Transcript From Quarks/Gluons to Strings: String/Gauge Duality

High Energy Scattering and
String/Gauge Duality
ISMD2004, Sonoma State, California
Chung-I Tan, Brown University
Confronting String Theory with High Energy Scattering
•
•
•
•
•
Evidence for QCD as an Effective String Theory
Maldacena’s Counter Revolution: Gauge/Gravity Duality
Re-discovery of QCD String and Extra Dimensions
Glueball Spectrum; Hard vs Soft Pomeron in AdS Background
Stretched Strings
QCD in Strong Coupling and HE Scattering
Some References
•
•
•
•
•
•
•
Maldacena, hep-th/9711200, The Large N Limit of Superconformal Field
Theory and Supergravity
Brower, Mathur, Tan, hep-th/0003115, Glueball Spectrum for QCD from
AdS Supergravity duality
Polchinski and Strassler, hep-th/010974, Hard Scattering and Gauge/String
Duality; hep-th/0219211, Deep Inelastic Scattering and Gauge/String
Duality; Polchinski and Susskind, hep-th/0011204, String Theory and the
Size of hadrons
Brodsky and de Teramond, hep-th/0310227, Light-Front Hadron Dynamics
and AdS/CFT Correspondence
Brower and Tan, hep-th/0207144, Hard Scattering in the M-theory dual for
the QCD String
Brower, Lowe and Tan, hep-th/0211201, Hagedorn transition for strings on
pp-wave ad tori with chemical potentials
Brower and Tan, in preparation, Stretched String in an AdS Black hole
Background and Light cone quantization
I. Evidence for QCD as String
Asymptotic Freedom
Confinement
Force at Long Distance--Constant
Tension/Linear Potential,
Coupling increasing, Quarks and
Gluons strongly bound <==>
“Stringy Behavior”
String Modes: Vibrational Modes and
Rotational Modes
Tests for String Excitations
• Open String
<===>
• Closed String <===>
• String Junction <===>
Mesons
Glueballs
Baryons
• Open String with Fixed Sources
Phenomenological tests <====> ISMD proceedings last 30 years
Can be tested using Lattice Gauge Theory:
Discrete sites ---> Finite Variables Positive Euclidean Action
---> Numerical Simulation
II. String/Gauge Duality
Early String Theories ==> Theory of Everything
Maldacena ==> New Era for YM Theories
“Failure” of Superstrings in Flat-Spacetime
(1973~1998)
•
•
•
•
Zero mass states: (gauge/graviton)
Extra dimensions: D=4+6=10
Supersymmetries:
No Hard Scattering:
AdS/CFT Duality (1998--)
4d YM Theories at weak coupling is dual to
higher dim String Theories with deformed AdS
Background.
•
•
•
•
•
Strong-Weak Dual
Example: Ising-model--(high-low temp.)
4d “Gauge” dual to “Geometry” in higher dim
Absence of 4 dim-Graviton
Wilson loop expectation ~ sum over surfaces in the bulk
with loop in boundary
Maldacena’s String Counter Revolution
Open String <=====> Closed String
duality
YM Wilson Loops <---> Expectation Values in Bulk!
Wilson Loop in AdS/CFT
Surface into AdS space
For rectangular loop: Extension into minimum surface in r-dir
Zero Tension for QCD String---> Unacceptable
III. Re-discovering QCD String
Witten’s proposal---AdS/BH metric
Breaking Supersymmetry
•Deformed AdS Space, Massive Graviton
•Soft Pomeron and Tensor Glueball
•QCD String Tension
Witten Proposal
AdS/BH metric
VI. QCD in Strong Coupling and
HE Scattering
• Stretched String--Transverse Vibrational Modes in deformed AdS.
• IR Physics--Glueballs as AdS Gravitons at strong coupling.
• UV Physics--Parton counting rules for wide-angle scattering.
• Stringy Deconfinement. Fat vs Thin Strings.
• Hagedorn Transition (aka deconfinement ?)
• Pomeron as Massive Graviton.
• BFKL Pomeron, Regge, DIS, Froissart Bound, ………..
QCD after Brane Revolution
New chapter for non-perturbative QCD
Weak Coupling --> Perturbative QCD :
Confinement
--> Dual description in AdS :
unchanged
weak field approx.
• Effective degrees of freedom---massless fields of type IIA String theory
• Confinement by AdS/BH background
Physics at Different QCD Scales in
AdS Space
Tensor Glueball/Graviton Wave
functions
n=0
n=8
n=1
rmin
Potential
n=3
r
Randall-Sundram graviton
Glueball Spectrum
R. Brower, S. Mathur, and C-I Tan, hep-th/0003115, “Glueball Spectrum of QCD
from AdS Supergravity Duality”.
QCD Rutherford Experiment
At WIDE ANGLES QCD exhibits power law behavior:
where n= i ni is the number of ``partons'' in external lines.
The OPE gives
in terms of the lowest twist ti.
Actually QCD is only conformal up to small asymptotic freedom logs.
Wide Angle Scattering
The 2-to-m glueball scattering amplitude T(p1,p2,L, pm+2) for plane wave glueball:
scatter via the string(M-theory) amplitude: A(pi, ri, Xi) in the 10-d (or 11-d) bulk space
(x,r,Y):
We now discuss two different approaches to the QCD string that both give the
correct parton scaling formula.
• AdS5 £ X with IR cut-off on r > rmin or 10-d IIB string theory
• AdS7 £ S4$ Black Hole with horizon r = rmin or 11-d M-theory.
This is a check on the underlining universality of Maldacena's duality conjecture.
10-d String theory Approach
Due to the Red Shift in the Warped Co-ordinate , D s = (R/r) D x ,a plane wave
glueball, exp[i x p], scatters with a local proper momentum,
String is UV shifted in the YM’s IR. (This is the so called UV/IR connection.)
THUS wide angle scattering IS exponentially suppressed in the region r2 [rmin,rscatt]
HOWEVER there is a small remaining amplitude at large r that that gives the
correct conformal scaling of the naive parton model!
E.g for a scalar glueball f » r-4 corresponding to ni = 4 for the YM operator,
Tr[F2], in exact agreement with the parton result.
Summary on Hard Scattering
(1) AdS5 Hard Scattering (Polchinski-Strassler):
WHY is it same QCD perturbative result with g2N ! (g2N)^2?
(2) AdS7 Hard Scattering (Brower-Tan):
WHY does this only depend on the string tension?
(3) Compared with lowest order perturbative results:
string tension
AdS5, e.g. Polchinski/Strassler
AdS7 M-theory, e.g., Brower/Tan
Soft vs Hard Regge Scattering
Similar arguments can be applied to the Regge limit: s >> -t
Dominant scattering at large r, gives a BFKL-like Pomeron with almost flat
``trajectory'' (actually a cut in the j-plane)
The IR region, r ' rmin, gives soft Regge pole with slope a'qcd » a' R3/r3min
The ``shrinkage'' of the Regge peak is caused the soft stringy ``form factor''
in impact parameter:
Pomeron in Gauge/Gravity Duality
• Pomeron as Massive Graviton!
• Intercept “lowered” from 2!
• Interpolate with BFKL due to “hard” at AdS Bdry.
Stretched String in deformed AdS
background
• Lattice evidence for string excitations
• Stretched String Excitations in AdS
M. Luscher and P. Weisz, J. H. E.
Physics, 07 (2002) 328
K. J. Junge, J. Kuti, C. Morningstar,
hep-lat/0207004
Wilson Loop as Surface Sum
Searching for evidence of String!
Minimum Surface for Wilson Loop
QCD String a la Polyakov
Energy of Transverse Oscillations
Linear Potential:
Excited states (Semi-classical limit)
At large L:
r0(z) ! 1, r(z) ! 1
except near end
points:
r0(z) = V2(z)/V2(0)
Radial (longitudinal) Mode
• Near r = rmin (or y = 0), V(y) ' r2min + const y2
THE SPECTRUM IS GAUGE INVARIANT
•Choose a gauge with fluctuations only in longitudinal (X3) or radial (Y) or
normal to classical surface, etc
X3 = z + x or Y = ycl(z) + x , etc
except at end points
D E = (d+1) MGB + O(1/L)
V. Future Developments
• Improved Confining Background
• Challenges: BFKL, Froissart bound,
Massless Quarks, Chiral Symmetry, ……..
• …………..
• Beyond strong coupling or Strong Gravity.
See you at next ISMD!