Non-perturbative renormalization and running of Delta F=2 four-fermion operators in the SF scheme
aa r X i v : . [ h e p - l a t ] D ec Non-perturbative renormalization and running of D F = four-fermion operators in the SF scheme Mauro Papinutto ∗ Dipartimento di Fisica, "Sapienza" Università di Roma, and INFN, Sezione di Roma,Piazzale Aldo Moro 2, I-00185 Roma, ITALY.E-mail: [email protected]
Carlos Pena, David Preti
Instituto de Física Teórica UAM/CSIC and Departamento de Física Teórica, UniversidadAutónoma de Madrid, Cantoblanco E-28049 Madrid, SpainE-mail: [email protected], [email protected]
We present preliminary results of a non-perturbative study of the scale-dependent renormaliza-tion constants of a complete basis of D F = a ) improved Wilson fermions and our gauge configurations containtwo flavors of massless sea quarks. The mixing pattern of these operators is the same as for aregularization that preserves chiral symmetry, in particular there is a "physical" mixing betweensome of the operators. The renormalization group running matrix is computed in the contin-uum limit for a family of Schrödinger Functional (SF) schemes through finite volume recursivetechniques. We compute non-perturbatively the relation between the renormalization group in-variant operators and their counterparts renormalized in the SF at a low energy scale, togetherwith the non-perturbative matching matrix between the lattice regularized theory and the variousSF schemes. The 32nd International Symposium on Lattice Field Theory,23-28 June, 2014Columbia University New York, NY ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ on-perturbative renormalization of D F = four-fermion operators Mauro Papinutto
1. Introduction
Flavor physics processes play a major role in the indirect search for New Physics (NP), becausethey are sensitive to the exchange of virtual NP particles through loop effects. These processesvanish at tree level in the SM and, despite the fact they are loop mediated and in some cases alsoCKM or helicity suppressed, may be theoretically very clean. Among them D F = D F = Q ± = ¯ f g m q ¯ f g m q ′ + ¯ f g m g q ¯ f g m g q ′ ± ( q ↔ q ′ ) Q ± = ¯ f g m q ¯ f g m g q ′ + ¯ f g m g q ¯ f g m q ′ ± ( q ↔ q ′ ) Q ± = ¯ f g m q ¯ f g m q ′ − ¯ f g m g q ¯ f g m g q ′ ± ( q ↔ q ′ ) Q ± = ¯ f g m q ¯ f g m g q ′ − ¯ f g m g q ¯ f g m q ′ ± ( q ↔ q ′ ) Q ± = ¯ f q ¯ f q ′ − ¯ f g q ¯ f g q ′ ± ( q ↔ q ′ ) Q ± = ¯ f g q ¯ f q ′ − ¯ f q ¯ f g q ′ ± ( q ↔ q ′ ) Q ± = ¯ f q ¯ f q ′ + ¯ f g q ¯ f g q ′ ± ( q ↔ q ′ ) Q ± = ¯ f g q ¯ f q ′ + ¯ f q ¯ f g q ′ ± ( q ↔ q ′ ) Q ± = f s mn q ¯ f s mn q ′ ± ( q ↔ q ′ ) Q ± = f s mn g q ¯ f s mn g q ′ ± ( q ↔ q ′ ) where the flavours q and q ′ can be thought of as two copy of the same flavour and will be setto be identical in the end. This allows to define " + " and " − " operators which are even or oddunder the switching symmetry q ↔ q ′ and thus do not mix with each other. In the following wewill drop the superscript " ± " and our discussion will apply to both " + " and " − " sectors separately.Moreover, parity symmetry prevents PE operators from mixing with PO ones. Finally we note thatthe operators Q , Q , Q , Q , Q , Q , Q , Q appear only in extensions beyond the SM.In a regularisation that preserves chiral symmetry, Q mixes under renormalization with Q ,and similarly Q with Q . The same mixing pattern is valid for Q and Q and for Q and Q ,which share the same chiral properties of the corresponding PE operators. The corresponding2 × Z and Z ) have large LO anomalousdimensions, and the same is true at NLO (even though this is a scheme-dependent statement).This poses some issues about the use of perturbation theory (PT) to compute the renormalizationfactors and/or the Renormalization Group (RG) running down to renormalization scales at whichthe operators are usually renormalized on the lattice.At present, few computations of these matrix elements in the D S = − − m had ∼ L QCD up to a scale m pt of the order of the W boson mass M W , where matching with PT atNLO is under much better control.Due to the explicit breaking of chiral symmetry with Wilson fermions, the renormalizationpattern of composite operators can be considerably more complex than in a chirality-preserving2 on-perturbative renormalization of D F = four-fermion operators Mauro Papinutto F sk g i , g g i , g g i , g f g g k g i g i Figure 1:
Left: 4-point correlators F sk of the operator Q k with the source s . Right: boundary-to-boundarycorrelators k ( g i on the boundaries) and f ( g on the boundaries). regularisation, because of the mixing with operators of different naïve chirality. While the 5 PEoperators Q i all mix with each other, it has been shown [4] that in the PO sector the mixing patternis the same as in a chirality-preserving regularisation, i.e. the one described above.As a consequence one can think of two possible strategies to avoid the spurious mixing in thePE sector when using non-perturatively O( a ) improved Wilson sea-quarks: • use twisted mass QCD at maximal twist for f , q and q ′ as explained in [5]. This setup isautomatically O( a ) improved but not unitary; • use Ward identities which relate the correlators of PE operators to those of PO ones as ex-plained in [6]. This setup is unitary but not automatically O( a ) improved.In both strategies one has to compute the renormalized matrix elements of PO operators whichpresent only the "physical" scale dependent mixing.In the present work we focus on the two matrices k = , k = ,
5, the renormalizationof the operator k = m ∈ [ L QCD , M W ] . In this exploratory study we have used non-perturbatively O( a ) improved Wilsonfermions with 2 massless sea flavors.
2. Non-perturbative renormalization in the SF scheme
By using the SF on a volume L we compute the 4-point correlators F sk of the operator Q k with the source s made by one of the five possible combinations of three g i and g bilinears on theboundaries. We also compute the correlators of two boundary bilinears with a g structure ( f ) orwith a g i structure ( k ), see [7] for details. These correlators are schematically represented in Fig. 1From the correlators we build the ratios A sk ; a ( L / ) = F sk ( L / ) f / − a k a , (2.1)where a ∈ { , , / } and s ∈ { , . . . , } . We impose renormalization conditions in the chiral limit(i.e. at bare mass m = m cr ) on each of the 2 × s , s for each value of a (to simplify the notation we avoid labelling Z with the indices s , s , a ): Z Z Z Z ! A s a A s a A s a A s a ! = A s a A s a A s a A s a ! g = (2.2)3 on-perturbative renormalization of D F = four-fermion operators Mauro Papinutto and similarly for the k = , ( s , s , a ) to make sense, one needs to check thatdet A s a A s a A s a A s a ! g = = k = , ( s , s ) ∈ { ( , ) , ( , ) , ( , ) , ( , ) , ( , ) , ( , ) } . Consideringthe fact that we allow for three values of a , we have in all 18 non-redundant conditions. Each ofthese conditions fixes the two renormalization matrices Z ( g , a m ) at the scale m = / L .From the renormalization constants we build the Step Scaling Functions (SSF) for the twomatrices Z (where we recall also the definition for the SSF of the coupling and where U ( m , m ) is the RG-evolution between the scales m and m ): S Q ( u , a / L ) ≡ Z ( g , L / a ) [ Z ( g , L / a )] − (cid:12)(cid:12)(cid:12) m = m cr u ≡ ¯ g ( L ) , s Q ( u ) = U ( m / , m ) (cid:12)(cid:12)(cid:12) m = / L = lim a → S Q ( u , a / L ) , (2.4) s ( u ) ≡ ¯ g ( L ) , u ≡ ¯ g ( L ) . The SSF s ( u ) has been computed in previous works by the ALPHA Coll. We have computed herethe two matrices s Q ( u ) ( k = , k = ,
5) for the 18 schemes and for 6 values of the couplingin the range u = . u = . a ) improved, so the continuum limit extrapolation is linear in a / L andhas been performed using lattices with L / a = { , , } and 2 L / a = { , , } by tuning the b values at each L to obtain the chosen value of u .Having the continuum limit of s Q ( u ) for u ∈ [ . , . ] , we can perform a fit accordingto a power series expansion s Q ( u ) = + s u + s u + s u + . . . , where the s i are 2 × g ( g ) = − g ¥ (cid:229) n = g n g n , b ( g ) = − g ¥ (cid:229) n = b n g n , s = g ln 2 , s = g ln 2 + b g ( ln 2 ) + g ( ln 2 ) , (2.5)where g is the NLO ADM in the SF scheme (we denote it by g SF1 ). The latter can be obtainedfrom the value g ref1 already known in a reference scheme through the following two-loop matchingrelations: g SF1 = g ref1 + [ c ( ) SF , ref , g ] + b c ( ) SF , ref + b l l ¶¶l c ( ) SF , ref − g ( ) c ( ) g , ¯ g = c g ( g ref ) g , ( Q ) SFR = c SF , ref ( g ref )( Q ) refR , (2.6) c ( g ) = + ¥ (cid:229) k = g k c ( k ) , where l is the gauge fixing parameter and b l is the beta function for the renormalized gauge fixingparameter l ( m ) (This is needed e.g. if we use as reference scheme the RI-MOM which dependson the gauge chosen. If we use MS there is no dependence upon the gauge.) S. Sint, unpublished notes, 2001 on-perturbative renormalization of D F = four-fermion operators Mauro Papinutto u [ s ] g SF g SF u [ s ] g SF g SF u [ s ] g SF g SF u [ s ] g SF g SF u [ s ] g SF g SF u [ s ] g SF g SF Figure 2:
First row: elements 22, 32, 33 of s Q ( u ) . Second row: elements 44, 54, 55 of s Q ( u ) . The matching coefficient c ( ) SF , ref = c ( ) SF , lat − c ( ) ref , lat involves the respective one-loop matchingmatrices c ( ) SF , lat and c ( ) ref , lat between the bare lattice operator and either the SF or the referencescheme. In the present work we have computed the matching matrix c ( ) SF , lat in perturbation theoryat one-loop. c ( ) ref , lat can be extracted from the literature (Ref. [9] for the RI-MOM scheme whileRefs. [10, 11, 12] for the MS scheme ). g ref1 can be found in [8] both in the RI-MOM and MS case, while c ( ) g is given in [13]. Havingall these ingredients we have computed g SF1 , for both 2 × g SF1 we can easily compute s and s from Eq. 2.5 and then perform a fit of the twomatrices s Q ( u ) by keeping s as a matrix of free parameters. As an example, results for someelements of s Q ( u ) in the a = / ( s , s ) = ( , ) scheme are presented in Fig. 2. In general,several of these elements differ substantially at the largest couplings from the LO and NLO PTresults, independently of the scheme chosen.
3. Non-perturbative renormalization group running
Once the s Q has been fitted on the whole range of couplings, the non-perturbative runningcan be obtained from the scale m had = / L max to the scale m pt = n m had where n is the number ofsteps performed and where L max is such that s − ( ¯ g ( L max )) belongs to the upper end of the rangeof couplings simulated: U ( m pt , m had ) = (cid:2) s Q ( u ) · · · s Q ( u n ) (cid:3) − , u i = ¯ g ( i m had ) . (3.1)In the present case, with 7 steps we have m had ≈ .
49 GeV while m pt ≈
63 GeV, where one expectsto safely match with the perturbative RG-evolution at NLO. We are grateful to S. Sharpe for having converted for us the MS scheme used in [10] to the one defined in [8]. on-perturbative renormalization of D F = four-fermion operators Mauro Papinutto m / L [ ˆ c ] ( µ ) g SF g SF −0.4−0.200.20.40.60.81 m / L [ ˆ c ] ( µ ) g SF g SF −0.06−0.05−0.04−0.03−0.02−0.0100.01 m / L [ ˆ c ] ( µ ) g SF g SF m / L [ ˆ c ] ( µ ) g SF g SF m / L [ ˆ c ] ( µ ) g SF g SF −8−6−4−20246 m / L [ ˆ c ] ( µ ) g SF g SF Figure 3:
First row: elements 22, 23, 32 of ˜ U ( m ) . Second row: elements 44, 45, 54 of ˜ U ( m ) . If operators mix, the RG-evolution is formally obtained by using U ( m , m ) = T exp (cid:26) Z ¯ g ( m ) ¯ g ( m ) g ( g ) b ( g ) d g (cid:27) . (3.2)We write the RG-evolution by separating the LO part and defining the function W ( m ) which canbe thought as containing contributions beyond LO: U ( m , m ) ≡ [ W ( m )] − U ( m , m ) LO W ( m ) , U ( m , m ) LO = (cid:20) ¯ g ( m ) ¯ g ( m ) (cid:21) g b , (3.3)where W ( m ) satisfies a new RG-equation and is regular in the UV: lim m → ¥ W ( m ) = The RGI operators are easily defined using the above form: Q RGI ≡ ˜ U ( m )( Q ( m )) R = (cid:20) ¯ g ( m ) p (cid:21) − g b W ( m )( Q ( m )) R . (3.4)This formula is still valid non-perturbatively. One can use it to perform the matching at m pt with the NLO perturbative evolution: Q RGI = (cid:20) ¯ g ( m ) p (cid:21) − g b W ( m pt ) U ( m pt , m had )( Q ( m had )) R , (3.5)by expanding W ( m ) in perturbation theory W ( m ) ≃ + ¯ g ( m ) J ( m ) + O ( ¯ g ) . J depends on the ADMat the NLO g and satisfies: ¶¶m J ( m ) = , J − (cid:20) g b , J (cid:21) = b b g − b g . (3.6)6 on-perturbative renormalization of D F = four-fermion operators Mauro Papinutto
Eq. 3.6 has been solved to obtain J in the SF scheme and compute the running ˜ U ( m ) definedby Eq. 3.4,3.5. As an example, results for some elements of ˜ U ( m ) in the a = / ( s , s ) =( , ) scheme are presented in Fig. 3 against the LO and the NLO perturbative results. Again, ingeneral several of these elements differ substantially at the lowest scales from the NLO PT results,independently of the scheme chosen.The total RGI renormalization matrix is defined from Q RGI ≡ Z RGI ( g ) Q ( g ) where Z RGI ( g ) = ˜ U ( m pt ) U ( m pt , m had ) Z ( g , a m had ) (3.7)and Z ( g , a m had ) is the non-perturbative renormalization constant matrix computed at the hadronicscale. Z ( g , L / a ) has been computed at three values of b ∈ { . , . , . } useful for largevolume simulations, on three volumes for each b ( L / a = { , , } ). By interpolating to L max forwhich ¯ g ( L max ) = .
61 one gets Z ( g , a m had ) for each b .
4. Conclusions
Thanks to the use of SF schemes, we have performed a first exploratory study of the non-perturbative RG-running of four-quark operators in the presence of mixing on a wide range of scaleswhich varies over 2 orders of magnitudes. Non-perturbative effects seem dangerously sizeable atscales of 2-3 GeV. Despite the dependence on the scheme, we are trying to understand whether thisobservation can be at the origin of the discrepancies found in the literature [1, 2, 3] where the NLOperturbative RG-running in the RI-MOM or MS scheme have been used. The same strategy usedhere is immediately portable to N f = + c SFscheme [14] one would gain automatic O ( a ) improvement and only need 3-point functions insteadof 4-point functions, with a consequent reduction of statistical fluctuations. References [1] P. A. Boyle et al. [RBC and UKQCD Coll.], Phys. Rev. D (2012) 054028.[2] V. Bertone et al. [ETM Coll.], JHEP (2013) 089 [Erratum-ibid. (2013) 143].[3] T. Bae et al. [SWME Coll.], Phys. Rev. D (2013) 7, 071503.[4] A. Donini, V. Giménez, G. Martinelli, M. Talevi and A. Vladikas, Eur. Phys. J. C (1999) 121.[5] R. Frezzotti et al. [ALPHA Coll.], JHEP (2001) 058; R. Frezzotti and G. C. Rossi, JHEP (2004) 070.[6] D. Bécirevi´c et al. , Phys. Lett. B (2000) 74.[7] F. Palombi, C. Pena and S. Sint, JHEP (2006) 089; M. Guagnelli et al. , JHEP (2006) 088.[8] A. J. Buras, M. Misiak and J. Urban, Nucl. Phys. B (2000) 397.[9] M. Constantinou et al. , Phys. Rev. D (2011) 074503.[10] R. Gupta, T. Bhattacharya and S. Sharpe, Phys. Rev. D (1997) 4036;[11] J. Kim et al. , Phys. Rev. D (2014) 014504;[12] S. Capitani et al. , Nucl. Phys. B (2001) 183.[13] S. Sint and R. Sommer, Nucl. Phys. B (1996) 71.[14] S. Sint, Nucl. Phys. B (2011) 491.(2011) 491.