Nucl.Phys.B v.795

Nuclear Physics B 795 (2008) 126www.elsevier.com/locate/nuclphysb

NNLO vertex corrections in charmless hadronicB decays: Imaginary part

Guido Bell a,b,a Arnold Sommerfeld Center for Theoretical Physics, Department fr Physik,

Ludwig-Maximilians-Universitt Mnchen, Theresienstrae 37, D-80333 Mnchen, Germanyb Institut fr Theoretische Teilchenphysik, Universitt Karlsruhe, D-76128 Karlsruhe, GermanyReceived 11 June 2007; received in revised form 10 August 2007; accepted 14 September 2007

Available online 20 September 2007

Abstract

We compute the imaginary part of the 2-loop vertex corrections in the QCD factorization frameworkfor hadronic two-body decays as B . This completes the NNLO calculation of the imaginary part ofthe topological tree amplitudes and represents an important step towards an NNLO prediction of direct CPasymmetries in QCD factorization. Concerning the technical aspects, we find that soft and collinear infrareddivergences cancel in the hard-scattering kernels which demonstrates factorization at the 2-loop order. Allresults are obtained analytically including the dependence on the charm quark mass. The numerical impactof the NNLO corrections is found to be significant, in particular they lead to an enhancement of the strongphase of the colour-suppressed tree amplitude. 2007 Elsevier B.V. All rights reserved.

PACS: 12.38.Bx; 13.25.Hw

Keywords: B-meson decays; Factorization; NNLO computations

1. Introduction

Charmless hadronic B decays provide important information on the unitarity triangle whichmay help to reveal the nature of flavour mixing and CP violation. In order to exploit the richamount of data that is currently being collected at the B factories, a quantitative control of the

* Correspondence address: Institut fr Theoretische Teilchenphysik, Universitt Karlsruhe, D-76128 Karlsruhe,Germany.

E-mail address: [email protected]/$ see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2007.09.006

2 G. Bell / Nuclear Physics B 795 (2008) 126

underlying strong-interaction effects is highly desirable. In the QCD factorization framework [1]the hadronic matrix elements of the operators in the effective weak Hamiltonian simplify consid-erably in the heavy-quark limit. Schematically,

M1M2|Qi |B FBM1+ (0)fM2

duT Ii (u)M2(u)

(1)+ fBfM1fM2

ddv duT IIi (, v,u)B()M1(v)M2(u),

where the non-perturbative strong-interaction effects are encoded in a form factor FBM1+ atq2 = 0, decay constants fM and light-cone distribution amplitudes M . The short-distance ker-nels T Ii =O(1) and T IIi =O(s) provide the basis for a systematic implementation of radiativecorrections; the former contain the short-distance interactions that do not involve the spectatorantiquark from the decaying B meson (vertex corrections) and the latter describe the ones withthe spectator antiquark (spectator scattering).

The next-to-leading order (NLO) corrections to the kernels T I,IIi , which constitute an O(s)correction to naive factorization, are already known from [1]. Recently, the next-to-next-to-leading order (NNLO) corrections to T IIi have been computed [26]. Due to the interactionwith the soft spectator antiquark, the spectator scattering term receives contributions from thehard scale mb and from an intermediate (hard-collinear) scale (QCDmb)1/2. Both typesof contributions are now available at O(2s ) (1-loop), indicating that the NNLO corrections arenumerically important.

In this work we compute NNLO corrections to T Ii for the so-called topological tree ampli-tudes (which arise from the insertion of currentcurrent operators). In contrast to the spectatorscattering term, the vertex corrections are dominated solely by hard effects and amount to a 2-loop calculation. In particular, we address the imaginary part of the hard-scattering kernels whichis the origin of a strong rescattering phase shift that blurs the information on the weak phases. Asan imaginary part is first generated at O(s), higher order perturbative corrections are expectedto significantly influence the pattern of strong phases and hence direct CP asymmetries. Our cal-culation represents an important step towards an NNLO prediction of direct CP asymmetries inQCD factorization.

The outline of this paper is as follows: In Section 2 we present our strategy for the calculationof the topological tree amplitudes by introducing two different operator bases. Section 3 containsthe technical aspects of the 2-loop calculation. In Section 4 we show how to extract the hard-scattering kernels from the hadronic matrix elements. Our analytical results can be found inSection 5. The numerical impact of the NNLO vertex corrections is investigated in Section 6 andwe finally conclude in Section 7. A more detailed presentation of the considered calculation canbe found in [7].

2. Choice of operator basis

In view of the calculation of topological tree amplitudes, we restrict our attention to thecurrentcurrent operators of the effective weak Hamiltonian for b u transitions

(2)Heff = GF2VubV

ud(C1Q1 +C2Q2)+ h.c.Due to the fact that we work within Dimensional Regularization (DR), we also have to considerevanescent operators [8]. These non-physical operators vanish in four dimensions but contribute

G. Bell / Nuclear Physics B 795 (2008) 126 3

at intermediate steps of the calculation in d = 4 2 dimensions. As the imaginary part has ef-fectively 1-loop complexity with respect to renormalization at O(2s ), the considered calculationonly requires 1-loop evanescent operators. For our purposes the complete operator basis is thusgiven by

Q1 =[ui

Lbi][dj Luj ],

Q2 =[ui

Lbj][dj Lui],

E1 =[ui

Lbi][dj Luj ] (16 4)Q1,

(3)E2 =[ui

Lbj][dj Lui] (16 4)Q2,

where i, j are colour indices and L = 1 5. The operator basis in (3) has been used in allprevious calculations within QCD factorization [14]. We refer to this basis as the traditionalbasis for convenience and denote the corresponding Wilson coefficients and operators with atilde.

It has been argued by Chetyrkin, Misiak and Mnz (CMM) that one should use a differentoperator basis in order to perform multi-loop calculations [9]. Although the deeper reason isrelated to the penguin operators which we do not consider here, we prefer to introduce the CMMbasis in view of future extensions of our work. This basis allows to consistently use DR witha naive anticommuting 5 to all orders in perturbation theory. In the CMM basis the currentcurrent operators and corresponding 1-loop evanescent operators read (indicated by a hat)

Q1 =[ui

LT Aij bj][dkLT

Akl ul

],

Q2 =[ui

Lbi][dj Luj ],

E1 =[ui

LT Aij bj][dkLT

Akl ul

] 16Q1,(4)E2 =

[ui

Lbit][dj Luj ] 16Q2,

with colour matrices T A and colour indices i, j , k, l.Comparing the operator bases in (3) and (4) we observe two differences: First, the two bases

use different colour decompositions which is a rather trivial point. More importantly, they containslightly different definitions of evanescent operators. Whereas the definitions in the CMM basiscorrespond to the simplest prescription to define evanescent operators, subleading terms of O()appear in the one of the traditional basis. These terms have been properly adjusted such that Fierzsymmetry holds to 1-loop order in d dimensions.

We follow the notation of [10] and express the hadronic matrix elements of the effectiveweak Hamiltonian in terms of topological amplitudes i(M1M2). E.g., the B 0 decayamplitude is written as

(5)

20

HeffB= VubV ud[1()+ 2()]A,where A = iGF /

2m2BF

B+ (0)f . The amplitude 1(M1M2) is called the colour-allowedtree amplitude which corresponds to the flavour content [qsb] of the decaying B meson, [qsu] ofthe recoil meson M1 and [ud] of the emitted meson M2. The colour-suppressed tree amplitude2(M1M2) then belongs to the flavour contents [qsb], [qsd] and [uu], respectively. For moredetails concerning the definition of the topological amplitudes we refer to Section 2.2 in [10].According to this definition, the left (right) diagram in Fig. 1 contributes to the tree amplitude1 (2). On the technical level these two insertions of a four-quark operator correspond to two


Fig. 1. Generic 1-loop diagram with different insertions of a four-quark operator Qi . The upper lines go into the emittedmeson M2, the quark to the right of the vertex and the spectator antiquark in the B meson (not drawn) form the recoilmeson M1.

different calculations. Instead of performing both calculations explicitly, we may alternativelycompute the amplitude 2 by inserting Fierz reordered operators into the left diagram of Fig. 1.To do so, it is essential to work with an operator basis that respects Fierz symmetry in d dimen-sions. As we have argued above, Fierz symmetry indeed holds to 1-loop order in the traditionalbasis from (3) which allows us to derive 2 directly from 1.

We conclude that the CMM basis is the appropriate choice for a 2-loop calculation whereasthe traditional basis provides a short-cut for the derivation of the colour-suppressed amplitude.We therefore pursue the following strategy for the calculation of the imaginary part of the NNLOvertex corrections: We perform the explicit 2-loop calculation in the CMM basis using the firsttype of operator insertion in the left diagram from Fig. 1. From this we obtain 1(Ci). We thentransform this expression into the traditional basis which yields 1(Ci) and finally apply Fierzsymmetry arguments to derive 2(Ci) from 1(Ci) by simply exchanging C1 C2.

3. 2-loop calculation

The core of the considered calculation consists in the computation of the matrix elements

(6)Q1,2 u(p)d(q1)u(q2)

Q1,2b(p)to O(2s ) which represents a 2-loop calculation. As will be described in Section 4.2, only(naively) non-factorizable diagrams with at least one gluon connecting the two currents in theleft diagram of Fig. 1 have to be considered here. The full NNLO calculation thus involves the2-loop diagrams depicted in Fig. 2, but only about half of the diagrams give rise to an imaginarypart. It is an easy task to identify this subset of diagrams since the generation of an imaginarypart is always related to final state interactions.

In our calculation we treat the partons on-shell and write q1 = uq , q2 = uq and p = p qsatisfying q2 = 0 and p2 = 2p q = m2b (with u 1 u). We use DR for the regulariza-tion of ultraviolet (UV) and infrared (IR) divergences and an anticommuting 5 accordingto the NDR scheme. We stress that we do not perform any projection onto the bound statesin the partonic calculation. We instead treat the two currents in the four-quark operators in-dependently and make use of the equations of motion in order to simplify the Dirac struc-tures of the diagrams. In order to calculate the large number of 2-loop integrals we proceedas follows: Using a general tensor decomposition of the loop integrals, we essentially dealwith the calculation of scalar integrals. With the help of an automatized reduction algorithm,we are able to express several thousands of scalar integrals in terms of a small set of so-

called master integrals (MIs). The most difficult part finally consists in the calculation of theseMIs.


Fig. 2. Full set of (naively) non-factorizable 2-loop diagrams. The bubble in the last four diagrams represents the 1-loopgluon self-energy. Only diagrams with final state interactions, i.e., with at least one gluon connecting the line to the rightof the vertex with one of the upper lines, give rise to an imaginary part.

In the remainder of this section we present some techniques that we have found useful for theconsidered calculation. We sketch the basic ideas of the aforementioned reduction algorithm and

discuss several techniques for the calculation of the MIs. We refer to the references quoted in thefollowing subsections for more detailed descriptions (see also [7]).


3.1. Reduction to master integrals

Any scalar 2-loop integral in our calculation can be expressed as

(7)I (u) =

ddk dd lSm11 SmssPn11 P

npp

,

where the Si are scalar products of a loop momentum with an external momentum or of two loopmomenta. The Pi denote the denominators of propagators and the exponents fulfil ni,mi 0.The scalar integrals themselves depend on the convolution variable u in the factorization for-mula (1). Very few integrals, arising from diagrams with a charm quark in a closed fermionloop, depend in addition on the ratio z mc/mb . We have suppressed this dependence in (7) forsimplicity.

Notice that an integral has different representations in terms of {S,P, n,m} because of thefreedom to shift loop momenta in DR. It is the underlying topology, i.e., the interconnectionof propagators and external momenta, which uniquely defines the integral. In the following weloosely use the word topology in order to classify the integrals. An integral with t differentpropagators Pi with ni > 0 is called a t -topology. The integrals in the considered calculationhave topology t 6.

The reduction algorithm makes use of various identities which relate integrals with differentexponents {n,m}. The most important class of identities are the integration-by-parts identi-ties [11] which follow from the fact that surface terms vanish in DR

(8)

ddk ddl

v

Sm11 SmssPn11 P

npp

= 0, v {k, l}.

In order to obtain scalar identities we may contract (8) with any loop or external momentumunder the integral before performing the derivative. In our case of two loop and two externalmomenta we generate in this way eight identities from each integral.

A second class of identities, called Lorentz-invariance identities [12], exploits the fact thatthe integrals in (7) transform as scalars under a Lorentz-transformation of the external momenta.In this way we may generate up to six identities from each integral depending on the number ofexternal momenta. In our example with only two linearly independent external momenta p and qthere is only one such identity given by

(9)

ddk ddl

[p q

(p

p q

q

)+ q2p

q p2q

p

]Sm11 SmssPn11 P

npp

= 0.

In total we obtain nine identities from a given integral, each of the identities containing theintegral itself, simpler integrals with smaller {n,m} and more complicated integrals with larger{n,m}. It is important that the number of identities grows faster than the number of unknownintegrals for increasing {n,m}. Hence, for large enough {n,m} the system of equations becomes(apparently) overconstrained and can be used to express more complicated integrals in terms ofsimpler ones. Not all of the identities being linearly independent, some integrals turn out to beirreducible to which we refer as MIs.

In the considered calculation we typically deal with systems of equations made of severalthousands equations. The solution being straightforward, the runtime of the reduction algorithm

depends strongly on the order in which the equations are solved. As a guideline for an efficientimplementation we have followed the algorithm from [13].


Fig. 3. Scalar master integrals that appear in our calculation. We use dashed lines for massless propagators and double(wavy) lines for the ones with mass mb (mc). Dashed/solid/double external lines correspond to virtualities 0/um2b /m2b ,respectively. Dotted propagators are taken to be squared.

The reduction algorithm enables us to express the diagrams of Fig. 2 as linear combina-tions of MIs which are multiplied by some Dirac structures. As the coefficients in these linearcombinations are real, we may extract the imaginary part of a diagram at the level of theMIs which is a much simpler task than for the full diagrams. As depicted in Fig. 3, we find14 MIs that contribute to the calculation of the imaginary part of the NNLO vertex correc-tions.

3.2. Calculation of master integrals

Some MIs in Fig. 3 can be solved easily, e.g., with the help of Feynman parameters. Forthe more complicated MIs the method of differential equations [14] in combination with theformalism of Harmonic Polylogarithms (HPLs) [15] turned out to be very useful. In this sectionwe give brief reviews of these techniques and conclude with a comment on the calculation of theboundary conditions to the differential equations.

The analytical results for the MIs from Fig. 3 can be found in Appendix A.1 of [7]. As anindependent check of our results we evaluated the MIs numerically using the method of sectordecomposition [16].

3.2.1. Method of differential equationsThe MIs are functions of the physical scales of the process which are given by scalar productsof the external momenta and masses of the particles. In our calculation the MIs depend on thedimensionless variable u as in (7).


For a given MI we perform the derivative with respect to u and interchange the order ofintegration and derivation

(10)u

MIi (u) =

ddk dd l

u

Sm11 SmssPn11 P

npp

.

The right-hand side being of the same type as Eqs. (8) and (9), this procedure again leads to a sumof various integrals with different exponents {n,m}. With the help of the reduction algorithm,these integrals can be expressed in terms of MIs which yields a differential equation of the form

(11)u

MIi (u) = a(u;d)MIi (u)+j =i

bj (u;d)MIj (u),

where we indicated that the coefficients a and bj may depend on the dimension d . The inhomo-geneity of the differential equation typically contains MIs of subtopologies which are supposedto be known in a bottomup approach. In case of the MIs from the third line of Fig. 3, one MIin the inhomogeneous part is of the same topology as the MI on the left-hand side of (11) andthus unknown. Writing down the differential equation for this MI, we find that we are left with acoupled system of linear, first order differential equations.

We are looking for a solution of the differential equation in terms of an expansion

(12)MIi (u) =j

cij (u)

j.

Expanding (11) then gives much simpler differential equations for the coefficients cij which canbe solved order by order in . In addition, in the case where we were left with a coupled systemof differential equations, the system turns out to decouple in the expansion. The solution of thehomogeneous equations is in general straightforward. The inhomogeneous equations can thenbe addressed with the method of the variation of the constant. This in turn leads to indefiniteintegrals over the inhomogeneities which typically contain products of rational functions withlogarithms or related functions as dilogarithms. With the help of the formalism of HPLs theseintegrations simplify substantially.

3.2.2. Harmonic polylogarithmsThe formalism of HPLs [15] allows to rewrite the integrations mentioned at the end of the last

section in terms of familiar transcendental functions which are defined by repeated integrationover a set of basic functions. We briefly summarize their basic features here, focussing on theproperties that are relevant for our calculation.

The HPLs, denoted by H( mw;x), are described by a w-dimensional vector mw of parametersand by its argument x. We restrict our attention to the parameters 0 and 1 in the following. Thebasic definitions of the HPLs are for weight w = 1

(13)H(0;x) lnx, H(1;x) ln(1 x)and for weight w > 1

(14)H(a, mw1;x) xdx f (a;x)H( mw1;x),0


where the basic functions f (a;x) are given by(15)f (0;x) =

xH(0;x) = 1

x, f (1;x) =

xH(1;x) = 1

1 x .

In the case of mw = 0, the definition in (14) does not apply and the HPLs read(16)H(0, . . . ,0;x) 1

w! lnw x.

The HPLs form a closed and linearly independent set under integrations over the basic functionsf (a;x) and fulfil an algebra such that a product of two HPLs of weight w1 and w2 gives a linearcombination of HPLs of weight w = w1 +w2.

As described above, the solution of the differential equations leads to integrals over productsof some rational functions with some transcendental functions as logarithms or dilogarithms.More precisely, we find, e.g., integrals of the type

(17)xdx

{1

1 x ,1x2

,1}H( mw;x).

It turns out that all these integrals can be expressed as linear combinations of HPLs of weightw+1. This is obvious for the first integral as it simply corresponds to the definition of an HPL, cf.(14) with a = 1. For the other integrals in (17), an integration-by-parts leads either to a recurrencerelation or again to integrals of the type (14). Not all integrals in our calculation fall into thesimple pattern (17), but a large part of this calculation can be performed along these lines.

In the considered calculation we encounter HPLs of weight w 3. Our results can be ex-pressed in terms of the following minimal set of HPLs

H(0;x) = lnx, H(1;x) = ln(1 x),(18)H(0,1;x) = Li2(x), H(0,0,1;x) = Li3(x), H(0,1,1;x) = S1,2(x).

The situation is more complicated for the last two MIs in the third line of Fig. 3 where the internalcharm quark introduces a new scale. However, a closer look reveals that these MIs depend ontwo physical scales only, namely um2b and m2c = z2m2b . The MIs can then be solved within theformalism of HPLs in terms of the ratio z2/u if we allow for more complicated argumentsof the HPLs as, e.g., 12 (1

1 + 4 ).

3.2.3. Boundary conditionsA unique solution of a differential equation requires the knowledge of its boundary conditions.

In the considered calculation the boundary conditions typically represent single-scale integralscorresponding to u = 0 or 1. It is of crucial importance that the integral has a smooth limit at thechosen point such that setting u = 0 or u = 1 does not modify the divergence structure introducedin (12).

In some cases the methods described so far can also be applied for the calculation of theboundary conditions since setting u = 0 or 1 leads to simpler topologies which may turn out tobe reducible. If so, the reduction algorithm can be used to express the integral in terms of knownMIs. If not, a different strategy is mandatory. In this case we tried to calculate the integral withthe help of Feynman parameters and managed in some cases to express the integral in termsof hypergeometric functions which we could expand in with the help of the MATHEMATICA

package HYPEXP [17]. Finally, the most difficult single-scale integrals could be calculated withMellinBarnes techniques [18].


4. Renormalization and IR subtractions

The matrix elements Qi which we obtained from computing the 2-loop diagrams in Fig. 2are ultraviolet (UV) and infrared (IR) divergent. In this section we show how to extract the finitehard-scattering kernels T Ii from these matrix elements.

4.1. Renormalization

The renormalization procedure involves standard QCD counterterms, which amount to thecalculation of various 1-loop diagrams, as well as counterterms from the effective Hamiltonian.We write the renormalized matrix elements as

(19)Qiren = ZZij Qj bare,where Z Z1/2b Z3/2q contains the wave-function renormalization factors Zb of the b-quark andZq of the massless quarks, whereas Z is the operator renormalization matrix in the effectivetheory. We introduce the following notation for the perturbative expansions of these quantities

Qiren/bare =k=0

(s

4

)kQi(k)ren/bare,

(20)Z = 1 +k=1

(s

4

)kZ

(k) , Zij = ij +

k=1

(s

4

)kZ

(k)ij

and rewrite (19) in perturbation theory up to NNLO which yieldsQi(0)ren = Qi(0)bare,Qi(1)ren = Qi(1)bare +

[Z

(1)ij +Z(1) ij

]Qj (0)bare,Qi(2)ren = Qi(2)bare +

[Z

(1)ij +Z(1) ij

]Qj (1)bare(21)+ [Z(2)ij +Z(1) Z(1)ij +Z(2) ij ]Qj (0)bare.

The full calculation thus requires the operator renormalization matrices Z(1,2). For the calculationof the imaginary part, the terms proportional to the tree level matrix elements do not contributeand Z(2) drops out in (21) as expected for an effective 1-loop calculation.

Mass and wave function renormalization are found to be higher order effects. For the renor-malization of the coupling constant we use

(22)Zg = 1 s4(

112

13nf

)+O(2s )

according to the MS-scheme. The expression for the 1-loop renormalization matrix Z(1) can befound, e.g., in [19] and reads

(23)Z(1) =(2 43 512 29

6 0 1 0

)1,

where the two lines correspond to the basis of physical operators {Q1,Q2} and the four columnsto the extended basis {Q1, Q2, E1, E2} including the evanescent operators E1,2 defined in (4).


4.2. Factorization in NNLO

In this section it will be convenient to introduce the following short-hand notation for thefactorization formula (1)

(24)Qiren = F Ti + ,where F denotes the B M1 form factor, Ti the hard-scattering kernels T Ii and the productof the decay constant fM2 and the distribution amplitude M2 . The convolution in (1) has beenrepresented by the symbol and the ellipsis contain the terms from spectator scattering whichwe disregard in the following.

Formally, we may introduce the perturbative expansions

(25)F =k=0

(s

4

)kF (k), Ti =

k=0

(s

4

)kT

(k)i , =

k=0

(s

4

)k(k).

Up to NNLO the expansion of (24) then yields

Qi(0)ren = F (0) T (0)i (0),Qi(1)ren = F (0) T (1)i (0) + F (1) T (0)i (0) + F (0) T (0)i (1),Qi(2)ren = F (0) T (2)i (0) + F (1) T (1)i (0) + F (0) T (1)i (1)

(26)+ F (2) T (0)i (0) + F (1) T (0)i (1) + F (0) T (0)i (2).In LO the comparison of (21) and (26) gives the trivial relation

(27)Qi(0) Qi(0)bare = F (0) T (0)i (0)

which states that the LO kernels T (0)i can be computed from the tree level diagram in Fig. 4(a). Inorder to address higher order terms we split the matrix elements into its (naively) factorizable (f)and non-factorizable (nf) contributions

(28)Qi(k)bare Qi(k)f + Qi(k)nf .The corresponding 1-loop diagrams are shown in Figs. 4(b) and 4(c) respectively. To this order(21) and (26) lead to

Qi(1)f + Qi(1)nf +[Z

(1)ij +Z(1) ij

]Qj (0)(29)= F (0) T (1)i (0) + F (1) T (0)i (0) + F (0) T (0)i (1),

which splits into

(30)Qi(1)nf + Z(1)ij Qj (0) = F (0) T (1)i (0)

for the calculation of the NLO kernels T (1)i and

(31)Qi(1)f +Z(1) Qi(0) = F (1) T (0)i (0) + F (0) T (0)i (1),

which shows that the factorizable diagrams and the wave-function renormalization are absorbedby the form factor and wave function corrections F (1) and (1).


Fig. 4. Tree level diagram (a), naively factorizable (b) and non-factorizable (c) NLO diagrams.

This suggests in NNLO the following structure

Qi(2)f +Z(1) Qi(1)f +Z(2) Qi(0)(32)= F (2) T (0)i (0) + F (1) T (0)i (1) + F (0) T (0)i (2).

These terms are thus irrelevant for the calculation of the NNLO kernels T (2)i which justifies thatwe could restrict our attention to the non-factorizable 2-loop diagrams from Fig. 2. In NNLO theremaining terms from (21) and (26) contain non-trivial IR subtractions

Qi(2)nf +Z(1) Qi(1)nf + Z(1)ij[Qj (1)nf + Qj (1)f ]+ [Z(2)ij +Z(1) Z(1)ij ]Qj (0)

(33)= F (0) T (2)i (0) + F (1) T (1)i (0) + F (0) T (1)i (1).This equation can be simplified further when we make the wave function renormalization factorsin the form factor and the distribution amplitude explicit

(34)F = Z1/2b Z1/2q Famp, = Zqamp.Notice that the resulting amputated form factor Famp and wave function amp contain UV-divergences by construction. Using (30), we see that the wave function renormalization factorscancel and arrive at the final formula for the calculation of the NNLO kernels T (2)i

Qi(2)nf + Z(1)ij[Qj (1)nf + Qj (1)f ]+ Z(2)ij Qj (0)

(35)= F (0) T (2)i (0) + F (1)amp T (1)i (0) + F (0) T (1)i (1)amp.As the tree level matrix elements and the factorizable 1-loop diagrams do not give rise to animaginary part, these terms can be disregarded in the present calculation.

4.3. IR subtractions

We now consider the IR subtractions on the right-hand side of (35) in some detail. Let us firstaddress the NLO kernels T (1)i which can be determined from Eq. (30). The renormalization inthe evanescent sector implies that the left-hand side of (30) is free of contributions from evanes-cent operators up to the finite order 0. However, as the NLO kernels enter (35) in combinationwith the form factor correction F (1)amp which contains double (soft and collinear) IR divergences,

2the NLO kernels are required here up to O( ). Concerning the subleading terms of O(), theevanescent operators do not drop out on the left-hand side of (30) and we therefore have to extend


the factorization formula on the right-hand side to include these evanescent structures as well.Schematically,

(36)Qi(1)nf + Z(1)ij Qj (0) = F (0) T (1)i (0) + F (0)E T (1)i,E (0)Ewith a kernel T (1)i,E =O(1) and an evanescent tree level matrix element F (0)E (0)E =O().1 Simi-larly, the right-hand side of (35) has to be modified to include these evanescent structures

Qi(2)nf + Z(1)ij[Qj (1)nf + Qj (1)f ]+ Z(2)ij Qj (0)

= F (0) T (2)i (0) + F (1)amp T (1)i (0) + F (0) T (1)i (1)amp(37)+F (0)E T (2)i,E (0)E + F (1)amp,E T (1)i,E (0)E + F (0)E T (1)i,E (1)amp,E.

Notice that the term with the kernel T (2)i,E =O(1) is not required to extract the finite piece of thephysical NNLO kernel T (2)i .

From the calculation of the 1-loop diagrams in Fig. 4(c), we find that the NLO kernels vanishin the colour-singlet case, T (1)2 = T (1)2,E = 0, whereas the imaginary part of the colour-octet kernelsis given by

1

ImT (1)1 (u) =CF

2Nc

{(3 2 lnu+ 2 ln u)

(1 + L+ 1

22L2

) (11 3 ln u ln2 u+ ln2 u)( + 2L)+[

32

4 26 +

(2 +

2

2

)lnu+

(9

2

2

)ln u

32

ln2 u 13(ln3 u ln3 u)]2 +O(3)},

(38)1

ImT (1)1,E(u) = CF

4Nc

{1 + L+

(83

12

lnu 12

ln u) +O(2)},

where L ln2/m2b and we recall that u 1 u.

4.3.1. Form factor subtractionsWe now address the form factor corrections which require the calculation of the diagram in

Fig. 5 (for on-shell quarks) and its counterterm. According to the definition of Famp in (34), wedo not have to consider the wave function renormalization of the quark fields.

We again have to compute the corrections for physical and evanescent operators. Concerningthe physical operator, the counterterm is found to vanish which reflects the conservation of thevector current. The evaluation of the 1-loop diagram in Fig. 5 gives

(39)F (1)amp(0) = CF(eE2

m2b

) ()

1 + 22(1 2) F

(0)(0),

1 In the notation of [2], the right-hand side of (36) corresponds to matrix elements of SCETI operators of the form[(Wc1)1hv][(Wc2)2(Wc2)]. In NNLO we match onto two SCETI operators with Dirac-structures 1 2 given

/n /n 1 1by O1 = /n+L 2 L and O2 = /n+L 2 L (in our notation p = 2mbn and q = 2mbn+). Thematrix element of O1 defines our structure F(0)(0) and the evanescent combination 3O212O1 defines F

(0)E

(0)E

.


Fig. 5. 1-loop contribution to the form factor correction F(1)amp.

Fig. 6. 1-loop contributions to the wave function correction (1)amp. The dashed line indicates the Wilson-line connectingthe quark and antiquark fields.

which contains double IR singularities as mentioned at the beginning of this section. On theother hand, the 1-loop diagram with an insertion of the evanescent operator yields a contributionproportional to the evanescent and the physical operators. We now have to adjust the countertermsuch that the renormalized (IR-finite) matrix element of the evanescent operator vanishes (whichensures that the evanescent structures disappear in the final factorization formula). We obtain

F(1)amp,E

(0)E = CF

[(eE2

m2b

) ()

24(1 + )(1 )2 24

]F (0)(0)

(40)CF(eE2

m2b

) ()

1 3 + 2 + 33 + 24(1 2)(1 )2 F

(0)E

(0)E .

The form factor subtractions in (37) then follow from combining (38), (39) and (40). We em-phasize that the corrections related to the evanescent operator do not induce a contribution to thephysical NNLO kernel T (2)1 in this case since

(41)1F

(1)amp,E ImT

(1)1,E

(0)E

[O()]F (0)(0).4.3.2. Wave function subtractions

Concerning the wave function corrections we are left with the calculation of the diagramsin Fig. 6 for collinear and on-shell partons with momenta uq and uq . However, as in our set-up q2 = 0 all these diagrams vanish due to scaleless integrals in DR. We conclude that the wavefunction corrections are determined entirely by the counterterms. We compute these countertermsby calculating the diagrams from Fig. 6 with an (IR-finite) off-shell regularization prescription inorder to isolate the UV-divergences (which are independent of the IR regulator). The countertermfor the physical operator is found to be

(42)F (0)(1)amp(u) = 2CF

1dwV (u,w)F (0)(0)(w)0


with the familiar EfremovRadyushkinBrodskyLepage (ERBL) kernel [20]

(43)V (u,w) =[(w u) u

w

(1 + 1

w u)

+ (uw) uw

(1 + 1

w u)]

+,

where the plus-distribution is defined as [f (u,w)]+ = f (u,w) (u w) 1

0 dv f (v,w). Forthe evanescent operator we obtain

F(0)E

(1)amp,E(u)

(44)= 2CF

10

dw[24VE(u,w)F (0)(0)(w)+ V (u,w)F (0)E (0)E (w)

],

where VE(u,w) denotes the spin-dependent part of the ERBL kernel given by

(45)VE(u,w) = (w u) uw

+ (uw) uw

.

Notice that the evanescent operators do induce a finite contribution to the physical NNLO ker-nel T (2)1 in this case as the convolution with the corresponding NLO kernel implies

(46)1F

(0)E ImT

(1)1,E

(1)amp,E

[6C2FNc

+O()]F (0)(0).

We finally quote the result for the convolution with the physical NLO kernel1F (0) ImT (1)1

(1)amp

= C2F

Nc

{[2

3+ lnu

u ln u

u+ ln2 u 2 lnu ln u ln2 u 4 Li2(u)

](1

+L)

+ 2

2 15

4 23 + 5u 42

(lnuu

+ ln uu

)

2

3ln u

3 Li2(u) 12u ln2 u+ 1 3u

2uln2 u 2

3ln3 u+ ln2 u ln u

(47)+ 23

ln3 u+ 2 ln uLi2(u)+ 2 Li3(u)+ 2 S1,2(u)+O()}F (0)(0).

5. Vertex corrections in NNLO

We now have assembled all pieces required for the calculation of the NNLO kernels T (2)ifrom (37). We indeed observe that all UV and IR divergences cancel in the hard-scattering ker-nels as predicted by the QCD factorization framework. Since this is the result of a complicatedsubtraction procedure, this can also be seen as a very stringent cross-check of our calculation.

5.1. 1 in CMM basis

The procedure outlined so far leads to the colour-allowed tree amplitude in the CMM operatorbasis defined in (4). We write [ ](48)1(M1M2) = C2 + s4CF

2NcC1V

(1) + s4

(C1V

(2)1 + C2V (2)2

)+O(2s ) + ,


where the ellipsis denote the terms from spectator scattering which are irrelevant for our pur-poses. In the CMM basis, the imaginary part of the vertex corrections V (1,2) takes the form

1

Im V (1) 1

0

dug1(u)M2(u),

1

Im V (2)1 1

0

du

{[(293

CA 23nf)g1(u)+CFh1(u)

]ln

2

m2b

+CFh2(u)+CAh3(u)+ (nf 2)h4(u;0)+ h4(u; z)+ h4(u;1)}M2(u),

(49)1

Im V (2)2 1

0

du

{6g1(u) ln

2

m2b+ h0(u)

}M2(u).

In writing (49) we have made the dependence on the renormalization scale explicit and disen-tangled contributions that belong to different colour structures. The NLO kernel g1(u) is givenby

(50)g1(u) = 3 2 lnu+ 2 ln uand the NNLO kernels hi(u) will be specified below. The kernel h4(u; zf ) stems from diagramswith a closed fermion loop and depends on the mass of the internal quark through zf = mf /mb .We keep a non-zero charm quark mass and write z zc = mc/mb for simplicity.

The NNLO kernels were so far unknown. They are found in this work to be

h0(u) =[

1554

+ 43 + 4 Li3(u) 4 S1,2(u) 12 lnuLi2(u)+ 43 ln3 u 6 ln2 u ln u

+ 2 u2

uLi2(u) 5 3u+ 3u

2 2u32u

ln2 u+ 3 2u4

2uulnu ln u

(

4 11u+ 2u2u

+ 42

3

)lnu (5 + 6u

2 12u4)224uu

+ (u u)]

+[

3 u+ 7u22u

ln2 u 11 10u2

4uuLi2(u)+ 1 14u

2

4ulnu ln u

+ 46 51uu

lnu (41 42u2)2

24u (u u)

],

h1(u) = 36 +[

2 ln2 u 4 Li2(u)+ 2(13 12u)u

lnu (u u)],

h2(u) =[

79 + 323 16 Li3(u) 32 S1,2(u)+ 83 ln3 u+ 2(4 u

2)

uLi2(u)

13 9u+ 6u2 4u3

2uln2 u+ 17 6u

2 8u44uu

lnu ln u( 2 2) 2 4 2 ]

2 5 11u+ 2u

u+ 4

3lnu (1 + 14u 8u )

8uu+ (u u)


+[

4 Li3(u)+ 4 S1,2(u) 23 ln3 u+ 2 ln2 u ln u 9 14u

2

uuLi2(u)

+ 13 11u+ 14u2

2uln2 u+ 5 7u

2

ulnu ln u

+ 4(

24 23uu

+ 2

3

)lnu (26 21u

2)2

6u (u u)

],

h3(u) =[1379

24 123 + 6 Li3(u)+ 12 S1,2(u) ln3 u 4 u

2

uLi2(u)

+ 9 2u+ 6u2 4u3

4uln2 u 7 + 4u

2 4u44uu

lnu ln u

+(

41 66u+ 8u24u

+ 2)

lnu+ (1 + 6u2 4u4)28uu

+ (u u)]

+[2 Li3(u)+ 4 S1,2(u)+ 4 lnuLi2(u)+ 13 ln

3 u+ 15 26u2

4uuLi2(u)

+ 11 14u 42u2

12uln2 u 11 14u

2

4ulnu ln u

(

2165 2156u36u

2

3

)lnu+ (53 42u

2)2

24u (u u)

],

h4(u; z) =[

176

+ 7u

22 ln2 x1x2

+ u

ln z2 (1 + 2) lnu

+ (2(1 + 4)x1 + 4x2) lnx1 (4x1 + 2(1 + 4)x2) lnx2 + (u u)]

+[

94z2

9u 2(1 3

2)

3ln2

x1x2

43

lnu ln z2 + (1 2u)(6u u2)

9uuln z2

+ 12 + 29 + 22

9lnu 2

9((1 + 4)(6 + 5)x1 6

(1 32)x2) lnx1

(51) 29(6(1 32)x1 (1 + 4)(6 + 5)x2) lnx2 (u u)

],

where the last kernel has been given in terms of

(52)x1 12 (

1 + 4 1), x2 12 (

1 + 4 + 1), z2

u.

In the massless limit h4(u; z) becomes

(53)h4(u;0) = 173 23

ln2 u+ 23

ln2 u+ 209

lnu 389

ln u.

5.2. 1 and 2 in traditional basisFollowing our strategy from Section 2, we compute the colour-suppressed amplitude 2 byrewriting the colour-allowed amplitude 1 in the traditional operator basis from (3). Manifest


Fierz symmetry in this basis relates the two amplitudes via

i(M1M2) = Ci + Ci1Nc

(54)+ s4

CF

Nc

[Ci1V (1) + s4

(Ci V

(2)1 + Ci1V (2)2

)+O(2s )]

+ ,

where the upper (lower) signs apply for i = 1 (i = 2) and the ellipsis correspond to the termsfrom spectator scattering. In order to derive V (1,2) we have to transform the Wilson coefficientsin the CMM basis Ci into the ones of the traditional basis Ci . To NLL approximation this trans-formation can be found, e.g., in [9] and reads

C1 = 2C2 + s4(

4C1 + 143 C2)

+O(2s ),(55)C2 = C1 + 13 C2 +

s

4

(169

C2

)+O(2s ).

Combining (48), (54) and (55) we obtain

1

Im V (1) = 1

Im V (1) =1

0

dug1(u)M2(u),

1

Im V (2)1 =1

Im[

12V

(2)2 + 2V (1)

]

=1

0

du

{3g1(u) ln

2

m2b+ 2g1(u)+ 12h0(u)

}M2(u),

1

Im V (2)2 =1

Im[V

(2)1 +

(CA

2CF

)V

(2)2 + (4CF CA)V (1)

]

=1

0

du

{[(203

CA + 6CF 23nf)g1(u)+CFh1(u)

]ln

2

m2b

+CF[h2(u) h0(u)+ 4g1(u)

]+CA[h3(u)+ 12h0(u) g1(u)

]

(56)+ (nf 2)h4(u;0)+ h4(u; z)+ h4(u;1)}M2(u).

Eq. (56) represents the central result of our analysis, specifying the imaginary part of the colour-allowed tree amplitude 1 and the colour-suppressed tree amplitude 2 according to (54). Theexpression for V (1) is in agreement with [1], whereas the expressions for V (2)1,2 are new. Thekernels g1(u) and hi(u) can be found in (50) and (51), respectively. The terms proportional

to nf have already been considered in the analysis of the large 0-limit in [21]. Our results arein agreement with these findings.


5.3. Convolutions in Gegenbauer expansion

Our results in (56) have been given in terms of convolutions of hard-scattering kernels withthe light-cone distribution amplitude of the emitted meson M2. We may explicitly calculate theseconvolution integrals by expanding the distribution amplitude into the eigenfunctions of the 1-loop evolution kernel

(57)M2(u) = 6uu[

1 +n=1

aM2n C(3/2)n (2u 1)

],

where aM2n and C(3/2)n are the Gegenbauer moments and polynomials, respectively. We truncatethe Gegenbauer expansion at n = 2 and perform the convolution integrals analytically. We find

10

dug1(u)M2(u) = 3 3aM21 ,

10

duh0(u)M2(u) =133312

+ 472

45 163 +

(154

+ 232

5

)aM21

(

17330

+ 182

35

)aM22 ,

10

duh1(u)M2(u) = 36 + 28aM21 ,

10

duh2(u)M2(u) =1369

6+ 139

2

45 323

(176

512

5

)aM21

(

10315

+ 712

35

)aM22 ,

10

duh3(u)M2(u) = 481

3+ 7

2

30+ 123

(64312

+ 112

10

)aM21

(

153180

1692

70

)aM22 ,

H4(z) 1

0

duh4(u; z)M2(u)

= 223

+ 148z2 96z4F(z) 36z4 ln2 y1y2

2[1 (2y1 + 1)(1 + 22z2)] ln y1y2

4 lny2{

+ 7 + 164z2 + 180z4 + 144z6 288z4F(z)+ 12z4(3 + 16z2 + 12z4) ln2 y1

y2


2[1 (2y1 + 1)(1 + 22z2 + 84z4 + 72z6)] ln y1y2

4 lny2}aM21

+{

35

+ 244z2 + 1483

z4 640z6 960z8 + 24z4(1 30z4 40z6) ln2 y1y2

(58) 576z4F(z)+ 8z2(2y1 + 1)(6 + 11z2 70z4 120z6) ln y1

y2

}aM22 ,

where we defined

y1 12(

1 + 4z2 1), y2 12(

1 + 4z2 + 1),(59)F(z) Li3(y1) S1,2(y1) lny1 Li2(y1)+ 12 lny1 ln

2 y2 112 ln3 z2 + 3.

In the massless limit the function H4(zf ) simplifies to

(60)H4(0) = 223 + 7aM21 +

35aM22 .

The finiteness of the convolution integrals in (58) completes the explicit factorization proof ofthe imaginary part of the NNLO vertex corrections.

We summarize our results for the vertex corrections in the Gegenbauer representation of thelight-cone distribution amplitude of the meson M2 (with CF = 43 , CA = 3, nf = 5)

1

Im V (1) = 3 3aM21 ,1

Im V (2)1 =(9 + 9aM21

)ln

2

m2b+ 1189

24+ 47

2

90 83

(

338

232

10

)aM21

(17360

+ 92

35

)aM22 ,

1

Im V (2)2 = (

26 + 1103

aM21

)ln

2

m2b 10 315

72+ 674

2

135 28

33

(

10 79372

1662

15

)aM21

(315548

1872

42

)aM22

(61)+H4(z)+H4(1),with H4(zf ) given in (58). In order to illustrate the relative importance of the individual contri-butions, we set = mb and z = mc/mb = 0.3 which yields

Im V (1) = 9.425 9.43aM21 ,Im V (2)1 = 141.621 + 58.36aM21 17.03aM22 ,

(62)Im V (2)2 = 317.940 115.62aM21 68.31aM22 .We thus find large coefficients for the NNLO vertex corrections and expect only a minor impactof the higher Gegenbauer moments, in particular in the symmetric case with aM21 = 0. Notice thatall contributions add constructively in 1,2 due to the relative signs of the Wilson coefficients,C 1.1 and C 0.2. In the case of the contribution from V (2) is found to exceed the1 2 1 1formally leading contribution V (1) due to the fact that the latter is multiplied by the small Wilson


coefficient C2. Concerning 2 the NNLO vertex corrections are also substantial, roughly sayingthey amount to a 50% correction. In both cases the impact of the charm quark mass is small, wefind a correction of 3% compared to the massless case. A more detailed numerical analysisincluding the contributions from spectator scattering will be given in the following section.

We finally remark that the large 0-limit considered in [21] fails to reproduce the imaginarypart of 1 as it completely misses the leading contribution from V (2)1 . In the case of 2 theapproximation turns out to be reasonably good with a deviation of 10% compared to the fullNNLO result.

6. Numerical analysis

6.1. Implementation of spectator scattering

In the numerical analysis we combine our results with the NNLO corrections from 1-loopspectator scattering obtained in [26]. In contrast to the vertex corrections considered in thiswork, the spectator term receives contributions from the hard scale h mb and the hard-collinear scale hc (QCDmb)1/2. According to this, the kernels T IIi from (1) factorize intohard functions H IIi and a (real) hard-collinear jet-function J||. Evaluating both kernels at thesame scale would imply parametrically large logarithms which may spoil the convergence ofthe perturbative expansion.

In order to resum these logarithms we follow Ref. [2] and perform the substitution

Ci()TIIi () [fBB ]() M1() M2()

Ci(h)H IIi (h) U||(h,hc) J||(hc) [fBB ](hc)(63) M1(hc) M2(h),

where U|| = eSU|| consists of a universal Sudakov factor S and a non-local evolution kernel U||.As an imaginary part is first generated at O(2s ) in the spectator term, we implement the resum-mation in the leading-logarithmic (LL) approximation. In the traditional operator basis from (3)the respective imaginary part takes the form

Imi(M1M2)spec

= s(h)s(hc)CF4N2c

9fM1 fB(hc)mbF

BM1+ (0)B(hc)

n,m

aM1m (hc)aM2n (h)

(64) Im[Ci1(h)Rmn1 (h,hc)+ Ci(h)Rmn2 (h,hc)]+O(3s ),where we made the scale dependence of the parameters explicit and introduced the first inversemoment of the B meson light-cone distribution amplitude 1B . We further wrote 1 = aM0 () inorder to simplify the notation. In (64) the resummation is encoded in

Rmni (h,hc)

(65)= 1eS(h,hc)1du6uuC(3/2)n (2u 1)

1dzCm(z;h,hc)ri(u, z),9

0 0


with the Sudakov factor S given in LL approximation in Eq. (106) of [6] and the kernels ri inEqs. (38) and (39) of [2]. Following [4] we defined

(66)Cm(z;h,hc) =1

0

dv 6vC(3/2)m (2v 1)U||(v, z;h,hc),

which can be computed by solving numerically the integro-differential equation

(67)dd ln

Cm(z;,hc) = 1

0

dw ||(z, w)Cm(w;,hc)

with initial condition Cm(z;hc,hc) = 6zC(3/2)m (2z 1) and || from Eq. (99) of [6].In order to illustrate the numerical importance of the resummation we compare the values of

the imaginary part of Rmni for m,n 2 and h = hc (line I, without resummation) and h =4.8 GeV, hc = 1.5 GeV (line II, with resummation) in Table 1. We observe that the resummationleads to a suppression of the spectator term of 10% due to the universal Sudakov factor (eS 0.89 for our choice of input parameters). The resummation effects induced by U|| turn out to beof minor numerical importance.

According to (64) we must evolve the Gegenbauer moments of the mesons M1 and M2 to thehard-collinear and the hard scale, respectively. In LL approximation the Gegenbauer momentsdo not mix and the evolution reads

(68)aMi () =(s(0)

s()

)i/20aMi (0)

with anomalous dimensions 1 = 649 and 2 = 1009 .We are left with the evolution of the B meson parameters to the hard-collinear scale. We

convert the HQET decay constant fB into the physical one fB using the LL relation

(69)fB() =(

s()

s(mb)

)2/0fB.

The evolution of B is more complicated. The solution of the integro-differential equation, whichgoverns the LL evolution of the B meson light-cone distribution amplitude, can be found in [22].Here we adopt a model-description for the B meson distribution amplitude to generate the evo-lution of B . We take the model from [22] which has the correct asymptotic behaviour and isalmost form-invariant under the evolution.

Finally we implement the BBNS model from [1] in order to estimate the size of power cor-rections to the factorization formula (1). This results in an additional contribution from spectator

Table 1Numerical values of the imaginary part of Rmn

i(h,hc) from (65) for h = hc (line I, without resummation) and

h = 4.8 GeV, hc = 1.5 GeV (line II, with resummation)R00i

R01i

R02i

R10i

R11i

R12i

R20i

R21i

R22i

i = 1 (I) 11.0 23.2 29.4 14.1 23.8 30.8 15.1 23.8 31.3i = 1 (II) 9.88 20.9 26.5 12.5 21.5 27.8 13.4 21.5 28.2i = 2 (I) 5.29 8.43 8.24 6.58 11.0 11.3 7.04 12.0 12.6

i = 2 (II) 4.77 7.60 7.43 5.85 9.72 9.98 6.27 10.6 11.2


scattering related to subleading projections on the light-cone distribution amplitudes of the lightmesons. It is given by

(70)Imi(M1M2)power =

sCF

N2c

3fM1 fBmbF

BM1+ (0)BCi1rM1 M2 Im[XH ],

where rM () = 2m2M/mb()/(mq + mq )(), M = 1 +

n(1)naMn and XH parameterizesan endpoint-divergent convolution integral. The latter is written as

(71)XH =(1 + HeiH

)ln

mB

h

which may generate an imaginary part due to soft rescattering of the final state mesons. We takelnmB/h 2.3, H = 1 and allow for an arbitrary phase H .

6.2. Tree amplitudes in NNLO

The numerical implementation of the vertex corrections from (54) is easier since they canbe evaluated at the hard scale h mb and depend only on few parameters. As can be readoff from (61), the first Gegenbauer moment of the meson M2 enters the leading term V (1). Wetherefore require its next-to-leading logarithmic (NLL) evolution (which can be found in [23])but as we restrict our attention to B decays in the following the first moment does notcontribute at all. Since the second Gegenbauer moment does not enter V (1), it is only required inLL approximation given by (68).

Our input parameters for the B tree amplitudes are summarized in Table 2. The valuefor the B meson decay constant is supported by QCD sum rule calculations [24] and recent latticeresults [25]. The form factor FB+ (at large recoil) has been addressed in the light-cone sum rule(LCSR) approach [26]. As we implement the model from [22] for the B meson distributionamplitude, we take the respective value for B . This value is somewhat larger than the one usedin previous QCD factorization analyses [14], but it is supported by a QCD sum rule and anLCSR calculation [27]. The value for the second Gegenbauer moment of the pion can be inferredfrom an LCSR analysis [28] and lattice results [29].

In order to estimate the size of higher-order perturbative corrections we vary the hard scalein the range h = 4.8+4.82.4 GeV and the hard-collinear scale independently between hc =1.5+0.90.5 GeV. Throughout we use 2-loop running of s with nf = 5 (nf = 4) for quantities thatare evaluated at the hard scale h (hard-collinear scale hc). The quark masses are interpreted aspole masses except for those entering rM .

Table 2Theoretical input parameters (in units of GeV or dimensionless)Parameter Value Parameter Value

(5)MS

0.225 f 0.131mb 4.8 fB 0.21 0.02mc 1.6 0.2 FB+ (0) 0.25 0.05mb(mb) 4.2 B(1 GeV) 0.48 0.12

(mu + md )(2 GeV) 0.008 0.002 a2 (1 GeV) 0.25 0.2


With these input parameters the complete NNLO result for the imaginary part of the topolog-ical tree amplitudes is found to be

Im1() = 0.012|V (1) + 0.031|V (2) 0.012|S(2)= 0.031 0.015 (scale) 0.006 (param) 0.010 (power)= 0.031 0.019,

Im2() = 0.077|V (1) 0.052|V (2) + 0.020|S(2)= 0.109 0.023 (scale) 0.010 (param) 0.045 (power)

(72)= 0.109 0.052.In these expressions we disentangled the contributions from the NLO (1-loop) vertex correc-tions V (1), NNLO (2-loop) vertex corrections V (2) and NNLO (1-loop) spectator scattering S(2).In both cases the NNLO corrections are found to be important, although small in absolute terms.For the imaginary part of 1 the NNLO corrections exceed the formally leading NLO resultwhich can be explained by the fact that the latter is multiplied by the small Wilson coefficientC2, cf. the discussion after (62). We further observe that the individual NNLO corrections comewith opposite signs which leads to a partial cancellation in their sum. The phenomenologicallymost important consequence of our calculation may be the enhancement of the imaginary part ofthe colour-suppressed tree amplitude 2.

In our error estimate in (72) we distinguished between uncertainties which originate fromthe variation of the hard and the hard-collinear scale (scale), from the variation of the inputparameters in Table 2 (param) and from the BBNS model which we used to estimate the sizeof power-corrections (power). By now we expect the power corrections to be the main limitingfactor for an accurate determination of the amplitudes. However, although the inclusion of NNLOcorrections has reduced the dependence on the renormalization scales, it still remains sizeable(in particular for h). For our final error estimate in the third line of each amplitude, we addedall uncertainties in quadrature.

Finally we remark that our numerical values for the NNLO spectator terms are much smallerthan the ones quoted in [2]. This is partly related to the fact that we use different hadronicinput parameters, in particular a much larger value for B . In addition to this, the authors of [2]essentially evaluate the hard functions H IIi at the hard-collinear scale in order to partly implementthe (unknown) NLL resummation of parametrically large logarithms. The NLL approximation isindeed required for the real part of the amplitudes, but as long as we concentrate on the imaginarypart it is consistent to work in the LL approximation as discussed in Section 6.1. As the spectatorterm is the main source for the uncertainties from the hadronic input parameters, we also obtainsmaller error bars than [2].

7. Conclusion

We computed the imaginary part of the 2-loop vertex corrections to the topological tree ampli-tudes in charmless hadronic B decays. Together with the 1-loop spectator scattering contributionsconsidered in [26], the imaginary part of the tree amplitudes is now completely determined atNNLO in QCD factorization.

Among the technical issues we showed that soft and collinear infrared divergences cancel

in the hard-scattering kernels and that the resulting convolutions are finite, which demonstratesfactorization at the 2-loop order. In our numerical analysis we found that the NNLO corrections


are significant, in particular they enhance the strong phase of the colour-suppressed tree ampli-tude 2. Further improvements of the calculation still require a better understanding of powercorrections to the factorization formula.

Our calculation represents an important step towards an NNLO prediction of direct CP asym-metries in QCD factorization. As the topological penguin amplitudes, which also affect the directCP asymmetries, have not yet been computed completely at NNLO (the contribution from spec-tator scattering can be found in [4]), we refrained from discussing them already in this work.Moreover, the strategy outlined in this work may also be applied for the calculation of the realpart of the topological tree amplitudes, which is, however, technically more involved [7,30].

Acknowledgements

It is a pleasure to thank Gerhard Buchalla for his continuous help and guidance and for help-ful comments on the manuscript. I am also grateful to Volker Pilipp and Sebastian Jger forinteresting discussions. This work was supported in part by the GermanIsraeli Foundation forScientific Research and Development under Grant G-698-22.7/2001 and by the DFG Sonder-forschungsbereich/Transregio 9.

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Nuclear Physics B 795 (2008) 2751www.elsevier.com/locate/nuclphysb

The supermembrane with central charges:(2 + 1)D NCSYM, confinement and phase transition

L. Boulton a,, M.P. Garcia del Moral b,c,d, A. Restuccia ea Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University,

Edinburgh EH14 2AS, United Kingdomb Perimeter Institute for Theoretical Physics, Waterloo, Canada, Ontario N2L 2Y5, Canada

c Department of Physics and Astronomy, MacMaster University, 1280 Main Street West,Hamilton, Ontario, L8S 4M1, Canada

d DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdome Departamento de Fsica, Universidad Simn Bolvar, Apartado 89000, Caracas 1080-A, Venezuela

Received 27 July 2007; accepted 12 November 2007

Available online 21 November 2007

Abstract

The spectrum of the bosonic sector of the D = 11 supermembrane with central charges is shown to bediscrete and with finite multiplicities, hence containing a mass gap. The result extends to the exact theory ourprevious proof of the similar property for the SU(N) regularised model and strongly suggest discretenessof the spectrum for the complete Hamiltonian of the supermembrane with central charges. This theory isa quantum equivalent to a symplectic non-commutative super-YangMills in 2 + 1 dimensions, where thespace-like sector is a Riemann surface of positive genus. In this context, it is argued how the theory in 4Dexhibits confinement in the N = 1 supermembrane with central charges phase and how the theory enters inthe quarkgluon plasma phase through the spontaneous breaking of the centre. This phase is interpreted interms of the compactified supermembrane without central charges. 2007 Elsevier B.V. All rights reserved.

PACS: 04.65; 11.25

Keywords: Supermembrane with central charges; Non-commutative super-YangMills theory; Bosonic sector; Spectrum

* Corresponding author.E-mail addresses: [email protected] (L. Boulton), [email protected],

[email protected] (M.P. Garcia del Moral), [email protected] (A. Restuccia).0550-3213/$ see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysb.2007.11.011

28 L. Boulton et al. / Nuclear Physics B 795 (2008) 2751

1. Introduction

1.1. Motivation

An important step towards a better understanding of the non-perturbative approach to super-string theory is the non-perturbative treatment of D = 11 supermembranes [1]. The quantisationof the latter, when it is embedded on Minkowski spacetime, was studied in [25] in terms ofa quantum mechanical maximally supersymmetric SU(N) YangMills matrix models. This wasalso considered in a different context in [6]. In the seminal work [2], it was shown that thespectrum of the SU(N) regularised supersymmetric Hamiltonian is continuous, consisting of theinterval [0,). Remarkably, the spectrum of the corresponding bosonic Hamiltonian, equivalentto the dimensional reduction of D = 10 super-YangMills to 0 + 1 spacetime, is discrete [7,8].However its configuration space comprises singular string-like spikes, which, along with super-symmetry, render the spectrum continuous.

The validity of the SU(N) regularisation is justified by the fact that the structure constantsof the area-preserving diffeomorphisms, the gauge symmetry of the supermembrane in the lightcone gauge, are equal to the large N limit of the SU(N) structure constants. The characterisationof the spectrum was performed in the SU(N) regularised model, but we are not aware of anyresult concerning the large N limit of the spectrum.

The supermembrane was interpreted as a many-body object. In this interpretation, the string-like spikes may connect different membranes without changing the energy of the system, indistinction to the standard situation in string theory. For a review see [10]. The supermembraneembedded on a target space with a compact sector was analysed in [9]. Although an SU(N)regularisation was not obtained, it was argued that the same qualitative features of the spectrumremain valid.

The D = 11 supermembrane with non-trivial central charges was introduced in [11]. Theconfiguration space of this model is restricted by a topological condition. This restriction impliesthe existence of a non-trivial central charge on the SUSY algebra of the supermembrane. Froma geometrical point of view, the topological condition determines a non-trivial U(1) principalbundle over the world volume whose canonical connections, U(1) monopoles, define minimalimmersions into the compactified sector of the target space. These immersions describe the wrap-ping of the supermembrane on a calibrated submanifold of the target space. An interesting caseis M4 T d (S1)72g , where M4 represents a Minkowski spacetime in the light-cone gaugeand g = 1,2,3 is the genus of the 2-torus. See [16] for the case M7 T 4. The resulting theorypreserves an N = 1 supersymmetry in those dimensions.

The supermembrane with non-trivial central charges does not contain string-like spikes andit admits an SU(N) regularisation [12]. The bosonic potential of this model increases towardsinfinity as we move away from zero in the configuration space, ensuring a compact resolvent forthe bosonic Hamiltonian [13]. The spectrum of the regularised Hamiltonian is discrete, with finitemultiplicity [14] and its heat kernel can be defined rigorously by a process described in [15].

For a g 2 base manifold, this supermembrane admits an interpretation in terms of an or-thogonal intersection of a suitable number of genus 1 supermembranes with non-trivial centralcharge [16]. In the type IIA picture, the theory compactified on a T 2, i.e. g = 1, may be viewed asa bundle of D2D0 bound state theories where D0 monopole charges are induced by non-constantfluxes on the D2 [17]. Extensions to SU(N) interacting supermembranes may be considered asin [18].In the cases where a minimal immersion from the base manifold into the compact sectorof the target space can be established (the former is a Riemann surface of genus g), the topo-

L. Boulton et al. / Nuclear Physics B 795 (2008) 2751 29

logical restriction can be solved, and the supermembrane with non-trivial central charges isequivalent (as a quantum field theory) to a symplectic non-commutative super-YangMills theory(NCSYM) [19,20]. This is the case, for instance, when the base manifold is a genus g Riemannsurface and the compact sector of the target space is a flat T 2g torus. The symplectic structure isdetermined by the minimal immersion, and describes the curvature of U(1) monopole connec-tions.

The BFSS conjecture takes the D0 action as a fundamental action [31]. This is precisely thesame Hamiltonian than the SU(N) regularised Hamiltonian of the D = 11 supermembrane inthe light cone gauge [2]. Its extension to interesting compactified target spaces as the BMNmodel [32] includes ChernSimons and mass terms, preserving the PP wave supersymmetry,allowing the existence of stable vacuum solutions. In the same sense, the topological conditionon the D = 11 supermembrane with central charges give rise to mass terms which render thetheory stable. However, the topological properties of the base manifold plays a fundamental rolein the large N limit analysis of the regularised supermembrane. This geometrical structure ismissing in the matrix model, where no rigorous analysis of the large N limit has been obtained.

Non-commutative YangMills theories have been considered as toy models for gravity [21]and [22]. The relation between supergravity and non-commutative YangMills become naturalin the context of supermembranes, since they are embedded on a target space which must be asolution of D = 11 supergravity. Moreover the supermultiplet of D = 11 supergravity has beenconjectured to be the ground state of the theory.

In the context of string theory, NCSYM appears in a very natural manner by wrapping D-branes [23]. A super-YangMills theory on a non-commutative torus is naturally related tothe compactification of a matrix theory on a dual torus with a constant C3 field, see for ex-ample, [2427]. The non-commutative YangMills theory in a flat space with a rational non-commutative parameter is related to ordinary YangMills theories with magnetic flux throughMorita equivalence, [28]. By comparing ordinary YangMills theories and non-commutativeones, it was found in [29] that both theories share the same degrees of freedom in the IR limit.However these degrees are redistributed differently in both theories in the UV limit.

It was argued in [27,30] that it should be the non-commutative theory the better suited fordescribing the IR limit and the commutative theory the one that best recovers the UV limit.

YangMills theories with boundary modelling QCD where extensively studied under thecommon name of bag models [41]. Super-YangMills theories behave differently at extremalenergies since they are either in the confined or in the screening phase. For the bosonic theory,the confined phase occurs at low temperatures where vector-like gluons form singlet bound statesof colour called glueballs, see [33].

The low temperature regime is characterised by the dynamics of gluons. At high temperaturesthe gluons do no give rise to bound states but rather to a plasma that constitute the screeningphase. If the fermions are introduced in the theory, at low energies they feel a binding forceand at high energy they remain free [35]. These two extremal regimes are thought in generalto be separated by a phase transition occurring when a global symmetry of the theory breaks.This phase transition is related to the spontaneous breaking of the centre, [35,36]. Furthermore,according to [35,36], this symmetry is conjectured to be of a topological nature due to magneticmonopoles or instantons.

The role of the centre has been considered to a large extent, cf. e.g. [33,37,38]. According

to the results of [39], the instanton gas picture is only appropriate in those cases where thetopological charge is discrete. Otherwise the correct picture would be the monopole. It is this


latter case, where the confinement should appear, [39], as it is for D = 4 and N = 1 in thesupermembrane with central charges.

Connections between membrane theory and YangMills theories have been considered in adifferent context in [40]. In [42] the critical behaviour of a gauge theory in the de-confined phaseis related with the behaviour of a scalar which has a symmetry induced by the centre of thegroup. The phase transition occurs when the topological defect is metastable and decays as aconsequence of quantum processes.

1.2. Aims and scopes of the present paper

Our main goal is to demonstrate that the bosonic Hamiltonian of the supermembrane with non-trivial central charges has a discrete spectrum with finite multiplicity. Based on this observationwe will show strong evidence confirming that the spectrum of the supersymmetric Hamiltonianpossesses the same qualitative properties. Consequently, the N = 1 NCSYM theory in (2+1)D isequivalent to the supermembrane with non-trivial central charge has the same spectral properties.

We also prove that, in the semiclassical regime, the large N limit of the eigenvalues of the reg-ularised Hamiltonian converge to those of the supermembrane with central charge. The spectrumof the Hamiltonian exhibits a mass gap. The scalar fields acquire a mass induced by the centralcharge of the SUSY algebra. One may also argue that this mass is induced by the centre Z(2) ofthe symplectic group associated to the central charge. Supermembranes with central charge andN = 1 in 4D may then be interpreted as the IR phase of a more general theory. This is achievedby increasing the energy the centre breaks spontaneously, so that the transition phase occurs ina screening phase. This should correspond to an N = 4 compactified supermembrane withoutcentral charges.

In Section 2, we describe the Hamiltonian of the supermembrane with central charges and itssemiclassical regime. In Section 3 we establish the main results of the present paper, a crucial op-erator bounds connecting the exact bosonic Hamiltonian and the semiclassical Hamiltonian. Thisoperator bound, see (22), allows us to deduce spectral properties of the exact bosonic Hamiltonianfrom those of the semiclassical one discussed in Section 2. Natural questions arises concerningthe large N limit of the regularised Hamiltonian. We address these questions in Section 4. Inthe final Section 5 we consider confinement properties of the theory in terms of the centre ofthe group both at the exact and the regularised level. We also discuss the transition phase to de-confinement and give an interpretation in terms of the supermembrane. An appendix is included,where we explicitly compute the SU(N) gauge symmetry in the regularisation model.

2. The supermembranes with central charges

In this section we analyse the exact action of the supermembrane with central charges as wellas its semiclassical approximation.

Let the D = 11 supermembrane be defined in terms of a base manifold, a g = 1 Riemannsurface , and a target space M9 S1 S1. Consider its formulation in the light cone gaugewhere the directions X+, X, P+ and P have been removed in the standard manner [4]. Thecanonically reduced Hamiltonian has the expression

W

(1(

PM)2

+ 1{XM,XN}2 + (fermionic terms))

2 W 4


subject to the condition

(1)C

PMW

dXM = 0.

Here and below M,N = 1, . . . ,9. The integral in (1) is the generator of an area preserving dif-feomorphism of for C any given closed path. This constraint may be expressed as a localcondition

{PM,X

M} aba

(PMW

)bX

M = 0,

which generates area preserving diffeomorphisms associated to exact one-forms coupled to theglobal constraint

(2)Ci

PMW

dXM = 0, i = 1,2,

where C1 and C2 form of a basis of homology on which generates area preserving diffeomor-phisms associated to harmonic one-forms.

The scalar densityW is present in expression (2) as a consequence of the gauge fixing

procedure and it is preserved by the diffeomorphisms mentioned above. Let us now impose sometopological restrictions on the configuration space which completely characterise the D = 11supermembrane with non-trivial central charge generated by the wrapping on the compact sectorof the target space. All maps from the base space , must satisfy

Ci

dXr = 2Sri Rr , r = 1,2,

(3)Ci

dXm = 0, m= 3, . . . ,9,

for i = 1,2 and

(4)

dXr dXs = rs(22R1R2)n,where n = detSri is fixed, each entry Sri is integer, and R1 and R2 denote the radii of the targetcomponent S1 S1. Note that (3) describe maps from to S1 S1 with dXm exact one-formsand dXr non-trivial closed one-forms. The only restriction upon these maps is the assumptionthat n is fixed. The term on the left side of (4) describes the central charge of the supersymmetricalgebra. As we shall see next, the factor R1R2(2)2 is the area of in the induced metric.

The general map satisfying (3)(4) can be constructed explicitly. Any closed one-form dXrmay be expressed as the sum of a harmonic and an exact form,

(5)dXr = Lrs dXs + rs dAs, s, r = 1,2,where Lr are real numbers and dX is a canonical basis of harmonic one-forms over . The termsdXs , s = 1,2, is found by considering the (unique) holomorphic one-form , normalised with


respect to the elements of the homology basis Ci , defined by:C1

w = 1,C2

w =,

where is the period of in the basis given by Ci . By construction, the imaginary part of ispositive. Let [16]

= dX1 + i dX2and define

dXr = (M1 dX)r ,where the constant matrix M is given by

M =(

1 Re0 Im

).

Then [16]Ci

dXr = ri

and

dXr dXs = rs .

If (3) is to be satisfied, necessarilyLrs = 2RrSrs .

Condition (4) impliesSrs S

tu

su = nrt .Define the scalar density

W by

abaXrbX

srs =W

where a /a , a = 1,2, a being local coordinates on . Thenrs dXr dXs =

W d 1 d 2.

A change of the canonical basis of homology over , implies varying the corresponding har-monic one-form dXr T rs dXs , where T SL(2,Z), that is

T rs Ttu

su = rt ,T rs integers. As the density

W remains invariant under these transformations, they are area-

preserving diffeomorphisms disconnected from the identity. The theory is then invariant under

SL(2,Z). The degrees of freedom are expressed in terms of Ar and the discrete set of integersdescribed by the harmonic one-forms. We can always fix these gauge transformations, within the


conformal class of the target torus, by

Sri = lr ri , l1l2 = n.Therefore

dXr = 2Rr lr dXr + rs dAs.After the gauge fixing there is a residual invariance Z(2).

The complete expression for the Hamiltonian of the D = 11 supermembrane subject to thetopological conditions (3) and (4) turns out to be [12,14,19],

H =

W d 1 d 2[

12

(PmW

)2+ 1

2

(rW

)2+ 1

4{Xm,Xn

}2(6)+ 1

2(DrXm)2 + 14 (Frs)2 +

({Pm,X

m}+Drr)

]+ (fermionic term),

where [12,19]

DrXm =DrXm +{Ar,X

m},

Drr = 2Rr lr ab

W

aXrb

(rW

)+ [Ar,r],

Frs =DrAs DsAr + [Ar,As]and Dr = 2Rrlr ab

WaX

rb . The associated mass operator is (mass)2 = Z2 +H , where Z2 =18 ((2)

2nR1R2)2 is the central charge.

2.1. Semiclassical regime of the bosonic Hamiltonian

The semiclassical approximation of the theory is obtained by considering only the quadraticterms in the above expression for the Hamiltonian. Let

Hsc =

W d 1 d 2[

12

(PmW

)2+ 1

2

(rW

)2

+ 12(DrX

m)2 + 1

4(Frs)2 +Drr

]+ (fermionic terms),

where, in the semiclassical approximation,

(7)Frs =DrAs DsAr.The general solution to the constraint Drr = 0 is

r = rs2RslsabaXsb(

W

),

where is a scalar density.r The kinetic term Ar may be rewritten, once integration by parts has been carried out, as

pq where p = W 12rsFrs and q = W . Thus


H =

d 1 d 2 W[

12

(rW

)2+ 1

4(Frs)2 +Drr

]

= 1k

d 1 d 2 W

[12

(pW

)2+ 1

2(Drq)

2]

which coincides with the contribution to the Hamiltonian of the transverse modes Xm, m =3, . . . ,9.

The above shows that, from a gauge independent point of view, the complete bosonic Hamil-tonian in the semiclassical approximation is

HBsc =

d 1 d 2 W[

12

(PMW

)2+ 1

2(DrX

M)2]

,

where M = 1, . . . ,8. If we now express XM and PM/W in terms of a complete orthonormalbasis of scalar symmetries over , we obtain

(8)XM =XMA () exp[2i

(arX

r)]( ),

(9)PM

W

= MA () exp[2i

(arX

r)]( ),

where A = (a1, a2). Thus, the bosonic contribution in the semiclassical Hamiltonian takes theform

HBsc =12[(MA

)2 +2A(XmA)2].The spectrum of HBsc is then characterised in the following fashion. For any finite subset of

N N there is an eigenvalue

(10) =A

A, A = 2(

R1l1a2)2 + (R2l2a1)2.

This expression coincides with the particular case considered in [47] where the semiclassicalquantization of the supermembrane over a fixed background was considered. The model in [47]correspond to a gauge fixed version of the supermembrane with central charges in the semiclas-sical approximation. By virtue of (10), for any given energy level E, there only exists a finitenumber of eigenvalues of HBsc below E. The spectrum is therefore discrete.

This is the expression of the eigenvalues when the zero point energy has been eliminated. It isautomatically cancelled when the semiclassical supersymmetric Hamiltonian is considered. Thisproperty was first demonstrated in [47] and it coincides exactly with the one for the semiclassicalsupermembrane with central charges.

3. Operator bounds on the exact bosonic Hamiltonian

According to the results reported in [13], the bosonic regularised Hamiltonian of the D = 11supermembrane with central charge, HBN , relates to its semiclassical approximation, H

Bsc,N , by

means of the following operator inequality:(11)HBN CNHBsc,N .


Here N denotes the size of the truncation in the Fourier basis of and CN is a positive constantdependent on the size of the truncation. A crucial step in the proof of (11), relies heavily on thecompactness of the unit ball of the configuration space which happens to be finite dimensional.In this section we show that the same operator relation holds true for the exact bosonic Hamil-tonians, i.e. the case N = , see Theorem 1. The main source of difficulties in the proof of thisassertion lies in the fact that now the unit ball of the configuration space does not possess theproperty of being compact. We overcome these difficulties by carrying out a detailed analysis ofeach term involved in the expansion of the potential term of HB .

3.1. The configuration space and the gauge fixing condition

We firstly construct the configuration space for the supermembrane with central charges in arigorous fashion and prescribe suitable gauge fixing condition.

Constant functions are harmonic, therefore the constant modes of the fields Xm and Ar be-long to the harmonic sector in (5). In addition only derivatives of Xm and Ar appear in all theexpressions. Let H1 denote the Hilbert space obtained by completing C1(), modulo locallyconstant functions, with respect to the norm

u2 =

d2Wgabaubu,

where gab is the inverse of the metric gab = aXrbXr induced over by the minimal immer-sion Xr . Below we employ the following convention: for a field u,

u =

d2Wu.

For simplicity we will consider below only the case n= 1 and R1 =R2 = 1.Note that

DruDru= ab

W

aXrcu

bdW

bXrd u

= gcdcud uand

{u,w} ab

waubw = rsDruDsw,

so that

u2 = DruDru.Let

(12)u4,2 =(Dru2 + DrDsu2)1/4.

Below and elsewhere Xm, Ar are assumed to be members of the configuration space H4,2 of1 4,2fields u H such that u4,2 < . The left-hand side of (12) is a well defined norm in H .

The latter is a linear space, however we do not make any assumption about completeness.


The potential, V , of the bosonic sector of the supermembrane with central charges is welldefined in H4,2 as

(13)V =DrXmDrXm + 14FrsFrs +

14{Xm,Xn

}2.

The introduction of the constrained space H4,2 H1 is justified by the fact that homogeneousterms of order 4 are present on the right-hand side of (13), so V is not well defined in the wholespace H1.

The following gauge fixing conditions are equivalent to those considered in [1214],

(14)D1A1 = 0 if D1A2 = 0 then A2 = 0.These are obtained by expressing the fields in terms of an orthonormal basis over . Integra-tion by parts yields D2A1D1A2 = 0. Similarly we also have D2A1{A1,A2} = 0. Note that(D1A2)2 = 0, implies A2 = 0.

3.2. The uniform quadratic bound for the bosonic potential

Let 2 be the potential term of HBsc , so that

2 = DrXmDrXm + (D1A2)2 + (D2A1)2.We may rewrite

V = 2 + 2B + A2,where

B = DrXm{Ar,Xm}+D1A2{A1,A2},A2 = {A1,Xm}2 + {A2,Xm}2 + {A1,A2}2 + {Xm,Xn}2.

Theorem 1. There exists a constant 0


Let

P(R)= 1 + 2bR + a2R2be the real polynomial whose variable is R 0. The existence of a constant C > 0 satisfying(15) is equivalent to the condition

(17)inf(Xm,Ar )4,2=1

[infR0

P(R)] C.

Note that B is the inner product in H1 of the field(D1X

m,D2Xm,D1A2,0

)and the field({

A1,Xm},{A2,X

m}, {A1,A2},

{Xm,Xn

}),

while A is the norm of the latter. Thus A2 = 0 yields B = 0, so the condition a2 = 0 impliesP(R)= 1. Hence

inf(Xm,Ar )4,2=1

[infR0

P(R)]= min

[1, inf

a =0

(1 b

2

a2

)]

= infa =0

(1 b

2

a2

)in (17).

The validity of the following lemma will immediately ensure (17), hence Theorem 1.

Lemma 2. Let a and b be the quantities defined by (16). Then

infa =0

(1 b

2

a2

)> 0.

Proof. We proceed by contradiction. Suppose that

infa =0

(1 b

2

a2

)= 0.

Then we can find sequences (Xm)j , (Ar)j H4,2, such that ((Xm)j , (Ar)j )4,2 = 1 and

(18)b2j

a2j 1

as j , where aj , bj are the quantities defined by (16) for the fields at the j th place of thesequences.

By construction

b2

a2= DrX

m{Ar,Xm} +D1A2{A1,A2} + (D2A1)0 + 0{Xm,Xn}22A2

.

1Thus, for each j , the left-hand side of (18) is the inner product of two unit vectors in H . Byvirtue of the CauchySchwarz inequality, these two vectors should become increasingly parallel


as j . Since the quantities a2 and b2 remain constant if we multiply the field (Xm,Ar) bya constant, without loss of generality we can chose our sequences (Xm)j , (Ar)j , such that

(19)(

Dr(Xm)j

j {(Ar)j , (X

m)j }Aj

)2 0,

(20)(

D1(A2)jj

{(A1)j , (A2)j }Aj

)2 0,

(21)(

D2(A1)jj

)2 0 and

( {(Xm)j , (Xn)j }Aj

)2 0.

When r = 1, the left side of (19), is(D1(Xm)j

j D2(A1)jD1(X

m)j

Aj

)2

(

D1(Xm)jj

)2 2

D1(Xm)j

j

D2(A1)jD1(Xm)jAj

.By virtue of the CauchySchwarz inequality,

D1(Xm)j

j

D2(A1)jD1(Xm)jAj

=(D1

(Xm

)j

)2 D2(A1)jjAj

(D1

(Xm

)j

)41/2(D2(A1)jj

)21/2 1Aj

(

D2(A1)jj

)21/2 1Aj

.

Furthermore, analogous results also hold true for the left sides of (19) when r = 2 and the leftside of (20). Hence, if Aj 0, the above, along with (19) and (20), imply

(Dr(Xm)j )2 + (D1(A2)j )2 + (D2(A1)j )22j

0

which is impossible.It is only left showing that the case Aj 0 also leads to a contradiction. We proceed as

follows.Let denote the Laplacian operator acting on L2(). Integration by parts yields

Dr(D1

(Xm

)j

)2Dr

(D1

(Xm

)j

)2= [()1/2(D1(Xm)j )2]2.Then, (Wm)j = ()1/2(D1(Xm)j )2 L2(). Since (Xm)j4,2 1, (Wm)jL2 1. Now()1/2 is a compact operator and(

D1(Xm

)j

)2 = ()1/2(Wm)j.

Thus (D1(Xm)j )2 has a subsequence which is convergent in L2 to an accumulation point,say Ym1 L2().

m 2 2 2Similarly (D2(X )j ) , (D1(A2)j ) and (D2(A1)j ) have corresponding subsequences, con-vergent in L2 to accumulation points, Ym2 , Z1 and Z2 in L2(). Furthermore Ymr and Zr lie


on L4(), so that we can evaluate P(R) at ((Ymr )1/2, (Zr)1/2). As A2 = 0 when we evaluate atthese limit fields, in fact P(R) achieves the constant value 1. But, since b and a2 are continuousin (DrXm,D1A2,D2A1)

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Nucl.Phys.B v.795