Aspects of heavy quark physics Citation Boyd, Charles Glenn (1991) Aspects of heavy quark physics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BZR0-2W44. https://resolver.caltech.edu/CaltechETD:etd-09142005-104947 Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. The [...] Wisgur corrections to semileptonic decay form factors for the [...] system are enumerated, and a general theorem on the vanishing of all [...] corrections at threshold is derived. The contribution of charged higgs scalars to the neutron electric dipole moment in multi-higgs models is also examined, and found to be near present experimental limits. Item Type: Thesis (Dissertation (Ph.D.)) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Wise, Mark B. Group: Caltech Theory Thesis Committee: Unknown, Unknown Defense Date: 13 May 1991 Record Number: CaltechETD:etd-09142005-104947 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-09142005-104947 DOI: 10.7907/BZR0-2W44 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 3528 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 14 Sep 2005 Last Modified: 21 Dec 2019 03:00 Thesis Files Preview PDF (Boyd_cg_1991.pdf) - Final Version See Usage Policy. 3MB Repository Staff Only: item control page

ASPECTS OF HEAVY QUARK PHYSICS Thesis by Charles Glenn Boyd In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1991 (Defended May 13, 1991) ii To my Mother and Father, for getting me to this point in spacetime ili I, ACKNOWLEDGEMENTS . A great many people have helped enrich my graduate life; I am grateful to those both listed and unlisted herein. To the best thesis advisor I know, my special thanks go to Mark Wise for giv- ing me a great many ideas, putting up with my slow periods, devoting remarkable amounts of time to me, creating an informal and supportive atmosphere, having bad taste in movies, and being an overall enjoyable personality. I could not have chosen my advisor more wisely. For teaching me field theory and answering every question in any subject I could think up, my gratitude goes to John Preskill. My appreciation also goes to all of the Davids: To David Politzer for many novel and enlightening conversations, to David Hitlin for explaining the experimental end, to David Brahm for a rewarding collaboration and friendship (and especially for resuscitating this thesis with jytex), and to David Montano for support and companionship when I needed it. My fondest thanks also go to Mercedes David for blissful distraction from physics, and to Victor and Sarai David for their continued friendship. For stimulating and enjoyable arguments and for frequent aid with the computer, my appreciation goes to Eric Raiten, Gil Rivlis, John Feiler, Jim Hughes, and Alex Ridgway, as well as to Darin Beigie, for just kicking around in his spare time. Thank you, Steve Frautschi and Helen Tuck, for smoothing out the administrative end. To the other muffins: I had a great time with Becky Truman, Mark Muldoon, and Steve Otto while on Allen Street. Thanks go to Alice Quillen for absurdly long iv but. equally wonderful conversations, to Janis McKenna for showing me the great . outdoors, and to Kirsten Lewis for loving weiner dogs and dancing to insanely happy ‘Disney music. To William and Sally Hurt belong my particular thanks for providing a warm and beautiful surrounding for negligible returns. Of course, I owe most to my family. I cannot thank my mother adequately for years of love and sacrifice, or my sister Karen for her unstinting support. My love goes to them, as well as to Jim, Brandon, and James. Finally, I wish to express my long-standing gratitude to the Albany gang: to Roy Lobdell, Bryan Mundell, and Don Chandler: Many of my happiest memories are shared with you guys. My admiration and affection go to Karen-Jo Wills, for being altogether together, and to Georgette Janner, who will always occupy a special place in my heart. For introducing excitement and randomness into life, and for being the best of friends, my continued devotion and best wishes go to Jeff Van Order. I hope that I have returned, or can someday return, a measure of my happiness to those who nurtured it in me. Vv ABSTRACT The + Wisgur corrections to semileptonic decay form factors for the Q, > 0,e7 system are enumerated, and a general theorem on the vanishing of all 2. corrections at threshold is derived. The contribution of charged higgs scalars to the neutron electric dipole moment in multi-higgs models is also examined, and found to be near present experimental limits. vi CONTENTS Dedication ............0. 004 204 Acknowledgements Abstract Contents I. Aspects of Heavy Quark Physics II. Introduction to Heavy Quark Symmetry. . 1. Ideas behing the Heavy Quark Expansion .... . 2. Derivation of the Effective Lagrangian . 3. B — D Semileptonic Decay in the Wisgur Limit III. Semileptonic Decay of Q, ......2.2.~. 1. Introduction . . . 2. The G*” Contribution . 3. The J Contribution ........2. 4. Normalization at Threshold 5. Conclusions 6. Acknowledgements 7. Tables of Form Factors .........202., IV. Vanishing of + Corrections at Threshold 1. Introduction . 2. Luke’s Theorem: A First View . 3. The P Lemma. . . uu ii . Vi 12 12 15 16 17 18 19 20 vii A, The Ademolio-Gatto Theorem... 2. eee ee 28 5. The Wigner: Eckart Theorem... 2... 1 eee ee 30 6. Luke’s Theorem for Scalars ............-2202-20204--4 31 7. Conclusions . 2... 33 8. Acknowledgements Se 33 V. The Neutron Electric Dipole Moment ..........2..2.. 35 1. Standard Model Predictions . . . 2... oe 35 2. Multi-Higgs Physics. 2. 2 1. ee 35 VI. Effective Hamiltonian for the Neutron Electric Dipole Moment . 37 1. Figures. 2 ww we ee 45 VII. Outlook . 2... 1. ee 48 VIII. References ..... 1... ee ee ee 50 1 I. ASPECTS OF HEAVY QUARK PHYSICS In the Standard Model and its generalizations, charge conjugation and parity vio- lating (CP violating) processes arise because of complex couplings involving the higgs, fermion, and other fields. In the Minimal Standard Model, the complex couplings in the quark mass matrices are shifted into the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix by field redefinitions. Extensions of the Standard Model typically have new CP violating complex couplings (e.g., higgs mass matrices in multi-higgs models, or squark mass and kinetic energy terms in supersymmetric models), which cannot be incorporated into the CKM matrix. Processes involving CP violation therefore offer the possibility of distinguishing between Standard and non-Standard Model physics once the CKM mixing angles are sufficiently well determined. Theoretical methods allowing more precise experimental extraction of particularly important CKM angles (such as V,, and V,,) set the stage for sensitive tests of physics beyond the Standard Model; the heavy quark symmetry of Isgur and Wisel!~‘l, to be explored in Chapters I-IV, is such a method. Hypothetical heavy quark processes leading to CP violation too large to be explained by the CKM matrix also provide mechanisms for probing physics beyond the Standard Model; the contribution to the neutron electric dipole moment of Weinberg’s!] three-gluon operator, examined in Chapters V and VI, is one such process. Although both topics above feature heavy quarks, they cast the quarks in very different roles. The heavy quark symmetry arises as a simplification of Quantum Chromodynamics (QCD) in the large-mass limit, while the importance of the top and bottom quarks to electric dipole moments is due to the large coupling of massive quarks to higgs particles. Because the top quark is expected to decay via W bosons before it has time to hadronize, the heavy quark symmetry will not be useful in predicting quantities pertaining to top physics. The bottom quark, on the other hand, appears ideally suited for the application of the large mass limit to QCD, as done in Chapters II through IV. 3 II. INTRODUCTION TO HEAVY QUARK SYMMETRY 1. Ideas behind the Heavy Quark Expansion The basic idea behind the heavy quark formalism is that QCD displays new symmetries in the limit of infinitely heavy quark masses. We can make this more precise by considering a meson, say a B meson, and letting the mass of the bottom quark m, go to infinity with the meson four-velocity v” fixed. In this limit, the heavy quark carries nearly all of the meson’s momentum, and we can write the heavy quark momentum as p“ = mgv“ + k#. The residual momentum k* measures how far the heavy quark is off-shell, and should be of order the QCD scale Aggp. One pictures the meson as consisting of a bowling ball (the heavy quark) and many ping pong balls (the light quarks and gluons). Eventually we will make an effective Lagrangian to describe the dynamics of this system, but already we can see some of the salient features such a description will have. Since the light degrees of freedom can transfer only order Ago p of momentum to the heavy quark, the four-velocity of the heavy quark will be conserved. The effective Lagrangian will therefore have a velocity superselection rule specifying that heavy quarks of different velocities do not interact with each other via nonperturbative strong physics. It is important to note that a << 1 always holds, since the transfer of large momentum by an infinite number of soft gluons results in the disintegration of the meson and is therefore irrelevant to our problem. Another feature of the m,— oo limit appears when we go to the meson rest frame, where the heavy quark acts as a static color source. Here, the heavy quark flavor is 4 itrelevant to dynamics. Further, in the presence of the color magnetic field that is due to the light degrees of freedom, a gluon with momentum Agcp cannot flip the spin of the infinitely heavy quark. In the infinite mass limit, then, the spin of the heavy quark is conserved. The resulting SU(2N) spin-flavor symmetry (dubbed the Wisgur symmetry after M. Wise and N. Isgur) will be manifested in the effective Lagrangian we derive in the next section, but first we look at some immediate experimental implications. To the extent that the b and c quarks are very heavy, nonperturbative QCD treats them the same. The light degrees of freedom in a B or corresponding D meson will thus be in the same state. This implies, for flavor-changing current J" acting on states normalized to twice their mass, that (OlJ*|B(w)) _ (OlI*1D(v)) = 2.1.1 ins Bri (2.1.1) Defining the decay constants fy by (OlJ"|X) = fxk, (2.1.2) where py is the momentum of meson X, we see that the decay constants scale as fa =4/ mad: (2.1.3) Preliminary lattice calculations make (2.1.3) suspectl§l, Whether the B and D mesons are heavy enough for the above relation to apply with any accuracy must wait for more lattice calculations or experiments, although we will see how to estimate errors in the next section. Another implication comes from the spin symmetry, which relates matrix ele- ments involving vector and pseudoscalar mesons. The spin operator for charm quarks 5 5, = : f{d@zclozc, whose commutation relations with flavor-changing currents are known, is a symmetry operator of the effective Hamiltonian. Writing the spin part of the D meson wave function as |D) = | T|) — | |) and of the vector meson as |[D*) =| T1) + | 11), we see that the spin operator changes the pseudoscaler into a vector meson: S,|D) = $|D*). Then, for example, (D|V,|B) = 2(D"|[S3, V3]|B) = —(D"|Ao|B), (2.1.4) where V,, A, are the # components of the vector and the axial flavor-changing cur- rents, respectively. Similarly to the pseudoscalar case above, this relation holds be- cause the state of the light degrees of freedom for the D and D* are identical for infinite c quark mass. A variety of relations can be derived by using commutation relations such as those above, but a simpler method exists. By examining the QCD Lagrangian in the large-mass limit, an effective field theory that duplicates the QCD results, order by A order in -222, can be constructed), 2. Derivation of the Effective Lagrangian The spin-flavor symmetry arises because nonperturbative gluons transfer mo- mentum that is small compared to the mass of the heavy quark. In principle, one arrives at the effective Lagrangian £.¢ for nonperturbative, strong interactions by integrating hard gluons out of the QCD Lagrangian. In practice, one matches matrix elements in the two theories to determine the coupling constants in £,_. For example, the QCD propagator for a heavy quark with momentum p“ = mgv“ + k* is p+m) _ impth+m) 1+4 i Agen. ; pe—m? —— Imv-k +k? = ( 9 rz t+ Ol m ). (2.2.1) 6 Because pair creation of heavy quarks is suppressed, we need use only the positive- frequency component of the heavy quark field. Then #Q = Q+o( 4822), Since heavy quarks are always nearly on-shell, the projection operator Li in the propagator may be set to one. Thus, the leading-order, heavy quark propagator is sy. Similarly, since ws fb A QAP) = AGP )Q + O(-2E*), (2.2.2) the gluon vertex to order one is iglv,, (2.2.3) where g is the strong coupling constant and T* is a SU(3) generator. These Feynman rules are reproduced to lowest order by £, = ih,v-Dh, if we take h, to be a rescaled heavy quark field with residual momentum k. The Lagrangian can be made Lorentz-invariant by integrating over all velocities dv Lott = | Sat. (2.2.4) This evasion of the No-Go Theorem, which forbids mixing of internal and spacetime symmetries, is accomplished by using an infinite number of fields, one for each four- velocity. For additional flavors, one simply sums over the flavor index, making explicit the spin and flavor symmetry. A simple method for deriving from QCD the oc Ager) symmetry-breaking cor- rections to £,q is to define the rescaled quark field in terms of the on-shell QCD heavy quark field c(x) by shifting away its large momentum dependence and projecting out the component that behaves like a particle of momentum mv : Ha) = FF playermers (2.2.5) , Since we are not interested in antiquarks, we will drop the negative frequency com- ponent of c(x) by hand. Inverting the definition gives a + expansion for the QCD field \ — eT IM VT 1— olan) = eimere( of ip by = sate] hy. (2.2.6) 1p — Th, =e Making this substitution in the QCD Lagrangian gives Z(c)- c 1 Zhe c Log = hiv’. Da — aa [D? + 49,0,,G"] hD + (bveve,v’), (2.2.7) c in which G#” denotes the gluon field-strength tensor, and we have treated the c and b quarks as heavy. The form of the b quark correction is the same as that of the c quarks, but the coefficients will be different at the charm-mass scale because of renormalization from the b to c scale. Since oO Assz2) corrections are dominated by the charm quark, we will not need the O(*82,) corrections. To examine weak meson decays at first order will, however, require the O(A222) corrections to flavor-changing currents. Substituting wanes (2.2.6) into vector and axial currents gives! Veena) = [2 saa)] iY fe 5 Bot Hi(ayertomrmere Abag(t) = | 28h) * He [a — The above results are valid to yt, at the charm mass scale. Since relations (2.2.8) i“ | he) (x)eWilmor—mev')-e between physical quantities are renormalization point-independent, our computations at this scale will be relevant for predictions at any scale. The factor involving strong coupling constants comes from strong renormalization of the Lagrangian between the b and c mass scales, and is numerically equal to about 1.1 . An immediate check on the validity of the = expansion for b and c quarks can now be made by examining the B* to B versus D* to D mass differences. At leading- 8 order we expect degenerate masses for the pseudoscalar and vector mesons, with corrections of order Agcp. Since the B mass splitting is about 45 MeV and the D mass splitting is 145 MeV, this agrees with leading order predictions and a value of Agcp = 200 — 300MeV. At first order, the mass splitting scales as + times a strong renormalization factor of 1.15, so we expect the mass splitting ratio to be = . 2.2. Mp+—-Mp 1.15mzg (2.2.9) In this case, the leading Agen correction is less than 10%. This prediction is a simple one which is duplicated by the nonrelativistic quark model. Less obvious, more useful predictions arise when we apply the heavy quark formalism to semileptonic decays. 3. B — D Semileptonic Decay in the Wisgur Limit Possibly the most important semileptonic decay to consider is B — D. The rate for this decay is proportional to the Kobayashi-Maskawa-Cabbibo mixing angle V.,, but because the hadronic part of the matrix element is unknown, it is difficult to extract the mixing angle from experimental data. Current models of the hadronic matrix element leave a theoretical uncertainty in V,, of around 30%, although even the error is difficult to estimate. The systematic, QCD-based, Sacp. expansion can do better than this, and is expected eventually to allow the extraction of V,, with errors of o(*2s2), a few percent. Although the calculation has been done to O(Ase2 yi, we will reproduce here only the leading-order analysis!!. The B — D decay factorizes (up to electromagnetic corrections) into hadronic 9 and leptonic matrix elements. A typical hadronic matrix element such as H* = (D(v')|V“,,(0)|B(v)) (2.3.1) weak is constrained by the Wisgur symmetry to transform under spin-flavor rotations in a manner dictated by the Wigner-Eckart Theorem and the transformation properties of the heavy quarks in B, D and V“. Further, this matrix element will be related to fd weak? transition elements involving B* or D*, as well as A because the light degrees of freedom for the B, B*, D, and D* mesons are identical. Lorentz invariance allows the experimentally relevant matrix elements to be pa- rameterized by six form factors, which may be defined by (D(v')|V"|B(o)) = fy(v tu! + f_(v —v')*, (2.3.2) (D*(v'), e|A"|B(v)) = age + ape + v(v + v')# + a_e-v(v —v')4, (2.3.3) (D*(v'), |V"|B(v)) = get" eh vgus, (2.3.4) where e* is the polarization vector for the D* meson, and V“ and A“ are weak vector and axial currents. In contrast, Wisgur symmetry will permit all six of these form factors to be written in terms of one unknown function €(v-v’). A useful mnemonic for doing the group theory that leads to this result is to construct interpolating fields B= hl (2.3.5) a and D> = Al) d3Vastss (2.3.6) 10 where the Dirac indices a, 8,6 are implicitly summed, and I,t represent the light degrees of freedom. Using these fields and extracting the Dirac structures that are relevant to heavy quark spin transformations, we may write schematically (D*(v'AOPRO|B(w)) = (Oty? fAOAY TAO ROTO) = (OltalplO)i> ¢ ETH], 5. Here we have contracted the heavy quark fields and have retained all momentum integrations and unknown strong interaction physics in a nonperturbative matrix element, which can be parameterized by Pag = (O|tals|0) = Abdig + Btag + Chia, (2.3.8) in which A,B, and C are arbitrary functions of v-v’. However, since the three functions simply add together after the contraction in Equation (2.3.7), P,, reduces to Since the light degrees of freedom are the same for all of the matrix elements under consideration, similar treatments will give other matrix elements in terms of the same unknown function €, dubbed the Isgur-Wise function. The results arel®l (Dv) |V"|B(o)) = Ymemg€(v - v')(v + v')" (D*(v'), |A*|B(v)) = Ying (v-v' eM + v-v!)—vlv-e]- (2.3.10) (D*(v'), €[V"|B(v)) = —i fing E(v - vet vgus The reduction of six form factors to one unknown function is an impressive feat of heavy quark symmetry, but it is not the best. Since @y°c is the generator of charm- quark number, it acts trivially on states. However, Wisgur symmetry also tells us its li matrix element in terms of the Isgur-Wise function. Since (D(v)|e9°e|D(v)) = 2m,v® c =m,.€(v- v) Tr [tt yo 14) (2.3.11) 2 = 2m,v°€(1), the Isgur-Wise function is normalized to one at threshold (v = v’). The significance is that B — D decay rates at threshold are now completely predicted in terms of one unknown quantity, the mixing angle V,. This allows extraction of V,, from experimental data. Previously, theoretical errors were comparable to or larger than current exper- imental data. Currently, using the heavy quark formalism, theoretical errors are expected to be a few percent, much better than existing experimental errors. Much higher production of B mesons is expected in the next few years (e.g., at Cleo, B factories, etc.), so it seems likely that we will know V,, quite well, quite soon. The reason we expect only a few percent theoretical error is that somewhat surprisingly, the O(A9e2) corrections to the normalization of € at threshold also 910] that similar statements can be made about heavy vanish!l! Indeed, it turns out! Lambda and Omega systems. We demonstrate the techniques of O( Agcp ) calculations and the vanishing of threshold corrections for the Omega system in the next Chapter. 12 III. Semileptonic Decay of 20,* 1. Introduction Recent advances in heavy quark physics!!~*] increase the predictive power of QCD and allow the determination of previously unknown, strong matrix elements. This is accomplished by treating the heavy quark as infinitely massive compared to the QCD scale. In this limit, QCD contains an additional SU(2N) spin-flavor symmetry!], where N is the number of heavy quarks. This symmetry has been exploited to derive relations among form factorst in numerous systemsl!!?14,7] , including B > D, A, > A,, and Q, — Q,. The calculations can be understood in terms of an effective field theory, which incorporates SU(2N) breaking terms as perturbations in a 1/m expansion. Because we expect the expansion parameter Aggp/m, ®& 1/5, it is desirable to include the 1/m, corrections. This is particularly important for extraction of the Kobayashi-Maskawa mixing angle V,, from baryon systems, where the relevant corrections may be significantly larger (typically a factor of two compared to mesons with similar light-quark content). It is a testament to the power of the above method that predictive power remains even at O(;-). Indeed, for the B + D and A, - A, systems, all O(5-) corrections vanish at threshold!"*l, allowing the possibility of determining V,, to this order. *from G. Boyd and D. Brahm, Phys. Lett. B254:468 (1991). tThe heavy quark symmetry was present in earlier phenomenological models, such as that of Ref. [11]. 13 In this letter, we calculate the O(5-) corrections to the 2, — Q, weak form factors. As in the two cases mentioned above, relations between form factors can be derived at this order, and all O( a) corrections vanish at threshold. The nor- malization and much of the notation we use will follow that of Reference [14]. After recapitulating the leading order results and displaying the O(=-) effective Lagrangian and flavor changing current, we consider the various unknown matrix elements both at and away from threshold. The results are tabulated in Tables 1 and 2. We will denote the 2 states by Q”, with M = 1 corresponding to 2 and M = 2 to 0*. The tensors By that describe the Q™ states are Bi(v,s) = B+ YY )Isulv,s), Biv, s) = u,(v, 8). (3.1.1) Here, u, is the Rarita-Schwinger spinor for the 0*, and flavor indices have been suppressed. The tensors obey YBM(v,s) = BM(v,s), v'BM(v,s) = 0, 7" Bi =0 (3.1.2) The vector V" and axial A” current form factors are defined by (2,(v, s))IV"1%4(v, 8)) = ale, s!) [Fa + Fo" + Pye] u(v, s) (Q.(v', s')A"|Q(v, 8)) = U(v', 8") [Gyay"ys + Guys + Ggv'"75] u(v, 8) (AE (04, s)]V"|O,(v, 8)) = uy(0', 8") [No y"95 + Nyv*v!ys + Ngv*vly5 + Nygy5] u(v, s) O*(v', s’)[A“|Q v,s)) =ty(v", 3) [K ryt + K,v*v" + Kav*v! + KK. g" u(v, 8 € é \ 1 2 4 The leading-order results can be parameterized by two unknown functions!'4l (see the first two columns of Table 1): (OM (v!, s')|hYP PAY OH (v, 8) = C BM (w', s' PBN [—g"” E(w) + v"v"’&(w)] (3.1.4) 14 ~6/25 C= (ey = 11. (3.1.5) a,(m,) The coefficient C is the strong renormalization correction evaluated at the scale p = m,. We are free to evaluate all matrix elements at pp = m, because relations between physical quantities are independent of the subtraction scale. To leading order, the form factors are uniquely determined at threshold by €,(1) = 1. To find the leading 1/m, corrections, we start with the effective Lagrangian") Le= hiv. DhO + hiv! DAY + she [(v’- D)? — D? — $967 yor” | ni) (3.1.6) c in which G*” denotes the gluon field-strength tensor, and the flavor-changing currents 7H CAO lay t Fan p) c (3.1.7) Ele 2 — At = Chi) fe - im r6| hy at the scale 4 = m,. The (v’- D)? term appearing in the interaction Lagrangian L’ (the third term of Equation (3.1.6) ) can be ignored because the equations of motion imply that an insertion of (v’ Dh? in a physical amplitude suppresses the amplitude by O(;-)- Insertions of the interaction Lagrangian with the dimension-three part of the currents give rise to the unknown, time-ordered matrix elements IMN(w:P) = i (2M (v', s’)|T { | dtx (20,40 a) (x) (20 rn”) co} |QN (v, s)) ce MN (w;P) = — (OM (o', 802 | atx (KO D2RO) (x) (AOPR®) (0) § [ON (v, 8 3 . Im ¢ ( ) v v u v b ) (3.1.8) c [™ N Since D? transforms trivially under the spin symmetries, [// contributes as does the right hand side of Equation (3.1.4), but with new functions €,(w) and €,(w). 15 The dimension-four part of the currents introduces —| vaste) « IM (w;7°D), = —— (OM (o', sO ED yPV IOP (v, 8)). (3.1.9) 2m c 2. The GY’ Contribution The Isgur-Wise symmetry constrains the form of IM to be 1+ 2 bod IM (wT) = BY (v', s') 0°? —— PBN (v,s) MY. (3.2.1) where My, is the most general tensor (antisymmetric in A, p), which can be con- structed from v,, vg and g,g. Myy may not contain v!, or v, because of Eq. (3.1.2). Further, since [y,, ¢](1 + #4) = 2(1 — ¥)(v, + ¥,), v4 and v!, are also disallowed. We may therefore write Myy =~} [mw)ghge + na(w)gho'’vy + ns(w)gko"v,| (3.2.2) without loss of generality. Of these three new functions, only 7, contributes at w = 1. Some algebra gives T}'(w;T) = a(v', sas{—m [oo +e + AP t+ wD] + mp [$+ w)P(H' + v)] — 73 [(w? — 1a, Lt — (P+ w)P — (1 + wh ¥'] fasu(v, 8) TP (w;P) = a,(v', 8) {Sy Po" + vo") + Snv'T(f' + w) + Grav" [(w— #)y’Ty, —P - AP ¥'] }asulo,s) (3.2.3) The functions 7, and 7, contribute as do €, and €, respectively, but with a propor- tionality constant dependent on the final spin state. They can therefore be eliminated from the form factors for either decay by redefinitions of €, and £,, although we will not do so at this time. 16 3. The J) Contribution To examine 1! (w;7*T), (see Eq. (3.1.9)), we first look at ; . —| 7 {(c)> y INN (wD), = 5 (OM (0, OED PHM ON (0, 8)) m (3.3.1) = BM(v',s')T BN(v, s) PY where the most general form for P,’” is In the w — 1 limit, the part of Pi” that gives a nonvanishing contribution to I¥%(1;T) depends only on («, + %4). However, since v° IM" (wT), = 0, WK, + ky + ke = 0, Wks + Ky = 0, (3.3.3) so I}!% gives no contribution to the current matrix elements at threshold. Our six unknown functions reduce to two after application of the usual!™! trick (2.0, 819, HTH )[O(v,5)) = 5 c meee = (vv), (Q(0', say TAY? |4(v, 8)) = I}'(w;T), + 22'(w;P), (3.3.4) where IM" (w; L),, is defined identically to 17% (w; L),,, except that the D, acts on the bottom quark field, and to the order we are working, Q = Mg-M, = Mg—m, & 1 GeV. Dotting Equation (3.3.4) with v“, using Equation (3.1.4), and noting v4 /MN (w; Pr), = 0, we get = 3m,(1 + w)7 “4s 3m.(1 + w)" Q1 -—w kz = OU =e) ¢ — (ky + wk) Kg = —(wWk, +k). (3.3.5) 3m... 17 A similar argument shows that I! (w;T), like its companion, gives no contribution at w=1. Thus, the dimension-four part of the current gives vanishing contribution to QN > OQ” matrix elements at threshold. 4. Normalization at Threshold By comparing the vector current to the QCD charm-quark symmetry generator, we get two normalization conditions at threshold (there is no implied sum over M);: (2 (v, 8)|eyelO™(v, 8)) = 1 = (QM (uv, 5)/T{(1 + iL/)hO yh} 0" (v, 8)) — (OM (v, MOD 7° — 9° D OOM (v, 8)) = €,(1) + 27 (157°) + 20 (159°) + 2 (5947) — EM), (3.4.1) Substituting J}*(1; 7°) = —3m,(1), H?(137°) = 3m(1), AM (y 9) = (1; 9°) = 0, and IM@™(1; 7°) = €,(1), these conditions reduce to £,(1) = 1, &(1) - 3, (1) = 0, €,(1) + 3m, (1) = 0. (3.4.2) Thus 7,(1) = &(1) = 0, and all O(;-) corrections vanish at threshold. In another paper"4 we explore the generality of this result. The distinct measurable quantities at threshold to O(—;) are [C is defined in (3.1.5)] LL. me RY +AG)+ A) =C, G()= "kc, Kl) = 4c. (3.4.3) 18 5. Conclusions Since €,(1) = 0, we may absorb the contribution of 1" (w;T) into the definitions of £,(w) and €,(w) without affecting their normalization at v = v’. Then away from w = 1, we have seven unknown functions and one dimensional constant describing fourteen weak decay form factors. The results for the vector and axial currents are listed in Table 1. Recall that 7; and x; are O(Agc¢p/m,), while €; are O(1). The relevant form factor is C= 1.1 times the sum of the nine entries; e.g., Fy = 1x [380+ 24 (w) + 2+ PG (w) + wn (w) + 1 — w?)n(w) + (wt L)Ky(w) + w(w + 1)K,(w)| (3.5.1) We can simplify these results by defining €) = €; — 3n,; and €/ = €, + 3, for 7 = 1,2. The new relations, correct to O(=-), are listed in Table 2. Predictions for form-factor relations are easier to construct in the Q* system since fewer nonperturbative functions contribute. For example, measurements of N, over a range of recoil momenta and a measurement of K, at a single kinematic point determine é/(w) and (2. Values of K, at subsequent kinematic points are then predicted by the results in Table 2. At threshold, the O(=-) form-factor predictions coincide with the O(1) predic- tions. Perturbative corrections have been calculated!*] and are easily incorporated into the results of this paper. Thus, if the Q, is seen in the near future, threshold measurements of the decays discussed here should yield a value for the Kobayashi- Maskawa angle V,, with theoretical uncertainties of order ae While these uncer- tainties are likely to be larger in the present system than in, say, the B-D system, alternate systems capable of extracting V,, have additional importance in understand- 19 ing higher-order corrections. The existence of three separate systems in which rates are predicted to O(+-) provides a useful laboratory for both theoretical and experi- mental physics. 6. Acknowledgements The authors wish to thank Mark Wise for helpful suggestions and illuminating discussions, as well as David Politzer, David Hitlin, David Montano, Jim Hughes, John Feiler, and Alex Ridgway for stimulating conversations. 20 7. Tables of Form Factors <i & gh-€, wb UB "2 13 Ky Ke Fol oe = “2 ww? 1 w 1—w? 0 w+l w(w+l) G, | “¥ wih oie (w -1)? w l-w? 0 w-1 w(w—1) 2(1—w _ ‘ F, 5 F x Het 0 -2 2w-2 1- w 1 G, 3 3 wai) 0 -~2 2w+2 wl —w ~~] Fy 2 20- 2 2-2w -2 Qw-2 1-w -2-w -1-2w G, | 72 2@ 2kew Qw-2 2 -2w-2 -w-1 2-w Ww ~1 wed ~1 ~fi V3(w-1) -V3 ~/fBw Ny V3 “V3” V3 0 3 Ste 0 “8 iu ie Th —w ~ 3B(w+1 ~ ~ ay NK) RB AR Yair 0 So 00 MR Ge 2 N, 0 0 Fier 0 0 0 V3 -V3 V3 K, | 0 0 wa 0 0 0 -Vv3 VB v3 Nz 0 a 0 0 0 v3 VB VB OV K 0 a 0 0 0 -VB V3 VW V3 >) —2 Ny B 0 a, 0 -V3 0 0 0 0 Ky | = 0 2% 0 V3 0 0 0 0 Table 1: Form Factors from Original Functions The form factors are given by C = 1.1 times the sum of the entries; see Eq. (3.5.1). 21 & 2 Bare roe Ns Ky Ky F, “2 wrod a w — 1 0 wtl w(w+l) G, 72 = gic (w—1)? 0 w-1l w(w-1) 2(1—w 2 F, 2 2 FS 0 1-w w 1 ~ 20-1 y G, - 2 = Her 0 wt —w —1 F; 2 28 3 2-20 l-w -2-w 1-2 Gs “2 awit) ie 2w-2 —w-l 2—w 2w—1 1 eer iE Ns Ky Ky ie awn] —1 mf 3 om Vf Bur Ky a a Vile 0 0 5 3 N, 0 0 wos 0 V3 -v3 V3 ta 2 K, 0 0 aem 0 -V3 v3 V3 Nz 0 a 0 0 V3 V3 —V3 K, 0 4 0 0 V3 VB v3 ~2 =2 N, B 0 a 0 0 0 0 K, a 0 Bee 0 0 0 0 _ Table 2: Form Factors from Shifted Functions For each decay separately, the functions 1, and 7, can be absorbed into &, and &,, respectively, without changing the normalization condition €,(1) = 1. We define £/ = €; — 3n; and €! = € + Sj. 22 IV. VANISHING OF = CORRECTIONS AT THRESHOLD* 1. Introduction Recent advances in heavy quark physics!!-?"7 increase the predictive power of QCD and allow the determination of previously unknown hadronic transition ampli- tudes. To lowest order, this is accomplished by exploiting the SU(2N) spin-flavor symmetry!) present in a system with N heavy quarks to derive relations among form factors in numerous systems!!?44), including BD, A, A,, and Q, > 2. Higher-order corrections can be calculated and understood in terms of an effective field theory?"5], which incorporates SU(2N) breaking terms as perturbations in a A = Agep/m expansion. In all three systems, it is found!) that all O(A) correc- tions vanish at threshold. We show that this is a general result that applies to any hadronic transition amplitude in which the initial and final particles are composed of a heavy quark or scalar plus light degrees-of-freedom in an arbitrary spin state. This result holds because the leading order corrections to the effective current make no contribution-at threshold. This allows use of the Ademollo-Gatto Theorem!!l, which says that the leading corrections to the effective Lagrangian also make no contribution. The outline of this chapter is as follows: We first write down the effective La- grangian and current, and make explicit the SU(4) symmetry present to leading order. We examine the consequences of normalization conditions at threshold and compare *From C. Boyd and D. Brahm, Phys. Lett. B257:393(1991). 23 them to similar conditions in chiral SU(3),,,,,- In Section 3 we remove obstacles to forming an SU(4) current algebra valid to O(A), namely, the current corrections. We then apply the methods of Ademollo and Gatto in Sections 4 and 5, concluding that all O(A) corrections vanish at threshold. We call this result Luke’s Theorem. The theorem is generalized to heavy scalars in Section 6. Finally, we summarize our findings in Section 7. 2. Luke’s Theorem: A First View The effective theory is written in the fields ale ) satisfyin phi? ) — Al) given by ying & ho = (4) ecient = (1 — gD )ceimere (4.2.1) so that c= erimere(y — By AHO = evimeve (BA (4.2.2) n=O and similarly for ne), By inserting the + expansion into the QCD Lagrangian, we derive the effective Lagrangian to O(A)*, £! = hiv!’ - DRO — 1 Fle 4 Li c 5 A? [D? + $950 6" | hy Ab?) 4. (b,vec,v') (4.2.3) Cc in which G*” denotes the gluon field-strength tensor, and the flavor-changing currents become, to Of), Vireak (©) = cal” 0 G — - AM) (x )e~mov—mev')-e (4.2.4) Avveak() = E etm) | ni) [ats “5 A) (g)e~Mrvemmer')-e *QOur definition (4.2.1) differs slightly from that of References [2,15,9], but. this alters the Lagrangian only by terms which vanish by the equations of motion. 24 at the scale = m,. Conservation of charm states that the matrix element of @y°%c is unity order by order in +. Schematically, (e7°c) = 1 = (hyPh) + E(iLhyh + hilpyh) (4.2.5) between identical states. This was first used by Lukel’l to show the vanishing of O(A) corrections to the B-D threshold amplitudes. In general, a vanishing JP contribution implies a vanishing £’ contribution to the vector transition element between states of identical spin, but does not obviously rule out nonzero contributions to transitions involving spin flip. Rather than try to rule out such contributions with the usual tensor methods, we will approach the problem using current algebra. We may rewrite the effective Lagrangian as L=Viv- DU —- UMD, (4.2.6) which has the SU(4) Noether currents ji = Vy"t, V —iVM7"t,¥, (4.2.7) ; = Se) 7 (OF) SB ‘ where M = 7 diag| =, ma) ao anh U = (AO AO, AO A), ney is the rescaled q¢ field with spin up (down), and the ¢, are the SU(4) generators. To leading order, the Noether currents form a SU(4) algebra Lel@), Io Met. = fared (BOE — 9). (4.2.8) It is convenient to work in the meson rest frame, where the relation between effective currents and QCD currents J (assembled from QCD fields c and b) which change heavy flavor from q; tog; is J(z) = eo may ay) j(v). There is a strong renormalization factor [see Eq. (4.2.4)}, which will remain implicit. 25 The heavy quark symmetry is strongly reminiscent of the approximate chiral (u,d,s) light quark symmetry. In chiral SU(3), the current algebra was used by Ademollo and Gatto!!®l to show that the vector current transition amplitudes are uncorrected to O05) at q? = 0: No analogous statement can be made for axial chiral currents because the SU(3),.:.) is spontaneously broken, a hindrance that is absent in the heavy quark case. On the other hand, the heavy quark symmetry- breaking term involves derivatives of the fields, which alters the Noether current at O(A). We address this issue first. 3. The J Lemma In order to apply the method of Ademollo and Gatto to O(A) corrections in the Lagrangian, we must first show that the J corrections to the current give no contribution to threshold transition elements. We will do this for baryons and mesons separately. A useful mnemonic for doing the SU(4) group theory is to construct interpolating fields for the external states, contract the heavy fields with the current, and shift all information about the light degrees of freedom into a nonperturbative matrix element. The interpolating field for a spin n — 4 baryon may be written B(v) = [eatatn pe ue) (ye) (4.3.1) Hy Hg sHy? where / represents the light degrees of freedom, Uwe) sg describes the polar- ization of the baryon, r is summed over heavy quark spins, the j; are Lorentz indices, and a, are Dirac indices. The light polarization tensor satisfies vie, 4... = 0. Hy ty The spi n+ $ baryon with heavy quark spin parallel to the same light degrees of freedom has an interpolating field Br(v) = [ten fe alintin ul (OE es estg din ga (4.3.2) where Dirac indices are now suppressed. A flavor-changing transition amplitude between spin n ~ $ baryons B, and B, thus contains l= (B,(v he Dy Thi |B,(v)) = Pirie Ch ebin Can gsiton u,(v') HEP fy, (v), where I is a collection of gamma matrices, and the nonperturbative matrix element may be schematically denoted by Pivertan — (Of lta Hn Dy [nti Han 10), (4.3.4) This is the J) contribution. At threshold (v = v'), the index \ can be carried only by vy, or g\', but in the latter case at least one of the remaining upper indices must be carried by vs, and from v's Ey. = 0 this vanishes. Thus Py?" = vy Xan, The equations of motion imply that an on-shell transition amplitude with an insertion of v- Dh is suppressed by O(A). Then v*J, = 0, forcing X"1"¥2" = 0. Since the nonperturbative matrix elements for transitions involving B* are the same, the J) contribution to any transition element in the B — B* system vanishes at threshold. If one sums.over initial and final spins, the above argument extends trivially to meson systems. Amplitudes between states of definite polarization require a bit more work. The light. degrees of freedom, with half-integral spin, can be written using only Dirac indices. For example, the interpolating field for a spin-n meson in which the 27 light degrees of freedom carry spin n + ; is written AP mee JOP Q HA. wm by oa lt v Aan Al ox | " © hy db Facy Oy Varn 1 on uge (4.3.5) The corresponding spin rn + 1 meson is NP Poy Gan, molt ln Mrpi ad pu ‘492 Al om l , " [uy fiat Vapor Tada 1%2n (4 nt i h Jae (4.3.6) At v = v’, an amplitude between spin n mesons looks like Th = (M(v')|h" DERM (0) (4.3.7) ee JIE 1G By Bg ow ; By akg 149 po Lite = PY Chay ten Magy Man 101 ey Vaan yey, ( 2 1 2 ) AB If one of the particles is spin n + 1, the P is replaced by I'y*+5, and one of the polarization tensors acquires an extra index «, This will make no difference to the argument presented below. Og By Bo The nonperturbative matrix element Py?’ now carries Dirac indices, which must be paired in Kronecker deltas Sasa, or contracted velocities j asa," Tt will turn out that a # is always equivalent to some combination of Kronecker deltas. The free index A can be carried only by vy or 7, so the amplitude will be a sum of terms, each of which contains either vy or 4). As in the baryon case, v\J{ = 0 implies / \ = 0 if I\ is proportional to v,. We proceed to show that this is the case. A given term in the amplitude containing 7, will be a product of traces of gamma matrices 7", 7,, contracted velocities #, and Pt. P+, where P* = it are velocity projection operators. The gamma matrices may be transposed, but this does not obstruct our proof. This can be seen by choosing the Majorana basis in which y* = (7*)", for spatial index &, and then going to the rest frame of the decaying meson. In this frame the naught cormponents of the polarization tensors are zero, so the + 28 have only spatial indices. The argument below then gives J) = 0 in the rest frame, and therefore in all frames. A term containing 7, will be either of the form or of the forin Cyjnlpyee Ely] oe Trfye ay Trly say ee PEPPTR. (4.3.9) Al! gamma matrices wrilten without Lorentz indices are contracted with polar- ization tensors. There are no insertions of ¢ because any trace with a contracted velocity # vanishes unless the trace contains either 4,, im which case the # converts the 7, into v,, or P*TP*, in which case the ¥ anticommutes through until it is absorbed by a projection operator. Since the trace in line (4.3.8) containing 7, contains an odd number of contracted gamma matrices, the trace with projection operators also contains an odd number of contracted gamma matrices. However, “P+ = P~+" + v", so this trace vanishes. The last trace in line (4.3.9) contains an even number of contracted gamma matrices. By anticommuting iif around the trace, we convert 7 into v,, showing that J, is proportional to vy. Then v*/\ = 0 forces 1, = 0, completing our proof. The above argument also implies the vanishing of some types of O( \*) threshold corrections, such as insertions of £’ with the O(A) part of the current (L'DY), although this appears academic at present. 4, The Ademollo-Gatto Theorem The heavy quark symmetry induces an SU(4)@5! among hadron quadruplets, such 29 as {.B, D, B*, D*}, We study, as an example, the vector-current matrix element for BD, (DW) [VE lB) = Fy (eee) a + 0)" + F(u-ve'\u = v')4, (4.4.1) weak We want to find F,(1) to O(A). We use the SU(2)yecio, SUbalgebra consisting of Qt = [ee qh erie, Qz = fe vagh () (ah (x), K,= [eH troy ho —hO(x)yPh)(a)). (4.4.2) Comparing to (4.2.4) and using the J) lemma, we see that between states of the same multiplet, Q7 is the charge corresponding to Vii. to O(A), up to a phase. To the same order, = (tb — ete), We sandwich the vector commutation relation (Qt, Qo] = Ab, (4.4.3) between (B(v')| and |B(v)), and on the left side insert a complete set of states. It is convenient to work in the meson rest-frame (where v® = 1), and in a volume V : then states are normalized (BiB) = 2V, and we get 50 S (Blo Q# ale”) (a(v"|Q7[Blv)) — (Qt Q7) = (Ble) |K,|B(v)). (4.4.4) To O(A), the right side equals V. The matrix elements on the left side can be written (al[H eft» Q*)|B) ™ A (al[d Ip Q*]|B) Bug l@) ~ Eag(B) Legla) — Eng (BY? {a|Q*|B) = (4.4.5) where Hyg is the symmetry-breaking part of the effective Hamiltonian Hg, and Loo is the eigenvalue of Hag: ¢.9., Eg (B) = 11g —m,. When fa) is not in the quadruplet, 30 the matrix element is O(A), and since these matrix elements appear in pairs ~ one insertion of C’ to leave the quadruplet and another to return - the contribution is O(A*). Therefore, we keep only intermediate states within the quadruplet, of which only |D) contributes. The left side of Eq. (4.4.4) thus becomes $7 MPOQT IBC). (4.4.6) Eq. (4.4.4) then becomes solVIV RP =V, (4.4.7) from which we conclude that Fi) =14+ 0’). (4.4.8) Alternately, we could study the B-+ D* transition, using QP = [ee gira, Qe- = [eager We), (4.4.9) which satisfy the axial commutation relation Now the form factor of the weak axial current A® is shown to be normalized to unity at threshold and uncorrected to O(A). 5. The Wigner-Eckart Theorem Ye We generalize to an arbitrary mesonic or baryonic quadruplet {B,C B*,C*}, with highest spin s+1. For given current and external states, the transition element 3t is parameterized by. velocity-dependent form factors which reduce at threshold to a single, reduced matrix element (Cl Vrcarcl B) = Fac. (CO VMocan |B) = 0, (C*|Vereak |B”) aad Paice, (4.5.1) (CA veaxl 2) = CBee (CPA veak lB) = Gages, (C™A weak |B”) = Garo A transition amplitude is the product of a reduced matrix element times a Clebsch- Gordan coefficient, e.g., (C*(s+1)|A*| B*(s+1)) = yew (S+1,5+1]1,0; 3+1, 941) (4.5.2) in the standard notation!4l, The Clebsch-Gordan coefficients are trivial in the vector case, since VV. is spin-0. We can sandwich the vector commutation relation between |B) states to show that Fge = 1, and between |B*) states to show that Fy.c. = 1. The sandwiched axial commutation relation between |B*(s-+1)) states gives, from Eq. (4.5.2), Gawcr = (s+1,s+1[1,0; st+1,s+1)~', while between |B*(s)) states and between |B(s)) states we get, respectively, s+1,s{1,0; s,8)Gyel? + |(s,5]1,0; 5.5 Geel =1 BC B A®|B"(in))" 3 (C*(m)|A?|.B(m’)) in the second term of where we have used (C'(im’) the first equation. Thus, all the reduced matrix elements are determined to O(A). G6. Luke’s Theorem for Scalars The above analysis is not limited to heavy fermions. For a heavy particle of arbitrary spin, one expects the leading term in the effective Lagrangian to contain 32 v- D, and higher-order terms to be quadratic in derivatives. These are precisely the conditions employed is the J lenima to demonstrate the nonrenormalization of the effective current at threshold. The Ademollo-Gatto theorem can then be applied to the relevant spin-flavor symmetry, The case of a heavy color triplet scalar has been done to leading order by Georgi and Wise?4, They use effective scalar fields \, = e's" y in terms of which the effective Lagrangian is £L, = V,Miv- DV, —U,NY,,. (4.6.1) The heavy quark fields h, have been assembled with the scalars in W = (h,, y*), the kinetic-energy term involves Af = diag{1,2m,], and the breaking term is N = diag( 2% iD). 2m, At leading order, the above Lagrangian is invariant under SU(3) @ U(1) trans- formations 6w,=i{ . w., (4.6.2) € f $s 27n x where 5* are the SU(2),.,,, generators on the quark and E is an infinitesimal spinor obeying 4 = E. The Noether current is jh = WV, Mo"L WV, + ih Nt, (4.6.3) 1 - ar : nytt 1. ; : where the t, are the SU(3) generators and V = diag[ 2, iD*]. Since the equations & of motion imply that »- Dy = 0 when inserted in a physical matrix element, the current is uncorrected at threshold by the J lemma. The Ademollo-Gatto theorem then asserts vanishing £' corrections at threshold. 33 7. Conclusions To lowest order, threshold transitions between heavy quark systems can be de- scribed by reduced matrix elements with known normalization. In the B-D system, for example, the only such element is the Isgur-Wise function!], and this normaliza- tion allows extraction of the Kobayashi-Maskawa mixing angle V,, with corrections of O(A). Luke! has shown that this normalization is unchanged even at O(A), allowing correspondingly reliable extraction of V,,. We have shown that these matrix elements are uncorrected at O(A) in general, for heavy quark systems of arbitrary spin. This result includes both the correction to the currents, by our J) lemma, and corrections arising from insertions of the Lagrangian (L’), by the Ademollo-Gatto Theorem. We have generalized it to heavy scalar particles, and we expect it to apply for heavy particles with general spin. The statement that O(A) corrections vanish for arbitrary spin systems is Luke’s Theorem. Luke’s Theorem unifies existing O(A) calculations. However, while it applies in principle to any heavy quark system, in practice the experimentally accessible systems are those already examined. Its present value is thus largely pedagogical, unless new particles (squarks, technibosons, heavy four-quark bags, etc.) are discovered, a prospect with perhaps a glimmer of hope. 8. Acknowledgements The authors wish to thank Adam Falk, David Montano, David Politzer, John Preskill, Alex Ridgway, and John Feiler for stimulating conversations. We especially thank Mark Wise for introducing to us the interpolating field method, as well as 34 postulating the result of the present paper. 35 V. THE NEUTRON ELECTRIC DIPOLE MOMENT 1. Standard Model Predictions The Wisgur symmetry is unnecessary for analysis of some heavy quark processes. Contributions of top quark loops to the neutron electric dipole moment (nedm) can be studied perturbatively. Such studies are sensitive to new physics at the weak scale. In the Standard Model with 0@9¢p = 0, the only CP-violating parameter is a complex phase in the CKM matrix. Since electric dipole moments are CP-violating quantities, Standard Model generation of a nedm involves W bosons. Unitarity of the CKM matrix requires at least two W loops, so that a nedm is order a2... It turns out!4l that the nedm is suppressed by an additional factor of Qstrong( My). The end result is that nedm generation via edm’s of fundamental quarks in the Standard Model is predicted to be less than about 107-*4e- cm. A larger contribution comes from baryon-meson loops!*), but these are also small: |d| < 10-84e- em. The current experimental limit on the nedm d is!4l |d| < 10-%5e- em. A mea- surement in the near future of a nonzero nedm would therefore signal the existence of physics beyond that of the Standard Model. 2. Multi-Higgs Physics Until recently, most extensions of the Standard Model were believed to predict nedm’s much smaller than the experimental limit (although one exception was the Minimal Supersymmetric Model, which was actually constrained by experiment!?*)), 36 The reason is that most CP-violating operators one can write down for an effective theory below the W and Higgs masses are suppressed. by either small CKM angles or powers of light quark to Higgs mass ratios. The exception is the three-gluon operator GGG, where G is the gluon field- strength tensor and G' is its dual. This operator, as noted by Weinbergl“l, is unsup- pressed by powers of light quark masses and. will typically dominate contributions to the nedm. It can appear in the low-energy effective Lagrangian of theories with several Higgs particles. Multi-Higgs theories generally contain additional complex couplings in the Higgs mass matrix if there are three or more Higgs doublets in the theory, or if there are two Higgs doublets and additional Higgs singlets. When the heavy particles (top quarks, W and Z bosons, and Higgs) are integrated out of the theory, the operators with complex couplings induce nonrenormalizable CP- violating operators such as GGG. In two-Higgs models with additional singlets, only neutral Iliggs contribute to GGG, which in turn contributes to the wavefunction of the neutron nonperturbatively. Using naive dimensional analysisl*4] to estimate the effect of the three-gluon operator on the neutron, one arrives at a predicted nedm very near current experimental thresholds. In theories with more than two Higgs doublets, charged Higgs can also induce the three-gluon operator. Indeed, over much of parameter space, charged Higgs processes dominate?7l, This is the topic of the next chapter. 37 VI. Effective Hamiltonian for the Electric Dipole Moment of the Neutron* The standard model with three generations of quarks and leptons and a single Higgs doublet has only two sources of CP violation: the vacuum angle for the strong (28) and the phase 6 in the Kobayashi-Maskawa matrix, The strin- interactions Igcp gent experimental limit on the electric dipole moment of the neutron, d, < 107” e— cm], gives rise to the bound Ogc¢p S 107° on the strong interaction vacuum angle®4, The phase 6, however, is not restricted by the present limit on the electric dipole mo- ment of the neutron, and in the minimal standard model it must be the source of the CP violation observed in kaon decays!*4), Understanding the conservation of CP by the strong interactions is an important problem in particle physics. Speculative explanations for the smallness of Ogqp do exist. For example, there could be a U(1) Peccei-Quinn symmetry] that is sponta- neously broken at a large energy scale. This converts Oggp to a dynamical variable, which is determined to be near zero by minimizing the vacuum energy. Alternatively, if quantum fluctuations in the topology of spacetime occur, the wave function of the universe may be infinitely strongly peaked on the subspace of universes where Og¢p is near zero", In the standard model only one Higgs doublet is required spontaneously to break 5U(3)xSU(2)xU(1) to the low-energy gauge group S5U(3)x U(1) and to give mass to the quarks and leptons. However, there is no compelling physical reason for such a minimal Higgs sector. Extensions of the standard model motivated by the hierarchy *C.G. Boyd, A. Gupta, S. Trivedi and M. Wise, Phys. Lett.B241:584 (1990). 38 puzzle typically have a much more complicated Higgs sector. One of the simplest extensions of the Higgs sector is the addition of more doublets. With one doublet the Higgs sector contributes a single real scalar to the spectrum of the theory. It automatically has no flavor-changing couplings to quarks; its coupling to quarks is proportional to their mass matrices so the same transformation that diagonalizes the quark mass matrices diagonalizes its couplings. With n doublets the physical degrees of freedom arising from the Higgs sector are (2n — 1) neutral scalars and (n — 1) charged scalars. It is no longer automatic that the tree level couplings of the neutral scalars are flavor diagonal. However, flavor-changing tree level couplings of the neutral scalars are absent if the up-type quark Yukawa couplings involve only one of the doublets and the down-type quark Yukawa couplings also involve only one of the doublets!*4], We consider a model of this type with n doublets H;,j = 1,...n where H, gives mass to the up-type quarks, H, gives mass to the down-type quarks, and the remaining n — 2 doublets do not Yukawa couple to the quarks. For three or more doublets, there are new phases in the Higgs sector that contribute to CP violation, In this Chapter we consider the influence of these new phases on the electric dipole moment of the neutron. In this model the coupling of the charged Higgs H®) to the quarks is given by 39 the Lagrangian density (“" £a~ ee WMG,2D) My VL) | 8 1 \o/ 6.0. sz WSP(a,2,1) V Mp(i+25)| + | + he. 2 7, where Ho H,=({ ° , AM (6.0.2) HO) and (H a) =v;. In Eq. (6.0.1), V is the Kobayashi-Maskawa matrix and m, O 0 m, 0 0 0 0 m, 0 0 m The quark fields in Eq. (6.0.1) are mass eigenstate fields, but that is not true of the Higgs fields. The mass eigenstate charged scalars gh) are related to the H mH by anxXn unitary transformation go = S- Yip HO) = S- yA Ho, (6.0.4) k=l k= One of the fields gh?) is the Goldstone boson associated with the spontaneous break- down of SU(2)xU(1) to U(1),,,,. Without loss of generality, we take it to be oh), SO that Ye = 9. v,/V2 My. (6.0.5) 40 Combining Eqs. (6.0.1), (6.0.4) and (6.0.5) gives!*6 d n (+) > 92 Pk 7) aa FP / : cay Pe J (2) aan mM, va- dy BV5 My (Fi) (ed Me Va) | i (6.0.6) + (3) (w,é,t) V Mp(1 +75) | s + hic. 21 b In unitary gauge the field of) becomes the longitudinal component of the W-boson field and so it is omitted from Eq. ( 6.0.6). (Its couplings are independent of Y.) For n > 3 the matrix Y contains phases that cannot be removed by redefining the phases of the fields gh) and hence are a source of CP violation. Integrating out the scalar degrees of freedom (we assume that none of the scalars is significantly lighter than the W-boson), the top-quark and the W- and Z-gauge bosons, generates an effective Hamiltonian for CP violating quantities. In the case of the electric dipole moment of the neutron, one is interested in flavor-singlet operators that violate parity P and time-reversal 7’. In addition to renormalizing the value of the vacuum angle Ogqp and the Kobayashi-Maskawa phase 6, this procedure will generate nonrenormalizable operators whose matrix elements contribute to the electric dipole moment of the neutron. The nonrenormalizable operators of dimension six that can contribute to the electric dipole moment of the neutron (and are not suppressed by light up, down and strange quark masses) are O, = ig? Tr[G,, G,? Gy] &’” (6.0.7a) O,=g m bT* 0, b GS, br? (6.0.7b) O,=em, bo, b Fy, (6.0.7c) Al , where F' is the electromagnetic field-strength tensor. In Eq. (6.0.7) we have included a factor of m, with O, and Og, since a factor of m, must accompany a (single) right-handed bottom quark. This factor increases the dimension of O, and O, by one unit. We have neglected other dimension-six operators (e.g., the color-electric dipole moment of the charm quark and four-quark operators) whose coefficients are suppressed (in the multi-Higgs models considered here) by factors of (m,/My)? or (m,/My,)?. The effective Hamiltonian density is 3 Hee = », CH) OH) (6.0.8) i=] The coefficients C;(My,) have a perturbative expansion in a,( My). In Ref. [4] it was noted that the operator O, gives a contribution to the electric dipole moment of the neutron that is not suppressed by light (compared with the QCD scale) quark masses. Its coefficient C;(My) has been computed and arises from 437]| Charged scalar exchange does two-loop graphs involving neutral scalar exchange! not contribute (again neglecting terms suppressed by (m,/My )? and (m./My,)?) to Cy (My). The coefficient C,(My,) can be computed from the one-loop graph in Fig. 1. In this case, the contribution from charged scalar exchange dominates over neutral scalar exchange. To compute C,,(.My) we work off-shell, matching the amputated one- particle-irreducible 6 —+ b+ gluon Green function in the complete theory with that in the effective theory. Since we are working off-shell, the graph in Fig. 1 can contribute not only to the coefficient of O, but also to the coefficient of the operator O, = bPP+y;b, which becomes a mass term when the equations of motion are applied. As noted in Ref. [38], the contribution of O, can be isolated by focussing on the Lorentz structure 7, ~ ys, where ps is the gluon vector index and k is its four-momentum. 42 Using this procedure (and the approximation |V,,|? ~ 1), we find that Gp k C(My) = V2 a 3 tm (32 #) (=) } , (6.0.9) 1 tn + pe (2-3) ** V(@,— 18" Ae, — 12 we where t, = mel. e020) With a subtraction point equal to My (the W-boson mass), there are large log- arithms of My divided by the QCD scale in the perturbative expansion of the matrix elements of the operators O,,O, and O;. These can be transferred from the matrix elements of the operators to their coefficients C; by using the renormalization group equations to move y down to the scale of the strong interactions. The anomalous dimension for O, was calculated in Ref. [39] and the anomalous dimension for O, was calculated in ref. [40]. Using these results, it follows that for My > p > m,, h 54/23 . Cy(") = late) C\(My) , (6.0.11a) M,») 114/23 | Cy() = Feral C(My) (6.0.11b) Since C',(My,) arises at two loops while C,( My) arises at one loop, we have neglected in Eq. (6.0.1la) possible mixing under renormalization of O, with O, (operator O, cannot mix with O, because with the factor of m, removed, it is only dimension five). At the scale = m,, it is appropriate to go over to an effective four-quark theory. In the effective four-quark theory, only the operator O, survives. However, because the coefficient C,(My,) arises at two-loops, it is important to include the contribution to the coefficient of O, (in the effective four-quark theory) that comes from matching 43 the one-loop Feynman diagrams in Figs. 2 and 3 with the tree level amputated three- gluon Greens function of O,. In Figs. 2 and 3 the shaded square denotes an insertion of O,. Explicit calculation of Fig. 2 gives g° d r X a) m(7 (T/,7°}) [2«9" @ Py TyPy + 3° 4 Py Ty My ve v vd 2 + Ey \ PT, Pg — €, \ PT, Te + €, \5 Y)\ Py Pa ’ (6.0.12) + 26,"\5 0) Py Py — egy Xa (Dale? + 27 p) + ry (p? +2" p)))] while explicit calculation of Fig. 3 gives 2 ~55 Te(T4{7!, r)) Cay A (nv? +2p-r) + py(r? + 2p- r)) ; (6.0.13) Comparing the sum of (12) and (13) with the amputated three gluon tree level Greens function of O, gives that in the effective four-quark theory, C,(m,) = fest) {Cu(My}+ oo [ese Cx(My)} . (6.0.14) Note that the operator O; does not match onto O, at one-loop, so its effects can be neglected (they will be suppressed by powers of the QCD scale divided by the bottom quark mass). Finally, moving the subtraction point y below the charm-quark mass, we go over to an effective three-quark theory where a,(My) 54/23 «,(m,) 54/25 «,(m,) 54/27 a,(™m,) a,(m,) a,,({L) { 1 oan ~40/23 (6.0.15) Eqs. (6.0.15) and (6.0.9) are the main results of this letter. The strong interaction corrections suppress the effects of C,;(My,) relative to those of C,(My,) by the factor [a,(My)/a,(m,)]*°/?? ~ 0.3. Also, there are special cases, for example, when |¥2,| << 1, where a further enhancement of C,(My,) occurs. 44 Towards the completion of this work we received a copy of Ref. [41], where C,(My,) is computed in minimal low-energy supergravity models. There it is also noted that at the weak scale, the color-electric dipole moment of heavy quarks (com- pared with the QCD scale) should, in principle, be included in the effective Hamilto- nian. Similar work to that presented here has been done independently by M. Dine and W. Fishler (CCNY-HEP-89/21 and UTTG-03-90) and E. Braaten, C.S. Li and T.C. Yuan), We thank them for discussing their work with us prior to its publica- tion. AS Fig. 1. Feynman diagram that determines the value of C,(My). FIGURE 1 46 Fig. 2. Some of the Feynman diagrams that determine the matching between C,(m,) in the effective five-quark theory and C,(m,) in the effective four-quark theory. + permutations FIGURE 2 47 Fig. 3. Remainder of the Feynman diagrams that determine the matching between C,(m,) in the effective five-quark theory and C,(m,) in the effective four-quark theory. + permutations FIGURE 3 48 VII. Outlook What have we learned from the preceding chapters? For the neutron electric dipole moment, we know that a value of 10~*°¢-em is incompatible with the Minimal Standard Model, but is compatible with multi-Higgs models with or without charged Iliggs exchange. The effect of charged Higgs on the neutron electric dipole moment is determined by integrating the heavy Higgs out of the theory, thus generating a color magnetic moment for the b quark. After renormalizing down to the b mass scale and integrating out the b quark, Weinberg’s three gluon operator is induced. This operator then feeds into the neutron wavefunction to produce an electric dipole moment. In addition to simple multi-Higgs extensions of the Standard Model, left-right symmetric and supersymmetric models can induce similarly large dipole moments, so the measurement of a nedm does little more than rule out the Standard Model. It is difficult to differentiate between competing models using only the neutron clectric dipole moment. One reason for this difficulty is the need for some estimation scheme, such as naive dimensional analysis, to determine the effect of the Weinberg operator on the neutron wavefunction, This problem is not present for the electron electric dipole moment, to which operators similar to Weinberg’s can induce a moment near current experimental limits!4], This allows quantitative information about new physics to be extracted from the electron electric dipole moment, although it will still be difficult to rule out competing models because of the presence of free parameters in most, theories, 49 More precise statements about physics beyond the Standard Model can be made from CP-violating processes once the CKM angles are determined accurately. A use- ful tool for theoretical extraction of the angle V,, from semileptonic flavor-changing decays is the Isgur-Wise heavy quark expansion (which has many other useful appli- cations, of course). 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