62
G. Altarelli and S. Forte
quadratic dependence on m t (and on other possible widely broken isospin multiplets
from new physics) arises because, in spontaneously broken gauge theories, heavy
virtual particles do not decouple. On the contrary, in QED or QCD, the running
of α and α s at a scale Q is not affected by heavy quarks with mass M Q.
According to an intuitive decoupling theorem [43], diagrams with heavy virtual
particles of mass M can be ignored at Q M provided that the couplings do not
grow with M and that the theory with no heavy particles is still renormalizable.
In the spontaneously broken EW gauge theories both requirements are violated.
First, one important difference with respect to unbroken gauge theories is in the
longitudinal modes of weak gauge bosons. These modes are generated by the Higgs
mechanism, and their couplings grow with masses (as is also the case for the
physical Higgs couplings). Second the theory without the top quark is no more
renormalizable because the gauge symmetry is broken as the (t,b) doublet would
not be complete (also the chiral anomaly would not be completely cancelled).
With the observed value of m t the quantitative importance of the terms of order
G F m 2
t /4π 2 √
2 is substancial but not dominant (they are enhanced by a factor
m 2
t /m 2
W ∼ 5 with respect to ordinary terms). Both the large logarithms and the
G F m 2
t terms have a simple structure and are to a large extent universal, i.e. common
to a wide class of processes. In particular the G F m 2
t terms appear in vacuum
polarization diagrams which are universal (virtual loops inserted in gauge boson
internal lines are independent of the nature of the vertices on each side of the
propagator) and in the Z → b ¯
b vertex which is not. This vertex is specifically
sensitive to the top quark which, being the partner of the b quark in a doublet, runs in
the loop. Instead all types of heavy particles could in principle contribute to vacuum
polarization diagrams. The study of universal vacuum polarization contributions,
also called “oblique” corrections, and of top enhanced terms is important for
an understanding of the pattern of radiative corrections. More in general, the
important consequence of non decoupling is that precision tests of the electroweak
theory may apriori be sensitive to new physics even if the new particles are too
heavy for their direct production, but aposteriori no signal of deviation has clearly
emerged.
While radiative corrections are quite sensitive to the top mass, they are unfortunately much less dependent on the Higgs mass. If they were sufficiently sensitive
by now we would precisely know the mass of the Higgs. But the dependence
of one loop diagrams on m H is only logarithmic: ∼ G F m 2
W log(m 2
H /m 2
W ).
Quadratic terms ∼ G 2
F m 2
H only appear at two loops [44] and are too small to
be detectable. The difference with the top case is that the splitting m 2
t − m 2
b
is a direct breaking of the gauge symmetry that already affects the 1- loop
corrections, while the Higgs couplings are “custodial” SU(2) symmetric in lowest
order.
G. Altarelli and S. Forte
quadratic dependence on m t (and on other possible widely broken isospin multiplets
from new physics) arises because, in spontaneously broken gauge theories, heavy
virtual particles do not decouple. On the contrary, in QED or QCD, the running
of α and α s at a scale Q is not affected by heavy quarks with mass M Q.
According to an intuitive decoupling theorem [43], diagrams with heavy virtual
particles of mass M can be ignored at Q M provided that the couplings do not
grow with M and that the theory with no heavy particles is still renormalizable.
In the spontaneously broken EW gauge theories both requirements are violated.
First, one important difference with respect to unbroken gauge theories is in the
longitudinal modes of weak gauge bosons. These modes are generated by the Higgs
mechanism, and their couplings grow with masses (as is also the case for the
physical Higgs couplings). Second the theory without the top quark is no more
renormalizable because the gauge symmetry is broken as the (t,b) doublet would
not be complete (also the chiral anomaly would not be completely cancelled).
With the observed value of m t the quantitative importance of the terms of order
G F m 2
t /4π 2 √
2 is substancial but not dominant (they are enhanced by a factor
m 2
t /m 2
W ∼ 5 with respect to ordinary terms). Both the large logarithms and the
G F m 2
t terms have a simple structure and are to a large extent universal, i.e. common
to a wide class of processes. In particular the G F m 2
t terms appear in vacuum
polarization diagrams which are universal (virtual loops inserted in gauge boson
internal lines are independent of the nature of the vertices on each side of the
propagator) and in the Z → b ¯
b vertex which is not. This vertex is specifically
sensitive to the top quark which, being the partner of the b quark in a doublet, runs in
the loop. Instead all types of heavy particles could in principle contribute to vacuum
polarization diagrams. The study of universal vacuum polarization contributions,
also called “oblique” corrections, and of top enhanced terms is important for
an understanding of the pattern of radiative corrections. More in general, the
important consequence of non decoupling is that precision tests of the electroweak
theory may apriori be sensitive to new physics even if the new particles are too
heavy for their direct production, but aposteriori no signal of deviation has clearly
emerged.
While radiative corrections are quite sensitive to the top mass, they are unfortunately much less dependent on the Higgs mass. If they were sufficiently sensitive
by now we would precisely know the mass of the Higgs. But the dependence
of one loop diagrams on m H is only logarithmic: ∼ G F m 2
W log(m 2
H /m 2
W ).
Quadratic terms ∼ G 2
F m 2
H only appear at two loops [44] and are too small to
be detectable. The difference with the top case is that the splitting m 2
t − m 2
b
is a direct breaking of the gauge symmetry that already affects the 1- loop
corrections, while the Higgs couplings are “custodial” SU(2) symmetric in lowest
order.