Computing the Adler function from the vacuum polarization Hanno Horch Institute for Nuclear Physics, University of Mainz In collaboration with M. Della Morte, G. Herdoiza, B. Jäger, A. Jüttner, H. Wittig
Outline 1 Adler function Methods to compute the Adler function Numerical results for the Adler function 2 Ward identity Partially twisted boundary conditions Vacuum polarization Numerical results 3 Conclusions
Adler function The Adler function is dened as 1 D(q 2 ) q 2 = 3π α d dq 2 α had(q 2 ), and it can be measured in e + e annihilation experiments. The Adler function is related to the vacuum polarization by D(q 2 ) = 12π 2 q 2 dπ(q2 ) d(q 2 ). The vacuum polarization tensor can be computed by Π µν (q 2 ) = d 4 xe iqx J µ (x)j ν (). From Euclidean invariance and current conservation one nds Π µν (q 2 ) = ( g µν q 2 q µ q ν ) Π(q 2 ). µ ν 1 Adler, Phys. Rev. D 1, 3714, 1974
Methods to compute the Adler function Fit to Π(q 2 ) Fit an ansatz to Π(q 2 ), and compute the derivative of the t function. We use the Pade ansatz ( Π fit (q 2 ) = c + q 2 c1 q 2 + c 2 + c ) 3 2 q 2 + c 2, 4 d dq 2 Π fit(q 2 c 1 c 2 2 ) = (c 2 2 + q2 ) 2 + c 3 c 2 4 (c 2 4 + q2 ) 2. Numerical derivative of Π(q 2 ) We use linear ts with varying ranges to approximate the derivative of Π(q 2 ). In our study we consider the CLS-ensembles given in the table below. Label V β a[fm] m π [MeV] m π L N cfg A5 64 32 3 5.2.79 312 4. 25 E5 64 32 3 5.3.63 451 4.7 168 F6 96 48 3 5.3.63 324 5. 217 N6 96 48 3 5.5.5 34 4. 173 cf. Capitani et al, arxiv:111.6365, 211.
Procedures for the numerical derivative 1 Procedure I at each q 2 perform a linear t Π [l] fit (q2 ) = a l + b l q 2, repeat these ts for several t ranges ɛ [.5, 1.]GeV 2, search for a region in ɛ where variations in b l are small. Π(q 2 ).14.12.1.8.6.4 Π(q 2 ) on N6, a =.5fm.9.85.8.75.7.1.2.3.4.5.6 1 2 3 4 5 6 7 8 9 Della Morte et al, arxiv:1112.2894, 212
Procedures for the numerical derivative 2 Procedure II at each q 2 we t the two functions Π [l] fit (q2 ) = a l + b l ln ( q 2), Π [q] fit (q2 ) = a q + b q ln ( q 2) + c q ( ln ( q 2 )) 2, repeat these ts for several t ranges ɛ [.5, 1.]GeV 2, apply cuts to the ts, such as removing ts with a large curvature c q, from the ts that survive pick the result, where the coecients b l and b q are similar. Π(q 2 ).21.18 Π(q 2 ) on N6, a =.5fm.15.12.9.6.3.1.1.1 1 1
Comparison of the dierent methods on N6 D(q 2 ) 2 1.8 1.6 1.4 1.2 1.8.6.4.2 1 2 3 4 5 6 7 8 9 Pade procedure I (linear) procedure II (quadratic) The dierent methods agree within errors for a large range of momentum transfers, but for very small and large values of q 2 we nd deviations for procedure I.
Scaling of the Adler function 1 1.8 1.6 1.4 1.2 D(q 2 ) 1.8.6.4.2 2 4 6 8 1 The phenomenological curve was provided by Meyer et al., arxiv:136.2532, 213. q 2 m 2 ρ phenomenology a =.5fm a =.63fm a =.79fm
Scaling of the Adler function 2 1.9 1.8 linear t in a linear t in a 2 phenomenology 1.7 7) D( q2 m 2 ρ 1.6 1.5 1.4 1.3.1.2.3.4.5.6.7.8 a[fm]
1 Adler function Methods to compute the Adler function Numerical results for the Adler function 2 Ward identity Partially twisted boundary conditions Vacuum polarization Numerical results 3 Conclusions
Partially twisted boundary conditions 1 We use partially twisted boundary conditions, cf. Sachrajda, Villadoro, Physics Letters B, 25, Bedaque, Physics Letters B, 24, de Divitiis et al, Physics Letters B, 24, which allow to tune the momenta to Ψ(x i + L) = e iθi Ψ(x i ), ˆp µ = 2sin ( πn µ L µ ) Θ(1) µ Θ (2) µ 2L µ In simulations the twist is interpreted as a constant background eld, where B =, and B i = U Θ µ (x) = U µ (x)e iabµ, ( ) B (1) i B (2) with B (j) i i ( ) = Θ(j) i q L, and Ψ(x) = (1) q (2) for N f = 2.
Partially twisted boundary conditions 1 From the variation of the Wilson action in the presence of twisted boundary conditions using a avor transformation, Ψ(x) Ψ (x) = e iαa τa 2 Ψ(x), Ψ(x) Ψ (x) = Ψ(x)e iαa τa 2, one nds that [ ] B µ, τ a 2 = is required to dene a conserved current 1. There are several ways to fulll this condition. (a) B µ = B µ 1, both quark elds are twisted by the same angle. Since only the dierence of the twists is relevant for the momentum this would remove the eect the twisted boundary conditions have. (b) B µ = Θµ L L, in the innite volume limit we recover the conserved current. 1 cf. Aubin et al, arxiv:137.471, 213 for a similar discussion.
Vacuum polarization To compute the vacuum polarization tensor Π l,p µν(q 2 ) = d 4 xe iqx J µ (l) (x)j ν (ps) (), we use the vector currents given by J (l) µ (x) = Q f Ψ(x)γµ Ψ(x), J µ (ps) (x) = Q f [ Ψ(x + µ)u 2 µ (x)e iabµ (γ µ + 1)Ψ(x) + Ψ(x)U µ (x)e iabµ (γ µ 1)Ψ(x + µ) ]. µ ν The Ward identity of the vacuum polarization in the innite volume is given by q ν Π l,p µν(q 2 ) =, q µ Π l,p µν(q 2 ) =.
Ward identity on E5 and F6 (point-split) As the lattice volume is increased the violation of the Ward identity vanishes for Θ. 6e-14 Θ x =. 2e-5 Θ x = 9 1 π q 2 [GeV 3 ] q νπ l,p ν (q2 ) 4e-14 2e-14 1.5e-5 1e-5 5e-6 2 4 6 8 1 12 2 4 6 8 1 12 E5, a =.63fm, L= 2.fm F6, a =.63fm, L= 3.fm
Eect of the Violation on Π(q 2 ) There are dierent approaches to quantify the eect of the violation of the WI on Π(q 2 ), A (p) µ (q 2 ν ) = q νπ µν q ν Π l,p µν(q 2 ), B µ (p) (q 2 q ) = 2 q µ Π µµ q µ q 2 Π(q 2. ) q 2 A (p) µ (q 2 ) is similar to a ratio used by Aubin et al, arxiv:137.471, 213. A (p) B (p) Θ = 9 1 π.8.6.4.2 -.2 -.4 q 2 2 4 6 8 1 12 1.8e-6 1.6e-6 1.4e-6 1.2e-6 1e-6 8e-7 6e-7 4e-7 2e-7-2e-7 2 4 6 8 1 12 q 2 E5, a =.63fm, L= 2.fm F6, a =.63fm, L= 3.fm
1 Adler function Methods to compute the Adler function Numerical results for the Adler function 2 Ward identity Partially twisted boundary conditions Vacuum polarization Numerical results 3 Conclusions
Conclusions Adler function From the vacuum polarization we can obtain the Adler function with dierent methods, which agree within errors for a large range of q 2. As we approach the continuum limit our results approach the phenomenological result. Ward identity The Ward identity q ν Π µν = is not fullled at nite size. For our largest twist angle Θ = 9π 1 the violation of the Ward identity for the point-split current is of O ( 1 5). The eect the violation has on the vacuum polarization is below the current sensitivity and thus negligible.
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Ward identity on E5 and F6 (point-split)[x] Θ x =. Θ x = 9 1 π 8e-14 4.5e-5 4e-5 q 2 [GeV 3 ] 6e-14 4e-14 3.5e-5 3e-5 2.5e-5 q νπ l,p 1ν (q2 ) 2e-14 2e-5 1.5e-5 1e-5 5e-6 2 4 6 8 1 12 2 4 6 8 1 12 E5, a =.63fm, L= 2.fm F6, a =.63fm, L= 3.fm
Ward identity on N6, β = 5.5, a =.5fm (point-split) For larger twist angles Θ the violation of the Ward identity becomes more severe. q 2 [GeV 3 ] q νπ l,p ν (q2 ) 3.5e-6 3e-6 2.5e-6 2e-6 1.5e-6 1e-6 5e-7-5e-7 Θ = 9 1 π Θ = 1 2 π Θ = 1 5 π 3 3.2 3.4 3.6 3.8 4
Ward identity on N6, F6, and A5 (local) As the continuum limit is approached the violation of the Ward identity due to the local current diminishes. Θ =. Θ = 9 1 π.9.9.8.8 q 2[GeV3 ] q µπ l,p µ (q2 ).7.6.5.4.3.2.1.7.6.5.4.3.2.1 -.1 1 2 3 4 5 6 7 8 -.1 1 2 3 4 5 6 7 8 9 F6, a =.63fm, L= 3.fm N6, a =.5fm, L= 2.4fm A5, a =.79fm, L= 2.5fm