liuzuo

Z Prime in Stueckelberg ModelsZuowei Liu

C.N. Yang Institute for Theoretical PhysicsStony Brook University

July 15, 2011 @ WeihaiZ Factory Conference

Outline•Stueckelberg models (St models)

•Z and Z Prime bosons

•Tevatron and LHC signatures, narrow resonance

•Kinetic mixing and mass mixing

•Hidden sector milli-charged dark matter

•PAMELA and Breit-Wigner enhancement

•CDF Wjj anomaly

Vector Boson Mass Growth without the Higgs Mechanism

L = −1

4FµνF

µν + gAµJµ − 1

2(mAµ + ∂µσ)

2

Stueckelberg Lagrangian

δAµ = ∂µλ

δσ = −mλ.

is invariant under the following gauge transformation

E.C.G. Stueckelberg, Helv. Phys. Acta 11, 225(1938)

See e.g. also Ogievetskii, Polubarinov, Kalb, Ramond, Kors, Nath, Feldman,

ZL, Cheung, Yuan, Kumar, Rajaraman, Wells, Zhang, Wang, Mambrini, ...

LSt = −1

4CµνC

µν + gXCµJµX − 1

2(∂µσ +M1Cµ +M2Bµ)

2

invariant under

δY Bµ = ∂µλY , δY σ = −M2λY , U(1)Y

δXCµ = ∂µλX , δXσ = −M1λX , U(1)X

• Cµ is the gauge field for U(1)X ; Bµ is the gauge field for U(1)Y

• σ is the axion; gX is the gauge coupling (gX ∼ gY )

• visible matter, JµX = qγµqQX ; hidden matter, Jµ

X = χγµχQX

• two mass parameters (M1, M2), or (M1, ≡ M2/M1)

Total Lagrangian LStSM = LSM + LSt

B. Kors and P. Nath, Phys. Lett. B 586, 366 (2004)

Stueckelberg Extension of Standard Model - StSM

Neutral Sector Mass MixingMass-squared matrix M2 for the neutral vector bosons Vµ = (Cµ, Bµ, A3

µ)

−1

2

3

a,b=1

VµaM2abV

µb

where M2 =

M2

1 M21 0

M21 M2

1 2 + 1

4g2Y v

2 − 14gY g2v

2

0 − 14gY g2v

2 14g

22v

2

Since Det(M2) = 0, we have one massless mode (photon) and two massivemodes (Z boson and Z boson).

The two non-vanishing eigenvalues are

M2± =

1

2

M2

1 (1 + 2) +v2

4(g2Y + g22)

±

M21 (1 + 2) +

v2

4(g2Y + g22)

2−M2

1 v2g2Y + g22(1 + 2)

when → 0, decouple occors.

Mass Matrix Diagonalization

OTM

2O = Diag(MZ ,MZ ,Mγ = 0)

The orthogonal matrix O can be parameterized as

O =

cψcφ − sθsφsψ −sψcφ − sθsφcψ −cθsφ

cψsφ + sθcφsψ −sψsφ + sθcφcψ cθcφ

−cθsψ −cθcψ sθ

wheretφ = M2/M1 =

tθ = gY cφ/g2

tψ = tθtφM2W (cθ

M2

Z −M2W

1 + t2θ

)−1

when → 0, φ → 0 and ψ → 0.

Electric Charge ModificationAfter diagonalizing the neutral sector mass matrix and re-writting the in-

teraction Lagrangian in the mass eigen bases Eµ = (Z µ, Zµ, Aγ

µ), one finds thatthere is a modification of the electric charge e, the coupling that appears in thephoton interaction term

eAγµJ

µem = eAγ

µ

JµY + J3µ

2

SM1

e2=

1

g22+

1

g2Y

StSM1

e2=

1

g22+

1 + 2

g2Y

Thus, in StSM, the Stueckelberg mechanism effectively changes gY to gY /√1 + 2.

To conserve the SM relation, one may need introduce the following relation

gY = gSMY

1 + 2

Z Prime interacts with fermions

• SM fermions, JµX(visible) = fγµfQf

X ;

• Hidden fermions, JµX(hidden) = χγµχQχ

X

Direct couplings via additional gauge boson C

Indirect couplings via mixing with B and A3

gauge interaction strength

suppressed by small mixing angles

V and A couplings to fermionsmodified Z boson vector and axial-vector couplings

Z Prime boson vector and axial-vector couplings

when → 0, (vf , af ) recover the SM values, and (vf , af ) → 0.

We assume NO direct coupling between SM fermions and the additional gauge boson here.

vf = cosψ(1− sin θ tanψ)T 3

f − 2 sin2 θ (1− csc θ tanψ)Qf

af = cosψ [1− sin θ tanψ]T 3f

vf = − cosψ(tanψ + sin θ)T 3

f − 2 sin2 θ ( csc θ + tanψ)Qf

af = − cosψ [tanψ + sin θ]T 3f

Examples of indirect constraints• Z mass shift

∆MZ = MZ()−MZ( = 0) ∝ 2

• LEPII contact interactions are parameterised by an effective Lagrangian

Leff added to SM

Leff =4π

(1 + δef )Λ±fAB

eAγ

µeA

fBγµfB

where δef=0(1), for f = e(f = e), and A, B can be L,R, V,A.

• Z Pole Observables

pull(Oi) =O

experimentali −Otheoretic

i

∆Oi

• Other constraints ...

LEP constraints - Z pole observablesTable 1: ∼ 0.03, and M1 = 200 GeV. χ2 =

PULL2.

Quantity Experiment ±∆ LEP FIT St FIT LEP PULL St PULLΓZ [GeV] 2.4952 ± 0.0023 2.4956 2.4956 -0.17 -0.17σhad [nb] 41.541 ± 0.037 41.476 41.469 1.76 1.95

Re 20.804 ± 0.050 20.744 20.750 1.20 1.08Rµ 20.785 ± 0.033 20.745 20.750 1.21 1.06Rτ 20.764 ± 0.045 20.792 20.796 -0.62 -0.71Rb 0.21643 ± 0.00072 0.21583 0.21576 0.83 0.93Rc 0.1686 ± 0.0047 0.17225 0.17111 -0.78 -0.53

A(0,e)FB 0.0145 ± 0.0025 0.01627 0.01633 -0.71 -0.73

A(0,µ)FB 0.0169 ± 0.0013 0.01627 0.01633 0.48 0.44

A(0,τ)FB 0.0188 ± 0.0017 0.01627 0.01633 1.49 1.45

A(0,b)FB 0.0991 ± 0.0016 0.10324 0.10344 -2.59 -2.71

A(0,c)FB 0.0708 ± 0.0035 0.07378 0.07394 -0.85 -0.90

A(0,s)FB 0.098 ± 0.011 0.10335 0.10355 -0.49 -0.50Ae 0.1515 ± 0.0019 0.1473 0.1476 2.21 2.05Aµ 0.142 ± 0.015 0.1473 0.1476 -0.35 -0.37Aτ 0.143 ± 0.004 0.1473 0.1476 -1.08 -1.15Ab 0.923 ± 0.020 0.93462 0.93464 -0.58 -0.58Ac 0.671 ± 0.027 0.66798 0.66812 0.11 0.11As 0.895 ± 0.091 0.93569 0.93571 -0.45 -0.45

χ2 =25.0 χ2 =25.2

D. Feldman, ZL and P. Nath, PRD 75, 115001 (2007)

200 500 1000 1500 2000 2500 30000

2

4

6

8

10

12

14

16

18

StSM Z Prime Mass [Gev]

Bra

nchi

ng F

ract

ion

(%)

u

t

e

d

W

Z Prime Decay

Branching Ratios

Enhanced leptonic branching ratios due to the mixing with hypercharge.

D. Feldman, ZL and P. Nath, JHEP 0611, 007 (2006)

NO direct couplings

between Z’ and

fermions

Dominant invisible decays

Branching ratios for Z with gX = g2, = 0.03, and mχ = 60 GeV.

10-4

10-3

10-2

10-1

100

200 300 400 500 600 700 800 900 1000

bran

chin

g ra

tio

mZ’ (GeV)

Z’ -> -

u+d+s+c+be + µ

t

K. Cheung and T. C. Yuan, JHEP 0703, 120 (2007)

Dilepton final states in Drell-Yan process

γ/Z/Z

Narrow resonance in dilepton final states

Reconstruct dileptons

invariant di-electron mass (GeV)300 350 400 450 500 550 600

Even

ts

0

20

40

60

80

100

120

140

160

180

StSM Z’ Signal

-1L = 5 fb = 14 TeVs

L1 Triggers

LHC

= 0.06

Probing a very narrow Z prime boson @ Tevatron

200 250 300 350 400 450 500 550 600 650 700 750 80010 3

10 2

10 1

100

StSM Z’ Mass [GeV]

Br(

Z’

l+ l)

[pb]

StSM constrained by EWStSM = .05StSM = .04StSM = .03StSM = .02CDF µ+ µ 95% C.L.CDF e+ e 95% C.L.CDF l+ l 95% C.L.D0 µ+ µ 95% C.L.D0 (e+ e + ) 95% C.L.

• D0 Run II 246 275 pb 1 • CDF Run II 200 pb 1

Stueckelberg Z’ Signals

D. Feldman, ZL and P. Nath, Phys. Rev. Lett. 97, 021801 (2006)

200 250 300 350 400 450 500 550 600 650 700 750 8000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

StSM Z Prime Mass [Gev]

Region I : Excluded by EW Constraints

Region VI : Unconstrained

Electroweak Constraint246~275 pb^( 1) HS=0246~275 pb^( 1) HS=VS8 fb^( 1) HS=08 fb^( 1) HS=VSI Excluded by EWIIIIIIVVVI Unconstrained

Exclusion curves in

the parameter

space

D. Feldman, ZL and P. Nath, Phys. Rev. Lett. 97, 021801 (2006)

(GeV)Z’M200 300 400 500 600 700 800 900 1000 1100

ee)

(fb)

BR

(Z’

× Z

’) p

(p

1

10

210

ObservedExpected

1±Expected 2±Expected

SSM Z’ = 0.02 StSMZ’ = 0.03 StSMZ’ = 0.04 StSMZ’ = 0.05 StSMZ’ = 0.06 StSMZ’

-1DØ, 5.4 fb

95% CL

New Tevatron Limits on Stueckelberg Z’

D0 Collaboration, arXiv:1008.2023

M [TeV]0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

B [p

b]

-110

1

10Expected limit

1±Expected 2±Expected

Observed limitSSMZ’

Z’Z’

ATLAS

llZ’ = 7 TeVs

-1 L dt ~ 40 pb

400 600 800 1000 1200 14000

0.2

0.4

0.6

M [GeV]

-3 1

0!

"R

68% expected

95% expected

=0.1Pl

M k/KK

G

=0.05Pl

M k/KK

G

SSMZ'

#Z'

95% C.L. limit

-1dt = 40.0pbL $CMS,

c)

+ee-µ+µ

CMS Collaboration, arXiv:1103.0981

ATLAS Collaboration, arXiv:1103.6218

200 250 300 350 400 450 500 550 600 650 700 750 80010 3

10 2

10 1

100

StSM Z Prime Mass [GeV]

Br(

Z’

l+ l

) [pb

]

pp Z’ l+ lLHC s1/2 = 14 TeV

Excluded by Tevatron Data

Area Above Top LineExcluded by LEP Data

The dilepton production cross section @ LHC

1000 1500 2000 2500 3000 3500

10 3

10 2

10 1

100

101

StSM Z Prime Mass [GeV]

Br(

Z’

l+ l)

[fb]

StSM Z’ Production Cross Sections

pp Z’ l+ lLHC s1/2 = 14 TeV

Area Above Top LineExcluded by LEP Data

Feldman, ZL, and Nath, JHEP 2006

ranging from 0.01 to 0.06

Kinetic Mixing

Hidden Sector

Milli-charge and paraphoton

Dark Matter

ΩDM = 0.23± 0.04

Relic Abundance

G. Jungman et al. JPhysics Reports 267 (1996) 195-373 221

Using the above relations (H = 1.66g$‘2 T 2/mpl and the freezeout condition r = Y~~(G~z~) = H), we

find

(n&)0 = (n&f = 1001(m,m~~g~‘2 +JA+)

N 10-S/[(m,/GeV)((~A~)/10-27 cm3 s-‘)I, (3.3)

where the subscript f denotes the value at freezeout and the subscript 0 denotes the value today.

The current entropy density is so N 4000 cmm3, and the critical density today is

pC II 10-5h2 GeVcmp3, where h is the Hubble constant in units of 100 km s-l Mpc-‘, so the

present mass density in units of the critical density is given by

0,h2 = mxn,/p, N (3 x 1O-27 cm3 C1/(oAv)) . (3.4)

The result is independent of the mass of the WIMP (except for logarithmic corrections), and is

inversely proportional to its annihilation cross section.

Fig. 4 shows numerical solutions to the Boltzmann equation. The equilibrium (solid line) and

actual (dashed lines) abundances per comoving volume are plotted as a function of x = m,/T

0 .01

0 .001

0.0001

10-b

,h 10-s

-; 10-7

c aJ 10-a a

2

10-Q

p lo-‘9

$ lo-”

z 10-m

F! lo-‘3

10 100

x=m/T (time +)

Fig. 4. Comoving number density of a WIMP in the early Universe. The dashed curves are the actual abundance, and

the solid curve is the equilibrium abundance. From [31].

See e.g. Jungman etal

Relic Abundance



find

(n&)0 = (n&f = 1001(m,m~~g~‘2 +JA+)

N 10-S/[(m,/GeV)((~A~)/10-27 cm3 s-‘)I, (3.3)





0,h2 = mxn,/p, N (3 x 1O-27 cm3 C1/(oAv)) . (3.4)





0 .01

0 .001

0.0001

10-b

,h 10-s

-; 10-7

c aJ 10-a a

2

10-Q

p lo-‘9

$ lo-”

z 10-m

F! lo-‘3

10 100

x=m/T (time +)



(1) DM in thermal equilibrium with background (1)


Relic Abundance



find

(n&)0 = (n&f = 1001(m,m~~g~‘2 +JA+)

N 10-S/[(m,/GeV)((~A~)/10-27 cm3 s-‘)I, (3.3)





0,h2 = mxn,/p, N (3 x 1O-27 cm3 C1/(oAv)) . (3.4)





0 .01

0 .001

0.0001

10-b

,h 10-s

-; 10-7

c aJ 10-a a

2

10-Q

p lo-‘9

$ lo-”

z 10-m

F! lo-‘3

10 100

x=m/T (time +)




(2) Universe cools down, DM stays in equilibrium

(2)


Relic Abundance



find

(n&)0 = (n&f = 1001(m,m~~g~‘2 +JA+)

N 10-S/[(m,/GeV)((~A~)/10-27 cm3 s-‘)I, (3.3)





0,h2 = mxn,/p, N (3 x 1O-27 cm3 C1/(oAv)) . (3.4)





0 .01

0 .001

0.0001

10-b

,h 10-s

-; 10-7

c aJ 10-a a

2

10-Q

p lo-‘9

$ lo-”

z 10-m

F! lo-‘3

10 100

x=m/T (time +)





(2)

(3) DM freezes out, decoupling from the background

(3)


Relic Abundance



find

(n&)0 = (n&f = 1001(m,m~~g~‘2 +JA+)

N 10-S/[(m,/GeV)((~A~)/10-27 cm3 s-‘)I, (3.3)





0,h2 = mxn,/p, N (3 x 1O-27 cm3 C1/(oAv)) . (3.4)





0 .01

0 .001

0.0001

10-b

,h 10-s

-; 10-7

c aJ 10-a a

2

10-Q

p lo-‘9

$ lo-”

z 10-m

F! lo-‘3

10 100

x=m/T (time +)





(2)


(3)

(4) DM number stays the same, forming halos, etc.

(4)


Relic Abundance



find

(n&)0 = (n&f = 1001(m,m~~g~‘2 +JA+)

N 10-S/[(m,/GeV)((~A~)/10-27 cm3 s-‘)I, (3.3)





0,h2 = mxn,/p, N (3 x 1O-27 cm3 C1/(oAv)) . (3.4)





0 .01

0 .001

0.0001

10-b

,h 10-s

-; 10-7

c aJ 10-a a

2

10-Q

p lo-‘9

$ lo-”

z 10-m

F! lo-‘3

10 100

x=m/T (time +)





(2)


(3)

(4) DM number stays the same, forming halos, etc.

(4)


σv ∼ pb

milli-charged DM relic density in Stueckelberg Models

150

200

250

300

350

400

450

500

50 100 150 200 250

mZ’

(G

eV)

m (GeV)

(d)

v = 0.95 +- 0.16 pb

gX = g2, = 0.03

0.1

1

10

100

0 50 100 150 200 250 300

v (p

b)

m (GeV)

gX = g2

= 0.03

(c)

K. Cheung and T. C. Yuan, JHEP 0703, 120 (2007)

Dark Matter can also annihilate via Z pole

Energy (GeV)1 10 100

))-(e

)+

+(e

) / (

+(e

Posi

tron

frac

tion

0.01

0.02

0.1

0.2

0.3

PAMELA

PAMELA

PAMELA collaboration, Nature 2009

Rising positron fraction from 10 to 100 GeV !


))-(e

)+

+(e

) / (

+(e

Posi

tron

frac

tion

0.01

0.02

0.1

0.2

0.3

PAMELA

PAMELA


background by Moskalenko & Strong

positron excess

solar modulation



))-(e

)+

+(e

) / (

+(e

Posi

tron

frac

tion

0.01

0.02

0.1

0.2

0.3

PAMELA

PAMELA


background by Moskalenko & Strong

positron excess

solar modulation


σvPAMELA σvRD ∼ pb

Breit-Wigner enhancement

Feldman, ZL and Nath, arXiv:0810.5762Ibe, Murayama and Yanagida, arXiv:0812.0072Guo and Wu, arXiv:0901.1450

Breit-Wigner enhancementhalo annihilation



freeze-out



freeze-out

Large enhancement from freeze-out to halo cross section



freeze-out


Annihilation on the RHS of the pole


Breit-Wigner enhancement

Narrow resonance is required for large enhancement

halo annihilation

freeze-out


Annihilation on the RHS of the pole


Posi

tron

Fra

ctio

n

Positron Energy Ee+ GeV

100 101 10210 2

10 1

StSM (A)StKM (B)StSM (C)PAMELA DataAMS 01Heat Combined

Explaining PAMELA positron excess in

StSM via Z’ narrow

resonance

(, δ, MZ , MD)(A) (0.01, 0, 298, 151.5)(B) (0.01, 0.03, 298, 151.5)(C) (0.005, 0, 297.9, 150.0)

Feldman, ZL and Nath,Phys.Rev.D79:063509,2009,arXiv:0810.5762

CDF Wjj anomaly

]2 [GeV/cjjM100 200

)2Ev

ents

/(8 G

eV/c

-50

0

50

100

150) -1Bkg Sub Data (4.3 fb

Gaussian

WW+WZ

) -1Bkg Sub Data (4.3 fb

Gaussian

WW+WZ

(a)

CDF collaboration,arXiv:1104.0699

NOT seen in D0D0,arXiv:1106.1921

3.2σ excess with 4.3 fb−1

now close to 5σ excess with 7.3 fb−1

M ∼ 144 GeVσ ∼ 4 pb

Explaining CDF Wjj excess in StSM

For baryonic Z , see e.g. also Cheung etal; Buckley etal; Wang etal

ZL, Nath, and Peim, arXiv:1105.4371, to appear in PLB

s-channel process is suppressed by small mixing angles.

t-channel process via direct gauge couplings is important

JµX = qγµqQX

Z’ leptonic branching ratios are naturally suppressed, but still detectable

Z

Z

W

Summary• We propose a new class of models in which the symmetry is

broken by a mixture of Stueckelberg and Higgs mechanisms.

• Stueckelberg Z’ can be much lighter than traditional Z’s. Upgraded Z-factories may detect the effects of the Stueckelberg gauge boson on the Z-pole observables!

• Kinetic mixings can become invisible in the Stueckelberg extensions.

• Milli-charged hidden sector dark matter can generate the PAMELA positron excess through the very narrow Stueckelberg Z’ resonance via Breit-Wigner enhancement.

• The Stueckelberg models produce new signatures in the form of narrow resonances which should be detectable at LHC.

• A baryonic Stueckelberg Z’ can explain the CDF Wjj anomaly.

Thanks

liuzuo

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