Semi-leptonic B decay spectra by Dressed Gluon Exponentiation

Last modified: June 21 2008

This webpage describes how to get and run the program used for the calculations of the inclusive B -> X_u l nu spectrum by Dressed Gluon Exponentiation (DGE). The first version was released with [hep-ph/0509360] by JR Andersen and E Gardi.

This is Version 2.0 , released on June 21 2008 by E Gardi .
It is released together with the La Thuile 2008 review [arXiv:0806.4524] which contains first numerical results to O(beta0 alpha_s^2).

When using this program please cite the following three papers:
1) JR Andersen, E Gardi, Inclusive spectra in charmless semileptonic B decays by Dressed Gluon Exponentiation [hep-ph/0509360] JHEP01(2006)097
2) JR Andersen, E Gardi, Radiative B decay spectrum: DGE at NNLO. [hep-ph/0609250] JHEP01(2007)029
3) E Gardi, On the determination of |Vub| from inclusive semileptonic B decays [arXiv:0806.4524]
A more complete list of references and some comments regarding previous versions appear below.

There are several differences between the present version (Version 2.0) and the previous one (Version 1.2). Most importantly, the way the resummation is matched into the fixed-order result is different; the details will be reported in a forthcoming publication.
In the new version:
1) The calculation is improved by matching to O(beta_0 alpha_s^2) (NNLO) using results form
Paolo Gambino, Einan Gardi and Giovanni Ridolfi,
Running-coupling effects in the triple-differential charmless semileptonic decay width, [hep-ph/0610140] JHEP 0612 (2006) 036
A NLO mode still exists and can be used to gauge the uncertainty.
2) The calculation at each point in phase space is now normalized perturbatively as 1/Gamma_total dGamma/dP^+dP^-dE_l
3) A possibility has been added to perform phase space integration with a variety of cuts using a Monte Carlo method. In this way the partial branching fraction Delta B(B-> X_u l nu; cut) corresponding to different cuts can be computed in one go (each corresponds to a line in the input file cuts.txt).
4) A possibility has been added to place a cut on the lepton energy after boosting it to the Upsilon 4S frame (previously there was a separate code correcting for the boost, which is not needed now).

Note that the calculation of the Sudakov factor as well as the power corrections have not changed compared to Version 1.2. These issues are described in [hep-ph/0509360] and [hep-ph/0609250], respectively.


How to obtain the program

The C++ program calculating the spectrum can be obtained here.
The tar.gz file should be untarred and gunzip'ed:

 tar xzvf src_package_2.0.tar.gz

The Makefile is setup to use gcc ver4. If you are using gcc version 3, please then uncomment the lines 6 and 7 and comment out lines 9 and 10 of the Makefile. The program calcB2uslm can now be compiled by issuing the commands

 cd version_2.0

The programme requires access to the CERN libraries, and assumes these are installed in the directory


If your libraries are installed in another directory, please change line 10 (or line 7 for gcc ver. 3) in the Makefile accordingly. After compilation the program can be run by the command


followed by the appropriate arguments. To get a list of allowed input, please run the program with no arguments.


Running the program:

The program expects at least 7 input parameters as well as additional parameters depending on the Mode:  

calcB2uslm par1
par2 par3 par4 par5 par6 par7 [par8 par9 [par10 [par 11 [par 12 [par 13[par 14[par 15[par 16]]]]]]]]
par1: alpha_s(Mz)
par2: m_b^MSbar(m_b^MSbar) [GeV]
par3: Nf
par4: 3/2
Renormalon Residue
par5: f^{pv}
par6: Matching
par7: Mode
par8: Mode Par1
par9: Mode Par2
par10: Mode Par3
par11: Mode Par4
par12: Mode Par5
par13: Mode Par6
par14: Mode Par7
par15: Mode Par8
par16: Mode Par9


The Mode parameters will be explained in the following.

Extracting Vub (Mode 35):

Let us define the fraction of events R_cut as the partial BF within the region of measurement, divided by the total b->u BF; see eq. (1.1) in hep-ph/0509360.
The program uses Monte Carlo integration of the triple differential width to compute R_cut within the region of the phase space that is specified by the following cuts:

El` > E0` [GeV] (no cut: 0)  A cut of the lepton energy in the Upsilon 4S frame
El > E0 [GeV] (no cut: 0)
Mx=sqrt(P+P-) < Mx-cut [GeV] (no cut: 5.279 GeV)
q^2 > q^2_min [GeV^2] (no cut: 0)
q^2 < q^2_max [GeV^2] (no cut: 27.8678 GeV^2)
P+ < P+max [GeV] (no cut: 5.279 )
s^h_max < s^h_max_cut [GeV^2] (no cut: 27.8678 GeV^2)

For example, to calculate the central value for R_cut with the following cuts:

El` > 1 GeV 
El > 0 GeV
M_x=sqrt(P+P-) < 1.7 GeV
q^2 > 0 GeV^2   [no effective q^2-cut]
q^2 < 27.8678 GeV^2   [no effective q^2-cut]
P+ < 5.279 GeV  [no effective P+-cut]
s^h_max < 27.8678 [no effective s^h_max-cut]

using matching coefficient at O(beta_0 alpha_s^2) with renormalization scale mu=mb and with the central values for alpha_s(M_z)=0.1176 and m_b^MSbar(m_b^MSbar)=4.2 GeV (PDG 2007) and 4 light flavours (Nf=4) one would run:

./calcB2uslm .1176 4.2 4 1.0 0.0 5 35 100000 1.0 cuts.txt > central_NNLO.out 


is an input text file containing the cut limits on the
kinematic parameters listed above in one line as follows:
1.0 0.0 1.7 0.0 27.8678 5.279 27.8678
corresponding to
E0` [GeV] E0[GeV] Mx[GeV] q^2_min[GeV^2] q^2_max[GeV^2] P^{+}_max[GeV] s^h_max[GeV^2] 

The meaning of the other parameters is as follows:  

./calcB2uslm alpha_s(M_Z) m_b^MSbar(m_b^MSbar) Nf C_{3/2} f^{pv} matching_order(NNLO=5) 35 number_of_points  mu_over_mb cuts.txt

The mode choice ‘35’ tells the program to calculate R_cut.
For standard studies use: Nf=4, C_{3/2}=1, f^{pv}=0 and matching_order=5 (NNLO). To revert to NLO use matching_order=6.
The last three parameters in the command above are mode parameters:

is the number of points in phase space to be generated: 100000 gives good statistical error. No importance sampling is used in this version.
sets the renormalization scale in the matching coefficient mu/mb (default 1.0)
the name of the text file containing the required cuts, each measurement would be a line, the file may contain many lines.

An estimate of the theoretical uncertainty in R_cut can be obtained by varying the following input values:
1) alpha_s within the quoted uncertainty; PDG 2007 values: alpha_s^MSbar(M_Z)=0.1176\pm 0.0020
2) mb within the quoted uncertainty; (conservative) PDG 2007 values: mb^MSbar(mb^MSbar)=4.20\pm 0.07 GeV
3) changing the choice of renormalization scale factor (mu/mb) between 0.5 and 2.0 (the default is 1.0).
4) The value of C_{3/2} and f^{pv}, which parametrise non-perturbative effects in the quark distribution in the B meson. These parameters can be determined from fits to the photon energy spectrum in B->X_s \gamma or fits to the B-> X_u l nu spectrum. The default values for (C_{3/2},f^{pv}) are (1.0,0.0). One can estimate the uncertainty (see [hep-ph/0609250]) by changing these parameters to C_{3/2},f^{pv})=(6.2,0.3). A conservative estimate is then obtained by taking the error as \pm the difference with respect to the default.
5) Weak Annihilation effect is presently assumed to increase the width by at most 2%. It is located at q^2=mb^2. The effect is not included in the program and the uncertainty should be added.

|V_ub| can be obtained from an experimental measurement of the partial branching fraction Delta B_cut (see e.g. eq. (4.8) in hep-ph/0509360)

Delta B(B-> X_u l nu; cut) = R_cut * tau_B * Gamma_total(B-> X_u l nu)

where the lhs is the measured value and the factors on the rhs are:
a) R_cut computed by the program.
b) tau_B, the B lifetime. The present (PDG 2008) world average is tau_B=1.573 ps,
c) The total width was obtained in Section 2 of JHEP01(2006)097.
Updating the latter to the present (conservative) estimate mb^MSbar(mb^MSbar)=4.20\pm 0.07, one has:

Gamma_total(B-> X_u l nu)/|V_ub|^2 = 67.3 \pm 5.4 ps^{-1}

Note that the error here is dominated by the uncertainty in mb, so it should be taken as fully correlated with the mb value used in R_cut.

References and Previous versions

The original program for computing the triple differential width in charmless semileptonic decays was written for the calculations reported in
Inclusive spectra in charmless semileptonic B decays by Dressed Gluon Exponentiation
Jeppe R. Andersen, Einan Gardi

The calculation method is similar to the one applied in the case of the radiative B decay, as first reported in
Taming the anti-B -> X_s gamma spectrum by Dressed Gluon Exponentiation
Jeppe R. Andersen, Einan Gardi
Cavendish-hep-05-05 (Feb 2005) 50p.
[HEP-PH 0502159]

The charmless semileptonic decay program was thereafter improved (Version 1.2 released Dec 5, 2006) to include the parametrisation of non-perturbative corrections as discussed in
Radiative B decay spectrum: DGE at NNLO.
Jeppe R. Andersen, Einan Gardi,
and added the possibility of a cut in s^h_max.

The present version is Version 2.0, see above. It has been first reported in the La Thuile 2008 proceedings:
On the determination of |Vub| from inclusive semileptonic B decays
Einan Gardi
Edinburgh 2008/20
Presented at "Rencontres de Physique de la Vallee d'Aoste", La Thuile, Italy, Feb. 2008.

Version 2.0 includes the full O(beta_0 alpha_s^2) corrections based on the perturbative calculation in
Running-coupling effects in the triple-differential charmless semileptonic decay width
Paolo Gambino, Einan Gardi and Giovanni Ridolfi,
Cavendish-hep-06-22, DFTT-25-2006, GEF-TH-16-2006
JHEP 0612 (2006) 036