fulltext

NOTE BREVI

The effective weak nonleptonic Hamiltonian under both QCD

and one-loop QED corrections

H. WHITE

Department of Mathematics, University of the West Indies - Mona, Jamaica (ricevuto il 14 Luglio 1997; approvato il 22 Settembre 1997)

Summary. — We obtain the effective Hamiltonian for weak nonleptonic decays

under a) QCD leading logarithmic corrections to all orders, b) QCD two-loop nonleading corrections, c) one-loop QED corrections, as in BURASA. J., JAMINM., LAUTENBACHERM. and WEISZP., Nucl. Phys. B, 400 (1993) 75.

PACS 12.28.Cy – Summation of perturbation theory.

PACS 13.40.Ks – Electromagnetic corrections to strong- and weak-interaction pro-cesses.

In ref. [1], the two-loop anomalous dimension matrix for operators relevant to electroweak nonleptonic processes was calculated, and here we proceed to obtain a suitable expression for the effective Hamiltonian for such processes.

We first study the relevant b-functions for the QCD and QED vertices [2]. We define a * 4g2

O4 p , ag4 e2O4 p , where g and e are the renormalized QCD and QED coupling constants, respectively. From [2], the b-functions are given by

da * dt 4 b(a * , ag) , dag dt 4 b g_{(a * , a} g) , where

.

/

´

b * (a * , ag) 4b(a*)2ag]bcga * 1b1cg(a * )2( , bg_{(a * , a} g) 42ag]bgca * 1b gc 2 (a * )2( , b(a) 4b(a, 0) 42ba22 b1a3

(1)

and t 4 ln( p2

O 2 m2).

Here we have omitted all terms in ag2 since we only consider one-loop QED

corrections. As a result we can take for a * in bg_{(a * , a}

g) the pure QCD coupling

constant, i.e. the solution of the equation da

dt 4 b(a , 0 ) . (2)

The equation dagOdt 4 bg(a , ag) thus possesses the solution

ag4 ag0exp

y

2



a0 a ]bgca 81b2gc(a 8) 2 ( da 8 b(a 8)

z

, (3)

where a and ag take the values a0 and ag0, respectively, when t 40. The term within the integral sign is

bgc ba 1

g

b2gc b 2 bgc_b 1 b2

h

1 O(a) . (4)

Thus the solution (3) reduces to

ag4 a0gLaA[ 1 1B(a2a0) ] , (5)

where L(a) 4a0Oa , A 4 bObgc, B 4

(

b2

gc

Ob 2 (bgcb1) Ob2

)

and the expression in square brackets is of order a and a0.

This is the expression for the running QED coupling constant in the presence of QCD corrections alone.

We now consider the solution of da * Odt4b(a*, ag) in (1), and express a * in the form

a * 4a1ag0H(a) , (6)

where a is the solution to (2). The equation for a * gives da dt 1 ag 0 dH dt 4 b(a * , ag) 4b(a*)1agb I (a * ) 4

4 b(a) 1 b 8 (a) a0gH(a) 1agbI(a) 1O(ag0 2) , where from (1) bI (a)4 2(bcga 1b1cga2) . (7) Thus dH dt 2 b 8 (a) H 4 ag a0g bI_(a) (8) or b(a) dH da 2 b 8 (a) H 4 ag ag0 bI_{(a) .}

This equation can be reduced to the form d da

y

H(a) b(a)

z

4 ag ag0 bI_(a) b2_(a) , which possesses the solution

H(a) b(a) 4 H(a0) b(a0) 1



a0 a a 8g a0g bI(a 8) da8 b2 (a 8) . (9)

We assume here that when t 40, a*04 a0, i.e. H(a0) 40. Thus H(a) 4b(a)



a0 a a 8g a0g bI (a 8) da8 b2 (a 8) . (10)

Using the expressions for b(a) and bI_{(a) given in (1) and (7) we note that}

bI(a) b2_(a) 4 2(bcga 1b1cga2) (ba2 1 b1a3)2 4 2b cg b2_a3

k

1 1

g

b1cg bcg 2 2 b1 b

h

a

l

. (11)

Using the expression for agin (5), (10) reduces to

H(a) 4b(a) I(a) ,

where

.

/

´

I(a) 4

g

2b cg b2

h

y

1 2Ka0 (A 12) a2 0 ] 1 2 LA 12(a)(1 K]12L A 11_(a)( (A 11) a0

z

, K 4b1cgObcg2 2 b1Ob . (12)

We now consider the expression for the effective Hamiltonian Hefffor the processes discussed in ref. [1] in terms of the local operators listed there. If these operators are denoted by Op, then Heff4

!

p cp(t , a0, a 0 g) Op4 c(t , a0, a0g)lO , where c(t , a0, a0g) 4c[0, a , ag] P exp

y

2



0 t G(a * , ag) dt

z

(13)

and t 4 ln(Mw2Om3), t 4 ln( p2O 2 m2), a , ag are the values of a , ag when t 4t, i.e.

p2

4 2 m2, Mwis the mass of the weak bosons which generate the four-quark operators of ref. [1].

G(a * , ag) is the anomalous dimension matrix for the operators. It admits the following expansion to orders a *g and a2:

The QCD one-loop term a

*B takes into consideration that the QCD coupling constant a * in (1) is not the same for the four quarks. As a result it is of the form

a_{*B 4aB1a}g0H(a) B * , where B * differs from B.

We express G(a * , ag) in the form 2aB1R, where

R 42a0gH(a) B * 2agBg1 a2Gcc21 aagGcg2 (14)

Pexpin (13) denotes the path-ordered exponential. The expression in square brackets in eq. (13) is



a0 a G(a * , ag) b(a) da . (15)

Using (14) and the approximation 1 b(a) 4 21 ba2 1 b1a3 4 2 1 ba2

m

1 2 b1 b a

n

.

The integral in (15) reduces to

P exp

y



a0 a

k

2 1 baB 1R A

l

_da

z

_, (16) where RA_{4 2a}0gI(a) B * 1 ag ba2B g 1 1 b

g

G c 21 b1 b B

h

1 ag ba

g

G cg 2 1 b1 b B g

h

(15) now becomes P exp

y



a0 a

k

2 1 ba B 1R A

l

_{da 8}

z

4 G(a) P exp

y



a0 a G21_{(a) R}AG(a) da

z

_, (17) where G_{(a) 4exp}

y



a0 a

k

2 1 ba B

l

da

z

4 [L(a) ] b21_B . We let the (distinct) eigenvalues of b21_{B be d}

i so that b21B can be expressed in the form

b21_{B 4}

!

_d

iPi, where Piare projection matrix operators satisfying

Pi24 Pi, PiPj4 0 (i c j) ,

!

i

Pi4 I . (18)

Hence G(a) in (17) is given by

G_{(a) 4 [L(a) ]}b21_B 4

!

i Ldi_{(a) P} i (19)

and G21_{(a) 4}

!

i L2di_{(a) P} i. Using (19), (17) becomes

!

i L2di_{(a) P} i1

!

i Ldi_(a)



a0 a L2di1 dk_{(a) P} iRAPkda . (20)

Using (16), the second term in (20) can be expressed in the form

!

i PiSikPk, where Sik4 T g ikBg1 TikcB * 1Tikcc

g

Gcc21 bi b B

h

1 T cg ik

g

Gcg1 b1 b B g

h

(21)

and the coefficients Tikg, T

c

iketc. can be expressed in terms of one single integral

Iik (A) 4Ldi_(a)



a0 a L2di1 dk1 A_{(a) da 4} aL dk1 A (a) 2a0Ldi(a) di2 dk2 A 1 1 . (22)

The expressions for Tikg, T

c ik etc. are

.

`

`

`

`

`

/

`

`

`

`

`

´

Tikc4 ag 0 Ldi_(a)



a0 a L2di1 dk_{(a) I(a) da 4} 4

g

2 ag 0_bcg b2

h

y

12Ka0 (A12) a20 ]I( 0 )ik2I(A12)ik (1 K (A11) a0 ]Iik( 0 )2Iik (A11)(

z

, Tikg4 1 bL di_(a)



a0 a L2di1 dk_(a) ag a2da 4 4 ag 0 b

k

1 a2 0 Iik (A 12)(12Ba0) 1 1 a0 BIik (A 11)

l

, Tcg ik4 1 bL di_(a)



a0 a L2di1dk_(a) ag a da4 ag0 b

k

1 a0 Iik

(A12)(12Ba0)1BIik(A)

l

, Tcc ik4 1 bL di_(a)



a0 a L2di1 dk_{(a) da 4} 1 bI ik_{( 0 ) .} (23)

In the calculation of the one-loop diagrams, a * is not the same for all four quarks p, n, c, l . We denote by g₁the ppAmand ccAm coupling constant and by g2the nnAm and

Fig. 1.

The graphs in fig. 1 represent one-loop QCD corrections to four-quark operators. Graphs a), b) and c) involve the factors g2

1, g1g2and g 2

2, respectively. From (1), we can express a *_{64 ( 1 O4 p) g}2

6in the form

a_{6* 4a1l6}ag0H(a) and a normalization can be fixed for H(a) such that

l24 1 for quark charge 2 1 O3 , l_{14 4 for quark change 2 O3 .}

Thus g1g24

k

( 4 p) 2 a1a2C a 1 1 2(l11 l2) ag 0 H(a) 4a1 5 2ag 0 H(a) . (24)

Thus the QCD one loop matrix is of the form

aB 1a0gH(a) B * . (25)

Similarly the coefficient c( 0 , a *₆, ag) can be calculated from the relevant one-loop diagrams and expressed in the form

c0

1 aci1 a0gH(a) c * 1agcg4 c01 d(a , ag) . (26)

Discarding terms of order a2

0LM(a), a0a(LM(a) etc., the effective Hamiltonian of (13) is Heff4 [

!

L di_{(a) c}0 lPi1

!

L di_{(a) d(a , a} g)lPi1

!

i , k c_l0_P iSikPk]lO . (27)

The terms independent of a0

g in (27) yield the effective Hamiltonian of ref. [3], (2.18), (6.19) and (6.20). The remaining terms in a0g yield the effective Hamiltonian for the processes discussed in ref. [1].

R E F E R E N C E S

[1] BURASA. J., JAMINM., LAUTENBACHERM. and WEISZP., Nucl. Phys. B, 400 (1993) 75. [2] WHITEH., Nuovo Cimento A, 108 (1995) 643.