We check to what extent the assumptions of the vacuum insertion
method, as used in the theory o f K —* tttt decays, can be derived from the
high N c approximation to the chiral perturbation theory. We find that,
besides the well-known problem of Fierz terms, only the assumption for
the K 13 formfactor ( /_ = 0) does not follow. This assumption, however,
affects the penguin contributions by less than four per cent and the nonpenguin
contributions by less than two percent[…

Białas, Piotr

Zalewski, K.

Acta Physica Polonica B

English

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Vacuum Insertion Method from Chiral Nc \to \infty Approach for the \Delta S = 1 Effective Hamiltonian  Author: Piotr Białas, K. Zalewski  Citation style: Białas Piotr, Zalewski K.. (1991). Vacuum Insertion Method from Chiral Nc \to \infty Approach for the \Delta S = 1 Effective Hamiltonian. "Acta Physica Polonica. B" (1991, no. 3, s. 319-324). Vol. 22(1991) A C TA  PH YSIC A  PO LO N ICA B No 3V A C U U M  IN S E R T IO N  M E T H O D  F R O M  C H IR A L  N c -> oo A P P R O A C H  F O R  T H E  A S  =  1 E F F E C T IV EH A M IL T O N IA NP .  B i a ł a s , K .  Z a l e w s k i *Institute o f  Nuclear Physics Radzikowskiego 152, 31-342 Cracow, Poland(Received October 29, 1990)We check to what extent the assumptions of the vacuum insertion method, as used in the theory o f  K  —* tttt decays, can be derived from the high N c approximation to the chiral perturbation theory. We find that, besides the well-known problem of Fierz terms, only the assumption for the K 13 formfactor ( /_  =  0) does not follow. This assumption, however, affects the penguin contributions by less than four per cent and the non­penguin contributions by less than two percent.PACS numbers: 11.15.Pg, 11.30.RdThis note is concerned with the matrix elements < -7r7r|jfiTeir| Ji" > occur­ring in the theory of the processesK °  -» ir°ir°, (1)K°  —> 7T + 7T_ , (2)K + —> ir+w° . (3)It is known [1], [2] that, whether one uses the vacuum insertion method (VIM) [3] or whether one uses the leading in l / N c term of the chiral pertur- bative expansion (CHNC) [2 ], one finds very similar results for these matrix elements. We discuss to what extent tills fact reflects assumptions common to both approaches and why the differences between the two affect little <  ww\H<.a\K > in the corresponding approximations.* Work supported in part by the G'PBP grant 01.03.For the matrix elements of the effective Hamiltonian one uses the ap­proximation< irn\HeiI\K  > =  Y v )  < ‘ir‘jr\Qi(fl ) \K  > > (4)1where the summation extends over all the operators Q, of dimension six, which can be constructed from the quark fields and have the relevant matrix elements nonzero. One possible choice of these operators [4] isQi =  {sd)v_A(uu)v_A ,Q2 =  (su )v-A(ud)v _A ,Q3 = (sd)v_A(qq)v_A ,Q 4 =  (sq)v-A(qd)v_A ,Qb =  (sd)v_A(qq)v+A ,Qo = - 8 (shqR){qRdL) , (5)where the summation over q -■ u, d, s and over colour in each bracket is implied. One notes the identityQi — Qi — Qi +  Qa • (6 )Another set [5], which has simpler symmetry properties under SU(3) - flavour, isOt = \ Q i - \ Q »0 2 =  — \Qi +  \Q i  +  \Qz -,^34 — |Q l  +  |Q 2 -  |i?3 >^ 5  =  ~ 2 K Q s +  \Q® ’Oa = \Q 5. (7)Actually in [5] instead of the operator 0 34 two other operators have beenintroduced:O34 =  03 -(- 5 O4 , (8 )where O3 corresponds to A I  =  |  and 0 4 to A I  = | .  When the scale pt changes, however, the operators 0 3 and 0 4 do not mix with other operators (multiplicative renormalization) and have the same anomalous dimensions. Therefore, operator 0 34 évoluâtes without changing its composition (multi­plicative renormalization again) fixed at ¡x — M w -In CHNC an explicit calculation shows that the nonzero matrix elements of the operators Q i ^  can be rewritten as follows< 7r°7r°|Q ^ K 0 >  =  2 < ir°|(5(i)v,_ il |fl'0 X  *°\{uu)v-A)\0 > ,< ir+n~\Q2\K°  >  = <  7r“ i ( s i i ) v - A | f f °  > <  7r+ | ( u d ) iz _ vi ) | 0  >  ,<  7r+ 7r0 | Q 1 |h T +  >  = <  w + \ ( s d ) v _ A \ I { + > <  7 r ° | ( u u ) K - y l ) | 0  >  ,<  7T + X0 |Q 2|- ^  + >  = <  X ° |(s u ) v -..a | jK’ + > <  7T+ |(«c£)v^ —^ l) |0  >  ,<  ir+w~\Qo\K° >  =  - 8 [ <  7r“ | ( 5 Lu n ) | / f °  > <  7r+ | ( u RtiL) |0  >+  < x +x~ |(d R<iL)|0 > <  0 |(sLdR)|Iifo > ] ,< x°7r°|Q6|/i:0 > =  —8[< 0 |(sndR) |/f °  > <  » ox o|(2n dL)|0 >+  < x°|(?i<dR)|Jif0 X  x°|(dRdL)|0 > ] . (9)The factor two in the first formula corresponds to two possible allotments of the 7r0,s. In VIM the products on the right hand sides contribute to t he matrix elements on the left hand sides by definition, but then additional “Fierz terms”A < inr\Qi\K >  =  A- < -k t \Q2\K > ,A < inr|Qa|A' > =  +- < TnrlQ^A > ,A  < ftr\Q3\K  >  =  T- < tck\Qa\K  >  ,A < 7nr|Q4|Jif >  =  A- <  7r7r|Q3|/<r > ,A <  x x \ Q , \ K  > = jf; < xir\Q6\K > ,A <  Tnr\Q(j\K >  =  A- <  wtc\Qs\K  >  , (10)are added. These terms are of course negligible in the N c —> oo limit. For N c finite the matrix elements on the right-hand-sides are interpreted as in (9). Moreover, for i =  1 ,2 ,3 ,4  one could expect in the vacuum insertion method terms of the form<  *x\{q<i}v-A\Q > <  0 | ( g g ) v - ^ | / f  >  , (11)where the flavour indices of the quarks have not been explicitly written. These terms vanish, however, because < 0 | j i | / f  > ~  pk = P*l +27,,, while only the vector current contributes to < x x |j£ |0  > . Thus the product is proportional to the divergence of the vector current, which is negligible. In CHNC < xx|jx,|0 > ~  py, — py, ,  which also makes the product vanish.The six matrix elements occurring in (9) and (10) can be reduced to two by using the Eckart-Wigner theorem. The currents -(sd )^_A  and ( s u )^ ^  form an isospin doublet. Thus—y/2 <  x°|sd|iiT0 >  = <  x _ |iu|Jir0 >=  -  <  T+\sd\K+ >= y/2 <  x ° |su |A + > , (12)where the subscripts V  -  A  have been omitted. Since qq is an isosinglet, we can defineY  = <  w+ir-\Q6\K° > = <  TrVlQejJiT0 > . (13)Here following [2] we have chosen the sign of the |tt°w° >  component of theisosinglet opposite _to that given by the Condon-Shortley convention. The currents u d ,  (uu — d d ) / y / 2, d u  form an isotriplet. Thus\/2  <  7T°|(tm|0 > =  — < 7r+ |uci|0 >  . (14)Combining this with (12), using (9) and keeping in mind the phase conven­tion for the two 7T° state we can defineX  = <  7r+7r-|Q2|jr° > =  V2 < w+w°\Ql \K + >= v/2 < k +tt°\Q2\K + > =  -  < x V lQ x I if0 > . (15)These reductions are common to VIM and CHNC.The matrix element X  is calculated as follows. The matrix elements (14) are expressed in both approaches by the pion decay constant< 7r+ |(ud)v~A |0 > =  -iV2F„pT , (16)where Fr =  93M eV  is the physical pion decay constant. For the matrixelements (12) the general formula involves two formfactors:<  v~\jL\K° > =  (pK +  pr ) f + +  (pK - p w) f_ . (17)In VIM one puts /+  =  1 and /_  =  0, while in CHNC the calculation yields/+  =  1, butm i- — m l  ./ - =  A 2 ' (18) x .The experimental estimates [6] are / _ / / +  =  —0.35 ±  0.15 for K + and / _ / / +  =  —0.11 ±  0.09 for iif£. In VIM, therefore,X  = - i \ /2 F T(m2K -  m l ) , (19)while in CHNCX  =  - i V ^ F n i m x  -  m l )  ^1 +  , (20)which differs from the previous formula by less than two per cent.The evaluation of the matrix elements of operator Qe is based on the following identities(Pk  -  P r ) <  7r-|(au)K_x |/i'° >  =  2 (m, -  m d) <  x " |(sRuL)|/ir0 >  ,(21)Pit <  tt* |(ud)y_,t|{) >  =  2(tnu +  m d) <  x+juadLjIO >  , (22)m 2 ( l  +  ^ f-)  =  2(m „ -I- m d) <  7r+7r-|(dn i/L)|0 >  ,(2 3 )<  0 |( sd )v - ,i |t fo*> =  iV2FKpK , (24)Pk  < 0 \{ s d )v „A \ K °  > -  2 ( m ,  +  m d) <  O l ^ d a ) ! # 0 > , ( 2 5 )which hold in both VIM and CIINC provided ACT from VIM is identified with Ax from CHNC. Substituting these formulae into (9) one finds using/ - =  or  =  (26)or using (18)A2XY  =  - i V 2 f ^ ^ ^ .  (27)xThe two expressions for Y  differ by less than four per cent. Here.  -  2 ,< -  -  . (28 )m ,  +  m d m u +  m dIn order to reproduce the phases from [2] one has to absorb the phase factor —i into e.g. the state |K  > . The difference between formulae (26) and (27) again follows from the different assumptions about the formfactor /_  in the two approaches.To summarize: comparing the derivations of the matrix elements for kaon decays < irn\Qi\K >  in CHNC and VIM (with the Fierz terms re­jected) one finds that not only the final results are similar, hut also most of the assumptions of VIM can be derived in the framework of CHNC. The only relevant difference in assumptions is that about the formfactor The corresponding contributions, however, are multiplied by m 2 and therefore numerically the two sets of formulae are ahnost equivalent. VIM could of course be updated by inserting the experimental value of / _ .  Since, how­ever, the ratio of the A I  =  1 /2  to the A /  =  3 /2  amplitudes for K  —» 7T7t decays is underestimated by about a factor of four in the approach described here, this small correction does not seem very important. Let us also note that the Fierz terms, omitted in the leading term of CHNC, reappear in the next approximation of tills method.REFERENCES[1] A.J. Buras, J.-M . Gerard, R. Riikl, Nucl. Phys. B 2 6 8 , 16 (1986).[2] W .A  Bardeen, A .J. Buias, J.-M. Gérard, Phys. Lett. 1 8 0 B , 133 (1986).[3] M.K. Gaillard, B .W . Lee, J.L. Rosner, Rev. Mod. Phys. 47 , 277 (1975).[4] F.J. Gilman, M.B. Wise, Phys. Rev. D 2 0 , 2392 (1979).[5] M.A. Shifman, A.I. Vainshtein, V.I Zakharov, Nucl. Phys. B 1 2 0 , 316 (1977).[6] Particle D ata Group, Phys. Lett. B 2 0 4 , (1988).

Vacuum Insertion Method from Chiral Nc \to \infty Approach for the \Delta S = 1 Effective Hamiltonian

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Vacuum Insertion Method from Chiral Nc \to \infty Approach for the \Delta S = 1 Effective Hamiltonian

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