the importance of the tev scale sally dawson lecture 3 fnal lhc workshop, 2006
DESCRIPTION
Issues with the Standard Model Unitarity Landau pole Triviality Dependence of Higgs mass on high scale physicsTRANSCRIPT
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The Importance of the TeV Scale
Sally DawsonLecture 3FNAL LHC Workshop, 2006
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The Standard Model Works
Any discussion of the Standard Model has to start with its successThis is unlikely to be an accident!
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Issues with the Standard Model Unitarity Landau pole Triviality Dependence of Higgs mass on high scale
physics
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Unitarity Consider 2 2 elastic scattering
Partial wave decomposition of amplitude
al are the spin l partial waves
2
2641 Asd
d
0
)(cos)12(16l
ll aPlA
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Unitarity Pl(cos) are Legendre polynomials:
)(cos)(coscos)12()12(8 1
1
*
00
llllll
PPdaalls
1
1
,
122
)()(l
xPxdxP llll
0
2)12(16l
lals
Sum of positive definite terms
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More on Unitarity
Optical theorem
Unitarity requirement:
0
2)12(16)0(Im1l
lals
As
2)Im( ll aa
21)Re( la
Optical theorem derived assuming only conservation of probability
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More on Unitarity
Idea: Use unitarity to limit parameters of theory
Cross sections which grow with energy always violate unitarity at some energy scale
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Consider W+W- pair production
Example: W+W-
t-channel amplitude:
In center-of-mass frame:
(p)
(q)
e(k) k=p-p+=p--q
)()()()1()1()(8
)( 525
2 pppu
kkqvgiWWvvAt
W+(p+)
W-(p-)
cos,sin,0,12
cos,sin,0,12
1,0,0,12
1,0,0,12
WW
WW
sp
sp
sq
sp
sM
ppkt
qps
WW /41
)(
)(
2
22
2
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W+W- pair production, 2
Interesting physics is in the longitudinal W sector:
Use Dirac Equation: pu(p)=0
sMO
Mp W
W
2
)()1()(4
)( 52
2
pukqvMgiWWvvA
WLLt
sMOsGWWvvA W
FLLt
22222
sin2)(
Grows with energy
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Feynman Rules for Gauge Boson Vertices
gppgppgpppppV ZZZ )()()(),,(
),,( ZWWV pppVig WWWZ
WW
eg
eg
cot
p-
p+
pZ
ggggggigWWVV 2
WWWZZ
WZWW
WW
eg
eg
eg
22
2
2
cot
cot
ggggggig 22
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W+W- pair production, 3 SM has additional contribution from s-channel Z exchange
For longitudinal W’s
)()()()()()()1()()(4
)( 252
2
ppkpgkpgppg
Mkkgpuqv
MsgiWWA
ZZs
)()1)()((4
)( 52
2
puppqvMgiWWA
WLLs
)()1()(4
)( 52
2
pukqvMgiWWvvA
WLLs
Contributions which grow with energy cancel between t- and s- channel diagrams
Depends on special form of 3-gauge boson couplings
Z(k)
W+(p+)
W-(p-)
(p)
(q)
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No deviations from SM at LEP2
LEP EWWG, hep-ex/0312023
No evidence for Non-SM 3 gauge boson vertices
Contribution which grows like me
2s cancels between Higgs diagram and others
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Example: W+W-W+W-
Recall scalar potential (Include Goldstone Bosons)
Consider Goldstone boson scattering: +-+
2222
2
222
22
28
222
zhvM
zhhv
MhMV
h
hh
2
22
2
22
2
2
2)(
h
h
h
hh
Msi
vMi
Mti
vMi
vMiiA
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+-+-
Two interesting limits: s, t >> Mh
2
s, t << Mh2
2
2
2)(vMA h
2)(vuA
2
200 8 v
Ma h
200 32 v
sa
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Use Unitarity to Bound Higgs
High energy limit:
Heavy Higgs limit
21)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
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Electroweak Equivalence Theorem
2
2
1111 )......()()()......(
EMO
AiiVVVVA
W
NNNNN
LLNLL
This is a statement about scattering amplitudes, NOT individual Feynman diagrams
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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
42hhMV h
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Consider hhhh
Real scattering, s+t+u=4Mh2
Consider momentum space-like and off-shell: s=t=u=Q2<0
Tree level: iA0=-6i
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hhhh, #2 One loop:
A=A0+As+At+Au
)1()()4(89
)()2(21)6(
2222
2
222222
xxQM
Mqpki
MkikdiiA
h
hh
n
s
...)1()()4(
16916 222
2
xxQMA h
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hhhh, #3
Sum the geometric series to define running coupling
(Q) blows up as Q (called Landau pole)
)(6log
891
6
2
Q
MQ
A
h
...log16
916 2
2
2
hMQA
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hhhh, #4
This is independent of starting point BUT…. Without 4 interactions, theory is non-
interacting Require quartic coupling be finite
0)(
1
Q
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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
vQvM h
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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 gaugeggdtd
tt
2
2
logQt
vMg t
t
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Does Spontaneous Symmetry Breaking Happen? SM requires spontaneous symmetry
This requires
For small
Solve
)0()( VvV
42 1616 tgdtd
2
2
2
4
log43)()(
vgv t
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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
23
vvM h
GeVM h 130
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Bounds on SM Higgs Boson If SM valid up to
Planck scale, only a small range of allowed Higgs Masses
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More Problems We often say that the SM cannot be the entire
story because of the quadratic divergences of the Higgs Boson mass
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Masses at one-loop
First consider a fermion coupled to a massive complex Higgs scalar
Assume symmetry breaking as in SM:
..)( 22chmiL RLFs
22)( vmvh F
F
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Masses at one-loop, #2
Calculate mass renormalization for
.....log32
32
2
2
2
F
FFF m
mm
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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
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Scalars are very different
Mh diverges quadratically! This implies quadratic sensitivity to high
mass scales
22
2
1
22
222
2222
11)2
2(
log8
)(
OmmImm
mmmmM
F
ssF
FFs
FsSh
1
01 )1(1log)( xaxdxaI
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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
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2
22222
22
GeV200TeV 0.7
123624
thZWF
h MMMMGM
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
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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
(...)120
2
hh MM
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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
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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