Nucl.Phys.B v.791

Nuclear Physics B 791 (2008) 119

Higgs boson production at the LHC:Transverse-momentum resummation and rapidity

dependenceGiuseppe Bozzi a, Stefano Catani b, Daniel de Florian c,

Massimiliano Grazzini b,a Institut fr Theoretische Physik, Universitt Karlsruhe, PO Box 6980, D-76128 Karlsruhe, Germany

b INFN, Sezione di Firenze and Dipartimento di Fisica, Universit di Firenze, I-50019 Sesto Fiorentino, Florence, Italyc Departamento de Fsica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,

(1428) Pabelln 1 Ciudad Universitaria, Capital Federal, ArgentinaReceived 30 May 2007; accepted 24 September 2007

Available online 10 October 2007

This paper is dedicated to the memory of Jiro Kodaira, great friend and distinguished colleague

Abstract

We consider Higgs boson production by gluon fusion in hadron collisions. We study the doubly-differential transverse-momentum (qT ) and rapidity (y) distribution of the Higgs boson in perturbativeQCD. In the region of small qT (qT MH , MH being the mass of the Higgs boson), we include the ef-fect of logarithmically-enhanced contributions due to multiparton radiation to all perturbative orders. Weuse the impact parameter and double Mellin moments to implement and factorize the multiparton kine-matics constraint of transverse- and longitudinal-momentum conservation. The logarithmic terms are thensystematically resummed in exponential form. At small qT , we perform the all-order resummation of largelogarithms up to next-to-next-to-leading logarithmic accuracy, while at large qT (qT MH ), we applya matching procedure that recovers the fixed-order perturbation theory up to next-to-leading order. Wepresent quantitative results for the differential cross section in qT and y at the LHC, and we comment onthe comparison with the qT cross section integrated over y. 2007 Elsevier B.V. All rights reserved.

PACS: 12.38.Cy; 14.80.Bn

Keywords: Higgs; QCD; Hadronic colliders; LHC

* Corresponding author.E-mail address: [email protected] (M. Grazzini).0550-3213/$ see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2007.09.034

2 G. Bozzi et al. / Nuclear Physics B 791 (2008) 119

1. Introduction

The search for the Higgs boson [1] and the study of its properties (mass, couplings, decaywidths) at hadron colliders require a detailed understanding of its production mechanisms. Thisdemands reliable computations of related quantities, such as production cross sections and theassociated distributions in rapidity and transverse momentum. In this paper we consider the pro-duction of the Standard Model (SM) Higgs boson by the gluon fusion mechanism.

The gluon fusion process gg H , through a heavy-quark (mainly, top-quark) loop, is themain production mechanism of the SM Higgs boson H at hadron colliders. When combinedwith the decay channels H and H ZZ, this production mechanism is one of the mostimportant for Higgs boson searches and studies over the entire range, 100 GeVMH 1 TeV,of Higgs boson mass MH to be investigated at the LHC [2]. In the mass range 140 GeV MH 180 GeV, the gluon fusion process, followed by the decay H WW +, canbe exploited as main discovery channel at the LHC and also at the Tevatron [3], provided thebackground from t t production is suppressed by applying a veto cut on the transverse momentaof the jets accompanying the final-state leptons.

The dynamics of the gluon fusion mechanism is controlled by strong interactions. Detailedstudies of the effect of QCD radiative corrections are thus necessary to obtain accurate theoreticalpredictions.

In QCD perturbation theory, the leading order (LO) contribution to the total cross sectionfor Higgs boson production by gluon fusion is proportional to 2S, S being the QCD coupling.The QCD radiative corrections to the total cross section are known at the next-to-leading order(NLO) [47] and at the next-to-next-to-leading order (NNLO) [812]. The Higgs boson rapiditydistribution is also known at the NLO [13] and at the NNLO [14,15]. The effects of a jet vetohave been studied up to the NNLO [11,14,15]. We recall that all the results at NNLO have beenobtained by using the large-Mt approximation, Mt being the mass of the top quark. This approx-imation is justified by the fact that the bulk of the QCD radiative corrections to the total crosssection is due to virtual and soft-gluon contributions [911,16,17]. The soft-gluon dominancealso implies that higher-order perturbative contributions can reliably be estimated by applyingresummation methods [9] of threshold logarithms, a type of logarithmically-enhanced terms dueto multiple soft-gluon emission. In Ref. [17], the NNLO calculation of the total cross sectionis supplemented with threshold resummation at the next-to-next-to-leading logarithmic (NNLL)level; the residual perturbative uncertainty at the LHC is estimated to be at the level of betterthan 10%. The NNLL + NNLO results [17] are nicely confirmed by the more recent com-putation [1820] of the soft-gluon terms at N3LO; the quantitative effect [18] of the additional(i.e., beyond the NNLL order) single-logarithmic term at N3LO is consistent with the estimateduncertainty at NNLL + NNLO. The effect of threshold logarithms on the rapidity distribution ofthe Higgs boson has been considered in Ref. [21].

The gluon fusion mechanism at O(2S) produces a Higgs boson with a vanishing transversemomentum qT . A large (or, however, non-vanishing) value of qT can be obtained only startingfrom O(3S), when the Higgs boson is accompanied by at least one recoiling parton in the finalstate. This mismatch by a power of S is a preliminary indication of the fact that the small-qTand large-qT regions are controlled by different dynamics regimes.

The large-qT region is identified by the condition qT MH . In this region, the perturba-tive series is controlled by a small expansion parameter, S(M2H ), and calculations based on the

3truncation of the series at a fixed order in S are theoretically justified. The LO, i.e., O(S), cal-culation is reported in Ref. [22]. The results of Ref. [22] and the higher-order studies of Refs. [23,

G. Bozzi et al. / Nuclear Physics B 791 (2008) 119 3

24] show that the large-Mt approximation is sufficiently accurate also in the case of the qT dis-tribution when qT MH , provided qT Mt . Using the large-Mt approximation, the NLO QCDcomputation of the qT distribution of the SM Higgs boson is presented in Refs. [14,15,2527].QCD corrections beyond the NLO are evaluated in Ref. [28], by implementing threshold resum-mation at the next-to-leading logarithmic (NLL) level. The results of the numerical programs ofRefs. [14,15] can also be safely (i.e., without encountering infrared divergences) extended fromlarge values of qT to qT = 0: in the small-qT region these programs evaluate the qT distributionup to NNLO.

In the small-qT region (qT MH ), where the bulk of events is produced, the convergenceof the fixed-order expansion is definitely spoiled, since the coefficients of the perturbative seriesin S(M2H ) are enhanced by powers of large logarithmic terms, ln

m(M2H/q2T ). The logarithmic

terms are produced by multiple emission of soft and collinear partons (i.e., partons with lowtransverse momentum). To obtain reliable perturbative predictions, these terms have to be re-summed to all orders in S. The method to systematically perform all-order resummation ofclasses of logarithmically-enhanced terms at small qT is known [2936]. In the case of the SMHiggs boson, resummation has been explicitly worked out at leading logarithmic (LL), NLL [35,37] and NNLL [38] level.

The fixed-order and resummed approaches at small and large values of qT can then be matchedat intermediate values of qT , to obtain QCD predictions for the entire range of transverse mo-menta. Phenomenological studies of the SM Higgs boson qT distribution at the LHC have beenperformed in Refs. [3945], by combining resummed and fixed-order perturbation theory at dif-ferent levels of theoretical accuracy. Other recent studies of various kinematical distributions ofthe SM Higgs boson at the LHC are presented in Refs. [4650].

In Refs. [41,44] we studied the Higgs boson qT distribution integrated over the rapidity. Inthe small-qT region, the logarithmic terms were systematically resummed in exponential formby working in impact-parameter and Mellin-moment space. A constraint of perturbative unitar-ity was imposed on the resummed terms, to the purpose of reducing the effect of unjustifiedhigher-order contributions at large values of qT and, especially, at intermediate values of qT .This constraint thus decreases the uncertainty in the matching procedure of the resummed andfixed-order contributions. Our best theoretical predictions were obtained by matching NNLL re-summation at small qT and NLO perturbation theory at large qT . NNLL resummation includesthe complete NNLO result at small qT , and the unitarity constraint assures that the total crosssection at NNLO is recovered upon integration over qT of the transverse-momentum spectrum.Considering SM Higgs boson production at the LHC, we concluded [44] that the residual pertur-bative QCD uncertainty of the NNLL + NLO result is uniformly of about 10% from small tointermediate values of transverse momenta.

In this paper we extend our study to include the dependence on the rapidity of the Higgsboson. Using the impact parameter and double Mellin moments, we can perform the extensionby maintaining all the main features of the resummation formalism of Refs. [36,44]. We are thenable to present results up to NNLL + NLO accuracy for the doubly-differential cross sectionin qT and rapidity at the LHC.

The paper is organized as follows. In Section 2 we recall the main aspects of the resummationformalism, and we illustrate the steps that are necessary to include the dependence on the rapidityin the qT resummed formulae. In Section 3 we apply the formalism to the production of the SM

Higgs boson at the LHC, and we perform quantitative studies on the qT and rapidity dependenceof the doubly-differential cross section. Some concluding remarks are presented in Section 4.


Additional technical details on the double Mellin moments of the resummation formulae aregiven in Appendix A.

2. Rapidity dependence in qT resummation

We consider the inclusive hard-scattering process

(1)h1(p1) + h2(p2) H(y,qT ,MH) + X,where the collision of the two hadrons h1 and h2 with momenta p1 and p2 produces the Higgsboson H , accompanied by an arbitrary and undetected final state X. The centre-of-mass energyof the colliding hadrons is denoted by

s. The rapidity, y, of the Higgs boson is defined in the

centre-of-mass frame of the colliding hadrons, and the forward direction (y > 0) is identified bythe direction of the momentum p1.

According to the QCD factorization theorem, the doubly-differential cross section for thisprocess is

d

dy dq2T(y, qT ,MH , s) =

a1,a2

10

dx1

10

dx2 fa1/h1(x1,

2F

)fa2/h2

(x2,

2F

)

(2) da1a2dy dq2T

(y, qT ,MH , s;S

(2R

),2R,

2F

),

where fa/h(x,2F ) (a = qf , qf , g) are the parton densities of the colliding hadrons at the fac-torization scale F , dab are the partonic cross sections, and R is the renormalization scale.Throughout the paper we use parton densities as defined in the MS factorization scheme, andS(q2) is the QCD running coupling in the MS renormalization scheme. The rapidity, y, and thecentre-of-mass energy, s, of the partonic cross section (subprocess) are related to the correspond-ing hadronic variables y and s:

(3)y = y 12

lnx1x2

, s = x1x2s,

with the kinematical boundary |y| < lns/M2 (|y| < lns/M2 ) and s > M2 (s > M2).The partonic cross section dab is computable in QCD perturbation theory. Its power series

expansion in S contains the logarithmically-enhanced terms, (nS/q2T ) ln

m(M2H/q2T ), that we

want to resum. To this purpose, we use the general (process-independent) strategy and the for-malism described in detail in Ref. [44]. The only difference with respect to Ref. [44] is that theresummation is now performed at fixed values of the rapidity y, rather than after integration overthe rapidity phase space. In the following we briefly recall the main steps of the resummationformalism, and we point out explicitly the differences with respect to Ref. [44].

We first rewrite (see Section 2.1 in Ref. [44]) the partonic cross section as the sum of twoterms,

(4)da1a2dy dq2T

= d(res.)a1a2

dy dq2T+ d

(fin.)a1a2

dy dq2T.

The logarithmically-enhanced contributions are embodied in the resummed component d (res.)a1a2 .(fin.)The finite component da1a2 is free of such contributions, and it can be computed by trunca-

tion of the perturbative series at a given fixed order (LO, NLO and so forth). In practice, after


having evaluated da1a2 and its resummed component at a given perturbative order, the finitecomponent d (fin.)a1a2 is obtained by the matching procedure described in Sections 2.1 and 2.4 ofRef. [44].

The resummation procedure of the logarithmic terms has to be carried out [3034] in theimpact-parameter space, to correctly take into account the kinematics constraint of transverse-momentum conservation. The resummed component of the partonic cross section is then obtainedby performing the inverse Fourier (Bessel) transformation with respect to the impact parameter b.We write1

(5)d(res.)a1a2

dy dq2T(y, qT ,MH , s;S) = M

2H

s

0

dbb

2J0(bqT )Wa1a2(y, b,MH , s;S),

where J0(x) is the 0th-order Bessel function, and the factor W embodies the all-order depen-dence on the large logarithms ln(MHb)2 at large b, which correspond to the qT -space termsln(M2H/q

2T ) (the limit qT MH corresponds to MHb 1, since b is the variable conjugate

to qT ).In the case of the qT cross section integrated over the rapidity, the resummation of the large

logarithms is better expressed [36,44] by defining the N -moments WN of W with respect toz = M2H/s at fixed MH . In the present case, where the rapidity is fixed, it is convenient (see,e.g., Refs. [51,52]) to consider double (N1,N2)-moments with respect to the two variablesz1 = e+yMH/

s and z2 = eyMH/

s at fixed MH (note that 0 < zi < 1). We thus introduce

W(N1,N2) as follows:

(6)W(N1,N2)a1a2 (b,MH ;S) =1

0

dz1 zN111

10

dz2 zN212 Wa1a2(y, b,MH , s;S).

More generally, any function h(y; z) of the variables y (|y| < lnz ) and z (0 < z < 1) canbe considered as a function of the two variables z1 = e+yz and z2 = eyz. Thus, throughoutthe paper, the (N1,N2)-moments h(N1,N2) of the function h(y; z) are defined as

(7)h(N1,N2) 1

0

dz1 zN111

10

dz2 zN212 h(y; z), where y =

12

lnz1z2

, z = z1z2.

Note that the double Mellin moments can also be obtained (see, e.g., Ref. [53]) by introducinga Fourier transformation with respect to y (with conjugate variable = i(N2 N1)) and thenperforming a Mellin transformation with respect to z (with conjugate variable N = (N1 +N2)/2):

(8)h(N1,N2) =1

0

dz zN1+

dy eiyh(y; z), where N1 = N + i/2, N2 = N i/2.

The convolution structure of the QCD factorization formula (2) is readily diagonalized byconsidering (N1,N2)-moments:

(9)d (N1,N2) =a1,a2

fa1/h1,N1+1fa2/h2,N2+1 d (N1,N2)a1a2 ,1 In the following equations, the functional dependence on the scales R and F is understood.


where fa/h,N = 1

0 dx xN1fa/h(x) are the customary N -moments of the parton distribu-

tions.The use of Mellin moments also simplifies the resummation structure of the logarithmic terms

in d (res.)(N1,N2)a1a2 . The perturbative factorW(N1,N2)a1a2 can indeed be organized in exponential formas follows:

(10)W(N1,N2)(b,MH ;S) =H(N1,N2)(MH ,S) exp{G(N1,N2)(S, L)},

where

(11)L = ln(M2Hb

2

b20+ 1

),

b0 = 2eE (E = 0.5772 . . . is the Euler number) and, to simplify the notation, the dependenceon the flavour indices has been understood.

The structure of Eq. (10) is in close analogy to the cases of soft-gluon resummed calculationsfor hadronic event shapes in hard-scattering processes [54] and for threshold contributions tohadronic cross sections [51,55,56]. The functionH(N1,N2) (which is process dependent) does notdepend on the impact parameter b and, therefore, its evaluation does not require resummation oflarge logarithmic terms. It can be expanded in powers of S as

(12)H(N1,N2)(MH ,S) = 0(S,MH )[

1 + SH(N1,N2)(1) +

(S

)2H(N1,N2)(2) +

],

where 0(S,MH ) is the lowest-order partonic cross section for Higgs boson production. Theform factor exp{G} is process independent2; it includes the complete dependence on b and, inparticular, it contains all the terms that order-by-order in S are logarithmically divergent whenb . The functional dependence on b is expressed through the large logarithmic terms nSLmwith 1m 2n. More importantly, all the logarithmic contributions to G with n + 2m 2nare vanishing. Thus, the exponent G can systematically be expanded in powers of S, at fixedvalue of = SL, as follows:

(13)G(N1,N2)(S, L) = Lg(1)(SL) + g(2)(N1,N2)(SL) + S

g(3)(N1,N2)(SL) + .

The term Lg(1) collects the leading logarithmic (LL) contributions nSLn+1; the function g(2)resums the next-to-leading logarithmic (NLL) contributions nSLn; g(3) controls the next-to-next-to-leading logarithmic (NNLL) terms nSLn1, and so forth.

Note that we use the logarithmic variable L (see Eq. (11)) to parametrize and organize theresummation of the large logarithms ln(MHb)2. We recall the main motivations [44] for thischoice. In the resummation region MHb 1, we have L ln(MHb)2 and the use of the vari-able L is fully legitimate to arbitrary logarithmic accuracy. When MHb 1, we have L 0(whereas3 ln(MHb)2 !) and exp{G(S, L)} 1. Therefore, the use of L reduces the effect

2 More precisely, it depends only on the flavour of the colliding partons (see Appendix A).3 As shown in Appendix B of Ref. [44] (see Eqs. (131) and (132) therein), after inverse Fourier transformation to

qT space, the b-dependent functions lnn(MHb)2 and Ln lead to quite different behaviours at large qT . When qT MH ,2 n1 n 2the behaviour (1/qT) ln (qT /MH ) (which is not integrable when qT ) produced by ln (MH b) is damped (and

made integrable) by the extra factor qT /MH exp(b0qT /MH ) produced in the case of Ln.


produced by the resummed contributions in the small-b region (i.e., at large and intermediate val-ues of qT ), where the large-b resummation approach is not justified. In particular, setting b = 0(which corresponds to integrate over the entire qT range) we have exp{G(S, L)} = 1: this prop-erty can be interpreted [44] as a constraint of perturbative unitarity on the total cross section; thedynamics of the all-order recoil effects, which are resummed in the form factor exp{G(S, L)},produces a smearing of the fixed-order qT distribution of the Higgs boson without affecting itstotal production rate.

The resummation formulae (10), (12) and (13) can be worked out at any given (and arbitrary)logarithmic accuracy since the functions H and G can explicitly be expressed (see Ref. [44]) interms of few perturbatively-computable coefficients denoted by A(n), B(n), H(n), C(n)N ,

(n)N . The

key role of these coefficients to fully determine the structure of transverse-momentum resum-mation was first formalized by Collins, Soper and Sterman [32,34,36]. The present status of thecalculation of these coefficients for Higgs boson production is recalled in Section 3.

In the case of the qT cross section integrated over the rapidity, Eq. (10) is still valid,provided the double (N1,N2)-moments are replaced by the corresponding single N -momentsWN,HN,GN (see Section 2.2 in Ref. [44]). The relation between double and single momentscan easily be understood by inspection of Eqs. (6)(8). We see that setting = 0 in Eq. (8) isexactly equivalent to integrate the cross section over the rapidity. Therefore, the functions WN ,HN , GN in Ref. [44] are obtained by simply setting N1 = N2 = N in the corresponding functionsW(N1,N2),H(N1,N2),G(N1,N2) of Eq. (10).

Moreover, from the results presented in Ref. [44], we can straightforwardly obtain the func-tions H(N1,N2) and G(N1,N2) from the functions HN and GN . Roughly speaking, we simply have

(14)G(N1,N2) = 12(GN1 + GN2), H(N1,N2) = [HN1HN2 ]1/2.

More precisely, these equalities are valid in the simplified case where there is a single speciesof partons (e.g., only gluons). In the following we comment on the physical picture that leads toEq. (14). The generalization to considering more species of partons does not require any furtherconceptual steps: it just involves algebraic complications related to the treatment of the flavourindices. The multiflavour case is briefly illustrated in Appendix A.

In the small-qT (large-b) region that we are considering, the kinematics of the Higgs bo-son is fully determined by the radiation of soft and collinear partons from the colliding par-tons (hadrons) in the initial state. The radiation of soft partons cannot affect the rapidity ofthe Higgs bosons. On the contrary, the radiation of partons that are collinear to p1 (p2), i.e.,in the forward (backward) region, decreases (increases) the rapidity of the Higgs boson asa consequence of longitudinal-momentum conservation (see Eq. (3)). Since the emissions ofcollinear partons from p1 and p2 are dynamically uncorrelated (factorized from each other),correlations arise only from kinematics. The use of the (N1,N2)-moments exactly factorizes(see Eqs. (2) and (9)) the kinematical constraint of longitudinal-momentum conservation. Itfollows that the (N1,N2)-dependence of W(N1,N2) is given by the product of two functions(say, W(N1,N2) =M(N1)1 M(N2)2 ) that depends only on N1 or N2, respectively. If all the par-tons have the same flavour, the two functions should be equal, and Eq. (14) directly follows from[W(N1,N2)]N1=N2=N =WN .

The formalism illustrated in this section defines a systematic order-by-order (in extended

sense) expansion [44] of Eq. (4): it can be used to obtain predictions with uniform perturbativeaccuracy from the small-qT region to the large-qT region. The various orders of this expansion


are denoted4 as LL, NLL + LO, NNLL + NLO, etc., where the first label (LL, NLL, NNLL,. . .) refers to the logarithmic accuracy at small qT and the second label (LO, NLO, . . .) refers tothe customary perturbative order5 at large qT . To be precise, the NLL + LO term of Eq. (4) isobtained by including the functions g(1), g(2) and the coefficient H(1) (see Eqs. (13) and (12))in the resummed component, and by expanding the finite (i.e., large-qT ) component up to its LOterm. At NNLL + NLO accuracy, the resummed component includes also the function g(3)N andthe coefficientH(2) (see Eqs. (13) and (12)), while the finite component is expanded up to NLO.It is worthwhile noticing that the NNLL + NLO (NLL + LO) result includes the full NNLO(NLO) perturbative contribution in the small-qT region.

We recall [44] that, due to our actual definition of the logarithmic parameter L in Eq. (10)and to our matching procedure with the perturbative expansion at large qT , the integral overqT of the qT cross section exactly reproduces the customary fixed-order calculation of the totalcross section. This feature is not affected by keeping the rapidity fixed. Therefore, the NNLO(NLO) result for total cross section at fixed y is exactly recovered upon integration over qT ofthe NNLL + NLO (NLL + LO) qT spectrum at fixed y.

Within our formalism, resummation is directly implemented, at fixed MH , in the space ofthe conjugate variables N1, N2 and b. To obtain the cross section in Eq. (2), as function of thekinematical variables s, y and qT , we have to perform inverse integral transformations. Theseintegrals are carried out numerically. We recall [44] that the resummed form factor (i.e., each ofthe functions g(k)(SL) in Eq. (13)) is singular at the value of b where S(2R)L = /0 (0 isthe first-order coefficient of the QCD function). This singularity has its origin from the presenceof the Landau pole in the running of the QCD coupling S(q2) at low scales. When performingthe inverse Fourier (Bessel) transformation with respect to the impact parameter b (see Eq. (5)),we deal with this singularity by using a minimal prescription [56,57]: the singularity is avoidedby deforming the integration contour in the complex b space (see Ref. [57]). We note that theposition of the singularity is completely independent of the values of N1 and N2. Thus, theinversion of the Mellin moments is performed in the customary way (in Mellin space there are nosingularities for sufficiently-large values of ReN1 and ReN2). In this respect, going from singleN -moments (as in Ref. [44]) to double (N1,N2)-moments (as in the present case, where therapidity is kept fixed) is completely straightforward, with no additional (practical or conceptual)complications.

3. Higgs boson production at the LHC

In this section we apply the resummation formalism of Section 2 to the production of theStandard Model Higgs boson at the LHC. We closely follow our previous study of the singledifferential (with respect to qT ) cross section, with the same choice of parameters as stated inSection 3 of Ref. [44]. Therefore, the integration over y of the double differential (with respectto y and qT ) cross sections presented in this section returns the qT cross sections of Ref. [44].As a cross-check of the actual implementation of the calculation, we have verified that after inte-gration over the rapidity the numerical results in Ref. [44] are reobtained within a high accuracy.

4 In the literature on qT resummation, other authors sometime use the same labels (NLL, NLO and so forth) with ameaning that is different from ours.

5 We recall that the LO term at small qT (i.e., including the region where qT = 0) is proportional to 2S, whereas the

LO term at large qT is proportional to 3S. This mismatch of one power of S (and the ensuing mismatch of notation)persists at each higher order (NLO, NNLO, . . . ).


As in Refs. [17,44], we use an improved version [16] of the large-Mt approximation. Thecross section is first computed by using the large-Mt approximation. Then, it is rescaled by a Bornlevel factor, such as to include the exact lowest-order dependence on the masses, Mt and Mb ,of the top and bottom6 quarks, which circulates in the heavy-quark loop that couples to theHiggs boson. We use the values Mt = 175 GeV and Mb = 4.75 GeV. As discussed in Ref. [17]and recalled in Section 1, this version of the large-Mt approximation is expected to produce anuncertainty that is smaller than the uncertainties from yet uncalculated perturbative terms fromhigher orders.

For the sake of brevity, we present quantitative results only at NNLL + NLO accuracy, whichis the highest accuracy that can be achieved by using the present knowledge of exact perturbativeQCD contributions (resummation coefficients and fixed-order calculations [2527]). We use theMRST2004 set [58] of parton distribution functions at NNLO. The use of NNLO parton densitiesconsistently matches the NNLL (NNLO) accuracy of our partonic cross section in the region ofsmall and intermediate values of qT .

Resummation up to the NLL level is under control from the knowledge of the perturbativecoefficients A(1), B(1), A(2) [35] and H(1) [37]. To reach the NNLL + NLO accuracy, the formfactor function G(N1,N2) in Eq. (13) must include the contribution from g(3)(N1,N2) (which iscontrolled by the coefficients B(2) [38] and A(3) [59]), and the coefficient function H(N1,N2) inEq. (12) has to be evaluated up to its second-order term H(2)(N1,N2). In Ref. [44] we exploitedthe unitarity constraint G(S, L)|b=0 = 0 to numerically derive an approximated form of thecoefficientH(2) from the NNLO calculation [12] of the total cross section. The recent calculationof Ref. [15], which is based on the complete evaluation ofH(2)(N1,N2) in analytic form, allows usto gauge the quality of the approximated form. We find that the use of theH(2) of Ref. [44] leadsto differences of about 1% with respect to the exact computation of the rapidity cross section atNNLO.

All the numerical results in this section are obtained by fixing the renormalization and fac-torization scales at the value R = F = MH . The resummation scale Q (the auxiliary scaleintroduced in Ref. [44] to gauge the effect of yet uncalculated logarithmic terms at higher or-ders) is also fixed at the value Q = MH . The mass of the Higgs boson is set at the valueMH = 125 GeV.

We start our presentation of the predictions for Higgs boson production at the LHC by con-sidering the qT dependence of the cross section at fixed values of the rapidity. In Fig. 1, we sety = 0 and we compare the customary (when qT > 0) NLO calculation (dashed line) with theresummed NNLL + NLO calculation (solid line).

As expected, the NLO result diverges to as qT 0 and, at small values of qT , it has anunphysical peak that is produced by the numerical compensation of negative leading logarithmicand positive subleading logarithmic contributions. The presence of this peak is not accidental. Atlarge qT , the perturbative expansion at any fixed order has no pathological behaviour: it leads to apositive cross section, whose value decreases as qT increases. When qT 0, instead, any fixed-order calculation diverges alternatively to depending on the perturbative order. Therefore,

6 We note that the Born level cross section is not insensitive to the contribution of the bottom quark. Adding thebottom-quark loop to the top-quark loop in the scattering amplitude produces a non-negligible interference effect inthe squared amplitude. The relative effect of the bottom quark decreases the Born level cross section by about 11% if

MH = 125 GeV, and by about 3% if MH = 300 GeV. If MH 500 GeV, the relative effect of the bottom quark isalways smaller than 1%.


Fig. 1. The qT spectrum at the LHC with MH = 125 GeV and y = 0: results at NNLL + NLO (solid line) and NLO(dashed line) accuracy. The inset plot shows the ratio K (see Eq. (15)) of the corresponding qT cross sections, fixingy = 0 (solid line) and integrating them over the full rapidity range (dashed line).

to go smoothly from the large-qT behaviour to the small-qT limit, the NLO (or N3LO, and soforth) calculation of the cross section has to show at least one peak in the intermediate-qT region.

We recall once more that the label NLO in Fig. 1 refers to (and originates from) the pertur-bative expansion at large qT . To avoid possible misunderstandings (coming from such a label)when interpreting the dashed (NLO) curve in the small-qT region, we point out that, the onlydifference produced in Fig. 1 by the NNLO calculation at small qT (this calculation can be car-ried out, for example, by using the NNLO codes of Refs. [14,15]) is a spike around the pointqT = 0. More precisely, as long as qT = 0, the dashed curve is exactly the result of the NNLOcalculation of the qT cross section at small qT . The only difference introduced in the plot by thisNNLO calculation would occur in the first bin (with arbitrarily small size) that includes the pointqT = 0. The NNLO value of the qT cross section in this first bin is positive and fixed by the valueof the NNLO total cross section.7 Of course, owing to the increasingly negative behaviour of theqT distribution when qT 0, the NNLO value of the qT cross section in the first bin increasesby decreasing the size of that bin.

The resummed NNLL + NLO result in Fig. 1 is physically well behaved at small qT (itvanishes as qT 0 and has a kinematical peak at qT 12 GeV), and it converges to the expectedNLO result only when qT is definitely large (qT MH ).7 By definition, the integral over qT of d2/(dqT dy) at NNLO is equal to d/dy at NNLO.


Fig. 2. The qT spectrum at the LHC with MH = 125 GeV and y = 2: results at NNLL + NLO (solid line) and NLO(dashed line) accuracy. The inset plot shows the ratio K (see Eq. (15)) of the corresponding qT cross sections, fixingy = 2 (solid line) and integrating them over the full rapidity range (dashed line).

To quantify more clearly the effect of the resummation on the NLO result, the value at y = 0of the qT dependent K-factor,

(15)K(qT , y) = dNNLL+NLO/(dqT dy)dNLO/(dqT dy)

,

is shown in the inset plot of Fig. 1. The dashed line shows the analogous K-factor as computedfrom the ratio of the rapidity integrated cross sections. The similarity between these two K-factorsis a first indication of the mild rapidity dependence of the resummation effects. By inspection ofthe inset plot, we note that NNLL resummation is relevant not only at small qT , but also in theintermediate-qT region: as soon as qT 80 GeV, the resummation effects are larger than 20%.Of course, the fact that K 1 at qT 24 GeV is purely accidental: it simply follows from theunphysical behaviour of the fixed-order perturbative expansion at small qT .

Considering other values of the rapidity, from the central to the off-central rapidity region, wefind the same features as observed at y = 0. Our results of the qT spectrum at y = 2 are presentedin Fig. 2. The NNLL + NLO spectrum has a peak at qT 11 GeV. As happens in the case of theqT distribution integrated over y, the effect of NNLL resummation is definitely non-negligiblestarting from relatively-high values of qT . For example, at qT = 50 GeV the NNLL + NLOresult is about 30% higher than the NLO result.To analyze the rapidity dependence in more detail, we study the doubly-differential crosssection at fixed values of qT . In Figs. 3 and 4, we show quantitative results at two typical values


Fig. 3. The rapidity spectrum at the LHC with MH = 125 GeV and qT = 15 GeV: results at NNLL + NLO (solid line)and NLO (dashed line) accuracy. The inset plot shows the K-factor as defined in Eq. (15).

of the transverse momentum, qT = 15 GeV and qT = 40 GeV, in the small-qT and intermediate-qT region, respectively.

Fig. 3 shows the rapidity distribution at NNLL + NLO (solid line) and NLO (dashes) accuracywhen qT = 15 GeV. At this value of qT , the effect of NNLL resummation reduces the crosssection. For example, when y = 0 the reduction effect is about 25%. As can be observed in theinset plot, the relative contribution from the resummed logarithmic terms is rather constant in thecentral rapidity region, and its dependence on y only appears in forward (and backward) region,where the cross section is quite small.

When qT = 40 GeV (see Fig. 4), instead, the effect of NNLL resummation increases theabsolute value of the cross section. For example, when y = 0 the NLO cross section is increasedby about 22%. Nonetheless, as for the relative effect of resummation and the rapidity dependenceof the K-factor, we observe features that are very similar to those in Fig. 3. The resummationeffects have a very mild dependence on y in the central and (moderately) off-central regions,and this explains the remarkable similarity between the solid and dashed lines in the inset plotof Figs. 1 and 2. Since the kinematical region where |y| 2 accounts for most of the total crosssection, when comparing the ratio K(qT , y) to the analogous ratio of the y-integrated crosssections, hardly any differences are expected, unless the large-rapidity region is explored.

The mild rapidity dependence of the qT shape of the resummed results can be studied with afiner resolution by defining the following ratio:

2

(16)R(qT ;y) = d /(dqT dy)

d/dqT.


Fig. 4. The rapidity spectrum at the LHC with MH = 125 GeV and qT = 40 GeV: results at NNLL + NLO (solid line)and NLO (dashed line) accuracy. The inset plot shows the K-factor as defined in Eq. (15).

This ratio gives the doubly-differential cross section normalized to the qT cross section integratedover the full rapidity range. For comparison, we consider also the qT -integrated version of thecross section ratio in Eq. (16), and we define the ratio

(17)Ry = d/dy

of the rapidity cross section d/dy over the total cross section .We have computed the ratio in Eq. (16) by using the resummed qT cross sections at NNLL +

NLO accuracy. The results, as a function of qT , are presented in Fig. 5 (solid lines) at two dif-ferent values, y = 0 and y = 2, of the rapidity. The results of the analogous (qT -independent)ratio Ry (computed8 at NNLO with the numerical programs of Refs. [14,15]) at the corre-sponding values of rapidity are also reported (dotted lines) in Fig. 5. The dashed lines in Fig. 5correspond to the computation of Eq. (16) by using the qT cross sections at NLO: we see that thedashed and solid lines are very similar (as expected from the similarity of the dashed and solidlines in the inset plot of Figs. 1 and 2). As discussed below, the results in Fig. 5 show that thecross section decreases and the qT spectrum softens when the rapidity increases.

8 The numerical accuracy of this computation is better than about 2%3%. Owing to the unitarity constraint in our

resummation formalism, the same result (with a similar numerical accuracy) can be obtained by integration over qT ofthe resummed qT cross sections.


Fig. 5. The rescaled qT spectrum (as defined by the ratio R(qT ;y) in Eq. (16)) at the LHC with MH = 125 GeV. Thesolid (dashed) lines correspond to the NNLL + NLO (NLO) results at two different values of the rapidity: y = 0 (upper)and y = 2 (lower). The dotted lines refer to the corresponding values of the ratio Ry (see Eq. (17)).

We observe that the lines at y = 0 lie above the lines at y = 2; this is just a consequence ofthe fact that the cross sections (both at fixed qT and after integration over qT ) decrease wheny increases.

At fixed y, R(qT ;y) is not constant: it depends (though very slightly) on qT . We note that thecorresponding upper and lower lines in Fig. 5 have different slopes with respect to qT : fixing qT ,the qT slope of R(qT ;y) decreases from positive to negative values as y increases from y = 0 toy = 2, thus showing that the qT spectrum becomes slightly softer at larger rapidity. In general, as|y| increases, the hardness of the qT shape of d2/(dqT dy) decreases. Since the cross sectiondecreases by increasing the rapidity, the hardness of d/dqT (the denominator in Eq. (16)) isintermediate between the values of the hardness of d2/(dqT dy) (the numerator in Eq. (16)) aty = 0 and at large |y|. As a consequence, the qT slope of R(qT ;y) is necessarily positive wheny = 0. Note that the qT slope is already negative when y = 2 (Fig. 5): this is a consequence ofthe fact that the bulk of the cross section is in the rapidity region |y| 2.

Our qualitative illustration of the results in Fig. 5 can be accompanied by some quantitativeobservations. We note that the rapidity dependence of the cross sections is sizeable: going fromy = 0 to y = 2, the ratio Ry decreases by about 43%; comparable variations affect the ratioR(qT ;y), which is not very different from Ry and it is slowly dependent on qT . Indeed, atfixed y, the ratio R(qT ;y) at NNLL + NLO accuracy has a small and nearly constant slopefrom low values of qT around the peak (say, qT 10 GeV) to qT = 100 GeV; varying qT in

this region, R(qT ;y) increases by about 11% when y = 0, and it decreases by about 16% wheny = 2. In the same range of qT and y, the values of R(qT ;y) at NNLL + NLO (solid lines)


and at NLO (dashed lines) are very similar: although this is expected at large qT , the differencesnever exceed the level of about 4% even at values of qT as low as qT 10 GeV.

In summary, the results in Fig. 5 show that, when |y| increases from the central to the (moder-ately) off-central region, the cross sections vary more in absolute value than in qT shape. Thesefeatures deserve some words of discussion.

We first consider the total cross section and the rapidity cross section d/dy. We recall (seeSection 1) that the value of these cross sections is sizeably affected by QCD radiative corrections.The bulk of the effect is due to the radiation of virtual and soft gluons, and they cannot affect therapidity of the Higgs boson. As a consequence, the ratio Ry has little sensitivity to perturbativeQCD corrections. The decreases of Ry as |y| increases is mainly driven by the decrease of thegluon density fg(x,M2H ) as x increases. Considering the large-qT region, similar arguments ap-ply to the qT cross sections d/dqT and d/(dqT dy), and similar conclusions apply to the ratioR(qT , y). In the small-qT region, we have to consider the additional and large effect producedon the qT cross sections by the logarithmically-enhanced terms lnm(M2H/q

2T ). These terms are

due to the radiation of soft and collinear partons. As already discussed in Section 2, the rapidityof the Higgs boson can be varied only by collinear radiation, while soft radiation can only leadto on overall (independent of y) rescaling of the qT cross sections. At the LL level, only softradiation contributes (the LL function g(1) in Eq. (13) does not depend on N1 and N2) and all thelogarithmic terms cancel in the ratio R(qT , y). The y sensitivity of R(qT , y) starts at the NLLlevel. The corrections produced on the dominant soft-gluon effects by the collinear radiation arephysically [29] well approximated by varying the scale of the gluon density from MHto qT . As a consequence, the variations of the hardness of the qT cross sections are mainlydriven by d lnfg(x, q2T )/d lnq2T , the amount of scaling violation of the gluon density. Since thescaling violation decreases as x increases, the hardness of d/(dqT dy) decreases and the qTspectrum softens as |y| increases. Note that, by increasing x, the gluon density decreases fasterthan its scaling violation: this explains why d/(dqT dy) varies more in absolute value than inqT shape when |y| increases.

We conclude this section with some comments about the theoretical uncertainties on thedoubly-differential cross section d/(dqT dy) at NNLL + NLO accuracy. In Ref. [44] theperturbative QCD uncertainties on d/dqT were investigated by comparing the results atNNLL + NLO and NLL + LO accuracies and by performing scale variations at NNLL + NLOlevel. We also considered the inclusion of non-perturbative contributions, and we found that theylead to small corrections provided qT is not very small. From these studies we concluded that theNNLL + NLO result has a QCD uncertainty of about 10% in the region from small (aroundthe peak of the qT distribution) to intermediate (say, roughly, qT MH/3) values of transversemomenta. Similar studies can be carried out in the case of the doubly-differential cross sectiond/(dqT dy). These studies are not reported here since their results and the ensuing conclusionsare very similar to those in Ref. [44]. The reason for this similarity is a feature that we havepointed out throughout this section: the qT resummation effects have a very mild dependence onthe rapidity and, thus, they are almost unchanged when comparing d/(dqT dy) with d/dqT(equivalently, they largely cancel in the ratio in the cross section ratio of Eq. (16)).

4. Summary

We have considered the resummation of the logarithmically-enhanced QCD contributions that

appear at small transverse momenta when computing the qT spectrum of a Higgs boson producedin hadron collisions. In our previous work on the subject [41,44], the rapidity of the Higgs boson


was integrated over: resummation was implemented by using a formalism based on a transformto impact parameter and Mellin moment space. In this paper we have extended the resummationformalism to the case in which the rapidity is kept fixed, and we have considered the doubly-differential cross section with respect to the transverse momentum and the rapidity. We haveshown that this extension can be carried out without substantial complications: it is sufficient toenlarge the conjugate space by introducing a suitably-defined double Mellin transformation.

The main aspects of our method [36,44], which are recalled here, are unchanged by the in-clusion of the rapidity dependence. The resummation is performed at the level of the partoniccross section, and the parton densities are factorized as in the customary fixed-order calcula-tions. The formalism is completely general and it can be applied to other processes: the largelogarithmic contributions are universal and, thus, they are systematically exponentiated in aprocess-independent form (see Eqs. (10) and (13)); the process-dependent part is factorized inthe hard-scattering coefficient H. A constraint of perturbative unitarity is imposed on the re-summed terms (see Eq. (11)), so that the qT smearing produced by the resummation does notchange the total production rate. This constraint reduces the effect of unjustified higher-ordercontributions at intermediate qT and facilitates the matching procedure with the complete fixed-order calculations at large qT . In particular, when the rapidity is kept fixed, the integration overqT of d/(dqT dy) at NNLL + NLO accuracy returns d/dy at NNLO.

We have presented numerical results for Higgs boson production at the LHC. Comparingfixed-order and resummed calculations, we find that the resummation effects are large at small qT(as expected) and still sizeable at intermediate qT . The inclusion of the rapidity dependence haslittle quantitative impact on this picture since, as we have shown, the qT resummation effects aremildly dependent on the rapidity. Going from the central to the (moderately) off-central rapidityregion, the qT shape of the spectrum slightly softens. In the range from small to intermediate val-ues of qT , the residual perturbative uncertainty of the NNLL + NLO predictions for d/(dqT dy)is comparable to that of advanced (NNLO or NNLL + NNLO) calculations of the qT inclusivecross sections d/dy and .

Acknowledgements

The work of D.dF. was supported in part by CONICET. D.dF. wishes to thank the PhysicsDepartment of the University of Florence and INFN for support and hospitality while this workwas completed.

Appendix A

In this appendix we present the structure of the resummation formula (10) by explicitly in-cluding the dependence on the flavour indices of the colliding partons.

In the context of our resummation formalism, a detailed derivation of exponentiation in themultiflavour case is illustrated in Appendix A of Ref. [44]. Considering a generic LO partonicsubprocess c + c F (F = H and c = c = g in the specific case of Higgs boson production bygluon fusion), and performing qT resummation after integration over the rapidity, the resummedcomponent d (res.)a1a2 /dq2T of the partonic cross section is controlled by the N -momentsWFa1a2,N .The final exponentiated result for these N -moments is given by the master formulae (106)(108)of Ref. [44]. We recall the master formula (106) in the following form: {I },F { } (A.1)WFa1a2,N (b,M;S) =

{I }Ha1a2,N (M,S) exp G{I },N (S, L) ,


where the sum extends over the following set of flavour indices:

(A.2){I } = c, c, ii , i2, b1, b2,and, for simplicity, the functional dependence on various scales (such as the renormalization andfactorization scales) is understood. The functions G{I },N and H{I },Fa1a2,N are given in the masterformulae (107) and (108), respectively. In the present paper, qT resummation is performed atfixed values of the rapidity, and the double (N1,N2)-momentsW(N1,N2)Fa1a2 in Eq. (6) replace theN -moments WFa1a2,N of Ref. [44]. The generalization of Eq. (10) to the multiflavour case isstraightforwardly obtained from Eq. (A.1) by the simple replacement N (N1,N2):

(A.3)W(N1,N2)Fa1a2 (b,M;S) ={I }H{I },(N1,N2)Fa1a2 (M,S) exp

{G(N1,N2){I } (S, L)}.

The exponent G(N1,N2){I } of the process-independent form factor and the process-dependent hardfactor H{I },(N1,N2)Fa1a2 are

(A.4)G(N1,N2){I } = Gc + Gi1,N1 + Gcb1,N1 + Gi2,N2 + Gcb2,N2,

(A.5)H{I },(N1,N2)Fa1a2 = (0)cc,FHFc ScCcb1,N1

[E

(i1)N1

V 1N1 UN1]b1a1

Ccb2,N2[E

(i2)N2

V 1N2 UN2]b2a2

.

The expressions in Eqs. (A.4) and (A.5) are completely analogous to the master formulae (107)and (108) in Ref. [44] (the functional dependence on the scales M,R,F and Q is explicitlydenoted in those formulae). In particular, we note that the dependence of G(N1,N2) and H(N1,N2)on the Mellin variables N1 and N2 is completely factorized: each of terms on the right-hand sideof Eqs. (A.4) and (A.5) depends only on one Mellin variable (either N1 or N2). This factorizedstructure is completely consistent with Eq. (14) and with the physical picture discussed belowEq. (14); the dependence on N1 (N2) follows the longitudinal-momentum flow and the flavourflow a1 b1 i1 c (a2 b2 i2 c) that are produced by collinear radiation from theinitial-state parton with momentum p1 (p2). The various Mellin functions (Gi,N ,E(i)N ,UN andso forth) in Eqs. (A.4) and (A.5) can be found in Ref. [44].

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Nuclear Physics B 791 (2008) 2059

Long wavelength limit of evolution of cosmologicalperturbations in the universe where scalar fields

and fluids coexistTakashi Hamazaki

Kamiyugi 3-3-4-606 Hachioji-city, Tokyo 192-0373, JapanReceived 7 July 2007; accepted 24 September 2007

Available online 29 September 2007

Abstract

We present the LWL formula which represents the long wavelength limit of the solutions of evolutionequations of cosmological perturbations in terms of the exactly homogeneous solutions in the most generalcase where multiple scalar fields and multiple perfect fluids coexist. We find the conserved quantity whichhas origin in the adiabatic decaying mode, and by regarding this quantity as the source term we determinethe correction term which corrects the discrepancy between the exactly homogeneous perturbations and thek 0 limit of the evolutions of cosmological perturbations. This LWL formula is useful for investigat-ing the evolutions of cosmological perturbations in the early stage of our universe such as reheating afterinflation and the curvaton decay in the curvaton scenario. When we extract the long wavelength limits ofevolutions of cosmological perturbations from the exactly homogeneous perturbations by the LWL formula,it is more convenient to describe the corresponding exactly homogeneous system with not the cosmologicaltime but the scale factor as the evolution parameter. By applying the LWL formula to the reheating modeland the curvaton model with multiple scalar fields and multiple radiation fluids, we obtain the S formularepresenting the final amplitude of the Bardeen parameter in terms of the initial adiabatic and isocurvatureperturbations. 2007 Elsevier B.V. All rights reserved.

PACS: 98.80.Cq

Keywords: Cosmological perturbations; Long wavelength limit; Reheating; Curvaton

E-mail address: [email protected]/$ see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2007.09.028

T. Hamazaki / Nuclear Physics B 791 (2008) 2059 21

1. Introduction and summary

Recently we come to be required to investigate the evolution of cosmological perturbations inthe very early universe [1,2]. According to the inflationary scenarios and the curvaton scenario, inthe early universe the wavelength of cosmological perturbations responsible for the present cos-mic structures such as galaxies and clusters of galaxies is much larger than the horizon scales.Therefore the methods for researching the cosmological perturbations on superhorizon scaleshave been sought. In this context, Nambu and Taruya pointed out that there exists an LWL for-mula representing the k 0 limit of the cosmological perturbations in terms of the exactlyhomogeneous perturbations [3]. Soon later in the multiple scalar fields system [4,5], in the multi-ple fluids system [4] the complete LWL formulae were constructed. Since the evolution equationsof the corresponding exactly homogeneous universe look simpler than the evolution equationsof cosmological perturbations, the LWL formula brought about great simplification. For this rea-son, the LWL formula was used for investigating the evolution of cosmological perturbation onsuperhorizon scales by several authors [3,58]. Especially the LWL formula was used effectivelyin the analysis of the system containing oscillatory scalar fields. By using such LWL formula,in single oscillatory scalar field system [6] and in nonresonant multiple oscillatory scalar fieldssystem [7] it was shown that the Bardeen parameter is conserved, and in resonant multiple os-cillatory scalar fields system [8] it was shown that the cosmological perturbation including theBardeen parameter can grow. In the papers [7,8], the averaging method representing the averag-ing over the fast changing angle variables was used for investigating the corresponding exactlyhomogeneous system, and by the LWL formula the evolution of the cosmological perturbationin the long wavelength limit was constructed from the corresponding exactly homogeneous per-turbation. In addition, the viewpoint that the evolutions of the cosmological perturbations onsuperhorizon scales are governed by the stability and/or instability of the corresponding exactlyhomogeneous universe [7,8] is physically important. In this context, the phase space of the corre-sponding exactly homogeneous system was investigated in detail and the role of the fixed pointsin the phase space in the stability and instability of cosmological perturbation was discussed [8].

By replacing the oscillating scalar fields with the dust fluids, the evolution of the cosmologicalperturbations during reheating after the inflation [9] and in the curvaton scenario [10] were inves-tigated. These authors treated the system dominated by dust-like scalar field fluid and radiationand investigated the influence of the entropy perturbation originating from the multicomponentproperty to the evolution of the total curvature perturbation variables such as the Bardeen pa-rameter. The purpose of these analyses was to determine the initial perturbation of the presentFriedmann universe in terms of the early stage seed perturbation. Although it was shown partiallythat this replacement is physically reasonable [6,9], it is dangerous that there can be possibility tomiss the instabilities characteristic to the rapidly oscillating scalar fields [8,1114]. The instabil-ity of the oscillatory scalar fields has been investigated [1518]. Therefore the direct investigationof the system where the scalar fields and fluids coexist becomes more and more necessary andthe LWL formula of the scalar-fluid composite system is required eagerly. If we have such LWLformula, by investigating the deformation of the phase space of the corresponding exactly homo-geneous system due to the dissipation of the oscillatory scalar fields into fluids we will be ableto interpret the evolution of cosmological perturbation more physically.

This paper is organized as follows. In Section 2, we construct the LWL formula as for suchscalar-fluid composite system based on the philosophy of the paper [4]. As shown in the paper

[4], the discrepancy exists between the evolution equations of the cosmological perturbationsin the k 0 limit and the evolution equations of the exactly homogeneous perturbations be-

22 T. Hamazaki / Nuclear Physics B 791 (2008) 2059

cause the former contains k2 = O(1) terms and the latter does not, therefore this discrepancyshould be corrected by the correction term which contributes the adiabatic decaying mode, butany general methods for determining such correction term have not been presented yet, and onlyin the multiple scalar fields system such correction term was determined. We show in the k 0limit the existence of the conserved quantity which has origin in the adiabatic decaying modeand which is related with k2 . By regarding this conserved quantity as the source term, andobtaining the special solution A, we correct the exactly homogeneous perturbation A and weobtain the complete LWL formula A = A + A in the most general scalar-fluid composite sys-tem. In Section 3, we point out that it is more appropriate to use the scale factor a rather thanthe cosmological time t as the evolution parameter when we use the LWL formula. As for thescalar quantity T , we use the perturbation variable DT , D is the operator which maps the exactlyhomogeneous scalar quantity T to the gauge invariant perturbation variable representing the Tfluctuation in the flat slice. D defined in this way can be interpreted as a kind of derivative oper-ator. In fact, the exactly homogeneous part (DT ) can be expressed as the derivative of T withrespect to the solution constant with the scale factor a fixed. In order to investigate the exactlyhomogeneous system containing oscillatory scalar fields, we use the action angle variables Ia , a .By using D defined in this way, we can define the action angle perturbation variables DIa , Dawhose exactly homogeneous parts are given as the derivatives of Ia , a with solution constant Cwith the scale factor a fixed. When we use the derivative operator D and the LWL formulae,it is essential to use the scale factor a as the evolution parameter. In Section 4, we apply theLWL formulae to the non-interacting multicomponents system and discuss the long wavelengthlimit of the evolution of the Bardeen parameter. In Section 5, we apply the averaging method bywhich the system is averaged over the fast changing angle variables to the decaying scalar fieldswhich have been discussed in the reheating model and the curvaton model. By evaluating thecorrections produced by the averaging process and the errors produced by the truncation of thesufficient reduced angle variables dependent part, the validity of the averaging method is estab-lished. In Sections 6, 7, we apply the LWL formula and the averaging method to the interactingmulticomponents model such as the reheating model, the curvaton model, respectively. We as-sume that the multiple scalar fields and the multiple radiation fluid components exist. In thesemodels, we construct the S formulae representing the final amplitude of the Bardeen parameter interms of the initial adiabatic and isocurvature perturbations. In our previous paper [8], the evolu-tionary behaviors of cosmological perturbations in the early universe where multiple oscillatoryscalar fields interact with each other have been investigated. This S formula gives the informa-tion about how the cosmological perturbations which grew in such early era are transmitted intothe radiation energy density perturbations through the energy transfer from the scalar fields intothe radiation fluids. We present the necessary condition for the initial entropic perturbations pro-duced in the early era to survive until the late radiation dominant universe. Section 8 is devotedto discussions containing consideration of the case where the decay rate depends on the physicalquantities. In Appendices A and B, the proofs of the propositions presented in Section 5 and theevaluations of the useful mathematical formulae used in Section 6 are contained.

In this paper, we consider the case where the homogeneous scalar fields obey the phenomeno-logical evolution equations as

(1.1) + 3H + U

+ S = 0.

The interactions between scalar fields are described by the interaction potential U , while theinteractions between scalar fields and fluids are described by S. This analysis includes the well-


known case S = [19,20] and the general case where S is an arbitrary analytic function of , which was discussed in the paper [21] but whose perturbations have not been investigated yet.Another supplemental purpose of our paper is to present the evolution equations of cosmologicalperturbations corresponding to the homogeneous system with various source term S especiallydependent on .

The notations used in this paper are based on the review [22] and the paper [4].

2. Derivation of the LWL formula

We give the definitions and the evolution equations as for the background and the perturbationvariables. Based on these notations, in the most general model where the multiple scalar fieldsand the multiple perfect fluid components interact, we give the LWL formula representing theevolutions of the perturbations variables in terms of the exactly homogeneous solutions.

We consider perturbations on a spatially flat RobertsonWalker universe given by

ds2 = (1 + 2AY)dt2 2aBYj dt dxj(2.1)+ a2[(1 + 2HLY)jk + 2HT Yjk]dxj dxk,

where Y , Yj and Yjk are harmonic scalar, vector and tensor for a scalar perturbation with wavevector k on flat three-space:

(2.2)Y := eikx, Yj := i kjkY, Yjk :=

(13jk kj kk

k2

)Y.

By using the gauge dependent variablesR and g representing the spatial curvature perturbationand the shear, respectively:

(2.3)R := HL + 13HT , g :=a

kHT B,

we can define two independent gauge invariant variables:

(2.4)A := A(RH

), :=R aH

kg.

In order to define the matter perturbation variables, we will consider the scalar quantity pertur-bation variables generally. As for covariant scalar quantity T = T + T Y , we define the gaugeinvariant perturbation variable representing the T fluctuation in the flat slice:

(2.5)DT := T THR.

Next we consider the covariant scalar quantity T2 whose background quantity is the time deriva-tive of T : T . The extension of T into the covariant scalar quantity T2 is not unique. For example,

(2.6)T2 = sgn(0T )[gT T ]1/2,

and

(2.7)T2 = nT ,where n is an arbitrary vector field satisfying(2.8)nn = 1,


have the same T as the background part. But these different T2s give the unique perturbationpart: DT2 = (DT ) TA. Therefore we can define DT by

(2.9)DT := (DT ) TA.We consider the universe where the scalar fields a (1 a NS) and the fluids ,P (1 Nf ) coexist, whose energymomentum tensor is divided into A = (S,f ) parts where Srepresents the multiple scalar fields, f represents the multiple fluids. The energymomentumtensor of f parts are further divided into individual fluids parts . On the other hand, the energymomentum tensor of S part cannot be divided into individual scalar fields parts a, since theinteraction potential U contains the terms consisting of plural scalar fields a :

(2.10)T =(T

)S+ (T )f = (T )S +

T ,

(2.11)0 = (Q)S + (Q)f = (Q)S +

Q,

where the energymomentum transfer vector QA is defined by

(2.12) T A = QA = QAu + fA,where u is the four velocity of the whole matter system and the momentum transfer fAsatisfies ufA = 0. For the scalar perturbation, the energymomentum tensor and the energymomentum transfer vector of each individual component are expressed as

(2.13)T 0A0 = (A + AY ),(2.14)T 0Aj = a(A + PA)(vA B)Yj ,(2.15)T jAk =

(PA

jk + PAYjk +TAY jk

),

and

(2.16)QA0 = [QA + (QAA+ QA)Y

],

(2.17)QAj = a[QA(v B)+ FcA

]Yj ,

where A, PA and QA are the background quantities of the energy density, the pressure and theenergy transfer of the individual component A, respectively. The anisotropic pressure perturba-tion TA and the momentum transfer perturbation FcA are already gauge invariant. As for thescalar quantities T = (A,PA,QA), we use DT as the gauge invariant perturbation variables.As for the gauge invariant velocity perturbation variable, we use

(2.18)ZA :=R aHk

(vA B).The energymomentum tensor of scalar fields part is given by

(2.19)(T )S = 12( + 2U) .

Since divergence of the energymomentum tensor is given by

() ( U ) (2.20)T S = a a a,


in order that the phenomenological equations of motion of the scalar fields become

(2.21)a Ua

= Sa,

we assume that

(2.22)(Q)S = Saa.By using the scalar fields background variables a , a , Sa and the corresponding perturbationvariables Da , Da , DSa , the background part of the fluid variables is given by

(2.23)S = 12 ()2 +U,

(2.24)PS = 12 ()2 U,

(2.25)hS = ()2,(2.26)QS = S ,

and the perturbation part of fluid variables is given by

(2.27)(D)S = S

D + S

D,

(2.28)(DP )S = PS

D + PS

D,(2.29)(hZ)S = H D,(2.30)(T )S = 0,(2.31)(DQ)S = S D DS,(2.32)(aFc)S = Sa

(kDa k

HaZ

).

When the source of the scalar field a , Sa is given as functions of the covariant scalar quantitiesT and T2 whose background part is T , that is Sa = Sa(T , T2), DSa is given by

(2.33)DSa = SaT

DT + SaT

DT .In such case, (DQ)S can be written as

(2.34)(DQ)S = QST

DT + QST

DT ,

which is assumed from now on. In the same way as the individual components T A , as for thetotal energymomentum tensor T = A T A , we can define the gauge invariant perturbationvariables such as D, DP , hZ and T . From (2.10), (2.11), we obtain the background equationsas

(2.35) = S +

,

(2.36)P = PS +

P,


(2.37)h = hS +

h,

(2.38)0 = QS +

Q,

and perturbation equations as

(2.39)D = DS +

D,

(2.40)DP = DPS +

DP,

(2.41)hZ = (hZ)S +

hZ,

(2.42)T = (T )S +

T ,

(2.43)0 = (DQ)S +

DQ,

(2.44)0 = (Fc)S +

Fc.

This Z is known as the Bardeen parameter [2224]. In the long wavelength limit, the Bardeenparameter is conserved in the case where the entropy perturbations are negligible. But in varioussystems it was reported that the entropy perturbations cannot be neglected [2,8,11,12], so inthe present paper we will investigate the evolutionary behavior of the Bardeen parameter morecarefully. Until now, as for the gauge invariant scalar quantity perturbation variables, we use D.But traditionally most scalar quantity perturbation variables have been written without using D:

(2.45)Ya := Da, g := D, PL := DP, QEg := DQ.This Ya has been called the SasakiMukhanov variable [25,26].

In terms of the gauge independent variables defined above, we give the evolution equationsof cosmological perturbations. From (2.21), the background and the perturbation parts can bewritten as

(2.46) + 3H + U

+ S = 0,

(2.47)L1(DT ,A) = k2

a2D k

2

a2

H,

where

(2.48)L1(DT ,A) = (D) + 3H(D) + 2U

D +DS A+ 2

(U

+ S

)A.

As for the fluid components, T = Q gives the background equations as(2.49) = 3Hh +Q,

and the perturbation equations as(2.50)L2(DT,A) = k2

a2Hh( Z),


(2.51)(hZ

H

)+ 3hZ + hA+DP 23T =

a

kFc + Q

HZ,

where

(2.52)L2(DT ,A) = (D) + 3HD + 3HDP QADQ.G

= 2T gives the background equations as

(2.53)H 2 = 2

3,

(2.54) = 3Hh,(2.55)H = 3

2(1 +w)H 2,

and the perturbation equations as

(2.56)L3(DT ,A) = 2 k2

3a2H 2,

(2.57)L4(DT ,A) = 2

3T k

2

a2,

(2.58)A+ 32(1 +w)Z = 0,

(2.59)A+ 1a

(a

H

)=

2

k2a2T ,

where

(2.60)L3(DT ,A) = 2A+D,(2.61)L4(DT ,A) = H A+ 2HA

2

2(D +DP).

The dynamical perturbation variables are classified into two groups, that is, what has analogywith the exactly homogeneous perturbations and what is not related with the exactly homo-geneous perturbations at all. The dynamical perturbation variables of the former type are DTrepresenting the scalar quantity T = (,P,,Q,S) perturbation in the flat slice, DT and themetric perturbation variable A. The dynamical perturbation variables of the latter type are theNewtonian gravitational potential and ZA, FcA, TA which have vector or tensor origin.In the above Li (i = 1, . . . ,4) equations, the former type dynamical perturbation variables arecontained in the left-hand side while the latter type perturbation variables are collected in theright-hand side. The exactly homogeneous perturbations DT and A corresponding to DTand A, respectively are constructed as

(2.62)(DT ) :=(T

C

)t

THR,

(2.63)A := (R

H

),

1(

a)(2.64)R :=a C t

,


where C is the solution constant of the background solution and the subscript t implies thatthe derivative with respect to C is performed with the cosmological time t fixed. On the otherhand, the dynamical perturbation variables of the latter type such as , ZA, FcA and TA do nothave exactly homogeneous counterparts. The evolution equations of cosmological perturbationscontaining Li (i = 1, . . . ,4) have analogy in the exactly homogeneous perturbation equations. Infact, the variations of the exactly homogeneous equations (2.46), (2.49), (2.53) and (2.55) give

(2.65)Li(DT ,A)= 0 (i = 1, . . . ,4),

respectively. The only difference between the exactly homogeneous perturbation Li (i =1, . . . ,4) equations and the actual k = 0 cosmological perturbation Li (i = 1, . . . ,4) equationsis that k2 terms exist in the latter but k2 terms do not exist in the former. Then the effect ofthe source term k2 is corrected in the following way. In performing the correction process, it isimportant to notice that the source terms k2 can be represented in terms of conserved quantitywhich has origin in the universal adiabatic decaying mode. In fact, as for f defined by

(2.66)f = a3H(A+ 1

2g

)= k

2

3Ha,

using (2.50), (2.56), (2.57) yields

(2.67)dfdt

= a3H 2wT + 12ak2(1 +w)Z.

When we assume that for k 0 limit(2.68)T 0, kZ 0,

are satisfied, the quantity f is conserved, whose value is written as c. Therefore for k 0 limit,

(2.69)k2 3Ha

c = O(1).This expression of is well known as that of the universal adiabatic decaying mode [4]. Inthe Li (i = 1, . . . ,4) equations containing DT , A, the Newtonian potential appears onlyin the form k2 , that is, accompanied by k2. When we assume that DT = O(1), A = O(1),k2 behaves as O(1). Since in the linear perturbation, the scale of the perturbation variables isarbitrary, the fact that = O(1/k2) does not imply the breakdown of the linear perturbation. Ifone wants to get = O(1), one simply assumes that DT = O(k2),A= O(k2). But as explainedlater, we cannot assume that is vanishing, since c defined by (2.69) must satisfy the constraint(2.80). Therefore in the k 0 limit where (2.68), (2.69) are satisfied, (2.47), (2.50), (2.56),(2.57) can be written as

(2.70)L1a(DT ,A) = 3aa3

c,

(2.71)L2(DT,A) = 3ha3

c,

(2.72)L3(DT ,A) = 2a3H

c,

3H (2.73)L4(DT ,A) = a3

c.


It can be verified that above four sets of Eqs. (2.70), (2.71), (2.72) and (2.73) are satisfied by

(2.74)A= 32(1 +w)g + g

H,

(2.75)DT = TH

g,

where

(2.76)g = ct0

dt1a3

.

This special solution for A = (DT ,A) is written as A. Since the variation of the exactly ho-mogeneous solution A satisfies (2.65), the general solutions of (2.70), (2.71), (2.72), (2.73)A = (DT ,A) can be expressed as

(2.77)A = A +A.The perturbation equations except Li (i = 1, . . . ,4) equations have vector origin, that is, theyare derived from the space component of the Einstein equations. Therefore these perturbationequations do not have any analogy with the exactly homogeneous perturbation equations. Asexplained in the paper [4], these perturbation equations determine the evolutions of the dynamicalperturbation variables which have vector or tensor origin, that is, which have no correspondencewith the exactly homogeneous perturbations, or give the constraint which should be satisfiedin order that the exactly homogeneous perturbations become the k 0 limit of evolutions ofcosmological perturbations. Therefore (2.51), (2.58) can be interpreted as the decision of theevolution of the variables Z which is not related to the exactly homogeneous solution at all interms of A = (DT ,A), the constraint to the exactly homogeneous perturbations, respectively.Integrating (2.51) yields

(2.78)hZ Ha3

[C +

t0

dt a3(hADP a

kFc + Q

HZ

)].

By summing (2.78) with respect to all the fluid components, we obtain

(hZ)f = Ha3

[

C +t0

dt a3(hfA (DP )f S D)

]

(2.79)= Ha3

[

C +(2

3a3

HA+ a3 D

)(2

3a3

HA+ a3 D

)0

].

Therefore (2.58) gives the constraint between C , c defined by (2.69), and 2NS + Nf solutionconstants of the exactly homogeneous perturbation as

(2.80)

C + 22

c (2

3a3

HA+ a3 D

)0= 0.

Integrating (2.59) givesH( )(2.81) =a

Ct t0

dt aA ,


where the first term containing Ct is well-known universal adiabatic decaying mode [4] and bycomparing with (2.69) we obtain

(2.82)Ct = 3k2

c.

If we assume c = 0, since (2.80) gives one constraint relation, we obtain 2NS +2Nf 1 solutionsand the Newtonian potential is obtained by (2.81) with Ct = 0:

(2.83) = Ha

t0

dt aA.

If we assume that c is nonvanishing, since c = O(1), A O(1), therefore(2.84)A

Ct O(k2),

we obtain the k 0 limit of the universal adiabatic decaying mode [4]:(2.85)1

3k2 H

ac,

which is consistent with (2.69). Then we have obtained the long wavelength limit of all thesolutions to the evolution equations of cosmological perturbations.

3. Use of the scale factor as the evolution parameter

As the gauge invariant variable representing the fluctuation of the scalar quantity T , we adoptDT defined by (2.5) which represents the T fluctuation in the flat slice, since it is the easiest tosee the correspondence with the exactly homogeneous perturbation of T . While until now wedescribed the exactly homogeneous variables as functions of t , C where t is the cosmologicaltime and Cs are solution constants, we can describe the exactly homogeneous variables as func-tions of a, C where a is the scale factor. For an arbitrary scalar quantity such as , P , S, Q, ,from (2.62), (2.64), (DT ) can be written as the partial derivative of the corresponding exactlyhomogeneous scalar quantity T with respect to solution constant C with the scale factor a fixed:

(3.1)(DT ) =(T

C

)a

.

Since

(3.2)1a

(

Ca

)a

= 12

(

C

)a

= 12

(D) = A,

this property of D also holds as for the time derivative of the scalar quantity T :

(3.3)(DT ) =(

CT

)a

.

Therefore the operator D defined by (2.5) can be interpreted as a kind of derivative operator, thatis, the derivative with respect to the solution constant C with a fixed. Because of this derivativeproperty of D, as for the scalar quantities T = (,P,S,Q) which are functions of , , we canunderstand(3.4)DT = T

D + T

D,


easily. The contribution to DT from the adiabatic decaying mode is given by the same form asthat of DT :

(3.5)(DT ) = TH

g,

where g is defined by (2.76).As seen from the above discussion, in order to derive the long wavelength limit of cosmo-

logical perturbations from the corresponding exactly homogeneous system by using the LWLformula, it is more appropriate to use as the evolution parameter the scale factor a than thecosmological time t . For example, as for the scalar-fluid composite system, the correspondingexactly homogeneous expressions are obtained by solving the first order differential equationssetting a , pa := a3a , as independent variables and the scale factor a as the evolution para-meter:

ad

daa = 1

H

pa

a3,

ad

dapa = a

3

H

U

a a

3

HSa,

(3.6)a dda

= 3h + QH

,

replacing H with the right-hand side of the Hubble law:

(3.7)H 2 = 2

3

[1

2a6a

p2a +U()+

].

While under use t as the evolution parameter our system is the constrained system with theHamiltonian constraint (3.7), under use a as the evolution parameter our system becomes theunconstrained system with respect to independent variables a , pa := a3a , .

For some time, we consider the system consisting of multiple scalar fields a only. Since theevolutions of a , pa := a3a can be described in terms of the Hamilton equations of motion,the evolutions of the corresponding perturbation variables Ya = Da , Pa := a3Da can also bewritten in terms of the Hamilton equations of motion:

(3.8)dYadt

= HPa

,dPa

dt= H

Ya,

whose Hamiltonian is given by

(3.9)H = 12a3

PaPa + a3

2VabYaYb +

2

2HabPaYb,

(3.10)Vab = 2U

ab+ 3

2

2ab +

2

2H

(U

ab + a U

b

)+ k

2

a2ab.

Although the evolutions of Ya , a3Ya can be formularized as the Hamiltonian dynamical system,Ya , Pa := a3Da are more convenient to see the correspondence with the long wavelength limitexpressions of cosmological perturbations given in terms of the derivative of a , pa := a3a withrespect to the solution constant with a fixed than Ya , a3Ya . Therefore this set of variables Ya ,

3 Pa := a Da is useful as the starting point from which to derive the finite wavelength correctionaround the long wavelength limit calculated in the previous papers [7,8]. The connection between


the old canonical variables Ya := Ya , Pa := a3Ya and the new canonical variables Ya , Pa :=a3Da are given by the canonical transformation defined by the generating function:

(3.11)W = YaPa + 34a3H

abYaYb.

When we treat the oscillatory scalar fields, the action angle variables Ia , a are useful [7,8]:

a = 1a3/2

2Iama

cos a,

(3.12)pa = a3/2

2maIa sin a,

where ma is the mass of the scalar field a . The action angle variables obey the evolution equationas

(3.13)a dda

Ia = a3

H

Uinta

+ a3/2

H

2Iama

sin aSa + 3Ia cos 2a,

(3.14)a dda

a = maH

+ a3

H

UintIa

+ a3/2

H

12maIa

cos aSa 32 sin 2a.

In order to investigate the cosmological perturbations in the universe containing oscillatory scalarfields, by using D defined in the above we define the action angle perturbation variables DIa ,Da starting from Ya := Da , Pa := a3Da . In the universe dominated by scalar fields only, bothYa , Pa and DIa , Da are pairs of canonical conjugate variables. These perturbation variablespairs are appropriate in order to interpret the effect of the dissipation from the scalar fields to theperfect fluids as the deviation from the Hamiltonian dynamical system. DIa , Da are defined bythe following expressions:

Ya = D[

1a3/2

2Iama

cos a

],

(3.15)Pa = D[a3/22maIa sin a],

where D in the right-hand side is interpreted as

(3.16)D =a

DIa

Ia+a

Da

a.

The expressions obtained from variations of Ia , a with a fixed in the previous papers [7,8] arethe long wavelength limits of DIa , Da defined by (3.15). In fact, the LWL formulae

(3.17)DIa =(Ia

C

)a

+(Ia

H 3Ia

)g,

(3.18)Da =(a

C

)a

+ aH

g,

where g is defined in (2.76) hold. While the third term in the right-hand side of (3.17) appearsbecause of the scale factor a dependence of the transformation law from a , a to Ia , a , the

parts of (3.17) and (3.18) reflect the fact that D is the derivative operator with respect to thesolution constant with the scale factor a fixed.


In order to solve the dynamics of the system containing the oscillatory scalar fields, we arerequired to perform the averaging over the fast changing angle variables a [7,8]. If we use thecosmological time t as the evolution parameter, our system is a constrained system. Therefore wemust check that our averaging procedure is consistent with the constraint and this process is rathercumbersome. But if we use the scale factor a as the evolution parameter, our system becomesunconstrained system, so the definition of the averag

Nucl.Phys.B v.791

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