Effects of subleading operators in the heavy quark effective theory

Volume 252, number 3

PHYSICS LETTERS B

20 December 1990

Effects of subleading operators in the heavy quark effective theory Michael E. Luke

1

Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA

Received 11 September 1990

The heavy quark effective theory is a powerful approach for calculating form factors for hadronic processes involving heavy (c and b) quarks. In this note, we investigate the leading order effects of the finite charm quark mass on semileptonic B decays and ec production. We find that the matrix elements of dimension-four operators suppressed by one power of AQcD/m¢ in the weak hadronic current may be parameterized by a single new non-perturbative function and one unknown dimensionful constant. Dimension-five operators in the effective lagrangian which break the spin-flavour SU (4) of the lowest order lagrangian introduce corrections of the same order of magnitude; these require the introduction of three more non-perturbative form factors. However, when the initial and final velocities of the heavy quark are equal, the AocD/mc corrections vanish and the absolutely normalized lowest order predictions remain valid.

1. In~oducfion The dynamics of mesons containing one heavy (b or c) quark simplify dramatically in the limit mq--. ov [ 15 ] when the heavy quark may be treated as a static colour source. In this limit, the spin o f the heavy quark decouples and Q C D w i t h f h e a v y flavours has an SU (2J) symmetry corresponding to rotations in spin and heavy flavour space. This symmetry allows one to derive a number o f useful relations between hadronic form factors for weak interactions; in particular, the myriad o f form factors required to parameterize b--,c decays are all determined in this limit by a single function o f the m o m e n t u m transfer, the "Isgur-Wise" function [ 3 ]. This idea has recently been cast in the framework o f a Lorentz covariant effective field theory by Georgi [6]. However, as in any effective theory, there are non-renormalizable interactions in the effective lagrangian which are suppressed by powers o f m o m e n t u m over the mass of the heavy particles in the theory. In addition, there are higher-dimensional operators in the weak hadronic current in the effective theory which arise from the matching conditions. The validity o f the heavy quark approximation for b--,c decays and ec production is therefore limited by the fact that the charm quark is not particularly heavy compared to the typical m o m e n t u m scale inside a meson, AQCD~ 300 GeV: the leading non-renormalizable terms are only suppressed by AQCD/mc ~ 0.2. This is the same order of magnitude as strong interaction corrections [ 7,8 ], and so should be taken into account [ 9,10 ]. However, because Q C D is a strongly interacting theory, the hadronic matrix elements of higher dimension operators in the weak current must be parameterized by additional nonperturbative form factors beyond the universal Isgur-Wise function. In addition, higher dimension operators in the effective lagrangian containing explicit powers o f the quark mass break the SU ( 2 f ) v symmetry which was the source of much o f the simplicity o f the heavy quark picture. These effects may too be parameterized by additional form factors, but we see that in general, beyond leading order much o f the predictive power of the heavy quark effective theory is lost. This paper is divided into two sections. In the first, after a brief review of the effective theory, we identify the leading higher dimension operators in the weak current and use the renormalization group to scale the current Email address: [email protected], [email protected] or [email protected] 0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )

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to the QCD scale. In the second, we will use the spin-flavour SU (4) transformation properties of the current and symmetry-breaking terms to parameterize the hadronic matrix elements of the weak current. This requires the introduction of four additional form factors which cannot be calculated in perturbation theory (although they are, in principle, calculable using lattice gauge theory). However, by demanding that the matrix elements (D(v) Ic7°cl D(v) ) and (D* (v, e) lee°cl D*(v, e) ) be properly normalized we find that the O(./IQCD/mc) corrections vanish when the velocities of the initial and final mesons are equal.

2. The weak current in the low energy theory

2.1. The effective lagrangian In the low energy effective theory heavy quarks moving with different velocities decouple: this is Georgi's "velocity superselection rule" [ 6 ]. The effective lagrangian

L#~rr= I"day i [ [ q ( v ) ~ u D U h q ( v ) .Iv 2v°

(2.1)

contains an infinite number of degrees of freedom, one heavy quark and antiquark field for each possible velocity. We may project out the particle and antiparticle states

htq+)(v)=P+hq=(~--~)hq.

(2.2)

In this work we will only be interested in heavy quark states, so we will drop the ( + ) superscript and simply denote an effective heavy quark field with flavour q and velocity v by hq(v), and the corresponding Dirac bispinor by uh, (v). The mass dependent piece of the heavy quark momentum has been removed from the effective quark field by a field redefinition,

hq( v , X) = exp ( imi (:vux u) q/(x ) ,

(2.3)

so that in momentum space,

0 uhq(v) = - ikuhq(v) .

(2,4)

Here, ku=p u - mqv ~ is the "residual" momentum of the quark which arises due to the coarse-graining of allowable heavy quark velocities. If the heavy quark is in a meson moving with a velocity vu, k measures the degree to which the quark is off-shell, which is typically of order AQCD. For fheavy quark flavours, the lagrangian (2.1) has an independent SU (2f)~ symmetry for each quark and antiquark field with a given velocity v, corresponding to rotations in spin and heavy flavour space. This symmetry is broken by subleading, non-renormalizable terms in L#~rfwhich are suppressed by powers of kU/mq; the leading terms are [ 9,10 ]: 1 0/3 ( '/'~ ) 5f'_ Ot,mq(l~)~q(V) (iv.D)Zhq(v) + az(/t)mq~q(V) (iD)Zhq(v),' ~g----~q t:iq(v)tr~'"Gu~hq(v) ,

(2.5)

where 3 ( ~ ( ~ ) , ~ - 8,25,

az(,U)=½, ot~(,u)=- ~1 \( ~O/S(/2 ] ) ~--9/25

(2.6)

oq(lt) = 1- ~ koqkmq)/

and Gu, is the gluon field strength. We will retain these terms only for the c quark, ignoring corrections of order

AQCD/mb. Furthermore, as discussed in ref. [9 ], we may use the equation of motion 448

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(D'v)hq(v) =O(D2/mq)

(2.7)

to show that matrix elements containing a single insertion h-q(v) (iv. D)2hq(v) vanish to this order, so we may neglect this operator.

2. 2. Matching conditions In the effective theory renormalized at a scale rob>/~> me, the c quark is treated as light while the b quark is heavy. The weak current responsible for b--. c decay is J~c(fi) =C(lt )e)~u( 1 + Ys )hb( V' ) +...

(2.8)

(we are using the convention ofref. [7 ] for the sign of Ys), where

( a s ( p ) ,]6/25 c(u) = \ ~ j

(2.9)

takes into account the strong interaction scaling of the current [ 1,3 ] and the omitted terms are suppressed by powers of as or DU/mb. The leading terms of the first type have been calculated [7,8 ] while the latter will be neglected compared to the DU/mc corrections to be discussed here. We match the weak current in the effective theory below the c scale to this by demanding that it be physically equivalent, that is, it gives the same matrix elements, at the matching scale. This requires the addition of higher dimension operators to the weak current below the c scale. It has the general form

J~(I.t) =Co(/t)h~ (v)TU( 1 +Ts)hb(v') + 1 G(U) (9~u(/t) +-.. mc

(2.10)

where the (9~' are dimension-four operators with the general from

(9,u ~h~(v)( -1D.I'i)Uhb(V ' ) ,

(2.11 )

and the I'i are some combination of 7 matrices. The matching condition

( c(p) [j~lhb(v' ) ) = (he(v)[J~lho(v'))

(2.12)

gives the relation

C(mc)ac(p)yU(1 +Ys)hb(V' )=C(m~)ahc(v) (1 + ~-~m~)yu(1 +ys)ho(v' ) =Co(mc)ahc(V)yU( l + ys)Uho(V, ) + __1 ci(mc)ahc(v)(k'Fi) UUhb(V'). me

(2.13)

In the first line, we have used the equation of motion

(l~-m¢)uc=rnc( ¢ - 1 + l~/mc)uc=O

(2.14)

to write Uc= ( ~ - ' ~ ) U ~ + ( ~ - ~ ) U c = ( 1 + ~ m c ) ( - ~ ) u c + O ( k Z / m ~ ) =

( 1 + 2~mc)uh,(v)+O(k2/mZc).

(2.15,

Defining the operators --

- i

--

¢~-h~(v)~,U(l+Ts)hb(v'), (~f=- -f~m hc(v)D~,U(l+ys)ho(v)'_

(2.16) 449

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gives the matching conditions Co(mc)=Cl(mc)=C(mc)

(2.17)

•

2. 3. Renormalization group

To take matrix elements of the current between hadronic states, the current should be renormalized at a typical hadronic scale, # ~AQcD. Invariance of the physics d

uUu (c,(u)~'(u) )=o

(2.18)

gives the renormalization group equation for the ci, d I.t-d-~ ci + yj~cj =O .

(2.19)

The anomalous dimension matrix Yu is straightforward to evaluate; the calculation is presented in detail in ref. [ 9 ] and will not be repeated here. The renormalization group scaling of the current is simplified drastically by applying the equation of motion (2.7); this allows us to ignore mixing of (.0oand (91with operators of the form ~ D. vFhb, as they have vanishing matrix elements when evaluated between physical states [ 11 ]. Furthermore, performing a parts integration of the interaction term e i ~ (v) ( - i [ ) u ) F h b ( V ' ) L u = - e i ~ ( v ) ( - iI) u) Fhb ( v' ) L u + iDUei° [ ~ (v) hb ( v ' ) L u ] + i ei°~ ( v ) hb ( v' ) DUL u + total divergence, (2.20)

(9- ( m b v - - m c v ' ) . x ,

where L u is the weak leptonic current, allows us to apply (2.7) to the b quark and make the replacement -i~

( v)[).v' I'hb( v' ) = A ( v' v - 1 ) ~ ( v)Fhb( v' ) ,

(2.21)

where mo - mc =mB -- mb --=-~~O (AQco) ,

(2.22)

due to the binding of the light quark in the meson. This is an effect of the lowest order lagrangian and so is the same for both heavy flavours; by the same token, to this order mo = mo.. The non-renormalizable symmetry breaking operators (2.5) introduce corrections of order A ~cD/mc which we have neglected. Note that the field redefinition (2.3) has removed the large momenta from the quark fields, and so they reappear as an explicit phase ei° in the weak hamiltonian. Since we are only interested in taking matrix elements of the weak hamiltonian, it is convenient to treat the substitution (2.21 ) as an operator identity. The mixing of (9owith an operator of this form may then be treated as an O (.~/mc) term in the anomalous dimensions of 60o.The solution to (2.19 ) is then

((Xs(]'/) ~8ai/27

c i ( l l ) = C ( m ¢ ) \ a- s ( -m ¢ ) }

,

i=0, 1

(2.23)

where we have defined ao = 1 - v ' v r ( v ' v ) +

and 450

.,i v ' v - r ( v ' v ) m~ v' v+ 1

,

al - 1 - v ' v r ( v ' v )

,

(2.24)

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1

r(v'v) = ~/(v,v)2 , 1 ln[v'v+x/(v'v)2--1 ] •

20 December 1990 (2.25)

The appropriate renormalization scale satisfies cz~(#) ~ 1.

2.4. Weak neutral current It is straightforward to extend the analysis of the previous two sections to the neutral current (Fc. We will only state the results here. The appropriate matching condition at Ft= me is

J~(mc)=h~(v)Fh~(v' ) - ~ = ~ ( v ) F ( l + ~ m"4¢ ( 1 - 0 )

h~(v)[~)F-l~]h~(v' ) )

h~(v')+ --~1 2m * ~ ( ' ) { - i l ~ ,/-3h~(v')

(2.26)

and the solution to the RGE is

d ) c;(u) 'J¢U(lt ) =C'o(lt )h~ ( v)F(1 + ~ (1-~) h¢(v')+ -T~m¢~(v)(-i~,/3hAo'),

(2.27)

where

/

. x \8a~/27

C'o(U)= \ ~ /

a'o= 1-v' vr(v' v) + 2

m~

(O~s(fl) )8a1/27 v'v-r(v'v) v' v+ l ' c; (u) = k ~ )

(2.28)

Once again, these are not operator statements and are only valid when taking appropriate matrix elements.

3. Hadronic matrix elements The matrix elements of the ~ ~ between hadronic states cannot be calculated in perturbation theory. However, the operators and states may be classified into irreducible multiplets of the SU (4) v® SU ( 4 ) v, spin-flavour symmetry of the two quark fields in the current; the group theory then restricts the number of independent form factors required to parameterize the matrix elements. The effects of symmetry breaking terms in the lagrangian (2.5) may also be taken into account by considering their transformation properties under the symmetry group. Following the notation of ref. [ 7 ], we associate with each meson state and current a matrix in the space of Dirac indices ® flavour:

,B(v' ) )--,B(v' )=( l +,' )/2® (~) , (D(v)J~[)(v)=(l+#)/2®(l hc(v)Vhb(V')-,P=r®(~

0),

(D*(v, ~)l-~I3*(v, e)=~,5~*(1+#)/2®(1 0 ) ,

~).

(3.1)

The matrix elements of (9~ need only be calculated to leading order, so we can neglect.symmetry breaking terms in L~ff. The most general form for the matrix elements of the operator G~'=-ih-i OaFh2 between two mesons M~ and MEthat is consistent with Lorentz invariance and the spin-flavour symmetry is (M~ (v)

lGUl ME(V' ) ) = ~

tr ({u (v,

v', #)M, (v)l~M2(v') ),

(3.2)

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where

~'(v, v', #) =~+ (v'v, #) (v~'+ v '~') +~_ (v'v, It) (v~'-v '~) + ~3(v' v, # ) ~

(3.3)

is the most general expression with the correct transformation properties. ~ is a kinematic factor which reflects the normalization of the states, so we take them to be the actual masses of the mesons, not the mass of the corresponding quark. The ~'s have the dimensions of mass and are expected to be of order AQCO. They cannot be calculated in perturbation theory. Their dependence on the renormalization point It is cancelled by that of the coefficients ci so that physical matrix elements are independent of It. The equation of motion (2.7) introduces the constraint

{3=-~+(v'v, it)(l+v'v)-{_(v'v, it)(1-v'v) ,

(3.4)

while (2.21) relates {_ (v'v, It) to the Isgur-Wise function {(v' v) =Co(It)~(v'v, It) of refs. [3,7 ],

(3.5)

& ( v ' v , It)=½~i¢o(V'V, It) ,

so that to leading order the matrix elements of (~1 are all determined by one unknown function, ~+ (v' v, It) and one dimensionful constant,/i, in addition to ~o(V'v, It). The matrix elements of (90which were calculated to leading order in refs. [3,7 ] will be modified by the presence of the symmetry breaking terms (2.5) in the effective lagrangian. In the notation of (3.1), they are proportional to the 8 × 8 matrices O,=(1/:

c 00), Q~"=( (l/mc)P+a~'VP+ 0

00)'

(3.6)

and "transform" as a 15~ 1 of SU(4)o. (An analogous set of matrices ~9'~ and O~'~ break the SU(4)~, of the other quark in the current. ) Once again, we will ignore the terms suppressed by 1/mb, although they are trivial to take into account using this notation. We simply replace the zeros in the lower right-hand corner of O1 and O~~ by 1/mb and ( 1/mb) [as(rn~)/as(mb) ] -9/25p+a/~PP+,respectively. The leading corrections to the matrix elements calculated by refs. [ 3,7 ] are linear in the Oi's. Only one nontrivial SU (4)~× SU (4)~, invariant can be constructed out of each Og, two mesons and the weak current. The corresponding reduced matrix element M a has two Dirac indices, corresponding to the light degrees of freedom in the mesons, which must be contracted to give a matrix element with the correct Lorentz properties. For terms containing O1 the only possibility is tr (M); however, for M ~ ---A~rl(v) O~ v/~t2 ( v' ) there are two non-vanishing possibilities: tr(v;,7~M ~'~) and tr(a~,dkP~). All other possibilities, such as tr(vu7~AP'~), vanish because P+ v,,a.""P+ ~P+ OW-v.")P+ =0. The most general form for the matrix elements of G=hl (v)Fh2(v') which satisfies (Ml (v)Ihl (v)Fh2(v')IM2(v')) = (M2(v')Ih2 (v')7°F*y°hl (v)IM1 ( v ) ) *

(3.7)

is therefore (M1 (v)I GI M2(v' ) ) = x / ~ m2 Co(it){~o(V' v, It) tr(~q (v)i~ff12(v')) +Zl (v' v, It) tr (3~1 (v) [O1/~+/~#'~ I~Q2(v ' ) ) +iz2(v'v, It) tr(v~M,(v)O~' ~ 2 ( v

)--vJ~l(V)~O'~2(v'

))

(3.8)

+)~3(v'v, It) tr (cr,~Ql (v) [ ~9~/~+ P O ~ ]A~¢2(v ' ) )}. For G=SFb, P@'~=i~@'~~ =0, so three of the terms in (3.8) vanish. This simply reflects the fact that we have ignored symmetry-breaking terms suppressed by 1~rob. For G=~Fc, these terms contribute. T invariance requires that the Z's all be real. 452

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Since Oi breaks the heavy flavour SU (2) but leaves unbroken the c quark spin SU (2), Xl (v'v, Iz) just gives a common contribution to D and D* final states of the same form as the leading order term, while the spin symmetry violating effects are parameterized by X2( v' v,/z) and Z3( v' v,/z). Applying this to the vector and axial vector pieces of the weak current (2.10) J~c = Vbc u +Abe u gives
:

Co, ,

l [X~(v'v,l~)-2(v'v-1)X2(v'v, ll)+6X3(v'v,l~) ]) [vU+v'u] m----~

1

+ - - [(v'v+l)~÷(v'v,~)-½(v'v-2).~o(v'v,~)] [vJ'-v'Ul mc

]

(3.9) ~

< D*(v, ~)1V~c IB(v') )

=--i~/mD*m'c°(lt)[(l+2-~c)~°(v'v'It)+

l---[X'(v'v'lt)--2Z3(v'v'lt)]]

(3.10)

and

u , ))=~/mD.mBco(lt) {[( (O*(v,e)lAb¢lB(v

1+

(v'v--I)A.~ ~,o(V'V,lt)+ 1 [Xl(V'V,lt)-2X3(v'v, lt)] 1 e*u 2me -~

mc

_ 1_~[2X2(v'v, I1) +~+ (v'v, It) - ½.,'r~o(v'v,I~) ]e*v'v'u~. mc J

(3.11 )

For the neutral current JS~ the result is =mDC;(/.t)

(

l ~o(V'V,It)+ mcc [2Xl(v'v, lt)-4(v'v-l)x2(v' V,a)+I2x3(V'V,a)]) (vU+v'U),


,

mc

(3.12)

(v'v-3)A) ~(v'v'#)2mc

[2XI(v'v,~)--2(v'v-I)z2(v'v,~)+4Z3(v'v,~)+(I+v'v)~+(v'v,~)]

] .up~P.*~,,,

(3.13)

and

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=m~Dm~.C'o(~)f[(l_

20 December 1990

( r / v - 3 ) ( v ' v - 1 ) A ) ~ ( v ' v , li) 2me

q + 1- - {2X~(v'v, lt) - 2 ( v ' v - 1)Xz(v' v, t~) +4X3(v'v, P) + [ (v'v) 2 - 1 ]~+ (v'v, lz)}/~*" me A _[(l

(V'V-2)A-~ 2m~ ]~o(V'V, li)

,

]

+ - - [2X~(v'v, IJ)-2v'vx2(v'v, lx)+4)C3(v'v,#)+ (v'v+2)~+(v'v, lz)] e*v'v ~ me

_ lmc[2X2(v'v'It)+{+(v'v'#)-½X{°(v'v'lt)IE*v'v'~'} "

(3.14)

In all of these expressions, we have used the fact c~(/z) = Co(g) + O (AQcD/mc), allowing us to take c~(/t) =Co(g) to this order. We have distinguished between mD and roD., although strictly speaking this is an O(AocD/mc) 2 effect, as mentioned earlier. For arbitrary v and v', not much remains of the elegance of the heavy quark picture. We have incalculable corrections of order AQCD/mc to all the leading order relations. For example, there is a contribution to the amplitude for the axial current to produce a B meson in a D-wave in the rest frame of the D* meson, which vanished to lowest order. This is proportional to the sum of the form factors a . and a_ in the notation of ref. [ 3 ], and is given by a_________~ a+ + =2--me (2X2(v'v,l.t)+~+(v'V, a+

mb

la)-½A~(v v,#) J mcF.o(V'V,lt) "

~,

(3.15)

This is of the same order of magnitude as the leading strong interaction contributions [7,8]. When v=v', all O(AqcD/mc) effects except those proportional to Z~ (v'v, #) and Z3(v'v, #) vanish in ( 3 . 9 ) (3.14). Furthermore, we can fix these by requiring that the vector current be normalized correctly, since at v = v' it is a symmetry current corresponding to c quark number. With our normalization, this gives ( D ( v ) ICT°clD(v)

) = 2mDVO=2mD v° (1 + _~1 [2z, (1,/~) + 1223(1,/~) 1~, / \ me

(D*(v, ,)leY°cl D*(v, e) )

=2mD.V°=2mD.V ° (1 + 1 [2X~(1,/t) -4Z3(1, # ) ] ~ , \ me

(3.16)

/

leading to the requirement X~(1,p) = X s ( 1 , p ) = 0 •

(3.17)

This means that at the symmetry point v= v' there are no O(AQcD/mc) corrections to the lowest order result beyond the trivial effect of using the meson masses instead of the corresponding heavy quark mass in the normalization of the states:

=2C(m¢)x~oms v ~',
e *~', (3.18)

This situation corresponds to the maximum momentum being transferred to the leptons. The absolutely normalized predictions which are valid in the heavy quark limit survive to first order in AQCD/m¢. 454

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Acknowledgement I a m grateful to H. Georgi a n d A. F a l k for useful discussions, a n d to B. H o l d o m a n d the Physics D e p a r t m e n t o f the U n i v e r s i t y o f T o r o n t o for their h o s p i t a l i t y while part o f this w o r k was completed. T h i s w o r k was supp o r t e d in part b y the N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t # P H Y - 8 7 1 4 6 5 4 a n d by a N a t u r a l Sciences a n d E n g i n e e r i n g R e s e a r c h C o u n c i l o f C a n a d a p o s t g r a d u a t e fellowship.

References [ 1 ] H.D. Politzer and M.B. Wise, Phys. Lett. B 206 ( 1988 ) 681; B 208 ( 1988 ) 504. [2] M.B. Voloshin and M.A. Shifman, Yad Fiz. 45 (1987) 463 [Sov. J. Nucl. Phys. 45 (1987) 292]; Soy. J. Nucl. Phys. 47 (1988) 511. [3] N. Isgur and M.B. Wise, Phys. Lett. B 232 (1989) 113; B 237 (1990) 527. [4] E. Eichten and B. Hill, Phys. Lett. B 234 (1990) 511. [5] B. Grinstein, Nucl. Phys. B 339 (1990) 253. [6] H. Georgi, Phys. Lett. B 240 (1990) 447. [7] A.F. Falk, H. Georgi, B. Grinstein and M.B. Wise, Nucl. Phys. B 343 (1990) 1. [8] A.F. Falk and B. Grinstein, Phys. Lett. B 247 (1990) 406; Harvard and Valencia preprint HUTP-90/A042, FTUV-90/32, IFIC/ 90-28. [9] A.F. Falk, B. Grinstein and M. Luke, Harvard preprint HUTP-90/A044. [ 10] E. Eichten and B. Hill, Fermilab preprint FERMILAB-PUB-90/54T. [ 11 ] H.D. Politzer, Nucl. Phys. B 172 (1980) 349.

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