7 Wilson Coefficients and Hard Dynamics

Transcription

1 7 Wilson Coefficients an Har Dynamics 7 WILSON COEFFICIENTS AND HARD DYNAMICS We now turn to the ynamics of SCET at one loop. An interesting aspect of loops in the effective theory is that often a full QCD loop graph has more than one counterpart with similar topology in SCET. We will compare the SCET one loop calculation for a single har interaction current with the one loop calculation in QCD. Our goal is to unerstan the IR an UV ivergences in SCET an the corresponing logarithms, as well as unerstaning how the terms not associate to ivergences are treate. In our analysis we will use the same regulator for infrare ivergences, an show that the IR ivergences in QCD an SCET exactly agree, which is a valiation check on the EFT. The ifference etermines the Wilson coefficient for the SCET operator that encoes the har ynamics. This matching result is inepenent of the choice of infrare regulator as long as the same regulator is use in the full an effective theories. Finally, the SCET calculation contains aitional UV ivergences, beyon those in full QCD, an the renormalization an anomalous imension etermine from these ivergences will sum up ouble Suakov logarithms. We will give two examples of matching QCD onto SCET, the b sγ transition, an e e - jets. The first example has the avantage of involving only one collinear sector, but the isavantage of requiring some familiarity with Heavy Quark Effective theory for the treatment of the b quark an involving contributions from two Dirac structures. The secon example only involves jets with a single Dirac structure, but has two collinear sectors. In both cases we will use Feynman gauge for all gluons, an imensional regularization with = 4 ɛ for all UV ivergences enoting them as 1/ɛ. To regulate the IR ivergences we will take the strange quark offshell, p 0. For IR ivergences associate purely with the heavy quark we will use imensional regularization enoting them 1/ɛ IR to istinguish from the UV ivergences. 7.1 b sγ, SCET Loops an Divergences As a 1-loop example consier the heavy-to-light currents for b sγ. Although there are several operators in the full electroweak Hamiltonian, for simplicity we will just consier the ominant ipole operator QCD F µν where F µν is the photon fiel strength an the quark tensor current is J µν J QCD = s Γb, Γ = σ µν P R. 7.1 In SCET the corresponing current for the original Lagrangian, prior to making the Y n fiel reefinition was J SCET = ξ n W Γh v C v n P = ω Cω χ n,ω Γh v. 7. In general because of the presense of the vectors v µ an n µ there can be a larger basis of Dirac structures Γ for the SCET current we will see below that at one-loop there are in fact two non-zero structures for the SCET tensor current. Note that the factor of v n makes it clear that the current preserves type-iii RPI. We will set v n = 1 in the following. Together with the QCD an leaing orer SCET Lagrangians, we can carry out loop calculations with these two currents. First lets consier loop corrections in QCD. We have a wavefunction renormalization graph for the heavy quark enote b, an one for the massless strange quark enote q: b q 5

2 7.1 b sγ, SCET Loops an Divergences 7 WILSON COEFFICIENTS AND HARD DYNAMICS This gives the wavefunction renormalization factors Z ψb an Z ψ respectively. In the on-shell scheme which inclues both the UV ivergences an the finite resiues these Z-factors are Z ψb = 1 s C F 1 µ 3 ln 4 ɛ ɛ IR m, b Z ψ = 1 s C F 1 ɛ ln p 1 µ. 7.3 If one instea uses MS for the wavefunction renormalization factors, then the finite resiues still show up in the final result for the S-matrix element ue to the LSZ formula. The remaining iagram is a vertex graph for the tensor current J QCD. At tree level the matrix element gives while the one-loop iagram V qc 0 = u pp R iσ µν s u b p b 7.4 p b p gives V 1 qc = s C F ln p p m ln b m b ɛ 1 p ln µ ln µ ω 3 ln µ f 1 1 qˆ ū s P R iσ µν u b m b s C F p f 1 qˆ µ γ ν p ν γ µ ū s P R ub, 7.5 m b where we have kept p = 0 only for the IR singularities, an set it to zero whenever it is not neee to regulate an IR ivergence. The variable ˆq = p b p /m b = 1 p b p/m b an the functions appearing in Eq. 7.5 are f 1 x = lnx lnx Li 1 x π 4, f x = lnx x 1 x Unlike for the conserve vector current, in QCD for the tensor current the sum of vertex an wavefunction graphs still contains a 1/E UV ivergence. Hence this QCD local current operator requires an aitional counterterm not relate to strong coupling renormalization, an it is given by s C F 1 Z tensor = E Aing together the QCD vertex graph an the contributions from the three Z s, an replacing the kinematic variable ˆq = 1 n p/m b = 1 ω/m b, the sum gives 1 QCD Sum = Vqc 1 1 Z ψ b 1 Z ψ 1 Ztensor 1 1 Vqc 0 s C F = u s Γu b ln p ω 3 p ln ω 1 ln µ ω 5 ɛ IR ω f 1 m b sc F ω p f ū s P R m µ γ ν p ν γ µ ub, 7.8 b m b 53

3 7.1 b sγ, SCET Loops an Divergences 7 WILSON COEFFICIENTS AND HARD DYNAMICS Next consier the ultrasoft loops in SCET. In Feynman gauge the ultrasoft wavefunction renormalization of the collinear quark vanishes, since the couplings are both proportional to n µ, an n = 0. The ultrasoft wavefunction renormalization of the heavy quark is just the HQET wavefunction renormalization. We summarize these two results as: Z us ξ n n µ n µ = 0, sc F h v = 1 ɛ ɛ. IR Z us 7.9 We can alreay note that the 1/E IR pole in Z h us v matches up with the IR pole in Z ψb in full QCD an this is the only IR ivergence that we are regulating with imensional regularization. In aition to wavefunction renormalization there is an ultrasoft vertex iagram for the SCET current. Using the on-shell conition v p b = 0 for the incoming b-quark, an the SCET propagator from Eq for a line with injecte ultrasoft momentum, we have us = s C F V 1 where the tree level SCET amplitue is = V 1 us = ig ic F ū n Γu v 1 ɛ µ n p ɛ ln p ln µ n p i0 p i0 k µ ɛ ι ɛ n v v k i0n k p /n p i0k i0 3π 0 Vscet, V 0 = u n Γu v, 7.11 scet an ι E = E e Eγ E ensures that the scale µ has the appropriate normalization for the MS scheme. Note that this graph is inepenent of the current s Dirac structure Γ. On the heavy quark sie the heavyquark propagator gives a P v = 1 v//, but this commutes with the HQET vertex Feynman rule an hence yiels a projector on the HQET spinor, P v u v = u v. On the light quark sie the propagator gives a n// an the vertex gives a n // to yiel the projector P n = n/n //4 acting on the light-quark spinor, P n u n = u n. Hence whatever Γ is inserte at the current vertex is also the Dirac structure that appears between spinors in the answer for the loop graph. For this heavy-to-light current this feature is actually true for all loop iagrams in SCET, the spin structure of the current is preserve by loops iagrams in the EFT. For ultrasoft iagrams it happens by a simple generalization of the arguments above, while for collinear iagrams the interactions only appear on the collinear quark sie of the Γ, so we just nee to know that they o not inuce aitional Dirac matrices. This is ensure by chirality conservation in the EFT. Lets finally consier the one loop iagrams with a collinear gluon. There is no wavefunction renormalization iagram for the heavy quark, since the collinear gluon oes not couple to it. There is a wavefunction renormalization graph for the light-collinear quark =... = n/ p C F s 1 p ln n p ɛ C 1 µ, so Z ξn = 1 F s 1 p ln ɛ µ We have not written out the SCET loop integran, but it follows in a straightforwar manner from using the collinear quark an gluon propagators an vertex Feynman rules from Fig.. Note that the result for Z ξn is the same as the full theory Z ψ. This occurs because for the wavefunction graph there is no connection 54

4 7.1 b sγ, SCET Loops an Divergences 7 WILSON COEFFICIENTS AND HARD DYNAMICS to the ultrasoft moes or the har prouction vertex, an by itself a single collinear sector is just a booste version of full QCD an Z ψ is inepenent of this boost. There are also no subtelties relate to zero-bin subtractions for this graph the subtraction integrans are power suppresse an therefore the subtraction vanishes. There is also a iagram generate by the two-quark two-gluon Feynman rule, but this tapole type iagram vanishes with our choice of regulators. There is also a tapole type iagram where two gluons are taken out of the Wilson lines in the vertex, which also vanishes, ie. = 0, = The last iagram we must consier is the collinear vertex graph with an attachment from the Wilson line going to the collinear quark propagator, k p k p = Vn 1 = ig k r n n n p l k l C F u n Γu v µ ɛ ι ɛ n k l k k p k l = 0 k l = p l = ig C F ˆ u n Γu v V 1 n µ µ µ c c r Here each momentum has been split into label an resiual components k = k µ, k µ r an p = p, p. There are no -momenta in the label components, an the only resiual component for the external p is its -momentum. For reasons that will soon become apparent, we have use a short han notation for the relativistic collinear gluon an quark propagators, which in fact contain a mixture of label an resiual momenta, k = k k p k, k p = k p k p p r k p c c r r c c c c, 7.15 an are homogeneous in the power counting with k p λ. We have also introuce the notation with a hat, Vˆ 1 n, for the collinear loop integran. In general in collinear loop integrals there can be a nontrivial interplay between the Wilson coefficients an the large collinear loop integration, because both epen on a momentum that is the same size in the power counting, namely the large minus momenta, k Q. When matching at one-loop, O s, in some cases the tree level har matching coefficient we insert might be inepenent of the loop momentum k. In this case we can insert it back into the calculation only at the en. Even in this case it must be inclue when consiering the renormalization group evolution, because the sharing of large momenta can lea to convolutions in the RG evolution equations. We will meet an example of this type later on when we iscuss the running of parton istributions for a collinear proton. For our example of the heavy-to-light current for b sγ, things are actually simple for a ifferent reason. The SCET operator in Eq. 7. contains only a single gauge invariant prouct of collinear fiels, ξ nw, an the Wilson coefficient only epens on the overall outgoing momentum of this prouct. Therefore if we inclue a coefficient into our iagram in Eq it gives only epenence on the total external momentum This result remains true for collinear loop iagrams at higher orers, so the coefficient can always be treate as multiplicative for this current, an the coefficient is always evaluate with the total -momentum C n p k n k = C n p

5 7.1 b sγ, SCET Loops an Divergences 7 WILSON COEFFICIENTS AND HARD DYNAMICS of the collinear jet, which in this case is n p = m b. Inee, even when we have collinear fiels for multiple irections, the large momentum are still fixe by the external kinematics as long as we have only onegauge invariant prouct of collinear fiels in each irection. In this case the Wilson coefficient for the har ynamics remains multiplicative in momentum space. An we remark that this is the case that is preominantly stuie for amplitues for LHC processes with an exclusive number of jets. In general the coefficient will still be a matrix in color space once we have enough colore particles to give more than one possibility for making an overall color singlet 4 particles. There is only one possibility for the current example an hence no matrix in color space. When we have more than one block of gauge invariant collinear fiels in the same collinear irection then this will no longer be true, there will be momentum convolutions between the har coefficient C an the collinear parts of the SCET operator. To perform the collinear loop integration in Eq we shoul follow the rules from section 4.5 on combining label an resiual momenta. As a first pass we will ignore the 0-bin restrictions k c = 0, p c. In this case we can apply the simple rule from Eq Results following this rule in SCET I are often calle the naive collinear integrals. Since only momenta of external collinear particles appear in the loop integran the multipole expansion is trivial for this integral, an this gives the same result that we woul have obtaine by ignoring the split into label an resiual momenta from the start: ˆV 1 naive n = µ ɛ ι ɛ k n n n p k n kk k p i = ɛ ɛ µ µ ɛ ln p ln p µ ln p 4 π This result for the loop integral can be obtaine either with stanar Feynman parameter rules or by contour integration in k or k. Feynman parameter tricks an other equations that are useful for oing loop integrals in SCET are summarize in Appenix E. Having assemble results for all the SCET loop graphs we can now a them up to obtain the bare SCET result 1 Sum SCET = Vus 1 Vn 1 Zus h v 1 1 Z ξ n 1 Vscet 0, 7.18 an then compare with the full QCD calculation, setting the renormalize coupling g = s µ. For the moment we still will label our SCET result as naive since it ignores the 0-bin restrictions. If we examine the IR ivergences encoe in the ln p factors an the 1/E IR from the heavy quark wavefunction renormalization then we fin for Γ = P R iσ µν that at leaing orer V qc 0 = V scet 0 an Sum QCD ren Sum SCET naive = s C F = s C F p ln 3 p ln m b Thus the results match up in the IR as long as the remaining 1/E terms in the SCET result can be interprette as UV ivergences. To obtain this result for the sum of the SCET iagrams there is an important cancellation between the collinear an ultrasoft iagrams, ln p /µ /E ln p /µn p/e = ln n p/µ/e = lnµ/m b /E. The cancellation of the ln p epenence in this 1/E pole is crucial both to match the IR ivergences correctly in QCD, an in orer for the remaining 1/E pole to possibly have an ultraviolet interpretation. The remaining epenence on n p = m b in the 1/E pole is fine because this m b p ln 3 p ln m b m b 1... ɛ IR V 0 scet..., 1 ɛ IR 1 ɛ 5 ɛ ɛ ln µ m b... Vscet

6 7.1 b sγ, SCET Loops an Divergences 7 WILSON COEFFICIENTS AND HARD DYNAMICS is the large momentum that the Wilson coefficient anyway epens on. This same cancellation also has a reflection in the ouble logarithms where the lnµ epenence cancels out from the ln p epenent term. Again this cancellation is important for the matching of IR ivergences with the full theory. The final catch is relate to our use of the naive collinear integran is the interpretation of the 1/E poles from the collinear loop integral. The 1/E ivergences from the ultrasoft vertex iagram are clearly etermine to be of UV origin from large eucliean momenta or large light-like momenta. However in the collinear vertex iagram with the naive integral one of the ivergences actually comes from n k 0, an hence is of IR origin. This IR region is actually alreay correctly accounte for by the ultrasoft iagram where the heavy quark propagator is time-like, v k i0, as it shoul be in the infrare region. In this region the original propagator oes not behave like n k. The n k term which comes from the collinear Wilson line W is instea the appropriate approximation for large n k, rather than small n k. Thus the issue with the naive collinear loop integral for the vertex iagram is that is ouble counts an IR region accounte for by the ultrasoft iagram. This ouble accounting is remove once we properly consier the 0-bin subtraction contributions. Therefore we apply now the rule with the 0-bin subtractions k c = 0, p c using Eq.4.4 to obtain It is easy to see where the 0-bin integran comes from because it can be obtaine from the appropriate ultrasoft scaling limit of the naive collinear integran. For k c = 0 we have a subtraction for the region k c λ where we only keep terms up to those scaling as λ 8, which gives precisely the integran in 1,0bin Eq. 7.0 enote as Vˆn. The terms with n k an n k in the enominator count as λ, while the term with k λ 4 to give the eight powers that compensate the k λ 8 for the subtraction. Note that we have kept the offshellness 0 = p λ since it is the same orer as the n pn k term. The other subtraction is k c = p c so we have the subtraction region k c p c λ. For this case one of the factors in the enominator is n k n p λ 0 an there is suppresion from the numerator as well so there is no contribution at Oλ 8. Being more careful about the UV 1/E an IR 1/E IR ivergences we fin ˆV 1,naive i n = ɛ IR ɛ ɛ ln µ ɛ IR p ɛ ln µ ɛ IR n p ˆV 1,0bin i n = ɛ 1 µ ln ɛ IR ɛ p ln µ, n p V 1 s C F n = ɛ ɛ µ µ ɛ ln p ln µ p ln p So we see that the subtraction cancels the n q 0 IR singularities 1/E IR in the first line. The UV ivergences arising from n q are inepenent of the IR regulator an just epen on the UV regulator E. Since the 0-bin contribution is scaleless with our choice of regulators, taking E IR = E an ignoring this subtraction woul give us the correct answer. Nevertheless, even with this regulator the 0-bin contribution is still important to obtain the correct physical interpretation for the ivergences. Since the final result after subtracting the 0-bin contribution is the same as in Eq with the 1/E poles all now known to be UV, we can etermine the appropriate UV counterterm to renormalize the SCET current. Defining ˆV 1 n = µ ɛ ι ɛ k n n n p k n kk k p n n n p = Vˆ 1,naive ˆV 1,0bin. 7.0 n kk n p n k p n n µ µ ln ln p p 4 π, 4 π. 7.1 C bare ω, E = Z C µ, ω, ECµ, ω = C Z C 1C, 7. k, Ω k For other less inclusive calculations or for other choices of regulators such as Ω k Λ Λ the subtractions are even more crucial to obtain the correct result an have the UV ivergences inepenent of the IR regulator. 57

7 7. e e -jets, SCET Loops 7 WILSON COEFFICIENTS AND HARD DYNAMICS an aing the counterterm graph with Z C 1C to cancel the 1/E poles in MS gives From these two results we see that the renormalize QCD an SCET have the same infrare ivergences. The ifference of these results is etermine by ultraviolet physics an etermines the one-loop matching result for the MS Wilson coefficients C 1 µ, ω, m b an C µ, ω, m b that multiply the SCET operator in Eq. 7. for the Dirac structures Γ = Γ µ γ ν ν γ µ 1 = P R iσ µν an Γ = Γ = P R n n respectively. Only the Dirac structure Γ 1 was present at tree-level, while Γ is generate at one-loop. Taking the ifference of the above two results an simplifying we fin C C 1 µ, ω, m b = 1 F s µ µ ln µ ω 11π 5 ln f 1, ω ω m b 1 C F s µ ω ω C µ, ω, m b = f. 7.5 m b m b 7. e e -jets, SCET Loops In this section we perform the matching from QCD onto SCET for the process e e -jets. This matching will be inepenent of the etails of the kinematical constraints that are use to enforce that we really are restricting ourselves to have only jets in the final state, which will all be containe in the long istance ynamics of the effective theory. Inee, the fact that we can successfully carry out this matching at the amplitue level makes it clear that it oes not epen on which constraints we put on the phase space of the -jet final state. Once again, it will also be inepenent of the choice of IR regulator as long as the same regulator is use in both the QCD an SCET calculations. We will use Feynman gauge in both QCD an SCET, an take = 4 E to regulate UV ivergences an offshellness for the quark an antiquark, p q = p q = p = 0, to regulate all IR ivergences. C Z C µ, ω, ɛ = 1 F s µ 1 1 µ 5 ɛ ln ɛ ω O ɛ s. 7.3 Where by momentum conservation ω = m b. We can now a up the collinear an ultrasoft loop graphs to obtain the final renormalize SCET result, an compare with the renormalize QCD result Sum QCD ren = s C F 1 ln p ɛ IR ω 3 p µ ω ln ω ln f Vscet ω m b s C F ω p µ γ ν p ν γ µ f ū s P R ub, m b m b ren 1 Sum SCET = Vus 1 Vn 1 Zus h v 1 1 Z ξ n 1 Z C 1 Vscet 0 = s C F 1 ln p ɛ IR ω 3 p ln µ ln µ 11π ω 1 7 Vscet In full QCD, the prouction of harons in e e collisions occurs via an s-channel exchange of a virtual photon or a Z boson. The coupling is either via a vector or an axial vector current an is therefore given by J QCD = q Γ i q, Γ V = g V γ µ, Γ A = g A γ µ γ 5, 7. where g V,A contain the electroweak couplings for the photon or Z-boson for a virtual photon g V = e q the electromagnetic charge of the quark q, an g A = 0. In SCET the current involves collinear quarks in the 58

8 7. e e -jets, SCET Loops 7 WILSON COEFFICIENTS AND HARD DYNAMICS back-to-back n an n irections J SCET = ξ n W n Γ i C P n, P n, µ W n ξ n = ω ω Cω, ω χ n,ω Γi χ n,ω. 7.7 By reparametrization invariance of type-iii the epenence on the label operators can only be in the combination ωω ' insie C, so C ω, ω = C ωω. 7.8 Finally in the CM frame momentum conservation fixes ω = ω ' = Q, the CM energy of the e e pair, so wee can write J SCET = CQ ξ n W n Γ i W n ξ n, 7.9 an the matching calculation in this section will etermine the renormalize MS Wilson coefficient CQ, µ. In this case there is only one relevant Dirac structure Γ i in SCET for each of the vector an axial-vector currents. We again begin by calculating the full theory iagrams. As in the case of B X s γ we nee the wave function contributions for the light quarks, in this case one for the quark an one for the anti-quark. Both wave function contributions are the same as the results obtaine before Z ψ = 1 s C F 1 p ln ɛ µ The remaining vertex graph can again be calculate in a straightforwar manner. At tree level we fin while the one loop vertex iagram n V 0 = ūp n Γ i v p 7.31 qc n n p q p q gives qc = µ ɛ ι ɛ k i ig ūp q γ T A p/ q k/ ip/ q k/ i π Γ i p q k igγ T A vp q p q k k = ig C F µ ɛ k π ūp q γ p/ q k/γ i p/ q k/ γ vpq p q k p q k k s C F 1 p p = ln 5 ln ɛ Q Q ln Q i0 µ π 1 ūp q Γ i vp q V 1 Here ι E = E e Eγ E ensures that the scale µ has the appropriate normalization for the MS scheme. Aing the QCD iagrams we fin 1 QCD Sum = Vqc 1 Z ψ 1 Vqc 0 s C F p = ln p 4 Q ln Q π ln Q µ ūp q Γ i vp q

9 7. e e -jets, SCET Loops 7 WILSON COEFFICIENTS AND HARD DYNAMICS As before, we next consier the loops in SCET. The wave function renormalization for the collinear quark is the same as in the previous section, an we fin C ξ = 0, Z ξ = 1 F s 1 p ln ɛ µ Z us The tree level amplitue in SCET is V 0 = p q Γ i v p, an to leaing orer V 0 = V 0 The scet u n n q qc scet. ultrasoft vertex graph in SCET involves an exchange between the n-collinear an n-collinear quarks, an is given by usoft = µ ɛ ι ɛ k n / in / ū n π ig n T A n p q in / n p q n / i Γi n p q n k p ig q n p q n k p n T A v n q k = ig C F µ ɛ ι ɛ n/n / n/n/ k n n ū n Γ i 4 4 v n π n k p q n p q n k p q n p q k s C F ɛ p4 p4 = ln ɛ µ ln Q µ Q π ūnp q Γ i v n p q V 1 There are two possible collinear vertex graphs which involve a contraction between the W n n A n Wilson line an a n-collinear quark, an another between the W n n A n Wilson line an the n-collinear quark For the first iagram, we fin Vcoll 1 = µ ɛ ι ɛ k γ ig ū n π n p/ p/ k/ γ p/ p/ k/ n /T A n p n p k n pn p k n/ n p k i p k g n n k T A i Γi v n k = ig C F µ ɛ ι ɛ k n n n p k ū π nγ i v n n k p k k s C F ɛ ɛ p p p = ln ln ln ɛ µ µ µ 4 π ūnp q Γ i v n p q. 7.3 One can easily show that the secon collinear vertex iagram gives the same result as the first iagram. Furthermore the collinear integral here is ientical to the one for b sγ in Eq The result in Eq. 7.3 is for the naive integran, since it oes not inclue the 0-bin subtraction contribution. But the 0

10 7. e e -jets, SCET Loops 7 WILSON COEFFICIENTS AND HARD DYNAMICS 0-bin subtraction terms here are scaleless as in Eq. 7.1, an hence the final result in Eq. 7.3 is correct with the interpretation of the 1/E ivergences as UV. Aing the SCET iagrams we fin after some straightforwar manipulations 1 SCET Sum = Vusoft 1 V coll 1 Z ξ 1 Vscet s C F 3ɛ ɛ Q = ln ɛ µ ln µ p ln µ Q µ 4 ln p4 p 8 5π ūnγ i v n s C F 3ɛ ɛ Q = ln ɛ µ ln p Q ln Q µ 4 ln p Q 4 ln Q µ 8 5π ūnγ i v n. Comparing the lnp epenence in the final line to the QCD amplitue in Eq We can see that SCET reprouces all IR ivergences of the form ln p /Q, an that the matching coefficient is therefore inepenent of IR ivergences as it shoul. However, while the matrix element of the full QCD current is UV finite since it is a conserve current, the matrix element in the effective theory is UV ivergent an therefore nees to be renormalize. Defining a renormalize coupling by CQ, E = Z C µ, Q, ECµ, Q = C Z C 1C 7.38 the renormalization constant that cancels the ivergences in Eq is C F s µ Z C = 1 ɛ 3 ɛ Q ɛ ln i0. µ Taking the ifference between the renormalize matrix elements in full QCD an SCET, ren s C F QCD sum = ln p p 4 Q ln Q ln Q µ π ūp n Γ i vp n, ren s C F p SCET sum = ln Q ln Q µ 4 ln p Q 4 ln Q µ 8 5π ūnγ i v n, we obtain the matching result for Wilson coefficient of the operator in Eq. 7.9 at one-loop orer C F s µ Q Cµ, Q = 1 ln i0 Q i0 µ 3 ln µ 8 π Note that the only momentum epenence in the Wilson coefficient is in logarithms of the ratio of the renormalization scale to the har scale Q. This epenence signals that it captures offshell physics from the har scale Q that we are integrating out. If we choose the renormalization scale to be equal to Q, we fin that all logarithms vanish C F s Q 7π CQ, Q = iπ. 7.4 Sometimes it is useful to avoi inucing large factors of π in the non-logarithmic terms, which can be accomplishe by using a complex scale, µ = iq. Here this gives CF s iq C iq, Q = 1 8 π For ijet observables escribe by the current in Eq. 7.9 the cross section is obtaine by squaring the amplitue, an will epen on a har function efine by Hµ, Q = Cµ, Q Thus the imaginary contributions in Cµ, Q cancel out for these observables

11 7.3 Summing Suakov Logarithms 7 WILSON COEFFICIENTS AND HARD DYNAMICS 7.3 Summing Suakov Logarithms With the information from either of the last two sections, we can calculate the anomalous imensions of the opertors or Wilson coefficients. Taking 0 = µ C bare ɛ = µ ZCµ, ɛcµ = µ Z C µ, ɛ Cµ Z C µ, ɛ µ Cµ, 7.45 µ µ µ µ we see that the anomalous imension is efine by a erivative of the counterterm µ Cµ = Zc 1 µ, ɛµ Z c µ, ɛ Cµ γ C µcµ. 7.4 µ µ To calculate the µ erivative we shoul recall the result for the erivative of the strong coupling in imensions µ s µ, E = E s µ, E β s, 7.47 µ where β s is the stanar = 4 QCD beta function written in terms of s µ, E. Lets apply this to our two examples in turn. The counterterm for the b sγ current is γ Z C = 1 s µc F 1 ɛ ɛ ln µ ω 5. ɛ where we ifferentiate both s µ an the explicit lnµ, noting that the 1/E terms cancel to yiel a well efine anomalous imension in the E 0 limit which is given on the last line. Similarly, the counterterm for the e e ijets current is jet C F s µ Z C = 1 ɛ 3 ɛ Q ɛ ln i0, µ Again in the last line we have taken the E 0 limit. Note the similarity in the form of the anomalous imensions for our two examples of Wilson coefficients. Both anomalous imension equations for Cµ are homogeneous linear ifferential equations because in both cases the operator mixes back into itself Using the efinition of γ C in Eq. 7.4 we fin γ 1 γ C µ, ω, ɛ = Z γ µ C µ Zγ C = µ C F s µ, ɛ 1 µ ɛ ɛ ln µ ω 5 ɛ CF s µ, ɛ = 4 ln µ ɛ ω 5 O ɛ s, γ γ C µ, ω = s µ 4C F ln µ 5C F, 7.49 ω 7.50 so the anomalous imension is obtaine by jet 1 γ C µ, Q, ɛ = µ Z jet µ Zjet C = µ C F s µ, ɛ µ ɛ 3 ɛ ɛ ln µ Q i0 C C F s µ, ɛ 4 = ɛ 4 ln µ Q 4 O i0 ɛ s, jet γ C µ, Q = s µ µ 4C F ln Q C F i0

12 7.3 Summing Suakov Logarithms 7 WILSON COEFFICIENTS AND HARD DYNAMICS An interesting feature of anomalous imensions in SCET is the presence of a single logarithm, lnµ. It can be shown by the consistency of SCET, or by consistency of top-own versus bottom-up evolution using a factorization theorem for a process with Suakov logarithms, that no terms with more than a single logarithm can appear in anomalous imensions. The coefficient of this single logarithm is relate to the cusp anomalous imension that governs the renormalization of Wilson lines that meet at a cusp angle β ij between lines along the four vectors n i an n j, where cosh β ij = n i n j / n i n j. In the light-like limit n i, nj 0 we have β ij. The cusp anomalous imension is linear in β ij in this limit, which yiels a logarithmic epenence on n i n j / n i n j since cosh β ij c e β ij /. This single logarithm is the same one encountere in Eqs an 7.50, where the ivergence has been hanle by the renormalization proceure, an hence has become a lnµ. Inee, if we consier making the BPS fiel reefinition for the ijet current we get Y n Y n, so it is clear that our ultrasoft iagrams involve two light-like Wilson lines meeting at a cusp. In the case of the collinear iagrams we have a Wilson line W n that meets up with a collinear quark ξ n, an in oing so also effectively forms a cusp. The all orers form for the anomalous imension of our two example currents is µ γ C µ, ω = a C Γ cusp s µ ln γ C s µ, ω C Γ cusp s = s kγ cusp s kγ k, γ C s = k C, 7.5 k=1 k=1 where Γ cups s is calle the cusp-anomalous imension, an the one-loop result has Γ cusp 1 = 4. The constant prefactor a C, the imensionful variable ω C, an the non-cusp anomalous imension γ C s all epen on the particular current uner consieration. In orer to solve the anomalous imension equation we shoul ecie what terms must be kept at each orer in perturbation theory that we woul like to consier. Counting s lnµ 1, the correct grouping for obtaining the leaing-log LL, next-to-leaing log NLL, etc., results is γ C µ, ω s lnµ s LL s lnµ NLL s s 3 lnµ NNLL Thus we see that the cusp-anomalous imension with the lnµ is require at one-higher orer than the non-cusp anomalous imension. Typically this is not a problem ue to the universal form of the cusp contribution, an the fact that its coefficients are known to 3-loop orer for QCD, that is up to Γ cusp 3. To solve the first orer ifferential equation involving γ C we also must specify a bounary conition for Cµ, ω. At both LL an NLL orer the tree-level bounary conition suffices, while at NNLL we nee the one-loop bounary conition, etc. Lets solve the generic anomalous imension at LL orer where sµ µ µ ln Cµ, ω = 4a C ln µ ω = a C s µ π µ ln. ω 7.54 This equation may be solve for specific quantum fiel theories. For QED without massless fermions the coupling oes not run, an with the tree-level bounary conition Cµ = ω, ω = 1 O s we have Cµ, ω = exp a C ln µ π ω This result involves an exponential of a ouble logarithm, an is often referre to as the Suakov form factor. The suppression encoe in this result is relate to the restrictions in phase space that are intrinsic for the allowe types of raiation that our operators can emit. The Suakov form factor also gives the 3

13 7.3 Summing Suakov Logarithms 7 WILSON COEFFICIENTS AND HARD DYNAMICS probability of evolving without branching in a parton shower. For QCD we must also account for the running of the coupling, an at LL orer we can use the LL β-function, β µ s µ = s µ, β 0 = C A T F n f. 7.5 µ π 3 3 Together Eqs an 7.5 are a couple set of ifferential equations. The easiest way to solve these two equations is to use the secon one to implement a change of variable for the first by noting that s π s µ π sµ ln µ = =, ln = β s β 0 ω Using the more generic bounary conition which fixes the coefficient at the scale µ 0, Cµ 0, ω = 1O s we then have π sµ s a C s s ln Cµ, ω = β 0 sµ 0 s π sω a sµ C s = β0 1 1 sµ 0 s s s ω ac 1 = β0 s µ 1 s µ 0 1 s ω ln s µ s µ 0 a C 1 = β0 sµ 0 z 1 ln z a C ω ln ln z, 7.58 β 0 µ 0 where in the last line we use 1/ s ω = 1/ s µ 0 β 0 lnω/µ 0, an efine π s µ z s µ 0 This result sums the infinite tower of leaing-logarithms in the exponent which are of the form, C exp s L s L 3 s 3 L 4..., where the coefficients here are schematic an L = lnµ/µ 0 is a potentially large logarithm. Again this result is calle the Suakov form factor with a running coupling. Note that the form of the series obtaine by expaning in the argument of the exponent is much simpler than what we woul obtain by expaning the exponent itself. At each orer in resumme perturbation theory the terms that are etermine by solving the anomalous imension equation can be classifie by the simpler series that appears in the exponential as follows ln C L s s s L k L k s L k A natural question to ask is how generic are the two examples treate so far in this section? It turns out that much of the structure here is quite generic for cases like our examples, where the ω variables are fixe by external kinematics. This will occur for any operator that involves only one builing block, χ n or B µ n s The solution is therefore a C 1 ω Cµ, ω = exp β0 sµ 0 z 1 ln z ac ln z/β µ k LL k, for each collinear irection n. For example, with four collinear irections we have the operator ω 1 ω ω 3 ω 4 Cω 1, ω, ω 3, ω 4 χ n1,ω 1 Γ µν B µ n,ω Bn ν 3,ω 3 χ n4,ω 4 7. NLL k β 0 0 sω NNLL 4

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