We present an accurate lattice QCD computation of the b-quark mass, the B and
Bs decay constants, the B-mixing bag-parameters for the full four-fermion
operator basis, as well as estimates for \xi and f_{Bq}\sqrt{B_q} extrapolated
to the continuum limit and the physical pion mass. We have used Nf = 2
dynamical quark gauge configurations at four values of the lattice spacing
generated by ETMC. Extrapolation in the heavy quark mass from the charm to the
bottom quark region has been carried out using ratios of physical quantities
computed at nearby quark masses, having an exactly known infinite mass limit.Comment: 7 pages, 4 figures, presented at the 31st International Symposium on
  Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, German

Carrasco, N

Ciuchini, M

Dimopoulos, P

Frezzotti

Gimenez, V

Herdoiza, G

Lubicz, V

Michael, C

Picca, E

Rossi, GC

Sanfilippo, F

Shindler, A

Silvestrini, L

Simula, S

Tarantino, C

English

arXiv

We present an accurate lattice QCD computation of the b-quark mass, the B and Bs decay constants, the B-mixing bag-parameters for the full four-fermion operator basis, as well as estimates for \xi and f_{Bq}\sqrt{B_q} extrapolated to the continuum limit and the physical pion mass. We have used Nf = 2 dynamical quark gauge configurations at four values of the lattice spacing generated by ETMC. Extrapolation in the heavy quark mass from the charm to the bottom quark region has been carried out using ratios of physical quantities computed at nearby quark masses, having an exactly known infinite mass limit

Tarantino, C.

FREZZOTTI, ROBERTO

ROSSI, GIANCARLO

B-physics computations from Nf=2 tmQCD

We present an accurate lattice QCD computation of the b-quark mass, the B and Bs decay constants, the B-mixing bag-parameters for the full four-fermion operator basis, as well as estimates for \xi and f_{Bq}\sqrt{B_q} extrapolated to the continuum limit and the physical pion mass. We have used Nf = 2 dynamical quark gauge configurations at four values of the lattice spacing generated by ETMC. Extrapolation in the heavy quark mass from the charm to the bottom quark region has been carried out using ratios of physical quantities computed at nearby quark masses, having an exactly known infinite mass limit.We present an accurate lattice QCD computation of the b-quark mass, the B and Bs decay constants, the B-mixing bag-parameters for the full four-fermion operator basis, as well as estimates for \xi and f_{Bq}\sqrt{B_q} extrapolated to the continuum limit and the physical pion mass. We have used Nf = 2 dynamical quark gauge configurations at four values of the lattice spacing generated by ETMC. Extrapolation in the heavy quark mass from the charm to the bottom quark region has been carried out using ratios of physical quantities computed at nearby quark masses, having an exactly known infinite mass limit

Carrasco, N.

Ciuchini, M.

Dimopoulos, P.

Frezzotti, R.

Gimenez, V.

Herdoiza, G.

Lubicz, V.

Michael, C.

Picca, E.

Rossi, G.C.

Sanfilippo, F.

Shindler, A.

Silvestrini, L.

Simula, S.

CERN Document Server

Archivio della Ricerca - Università di Roma Tor vergata

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B-physics computations from N f = 2 tmQCD
N. Carrasco(a), M. Ciuchini(b), P. Dimopoulos∗(c,d), R. Frezzotti(d), V. Giménez(a) G.
Herdoiza( f ), V. Lubicz(g,b), C. Michael(h), E. Picca(g), G.C. Rossi(d), F. Sanfilippo(i),
A. Shindler( j), L. Silvestrini(k), S. Simula(b), C. Tarantino(g,b)
(a) Departament de Física Teòrica and IFIC, Univ. de València-CSIC Dr. Moliner 50, E-46100
València, Spain, E-mail: nuria.carrasco@uv.es, vicente.gimenez@uv.es
(b) INFN, Sezione di Roma Tre,c/o Dipartimento di Fisica, Università Roma Tre, Via della Vasca
Navale 84, I-00146 Rome, Italy, E-mail: {ciuchini, simula}@roma3.infn.it
(c) Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi,
Compendio del Viminale, Piazza del Viminale 1, I-00184 Rome, Italy,
(d) Dipartimento di Fisica, Università di Roma “Tor Vergata” and INFN Sezione “Tor Vergata",
Via della Ricerca Scientifica 1, I-00133 Rome, Italy,
E-mail: {dimopoulos,frezzotti,rossig}@roma2.infn.it
( f ) PRISMA Cluster of Excellence, Institut für Kernphysik,Johannes Gutenberg-Universität,
D-55099 Mainz, Germany, E-mail: herdoiza@kph.uni-mainz.de
(g) Dipartimento di Fisica, Università Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy,
E-mail: e.picca88@gmail.com, {lubicz, tarantino}@fis.uniroma3.it,
(h) Theoretical Physics Division, Department of Mathematical Sciences, The University of
Liverpool, Liverpool L69 3BX, UK, E-mail: C.Michael@liverpool.ac.uk
(i) Laboratoire de Physique Théorique (Bât. 210), Université Paris Sud, F-91405 Orasay-Cedex,
France, E-mail: francesco.sanfilippo@th.u-psud.fr
( j) CERN, Physics Department, 1211 Geneva 23, Switzerland
(k) INFN, Sezione di Roma, c/o Dipartimento di Fisica, Sapienza, Università di Roma, Piazzale
A. Moro, I-00185 Rome, Italy, E-mail: Luca.Silvestrini@roma1.infn.it
We present an accurate lattice QCD computation of the b-quark mass, the B and Bs decay con-
stants, the B-mixing bag-parameters for the full four-fermion operator basis, as well as estimates
for ξ and fBq√Bq extrapolated to the continuum limit and the physical pion mass. We have used
N f = 2 dynamical quark gauge configurations at four values of the lattice spacing generated by
ETMC. Extrapolation in the heavy quark mass from the charm to the bottom quark region has
been carried out using ratios of physical quantities computed at nearby quark masses, having an
exactly known infinite mass limit.
31st International Symposium on Lattice Field Theory - LATTICE 2013
July 29 - August 3, 2013
Mainz, Germany
∗Speaker.
c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/
B-physics computations from N f = 2 tmQCD P. Dimopoulos
1. Introduction
Many precision tests of the Standard Model (SM) and some proposals regarding stringent
constraints on New Physics (NP) generalisations of the SM involve accurate studies of processes
in the b-quark region. Experimental measurements on the leptonic decays B → τντ and B0(s) →
µ+µ− need precise input information of the pseudoscalar decay constants fB and fBs from lattice
computations. Neutral B-meson oscillations are described, in the SM, by a single four-fermion
operator whose matrix element (bag parameter) can be determined in lattice simulations. B-meson
mixing, besides its crucial role in the Unitarity Triangle (UT) analysis, can provide important clues
for detecting NP effects. The knowlegde of the values of the bag parameters of the complete four-
fermion operator basis is required to obtain predictions about the NP scale for physics beyond the
SM.
In these proceedings we report on computations of a number of B-physics hadronic observ-
ables performed within the European Twisted Mass Collaboration (ETMC). We have computed in
the continuum limit, by making use of data at four values of the lattice spacing, the value of the b-
quark mass, the pseudoscalar meson decay constants, fB and fBs as well as their ratio. Moreover we
have determined the bag parameters corresponding to the full four-fermion operator basis, as well
as other interesting phenomenological quantities like ξ and fBq
√
B(q)i (i = 1, . . . ,5 and q = d/s).
For a more detailed discussion of methods and results of this study the reader is referred to a re-
cent ETMC publication [1] as well as to [2, 3]. First ETMC results of the b-quark mass using
N f = 2+1+1 gauge ensembles can be found in [4].
2. Lattice setup and computational details
We have employed N f = 2 dynamical quark gauge configurations generated by ETMC ([5] –
[7]) using Wilson twisted mass action tuned to maximal twist [8] at four values of the inverse bare
gauge coupling, β . The lattice spacing values lie in the interval [0.05, 0.1] fm. Light quark mass
values of the degenerate u/d quark produce light pseudoscalar mesons (“pions") in the range 280≤
mPS ≤ 500 MeV. We recall that the maximally twisted fermionic action offers the advantage of
automatic O(a) improvement for all the physical observables computed on the lattice [8]. Strange
and charm quarks are treated in the quenched approximation. We have computed 2- and 3-point
correlation functions using valence quark masses whose range is extended from the light sea quark
mass up to 2.5-3 times the charm quark mass. The values of the light valence quark mass are set
equal to the light sea ones. Renormalised quark masses are determined from the bare ones using
the renormalisation constant (RC) Zµ = Z−1P , µR = µbare/ZP. The ZP values have been computed
using RI/MOM techniques in [9]. Details of the ETMC determinations of light, strange and charm
quark mass using N f = 2 gauge ensembles are given in [10].
We have employed smeared interpolating operators in combination with APE smeared gauge
links to compute 2- and 3- point correlation functions. With respect to the local fields the use
of smeared ones reduce the overlap with the excited states and increased the projection onto the
lowest energy eigenstate. We are able thus to extract heavy-light meson masses and matrix ele-
ments at relatively small Euclidean time separations. Moreover, as it is shown in [1], even better
projection onto the ground state is achieved by employing a linear combination of smeared and lo-
2
B-physics computations from N f = 2 tmQCD P. Dimopoulos
cal interpolating fields. The optimal linear combination is determined by tuning the superposition
parameter.
3. Computation of the b-quark mass and decay constants fB and fBs
For the determination of B-physics observables for which the asymptotic behaviour is known
from HQET, it is in general possible to use the ratio method described in ([1] – [3]). We briefly
recall the computation for the b-quark mass. HQET suggests that in the static limit the heavy-light
pseudoscalar meson mass obeys
lim
µpoleh →∞
(
Mhℓ/µpoleh
)
= const. (3.1)
We consider a set of heavy quark masses (µ(1)h ,µ
(2)
h , · · · ,µ
(N)
h ) having a fixed ratio, λ , between
any two successive values: µ(n)h = λ µ
(n−1)
h . Throughout this work the “overline" notation for the
quark masses denotes that they are renormalised in the MS scheme at 3 GeV. We construct properly
normalised ratios namely, y(µ(n)h ,λ ; µ ℓ,a), of the quantity in the lhs of Eq. (3.1) at nearby values
of the heavy quark mass. Taking the static and continuum limit one gets the exact equation
lim
µh→∞
lim
a→0
y(µ (n)h ,λ ; µ ℓ,a) = 1 . (3.2)
At each value of µ(n)h we perform a combined chiral and continuum fit on the ratios to extract the
quantity y(µh) ≡ y(µh,λ ; µu/d ,a = 0). Discretisation effects in ratios are well under control. In
Fig. 1(a) we show the data and the fit for the ratio that corresponds to the largest simulated pair of
the heavy quark mass values. We describe the dependence of the ratio y(µh) on µh using the fit
ansatz
y(µh) = 1+
η1
µh
+
η2
µ2h
, (3.3)
in which the constraint limµh→∞ y(µh) = 1 is implemented. The fit parameters could be, in general,
functions of log(µh). However in the range of the currently explored heavy quark mass values this
logarithmic dependence can be safely neglected. Data and fit are shown in Fig. 1(b). The final step
consists in determining the b-quark mass value from the chain equation
y(µ (2)h )y(µ
(3)
h ) . . . y(µ
(K+1)
h ) = λ−K
Mhu/d(µ(K+1)h )
Mhu/d(µ(1)h )
·
[ ρ(µ(1)h ,µ)
ρ(µ(K+1)h ,µ)
]
, (3.4)
where the function ρ(µ(n)h ,µ), that is known up to N3LO in perturbation theory, relates the MS
renormalised quark mass (at the scale of µ = 3 GeV) to the pole mass: µpoleh = ρ(µh,µ)µh(µ).
In Eq. (3.4) λ , K and µ (1)h have been chosen in such a way that Mhu/d(µ (K+1)h ) coincides with
the experimental value of the B-meson mass, MB = 5.279 GeV. The quantity Mhu/d(µ(1)h ) is the
result of the combined chiral and continuum fit of pseudoscalar meson mass values evaluated at
the reference heavy quark mass, µ(1)h , that lies close to the charm quark mass. For (µ
(1)
h ,λ ) =
(1.05 GeV, 1.1784), Eq. (3.4) is satisfied for K = Kb = 9. The b-quark mass in the MS scheme at
3 GeV is
mb(mb,MS) = 4.29(9)(8)[12] GeV, (3.5)
3
B-physics computations from N f = 2 tmQCD P. Dimopoulos
where the first two errors are statistical and systematic, respectively. Adding them in quadrature
gives the total error in the square brackets. Having calculated the value of the b-quark mass, we
CL - phys. point
β = 4.20
β = 4.05
β = 3.90
β = 3.80
µℓ (GeV)
y
(µ
(8
)
h
)
0.070.060.050.040.030.020.010.00
1.02
1.01
1.00
0.99
0.98
0.97
0.96
(a)
µ¯−1b
1/µh (GeV
−1)
y
(µ
h
)
0.90.80.70.60.50.40.30.20.10.0
1.04
1.03
1.02
1.01
1.00
0.99
0.98
(b)
Figure 1: (a) Combined chiral and continuum fit of the ratio of heavy-light pseudoscalar meson mass for
our largest value of the heavy quark mass (empty black circle is our estimate at the physical u/d quark mass
in the continuum limit). (b) y(µh) against 1/µh using the fit ansatz (3.3) (χ2/d.o. f .= 0.3).
follow analogous strategies to compute the heavy-strange pseudoscalar decay constant, fBs, and the
value of the ratio of the heavy-strange and the heavy-light decay constants, fBs/ fB. We then get
our estimate for the heavy-light pseudoscalar decay constant from fB = fBs / ( fBs/ fB). Statistical
errors are estimated using the bootstrap method. Our results read
fBs = 228(5)(6)[8] MeV, fB = 189(4)(7)[8] MeV, fBs/ fB = 1.206(10)(22)[24]. (3.6)
As a byproduct of our analysis we obtain the values of the pseudoscalar decay constants for the Ds
and D mesons as well as for their ratio:
fDs = 250(5)(5)[7] MeV, fD = 208(4)(6)[7] MeV, fDs/ fD = 1.201(7)(20)[21] (3.7)
Error coding is as in Eq. (3.5).
4. Computation of bag parameters and ξ
The ∆B = 2 effective weak Hamiltonian in its most general form reads
H
∆B=2
eff =
1
4
5
∑
i=1
CiOi +
1
4
3
∑
i=1
˜Ci ˜Oi, (4.1)
with
O1 = [b
α γµ(1− γ5)qα ][b
β γµ(1− γ5)qβ ], O2 = [b
α
(1− γ5)qα ][b
β
(1− γ5)qβ ],
O3 = [b
α
(1− γ5)qβ ][b
β
(1− γ5)qα ], O4 = [b
α
(1− γ5)qα ][b
β
(1+ γ5)qβ ],
O5 = [b
α
(1− γ5)qβ ][b
β
(1+ γ5)qα ], ˜O1 = [b
αγµ(1+ γ5)qα ][b
β γµ(1+ γ5)qβ ]
˜O2 = [b
α
(1+ γ5)qα ][b
β
(1+ γ5)qβ ], ˜O3 = [b
α
(1+ γ5)qβ ][b
β
(1+ γ5)qα ]. (4.2)
4
B-physics computations from N f = 2 tmQCD P. Dimopoulos
In Eqs (4.2) q ≡ d or s, α and β denote color indices and spin indices are implicitly contracted
within square brackets. The Wilson coefficients Ci and ˜Ci have an implicit renormalization scale
dependence which is compensated by the scale dependence of the renormalization constants of the
corresponding operators. The parity-even parts of the operators Oi and ˜Oi are identical. Due to
parity conservation in strong interactions it is sufficient to consider the matrix elements where only
the parity-even components of the operators Oi enter. In the SM only the matrix element of the
operator O1 enters. The bag parameters, Bi (i = 1, . . . ,5), are defined through the equations
〈B0q|O1(µ)|B0q〉 = C1 B
(q)
1 (µ) m2Bq f 2Bq (4.3)
〈B0q|Oi(µ)|B0q〉 = Ci B
(q)
i (µ) [
m2Bq fBq
mb(µ)+mq(µ) ]
2
for i = 2, . . . ,5, (4.4)
with Ci = (8/3, −5/3, 1/3, 2, 2/3). We have used a mixed fermionic action setup to evaluate
the four-fermion matrix elements as it offers the advantage that matrix elements are at the same
time O(a)-improved and free of wrong chirality mixing effects [11]. Four-fermion RCs have been
computed using RI/MOM techniques in [12].
Since in the static limit each of the five bag parameters is a constant, we can apply the ratio
method adopting a strategy analogous to the one used for the b-quark mass. For more details on
the computation of ratios of bag parameters the reader is referred to Ref. [1]. In Table 1 we report
our results in the MS scheme of Ref. [13] at the scale of the b-quark mass.
(MS, mb)
B(d)1 B
(d)
2 B
(d)
3 B
(d)
4 B
(d)
5
0.85(3)(2)[4] 0.72(3)(1)[3] 0.88(12)(6)[13] 0.95(4)(3)[5] 1.47(8)(9)[12]
B(s)1 B
(s)
2 B
(s)
3 B
(s)
4 B
(s)
5
0.86(3)(1)[3] 0.73(3)(1)[3] 0.89(10)(7)[12] 0.93(4)(1)[4] 1.57(7)(8)[11]
Table 1: Continuum limit results for B(d)i and B
(s)
i (i = 1, . . . ,5), renormalized in the MS scheme of Ref. [13]
at the scale of the b-quark mass. Error coding is as in Eq. (3.5).
We also apply the ratio method to compute the SU(3)-breaking ratios B(s)1 /B(d)1 and the im-
portant ξ = ( fBs/ fBd)(B(s)1 /B(d)1 )1/2 parameter. In Fig. 2(a) and (b) we display the dependence on
the inverse heavy quark mass of the appropriate ratios for B(s)1 /B
(d)
1 and ξ , denoted as ζω(µh) and
ζξ (µh), respectively. For B(s)1 /B(d)1 and ξ we obtain (see Ref. [1] for details)
B(s)1 /B
(d)
1 = 1.007(15)(14)[21], ξ = 1.225(16)(26)[31]. (4.5)
Our results for the quantities fBq
√
B(q)i (i= 1, . . . ,5 and q= d/s) can be found in Table 4 of Ref. [1].
5. Model-independent constraints on ∆B = 2 operators and NP scale from UTA
The NP generalisation of the UT analysis is carried out by including matrix elements that,
though absent in the SM, may appear in the theoretical parametrisation of various observables in
5
B-physics computations from N f = 2 tmQCD P. Dimopoulos
µ¯−1b
1/µh (GeV
−1)
ζ ω
(µ
h
)
0.90.80.70.60.50.40.30.20.10.0
1.005
1.000
0.995
0.990
(a)
µ¯−1b
1/µh (GeV
−1)
ζ ξ
(µ
h
)
0.90.80.70.60.50.40.30.20.10.0
1.010
1.005
1.000
0.995
(b)
Figure 2: ζω(µh) and ζξ (µh) against 1/µh are shown in panels (a) and (b), respectively. For both cases
the fit function has a polynomial form of the type given by Eq. (3.3) (blue curve). In both panels a fit of the
form ζ (µh) = 1+η/µh is also performed (black dashed straight line). The vertical black thin line marks
the position of 1/µb.
extensions of the SM. We employ our unquenched lattice QCD results for the bag parameters of the
full basis of the ∆B = 2 four-fermion operators to update the UT analysis beyond the SM presented
in Ref. [14]. The effective weak Hamiltonian, Eq. (4.1), is parameterized by Wilson coefficients of
the form Ci(Λ) = (Fi Li)/(Λ2) with i = 2, . . . ,5. We denote by Fi the (generally complex) relevant
NP flavor coupling, by Li a (loop) factor which depends on the interactions that generate Ci(Λ), and
Λ is the scale of NP, i.e. the typical mass of new particles mediating ∆B = 2 transitions. Hence,
the phenomenologically allowed range for each of the Wilson coefficients can be translated into a
lower bound on Λ. Other assumptions on the flavor structure of NP correspond to different choices
of the Fi functions.
Following Ref. [14], we derive the lower bounds on the NP scale Λ by setting Li = 1 which
corresponds to strongly-interacting and/or tree-level coupled NP. Two other interesting possibilities
are provided by loop-mediated NP contributions with Li proportional to either α2s or α2W . The
first case corresponds for example to gluino exchange in the minimal supersymmetric SM, while
the latter applies to all models with SM-like loop-mediated weak interactions. To obtain a lower
bound on Λ entailed by loop-mediated contributions, one simply has to multiply the bounds we
quote below by αs(Λ)∼ 0.1 or αW ∼ 0.03, respectively.
Our analysis is performed (as in [14]) by switching on one coefficient at a time in each sector,
thus excluding the possibility of having accidental cancellations among contributions of different
operators. Thus in the case of accidental cancellations the bounds will be weaker.
In Tables 2 and 3 we collect the results for the upper bounds on the |CBdi | and |C
Bs
i | coefficients
and the corresponding lower bounds on the NP scale Λ. The superscript Bd or Bs is to remind that
we are reporting the bounds coming from the Bd- and Bs-meson sectors we are here analyzing.
In the present analysis both experimental and theoretical inputs have been updated with respect
to Ref. [14] (see Ref. [15]). The constraints on the Wilson coefficients of non-standard operators
and, consequently, on the NP scale turn out to be significantly more stringent than in Ref. [14], in
B-physics computations from N f = 2 tmQCD P. Dimopoulos
particular for the Bs sector. Comparing with the results of the UT–analysis in Ref [12], we notice
that (at least for generic NP models with unconstrained flavour structure) the bounds on the NP
scale coming from K0– ¯K0 matrix elements turn out to be the most stringent ones.
95% upper limit Lower limit on Λ
(GeV−2) (TeV)
|CBd1 | 4.7 ·10−12 4.6 ·102
|CBd2 | 3.0 ·10−13 1.8 ·103
|CBd3 | 1.1 ·10−12 9.5 ·102
|CBd4 | 9.5 ·10−14 3.2 ·103
|CBd5 | 2.7 ·10−13 1.9 ·103
Table 2: 95% upper bounds for the |CBdi | coef-
ficients and the corresponding lower bounds on
the NP scale, Λ, for a strongly interacting NP
with generic flavor structure (Li = Fi = 1).
95% upper limit Lower limit on Λ
(GeV−2) (TeV)
|CBs1 | 5.6 ·10−11 1.3 ·102
|CBs2 | 4.9 ·10−12 4.5 ·102
|CBs3 | 1.8 ·10−11 2.3 ·102
|CBs4 | 1.6 ·10−12 7.9 ·102
|CBs5 | 4.5 ·10−12 4.7 ·102
Table 3: 95% upper bounds for the |CBsi | coef-
ficients and the corresponding lower bounds on
the NP scale, Λ, for a strongly interacting NP
with generic flavor structure (Li = Fi = 1).
Acknowledgements
G. H. acknowledges the support by DFG (SFB 1044). We acknowledge computer time made avail-
able to us on the Altix system at the HLRN supercomputing service in Berlin under the project
"B-physics from lattice QCD simulations". Part of this work has been completed thanks to allo-
cation of CPU time on BlueGene/Q -Fermi based on the agreement between INFN and CINECA
and the specific initiative INFN-RM123. We acknowledge partial support from ERC Ideas Starting
Grant n. 279972 “NPFlavour” and ERC Ideas Advanced Grant n. 267985 “DaMeSyFla".
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B-physics computations from Nf=2 tmQCD

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