Heavy neutrinos from gluon fusion

Richard Ruiz; Michael Spannowsky; Philip Waite

arXiv:1706.02298·hep-ph·September 29, 2017

Heavy neutrinos from gluon fusion

Richard Ruiz, Michael Spannowsky, Philip Waite

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TL;DR

This paper demonstrates that gluon fusion is a significant production mechanism for heavy neutrinos at high-energy colliders, with corrections greatly enhancing predicted cross sections, challenging previous assumptions about its relevance.

Contribution

It provides the first detailed resummation of soft gluon corrections up to N$^3$LL for heavy neutrino production via gluon fusion, showing its importance at collider energies.

Findings

01

Gluon fusion cross section increases by 160-260% with corrections.

02

At 14 TeV, gluon fusion rivals Drell-Yan production.

03

Gluon fusion overtakes Drell-Yan at energies above 20-25 TeV.

Abstract

Heavy neutrinos, a key prediction of many standard model extensions, remain some of the most searched-for objects at collider experiments. In this context, we revisit the premise that the gluon fusion production mechanism, $g g \to Z^{*} / h^{*} \to N ν_{ℓ}$ , is phenomenologically irrelevant at the CERN LHC and report the impact of soft gluon corrections to the production cross section. We resum threshold logarithms up to next-to-next-to-next-to-leading logarithmic accuracy (N $^{3}$ LL), thus capturing the dominant contributions to the inclusive cross section up to next-to-next-to-leading order (N $^{2}$ LO). For $m_{N} > 150$ GeV and collider energies $s = 7 - 100$ TeV, corrections to the Born rates span $+ 160$ to $+ 260%$ . At $s$ =14 TeV, the resummed channel is roughly equal in size to the widely-believed-to-be-dominant charged current Drell-Yan process and overtakes it outright at…

Tables3

Table 1. Table 1: Heavy N 𝑁 N production cross sections via the GF mode at various accuracies, divided by active-heavy mixing | V ℓ N | 2 superscript subscript 𝑉 ℓ 𝑁 2 |V_{\ell N}|^{2} , scale dependence ( % ) (\%) , and K 𝐾 K -factor, for representative m N subscript 𝑚 𝑁 m_{N} and s 𝑠 \sqrt{s} .

\sqrt{s}

14 TeV

33 TeV

100 TeV

m_{N}

300 GeV

500 GeV

300 GeV

500 GeV

300 GeV

1 TeV

σ / {| V_{ℓ ​ N} |}^{2}

[fb]

σ

[fb]

K

σ

[fb]

K

σ

[fb]

K

σ

[fb]

K

σ

[fb]

K

σ

[fb]

K

GF LO

65.4

\dots

13.5

\dots

415

\dots

115

\dots

2.84

\times 10^{3}

\dots

154

\dots

GF NLL

65.9

{}_{- 26 %}^{+ 14 %}

1.01

13.7

{}_{- 27 %}^{+ 17 %}

1.01

414

{}_{- 23 %}^{+ 8 %}

1.00

115

{}_{- 24 %}^{+ 11 %}

1.00

2.83

{}_{- 18 %}^{+ 2 %}\times 10^{3}

1.00

154

{}_{- 21 %}^{+ 8 %}

1.00

GF

N^{2} ​ LL

170

{}_{- 7 %}^{< 1 %}

2.61

34.9

{}_{- 8 %}^{< 1 %}

2.59

1.03

{}_{- 9 %}^{< 1 %}\times 10^{3}

2.49

281

{}_{- 7 %}^{< 1 %}

2.45

6.83

{}_{- 13 %}^{+ 2 %}\times 10^{3}

2.40

351

{}_{- 8 %}^{< 1 %}

2.28

GF

N^{3} ​ LL

202

{}_{- 11 %}^{+ 5 %}

3.09

41.3

{}_{- 9 %}^{+ 3 %}

3.06

1.21

{}_{- 13 %}^{+ 8 %}\times 10^{3}

2.92

327

{}_{- 11 %}^{+ 6 %}

2.85

7.88

{}_{- 16 %}^{+ 13 %}\times 10^{3}

2.77

401

{}_{- 11 %}^{+ 8 %}

2.60

Table 2. Table 2: Total hadronic cross sections for g g → Z ∗ / h ∗ → N ν ℓ → 𝑔 𝑔 superscript 𝑍 superscript ℎ → 𝑁 subscript 𝜈 ℓ gg\to Z^{*}/h^{*}\to N\nu_{\ell} at various accuracies, divided by active-heavy mixing | V ℓ N | 2 superscript subscript 𝑉 ℓ 𝑁 2 |V_{\ell N}|^{2} , for representative collider energies s 𝑠 \sqrt{s} .

$\sqrt{s} = 13$ TeV
$m_{N}$ [GeV]	$σ^{LO} / {\| V_{ℓ N} \|}^{2}$ [fb]	$σ^{NLL} / {\| V_{ℓ N} \|}^{2}$ [fb]	$σ^{N^{2} LL} / {\| V_{ℓ N} \|}^{2}$ [fb]	$σ^{N^{3} LL} / {\| V_{ℓ N} \|}^{2}$ [fb]
150	0.9097E+02	0.9131E+02 ${}_{- 25.4 %}^{+ 11.8 %}$	0.2419E+03 ${}_{- 7.2 %}^{+, < 0.1 %}$	0.2867E+03 ${}_{- 12.3 %}^{+ 6.5 %}$
175	0.8811E+02	0.8868E+02 ${}_{- 25.8 %}^{+ 12.4 %}$	0.2334E+03 ${}_{- 6.8 %}^{+, < 0.1 %}$	0.2770E+03 ${}_{- 12.0 %}^{+ 5.6 %}$
200	0.8510E+02	0.8560E+02 ${}_{- 25.9 %}^{+ 12.9 %}$	0.2243E+03 ${}_{- 6.4 %}^{+, < 0.1 %}$	0.2665E+03 ${}_{- 11.6 %}^{+ 5.5 %}$
225	0.8020E+02	0.8075E+02 ${}_{- 26.0 %}^{+ 13.3 %}$	0.2110E+03 ${}_{- 6.7 %}^{+, < 0.1 %}$	0.2504E+03 ${}_{- 11.3 %}^{+ 5.2 %}$
250	0.7330E+02	0.7379E+02 ${}_{- 26.2 %}^{+ 13.7 %}$	0.1924E+03 ${}_{- 6.9 %}^{+, < 0.1 %}$	0.2282E+03 ${}_{- 11.1 %}^{+ 5.0 %}$
275	0.6451E+02	0.6496E+02 ${}_{- 26.2 %}^{+ 14.1 %}$	0.1690E+03 ${}_{- 7.0 %}^{+, < 0.1 %}$	0.2008E+03 ${}_{- 10.8 %}^{+ 4.9 %}$
300	0.5467E+02	0.5507E+02 ${}_{- 26.4 %}^{+ 14.6 %}$	0.1430E+03 ${}_{- 7.2 %}^{+, < 0.1 %}$	0.1697E+03 ${}_{- 10.4 %}^{+ 4.6 %}$
325	0.4500E+02	0.4515E+02 ${}_{- 26.3 %}^{+ 15.0 %}$	0.1171E+03 ${}_{- 7.2 %}^{+, < 0.1 %}$	0.1388E+03 ${}_{- 10.2 %}^{+ 4.7 %}$
350	0.3629E+02	0.3640E+02 ${}_{- 26.5 %}^{+ 15.6 %}$	0.9423E+02 ${}_{- 7.5 %}^{+, < 0.1 %}$	0.1117E+03 ${}_{- 9.9 %}^{+ 4.4 %}$
375	0.2922E+02	0.2947E+02 ${}_{- 26.7 %}^{+ 15.6 %}$	0.7613E+02 ${}_{- 7.7 %}^{+, < 0.1 %}$	0.9010E+02 ${}_{- 9.7 %}^{+ 4.2 %}$
400	0.2370E+02	0.2396E+02 ${}_{- 26.7 %}^{+ 16.1 %}$	0.6185E+02 ${}_{- 7.8 %}^{+, < 0.1 %}$	0.7328E+02 ${}_{- 9.6 %}^{+ 3.9 %}$
450	0.1593E+02	0.1612E+02 ${}_{- 27.0 %}^{+ 16.8 %}$	0.4147E+02 ${}_{- 8.2 %}^{+, < 0.1 %}$	0.4910E+02 ${}_{- 9.3 %}^{+ 3.4 %}$
500	0.1089E+02	0.1106E+02 ${}_{- 27.1 %}^{+ 17.2 %}$	0.2838E+02 ${}_{- 8.4 %}^{+, < 0.1 %}$	0.3355E+02 ${}_{- 8.9 %}^{+ 3.4 %}$
550	0.7617E+01	0.7747E+01 ${}_{- 27.3 %}^{+ 17.5 %}$	0.1984E+02 ${}_{- 8.6 %}^{+, < 0.1 %}$	0.2351E+02 ${}_{- 8.8 %}^{+ 3.2 %}$
600	0.5413E+01	0.5518E+01 ${}_{- 27.3 %}^{+ 18.0 %}$	0.1410E+02 ${}_{- 8.7 %}^{+, < 0.1 %}$	0.1670E+02 ${}_{- 8.5 %}^{+ 3.0 %}$
$\sqrt{s} = 14$ TeV
150	0.1065E+03	0.1069E+03 ${}_{- 25.0 %}^{+ 11.2 %}$	0.2824E+03 ${}_{- 7.5 %}^{+, < 0.1 %}$	0.3348E+03 ${}_{- 12.7 %}^{+ 6.7 %}$
175	0.1039E+03	0.1043E+03 ${}_{- 25.3 %}^{+ 11.8 %}$	0.2733E+03 ${}_{- 7.2 %}^{+, < 0.1 %}$	0.3239E+03 ${}_{- 12.2 %}^{+ 6.1 %}$
200	0.1006E+03	0.1012E+03 ${}_{- 25.5 %}^{+ 12.4 %}$	0.2641E+03 ${}_{- 6.7 %}^{+, < 0.1 %}$	0.3132E+03 ${}_{- 11.9 %}^{+ 5.7 %}$
225	0.9522E+02	0.9567E+02 ${}_{- 25.7 %}^{+ 13.0 %}$	0.2490E+03 ${}_{- 6.3 %}^{+, < 0.1 %}$	0.2952E+03 ${}_{- 11.5 %}^{+ 5.4 %}$
250	0.8711E+02	0.8776E+02 ${}_{- 25.9 %}^{+ 13.2 %}$	0.2279E+03 ${}_{- 6.7 %}^{+, < 0.1 %}$	0.2702E+03 ${}_{- 11.3 %}^{+ 5.1 %}$
275	0.7693E+02	0.7746E+02 ${}_{- 25.9 %}^{+ 13.6 %}$	0.2008E+03 ${}_{- 6.7 %}^{+, < 0.1 %}$	0.2380E+03 ${}_{- 11.0 %}^{+ 5.1 %}$
300	0.6538E+02	0.6585E+02 ${}_{- 26.0 %}^{+ 13.9 %}$	0.1704E+03 ${}_{- 6.8 %}^{+, < 0.1 %}$	0.2018E+03 ${}_{- 10.7 %}^{+ 5.0 %}$
325	0.5386E+02	0.5425E+02 ${}_{- 26.1 %}^{+ 14.4 %}$	0.1401E+03 ${}_{- 7.0 %}^{+, < 0.1 %}$	0.1659E+03 ${}_{- 10.5 %}^{+ 4.7 %}$
350	0.4377E+02	0.4395E+02 ${}_{- 26.2 %}^{+ 14.9 %}$	0.1133E+03 ${}_{- 7.1 %}^{+, < 0.1 %}$	0.1341E+03 ${}_{- 10.2 %}^{+ 4.6 %}$
375	0.3557E+02	0.3565E+02 ${}_{- 26.2 %}^{+ 15.5 %}$	0.9174E+02 ${}_{- 7.2 %}^{+, < 0.1 %}$	0.1085E+03 ${}_{- 9.8 %}^{+ 4.5 %}$
400	0.2900E+02	0.2913E+02 ${}_{- 26.5 %}^{+ 15.6 %}$	0.7480E+02 ${}_{- 7.5 %}^{+, < 0.1 %}$	0.8841E+02 ${}_{- 9.7 %}^{+ 4.2 %}$
450	0.1951E+02	0.1975E+02 ${}_{- 26.7 %}^{+ 16.2 %}$	0.5061E+02 ${}_{- 7.9 %}^{+, < 0.1 %}$	0.5985E+02 ${}_{- 9.5 %}^{+ 3.8 %}$
500	0.1351E+02	0.1367E+02 ${}_{- 26.9 %}^{+ 16.8 %}$	0.3493E+02 ${}_{- 8.1 %}^{+, < 0.1 %}$	0.4128E+02 ${}_{- 9.2 %}^{+ 3.4 %}$
550	0.9494E+01	0.9636E+01 ${}_{- 27.0 %}^{+ 17.2 %}$	0.2456E+02 ${}_{- 8.3 %}^{+, < 0.1 %}$	0.2899E+02 ${}_{- 8.8 %}^{+ 3.5 %}$
600	0.6807E+01	0.6922E+01 ${}_{- 27.1 %}^{+ 17.4 %}$	0.1762E+02 ${}_{- 8.5 %}^{+, < 0.1 %}$	0.2083E+02 ${}_{- 8.7 %}^{+ 3.2 %}$

Table 3. Table 3: Total hadronic cross sections for g g → Z ∗ / h ∗ → N ν ℓ → 𝑔 𝑔 superscript 𝑍 superscript ℎ → 𝑁 subscript 𝜈 ℓ gg\to Z^{*}/h^{*}\to N\nu_{\ell} at various accuracies, divided by active-heavy mixing | V ℓ N | 2 superscript subscript 𝑉 ℓ 𝑁 2 |V_{\ell N}|^{2} , for representative collider energies s 𝑠 \sqrt{s} .

$m_{N}$ [GeV]	$σ^{LO} / {\| V_{ℓ N} \|}^{2}$ [fb]	$σ^{NLL} / {\| V_{ℓ N} \|}^{2}$ [fb]	$σ^{N^{2} LL} / {\| V_{ℓ N} \|}^{2}$ [fb]	$σ^{N^{3} LL} / {\| V_{ℓ N} \|}^{2}$ [fb]
$\sqrt{s} = 33$ TeV
150	0.5547E+03	0.5510E+03 ${}_{- 21.3 %}^{+ 5.8 %}$	0.1408E+04 ${}_{- 10.8 %}^{+, < 0.1 %}$	0.1658E+04 ${}_{- 15.1 %}^{+ 10.3 %}$
200	0.5672E+03	0.5663E+03 ${}_{- 21.9 %}^{+ 6.3 %}$	0.1428E+04 ${}_{- 10.2 %}^{+, < 0.1 %}$	0.1674E+04 ${}_{- 14.1 %}^{+ 9.7 %}$
300	0.4146E+03	0.4144E+03 ${}_{- 22.5 %}^{+ 8.3 %}$	0.1033E+04 ${}_{- 8.8 %}^{+, < 0.1 %}$	0.1209E+04 ${}_{- 13.1 %}^{+ 8.3 %}$
400	0.2132E+03	0.2127E+03 ${}_{- 23.1 %}^{+ 10.0 %}$	0.5250E+03 ${}_{- 7.4 %}^{+, < 0.1 %}$	0.6134E+03 ${}_{- 11.9 %}^{+ 7.1 %}$
500	0.1147E+03	0.1146E+03 ${}_{- 23.6 %}^{+ 11.1 %}$	0.2806E+03 ${}_{- 6.6 %}^{+, < 0.1 %}$	0.3273E+03 ${}_{- 11.1 %}^{+ 6.3 %}$
600	0.6606E+02	0.6615E+02 ${}_{- 23.9 %}^{+ 12.0 %}$	0.1608E+03 ${}_{- 5.9 %}^{+, < 0.1 %}$	0.1873E+03 ${}_{- 10.5 %}^{+ 5.6 %}$
700	0.4011E+02	0.4025E+02 ${}_{- 24.2 %}^{+ 13.0 %}$	0.9747E+02 ${}_{- 5.5 %}^{+, < 0.1 %}$	0.1135E+03 ${}_{- 9.9 %}^{+ 5.2 %}$
800	0.2550E+02	0.2564E+02 ${}_{- 24.4 %}^{+ 13.6 %}$	0.6183E+02 ${}_{- 5.9 %}^{+, < 0.1 %}$	0.7191E+02 ${}_{- 9.5 %}^{+ 4.8 %}$
900	0.1681E+02	0.1690E+02 ${}_{- 24.7 %}^{+ 14.2 %}$	0.4066E+02 ${}_{- 6.3 %}^{+, < 0.1 %}$	0.4733E+02 ${}_{- 9.3 %}^{+ 4.4 %}$
1000	0.1141E+02	0.1147E+02 ${}_{- 24.7 %}^{+ 14.8 %}$	0.2753E+02 ${}_{- 6.4 %}^{+, < 0.1 %}$	0.3204E+02 ${}_{- 9.0 %}^{+ 4.2 %}$
1100	0.7916E+01	0.7981E+01 ${}_{- 24.9 %}^{+ 15.2 %}$	0.1912E+02 ${}_{- 6.6 %}^{+, < 0.1 %}$	0.2228E+02 ${}_{- 8.7 %}^{+ 4.0 %}$
1200	0.5625E+01	0.5685E+01 ${}_{- 25.0 %}^{+ 15.4 %}$	0.1360E+02 ${}_{- 6.8 %}^{+, < 0.1 %}$	0.1582E+02 ${}_{- 8.6 %}^{+ 3.8 %}$
1300	0.4066E+01	0.4115E+01 ${}_{- 25.1 %}^{+ 15.8 %}$	0.9831E+01 ${}_{- 7.0 %}^{+, < 0.1 %}$	0.1144E+02 ${}_{- 8.4 %}^{+ 3.7 %}$
1400	0.2985E+01	0.3031E+01 ${}_{- 25.2 %}^{+ 16.0 %}$	0.7234E+01 ${}_{- 7.1 %}^{+, < 0.1 %}$	0.8425E+01 ${}_{- 8.4 %}^{+ 3.4 %}$
1500	0.2225E+01	0.2261E+01 ${}_{- 25.2 %}^{+ 16.3 %}$	0.5389E+01 ${}_{- 7.1 %}^{+, < 0.1 %}$	0.6281E+01 ${}_{- 8.2 %}^{+ 3.5 %}$
$\sqrt{s} = 100$ TeV
150	0.3230E+04	0.3223E+04 ${}_{- 16.8 %}^{+, < 0.1 %}$	0.7967E+04 ${}_{- 15.5 %}^{+ 3.9 %}$	0.9244E+04 ${}_{- 18.8 %}^{+ 15.2 %}$
200	0.3516E+04	0.3507E+04 ${}_{- 17.2 %}^{+ 0.1 %}$	0.8569E+04 ${}_{- 14.3 %}^{+ 3.2 %}$	0.9922E+04 ${}_{- 17.6 %}^{+ 14.1 %}$
300	0.2839E+04	0.2832E+04 ${}_{- 17.9 %}^{+ 1.8 %}$	0.6825E+04 ${}_{- 12.9 %}^{+ 2.2 %}$	0.7876E+04 ${}_{- 16.2 %}^{+ 13.0 %}$
400	0.1648E+04	0.1645E+04 ${}_{- 18.7 %}^{+ 3.3 %}$	0.3909E+04 ${}_{- 11.6 %}^{+ 1.0 %}$	0.4493E+04 ${}_{- 15.0 %}^{+ 11.6 %}$
500	0.9911E+03	0.9913E+03 ${}_{- 19.2 %}^{+ 4.7 %}$	0.2330E+04 ${}_{- 10.5 %}^{+ 0.2 %}$	0.2674E+04 ${}_{- 14.0 %}^{+ 10.5 %}$
600	0.6330E+03	0.6322E+03 ${}_{- 19.7 %}^{+ 5.7 %}$	0.1473E+04 ${}_{- 9.6 %}^{+, < 0.1 %}$	0.1688E+04 ${}_{- 13.1 %}^{+ 9.8 %}$
700	0.4242E+03	0.4220E+03 ${}_{- 20.0 %}^{+ 6.6 %}$	0.9775E+03 ${}_{- 8.9 %}^{+, < 0.1 %}$	0.1119E+04 ${}_{- 12.6 %}^{+ 9.0 %}$
800	0.2937E+03	0.2938E+03 ${}_{- 20.4 %}^{+ 7.3 %}$	0.6768E+03 ${}_{- 8.4 %}^{+, < 0.1 %}$	0.7728E+03 ${}_{- 11.9 %}^{+ 8.3 %}$
900	0.2101E+03	0.2101E+03 ${}_{- 20.5 %}^{+ 7.8 %}$	0.4817E+03 ${}_{- 8.0 %}^{+, < 0.1 %}$	0.5498E+03 ${}_{- 11.6 %}^{+ 8.1 %}$
1000	0.1541E+03	0.1538E+03 ${}_{- 20.8 %}^{+ 8.4 %}$	0.3513E+03 ${}_{- 7.5 %}^{+, < 0.1 %}$	0.4010E+03 ${}_{- 11.2 %}^{+ 7.8 %}$
1100	0.1152E+03	0.1153E+03 ${}_{- 21.0 %}^{+ 9.0 %}$	0.2624E+03 ${}_{- 7.1 %}^{+, < 0.1 %}$	0.2996E+03 ${}_{- 10.9 %}^{+ 7.5 %}$
1200	0.8800E+02	0.8830E+02 ${}_{- 21.3 %}^{+ 9.1 %}$	0.2006E+03 ${}_{- 7.1 %}^{+, < 0.1 %}$	0.2291E+03 ${}_{- 11.0 %}^{+ 6.9 %}$
1300	0.6822E+02	0.6839E+02 ${}_{- 21.4 %}^{+ 9.8 %}$	0.1550E+03 ${}_{- 6.5 %}^{+, < 0.1 %}$	0.1771E+03 ${}_{- 10.4 %}^{+ 6.6 %}$
1400	0.5369E+02	0.5371E+02 ${}_{- 21.5 %}^{+ 10.1 %}$	0.1214E+03 ${}_{- 6.2 %}^{+, < 0.1 %}$	0.1384E+03 ${}_{- 10.1 %}^{+ 6.6 %}$
1500	0.4272E+02	0.4278E+02 ${}_{- 21.7 %}^{+ 10.4 %}$	0.9653E+02 ${}_{- 6.0 %}^{+, < 0.1 %}$	0.1102E+03 ${}_{- 10.1 %}^{+ 6.2 %}$

Equations66

q \overline{q^{'}} \to W^{\pm (*)} \to N ℓ^{\pm}, q \in {u, c, d, s, b},

q \overline{q^{'}} \to W^{\pm (*)} \to N ℓ^{\pm}, q \in {u, c, d, s, b},

q γ ⟶ W γ \to N ℓ N ℓ^{\pm} q^{'} .

q γ ⟶ W γ \to N ℓ N ℓ^{\pm} q^{'} .

g g \to Z^{*} / h^{*} \to N ν_{ℓ} (-) .

g g \to Z^{*} / h^{*} \to N ν_{ℓ} (-) .

K^{N^{3} LL} = σ^{N^{3} LL} / σ^{LO}

K^{N^{3} LL} = σ^{N^{3} LL} / σ^{LO}

K^{N^{2} LL} = σ^{N^{2} LL} / σ^{LO}

(ν_{L i} N_{R j}^{c}) = (U_{3 \times 3} X_{n \times 3} V_{3 \times n} Y_{n \times n}) (ν_{m} N_{m^{'}}^{c}) .

(ν_{L i} N_{R j}^{c}) = (U_{3 \times 3} X_{n \times 3} V_{3 \times n} Y_{n \times n}) (ν_{m} N_{m^{'}}^{c}) .

ν_{ℓ} = m = 1 \sum 3 U_{ℓ m} ν_{m} + m^{'} = 1 \sum n V_{ℓ m^{'}} N_{m^{'}}^{c} .

ν_{ℓ} = m = 1 \sum 3 U_{ℓ m} ν_{m} + m^{'} = 1 \sum n V_{ℓ m^{'}} N_{m^{'}}^{c} .

L_{Int.} =

L_{Int.} =

-

-

σ (pp \to N + X) /∣ V_{ℓ N} ∣^{2} = σ_{0} (pp \to N + X) .

σ (pp \to N + X) /∣ V_{ℓ N} ∣^{2} = σ_{0} (pp \to N + X) .

σ^{FO}

σ^{FO}

\overset{σ}{^}^{Z}

\overset{σ}{^}^{Z}

\overset{σ}{^}^{h}

F_{Z} (Q^{2})

F_{Z} (Q^{2})

F_{h} (Q^{2})

f (r)

σ^{FO} \to σ^{Res} : Δ_{ij}^{β, FO} (z) \to Δ_{ij}^{β, Res} (z) .

σ^{FO} \to σ^{Res} : Δ_{ij}^{β, FO} (z) \to Δ_{ij}^{β, Res} (z) .

Δ_{g g}^{β, Res} (z) = ∣ C_{β} (Q^{2}, μ_{h}^{2}) ∣^{2} U (Q^{2}, μ_{α}^{2}, μ_{h}^{2}, μ_{s}^{2}, μ_{f}^{2})

Δ_{g g}^{β, Res} (z) = ∣ C_{β} (Q^{2}, μ_{h}^{2}) ∣^{2} U (Q^{2}, μ_{α}^{2}, μ_{h}^{2}, μ_{s}^{2}, μ_{f}^{2})

\times \frac{z z ^{- η}}{( 1 - z ) ^{1 - 2 η}} \tilde{s}_{Higgs} (lo g \frac{Q ^{2} ( 1 - z ) ^{2}}{μ _{s}^{2} z} + \partial_{η}, μ_{s}) \frac{e ^{- 2 γ_{E} η}}{Γ ( 2 η )} .

C_{h} (Q^{2}, μ_{h}^{2})

C_{h} (Q^{2}, μ_{h}^{2})

C_{X} (Q^{2}, μ^{2})

C_{Z} (Q^{2}, μ_{h}^{2})

C_{Z} (Q^{2}, μ_{h}^{2})

\frac{d}{d lo g μ} C_{β} (Q^{2}, μ^{2}) =

\frac{d}{d lo g μ} C_{β} (Q^{2}, μ^{2}) =

[Γ_{cusp}^{A} lo g (\frac{Q ^{2}}{μ ^{2}}) + γ^{S} + γ^{t}] C_{β} (Q^{2}, μ^{2}),

U (Q^{2}, μ_{α}^{2}, μ_{h}^{2}, μ_{s}^{2}, μ_{f}^{2}) = \frac{α _{s}^{2} ( μ _{s}^{2} )}{α _{s}^{2} ( μ _{f}^{2} )} [\frac{β ( α _{s} ( μ _{s}^{2} )) / α _{s}^{2} ( μ _{s}^{2} )}{β ( α _{s} ( μ _{α}^{2} )) / α _{s}^{2} ( μ _{α}^{2} )}]^{2}

U (Q^{2}, μ_{α}^{2}, μ_{h}^{2}, μ_{s}^{2}, μ_{f}^{2}) = \frac{α _{s}^{2} ( μ _{s}^{2} )}{α _{s}^{2} ( μ _{f}^{2} )} [\frac{β ( α _{s} ( μ _{s}^{2} )) / α _{s}^{2} ( μ _{s}^{2} )}{β ( α _{s} ( μ _{α}^{2} )) / α _{s}^{2} ( μ _{α}^{2} )}]^{2}

(\frac{Q ^{2}}{μ _{h}^{2}})^{- 2 a_{Γ} (μ_{h}^{2}, μ_{s}^{2})} e^{4 S (μ_{h}^{2}, μ_{s}^{2}) - 2 a_{γ^{S}} (μ_{h}^{2}, μ_{s}^{2}) + 4 a_{γ^{B}} (μ_{s}^{2}, μ_{f}^{2})},

m_{b} = 0 GeV, m_{t} = 173.2 GeV, m_{h} = 125.7 GeV .

m_{b} = 0 GeV, m_{t} = 173.2 GeV, m_{h} = 125.7 GeV .

μ_{r}, μ_{f}, μ_{t}, μ_{h} = Q and μ_{s} = Q (1 - τ) / (1 + 7 τ),

μ_{r}, μ_{f}, μ_{t}, μ_{h} = Q and μ_{s} = Q (1 - τ) / (1 + 7 τ),

K^{N^{j} LO + N^{k} LL} \equiv σ^{N^{j} LO + N^{k} LL} / σ^{LO} .

K^{N^{j} LO + N^{k} LL} \equiv σ^{N^{j} LO + N^{k} LL} / σ^{LO} .

:

:

GF N^{3} LL

:

:

GF N^{3} LL

GF N^{3} LL

GF N^{2} LL

GF NLL

GF N^{3} LL

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Full text

IPPP/17/48

Heavy neutrinos from gluon fusion

Richard Ruiz

[email protected]

Institute for Particle Physics Phenomenology, Department of Physics, Durham University, Durham DH1 3LE, U.K.

Michael Spannowsky

[email protected]

Institute for Particle Physics Phenomenology, Department of Physics, Durham University, Durham DH1 3LE, U.K.

Philip Waite

[email protected]

Institute for Particle Physics Phenomenology, Department of Physics, Durham University, Durham DH1 3LE, U.K.

Abstract

Heavy neutrinos, a key prediction of many standard model extensions, remain some of the most searched-for objects at collider experiments. In this context, we revisit the premise that the gluon fusion production mechanism, $gg\to Z^{*}/h^{*}\to N\nu_{\ell}$ , is phenomenologically irrelevant at the CERN LHC and report the impact of soft gluon corrections to the production cross section. We resum threshold logarithms up to next-to-next-to-next-to-leading logarithmic accuracy (N3LL), thus capturing the dominant contributions to the inclusive cross section up to next-to-next-to-leading order (N2LO). For $m_{N}>150$ GeV and collider energies $\sqrt{s}=7-100$ TeV, corrections to the Born rates span $+160$ to $+260\%$ . At $\sqrt{s}$ =14 TeV, the resummed channel is roughly equal in size to the widely-believed-to-be-dominant charged current Drell-Yan process and overtakes it outright at $\sqrt{s}\gtrsim 20-25$ TeV. Results are independent of the precise nature/mixing of $N$ and hold generically for other low-scale seesaws. Findings are also expected to hold for other exotic leptons and broken axial-vector currents, particularly as the $Z^{*}$ contribution identically reduces to that of a pseudoscalar.

I Introduction

In stark contrast to the standard model (SM) of particle physics, neutrinos have nonzero mass, and misaligned flavor and mass eigenstates Ahmad:2002jz ; Ashie:2005ik . Hence, the origins of their sub-eV masses and large mixing angles are two of the most pressing open questions in particle physics today. In light of recent evidence for the Higgs mechanism’s role in generating charged lepton masses Aad:2015vsa ; Chatrchyan:2014nva , we argue that the existence of neutrino Dirac masses comparable to other elementary fermions’ Dirac masses is an increasingly likely prospect. If this is the case, then observed neutrino phenomenology can be accommodated by low-scale seesaw mechanisms, such as the Inverse Mohapatra:1986aw ; Mohapatra:1986bd ; Bernabeu:1987gr or Linear Akhmedov:1995ip ; Akhmedov:1995vm seesaw models.

In such seesaw scenarios, TeV-scale heavy neutrinos’ mass eigenstates ( $N$ ) can couple to electroweak (EW) bosons with sizable Antusch:2014woa ; Fernandez-Martinez:2016lgt active-sterile mixing, but at the same time do not decouple from Large Hadron Collider (LHC) phenomenology Pilaftsis:1991ug ; Kersten:2007vk due to their pseudo-Dirac nature. Subsequently, low-scale seesaw mechanisms can be tested at the LHC with $\mathcal{O}(100-1000){\rm~{}fb^{-1}}$ Arkani-Hamed:2015vfh ; Golling:2016gvc ; Antusch:2016ejd ; Arganda:2015ija ; Arganda:2014dta ; Baglio:2016bop , demonstrating the sensitivity and complementarity of collider and oscillation experiments.

Hadron collider investigations of heavy $N$ typically rely on the charged current (CC) Drell-Yan (DY) process Keung:1983uu , shown in Fig. 1(a) and given by

[TABLE]

or the sizable vector boson fusion (VBF) channel Datta:1993nm ; Dev:2013wba ; Alva:2014gxa ; Degrande:2016aje ,

[TABLE]

As seen in Fig. 1(c), VBF is driven by the $W\gamma\to N\ell$ subprocess and receives longitudinal $W$ enhancements for $m_{N}\gg M_{W}$ Alva:2014gxa . Notable is the renewed interest Hessler:2014ssa ; Degrande:2016aje in the gluon fusion (GF) process Degrande:2016aje ; Willenbrock:1985tj ; Dicus:1991wj ; Hessler:2014ssa , shown in Fig. 1(b),

[TABLE]

Variants of this process have been studied recently in BhupalDev:2012zg ; Batell:2015aha ; Gago:2015vma ; Accomando:2016rpc ; Nemevsek:2016enw ; Caputo:2017pit ; Das:2017zjc . While GF proceeds anomalously through off-shell $Z^{*}/h^{*}$ bosons and is formally an $\mathcal{O}(\alpha_{s}^{2})$ correction to the neutral current (NC) DY process, $q\overline{q}\to Z^{*}\to N\overset{(-)}{\nu_{\ell}},$ the channel’s cross section is known to surpass the DY and VBF rates for collider energies $\sqrt{s}\gtrsim 30-40$ TeV Willenbrock:1985tj ; Hessler:2014ssa ; Degrande:2016aje . At 14 TeV, GF is factors smaller than the DY channels. These conclusions are noteworthy, as they rely on the GF rate at leading-order (LO) accuracy being a good estimate of the total cross section. However, at LO, the GF rates for the SM Higgs boson Dawson:1990zj ; Spira:1995rr ; Spira:1997dg ; Harlander:2002wh ; Ravindran:2003um ; Catani:2014uta ; Anastasiou:2015ema , heavy scalars Spira:1997dg , and pseudoscalars Harlander:2002vv ; Anastasiou:2002wq ; Ahmed:2016otz are greatly underestimated.

In light of this, we report, for the first time, the impact of soft gluon corrections to heavy $N$ production in GF. We resum threshold logarithms up to next-to-next-to-next-to-leading logarithmic accuracy (N3LL). For GF, this captures the leading contributions to the inclusive cross section $(\sigma)$ up to next-to-next-to-leading order (N2LO) Bonvini:2014qga . Our findings have immediate impact on searches at hadron colliders, and thereby challenge the paradigm that GF is phenomenologically irrelevant for the discovery and study of heavy $N$ at the LHC.

For $m_{N}=150-1000{\rm~{}GeV}$ , $\sqrt{s}=7-100$ TeV, and scale choices comparable to the hard process, we report

[TABLE]

We find that GF dominates over the DY-like processes of Eq. (1) for $m_{N}=500-1000$ GeV at ${\color[rgb]{0,0,0}\sqrt{s}\gtrsim 20-25}$ TeV. The corrections exhibit perturbative convergence and are consistent with those for Higgs and heavy (pseudo)scalar production Dawson:1990zj ; Spira:1995rr ; Harlander:2002wh ; Ravindran:2003um ; Catani:2014uta ; Spira:1997dg ; Harlander:2002vv ; Anastasiou:2002wq ; Anastasiou:2015ema ; Ahmed:2016otz . Our results are independent of the precise nature/mixing of $N$ and hold generically for other low-scale seesaws. Results are also expected to hold for other exotic leptons, e.g. triplet leptons in the Type III seesaw and other colorless, axial-vector currents.

This report continues as follows: We first describe our phenomenological heavy $N$ model, then present the resummation formalism employed, emphasizing a new treatment of the $Z^{*}$ current. After summarizing our computational setup, we present our results and conclude.

II Heavy Neutrino Model

Throughout this study, we adopt the neutrino mixing formalism of Atre:2009rg : For $i~{}(m)=1,\dots,3$ left-handed (light) states and $j~{}(m^{\prime})=1,\dots,n$ right-handed (heavy) states, chiral neutrinos can be rotated into mass eigenstates by

[TABLE]

After further rotating the charged leptons into the mass basis, the flavor state $\nu_{\ell}$ in the mass basis is explicitly

[TABLE]

$U_{\ell m}$ is the observed light neutrino mixing matrix and $V_{\ell m^{\prime}}$ parametrizes active-heavy mixing. For EW-scale $N_{m^{\prime}}$ , the latter is constrained by precision EW data to be $|V_{\ell N}|\lesssim 10^{-2}-10^{-1}$ Antusch:2014woa ; Fernandez-Martinez:2016lgt . For simplicity, we consider only the lightest heavy state, denoted $N$ .

In the mass basis, the EW interaction Lagrangian is

[TABLE]

At the production level, $|V_{\ell N}|$ factorizes out of cross sections, a result that holds at all orders in $\alpha_{s}$ Ruiz:2015zca ; Degrande:2016aje . This allows one to define Han:2006ip a “bare” cross section $\sigma_{0}$ in which one sets $|V_{\ell N}|=1$ . Subsequently, flavor model-independent cross sections are given by

[TABLE]

Due to this factorization, the QCD corrections we present are universal across low-scale seesaws that feature $N$ .

III Threshold Resummation Formalism

We now summarize our resummation formalism and the special consideration of the $Z^{*}$ mediator in GF.

For a color-singlet final state $V$ , the inclusive $pp\to V+X$ fixed-order (FO) cross section is given generically by the Collinear Factorization Theorem111The equivalent measure: $\int_{\tau_{0}}^{1}d\xi_{1}\int_{\tau_{0}/\xi_{1}}^{1}dx\int_{x}^{1}dz/z$ , with $\xi_{2}=x/z$ , may lead to faster numerical convergence for some processes. Collins:1984kg ; Collins:1985ue ; Collins:2011zzd ,

[TABLE]

That is, the hadronic scattering rate $\sigma$ is the convolution $(\otimes)$ of parton distribution functions (PDFs) $f$ , the soft coefficient function $\Delta$ , and the partonic-level $ij\to V$ hard scattering rate $\hat{\sigma}$ , which occurs at the hard scattering scale $Q=\sqrt{p_{V}^{2}}$ . Scale dependence of these quantities is implied but made explicit below. $f_{i/p}(\xi,\mu_{f})$ are the likelihoods of observing parton $i$ in $p$ carrying longitudinal momentum $p_{z}^{i}=\xi p_{z}^{p}\gg p_{T}^{i}$ , when DGLAP-evolved Gribov:1972ri ; Dokshitzer:1977sg ; Altarelli:1977zs to a factorization scale $\mu_{f}$ , generating the partonic scale $\sqrt{\hat{s}}=\sqrt{\xi_{1}\xi_{2}s}$ . $\Delta^{\beta,{\rm FO}}_{ij}(z)=\delta(1-z)+\mathcal{O}(\alpha_{s})$ accounts for soft gluons carrying a momentum fraction $(1-z)$ , with $z=Q^{2}/\hat{s}$ , emitted in the $ij\to A$ transition $(\hat{\sigma}^{\beta})$ via a color/Lorentz structure labeled as $\beta$ . Above, $\tau_{0}=\min\{Q^{2}\}/s$ is the kinematic threshold below which $ij\to A$ is kinematically forbidden, and $\tau=Q^{2}/s=\xi_{1}\xi_{2}z$ is similarly the hard threshold.

For the $gg\to Z^{*}/h^{*}\to N\nu_{\ell}$ process, with $\beta\in\{Z,h\}$ , $Q^{2}=(p_{N}+p_{\nu})^{2}>m_{h}^{2},M_{Z}^{2}$ and $\tau_{0}\approx m_{N}^{2}/s$ , the hard partonic-level Born cross sections are222We note that the expression for $\hat{\sigma}^{h}$ in Hessler:2014ssa contains typographic errors. Willenbrock:1985tj ; Hessler:2014ssa

[TABLE]

where $r_{X}=m_{X}^{2}/Q^{2}$ . For quarks with weak isospin charge $(T_{L}^{3})_{q}=\pm 1/2$ , the $Z/h$ one-loop form factors are

[TABLE]

A few remarks: $(i)$ While we use the fully integrated $\hat{\sigma}^{\beta}$ , the resummation formalism we employ Becher:2006mr ; Becher:2006nr operates in momentum space. Hence, phase-space cuts on an $n$ -body final state can be implemented if one starts from the differential $d\hat{\sigma}^{\beta}$ . $(ii)$ The $Z^{*}/h^{*}$ contributions add incoherently due to the (anti)symmetric nature of the $(Z^{*})h^{*}$ coupling Willenbrock:1985tj . $(iii)$ The similarity of $\hat{\sigma}^{Z}$ and $\hat{\sigma}^{h}$ follows from the fact that, after summing over SU $(2)_{L}$ doublet constituents, the net $Z^{*}$ contribution is a pseudoscalar-like coupling proportional to quark Yukawa couplings. This is in accordance with the Goldstone Equivalence Theorem, pointed out first for the $gg\to N\nu_{\ell}$ process in Ref. Degrande:2016aje . Subsequently, this asymptotic behavior means one can further simplify the original expressions of Refs. Willenbrock:1985tj ; Hessler:2014ssa to those above.

The axial-vector-pseudoscalar correspondence, however, is more general: For a massive, colorless vector $V(q_{\mu})$ participating in the loop process $gg\to V^{*}$ , the most general current-propagator contraction (in the unitary gauge) is of the form $\Gamma^{\mu}\Pi_{\mu\nu}\sim(g_{V}\gamma^{\mu}+g_{A}\gamma^{\mu}\gamma^{5})(g_{\mu\nu}-q_{\mu}q_{\nu}/M_{V}^{2})$ . By $C$ -symmetry (Furry’s theorem), the vector current $g_{V}\gamma^{\mu}$ vanishes; by angular momentum conservation (Landau-Yang theorem), the transverse polarization $g_{\mu\nu}$ does not contribute. Hence, $\Gamma^{\mu}\Pi_{\mu\nu}\sim g_{A}\gamma^{\mu}\gamma^{5}q_{\mu}q_{\nu}/M_{V}^{2}$ . After decomposing quark propagators in the triangle loop via spinor completeness relations and exploiting the Dirac equation, one finds $\Gamma^{\mu}\Pi_{\mu\nu}\sim\gamma^{5}(2m_{f}q_{\nu}/M_{V}^{2})$ . That is, a pseudoscalar coupling proportional to the quark mass $m_{f}$ .

Moreover, emissions of soft gluons off fermions do not change the loop’s structure due to soft factorization. Therefore, one may approximate soft QCD corrections to the $gg\to V^{*}$ subprocess for $V^{*}$ possessing axial-vector couplings to fermions with those corrections for a pseudoscalar. This is a main finding of this work and was not observed in previous resummations of $gg\to Z^{*}$ .

As $Q$ approaches the partonic threshold $\sqrt{\hat{s}}$ , accompanying gluon radiation is forced to be soft, with $E_{g}\sim\sqrt{\hat{s}}(1-z)$ . This generates numerically large phase-space logarithms of the form $\log(1-z)$ that spoil the perturbative convergence of Eq. (11). Threshold logarithms, however, factorize and can be resummed to all orders in ${\alpha_{s}\log(1-z)}$ via exponentiation Catani:1989ne ; Sterman:1986aj ; Catani:1990rp ; Contopanagos:1996nh ; Forte:2002ni .

We perform this resummation by working in the Soft-Collinear Effective Theory (SCET) framework Bauer:2000yr ; Bauer:2001yt ; Beneke:2002ph . This permits Eq. (11) to be factorized directly in momentum space Becher:2006mr ; Becher:2006nr by segmenting and regulating divergent regions of phase space with hard and soft scales, $\mu_{h}$ and $\mu_{s}$ . (This is unlike perturbative QCD where one works in Mellin space Catani:1989ne ; Sterman:1986aj ; Catani:1990rp .) Scale invariance of physical observables then implies that factored components can also be independently renormalization group (RG)-evolved and matched via exponentiation Catani:1989ne ; Sterman:1986aj ; Catani:1990rp ; Contopanagos:1996nh ; Forte:2002ni . Thus, numerically large quantities are replaced with perturbative ones regulated by $\mu_{h}$ and $\mu_{s}$ and with RG-evolution coefficients that run $\mu_{h}$ and $\mu_{s}$ to $\mu_{f}$ and $Q$ .

In practice, the resummation procedure reduces to replacing the soft coefficient function $\Delta^{\beta,{\rm FO}}_{ij}(z)$ in Eq. (11):

[TABLE]

For GF production of heavy leptons via $s$ -channel pseudoscalar $(\beta=Z)$ and scalar $(\beta=h)$ mediators, as given in Fig. 1(b), the SCET-based soft coefficient $\Delta_{gg}^{\beta,\text{Res}}$ in the notation of Becher:2007ty ; Ahrens:2008nc is

[TABLE]

$C_{\beta}$ is the process-dependent, so-called hard function and accounts for (hard) virtual corrections to the hard process. For $\beta=h$ , the function is given by the two-step SCET matching coefficients $C_{t}$ and $C_{S}$ of Ahrens:2008nc , with

[TABLE]

where $X\in\{t,S\}$ . The product of $C_{t}$ and $C_{S}$ , which can be expanded individually as power series in $\left(\alpha_{s}/4\pi\right)$ , is equivalent to a one-step SCET matching procedure when setting $\mu_{t}=\mu_{h}$ Berger:2010xi . For $\beta=Z$ , the one-step matching hard function can also be expanded as a power series. In the notation of Ahmed:2015qpa ; Ahmed:2016otz , this is

[TABLE]

We briefly note that the $\log(\mu_{h}^{2}/m_{t}^{2})$ term that appears in $C_{g,2}^{A,{\rm eff}}$ of Ahmed:2015qpa should be replaced with $\log(Q^{2}/m_{t}^{2})$ in order to preserve the scale independence of the total cross section, a physical observable Ahmed:2016otz . With this modification, both $C_{h}$ and $C_{Z}$ satisfy the evolution equation,

[TABLE]

for anomalous dimensions $\Gamma^{A}_{\rm cusp},\gamma^{S,t},$ as given in Becher:2007ty ; Ahrens:2008nc .

The soft scalar function $\tilde{s}_{\mathrm{Higgs}}$ describes (soft) radiation off incoming gluons and hence is universal for scalars and pseudoscalars. The derivatives in $\tilde{s}_{\mathrm{Higgs}}$ are regular partial derivatives that act to the right, before $\eta\equiv 2a_{\Gamma}(\mu_{s}^{2},\mu_{f}^{2})$ is evaluated numerically.

We include an additional factor of $\sqrt{z}$ in Eq. (III) with respect to Becher:2007ty ; Ahrens:2008nc . As noted in Becher:2007ty ; Ahrens:2008nc ; Ahrens:2011mw ; Broggio:2015lya , the inclusion of the factor accounts precisely for power corrections that are manifest in the traditional QCD/Mellin-space resummation formalism. Numerically, we find this increases our total normalization by ${\color[rgb]{0,0,0}\mathcal{O}(5-10)\%}$ at ${\rm N}^{3}{\rm LL}$ and our residual scale dependence by ${\color[rgb]{0,0,0}\mathcal{O}(5\%)}$ .

RG-running is described by the evolution function,

[TABLE]

where $\mu_{\alpha}=\mu_{t}~{}(\mu_{h})$ for $\beta=h~{}(Z)$ .

For definitions and explicit expressions of the quantities in Eqs. (III)–(23) up to $\mathcal{O}(\alpha_{s}^{2})$ , see Ahrens:2008nc ; Becher:2007ty . Mappings between ${\rm N}^{k}{\rm LL}$ accuracy and required ingredients can be found in Becher:2007ty ; Bonvini:2014tea . At ${\rm N}^{3}{\rm LL}$ , one needs at two loops: $C_{\beta}$ for both pseudoscalar Ahmed:2015qpa and scalar Harlander:2000mg ; Ahrens:2008nc , as well as $\tilde{s}_{\mathrm{Higgs}}$ Idilbi:2006dg ; Ahrens:2008nc ; Becher:2007ty . Note that while the results of Ahmed:2015qpa ; Harlander:2000mg ; Ahrens:2008nc are derived in the heavy top limit, $\mathcal{O}(Q^{2}/m_{t}^{2})$ corrections to inclusive (pseudo)scalar cross sections are known to be $\mathcal{O}(1-10\%)$ Spira:1997dg ; Caola:2011wq , even for $Q^{2}\gg m_{t}^{2}$ , justifying their use in our calculation.

IV Computational Setup

For the DY and VBF channels, we use the methodology of Degrande:2016aje to compute inclusive cross sections and uncertainties at NLO in QCD, but with the following exceptions: we use the NNPDF 3.0 QED NLO PDF set Ball:2014uwa ; Ball:2013hta and do not apply phase-space cuts to the CC DY process. Scale choices and regulating VBF cuts are unchanged. For GF, we adopt the additional SM inputs Agashe:2014kda :

[TABLE]

To best match the accuracy of the resummation calculation, we use the NNPDF 3.0 NNLO+NNLL PDF set Bonvini:2015ira ; while the set’s uncertainties are sizable, the use of a FO PDF set would formally double-count initial-state gluons. Cross sections are calculated using in-house code with Monte Carlo integration performed via the CUBA libraries Hahn:2004fe , and checked at LO against Hessler:2014ssa ; Degrande:2016aje . The soft coefficient function $\Delta^{\beta,{\rm Res}}_{gg}$ is checked against Bonvini:2014joa ; Bonvini:2014tea ; Ahmed:2016otz .

To minimize the numerical impact of missing QCD corrections, we follow Becher:2007ty ; Ahrens:2008nc and choose the scale scheme

[TABLE]

for both the Born and resummed GF calculations. For GF, we report the scale dependence associated with simultaneously varying $\mu_{f},\mu_{r},$ and $\mu_{s}$ over $0.5\leq\mu_{X}/\mu_{\rm Default}\leq 2$ . While the $\mu_{s}$ dependence itself is numerically small, we vary it jointly with $\mu_{f}$ to ensure that the subtraction terms required for numerical evaluation lead sufficiently to numerical convergence; see Ahrens:2008nc ; Becher:2007ty for more details. Missing FO terms that would otherwise stabilize $\mu_{f}$ represents the largest source of uncertainty. Indeed, we find other scale uncertainties to be relatively small owing to our high logarithmic accuracy.

In the following, we report only residual scale dependence. For studies on PDF uncertainties in heavy $N$ production, see Alva:2014gxa ; Degrande:2016aje , and for threshold-improved PDF uncertainties, see Beenakker:2015rna ; Mitra:2016kov . PDFs and $\alpha_{s}(\mu)$ are evaluated using the LHAPDF 6 libraries Buckley:2014ana .

V Results

At $\sqrt{s}=14{\rm~{}TeV}$ and as a function of heavy $N$ mass, we show in Fig. 2(L) the inclusive $N$ production cross section (divided by active-heavy mixing $|V_{\ell N}|^{2}$ ) for the CC DY and VBF processes at NLO, for GF at LO, andfor GF at ${\rm N}^{3}{\rm LL}$ . The thickness of each curve corresponds to the residual scale dependence; no scale dependence is shown for GF at LO. In the lower panel is the ratio of the resummed and Born GF rates. We quantify the impact of QCD corrections with a $K$ -factor, defined generically as

[TABLE]

For $m_{N}=150-1000$ GeV, cross sections span roughly

[TABLE]

For GF, $K$ -factors and uncertainties span approximately

[TABLE]

These rates should be compared with DY (VBF) $K$ -factors of $K^{\rm NLO}_{\rm DY~{}(VBF)}=1.15-1.25~{}(0.98-1.06)$ and uncertainties of $(\delta\sigma/\sigma)^{\rm DY~{}(VBF)}=\pm 1-5~{}(5-11)\%$ Degrande:2016aje .

For the mass range under consideration, one observes unambiguously that the resummed GF rates at ${\rm N}^{2}{\rm LL}$ and ${\rm N}^{3}{\rm LL}$ are markedly larger than the LO rate, with $K\gtrsim{\color[rgb]{0,0,0}2-2.5}$ and notably independent of $m_{N}$ . This is unlike NLL, where $K\sim 1$ , since one essentially runs only $\alpha_{s}(\mu)$ and $C_{\beta},~{}\tilde{s}_{\rm Higgs}\sim 1$ ; here, the uncertainty simply corresponds to varying $\alpha_{s}$ . We find $\sigma^{{\rm N}^{3}{\rm LL}}/\sigma^{{\rm N}^{2}{\rm LL}}\sim{\color[rgb]{0,0,0}1.1-1.2}$ , indicating convergence of the perturbative series.

As previously stated, the residual uncertainty at ${\rm N}^{3}{\rm LL}$ stems from missing FO contributions. Such terms, likely positive definite Bonvini:2014qga , consist of hard, initial state radiation (ISR) with $p_{T}^{j}\gtrsim\mu_{f}=Q$ , which are not, by construction, included in the DGLAP-evolution of the PDFs. The sizes of the ${\rm N}^{2}{\rm LL}$ and ${\rm N}^{3}{\rm LL}$ corrections are, in part, due to our scale choices and the desire to minimize the importance of missing QCD corrections. Choosing alternative, less intuitive scales for the Born process can, of course, lead to smaller $K$ -factors, but also to larger ones. At both ${\rm N}^{2}{\rm LL}$ and ${\rm N}^{3}{\rm LL}$ , the size of the uncertainty band is due to a residual $\mu_{f}$ dependence, and requires matching to hard ISR from FO contributions to be reduced. Moreover, these corrections are in line with those for Higgs and heavy (pseudo)scalar production Dawson:1990zj ; Spira:1995rr ; Harlander:2002wh ; Spira:1997dg ; Harlander:2002vv ; Anastasiou:2002wq ; Anastasiou:2015ema ; Ahmed:2016otz .

In comparison with other heavy $N$ production modes, we find for $m_{N}\gtrsim{\color[rgb]{0,0,0}300}{\rm~{}GeV}$ that the GF rate is now comparable to the CC and NC DY (not shown for clarity) rates. When basic fiducial cuts are applied on the charged lepton in the CC processes the combined GF+NC DY rate is slightly larger than the combined VBF+CC DY channel. For $m_{N}\lesssim 600{\rm~{}GeV}$ , i.e., masses that are most relevant for LHC phenomenology due to mixing-suppression Atre:2009rg ; Alva:2014gxa , the GF channel is factors larger than the VBF mechanism, indicating its potential importance at the LHC and its upgrades/successors.

We briefly note that the relative importance of the VBF mechanism found in Fig. 2 is considerably less than what has been found in previous investigations, e.g., Ref. Dev:2013wba and follow-up works by the same authors. It was shown in Ref. Alva:2014gxa that the findings of Ref. Dev:2013wba were qualitatively and quantitatively incorrect: Their claimed “ $t$ -channel enhancement” are in reality due to several poorly/unregulated QCD and QED collinear divergences. Numerically, their cross sections were overestimated by $100\times$ in some instances. We refer readers to Refs. Alva:2014gxa ; Ruiz:2015zca ; Degrande:2016aje for correct, all-orders/resummed treatments of these contributions; to Ref. Alva:2014gxa ; Arganda:2015ija for a quantitative assessment of $W\gamma$ scattering in heavy $N$ searches; and to Ref. Degrande:2016aje for non-expert-friendly infrared- and collinear-safe collider definitions for such processes.

In Fig. 2(R), we plot as a function of $\sqrt{s}$ for representative $m_{N}$ the summed GF + NC DY channels as well as the summed CC DY + VBF channels. We add the channels incoherently, as GF is formally a noninterfering $\mathcal{O}(\alpha_{s}^{2})$ correction to NC DY, and similarly VBF is a noninterfering $\mathcal{O}(\alpha)$ correction to CC DY. Relative uncertainties are added in quadrature.

We observe for $m_{N}=500-1000$ GeV that the inclusive production rate of $N\nu_{\ell}$ overtakes the inclusive $N\ell^{\pm}$ production at ${\color[rgb]{0,0,0}\sqrt{s}\gtrsim 15-30}$ TeV. For $\sqrt{s}=33~{}(100){\rm~{}TeV}$ , this difference is a factor of ${\color[rgb]{0,0,0}1-1.6~{}(2.5-2.7)}$ and is driven by the GF rate, for which the luminosity grows much faster than the $q\overline{q^{\prime}}$ (DY) and $qq$ (VBF) luminosities with increasing collider energies. While not shown, we find for $m_{N}=500-1000{\rm~{}GeV}$ that the GF rate individually exceeds the CC DY rate for $\sqrt{s}{\color[rgb]{0,0,0}\gtrsim 20-25{\rm~{}TeV}}$ . For increasing $\sqrt{s}$ , we find that the resummation has a smaller impact on the total GF rate, with

[TABLE]

This drop in $K$ is again due to an increasing importance of hard ISR, and similarly leads to a sizable residual $\mu_{f}$ dependence. In checks against heavy (pseudo)scalar production Bonvini:2014joa ; Bonvini:2014tea ; Ahmed:2016otz , we find similar results, and that the importance of FO corrections is $\mathcal{O}(+10\%)$ . Such corrections would likely push net $K$ -factors for the $gg\to N\nu_{\ell}$ process to $K\sim 3$ , confirming the conjecture of Hessler:2014ssa . We summarize our results in Table 1.

Due to the severe model-dependence of $V_{\ell N}$ as well as the associated phenomenology, an investigation into which is well beyond the scope of this study, we defer further interpretation of our results to future studies.

Usage: For the use of these results in studies, we advocate LO+parton shower event generation following Degrande:2016aje . Total inclusive rates should then be normalized to those tabulated in Tables 2 and 3. The flatness of the resummed $K$ -factors means interpolation to unlisted $m_{N}$ is reliable.

VI Summary and Conclusion

The existence of tiny neutrino masses and large mixing is unambiguous evidence for physics beyond the SM. In light of Higgs boson data, the prospect of neutrino Dirac masses existing is increasingly likely. Low-scale seesaw models with TeV-scale heavy neutrinos that couple appreciably to EW bosons are scenarios that can accommodate these seemingly contradictory observations, and still give rise to LHC phenomenology.

In this context, we have evaluated, for the first time, soft corrections to the GF production mode $gg\to Z^{*}/h^{*}\to N\nu_{\ell}$ . This was made possible by a new treatment treatment of the $gg\to Z^{*}$ subprocess. For $m_{N}=150-1000{\rm~{}GeV}$ and $\sqrt{s}=7-100$ TeV, we report:

[TABLE]

We find that GF dominates over DY-like processes of Eq. (1) for $m_{N}=500-1000$ GeV at ${\color[rgb]{0,0,0}\sqrt{s}\gtrsim 20-25}$ TeV. Corrections exhibit perturbative convergence and are consistent with Higgs and heavy (pseudo)scalar production. Moreover, our results are independent of the precise nature/mixing of $N$ , and are expected to hold for other exotic leptons as well as other colorless, broken axial-vector currents one finds in other seesaw scenarios.

Acknowledgements.

Acknowledgements: M. Bonvini, S. Dawson, C. Fabrizio, E. Molinaro, B. Pecjak, L. Rottoli, D. Scott, C.-F. Tamarit, and C. Weiland are thanked for discussions. This work was funded in part by the UK STFC, and the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement 674896 (Elusives ITN). RR acknowledges the CERN Theory group’s generous hospitality during the completion of this work. *

Appendix A Appendix

In Tables 2-3, we list total hadronic cross sections for $gg\to Z^{*}/h^{*}\to N\nu_{\ell}$ at various accuracies, divided by active-heavy mixing $|V_{\ell N}|^{2}$ , for representative collider energies $\sqrt{s}=13,~{}14,~{}33,$ and 100 TeV.

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