what do we know about the standard model? sally dawson lecture 4 tasi, 2006
TRANSCRIPT
What do we know about the Standard Model?
Sally Dawson
Lecture 4
TASI, 2006
The Standard Model Works
Any discussion of the Standard Model has to start with its success
This is unlikely to be an accident!
Unitarity Consider 2 2 elastic scattering
Partial wave decomposition of amplitude
al are the spin l partial waves
2
264
1A
sd
d
0
)(cos)12(16l
ll aPlA
Unitarity Pl(cos) are Legendre polynomials:
)(cos)(coscos)12()12(8 1
1
*
00
llllll
PPdaalls
1
1
,
12
2)()(
lxPxdxP ll
ll
0
2)12(
16
llal
s
Sum of positive definite terms
More on Unitarity
Optical theorem
Unitarity requirement:
0
2)12(
16)0(Im
1
llal
sA
s
2)Im( ll aa
2
1)Re( la
Optical theorem derived assuming only conservation of probability
More on Unitarity
Idea: Use unitarity to limit parameters of theory
Cross sections which grow with energy always violate unitarity at some energy scale
Example 1: W+W-W+W-
Recall scalar potential (Include Goldstone Bosons in Unitarity gauge)
Consider Goldstone boson scattering:
+-+
2222
2
222
22
28
222
zhv
M
zhhv
Mh
MV
h
hh
2
22
2
22
2
2
2)(
h
h
h
hh
Ms
i
v
Mi
Mt
i
v
Mi
v
MiiA
+-+-
Two interesting limits: s, t >> Mh
2
s, t << Mh2
2
2
2)(v
MA h
2)(
v
uA
2
200 8 v
Ma h
200 32 v
sa
Use Unitarity to Bound Higgs
High energy limit:
Heavy Higgs limit
2
1)Re( la
2
200 8 v
Ma h
200 32 v
sa
Mh < 800 GeV
Ec 1.7 TeV
New physics at the TeV scale
Can get more stringent bound from coupled channel analysis
Electroweak Equivalence Theorem
2
2
1111 )......()()()......(
E
MO
AiiVVVVA
W
NNNNN
LLNLL
This is a statement about scattering amplitudes, NOT individual Feynman diagrams
Plausibility argument for Electroweak Equivalence Theorem Compute (hWL
+WL-) for Mh>>MW
W
h
WW
W
M
Mig
M
ppigM
gigMiA
2
2
2
)(
28)(
3
h
MGWWh hF
LL
(hWW) Mh
for Mh 1.4 TeV
WWL M
ppp
M ),0,0,(
10
Landau Pole
Mh is a free parameter in the Standard Model Can we derive limits on the basis of
consistency? Consider a scalar potential:
This is potential at electroweak scale Parameters evolve with energy in a calculable
way
422
42hh
MV h
Consider hhhh
Real scattering, s+t+u=4Mh2
Consider momentum space-like and off-shell: s=t=u=Q2<0
Tree level: iA0=-6i
hhhh, #2
One loop:
A=A0+As+At+Au
)1()()4(8
9
)()2(2
1)6(
2222
2
222222
xxQM
Mqpk
i
Mk
ikdiiA
h
hh
n
s
...)1()()4(
16
916 222
2
xxQMA h
hhhh, #3
Sum the geometric series to define running coupling
(Q) blows up as Q (called Landau pole)
)(6
log89
1
6
2
Q
MQ
A
h
...log16
916
2
2
2
hM
QA
hhhh, #4
This is independent of starting point BUT…. Without 4 interactions, theory is non-
interacting Require quartic coupling be finite
0)(
1
Q
hhhh, #5
Use =Mh2/(2v2) and approximate log(Q/Mh)
log(Q/v) Requirement for 1/(Q)>0 gives upper limit on Mh
Assume theory is valid to 1016 GeV Gives upper limit on Mh< 180 GeV
Can add fermions, gauge bosons, etc.
2
2
222
log9
32
vQ
vM h
High Energy Behavior of
Renormalization group scaling
Large (Heavy Higgs): self coupling causes to grow with scale
Small (Light Higgs): coupling to top quark causes to become negative
Q
Qlog(...)
)(
1
)(
1
)(12121216 4222 gaugeggdt
dtt
2
2
logQ
tv
Mg tt
Does Spontaneous Symmetry Breaking Happen? SM requires spontaneous symmetry
This requires
For small
Solve
)0()( VvV
42 1616 tgdt
d
2
2
2
4
log4
3)()(
v
gv t
Does Spontaneous Symmetry Breaking Happen? (#2) () >0 gives lower bound on Mh
If Standard Model valid to 1016 GeV
For any given scale, , there is a theoretically consistent range for Mh
2
2
2
22 log
2
3
v
vM h
GeVM h 130
Bounds on SM Higgs Boson
If SM valid up to Planck scale, only a small range of allowed Higgs Masses
Problems with the Higgs Mechanism We often say that the SM cannot be the entire
story because of the quadratic divergences of the Higgs Boson mass
Masses at one-loop
First consider a fermion coupled to a massive complex Higgs scalar
Assume symmetry breaking as in SM:
..)(22
chmiL RLFs
22
)( vm
vh FF
Masses at one-loop, #2
Calculate mass renormalization for
])][([)2()(
2)(
22224
42
2
sF
FFF mpkmk
mkkdi
ipi
Renormalized fermion mass
Do integral in Euclidean space
2
0
224
22
24
220
2
44
40
)()( yfdyykfkd
kkkkkk
kidkd
ikk
EE
E
E
222224
1
04
2
)]1([
)1(
32
)(
xmxmk
xmkddxi
pm
sF
FF
mpFFF
Renormalized fermion mass, #2 Renormalization of fermion mass:
.....log32
3
)]1([)1(
32
2
2
2
2
1
0 0
22222
22
F
FF
sF
FFF
m
m
xmxmy
dyyxdx
mm
Symmetry and the fermion mass mF mF
mF=0, then quantum corrections vanish
When mF=0, Lagrangian is invariant under LeiLL
ReiRR
mF0 increases the symmetry of the threoy Yukawa coupling (proportional to mass) breaks
symmetry and so corrections mF
Scalars are very different
Mh diverges quadratically! This implies quadratic sensitivity to high
mass scales
]))[((
)))(((
)2()(2
)( 22224
4
2
2
2FF
FFF
s mpkmk
mpkmkTrkdi
ipi
22
2
1
22
222
2222
11)
22(
log8
)(
Om
mI
mm
mmmmM
F
ssF
FFs
FsSh
1
0
1 )1(1log)( xaxdxaI
Scalars (#2) Mh diverges quadratically! Requires large cancellations (hierarchy
problem) Can do this in Quantum Field Theory h does not obey decoupling theorem
Says that effects of heavy particles decouple as M
Mh0 doesn’t increase symmetry of theory Nothing protects Higgs mass from large
corrections
2
22222
2
2
GeV200TeV 0.7
123624
thZWF
h MMMMG
M
Mh 200 GeV requires large cancellations
• Higgs mass grows with scale of new physics, • No additional symmetry for Mh=0, no protection
from large corrections
h h
Light Scalars are Unnatural
What’s the problem?
Compute Mh in dimensional regularization and absorb infinities into definition of Mh
Perfectly valid approach Except we know there is a high scale
(...)12
02
hh MM
Try to cancel quadratic divergences by adding new particles SUSY models add scalars with same
quantum numbers as fermions, but different spin
Little Higgs models cancel quadratic divergences with new particles with same spin
We expect something at the TeV scale If it’s a SM Higgs then we have to think hard
about what the quadratic divergences are telling us
SM Higgs mass is highly restricted by requirement of theoretical consistency