Inadequacy of zero-width approximation for a light Higgs boson signal
In the Higgs search at the LHC, a light Higgs boson ( GeV 130 GeV) is not excluded by experimental data. In this mass range, the width of the Standard Model Higgs boson is more than four orders of magnitude smaller than its mass. The zero-width approximation is hence expected to be an excellent approximation. We show that this is not always the case. The inclusion of off-shell contributions is essential to obtain an accurate Higgs signal normalisation at the precision level. For , , corrections occur due to an enhanced Higgs signal in the region , where also sizable Higgs-continuum interference occurs. We discuss how experimental selection cuts can be used to exclude this region in search channels where the Higgs invariant mass cannot be reconstructed. We note that the decay modes in weak boson fusion are similarly affected.
Keywords:Higgs Physics, QCD, Hadron-Hadron Scattering
A key objective of current particle physics research is the experimental confirmation of a theoretically consistent description of elementary particle masses. In the Standard Model (SM), this is achieved through the Higgs mechanism Higgs:1964ia (); Higgs:1964pj (); Higgs:1966ev (); Englert:1964et (); Guralnik:1964eu (), which predicts the existence of one physical Higgs boson. Searches for the SM Higgs boson have been carried out at the Large Electron Positron Collider (LEP) and the Tevatron, which resulted in a lower Higgs mass bound of GeV Barate:2003sz (), and the exclusion of GeV and GeV :2012zzl (), respectively.111All bounds and exclusion limits are at 95% C.L. Higher sensitivity is attainable at the Large Hadron Collider (LHC), which was built as Higgs discovery machine. The combined analysis of the CMS 2011 data of 4.6–fb at TeV excludes GeV Chatrchyan:2012tx (). Similarly, a recent ATLAS study excludes GeV, GeV and GeV :2012an (). A light Higgs boson is therefore not excluded by experimental data. In fact, in a seminar on 4th July 2012 at CERN, ATLAS and CMS have presented evidence that a SM-like Higgs boson with – GeV has been observed at the level. It is therefore important to examine the accuracy of theoretical predictions for Higgs production and decay at the LHC that are used in experimental analyses for light Higgs masses.
For light Higgs masses, the loop-induced gluon-fusion production () Georgi:1977gs () dominates. Next-to-leading order (NLO) QCD corrections have been calculated in the heavy-top limit Dawson:1990zj () as well as with finite and mass effects Djouadi:1991tka (); Graudenz:1992pv (); Spira:1995rr (). NLO corrections of 80–100% at the LHC motivated the calculation of next-to-next-to-leading order (NNLO) QCD corrections in the heavy-top limit Harlander:2002wh (); Anastasiou:2002yz (); Ravindran:2003um () enhanced by soft-gluon resummation at next-to-next-to-leading logarithmic (NNLL) level Catani:2003zt (); deFlorian:2011xf () and beyond Moch:2005ky (); Laenen:2005uz (); Idilbi:2005ni (); Ravindran:2005vv (); Ravindran:2006cg (); Ahrens:2008nc (). Fully differential calculations have been presented in Refs. Anastasiou:2004xq (); Catani:2007vq (). The accuracy of the approximation at NNLO has been investigated in Refs. Marzani:2008az (); Harlander:2009bw (); Pak:2009bx (); Harlander:2009mq (); Pak:2009dg (); Harlander:2009my ().222Scale, PDF, strong coupling and heavy-top-limit uncertainties have recently been reappraised in Ref. Baglio:2010ae (). In addition to higher-order QCD corrections, electroweak (EW) corrections have been computed up to two loops Djouadi:1994ge (); Aglietti:2004nj (); Degrassi:2004mx (); Actis:2008ug (); Actis:2008ts (); Ahrens:2010rs (); Keung:2009bs (); Brein:2010xj () and found to be at the 1–5% level. Mixed QCD-EW effects have also been calculated Anastasiou:2008tj (). Refined calculations/updated cross sections for have been presented in Refs. deFlorian:2009hc (); Dittmaier:2011ti (); Anastasiou:2011pi (); Anastasiou:2012hx (); deFlorian:2012yg (). Kinematic distributions and NNLO cross sections with experimental selection cuts have also been studied extensively for 4 leptons () Anastasiou:2007mz (); Grazzini:2008tf () and all other important decay modes (see Ref. Dittmaier:2012vm () and references therein). NLO EW corrections to 4 leptons have been calculated in Refs. Bredenstein:2006rh (); Denner:2011vt ().
The proper theoretical description of the Higgs boson line shape is an essential ingredient for heavy Higgs searches and has been studied in detail in Ref. Goria:2011wa (). A comparison of the zero-width approximation (ZWA, see below) and finite-width Higgs propagator schemes for inclusive Higgs production and decay can also be found in Refs. Anastasiou:2011pi (); Buehler:2012zf (); Anastasiou:2012hx ().333The accuracy of the ZWA in the context of beyond-the-SM physics has been studied in Refs. Berdine:2007uv (); Kauer:2007zc (); Kauer:2007nt (); Uhlemann:2008pm (). In the light Higgs mass range the on-shell width of the SM Higgs boson is more than four orders of magnitude smaller than its mass, for instance MeV for a mass of GeV.444Width computed with HTO, see Section 2. The ZWA a.k.a. narrow-width approximation, which factorizes the Higgs cross section into on-shell production and on-shell decay when approaches zero, is expected to be excellent well below the and thresholds with an error estimate of . For Higgs production in gluon fusion, we show in Sections 2 and 3 that this is not always the case. For , we find that the deviation between ZWA and off-shell results is particularly large. We therefore take into account the resonance-continuum interference (see Fig. 1, left and right), which was studied in Refs. Glover:1988fe (); Glover:1988rg (); Binoth:2006mf (); Accomando:2007xc (); Campbell:2011cu (); Kauer:2012ma () and for related processes in Refs. Dixon:2003yb (); Dixon:2008xc (); Accomando:2011eu (). For studies of the continuum background (see Fig. 1, center and right), we refer the reader to Refs. Campbell:2011bn (); Frederix:2011ss (); Melia:2012zg (); Agrawal:2012df () and references therein.
The paper is organised as follows: In Section 2, we briefly review the zero-width approximation and why it can be inadequate. We then present and discuss inclusive results in ZWA and with off-shell effects for the processes all and with GeV including Higgs-continuum interference effects. In Section 3, we extend our ZWA v. off-shell analysis by considering experimental Higgs search procedures, selection criteria and transverse mass observables for all 4 leptons search channels. Higgs-continuum interference effects are again included. Conclusions are given in Section 4.
2 Inclusive analysis
In the SM, the common belief is that for a light Higgs boson the product of on-shell production cross-section (say in gluon-gluon fusion, ) and branching ratios reproduces the correct result to great accuracy. The expectation is based on the well-known result
where is the virtuality of the Higgs boson, and are the on-shell Higgs mass and width and denotes the principal value (understood as a distribution). Furthermore, is connected to the th derivative of the delta-function by and the expansion is in terms of the coupling constant, up to a given order .
In general, the ZWA can be applied to predict the probability for resonant scattering processes when the total decay width of the resonant particle is much smaller than its mass . Note that both concepts, on-shell mass and width, are ill-defined for an unstable particle and should be replaced with the complex pole, which is a property of the -matrix, gauge-parameter independent to all orders of perturbation theory. Nevertheless, let us continue with our qualitative argument: in the limit , the mod-squared propagator
with 4-momentum approaches the delta-function limit of Eq. (1), i.e.
The scattering cross-section thus approximately decouples into on-shell production () and decay as shown in Eqs. (4)–(6), where is the total 4-momentum squared, argument based on the scalar nature of the resonance. Based on the scales occurring in , the conventional error estimate is . This will not be accurate when the dependence of or is strong enough to compete with the dependence of . An interesting example is , where above . We note that similar effects have been observed for processes in SM extensions Berdine:2007uv (); Kauer:2007zc (); Kauer:2007nt (); Uhlemann:2008pm ().
An important observation is that the Breit-Wigner distribution does not drop off nearly as fast as, for instance, a Gaussian. The relative contribution of the tail more than widths from the peak can be estimated as , because Kauer:2001sp ()
Since the width of a light Higgs is so small, corresponds to only a few GeV, beyond which one would expect less than of the signal cross section.
A potential worry, addressed in this paper, is: to which level of accuracy does the ZWA approximate the full off-shell result, given that at the on-shell width (very close to the imaginary part of the complex pole) is . When searching for the Higgs boson around one should not care about the region but, due to limited statistics, theory predictions for the normalisation in are used over the entire spectrum in the invariant mass.
Therefore, the question is not to dispute that off-shell effects are depressed by a factor , as shown in Eq. (1), but to move away from the peak in the invariant mass distribution and look at the behavior of the distribution, no matter how small it is compared to the peak; is it really decreasing with ? Is there a plateau? For how long is the plateau lasting? How does that affect the total cross-section if no cut is made?
where is the Higgs complex pole, parametrized by . Note that is not the on-shell width, although the numerical difference is tiny for low values of , as shown in Ref. Goria:2011wa ().
The production cross-section, , is computed with NNLO QCD corrections (see Ref. Dittmaier:2012vm ()) and NLO EW ones Actis:2008ug (). The partial decay width of the off-shell Higgs boson of virtuality (), is computed at NLO with leading NNLO effects in the limit of large Higgs boson mass, see Ref. Bredenstein:2007ec (). Numerical results in this section are obtained with the program HTO (G. Passarino, unpublished) that allows for the study of the Higgs boson lineshape, in gluon-gluon fusion, using complex poles. Our results refer to and are based on the MSTW2008 PDF sets Martin:2009iq (). They are implemented according to the OFFP scheme, see Eq. (45) of Ref. Goria:2011wa (). Furthermore, we set the renormalization and factorization scale to the Higgs virtuality.
Away (but not too far away) from the narrow peak the propagator and the off-shell width behave like
above threshold with a sharp increase just below it (it increases from at to at ). Our result for the () invariant mass distribution is shown in Fig. 2. It confirms that, above the peak, the distribution is decreasing until the effects of the threshold become effective with a visible increase followed by a plateau, by another jump at the -threshold, beyond which the signal distribution decreases almost linearly (on a logarithmic scale). For the effect is drastically reduced and confined to the region between GeV and GeV, where the distribution is already five orders of magnitude below the peak.
What is the net effect on the total cross-section? We show it for in Table 1 where the contribution above the -threshold amounts to . We have checked that the effect does not depend on the propagator function, complex-pole propagator or Breit-Wigner distribution. The size of the effect is related to the shape of the distribution function. The complex-mass scheme can be translated into a more familiar language by introducing the Bar-scheme Goria:2011wa (). Performing the well-known transformation
a remarkable identity follows (defining the so-called Bar-scheme):
showing that the complex-pole scheme is equivalent to introducing a running width in the propagator with parameters that are not the on-shell ones. Special attention goes to the numerator in Eq. (11) which is essential in providing the right asymptotic behavior when , as needed for cancellations with the rest of the amplitude. Therefore, it is not advisable to use a naive, running-width Breit-Wigner distribution or to use a propagator with .
In Table 2, we present the invariant mass distribution integrated bin-by-bin.
If we take the ZWA value for the production cross-section at and for () and use the branching ratio into of we obtain a ZWA result of with a difference w.r.t. the off-shell result, fully compatible with the effect coming from the high-energy side of the resonance. In Table 1, we also see that the effect is much less evident if we sum over all final states with a net effect of only . This agrees well with the deviation of 0.5% between ZWA and fixed-width Breit-Wigner scheme (FWBW) given in Table 1 of Ref. Anastasiou:2011pi () for GeV. At GeV, de Florian-Grazzini obtain a 0.3%–0.4% deviation between ZWA and CPS (or FWBW) with “pure massless NNLO,” i.e. without resummation, heavy quark effects and EW corrections, and a slightly smaller deviation for the full calculation deFlorianPrivComm (). For all, one can thus expect deviations of depending on the particular implementation of the calculation.
Of course, the signal per se is not a physical observable and one should always include background and interference. In Fig. 3 we show the complete LO result for calculated with HTO with a cut of on the transverse momentum of the . The large destructive effects of the interference above the resonant peak wash out the peculiar structure of the signal distribution. If one includes the region in the analysis then the conclusion is: interference effects are relevant also for the low Higgs mass region, at least for the final state. It is worth noting again that the discussed effect on the signal has nothing to do with effects; above the -threshold the distribution is higher than expected (although tiny w.r.t. the narrow peak) and stays roughly constant up to the -threshold after which we observe an almost linear decrease. This is why the total cross-section is affected (in the final state) at the level.
To conclude our inclusive analysis, we note that our findings are driven by the interplay between the -dependence of the Higgs propagator and the decay matrix element. They should hence not only apply to Higgs production in gluon fusion, but also to Higgs production in weak boson fusion (WBF). The enhancement for above may even be stronger in WBF, because decreases less rapidly than with increasing Higgs invariant mass.555Preliminary results for inclusive WBF Higgs production reveal effects similar to all, yielding a deviation of 1% between ZWA and FWBW (but no difference between FWBW and CPS as expected for GeV) RebuzziPrivComm (); VBFTwiki ().
3 Analysis with experimental selection cuts
In this section, we adopt the common selection cuts definition between ATLAS and CMS for processes () HWWcuts (); HZZcuts () and calculate parton-level leptons cross sections at LO using gg2VV gg2VV () based on Refs. Binoth:2005ua (); Binoth:2006mf (); Binoth:2008pr (); Kauer:2012ma (); Hahn:1998yk (); Hahn:2000kx (), with Higgs in ZWA as well as off-shell including interference with continuum production (where contributions are also included).666All cross sections are evaluated with a GeV cut. This technical cut prevents numerical instabilities when evaluating the continuum amplitude.All results are given for a single lepton flavour combination. No flavour summation is carried out for charged leptons or neutrinos. As input parameters, we use the specification of the LHC Higgs Cross Section Working Group in App. A of Ref. Dittmaier:2011ti () with NLO and scheme. Finite top and bottom quark mass effects are included. Lepton masses are neglected. We consider the Higgs masses GeV and GeV with GeV and GeV, respectively. The Higgs widths have been calculated with HDECAY Djouadi:1997yw (). The fixed-width prescription is used for Higgs and weak boson propagators. The renormalisation and factorisation scales are set to . The PDF set MSTW2008NNLO Martin:2009iq () with 3-loop running for and is used. The CKM matrix is set to the unit matrix, which causes a negligible error Kauer:2012ma ().
The accuracy of the ZWA Higgs cross section and the impact of off-shell effects is assessed with the ratio
To facilitate comparison with off-shell distributions, we define the ZWA distribution as suggested by Eq. (5):
Each signal process leptons (with amplitude ) and corresponding continuum background process leptons (with amplitude ) have identical initial and final states. Hence interference occurs, and the distinction between signal and background cross sections becomes blurred:
We assess interference effects using a -inspired interference measure,
and a -inspired measure,
In the following, charged leptons are denoted by .
3.1 and at GeV
The same- and different-flavour 4-charged-lepton channels have been analysed by ATLAS ATLAS:2012ac () and CMS Chatrchyan:2012dg () for Higgs masses in the range 110–600 GeV. In these search channels, the invariant mass of the intermediate Higgs () can be reconstructed. The spectrum is hence used as the discriminant variable in the final stage of the analysis, and the test statistic is evaluated with a binned maximum-likelihood fit of signal and background models to the observed distribution. For light Higgs masses, the observed distribution is dominated by experimental resolution effects and for example fitted as Gaussian with a standard deviation of 2–2.5 GeV (or similar bin sizes are used). Since the width of a light SM Higgs boson is 2–3 orders of magnitude smaller, one would expect that the ZWA is highly accurate. According to Eq. (7), the constraints on mentioned above introduce an error of order 0.1%. Invariant masses above , where large deviations from the Breit-Wigner shape occur, are excluded by the experimental procedure. Higgs-continuum interference effects are negligible. For illustration, we compute the Higgs cross section in ZWA and off-shell including continuum interference in the vicinity of , more precisely GeV. To take into account the detector acceptance, we require GeV and . Leptons are separated using . Following Ref. ATLAS:2012ac (), we apply the cuts 76 GeV GeV and GeV GeV. The invariant mass of the same-flavour, opposite-sign lepton pair closest to is denoted by . denotes the invariant mass of the remaining lepton pair. The singularity for vanishing virtuality is excluded by requiring GeV.777This cut is induced by the phase space generation. The results are displayed in Table 3.
|[fb], , TeV, GeV||ZWA||interference|
3.2 at GeV
The search channel has been analysed by ATLAS ATLAS_lvlv () and CMS Chatrchyan:2012ty () for Higgs masses in the range 110–600 GeV. We apply the standard cuts GeV, , GeV and GeV. As Higgs search selection cuts, we apply the standard cuts and in addition 50 GeV and . Since cannot be reconstructed, ATLAS and CMS also use transverse mass observables that aim at approximating . Ref. ATLAS_lvlv () uses the transverse mass definition888 In the absence of additional observed final state particles, the expressions for simplify due to .
and applies a cut for GeV. Ref. Chatrchyan:2012ty () uses the transverse mass definition
where is the angle between and , and applies a cut for GeV.
Cross sections are presented in Table 4. When standard cuts are applied, the phase space region where GeV, or equivalently , contributes 16% to the off-shell Higgs cross section. The error of the ZWA exceeds 15%, and interference effects are of . Figs. 4 and 5 illustrate that the region with is responsible for the inaccuracy of the ZWA as well as the unexpectedly large interference effects, in agreement with our discussion in Section 2. Fig. 6 demonstrates that finite-width effects and Higgs-continuum interference are negligible in the resonance region, i.e. , for a Higgs mass of 125 GeV. The Higgs search selection has additional cuts, in particular an upper bound on the invariant mass of the observed dilepton system, which significantly reduce the contribution from the region with , as seen in Fig. 7. The result is a substantial mitigation of the finite-width and interference effects as seen in Table 4.
We now consider the impact of cuts on transverse mass observables, which are designed to have the physical mass of the decaying parent particle (the invariant mass in the off-shell case) as upper bound Barr:2009mx ().999We note that is referred to as in Ref. Barr:2009mx (). For the process considered here, this is shown in Fig. 8. Evidently, imposing a cut is an effective means to suppress interference effects. This was first noticed and studied for the variable in Ref. Campbell:2011cu (). In Table 4, one can see that both, and , are suitable transverse mass variables, with being slightly more effective. This is also borne out by the transverse mass distributions in Figs. 9 and 10. With regard to the ZWA, Table 4 shows that the application of the or cut reduces the ZWA error to the sub-percent level.
|[fb], , TeV, GeV||ZWA||interference|
|Higgs search cuts||1.950(1)||1.980(1)||2.705(2)||4.497(3)||0.9850(7)||0.9599(7)||0.905(2)|
3.3 at GeV
The search channel has been analysed by ATLAS ATLAS_2l2v () and CMS Chatrchyan:2012ft () for Higgs masses in the range 200–600 GeV. In this section we focus on the lowest studied Higgs mass of GeV with . Note that is slightly above the pair production threshold. A clean separation of the Higgs resonance and the region with large continuum background is thus not possible. We apply the Higgs search cuts GeV, , GeV GeV, GeV and . Refs. ATLAS_2l2v () and Chatrchyan:2012ft () use a transverse mass distribution as final discriminant in searching for an excess of data over the SM background expectation. Ref. Chatrchyan:2012ft () employs the transverse mass variable first proposed in Ref. Rainwater:1999sd () for the weak boson fusion channel:
with defined in Eq. (18) and
Note that , unlike and , does not have a kinematic edge at . The variable used in Ref. ATLAS_2l2v () is obtained by replacing with in the definition of , which causes only minor differences for . No cut is applied in the analyses.
In Table 5, cross section results are given. The ZWA is accurate at the percent level. Fig. 11 reveals that the off-shell enhancement of the high tail is moderate. Higgs-continuum interference is constructive and of . Significant interference occurs in the vicinity of the Higgs resonance as shown in Figs. 12 and 13. interference effects are comparable to interference effects for similar Higgs masses Binoth:2006mf (); Campbell:2011cu (). The distributions displayed in Fig. 14 show that sizable ZWA deviations occur at the differential level.
|[fb], , TeV, GeV||ZWA||interference|
3.4 at GeV
Given the rapid increase in integrated luminosity at the LHC, the mode could also be of interest at GeV. We therefore extend our study to this Higgs mass. The following selection cuts are applied: GeV, , GeV GeV and GeV. As seen in Fig. 15, the off-shell enhancement of the high tail is particularly pronounced. In Table 6, cross section results are given. The phase space region where GeV, or equivalently , contributes 37% to the off-shell Higgs cross section. The ZWA underestimates the Higgs cross section by a similar amount. Figs. 15 and 16 illustrate that the region with is also responsible for interference effects of . Fig. 17 demonstrates that finite-width effects and Higgs-continuum interference are negligible in the resonance region.
To mitigate the impact of the region with large ZWA deviations and Higgs-continuum interference, we propose to employ a cut. With this cut, the off-shell and interference effects () are reduced to the 2% level. The distribution displayed in Fig. 18 shows that the contamination of the region from the interference-inducing region is more severe than in the case (see Fig. 9).
|[fb], , TeV, GeV||ZWA||interference|
In the Higgs search at the LHC, a light Higgs boson is not excluded by experimental data. In the mass range GeV 130 GeV, one has for the SM Higgs boson. We have shown for inclusive cross sections and cross sections with experimental selection cuts that the ZWA is in general not adequate and the error estimate is not reliable for a light Higgs boson. The inclusion of off-shell contributions is essential to obtain an accurate Higgs signal normalisation at the 1% precision level. We have traced this back to the dependence of the decay (and to a lesser degree production) matrix element on the Higgs virtuality . For the decay modes we find that above the weak-boson pair production threshold the dependence of the decay matrix element compensates the -dependence of the Higgs propagator, which results in a significantly enhanced off-shell cross section in comparison to the ZWA cross section, when this phase space region, which is also affected by sizable Higgs-continuum interference, contributes. As shape of the enhancement above , we find a “plateau” up to the -threshold and an exponential decrease beyond it. The total cross section thus receives an off-shell correction. We have further illustrated that the region above is responsible for Higgs-continuum interference effects, which, due to the off-shell enhanced tail, can have a significant impact even for . We find that in the vicinity of the Higgs resonance finite-width and Higgs-continuum interference effects are negligible for , while for GeV this is not the case.
For weak boson decays that permit the reconstruction of the invariant Higgs mass, the enhanced region is eliminated by the experimental procedure as long as . For channels where the Higgs invariant mass cannot be reconstructed, we have illustrated that search selection cuts for a light Higgs boson reduce the contribution of the off-shell enhanced tail, and that the tail can effectively be excluded for the as well as decay mode by using transverse mass observables () that approximate the Higgs invariant mass and applying a cut. We predict that the weak boson fusion channels also exhibit an off-shell enhanced tail, since the effect is primarily caused by the Higgs decay matrix element. It is worth noting that we make no statement about the observability of a large invariant mass signal due to a low-mass Higgs boson, which is hampered by the huge background and interference effects.
After the -observation of a SM-like Higgs signal at – GeV reported by ATLAS and CMS in a recent seminar, the next step in the analysis will be the extraction of the Higgs couplings and properties. This study will initially be performed using the ZWA with a consequent error of on the couplings. Although this is still tolerable with current statistics, the results presented above make it clear that off-shell effects have to be included in future analyses.
In summary, we have elucidated the inadequacy of the ZWA in general and the existence of an enhanced tail in the Higgs invariant mass distribution of a light Higgs boson that decays to a weak-boson pair in particular, which makes off-shell calculations mandatory and will lead to significant errors when the ZWA is used, unless the affected region of invariant masses above the weak-boson-pair threshold is excluded with selection cuts. The latter is also motivated by the fact that intuitively one would not like to assign events with large invariant mass to a low-mass Higgs signal, not least because they are affected by large signal-background interference. As consequence of our findings, we recommend that the explicit or implicit application of the ZWA in experimental studies be identified and corrected for.
We would like to thank the conveners and members of the LHC Higgs Cross Section Working Group for stimulating and informative discussions. N.K. would like to thank M. Rodgers for useful comparisons. G.P.’s work is supported by MIUR under contract 2001023713006 and by Compagnia di San Paolo under contract ORTO11TPXK. Financial support from the Higher Education Funding Council for England, the Science and Technology Facilities Council and the Institute for Particle Physics Phenomenology, Durham, is gratefully acknowledged by N.K.
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