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The rho-parameter and top quark mass constraints Peters, Daniel John 1990

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T H E RHO-PARAMETER AND TOP QUARK MASS CONSTRAINTS By Daniel John Peters B.Sc, Dalhousie University, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1990 © Daniel John Peters, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ?HYsic-s The University of British Columbia Vancouver, Canada Date H Oc/roged \110 DE-6 (2/88) Abstract Experimental determinations of limits on radiative effects in the Standard Model can be used to place limits on the mass of the top quark. This is exemplified by the comparison of the /)-parameter, as determined by neutrino-electron scattering experiments, to the radiative corrections to that quantity calculated to order g2. It is suggested that an extension of the calculation to order g4 might significantly affect these constraints. Some partial results of such a calculation are given. ii Table of Contents Abstract ii List of Figures iv Acknowledgements v 1 Neutrino-electron Scattering and the p-parameter 1 2 Constraints on rat from Corrections to p 8 2.1 Comments on Corrections to xw 9 2.2 One-loop corrections to p 10 2.3 Constraints on mt 16 3 Some Two-loop Calculations 18 Bibliography 26 Appendices 26 A Feynman Rules 27 A.l Vertices including fermions 27 A.2 Purely bosonic vertices 28 iii List of Figures 1.1 Tree-level diagrams for neutrino-electron scattering 2 2.1 (a) Z-7 transition contribution to i/M — e scattering; (b) Quark loop as part of the Z-7 transition 9 2.2 One-loop internal self-energy corrections to — e scattering 11 2.3 A graph of p as a function of mt (in GeV) for MH = 42 GeV (upper curve) and MH = 1000 GeV (lower curve) 16 iv Acknowledgements I wish to thank my supervisor, Dr. John N. Ng, for his help and his great patience. I also want to thank my parents for their love and encouragement. v Chapter 1 Neutrino-electron Scattering and the p-parameter The p-parameter, a measure of weak isospin breaking in the standard model, is a conve-nient device for dealing with radiative corrections. It is usually defined by _ Mw n n 9 ~ M\ cos2 ew 1 j where Mw, Mz are the masses of the W,Z bosons and 6W is the weak angle. This definition needs further clarification in order to be satisfactory. For one thing, at the lowest order Lagrangian level we have cosflu, = Mw/Mz, and this has led some authors to define the renormalized 8W by the latter expression with Mw,Mz referring to physical masses. If, in this case, the masses in the definition of p were physical then p would be exactly 1, and the notion of a "correction" to p would be absurd. The p-parameter must be defined by the process used to measure it. Here, we will consider the processes i/Me —• v^e (RI), PMe —• PMe (R2), and i/Me —• vep (R3). The tree-level diagrams are shown in figure 1.1, where the kinematics are also defined. The value of p can be extracted from the cross-sections for these processes. At this point some definitions and conventions should be mentioned. The abbrevia-tions sw = sin9w, Cu, = cosflu,, xw = s2, L = (1 — 75)/2, and R = (1 + 75)/2 will be used. The convention used for the Dirac 7-matrices is that of Bjorken and Drell [1]. All calculations will be done in the renormalizable't Hooft-Feynman gauge, for maximum simplicity of the W,Z propagators. But this requires the consideration of the charged and neutral "would-be-Goldstone" bosons, which are respectively denoted by <^ ± and x, and 1 Chapter 1. Neutrino-electron Scattering and the p-paxaxneter 2 Figure 1.1: Tree-level diagrams for neutrino-electron scattering have respectively the same masses as the W and Z bosons. Feynman diagrams involving the couplings of (ft*1, x, or Higgs bosons to fermions other than the top quark will in most cases be neglected because these coupling strengths are proportional to mj/Mwi where mj is the fermion's mass. The (tree-level) amplitude for Rl is —ig2 _ 1 using the fact that \q2\ <C Af| (where q = pz — pi = p2 — Pi)- (It can be seen from what follows that the maximum value of |q 2 | is about 2meE\, so for this approximation not to be valid, E\ must be comparable to about 8 PeV.) We will need to find l-M^I2, which is just |.Mjy|2 summed and averaged over the appropriate spin states. One might expect to get jSapiDB |.Mjv|2> since there are two initial spin \ particles, but this assumes that the beams are unpolarized. In fact, the incoming neutrinos are all left-handed, but since right-handed neutrinos do not interact, we need only average over the target electron spins. Thus we have Wrf = \ £ \MN\2 spins Chapter 1. Neutrino-electron Scattering and the p-parameter 3 spins xue(p4)7M((xti; - + xwR)ue(p2)ue(p2)'ru((xw - + xwR)ue(p4) 2^M* ? l^MYLu^p^u^p^Lu^) +x2wTr(^47^27^) + m2exw(xw - ^Trfr^) ] (PsPi + PzPx ~ Pi' Ps9^ + ie^PspPia) * [(2afw - xw + ^){P4nP2t> + P4uP2n - P2 • P49nv) +(- - xw)ieaiipvp°p2 + 2m\xw(xw — -)g^] 94 1 ^-^|[(2a; 2 y - X W + -)((px • p2)(p3 • P4) + (Pl • P*)(p2 • Pa)) +(\ ~ x»)((Pi • P2){pz • PA) - (pi • Pi)(P2 • Pz)) + 2m2exw{^ - xw)p! • p3] - ^ M4"[(x™ - 2^ ( p i •p 2)(P 3 , p 4) + ^ w C P i -P4)(P2-P3) - mexw(xw - -)pa -p3] (1.3) where we have used e^ea^p^plpzpPl = 2[(pi • p4)(/>2 • P3) - (pi • P 2 X P 3 • P4)]. The cross-section is 1 dPp4 \2\r\/n r? * 47r22E/((Pl+P2-p4)2)0(^l + ^ 2 - ^ ) i ^ | ^ | 2 . (1.4) Eventually <f<7jv will be found in terms of dE4, and the only values of E4 which will be considered in the subsequent integration are those for which Q(Ei + E2 — E4) = 1; therefore there is no need to write the ©-function explicitly. The argument of the 8-function is s + m2 — 2(pi + P2) • P4> where s is the invariant Mandelstam variable s — (pi +P2)2- In the lab frame (p2 — 0) we have p2 • p4 = meE and pi • p4 = Ei(E — pcos $) where E = E4i p = |p4|, and 0 is the angle between pi and p4. Using d?p4 = p2dpdCt and Chapter 1. Neutrino-electron Scattering and the p-parameter 4 pdp = E dE we get dcrN = J d f d n 6{a + m2e- 2(E1 + me)E + 2Expcos 0)\MN\2. (1.5) 6lni p\ • P2 Integrating over all angles we get dE ""-tSBZTjMsP (1-6) where cos 0 is now (s+m2—2(Ei+me)E)/2Eip. One can easily show that px -p2 = P3-P4 = meEi, pi • p 4 = pi • pz = me(me + Ex — £), px • pz = me(E — me), and s = 2meEi + m2. Therefore d < T N = l e^v A f | [ ( g w ~ l ^ 1 ' + x " ( m ° + E l " E ) 2 " g w ( 3 ! w ~ l ) m e ( E ~ m e ) ] (1.7) The total cross-section is the integral of this expression from the minimum outgoing energy, m e, to the maximum, ^(s/me + ml/s). Assuming that E\ >• me, as is always the case in such experiments, we can treat me as zero in the square bracket above, treat the limits of integration as 0 and E\, and say s = 2meE\. So 2G2FmeP2 1 2 3 1 2 3 °N = ——p—[Kxw --) + g^-^lJ = ^V(f 4-*« + £) (1.8) The amplitude for R2 is —ig2 _ 1 MA = 2c2 M2*v*Ml*LvtoM*'(P*)l*((Xw ~ ^ L + x*R)u*fa)- (1-9) Calculating the sum over spins, it becomes obvious that the only difference between l-M^I2 and LMJV| 2 1 8 * n a * * n e r ° l e s 0 1 Pi and P3 are reversed. A brief look at the last line Chapter 1. Neutrino-electron Scattering and the p-parameter 5 of equation 1.3 shows that swapping p\ and p 3 is equivalent to swapping (xw — j) 2 and a;2. Therefore OA = ^ V ^ z 2 - + ^) (1.10) and the value of xw can be found from the ratio (1.11) O-A _ fx2 -j-Xw + Tl _ 16a:2 - 4xw + 1 Proceeding in much the same way, the amplitude for the charged-current reaction (R3) is found to be -ig2 _ Mc = ^^u^p^Lu^ip^u^ipJ^Luefa) (1.12) so the squared amplitude, summed and averaged over spins, is \McV = 5 £ l^tc l 2 = grOn • j»2)(P8 - lu) (i.i3) spins The differential cross-Bection (integrated over p4 first, rather than p3), is 1 <Pl*c„ . ^!^ 1 Working in the lab frame again, we get p2 -pz — meE and pi -pz = E\(E — pcos 5) where this time E = Ez, p = \pz\, and 0 is the angle between p\ and p*3. Now we have dec = 3 3 ^ ^ 6 ( 3 + mj - 2(£ x + me)E + 2E l Pcos d)\Mrf (1.15) which, upon angular integration, gives us ^ _ _ ^ _ T X r i 2 S*P3 • P4 1 R . ^ = 3 2 ^ l P l . p 2 ^ c | = le^A^T ( L 1 6 ) where cos r? is now (s + m 2 — 2(£i + me)E)/2Eip. This time p 3 • p 4 = m e £ i — | (m 2 — m2), and 5 is of course the same as it was in (RI). Notice that doc/dE is independent of E, so the integration is trivial. The minimum and maximum energies are respectively Chapter 1. Neutrino-electron Scattering and the p-parameter 6 (m*+ml)/2rne and ^(s/me+myrtle/s). Also, E\ must be at least (m2 — ml)/2me, which is much greater than m ,^ so these energies become in effect m2l/2me and E\. In the end we get ac = ^s(l - ^ ) 2 . (1.17) 7T S The ratio of these two cross-sections is ix2 -x + i V c ' 9 (l-mllsf so, having found £„, from the ratio <TA/O~N, we can extract p from the ratio O~NI<TC- This is to be regarded as the definition of p. Any deviations from the tree-level "prediction" of p = 1 will then be compared with radiative corrections to these scattering processes calculated in the standard model. The most recent measurement of xw (which uses the ratio O~A/O~N as indicated in equation 1.11) was given by the CHARM collaboration [2]. The value is xw = 0.211 ± 0.035(stat.) ± 0.011(syst.) . Their result for the cross section of process Rl is O-N/E! = (2.2 ± 0.4(stat.) ± 0.4(syst.)) • lO-^cn^GeV - 1 . For the charged-current process a later CHARM II result [3] is = (17.62 ± 1.32) • l O ^ W G e V " 1 . E^l-ml/sY This leads via equation 1.18 to p = 1.13 ± 0 . 1 6 . (1.19) The large uncertainty in this result is due to the low statistics of these experiments. The neutral-current results of ref. [2] are based on 83 ± 16 neutrino events and 112 ± 21 Chapter 1. Neutrino-electron Scattering and the p-parameter 7 antineutrino events, and the charged-current results of ref. [3] are based on 4808 events. To improve the precision of this kind of measurement of p to ~ ±1% one would need high flux neutrino beams. Chapter 2 Constraints on mt from Corrections to p The top quark and the Higgs boson have not been observed as yet; moreover, their masses are free parameters in the standard model. Constraints can be placed on these masses by measurements of radiative corrections. It is the aim of this chapter to show how the mass of the top quark can be constrained by a measurement of the p-parameter. As mentioned in the first chapter, deviations from p = 1 are attributed to radiative corrections to the scattering amplitudes described there. In this chapter we are concerned with corrections to order g2. Even to this order in perturbation there are many Feynman diagrams, all of which would need to be calculated in a complete discussion. The only ones which involve the top quark or the Higgs boson are the internal self-energies and the Z-photon transition. A calculation of the one-loop diagrams in question will lead to the dependence of p on the top mass m* and the Higgs mass MH- In the end we will have p = 1 + (Ap) t 0p + ( A p ) f f i g g s + ( A p ) o t h e r , (2.1) where all the mt-dependence will be in (Ap)^Qp and all the M/j-dependence will be in (Ap)jjjggS. (This separation is in fact only possible up to one-loop. To higher orders, there are Feynman diagrams which involve both the top quark and the Higgs boson, which means that some terms in Ap will include both m t and MJJ.) Before the loop calculations are done, the connection between their results and the corrections to p is discussed. 8 Chapter 2. Constraints on mt from Corrections to p 9 Figure 2.1: (a) Z-7 transition contribution to i/M — e scattering; (b) Quark loop as part of the Z-7 transition 2.1 Comments on Corrections to xw Many of the corrections to the scattering amplitude represented by Feynman diagrams are corrections to g or to xw and not to p, the most obvious examples being vertex corrections. A less obvious example is the Z-photon transition (figure 2.1). The fact that the electron couples to a photon rather than a Z indicates a correction to the couplings. In fact, it is best to treat this as part of the neutrino-photon vertex; in other words, part of the neutrino charge radius. Let us briefly examine the quark-dependent part of the Z-photon transition. It has the form A(q2g'il' — q^q"), due to gauge invariance. The q^q" term will not contribute, because the polarization tensor in the photon propagator connects the momentum q to the Dirac matrix in the photon-electron coupling, giving something of the form ue{p4)fi u e (p2) , which is zero by the Dirac equation, using the fact that q = p2 — p^. This loop diagram will give a correction to the amplitude of i 2s 1 {AMN)Z^ = ^ ^ « ^ ( p 3 ) 7 / i 2 i u ^ ( P 0 ( - ^ ( 0 ) ) S e ( p 4 ) 7 M " e ( p 2 ) (2.2) (using e = gs). To find A(0) we need to calculate Ez7(<72). Its mrdependence (from the Chapter 2. Constraints on mt from Corrections to p 10 quark loop in figure 2.1(b)) is (Y \ - 3 g 2 s Q « f ^ + ™t)Y(cLL + cRR){y-4 + mT)Y\ K^hop - ^ A* J ( 2 ? R ) N { K 2 _ M 2 ) ( ( A ; _ Q ) 2 _ ^ —i :3g2sQ(cL + cR) ri 2(m2<T - *(1 - x ) (2 g y - <?2<T)) (47r)»/2cu; W O 1 (m2t - q2x{l - x))*/2 V (2-n)(m t 2 - 92x(l - X))JT £ } ] (m? - 92:r(l - x))°/2 v 2' _ Zg2sQ(cL+cR) a , Z1 , v 2 , _ m ; - ^ ( l - i ) , (2.3) y4(g2) is simply the above expression without the g2^" — q^qv factor. So A ( 0 ) = _ , . ? ! t o ) ( H _ i £ _ l n ^ , ( , 4 ) It has been pointed out [4] that taking radiative corrections into account in the de-termination of xw from O~N/O~C will affect the value only by ~ 1%. The fact that A(0) depends only logarithmically on mt means that this will still be true for a wide range of values of mt. Thus we can ignore corrections to xw for purposes of constraining mt from radiative corrections to p. 2.2 One-loop corrections to p In this section all the one-loop corrections to p are presented. Those which depend on mt and MH are calculated explicitly from the relevant one-loop diagrams. All calculations are done using the technique of dimensional regularization, where the number of dimensions of spacetime is n = 4 — e, and p, is an arbitrary quantity with units of mass. The self-energy of the internal W or Z boson (Figure 2.2) has the form A(q2)gliV + B{<l2)<f(r'm general; here we need only work it out for the case q = 0. Let S denote A(0), then the self-energy is Y.g'"'. So the scattering diagram with the self-energy included is simply i E / M 2 times the tree diagram. Chapter 2. Constraints on mt from Corrections to p 11 ^ ^ t z z z z z x z ( a > f % < b > ' / ' \ < 0 > » * V W W W v vwww z z z z (d) „ (e) H H i r S J Y ^ J ^ W A A A A A A , m A A A / w w w w w w <t> w (f) (g) ^ (n) V W W W v V W W W w W W , . 4 w (I) (J) Figure 2.2: One-loop internal self-energy corrections to i /^ - e scattering Chapter 2. Constraints on mt from Corrections to p 12 So, thinking specifically of the neutral-current process, the Z-self-energy gives a rel-ative correction of to MN] therefore a relative correction of 2Re(i£.z/M|) to jjVfjvl2, to °"JV> to O~N/O~C, and to p2; therefore a correction of Re(i£2:/M§) to p. (Since p = 1 at tree level, there is no need to distinguish between relative and absolute correc-tions to p.) The correction to p from the W-self-energy in the charged-current process is similarly — R<e(i%w/Mw). One part of the internal self-energies is the tadpole diagram in which a Higgs boson connects the vector boson to a quark loop (Figure 2.2 (d),(i)). The W-W-Higgs vertex has the same Feynman rule as the Z-Z-Higgs vertex except that the former has an extra factor of c 2; this, along with Ap = Re(iS^/Af| — iY,wlMw) and the fact that the tadpole itself is independent of that to which it is attached, makes it clear that the contributions to p of these diagrams cancel. The same is true of the tailless tadpole diagram in which a Higgs loop attaches directly to the vector boson with a 4-boson coupling (Figure 2.2 (e),(j)): the W-W-Higgs-Higgs vertex is <?w times the strength of the Z-Z-Higgs-Higgs vertex, and the contributions to p cancel. This reduces the number of diagrams involving the top quark or the Higgs boson which we need to calculate from ten to six. This also makes it possible to separate the Ap terms as described at the beginning of the chapter, since there remain no diagrams which involve both of these two particles. (Ap)^0p is calculated first. The Z-self energy needs only be calculated in the case q = 0; the relevant part (from the simple quark loop, Figure 2.2(a)) of its dependence on mt is Zg2 , f Ph Tv[^ + mt)r(cLL + cRR)^+ mt)Y{cLL + cRR)} (Sz)top = - X M J 7 ^ (2ir)» (Jfc2 - m 2) 2 • 3g2 Am2cLcR e _ (2 - n)m2(cLL + cRR)^ t e u (l*)»'2c2/ { (m?)«/a V (m2)'/2 U 1 + 2))9 • 3g 2 (c £ -c f l ) 2 m 2 ,2 m2 = » ^ ( I - 7 * - l n i ^ ( 2 ' 5 ) Chapter 2. Constraints on mt from Corrections to p 13 This leads to a contribution to p of Sg2 m2 2 m2 327r2c2Mfve 4 ^ ' where (ci, — CR)2 — j has been used. In the W-self-energy (which also needs only to be calculated in the case q = 0), the relevant quark loops (Figure 2.2(f)) include the top quark and the various down-type quarks (d, s, b) with the contributions of these loops weighted by elements of the Kobayashi-Maskawa mixing matrix (which appear in the Feynman rule for the quark-W coupling). The masses of all these quarks are so small in relation to mt that they can all be treated safely as zero (the experimental minimum value for mt is currently 89 GeV [5]). The effect of this approximation is that the t-b, t-s, and t-d loops are identical but for their mixing factors, the absolute squares of which add to 1 due to unitarity. So we can consider the top quark to appear in only one kind of quark loop in the W-self-energy, along with a massless quark, and we can ignore all KM mixing factors. Thus the relevant part of the mt-dependence of the W-self-energy is 3</2 , r dnk Tr[(y + mtyfLlj-fL] (WJtop - " 2 A* J ^ 2 — 2 ) " 2(47r)»/2^ Jo (m?x)«/2 9 i Jo e 4wu2 167r2 np which leads to a contribution to p of _3£2__m|_.2 _ _ m2 1 327r2M^e l E 4TT//2 + 2 ) -(--K-lnj-h + j r (2.6) Combining the results, we get . . , 3o2 m? „ Chapter 2. Constraints on mt from Corrections to p 14 where we have used the fact that Mw = c^Mz to 0(1). Notice that this correction to p is positive. Next, we need to find (A/>)Higgs- There are four different loops to calculate; two in each of and Y>w (which only need to be calculated for the case q = 0). The loop which consists of a Z and a Higgs (Figure 2.2(b)) is g2Ml , f d"k \ Z)2H c l P J (27r)„ { k 2 _ M ] i ) { k 2 _ M 2 ) 92MZ ... / i r(f)g^ 92 2 , fi "l^Y^cl1" Jo (M2Hx + M2Z{\ -x))*!2 whence p gets a contribution of (2.8) 92 ,2 , , Ml Mh , Ml. —2 ( -vc- + 1 - In — - In ——) 167r2c^£ m47r//2 Ml-Ml Ml' The loop which consists of a x a n < l a Higgs (Figure 2.2(c)) is K Z ) x H c\ J (2x)« (*2 - M£)(fc2 _ M | ) = -|r(-i + f )<r (47r)"/2c2 M 7o (M&z + Af|(l - aOJ-i+'/a (2.9) and this gives p a contribution of 9j_(n , 3&*,2 _ 3 M l M | _ _ JU|_ 647r2ciU M l n e 7 2 Mjj - M2Z ^p2 M2Z{M2H - M2Z) 4*^' Chapter 2. Constraints on mt from Corrections to p 15 There corresponding loops in the W-self-energy (Figure 2.2(g),(h)) are (2JT)» (fc2 - MH)(k2 - M&) _ g2M2w2 M2W Ml Ml ~ - , l 6 ^ ( 7 " 7 f i + 1 ~ b 4 ^ - A ^ - A ^ b j i ^ ) ( 2 - 1 0 ) and (Y \ _ N2 f d^k k^k"  K W H H ~ 9 J (2n)n(k*-Ml)(k2-M&) - i g 2 ((M2±M2 \(2 „ 1 3 ^ 1 Mw , M2W Ml Ml (2.11) (these integrals are easily done by comparison with the Z-self-energy calculations which are done above). The contributions to p from these two loops are -TW{l-™ + 1 - l n ^ - M l - M ^ * ^ ^ 92 ( n , Ml. .2 _ 3 M2W M2W Ml M2H 64TT 2 U + M 2 , A e 7 £ 2 Af2, - 4TT//2 M ^ ( M £ - M 2 , ) 4*^" Adding these four results, and subtracting 3g2 xw 2 5 M i 2 6 ^ ( ^ ( 7 - ^ + 6 4 ^ ) +1" O , which is independent of MH and can therefore be shifted into (Ap)0\]leT, we get, after a bit of algebra, , A . 3#2 , Ml i Ml 1 Ml Ml, , x (^)Higgs = 6 ^ ( M ^ _ M ^ l n ^ - clM2H-M2zln Ml} ( 2 J 2 ) From this we can see that the asymptotic form of (Ap)jjjggS as MH becomes large is logarithmic. Chapter 2. Constraints on mt from Corrections to p 16 MH = 1000 GeV (lower curve), using xw = 0.229 The remainder of Ap is [6] * * x j 2 . l , 5 15 1 , 14 , , 1 , 17 25 n t , 2 (Mother = I ^ l ^ - T X w _ ^ + y X J + ^ ( 4 ^ - T _ 2 l w + 4 1 J l n C -- i { l - r w ~ x l ) l n z ~ ~ l ] > ( 2 J 3 ) which is clearly independent of mt and MH• This contribution arises from loops made of gauge bosons. Such loops exist because non-abelian gauge theories allow for triple and quadruple gauge boson couplings (see Appendix). 2.3 Constraints on mt Figure 2.3 illustrates the dependence of p on m< and MH- The graph starts at m*=89 GeV, which is the current experimental minimum value [5]. The chosen values for MH are 42 GeV (which is the current experimental minimum value [7]) and 1000 GeV (which has been argued as a maximum value for MH on the grounds that perturbation theory would break down otherwise [8]). As can be seen from the figure, p changes very little if mt is fixed and MH varies from 42 to 1000 GeV. So a measurement of p can tell us very little about MH, but, if the measurement is sufficiently precise, mt can be constrained. Chapter 2. Constraints on mt from Corrections to p 17 Using the experimental results mentioned in chapter 1, the maximum value of m t is found from the maximum experimental value of p, 1.29, and the minimum value of (A/?)jjjggS + (Ap)0ihev, which is -0.023 (corresponding to xw = 0.174 and MH = 1000 GeV). This gives (Ap)^0p = 0.31 and therefore mt « 1000 GeV. No minimum value for mt is obtained. The best experimental determinations of p and xw are from the consideration of a vari-ety of measurements, including vp —> vp, deep inelastic scattering, and the masses of the W and Z [9]. These results are p = 0.998±0.0086 and xw = 0.229±0.0064. Here the max-imum experimental value of p is 1.007 and the minimum value of (A/')Higgs"Ki^V)other * s -0.020 (corresponding to xw = 0.223 and MH = 1000 GeV). This gives (A/>) t o p = 0.027, from which m% gets the (better) maximum value of 290 GeV. A minimum value can also be found. The minimum experimental value of p is 0.989 and the maximum value of (Ap)jjjggS + ( A / 9 ) o t n e r is —0.017 (corresponding to xw = 0.235 and MJJ = 42 GeV). This gives (Ap)^0^ = 0.006, and thus a minimum value for mt of about 140 GeV. Chapter 3 Some Two-loop Calculations In this chapter some results of two-loop contributions to the W-self-energy are displayed. It is hoped that all the two-loop calculations relevant to A p will be completed so that modifications may be made to existing predictions of mt. One might be tempted to argue that the 0(g4) corrections are usually much smaller than the 0(g2) ones, and therefore of no significance (considering the uncertainties in-volved). But in the case of the W,Z-self-energies and the Z-photon transition there are two-loop diagrams in which Higgs particles couple to top quarks, and since the strength of this coupling is grrit/2Mw, a sufficiently large value of mt could make the two-loop contributions comparable in size to the one-loop ones. In particular, the asymptotic form of Ap for large mt might be m4 rather than m\ as it is to order g2. Certainly some of the diagrams give m4 terms, and it would come as a great surprise if such terms all cancelled out in the end. The complete calculation of the 0(g4) corrections would be very lengthy, but if we confine our calculations to the W,Z-self-energies and the Z-photon transition, we would likely get an accurate indication of what the result of the complete 0(g4) cal-culation would give us. We would certainly find all the m4 terms this way, since no other types of corrections include the top-Higgs couplings. In the case of zero external momentum (i.e. q = 0) the two-loop integrals can always be written in terms of the functions and J^2'1) which are defined by (3.1) 18 Chapter 3. Some Two-loop Calculations 19 and d"k r d"! / ( 2 ' 1 ) ( m i ' m 2 ' m 3 ) = / ( 2 ^ I (2x)n(*»-m?)^-4((ifc-0a-mi)- ( 3 , 2 ) This decomposition requires three tricks. The first is to use the fact that, when there is no external momentum, numerators of aM6" are equivalent to £a • bg1*" where a = k or / and 6 = k or /. The second is to regroup the terms of the numerator so that they will individually cancel with factors of the denominator and leave the new numerators free of momenta (i.e., partial fractions). The third is to relate the function i^1,1), defined by /d^k f cPl 1 ( 2 ^ J (2ir)"(k*-ml)(P-ml)((k-iy-miy ( 3 > 3 ) to I2'1 via f ( 1 , 1 ) (mi , m 2 , m3) = —^—(mj/ ( 2 , 1 )(mi,m2, ra3) + m o / ' 2 , 1 ' ^ , m 3 , mi) n — 6 + m I/^ Cma.mx.mj)). (3.4) This last formula results from integration of k by parts (insertion of dk^/dk* = n) [10]. can be calculated very simply, giving ^ - S ( ! + ( i - 7 E - i n £ ' The 0(e) term must be calculated because products of two functions sometimes arise in the decompositions of the two-loop integrals. The calculation of i^2,1) (which is much more complicated) gives 1 ,2 \,_ „ _ m? m 2 - " 2 5 6 ^ [ ? + - ( l - 2 7 E - 2 1 n ^ ) + ( ( 2 7 E - l ) l » £+h?% 3 7r2 , m L . Chapter 3. Some Two-loop Calculations 20 where J(a,6) 1, , , 1 — a — 6 r l , , 2a 1 + -lnaln6-f- = — - h a 7= 2 VA 4 l - a - b - s / \ l l 2 2b + - In2 = 4 l-a-b-y/X , TT2 1 6, , , l + a - 6 + \ / X . , l - a + 6 - v X , ~6~ + 4 l n ^ l + , - 6 - V X ) ( l - a + 6 + VX)) + Li2(: - 2 a 0 + -26 A = 1— a — b — s/XJ ' z v l — a — b — yf\ 1 - 2a - 26 - 2a6 + a 2 + 62, )], (3.7) and Li2 is the dilogarithm or Spence function, defined by r* ln(l - y) Li : dy. (3.8) Some of the results of two-loop calculations that have been done are given below. Here, CL and CR refer to the left- and right-handed parts of the Z-top coupling, and c^ ,/ and CR/ refer to those of the Z-bottom coupling. Q and Qf refer to the top and bottom charges in units of e. The answers given are each to be multiplied by a colour factor of 3 in order to be correct. As in chapter 2, the masses of non-top quarks are treated as zero. = 2ig4[—cL\{H^{mt)f fl 17lt + (——(i - - f K -n - 2 , _ Ml. n - 1 . •CR) 1 n rrif M2W n - M2Z - Ml n - 1 . 1 -H^(mt)H^(Mz) n ™2 n - 2 ( m 2 - A f f ) 2 n n - 1 C f l ) 5^1ie i f ( , ) ( m ' ) * ( , l ( A ^ ) n •™2 -CL - n iz - mW (m 2 - M2)cR) M2Z-M2W + ( n — 2 (m 2 - M^r)2 n - 1 2 2^ > x n n M i - M 2 , / ^ ( M ^ m ^ O ) •^• 1 )(M W ) m„0) Chapter 3. Some Two-loop Calculations 21 ,n — 2,. Ml. n —1 . Ml wnw.r \ " < — ^ " Tnf ~ — ^ M ^ M l 1 1)(M*m«m*) I^\Mw,mt,mt)]g^ (3-9) = 2ig4 CL/(1 +-f)—5 — T n m\ Mz — Mw H^\mt)H{1\Mz) + 4 ± ^ £ _ 7 ( u ) ( M l v , m < ) 0 ) + Mi m?(Mz-Mwy m\{M\ - Mw) 7(1,1)(Afz,0,0) Af, m?t(Mz-Mw) t (3.10) = 2zyx^[^(/YW(mt))2 - U^r + , )# ( 1 )(m,)# ( 1 )(^HO - ( 3n - 7 ^ ^ j ^ ^ ^ ) _ I i^j ( i . i ) ( m | | o | o) n nMw + ^ ( ^ + n - 3 - (n - 2)^%-)I™(Mw, mu 0)]<T mi (3.11) = V x . ^ g / K ^ + ±)HW(mt)HW(Mw) + ^I{X*\rnu 0,0) ' Mw m\ ~ ( m L ? M f ) 2 / ( 1 ' 1 ) ( M v y > m < , 0 ) + ^ / ( M ^ > M ) ] < T m2 (3.12) Chapter 3. Some Two-loop Calculations 22 4 n - 2 W • ~ ' s - — [ - m ^ W ) H W ( m ' ) i H W ( M H ) ~ H m ( M w ) ) + 2 < S ^ ) / ( u ) ( ^ ' m " 0 ) - i K t / ( u ) ( M " " m " 0 ) l 9 " ' ( 3 - 1 3 ) 4 l2(A*2 - JWjj,) if(,)(ro,)(#(1)(Mz) - HW(Mw)) + (<*(m? - - M l ) - c L - ^ - m 2 ) M , _ J ' 1 ' 1 ^ ^ m«, mt) - (cR{m2 - ^M^) - C L ^ - ^ m 2 ) M 2 I{1'l)(Mw,rnumt) = v * — C L ' [ M l - M ^ •H^m^H^iMz) - HW(Mw)) + 3 ^ 3 ; / ( 1 , 1 ) ( M z ' m t ' 0 ) " ^ ^ 7 ( 1 , 1 ) ^ m < ' ° ) M7 + Ml-Mw /(U>(M Z ,0,0)-Ml-Mw Af,2 • / ( 1 ' 1 ) ( ^Hr ,0 ,0 ) ]^ (3.15) Chapter 3. Some Two-loop Calculations 23 W * / W 1 l r [ ^ ( 1 ) ^ 2 - c i ( M i - M ^ ) H^(mt)H^(Mz) Ml -MV 2m 2 ~ My/ Ml -Mw ~ Mw Ml-Mw' cl(Ml-Mwy cl(Ml - Mw) = 2ig4xwQ[^HW(mt)HU(Mw) + J j ^ j<w>(TOf, 0,0) 0) = _ t y X w ! L _ 2 g , [ ^ ( i ) ( m < ) ^ ( i ) ( M l v ) _ i / ( u ) ( r a j i o , 0 ) + - l ) / ( 1 , 1 ) ( ^ , m t , 0 ) + ' ^ ( J l f H r . O . O ) ] ^ - ' (3.16) (3.17) (3.18) Chapter 3. Some Two-loop Calculations 24 \ H t t W (m2 - M^)(2m2 - Mw)  + Mw(Mjj - Mw) 1 \Mw,mtM9 (3.19) c?.. n C i ' ^ « 2 " 7 ^ " (? " l ) ^ ) ( ^ ( 1 ) ( m O ) 2 - ^ / ^ ( m , ) * ^(Mz) /»if 77iy> 777,/ Af 2 M2 + ( ( n " 2 ) ( 1 " 2mf ) C f l " ( ( 2 " W ~ 4 + m<> m<) M2 M2 + (2(1 - - f )cL - (n - 2)c*)(l - - f )/<1'1>(Jlfz,mt,0) - ^{Mlm]cL + ( £ - l)c f i)/( 1' 1)(Mz,0,0)]^ (3.20) = »5 n QQ'lzti* ~ f )(#(V,))2 - 2/<1'1>(mt> m „ 0) + (4-n)/(1'1)(mt,0,0)]5'"' (3.21) Many of these diagrams clearly have no m\ terms. Nevertheless, all of the diagrams Chapter 3. Some Two-loop Calculations 25 which include the top quark must be included in order to maintain gauge invariance in the full calculation. This full calculation is in progress. Bibliography [1] J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, (1964) McGraw Hill, New York [2] CHARM collaboration, Z. Phys. C 41 (1989) 567 [3] CHARM II collaboration, preprint CERN-EF/90-76 (1990) [4] D. Y. Bardin and 0. M. Dokuchaeva, Sov. J. Nucl. Phys. 36 (1982) 282 [5] CDF collaboration, Phys. Rev. Lett. 64 (1990) 142 [6] W. J. Marciano and A. Sirlin, Phys. Rev. D 22 (1980) 2695 [7] ALEPH collaboration, preprint CERN-PPE/90-101 [8] B. W. Lee, C. Quigg, and H. B. Thacker, Phys. Rev. Lett. 38 (1977) 883 [9] U. Amaldi et al., Phys. Rev. D 36 (1987) 1385 [10] J. van der Bij and M. Veltman, Nucl. Phys B 231 (1984) 205 [11] T.-P. Cheng and L.-F. Li, Gauge Theory of Elementary Particle Physics, (1984) Oxford University Press 26 Appendix A Feynman Rules The Feynman rules for vertices in the standard electroweak theory are listed below. Several of these were used in the preceding chapters, and many more will need to be used in the complete two-loop calculation. They can be worked out from the standard model Lagrangian (see, for example, [11]). A . l Vertices including fermions In each diagram, u can be replaced by c or t, and d can be replaced by s or b, with the corresponding change in the Feynman rule. Similarly, e" can be replaced by [i~ or T~, provided that ve is correspondingly changed to i / M or vT. The symbol / refers to any of the quarks or leptons. 27 Appendix A. Feynman Rules 28 H 2MW = g m d -* 2MW e X 9™° -.5 2MW 1 u 2MW u *0 V2M, w •Uud(TndL — muR) Here Qf, c£, and are given by the following table: <f, 5,6 C , / i , T 2 3 _ 1 3 0 -1 1 _ 2 2 3 X « U 2 i 3 xu) 1 2 ~2 "f" 4 - * - X 3 0 and (/ is the Kobayashi-Maskawa mixing matrix. tgm, —L V2MW e A.2 Purely bosonic vertices In these diagrams, fci, fc2, &3 refer to the momenta of the particles. All particles are con-sidered to be entering the vertex for purposes of determining the sign of the momentum or charge. The following definitions are used: C^piht h , fa) = - k2)pglu, + (k2 - h)»gi,P + (fa - fa)^^ Siuf,po — 2gflvgp<T — g^pdi/c — g^gup Appendix A. Feynman Rules Appendix A. Feynman Rules 31 f.k, y X f . k , = igsw(k2 - fci)A (1 - 2xu;)(fc2 -

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