work with a. gehrmann-de ridder, t. gehrmann, e.w.n. glover, a....

hadron plane

x

y

z

lepton plane

p1 p2

k1

��

✓ �

1

Angular Coefficients in Z-boson production

work with A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. HussarXiv:1708.00008

Rhorry Gauld Moriond QCD - 23.03.2018

2

Outline

p

p

V

`

¯p1

p2

q

k1

k2

X

Rhorry Gauld, ETH Zurich

• Introduction - motivation - framework

• Physics results- Comparison to ATLAS/CMS data - Assessing Lam-Tung violation

• Conclusion

3

Angular coefficients in Z production

kµ1,2 =

pq2

2(1,± sin ✓ cos�,± sin ✓ sin�,± cos ✓)T

Defining lepton kinematics in V(q) rest frame

d�

d4q cos ✓ d�=

3

16⇡

d�unpol.

d4q

⇢(1 + cos2 ✓) +

1

2A0 (1� 3 cos2 ✓)

+A1 sin(2✓) cos�+1

2A2 sin2 ✓ cos(2�)

+A3 sin ✓ cos�+A4 cos ✓ +A5 sin2 ✓ sin(2�)

+A6 sin(2✓) sin�+A7 sin ✓ sin�

�

Decompose cross section in terms of spherical polynomials

l = 0 m = 0

l = 1 m = �1, 0, 1

l = 2 m = �2,�1, 0, 1, 2

Total of 9 terms Encode QCD dynamics Lepton pair kinematics

A0,..,7(q)

fi(✓,�)

fi(✓,�)

p(p1) + p(p2) ! V (q) +X ! `(k1) + ¯(k2) +X<latexit sha1_base64="75kbhH4ewhHGJyv2LUr98zLREk0=">AAACJHicbZDLSsNAFIYn9VbrLerSzWARUgolKYIKLopuXFYwbaEJYTKdtkMnF2cmQgl9GTe+ihsXVly48VmcpFlo64GBj//8hzPn92NGhTTNL620tr6xuVXeruzs7u0f6IdHHRElHBMbRyziPR8JwmhIbEklI72YExT4jHT9yW3W7z4RLmgUPshpTNwAjUI6pBhJJXn6dWzEnlWDdZhBswYdGcGO8Vir93J0CGPGZOFwfMTTTJgppakcnl41G2ZecBWsAqqgqLanz51BhJOAhBIzJETfMmPppohLihmZVZxEkBjhCRqRvsIQBUS4aX7lDJ4pZQCHEVcvlDBXf0+kKBBiGvjKGSA5Fsu9TPyv10/k8NJNaRgnkoR4sWiYMKjOzyKDA8oJlmyqAGFO1V8hHiOOsFTBVlQI1vLJq2A3G1cN8/682rop0iiDE3AKDGCBC9ACd6ANbIDBM3gF72CuvWhv2of2ubCWtGLmGPwp7fsHhwGgnw==</latexit><latexit sha1_base64="75kbhH4ewhHGJyv2LUr98zLREk0=">AAACJHicbZDLSsNAFIYn9VbrLerSzWARUgolKYIKLopuXFYwbaEJYTKdtkMnF2cmQgl9GTe+ihsXVly48VmcpFlo64GBj//8hzPn92NGhTTNL620tr6xuVXeruzs7u0f6IdHHRElHBMbRyziPR8JwmhIbEklI72YExT4jHT9yW3W7z4RLmgUPshpTNwAjUI6pBhJJXn6dWzEnlWDdZhBswYdGcGO8Vir93J0CGPGZOFwfMTTTJgppakcnl41G2ZecBWsAqqgqLanz51BhJOAhBIzJETfMmPppohLihmZVZxEkBjhCRqRvsIQBUS4aX7lDJ4pZQCHEVcvlDBXf0+kKBBiGvjKGSA5Fsu9TPyv10/k8NJNaRgnkoR4sWiYMKjOzyKDA8oJlmyqAGFO1V8hHiOOsFTBVlQI1vLJq2A3G1cN8/682rop0iiDE3AKDGCBC9ACd6ANbIDBM3gF72CuvWhv2of2ubCWtGLmGPwp7fsHhwGgnw==</latexit><latexit sha1_base64="75kbhH4ewhHGJyv2LUr98zLREk0=">AAACJHicbZDLSsNAFIYn9VbrLerSzWARUgolKYIKLopuXFYwbaEJYTKdtkMnF2cmQgl9GTe+ihsXVly48VmcpFlo64GBj//8hzPn92NGhTTNL620tr6xuVXeruzs7u0f6IdHHRElHBMbRyziPR8JwmhIbEklI72YExT4jHT9yW3W7z4RLmgUPshpTNwAjUI6pBhJJXn6dWzEnlWDdZhBswYdGcGO8Vir93J0CGPGZOFwfMTTTJgppakcnl41G2ZecBWsAqqgqLanz51BhJOAhBIzJETfMmPppohLihmZVZxEkBjhCRqRvsIQBUS4aX7lDJ4pZQCHEVcvlDBXf0+kKBBiGvjKGSA5Fsu9TPyv10/k8NJNaRgnkoR4sWiYMKjOzyKDA8oJlmyqAGFO1V8hHiOOsFTBVlQI1vLJq2A3G1cN8/682rop0iiDE3AKDGCBC9ACd6ANbIDBM3gF72CuvWhv2of2ubCWtGLmGPwp7fsHhwGgnw==</latexit>


Tension observed for

|ll

|y0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

| [pb

]ll

/d|y

σd

20406080

100120140160180200

DataPrediction (CT10nnlo)

ATLAS-1 = 7 TeV, 4.6 fbs

Z+X→pp

(a)

|lη|

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

| [pb

]lη

/d|

σd

250300350400450500550600650700750

)+Data (W)−Data (W

Prediction (CT10nnlo)


+X±W→pp

(b)

Figure 3: (a) Di↵erential Z-boson cross section as a function of boson rapidity, and (b) di↵erential W+ and W� crosssections as a function of charged decay-lepton pseudorapidity at

ps = 7 TeV [41]. The measured cross sections are

compared to the Powheg+Pythia 8 predictions, corrected to NNLO using DYNNLO with the CT10nnlo PDF set.The error bars show the total experimental uncertainties, including luminosity uncertainty, and the bands show thePDF uncertainties of the predictions.

[GeV]llT

p0 20 40 60 80 100

0A

0

0.2

0.4

0.6

0.8

1

1.2DataDYNNLO (CT10nnlo)


Z+X→pp

(a)

[GeV]llT

p0 20 40 60 80 100

2A

0

0.2

0.4

0.6

0.8

1

1.2DataDYNNLO (CT10nnlo)


Z+X→pp

(b)

Figure 4: The (a) A0 and (b) A2 angular coe�cients in Z-boson events as a function of p``T [42]. The measuredcoe�cients are compared to the DYNNLO predictions using the CT10nnlo PDF set. The error bars show the totalexperimental uncertainties, and the bands show the uncertainties assigned to the DYNNLO predictions.

17

4

Angular coefficients in Z production

[GeV]ZT

p1 10 210

0A

0.2−

0

0.2

0.4

0.6

0.8

1

1.2ATLAS

-18 TeV, 20.3 fbDataDYNNLO (NNLO)DYNNLO (NLO)POWHEGBOXPOWHEGBOX+PYTHIA8POWHEGBOX+HERWIG

[GeV]ZT

p1 10 210

(Dat

a)0

(The

ory)

- A

0A

0.1−

0.05−

0

0.05

0.1

0.15

0.2

0.25

DataDYNNLO (NNLO)DYNNLO (NLO)POWHEGBOXPOWHEGBOX+PYTHIA8POWHEGBOX+HERWIG

ATLAS-18 TeV, 20.3 fb

[GeV]ZT

p1 10 210

2A

0.2−

0

0.2

0.4

0.6

0.8

1

1.2ATLAS

-18 TeV, 20.3 fbDataDYNNLO (NNLO)DYNNLO (NLO)POWHEGBOXPOWHEGBOX+PYTHIA8POWHEGBOX+HERWIG

[GeV]ZT

p1 10 210

(Dat

a)2

(The

ory)

- A

2A

0.1−

0.05−

0

0.05

0.1

0.15

0.2

0.25

0.3



[GeV]ZT

p1 10 210

2-A 0A

0.1−

0

0.1

0.2

0.3

0.4



[GeV]ZT

p1 10 210

(Dat

a)2

-A 0(T

heor

y) -

A2

-A 0A

0.2−

0.1−

0

0.1

0.2

0.3



[GeV]ZT

p1 10 210

1A

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18



[GeV]ZT

p1 10 210

(Dat

a)1

(The

ory)

- A

1A

0.06−

0.04−

0.02−

0

0.02

0.04

0.06

0.08

0.1

0.12



Figure 19: Distributions of the angular coe�cients A0, A2, A0 � A2 and A1 (from top to bottom) as a function of pZT.

The results from the yZ-integrated measurements are compared to the DYNNLO predictions at NLO and at NNLO,as well as to those from PowhegBox + Pythia8 and PowhegBox + Herwig (left). The di↵erences between thecalculations and the data are also shown (right), with the shaded band around zero representing the total uncertaintyin the measurements. The error bars for the calculations show the total uncertainty for DYNNLO, but only thestatistical uncertainties for PowhegBox.

44

A0 �A2

(weighting of NLO-PS / Fixed-Order)

• Retain full kinematic dependence of leptons (corresponding QCD dynamics)

• Also important testing ground for W production (for MW extraction)

the three-dimensional boson production phase space, defined by the variables m, pT, and y, and the two-dimensional boson decay phase space, defined by the variables ✓ and �. Accordingly, a prediction ofthe kinematic distributions of vector bosons and their decay products can be transformed into anotherprediction by applying separate reweighting of the three-dimensional boson production phase-space dis-tributions, followed by a reweighting of the angular decay distributions.

The reweighting is performed in several steps. First, the inclusive rapidity distribution is reweightedaccording to the NNLO QCD predictions evaluated with DYNNLO. Then, at a given rapidity, the vector-boson transverse-momentum shape is reweighted to the Pythia 8 prediction with the AZ tune. This pro-cedure provides the transverse-momentum distribution of vector bosons predicted by Pythia 8, preservingthe rapidity distribution at NNLO. Finally, at given rapidity and transverse momentum, the angular vari-ables are reweighted according to:

w =1 + cos2 ✓ +

Pi A0i(pT, y) Pi(cos ✓, �)

1 + cos2 ✓ +P

i Ai(pT, y) Pi(cos ✓, �),

where A0i are the angular coe�cients evaluated at O(↵2s ), and Ai are the angular coe�cients of the

Powheg+Pythia 8 samples. This reweighting procedure neglects the small dependence of the two-dimensional (pT,y) distribution and of the angular coe�cients on the final state invariant mass. Theprocedure is used to include the corrections described in Sections 6.2 and 6.3, as well as to estimate theimpact of the QCD modelling uncertainties described in Section 6.5.

The validity of the reweighting procedure is tested at particle level by generating independent W-bosonsamples using the CT10nnlo and NNPDF3.0 [96] NNLO PDF sets, and the same value of mW . Therelevant kinematic distributions are calculated for both samples and used to reweight the CT10nnlo sampleto the NNPDF3.0 one. The procedure described in Section 2.2 is then used to determine the value of mWby fitting the NNPDF3.0 sample using templates from the reweighted CT10nnlo sample. The fitted valueagrees with the input value within 1.5±2.0 MeV. The statistical precision of this test is used to assign theassociated systematic uncertainty.

The resulting model is tested by comparing the predicted Z-boson di↵erential cross section as a functionof rapidity, the W-boson di↵erential cross section as a function of lepton pseudorapidity, and the angu-lar coe�cients in Z-boson events, to the corresponding ATLAS measurements [41, 42]. The comparisonwith the measured W and Z cross sections is shown in Figure 3. Satisfactory greement between the meas-urements and the theoretical predictions is observed. A �2 compatibility test is performed for the threedistributions simultaneously, including the correlations between the uncertainties. The compatibility testyields a �2/dof value of 45/34. Other NNLO PDF sets such as NNPDF3.0, CT14 [97], MMHT2014 [98],and ABM12 [99] are in worse agreement with these distributions. Based on the quantitative comparisonsperformed in Ref. [41], only CT10nnlo, CT14 and MMHT2014 are considered further. The better agree-ment obtained with CT10nnlo can be ascribed to the weaker suppression of the strange quark densitycompared to the u- and d-quark sea densities in this PDF set.

The predictions of the angular coe�cients in Z-boson events are compared to the ATLAS measurement atp

s = 8 TeV [42]. Good greement between the measurements and DYNNLO is observed for the relevantcoe�cients, except for A2, where the measurement is significantly below the prediction. As an example,Figure 4 shows the comparison for A0 and A2 as a function of pZ

T. For A2, an additional source ofuncertainty in the theoretical prediction is considered to account for the observed disagreement with data,as discussed in Section 6.5.3.

16

ATLAS - arXiv:1701.07240ATLAS - arXiv:1606.00689


5

$$$$X.$Chen,$J.$Cruz$Mar)nez,$J.$Currie,$AG,$T.$Gehrmann,$E.W.N.$Glover,$$$$$$$$$$$$$$$$$$$$$$A.Huss,$$M.$Jacquier,$T.$Morgan,$J.$Niehues,$J.$Pires$•  Common$infrastructure$for$the$implementa)on$of$NNLO$

correc)ons$using$antenna$subtrac)on$for$:$$

•  +$….$

21$

NNLOJET$

Aude$Gehrmann8De$Ridder$$$Par)cle$Physics$Colloquium,$Karlsruhe,$07.07.2016$

NNLOJET

_ ___ ____ ____ ____________

/ |/ / |/ / / / __ \__ / / __/_ __/

/ / / /__/ /_/ / // / _/ / /

/_/|_/_/|_/____/\____/\___/___/ /_/

X. Chen, J. Cruz-Martinez, J. Currie, A. Gehrmann–De Ridder, T. Gehrmann,E.W.N. Glover, AH, M. Jaquier, T. Morgan, J. Niehues, J. Pires

Common code base for NNLO corrections using Antenna SubtractionI pp ! Z/�⇤ ! `+`� + 0, 1 jetsI pp ! H ! �� + 0, 1, 2 jetsI pp ! dijetsI ep ! 2 jets (talk by J. Niehues)I . . .

I Fully differential parton-level event generatorI Work in progress: Interface to APPLgrid, fastNLO

6/23

pp ! dijets

pp ! H !! �� + 0, 1, 2 jets

pp ! Z/�⇤ ! l+l� + 0, 1 jets

ep ! 2(+1) jets

X. Chen, J. Cruz-Martinez, RG, A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss, I. Majer, J. Niehues, J. Pires, D. Walker + J. Currie, T. Morgan

[CERN, IPPP Durham, Zurich (ETH, UZH), Lisbon (CFTP)]

Processes:

pp ! V ! ll + 0, 1 jets

pp ! H + 0, 1, 2 jets

pp ! dijets

ep ! 1, 2 jets

ee ! 3 jets

...

Common framework for NNLO corrections

• parton-level Monte Carlo generator

• basis: Antenna subtraction formalism

• implementation strongly follows:

• In progress: APPLfast-NNLO interface

Gehrmann(-De Ridder), Glover - arXiv:0505111

PDF fitting with full NNLO calculations

Currie, Glover, Wells - arXiv:1301.4693


6

$$$$X.$Chen,$J.$Cruz$Mar)nez,$J.$Currie,$AG,$T.$Gehrmann,$E.W.N.$Glover,$$$$$$$$$$$$$$$$$$$$$$A.Huss,$$M.$Jacquier,$T.$Morgan,$J.$Niehues,$J.$Pires$•  Common$infrastructure$for$the$implementa)on$of$NNLO$

correc)ons$using$antenna$subtrac)on$for$:$$

•  +$….$

21$

NNLOJET$

Aude$Gehrmann8De$Ridder$$$Par)cle$Physics$Colloquium,$Karlsruhe,$07.07.2016$

NNLOJET

_ ___ ____ ____ ____________

/ |/ / |/ / / / __ \__ / / __/_ __/

/ / / /__/ /_/ / // / _/ / /

/_/|_/_/|_/____/\____/\___/___/ /_/

X. Chen, J. Cruz-Martinez, J. Currie, A. Gehrmann–De Ridder, T. Gehrmann,E.W.N. Glover, AH, M. Jaquier, T. Morgan, J. Niehues, J. Pires

Common code base for NNLO corrections using Antenna SubtractionI pp ! Z/�⇤ ! `+`� + 0, 1 jetsI pp ! H ! �� + 0, 1, 2 jetsI pp ! dijetsI ep ! 2 jets (talk by J. Niehues)I . . .

I Fully differential parton-level event generatorI Work in progress: Interface to APPLgrid, fastNLO

6/23

pp ! dijets

pp ! H !! �� + 0, 1, 2 jets

pp ! Z/�⇤ ! l+l� + 0, 1 jets

ep ! 2(+1) jetsProcesses:

pp ! V ! ll + 0, 1 jets

pp ! H + 0, 1, 2 jets

pp ! dijets

ep ! 1, 2 jets

ee ! 3 jets

...

Common framework for NNLO corrections

• parton-level Monte Carlo generator

• basis: Antenna subtraction formalism

• implementation strongly follows:

• In progress: APPLfast-NNLO interface

Gehrmann(-De Ridder), Glover - arXiv:0505111

PDF fitting with full NNLO calculations

Currie, Glover, Wells - arXiv:1301.4693

Gehrmann-De Ridder et al., arXiv:1507.02850

pT,Z spectrum at NNLO<latexit sha1_base64="8j7X8dFbDsmlbC8qKPVODc4Fs/I=">AAACBXicbVDNSsNAGNz4W+tf1KMIi0XwICURQb0VvXiQWqGxxTaEzXbTLt1Nwu5GKCG9ePFVvHhQ8eo7ePNt3LY5aOvAwjDzfXw748eMSmVZ38bc/MLi0nJhpbi6tr6xaW5t38koEZg4OGKRaPpIEkZD4iiqGGnGgiDuM9Lw+5cjv/FAhKRRWFeDmLgcdUMaUIyUljxzL/bS+tF9lrYFH8qYYCUSPkRqWK1e32SeWbLK1hhwltg5KYEcNc/8ancinHASKsyQlC3bipWbIqEoZiQrthNJYoT7qEtamoaIE+mm4xgZPNBKBwaR0C9UcKz+3kgRl3LAfT3JkerJaW8k/ue1EhWcuSkN40SREE8OBQmDKoKjTmCHCh2cDTRBWFD9V4h7SCCsdHNFXYI9HXmWOMfl87J1e1KqXORtFMAu2AeHwAanoAKuQA04AINH8AxewZvxZLwY78bHZHTOyHd2wB8Ynz+hrplt</latexit><latexit sha1_base64="8j7X8dFbDsmlbC8qKPVODc4Fs/I=">AAACBXicbVDNSsNAGNz4W+tf1KMIi0XwICURQb0VvXiQWqGxxTaEzXbTLt1Nwu5GKCG9ePFVvHhQ8eo7ePNt3LY5aOvAwjDzfXw748eMSmVZ38bc/MLi0nJhpbi6tr6xaW5t38koEZg4OGKRaPpIEkZD4iiqGGnGgiDuM9Lw+5cjv/FAhKRRWFeDmLgcdUMaUIyUljxzL/bS+tF9lrYFH8qYYCUSPkRqWK1e32SeWbLK1hhwltg5KYEcNc/8ancinHASKsyQlC3bipWbIqEoZiQrthNJYoT7qEtamoaIE+mm4xgZPNBKBwaR0C9UcKz+3kgRl3LAfT3JkerJaW8k/ue1EhWcuSkN40SREE8OBQmDKoKjTmCHCh2cDTRBWFD9V4h7SCCsdHNFXYI9HXmWOMfl87J1e1KqXORtFMAu2AeHwAanoAKuQA04AINH8AxewZvxZLwY78bHZHTOyHd2wB8Ynz+hrplt</latexit><latexit sha1_base64="8j7X8dFbDsmlbC8qKPVODc4Fs/I=">AAACBXicbVDNSsNAGNz4W+tf1KMIi0XwICURQb0VvXiQWqGxxTaEzXbTLt1Nwu5GKCG9ePFVvHhQ8eo7ePNt3LY5aOvAwjDzfXw748eMSmVZ38bc/MLi0nJhpbi6tr6xaW5t38koEZg4OGKRaPpIEkZD4iiqGGnGgiDuM9Lw+5cjv/FAhKRRWFeDmLgcdUMaUIyUljxzL/bS+tF9lrYFH8qYYCUSPkRqWK1e32SeWbLK1hhwltg5KYEcNc/8ancinHASKsyQlC3bipWbIqEoZiQrthNJYoT7qEtamoaIE+mm4xgZPNBKBwaR0C9UcKz+3kgRl3LAfT3JkerJaW8k/ue1EhWcuSkN40SREE8OBQmDKoKjTmCHCh2cDTRBWFD9V4h7SCCsdHNFXYI9HXmWOMfl87J1e1KqXORtFMAu2AeHwAanoAKuQA04AINH8AxewZvxZLwY78bHZHTOyHd2wB8Ynz+hrplt</latexit>


X. Chen, J. Cruz-Martinez, RG, A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss, I. Majer, J. Niehues, J. Pires, D. Walker + J. Currie, T. Morgan

[CERN, IPPP Durham, Zurich (ETH, UZH), Lisbon (CFTP)]

p

p

V

`

¯p1

p2

q

k1

k2

X(there is also N-jettiness calculation, Boughezal et al. - arXiv:1512.01291)

• spherical polynomials form an orthonormal basis

• perform projection (numerically) to obtain their coefficients

practically: run pp > ll+X, perform projection in Collins-Soper frame fill histograms (w.r.t. ) weighted by

reality: integrating highly oscillating functions…solutions: clever reweighting of P.S. + many integrand evaluations..

7

Predictions of angular coefficients

for the non-vanishing diagonal components of the hadronic tensor. We observe that the

Lam–Tung relation is equivalent to A0 � A2 = 0. Note that the result of Eq. (2.11) is not

frame independent but only holds if both the z- and x-axis in the lepton-pair rest frame lie

in the hadronic event plane. This condition enters in the step where the form factors Hi are

expressed in terms of the diagonal Hij components using Eq. (2.12) and can be understood

by inspecting the covariant formulation of Eq. (2.2) in the lepton-pair rest frame: The only

form factors that contribute to the trace of the hadronic tensor on the r.h.s. of Eq. (2.2)

are H1,...,4. The tensor structures multiplying H2,3,4 only involve momenta lying inside the

hadronic plane and it is solely the tensor gµ⌫ multiplying the form factor H1 which has a

non-vanishing component orthogonal to it. The Lam–Tung relation therefore distinguishes

the direction perpendicular to the hadronic plane and can be interpreted as a statement

about the current–current correlation of the hadronic tensor in this direction.6

Making use of the completeness of the spherical harmonics, the angular coe�cients

appearing in the decomposition provided in Eq. (2.5) can be extracted through the pro-

jectors

A0 = 4� 10⌦cos2 ✓

↵, A1 = 5 hsin(2✓) cos�i , A2 = 10

⌦sin2 ✓ cos(2�)

↵,

A3 = 4 hsin ✓ cos�i , A4 = 4 hcos ✓i , A5 = 5⌦sin2 ✓ sin(2�)

↵,

A6 = 5 hsin(2✓) sin�i , A7 = 4 hsin ✓ sin�i , (2.13)

where h. . .i denotes taking the (normalised) weighted average over the angular variables ✓,

� and is defined as

hf(✓,�)i ⌘

R 1�1 d cos ✓

R 2⇡0 d� d�(✓,�) f(✓,�)

R 1�1 d cos ✓

R 2⇡0 d� d�(✓,�)

. (2.14)

The dominant angular coe�cients are A0,...,4, while A5,6,7 vanish at O(↵s) and only

receive small O�↵2s

�corrections from the absorptive parts of the one-loop amplitudes in

Z+ jet production. We therefore will not discuss the coe�cients A5,6,7 in the following. In

the case of pure �⇤ exchange, the relevant coe�cients are the parity-conserving coe�cients

A0,1,2. A3 and A4, on the other hand, are odd under parity and proportional to the product

of vector- and axial-vector-couplings of the gauge boson to the fermions. As such, they

are sensitive to the relative rate of incoming down- and up-type quark fluxes as well as

the weak mixing angle sw. All the coe�cients Ai vanish in the limit pT,Z ! 0 with the

exception of A4, which is finite in this limit and directly related to the forward–backward

asymmetry.

One of the goals of this work is to assess the compatibility of the observed extent of

Lam–Tung violation with that expected in predictions based on pQCD. This can be done

by directly studying the pT,Z distribution for the di↵erence of the angular coe�cients A0

and A2. Here, we propose a new observable

�LT⌘ 1�

A2

A0, (2.15)

6For hypothetical spin-0 partons, the current correlator would be completely confined within the hadronic

event plane, which then yields for Eq. (2.2): H1 = 0.

– 7 –

f(✓,�)

A0 = 4� 10⌦cos2 ✓

↵, A1 = 5 hsin(2✓) cos�i , A2 = 10

⌦sin2 ✓ cos(2�)

↵,

A3 = 4 hsin ✓ cos�i , A4 = 4 hcos ✓i (relevant angular coefficients)

m``, y``, p``T

(lab-frame)

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8

study the region:accuracy: NNLO (from Z+jet @ NNLO) input scheme: -scheme PDF set: PDF4LHC NNLO asmz0118scale:

pT,Z > 10 GeV

Gµ

µ0 =q

m2`` + p2T,``

Scale variation (arbitrary stuff)

Numerical set-up (boring stuff)

If you correlate numerator and denominator, `artificial’ cancellation

1/2 < µF /µR < 2

No dependence at all at LOµR

We independently vary in numerator/denominator, such that

1/2 < µia/µ

jb < 2

i, j = num. or den.a,b = fac. or ren.

31-point scale variation


210 [GeV]Z

TP

0

0.02

0.04

0.06

0.08

0.1

0.12

0.141A

ATLAS dataLONLONNLO

NNLOJET = 8 TeVs inclusiveZ

Z+X, y→pp

[GeV]T, Z

p

0.5

1

1.5

Rat

io to

NLO

100 50020

210 [GeV]Z

TP

0

0.2

0.4

0.6

0.8

10A ATLAS data

LONLONNLO


Z+X, y→pp

[GeV]T, Z

p0.6

0.81

1.2

1.4

Rat

io to

NLO

100 50020

9

Predictions vs. data (ATLAS)

A0

ATLAS data (8 TeV): y`¯ inclusive, m`¯ 2 [80, 100] GeV<latexit sha1_base64="nEWNtasxSG+yLlYrb7ywtHEehyg=">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</latexit><latexit sha1_base64="nEWNtasxSG+yLlYrb7ywtHEehyg=">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</latexit><latexit sha1_base64="nEWNtasxSG+yLlYrb7ywtHEehyg=">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</latexit>

A1


10

210 [GeV]Z

TP

0

0.02

0.04

0.06

0.08

0.1

0.12

0.143A

ATLAS dataLONLONNLO


Z+X, y→pp

[GeV]T, Z

p

0

0.51

1.52

Rat

io to

NLO

100 50020

210 [GeV]Z

TP

0

0.02

0.04

0.06

0.08

0.1

4A ATLAS data

LONLONNLO


Z+X, y→pp

[GeV]T, Z

p

0.5

1

1.5

Rat

io to

NLO

100 50020


A3 A4



11



210 [GeV]Z

TP

0

0.2

0.4

0.6

0.8

12A ATLAS data

LONLONNLO


Z+X, y→pp

[GeV]T, Z

p0.6

0.81

1.2

1.4

Rat

io to

NLO

100 50020

A2


12

Predictions vs. data (CMS)

10 210 [GeV]Z

TP

0

0.2

0.4

0.6

0.8

10A CMS data

LONLONNLO

NNLOJET = 8 TeVs [0.0,1.0]∈| Z

Z+X, |y→pp

[GeV]T, Z

p0.6

0.8

1

1.2

1.4

Rat

io to

NLO

10010

10 210 [GeV]Z

TP

0.02−

0.01−

00.010.020.030.040.050.060.071

A

CMS dataLONLONNLO


Z+X, |y→pp

[GeV]T, Z

p0

0.5

1

1.5

2

Rat

io to

NLO

10010

10 210 [GeV]Z

TP

0

0.02

0.04

0.06

0.08

0.1

0.121A

CMS dataLONLONNLO


Z+X, |y→pp

[GeV]T, Z

p

0.60.8

11.21.4

Rat

io to

NLO

10010

10 210 [GeV]Z

TP

0

0.2

0.4

0.6

0.8

12A CMS data

LONLONNLO


Z+X, |y→pp

[GeV]T, Z

p0.6

0.8

1

1.2

1.4

Rat

io to

NLO

10010

10 210 [GeV]Z

TP

0

0.2

0.4

0.6

0.8

12A CMS data

LONLONNLO


Z+X, |y→pp

[GeV]T, Z

p0.6

0.8

1

1.2

1.4

Rat

io to

NLO

10010

10 210 [GeV]Z

TP

0

0.2

0.4

0.6

0.8

10A CMS data

LONLONNLO


Z+X, |y→pp

[GeV]T, Z

p0.6

0.8

1

1.2

1.4

Rat

io to

NLO

10010

A2

A2

A1

A1A0

A0


13

Predictions vs. data (CMS)

10 210

0

0.2

0.4

0.6

0.8

10A CMS data

LONLONNLO


Z+X, |y→pp

[GeV]T, Z

p0.6

0.8

1

1.2

1.4

Rat

io to

NLO

10010

10 210

0.02−

0.01−

00.010.020.030.040.050.060.071

A

CMS dataLONLONNLO


Z+X, |y→pp

[GeV]T, Z

p0

0.5

1

1.5

2

Rat

io to

NLO

10010

10 210

0

0.02

0.04

0.06

0.08

0.1

0.121A

CMS dataLONLONNLO


Z+X, |y→pp

[GeV]T, Z

p

0.60.8

11.21.4

Rat

io to

NLO

10010

10 210

0

0.2

0.4

0.6

0.8

12A CMS data

LONLONNLO


Z+X, |y→pp

[GeV]T, Z

p0.6

0.8

1

1.2

1.4

Rat

io to

NLO

10010

10 210

0

0.2

0.4

0.6

0.8

12A CMS data

LONLONNLO


Z+X, |y→pp

[GeV]T, Z

p0.6

0.8

1

1.2

1.4

Rat

io to

NLO

10010

10 210

0

0.2

0.4

0.6

0.8

10A CMS data

LONLONNLO


Z+X, |y→pp

[GeV]T, Z

p0.6

0.8

1

1.2

1.4

Rat

io to

NLO

10010

A1

A1A0

A0

10 210 [GeV]Z

TP

0

0.2

0.4

0.6

0.8

12A CMS data

LONLONNLO


Z+X, |y→pp

[GeV]T, Z

p0.6

0.8

1

1.2

1.4

Rat

io to

NLO

10010

A2


14

Assessing Lam-Tung violationIn Collins-Soper reference frame, a relation (in FO) between :

Shown by Lam, Tung for DY(’78,’79,’80), known as Lam-Tung relation

A0, A2

A0 = A2, valid to O(↵s)

�LT = 1� A2

A0

However, in FO this relation is broken at , by real and virtualO(↵2s)

(dependence on unpol. sigma cancels)

Can quantify agreement with data with chi-squared test to

Section 2, the extent of the breaking of the Lam–Tung relation can be assessed by measuring

the pT,Z distribution for the di↵erence of the angular coe�cients A0 and A2, or equivalently

through the normalised observable �LT. The latter has the benefit of better exposing the

violation of the Lam–Tung relation in the lower pT,Z range, where the angular coe�cients

A0 and A2 are individually relatively small. In the following, we discuss the corrections to

(A0�A2) and quantify the consistency of the data and the predictions by performing a �2

test. We then present results for �LT and perform a comparison to data, where the data

points for the latter are obtained by re-expressing �LT in terms of the measured angular

coe�cients.

Before comparing to data, it is important to comment on the expected accuracy of our

theoretical predictions for these two observables. As for the individual angular coe�cients,

our theoretical predictions for (A0�A2) are obtained from the computation of the produc-

tion process for Z + jet at O(↵3s ). While the O(↵3

s ) contributions comprise genuine NNLO

corrections to the individual Ai coe�cients as demonstrated throughout this section, the

prediction degrades to an NLO-accurate description for the di↵erence (A0�A2) and �LT.8

For consistency with the rest of the paper, we will continue to refer to the corrections of

order O(↵3s ) as “NNLO corrections” and similarly label the figures in this section as NNLO

predictions.

Figure 8 shows the pT,Z distribution for (A0 � A2), where the ATLAS data is rep-

resented by black points, and is compared to LO (blue), NLO (green), and NNLO (red)

theoretical predictions. [ The NNLO corrections are observed to be large and

positive, amounting to +40% at moderate pT,Z values, and provide an improved

description of the ATLAS data. It is worth noting that while a reduction of

the absolute scale uncertainties is already observed at NNLO with respect to

NLO, the relative uncertainty is in fact reduced by almost a factor of two across

the shown pT,Z range. ] This is a reflection of the fact that the computation of the

Z + jet-production process at O(↵3s ) at finite pT,Z used to predict the di↵erence (A0 �A2)

is only NLO accurate and therefore yields corrections and remaining scale uncertainties

which are typical for NLO e↵ects. In Fig. 8, we have also chosen to include the regularised

ATLAS data (indicated by the grey fill). As discussed towards the start of this section,

large bin-to-bin correlations are introduced in the regularisation procedure of the ATLAS

data (see for example Fig. 24 of Ref. [33]). We believe that this demonstrates how a visual

comparison of the theory prediction with respect to the regularised data (in this case at

least) can lead one to overestimate the disagreement between theory and data.

As an alternative to a visual comparison, the quality of the theoretical description of

the data can again be quantified by performing a �2 test according to

�2 =NdataX

i,j

(Oiexp �Oi

th.)��1ij (Oj

exp �Ojth.), (3.2)

where Oiexp and Oi

th. are respectively the central value of the experimental and theor-

etical predictions for data point i, and ��1ij is the inverse covariance matrix. In this

8In the sense that the first non-trivial prediction for these observables begins at O(↵

2s ).

– 16 –

A0 �A2

A0 �A2 6= 0


15

210 [GeV]Z

TP

0

0.05

0.1

0.15

0.2

0.252 A− 0

A

ATLAS data (reg.)ATLAS dataLONLONNLO


Z+X, y→pp

[GeV]T, Z

p0

1

2

3

Rat

io to

NLO

100 50020

same binning choice as the data. The results are

NLO (ATLAS): �2/Ndata = 185.8/38 = 4.89 ,

NNLO (ATLAS): �2/Ndata = 68.3/38 = 1.80 .

This test indeed demonstrates that the NLO predictions give a poor description of the data

in the considered pT,Z range, a point that was also highlighted in the experimental ana-

lysis [33]. This tension is largely reduced with the inclusion of the NNLO corrections, and

from closer inspection of Fig. 3, can be mainly attributed to the large negative corrections

to the A2 distribution.

The corresponding pT,Z distributions for the CMS measurement are shown in Fig. 9 for

the rapidity bins |yZ| 2 [0.0, 1.0] (left) and |yZ| 2 [1.0, 2.1] (right). The NNLO corrections

to (A0 � A2) exhibit a similar behaviour for the CMS kinematic selections, and again

improve the description of data. This agreement can also be quantified by performing a

�2 test, where in this case the test is performed directly on the (A0 � A2) distribution as

no covariance matrix for these Ai coe�cients is publicly available. In total 14 data points

are considered, corresponding to seven pT,Z bins for each rapidity selection. The results,

assuming uncorrelated bins, are

NLO (CMS): �2/Ndata = 24.5/14 = 1.75 ,

NNLO (CMS): �2/Ndata = 14.2/14 = 1.01 .

Similar to the findings for the ATLAS data, the description of the CMS data is substantially

improved at NNLO.

As discussed previously, it is also informative to express the data in terms of the new

obserable �LT as defined in Eq. (2.15). This comparison is performed in Figs. 10 and 11 for

the ATLAS and CMS measurements, respectively, where the data has been re-expressed in

terms of this quantity.9 It is found that the extent of the Lam–Tung violation observed in

data is consistently described by the NNLO predictions. While there is some tendency for

the data to prefer a stronger Lam–Tung violation for pT,Z > 40 GeV, more precise data is

required to confirm this behaviour.

4 Conclusions and outlook

Using our calculation of the Z+ jet process at NNLO [36], we have computed the pT,Z dis-

tributions for the angular coe�cients in Z-boson production to O(↵3s ). We have focussed

on the phenomenologically most relevant angular coe�cients Ai=0,...,4 for pp collisions atps = 8 TeV and have compared them with available LHC data. [ The NNLO correc-

tions are observed to have an important impact on the predicted distributions

for A0, A1, and A2, resulting in a substantial reduction of the uncertainty of the

predictions, as well as altering the shapes of these distributions. Of particular

note is that the corrections to the A2 distribution are both large and negative

9We omit the lower panels with the K-factors in these figures, as they are almost identical to the case

of (A0 �A2) shown in Figs. 8 and 9 due to the small corrections to the A0 coe�cient.

– 18 –

Assessing Lam-Tung violation


16

10 210 [GeV]Z

TP

0

0.05

0.1

0.15

0.2

0.252 A− 0

A

CMS dataLONLONNLO


Z+X, |y→pp

[GeV]T, Z

p0.5

1

1.5

2

Rat

io to

NLO

10010

10 210 [GeV]Z

TP

0

0.05

0.1

0.15

0.2

0.252 A− 0

A

CMS dataLONLONNLO


Z+X, |y→pp

[GeV]T, Z

p0.5

1

1.5

2

Rat

io to

NLO

10010


NLO (ATLAS): �2/Ndata = 185.8/38 = 4.89 ,

NNLO (ATLAS): �2/Ndata = 68.3/38 = 1.80 .














NLO (CMS): �2/Ndata = 24.5/14 = 1.75 ,

NNLO (CMS): �2/Ndata = 14.2/14 = 1.01 .


improved at NNLO.


















– 18 –

Assessing Lam-Tung violation


17

[GeV]T, Z

p0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

LTΔ


Z+X, y→pp

0A2A

− = 1 LTΔ ATLAS dataNLONNLO

100 50020

�LT = 1� A2

A0

Assessing Lam-Tung violation (ATLAS)


•NNLO QCD necessary to describe data (NLO QCD inadequate)

•Impacts other measurements (e.g. MW, ATLAS arXiv:1701.07240)

•We can provide NNLO predictions for W/Z distributions

18

Summary & Conclusions

the three-dimensional boson production phase space, defined by the variables m, pT, and y, and the two-dimensional boson decay phase space, defined by the variables ✓ and �. Accordingly, a prediction ofthe kinematic distributions of vector bosons and their decay products can be transformed into anotherprediction by applying separate reweighting of the three-dimensional boson production phase-space dis-tributions, followed by a reweighting of the angular decay distributions.

The reweighting is performed in several steps. First, the inclusive rapidity distribution is reweightedaccording to the NNLO QCD predictions evaluated with DYNNLO. Then, at a given rapidity, the vector-boson transverse-momentum shape is reweighted to the Pythia 8 prediction with the AZ tune. This pro-cedure provides the transverse-momentum distribution of vector bosons predicted by Pythia 8, preservingthe rapidity distribution at NNLO. Finally, at given rapidity and transverse momentum, the angular vari-ables are reweighted according to:

w =1 + cos2 ✓ +

Pi A0i(pT, y) Pi(cos ✓, �)

1 + cos2 ✓ +P

i Ai(pT, y) Pi(cos ✓, �),

where A0i are the angular coe�cients evaluated at O(↵2s ), and Ai are the angular coe�cients of the

Powheg+Pythia 8 samples. This reweighting procedure neglects the small dependence of the two-dimensional (pT,y) distribution and of the angular coe�cients on the final state invariant mass. Theprocedure is used to include the corrections described in Sections 6.2 and 6.3, as well as to estimate theimpact of the QCD modelling uncertainties described in Section 6.5.

The validity of the reweighting procedure is tested at particle level by generating independent W-bosonsamples using the CT10nnlo and NNPDF3.0 [96] NNLO PDF sets, and the same value of mW . Therelevant kinematic distributions are calculated for both samples and used to reweight the CT10nnlo sampleto the NNPDF3.0 one. The procedure described in Section 2.2 is then used to determine the value of mWby fitting the NNPDF3.0 sample using templates from the reweighted CT10nnlo sample. The fitted valueagrees with the input value within 1.5±2.0 MeV. The statistical precision of this test is used to assign theassociated systematic uncertainty.

The resulting model is tested by comparing the predicted Z-boson di↵erential cross section as a functionof rapidity, the W-boson di↵erential cross section as a function of lepton pseudorapidity, and the angu-lar coe�cients in Z-boson events, to the corresponding ATLAS measurements [41, 42]. The comparisonwith the measured W and Z cross sections is shown in Figure 3. Satisfactory greement between the meas-urements and the theoretical predictions is observed. A �2 compatibility test is performed for the threedistributions simultaneously, including the correlations between the uncertainties. The compatibility testyields a �2/dof value of 45/34. Other NNLO PDF sets such as NNPDF3.0, CT14 [97], MMHT2014 [98],and ABM12 [99] are in worse agreement with these distributions. Based on the quantitative comparisonsperformed in Ref. [41], only CT10nnlo, CT14 and MMHT2014 are considered further. The better agree-ment obtained with CT10nnlo can be ascribed to the weaker suppression of the strange quark densitycompared to the u- and d-quark sea densities in this PDF set.

The predictions of the angular coe�cients in Z-boson events are compared to the ATLAS measurement atp

s = 8 TeV [42]. Good greement between the measurements and DYNNLO is observed for the relevantcoe�cients, except for A2, where the measurement is significantly below the prediction. As an example,Figure 4 shows the comparison for A0 and A2 as a function of pZ

T. For A2, an additional source ofuncertainty in the theoretical prediction is considered to account for the observed disagreement with data,as discussed in Section 6.5.3.

16

(for future MW extractions)

1) Shapes of distributions altered @ NNLO

2) Shapes of distributions not altered @ NNLO (Normalisation better determined)

A0, A1, A2

A3, A4


NLO (ATLAS): �2/Ndata = 185.8/38 = 4.89 ,

NNLO (ATLAS): �2/Ndata = 68.3/38 = 1.80 .














NLO (CMS): �2/Ndata = 24.5/14 = 1.75 ,

NNLO (CMS): �2/Ndata = 14.2/14 = 1.01 .


improved at NNLO.


















– 18 –

pT<latexit sha1_base64="Al0SeYEHRaFo2S1gfdvFx14MR60=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rNLbQhrLZbtqlm03YnQgl9Dd48aDi1T/kzX/jts1BWx8MPN6bYWZemEph0HW/ndLa+sbmVnm7srO7t39QPTx6NEmmGfdZIhPdCanhUijuo0DJO6nmNA4lb4fju5nffuLaiES1cJLyIKZDJSLBKFrJT/t5a9qv1ty6OwdZJV5BalCg2a9+9QYJy2KukElqTNdzUwxyqlEwyaeVXmZ4StmYDnnXUkVjboJ8fuyUnFllQKJE21JI5urviZzGxkzi0HbGFEdm2ZuJ/3ndDKPrIBcqzZArtlgUZZJgQmafk4HQnKGcWEKZFvZWwkZUU4Y2n4oNwVt+eZX4F/WbuvtwWWvcFmmU4QRO4Rw8uIIG3EMTfGAg4Ble4c1Rzovz7nwsWktOMXMMf+B8/gBloo6a</latexit><latexit sha1_base64="Al0SeYEHRaFo2S1gfdvFx14MR60=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rNLbQhrLZbtqlm03YnQgl9Dd48aDi1T/kzX/jts1BWx8MPN6bYWZemEph0HW/ndLa+sbmVnm7srO7t39QPTx6NEmmGfdZIhPdCanhUijuo0DJO6nmNA4lb4fju5nffuLaiES1cJLyIKZDJSLBKFrJT/t5a9qv1ty6OwdZJV5BalCg2a9+9QYJy2KukElqTNdzUwxyqlEwyaeVXmZ4StmYDnnXUkVjboJ8fuyUnFllQKJE21JI5urviZzGxkzi0HbGFEdm2ZuJ/3ndDKPrIBcqzZArtlgUZZJgQmafk4HQnKGcWEKZFvZWwkZUU4Y2n4oNwVt+eZX4F/WbuvtwWWvcFmmU4QRO4Rw8uIIG3EMTfGAg4Ble4c1Rzovz7nwsWktOMXMMf+B8/gBloo6a</latexit><latexit sha1_base64="Al0SeYEHRaFo2S1gfdvFx14MR60=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rNLbQhrLZbtqlm03YnQgl9Dd48aDi1T/kzX/jts1BWx8MPN6bYWZemEph0HW/ndLa+sbmVnm7srO7t39QPTx6NEmmGfdZIhPdCanhUijuo0DJO6nmNA4lb4fju5nffuLaiES1cJLyIKZDJSLBKFrJT/t5a9qv1ty6OwdZJV5BalCg2a9+9QYJy2KukElqTNdzUwxyqlEwyaeVXmZ4StmYDnnXUkVjboJ8fuyUnFllQKJE21JI5urviZzGxkzi0HbGFEdm2ZuJ/3ndDKPrIBcqzZArtlgUZZJgQmafk4HQnKGcWEKZFvZWwkZUU4Y2n4oNwVt+eZX4F/WbuvtwWWvcFmmU4QRO4Rw8uIIG3EMTfGAg4Ble4c1Rzovz7nwsWktOMXMMf+B8/gBloo6a</latexit>


19

Angular coefficients for W

kµ1,2 =

pq2

2(1,± sin ✓ cos�,± sin ✓ sin�,± cos ✓)T

Defining lepton kinematics in V(q) rest frame

d�

d4q cos ✓ d�=

3

16⇡

d�unpol.

d4q

⇢(1 + cos2 ✓) +

1

2A0 (1� 3 cos2 ✓)

+A1 sin(2✓) cos�+1

2A2 sin2 ✓ cos(2�)

+A3 sin ✓ cos�+A4 cos ✓ +A5 sin2 ✓ sin(2�)

+A6 sin(2✓) sin�+A7 sin ✓ sin�

�

Decompose cross section in terms of spherical polynomials fi(✓,�)

p(p1) + p(p2) ! V (q) +X ! `(k1) + ¯(k2) +X<latexit sha1_base64="75kbhH4ewhHGJyv2LUr98zLREk0=">AAACJHicbZDLSsNAFIYn9VbrLerSzWARUgolKYIKLopuXFYwbaEJYTKdtkMnF2cmQgl9GTe+ihsXVly48VmcpFlo64GBj//8hzPn92NGhTTNL620tr6xuVXeruzs7u0f6IdHHRElHBMbRyziPR8JwmhIbEklI72YExT4jHT9yW3W7z4RLmgUPshpTNwAjUI6pBhJJXn6dWzEnlWDdZhBswYdGcGO8Vir93J0CGPGZOFwfMTTTJgppakcnl41G2ZecBWsAqqgqLanz51BhJOAhBIzJETfMmPppohLihmZVZxEkBjhCRqRvsIQBUS4aX7lDJ4pZQCHEVcvlDBXf0+kKBBiGvjKGSA5Fsu9TPyv10/k8NJNaRgnkoR4sWiYMKjOzyKDA8oJlmyqAGFO1V8hHiOOsFTBVlQI1vLJq2A3G1cN8/682rop0iiDE3AKDGCBC9ACd6ANbIDBM3gF72CuvWhv2of2ubCWtGLmGPwp7fsHhwGgnw==</latexit><latexit sha1_base64="75kbhH4ewhHGJyv2LUr98zLREk0=">AAACJHicbZDLSsNAFIYn9VbrLerSzWARUgolKYIKLopuXFYwbaEJYTKdtkMnF2cmQgl9GTe+ihsXVly48VmcpFlo64GBj//8hzPn92NGhTTNL620tr6xuVXeruzs7u0f6IdHHRElHBMbRyziPR8JwmhIbEklI72YExT4jHT9yW3W7z4RLmgUPshpTNwAjUI6pBhJJXn6dWzEnlWDdZhBswYdGcGO8Vir93J0CGPGZOFwfMTTTJgppakcnl41G2ZecBWsAqqgqLanz51BhJOAhBIzJETfMmPppohLihmZVZxEkBjhCRqRvsIQBUS4aX7lDJ4pZQCHEVcvlDBXf0+kKBBiGvjKGSA5Fsu9TPyv10/k8NJNaRgnkoR4sWiYMKjOzyKDA8oJlmyqAGFO1V8hHiOOsFTBVlQI1vLJq2A3G1cN8/682rop0iiDE3AKDGCBC9ACd6ANbIDBM3gF72CuvWhv2of2ubCWtGLmGPwp7fsHhwGgnw==</latexit><latexit sha1_base64="75kbhH4ewhHGJyv2LUr98zLREk0=">AAACJHicbZDLSsNAFIYn9VbrLerSzWARUgolKYIKLopuXFYwbaEJYTKdtkMnF2cmQgl9GTe+ihsXVly48VmcpFlo64GBj//8hzPn92NGhTTNL620tr6xuVXeruzs7u0f6IdHHRElHBMbRyziPR8JwmhIbEklI72YExT4jHT9yW3W7z4RLmgUPshpTNwAjUI6pBhJJXn6dWzEnlWDdZhBswYdGcGO8Vir93J0CGPGZOFwfMTTTJgppakcnl41G2ZecBWsAqqgqLanz51BhJOAhBIzJETfMmPppohLihmZVZxEkBjhCRqRvsIQBUS4aX7lDJ4pZQCHEVcvlDBXf0+kKBBiGvjKGSA5Fsu9TPyv10/k8NJNaRgnkoR4sWiYMKjOzyKDA8oJlmyqAGFO1V8hHiOOsFTBVlQI1vLJq2A3G1cN8/682rop0iiDE3AKDGCBC9ACd6ANbIDBM3gF72CuvWhv2of2ubCWtGLmGPwp7fsHhwGgnw==</latexit>

~pT,⌫ pµ`<latexit sha1_base64="XOntL/28E6FYLwADnwylGOpmTKw=">AAACDnicbVC7TgJBFJ3FF+ILtbSZSDQWhizGRO2INpaYsELCIpkdLjBhdnaZBwnZ7B/Y+Cs2Fmpsre38G4dHoeBJbnJyzr25954g5kxp1/12MkvLK6tr2fXcxubW9k5+d+9eRUZS8GjEI1kPiALOBHiaaQ71WAIJAw61oH8z9mtDkIpFoqpHMTRD0hWswyjRVmrlj/0h0CROW0n11BcmxdgfDAxp4/gh8UNjdR84T1v5glt0J8CLpDQjBTRDpZX/8tsRNSEITTlRqlFyY91MiNSMckhzvlEQE9onXWhYKkgIqplM/knxkVXauBNJW0Ljifp7IiGhUqMwsJ0h0T01743F/7yG0Z3LZsJEbDQIOl3UMRzrCI/DwW0mgWo+soRQyeytmPaIJFTbCHM2hNL8y4vEOyteFd2780L5epZGFh2gQ3SCSugCldEtqiAPUfSIntErenOenBfn3fmYtmac2cw++gPn8wcHZpze</latexit><latexit sha1_base64="XOntL/28E6FYLwADnwylGOpmTKw=">AAACDnicbVC7TgJBFJ3FF+ILtbSZSDQWhizGRO2INpaYsELCIpkdLjBhdnaZBwnZ7B/Y+Cs2Fmpsre38G4dHoeBJbnJyzr25954g5kxp1/12MkvLK6tr2fXcxubW9k5+d+9eRUZS8GjEI1kPiALOBHiaaQ71WAIJAw61oH8z9mtDkIpFoqpHMTRD0hWswyjRVmrlj/0h0CROW0n11BcmxdgfDAxp4/gh8UNjdR84T1v5glt0J8CLpDQjBTRDpZX/8tsRNSEITTlRqlFyY91MiNSMckhzvlEQE9onXWhYKkgIqplM/knxkVXauBNJW0Ljifp7IiGhUqMwsJ0h0T01743F/7yG0Z3LZsJEbDQIOl3UMRzrCI/DwW0mgWo+soRQyeytmPaIJFTbCHM2hNL8y4vEOyteFd2780L5epZGFh2gQ3SCSugCldEtqiAPUfSIntErenOenBfn3fmYtmac2cw++gPn8wcHZpze</latexit><latexit sha1_base64="XOntL/28E6FYLwADnwylGOpmTKw=">AAACDnicbVC7TgJBFJ3FF+ILtbSZSDQWhizGRO2INpaYsELCIpkdLjBhdnaZBwnZ7B/Y+Cs2Fmpsre38G4dHoeBJbnJyzr25954g5kxp1/12MkvLK6tr2fXcxubW9k5+d+9eRUZS8GjEI1kPiALOBHiaaQ71WAIJAw61oH8z9mtDkIpFoqpHMTRD0hWswyjRVmrlj/0h0CROW0n11BcmxdgfDAxp4/gh8UNjdR84T1v5glt0J8CLpDQjBTRDpZX/8tsRNSEITTlRqlFyY91MiNSMckhzvlEQE9onXWhYKkgIqplM/knxkVXauBNJW0Ljifp7IiGhUqMwsJ0h0T01743F/7yG0Z3LZsJEbDQIOl3UMRzrCI/DwW0mgWo+soRQyeytmPaIJFTbCHM2hNL8y4vEOyteFd2780L5epZGFh2gQ3SCSugCldEtqiAPUfSIntErenOenBfn3fmYtmac2cw++gPn8wcHZpze</latexit>

(p⌫ + p`)2 = m2

W ! |pz,⌫ |<latexit sha1_base64="SYB7OSEs7NhZRR3fofEjXwit38w=">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</latexit><latexit sha1_base64="SYB7OSEs7NhZRR3fofEjXwit38w=">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</latexit><latexit sha1_base64="SYB7OSEs7NhZRR3fofEjXwit38w=">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</latexit>

✓ ⌘ ⇡ � ✓<latexit sha1_base64="P56FL47cBXL///+T7t9J1APCpfo=">AAACAnicbVA9SwNBEN3zM8avUzttFoNgY7gTQe2CNpYRjAnkQtjbTJIle3vn7lwgHAEb/4qNhYqtv8LOf+Pmo9DEBwOP92aYmRcmUhj0vG9nYXFpeWU1t5Zf39jc2nZ3du9NnGoOFR7LWNdCZkAKBRUUKKGWaGBRKKEa9q5HfrUP2ohY3eEggUbEOkq0BWdopaa7H2AXkNEAHlLRp0Ei6AmdaE234BW9Meg88aekQKYoN92voBXzNAKFXDJj6r6XYCNjGgWXMMwHqYGE8R7rQN1SxSIwjWz8w5AeWaVF27G2pZCO1d8TGYuMGUSh7YwYds2sNxL/8+opti8amVBJiqD4ZFE7lRRjOgqEtoQGjnJgCeNa2Fsp7zLNONrY8jYEf/bleVI5LV4WvduzQulqmkaOHJBDckx8ck5K5IaUSYVw8kieySt5c56cF+fd+Zi0LjjTmT3yB87nDzoFls4=</latexit><latexit sha1_base64="P56FL47cBXL///+T7t9J1APCpfo=">AAACAnicbVA9SwNBEN3zM8avUzttFoNgY7gTQe2CNpYRjAnkQtjbTJIle3vn7lwgHAEb/4qNhYqtv8LOf+Pmo9DEBwOP92aYmRcmUhj0vG9nYXFpeWU1t5Zf39jc2nZ3du9NnGoOFR7LWNdCZkAKBRUUKKGWaGBRKKEa9q5HfrUP2ohY3eEggUbEOkq0BWdopaa7H2AXkNEAHlLRp0Ei6AmdaE234BW9Meg88aekQKYoN92voBXzNAKFXDJj6r6XYCNjGgWXMMwHqYGE8R7rQN1SxSIwjWz8w5AeWaVF27G2pZCO1d8TGYuMGUSh7YwYds2sNxL/8+opti8amVBJiqD4ZFE7lRRjOgqEtoQGjnJgCeNa2Fsp7zLNONrY8jYEf/bleVI5LV4WvduzQulqmkaOHJBDckx8ck5K5IaUSYVw8kieySt5c56cF+fd+Zi0LjjTmT3yB87nDzoFls4=</latexit><latexit sha1_base64="P56FL47cBXL///+T7t9J1APCpfo=">AAACAnicbVA9SwNBEN3zM8avUzttFoNgY7gTQe2CNpYRjAnkQtjbTJIle3vn7lwgHAEb/4qNhYqtv8LOf+Pmo9DEBwOP92aYmRcmUhj0vG9nYXFpeWU1t5Zf39jc2nZ3du9NnGoOFR7LWNdCZkAKBRUUKKGWaGBRKKEa9q5HfrUP2ohY3eEggUbEOkq0BWdopaa7H2AXkNEAHlLRp0Ei6AmdaE234BW9Meg88aekQKYoN92voBXzNAKFXDJj6r6XYCNjGgWXMMwHqYGE8R7rQN1SxSIwjWz8w5AeWaVF27G2pZCO1d8TGYuMGUSh7YwYds2sNxL/8+opti8amVBJiqD4ZFE7lRRjOgqEtoQGjnJgCeNa2Fsp7zLNONrY8jYEf/bleVI5LV4WvduzQulqmkaOHJBDckx8ck5K5IaUSYVw8kieySt5c56cF+fd+Zi0LjjTmT3yB87nDzoFls4=</latexit>

20

ATLAS, unregularised data

[GeV]ZT

p1 10 210

2-A 0A

-0.1

0

0.1

0.2

0.3UnregularisedRegularised

-18 TeV, 20.3 fbATLAS

-integratedZ: yCC

µµ+CCee

Figure 28: For the eeCC+µµCC channel in the yZ-integrated configuration, overlays of regularised with unregularisedresults are shown for A0 � A2.

57

21

NNLO Subtraction

�NNLO =

Z

�n+2

�d�RR

NNLO� d�S

NNLO

�

+

Z

�n+1

�d�RV

NNLO� d�T

NNLO

�

+

Z

�n+0

�d�V V

NNLO� d�U

NNLO

�

X= Finite� 0

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Each line individually finite, can be integrated in 4-d

mimic unresolved

explicit pole cancellation

22

[GeV]T, Z

p0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

LTΔ


Z+X, |y→pp

0A2A

− = 1 LTΔ CMS dataNLONNLO

10010 [GeV]

T, Zp

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

LTΔ


Z+X, |y→pp

0A2A

− = 1 LTΔ CMS dataNLONNLO

10010

�LT = 1� A2

A0

Assessing Lam-Tung violation (CMS)

23

(un)correlated uncertainties

210 [GeV]Z

TP

0

0.2

0.4

0.6

0.8

12A

NNLO central

NLO uncorrelated

NLO correlated


Z+X, y→pp

[GeV]T, Z

p

0.8

1

1.2

Rat

io to

NLO

100 50020

210 [GeV]Z

TP

0

0.2

0.4

0.6

0.8

12A

NLO central

NNLO uncorrelated

NNLO correlated


Z+X, y→pp

[GeV]T, Z

p

0.8

1

1.2

Rat

io to

NLO

100 50020

NLO NNLO

( )

Angular coefficients first measured in low-mass Dell-Yan

* Measurements by Na10/E615 in late 80s (Pion beams on Tungsten)* Measurements by NuSea ’06/’08 (pp and pd collisions)

At LHC and TeVatron, measurements performed around Z-boson mass* CDF measurement - arXiv:1103.5699 * ATLAS/CMS measurements - arXiv:1606.00689 / arXiv:1504.03512

Measurement of angular coefficients (polarisation) in W production * ATLAS/CMS measurements - arXiv:1203.2165 / arXiv:1104.3829

24

History lessonA0, A1, A2

A3, A4(sensitive to )

(see ref. [27-30] of arXiv:1708.00008)

pT,W > 30/50 GeV

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6 7 Results

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Figure 1: Relative uncertainties in percent of the normalised fiducial cross section measure-ment. Each plot shows the qT dependence in the indicated ranges of |y|.

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Figure 2: Relative uncertainties in percent of the absolute fiducial cross section measurement.The 2.6% uncertainty in the luminosity is not included. Each plot shows the qT dependence inthe indicated ranges of |y|.

five bins in |y| and the last plot shows the qT dependence integrated over |y|. In the bottompanels the ratio of the FEWZ prediction to data is shown. The vertical error bars represent thestatistical uncertainties of data and simulation. The red-hatched bands drawn at the pointsrepresent the systematic uncertainties of the measurement only. The scale uncertainties areindicated by the grey-shaded areas and the PDF uncertainties by the light-hatched bands. Thescale uncertainties are estimated from the envelope of the following combinations of variationsof the factorisation µF and the renormalisation µR scales: (2µF,2µR), (0.5µF,0.5µR), (2µF,µR),

Experimentally

Anastasiou et al. - arXiv:03012266

CMS - arXiv:1504.03511

• Known to NNLO QCD ( and )

• Measured lepton distributions sensitive to:

pT,Z<latexit sha1_base64="3CpRDK+a8OuxSATVMol963168dM=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBg5REBPVW9OKxQmOLbSib7aZdutksuxuhhPwILx5UvPp/vPlv3LY5aOuDgcd7M8zMCyVn2rjut1NaWV1b3yhvVra2d3b3qvsHDzpJFaE+SXiiOiHWlDNBfcMMpx2pKI5DTtvh+Hbqt5+o0iwRLTORNIjxULCIEWys1Jb9rHX2mPerNbfuzoCWiVeQGhRo9qtfvUFC0pgKQzjWuuu50gQZVoYRTvNKL9VUYjLGQ9q1VOCY6iCbnZujE6sMUJQoW8Kgmfp7IsOx1pM4tJ0xNiO96E3F/7xuaqKrIGNCpoYKMl8UpRyZBE1/RwOmKDF8YgkmitlbERlhhYmxCVVsCN7iy8vEP69f1937i1rjpkijDEdwDKfgwSU04A6a4AOBMTzDK7w50nlx3p2PeWvJKWYO4Q+czx9+CY80</latexit><latexit sha1_base64="3CpRDK+a8OuxSATVMol963168dM=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBg5REBPVW9OKxQmOLbSib7aZdutksuxuhhPwILx5UvPp/vPlv3LY5aOuDgcd7M8zMCyVn2rjut1NaWV1b3yhvVra2d3b3qvsHDzpJFaE+SXiiOiHWlDNBfcMMpx2pKI5DTtvh+Hbqt5+o0iwRLTORNIjxULCIEWys1Jb9rHX2mPerNbfuzoCWiVeQGhRo9qtfvUFC0pgKQzjWuuu50gQZVoYRTvNKL9VUYjLGQ9q1VOCY6iCbnZujE6sMUJQoW8Kgmfp7IsOx1pM4tJ0xNiO96E3F/7xuaqKrIGNCpoYKMl8UpRyZBE1/RwOmKDF8YgkmitlbERlhhYmxCVVsCN7iy8vEP69f1937i1rjpkijDEdwDKfgwSU04A6a4AOBMTzDK7w50nlx3p2PeWvJKWYO4Q+czx9+CY80</latexit><latexit sha1_base64="3CpRDK+a8OuxSATVMol963168dM=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBg5REBPVW9OKxQmOLbSib7aZdutksuxuhhPwILx5UvPp/vPlv3LY5aOuDgcd7M8zMCyVn2rjut1NaWV1b3yhvVra2d3b3qvsHDzpJFaE+SXiiOiHWlDNBfcMMpx2pKI5DTtvh+Hbqt5+o0iwRLTORNIjxULCIEWys1Jb9rHX2mPerNbfuzoCWiVeQGhRo9qtfvUFC0pgKQzjWuuu50gQZVoYRTvNKL9VUYjLGQ9q1VOCY6iCbnZujE6sMUJQoW8Kgmfp7IsOx1pM4tJ0xNiO96E3F/7xuaqKrIGNCpoYKMl8UpRyZBE1/RwOmKDF8YgkmitlbERlhhYmxCVVsCN7iy8vEP69f1937i1rjpkijDEdwDKfgwSU04A6a4AOBMTzDK7w50nlx3p2PeWvJKWYO4Q+czx9+CY80</latexit>

Typically* scale uncertainties sub-leading

Choice of input PDFsElectroweak input parameters (s2w, …)

*Observable dependent

• Excellent performance of LHC experiments

• CMS example: Normalised distribution

Sub-percent precision achieved in Run-I

pT,Z<latexit sha1_base64="3CpRDK+a8OuxSATVMol963168dM=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBg5REBPVW9OKxQmOLbSib7aZdutksuxuhhPwILx5UvPp/vPlv3LY5aOuDgcd7M8zMCyVn2rjut1NaWV1b3yhvVra2d3b3qvsHDzpJFaE+SXiiOiHWlDNBfcMMpx2pKI5DTtvh+Hbqt5+o0iwRLTORNIjxULCIEWys1Jb9rHX2mPerNbfuzoCWiVeQGhRo9qtfvUFC0pgKQzjWuuu50gQZVoYRTvNKL9VUYjLGQ9q1VOCY6iCbnZujE6sMUJQoW8Kgmfp7IsOx1pM4tJ0xNiO96E3F/7xuaqKrIGNCpoYKMl8UpRyZBE1/RwOmKDF8YgkmitlbERlhhYmxCVVsCN7iy8vEP69f1937i1rjpkijDEdwDKfgwSU04A6a4AOBMTzDK7w50nlx3p2PeWvJKWYO4Q+czx9+CY80</latexit><latexit sha1_base64="3CpRDK+a8OuxSATVMol963168dM=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBg5REBPVW9OKxQmOLbSib7aZdutksuxuhhPwILx5UvPp/vPlv3LY5aOuDgcd7M8zMCyVn2rjut1NaWV1b3yhvVra2d3b3qvsHDzpJFaE+SXiiOiHWlDNBfcMMpx2pKI5DTtvh+Hbqt5+o0iwRLTORNIjxULCIEWys1Jb9rHX2mPerNbfuzoCWiVeQGhRo9qtfvUFC0pgKQzjWuuu50gQZVoYRTvNKL9VUYjLGQ9q1VOCY6iCbnZujE6sMUJQoW8Kgmfp7IsOx1pM4tJ0xNiO96E3F/7xuaqKrIGNCpoYKMl8UpRyZBE1/RwOmKDF8YgkmitlbERlhhYmxCVVsCN7iy8vEP69f1937i1rjpkijDEdwDKfgwSU04A6a4AOBMTzDK7w50nlx3p2PeWvJKWYO4Q+czx9+CY80</latexit><latexit sha1_base64="3CpRDK+a8OuxSATVMol963168dM=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBg5REBPVW9OKxQmOLbSib7aZdutksuxuhhPwILx5UvPp/vPlv3LY5aOuDgcd7M8zMCyVn2rjut1NaWV1b3yhvVra2d3b3qvsHDzpJFaE+SXiiOiHWlDNBfcMMpx2pKI5DTtvh+Hbqt5+o0iwRLTORNIjxULCIEWys1Jb9rHX2mPerNbfuzoCWiVeQGhRo9qtfvUFC0pgKQzjWuuu50gQZVoYRTvNKL9VUYjLGQ9q1VOCY6iCbnZujE6sMUJQoW8Kgmfp7IsOx1pM4tJ0xNiO96E3F/7xuaqKrIGNCpoYKMl8UpRyZBE1/RwOmKDF8YgkmitlbERlhhYmxCVVsCN7iy8vEP69f1937i1rjpkijDEdwDKfgwSU04A6a4AOBMTzDK7w50nlx3p2PeWvJKWYO4Q+czx9+CY80</latexit>

Extremely clean + lots of stats

Theoretically


26Rhorry Gauld, ETH Zurich

Looking forward (LHCb)

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Noname manuscript No.

(will be inserted by the editor)

Prospects for improving the LHC W boson mass measurement with

forward muons

Giuseppe Bozzia,1

, Luca Citellib,2

, Mika Vesterinenc,4

,

Alessandro Vicinid,3

1 Dipartimento di Fisica, Universita degli Studi di Milano and INFN, Sezione di Milano, Via Celoria 16, 20133 Milano(Italy)

2Dipartimento di Fisica, Universita degli Studi di Milano, Via Celoria 16, 20133 Milano (Italy)3Dipartimento di Fisica, Universita degli Studi di Milano and INFN, Sezione di Milano and TIF Lab, Via Celoria 16,

20133 Milano (Italy)4Physikalisches Institut, Ruprecht-Karls-Universitat, Im Neuenheimer Feld 226 69120 Heidelberg (Deutschland)

the date of receipt and acceptance should be inserted later

Abstract Measurements of the W boson mass are planned by the ATLAS and CMS experiments, butfor the time being, these may be unable to compete with the current world average precision of 15 MeV,due to uncertainties in the PDFs. We discuss the potential of a measurement by the LHCb experimentbased on the charged lepton transverse momentum p`T spectrum in W ! µ⌫ decays. The unique forwardacceptance of LHCb means that the PDF uncertainties would be anti-correlated with those of p`T basedmeasurements by ATLAS and CMS. We compute an average of ATLAS, CMS and LHCb measurementsof mW from the p`T distribution. Considering PDF uncertainties, this average is a factor of 1.3 moreprecise than an average of ATLAS and CMS alone. Despite the relatively low rate of W production inLHCb, we estimate that with the Run-II dataset, a measurement could be performed with su�cientexperimental precision to exploit this anti-correlation in PDF uncertainties. The modelling of the lepton-pair transverse momentum distribution in the neutral current Drell-Yan process could be a limiting factorof this measurement and will deserve further studies.

1 Introduction

The Standard Model precisely relates the mass of the W boson to the more precisely measured Zboson mass, fine structure constant and Fermi constant. The resulting indirect constraint on the Wboson mass from a global fit [1,2,3] to experimental data is roughly a factor of two more precise thanthe direct measurement, mW = 80.385 ± 0.015 GeV [4], leaving room for new physics, for examplein supersymmetry [5]. The world average for mW is dominated by measurements from the FermilabTevatron collider experiments, CDF [6,7] and D0 [8,9]. The CDF and D0 measurements used 2.1 fb�1

and 4.9 fb�1, respectively, out of the roughly 10 fb�1 Tevatron Run-II dataset. Updates from bothexperiments are therefore highly anticipated. The current measurements are mostly limited by statisticaluncertainties; either directly through limitedW samples or indirectly through limited calibration samples.The uncertainty due to the parton distribution functions (PDFs) is around 10 MeV, with some variationbetween experiment, lepton flavour and fit variable. The Tevatron measurements used three di↵erent fitvariables to extract mW :

1. The transverse mass, mWT =

q2p`T�ET (1� cos�), where p`T is the charged lepton transverse momen-

tum,�ET is the missing transverse energy measured by the calorimeter, which estimates the neutrinotransverse momentum; and � is the azimuthal opening angle between the neutrino and charged lepton.

2. The charged lepton transverse momentum p`T itself,3. The missing transverse energy�ET itself.

aemail: [email protected]: [email protected]: [email protected]: [email protected]

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Table 5 The uncertainties on di↵erent LHC averages for mW . The separate experimental and PDF uncertainties arelisted, as are the weights that minimise the total uncertainty.

�mW (MeV)Scenario Experiments Tot Exp PDF ↵

Default 2⇥GPD + LHCb 9.0 4.7 7.7 (0.30, 0.44, 0.22, 0.04)

Default 1⇥GPD + LHCb 10.1 6.5 7.7 (0.31, 0.40, 0.25, 0.04)Default 2⇥GPD 12.0 5.8 10.5 (0.28, 0.72, 0, 0)

PDF4LHC(3-sets) 2⇥GPD + LHCb 13.6 4.8 12.7 (0.43, 0.41, 0.12, 0.04)

PDF4LHC(3-sets) 1⇥GPD + LHCb 14.6 7.3 12.7 (0.43, 0.40, 0.12, 0.04)PDF4LHC(3-sets) 2⇥GPD 17.7 5.5 16.9 (0.50, 0.50, 0, 0)

�LHCbexp = 0 2⇥GPD + LHCb 8.7 4.0 7.7 (0.31, 0.41, 0.24, 0.04)

�LHCbexp = 0 1⇥GPD + LHCb 9.8 5.9 7.9 (0.31, 0.37, 0.28, 0.04)�LHCbexp = 0 2⇥GPD 12.0 5.8 10.5 (0.28, 0.72, 0, 0)

�GPDexp = 0 2⇥GPD + LHCb 7.9 1.9 7.7 (0.29, 0.48, 0.19, 0.04)

�GPDexp = 0 1⇥GPD + LHCb 7.9 1.9 7.7 (0.29, 0.48, 0.19, 0.04)�GPDexp = 0 2⇥GPD 10.5 0.1 10.5 (0.26, 0.74, 0, 0)

�PDF = 0 2⇥GPD + LHCb 4.6 4.6 0.0 (0.34, 0.34, 0.22, 0.10)

�PDF = 0 1⇥GPD + LHCb 5.8 5.8 0.0 (0.23, 0.23, 0.37, 0.17)�PDF = 0 2⇥GPD 5.5 5.5 0.0 (0.50, 0.50, 0, 0)

�LHCbexp ⇥ 2 2⇥GPD + LHCb 9.6 5.6 7.7 (0.29, 0.50, 0.17, 0.04)

�LHCbexp ⇥ 2 1⇥GPD + LHCb 10.8 7.6 7.7 (0.30, 0.46, 0.20, 0.05)�LHCbexp ⇥ 2 2⇥GPD 12.0 5.8 10.5 (0.28, 0.72, 0, 0)

�GPDexp ⇥ 2 2⇥GPD + LHCb 11.2 7.9 8.0 (0.32, 0.35, 0.29, 0.04)

�GPDexp ⇥ 2 1⇥GPD + LHCb 13.9 10.5 9.0 (0.31, 0.26, 0.37, 0.05)�GPDexp ⇥ 2 2⇥GPD 15.6 11.5 10.6 (0.32, 0.68, 0, 0)

�PDF ⇥ 2 2⇥GPD + LHCb 16.0 4.7 15.3 (0.30, 0.45, 0.21, 0.04)

�PDF ⇥ 2 1⇥GPD + LHCb 16.7 6.7 15.3 (0.30, 0.44, 0.22, 0.04)�PDF ⇥ 2 2⇥GPD 21.7 5.9 20.9 (0.27, 0.73, 0, 0)

5 Uncertainties stemming from the pWT modelling

Another source of theoretical uncertainty that we have overlooked so far is the pWT model. This stronglya↵ects the preparation of the templates that are used to fit the data and eventually to extract mW . Thepresence, at low lepton-pair transverse momenta, of large logarithmically enhanced QCD correctionsand the role, in the same kinematic region, of non-perturbative e↵ects have been discussed in Refs. [30,31], but the dependence of the resulting model on the acceptance cuts has never been investigated indetail and will deserve a dedicated study. The p`T is more sensitive to this than mW

T . At the Tevatron,the uncertainty from the pWT model on the p`T fit was around 5 MeV, but perturbative QCD scaleuncertainties should also be taken into account. To a first approximation the results of the present noteare independent of the uncertainty stemming from on the pWT modelling and will hopefully be confirmed ifthe latter will turn out to be under control. On a longer term perspective we will need a global analysis ofall the non-perturbative elements active in the proton description: the PDFs uncertainties, in particularthe role of heavy quarks in the proton [32,12], and the description of the intrinsic transverse momentumof the partons. The inclusion of all the di↵erent Drell-Yan channels (neutral current, W+ and W�) inthe di↵erent acceptance regions of the LHC experiments might have an impact on a systematic reductionof all these uncertainties.

6 Summary

Improving the precision on mW remains a priority in particle physics. At the LHC, there is no shortage ofW production, but there are potentially limiting PDF uncertainties on the anticipated measurements byATLAS and CMS, which cover central lepton pseudorapidities, |⌘| . 2.5. We show that a measurementin the forward acceptance of the LHCb experiment, 2 < ⌘ < 4.5, would have a PDF uncertainty thatis highly anti-correlated with those of ATLAS and CMS. In this paper we study the measurement of

8

work with a. gehrmann-de ridder, t. gehrmann, e.w.n. glover, a....

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