aa r X i v : . [ h e p - e x ] J un DESY-15-085
Charm, Beauty and Top at HERA
O. Behnke, A. Geiser, M. Lisovyi ∗ ,DESY, Hamburg, Germany ∗ now at Physikalisches Institut, Universit¨at Heidelberg, Heidelberg, Germany October 3, 2018
Abstract
Results on open charm and beauty production and on the search for top production in high-energy electron-proton collisions at HERA are reviewed. This includes a discussion of relevanttheoretical aspects, a summary of the available measurements and measurement techniques, andtheir impact on improved understanding of QCD and its parameters, such as parton density func-tions and charm- and beauty-quark masses. The impact of these results on measurements at theLHC and elsewhere is also addressed.
Contents
Charm and Beauty detection at HERA 30 p r elT . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Charm and Beauty with inclusive lifetime tagging . . . . . . . . . . . . . . . . . . . . . 344.4 Charm and Beauty with double tagging . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 D ∗ inclusive measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.1.1 Charm total cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.1.2 D ∗ single-differential cross sections . . . . . . . . . . . . . . . . . . . . . . . . . 446.1.3 D ∗ double-differential cross sections . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 Inclusive measurements using other tagging methods . . . . . . . . . . . . . . . . . . . 476.3 Studies with a D ∗ and one other hard parton . . . . . . . . . . . . . . . . . . . . . . . . 496.4 Parton-parton-correlation studies in charm-tagged events . . . . . . . . . . . . . . . . . 506.4.1 x obsγ studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.4.2 Azimuthal correlations ∆ φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.4.3 Study of hard-scattering angle cos θ ∗ . . . . . . . . . . . . . . . . . . . . . . . . 546.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 b ¯ b and jet-jet correlations . . . . . . . . . . . . . . . . . . . . . . . . . 617.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
10 Summary and outlook 91 Introduction
HERA was the first and so far only high energy electron -proton collider. The production of heavy-quarkfinal states in deeply inelastic scattering (DIS) and photoproduction ( γp ) from ep interactions at HERA(Fig. 1) originally was [1] and still is (this review) one of the main topics of interest of HERA-relatedphysics, and of Quantum Chromodynamics (QCD) in general.(a) (b)Figure 1: (a) The dominant production process for charm and beauty quarks in ep collisions at HERA,the boson-gluon fusion (BGF) reaction. (b) The simplest Quark-Parton-Model diagram for deeplyinelastic scattering on a light quark.A quark is defined to be “heavy” if its mass is significantly larger than the QCD scale parameterΛ QCD ∼
250 MeV. The heavy quarks kinematically accessible at HERA are the charm and beautyquarks, which are the main topic of this review. At the time of the proposal of the HERA collider andexperiments in the 1980’s [2], a search for the top quark was one of the major goals [3]. This influencedparts of the detector design: if at all, top quarks would be produced boosted into the proton direction,and top-quark mass reconstruction from hadronic final states would profit from an excellent hadronicenergy resolution. As we know today, top-quark pair production was out of the kinematic reach ofthe HERA collider. Single top-quark production is kinematically possible, but strongly suppressed byStandard Model couplings. This allows the search for non-Standard Model top-production processeswhich will be covered in Section 5.Charm production at HERA, in particular in deeply inelastic scattering, was realised from veryearly on to be of particular interest for the understanding of QCD [1, 4]. Up to one third of the HERAcross section is expected to originate from processes with charm quarks in the final state: assuming“democratic” contributions from all quark flavours, which is a reasonable assumption at very highmomentum transfers, this fraction f ( c ) can be approximated by the ratio of photon couplings in Fig.1, which are proportional to the square of the charges Q q , q = u, d, s, c, b of the kinematically accessiblequark flavours: f ( c ) ∼ Q c Q d + Q u + Q s + Q c + Q b = 411 ≃ . , (1)while a similar approximation for beauty yields f ( b ) ∼ ≃ .
09. In general, the impact of beauty oninclusive cross sections at HERA is thus smaller than the impact of charm.At momentum transfers large enough for these approximations to be meaningful, charm and beautycan be treated as an integral part of the “quark-antiquark sea” inside the proton (Fig. 2), similar to the Throughout this document, the term “electron” includes positrons, unless explicitly stated otherwise. . e + p g c, b Figure 2: Quark-parton-model view of heavy flavour production in ep collisions at HERA.Different approaches to the theoretical treatment of charm- and beauty-quark production at HERAare discussed in Section 2. All these treatments crucially make use of the fact that the heavy-quarkmass acts as a kinematic cut-off parameter in most of the QCD processes in which heavy quarks occur.Furthermore, the fact that the heavy-quark mass is “large” compared to the QCD scale Λ QCD ∼ .
25 GeV allows the usage of this mass as a “hard scale” in QCD perturbation theory, appropriatelytaking into account quark mass effects in perturbative calculations (Fig. 3). On the other hand,the “smallness” in particular of the charm-quark mass with respect to other scales appearing in theperturbative expansion, such as the virtuality of the photon, Q , or the transverse momentum of a jetor a quark, p T , can give rise to potentially large logarithmic corrections, e.g. of the form ∼ [ α s ln( p T /m Q )] n or ∼ [ α s ln( Q /m Q )] n (2)where n is the order of the logarithmic expansion, α s is the strong coupling constant, and m Q, Q = c,b isthe heavy-quark mass. The size and treatment of these corrections is one of the issues to be investigated.During the lifetime of the HERA collider, of order 10 charm and 10 beauty events should havebeen produced in the H1 and ZEUS detectors, of which O (10%) have been recorded to tape via bothinclusive and dedicated triggers. HERA can thus truly be considered to be a charm factory. Further-more, the fact that charm- and beauty-quark production at HERA can be studied essentially over itscomplete kinematic range, from the c ¯ c - or b ¯ b -mass threshold up to squared momentum transfers oforder 1000 GeV , offers the opportunity to treat HERA as a “QCD laboratory” to test the differentpossible theoretical approaches to heavy-quark production against experimental data. Many such testsare presented in Sections 6 and 8.In particular, the charm-production measurements can be used to constrain important QCD pa-rameters, such as the charm-quark mass and its running, and has important consequences for the4igure 3: Possible hard scales in the boson-gluon-fusion process. For the explanation of the symbolssee text.determination of other parameters like the QCD strong coupling constant, α s . The measurements canalso be used to constrain the charm fragmentation parameters, to constrain the flavour compositionof virtual quarks in the proton, and to determine or cross-check the gluon distribution inside the pro-ton. Such measurements and results are discussed in Section 9. Finally, the HERA charm results havea significant impact on measurements and theoretical predictions for many QCD-related processes athadron colliders, such as the LHC. For instance, the resulting constraints on the flavour composition ofquarks in the proton reduce the uncertainties of the LHC W - and Z -production cross sections, and theconstraints on the gluon content of the proton are an important ingredient for the determination of theHiggs Yukawa coupling to top quarks from the dominant gluon-gluon-fusion Higgs-production processvia an intermediate top quark loop. Such cross-correlations are also discussed in detail in Section 9.Beauty production at HERA (Sections 7 and 8) offers further complementary insight into the the-oretical intricacies of heavy-flavour production in QCD. Due to its higher mass ( m b ∼ b ¯ b -mass threshold. This offers a particularly sensitive handle on the treatment of mass effects inQCD but also requires a particularly careful treatment of these mass effects in order to obtain reliablepredictions. Finally, the coupling of the photon to b quarks is four times smaller than the coupling tocharm quarks (Eq. (1)), and the higher b -quark mass yields a strong kinematic suppression. Thereforein practice, depending on the region of phase space probed, the b -production cross section at HERAis about 1 – 2 orders of magnitude smaller than the cross section for charm production. This makesseparation from the background and accumulation of a significant amount of statistics experimentallymuch more challenging. Also, the experimental analyses of beauty are often not fully separable fromthose of charm production. One of the highlights is the measurement of the beauty-quark mass (Section9). Others are the measurement of the total beauty-production cross section at HERA (Section 7), andthe potential impact of HERA measurements on b -quark-initiated production processes at the LHC(Section 9).Last but not least, most of the results presented depend on a good understanding of the performanceof the HERA machine and the HERA detectors, as well as on mastering the heavy-flavour detectiontechniques. Unfortunately, only a small fraction of the original data can make it through the variousevent filtering and reconstruction stages. These aspects will be addressed in Sections 3 and 4.Except for the shortest ones, each section will start with a brief introduction and close with asummary, such that a reader less interested in the details may decide to skip the reading of the more5etailed parts of the section.Some of the material in this review has been adapted from an earlier unpublished review [7] of oneof the authors. Further complementary information, in particular on charmonium and bottomoniumproduction or diffractive charm production, which are not covered by this review, is available elsewhere[8, 9, 10, 11, 12]. The broader context of other physics topics can be explored in a more general reviewon collider physics at HERA [13]. 6 Theory of heavy-flavour production at HERA
This section describes the different theoretical approaches to charm and beauty cross-section predictions,which will be needed later in the discussion of • the Monte-Carlo (MC) based acceptance corrections for the data sets used to obtain cross sections; • the extrapolation of different measurements to a common phase space, such that they can becompared or combined; • the comparison of different theory predictions to the measured cross sections; • the parton-density fits including the heavy-flavour data; • the fits of the charm and beauty masses and their running.Since there is a large overlap between the theoretical approaches for these different purposes they willbe discusssed in a common framework in the following. The measurements of heavy-quark production at HERA have been restricted, for statistical reasons,to neutral current events (exchange of a neutral boson) and to the kinematic region of the negativefour-momentum transfer squared Q < ∼ , where photon exchange dominates and Z exchangecan be neglected. Figure 4 illustrates the event kinematic variables for ep scattering with heavy-quarkproduction via the boson (i.e. photon) gluon fusion process (see also Fig. 1). e(k) e(k’) g (q) c, bxP c, bx g Pp(P) Xs Q W g p Figure 4: Illustration of event kinematic variables for ep scattering at HERA with heavy-quark pro-duction via the boson-gluon-fusion process.The four-momenta of the incoming electron k , the outgoing electron k ′ and the proton P can beused to define the following Lorentz-invariant variables: s = ( k + P ) (3) Q = − q = − ( k − k ′ ) (4)7 = Q P · q (5) y = P · qP · k (6) W γp = ( P + q ) (7)Here √ s is the centre-of-mass energy of the ep system and Q is the photon virtuality. W γp is thecentre-of-mass energy of the γ ( ∗ ) p system. In the simple Quark Parton Model [14] (QPM) the Bjorkenscaling variable x describes the proton momentum fraction carried by the scattered parton (Figs. 1(b)and 2). The inelasticity, y , gives the fraction of the electron energy taken by the photon in the protonrest frame. Only three of these five kinematic variables are independent. Neglecting the masses of theelectron and the proton the following relations between these quantities hold: Q = s · x · y (8) W γp = y · s − Q (9)In the full QCD case this picture becomes more complicated, as illustrated in Fig. 4, where the protonmomentum fraction x g carried by the gluon does not coincide any longer with Bjorken x . However,Eqs. (3) – (9) remain mathematically valid.The ep scattering events are classified by the photon virtuality Q . The regime of small Q ≈ is called photoproduction ( γp ) and the regime Q ∼ > is called Deeply Inelastic Scattering (DIS).More details on inclusive DIS results and proton structure can be found elsewhere [15, 16, 17, 18, 19]. In fixed-order perturbative QCD (pQCD) the calculation of any parton-level cross section in ep , γp , ¯ pp or pp collisions can be expressed as σ ( ab ) = Z dx a dx b f ap a ( x a , µ a ) f bp b ( x b , µ F )ˆ σ p a p b ( x a P a , x b P b , µ a , µ F , α s ( µ R )) (10)where a = e, γ, ¯ p or p is one incoming beam particle, and the other, b , is a proton. p a is a “parton”taken from a , e.g. the electron or photon itself (DIS and real photons), a slightly virtual photonradiated from the electron (photoproduction), or a gluon or quark from the structure of a real photonor (anti)proton. p b is a parton taken from the proton, i.e. a gluon or quark. x a and x b represent therespective momentum fractions of these partons with respect to their “parent” momenta P a and P b .Note that these correspond to Bjorken x and y as defined in the previous section only in the caseof the quark-parton-model approximation to deeply inelastic ep scattering, while they have a differentmeaning in other cases. For example, in Fig. 4, the quantity x g (rather than x ) corresponds to x b asdefined in Eq. (10), while x a = y in the photoproduction interpretation, and x a = 1 in the hard DISinterpretation. f ap a and f bp b are the probability density functions, or parton density functions (PDFs),which give the probability, e.g. in the f bp b case, to find a parton of type p b with momentum fraction x b ina proton. ˆ σ p a p b represents the cross section for the partonic hard scattering reaction. This is sometimessplit into the so-called hard process, i.e. the part of the reaction with the highest momentum transfer,and so-called initial state (i.e. occurring before the hard process) or final state (i.e. occurring after thehard process) radiation (Fig. 5). Part of the initial state radiation can also be absorbed into the partondensity definition. Three energy scales µ a , µ F and µ R appear in the expansion given by Eq. (10), whichis also called factorisation, because the cross section is separated into semi-independent factors.The renormalisation scale µ R determines the scale at which the value of the strong coupling constant α s is evaluated, i.e. it is the reference point around which the perturbative Taylor expansion of QCD and using the improved Weizs¨acker-Williams approximation [20] in the case of y . α s , the resultdoes not depend on the choice of this scale. After truncation of the series to finite order, the neglectedhigher-order corrections arising from the contribution of a particular subprocess are minimised if thisscale is chosen to be close to the physical scale of the momentum transfer in this subprocess. Since athigh enough perturbative order there are always different subprocesses with differing physical scales (seee.g. Figs. 3, 5), no single scale choice can universally cover all such scales. The variation of the crosssection with respect to a variation of the renormalisation scale is used to estimate the uncertainty dueto the finite-order truncation of this perturbative series. Some further aspects concerning the choice ofthis scale are discussed in Section 2.11.The factorisation scale µ F = µ b determines at which scale the proton PDFs are evaluated. Bydefault, any initial state radiation (lower blob in Fig. 5) with a momentum transfer smaller than thefactorisation scale will be absorbed into the (usually collinear) PDF definition. In contrast, any initialstate radiation with a momentum transfer larger than this scale, and all final state radiation downto the fragmentation scale (see Section 2.10), will be considered as part of the matrix element, withcorrect (noncollinear) kinematics. On one hand, the choice of a lower factorisation scale therefore givesa more detailed description of the initial state radiation kinematics at a given order. On the otherhand, the explicit treatment of initial state QCD radiation in the matrix element “uses up” a powerof α s that would otherwise have been available for a real radiation elsewhere in the process, or for avirtual correction. This effectively reduces the order of the calculation with respect to the case wherethe same radiation is absorbed into the PDF definition, and therefore reduces the overall accuracy of thecalculation. Empirically, choosing a factorisation scale equal to or at least similar to the renormalisationscale has been found to be a good compromise.The third scale, µ a , is conceptually the same as µ F in the ¯ p , p and resolved γ cases, and therefore9aken to be equal to it, while in the electron and direct photon case it is the scale at which the electro-magnetic coupling α is evaluated for the electromagnetic part of the matrix element (see Section 2.9). In general, the terminology “leading order” (LO), next-to-leading order (NLO), etc. for a perturbativeQCD expansion is not unique. It can either refer to a specific power of the strong coupling constant α s or to a specific number of loops in the perturbative expansion of the matrix elements and/or partonsplitting functions contributing to a given process. In order to be precise, this additional informationthus needs to be quoted explicitly.At leading (0-loop) order in QCD, as implemented in the form of tree-level 2 → ep collisions is dominatedby boson-gluon fusion (Fig. 6(a)), complemented by other diagrams (Figs. 6(b-d)). Since a c ¯ c or b ¯ b pair is being produced (collectively referred to as Q ¯ Q ), there is a natural lower cut-off 2 m Q for the massof the hadronic final state. (a) p g bb (b)( ) (d) (a) (b) p g gb ( ) (d)(a) (b)( ) p g bg (d) g pe bb (a) (b)( ) (d) direct- γ resolved- γ (a) γg -fusion (b) hadron-like (c) excitation (d) excitationFigure 6: Beauty production processes in leading order (0 loop) QCD as implemented e.g. in PYTHIA[21]. Note that in (c) and (d) the “photon remnant” (arrow arising from the photon) contains a ¯ b quark.In the so-called massless approach (Zero-Mass Variable-Flavour-Number Scheme, ZMVFNS), inwhich the heavy quark mass is set to 0 for the computation of the matrix elements and kinematics, thisnatural cut-off is replaced by an artificial cut-off (“flavour threshold”) at Q ∼ m Q for deeply inelasticscattering, or p T ∼ m Q for photoproduction. Below this threshold, which is often applied at the level ofthe factorisation scale, the heavy-flavour production cross section is (unphysically) assumed to vanish.Above this threshold, heavy quarks are assumed to occur as massless partons in the proton, like the u, d, and s quarks (Figs. 1(b) and 2). Except for cases in which both final state charm quarks havelarge transverse momenta p T > µ F , the gluon splitting to Q ¯ Q in Fig. 6(a) is thus assumed to happeninside the proton, and to be part of the evolution of the parton density functions. The running of α s is calculated using 3 flavours ( u, d, s ) below the renormalisation scale m c , using 4 flavours (includingcharm) between m c and m b , and using 5 flavours above the scale m b . This results in a quark-parton-model-like scattering of the electron off a heavy quark “in the proton” (Fig. 2), defining the conceptof the heavy-quark PDF. In this picture the leading-order process is now an O ( α s ) process, while theboson-gluon-fusion graph (Fig. 1(a), with both heavy quarks at high p T ) is treated as part of the O ( α s )next-to-leading order corrections. This illustrates the partially ambigous meaning of terms like LO,NLO, etc., discussed at the start of this subsection.Higher order corrections can be applied either by explicitly including them into the calculation of thematrix elements and/or splitting functions, or, if they are to be applied at tree level only, by adding anadditional so-called parton shower step. In the latter case, also referred to as leading order plus leading10og parton shower (LO+PS), the outgoing and incoming partons of the core “hard” matrix elements areevolved forward or backwards using splitting functions as they are applied during the PDF evolution.Most MCs (e.g. PYTHIA [21], HERWIG [22] and RAPGAP [23]) use the standard DGLAP evolution,as implemented e.g. in JETSET [24] for this purpose. However, in contrast to the PDF evolution, finitetransverse momenta are assigned to the partons. Some MCs (e.g. ARIADNE [25]) use a colour dipolemodel for this evolution, while others use BFKL [26] or CCFM [27] inspired so-called k t factorisation(e.g. CASCADE [28]). In the context of such MCs, the first diagram in Fig. 6 is referred to as directproduction or flavour creation, and the third and fourth are referred to as flavour excitation (in thephoton). Either the second only (e.g. PYTHIA) or collectively the last three (e.g. HERWIG) arebeing referred to as resolved photon processes. The second can uniquely be referred to as a hadron-likeresolved-photon process.For the explicit generation of heavy-flavour final states in such LO+PS MCs, the boson-gluon-fusiondiagram (Fig. 1(a)) is optionally treated using massive matrix elements, while for all other diagramsthe massless treatment remains the only available option. The heavy-quark masses appear in theoretical QCD calculations in several ways. Their physical defini-tion arises from their appearance as parameters in the QCD Lagrangian. The exact value of the massesdepends on the renormalisation scheme applied. In the
M S scheme, the masses are defined as pertur-bative scale-dependent running parameters (
M S running mass), similar to the running strong couplingconstant. In the on-shell mass renormalisation scheme, the masses are defined as the poles of the quarkpropagator (pole mass), similar to the usual definition of the lepton masses. This is also the definitionwhich one would naively expect to enter phase space calculations. However, since quarks do not existas free particles, and since the definition of the propagator pole inevitably involves contributions fromthe nonperturbative region, the pole mass definition has an intrinsic uncertainty of order Λ
QCD [29].At next-to-leading (one loop) order in perturbation theory, the relation between the pole and runningmass definitions can be expressed as [30] m Q ( m Q ) = m poleQ (1 − α s ( m Q )3 π ) , (11)where the running mass has been expressed in terms of its value at “its own scale”. Its scale dependencecan be expressed as [30] m Q ( µ ) = m Q ( m Q )(1 − α s ( µ ) π ln µ m Q ) , (12)or alternatively as [31] m Q ( µ ) = m Q ( m Q ) ( α s ( µ ) π ) β ( α s ( m Q ) π ) β , (13)with β = . Higher order expressions can also be found in the quoted references.At leading (0 loop) order, the difference between the two definitions vanishes. Finally, in the contextof so-called massless schemes, the “mass” is defined as a kinematic cutoff parameter in certain parts ofthe theory calculations.The pole-mass definition has been used in most QCD calculations relevant for this review. In recentvariants of the ABKM [32] and ACOT [33] schemes, the M S -running-mass definition is used instead.The latter has the advantage of reducing the sensitivity of the cross sections to higher order corrections,and improving the theoretical precision of the mass definition [32].11 .5 The zero-mass variable-flavour-number scheme
In its “NLO” variant, including one-loop virtual corrections (Fig. 7)(b)), the ZMVFNS has been usedfor most NLO variable-flavour parton-density fits up to a few years ago, such as CTEQ6M [35], ZEUS-S[36], H1 [37], NNPDF2.0 [38].(a) (b) (c) (d)Figure 7: Leading order ( O ( α s )) (a) and selection of next to leading order ( O ( α s )) (b)–(d) processesfor heavy flavour production in DIS in the massless scheme. For (b), only the interference term with(a) contributes at this order.One of its advantages is that e.g. next-to-leading-log (NLL) resummation of terms proportional to log ( Q /m Q ) can be applied to all orders, avoiding the problem that such logs could spoil the convergenceof the perturbation series at high momentum transfers. However, it is clear that this simplified approachcan not give the correct answer for processes near the “flavour threshold”. This has been verifiedexperimentally e.g. for the DIS case [39] (Section 8.2). Also, it was found that neglecting the charmmass in the cross section calculations used for the PDF extraction can result in untolerably large effectson theoretical predictions even at high scales, such as W and Z production at the LHC [40]. Allmore recent PDF approaches [41, 42, 43, 44, 45, 46, 47] therefore include at least a partial explicitconsideration of the charm mass in the matrix elements (Sections 2.6 and 2.7). Nevertheless, sincehigher orders are more easily calculable in this scheme, the massless approach can offer advantages e.g.in high-energy charm-photoproduction processes [48, 49] in which the consideration of an extra orderof α s in the final state allows a reduction of the theoretical uncertainty (Section 6). The fixed-flavour-number scheme (FFNS) treats the heavy-quark masses explicitly and follows a rigourousquantum field theory ansatz. Full NLO (one loop) calculations of heavy-flavour production in thisscheme exist for DIS [50, 51, 52, 53, 54, 55, 46, 47], for photoproduction [56, 57, 58, 59] and forhadroproduction [60]. Some partial NNLO (two-loop) calculations are also available [61, 62]. In thisscheme, heavy flavours are treated as massive at all scales, and never appear as an active flavour in theproton. In the case in which all heavy flavours are treated as massive, the number of light flavours inthe PDFs is thus fixed to 3, and charm as well as beauty are always produced in the matrix element(Fig. 8). So-called flavour excitation processes (Fig. 6(c,d)), which are often classified as leading order( O ( α s )) QCD in partially massless MC approaches [21, 22, 23] of charm or beauty production, appearas O ( α s ) NLO corrections in the fully massive approach (Fig. 8).There are several variants of the FFNS for heavy-flavour production in DIS (see e.g. remarks inappendix of [63]). In one approach, here called FFNS A, the α s evolution used together with the 3 flavourPDFs is also restricted to 3 flavours. Thus, the small contribution from heavy flavour loops (Fig. 9(a)) MRST98 [34] is a notable early exception. ′ ) [52, 54]. In either variant, this leads to a lower effective value of α s than in themassless scheme when evolved to high reference scales, e.g. α s ( M Z ). This is one of the consequencesof the non-resummation of log( Q /m Q ) terms, and is partially compensated e.g. by a conceptuallylarger gluon PDF. Despite the conceptual disadvantage of not allowing all order resummation of masslogarithms, this scheme yields very reasonable agreement with charm and beauty data at HERA up tothe highest Q and p T (Sections 6–8). At HERA energies, the numerical differences between schemesA and A ′ are of order 1%, and therefore almost negligible compared to the current data precision.(a) (b) (c)Figure 9: Heavy Flavour loop correction (a) and gluon splitting (b,c) processes in the massive scheme.The thick (thin) lines indicate heavy (light) flavours.In the FFNS B approach , which was widely used in the early nineties [50, 51, 56, 60, 68], the runningof α s is calculated by incrementing the number of flavours when crossing a flavour threshold, like inthe variable-flavour-number scheme. The class of logs corresponding to this running is thus resummedboth in the α s and in the PDF evolution, and the “missing” heavy flavour log resummation is restrictedto other cases like gluon splitting and vertex corrections in the matrix elements. This approach ispossible since most of the loop and leg corrections which diverge in the massless case, but compensateeach other to yield finite contributions, remain separately finite in the massive case. They can thusbe separated. At one-loop order, the A and B approaches differ by the way a virtual heavy flavourcorrection in the BGF matrix element (Fig. 9(a)) (which is missing in the A ′ approach) is treated. TheFFNS B scheme conceptually yields a value of α s at high scales which is the same as the one from thevariable-flavour approach, and is probably less sensitive to “missing logs” at very high scales than theFFNS A approach. Most NLO photoproduction [56] and hadroproduction [60] calculations, as well as Elsewhere [64] this is sometimes called the mixed flavour number scheme. For a discussion see [65][66]. A recent newvariant of it [67] is referred to as the “doped” scheme. ′ scheme can not converge to the exact result since heavy flavour loop correctionsare completely missing, but, as stated earlier, the practical consequences at HERA energies are small.It can however serve as a useful ingredient to variable flavour number scheme calculations (Section 2.7).There are also other differences. In the ABKM approach, final state gluon splitting (Fig. 9(b,c)) isconceptually treated as part of the light-flavour contribution, while it is treated as part of heavy flavourproduction in many others [50, 51, 56, 60] (see Section 2.8), as well as in the measured cross sections,since it can hardly be distinguished experimentally. This difference is small [69] in most regions of phasespace, but might need to be accounted for when comparing data and theory.Finally, FFNS calculations in DIS are currently available in leading order ( O ( α s )), or NLO ( O ( α s ))[46, 50, 51]. Partial NNLO (two-loop, O ( α s )) calculations also exist, based on a full calculation ofthe O ( α s ) log and the O ( α s ) log terms, and the leading term from threshold resummation for the O ( α s ) constant term [61]. Further NNLO corrections for the high scale limit [16] have not yet beenimplemented in practice. Actually, both the NLO and partial NNLO ABKM heavy flavour calculationsuse the PDFs from their NNLO (two loop, O ( α s ) in the matrix elements) fit to the inclusive data [46].Some of the differences between the calculations discussed in this section and in the next two sectionsare also summarised in Table 1, using the example of reduced charm cross sections in DIS. An alternative to the fixed-flavour-number approach is given by the so-called general-mass variable-flavour-number schemes (GMVFNS) [70, 71, 72, 73]. In these schemes, charm production is treatedin the FFNS approach in the low- Q region, where the mass effects are largest, and in the masslessapproach at very high scales, where the effect of resummation is most noticeable. At intermediate scales(in practice often at all scales above the “flavour threshold”), an interpolation is made between the twoschemes, avoiding double-counting of common terms, while making a continous interpolation betweendiffering terms. This scheme combines the advantages of the two previous schemes, while introducingsome level of arbitrariness in the treatment of the interpolation. g* ccc -c X( ) g* cc X -c G c-c X g* G - + Figure 10: Leading order (0-loop) diagrams for charm production in DIS in the variable flavour numberscheme: On the left the QPM diagram is shown, on the right the BGF diagram and in the middle the“subtraction diagram” [74]. The vertex correction loop diagram (Fig. 7(b)), which also contributes tothis order in α s , is not shown.One of the most constrained schemes is the BMSN scheme [73] used by the VFNS approach ofABKM [62]. At NLO, it interpolates between O ( α s ) charm matrix elements in the FFNS part using theABKM FFNS scheme, and O ( α s ) matrix elements in the massless part. It has no tuneable parameters,and (currently) uses the pole-mass definition for the FFNS part. In contrast to most other GMVFNSschemes, the switch to a larger number of flavours should not be made at the “flavour threshold”,but at a scale which is high enough that additional semi-arbitrary kinematic correction terms are notrequired. In practice, the 3-flavour scheme is used for processes at HERA energies, while the 4- or5-flavour schemes are recommended for applications at the LHC.14 heory Scheme Ref. F L ) m c PDF Massive / F L Massless F α s ( m Z ) Scaledef. [GeV] ( Q ∼ < m c ) ( Q ≫ m c ) ( n f = 5)MSTW08 NLO RT standard [75] F c L ) . O ( α s ) O ( α s ) O ( α s ) 0 . Q MSTW08 NNLO O ( α s ) approx.- O ( α s ) O ( α s ) 0 . O ( α s ) O ( α s ) O ( α s ) 0 . O ( α s ) approx.- O ( α s ) O ( α s ) 0 . F c L ) . O ( α s ) O ( α s ) O ( α s ) 0 . Q NNPDF2.1 FONLL A FONLL A [77] n.a. √ O ( α s ) O ( α s ) O ( α s ) 0 . Q NNPDF2.1 FONLL B FONLL B F c L ) √ O ( α s ) O ( α s ) / O ( α s ) O ( α s )NNPDF2.1 FONLL C FONLL C F c L ) √ O ( α s ) O ( α s ) O ( α s )CT10 NLO S-ACOT- χ [55] n.a. 1 . O ( α s ) O ( α s ) O ( α s ) 0 . p Q + m c CT10 NNLO [78] F c ¯ c L ) . O ( α s ) O ( α s ) O ( α s )ABKM09 NLO FFNS A [46] F c ¯ c L ) .
18 (
M S ) O ( α s ) O ( α s ) - 0.1135 p Q + 4 m c ABKM09 NNLO O ( α s ) approx.- O ( α s ) -HVQDIS+ZEUS S FFNS B [51] F c L ) . O ( α s ) O ( α s ) - 0.118 p Q + 4 m c Table 1: Selected calculations for reduced charm cross sections in DIS from different theory groups as used in this review. The table showsthe heavy flavour scheme used and the corresponding reference, the respective F L ) definition (Section 2.8), the value and type of charmmass used (Section 2.4), the order in α S of the PDF part and the massive and massless parts of the calculation (and of the massless partof F L , which, except for FONLL B, is usually taken to be the same as for the massive part), the value of α s , the renormalisation andfactorisation scale. The distinction between the two possible F L ) definitions is not applicable (n.a.) for O ( α s ) calculations, or in photo-or hadroproduction. Usually, the order of the PDF part is used to define the label LO, NLO, or NNLO. he NLO version of the TR scheme [71] combines the O ( α s ) charm matrix elements in the FFNSA ′ scheme with the O ( α s ) matrix elements of the massless scheme, requiring continuity of the physicalobservables in the threshold region. In this case the usage of the A ′ scheme is fully appropriate, since themissing terms will be taken care of by the massless and interpolation terms. Several variants exist for theinterpolation, including the so-called standard scheme used e.g. in MSTW08 [43] and HERAPDF1.0[41], and the optimised scheme preferred for more recent versions, since it avoids a kink in the Q dependence of the cross section [76]. Both of these variants also exist in a partial NNLO approach [75],including approximate O ( α s ) threshold resummation terms for the FFNS part, and a full O ( α s ) NNLOcalculation for the massless part. They all use the pole mass definition for the FFNS part.The ACOT [70] scheme, used by CTEQ [44], also exists in several variants. At NLO, O ( α s ) (i.e.leading order) FFNS matrix elements are interpolated to O ( α s ) (now NLO) massless matrix elements .Due to the LO FFNS treatment, there is no difference between the pole-mass and running-mass schemes.The interpolation is made in two variants: the S-ACOT approach [79], and the ACOT- χ approach [80].The NNLO variant of CT10 [81] uses both FFNS and massless matrix elements at O ( α s ), in the S-ACOTscheme.The FONLL scheme [69] has 3 variants. The FONLL A approach, used by NNPDF2.1 [45] isequivalent [69] to the CTEQ S-ACOT approach, and uses O ( α s ) FFNS heavy-quark matrix elementsat NLO. FONLL B and C both use O ( α s ) FFNS heavy-quark matrix elements. FONLL B uses O ( α s )matrix elements for light quarks, like MSTW, while FONLL C uses O ( α s ) matrix elements for lightquarks like ABKM. However, they differ from the latter in the way they treat the interpolation terms.The FONLL C scheme is also similar [69] to the CTEQ S-ACOT NNLO scheme. A full NNLO versionof the FONLL A approach also exists [77, 82]. Final state gluon splitting is not included in the charmcross-section predictions for any of these schemes.The ABM group uses the BSMN approach [73] to generate a GMVFNS scheme out of their FFNS3-, 4- and 5-flavour PDFs [62].For photoproduction, a GMVFNS calculation [83] exists for single inclusive cross sections.All GMVFNS variants use the variable-flavour approach for the running of α s . Although the massis unambigously defined in the massive part of the calculation (usually the pole mass), the partialarbitrariness in the treatment of the interpolation terms (Fig. 10) prevents a clean interpretationof the charm and beauty quark masses in terms of a single renormalisation scheme. Therefore, incontrast to the pure FFNS treatment, the charm mass appearing in VFNS schemes can be treatedas an effective mass parameter [84]. We will use the symbols M c and M b for these effective massparameters. Alternatively, the presence of the interpolation terms can be included as an additionaluncertainty on the respective mass definition [33]. In analogy to the inclusive neutral current DIS cross section, the cross sections for heavy-quark pro-duction in DIS can be expressed in terms of the heavy-quark contributions to the inclusive structurefunctions [17] F , F L and F , denoted by F Q ¯ Q , F Q ¯ QL and F Q ¯ Q ( Q = c, b ): dσ Q ¯ Q ( e ± p ) dx dQ = 2 πα x Q (cid:16) (1 + (1 − y ) ) F Q ¯ Q − y F Q ¯ QL ∓ x (1 − (1 − y ) ) F Q ¯ Q (cid:17) , (14)where α is the electromagnetic coupling constant. The structure function F Q ¯ Q makes the dominantcontribution to the neutral current scattering in the kinematic regime accessible at HERA. F Q ¯ Q contains also referred to as RT The fact that similar matrix elements are denoted by different labels concerning their effective order in differentcontext is very confusing, but unavoidable due to different definitions of the truncation of the perturbative QCD series. γZ interference and Z exchange, therefore for the region Q ≪ M Z , which wasstudied at HERA, this contribution is suppressed and can be neglected. The longitudinal heavy-quarkstructure function F Q ¯ QL parametrises the contribution from coupling to the longitudinally polarisedphotons. The contribution of F Q ¯ QL to the ep cross section is suppressed for y ≪
1, but can be up to afew percent in the kinematic region of the heavy-quark measurements at HERA and thus can not beneglected.For both electron and positron beams, neglecting the F Q ¯ Q contribution, the reduced heavy-quarkcross section, σ Q ¯ Q red , is defined as σ Q ¯ Qred ( x, Q ) = dσ Q ¯ Q ( e ± p ) dx dQ · x Q πα Y + = F Q ¯ Q − y Y + F Q ¯ QL , (15)where Y ± = (1 ± (1 − y ) ). Thus, σ Q ¯ Q red and F Q ¯ Q only differ by a small F Q ¯ QL correction at high y [85].In the Quark-Parton Model, the structure functions depend on Q only and can be directly relatedto the parton density functions. In the QCD case, and in particular for heavy flavour production, thiscorrelation is strongly diluted, and the structure functions depend on both x and Q . More informationon the general case can be found e.g. in [17, 41].Using the example of the charm case [84], the above definition of F c ¯ c L ) ( x, Q ) (also denoted as ˜ F c [69]or F c,SI [82]) is suited for measurements in which charm is explicitly detected. It differs from what issometimes used in theoretical calculations in which F c L ) ( x, Q ) [69, 75, 86] is defined as the contributionto the inclusive F L ) ( x, Q ) in which the virtual photon couples directly to a c or ¯ c quark. The latterexcludes contributions from final state gluon splitting to a c ¯ c pair in events where the photon couplesdirectly to a light quark, and contributions from events in which the photon is replaced by a gluonfrom a hadron-like resolved photon. As shown in table 1 of [69], the gluon splitting contribution isexpected to be small enough to allow a reasonable comparison of the experimental results to theoreticalpredictions using this definition. The hadron-like resolved photon contribution is expected to be heavilysuppressed at high Q , but might not be completely negligible in the low Q region. From the point ofview of pQCD it appears at O ( α s ) and it is neglected in all theoretical DIS calculations used in thisreview. In addition to the different QCD schemes discussed above, predictions of charm production can alsodiffer through their treatment of QED corrections. Some of these corrections, e.g. collinear photonradiation from the initial state electron before the hard interaction (Fig. 11) can actually be large (oforder α ln Q max m e ), and can influence the definition of the Q , x and y variables [17]. For photoproductioncalculations, the improved Weizs¨acker-Williams approximation [20] can be used to parametrise thephoton spectrum arising from the incoming electron.For acceptance corrections (and partially for visible cross sections), predictions including full LOQED radiative corrections, as implemented e.g. in HERACLES [87], are used. For more sophisticatedpurposes, an NLO version of these corrections is available in the HECTOR package [88]. At the level ofthe DIS structure functions or σ r it is customary to translate the measured cross sections to so-calledBorn-level cross sections, i.e. cross sections in which all QED corrections have been removed, to easecomparison of the data with pure QCD predictions. There is one potential exception: the fine structure“constant” α can be used in two different ways. • as a genuine atomic scale constant α = . , i.e. all virtual QED corrections are removed, too; • as running α in the M S scheme, i.e. the respective relevant virtual corrections are kept, and atypical value for HERA kinematics is then α ≃ [89].17 + p g g c, bc, b (cid:214) a s g Figure 11: BGF diagram with initial state photon radiation.The difference between these two approaches in QED has some remote similarity to the differencebetween the FFNS A and B schemes in perturbative QCD (Section 2.6), treating all quarks and leptonsas “heavy” with respect to the atomic scale.
Equation (10) allows one to make predictions of heavy-quark production with partons in the final state.However, cross sections are measured and reported mostly in terms of heavy-flavour hadrons, leptonsfrom their decay, or collimated jets of hadrons. Therefore such predictions have to be supplementedwith a fragmentation or hadronisation model.In analogy to e + e − collisions [90], the factorised cross section for the production of a heavy-quarkhadron H as a function of transverse momentum p HT can be written as: dσ H dp HT ( p HT ) = Z dp QT p QT dσ Q dp QT ( p QT , µ f ) D HQ ( p HT p QT , µ f ) · f ( Q → H ) , (16)where σ Q ( x, µ f ) is the production cross section for heavy quarks (Eq. (10)), D HQ is the fragmentationfunction, µ f is the fragmentation scale and f ( Q → H ) in the fragmentation fraction. The latter isdefined as the probability of the given hadron H to originate from the heavy quark Q . The fragmentationfunction defines the probability for the final-state hadron to carry the fraction z = p HT /p QT of the heavy-quark momentum. The fragmentation function is defined similarly to the PDFs. In the “massless”approximation, it is defined at a starting scale and has to be evolved to a characteristic scale µ f of the process using perturbative QCD. In the massive fixed flavour approach, this evolution can beconceptually absorbed into the pole mass definition.Fragmentation fractions as well as the starting parametrisation of the fragmentation function cannot be calculated perturbatively. Thus they have to be extracted from data. Comprehensive phe-nomenological analyses of the charm and beauty fragmentation functions in e + e − collisions have beenperformed [91, 92, 93]. While the QCD evolution is process-dependent, the non-perturbative ingredientsof the fragmentation model are assumed to be universal . Comparing measurements from HERA andresults from e + e − colliders one can test this universality.However, the tools that are available for O ( α s ) fixed-order calculations of the heavy-quark produc-tion cross sections in ep collisions, which are mostly used for exclusive final states, do not comprise a The non-perturbative fragmentation function is universal only if it is accompanied by appropriate evolution. D NP( z ) is used in conjunction with the parton-level cross sections. The para-metric forms of the independent fragmentation functions most commonly used at HERA are due toPeterson [94]: D NP( z ) ∝ z (1 − /z − ε/ (1 − z )) , Kartvelishvili [95]: D NP( z ) ∝ z α (1 − z )and to the Bowler modification of the symmetric Lund [96] parametrisation: D NP( z ) ∝ /z r Q bm Q (1 − z ) a exp ( − b ( m H + p T ) /z ) , where ǫ , α , a , b and r Q are free parameters that depend on the heavy-flavour hadron species and haveto be extracted from data. Since no QCD evolution is applied, the corresponding parameters might bescale- and process-dependent.The recent GMVFNS NLO predictions for charm photoproduction [83] incorporate a perturbativefragmentation function and have been tested against data (Section 6). For many cross-section predictions the dominant contribution to the theoretical uncertainty arises fromthe variation of the renormalisation and factorisation scales by a factor 2 around some suitably chosendefault scale. Such a variation is intended to reflect the uncertainty due to uncalculated higher orders. Itmight therefore be useful to consider some phenomenological aspects of these scale choices as consideredin a mini-review on beauty production at HERA and elsewhere [97], focusing in particular on the choiceof the renormalisation scale.Ideally, in a QCD calculation to all orders, the result of the perturbative expansion does not dependon the choice of this scale. In practice, a dependence arises from the truncation of the perturbativeseries. Since this is an artefact of the truncation, rather than a physical effect, the optimal scale cannot be “measured” from the data. Thus, it must be obtained phenomenologically.Traditionally, there have been several options to choose the “optimal” scale, e.g. • The “natural” scale of the process. This is usually taken to be the transverse energy, E T , of thejet for jet measurements, the mass, m , of a heavy particle for the total production cross section ofthis particle, or the combination q m + p T for differential cross sections of such a particle. Often,this is the only option considered. The choice of this natural scale is based on common sense, andon the hope that this will minimise the occurrance of large logs of the kind described above, forthe central hard process. However, higher order subprocesses such as additional gluon radiationoften occur at significantly smaller scales, such that this choice might not always be optimal. • The principle of fastest apparent convergence (FAC) [98]. The only way to reliably evaluateuncalculated higher orders is to actually do the higher-order calculation. Unfortunately, this isoften not possible. Instead, one could hope that a scale choice which makes the leading-orderprediction identical to the next-to-leading-order one would also minimise the NNLO corrections.This principle, which can be found in many QCD textbooks, can not be proven. However, recentactual NNLO calculations might indicate that it works phenomenologically after all (see below). • The principle of minimal sensitivity (PMS) [99]. The idea is that when the derivative of the crosssection with respect to the NLO scale variation vanishes, the NNLO corrections will presumably19 m r / M H s (pp → H+X) [pb] M H = 120 GeV LONLON LON LO approx Figure 12:
Scale dependence of the total cross section for beauty production at HERA-B [100] (left) and forHiggs production at the LHC [101] (right). also be small. Again, there is no proof that this textbook principle should work, but actual NNLOcalculations might indicate that it does (see below).To illustrate these principles, consider two examples. First, the prediction for the total cross sectionfor beauty production at HERA-B [100] (Fig. 12). The natural scale for this case is the b -quark mass, µ = m b , and all scales are expressed as a fraction of this reference scale. Inspecting Fig. 12, one findsthat both the PMS and FAC principles, applied to the NLO prediction and to the comparison withLO (NLO stability), would yield an optimal scale of about half the natural scale. The same conclusionwould be obtained by using the NLO+NLL prediction, including resummation, and comparing it toeither the LO or the NLO prediction (NLO+NLL stability).Second, the prediction for Higgs production at the LHC [101] (Fig. 12). The reference scale is nowthe Higgs mass ( µ = m H ). However, inspecting the behaviour of the LO and NLO predictions, neitherthe FAC nor the PMS principle would yield a useful result in this case, since the two predictions do notcross, and the NLO prediction does not have a maximaum or minimum. This situation occurs ratherfrequently, and is also true for b production at HERA. Fortunately, in the case of Higgs production, theNNLO and even approximate NNNLO predictions have actually been calculated (Fig 12). Applying theFAC and PMS prescriptions to these instead (NNLO stability), again a scale significantly lower thanthe default scale would be favoured. This might indicate that choosing a scale which is smaller thanthe default one makes sense even if the FAC and PMS principles do not yield useful values at NLO.Beyond these examples, a more general study is needed to phenomenologically validate this approach.To avoid additional complications arising from a multiple-scale problem caused by e.g. the scale Q atHERA or the scale M Z at LEP, the study was limited to cross sections for photoproduction at HERA, orhadroproduction at fixed-target energies, the Tevatron, and LHC. The somewhat arbitrary selection ofprocesses includes beauty production at the Sp ¯ pS [102, 103], the Tevatron [103], and HERA-B [100], topproduction at the Tevatron [100, 103], direct photon production at fixed target [104], Z [105] and Higgs[101] production at the LHC, jets at HERA [106] and at the Tevatron [107]. This selection is obviouslynot complete, and many further calculations, in particular NNLO calculations, have been achieved sincethis study [97] was originally made. However, it is not biased in the sense that all processes that wereoriginally considered were included, and none were discarded. Clearly, a quantitative update of thisstudy would be useful, but was not yet done. Qualitatively, all newer predictions which the authorshave been made aware of either confirm this conclusion, or at least do not significantly contradict it.In each case the natural scale as defined above was used as a reference. In addition, wherever20igure 13: Summary of optimised scales derived as described in the text. possible, the optimal scales from both the FAC and PMS principles, evaluated at NLO (NLO stability),NLO+NLL (NLO+NLL stability), and/or NNLO/NNNLO (NNLO stability) were evaluated separately.Figure 13 shows the result of this evaluation. Each crossing point, maximum, or minimum in Fig. 12yields one entry into this figure, and similarly for all the other processes. The conclusion is that the FACand PMS principles tend to favour scales which are around 25-60% of the natural scale. Amazingly, thisseems to be independent of whether these principles are applied at NLO, NLO+NLL, or NNLO level.For the jet [107] or b-jet [108] cross sections at the Tevatron, it has in part already become customaryto use half the natural scale as the central scale.Using the natural scale as the default and varying it by a factor two, which is the choice adoptedfor most data/theory comparisons, covers only about half the entries, while the other half lies entirelybelow this range. Instead, using half the natural scale as the default and varying it by a factor two,thus still including the natural scale in the variation, covers about 95% of all the entries.This yields the following conclusions. • Obviously, whenever an NNLO calculation is available, it should be used. • Whenever possible, a dedicated scale study should be made for each process for the kinematicrange in question. Although there is no proof that the FAC and PMS principles should work,in practice they seem to give self-consistent and almost universal answers for processes at fixedtarget energies, HERA, the Tevatron, and the LHC. • In the absence of either of the above, the default scale should be chosen to be half the naturalscale, rather than the natural scale, in particular before claiming a discrepancy between data andtheory. Empirically, this should enhance the chance that the NNLO calculation, when it becomesavailable, will actually lie within the quoted error band. To evaluate uncertainties, the customaryvariation of the central scale by a factor 2 up and down remains unaffected by this choice.The latter principle has already been applied to a few of the results covered in this review. Ofcourse, choosing the natural scale as the central value, which is still the default for most calculations(or making any other reasonable scale choice), is perfectly legitimate and should also describe the datawithin the theoretical uncertainties. However, if it does not, it might be useful to consider alternativechoices as discussed above before claiming evidence for the failure of QCD, and hence for new physics.21urther complementary information, in particular on the related theory aspects, is available else-where [109].
The theory of heavy flavour production in the framework of perturbative QCD, and in particular theoccurrence of different possibilities to treat the heavy quark masses in the PDF, matrix element andfragmentation parts of the calculation, introduces a significant level of complexity into the correspondingQCD calculations, in addition to the usual scheme and scale choices. Confronting different choices withdata can be helpful to understand the effects of different ways to truncate the perturbative series.The majority of the available MC calculations for the analysis of HERA data is based on leadingorder (plus parton shower) approaches, combining a massive approach for the core boson-gluon fusionprocess, and the massless approach for tree level higher order corrections. It will be demonstrated in thelater chapters that this is fully adequate for acceptance corrections. For comparisons of the measureddifferential cross sections with QCD predictions, a next-to-leading order massive approach (fixed flavourscheme) is the state of the art.In some cases massless calculations are still in use, e.g. to facilitate the perturbative treatment offragmentation, or to implement resummation of some of the logarithms arising when the mass competeswith other hard scales occurring in a process. In particular for the prediction of the inclusive heavyflavour structure functions in DIS, a variety of so-called general-mass variable-flavour schemes areavailable, merging massive calculations at low scales with massless calculations at high scales. Theseare particularly useful for the extraction of PDFs over very large ranges in energy scale. Partial NNLOcalculations are also available for such inclusive quantities, both in the fixed and variable flavour numberschemes. Due to the absence of extra semi-arbitrary parameters, the fixed flavour number scheme isparticularly well suited for the extraction of QCD parameters like the heavy quark masses.In general, QED corrections are nonnegligible, and available both at leading and next-to-leadingorder. Since α is much smaller than α s , the leading order precision is often sufficient. Several compet-ing fragmentation models are in use, and the perturbative treatment of fragmentation in the massiveapproach is still in its infancy.Since higher order corrections are large, the uncertainties reflected by the QCD scale variations areoften dominant. Until full NNLO calculations become available, a careful consideration of the choice ofthese scales can be helpful to avoid premature conclusions concerning potential discrepancies betweenthe theory predictions and the data. 22 The HERA collider and experiments
In this section, the HERA collider, the H1 and ZEUS experiments, as well as the reconstruction of thedata from these experiments will be briefly described, with focus on aspects relevant for heavy flavourproduction.
HERA (German: Hadron-Elektron-Ring-Anlage) was the first and so far the only electron–protoncollider. It was located at DESY in Hamburg, Germany. The circumference of the HERA ring (seeFig. 14) was 6 . ep storage rings with the H1 and ZEUS experimentsprotons were accelerated in two separate rings to final energies of 27 . − . √ s = 318 −
319 GeV (300 GeV before1998). Both beams were stored in 180 bunches. The bunch-crossing rate was 10 MHz. Electrons andprotons collided in two interaction regions, where the H1 and ZEUS detectors were located.In the years 2001 – 2002 the HERA collider was upgraded to increase the instantaneous luminosity.At the same time a number of upgrades of the H1 and ZEUS detectors were put in place, as describedbelow. Therefore, the data taking was subdivided into two phases: “HERA I” and “HERA II” corre-sponding to the data taking periods 1992 – 2000 and 2003 – 2007, respectively. In 2007, a few monthswere dedicated to data taking at lower centre-of-mass energies.
The H1 and ZEUS detectors were typical modern multi-purpose collider experiments and are describedin detail in [110, 111, 112, 113]. Figure 15 visualises the layout of the H1 and ZEUS detectors . Due tosignificantly higher energy of the protons there was more detector hardware installed in the direction The right-handed Cartesian coordinate system used at H1 and ZEUS has the Z axis pointing in the nominal protonbeam direction, referred to as the “forward direction”, and the X axis pointing towards the centre of HERA. Its coordinate ± p ❄❈❈❈❈❈❲ LAR calor. ❄❄ Central Jet Ch. ✁✁✁✁✁✁✁✕
Forward tracker ✄✄✄✄✄✄✎ ❄
SpaCal
ZR View e ± p ✻ MVD (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✒
CTD ❄✑✑✑✑✑✑✑✑✰ ❍❍❍❍❍❍❍❍❍❍❥
CAL ✘✘✘✘✘✘✘✘✿
STT
Figure 15: The rz -view of the tracking system and calorimeters of the H1 (top) and ZEUS (bottom)detectors. The layout of the central silicon tracker (CST) and the microvertex detector (MVD) areshown separately below. The electron beam enters from the left, while the proton beam enters fromthe right. 24 CST MVD θ -coverage [30 ◦ , ◦ ] [7 ◦ , ◦ ] η -coverage [ − . , .
3] [ − . , . σ ( δ ) 43 ⊕ /p T µ m 46 ⊕ /p T µ m Drift chambers:
CTD CTD θ -coverage [20 ◦ , ◦ ] [15 ◦ , ◦ ] η -coverage [ − . , .
74] [ − . , . σ ( p T ) /p T . p T ⊕ .
015 0 . p T ⊕ . ⊕ . /p T Calorimeters:
LAr CAL (HERA I) θ -coverage [4 ◦ , ◦ ] [2 . ◦ , . ◦ ] η -coverage [ − . , .
35] [ − . , . σ ( E ) /E . / √ E ⊕ .
01 0 . / √ E Hadronic σ ( E ) /E . / √ E ⊕ .
02 0 . / √ E SpaCal θ -coverage [153 ◦ , ◦ ] η -coverage [ − . , − . σ ( E ) /E . / √ E ⊕ . Muon systems:
CMD R/B/FMUON+BAC θ -coverage [4 ◦ , ◦ ] [5 ◦ , ◦ ] η -coverage [ − . , .
4] [ − . , . p T and energies E are in units of GeV. δ is the transverse distance of closest approach of tracks to the nominal vertex. σ ( δ ) is the resolution of δ , averaged over the azimuthal distribution of tracks.of the outgoing proton beam. The figures show the key parts of the main detectors that were used fortagging and reconstruction of heavy-flavour events. Some of the most important benchmarks of theH1 and ZEUS detectors, such as polar angle coverage and momentum, the energy resolution and theresolution of the impact parameter δ , are listed in Table 2. In the following the main components ofthe H1 and ZEUS detectors are discussed with emphasis on the advantages of the respective designs: Tracking Chambers:
Tracks from charged particles were reconstructed based on the position mea-surements in the large
Central Drift Chambers . The pulse height on the sense wires was used tomeasure the energy loss in the detector medium, dE/dx . The dE/dx measurements were used for par-ticle identification, distinguishing between electrons, pions, kaons and protons in a limited momentum origin is at the nominal center of the respective detector, which coincided with the nominal interaction point in the HERA Iperiod. The pseudorapidity is defined as η = − ln (tan( θ/ θ , is measured with respect to theproton beam direction. The x - y - or r - φ -plane is also denoted as the transverse plane. Also referred to as the transverse distance of closest approach to the nominal vertex • The H1 tracking detector comprised two chambers CJC1 and CJC2 [111] while the ZEUS CTD [114]was a single chamber that was divided into nine superlayers. • For ZEUS a superconducting coil surrounded the tracking detectors and provides a magnetic fieldof 1.43 Tesla. This is considerably higher than the 1.15 Tesla delivered by the H1 superconductingcoil, situated outside the calorimeter. In both experiments the magnetic field within the trackingsystem was parallel to the Z axis. • Four of the nine superlayers of the ZEUS chambers were equipped with stereo wires, which weretilted ∼ ◦ with respect to the beam axis. This provided z -measurement points for tracks witha resolution of ∼ . Z -position measurement was obtained by the division of the charges recorded at both wireends, yielding a moderate resolution of a few centimetres. Two additional Z -drift-chambers wereinstalled to provide for each track a few Z -measurement points with typically 300 µ m resolution.In the forward region H1 and ZEUS have installed a Forward Tracking Detector (FTD) [115]and a
Straw Tube Tracker (STT) [116], respectively, that are based on drift-chambers. Their mainpurpose is to extend the polar angular coverage to angles smaller than 20 ◦ , outside the acceptanceof the central drift chambers. However, for both experiments these detectors have not been used formomentum reconstruction due to a large amount of dead material in front of them. Nevertheless, theforward detectors were partially used in the pattern recognition. Vertex-detector:
The
Central Silicon Tracker (CST) [117] and the
Micro Vertex Detector (MVD) [118] were located in the heart of the H1 and ZEUS experiments. The MVD was installed onlyfor the HERA II data taking. The vertex detectors allowed the determination of trajectories of chargedparticles in the vicinity of the primary vertex. The achieved precision was sufficient to resolve verticesfrom secondary decays. This is essential for the tagging of weakly-decaying heavy-flavour hadrons witha typical lifetime cτ ≃
100 – 300 µ m.The CST (MVD) consisted of two (three) 36 cm (63 cm) long concentric cylindrical layers of double-sided silicon-strip detectors The innermost layer of the CST and MVD was located at 57 . ∼
45 mm, respectively. The most important benchmarking parameters are given in Table 2. Thefollowing intrinsic hit resolutions were achieved: ∼ µ m for the CST and ∼ µ m for the MVD.The CST had a somewhat better average transverse impact parameter resolution mainly due to lessmaterial, hence less multiple scattering, but was essentially restricted to track reconstruction in thetransverse plane. The MVD contained four wheels of double-sided silicon-strip detectors in the forwardregion that extended the polar-angle coverage from 20 ◦ to 7 ◦ . Furthermore, it allowed 3D standalonepattern recognition. At H1 the CST was supplemented with the forward and backward strip detectors,FST and BST, that extended the polar-angle coverage of CST to [7 ◦ , ◦ ]. However, these were notused in H1 heavy-flavour analyses. Calorimeters:
The tracking detectors were surrounded by calorimeter systems, which covered almostthe full solid angle. Their main tasks were to identify and measure the scattered electron, to reconstructthe hadronic final state (e.g. jets) and photons and to separate electrons from hadrons. At H1 a fine-grain liquid-argon (LAr) sandwich calorimeter [111, 119] was installed in the central and forward region.It was supplemented in the backward region with the lead-scintillating fibre calorimeter SpaCal [112,120]. In the ZEUS detector the solenoid was surrounded by a high-resolution compensating uranium–scintillator calorimeter (CAL) [121]. The calorimeters had inner electromagnetic and outer hadronicsections. The electron and hadron energy scales of the calorimeters were known at the level of 1% and26%, respectively. The calorimeters were calibrated from the data using kinematic constraints. Overall,both calorimeter systems performed very well.Additionally, for detection of very-low- Q events, where the electron is scattered at a small angle,ZEUS installed the beampipe calorimeter (BPC) [122]. This calorimeter was in operation in HERA Iand was located just 4 . . < Q < . Electron taggers:
Both H1 and ZEUS were equipped with special detectors, called electron taggers,which were able to detect electrons scattered at very small angles. Especially during the HERA I period,these could be used to explicitly identify photoproduction events in specific ranges of W . Muon systems:
To identify muons both experiments installed large arrays of limited-streamer tubes [111,123] inside and outside the magnetic return yoke (not shown), which covered a wide range in polar an-gle and measured muons efficiently for transverse momenta above ∼ p T . The return yoke of the ZEUS detector was also equipped with drift tubesproviding complementary muon identification and serving as a backing calorimeter (BAC). A forwardmuon system completed the coverage of the tracking detectors. The H1 detector had a similar muoncoverage, including the usage of the liquid argon calorimeter as a tracking calorimeter. Luminosity measurement:
In both experiments the luminosity was measured using the photonbremsstrahlung process ep → eγp . The photons were detected by dedicated detectors [124, 125] about100 m away from the interaction points in the e -beam direction. In addition, H1 also used the SpaCalto measure the large-angle QED compton scattering [126]. The ultimate precision of the luminositymeasurement by H1 (ZEUS) is 2 .
3% (1 . .
5% (2 . Trigger and readout system:
Both H1 [127] and ZEUS [113, 128] have used a multi-level triggersystem to select interesting ep events online and to suppress background from beam–gas interactions.The H1 trigger system consisted of two hardware layers and one software filter. It was supplemented forthe HERA II period by an additional track trigger. The ZEUS trigger was based on one hardware andtwo software levels. The first two levels mostly operated with the energy sums in the calorimeter, timingand limited tracking information. On the third level a complete reconstruction of the event is performed,using a simplified version of the offline reconstruction software, to select more sophisticated objects likejets, tracks and even D-meson candidates. The triggers reduced the rate from the nominal HERAbunch-crossing rate 10 MHz to the storage rate ∼
10 Hz. While the topology of DIS events allowedtriggering on the scattered electron inclusively already at the first level of the trigger chain, triggeringon heavy-flavour photoproduction was more challenging and required reconstruction of leptons, tracks,hadronic activity in the calorimeter or even explicitly charm hadrons. Both experiments had capabilitiesto include limited tracking information already at the first trigger level, for instance on the number oftracks and the fraction which originates from the ep interaction vertex. Various heavy-flavour tagging techniques (see Section 4) exploit different measured quantities, liketracks, vertices, energy-flow objects, jets and muons. The reconstruction of these quantities is describedin the following: 27igure 16: Helix parameterisation in the track-fit procedure. The initial parametrisation at the pointof closest approach as well as effects of multiple scattering are shown.
Tracking:
Tracks were reconstructed combining hits from the central tracking chambers, silicon-stripdetectors and forward/backward trackers for high | η | . In both experiments tracks were parametrisedwith a helix defined by 5 parameters (Fig. 16 shows the r - φ -view): the curvature κ = Q/R , which isthe signed inverse radius, D , the dca distance of closest approach in the XY plane, φ , the azimuthangle, z , the distance of closest approach along the Z axis, and the polar angle θ . To account formultiple scattering and energy loss in the material along the track trajectory and for inhomogeneities ofthe magnetic field, the track parametrisation was refined in a track-refit process based on a broken-linesalgorithm [129] in H1 and a Kalman filter [130] in ZEUS. Additionally, the parameterisation of all tracksthat were fitted to the primary or to a secondary vertex (see below) was further improved by a trackrefit using the vertex position as a constraint.Analyses that aimed at the best tracking precision used tracks with typically | η | < .
7. This coveragecorresponds to the region where efficiencies and resolutions are high and well known. Performancebenchmarking for the two tracking systems is shown in Table 2.At low momenta, the dE/dx measurement for each track allowed the separation of pions, kaons andprotons, while at high momenta electron/hadron separation was possible to some degree.
Vertexing:
Reconstructed tracks were used as an input for the primary vertex in each event. Inaddition, if silicon-strip information was available, the time-averaged mean XY position of the ep interaction region, the beam spot, was used to further confine the position of the primary vertex in theevent. The beam spot was measured by the experiments as a function of time for each ∼ σ x = 145 µ m and σ y = 25 µ m in H1HERA I data, σ x = 110 µ m and σ y = 30 µ m in H1 HERA II data, and σ x = 85 µ m and σ y = 23 µ m inZEUS HERA II data. The beam-spot size along the Z axis was much larger, σ Z ∼
10 cm, and thereforewas not used as a constraint. In H1 the vertex fitting was performed in the XY plane , while ZEUSdid a full vertex fit in XY Z . Nevertheless, both experiments used only the XY projections of decaysin heavy-flavour analyses due to superior resolution.In addition, in the context of some ZEUS heavy-flavour analyses, selected tracks were removed fromthe primary vertex fit and the fit was re-done. Combinations of such tracks were fitted to a displacedsecondary vertex that was associated with a decay of a heavy-flavour hadron. Procedures similar tothose used in the primary-vertex fit are used to fit the secondary vertices as well. Alternatively,combinations of impact parameters of several tracks were used in H1. Secondary vertices give animportant handle: • to test the hypothesis that selected tracks originate from a decay of the same particle by evaluationof the χ of the secondary vertex; An iterative procedure to determine the Z position of the primary vertex was used. For obvious reasons, no beamspot constraint is used. to evaluate the flight distance of that particle, which is related to the particle cτ .This will play an essential role (together with the track impact parameter) in the heavy-flavour lifetimetagging (cf. Section 4). Electron reconstruction:
Electron identification was needed to reconstruct DIS events as well asto measure semi-leptonic decays of beauty and charm quarks. Electrons were separated from hadronsusing the shape of clusters in the calorimeter and dE/dx information from the central drift chambers.A typical phase-space coverage for electrons in a beauty-production measurement in the semi-leptonicelectron decay channel was p T > p T > . − < η < . | η | < .
5) for the H1 (ZEUS)measurements.
Muon reconstruction:
Muons were identified by combining information from the tracking systems,calorimeters and muon chambers. For p T > . − p > − . < η < . Hadronic system:
Energy flow objects (EFOs) were used in both experiments to reconstruct thehadronic final state [132, 133]. These objects were based on a combination of information from thecalorimeter and the tracking system optimising energy resolution. Track information is superior forlow-energy EFOs, while the calorimeter measurement is preferred at high energy as well as to measureneutral particles.Jets at HERA have been reconstructed with the inclusive k T clustering algorithm [134, 135] . The R parameter was set to R = 1, which is larger than the values used typically at pp and p ¯ p experiments(0 . . k T , anti- k T [138] and SIScone [139] algorithms produce very similar measurement resultsand that the precision of NLO QCD calculations for the anti- k T algorithm is very similar to that of the k T algorithm.The final precision of the jet energy scale uncertainty of the H1 and ZEUS calorimeters was 1 – 2%(see [140] for a recent review of jet results from HERA). HERA was the first and so far only high energy ep collider. The results discussed in this review wereobtained with the H1 and ZEUS detector in two different running periods, denoted “HERA I” and“HERA II”. The main detector parts relevant for the detection and reconstruction of heavy flavourevents were the electromagnetic part of the calorimeters for the reconstruction of the scattered electron(if detectable), the calorimeters and tracking systems for the reconstruction of the decay products ofheavy flavoured particles, and the muon systems for the detection of semileptonic decay final states. Mostly in the longitudinally-invariant mode with the massless P T and massive E T recombination schemes [136] inthe H1 and ZEUS experiments, respectively Charm and Beauty detection at HERA
The large charm- and beauty-quark masses result in kinematical suppression of their production com-pared to the light-flavour cross sections. Therefore, special techniques have to be employed to separatecharm and beauty “signal” from the dominating light-flavour “background”. These techniques utilisedistinct properties of the charm and beauty hadrons (see Fig. 17 for illustration): ) (MeV) p )-M(K s pp M(K
140 150 160 170 C o m b i na t i on s / . M e V -1 ZEUS 372 pb < 1 GeV
130 < W < 300 GeV, Q )| < 1.6 and + (D* h ) > 3.8 GeV, | + (D* T p )| > 1.6 (D h ) < 3.8 GeV or | (D T p + Background mod GaussWrong charge BackgroundBackground 59 – ) = 2139 + (D* add N ZEUS / cm d -0.1 0 0.1 E n t r i e s H1 / cm d -0.1 0 0.1 E n t r i e s H1 Data b c uds sum / cm d -0.1 0 0.1 E n t r i e s / cm d -0.1 0 0.1 E n t r i e s H1 Data b c uds sum / cm d -0.1 0 0.1 E n t r i e s (a) p relt [ GeV ] E n t r i es Databb - cc - udsSum H1 Q < Figure 17: Overview of tagging methods for heavy-flavour events. Each method is accompanied by anillustrative distribution [141, 142, 143].1.
Flavour tagging.
The tagging of the quark flavour is done either by full reconstruction ofdecays of heavy-flavoured hadrons or by lepton tagging from the semi-leptonic decays of thosehadrons. The former was used at HERA only for charm tagging, since low production rates andsmall branching ratios for useful decays led to insufficient statistics for fully-reconstructed beautyhadrons. The latter was mostly used for beauty tagging, since b -hadron decays produce leptonswith sufficiently high momenta for efficient identification in the detectors.2. Lifetime tagging.
This method exploits the relatively long lifetimes of weakly-decaying heavy-flavour hadrons through the reconstruction of tracks with large impact parameter δ or displacedsecondary vertices. In addition, the information about the flight direction, extracted from eitherthe track, hadron or jet momentum, can be used to construct a signed impact parameter or a signeddecay length (see later). This is a powerful tool to separate charm and beauty from light-flavourevents, in which tracks mostly originate from the primary vertex.30. Mass tagging.
The tagging using the mass of the heavy quark or meson is performed eitherexplicitly by a full reconstruction of the mass from all decay products (also see flavour tagging), bya partial reconstruction via the mass of a jet or of all tracks at a secondary vertex, or indirectly bymeasuring the relative transverse momentum of a particle with respect to the axis of the associatedjet, p rel T . The latter was mostly used to separate beauty events from production of other flavours,since the large quark mass produces large p rel T values.The above methods are all based on measuring the decay particles of one heavy quark ( single tag ).Several methods can be combined to increase the purity at the cost of statistics. Both heavy quarks( double tag ) in an event can be tagged, by applying one method to tag one heavy quark and another (orthe same) for the other heavy quark. This allows a more detailed study of the heavy-quark productionmechanisms, but the double tagging efficiencies are low.In the following the different tagging methods are discussed in more detail with emphasis on theadvantages and disadvantages of each method. Most of the HERA charm results have been made using the golden decay channel D ∗ + → D π + s → ( K − π + ) π + s (see Fig. 18) . Occasionally, also the D → K − π + π − π + decay channel was used to increasestatistics. Due to the small energy release in the decay D ∗ + → D π + s ( M ( D ∗ + ) − M ( D ) − M ( π + ) ≈ D ∗ − D mass difference is strongly enhanced, providing an excellent signal to background ratio. This also leadsto a small momentum of the produced pion, which is often called the “slow” pion, π s . The capability ofa detector to measure very-low-momentum tracks defines the accessible region of p T ( π s ) and thereforethe phase space of the D ∗ : a typical p T ( π s ) > . p T ( D ∗ ) > . p T ( D ∗ ) cut was raised to suppress the combinatorial background, which rises at low p T (cf.Fig. 18 (a) and (b)). For the signal extraction, the observable ∆ M = M ( K − π + π + s ) − M ( K − π + ) waschosen. The number of signal events is determined either by counting the number of events in the peakregion after subtracting the combinatorial background, which is estimated from the ∆ M distributionof “wrong charge” K + π + π − s combinations, by fitting the spectrum with a Gaussian-like shape for thesignal and a phenomenological function for the background, or by a combination of these two methods.Besides the D ∗ golden decay channel, the D → K − π + , D + → K − π + π + , D + s → K + K − π + andΛ + c → pK − π + decays of charm hadrons were used to tag charm in the events. These charm hadronsfeature much larger background (see Fig. 18(c) for an example of a D + measurement). However, alifetime tag can be added, exploiting the relatively large values cτ ∼
100 – 300 µ m for the weakly-decaying charm hadrons. The decay length is reconstructed by fitting a displaced secondary vertex toselected tracks of decay products (see Section 4.3 for more details). This combined approach allowed tosignificantly reduce combinatorial background, most noticeably for the D + that has the largest lifetime(cf. Fig. 18(c) and (d)), to a level that is still somewhat worse than but comparable to the D ∗ .Nevertheless, due to the low boost, lifetime tagging is inefficient in the phase space p T ( D ) < m ( D ) thatis also not accessible with D ∗ ’s. In this region it is beneficial to study particular decay channels withneutral strange hadrons K S or Λ, for example D + → K S π + .The H1 and ZEUS experiments achieved a similarly good signal mass-peak resolution for charmhadrons. Note that the distributions in Fig. 18 can not be compared directly due to different kinematicregions.In summary, the advantages (+) and disadvantages ( − ) of full hadron reconstruction are: Throughout the paper the D ∗ + is mostly referred to as D ∗ . The full reconstruction of D ∗ ’s at HERA was impossiblesince the resulting decay photon or π could not be reliably measured. [GeV] p ) - m(K pp m(K E n t r i es / . M e V H1 dataWC comb.Sig+Bg Fit 343 D* – H1 (a) p T ( D ∗ ) > .
25 GeV ) (MeV) p )-M(K s pp M(K
140 150 160 170 C o m b i na t i on s / . M e V -1 ZEUS 372 pb < 1 GeV
130 < W < 300 GeV, Q )| < 1.6 and + (D* h ) > 3.8 GeV, | + (D* T p )| > 1.6 (D h ) < 3.8 GeV or | (D T p + Background mod GaussWrong charge BackgroundBackground 59 – ) = 2139 + (D* add N ZEUS (b)
ZEUS pp ) (GeV) C o m b i na t i on s pe r M e V ZEUS 1999-2000Reflections subtractedGauss mod + Backgr.130 < W <
300 GeV, Q < p T (D ± ) > h (D ± )| < ± ) = 8950 ± (c) ) (MeV) pp M(K C o m b i na t i on s / . M e V ZEUS -1 ZEUS 372 pb < 1 GeV
130 < W < 300 GeV, Q)| < 1.6 + (D h ) > 3.8 GeV, | + (D T p + Background sum GaussBackground 324 – ) = 18917 + N(D (d)
Figure 18: Full reconstruction of mass spectra for charm mesons in the data. The reconstructed massdifference ∆ M for D ∗ (a) [144] and (b) [143] and the D + mass (c) [145] and (d) [143] are shown.+ The full reconstruction allows an accurate determination of the momentum of the charm hadron,which is correlated with the kinematics of the charm quark and can be used to study the frag-mentation process.+ An excellent signal to background ratio of ∼ D ∗ golden decaychannel or the reconstruction of other weakly-decaying mesons including a lifetime tag (Fig. 18).+ The combinatorial background can be parametrised with an empirical function and does notdepend on Monte Carlo simulations of the light-flavour background.+ The signal mass peak is a clear signature which can be used in the online filtering of events toreduce the trigger rates. This requires the usage of advanced tracking information in the triggerlogic. − The typical probabilities for a charm quark to hadronise into a specific D meson are ≈ .
15 – 0 . BR , for the commonly used decay channels are ≈ .
05 – 0 .
10; thus, only ∼ c quarks can be tagged. The kinematic and geometric acceptances reduce thevisible fraction even further, in particular in the case of the additional use of lifetime tagging,which reduces the detection efficiency by a factor ∼ -0.1 -0.05 0 0.05 0.1 0.15 Impact Parameter d [ cm ] E n t r i es Databb - cc - udsSum H1 Q < p relt [ GeV ] E n t r i es Databb - cc - udsSum H1 Q < Figure 19: Distributions of the signed impact parameter δ of the muon track (left) and the transversemuon momentum p rel T relative to the axis of the associated jet (right) for the photoproduction eventsample of the H1 beauty analysis [142]. The data (dots) are compared to Monte Carlo predictions(solid line). Separate contributions of events arising from b -quark (dashed line), c -quarks (dotted line)and the light-quark (dash-dotted line) production are shown. r elT A well established method to identify beauty quarks is to select a muon with high transverse mo-mentum of typically above 1 . − b -quark decay, which is associated to a jetthat represents the beauty quark and consists of the muon and further final-state particles. The back-ground to this signature is composed of charm production with a genuine muon from semi-leptonicdecays and of light-flavour events with a hadron misidentified as a muon (mainly due to in-flight π + and K + decays and hadronic energy leakage). To separate beauty from the background contributionsfor a single tag, the p rel T observable is used which, due to the large beauty-quark mass, extends to muchlarger values than for the other sources. Additionally, a lifetime tag can be added by reconstructing thesigned impact parameter, δ , of the muon track (see Section 4.3 for more details). This further improvesbeauty separation and also allows charm-event tagging, since long-lived heavy-flavour hadrons lead tolarger δ values than in light-flavour events. Figure 19 shows the distribution of both variables. Thebeauty component has a distinct shape in both that allows one to disentangle it from the others. Thefractions of light-flavour, charm and beauty events in the data are determined from template fits tothe discriminating variables extracting template shapes from Monte Carlo simulations with data-drivencorrections. In some analyses also the missing energy in the detector, which is associated with theundetected neutrino, was considered to improve charm/light-flavour separation.A similar technique can be followed using electrons. This allows going down to p T ( e ) > η region is narrower than for muons (see Section 3) and the lepton signature ismore complex.In summary, the advantages (+) and disadvantages ( − ) of the use of semileptonic hadron decaysare: Also charm, but less efficiently.
33 The relatively large branching ratio BR ( b → ℓX ) ∼
21% [146], which includes b → cX → ℓX andother cascade decays, provides a reasonable tagging efficiency for b quarks.+ Muon tagging extends the phase space of heavy-quark measurements, due to additional coverageoutside of the polar acceptance of the central tracking systems. Thus measurements of beautyproduction in the forward and backward regions are possible with muons.+ Semi-isolated leptons provide a clean experimental signature for the trigger system. They allowone to efficiently select beauty events suppressing charm and light-flavour production by p T ( ℓ )cuts. − The requirement of a jet associated with the lepton in order to use p rel T or lifetime tagging cutsinto the low- p T phase space of the b quarks. − The usage of semi-leptonic tagging for charm studies is very complicated due to weak separationpower.
The aforementioned tagging methods suffer from the fact that only a fraction of the charm or beautyquark decays ends up in the selected final state. This can be avoided by using an inclusive taggingmethod , based on the long lifetime of charm and beauty quarks: cτ c ≃
100 – 300 µ m and cτ b ≃ µ m.This approach relies on silicon-strip detectors to accurately measure the track parametrisation in thevicinity of the interaction vertex (see Section 3). Therefore, it was not yet available in ZEUS for theHERA I data set and was pioneered at HERA by the H1 collaboration. Lifetime tagging can be basedeither on impact parameters of individual tracks or on decay lengths of displaced secondary verticesfitted to selected tracks. Both techniques typically use tracks with transverse momenta p T > . XY plane, since the profile of the interactionregion and also the detector layouts do not allow for sufficiently high resolution of the tracks in thecoordinate along the beam line.The impact parameter distribution allows the separation of long-lived heavy-flavour hadrons fromshort-lived light-flavour hadrons. Figure 20 illustrates how the signed impact parameter is defined withrespect to the jet to which the track is associated. A positive sign is assigned to δ if the angle α betweenthe jet axis and the line joining the primary vertex and the point of closest approach is less than 90 ◦ ,and a negative sign otherwise. Figure 19(left) shows the distribution of the signed impact parameter ofmuon tracks in the data and the flavour decomposition in the Monte Carlo simulations. The light-flavourcomponent is characterised by a very small lifetime and the observed symmetric distribution is causedby the detector resolution. In contrast, the charm and beauty contributions exhibit a pronounced tailfor large positive δ values. To further improve the separation power, the impact parameter significance S = δ/σ ( δ ) can be used. This allows the rejection of candidates with an insignificant measurement ofthe impact parameter.In the vertexing approach, a displaced secondary vertex is fitted to all tracks that are associatedto a selected jet . The distance between the primary and the secondary vertex is sensitive to thelifetime of the hadron that initiated the jet. Similar to the impact parameter, the flight direction canbe introduced to form the signed decay length as used by H1 or the projected decay length in ZEUS.The former is defined similarly to the signed impact parameter, whereas the latter is defined as l = ( ~r SV − ~r PV ) · ~p | ~p | Here, a jet can be either a real jet in the detector or a set of tracks corresponding to a D -meson candidate. et axistrack jet axistrack Figure 20: Illustration of the positive and negative signed impact parameter (left). A reconstructedvertex for a heavy-flavour decay shown in the XY plane [147] (right). The errors of the primaryand secondary vertex positions (shaded ellipses) have been blown up by a factor of 10 for illustrativepurposes.(see Fig. 20), i.e. the projection of the vector from the primary to the secondary vertex on the jetmomentum. Finally, the ratio of such a quantity over its uncertainty, S , provides the optimal sepa-ration power. A kinematic reconstruction of the mass of the vertex that corresponds to the jet, m vtx ,provides an additional handle on flavour separation, since contributions of light-flavours, charm andbeauty are expected to populate predominantly the small ( m vtx ≪ m D ), medium ( m vtx < ∼ m D ) andlarge ( m D < m vtx < ∼ m B ) mass domains, respectively. Figure 21 illustrates the lifetime tagging withsecondary vetrices. The light-flavour contribution is symmetric around zero, while the charm andbeauty components exhibit a pronounced asymmetry in the region of large decay-length significance .The beauty contribution dominates at large vertex mass and large significance values.Lifetime tagging can be used either as an add-on to other tagging techniques (as described before) oras a separate tagging tool. The dominant background is light-flavour production, which is symmetric inthe signed impact parameter significance or the projected decay length significance. The contents of thenegative bins of the significance distribution can be subtracted from the contents of the correspondingpositive bins, yielding a subtracted significance distribution. This way, the contribution from light-flavour quarks is minimised.The H1 collaboration has chosen to use a combination of the signed impact parameter significanceof individual tracks and the signed decay-length significance to tag heavy-flavour production. Eventsare exclusively categorised according to the number of tracks in the event. The significances S , S and S are defined as the significance of the track with the highest, second highest and the thirdhighest absolute significance, respectively. The S and S significance distributions (Fig. 22) are usedfor events with one and two selected tracks, respectively. For events with three or a higher numberof tracks various sensitive variables including S , S , S and the signed decay-length significance ofthe reconstructed secondary vertex are combined using an artificial neural network. In general, S has a better discrimination between light- and heavy-flavour contributions than S , since the chance ofreconstructing two high significance tracks is further reduced for light-flavour. The neural network (Fig.22(c)) provides separation between c and b events. For all distributions the negative part was subtractedfrom the positive one to minimise the light-flavour component and a least-squares fit was performedsimultaneously to all three distribution. The charm and residual light-flavour components were found tobe very strongly anti-correlated in such fits (typical correlation coefficients are C lc < − . -20 -15 -10 -5 0 5 10 15 20 E n t r i es S -20 -15 -10 -5 0 5 10 15 20 E n t r i es S -20 -15 -10 -5 0 5 10 15 20 E n t r i es S -20 -15 -10 -5 0 5 10 15 20 E n t r i es S -20 -15 -10 -5 0 5 10 15 20 E n t r i es S -20 -15 -10 -5 0 5 10 15 20 E n t r i es S -20 -15 -10 -5 0 5 10 15 20 E n t r i es S -20 -15 -10 -5 0 5 10 15 20 E n t r i es ZEUS < 1.4 GeV vtx vtx vtx vtx
No restriction on m -1 ZEUS 354 pbMonte CarloLFCharmBeauty
Figure 21: Distributions of the decay-length significance for different ranges of the vertex mass m vtx .[148]. The data (points) are compared to Monte Carlo simulations (filled area). The individual contri-butions of beauty (dashed line), charm (dotted line) and light flavours (solid line) are shown.correlation with beauty is weaker due to the more distinct shape of the beauty distributions ( C cb ≈ − . C lb ≈ . χ fit of the subtractedsignificance distribution was performed in the three vertex-mass bins simultaneously. The correlationpattern between components was found to be very similar to the one in the H1 analyses. Figure 23shows the subtracted significance distributions. With optimised cuts after subtraction one can get sam-ples with very high charm and beauty enrichment of roughly 80% and 90%, respectively. Such selectionresulted in ∼ ∼ − ) of this inclusive lifetime tagging methodare:+ This tagging method gives access to the largest statistics due to the inclusive selection of the finalstate.+ The technique provides strong discrimination power and is often combined with other taggingmethods.+ With the applied track minimal transverse momentum cut of 0 . ep scattering. The additional typical jet cut E jet T > ∼ m b retains a high acceptance for beauty production near threshold. − The total achieved effective signal to background ratio is typically not better than 1:10 for bothcharm and beauty. This can be estimated from the numbers of charm and beauty events in thepositive subtracted significance spectra, which effectively represent the numbers of tagged events,36
Track significance S ) ) - n ( - S n ( S Track significance S ) ) - n ( - S n ( S
10 H1 DataTotal MCudscb Track significance S ) ) - n ( - S n ( S
10 (a) Track significance S ) ) - n ( - S n ( S Track significance S ) ) - n ( - S n ( S
10 H1 DataTotal MCudscb Track significance S ) ) - n ( - S n ( S
10 (b)
NN Output n ( NN O u t pu t) - n ( - NN O u t pu t) NN Output n ( NN O u t pu t) - n ( - NN O u t pu t)
10 H1 DataTotal MCudscb
NN Output n ( NN O u t pu t) - n ( - NN O u t pu t)
10 (c)
Figure 22: The subtracted distributions of (a) S , (b) S and (c) the neural-network output [149].The beauty- (dashed line), charm- (dotted line) and light-flavour (dashed-dotted line) contributions areshown.and from the errors achieved for the charm and beauty components in the fit. − The method requires the track resolutions and efficiencies to be thoroughly scrutinised. − With the typical cuts on the jet transverse momentum one actually cuts strongly into the kinematicphase space for charm. A requirement of a jet with E jet T > p DT > . ∼
60% of the quark transverse momentum is transfered to the D meson. The various heavy-flavour tagging methods outlined above can be combined, aiming towards taggingboth heavy quarks in the event. At HERA, D ∗ µ combinations were used to tag both charm and beautyevents, while a di-lepton tag was used for beauty only. The usage of two flavour tags significantly reducesthe light-flavour background, which allows omitting any additional mass or lifetime tags. Furthermore,it gives access to correlations between the quarks of heavy flavour pair.In the example of the photon-gluon fusion process, γg → c ¯ c or b ¯ b , the two heavy quarks are producedback-to-back in the γg frame as illustrated in Fig. 24 for the beauty case. Unlike-sign combinations suchas µ + µ − or D ∗ + µ − can be produced from either the same or different b quarks, while like-sign combina-tions originate always from different b quarks (combination of b and ¯ c decays + charged conjugate, or B − ¯ B mixing). In charm events only oppositely-charged combinations are produced. In analyses using D ∗ µ tags the charm and beauty components were separated based on the sign combination and theangular distance in azimuth. In charm events mostly back-to-back unlike-sign D ∗ and µ are produced,while for beauty both like- and unlike-sign combinations are possible and more complicated angulardistributions arise. The di-lepton analyses have also used the information about charge combinationand angular separation between leptons, and additionally the mass of the lepton pair.In summary, the advantages (+) and disadvantages ( − ) of double tagging are:+ For a large fraction of the events double tagging gives access to the kinematics of both heavyquarks. This information can be used to investigate the c ¯ c and b ¯ b production processes in detail.+ Since light flavours are efficiently suppressed by the requirement of two flavour tags, the leptonscan be selected in transverse momentum down to ∼ p rel T tag that was discussed inSection 4.2. Light-flavour production is suppressed by D ∗ reconstruction S| E n t r i es |S| E n t r i es |S| E n t r i es |S| E n t r i es |S| E n t r i es |S| E n t r i es |S| E n t r i es |S| E n t r i es ZEUS < 1.4 GeV vtx vtx vtx vtx
No restriction on m -1 ZEUS 354 pbMonte CarloLFCharmBeauty
Figure 23: The subtracted distributions of the decay-length significance for different ranges of thevertex mass m vtx [148]. The data (points) are compared to Monte Carlo simulations (filled area).Individual contributions of beauty (dashed line), charm (dotted line) and light flavours (solid line) areshown. D * +−+ _ − _ Figure 24: Various possible ways to produce D ∗ µ or a µµ pair from the decays of the b - and the ¯ b -quarkproduced in the photon gluon fusion process at HERA. − The total tagging efficiency is very low. − The lepton tagging is well suited for the measurement of beauty production but has relativelysmall acceptance for charm, where, due to softer fragmentation, the leptons take a smaller fractionof the quark transverse momentum than in the beauty case. − Due to their low p T , the correlation of the D ∗ and the µ momenta with those of the parent quarksis not as good as for jets. Various heavy-flavour tagging methods have been used at HERA. Each of them has advantages anddisadvantages, which results in different tags being optimal for different purposes. The most commonlyused tags have been D ∗ reconstruction and inclusive lifetime tags for charm and lepton + p rel T and inclu-sive lifetime tags for beauty. Whenever possible, a comparison between (and potentially a combination38f) measurements performed with different techniques allows improved constraints on the measurements,due to cross-calibration of systematics of different nature for independent tags. Often a combinationof tags yields an increased purity of the heavy-flavour sample, at the cost of reduced efficiency andadditional systematics. In general, the choice of the tagging method(s) is a trade-off between statisticaland systematic uncertainties. 39 Search for single top-quark production
Already before the start of HERA data taking, it became clear from the lower limits of order 70 – 80 GeVon the top-quark mass obtained from ¯ pp collisions by the UA1/UA2 [150] and CDF [151] collaborations,that top-quark pair production would probably be outside the kinematic reach of HERA. This wasconfirmed by indirect constraints from LEP [152] and by the direct observation of the top quark at theTevatron at a mass of 174 GeV [153]. Single top quark production in the charged current reaction [154] e + + b → ¯ ν e + t (and its charged conjugate) remained kinematically possible, but the expected Standard Model crosssection of less than 1 fb [155] is too small to be experimentally accessible. This is due to the fact thatthe occurrence of initial state b quarks is strongly suppressed at high x , since it would need to originatefrom the splitting of ultra-high- x gluons in the proton (in analogy to Fig. 2), which are known to bevery rare [15]. Charged current reactions on light initial state quarks are strongly suppressed by thevery small corresponding CKM matrix elements [156].If at all, single top quarks could thus be produced at HERA only via a process beyond the StandardModel [157]. One such process is the transition of a u quark into a t quark via a flavour-changingneutral current [158] (Fig. 25) caused by non-Standard Model couplings of the photon or Z boson.Figure 25: Feynman graph for anomalous single top production [159].This possibility was investigated in particular due to an excess observed by the H1 collaboration inthe single isolated lepton + jets final state [160], which was however not confirmed by a correspondingZEUS analysis [161], and greatly reduced in significance by a common analysis of the ZEUS and H1data [162]. g u Br -3 -2 -1 uZ B r -2 -1 exc l ud e d b y Z E U S ZEUS -1 ZEUS 0.5 fb ,u(c)Z g u(c) fi ALEPH t ,u(c)Z g u(c) fi CDF t g u fi H1 t u(c)Z fi D0 t g u Br -3 -2 -1 uZ B r -2 -1 Figure 26: Limits on anomalous couplings for single top production, translated to branching fractions(Br) for top decay into uZ or uγ [159]. 40o significant signal was observed. Fig. 26 shows the exclusion contours obtained by the H1 [163]and ZEUS [159] collaborations for the anomalous couplings to photons or Z bosons for this process,translated to branching ratios for anomalous top quark decays. The H1 limit is a bit looser than thethe one from ZEUS due to the small excess mentioned above. These limits are competitive with limitsobtained from other colliders, also shown in Fig. 26, and currently represent the best limit for theanomalous photon coupling. They can be improved further in future analyses using data from the LHC[164]. Single top quarks were also searched for in the full-hadronic top decay channel, with a similarsensitivity as for the lepton channel [165]. However, the corresponding analysis was performed on theHERA I data set only, so the resulting limit is no longer competititive.41 Charm photoproduction
The study of open charm- and beauty-production cross sections provides stringent tests of perturbativeQCD from several perspectives. On one hand, the size of the charm- and beauty-quark masses ensuresthat for all such final states the production cross sections are in the perturbatively calculable regime,since m c , m b ≫ Λ QCD . On the other hand, the QCD scales obtained from these masses compete withother potential scales like the quark tranverse momentum or, in the DIS case, the virtuality of theexchanged photon (Fig. 3). The treatment of such a multi-scale problem is theoretically challengingand a comparison of different theoretical schemes to data can shed light on the strengths and weak-nesses of the respective perturbative approximations. Furthermore, from the theoretical point of view,the presence of final-state heavy-flavour hadrons ensures that these processes will not interfere withcorresponding final states involving only gluons and light quarks, and that they can thus be treatedindependently for each flavour. Thus, in the following, charm and beauty production will be treatedseparately.In general, photoproduction processes have many similarities to corresponding processes in hadropro-duction, “simply” interchanging the incoming quasi-real photon with a quasi-real gluon. Since the S p ¯ p Sand Tevatron p ¯ p colliders went into operation almost a decade before the HERA collider, perturbativeNLO QCD calculations have often first been obtained for hadroproduction, although the incoming pho-ton diagrams are somewhat easier to calculate. Furthermore, since the virtuality of the incoming photon( Q < ) is of the order of typical hadron masses or lower, the photon can have a hadron-likestructure (“hadron-like resolved photon”, Fig. 6b). Thus, the cross sections get a contribution from theconvolution of perturbative hadroproduction diagrams with this photon structure. This in turn com-plicates the photoproduction cross-section calculations. However, this hadron-like photon contributionis small (of order 10% or less) in the case of the treatment of heavy-flavour production in the massivescheme.An important variable for photoproduction analyses at HERA is the event kinematic observable y ,which represents the fraction of the electron energy entering the hard interaction (Section 2.1). Thisvariable can be reconstructed from the hadronic final state in the main detector (Section 3). An overviewof all charm photoproduction measurements in H1 and ZEUS is given in Table 3. D ∗ inclusive measurements Despite the large loss in statistics through fragmentation fractions and branching ratios, it is clear fromTable 3 that the very clean explicit reconstruction of D ∗ -meson final states offers the best effectivesignal sensitivity for charm photoproduction. In the very first ZEUS [166] and H1 [167] measurements on open-charm production (entries 1 and 2 inTable 3) the inclusive D ∗ results were extrapolated to obtain total charm-photoproduction cross sections.The results are shown in Fig. 27(left) as a function of the photon-proton centre-of-mass energy W γp .Measurements from fixed-target experiments from the pre-HERA era are also shown. At HERA, both W γp values and the observed cross sections are roughly one order of magnitude larger. The steep cross-section rise reflects the fact that with increasing W γp gluons with smaller and smaller proton momentumfractions are accessible for charm production via the photon-gluon-fusion process (Fig. 4). The datain Fig. 27 are compared to a massive scheme NLO prediction [58], which is able to describe both thefixed-target data at lower W γp and the HERA data at higher W γp . Despite the large uncertainties,this demonstrated early on that the basic charm-production mechanism in photoproduction is at leastreasonably well understood. 42 o. Analysis c-Tag Ref. Exp. Data L [ pb − ] Q [ GeV ] y Particle p T [ GeV ] η Events effect.s:b bgfreeevents1 D ∗ incl. Kππ s [166] ZEUS 93 . < . , . D ∗ > . − . , .
5] 48 ±
11 1 : 1 . D ∗ tagged Kππ s [167] H1 94 . < .
01 [0 . , . D ∗ > . − . , .
0] 119 ±
16 1 : 1 . incl. . < . , .
80] 97 ±
15 1 : 1 . D ∗ incl. Kππ s [168] ZEUS 94 . < . , . D ∗ > − . , .
0] 152 ±
16 1 : 0 . K ππ s ±
29 1 : 3 . D ∗ tagged Kππ s [169] H1 95-96 . < .
009 [0 . , . D ∗ > − . , .
5] 299 ± n.a. . < .
01 [0 . , . > . y ( D ∗ )) 489 ± n.a. D ∗ incl. Kππ s [170] ZEUS 96-97 < . , . D ∗ > − . , .
5] 3702 ±
136 1 : 4 . K ππ s > ±
108 1 : 7 . D ∗ + dijet Kππ s D ∗ Jet1(2) > > − . , . − . , .
4] 587 ±
41 1 : 1 . Kππ s [171] ZEUS 98-00 < . , . D ∗ [1 . ,
20] [ − . , .
6] 10350 ±
190 1 : 2 . D ∗ tagged Kππ s [172] H1 99-00 < .
01 [0 . , . D ∗ > − . , .
5] 1166 ±
82 1 : 4 . +jet Jet > − . , .
5] 592 ±
57 1 : 4 . +dijet Jet 1(2) > − . , .
5] 496 ±
53 1 : 4 . D ∗ + dijet Kππ s [173] ZEUS 96-00 < . , . D ∗ Jet1(2) > > − . , . − . , .
9] 1092 ±
43 1 : 0 . D ∗ + jet Kππ s [174] ZEUS 98-00 < . , . D ∗ > − . , .
5] 4891 ±
113 1 : 1 . + dijet Jet1(2) > − . , .
4] 1692 ±
70 1 : 1 .
10 lifet.+dijet imp.par. [141] H1 99-00 < . , . TrackJet1(2) > . > − . , . − . , .
3] 4600 ±
460 1 : 45 100 D ∗ + µ Kππ s + µ [175] H1 98-00 < . , . D ∗ µ > . p > − . , . − . , .
74] 53 ±
13 1 : 2 . e + dijet e + E T [176] ZEUS 96-00 < . , . e Jet1(2) > . > − . , . − . , . ∼ n.a.
13 lifet.+dijet sec. vtx. [177] ZEUS 05 < . , . tracksJet1(2) > . > − . , . − . , . ∼ n.a. µ + dijet µ +imp.par. [178] H1 06-07 < . . , . µ Jet1(2) > . > − . , . − . , .
5] 3315 ±
170 1 : 7 . D ∗ incl Kππ s [179] H1 06-07
31 – 93 < . , . D ∗ > . − . , .
5] 8232 ±
164 1 : 2 . +dijet Jet 1(2) > . − . , .
9] 3937 ±
114 1 : 2 . D ∗ incl Kππ s [180] ZEUS 06-07 < . , . D ∗ [1 . ,
20] [ − . , .
6] 12256 ±
191 1 : 2 . MER 07 . ±
37 1 : 2 . LER 07 . ±
49 1 : 1 . Table 3:
Charm photoproduction cross-section measurements at HERA. Information is given for each analysis on the charm tagging method, theexperiment, the data taking period, integrated luminosity, Q and y ranges and the cuts on transverse momenta and pseudorapidities of selected final stateparticles. The last three columns provide information on the number of tagged charm events, the effective signal-to-background ratio and the equivalent number ofbackground-free events. The centre-of-mass energy of all data taken up to 1997 (6 th column) was 300 GeV, while it was 318 – 319 GeV for all subsequent runs, withthe exception of the analyses marked “MER” and “LER” (entry 16), for which the data were taken at 251 and 225 GeV. -1
110 10 10 W g p (GeV) s ( g p fi cc - X ) ( m b ) E691PECSLACWA4EMCBFPCIF H1 tagged dataH1 untagged dataZEUS
QCD (MRSG + GRV) upper m /m c = 0.5 lower m /m c = 2 (GeV)s
240 260 280 300 320 s R ZEUS
D* T D* h | < 1 GeV Q = æ W Æ
136 GeV 152 GeV 192 GeV D* X fi ZEUS epNLO QCD
Figure 27: Left: Total charm-photoproduction cross section as a function of centre-of-mass energy W γp [167]. The data shown are from the first H1 and ZEUS publications on open charm production and fromprevious fixed-target experiments. Right: Inclusive charm-photoproduction cross section as a functionof ep centre-of-mass energy [180], normalised to the cross section at 318 GeV.Figure 27(right) shows the latest HERA measurement in photoproduction [180] (entry 16 in Table3), focusing on the dependence of the inclusive visible cross section on the centre-of-mass energy. Thismakes use of the very last HERA running period, in which the proton beam energy was lowered. Thisresult was obtained and published based on a ZEUS master thesis [182], which was made possible by thestrong simplification of the data format and calibration procedure implemented as part of a long-termhigh energy physics data preservation project [183]. The result is presented as a ratio to the highestcentre-of-mass energy cross section, such that both experimental and theoretical correlated uncertaintiescancel. While the data uncertainties remain dominated by statistical uncertainties (inner error bars),the theoretical uncertainties are dramatically reduced with respect to the absolute predictions in Fig.27. The massive NLO prediction [59] agrees well with the data, indicating that the extrapolation of theenergy dependence to even higher centre-of-mass energies such as those at a future LHeC collider [184]can be reliably predicted. In addition, since different centre-of-mass energies correspond to different x ranges, such a ratio potentially provides constraints on the gluon PDF in the proton. D ∗ single-differential cross sections Figure 28 shows the results for the ZEUS HERA I [171] and H1 HERA II [179] D ∗ analyses (entries 6and 15 of Table 3) as a function of the D ∗ transverse momentum and pseudorapidity. These results havebeen selected since the data samples used in these analyses are among those with the highest statisticalsignificance of all heavy flavour measurements at HERA, as can be seen from the last column of Table 3(entry 6). The data span a large kinematic range from p T ( D ∗ ) = 1 . ∼ m c to p t = 20 GeV ≫ m c .Over this range the cross section falls off by about four orders of magnitude. The measurements are com-pared to five NLO predictions: massive fixed-flavour scheme (NLO,FMNR) calculations from Frixione et al. [58], a variant of these calculations matched to parton showers (MC@NLO) [185], massless scheme(NLL) predictions from Kniehl et al. [48], general mass variable flavour scheme (FONLL) calculationsfrom Cacciari et al. [181], and a different GMVFNS variant (GMVFNS) from Kniehl et al. [83]. Bothcalculations from Kniehl et al. include a perturbative treatment of the charm fragmentation function.At first glance, all five predictions are able to describe the spectrum over the complete p T ( D ∗ ) rangewithin a factor of two. However, looking more in detail, one observes:44 [ nb / G e V ] T / dp s d -1 H1 DataFMNRGMVFNSMC@NLO (D*) [GeV] T p no r m R [ nb ] h / d s d H1 DataFMNRGMVFNSMC@NLO (D*) h -1.5 -1 -0.5 0 0.5 1 1.5 no r m R Figure 28: D ∗ single differential cross sections in photoproduction as function of the D ∗ transversemomentum (left) and pseudorapidity (right), from ZEUS [171] (top and center) and H1 [179] (bottom).The measurements are compared to five NLO predictions: the massive scheme calculations from Frixione et al. [58] without (NLO, FMNR) and with (MC@NLO) [185] interface to LL parton showering, themassless scheme predictions from Kniehl et al. [48] (NLL) and the general mass variable flavour schemecalculations from Cacciari et al. [181] (FONLL) and Kniehl et al. [83] (GMVFNS). The NLL andGMVFNS predictions include a perturbative treatment of D ∗ fragmentation.45. The uncertainty of the measurements is generally much smaller than those of the theory, domi-nated by QCD scale variations, the variation of the charm mass, and the variation of the charmfragmentation parameters. Especially for low transverse momenta p T ( D ∗ ) < p T , and for the η distribution, which is dominated by the low p T D ∗ contribution. Themeasured cross sections are higher than the central prediction, but the predictions are consistentwith the data within the large uncertainties. At high transverse momenta, contrary to manypeople’s expectations originally based on leading-order studies [186], the FONLL prediction isactually lower than the NLO prediction. Thus the final state resummation corrections originatingfrom higher-order log terms in the massless part of the calculation reduce the prediction, ratherthan enhancing it. The data are closer to the pure NLO prediction. At least within the HERAkinematic regime there is thus no evidence for the claim [186] that the massive fixed-order calcu-lation should fail at large values of charm transverse momentum unless final state resummationcorrections are applied. Both predictions give a reasonable but not perfect description of theshape of the η distribution.3. The massless NLL prediction, which, in contrast to the massive predictions discussed in theprevious item, incorporates a proper perturbative treatment of charm fragmentation [49], fits thedata well at low p T , while it is a bit too low for high p T , where it is expected to work best.As expected, it is similar to the FONLL prediction in this region. The theoretically superiortreatment of fragmentation does not lead to a smaller uncertainty, as can be seen from the η distribution. Also, the shape of the η distribution is a bit less well described than with themassive prediction. In this approximation, a large fraction of the cross section arises from the(massless) charm contribution to the photon parton density function (using the AFG [187] or GRV[188] parametrisations), in contrast to the “direct” contribution, which is also shown separately.4. The partially massive GMVFNS prediction, which incorporates a perturbative treatment of thecharm fragmentation function, has a larger uncertainty than the traditional massive predictions,similar to the NLL prediction. The shape of this prediction describes the data better than theNLL prediction.5. The MC@NLO prediction has the same core parton-level cross section as the NLO/FMNR pre-dictions by definition. The differences seen w.r.t. the latter must thus arise from the addition ofthe HERWIG-type parton showers and the different fragmentation treatment. It exhibits slightysmaller uncertainties, but, surprisingly, fits the data less well than the original NLO/FMNR pre-dictions. This offers room for potential retuning of some of the MC parameters entering thiscalculation.A similar inclusive D ∗ photoproduction measurement as the above is available from H1 [172], per-formed in a more restricted W γp region (entry 7 in table 3) with a roughly ten times smaller datasample. The narrower kinemetic range and smaller statistics are due to explicit detection of the elec-tron scattered at very low angles in dedicated forward electron taggers (section 3.2), which was part ofthe trigger requirement. The conclusions are very similar.ZEUS has also recorded such tagged photoproduction samples, but they were found not to bestatistically competitive with results from data sets triggered on inclusive D ∗ production.46 .1.3 D ∗ double-differential cross sections Double-differential cross-section measurements as a function of the D ∗ transverse momentum and pseu-dorapidity have been performed by ZEUS in [171] and also in a previous charm milestone paper [168](entries 6 and 5 in Table 3), and by H1 [179] (entry 15 in Table 3). The results of the latter areshown in Fig. 29, together with some of the predictions already discussed for the single-differentialcase. In general, the conclusions are similar to those from the single-differential cross sections. At high (D*) h -1 0 1 ( D * )[ nb / G e V ] h d T / dp s d (D*) < 2.5 GeV T p £ (D*) h -1 0 1 ( D * )[ nb / G e V ] h d T / dp s d H1 (D*) < 4.5 GeV T p £ (D*) h -1 0 1 ( D * )[ nb / G e V ] h d T / dp s d (D*) < 12.5 GeV T p £ DataFMNRGMVFNSMC@NLO
Figure 29: D ∗ double differential cross sections in photoproduction as function of the D ∗ transversemomentum and pseudorapidity from H1 [179]. The measurements are compared to three out of the fiveNLO predictions also shown in Fig. 28. p T the uncertainty of the theory predictions reduces as expected, such that the comparisons becomemore meaningful. Reasonable agreement with the data is observed for all predictions in this high- p T region, while MC@NLO undershoots the data at low p T and η , similar to what was observed in thesingle-differential case (Fig. 28). Although the D ∗ channel generally yields the best signal-to-background ratio and therefore the besteffective overall statistics (last column of Table 3), the small branching ratio limits the statistics inregions in which the cross section is small. In such regions, more inclusive tagging techniques can be anadvantage. Furthermore, the consistency of results obtained with different tagging methods enhancesconfidence in the results.H1 has performed a measurement based on inclusive lifetime tagging [141] (entry 10 in table 3),which extends to the highest charm transverse momenta p cT = 35 GeV reached so far. Here events47ith two jets in the central rapidity region are used (cuts are listed in table 3). Due to the highjet transverse momenta the events are efficiently triggered using the deposits of the jet particles inthe calorimeter. An inclusive lifetime tagging is applied, based on the displaced impact parametersof jet-associated charged tracks from charm and beauty decays. Details of the tagging method arediscussed in Section 4. Figure 30(left) shows the measured charm-production cross sections as functionof the transverse momentum of the leading jet. The data are compared to a massive scheme NLO / GeV jett p
15 20 25 30 35 [ pb / G e V ] j e t t / dp s d DataPythiaCascade had ˜ NLO QCD ejjX fi X c ec fi ep / GeV jett p
15 20 25 30 35 [ pb / G e V ] j e t t / dp s d DataPythiaCascade had ˜ NLO QCD ejjX fi X c ec fi ep CHARM / GeV jett p
15 20 25 30 35 [ pb / G e V ] j e t t / dp s d DataPythiaCascade had ˜ NLO QCD ejjX fi X c ec fi ep CHARM (a) / GeV jett p
15 20 25 30 35 [ pb / G e V ] j e t t / dp s d DataPythiaCascade had ˜ NLO QCD ejjX fi X c ec fi ep CHARM (a) H1 / GeV jett p
15 20 25 30 35 [ pb / G e V ] j e t t / dp s d DataPythiaCascade had ˜ NLO QCD ejjX fi X c ec fi ep CHARM (a) H1 / GeV jett p
15 20 25 30 35 [ pb / G e V ] j e t t / dp s d ( pb / G e V ) c - j e t T / dp s d
10 5 10 15 20 25 ( pb / G e V ) c - j e t T / dp s d e fi ZEUS 96-00 c ZEUS 98-00 D* jets c jet -1 ZEUS 133 pb
NLO QCD )/4 +p =(m m +p =m m ZEUS ecX) fi (ep c-jetT /dp s d | < 1.5 c-jet h , 0.2 < y < 0.8, | < 1 GeV Q (GeV) c-jetT p t h s / m eas s (GeV) c-jetT p t h s / m eas s Figure 30: Left: Differential cross sections for the process ep → ec ¯ cX → ejjX as function of thetransverse momentum p jet t of the leading jet, from the H1 analysis [141]. The data are compared toan NLO calculation [58] in the massive scheme, and to LO+PS MC predictions from PYTHIA [21]and CASCADE [28]. Right: Summary of differential c -quark jet cross sections as a function of the jettransverse momentum, as measured by the ZEUS collaboration [177]. The data are compared to anNLO calculation [58] in the massive scheme, for two different QCD scale choices.prediction [58], which describes the data reasonably and equally well up to the highest jet transversemomenta. To compare this result with the above D ∗ measurement (Fig. 28) one has to take intoaccount that the jet gives a direct approximation of the charm quark kinematics, while on averagethe D ∗ takes only about 70% of the charm quark momentum in the fragmentation (after cuts). Thus,the kinematic range tested with the leading jet p T from 11 to 35 GeV roughly corresponds to a D ∗ transverse-momentum region from 8 to 25 GeV. For D ∗ transverse momenta from 8 GeV up to thehighest covered value of 20 GeV the D ∗ data are similarly well described by the NLO calculation asthe dijet data at their correspondingly higher momenta. So the two independent measurements usingdifferent tagging techniques give consistent results.A similar and more direct comparison is shown in Fig. 30(right) for several measurements from ZEUS(entries 8,12,13 in table 3). Here, the measurements have already been translated to cross sections forinclusive c -jet production. The results obtained from D ∗ and inclusive-vertex tagging agree well witheach other and with theory. Since in the core of a jet electrons are not easily separated from π/π overlaps, charm tagging using semileptonic decays into electrons is experimentally difficult and thecorresponding c → e result, which was a byproduct of an analysis focusing on beauty production, might Several other such charm analyses were eventually not published due to insufficient control of systematics. m + p T (dashed line in Fig. 30(right)) should be considered. Good agreement is observedbetween the results of the two experiments. D ∗ and one other hard parton To obtain more information on the charm-photoproduction process, one possibility is to require thepresence of a jet in the final state in addition to the D ∗ , which is not associated to the D ∗ . This meansthat the jet and the D ∗ are well separated in their directions and that the jet tags another hard partonin the process. This parton can be the other charm quark or a gluon or light quark. In one analysis[172] a very soft jet momentum cut p T > | η | < .
5, thus covering the same kinematic range as the D ∗ s. In Fig. 31 thedifferential cross sections are shown as function of the pseudorapidities of the D ∗ and the jet. For the ( D * ) [ nb ] h / d s d p: D* + other jet g H1 Data Had ˜ FMNR FMNR Had ˜ ZMVFNS p: D* + other jet g H1 (D*) h -1.5 -1 -0.5 0 0.5 1 1.5 R Data Had ˜ FMNR FMNR Had ˜ ZMVFNS ( j e t) [ nb ] h / d s d p: D* + other jet g H1 Data Had ˜ FMNR FMNR Had ˜ ZMVFNS p: D* + other jet g H1 (jet) h -1.5 -1 -0.5 0 0.5 1 1.5 R Data Had ˜ FMNR FMNR Had ˜ ZMVFNS
Figure 31: D ∗ + jet cross sections as function of the pseudorapidities of the D ∗ (left) and the jet(right), from the H1 analysis [172]. The measurements are compared to two NLO predictions, themassive scheme (FMNR) calculations from Frixione et al. [58] and the massless scheme (ZMVFNS)predictions from Heinrich and Kniehl [48].leading-order boson-gluon-fusion process it is expected that the D ∗ tags one charm quark and the jet theother. Since similar momentum cuts are applied for the D ∗ and the jet, one would expect very similarpseudorapidity distributions for the D ∗ and the jet. However, the observed pseudorapidity spectrum forthe jet (Fig. 31) is significantly shifted towards the more forward direction compared to that of the D ∗ .This indicates that, as expected from higher-order contributions, the jet often tags another parton, i.e.a gluon or a light quark. This effect is predicted by the massive and massless scheme NLO calculationsto which the data are compared in Fig. 31, and these calculations describe the data reasonably well.Also, the additional jet requirement significantly reduces the theoretical uncertainties w.r.t. Fig. 28.In addition to jets not associated to the D ∗ , the corresponding ZEUS measurement [174] also selectedevents in which the D ∗ is associated to the jet. In the latter case, one does obtain information only aboutone hard parton in the event, which is a charm quark. Furthermore, the jet tranverse momentum cut p jetT > − . < η jet < . E T spectra for D ∗ -tagged jets (from charm quarks) and untagged jets (from charm, gluons, or lightquarks) are similar. The pseudorapity distributions for D ∗ -tagged and untagged jets (Fig. 32) show49ifferences consistent with those of the H1 analysis. As expected, the average jet pseudorapity increaseswith increasing jet E T . Again, the theoretical uncertainties are reduced with respect to those of Fig.29. At high jet E T , the shape of the massive calculation describes the data somewhat better than themassless one. ZEUS -tagged jet * D >6 GeV jetT
EZEUS 98-00Jet energy scale uncertainty<9 GeV jetT jetT
E Untagged jets >6 GeV jetT E NLO QCD (massive) had. ˜ NLO QCD (massive) Beauty
Untagged jets >6 GeV jetT E NLO QCD (massless) had. ˜ NLO QCD (massless) * D fi NLO QCD (massless) no b <9 GeV jetT jetT jetT
E >9 GeV jetT E e ’ + D * + j e t + X ) ( nb ) fi ( e p j e t h / d s d jet h -1 0 1 2 jet h -1 0 1 2 jet h -1 0 1 2 Figure 32: D ∗ + jet cross sections as a function of the pseudorapidities of D ∗ -tagged (left) and untagged(center and right) jets, from the ZEUS analysis [174]. The measurements are compared to two NLOpredictions, the massive scheme calculations from Frixione et al. [58] and the massless scheme predictionsfrom Heinrich and Kniehl [48].The selection of events with a D ∗ and a muon from a semileptonic charm decay, not associated withthe D ∗ [175], allows explicit tagging of both charm quarks. The small statistics (entry 11 in Table 3,where also the visible phase space cuts are given) do not allow differential distributions, but the observedtotal visible cross section for D ∗ µ production from double-tagged c ¯ c final states of 250 ± ±
40 pb isconsistent with the prediction from the massive NLO calculation [56] of 256 +159 − pb. Analyses using tagged charm events with two identified hard partons in the final state and studying thecorrelations of the two partons [170, 172, 173, 174, 141, 178, 179] provide the most detailed informa-tion on the charm-production mechanism. Similar to the previous subsections, there are two differentexperimental approaches: • The D ∗ tag is used for charm tagging. For the two hard partons either the reconstructed D ∗ plusan additional non-associated jet are used [172, 174], or alternatively two jets are identified, oneof which is tagged by the D ∗ [170, 173, 174, 179].50 Alternatively, dijet events are selected and one jet is tagged as a charm jet using the displacedimpact parameters of jet-associated charged tracks [141] or by a muon from a charm semileptonicdecay [178].With the two identified partons three correlation observables are constructed, which will be discussedin the following:1. The observable x obsγ , which allows the separation, in the leading order picture, of direct- andresolved-photon interactions. In the NLO picture, it separates 3-parton from 2-parton final states.2. The azimuthal correlation ∆ φ of the two partons, which is sensitive to higher-order effects. Com-bined with x obsγ , it can distinguish between 2-parton, 3-parton and 4-parton final states.3. The hard-scattering angle cosθ ∗ of the two partons, which allows the distinction of contributionswith quark or gluon propagators in the hard scattering. x obsγ studies The case of two jets is assumed in the following for the two hardest partons. The observable x obsγ isdefined as x obsγ = P Jet ( E − p Z ) + P Jet ( E − p Z ) P h ( E − p Z ) . (17)The sums in the numerator run over the particles associated with the two jets and those in the denom-inator over all detected hadronic final state particles. E and p Z denote the particle energy, and themomentum parallel to the proton beam, respectively.In the leading-order pQCD picture (Section 2.3, 2 partons + potential photon remnant + protonremnant) this variable is an estimator of the fraction of the photon energy entering the hard interaction.For the direct boson-gluon-fusion process (Fig. 6(a)) x obsγ approaches unity, as the hadronic final stateconsists of only the two hard jets and the proton remnant in the forward region, which contributes littleto P h ( E − p Z ). In resolved processes (Figs. 6(b-d)) the photon remnant significantly contributes tothe denominator but not to the numerator, so x obsγ can be small. The addition of parton showering cansomewhat dilute this simple picture. x obsγ is also smaller than unity for next-to-leading-order processes with a third hard outgoing parton(Fig. 8). In the massive NLO case for charm production this often coincides with the other quarkoriginating from initial-state photon splitting into a c ¯ c pair, which would be classified as a photonremnant in the leading-order picture. Since in the fixed-flavour NLO case there are at most threepartons, x obsγ separates 2-parton from 3-parton final states. In the variable-flavour NLO case the twopictures described above get mixed, since in the case of an initial-state c quark from the photon theother c quark can be a fourth hard parton. Thus, in general, the observable x obsγ is sensitive to theresolved-photon structure (if any) and to tree-level higher-order processes (if any).One of the milestone papers on charm photoproduction at HERA was the ZEUS analysis [170],where x obsγ studies are performed using events with a D ∗ and two jets. The jets are required to havetransverse momenta p jet T > | η jet | < .
4. In mostevents the D ∗ is associated to one of the two jets. Figure 33 shows the measured single-differentialcross section as a function of x obsγ . A peak at large x obsγ > .
75 is observed, which reflects the direct-photon/2-hard-parton component. Roughly 50% of the data are observed at x obsγ < .
75, indicatinglarge contributions from resolved-photon/3-hard-parton or other higher-order contributions.In the lower plot in Fig. 33 the data are compared to predictions from a massive scheme NLOcalculation [58]. Not all theoretical uncertainties are shown here. The calculation has a tendency tounderestimate the data cross sections at x obsγ < .
75, where it is effectively a leading order calculation.This might indicate the need for even higher-order corrections.51
ZEUS 1996+97 x g OBS d s / d x g O BS ( nb ) (a) DataHerwig: direct + resolvedHerwig: directHerwig: resolvedHerwig: resolved withoutcharm excitation x g OBS d s / d x g O BS ( nb ) (b)(b) DataMassive NLO, parton level, e =0.02 m R = 1.0 m ⊥ , m c = 1.5 GeV m R = 0.5 m ⊥ , m c = 1.2 GeV Figure 33: Differential cross section as a function of x obsγ for dijet events with an associated D ∗ meson,from a ZEUS analysis [170]. The shaded band indicates the energy scale uncertainty. The same dataare compared in the lower plot to an NLO calculation [58], and in the upper plot to Monte Carlopredictions from HERWIG [22] with direct and resolved photon contributions shown separately. Thelatter is dominated by the charm excitation component.A much better shape description is obtained with the LO+PS HERWIG [22] Monte Carlo programas shown in the upper plot of Fig. 33. In this calculation a large part of the NLO photon splittingdiagram in Fig. 7(c) is included in the form of a charm excitation component, where the charm quark istreated as a massless constituent of the resolved photon, as shown in Figs. 6(c) and 6(d). This gives thedominant contributions for x obsγ < .
75. Combined with parton showering, which also pulls the “direct”contribution towards lower x obsγ values, this provides a reasonable data description. This LO+PS MCapproach thus provides an effective way to describe the small x obsγ region, although the charm quark istreated as massless in a kinematic region where this is probably not a good approximation. Note thatthe total cross section with D ∗ + dijets is only about 18% of the D ∗ cross section without the dijets,also measured in [170], for the same D ∗ cuts applied ( p T ( D ∗ ) > | η ( D ∗ ) | < . x obsγ < .
75 region in the D ∗ + dijet sample contributes only a relatively smallpart to the inclusive- D ∗ cross section.Another ZEUS analysis [174] using events with a D ∗ and at least one jet compares the measured x obsγ cross sections to both massive and massless scheme NLO calculations. Here the D ∗ and a jet, towhich the D ∗ is not associated, are taken as estimators for the two leading partons and used for the x obsγ reconstruction in Eq. (17). The jet is required to have transverse momentum p T > − . < η < .
4. Figure 34 shows the differential cross sections as a function of Note that the available massless scheme calculations [48] provide only cross sections for a D ∗ + jet final state butnot for two jets. obsγ . In the left (right) plot the data are compared to the massive (massless) scheme NLO predictions.Both predictions are a bit too low for the 3-or-more-parton final state region x obsγ < .
75, but are stillcompatible with the data within their uncertainties. Note that in the massless calculation, which absorbsthe initial state photon splitting to c ¯ c into the photon PDF, this contribution is effectively calculated toNLO (one-loop virtual corrections), while it is only calculated to LO (0 loop) in the massive case. Thispartially explains why the uncertainty of the massless calculation is much smaller in this region. Athigh x obsγ both calculations are effectively NLO (1-loop) calculations, and the uncertainties are similar.Figure 34: Differential cross sections as a function of x obsγ for events with a jet and a D ∗ meson,which is not associated to the jet, from the ZEUS analysis [174]. In the left (right) upper plots thedata are compared to a NLO calculation in the massive [58] (massless [48]) scheme. The bottom plotshows a comparison to the PYTHIA and HERWIG MC, which were used to calculate the hadronisationcorrections in the upper left plot. ∆ φ In a ZEUS analysis [174] using events with a D ∗ and two jets (entry 9 in Table 3), and in the H1measurement [172] with a D ∗ and a non-associated jet (entry 7), the azimuthal correlation of thetwo hard partons is investigated. In the leading-order picture of direct-photon interactions (Fig. 6a),the two charm quarks are produced back-to-back in the azimuthal plane of the lab frame, i.e. with∆ φ = 180 ◦ . Smaller ∆ φ can be due to higher-order processes, such as gluon radiation, or due to a non-zero transverse momentum of the partons that enter the hard interaction, e.g. from a flavour-excitationprocess in which the c quark gets a finite transverse momentum in the backwards parton showering step.53n the ZEUS analysis jets were selected with harder transverse-momentum requirements but in a wider η range than in the H1 analysis (cf. entries 9 and 7 in Table 3). Figure 35 shows the differential crosssections as a function of the azimuthal difference ∆ φ between the D ∗ and the jet for the H1 analysisand between the two jets for the ZEUS measurement. The H1 result is shown in the two rightmostplots. The cross sections are highest for ∆ φ = 180 ◦ , i.e. for the back-to-back configuration, and drop offtowards smaller angles. NLO calculations in the massive scheme [58] and in the massless scheme [48] arecompared to the data. Both calculations drop off more steeply than the data towards smaller openingangles. Below ∆ φ ≈ ◦ the two calculations predict very small contributions, while there are stillsizeable ones in the data. A better description is obtained with the LO+PS programs PYTHIA [21]and CASCADE [28]. PYTHIA includes charm excitation processes in resolved-photon events, whichgive the dominant contribution for ∆ φ < ◦ and provide a reasonable data description in this region.The results of the ZEUS analysis [174] are shown in the left and central plots of Fig. 35. Herethe azimuthal correlation is measured separately in the 2-parton region x obsγ > .
75 and in the 3-or-more-parton region x obsγ < .
75. The data are compared to an NLO calculation [58] in the massivescheme. For the high- x obsγ region the description is satisfactory. However, in the low- x obsγ region theNLO calculation is clearly falling below the data for ∆ φ < ◦ . This is straightforward to understandsince a 3-parton final state can not produce an angle between the two leading p T partons of less than120 ◦ . Correspondingly, at least four partons are needed to populate this region. A massive NLOcalculation produces at most three, so an NNLO calculation is needed to fill the gap. Again, a bettershape description is obtained by PYTHIA and HERWIG, which can provide several extra partonsthrough parton showering (of which flavour excitation is a part). Thus, the conclusion is again verysimilar to the above studies with the x obsγ observable: the NLO calculation is missing a component inthe data, which can be effectively described by a LO+PS calculation. As to be expected from thisexplanation, in the two lower plots the MC@NLO calculation, which complements the 2- and 3-partonNLO matrix elements by parton showering, is able to describe these data well.An H1 analysis using µ +dijet final states [178] (entry 14 in table 3) further supports these conclu-sions. cos θ ∗ In a dedicated analysis [173], using events with a D ∗ and two jets (entry 8 in Table 3), ZEUS hasinvestigated the scattering angle θ ∗ of the charm quark with respect to the proton direction in the dijetrest frame. The charm quark is identified by the jet to which the reconstructed D ∗ is associated. The cos θ ∗ distribution strongly reflects the type of the propagator particle exchanged in the 2 → • For a charm quark propagator cos θ ∗ should follow a (1 − | cos ( θ ∗ ) | ) − distribution. The directphoton (Fig. 6(a)) and the resolved process with a gluon from the photon structure (Fig. 6(b))belong to this class of processes and also one of the charm excitation diagrams (Fig. 6(c)). • For a gluon propagator cos θ ∗ should follow a (1 − | cos ( θ ∗ ) | ) − distribution, i.e. a much steeperrise for | cos θ ∗ | →
1. For leading-order processes only the charm-excitation mechanism providessuch a contribution (Fig. 6(d)).The main idea of the analysis is to look for such effects directly in the data. Special cuts are appliedin order to ensure a flat acceptance for the cos θ ∗ distribution over a wide range, extending to as largevalues of | cos θ ∗ | as possible. The invariant mass of the two jets is required to be above 18 GeV. The However, in the 3-parton topology one of the leading jets can escape outside of the kinematic region of the mea-surement. Thus, the softest jet is used instead, which leads to strongly suppressed but non-zero charm contribution for∆ φ < ◦ EUS -4 -3 -2 -1 >0.75 obs g x ZEUS 98-00Jet energy scale uncertainty -4 -3 -2 -1 <0.75 obs g x NLO QCD (massive) had. ˜ NLO QCD (massive) Beauty -4 -3 -2 -1 >0.75 obs g x · HERWIG 1.5 · PYTHIA -4 -3 -2 -1 <0.75 obs g x e ’ + D * + jj + X ) ( nb / r a d . ) fi ( e p jj fD / d s d (rad.) jj fD (rad.) jj fD ] (cid:176) ( D * , j e t) [ nb / f D / d s d -3 -2 -1 p: D* + other jet g H1 DataCascade 1.2Pythia 6.2Pythia 6.2 (dir.) p: D* + other jet g H1 ] (cid:176) (D*, jet) [ f D R DataCascade 1.2Pythia 6.2Pythia 6.2 (dir.) [deg] jj fD [ pb / d e g ] jj fD / d s d -1 X m ejj fi Xc ec fi H1 Cross section: ep >0.75 obs g x [deg] jj fD [ pb / d e g ] jj fD / d s d -1 H1 Data 06/07PYTHIACASCADEHERWIGMC@NLO [deg] jj fD [ pb / d e g ] jj fD / d s d -2 -1 X m ejj fi Xc ec fi H1 Cross section: ep £ obs g x [deg] jj fD [ pb / d e g ] jj fD / d s d -2 -1 H1 Data 06/07PYTHIACASCADEHERWIGMC@NLO ] (cid:176) ( D * ,j e t) [ nb / f D / d s d -3 -2 -1 p: D* + other jet g H1 Data Had ˜ FMNR FMNR Had ˜ ZMVFNS p: D* + other jet g H1 Data Had ˜ FMNR FMNR Had ˜ ZMVFNS p: D* + other jet g H1 ] (cid:176) (D*, jet) [ f D R Data Had ˜ FMNR FMNR Had ˜ ZMVFNS
Figure 35: Azimuthal differences of two outgoing hard partons in charm events. The upper rowshows the D ∗ -tagged jet-jet azimuthal difference from ZEUS [174] separately for the high (left) andlow (middle) x obsγ region and the D ∗ -other jet azimuthal difference from H1 [172] without cut on x obsγ (right). Also shown are massive (NLO/FMNR)[56] and massless (ZMVFNS)[48] NLO predictions. Themiddle row shows the same data compared to HERWIG[22], PYTHIA[21] and CASCADE[28] LO+PSMC predictions. The bottom row shows the muon-tagged jet-jet azimuthal difference from H1 [178]compared to these same MCs and to the NLO+PS MC@NLO [185] prediction.55verage pseudorapidity of the two jets, defined as η jet + η jet is required to be smaller than 0.7. Notethat these cuts yield a much smaller (but still sizeable) contribution at x obsγ < .
75 than the one shownin Fig. 33, mainly because they implicitly restrict the two jets to the pseudorapidity region η jet < . cosθ ∗ , separatelyfor x obsγ < .
75 and x obsγ > .
75. In the lower half of Fig. 36 the data are compared to massive schemeFigure 36: Differential cross sections as function of cos θ ∗ for dijet events with an associated D ∗ meson [173]. Results are given separately for samples enriched in direct ( x obsγ > .
75) and resolvedphoton events ( x obsγ < . x obsγ > .
75 region the NLO calculation provides a good description ofthe data over the whole range of cos θ ∗ , with reasonably small uncertainty. The relatively shallow cos θ ∗ dependence is consistent with the expectation that this region is dominated by the boson-gluon-fusiongraph, where the propagator particle is a charm quark, and for which the prediction is stabilised atNLO by 1-loop virtual corrections. In the x obsγ < .
75 region a much stronger rise is visible towardsmore negative cos θ ∗ values, and the central region is more strongly depleted. This can be interpreted asa direct proof for sizeable contributions from gluon propagator exchanges such as the charm excitationprocess (Fig. 6(d)), which at NLO is a tree-level process (Fig. 8(d,e) ). Correspondingly, the NLO56ncertainty is much larger. The upper edge of the uncertainty band describes the data reasonably. Thestrong asymmetry can be attributed to the fact that the charm jet will preferentially be correlated withthe incoming photon direction.The plots in the upper half of Fig. 36 show that the PYTHIA and HERWIG LO+PS MCs withtheir large excitation contributions are able to describe the data well everywhere. This is particularlytrue for PYTHIA. For the NLO calculation this means that contributions beyond NLO would probablyfurther improve the description. On the other hand, the CASCADE MC, which attempts to describehard higher-order topologies by allowing initial state partons to have sizeable transverse momentum,reasonably describes the shapes, but fails to describe the relative normalisation of the low- and high- x obsγ regions. The charm mass provides a semi-hard QCD scale which already allows the application of perturbativecalculations to all of phase space, but which also competes with other, often even harder perturbativescales. Total cross sections for charm photoproduction are reasonably described by such calculations.Single-differential cross sections already provide a good handle to test the applicability of different QCDapproximations, although the theoretical uncertainties are mostly much larger than the experimentalones. The theory predictions agree with the data up to the highest accessible transverse momenta,showing no indications that final state resummation corrections are needed for massive calculations.Double-differential cross sections, in particular those including jets, reveal a partial failure of themassive scheme NLO predictions for the three independent parton-parton kinematic observables x obsγ , cos θ ∗ and ∆ φ , which were studied in charm events with a D ∗ and one or two jets. For certain kinematicregions this can be traced back to the absence of 4-or-more-parton final states in the calculation. Thepartially large theoretical uncertainties can be explained by the absence of stabilizing virtual correctionsfor 3-parton final states at this order. The NLO calculations in the massless scheme, where available, domostly not provide a better description for the observables. The LO+PS MCs PYTHIA and HERWIG,which are often used for acceptance corrections, are able to describe all topologies reasonably, ofteneven very well. The CASCADE MC performs somewhat less well on average.57 Beauty photoproduction
From the theoretical point of view, the only differences between charm photoproduction as discussed inthe previous section and beauty photoproduction are the beauty-quark mass and electric charge. Thefirst suppresses the cross section w.r.t. charm at low values of transverse momentum, while the secondsuppresses it by about a factor 4 everywhere (Eq. 1). Experimentally, the signal-to-background ratio isthus more challenging, and the available statistics is smaller. Together with the small branching ratioto specific final states, this precludes any attempt to use fully-reconstructed beauty-hadron final statesat HERA. On the other hand, the higher mass and longer lifetime compared to charm hadrons increasesthe tagging efficiency for inclusive tagging methods.Table 4 summarises all H1 and ZEUS beauty photoproduction measurements. For the reasonsexplained above, the first such measurements (entries 1 and 2) came several years after the first mea-surements of charm, focused on beauty jet production, and were severely limited by statistics.
Due to their high mass, even beauty quarks at rest in the centre-of-mass system of the partonic in-teraction still produce reasonably high-momentum muons or electrons, which can be detected, whendecaying semileptonically. The forward and backward muon systems allow the detection of such beautyquarks even when they are strongly boosted along the beam direction. Furthermore, the requirementof two such muons, i.e. a double tag, strongly reduces both the light flavour and charm backgrounds.In a ZEUS analysis [194] these properties were used to measure the total cross section for beautyproduction in ep collisions without any cuts, i.e. including both photoproduction and DIS, by pushingthe measureable muon phase space to the limit (entry 12 of Table 4). After correcting for muonacceptance and semileptonic branching ratios the resulting total cross section for b ¯ b pair production in ep collisions at HERA for √ s = 318 GeV was determined to be σ tot ( ep → b ¯ bX ) = 13 . ± . . ) +4 . − . (syst . ) nb , (18)where the first uncertainty is statistical and the second systematic. The total cross section predicted bynext-to-leading-order QCD calculations was obtained in the massive approach by adding the predictionsfrom FMNR [56] and HVQDIS [51] for Q less than or larger than 1 GeV , respectively. The resultingcross section for √ s = 318 GeV, using the scale choice µ = q m b + p T b σ NLOtot ( ep → b ¯ bX ) = 7 . +4 . − . nb (19)is a factor 1.8 lower than the measured value, although compatible within the large uncertainties.Compareable measurements were obtained in reduced regions of phase space from D ∗ + muon anddielectron final states (entries 10, 11 and 13 in table 4), and similar results were obtained for the ratioof measured to predicted cross sections (see also corresponding entries in Fig. 37). Since D ∗ mesonsand semi-isolated electrons could only be measured in the more central rapidity range, total b ¯ b crosssections were not extracted. In order to make them compareable with each other, in Fig. 37 almost all available beauty-photopro-duction cross sections have been translated, using NLO massive QCD calculations, into cross sectionsfor inclusive b -quark production as a function of p T b in the kinematic range Q < , 0 . < y < . entries 10,5,4,13,2,11,3,6,7,12,8 in table 4, following the order in the figure legend o. Analysis b Tag Ref. Exp. Data L [ pb − ] Q [ GeV ] y Particle p T [ GeV ] η Events bgfreeevents1 µ + dijets µ + p relT [189] H1 96 . < . , . µ jet1(2) > > − . , .
1] 470 ±
43 120 e + dijets e + p relT [190] ZEUS 96-97 . < . , . e jet1(2) > . > − . , . − . , .
4] 140 ±
35 16 µ + dijets µ + p relT [191] ZEUS 96-00 < . , . µ jet1(2) > . > − . , . − . , .
5] 834 ±
65 165 µ + dijets µ + p relT + δ [142] H1 99-00 < . , . µ jet1(2) > . > − . , . − . , .
5] 1745 128 < . , . TrackJet1(2) > . > − . , . − . , . ∼ e + dijets e + p relT + E T [176] ZEUS 96-00 < . , . e Jet1(2) > . > − . , . − . , . ∼ µ +dijets µ + p relT + δ [192] ZEUS 05 < . , . µ Jet1(2) > . > − . , . − . , .
5] 7351 122 < . , . tracksJet1(2) > . > − . , . − . , . ∼ µ + dijets µ +imp.par. [178] H1 06-07 < . . , . µ Jet1(2) > . > − . , . − . , .
5] 6807 425 D ∗ + µ Kππ s + µ [175] H1 98-00 < . , . D ∗ µ > . p > − . , . − . , .
74] 56 ±
17 15 D ∗ + µ Kππ s + µ [193] ZEUS 96-00 < . , . D ∗ µ > . > . − . , . − . , .
3] 232 16
12 dimuon µ + µ [194] ZEUS 96-00 all all µ > . .
75) [ − . , .
5] 4146 86
13 dielectron e + e [195] H1 07 < . , . e > − . , . ∼ Table 4:
Beauty photoproduction cross-section measurements at HERA. Information is given for each analysis on the beauty tagging method, theexperiment, the data taking period, integrated luminosity, Q and y ranges and the cuts on transverse momenta and pseudorapidities of selected final stateparticles. The last two columns provide information on the number of events in the analysis (number of signal events if an uncertainty is given) and the equivalentnumber of background-free events. The centre-of-mass energy of all data taken up to 1997 (6 th column) was 300 GeV, while it was 318 – 319 GeV for all subsequentruns. ( pb / G e V ) b T / dp s d -1
10 0 5 10 15 20 25 30 ( pb / G e V ) b T / dp s d -1 m H1 97-00 D*H1 99-00 b jet jet m fi H1 99-00 b ee fi H1 b e fi ZEUS 96-97 b m D* fi ZEUS 96-00 b m fi ZEUS 96-00 b e fi ZEUS 96-00 b jet m fi ZEUS 05 b mm fi bb -1 ZEUS 114 pb b jet -1 ZEUS 133 pb
NLO QCD )/4 +p =(m m +p =m m HERA ebX) fi (ep bT /dp s d | < 2 b h , 0.2 < y < 0.8, | < 1 GeV Q (GeV) bT p t h s / m eas s (GeV) bT p t h s / m eas s Figure 37: Differential cross section as a function of the transverse momentum of b quarks for thekinematic range indicated in the figure. The bottom inset shows the ratio of the measured crosssections to the central NLO QCD prediction. For more details see the text.and beauty pseudorapidity | η b | <
2. Each data point is displayed at the centroid of the p T distributionof the b quarks entering the measurement bin of the respective analysis, which is mostly a bin in b -jet E T , or, where not available, a bin in muon or electron p T . The b -quark p T rather than the b -jet p T has been chosen here because the measurements extend down to very low p T at which jets can not beusefully defined any more. Two massive NLO [56] theory predictions are given: one with scale choice µ = q m b + p T b (dashed), and one with scale choice µ = q m b + p T b /
2. The full theory uncertainty bandis shown for the latter (for the scale choice see also Section 2.11). It is dominated by the scale variations(independent variation of renormalisation and factorisation scales by factor 2) and by the variation ofthe pole mass ( m b = 4 . ± .
25 GeV). Where not provided directly in the original publications, the datapoints were obtained using the data/theory ratio of the respective original measurements and rescalingthem to the theory prediction in Fig. 37, properly accounting for differences in the respective theorycalculation settings.Within the large uncertainties, reasonable agreement between theory and data is observed over thecomplete p T range covering 3 orders of magnitude in the cross section. In particular, as in the charmcase, there is no indication for a failure of the predictions at large p T . There might be a trend thaton average, the measurements of the double tagging analyses ( D ∗ µ , ee and µµ ), which were alreadybriefly discussed in the total cross-section subscetion, tend to lie a bit above the other measurementswhich typically require dijet final states. The effect is not very significant, but if taken serious, mightindicate that the contribution of b quarks not associated to jets might be underestimated by the theory.Unfortunately, currently no measurement is available which directly tests this hypothesis by consideringboth topologies in a single analysis framework. 60ll available beauty photoproduction results are represented in this plot, except the results of thevery first H1 analysis [189] (entry 1 in Table 4), which has been declared superseded by a more recentanalysis [142], and the results of one of the latest H1 analyses [178] (entry 9), for which no comparisonto pure NLO predictions was provided.Double-differential cross sections have not been measured so far. In the future they could best beextracted using the inclusive vertexing approach [177], which offers the best effective statistics (entry 8in Table 4).Figure 38: Single-differential cross sections for muons from b ¯ b decays to dimuons, as a function ofthe muon transverse momentum (left), pseudorapidity (center), and dimuon azimuthal angle difference(right), from ZEUS [194]. The measurements are compared to massive NLO predictions with the samesettings as the band in Fig. 37, and to the PYTHIA [21] prediction scaled to the data. b ¯ b and jet-jet correlations Several results give insight into correlations between two final state partons in b ¯ b events. The ZEUSanalysis of dimuon final states (Fig. 38, entry 11 in Table 4) studies the azimuthal angle differencebetween muons originating from different b quarks, in addition to single-differential distributions. Boththe massive NLO predictions and the PYTHIA MC predictions used for acceptance correction showreasonable agreement with the data, in particular in shape.An H1 analysis of dijet final states in which one of the jets is tagged by a muon from a semilep-tonic b decay (Fig. 39, entry 9 in Table 4) studies the x obsγ and ∆ φ variables described earlier in thecharm section. The MC@NLO prediction describes the data, except in the high x obsγ , high ∆ φ bin. obs g x [ pb ] g / d x s d obs g x [ pb ] g / d x s d X m ejj fi Xb eb fi H1 Cross section: ep
H1 Data 06/07PYTHIACASCADEHERWIGMC@NLO obs g x [ pb ] g / d x s d [deg] jj fD [ pb / d e g ] jj fD / d s d -2 -1 [deg] jj fD [ pb / d e g ] jj fD / d s d -2 -1 X m ejj fi Xb eb fi H1 Cross section: ep >0.75 obs g x H1 Data 06/07PYTHIACASCADEHERWIGMC@NLO [deg] jj fD [ pb / d e g ] jj fD / d s d -2 -1 [deg] jj fD [ pb / d e g ] jj fD / d s d -2 -1 [deg] jj fD [ pb / d e g ] jj fD / d s d -2 -1 X m ejj fi Xb eb fi H1 Cross section: ep £ obs g x H1 Data 06/07PYTHIACASCADEHERWIGMC@NLO [deg] jj fD [ pb / d e g ] jj fD / d s d -2 -1 Figure 39: Beauty cross sections as a function of x obsγ (left) and as a function of the jet-jet azimuthalangle difference ∆ φ for high (center) and low (right) values of x obsγ , from a recent H1 analysis [178].The measurements are compared to MC@NLO [185] predictions, as well as to predictions from thePYTHIA [21], HERWIG [22] and CASCADE [28] LO+PS Monte Carlos.61he agreement is thus slightly worse than in the charm case (Fig. 35). PYTHIA agrees everywhere,HERWIG describes the shape but not the normalisation, and CASCADE fails for both. The latterfinding is again in qualitative agreement with the charm result. Thus the PYTHIA or HERWIG MCsshould preferentially be used for acceptance corrections.A similar analysis by ZEUS [192] (not shown) compares the data directly to the massive NLOpredictions. Not surprisingly, these predictions fail in the same kinematic regions as for charm (Fig.35), for the same reasons as discussed there.Several other analyses [190, 191, 142] also studied x obsγ , with less statistics than but similar conclu-sions as for charm. Due do the suppression of the cross section by mass and charge, and small branching ratios to exclusivefinal states, only inclusive or semi-inclusive tagging methods can be used at HERA to measure beautyproduction. The reasonable acceptance for the detection of b hadron decays down to 0 transverse mo-mentum and the coverage of almost the full physically relevant rapidity range allowed the measurementof the total beauty production cross section at HERA. This cross section is higher than, but still com-patible with, NLO QCD predictions. Several single differential beauty photoproduction cross sectionshave also been measured. The measurements from H1 and ZEUS and from different final states agreereasonably well with each other and with QCD predictions from threshold up to the highest accessibletransverse momenta. Double differential cross sections have not yet been measured.62 Charm and beauty production in DIS
In the previous Sections 6 and 7 heavy-flavour production in ep collisions with the exchange of quasi-real photons was discussed. The production of charm and beauty quarks was also studied in thedeeply inelastic scattering regime, which corresponds to photon virtualities Q ∼ > . Large photonvirtuality provides an additional hard scale in the calculations and allows probing the parton dynamicsinside the proton with high resolution. An overview of all measurements is given in Table 5 and Table 6for charm and beauty production, respectively. Already in the first H1 [196] and ZEUS [197] measurements in DIS (entries 1 and 2 in Table 5) boson-gluon fusion was clearly identified to be the dominant production mechanism for charm quarks. Thiswas investigated using the distribution of the fractional momentum of D ∗ mesons in the γp system, x D = p ∗ D W γp , where p ∗ D denotes the D ∗ momentum measured in the γ ∗ p frame. The data were used todisentangle between BGF and QPM-like models (cf. Fig. 1(a) and Fig. 2). The BGF process producesa c ¯ c pair that recoils against the proton remnant in the γ ∗ p frame, while the (massless) QPM producesa single charm quark recoiling against the proton remnant (which contains the other charm quark).Since the D meson carries a large fraction x D of the charm quark momentum, the former model shouldlead to significantly softer distribution in x D . Figure 40 shows a comparison of the two models againstthe ZEUS data. The observed shape of the cross section in the data proves that BGF is the dominantFigure 40: Normalised differential D ∗ -production cross section as a function of x ∗ D [197]. The mea-surement was performed for 5 < Q <
100 GeV . The points show the data, while solid and dashedlines show the BGF (PGF) and QPM predictions.charm-production process in DIS at HERA. This was quantified in [196] in the leading order QCDpicture by setting an upper limit for the fraction of the QPM-like contribution f ( QP M ) to charm DISproduction to be below 0 .
05 at 95% C.L.
Transition from photoproduction to DIS:
The ZEUS collaboration has studied [202] (entry 9 inTable 5) charm production in the intermediate Q region between photoproduction and DIS: 0 . . − . , .
5] 103 ±
13 1 : 0 . D incl. Kπ D > . ±
19 1 : 1 . D ∗ incl. Kππ s [197] ZEUS 94 , < . D ∗ [1 . , .
0] [ − . , .
5] 122 ±
17 1 : 1 . D ∗ incl. Kππ s [169] H1 95-96
10 [2 , . , . D ∗ [1 . ,
15] [ − . , .
5] 583 ±
35 1 : 1 . D ∗ incl. Kππ s [198] ZEUS 96-97
37 [1 , . , . D ∗ [1 . ,
15] [ − . , .
5] 2064 ±
72 1 : 1 . Kππππ s [2 . ,
15] 1277 ±
124 1 : 11 106 D ∗ incl. Kππ s [199] H1 97
18 [1 , . , . D ∗ > . − . , .
5] 973 ±
40 1 : 0 . D ∗ incl. Kππ s [200] ZEUS 98-00
82 [1 . , . , . D ∗ [1 . ,
15] [ − . , .
5] 5545 ±
129 1 : 2 1850 D incl. D mes. + S [147] H1 99-00
48 [2 , . , . D mesons > . − . , . n.a. n.a. D ∗ incl. Kππ s [201] H1 99-00
47 [2 , . , . D ∗ [1 . ,
15] [ − . , .
5] 2604 ±
77 1 : 1 . + dijet Jet1(2) > − , .
5] 668 ±
49 1 : 2 . D ∗ incl. Kππ s [202] ZEUS 98-00
82 [0 . , .
7] [0 . , . D ∗ [1 . ,
9] [ − . , .
5] 253 ±
25 1 : 1 . D incl. D mes. [203] ZEUS 98-00
82 [1 . , . , . D mesons > − . , . n.a. n.a. D + incl. Kππ [204] ZEUS 96-00
120 [1 . , . , . D + [0 ,
10] [ − . , .
6] 691 ±
107 1 : 16 42Λ c incl. pK S Λ c ±
25 1 : 7 10Λ π + ±
34 1 : 13 6
12 incl. lifet. imp. par. [205] H1 99-00 >
150 [0 . , . Track > . − . , . ∼
13 incl. lifet. imp. par. [206] H1 99-00
57 [6 , . , . Track > . − . , . ∼ D incl. Kπ + S [207] ZEUS 05
134 [5 , . , . D [1 . ,
15] [ − . , .
6] 8274 ±
352 1 : 14 550 µ + jet µ + p rel T + δ + E T [208] ZEUS 05 >
20 [0 . , . µ > . − . , . ∼ D ∗ incl. Kππ s [209] H1 04-07
351 [100 , . , . D ∗ [1 . ,
15] [ − . , . ∼
600 1 : 7 260 D ∗ incl. Kππ s [144] H1 04-07
348 [5 , . , . D ∗ > .
25 [ − . , .
8] 24705 ±
343 1 : 3 . D ∗ incl. Kππ s [210] ZEUS 04-07
363 [5 , . , . D ∗ [1 . ,
20] [ − . , .
5] 12893 ±
185 1 : 2 . D + incl. Kππ + S [211] ZEUS 04-07
354 [5 , . , . D + [1 . ,
15] [ − . , .
6] 8356 ±
198 1 : 3 .
20 incl. lifet. δ + S [149] H1 06-07
189 [5 , n.a. Track > . − . , . ∼ n.a. n.a.21 incl. lifet. jet + δ + S [212] H1 06-07 > . , . Jet > − . , . ∼
22 incl. lifet. jet + S [148] ZEUS 04-07
354 [5 , . , . Jet > . − . , . ∼ Table 5:
Charm DIS measurements at HERA. Information is given for each analysis on the charm tagging method, the experiment,the data taking period, integrated luminosity, Q and y ranges and the cuts on transverse momenta and pseudorapidities of selected finalstate particles. The last three columns provide information on the number of tagged charm events, the effective signal-to-background ratioand the equivalent number of background-free events. The centre-of-mass energy of all data taken up to 1997 (6 th column) was 300 GeV,while it was 318 – 319 GeV for all subsequent runs. o. Analysis c-Tag Ref. Exp. Data L [ pb − ] Q [GeV ] y Particle p T [GeV] η Events effect.s:b bgfreeevents1 µ + jet µ + p rel T [213] ZEUS 99-00 > . , . µ Jet > E BrT > − . , . − , . ∼
290 1 : 4 . µ + jet µ + p rel T [142] H1 99-00
50 [2 , . , . µ Jet > . p BrT > − . , . − . , . ∼
230 1 : 2 . µ + jet µ + p rel T [214] ZEUS 96-00 > . , . µ Jet > . > > − . − , . ∼ µ + jet µ + p rel T + δ + E T [208] ZEUS 05 >
20 [0 . , . µ > . − . , . ∼ e + jet e + p rel T + δ + E T [215] ZEUS 04-07 >
10 [0 . , . e [0 . ,
8] [ − . , . ∼ D ∗ + µ Kππ s + µ [193] ZEUS 96-00 > . , . D ∗ µ > . > . − . , . − . , . ∼
11 1 : 1 4 >
150 [0 . , . Track > . − . , . ∼
760 1 : 16 45
57 [6 , . , . Track > . − . , . ∼ δ + S [149] H1 06-07
189 [5 , n.a. Track > . − . , . ∼ n.a. n.a.10 incl. lifet. jet + δ + S [212] H1 06-07 > . , . Jet > − . , . ∼
11 incl. lifet. jet + S [148] ZEUS 04-07
354 [5 , . , . Jet > − . , . ∼ Table 6:
Beauty DIS measurements at HERA. Information is given for each analysis on the beauty tagging method, the experiment, thedata taking period, integrated luminosity, Q and y ranges and the cuts on transverse momenta and pseudorapidities of selected final stateparticles. The last three columns provide information on the estimated number of tagged beauty events, the effective signal-to-backgroundratio and the equivalent number of background-free events. The centre-of-mass energy of all data taken up to 1997 (6 th column) was300 GeV, while it was 318 – 319 GeV for all subsequent runs. The ”Br” label in the superscript refers to measurements in the Breit frame(see text). EUS -4 -3 -2 -1
110 10 -1 Q (GeV ) d s / d Q ( nb / G e V ) ZEUS BPCZEUS DIS 98-00HVQDIS e p → D * X (a) ZEUS -3 -2 -1 Q (GeV ) sg p ( W = G e V ) ( nb ) ZEUS BPCZEUS DIS 98-00ZEUS Photoproduction 96-97 g p → D * X (b) Figure 41: (a) Differential ep cross section for D ∗ production as a function of Q [202] in the kinematicregion indicated in Table 5. The measurements [202, 200] are compared to massive-scheme NLO QCDpredictions (HVQDIS) [51]. (b) Differential γp cross section for D ∗ production as a function of Q [202].The D ∗ data are shown in the photoproduction [170], transition [202] and DIS [200] regions. The curveshows a fit to the data (see text).to these very-low- Q as well as Q > . data [200] (entry 6 in Table 5). The calculations providea remarkable description of the drop of the measured cross sections over 5 orders of magnitude from Q = 0 .
05 GeV ≪ m c to Q = 100 GeV ≫ m c . The slope of dσ/dQ changes with Q : it issteeper at high Q > m c , where it is mainly dictated by the photon-propagator dependence 1 /Q ,than at low Q < m c , where an asymptotic 1 /Q dependence is expected. To study this further, themeasured D ∗ electroproduction cross sections were converted into γ ∗ p cross sections using the photonflux in the improved Weizs¨acker-Williams approximation (see Section 2.9). Figure 41(b) shows theconverted DIS as well as the photoproduction cross sections [170] (entry 5 in Table 3). The very-low- Q measurements are consistent with the photoproduction cross section. The data were fitted with afunction σ γp ( Q ) ∝ M / ( Q + M ). The extracted value was M = 13 ± , which is close to 4 m c and is significantly larger than the value obtained from inclusive data, M = 0 . ± .
04 GeV ≃ m ρ [122]. Performance of the ZMVFNS: D ∗ -production single-differential cross sections in DIS have alsobeen used to test available calculations in the massive and massless schemes. Figure 42 shows a com-parison of the most precise measurements from H1 [209, 144] (entries 16 and 17 in Table 5) to NLOQCD calculations. Since the ZMVFNS calculation is only valid in the regime where the charm-quarkmass can be neglected, an additional restriction is needed on the D ∗ transverse momentum in the γ ∗ p frame, p ∗ T ( D ∗ ) > γp system (see Fig. 28). The inelasticity is correlated with the centre-of-mass energy in the γ ∗ p frame, W γp (see Eq. (9)), thus the low- y region corresponds to the low- W γp region. Therefore, as expected,the ZMVFNS predictions deviate significantly from the massive-scheme calculations at low y , where W γp is not ≫ m c , and come close to the FFNS calculations at high y . At low y the massless-schemecalculations clearly fail to describe the data, while massive predictions are in agreement with the mea-surement in the whole y range. Also for Q >
100 GeV the massive predictions describe the data wellwithin uncertainties, whereas the massless approach significantly overestimates the charm cross section. The actual kinematic threshold for a D ∗ − D meson pair with only the D ∗ detected is ( p m ( D ∗ ) + p T ( D ∗ ) + m D ) − p T ( D ∗ ) = 17 GeV . .1 0.2 0.3 0.4 0.5 0.6 0.7 / d y [ nb ] s d / d y [ nb ] s d H1 dataZM-VFNS (CTEQ6.6M)HVQDIS (CT10f3) / d y [ nb ] s d D* in DIS H1 y no r m R y no r m R (a) Q [GeV ] d s v i s / d Q [ nb G e V - ] H1 -5 -4 -3 H1 dataHVQDISZMVFNSp* T (D*) > (b) Figure 42: Differential D ∗ -production cross section as a function of y for Q <
100 GeV [144] (a) and asa function of Q for Q >
100 GeV [209] (b). Also shown are the massive NLO prediction (HVQDIS) [51]and the massless NLO prediction (ZMVFNS) [48, 216]. The ratio shown on the left, R norm , representsthe ratio of individually-normalised distributions to the data, thus allowing a comparison of shapesonly.The data also clearly establish that the ZMVFNS fails to describe heavy-flavour production in DIS atHERA. Similar conclusions were drawn in [201] (entry 8 in Table 5), but with a lower precision of thedata. Event and heavy-flavour kinematics:
Most of the analyses summarised in Tables 5 and 6 studiedevent, charm and beauty kinematics differentially in the respective fiducial phase spaces. The mostprecise D ∗ measurements [209, 144, 210] (entries 16–18 in Table 5) were combined [217] to obtain themost precise charm differential cross sections with essentially no theory uncertainty due to extrapolationto a common phase space. The combination was done with a careful treatment of correlations. Asexpected, the individual measurements were found to be consistent. The uncorrelated uncertainties werereduced due to effective doubling of statistics, while the correlated systematic uncertainties were reducedthrough cross-calibration effects between the two experiments. Figure 43(a)–(c) shows a comparison ofmassive-scheme NLO QCD predictions [51] to the D ∗ combined single-differential cross sections. Thepredictions describe the data very well within uncertainties. However, the data reach 5% precision overa large fraction of the measured phase-space, whereas the typical theory uncertainty ranges from 30%at low Q to 10% at high Q . The theory uncertainty is dominated by the independent variation of the µ R and µ F scales, the uncertainty on the charm-quark pole mass and variations of the fragmentationmodel. Therefore, higher-order massive-scheme NNLO calculations and an improved fragmentationmodel for these predictions are needed to fully exploit the potential of these data. In addition, theoryuncertainties were studied in detail and a “customised” prediction was obtained by a variation of thetheory parameters within their uncertainties, to show that the calculations can simultanously providea good description of the shape and normalisation of all measured distributions with a single set ofparameters. This led to a renormalisation scale reduced by a factor 2 (see also Section 2.11), thecharm-quark pole mass reduced to m c = 1 . D mesons as well as of leptons from heavy-flavourdecays and of heavy flavour jets were measured. In particular, Fig. 43(d) shows the D + differential67 (D*) h -1.5 -1 -0.5 0 0.5 1 1.5 ( D * ) ( nb ) h / d s d HERA-IINLO QCDNLO QCD customised – D* fi NLO QCD b < 1000 GeV T p(D*)| < 1.5 h | X H1 and ZEUS – eD* fi ep (a) ( D * ) ( nb / G e V ) T / dp s d -4 -3 -2 -1 (D*) (GeV) T p2 3 4 5 6 7 8 9 10 20 r a t i o t o H E RA HERA-IINLO QCDNLO QCD customised – D* fi NLO QCD b < 1000 GeV T p(D*)| < 1.5 h | X H1 and ZEUS – eD* fi ep (b) ) ( nb / G e V / d Q s d -5 -4 -3 -2 -1 ) (GeV Q10 r a t i o t o H E RA HERA-IINLO QCDNLO QCD customised – D* fi NLO QCD b < 1000 GeV T p(D*)| < 1.5 h | X H1 and ZEUS – eD* fi ep (c) (d) Figure 43: Differential D ∗ -production cross section [217] as a function of (a) η ( D ∗ ), (b) p T ( D ∗ ) and(c) Q . The data points are shown with uncorrelated (inner error bars) and total (outer error bars)uncertainties. Also shown are the NLO QCD predictions (HVQDIS) [51] with theory uncertaintiesindicated by the band. The beauty-production contribution is included in the cross section definitionand is plotted separately. A customised NLO calculation (see text) is also shown. (d) Differential D + cross section as a function of p T ( D + ) down to p T ( D + ) = 0 GeV [204].cross section [204] (entry 11 in Table 5) measured down to p T ( D + ) = 0 GeV. The measurement wasdone in the D + → K S π + decay channel. The presence of a neutral strange hadron in the decay resultedin a reasonable signal-to-background ratio even at very low transverse momentum of the D + . Thedata were found to be described by the massive NLO QCD calculations within about two standarddeviations.Furthermore, parton-parton correlations have been studied in D ∗ -tagged events [201] (entry 8 inTable 5). The conclusions are similar to those obtained from the respective photoproduction measure-ments. In general the massive QCD calculations provide a good description apart from the region ofsmall ∆ φ and very large | ∆ η | between the two leading jets in the event (not shown).68 (GeV) jetT E ( pb / G e V ) j e t T / d E s d -1 -1 ZEUS 354 pb had C · HVQDIS+ZEUS-S had C · HVQDIS+ABKM Rapgap x 1.49
ZEUS e jet X fi X b e b fi ep (GeV) jetT E D a t a / H V Q D I S ′ ′ ′ (a) jet h -1.5 -1 -0.5 0 0.5 1 1.5 2 ( pb ) j e t h / d s d -1 ZEUS 354 pb had C · HVQDIS+ZEUS-S had C · HVQDIS+ABKM Rapgap x 1.49
ZEUS e jet X fi X b e b fi ep jet h -1.5 -1 -0.5 0 0.5 1 1.5 2 D a t a / H V Q D I S ′ ′ ′ (b) Figure 44: Differential cross section for inclusive-jet production in beauty DIS events as a functionof E jet T (a) and η jet (b) [148]. The data points are shown with statistical (inner error bars) and total(outer error bars) uncertainties. Also shown are the NLO QCD predictions (HVQDIS) [51], correctedfor hadronisation effects, with theory uncertainties indicated by the band. The dashed line shows theprediction from the RAPGAP MC generator [23] scaled to the measured integrated cross section.Inclusive lifetime tagging (entries 21, 22 in Table 5) allowed the extension of the kinematic range ofcharm measurements up to E jet T = 35 GeV (not shown), which roughly corresponds to p T ( D ) ≈
20 GeV,where the statistics of fully reconstructed charm mesons becomes poor. Good agreement is againobserved.Figure 44 shows the corresponding single-differential jet cross sections for beauty production inDIS [148] (entries 11 and 22 in Tables 6 and 5, respectively). The lifetime-tagging technique togetherwith the reconstruction of the vertex mass were used to extract charm- and beauty-jet cross sectionssimultaneously. This measurement was selected since it has the highest statistical significance forbeauty-quark production, as can be seen from the last column of Table 6. The typical precision reachedin the data is 10 – 20% and is comparable to the theory uncertainties. The massive-scheme NLO QCDcalculations provide a good description of the shape and normalisation of the measured cross sections.
The large collected data samples allowed measurements of double-differential heavy-flavour cross sec-tions, to study the correlations between various kinematic variables.The H1 collaboration has studied [201] (entry 8 in Table 5) the cross section as a function of x obsγ in different Q ranges, complementing the measurements in the photoproduction regime discussed inSection 6.4.1. It was shown that the amount of higher order contributions included in the massiveNLO calculations, including topologies which would be called “flavour excitation” in the leading orderpicture, is enough to describe the data for different Q , while the BGF-only component of the RAPGAP69 D*) h -1.5 -1 -0.5 0 0.5 1 1.500.20.40.60.81 (D*) h -1.5 -1 -0.5 0 0.5 1 1.500.20.40.60.81 *(D*) < 1.25 GeV T * [ nb / G e V ] T dp h / d s d H1 (D*) h -1.5 -1 -0.5 0 0.5 1 1.500.511.5 (D*) h -1.5 -1 -0.5 0 0.5 1 1.500.511.5 *(D*) < 2.00 GeV T (D*) h -1.5 -1 -0.5 0 0.5 1 1.500.20.40.60.81 (D*) h -1.5 -1 -0.5 0 0.5 1 1.500.20.40.60.81 *(D*) < 3.00 GeV T (D*) h -1.5 -1 -0.5 0 0.5 1 1.500.050.10.150.2 (D*) h -1.5 -1 -0.5 0 0.5 1 1.500.050.10.150.2 *(D*) < 6.00 GeV T (D*) h -1.5 -1 -0.5 0 0.5 1 1.500.0020.0040.0060.0080.01 (D*) h -1.5 -1 -0.5 0 0.5 1 1.500.0020.0040.0060.0080.01 *(D*) < 20.00 GeV T H1 dataHVQDIS (MSTW2008f3)HVQDIS (CT10f3)
D* in DIS
Figure 45: Double-differential D ∗ cross section as a function of p ∗ T ( D ∗ ) and η ( D ∗ ) [144]. The datapoints are shown with statistical (inner error bars) and total (outer error bars) uncertainties. NLOQCD calculation (HVQDIS) [51] with two different proton parton densities are compared to the data.Theoretical uncertainties are indicated by the bands.Monte Carlo can describe the measurement for Q > after rescaling, but fails to describethe shape observed in the data at lower Q (not shown here). This is to be expected, since in this“photoproduction-like” region (see Fig. 41) the “flavour excitation” component will then be missing.Using the full HERA II data sample, cross sections have been measured by H1 [144] (entry 17in Table 5) as a function of the D ∗ pseudorapidity in the laboratory frame, η ( D ∗ ), in bins of the D ∗ transverse momentum in the γp centre-of-mass frame, p ∗ T ( D ∗ ). Figure 45 shows a comparison of themassive NLO QCD predictions to the data. At large p ∗ T ( D ∗ ), D ∗ production in the backward regionis very suppressed, while at low p ∗ T ( D ∗ ) the η ( D ∗ ) distribution is rather flat in the phase space ofthe measurement. The massive-scheme NLO predictions provide a good description of the data. Thepredictions depend only very little on the proton PDFs used for the calculation.In most of the analyses summarised in Tables 5 and 6 the double-differential cross sections in Q and y or Q and x were also measured. These measurements allowed dedicated studies of the inclusive heavy-flavour event kinematics which can be expressed in terms of the charm reduced cross sections, or of thecharm contribution to the structure function F (see Section 8.4). Figure 46 shows the combined double-differential D ∗ cross sections as a function of Q and y [217]. Massive-scheme NLO QCD predictionsprovide a good description of these cross sections in the full range in Q between 1 . and 1000 GeV .70 d y ( nb / G e V / d Q s d y H1 and ZEUS -2 -1 -1 -2 -1 -1 <3.5 GeV -2 -1 -1 -2 -1 -1 <5.5 GeV -2 -1 -1 -2 -1 -1 <9 GeV -2 -1 -1 -2 -1 -1 <14 GeV -2 -1 -1 -2 -1 -1 <23 GeV -2 -1 -1 -2 -1 -1 <45 GeV -2 -1 -1 -2 -1 -1 <100 GeV -2 -1 -1 -2 -1 -1 <158 GeV -2 -1 -1 -2 -1 -1 <251 GeV -2 -1 -1 -3 · -2 -1 -1 -3 · <1000 GeV D*X fi HERA ep NLO QCDNLO QCD customised D* fi NLO QCD b y < 1000 GeV T p (D*)| < 1.5 h | y Figure 46: Double-differential D ∗ cross section as a function of Q and y [217]. The data pointsare shown with uncorrelated (inner error bars) and total (outer error bars) uncertainties. Also shownare the NLO QCD predictions (HVQDIS) [51] with theory uncertainties indicated by the band. Thebeauty-production contribution is included in the cross-section definition and is plotted separately.The theoretical uncertainties decrease with increasing Q . For Q ∼ <
50 GeV the theoretical uncertaintiesare larger than those of the measured cross sections. Similar to the single-differential distributions shownin Fig. 43, the theoretical uncertainties are dominated by the scale variations, the uncertainty on thecharm-quark pole mass and the variation of the fragmentation model. A higher-order calculation withimproved fragmentation model is needed to achieve a theoretical precision similar to the data. The measured double-differential DIS cross sections of heavy-flavour production as a function of Q and y or Q and x were used to extract the heavy-flavour reduced cross sections, σ Q ¯ Q red , or the heavy-flavour contribution to the proton structure function F , F Q ¯ Q , where Q is either c or b . As discussed inSection 2.8, the inclusive double-differential cross sections of heavy-flavour production can be expressedin terms of σ Q ¯ Q red or F Q ¯ Q and F Q ¯ QL . In measurements of F Q ¯ Q the small contribution arising from F Q ¯ QL wassubtracted relying on theory, corresponding to a correction of up to 4%. The extraction from themeasured cross sections requires an extrapolation from the experimentally accessible kinematic region71n p T and η and a particular final state to the full phase space of heavy quarks. The extrapolationwas done either using the massive-scheme NLO QCD calculations or LO+PS Monte Carlo simulations.Since this procedure relies on the description of kinematic distributions by predictions, a non-negligibletheoretical uncertainty was introduced. This additional uncertainty was estimated by varying theparameters in the calculations which affect the shapes of the kinematic distributions.The σ Q ¯ Q red and F Q ¯ Q values extracted from measurements performed with different experimental tech-niques and different detectors can be directly compared. Such measurements are complementary toeach other due to different dominant sources of systematics, mostly independent statistics and differentkinematic coverage, resulting in somewhat different theoretical uncertainties due to extrapolation. Forinstance, for the σ c ¯ c red measurements the dominant systematics in the H1 inclusive vertexing analysis (en-try 20 in Table 5) is due to the treatment of the light-flavour component, while in the H1 D ∗ HERA IImeasurement (entry 17 in Table 5) the dominant systematics is due to the modelling of the trackingefficiency. The ZEUS analysis of charm semileptonic decays (entry 15 in Table 5) has yet completelydifferent systematics. Therefore, a combination of measurements with such different techniques allowsa significant reduction not only of statistical and uncorrelated but also of correlated systematic andextrapolation uncertainties. s r e d cc_ H1 VTXH1 D* HERA-II H1 D* HERA-IZEUS c → m X ZEUS D* 98-00ZEUS D* 96-97 ZEUS D ZEUS D + H1 and ZEUS Q =2.5 GeV Q =5 GeV Q =7 GeV Q =12 GeV Q =18 GeV Q =32 GeV Q =60 GeV Q =120 GeV Q =200 GeV Q =350 GeV -4 -3 -2 Q =650 GeV -4 -3 -2 Q =2000 GeV HERA -4 -3 -2 x Figure 47: Combined reduced cross sections [84] σ c ¯ c red (filled circles) as a function of x for fixed values of Q . The input data are shown with various other symbols as explained in the legend. The error barsrepresent the total uncertainty including uncorrelated, correlated and procedural uncertainties addedin quadrature. For presentation purposes each individual measurement was shifted in x .72igure 47 shows a comparison of H1 and ZEUS measurements of the charm reduced cross sections as well as the milestone result of their combination [84]. The combination accounts for correlations of thesystematic uncertainties among the different input data sets. The individual σ c ¯ c red measurements showgood consistency, with a χ value of 62 for 103 degrees of freedom. The combined data are significantlymore precise than any of the input data sets. Figure 47 also highlights the advantages of differenttagging techniques: while D ∗ has superior precision at low Q due to better signal-to-background ratio,the inclusive vertexing analysis with lifetime tagging dominates at high Q due to the larger accessiblestatistics. The final total precision of the combined charm reduced cross sections is 10% on average andreaches 6% at low x and medium Q . This corresponds to a factor 2 improvement over the most precisedata set in the combination. =2.5 GeV Q = 5 GeV Q = 7 GeV Q -3 =12 GeV Q -3 =18 GeV Q -3 =32 GeV Q =60 GeV Q =120 GeV Q =200 GeV Q =350 GeV Q =650 GeV Q =2000 GeV Q r e d c c s -4 -3 -2 -4 -3 -2 x -4 -3 -2 HERA HERAPDF1.5
H1 and ZEUS ) (GeV Q + . i bb F -1 ZEUS vtx 354 pb -1 ZEUS e 363 pb -1
114 pb m ZEUS -1 +vtx 126 pb m ZEUS -1 H1 vtx 246 pbHERAPDF 1.5ABKM NNLOMSTW08 NLOMSTW08 NNLOCTEQ6.6 NLOJR09 x=0.032 i=0x=0.013 i=1x=0.005 i=2x=0.002 i=3x=0.0013 i=4x=0.0005 i=5x=0.0002 i=6x=0.00013 i=7
Figure 48:
Left:
Combined σ c ¯ c red [84] (filled circles) as a function of x for fixed values of Q . The errorbars represent the total uncertainty including uncorrelated, correlated and procedural uncertaintiesadded in quadrature. The data are compared to the NLO predictions based on HERAPDF1.5 [42]in the TR standard GMVFNS [71]. The line represents the prediction using M c = 1 . M c in therange 1 . < M c < .
65 GeV.
Right:
Measurements of F b ¯ b [148, 149, 208, 214, 215] (various symbols)as a function of Q at fixed values of x . The inner error bars are the statistical uncertainties, whilethe outer error bars are the statistical, systematic and extrapolation uncertainties added in quadrature.The data are compared to several NLO and NNLO predictions, including HERAPDF1.5 [42] in the TRstandard GMVFNS [71]. The uncertainty band shows the full PDF uncertainty which is dominated bythe variation of M b . entries 4,6,8,12,13,14,15,16,17,20 in Table 5 H1 CHARM AND BEAUTY CROSS SECTION FRACTION -3 -2 -1 f qq _ x = 0.0002 H1 Data f cc_ f bb_ x = 0.00032MSTW08 NNLOf cc_ f bb_ x = 0.0005 -3 -2 -1 x = 0.0008 x = 0.0013 x = 0.002 -3 -2 -1 x = 0.0032 x = 0.005 x = 0.013 -3 -2 -1
10 10 Q / GeV x = 0.02
10 10 Q / GeV x = 0.032
10 10 Q / GeV x = 0.05 - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - – - - - - - - - - - - - - - - - - -- - - - - - - - - – - - - - - - - - - - - - - - - - -- - - - - - - - - – - - - - - - - - - - - - - - - - -. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . -4 -3 -2 -4 -3 -2 -4 -3 -2 Q = 2.5 GeV s cc r e d / s r e d Q = 5 GeV Q = 7 GeV Q = 12 GeV Q = 18 GeV Q = 32 GeV Q = 60 GeV Q = 120 GeV Q = 200 GeV x Q = 350 GeV Q = 650 GeV x HERAHERAPDF1.5 - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -Figure 49: Fraction of charm and beauty contributions to the inclusive DIS cross section as a func-tion of x and Q [206, 84, 218]. Also shown (curves) is a GMVFNS prediction by MRST [219] andHERAPDF1.5. The dashed and dash-dotted lines are the asymptotic limits for charm and beauty fromEq. (1).Additionally, new statistically independent measurements of charm production have been published(entries 18,19 and 22 in Table 5), in particular the ZEUS D ∗ measurement with HERA II data. InFig. 48 the combined σ c ¯ c red [84] and individual measurements of F b ¯ b (entries 3–5, 7–9, 11 in Table 6) arecompared to NLO and NNLO QCD predictions.The beauty measurements are all in good agreement with each other and the most precise data wereobtained with inclusive lifetime tagging. The NLO QCD prediction in the GMVFNS approach based onHERAPDF1.5 [42] is common for the two comparisons. The good agreement between these predictionsand the heavy-flavour data shows that the gluon density, which in HERAPDF1.5 is extracted from thescaling violations of F , is adequate for the description of these gluon-induced production processes.Other GMVFNS predictions were also compared to the combined charm reduced cross sections (notshown). The best description of the data was provided by predictions including partial O ( α s ) cor-rections, while predictions including O ( α s ) terms agreed well with the data and predictions including O ( α s ) have shown the largest deviations [84]. The theoretical uncertainty for σ c ¯ c red and F b ¯ b increases atlow Q and is dominated by the M c variation. This indicates that the low- Q data are sensitive tothe value of the heavy-quark mass used in the calculation, which was exploited to extract the optimal M c values for different GMVFNS schemes as well as to measure the running heavy-quark masses (seeSection 9.2).Fig. 49 shows the fraction of the heavy-flavour component in the total inclusive DIS cross section: f q ¯ q = F q ¯ q /F and σ c ¯ c red / σ red . As expected, the heavy-flavour fractions increase with increasing Q .For x ∼ < .
01, the asymptotic limit is approached towards Q ∼ m c ∼
100 GeV for charm and Q ∼ m b ∼ for beauty. The charm and beauty fractions in the high- Q data come close to4 /
11 and 1 /
11, respectively, stressing the importance of the heavy-flavour component for the descriptionof inclusive DIS. The observed suppression of the heavy-flavour fractions for x ∼ > .
01 originates from the74ising importance of the valence-quark contribution to the inclusive DIS cross section in this kinematicdomain.
Large photon virtuality Q provides an additional hard scale in the QCD calculations of heavy flavourproduction and allows probing the parton dynamics inside the proton more directly than photoproduc-tion. The dominant contribution to the charm and beauty cross sections arises from photon-gluon fusion.For Q ≫ m Q , where the photon virtuality is the dominant hard scale, the cross-section behaviour issimilar to the one of the inclusive cross section for deeply inelastic scattering. At high Q and low x ,the naively expected charm and beauty contributions of 4 /
11 and 1 /
11 are asymptotically approached.For Q ≪ m Q , where the quark mass is the dominant hard scale, the cross section behaves essentiallylike photoproduction, i.e. the photon can be approximated to be quasi-real. NLO QCD predictionsusing zero-mass schemes (ZMVFNS) fail to describe the data in the vicinity of or below the so-called“flavour threshold” at Q ∼ m Q . NLO QCD predictions in the massive scheme (FFNS) give a gooddescription of heavy flavour production at HERA over the complete accessible kinematic range. NLOpredictions in variable flavour number schemes (GMVFNS) are only available for inclusive quantitities,and perform about equally well. There is no indication for the need of resummation of ln Q /m Q termsat HERA energies. In particular for charm, the uncertainties from QCD corrections beyond NLO andfrom the modelling of fragmentation are considerably larger than the experimental uncertainties of themeasured cross sections. Improved QCD calculations would therefore be highly welcome.75 Measurement of QCD parameters, proton structure, andimpact on LHC and other experiments
So far the emphasis was on direct cross-section measurements from the HERA data and on the compar-ison to and performance of different theoretical approximations for the perturbative QCD expansion.In this section the extraction of more fundamental QCD parameters and parametrisations will be dis-cussed, which are of direct relevance to all high energy physics processes and to the Standard Model ofparticle physics in general.
As outlined in Section 2.10, fragmentation fractions, i.e. the probability of a quark of a given flavour toform a specific final state hadron, and fragmentation functions, parametrising the fraction of the energyor momentum of the final state quark which will be taken by the final state hadron, are essential torelate theoretical QCD calculations at parton level to measurable hadronic final states.Studies of the fragmentation process are based on a complete reconstruction of the final-state hadron.The statistics accessible at HERA for fully-reconstructed beauty hadrons is extremely low due to themoderate beauty-production cross section and small branching ratios. On the other hand, HERA is effec-tively a charm factory, with about 10 charm events recorded to tape. Therefore, only the fragmentationof charm quarks has been studied by H1 and ZEUS. Charm fragmentation has been studied in both theDIS and photoproduction regimes. A comparison between these results and e + e − measurements pro-vides a so far unique test of the fragmentation universality in the heavy-flavour sector for colour-neutral(electromagnetic) vs. coloured (strongly interacting) initial states. The explicit reconstruction of a D ∗ meson in the final state has the optimal signal sensitivity of allfully reconstructed charm final states (cf. Table 3 and Table 5). Thus, it has been used for studies ofthe non-perturbative charm fragmentation function (Section 2.10). Since the momentum of the charmquark is not measured in the detector, the fragmentation function is not a directly accessible quantity.It can be approximated either by jets to which a reconstructed D ∗ meson is associated (for high- p T events) or by the overall energy flow in the event hemisphere around the D ∗ (for production close tothe kinematic threshold). Parameters of the fragmentation function were extracted from the data byfitting corresponding predictions to the measured normalised differential cross sections as a function of z jet = ( E + P ) D ( E + P ) jet and z hem = ( E + P ) D P hem ( E + P )for the jet and hemisphere methods, respectively. The tuning of the fragmentation parameters wasdone based on Monte Carlo simulations [21, 23] with similar JETSET [24] settings or on NLO QCDcalculations using the same “heavy quark” definition and similar schemes for the cancellation of collinearand infrared divergences in photoproduction and DIS [56, 51]. Due to the heavy quark masses most of these terms are not really divergent. Nevertheless, events with “similar”topologies and large but almost cancelling weights are produced in correlated groups. EUS z / s d s / d z ZEUS (120 pb -1 )FMNR · C PYThad (Kartvelishvili a =2.67 +0.25- 0.31 )FMNR · C PYThad (Kartvelishvili a =1.2)FMNR · C PYThad (Kartvelishvili a =4.0) (a) j e t / d z s d s / j e t / d z s d s / jet sample – D* H1 Data (parton level) = 3.8 + 0.3 a HVQDIS = 3.8 - 0.3 a HVQDIS jet z R (b) h e m / d z s d s / h e m / d z s d s / jet sample – No D*
H1 Data (parton level) = 6.1 + 0.9 a HVQDIS = 6.1 - 0.8 a HVQDIS hem z R (c) h e m / d z s d s / h e m / d z s d s / jet sample – No D*
H1 Data = 10.3 + 1.9 a RAPGAP = 10.3 - 1.6 a RAPGAP = 4.4 a RAPGAP hem z R (d) Figure 50: The normalised D ∗ cross sections as a function of z from the (a) ZEUS [221] and (b), (c), (d)H1 [220] analyses. The hard (b) and the threshold (c-d) regions are shown from the H1 measurement.The statistical (inner error bars) and the statistical and systematic uncertainties added in quadrature(outer error bars) are shown separately. The data are compared to the NLO QCD predictions (a-c) fromHVQDIS [51] and FMNR [56] as well as to the RAPGAP MC [23] (d) based on the string fragmentationmodel. The parameter of the Kartvelishvili fragmentation function was tuned to the data in each case.The H1 [220] and ZEUS [221] measurements were done in the DIS and photoproduction regimes,respectively, utilising HERA I data sets. The H1 experiment investigated both the high p T and thethreshold regions, in order to cover the largest possible phase space, while ZEUS restricted the measure-ment to the high- p T regime, in order to reach small z values without strongly biasing the distributions,and in order to minimise perturbative fragmentation factorisation effects within the data set. This ledto H1 selecting jets with E ∗ T > γ ∗ p rest-frame and ZEUS cutting on E T > p T high p T ˆ s, GeV ∼ ∼ ∼ z method hem hem jet jetKartvelishvili [95] α . +0 . − . . +0 . − . . +0 . − . . +0 . − . ( χ /ndof ) (37 . /
4) (4 . /
4) (4 . /
3) n.a.Peterson [94] ε . +0 . − . . +0 . − . . +0 . − . . +0 . − . ( χ /ndof ) (38 . /
4) (18 . /
4) (23 . /
3) n.a.Table 7: Parameters of fragmentation function extracted for the NLO QCD predictions by H1 andZEUS.functions for the HVQDIS and FMNR NLO calculations, which should also be applicable to the concep-tually similar MNR [60] calculations in hadroproduction, are presented in Table 7. A few observationscan be made from the distributions: • As expected, due to the high quark mass, a charm meson retains a large fraction of the momentumof a charm quark. Therefore, the charm fragmentation is much harder than that of light hadrons. • The fragmentation of charm quarks to D ∗ mesons near to the kinematic threshold is harder than inthe region away from the threshold (cf. Fig. 50 (b) and (c)). This can be qualitatively understoodas a consequence of the fact that the phase space available for the production of additionalparticles is smaller near threshold. As a result, the fragmentation parameters extracted in thetwo kinematic regions are significantly different. This leads to the conclusion that the differentkinematic regimes can not be described simultaneously within the framework of the independentfragmentation function. This is to be expected, since the nonperturbative phase space suppressionis incompletely modeled in this approach. • NLO QCD calculations in conjunction with an independent fragmentation fail to describe thedata close to the kinematic threshold: χ /ndof ≈ / z distribution in the data: χ /ndof ≈ / • The Peterson fragmentation provides a much worse description of the data than the Kartvelishvilifunction. This has also been observed elsewhere [222]. • The jet and hemisphere methods in the region where both are applicable, i.e. away from thethreshold, yield similar results for the Kartvelishvili parametrisation, while they remain some-what different in the Peterson case. Thus again, the Kartvelishvili parameterisation seems to bepreferred.For some recent measurements [84, 144, 204, 209, 210, 211, 217], these results have been usedexplicitly to model the fragmentation for the comparison of theoretical predictions to the charm HERAdata (see Section 8) and for the extrapolation to the full phase space in the context of the extractionof the charm reduced cross sections (see Section 8 and Section 9.2). This has shown that there are781 [220] ZEUS [221]kinematics threshold high p T high p T ˆ s, GeV ∼ ∼ ∼ z method hem hem jet jetKartvelishvili α . +1 . − . . +0 . − . . +0 . − . n.a.( χ /ndof ) (37 . /
4) (4 . /
4) (4 . /
3) n.a.Peterson ε . +0 . − . . +0 . − . . +0 . − . . +0 . − . ( χ /ndof ) (38 . /
4) (18 . /
4) (23 . /
3) n.a.Table 8: Parameters of fragmentation function extracted by H1 and ZEUS for the fragmentation modelin PYTHIA with other parameter settings set to the default values. Note, that also a set of fragmen-tation parameters was extracted by H1 [220] for the ALEPH PYTHIA tune [223].significant theory uncertainties due to fragmentation. A consistent phenomenological reanalysis of theH1 and ZEUS data is needed in order to resolve the differences observed in different kinematic domains,which originate from neglecting perturbative evolution and phase space effects, hopefully resulting inan important reduction of the related theory uncertainties. It is worth mentioning that the completeHERA II dataset, which has not yet been analysed in this context, is available in principle for thispurpose.Table 8 shows the equivalent results extracted from LO+PS MCs using the “default” JETSETsettings as used e.g. by the PYTHIA and RAPGAP MCs. In this case a perturbative evolution of thefragmentation function is partially included through the parton showering, and phase space correctionsare applied. Despite the poor χ , the Peterson parameters extracted from the intermediate and high p T jet samples now agree with each other, as well as with the corresponding default parameter 0.05extracted from e + e − collisions [221]. This confirms the universality of the nonperturbative part offragmentation. Kartvelishvili parameters are unfortunately not available for all data sets and canhence not be compared. Near threshold, even the MC model does not yield the same fragmentationparameters, and the χ is generally bad. This indicates that still not all threshold effects might havebeen fully accounted for. The fractions of c quarks hadronising into a particular charm hadron, f ( c → D, Λ c ), have been measuredby H1 and ZEUS in the DIS [147, 203, 204] and photoproduction [145, 143] regimes. The measure-ments were done for D + , D , D ∗ + , D + s and Λ c based on full reconstruction of the charm-hadron decays.The fragmentation fractions were extracted from integrated visible cross sections. The typical fidu-cial phase space of the charm hadrons was defined by p T ( D, Λ c ) > η ( D, Λ c ) < .
6. Thefragmentation fractions were extracted with the additional constraint that the sum of the fractions forall weakly-decaying open-charm hadrons (i.e. the ground states from the point of view of strong andelectromagentic interactions) has to be equal to unity. This was done by a constrained fit in H1 [147]and by an advanced procedure called equivalent phase space treatment in ZEUS [203]. In addition todirect production, such experimentally measured fragmentation fractions include also all possible decaychains of excited charm hadrons.Figure 51 shows a compilation of all available charm fragmentation fraction measurements. TheHERA data are compared to an average of e + e − measurements [224, 225]. To allow a direct comparison,79 C ha r m f r ag m en t a t i on f r a c t i on s p g ZEUSHERA II p g ZEUSHERA I ep DISZEUSHERA I ep DISH1 - e + e ) D fi f (c ) + D fi f (c ) + D* fi f (c ) s D fi f (c ) c Λ → f (c Figure 51: Fractions of charm quarks hadronising into a particular charm hadron [143]. Measurementsfrom HERA are compared to the combined e + e − data. Different hadron species are shown with differentmarker types.all measurements have been corrected [226] to the decay branching fractions from PDG 2010 [227].The HERA data reach very high precision, benefiting from a partial cancellation of some systematicuncertainties in the ratio. In particular, the recent ZEUS measurement [143] is based on the fullHERA II data sample and made use of the finalised tracking with lifetime tagging for D , D + and D + s .This allowed to reduce both statistical and systematic uncertainties. The ultimate precision achievedwith ZEUS HERA II data alone is fully competitive with the precision of the e + e − average from severalexperiments. All data from DIS, photoproduction and e + e − collisions are in agreement within thehigh accuracy of the data. This demonstrates that the charm fragmentation fractions are independentof the production mechanism, and therefore supports the hypothesis of universality of heavy-quarkfragmentation. The agreement between the fragmentation fractions has been checked quantitatively inthe context of a combination [226].In addition to the fragmentation fractions, various charm fragmentation ratios were extracted: theratio of the neutral to charged D -meson production rates, R u/d , the fraction of the charged D mesonsproduced in a vector state, P dV , and the strangeness-suppression factor, γ s . Figure 52 shows a compar-ison of HERA measurements [143, 145, 147, 203, 204] with results obtained in e + e − collisions (numbersquoted in [203] using average from [224]) and hadroproduction by CDF [228], ALICE [229, 230] and AT-LAS [225]. Also shown is a global average of P dV results from e + e − , photo- and hadroproduction [231] ,which also includes some e + e − , ep and CDF data shown separately. Note, that the uncertainty of theaverage is driven by the π − – A result from the WA92 experiment [232], which is quoted with statisticaluncertainty only, i.e. treating all systematic uncertainties, including the branching-ratio uncertainty, ascorrelated between D + and D ∗ + . The fragmentation ratios extracted from HERA, e + e − and hadropro- The paper as well as other recent measurements [228, 229, 225] report P V values, which correspond to the fractionof charged and neutral D mesons produced in the vector state. However the measurements rely on isospin symmetryassumption, which makes P V identical to P dV . u/d R Vd P s γ ZEUS HERA I DIS ZEUS HERA I PHP - e + eH1 HERA I DIS ZEUS HERA II PHP combination dV PALICE 7 TeV ATLAS 7 TeV CDF
Figure 52: Fragmentation ratios R u/d , P dV , γ s measured at HERA and elsewhere [147, 203, 204, 145,143, 224, 228, 229, 230, 225]. Different measurements are shown with different marker types. The errorbars indicate the statistical and systematic uncertainties added in quadrature. The branching-ratiouncertainties are not shown due to the high degree of correlation between experiments and can befound in the original papers. The filled band shows the result of the P dV combination [231].duction data agree within experimental uncertainties. The ultimate precision achieved with the fullHERA II data set is competitive with the most precise measurements in other experiments. Vari-ous simple theory expectations can be tested against the data. The R u/d measurements are slightlyabove, but still in agreement within uncertainties, with the isospin invariance expectation of unity. The P dV measurements are smaller than the naive spin-counting expectation 0 .
75 and the string fragmentationprediction 0 .
66 [233, 234].Excited charm mesons have also been studied with the ZEUS detector using the HERA I [235]and HERA II [236] datasets. Some parameters of the orbitally-excited charm states D (2420) , + with J P = 1 + and D ∗ (2460) , + with J P = 2 + as well as charm-strange state D s (2536) + were measured. Themasses and widths were found to be in good agreement between the two measurements and with thePDG average. The helicity parameters h for D (2420) and D s (2536) + were found to be in agreementwith e + e − measurements. The measured D (2420) parameter was found to prefer a mixture of S and D waves in the decay to D ∗ + π − , although it is also consistent with a pure D wave. In addition,fragmentation fractions and ratios of branching ratios were extracted. For some parameters HERAcan provide important or even so far unique information. For example, the fragmentation fractionsfor the studied excited mesons are so far very poorly experimentally determined. The D (2420) + and D (2420) + fragmentation fractions were measured for the first time [236]: f ( c → D +1 ) = 4 . ± . . ) +2 . − . (syst . )% ,f ( c → D ∗ +2 ) = 3 . ± . . ) +0 . − . (syst . )% . .2 Measurement of parton density functions The gluon PDF at low- and medium- x values is mostly constrained by the scaling violations of the inclu-sive structure function F . In contrast, heavy-quark production at HERA provides a direct probe of thegluon momentum distribution in the proton through the γ ∗ g → c ¯ c process. Such direct measurementsare complementary to the indirect approach.Already the very early charm measurements were used to directly extract the gluon PDF, as wasdone by the H1 collaboration in [169]. The gluon densities extracted from the charm data were foundto be in agreement with the result of a QCD analysis of inclusive F measurements, although the charmmeasurement was limited by statistics.The recent combined charm DIS data [84] were also used in a QCD analysis [84] together with thecombined inclusive HERA I DIS cross sections [218]. Only the data with Q > . were used inthe analysis to assure applicability of pQCD calculations. The analysis was performed at NLO using theHERAFitter package [237, 218, 238] and closely followed the HERAPDF1.0 prescription [218]. Variousimplementations of the NLO GMVFNS approach were used and the role of the value of the charm quarkmass parameter (see Section 2.7), M c , was studied. For each heavy-flavour scheme a number of PDFfits was performed to scan χ of the PDF fit as a function of M c . From the scan the optimal value, M opt c , of the charm-quark mass parameter in a given scheme was determined by the minimum of the χ and the corresponding fit uncertainty was evaluated from the ∆ χ = 1 variation. The procedureis illustrated in Fig. 53(a) which shows the fit to the inclusive DIS data alone and together with σ c ¯ c red .The inclusive DIS cross sections alone only weakly constrain M opt c , indicated by the shallowness of the χ ( M c ) distribution. The charm DIS cross sections provide the required constraint to extract M opt c .Additionally, for each GMVFNS approach the model and parametrisation assumptions in the fits werevaried one-by-one and the corresponding χ scan as a function of M c was repeated. The differencebetween M opt c obtained with the default assumptions and the result of each variation was taken as thecorresponding source of uncertainty. The dominant contribution to the uncertainty was found to comefrom the variation of the minimum Q value for inclusive DIS data used in the fit.Figure 53(b) shows the χ distributions as a function of M opt c obtained from fits to the inclusiveHERA I data and the combined σ c ¯ c red for all variable-flavour-number schemes considered. All schemesyield similar minimal χ values, however at quite different values of M opt c . The resulting values of M opt c are given in Table 9 together with the evaluated uncertainties, the minimal total χ values and the χ contribution from the charm data. The ACOT-full scheme provides the best global description ofthe inclusive and charm data together, while the RT optimised scheme yields the best description ofthe charm data alone. The fits in the S-ACOT- χ scheme result in a very low value of M opt c comparedto other approaches. Since this scheme only includes a leading-order approximation of heavy-flavourproduction at the order considered here (see Table 1), effectively no distinction is made between poleor running mass. All NLO VFNS predictions using corresponding M opt c values for each scheme providea similarly good description of the σ c ¯ c red data [84].Figure 54 shows the PDFs extracted from the fit to the inclusive DIS data alone and together withthe σ c ¯ c red data in the RT optimised VFNS . A comparison of the extracted PDF uncertainties yields thefollowing conclusions about the impact of the σ c ¯ c red data [84]: • the uncertainty on the gluon PDF was reduced, mostly due to a reduction of the parametrisationuncertainty due to the additional constraints that the charm data introduce due to the BGFprocess; This minimisation uncertainty is usually referred to as the “experimental” uncertainty in the HERAPDF context [218,84]. However, it can absorb some other sources of uncertainties, e.g. variations of PDF parametrisation. Therefore, themore general term “fit uncertainty” is used here. Similar observations were made with other schemes. [GeV] c M ) c ( M χ HERA-I inclusiveCharm + HERA-I inclusive
H1 and ZEUS
RT standard ± =1.50 optc M (a) [GeV] c M ) c ( M χ RT standardRT optimisedACOT-full χ S-ACOT-ZM-VFNS optC M H1 and ZEUS
Charm + HERA-I inclusive (b)
Figure 53: (a) The values of χ ( M c ) for the PDF fit to the combined HERA DIS data [84] in the RTstandard scheme [71]. The open symbols indicate the results of the fit to inclusive DIS data only. Theresults of the fit including the combined charm data are shown by filled symbols. (b) The values of χ ( M c ) for the PDF fit to the combined HERA inclusive DIS and charm measurements [84]. Differentheavy flavour schemes are used in the fit and presented by lines with different styles. The values of M opt c for each scheme are indicated by the stars. -4 -3 -2 -1
10 1 HERAPDF1.0 13p HERAPDF1.0 + charm x x f = 10 GeV Q v xu v xd 0.05) × xg ( H1 and ZEUS (a) -4 -3 -2 -1
10 1 HERAPDF1.0 13p HERAPDF1.0 + charm x x f = 10 GeV Q cxuxsx 1.1) × (dx H1 and ZEUS (b)
Figure 54: Parton density functions [84] x · f ( x, Q ) with f = g, u v , d v , u, d, s, c for (a) valence quarksand gluon and for (b) sea anti-quarks obtained from the combined QCD analysis of the inclusive DISdata and σ c ¯ c red (dark shaded bands) in the RT optimised scheme as a function of x at Q = 10 GeV . Notethat, somewhat confusingly but following common practice, here the variable x refers to x b in Eq. 10,rather than to Bjorken x . For comparison the results of the QCD analysis of the inclusive DIS dataonly are also shown (light shaded bands). The gluon distribution function is scaled by a factor 0 .
05 andthe xd distribution function is scaled by a factor 1 . M opt c χ /n dof χ /n dp [GeV] σ NC,CC red + σ c ¯ c red σ c ¯ c red RT standard 1 . ± . fit ± . mod ⊕ param ⊕ α s . /
626 49 . / . ± . fit ± . mod ⊕ param ⊕ α s . /
626 45 . / . ± . fit ± . mod ⊕ param ⊕ α s . /
626 53 . / χ . ± . fit ± . mod ⊕ param ⊕ α s . /
626 50 . / . ± . fit ± . mod ⊕ param ⊕ α s . /
626 55 . / M opt c as determined from the M c scans in differentheavy flavour schemes [84]. The uncertainties of the minimisation procedure are denoted as “fit”,the model, parametrisation and α s uncertainties were added in quadrature and are represented by“mod ⊕ param ⊕ α s ”. The corresponding global and partial χ are presented per degrees of freedom, n dof , and per number of data points, n dp , respectively. • the uncertainty on the charm-quark PDF is considerably reduced due to the constrained rangeof M c . The M c variation was set to 1 . < M c < .
65 GeV for the fit to the inclusive data onlyand was defined by the evaluated total uncertainties as given in Table 9 for the fit including thecharm data; • the uncertainty on the up-quark sea PDF was correspondingly reduced, because the inclusive dataconstrain the sum of up- and charm-quark sea; • the uncertainty on the down-quark sea was also reduced because it was constrained to be equalto the up-quark sea at low x ; • the uncertainties on the valence-quark and strange-quark sea PDFs were almost unaffected; • the central PDFs were not altered significantly and were found to be within the uncertainties ofthe PDFs based on inclusive data only. This reflects the good description of the charm data bythe default PDFs (section 8.4).By now, the combined charm reduced cross sections [84] have been used in QCD analyses by variousPDF-fitting groups [239, 240, 33, 241, 242, 243, 244]. They are an important ingredient to constrainthe proton flavour composition (see also next section) and to stabilise its gluon content. The latter isespecially important for Higgs production at the LHC, for which the dominant process is gluon-gluonfusion via an intermediate top-quark loop. Measuring this process precisely, in combination with aprecise knowledge of the gluon content of the proton, allows the extraction of a precise measurement ofthe Higgs-top Yukawa coupling.Instead focusing on the low- x range, the HERA charm and beauty data have recently been usedin conjunction with charm and beauty data from LHCb to constrain the gluon distribution down to x ∼ × − [245]. This region is particularly relevant for the prediction of cross sections for processesoccurring in cosmic ray interactions. In the previous section it was outlined how the inclusion of charm data into GMVNFS PDF fits, andin particular the constraint on the charm quark mass parameter derived from these data, imposesconstraints on the gluon content (relevant e.g. for Higgs production) and on the flavour composition of84he quarks in the proton. This, in turn, affects theoretical predictions for processes which are sensitiveto this flavour composition, such as the production for W and Z bosons at LHC.Figure 55 shows NLO predictions for W and Z production at LHC for PDFs extracted in differentheavy flavour schemes as discussed in Section 9.2, as a function of the charm quark mass parameter M c used in the PDF fit. For fixed M c , these predictions differ by about 7%. The dependence on M c isopposite to what one would naively expect (see also Fig. 54). A higher charm mass leads to less charmin the proton (fewer gluons split) but to a higher gluon density. This in turn increases the amount of u and d sea quarks in the proton, even more so since the total sea is constrained by the inclusive protonstructure functions. The larger number of u and d quarks overcompensates the smaller number of c quarks and leads to an increase of the W and Z cross sections as shown in Fig. 55. The fit in Fig. 54actually led to a smaller charm mass than the default, therefore the effect on the PDF was opposite. [GeV] c M [ nb ] + W σ optC M RT standard RT optimised ACOT-full χ S-ACOT- ZM-VFNS = 7 TeVsCharm + HERA-I inclusive
H1 and ZEUS (a) [GeV] c M [ nb ] - W σ optC M RT standard RT optimised ACOT-full χ S-ACOT- ZM-VFNS = 7 TeVsCharm + HERA-I inclusive
H1 and ZEUS (b) [GeV] c M [ nb ] Z σ optC M RT standard RT optimised ACOT-full χ S-ACOT- ZM-VFNS = 7 TeVsCharm + HERA-I inclusive
H1 and ZEUS (c)
Figure 55: NLO predictions for (a) W + , (b) W − and (c) Z production cross sections at theLHC for √ s = 7 TeV as a function of M c used in the corresponding PDF fit [84]. The differentlines represent predictions for different implementations of the VFNS. The predictions obtainedwith PDFs evaluated with the M opt c values for each scheme are indicated by the stars. Thehorizontal dashed lines show the resulting spread of the predictions when choosing M c = M opt c .The stars in Fig. 55 indicate the cross section predictions for the optimal mass for each heavy flavourscheme, as extracted from the charm data in Fig. 53(b). All predictions then coincide to within 2%,85ndependent of the heavy flavour scheme used. This demonstrates that using the optimal mass for eachscheme which best fits the HERA charm data stabilises the flavour composition in the proton, and leadsto a reduction of this contribution to the cross section uncertainty by about a factor 3. To minimisethe uncertaities arising from the charm and beauty masses, for GMVFNS schemes it is thus stronglyrecommended to use the optimal mass parameters as derived from the heavy-flavour structure-functiondata rather than a mass obtained from external considerations.A similar analysis for beauty, which remains to be done, will in addition yield experimental con-straints on the b PDF in the framework of 5-flavour PDFs for LHC, which are so far constrained bytheory only. This in turn will be relevant e.g. for a future measurement of the Higgs- b Yukawa couplingfrom associated Higgs- b ¯ b production. The sensitivity of the HERA reduced charm cross sections to the charm-quark mass, already partiallystudied in Section 9.2, can be used to measure the charm quark mass appearing in perturbative QCD,whose value depends on the renormalisation scheme within which it is being evaluated. The two massdefinitions which are most commonly used are the pole mass and the
M S running mass (Section 2.4).Since the
M S mass is perturbatively better defined, recent charm mass measurements concentrate onthis renormalization scheme. The FFNS scheme (Section 2.6) is most suited for this evaluation, sinceit fully accounts for mass effects without any additional free parameters. FFNS calculations of thereduced cross section in this scheme exist at NLO and partial NNLO [62, 32]. All results quoted in thefollowing are obtained from these calculations unless otherwise quoted.The first determination of the
M S charm-quark mass [246], from a subset of D ∗ charm data fromthe H1 collaboration, in which also the details of the theoretical framework are given, obtained m c ( m c ) = 1 . ± . +0 . − . (scale) GeV (20)at NLO, and m c ( m c ) = 1 . ± . +0 . − . (scale) ± . Q > . , to obtain the NLO measurement [84] m c ( m c ) = 1 . ± . ± . ± . ± . α s ) GeV . (22)This result has a slightly more elaborate evaluation of the uncertainties related to the data extrapolationas well as other model and parametrisation uncertainties, while it does not include uncertainties on thenormalisation of the cross section predictions due to QCD scale variations.The same data where then used by the ABM group and collaborators [239] to reobtain similarevaluations, m c ( m c ) = 1 . ± . +0 . − . (scale) GeV (23)at NLO, and m c ( m c ) = 1 . ± . +0 . − . (scale) +0 . − . (theory) GeV (24)at partial NNLO. The smaller central NLO value and the smaller uncertainty are mainly due to thefact that the charm data from the lowest Q bin were included. A correlated measurement of m c ( m c )86 σ r e d cc_ H1 and ZEUS Q =2.5 GeV Q =5 GeV Q =7 GeV Q =12 GeV Q =18 GeV Q =32 GeV Q =60 GeV Q =120 GeV Q =200 GeV Q =350 GeV -4 -3 -2 Q =650 GeV -4 -3 -2 Q =2000 GeV HERA
ABM09NNLO MS ABM09NLO MS -4 -3 -2 x Figure 56: Combined reduced cross sections [84] filled circles as a function of x for fixed values of Q . The error bars represent the total uncertainty including uncorrelated, correlated and proceduraluncertainties added in quadrature. The data are compared to predictions of the ABM group at NLO(hashed band) and NNLO (shaded band) in FFNS using the MS definition for the charm quark mass. ) [GeV] c (m c m χ FF (ABM)
H1 and ZEUS
Charm + HERA-I inclusive ± )=1.26 c (m c m Figure 57: (left) The values of χ for the PDF fit [84] to the combined HERA DIS data including charmmeasurements as a function of the running charm quark mass m c ( m c ). The FFNS ABM scheme is used,where the charm quark mass is defined in the MS scheme. (right) The values of mc(mc) obtained inthe NLO and NNLO variants of the ABM analysis [247] with the value of α s ( M Z ) fixed. The positionof the star displays the result with the value of α s ( M Z ) fitted.87nd the strong coupling constant [247] was also obtained (Fig. 57(right)). In particular for the NLOcase, the correlation between m c ( m c ) and α s is non-negligible.They were also used by the CTEQ group [33] to derive the M S mass in the context of the S-ACOT- χ VFNS, using charm matrix elements to one-loop order in the massive part of the calculation. The result m c ( m c ) = 1 . +0 . − . GeV (25)exhibits a larger uncertainty than the previous extractions due to the additional uncertainty from thevariation of the (single) free parameter of this VFNS scheme, and due to conversions between the poleand running masses in the extraction process.All these results from a predominantly space-like perturbative process are consistent with each otherand with the world average [146] m c ( m c ) = 1 . ± .
025 GeV (26)obtained from lattice QCD and time-like processes. This is a highly nontrivial triumph of QCD. Someof the above measurements are now included in the latest world average [30], and further improvementson both the experimental and theoretical sides have the potential to further improve the correspondingprecision.In a recent preliminary result [248] the same data have again been used to determine the actualscale dependence (‘running’) of the charm-quark mass in the
M S scheme, according to Eq. (13). Forthis purpose, the charm data were subdivided into 6 different Q ranges, for which the charm mass wasextracted separately at the scale < Q > +4 m c , where < Q > is the average of each range. The resultis shown in Fig. 58. This is the first explicit measurement of the scale dependence of the charm quarkmass. [GeV] µ ) [ G e V ] µ ( c m H1 and ZEUS preliminary
HERA (prel.)PDG with evolved uncertainty
Figure 58: Measured charm mass m c ( µ ) in the M S running mass scheme as a function of the scale µ as defined in the text (black points). The red point at scale m c is the PDG world average [146] and theband is its expected running [31]. Using the same approach as outlined above for charm, the ZEUS collaboration has used a fit (Fig. 59)to the beauty reduced-cross-section data [148] to extract the value of the beauty-quark
M S runningmass at NLO, m b ( m b ) = 4 . ± . +0 . − . (mod) +0 . − . (param) +0 . − . (theo) GeV , (27)88 -3 -3 x -4 -3 -2 x -4 -3 -2 x -4 -3 -2 bb r σ -1 ZEUS 354 pb=4.07 GeV (best fit) b QCD fit, m =3.93 GeV b QCD fit, m =4.21 GeV b QCD fit, m
ZEUS = 6.5 GeV Q = 12 GeV Q = 25 GeV Q = 30 GeV Q = 80 GeV Q = 160 GeV Q = 600 GeV Q Figure 59: Reduced beauty cross section (filled symbols) as a function of x for seven different values of Q [148]. Also shown are the results of a QCD fit for different values of the M S running mass m b ( m b ).where the theoretical uncertainty is dominated by the scale variation uncertainty. This is the first suchextraction from HERA data, and agrees well with the world average [146] m b ( m b ) = 4 . ± .
03 GeV . (28)Figure 60 shows this result, translated to the scale 4 m b , compared to the PDG value and its expectedrunning and to values extracted from LEP data at the scale M Z . The expected running of the M S beauty-quark mass is confirmed. This is a nontrivial test of the basics of QCD. [GeV] µ ) [ G e V ] µ ( b m ZEUS
PDG with evolved uncertaintyZEUSDELPHI 3-jetsDELPHI 4-jets NLOALEPHOPALSLD
Figure 60: Measured beauty mass m b ( µ ) in the M S running mass scheme as a function of the scale µ from HERA [148] and LEP [249] data. The red point at scale m b is the PDG world average [146] andthe band is its expected running [31]. 89 .6 Summary Heavy flavour physics at HERA yields many results which are of interest for particle physics in general.The usage of HERA as a “charm factory” generates world-class information on charm fragmentationfunctions and fragmentation fractions and allows tests of the fragmentation universality. The constraintson PDFs from charm data, and to a lesser extent also from beauty data, help to reduce uncertaintiesfor important cross sections at LHC, such as heavy flavour, W/Z and Higgs production. Constraintson the latter are important for the measurement of the Higgs Yukawa couplings. More directly, thecharm and beauty DIS data have been used to extract well defined measurements of the charm- andbeauty-quark masses, which enter the world average. The running of the charm-quark mass has beenmeasured for the first time ever. By comparing with LEP data, the running of the beauty quark masshas also been confirmed. In general, the good agreement of QCD predictions with the data support theapplicability of the HERA results to all particle physics applications for which they might be directlyor indirectly relevant. 90
Charm and beauty production at HERA are a great laboratory to test the theory of heavy flavour pro-duction in the framework of perturbative QCD and to measure some of its parameters. The occurrenceof different possibilities to treat the heavy quark masses in the PDF, matrix element and fragmenta-tion parts of the calculation introduces a significant level of complexity into the corresponding QCDcalculations, in addition to the usual scheme and scale choices. Confronting such different choices withdata can be helpful to understand the effects of different ways to truncate the perturbative series andto evaluate their impact on the measurement of fundamental parameters, both at HERA and at othercolliders.HERA was the first and so far only high energy ep collider. The heavy flavour results discussed inthis review were obtained with the H1 and ZEUS detectors which were well suited for the detection ofheavy flavoured particles. Adding the luminosities from the two collider experiments, a total luminosityof about 1 fb − was collected.The availability of many different charm and beauty tagging methods allows results to be obtainedthrough several different final states with different systematics. In addition to the statistical benefit fromcombining different samples, such combinations also profit from cross calibrations of the systematicsfrom different methods and experiments.Due to the high top mass, the only top final state which might have been detectable at HERA issingle top production with non-Standard-Model couplings. No signal is seen, and the coupling limitsderived are competitive.The charm (beauty) quark masses provide semi-hard (hard) QCD scales which allow the succesfulapplication of perturbative calculations over the complete phase space. However, these masses alsocompete with other, often even harder perturbative scales. Total cross sections for charm photoproduc-tion and the total cross section for beauty production (including photoproduction and deeply inelas-tic scattering) are reasonably described by perturbative calculations at next-to-leading order (NLO).Single-differential cross sections already provide a good handle to test the applicability of different QCDapproximations, although the theoretical uncertainties are mostly much larger than the experimentalones. The theory predictions agree with the data up to the highest accessible transverse momentaor photon vitualities, showing no indications that final state resummation corrections are needed formassive calculations in the HERA kinematical domain. Double-differential cross sections, in particu-lar those including jets, reveal a partial failure of the massive scheme NLO predictions for kinematicobservables which would need final states with four or more partons in the calculation. Althoughstatistics and therefore precision is higher for photoproduction, qualitatively very similar conclusionsare obtained for photoproduction and deeply inelastic scattering (DIS). The NLO calculations in themassless scheme, where available, do mostly not provide a better description for the observables, andclearly fail for some DIS observables. The LO+PS MCs PYTHIA and HERWIG, which are often usedfor acceptance corrections, are able to describe all topologies reasonably, often even very well. TheCASCADE k t -factorisation MC performs somewhat less well on average.In DIS, the large photon virtuality Q provides an additional hard scale in the QCD calculations ofheavy flavour production and allows probing the parton dynamics inside the proton more directly than inphotoproduction. The dominant contribution to the charm and beauty cross sections arises from photon-gluon fusion. For Q ≫ m Q , where the photon virtuality is the dominant hard scale, the cross-sectionbehaviour is similar to the one of the inclusive cross section for deeply inelastic scattering. At high Q and low x , the naively expected charm and beauty contributions of 4 /
11 and 1 /
11 are asymptoticallyapproached. NLO QCD predictions in the massive scheme (FFNS) give a good description of heavyflavour production at HERA in DIS over the complete accessible kinematic range. NLO predictionsin variable-flavour-number schemes (GMVFNS) are only available for inclusive quantities, and performabout equally well. 91n particular for charm, the uncertainties from QCD corrections beyond NLO and from the modellingof fragmentation are considerably larger than the experimental uncertainties of the measured crosssections. Improved QCD calculations would therefore be highly welcome.Heavy flavour physics at HERA yields many results which are of interest for particle physics ingeneral. The usage of HERA as a “charm factory” generates world-class information on charm frag-mentation functions and fragmentation fractions and allows tests of the fragmentation universality. Theconstraints on proton parton distribution functions (PDFs) from charm and beauty data help to reduceuncertainties for important cross sections at the LHC, such as heavy flavour, W/Z and Higgs produc-tion. Constraints on the latter are important for the measurement of the Higgs Yukawa couplings.More directly, the charm and beauty DIS data have been used to extract well defined measurements ofthe charm- and beauty-quark masses, which enter the world average. The running of the charm-quarkmass has been measured for the first time ever, and the running of the beauty quark mass has beenconfirmed.In general, the good agreement of QCD predictions with the HERA data support the applicabilityof the QCD results derived from these data to all particle physics applications for which they mightbe directly or indirectly relevant. Some of the most important HERA heavy flavour results have beenobtained during the last 2-3 years. Even 8 years after the end of data taking the potential of the HERAheavy flavour data has still not been fully used in all cases, so there is room for significant furtherimprovements, in particular also on the theory side, hoping e.g. for differential NNLO calculations in ep collisions, similar to those which have recently started to appear for the pp case. Acknowledgements
This review is a partial summary of the work of perhaps a thousand technicians, engineers and physicistsfor more than two decades. It is our great pleasure to thank the many collegues who have contributedto the results presented here. We thank O. Kuprash and L. Schalow for technical contributions to thisreview. 92 eferences [1] A. Ali et al. , “Heavy Quark Physics at HERA”, in
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