Z bb in Higgsless Models PRD 79 , 075016 (2009) [ arXiv:0902.3910 ] - - PowerPoint PPT Presentation

Z → bb in Higgsless Models

PRD 79, 075016 (2009) [arXiv:0902.3910]

Ken Hsieh

Pheno 09 Madison, Wisconsin May 11, 2009

In collaboration with: Tomohiro Abe (Nagoya University) Sekhar Chivukula (Michigan State University) Neil Christensen (Michigan State University) Shinya Matsuzaki (University of North Carolina) Elizabeth Simmons (Michigan State University) Masaharu Tanabashi (Nagoya University)

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

Why Calculate Z → bLbL?

The top quark is special in the SM model. Solutions to the hierarchy problem necessarily involve modifications / additions to the top sector, and the top quark mass sets a scale for flavor hierarchy. Through SU(2)L, the coupling between Z and bL feels these modifications and the well-measured Z-pole

• bservables (ex: Γ(Z → bLbL) and Ab

LR) serve as strong

constraints. In this talk, we elucidate the gaugeless technique of calculating flavor-dependent corrections to Z → bLbL, and apply this technique to the Three-Site Higgsless Model. The reference contains many other ways of understanding the result of this calculation.

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

Z → bLbL in the SM

The brute-force calculation is straightforward (mb = 0, m2

W ≪ m2 t ):

= ✐ gZm2

t

192π2v2

• 6
• 1

ε + ln µ2 m2

t

• (4s2

W − 4) + (20s2 W − 12)

• = ✐

gZm2

t

192π2v2

• 6
• 1

ε + ln µ2 m2

t

• (7 − 6s2

W) + (39 − 30s2 W)

• = ✐

gZm2

t

192π2v2

• 6
• 1

ε + ln µ2 m2

t

• (2s2

W − 3) + (10s2 W − 15)

• ✐

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

Z → bLbL in the SM

The brute-force calculation is straightforward (mb = 0, m2

W ≪ m2 t ):

= ✐ gZm2

t

192π2v2

• 6
• 1

ε + ln µ2 m2

t

• (4s2

W − 4) + (20s2 W − 12)

• = ✐

gZm2

t

192π2v2

• 6
• 1

ε + ln µ2 m2

t

• (7 − 6s2

W) + (39 − 30s2 W)

• = ✐

gZm2

t

192π2v2

• 6
• 1

ε + ln µ2 m2

t

• (2s2

W − 3) + (10s2 W − 15)

• = ✐ gZm2

t

16π2v2 ⇐ The result is finite, as expected, because there is no flavor-dependent counter- term in the SM.

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

The Gaugeless Calculation

Barbieri, Beccaria, Ciafaloni, Curi, and Vicere.

• Phys. Lett. B288, 95 (1992). Nucl. Phys. B409, 105 (1993).

The key idea: treat Z-boson as external field that couples to Jµ

3 − s2 θJµ Q.

      −→   Z

b b

π Z MZ   For the gaugeless calculation, we compute the coefficient to the op- erator ∂ µπZbLγµbL and relate to Z → bLbL through a Ward-Takahashi identity       ∼ MZ    

pµ

+    Wavefunction renormalization contributions   

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

πZ → bLbL in the SM

This is easier than the Z → bLbL calculation: MZ      

pµ

= ✐ gZm2

t

16π2v2 MZ      

pµ

= 0 (No πππ-vertex) MZ   

t b b

π W π Z

b

 

pµ

= 0 (No πbLbR-vertex when mb = 0.) ✐

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

πZ → bLbL in the SM

This is easier than the Z → bLbL calculation: MZ      

pµ

= ✐ gZm2

t

16π2v2 MZ      

pµ

= 0 (No πππ-vertex) MZ   

t b b

π W π Z

b

 

pµ

= 0 (No πbLbR-vertex when mb = 0.) MZ

• pµ

= ✐ gZm2

t

16π2v2 ⇐ There is only one, finite diagram in the SM.

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

The Gaugeless Limit

Barbieri, Beccaria, Ciafaloni, Curi, and Vicere.

• Phys. Lett. B288, 95 (1992). Nucl. Phys. B409, 105 (1993).

In the gaugeless limit, we treat the massive Z-boson as an external, clas- sical field that couples to the conserved (before EWSB) current Jµ = −MZ∂ µπZ + ˆ Jµ, ˆ Jµ = gL

ZbbbLγµPLbL + gR ZbbbRγµPRbR + ··· .

The Ward-Takahashi identity then reads ∂ x

µ

• Tˆ

Jµ(x)b(y)b(z)

• = MZ
• T(xπZ(x))b(y)b(z)
• − δ(x − y)
• gL

ZbbPL + gR ZbbPR

• b(x)b(z)
• + δ(x − z)
• b(y)b(x)
• gL

ZbbPR + gR ZbbPL

• ,

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

The Gaugeless Limit - cont.

In terms of amputated Green’s functions, we have ✐(p1 + p2)µ

• ˆ

Jµ(p1 + p2)b(p2)b(p1)

• 1PI

= −✐MZ

• πZ(p1 + p2)b(p2)b(p1)
• 1PI

− S−1

bb (p1)

• gL

ZbbPL + gR ZbbPR

• +
• gL

ZbbPR + gR ZbbPL

• S−1

bb (−p2),

✐ ✐

✶ ✶

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

The Gaugeless Limit - cont.

In terms of amputated Green’s functions, we have ✐(p1 + p2)µ

• ˆ

Jµ(p1 + p2)b(p2)b(p1)

• 1PI

= −✐MZ

• πZ(p1 + p2)b(p2)b(p1)
• 1PI

− S−1

bb (p1)

• gL

ZbbPL + gR ZbbPR

• +
• gL

ZbbPR + gR ZbbPL

• S−1

bb (−p2),

We can organize the Ward-Takahashi identity with the Lorentz structure of the Green’s functions ✐ ✐

✶ ✶

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

The Gaugeless Limit - cont.

In terms of amputated Green’s functions, we have ✐(p1 + p2)µ

• ˆ

Jµ(p1 + p2)b(p2)b(p1)

• 1PI

= −✐MZ

• πZ(p1 + p2)b(p2)b(p1)
• 1PI

− S−1

bb (p1)

• gL

ZbbPL + gR ZbbPR

• +
• gL

ZbbPR + gR ZbbPL

• S−1

bb (−p2),

We can organize the Ward-Takahashi identity with the Lorentz structure of the Green’s functions ✐(p /1 + p /2)    

p2 p1

b b Jµ

   

γµ

= −✐MZ(p /1 + p /2)   

p2 p1 p1+p2

π Z b b

 

p

−

• −1

p p

/1

• gL

ZbbPL + gR ZbbPR

• +
• gL

ZbbPR + gR ZbbPL

• −1

p

p /2, At tree level, this gives the gauge couplings of the Z-boson.

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

The Gaugeless Limit - cont.

In terms of amputated Green’s functions, we have ✐(p1 + p2)µ

• ˆ

Jµ(p1 + p2)b(p2)b(p1)

• 1PI

= −✐MZ

• πZ(p1 + p2)b(p2)b(p1)
• 1PI

− S−1

bb (p1)

• gL

ZbbPL + gR ZbbPR

• +
• gL

ZbbPR + gR ZbbPL

• S−1

bb (−p2),

We can organize the Ward-Takahashi identity with the Lorentz structure of the Green’s functions ✐(p1 + p2)µ    

p2 p1

b b Jµ

   

γµγ5

(= 0) = −✐MZ   

p2 p1 p1+p2

π Z b b

 

γ5

−

• −1

✶

• gL

ZbbPL + gR ZbbPR

• +
• gL

ZbbPR + gR ZbbPL

• −1

✶

, At tree level, this relates the πZbb coupling to mb.

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

The Ward Identity at Loop-Level

The one-loop ZbLbL coupling is given by ✐ g

L,1-loop Zbb

= ✐    

p2 p1

b b Jµ

   

γµ

✐ ✐ ✐ ✐ ✐

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

The Ward Identity at Loop-Level

The one-loop ZbLbL coupling is given by ✐ g

L,1-loop Zbb

= ✐

• ZL

b

   

p2 p1

b b Jµ

   

γµ

• ZL

b

✐ ✐ ✐ ✐ ✐

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

The Ward Identity at Loop-Level

The one-loop ZbLbL coupling is given by ✐ g

L,1-loop Zbb

= ✐

• ZL

b

   

p2 p1

b b Jµ

   

γµ

• ZL

b

= − ✐ MZ

• ZL

b

  

p2 p1 p1+p2

π Z b b

 

p

• ZL

b −

• gL

Zbb

• ZL

b

• −1

p

• ZL

b

✐ ✐ ✐ ✐

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

The Ward Identity at Loop-Level

The one-loop ZbLbL coupling is given by ✐ g

L,1-loop Zbb

= ✐

• ZL

b

   

p2 p1

b b Jµ

   

γµ

• ZL

b

= − ✐ MZ

• ZL

b

  

p2 p1 p1+p2

π Z b b

 

p

• ZL

b −

• gL

Zbb

• ZL

b

• −1

p

• ZL

b

= − ✐ MZ   

p2 p1 p1+p2

π Z b b

 

p

+ ✐

• g

L,tree Zbb

• 1 + 1

2δZL

b

• 1 − δZL

b

• 1 + 1

2δZL

b

• ✐

✐

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

The Ward Identity at Loop-Level

The one-loop ZbLbL coupling is given by ✐ g

L,1-loop Zbb

= ✐

• ZL

b

   

p2 p1

b b Jµ

   

γµ

• ZL

b

= − ✐ MZ

• ZL

b

  

p2 p1 p1+p2

π Z b b

 

p

• ZL

b −

• gL

Zbb

• ZL

b

• −1

p

• ZL

b

= − ✐ MZ   

p2 p1 p1+p2

π Z b b

 

p

+ ✐

• g

L,tree Zbb

• 1 + 1

2δZL

b

• 1 − δZL

b

• 1 + 1

2δZL

b

• =✐ g

L,tree Zbb − ✐MZ

  

p2 p1 p1+p2

π Z b b

 

p

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

The Ward Identity at Loop-Level

The one-loop ZbLbL coupling is given by ✐ g

L,1-loop Zbb

= ✐

• ZL

b

   

p2 p1

b b Jµ

   

γµ

• ZL

b

= − ✐ MZ

• ZL

b

  

p2 p1 p1+p2

π Z b b

 

p

• ZL

b −

• gL

Zbb

• ZL

b

• −1

p

• ZL

b

= − ✐ MZ   

p2 p1 p1+p2

π Z b b

 

p

+ ✐

• g

L,tree Zbb

• 1 + 1

2δZL

b

• 1 − δZL

b

• 1 + 1

2δZL

b

• =✐ g

L,tree Zbb − ✐MZ

  

p2 p1 p1+p2

π Z b b

 

p

⇐ This justifies the earlier calculation.

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

The Three-Site Higgsless Model - Set up

Chivukula, Coleppa, Di Chiara, Simmons, He, Kurachi, and Tanabashi. PRD 74, 075011 (2006). *Additional references in the paper.

Three-site → maximally deconstructed. Higgsless → non-linear Σ-fields. Implements ideal delocalization: αS = 0 at tree-level.

G = f 2

1

4 Tr

• DµΣ1

† DµΣ1

• +

f 2

2

4 Tr

• DµΣ2

† DµΣ2

• ,

Σ = Exp

• ✐

πA

1

f1 σA

• ,

DµΣ1 = ∂ µΣ1 + ✐˜ g

• Wµ

H · Σ1

• − ✐g
• Σ1 · Wµ

L

• ,

−F = MD

• εLψL0Σ1ψR1 + ψL1ψR1 + εRψL1Σ2ψR2
• Ken Hsieh - Michigan State University

Z → bb in Higgsless Models (arXiv:0902.3910)

The Three-Site Higgsless Model - Spectra and Features

Spectra and mass-generation of the Three-Site model: Additional gauge bosons MW′,Z′ ∼ ˜ gf > 400 GeV Additional fermions MF ∼ MD 3 TeV. SM fermion masses generated through a seesaw-type mechanism (Mf ∼ εLεRMD). Flavor-dependencies stored in εR: εL is flavor-blind at tree-level and crucial for implementing ideal-delocalization. The new ingredients to Z → bLbL in 3SHM The physical states are no longer gauge eigenstates, even at tree-level. The presence of off-diagonal couplings to the Z (e.g. ZµBLγµbL). We can not take the gaugeless limit for SU(2)h: it remains gauged.

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

The Ward-Takahashi Identity in 3-Site Higgsless Model

The current now contains extra contributions ... Jµ

3S = ˆ

Jµ

3S − MZ∂ µπZ,

ˆ Jµ

3S = bγµgZbbb +

• BγµgZBbb + h.c.
• + ··· .

gZbb ≡ gL

ZbbPL + gR ZbbPR,

˜ gZbb ≡ gL

ZbbPR + gR ZbbPL,

and so does the Ward-Takahashi identity ∂ x

µ

• Tˆ

Jµ

3S(x)b(y)b(z)

• = MZ
• T(xπZ(x))b(y)b(z)
• − δ(x − y)
• gZbb
• b(x)b(z)
• + gZBb
• B(x)b(z)
• + δ(x − z)
• b(y)b(x)
• ˜

gZbb +

• b(y)B(x)
• ˜

gZBb

• .

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

Amputation with Kinetic Mixing

The Ward-Takahashi identity is a relationship among full Green’s functions: ∂ x

µ

• Tˆ

Jµ

3S(x)b(y)b(z)

• = MZ
• T(xπZ(x))b(y)b(z)
• − δ(x − y)
• gZbb
• b(x)b(z)
• + gZBb
• B(x)b(z)
• + δ(x − z)
• b(y)b(x)
• ˜

gZbb +

• b(y)B(x)
• ˜

gZBb

• .

With kinetic-mixing, we have to properly amputate full Green’s functions.

• =
• b
• +
• B

b

• +
• b

B

• +
• B

b

• B

b B

• Ken Hsieh - Michigan State University

Z → bb in Higgsless Models (arXiv:0902.3910)

Some Intermediate Steps

Compared to the SM, we have many more terms ... ✐(p1 + p2)µ

• b

+

• −1

B b

• +
• b

B

• −1

= − ✐MZ

• b

+

• −1

B b

• +
• b

B

• b

B

• −1

−

• −1 gZbb −
• −1 gZBb
• b

B

• −1

+ ˜ gZbb

• −1 +
• −1

B b

• ˜

gZBb

• −1 ,

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

Some Intermediate Steps

Compared to the SM, we have many more terms ... but they are all one-loop at leading order. ✐(p1 + p2)µ

• b

+

• −1

B b

• +
• b

B

• −1

= − ✐MZ

• b

+

• −1

B b

• +
• b

B

• b

B

• −1

−

• −1 gZbb −
• −1 gZBb
• b

B

• −1

+ ˜ gZbb

• −1 +
• −1

B b

• ˜

gZBb

• −1 ,

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

Kinetic Mixing

The final result ✐ g

L,1-loop Zbb,3S = ✐ g L,tree Zbb − ✐MZ

      

p2 p1 p1+p2

π Z b b

 

γµ

− gπZBLbL MB

• p

B

• p

   , = g

L,tree Zbb + gZ

m2

t

16π2v2

• 1 + 1

8 ln Λ2 M2

D

• .

Some features: Only kinetic-mixing (not mass-mixing) enters the calculation: we are after the coefficient of the operator (∂µπZ)bLγµbL. Only ln(Λ2/M2) and not ln(M2/m2

t ):

We can understand ln(Λ2/M2) from RGE effects of εL, and there is no flavor-non-universal operator that gives further scaling below M.

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

Other Checks & Conclusions

In the paper, we obtain the same result and further insights via several other calculations: direct calculation in unitary gauge, renormalization group analysis, chiral current.

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)

Other Checks & Conclusions

In the paper, we obtain the same result and further insights via several other calculations: direct calculation in unitary gauge, renormalization group analysis, chiral current. We can also apply this technique to other models and derive constraints on the relevant parameters. One particular model is the general Universal Extra Dimension model (in progress: N. Chris-

tensen, T. Flacke, K. Hsieh, and A. Menon).

Ken Hsieh - Michigan State University Z → bb in Higgsless Models (arXiv:0902.3910)