A determination of m c (m c ) from HERA data using a matched heavy-flavor scheme

Journal of High Energy Physics, Aug 2016

The charm quark mass is one of the fundamental parameters of the Standard Model Lagrangian. In this work we present a determination of the \( \overline{\mathrm{MS}} \) charm mass from a fit to the inclusive and charm HERA deep-inelastic structure function data. The analysis is performed within the xFitter framework, with structure functions computed in the FONLL general-mass scheme as implemented in APFEL. In the case of the FONLL-C scheme, we obtain m c (m c ) = 1.335 ± 0.043(exp) − 0.000 + 0.019 (param) − 0.008 + 0.011 (mod) − 0.008 + 0.003 (th) GeV. We also perform an analogous determination in the fixed-flavor-number scheme at next-to-leading order, finding m c (m c ) = 1.318 ± 0.054(exp) − 0.010 + 0.011 (param) − 0.004 + 0.015 (mod) − 0.004 + 0.045 (th) GeV, compatible with the FONLL-C value. Our results are consistent with previous determinations from DIS data as well as with the PDG world average.

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A determination of m c (m c ) from HERA data using a matched heavy-flavor scheme

Received: May determination of Valerio Bertone 0 2 3 5 7 Stefano Camarda 0 2 5 Amanda Cooper-Sarkar 0 2 5 7 Alexandre Glazov 0 2 5 6 Voica Radescu 0 2 5 7 Juan Rojo 0 2 5 7 Andrey Sapronov 0 1 2 5 Oleksandr Zenaiev 0 2 5 6 Achim Geiser 0 2 5 6 0 Joliot-Curie 6 , 141980, Dubna, Moscow Region , Russia 1 Joint Institute for Nuclear Research , JINR 2 st. Podchorozych 1 , 30-084 Cracow , Poland 3 Keble Road, University of Oxford , OX1 3NP Oxford , U.K 4 T. Kosciuszko Cracow University of Technology, Institute of Physics 5 Agnieszka Luszczak 6 Deutsches Elektronen-Synchrotron (DESY) , Notkestrasse 85, D-22607 Hamburg , Germany 7 Rudolf Peierls Centre for Theoretical Physics The charm quark mass is one of the fundamental parameters of the Standard Model Lagrangian. In this work we present a determination of the MS charm mass from a t to the inclusive and charm HERA deep-inelastic structure function data. The analysis is performed within the xFitter framework, with structure functions computed in the FONLL general-mass scheme as implemented in APFEL. In the case of the FONLL-C scheme, we obtain mc(mc) = 1:335 0:043(exp)+00::001090(param)+00::001018(mod)+00::003038(th) GeV. We also perform an analogous determination in the xed- avor-number scheme at next-to-leading order, nding mc(mc) = 1:318 0:054(exp)+00::001110(param)+00::001159(mod)+00::004054(th) GeV, compatible with the FONLL-C value. Our results are consistent with previous determinations from DIS data as well as with the PDG world average. Deep Inelastic Scattering (Phenomenology); QCD Phenomenology - A mc(mc) from 1 Introduction 2 FONLL with MS heavy-quark masses 2.1 Implementation 2.1.1 2.1.2 2.1.3 2.1.4 MS mass vs. pole mass Solution of the RGE for the running of the MS mass Matching conditions Structure functions 2.2 Benchmark QCD t settings 3 4 Results 4.1 4.2 4.3 4.4 5 Conclusions 1 Introduction Comparison to other results Cross-checks Discussion on the Q2min dependence of the mass determination Discussion on the sensitivity to mc(mc) of the inclusive data The masses of the heavy quarks, charm, bottom and top, are fundamental parameters of the Standard Model [1]. A precise determination of their values is of utmost importance; as an example, the fate of the electroweak vacuum depends crucially on the exact value of mt [2]. In the case of the charm quark, since its mass mc is larger than the scale QCD of Quantum Chromodynamics (QCD), its value is a direct input of many perturbative calculations involving charm quarks in the initial and/or in the nal state. Di erences in the value of the charm quark mass and in the treatment of its e ects in deep-inelastic-scattering structure functions can lead to di erences in modern analyses of parton distribution functions (PDFs) [3{7], with implications for precision phenomenology at the Large Hadron Collider (LHC). As a consequence, a high-precision determination of 0:025 GeV [8], where the result is dominated by high-precision data from charm production in e+e collisions. It is therefore interesting to provide alternative determinations of the charm mass from other processes, both to test the robustness of the global average and to attempt to further reduce the present uncertainty. A process directly sensitive to the charm mass is open-charm production in leptonproton deep-inelastic scattering (DIS). This process has been measured with high accuracy at the HERA collider and the results of di erent measurements implying various charmtagging techniques are combined [9]. The charm contribution to the inclusive structure functions can be determined through the measurement of the charm-pair production cross section. In addition, the nal combination of inclusive measurements from Runs I and II at HERA has been recently presented in [7]. DIS structure functions can be described using a variety of theoretical schemes, including the xed- avor number (FFN) scheme, where charm mass e ects are included to a xed perturbative order, the zero-mass variable- avor number (ZM-VFN) scheme that neglects power-suppressed terms in the charm mass but resums to all orders large collinear logarithms, and the so-called matched general-mass variable- avor-number (GMVFN) schemes, which interpolate smoothly between the two regimes. A recent discussion and summary of the application of these schemes to heavy- avor data at HERA can be found e.g. in [10]. Examples of matched general-mass schemes in electro-, photo- and hadroproduction include FONLL [11{13], TR [14{16], ACOT [17], and a scheme generically referred to as GMVFNS [18{22]. In this work we will mostly concentrate on the FONLL scheme and on its implications for the determination of the charm mass. For the sake of comparison with previous studies [9, 23{25], a determination of the charm mass in the FFN scheme at NLO is also performed. The original formulation of the FONLL general-mass scheme for DIS structure functions was derived in the pole (on-shell) heavy quark scheme [11]. In ref. [26] it was shown how DIS structure functions in the FFN scheme can be modi ed to include MS heavyquark masses. The same scheme conversion can be applied to any GM-VFN scheme, and in this work we provide the relevant expressions for FONLL structure functions with MS running masses. The main advantage of the use of MS masses is the possibility of direct connection with the precise determinations from low-energy experimental data [8]. In this work we will use the xFitter open-source framework [27] (previously known as HERAfitter) to extract the MS charm mass from a PDF t to the most up-to-date inclusive and charm data from HERA. Structure functions are computed using the FONLL scheme as implemented in the APFEL [28] code. Our results have been obtained employing the most accurate perturbative calculations presently available and will include a detailed characterization of the di erent sources of uncertainties on mc(mc) from data, theory and tting methodology. As we will show, the results are consistent with the global PDG average as well as with previous determinations based on the FFN [9, 23{25] and in the S-ACOT [29] schemes.1 The uncertainty in our results turns out to be competitive with that of previous determinations based on DIS structure functions. 1See also [30] for a recent determination of the pole charm mass from a global PDF t. { 2 { HJEP08(216)5 The outline of this paper is the following. In section 2 we discuss how FONLL can be formulated in terms of MS masses and present a benchmark of its implementation in APFEL. In section 3 we describe the settings of the PDF ts and the treatment of the uncertainties. Results for the determination of mc(mc) are presented in section 4, where we also compare with previous determinations. We conclude and discuss possible next steps in section 5. 2 FONLL with MS heavy-quark masses In this section we discuss how the FONLL general-mass variable- avor-number scheme for DIS structure functions can be expressed in terms of MS heavy-quark masses. We also describe the subsequent implementation in the public code APFEL, and present a number of benchmark comparisons with other public codes. In general, higher-order calculations are a ected by ambiguities in the prediction for the physical quantities due to the choice of the subtraction scheme used to remove divergences. In fact, di erent prescriptions imply di erent numerical values of the parameters of the underlying theory. As far as the mass parameters are concerned, the pole mass de nition is usually more common in the calculation of massive higher-order QCD corrections to heavy-quark production processes. The main reason for this is that the pole mass is, by its own de nition, more closely connected to what is measured in the experiments. On the other hand, it is well known that observables expressed in terms of the pole mass present a slow perturbative convergence. This is caused by the fact that the pole mass de nition su ers from non-perturbative e ects which result in an intrinsic uncertainty of order QCD [31]. The MS scheme, which stands for modi ed minimal subtraction scheme, is instead free of such ambiguities and as a matter of fact massive computations expressed in terms of heavyquark masses normalized in this scheme present a better perturbative convergence [26]. As a consequence, the results obtained in the MS scheme are more appropriate to achieve a reliable determination of the numerical value of the charm mass. The FONLL scheme, as any other GM-VFN scheme, aims at improving the accuracy of xed-order calculations at high scales by matching them to resummed computations. In DIS this results in the combination of massive ( xed-order) calculations, that are more reliable at scales closer to the heavy-quark masses, with resummed calculations that are instead more accurate at scales much larger than the heavy-quark masses. However, in the original derivation, the massive component of the FONLL scheme was expressed in terms of the pole masses [11]. It is then one of the goals of this paper to provide a full formulation of the FONLL scheme applied to DIS structure functions in terms of MS masses. A detailed discussion on such a formulation is given below in section 2.1. Here, we limit ourselves to describing the main steps needed. We point out that our procedure is entirely driven by the requirement of de ning DIS structure functions in terms of the MS masses order-by-order in perturbation theory. Thus, the goal of the current work is to completely replace the pole mass with the MS mass as a fundamental object in a variable avour number scheme. As discussed in ref. [29], other approaches to the same procedure exist which formally di er by subleading { 3 { contributions. Numerically, such contributions might be non-negligible raising the question of the optimal perturbative convergence at higher orders of the QCD series in terms of MS masses. Only the inclusion of higher-order corrections will help clarify this aspect. The generic form of the DIS structure functions in the FONLL approach applied to charm production is: F (x; Q; mc) = F (3)(x; Q; mc) + F (d)(x; Q; mc) F (d)(x; Q; mc) = F (4)(x; Q) F (3;0)(x; Q; mc) ; where x, Q, and mc are the Bjorken variable, the virtuality of the photon, and the mass of the charm quark, respectively. In eq. (2.1) the three- avor structure function F (3) is evaluated retaining the full charm-mass dependence and with no charm in the initial state. The four- avor structure function F (4) is instead computed by setting mc to zero and allowing for charm in the initial state, and its associated PDF reabsorbs the mass (collinear) divergences which are in turn resummed by means of the DGLAP evolution. Finally, F (3;0) represents the massless limit of F (3) where all the massive power corrections are set to zero and only the logarithmically enhanced terms are retained. This last term is meant to subtract the double counting terms resulting from the sum of F (3) and F (4). In fact, the role of F (3;0) is twofold: for Q mc, by de nition F (3) and F (3;0) tend to the same value so that the FONLL structure function reduces to F (4). By contrast, in the region where Q ' mc it can be shown that F (d) becomes subleading in FONLL structure function to F (3) up to terms beyond the nominal perturbative accuracy. s reducing the It should be noticed that, even though F (d) in eq. (2.1) becomes subleading in the low-energy region, it might become numerically relevant and it is advisable to suppress it. To this end, the term F (d) in eq. (2.1) is usually replaced by: F (d0)(x; Q; mc) = D(Q; mc)F (d)(x; Q; mc) ; (2.1) particular functional form of the damping factor given in eq. (2.3) is somewhat arbitrary. In fact, any function D such that F (d0) and F (d) only di er by power-suppressed terms, namely: D(Q; mc) = 1 + O mc2 Q2 ; is a formally suitable choice. In the results section we will also consider the e ect of varying the functional form of the damping factor in order to estimate the associated theoretical uncertainty on mc(mc). Given the possible di erent perturbative structure of the elements that compose the FONLL structure function in eq. (2.1), two possibilities for the de nition of the perturbative { 4 { ordering are possible: the relative and the absolute de nitions. In the relative de nition F (4) and F (3) are combined using the same relative perturbative accuracy, that is LO with LO, NLO with NLO, and so on. The absolute de nition, instead, is such that LO refers to O( s0) (parton model), NLO to O( s), and so forth. This issue is relevant in the neutralcurrent case where the lowest non-vanishing order is O( s0) for F (4) and O( s) for F (3)2 such that the relative and absolute orderings lead to di erent prescriptions. Beyond LO, there are currently three possible variants of the FONLL scheme, all of them implemented in APFEL: the FONLL-A variant adopts the absolute ordering at O( s) and thus only terms up to this accuracy are included. This variant is formally NLO and thus also PDFs should be evolved using the same accuracy in the DGLAP evolution. The FONLL-B variant is instead computed using the relative ordering at NLO. Therefore, F (4) is computed at O( s) and combined with F (3) at O( s2). F (3;0) is instead computed dropping the non-logarithmic O( s2) term to match the accuracy of F (4) in the low-energy region. PDFs are again evolved at NLO. Finally, the FONLL-C scheme adopts the absolute ordering at O( s2). This is formally a NNLO scheme thus PDFs should be evolved using the same accuracy. Presently, no other variant beyond FONLL-C can be pursued because the O( s3) massive coe cient functions are not known yet. Approximate NNLO corrections valid near the partonic threshold, in the high-energy (small-x) limit, and at high scales Q2 m2 have been derived in ref. [34] and they are currently employed by the ABM group to determine NNLO PDFs [6]. As clear from the description above, the computations for the three- avor structure functions F (3) and F (3;0) depend explicitly on the charm mass, while F (4) does not. In addition, as already mentioned, the expressions needed to compute F (3) and F (3;0) are usually given in terms of the pole mass. As a consequence, one of the steps required to achieve a full formulation of the FONLL structure functions in terms of MS masses is the adaptation of the heavy- avor contributions to the structure functions. A thorough explanation of the procedure adopted to perform such transformation can be found in ref. [26] for both neutral- and charged-current structure functions. In section 2.1 we re-derive the main formulae and report the full expressions for the relevant coe cient functions. It should be pointed out that the derivation presented in ref. [26] is performed assuming R = mc(mc), R being the renormalisation scale, and the renormalisation scale dependence of s is restored only at the end using the expansion of the solution of the relative RG equation. Such a procedure implies that the heavy-quark mass is not subject to the relative RG equation: in other words, the mass running is not expressed explicitly. The reason is that in the running of the heavy-quark mass in MS one can resum logarithms of R=mc(mc) and this 2This is strictly true only if the heavy-quark PDFs are dynamically generated via gluon splitting. In leading to a \realignment" of the perturbative structure between F (4) and F (3) (see refs. [32, 33]). fact, the presence of an intrinsic heavy-quark component would introduce a O( s0) contribution also in F (3) { 5 { is not required in a xed-order calculation. On the contrary, when dealing with a GM-VFN scheme like FONLL, such a resummation is an important ingredient and thus should be consistently incorporated into the derivation. For this reason, the transition from pole to MS masses of the massive structure functions presented in section 2.1 is done at the generic renormalisation scale R and the connection between mc(mc) and mc( R) is established solving the appropriate RG equation. A further complication that arises in FONLL as a VFN scheme is the fact that the involved running quantities, that is PDFs, s and the mass itself, have to be properly matched when crossing a heavy-quark threshold in their evolution. The matching conditions for PDFs and s are presently known up to O( s2) [35] and O( s3) [36], respectively, but those for PDFs are given in terms of the pole mass. In the next section we will show how to express them in terms of the MS mass up to the relevant accuracy. As far as the matching of the mass is concerned, the expressions for the matching conditions are given in ref. [37] up to O( s3) also in terms of MS mass. 2.1 Implementation In this section we will describe in some detail the implementation of the FONLL scheme in terms of the MS heavy-quark masses in APFEL. Starting from the more usual de nition of structure functions in terms of pole masses, our goal is to consistently replace them with the MS mass de nition. 2.1.1 MS mass vs. pole mass The (scale independent) pole mass M and the (scale dependent) MS mass m( ) arise from two di erent renormalization procedures and, as already mentioned, in perturbation theory they can be expressed one in terms of the other. The relation connecting pole and MS mass de nitions has been computed in ref. [31] up to four loops. However, in the following we will only need to go up to one loop and thus we report here the corresponding relation: In the following we will use eq. (2.5) to replace the pole mass M with the MS mass m( ). with: and: where CF = 4=3 is one of the usual QCD color factors. Moreover, we have de ned: M m( ) = 1 + h(1)as + O(as2) ; h(1)( ; m( )) = CF (4 + 3L m) ; as as( ) = s( ) 4 ; { 6 { (2.5) (2.6) (2.7) (2.8) In order to evaluate the running of m( ) with the renormalization scale we have to solve the corresponding renormalization-group equation (RGE): 2 dm d 2 = m( ) m(as) = 1 n=0 m( ) X m(n)asn+1 ; whose rst three coe cients can be taken from ref. [38]:3 m(0) = 4 ; m(1) = 202 3 In order to get an analytical expression out of eq. (2.14), one can expand the integrand in the r.h.s. using the perturbative expansions of m(as) and (as) given in eqs. (2.9) and (2.11). This allows us to solve the integral analytically, obtaining: 1 + (c1 1 + (c1 b1c0)a + 12 [c2 b1c0)a0 + 12 [c2 c1b1 c1b1 b2c0 + b21c0 + (c1 b2c0 + b21c0 + (c1 b1c0)2]a2 b1c0)2]a20 ; 3The following expressions have been adjusted taking into account our de nition of as which di ers by a factor of 4 with respect to that of ref. [38]. { 7 { where Nf is the number of active avors. In addition, the RGE for the running of s reads: Combining eqs. (2.9) and (2.11) we obtain the following di erential equation: whose solution is: m( ) = m( 0) a c0 a0 2 ddas2 = (as) = 1 X n=0 nasn+2 ; "Z as( ) m(as) das# : as( 0). Eq. (2.15) represents the NNLO solution of the RGE for the MS mass m( ). Of course, the NLO and the LO solutions can be easily extracted from eq. (2.15) just by disregarding the terms proportional to a2 and a20 for the NLO solution and also the terms proportional to a and a0 for the LO solution.4 2.1.3 Matching conditions When working in the context of a VFN scheme, all running quantities are often required to cross heavy-quark thresholds when evolving from one scale to another. Such a transition in turn requires the matching di erent factorization schemes whose content of active avors di ers by one unit. In other words, if the perturbative evolution leads from an energy region where (by de nition) there are Nf 1 active avors to another region where there are Nf active avors, the two regions must be consistently connected and such a connection can be evaluated perturbatively. This goes under the name of matching conditions. In general, matching conditions give rise to discontinuities of the running quantities at the matching scales and in the following we will report the matching conditions up to NNLO in terms of the MS heavy-quark thresholds for: s( ), m( ) and PDFs. Matching of s( ). The matching conditions for s were evaluated in ref. [36] to three loops. We report here the relation up to two loops (again taking into account the factor 4 coming from the di erent de nitions of a): a(Nf 1)( ) a(Nf )( ) = 1 L M a(Nf )( ) + 9 M being the pole mass of the n-th avor. From eq. (2.5) we can easily infer that: ln M 2 = ln m2( ) + 2 ln[1 + h(1)( )a(Nf )( )] = ln m2( ) + 2h(1)( )a(Nf )( ) + O([a(Nf )]2) : Therefore, it is straightforward to see that: L M = L m 2h(1)a(Nf ) = L m + 8L m a(Nf ) ; 32 3 a(Nf 1)( ) a(Nf )( ) = 1 L ma(Nf )( ) + 9 consistently with eq. (20) of ref. [37]. 4In order to be consistent, the evaluation of a and a0 eq. (2.15) must be performed at the same perturbative order of m( ). So, for instance, if one wants to evaluate the NNLO running of m( ) also the value of a and a0 must be computed using the NNLO running. 2 3 2 3 { 8 { (2.18) (2.19) In order to simplify this expression, it is a common procedure to perform the matching at the point where the logarithms vanish. In this particular case, choosing which can be easily inverted obtaining: heavy-quark thresholds. In particular, one needs to match the (Nf for the mass mq( ), with q = c; b; t, at the threshold mh( ), where h = c; b; t. From ref. [37] HJEP08(216)5 Exactly as before, if we choose to match the two schemes at the scale = mh( ) = mh(mh), the logarithmic terms vanish and we are left with: we read: m(qNf )(mh) = 1 27 89 [a(Nf 1)(mh)]2 m(qNf 1)(mh) : { 9 { (2.21) (2.22) (2.23) (2.24) (2.25) (2.26) (2.27) Matching of PDFs. To conclude the section on the matching conditions, we nally consider PDFs. One can write the singlet and the gluon in the (Nf )-scheme in terms of singlet and gluon in the (Nf 1)-scheme at any scale as follows: (Nf ) NS;(2) + A~hSq;(2)] as2Agq;h S;(2) asA~hSg;(1) + as2A~hSg;(2) ! 1 + asAgSg;(;1h) + as2Agg;h S;(2) where the form of the functions entering the transformation matrix above are given in appendix B of ref. [39] in terms of the pole mass. We omit the matching conditions for the non-singlet PDF combinations because they have no O(as) correction and the rst correction appears at O(as2). This leaves the conversion from the pole to the MS mass scheme una ected up to NNLO. In order to replace the pole mass M with the MS mass m( ), we just have to plug eq. (2.19) into eq. (2.27). In doing so, only the O(as) terms proportional to ln( 2=M 2) play a role in the conversion up to NNLO. Since the functions A~S;(1) and AgSg;(;1h) can be written as: hg replacing M with m in eq. (2.28) using eq. (2.19) leads to: Therefore eq. (2.27) in terms of m becomes: (2.28) (2.29) (2.30) (2.31) (2.32) (Nf ) As usual, we choose to match the (Nf )-scheme to the (Nf m(m) so that all the logarithmic terms vanish, obtaining: (Nf ) malization scale, which we will now denote as R. The scale that explicitly appears in the PDFs is instead the factorization scale, which we will now denote with F . In principle, renormalization and factorization scales are di erent but one usually takes them to be proportional to each other, as R = F , where can be any real number.5 The most common choice when matching the (Nf 1)-scheme to the (Nf )-scheme is to set F equal to heavy-quark thresholds (Mc, Mb and Mt in the pole-mass scheme and mc(mc), mb(mb) and mt(mt) in the MS scheme). In doing so, the logarithmic terms in the PDF matching conditions are assured to vanish. However, if is di erent from one, the logarithmic terms in the matching conditions for as( R) and mq( R) do not vanish anymore. In the following we will show how the matching conditions for as and mq change for 6= 1. 5It should be noticed that in the case 6= 1 PDFs acquire an implicit dependence on R that comes from a rede nition of the splitting functions that in turn derives from the expansion of s( R) around R = F . Let us start with s. Inverting eq. (2.20) we obtain: a(Nf )( R) a(Nf 1)( R) = 1 + c1a(Nf 1)( R) + c2[a(Nf 1)( R)]2 ; As usual, the matching scale is chosen to be F = m(m), so that: But using eq. (2.14), it is easy to see that: L m ! [1 + m(0)as( m)] ln : not matter whether one uses as subleading up to NNLO. It should be noticed that in the eq. (2.38), since a(sNf 1) = a(sNf ) + O([a(sNf )]2), it does (Nf )( m) or as ( m) because the di erence would be Therefore, setting = R = m(m) = m in eq. (2.20) and using eq. (2.38), one gets: a(Nf 1)( m) = a(Nf )( m) 1 ln a(Nf )( m) whose inverse is: a(Nf )( m) = a(Nf 1)( m) 1 + ln a(Nf 1)( m) + 4 9 ln2 4 9 4 3 4 3 ln2 2 3 2 3 2 3 2 3 20 9 ln + 89 27 [a(Nf 1)( m)]2 : Now let us turn to mq. In this case there is not much to do. In fact, for an arbitrary matching point the matching condition of the MS mass starts at O( s2) (cfr. eq. (2.23)), therefore writing L m in terms of ln would give rise to subleading terms up to NNLO (see eq. (2.38)). As a consequence, we have that: m(qNf )( mh) = 1 We nally turn to discuss how the DIS massive structure functions change when expressing them in terms of the MS masses. We will rst consider the neutral-current (NC) massive structure functions up to O( s2), which is the highest perturbative order at which corrections are known exactly, and then we will consider the charged-current (CC) massive structure functions again up to the highest perturbative order exactly known,6 that is O( s). In order to shorten the notation, we will adopt the following de nitions: M = pole mass; m m( ) = MS mass; as as( ); h(l) h(l)( ; m( )) : Neutral current. Dropping all the unnecessary dependences, the NC massive structure functions up to O(as2) have the form: HJEP08(216)5 F = asF (0)(M ) + as2F (1)(M ) + O(as3) : The goal is to replace explicitly the pole mass M with the MS mass m using eq. (2.5). To this end, following the procedure adopted in refs. [26, 41], we expand F (0)(M ) and F (1)(M ) around M = m: so that, up to O(as2), what we need is: Finally, we have that: F (l)(M ) = n=0 X1 1 dnF (l) n! dM n M=m (M m)n ; F (0)(M ) = F (0)(m) + asmh(1) dF (0) F (1)(M ) = F (1)(m) : M=m ; F = asF (0)(m) + as2 F (1)(m) + mh(1) dF (0) M=m # : (2.43) (2.44) (2.45) (2.46) (2.47) (2.49) F (0)(M ) = x where g is the gluon distribution and we have used the following de nitions: 1 1+ 4QM22 ; xmax(M ) = (z; M ) = 1; (M ) = (M ) = : (2.48) Q2 M 2 ; 2 M 2 that: De ning: G(z; M ) = Cg(0)( (z; M ); (M ); (M )) ; 6In a recent publication [40] the O( s2) corrections (NNLO) to charm production in CC DIS were presented. However, no analytical expression was provided. We now need to evaluate explicitly the derivative in eq. (2.46). First of all we observe " z g x z 1 Q2 4M 2 1 z x z g x z the derivative of eq. (2.47) can be written as: dF (0) Z xmax(M) x dxmax G(xmax; M ) = dz dxmax G(xmax; M ) : dM dM It can be shown that the boundary term in eq. (2.52) vanishes (see ref. [26]), thus it can thus: be omitted. we have that: Finally, considering that: Gathering all pieces and taking into account that: dF (0) dM M=m F (1)(M ) = = x = x X i=q;q;g x x x " Z xmax(M) dz Z xmax(m) dz z x z g z g x z M=m M=m : x q z i z x Ci(1)(z; M ) (2.50) (2.51) (2.52) (2.53) (2.54) (2.55) !# + (2.56) (2.57) and using eqs. (2.46) and (2.54), one can explicitly write down the full structure of the massive structure functions (F2 and FL) in terms of MS masses up to O( s2) as follows: F = x Z xmax(m) dz x + X x i=q;q z x g x z " M=m In order to carry out the implementation, we need to evaluate explicitly the derivative of Cg(0) in eq. (2.56) and this must be done separately for F2 and FL. We consider F2 rst. The explicit expression of C2(0;g) is the following: C2(0;g)(z; Q2; M 2) = TRn2(1 6 4 2)I2( ; z) 2(1 2 )I1( ; z) + I0( ; z)+ 4(2 )J2( ; z) + 4(2 )J1( ; z) J0( ; z)o ; Iq( ; z) = zq ln Jq( ; x) = zqv ; 1 + v 1 v = Q2 1 = = h v2 " 4(2 2 v 1 ( 2(1 v 2 v ) 1 2zq v2 ; From the de nitions in eq. (2.60), we obtain: HJEP08(216)5 4 : 1 z z 4J1 i (2.58) (2.59) (2.60) (2.61) (2.62) (2.63) (2.64) (2.65) with: Therefore: To nd the explicit expression, we just need to evaluate the derivative of Iq and Jq starting from eqs. (2.58) and (2.59) which is easily done: M TR 2 2( 6 8 )I2 + 4I1 + 4J2 2(1 6 2(1 2 ) In the end we get: M TR 4 ( 6 8 )z2 + 2z ln + 8 z(z 1)v 1 + v 1 v 6 4(2 4 2)z2 2(1 2 )z + 1 )z2 + 4(2 )z 1 ) : The implementation of the FONLL scheme given in eq. (2.1) requires the massless limit of the massive structure functions. In practice this means that one needs to compute the limit M ! 0 of the massive coe cient functions retaining the logarithmic enhanced terms. In order to apply this recipe to eq. (2.64), we observe that: M!!0 0 ; v M!!0 1 ; and that: 1 + v 1 v M!!0 ln M 2z z) ; 2 M = TR 2z2 2z + 1 : We now turn to consider FL. In this case the the gluon coe cient function takes the Therefore, using eq. (2.61), we immediately get: M TR It is nally easy to realize that: = 0 : Charged current. In this section we consider the CC massive structure functions. The treatment follows the exact same steps as the NC structure functions, with the only difference being that in the CC case the rst non-vanishing term is O(as0). This means that, truncating the perturbative expansion at O(as), we have: Fk = Fk(0)(M ) + asFk(1)(M ) + O(as2) ; with k = 2; 3; L. Therefore, expanding F (0) and F (1) around M = m and keeping only the terms up to O(as), one obtains: " Fk = Fk(0)(m) + as Fk(1)(m) + mh(1) dFk(0) # : dM M=m The leading-order contribution can be written as follows: CL(0;g) z; Q2; M 2 = TR [ 8 I2( ; z) 4J2( ; z) + 4J1( ; z)] : (2.68) where we have also de ned: Fk(0)(M ) = bk(M )s0( (M )) ; = x 1 + | x and 8 < >> b2 = b3 = 1 > >: bL = (1 ) ; s0 = 2jVcsj2s + 2jVcdj2d : (2.66) (2.67) (2.69) (2.70) (2.71) (2.72) (2.73) (2.74) (2.75) d bk ) ) ) d d 1 ; M=m M=m ; ; (2.76) (2.77) (2.78) mh(1) dFk(0) dM M=m = mh(1) d dFk(0) dM that can be conveniently rewritten as: so that, using eq. (2.74), we have that: mh(1) dFk(0) dM = 2h(1)(1 ) dbk d d d + bk( ) s0( ) M=m mh(1) dF2(0) mh(1) dF3(0) mh(1) dFL(0) dM dM dM M=m M=m M=m = 2h(1)(1 = 2h(1)(1 = 2h(1)(1 Finally, we notice that in the massless limit, where ! 1, all expressions in eq. (2.78) vanish, with the consequence that the CC massive structure functions up to O(as) in terms of the pole mass M or the MS mass m are exactly the same. 2.2 In order to validate the implementation in APFEL, we have benchmarked it against public codes. To the best of our knowledge, there exist no public codes able to compute structure functions in the FONLL scheme with MS masses. For this reason the best we could do is to benchmark the various ingredients separately. As a rst step, we present the benchmark of the running of PDFs, s and mc7 in the VFN scheme with MS heavy-quark thresholds. The di erence with respect to the more common pole-mass formulation arises from the fact that the matching of the evolutions at the heavy-quark thresholds needs to be adapted to take into account the di erent scheme used to renormalize the masses. The full set of such matching conditions for PDFs, s and mc has been collected in section 2.1. We start with the DGLAP PDF evolution in the VFN scheme with MS heavy-quark thresholds. A careful benchmark was already presented in the original APFEL publication. In particular, the APFEL evolution has been checked against the HOPPET code [42] v1.1.5, nding a very good agreement at the O 10 4 level or better. Since then, APFEL has undergone several changes and improvements and thus we repeated the benchmark using the same settings and nding the same level of agreement with HOPPET, as shown in gure 1 for a representative set of combinations of PDFs.8 7The running of mb and mt has also been checked nding the same lavel of accuracy found for mc. 8We observe that, thanks to a better interpolation strategy, the predictions at the transition regions between internal x-space subgrids is now smoother. L E F P A f( x 100 10 1 xg(x,μF) xuv(x,μF) xdv(x,μF) xs+(x,μF) μF = 100 GeV 10−4 10−3 x 10−2 10−1 HJEP08(216)5 at NNLO with MS heavy-quark thresholds. The evolution settings, i.e. initial scale PDFs, reference value of s, and heavy-quark thresholds, are the same as used in the Les Houches PDF evolution benchmark [43]. The upper inset shows the gluon PDF xg, the valence up and down PDFs xuv xu xu and xdv xd xd, respectively, and the total strangeness xs+ xs + xs at F = 100 GeV as functions of the Bjorken variable x as returned by APFEL. In the lower inset the ratio to HOPPET is displayed showing a relative di erence of 10 4 or better all over the considered range. Although the benchmark of the DGLAP evolution already provides an indirect check of the evolution of s, we have also performed a direct check of the VFNS evolution with MS heavy-quark thresholds of s along with the evolution of the MS charm mass. To this end, we have used the CRunDec code [44], which is the C++ version of the Mathematica package RunDec [37]. In gure 2 we show the comparison between APFEL and CRunDec for the three-loop evolution (NNLO) of the strong coupling s (left plot) and the charm mass mc (right plot). As is clear from the lower insets, the agreement between the two codes is excellent. Also the one- and two-loop evolutions have been checked nding the same level of agreement. Finally, we benchmarked the implementation of massive DIS structure functions (i.e. F (3) in eq. (2.1)) with MS masses against the public code OPENQCDRAD v1.6 [45]. OPENQCDRAD implements DIS structure functions in terms of the MS heavy-quark masses following the formalism discussed in ref. [26]. However, as already mentioned above, such a procedure does not directly correspond to what is needed for the implementation of the FONLL scheme. In order to make the comparison with OPENQCDRAD possible, we have implemented in APFEL a variant of the FONLL scheme with MS masses where, as done in OPENQCDRAD , the RG running of the heavy-quark masses is expanded and truncated to the appropriate order. In gure 3 we show the comparison between APFEL and OPENQCDRAD for the exclusive charm neutral-current structure functions F2c (left plot) and FLc (right plot) at O( s2) for three di erent values of Q2 and over a wide range of x. As is clear from the lower ratio plots, the agreement is typically at the per-mil level except in the very large-x region 1.3 0.8 e D1.000001 n /CR1.000000 E F0.999999 P tion with MS heavy-quark thresholds of the strong coupling mc (right plot). The evolution settings are: (nf =3)(p2 GeV) = 0:35, m(cnf =4)(mc) = p s 2 GeV, s (left plots) and the MS charm mass and m(nf =5)(mb) = 4:5 GeV. The upper insets show the strong coupling s (left) and the charm b mass mc (right) as functions of the renormalization scale R as returned by APFEL. In the lower insets the ratios to CRunDec are displayed showing a relative di erence well below 10 6 over the complete range considered. 3APFEL v2.7.0 vs. OpenQCDrad v1.6, F2c(x,Q) at O(αs2) in the FFNS 0.6APFEL v2.7.0 vs. OpenQCDrad v1.6, FLc(x,Q) at O(αs2) in the FFNS A 2 5 50 100 2 5 50 100 10 μR [GeV] 20 10 L E E F F P P 10−2 10−1 A A 10−4 10−3 x 10−2 10−1 charm structure functions with MS heavy-quark masses at O( s2). As an input PDF set we have used MSTW2008nlo68cl nf3 [46] from which also the numerical values of s and mc are taken. The upper insets show F2c (left) and FLc (right) as functions of x for Q2 = 10; 100; 1000 GeV2 as returned by APFEL. In the lower insets the ratios to OPENQCDRAD are displayed showing a relative di erence at the per-mil level except in the very large-x region where, due to the smallness of the predictions, the relative di erences tend to increase but maintain a good level of absolute accuracy. where, due to the smallness of the predictions, the relative di erence tends to increase but maintains a good level of absolute accuracy. To conclude this section, we observe that, referring to eq. (2.1), the introduction of the MS masses does not a ect the four- avor structure function F (4). The structure function F (3;0) is instead a ected by the transition from pole to MS masses. Since we are not aware of any public code that computes such structure functions, a direct bechmark has not been possible. However, as a sanity check we have checked that F (3;0) and F (3) for large values of Q2 tend to the same value, as the de nition of F (3;0) requires. 3 QCD ts were performed to the combined H1 and ZEUS charm production crosssection measurements [9] together with the combined HERA1+2 H1 and ZEUS inclusive DIS cross-section data [7], accounting for all given sources of systematic uncertainties. The kinematic region covered by HERA is constrained by the invariant mass of the hadronic system of W > 15 GeV and the Bjorken scaling variable of x < 0:65, therefore target mass corrections are expected to have negligible e ects and are not discussed in this paper. The settings of the QCD ts in xFitter closely follow those used for the HERAPDF2.0 PDF extraction [7], with a few di erences related to the speci cs of the current analysis which are motivated in the following. The nominal result is extracted using the FONLL-C variant of the FONLL scheme discussed in section 2. It should be pointed out that, while being accurate at NNLO for the inclusive DIS cross sections, the sensitivity to mass corrections of the FONLL-C scheme is actually NLO. The reason is that at O( s0) the FONLL scheme reduces to the parton model which is insensitive to heavy-quark mass e ects. Therefore, the rst mass-sensitive term is O( s) which is the accuracy of the FONLL-A scheme which would thus provide a LO determination of the charm mass. Both the FONLL-B and the FONLL-C schemes, instead, include the O( s2) massive corrections and thus would both produce determinations of the mass of the charm accurate at NLO. The advantage of FONLL-C with respect to FONLL-B is that it is accurate at O( s2) also in the massless sector and thus it is supposed to provide a better description of the data. In other words, FONLL-C is the most accurate variant of the FONLL scheme presently available and as such it will be employed for our determination of mc(mc). The result obtained in the FONLL scheme is accompanied by an analogous determination of mc(mc) obtained using the FFN scheme with MS masses [6] at NLO. Access to the structure functions calculated with the FFN scheme is possible via the xFitter interface to the OPENQCDRAD program [45] using the QCDNUM program for the PDF evolution [47]. The procedure to determine the MS charm mass follows closely the methodology described in ref. [9]. It involves a series of ts in each of which a set of PDFs is determined corresponding to numerical values of charm mass ranging between mc(mc) = 1:15 GeV and mc(mc) = 1:60 GeV with steps of 0:05 GeV. For each value of mc(mc) a value of global 2 is obtained. The best t value of mc(mc) is determined from the minimum of the parabolic t to the resulting 2 distribution and the associated 1uncertainty, which re ects the sensitivity of the data set to the charm mass, is determined as the 2 = 1 variation around the minimum. We now discuss the settings of the nominal ts and the variations that we performed to assess the di erent sources of uncertainty deriving from: the PDF parametrization, the model parameters, and the theoretical assumptions. The assumption that heavy-quark PDFs are dynamically generated via gluon splitting at the respective thresholds requires that the starting scale Q0 at which PDFs are parametrized is below the charm threshold, which in turn is identi ed with mc(mc). Given the range in which the scan of mc(mc) is done (from 1.1 to 1.6 GeV), we have chosen to set Q0 = 1 GeV to allow all ts to be parametrized at the same starting scale. The combinations and the relative functional forms of the initial scale PDFs have been chosen following the parametrization scan procedure as performed for the HERAPDF2.0 determination [7], and the optimal con guration has been found to be: xg(x) = xuv(x) = xu(x) xdv(x) = xd(x) xU (x) = = AgxBg (1 x)Cg A0gxBg0 (1 x)25; xu(x) = Auv xBuv (1 xd(x) = Adv xBdv (1 xu(x) = AU xBU (1 x)Cuv (1 + Euv x2); x)Cdv ; x)CU (1 + DU x); xD(x) = xd(x) + xs(x) = ADxBD (1 x)CD : There are 14 free parameters, since additional constraints were applied as follows. The QCD sum rules are imposed at the starting scale and constrain the normalisation parameters Ag, Auv , Adv . The light-sea quark parameters that a ect the low-x kinematic region BU and BD, as well as the normalisation parameters AU and AD, are constrained by the requirement that u ! d as x ! 0, leading to the following constraints: BU = BD; AU = AD(1 fs); (3.1) (3.2) (3.3) with fs being the strangeness fraction of D assumed at the starting scale, i.e. fs = s=D, because HERA data alone are not able to provide a precise light-sea avor separation. The strangeness fraction for the nominal ts is set to fs = 0:4, as in the HERAPDF2.0 analysis [7]. In order to estimate the uncertainty associated to the PDF parametrization, we have considered the following variations with respect to the nominal con guration: we have moved up the initial scale Q0 from 1 to restricted the mc(mc) range in which we did the scan because we could not use values of the charm mass such that mc(mc) < p1:5 GeV. We were however able to perform the parabolic t in order to nd the best t value of mc(mc). This complication does not arise in the FFN scheme in which there is no threshold crossing. p1:5 GeV. In the FONLL scheme, this In the xuv distribution we have included an additional linear term so that the last factor in second line of eq. (3.1) reads (1 + Duv x + Euv x2). After trying di erent variations of the parametrization, we found that this particular choice leads to the largest di erences. Moving to the model parameters, the values of the bottom and top quark masses for the nominal ts are chosen to be equal to the PDG values, de ned in the MS scheme, i.e. mb(mb) = 4:18 GeV and mt(mt) = 160 GeV [8]. The value of the strong coupling is set to s(MZ ) = 0:118. It should be pointed out that this value of s assumes 5 active avors. For the FFN scheme ts, though, one needs to use the value of s with 3 active avors. In order to nd this value one has to evolve s(MZ ) down to below mc(mc) in the VFN scheme and evolve back to MZ with 3 active avors. We have computed the value of s with 3 active avors for each of the values of mc(mc) considered. The uncertainty associated to model parameters will be estimated by considering the following variations: the bottom mass has been moved up and down by 0.25 GeV, i.e. mb(mb) = 3:93 GeV and mb(mb) = 4:43 GeV. The magnitude of the variation is actually much larger than the present uncertainty on the bottom mass and thus our choice is meant to provide a conservative estimate of the associated uncertainty. The variation of the strong coupling follows the recent PDF4LHC prescription [48]. In particular, we have considered the conservative variation up and down by 0.0015 with respect to the nominal value, i.e. s(MZ ) = 0:1165 and s(MZ ) = 0:1195. Finally, we considered the value of the strangeness fraction introduced in eq. (3.3) as being a model parameter and we have thus varied it up and down by 0.1 around the nominal value considering fs = 0:3 and fs = 0:5. We nally turn to the theory assumptions and their variations. These mostly concern unknown higher-order corrections and the most common way to estimate them is by varying the renormalization and the factorization scales R and F . As nominal scales in our analysis we have chosen 2R = 2F = Q2 for both the FONLL9 and the FFN scheme analyses. Another possible source of theoretical uncertainty in the FONLL scheme is the presence of the damping factor discussed in section 2 which is meant to suppress unwanted subleading terms and whose explicit form in the nominal ts is given in eq. (2.3). The theoretical uncertainty associated to the missing higher-order corrections has been estimated as follows: the factorization and renormalization scales were varied by a factor 2 up and down with respect to the nominal values, that is choosing 2R = 2F = Q2=2 and 2R = 2Q2. Such variations have been applied only to the heavy-quark components of the 2F = structure functions, while the light part has been left unchanged. The reason for this is that, in order to estimate the theoretical uncertainty associated to the determination of mc(mc), we want to perform scale variations only in the part of the calculation sensitive to this parameter, which is clearly the charm structure function (for consistency, the same variation was applied also to the bottom structure functions). As already mentioned, the FONLL damping factor represents a further source of uncertainty. It has the role of suppressing unwanted subleading terms but the particular 9A scale choice involving the heavy-quark mass would lead to technical complications with the FONLL matching as implemented in APFEL. However, we have checked that the more commonly used scales 2F = Q2 +4mc(mc)2 produce a very marginal di erence in the determination of mc(mc) in the FFN scheme. 2R = way in which this suppression is implemented is somewhat arbitrary. To assess the impact of our particular choice on the determination of mc(mc), we have changed the suppression power around the nominal one, considering the following functional form: Dp(Q; mc) = (Q2 In addition, to assure the applicability of perturbative QCD and to keep higher-twist corrections under control, a cut on Q2 is imposed on the tted data. Our nominal cut is Q2 > Q2min = 3:5 GeV2. The choice of the value of Q2min requires some care; an extensive discussion on the impact of varying it on the determination of mc(mc) is given in section 4.3. To conclude this section, we observe that the self-consistency of the input data set and the good control of the systematic uncertainties enable the determination of the experimental uncertainties in the PDF ts using the tolerance criterion of In this section we will present the result for our the determination of the value mc(mc) in the MS renormalization scheme using the FONLL scheme with its associated set of The parabolic t to the global 2 as a function of mc(mc) is shown in gure 4 and yields a best t value and its 1- experimental uncertainty equal to mc(mc) = 1:335 0:043 GeV. An estimate of the parametric, model, and the theoretical uncertainties, performed following the procedure described in section 3, is summarised in the second column of table 1 and leads to our nal result: mc(mc) = 1:335 0:043(exp)+00::001090(param)+00::001018(mod)+00::003038(th) GeV. (4.1) An illustration of the deviations, again determined through parabolic ts, caused by the variations employed to determine the parametric, model, and theoretical uncertainties is given in gure 5. After we have determined the best t value of the charm mass in eq. (4.1), we have used the central value to perform a further t in the FONLL-C scheme (nominal t). In table 2 we report the partial 2's over the number of data points for each subset along with the total correlated 2, the logarithmic penalty, and the total 2 per degree of freedom. As an illustration, the singlet and the gluon PDFs extracted from the nominal ts are compared with other GM-VFNS PDF sets: CT14 [5], HERAPDF2.0 [7], MMHT14 [50], NNPDF3.0 [3]. They are shown in gure 6 at the scale Q2 = 10 GeV2, where the the experimental uncertainties from the nominal ts on PDFs are estimated using Monte Carlo procedure with the root mean square estimated from 500 replica. An overall good agreement is observed. The FONLL determination of mc(mc) presented above is supported by an analogous determination in the FFN scheme at NLO. The corresponding parabolic t with the associated experimental uncertainty is shown in gure 7. Also in this case a full characterization FONLL−C mc(mc) = 1.335 ± 0.043 GeV 1 1.2 1.4 1.6 mc(mc) [GeV] 2F = 2R = 2 Q2 2F = 2R = Q2=2 FONLL Damping power = 1 FONLL Damping power = 4 with the respective results obtained in the FONLL-C scheme and in the FFN scheme at NLO. FONLL-C 1:335 0:043 1:354 [+0:019] 1:340 [+0:005] 1:338 [+0:003] 1:332 [ 0:003] 1:330 [ 0:005] 1:343 [+0:008] 1:342 [+0:007] 1:329 [ 0:006] 1:347 [+0:012] 1:361 [+0:026] 1:352 [+0:017] 1:327 [ 0:008] 1:318 0:054 2 1540 χ1520 1500 1480 1460 1440 1420 1400 1380 1360 1340 HERA−I+II inclusive + charm FONLL−C Damping power = 1 mc(mc) = 1.352 ± 0.041 GeV FONLL−C Damping power = 4 mc(mc) = 1.327 ± 0.044 GeV 1 1.2 1.4 2's as functions of mc(mc) in the FONLL-C scheme for all variations performed to estimate the non-experimental uncertainties on mc(mc). Data Set Charm cross section H1-ZEUS combined HERA1+2 CCep HERA1+2 CCem HERA1+2 NCem HERA1+2 NCep 820 HERA1+2 NCep 920 HERA1+2 NCep 460 HERA1+2 NCep 575 Correlated Log penalty 2 Total 2 / d.o.f. 2 mass mc(mc) = 1:335 GeV. The partial 2's per data point along with the total correlated logarithmic penalty, and the total 2 / d.o.f. are reported, as de ned in ref. [49]. ,(xQ 5 Σ x 4 3 2 1 0 10­4 HJEP08(216)5 Q2 = 10 GeV2 NNPDF3.0 MMHT2014 CT14 HERAPDF20_EIG FONLL­C (MCerr) 10­3 10­2 10­1 10­4 10­3 10­2 10­1 x 1 butions from the nominal FONLL-C t with other PDF sets determined using GM-VFN schemes: HERAPDF2.0, CT14, MMHT14, NNPDF3.0. Q , x (xg 20 15 10 5 0 44 / 47 43 / 39 55 / 42 218 / 159 67 / 70 439 / 377 220 / 204 219 / 254 104 +12 1420 / 1178 1460 1400 1380 1 HERA­I+II inclusive + charm FF (MS) mc(mc)=1.318 ± 0.054 GeV mc(m ) [GeV] of the non-experimental uncertainty has beed achieved by carrying out the same parametric, model, and theory variations (except for the variation of the damping factor which is speci c of the FONLL scheme). The results of the variation in the FFN scheme are reported in the third column of table 1. The nal result is: mc(mc) = 1:318 0:054(exp)+00::001110(param)+00::001159(mod)+00::004054(th) GeV ; (4.2) which is in agreement with the FONLL determination given in eq. (4.1). It is interesting to notice that we observe a reduced scale dependence in the FONLL scheme as compared to the FFN scheme. We ascribe this e ect to the fact that the leading contributions in the FONLL scheme involve both gluon- and quark-initiated processes; typically the contributions from gluon processes decrease with the scale, while the contributions from quark processes tend to increase. Conversely, the FFN scheme is mostly driven by gluon processes the contributions of which (along with s) tend to be monotonic in leading to larger scale variations.10 As discussed section 2.1.3, the running of the MS heavy-quark masses in the VFN scheme, exactly like the running of s and PDFs, is not univocally de ned at the heavyquark thresholds due to the presence of the so-called matching conditions. In particular, when giving the value of the mass at one of the heavy-quark thresholds, one should also 10We thank Fred Olness for this interesting observation. specify whether this corresponds to the value immediately below or above the threshold itself. This is typically done by complementing the value with the number of active avors used in the computation. In fact, in general m(cNf =3)(mc) 6= m(cNf =4)(mc). On theoretical grounds, this di erence is relevant when comparing a determination obtained in a VFN scheme like FONLL with a determination obtained in the (Nf = 3) FFN scheme: in the latter one automatically determines m(cNf =3)(mc), while in the former it is more natural to extract m(cNf =4)(mc). However, eqs. (2.25) and (2.26) tell us how the two values are connected up to O( s2) and applying eq. (2.25) to the central value eq. (4.1) one gets m(cNf =3)(mc) = 1:339 GeV, that is a di erence of 0.004 GeV as compared to the nominal value which is well within the current uncertainty on mc(mc). We can then conclude that, even though providing a value mc(mc) is ambiguous if the number of active avors is not speci ed, the magnitude of the ambiguity is currently not large enough to signi cantly a ect the current determinations. Comparison to other results It is interesting to compare our results with the past determinations of MS charm mass mc(mc) using a similar methodology (also see ref. [10, 25, 29] for previous comparisons). The analysis of ref. [24] was performed in the ABM11 framework [51] using the FFN scheme at NLO and at approximate NNLO and based on world data for DIS from HERA, and xed-target DIS experiments and Tevatron Drell-Yan data. While the analysis in ref. [24] was performed including the same exclusive charm cross-section data used in this study, it did not include the HERA1+2 combined inclusive cross-section data set which was not available at the time, but used instead the HERA combined data from run 1 only. An earlier analysis [23] used a partial charm dataset only, with correspondingly larger uncertainties, while a subsequent analysis [25] investigated the correlation between the measurement of mc(mc) and the strong coupling constant. The analysis of ref. [29] is instead based on the CT10NNLO global analysis, and uses the S-ACOT GM-VFN scheme discussed, e.g., in ref. [17]. It is based on a slightly wider data set as it includes LHC jet production data and also a set of older F2c measurements at HERA [52] that are not included in the more recent combined charm data. The authors of ref. [29] provide a set of four determinations deriving from di erent strategies to convert the pole-mass de nition into MS. They also provide a separate estimate of the uncertainty due to the O( s3) corrections for one of the four strategies essentially by varying the parameter that governs a generalized version of the rescaling variable . Finally, a determination of the charm mass mc(mc) was produced by the H1 and ZEUS collaborations in the framework of the HERAPDF QCD analysis in the same publication in which the charm cross-section measurements employed in our study were presented [9]. That determination also used only the HERA combined inclusive data from run 1 [53]. In table 3 we report the numerical values for the mc(mc) determinations listed above along with our results and the world average value [54]. A short clari cation about the nomenclature of the uncertainties reported in table 3 is in order. In section 3 we discussed extensively the meaning of the uncertainties associated to our determinations. In doing so, HJEP08(216)5 scheme 1:335 1:318 1:26 1:24 1:15 0:05(exp) 0:03(mod) 0:02(param) 0:02( s) 0:03(exp)+00::0032(scale)+00::0007(th) (approx. NNLO) 0:04(exp)+00::0040(scale) (NLO) 1:12+00::0151 (strategy 1) 1:18+00::0151 (strategy 2) 1:19+00::0165 (strategy 3) 1:24+00::0165 (strategy 4) 1:275 0:025 minations extracted in this work. The PDG world average value is also reported for reference. we tried to be consistent with the previous determinations, nevertheless some di erences remain. As far as the determination in ref. [9] is concerned, while their de nition of \(exp)" and \(param)" essentially coincides with ours, their \(model)" uncertainty includes the variation of the cut in Q2 (that we will discuss separately in section 4.3) but does not include the s variation, which is instead quoted separately. In addition, the authors do not quote any scale variation uncertainty. The nomenclature of ref. [24] is also di erent from ours. Apart from the common \(exp)" uncertainty, for the NLO determination the authors only quote the \(scale)" uncertainty, which essentially coincides with our \(th)" (even though the FONLL \(th)" uncertainty also accounts for the variation of the damping factor), while for the approximate NNLO determination they also quote a \(th)" uncertainty which, di erently from our nomenclature, accounts for the uncertainty on the approximated expressions used at O( s3). Finally, the determinations in ref. [29] only quote the experimental uncertainty (the asymmetric uncertainties are due to the use of a generic second-degree polynomial to t the 2 pro les). A graphical representation of table 3 is shown in gure 8 where the inner error bars display the experimental uncertainty while the outer error bars (when present) are obtained as a sum in quadrature of all uncertainty sources. The blue vertical band represents the world average and provides a reference for all other determinations. It is clear that, while the spread of the current determinations of mc(mc) from DIS data covers a pretty large range, they are generally in agreement with the world average. As far as our determinations in particular are concerned, we observe that, apart from being consistent with each other and with the world average, they also present competitive uncertainties. This is particularly relevant for the FONLL determination because this is the rst time that this scheme is employed for a direct determination of the charm mass. Figure 8 shows that our determinations tend to be larger than the world average while most of the previous determinations place themselves below it. Detailed investigations show that the largest contribution to this di erence arises from the use of to the new FF (this work) FF (HERA) FF (Alekhin et al. approx. NNLO) FF (Alekhin et al. NLO) S−ACOT−χ (CT10 str. 1) S−ACOT−χ (CT10 str. 2) S−ACOT−χ (CT10 str. 3) S−ACOT−χ (CT10 str. 4) World average display the experimental uncertainty while the outer error bars (when present) are obtained as a sum in quadrature of all uncertainty sources. The blue vertical band represents the world average and provides a reference for all other determinations. combined HERA1+2 combined inclusive cross section measurements that are employed for the rst time to determine the charm mass and that, as we will discuss in section 4.3, tend to prefer larger values of mc(mc). 4.2 It is worth mentioning that we have also employed the variants A and B of the FONLL scheme discussed in section 2 to determine mc(mc). While the FONLL-A scheme is accurate to LO in the massive sector and thus does not produce a reliable determination of the charm mass, the FONLL-B has the same formal accuracy in the massive sector as FONLL-C and indeed it leads to a determination comparable to that given in eq. (4.1) both for the central value and the uncertainties. It is interesting to notice that the FONLL-B scheme in the low-energy region resembles very closely the FFN scheme at NLO. In particular, both schemes are accurate to O( s2) in the massive sector and to O( s) in the light sector. As a matter of fact, we nd that the experimental uncertainty associated to the FONLL-B determination is very close to the FFN one quoted in eq. (4.2), which in turn is around 20% larger than that associated to the FONLL-C determination. This suggests that the O( s2) corrections to the light sector that are present in the FONLL-C scheme, which depend on the heavy-quark mass by means of diagrams in which a gluon splits into a pair of heavy quarks, provide a further constraint on mc(mc). Finally, we have also attempted a determination in the FFN scheme using the approximate NNLO massive structure functions as implemented in OPENQCDRAD. However, we did not pursue a full characterization of the uncertainties because we believe that this determination, while giving a quantitative indication of the e ect of the NNLO corrections, cannot claim an NNLO accuracy and thus does not add anything to our NLO determinations. Discussion on the Q2min dependence of the mass determination Our determination of mc(mc) given in eq. (4.1) was obtained cutting o all data with Q2 < Q2min = 3:5 GeV2. The necessity of such a cut stems from the fact that low-energy data are hard to describe for two main reasons: the large value of s with consequent large higher-order corrections, and sizable higher-twist corrections. In addition, as pointed out in ref. [55], the low-Q2 region (low-x, in fact) might be a ected by deviations from the xed-order DGLAP evolution whose description might require small-x perturbative resummation. The dependence on Q2min of ts to HERA data has already been discussed in the context of the inclusive measurements only. In this section, we will address this issue considering also the HERA charm production data. The particular value of Q2min used in our analysis (3.5 GeV2) was determined by requiring a good t quality but maintaining a good sensitivity to mc(mc). This is illustrated in gure 9 where the global 2 per degree of freedom is plotted as a function of Q2min in the left panel while the best t of mc(mc) is plotted as a function of Q2min in the right panel. Looking at the left panel it is clear that, as expected, the global 2 improves as more and more low-energy data are excluded from the t. On the other hand, the right plot shows that the experimental uncertainty associated to mc(mc) gets larger and larger as Q2min increases indicating that, again as expected, the sensitivity to mc(mc) deteriorates if low-energy data are excluded. In the light of the plots in gure 9, we conclude that Q2min = 3:5 GeV2 represents a good compromise between a good description of the full data set and a good sensitivity to mc(mc). In this context, it is interesting to look at the behaviour of the partial 2's as a function of Q2min of the charm and inclusive cross-section data separately to assess in a more speci c way which nominal value of Q2min is more convenient. Since the meaning of \degrees of freedom" is unclear for a subset of the full data set, in order to quantify the degree of improvement in the partial 2's, we consider the following quantity: 2 Npoints (Q2min) = 2(Q2min) Npoints(Q2min) 2(Q2min = 2:5 GeV2) Npoints(Q2min = 2:5 GeV2) ; (4.3) which provides an estimate of the improvement of the 2 per data point with respect to our lowest cut Q2min = 2:5 GeV2. If for a given value of Q2min this quantity is larger than one, this means that that speci c cut leads to an improvement of the 2 which is larger than the degrees of freedom subtracted by excluding a given number of data points and thus the excluded data points with respect of the reference cut (2.5 GeV2) are poorly described. On the contrary, if the quantity in eq. (4.3) is smaller than one, this means that the excluded HJEP08(216)5 Global dataset, FONLL−C 1.4 m ( dependence of the global best t value of mc(mc) with the associated experimental uncertainty as a function of Q2min. Both plots have been obtained using the FONLL-C scheme. data points are better described than the tted ones. In the left panel of gure 10 we show the behaviour of the contribution to the global 2 = Npoints originating from the charm data points only. It is clear that any cut between 3.5 and 5 GeV2 improves drastically the partial 2 while cuts above 5 GeV2 either cause a much less signi cant improvement or even lead to a deterioration. This provides a further con rmation of the fact that our nominal cut (3.5 GeV2) is a sensible choice. It is also interesting to look at the best t values of mc(mc) and the relative uncertainty preferred by a given subset as a function of Q2min to quantify the sensitivity to mc(mc) as more and more data are excluded from the t. This is plotted in the right panel of gure 10 for the charm cross-section data. It is clear that this particular subset of data tends to prefer values of mc(mc) around 1.23 GeV which is substantially lower than the global value given in eq. (4.1). The stability of the central value of mc(mc) for di erent values of Q2min is remarkable and, as expected, the experimental uncertainty tends to increase for larger value of Q2min indicating a loss of sensitivity. Finally, we have done the same exercise for the HERA1+2 inclusive cross-section data and in gure 11 we present the relative plots. In the left panel we observe that the this subset improves essentially monotonically as Q2min increases while from the right panel it is clear that the preferred value of mc(mc) of the inclusive cross sections is substantially larger than that preferred by the charm cross sections with, again, uncertainties than become broader for larger values of Q2min. It is nally clear that our best value for mc(mc) quoted in eq. (4.1) is a compromise between the lower value preferred by the exclusive 2 of charm data and the larger value preferred by the inclusive data. 4.4 Discussion on the sensitivity to mc(mc) of the inclusive data It is clear from the right panels of gures 10 and 11 that the exclusive charm and inclusive data subsets prefer somewhat di erent values of mc(mc). However, the values shown in these gures are clearly correlated because they were obtained in a simultaneous t to all data. In order to investigate a possible tension, we have performed a t to the inclusive data H1−ZEUS combined charm cross sections, FONLL−C H1−ZEUS combined charm cross sections, FONLL−C m ( )c (mc 1.4 1.55 2= Npoints as a function of Q2min for the charm cross-section subset. Right plot: dependence of the best t value of mc(mc) with the associated experimental uncertainty as a function of Q2min for the charm cross-section subset in the combined t. Both plots have been obtained using the FONLL-C scheme. HERA1+2 combined inclusive cross sections, FONLL−C HERA1+2 combined inclusive cross sections, FONLL−C 2 s t ino 1.4 p 1.6 0.8 Qmin2 [GeV2] 20 30 2 3 5 20 30 10 the low Q2 part of the inclusive data. only using both the FONLL-C and FFN schemes. The 2 pro les are shown in gure 12. In contrast to gures 4 and 7, in both schemes the scan in mc(mc) of the ts to inclusive data only yielded a shallow 2 dependences with a minimum around 1:7 GeV. This demonstrates that the inclusive data alone cannot constrain mc(mc) reasonably well, but also why this data exerts an upwards pull on the mc(mc) value in the combined t. Furthermore, since gures 9, 10, and 11 in section 4.3 present an overall remarkable stability of the central value of mc(mc) for di erent values of Q2min, the observed feature cannot be attributed to 5 In this work we have presented a new determination of the MS charm quark mass mc(mc) obtained by tting HERA charm and inclusive DIS data. In particular, we included in our ts the combined H1 and ZEUS charm production cross-section measurements [9] and the nal combination of HERA1+2 H1 and ZEUS inclusive DIS cross-section data [7], the latter being used in this work for the rst time for the extraction of the charm mass. Our 2 χ1440 1420 1360 1300 HERA−I+II inclusive data only red circles indicate the FONLL-C scheme while the blue squares the FFN scheme at NLO. determination is based on the FONLL general-mass variable- avor-number scheme, and has required the generalization of the FONLL structure functions, originally constructed in the pole-mass scheme, in terms of MS heavy quark masses. A detailed estimate of the various sources of uncertainty that a ect our determination of mc(mc) has been performed. In particular, we estimated the uncertainties due to the choice of the PDF parametrization, the model parameters used as input for the theoretical computations, and the missing higher-order corrections. We found that those sources of uncertainty are smaller than the experimental uncertainty, resulting in a competitive determination of the charm mass. We complemented the FONLL extraction of the charm mass with an analogous determination based on the xed- avour number scheme at next-to-leading order, nding a good agreement between the two. In addition, we compared our results with previous determinations also based on ts to DIS data and with the PDG world average nding again a generally good agreement. We nd that the values extracted in this work, although compatible within uncertainties, tend to be slightly higher than previous determinations from HERA data. This feature seems to be associated to the nal HERA1+2 combined inclusive dataset, which tends to prefer larger values of mc(mc) as compared to the charm structure function data, and thus increases the best- t value. In the future, it would be interesting to repeat the FONLL determination in the context of a global PDF analysis, since, in addition to the inclusive and charm HERA data, other experiments are expected to have some sensitivity to the value of the MS charm mass. In addition, the use of a wider dataset might lead to a reduction of the experimental uncertainties of the mc(mc) determination. Moreover, our analysis is based on the standard assumption that the charm PDF is dynamically generated by collinear splitting from gluons and light quarks. In this respect, it would be useful to redo the determination of mc(mc) in the presence of a possible non-perturbative charm PDF, for which the generalized FONLL structure functions accounting for a tted heavy quark PDF are available [32]. Acknowledgments We would like to thank Fred Olness for the careful reading of the manuscript and for providing helpful comments and suggestions. J. R. is supported by an STFC Rutherford Fellowship and Grant ST/K005227/1 and ST/M003787/1. V. B. and J. R. are supported by the European Research Council Starting Grant \PDF4BSM". We are grateful to the DESY IT department for their support of the xFitter developers. Open Access. 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Valerio Bertone, Stefano Camarda, Amanda Cooper-Sarkar. A determination of m c (m c ) from HERA data using a matched heavy-flavor scheme, Journal of High Energy Physics, 2016, 50, DOI: 10.1007/JHEP08(2016)050