Worldsheet-Induced Corrections to the Holographic Veneziano - PowerPoint PPT Presentation

Worldsheet-Induced Corrections to the Holographic Veneziano Amplitude Edwin Ireson Department of Physics, Swansea University Singleton Park SA28PP Swansea, U.K. 10/11/2016 Based on: A. Armoni (1509.03077) A.Armoni & E.Ireson (1607.04422,1611.00342) Edwin Ireson (Swansea University) Holographic Veneziano Corrections 10/11/2016 1 / 20

Introduction 1.Introduction Explaining the linear Regge trajectory of mesons helped birth String Theory through the Veneziano amplitude, the 4-point open string amplitude in flat space. Improving upon this relation is a long-standing goal: asymptotic freedom forbids the Regge trajectory to stay linear at all energies. Holography provides a bridge between the two setups but involves highly-curved backgrounds in which the Veneziano amplitude is not easily seen. Some work has been done already to recover it. In pure field theory, can prove that assuming area-law behaved (i.e. confining) Wilson loops at all energies reproduces Veneziano behaviour (Makeenko, Olesen). We notice that in some string backgrounds, can force any worldsheet hanging from a Wilson loop at the boundary to exhibit area-law behaviour, recovering the Veneziano amplitude. This destroys almost all contributions from holographic coordinates, but can be systematically be improved upon. Q: How does curvature of a ”realistic” string theory affect the Regge trajectory of mesons in QCD? Does it match observed phenomena? Edwin Ireson (Swansea University) Holographic Veneziano Corrections 10/11/2016 2 / 20

The Worldline Formalism 2. The Worldline Formalism First we need a set-up reproducing the Veneziano amplitude in holography. To map the QCD amplitude into holography we use the Worldline Formalism. Rewrite the path integral as a sum over Wilson loops: � ∞ � � � − N f dT Z = DA exp( − S YM ) exp 2 Tr T W T [ A ] (1) 0 � � T x µ ˙ x µ + ψ µ ˙ � T DxD ψ e − 1 x µ A µ − 1 0 d τ ˙ ψ µ e i 0 d τ ˙ 2 ψ µ F µν ψ µ W T [ A ] = (2) 2 In the large N c , fixed N f linearise the exponential. Inserting 4 meson operators then restricts the Wilson loops to pass through those 4 points. � 4 � � ∞ � � − N f dT � � � q ¯ q ( x i ) = DA exp( − S YM ) 2 Tr T W T [ A ] (3) � x 1 , x 2 , x 3 , x 4 0 i =1 At this stage, assuming the Wilson loops are all area-behaved yields a Veneziano amplitude, can prove without string theory, but strings give a framework for improvement. Edwin Ireson (Swansea University) Holographic Veneziano Corrections 10/11/2016 3 / 20

The Worldline Formalism This quantity we map to a computation in String Theory via the gauge-gravity duality. For a suitable dual to Yang-Mills, of target space metric G MN : 4 � � � � � � [ DX ] W exp ( ik µ d 2 σ G MN ∂ α X M ∂ α X N A ( k 1 ... 4 ) = i X µ ( σ i )) exp − d σ i i =1 (4) Where W ( σ i ) exp ( ik µ i X µ ( σ i )) is a generic ansatz for the meson operator in that space, where W is usually unknown. The flat open string 4-point amplitude reproduces the Veneziano so set the space up such that the strings are mostly flat. This relies on several assumptions: No quark masses, No higher genus corrections g αβ = η αβ , Dual background exhibits confinement (known conditions on G MN ), Ignore additional compact coordinates unrelated to holographic direction, Implement by hand the effects of W on the amplitude. Edwin Ireson (Swansea University) Holographic Veneziano Corrections 10/11/2016 4 / 20

The Worldline Formalism Impose that the characteristic depth of the space is infinitely small, such that all Wilson loops exhibit an area law Boundary Horizon Figure: A confining string worldsheet accreting on the end of space. Edwin Ireson (Swansea University) Holographic Veneziano Corrections 10/11/2016 5 / 20

The Worldline Formalism Practical example: Witten’s model of D 4 branes wrapped around a circle. Where f ( U ) = 1 − U 3 U 3 , KK � 3 / 2 � dU 2 � 3 / 2 � � U � R � ds 2 = dX 2 + d τ 2 f ( U ) f ( U ) + U 2 d Ω 4 � + (5) R U Taking U KK → ∞ , U KK = cst. = λ , dU = 0 brings the end of space R (Euclidean horizon) up towards the boundary: ds 2 = λ 3 / 2 dX 2 + . . . (6) This makes loop size far exceed depth of space, ”most” loops confine. Free worldsheet action, obtain Veneziano amplitude. Can relax our assumptions a little. Instead of assuming depth of space infinitely small, we take it to be a small finite parameter with which we build interactions, by expanding the metric order by order, turning on interactions between X and U . We still need to neglect contributions from the edges of the sheet, very subleading in the classical action. Edwin Ireson (Swansea University) Holographic Veneziano Corrections 10/11/2016 6 / 20

Preparing the worldsheet field theory 2. Preparing the worldsheet field theory To prepare a workable worldsheet field theory, we need to perform the following steps. Creating the interaction terms would be easier if the action was not singular. Some thought is required to find the correct way to regularise it. � 3 / 2 �� � � � 3 / 2 � 1 − U 3 � U � �� � R 1 ds 2 = dx 2 + d τ 2 dU 2 + U 2 d Ω 4 KK + 1 − U 3 U 3 R U KK U 3 A good change of coordinates needs to make this metric regular around the origin, but also to have a regular and non-vanishing Jacobian, because the � parametrisation-invariant NLSM measure is [ DX ] = det( G ) DX . At some point, expand metric locally around end of space, creating an interacting QFT for modes on the worldsheet. Edwin Ireson (Swansea University) Holographic Veneziano Corrections 10/11/2016 7 / 20

Preparing the worldsheet field theory u 2 For regularity of the metric, change coordinates. writing U = U KK (1 + KK ) U 2 this regularises the coordinate system around the horizon. The form is standard for such Euclidean black hole-like metrics. For regularity of the determinant of the metric, we use a Kruskal-like procedure. Branes wrap a compact direction, resulting in a cone-like (”cigar”) submanifold with vanishing subdeterminant at u = 0: ds 2 = · · · a ( u 2 ) du 2 + u 2 b ( u 2 ) d τ 2 + · · · = C ( Y 2 + Z 2 ) dY 2 + dZ 2 � � + · · · (7) The Kruskal procedure ”unwraps” the warped cone to warped Cartesian � coordinates, but crucially, requires to compute Tortoise coordinate, G UU dU . Very impractical to do globally given form of integrand. We only need information from the metric locally around u = 0: in an expansion around U KK assuming small fluctuations, change variables from ( u , τ ) to Kruskal-like coordinates ( Y , Z ). This naturally preserves shift symmetry in τ i.e. a U (1) global symmetry. Edwin Ireson (Swansea University) Holographic Veneziano Corrections 10/11/2016 8 / 20

Preparing the worldsheet field theory With a metric expanded to first order the exact change is Y 2 + Z 2 u 2 u 2 = exp( ) (8) 2 U 2 2 U 2 2 U 2 KK KK KK Relation can be inverted and new metric expanded to first order to obtain an effective Lagrangian for the fluctuations. Defining λ = U KK and the doublet Υ = ( Y , Z ), the (bosonic) Lagrangian in R these coordinates is then 1 + 3Υ 2 � � 4 L = λ 3 / 2 ∂ α X µ ∂ α X µ + 3 λ 3 / 2 ∂ α Υ · ∂ α Υ + . . . (9) 2 U 2 KK But the integration measure also has to change, as it is proportional to det( G ): by this change of coordinates 1 + 6Υ 2 � � 16 det( G ) = U 8 + . . . (10) KK 9 λ 3 U 2 KK This can be exponentiated to give a small mass to the new radial field. Edwin Ireson (Swansea University) Holographic Veneziano Corrections 10/11/2016 9 / 20

Preparing the worldsheet field theory Some comments about the path integration: � 4 � �� k i · X ( σ i ) Compute the string 4-point function exp � , by the usual i =1 4 k i · X ( σ ) δ ( σ − σ i ). trick of writing this as a current J ( σ ) = � i =1 Then, sufficient to compute partition function with a non-zero current. Introduces a 1-leg vertex in the Feynman rules (ending a propagator with a Fourier kernel), care taken for loop order vs. expansion order: Despite worldsheet having supersymmetry, ignore superpartners. They communicate less directly with the X fields than Υ does. Edwin Ireson (Swansea University) Holographic Veneziano Corrections 10/11/2016 10 / 20

Computing and analysing the correction 3. Computing and analysing the correction 2D QFTs have technical particularities, notably related to regularisation. Propagator is UV divergent in two dimensions. We use analytic regularisation: introducing an arbitrary mass scale µ , � � � 1 � 1 d x µ x = lim (11) p 2 + m 2 ( p 2 + m 2 ) 1+ x dx x → 0 AR This also method also works in the massless case, dealing with the IR divergence identically. With correct variant of MS , difficulties are dealt with automatically. Edwin Ireson (Swansea University) Holographic Veneziano Corrections 10/11/2016 11 / 20