Lower Bounds Higgs Mass Vacuum Stability

Lower bounds on Higgs mass from vacuum
stability constraints
Subham Dutta Chowdhury
December 8, 2014
Term Paper for HE-397
Subham Dutta Chowdhury Lower bounds on Higgs mass from vacuum stability constraints 1/19

Outline
Introduction
The standard model Lagrangian
Spontaneous Symmetry Breaking
Renormalization Constraints
Beta functions
Diagrams (Gauge bosons)
Diagrams (Fermions)
Renormalized Coupling
Bounds on Higgs mass
Bibliography

The standard model Lagrangian
The Standard model Lagrangian is given by
L = (D)y(D)
1
4
WW
1
4
FF + il0
il = Dl0
il
i q0
il + i u0
il = Dq0
iR + i d0
iR = Du0
iR = Dd0
iR
+Lyukawa + 2(y) (y)2 (1)
Where,
Lyukawa = f(e)
ij
jR + f(d)
l0
ile0
ij
jR + f(u)
q0
ild0
ij
q0
il
~u0
jR
D = (@ i
g1
2
W: i
g2
2
B) (2)
Spontaneous symmetry breaking gives masses to the vector bosons, higgs
and the fermions.
The essential point to be noted is that the mechanism depends on the
choice of a stable vacuum given by
hyi =
()2
2
=
v2
2
(3)

Spontaneous Symmetry breaking
The spontaneous symmetry breaking via the higgs mechanism gives rise
to the following scalar-boson interaction as well as vector boson masses.
For convenience we can choose the unitary gauge:-
L =

0 (x)+v p
2

(
g1
2
W: +
g2
2
B)(
g1
2
W: +
g2
2
B)

0
(x)+v p
2
!
(4)
For the fermion masses, which are derived from the yukawa couplings, we
have,
Lyukawa = fij 0
iL

0
(x)+v p
2
!
0
jR
(5)

Spontaneous Symmetry Breaking
We must take note of the fact that we have implicitly assumed that the
quartic coupling is positive. If we let 0 we have no minima.
(a) 0 (b) 0
Figure: Various form of potentials

The vector boson masses and fermion masses are given by
Mw =
g2
1v2
4
Mz =
2v2
g2
1v2 + g2
4
Mf =
fijv
2
(6)
The quantity v depends on .
But this quantity is a running coupling.

If the coupling becomes negative we lose our stable vacuum. The
corrections to the coupling are provided by the gauge boson-scalar and
fermion-scalar interactions.
The relevant couplings are derived from the Lagrangian after imposing
the unitary gauge.
gWW =
2M2w
g
v2
gZZ =
2M2
z g
v2
gWW =
2M2w
g
v
gZZ =
2M2
z g
v
gff =
p
2Mf
v

(7)

Beta functions
The beta functions are derived by requiring that the bare coupling
constants do not depend on the

ctitious mass parameter . This leads
to
@
@ log

2
v2
=
1
162 (122 + 6g2
f
3
2
(g2
1 + g2
2)
3g4
f +
3
16
(2g4
1 + (g2
1 + g2
2)2)) (8)
The beta function has been derived taking into account all possible forms
of corrections to the coupling constant

Beta Functions
This beta function can be arranged into the following form.
d
d
=
3
42 ( +)( )
(9)
Taking a look at the beta function expression we realize that if
f + 3
3g4
16 (2g4
1 + (g2
1 + g2
2)2) 0 then we have 0 +. Thus
is an ultraviolet-stable

xed point.
for 0 (v2) +, we always have ! . Thus it becomes negative
and the symmetry breaking condition is spoilt.

Beta Functions
The problem becomes signi

cant when we are looking at small values of
. Hence for all practical purposes we can drop the dependent terms in
the beta functions.
We get,
d
d log

q2
v2
'
1
162 (3g4
f +
3
16
(2g4
1 + (g2
1 + g2
2)2)) (10)
This is the expression one gets by directly evaluating the correction
diagrams as mentioned in the forthcoming slides.

Diagrams (Gauge bosons)
The relevant diagrams for the correction to the vertex is given below.
Note that this is the gauge boson correction
(a) s-channel
(b) t-channel
(c) u-channel
Figure: Gauge boson corrections to the scalar vertex

Diagrams (Fermions)
The diagram corresponding to one-loop correction by fermions is given.
Note that only the top quark contribution is taken since top quark has
the heaviest mass and yukawa couplings are proportional to quark masses.
Figure: fermion corrections to the scalar vertex

Renormalized coupling
The renormalized scalar coupling is given by
Z = 1 +
g4
1
322
+
(g2
1 + g2
2)2
642

g4
f
82
(11)
The bare coupling is given by,
0 = Z2
Z ~ (12)
Where, we have used the MS scheme in dimensional regularisation.
The beta function becomes
d
d log

q2
v2
=
1
162 (3g4
f +
3
16
(2g4
1 + (g2
1 + g2
2)2)) (13)

We get after solving the renormalization equation (where the running of
coupling constants gf , g1, g2 have been neglected),
() = (2) +
1
162 (3g4
f +
3
16
(2g4
1 + (g2
1 + g2
2)2)) log

2
v2

(14)
To ensure () remains positive we need (since M2
h = 2v2),
M2
h
v2
82 (3g4
f
3
16
(2g4
1 + (g2
1 + g2
2)2)) log

2
v2

(15)
Requiring the other couplings to also evolve, the lower bound has been
obtained numerically for N = 3, N = 8 cases.

With increase of gf we have the upper and lower bounds coinciding. This
can be understood as follows. With the increase of gf , the rhs of the
beta function equation becomes more unstable. The range of initial
values of narrows down.
As Mt (since gf =
p
2Mt
v ) reaches its upper bound, higgs mass is
essentially determined to be 220 GeV (N = 3), 280 Gev(N = 8).

Figure: Bounds on higgs mass as a function of cuto at a

Lower Bounds Higgs Mass Vacuum Stability

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