The AdS/CFT/Unparticle

Correspondence

Giacomo Caccipaglia, Guido MarandellaJamison Galloway, John McRaven

and work in progress David Stancato

hep-ph/0708.0005, 0804.0424, 0805.0799

Outline

unmotivation

IR cutoff

AdS/CFT/unparticle correspondence

gauge interactions

LHC applications

Hierarchy Problem Now

SUSY Technicolor

Hierarchy Problem Now

Extra Dimensions

SUSY Technicolor

flat curved

Hierarchy Problem Now

Extra Dimensions

SUSY Technicolor

flat curved

Hierarchy Problem Now

Extra Dimensions

SUSY Technicolor

HiggslessRSsmall

large

discrete

Little Higgs

MCH

gaugephobic

flat curved

Hierarchy Problem Now

Extra Dimensions

SUSY Technicolor

HiggslessRSsmall

large

discrete

Little Higgs

MCH

gaugephobicAll of these are fine-tuned!

Looking Under the Lamppost

Looking Under the Lamppost

quirks/hidden valleys/unparticles

New Sector

SM New

Mediator

Quirk/Hidden Valley/Unparticle Model

X

CFT, no confinement unparticles

QCD-like confinement hidden valley

stringy confinement quirks n=0

n=few

n=many

X is a heavy fermion with both SM and New gauge couplings

n fermionsNewSM

UnparticlesGeorgi:

a different way to calculate in CFT’s

phase space looks like a fractional number of particles

Georgi hep-ph/0703260, 0704.2457

!(p, d) !!

d4x eipx"0|TO(x)O†(0)|0#

=Ad

2!

! !

0(M2)d"2 i

p2 $M2 + i"dM2

Ad =16!5/2

(2!)2d

!(d + 1/2)!(d! 1)!(2d)

= iAd

2

!!p2 ! i!

"d!2

sin d"

unparticle propagator

d!(p, d) = Ad !!p0

"!!p2

" !p2

"d!2

d!(p, 1) = 2! "!p0

"#(p2)

unparticle phase space

p22 = 0

p2 = (p1 + p2)2 != 0

p21 = 0

p2i = 0

p2 =

!"

i

pi

#2

!= 0

CFT 101

discuss, this can have a profound impact on their mo-mentum dependence.

The spectral density !(p2, µ) defined in (7) obeys therenormalization-group evolution equation [50]

d!(p2, µ)

d lnµ= !

!2!cusp(µ) ln

p2

µ2+ 2"J(µ)

"!(p2, µ)

! 2!cusp(µ)

# p2

0

dp!2!(p!2, µ) ! !(p2, µ)

p2 ! p!2. (8)

The quantities !cusp and "J are anomalous dimensions,which depend on the renormalization scale only throughthe running coupling #s(µ). Their perturbative expan-sions are known to three-loop order. In particular, !cusp

is the cusp anomalous dimension of Wilson loops withlight-like segments [56], which plays a central role in thephysics of soft-gluon interactions (see e.g. [57]). We stressthat the form of the evolution kernel in (8) is exact; itssimplicity is a consequence of dimensonal analysis com-bined with some magic properties of Wilson lines.

The exact solution to the evolution equation was ob-tained in [54]. It can be written in the form

!(p2, µ0) = N(M, µ0)$p2

%!"1

" &j'

lnp2

M2+ $!, M

( e""E!

!(%), (9)

where $! denotes a derivative with respect to the quantity%, which is then identified with

% =

# M2

µ20

d&2

&2!cusp(&) . (10)

The normalization factor N has scaling dimension !2%and is given by

lnN(M, µ0) =

# M2

µ20

d&2

&2

!!cusp(&) ln

1

&2+ "J(&)

". (11)

This quantity is momentum-independent and will thusbe irrelevant to our discussion. The function &j(x, M) hasa perturbative expansion free of large logarithms. It isobtained from the Laplace transform

&j(x, M) =

# #

0

dp2 e"p2/s !(p2, M) , (12)

where s = ex+"EM2. At one-loop order [58]

&j(x, M) = 1 +CF #s(M)

4'

)2x2 ! 3x + 7 !

2'2

3

*. (13)

The two-loop expression for this function can be foundin [50].

When the tree-level approximation &j = 1 is used in(9), the result exactly coincides with the unparticle spec-tral density (2). The terms of order #s(M) in &j lead to

0 20 40 60 80 1000

0.05

0.1

0.15

0.2

0.25

0.3

p2 [GeV2]

!(p2

)/N

[GeV

!1]

1 2 5 10 20 50 100

0.05

0.1

0.2

0.5

p2 [GeV2]!(

p2)/

N[G

eV!

1]

FIG. 1. Comparison of the unparticle spectral density (2)(dashed) and the spectral density (9) of a massless quark jetat next-to-leading order in QCD (solid). We use parametersM = 10 GeV and ! = 0.5. The right plot shows the sameresults on logarithmic scales.

logarithmic modifications of the simple power form. Inthe “unparticle language” they would indicate a smallbreaking of conformal invariance, which is unavoidable ifthe unparticle sector is coupled to the Standard Model.Therefore, our result (9) shares all features of a realisticmodel for the spectral function of the unparticles of aconformal sector coupled to the Standard Model. In Fig-ure 1 we compare the results (2) and (9) for a particularset of input parameters.

In our “interacting particle model” for unparticlestates the exponent % = dU ! 1 is expressed as an in-tegral over the cusp anomalous dimension, see (10). In atheory such as QCD the numerical value of % can be O(1)provided the scales µ0 and M are widely separated. Thisis because the perturbative smallness of the cusp anoma-lous dimension is overcome by the logarithmic integra-tion over scales. In leading logarithmic approximationone finds

% #!0

(0ln

#s(µ0)

#s(M)(14)

with !0 = 4CF = 16/3 and (0 = 113

CA! 23nf = 23/3 (for

nf = 5 light flavors). Considering the case M = 10GeVas an example, we obtain % = 0.5 for µ # 1.2GeV. Otherexamples of jet functions have a similar functional formbut di"erent values of %. For the example of a gluon jetthe one-loop coe#cient !0 = 4CA is a factor 9/4 largerthan in the case of a quark jet (for Nc = 3), leading toeven larger % values.

The discussion above may be generalized to the case ofmassive QCD jets. If the quark field ) in (5) has mass m,then relations (5)–(8) remain valid, but the solution (9)must be modified. In this case it is no longer possible towrite the solution in closed form, however a perturbativeexpansion of the resummed spectral function can still beobtained [59,60]. At one-loop order one finds

3

Quarks are Unparticles

Neubert hep-ph/0708.0036

Why (broken) CFT’sare Interesting

unparticles must be equivalent to RS2

IR cutoff at TeV turns RS2 to RS1

a new type of IR cutoff could lead to new approaches to the hierarchy problem

new phenomenology for LHC

!(p, µ, d) !!

d4x eipx"0|TO(x)O†(0)|0#|µ

=Ad

2!

! !

µ2(M2 $ µ2)d"2 i

p2 $M2 + i"dM2

= iAd

2

!µ2 ! p2 ! i!

"d!2

sin d"

!(p, µ, 1) =i

p2 ! µ2 + i!

IR cutoff propagator

Fox, Rajaraman, Shirman hep-ph/0705.3092

M

!

!(p) ! (µ2 " p2 " i!)d!2

Spectral DensitiesRS1Unparticle/RS2

M

!

Massive Unparticle

M

!

ds2 =R2

z2

!dx2

µ ! dz2"

z > !

Sbulk =12

!d4x dz

!g(g!"!!"!"" + m2"2)

d[O] = 2± ! = 2±!

4 + m2R2

AdS/CFT

!(p, z) = az2J!(pz) + bz2J!!(pz)

!(p, ") = "!!R!3/2 !0(p)

d = 2± !

S =12

!d4p

(2!)4"0(!p)"0(p)K(p)

K(p) = (2! !)"!2! + b p2! + c p2"2!2! + . . .

S =12

!d4x dz !z

"R3

z3"!z"

#

AdS/CFT

d = 2! !

A

!0

S! =12

!d4p

(2!)4A(!p) K"1A(p)

S! =12

!d4p

(2!)4"0(!p)K"0(p) +

12

!d4p

(2!)4"0(p)A(p)

AdS/CFT

Klebanov, Witten hep-th/9905104

is the fieldis the source

d = 2! !

S! =12

!d4p

(2!)4A(!p) K"1A(p)

S! =12

!d4p

(2!)4"0(!p)K"0(p) +

12

!d4p

(2!)4"0(p)A(p)

AdS/CFT

!O(p!)O(p)" # !2S!

!A(p!) !A(p)# !(4)(p + p!)

(2")4(p2)d"2

Klebanov, Witten hep-th/9905104

z5!z

!1z3

!z"

"! z2(p2 ! µ2)"!m2R2" = 0

H = µz2

Sint =12

!d4x dz

!gH!!

!O(p!)O(p)" # !(4)(p + p!)(2")4

(p2 $ µ2)d"2

AdS/CFT/UnparticlesIR Cutoff

TMNbrane ! |gMN!|

!zIR

R

"2 µ4

R6z5IR !

!zIR

R

"224

M3!

R2

zIR ! 1µ

!24 R4µM3

!"1/5

M! = 1019 GeV, R"1 = 109 GeV, µ = 100GeV

1zIR! 10!23 GeV

IR Cutoff Stability

!F (p, m, d) !!

d4x eipx"0|T#(x)#†(0)|0$

= iAd!1/2

2!

! "

m2(M2 %m2)d!5/2 & p + m

p2 %M2 + i"dM2

= iAd!1/2

2 cos(d!)(& p + m)

"m2 % p2 % i"

#d!5/2

!F (p,m, 3/2) =i(! p + m)

p2 "m2 + i!

Unfermion propagator

Sbulk,f =!

d5x

"R

z

#4 $!i!̄"̄µ#µ!! i$"µ#µ$̄

+12($"##z !! !̄

"##z $̄) +

c

z

%$! + !̄$̄

&#

! = g(pz)!4

"̄ = f(pz)"̄4

g(pz) = z52

!A(p)Jc+ 1

2(pz) + B(p)J!c! 1

2(pz)

"

f(pz) = z52

!A(p)Jc! 1

2(pz) + B(p)J!c+ 1

2(pz)

"

AdS/CFT/Unfermions

!(p, ") = 0, #(p, ") = #0(p)

!!L(p!)!L(p)" # !(4)(p + p!)(2")4

i#µpµ(p2)d"5/2

S! =!

d4p

(2!)4"̄0

"i#̄µpµ(p2)3/2"d

#"0

S =!

d4p

(2!)4"0

"i#µpµ(p2)d!5/2

#"̄0

c <12 dL = 2! c

AdS/CFT/Unfermions

AdS/CFT/Unfermions

Cacciapaglia, Marandella, JT hep-th/0802.2946

c < !12

c <12

c >12

c > !12

dR =32

dL = 2! c

dL =32

dR = 2 + c

H = m z

cL = !cR

p!!

p2 "m2

Sint =!

d4x dz

"R

z

#5

H (!L"R + !R"L + h.c.)

AdS/CFT/Unfermions

S =!

d4p

(2!)4"†(p)

"µ2 ! p2

#2!d"(p)

F (x! y) =!!2 ! µ2

"2!d"(x! y)

Effective Action

S =!

d4xd4y !†(x)F (x! y)!(y)

F (x! y)" F (x! y)W (x, y)

Minimal Gauge Coupling

W (x, y) = P exp!!igT a

" y

xAa

µdwµ

#

...

cf Mandelstam Ann Phys 19 (1962) 1

= !igT a 2p! + q!

2p · q + q2

!"µ2 ! (p + q)2

#2!d !"µ2 ! p2

#2!d$

Gauge Vertex

! 14g2

5

! !

!d4x dz

"R

z

#!(z)F aMNF a

MN

!(z) = e!mz

1g24

=R

g25

! !

!

e"mz

z! R

g25

["!E " log(m")]

1g24

! R

g25

log!

!UV

m

"

AdS Gauge Fields

dilaton:

f !!(z)!!

m +1z

"f !(z) + p2f(z) = 0

zIR = 100/m zIR = 200/m

Gauge KK Modes

0.2 0.4 0.6 0.8 1.0 1.2 p!m

!1.0

!0.5

0.5

1.0

0.2 0.4 0.6 0.8 1.0 1.2 p!m

!1.0

!0.5

0.5

1.0

spectrum

unFermion Anomaliesp

p ! k2

p ! q

!!

!µ

!"!5

k1

k2q

p

p ! q

!!!5

!µ

!"

p ! k1

q

k1

k2

p

k1p ! q

!µ!

k2!"!5q

q k1

p

!!µ!5p ! k2

!"

k2

q

p

k1

!µ

p ! k1 k2

!!"!5

p

k1

k2

q

!!µ"!5

Galloway, McRaven, JT hep-th/0805.079

independent of d

unquark production

!d

!1

!!!!diag. 1

=d(2! d)2(4! d)

3

!d

!1

!!!!diag. 2

=(d! 1)(d! 2)(d2 ! 5d + 3)

3

!d(m) = (2! d)

!"

1! 4m2

q2

#3

!1(m = 0)

unquark production

d < 2

!unparticle = (2! d)!particle

Colored Unproduction

d < 2

R-Hadrons, anomalous jets/E loss

Cacciapaglia, Marandella, JT hep-ph/0708.0005

2 jets +

Pair production

is aligned to visible

!pT

!pT

CFT stuff radiationnot aligned!pT

pT

Anomalous Jets

QCD radiation

Hard jet + 2 jets +

in opposite direction to the hard jet

!pT

!pT

Detailed calculation and simulation needed (background)

Anomalous Jets

S = !!

d4x H†(!2 + µ2)2!dH

unHiggs Model

unHiggs Model

S = !!

d4x H†(D2 + µ2)2!dH + !ttRH

!d!1

"tb

#

L

+ h.c.

H =1!2eiT a!a/vd

!0

vd + h

"

!!

d4x !

"H†H

!2d!2! V 2

2

#2

unHiggs Model

S = !!

d4x H†(D2 + µ2)2!dH + !ttRH

!d!1

"tb

#

L

+ h.c.

m2h !

!4!2d

16!2

LY = !t tRH

!d!1

!tb

"

L

m2h = 3

!!t

!d!1

"2 !2

16"2= 3

#mt

V

$2 !4!2d

16"2

Mass Divergence

Solve the little hierarchy problem?

!g2Aa!Ab

""H†#T aT b"H#!

g!"(d ! 2)µ2!2d

!q!q"

q2

"(d ! 2)µ2!2d !

#µ2 ! q2

$2!d !#µ2

$2!d

q2

%&

M2W =

g2(2! d)µ2!2dv2d

4

unHiggs and MW

Mh ! "ig4

4M2W (2" d)µ2!2d

("s)2!d

WW Scattering

unHiggs exchange is insufficient to unitarize WW scattering

at large s

Mhh = !ig2

4M2W

!s +

(!s)2!d

(2! d)µ2!2d

"

WW Scattering

unHiggs 6 point vertex does unitarize WW scattering

!2 ! "(e+e! " HZ)"SM

unHiggs at LEP

1 1.2 1.4 1.6 1.8

d

0.1

0.2

0.3

0.4

0.5

0.6

!2MUnH " 50 GeV

MUnH " 100 GeV

10-2

10-1

1

20 40 60 80 100 120

mH

(GeV/c2)

95%

CL

lim

it o

n !

2

LEP"s = 91-210 GeV

Observed

Expected for background

(a)

a light unHiggs could have been missed at LEP

Conclusionsmassive unparticles with gauge interactions

are a new type of BSM physics

there are new kinds of LHC signals andnew ways to break electroweak symmetry

d!(p, µ, d) = Ad !!p0

"!!p2 ! µ2

" !p2 ! µ2

"d!2

d!(p, µ, 1) = 2! "!p0

"#(p2 ! µ2)

Massive Unparticle phase space

ig!a!(p, q) ! 2p! + q!

2p · q + q2

!"µ2 " (p + q)2

#2!d ""µ2 " p2

#2!d$

iqµ!aµ = "!1(p + q, m, d)T a ! T a"!1(p, m, d)

Ward-Takahashi Identity

unquark production

unquark production

divergent!

qq production: 1 loop

qq production: 1 loop

qq production: 1 loop

divergent!

qq production: 1 loop

qq production: 1 loop

divergent!

finite cross-section

+

unquark production

unquark production

Im!q2 ln(!q2/!2)

"= q2!

qq

!d4p

(2!)4q2 ! 4p.q + 4p2

(q2 ! 2p.q)2

"2!

#(p! q)2

p2

$2!d

!#

p2

(p! q)2

$2!d%

unquark production

lnZ = !12

lnDet(D2 + m2)2!d

= !12Tr ln(D2 + m2)2!d

= !12(2! d)Tr ln(D2 + m2)

In General

!GB(q) = ! i

(µ2 ! q2 ! i!)2!d ! µ4!2d

!ab!"(q) = !g2"H†#T aT b"H# q!q"

q4

$!"

µ2 ! q2#2!d !

"µ2

#2!d$2

"GB(q)

unHiggs!

gauge invariance maintained

!

g!!H†"Aa

!T a!#!†Aa!T a!H"

"

$#!

µ2 # q2"2!d #

!µ2

"2!d$q!/q2

unHiggs