8 Deep Inelastic Scattering

Transcription

1 8 DEEP INELASTIC SCATTERING an again the ω i s will be fixe by momenta that are external to collinear loops. An example where this woul not be true is if we ha the same collinear irection n in two or more of our builing blocks, such as ω ω Cω, ω ] χ n,ω n / χ n,ω For this operator one combination of ω an ω will be fixe by momentum conservation, while the other combination will involve collinear loop momenta. This will lea to anomalous imension equations of a more complicate form, involving convolutions such as µ Cµ, ω = µ ω γµ, ω, ω Cµ, ω Inee, the operator in Eq is responsible for several classic evolution equations: i DIS where we have DGLAP evolution for the parton istribution functions f i/p ξ, ii har exclusive processes like γ π π where we have Brosky-Lepage evolution for the light-cone meson istributions φ π x, an iii the eeply virtual Compton scattering process γ p γp ' where the evolution is a combination of both of theses. It is interesting that all of these processes are sensitive to ifferent proections of the evolution of the single operator given in Eq We will carry out an example of an evolution equation with a convolution in the next section, where we consier DIS an the DGLAP equation. 8 Deep Inelastic Scattering RUGH DIS is a rich subect, so for the purpose of these notes we will treat only aspects relate to factorization an the renormalization group evolution with SCET. In particular we will emonstrate the factorization of momentum by showing that the forwar DIS scattering amplitue can be written as an integral over har coefficients times parton istribution functions. 8. Factorization of Amplitue The scattering process is epicte in the figure. The har scale of the process is efine by the photon Figure : Deep Inelastic Scattering momentum q µ q = 8. 65

2 8. Factorization of Amplitue 8 DEEP INELASTIC SCATTERING an satisfies» Λ. ur Borken variable x is efine in the stanar way an with momentum conservation efine by p µ q µ = p µ, we have X x = 8. p q X p p = x m. 8.3 x With this result we may etermine the various energy regions of the process Describe Parton Variables We will consier our scattering process in the stanar PE region so that the final state has p of X orer an can consequently be integrate out. Conversely, the proton with its comparatively small invariant mass p Λ may be treate as a collinear fiel. We analyze the process in the Breit Frame in which the perpenicular momentum component of q µ is zero with The proton an final state momentum are then Regions Description x = p X Stanar PE Region x Λ = p X Λ Enpoint Region x Λ = p X Λ Resonance Region µ q = n µ n µ. 8.4 µ µ The cross section for DIS in terms of leptonic an haronic tensors is n n p µ = n p n p 8.5 µ µ n n m p = n p n p 8.6 n µ = x 8.7 µ µ µ p X = p q 8.8 n µ n µ x =. 8.9 x 3 k ' πe 4 σ = L µν k, k ' W µν p, q 8. p k ' π 3 s 4 where k an k ' are the incoming an outgoing lepton momenta, respectively, an we have efine q k ' k, an s p k. L µν k, k ' is the leptonic tensor compute using stanar FT methos an W µν p, q is the haronic tensor which will occupy us in this section. W µν is relate to the imaginary part of the DIS scattering amplitue by W µν p, q = ImT µν 8. π where T µν p, q = p ˆT µν q p ˆTµν q = i 4 xe iqx T J µ xj ν ]

3 8. Factorization of Amplitue 8 DEEP INELASTIC SCATTERING Taking J µ to be an electromagnetic current, we may write q µq ν q T µ,ν p, q = g µν T q x, µ p µ p x which satisfies current conservation, P, C, an T symmetries. Matching the Tˆµν q onto the most general leaing orer SCET operator for collinear fiels in the n µ irection an satisfying current conservation qµ T ˆ µν we have µν ˆT µν g g nµ n µ n ν n ν g 8.4 where with igb λ an P ± efine as n / n,p n,p = ξ i W C P, P W ξ i 8.5 g = TrW λ B W C g P, P W B λ W ] 8.6 q ν Tx,. 8.3 x 8.7 λ igb in D n, idn, λ ], P ± = P ±P. 8.8 The subscripts in are arbitrary labels, similar to those foun in 8.3, which ifferentiate the two g parts of of Tˆµν. The superscript efines the flavor u,, s, etc. of quarks an the superscript g in g stans for a gluon. In accor with their labels, will lea to the quark an anti-quark PDF an will lea to the gluon PDF. The placement of factors of is one in orer to yiel imensionless Wilson coefficients. The fact that these Wilson coefficients are imensionless can be unersto by realizing that accoring to 8., Tˆµν has mass imension. In 8.4 there are both quark an gluon operators. However, with Tˆµν efine in terms of an electromagnetic current we can focus on the quarks an treat the gluons as an higher orer contribution so that Tˆµν becomes µν g n Tˆµν ˆ µ µ n µ n ν n ν. 8.9 Returning to the quark operator, we may introuce a convolution to separate the har coefficients from the long istance operators = ω ω C ω, ω ξ n W ω δω P / n W ξ n ω δω P] 8. where ω ± = ω ± ω. ur hope is to connect this operator to the PDF as a clear emonstration of factorization. The PDF for quarks is given by iξn py fi/pξ = ye p ξyw y, ynξ / y p 8. an the PDF for anti-quarks is simply f i/p ξ = f i/p ξ. In momentum space, we can write the matrix element in 8. as p ξyw y, y/ nξy p = p ξ n W ω n/ W ξ n ω p 8. = 4n p ξ δω 8.3 δω ξn pf i/p ξ δω n pf i/p ξ]. 8.4 ν 67

4 8. Factorization of Amplitue 8 DEEP INELASTIC SCATTERING The elta function over ω sets ω = ω. The other set of elta functions ensure that for ω > we use quark PDF f i/p z. an for ω < we use anti-quark PDF f i/p z. Using these results we may rewrite our operator i incluing averages as p p = 4 = 4 i Now, by charge conugation invariance reference, we have C ω, ω = Cω, ω so that the final form of the average matrix element is We note that although we are using SCET II no soft gluons have appeare in our analysis. This fact can be unerstoo by observing that our original operator = ξ i / W C P P W ξ i, n / = n,p n,p is a color singlet an therefore ecouples from any color-charge changing i.e. gluon interactions. With 8.9 we have the necessary result for a emonstration of factorization. Now all that is left to o is perform the matching of the full fiel theoretic operators T x, an T x, onto the operators. Recalling our formula for T µν in terms of Tˆµν, we have This is the SCET amplitue. The CD amplitue is Tµ,ν SCET qµq ν p, q = g µν q ωω C ω, ω ξ n W ω δω P n/ W ξ n ω δω P] 8.5 ω ω C ω, ω 4n p 8.6 ξ δω δω ξn pf i/p ξ δω n pf i/p ξ] 8.7 = n p Cn i pξ, f i/p ξ C n i pξ, f i/p ξ]. 8.8 i p p = n p Cn i pξ, f i/p zξ f i/p zξ]. 8.9 T µν = p T ˆ µν p 8.3 µν g = µ i 4n n ν i p p p p. 8.3 T x, Writing this result in light-cone coorinates an using the War Ientity q ν L µν = q µ =, an the q fact that all terms proportional to µ nµ n µ = become zero upon contraction with L µν, we have p µ q µ x Tµν CD = g µν Tx, n µν 4x T x, T x, p ν q ν Tx, 8.3 x L µν

5 8. Factorization of Amplitue 8 DEEP INELASTIC SCATTERING We refer the reaer to?] for a full erivation of this result. Matching T CD onto T SCET, yiels the relations i p p = T x, 8.34 i p p = T x, 4x T x, 8.35 which, upon inversion, gives T x, i = p p 8.36 = ξc i x n pξ, f i/p ξ f i/p ξ] 8.37 T x, 8x = p 3 4x = i x p i p 3 p 8.38 ξ 4C n pξ, C n pξ, f i/p ξ f i/p ξ] where in the Breit Frame x = p q = n pn q = n p. With the efinition H z C z,,,,, µ,, 8.4 where the har scale an the µ epenence has been mae explicit, we have the final result T x, = ξ H ξ x f x i/p ξ f i/p ξ] 8.4 T x, ] 4x ξ = ξ 4H H ξ f x x i/p ξ f i/p ξ] 8.43 where the sum over i is implicit. Remarks This result represents the general to all orers in α s factorization for DIS. As promise we have the computable har coefficients H i weighte by the universal non-perturbative PDFs f i/p an f i/p. The coefficients C are imensionless an can therefore only have α s µ lnµ/ epenence on. This result is in accor with Borken Scaling. The µ in H i µ an f i/p µ is typically calle the factorization scale µ = µ F. There is also the renormalization scale as in α s µ R. In SCET µ is both the renormalization an factorization scale, since the same parameter µ is responsible for the running of the EFT coupling α s µ an for the EFT coupling C µ. When we consier the tree level matching onto the wilson coefficients we fin that C = implying the Callan-Gross relation W = 8.44 W 4x 69

6 8. Renormalization of PDF 8 DEEP INELASTIC SCATTERING an that ] C ω = e i ω iɛ ω 8.45 iɛ ξ H = e i δ x Renormalization of PDF RUGH In this section we calculate the anomalous imension of the parton istribution function. We efine the PDF as f q ξ = p n χ n n/ χ n,ω p n 8.47 where ω = ξ n p n >. Since we have a forwar matrix element there is no nee to consier a momentum label ω ' on χ n, by momentum conservation it woul be fixe to ω ' = ω. We renormalize our PDF in our EFT framework with imensional regularization, noting that there are only collinear fiels an no ultrasoft interactions for this example. Collinear loop processes can change ω or ξ an also the type of parton. The renormalize PDF operators are given in terms of bare operators as fi bare ξ = ξ Z i ξ, ξ f ξ, µ The µ inepenence of the bare operators f i bare ξ yiels an RGE for the renormalize operators in MS, µ f i ξ, µ = ξ γ i ξ, ξ f ξ, µ 8.49 µ where γ i = At -loop we can take Z ξ, ξ '' = δ ii δξ ξ '' so that ii -loop γ i ξ Z ii ξ, ξ µ Z i ξ, ξ. 8.5 µ = µ Zi ξ, ξ ] -loop µ 8.5 Computing the PDF at tree level, we obtain n/ = u n u n δω p = δ ω/p 8.5 ' -n " p At the -loop level there are multiple contributions the first contribution yiels the computation p = ig C F l µ ɛ e ɛγ E l δl l i] l p ω 8.53 i] 4π ɛ g A ɛ = 4π ɛ Γɛe ɛγ E zθzθ z µ 8.54 α s C F = zθzθ z π ɛ ln A/µ ]

7 8. Renormalization of PDF 8 DEEP INELASTIC SCATTERING where A = p p z z with z = ω/p The next contribution is given by =ig n/ l u /n n C n lnu / {}}{{}}{ n F l p l l p δl ω δp ω] 8.56 C F α s µ z = e ɛγ E θzθ z p p z i ɛ Γɛ π z ɛ µ 8.57 p p ] ɛ z i Γ ɛγ ɛ δ z µ 8.58 Γ ɛ We can simplify this result with use of the istribtuion ientity. real virtual θ z δ z = L z EL z 8.59 z E E where the plus function L n x is efine as θ x ln n ] x L n x = x an satisfies the following ientities 8.6 ln n x x L n x =, L n xgx = x gx g]. 8.6 x With this replacement we fin that the /E terms in the real an virtual terms cancel an the remaining /E is UV ivergent. In the en the explicit contribution of this process is C F α s µ = { δ z zθzl z} π ɛ ln µ 8.6 p p z i ] π zl zθz δ z The last conrtibution to the renormalize PDF is the wavefunction renormalization of the external fermions. αsc F F ig = δ zz ψ = π 4ɛ 4 4 ln µ ] δ p p z 8.64 i There are aitional contributions from iagrams such as those in, but we will ignore these by assuming that the operator is not a flavor singlet. Summing the various contributions, we have { C F α s µ 3 Sum = δ z zθzl z π 4 } z θzθ z ɛ ln µ ] p p finite function of z z i C F α s µ = z π z ɛ ln µ p p finite function of z z i }{{} Determines Zqq -loop

8 8.3 General Discussion on Appearance of Convolutions in SCET I an SCET II If we let the total momentum of the haronic state be ˆp. Then efine p /pˆ -loop Then our Z qq becomes = ξ. So that ω ξpˆ ξ z = = = 8.66 p ξ ' pˆ ξ ' qq = δ z α sµ z C F θzθ z ɛ π z Z -loop An usng γ i = µ Z i z, µ, µ α s µ = Eα s µ βα s µ] 8.68 µ µ we then obtain the our final result C γ qq ξ, ξ F α s µ θξ ξθ ξ z = π ξ 8.69 z which is the Aliterelli - Parisi DGLAP quark anomalous imension at one-loop General Discussion on Appearance of Convolutions in SCET I an SCET II 9 Diet Prouction, e e ets RUGH The prouction of ets at an e e collier has historically been very important. Measurements of various et in e e collisions were use to valiate CD as the correct theory of the strong interaction, an to this ay, even years after the LEP has been turne off, measurements of event shape istributions are being use to stuy the nature of the strong interaction an to etermine funamental constants of nature such as the coupling constant of the strong interaction. The ominant kinematical situation in e e ets is to prouce two ets, but of course a larger number of ets can be obtaine by the emission of aitional har strongly interacting particles. In this section we will iscuss the prouction of two ets in e e collisions, which is to say the prouction of energetic particles in two back-to-back irections, accompanie only by usoft raiation in arbitrary regions of phase space. Clearly, the question whether we have or more ets has to be etermine on an event by event basis, an there are many possible observables which can istinguish -et events from events with more than ets. The most natural efinition might be to use a et fining algorithm, an select those events with exactly two har ets as efine by this algorithm. However, there is another set of observables which can be use to ientify -et events, an which are much easier to analyze theoretically. This class of observables are calle event shapes, with the most well known event shape variable being thrust. In this section, we will only iscuss the thrust istribution in e e collisions, but it shoul be clear from the iscussion how one can exten the results to other event shape variables or other -et observables. 9. Kinematics, Expansions, an Regions The thrust of an event is efine as follows: i p T = max i n T nt i p i 9. 7

9 MIT pencourseware Effective Fiel Theory Spring 3 For information about citing these materials or our Terms of Use, visit: