a r X i v :h e p -p h /9212266v 1 16 D e c 1992
SLAC–PUB–6017WIS–92/99/Dec–PH
年月日英文
December 1992
T/E
The Subleading Isgur-Wi Form Factor χ3(v ·v ′)
to Order αs in QCD Sum Rules
Matthias Neubert
Stanford Linear Accelerator Center
Stanford University,Stanford,California 94309
Zoltan Ligeti and Yof Nir
Weizmann Institute of Science
Physics Department,Rehovot 76100,Israel
We calculate the contributions arising at order αs in the QCD sum rule for the spin-symmetry violating universal function χ3(v ·v ′),which appears at order 1/m Q in the heavy quark expansion of meson form factors.In particular,we derive the two-loop perturbative contribution to the sum rule.Over the kinematic range accessible in B →D (∗)ℓνdecays,we find that χ3(v ·v ′)does not exceed the level of ∼1%,indicating that power corrections induced by the chromo-magnetic operator in the heavy quark expansion are small.
(submitted to Physical Review D)
I.INTRODUCTION瓶子的英文
In the heavy quark effective theory(HQET),the hadronic matrix elements describing the mileptonic decays M(v)→M′(v′)ℓν,where M and M′are pudoscalar or vector mesons containing a heavy quark,can be systematically expanded in inver powers of the heavy quark mass[1–5].The coefficients in this expansion are m Q-independent,universal functions of the kinematic variable y=v·v′.The so-called Isgur-Wi form factors characterize the properties of the cloud of light quarks and gluons surrounding the heavy quarks,which act as static color sources.At leading order,a
2012年12月英语六级听力
single functionξ(y)suffices to parameterize all matrix elements[6].This is expresd in the compact trace formula[5,7] M′(v′)|J(0)|M(v) =−ξ(y)tr{
国家公务员用书
(2)
m M P+ −γ5;pudoscalar meson
/ǫ;vector meson
is a spin wave function that describes correctly the transformation properties(under boosts and heavy quark spin rotations)of the meson states in the effective theory.P+=1
g s
2m Q O mag,O mag=
M′(v′)ΓP+iσαβM(v) .(4)
The mass parameter¯Λts the canonical scale for power corrections in HQET.In the m Q→∞limit,it measures thefinite mass difference between a heavy meson and the heavy quark that it contains[11].By factoring out this parameter,χαβ(v,v′)becomes dimensionless.The most general decomposition of this form factor involves two real,scalar functionsχ2(y)andχ3(y)defined by[10]
χαβ(v,v′)=(v′αγβ−v′βγα)χ2(y)−2iσαβχ3(y).(5)
Irrespective of the structure of the current J ,the form factor χ3(y )appears always in the following combination with ξ(y ):
ξ(y )+2Z ¯Λ
d M m Q ′
χ3(y ),(6)
where d P =3for a pudoscalar and d V =−1for a vector meson.It thus effectively
renormalizes the leading Isgur-Wi function,prerving its normalization at y =1since χ3(1)=0according to Luke’s theorem [10].Eq.(6)shows that knowledge of χ3(y )is needed if one wants to relate process which are connected by the spin symmetry,such as B →D ℓνand B →D ∗ℓν.
Being hadronic form factors,the universal functions in HQET can only be investigated using nonperturbative methods.QCD sum rules have become very popular for this purpo.They have been reformulated in the context of the effective theory and have been applied to the study of meson decay constants and the Isgur-Wi functions both in leading and next-to-leading order in the 1/m Q expansion [12–21].In particular,it has been shown that very simple predictions for the spin-symmetry violating form factors are obtained when terms of order αs are neglected,namely [17]
代沟英文χ2(y )=0,χ3(y )∝ ¯q g s σαβG αβq [1−ξ(y )].
(7)
In this approach χ3(y )is proportional to the mixed quark-gluon condensate,and it was
estimated that χ3(y )∼1%for large recoil (y ∼1.5).In a recent work we have refined the prediction for χ2(y )by including contributions of order αs in the sum rule analysis [20].We found that the are as important as the contribution of the mixed condensate in (7).It is,therefore,worthwhile to include such effects also in the analysis of χ3(y ).This is the purpo of this article.
II.DERIV ATION OF THE SUM RULE
The QCD sum rule analysis of the functions χ2(y )and χ3(y )is very similar.We shall,therefore,only briefly sketch the general procedure and refer for details to Refs.[17,20].Our starting point is the correlator
d x d x ′
d z e
英语夏令营
i (k ′·x ′−k ·x )
0|T
[¯
q ΓM ′P ′
+
ΓP +iσαβP +ΓM
+Ξ3(ω,ω′,y )tr
2σαβ
2
(1+/v ′),and we omit the velocity labels in h and h ′for simplicity.The heavy-light currents interpolate pudoscalar or vector mesons,depending on the choice ΓM =−γ5
or ΓM =γµ−v µ,respectively.The external momenta k and k ′in (8)are the “residual”off-shell momenta of the heavy quarks.Due to the pha redefinition of the effective heavy quark fields in HQET,they are related to the total momenta P and P ′by k =P −m Q v and k ′=P ′−m Q ′v ′[3].
The coefficient functions Ξi are analytic in ω=2v ·k and ω′=2v ′·k ′,with discontinuities for positive val
ues of the variables.They can be saturated by intermediate states which couple to the heavy-light currents.In particular,there is a double-pole contribution from the ground-state mesons M and M ′.To leading order in the 1/m Q expansion the pole position
is at ω=ω′=2¯Λ.
In the ca of Ξ2,the residue of the pole is proportional to the universal function χ2(y ).For Ξ3the situation is more complicated,however,since inrtions of the chromo-magnetic operator not only renormalize the leading Isgur-Wi function,but also the coupling of the heavy mesons to the interpolating heavy-light currents (i.e.,the meson decay constants)and the physical meson mass,which define the position of the pole.一对一辅导机构排名
1The correct expression for the pole contribution to Ξ3is [17]
Ξpole 3(ω,ω′
,y )
=
F 2
(ω−2¯Λ
+iǫ)
.(9)
幻觉 英文Here F is the analog of the meson decay constant in the effective theory (F ∼f M
√
m Q
δΛ2+...
aeroplanes
,
0|j (0)|M (v ) =
iF
2
G 2tr
2σαβ
ΓP +σαβM (v )
,
where the ellips reprent spin-symmetry conrving or higher order power corrections,and j =¯q Γh (v ).In terms of the vector–pudoscalar mass splitting,the parameter δΛ2is
given by m 2V −m 2P =−8¯ΛδΛ2.
For not too small,negative values of ωand ω′,the coefficient function Ξ3can be approx-imated as a perturbative ries in αs ,supplemented by the leading power corrections in 1/ωand 1/ω′,which are proportional to vacuum expectation values of local quark-gluon opera-tors,the so-called condensates [22].This is how nonperturbative corrections are incorporated in this approach.The idea of QCD sum rules is to match this theoretical reprentation of Ξ3to the phenomenological pole contribution given in (9).To this end,one first writes the theoretical expression in terms of a double dispersion integral,
Ξth 3(ω,ω′
,y )
=
d νd ν
′
ρth 3(ν,ν′
,y )
1There
are no such additional terms for Ξ2becau of the peculiar trace structure associated with
this coefficient function.
possible subtraction terms.Becau of theflavor symmetry it is natural to t the Borel parameters associated withωandω′equal:τ=τ′=2T.One then introduces new variables ω±=1
2T ξ(y) F2e−2¯Λ/T=ω0
dω+e−ω+/T ρth3(ω+,y)≡K(T,ω0,y).(12)
The effective spectral density ρth3aris after integration of the double spectral density over ω−.Note that for each contribution to it the dependence onω+is known on dimensional
grounds.It thus suffices to calculate directly the Borel transform of the individual con-tributions toΞth3,corresponding to the limitω0→∞in(12).Theω0-dependence can be recovered at the end of the calculation.
When terms of orderαs are neglected,contributions to the sum rule forΞ3can only be proportional to condensates involving the gluonfield,since there is no way to contract the gluon contained in O mag.The leading power correction of this type is reprented by the diagram shown in Fig.1(d).It is proportional to the mixed quark-gluon condensate and,as shown in Ref.[17],leads to(7).Here we are interested in the additional contributions arising at orderαs.They are shown in Fig.1(a)-(c).Besides a two-loop perturbative contribution, one encounters further nonperturbative corrections proportional to the quark and the gluon condensate.
Let usfirst prent the result for the nonperturbative power corrections.Wefind
K cond(T,ω0,y)=αs ¯q q T
T + αs GG y+1
sprinting的意思
− ¯q g sσαβGαβq
√y2−1),
δn(x)=
1
(4π)D
×
1
dλλ1−D
∞
λ
d u1
∞
1/λ
d u2
(u1u2−1)D/2−2
where C F=(N2c−1)/2N c,and D is the dimension of space-time.For D=4,the integrand diverges asλ→0.To regulate the integral,we assume D<2and u a triple integration by parts inλto obtain an expression which can be analytically continued to the vicinity of D=4.Next we t D=4+2ǫ,expand inǫ,write the result as an integral overω+,and introduce back the continuum threshold.This gives
K pert(T,ω0,y)=−αs
y+1 2
ω0
dω+ω3+e−ω+/T(16)
× 12−23
∂µ+
3αs
9π
¯Λ,(17)
which shows that divergences ari at orderαs.At this order,the renormalization of the sum rule is thus accomplished by a renormalization of the“bare”parameter G2in(12).In the
9π
¯Λ 1µ2 +O(g3s).(18)
Hence a counterterm proportional to¯Λξ(y)has to be added to the bracket on the left-hand side of the sum rule(12).To evaluate its effect on the right-hand side,we note that in D dimensions[17]
¯Λξ(y)F2e−2¯Λ/T=3
y+1 2
ω0
dω+ω3+e−ω+/T(19)
× 1+ǫ γE−ln4π+2lnω+−ln y+1
2T ξ(y) F2e−2¯Λ/T
=αs
y+1 2
ω0
dω+ω3+e−ω+/T 2lnµ6+ y r(y)−1+ln y+1
