
iγ 2Dirac
γ 0 , with
Bjorken
conventp
representation
CDrell
=
iγ [21]
γ 0 , phase
with
Bjorken
Standard Model as an effective field theory
The
light particles exist, like axions or sterile neutrinos.
Standard
as theories
an effective
field
theory
In mostModel
of beyond-SM
that have been
considered
to date, reduction
The
low energies proceeds via decoupling of heavy particles with masses of order Λ o
a decoupling at the perturbative level is described by the Appelquist-Carazzon
effective
Lagrangian
up to operator
dimension operators
D=6 reads:
This inevitably
leads to appearance
of higher-dimensional
in the SM La
are suppressed by powers of Λ
" #
!
!
1
1
1
(5) (5)
(6) (6)
(4)
Ck Qk + 2
Ck Qk + O
,
LSM = LSM +
3
Λ k
Λ k
Λ
(4)
where LSM is the usual “renormalizable” part of the SM Lagrangian. It contains
X
ϕ and ϕ D
ψ ϕ
(n)
1
and
-four
operators
only.
In
the
remaining
terms,
Q
denote
59 operator (2499
flavordimension-n
q. numbers) o
k incl.
Q
f
G G G
Q
(ϕ ϕ)
Q
(ϕ ϕ)(¯l e ϕ)
!(n)
Q
f CG
G stand
G
Q for (ϕthe
ϕ)!(ϕcorresponding
ϕ)
Q
(ϕ ϕ)(q̄ udimensionless
ϕ)
!
coupling constants (Wilson coeffi
k
"
# "
#
ϕD ϕ ϕD ϕ
Q
ε W W W
Q
Q
(ϕ ϕ)(q̄ d ϕ)
(n)
Buchmüller,
Wyler
(1986)
high-energy theory is specified, all the coefficients Ck can be d
$ W underlying
Q
ε the
W
W
Hagiwara et al. (1987 & 1993)
outψ Xϕ
the heavy fields.
Xintegrating
ϕ
ψ ϕ D
Grzadkowski, Iskrzynski, Misiak, Rosiek (2010)
¯l σ e )τ ϕW
¯l γ l )
Q
ϕ ϕ GOur
G
Q
(
Q
(ϕ
i
D
ϕ)(
goal in this paper is to find a complete set of independent operators of dim
!
Q
ϕ ϕG G
Q
(¯l σ e )ϕB
Q
(ϕ i D ϕ)(¯l τ γ l )
that are built
out of! Gthe QSM (ϕfields
and are consistent with the SM gauge symmetr
Q
ϕ ϕW W
Q
(q̄ σ T u )ϕ
i D ϕ)(ē γ e )
$ Won the
analysis
ofi D such
by Buchmüller and Wyler [3] but r
Q
ϕrely
ϕW
Q original
(q̄ σ u )τ ϕ
!W
Q
(ϕ
ϕ)(q̄ γ q operators
)
Q
ϕ ϕB B
Q
(q̄ σ u )ϕ
!B
Q
(ϕ i D ϕ)(q̄ τ γ q )
the
full classification
once
again
from the outset. One of the reasons for repeatin
!
Q
ϕ ϕB B
Q
(q̄ σ T d )ϕ G
Q
(ϕ i D ϕ)(ū γ u )
is the factQ that
many linear
combinations of operators listed in Ref. [3] vanish by
Q
ϕ τ ϕW B
(q̄ σ d )τ ϕ W
Q
(ϕ i D ϕ)(d¯ γ d )
$Motion
operators
Q
ϕOf
τ ϕW
B
Q (EOMs).
(q̄ σ d )ϕ B Such
Q
i(ϕ
! D ϕ)(ū γ d ) are redundant, i.e. they give no contributi
matrix
elements,
both
perturbation
theory (to all orders) and beyond [4–9].
Table
2: Dimension-six
operators other
than the in
four-fermion
ones.
Operators
other
than
four-fermion
operators
presence
several EOM-vanishing
combinations in Ref. [3] has been already p
The complete
set of
of dimension-five
and -six operators
3
G
!
G
W
"
W
ABC
Aν
µ
Bρ
ν
Cµ
ρ
ϕ
ABC
Aν
µ
Bρ
ν
Cµ
ρ
ϕ!
IJK
Iν
µ
Jρ
ν
Kµ
ρ
IJK
Iν
µ
Jρ
ν
Kµ
ρ
2
ϕG
!
ϕG
ϕW
"
ϕW
ϕB
!
ϕB
ϕW B
"B
ϕW
3
6
3
†
†
2
†
⋆
µ
†
Aµν
†
A
µν
Aµν
†
I
µν
Iµν
†
I
µν
Iµν
µν
µν
µν
µν
† I
I
µν
µν
† I
I
µν
µν
eW
eB
p
uG
p
uW
p
uB
p
dW
p
dB
µν
r
A
r
µν
p
dG
µν
µν
µν
µν
p
µν
I
r
r
r
(1)
ϕl
(3)
ϕl
µν
r
A
µν
ϕe
I
I
µν
(1)
ϕq
r
A
I
µν
µν
(3)
ϕq
r
A
µν
ϕu
I
I
µν
ϕd
µν
ϕud
p r
†
dϕ
2
µν
p
p r
†
uϕ
µ
3
†
eϕ
2
A
µν
†
2
†
2
†
†
ϕD
4
p r
2
↔
µ
↔
†
I
µ
↔
†
µ
↔
†
µ
↔
†
I
µ
↔
†
µ
↔
†
µ
†
†
µ
p
µ
r
I µ
p
p
p
µ
µ
r
r
r
I µ
p
p
p
p
µ
µ
µ
r
r
r
r
light particles exist, like axions or sterile neutrinos.
Standard
as theories
an effective
field
theory
In mostModel
of beyond-SM
that have been
considered
to date, reduction
low energies proceeds via decoupling of heavy particles with masses of order Λ o
a decoupling at the perturbative level is described by the Appelquist-Carazzon
effective
Lagrangian
up to operator
dimension operators
D=6 reads:
This inevitably
leads to appearance
of higher-dimensional
in the SM La
are suppressed by powers of Λ
" #
!
!
1
1
1
(5) (5)
(6) (6)
(4)
Ck Qk + 2
Ck Qk + O
,
LSM = LSM +
3
Λ k
Λ k
Λ
The
(4)
where LSM is the usual “renormalizable” part of the SM Lagrangian. It contains
(L̄L)(L̄L)
(R̄R)(R̄R)
(n)
1 (L̄L)(R̄R)
and
-four
operators
only.
In
the
remaining
terms,
Q
denote
Q
(¯l γ l )(¯l γ l )
Q
(ē γ e )(ē γ e )
Q
(¯l γ l )(ē γ e )
59 operator (2499
flavordimension-n
q. numbers) o
k incl.
Q
(q̄ γ q (n)
q)
Q
γ u )(ū γ u )
Q
(¯l γ l )(ū γ u )
Ck)(q̄ γ stand
for(ū¯ the
corresponding
dimensionless coupling constants (Wilson coeffi
¯
¯
¯
Q
(q̄ γ τ q )(q̄ γ τ q )
Q
(d γ d )(d γ d )
Q
(l γ l )(d γ d )
(n)
Buchmüller,
Wyler
(1986)
Q
(¯l the
γ l )(q̄ γunderlying
q)
Q
(ē γ high-energy
e )(ū γ u )
Q
(q̄
γ
q
)(ē
γ
e
)
theory is specified, all the coefficients Ck can be d
Hagiwara et al. (1987 & 1993)
Q
(¯l γ τ l )(q̄ γ τ q )
Q
(ē γ e )(d¯ γ d )
Q
(q̄ γ q )(ū γ u )
integrating
out
the heavy
fields.
Q
(ū γ u )(d¯ γ d )
Q
(q̄ γ T q )(ū γ T u )
Grzadkowski, Iskrzynski, Misiak, Rosiek (2010)
Our goal
inγ this
Q
(ū
T u )(d¯ γpaper
T d ) Q is to
(q̄ γfind
q )(d¯ γ a
d ) complete set of independent operators of dim
γ T q )(d¯ γ T d )
that are built out of the QSM(q̄ fields
and are consistent with the SM gauge symmetr
(L̄R)(R̄L) and (L̄R)(L̄R)
B-violating
Flavor by
observables
are and
crucial
in order
to r
!
" ! of such
"
rely
on
the
original
analysis
operators
Buchmüller
Wyler
[3]
but
¯
¯
Q
(l e )(d q )
Q
ε ε (d ) Cu (q ) Cl
explore this enormous parameter space!
!
"!
"
Q
(q̄the
u )ε (q̄full
d)
Q
ε ε (q
) Cq again
(u ) Ce from the outset. One of the reasons for repeatin
classification
once
!
"!
"
Q
(q̄ T u )ε (q̄ T d ) Q
ε ε ε
(q ) Cq
(q ) Cl
is the factQ thatε many
linear
combinations
of operators listed in Ref. [3] vanish by
!
"!
"
Q
(¯l e )ε (q̄ u )
(τ ε) (τ ε)
(q ) Cq
(q ) Cl
!
"!
"
Motion
(EOMs).
are redundant, i.e. they give no contributi
Q
(¯l σ Of
e )ε (q̄
σ u) Q
ε
(dSuch
) Cu (uoperators
) Ce
matrix
elements,
Table 3: Four-fermionboth
operators. in perturbation theory (to all orders) and beyond [4–9].
Four-fermion
operators
presence
of several
combinations in Ref. [3] has been already p
isospin and colour
indices in the upper
part of Tab. 3. EOM-vanishing
In the lower-left block of that table,
ll
p µ r
s
(1)
qq
p µ r
s
(3)
qq
p µ
(1)
lq
(3)
lq
I
r
s
p µ r
s
p µ
I
r
s
µ
t
µ
t
µ I
µ
t
t
µ I
t
µ
ee
p µ r
s
uu
p µ r
s
dd
p µ r
s
eu
p µ r
s
ed
p µ r
s
(1)
ud
(8)
ud
p µ r
p µ
A
le
p µ r
s
t
lu
p µ r
s
t
ld
p µ r
s
t
qe
p µ r
s
t
(1)
qu
p µ r
s
t
(8)
qu
µ
µ
µ
µ
s
r
t
s
µ
µ
A
p µ
(1)
qd
t
(8)
qd
j
s t
j
p r
ledq
(1)
quqd
jk
k
s t
r
jk
k
s
(1)
lequ
j
p r
jk
k
s t
(3)
qqq
(3)
lequ
j
p µν r
jk
k µν
t
s
duu
(8)
quqd
j
p r
j
p
A
αβγ
duq
A
αβγ
qqu
t
(1)
qqq
αβγ
αβγ
I
jk
α T
p
jk
αj T
p
jk mn
jk
αβγ
I
βk
r
αj T
p
mn
α T
p
s
p µ r
s
αj T
p
A
r
γj T
s
k
t
γ T
s
t
βk
r
β
r
r
p µ
β
r
A
γm T
s
βk
r
γ T
s
s
µ
µ
µ
µ
µ
µ
µ
µ
t
t
t
t
t
A
t
t
A
t
n
t
γm T
s
n
t
t
colour indices are still contracted within the brackets, while the isospin ones are made explicit.
Standard Model as an effective field theory
We will discuss a couple of interesting aspects of SM physics
from the perspective of this construction:
• neutrino masses and the see-saw mechanism
• effective weak interactions in the quark sector
• anomalous magnetic moment of the muon
• proton decay
• conservation of baryon and lepton numbers
(accidental symmetries)
• Higgs production at the LHC
Neutrino masses and the see-saw mechanism
The discovery of non-zero neutrino masses is often described
as a departure from the SM But this is no longer true if we consider the SM as an effective
low-energy theory
Without a right-handed neutrino (which indeed is not part of
the SM), it is impossible to write a neutrino mass term at the
level of relevant or marginal operators
However, it is possible to write a gauge-invariant neutrino mass
term at the level of irrelevant operators of dimension ≥5:
Lneutrino
mass
g ˜T
=
lL
M
⇤
C ˜ lL
Neutrino masses and the see-saw mechanism
However, it is possible to write a gauge-invariant neutrino mass
term at the level of irrelevant operators of dimension ≥5:
!
!
Lneutrino
mass
g ˜T
=
lL
M
⇤
C ˜ lL
After electroweak symmetry breaking, this gives rise to a
Majorana mass term of the form:
!
!
Lneutrino
mass
=
v2 g T
⌫˜L C ⌫L
2M
The SM as an effective field theory predicts that neutrinos
2
should be massive, with m⌫ ⇠ v /M suppressed by the
fundamental scale of some BSM physics
Neutrino masses and the see-saw mechanism
Experiments hints at the fact that the fundamental scale relevant
for the generation of neutrino masses is very heavy, 14
!
M ⇠ 10
GeV
which is not far from the scale of grand unification
Extensions of the SM containing heavy,
right-handed neutrinos (with masses that
are naturally of order M) provide explicit
examples of fundamental theories which
yield such a Majorana mass term when
the heavy, right-handed neutrinos are
integrated out (see-saw mechanism)
Weak interactions at low energies (flavor physics)
Fermi’s description of the weak interactions at low energy is a
prime example of an effective field theory, which has provided
first evidence for the scale of electroweak symmetry breaking
At the low energies relevant for neutron β-decay, kaon physics,
charm physics or B-meson physics (few MeV - few GeV), we can
integrate out the heavy W and Z bosons as well as the top-quark
and Higgs boson from the SM
This gives rise to a low-energy effective theory containing
4-fermion interactions (Fermi theory) and dipole interactions
between fermions and the photon and gluon
This effective Lagrangian successfully describes the huge
phenomenology of flavor-changing processes
Weak interactions at low energies (flavor physics)
Example:
Lagrangian
for b→s pseudoscalar
FCNC transitions
all other B Effective
decays into two
light, flavour-nonsinglet
mesons. Using
(see
Buras
lectures
derivation)
unitarity
relation
−λt = λufor
+ λa
write
c , we
the
" into two
# mesons. Using the
all otherG B !
decays
pseudoscalar
⇤ light, flavour-nonsinglet
!
F
p = pVps Vpb p (CKM matrix elements)
√
λ
C
Ci Qi + C7γ Q7γ + C8g Q8g + h.c. ,
(1)
H
=
2Q
eff
unitarity
relationp −λ1tQ=1 +
λuC+
λ2c ,+we write
2 p=u,c
i=3,...,10
GF
√ left-handed
where H = the
from
λp C1 current–current
Qp1 + C2 Qp2 + operators
Ci Qarising
Q7γW+-boson
C8g Qexchange,
i + C7γ
8g + h.c. ,
p=u,c
i=3,...,10 operators, and Q7γ and Q8g are the
Q3,...,6 and Q7,...,102 are
QCD and electroweak penguin
Qp1,2
eff are
!
"
#
!
electromagnetic
and chromomagnetic dipole operators. They are given by
p
(1)
where Q1,2 are the left-handed current–current operators arising from W -boson exchange
p
p
=
(p̄b)
(s̄p)
,
Q
(p̄i bj )V −A
(s̄j pi )V −A ,and Q7γ and Q8g are the
Q3,...,6Qand
Q
are
QCD
and
electroweak
penguin
operators,
V
−A
V
−A
1
2 =
7,...,10
!chromomagnetic dipole operators.!
electromagnetic
and
They
by
Q = (s̄b)
(q̄q)
,
Q = (s̄ b )
(q̄ qare
) given
,
3
V −A
q
V −A
4
p
Q
= (p̄b)
Q5 =1(s̄b)
(q̄q)VV +A
−A,,
V −A V −A
q (s̄p)
3
= (s̄b)
q (q̄q)
V −A
, ,
Q7 Q
=3(s̄b)
V +A
V −A V −A
q 2 eq (q̄q)
! !
Q9 =5(s̄b)V −A V −Aq 32 eq (q̄q)
,
V −A
q
V +A
Q = (s̄b)
Q7γ
where
(q̄q)
q
j i V −A
p
Q
(s̄i )jVp+A
Q6 = (s̄i2bj=
)V (p̄
−Aj q
i )V,−A ,
−Ai bj )V
q (q̄
!
! !
i j V −A
,
!
!
3
Qi4bj=
(s̄i bj )Vq −A
Q8 = (s̄
)V −A
e (q̄ qq )(q̄j qi ),V −A ,
2 q j i V +A
!
!
3
(s̄i6bj )V −Ai j Vq −A
e (q̄ q )
,
2 q j qi V j−Ai V +A
!
Q10 = Q = (s̄ b )
(q̄ q )
,
!
!
−g
−e
3µν
3
s
µν
e
(q̄q)
,
Q
=
(s̄
b
)
ebq (q̄
,
Q
mb s̄σ
(1
+
γ
)F
b
,
Q
=
m
s̄σ
(1
+
γ
)G
,
(2)
=7 =2 (s̄b)
q
V
+A
8
i
j
V
−A
q
j qi )V +A
V −A
q
µν
8g
b
µν
5
25
2
8π
8π 2
3
Q
=
(s̄b)
,
9 = q̄ γ V(1
−A
q 2,ei,
q (q̄q)
(q̄1 q2 )V ±A
± γ5)q
j areV −A
colour
1 µ
2
!
3
Q
=
(s̄
b
)
eq (q̄j qi )ofV −A
10 e are
i the
j V −A
indices,
electricq 2charges
the,
q
!
Weak interactions at low energies (flavor physics)
Example: Effective Lagrangian for b→s FCNC transitions
(see Buras lectures for a derivation)
u
s
u
s
d
s
W
W
g
u,c,t
W
u,c,t
g
d
u
d
u
q
q
(a)
SM diagrams involving virtual
heavy-particle exchanges
contributing to the low-energy
effective weak Lagrangian
d
s
(b)
d
s
W
b
s
u,c,t
u,c,t
u,c,t
W
γ,Z
q
W
W
t
t
g,γ
γ,Z
q
q
q
(c)
d
(d)
b,s
d
s
W
W
u,c,t
t
t
u,c,t
γ,Z
b,s
W
(e)
d
l
l
(f)
Weak interactions at low energies (flavor physics)
Example: Effective Lagrangian for b→s FCNC transitions
(see Buras lectures for a derivation)
From the fact that the leading operators in the low-energy
effective theory have dimension 6, it follows that the
corresponding couplings are irrelevant and proportional to MW2,
indeed: !
!
GF
g22
p =
2
8M
2
W
The strong suppression of these contributions at low energies
explains why we refer to these interactions as the weak
interactions, even though the coupling constants of the
SU(2)L⊗U(1)Y electroweak interactions is about as large as the
electromagnetic coupling constant
discuss the most important experimental observables,
⇤ and3,
uncertainties.
the numerical
analysis.
start
• We use the information
on BIn
!section
Kuncertainties.
Bswe
! perform
form
factors
from
most precise
LCSR
In section
3,
wethe
perform
the We
numerica
which
of theoretical
uncertainties,
if underestimated,
could
account
calculation [13,
16], sources
taking into
account
the correlations
between
the uncertainties
of
which all
sources
of theoretical
uncertainties,
if underesti
within
thedi↵erent
SM.even
Weq 2then
proceed
model-independent
di↵erent formeven
factors
and at
values.
ThisSM.
iswith
particularly
important
to estimate
within
the
Wea then
proceed
with
aanalysis
modelstudying
the
allowed
regions
for
the
NP
Wilson
coefficients.
In
section
4, we
the uncertainties in angular observables
that
ratios
of form
factors.
studying
theinvolve
allowed
regions
for the
NP Wilson coeffici
model-independent findings
imply for the findings
Minimalimply
Supersymmetric
Standard
model-independent
for the Minimal
Supe
Our paper is organized
as
follows.
In
section
2,
we
define
the
e↵ective
Hamiltonian
and
for models with a new
neutral
We summarize
and conclu
forheavy
models
with gauge
a new boson.
heavy neutral
gauge boson.
We
discuss the most important
experimental
observables,
detailing
our treatment
of theoretical
Several appendices
contain
all our SM
predictions
for the observables
of int
Several appendices contain all our SM⇤predictions for
uncertainties. In section
3, we of
perform
the numerical
analysis.
We start
by investigating
our treatment
form factors,
and plotss
of constraints
on Wilson
coefficients.
our treatment of form
factors, and plots of constraints
which sources of theoretical uncertainties, if underestimated, could account for the tension
(⇤) +
even within the SM. We then proceed with a model-independent analysis beyond the SM,
2. Observables Wilson
and
uncertainties
studying the allowed regions for the NP2.
coefficients. Inand
section
4, we discuss what the
Observables
uncertainties
model-independent2.1.
findings
imply for
the Minimal Supersymmetric Standard Model as well as
E↵ective
Hamiltonian
2.1. boson.
E↵ective
Hamiltonian
for models with a new heavy neutral gauge
We summarize
and conclude in section 5.
e↵ective
Hamiltonian
for b ! s transitions
can beofwritten
as details on
4
Several
appendicesThe
contain
all our
SM The
predictions
the observables
4
e↵ectivefor
Hamiltonian
for b ! sinterest,
transitions
can be
2
our treatment of form factors, and plots of constraints
on
Wilson
coefficients.
X
4 GF
0 0
⇤ e
X
p
H
=
V
V
(C
O
+
C
O⇤i ) e+2 h.c.
4
G
i
i
F
e↵
tb ts
3
i
3
2
p Vtb Vts
16⇡
He↵ =
(Ci O
2
i
16⇡ 2
2
i
2. 2Observables
and
uncertainties
2
and we consider NP e↵ects in the following set of dimension-6 operators,
SMwe
operators:
and
consider NP e↵ects in the following set of dime
m
mb
2.1.1 E↵ective Hamiltonian O = b (s̄ P b)F µ⌫ ,
0
O
=
(s̄ µ⌫ PL b)F µ⌫
1
m
7
µ⌫ R
7 µ⌫
b
e
O7 =
(s̄ µ⌫ PR b)F , e
O
The e↵ective Hamiltonian for b ! s transitions can
be
written
as
e
µ
0
µ
O9 = (s̄ µ PL b)(`¯ `) ,
O
= (s̄ µ PR b)(`¯ `) ,
µ9
0
¯
0
O
2 X µ O9 = (s̄ µ PL b)(` 0`) ,
µ
4G
e
¯
¯
F
0
0
⇤
O
=
(s̄
P
b)(
`
`)
,
O
=
(s̄
P
b)(
`
`
p 10Vtb Vts µ L2
He↵ =
(Ci5O
Oi +=C(s̄
h.c.`¯ µ10 `) , µ R
(1)5O
i Oiµ)P+
b)(
10
5
L
16⇡
2
-1
-1
i
the complete
-4
-3
-2 Of -1
0
1 set of dimension-6 operators invariant under the strong and
-4
-3
-2
-1
0
1
Of
the
complete
set of
dimension-6
operators invariant
NP
Opposite
chirality
operators:
gauge
groups,
this
set
does
not
include
and we consider Re(C
NP
e↵ects
in
the
following
set
of
dimension-6
operators,
)
Re(C9NP )
9
gauge groups, this set does not include
m
mb
b
µ⌫
0
•NPFour-quark
operators
(includingOcurrent-current,
and (2)
elec
NP )-Re(C
0 ) plane
NP
NP ) plane
NP
0
NP
O
=
(s̄
P
b)F
,
=
(s̄ µ⌫ PLQCD
b)F µ⌫penguin,
,
Figure 5:Figure
Allowed
regions
in
the
Re(C
(left)
and
the
Re(C
)-Re(C
5: Allowed regions in the
7
9 Re(C9 )-Re(C
9
10 µ⌫ 10R) plane
9 ) plane (left) and the79Re(C9 )-Re(C
• Four-quark
(including
current-current,
e operators).
e
These operators
only operators
contribute
to the observables
considerQ
(right). (right).
The blue
contours
correspond
to the 1toand
best
regions
from
the
The
blue contours
correspond
the 21 and
2 fitbest
fit regions
fromglobal
the global
operators).
operators
contribute
to(3)th
O
=if (s̄
Ponly
b)(
`¯ µ `)mixing
,
O90These
= (s̄listed
b)(`¯ µonly
`) , through
µbranching
µ PRabove
Lthrough
sis
into
the operators
and
higher
o
fit. The fit.
green
redand
contours
correspond
to the to
1 and
regions
onlyif
Theand
green
red contours
correspond
the 12 and
2 9regions
branching
⇤
+
⇤
+
0 mixing into the
sis through
listed above
ratio data
or data
only or
data
ondata
B !onKBµ!µK angular
observables
intoPaccount.
ratio
only
µ µ angular
observables
into account.
¯ µ 5 `) ,
Ois10taken
=is(s̄taken
O10
= (s̄ µ PR b)(`¯ µoperators
(4)
µ L b)(`
5 `) ,
Weak interactions at low energies (flavor physics)
Re(C9 )
NP
Re(C10
)
A global analysis of experimental data on B ! X , B ! K ,
and Bd! K µ µ decay distributions provides information
about various operator coefficients (defined to vanish in SM):
Altmannshofer, Straub:1411.3161
Of the complete set of dimension-6 operators invariant under the strong and electromagnetic
tension
SM prediction
tension with
thewith
SM the
prediction
gauge groups, this set does not include
3
3
RK =
• Four-quark operators (including current-current, QCD penguin, and electroweak penguin
operators).
These
operators
only contribute to the observables considered in this analyThe theoretical
of the
SM prediction
is completely
negligible
compared
the current
The theoretical
error oferror
the SM
prediction
is completely
negligible
compared
to thetocurrent
sisprediction
throughand
mixing
into the operators
listed above and through higher order corrections.
experimental uncertainties. The tension between the SM
the experimental
data
+
BR(B
BR(B !
Kµ+!
µ Kµ
)[1,6]µ )[1,6] +0.090 +0.090
SM
RK =
= 0.745 0.036
± ,0.036
,
RK
+ e=) 0.745 0.074 ±0.074
+
BR(B
!
Ke
[1,6]
BR(B ! Ke e )[1,6]
SM
'RK
1.00'. 1.00 .
A first hint of New Physics?
(23) (23)
experimental uncertainties. The tension between the SM prediction and the experimental data
is driven by the reduced B +
! Kµ+ µ branching ratio, while the measured B !+ Ke+ e
Anomalous magnetic moment of the muon
In a celebrated calculation that was the birth of modern QFT,
Schwinger computed the anomalous magnetic moment of the
electron in 1948 and found:
!
!
ge
ge 2
↵
µe =
=
+ ...
, with ae =
2me
2
2⇡
How will this result be affected if the SM is considered as an
effective field theory?
Anomalous magnetic moment of the muon
In a celebrated calculation that was the birth of modern QFT,
Schwinger computed the anomalous magnetic moment of the
electron in 1948 and found:
!
!
ge
ge 2
↵
µe =
=
+ ...
, with ae =
2me
2
2⇡
How will this result be affected if the SM is considered as an
effective field theory?
Add unique dimension-5 operator ( = 5 ,
gv ¯
M2
µ⌫ F
=
µ⌫
factor v required by EWSB
1):
Anomalous magnetic moment of the muon
In a celebrated calculation that was the birth of modern QFT,
Schwinger computed the anomalous magnetic moment of the
electron in 1948 and found:
!
!
ge
ge 2
↵
µe =
=
+ ...
, with ae =
2me
2
2⇡
This adds g/M to µe and hence:
!
!
↵
gme v
ae =
+
+ ...
2
2⇡
M
me the additional term will be very small, and
As long as M
by comparing a measurement of µe with theory we can
constrain M
Anomalous magnetic moment of the muon
Analogous discussion (with me replaced by mµ ) holds for the
muon
In this case, there is presently a 3.6 σ discrepancy between
theory and experiment:
!
aSM
µ
!
aexp
⇡
µ
2.8 · 10
9
Interpreting this effect in terms of our irrelevant operator implies
that:
contains loop factor (small)
M⇠
p
g ⇥ 100 TeV
One of the best hints for BSM physics!
Proton decay
Suppose you know the gauge symmetry SU(3)c⊗SU(2)L⊗U(1)Y
of the SM but nothing else (no GUTs). What could you say
about proton decay?
The effective Lagrangian must contain at least three quark
fields (change baryon number by 1 unit) and one lepton field
(change lepton number by 1 unit)
Hence:
!
Lproton
decay
g
⇠ 2 qqq`
M
Since the lowest-dimension operators have dimension 6
(corresponding to i = 2 ), the proton can be made sufficiently
long-lived by raising the fundamental scale M into the 1016 GeV
range
Proton decay
Now imagine that you do not know about the existence of
quarks (no one has seen any) but you do know about protons
and pions
Then an effective Lagrangian giving proton decay could be:
!
Lproton
decay
⇠ g ⇡ ¯e
p
This is a marginal operator, and hence proton decay would not
be suppressed by any large mass scale!
In some sense, we see that the longevity of the proton provides
a hint for a substructure of the proton: replacing a fundamental
field by a composite of several fields raises the dimension of
the operators and hence gives rise to additional suppression
Proton decay
The same trick can be applied to other fine-tuning problems
For example, the hierarchy problem can be solved by supposing
that the Higgs boson is not an elementary scalar particle but
instead a composite of a pair of elementary fermions
If this is the case, then the Higgs mass term corresponds to a
4-fermion operator, which is irrelevant
This is the main idea of composite Higgs and technicolor
theories
Baryon and lepton number conservation
In the construction of the SM, the conservation of baryon and
lepton number is not imposed as a condition There are no corresponding U(1) symmetries of the Lagrangian
How can we understand that in nature we have not seen any
hints of baryon- or lepton-number violating processes?
Baryon and lepton number conservation
In the construction of the SM, the conservation of baryon and
lepton number is not imposed as a condition There are no corresponding U(1) symmetries of the Lagrangian
How can we understand that in nature we have not seen any
hints of baryon- or lepton-number violating processes?
The answer is that it is impossible to construct any relevant or
marginal operator that would respect the gauge symmetries of
the SM and violate baryon or lepton number!
Hence, at the level of renormalizable interactions, baryon- and
lepton-number conservation are accidental symmetries of the
SM
Higgs production at the LHC
The protons collided at the LHC contain only light quarks (u,d,
and a little bit of s), which in the SM have negligible couplings to
the Higgs boson, and gluons, which do not couple to the Higgs
boson at all
How, then, is the Higgs boson produced in pp collisions at the
LHC?
Higgs production at the LHC
The protons collided at the LHC contain only light quarks (u,d,
and a little bit of s), which in the SM have negligible couplings to
the Higgs boson, and gluons, which do not couple to the Higgs
boson at all
How, then, is the Higgs boson produced in pp collisions at the
LHC?
We can gain insight by assuming (as seems to be the case) that
the Higgs boson is lighter than the top quark
We can then construct an effective low-energy theory for Higgs
physics, in which the top quark is integrated out
Higgs production at the LHC
In this effective low-energy theory, direct couplings of the Higgs
boson to pairs of gluons and photons arise at the level of
irrelevant dimension-5 operators, with coefficients that scale
like 1/mt , e.g.:
!
!
Lhgg
yt
↵s
a
µ⌫,a
p
=
h Gµ⌫ G
2mt 12⇡
These operators appear first at one-loop
order, via the exchange of a virtual top-quark
g
t
q (n)
h
qt
(n)
q (n)
t
g
The effective hgg interaction provides the
dominant production mechanism for the Higgs
Figure 1: Effective couplings of the Higgs boson to two gluons
boson in gluon-gluon fusionofat
the LHC
KK quarks.
that it has unit area. The η-dependence of the Yukawa coupling
Summary
Effective field theories are a very powerful tool in quantum field
theory
The are of great practical use, but also provide the conceptual
tools to understand scale separation (factorization) and
renormalization in a physical and systematic way
Effective field theories are abundant, since any QFT can be
considered as an effective low-energy theory of some more
fundamental theory, which is often not yet known
Because of this fact, effective field theories provide the tools to
perform indirect searches for new physics beyond the
Standard Model
Contact: [email protected]
activities 2015
Mainz Institute for
Theoretical Physics
ScientiFic ProgrAmS
JgU campus mainz
effective theories and Dark matter
crossroads of neutrino Physics
Vincenzo Cirigliano LAnL, Richard J. Hill Univ. chicago,
Achim Schwenk, tU Darmstadt, Tim M.P. Tait Uc irvine
Steen Hannestad Aarhus Univ., Patrick Huber Virginia tech,
Alexei Smirnov mPi-K Heidelberg, Joachim Kopp JgU mainz
march 16-27, 2015
July 20-August 14, 2015
Amplitudes, motives and Beyond
Fundamental Parameters from Lattice QcD
Francis Brown iHeS Paris, Marcus Spradlin Brown Univ.,
Don Zagier mPi-m Bonn, Stefan Weinzierl JgU mainz,
Stefan Müller-Stach JgU mainz
Gilberto Colangelo Bern Univ., Heiko Lacker HU Berlin,
Georg von Hippel JgU mainz, Hartmut Wittig JgU mainz
August 31-September 11, 2015
may 26-June 12, 2015
Stringy geometry
Higher orders and Jets at the LHc
Matteo Cacciari LPtHe Paris, Paolo Nason inFn milan,
Giulia Zanderighi cern & Univ. oxford
Eric Bergshoeff Univ. groningen, Gianfranco Pradisi
University of rome „tor Vergata“, Fabio Riccioni inFn rome
„La Sapienza“, Gabriele Honecker JgU mainz
June 29-July 17, 2015
September 14-25, 2015
toPicAL WorKSHoPS
JgU campus mainz
challenges in Semileptonic B Decays
the Ultra-Light Frontier
Paolo Gambino Univ. turin, Marcello Rotondo inFn Padua,
Christoph Schwanda ÖAW Vienna, Andreas Kronfeld Fermilab,
Sascha Turczyk JgU mainz
Surjeet Rajendran Univ. Stanford, Dmitry Budker JgU mainz
April 20-24, 2015
Quantum Vacuum and gravitation
Higgs Pair Production at colliders
Manuel Asorey Univ. Zaragoza, Emil Mottola LAnL,
Ilya Shapiro Univ. Juiz de Fora, Andreas Wipf Univ. Jena
June 15-19, 2015
Daniel de Florian Univ. Buenos Aires, Christophe Grojean cern,
Fabio Maltoni Univ. Leuven, Aleandro Nisati inFn rome,
José Zurita JgU mainz
June 22-26, 2015
April 27-30 2015
For more details: http://www.mitp.uni-mainz.de
Applications via http://indico.mitp.uni-mainz.de
mainz institute for theoretical Physics
PRISMA Cluster of Excellence
Johannes Gutenberg-Universität Mainz, Germany
www.mitp.uni-mainz.de