Non-perturbative renormalization of tensor bilinears in Schrödinger Functional schemes

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1 Non-pertrbative renormalization of tensor bilinears in Schrödinger Fnctional schemes Patrick Fritzsch, a Carlos Pena, a,b a a Institto de Física Teórica UAM/CSIC, Universidad Atónoma de Madrid, C/ Nicolás Cabrera 3-5, Cantoblanco, Madrid 2849 b Departamento de Física Teórica, Universidad Atónoma de Madrid, Cantoblanco, Madrid p.fritzsch@csic.es, carlos.pena@am.es, david.preti@csic.es We present preliminary reslt for the stdy of the renormalization grop evoltion of tensor bilinears in Schrödinger Fnctional (SF) schemes for N f = and N f = 2 QCD with non-pertrbatively O(a)-improved Wilson fermions. First N f = 2+ reslts (proceeding in parallel with the ongoing comptation of the rnning qark masses []) are also discssed. A one-loop pertrbative calclation of the discretisation effects for the relevant step scaling fnctions has been carried ot for both Wilson and O(a)-improved actions and for a large nmber of lattice resoltions. We also calclate the two-loop anomalos dimension in SF schemes for tensor crrents throgh a scheme matching procedre with RI and MS. Thanks to the SF iterative procedre the non-pertrbative rnning over two orders of magnitde in energy scales, as well as the corresponding Renormalization Grop Invariant operators, have been determined. PoS(LATTICE 25)25 The 33rd International Symposim on Lattice Field Theory 4-8 Jly 25 Kobe International Conference Center, Kobe, Japan Speaker. c Copyright owned by the athor(s) nder the terms of the Creative Commons Attribtion-NonCommercial-NoDerivatives 4. International License (CC BY-NC-ND 4.).

2 . Introdction Tensor crrents play an important rôle in interesting processes throgh which the consistency of the Standard Model (SM) is being crrently probed, as e.g. rare meson decays (see e.g. [2, 3] and references therein) or precision measrements of β-decays and limits on the netron electric dipole moment (see e.g. [4]). Considering the tensor operator for two generical (and formally distinct) flavors ψ,ψ 2 as T µν = iψ (x)σ µν ψ 2 (x), its O(a) improvement in the chiral limit is achieved by considering the combination T I µν = T µν + ac T ( µ V ν ν V µ ) (.) where the coefficient c T was compted at -loop within the Schrödinger Fnctional (SF) in [5] and reads c T =.896()C F g 2 + O(g4 ), for C F = (N 2 )/2N and N colors. At vanishing spatial momentm, the only non-vanishing two-point fnctions with bondary operators allowed by the SF bondary conditions involve the "electric" components (see Eq. (2.) below) ( Tk I = T k + ac T V k ) k V (.2) where the second term in the parenthesis of the rhs vanishes when inserted in Eq. (2.). This operator renormalizes mltiplicatively; i.e. the corresponding operator insertion in any on-shell renormalized correlation fnction is given by Ō(x, µ) = lim a Z(g,aµ)O(x,g ), (.3) where g, a are the bare copling and the lattice spacing respectively and µ is the renormalization scale. The renormalization grop rnning is described by the Callan-Symanzik eqations µ Ō(x, µ) = γ(ḡ(µ))ō(x, µ), µ µ ḡ(µ) = β(ḡ(µ)). (.4) µ Since we will work in a mass independent scheme (i.e. renormalization conditions will be imposed in the chiral limit), the β-fnction and all anomalos dimensions will only depend on the renormalized copling ḡ(µ) and they can be expanded pertrbatively in powers of g as β(g) g g 3 (b + b g 2 + b 2 g ), with niversal coefficients b, b, γ given by { 2 3 N f b = (4π) 2 }, b = (4π) 4 All the other coefficients of the expansions are scheme dependent. γ(g) g g 2 (γ + γ g 2 + γ 2 g ), (.5) { 2 38 } 3 N f, γ = 2C F (4π) 2. (.6) PoS(LATTICE 25)25 2. Renormalization in SF schemes In order to impose renormalization conditions we introdce a SF two-point fnction of the tensor crrent with bondary sorces of the form k T (x ) = a6 2 T k (x )( ζ (y)γ k ζ (z)) (2.) y,z 2

3 and the respective improved correlator is then k I T (x ) = k T (x ) + ac T k V x. In order to avoid extra divergences arising from the bondary the correlator k T can be normalized with bondary-tobondary correlators k = a2 6L 6 ( ζ (y )γ k ζ (z ))( ζ (y)γ k ζ (z)) (2.2) y,z,y,z f = a2 6L 6 ( ζ (y )γ 5 ζ (z ))( ζ (y)γ 5 ζ (z)). (2.3) y,z,y,z The (mass independent) renormalization conditions then read Z (α) T (g,l/a) ki T (L/2) f /2 α = k T (L/2) k α f /2 α k α m =m c,g =. (2.4) The freedom in the choice of α in (2.4) and in the angle θ entering spatial bondary conditions [6] define a class of renormalization schemes. In the present work we have considered α =,/2 and θ =,.5,. Following the standard SF iterative renormalization procedre [7], we define step scaling fnctions (SSF) as Σ (α) Z(α) T (,a/l) = T (g,a/2l) Z (α) T (g (2.5),a/L) The continm SSF for a bilinear correlator [7, 8] and for the copling [9, ] are defined respectively by σ T (g 2 ) = exp { σ(g 2 ) g } dg γ(g ) β(g ) σ(g 2 ) log(2) = dg g β(g ) where the index α has been sppressed. 3. Pertrbative one-loop comptation σ T () = lim a Σ T (,a/l), (2.6) σ() = ḡ 2 (2L), = ḡ 2 (L), (2.7) We can expand all the correlators entering Eq. (2.4), as well as renormalization constants, in powers of g 2 X = n= g 2n X (n), (3.) where X in Eq. (3.) can be either Z T, k T, k V, f and k. The one-loop improvement for the k T (x ) now reads kt I (x ) = k () T (x ) + g 2 k () T (x ) + ag 2 c () T k () V (x) x + O(ag 4 ) (3.2) and the renormalization constant for α = is given by { k () T (x ) Z () T (x,l/a) = k () T (x ) f () 2 f () + ac () T } k () V (x) x k () T (x ) where the notation F () = F () + F () bi + m () c m F () stands for one-loop coefficients where the contribtion related to the critical mass and the bondary conter terms, have been sbtracted [], (3.3) PoS(LATTICE 25)25 3

4 2.5.5 δ k, f, csw= δ k, k, csw= δ, f, csw= k δ, k, csw= k k δ k theta= theta=.5, f theta=.5, k theta=., f theta=., k a/l a/l Figre : -loop lattice artefacts in the SSFs. On the left, the three lines correspond to the SW action for three vales of θ angle and the two different markers correspond to the two schemes α =,/2 (note that at -loop f = k for θ = ). On the right, a comparison between ctoff effects from Wilson and SW actions at θ =.5. except for the term coming with c t. The -loop critical mass m () c is taken from [2]. Following [3] and [], in order to stdy the approach to the continm limit of the SSFs we define the relative deviation k (g 2,L/a) = Σ T (g 2,L/a) =g 2 SF (L) σ T () = k( )δ k g 2 + O(g 4 σ T () ) (3.4) where the one-loop coefficient (see Fig. ()) is given by δ k = Z() T (2L/a) Z() T (L/a), k( ) = γ ln(2) (3.5) k( ) The dependence on a/l of the -loop renormalization constant can be described according to [4] as Z () a ) { ( )} ν L T (L/a) r ν + s ν log. (3.6) ν=( L a In order to assess the systematic ncertainties in fit coefficients, we tried several fit ansaetze by trncating the series at different orders and changing the minimal vale in the fit range [(L/a) min, L/a = 48] as showed in Fig.2. We checked that, within systematic ncertainties, the correct vale of the LO anomalos dimension s = γ is reprodced; after that, the term with s can be sbtracted to improve the fitting precision. In the errors on the fit parameters the systematics related to the choice of the ansatz has been taken into accont in a conservative way. In order to extract the twoloops anomalos dimension in the SF scheme for the tensor bilinear, we sed a scheme-matching procedre with a given reference scheme where γ is known. This bypasses a direct two-loop calclation in the SF. Since tensor bilinears renormalize mltiplicatively, the same strategy employed for qark masses [3] can be sed. SF NLO anomalos dimension (listed in Table ) can be written as γ () SF = γ() re f + 2b χ () γ χ () g (3.7) where χ g () is related to a scheme matching for the copling, and χ () is related to the finite part of Eq. (3.6). In fact χ () = χ () SF,re f = χ() SF,lat χ() re f,lat, χ g () = 2b log(µl) 4π (c, + c, N f ) (3.8) PoS(LATTICE 25)25 4

5 .5 Z (), theta=. Z () log(l/a), theta= a/l Z () f, theta=.5 Z () f log(l/a), theta=.5 Z () k, theta=.5 Z () k log(l/a), theta=.5 Z () f, theta=. Z () f log(l/a), theta=. Z () k, theta=. Z () k log(l/a), theta=. Figre 2: Z () T at -loop for varios vales of θ and for α (in particlar the two choices of α correspond to scheme labeled f and k ). The behavior of the renormalization constant after the sbtraction of the leading logaritmic divergence is also displayed. and χ () SF,lat = r. Since Eq. (3.7) is independent on the choice of the ref scheme, we have crosschecked or reslts sing both MS and RI schemes ([5], [6] for finite parts, and [7] for the 2-loops anomalos dimension in both schemes). 4. Non-pertrbative renormalization and rnning Or non-pertrbative comptation has been carried ot for both N f =,2 and is ongoing for N f = 3 in parallel with the mass []. Once the renormalization constants given by imposing the condition (2.4) have been compted on a given lattice of size L/a and the doble lattice of size 2L/a, eq. (2.5) is sed to obtain the non-pertrbative vale of the SSF. We have compted here the SSFs for 4 vales for the copling in the range = [.8873,3.48] for qenched data, and for 6 coplings in the range = [.9793,3.334] for N f = 2 (see Fig.3). Since we did not implement O(a) improvement of the tensor operator in the qenched case, the continm extrapolation is linear in a/l and has been performed sing lattices with L/a = {6,8,2,6} and 2L/a = {2,6,24,32}. In the final analysis the smallest lattice has been discarded becase it is affected by large ctoff effects. In order to redce the ncertainty on the extrapolation we have performed a constrained fit between data from an O(a)-improved action and an nimproved one after testing niversality of the continm limit. Regarding N f = 2, since both action and operator are O(a)-improved, the continm limit is approached qadratically. In this case the extrapolation has been performed sing L/a = {6,8,2} and 2L/a = {2,6,24}. Once in the continm we adopted a polynomial fit ansatz for the SSFs of the form PoS(LATTICE 25)25 σ T () = + σ () + σ (2) 2 + σ (3) 3 + O( 4 ) (4.) scheme s SF (α = ) SF (α = /2) MS γ s/γ.43(33).32()n f.3767(33).()n f.9.9n f Table : NLO anomalos dimension in the SF scheme for the two schemes corresponding to α =,/2 and θ =.5, compared with the anomalos dimensions in continm schemes. 5

6 T f. f Figre 3: SSFs for N f = (left figre), N f = 2 (center figre), preliminary N f = 3 (on the right). Dashed black line is the LO approximation, dashed ble line is the NLO approximation, red solid line is the fit inclding the cbic term. Ble markers are non-pertrbative points. In the plot on the right red and ble points correspond to nimproved and -loop improved SSF respectively. where the first two coefficients are kept fixed to their pertrbative vales [ ] σ () = γ log(2), σ (2) = γ SF log(2) + 2 (γ ) 2 + b γ (log(2)) 2 (4.2) and the cbic coefficient has been fitted for both qenched and two-flavor data. The rnning between two scale µ and µ 2 is given by eq. (.4) and can be written as { ḡ(µ2 ) U(µ 2, µ ) = exp dg γ(g) } Z(g,aµ 2 ) = lim ḡ(µ ) β(g) a Z(g,aµ ). (4.3) Once SSF has been fitted on the range of coplings, the non-pertrbative rnning can be obtained. The evoltion coefficient is compted non-pertrbatively as a prodct of SSFs U(µ pt, µ had ) = n i= [σ T ( i )] with i = ḡ 2 (2 i µ had ). For both N f = and N f = 2 we have compted n = 7 nonpertrbative steps (i.e. achieving a factor 2 7 in the ratio between µ pt and µ had, see Fig.4), connecting an hadronic scale µ had = 275MeV (484MeV for N f = 2) p to 35GeV ( 62GeV) an high energy scale, where pertrbation theory is spposed to be safe. At those scales (compted with Λ SF for N f = and N f = 2 from [, 8] respectively) the NP evoltion is matched with pertrbation theory at NLO, ĉ(µ had ) = ĉ(µ pt )U(µ pt, µ had ) where ĉ is defined as ĉ(µ) = O [ḡ2 ] γ { RGI O(µ) = ZRGI T Z T (µ) = 2b (µ) ḡ(µ) ( γ(g) exp dg 4π β(g) γ )} b g (4.4) PoS(LATTICE 25)25 Eqivalently, the total RGI renormalization constant is defined as 5. Conclsions Z RGI T (g ) = ĉ(µ pt )U(µ pt, µ had )Z T (g, µ had ). (4.5) On the pertrbative side we have analysed ctoff effects of the SSF for varios SF schemes, and from the finite parte of the -loop renormalization constant we have been able to give the first preliminary determination of the NLO anomalos dimension in SF schemes for the tensor crrents, which are the only qark bilinears with a non-trivial anomalos dimension independent from that of qark masses. Moreover, thanks to the non-pertrbative lattice comptation for both N f = 6

7 ORGI/O(µ).2..9 /2.8 2/2 2/ µ/ ORGI/O(µ) /2.8 2/2 2/ µ/ Figre 4: The red dashed line is the LO rnning coefficient ĉ while the dashed ble line and solid black line are NLO approximation the first with NLO-γ and NLO-β, the latter with NLO-γ and NNLO-β. Reslts for N f = (on the left) and N f = 2 (on the right) are provided in a single scheme (α = ). and N f = 2 we have compted the non-pertrbative SSF in the continm throgh which with 7 recrsion steps the rnning over more than 2 orders of magnitde is compted, from an hadronic scale p to a pertrbative one. Despite the dependence of the rnning on the scheme and on N f, since the correction given to the rnning by the NLO anomalos dimension respect to the LO is large, we still observe large systematics de to the matching with pertrbation theory on the scale of 2-3GeV. The same strategy, briefly explained here, is being applied to N f = 3 simlations that will allow a more physical and precise determination of both the rnning and the RGI. References [] I. Campos at al., PoS LATTICE25 (25) [arxiv: [hep-lat]]. [2] M. Artso et al., Er. Phys. J. C 57, 39 (28) [arxiv:8.833 [hep-ph]]. [3] M. Antonelli et al., Phys. Rept. 494, 97 (2) [arxiv: [hep-ph]]. [4] T. Bhattacharya et al. Phys. Rev. D 85, 5452 (22) [arxiv:.6448 [hep-ph]]. [5] S. Sint and P. Weisz, Ncl. Phys. Proc. Sppl. 63, 856 (998) [hep-lat/97996]. [] M. Lüscher et al. Ncl. Phys. B 43, 48 (994) [hep-lat/9395]. [] F. Palombi, C. Pena and S. Sint, JHEP 63, 89 (26) [hep-lat/553]. [2] H. Panagopolos and Y. Proestos, Phys. Rev. D 65, 45 (22) [hep-lat/82]. [3] S. Sint et al. [ALPHA Collaboration], Ncl. Phys. B 545, 529 (999) [hep-lat/9883]. [4] A. Bode et al. [ALPHA Collaboration], Ncl. Phys. B 576, 57 (2) [hep-lat/998]. PoS(LATTICE 25)25 [6] S. Sint, Ncl. Phys. B 42, 35 (994) [heplat/93279]. [7] S. Capitani et al. [ALPHA Collaboration], Ncl. Phys. B 544, 669 (999) [hep-lat/9863]. [8] M. Della Morte et al. [ALPHA Collaboration], Ncl. Phys. B 729, 7 (25) [hep-lat/5735]. [9] A. Bode et al. [ALPHA Collaboration], Phys. Lett. B 55, 49 (2) [hep-lat/53]. [5] A. Skoropathis and H. Panagopolos, Phys. Rev. D 79, 9458 (29) [arxiv: [hep-lat]]. [6] S. Capitani et al. Ncl. Phys. B 593, 83 (2) [hep-lat/74]. [7] J. A. Gracey, Ncl. Phys. B 667, 242 (23) [hepph/3663]. [8] P. Fritzsch et al. Ncl. Phys. B 865, 397 (22) [arxiv: [hep-lat]]. 7