Ad Lucem: QED parton distribution functions in the MMHT framework
Abstract
We present the MMHT2015qed PDF set, resulting from the inclusion of QED corrections to the existing set of MMHT Parton Distribution Functions (PDFs), and which contain the photon PDF of the proton. Adopting an input distribution from the LUXqed formulation, we discuss our methods of including QED effects for the full, coupled DGLAP evolution of all partons with QED at \({\mathcal {O}}(\alpha )\), \({\mathcal {O}}(\alpha \alpha _{S})\), \({\mathcal {O}}(\alpha ^2)\). While we find consistency for the photon PDF of the proton with other recent sets, building on this we also present a set of QED corrected neutron PDFs and provide the photon PDF separated into its elastic and inelastic contributions. The effect of QED corrections on the other partons and the fit quality is investigated, and the sources of uncertainty for the photon are outlined. Finally we explore the phenomenological implications of this set, giving the partonic luminosities for both the elastic and inelastic contributions to the photon and the effect of our photon PDF on fits to high mass Drell–Yan production, including the photon-initiated channel.
1 Introduction
The precision physics program at the Large Hadron Collider (LHC) aims to observe processes at an unprecedented level of accuracy and experimental sensitivity. As part of these efforts, the analyses conducted by the LHC experimental collaborations are increasingly undertaken with theoretical cross section predictions at next-to-next-to-leading order (NNLO) in QCD, which includes \({\mathcal {O}}(\alpha _S^2)\) corrections. At this level of precision, it is expected that electroweak (EW) corrections, including those with photon–initiated (PI) processes, will begin to have observable effects as \(\alpha _\mathrm{QED} \sim \alpha _{S}^2\) at the typical scales being probed at the LHC. These should therefore be incorporated in theoretical predictions. In particular, electroweak corrected partonic cross sections should be calculated with corresponding Parton Distribution Functions (PDFs) produced at NLO and NNLO in QCD and the appropriate order in QED. This is achieved primarily by modifying the DGLAP [1, 2, 3, 4, 5] factorisation scale evolution of the PDFs to include QED parton splittings. The most significant effect of this change is the necessary inclusion of the photon as a constituent parton of the proton. Subsequently one can also begin to calculate the effect of PI sub-processes as corrections to the leading QCD cross section for processes such as Drell–Yan [6], EW boson–boson scattering [7] and Higgs production with an associated EW boson [8], which are expected to be sensitive to these effects. In a different context, semi-exclusive [9] and exclusive production of states with EW couplings are also related to the photon content of the proton, albeit not directly to the inclusive photon PDF. Here, PI processes play an important role, see e.g. [10, 11] for recent studies in the context of compressed SUSY scenarios.
MRST provided the first such publicly available QED set [12], modelling the photon at the input scale as arising radiatively from the quarks (and their respective charges) below input, with DGLAP splitting kernels at \({\mathcal {O}}(\alpha )\) in QED. Other such sets were subsequently developed that either adopted similar phenomenological models [13], or sought to constrain the photon in an analogous way to other partons by fits to Drell–Yan data [14, 15], first developed by the NNPDF Collaboration. These early sets saw relatively large discrepancies between photon PDFs. Large modelling uncertainties persisted due to the freedom in the choice of scale above which photons are produced radiatively, modelled in the MRST set as the difference between the current and constituent quark masses, while the approach taken by the CTEQ14QED set [13] was to attempt to fit a parameterisation based on the total momentum carried by the photon from \(ep\rightarrow e\gamma + X\) data. In the case of NNPDF2.3QED [14], the constraints available directly from data were rather weak, due to the small size of the PI contributions. This lead to large photon PDF errors, with a \({\mathcal {O}}(100\%)\) uncertainty at high x. In all cases the available data was unable to constrain the photon to a high degree of accuracy.
A final significant drawback of these early sets was that the majority did not account for the contribution to the photon PDF from elastic scattering, in which the proton coherently emits electromagnetic radiation without disintegration, in contrast to photon contributions previously accounted for from inelastic scattering processes, assumed to arise from quark splittings. This distinction between the elastic and inelastic photon emission was one that was seldom systematically treated, if considered at all.
Significant strides have been made in recent years to overcome these deficiencies. First, more accurate determinations of the photon distribution at input have been developed by making use of the experimentally well determined elastic form factors of the proton, as in [16] and further developed in [9, 17]. More precisely, the photon PDF corresponds to the flux of emitted photons within the context of the equivalent photon approximation, and as discussed in some of the early work on this [18], the contributions from elastic and inelastic emission to the photon PDF are directly related to the corresponding structure functions (\(F_{1,2}^{ el}\), \(F_{1,2}^{inel}\)) probed in lepton–proton scattering. This idea has been revived in various works over the previous decades [19, 20, 21, 22], and has most recently been demonstrated within a rigorous and precise theoretical framework by the LUXqed group [23, 24], where the first publicly available photon PDF applying this approach was also provided.
As the elastic and inelastic proton structure functions have been determined experimentally to high precision, this has in turn allowed for the determination of the elastic and inelastic contributions to the photon to the level of a few percent. In addition to these developments, QED DGLAP splitting kernels have now been calculated to \({\mathcal {O}}(\alpha \alpha _S)\) [25] and \({\mathcal {O}}(\alpha ^2)\) [26], whose effects, as shown in Sect. 2.3, are not insignificant to the evolution of the photon and other partons. In light of this, a greater confidence may be had regarding the effects of QED modified partons and their impact on cross section calculations. One PDF set including a photon distribution which is based on the LUXqed approach has recently been produced by the NNPDF group [38].
In this paper, we outline the efforts undertaken by the MMHT group to develop a fully consistent set of QED partons, adopting the LUXqed formulation at input scale \(Q_0\) for the photon. QED splitting kernels to \({\mathcal {O}}(\alpha )\), \({\mathcal {O}}(\alpha _S)\) and \({\mathcal {O}}(\alpha ^2)\) are incorporated into the DGLAP evolution and the effect of this is explored. Furthermore, we also adopt a model for higher–twist (HT) effects in the quarks at low \(Q^2\), as the evolution of the photon PDF is sensitive to these corrections, due to a lower input scale used in comparison to that of other PDF sets.
As well as the conventional set of QED altered PDFs, we provide grids for the photon PDF separated into its elastic and inelastic components, as well as a consistent set of QED corrected neutron PDFs. Although the phenomenological implications of a neutron set are limited, their production is necessary for a consistent fit to deuteron and nuclear fixed target data from neutrino (\(\nu \)N) DIS scattering experiments used to constrain the PDFs. The QED corrected neutron PDFs of MRST [12] provided isospin violating partons, with \(u_{(p)} \ne d_{(n)}\), and these were seen to reduce the NuTeV \(\sin ^2\theta _W\) anomaly [27]. The breaking of isospin symmetry may also have implications for the development of nuclear PDFs, and our current treatment develops this earlier approach, providing new predictions for the magnitude of isospin violation.
Finally, we will explore the phenomenological consequences of this set, demonstrating the effects of QED incorporation on \(F_2(x,Q^2)\) as calculated from PDFs, the partonic luminosities as a function of centre-of-mass (CoM) energy and the change in fit quality after refitting the partons with QED. We also explore the consequences of fitting to ATLAS high-mass Drell–Yan data [28], with both QED effects and PI corrections to the cross section produced by our set. We find that the effect of a fully coupled QED DGLAP evolution is non-negligible on the gluon and quark PDFs.
2 Including QED effects in the MMHT framework
In this section we describe how the MMHT framework has been modified to incorporate the QED splitting kernels in DGLAP evolution and the form we take for the input distribution of the photon, and discuss their effect on the final set of partons and the corresponding PDF uncertainties.
2.1 Baseline QCD fit
Throughout this paper, in order to meaningfully interpret the effects of including QED effects, we will compare the new partons to a baseline set of PDFs evolved and fit solely with QCD kernels (at, unless explicitly stated, NNLO). However, this set differs from the most recent public release of partons, MMHT2014 [29]. In particular, this more closely corresponds to the set described in [30], where the HERA Run I + II combined cross section data [31] have been included in the fit. Furthermore, we now include some additional data on \(t{\bar{t}}\) production (\(\sigma (t{\bar{t}})\)) from the ATLAS and CMS collaborations. In addition, further small amendments have been made to the NLO and NNLO QCD kernels in the evolution, as detailed in Sect. 2.3. Hence, we refer to this as the MMHT2015 PDF set and the PDFs with the QED effects included as MMHT2015qed.
2.2 Input photon distribution
Turning to \(F_{2, L}^{( inel)}\), this displays two distinct modes of behaviour. For the continuum \(W^2 \gtrsim 4\) \(\hbox {GeV}^{2}\) region, the \(x, Q^2\) dependence of \(F_{2,L}\) is seen to be relatively smooth, while in the resonance \(W^2 \lesssim 3\) \(\hbox {GeV}^{2}\) region, various Breit-Wigner type resonances contribute, due to the presence of hadronic excited states such as the \(\Delta \) and associated modes. To describe both of these regions, two different fits are used above and below a threshold of \(W_{\mathrm{cut}}^2 = 3.5\) \(\hbox {GeV}^{2}\). For the continuum (\(W^2 \ge W_{\mathrm{cut}}^2\)) region, we use the HERMES GD11-P [34] fit, while for the resonance (\(W^2 < W_{\mathrm{cut}}^2\)) region we take a fit to data from the CLAS collaboration [35].
The structure functions themselves exhibit enhanced sensitivity to particular effects at lower starting scales (1 GeV in the MMHT framework, in comparison to 10 GeV adopted by LUXqed) such as proton mass corrections \({\mathcal {O}}(m_p^2/Q^2)\) and higher twist terms. Hence, modifications are made to account for these during the evolution, as discussed in Sect. 2.3.
2.3 Modifications to DGLAP evolution
2.3.1 PDF basis
However, we note that the LUXqed PDF set [24] does include this contribution in the DGLAP evolution used to develop their \(x\gamma (x,Q^2)\). Since the right hand side of Eq. 24 is a \(\delta (1-x)\) term multiplied by a negative coefficient, the extra contributions from the lepton splitting terms in DGLAP are anticipated to slightly reduce the magnitude of a photon whose evolution accounts for them (as one anticipates from the process \(\gamma \rightarrow l{\bar{l}}\)).
Note that this sum over valence-like non-singlet distributions corresponded to Eq. (18) in the original MMHT framework, which neglected the strange, charm and bottom distributions due to their small relative size. With the release of the set described in this paper, the contribution from these off-diagonal splittings for all flavours are now included, which represent minor changes, \({\mathcal {O}}(10^{-5})\), in a like-for-like comparison with the original MMHT partons purely in QCD.
2.3.2 Target mass and higher twist corrections
As previously noted, MMHT2015qed differs in its production of a photon PDF from other contemporary sets in adopting a straightforward evolution in \(Q^2\) space, from a starting scale of \(Q_0 = 1\) GeV. However, at low scales such as these, target mass corrections, which account for the finite mass of the proton, and higher twist terms have non-negligible contributions to \(F_{2,L}\). Above \(Q_0\), the \(F_2^{(inel)}\) contributions to \(\gamma ^{(inel)}\), as in Eq. (4), are modelled by the parton splittings in DGLAP, which require some modification to capture the relevant behaviour at high x.
This is of interest because the parameter \(A'_2\) is not well determined, and in [41], is fit loosely to structure function data to yield a value of \(A'_2 = -0.2\) \(\hbox {GeV}^{2}\). As discussed above, data sensitive to renormalon contributions are typically excluded in global fits, to remove any sensitivity to such non–perturbative effects. In particular, in MMHT kinematic cuts of \(W^2 > 15\) \(\hbox {GeV}^{2}\) (and \(W^2 > 20\) \(\hbox {GeV}^{2}\) at LO) are taken, while for those data sets relating to \(\nu ({\bar{\nu }})N\) experiments to measure \(xF_3\) a more stringent cut of \(W^2 > 25\) GeV is imposed [45]. However, with the aim of determining a more precise value of \(A'_2\) we have relaxed these constraints, lowering the threshold to \(W^2 > 5\) \(\hbox {GeV}^{2}\) and modifying \(F_{2}\) and \(F_{3}\) to include the relevant renormalon contributions as in [41], i.e. with modifications of the form shown in Eq. (30).
In Fig. 2 the fit quality for different values of \(A'_2\) is shown. We find that \(A'_2 = -(0.3\pm 0.1)\) \(\hbox {GeV}^{2}\), with uncertainties determined from a generous \(\Delta \chi ^2 = \pm 10\) variation in the fit (to one significant figure). This is motivated by the dynamical tolerance scheme used in our framework, as outlined in Sect. 6 of [45], where it was found that in order to provide reasonable uncertainties when fitting to many disparate data sets in tension with one another, one typically requires tolerances \(T = \sqrt{\Delta \chi ^2_{global}} \sim 3\) rather than the \(T = 1\) one would obtain from a standard ‘parameter-fitting’ criterion. We note this choice also corresponds to the fixed tolerance uncertainty schemes adopted by early CTEQ sets [46]. The uncertainty on this is then propagated as an independent source of uncertainty for the photon, as discussed in Sect. 4.3. This represents a slightly larger renormalon contribution than predicted from [41], though the data are unable to provide significant constraints in either case.
As seen in Fig. 3, the target mass corrections lead to a \(\sim 3\%\) increase in the photon at high x, while the renormalon contributions, which provide an increasingly positive contribution to \(F_2\) at high x, correspondingly enhance the photon at moderate to high x. Note that the turn around in both figures at \(x \simeq 0.5\) occurs due to the previously mentioned effective kinematic cut on all photon contributions at high x and low \(Q^2\). This cut itself is also a function of the proton mass \(m_p\), though for our purposes we consider the kinematic cut imposed due to the target mass (i.e. the cut in x) as independent from the term introduced in the evolution and it is seen that the two have opposite effects on the high x photon, with the kinematic cut ultimately dominating and the effect of the corrections to the splitting function and the renormalon contribution being suppressed as \(x \rightarrow 1\).
2.4 Separation of elastic and inelastic components
As noted in Sect. 2.2, the photon PDF actually comprises of two component distributions, \(\gamma (x,Q^2) = \gamma ^{(el)}(x,Q^2) + \gamma ^{(inel)}(x,Q^2)\), which represent photon contributions from elastic and inelastic proton scattering events, respectively. Separating \(\gamma ^{(el)}\) and \(\gamma ^{(inel)}\) from one another while consistently performing the evolution for all the partons required certain changes to be made from the standard procedure for performing DGLAP, due to the fact that the generation of \(\gamma ^{(el)}\) in the evolution is independent of parton splittings, as detailed below.
Although the provisions outlined above are needed for the evolutions of \(\gamma ^{(el)}\) and \(\gamma ^{(inel)}\), i.e. those contributions from splitting functions of the form \(P_{\gamma \{q,{\bar{q}},g,\gamma \}}\), the treatment for the rest of the partons remains broadly unchanged. Since the quark, antiquark and gluon contributions from \(P_{\{q,{\bar{q}},g,\gamma \}\gamma }\) splittings do not distinguish between \(\gamma ^{(el)}\) and \(\gamma ^{(inel)}\), the entire photon contribution, \(\gamma (x,Q^2) = \gamma ^{(el)}(x,Q^2) + \gamma ^{(inel)}(x,Q^2)\), is passed to the relevant splitting kernels during evolution.
2.5 Momentum conservation
The inclusion of the photon PDF requires that the photon be included in the momentum sum rule (14), naturally leading to a redistribution of momentum in the other partons in order to obey Eq. (14) at input. However, due to the procedure adopted for the inclusion of \(\gamma ^{(el)}\), outlined in the previous section, as well as higher twist terms, this equation is not strictly obeyed during the evolution. This reflects the discrepancy between effects of non-perturbative corrections, such as that of target masses, and the parton model. In this section we outline the consequences of such changes.
In particular, Fig. 7 (right) indicates that the reduction to the total momentum carried by the photon is, as anticipated, most strongly affected by the kinematic cut at low scales until \(Q^2\sim 10\) \(\hbox {GeV}^{2}\) (with total changes of less than 1%). Since the overall momentum carried by the photon is small, \(\sim 2-3\times 10^{-3}\), at low scales where momentum violating effects are most prevalent, this leads to the minuscule amount of change observed in the total momentum of the partons.
We now discuss other effects during the evolution which contribute to violation of the momentum sum rule. The momentum sum rule is constrained to be obeyed by all the partons at the input scale, and both the inelastic and elastic photons are considered when imposing the momentum sum rule for the parameterisation of the quarks, as in Eq. (14). However, above the input scale the contribution to \(\gamma ^{(el)}\) that comes from the second term in (33), that is due to elastic photon emission, will lead to some momentum sum rule violation, as this contribution does not originate from standard DGLAP evolution, and is not balanced by a corresponding loss of quark and antiquark momentum, i.e., any \(\gamma \) contribution from the quarks during evolution, e.g. \(q \rightarrow q+\gamma \) is absorbed into the definition of \(\gamma ^{(inel)}\). (\(\gamma ^{(el)}\) is not entirely decoupled from the evolution of the quarks, since \(\gamma ^{(el)} \rightarrow q\bar{q}\) splitting are still permissible.) In practice, this effect is negligible, with momentum violating effects of \(O(10^{-4})\) observed in the sum rule during evolution, and in fact stabilises at higher \(Q^2\) where the elastic contribution is less significant.
Likewise, other higher twist terms included in the evolution for the purposes of QED lead to small amounts of momentum violation. Since the quark distributions, \(q_i(x,Q^2)\), passed to both \(P_{q,q}^{(0,1)}(x)\) and \(P_{\gamma ,q}^{(0,1)}(x)\) differ due to the inclusion of renormalon corrections for the latter but not the former, this aspect of the evolution also invalidates momentum violation to a small degree, also shown in Fig. 8, creating a small amount of violation of \({\mathcal {O}}(2\times 10^{-5})\).
3 QED neutron PDFs
3.1 Modified DGLAP evolution
For the valence distributions, in primary, the magnitude of isospin violation is seen to be a few percent, becoming significant especially at low and high x, where all distributions tend towards 0, as shown in Fig. 9. Of note is the fact that the discrepancy between the predicted ratio of valence quarks and the naïve isospin assumption remains at the \(\sim 1\%\) level, even for the peak of the valence distributions (at \(x\sim \frac{1}{3}\), \(x\sim \frac{2}{3})\)). This effect is seen to increase during the evolution, with differences of \(\sim 5\%\) at \(Q = 100\) \(\hbox {GeV}^{2}\).
Finally, although the primary interest in this paper for the development of QED corrected neutron PDFs is to provide a manner of relating the PDFs to deuterium scattering experiments used to constrain the partons, we also wish to highlight the potential relevance of this set in the determination of nuclear PDFs. In particular, the assumption made in modern determinations of nuclear PDFs (such as those of EPPS [47] and nCTEQ [48]) is to fit to data with the assumption that the u and d quark type distributions in the neutron and proton are related to one another by isospin symmetry. With the development of this set, we propose that this assumption need not be applied strictly and that with the introduction of QED effects, the small amounts of isospin violation shown in Fig. 9 may be of relevance when the determination of nuclear PDFs reach the \({\mathcal {O}}(5\%)\) level. While current determinations do not reach this level of precision, a QED corrected relationship between proton and neutron PDFs may provide better fits to the available data, and is of interest given that recent work has begun to adopt quark flavour dependence in fits [47].
3.2 The photon PDF of the neutron
As in the case of the proton, there is also a corresponding photon PDF of the neutron, \(\gamma _{(n)}(x,Q^2)\), which should in general be included. At input, the expression for this is adapted from that of the proton, Eq. (4), with the proton mass replaced by that of the neutron and the relevant form factors substituted or approximated in the manner discussed below.
4 Results
4.1 Changes to PDFs due to QED corrections
Here we show the changes in the parton distributions that are produced as a result of the changes given in the preceding sections. We include the \({\mathcal {O}}(\alpha )\), \({\mathcal {O}}(\alpha \alpha _S)\) and \({\mathcal {O}}(\alpha ^2)\) QED corrections, unless otherwise stated, and compare against the baseline PDF set without QED effects described in Sect. 2.1.
In Fig. 15 (right) we show the effect on the quarks of refitting. We can see that the exaggerated effects of the evolution at low x are compensated by the other parameters of the gluon, as discussed above. On the other hand, the behaviour of the partons at high x, which shows a small reduction in the singlet distributions are a genuine effect due to the inclusion of the QED contribution to \(P_{qq}\). In particular, this reduction is primarily a natural consequence of the \(q\rightarrow q+\gamma \) emission, which at high x has the effect of reducing the quark singlet momenta, with corresponding increases in \(x\gamma (x,Q^2)\).
We note that although the s distribution experiences a larger magnitude of change due to QED than that of the other partons, this effect is a consequence of the \(s+{\bar{s}}\) distribution being less well constrained by the data, and therefore more sensitive to the effects of refitting, rather than having an enhanced sensitivity to the effects of QED.
In Figs. 16, 17 and 18 we show the ratio of the PDFs with and without QED effects, including the corresponding PDF uncertainties. We can see that upon refitting the singlet (\(q+{\bar{q}}\)) and gluon PDFs all lie within the PDF uncertainties of the pure QCD fit, with the central values and uncertainties remaining only modestly affected, with \({\mathcal {O}}(2\%)\) reduction for the \(s+{\bar{s}}\) distribution, (with a slight increase in the reduction at high x, due to the effect of QED splittings mentioned above). The up valence quark, \(u_V\) and to a lesser extent the down valence quark \(d_V\), are most sensitive to QED effects, with a \({\mathcal {O}}(2-5\%)\) change at low x in their central values, though this is relatively marginal given the large uncertainties (\(\sim 20\%\)) in the valence quark PDFs in this region.
In Fig. 19 we see the details of the momentum carried by each of the partons as a function of \(Q^2\) for both the proton and neutron. At input the fractional momentum carried by the photon in the proton is 0.00196, and this increases to about 0.007 at very high \(Q^2\). In the neutron the input figure is much smaller, i.e. 0.0003, but the rate of increase at higher \(Q^2\) is comparable to the proton, though a little lower due to the dominant radiation at high x being from down quarks rather than up quarks.
In Fig. 20 we show the effect of the higher and mixed order corrections to the evolution on \(x\gamma (x,Q^2)\). We can see that the \({\mathcal {O}}(\alpha \alpha _S)\) and \({\mathcal {O}}(\alpha ^2)\) kernels are seen to reduce the photon distribution by \(\sim 1-3\%\), particularly at high x. The effect induced by the \({\mathcal {O}}(\alpha ^2)\) kernels is of \({\mathcal {O}}(0.5-1\%)\), and further changes associated with the exclusion of yet higher orders in perturbation theory are expected to be even smaller. Since other sources of uncertainty, discussed in Sect. 4.3 are somewhat larger, it is not thought that such scale uncertainties will be significant for the photon at the level of accuracy being discussed in this paper.
4.2 Results of global fits with QED corrections
4.2.1 The quality of the global fits
The total \(\chi ^2\) for partons with the effects of QED, both prior to and after refitting the parton parameters, at NLO and NNLO. Before the fit, the parameters derived from the QCD fits described in Sect. 2.1 are used. The NLO fit contains 3609 data points, while the NNLO contains a total of 3276 (since the later omit some jet data). The numbers in brackets show the change in \(\chi ^2\) due to the inclusion of the QED corrections
Change in \(\chi ^2\) due to QED evolution compared to MMHT14+HERA I+II | |||
---|---|---|---|
NLO before fit | NLO after fit | NNLO before fit | NNLO after fit |
4180 (+41) | 4151 (+12) | 3574 (+42) | 3539 (+7) |
The \(\chi ^2\) breakdown showing \(\chi ^2/N_{pts}\) by data set for NNLO QCD and NNLO QCD + QED PDF fits
Data set | \(\chi ^2/N_{pts}\) NNLO after fit (QCD) | \(\chi ^2/N_{pts}\) NNLO after fit(QCD+QED) |
---|---|---|
BCDMS \(\mu p\) \(F_2\) [52] | 178/163 | 182/163 (+4) |
BCDMS \(\mu d\) \(F_2\) [53] | 142/151 | 144/151 (+2) |
NMC \(\mu p\) \(F_2\) [54] | 124/123 | 125/123 |
NMC \(\mu d\) \(F_2\) [54] | 108/123 | 108/123 |
NMC \(\mu n/\mu p\) \(F_2\) [55] | 128/148 | 127/148 |
E665 \(\mu p\) \(F_2\) [56] | 65/53 | 65/53 |
E665 \(\mu d\) \(F_2\) [56] | 61/53 | 61/53 |
31/37 | 31/37 | |
26/38 | 25/38 | |
66/57 | 66/57 | |
E866/NuSea pp DY [62] | 224/184 | 223/184 |
E866/NuSea pd / pp DY [63] | 11/15 | 11/15 |
NuTeV \(\nu N\) \(F_2\) [64] | 37/53 | 36/53 (-1) |
CHORUS \(\nu N\) \(F_2\) [44] | 29/42 | 29/42 |
NuTeV \(\nu N\) \(xF_3\) [64] | 31/42 | 31/42 |
CHORUS \(\nu N\) \(xF_3\) [44] | 19/28 | 19/28 |
CCFR \(\nu N \rightarrow \mu \mu X\) [65] | 77/86 | 78/86 |
NuTeV \(\nu N \rightarrow \mu \mu X\) [65] | 42/40 | 41/40 |
HERA I+II CC \(e^+ p\) [31] | 52/39 | 52/39 |
HERA I+II CC \(e^- p\) [31] | 63/42 | 65/42 (+2) |
HERA I+II NC \(e^+p\) 920 GeV [31] | 510/402 | 510/402 |
HERA I+II NC \(e^-p\) 920 GeV [31] | 239/159 | 240/159 (+1) |
HERA I+II NC \(e^+p\) 820 GeV [31] | 88/75 | 88/75 |
HERA I+II NC \(e^-p\) 575 GeV [31] | 261/259 | 262/259 |
HERA I+II NC \(e^-p\) 460 GeV [31] | 246/209 | 246/209 |
HERA ep \(F_2^{charm}\) [66] | 80/52 | 80/52 |
DØ II \(p{\bar{p}}\) incl. jets [67] | 117/110 | 117/110 |
CDF II \(p{\bar{p}}\) incl. jets [68] | 60/76 | 60/76 |
CDF II W asm. [69] | 16/13 | 15/13 |
DØ II \(W \rightarrow \nu e\) asym. [70] | 31/12 | 30/12 |
DØ II \(W \rightarrow \nu \mu \) asym. [71] | 16/10 | 16/10 |
DØ II Z rap. [72] | 17/28 | 17/28 |
CDF Z rap. [73] | 40/28 | 40/28 |
ATLAS \(W^+\), \(W^-\), Z [74] | 41/30 | 41/30 |
CMS W asymm \(p_T > 35\) GeV [75] | 7/11 | 7/11 |
CMS asymm \(p_T>\) 25 GeV, 30 GeV [76] | 8/24 | 8/24 |
LHCb \(Z\rightarrow e^+ e^-\) [77] | 22/9 | 22/9 |
LHCb W asymm \(p_T > 20\) GeV [78] | 14/10 | 13/10 |
CMS \(Z\rightarrow e^+ e^-\) [79] | 23/35 | 22/35 |
ATLAS high-mass Drell–Yan [6] | 17/13 | 18/13 |
CMS double diff. Drell–Yan [80] | 152/132 | 152/132 |
Tevatron, ATLAS, CMS \(\sigma _{t{\bar{t}}}\)* [81, 82, 83, 84, 85, 86, 87] | 14/18 | 14/18 |
All data | 3532/3276 | 3539/3276 (+7) |
4.2.2 The photon PDF of the proton
Another reason why we anticipate that the \(x\gamma (x,Q^2)\) as outlined in this work may be somewhat greater in value, in an intermediate range in x, compared to that of LUXqed is due to the exclusion of lepton splitting contributions in our DGLAP evolution, which are included in the evolution used to develop the LUXqed set. In Sect. 2.3 we explicitly neglected the sum over lepton charges in Eq. 25. In general, since \(\gamma \rightarrow l{\bar{l}}\) splittings should reduce the photon distribution (nearly uniformly since it occurs as a coefficient to \(\delta (1-x)\) in \(P_{\gamma \gamma }\)), one expects that excluding this term should lead to a somewhat increased photon. To estimate the effect of including this term, in Fig. 24 we draw a comparison to \(x\gamma (x,Q^2)\) evolved with \({\mathcal {O}}(\alpha )\) lepton splittings included in evolution and as anticipated find that this term does lead to a \({\mathcal {O}}(1-2\%)\) reduction, which becomes more pronounced at higher \(Q^2\). Along with the ratio of the charged singlet used in the evolution, neglecting lepton splittings^{1} leads to an independent source of enhancement for our \(x\gamma (x,Q^2)\), further accounting for the difference seen in Fig. 22.
Common to all the sets are errors of \({\mathcal {O}}(1\%)\), displaying the remarkable improvements in accuracy seen in photon PDFs developed on the strategy outlined in this paper and that of [17] and [23], in comparison to that of older sets. A full breakdown of the contributing sources of error are explored in Sect. 4.3.
4.2.3 QED corrected structure functions
In Fig. 25 we show the effect of these changes with and without refitting. Again, the sensitivity introduced by the gluon parameterisation is seen to have an effect at low x, reducing \(F_{2,3}\) somewhat, while after fitting, the CC structure functions \(F_{2,3}\) are moderately decreased at low x. In the NC case however, \(F_2\) is generally reduced by \({\mathcal {O}}(0.5\%)\), as anticipated by the fact that the introduction of QED in the evolution is seen in general to diminish the quark singlet content, see Fig. 15.
4.2.4 Effects of QED on \(\alpha _S\) determination in the global PDF fit
In addition to the fit described above, we have also performed a simultaneous fit to the strong coupling , \(\alpha _S(M_Z)\). The value typically used during the evolution and the comparison to data is taken as a fixed value \(\alpha _S(M_Z) = 0.118\), which reflects a combination of both the best fit value exclusively from our fit to data, and the independent inclusion of the world average of \(\alpha _S(M_Z) = 0.1181 \pm 0.0011\) [89], as discussed in Sect. 5.1 of [29].
In principle, one might expect that the value of \(\alpha _S(M_Z)\) found after refitting with the effects of QED included will be somewhat less than that in a pure QCD fit. This is because at leading order, the effect on the \(q+{\bar{q}}\) distributions during the evolution, particularly at high x, is due to gluon emission, \(q\rightarrow q g\), which leads to a slight reduction of the singlet. In a pure QCD fit, the parameters that provide the best fit are a combination of both the input distribution and a value of \(\alpha _S(M_Z)\) which drives gluon emission at a rate (determined by \(P_{qq}^{(QCD)}\)) in the evolution such that the PDFs at higher scales are best fit to the data.
This was also investigated in the development of the original MRST QED set [12], where it was found that despite the above considerations, between the pure QCD and QCD+QED fit, \(\alpha _S(M_Z)\) remained essentially unchanged. The reason found for this was that the fit (especially the NMC and HERA data) preferred a larger value for the gluon at low x, which is sensitive to \(\alpha _S(M_Z)\) since \(d F_2/d \ln Q^2 \propto \alpha _S P_{qg}\otimes g(Q^2)\). However, the momentum carried by the photon detracts from that carried by the small-x gluon and as a result, the change to the gluon at small x has a tendency to require a larger value of \(\alpha _S(M_Z)\) than would otherwise be obtained. This pulls in a direction opposite to the reduction of \(\alpha _S(M_Z)\) as described above, and reduces the magnitude by which one might anticipate a change after refitting with the effects of QED.
With the updated QED parton framework, we find that \(\alpha _S(M_Z)\) experiences a reduction from 0.1181 in the pure NNLO QCD case to 0.1180 in the fit with QED, while at NLO the result is unchanged within the numerical precision of the fit. Although at NNLO this does represent a small reduction, in neither case is allowing \(\alpha _S\) to be free seen to improve the total fits by any significant degree, with \(\Delta \chi ^2 < 1\). However, in future global fits, the inclusion of QED effects in the partons may come to be significant as the accuracy of such measurements are improved.
4.2.5 Photon-photon luminosity
A sense of the relevance of the photon PDF to particle production at colliders such as the LHC may be determined from an inspection of the \(\gamma \gamma \) luminosity expected at these energies (14 TeV), shown in Fig. 26. As seen in Fig. 22, our photon and that of other sets based on the LUXqed formulation show good agreement, and therefore our predicted \(\gamma \gamma \) luminosity, \(\mathrm{d} L_{\gamma \gamma }/\mathrm{d}\ln M^2\), bears a strong resemblance to others in the literature (see e.g. Fig. 19 in [24]). Also shown in Fig. 27 is the expected luminosity for a High-Energy LHC proposal with (CoM) energy \(\sqrt{s} = \)27 TeV, and a Future Circular Collider with \(\sqrt{s} = \)100 TeV, where the total \(\gamma \gamma \) luminosity is comparable to that of \(\Sigma _i (q_i{\bar{q}}_i+{\bar{q}}_iq_i)\) at present LHC CoM energies (14 TeV).
The cross section for this CEP process can be calculated within the so–called Equivalent Photon Approximation [18], in which the photon flux associated with the colliding beam of charged particles may be expressed in terms of the elastic structure functions \(F_{2,L}^{(el)}\), in a manner similar to that considered in this paper. The \(\gamma ^{(el)}\gamma ^{(el)}\) luminosity, represented in Fig. 26, corresponds to precisely the luminosity that could be delivered in this approach.
However, this interpretation must be qualified with an important caveat, which is that for an exclusive production process, where both protons remain intact after scattering, one needs to multiply the final result obtained from the naïve use of \(\gamma ^{(el)}\) as an incoming parton by a ‘soft survival’ factor, corresponding to the probability of no additional particle production due to multi-particle interactions (MPI) [94]. Furthermore, the luminosities shown in Fig. 26 can not directly be applied to the calculation of cross sections for more exclusive final states, such as when explicit cuts are placed on the presence of additional tracks within the central portion of the detector, but require suitable modification as in [9].
4.3 Uncertainties on the photon PDF
Our treatment of the contributions to the photon PDF uncertainty are in some cases identical to LUXqed. However, as discussed in Sect. 2.3, due to the lower starting scale adopted in our evolution procedure, we also include higher twist corrections in the form of a renormalon model, for which the undetermined coefficient \(A'_2\) in Eq. (30) is fit to the data, introducing an independent source of uncertainty.
Elastic: The uncertainty contributions from the A1 fit for \(F_2^{(el)}\) are twofold. In particular, the fits provided by A1 are given in the unpolarized and polarised forms, where the latter accounts for potential two photon exchange (TPE) processes between the lepton probe and the proton in DIS experiments. Following the approach of LUXqed, we use the latter for our estimate precisely because it provides constraints on TPE. As well as the intrinsic uncertainty provided by the A1 collaboration for this fit \(\delta (F_2^{(el)})_a\), similarly to LUX, we adopt the symmetrised difference between the polarised and unpolarized fit as an independent source of error, \(\delta (F_2^{(el)})_b\). The total uncertainty on \(F_2^{(el)}\) is then simply given by the sum of these two contributions in quadrature.
- R: The contributions from \(F_{L}\) are modelled in precisely the same manner as that of LUXqed, using the parameterisation of the form:where \(R_{L/T} = \sigma _L(x,Q^2)/\sigma _T(x,Q^2)\) represents the ratio between the absorption cross sections for longitudinal and transversely polarised photons. Our expression for this ratio is provided by the LUXqed group, who, following the procedure used by the HERMES collaboration [35], in turn adapt the expression from the \(R_{1998}\) fit [36] provided by the E143 Collaboration for use in low \(Q^2\) regions and assign it a conservative \(\pm 50\%\) uncertainty, which we also adopt.$$\begin{aligned} F_L(x,Q^2)\, {=}\,F_2(x,Q^2)\Big (1{+}\frac{4m_p^2x^2}{Q^2}\Big )\frac{R_{L/T}(x,Q^2)}{1{+}R_{L/T}(x,Q^2)},\nonumber \\ \end{aligned}$$(62)
\(\mathbf{W }^{{\mathbf{2 }}}\): As discussed in Sect. 2.2, two distinct fits for \(F_2^{(inel)}\) are used above (HERMES [35]) and below (CLAS [34] and Cristy-Bosted [95]) a threshold of \(W_\mathrm{cut}^2 = 3.5\) \(\hbox {GeV}^{2}\). Since \(W_{\mathrm{cut}}^2\) is defined somewhat arbitrarily and theoretically induces some small amount of discontinuity in the contributions to \(\gamma ^{(inel)}\), we treat the cut value as an independent source of uncertainty, varying it the region \(3< W_{\mathrm{cut}}^2 < 4\) \(\hbox {GeV}^2\). Even with this relatively conservative approach, the uncertainty on \(W_\mathrm{cut}^2\) is seen to be vastly dominated by other sources.
Resonance: The uncertainty of \(F_2^{(inel)}\) in the resonance region is taken as the symmetrised difference between the CLAS fit, which is used as the standard for our input, and that of the Cristy-Bosted, similar to the procedure used by LUXqed.
Continuum: The uncertainty of \(F_2^{(inel)}\) in the continuum region is adapted directly from the uncertainty bands of the GDP-11 fit provided by the HERMES collaboration. This is a different type of uncertainty estimate from this source as that adopted by LUXqed, who vary the scale at which \(F_2\) goes from being described by the GDP-11 fit to calculated in terms of the PDFs. However, each estimation of uncertainty is very small.
Renormalon: For the fitting and uncertainty of the coefficient \(A'_2\) in Eq. (30), we implemented the original renormalon model of [41] into the calculation of the structure functions themselves, \(F_{2,3}\), as used in the fit . \(A'_2\) was then varied to induce a \(\Delta \chi ^2 = \pm 10\) change in the overall fit quality of the partons (as seen in Fig. 2 in Sect. 2.3), creating a generous uncertainty band of \(-0.4< A'_2 < -0.2\), with a best fit value of -0.3. We note that our global fit to the data favours a renormalon contribution \(\sim 50\%\) greater than the value used in the original model by Dasgupta and Webber [41]. At high x, this is seen to be a comparable source of uncertainty with that of \(\delta (F_2^{(el)})\). Unlike all other terms discussed so far, the uncertainty in \(A'_2\) enters during the evolution, rather than at input.
PDFs: Above the input scale \(Q_0^2 = 1\) \(\hbox {GeV}^{2}\) the \(\gamma ^{(inel)}\) contributions are modelled solely from the splittings of other partons during the DGLAP evolution. Hence, the intrinsic uncertainty on the other PDFs propagate into the form of the photon PDF as it evolves. This reflects the standard 50 eigenvector uncertainties associated with the fit of the free parameters in the MMHT parameterisation (see Eqs. (11) and (13)), which generate the uncertainty bands for all flavours of parton (\(q, {\bar{q}}, g\)), naturally generating uncertainties in the photon during splittings of the form \(q\rightarrow q\gamma \) and \(g\rightarrow q{\bar{q}}\gamma \). At low x, as is the case of LUXqed, this dominates as the primary source of uncertainty.
It is noted that we do not account for the uncertainty that arises from the Higher Order (HO) terms missing from the QCD components of the evolution, as estimated in LUXqed. Although we have given an indication of the magnitude of the change in order from QCD (from NLO to NNLO) in the evolution in Fig. 21 of the previous section (which broadly corresponds to the (HO) band in Fig. 15 of [24]), we do not treat this difference as an independent source of uncertainty, since PDFs have typically been provided at both NLO and NNLO in QCD, each with independently derived uncertainty bands. Despite not being included as a default, recent work [96] has begun to explore the possibility of incorporating such uncertainties into the PDF fitting framework of MMHT in a standard manner.
Overall, we note the similarity between the form of our uncertainty with others, being less than 2% for \(10^{-5}< x < 0.5\), demonstrating a drastic improvement with early photon PDF sets such as MRST2004QED [12] and NNPDF2.3 [14].
We provide the photon PDF along with the quark, antiquark and gluon PDFs in grids which also contain all information about the uncertainties. PDF sets are typically provided as grids in the LHAPDF6 format, with each grid representing either the central value of the PDFs, or the PDFs at a given ± eigenvector direction in the independent parameter space PDFs. As noted above, as well as the uncertainties that are routinely given in such sets associated with the non-photon PDF parameters, the set that is produced as a result of the work described here now contains uncertainties associated with the photon parameters at input and the \(A'_2\) parameter for the renormalon in the evolution. The grids will be discussed in more detail in the Appendix.
5 High mass Drell–Yan
5.1 QED and photon PDF sensitivity in high mass Drell–Yan
In order to explore the phenomenological implications of our photon PDF set, we calculate the effects on the double differential cross section for lepton pair (Drell–Yan) production at the LHC. This process is of particular interest, since the effects of QED, especially in the partons, is expected to be of non-negligible significance, particularly due the inclusion of \(x\gamma (x,Q^2)\) as a contribution to the cross section. Below, we will consider the impact of both including QED effects in the evolution of the PDFs as well as the addition of photon-initiated (PI) contributions, as shown in Fig. 32, where the photon PDF enters as a direct input for the colliding partons.
5.2 Comparison with ATLAS Drell–Yan data
In order to gauge the magnitude (and phenomenological significance) of these effects we compare to data provided by the ATLAS collaboration [28] for high mass (116 GeV \(< m_{ll}<\) 1500 GeV) Drell–Yan lepton pair production. The focus on production at high mass is chosen in order to reduce the effects of the Z production peak, \(Q \sim M_Z = 91\) GeV, since the relative contribution of the PI processes are greater in the regions dominated by the \(\gamma \) channel. Therefore, the effects of PI contributions are anticipated to be more readily observable at low, \(m_{ll} \ll M_Z^2\), or high, \(m_{ll} \gg M_Z^2\), lepton pair invariant masses.
ATLAS provides double differential cross section measurements in 5 bins of lepton pair invariant mass, \(m_{ll}\) and 12 or 6 pseudo-rapidity \(\eta \) bins, depending on the mass region. Fig. 33 shows as a comparison for a range of cases: (a) a standard QCD fit partons at NNLO as outlined in Sect. 2.1, (b) with QED modified partons to provide cross section calculations at NNLO in QCD and (c) with QED modified partons and additional contributions to the cross section from \({\mathcal {O}}(\alpha )\) photon initiated processes as shown in Fig. 32.
To calculate cross sections, we use grids provided by the xFitter collaboration [15], at NLO in QCD (generated with MadGraph5\(\_\)aMC@NLO [97], aMCfast [98] and FEWZ [99]), and including PI processes at LO in QED. NNLO QCD corrections are included via K-factors. Such grids were developed and used in [15] with the aim of determining \(x\gamma (x,Q^2)\) from the same ATLAS data. These are then interfaced with a modified version of APPLgrid that we have adapted to include \(\gamma \gamma \) processes for the final calculation.
In the following analysis it is emphasised that the contributions of PI processes implemented in the comparison to data will be most sensitive to \(x\gamma ^{(inel)}(x,Q^2)\), due to the prevalence of this contribution in comparison to \(x\gamma ^{(el)}(x,Q^2)\) at higher scales (as was seen in the lower part of Fig. 5 in Sect. 2.4).
First, it is observed that the addition of QED in the process of DGLAP leads to a tendency to decrease the dominantly \(q{\bar{q}}\) contribution to the cross section, increasingly so at higher rapidity. This is expected, as from Fig. 15 one observes that the quarks experience a reduction at high x of \(\sim 1\%\) due to \(q \rightarrow q + \gamma \) type splittings. Second, the inclusion of PI contributions to the cross section is seen, as expected, to lead to an increase in the cross section relative to the QED corrected partons across all bins, as the inclusion of \(x\gamma (x,Q^2)\) opens up a new channel for lepton pair production, unaccounted for in pure QCD calculations. Since the magnitude of the photon PDF is seen to become larger at low x, particularly at high scales (\(Q^2 = 10^4 \sim 10^8\) \(\hbox {GeV}^{2}\)) and \(\eta \simeq \frac{1}{2}\ln {(x_1/x_2)}\) where 1 and 2 denote the incoming photons, the predominance of the photon at low x manifests as an enhanced cross section contribution in the lower and intermediate \(\eta \) bins, an effect seen to hold across all mass bins. At high rapidities the smallness of the large-x photon makes this photon contribution smaller than the decrease due to the quark suppression noted above.
At high \(\eta \), however, the change due to QED effects in the evolution is seen to be comparable in magnitude to that of PI contributions. In particular, we wish to highlight that for precision calculations of electroweak effects, one requires that all the partons be consistently treated (i.e. to contain all QED splittings for the quarks and gluons in an interdependent and coupled fashion) with QED in the evolution, as well as including the photon for a consistent treatment. This is especially noteworthy since the general trend of the partons after refitting with QED has an opposing effect on the cross section compared to that of PI contributions (due to a reduction of the total quark singlet), and as such, neglecting them can in principle lead to an over-estimation of the cross section where PI contributions are simply added on top of the standard QCD result, without the compensating effect in the other partons.
In fact, at high \(x,\eta \), where PI contributions are less relatively important as \(x\gamma (x,Q^2)\) rapidly diminishes, the effect of refitting the partons with QED is such that even the inclusion of PI contributions after accounting for QED in the evolution leads to a cross section less than that of the standard NNLO QCD prediction. In other words, the reduction of the total quark singlet content has a greater impact than the additional cross section contributions that are available from PI processes.
5.3 Including the ATLAS data in the global fit
In the aforementioned analysis, the cross section calculations are performed using a set of PDFs which has not included the Drell–Yan data from ATLAS itself in the global fit for the determination of parton parameters. In the remainder of this section, we discuss the effects of including these data in the fit itself and the subsequent effect on the recalculation of the cross section. In Fig. 34 we present the ratio of the cross section calculation from the QED corrected partons, including the contributions of PI processes, both before and after refitting to the data with these effects. We can see that there is no substantial improvement in data description after refitting.
A table denoting how the numbering of the grid files (produced in the LHAPDF6 format) corresponds to the uncertainties listed in the text
File number index \(\{x\}\) | Corresponding Uncertainty |
---|---|
01-50 | The standard PDF uncertainties associated with the \(q+{\bar{q}}\), \(q-{\bar{q}}\) and g distributions for all flavours |
51-52 | The uncertainty contributions from \(A'_2\) (51: -0.4, 52:-0.2) |
53-54 | The uncertainty contributions from the Continuum contributions (53: Upper band, 54: Lower band) |
55-56 | The uncertainty contributions from the Resonance contributions (53: Upper band, 54: Lower band) |
57-58 | The uncertainty contributions from \(W_{\mathrm{cut}}\) (57: 3 \(\hbox {GeV}^{2}\), 58: 4 \(\hbox {GeV}^{2}\)) |
59-60 | The uncertainty contributions from R (59: +50%, 60: -50%) |
61-62 | The uncertainty contributions from the Elastic contributions (53: Upper band, 54: Lower band) |
6 Conclusions
In this paper, we have presented the updated MMHT partons, modified to include the effects of QED in their evolution. Our resultant photon PDF, \(x\gamma (x,Q^2)\), based on a similar methodology for the input to that of LUXqed is seen to closely resemble others in the literature, despite several modifications made to take into account our lower starting scale for the evolution and the fact that we use our own PDFs.
We have also outlined the procedure developed to provide an approximate QED corrected DGLAP evolution for the PDFs of the neutron, leading to a neutron photon PDF and isospin violating valence quark PDFs, which may hold significance for the future development of neutron PDFs. The photon PDF of the neutron is seen to be of a similar magnitude to that of the proton at higher \(Q^2\). We provide the PDFs in grids which contain the central sets and uncertainties. PDF sets are provided as grids in the LHAPDF6 format. Details are contained in the Appendix.
Finally, although the fit quality remains broadly unchanged after refitting with these effects, we have observed that for the process of high-mass Drell–Yan production, the effects of both photon initiated processes, as well as changes in the quark and antiquark PDFs due to the effects of evolution, may become significant with the advent of precision measurements in this kinematic region and that the effects of QED in the evolution may be as significant as that of the photon, highlighting a need for a fully consistent set of QED corrected partons.
Footnotes
- 1.
Note that excluding this term is still a reasonable approximation given that a fully consistent treatment with a coupled DGLAP evolution would require the development of lepton PDF distributions which as discussed in [39] are found to have a negligible impact on the evolution of the PDFs on the whole.
Notes
Acknowledgements
We would like to thank Valerio Bertone and Francesco Giuli. We would also like to thank Markus Diehl for illuminating discussions. LHL thanks the Science and Technology Facilities Council (STFC) for support via grant awards ST/P004547/1. RST thanks the Science and Technology Facilities Council (STFC) for support via grant award ST/P000274/1. RN thanks the Elizabeth Spreadbury Fund.
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