A subleading operator basis and matching for gg → H

Journal of High Energy Physics, Jul 2017

The Soft Collinear Effective Theory (SCET) is a powerful framework for studying factorization of amplitudes and cross sections in QCD. While factorization at leading power has been well studied, much less is known at subleading powers in the λ ≪ 1 expansion. In SCET subleading soft and collinear corrections to a hard scattering process are described by power suppressed operators, which must be fixed case by case, and by well established power suppressed Lagrangians, which correct the leading power dynamics of soft and collinear radiation. Here we present a complete basis of power suppressed operators for gg → H, classifying all operators which contribute to the cross section at \( \mathcal{O}\left({\lambda}^2\right) \), and showing how helicity selection rules significantly simplify the construction of the operator basis. We perform matching calculations to determine the tree level Wilson coefficients of our operators. These results are useful for studies of power corrections in both resummed and fixed order perturbation theory, and for understanding the factorization properties of gauge theory amplitudes and cross sections at subleading power. As one example, our basis of operators can be used to analytically compute power corrections for N -jettiness subtractions for gg induced color singlet production at the LHC.

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A subleading operator basis and matching for gg → H

Revised: June A subleading operator basis and matching for gg Ian Moult 0 1 2 4 Iain W. Stewart 0 1 3 Gherardo Vita 0 1 3 0 Cambridge , MA 02139 , U.S.A 1 Berkeley , CA 94720 , U.S.A 2 Theoretical Physics Group, Lawrence Berkeley National Laboratory 3 Center for Theoretical Physics, Massachusetts Institute of Technology , USA 4 Berkeley Center for Theoretical Physics, University of California , USA The Soft Collinear E ective Theory (SCET) is a powerful framework for studying factorization of amplitudes and cross sections in QCD. While factorization at leading power has been well studied, much less is known at subleading powers in the sion. In SCET subleading soft and collinear corrections to a hard scattering process are described by power suppressed operators, which must be xed case by case, and by well established power suppressed Lagrangians, which correct the leading power dynamics of soft and collinear radiation. Here we present a complete basis of power suppressed operators for gg ! H, classifying all operators which contribute to the cross section at O( 2), and showing how helicity selection rules signi cantly simplify the construction of the operator basis. We perform matching calculations to determine the tree level Wilson coe cients of our operators. These results are useful for studies of power corrections in both resummed E ective Field Theories; Higgs Physics; Perturbative QCD; Resummation - HJEP07(21)6 and xed order perturbation theory, and for understanding the factorization properties of gauge theory amplitudes and cross sections at subleading power. As one example, our basis of operators can be used to analytically compute power corrections for N -jettiness subtractions for gg induced color singlet production at the LHC. 1 Introduction Helicity operators in SCET 2 3 2.1 2.2 3.1 3.2 3.3 Operator basis SCET Helicity operators Leading power Subleading power Subsubleading power 3.4 Cross section contributions and factorization 4 Matching 4.1 4.2 4.3 Leading power matching Subleading power matching Subsubleading power matching 3.3.1 3.3.2 3.3.3 3.4.1 3.4.2 3.4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.3.6 Collinear eld insertions P? insertions Ultrasoft insertions Vanishing at O( ) Relevant operators at O( 2) Comparison with q q Ultrasoft derivative qqg ggg qqgg gggg Ultrasoft gluon 5 Conclusions A Generalized basis with P?n; P?n 6= 0 A.1 Operators A.2 Matching B Useful Feynman rules { i { Introduction Factorization theorems play an important role in understanding the all orders behavior of observables in Quantum Chromodynamics (QCD). While typically formulated at leading power, the structure of subleading power corrections is of signi cant theoretical and practical interest. A convenient formalism for studying factorization in QCD is the Soft Collinear E ective Theory (SCET) [1{4], an e ective eld theory describing the soft and collinear limits of QCD. SCET allows for a systematic power expansion in 1 at the level of the Lagrangian, and simpli es many aspects of factorization proofs [5]. SCET has been used to study power corrections at the level of the amplitude [6] and to derive factorization theorems at subleading power for B decays [7{13]. More recently, progress has been made towards understanding subleading power corrections for event shape observables [14{17]. In this paper, we focus on the power suppressed hard scattering operators describing the gluon initiated production (or decay) of a color singlet scalar. We present a complete operator basis to O( 2) in the SCET power expansion using operators of de nite helicity [17{19], and discuss how helicity selection rules simplify the structure of the basis. We also classify all operators which can contribute at the cross section level at O( 2), and discuss the structure of interference terms between di erent operators in the squared matrix element. We then perform the tree level matching onto our operators. These results can be used to study subleading power corrections either in xed order, or resummed perturbation theory, and compliment our recent analysis for the case of qq initiated production [17]. We will consider the production of a color singlet nal state, which we take for concreteness to be the Higgs, with the underlying hard Born process ga(qa) gb(qb) ! H(q1) ; where ga;b denote the colliding gluons, and H the outgoing Higgs particle. We work in the Higgs e ective theory, with an e ective Higgs gluon coupling Lhard = C1(mt; s) G 12 v G H ; p obtained from integrating out the top quark. Here v = ( 2GF ) 1=2 = 246 GeV, and the matching coe cient is known to O( s3) [20]. The active-parton exclusive jet cross section corresponding to eq. (1.1) can be proven to factorize for a variety of jet resolution variables. For concreteness we will take the case of beam thrust, B. The leading power factorized expression for the beam thrust cross section can be written schematically in the form [21] (1.1) (1.2) d (0) dxa dxb d (qa + qb; q1) M (fq1g) Hg(0)(fqig) hBg(0)Bg(0)i Sg(0) ; (1.3) where the xa;b denote the momentum fractions of the incoming partons, d denotes the Lorentz-invariant phase space for the Born process in eq. (1.1), and M (fqig) denotes the measurement made on the color singlet nal state.1 The dependence on the underlying hard 1By referring to active-parton factorization we imply that this formula ignores contributions from proton spectator interactions [22] that occur through the Glauber Lagrangian of ref. [23]. There are also perturbative corrections at O( s4) that are described by a single function Bgg in place of BgBg [23, 24]. { 1 { interaction is encoded in the hard function Hb (fqig) and the trace is over color. The soft function Sb describes soft radiation, and the beam functions Bi describe energetic initialstate radiation along the beam directions [25]. The factorization theorem of eq. (1.3) allows logarithms of B to be resummed to all orders through the renormalization group evolution of the hard, beam and soft functions. The factorization formula in eq. (1.3) captures all terms in the cross section scaling as B 1, including delta function terms. More generally the cross section can be expanded in powers of B as, d Here the superscript refers to the suppression in powers of p B relative to the leading power cross section. This particular convention is chosen due to the power expansion in SCET, where one typically takes the SCET power counting parameter orders in eq. (1.4) are expected to vanish, and we will show this explicitly for d (1)=d B. The rst non-vanishing power correction to the cross section then arises from d (2)=d B, to scale like 2 B. Odd which contains all terms that scale like O( B0 ). It is generally expected that the power corrections in eq. (1.4) obey a factorization formula similar to that of eq. (1.3). Schematically, d (n) d B = dxa dxb d (qa + qb; q1) M (fq1g) X H(nHj) j j j hB(nBj)B(n0Bj)i j j S(nSj); (1.5) where j sums over the multiple contributions that appear at each order, nHj + nBj + n0Bj + nSj = n, and denotes a set of convolutions, whose detailed structure has not been speci ed and is known to be more complicated than typical leading power factorization theorems. We also let include nontrivial color contractions. The derivation of such a formula would enable for the resummation of subleading power logarithms using the renormalization group evolution of the di erent functions appearing in eq. (1.5), allowing for an all orders understanding of power corrections to the soft and collinear limits. To derive a factorization theorem in SCET, QCD is matched onto SCET, which consists of hard scattering operators in Lhard and a Lagrangian Ldyn describing the dynamics of soft and collinear radiation (1.4) The dynamical Lagrangian can be divided into two parts LSCET = Lhard + Ldyn : Ldyn = Lfact + L(G0) : (1.6) (1.7) Here L(G0) is the leading power Glauber Lagrangian determined in ref. [23] which couples together soft and collinear elds in an apriori non-factorizable manner, and Lfact includes both the leading interactions which can be factorized into independent soft and collinear Lagrangians, and subleading power interactions which are factorizable as products of soft and collinear elds. Our focus here is on determining the subleading power Lhard for gg ! H, and Ldyn only plays a minor role when we carry out explicit matching calculations (and L(G0) does not play a role at all since these matching calculations are tree level). { 2 { The hard scattering operators are process dependent, while the Lagrangian Ldyn is universal and the relevant terms for our analysis are known in SCET to O( 2) in the power expansion [26{31]. A eld rede nition can be performed in the e ective theory [5] which allows for the decoupling of leading power soft and collinear interactions in Lfact. If L(G0) is proven to be irrelevant, then the Hilbert spaces for the soft and collinear dynamics are factorized, and a series of algebraic manipulations can be used to write the cross section as a product of squared matrix elements, each involving only collinear or soft elds. This provides a eld theoretic de nition of each of the functions appearing in eq. (1.5) in terms of hard scattering operators and Lagrangian insertions in SCET. Since the Lagrangian insertions are universal, the remaining ingredient which is required to derive a subleading power factorization theorem for the gg ! H process is a complete basis of subleading power hard scattering operators. The derivation of a basis, which is the goal of this paper, provides the groundwork for a systematic study of power corrections for color singlet production through gluon fusion. An important application of the results presented in this paper is to the calculation of subleading power corrections to event shape observables for gg ! H, such as 0-jettiness [21]. Recently, there has been considerable interest in the use of event shape observables for performing NNLO xed order subtractions using the qT [32] or N -jettiness [33, 34] subtraction schemes. These ideas have been applied to color singlet production [35{45], to the production of a single jet in association with a color singlet particle [33, 46{48], and to inclusive photon production [49]. By analytically computing the power corrections for the subtractions, their stability and numerical accuracy can be signi cantly improved. This was shown explicitly in [16] with the SCET based analytic calculation of the leading power corrections for 0-jettiness for qq initiated Drell Yan like production of a color singlet, and it would be interesting to extend this calculation to gg ! H. For a direct calculation of the power corrections in QCD, see [50]. An outline of this paper is as follows. In section 2 we provide a brief review of SCET and of the helicity building blocks required for constructing subleading operators in SCET. In section 3 we present a complete basis of operators to O( 2) for the gluon initiated production of a color singlet, and carefully classify which operators can contribute to the cross section at O( 2). In section 4 we perform the tree level matching to the relevant operators. We conclude and discuss directions for future study in section 5. 2 Helicity operators in SCET In this section we brie y review salient features of SCET, as well as the use of helicity operators in SCET. Reviews of SCET can be found in refs. [51, 52], and more detailed discussions on the use of helicity operators can be found in refs. [17{19]. 2.1 SCET SCET is an e ective eld theory of QCD describing the interactions of collinear and soft particles in the presence of a hard interaction [1{5]. Collinear particles are characterized by a large momentum along a particular light-like direction, while soft particles are characterized by having a small momentum with homogenous scaling of all its components. For { 3 { each jet direction present in the problem we de ne two light-like reference vectors ni and ni such that ni2 = ni2 = 0 and ni ni = 2. We can then write any four-momentum p as n n p = ni p i + ni p i + pni? 2 2 : A particle with momentum p close to the ~ni direction will be referred to as ni-collinear. In lightcone coordinates its momenta scale like (ni p; ni p; pni?) 1 is a formal power counting parameter determined by the measurements or kinematic restrictions imposed on the QCD radiation. The choice of reference vectors is not unique, and any two reference vectors, ni and n0i, with ni n0 i O( 2) describe the same physics. The freedom in the choice of ni is represented in the e ective theory as a symmetry known as reparametrization invariance (RPI) [26, 27]. More explicitly, there are three classes of ni p ( 2; 1; ). Here RPI transformations under which the EFT is invariant RPI-I ni ! ni + ni ! ni ? RPI-II ni ! ni ni ! ni + ? RPI-III ni ! e ni ni ! e ni : ? (2.1) (2.2) 0, and The transformation parameters are assigned the power counting , ? 0 . Additionally, while in nitesimal, and satisfy ni can be a ? = ni nite parameter, the parameters ? and ? are ? = ni ? = ni ? = 0. RPI symmetries can be used to relate operators at di erent orders in the power expansion, and will be used in this paper to relate the Wilson coe cients of several subleading power operators to the leading power Wilson coe cients for the gg ! H process. Furthermore, the RPIIII symmetry will constrain the form of the Wilson coe cients of our subleading power operators. At tree level the Wilson coe cients are simply rational functions of the large momentum components of the elds appearing in the operator, which must satisfy the rescaling symmetries of RPI-III. SCET is constructed by decomposing momenta into label and residual components p = p~ + k = ni p~ i + p~ni? 2 + k : n (2.3) The momenta ni p~ Q and p~ni? Q is a typical scale of the hard interaction, while k describing uctuations about the label momentum. Fields with momenta of de nite scaling are obtained by performing a multipole expansion. Explicitly, the e ective theory consists of collinear quark and gluon elds for each collinear direction, as well as soft quark and gluon elds. Independent gauge symmetries are enforced for each set of elds, which have support for the corresponding momenta carried by that eld [31]. The leading power gauge symmetry is exact, and is not corrected at subleading powers. In SCET, elds for ni-collinear quarks and gluons, ni;p~(x) and Ani;p~(x), are labeled by their collinear direction ni and their large momentum p~. The collinear elds are written in a mixed representation, namely they are written in position space with respect to the residual momentum and in momentum space with respect to the large momentum components. Q are referred to as the label components, where 2Q is a small residual momentum { 4 { Operator Power Counting Bni? ni qus 3 Dus 2 Derivatives acting on collinear elds give the residual momentum dependence, which scales k 2Q, whereas the label momentum operator P gives the label momentum component. It acts on a collinear eld as P ni;p~ = p~ ni;p~. Note that we do not need an explicit ni label on the label momentum operator, since it is implied by the eld that the label momentum operator is acting on. We will use the shorthand notation P = ni P. We will often suppress the explicit momentum labels on the collinear elds, keeping only the label of the collinear sector, ni. Of particular relevance for the construction of subleading power operators is the P? operator, which identi es the O( ) perp momenta between two collinear elds within a collinear sector. Soft degrees of freedom are described in SCET by quark and gluon elds qus(x) and Aus(x). In this paper we will restrict ourselves to the SCETI theory where the soft degrees of freedom are referred to as ultrasoft so as to distinguish them from the soft modes of SCETII [53]. The operators we construct are also applicable in the SCETII theory, but additional soft operators would be required. For a more detailed discussion see ref. [17]. 2Q, but do not carry label momenta, since they are not associated with any collinear direction. Correspondingly, they also do not carry a collinear sector label. The ultrasoft elds are able to exchange residual momenta between distinct collinear sectors while remaining on-shell. SCET is constructed such that manifest power counting in the expansion parameter is maintained at every stage of a calculation. All elds have a de nite power counting [3], shown in table 1, and the SCET Lagrangian is expanded as a power series in LSCET = Lhard + Ldyn = X the hard scattering operators O(i), and are determined by an explicit matching calculation. The hard scattering operators encode all process dependence, while the L dynamics of ultrasoft and collinear modes in the e ective theory, and are universal. The terms we need are explicitly known to O( 2), and can be found in a summarized form in [51]. Finally, L(G0) is the leading power Glauber Lagrangian [23], which describes the leading power coupling of soft and collinear degrees of freedom through potential operators. (i) describe the In this paper we will be interested in subleading power hard scattering operators, in particular, Lhard and L(h2a)rd. The hard e ective Lagrangian at each power is given by a (1) product of hard scattering operators and Wilson coe cients, (j) { 5 { The appropriate collinear sectors fnig are determined by directions found in the collinear states of the hard process being considered. If there is a direction n01 in the state then we sum over the cases where each of n1, : : :, n4 is set equal to this n0 .2 The sum over A; in 1 eq. (2.5) runs over the full basis of operators that appear at this order, which are speci ed by either explicit labels A and/or helicity labels on the operators and coe cients. The C~ (j) are also vectors in the color subspace in which the O( j ) hard scattering operators A O~ A(j)y are decomposed. Explicitly, in terms of color indices, we follow the notation of ref. [18] and have O~ y + : ( : ::: : )[ : ] = Oa1 n + : ( : ::: : )[ : ] T a1 n ; Ca1 + : (n: ::: : )[ : ] = X C+k : ( : ::: : )[ : ] k T a1 k n T a1 n C~+ : ( : ::: : )[ : ] : (2.6) n is a row vector of color structures that spans the color conserving subspace. The ai are adjoint indices and the i are fundamental indices. The color structures do not necessarily have to be independent, but must be complete. Hard scattering operators involving collinear elds are constructed out of products of elds and Wilson lines that are invariant under collinear gauge transformations [2, 3]. The eld building blocks for these operators are collinear gauge-invariant quark and gluon elds, de ned as h 1 h g ni;!(x) = (! Pni ) Wnyi (x) ni (x) ; i Bni?;!(x) = (! + Pni ) Wnyi (x) iDni? Wni (x) : i For this particular de nition of ni;!, we have ! > 0 for an incoming quark and ! < 0 for an outgoing antiquark. For Bni;!?, ! > 0 (! < 0) corresponds to outgoing (incoming) gluons. The covariant derivative in eq. (2.7) is given by, (2.7) (2.8) (2.9) and the collinear Wilson line is de ned as iDni? = Pni? + gAni? ; Wni (x) = " X perms exp g Pni n Ani (x) # : distinct classes fnig and fnjg have ni nj 2 The emissions summed in the Wilson lines are O( 0) in the power counting. The square brackets indicate that the label momentum operators act only on the elds in the Wilson line. The collinear Wilson line, Wni (x), is localized with respect to the residual position x, so that ni;!(x) and Bni;!(x) can be treated as local quark and gluon elds from the perspective of the ultrasoft degrees of freedom. All operators in the theory must be invariant under ultrasoft gauge transformations. Collinear elds transform under ultrasoft gauge transformations as background elds of 2Technically the ni in fnig are representatives of an equivalence class determined by demanding that { 6 { the appropriate representation. Dependence on the ultrasoft degrees of freedom enters the operators through the ultrasoft quark eld qus, and the ultrasoft covariant derivative Dus, de ned as (2.10) Other operators, such as the ultrasoft gluon eld strength, can be constructed from the ultrasoft covariant derivative. The power counting for these operators is shown in table 1. The complete set of collinear and ultrasoft building blocks is summarized in table 1. These can be combined, along with Lorentz and Dirac structures, to construct a basis of hard scattering operators at any order in the SCET power counting. All other eld and derivative combinations can be reduced to this set by the use of equations of motion and operator relations [54]. As shown in table 1, both the collinear quark and collinear gluon building block elds scale as O( ). Therefore, while for most jet processes only a single collinear eld appears in each sector at leading power, subleading power operators can involve multiple collinear elds in the same collinear sector, as well as P? insertions. The scaling of an operator is simply obtained by adding up the powers for the building blocks it contains. This implies that at higher powers hard scattering operators involve more and more elds, or derivative insertions, leading to any increasingly complicated structure. Furthermore, to ensure that the e ective theory completely reproduces all IR limits of the full theory, as well as to guarantee that the renormalization group evolution of the operators is closed, it is essential that operator bases in SCET are complete, namely all operators consistent with the symmetries of the problem must be included. Enumerating a minimal basis of operators becomes di cult at subleading power, and it is essential to be able to e ciently identify independent operators, as well as to make manifest all symmetries of the problem. 2.2 Helicity operators An e cient approach to simplify operator bases in SCET is to use operators of de nite helicity [17{19]. This general philosophy is well known from the study of on-shell scattering amplitudes, where it leads to compact expressions, removes gauge redundancies, and makes symmetries manifest. The use of helicities is also natural in SCET since the e ective theory is formulated as an expansion about identi ed light like directions with respect to which helicities are naturally de ned, and collinear elds carry these directions as labels. Furthermore, since SCET is formulated in terms of collinear gauge invariant elds, see eq. (2.7), one can naturally project onto physical polarizations. SCET helicity operators were introduced in [18] where they were used to study leading power processes with high multiplicities. This was extended to subleading power in [19] where it was shown that the use of helicity operators is also convenient when multiple elds appear in the same collinear sector. In this section we brie y review SCET helicity operators, since we will use them to simplify the structure of the subleading power basis for gg ! H. We will follow the notation and conventions of [17{19]. A summary of the complete set of operators that we will use is given in table 2. { 7 { HJEP07(21)6 Power counting: Equation: a (2.14) 2 (2.15) (2.16) 2 2 (2.23) 2 Field: Power counting: Equation: a scattering operators for gg ! H, together with their power counting order in the -expansion, and the equation numbers where their de nitions may be found. The building blocks also include the conjugate currents J y in cases where they are distinct from the ones shown. We de ne collinear gluon and quark elds of de nite helicity as Bi a = i = 1 " 2 (ni; ni) Bnai?;!i ; 5 ni; !i ; i = ni; !i 1 2 5 : Here a, , and are adjoint, 3, and 3 color indices respectively, and the !i labels on both the gluon and quark building blocks are taken to be outgoing, which is also used for our helicity convention. Using the standard spinor helicity notation (see e.g. [55] for an introduction) p j i p h j jp+i = 2 1 + 5 u(p) ; h p j = sgn(p0) u(p) with p lightlike, the polarization vector of an outgoing gluon with momentum p can be written "+(p; k) = hp+j jk+i p ; p2[kp] ; where k 6= p is an arbitrary light-like reference vector, chosen to be ni in eq. (2.11a). Since fermions always arise in pairs, we can de ne currents with de nite helicities. Here we will restrict to the case of two back to back directions, n and n, as is relevant for gg ! H. A more general discussion can be found in refs. [17, 19]. We de ne helicity currents where the quarks are in opposite collinear sectors, h = 1 : !n !n [nn] n+ n ; (J y)nn0 = p 2 !n !nhnni n n+ ; { 8 { (2.11a) (2.11b) (2.12) (2.13) (2.14) as well as where the quarks are in the same collinear sector, h = vectors ni; ni, so it is naturally written as Here i can be either n or n. All of these currents are manifestly invariant under the RPIIII symmetry of SCET. The Feynman rules for all currents are very simple, and are given in [17]. Note that the operators Jnn , Ji0 , and Ji0 have quarks of the same chirality, and hence are the ones that will be generated by vector gauge bosons. At subleading power one must also consider insertions of the Pi? operator. Note that we can drop the explicit i index on the P? operator, as it is implied by the eld that the operator acts on the perpendicular subspace de ned by the P+?(ni; ni) = (ni; ni) P? ; P?(ni; ni) = +(ni; ni) P? : (2.16) The P? operator carry helicity h = 1. We use square brackets to denote which elds are acted upon by the P? operator, for example Bi+ P+?Bi Bi , indicates that the P+? operator acts only on the middle eld, whereas for currents, we use a curly bracket notation P?Ji0 = Ji0 (P?)y = 1 1 2p! ! 2p! ! hP? i+in=i i+ ; h i+n=i i+(P?)y ; i to indicate which of the elds within the current is acted on. eld rede nition. The BPS eld rede nition is de ned by [5] To work with gauge invariant ultrasoft gluon elds, we construct our basis post BPS Bn? ! YnabBnb?; a n ! Y n n ; and is performed in each collinear sector. Here Yn, Yn are fundamental and adjoint ultrasoft Wilson lines. For a general representation, r, the ultrasoft Wilson line is de ned by 3 Yn(r)(x) = P exp 4ig ds n Aaus(x + sn)T(ar)5 ; where P denotes path ordering. The BPS eld rede nition has the e ect of decoupling ultrasoft and collinear degrees of freedom at leading power [5], and it accounts for the full physical path of ultrasoft Wilson lines [56, 57]. The BPS eld rede nition introduces ultrasoft Wilson lines into the hard scattering operators. These Wilson lines can be arranged with the ultrasoft elds to de ne ultrasoft 2 0 Z 1 { 9 { (2.17) (2.18) (2.19) gauge invariant building blocks. In particular, the gauge covariant derivative in an arbitrary representation, r, can be sandwiched by Wilson lines and decomposed as Yn(ir) yiDu(rs) Yn(ir) = i@us + [Yn(ir) yiDu(rs) Yn(ir)] = i@us + T(ar)gBuas(i) : Here we have de ned the ultrasoft gauge invariant gluon eld by gBuas(i) = 1 ni iGbus Ynbai : In the above equations the derivatives act only within the square brackets. Note from a eq. (2.21), that ni Bus(i) = 0. The Wilson lines which remain after this procedure can be absorbed into a generalized color structure, TBPS (see [19] for more details). Determining a complete basis of color structures is straightforward, and detailed examples are given in [17]. Having de ned gauge invariant ultrasoft gluon elds, we can now de ne ultrasoft gauge invariant gluon helicity elds and derivative operators which mimic their collinear counterparts. For the ultrasoft gluon helicity elds we de ne the three building blocks a and similarly for the ultrasoft derivative operators = " Unlike for the gauge invariant collinear gluon elds, for the ultrasoft gauge invariant gluon eld we use three building block elds to describe the two physical degrees of freedom because the ultrasoft gluons are not fundamentally associated with any direction. Without making a further gauge choice, their polarization vectors do not lie in the perpendicular space of any xed external reference vector. When inserting ultrasoft derivatives into operators we will use the same curly bracket notation de ned for the P? operators in eq. (2.17). Gauge invariant ultrasoft quark elds can also appear explicitly in operator bases at subleading powers. From table 1 we see that they power count as O( 3), and are therefore not relevant for our construction of an O( 2) operator basis. Details on the structure of subleading power helicity operators involving ultrasoft quarks can be found in [17]. It is important to emphasize that although ultrasoft quarks do not appear in the hard scattering operators at O( 2) they do appear in the calculation of cross sections or amplitudes at O( 2) due to subleading power Lagrangian insertions in the e ective (examples where they play an important role for factorization in B-decays include both exclusive decays [53, 58, 59] and inclusive decays [7, 8, 10]). Such ultrasoft quark contributions also played an important role in the recent subleading power perturbative SCET calculation of ref. [16]. Finally, we note that the helicity operator basis presented in this section only provides a complete basis in d = 4, and we have not discussed evanescent operators [60{62]. An extension of our basis to include evanescent operators would depend on the regularization scheme. However, in general additional building block elds would be required, for example an scalar gluon Ba to encode the ( 2 ) transverse degrees of freedom of the gluon. As in (2.20) (2.21) standard loop calculations, we expect that the evanescent operators at each loop order could be straightforwardly identi ed and treated. Since we do not perform a one-loop matching to our operators, we leave a complete treatment of evanescent operators to future work. 3 Operator basis In this section we enumerate a complete basis of power suppressed operators up to O( 2) for the process gg ! H. The organization of the operator basis in terms of helicity operators will make manifest a number of symmetries arising from helicity conservation, greatly reducing the operator basis. Helicity conservation is particularly powerful in this case due to the spin-0 nature of the Higgs. The complete basis of eld structures is summarized in table 3. In section 3.4 we will show which operators contribute to the cross section at O( 2). These operators are indicated with a check mark in the table. Examining eq. (2.5) we see that the hard Lagrangian in SCET is written as a sum over label momenta of the hard operators. For the special case of two back-to-back collinear sectors this reduces to (j) up to the swap of n $ n. This means that when writing an operator with di erent eld structures in the two collinear sectors we are free to make an arbitrary choice for which is labeled n and which n, and this choice can be made independently for each operator. When squaring matrix elements, all possible interferences are properly incorporated by the sum over directions in eq. (3.1). As discussed in section 1, we will work in the Higgs e ective theory with a Higgs gluon coupling given by the e ective Lagrangian in eq. (1.2). We therefore do not consider operators generated by a direct coupling of quarks to the Higgs. All quarks in the nal state are produced by gluon splittings. The extension to include operators involving quarks coupling directly to Higgs, as relevant for H ! bb, is straightforward using the helicity building blocks given in section 2.2. 3.1 Leading power The leading power operators for gg ! H in the Higgs e ective theory are well known. Due to the fact that the Higgs is spin zero, the only two operators are gngn : OB(0+)a+b = Bn+ Bn+ H ; a b O(0)ab = Bn Bn a b B H : (3.2) Order Category Operators (equation number) # helicity con gs color O( 0) O( ) O( 2) Hgg Hqqg HqqQQ Hqqqq Hqqgg Hgggg P? Ultrasoft O(0)ab = Bn 1 Bn 1 a a B 1 1 OB(1n);an 1( i) = Bn;n 1 Jnn j a Oq(2Q)1( 1; 2) = J(q)n 1 J(Q)n 2 H (3.2) = SBna 1 Bn 2 Bn 3 Bn 4 b c d = SBna 1 Bn 2 Bn 3 Bn 4 b c d H (3.14) H (3.16) O(2)a P a 1( 2)[ P ] = Bn 1 fJn 2 (P?P )yg H (3.27) a b c = Bus(n) 1 Bn 2 Bn 3 a b c = Bus(n) 1 Bn 2 Bn 3 H (3.43) H (3.45) O(2) OB(2()uasb(cn)) 1: 2 3 OB(2()uasb(cn)) 1: 2 3 O(2)ab 2 4 4 2 2 3 1 4 2 3 2 4 4 2 2 4 2 2 4 1 1 2 2 2 2 2 3 3 9 9 1 2 1 1 1 2 2 1 X X X X X X X X X represents a symmetry factor present for some cases, and detailed lists of operators can be found in the indicated equation. The number of allowed helicity con gurations are summarized in the fourth column. The nal column indicates which operators contribute to the cross section up to O( 2) in the power expansion, as discussed in section 3.4. Counting the helicity con gurations there are a total of 53 operators, of which only 28 contribute to the cross section at O( 2). Of those 28, only 24 have non zero Wilson coe cients at tree level since the operators in eq. (3.27) are absent at this order. These numbers do not include the number of distinct color con gurations which are indicated in the 5th column. Here the purple circled denotes that this is a hard scattering operator in the e ective theory, while the dashed circles indicate which elds are in each collinear sector. Note that here we have opted not to include a symmetry factor at the level of the operator. We will include symmetry factors in the operator only when there is an exchange symmetry within a given collinear sector. We assume that overall symmetry factors which involve exchanging particles from di erent collinear sectors are taken into account at the phase HJEP07(21)6 space level. The color basis here is one-dimensional, and we take it to be T ab = ab ; TBaPbS = T Yn Yn ab = T Yn Yn ba : (3.3) Due to the spin zero nature of the Higgs, the O( ) operators are highly constrained. To simplify the operator basis we will work in the center of mass frame and we will further choose our n and n axes so that the total label ? momentum of each collinear sector vanishes. This is possible in an SCETI theory since the ultrasoft sector does not carry label momentum, and it implies that we do not need to include operators where the P? operator acts on a sector with a single collinear operator must therefore come from an explicit collinear eld. eld. At O( ) the suppression in the There are two possibilities for the collinear eld content of the operators, either three collinear gluon elds, or two collinear quark elds and a collinear gluon eld. Interestingly, the helicity selection rules immediately eliminate the possibility of O( ) operators with three collinear gluon elds, since they cannot sum to a zero helicity state. We therefore only need to consider operators involving two collinear quark elds and a collinear gluon eld. The helicity structure of these operators is also constrained. In particular, to cancel the spin of the collinear gluon eld, the collinear quark current must have helicity the quark-antiquark pair arises from a gluon splitting, since we are considering gluon fusion in the Higgs EFT, and therefore both have the same chirality. Together this implies that the quarks are described by the current Jnn . The only two operators in the basis at O( ) are HJEP07(21)6 qn(qg)n : taken to be n, and (qg)nqn : OB(1n)+a(+) = Bna+ Jnn + H ; OB(1n)a ( ) = Bna Jnn H ; for the case that the gluon eld is in the same sector as the antiquark eld, which we have OB(1n)a (+) = Bna Jnn + H ; OB(1n)+a( ) = Bna+ Jnn H ; for the case that the gluon eld is in the same direction as the collinear quark eld. In both cases the color basis is one-dimensional T a = T a . After the BPS eld rede nition we have a ; (3.4) (3.5) (3.6) for eqs. (3.4) and (3.5) respectively. 3.3 At O( 2) the allowed operators can include either additional collinear eld insertions, insertions of the P? operator, or ultrasoft eld insertions. We will treat each of these cases in turn. We begin by considering operators involving only collinear eld insertions. At O( 2) the operator can have four collinear elds. These operators can be composed purely of collinear gluon elds, purely of collinear quark elds, or of two collinear gluon elds and a collinear quark current. In each of these cases helicity selection rules will restrict the possible helicity combinations of the operators. Two quark-two gluon operators. We begin by considering operators involving two collinear quark elds and two collinear gluon elds, which are again severely constrained by the helicity selection rules. Since the two gluons elds can give either helicity 0 or 2, the only way to achieve a total spin zero is if the quark elds must be in a helicity zero con guration. Furthermore, since they arise from a gluon splitting they must have the same chirality. This implies that all operators must involve only the currents Jn 0 or Jn 0 , where we have taken without loss of generality that the two quarks are in the n-collinear sector, as per the discussion below eq. (3.1). The gluons can then either be in opposite collinear sectors, or in the same collinear sector. The color basis before BPS eld rede nition is identical for the two cases. It is three dimensional, and we take as a basis T ab = (T aT b) ; (T bT a) ; tr[T aT b] : In the case that the two collinear gluons are in opposite collinear sectors a basis of helicity operators is given by (gqq)n(g)n : OB(21)+ab+(0) = Bn+ Bnb+ Jn 0 H ; a OB(21)ab (0) = Bn Bnb Jn 0 H ; a OB(21)+ab+(0) = Bn+ Bnb+ Jn 0 H ; a OB(21)ab (0) = Bn Bnb Jn 0 H : a The color basis after BPS eld rede nition is given by TBaPbS = (YnT Yn)cb(T aT c) ; (YnT Yn)cb(T cT a) ; TF (YnT Yn)ab ; (3.9) In the case that the two gluons are in the same collinear sector a basis of helicity where we have used tr[T aT b] = TF ab. operators is given by (qq)n(gg)n : OB(22)+ab (0) = Bn+ Bnb Jn 0 H ; a The color basis after BPS eld rede nition is OB(22)+ab (0) = Bn+ Bnb Jn 0 H : a TBaPbS = (YnyYnT aT bYnyYn) ; (YnyYnT bT aYnyYn) ; tr[T aT b] : (3.7) (3.8) (3.10) (3.11) HJEP07(21)6 Four gluon operators. Operators involving four collinear gluon elds can have either two collinear gluon elds in each sector, or three collinear gluon elds in one sector. A basis of color structures before BPS eld rede nition is given by T abcd = B B B B B B Btr[acdb] + tr[abdc]C Btr[adbc] + tr[acbd]C 12 BBBBBBtttrrr[[[aaabcdcdbdcb]]] tr[adcb]CC tr[abdc]CC : tr[acbd]CC B B B B 1 ab cd 1 2 (YnT Yn)ac(YnT Yn)bd 1 T T 2 (Yn Yn)ad(Yn Yn)bc not hold for Nc > 3. helicity operators is given by (gg)n(gg)n : O4(2g)1a+bc+d++ = O4(2g)1abcd = 1 a b c d 4 Bn+Bn+Bn+Bn+ H ; b 1 a 4 Bn Bn Bn Bn c d Here we have used a simpli ed notation, writing only the adjoint indices of the color matrices appearing in the trace. For example, tr[abcd] tr[T aT bT cT d]. The color bases after BPS eld rede nition will be given separately for each case. For the speci c case of SU(Nc) with Nc = 3 we could further reduce the color basis by using the relation tr[abcd + dcba] + tr[acdb + bdca] + tr[adbc + cbda] = tr[ab]tr[cd] + tr[ac]tr[db] + tr[ad]tr[bc] : (3.13) We choose not to do this, as it makes the structure more complicated, and because it does In the case that there are two collinear gluon elds in each collinear sector, a basis of O4(2g)1a+bcd+ a b c d = Bn+Bn Bn+Bn The spin zero nature of the Higgs implies that a number of helicity con gurations do not contribute, and therefore are not included in our basis operators here. The color basis after BPS eld rede nition is given by TBaPbcSd = B(tr[T a0 T c0 T d0 T b0 ] + tr[T b0 T d0 T c0 T a0 ])Yn B(tr[T a0 T d0 T b0 T c0 ] + tr[T c0 T b0 T d0 T a0 ])Yn a0a a0a 0(tr[T a0 T b0 T c0 T d0 ] + tr[T d0 T c0 T b0 T a0 ])Yna0a b0b c0c d0d1T Yn Yn Yn 12 BBBBBB(((tttrrr[[[TTT aaa000 TTT bcd000TTTcdb000TTTdcb000 ]]] The other relevant case has three gluons in one sector, which we take to be the n collinear sector. The basis of operators is then given by (g)n(ggg)n : O4(2g)2a+bc+d+ = 1 a 2 Bn+Bn+Bn+Bn b c d H ; O4(2g)2abc+d = 1 a 2 Bn Bn+Bn Bn b c d H : (3.16) In this case, the post-BPS color basis is given by 0(tr[T a0 T b0 T c0 T d0 ] + tr[T d0 T c0 T b0 T a0 ])Yna0a b0b c0c d0d1T Yn Yn Yn 12 BBBBBB(((tttrrr[[[TTT aaa000 TTT bcd000TTTcdb000TTTdcb000 ]]] Yn Yn Yn Yn d0dC d0dCC d0dCC d0dCCC : d0dCC C C C C A 1 2 (YnT Yn)ab cd 1 T 2 (Yn Yn)ac bd 1 T 2 (Yn Yn)ad bc The helicity basis has made extremely simple the task of writing down a complete and minimal basis of four gluon operators, which would be much more di cult using traditional Lorentz structures. The helicity operators also make it simple to implement the constraints arising from the spin zero nature of the Higgs. Four quark operators. We now consider the case of operators involving four collinear quark elds. These operators are again highly constrained by the helicity selection rules and chirality conservation, since each quark-antiquark pair was produced from a gluon splitting. In particular, these two constraints imply that there are no operators with nonvanishing Wilson coe cients with three quarks in one collinear sector. Therefore, we need only consider the cases where there are two quarks in each collinear sector. When constructing the operator basis we must also treat separately the case of identical quark avors Hqqqq and distinct quark avors HqqQQ. For the case of distinct quark avors HqqQQ we will have a q $ Q symmetry for the operators. Furthermore the two quarks of avor q, and the two quarks of avor Q, are necessarily of the same chirality. In the case that both quarks of the same avor appear in the same current, the current will be labeled by the avor. Otherwise, the current will be labeled with (qQ) or (Qq) appropriately. For all these cases, the color basis is T = ; : (3.18) We will give results for the corresponding TBPS basis as we consider each case below. For the case of operators with distinct quark avors HqqQQ and two collinear quarks in each of the n and n sectors there are three possibilities. There is either a quark antiquark pair of the same avor in each sector (e.g. (qq)n(QQ)n), a quark and an antiquark of distinct avors in the same sector (e.g. (qQ)n(Qq)n), or two quarks with distinct avors in the same sector (e.g. (qQ)n(qQ)n). In the case that there is a quark anti-quark pair of the same avor in each sector, the basis of helicity operators is (qq)n(QQ)n : Oq(2Q)1(0;0) = J(q)n0 J(Q)n0 H ; Oq(2Q)1(0;0) = J(q)n0 J(Q)n0 H ; Oq(2Q)1(0;0) = J(q)n0 J(Q)n0 H ; Oq(2Q)1(0;0) = J(q)n0 J(Q)n0 H ; (3.19) where we have chosen the q quark to be in the n sector. Since all the operators have total helicity 0 along the n^ direction, there are only chirality constraints here and no constraints from angular momentum conservation. In the case that there is a quark antiquark of distinct avors in the same sector, chirality and angular momentum conservation constrains the basis to be (qQ)n(Qq)n : Oq(2Q)2(0;0) = J(qQ)n0 J(Qq)n0 H ; Oq(2Q)2(0;0) = J(qQ)n0 J(Qq)n0 H : For the operators in eqs. (3.19) and (3.20) the color basis after BPS eld rede nition is When there are two quarks of distinct avors in the same sector the basis of helicity operators is constrained by chirality and reduced further to just two operators by angular momentum conservation, giving (qQ)n(qQ)n : Oq(2Q)3(+; ) = J(q)nn+ J(Q)nn H ; Oq(2Q)3( ;+) = J(q)nn J(Q)nn+ H : For the operators in eq. (3.22) the color basis after BPS eld rede nition is ; hYnyYni hYnyYni : In the cases considered in eqs. (3.19) and (3.20) where there is a quark and antiquark eld in the same collinear sector, we have chosen to work in a basis using Ji0 and Ji0 which contain only elds in a single collinear sector. One could also construct an alternate form for the basis, for example using the currents Jnn . From the point of view of factorization our basis is the most convenient since the elds in the n and n-collinear sectors are only connected by color indices, which will simplify later steps of factorization proofs. In the (3.20) (3.21) (3.22) (3.23) for our basis. distinct operators include (qq)n(qq)n : following, we will whenever possible use this logic when deciding between equivalent choices For identical quark avors the operators are similar to those in eqs. (3.19), (3.22). The Oq(2q)1(0;0) = 1 4 J(q)n0 J(q)n0 H ; Oq(2q)1(0;0) = J(q)n0 J(q)n0 H ; Oq(2q)1(0;0) = 1 4 J(q)n0 J(q)n0 H ; (qq)n(qq)n : two operators Note that in eq. (3.24) there are only three operators due to the equivalence between the Oq(2q)3(+; ) = J(q)nn+ J(q)nn H : X J(q)n0 J(q)n0 H X J(q)n0 J(q)n0 H ; n n due to the fact that the n label is summed over, as in eq. (3.1). We also have the same color bases as in eqs. (3.21) and (3.23) for Oq(2q)1 and Oq(2q)3 respectively. 3.3.2 P? insertions Since we have chosen to work in a frame where the total ? momentum of each collinear sector vanishes, operators involving explicit insertions of the P? operator rst appear at O( 2). The P? operator can act only in a collinear sector composed of two or more elds. At O( 2), there are then only two possibilities, namely that the P? operator is inserted into an operator involving two quark elds and a gluon eld, or it is inserted into an operator involving three gluon elds. In the case that the P? operator is inserted into an operator involving two quark elds and a gluon eld, the helicity structure of the operator is highly constrained. In particular, the quark elds must be in a helicity zero con guration. Combined with the fact that they must have the same chirality, this implies that all operators must involve only the currents Jn 0 or Jn 0 . Here we have again taken without loss of generality that the two quarks are in the n-collinear sector. A basis of operators is then given by O(2)a P (3.24) HJEP07(21)6 (3.25) (3.26) where the color structures that appear at tree level are the rst components of the color basis of eqs. (3.44) and (3.46). In terms of helicity operators, (2) (2) (2) (2) OB(us(n))0:++ = OB(us(n))0: OB(us(n))0:++ = = = OB(us(n))0: 2g if abd 2g if abd 2g if abd 2g if abd T T T T Yn Yn Yn Yn Yn Yn Yn Yn dc dc dc dc !2Bna+;!1 Bn+;!2 Bus(n)0H ; b c !2Bna ;!1 Bn ;!2 Bus(n)0H ; b c !1Bna+;!1 Bn+;!2 Bus(n)0H ; b c !1Bna ;!1 Bn ;!2 Bus(n)0H : b c This agrees with the relation derived from RPI symmetry, given in eq. (3.47). For convenience, we also give the Feynman rule for the contribution of the hard scattering operators to a single ultrasoft emission both before BPS eld rede nition (4.42) (4.43) HJEP07(21)6 = 2gf abc!1g? n 2gf abc!2g? n ; as well as after BPS eld rede nition = 2gf abc !1 n = 2gf abc n !1 + n p3 n p3 n p3 n p3 n !2 !2 n n !2 + n p3 n p3 n p3 n p3 n !1 : (4.44) Note that the contribution from hard scattering operators before the BPS eld rede nition is local, but not gauge invariant, since before BPS eld rede nition there are also SCET T -product diagrams involving. After BPS eld rede nition, the contribution from the hard scattering operators is gauge invariant, but at the cost of locality. However, as emphasized in [17], the form of the non-locality is dictated entirely by the BPS eld rede nition, and is therefore not problematic. It is therefore advantageous to work in terms of the ultrasoft gauge invariant building blocks, so that the contributions from the hard scattering operators alone are gauge invariant. Note also that here we have restricted the ? momentum of the two collinear particles to vanish for simplicity. Furthermore, because of the ultrasoft wilson lines in the color structure of eq. (3.44), there are also Feynman rules with multiple ultrasoft emissions. This is analogous to the familiar case of the B? operator which has Feynman rules for the emission of multiple collinear gluons. 4.3.5 qqgg A basis for the operators involving two collinear quark and two collinear gluon elds was given in section 3.3.1. In section 3.4.2 it was argued that the only non-vanishing contributions to the cross section at O( 2) arise from operators with the two collinear quarks and a collinear gluon in one sector, recoiling against a collinear gluon in the other sector. In performing the matching to these operators there are potentially T -product terms from the three gluon O( 2) operator of section 4.3.3, where one of the gluons splits into a qq pair. By choosing the momentum n n n n we see from eq. (4.33) that all SCET T -product contributions vanish, so that the result must be reproduced by hard scattering operators in SCET. Expanding the QCD diagrams to O( 2), we nd that all the contributions from the two gluon vertex in the Higgs e ective theory vanish HJEP07(21)6 O( 2) O( 2) This result might be anticipated from the structure of the diagrams. However, there is a non-vanishing contribution from the three-gluon vertex in the Higgs e ective theory O( 2) = 4g2f abc!4 3? In terms of standard Lorentz and Dirac structures the corresponding hard scattering operator is given by OB(21) = 4g2if abc!4 b c (!1 + !2)2 Bn?;!4 Bn?;!3 n;!1 T a n= 2 n; !2 H : Projected onto the helicity operator basis of eq. (3.8), and using the color basis of eq. (3.7), O( 2) = 0 (4.46) (4.47) (4.48) For convenience, we also give the Feynman rule for the operator = 4g2f abcT a!4 4?n !4 = n 2 : (4.50) Again, this contains additional terms not present in the matching calculation, and it is straightforward to check that they are necessary to satisfy the required Ward identities. Finally, we consider the matching to the operators involving four collinear gluon elds. A basis of such operators was given in eq. (3.16). In section 3.4.2 it was argued that to contribute to the cross section at O( 2), there must be three collinear gluons in the same sector. For concreteness, we take this to be the n sector. The operators with three gluons in the n sector can be obtained by crossing n $ n. To perform the matching we choose the momenta as n n n n n With this choice, each particle in the n sector is on-shell, but the sum of any two of their momenta is o -shell, pi2 = 0 ; (p1 + pj )2 O(1) ; (pj + pk)2 O( 2) ; j; k = 2; 3; 4 ; j 6= k ; (4.52) which regulates all propagators. This particular choice of momenta is convenient since it simpli es T -product contributions from SCET. Furthermore, we take the external polarizations to be purely perpendicular, i.e. i = i?. All of the four gluon operators give a non-vanishing contribution to the four-gluon matrix element for this choice of polarization, allowing their Wilson coe cients to be obtained. In computing the full theory diagrams for the matching it is convenient to separate the diagrams into those involving on-shell propagators, which will be partially reproduced by T -product terms in SCET, and diagrams involving only o -shell propagators. Since the four gluon operators obtain their power suppression entirely from the elds, for diagrams involving only o -shell propagators the residual momenta in eq. (4.51) can be ignored, as they contribute only power suppressed contributions. Diagrams with on-shell propagators are regulated by the residual momenta in eq. (4.51). We begin by considering the expansion of the full theory diagrams that don't involve any on-shell propagators. In this case, all ? momenta can be set to zero, and the result will be purely local. The relevant QCD diagrams expanded to O( 2) arise from the four gluon vertex in the Higgs e ective theory, O( 2) = 4ig2(f eabf ecd + f eadf ecb) 1? 3? 2? 4? + 4ig2(f eacf ebd + f eadf ebc) 1? + 4ig2(f eabf edc + f eacf edb) 1? 4? 2? 2? 3? 3? 4? ; (4.53) B B 0 B 1 A 1 A 0 B + permsC 1 A O( 2) = 2ig2 !2 !3 + !4 hf baef cde( 3? + f bcef ade( 1? +f bdef ace( 1? or sequential emissions with two o -shell propagators (4.55) (4.56) from a splitting o of the three gluon vertex, + permsCC O( 2) = 2ig2 !3 !2 !4 f abef cde 1? 4? 3? 2? + [(2; d) $ (4; b)] + [(3; c) $ (4; b)] ; (4.54) and from multiple emissions o of the two gluon vertex, either using the four gluon vertex with a single o -shell propagator + permsC O( 2) = 2ig2 !2!3 In the last case we have not explicitly listed the permutations, since all possible permutations are required. We now consider the expansion of the full theory diagrams involving on-shell propagators. These will generically involve both local and non-local pieces. The non-local pieces will be directly reproduced by T -products in the e ective theory. The rst class of diagrams involving on-shell propagators are those with all propagators on-shell. Here, at tree level, the dynamics occurs entirely within a single collinear sector. The two relevant QCD diagrams expanded to O( 2) are O( 2) = 0 ; 0 B 1 A O( 2) + permsC = 0 ; (4.57) both of which have vanishing subleading power contributions. plify the results, we will often use the relation Next, we consider diagrams involving both on-shell and o -shell propagators. To simp 2 !3 !2 ; (4.58) which will allow us to write the result in terms of a local term, which is just a rational function of the label momenta, and a non-local term, which explicitly contains the on-shell 0 B 0 B B as well as non-local contributions, O( 2) propagator. These non-local terms will be cancelled by the T -product diagrams in SCET. For a rst class of diagrams, where we have a nearly on-shell splitting in the n-collinear sector, we have both a local term 0 B 1 C A O( 2) = 4ig2f aedf bce (!3 !4) (!3 + !4) 1? 2? 3? 4? ; (4.59) when the splitting is into the particles 3 and 4, as well as a term that has both local and non-local pieces As will be discussed in more detail when we consider the corresponding diagrams in the EFT, the rst permutation is purely local, since there is no corresponding T -product term in the e ective theory, and thus it must be fully reproduced by a hard scattering operator. This particular splitting allows a slight simpli cation in the calculation of the SCET diagrams. For a second class of diagrams, where we have an on-shell splitting emitted from an o -shell leg, we again have a purely local term 0 B B 1 C C A O( 2) = 0 ; = 4ig2f aebf dce 2(!2 + !3) p !4 1?p? 2? 3? 4? (2!3 + !4) 1? 4? 2? 3? p?] : = 2ig2f aebf dce 4!4 (!2 + !3)(p2 + p3)2 !3(!2 !3)(!2 + !3 + !4)2 !2!4(!2 + !3)2 p ? 1?p? 2? 3? 4? 1? 4? 2? 3? Again, we see the same pattern, that the rst permutation gives rise to a purely local term, while the second two permutations give rise to both local and non-local terms. Finally, we have the diagrams involving the three gluon vertex in the Higgs e ective theory. We again have a local contribution and a non-local contribution = 2ig2f adef ebc !2(!3 !4) (!3 + !4)2 1? 2? 3? 4 ; (4.63) 0 B 0 B = 2ig2f aebf dce 1 C A O( 2) + 8 !2!4 1 C A O( 2) 1?p? !2!3 2? 3? 4? 1? 4? 2? 3? p?] : The non-local terms in the above expansions must be reproduced by T -product terms in the e ective theory. First, there are potential contributions from OPB Feynman rule for Bn;?, which is given in appendix B. Such contributions give vanishing overlap for our choice of ? polarizations. There are however T -product contributions aris(2) , with the two gluon ing from the three gluon OPB (2) operator, with an L rule for the OPB (2) vertex was given in eq. (4.33). Since the OPB (2) operator has an explicit P? insertion, it vanishes in the case that either of the particles in the n sector has no perpendicular momentum. This is why our particular choice of momenta for the matching simpli es the structure of the T -products. The two non-vanishing permutations are given by (0) insertion. The three gluon Feynman + (4.64) HJEP07(21)6 (4.65) 1 A non-loc. (4.66) 8ig2f abef ecd (!2 + !3 + !4)2 (!3 + !2)!4 !3 (!2 + !3) 1? 4? 3? 2? p ? 1?p? 2? 3? 4? (p2 + p3)2 which consists both of a local and a non-local term. The non-local terms exactly reproduce the ones obtained in the QCD expansion 0 B B C C A 0 + +permsC = 8ig2p 1?p? 2? 3? 4? (p2 + p3)2 (!3 + !2)!4 f abef ecd (!2 + !3 + !4)2 + [3 $ 4; b $ c] : While it is of course necessary that the EFT reproduces all such non-local terms, this is also a highly non trivial cross check of both the three and four gluon matching. The matching coe cients for the hard scattering operators are given by the remaining local terms. Before presenting the result we brie y comment on the organization of the color structure. All diagrams are proportional to f abef cde, f acef bde or f adef bce, which are related by the Jacobi identity f abef cde = f acef bde f adef bce. A basis in terms of structure constants can easily be related to the trace basis of (3.12) using f acef bde = tr[abdc] + tr[acdb] f adef bce = tr[abcd] + tr[adcb] tr[acbd] tr[acbd] tr[adbc] = e2 tr[adbc] = e1 e3 ; e3 ; (4.67) where ei is the i-th element of the basis in (3.12). We nd it most convenient to write the Wilson coe cient in the (f acef bde; f adef bce) basis. After subtracting the local piece of the SCET T product of (4.65) from the full theory graphs, and manipulating the result to bring it into a compact form, we nd the following operator O4(2g) = 16 sf adef bce(Bna?;!i Bn?;!j )(Bnc?;!k Bn?;!` ) 3+ b d !j3 + !k3 + !`3 + !j !k!` (!j + !k)(!j + !`)(!k + !`) : (4.68) The Wilson coe cient is manifestly RPI-III invariant. When the matrix element of this operator is taken we are forced to sum over permutations which gives the proper Bose symmetric result, as well as inducing terms with other color structures. In terms of the helicity operators of eq. (3.16), we have O4(2g) = 16 sf adef bce 3 + !j3 + !k3 + !`3 + !j!k!` (!j + !k)(!j + !`)(!k + !`) ! h a Bn+;!i Bn+;!j Bn+;!k Bn ;!` + Bna+;!i Bn+;!j Bn ;!k Bn+;!` b c d b c d + Bna ;!i Bn ;!j Bn+;!k Bn ;!` + Bna ;!i Bn ;!j Bn ;!k Bn+;!` b c d b c d i s 3 + !j3 +!k3 +!`3 +!j!k!` (!j +!k)(!j +!`)(!k +!`) h(f adef bce +f acef bde)Bna+;!i Bn+;!j Bn+;!k Bn ;!` b c d (f adef bce + f abef cde)Bna ;!i Bn+;!j Bn ;!k Bn ;!` : b c d i (4.69) We see that all the helicity selection rules are satis ed in the tree level matching, as expected. We have also checked the result using the automatic FeynArts [66] and FeynRules implementation of the HiggsE ectiveTheory [67]. For more complicated calculations at subleading power in SCET it would be interesting to fully automate the computation of Feynman diagrams involving power suppressed SCET operators and Lagrangians. The four gluon operators derived in this section can be used to study O( s2) collinear contributions at O( 2). It would be interesting to understand in more detail the universality of collinear splittings at subleading power, as well as collinear factorization properties. For some recent work in this direction from a di erent perspective, see [68, 69]. The behavior of these Wilson coe cients is also quite interesting. They exhibit a singularity as any pair of collinear particles simultaneously have their energy approach zero. This was also observed in the Wilson coe cients for operators describing the subleading collinear limits of two gluons emitted o of a qq vertex [17]. 5 In this paper we have presented a complete basis of operators at O( 2) in the SCET expansion for color singlet production of a scalar through gluon fusion, as relevant for gg ! H. To derive a minimal basis we used operators of de nite helicities, which allowed us to signi cantly reduce the number of operators in the basis. This simpli cation is due to helicity selection rules which are particularly constraining due to the scalar nature of the produced particle. We also classi ed all possible operators which could contribute to the cross section at O( 2). In performing this classi cation the use of a helicity basis again played an important role, allowing us to see from simple helicity selection rules which operators could contribute. While the total number of subleading power operators is large, the number that contribute at the cross section level is smaller. We compared the structure of the contributions to the case of a quark current, q q, nding interesting similarities, despite a slightly di erent organization in the e ective theory. A signi cant portion of this paper was devoted to a tree level calculation of the Wilson coe cients of the subleading power operators which can contribute to the cross section at O( 2). The Wilson coe cients obtained in this matching will allow for a study of the power corrections at NLO and for the study of the leading logarithmic renormalization group structure at subleading power. An initial investigation of the renormalization group properties of several subleading power operators relevant for the case of e+e ! qq was considered in [15]. A number of directions exist for future study, with the goal of understanding factorization at subleading power. In particular, one would like to combine the hard scattering operators derived in this paper with the subleading SCET Lagrangians to derive a complete factorization theorem at subleading power for a physical event shape observable. Combined with the operators in [17], all necessary ingredients are now available to construct such a subleading factorization for thrust for qq or gg dijets in e+e collisions. This would also allow for a test of the universality of the structure of the subleading factorization. The operators of this paper can also be used to study threshold resummation, where power corrections of O((1 z)0) have received considerable attention [70{81], particularly for the qq channel, but it would be interesting to extend this to the gg case. An interesting application of current relevance of the results presented in this paper is to the calculation of xed order power corrections for NNLO event shape based subtractions. Gaining analytic control over power corrections can signi cantly improve the performance and stability of such subtraction schemes. This has been studied for qq initiated Drell Yan production to NNLO in [16] using a subleading power operator basis in SCET (see also [50] for a direct calculation in QCD). Combined with the results for the operator basis and matching for qq initiated processes given in [17], the operator basis presented in this paper will allow for the systematic study of power corrections for color singlet production and decay. Acknowledgments We thank the Erwin Schrodinger Institute and the organizers of the \Challenges and Concepts for Field Theory and Applications in the Era of LHC Run-2" workshop for hospitality and support while portions of this work were completed. This work was supported in part by the O ce of Nuclear Physics of the U.S. Department of Energy under the Grant No. DESC0011090, by the O ce of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231, and the LDRD Program of LBNL. I.S. was also supported by the Simons Foundation through the Investigator grant 327942. A Generalized basis with P?n; P?n 6= 0 In the main text we presented a complete basis of operators to O( 2) in a frame where the total P? in each collinear sector is restricted to be zero. In this section we extend the basis, giving the additional operators present when the individual collinear sectors have non-vanishing P?. We then perform a tree level matching calculation to those operators which can contribute to the cross section at O( 2). While all these operators are xed by RPI, we choose to nd their coe cients by simply performing the tree level matching with more general kinematics. A.1 Operators We begin by noting that operators involving two collinear gluon elds with a single insertion of the P? operator are eliminated by the helicity selection rules. Operators involving two collinear gluon elds must therefore have two insertions of the P? operator. A basis of helicity operators involving two insertions of the P? operator, where one P? operator acts in each collinear sector, is given by (P?gn)(P?gn) : PBP+ [ :+] = [P? Bna+] [P?+Bnb ] H ; PBP++[+:+] = [P?+Bna+] [P?+Bnb+] H ; PBP +[+: ] = [P?+Bna ] [P? Bnb+] H ; [ : ] = [P? Bna ] [P? Bnb ] H : (A.1) When both P? operators act on the same collinear sector, which we take to be the n-collinear sector, then we have (P?P?gn)gn : BPP++[ +] = [P? P?+Bna+] [Bnb+] H ; BPP++[++] = [P?+P?+Bna ] [Bnb+] H ; Note that we have used up our freedom to integrate by parts by never having the P? operator act on the H eld. The color basis for all these operators before and after the BPS eld rede nition is the same as given in eq. (3.3). We also must consider the generalization of the operators involving three gluon or quark elds to generic ? momentum in the collinear sectors. As discussed in the text surrounding eq. (3.27), in the case that the P? operator is inserted into an operator involving two quark elds and a gluon eld, the helicity structure of the operator is highly constrained. In particular, the quark elds must be in a helicity zero con guration, and also have the same chirality. This implies that all operators must involve only the currents Jn 0 or Jn 0 . Here we have taken without loss of generality that the two quarks are in the n-collinear sector. The basis of O( 2) operators for the case that the P? operator acts on the n sector, is a P +(0)[+] = Bn+ Jn 0 P? H ; H ; H ; H ; O(2)a P O(2)a P P P O(2)a O(2)a H ; H ; H ; H ; which replaces the four operators in eq. (3.27). For the case that the P? operator acts on the n sector the basis is (P?g)n(qq)n : O(2)a P +(0)[+] = [P? Bna+] Jn 0 H ; O(2)a P +(0)[+] = [P? Bna+] Jn 0 H ; O(2)a O(2)a P P (0)[ ] = [P?+Bna ] Jn 0 H ; (0)[ ] = [P?+Bna ] Jn 0 H : The color basis for all these operators (before and after the BPS eld rede nition) is the same as given in eqs. (3.28) and (3.29). The nal case we must consider are the generalized versions of eq. (3.30), which involve the insertion of a single P? operator into an operator involving three collinear gluon elds. In this case a basis of O( 2) operators for the case that the P? operator acts in the n-collinear sector is given by (g)n(gg P?)n : + c PB O(2)abc PB a b [+] = Bn Bn a b +[ ] = Bn Bn O(2)abc + c P? Bn c P? Bn+ H ; H ; H ; where these six operators replace the four in eq. (3.30). In addition we have operators for the case that the P? acts in the n-collinear sector, (P?g)n(gg)n : H ; (A.6) The color basis for all of these operators is the same as given in eq. (3.31). A.2 We now consider the matching to these operators. We begin with the matching to the operators involving two P? insertions into the leading power operator. We use a two gluon nal state and take the kinematics as + p1? + p2? : n n We nd at O( 2) (A.7) (A.8) (A.9) (A.10) O( 2 = 4i abp1? p2? 3? 4? : This is recognized as the tree level matrix element of the operator (2) OPBP a b 4 abg g [P?Bn?;!1 ][P?Bn?;!2 ] ; or in terms of helicity operators, 4[P? Bna+] [P? Bnb+] H ; 4[P?+Bna+] [P?+Bnb+] H ; We see that not all possible helicity combinations appear in the tree level matching. Furthermore, the operators of eq. (A.2) where both P? insertions are in the same collinear sector do not appear at this order. We now consider the matching to the operators of eqs. (A.3) and (A.4). We can simplify the matching by performing it in two steps. First, to extract the Wilson coe cient of the operator involving the action of the P? on the collinear gluon eld we take our kinematics as n n n n n + p? + p1r 2 With this choice, all subleading Lagrangian insertions vanish, for similar reasons as for the gqq matching discussed in the text, as do insertions of the operator of eq. (A.9), so that the result must be reproduced by hard scattering operators. Expanding the QCD result, we nd (A.12) (A.13) (A.14) matching by taking + p1? n + p2? n where, unlike in the text, we have allowed for a generic ? momentum in the n-collinear sector. Note that for this con guration it is still true that subleading T -products vanish,f or similar reasons as for the gqq matching discussed in the text, at least at this order. Only the operator of eq. (A.9) appeared in the matching, however its contribution vanishes for this matching con guration. Expanding the full theory result we nd To extract the operators where the P? acts in the n-collinear sector we simplify the HJEP07(21)6 O( 2) O( 2) = 0 ; 4gf abc !3 !2 !2 !3 ( 1 3)(p1? 2) ( 1 2)(p1? 3) ; = 0 ; (A.16) = 4gf abc [( 1 2)(p1? 3) ( 1 3)(p1? just as was the case when the ? momenta in each sector were restricted to vanish. Finally, we must consider the matching with general ? momenta to the three gluon operators. Again, we can perform the matching in two steps. In the rst step we take the momenta as n n n + p1? + p1r 2 to isolate the action of the operator with an insertion of the P? operator in the n-collinear sector. The QCD amplitudes expanded to this order are 0 B B C A O( 2) O( 2) O( 2) There are no SCET contributions at this order, since for our choice of kinematics there is no perpendicular momentum owing in the n leg. Therefore, the hard scattering operators which appear in the tree level matching are OPB (2) = 4gif abc 1 + !2 !3 Bn?;!2 [Bna?;!1 P?y] Bn?;!3 : b c In the second step of the matching we can take the kinematics as ; + p3? + p3r 2 ; n the n-collinear sector. Expanding the relevant QCD diagrams to O( 2), we nd which allows us to determine the Wilson coe cients of the operators with a P? acting in HJEP07(21)6 0 B B B O( 2) = 0 ; = 4gf abc 1) ( 1 2)(p2;? 3)] + ( 1 2)(p3;? 3) (2 $ 3) ; ( 1 3)(p2;? 2) ( 1 2) (p2;? 3)+( 2 3)(p2;? 1) (2 $ 3) : There are no SCET T -product contributions, so that these must be exactly reproduced by hard scattering operators in the e ective theory. We therefore nd the following operators 4g 1 + (2) OPB1 = (2) (2) OPB2 = 4g 2 + OPB3 = 4g 2 + !3 !2 !3 !2 !3 !2 if abc a Bn?;!1 if abc a h c if abc a Bn?;!1 b P?Bn?;!2 i c Bn?;!3 H ; h c B?n;!2 P? y i b Bn?;!3 H ; Bn?;!1 B?n;!3 P? Bn?;!2 h b i H : (A.20) These can be projected onto de nite helicities following eq. (4.32). B Useful Feynman rules In this appendix we summarize for convenience several useful Feynman rules used in the text, both from the Higgs e ective theory, and from SCET. The Feynman rules in the Higgs e ective theory with 1 C C C A !3 !2 O( 2) Ohard = G G H ; (A.17) (A.18) (A.19) (B.1) are well known, and are given by 4i ab(p1 p2g p1p2) ; 4gf deg(p1g 4gf ged(p3g 4gf egd(p2 g p1 g ) p3g ) p2g ) ; = 4ig2(f adf f aeg + f aef f adg)g g + 4ig2(f adef afg + f adgf afe)g g + 4ig2(f adef agf + f adf f age)g g : Before presenting the subleading power Feynman rules in SCET, we begin by brie y reviewing the Lagrangian, and gauge xing for the collinear gluons. The gauge covariant derivatives that we will use to write the Lagrangian are de ned by n 2 iDns = iDn + gn Aus ; gn Aus ; 2 2 P + P? ; 2 (B.2) (B.4) (B.5) (B.6) (B.7) (B.8) (B.9) Ln (0) = n in Dns + iD= n?Wn L(n0g) = 1 2g2 tr ([iDns; iDns])2 1 Pn 1 WnyiD= n? = n 2 n ; + 2tr cn[i@ns; [iDns; cn]] ; and the ultrasoft Lagrangian, L(u0s), is simply the QCD Lagrangian. We have used a covariant gauge with gauge xing parameter for the collinear gluons. and their gauge invariant versions are given by The leading power SCET Lagrangian can be written as and iDn = WnyiDnWn ; iDn? = WnyiDn?Wn = Pn? + gBn? ; iDns = WnyiDnsWn : L (0) = Ln (0) + L(n0g) + L(u0s) ; The O( ) Lagrangian can be written L(1) = L(1n) + L(A1n) + L(1n)qus ; L(1n) = n iD= us? L(A1n) = 2 1 iD= n? + iD= n? P 1 P iD= us? = n 2 n ; 1 g2 Tr iDns; iDn? iDns ; iDu?s + 2 Tr [iDus?; An? ][i@ns; An ] L(1n)qus = ngB=n?qus + h.c.. Finally, the O( 2) Lagrangian can be written as [26, 30, 31] L(2) = L(2n) + L(A2n) + L(2n)qus ; igB=n?qus + h.c. ; = n 2 n ; 1 1 1 where 1 P L nqus = n 2 [Wnyin DWn]qus + n n=2 iD= n? (2) n= 1 P L(n2) = n iD= us? iD= us? iD= n? (P)2 in Dus iD= n? L(n2g) = L(g2f) = 1 1 1 g2 Tr [iDns; iDu?s ][iDns ; iDu?s ] + g2 Tr [iDus?; iDus?][iDn? ; iDn? ] + g2 Tr ([iDns ; in Dns][iDns ; in Dus]) + g2 Tr [iDus?; iDn? ][iDn? ; iDus?] ; Tr [iDus?; An? ][iDus?; An? ] + Tr ([in Dus; n An][i@ns; An ]) + 2Tr cn[iDus?; [WniDu?s Wny; cn]] + Tr (cn[in Dus; [in Dns; cn]]) + Tr cn[P; [Wnin DusWny; cn]] : Using these Lagrangians, one can derive the required Feynman rules for the calculations described in the text. The O( ) Feynman rule for the emission of a ultrasoft gluon from a collinear gluon in a general covariant gauge, speci ed by a gauge xing parameter , is given by (B.10) (B.11) (B.12) (B.13) gf abc g 2g p n? + g p n 1 p 1 + 1 n ps 2 n p2nn n pn p2nn n pn p n pn 1 1 2 n pn + n pn + n n n ps (B.14) and the O( ) propagator correction to the gluon propagator is given by 4i abg q ? qr? + 2i 1 1 ab qr?q + q qr? : (B.15) For the matching calculation for the operators involving an ultrasoft derivative in section 4.3.1, we also needed the O( 2) corrections to the propagator, which is given by (q n n qr + q n n qr) + where the dots indicate the other tensor components in the light cone basis, which are not relevant for the current discussion. For simplicity, the matching was performed using a ? polarized gluon. In the n-collinear sector, the leading power hard scattering operator produces only n, and ? polarized gluons. Therefore, only the ? ? and n ? components of the propagator are needed. In the matching, the ? ? term vanishes since it proportional to the residual ? momentum, which is set to zero, and the n ? term vanishes for a ? polarized gluon, due to the gluons equation of motion, q? ? = 0. At O( 2), the individual propagator and emission factors are su ciently complicated that it is also convenient to give the complete result for the matrix element = h0jT fBn?(0); L(A2n) gj n; pn; s; psi =1 = = if abc n n pn n ps 2 s ps g g ? ? g g ? ? ; (B.17) where we have restricted to = 1 for simplicity. Since we have also matched to operators involving collinear quarks, we also summarize the subleading power Feynman rules involving collinear quark. The Feynman rules for the correction to a collinear quark propagator are given by = i = i 2 n= 2p? pr? ; n p n= pr2? ; 2 n p (B.18) (B.19) n p ?p=r? + p=0r? n p0 p=r?p=? n n qn p p=0 p=0 n qn p0 ? r? n p0r?p= = n qn p0 ? n + p=0 p= n qn p0 ? r? n n p0r2? n p0 p= n pr ? ? (n p)2 p = ? ? (n p0)2 n pr n p=0 p= n pr n q(n p)2 + ? ? n p=0 p= n pr ! n= ? ? n q(n p0)2 2 : We can see that each term in the power suppressed collinear Lagrangian insertions are proportional to either pr?, or n pr. At tree level, and in the absence of ultrasoft particles, one can use RPI to set all these terms to zero. This was used extensively to simplify our matching calculations. For convenience we also give the expansion of the Wilson lines and collinear gluon eld to two emissions. The collinear Wilson lines are de ned by and the Feynman rules for the emission of a collinear gluon are given by = igT a n + n p p = 0 ? n p0 p= p=0 n pn p0 ? n ! n= 2 (B.21) 2 ; (B.22) (B.23) (B.24) (B.25) (B.26) igT a igT a n An(x) g P T bT a T bT a Wn = X exp perms Expanded to two gluons, both with incoming momentum, we nd = g A?akT a k n AankT a n k Bn? 1 h g WnyiDn?Wni : + g2(T aT b T aT b T bT a) n Aank1A?bk2 n k1 T bT a T aT b T aT b n k1(n k1 + n k2) n k2(n k1 + n k2) n k1(n k1 + n k2) n k2(n k1 + n k2) n Aank1n Abnk2 ; n Aank1n Abnk2 : 2! 2! Wn = 1 Wny = 1 + n k n k gT an Aank + g2 gT an Aank + g2 The collinear gluon eld is de ned as + g2(k1? +k2?) n k1(n k1 + n k2) n k2(n k1 + n k2) n Aank1n Abnk2 : 2! In both cases, at least one of the gluons in the two gluon expansion is not transversely polarized. Such terms can therefore be eliminated in matching calculations by choosing particular polarizations, as was done in the text. Open Access. 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Ian Moult, Iain W. Stewart, Gherardo Vita. A subleading operator basis and matching for gg → H, Journal of High Energy Physics, 2017, 67, DOI: 10.1007/JHEP07(2017)067