Lecture XI

Particle Physics 2011 

Luis Anchordoqui 

Thursday, November 17, 2011

Radiative Corrections 

As a rule ☛ size of radiative corrections to a given process is 

determined by discrepancy between various mass & energy scales involved 

For a wide class of low-energy and 

-boson observables 

dominant effects originate entirely in gauge boson propagators 

➊●⦁ 

●⦁ 

➋ ∆ρ 

●⦁ 

➌ 

●⦁ 

➍ 

(oblique corrections) 

and can be parametrized in terms of 4 electroweak parameters: 

∆α 

Z 

∆α, ∆ρ, ∆r, and ∆κ 

determines running fine structure constant at 

α(m Z )/α = (1 − ∆α) −1 

Z 

boson scale 

measures quantum corrections of NC/CC amplitudes at low energy 

∆r embodies non-photonic corrections to muon lifetime 

∆κ 

controls effective weak mixing angle 

that occurs in ratio of 

Thursday, November 17, 2011 

sin 2 ¯θw = sin 2 θ w (1 + ∆κ) 

Zf ¯f 

vector and axial-vector couplings 

i.e. c f V /cf A =1− 4|Q f | sin 2 ¯θw

Chess Notation 

Today’s class ☛ intro to theory of electroweak radiative corrections 

and its role in testing SM, predicting top mass, constraining Higgs mass, 

and searching for deviations that may signal presence of new physics 

Implementing such a program 

can be first formulated from point of view of experimentalist 

Introducing notation ☛ 

sin 2 θ w = s 2 =1− c 2 , m 2 W ≡ w, m 2 Z ≡ z 

Electroweak theory predicts at Born level that: 

w 

z 

=1− s2 

♜ 

σ(ν µ e) 

σ(¯ν µ e) = 3 − 12s2 + 16s 4 

♚ 

1 − 4s 2 + 16s 4 

πα 1 

√ 

2 GF 

w = s2 

♟ 

Γ(Z → f ¯f) 

m Z 

= α 3 C F 

((c f V )2 +(c f A )2) 

♝ 

A LR ≃ A τ ≃ 

[ 4 

3 A FB 

] 1/2 

≃ 2(1 − 4s 2 ) 

♛ 


Chessboard 

Chess Eqs. represent incomplete list of experiments 

capable of measuring sin 2 θ w 

Validity of SM requires that each measurement yields same value of s 2 

〡 

Ratio of 

ν µ 

scattering on left- and right-handed electrons 

which is a function of 

sin 2 θ w 

only ☛ 

♚ 

〢 

Measurement of weak boson masses ☛ 

♜ 

〣 

m W ,α G F ♟ 

Combination of , and as determined by muon lifetime ☛ 

ⅠⅤ 

Ⅴ 

Partial widths of 

with vector 

and color factor 

Z 

into a fermion pair 

and axial coupling 

c f V 

and 

c f A 

C F = 3 (1) for quarks (leptons) ☛ 

Various asymmetries measured at Z-factories ☛ ♛ 

♝ 


Corrections to 

Z 

propagator 

After inclusion of 

O(α) 

corrections 

sin 2 θ w values obtained from different methods will no longer be same 

ple, 

because 

the diagram 

radiative example, the corrections diagram modify chess eqs. in different ways 

E.G. 

diagram ☛ 

Z 

W W 

modifies -channel (5.5.92) 

(5.5.92) 

W (5.5.92) 

propagator measured by ♚ 

W 

fies 

It 

the 

does 

t-channel 

not It does 

Z propagator 

-- not, however, contribute 

measured 

-- contribute to O(α) shifts in 

by (5.5.87); see 

to the W, Z masses 

also (5.1.27). 

shifts in masses 

f 

W 

es not, however, contribute to O(α) shiftsf 

in the W, Z masses 

Z 

W 

example, the diagram 

f 

+ + + 

W. . . 

W 

W 

H 

+ + + . . . 

(5.5.93) 

+ + + . . . 

W 

W 

modifies the t-channel Z propagator measured by (5.5.87); see also (5.1.27). 

modifies It does not, the t-channel however, contribute Z propagator O(α) measured shiftsby in (5.5.87); the W, Zsee masses also (5.1.27). 

Z 

Z 

W 

W 

Z 

Z 

Z 

+ + + . . . 

f 

f 

W 

t 

(5.5.93) 

(5.5.93) 

f 

W 

H 

W 

(5.5.94) 

which yield an improved sin 2 θ w 

sin 2 value θvia w (5.5.88). value There viais no♜ 

real mystery (5.5.94) 

which here. yield Afteran inclusion improved of sin O(α) 2 θ w contributions value via (5.5.88). in Eqs. There (5.5.87)–(5.5.91), is no real mystery they 

which yield an improved 


W 

H 

O(α) 

W, Z

Weinberg Angle 

Experimentalist has to make a choice and define Weinberg angle to 

All other experiments should be reformulated in terms of preferred 

What this choice should be is no longer a matter of debate 

we will define 

sin 2 θ w 

in terms of physical masses of weak bosons 

O(α) 

s 2 

i.e. 

sin 2 θ w ≡ 1 − m2 W 

m 2 Z 

=0.23122(15) ☎ 

This choice is particularly useful in that one can estimate 

radiative corrections in terms of renormalization group 

which has been previously introduced 

O(α) 

corrections can be qualitatively understood 

in terms of loop corrections to vector-boson propagators 

 

and 

 

A most straightforward test of theory is now obtained 

by fixing 

sin 2 θ w 

in terms of measured weak boson masses 

and verifying that its value coincides with other measurements of 

θ w 


Free Parameters 

All UV divergences in QED can be absorbed in two parameters: 

α 

and 

m e 

List of parameters to be fixed by experiment in electroweak theory includes 

α, m W ,m Z ,m H ,m f 

 

g, g ′ , λ, µ, Y f 

weak mixing angle does not appear in list of parameters: 

its value is automatically determined by via ☎ 

Traditionally ☛ SM Lagrangian is determined in terms of 

There is in fact a direct translation between sets and ☕ 

☕ 

m W ,m Z 

g 2 = e 2 

z 

z − w 

g ′2 = e 2 z w 

λ = e 2 

zm 2 H 

8w(z − w) 

Y f = e 2 

zm 2 f 

2w(z − w) 


Y f = e 2 

2w(z − w) . (5.5.101) 

Electromagnetic Radiative Corrections 

As an example we will show how the relation (5.5.89) zm 

Show how relation ♟ is calculated to O(α) 

2 is calculated to 

Y 

O(α) in terms of the weak angle θ 

θ 

w defined by f = e 

(5.5.88). 2 f 

example ☛ 

The origin of the 

relation in terms (5.5.89) of weak is the angle muon’s lifetime w defined which, to by leading ♜ order, is given by 

Origin of relation the diagram ♟ is lifetime ☛ to leading order is given by diagram 

µ 

and 

f 

H 

λ = e 2 

8w(z − w) , (5 

2w(z − w) . (5 

As an example we will show how the relation (5.5.89) is calcula 

O(α) in terms of the weak angle θ w defined by (5.5.88). The origin 

relation (5.5.89) is the muon’s lifetime which, to leading order, is gi 

the diagram 

Γ (0) 

µ = W 

Γ (0) 

µ = W 

(5.5.102) 

(5 

In Fermi theory In Fermi ☛ theory, electromagnetic radiative corrections must be included to 

must be included obtain the to result obtain to O(α). result Symbolically, to O(α) 

Symbolically Γ (1) 

where 

In Fermi theory, electromagnetic radiative corrections must be inclu 

obtain the result to O(α). Symbolically, 

µ = G F 

√ [1 + photonic Γ (1) 

µ = G F 

corrections] , (5.5.103) 

2 

Γ (1) 

√ 

2 

[1 + photonic corrections] , (5 

µ = G F 

√ [1 + photonic corrections] 

2 

where 

 

where 

photonic corrections = + . . . (5 

photonic corrections = 

photonic corrections = + . . . (5.5.104) 


Electroweak Radiative Corrections 

] 

[ 

Γ (1) 

µ = e2 + box 

1 + photonic corrections 

electroweak In electroweak theory, on thetheory 

other hand, 

8s 2 c 2 z 

where 

+ propagator 

[ 

[ 

In electroweak Γ (1) theory, on the other 

Γ (1) hand, 

µ = e2 

+ vertex 

µ = e2 


8s 2 c 2 z 

f ] 

[ 

8s 2 c 2 1 + photonic corrections 

z 

Γ (1) 

µ = e2 + propagator 

+ box 


8s 2 c 2 z+ vertex 

propagator 

(5.5.105) 

= + 

where + propagator ♬ 

] 

+ box+ propagator 

(5.5.105) 

+ vertex 

+ vertex f 

ere 

] 

] 

+ box 

H 

propagator + = box + 

f 

t 

where 

where 

+ 

propagator = + 

f 

t 

H 

propagator = + 

+ 

H 

+ 

+ 

H 

In electroweak theory, on the other + vertex hand, 

+. . . 

vertex = + . . . 

+. . . (5.5.106) 

+. . . (5.5.106) Z 

and 

vertex = + . . . (5.5.107) 

vertex = + . . . 

vertex = + . . . (5.5.107) box = + . . . 


Z 

and 

and 

Z 

Equating (5.5.103) and (5.5.105) we obtain 

W 

box = + . . . 

t 

t 

+. . .

Electroweak Model at Born Level@ 

e threshold for producing charmed par- 

. Above the threshold for all five quark 

1 as predicted. These measurements conuark, 

because R = 11 would be reduced 

3 

e color. 

with 

ified when interpreted in the context of 

the (leading order) process e + e − → q¯q. 

grams where the quark and/or antiquark 

 

Equating and ♬ we obtain G F = πα 

Before discussing status of measurements of 

σ e + e − →q ¯q 

α = e2 

4π 

∆r =∆α − c2 

to leading order ☛ ∆r =0 

using 

σ e + e − →µ + µ − (3.5.86) 

coupling α, 

1 

√ (1 + ∆r) 

2 ws2 ∆r ☛ 

s 2 ∆ρ +∆ rem 

note that 

m W = gv 

2 = g 

2 √ 2 λ m H and m Z = m W 

cos θ w 

♒ 

 

♒ 

(3.5.87) 

reduces to Born relation 

♟ 


nate this quantity. We specifically isolated the fermions which are responsible 

the electroweak radiative corrections are gathered in ∆r. Notation (5.5.110) 

for Dominant Terms 

recognizes the running the fact of αthat from in the theOMS muon scheme, to thevacuum Z mass, polarization loops dominate 

this quantity. We specifically isolated the fermions which are 

Electroweak radiative corrections are gathered in ∆r 

responsible 

for the running of α from the muon to the Zf 

mass, 

We specifically isolated fermions which are responsible for running of 

∆α = ∑ f 

(5.5.111) 

f 

∆α = ∑ f 

(5.5.111) 

as well as the third generation, heavy quark diagram 

as well as third generation heavy quark diagram 


∆ρ = t 

W 

∆ρ = 

W 

t 

(5.5.112) 

(5.5.112) 

∆ rem 

Other contributions are small and are grouped in remainder 

Other contributions are small in the OMS scheme and are grouped in the 

To “remainder” Other can extent contributions imagine ∆ rem that . summing are ∆small rem the inis the series small OMS ☛ one scheme and can areimagine grouped insumming the series 

“remainder” Before discussing ∆ rem . the status of measurements of ∆r, we make several comments. 

Before To discussing leading order the status ∆r = of0 measurements of ∆r, we make several comments. 

+ 

and, using (3.5.87) and (2.4.88), (5.5.109) 

+ . . . 

reduces to 

To 

the 

leading 

Born 

order 

relation 

∆r = 

(5.5.89). 

0 and, using 

The full 

(3.5.87) 

order 

and 

α calculation 

(2.4.88), (5.5.109) 

of ∆r will 

reduces to the Born relation (5.5.89). The full order α calculation of ∆r will 

not be presented here. We have attempted to describe the full formalism in 

not be presented here. We have attempted to describe the full formalism in (5.5.113) 

by a relatively replacement accessible (1 way + elsewhere. ∆r) → 

a relatively accessible way elsewhere. 23 (1 23 To− the∆r) extent −1 

by the replacement (1 + ∆r) → 

that ∆ 

To(1the extent ∆r) −1 that in (5.5.109). 

in 

∆ rem ♒ is small, one 

rem is small, one 


∆ρ 

recognizes the fact that in the OMS scheme, vacuum polarization ∆α = loops ∑ dominate 

this quantity. We specifically isolated the fermions which are responsible 

f 

for the running of α from the muon to the Z mass, 

We already discussed running of 

α 

from small lepton masses to 

f 

m Z 


∆α = ∑ (5.5.111) 

f 

t 

Other large contribution 

∆ρ 

which represents loop ☛ 


∆ρ = 

W 

Its value is given by 

∆ρ = α 4π 

with 

N C =3 

z 

ω(z − ω) N C|U tb | 2 [ m 2 t F (m 2 t ,m 2 b)+m 2 bF (m 2 b,m 2 ] 

t ) Other contributions are small in the OMS scheme and are grouped in the 

F (m 2 1,m 2 2)= 

∆ρ = 

where is number of colors and U tb is CKM matrix element 


|U tb | 2 ≃ 1 

t 

is our primary focus here 

Other contributions are small in the OMS scheme 

(5.5.112) 

“remainder” W ∆ rem . 

Before discussing the status of measurements of ∆ 

ments. To leading order ∆r = 0 and, using (3.5.87 

reduces to the Born relation (5.5.89). The full order 

 

not be presented here. We have attempted to descr 

a relatively accessible way elsewhere. 23 To the exten 

“remainder” ∆ rem . 

Before discussing ∫ the status of measurements 1 

[ of ∆r, we make several 

dx x ln m 2 1(1 − x)+m 2 ] 

comments. 

To leading order ∆r = 0 and, using (3.5.87) and (2.4.88), (5.5.109) 

reduces to the Born relation (5.5.89). 23 F. The Halzen fulland order D. αA. calculation Morris, Phys. 

2x of ∆r Lett. will ☢B 237, 107 (199 

not be presented here. 0 We have F. attempted Halzen and to describe B. A. Kniehl, the full Nucl. formalism Phys. Bin 

353, 567 (199 

a relatively accessible way elsewhere. M. L. 23 Stong, To thePhys. extent Lett. thatB ∆277, rem is 503 small, (1992); oneF. Halzen, B. 

Z. Phys. C 58, 119 (1993); M. C. Gonzalez-Garcia, F. Halze 

23 F. Halzen and D. A. Morris, Phys. Lett. B 237, 107 (1990); Part. World 2, 10 (1991); 

Lett. 322, 233 (1994). 

F. Halzen and B. A. Kniehl, Nucl. Phys. B 353, 567 (1991); F. Halzen, P. Roy and 

M. L. Stong, Phys. Lett. B 277, 503 (1992); F. Halzen, B. A. Kniehl and M. L. Stong, 

174

Fingeprints of Electroweak theory 

e third generation, heavy quark diagram 

Diagram 

In QED 

ibutions are small in the OMS scheme and are grouped in the 

only equal mass fermions and antifermions appear in neutral photon loops 

∆ rem . 

and diagrams of this type are not possible 

iscussing the status of measurements of ∆r, we make several comleading 

Rewrite order 

∆r and = 0 and, ☢ 

using form (3.5.87) and (2.4.88), (5.5.109) 

he Born relation (5.5.89). The full 

∆ρ = G order [ α calculation of ∆r ] 

F 

m 2 t + m 2 b − 2m2 will 

ented here. We have attempted to describe the full formalism in b m2 t 

accessible way elsewhere. 23 To the extent that ∆ rem is small, one 

m 2 t /z 

∆ρ = 

≃ 

4π 

G F 

4π m2 t ≃ 3α 

16π 

m 2 t − m 2 b 

1 m 2 t 

c 2 s 2 z 

ln m2 t 

m 2 b 

contribution to an observable is forbidden in QED and QCD 

where virtual particle effects are suppressed by inverse powers of their masses 

⌚ embodies this 174 requirement because ɛ =0 for photon loops 

Conversely ☛ appearance of 

W 

t 

has important property that 

m 2 t /z 

term 

(5.5.112) 

and D. A. Morris, Phys. Lett. B 237, 107 (1990); Part. World 2, 10 (1991); 

d B. A. Kniehl, Nucl. Phys. B 353, 567 (1991); F. Halzen, P. Roy and 

Phys. Lett. B 277, 503 (1992); F. Halzen, B. A. Kniehl and M. L. Stong, 

, 119 (1993); M. C. Gonzalez-Garcia, F. Halzen and R. A. Vazquez, Phys. 

233 (1994). 


defining 

m t = m b + ɛ 

☛ 

∆ρ ≃ G F 

3π 2 ɛ 

❃ 

is a characteristic feature of electroweak theory 

⌚

We are now ready to illustrate that ... 

 

 

 

We first determine experimental value of 

Using 

G F 

√ 

2 

= 

g2 

8m 2 W 

and 

∆r exp ≃ 1 − (37.281 GeV) 2 

We next recall 

Crucial point is that 

O(α) 

∆α ≃ 1 − 

∆r exp ≠∆α 

∆r 

from 

e = g sin θ w 

♒ 

z 

ω(z − ω) ≃ 0.035 

α(0) 

α(m 2 Z ) ≃ 1 − 128 

137 ≃ 0.066 

(◬ ≠⌛) 

SM relation requires a non-vanishing 

value of 

∆ρ ≠0 

◬ 

⌛ 

∆ρ 

 

 

Using ❃ we obtain that 

∆ρ =0.0086 

and yields 

(∆r) calculated =∆α − c2 

∆ρ =0.037 

s2 in agreement with experimental value 

We leave it as an exercise to insert errors into calculation 

and show that argument survives a straightforward statistical analysis 


to insert errors into the calculation and show that our argument survives a 

straightforward Constrains statistical analysis. on Higgs Mass 

The Higgs particle makes a contribution to ∆r: 

Higgs particle makes a contribution to 

∆r 

∆h = 

∆h = 

W 

H 

= 11α 

48π 

48π 

1 

1 

c ln m2 H 

2 z . (5.5.121) 

c 2 ln m2 H 

z 

From 

we obtain that 

From (5.3.59) we obtain that ∆h < 0.0006, a contribution too small to 

contribution be sensed too small by the simple to be analysis sensed presented by simple above. analysis The quantity presented ∆r is in above 

principle Quantity sensitive ∆r is to in theprinciple Higgs mass. sensitive More sophisticated to Higgs analyses mass which 

include the dominant O(α 2 ) corrections are now yielding weak, but definite, 

constraints on the value of m H . 

Other measurements support the electroweak model’s radiative correction 

associated 

program 

with the 

is 

t¯b 

however 

loop ∆ρ. 

far 

Recall 

from 

that charged 

complete 

weak currents couple with 

problem strength Gcan F , while be quantified neutral currents by couple rewriting as ρG F ; ◬ see and (5.1.10) and as(5.1.23). 

Radiative corrections predicted by SM have successfully confronted experiment 

Using 

Z 

∆r exp = F (m W 176 ,m t ,m H ) 

-pole measurements of SLD and LEP1 

electroweak radiative corrections are evaluated 

to predict masses of top quark and 


114.4 GeV

(m W ,m t ) 

plane 

July 2008 

80.5 

LEP2 and Tevatron (prel.) 

LEP1 and SLD 

68% CL 

m W 

[GeV] 

80.4 

80.3 

!" 

m H 

[GeV] 

114 300 1000 

150 175 200 

m t 

[GeV] 

m t ,m W 

Contour curves of 68% CL in ( 

) plane 

for 

Figure 

direct 

5.5: Contour 

measurements 

curves of 68% CL 

and 

in the 

indirect 

(m t ,m W ) 

determinations 

plane for direct measurements 

and the indirect determinations. 

mThe band shows the correlation 

Band shows correlation between W and m expected in SM 

between m W and m t expected in the standard model. t 


Neutrino Oscillation 

Convincing experimental evidence exists 

for oscillatory transitions 

ν α ⇋ ν β 

Simplest and most direct interpretation of atmospheric data 

is that of muon neutrino oscillations 

Evidence of atmospheric ν µ disappearing is now at > 15σ 

most likely converting to ν τ 

Angular distribution of contained events shows that 

for 

deficit comes mainly from 

E ν ∼ 1 GeV 

between different neutrino flavors 

L atm ∼ 10 2 − 10 4 km 

These results have been confirmed by KEK-to-Kamioka (K2K) experiment 

which observes disappearance of accelerator ν µ at a distance of 

Data collected by the Sudbury Neutrino Observatory (SNO) 

in conjunction with data from Super-Kamiokande (SK) 

show that solar ν e convert to ν µ or ν τ with CL of more than 

7σ 

250 km 

KamLAND Collaboration has measured flux of ν e from distant reactors 

and find that ν e disappear over distances of about 180 km 

All these data suggest that neutrino eigenstates that travel through space 

are not flavor states that we measured through weak force 

but rather mass eigenstates 


Lepton Flavor Mixing 

|ν α 〉 |ν i 〉 

U 

Flavor eigenstates and mass eigenstates 

are related by a unitary transformation 

|ν α 〉 = ∑ i 

U αi |ν i 〉 ⇔ |ν i 〉 = ∑ α 

(U † ) iα |ν α 〉 = ∑ α 

(i.e. mixing matrix) 

U ∗ αi|ν α 〉 

with 

U † U = I, i.e., ∑ U αi Uβi ∗ = δ αβ and 

i 

For antineutrinos one has to replace U αi by 

i.e. 

|¯ν α 〉 = ∑ Uαi|¯ν ∗ i 〉 

i 

⨸ 

∑ 

i 

Uαi 

∗ 

U αi U ∗ αj = δ ij 

♁ 

Number of parameters of an unitary matrix 

It is easy to see that relative phases of n 2 

is 

neutrino states 

can be redefined such that 

n × n 

2n − 1 

(n − 1) 2 

1 

2 n(n − 1) weak mixing angles of an n 

and 

1 

2 


For these it is convenient to take: 

(n − 1)(n − 2) 

CP 

-violating phases 

2n 

independent parameters are left 

-dimensional rotation

Hamiltonian Mechanics 

⚈ 

Being eigenstates of mass matrix ☛ states 

|ν i 〉 

(i.e. they have the time dependence) 

are stationary states 

|ν i (t)〉 = e −iE it |ν i 〉 

where 

with 

E ≈ p 

E i = 

√ 

☛  here it is assumed that 

p 2 + m 2 i ≈ p + m2 i 

2p ≈ E + m2 i 

2E 

is total neutrino energy 

neutrinos are stable 

 

⚈ 

A pure flavor state 

|ν α 〉 = ∑ i 

U αi |ν i 〉 present at t =0 

develops with time into state 

|ν(t)〉 = ∑ i 

U αi e −iE it |ν i 〉 = ∑ i,β 

U αi U ∗ βie −iE it |ν β 〉 


Transition Amplitude 

Time dependent transition amplitude 

for transition from flavor 

ν α 

to flavor 

ν β 

is 

A(ν α → ν β ) ≡〈ν β |ν(t)〉 = ∑ i 

U αi U ∗ βie −iE it 

 

= ∑ i,j 

U αi δ ij e −iE it (U † ) jβ 

with D ij = δ ij e −iE it 

(diagonal matrix) 

= (UDU † ) αβ 

An equivalent expression for transition amplitude is obtained by 

inserting into and extracting an overall phase factor e −iEt 

A(ν α → ν β ,t) = ∑ i 

U αi Uβi ∗ e − im2 i t 

2E 

= ∑ i 

U αi Uβi ∗ e − im2 i L 

2E 

L = ct 

c =1 

where (recall ) is distance of detector 

in which ν β is observed from ν α source 


i → j 

For an arbitrary chosen fixed 

Transition Amplitude 

j 

transition amplitude becomes 

Ã(ν α → ν β ,t) = e iE jt A(ν α → ν β ,t) 

☤ 

= ∑ i 

U αi U ∗ βi e −i(E i−E j ) t 

= δ αβ + ∑ i 

U αi U ∗ βi 

[ 

] 

e −i(E i−E j )t − 1 

= δ αβ + ∑ i≠j 

U αi U ∗ βi 

[ 

e 

−i∆ ij 

− 1 ] 

with 

∆ ij =(E i − E j ) = 1.27 δm2 ij L 

E 

L is measured in km in GeV and δm 2 ij = m 2 i − m 2 j in eV 2 


E 

when 

(in ☤unitarity relation ♁ has been used)

Properties of Transition Amplitude 

✔ 

Transition amplitudes are thus given by 

(n − 1) 2 

n(n − 1) 

independent parameters of unitary matrix 

(which determines sizes of oscillations) 

real parameters 

and 

n − 1 

mass square differences 

(which determine frequencies of 

oscillations) 

✔ 

If 

CP 

is conserved in neutrino oscillations 

all -violating phases vanish and U αi are real 

i.e. 

U 

CP 

is an orthogonal matrix 

U −1 = U T with 

1n(n − 1) 

Number of parameters for transition amplitude is then 

2 

parameters 

1 

2 

(n − 1)(n + 2) 

✔ 

Using ⨸ we obtain amplitudes for transitions between antineutrinos 

A(¯ν α → ¯ν β ; t) = ∑ i 

U ∗ αiU βi e −iE it 

 


This follows directly from 

and 

CPT 

Comparing and following relation holds for 

C 

P 

transformations between neutrinos and antineutrinos 

CP T 

theorem: 

provides necessary flip from left-handed neutrino 

T 

A(¯ν α → ¯ν β )=A(ν β → ν α ) ≠ A(ν α → ν β ) 

changes particle into antiparticle 

to right-handed antineutrino and vice versa 

reverses arrow indicating transition 

If CP is conserved ☛ and are real in and 

U αi 

U βi 

That is ☛ if time reversal invariance holds one has 

A(¯ν α → ¯ν β )=A(ν α → ν β )=A(¯ν β → ¯ν α )=A(ν β → ν α ) 

Therefore ☛ 

CP 

e.g. by comparing oscillations 

violation can be searched for 

ν α → ν β 

and 

ν β → ν α 


Transition Probability 

Transition probabilities are obtained by squaring moduli of amplitudes 

P να →ν β 

= 

∑ 

∣ 

i 

U αi Uβie ∗ −iE it 

∣ 

2 

⌘ 

= δ αβ − 4 ∑ i>j 

Re(U ∗ αi U βi U αj U ∗ βj) sin 2 ∆ ij 

+ 2 ∑ i>j 

Im(U ∗ αi U βi U αj U ∗ βj) sin 2∆ ij 


Matching Experimental data 

✻ 

✻ 

In standard treatment of neutrino oscillations 

flavor eigenstates 

|ν α 〉(α = e, µ, τ) 

are expanded in terms of 3 mass eigenstates |ν i 〉(i =1, 2, 3) 

Atmospheric neutrino data suggest corresponding 

corresponding oscillation phase must be maximal ∆ atm ∼ 1 

which requires 

δm 2 atm ∼ 10 −4 − 10 −2 eV 2 

✻ 

Assuming 

all upgoing multi-GeV ν µ 

while none of downgoing ones do 

oscillate into different flavor 

observed up-down asymmetry 

leads to a mixing angle very close 

to maximal 

sin 2 2θ atm > 0.85 

✻ 

✻ 

Combined analysis of atmospheric neutrinos with K2K 

leads to a best fit-point and 

δm 2 atm =2.2 +0.6 

−0.4 × 10−3 eV 2 and tan 2 θ atm =1 +0.35 

1σ 

ranges 

On the other hand ☛ reactor data suggest 


|U e3 | 2 ≪ 1 

−0.26

Mass eigenstates 

To simplify... 

|U we use fact that e3 | 2 is nearly zero to ignore possible CP 

and assume that elements of 

With this in mind ☛ we can define a mass basis as follows 

U 

are real 

violation 

|ν 1 〉 = sin θ ⊙ |ν ⋆ 〉 + cos θ ⊙ |ν e 〉 

|ν 2 〉 = cos θ ⊙ |ν ⋆ 〉−sin θ ⊙ |ν e 〉 

where 

θ ⊙ 

is solar mixing angle 

|ν 3 〉 = 1 √ 

2 

(|ν µ 〉 + |ν τ 〉)

Flavor Eigenstates 

Inversion of neutrino mass-to-flavor mixing matrix leads to 

|ν e 〉 = cos θ ⊙ |ν 1 〉−sin θ ⊙ |ν 2 〉 & |ν ⋆ 〉 = sin θ ⊙ |ν 1 〉 + cos θ ⊙ |ν 2 〉 

by adding

For 

Far-out neutrinos 

∆ ij ≫ 1 phases will be erased by uncertainties in L and E 

(as would be case for neutrinos propagating over cosmic distances), 

averaging over 

sin 2 ∆ ij 

in ⌘ we obtain 

P (ν α → ν β )=δ αβ − 2 ∑ i>j 

U αi U βi U αj U βj 

⌬ 

using 

2 ∑ 1>j 

= ∑ i,j 

− ∑ i=j 

, 

⌬ can be re-written as 

P (ν α → ν β ) = δ αβ − ∑ i,j 

U αi U βi U αj U βj + ∑ i 

U αi U βi U αi U βi 

⍒ 

= δ αβ − 

( ∑ 

i 

U αi U βi 

) 2 

+ ∑ i 

U 2 αiU 2 βi 

Since 

δ αβ = δ 2 αβ 

first and second terms in ⍒ cancel each other 

P (ν α → ν β )= ∑ i 

U 2 αi U 2 βi 


Average Probability 

Probabilities for flavor oscillation are then 

P (ν µ → ν µ )=P (ν µ → ν τ )= 1 4 (cos4 θ ⊙ + sin 4 θ ⊙ + 1) 

P (ν µ → ν e )=P (ν e → ν µ )=P (ν e → ν τ ) = sin 2 θ ⊙ 

cos 2 θ ⊙ 

P (ν e → ν e ) = cos 4 θ ⊙ + sin 4 θ ⊙ 

Let ratios of neutrino flavors at production in cosmic sources 

be written as 

w e : w µ : w τ 

so that relative fluxes 

with 

∑ 

α 

w α =1 

of each mass eigenstate are w j = ∑ α 

ω α U 2 αj 

We conclude that probability of measuring on Earth a flavor 


P να detected = ∑ j 

∑ 

Uαj 

2 β 

w β U 2 βj 

α 

is

Earhtly Ratios 

⍜ Any initial flavor ratio that contains w e =1/3 

will arrive at Earth with equipartition on 3 flavors 

⍜ Cosmic neutrinos arise dominantly from decay of charged pions 

their initial flavor ratios of nearly 

1 : 2 : 0 

should arrive at Earth democratically distributed 

⍜ Fairly robust prediction of 1 : 1 : 1 cosmic neutrino flavor ratios 

⍜ Prediction for pure ¯ν e source originating via neutron β -decay 

has different implications for flavor ratios 

yields Earthly ratios 

w e =1 ∼ 5 : 2 : 2 

⍜ Such a unique ratio would appear above 

in direction of neutron source 

1 : 1 : 1 

background 

⍜ Flavor identification of neutrinos on a statistical basis 

becomes possible at IceCube 

opening up a window for discoveries in particle physics 

not otherwise accessible to experiment 


the Vampire 

❖ 

❖ 

❖ 

❖ 

In SM masses arise from Yukawa interactions 

which couple right-handed fermion with its left-handed doublet 

after spontaneous symmetry breaking 

Because no right-handed neutrinos exist in SM 

Yukawa interactions leave neutrinos massless 

One may wonder if neutrino masses could arise from loop corrections 

or even by non-perturbative effects 

but this cannot happen 

because any neutrino mass term that can be constructed with SM fields 

would violate total lepton symmetry 

To introduce a neutrino mass term 

we must either extend particle content 

or else abandon gauge invariance and/or renormalizability 


How to kill a Vampire 

♰ 

We keep gauge symmetry 

and introduce arbitrary number 

right-handed neutrino states 

(singlets under hypercharge) 

m 

ν Rs (1, 1) 0 

of additional 

With particle contents of SM 

and addition of an arbitrary number of right-handed neutrinos 

one can construct two types of mass terms 

that arise from gauge invariant renormalizable operators 

m 

−L Mν = ∑ 

α=e,µ,τ 

m∑ 

i=1 

M D 

iα ¯ν Ri ν Lα + 1 2 M N ij ¯ν Ri ν c Rj + h.c. 

♰ indicates a charge conjugated field 

ν C (ν c = C ¯ν T ) 

♰ M D is a complex m × 3 matrix 

and M N 


is a symmetric matrix of dimension m × m

Dirac Neutrinos 

⎊ 

Forcing 

M N =0 

leads to a Dirac mass term 

which is generated from Yukawa interactions 

after spontaneous electroweak symmetry breaking 

Y ν 

iα ¯ν Ri ¯φ† L Lα ⇒ M D iα = Y ν 

iα 

v √2 

similarly to charged fermion masses 

⎊ 

⎊ 

⎊ 

⎊ 

Such a mass term conserves total lepton number 

but it breaks the lepton flavor number symmetries 

For 

m =3 

we can identify the hypercharge singlets 

with right-handed component of 4-spinor neutrino field 

Y 3 × 3 

Since matrix is (in general) a complex matrix 

flavor neutrino fields ν e ,ν µ and ν τ do not have a definite mass 

Massive neutrino fields are obtained via diagonalization of L Mν 


If 

M N ≠0 

Majorana neutrinos 

☛ neutrino masses 

receive important contribution from Majorana mass term 

Such a term is singlet of SM gauge group 

Therefore ☛ it can appear as bare mass term 

Furthermore ☛ since it involves two neutrino fields 

it breaks lepton number by two units 

More generally ☛ such a term is allowed 

only if neutrinos carry no additive conserved charge 

This is reason that such terms are not allowed 

for any charged fermions which (by definition) carry U(1) EM 


Majorana neutrinos (cont’d) 

Dirac mass terms are protected by symmetries of SM 

(neutrino masses 

∝ 

EW symmetry-breaking scale) 

Majorana mass term is SM singlet 

elements of M N are not protected by SM symmetries 

It is plausible that Majorana mass term 

is generated by new physics beyond SM 

and right-handed chiral neutrino fields ν R 

belong to nontrivial multiplets of symmetries of high energy theory 


Title 

Title 

SM accuracy has been shown 

in a variety of experiments 

to an amazing level 

☛ some observables 

beyond even one part in a million 

Saga of SM is still exhilarating 

because it leaves all questions 

of consequence unanswered 

Most evident of unanswered questions is why weak interactions are weak 

In gauge theory 

natural values for m W 

are zero or Planck mass 

SM does not contain physics that dictates 

why its actual value is of order 100 GeV 

Thursday, November 17, 

onday, November 14, 

Monday, 2011 

2011 

November 14, 2011

The Ugly... 

e 

u d 

s 

µ 

c 

τ 

b 

t 

10 −4 

10 −3 

10 −2 

ν 1 

ν 2 

ν 3 

m [eV] 

10 −1 

10 0 

10 1 

10 2 

10 3 

10 4 

10 5 

10 6 

10 7 

10 8 

10 9 

10 10 

10 11 

10 12 

Figure 5.6: Order of magnitude of the masses of quarks and leptons. 

Order of magnitude of masses of quarks & leptons 

ith real positive masses m k . 

ritten as 

The resulting diagonal mass term can b 

−L Mν = 

3∑ 

m k¯ν Rk ν Lk + h.c. = 

3∑ 

m k ν k ν k (5.6.156 


k=1 

k=1

The Good... 

Already in 1934 Fermi provided an answer 

with a theory that prescribed a quantitative relation 

between fine structure constant and weak coupling 

G F ∼ α/m 2 W 

Although Fermi adjusted m W 

to accommodate strength and range of nuclear radioactive decays 

one can readily obtain a value of m W of 

from observed 

µ 

Answer is off by a factor of 2 

40 GeV 

decay rate for which proportionality factor is 

because discovery of parity violation and neutral currents 

was in future and introduces an additional factor 1 − m 2 W /m 2 Z 

[ ] [ ] 

πα 1 

G F = √ 

2m 

2 

W 

1 − m 2 (1 + ∆r) 

W /m2 Z 

π/ √ 2 

Fermi could certainly not have anticipated 

that we now have a renormalizable gauge theory 

that allows us to calculate radiative correction 

∆r 

to his formula 


der el sorted predictions dimensional to rather with operators extreme too highin lengths precision, Weinberg’s by proposing pushing effective Λ to that Lagrangian 10the TeV. factor 

ection esorted rder dimensional (2m to rather 

Higgs 2 operators extreme 

mass 

lengths in Weinberg’s by proposing 

Radiative effective that Lagrangian the factor 

Corrections 

rection resorted(2m to rather 2 W + m2 

W extreme lengths by proposing that the factor 

One of most + Z + m2 

m2 Z + H − 4m2 

acute m2 H − t ) must vanish; exactly! This 

ion. The problem is now “solved” 4m2 t ) must 

problems because 

vanish; 

connected scales 

exactly! 

as 

This 

large with asultraviolet divergences 

tion. rrection The (2mproblem 2 W 

concerns + m2 is 

Z 

radiative + now m2 “solved” 

H − 4m2 t ) must because vanish; scales exactly! as large This as 

accommodated by the observations 

corrections 

once one 

to 

eliminates 

mass appearing 

the 

in Higgs potential 

ition. accommodated The problemby isthe now observations “solved” because once one 

V = µ 2 scales eliminates 

ΦΦ † as large the 

make this stick to all orders and for Λ ≤ 10 TeV, this requires 

as 

emake accommodated this stick toby allthe orders observations and for Λonce ≤ 10one TeV, eliminates this 

+ 

requires 

λ(Φ the † Φ) 2 

n make 

f, W, 

this stick to all orders 

self-coupling 

and for Λ ≤ 10 

radiative 

TeV, this requires 

and H 

corrections to Higgs mass are 

H 

H 

f 

f 

H 

H 

i Y f 

2 

2 

∫ Λ 

d 4 k 

(2π) 4 tr( 

i 

̸ k − m f 

i 

̸ k+ ̸ p − m f 

) ∼ −Λ 2 tr(I) 

Y 2 

f 

32π 2 

H 

H 

W,Z 

W,Z 

HH 

H 

i g2 

4 

∫ Λ 

d 4 k 1 

(2π) 4 k 2 − m 2 W 

∼ Λ 2 g 2 

64π 2 

Z 

m W → m Z 

g 2 → g 2 + g ′ 2 

H 

H 

H 

H 

H 

H 

H 

H 

i6λ 

∫ Λ 

d 4 k 1 

(2π) 4 k 2 − m 2 H 

∼ Λ 2 3λ 

8π 2 

m 2 W = 1 4 g2 v 2 , v = 246 GeV, m 2 Z = 1 4 (g2 + g ′2 )v 2 , m 2 f = 1 2 Y 2 

f v 2 

Y f 

is Yukawa coupling 

m 2 H =2λv 2 , g g ′ SU(2) L × U(1) Y 

λ is quartic Higgs coupling and Λ is a cutoff 


and are gauge couplings

Difference between bare and renormalized masses is 

∆µ 2 = 

Veltman Condition 

⎛ 

1 

⎝9g 2 

64π 2 +3g ′2 + 24 λ − 8 ∑ f 

N f Y 2 

f 

⎞ 

⎠ Λ 2 

≃ 

3 

16π 2 v 2 (2m2 W + m 2 Z + m 2 H − 4m 2 t )Λ 2 

SM works amazingly well by fixing 

at electroweak scale 

generally assumed ☛ this indicates existence of new physics beyond SM 

Following Weinberg 

Λ 

L(m W )=|µ 2 | H † H + 1 4 λ(H† H) 2 + L gauge 

SM 

+ LYukawa SM + 1 Λ L5 + 1 Λ 2 L6 + . . . 

where operators of higher dimension parametrize physics beyond SM 

Some have resorted to rather extreme lengths by proposing that 

factor multiplying unruly quadratic correction 

(2m 2 W + m 2 Z + m 2 H − 4m 2 t ) 

must vanish exactly! 

By most conservative estimates this new physics is within our reach 


New Physics at the TeV Scale? 

Avoiding fine tuning requires 

E.G. for m H = 115 − 200 GeV, 

∆µ 2 ∣ ∣∣∣ 

∣ µ 2 = δv2 

v 2 

Λ 2 − 3 

to be revealed by CERN LHC 

≤ 10 ⇒ Λ = 2 − 3 TeV 

we have implicity used 

v 2 = −µ 2 /λ 

[valid in approximation 

disregarding terms beyond 

Let's contemplate possibilities 

O(H 4 ) 

in Higgs potential] 

⌾ 

Veltman condition happens to be satisfied 

and this would leave particle physics with an ugly fine tuning problem 

⌾ 

⌾ 

This is very unlikely ☛ LHC must reveal Higgs physics 

already observed via radiative correction 

or at least discover physics that implements Veltman condition 

It must appear at 

2 − 3TeV 

even though higher scales can be rationalized 

when accommodating selected experiments 


SUSY 

Supersymmetry predicts that interactions exist 

that would change fermions into bosons and vice versa 

and that all known fermions (bosons) have SUSY boson (fermion) partner 

SUSY offers solution of bad behavior of radiative corrections in SM 

As for every boson there is a companion fermion 

bad divergence associated with Higgs loop 

cancelled by a fermion loop with opposite sign 

H 

Figure 1: Supersymmetry offers a neat solution of the bad behavior of radiative corrections in the 

standard model. As for every boson there is a companion fermion, the bad divergence associated 

with the Higgs loop is cancelled by a fermion loop with opposite sign. 

even though it elegantly controls quadratic divergence 

by cancellation of boson and fermion contributions 

Let’s contemplate the possibilities. The Veltman condition happens 

2 − 

to 

3TeV 

be satisfied and this 

would leave particle physics with an ugly fine tuning problem. This is very unlikely; LHC must 

reveal the Higgs physics already observed via radiative correction, or at least discover the physics 

it is already fine-tuned at a scale of 


The Bad... 

Next class 

U have a test! 



Happy thanksgiving

Lecture XI

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?