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SU(2)LQ SU(2)RQ: in the techniquark sector, communicated by a nonrenor- × malizable interaction. Note that breaking the full sym- QL LQQL ,QR RQQR . (2) metry SU(2) SU(2) SU(2) SU(2) → → LQ × RQ × LT × RT → SU(2)L SU(2)R SU(2) requires both symmetry It also has a “hidden sector” with techniquarks TL and × → breaking parameters: the condensate T LTR and the TR, transforming under a separate global chiral symme- spurion α/M 2. h i try SU(2) SU(2) , LT × RT Expanding the interaction (6) to linear order in Π, T L LT T L , T R RT T R . (3) m → → mQ Q + i Q ΠQ + ..., (7) L R f L R The two sectors are coupled by a nonrenormalizable con- tact term that arises from extended technicolor at a high we see that the chiral symmetry breaking mass term is scale M, related to the Goldstone boson coupling via m = gπQQf. α This is the famous Goldberger-Treiman relation. It fol- Q T T Q . (4) M 2 L L R R lows solely from the fact that the chiral symmetry is bro- ken spontaneously. Note that the Goldstone-quark cou- The parameter α/M 2 may be viewed as a “spu- pling is suppressed by m/f f 2/M 2. rion,” in the sense that it is a constant that carries It is instructive to consider∼ the power counting in SU(2)4 quantum numbers. The ETC interaction (4) this theory from a top-down, “Wilsonian” point of view. breaks SU(2) SU(2) SU(2) and SU(2) From this perspective, one starts with a theory defined LQ × LT → L RQ × SU(2)RT SU(2)R, leaving the SU(2)L SU(2)R sub- at short distances and then progressively integrates out group intact.→ We put scare quotes around× “spurion” be- modes of longer and longer wavelength to arrive at an cause we do not necessarily view α as representing the effective theory at lower energies. At each energy scale, vacuum expectation value of a field. This spurion (ac- the relevant symmetries are those that are unbroken at knowledging this possible abuse of terminology, we now that scale. The natural sizes of the operator coefficients drop the quotes) carries a power counting dimension of are just the magnitudes that are generated by this proce- 1/M 2; most importantly, it alone controls the SU(2) dure. Delicate cancellations between coefficients present LQ × SU(2)RQ SU(2)LT SU(2)RT SU(2)L SU(2)R at short distances and corrections generated by Wilso- symmetry× breaking in× the low energy→ effective× theory. nian running are considered to be unnatural. At some scale f, the technicolor interaction becomes Above the ETC scale M, we assume that we have a 3 strong and the techniquarks condense, T LTR = f . (In theory in which the techniquark and quark fields inter- this paper we are interested only in theh generali question act with each other freely. (The theory could be non- of power counting the mass scales present in the theory, renormalizable, but any nonrenormalizable operators at so we drop all factors of 4π. Of course, the correct count- that scale give even smaller contributions at low energies ing of such loop factors may be phenomenologically im- than the ones we are interested in.) When we integrate portant in a particular theory.) This condensate breaks out modes at and above the scale M, including the ETC the SU(2)LT SU(2)RT symmetry to its diagonal sub- sector of the theory, we generate a tower of operators sup- group, and produces× three Goldstone bosons in the ef- pressed by powers of 1/M. Below M, these nonrenormal- fective theory below f. The product T LTR is replaced izable operators provide the only interactions between by a composite field containing the Goldstone multiplet the quark and techniquark sectors. Π= πaT a, We now integrate out momenta between M and the lower scale f. All modes have energies below M, so no T T f 2Σ= f 3eiΠ/f . (5) L R → powers of M are induced in the numerators of the op- erator coefficients. Thus the organization of operators This is in accord with the general construction of Callan, according to power counting in 1/M is preserved. Of Coleman, Wess and Zumino [4, 5], which follows sim- course, this is the standard behavior of an effective field ply from the structure of the spontaneously broken sym- † theory. metry. The field Σ transforms as Σ LT ΣRT under The final step is to run the theory from f to m SU(2) SU(2) . In terms of Σ,→ the ETC interac- 3 2 ∼ LT × RT f /M . In this momentum regime, the degrees of free- tion becomes dom of the techniquark sector are those of a theory with 2 spontaneously broken SU(2) SU(2) . They contain αf LT × RT Q ΣQR . (6) the massless nonlinearly self-coupled Goldstone multiplet M 2 L Σ. There may be other light particles in the techniquark The interactions of the quarks with the Goldstones are sector, but whether they are strongly or weakly coupled found by expanding Σ. The leading term gives rise to the below f, they do not change the Goldstone boson power quark mass m = αf 3/M 2, which breaks the quark sector counting in the quark sector. global symmetry SU(2)LQ SU(2)RQ to the diagonal The Goldstone multiplet contains the chiral symmetry SU(2) subgroup. The breakdown× of chiral symmetry in breaking term f. This chiral symmetry breaking is com- the quark sector is the result of chiral symmetry breaking municated to the quark sector through 1/M suppressed 3 operators that break SU(2) SU(2) SU(2) in the effective theory receive contributions from quark LQ × RQ × LT × SU(2)RT SU(2)L SU(2)R. Because of these op- sector modes with momenta in the entire range from m erators, the→ quark sector× “sees” a techniquark sector to M. in which chiral symmetry is realized nonlinearly. The These are the conditions for evaluating loops that are weakly coupled spurion insertions transmit the chiral consistent with the rules for power counting given earlier. symmetry breaking to the quark sector. Such consistency conditions are analogous to those given Running from f down to m, it is impossible to gener- by Georgi and Manohar [6] for the effective theory of ate effects that produce positive powers of the scale M. chiral quarks and pions. The effective lagrangian for the field Σ is not calculable Note that under the chiral SU(2)L SU(2)R, the quark × perturbatively, but at the scale m it remains a nonlin- fields transform linearly and the Goldstone fields nonlin- ear function of Π and ∂. (The back reaction from the early. In this formulation, the decoupling of the Gold- quark sector could induce corrections suppressed by pow- stone bosons as M is manifest, but the Goldstone → ∞ ers of 1/M, but these are comparatively tiny and unim- bosons are not explicitly derivatively coupled. (The ab- portant.) Meanwhile, the quark sector modes that are sence of a momentum-independent contact interaction integrated out are no different than they were between f can be shown using the equations of motion.) However, and M. Running from f to m changes the coefficients of one can perform a nonlinear field redefinition, leaving the the nonrenormalizable operators, but it does not change physical content of the effective theory invariant, that the power counting in M and f. makes the derivative coupling manifest [4, 5]. This is 2 From the Wilsonian point of view, the effective la- done by writing Σ = ξ , and redefining grangian at the scale m receives contributions from a va- Q = ξ†Q , Q = ξQ . (8) riety of short distance modes that have been integrated L L R R out of the theory. These include quark modes with mo- After this transformationb the symmetryb breaking mass menta between m and M, Goldstone modes between m term is simply mQLQR, but the kinetic terms now take and f, and techniquark modes between f and M. Of the form course, we would not actually calculate with a Wilsonian b b effective action containing a strict ultraviolet cutoff. In- m m † QL γ ∂mQL = QL γ ∂m + ξ ∂mξ QL , stead, we would use a momentum-independent scheme m m †
such as dimensional regularization and minimal subtrac- QR γ ∂mQR = QbR γ ∂m + ξ∂mξ Qb R . (9)
tion. For that reason, our effective theory would have For our purposes, this formulationb obscures theb essential loop divergences that must be subtracted. The value of points. The Goldberger-Treiman relation is hidden, and the Wilsonian perspective is the insight it gives into the is only revealed when the equations of motion are applied physical meaning of these divergences. to matrix elements. The decoupling of the Goldstone To determine the correct power counting, it is useful bosons as M is not manifest. It is also difficult to to recall the various types of ultraviolet behavior that identify the appropriate→ ∞ cutoffs for the different types of can appear when the operators in the effective theory, loop diagrams. some of which are nonrenormalizable, are put into loops. Ultraviolet convergent diagrams are dominated by mo- menta at or below the scale of the effective theory, and III. POWER COUNTING IN MODELS WITH A are calculable. Logarithmically divergent diagrams re- HIDDEN SECTOR ceive contributions from momenta at all scales and are interpreted as controlling the scale dependence of the ef- The ETC model described above is directly relevant to fective operators. Finally, power divergent diagrams are our discussion of hidden sector supersymmetric models. dominated by momenta at the cutoff scale; they contain In each case, the symmetry is broken in a “hidden” sector no information about the low energy dynamics. These and then communicated to a “visible” sector via interac- divergences are cancelled by matching corrections at the tions suppressed by powers of 1/M. Recalling that the high scale. The corresponding counterterms necessarily quark and techniquark parts of the ETC model play the obey a consistent power counting in inverse powers of the roles, respectively, of the visible and hidden sectors, we cutoff. note that: What is the power counting in the case of our ETC toy The global symmetry of the renormalizable interac- model? The essential question is the scale at which power tions• is enhanced to separate global symmetries in the divergences should be cut off. The answer depends on the visible and hidden sectors. fields that appear in the loops. Goldstone loops are cut Nonrenormalizable operators, suppressed by the off at f because above this scale, the Goldstones must be scale• M, connect the visible and hidden sectors and break replaced by the techniquarks as the appropriate degrees the enhanced global symmetry. Below the scale M, the of freedom. On the other hand, loops that contain only coefficients of these operators may be viewed as spurion quarks run all the way up to the messenger scale M. The fields with a definite power counting. The decoupling of power divergences determine the power counting of the the two sectors as M is associated with the restora- counterterms, and we have argued that the coefficients tion of the enhanced→ symmetry. ∞ 4

Below the scale f, there is a Goldberger-Treiman breaking in the presence of the spurion. relation• between the symmetry breaking terms in the Hence supersymmetry breaking in the visible sector is visible sector and the coupling of visible sector fields to always accompanied by a spurion that contains explicit the Goldstone bosons. Consistent with the Goldberger- factors of 1/M. Since the spurion terms vanish as M Treiman relation, the Goldstone bosons decouple from for any value of F , supersymmetry breaking effects can→ the visible sector as M . always∞ be decoupled. To preserve this decoupling, any → ∞ The effective theory below f is constructed so that it consistent power counting of supersymmetry breaking at is• consistent and stable under renormalization; the sup- low energies must contain a consistent counting in powers pression of the Π field by f is consistent with cutting off of 1/M. Goldstone loop momenta at the scale f. Conversely an effective theory that is not constructed consistently will not be stable under radiative corrections. The breaking IV. POWER COUNTING IN MODELS WITH of the entire global symmetry must be power counted BROKEN SUPERSYMMETRY correctly term by term. In loops with Goldstone bosons, momenta are cut off We now turn to the construction of the effective la- at• the chiral symmetry breaking scale f. In quark loops, grangian for spontaneously broken supersymmetry. We momenta are cut off at the messenger scale M. assume that supersymmetry is broken in a hidden sec- tor by some chiral superfield that acquires a nonzero F One could integrate the techniquark sector out of the term Fθθ. The precise nature of this field, whether it is theory• entirely, producing a theory of only quark fields. elementary or composite, is irrelevant to the low energy (This theory would be nonlocal because the Goldstones physics of the visible sector. We do know one other thing are massless.) The ultraviolet cutoff for such a theory about it, however: spontaneous breaking of global super- would be M. What is unusual is that the coefficients symmetry implies the existence of a massless Goldstino, of nonrenormalizable operators, suppressed by powers of which by the construction of Wess and Samuel [8] may 1/M, typically depend on hidden sector dynamics at the be assembled into an analog of Σ from the ETC example lower scale f, or on logarithms of f/M.1 The point is above. This field is S = F ΘΘ, where that because the global symmetry of the entire theory is enhanced as M , one can consistently count pow- 1 2i 2 → ∞ Θ = θ + G + θσmG ∂ G + ... (10) ers of 1/M in the visible sector, even below the scale α α F α F 2 m α f at which dynamics in the hidden sector may become strongly coupled. is a chiral superfield that contains the Goldstone fermion These features all have analogs in the case of super- G. symmetry breaking. First, consider the global symme- Given a visible sector chiral superfield Φ, one may write tries of a supersymmetric theory with visible and hid- an interaction term, analogous to (4), that connects the den sectors coupled only by interactions suppressed by visible and hidden sectors, a messenger scale M. As M , the two sectors de- α 4 2 4 couple from each other, and the→ ∞ global supersymmetry d θSS†Φ†Φ= m d θ ΘΘΘΘΦ†Φ . (11) M 2 Z Z is enhanced to SUSY V SUSY H . (To be clear, this not × 2 2 2 is N = 2 supersymmetry, because the generators of Here m = αF /M is a soft supersymmetry breaking SUSY SUSY 2 V and H close into Hamiltonians acting, re- mass. The coefficient α/M of this operator is the spu- spectively, only on the visible and hidden sector fields. rion that breaks SUSY V SUSY H SUSY . It is clear But these two Hamiltonians commute in the decoupling that as M the enhanced× symmetry→ is restored. limit; in this limit, and neglecting gravity, the hidden and Writing Φ =→A ∞+ θχ + ... , and expanding Θ to lin- visible sectors can actually be thought of as functions of ear order in G, we see that this term also gives rise to independent sets of spacetime coordinates!) a Goldberger-Treiman relation for spontaneously broken The terms suppressed by M break the global symme- supersymmetry. Here, the soft mass term m2A∗A is re- 2 ∗ try to ordinary supersymmetry, SUSY V SUSY H lated to the Yukawa coupling (m /F )A χG. × → SUSY ; they should be thought of as containing a spu- The Goldberger-Treiman relation follows from the fact rion field that transforms under SUSY V SUSY H and that supersymmetry is spontaneously broken. There is × acquires a vacuum expectation value that leaves only or- an analogous relation for every operator that communi- dinary supersymmetry unbroken. If supersymmetry is cates supersymmetry breaking to the visible sector [9]. broken in the hidden sector at some scale √F , it is re- Such terms include soft masses, ally SUSY that is broken. It only becomes SUSY H V 2 1 4 † † 2 ∗ m0 ∗ d θSS Φ Φ m0 A A + A χG + ..., (12) M 2 Z ⇒ F

1 An example, in a supersymmetric context, of how such hidden 2 sector logarithms can appear in visible sector operators is pre- 1 4 † 2 2 2 B0 d θSS Φ B0 A + AχG + ..., (13) sented in Ref. [7]. M 2 Z ⇒ F 5

GG GG A* 2 ∂ A 2 |C0| A ~ M4 AA* A* 2 C0 F ~ AA*∂2A 2 F ∂ A* A

FIG. 2: Goldstino-induced operators renormalize the visible A A* ∂2A sector soft scalar masses. The loop momentum runs to the scale M. FIG. 1: New nonrenormalizable operators are generated by integrating out Goldstino loops. The Goldstino loop momenta such operators are missing at the scale M, they will be run to the scale √F . generated by loops between √F and M, with coefficients consistent with the power counting in 1/M. Note that in the effective theory, even in the absence gaugino masses, 2 of a superpotential, one can generate d θ terms from 4 1 2 α m1/2 mn supersymmetry breaking d θ termsR via the Giudice- d θ SW W m1 2 λλ+ λσ GF +..., M Z α ⇒ / F mn Masiero mechanism [11]. InR particular, one can produce (14) both supersymmetric, and scalar self-interactions, 1 F d4θS† Φ2 d2θ Φ2 , (18) 1 2 3 3 A0 2 M Z ⇒ M Z d θS Φ A0 A + A χG + ..., (15) M Z ⇒ F and nonsupersymmetric operators,

1 4 † 2 F 2 2 1 4 2 d θSS Φ d θS Φ . (19) d θSS† (Φ Φ† + h.c.) (16) M 2 Z ⇒ M 2 Z M 3 Z Terms originating in this way are suppressed by an ex- 2 ∗ C0 ∗ 2 2 C0 (A A + h.c.)+ AA χG + .... tra factor of F/M relative to d θ operators that arise F ⇒ directly at the scale M. R Because we are interested only in the power counting, Below √F , the effective theory contains Goldstinos we present these expansions schematically. Each term and visible sector fields. In this effective theory, the rules implicitly includes a SUSY SUSY breaking spurion for cutting off power divergent diagrams are the same as V × H analogous to α. The coefficients m0, m1/2, A0 and B0 in our ETC toy model. Goldstino loops are cut off at are all of order F/M, which we take to be of order the the scale √F , while loops involving visible sector fields 2 3 weak scale mW . The coefficient C0 is of order F /M run all the way up to the messenger scale M. In fact, if 2 ∼ mW /M; that is, it is suppressed compared to the others the Goldstinos are completely integrated out, the effec- 2 by an additional power of F/M mW /M. Note that tive theory of the visible sector fields is consistent to the the term ∼ scale M. As in the ETC model, this is true even though the supersymmetry breaking condensate is generated at 2 2 2 d θS Φ F A + A χG + ... (17) the lower scale √F . Z ⇒ These rules for treating power divergent diagrams en- is not allowed by our power counting since it does not force the consistent power counting of the soft symme- recover the larger SUSY V SUSY H symmetry as M try breaking terms. For example, they imply that the . × → suppression of the nonholomorphic trilinear term (16) ∞The operators (12) – (16) are generically present in any relative to the others is required by the consistency of theory of hidden sector supersymmetry breaking. Their the low energy theory. The Goldberger-Treiman relation coefficients are determined by matching at the scale M. plays the critical role in the argument, because the ex- In a given theory, however, one or more of the operators pansion of the nonholomorphic term produces not only 2 ∗ may be missing or suppressed because of physics at the the trilinear coupling C0A A , but also a term with four scale M. Goldstinos, If a d2θ “superpotential” term does not appear at C0 2 the scaleR M, it will not be generated perturbatively by GGGG AA∗∂ A. (20) F 4 loops between √F and M because of the supersymmet- ric nonrenormalization theorem [10]. Therefore the cor- We now contract the Goldstino lines, as shown in Fig. 1, responding operator will not appear in our effective the- and integrate the loops, each of which has a cubic diver- ory. By contrast, d4θ terms are not protected by the gence, up to √F . This induces a new counterterm, which nonrenormalizationR theorem. For that reason, we expect by consistency must scale as 4 our effective theory to contain all d θ operators consis- C0 AA∗∂2A. (21) tent with the symmetries of the theory.R If one or more F 6

u ∂u* u F χ χ ~ χ χ hu hd hu hd M q ∂2h * χ d q χq F2 FIG. 3: The nonrenormalizable operator (30) gives rise to ~ quhu a supersymmetric µ term of order F/M mW . The loop M3 ∼ 2 momentum runs to the scale M. ∂ q* hd

Finally, we form the product of this term with its Hermitian conjugate and compute the one loop diagram 2 q h shown in Fig. 2, allowing the derivatives ∂ to act on u the fields in the loop. The cutoff of this quartically di- vergent diagram is M; the result is a correction to the FIG. 4: Two nonrenormalizable operators generate a holo- 2 scalar mass, morphic scalar coupling suppressed by F/M . The loop mo- mentum runs to the scale M. 2 4 C0 M | | A∗A. (22) F 2 SUSY V SUSY H SUSY . Note that operator (27) 2 3 If C0 is of order F /M , as required by the power count- produces× the usual holomorphic→ A term when one takes ing of the effective theory, the contribution to the soft the θθ component of the expansion of S, while the oper- mass is of order the weak scale, ator (28) is a higher dimension object that is suppressed 2 and usually not included. 2 F 2 Let us use these terms to illustrate how “missing” op- δmA 2 mW , (23) ∼ M ∼ erators can be generated with the power counting as de- which is not destabilized. scribed above. As before, we keep track only of powers of the scales √F and M, not of loop factors of 4π. For our first example, we show how a weak scale µ term can arise V. IMPLICATIONS FOR THE MINIMAL even if it is missing at the messenger scale M. To do this, SUPERSYMMETRIC STANDARD MODEL we consider the Hermitian conjugate of the operator (28) and extract the following term with four Goldstinos: Let us now see what these considerations imply for the 1 m ∗ minimal supersymmetric standard model (MSSM). The GGGGχhd σ χ¯q∂mu¯ . (29) F 4M 3 MSSM has the superpotential Contracting the Goldstino lines and integrating the Gold- P = µHuHd + λu QUHu + λd QDHd . (24) stino momenta to √F , we infer the existence of the coun- terterm The soft supersymmetry breaking terms are generated at the matching scale M, and below √F take the forms (12) F χ σmχ¯ ∂ u¯∗ . (30) – (16) and their variants at higher order. In particular, M 3 hd q m these include the Higgs terms given by the operators We now form the product of this operator with the ordi- 1 4 † nary Yukawa interaction, as shown in Fig. 3. Integrating d θSS† (H†H + H H ) , (25) M 2 Z u u d d the quark and squark momenta to M, we induce a su- persymmetric Higgsino mass term of order F/M mW . More specifically, the absence of a Higgsino mass∼ of this 1 4 † 2 d θSS HuHd , (26) size would require that this loop effect be canceled by a M Z fine tuning at the messenger scale M. As a second example, we show that d4θ operators 2 1 2 induce a trilinear A term that is suppressedR by F/M d θS (QUH + QDH ) , (27) 2 M Z u d with respect to the d θ power counting. For this we again use the higherR order term (28), which gives rise to the scalar operator 1 4 † † † 3 d θSS (QUH + QDHu) . (28) Z d F 2 M qu¯ ∂ h∗ , (31) M 3 d In the low energy effective lagrangian, each of these terms appears proportional to a spurion that breaks exactly as in the discussion leading up to Eq. (21). In an 7 analogous fashion, we also generate the operator Because the MSSM contains no gauge singlets, the au- thors of Refs. [2] and [3] propose to take C0 to be of F 2 ∗ order F/M mW . However, we see from Eq. (34) that 4 q∂ q huhd (32) ∼ M if C0 mW , the Higgs mass correction from Goldstino loops∼ is of order from a higher order term in 1/M,

1 2 2 d4θSS†QQ†H H . (33) δm M . (35) M 4 Z U D hu,hd ∼ Connecting them as shown in Fig. 4, and integrating In other words, Goldstino loops destabilize the weak scale the squark momenta to M, we generate the holomorphic 2 3 if one does not use the correct power counting for the scalar trilinear quh¯ u, with coefficient of order F /M 2 symmetry breaking terms. Consistency requires that C0 m /M. Again, the precise statement is that a fine tun-∼ W must be taken to be of order F 2/M 3, implying that non- ing at the messenger scale would be required to avoid holomorphic trilinear terms must have coefficients of or- such a term in the low energy lagrangian. Of course, 2 der mW /M, rather than mW . this term is suppressed by mW /M compared to the weak scale; the result is interesting, even in principle, only in We close by emphasizing the very general nature of the absence of the lower order d2θ operator (27) (which this result. Just as the interactions with the Goldstone does not requiring a fine tuning).R bosons follow from the form of chiral symmetry breaking Finally, we emphasize that the correct power counting in extended technicolor, the interactions with Goldstinos in 1/M is essential for the consistency of the low energy follow immediately, via the Goldberger-Treiman relation, theory. In particular, if one operator is arbitrarily as- from the form of the soft terms in the MSSM. The two signed an anomalously large coefficient, Goldstino loop are inextricably linked because they share the physics of effects will drive up the other coefficients to restore the symmetry breaking and its communication through the proper counting. messenger sector. The special power of effective field the- As a phenomenologically important example, consider ory, when combined with nonlinear realizations, is that the operator (28), which gives rise to nonholomorphic this connection is revealed in a way that is simultaneously ∗ trilinear terms such as quh¯ d. Hall and Randall [2] and transparent, elegant and useful for phenomenology. Weinberg [3] pointed out that such terms, of the form 2 ∗ mW A A , are “soft” in the sense that they do not desta- bilize the hiercharchy in a theory (such as the MSSM) that has no gauge singlets. This observation is correct, but incomplete, since the Goldberger-Treiman relation Acknowledgments introduces Goldstino couplings that do destabilize the hierarchy in the presence of such a term. To see that this is so, suppose that the operator (28) This paper is dedicated to the memory of Julius Wess, is included with an arbitrary coefficient C0. By an ar- from whom, both personally and through his published gument identical to the one that leads to Eq. (22), there work, we learned so much. We are grateful to the Aspen must then be a counterterm of the form Center for Physics, where portions of this work were com- 2 4 pleted. This work was supported in part by the National C0 M | | (h∗ h + h∗h ) . (34) Science Foundation under grant NSF-PHY-0401513. F 2 u u d d

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