Contact interactions in string theory and a reformulation of QED

Transcription

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

2 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

3 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 Arχiv: v2 (Review)

4 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 Arχiv: v2 (Review)

5 Motivation It has been shown that the classical field strength tensor of Maxwell electrodynamics can be determined from a string theory perspective [4] : F c µν (x) = 4π 2 dσ µν (X) δ 4 (x X) (1) 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: Σ 4 Mansfield: Arχiv: v2

6 Motivation It has been shown that the classical field strength tensor of Maxwell electrodynamics can be determined from a string theory perspective [4] : F c µν (x) = 4π 2 dσ µν (X) δ 4 (x X) (1) 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. Σ 4 Mansfield: Arχiv: v2

7 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 D ( ) ( ) ] ΨΨ exp Ψ (γ D m) Ψ ( = log det (γ D) 2 + m 2) (2) 5 Nucl. Phys. B385

8 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

9 From worldlines to strings Using the stringy expression for F µν the classical free action for A becomes 1 d 4 x F µν (x) F µν (x) = q2 4 4 dσ µν (X) δ 4 (X X ) dσ µν (X ) (4) This splits into two terms: q 2 4 δ2 (0) A (Σ) + q2 dσ µν (X) δ 4 (X X ) dσ µν (X ) 4, (5) ξ ξ which consists of the Nambu-Goto action of bosonic string theory and a (non-local) contact interaction.

10 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 S = i Goal: S Poly [X i, g i ] + ij q 2 4 dσ µν i (X i ) δ 4 (X i X j ) dσ µν j (X j ) We want to show that the partition function of the string theory coincides with the expectation value of a product of Wilson loops (6)

11 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 S = i Goal: S Poly [X i, g i ] + ij q 2 4 dσ µν i (X i ) δ 4 (X i X j ) dσ µν j (X j ) We want to show that the partition function of the string theory coincides with the expectation value of a product of Wilson loops (6) N i=1 D(X i, g i ) DA e S = Z 0 N e S gf i e i dw i A (7)

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

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

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

15 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 q 2 ( ) d 2 θ i d 2 z i Di X [µ i D ix ν] i dx i θ i θi Ψ[µ i Ψν] i δ d (X i X j ) y ( i=0 ) d 2 θ j d 2 z j Dj X [µ j D jx ν] j dx j θ j θj Ψ[µ j Ψν] j (14) y j=0 We fix the worldsheet to the boundary by generalising the previous Dirichlet boundary conditions X µ y=0 = w µ, ( Ψ µ + Ψ µ) y=0 = h 1/4 ψ µ. (15)

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

17 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: ( G ɛ 0 = 1 + i 2 θ θ ) ( ) 2y f (17) y ɛ The cut-off ɛ introduces a scale into the system which breaks conformal invariance! Wick contractions yield a common exponential term with an expansion ( e πα k 2 G 0 = 1 + i 2 θ θ ) ( ) e πα k 2 f 2y ɛ. (18) y

18 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 q 2N N j=1 B dx j dx j ( e ikj (wj w j) dwj k 2 + dx j ( dw j dx j h j ik j ψ j ψ j ) ) h j ik j ψ jψ j (19) This is independent of the cut-off, ɛ, and the string tension, α.

19 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: 1 ɛ F µ1...νn+1 (k 1,.., k n+1 ) ( ɛ d 2 z n+1 : e ik X(zn+1) : y 2 n+1 ) α K 2 /4 where K = n+1 r=1 k r and F µ1...νn+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! (20)

20 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: K µν ɛ ( d 2 z n+1 : Ψ ɛ µ Ψ ν e ik X(zn+1) : y 2 n+1 ) α K 2 /4 (21) where K = n+1 r=1 k r and K µν holds the index structure, formed by integrating the insertions about a reference point z n+1. Its variation under the residual supersymmetry is proportional to the variation of the boundary term ɛ 1/2 dx exp(ik w).

21 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: K µν ɛ ( d 2 z n+1 : Ψ ɛ µ Ψ ν e ik X(zn+1) : y 2 n+1 ) α K 2 /4 (21) where K = n+1 r=1 k r and K µν holds the index structure, formed by integrating the insertions about a reference point z n+1. Its variation under the residual supersymmetry is proportional to the variation of the boundary term ɛ 1/2 dx exp(ik w). But we directly proved that this contribution vanishes.

22 Results for the spinning string The form of G means that that there are three important configurations of the insertions: When the insertions are also close to the boundary the supersymmetry allows us to constrain the form of the results it is possible to generate. There are two possible divergences 1 dx e ik X K ρ and dx ( Ψ + Ψ ) ρ e ik X (22) ɛ ɛ 1 4 There s also a possible finite piece invariant under the supersymmetry which would spoil the result: dx e ik X ( dx µ /dx + ik (Ψ + Ψ)(Ψ + Ψ) µ) (23) The generalised Gauss law comes to the rescue.

23 The classical action and the conformal scale The conformal scale of the worldsheet metric has decoupled from the calculation. So too has the scale on the string tension. They only appear in the prefactors exp ( S [X c ] S L [φ, χ]]) (24) We deal with these in turn: The tensionless limit α k 2 0 removes the dependence on the classical action There are a number of ways to handle the Liouville theory Appeal to it cancelling out when we normalise against the free theory partition function Assume the existence of further internal degrees of freedom to take us into a critical theory

24 The final result The structure provided by worldsheet supersymmetry ensured that no divergences or finite corrections were encountered so in this case the partition function coincides with the expectation value of a product of super Wilson-loops: n D(g, X, w, ψ, h, χ) j e S S B = j Z 0 n D(w, ψ, h, χ) j DA N e S A S B W [A]. (25) This is the result of our work. j j

25 Non-Abelian gauge theory We ve taken spinor QED as the field theory we wish to reformulate. We dealt with the Liouville mode in a slightly unsatisfactory way. What could these internal degrees of freedom be? We can use them to provide the extra details required for a field theory with a non-abelian symmetry. The super Wilson loop now takes the form { W [A] = P e ( dw A A τ A dξ ψ µ F A µν τ A ψ ν ) } (26) We have seen in a previous seminar how the path ordering, group representations and chirality can be dealt with in the worldline approach [7] by introducing further degrees of freedom ϕ and ϕ. How can we include them in the string theory? 7 Arχiv:

26 Non-Abelian gauge theory We introduce new superfields Y and Ỹ and modify the interaction q 2 d 2 θ i d 2 z i e Φ A DY i τabdỹ k i B D i X [µ i D ix ν] i δd (X i X j ) d 2 θ j d 2 z j Dj X [µ j D jx ν] j DY j R τrsdỹ k j S e Φ (27) The boundary contribution comes from the classical piece and provides factors of the form e φ/2 ϕ A τ k AB ϕb. The equations of motion for ϕ and ϕ are first order and get matched to boundary fields which impose the path ordering.

27 Non-Abelian gauge theory We stand to pick up other terms from the quantum fluctuations of Y and Ỹ : DY A 1 τ k ABDỸ B 1 DY R 2 τ l RSDỸ S 2 = D 1 D 2 G 12 D 1 D2 G 12 τ m AA, (28) but this vanishes if the generators of the symmetry group are traceless. We ve heard how this technique can be applied to the standard model. A natural question is whether the same procedure leads to familiar results when applied to other symmetry groups.

28 Unified theories I recently considered the groups SU(5) and SU(6) as candidate unified theories [8]. I computed the representations and chiralities that appear if the Wilson loop is taken to transform in the fundamental representation of each group. For W [A] transforming in the 5 of SU(5) the result is (tr (W 5) + tr (W 10 ) + 1) P L + (tr (W 5 ) + tr (W 10 ) + 1) P R. (29) For the 6 of SU(6) one finds (tr (W 6 ) + tr (W 20 ) + tr (W 6))P L + (tr (W 15 ) tr (W 15 ))P R (30) 8 Arχiv:

29 Conclusion We have presented a reformulation of spinor QED where the fundamental degrees of freedom generating the gauge interactions are tensionless spinning strings interacting on contact. This string theory is unusual in a number of ways 1. The string world-sheets correspond to the trajectories of lines of electric flux joined to charged particles. 2. It is off-shell and we have open string vertex operators integrated throughout the worldsheet. 3. The conformal scale decouples so there is no Weyl anomaly. In this way to model favours spinor matter. 4. The string length-scale is large compared to the size of the Wilson loops. 5. The non-abelian generalisation is natural and leads to an interesting worldline model.

30 Conclusion We have presented a reformulation of spinor QED where the fundamental degrees of freedom generating the gauge interactions are tensionless spinning strings interacting on contact. This string theory is unusual in a number of ways 1. The string world-sheets correspond to the trajectories of lines of electric flux joined to charged particles. 2. It is off-shell and we have open string vertex operators integrated throughout the worldsheet. 3. The conformal scale decouples so there is no Weyl anomaly. In this way to model favours spinor matter. 4. The string length-scale is large compared to the size of the Wilson loops. 5. The non-abelian generalisation is natural and leads to an interesting worldline model. Thank you for your attention.