Non-perturbative renormalization of tensor bilinears in Schrödinger Functional schemes1footnote 11footnote 1 IFT-UAM/CSIC-15-10, FTUAM-15-41

# Non-perturbative renormalization of tensor bilinears in Schrödinger Functional schemes111 Ift-Uam/csic-15-10, Ftuam-15-41

Patrick Fritzsch, Carlos Pena, David Preti
Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid,
C/ Nicolás Cabrera 13-15, Cantoblanco, Madrid 28049
Departamento de Física Teórica, Universidad Autónoma de Madrid,
Cantoblanco, Madrid 28049
E-mail: p.fritzsch@csic.es, carlos.pena@uam.es, david.preti@csic.es
Speaker.
###### Abstract

We present preliminary result for the study of the renormalization group evolution of tensor bilinears in Schrödinger Functional (SF) schemes for and QCD with non-perturbatively -improved Wilson fermions. First results (proceeding in parallel with the ongoing computation of the running quark masses [1]) are also discussed. A one-loop perturbative calculation of the discretisation effects for the relevant step scaling functions has been carried out for both Wilson and -improved actions and for a large number of lattice resolutions. We also calculate the two-loop anomalous dimension in SF schemes for tensor currents through a scheme matching procedure with RI and . Thanks to the SF iterative procedure the non-perturbative running over two orders of magnitude in energy scales, as well as the corresponding Renormalization Group Invariant operators, have been determined.

Non-perturbative renormalization of tensor bilinears in Schrödinger Functional schemesthanks: IFT-UAM/CSIC-15-10, FTUAM-15-41

Patrick Fritzsch, Carlos Pena, David Pretithanks: Speaker.

Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid,

C/ Nicolás Cabrera 13-15, Cantoblanco, Madrid 28049

Departamento de Física Teórica, Universidad Autónoma de Madrid,

Cantoblanco, Madrid 28049

\abstract@cs

The 33rd International Symposium on Lattice Field Theory 14 -18 July 2015 Kobe International Conference Center, Kobe, Japan

## 1 Introduction

Tensor currents play an important rôle in interesting processes through which the consistency of the Standard Model (SM) is being currently probed, as e.g. rare meson decays (see e.g. [2, 3] and references therein) or precision measurements of -decays and limits on the neutron electric dipole moment (see e.g. [4]). Considering the tensor operator for two generical (and formally distinct) flavours , as its improvement in the chiral limit is achieved by considering the combination

 TIμν=Tμν+acT(~∂μVν−~∂νVμ) (1.0)

where the coefficient was computed at 1-loop within the Schrödinger Functional (SF) in [5] and reads , for and N colours. At vanishing spatial momentum, the only non-vanishing two-point functions with boundary operators allowed by the SF boundary conditions involve the "electric" components (see Eq. (2) below)

 TI0k=T0k+acT(~∂0Vk−~∂kV0) (1.0)

where the second term in the parenthesis of the rhs vanishes when inserted in Eq. (2). This operator renormalizes multiplicatively; i.e. the corresponding operator insertion in any on-shell renormalized correlation function is given by

 ¯O(x,μ)=lima→0Z(g0,aμ)O(x,g0), (1.0)

where , are the bare coupling and the lattice spacing respectively and is the renormalization scale. The renormalization group running is described by the Callan-Symanzik equations

 μ∂∂μ¯O(x,μ)=γ(¯g(μ))¯O(x,μ),μ∂∂μ¯g(μ)=β(¯g(μ)).\specialhtml:\specialhtml: (1.0)

Since we will work in a mass independent scheme (i.e. renormalization conditions will be imposed in the chiral limit), the -function and all anomalous dimensions will only depend on the renormalized coupling and they can be expanded perturbatively in powers of as

 β(g)\lx@stackrelg∼0≈−g3(b0+b1g2+b2g4+…),γ(g)\lx@stackrelg∼0≈−g2(γ0+γ1g2+γ2g4+…), (1.0)

with universal coefficients , , given by

 b0=1(4π)2{11−23Nf},b1=1(4π)4{102−383Nf},γ0=2CF(4π)2. (1.0)

All the other coefficients of the expansions are scheme dependent.

## 2 Renormalization in SF schemes

In order to impose renormalization conditions we introduce a SF two-point function of the tensor current with boundary sources of the form

 kT(x0)=−a62∑y,z⟨T0k(x0)(¯ζ(y)γkζ(z))⟩\specialhtml:\specialhtml: (2.0)

and the respective improved correlator is then In order to avoid extra divergences arising from the boundary the correlator can be normalized with boundary-to-boundary correlators

 k1=−a126L6∑y,z,y′,z′⟨(¯ζ′(y′)γkζ′(z′))(¯ζ(y)γkζ(z))⟩ (2.0)
 f1=−a126L6∑y,z,y′,z′⟨(¯ζ′(y′)γ5ζ′(z′))(¯ζ(y)γ5ζ(z))⟩. (2.0)

The (mass independent) renormalization conditions then read

 Z(α)T(g0,L/a)kIT(L/2)f1/2−α1kα1=kT(L/2)f1/2−α1kα1∣∣ ∣∣m0=mc,g0=0.\specialhtml:\specialhtml: (2.0)

The freedom in the choice of in (2) and in the angle entering spatial boundary conditions [6] define a class of renormalization schemes. In the present work we have considered and . Following the standard SF iterative renormalization procedure [7], we define step scaling functions (SSF) as

 Σ(α)T(u,a/L)=Z(α)T(g0,a/2L)Z(α)T(g0,a/L)\specialhtml:\specialhtml: (2.0)

The continuum SSF for a bilinear correlator [7, 8] and for the coupling [9, 10] are defined respectively by

 σT(g2)=exp⎧⎨⎩∫√σ(g2)gdg′γ(g′)β(g′)⎫⎬⎭→σT(u)=lima→0ΣT(u,a/L), (2.0)
 −log(2)=∫√σ(g2)gdg′1β(g′)→σ(u)=¯g2(2L),u=¯g2(L), (2.0)

where the index has been suppressed.

## 3 Perturbative one-loop computation

We can expand all the correlators entering Eq. (2), as well as renormalization constants, in powers of

 X=∞∑n=0g2n0X(n),\specialhtml:\specialhtml: (3.0)

where in Eq. (3) can be either , , , and . The one-loop improvement for the now reads

 kIT(x0)=k(0)T(x0)+g20k(1)T(x0)+ag20c(1)T~∂0k(0)V(x)∣∣x0+O(ag40) (3.0)

and the renormalization constant for is given by

 Z(1)T(x0,L/a)=−⎧⎪⎨⎪⎩¯k(1)T(x0)k(0)T(x0)−¯f(1)12f(0)1+ac(1)T~∂0k(0)V(x)|x0k(0)T(x0)⎫⎪⎬⎪⎭ (3.0)

where the notation stands for one-loop coefficients where the contribution from boundary counter terms, as well as the one related to the critical mass, have been subtracted [11]. The 1-loop critical mass is taken from [12]. Following [13] and [11], in order to study the approach to the continuum limit of the SSFs we define the relative deviation

 Δk(g20,L/a)=ΣT(g20,L/a)|u=g2SF(L)−σT(u)σT(u)=δkg20+O(g40) (3.0)

where the one-loop coefficient (see Fig. (1)) is given by

 δk=Z(1)T(2L/a)−Z(1)T(L/a)γ0log(2)−1 (3.0)

The dependence on of the 1-loop renormalization constant can be described according to [14] as

 Z(1)T(L/a)≈∞∑ν=0(aL)ν{rν+sνlog(La)}.\specialhtml:\specialhtml: (3.0)

In order to assess the systematic uncertainties in fit coefficients, we tried several fit ansaetze by truncating the series at different orders and changing the minimal value in the fit range . We checked that, within systematic uncertainties, the correct value of the LO anomalous dimension is reproduced; after that, the term with can be subtracted 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 account in a conservative way. In order to extract the two-loops anomalous dimension in the SF scheme for the tensor bilinear, we used a scheme-matching procedure with a given reference scheme where is known. This bypasses a direct two-loop calculation in the SF. Since tensor bilinears renormalize multiplicatively, the same strategy employed for quark masses [13] can be used. SF NLO anomalous dimension can be written as

 γ(1)SF=γ(1)ref+2b0χ(1)−γ0χ(1)g\specialhtml:\specialhtml: (3.0)

where is related to a scheme matching for the coupling, and is related to the finite part of Eq. (3). In fact

 χ(1)=χ(1)SF,ref=χ(1)SF,lat−χ(1)ref,lat,χ(1)g=2b0log(μL)−14π(c1,0+c1,1Nf) (3.0)

and . Since Eq. (3) is independent on the choice of the “ref” scheme, we have crosschecked our results using both and schemes ([15], [16] for finite parts, and [17] for the two-loops anomalous dimension in both schemes).

## 4 Non-perturbative renormalization and running

Our non-perturbative computation has been carried out for both and is ongoing for in parallel with the mass [1]. Once the renormalization constants given by imposing the condition (2) have been computed on a given lattice of size and the double lattice of size , eq. (2) is used to obtain the non-perturbative value of the SSF. We have computed here the SSFs for values for the coupling in the range for quenched data, and for couplings in the range for . Since we did not implement improvement of the tensor operator in the quenched case, the continuum extrapolation is linear in and has been performed using lattices with and . In the final analysis the smallest lattice has been discarded because it is affected by large cutoff effects. In order to reduce the uncertainty on the extrapolation we have performed a constrained fit between data from an -improved action and an unimproved one after testing universality of the continuum limit. Regarding , since both action and operator are -improved, the continuum limit is approached quadratically. In this case the extrapolation has been performed using and . Once in the continuum we adopted a polynomial fit ansatz for the SSFs of the form

 σT(u)=1+σ(1)u+σ(2)u2+σ(3)u3+O(u4) (4.0)

where the first two coefficients are kept fixed to their perturbative values

 σ(1)=γ0log(2),σ(2)=γSF1log(2)+[12(γ0)2+b0γ0](log(2))2 (4.0)

and the cubic coefficient has been fitted for both quenched and two-flavour data.

The running between two scale and is given by eq. (1) and can be written as

 U(μ2,μ1)=exp{∫¯g(μ2)¯g(μ1)dgγ(g)β(g)}=lima→0Z(g0,aμ2)Z(g0,aμ1).\specialhtml:\specialhtml: (4.0)

Once SSF has been fitted on the range of couplings, the non-perturbative running can be obtained. The evolution coefficient is computed non-perturbatively as a product of SSFs with . For both and we have computed non-perturbative steps (i.e. achieving a factor in the ratio between and ), connecting an hadronic scale ( for ) up to () an high energy scale, where perturbation theory is supposed to be safe. At those scales (computed with for and from [10, 18] respectively) the NP evolution is matched with perturbation theory at NLO, where is defined as

 ^c(μ)=ORGIO(μ)=ZRGITZT(μ)=[¯g2(μ)4π]−γ02b0exp{−∫¯g(μ)0dg(γ(g)β(g)−γ0b0g)} (4.0)

Equivalently, the total RGI renormalization constant is defined as

 ZRGIT(g0)=^c(μpt)U(μpt,μhad)ZT(g0,μhad). (4.0)

## 5 Conclusions

On the perturbative side we have analysed cutoff effects of the SSF for various SF schemes, and from the finite parte of the 1-loop renormalization constant we have been able to give the first preliminary determination of the NLO anomalous dimension in SF schemes for the tensor currents, which are the only quark bilinears with a non-trivial anomalous dimension independent from that of quark masses. Moreover, thanks to the non-perturbative lattice computation for both and we have computed the non-perturbative SSF in the continuum through which with 7 recursion steps the running over more than 2 orders of magnitude is computed, from an hadronic scale up to a perturbative one. Despite the dependence of the running on the scheme and on , since the correction given to the running by the NLO anomalous dimension respect to the LO is large, we still observe large systematics due to the matching with perturbation theory on the scale of -. The same strategy, briefly explained here, is being applied to simulations that will allow a more physical and precise determination of both the running and the RGI.

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