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Published for SISSA by Springer

Received: September 2, 2016

Accepted: October 18, 2016 Published: October 20, 2016

Full top quark mass dependence in Higgs boson pair production at NLO

S. Borowka,a N. Greiner,a G. Heinrich,b S.P. Jones,b M. Kerner,b J. Schlenkb and T. Zirkeb

aInstitute for Physics, Universitat Zurich,

Winterthurerstr.190, 8057 Zurich,Switzerland

bMax-Planck-Institute for Physics,

Fohringer Ring 6, 80805 Munchen, Germany

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We study the e ects of the exact top quark mass-dependent two-loop corrections to Higgs boson pair production by gluon fusion at the LHC and at a 100 TeV hadron collider. We perform a detailed comparison of the full next-to-leading order result to various approximations at the level of di erential distributions and also analyse non-standard Higgs self-coupling scenarios. We nd that the di erent next-to-leading order approximations di er from the full result by up to 50 percent in relevant di erential distributions. This clearly stresses the importance of the full NLO result.

Keywords: NLO Computations, QCD Phenomenology

ArXiv ePrint: 1608.04798

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP10(2016)107

Web End =10.1007/JHEP10(2016)107

JHEP10(2016)107

Contents

1 Introduction 1

2 Details of the calculation 32.1 Amplitude structure 32.2 Leading order cross section 52.2.1 Heavy top limit 62.3 NLO cross section 62.3.1 Calculation of the virtual two-loop amplitude 72.3.2 Real radiation 142.4 Validation of the calculation and expansion in 1=m2t 152.4.1 Expansion in 1=m2t 152.4.2 Checks of the calculation 15

3 Phenomenological results 163.1 Setup and total cross sections 163.2 NLO distributions 163.3 Sensitivity to the triple Higgs coupling 24

4 Conclusions 26

1 Introduction

After the discovery of a boson [1, 2] whose characteristics have so far been consistent with the Standard Model Higgs boson, it is a primary goal of the LHC and future colliders to further scrutinize its properties. In particular, the form of the Higgs potential needs to be reconstructed by experimental measurements, in order to con rm the mechanism of electroweak symmetry breaking postulated by the Standard Model. One of the parameters entering the Higgs potential, the mass of the Higgs boson, already has been measured to an impressive precision [3]. The other parameter, the Higgs boson self-coupling, is more di cult to constrain, as it requires the production of at least two Higgs bosons. The cross sections for Higgs boson pair production at the LHC are about three orders of magnitude smaller than the ones for single Higgs production. The dominant production channel is the gluon fusion channel, as for single Higgs boson production at the LHC.

In the gluon fusion channel, there are two categories of contributions to di-Higgs production: either a virtual Higgs boson, produced by the same mechanism as in single Higgs production, is decaying into a Higgs boson pair, involving the self-coupling hhh, or the

two Higgs bosons are both directly radiated from a heavy quark. At leading order (LO),

{ 1 {

JHEP10(2016)107

these two mechanisms can be attributed to \triangle" and \box" contributions, respectively. However, at NLO, i.e. at the level of two-loop diagrams, the diagram topologies are more complicated, such that the association of \triangle diagrams" to diagrams containing the self-coupling hhh becomes invalid.

The Higgs boson pair production cross section is additionally suppressed by the fact that there is destructive interference between contributions containing the Higgs boson self-coupling and the ones containing only Yukawa couplings to heavy quarks, and that for larger values of p^s, the contributions with an s-channel virtual Higgs boson propagator are strongly suppressed.

Therefore, narrowing the window of possible values for the triple-Higgs coupling experimentally will have to wait until the high-luminosity run of the LHC [4{6], if Standard Model rates are assumed. However, the Higgs boson pair production rate could be modi- ed by physics beyond the Standard Model (BSM), and hence it is important to be able to distinguish BSM e ects from Standard Model higher order corrections. In this paper we will study the e ects of a modi ed Higgs boson self-coupling and show that the Higgs boson invariant mass distribution is quite sensitive to changes in hhh, as such changes

modify the interference pattern.

Both ATLAS and CMS have published measurements of Higgs boson pair production in the decay channels b b [7{10], b bb b [9, 11{14], W W , b bW W , +b b [9, 15{21].

Phenomenological studies about Higgs boson pair production and the feasibility of Higgs boson self-coupling measurements can be found e.g. in refs. [22{50].

The leading order (one-loop) calculation of Higgs boson pair production in gluon fusion has been performed in refs. [51{53]. NLO corrections were calculated in the mt ! 1 limit,

where the top quark degrees of freedom are integrated out, leading to point-like e ective couplings of gluons to Higgs bosons (\Higgs E ective Field Theory", HEFT).

Top quark mass e ects have been included in various approximations. Calculating the NLO corrections within the heavy top limit and then rescaling the result di erentially by a factor BFT=BHEFT, where BFT denotes the leading order matrix element squared in the full theory, is denoted \Born-improved HEFT" approximation. This calculation [54], implemented in the program Hpair, led to a K-factor of about two. In ref. [55], another approximation, called \FTapprox", was introduced, which contains the full top quark mass dependence in the real radiation, while the virtual part is calculated in the HEFT approximation and rescaled by the re-weighting factor BFT=BHEFT. The \FT[prime]approx" result [55]

in addition uses partial NLO results for the virtual part, i.e. it employs the exact results where they are known from single Higgs production. The \FTapprox" calculation leads

to a cross section which is about 10% smaller than the Born-improved NLO HEFT cross section. Using the \FT[prime]approx" procedure, the reduction is about 9% with respect to the

Born-improved NLO HEFT result. It was also found that top width e ects can reach up to

4% above the t t threshold [55]. At LO, a nite top width reduces the total cross section

at ps = 14 TeV by about 2%. In our calculation we do not include a nite top width.

In addition, the HEFT results at NLO and NNLO have been improved by an expansion in 1=m2t in refs. [56{59], with max = 6 at NLO, and max = 2 for the soft-virtual part at

NNLO [58]. In the latter reference it is also demonstrated that the sign of the nite top

{ 2 {

JHEP10(2016)107

g(p1)

H(p3)

g(p2) H(p4)

H(p4)

g(p2)

Figure 1. Diagrams contributing to the process gg ! hh at leading order.

mass corrections, amounting to about [notdef]10%, depends on whether the re-weighting factor

is applied at di erential level, i.e. before the integration over the partonic centre of mass energy, or at total cross section level.

The NNLO QCD corrections in the heavy top limit have been performed in refs. [57, 60, 61], and they have been supplemented by an expansion in 1=m2t in ref. [58] and by resummation at NLO+NNLL in ref. [62]. The most precise results within the in nite top mass approximation are NNLO+NNLL resummed results, calculated in ref. [63], leading to K-factors of about 1.2 relative to the Born-improved HEFT result. Very recently, fully di erential NNLO results in the HEFT approximation have become available [64].

As the di erent approximations partly led to corrections with opposite sign, there was a rather large uncertainty associated with the unknown e ect of the exact top quark mass dependence at NLO, which was estimated to be of the order of 10% at ps = 14 TeV.

The full NLO calculation which became available recently [65], revealed a 14% reduction of the total cross section compared to the Born improved HEFT at ps = 14 TeV and a 24% reduction at ps = 100 TeV.

At di erential distribution level, we found that the deviation from the Born-improved HEFT approximation can be as large as 50% in the tails of distributions like the Higgs boson pair invariant mass or Higgs boson transverse momentum distributions.

This paper is structured as follows. In section 2 we give details of the calculation, in particular about the calculation of the two-loop amplitude and about the 1=mt expansion which we also performed. In section 3 we discuss our phenomenological results. We study various distributions at ps = 14 TeV and ps = 100 TeV, comparing the full NLO result to di erent approximations. We also analyze the e ects of non-Standard Model values of the triple Higgs coupling.

2 Details of the calculation

2.1 Amplitude structure

The leading order diagrams contributing to the process gg ! hh are shown in gure 1. As

the cross section does not have a tree level contribution, the virtual contribution at next-to-leading order involves two-loop diagrams, and the NLO real radiation part involves one-loop diagrams up to pentagons.

{ 3 {

JHEP10(2016)107

The amplitude for the process g(p1; ) + g(p2; ) ! h(p3) + h(p4) can be decomposed into form factors as

Mab = ab (p1; n1) (p2; n2) M (2.1)

M =

F1(^s; ^t; m2h; m2t; D) T 1 + F2(^s; ^t; m2h; m2t; D) T 2 ;

where n1; n2 are arbitrary reference momenta for the two gluon polarization vectors ; . Colour indices are denoted by a; b and

s = (p1 + p2)2; ^

t = (p1 p3)2; ^

u = (p2 p3)2 : (2.2)

The decomposition into tensors carrying the Lorentz structure is not unique. It is however convenient to de ne the form factors such that [52]

M++ = M =

which is ful lled with the following de nitions

T 1 = g

T 2 = g + 1

u ^

t; m2h; m2t; D) : (2.5)

As the LO form factor F[triangle] only contains the triangle diagrams, which have no angular

momentum dependence, it can be attributed entirely to an s-wave contribution. The form factors F and F2 can be attributed to the spin-0 and spin-2 states of the scattering amplitude, respectively.

We can get an idea about the angular dependence of F1 and F2 by considering the partial wave decomposition of the scattering amplitude, which is independent of the loop order. It should be noted however that this analysis is valid for 2 ! 2 scattering. At NLO,

the cross section for the process gg ! HH also contains a 2 ! 3 scattering contribution

from the real radiation. Therefore the analysis of the angular dependence below does not apply to the full NLO cross section.

In general, for a scattering process a + b ! c + d with the corresponding helicities

a; : : : ; d, the partial wave decomposition reads [66{68]

h c d[notdef]T (E)[notdef]00 a b[angbracketright] = 16

XJ(2J + 1)[angbracketleft] c; d[notdef]T J(E)[notdef] a; b[angbracketright]ei(sisf)dJsi;sf ( ) ; (2.6)

{ 4 {

s 8v2

JHEP10(2016)107

s8v2 F1 (2.3)

M+ = M+ =

s8v2 F2 ;

p 1 p 2 p1 [notdef] p2

; (2.4)

p2T (p1 [notdef] p2) n

m2h p 1 p 2 2 (p1 [notdef] p3) p 3 p 2

2 (p2 [notdef] p3) p 3 p 1 + 2 (p1 [notdef] p2) p 3 p 3o

where p2T = (^

t m4h)=^

s ; T1 [notdef] T2 = D 4 ; T1 [notdef] T1 = T2 [notdef] T2 = D 2 :

At leading order, we can further split F1 into a \triangle" and a \box" contribution

F1(^s; ^

t; m2h; m2t; D) = F[triangle](^s; ^

t; m2h; m2t; D) + F (^s; ^

with si = a b and sf = c d, and where [angbracketleft] c d[notdef]T (E)[notdef]00 a b[angbracketright] denotes the transition

matrix element. Unitarity must hold for each partial wave independently, i.e. [notdef]T J[notdef] 1 : Thus the amplitude is decomposed into (orthogonal) Wigner d-functions dJsi;sf ( ), where J denotes the total angular momentum and si; sf the total spin of the initial and nal state, respectively. The structure of the amplitude is such that F1 only contributes to si = 0, while F2 only contributes to si = 2. F1 has a component proportional to d00;0( ) as well as components proportional to dJ0;0( ) with J 2, while the leading contribution to F2

starts at d22;0( ). The partial waves for J > 2 are suppressed. The d-functions dJ0;0( ) are proportional to the Legendre-Polynomials PJ(cos ). As P0(x) = 1; P2(x) = 12 (3x2 1)

and d22;0( ) sin2 , we can conclude that the leading angular dependence of F2 should

be sin2 . From the analytic expression for F2 at leading order [52], we can verify that

indeed F2 p2T = (^

u ^

t m4h)=^

s = ^s4 2h sin2 where 2h = 1 4m2h=^

Further, using again the fact that the leading contributions to the amplitude come from the lower partial waves in eq. (2.6), we also conclude that the contribution from F2 should be subleading with respect to F1 in most of the kinematic regions. Indeed we observe that the contribution of the form factor F2 to the virtual two-loop amplitude is suppressed as compared to F1.

2.2 Leading order cross section

The functions Fi at leading order with full mass dependence can be found e.g. in refs. [52, 53]. At LO, the \triangle" form factor has the simple form

F[triangle] = C[triangle]

F[triangle] ; C[triangle] =

JHEP10(2016)107

hhh

; hhh = 3m2h ; (2.7)

s m2h

2 + (4 m2q ^s)C0 = 2^sq [1 + (1 q)f(q)] ;

where = 1 in the Standard Model, q = 4 m2q=^

s and

F[triangle] = 4m2q

arcsin2 1

pq for q 1

f(q) =

(2.8)

log 1+p1

1p1q

2for q < 1

d4q

(q2 m2q)

(q + p1)2 m2q (q+ p1 + p2)2 m2q : (2.9)The partonic leading order cross section for gg ! hh can be written as

LO(gg ! hh) =

2s( R) 212v4(2)3^

C0 =

n[notdef]F1[notdef]2 + [notdef]F2[notdef]2o: (2.10)

The integration limits ^

t[notdef] are derived from a momentum parametrisation in the centre-of-mass frame, leading to ^

t[notdef] = m2h ^s2 (1 h), where 2h = 1 4m

s .

To obtain the hadronic cross section, we also have to integrate over the PDFs. De ning the luminosity function as

dLijd =

Xij

dxx fi(x; F )fj

x; F

; (2.11)

{ 5 {

the total cross section reads

LO =

0 d

dLgg d ^

LO(^s = s) ; (2.12)

where s is the square of the hadronic centre of mass energy, 0 = 4m2h=s, and F is the factorization scale.

2.2.1 Heavy top limit

In the mt ! 1 approximation the LO form factors are given by

F[triangle] !

43 ^

JHEP10(2016)107

s ; F !

43 ^

s ; F2 ! 0 ; (2.13)

which implies for the the e ective ggH and ggHH couplings ch and chh,1

ch = chh =

s 4

i3 + O

m2h 4m2t

: (2.14)

From the expressions above we can derive the following expression for the squared amplitude in the heavy top limit:

[notdef]M[notdef]2

43 m2h

s m2h

+ 2 m4h 2

: (2.15)

For = 1, this expression vanishes at the Higgs boson pair production threshold ^

s 4m2h.

This explains why near the threshold the contributions containing the triple Higgs boson coupling and the ones which do not contain an s-channel Higgs boson exchange almost cancel. On the other hand, if the triple Higgs boson coupling was di erent from the Standard Model value, for example equal to zero, this should be clearly seen from the behaviour of the mhh distribution. We investigate the e ects of non-standard values for the triple Higgs boson coupling in section 3.3.

2.3 NLO cross section

The NLO cross section is composed of various parts, which we discuss separately in the following.

NLO(pp ! hh) = LO + virt + rgg + rgq + rg q + rq q : (2.16)

The contributions from the real radiation, r, can be divided into four channels, according to the partons in the initial state. The q

q channel is infrared nite. Details are given in

section 2.3.2.

1Higher order corrections to these e ective couplings, and to couplings involving more than two Higgs bosons, can be found in ref. [69] and references therein.

(^s m2h)2

{ 6 {

2.3.1 Calculation of the virtual two-loop amplitude

Amplitude generation. For the virtual two-loop amplitude, we use projectors P j to achieve a separation into objects carrying the Lorentz structure T i and the form factors

F1 and F2,

P 1M =

s8v2 F1(^s; ^

t; m2h; m2t; D) ;

P 2M =

s8v2 F2(^s; ^

t; m2h; m2t; D) :

In D dimensions we can use the tensors T i, de ned in eqs. (2.4), to build the projectors

P 1 = 14

D 2

D 3

JHEP10(2016)107

T 1

1 4

D 4

D 3

T 2 ; (2.17)

P 2 =

1 4

D 4

D 3

T 1 + 14

D 2

D 3

T 2 : (2.18)

The virtual amplitude has been generated with an extension of the program GoSam [70, 71], where the diagrams are generated using Qgraf [72] and then further processed using Form [73, 74]. The two-loop extension of GoSam contains an automated python interface to Reduze [75], which implies that the user has to provide the integral families when running GoSam-2loop. The other input les needed by Reduze are generated automatically by GoSam-2loop, based on the kinematics of the given process. The reduction of the integrals occurring in the amplitude to master integrals should be performed separately, where in principle either of the codes Reduze [75], Fire5 [76] or LiteRed [77] can be used. Examples of two-loop diagrams contributing to Higgs boson pair production are shown in gure 2.

We would like to point out again that the distinction between \triangle diagrams" and \box diagrams" becomes ambiguous beyond the leading order. At two-loop and beyond there are diagrams which contain triangle sub-diagrams but which do not contain the Higgs boson self coupling, see gure 2(l).

Integral families and reduction. For the reduction of planar diagrams we have de ned ve integral families Fi. Each family contains nine propagators which allows irreducible

scalar products in the numerator to be written in terms of inverse propagators prior to reduction. In more detail, the occurring integrals have the form

I =

dDk1

dDk2 Nr1j1 : : : NrtjtNs1jt+1 : : : Nsntjn ; (2.19)

where the Nj denote propagators of the generic form 1=(k2 m2) with integer exponents

ri 1 and si 0. The maximal number of propagators forming denominators, i.e. with

positive exponents ri, in our case is tmax = 7, and we nd that integrals with up to four inverse propagators appear in the amplitude.

We chose a non-minimal set of integral families in favour of preserving symmetries as much as possible. The families are listed in table 1. The example diagrams shown in gure 2 can be assigned to the families as follows: diagrams (a), (j) and (l) to F1, diagrams

(b) and (c) to F2, diagrams (e), (h) and (k) to F3, diagram (d) to F4, diagram (i) to F5.

{ 7 {

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

Figure 2. Examples of two-loop diagrams entering the virtual amplitude.

The amplitude generation leads to about 10000 integrals before any symmetries are taken into account. After accounting for symmetries and after reduction (complete reduction of the planar sectors and partial reduction of the non-planar ones), we end up with 145 planar master integrals plus 70 non-planar integrals, and a further 112 integrals that di er by a crossing. As these integrals contain four independent mass scales, ^

s; ^

t; m2t; m2h, only

a small subset is known analytically. Besides the diagrams which are factorizing into two one-loop diagrams [59], the known integrals are the two-loop diagrams with two light-like legs and one massive leg, which enter single Higgs boson production, calculated e.g. in refs. [78{82], and the triangles with one light-like and two o -shell legs occurring in the two-loop calculation of H ! Z [83, 84]. However, we calculate all integrals numerically

using the program SecDec [85{87].

{ 8 {

JHEP10(2016)107

k21 m2t k21 m2t k21k22 m2t k22 m2t (k1 k2)2 m2t (k1 k2)2 (k1 k2)2 (k1 + p1)2(k1 + p1)2 m2t (k1 + p1)2 m2t (k2 + p1)2 m2t (k2 + p1)2 m2t (k2 + p1)2 m2t (k1 p2)2(k1 p2)2 m2t (k1 p3)2 m2t (k2 p2)2 m2t (k2 p2)2 m2t (k2 p3)2 m2t (k2 p2 p3)2 m2t (k1 p2 p3)2 m2t (k1 p2 p3)2 m2t (k1 + p1 + p3)2 (k2 p2 p3)2 m2t (k2 p2 p3)2 m2t (k2 + p1 p2)2

F4 F5

k21 m2t k21k22 k22 m2t

(k1 k2)2 m2t (k1 k2)2 m2t (k1 + p1)2 m2t (k1 + p1)2

(k2 + p1)2 (k2 + p1)2 m2t (k1 p2)2 m2t (k1 p3)2

(k2 p2)2 (k2 p3)2 m2t (k1 p2 p3)2 m2t (k1 p2 p3)2

(k2 p2 p3)2 (k2 p2 p3)2 m2t

Table 1. Integral families for the reduction of the planar diagrams. The non-planar integrals were computed as tensor integrals.

As the integral basis is not unique, we choose to have two set-ups, relying on di erent sets of basis integrals. This serves as a strong check of the calculation of the virtual amplitude. It has previously been noted that using a nite basis [88] along with sector decomposition can increase the precision obtained by numerical integration for a given number of sampling points [89]. We also observed that switching to a nite basis in some of the planar sectors turned out to be bene cial for the numerical evaluation of the master integrals.

A complete reduction could not be obtained for the non-planar 4-point integrals. The inverse propagators appearing in unreduced integrals were rewritten in terms of scalar products such that the resulting integrals had the lowest possible tensor rank. The tensor integrals (up to rank 4) were then directly computed with SecDec.

We would like to mention that non-planar diagrams also contribute to the leading colour coe cient. Therefore we could not identify a contribution which is both dominant and gauge invariant where only planar integrals contribute.

Renormalization. We expand the amplitude in a0 = 0=(4), where 0 is the bare QCD coupling. The bare amplitude can be written as

AB = a0A(1)B + a20A(2)B + O(a30); (2.20)

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JHEP10(2016)107

where the one- and two-loop coe cients are given by

A(1)B = S 2 0 h~

b(1)0 + ~b(1)1 + ~b(1)2

2 + O( 3)i

; (2.21)

A(2)B = S2 4 0 "~b(2)

2 +

~b(2)1

~b(2)0 + O( )#

: (2.22)

Here 20 is a parameter introduced in dimensional regularisation to maintain a dimensionless bare coupling and S = e E (4) ; with E the Euler constant. The one-loop amplitude is expanded to O( 2) as it appears multiplied by the Catani-Seymour insertion operator

stemming from the integrated dipoles, I, which has poles of O( 2).To renormalize the gluon wave function we must multiply the amplitude by (ZA)

JHEP10(2016)107

for each external gluon leg, where ZA is the gluon eld renormalization constant. We renormalize the QCD coupling using the relation

a0 = a Za

; a = s4 (2.23)

where s is the renormalized coupling and Za is the associated renormalization constant. Here R is the renormalization scale and the dependence of s on R is implicit. The top mass is renormalized by relating the bare top mass m2t0 to the renormalized top mass m2t via

m2t0 = m2t + a m2t: (2.24)

In practice, we compute top mass counter-term diagrams, treating a m2t as a counter-term insertion in top quark lines and renormalize the top Yukawa coupling using

yt0 =

2R 20

1 + a m2t m2t

yt: (2.25)

No Higgs wave function or mass renormalization is required as we compute only QCD corrections.

In our calculation we use conventional dimensional regularization (CDR) with D = 4 2 . We renormalize the top mass in the on-shell scheme and the QCD coupling in

the MS ve- avour scheme (Nf = 5) with the top quark loops in the gluon self-energy subtracted at zero momentum.

The one-loop renormalization constants are given to rst order in a by2

ZA = 1 + a ZA + O(a2); (2.26)

Za = S1

1 + a Za + O(a2) ; (2.27)

where

ZA =

m2t 2R

43 TR ;

Za =

1 0 + Zhqa ; 0 =

3 CA

43 TRNlightf ;

43 TR ; (2.28)

2Note that Za corresponds to the renormalization factor of the coupling gs squared, therefore it is twice the expression for Zgs found in the literature, see e.g. eq. (3.4) of ref. [90].

Zhqa =

m2t 2R

{ 10 {

and the mass counter-term in the on-shell scheme is given by

m2t =

3 4 + O( ) : (2.29)

The coe cients ~bi in (2.21), (2.22) contain integrals Ir;s(^s; ^

t; m2h; m2t), where r denotes the number of propagators in the denominator and s denotes the number of propagators in the numerator and therefore de nes the tensor rank of the integral, see also eq. (2.19). The integrals have mass dimension [Ir;s] = D L 2r + 2s, with L the number of loops.

We may therefore factor a dimensionful parameter M out of each integral such that they depend only on dimensionless ratios

Ir;s(^s; ^

t; m2h; m2t) = (M2)L (M2)2Lr+sIr;s

s M2 ;

m2t 2R

2 m2t CF

JHEP10(2016)107

t M2 ;

m2h M2 ;

m2t M2

: (2.30)

The renormalized amplitude may then be written as

Avirt =YngZ1 2 A

AB a0 ! a Za 2R= 20

; m2t0 ! m2t + a m2t

= aA(1) + a2

2 ZA + Za

A(1) + a2 m2tAct;(1) + a2A(2) + O(a3); (2.31)

A(1) =

2R M2

hb(1)0 + b(1)1 + b(1)2 2 + O( 3)i; (2.32)

Act;(1) =

2R M2

hc(1)0 + c(1)1 + O( 2)i; (2.33)

A(2) =

2R M2

"b(2)2 2 +b(2)1 + b(2)0 + O( )#; (2.34)

where

~b(L) = (M2)L b(L) ;(L) = (M2)L c(L): (2.35)

Since m2t contains poles of O( 1) the coe cient c of the top mass counter-term must be

expanded to O( ). It is obtained by the insertion of a mass counter-term into the heavy

quark propagators,

mab(p) = i ac

6p m

(i m)

i cb

6p m

; (2.36)

where a; b; c are colour indices in the fundamental representation. Alternatively, the mass counter-term can be obtained by taking the derivative of the one-loop amplitude with respect to m.

The coe cients b and c in (2.31) are calculated numerically. We have extracted the dependence of the coe cients on the renormalization scale and introduced a dependence on a new scale, M, which we keep xed in our numerics.

For the infrared singularities stemming from the unresolved real radiation, we use the Catani-Seymour subtraction scheme [91]. The infrared poles of the virtual amplitude are

{ 11 {

cancelled after combination with the I-operator, which is given by

Igg( ) = s 2

(4)

(1 )

CA 2 + 02 CA23 + 02 + Kg; (2.37)

where Kg is also de ned by the Catani-Seymour subtraction scheme [91]. Inserting the I-operator into the Born amplitude leads to3

AIRct =

2 [notdef] Igg( ) a A(1)

= a2

2R ^ s

[notdef] 2 [notdef]

2R ^ s

2R M2

1 2212 2 CA 2 + 0 CA223 + 0 + 2Kg

hb(1)0 + b(1)1 + b(1)2 2i; (2.38)

where we again have extracted a factor (M2) from the integrals contained in the one-loop amplitude. Using (2.31) and (2.38) we therefore have

Avirt + AIRct = aA(1) + a2

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[notdef]

2R M2

hc(1)0 + c(1)1

2R M2

m2t

"b(2)2 2 +b(2)1 + b(2)0#

+ hb(1)0 + b(1)1 + b(1)2 2i 2R ^ s

2 CA 2 + 0 + n. 0

= aA(1) + a2

2R M2

1 2

h2 CA b(1)0 + b(2)2i

+ 1

2 CA b(1)0 ln

2R^s + b(2)2 ln 2RM2 + b(2)1 6m2tCF c(1)0 + 2 CA b(1)1

+ b(1)0 0 ln

2R^s + ln 2RM2 b(2)1 ln 2Rm2t 6m2tCF c(1)0

+ 2 CA b(1)1 ln

2R^s + CA b(1)0 ln2

2R^s + b(2)22 ln2 2R M2

: (2.39)

By construction the double pole in must vanish, thus (2.39) implies

b(2)2 = 2 CA b(1)0 : (2.40)

Substituting the above relation back into (2.39) we see that the dependence on the renormalization scale R cancels in the single pole term. The dependence of the cross section on the factorization scale is encoded in the P and K terms of the Catani-Seymour framework [91].

3The factor of 12 is necessary to cancel the factor of 2 obtained from squaring Avirt + AIR ct to get the

cross section.

{ 12 {

+ nite non-logarithmic terms

Integration of the two-loop amplitude. To evaluate the two-loop integrals appearing in the amplitude we rst apply sector decomposition as implemented in SecDec. In the Euclidean region sector decomposition resolves singularities in the regulator , leaving only nite integrals over the Feynman parameters which can be evaluated numerically. In the physical region we treat the integrable singularities by contour deformation [86, 92{94]. To obtain the di erential cross section we have to evaluate integrals at phase space points very close to threshold, where no special treatment is necessary but numerical convergence is considerably harder to achieve.

After sector decomposition each loop-integral Ij can be written as a sum over sectors s which have a Laurent series starting at some -order emins

Ij( ) =

Xs Xe>emins eIj;s;e: (2.41)

For the numerical evaluation of the amplitude we structured the code such that the integrand of each sector-decomposed loop integral Ij;s;e is stored along with the Laurent series

of their coe cients aj appearing in the expressions for the amplitudes (2.32){(2.34). E.g. at two-loop we write the amplitude as

A(2) =

(2.43)

is minimal. (i)j;k is the error estimate of integral Ij including its coe cients in b(i)k and

is a Lagrange multiplier. Since the loop integrals can contribute to several results b(i)k, we apply the above optimization formula for each required order in and for both form factors. For each integral, we then use the maximum of the estimated number of required sampling points. Instead of directly evaluating each integral with the calculated number of sampling points, we limit the number of new sampling points and iterate this procedure to reach the desired accuracy, updating the estimated number of sampling points after each iteration. The desired accuracy for the nite part of the two-loop amplitude ("(2)0) is set to 3% for form factor F1 and (depending on the ratio F2=F1) to a value of 5-20% for form factor F2.

{ 13 {

JHEP10(2016)107

Xj;s;e

Ij;s;e [notdef] aj( ) (2.42)

and store aj as a vector containing the coe cients of Ij in the expressions for b(2)k, leading to the amplitude structure given in eq. (2.34).

Structuring the code this way allows us to dynamically set the number of sampling points used for each integral according to its contribution to the amplitude. After calculating each integral with a xed number of sampling points, we assume that the integration error j of the integrals scales as j / t j with the integration time tj. To e ciently

calculate the results b(i)k with a given relative accuracy "(i)k = (i)k=b(i)k, we estimate the required number of sampling points for each integral such that the total time

T (i)k =

Xjtj +

@( (i)k)2

Xj ( (i)j;k)2

g(p3)

H(p4)

(p5)

(p1)

(p2)

u(p3)

H(p4)

H(p5)

g(p1)

g(p2)

g(p3)

H(p5)

u(p1)

g(p2)

H(p4)

(p4)

JHEP10(2016)107

u(p1)

(p2)

g(p2)

g(p3)

g(p1)

H(p5)

Figure 3. Examples of diagrams contributing to the real radiation part at NLO. The diagrams in the second row do not lead to infrared singularities.

For the integration we use a quasi-Monte Carlo method based on a rank-one lattice rule [95{97]. For suitable integrands, this rule provides a convergence rate of O(1=n)

as opposed to Monte Carlo or adaptive Monte Carlo techniques, such as Vegas [98], which converge O(1=pn), where n is the number of sampling points. While we observe a

convergence rate of O(1=n) for most of the integrals, the convergence of some integrals is

worse and we therefore assume a scaling of j(tj) with exponent = 0:7 when estimating the number of required sampling points.

The integration rule is implemented in OpenCL 1.1 and a further (OpenMP threaded) C++ implementation is used as a partial cross-check. The 913 phase-space points at 14 TeV (1029 phase-space points at 100 TeV) used for the current publication were computed with

16 dual Nvidia Tesla K20X GPGPU nodes. More details on the numeric evaluation of the amplitudes can be found in refs. [99, 100].

2.3.2 Real radiation

As we calculate a process which is loop-induced, the NLO corrections involve two-loop integrals. But, for the real part only single-unresolved radiation can occur. This means that a standard NLO infrared subtraction scheme can be used. We use the Catani-Seymour dipole formalism [91], combined with a phase space restriction parameter to restrict the dipole subtraction to a limited region, as suggested in ref. [101].

There are four partonic channels for the real radiation contribution to the cross section:

r(gg ! hh + g); r(gq ! hh + q); r(g

q ! hh +

q); r(q

q ! hh + g) : (2.44)

Including all crossings, there are 78 real radiation diagrams. Infrared singularities only originate from initial state radiation, diagrams with extra gluons radiated from a heavy quark line are infrared nite, which implies that the q

q channel is nite. Example diagrams

are depicted in gure 3.

{ 14 {

2.4 Validation of the calculation and expansion in 1/m2t

2.4.1 Expansion in 1/m2t

We have calculated top mass corrections as an expansion in 1=m2t in the following way: we write the partonic di erential cross section as

exp;N =

X=0d^ ()

2; (2.45)

where 2

p^s; p^t; p^u; mh

, and determine the rst few terms (up to N = 3) of this asymptotic series. The case N = 0 reproduces to the usual e ective theory approach, without the need to calculate Wilson coe cients separately, however.

To generate the diagrams we again use qgraf [72]. The generation and expansion of the amplitude in small external momenta is then performed using q2e/exp [102, 103] and leads to two-loop vacuum integrals inserted into tree-level diagrams as well as one-loop vacuum integrals inserted into massless one-loop triangles. Whereas the vacuum integrals are evaluated with Matad [104], the massless integrals can be expressed in terms of a single one-loop bubble, which we achieve with the help of Reduze [75]. Again, the algebraic processing of the amplitude is done with Form [73, 74].

The exact and expanded matrix elements were combined in the following way: a series expansion for the virtual corrections was performed then rescaled with the exact born,

dV + dLO( ) I dVexp;N

dLO( )

dLOexp;N( )

JHEP10(2016)107

+ dLO( ) I

= dVexp;N + dLOexp;N( ) I

dLO( )

dLOexp;N( )

| {z }

= dVexp;N + dLOexp;N( ) I

dLO( = 0)

dLOexp;N( = 0)

+ O ( ) : (2.46)

The rst identity is valid because the colour structure of the exact and the expanded LO cross section are identical, and the second because the sum in the bracket is nite. Thus one needs to know only the dependence of the expanded LO cross section in this approximation.

There is some ambiguity when to do the rescaling, i.e. before or after the phase-space integration, and convolution with the PDFs. We opt to do it on a fully di erential level,i.e. the rescaling is done for each phase-space point individually.

2.4.2 Checks of the calculation

We have veri ed for all calculated phase space points that the coe cients of the poles in are zero within the numerical uncertainties. For a randomly chosen sample of phase-space points we have calculated the pole coe cients with higher accuracy and obtained a median cancellation of ve digits.

Our implementation of the virtual two-loop amplitude has been checked to be invariant under the interchange of ^

t and ^

u at various randomly selected phase-space points.

{ 15 {

Single Higgs boson production has been re-calculated with the same setup for the virtual corrections and compared to the results obtained with the program Sushi [105]. Further, the one-loop amplitude has been computed using an identical framework to the two-loop amplitude and has been checked against the result of ref. [52].

As a further cross-check we have also calculated top mass corrections as an expansion in 1=m2t as explained above. We have also compared to results provided to us by Jens

Ho for the orders N = 4; 5; 6 in the expansion above, worked out in [58]. The result of the comparison is shown in gure 4. One can see that below the 2mt threshold, where agreement is to be expected, the expansion converges towards the full result.

The computation of the mass counter-term diagrams has been cross-checked by expanding the one-loop amplitude about the bare top mass

A(1)B(m2t) = A(1)B(m2t0) a m2t

= A(1)B(m2t0) a m2tAct;(1)B(m2t0); (2.47)

where Act;(1) is the one-loop top quark mass counter-term.

On the real radiation side, we have veri ed the independence of the amplitude from the phase space restriction parameter . We have also varied the technical cut pminT in the range 102 pminT=p^s 106 to verify that the contribution to the total cross section is

stable and independent of the cut within the numerical accuracy.

Further, we have compared to the results of ref. [55] for the Born-improved HEFT and FTapprox approximations and found agreement within the numerical uncertainties [106].

3 Phenomenological results

3.1 Setup and total cross sections

We use the PDF4LHC15 nlo 100 pdfas [107{110] parton distribution functions, along with the corresponding value for s for both the NLO and the LO calculation. The masses have been set to mh = 125 GeV, mt = 173 GeV, and the top quark width has been set to zero.

We use no cuts except a technical cut in the real radiation of pminT = 104 p^s. The scale variation bands are the result of a 7-point scale variation [106] around the central scale 0 =

mhh=2, with R;F = cR;F 0, where cR; cF 2 [notdef]2; 1; 0:5[notdef], except that the extreme variations

(cR; cF ) = (2; 0:5) and (cR; cF ) = (0:5; 2) are omitted. The values we obtain for the total cross sections are shown in table 2. The full NLO result has a statistical uncertainty of0.3% at 14 TeV (0.16% at 100 TeV) stemming from the phase space integration and an additional uncertainty stemming from the numerical integration of the virtual amplitude of 0.04% at 14 TeV and 0.2% at 100 TeV. These uncertainties are not included in table 2, where only scale variation uncertainties are shown.

3.2 NLO distributions

In this section we show di erential distributions at ps = 14 TeV and ps = 100 TeV for various observables and compare to the approximate results in order to assess the e ect of

{ 16 {

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@@m2t A(1)B

(m2t)

m2t0

V[triangleleft]VN=0

VN=1[triangleleft]VN=0 VN=2[triangleleft]VN=0

VN=3[triangleleft]VN=0 VN=4[triangleleft]VN=0

VN=5[triangleleft]VN=0 VN=6[triangleleft]VN=0

1[triangleright]6 V [prime]N=1[triangleleft]V [prime]N=0 V [prime]N=2[triangleleft]V [prime]N=0

V [prime]N=3[triangleleft]V [prime]N=0 V [prime]N=4[triangleleft]V [prime]N=0

V [prime]N=5[triangleleft]V [prime]N=0 V [prime]N=6[triangleleft]V [prime]N=0

V[triangleleft]V [prime]N=0

JHEP10(2016)107

1[triangleright]3

1[triangleright]0

0[triangleright]7

300 350 400 450 500 550 mhh [GeV]

Figure 4. Comparison of the virtual part as de ned in eq. (2.46) with full top-quark mass dependence to various orders in a 1=m2t expansion. V [prime]N denotes the Born-improved HEFT result to order

N in the 1=m2t expansion, i.e. V [prime]N = VN BFT=BN. The results for the orders N = 4; 5; 6 have been provided to us by Jens Ho [58].

ps LO B-i. NLO HEFT NLO FTapprox NLO

14 TeV 19.85+27:6%20:5% 38.32+18:1%14:9% 34.26+14:7%13:2% 32.91+13:6%12:6%

100 TeV 731.3+20:9%15:9% 1511+16:0%13:0% 1220+11:9%10:7% 1149+10:8%10:0%

Table 2. Total cross sections at various centre of mass energies (in femtobarns). The uncertainty in percent is from 7-point scale variations as explained in the text. The central scale is mhh=2. We

used mt = 173 GeV, mh = 125 GeV. The PDF set is PDF4LHC15 nlo 100 pdfas.

the full top quark mass dependence at NLO. Results which are obtained within the e ective eld theory approach without reweighting by the leading order results in the full theory are always denoted by \basic HEFT", while \B-i. NLO HEFT" stands for the Born-improved NLO HEFT result, where the NLO corrections have been calculated in the mt ! 1 limit

and then a reweighting factor BFT=BHEFT is applied (on di erential level, BFT stands for the Born amplitude squared in the full theory).

We decided to take the same bin sizes as in ref. [64], such that the di erences to the e ective theory results can be exhibited most clearly. In gure 5 we show the Higgs boson pair invariant mass distribution mhh at ps = 14 TeV and ps = 100 TeV, comparing

the full NLO result to various approximations. In particular, we compare to the \basic HEFT" approximation at ps = 14 TeV, showing that it fails to describe the distribution. Comparing the results at 14 TeV and 100 TeV, we observe that the di erences of the full

{ 17 {

0[triangleright]20

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B-i[triangleright] NLO HEFT NLO FTapprox LO basic HEFT NLO basic HEFT NLO

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d[triangleleft]dm hh[fb[triangleleft]GeV]

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mhh [GeV]

300 400 500 600 700 800 900 1000

mhh [GeV]

(a) 14 TeV.

(b) 14 TeV, normalised.

0[triangleright]005

1[triangleleft](d[triangleleft]dm hh)[GeV1]

d[triangleleft]dm hh[fb[triangleleft]GeV]

B-i[triangleright] NLO HEFT NLO FTapprox LO basic HEFT NLO basic HEFT NLO

0[triangleright]004

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ratio

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0[triangleright]5

0[triangleright]50

300 400 500 600 700 800 900 1000

mhh [GeV]

300 400 500 600 700 800 900 1000

mhh [GeV]

(d) 100 TeV, normalised.

Figure 5. Higgs boson pair invariant mass distribution mhh at ps = 14 TeV and ps = 100 TeV for

absolute values (left panels) and normalised to the corresponding total cross section (right panels).

NLO result to the Born-improved HEFT and also to the FTapprox result are ampli ed at

100 TeV, as expected, as the HEFT approximation does not have the correct high energy behaviour. This scaling behaviour will be discussed more in detail below.

The K-factors are de ned as the ratio of the NLO curve of the colour in the upper part of the plot to the corresponding LO result. This means that the \basic HEFT" curve is divided by the \basic HEFT" LO result. This is why the purple and the blue curves in gures 5 (a) and (c) lie on top of each other. We see that the K-factor is far from being uniform for the mhh distribution, while the HEFT results suggest a uniform K-factor.

The ratio plots in gures 5 (b) and (d) are de ned as the ratio of the curves normalized to their total cross section, shown in the upper plot, to the corresponding leading order result. As the purple curve in the upper plot is normalised to the total cross section in the basic HEFT approximation, while the blue curve is normalized to the Born-improved HEFT total cross section, the blue and the purple curves in the ratio plot do not coincide.

The pT;h distribution shown in gure 6 denotes the distribution of the \single inclusive" Higgs boson transverse momentum, which denotes the transverse momentum distribution of

{ 18 {

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1[triangleleft](d[triangleleft]dp T [arrowhookleft]h)[GeV1]

0[triangleright]008

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0[triangleright]15

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1[triangleright]5

ratio

1[triangleright]5

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0 100 200 300 400 500

pT[arrowhookleft]h [GeV]

0 100 200 300 400 500

pT[arrowhookleft]h [GeV]

JHEP10(2016)107

(a) 14 TeV.

(b) 14 TeV, normalised.

0[triangleright]007

1[triangleleft](d[triangleleft]dp T [arrowhookleft]h)[GeV1]

B-i[triangleright] NLO HEFT NLO FTapprox LO basic HEFT NLO basic HEFT NLO

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1[triangleright]5

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0[triangleright]5

0 100 200 300 400 500 pT[arrowhookleft]h [GeV]

0 100 200 300 400 500

pT[arrowhookleft]h [GeV]

(d) 100 TeV, normalised.

Figure 6. Transverse momentum distribution of (any) Higgs boson at ps = 14 TeV and ps = 100 TeV.

any (randomly picked) Higgs boson. In contrast, gure 7 shows the transverse momentum distributions of the leading-pT (\harder") and subleading-pT (\softer") Higgs boson.

As in gure 5, the K-factors are de ned as the ratio of the NLO curve of the colour in the upper part of the plot to the corresponding LO result, which means that the \basic HEFT" curve is divided by the \basic HEFT" LO result. However, in contrast to gure 5, the purple and the blue curves in gures 6 (a) and (c) do not lie on top of each other any longer. This is because there is some arbitrariness in the way the real radiation contribution, which has 2 ! 3 kinematics, is rescaled at the di erential level, i.e. for each

individual phase space point, by the Born contribution, which has 2 ! 2 kinematics. We

use a mapping of the momenta which is a weighted average over the mappings used for the dipole subtraction terms for the re-weighting factor dLOFT=dLOHEFT in the real radiation part. This mapping preserves the mhh distribution, i.e. dLO calculated with the mapped momenta and calculated with the genuine 2 ! 2 kinematics coincide in the mhh case.

For the transverse momentum distributions however, the dependence of the di erential re-weighting factor on the momentum mappings is apparent, as the transverse momentum is closely related to the third nal state particle.

{ 19 {

It again becomes very clear that reweighting the basic HEFT result is indispensable in order to get at least somewhat close to the shape of the full NLO result. The pT;h

distribution in gure 6(a) shows that, while the Born-improved NLO HEFT result starts moving out of the scale variation band of the full NLO result at 14 TeV beyond pT;h mt, the FTapprox result stays within the scale uncertainty band of the full NLO result, (even though it is clear that it systematically overestimates the full result by about 20-30%). This is not surprising, as the tail of the pT;h distribution is to a large extent dominated by the real radiation contribution. At ps = 100 TeV, the FTapprox result leaves the scale

variation band of the full NLO result beyond pT;h 280 GeV, but still is much closer to

the full result than the Born-improved NLO HEFT result. The di erences of the latter to the full result are ampli ed at 100 TeV.

In any case, it is clear that the scale variation bands can only be indicative of missing higher order corrections in perturbation theory, while the top quark mass e ects (or the omission of the exact top quark mass dependence) are in a di erent category. Therefore one cannot expect that, for example, the NLO HEFT scale variation band would comprise the full NLO result. It is also worth mentioning that the \FT[prime]approx" approximation [55], where the partial two-loop results (known from single Higgs production) were included, turned out to be a worse approximation than \FTapprox", where the virtual part is given by the

Born-improved NLO HEFT result, as it lead to a larger cross section than the \FTapprox"

one, and the latter is still larger than the full result.Note that for 2 ! 2 scattering the transverse momentum of the Higgs boson is given by

p2T =

s4 2h sin2 . Therefore, at leading order, the pT;h transverse momentum distribution directly re ects the angular dependence of the virtual amplitude. However, at NLO, the angular dependence of the form factors is in uenced to a large extent by the real radiation. This can be seen from the distributions of the leading-pT (\harder") and subleading-pT (\softer") Higgs bosons shown in gure 7. The Higgs boson will pick up a large transverse momentum if it recoils against a hard jet, therefore the K-factor of the phardT;h grows in the tail of the distribution, which is dominated by 2 ! 3 kinematics.

Figure 8 shows the rapidity distributions of both the Higgs boson pair and the leading-pT Higgs boson. As the mass e ects are uniformly distributed over the whole rapidity range, the K-factors are close to uniform for these distributions, and the FTapprox result

is within 10% of the full result. In gure 9 we display the tails of the mhh and pT;h distributions on a logarithmic scale, in order to exhibit the scaling behaviour in the high energy limit. Using leading-log high energy resummation techniques, it can be shown [111] that at high transverse momentum, the di erential partonic cross section for single Higgs (+jets) production d=dpT;h 1=paT;h scales with a = 2 in the full theory, however with

a = 1 in the e ective theory. This behaviour also has been recently con rmed by a (leading order) calculation of Higgs + 1,2,3 jet production with full mass dependence [112]. In order to investigate the high energy scaling behaviour we tted a line to the tail of the leading order mhh distribution (with the luminosity factor set to one, plotted logarithmically), and found the following scaling behaviour: with full mass dependence, the scaling is as m3hh for d^

=dmhh i.e. the partonic cross section scales as ^

approximation the scaling is as mhh for d^

=dmhh i.e. the partonic cross section grows as ^

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0[triangleright]00

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T[arrowhookleft]h[fb[triangleleft]GeV]

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T[arrowhookleft]h[fb[triangleleft]GeV]

d[triangleleft]dphard

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0 100 200 300 400 500 phardT[arrowhookleft]h [GeV]

0 100 200 300 400 500 psoftT[arrowhookleft]h [GeV]

(d) 100 TeV, subleading pT .

Figure 7. Transverse momentum distribution of the leading-pT Higgs boson (left panels) and the subleading-pT Higgs boson (right panels) at ps = 14 TeV and ps = 100 TeV.

From gure 9 one can see that this relative di erence in the high-energy scaling behaviour between the full calculation and the basic HEFT approximation is similar at NLO.

In gure 10 we show distributions for an improved FTapprox, which is supplemented

with higher order terms in the expansion of the virtual amplitude in 1=m2t as given by eq. (2.46), dubbed \exp. virt." for \expanded virtuals". We see a trend similar to the one for the virtual (plus I-operator) part shown in gure 4.

In order to better account for missing higher order corrections it is desirable to combine the full NLO with NNLO results obtained in the HEFT, ideally on a di erential level. As a rst attempt to achieve this, we take the NNLO to NLO ratio from ref. [64] and calculate

d NLO-i.NNLO HEFT = d NLO d NNLO basic HEFTd NLO basic HEFT (3.1)

bin by bin, where \NLO-i. NNLO HEFT" stands for NLO-improved NNLO HEFT. Results for various distributions are shown in gure 11. The error band is the NLO-rescaled scale uncertainty of the NNLO basic HEFT distributions, and the error on the central value is due to the error on the full NLO result. Applying the same naive rescaling on the total cross

{ 21 {

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yhardh

(d) 100 TeV, rapidity of the leading-pT Higgs.

Figure 8. Rapidity distribution of the Higgs boson pair and the leading-pT Higgs boson at ps =14 TeV and ps = 100 TeV.

103

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(b) 14 TeV, scaling behaviour of pT,h.

Figure 9. Higgs boson pair invariant mass distribution (a) and transverse momentum distribution (b) at ps = 14 TeV on a logarithmic scale. The di erent high-energy scaling behaviour of the amplitude in the full and the basic HEFT calculation can be clearly seen in the tails of the distributions.

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(a) 14 TeV, scaling behaviour of mhh.

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Figure 10. Invariant mass distribution of the Higgs boson pair (a) and pT distribution of any Higgs (b) at ps = 14 TeV combining the full real emission with the virtual contribution expanded in 1=m2t up to order N. Note that N = 0 corresponds to FTapprox.

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Figure 11. Invariant mass (a) and rapidity distribution (b) of the Higgs boson pair and transverse momentum distribution of the leading-pT (c) and the subleading-pT Higgs boson (d) at ps = 14 TeV including the combination with the NNLO HEFT results from ref. [64] described in the main text.

{ 23 {

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JHEP10(2016)107

Figure 12. Total cross sections for various values of the triple Higgs coupling. Panel (b) zooms into the region around the minimum. The curves are the result of an interpolation of integer values for 2 [notdef]1; : : : ; 5[notdef].

section, one obtains NLO-i.NNLO HEFT = 38:67+5:2%7:6% for 14 TeV, where we have neglected

the numerical errors and simply quote the relative scale uncertainty given in ref. [64] for the NNLO basic HEFT result.

3.3 Sensitivity to the triple Higgs coupling

As already mentioned in section 2.1, the Higgs boson self-coupling in the Standard Model is quite special. Not only that it is completely determined in terms of the Higgs boson mass and VEV, but it also leads to the fact that at the double Higgs production threshold p^s = 2m2h, the LO cross section is almost vanishing, due to destructive interference between box and triangle contributions. Therefore a measurement of the Higgs boson self-coupling is a very sensitive probe of New Physics e ects.

A more complete analysis of such e ects would require an approach where further operators are taken into account, for example operators which mediate direct t

tHH couplings

(and Higgs-gluon couplings which can di er from the SM HEFT ones), see e.g. [35, 38, 40]. However, the conclusions drawn from the calculation of NLO corrections in the mt ! 1 limit to the extended set of EFT Wilson coe cients have to be taken with a grain of salt, as the full top quark mass dependence may a ect them considerably.

In this section we would like to focus on just a single line in the parameter space of possible non-SM Higgs couplings and investigate the behaviour of the mhh distribution under variations of , where we have de ned hhh = 3m2h , see eq. (2.7).

In gure 12 we show the total cross section as a function of . As already observed for the LO cross section [23], it has a minimum around = 2. Negative values, which are not excluded neither theoretically nor experimentally (within certain broad limits given e.g. by vacuum stability), do not lead to destructive interference and therefore result in a much larger cross section. For large positive values, 5, the total cross section is of comparable

size to the one for [similarequal] 0, but the shape of the mhh distribution is completely di erent. This

can be seen in gure 13, where we show the Higgs boson pair invariant mass distribution for various values of the Higgs boson self-coupling, at ps = 14 TeV and ps = 100 TeV. For = 5, the di erential cross section is mainly dominated by contributions containing the Higgs boson self coupling and peaks at low mhh values. In contrast, the = 0 case,

{ 24 {

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d[triangleleft]dm hh[fb[triangleleft]GeV]

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d[triangleleft]dm hh[fb[triangleleft]GeV]

LO[arrowhookleft] = 0 NLO[arrowhookleft] = 0 LO[arrowhookleft] = 1 NLO[arrowhookleft] = 1 LO[arrowhookleft] = 2 NLO[arrowhookleft] = 2

d[triangleleft]dm hh[fb[triangleleft]GeV]

LO[arrowhookleft] = 1

NLO[arrowhookleft] = 1

LO[arrowhookleft] = 3 NLO[arrowhookleft] = 3 LO[arrowhookleft] = 5 NLO[arrowhookleft] = 5

0[suppress]300 400 500 600 700 800 900 1000

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0[suppress]300 400 500 600 700 800 900 1000

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(d) 100 TeV.

Figure 13. NLO and LO results with full top quark mass dependence for the mhh distribution at14 TeV and 100 TeV, for various values of the triple Higgs coupling, where = 1 corresponds to the Standard Model value.

which does not contain any triple Higgs coupling contribution, peaks shortly beyond the 2mt threshold at mhh 400 GeV, as does the case = 1. In the latter case, however, the

total cross section is much larger. The case = 2 shows a dip at mhh 300 GeV, which is due to destructive interference e ects as mentioned above. At 100 TeV, the shape of the distributions is very similar. However, the fact that the cross sections are much larger can be exploited to place cuts which enlarge the sensitivity to the Higgs boson self coupling. For example, one can try to enhance the self-coupling contribution by cuts favouring highly boosted virtual Higgs bosons, decaying into a Higgs boson pair which could be detected in the b bb b channel. A highly boosted virtual Higgs boson must recoil against a high-pT jet. Therefore, an enhancement of the boosted component could be achieved by imposing a pminT;jet cut on the recoiling jet in Higgs boson pair plus jet production [113].4 An additional advantage of boosted Higgs bosons is the fact that they lend themselves to the use of the b bb b rather than the b b decay channel, as the decay channel into b-quarks is accessible through boosted techniques. This leads to a gain in the rate which easily makes up for the loss in statistics due to a high pminT;jet cut.

Figure 14 shows a comparison to the di erent approximations for various values of , as well as the K-factors. For all values of , the K-factors are far from being uniform, while the HEFT approximation suggests almost uniform K-factors for 1. For = 2,

we see a pronounced \interference dip" at mhh 330 GeV, which is present at LO already.

4We thank Michelangelo Mangano for pointing this out.

{ 25 {

We can get an idea about the destructive interference e ect by observing the following: in the basic HEFT approximation, the squared Born amplitude is given by eq. (2.15). This expression has a double zero at ^

s = m2h(1 + 3 ). Therefore, the re-weighting factor BFT=BHEFT can get large when BHEFT approaches zero, i.e. at p^s [similarequal] 330:72 GeV for = 2,

p^s [similarequal] 395:29 GeV for = 3, p^s [similarequal] 450:7 GeV for = 4 and 500 GeV for = 5. In the full

theory, the amplitude does not vanish completely at these points, but nonetheless also gets small, which should be the reason for the dips in the mhh distributions for = 2 and 3.

4 Conclusions

We have presented results of a fully di erential calculation of Higgs boson pair production in gluon fusion at NLO retaining the exact top quark mass dependence. For the total cross section at ps = 14 TeV, we found a reduction of 14% compared to the Born improved HEFT, and a 24% reduction at ps = 100 TeV. For di erential distributions, the mass e ects can be even larger. In the tails of the Higgs boson transverse momentum distributions, the di erences to the Born improved NLO HEFT approximation amount to more than 50%, while the FTapprox result, where the full top mass dependence is included only in the real radiation part, stays within 20% of the full result. The basic NLO HEFT approximation, where no reweighting by the Born result in the full theory is performed, fails to properly describe the shape of the mhh and pTh distributions, in particular in the tails of the distributions. To quantify this well-known fact, we have performed an analysis of the high-energy scaling behaviour.

We also studied the in uence of non-standard values for the Higgs boson self-coupling on the total cross sections and mhh distributions. As is known from leading order, there is destructive interference between various contributions to the cross section, and this feature persists at NLO. Varying hhh= SM leads to a minimum in the value for the total cross section around hhh= SM 2:3. The shape of the mhh distribution is rather sensitive to

variations of hhh, which alter the interference pattern. For example, at hhh = 0, the total cross section is almost as large as for hhh= SM = 5, but the shape of the distributions is

very di erent.

Further, we made a rst attempt to combine the full NLO results with the NNLO results calculated in the basic HEFT approximation [64] at di erential distribution level, which should lead to a \NLO-improved NNLO HEFT" result, which may still be improved in the near future in various directions, for example towards Higgs boson decays.

Acknowledgments

We are grateful to Andreas von Manteu el for his support with the use of Reduze and to Jens Ho for providing us results to compare to the 1=mt expansion. We also would like to thank Thomas Hahn, Stephan Jahn, Gionata Luisoni, Fabio Maltoni, Michelangelo Mangano and Magdalena Slawinska for useful discussions. This research was supported in part by the Research Executive Agency (REA) of the European Union under the Grant Agreement PITN-GA2012316704 (HiggsTools). S. Borowka gratefully acknowledges nancial support by the ERC Advanced Grant MC@NNLO (340983). NG was supported by

{ 26 {

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1[triangleright]0

d[triangleleft]dm hh[fb[triangleleft]GeV]

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(f) 14 TeV, = 5.

Figure 14. Higgs boson pair invariant mass distribution mhh at ps = 14 TeV for non-standard

values of the triple Higgs coupling.

the Swiss National Science Foundation under contract PZ00P2 154829. GH would like to acknowledge the Kavli Institute for Theoretical Physics (KITP) for their hospitality. We gratefully acknowledge support and resources provided by the Max Planck Computing and Data Facility (MPCDF).

{ 27 {

(e) 14 TeV, = 4.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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SISSA, Trieste, Italy 2016

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Abstract

We study the effects of the exact top quark mass-dependent two-loop corrections to Higgs boson pair production by gluon fusion at the LHC and at a 100 TeV hadron collider. We perform a detailed comparison of the full next-to-leading order result to various approximations at the level of differential distributions and also analyse non-standard Higgs self-coupling scenarios. We find that the different next-to-leading order approximations differ from the full result by up to 50 percent in relevant differential distributions. This clearly stresses the importance of the full NLO result.

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Title

Full top quark mass dependence in Higgs boson pair production at NLO

Author

Borowka, S; Greiner, N; Heinrich, G; Jones, Sp; Kerner, M; Schlenk, J; Zirke, T

Pages

1-35

Publication year

2016

Publication date

Oct 2016

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP10(2016)107

ProQuest document ID

1830831321

SISSA, Trieste, Italy 2016

Full top quark mass dependence in Higgs boson pair production at NLO

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