Weak interaction matrix elements in strong coupling lattice QCD

510

Nuclear Physics B (Proc. Suppl.) 4 (1988) 510-514 North-Holland, Amsterdam

WEAK INTERACTION

MATRIX ELEMENTS IN STRONG COUPLING LATTICE QCD

Apoorva PATEL Theory Division, CERN, Geneva, Switzerland Amplitudes of many non-leptonic processes require a non-perturbative calculation of matrix elements between pseudo-Goldstone boson states, after the standard perturbation theory in the Electro-Weak sector. Several such matrix elements are estimated using the techniques of strong coupling lattice QCD.

Though leptonic Electro-Weak processes are well

ements are to be evaluated at low energy scales.

understood in the framework of perturbation theory,

L,.,

= F_., c,(,,)

(1)

that is not the case with non-leptonic processes. The missing component in the latter case is the knowledge

Here both the effective operators Oi and the Wilson

of how non-perturbative QCD contributions modify the

coefficients Ci are defined using a particular regular-

amplitudes at low energy scales. Some well known ex-

isation scheme and they depend on the measurement

amples are :

scale #. The matrix elements of Li,,t between hadronic

1) K -* 2r non-leptonic decays and A I = 1/2 rule.

states however have no dependence either on the regu-

2) K ° - ~-o mixing and the CP violation parameter

larisation or the scale t~. The coefficients Ci are generally calculated using weak coupling perturbation the-

3) CP violation in K --* 2 r decays and the parameter

£t.

ory, while the quantities < Oi >-=< h tlOilh~ > are estimated by non-perturbative methods.

4) ~r+ - r ° mass difference. A common feature among the processes listed above

The lattice formulation of QCD provides an explicitly non-perturbative framework for evaluating the

is that they involve only the pseudo-Goldstone boson

< Oi >. In this regularisation, one can imagine lower-

fields. In such cases, the pattern of chiral symmetry

ing the scale t~ all the way down to the strong coupling

breaking restricts the form of the various amplitudes,

limit. The aim of this talk is to present the methodol-

and one hopes to achieve an understanding of the re-

ogy for determining the < O~ > in this limit. I use the

maining parameters in a theoretical framework.

staggered fermion formulation, which by preserving a

The strategy in the computation of such amplitudes 1 is to start with the standard SU(3)c × SU(2)L x U(1)y model, and carry out a perturbative expansion in the Electro-Weak interactions.

Then the energy scale is

gradually lowered evolving the various interactions us-

subset of the continuum chiral symmetries at all scales produces Goldstone bosons with the correct chiral behaviour. Several drawbacks of such an analysis are already known :

ing renormalisation group equations. When the energy

o One is working at the "wrong" fixed point of the

scale becomes comparable to a heavy field mass, the

theory. The weak coupling scaling behavior is lost

heavy field is integrated out using the Operator Prod-

and the hadron spectrum cannot be fit to better

uct Expansion, leaving behind a Lagrangian contain-

than ~ 30% accuracy.

ing effective interactions of the lighter fields. Contin-

o The complete set of continuum chiral symmetries

uing this process, finally one arrives at a Lagrangian

does not exist on the lattice and mixing between

containing renormalised interactions, whose matrix el-

different lattice operators takes place.

0920-5632/88/$03.50
A. Patel / Weak interaction matrix elements

511

o Perturbative calculations of C~ and operator mix-

correlationfunctions written in l-spinor loop Fierz ar-

ing cannot be trusted. They can only be regarded

rangement and containing 1-1ink currents. This selection of terms is based on a desire to have correlation

as a qualitative guide. These reasons make it clear that the results obtained

functions going over to the correct continuum limit as

in the strong coupling limit should be interpreted only

well as containing only the (partially)conserved vector

as trends and not as quantitative predictions. Never-

and axial vector latticecurrents. I optimisticallyhope

theless, there are certain welcome features in pursuing

that the conserved nature of the currents keeps the

this approach :

renormalisation and operator mixing problems under

• One still has a regulated theory of (partially) conserved currents and their corresponding charges.

control. To maintain the analogy with the continuum expressions, I use the naive fermion notation :

This has a certain degree of similarity with the chiral Lagrangian description of pseudo-Goldstone

I V~(~) =

~ [ ~(x)~u~C~)~Cx

boson interactions, and one can in fact compare the two sets of results. The regulated lattice the-

+

+

1

A~Cx ) = ~ [ ~ ( ~ ) ~ s U ~ ( ~ ) ~ (

+ ~)

1 • + ~)

ory actually has less number of free parameters

+ ~(~ + ~)~,~sU~(~)~°~(~) ]

and so more predictive power. • Chiral behaviour is not assumed but is a conse-

All the correlation functions considered here will be

quence. No restriction to small quark masses is

such that they can be spin diagonalised and a factor

necessary, and corrections to the leading chiral be-

proportional to the unit matrix can he dropped. The

haviour can be calculated.

description then changes in to the staggered fermion

• Unlike the Monte-Carlo calculations, it is easier

language. Also it remains transparent that the depar-

to work with an infinite lattice than a finite one.

ture of the lattice correlation functions from the con-

This makes it possible to use continuous momenta

tinuum ones is O(a2), modified by factors of In(a) in

and directly compute on-shell matrix elements.

asymptotically free QCD.

The ultimate interest of the lattice game is to extrap-

The strong coupling expansion is an expansion in

olate the results towards the weak coupling limit, and

powers of 1/N9 2 about the point g 2 = oo. So it can be

that job still has to be left to Monte-Carlo calculations.

thought of as a rearranged 1/N expansion. At g2 = oo

The 4 - F e r m i operators of the Weak interactions

the gauge field action is zero, and the correlation func-

have the structure of a product of two currents. These

tions can be expressed as diagrams containing random

obey both spinor and colour Fierz identities. The tran-

walks of colour singlet hadrons. These diagrams can be

scription of these operators on the lattice satisfying

classified according to the powers of N in their contri-

the spinor Fierz identities involves split-point currents

butions. As a simple illustration diagrams contributing

spread over a 2 4 hypercube; and to simultaneously

to < ~ b > at different orders in N are shown in Fig.1.

maintain the colour Fierz identities the split vertices

It is obvious that terms of arbitrarily high order in 1/N

of the current are joined by a gauge connection, which

contribute even at g~ = oo. I will limit myself to cal-

after a sum over all equal length paths is projected back

culating the first two non-trivial terms in such a 1/N

on to the SU(N) gauge group s. Such a process gives

series.

rise to many terms. Only some of them may contribute

The first step in the process is to calculate the lat-

to a given process as dictated by discrete symmetries,

tice meson propagator by resumming a hopping param-

and many of these contributions are equal in the con-

eter expansion. Then an n - p o i n t correlation function

tinuum limit.

of quark bilinears can be calculated by adding up all

In the strong coupling limit, I choose to work with

the mesonic Feynman diagrams. It is easier to carry

512

A. Patel / Weak interaction matrix elements The expressions in eq.(3) are evaluated at leading order in N. The Weak interaction matrix elements are O(N 2) in the same order. They completely factorise and produce results identical to the vacuum insertion values. Up to the same order the Wilson coefficients Ci(/~) are independent of the scale /~, and no QCD effects appear. In order to see interesting effects one has to go O(N)

0(1)

0(11

OIl/N)

to the next order in N.

Now the QCD effects show

up. New operators Oi enter the picture, giving rise to correlation functions of O(N).

An anomalous di-

mension matrix appears as well, producing renormalisation group scaling of the Wilson coefficients Ci(p). In addition to these there are 1/N corrections (e. g., from dynamical fermions) to the leading O(N 2) correm

Fig.1 : Various contributions to < ~b~b> at strong coupling: (a) mesonic screening, (b) dynamical fermions, (c) baryonic screening and (d) self-screened meson loop.

lation functions. However, due to the factorisability of the O(N 2) correlation functions, these 1/N effects can be completely absorbed in quantities like re,r, f,r and normalisation constants. Therefore, as long as these quantities are factorised out from the final amplitude, it remains consistent to use the expressions in eq.(3)

out the calculation in the momentum space, where the constraint of momentum conservation at every vertex is equivalent to summing over all positions of the corresponding vertex in position space. The results for 2 - p o i n t functions have been presented in ref.3. Using f~ to fix the lattice scale, the electromagnetic contribution to the pion mass difference was calculated to be, m r+2 -m~o2 __ (41.3 M e V ) 2, compared with the experimental value of (35.6 M e V ) 2. To keep the expressions

[a)

(b)

for the Weak interaction matrix elements simple, I will here ignore backtracking of random walks and use the l a r g e - d limits of the results in ref.3 : tc =

(m_~_)= 4sinh2

1

,

1 - - - 2d

(3)

4~21c~ exp(ika)

Fig.2 : (a) A quark level diagram for K ° - -go mixing, and (b) a diagram for K ° --* 2~r decay with a spectator quark.

Here x is the effective quark hopping parameter and

in all correlation functions. In other words, the 1/N corrections to the O ( N 2) terms change the physical value of the lattice spacing, but not the dimensionless

r~(k) denotes the momentum space ~ s - m e s o n prop-

numbers.

fK = - ~ - t

+ 6 8 ) , r~(k) = 1 - 2 ~ . E . c o s k .

agator with the first and last steps in any and a th direction respectively.

With this machinery in place, it is trivial to calculate the contribution of the AS = 2 operator to

A. Patel / Weak interaction matrix elements

K° - ~

513

mixing.

(KOI(~*)V--A(~*)V--AI~O) = 2f~
1

~) (4)

= 2sinh(V)

The 3 - p o i n t function involves only a tree graph shown in Fig.2(a). The K ~ 2~r decays require computations of 4 - p o i n t functions. The relevant 4 - F e r m i AS = 1 operators

• K°

~K"

(a)

{b)

(without integrating the charm quark out) are : 1

o ± = ~ [ ( ~ , ) v - ~ ( ~ , , ) v - ~ + (-~*)v-aC~,,)v-~

Fig.3 : Tree level meson diagrams for K + decay. (a) is

-(~.)v-.,,C~c)v-. :F (Q*)v-.C~)v-., l (5)

O(N ~) ~ a (H is O(N).

The initial condition on the Wilson coefficients at the

sum of the diagram, of Fig.5(~d) however does pro-

Weak scale is C+ = C_ = 14. Inclusion of the l - l o o p

duce a non-trivial contribution. With the charm quark

QCD renormalisation effects suppresses C+ and en-

in the internal loop, and k~ and PI~, respectively de-

hances C_ 5. Moreover,

noting the four momenta of the K ° and lr +, this contribution is :

C++C_

:

O(1) , C + - C _

c _ = (c+) (~+N)/o-N)

= O(N ) ,

f

d41

~°

(8)

(6)

First consider the decay K + ~ r % r °. This A I = 3/2 process again involves only tree level meson diagrams. They are shown in Fig.3(a-b), where the filled and unfilled boxes represent the 4 - F e r m i vertices created by the W - b o s o n exchange in the s and t channels respectively. The net result is : (g+lo+l~r+~r o) = f'rta.2V/_2 t ' ' K _ M~)(I - 6)(1 + ~)I (a) --

M~2~°

(b)

(7)

The ensuing prediction
Fig.4 : K ° decay d i a g r a m s h a v i n g K ° --* v a c u u m as a

result 0.010 G e V 3.

subprocess.

The next task is to evaluate the decay amplitude for K ° --* ~r+~ - . The tree diagrams contributing to this process are those of Fig.3(a) (with K +, ~r° replaced by K ° , r - ) and Fig.5(a).

In addition there are di-

agrams with an internal meson loop. The diagrams shown in Fig.4(a-b) add up to zero, clearly demonstrat-

(x + 2~.~o E . co~ t.)(1 - 2 ~ o E ~ co. t.) C1 + 2 ~ o E ~ co~(l - p,)~)(1 - 2 ~ o E . co~(l - k).) The integral over the lattice Brillouin zone can be re-

ing that graphs containing K ° ---* vac as a subprocees

duced to a three dimensional one using the residue the-

cannot contribute to an on-shell decay amplitude. The

orem, but then it has to be calculated numerically. I

514

A. Patel / Weak interaction matrix elements O ( N 2 ) , but the K ° --, ~r°r ° decay is O ( N ) .

obtain E ( u ) - E(c) ~ 0.04f~(M 2 - M~).

At this

The diagrams for the decay K ° --* r°Tr ° are topo-

stage there is no real clue as to the appropriate values of

logically the same, but appear with different coeffi-

the coefficients C+ (#) at the lattice momentum cut-off

cients. Putting everything together, the full matrix

scale # = 7fla. I can only work backwards and deter-

elements become

mine that C+ (/~) = 0.33 and C_ (#) = 10 bring the pre-

(K°]Lintllc+~r - )

dicted decay rates close to the experimental ones. The

f,r t 412 = - ~ , , 1 K - M2x) •

(9)

justification for such values of the Wilson coefficients at strong coupling is currently under investigation.

1

[ C_ (#)(1 - ~ ) ( 1 - 6 + ~'(lt¢ + 6) + 0.04) 1 + 0 + ( , ) ( 1 + ~ ) ( 1 - 6 - ~(1~ + 6) - 0.04) ]

The (Y - A)(V + A) penguin operators appear in the calculation of e' and in the effective theory where the charm quark is integrated out.

.°)

=

f~

IR.2

- M2)

Their contribu-

tion can be estimated by summing up the diagrams of

•

Fig.5(b-d) with the appropriate spinor structure. 1

[ C_(#)(1 - ~ ) ( 1 - 6 +

~---*(1~:+ 6 ) +0.04)

1

C+(#)(1 + ~ ) ( 1 - 6 + t%(11¢+ 6) + 0.04) ]

-

In principle the approach described here can be extended to the Wilson fermion case, with non-perturbative subtractions using operators of dimension < 6, along

These have to be compared with the experimental num-

the lines of ref.6. However, at large values of "a" typi-

bers 0.156 G e V s and 0.105 GeV s respectively.

cal of the strong coupling region, one has to correct for

The matrix elements in eqs.(7,9) have the correct

contamination due to operators of dimension > 6 as

behaviour in the chiral limit. Also it can be seen that

well. It is not known whether a consistent description

the K + --, 7r+Tr° and the K ° --~ 7r+~r- decays are

can be found in such a case. REFERENCES 1. A pedagogical discussion can be found in : H. Georgi, Weak Interactions and Modern Particle Theory (Benjamin/Cummings, 1984).

2. S. Sharpe, A. Patel, R. Gupta, G. Guralnik and G. Kilcup, Nucl. Phys. B286 (1987) 253. 3. O. Martin and A. Patel, Phys. Lett. 174B (1986) 94. la)

{b}

It)

Id)

4. A common factor of GF sin 8c cos 0c/V/2 has been taken out for brevity. 5. M. Gaillard and B. Lee, Phys.

Rev.

Lett.

33

(1974) 108. Fig.5 : Diagrams contributing to K ° (but not to K +)

decay. A11 are O ( N ) .

G. Altarelli and L. Maiani, Phys. Lett. 52B (1974) 351. 6. M. Bochicchio, L. Maiani, G. Martinelli, G. C. Rossi and M. Testa, Nucl. 331.

Phys.

B262 (1985)