QCD, Colliders & Jets - HW II Solutions. x, x

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1 QCD, Colliders & Jets - HW II Solutions. As discussed in the Lecture the parton distributions do not scale as in the naïve parton model but rather are epected to ehibit the scaling violation predicted by QCD. The structure of the epected renormaliation of the parton distribution functions is summaried in terms of the DGLAP (also often called the Altarelli- Parisi, i.e., the AP in DGLAP) splitting functions. As noted in Lecture the lowest order epressions for these functions are given by 3 Pqq CF, Pqg TR, Pgq CF, C A 4n ftr Pgg CA, 6 where the + notation means f f f d d a) Verify that the corresponding anomalous dimensions (the moments of these functions, d P qq ) have the forms. CF, k k, qg TR QCD Maria Laach 8 HW II Solutions

2 gq C F C n T. gg A f R k k 3 Solution: This is a straightforward eercise to performing the appropriate integrals. The anomalous dimension for the quark to quark process has the form qq F k, 3 C d CF d CF d k k k k k CF d k k k k k k CF d k k C F k 3 k CF k k CF. k k For the gluon to quark case we have QCD Maria Laach 8 HW II Solutions

3 T d qg R T d R TR TR TR. Similarly for quark to gluon we find gq CF d F C d CF CF C F Finally for gluon to gluon we have. QCD Maria Laach 8 3 HW II Solutions

4 gg A C 4n T d CA 6 C 4 A n ft R CA d 6 C 4 k k A n f T R C d k k 6 C 4n T CA k k 6 A f R nt f CA k k 3 nt f R CA. k k 3 R A f R Now consider the evolution of the singlet quark distribution given by the sum q q, i which mies with the gluon distribution via the evolution equation. In terms of the t ln Q we have moments with evolution variable QCD i tqq n fqg i d s. dt g gq gg g b) Verify that for = there are two eigenvalues to the above evolution equation and that the corresponding anomalous dimensions are (momentum QCD Maria Laach 8 4 HW II Solutions

5 conservation) and 6 9 n f 3 g and ng Solution: First we need we have (C F = 4/3, C A = 3, T R = ½) qq qg gq gg corresponding to the eigenvectors 3 6, respectively. f for the various channels. From the results of part a) nf nf Thus the anomalous dimension matri is 6 n f 9 3 A 6 n f 9 3, where det A and n f 6 Tr A. Thus the eigenvalues must satisfy det A, Tr A. 9 3 We easily deduce that the eigenvalues, with the appropriate identification by magnitude, are n f QCD Maria Laach 8 5 HW II Solutions

6 n f If we write the eigenvectors in the form Cg we want to solve for), we have the evolution equation d dt (with C the constants that. s Cg Cg Thus the matri equations we must solve is A T C. C nf n f C C 6 n f C These are easy to solve and we find the desired results C nf 3 3nf ; C Thus the first eigenfunction is g, i.e., total momentum, and the eigenvalue,, ust corresponds to total momentum conservation, i.e., the total momentum does not change with the scaling parameter t (and is always equal to ). c) Use the result of b) to find the momentum fraction in quarks and that in gluons at truly asymptotic values Q. Solution: We now use the results of part b) to consider the relative fraction of momentum in quarks and gluons at asymptotic scales. In particular, we want to QCD Maria Laach 8 6 HW II Solutions

7 focus on the eigenfunction above with the negative eigenvalue. The general form of the solution to the DGLAP equation tells us that the eigenfunction behaves as b s t Cg t Cg t s t,. t The asymptotic momentum fractions can be obtained from this result, in the form 3n, t C g, t f g, t, 6 and from momentum conservation,, t g, t 6, t. n 3 f Assuming that the (truly) asymptotic value of n f is 6, we have 9, t.59, n 6 8 g, t.47, 7, t 9.5. g, t 8 At truly asymptotic scales we epect slightly more momentum in quarks than gluons in reasonable agreement with current data. f n f QCD Maria Laach 8 7 HW II Solutions

8 . The evolution of the distribution functions tends to build up the gluon distribution at small, which will be important at the LHC. Here we consider this point in more detail. In the limit of small and very large Q the DGLAP equation is dominated by the small argument behavior of the splitting function P gg. a) Verify that in this limit the gluon distribution G(,t) = g(,t) satisfies the t ln Q ) equation (again QCD, 3 dg t s t dy G y t Solution: We start with the standard DGLAP result t dt d G, t s dy d y P gq y, t P qq g y, t. dt For the dominant contribution comes from soft gluon emission (recall the / bits in the splitting functions) where both y and. Hence the dominant soft y gluon term evolves from the previous soft gluon terms and lim yo g y. P Also we have gg y,. C A 6. So we can write t d s dy d G, t 6 y yg y, t dt y s t dy d 3 y G y, t. y Now we can make use of the following property of the delta function d y y y. y y QCD Maria Laach 8 8 HW II Solutions

9 Using this result in the previous equation yields the desired result in the small limit t d s dy G, t 3 G y, t. dt y b) Now use the -loop form for s and change variables to lnt and 4 b ln to show that the approimate equation we want to solve is dg, G dd,. Solution: With the -loop form of the coupling we have s 4 4. bot be Thus we can rewrite the previous result as d d dy t G, t G, G y,, dt d b y and then taking a derivative with respect to we find G, G,. b We can simplify this further with the variable defined above, 4 b ln with d4 b d. So finally we have the desired result, QCD Maria Laach 8 9 HW II Solutions

10 G,,. G c) Verify that at truly large values of both and this equation is solved by G, ~ e or 48 t g, t ~ ep ln ln g, t. b t Solution: In the limit of large and, i.e., ln t and ln / large, we have e s,, e e e e e e. Thus, as desired, the asymptotic solution of the differential equation is proportional to G, ~ e. So, substituting in terms of the original functions and variables and including an appropriate overall normaliation, we have the asymptotic result, QCD Maria Laach 8 HW II Solutions

11 48 t g, t ep ln ln g, t. b t d) To get a feeling for the sie of this enhancement assume that the gluon distribution at Q = 5 GeV is given by the following (fictitious) epression, g Note that it already has the / behavior at small. Now evaluate the above enhancement factor for Q = GeV at =. with QCD =. GeV. How much larger is the evolved distribution at this value, assuming that the above epression is relevant in the specified kinematic regime? (Take n f = 5 for this estimate.) Solution: This is ust a numerical eercise to get a handle on the sie of the enhancement do to the renormaliation of the gluon distribution. Starting with g, t , we want to evaluate the enhancement of the small behavior by the etra eponential factor above. With the chosen kinematic values we have t ln. ln ln ln t ln 5. For the choice =., ln / = 4.6 and with n f = 5 we obtain b and QCD Maria Laach 8 HW II Solutions

12 48 t b t 6.4 ep ln ln ep e This is a large enhancement indeed and it is even larger for smaller values. 3. Let us focus briefly on the Drell-Yan process, the production of a virtual photon in hadron-hadron collisions via the annihilation of a quark and antiquark. The short distance ( hard ) process is the time reversed version of the electron-positron annihilation process. Thus e e qq becomes qq, where the specific choice of lepton arises from the desire to employ a lepton pair that is easily detected. This cross section must then be convoluted with the appropriate parton distribution functions. In terms of the scaled virtual photon mass Q s and the photon rapidity y q q q q the cross section looks like ln, the scaling or parton model version of d 8 y y s g e, e, d dy 3 where the (LO) parton luminosity function has the form a b a b g a, b ef q f a q f b q f a q f b. 3 f The label a,b correspond to the incident hadrons. The eplicit factor of /3 is required because the conventional normaliation of the pdfs ( qq), includes an implicit sum over colors. Here the quark-antiquark pair that annihilates must be of the same color. Thus the annihilation occurs for only /3 of the possible pairs. Contrary to the collider physics we have focused on in the Lectures, consider now the case of pion beams incident on a nuclear target, i.e., composed of the canonical nucleon N = (p+n)/. If we focus on large Q s so that we can safely assume that the interaction is dominated by the valence quarks (and antiquarks), determine the epected (and observed) value of the ratio QCD Maria Laach 8 HW II Solutions

13 R DY N X N X. Solution: In the limit of valence quark dominance we have the nucleon represented by N uud udd du, the positive pion by ud and the negative pion by. Thus the elementary Drell-Yan process at large is dominated by uu annihilation in the negative pion case and dd annihilation in the positive pion case (i.e., the antiquark must come from the pion). Using the isospin symmetry of the strong interactions (i.e., QCD) we epect the relevant parton distributions to be identical, i.e., N N. d u u d Thus the only difference in the two cases (i.e., the pion beams) will arise from the electric charges of the relevant valence quarks,, R DY valence N X ed 9. N X e 4 u 4 9 This naïve prediction is in reasonable agreement with data. 4. In this eercise we want to become familiar with various features of collider kinematics. As noted in the Lecture the real rapidity and the pseudo-rapidity are defined by E p rapidity yj.5ln E p pseudo-rapidity lntan where the direction is the direction of the beam. QCD Maria Laach 8 3 HW II Solutions

14 a) Verify, as stated in the Lectures, that for any particle of mass M we can write E M p cosh y, p M p sinh y, p p p T y T T. Solution: The constraints to be satisfied are that E M p pt M and that p covers the full range from to. Clearly the state epression for the longitudinal momentum satisfies this last constraint for < y <. Then we need to check the form of the energy given p E M p p M p M p sinh y T T T M p cosh y M. T b) Prove that tanh cos and thus that is easy to measure. Solution: We ust use the definitions and calculate cot tan cos sin sinh e e tanh cosh e e cot tan cos sin cos. c) If particles are produced uniformly in longitudinal phase space with a differential distribution that looks like dn dp C E, with C a constant, find the corresponding distribution in y, dn dy. Solution: We ust need to use the definitions to find QCD Maria Laach 8 4 HW II Solutions

15 dp M pt d sinh y E M p cosh y T dy. Thus particles that are uniform in (relativistic) longitudinal phase are uniform in rapidity E dn dp dn C. dy This is a very good approimate description of the soft particles produced in the bulk of inelastic interactions (the ones not corresponding to some hard interaction). Their distribution is cut off rapidly (~ eponentially) in p T and is approimately flat in rapidity. d) Prove that the rapidity equals the pseudo rapidity, = y, for a massless particle (and thus approimately for a relativistic particle, E M). Solution: For massless particles E M p p and thus p pcos pcos tanh y cos tanh E E p y. In the last step we used the result of part b), true for any E/M value. e) Prove that for a Lorent transformation (boost) in the beam () direction, the rapidity, y, of every particle is shifted by a constant y, which is related to the boost velocity. Recall the form of such a boost to a reference frame moving in the direction with velocity u (with respect to the original frame and with c = ) QCD Maria Laach 8 5 HW II Solutions

16 QCD Maria Laach 8 6 HW II Solutions E E p p p E p p, p p, y y u,. Solution: We proceed by writing out the form of the boost in the notation of rapidities E M p cosh y M p cosh y T T E p M p cosh y sinh y p M p sinh y M p sinh y T T T M p sinh y cosh y. Trying the Ansat that y y y we immediately find the consistent solution that cosh y cosh y y cosh y cosh y sinh ysinh y E p cosh y sinh y T,, sinh y sinh y y sinh y cosh y cosh ysinh y sinh ycosh y cosh y, sinh y y sinh cosh y sinh y. Finally we note that this arithmetic depended only on the boost itself, not the original E, i.e., not the original rapidity. Hence the rapidity of EVERY particle is shifted by

17 the same amount. A Lorent transformation in the direction is ust a translation in rapidity (this is why invariant differential longitudinal phase space is proportional to dy) f) Consider a Z boson that is produced on-shell at the LHC in a qq annihilation process. The velocity of the Z boson is along the beam direction. What is the condition relating and, the momentum fractions of the quark and anti-quark. (Compare to the epressions in the previous eercise.) Solution: The usual (parton model) kinematics (see the previous eercise) tell us that s s EZ, p, Z, pt, Z, M Z M Z EZ p, Z s. s Constructing the rapidity of the Z we find y Z EZ p ln ln ln EZ p, Z, Z M s y e y Z y e e Z Z Z M s yz e e These relations between the momentum fractions (the s) and the rapidity (and the resonance mass) are ust what was stated in the previous eercise. Z y Z. 5. Demonstrate that both the cone algorithm (without seeds) and the k T algorithm are IRS at NLO in pqcd, i.e., show that the found et will have the same properties whether it contains a single parton or a pair of collinear partons with the same total momentum. Also show that the et is unchanged by the emission of a (vanishingly) QCD Maria Laach 8 7 HW II Solutions

18 soft gluon. This does not require a slick argument. The idea is ust to give you the opportunity to think through what it takes to be IRS. Solution: The point here is that the et algorithms sum up the momentum of obects (partons, hadrons, calorimeter towers) that are nearby each other, in some sense. Either they are obects inside an angular cone of sie R in the cone algorithm, or they are pair wise near each other in momentum space in the k T algorithm. Thus a collinear pair of partons will always end up either in the same cone, or merged first in the k T algorithm because they have d i =. For the case of a ero energy etra parton it may not end up in the same et with the non-ero energy parton, either because it is outside of the cone or because it has the minimum d i and is a separate et, but this does not matter. It is still true that the algorithm will find the same non-ero energy et (the other parton) as in the case of no etra emission. Thus the divergent parts of the real emission will contribute to the same et (found by the algorithm) as the virtual emission contribution and the divergences will cancel. Hence the et cross section defined by both the cone algorithm (without seeds) and the k T algorithm are IRS. 6. Use the Snowmass definition of the iterative cone algorithm (i.e., E T weighting instead of 4-vector addition) to show that the -parton phase space splits up as indicated in the figure in the Lecture. While this is really a -D problem in (y,), the fact that there are only partons, which effectively lie in a plane, means we can think of it as a -D problem, i.e., ust the separation d in that plane. Solution: The point here is to think about how the cone algorithm works in pqcd in order to understand some of the issues that have arisen. We can think in terms of a - D angular vector with components rapidity and aimuthal angle, y,. Now for y y, with d. If partons the angular separation is given by we choose, for simplicity, to measure the rapidities and aimuthal angles from the geometric center of the cone, the location of the E T weighted centroid is given by ET, y ET, y ET, ET, y y C,,, ET, ET, ET, E T, where we have chosen ET, ET,, ET, ET,. Thus the condition that the E T weighted centroid coincides with the geometric center of the cone is that QCD Maria Laach 8 8 HW II Solutions

19 C y, y, y, y,. Thus, as epected, the more energetic parton must be closer to the center of the cone than the less energetic one. So the condition that both partons are inside a cone of radius R centered at the E T weighted centroid is that C R. Thus the effective (- D) boundary for finding a single cone with both partons in side is given by y y, y,, d ( ) ( ) R. C This is what is meant to be illustrated in the figure. QCD Maria Laach 8 9 HW II Solutions