The holographic Pomeron and low-x physics

Miguel S. Costa

Faculdade de Ciências da Universidade do Porto

Holography and Extreme Chromodynamics - HoloQuark 2018Santiago de Compostela, July 2017

Regge behaviour in QCD

Regge behaviour in QCD

• Hadronic resonances fall in linear trajectoriesChapter 1. Introduction

Figure 1.2: The Chew–Frautschi plot. Spin J of the isospin I = 1 even parity mesons againsttheir mass squared. (From reference [4])

String Theory was discovered forty years ago as an attempt to understand hadronic physics.

By that time, QCD and String Theory competed as models of the strong force. Of course, this

QCD/String dispute was decided long ago in favor of QCD. However, the modern viewpoint

replaces dispute by duality, and rephrases the main question: Is QCD a String Theory?

1.1 Hadronic Spectrum & Strings

Although the fundamental particles of QCD are quarks and gluons, the confinement mechanism

disallows their direct observation. Instead, the observed spectrum is characterized by a long

list of colorless bound states of the fundamental particles. Most of these bound states are

unstable and are found as resonances in scattering experiments. At the present day, we are still

unable to accurately predict the observed hadronic spectrum directly from the QCD dynamics1.

Nevertheless, from a phenomenological perspective, the hadronic spectrum has several inspiring

features.

In figure 1.2 we plot the spin J of the lighter mesons against their mass squared m2. The

result is well modeled by a linear Regge trajectory

J = α!

m2"

= α(0) + α′m2 ,

where α(0) and α′ are known as the intercept and the Regge slope, respectively. In fact, most

1See [2] and [3] and references therein for attempts using the lattice formulation of QCD and the AdS/CFTcorrespondence.

2

J = j(t) = j(0) + ↵0t

t =

Regge behaviour in QCD

• Hadronic resonances fall in linear trajectoriesChapter 1. Introduction

Figure 1.2: The Chew–Frautschi plot. Spin J of the isospin I = 1 even parity mesons againsttheir mass squared. (From reference [4])

String Theory was discovered forty years ago as an attempt to understand hadronic physics.

By that time, QCD and String Theory competed as models of the strong force. Of course, this

QCD/String dispute was decided long ago in favor of QCD. However, the modern viewpoint

replaces dispute by duality, and rephrases the main question: Is QCD a String Theory?

1.1 Hadronic Spectrum & Strings

Although the fundamental particles of QCD are quarks and gluons, the confinement mechanism

disallows their direct observation. Instead, the observed spectrum is characterized by a long

list of colorless bound states of the fundamental particles. Most of these bound states are

unstable and are found as resonances in scattering experiments. At the present day, we are still

unable to accurately predict the observed hadronic spectrum directly from the QCD dynamics1.

Nevertheless, from a phenomenological perspective, the hadronic spectrum has several inspiring

features.

In figure 1.2 we plot the spin J of the lighter mesons against their mass squared m2. The

result is well modeled by a linear Regge trajectory

J = α!

m2"

= α(0) + α′m2 ,

where α(0) and α′ are known as the intercept and the Regge slope, respectively. In fact, most

1See [2] and [3] and references therein for attempts using the lattice formulation of QCD and the AdS/CFTcorrespondence.

2

J = j(t) = j(0) + ↵0t

t =

I = 1

(s � t)

1.1. Hadronic Spectrum & Strings

Figure 1.3: Regge trajectory determined from the large energy (20–200 GeV ) behavior of thedifferential cross section of the process π− + p → π0 + n. The straight line is obtained byextrapolating the trajectory in figure 1.2. (From reference [4])

of the hadronic resonances fall on approximately linear Regge trajectories with slopes around

1(GeV )−2 and different intercepts. A linear relation between spin and mass squared suggests a

description of the bound states as string like objects rotating at relativistic speeds. Indeed, the

spin of a classical open string with tension T rotating as a straight line segment, with endpoints

traveling at the speed of light, is given by α′ = (2πT )−1 times its energy squared2.

A related stringy feature of QCD is the high energy behavior of scattering amplitudes.

Experimentally, at large center–of–mass energy√

s, the hadronic scattering amplitudes show

Regge behavior

A(s, t) ∼ β(t)sα(t) ,

where t is the square of the momentum transferred. The appropriate Regge trajectory α(t)

that dominates a given scattering process is selected by the exchanged quantum numbers. For

example, the process

π− + p → π0 + n

is dominated by the exchange of isospin I = 1 even parity mesons, i. e. the Regge trajectory

in figure 1.2. In figure 1.3 we plot the Regge trajectory obtained from the behavior of the

differential cross section at large s. Elastic scattering is characterized by the exchange of the

vacuum quantum numbers. In this case the scattering amplitude is dominated by the Pomeron

2See section 2.1.3 of [5] for details.

3

even parity mesons exchanged for

⇡� + p ! ⇡0 + n

A(s, t) ⇠ �(t) s j(t)

Regge behaviour in QCD

• Hadronic resonances fall in linear trajectoriesChapter 1. Introduction

Figure 1.2: The Chew–Frautschi plot. Spin J of the isospin I = 1 even parity mesons againsttheir mass squared. (From reference [4])

String Theory was discovered forty years ago as an attempt to understand hadronic physics.

By that time, QCD and String Theory competed as models of the strong force. Of course, this

QCD/String dispute was decided long ago in favor of QCD. However, the modern viewpoint

replaces dispute by duality, and rephrases the main question: Is QCD a String Theory?

1.1 Hadronic Spectrum & Strings

Although the fundamental particles of QCD are quarks and gluons, the confinement mechanism

disallows their direct observation. Instead, the observed spectrum is characterized by a long

list of colorless bound states of the fundamental particles. Most of these bound states are

unstable and are found as resonances in scattering experiments. At the present day, we are still

unable to accurately predict the observed hadronic spectrum directly from the QCD dynamics1.

Nevertheless, from a phenomenological perspective, the hadronic spectrum has several inspiring

features.

In figure 1.2 we plot the spin J of the lighter mesons against their mass squared m2. The

result is well modeled by a linear Regge trajectory

J = α!

m2"

= α(0) + α′m2 ,

where α(0) and α′ are known as the intercept and the Regge slope, respectively. In fact, most

1See [2] and [3] and references therein for attempts using the lattice formulation of QCD and the AdS/CFTcorrespondence.

2

J = j(t) = j(0) + ↵0t

t =

I = 1

(s � t)

1.1. Hadronic Spectrum & Strings

Figure 1.3: Regge trajectory determined from the large energy (20–200 GeV ) behavior of thedifferential cross section of the process π− + p → π0 + n. The straight line is obtained byextrapolating the trajectory in figure 1.2. (From reference [4])

of the hadronic resonances fall on approximately linear Regge trajectories with slopes around

1(GeV )−2 and different intercepts. A linear relation between spin and mass squared suggests a

description of the bound states as string like objects rotating at relativistic speeds. Indeed, the

spin of a classical open string with tension T rotating as a straight line segment, with endpoints

traveling at the speed of light, is given by α′ = (2πT )−1 times its energy squared2.

A related stringy feature of QCD is the high energy behavior of scattering amplitudes.

Experimentally, at large center–of–mass energy√

s, the hadronic scattering amplitudes show

Regge behavior

A(s, t) ∼ β(t)sα(t) ,

where t is the square of the momentum transferred. The appropriate Regge trajectory α(t)

that dominates a given scattering process is selected by the exchanged quantum numbers. For

example, the process

π− + p → π0 + n

is dominated by the exchange of isospin I = 1 even parity mesons, i. e. the Regge trajectory

in figure 1.2. In figure 1.3 we plot the Regge trajectory obtained from the behavior of the

differential cross section at large s. Elastic scattering is characterized by the exchange of the

vacuum quantum numbers. In this case the scattering amplitude is dominated by the Pomeron

2See section 2.1.3 of [5] for details.

3

even parity mesons exchanged for

⇡� + p ! ⇡0 + n

A(s, t) ⇠ �(t) s j(t)

Total cross section

� ⇠ sj(0)�1

• For elastic scattering, exchanged trajectory has the vacuum quantum numbers.

(Soft) Pomeron trajectory

• For elastic scattering, exchanged trajectory has the vacuum quantum numbers.

(Soft) Pomeron trajectoryChapter 1. Introduction

Figure 1.4: Total cross sections for elastic scattering at high energy. The cross sections riseslowly due to pomeron exchange. (From reference [6])

trajectory[6, 4]

αP (t) ≃ 1, 08 + 0, 25 t , (GeV units) .

There is some evidence from lattice simulations that there are glueball states lying on this

trajectory starting from spin J = 2 [7, 8]. Furthermore, an even glueball state with spin 2

lying on the pomeron trajectory seems to have been found in experiments [9]. However, in real

QCD, glueball states mix with mesons and their identification is not clear [6]. An important

consequence of the pomeron intercept being larger than 1, is that hadrons effectively expand at

high energies. More precisely, the total cross section for elastic processes in QCD grows with

center–of–mass energy,

σ ∼ sαP (0)−1 ∼ s0.08 ,

as can be seen in figure 1.4. This expansion with energy reinforces the picture of hadrons as

stringlike objects. It is well known [10] that the average size of a fundamental string is given by

the divergent sum,

< R2 >∼ α′∞!

n=1

1

n,

coming from the contributions of zero point fluctuations of each string mode. However, in

a scattering experiment, only the modes with frequency smaller than the energy√

s can be

4

�P � 1.08 + 0.25 t (GeV units)

[Landshoff-Donnachie]

• For elastic scattering, exchanged trajectory has the vacuum quantum numbers.

Elastic cross sections in QCD

� � s�P (0)�1 � s0.08

(Soft) Pomeron trajectoryChapter 1. Introduction

Figure 1.4: Total cross sections for elastic scattering at high energy. The cross sections riseslowly due to pomeron exchange. (From reference [6])

trajectory[6, 4]

αP (t) ≃ 1, 08 + 0, 25 t , (GeV units) .

There is some evidence from lattice simulations that there are glueball states lying on this

trajectory starting from spin J = 2 [7, 8]. Furthermore, an even glueball state with spin 2

lying on the pomeron trajectory seems to have been found in experiments [9]. However, in real

QCD, glueball states mix with mesons and their identification is not clear [6]. An important

consequence of the pomeron intercept being larger than 1, is that hadrons effectively expand at

high energies. More precisely, the total cross section for elastic processes in QCD grows with

center–of–mass energy,

σ ∼ sαP (0)−1 ∼ s0.08 ,

as can be seen in figure 1.4. This expansion with energy reinforces the picture of hadrons as

stringlike objects. It is well known [10] that the average size of a fundamental string is given by

the divergent sum,

< R2 >∼ α′∞!

n=1

1

n,

coming from the contributions of zero point fluctuations of each string mode. However, in

a scattering experiment, only the modes with frequency smaller than the energy√

s can be

4

�P � 1.08 + 0.25 t (GeV units)

[Landshoff-Donnachie]

• For elastic scattering, exchanged trajectory has the vacuum quantum numbers.

J � 2Evidence from lattice QCD that there are glueballs on this trajectory with .

Elastic cross sections in QCD

� � s�P (0)�1 � s0.08

(Soft) Pomeron trajectoryChapter 1. Introduction

Figure 1.4: Total cross sections for elastic scattering at high energy. The cross sections riseslowly due to pomeron exchange. (From reference [6])

trajectory[6, 4]

αP (t) ≃ 1, 08 + 0, 25 t , (GeV units) .

There is some evidence from lattice simulations that there are glueball states lying on this

trajectory starting from spin J = 2 [7, 8]. Furthermore, an even glueball state with spin 2

lying on the pomeron trajectory seems to have been found in experiments [9]. However, in real

QCD, glueball states mix with mesons and their identification is not clear [6]. An important

consequence of the pomeron intercept being larger than 1, is that hadrons effectively expand at

high energies. More precisely, the total cross section for elastic processes in QCD grows with

center–of–mass energy,

σ ∼ sαP (0)−1 ∼ s0.08 ,

as can be seen in figure 1.4. This expansion with energy reinforces the picture of hadrons as

stringlike objects. It is well known [10] that the average size of a fundamental string is given by

the divergent sum,

< R2 >∼ α′∞!

n=1

1

n,

coming from the contributions of zero point fluctuations of each string mode. However, in

a scattering experiment, only the modes with frequency smaller than the energy√

s can be

4

�P � 1.08 + 0.25 t (GeV units)

[Landshoff-Donnachie]

Regge theory

Regge theory

• t-channel partial wave expansion A(s, t) =X

J

aJ(t)PJ

⇣1 + 2

s

t

⌘

Regge theory

��s

t

⇥J

• t-channel partial wave expansion A(s, t) =X

J

aJ(t)PJ

⇣1 + 2

s

t

⌘

Regge theory

��s

t

⇥J

• t-channel partial wave expansion A(s, t) =X

J

aJ(t)PJ

⇣1 + 2

s

t

⌘

• Exchange of spin field has pole at t = m2(J)J aJ(t) ⇥ r(J)t�m2(J)

Regge theory

��s

t

⇥J

• t-channel partial wave expansion A(s, t) =X

J

aJ(t)PJ

⇣1 + 2

s

t

⌘

• Exchange of spin field has pole at t = m2(J)J aJ(t) ⇥ r(J)t�m2(J)

• Sum exchanges in leading Regge trajectory and analytically continue in (Sommerfeld-Watson transform)

�

J

�⇥

C

dJ

2�i

�

sin(�J)

J

t = m2

2

4

6

8

J

Regge theory

��s

t

⇥J

• t-channel partial wave expansion A(s, t) =X

J

aJ(t)PJ

⇣1 + 2

s

t

⌘

• Exchange of spin field has pole at t = m2(J)J aJ(t) ⇥ r(J)t�m2(J)

• Sum exchanges in leading Regge trajectory and analytically continue in (Sommerfeld-Watson transform)

�

J

�⇥

C

dJ

2�i

�

sin(�J)

J

t = m2

2

4

6

8

J

J

� � �•2 4 6

• Pick leading pole

aJ(t) ⇥ �j�(t) r(j(t))J � j(t) j(t)

A(s, t) ⇡ �(t) s j(t)

Deep Inelastic Scattering (DIS)

P

X

γe� e�

• Pomeron enters also in diffractive processes. For example DIS, where electron interacts with proton via exchange of off-shell photon

Deep Inelastic Scattering (DIS)

• Optical theorem

�

X= Im

2

(t = 0)X

P

γ γ

P

γ

P

P

X

γe� e�

• Pomeron enters also in diffractive processes. For example DIS, where electron interacts with proton via exchange of off-shell photon

Deep Inelastic Scattering (DIS)

• Optical theorem

�

X= Im

2

(t = 0)X

P

γ γ

P

γ

P

P

X

γe� e�

• Pomeron enters also in diffractive processes. For example DIS, where electron interacts with proton via exchange of off-shell photon

• Hadronic tensor W ab(x, Q, t) = i

�d4y eiq·y⇤P |T{ja(y) jb(0)}|P ⇥⌅

s = � (q + P )2

s

q

P

Q2 = q2

s = � (q + P )2

s

q

P

• Bjorken

P

q

p

x

p = xP

Q2 = q2

s = � (q + P )2

s

q

P

• Bjorken

P

q

p

x

p = xP

Q2 = q2

large s xsmall�s � Q2

x

s = � (q + P )2

s

q

P

• Bjorken

P

q

p

x

p = xP

• Transverse resolution

1/Q

1/Q

Pγ

Q2 = q2

large s xsmall�s � Q2

x

Gluons dominate at small xfi

�x, Q2

⇥

• At low a steep - dependence is observedxx � 10�2� ⇠ x

1�j0

Gluons dominate at small xfi

�x, Q2

⇥

j0 ⇠ 1.1� 1.4

j 0�

1

Effective slop varies with Q

• At low a steep - dependence is observedxx � 10�2� ⇠ x

1�j0

Is it the same Regge trajectory? One or two pomerons (soft and hard)?

Gluons dominate at small xfi

�x, Q2

⇥

j0 ⇠ 1.1� 1.4

j 0�

1

Effective slop varies with Q

• At low a steep - dependence is observedxx � 10�2� ⇠ x

1�j0

Hard Pomeron [BFKL - Balitsky, Fadin, Kuraev & Lipatov]

Two gluon exchange with ladder interactions

(�s ln 1/x)nResums contributions

Valid for hard probes (conformal limit)

Q

Q � ⇤QCD

Hard pomeron is a cut in J-plane

j0 = 1 +12 ln 2

⇡↵s = 1 + 2.65↵s

J

� � �2 4 6j0

Hard Pomeron [BFKL - Balitsky, Fadin, Kuraev & Lipatov]

Two gluon exchange with ladder interactions

(�s ln 1/x)nResums contributions

Valid for hard probes (conformal limit)

Q

Q � ⇤QCD

Hard pomeron is a cut in J-plane

j0 = 1 +12 ln 2

⇡↵s = 1 + 2.65↵s

J

� � �2 4 6j0

• Breaking conformal symmetry, explains well DIS data outside the confining region [ Kowalski, Lipatov, Ross, Watt 10]Q > ⇤QCD

• Strong rise in , violating Froissart bound 1/x

� � m�

�ln s)2

Eventually growth slows down (multi-pomeron, eikonal resummation)

R

1/Q

• Perturbation theory will breakdown, even for small coupling, because there will be gluon saturation at very low x.

• Strong rise in , violating Froissart bound 1/x

� � m�

�ln s)2

Eventually growth slows down (multi-pomeron, eikonal resummation)

R

1/Q

• Perturbation theory will breakdown, even for small coupling, because there will be gluon saturation at very low x.

• AdS/CFT as starting point for pomeron physics

• Strong rise in , violating Froissart bound 1/x

� � m�

�ln s)2

Eventually growth slows down (multi-pomeron, eikonal resummation)

R

1/Q

• Perturbation theory will breakdown, even for small coupling, because there will be gluon saturation at very low x.

• AdS/CFT as starting point for pomeron physics

Graviton Regge trajectory dual to pomeron trajectory [Brower, Polchinski, Strassler, Tan 06]

• Strong rise in , violating Froissart bound 1/x

� � m�

�ln s)2

Eventually growth slows down (multi-pomeron, eikonal resummation)

Summary so far

• QCD exhibits Regge behaviour. Elastic scattering (and related processes) dominated by exchange of pomeron Regge trajectory.

Summary so far

• QCD exhibits Regge behaviour. Elastic scattering (and related processes) dominated by exchange of pomeron Regge trajectory.

• For scattering of hadronic states, data very well described with phenomenological soft pomeron. With hard probe growth is steeper, and data better described with BFKL hard pomeron. One or two pomerons? Confinement effects?

Summary so far

• QCD exhibits Regge behaviour. Elastic scattering (and related processes) dominated by exchange of pomeron Regge trajectory.

• Construct Regge theory for CFTs, or for the dual gravity (string theory). For the example of N=4 SYM results are valid at any coupling.

• For scattering of hadronic states, data very well described with phenomenological soft pomeron. With hard probe growth is steeper, and data better described with BFKL hard pomeron. One or two pomerons? Confinement effects?

Summary so far

• QCD exhibits Regge behaviour. Elastic scattering (and related processes) dominated by exchange of pomeron Regge trajectory.

• Once dual description of pomeron well understood, can apply to low x physics in QCD starting from holographic QCD description, including confining region.

• Construct Regge theory for CFTs, or for the dual gravity (string theory). For the example of N=4 SYM results are valid at any coupling.

• For scattering of hadronic states, data very well described with phenomenological soft pomeron. With hard probe growth is steeper, and data better described with BFKL hard pomeron. One or two pomerons? Confinement effects?

Regge Kinematics in CFTs [Cornalba, MSC, Penedones, Schiappa 06]

• Consider correlator with EMG current and scalar operator

Aab(yi) = ⇥ja(y1)O(y2) jb(y3)O(y4)⇤

ja = �a O

Regge Kinematics in CFTs [Cornalba, MSC, Penedones, Schiappa 06]

• Consider correlator with EMG current and scalar operator

Aab(yi) = ⇥ja(y1)O(y2) jb(y3)O(y4)⇤

ja = �a

=A(z, z)

(y13)2⇠(y24)2�

O

Regge Kinematics in CFTs [Cornalba, MSC, Penedones, Schiappa 06]

• Consider correlator with EMG current and scalar operator

Aab(yi) = ⇥ja(y1)O(y2) jb(y3)O(y4)⇤

ja = �a

=A(z, z)

(y13)2⇠(y24)2�

O

y+y�

y1 y2

y3y4

• Regge kinematics is Lorentzian. Analytically continue from Euclidean theory ( ) to on real axis.z = z⇤ z, z

z

z

0 1

• Regge limit

z, z ! 0 with

z

zfixed

Conformal Regge Theory [Cornalba 07; MSC, Penedones, Gonçalves 12]

• Expand in t-channel conformal partial waves

(J,�)j O

Oj

A(z, z) =X

k

C13kC24k G�k,Jk(z, z)

Conformal Regge Theory [Cornalba 07; MSC, Penedones, Gonçalves 12]

• Expand in t-channel conformal partial waves

(J,�)j O

Oj

A(z, z) =X

k

C13kC24k G�k,Jk(z, z)

• Euclidean OPE dominated by lowest dimension limz,z!0

G�,J ⇠ (zz)�2 fE(z/z)

Conformal Regge Theory [Cornalba 07; MSC, Penedones, Gonçalves 12]

• Expand in t-channel conformal partial waves

(J,�)j O

Oj

A(z, z) =X

k

C13kC24k G�k,Jk(z, z)

• Euclidean OPE dominated by lowest dimension limz,z!0

G�,J ⇠ (zz)�2 fE(z/z)

but after analytic continuation highest spin dominates limz,z!0

G�,J ⇠ (zz)1�J2 fL(z/z)

Conformal Regge Theory [Cornalba 07; MSC, Penedones, Gonçalves 12]

• Expand in t-channel conformal partial waves

(J,�)j O

Oj

A(z, z) =X

k

C13kC24k G�k,Jk(z, z)

• Euclidean OPE dominated by lowest dimension limz,z!0

G�,J ⇠ (zz)�2 fE(z/z)

but after analytic continuation highest spin dominates limz,z!0

G�,J ⇠ (zz)1�J2 fL(z/z)

• Spectral representation (h = d/2)

A(z, z) =X

J

Z 1

�1d⌫

C13JC24J

⌫2 + (�(J)� h)2F⌫,J(z, z)

Conformal Regge Theory [Cornalba 07; MSC, Penedones, Gonçalves 12]

• Expand in t-channel conformal partial waves

(J,�)j O

Oj

A(z, z) =X

k

C13kC24k G�k,Jk(z, z)

• Euclidean OPE dominated by lowest dimension limz,z!0

G�,J ⇠ (zz)�2 fE(z/z)

but after analytic continuation highest spin dominates limz,z!0

G�,J ⇠ (zz)1�J2 fL(z/z)

Regge limit ⇥ � 0 , � fixed

�2 = zz , e2⇢ =z

z

⇠X

J

�1�J

Zd⌫ ↵J(⌫) ⌦i⌫(⇢)

• Spectral representation (h = d/2)

A(z, z) =X

J

Z 1

�1d⌫

C13JC24J

⌫2 + (�(J)� h)2F⌫,J(z, z)

Conformal Regge Theory [Cornalba 07; MSC, Penedones, Gonçalves 12]

• Expand in t-channel conformal partial waves

(J,�)j O

Oj

A(z, z) =X

k

C13kC24k G�k,Jk(z, z)

• Euclidean OPE dominated by lowest dimension limz,z!0

G�,J ⇠ (zz)�2 fE(z/z)

but after analytic continuation highest spin dominates limz,z!0

G�,J ⇠ (zz)1�J2 fL(z/z)

Regge limit ⇥ � 0 , � fixed

�2 = zz , e2⇢ =z

z

⇠X

J

�1�J

Zd⌫ ↵J(⌫) ⌦i⌫(⇢)

• Spectral representation (h = d/2)

A(z, z) =X

J

Z 1

�1d⌫

C13JC24J

⌫2 + (�(J)� h)2F⌫,J(z, z)

Harmonic function on Hd�1 r2⌦i⌫ = �(⌫2 + h� 1)⌦i⌫( )

A(�, ⇢) = �

Zd⌫

Z

C

dJ

2⇡i

⇡

2 sin(⇡J)

⇣��J + (��)�J

⌘↵J(⌫)⌦i⌫(⇢)

• Sum over spin using Sommerfeld-Watson transform J

� � �•2 4 6

j(⌫)

A(�, ⇢) = �

Zd⌫

Z

C

dJ

2⇡i

⇡

2 sin(⇡J)

⇣��J + (��)�J

⌘↵J(⌫)⌦i⌫(⇢)

• Sum over spin using Sommerfeld-Watson transform J

� � �•2 4 6

j(⌫)

Regge pole for such that

⌫2 +��(j(⌫))� h

�2= 0

J = j(⌫)

↵J(⌫) =C13JC24J

⌫2 + (�(J)� h)2⇡ �

j0(⌫)C13j(⌫)C24j(⌫)

2⌫�J � j(⌫)

�

A(�, ⇢) = �

Zd⌫

Z

C

dJ

2⇡i

⇡

2 sin(⇡J)

⇣��J + (��)�J

⌘↵J(⌫)⌦i⌫(⇢)

• Sum over spin using Sommerfeld-Watson transform

A(�, ⇢) =

Z 1

�1d⌫ ↵(⌫)�1�j(⌫) ⌦i⌫(⇢)

J

� � �•2 4 6

j(⌫)

Regge pole for such that

⌫2 +��(j(⌫))� h

�2= 0

J = j(⌫)

↵J(⌫) =C13JC24J

⌫2 + (�(J)� h)2⇡ �

j0(⌫)C13j(⌫)C24j(⌫)

2⌫�J � j(⌫)

�

J = j(⇥, �) = j0(�)�D(�) ⇥2 + · · ·

A(�, ⇢) =

Zd⌫ ↵(⌫,�)�1�j(⌫,�) ⌦i⌫(⇢)

N=4 Super Yang Mills

• Correlation functions that exchange vacuum quantum numbers are dominated in Regge limit by exchange of pomeron/graviton Regge trajectory (twist 2)

� = �(J) or J = j(�)

2

4

1 2 4

⇠ ln J

BFKL

J

�� 2

GRAVITON

. . .

= i⌫

OJ ⇠ Tr�F↵�1D�2 . . . D�J�1F

↵�J

�

J = j(⇥, �) = j0(�)�D(�) ⇥2 + · · ·

A(�, ⇢) =

Zd⌫ ↵(⌫,�)�1�j(⌫,�) ⌦i⌫(⇢)

N=4 Super Yang Mills

• Correlation functions that exchange vacuum quantum numbers are dominated in Regge limit by exchange of pomeron/graviton Regge trajectory (twist 2)

1

2

3

4

g � 1Pomeron

• Weak coupling

� = �(J) or J = j(�)

2

4

1 2 4

⇠ ln J

BFKL

J

�� 2

GRAVITON

. . .

= i⌫

OJ ⇠ Tr�F↵�1D�2 . . . D�J�1F

↵�J

�

J = j(⇥, �) = j0(�)�D(�) ⇥2 + · · ·

A(�, ⇢) =

Zd⌫ ↵(⌫,�)�1�j(⌫,�) ⌦i⌫(⇢)

N=4 Super Yang Mills

• Correlation functions that exchange vacuum quantum numbers are dominated in Regge limit by exchange of pomeron/graviton Regge trajectory (twist 2)

1

2

3

4

g � 1Pomeron

• Weak coupling

2

1 3

4

(�, J)

g � 1Grav Reggeon

• Strong coupling

1

2

3

4

� = �(J) or J = j(�)

2

4

1 2 4

⇠ ln J

BFKL

J

�� 2

GRAVITON

. . .

= i⌫

OJ ⇠ Tr�F↵�1D�2 . . . D�J�1F

↵�J

�

Graviton/Pomeron Regge trajectory at strong coupling [BPST 06]

Graviton/Pomeron Regge trajectory at strong coupling [BPST 06]

• At strong coupling pomeron trajectory described by graviton Regge trajectory of string theory in AdS (large N, conformal theory)

O

ja jb

O

J�D2 �m2

�ha1...aJ = 0

m2 = �(�� 4)� Jwith

Exchange of spin J field in AdS (symmetric, traceless and transverse)

AdS scattering process

Graviton/Pomeron Regge trajectory at strong coupling [BPST 06]

• At strong coupling pomeron trajectory described by graviton Regge trajectory of string theory in AdS (large N, conformal theory)

O

ja jb

O

J�D2 �m2

�ha1...aJ = 0

m2 = �(�� 4)� Jwith

Exchange of spin J field in AdS (symmetric, traceless and transverse)

AdS scattering process

R2H3

l�L

• AdS impact parameter representation. In Regge limit [Cornalba, MSC, Penedones, Schiappa 07]

AJ (s, t) ⇡ iV J0J s

Zdl?e

iq?·l?Z

dz

z3dz0

z03�1(z)�3(z)�2(z

0)�4(z0)SJ�1GJ (L)

Graviton/Pomeron Regge trajectory at strong coupling [BPST 06]

• At strong coupling pomeron trajectory described by graviton Regge trajectory of string theory in AdS (large N, conformal theory)

O

ja jb

O

J�D2 �m2

�ha1...aJ = 0

m2 = �(�� 4)� Jwith

Exchange of spin J field in AdS (symmetric, traceless and transverse)

AdS scattering process

, AdS energy squaredS = zz0s , impact parametercoshL =

z2 + z02 + l2?2zz0

z

z0

R2H3

l�L

• AdS impact parameter representation. In Regge limit [Cornalba, MSC, Penedones, Schiappa 07]

AJ (s, t) ⇡ iV J0J s

Zdl?e

iq?·l?Z

dz

z3dz0

z03�1(z)�3(z)�2(z

0)�4(z0)SJ�1GJ (L)

AJ(s, t) ⇡ iV J0J s

Zdl?e

iq?·l?Z

dz

z3dz0

z03�1(z)�3(z)�2(z

0)�4(z)SJ�1GJ (L)

AJ(s, t) ⇡ iV J0J s

Zdl?e

iq?·l?Z

dz

z3dz0

z03�1(z)�3(z)�2(z

0)�4(z)SJ�1GJ (L)

• is the integrated propagator ( )GJ(L)

GJ(L) ⇠ i (zz0)(J�1)

Zdw+dw�⇧+···+.�···�(z, z

0, w)

w = x� x

0 = (w+, w

�, l?)

AJ(s, t) ⇡ iV J0J s

Zdl?e

iq?·l?Z

dz

z3dz0

z03�1(z)�3(z)�2(z

0)�4(z)SJ�1GJ (L)

• is the integrated propagator ( )GJ(L)

GJ(L) ⇠ i (zz0)(J�1)

Zdw+dw�⇧+···+.�···�(z, z

0, w)

w = x� x

0 = (w+, w

�, l?)

and obeys scalar propagator equation in transverse spaceh⇤H3 � 3��(�� 4)

iGJ (L) = ��H3(y, y

0)

R2H3

l�L

z

z0

y

y0

AJ(s, t) ⇡ iV J0J s

Zdl?e

iq?·l?Z

dz

z3dz0

z03�1(z)�3(z)�2(z

0)�4(z)SJ�1GJ (L)

• is the integrated propagator ( )GJ(L)

GJ(L) ⇠ i (zz0)(J�1)

Zdw+dw�⇧+···+.�···�(z, z

0, w)

w = x� x

0 = (w+, w

�, l?)

and obeys scalar propagator equation in transverse spaceh⇤H3 � 3��(�� 4)

iGJ (L) = ��H3(y, y

0)

R2H3

l�L

z

z0

y

y0

✓� d

dz2+ V (z)

◆= t (z)

, reduces to Schrodinger problem

V =

✓15

4+�(�� 4)

◆1

z2, with

z

VGJ(L) = eiq?·l?pz (z)

� = �(J)

AJ(s, t) ⇡ iV J0J s

Zdl?e

iq?·l?Z

dz

z3dz0

z03�1(z)�3(z)�2(z

0)�4(z)SJ�1GJ (L)

• is the integrated propagator ( )GJ(L)

GJ(L) ⇠ i (zz0)(J�1)

Zdw+dw�⇧+···+.�···�(z, z

0, w)

w = x� x

0 = (w+, w

�, l?)

and obeys scalar propagator equation in transverse spaceh⇤H3 � 3��(�� 4)

iGJ (L) = ��H3(y, y

0)

R2H3

l�L

z

z0

y

y0

✓� d

dz2+ V (z)

◆= t (z)

, reduces to Schrodinger problem

V =

✓15

4+�(�� 4)

◆1

z2, with

z

VGJ(L) = eiq?·l?pz (z)

Application to low physics in QCDx

Application to low physics in QCDx

P

X

γe� e�

• Deep inelastic scattering (DIS)[Hatta, Iancu, Mueller 07; Cornalba, MSC 08;Brower, Djuric, Sarcevic, Tan 10]

Application to low physics in QCD

Optical theorem�

X= Im

2

(t = 0)X

P

γ γ

P

γ

P

x

P

X

γe� e�

• Deep inelastic scattering (DIS)[Hatta, Iancu, Mueller 07; Cornalba, MSC 08;Brower, Djuric, Sarcevic, Tan 10]

Application to low physics in QCD

Optical theorem�

X= Im

2

(t = 0)X

P

γ γ

P

γ

P

x

• DVCS & VMP ��Q

P

d�

dt(Q, x, t) � |W |2

�tot(Q, x)P

�, V

[MSC, Djuric 12; MSC, Djuric, Evans 13]

P

X

γe� e�

• Deep inelastic scattering (DIS)[Hatta, Iancu, Mueller 07; Cornalba, MSC 08;Brower, Djuric, Sarcevic, Tan 10]

Hard and soft pomeron are distinct Regge trajectories [Donnachie, Landshoff]

• Explain DIS data with two Regge trajectories

�(Q2, x) / f0(Q

2)x�j0 + f1(Q2)x�j1

j0 = 1.26

j1 = 1.08

Hard and soft pomeron are distinct Regge trajectories [Donnachie, Landshoff]

• Explain DIS data with two Regge trajectories

�(Q2, x) / f0(Q

2)x�j0 + f1(Q2)x�j1

j0 = 1.26

j1 = 1.08

fk�Q2

�= Pk

�Q2

�'k

�Q2

�

Known function of and jkQ2

Wave function of a 1D Schrodinger problem in holographic direction

• Let us apply this idea to gauge/string duality

z ⇠ 1/QHolographic direction

[Bayona, MSC, Quevedo 17]

Hard and soft pomeron are distinct Regge trajectories [Donnachie, Landshoff]

It seems data “knows” about holographic QCD!!

• Explain DIS data with two Regge trajectories

�(Q2, x) / f0(Q

2)x�j0 + f1(Q2)x�j1

j0 = 1.26

j1 = 1.08

fk�Q2

�= Pk

�Q2

�'k

�Q2

�

Known function of and jkQ2

Wave function of a 1D Schrodinger problem in holographic direction

• Let us apply this idea to gauge/string duality

z ⇠ 1/QHolographic direction

[Bayona, MSC, Quevedo 17]

Hard and soft pomeron are distinct Regge trajectories [Donnachie, Landshoff]

DIS from gauge/string duality

• Hadronic tensor W ab(x, Q, t) = i

�d4y eiq·y⇤P |T{ja(y) jb(0)}|P ⇥⌅

DIS from gauge/string duality

• Hadronic tensor W ab(x, Q, t) = i

�d4y eiq·y⇤P |T{ja(y) jb(0)}|P ⇥⌅

• Strategy for DIS phenomenology: choose a holographic QCD model; analytically continue spin J equation in region below graviton pole .

W =

Zdzdz0�1(z)�3(z) KP (s, t, z, z

0) �2(z0)�4(z

0)

2

4

1 2 4

⇠ ln J

BFKL

J

�� 2

GRAVITON

. . .

= i⌫

(J < 2)

DIS from gauge/string duality

• Hadronic tensor W ab(x, Q, t) = i

�d4y eiq·y⇤P |T{ja(y) jb(0)}|P ⇥⌅

non-normalizable 0 < z < 1/Q

• Strategy for DIS phenomenology: choose a holographic QCD model; analytically continue spin J equation in region below graviton pole .

W =

Zdzdz0�1(z)�3(z) KP (s, t, z, z

0) �2(z0)�4(z

0)

2

4

1 2 4

⇠ ln J

BFKL

J

�� 2

GRAVITON

. . .

= i⌫

(J < 2)

DIS from gauge/string duality

• Hadronic tensor W ab(x, Q, t) = i

�d4y eiq·y⇤P |T{ja(y) jb(0)}|P ⇥⌅

non-normalizable 0 < z < 1/Q

normalizablez0 ⇠ 1/M

• Strategy for DIS phenomenology: choose a holographic QCD model; analytically continue spin J equation in region below graviton pole .

W =

Zdzdz0�1(z)�3(z) KP (s, t, z, z

0) �2(z0)�4(z

0)

2

4

1 2 4

⇠ ln J

BFKL

J

�� 2

GRAVITON

. . .

= i⌫

(J < 2)

DIS from gauge/string duality

• Hadronic tensor W ab(x, Q, t) = i

�d4y eiq·y⇤P |T{ja(y) jb(0)}|P ⇥⌅

non-normalizable 0 < z < 1/Q

Pomeron Kernel normalizablez0 ⇠ 1/M

• Strategy for DIS phenomenology: choose a holographic QCD model; analytically continue spin J equation in region below graviton pole .

W =

Zdzdz0�1(z)�3(z) KP (s, t, z, z

0) �2(z0)�4(z

0)

2

4

1 2 4

⇠ ln J

BFKL

J

�� 2

GRAVITON

. . .

= i⌫

(J < 2)

Holographic QCD

z

IRUV

x

↵

Boundary

ds

2 = e

2A(z)�dz

2 + ⌘↵�dx↵dx

��

� = �(z)

• QCD dual is a 5D theory with a graviton and a dilaton

AdS fields single trace operators

� $ F 2

gab $ T↵�

$

Holographic QCD

z

IRUV

x

↵

Boundary

ds

2 = e

2A(z)�dz

2 + ⌘↵�dx↵dx

��

� = �(z)

Constructed to match QCD perturbative beta function

Reproduces: heavy quark-antiquark linear potential; glueball spectrum from lattice simulations; thermodynamic properties of QGP (bulk viscosity, drag force and jet quenching parameters)

Judicious choice of potential with only 2 free parameters

[Gursoy, Kiritsis, Nitti 07]• Test our ideas with a 5D dilaton-gravity model

S =1

22

Zd

5x

p�g e

�2�hR+ 4(@�)2 + V (�)

i

• QCD dual is a 5D theory with a graviton and a dilaton

AdS fields single trace operators

� $ F 2

gab $ T↵�

$

Spin J field in holographic QCD

Spin J equation must: • In AdS limit reduce to �D2 �m2

�ha1...aJ = 0 m2 = �(�� 4)� J , � = �(J)

• For reproduce TT metric fluctuationsJ = 2

• Construct spin J field dual to gluon operatorDecompose symmetric, traceless, transverse field with respect to global boundary symmetry. Propagating modes have boundary indices

ha1...aJ SO(1, 3)h↵1...↵J

OJ ⇠ Tr�F↵�1D�2 . . . D�J�1F

↵�J

�

[Bayona, MSC, Djuric, Quevedo 15]

Spin J field in holographic QCD

Spin J equation must: • In AdS limit reduce to �D2 �m2

�ha1...aJ = 0 m2 = �(�� 4)� J , � = �(J)

• For reproduce TT metric fluctuationsJ = 2

• Construct spin J field dual to gluon operatorDecompose symmetric, traceless, transverse field with respect to global boundary symmetry. Propagating modes have boundary indices

ha1...aJ SO(1, 3)h↵1...↵J

OJ ⇠ Tr�F↵�1D�2 . . . D�J�1F

↵�J

�

[Bayona, MSC, Djuric, Quevedo 15]

Equation for propagating mode in effective field theory⇣r2 � 2�rz + JA2e�2A ��(�� 4)+

+(J � 2)e�2Aha �+ b �2 + c

⇣A� A2

⌘i⌘h↵1...↵J = 0

Spin J field in holographic QCD

Spin J equation must: • In AdS limit reduce to �D2 �m2

�ha1...aJ = 0 m2 = �(�� 4)� J , � = �(J)

• For reproduce TT metric fluctuationsJ = 2

• Construct spin J field dual to gluon operatorDecompose symmetric, traceless, transverse field with respect to global boundary symmetry. Propagating modes have boundary indices

ha1...aJ SO(1, 3)h↵1...↵J

OJ ⇠ Tr�F↵�1D�2 . . . D�J�1F

↵�J

�

[Bayona, MSC, Djuric, Quevedo 15]

BFKL

2

4

1 2 4

⇠ ln JJ

�� 2

GRAVITON

UV free theoryunitarity bound

IR described by graviton trajectory

�(�� 4) ⇡ 2

l2s(J � 2)

⇣1 + d e��/2

⌘+ e�4�/3

�J2 � 4

�{{Equation for propagating mode in effective field theory

⇣r2 � 2�rz + JA2e�2A ��(�� 4)+

+(J � 2)e�2Aha �+ b �2 + c

⇣A� A2

⌘i⌘h↵1...↵J = 0

Many Regge trajectories

O O

J

• Consider 5D exchange of spin J field in the Regge limit

AJ (s, t)j j

Many Regge trajectories

O O

J

• Consider 5D exchange of spin J field in the Regge limit

AJ (s, t)

is the FT of integrated propagatorGJ(z, z0, t)

GJ (z, z0, l?) ⇠ i e(1�J)(A+A0)

Zdw+dw�⇧+···+.�···�(z, z

0, w)

j j

Many Regge trajectories

O O

J

• Consider 5D exchange of spin J field in the Regge limit

AJ (s, t)

is the FT of integrated propagatorGJ(z, z0, t)

GJ (z, z0, l?) ⇠ i e(1�J)(A+A0)

Zdw+dw�⇧+···+.�···�(z, z

0, w)

Reduces to a Schrodinger problem (spectral representation)

z

V (z)

j j

• Sum over spin J exchanges in 5D dual theory

Poles in the J-plane at t = tn(J) ) J = jn(t)

X

J

!Z

dJ

sin⇡J

z

V (z)

J

� � �2 4 6

j2(t) j1(t)

j0(t)

• Sum over spin J exchanges in 5D dual theory

Poles in the J-plane at t = tn(J) ) J = jn(t)

X

J

!Z

dJ

sin⇡J

z

V (z)

J

� � �2 4 6

j2(t) j1(t)

j0(t)

• At the end of the day, structure function is of the form

F2

�x,Q

2�=

X

n

gn Q2jn(0)

P13

�Q

2�x

1�jn(0)

F2

�x,Q

2�=

X

n

gn Q2jn(0)

P13

�Q

2�x

1�jn(0)

F2

�x,Q

2�=

X

n

gn Q2jn(0)

P13

�Q

2�x

1�jn(0)

P13

�Q2

�=

Zdz P13

�Q2, z

�e(1�jn(0))A(z)eB(z) n

�jn(0), z

�

• Dependence on virtual photon wave function

F2

�x,Q

2�=

X

n

gn Q2jn(0)

P13

�Q

2�x

1�jn(0)

• Dependence on fixed target absorbed in coupling

gn = �2⇡2jn(0)jn(0)2jn(0)

j0n(0)

Zdz P24

�P 2, z

�e(1�jn(0))A(z)eB(z) ⇤

n

�jn(0), z

�

P13

�Q2

�=

Zdz P13

�Q2, z

�e(1�jn(0))A(z)eB(z) n

�jn(0), z

�

• Dependence on virtual photon wave function

Test model agains low x DIS data from HERA

Truncated data to region. Has 249 data points and large range in Q ⇣

0.1 < Q2 < 400 GeV2⌘

x < 0.01

(GeV)2

0.5

1.0

1.5

x

F2

1e−06 1e−05 1e−04 1e−03 1e−02

0.5

1.0

1.5

0.1

0.15

0.2

0.25

0.350.4

0.5

0.65

0.85

1.21.5

2

2.7

3.54.5

6.58.5

10

1215

18

22

27

35

45

60

70

90

120

150

200

250300

400

Test model agains low x DIS data from HERA

Kept the first 4 Regge trajectories (up to intercept of meson trajectory that will also contribute)

5 parameters from spin J equation; 4 parameters from coupling of each pomeron

Truncated data to region. Has 249 data points and large range in Q ⇣

0.1 < Q2 < 400 GeV2⌘

x < 0.01

0.5

1.0

1.5

x

F2

1e−06 1e−05 1e−04 1e−03 1e−02

0.5

1.0

1.5

0.1

0.15

0.2

0.25

0.350.4

0.5

0.65

0.85

1.21.5

2

2.7

3.54.5

6.58.5

10

1215

18

22

27

35

45

60

70

90

120

150

200

250300

400

Test model agains low x DIS data from HERA

Kept the first 4 Regge trajectories (up to intercept of meson trajectory that will also contribute)

5 parameters from spin J equation; 4 parameters from coupling of each pomeron

Truncated data to region. Has 249 data points and large range in Q ⇣

0.1 < Q2 < 400 GeV2⌘

x < 0.01

Parameters fixed with �2 = 1.7

GeV�1

0.2 0.5 2.0 5.0 20.0 100.0

0.0

0.1

0.2

0.3

0.4

0.5

Q2

ε

• Reproduced long sought running of effective exponent

� ⇠ f(Q)

✓1

x

◆✏eff (Q)

- consistent with universal behavior of soft pomeron 1.09 intercept observed for soft probes in elastic cross sections

(GeV)2

0.2 0.5 2.0 5.0 20.0 100.0

0.0

0.1

0.2

0.3

0.4

0.5

Q2

ε

• Reproduced long sought running of effective exponent

� ⇠ f(Q)

✓1

x

◆✏eff (Q)

- consistent with universal behavior of soft pomeron 1.09 intercept observed for soft probes in elastic cross sections

(GeV)2

−5 0 5 10 15 20 25 30

01

23

45

6

t

j(t)

1.17

1.09

0.969

0.900

• Regge trajectories consistent with lattice QCD glueball spectrum!

In green meson trajectories

[Meyer 05]

Shape matches [Caron-Huot, Komargodski, Sever, Zhiboedov et al 16]

(GeV)2

EMG current and Reggeon non-minimal coupling [Amorim, MSC, Quevedo 18]

• So far considered minimal coupling between U(1) gauge field and graviton trajectory. But for graviton perturbations in AdS there are two possible couplings

EMG current and Reggeon non-minimal coupling [Amorim, MSC, Quevedo 18]

• So far considered minimal coupling between U(1) gauge field and graviton trajectory. But for graviton perturbations in AdS there are two possible couplings

• Generalized to spin J field in graviton Regge trajectory [Robert talk]

EMG current and Reggeon non-minimal coupling [Amorim, MSC, Quevedo 18]

• So far considered minimal coupling between U(1) gauge field and graviton trajectory. But for graviton perturbations in AdS there are two possible couplings

• Generalized to spin J field in graviton Regge trajectory [Robert talk]

• Quality of fit improved significantly!

�2

d.o.f. = 1.1

x

F2

• Quality of fit improved significantly!

�2

d.o.f. = 1.1

x

F2

• Non-minimal coupling has dimensions and defines scale of 1-10 GeV. Matches order of magnitude of gap between spin 4 and 2 glueballs [CEMZ 14]

J

t (GeV)2

Concluding Remarks

Concluding Remarks

• Gauge/strings duality sheds light into long standing puzzle in QCD: the connection between hard and soft pomeron. They are just different Reggeons that arise form graviton Regge trajectory in dual 5D space.

Concluding Remarks

• Gauge/strings duality sheds light into long standing puzzle in QCD: the connection between hard and soft pomeron. They are just different Reggeons that arise form graviton Regge trajectory in dual 5D space.

• Test this picture against other processes such as DVCS and VMP.

Concluding Remarks

• Include meson trajectories.

• Gauge/strings duality sheds light into long standing puzzle in QCD: the connection between hard and soft pomeron. They are just different Reggeons that arise form graviton Regge trajectory in dual 5D space.

• Test this picture against other processes such as DVCS and VMP.

Concluding Remarks

• Include meson trajectories.

• Gauge/strings duality sheds light into long standing puzzle in QCD: the connection between hard and soft pomeron. They are just different Reggeons that arise form graviton Regge trajectory in dual 5D space.

• Test this picture against other processes such as DVCS and VMP.

• Coupling of Pomeron to gluon jets.

Concluding Remarks

• Include meson trajectories.

• Gauge/strings duality sheds light into long standing puzzle in QCD: the connection between hard and soft pomeron. They are just different Reggeons that arise form graviton Regge trajectory in dual 5D space.

• Test this picture against other processes such as DVCS and VMP.

• Coupling of Pomeron to gluon jets.

• How generic are our results? Should try other holographic QCD models...

THANK YOU