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Nuclear Physics B 888 (2014) 17-29

www.elsevier.com/locate/nuclphysb

Virtual corrections to Higgs boson pair production in the large top quark mass limit

Jonathan Grigo, Kirill Melnikov \ Matthias Steinhauser *

Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany Received 11 August 2014; accepted 5 September 2014 Available online 8 September 2014 Editor: Tommy Ohlsson

Abstract

We calculate the three-loop matching coefficient Cgg, required for a consistent description of Higgs boson pair production in gluon fusion through next-to-next-to-leading order QCD in the heavy top quark approximation. We also compute the gg ^ HH amplitude in mt approximation in the full theory

and show its consistency with an earlier computation in heavy-top effective theory. © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

1. Introduction

After the discovery of a Higgs boson at the LHC, detailed investigation of its properties becomes one of primary goals of ATLAS and CMS. Important among such studies is the exploration of the Higgs boson self-coupling X. In the Standard Model, this coupling is directly related to the Higgs field potential responsible for the symmetry breaking; in the broken phase, it induces couplings of three Higgs bosons between themselves.

Experimentally, information about X is obtained from the process of Higgs boson pair production [1,2] which will be accessible after the high-luminosity upgrade of the LHC. It is well understood by now that observation of Higgs boson pair production is difficult and requires both, new ideas on how to isolate the HH signal from the background, and accurate predictions for

* Corresponding author.

1 On leave of absence from Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MA, USA. http://dx.doi.org/10.1016/j.nuclphysb.2014.09.003

0550-3213/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

the Higgs pair production in the Standard Model. In the past year we have witnessed significant advances in both of these directions.

Indeed, building upon the early ideas of Refs. [3,4] it was suggested to study Higgs pair production in W+W-bb, yybb, bbbb, and bbx+ t- channels using substructure techniques [5-7], as well as utilize ratios of cross sections [8] for single and double Higgs production to reduce the theory uncertainty and obtain best sensitivity to Higgs boson self-couplings. It remains to be seen how these theoretical ideas will bare in real experimental searches, but the current consensus seems to be that the Higgs self-coupling can be measured with the accuracy between twenty and forty percent (see, e.g., Ref. [9]).

To interpret results of experimental measurements with this accuracy, one needs to ensure that Standard Model predictions for Higgs boson pair production are known with sufficient precision. Below we summarize the current status of theoretical computations of Higgs boson pair production in the Standard Model. The leading order predictions for gg ^ HH are known since long ago; they were computed in Refs. [1,2] where the exact dependence on all kinematic variables - primarily the top quark mass - has been taken into account. Improving on these results would have required the two-loop computations with massive internal (top quarks) and external (Higgs bosons) particles; currently, such computations are technically not feasible. Instead, a possible way forward is provided by studying the QCD corrections in the approximation where the top quark mass is taken to be much larger than all other kinematic invariants in the problem. Working to leading order in 1 /mt expansion, one can integrate out the top quark and obtain an effective theory where Higgs bosons couple directly to gluons. Within such theory, next-to-leading order (NLO) computations for pp ^ HH become feasible and have been performed in Ref. [10] in mt ^m approximation while finite 1/mt corrections were calculated in [11]. Recently, the next-to-next-to-leading order (NNLO) QCD corrections to pp ^ HH were computed in [12,13] in mt ^m approximation using the close analogy between pp ^ H and pp ^ HH production in effective theory. Soft-gluon resummations and the determination of dominant n2 terms have been considered in [14] at next-to-next-to-leading logarithmic order.

In spite of tremendous progress with fixed order computations for double Higgs production, we note that NNLO QCD result of Refs. [12,13] is formally not complete. Indeed, at the NNLO QCD accuracy for Higgs pair production, one needs the Wilson coefficient CHH which was not available when Refs. [12,13] were written. The goal of this paper is to perform the computation of the CHH Wilson coefficient and therefore provide the last missing ingredient required to describe the Higgs boson pair production through NNLO QCD in the large-mt approximation.

Before we proceed with the computation of the Wilson coefficient, a word of caution about the validity of large-mt approximation is in order. Indeed, it is well known that for Higgs pair production the mt ^m limit provides a poor description of both the total cross section and kinematic distributions. In such a situation it is far from clear that extending mt ^ m computations to NNLO, as was, e.g., done in Refs. [12,13], is a sensible way to estimate higher order corrections to Higgs boson pair production. Understanding the validity of this approach was the primary goal of Ref. [11]2 where it was shown that, for a properly chosen leading order cross section, the 1 /mt effects at NLO are moderate, in the 15-20 percent range. If we assume that the same remains true at NNLO, we conclude that mt ^ m NNLO QCD corrections can be used to provide a reliable estimate of NNLO QCD corrections with the full top quark mass dependence.

2 See also Refs. [15,16] for higher order terms in the expansion in the inverse top quark mass.

The remainder of the paper is organized as follows. In the next section we introduce the effective Lagrangian for single and double Higgs production in gluon fusion. In Section 3 we describe the matching calculation of CHH. In Section 4 we discuss the computation of the virtual corrections to the gg ^ HH cross section in the full theory which serves as the cross-check of some results presented in Ref. [12]. In Section 5 we present our conclusions.

2. Effective Lagrangian for Higgs pair production

The leading order effective Lagrangian that describes interactions of any number of Higgs bosons with gluons in mt limit is given by

Leff =-3^ 01lni1 + v)- (1)

In Eq. (1) H and v are the Higgs boson field and the vacuum expectation value, respectively, O\ = 1/4GapvGpv,a, where G^v is the gluon field strength tensor, and as is the strong coupling constant. This Lagrangian is modified in higher orders of perturbative QCD. To account for this, we restrict Eq. (1) to describe interactions of gluons with up to two Higgs bosons,3 and write

Leff =-Vc°H°0 + 2(V)2cHh00. (2)

The matching coefficients CH and CHH incorporate radiative effects of top quarks that are integrated out from the Standard Model; they are given by perturbative series in the strong coupling constant.

Superscripts "0" in Eq. (2) indicate that operator renormalization has not yet been performed, so that both C°H and C°HH as well as matrix elements involving 00 are ultraviolet divergent. Following Ref. [17] we can write C°X00° = C°X/ZOl x ZO100° = CxO1, X e {H, HH}, where [18]

Zo. = 1 - ^ + (¿ÏÏI x^ + oiOD. (3)

This procedure leads to finite coefficient functions CH and CHH. In Eq. (3) we used as = a(5) (p) to denote the MS strong coupling constant in a theory with five active flavors; we will use this notation throughout the paper. We also used standard notation j0 = 11CA/3 — 4Tpni/3 and = 34CA/3 -4CFTFnl -20CATFnl/3, where CA = Nc, CF = (Nc2 - 1)/(2Nc) and TF = 1/2 are SU(Nc) color factors and nl = 5 is the number of massless quarks.

It is convenient to introduce the perturbative expansion of CH and CHH via

Cx = -3n ZCXk^i^-S^) , X e{H,HH}, (4)

^ n>0 \ n /

with C(°) = CHH = 1. Note that equality of CH and CHH at leading order follows from the Lagrangian in Eq. (1). We have chosen to parametrize CH and CHH in terms of the five-flavor strong coupling constant.

3 For the matching coefficients we adopt the notation of Ref. [12]. This implies that Ch = 4C1 with C1 from Ref. [17].

3. Direct calculation of matching coefficients

Since CH and CHH are matching coefficients between full and effective theories, it is convenient to derive them as follows: compute amplitudes of any physical process that depends on one or both of them in full and effective theories and adjust CH and CHH in such a way that the two amplitudes agree. Of course, to determine CH and CHH independently, we need to consider two, rather than one, physical processes; we choose them to be (i) Higgs boson production in gluon fusion gg ^ H and (ii) Higgs boson pair production in gluon fusion gg ^ HH. The amplitude of the first process depends on CH. The amplitude of the second process depends on both CH and Chh.

We begin with the computation of CH and consider the process g(q1)g(q2) ^ H with q2 = q| = 0 and qi ■ q2 = m2H/2. We are interested in the behavior of this process in the limit qi ~ q2 ~ mH ^ mt where the scattering amplitude can be computed in both full and effective theories. The requirement that the two amplitudes are equal up to power-suppressed terms reads

lim —0- ^full(q1 ,q2,mH,mt) = Zqx Aeff(q1,q2,mH) + O(qi/mt,mH/mt). (5)

We now study this equation order-by-order in QCD perturbation theory. At leading order, the amplitude in the full theory is given by the one-loop triangle gg ^ H diagram which can be Taylor expanded in external gluon momenta. The amplitude in the effective theory follows from the Lagrangian (1) and reads

Aeff = — [(q1 ■ q2)61 ■ €2 - (€1 • qi)(^2 ■ q1)]. (6)

Upon equating full and effective theory amplitudes, we find CH = -as/(3n), which is the first term in the expansion of the result in Eq. (4).

At NLO, the situation changes for the following reasons. On one hand, loop corrections to gg ^ H amplitudes in the effective theory appear. On the other hand, Taylor expansion of gg ^ H amplitude in small momenta and the Higgs mass no longer gives correct full theory amplitude even in the limit q1 ~ q2 ~ mH ^ mt since non-analytic dependencies on s and m2H do, in general, appear.

To cure these problems, the large-mass expansion procedure [19] is applied to Feynman diagrams that contribute to the full theory amplitude. The large-mass expansion splits all loop momenta into soft k ~ q1 ~ q2 ~ mH and hard k ~ mt and allows systematic Taylor expansions of integrands in both of these regimes. Scaling of loop momenta determines scaling of integrals since ddk|soft ~ sd/2 and ddk|hard ~ mdt. Since the gg ^ H amplitude necessarily involves at least one loop of top quarks, only one of the two loop momenta can be soft. For the NLO amplitude in full theory this implies4

AMl = m-2€ ALO + s-€m-2€ Af + m-4€ ANLLO. (7)

We note that hard part of the amplitude AfuU is obtained by Taylor expansion of integrands of loop integrals in powers of q\2/mt and mH/mt; therefore, to obtain ANLo only two-loop vacuum integrals need to be computed. On the contrary, the soft part of the amplitude requires

4 We note that for the process gg ^ H, s and mH are equal.

computation of integrals of form-factor type which depend on external soft kinematic parameters. When quantum corrections are computed in the effective theory, only soft contributions are generated. Therefore

Keff = Ch Kg + ^ f + •••• (8)

Since we are interested in CH which, by construction, cannot depend on s, Eqs. (5), (7) and (8) can be matched provided that

CHZoi KG =1 m-2 KLGd + m-4 KNaLO) • (9)

In Eq. (9), Z30 is the decoupling constant of the gluon field (cf. Refs. [17,20]), which is needed for the (on-shell) wave function renormalization of external gluons induced by the top quark loops.

The result shown in Eq. (9) allows us to obtain the matching coefficient CH by ignoring all loop corrections to gg ^ H amplitude in the effective theory and by computing Taylor expansion of relevant diagrams in q\^2/mt and mH/mt in the full theory. Extension of the above discussion to NNLO is straightforward. We write

n 7 /<eff 1 / — 2e 4hard | ,„„—4e /(hard , ,„„—6e /(hard ) /im

CHZOi KLO = To \mt KLO + mt KNLO + mt KNNlG ' (10)

and solve for CH order by order in the strong coupling constant as.

Before we show the (known) result for CH, we would like to make a few technical remarks. First, we note that it may be inconvenient to deal with external gluon polarization vectors (cf. Eq. (6)) in multi-loop computations. If so, one can use an appropriate projection operator to avoid them. A convenient choice, that respects transversality of the gluon polarization vectors, is

^ + q'lq2 + q2ql J (11) 1 q1 ■ q2

which transforms the leading order amplitude in Eq. (6) into

Keff ^—<cH(d — 2)(qx ■ q2)- (12)

Second, we note that we first renormalize the top quark mass on-shell, and the strong coupling as in the MS scheme with six active flavors. We then apply the two-loop decoupling relations to transform aS6) to aS5). We note that in this relation the O(e) terms have to be kept at one-loop order since the two-loop term of CH has an 1 /e pole whereas the one-loop term is finite. The finite result for CH, obtained via C0/Zo1 is given by [17,21,22]

as \ /5 3 \a,

<h=—d1 + 5 — 3 C')n+

1063 2 5 25 27 2

-C A--CaTF--CACF +--Cr

576 A 96 A F 12 A F + 32 F

1 /7,11 \ u2

—12 «ftf +( -6 <a—<)ln m

47 5 1 u 2

+ niTfi--Ca--Cf +--Cf ln—T

+ 1 M 144 A 16 f + 2 f m2

as : + O ^a's'

'oocv----

(a) (b) (c)

Fig. 1. Effective-theory diagrams with ggH and ggHH operators contributing to the double Higgs boson production.

as \ 11 as = -—\ 1 + — ^ +

2777 19 /j2 + 7T ln^-r

(-S + J^M + °<4 (13)

288 16

where as = a(5)(/) is the MS coupling constant defined in the theory with nl = 5 massless flavors and mt is the pole mass of the top quark.

We are now in position to extend the above discussion in such a way that the computation of CHH becomes possible. To this end, we choose the gluon fusion process where two Higgs bosons are produced, g(q1)g(q2) ^ H(q3)H(q4). We then apply the same reasoning as for the single Higgs boson production and compare amplitudes for Agg^HH computed in full and effective theories assuming that q1 ~ q2 ~ q3 ~ q4 ~ mH ^ mt. We note, however, that there is a subtlety in this case that is related to the fact that pairs of Higgs bosons can not only be produced through the ggHH operator but also through one or two ggH operators in effective theory, see Fig. 1. This can occur in two different ways. For example, already at leading order, double Higgs production in the full theory receives contributions from a box diagram gg ^ HH and from a triangle diagram gg ^ H* where the virtual Higgs boson splits into a HH pair. The second contribution has nothing to do with the matching coefficient CHH. Our master formula that is based on equating amplitudes in full and effective theories automatically takes care of this since an identical contribution is also generated in the effective theory through a local interaction vertex ggH. Hence, diagrams with intermediate off-shell Higgs bosons cancel exactly between full and effective theory amplitudes so that at leading order only gg ^ HH box diagram in the full theory is needed to obtain the Wilson coefficient CHH. Similar subtleties occur in higher orders, see, e.g., Fig. 1(c). Nevertheless, separation of loop momenta into soft and hard and the understanding that effective theory loops are always soft allows us to consider only hard contributions in the full theory and equate them directly to products of matching coefficients and various tree amplitudes in the effective theory. We therefore obtain the following generalization of Eq. (10) valid in the case of Higgs pair production

CHHZO1 Atree, 1PI + CH ZO1 Abffe, 1PR,A.=0 + CHZO1 Aeffe, 1PR,A=0

— — i4hard+ 4hard -I- 4hard ^ (14)

= z 0 lA1PI + a1pr,a.=0 + A1PR,A.=0j. (14)

When writing Eq. (14) we introduced labels 1PI and 1PR, to denote one-particle reducible and one-particle irreducible contributions in both full and effective theory. Moreover, we separated various one-particle reducible contributions on both sides of Eq. (14) into those that involve and do not involve the triple Higgs boson coupling X. We also note that these one-particle reducible contributions contain poles in soft kinematic parameters, so that it is more appropriate to talk

about Laurent rather than Taylor expansion of full theory amplitudes in Eq. (14). However, all kinematic poles cancel exactly between the left-hand and the right-hand side of Eq. (14), as required by the consistency of effective theory.

We note that Eq. (14) can be immediately used for the computation of the matching coefficient CHH since this is the only unknown quantity there. However, before doing that, it is important to realize that Eq. (14) can be significantly simplified. Indeed, as the immediate generalization of the leading order discussion in the previous paragraph, we observe the exact matching between one-particle reducible contributions to Eq. (14) caused by nonvanishing triple Higgs boson coupling; this allows us to remove ipR x=0 and KPR = from both sides of Eq. (14).

It is natural to think that further simplifications are possible. For example, it is easy to imagine that Z20i CHKffe 1PR a.=0 and K^R A=0 should match exactly on the two sides of the equation and can be removed. Indeed, this is what happens through two loops but the two contributions do not match exactly at three loops leaving a remainder that gets re-absorbed into CHH matching coefficient. Finally, we want to point out that all calculations have been performed for arbitrary gauge parameter f which drops out in the final result, a strong check of the correctness of our calculation.

The final result for CHH that we obtain can be summarized as follows. Using the parametriza-tion of CH and CHH in Eq. (4), we find

C(1) _C(1) C(2) _C(2) _i_ a(2)

<HH = <H , <HH = <H + ^HH>

(2) 7 2 5 11 1 35 2ni

ahH = 8 <a — 6 CaTf — — CaCf + 2 CFTF + CfmTF = — + y, (15)

(2) (2)

where ni is the number of massless quarks. We note that the difference between CHH and CyH is significant. Indeed, for ni = 5 and u = mt, we find

AnH ^ 4-79, «6-15, (16)

which implies that C^H/Cff ^ 1^8. We note that in the computation of Refs. [12,13] it was

(2) (2) (2) assumed that 0 < CHH < 2CH ; Eq. (16) shows that our result for CHH is within this interval

but close to its upper boundary. The numerical effects on Chh = Ch on the cross section is

investigated in Section 5.

4. Virtual corrections to gg ^ HH production at NNLO

In the previous section we computed the matching coefficient CHH by comparing hard contributions in the full theory and tree contributions in the effective theory. In this way, we only had to compute vacuum bubble integrals to obtain CHH. However, we can calculate the full gg ^ HH amplitude in mt approximation if we account also for soft contributions in the full theory. Then we obtain the NNLO virtual corrections to gg ^ HH amplitude independent of effective theory computations.

How difficult is it to compute soft contributions through NNLO for the double Higgs production? It turns out that it is not so hard. Indeed, since we have to deal with at most three-loop diagrams in the full theory and since at least one of those three loops has to be hard, the most complicated soft integrals that need to be computed are two-loop three-point functions and one-loop four-point functions with all internal and two external lines massless. All such integrals are known which means that we can obtain full gg ^ HH amplitude from the full theory.

We consider production of the Higgs boson pair in gluon collisions g(q1) + g(q2) ^ H(q3) + H(q4) and introduce Mandelstam variables s = (qi + q2)2 = (q3 + q4)2, t = (qi - q3)2 = (q2 — q4)2 and u = (q1 — q4)2 = (q2 — q3)2. Gluons and Higgs bosons are on the mass shell, q2 2 = 0 and q|4 = m2H. We write virtual contributions to gg ^ HH differential cross section as

as dg„(1) / Ol 2k dt V 2n

2dof dt

+ O (o

where again as = a(5)(^). The leading order cross section in Eq. (17) can be written as

= £loN (C2O — 4€Clo + 4e2),

N = ( ^(1 — 1 + f2^2 — f32 ^3 + f4i ^4 + ,

a2[(tu — m4H)/sY

21132v4n 3(1 — e)2r(1 — e)(4n)-

We note that CLO is the sum of two leading order contributions to Higgs boson pair production cross section associated with box and triangle diagrams and that e -dependent factors in SLO originate from the d -dimensional two-particle phase space, and the average over gluon polarizations. The higher order e terms in Eq. (18) differ from such terms in Ref. [12] since the matching coefficients used in [12] are strictly four-dimensional. We can emulate this effect in our calculation and reproduce the results of Ref. [12]. Similar comments also apply to the NLO results given below.

Representative Feynman diagrams contributing to the amplitude Agg^HH can be found in Fig. 2. We compute the differential cross sections using large-mass expansion [19] with the help of the C++ program exp [23] that factorizes all integrals into hard (vacuum) and soft (two-loop three-point and one-loop four-point) integrals. As we already noticed, all such integrals can be computed in a straightforward way. Once this is done, we obtain perturbative results for the virtual corrections to the gg ^ HH cross section through NNLO in the heavy top approximation.

We note that, since virtual corrections are computed in the full theory, the results are made ultraviolet finite by means of standard renormalization procedure. In particular, no matching computations are required. Therefore, by comparing the result of the full theory computation with Ref. [12], one can independently verify the effective theory computations reported there and, at the same time, check the consistency of CHH computation described in the previous section.

To present the results for virtual corrections, we follow the standard practice and isolate infrared-divergent pieces using Catani's representation of scattering amplitudes [24]. For ultraviolet finite gg ^ HH scattering amplitude, we write

5 The definition of Clq is taken from Ref. [12], however, we set the width of the Higgs boson to zero.

'ÙÛÛ1

'ÛOCV—*

Fig. 2. Sample Feynman diagrams contributing to the amplitude Agg^HH.

Agg^HH = as

as , as

Ao + a- Al + a-

2n \2n

Ai = l(1]Ao + Aiifln,

,(1,2)

A2 = l(2)Ao + I(1)Ai + A2,fin.

The two operators Ig , ) depend on QCD color factors CA, CF and niTp, the Mandelstam variable s and the dimensional regularization parameter e. In the limit e ^ 0, lg1,2) develop 1 fe2 and 1/e4 singularities, respectively. On the other hand, A^,2),An contributions to NLO and NNLO amplitudes are finite. The exact form of lg1,2) operators can be found in Refs. [24,25]; we do not reproduce them here. Using the representation of scattering amplitude (20), we write the virtual contributions to gg ^ HH cross sections as

+ 2Re[lgi) ]da

(2) v, fin

(1) v, fin

+ 2Re[ I^-fi + {II^I2 + 2Re[ (I^) 2] + 2Re[ lg2)]}

It follows from Eq. (21) that all divergent contributions are proportional to either leading or NLO cross sections. Since the leading order cross section has already been given in Eq. (18), it is sufficient to provide results for finite NLO and NNLO contributions.

In the following we present our results in a way which allows for a simple comparison with Ref. [12]. Contributions to gg ^ HH amplitude split naturally into two classes - one that corresponds to only one effective vertex (ggH or ggHH; they occur after shrinking the vacuum bubbles to a point) and the other one that involves two ggH vertices. In the former case, all soft contributions are reducible to three-point functions and are proportional to the leading order amplitude CLO. Diagrams with two effective vertices start to contribute at NLO and the

---- ^jgire®----

o o o o o o

---- '05000----

Fig. 3. One-loop (a) and two-loop (b) form-factor contributions which lead to F(1) and F(2). Multiplying (c) and (d) with the LO amplitude leads to n(1) and

r(2) . y (2)

is obtained from squaring contribution (c).

corresponding one-loop corrections are needed at NNLO. For convenience we show sample diagrams up to NNLO in Fig. 3 where also the notation for the individual contributions is introduced. Following this classification, we write the finite contribution to the one-loop cross section as

(1) v, fin

F(1) + Clo

+ O (e

where the first term in square brackets is the contribution of diagrams with a single effective vertex and the second term is the contribution of all diagrams with two effective vertices. We perform a similar decomposition at NNLO and write

(2) v, fin

= £lo[Clof(2) + Clo^(2) + v(2)] + o(e),

where the new element V(2) is the contribution of NLO diagrams with two effective vertices [cf. Fig. 3(c)] squared.

In addition to soft contribution described so far, hard contributions also enter Eqs. (22) and (23). They can be computed directly using full theory diagrams without resorting to separating these hard contributions into CH and CHH. We can then combine CH and CHH results described in the previous section with the effective theory computation reported in Ref. [12] and compare the result with the full mt computation described in this section. The two results agree which provides a good consistency check for both, the effective theory computation and the calculation of the CHH Wilson coefficient reported in the previous section.

We conclude by showing full results for various quantities that enter Eqs. (22) and (23) from the full theory computation. We give results for arbitrary renormalization scale \x and separate contributions due to different color factors. We obtain

F(X) = 3 Ca[15 + 11Ls ] — 3CF — 3 LsniTF + A1 CA

—37 — y £2 + 12ft + 15L„

— 11Ls + y L2

— 3L„

+ 3 nlTF [14ft + 4Ls — 2L2 ]J

3 + 61ft — — &Ls — — £3 + 12ft Ls + 18ft — 31 Lm + — L2m

— 6Ls — — L2 + — L

+ 2 CF

29 2 — y — 9 £2 + 35 Lm — 3L2n

+ 3 nlTF

—7 £2 + 5ftLs + 2 £3 + L2S — 3 L3

r(1) = 1 - te\l + Z_H +

3 3 s tu stu

+ Ca[45 + 22Ls] - 45CF - 8LsmTF

+ e2 2ft - 101CF - -niTF[7ft + 2Ls + 2LmLs - L2]

+ 3Ca[210 + 154ft - 48ft + 22(2Ls + 2LmLs - Lf)]J

:F(2) = CA

23 827 83

--ft--ft + ~ ft + - Lm +--Ls +--L2

6 Z2 36 Z3 + 8Z4 + 2 m + 3 s + 12 s

+ 9C2F

+ CaCF

- 3ni TfCA

145 11 '

6 2 Lm 11 Ls

' 2255

+ n2 T2 217

4 2 22 3 L - TZ2

--Ca--CF

24 A 3 2

+ 40Ls + 22L2--ft +--ft

54 + s + s 6 Z2 + 3 Z3

- 3niTFCF[41 - 12Lm - 24ft], R(2) = -7C2 + 11CaCF - 8niCFTF + 3CA

- 8Cf - 9 TFni

+ 4Ls + Lt + Lu

476 11 4mH

— + — (4Ls + Lt + Lu) + —H 9 3 s

- CA( 1 , 2mH

2Li2 1 -

+ 4Li2

m2H\ ( m2H

+ 4Li2

+4ln( 1 - mk) m( - mk

+ 4ln( 1 - mH^\ lnf-^^ - 8ft - ln2f-u u u

(3stu)

'[mH(t + u)2 - 2mHtu(t + u)2 + 12u2(4s2 + (t + u}2)]

with Lm = ln(u2/m2), Ls = ln(u2/s), Lu = ln[u2/(—u)], Lt = ln[u2/(—t)]. For CA = 3,

CF = 4/3, TF = 1/2, u2 = s and e = 0 these results agree with the analytic expressions of ( 2) (2) ( 2) Ref. [12] provided that CHH — <h in Eq. (15) of that reference is replaced by AHH given in

our Eq. (15).

5. Conclusions

We computed the three-loop Wilson coefficient of a G2H2 operator that describes interactions of two Higgs bosons with gluons in the approximation that the top quark mass is infinitely large. This is the last missing ingredient that is required to perform consistent NNLO QCD computation of Higgs pair production in the large-mt limit. Our main result - the three-loop contribution to the Wilson coefficient CHH - is given in Eq. (15). We have also computed virtual corrections to Higgs pair production in gluon fusion in the full theory using asymptotic expansions in the inverse top quark mass and verified consistency of our CHH computation with the calculation of gg ^ HH virtual corrections within the effective field theory [12].

An interesting feature of the computed three-loop corrections is that they break the equality CH = CHH that persists through two loops. Therefore, their main effect is to change the relative contributions of the box and triangle diagrams to double Higgs production. Since box and triangle

contributions cancel exactly at the threshold for producing the two Higgs bosons, the relatively small difference between CH and CHH gets kinematically amplified.6 Indeed, using the relation between Higgs boson self-coupling, the vacuum expectation value and the Higgs boson mass 2X2 v — mH, we write the relative correction as

d^CH=Chh - d°Cn=Chh _ 2(s - mH) a(2)

daCn —Chh (s - 4mH) HH

(d \ 2 (s — m2 )

Trh --Hr> (25)

0.11/ (s - 4mH)

where AHH from Eq. (15) is used. The strong kinematic enhancement at the threshold s — 4m2H is evident. Numerically, assuming mH — 125 GeV and as — 0.11, the correction to cross section for gg ^ HH computed using CH — CHH approximation amounts to 6.4 percent at

— 270 GeV and 1.7 percent at — 400 GeV. The change in the total hadronic cross section

(2) (2)

pp ^ HH amounts to 1%, compared to the case CH) — CHH. While all these corrections are quite moderate, the change in threshold behavior is interesting and is qualitatively different from a relatively uniform enhancement of lower-order cross sections provided by soft QCD effects.

Acknowledgements

This work is supported by the Deutsche Forschungsgemeinschaft through grant STE 945/2-1 and by KIT through its distinguished researcher fellowship program.

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6 We note that this is very similar to what happens when Higgs boson self-coupling constant X is shifted away from its Standard Model value and/or when 1/mt corrections to box and triangle contributions are taken into account [11].

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