O(a^2) corrections to 1-loop matrix elements of 4-fermion operators with improved fermion/gluon actions
Martha Constantinou, Vittorio Lubicz, Haralambos Panagopoulos, Apostolos Skouroupathis, Fotos Stylianou
aa r X i v : . [ h e p - l a t ] J a n O ( a ) corrections to 1-loop matrix elements of4-fermion operators with improved fermion/gluonactions Martha Constantinou a ∗ , Vittorio Lubicz b , Haralambos Panagopoulos a , ApostolosSkouroupathis a ∗ , Fotos Stylianou † a ∗ a Department of Physics, University of CyprusP.O.Box 20537, Nicosia CY-1678, Cyprus b Dipartimento di Fisica, Università di Roma Tre and INFN, Sezione di Roma Tre,Via della Vasca Navale 84, I-00146 Roma, ItalyE-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] We calculate the corrections to the amputated Green’s functions of 4-fermion operators, in1-loop Lattice Perturbation theory. The novel aspect of our calculations is that they are carriedout to second order in the lattice spacing, O ( a ) .We employ the Wilson/clover action for massless fermions (also applicable for the twistedmass action in the chiral limit) and the Symanzik improved action for gluons. Our calculationshave been carried out in a general covariant gauge. Results have been obtained for several popularchoices of values for the Symanzik coefficients (Plaquette, Tree-level Symanzik, Iwasaki, TILWand DBW2 action).We pay particular attention to D F = F stands for flavour: S , C , B ). We study the mixing pattern of these operators, to O ( a ) , usingthe appropriate projectors. Our results for the corresponding renormalization matrices are givenas a function of a large number of parameters: coupling constant, clover parameter, number ofcolors, lattice spacing, external momentum and gauge parameter.The O ( a ) correction terms (along with our previous O ( a ) calculation of Z Y ) are essentialingredients for minimizing the lattice artifacts which are present in non-perturbative evaluationsof renormalization constants with the RI ′ -MOM method.A longer write-up of this work, including non-perturbative results, is in preparation together withmembers of the ETM Collaboration [1]. PACS: 11.15.Ha, 12.38.Gc, 11.10.Gh, 12.38.Bx
The XXVII International Symposium on Lattice Field TheoryJuly 26-31, 2009Peking University, Beijing, China ∗ Work supported in part by the Research Promotion Foundation of Cyprus (Proposal Nr: TEXN/0308/17,ENI S X/0506/17). † Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ( a ) corrections to 1-loop matrix elements of 4-fermion operators Fotos Stylianou
1. Introduction
A number of flavour-changing processes are currently under study in Lattice simulations.Among the most common examples are the decay K → pp and K – ¯ K oscillations. From ex-perimental evidence, we know that these weak processes violate the CP symmetry. In theory, thecalculation of the amount of CP violation in K – ¯ K oscillations requires the knowledge of B K .The Kaon B K parameter is obtained from the D S = B K = h ¯ K | ˆ O D S = | K i h ¯ K | ¯ s g m d | i h | ¯ s g m d | K i , (1.1)where s and d stand for strange and down quarks, and ˆ O D S = is the effective 4-quark interactionrenormalized operator, corresponding to the bare operator: O D S = = ( ¯ s g L m d )( ¯ s g L m d ) , g L m = g m ( − g ) . (1.2)The above operator splits into parity-even and parity-odd parts; in standard notation: O D S = = O D S = VV + AA − O D S = VA + AV . Since the above weak process is simulated in the framework of Lattice QCD,where Parity is a symmetry, the parity-odd part gives no contribution to the K – ¯ K matrix element.Thus, we conclude that B K can be extracted from the correlator ( x > y < C KOK ( x , y ) = h ( ¯ d g s )( x ) ˆ O D S = VV + AA ( )( ¯ d g s )( y ) i , O D S = VV + AA = ( ¯ s g m d )( ¯ s g m d ) + ( ¯ s g m g d )( ¯ s g m g d ) , (1.3) where O D S = VV + AA is the bare operator and ˆ O D S = VV + AA is the respective renormalized operator.In place of the operator in Eq. (1.3) it is advantageous to use a four-quark operator with adifferent flavour content ( s , d , s ′ , d ′ ), and with D S = D s + D s ′ =
2, namely [2]: O D S = VV + AA = ( ¯ s g m d )( ¯ s ′ g m d ′ ) + ( ¯ s g m g d )( ¯ s ′ g m g d ′ ) + ( ¯ s g m d ′ )( ¯ s ′ g m d ) + ( ¯ s g m g d ′ )( ¯ s ′ g m g d ) , (1.4)where now the correlator is given by: C K O K ′ ( x , y ) = h ( ¯ d g s )( x ) O D S = VV + AA ( )( ¯ d ′ g s ′ )( y ) i . Makinguse of Wick’s theorem one checks the equality: C K O K ′ ( x , y ) = C KOK ( x , y ) , which means that bothcorrelators contain the same physical information.The aforementioned matrix elements are very sensitive to various systematic errors. A majorissue facing Lattice Gauge Theory, since its early days, has been the reduction of effects inducedby the finiteness of lattice spacing a , in order to better approach the elusive continuum limit.In order to obtain reliable non-perturbative estimates of physical quantities (i.e. improving theaccuracy of B K ) it is essential to keep under control the O ( a ) systematic errors in simulations or,additionally, reduce the lattice artifacts in numerical results. Such a reduction, regarding renormal-ization functions, can be achieved by subtracting appropriately the O ( a ) perturbative correctionterms presented in this paper, from respective non-perturbative results.In this paper we calculate the amputated Green’s functions and the renormalization matricesof the complete basis of 20 four-fermion operators of dimension six which do not need powersubtractions (i.e. mixing occurs only with other operators of equal dimensions). The calculationsare carried out up to 1-loop in Lattice Perturbation theory and up to O ( a ) in lattice spacing. Ourresults are immediately applicable to other D F = D − ¯ D or B − ¯ B mixing. Let us also mention that in generic new physics models (i.e. beyondthe standard model), the complete basis of 4-fermion operators contributes to neutral meson mixingamplitudes; this is the case for instance of SUSY models (see e.g. [3]).2 ( a ) corrections to 1-loop matrix elements of 4-fermion operators Fotos Stylianou
2. Amputated Green’s functions of 4-fermion D S = D s + D s ′ = operators. In this work we evaluate, up to O ( a ) , the 1-loop matrix element of the 4-fermion operators : O XY ≡ ( ¯ s X d )( ¯ s ′ Y d ′ ) ≡ (cid:229) x (cid:229) c , d (cid:229) k , k , k , k (cid:16) ¯ s ck ( x ) X k k d ck ( x ) (cid:17)(cid:16) ¯ s ′ dk ( x ) Y k k d ′ dk ( x ) (cid:17) (2.1) O FXY ≡ ( ¯ s X d ′ )( ¯ s ′ Y d ) ≡ (cid:229) x (cid:229) c , d (cid:229) k , k , k , k (cid:16) ¯ s ck ( x ) X k k d ′ ck ( x ) (cid:17)(cid:16) ¯ s ′ dk ( x ) Y k k d dk ( x ) (cid:17) (2.2) with a generic initial state: ¯ d ′ a i ( p ) s ′ a i ( p ) | i , and a generic final state: h | ¯ d a i ( p ) s a i ( p ) . Spinindices are denoted by i , k , and color indices by a , c , d , while X and Y correspond to the followingset of products of the Dirac matrices: X , Y = { , g , g m , g m g , s mn , g s mn } ≡ { S , P , V , A , T , ˜ T } ; s mn = [ g m , g n ] . (2.3) Our calculations are performed using massless fermions described by the Wilson/clover action.By taking m f =
0, our results are identical also for the twisted mass action and the Osterwalder-Seiler action in the chiral limit (in the so called twisted mass basis). For gluons we employ a3-parameter family of Symanzik improved actions, which comprises all common gluon actions(Plaquette, tree-level Symanzik, Iwasaki, DBW2, Lüscher-Weisz). Conventions and notations forthe actions used, as well as algebraic manipulations involving the evaluation of 1-loop Feynmandiagrams (up to O ( a ) ), are described in detail in Ref. [4].To establish notation and normalization, let us first write the tree-level expression for the am-putated Green’s functions of the operators O XY and O FXY : L XYtree ( p , p , p , p , r s , r d , r s ′ , r d ′ ) a a a a i i i i = X i i Y i i d a a d a a , (2.4) ( L F ) XYtree ( p , p , p , p , r s , r d , r s ′ , r d ′ ) a a a a i i i i = − X i i Y i i d a a d a a , (2.5) where r is the Wilson parameter, one for each flavour.We continue with the first quantum corrections. There are twelve 1-loop diagrams that enterour 4-fermion calculation, six for each operator O XY , O FXY . The diagrams d − d corresponding tothe operator O XY are illustrated in Fig. 1. The other six diagrams, d F − d F , involved in the Green’sfunction of O FXY are similar to d − d , and may be obtained from d − d by interchanging thefermionic fields d and d ′ along with their momenta, color and spin indices, and respective Wilsonparameters.The only diagrams that need to be calculated from first principles are d , d and d , while therest can be expressed in terms of the first three. In particular, the expressions for the amputatedGreen’s functions L XYd − L XYd can be obtained via the following relations: L XYd ( p , p , p , p , r s , r d , r s ′ , r d ′ ) a a a a i i i i = (cid:16) L XYd ( − p , − p , − p , − p , r d , r s , r d ′ , r s ′ ) a a a a i i i i (cid:17) ⋆ , (2.6) L XYd ( p , p , p , p , r s , r d , r s ′ , r d ′ ) a a a a i i i i = L YXd ( p , p , p , p , r s ′ , r d ′ , r s , r d ) a a a a i i i i , (2.7) L XYd ( p , p , p , p , r s , r d , r s ′ , r d ′ ) a a a a i i i i = L YXd ( p , p , p , p , r s ′ , r d ′ , r s , r d ) a a a a i i i i . (2.8) Once we have constructed L XYd − L XYd we can use relation: ( L F ) XYd j ( p , p , p , p , r s , r d , r s ′ , r d ′ ) a a a a i i i i = − L XYd j ( p , p , p , p , r s , r d ′ , r s ′ , r d ) a a a a i i i i , (2.9) The superscript letter F stands for Fierz. ( a ) corrections to 1-loop matrix elements of 4-fermion operators Fotos Stylianou p ppp p ppp p ppp p ppp p ppp p ppp d d d d d d
Y X
Y X
Y X
Y X XY Y X ’’ ’’ ’’ ’’ ’’ ’’ sds s sd d s sd d s sd d s sd d s sd dd
Figure 1: O XY .Wavy (solid) lines represent gluons (fermions). to derive the expressions for ( L F ) XYd − ( L F ) XYd . From the amputated Green’s functions for all twelvediagrams we can write down the total 1-loop expressions for the operators O XY and O FXY : L XY − loop = (cid:229) j = L XYd j , ( L F ) XY − loop = (cid:229) j = ( L F ) XYd j . (2.10) In our algebraic expressions for the 1-loop amputated Green’s functions L XYd , L XYd and L XYd wekept the Wilson parameters for each quark field distinct, that is: r s , r d , r s ′ , r d ′ for the quark fields s , d , s ′ and d ′ respectively. For the required numerical integration of the algebraic expressions ofthe integrands, corresponding to each Feynman diagram, we are forced to choose the square of thevalue for each r parameter. As in all present day simulations, we set: r s = r d = r s ′ = r d ′ ≡ . (2.11)Concerning the external momenta p i (shown explicitly in Fig. 1) we have chosen to evaluate theamputated Green’s functions at the renormalization point: p = p = p = p ≡ p . (2.12)It is easy and not time consuming to repeat the calculations for other choices of Wilson parametersand for other renormalization prescriptions. The final 1-loop expressions for L XYd , L XYd and L XYd ,up to O ( a ) , are obtained as a function of: the coupling constant g , clover parameter c SW , numberof colors N c , lattice spacing a , external momentum p and gauge parameter l .The crucial point of our calculation is the correct extraction of the full O ( a ) dependencefrom loop integrands with strong IR divergences (convergent only beyond 6 dimensions). Thesingularities are isolated using the procedure explained in Ref. [4]. In order to reduce the numberof strong IR divergent integrals, appearing in diagram d , we have inserted the identity below intoselected 3-point functions: = c a p (cid:16) \ k + a p + \ k − a p − k + (cid:229) s sin ( k s ) sin ( ap s ) (cid:17) , (2.13) where ˆ q = (cid:229) m sin ( q m ) and k ( p ) is the loop (external) momentum. The common factor in Eq.(2.13) can be treated by Taylor expansion. For our calculations it was necessary only to O ( a ) : c a p = a p + (cid:229) s p s ( p ) + O ( a p ) . (2.14) ( a ) corrections to 1-loop matrix elements of 4-fermion operators Fotos Stylianou
Here we present one of the four integrals with strong IR divergences that enter in this calculation: Z p − p d k ( p ) sin ( k m ) sin ( k n ) ˆ k \ k + a p \ k − a p = d mn h . − ln ( a p ) p + a p (cid:16) . ( ) − ln ( a p ) p (cid:17) − a p m (cid:16) . ( ) + ln ( a p ) p (cid:17) + . a (cid:229) s p s p i + a p m p n h . a p − . ( ) + ln ( a p ) p − . ( p m + p n ) p + . (cid:229) s p s ( p ) i + O ( a p ) . The results for the other three integrals can be found in Ref. [4]. Integrands with simple IRdivergences (convergent beyond 4 dimensions) can be handled by well-known techniques.Due to lack of space we present only the results for L XYd and for the special choices: c SW = l = r s = r d = r s ′ = r d ′ =
1, and tree-level Symanzik action: L XYd ( p ) a a a a i i i i = g p (cid:18) d a a d a a − d a a d a a N c (cid:19) × (cid:26) X i i Y i i (cid:20) −
12 ln ( a p ) − . ( ) (cid:21) + (cid:229) m ( X g m ) i i ( Y g m ) i i [ − . ( )]+ (cid:229) m , n ( X g m g n ) i i ( Y g m g n ) i i (cid:20)
18 ln ( a p ) + . ( ) (cid:21) + (cid:229) m , n , r ( X g m g r ) i i ( Y g n g r ) i i (cid:20) . p m p n p (cid:21) + a ( L O ( a ) ) XYd + a ( L O ( a ) ) XYd (cid:27) , (2.15) where: ( L O ( a ) ) XYd = (cid:229) m (cid:16) ( X g m ) i i Y i i + X i i ( Y g m ) i i (cid:17) × (cid:20) ip m (cid:18) −
14 ln ( a p ) + . ( ) (cid:19)(cid:21) + (cid:229) m , n (cid:16) ( X g m g n ) i i ( Y i i g n ) + ( X g n ) i i ( Y g m g n ) i i (cid:17) × (cid:20) ip m (cid:18)
116 ln ( a p ) + . ( ) (cid:19)(cid:21) , (2.16) and: ( L O ( a ) ) XYd = X i i Y i i (cid:20) p (cid:18) − ( a p ) + . ( ) (cid:19) + . ( ) (cid:229) s p s p (cid:21) + (cid:229) m ( X g m ) i i ( Y g m ) i i (cid:20) p (cid:18) −
748 ln ( a p ) + . ( ) (cid:19) + . ( ) p m (cid:21) + (cid:229) m , n (cid:16) ( X g m g n ) i i Y i i + X i i ( Y g m g n ) i i (cid:17) × " . ( ) p n p m p + (cid:229) m , n ( X g m ) i i ( Y g n ) i i (cid:20) p m p n (cid:18) −
16 ln ( a p ) − . ( ) (cid:19)(cid:21) + (cid:229) m , n ( X g m g n ) i i ( Y g m g n ) i i (cid:20) p (cid:18) ( a p ) − . ( ) (cid:19) − . (cid:229) s p s p + p m (cid:18) − ( a p ) + . ( ) (cid:19)(cid:21) + (cid:229) m , n , r ( X g m g r ) i i ( Y g n g r ) i i (cid:20) p m p n (cid:18) ( a p ) − . ( ) + . (cid:229) s p s p (cid:19) − . ( p m p n + p m p n ) p − . ( ) p m p n p r p (cid:21) . (2.17) Similar expressions exist for L XYd and L XYd . 5 ( a ) corrections to 1-loop matrix elements of 4-fermion operators Fotos Stylianou
3. Mixing and Renormalization of O XY and O FXY on the lattice.
The matrix element h ¯ K | O D S = VV + AA | K i is very sensitive to various systematic errors. The mainroots of this problem are: a) O ( a ) systematic errors due to numerical integration, b) the operator O D S = VV + AA mixes with other 4-fermion D S = D S = D S = P , Charge conjugation C , Flavour exchange sym-metry S ≡ ( d ↔ d ′ ) , Flavour Switching symmetries S ′ ≡ ( s ↔ d , s ′ ↔ d ′ ) and S ′′ ≡ ( s ↔ d ′ , d ↔ s ′ ) ),with 4 degenerate quarks. This basis can be decomposed into smaller independent bases accordingto the discrete symmetries P , S , CPS ′ , CPS ′′ . Following the notation of Ref. [5] we have 10 ParityConserving operators, Q , ( P = + , S = ±
1) and 10 Parity Violating operators, Q , ( P = − , S = ± Q S = ± ≡ (cid:2) O VV ± O FVV (cid:3) + (cid:2) O AA ± O FAA (cid:3) , Q S = ± ≡ (cid:2) O VV ± O FVV (cid:3) − (cid:2) O AA ± O FAA (cid:3) , Q S = ± ≡ (cid:2) O SS ± O FSS (cid:3) − (cid:2) O PP ± O FPP (cid:3) , Q S = ± ≡ (cid:2) O SS ± O FSS (cid:3) + (cid:2) O PP ± O FPP (cid:3) , Q S = ± ≡ (cid:2) O T T ± O FTT (cid:3) , n Q S = ± ≡ (cid:2) O VA ± O FVA (cid:3) + (cid:2) O AV ± O FAV (cid:3) , ( Q S = ± ≡ (cid:2) O VA ± O FVA (cid:3) − (cid:2) O AV ± O FAV (cid:3) , Q S = ± ≡ (cid:2) O PS ± O FPS (cid:3) − (cid:2) O SP ± O FSP (cid:3) , ( Q S = ± ≡ (cid:2) O PS ± O FPS (cid:3) + (cid:2) O SP ± O FSP (cid:3) , Q S = ± ≡ (cid:2) O T ˜ T ± O FT ˜ T (cid:3) . Summation over all independent Lorentz indices (if any), of the Dirac matrices, is implied. Theoperators shown above are grouped together according to their mixing pattern. This implies thatthe renormalization matrices Z S = ± ( Z S = ± ), for the Parity Conserving (Violating) operators, havethe form: Z S = ± = Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z S = ± , Z S = ± = Z Z Z Z Z Z Z Z Z S = ± . (3.1) Now the renormalized Parity Conserving (Violating) operators, ˆ Q S = ± ( ˆ Q S = ± ), are definedvia the equations: ˆ Q S = ± l = Z S = ± lm · Q S = ± m , ˆ Q S = ± l = Z S = ± lm · Q S = ± m , (3.2)where l , m = , . . . , m is implied). The renormalized amputated Green’s functionsˆ L S = ± ( ˆ L S = ± ) corresponding to Q S = ± ( Q S = ± ), are given in terms of their bare counterparts L S = ± ( L S = ± ) through:ˆ L S = ± l = Z − Y Z S = ± lm · L S = ± m , ˆ L S = ± l = Z − Y Z S = ± lm · L S = ± m , (3.3)6 ( a ) corrections to 1-loop matrix elements of 4-fermion operators Fotos Stylianou where Z Y is the quark field renormalization constant.In order to calculate the renormalization matrices Z S = ± ( Z S = ± ), we make use of the appro-priate Parity Conserving (Violating) Projectors P S = ± ( P S = ± ): P S = ± ≡ + P VV + P AA N c ( N c ± ) , P S = ± ≡ + P VV − P AA ( N c − ) ± P SS − P PP N c ( N c − ) , P S = ± ≡ ± P VV − P AA N c ( N c − ) + P SS − P PP ( N c − ) , P S = ± ≡ + P SS + P PP N c ( N c − ) N c ± ∓ P T T N c ( N c − ) , P S = ± ≡ ∓ P SS + P PP N c ( N c − ) + P TT N c ( N c − ) N c ∓ , P S = ± ≡ − P VA + P AV N c ( N c ± ) , P S = ± ≡ − P VA − P AV ( N c − ) ∓ P SP − P PS N c ( N c − ) , P S = ± ≡ ∓ P VA − P AV N c ( N c − ) − P SP − P PS ( N c − ) , P S = ± ≡ + P SP + P PS N c ( N c − ) N c ± ∓ P T ˜ T N c ( N c − ) , P S = ± ≡ ∓ P SP + P PS N c ( N c − ) + P T ˜ T N c ( N c − ) N c ∓ , where P XY ≡ ( X i i ⊗ Y i i ) d a a d a a . Again, summation is implied over all independent Lorentzindices (if any) of the Dirac matrices. The above Projectors are chosen to obey the followingorthogonality conditions: Tr ( P S = ± l · L S = ± m ( tree ) ) = d lm , Tr ( P S = ± l · L S = ± m ( tree ) ) = d lm , (3.4)where the trace is taken over spin and color indices, and L S = ± ( tree ) , L S = ± ( tree ) are the tree-level amputatedGreen’s functions of the operators Q S = ± , Q S = ± respectively.We impose the renormalization conditions: Tr ( P S = ± l · ˆ L S = ± m ) = d lm , Tr ( P S = ± l · ˆ L S = ± m ) = d lm . (3.5)By inserting Eqs. (3.3) in the above relations, we obtain the renormalization matrices Z S = ± , Z S = ± in terms of known quantities: Z S = ± = Z Y h(cid:0) D S = ± (cid:1) T i − , Z S = ± = Z Y h(cid:0) D S = ± (cid:1) T i − , (3.6)where: D S = ± lm ≡ Tr ( P S = ± l · L S = ± m ) , D S = ± lm ≡ Tr ( P S = ± l · L S = ± m ) . (3.7)Note that D S = ± and D S = ± have the same matrix structure as Z S = ± and Z S = ± respectively.Due to lack of space we provide only the matrix D S =+ (Parity Violating P = −
1, Flavourexchange symmetry S = +
1) for the special choices: c SW = l = r s = r d = r s ′ = r d ′ = N c =
3, and tree-level Symanzik action: D S =+ = + D + D D D + D + D D D + D S =+ (3.8) ( a ) corrections to 1-loop matrix elements of 4-fermion operators Fotos Stylianou where: D = + g p (cid:20) . ( ) + ( a p ) + (cid:16) . ( ) − ( a p ) (cid:17) a p − . ( ) a (cid:229) s p s p (cid:21) , D = + g p (cid:20) . ( ) + ln ( a p ) + (cid:16) . ( ) − ( a p ) (cid:17) a p − . ( ) a (cid:229) s p s p (cid:21) , D = − g p (cid:20) . ( ) + (cid:16) . ( ) −
13 ln ( a p ) (cid:17) a p − . ( ) a (cid:229) s p s p (cid:21) , D = − g p (cid:20) . ( ) − ( a p ) + (cid:16) . ( ) −
76 ln ( a p ) (cid:17) a p − . ( ) a (cid:229) s p s p (cid:21) , D = + g p (cid:20) . ( ) − ( a p ) + (cid:16) . ( ) −
49 ln ( a p ) (cid:17) a p + . ( ) a (cid:229) s p s p (cid:21) , D = + g p (cid:20) . ( ) − ( a p ) − (cid:16) . ( ) −
118 ln ( a p ) (cid:17) a p + . ( ) a (cid:229) s p s p (cid:21) , D = + g p (cid:20) . ( ) − ( a p ) + (cid:16) . ( ) −
79 ln ( a p ) (cid:17) a p − . ( ) a (cid:229) s p s p (cid:21) , D = + g p (cid:20) . ( ) +
13 ln ( a p ) + (cid:16) . ( ) − ( a p ) (cid:17) a p − . ( ) a (cid:229) s p s p (cid:21) , D = + g p (cid:20) . ( ) +
173 ln ( a p ) + (cid:16) . ( ) − ( a p ) (cid:17) a p − . ( ) a (cid:229) s p s p (cid:21) . In order to obtain Z Y for a given renormalization prescription, one must make use of the inversefermion propagator, S − , calculated (up to 1-loop and up to O ( a ) for massless Wilson/cloverfermions and Symanzik gluons) in Ref. [4]. References [1] M. Constantinou, V. Lubicz, H. Panagopoulos, A. Skouroupathis, F. Stylianou and members of theETM Collaboration, in preparation.[2] R. Frezzotti and G.C. Rossi,
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