Lattice renormalization of the static quark derivative operator

Physics Letters B 632 (2006) 319–325 www.elsevier.com/locate/physletb

Lattice renormalization of the static quark derivative operator B. Blossier a,∗ , A. Le Yaouanc a , V. Morénas b , O. Pène a a Laboratoire de Physique Théorique (Bât.210), Université Paris XI-Sud, Centre d’Orsay, 91405 Orsay cedex, France b Laboratoire de Physique Corpusculaire, Université Blaise Pascal, CNRS/IN2P3, F-63000 Aubière cedex, France

Received 3 August 2005; received in revised form 7 October 2005; accepted 12 October 2005 Available online 24 October 2005 Editor: G.F. Giudice

Abstract We give the analytical expressions and the numerical values of radiative corrections to the covariant derivative operator on the static quark line, used for the lattice calculation of the Isgur–Wise form factors τ1/2 (1) and τ3/2 (1). These corrections induce an enhancement of renormalized quantities if an hypercubic blocking procedure is used for the Wilson line, while there is a reduction without such a procedure.  2005 Elsevier B.V. All rights reserved. PACS: 12.38.Gc; 12.39.Hg; 13.20.He

1. Introduction In a previous paper [1] we proposed a method to compute on the lattice, in the static limit of HQET, the Isgur–Wise form factors τ1/2 (1) and τ3/2 (1) which parameterize decays of B mesons into orbitally excited (P wave) D ∗∗ charmed mesons. Keep in mind that the zero recoil is the only definite limit of HQET on the lattice, because the Euclidean effective theory with a nonvanishing spatial momentum of the heavy quark is not lower bounded. Then it reveals impossible to calculate directly the τj ’s from the currents because the matrix elements vanish at zero recoil. To compute τ1/2 (1) and τ3/2 (1), we proposed to evaluate on the lattice the matrix ∗ ¯ ¯ elements H0∗ (v)|h(v)γ i γ5 Dj h(v)|H (v) and H2 (v)|h(v)γ i γ5 Dj h(v)|H (v), using the following equalities:     ¯ H0∗ (v)h(v)γ i γ5 Dj h(v) H (v) = igij (MH0∗ − MH )τ 1 (1),

(1)

√    ∗  ¯ H2∗ (v)h(v)γ i γ5 Dj h(v) H (v) = −i 3(MH2∗ − MH )τ 3 (1)ij ,

(2)





2

2

where Di is the covariant derivative (Di = ∂i + igAi ), MH , MH0∗ and MH2∗ are the mass of the 0− , 0+ and 2+ states respectively, and ij∗ is the polarization tensor. These relations are defined between renormalized quantities. Then we have to renormalize the matrix element of the derivative operator computed on the lattice. We explained that power and logarithmic divergences are not to be feared in the zero recoil limit. However finite renormalization is present and we want to establish the one-loop contributions to the derivative operator with the hypercubic blocking [2] of the Wilson line. We have to renormalize and to match onto the continuum the bare operator OijB = h¯ B γi γ 5 Dj hB , where hB is the bare heavy quark field. We choose the MOM scheme whose renormalization conditions are: (1) the renormalized heavy quark propagator is equal to the free one, and (2) the renormalized vertex function taken between renormalized external legs is the tree level one.

* Corresponding author.

E-mail address: [email protected] (B. Blossier). 0370-2693/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.10.026

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B. Blossier et al. / Physics Letters B 632 (2006) 319–325

The r.h.s. of Eqs. (1) and (2) are independent of the renormalization scale µ. Indeed, on the one hand, MH0∗ − MH ≡ Λ¯ 0+ − Λ¯ 0− and MH2∗ − MH ≡ Λ¯ 2+ − Λ¯ 0− are physical quantities. On the other hand, √ ¯ µ γ 5 b|B D0 |cγ ≡ g + (v + v  )µ + g − (v − v  )µ ≡ −τ1/2 (µ, w) w − 1F µ , √ mB mD0 √ √   µ F = w + 1 C15 (µ, w)a µ + w − 1 C25 (µ, w)v µ + C35 (µ, w)v  µ , √ where w 2 − 1 a µ = (v − v  )µ , and the Ci5 ’s are the matching coefficients between the QCD operator cγ ¯ µ γ 5 b and the HQET µ 5 µ 5  µ 5 operators c¯v  γ γ bv , c¯v  v γ bv and c¯v  v γ bv , respectively, √ D0 |cγ ¯ µ γ 5 b|B = −τ1/2 (µ, w)F µ −→ −τ1/2 (µ, 1) 2C15 (µ, 1)a µ . √ √ w→1 w − 1 mB mD0

(3)

C15 (µ, 1) ≡ C15 (1) [3] (C15 (1) ≡ ηA = 0.986 ± 0.005 [4]) and the l.h.s. of (3) is also independent of µ: thus τ1/2 (µ, 1) ≡ τ1/2 (1). We can use the same argument to prove the scale independence of τ3/2 (µ, 1). Consequently the scale µ will be omitted in the following. It is well known that the heavy quark self-energy diverges linearly in 1/a [5], so we introduce a mass counterterm δm to cancel this divergence. Numerically it is canceled nonperturbatively in the ratio between the three-point function and the two-point functions to obtain a matrix element, or in the difference between binding energies of heavy-light mesons. The bare heavy propagator on the lattice is  −a[δm + Σ(p)] n a a ≡ Z2h S R (p). = S B (p) = (4) 1 − e−ip4 a + aδm + aΣ(p) 1 − e−ip4 a n 1 − e−ip4 a By choosing the renormalization conditions R −1  S (p)ip →0 = ip4 , δm = −Σ(p4 = 0), 4

the constant Z2h is then

 dΣ  Z2h = 1 − . d(ip4 ) ip4 →0

The bare vertex function VijB (p) is defined as 
−1  −1 (p) eip·(x−y) hB (x)OijB (0)h¯ B (y) S B (p) VijB (p) = S B x,y


 −1 ZD R −1 S = (p) eip·(x−y) hR (x)OijR (0)h¯ R (y) S R (p), Z2h x,y

(5)

where OijB (0) = ZD OijR (0). We will see below that VijB (p) can be written as 5 R ¯ VijB (p) = (1 + δV )u(p)γ i γ pj u(p) ≡ (1 + δV )Vij (p).

(6)

δV is given by all the 1PI one-loop diagrams containing the vertex. −1 We obtain H ∗∗ |OijR |H  = ZD H ∗∗ |OijB |H  where ZD = Z2h (1 + δV ) and H ∗∗ |OijB |H  was computed on the lattice. This Letter is organized as follows: in Section 2 we recall the action we use for the heavy quark, we give the corresponding Feynman rules and we clarify our notations; in Section 3 we give the analytical expression for the heavy quark self-energy, in Section 4 we give the analytical expression for radiative corrections to the derivative operator in lattice HQET. We briefly conclude in Section 5.

2. Heavy quark action and Feynman rules The lattice HQET action for the static heavy quark is    ˆ ˆ + aδmh† (n)h(n) , h† (n) h(n) − U †,HYP (n − 4)h(n − 4) S HQET = a 3 4

n

(7)

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where U4HYP (n) is a link built from an hypercubic blocking. We will use in the rest of the Letter the following notations taken from [6–8]:

π/a

≡ p

−π/a

≡ −π/a

p

h(n) =

π/a

d 4p , (2π)4

d 3p , (2π)3

a

4



π

e

ipn

= δ(p),

≡

n

−π

k

d 4k , (2π)4

π

≡ k

−π

d 3k , (2π)3

eipn h(p), p a

a



a 2 g02 a Aµ (n)Abµ (n)T a T b + O g03 , 2!

a2g2 a a a = 1 + iag0 Bµ (n)T − Bµ (n)Bµb (n)T a T b + O g03 , 2!

Uµ (n) = eiag0 Aµ (n)T = 1 + iag0 Aaµ (n)T a − a

a

UµHYP (n) = eiag0 Bµ (n)T

a a Aµ (n) = eip(n+ 2 ) Aaµ (p),

a

Bµa (n) =

p

  a(p + p  )µ , cµ = cos 2    kλ W =2 sin2 . 2



Γλ = sin akλ ,

eip(n+ 2 ) Bµa (p), p

 a(p + p  )µ sµ = sin , 2

  kµ Mµ = cos , 2



 kµ Nµ = sin , 2

λ

In the Fourier space, the action is given at the order of O(g02 ) by



 a δ(q + p  − p)h† (p)B4a (q)T a h(p  )e−i(p4 +p4 ) 2 S HQET = a −1 h† (p) 1 − e−ip4 a h(p) + δmh† (p)h(p) + ig0 p

+



ag02 2!

p

The block gauge fields Bµ =

∞ 

p

Bµa

q

p p q  a

δ(q + r + p  − p)h† (p)B4a (q)B4b (r)T a T b h(p  )e−i(p4 +p4 ) 2 .

(8)

r

can be expressed in terms of the usual gauge fields

Bµ(n) ,

n=1 (n) Bµ

(1)

contains n factors of A. At NLO, it was shown that we only need Bµ [9]:  Bµ(1) (k) = hµν (k)Aν (k), hµν (k) = δµν Dµ (k) + (1 − δµν )Gµν (k),

where

ν

Dµ (k) = 1 − d1

 ρ =µ

Nρ2 + d2



Nρ2 Nσ2 − d3 Nρ2 Nσ2 Nτ2 ,

ρ<σ,ρ,σ =µ

  Nρ2 + Nσ2 Nρ2 Nσ2 Gµν (k) = Nµ Nν d1 − d2 + d3 , 2 3

d1 = (2/3)α1 1 + α2 (1 + α3 ) , d2 = (4/3)α1 α2 (1 + 2α3 ),

d3 = 8α1 α2 α3 .

Two sets of αi ’s have been chosen: (1) α1 = 0.75, α2 = 0.6, α3 = 0.3 (which has been chosen in our simulation and has been motivated in [2]) and (2) α1 = 1.0, α2 = 1.0, α3 = 0.5, motivated in [10]. We will label these two sets respectively by HYP1 and HYP2. The Feynman rules can be easily deduced (they must be completed by the application of hµν ):

−1 heavy quark propagator a 1 − e−ip4 a +  , a (p, p ) vertex Vµ,hhg

 a

−ig0 T a δµ4 e−i(p4 +p4 ) 2 ,  1  a ab (p, p ) − ag02 δµ4 T a , T b e−i(p4 +p4 ) 2 , vertex Vµν,hhgg 2

−1 2 gluon propagator in the Feynman gauge a δµν δ ab 2W + a 2 λ2 .

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B. Blossier et al. / Physics Letters B 632 (2006) 319–325

Note that p  and p are the in-going and the out-going fermion momenta, respectively. We also introduce an infrared regulator λ ab by introducing the anticommutator of the SU(3) generators, normalfor the gluon propagator. We symmetrize the vertex Vµν,hhgg  1 ized by a factor 2 . The gluon propagator and the vertices are defined with the A field. The coefficient 3i=1 h24i ≡ H (N4 ) will enter as a global multiplicative factor in all the integrals expressed below. We have chosen the Feynman gauge: since one calculates the renormalization of a gauge-invariant operator, the renormalization factor ZD is gauge invariant. 3. Heavy quark self-energy From (4) we have Σ(p) = −(F1 + F2 ), where F1 and F2 correspond to the diagrams shown in Fig. 1(a) and (b):

1 2 H (N4 ) e−i(k4 +2ap4 ) H (N4 ) e−i(k4 +2ap4 ) 4 = − g F1 = − g02 0 2 2 2 −i(k +ap ) 4 4 + 3a 3a 2W + a λ 1 − e N4 + E 2 1 − e−i(k4 +ap4 ) +  k

=−

1 2 g 3a 0

k

k

e−i(k4 +2ap4 )

1 H (N4 ) = − g02 (N4 + iE)(N4 − iE) 1 − e−i(k4 +ap4 ) +  3a

k

1 H (−iE) e−2iap4 . √ E 1 + E 2 eE  − e−iap4

(9)

Note that Latin indices are spatial and E2 =



Ni2 +

i

a 2 λ2 , 4

 2      d2  2 d3  2 2 Ni2 + d2 Ni2 Nj2 − d3 N12 N22 N32 + N42 Ni2 d1 − Nj + Nj , H (N4 ) = 1 − d1 2 3 i

i
i

j =i

j =i

E  = 2 argsh(E). In (9) we have eliminated properly the noncovariant pole k4 = −ap4 + i by closing the integration contour in the complex plane (k4 ) < 0 where there is the single pole N4 = −iE. Furthermore the integrals along the lines k4 = ±π + ik4 are equal, because the integrand is 2π -periodic. Finally we have in the limit ap4 → 0:  

1 H (−iE) 1 H (−iE) 4 2 4 2 1 1 + F1 = g 0 (10) g ip4 . √ √  +   3a 3 0 4E 1 + E 2 1 − eE 2E 1 + E 2 eE − 1 2 (eE − 1)2 k

k

We find 

 g02  f1 (αi )/a + ip4 2 ln a 2 λ2 + f2 (αi ) . 2 12π The tadpole diagram F2 is

g2 H (N4 ) 1 4g02 −iap4 H (N4 ) 1 4g02 F2 = − e = − (1/a − ip4 ) ≡ − 0 2 (1/a − ip4 )f3 (αi ). 2 3a 2W ap4 →0 2 3 2W 12π F1 ≡ −

k

(11)

(12)

k

The factor 1/2 is introduced to compensate the over-counting of the factor 2 in the Feynman rule of the 2-gluon vertex when a closed gluonic loop is computed. We can point out that the divergent part g02 σ0 (αi ), σ0 = f1 + f3 (13) 12π 2 a of the self-energy is smaller with the sets HYP1 and HYP2 of the αi ’s than with the corresponding contribution without “hypercubic” links [5], as shown in Table 1: σ0 (αi = 0) = 19.95, σ0 (HYP1) = 5.76 and σ0 (HYP2) = 4.20, in good agreement with computations made by the ALPHA Collaboration [10], which compares the pseudoscalar heavy-light meson effective energy with Σ0 (αi ) =

Fig. 1. Self-energy corrections: (a) sunset diagram; (b) tadpole diagram.

B. Blossier et al. / Physics Letters B 632 (2006) 319–325

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Table 1 Numerical values of the parameters f1 , f2 , f3 , f4 , σ0 , z2 , zd defined in Eqs. (11), (12), (18), (13), (14) and (19), respectively αi = 0

HYP1

HYP2

f1 f2 f3 f4

7.72 −12.25 12.23 −12.68

1.64 1.60 4.12 −10.32

−1.76 9.58 5.96 −8.18

σ0 z2 zd

19.95 24.48 11.80

5.76 2.52 −7.80

4.20 −3.62 −11.80

Fig. 2. Operator corrections.

different static heavy quark actions, and by Hasenfratz et al. [11]. Qualitatively, one expects that the hypercubic blocking reduces the radiative corrections since it amounts roughly to introduce an additional cut-off in the integrals. The wave function renormalization Z2h is Z2h (αi ) = 1 +



 g02  −2 ln a 2 λ2 + z2 (αi ) , 2 12π

z2 = f3 − f2 .

(14)

|z2 | is also reduced by the hypercubic blocking, as indicated on Table 1: z2 (αi = 0) = 24.48 [5], z2 (HYP1) = 2.52 and z2 (HYP2) = −3.62. 4. Derivative operator in lattice HQET We have to renormalize the operator OijB = h¯ B γi γ 5 Dj hB . Following [6], a4



OijB (n) = a 4

n

1  ¯B h (n)γi γ 5 Uj (n)hB (n + jˆ) − h¯ B (n)γi γ 5 Uj† (n − jˆ)hB (n − jˆ) 2a n

 =



a −1 δ(p − p  )h¯ B (p) iγi γ 5 sj hB (p  ) + ig0

δ(q + p  − p)h¯ B (p)γi γ 5 ci Aaj (q)T a hB (p  )

p p q

p p

−



iag02 2!

p p q

δ(q + r + p  − p)h¯ B (p)T a T b γi γ 5 sj Aaj (q)Abj (r)hB (p  ).

(15)

r

Note that we have chosen to not submit the covariant derivative to the hypercubic blocking. The vertex function VijB is obtained by writing VijB = Vij0 + Vij1 + Vij2 , corresponding to the diagrams (a), (b) and (c) in Fig. 2; k 0 5 Vijk (αi ) = u(p)γ ¯ i γ u(p)Vj (αi ), k = 0, 1, 2. The contribution Vij is then given by computing 4i Vj0 (αi ) = −

3a

g02 k

4i = − g02 3a

k

=−

4i 2 g 3a 0

k

H (N4 ) e−i(k4 +2ap4 ) sin(k + ap) j 2W + a 2 λ2 (1 − e−i(k4 +ap4 ) + )2 k4 +ap4 2  H (N4 ) e−i 2 −iap4 (Γ + ap cos k )e j j j 2W + a 2 λ2 1 − e−i(k4 +ap4 ) + 

H (N4 ) 1 (Γj + apj cos kj )(1 − iap4 ) . k4 +ap4 k4 +ap4 2W + a 2 λ2 [2i sin( ) + ei 2 ]2 2

(16)

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B. Blossier et al. / Physics Letters B 632 (2006) 319–325

We can get rid of the integrand proportional to Γj , because it is an odd term. It remains

cos kj H (N4 ) 4 . Vj0 (αi ) = − ig02 pj 3 2W + a 2 λ2 (2iN4 + M4 )2 k

The integrand has poles at N4 = ±iE, k4 = 2iargth( 2 ). Once again we close the integration contour around the single pole N4 = −iE. Since j is spatial, the “sail” diagram drawn on Fig. 2(b) does not give any contribution, thus Vij1 = 0. There is finally the tadpole contribution Vij2 which is given by computing Vj2 (αi ) = −

14 2 ig pj 2! 3 0

ig 2 1 = − 0 2 pj 12.23. 2W 12π

(17)

k −1 We have finally H ∗∗ |OijR |H  = ZD (αi )H ∗∗ |OijB |H (αi ) where   ZD (αi ) = Z2h (αi ) 1 + δV (αi ) , 



4g 2 g02 4g02 1 H (N4 ) cos kj H (−iE) cos kj 1 + − 12.23 = − δV (αi ) = − 0 √ 3 3 2 (2W + a 2 λ2 )(2iN4 + M4 )2 4W (2E)3 1 + E 2 12π 2 k

k

≡



g02 12π 2



 2 ln a 2 λ2 + f4 (αi ) ,

(18)

g02 (19) zd (αi ), zd = z2 + f4 . 12π 2 The numerical values of zd are indicated in Table 1. Remark that infrared divergences appearing in Z2h and 1 + δV cancel and there is no dependence on a, a further consequence of ¯  )γi γ5 Dj h(v)|H (v) at zero recoil. Note also that as already mentioned the the µ independence of the matrix element H0∗ (v  )|h(v tadpole diagram of the operator vertex is not smoothed by the hypercubic blocking; therefore it is quite large. On the other hand the tadpole contribution to the self-energy is smoothed by the blocking. The final result is a large positive correction to the renormalized −1 −1 (HYP1) = 1.07 and ZD (HYP2) = 1.10, thus the discrepancy between matrix element. By fixing g0 = 1 (β = 6.0), this gives ZD −1 −1 −1 ZD (HYP1) and ZD (HYP2) is small (3%); without hypercubic blocking one would obtain ZD (αi = 0) = 0.90. We can think of applying a boosting procedure; in the pure HYP case the boosting plaquette correction is very small [8] (equation 19 and below). On the other hand we have to take into account that the covariant derivative operator involves links without hypercubic blocking, therefore one should employ a different prescription for the diagram drawn on Fig. 2(c), possibly leading to a larger tadpole −1 (αi = 0), and therefore a larger positive correction. Anyhow this kind of recipe would not contribution from the operator to ZD −1 lead to ZD (αi = 0) significantly larger than 1.1. −1 With our exploratory lattice study and taking account ZD (HYP1), we find τ 1 (1) = 0.41(5)(?), τ 3 (1) = 0.57(10)(?) and τ 23 (1)− ZD (αi ) = 1 +

2

2

2

τ 21 (1) = 0.15(10), where systematics are unknown; one is then not too far (within 1σ ) from saturating by ground states the Uraltsev 2  (n) (n) sum rule [12] n |τ 3 (1)|2 − |τ 1 (1)|2 = 14 . However the relation µ2π − µ2G > 9∆2 τ 21 (1) [13], with ∆ ≡ MH0∗ − MH = 0.4 GeV 2

2

2

[1], µ2G = 0.35 GeV2 , leads to µ2π larger than 0.6 GeV2 , which is significantly above experimental determination by moments. 5. Conclusion ¯ i γ 5 Dj h in lattice HQET with In this Letter we have calculated the radiative corrections to the covariant derivative operator hγ an hypercubic blocking of the Wilson line defining the heavy quark propagator. This determines the renormalization of the operator which is used to estimate the Isgur–Wise functions between the ground state and the L = 1 excitations at zero recoil. We find that there is a global, but moderate, enhancement of τ 1 (1) and τ 3 (1) with respect to the bare quantities computed on the lattice in the 2 2 case where one introduces fat timelike links, while there is a reduction with a simple Wilson line. References [1] D. Be´cirevi´c, et al., Phys. Lett. B 609 (2005) 298. [2] A. Hasenfratz, F. Knechtli, Phys. Rev. D 64 (2001) 034504, hep-lat/0103029. [3] A.V. Manohar, M.B. Wise, in: T. Ericson, P.V. Landshoff (Eds.), Heavy Quark Physics, in: Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology, 2000, p. 96. [4] M. Neubert, Phys. Rep. 245 (1994) 259.

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