Nonrenormalization and Operator Mixing via On-Shell Methods
Zvi Bern,
1,2
Julio Parra-Martinez,
1
and Eric Sawyer
1
1
Mani L. Bhaumik Institute for Theoretical Physics, UCLA Department of Physics and Astronomy, Los Angeles, California 90095, USA
2
Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland
(Received 22 October 2019; accepted 20 November 2019; published 4 February 2020)
Using on-shell methods, we present a new perturbative nonrenormalization theorem for operator mixing
in massless four-dimensional quantum field theories. By examining how unitarity cuts of form factors
encode anomalous dimensions, we show that longer operators are often restricted from renormalizing
shorter operators at the first order where Feynman diagrams exist. The theorem applies quite generally and
depends only on the field content of the operators involved. We apply our theorem to operators of
dimension five through seven in the standard model effective field theory, including examples of nontrivial
zeros in the anomalous-dimension matrix at one through four loops. The zeros at two and higher loops go
beyond those previously explained using helicity selection rules. We also include explicit sample
calculations at two loops.
DOI: 10.1103/PhysRevLett.124.051601
Introduction.—A key challenge in particle physics is to
identify physics beyond the standard model. Because
current experimental data at colliders are well described
by the standard model, it is unclear which theoretical
direction will ultimately prove to be the one chosen by
nature. Therefore, it is important to quantify new physics
beyond the standard model in a systematic, model-
independent manner. The theoretical framework for
doing so is via effective field theories that extend the
standard model Lagrangian by adding higher-dimension
operators [1,2]
ΔL ¼
X
i
c
i
O
i
; ð1Þ
with coefficients c
i
suppressed by powers of a high-energy
scale Λ dictated by the dimension of O
i
. The resulting
theory, known as the standard model effective field theory
(SMEFT), is reviewed in Ref. [3].
As for all quantum field theories, renormalization
induces mixing of these operators. This can be parame-
trized by the renormalization group equation
16π
2
∂c
i
∂ log μ
¼ γ
UV
ij
c
j
; ð2Þ
where γ
UV
ij
is the anomalous-dimension matrix and μ
is the renormalization scale. Usually, γ
UV
ij
is calculated
perturbatively in the marginal couplings of the standard
model Lagrangian, which we will denote collectively as g.
The complete one-loop anomalous-dimension matrix for
operators up to dimension six has been computed in
Refs. [4,5]. These calculations reveal a number of vanish-
ing entries related to supersymmetry [6], which seem
surprising at first because there are valid diagrams that
can be written down. These zeros have been elegantly
explained [7] using tree-level helicity selection rules [8],
which set certain classes of tree-level amplitudes to zero.
The tree-level vanishings imply, through unitarity, that
certain logarithms and their associated anomalous dimen-
sions are not present. Although these selection rules are
reminiscent of supersymmetric ones, they hold for generic
massless quantum field theories in four dimensions.
Might it be possible that beyond one loop there are new
nontrivial zeros? At first sight, this seems rather unlikely
because the helicity selection rules fail to hold at loop level.
In this Letter, we show that, contrary to expectations, there
are, in fact, additional nontrivial zeros in the higher-loop
anomalous-dimension matrix. As in Ref. [7], our only
assumption is that the theory does not contain any relevant
couplings (e.g., masses). To state the new nonrenormaliza-
tion theorem, we define the length of an operator, l ðO Þ,as
the number of fundamental field insertions in O. Then the
statement of the theorem is as follows.
Theorem 1.—Consider operators O
s
and O
l
such that
lðO
l
Þ >lðO
s
Þ. O
l
can renormalize O
s
at L loops only
if L>lðO
l
Þ − lðO
s
Þ.
At fixed loop order, sufficiently long operators cannot
renormalize short operators because there would be too
many legs to form a diagram with the right structure.
Such zeros in the anomalous-dimension matrix are trivial.
As written above, the theorem applies nontrivially at
Published by the American Physical Society unde r the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI. Funded by SCOAP
3
.
PHYSICAL REVIEW LETTERS 124, 051601 (2020)
0031-9007=20=124(5)=051601(6) 051601-1 Published by the American Physical Society
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