A new determination of the charm mass from the non-analytic reconstruction of the heavy quark correlator
aa r X i v : . [ h e p - ph ] F e b A new determination of the charm mass from thenon-analytic reconstruction of the heavy quarkcorrelator
David Greynat ∗ † Departamento de Física Teórica, Facultad de CienciasUniversidad de Zaragoza50009 Zaragoza, SpainE-mail: [email protected]
Pere Masjuan
Institut für Kernphysik,Johannes Gutenberg-UniversitätJ.J. Becher-Weg 45D-55099 Mainz, GermanyE-mail: [email protected]
Using the new non-analytic reconstruction method obtained from Mellin-Barnes properties, onecan extract the value m c ( MS ) = . ± .
08 GeV from experimental data of the radiation-corrected measured hadronic cross section to the calculated lowest-order cross section for muonpair production in the heavy-quark approximation.
Xth Quark Confinement and the Hadron Spectrum,October 8-12, 2012TUM Campus Garching, Munich, Germany ∗ Speaker. † This work has been supported by the Spanish DGIID-DGA grant 2009-E24/2, the Spanish MICINN grantsFPA2009-09638 and CPAN-CSD2007-00042 and by the Deutsche Forschungsgemeinschaft DFG through the Collabo-rative Research Center "The Low-Energy Frontier of the Standard Model" (SFB 1044). c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ harm mass and non-analytic reconstruction
David Greynat
1. Introduction
An accurate determination of the charm mass plays an important role on the precise physicalevaluation of several observables, from K and B decays to CKM matrix elements and in latticeQCD. One of the usual techniques to extract the charm mass is to use the sum rules approach basedon the relation between the moments of the production rate R and the inverse power of the squaremass of the c quark, and the Padé method (see [1, 2]). This approach should confront the factthat one have to employ the moments of the integral of R over the whole energy range, which are global properties, even though they are only known up to a certain scale L (since we only knowexperimentally R in a finite window). We propose to wield the local properties of R through a new"non-analytic reconstruction" method [3, 4]. As we will show, this approach allows us to obtainlocal properties of the heavy quark correlators at each points of the spectrum with a systematicerror and then to find a value of the charm mass directly on a c regression on the experimentalpoints.
2. Details of the method
Let us consider the vector polarization function (cid:0) q m q n − q g mn (cid:1) P ( q ) = i Z d x e iqx (cid:10) (cid:12)(cid:12) T j m ( x ) j m ( ) (cid:12)(cid:12) (cid:11) , (2.1)with the current j m ( x ) = y ( x ) g m y ( x ) , which has a cut in the complex plane starting at q = m ,where m is the (pole) mass of the heavy quark considered. In QCD perturbation theory, it can beexpanded as P ( q ) = P ( ) + P ( ) ( q ) + (cid:16) a s p (cid:17) P ( ) ( q ) + (cid:16) a s p (cid:17) P ( ) ( q ) + (cid:16) a s p (cid:17) P ( ) ( q ) + O ( a s ) , (2.2)where only P ( ) and P ( ) are know analytically, (for z = q / m ) P ( ) ( z ) = p (cid:20) + z − ( − z )( + z ) z G ( z ) (cid:21) , (2.3)and P ( ) ( z ) = p (cid:20) + z − ( − z )( + z ) z G ( z ) + ( − z )( − z ) z G ( z ) − ( + z ) z (cid:18) + z ( − z ) ddz (cid:19) I ( z ) z (cid:21) , (2.4)in which we used the auxiliary functions, G ( z ) = u log uu − I ( z ) = h z + ( − u ) + ( u ) i − h ( − u ) + Li ( u ) i ln u − h ( + u ) + ln ( − u ) i ln u , (2.6)2 harm mass and non-analytic reconstruction David Greynat and u = p − / z − p − / z + . (2.7)As it has been shown [3, 4] even if the functions P ( ) and P ( ) are unknown analytically, onecan reconstruct them from their expansions around q → q → m (thresholdexpansion) and q → ¥ (OPE), as P ( k ) ( z ) = N ∗ k (cid:229) n = W ( k ) ( n ) w n + (cid:229) p ,ℓ ( − ) ℓ h a ( k ) p ,ℓ Li ( ℓ ) ( p , w ) − b ( k ) p ,ℓ Li ( ℓ ) ( p , − w ) i + E ( k ) ( N ∗ k , w ) . (2.8)Let emphasize a little this expression. First one defines the so-called conformal change ofvariable z = w ( + w ) , w = − √ − z + √ − z . (2.9)This change of variables maps the cut z plane into a unit disc in the w plane, as we can see onFigure 2.1. The physical cut z ∈ [ , ¥ [ is transformed into the circle | w | = z = w = z = w =
1, the limit z → + ¥ ± i e into w → − ± i e , and z → − ¥ into w → − − z Re z Im ω Re ω z = ω (1+ ω ) G − G + Figure 1:
Conformal mapping between z and w . For both functions P ( ) and P ( ) , Feynman diagrams calculations at q → N ∗ k (for k = , P ( k ) ( z ) = q → N ∗ k (cid:229) n = C ( k ) ( n ) z n + O (cid:16) z N ∗ k + (cid:17) = w → N ∗ k (cid:229) n = W ( k ) ( n ) w n + O (cid:16) w N ∗ k + (cid:17) , (2.10)where the relation between the two coefficients C ( k ) and W ( k ) ( n ) is W ( k ) ( n ) = ( − ) n n (cid:229) p = ( − ) p p G ( n + p ) G ( p ) G ( n + − p ) C ( k ) ( p ) , (2.11) C ( k ) ( n ) = − n G ( n ) n (cid:229) p = p G ( + n − p ) G ( + n + p ) W ( k ) ( p ) . (2.12)3 harm mass and non-analytic reconstruction David Greynat
The main part of the approximation in (2.8) lies on the combination of the polylogarithmsfunctions,Li ( ℓ ) ( s , w ) = d ℓ d s ℓ (cid:20) wG ( s ) Z d t − w t log s − (cid:18) t (cid:19)(cid:21) = | w | < ( − ) ℓ ¥ (cid:229) n = log ℓ nn s w n , (2.13)and the analytic evaluation of the coefficients a ( k ) p ,ℓ and b ( k ) p ,ℓ . In order to reconstruct P ( ) and P ( ) ,we collect here their corresponding coefficients (see [3, 4] for more details) a ( ) , = . a ( ) , = − . a ( ) , = . a ( ) , = . , b ( ) , = . b ( ) , = . b ( ) , = . b ( ) , = − . , (2.14) a ( ) − , = . a ( ) , = . a ( ) , = − . a ( ) , = . a ( ) , = . a ( ) , = . , b ( ) , = − . b ( ) , = . b ( ) , = − . b ( ) , = − . b ( ) , = . , b ( ) , = − . b ( ) , = . b ( ) , = . b ( ) , = − . . (2.15)At least, one gives the error functions E ( k ) , E ( ) ( N ∗ , w ) = " + ¥ (cid:229) n = N ∗ + log . nn w n (2.16) E ( ) ( N ∗ , w ) = " + − ¥ (cid:229) n = N ∗ + log nn w n , (2.17)which encode the systematic error from the reconstructions. There exists several experimental results for the e + e − in hadrons that one can use for the fittingof the c quark mass. Each of the experiments give the ratio R ( s ) of the radiation-corrected measuredhadronic cross section to the calculated lowest-order cross section for muon pair production, R ( s ) = s ( e + e − −→ hadrons ) s ( e + e − −→ m + m − ) = s ( e + e − −→ hadrons ) pa / s , (2.18)that has the experimental values shown in Fig. 2 .This Fig 2 shows that the complete spectrum is sensitive to resonances, as expected. It isobvious that a perturbative approach cannot take into account the resonances description, then one4 harm mass and non-analytic reconstruction David Greynat
Experiment ReferenceMARK I [5]PLUTO [6]CrystalBall (Run 1) [7]CrystalBall (Run 2) [7]MD1 [8]CLEO [9]CLEO [10, 11]BES [12]BES [13]CLEO [14]CLEO [15]
Table 1:
All different experimental sets considered for the fits. √ s (in GeV) R exp . Figure 2:
Collection of the different experimental sets for the V-V spectrum. has to make an arbitrary choice on where we assume that the continuum limit is reached or inother words, where the perturbative description is pertinent. Let’s choose the value of 5 GeV. Ofcourse the influence of the arbitrariness has to be discussed and taken account in the evaluation ofthe error but it is something depending on the perturbative and heavy-quark limit more than thereconstruction itself.The idea now is to perform a fit among all this data points to extract the perturbative mass m c of the c -quark. 5 harm mass and non-analytic reconstruction David Greynat
The first step in the fitting procedure is to choose the following expression for the running a s ( s ) , a s ( s ) = pb ln ( s / L ) (cid:20) − b b ln [ ln ( s / L )] ln ( s / L )+ b b ln ( s / L ) (cid:18) ln (cid:2) ln ( s / L ) (cid:3) − (cid:19) + b b b − ! , (2.19)where L is the energy scale and the b -function has coefficients b = − n f , b = − n f , b = − n f + n f , (2.20)and n f is the number of quarks with mass smaller than √ s / theoretical expression (2.18) is related to P ( q ) (2.2), up to a s , R th. ( s ) = "(cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) N c " + a s ( s ) p + . (cid:18) a s ( s ) p (cid:19) − . (cid:18) a s ( s ) p (cid:19) + p (cid:18) (cid:19) Im " P ( ) + a s ( s ) p P ( ) + (cid:18) a s ( s ) p (cid:19) P ( ) + C (cid:18) a s ( s ) p (cid:19) P ( ) (2.21)where all P ( k ) functions have the argument z = s m c , and N c is the number of colors.The goal of the analysis is to extract m c from the comparison between the value of R exp. and R th. . The usual method used is to built the moments associated to R from 0 to L and identifyingthe coefficients of the Taylor expansion that are proportional up to a factor to m − c . Instead of thisapproach, we propose to perform the analysis directly on the function itself, because thanks to thereconstruction method formula (2.8), its expression is available and its systematic error too (2.16).For this we will use a c -method with the assumption c ( m c ) . = N (cid:229) j = (cid:18) R exp. ( s j ) − R th. ( s j ) s exp. ( s j ) (cid:19) + (cid:18) R exp. ( s j ) − R th. ( s j ) s th. ( s j ) (cid:19) , (2.22)where the s j are the experimental energy points, the s exp. is the experimental error and the theoret-ical error s th. due the approximation of the reconstruction is given by s ( s ) = p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Im "(cid:18) a s ( s ) p (cid:19) E ( ) ( N ∗ , w ) + p C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Im "(cid:18) a s ( s ) p (cid:19) E ( ) ( N ∗ , w ) , (2.23)with w = − q − s m c + q − s m c . 6 harm mass and non-analytic reconstruction David Greynat
3. Results a s At a s order, one obtains after a regression procedure with a c / d.o.f. = . m c ( pole ) = . ± .
08 GeV , (3.1)that is translated into the MS mass as [16] m c ( MS ) = . ± .
08 GeV . (3.2)Assuming now that the mass m c obeys to a Gaussian density of probability, one can easilyreconstruct points by points the error generated on R th. by this hypothesis, taking into account thatthe relation between m c and R th. is highly non linear and non trivial for expressing the error. Wechoose then to use a Monte-Carlo approach to obtaining the mean value of R th. and its error asshown in Fig 3. ReconstructionCLEO 2007MD-1 1996ChrystalBall 1990¯ m c ( pole ) = . ± .
08 GeV c / d.o.f = . √ s (in GeV) R exp . , R th . Figure 3:
The reconstructed radiation-corrected measured hadronic cross section to the calculated lowest-order cross section for muon pair production.
4. Conclusions
We show that it is possible to extract the charm mass value after a c regression to the exper-imental data of the radiation-corrected measured hadronic cross section to the calculated lowest-order cross section for muon pair production using the non-analytic reconstruction of the heavy-quark correlators. We present here a preliminary result up to a s . The next step would include theorder a and a complete analysis of all different systematic contributions [17].7 harm mass and non-analytic reconstruction David Greynat
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