Brower, Polchinski, Strassler, Tan (BPST)The Pomeron and Gauge

ANOMALOUS DIMENSIONS, POMERON INTERCEPT AND ADS/CFT

Chung-I Tan, Brown University Non-Perturbative QCD Workshop Paris, June 10-13, 2013

Brower, Polchinski, Strassler, Tan (BPST)The Pomeron and Gauge/String Duality (2006) Richard Brower, Marko Djuri´c, Ina Sarcevi´cand Chung-I Tan: String-Gauge Dual Description of DIS and Small-x, 10.1007/JHEP 11(2010)051, arXiv:1007.2259 R. Brower, M. Costa, M. Djuric, T. Raben, and C-I Tan: “Conformal Pomeron and Odderon Intercepts at Strong Coupling” (to appear.) Outline

•QCD High Energy Scattering with AdS/CFT -- Universality •Consequence of Conformal Invariance •DIS at low-x -- Uniﬁcation •DGLAP (large Q) vs BFKL (small x) •OPE (Anomalous Dimensions) and Conformal Pomeron •Conformal Pomeron and Odderon Intercepts in strong coupling •Saturation, Conﬁnement, etc., and DIS •Summary and Outlook •Summary and Outlook Outline

•QCD High Energy Scattering with AdS/CFT -- Universality •Consequence of Conformal Invariance •DIS at low-x -- Uniﬁcation •DGLAP (large Q) vs BFKL (small x) •OPE (Anomalous Dimensions) and Conformal Pomeron •Conformal Pomeron and Odderon Intercepts in strong coupling •Saturation, Conﬁnement, etc. and DIS •Summary and Outlook •Summary and Outlook Outline

•QCD High Energy Scattering with AdS/CFT -- Universality •Consequence of Conformal Invariance •DIS at low-x -- Uniﬁcation •DGLAP (large Q) vs BFKL (small x) •OPE (Anomalous Dimensions) and Conformal Pomeron •Conformal Pomeron and Odderon Intercepts in strong coupling •Saturation, Conﬁnement, etc. and DIS •Summary and Outlook •Summary and Outlook Executive Summary:

Gauge/String Duality (AdS/CFT) 2-GLUONS GRAVITON

 Establishing “Pomeron” in QCD non-perturbatively,

 Uniﬁcation of Soft and Hard Physics in High Energy Collision

 New phenomenology based on “Large Pomeron intercept”, e.g., DIS at small-x: (DGLAP vs Pomeron), DVCS, Central Diffractive Higgs Production, etc.

HE scattering since AdS/CFT ADS BUILDING BLOCKS BLOCKS

For 2-to-2 2 3 3 iq (x x0) A(s, t)=g d bd b⇥ e ?· (z) (s, x x⇥,z,z⇥) (z⇥) 0 13 K 24 A(s, t)= Z . 13 KP 24

2 3 3 iq (x x0) A(s, t)=g d bd b⇥ e ?· (z) (s, x x⇥,z,z⇥) (z⇥) 0 13 K 24 Z e

3 2 5A(z) d b dzd x g(z)whereg(z)=det[gnm]= e ⌘ ? p For 2-to-3 A(s, s ,s ,t ,t )= V , 1 2 1 2 13 KP KP 24 e e BASIC BUILDING BLOCK

•Elastic Vertex:

•Pomeron/Graviton Propagator:

2 ij (zz0) dj 1+e j (s, b, z, z0)= s G (z,x?,z0,x0?; j) K R4 2i sin j j ✓ ◆ Z ✓ ◆ b (2 (j)) 1 e Gj(z,x?,z0,x0?)= , conformal: 4⇥zz0 sinh

(j)=2+p2 1/4 (j j ) 0 p conﬁnement: Gj(z,x?,z0,x0?; j) discrete sum HERA vs LHeC region: dots are H1-ZEUS small-x data points I. Gauge-String Duality: AdS/CFT Weak Coupling: ab a Gluons and Quarks: Aµ (x), f (x)

Gauge Invariant Operators: ¯(x)(x), ¯(x)Dµ(x) 2 3 S(x)=TrFµ⇥ (x),O(x)=TrF (x)

Tµ⇤ (x)=TrFµ(x)F⇤ (x), etc. (x)= TrF2 + ¯D + L ⇤ ··· Strong Coupling: (0) Metric tensor: Gmn(x)=gmn(x)+hmn(x) Anti-symmetric tensor (Kalb-Ramond ﬁelds): bmn(x) Dilaton, Axion, etc. (x),a(x), etc. Cmn (x) Other differential forms: ··· (x)= (G(x),b(x),C(x), ) L L ··· 4 d xi(x) i(x) e O CFT = string [i(x, z) z 0 i(x)] ⇤ R ⌅ Z |

Bulk Degrees of Freedom from type- IIB Supergravity on AdS5: Supergravity limit

Strong coupling

Conformal

Pomeron as Graviton in AdS

21 1 Introduction

Paper I: AdS5 Witten Diagrams at high energy. Eﬀective Lagragian. Remark on BPST Pomeron paper. Eikonal anzats.

Paper I: Box diagram and Shock wave eikonal sum

Here we reformulate the computaiton of Witten diagrams in AdS5 space with and without a IR cut-oﬀ suitable for the study of hight energy scattering. This provides a framework for going beyond the leading large N limit studied in BPST Regge limit in the extreme super gravity approximaiton.

• Draw all “Witten-Feynman” Diagrams in AdS5, 2 Basics • High Energy Dominated by Spin-2 Exchanges:

p + p p + p (2.1) 1 2 → 3 4

MN 2 One Graviton Exchange at High Energy S = dz√g ∂M φ(z)g ∂N φ(z) + ∆(∆ d)φ (z) (2.2) ! " − #

where d = 4, and the AdS5 background metric is Figure 91: The t-channel exchange graph d2z = dx dxµ + d2z (2.3) As in the past, we simplify the int2egral by µusing translation in0variance to translate x1 to 0, and then performing an inverszio0n."As a result, # 5 ∆1+∆3 2∆3 d z z0 A(w, x , x ) = x − I(w" x" ) , I(w) = G (w, z) (7.32) 1 3 13 13 5 ∆ 2∆3 Scalar propagator: | | − !H z0 z We now use the fact that G is a Green function and satisfies ( +∆(∆ d))G (w, z) = ∆ (5) w − ∆ Strong Coupling Pomeron has J =2 δ(w, z), so that φ∆(z)φ∆(w) = G∆ (z, w) (2.4) • $ % ∆1+∆3 w0 ( w + ∆(∆ d))I(w) = (7.33) • Need to consider finite. satisfies − w2∆3 2 2 ∆13 In 1terms of the scale invariant combination ζ = w0/w , we have I(w) = w0 fS(ζ), ∆13 = For QCD, needs confinement to introduce a scale. ∆ ∆ and the fuMnctiNon f now satisfies the following di(ff5er)ential equation 5 • 1 − ∂3 √gg ∂S + ∆(∆ d) G (z, w) = δ (z w) (2.5) M 2 N ∆ g 4ζ (ζ 1)fS"" + 4ζ[(∆13 + 1)ζ ∆13 + d/2 1]fS" (7.34) " − √ − − − # − − +(∆ ∆ )(∆ + ∆ d)f = ζ∆3 − 13 13 − S Making the change of variables σ = 1/ζ, we find that the new differential equation is Conformal Invariance lmeaandifesstlytoof tIhse ohymperegteormieetsric otyfpeAanDd isSsolv,edGby(5) is a function of ∆ ∆13 d ∆5 ∆13∆d 1 fS(ζ) = F − , − − ; ; 1 (7.35) " 2 2 2 − ζ # The other linearly independent solution2to the hypergeometric2 equation is singular as (x y) + (z0 w0) ζ 1, which is unacucept=able since the original integral was perfectly regular in this limit (2.6) (w→hich corresponds to w$ 0). − − → 2z w 0 0 k It is easier, however, to find the solutions in terms of a power series, fS(ζ) = k fSkζ . Upon substitution into (7.34), we find solutions that truncate to a finite number$of terms in ζ, provided ∆ +∆ ∆ is a positive integer. Notice that k need not take integer values, 1 3 − spin S field, rather k ∆3 must be integer. The series truncates from above at kmax = ∆3 1, so that f = 0 w−hen k ∆ , and − Sk ≥ 3 1 1 Γ(k)Γ(k + ∆13)Γ( 2 ∆1 + ∆3 ∆ )Γ( 2 ∆ + ∆1 + ∆3 d ) − −fSk−= −{ −− } { − − − − } (7.36) (d 1 24SΓ)(∆ )Γ(∆ )Γ(k2+ 1 +(1d+∆ 1 2∆S))Γ(k + 1 +(1d∆ 1+ 2∆S)d ) 2 ( ∂z0 z0 ∂1z0 +3 q z0 2 { 13 − } + z0 2 { 13 − m} )φ(u) = 0 (2.7) − Still under the assumption that ∆1 + ∆3 ∆ is a positive integer, the series also truncates from below at k = 1 (∆ ∆ ). − min 2 − 13 73

2 WHAT IS THE BARE POMERON ? LEADING 1/N TERM CYLINDER EXCHANGE

WEAK: TWO GLUON <=> STRONG: ADS GRAVITON

J =2

Jcut =1+1 1=1 1 12 1 S = d4xdz g(z) + + gMN@ @ 22 R R2 2 M N Z p ⇣ ⌘ F.E. Low. Phys. Rev. D 12 (1975), p. 163. AdS Witten Diagram: Adv. S. Nussinov. Phys. Rev. Lett. 34 (1975), p. Theor. Math. Physics 2 (1998)253 1286. Holographic Approach to QCD

•Spin-2 leads to too fast a rise for cross sections 2 Need to consider g N c finite • ⌘ •Graviton (Pomeron) becomes j-Plane singularity at

j0 =2 2/⇥ j0 :2 2 2/⇤ ⇥ •Comfinement: Particles and Regge trajectories • Brower, Polchinski, Strassler, and Tan: “The Pomeron and Gauge/String Duality,” hep-th/063115

HE scattering after AdS/CFT = 4 Strong vs Weak g2N N c j0

1.5 j0 =1.25? 1 j0 =2 j =1 0 Graviton 0.5 Two Gluon 2 4 6 8 BFKL QCD? αNBPST j = 1 + ln(2)g2N /⇡2 0 c j =2 2/ g2N 0 c p Outline

•QCD High Energy Scattering with AdS/CFT -- Universality •Consequence of Conformal Invariance •DIS at low-x -- Uniﬁcation •DGLAP (large Q) vs BFKL (small x) •OPE (Anomalous Dimensions) and Conformal Pomeron •Conformal Pomeron and Odderon Intercepts in strong coupling •Saturation, Conﬁnement, etc. and DIS •Summary and Outlook •Summary and Outlook II: Pomeron in the conformal Limit, OPE, and Anomalous Dimensions

0 Gmn = gmn + hmn

Massless modes of a closed string theory:

Need to keep higher string modes

As CFT, equivalence to OPE in strong coupling: using AdS CFT correlate function – coordinate representation (x )(x )(x )(x ) (u, v) h 1 2 3 4 i⇠······A (u, v)= a(,J)G (u, v) OPE: A (,J) X,J 2 2 2 2 x12x34 x23x14 u = 2 2 ,v= 2 2 x13x24 x13x24

Minkowski Limit: u 0,v1+O(pu) ! !

HE scatteringDynamics after AdS/CFT (J, ) ,J=0, 1, 2, = g2N ··· c QCD: EMERGENCE OF 5-DIM: ADS Full O(4, 2) Conformal Group

15 generators: Pµ,Mµ⌫ ,D,Kµ

SO(4, 2) = SO(1, 1) SO(3, 1) ⇥ Longitudinal Boost: SO(1, 1)

Maximal commuting subgroup: SO(3, 1)

6 generators iD M12,P1 iP2,K1 iK2 ± ± ⌥

HE scattering after AdS/CFT CFT correlate function – coordinate representation (x )(x )(x )(x ) (u, v) h 1 2 3 4 i⇠······A (u, v)= a(,J)G (u, v) OPE: A (,J) X,J Amplitude – in mixed representation SO(4, 2) = SO(1, 1) SO(3, 1) ⇥ i i 1 d 1 dj j ~ A(s, b)= (,j) s (,j)(b) i 2⇡i i 2⇡i A Y Z 1 Z 1 1 HE scatteringDynamics after AdS/CFT (,j) e A ⇠ (j, ) Symmetry Isometry AdS/CF T ===>

Full O(4, 2) Conformal Group as Isometries of AdS5 Space

+ 2 2 2 dx dx+(dx ) +dz ds = z2 ? SO(4, 2) = SO(1, 1) SO(3, 1) ⇥ S(3, 1) SL(2,C): “Mobius group” - ' as Isometries of the Euclidean (transverse) AdS3 Space

i i 1 d 1 dj j A(s, b)= dz dz0 (,j,z,z0) s (L(b,z,z0)) i 2⇡i i 2⇡ A Y Z Z Z 1 Z 1 HE scattering after AdS/CFT e Symmetry Isometry AdS/CF T ===>

Full O(4, 2) Conformal Group as Isometries of AdS5 Space

+ 2 2 2 dx dx+(dx ) +dz ds = z2 ? SO(4, 2) = SO(1, 1) SO(3, 1) ⇥ S(3, 1) SL(2,C): “Mobius group” - ' as Isometries of the Euclidean (transverse) AdS3 Space

i i 1 d 1 dj j A(s, b)= dz dz0 (,j,z,z0) s (L(b,z,z0)) i 2⇡i i 2⇡ A Y Z Z Z 1 Z 1 HE scattering after AdS/CFT e i i 1 d 1 dj j A(s, b)= dz dz0 (,j,z,z0) s (L(b,z,z0)) i 2⇡i i 2⇡ A Y Z Z Z 1 Z 1

e i⇡j 1+e AdS/CFT: (,j,z,z0)=1(z)2(z)3(z0)4(z0) (,j) A ⇥ sin ⇡j ⇥ A 1 (,j) Dynamics: A ⇠ (j, )

j A(s, b)= dzdz0 ⇧ (j) s (L ) i Y(j) (z,z0,b) j=0,2, Z X··· Anomalous Dimension: (j, )e (j, ) j 2 ⌘

In the limit , only j =2survives. !1 HE scattering after AdS/CFT 1 (,j) A ⇠ (j, )

String Theoretic Approach (BPST):

OPE ==> P omeron V ertex Operator

(L 1)V =(L¯ 1)V =0 0 P 0 P

Brower, Polchinski, Strassler, and Tan: “The Pomeron and Gauge/String Duality,” hep-th/063115 HE• scattering after AdS/CFT Strong Coupling Pomeron Propagator--Conformal Limit

• Use J-dependent Dimension

1/2 ∆ : 4 ∆(J) = 2 + [2√λ(J J0)] = 2 + ¯j (2.8) → − !

• BFKL-cut: 2 J0 = 2 (2.9) − √λ

Double-Mellin representation: 3 O•ne graviton Exchange dj d sje A(s, b b0,z,z0)= 2i 2i ···j j ( 2)2 Z Z 0 We begin by considering the one gravition exchange WDitten diagram in Fig. for scalar sources on the boundary of AdS at xi. The bluk co-ordiantes are written wM = (wµ, w0) where we reserve µ, ν, .. for the standard 4-d Minkowski or Euclidean co-ordinates after Wick rotation.

Freedman et al., (hep th/9903196), give the following expression for this diagram, −

Igrav(x1, x2, x3, x4) 2 gs MN M !N ! = dz√g dw√g T (x , x , z)G ! ! (z, w)T (x , x , w) (3.1) 4 " " 1 3 MNM N 2 4 where the bulk to bulk graviton propagator is given by

GMNM !N ! (z, w) = (∂M ∂M ! u ∂N ∂N ! u + ∂M ∂N ! u ∂N ∂M ! u)G(u) + gMN gM !N ! H(u) (3.2) gs is the string coupling, G(u) is the scalar propagator, and H(u) a linear function of G(u). By symmetry arguments they depend only on the geodisic distance for z = (zµ, z0) to w = (wµ, w0): (z w)µ(z w) + (z w )2 u = − − µ 0 − 0 (3.3) 2z0w0 T MN (x, x!, z) is the energy momenut tensor for a scalar particle from x to x! on the boundary (see ﬁgure).

We ﬁnd it more convenient to write Feynam rules in momentum space for the 4 ﬂat co-ordinates,

(2π)4δ4(p + p + p + p )T (1)(p , p , p , p ) = Π d4x e−ipixi I (x , x , x , x ) (3.4) 1 2 3 4 4 1 2 3 4 " i i grav 1 2 3 4

(1) Here the superscript for T4 reminds us that this is the one-graviton exchange contribution to the −(d+1) four-point amplitude. Since √g depends only on the AdS radial variable, e.g., at z, √g = z0 , one can carry out the Fourier transform, arriving at a simple “mixed-representation”,

(1) 1 MN M !N ! T (p , p , p , p ) = dz √g dw √g T˜ (p , p , z )G˜ ! ! (q, z , w )T˜ (p , p , w ) 4 1 2 3 4 4 " 0 " 0 1 3 0 MNM N 0 0 2 4 0 (3.5)

3 = 4 SYM Leading Twist (J)vsJ: N Anomalous Dimensions λ = 0 DGLAP (DIS moments)

(0,2) Tµ ν γ =0

λ = 0, BFKL

= g2N =0 j = j0 @ min Δ ANOMALOUS DIMENSIONS: (j, )=(j, ) j 2

2 =0

(j)=2+p2 g2N (j j ) c 0 qp

=2 1+ g2N(n 2)/2 n n q p

Energy-Momentum Conservation built-in automatically. Holographic Approach to QCD

2 Need to consider g N c finite • ⌘ •Graviton (Pomeron) becomes j-Plane singularity at

j0 =2 2/⇥ j0 :2 2 2/⇤ ⇥ •Confinement: Particles and Regge trajectories • Brower, Polchinski, Strassler, and Tan: “The Pomeron and Gauge/String Duality,” hep-th/063115

HE scattering after AdS/CFT Outline

•QCD High Energy Scattering with AdS/CFT -- Universality •Consequence of Conformal Invariance •DIS at low-x -- Uniﬁcation •DGLAP (large Q) vs BFKL (small x) •OPE (Anomalous Dimensions) and Conformal Pomeron •Conformal Pomeron and Odderon Intercepts in strong coupling •Saturation, Conﬁnement, etc. and DIS •Summary and Outlook •Summary and Outlook III. Deep Inelastic Scattering (DIS) at small-x Deep Inelastic Scattering (DIS)

Q2 F2(x, Q2) = 2 [⌅T (⇥p)+L (⇥p)] 4⇤ em Q2 x s

Q2 Small x : 0 s Optical T heorem

2 2 total(s, Q ) = (1/s)Im A(s, t = 0; Q )

HE scattering after AdS/CFT HERA vs LHeC region: dots are H1-ZEUS small-x data points ELASTIC V S DIS ADS BUILDING BLOCKS

2 3 3 A(s, x x0 )=g0 d bd b012(z)G(s, x x0 ,z,z0)34(z0) ? ? ? ? Z 1 (s)= ImA(s, 0) T s

for F2(x, Q)

1 (z) (z,Q)= [Qz)4(K2(Qz)+K2(Qz)] 13 ! ⇤⇤ z 0 1

3 2 5A(z) d b dzd x g(z)whereg(z)=det[gnm]= e ⌘ ? p DIS in String Theory continued

2 F2(x, Q ) from AdS/CFT 2 Q 1 1 ⇢ 2 Q (Q ) log ( ) Q 0 Q0 x Q0 F2 = c exp( Q ) Q 2 Q 1 2 1 0 log(Q ) ⇢ log(Q0 Q x ) 0 Q0 x 0 q I This is the expression we will use later for comparing to data. Let’s make a few comments about this function.

I c is a dimensionless normalization constant. I have grouped here all the constants the multiply F2, including the coupling constant that comes from , and only appears as product together with normalization. 2 1 (1 ⇢) I At any Q ﬁxed, we see that at small x the term ( x ) dominates. This leads to a violation of the Froissart bound.

Djuri´c— DIS after AdS/CFT Introduction 18/21 MOMENTS AND ANOMALOUS DIMENSION

2 1 n 2 2 M (Q )= dx x F (x, Q ) Q n n 0 2 ! R 2 =0

Simultaneous compatible large Q2 and small x evolutions! Energy-Momentum Conservation built-in automatically. MOMENTS AND ANOMALOUS DIMENSION

2 1 n 2 2 M (Q )= dx x F (x, Q ) Q n n 0 2 ! R 2 =0

(j)=2+p2 g2N (j j ) c 0 qp

=2 1+ g2N(n 2)/2 n n q p

Simultaneous compatible large Q2 and small x evolutions! Energy-Momentum Conservation built-in automatically. Outline

•QCD High Energy Scattering with AdS/CFT -- Universality •Consequence of Conformal Invariance •DIS at low-x -- Uniﬁcation •DGLAP (large Q) vs BFKL (small x) •OPE (Anomalous Dimensions) and Conformal Pomeron •Conformal Pomeron and Odderon Intercepts in strong coupling •Saturation, Conﬁnement, etc. and DIS •Summary and Outlook •Summary and Outlook IV: More on Pomeron and Odderon in the conformal Limit

Massless modes of a closed string theory: 0 metric tensor, Gmn = gmn + hmn Kolb-Ramond anti-sym. tensor, bmn = bnm dilaton, etc. , ⇥, ··· Gauge/String Duality: Conformal Limit

• C=+1: Pomeron <===> Graviton

• C=-1: Odderon <===> Kalb-Ramond Field

45 = 4 SYM Leading Twist (J)vsJ: N Anomalous Dimensions λ = 0 DGLAP (DIS moments)

(0,2) Tµ ν γ =0

λ = 0, BFKL

= g2N =0 j = j0 @ min Δ ANOMALOUS DIMENSION:

2 2 =0 (j)=2+p2 g Nc(j j0) =2 1+ g N(n 2)/2 n 2 n qp q Energy-Momentum Conservation built-inp automatically. Connection to Spin Chain in = 4 YM: N tr DS Z⌧ (S)2 = ⌧ 2 + a (⌧, )S + a (⌧, )S2 + 1 2 ··· e ⌧ =2, (S)=(S + 2) 2 a (2, )=2p 1+O(1/p) e 1 trFµ⌫ D⌫ D⌫0 F⌫0µ0 ··· a2(2, )=3/2+O(1/p) B.Basso, 1109.3154v2 S =0 BPS (S)2 4+2p S ! ' e POMERON AND ODDERON IN STRONG COUPLING: