lower bounds on higgs mass from vacuum instability constraints

Lower bounds on Higgs mass from vacuumstability 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 = (Dµφ)†(Dµφ)− 1

4WµνW

µν − 1

4FµνF

µν + il′il /Dl′il

iq′il /Dq′il + iu′iR /Du

′iR + id′iR /Dd

′iR

+Lyukawa + µ2(φ†φ)− λ(φ†φ)2 (1)

Where,

Lyukawa = f(e)ij l′ilφe

′jR + f

(d)ij q

′ilφd

′jR + f

(u)ij q′ilφu

′jR

Dµφ = (∂µ − ig12Wµ.σ − i

g22Bµ)φ (2)

� Spontaneous symmetry breaking gives masses to the vector bosons, higgsand the fermions.

� The essential point to be noted is that the mechanism depends on thechoice of a stable vacuum given by

〈φ†φ〉 =(µ)2

2λ=v2

2(3)


Spontaneous Symmetry breaking

� The spontaneous symmetry breaking via the higgs mechanism gives riseto the following scalar-boson interaction as well as vector boson masses.For convenience we can choose the unitary gauge:-

L =(

0 η(x)+v√2

)(g12Wµ.σ +

g22Bµ)(

g12Wµ.σ +

g22Bµ)

(0

η(x)+v√2

)(4)

� For the fermion masses, which are derived from the yukawa couplings, wehave,

Lyukawa = fijΨ′iL

(0

η(x)+v√2

)Ψ′jR

(5)


Spontaneous Symmetry Breaking

� We must take note of the fact that we have implicitly assumed that thequartic coupling λ is positive. If we let λ < 0 we have no minima.

(a) λ > 0 (b) λ < 0

Figure: Various form of potentials



� The vector boson masses and fermion masses are given by

Mw =g21v

2

4

Mz =g21v

2 + g22v2

4

Mf =fijv

2(6)

� The quantity ”v” depends on λ.

� But this quantity is a running coupling.



� If the coupling λ becomes negative we lose our stable vacuum. Thecorrections to the coupling are provided by the gauge boson-scalar andfermion-scalar interactions.

� The relevant couplings are derived from the Lagrangian after imposingthe unitary gauge.

gηηWW =−2M2

wgµνv2

gηηZZ =−2M2

z gµνv2

gηWW =−2M2

wgµνv

gηZZ =−2M2

z gµνv

gηff =−√

2Mf

v(7)


Beta functions

� The beta functions are derived by requiring that the bare couplingconstants do not depend on the fictitious mass parameter µ. This leadsto

∂λ

∂ log(µ2

v2

) =1

16π2(12λ2 + 6λg2f −

3

2λ(g21 + g22)

−3g4f +3

16(2g41 + (g21 + g22)2)) (8)

� The beta function has been derived taking into account all possible formsof corrections to the coupling constant


Beta Functions

� This beta function can be arranged into the following form.

dλ

dτ=

3

4π2(λ− λ+)(λ− λ−)

(9)

� Taking a look at the beta function expression we realize that if−3g4f + 3

16 (2g41 + (g21 + g22)2) < 0 then we have λ− < 0 < λ+. Thus λ−is an ultraviolet-stable fixed point.

� for 0 < λ(v2) < λ+, we always have λ→ λ−. Thus it becomes negativeand the symmetry breaking condition is spoilt.


Beta Functions

� The problem becomes significant when we are looking at small values ofλ. Hence for all practical purposes we can drop the λ dependent terms inthe beta functions.

� We get,

dλ

d log(q2

v2

) ' 1

16π2(−3g4f +

3

16(2g41 + (g21 + g22)2)) (10)

This is the expression one gets by directly evaluating the correctiondiagrams 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 hasthe 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

Zλ = 1 +g41

32π2λε+

(g21 + g22)2

64π2λε−

g4f8π2λε

(11)

� The bare coupling is given by,

λ0 = Z−2φ Zλµελ (12)

Where, we have used the MS scheme in dimensional regularisation.

� The beta function becomes

dλ

d log(q2

v2

) =1

16π2(−3g4f +

3

16(2g41 + (g21 + g22)2)) (13)



� We get after solving the renormalization equation (where the running ofcoupling constants gf , g1, g2 have been neglected),

λ(Λ) = λ(η2) +1

16π2(−3g4f +

3

16(2g41 + (g21 + g22)2)) log

(Λ2

v2

)(14)

� To ensure λ(Λ) remains positive we need (since M2h = 2v2λ),

M2h >

v2

8π2(3g4f −

3

16(2g41 + (g21 + g22)2)) log

(Λ2

v2

)(15)

� Requiring the other couplings to also evolve, the lower bound has beenobtained numerically for N = 3, N = 8 cases.



� With increase of gf we have the upper and lower bounds coinciding. Thiscan be understood as follows. With the increase of gf , the rhs of thebeta function equation becomes more unstable. The range of initialvalues of λ narrows down.

� As Mt (since gf =√2Mt

v ) reaches its upper bound, higgs mass isessentially determined to be 220 GeV (N = 3), 280 Gev(N = 8).


�

Figure: Bounds on higgs mass as a function of cutoff at a fixed fermion mass


(a) N=3 (b) N=8

Figure: Bounds on higgs mass as a function of top quark mass


References

� N.cabibbo, L.maiani, G.parisi, R.petronzio,”Bounds on the fermions andhiggs boson masses in Grand unified theories”, [Nucl. Phys B158(1979)295-305].

� L.maiani, G.parisi, R.petronzio, ”Bounds on the number and masses ofquarks and leptons”, [Nucl. Phys B136 (1978) 115]

� T.P cheng, E.Eichten, L.F li, ”Higgs Phenomenon in asymptotically freegauge theories” [Physical Review D9 (1975) 259].

� Yorikiyo Nagashima, ”Beyond The Standard Model Of ElementaryParticle Physics”.

� Thomas Hambye, Kurt Riesselmann, ”Matching conditions and Higgsmass upper bounds revisited”, [hep-ph/9610272]


Thank you.
