Contact interactions in string theory and a reformulation of QED - PowerPoint PPT Presentation

Contact interactions in string theory and a reformulation of QED James Edwards QFT Seminar November 2014 Based on arXiv:1409.4948 [hep-th] and arXiv:1410.3288 [hep-th]

Outline Introduction Worldline formalism From worldlines to strings Forming an interacting string theory Main result Spinor QED and the spinning string The supersymmetric model Results for the spinning string Discussion Decoupling of the conformal scale Generalisation to non-Abelian theory Unified theories Conclusion

Introduction There is a long history of association between string theories and gauge theories 1. Flux tubes in QCD 2. Nambu [ 1 ] and Polyakov loops [ 2 ] . 3. Bern-Kosower rules [ 3 ] . 4. ADS / CFT... 1 Phys Lett B 80 2 Nucl Phys B 164 3 Ar χ iv:0101036v2 (Review)

Introduction There is a long history of association between string theories and gauge theories 1. Flux tubes in QCD 2. Nambu [ 1 ] and Polyakov loops [ 2 ] . 3. Bern-Kosower rules [ 3 ] . 4. ADS / CFT... The work I shall present takes a complementary approach - a theory of interacting tensionless spinning strings provides the expectation value of a product of Wilson loops in spinor QED. 1 Phys Lett B 80 2 Nucl Phys B 164 3 Ar χ iv:0101036v2 (Review)

Motivation It has been shown that the classical field strength tensor of Maxwell electrodynamics can be determined from a string theory perspective [ 4 ] : �� � d Σ µν ( X ) δ 4 ( x − X ) F µν ( x ) = 4 π 2 (1) c Σ This describes the functional average of an operator over the configurations of a string bounded by the worldline of a particle / anti-particle pair. It has some remarkable properties: 4 Mansfield: Ar χ iv:1108.5094v2

Motivation It has been shown that the classical field strength tensor of Maxwell electrodynamics can be determined from a string theory perspective [ 4 ] : �� � d Σ µν ( X ) δ 4 ( x − X ) F µν ( x ) = 4 π 2 (1) c Σ This describes the functional average of an operator over the configurations of a string bounded by the worldline of a particle / anti-particle pair. It has some remarkable properties: ◮ The string theory is off-shell and not in the expected critical dimension. ◮ Vertex operators are integrated over the entire worldsheet. The key to understanding this is in the decoupling of the conformal scale worldsheet metric. 4 Mansfield: Ar χ iv:1108.5094v2

Worldline formalism The worldline formalism of quantum field theory relates the field theory to a set of one dimensional curves interpreted as the worldlines of particles described by a one dimensional theory. Strassler [ 5 ] reformulated scalar and spinor QED and derived the Bern-Kosower “Master Formula” without recourse to string theory. Integrating over matter fields gives effective action: �� �� � ¯ − ¯ � � Γ [ A ] QED = log ΨΨ exp Ψ ( γ · D − m ) Ψ D ( γ · D ) 2 + m 2 � � = − log det (2) 5 Nucl. Phys. B385

Worldline formalism The worldline formalism of quantum field theory relates the field theory to a set of one dimensional curves interpreted as the worldlines of particles described by a one dimensional theory. Strassler [ 6 ] reformulated scalar and spinor QED and derived the Bern-Kosower “Master Formula” without recourse to string theory. Integrating over matter fields gives effective action: � � ¯ − ¯ � � � Γ [ A ] QED = D ΨΨ exp Ψ ( γ · D − m ) Ψ � = D ( w, h, ψ, χ ) exp ( − S point ( w, h, ψ, χ )) W [ A ] (3) 6 Nucl. Phys. B385

From worldlines to strings Using the stringy expression for F µν the classical free action for A becomes d 4 x F µν ( x ) F µν ( x ) = q 2 − 1 � � � d Σ µν ( X ) δ 4 ( X − X ′ ) d Σ µν ( X ′ ) 4 4 (4) This splits into two terms: q 2 4 δ 2 (0) A (Σ) + q 2 � � � d Σ µν ( X ) δ 4 ( X − X ′ ) d Σ µν ( X ′ ) � , (5) � 4 � ξ � = ξ ′ which consists of the Nambu-Goto action of bosonic string theory and a (non-local) contact interaction.

The main results Take a set of curves { w i } and introduce bosonic strings whose endpoints are fixed to these curves. The strings interact via the action q 2 � � ( X i ) δ 4 ( X i − X j ) d Σ µν � � d Σ µν S = S Poly [ X i , g i ] + ( X j ) i j 4 i ij (6) Goal: We want to show that the partition function of the string theory coincides with the expectation value of a product of Wilson loops

The main results Take a set of curves { w i } and introduce bosonic strings whose endpoints are fixed to these curves. The strings interact via the action q 2 � � ( X i ) δ 4 ( X i − X j ) d Σ µν � � d Σ µν S = S Poly [ X i , g i ] + ( X j ) i j 4 i ij (6) Goal: We want to show that the partition function of the string theory coincides with the expectation value of a product of Wilson loops � D A N � D ( X i , g i ) e − S = � N e − S gf � � e − i dw i · A (7) Z 0 i =1 i

Spinor matter For spinor QED we deal with the super-Wilson loop √ � dw · A + 1 � hψ µ F µν ψ ν W [ A ] = dξ (8) 2 We generalise to the spinning string with gauge fixed action �� � 1 � D X µ D X µ − d 2 zd 2 θ ¯ dx ¯ S = Ψ · Ψ (9) 4 πα ′ y =0 where D = ∂ θ + θ∂ z , ¯ D = ¯ ∂ z + ¯ θ ¯ ∂ ¯ z and X is the superfield X µ = X µ + θ Ψ µ + ¯ θ ¯ Ψ µ θB µ ) (10) ( + θ ¯

Supersymmetry The worldline action on the boundaries of the spinning strings takes the form � 1 2 ψ µ F µν ψ ν √ � � � � dw � S B = 1 ψ · dψ dξ + χ dw dξ · A + 1 √ dξ · ψ + i h dξ 2 h 0 (11) and has a local supersymmetry parameterised by (Gramssmann) δα : √ � dw � δ α ψ = δα dξ − 1 δ α χ = 2 d δα δ α w = δαψ , √ 2 χψ , δ α h = δαχ , dξ h (12)

Supersymmetry The worldline action on the boundaries of the spinning strings takes the form � 1 2 ψ µ F µν ψ ν √ � � � � dw � S B = 1 ψ · dψ dξ + χ dw dξ · A + 1 √ dξ · ψ + i h dξ 2 h 0 (11) and has a local supersymmetry parameterised by (Gramssmann) δα : √ � dw � δ α ψ = δα dξ − 1 δ α χ = 2 d δα δ α w = δαψ , √ 2 χψ , δ α h = δαχ , dξ h (12) The gauge fixed action spinning string has a residual global supersymmetry parameterised by η � ∂ � ∂θ − θ ∂ ∂z + ∂ θ ∂ θ − ¯ δ X = η X (13) ∂ ¯ ∂ ¯ z

From strings to fields To reformulate the field theory we generalise the interaction and impose boundary conditions: ◮ The supersymmetric generalisation of the interaction term is � �� � � δ d ( X i − X j ) d 2 z i ¯ D i X [ µ i D i X ν ] dx i θ i ¯ θ i ¯ Ψ [ µ i Ψ ν ] q 2 d 2 θ i i − i y i =0 �� � � � d 2 z j ¯ D j X [ µ j D j X ν ] dx j θ j ¯ θ j ¯ Ψ [ µ j Ψ ν ] d 2 θ j × j − (14) j y j =0 ◮ We fix the worldsheet to the boundary by generalising the previous Dirichlet boundary conditions Ψ µ + ¯ y =0 = h 1 / 4 ψ µ . X µ | y =0 = w µ , Ψ µ �� � (15) �

Vertex operators We proceed by pertubatively expanding the interaction term which leads to the insertion of vertex operators inside the path integral: d d k � (2 π ) d e − ik · x 1 D X [ µ D X ν ] δ d ( X − X ′ ) ¯ D X ′ [ µ D X ′ ν ] = 4 V µν ( k ) V µν ( − k ) ¯ V µν ( k ) = ¯ D X [ µ D X ν ] e ik · X . (16) This seems to be inconsistent with the mass-shell condition required to avoid the Weyl anomaly!

Results for the spinning string The behaviour of the Green’s function at coincident points is important. Divergences in G require regularisation. We regulate it in a manner that preserves the residual supersymmetry: � 2 y � 1 + i θ ∂ � � 2 θ ¯ G ǫ √ ǫ 0 = f (17) ∂y The cut-off ǫ introduces a scale into the system which breaks conformal invariance! Wick contractions yield a common exponential term with an expansion � � 1 + i θ ∂ � � 2 y − πα ′ k 2 f e − πα ′ k 2 G 0 = 2 θ ¯ √ ǫ e . (18) ∂y

Results for the spinning string The form of G means that that there are three important configurations of the insertions: ◮ When the insertions are close to the boundary we find the super-Wilson loop N e ik j · ( w j − w ′ j ) � dw j � � � q 2 N � dx j dx ′ + h j ik j · ψ j ψ j · j k 2 dx j B j =1 � � dw ′ � j j ik j · ψ ′ j ψ ′ h ′ − (19) j dx ′ j ◮ This is independent of the cut-off, ǫ , and the string tension, α ′ .

Results for the spinning string The form of G means that that there are three important configurations of the insertions: ◮ When the insertions are close together in the bulk we find possible divergences: � α ′ K 2 / 4 1 � � ǫ d 2 z n +1 : e iK · X ( z n +1 ) : ˜ F µ 1 ...ν n +1 ( k 1 , .., k n +1 ) y 2 ǫ n +1 (20) where K = � n +1 r =1 k r and ˜ F µ 1 ...ν n +1 holds the index structure, formed by integrating the insertions about a reference point z n +1 . ◮ This is not supersymmetric so the coefficient must vanish!