✬ ✩ → → Generalized quark–antiquark potential at weak and strong coupling Nadav Drukker Based on arXiv:1105.5144 - N.D. and Valentina Forini Workshop on ”Large- N Gauge Theories” The Galileo Galilei Institute for Theoretical Physics Firenze June 6, 2011 → → ✫ ✪ 1
✬ ✩ Introduction and motivation • One of the most fundamental quantities in a quantum field theory is the potential between charged particles. • In gauge theories this is captured by a long rectangular Wilson loop, or a pair of antiparallel lines. ✫ ✪ Nadav Drukker 2 generalized potential
✬ ✩ Introduction and motivation • One of the most fundamental quantities in a quantum field theory is the potential between charged particles. • In gauge theories this is captured by a long rectangular Wilson loop, or a pair of antiparallel lines. • Such an object exists also in N = 4 SYM. – The Wilson loop calculates the potential between two W-bosons arising from a Higgs mechanism. – It is known to two–loop order in perturbation theory and classically and at one–loop in string theory. ✫ ✪ Nadav Drukker 2-a generalized potential
✬ ✩ Introduction and motivation • One of the most fundamental quantities in a quantum field theory is the potential between charged particles. • In gauge theories this is captured by a long rectangular Wilson loop, or a pair of antiparallel lines. • Such an object exists also in N = 4 SYM. – The Wilson loop calculates the potential between two W-bosons arising from a Higgs mechanism. – It is known to two–loop order in perturbation theory and classically and at one–loop in string theory. • Can we do any better? ✫ ✪ Nadav Drukker 2-b generalized potential
✬ ✩ Introduction and motivation • One of the most fundamental quantities in a quantum field theory is the potential between charged particles. • In gauge theories this is captured by a long rectangular Wilson loop, or a pair of antiparallel lines. • Such an object exists also in N = 4 SYM. – The Wilson loop calculates the potential between two W-bosons arising from a Higgs mechanism. – It is known to two–loop order in perturbation theory and classically and at one–loop in string theory. • Can we do any better? • Shouldn’t integrability allow us to calculate this for all values of the coupling (in the planar approximation)? ✫ ✪ Nadav Drukker 2-c generalized potential
✬ ✩ Wilson loops in N = 4 super Yang-Mills � � � � Maldacena Rey, Yee • The usual Wilson loop is �� � x µ ds W = Tr P exp iA µ ˙ • The most natural Wilson loops in N = 4 SYM include a coupling to the scalar fields �� � � x µ + | ˙ x | θ I Φ I � W = Tr P exp iA µ ˙ ds θ I do not have to be constant. • For a smooth loop and | θ I | = 1, these are finite observables. • The scalar coupling is natural for calculating the potential between W-bosons. ✫ ✪ Nadav Drukker 3 generalized potential
✬ ✩ Wilson loops in N = 4 super Yang-Mills � � � � Maldacena Rey, Yee • The usual Wilson loop is �� � x µ ds W = Tr P exp iA µ ˙ • The most natural Wilson loops in N = 4 SYM include a coupling to the scalar fields �� � � x µ + | ˙ x | θ I Φ I � W = Tr P exp iA µ ˙ ds θ I do not have to be constant. • For a smooth loop and | θ I | = 1, these are finite observables. • The scalar coupling is natural for calculating the potential between W-bosons. • For a pair of antiparallel lines � � � W � ≈ exp − T V ( L, λ ) • In a conformal theory we expect V ( L, λ ) = f ( λ ) L ✫ ✪ Nadav Drukker 3-a generalized potential
✬ ✩ • Explicit calculations at weak and at strong coupling: λ 2 − λ 8 π 2 L ln T L + · · · λ ≪ 1 4 πL + V ( L, λ ) = 4 π 2 √ λ � 1 − 1 . 3359 . . . � √ + · · · λ ≫ 1 4 ) 4 L Γ( 1 λ ✫ ✪ Nadav Drukker 4 generalized potential
✬ ✩ • Explicit calculations at weak and at strong coupling: λ 2 − λ 8 π 2 L ln 1 λ + · · · λ ≪ 1 4 πL + V ( L, λ ) = 4 π 2 √ λ � 1 − 1 . 3359 . . . � √ + · · · λ ≫ 1 4 ) 4 L Γ( 1 λ ✫ ✪ Nadav Drukker 5 generalized potential
✬ ✩ • Explicit calculations at weak and at strong coupling: λ 2 − λ 8 π 2 L ln 1 λ + · · · λ ≪ 1 4 πL + V ( L, λ ) = 4 π 2 √ λ � 1 − 1 . 3359 . . . � √ + · · · λ ≫ 1 4 ) 4 L Γ( 1 λ • Hard to guess how to connect these two regimes. • Could go to O ( λ 3 ) and O ( λ 4 ). • We will add extra parameters and study a larger family of observables. • Thus gather more information to help guess an exact interpolating function. ✫ ✪ Nadav Drukker 5-a generalized potential
✬ ✩ Outline • Introduction and motivation • Generalized quark-antiquark potential • Perturbation theory calculation • Classical string surfaces • One loop string determinants • Expansions in small angles • Summary ✫ ✪ Nadav Drukker 6 generalized potential
✬ ✩ Generalized quark-antiquark potential • The straight line and circular Wilson loop are 1 / 2 BPS. • Their expectation value is known exactly. ✫ ✪ Nadav Drukker 7 generalized potential
✬ ✩ Generalized quark-antiquark potential • The straight line and circular Wilson loop are 1 / 2 BPS. • Their expectation value is known exactly. • Can we somehow view the antiparallel lines as a deformation of the circle/line? ? → ✫ ✪ Nadav Drukker 7-a generalized potential
✬ ✩ • We take the following family of curves: 4 2 − 4 − 2 2 4 − 2 − 4 ✫ ✪ Nadav Drukker 8 generalized potential
✬ ✩ • We take the following family of curves: 4 • These are pairs of arcs with opening angle 2 π − φ . • φ = 0 is the 1 / 2 BPS circle. • φ → π gives the antiparallel lines. − 4 − 2 2 4 − 2 − 4 ✫ ✪ Nadav Drukker 8-a generalized potential
✬ ✩ • We take the following family of curves: 4 • These are pairs of arcs with opening angle 2 π − φ . • φ = 0 is the 1 / 2 BPS circle. • φ → π gives the antiparallel lines. − 4 − 2 2 4 • Can have each line couple to a different scalar field − 2 Φ 1 cos θ 2+Φ 2 sin θ Φ 1 cos θ 2 − Φ 2 sin θ and 2 2 − 4 • Gives another parameter: θ . ✫ ✪ Nadav Drukker 8-b generalized potential
✬ ✩ • We take the following family of curves: 4 • These are pairs of arcs with opening angle 2 π − φ . • φ = 0 is the 1 / 2 BPS circle. • φ → π gives the antiparallel lines. − 4 − 2 2 4 • Can have each line couple to a different scalar field − 2 Φ 1 cos θ 2+Φ 2 sin θ Φ 1 cos θ 2 − Φ 2 sin θ and 2 2 − 4 • Gives another parameter: θ . • Crucial point: Calculations are no harder than for the antiparallel case! ✫ ✪ Nadav Drukker 8-c generalized potential
✬ ✩ • By a conformal transformation which maps one cusp to infinity: 4 2 − 4 − 2 2 4 − 2 − 4 • This is a cusp in Euclidean space. • Taking φ = iu and u → ∞ gives the Lorenzian null cusp. ✫ ✪ Nadav Drukker 9 generalized potential
✬ ✩ • By the inverse exponential map we get the gauge theory on S 3 × R • These are parallel lines on S 3 × R . ✫ ✪ Nadav Drukker 10 generalized potential
✬ ✩ • From this last picture we expect � � � W � ≈ exp − T V ( φ, θ, λ ) • The same is true for the cusp in R 4 with T = log R ǫ ✫ ✪ Nadav Drukker 11 generalized potential
✬ ✩ • From this last picture we expect � � � W � ≈ exp − T V ( φ, θ, λ ) • The same is true for the cusp in R 4 with T = log R ǫ • This V ( φ, θ, λ ) is the generalization of V ( L, λ ) we study. • For φ → π it has a pole and the residue is LV ( L, λ ). ✫ ✪ Nadav Drukker 11-a generalized potential
✬ ✩ • From this last picture we expect � � � W � ≈ exp − T V ( φ, θ, λ ) • The same is true for the cusp in R 4 with T = log R ǫ • This V ( φ, θ, λ ) is the generalization of V ( L, λ ) we study. • For φ → π it has a pole and the residue is LV ( L, λ ). • Expanding at weak coupling ∞ � n � λ � V ( n ) ( φ, θ ) V ( φ, θ, λ ) = 16 π 2 n =1 • And at strong coupling √ � 4 π ∞ � l λ V ( l ) � √ V ( φ, θ, λ ) = AdS ( φ, θ ) 4 π λ l =0 ✫ ✪ Nadav Drukker 11-b generalized potential
✬ ✩ Weak coupling 1–loop graphs • Just the exchange of a gluon and scalar field • This graph is given by the integral � � ∂ λ � W � λ =0 = ds dt �− A ( s ) · A ( t ) + Φ( s ) · Φ( t ) � � � x µ ( t ) + θ I ( s ) θ I ( t ) ds dt − ˙ λ � x µ ( s ) ˙ = 8 π 2 | x ( s ) − x ( t ) | 2 λ cos θ − cos φ λ cos θ − cos φ φ log R � = ds dt s 2 + t 2 + 2 st cos φ = 8 π 2 8 π 2 sin φ ǫ • Therefore V (1) ( φ, θ ) = − 2 cos θ − cos φ φ sin φ ✫ ✪ Nadav Drukker 12 generalized potential
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