Making the most and the best of Unparticle

to accept a difficult situation and do as well as you can

to gain as much advantage and enjoyment as you can from sth

A survey of scale invariance and conformal group

Banks-Zaks fields as an example of non-trivial IR fixed point

Unparticle effective theory, propagator and vertex

The difficulty in giving unparticle SM gauge quantum number

Scale invariance breaking

Gauge interaction of unparticle

It has been a long dream that the current human size is not so special.

It will be business as usual after a scale transformation.

Scale Transformation

MM 3 AA 2

xx

The weight a unit area of bone could sustain has to change, too.

Scale transformation in QFT

xx

In high energy, the masses of all particles may be ignored and a scale invariant theory will emerge.

Fundamental particle masses will break scale symmetry

From studying scale invariant QFT, like massless free field, physicists found the theory is invariant under Minkowski space inversion I

2:

x

xxI

From translation T axxT : we can generate anew symmetry.

22

2

21:

xaax

xaxxITI

Special conformal transformation

22

2

21 xaax

xaxx

Special Conformal Transformation

Conformal group: Transformations that preserve the form of the metric up to a factor. )()()( 2 xgxxg

It preserve the angle between two 4 vectors22BA

BA

It includes Poincare transformations, scale transformation and …

It is widely believed that unitary interacting scale invariant theories are always invariant under the full conformal group. (only proven in 2D)

The properties of a scale invariant theory are usually determined by a set of operators which is eigenfunctions of the scaling operator D.

)()(')( xOxOxO ud

Conformal invariance severely restricts the two point function of these operators.

udx

xOO2

1)()0(

They are not eigenfunction of mass operator P2. 02, 22 iPPD

They have a continuous spectrum.

S. Coleman et al have shown that under some conditions, a scale invariant theory is also comformally invariant (including all renormalizable field theory)

Field theories generally exhibit scale invariant UV fixed point (often free) and scale invariant IR fixed point (often trivial, meaning non-interacting).

What if the IR fixed point is non-trivial?

In QFT, it is more complicated due to the presence of renormalization scale .

The coupling constant depends on μ.

lnd

dg

)(ln

fgFor dimensionless g we get a dimensionful parameter Λ.

Scale invariance is broken. Dimensional transmutation

unless ……. at some point 0 Fixed Point

At fixed point, scale invariance is recovered.

For an SU(3) gauge theory with NF massless Dirac fermion

22

5

12

3

01616

)(

ggg

FNRT )(3

4110

FNRTRC )()(420102 21

TNNF 4

33at *

TCNN FF

2420

102'at

*' NN F

FF NNN ' If * 0,0 10

Two loop β function has a non-trivial zero.

UVIR

non-trivial IR fixed point

This asymptotic free gauge theory with massless fermions has a

Banks-Saks (BS) Model

constant ,0, g

Scale invariance

*g

BZ Model

U

cgg

log*Dimensional transmutation

Ug

U

Scale invariant theory

BZ Model

U

Scale invariant theory

If it becomes strong interacting near IR fixed point,

massless fermions

operators which are eigenfunctions of scale transformation D

unparticles

Integrate out degrees of freedom which is usually of order U

They will stay since they have continuous spectra.

)()(')( xOxOxO ud

BSor

BSU

Udd

U OUBZ

BZO

OU is of dimension du

Unparticle propagator

4

422

2)()0(00)0()(0

Pd

PPOeOxO UiPx

UU

2222

)()0(0

U

U

d

dU PAPPO

Scale invariance almost determines unparticle propagator completely.

PPP

Pdnn

n )(

21insert 4

4

Scale invariance dictates the left scale with dimension 2dU.

This is identified as the phase space factor for n massless final particles.

2

1

23

402

1

44

22

n

jn

jjj

n

jj PA

pdpppP

Unparticle with dimension dU looks like a non-integer number dU of particles.

iMP

iMdM

A

OxOTexd

UUdd

UUiPx

22

22

0

2

4

2

0)0()(0

take time order and Fourier transformation

)(2

2)(

20)0()(0

224

422

0

2

4

422222

4

422

PMePd

MdMA

PdPAePMdM

PdPAeOxO

iPxd

d

d

diPx

d

diPx

UU

U

U

U

U

U

U

The usual factor for a normal field operator

22

sin

1

2

UU

d

U

d iPd

Ai

0

1

sin1

pdx

x

x p

well defined for negative P2

The cut has to chosen at positive timelike P2.

Unparticles have continuous spectra of masses.

For non-integer dU there is a cut in the space of P2.

The cut could be seen as a combination of continuous poles.

2P

This is really no surprise since unparticle has continuous mass spectrum.

Interaction with SM particles

through the exchange of a heavy particle of mass MU

BZSMkU

OOM

1

USMkU

ddU

U OOM

CUBZ

U

Non-renormalizable vertex

With unparticle vertex and propagator, very interesting phenomenology can be studied.

K. Cheung, W.Y. Keung and T.C. Yuan PRD 76 055003

C.H. Chen and C.Q Geng, PRD

continuous missing energy in real unparticle emission

UqqZ

K. Cheung, W.Y. Keung and T.C. Yuan, PRL

interference with SM through virtue unparticle exchange

Drell-Yan Process

Unparticles are a hidden sector, like heavenly god

They are so unlike us, normal, earthly particles.

Seeing them needs to be so rare that it’s called a miracle.

USMU

U OOM

To increase its importance in our earthly life and to teach us his message, God needs an incarnation, ie. becoming a human form.

Unparticles needs to be given SM gauge quantum numbers.

That miracle still doesn’t happen everyday means that scale invariance needs to be broken in the low energy and will manifest itself as energy gets higher.

Scale invariance most likely will be broken anyway.

P. Fox, A. Rajaraman and Y. Shirman PRD 2007

BZdU

OHM BZ

2

2

1Ud

U

ddU

U OHM

CBZ

UBZ

2

2

v

The scale invariance supposedly will be broken at

22

2

4 vM

U

BZ

U dU

d

U

UdU

U

conformal window UU E

For the window to be not too narrow, UM

Nonrenormalizable couplings will be suppressed.

Unparticles will be unaccessible.

vMvU 10,

vMvU 2,

M. Bander, J. Feng, A. Rajaraman and Y. Shirman 0706.2677

However, we don’t need non-renormalizable terms to access unparticle in case they are incarnated, ie. has SM quantum number.

Imagine the following scenario:

vU

vU 210

vMU 310

Make sure we didn’t see unparticle until LHC

The conformal window is about two degrees of magnitude

Non-renormalizable interaction could be ignored.

We ask the same question Howard asked:

How does a SM flavored or colored unparticle look like in collider?

Two hurdles to overcome:

How do we introduce scale invariance breaking effects?

How do we flavor or color a unparticle?

How do we introduce scale invariance breaking effects?

Parameterize the breaking with an infrared cutoff.

Parameterize the breaking with an infrared cutoff m.

222

2222

4

2

0)0()(0),,(

m

dU

ipxU

iMp

imMdM

A

OxTOexddmp

U

It reduced to Georgi’s unparticle propagator as 0m

and reduced to particle propagator with mass m as 1Ud

UdU

UU

imp

i

d

Admp

222sin2

),,(

2m

2P

How do we flavor or color a unparticle?

Gauge interaction of unparticles

How do we flavor or color a unparticle?

Gauge interaction of unparticles

The unparticle propagator naively will imply a non-local Lagrangian:

)()(2

sin2 2224

4

ppmppd

A

dS U

U

d

d

U

)()()(44 xyxFyydxdS

y

x

aa dwAigTPyxW exp),(For gauge symmetry,

insert a Wilson Line

Vertex of a Gluon coupled to two unparticles.

)(),()(),( yUyxWxUyxW

Vertex of two gluons and two unparticles

sUnpartcile* gqq

UP

U d2

Corresponding scalar particle pair production cross section

Option II My suggestion is to use the representation of unparticle as bulk field in extra dimensional model

iMP

iM

dMOxTOexd O

iPx

22

22

4 )(2

0)0()(0

The unparticle propagator contains a cut for timelike P.

A cut line can be decomposed into a collection of point poles with the gap goes to zero!

An unparticle may correspond to a collection of particles!

iMP

iM

dMOxTOexd O

iPx

22

22

4 )(2

0)0()(0

222 )()( u

u

dd MAMScale invariance

It suggests a collection of non-interacting particles created by operator O with continuous mass distribution. Start with a discrete form:

n n

niPx

iMP

iFOxTOexd

22

24 0)0()(0

22222 )0(0

MMOF nn

)(nfM n 0 mass gap

222 ud

nn MFScaling

Kaluza-Klein Modes of Bulk Field

• Bulk Fields contains Kaluza-Klein (K

K) States Ψn with wavefunctions: due to periodicity.

• The extra-dimension wave number

• E-p relation • KK states look like having 4D masse

sR

n

…….

yR

ni

e

02

2

55

nn R

n

Rnk /

02

22

R

npE

Towers of particles appear in extra dimensional models.

Kaluza-Klein Modes of Bulk Field

…….

0, R

Non-compact extra dimensions gives towers of particles with continuous mass distribution.

Assume only deconstructed unparticle see the non-compact extra dimension. Even gravity won’t see it.

Or the gravity is localized! RSII

22222 )0(0

MMOF nn

needs to scale with Mn2 222

ud

nn MF

Scaling

222 ud

nn MF

Assume that

1)0(0 O

222222 nnn MNMMF

density of states

For m extra dimensions

m

i

in R

nM

1

22

Density of states is proportional to the hyper sphere shell:

21222/122

122 )( n

m

nn

m

nn MdMMdMMN

12

mdu

ADD realization

W.Y. Keung ......5.2,0.2,5.1ud

Consider ADD with m extra dimensions and a bulk scalar field ),,( 21 myyyx

)0,0,0,()( xxO is a KK state

unparticle with dimension 3/2Bulk scalar field in 5 non-compact dimensions

need a ultraviolet cutoff

Bulk scalar field in 4 non-compact continuous dimensions and one non-compact discrete dimension

ADD realization of deconstruction

ADS-CFT

Consider one extra dimension: z

Need 4D Poincare symmetry: 222 )( dzdxdxzwds

xx 4D unparticle theory is scale invariant

In 5D, no longer conformal, it needs to corresponds to an isometry of the metric.

MNMN gg

zz z

R

22

22 R

z

dzdxdxds

AdS5 Anti-de-Sitter Space

AdS5 Anti-de-Sitter Space: the most symmetric spacetime with negative curvature

222 dydxdxeds ky

Conformally Flat frame

k

ez

ky

dyedz ky

22

22 R

z

dzdxdxds

kR

1

MNMNMN gzg 2~ It’s common to take out the dimensions from all the coordinates.

0 zy

zy

Boundary

Non-compact

Isometry in AdS5

22

22 R

z

dzdxdxds

zzxx , The metric does not change

MNMN gg

Conformal Symmetry on the boundary z=0 xx CFT

AdS-CFT Correspondence

0),( zx

unpartcile CFT in 4DNon-compact AdS5 model

What is the dimension of the unparticle that corresponds to a massive bulk scalar field?

Bulk scalar field in AdS5

2/225

4 mggdzdxS NMMN

),(lim)(0

zxzxO ud

z

Boundary operator in unparticle CFT

Dimension du is the solution of Eq.

)4(25 uu ddm

Bulk scalar field correlation function in AdS5

2/225

4 mggdzdxS NMMN

22

222 R

z

dzdxds

2225

25

42 Rmz

z

dzdxNS M

M

)',;('lim)',0(),('lim)0()(00

zzxGzzzzxzzOxO uuuu dd

z

dd

z

Two point function

)'()',;(25

5233 zzzzqGmzqzz

zz

If we care only the limit 0z

0)',;(25

35

zzqGm

zz

zz zG )4(2

5 m

ud

)',;('lim)0()(0

zzxGzzOxO uu dd

z

under scale transformation xx

)',;('lim)0()()0()(0

zzxGzzOxOOxO uu dd

z

)Z',Z;(Z'Zlim)',;('lim0Z

211112

0xGzzxGzz uuuuuu dddddd

z

)0()(2 OxOud

Isometry in AdS5

zzxx 11 ,

udxOxO 2)0()( udxOxO 2)0()(

22')',,(

zxx

zxxzG

n n

nn

iMq

zzzzqG

22

)'()()',;(

Consider the compact version

1,0z

udn zz )(

For large n 11 sin CzMz n

dn

u

Mass spectrumm

n z

CnM 2

udxOxO 2)0()(

)()(),( zxzx nn

n

KK expasion

0)(25

5233

zmzMz

zz

z nn

Same thing can be done for other Lorentz structure (there is a table of relations between dimensions and bulk masses in the Maldacena et al review)

0),( zx

unpartcile CFT in 4DNon-compact AdS5 model

Deconstruction

Non-exotic, but with gravity!

To localize gravity, may need a Planck brane at kz

1

An ultraviolet brane corresponds to unltraviolet cutoff in 4D.

RSII Unparticle theory cutoff at uM

0),( zx

unpartcile CFT in 4DNon-compact AdS5 model

Gauge interaction of unparticles

introduce bulk color gauge field and bulk gauge transformation

put SM on the brane, transforming under restricted localized gauge transformation

Bulk scalar field can be naturally colored or flavored!

0),( zx

IR brane introduce a natural infrared cutoff

An IR brane will naturally give an IR cutoff and hence a breaking of scale invariance

a bonus

Other deconstruction ( flat without gravity )

Fractal Theory Space Fractional Dimension