The Lange-Neubert evolution equation describes the scale dependence of the
wave function of a meson built of an infinitely heavy quark and light antiquark
at light-like separations, which is the hydrogen atom problem of QCD. It has
numerous applications to the studies of B-meson decays. We show that the kernel
of this equation can be written in a remarkably compact form, as a logarithm of
the generator of special conformal transformation in the light-ray direction.
This representation allows one to study solutions of this equation in a very
simple and mathematically consistent manner. Generalizing this result, we show
that all heavy-light evolution kernels that appear in the renormalization of
higher-twist B-meson distribution amplitudes can be written in the same form.Comment: 8 page

Braun, V. M.

Manashov, A. N.

English

arXiv

The Lange–Neubert evolution equation describes the scale dependence of the wave function of a meson built of an infinitely heavy quark and light antiquark at light-like separations, which is the hydrogen atom problem of QCD. It has numerous applications to the studies of B -meson decays. We show that the kernel of this equation can be written in a remarkably compact form, as a logarithm of the generator of special conformal transformation in the light-ray direction. This representation allows one to study solutions of this equation in a very simple and mathematically consistent manner. Generalizing this result, we show that all heavy–light evolution kernels that appear in the renormalization of higher-twist B -meson distribution amplitudes can be written in the same form

Braun, Vladimir M.

Manashov, Alexander

University of Regensburg Publication Server

Physics Letters B 731 (2014) 316–319Contents lists available at ScienceDirectPhysics Letters Bwww.elsevier.com/locate/physletbConformal symmetry of the Lange–Neubert evolution equationV.M. Braun a,∗, A.N. Manashov a,ba Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germanyb Department of Theoretical Physics, St. Petersburg University, 199034 St. Petersburg, Russiaa r t i c l e i n f o a b s t r a c tArticle history:Received 25 February 2014Accepted 26 February 2014Available online 4 March 2014Editor: B. GrinsteinThe Lange–Neubert evolution equation describes the scale dependence of the wave function of a mesonbuilt of an inﬁnitely heavy quark and light antiquark at light-like separations, which is the hydrogen atomproblem of QCD. It has numerous applications to the studies of B-meson decays. We show that the kernelof this equation can be written in a remarkably compact form, as a logarithm of the generator of specialconformal transformation in the light-ray direction. This representation allows one to study solutionsof this equation in a very simple and mathematically consistent manner. Generalizing this result, weshow that all heavy–light evolution kernels that appear in the renormalization of higher-twist B-mesondistribution amplitudes can be written in the same form.© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.1. Studies of heavy meson weak decays have been instrumen-tal to uncover the ﬂavor sector of the Standard Model and can bea gate to new physics at TeV scales, if it exists. Considerable ef-fort has been invested to understand the QCD dynamics of heavymeson decays in the heavy quark limit. The B-meson distributionamplitude (DA), ﬁrst introduced in [1], provides the key nonpertur-bative input in the QCD factorization approach [2] for weak decaysinvolving light hadrons in the ﬁnal state.Following an established convention, we deﬁne the B-mesonDA as the renormalized matrix element of the bilocal operatorbuilt of an effective heavy quark ﬁeld hv(0) and a light antiquarkq¯(zn) at a light-like separation:〈0|q¯(zn)/n[zn,0]Γ hv(0)|B¯(v)〉= − i2F (μ)Tr[γ5/nΓ P+]Φ+(z,μ) (1)with[zn,0] ≡ Pexp[ig1∫0dα nμAμ(αzn)]. (2)Here vμ is the heavy quark velocity, nμ is the light-like vector,n2 = 0, such that n · v = 1, P+ = 12 (1+ /v) is the projector on uppercomponents of the heavy quark spinor, Γ stands for an arbitraryDirac structure, |B¯(v)〉 is the B¯-meson state in the heavy quarkeffective theory (HQET) and F (μ) is the decay constant in HQET,* Corresponding author.http://dx.doi.org/10.1016/j.physletb.2014.02.0510370-2693/© 2014 The Authors. Published by Elsevier B.V. This is an open access articleSCOAP3.which is used for normalization. The effective heavy quark can berelated to the Wilson line through the following equation [3]:〈0|hv(0)|h, v〉 = [0,−v∞] = Pexp[ig0∫−∞dα vμAμ(αv)], (3)so that the operator in Eq. (1) can be viewed as a single light an-tiquark attached to the Wilson line with a cusp containing onelightlike and one timelike segment.The invariant function Φ+(z,μ) where z is a real number de-ﬁnes what is usually called the leading twist B-meson DA in posi-tion space. Its Fourier transform isφ+(k,μ) = 12π∞∫−∞dz eikzΦ+(z − i0,μ),Φ+(z,μ) =∞∫0dk e−ikzφ+(k,μ), (4)where in the ﬁrst equation the integration contour goes below thesingularities of Φ+(z,μ) that are located in the upper half-plane.The parameter μ is the renormalization (factorization) scale. Wetacitly imply using dimensional regularization with modiﬁed min-imum subtraction.The scale dependence of the DA is driven by the renormaliza-tion of the corresponding nonlocal operatorO+(z) = q¯(zn)/n[zn,0]Γ hv(0).under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded byV.M. Braun, A.N. Manashov / Physics Letters B 731 (2014) 316–319 317The corresponding one-loop Z -factor was computed by Lange andNeubert (LN) [4], giving rise to an evolution equation which isconvenient to write, for our purposes, as a renormalization groupequation for the operator O+(z) [5,6]:(μ∂∂μ+ β(g) ∂∂ g+ αsC FπH)O+(z,μ) = 0, (5)where[H f ](z) =1∫0dαα(f (z) − α¯ f (α¯z))+ ln(iμz) f (z) − 54f (z),α¯ ≡ 1− α. (6)This equation thus governs the scale dependence of the B-mesonDA in position space, Φ+(z,μ). It is fully equivalent to the originalLN equation for the DA in momentum space, φ+(k,μ), as it is easyto show by Fourier transformation.2. We will demonstrate that the LN kernel (6) can be written interms of the generators of collinear conformal transformationsS+ = z2∂z + 2 jz, S0 = z∂z + j, S− = −∂z, (7)where j = 1 is the conformal spin of the light quark. They satisfythe standard SL(2) commutation relations[S+, S−] = 2S0, [S0, S±] = ±S±. (8)The starting observation is that the integral operator H (LN kernel)can be written in a somewhat different form by studying its actionon the test functions f (z) = zp , z∂z f (z) = pf (z). Here and below∂z = ∂/∂z. In this way one obtains[H f ](z) =[ψ(z∂z + 2) − ψ(1) + ln(iμz) − 54]f (z). (9)Next, we use the identity for a fractional derivative (i∂z)a deﬁnedas the multiplication operator ka in momentum representation [7]:(i∂z)a = (iz)−a Γ (a − z∂z)Γ (−z∂z) . (10)It holds for the functions f (z) that are holomorphic in the lowercomplex half-plane m z < 0, z ∈C− , and vanish at inﬁnity. Fouriertransform for such functions goes over positive momenta f (z) =∫∞0 dk e−ikz f˜ (k), (i∂z)a f (z) =∫∞0 dk e−ikzka f˜ (k), corresponding inour case to positive values of the light-quark energy ω = k/2 inthe B-meson rest frame, cf. Eq. (4). Expanding this identity arounda = 0 one getsln(i∂z) = ψ(−z∂z) − ln(iz) (11)and making an inversion z → −1/zln(iz2∂z)= ψ(z∂z) + ln(iz). (12)Finally, since for any function f (z∂z)z = zf (z∂z + 1), we can writethis identity asz−2 ln(iz2∂z)z2 = ln[i(z2∂z + 2z)]= ln(i S+)= ψ(z∂z + 2) + ln(iz). (13)Comparing with Eq. (6) we see thatH= ln(iμ S+)− ψ(1) − 5 (14)4which is our main result. Note that the scale μ under the loga-rithm is necessary simply because S+ has dimension [mass]−1.Alternatively, the same expression can be derived starting fromthe commutation relations for the LN kernel obtained in Ref. [6]:[S+,H] = 0, [S0,H] = 1. (15)Since the problem has one degree of freedom — the light-conecoordinate of the light quark — it follows from [S+,H] = 0 thatthe operator H must be a function of S+ , H = h(S+). This func-tion can be found using the second commutation relation. LetS = S0 + 1. Then S+ = zS and the relation [S0,h(S+)] = 1 canbe written equivalently as [S,h(z S)] = 1. Taking into account that[S, z S] = z S one obtains an equation on the function h(s)s h′(s) = 1 ⇒ h(s) = ln s + constant, (16)reproducing the result in Eq. (14) up to a (scheme-dependent) con-stant.3. The main advantage of Eq. (14) is that diagonalization of thekernel H can be traded for a much simpler task of diagonalizationof the ﬁrst-order differential operator S+ (7). Eigenfunctions of S+take a simple form1Q s(z) = − 1z2eis/z, i S+Q s(z) = sQ s(z), (17)so thatHQ s(z) =[ln(μs) − ψ(1) − 54]Q s(z). (18)A further advantage is that one can use SL(2) representation theorymethods to work with these solutions, see e.g. Ref. [11] for a shortdiscussion of this technique. In particular one can make use of thestandard SL(2) invariant scalar product [12] (for spin j = 1)〈Φ|Ψ 〉 = 1π∫C−d2zΦ(z)Ψ (z), (19)where the (two-dimensional) integration goes over the lower half-plane C− , m z < 0. The generator i S+ is self-adjoint w.r.t. thisscalar product. The eigenfunctions (17) are orthogonal to eachother and form a complete set〈Q s′ |Q s〉 = 1s δ(s − s′),∞∫0ds s Q s(z)Q s(z′)= e−iπ(z − z¯′)2 . (20)The function on the r.h.s. of the completeness relation is calledreproducing kernel [13]. It acts as a unit operator so that for anyfunction holomorphic in the lower half-planeΨ (z) = 1π∫C−d2z′ e−iπ(z − z¯′)2 Ψ(z′). (21)Hence the B-meson DA (1) can be expanded asΦ+(z,μ) =∞∫0ds sη(s,μ)Q s(z)= − 1z2∞∫0ds seis/z η(s,μ), η(s,μ) = 〈Q s|Φ〉. (22)1 The sign is chosen such that Q s(z) are real and positive for z = −iτ , τ > 0.318 V.M. Braun, A.N. Manashov / Physics Letters B 731 (2014) 316–319The integration goes over all possible eigenvalues of the step-upgenerator S+ that corresponds to special conformal transforma-tions along the light-ray nμ . This representation is very similar tothe one suggested in Ref. [8].The scale-dependence of the coeﬃcients η(s,μ) is governed bythe renormalization-group equation(μ∂∂μ+ β(αs) ∂∂αs+ Γcusp(αs) ln(μss0))× F (μ)η(s,μ) = 0, (23)where s0 = e5/4−γE and Γcusp(αs) = αsπ CF + · · · is the cusp anoma-lous dimension [9,10].The solution of this equation takes the formF (μ)η(s,μ)= F (μ0)η(ξ,μ0) × exp{−μ∫μ0dττΓcusp(αs(τ ))ln(τss0)}= F (μ0)η(ξ,μ0)(μ0ss0)r(μ)B(μ), (24)wherer(μ) = −α(μ)∫α(μ0)dαβ(α)Γcusp(αs) = 2CFβ0ln(α(μ)α(μ0))+ · · · ,B(μ) = exp{−α(μ)∫α(μ0)dαβ(α)Γcusp(α)α∫α(μ0)dα′β(α′)}. (25)In practical applications the momentum (energy) representationfor the B-meson DA φ+(k,μ) as deﬁned in (4) is more convenient.This can be derived easily by observing that exponential functionse−ipz , p > 0 are mutually orthogonal and form a complete set w.r.t.the same scalar product〈e−ipz∣∣e−ip′z〉= 1pδ(p − p′). (26)HenceΦ+(z,μ) =∞∫0dp pe−ipz〈e−ipz′∣∣Φ+(z′,μ)〉=∞∫0dp pe−ipz∞∫0ds sη(s,μ)〈e−ipz′∣∣Q s(z′)〉 (27)and thereforeφ+(k,μ) = 12π∞∫−∞dz eikzΦ+(z − i0,μ)= k∞∫0ds sη(s,μ)〈e−ikz∣∣Q s(z)〉. (28)Using〈e−ikz∣∣Q s(z)〉= 1√ksJ1(2√ks)(29)we ﬁnally obtainφ+(k,μ) =∞∫0ds√ks J1(2√ks)η(s,μ), (30)where J1(x) is the Bessel function. The representation in Eq. (30)is equivalent to the one suggested by Bell, Feldmann, Wang andYip in Ref. [8], who noticed that the evolution equation is signiﬁ-cantly simpliﬁed in this manner. In their notation, cf. second linein Eq. (2.17), sη(s,μ) ≡ ρ+(1/s,μ).The orthogonality relation (26) combined with the projection(29) leads to a familiar relation for the Bessel functions∞∫0ds J1(2√ps ) J1(2√p′s)= δ(p − p′), (31)which can be used to invert Eq. (30) and express η(s,μ) in termsof φ+(k,μ).Note that the representation in (14) is valid for the evolutionkernel in momentum space as well, but the generator S+ has tobe taken in the adjoint representationS+ = i[k∂2k + 2 j∂k], j = 1. (32)The Bessel functions appearing in (29), (30) are eigenfunctions ofS+ , indeed:s〈e−ikz∣∣Q s(z)〉= 〈e−ikz∣∣i S+Q s(z)〉= 〈i S+e−ikz∣∣Q s(z)〉= iS+〈e−ikz∣∣Q s(z)〉. (33)Of particular interest for the QCD description of B-decays is thevalue of the ﬁrst negative momentλ−1B (μ) =∞∫0dkkφ+(k,μ) =∞∫0dτ Φ+(−iτ ,μ)=∞∫0dsη(s,μ). (34)As demonstrated in [8], QCD factorization expressions for B decayamplitudes can conveniently be written in terms of η(s,μ) as well,so that we do not dwell on this topic here.4. The same representation can be derived for arbitrary two-particle heavy–light one-loop kernels that contribute to the evo-lution equations for higher-twist B-meson DAs [6]. The differenceto the leading twist is that the two-particle evolution equationsare not closed: The two-particle, 2 → 2, kernels appear as partsof larger mixing matrices involving 2 → 3 parton transitions, how-ever, 3 → 2 transitions do not occur at the one-loop level.Explicit expressions for all 2 → 2 heavy–light kernels have beenderived in Ref. [6], see Section 3.2. They can be written in terms ofan integral operator[H j f ](z) =1∫0dαα[f (z) − α¯2 j−1 f (α¯z)]+ ln(iμz) f (z)− [σh + σ] f (z), (35)where j is the conformal spin of the light parton  (quark orgluon) and the constants σh = 1/2, σquark = 3/4, σgluon = β0/4Nc(β0 = 11/3Nc −2/3n f ) are related to the anomalous dimensions ofthe ﬁelds. Conformal spin of a parton is deﬁned as j = (d + s)/2where d is canonical dimension and s is spin projection on thelight cone, see [14]. For a quark j = 1 for the “plus” projectionV.M. Braun, A.N. Manashov / Physics Letters B 731 (2014) 316–319 319that contributes to the leading-twist B-meson DA (1), in whichcase (35) reproduces (6), and j = 1/2 for the “minus” projectionthat is relevant for the DA Φ−(z,μ), cf. [2]. In turn, for a gluonj = 3/2 for the leading-twist projection and j = 1 for the higher-twist.Following the above derivation for j = 1 we obtain the follow-ing representation for the kernel in the general case:H j = ln(iμS( j)+)− ψ(1) − σh − σ, (36)where the generator of special conformal transformations S( j)+ forspin j is deﬁned in Eq. (7). The eigenfunctions of S( j)+ have theformQ ( j)s (z) = e−iπ jz2 jeis/z, i S( j)+ Q( j)s (z) = s Q ( j)s (z). (37)They are orthogonal and form a complete set with respect to theSL(2) scalar product [13]〈Φ|Ψ 〉 j = 2 j − 1π∫C−D j zΦ(z)Ψ (z), (38)where D j z = d2z [i(z − z¯)]2 j−2. One obtains〈Q ( j)s∣∣Q ( j)s′ 〉 j = (2 j)s2 j−1 δ(s − s′),1(2 j)∞∫0ds s2 j−1Q ( j)s (z)Q( j)s(z′)= e−iπ j(z − z¯′)2 j . (39)The expression on the r.h.s. of the second integral deﬁnes the re-producing kernel for arbitrary spin j [13], i.e. for arbitrary function(holomorphic in the lower plane)Ψ (z) = 2 j − 1π∫C−D j z′e−iπ j(z − z¯′)2 j Ψ (z′). (40)The functions Q js (z) diagonalize the renormalization group ker-nelH j Q js (z) =[ln(μs) − ψ(1) − σh − σ]Q js (z) (41)so that it is natural to write matrix elements of generic heavy–lightoperators as an expansionΦ j(z,μ) =∞∫0ds s2 j−1η j(s,μ)Q ( j)s (z), (42)where Φ j(z,μ) is analogue of Φ+(z,μ) (1).The expansion coeﬃcients φ j(k,μ) appearing in the FouriertransformΦ j(z,μ) =∞∫0dk e−ikzφ j(k,μ) (43)can be found making use of the following relations:〈e−ikz∣∣e−ik′z〉 j = (2 j)k1−2 jδ(k − k′),〈e−ikz∣∣Q ( j)s 〉 j = (2 j)(ks)1/2− j J2 j−1(2√ks ). (44)In this way one obtainsφ j(p,μ) =∞∫0dsη j(s,μ)(sp)j−1/2 J2 j−1(2√ps ). (45)In particular for j = 1/2 corresponding to the B-meson DAφ−(k,μ) [2] the conformal expansion goes over Bessel functionsJ0(2√ks ) as compared to J1(2√ks) for the leading twist, cf. [8].5. To summarize, we have constructed a conformal expansionof the distribution amplitudes of heavy–light mesons in terms ofeigenfunctions of the generator of special conformal transforma-tions. This construction is similar in spirit to the well-known ex-pansion of DAs of light mesons in Gegenbauer polynomials whichare eigenfunctions of two-particle SL(2) Casimir operators, seee.g. [14]. Similar to the latter case, this expansion can serve as abasis for the construction of approximations of phenomenologicalrelevance.As we have shown, this structure is a consequence of the com-mutation relations (15) and it would be very interesting to ﬁndout whether these relations hold true to all orders in perturbationtheory for a conformal theory like N = 4 SYM. The consequencesof our results for the DAs of baryons made of one heavy and twolight quarks should be studied as well.AcknowledgementThis work was supported by the DFG, grant BR2021/5-2.References[1] A. Szczepaniak, E.M. Henley, S.J. Brodsky, Phys. Lett. B 243 (1990) 287;A.G. Grozin, M. Neubert, Phys. Rev. D 55 (1997) 272.[2] M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Phys. Rev. Lett. 83 (1999)1914;M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Nucl. Phys. B 591 (2000)313.[3] G.P. Korchemsky, A.V. Radyushkin, Phys. Lett. B 279 (1992) 359.[4] B.O. Lange, M. Neubert, Phys. Rev. Lett. 91 (2003) 102001.[5] V.M. Braun, D.Yu. Ivanov, G.P. Korchemsky, Phys. Rev. D 69 (2004) 034014.[6] M. Knödlseder, N. Offen, J. High Energy Phys. 1110 (2011) 069.[7] A.V. Belitsky, S.E. Derkachov, A.N. Manashov, arXiv:1401.7307 [hep-th].[8] G. Bell, T. Feldmann, Y.-M. Wang, M.W.Y. Yip, J. High Energy Phys. 1311 (2013)191.[9] A.M. Polyakov, Nucl. Phys. B 164 (1980) 171.[10] G.P. Korchemsky, A.V. Radyushkin, Nucl. Phys. B 283 (1987) 342.[11] S.E. Derkachov, G.P. Korchemsky, A.N. Manashov, J. High Energy Phys. 0307(2003) 047.[12] I.M. Gelfand, M.I. Graev, N.Ya. Vilenkin, Generalized Functions. vol. 5: IntegralGeometry and Representation Theory, Academic Press, New York, 1966.[13] B.C. Hall, Holomorphic methods in mathematical physics, Contemp. Math. 260(1999) 1–59, arXiv:quant-ph/9912054.[14] V.M. Braun, G.P. Korchemsky, D. Mueller, Prog. Part. Nucl. Phys. 51 (2003) 311.

Conformal symmetry of the Lange-Neubert evolution equation

arXiv:1402.5822v1  [hep-ph]  24 Feb 2014Conformal symmetry of the Lange-Neubertevolution equationV.M. Braun1 and A.N. Manashov1,21 Institut fu¨r Theoretische Physik, Universita¨t Regensburg, D-93040 Regensburg, Germany2 Department of Theoretical Physics, St.-Petersburg University, 199034, St.-Petersburg,RussiaAbstractThe Lange-Neubert evolution equation describes the scale dependence of the wavefunction of a meson built of an infinitely heavy quark and light antiquark at light-likeseparations, which is the hydrogen atom problem of QCD. It has numerous applica-tions to the studies of B-meson decays. We show that the kernel of this equation canbe written in a remarkably compact form, as a logarithm of the generator of specialconformal transformation in the light-ray direction. This representation allows oneto study solutions of this equation in a very simple and mathematically consistentmanner. Generalizing this result, we show that all heavy-light evolution kernels thatappear in the renormalization of higher-twist B-meson distribution amplitudes canbe written in the same form.1. Studies of heavy meson weak decays have been instrumental to uncover the flavor sectorof the Standard model and can be a gate to new physics at TeV scales, if it exists. Considerableeffort has been invested to understand the QCD dynamics of heavy meson decays in the heavyquark limit. The B-meson distribution amplitude (DA), first introduced in [1], provides thekey nonperturbative input in the QCD factorization approach [2] for weak decays involvinglight hadrons in the final state.Following an established convention we define the B-meson DA as the renormalized matrixelement of the bilocal operator built of an effective heavy quark field hv(0) and a light antiquarkq¯(zn) at a light-like separation:〈0|q¯(zn) 6n[zn, 0]Γhv(0)|B¯(v)〉 = − i2F (µ) Tr [γ5 6nΓP+] Φ+(z, µ) (1)with[zn, 0] ≡ Pexp[ig∫ 10dαnµAµ(αzn)]. (2)Here vµ is the heavy quark velocity, nµ is the light-like vector, n2 = 0, such that n · v = 1,P+ =12(1+ 6 v) is the projector on upper components of the heavy quark spinor, Γ stands foran arbitrary Dirac structure, |B¯(v)〉 is the B¯-meson state in the heavy quark effective theory1(HQET) and F (µ) is the decay constant in HQET, which is used for normalization. Theeffective heavy quark can be related to the Wilson line through the following equation [3]:〈0|hv(0)|h, v〉 = [0,−v∞] = Pexp[ig∫ 0−∞dα vµAµ(αv)], (3)so that the operator in Eq. (1) can be viewed as a single light antiquark attached to the Wilsonline with a cusp containing one lightlike and one timelike segment.The invariant function Φ+(z, µ) where z is a real number defines what is usually called theleading twist B-meson DA in position space. Its Fourier transform isφ+(k, µ) =12π∫∞−∞dz eikz Φ+(z − i0, µ) ,Φ+(z, µ) =∫∞0dk e−ikz φ+(k, µ) , (4)where in the first equation the integration contour goes below the singularities of Φ+(z, µ) thatare located in the upper-half plane. The parameter µ is the renormalization (factorization)scale. We tacitly imply using dimensional regularization with modified minimum subtraction.The scale dependence of the DA is driven by the renormalization of the correspondingnonlocal operatorO+(z) = q¯(zn) 6n [zn, 0]Γ hv(0).The corresponding one-loop Z-factor was computed by Lange and Neubert (LN) [4], giving rizeto an evolution equation which is convenient to write, for our purposes, as a renormalizationgroup equation for the operator O+(z) [5, 6]:(µ∂∂µ+ β(g)∂∂g+αsCFπH)O+(z, µ) = 0 , (5)where[Hf ](z) =∫ 10dαα(f(z)− α¯f(α¯z))+ ln(iµz) f(z)− 54f(z) , α¯ ≡ 1− α . (6)This equation thus governs the scale dependence of the B-meson DA in position space, Φ+(z, µ).It is fully equivalent to the original LN equation for the DA in momentum space, φ+(k, µ), asit is easy to show by Fourier transformation.2. We will demonstrate that the LN kernel (6) can be written in terms of the generatorsof collinear conformal transformationsS+ = z2∂z + 2jz , S0 = z∂z + j , S− = −∂z , (7)where j = 1 is the conformal spin of the light quark. They satisfy the standard SL(2) com-mutation relations[S+, S−] = 2S0 , [S0, S±] = ±S± . (8)2The starting observation is that the integral operator H (LN kernel) can be written in asomewhat different form by studying its action on the test functions f(z) = zp, z∂zf(z) =pf(z). Here and below ∂z = ∂/∂z. In this way one obtains[Hf ](z) =[ψ(z∂z + 2)− ψ(1) + ln(iµz)− 54]f(z) . (9)Next, we use the identity for a fractional derivative (i∂z)a defined as the multiplication operatorka in momentum representation [7]:(i∂z)a = (iz)−aΓ(a− z∂z)Γ(−z∂z) . (10)It holds for the functions f(z) that are holomorphic in the lower complex half-plane ℑmz < 0,z ∈ C−, and vanish at infinity. Fourier transform for such functions goes over positive momentaf(z) =∫∞0dk e−ikzf˜(k), (i∂z)af(z) =∫∞0dk e−ikzka f˜(k), corresponding in our case to positivevalues of the light-quark energy ω = k/2 in the B-meson rest frame, cf. Eq. (4). Expandingthis identity around a = 0 one getsln(i∂z) = ψ(−z∂z)− ln(iz) (11)and making an inversion z → −1/zln(iz2∂z) = ψ(z∂z) + ln(iz) . (12)Finally, since for any function f(z∂z)z = zf(z∂z + 1), we can write this identity asz−2 ln(iz2∂z)z2 = ln[i(z2∂z + 2z)]= ln(iS+) = ψ(z∂z + 2) + ln(iz) . (13)Comparing with Eq. (6) we see thatH = ln(iµ S+)− ψ(1)− 54(14)which is our main result. Note that the scale µ under the logarithm is necessary simply becauseS+ has dimension [mass]−1.Alternatively, the same expression can be derived starting from the commutation relationsfor the LN kernel obtained in Ref. [6]:[S+,H] = 0 , [S0,H] = 1 . (15)Since the problem has one degree of freedom — the light-cone coordinate of the light quark —it follows from [S+,H] = 0 that the operator H must be a function of S+, H = h(S+). Thisfunction can be found using the second commutation relation. Let S = S0+1. Then S+ = zSand the relation [S0, h(S+)] = 1 can be written equivalently as [S, h(z S)] = 1. Taking intoaccount that [S, z S] = z S one obtains an equation on the function h(s)s h′(s) = 1 =⇒ h(s) = ln s+ constant , (16)3reproducing the result in Eq. (14) up to a (scheme-dependent) constant.3. The main advantage of Eq. (14) is that diagonalization of the kernel H can be tradedfor a much simpler task of diagonalization of the first-order differential operator S+ (7). Eigen-functions of S+ take a simple form1Qs(z) = − 1z2eis/z , iS+Qs(z) = sQs(z) , (17)so thatHQs(z) =[ln(µ s)− ψ(1)− 54]Qs(z) . (18)A further advantage is that one can use SL(2) representation theory methods to work withthese solutions, see e.g. Ref. [11] for a short discussion of this technique. In particular one canmake use of the standard SL(2) invariant scalar product [12] (for spin j = 1)〈Φ|Ψ〉 = 1π∫C−d2zΦ(z) Ψ(z) , (19)where the (two-dimensional) integration goes over the lower half-plane C−, ℑmz < 0. Thegenerator iS+ is self-adjoint w.r.t. this scalar product. The eigenfunctions (17) are orthogonalto each other and form a complete set〈Qs′|Qs〉 = 1sδ(s− s′) ,∫∞0ds sQs(z)Qs(z′) =e−iπ(z − z¯′)2 . (20)The function on the r.h.s. of the completeness relation is called reproducing kernel [13]. Itacts as a unit operator so that for any function holomorphic in the lower half planeΨ(z) =1π∫C−d2z′e−iπ(z − z¯′)2Ψ(z′) . (21)Hence the B-meson DA (1) can be expanded asΦ+(z, µ) =∫∞0ds s η(s, µ)Qs(z) = − 1z2∫∞0ds s eis/z η(s, µ) , η(s, µ) = 〈Qs|Φ〉 . (22)The integration goes over all possible eigenvalues of the step-up generator S+ that correspondsto special conformal transformations along the light-ray nµ. This representation is very similarto the one suggested in Ref. [8].The scale-dependence of the coefficients η(s, µ) is governed by the renormalization-groupequation(µ∂∂µ+ β(αs)∂∂αs+ Γcusp(αs) ln(µ s/s0))F (µ)η(s, µ) = 0 , (23)1The sign is chosen such that Qs(z) are real and positive for z = −iτ , τ > 0.4where s0 = e5/4−γE and Γcusp(αs) =αsπCF + . . . is the cusp anomalous dimension [9, 10].The solution of this equation takes the formF (µ) η(s, µ) = F (µ0) η(ξ, µ0)× exp{−∫ µµ0dττΓcusp(αs(τ)) ln(τ s/s0)}= F (µ0) η(ξ, µ0)(µ0 ss0)r(µ)B(µ) , (24)wherer(µ) = −∫ α(µ)α(µ0)dαβ(α)Γcusp(αs) = 2CF/β0 ln(α(µ)α(µ0))+ . . . ,B(µ) = exp{−∫ α(µ)α(µ0)dαβ(α)Γcusp(α)∫ αα(µ0)dα′β(α′)}. (25)In practical applications the momentum (energy) representation for the B-meson DA φ+(k, µ)as defined in (4) is more convenient. This can be derived easily by observing that exponentialfunctions e−ipz, p > 0 are mutually orthogonal and form a complete set w.r.t. the same scalarproduct〈e−ipz|e−ip′z〉 = 1pδ(p− p′) . (26)HenceΦ+(z, µ) =∫∞0dp p e−ipz〈e−ipz|Φ+(z, µ)〉 =∫∞0dp p e−ipz∫∞0ds s η(s, µ) 〈e−ipz|Qs(z)〉 (27)and thereforeφ+(k, µ) =12π∫∞−∞dz eikz Φ+(z − i0, µ) = k∫∞0ds s η(s, µ) 〈e−ikz|Qs(z)〉 . (28)Using〈e−ikz|Qs(z)〉 = 1√ksJ1(2√ks) (29)we finally obtainφ+(k, µ) =∫∞0ds√ks J1(2√ks) η(s, µ) , (30)where J1(x) is the Bessel function. The representation in Eq. (30) is equivalent to the onesuggested by Bell, Feldmann, Wang and Yip in Ref. [8], who noticed that the evolution equationis significantly simplified in this manner. In their notation, cf. second line in Eq. (2.17),s η(s, µ) ≡ ρ+(1/s, µ).5The orthogonality relation (26) combined with the projection (29) leads to a familiar rela-tion for the Bessel functions∫∞0ds J1(2√ps) J1(2√p′s) = δ(p− p′) , (31)which can be used to invert Eq. (30) and express η(s, µ) in terms of φ+(k, µ).Note that the representation in (14) is valid for the evolution kernel in momentum spaceas well, but the generator S+ has to be taken in the adjoint representationS+ = i[k∂2k + 2j∂k], j = 1 . (32)The Bessel functions appearing in (29), (30) are eigenfunctions of S+, indeed:s〈e−ikz|Qs(z)〉 = 〈e−ikz|iS+Qs(z)〉 = 〈iS+e−ikz|Qs(z)〉 = iS+〈e−ikz|Qs(z)〉 . (33)Of particular interest for the QCD description of B-decays is the value of the first negativemomentλ−1B (µ) =∫∞0dkkφ+(k, µ) =∫∞0dτ Φ+(−iτ, µ) =∫∞0ds η(s, µ) . (34)As demonstrated in [8], QCD factorization expressions for B decay amplitudes can convenientlybe written in terms of η(s, µ) as well, so that we do not dwell on this topic here.5. The same representation can be derived for arbitrary two-particle heavy-light one-loopkernels that contribute to the evolution equations for higher-twist B-meson DAs [6]. Thedifference to the leading twist is that the two-particle evolution equations are not closed: Thetwo-particle, 2→ 2, kernels appear as parts of larger mixing matrices involving 2→ 3 partontransitions, however, 3→ 2 transitions do not occur at the one-loop level.Explicit expressions for all 2 → 2 heavy-light kernels have been derived in Ref. [6], seeSec. 3.2. They can be written in terms of an integral operator[Hjf ](z) =∫ 10dαα[f(z)− α¯2j−1f(α¯z)]+ ln(iµz)f(z)− [σh + σℓ]f(z) , (35)where j is the conformal spin of the light parton ℓ (quark or gluon) and the constants σh = 1/2,σquark = 3/4, σgluon = β0/4Nc (β0 = 11/3Nc− 2/3nf) are related to the anomalous dimensionsof the fields. Conformal spin of a parton is defined as j = (d + s)/2 where d is canonicaldimension and s is spin projection on the light cone, see [14]. For a quark j = 1 for the “plus”projection that contributes to the leading-twist B-meson DA (1), in which case (35) reproduces(6), and j = 1/2 for the “minus” projection that is relevant for the DA Φ−(z, µ), cf. [2]. Inturn, for a gluon j = 3/2 for the leading-twist projection and j = 1 for the higher-twist.Following the above derivation for j = 1 we obtain the following representation for thekernel in the general case:Hj = ln(iµS(j)+ )− ψ(1)− σh − σℓ , (36)6where the generator of special conformal transformations S(j)+ for spin j is defined in Eq. (7).The eigenfunctions of S(j)+ have the formQ(j)s (z) =e−iπjz2jeis/z , iS(j)+ Q(j)s (z) = sQ(j)s (z) . (37)They are orthogonal and form a complete set with respect to the SL(2) scalar product [13]〈Φ|Ψ〉j = 2j − 1π∫C−DjzΦ(z) Ψ(z) , (38)where Djz = d2z [i(z − z¯)]2j−2. One obtains〈Q(j)s |Q(j)s′ 〉j =Γ(2j)s2j−1δ(s− s′) , 1Γ(2j)∫∞0ds s2j−1Q(j)s (z)Q(j)s (z′) =e−iπj(z − z¯′)2j . (39)The expression on the r.h.s. of the second integral defines the reproducing kernel for arbitraryspin j [13], i.e. for arbitrary function (holomorphic in the lower plane)Ψ(z) =2j − 1π∫C−Djz e−iπj(z − z¯′)2j Ψ(z) . (40)The functions Qjs(z) diagonalize the renormalization group kernelHjQjs(z) =[ln(µs)− ψ(1)− σh − σℓ]Qjs(z) (41)so that it is natural to write matrix elements of generic heavy-light operators as an expansionΦj(z, µ) =∫∞0ds s2j−1ηj(s, µ)Q(j)s (z) , (42)where Φj(z, µ) is analogue of Φ+(z, µ) (1).The expansion coefficients φj(k, µ) appearing in the Fourier transformΦj(z, µ) =∫∞0dk e−ikz φj(k, µ) (43)can be found making use of the following relations:〈e−ikz|e−ik′z〉j = Γ(2j) k1−2j δ(k − k′) ,〈e−ikz|Q(j)s 〉j = Γ(2j) (ks)1/2−j J2j−1(2√ks) . (44)In this way one obtainsφj(p, µ) =∫∞0ds ηj(s, µ) (sp)j−1/2 J2j−1(2√ps) . (45)In particular for j = 1/2 corresponding to the B-meson DA φ−(k, µ) [2] the conformal ex-pansion goes over Bessel functions J0(2√ks) as compared to J1(2√ks) for the leading twist,cf. [8].76. To summarize, we have constructed a conformal expansion of the distribution ampli-tudes of heavy-light mesons in terms of eigenfunctions of the generator of special conformaltransformations. This construction is similar in spirit to the well-known expansion of DAsof light mesons in Gegenbauer polynomials which are eigenfunctions of two-particle SL(2)Casimir operators, see e.g. [5]. Similar to the latter case, this expansion can serve as a basisfor the construction of approximations of phenomenological relevance.As we have shown, this expansion is a consequence of the commutation relations (15) andit would be very interesting to find out whether these relations hold true to all orders inperturbation theory for a conformal theory like N = 4 SYM. The consequences of our resultsfor the DAs of baryons made of one heavy and two light quarks should be studied as well.7. AcknowledgementsThis work was supported by the DFG, grant BR2021/5-2.References[1] A. Szczepaniak, E. M. Henley and S. J. Brodsky, Phys. Lett. B 243 (1990) 287.[2] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. 83 (1999)1914; Nucl. Phys. B 591 (2000) 313.[3] G. P. Korchemsky and A. V. Radyushkin, Phys. Lett. B 279 (1992) 359.[4] B. O. Lange and M. Neubert, Phys. Rev. Lett. 91 (2003) 102001.[5] V. M. Braun, D. Yu. Ivanov and G. P. Korchemsky, Phys. Rev. D 69 (2004) 034014.[6] M. Kno¨dlseder and N. Offen, JHEP 1110 (2011) 069.[7] A. V. Belitsky, S. E. Derkachov and A. N. Manashov, arXiv:1401.7307 [hep-th].[8] G. Bell, T. Feldmann, Y.-M. Wang and M. W. Y. Yip, JHEP 1311 (2013) 191.[9] A. M. Polyakov, Nucl. Phys. B 164 (1980) 171.[10] G. P. Korchemsky and A. V. Radyushkin, Nucl. Phys. B 283 (1987) 342.[11] S. E. Derkachov, G. P. Korchemsky and A. N. Manashov, JHEP 0307 (2003) 047.[12] I.M. Gelfand, M.I. Graev, N.Ya. Vilenkin, Generalized functions. Vol. 5: Integral geom-etry and representation theory,” Academic Press (New York, 1966).[13] B.C. Hall. Holomorphic Methods in Mathematical Physics - Contemporary Mathematics,Volume 260, pp. 1-59. quant-ph/9912054[14] V. M. Braun, G. P. Korchemsky and D. Mueller, Prog. Part. Nucl. Phys. 51 (2003) 311.8

https://epub.uni-regensburg.de/32649/1/1402.5822v1.pdf

Conformal symmetry of the Lange-Neubert evolution equation

Abstract

Similar works

Full text

Available Versions

University of Regensburg Publication Server

University of Regensburg Publication Server