On the validity of the effective field theory approach to SM precision tests
Received: May
On the validity of the e ective eld theory approach to SM precision tests
Roberto Contino 0 1 2
Adam Falkowski 0 1
Florian Goertz 0 1 2
Christophe Grojean 0 1
Francesco Riva 0 1 2
Lausanne 0 1
Switzerland 0 1
Geneva 0 1
Switzerland 0 1
0 Notkestrasse 85 , D-22607 Hamburg , Germany
1 Bat. 210, Universite Paris-Sud , 91405 Orsay , France
2 Theoretical Physics Department , CERN
We discuss the conditions for an e ective eld theory (EFT) to give an adequate low-energy description of an underlying physics beyond the Standard Model (SM). Starting from the EFT where the SM is extended by dimension-6 operators, experimental data can be used without further assumptions to measure (or set limits on) the EFT parameters. The interpretation of these results requires instead a set of broad assumptions (e.g. power counting rules) on the UV dynamics. This allows one to establish, in a bottom-up approach, the validity range of the EFT description, and to assess the error associated with the truncation of the EFT series. We give a practical prescription on how experimental results could be reported, so that they admit a maximally broad range of theoretical interpretations. Namely, the experimental constraints on dimension-6 operators should be reported as functions of the kinematic variables that set the relevant energy scale of the studied process. This is especially important for hadron collider experiments where collisions probe a wide range of energy scales.
aInstitut de Theorie des Phenomenes Physiques
Models
1 Introduction
2
General discussion
3
4
5
2.1
2.2
2.3
2.4
Model-independent experimental results
EFT validity and interpretation of the results
On the necessity of a power counting
On the importance of loop corrections
Limitations of the D = 6 EFT
An explicit example
Summary A D = 6 versus D = 8 contributions to the likelyhood
1
Introduction
We consider an EFT where the SM is extended by a set of higher-dimensional operators,
and assume that it reproduces the low-energy limit of a more fundamental UV description.
The theory has the same eld content and the same linearly-realized SU(
3
)
SU(
2
)
U(
1
)
local symmetry as the SM. The di erence is the presence of operators with canonical
dimension D larger than 4. These are organized in a systematic expansion in D, where
each consecutive term is suppressed by a larger power of a high mass scale. Assuming
baryon and lepton number conservation, the Lagrangian takes the form [1{3]
Le = LSM + X c(
6
) (
6
) + X c(8) (8) +
i Oi j Oj
;
i
j
where each Oi
(D) is a gauge-invariant operator of dimension D and ci(D) is the corresponding
e ective coe cient. Each coe cient has dimension 4
D and scales like a given power of
the couplings of the UV theory; in particular, for an operator made of ni elds one has
c(D)
i
(coupling)ni 2
(high mass scale)D 4
its couplings. It follows from simple dimensional analysis after restoring ~ 6= 1 in the
Lagrangian since couplings, as well as elds, carry ~ dimensions [4{6] (see also refs. [7,
8]). An additional suppressing factor (coupling=4 )2L may arise with respect to the naive
scaling if the operator is rst generated at the Lth-loop order in the perturbative expansion.1
If no perturbative expansion is possible in the UV theory because it is maximally strongly
coupled, then eq. (1.2) gives a correct estimate of the size of the e ective coe cients by
setting coupling
di erent conclusions, and in refs. [14{18]; see also refs. [10, 19{30] for a discussion about
matching UV models to the EFT, which indirectly addresses the question of its validity).
We will discuss the following points:
Under what conditions does the EFT give a faithful description of the low-energy
phenomenology of some BSM theory?
When is it justi ed to truncate the EFT expansion at the level of dimension-6
operators? To what extent can experimental limits on dimension-6 operators be a ected
by the presence of dimension-8 operators? Are there physically important examples
where dimension-8 operators cannot be neglected?
When is it justi ed to calculate the EFT predictions at tree level? In what
circumstances may including 1-loop and/or real-emission corrections modify the predictions
in a relevant way?
It is important to realize that addressing the above questions cannot be done in a
completely model-independent way, but requires a number of (broad) assumptions about
the new physics. An illustrative example is that of the Fermi theory, which is an EFT
for the SM degrees of freedom below the weak scale after the W and Z bosons have been
integrated out. In this language, the weak interactions of the SM fermions are described
at leading order by 4-fermion operators of D=6, such as:
Le
c(
6
) (e PL e)(
PL ) + h:c: ;
c(
6
) =
g2=2
m2W
=
2
v2
:
(1.3)
1See for instance refs. [9, 10] for a discussion on whether a given operator can be generated at tree-level
or at loop-level.
{ 2 {
This operator captures several aspects of the low-energy (...truncated)