One-loop potential with scale invariance and effective operators
aa r X i v : . [ h e p - ph ] M a y One-loop potential with scale invariance andeffective operators
D. M. Ghilencea ∗ CERN Theory Division, CH-1211 Geneva 23, Switzerland and Theoretical Physics Department,National Institute of Physics and Nuclear Engineering (IFIN-HH) Bucharest, 077125 RomaniaE-mail: [email protected]
We study quantum corrections to the scalar potential in classically scale invariant theories, usinga manifestly scale invariant regularization. To this purpose, the subtraction scale m of the dimen-sional regularization is generated by spontaneous scale symmetry breaking, from a subtraction function of the fields, m ( f , s ) . This function is then uniquely determined from general princi-ples showing that it depends on the dilaton only, with m ( s ) ∼ s . The result is a scale invariantone-loop potential U for a higgs field f and dilaton s that contains an additional finite quantumcorrection D U ( f , s ) , beyond the Coleman Weinberg term. D U contains new, non-polynomialeffective operators like f / s whose quantum origin is explained. A flat direction is maintainedat the quantum level, the model has vanishing vacuum energy and the one-loop correction to themass of f remains small without tuning (of its self-coupling, etc) beyond the initial, classicaltuning (of the dilaton coupling) that enforces a hierarchy h s i ≫ h f i . The approach is useful tomodels that investigate scale symmetry at the quantum level. Proceedings of theCorfu Summer Institute 2015 "School and Workshops on Elementary Particle Physics and Gravity"1-27 September 2015, Corfu, Greece ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ ne-loop scale invariant potential and effective operators
D. M. Ghilencea
1. General considerations
Scale invariant theories [1] were often considered as an alternative to supersymmetry to ad-dress the hierarchy problem. Since such theories forbid the presence of dimensionful parameters inthe Lagrangian, scale symmetry be it classical or quantum, must be broken in the real world. Thisbreaking can be explicit or spontaneous. We investigate the latter case, since this preserves the UVbehaviour of the initial scale invariant theory. In the spontaneous breaking we have the dilaton s asthe Goldstone mode of this symmetry, with non-vanishing vacuum expectation value (vev) h s i 6 = h s i may be related to the Planck scale. We do not detailhow s acquires a vev, but simply search for solutions with h s i 6 =
0. All scales of the theory arethen related to h s i . A hierarchy of scales which are vev’s of the different fields present, can thenbe generated either by one initial (classical) tuning of the couplings to small values [2] or as in [3].The purpose of this talk based on [4] is to consider a classically scale invariant theory and toshow how to compute the quantum corrections to the scalar potential in a manifestly scale invariantway. Such approach is important since it preserves at the quantum level the initial symmetry of thetheory and its UV properties, relevant for the Higgs physics. Investigating scale invariant theoriesat quantum level is non-trivial because the regularization of their quantum corrections breaks thescale symmetry explicitly . Indeed, in its traditional form, the regularization, be it dimensionalregularization (DR) or some other scheme, introduces a subtraction scale m . Its presence ruinsexactly the symmetry that we want to study at the quantum level. To avoid this situation one shouldgenerate the subtraction scale in a dynamical way. Consider then replacing m of the DR scheme bya field-dependent subtraction function m ( s ) [17], see also more recent [18]. Having couplings ormasses that are field-dependent is something familiar in string theory. After spontaneous breakingof scale symmetry when h s i 6 =
0, the subtraction scale is generated as m ( h s i ) . Doing so hasimplications at the quantum level, presented below.Preserving the scale symmetry of the action during the UV regularization of the quantumcorrection is actually required in theories which are non-renormalizable, to avoid regularizationartefacts. Since some of these theories may be be non-renormalizable [19, 20], it is then worthexploring the consequence of a such regularization. This is also important for the naturalness prob-lem, as argued long ago by Bardeen [21]. The Standard Model (SM) with the higgs mass m h = . This gives a phenomenological motivation to studya scale invariant regularization of quantum corrections, with spontaneous breaking of scale sym-metry. This talk (based on [4]) continues previous similar studies [18, 19, 22], with some notable As a result of such explicit breaking the initial flat direction of the classically scale invariant theory is lifted; thereis extensive model building in this direction, see for example more recent [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. In DR the scale m relates the dimensionless (renormalized) coupling l ( r ) to the dimensionful one l once the d = d = − e ; for a quartic higgs coupling: l = m e (cid:0) l ( r ) + (cid:229) n a n / e n (cid:1) . m h remains quadratic in the scale (of “new physics”) generated by spontaneous scale symmetry breaking. ne-loop scale invariant potential and effective operators D. M. Ghilencea differences and with new results shown below.We consider a classical, scale invariant theory of a Higgs-like scalar f and the dilaton s , with L = ¶ m f¶ m f + ¶ m s¶ m s − V ( f , s ) (1.1)where V = l f f + l m f s + l s s (1.2)There exists a non-trivial solution with spontaneous breaking of scale symmetry h s i 6 =
0, (also h f i 6 =
0) provided that two conditions are met: firstly, l m = l f l s with l m < h f i h s i = − l m l f , ⇒ V = l f (cid:16) f + l m l f s (cid:17) (1.3)Then spontaneous scale symmetry breaking implies electroweak symmetry breaking at tree-level,with V min = s while the mass of f is m f = − l m ( − l m / l f ) h s i . Although we are not interestedin the exact value of h s i , since scale invariance is expected to be broken by Planck physics it islikely that h s i ∼ M Planck . To ensure a hierarchy of scales h f i ≪ h s i and that the mass of thehiggs-like f is near the electroweak scale, one can tune once the couplings (at the classical level) toenforce a relation l s ≪ | l m | ≪ l f [2, 3]. Such hierarchy of couplings is stable under the quantumcorrections of the renormalization group flow [2]. One would like to explore these issues at thequantum level in a scale invariant approach.More generally, in theories with scale symmetry, the potential for f and s has the form V = s W ( f / s ) . Assuming spontaneous breaking of this symmetry h s i 6 =
0, the two minimumconditions for V become W ′ ( x ) = W ( x ) = , x ≡ h f ih s i ; h s i , h f i 6 = . (1.4)One minimum condition fixes the ratio h f i / h s i while the second gives a relation among the cou-plings of the theory such as W ( x ) =
0. If these two equations for W have a solution x , then h f i (cid:181) h s i . A flat direction exists in the plane ( f , s ) with f / s = x along which the vacuumenergy vanishes V ( h f i , h s i ) =
0. These results can remain valid at the quantum level as well, pro-vided that the calculation of the quantum corrections is manifestly scale invariant, since then thepotential remains of the form V ∼ s ˜ W ( f / s ) where ˜ W is the quantum corrected W . Also, the flatdirection corresponding to the Goldstone mode (dilaton) remains flat at the quantum level (to allorders) if the calculation preserves the scale symmetry which is broken only spontaneously.3 ne-loop scale invariant potential and effective operators D. M. Ghilencea
2. Quantum scale invariance of the potential and effective operators
Let us then calculate the one-loop potential in a scale invariant regularization. To enforce di-mensionless couplings in the usual DR scheme, one replaces the quartic couplings l → lm − d ,with m the subtraction scale. This breaks classical scale symmetry. To avoid this, replace m byan unknown function of the fields, m ( f , s ) , whose vacuum expectation value generates dynami-cally the subtraction scale m ( h f i , h s i ) . This function is determined later, but is assumed to becontinuous and differentiable at all fields values. Then, in d = − e dimensions, the potential is˜ V ( f , s ) = m ( f , s ) − d V ( f , s ) (2.1)To be general, we also allowed a f -dependence of the subtraction function. There are now “evanes-cent” interactions between say s in m ( f , s ) and V ( f , s ) , that are absent in the limit d =
4. Theyare due to the scale symmetry in d = − e . The one-loop potential is then computed as usual, butwith ˜ V instead of original V : U = ˜ V ( f , s ) − i Z d d p ( p ) d Tr ln (cid:2) p − ˜ V ab + i e (cid:3) (2.2)Here ˜ V ab = ¶ ˜ V / ¶a¶b , with a , b = f , s . Up to O ( e ) terms˜ V ab = m e h V ab + em − N ab i , (2.3) N ab ≡ m ( m a V b + m b V a ) + ( mm ab − m a m b ) V , (2.4)where m a = ¶m / ¶a , m ab = ¶ m / ¶a¶b , V a = ¶ V / ¶a , and V ab = ¶ V / ¶a¶b , are field depen-dent quantities. Denote by M s the eigenvalues of the matrix V ab and by k ≡ p e / − g E , then U = m ( f , s ) e n V − p h (cid:229) s = f , s M s (cid:16) e − ln M s ( f , s ) km ( f , s ) (cid:17) + ( V ab N ba ) m ( f , s ) io (2.5)In the last term a summation over repeated indices (fields) is understood. The counterterms are d U c . t . ≡ m ( f , s ) e n d Z l f l f f + d Z l m l m f s + d Z l s l s s (cid:3) (2.6)from which one easily finds the coefficients d Z l j ≡ Z l j −
1; they have values identical to those ifthe theory were regularized with m =constant (the same is true about the beta functions of l j ). For after spontaneous scale symmetry breaking. The eigenvalues M s are the roots ( q ) of q − q ( V ff + V ss ) + ( V ff V ss − V fs ) = ne-loop scale invariant potential and effective operators D. M. Ghilencea example d Z l f = / ( p e )( l f + l m / l f ) , etc. The renormalized potential is then U ( f , s ) = V ( f , s ) + p n (cid:229) s = f , s M s ( f , s ) (cid:16) ln M s ( f , s ) m ( f , s ) − (cid:17) + D U ( f , s ) o D U = − m n V (cid:2) ( mm ff − m f ) V ff + ( mm fs − m f m s ) V fs + ( mm ss − m s ) V ss (cid:3) + m ( m f V ff + m s V fs ) V f + m ( m f V fs + m s V ss ) V s o (2.7)We recovered the usual Coleman-Weinberg term [23, 24] in a modified, scale invariant form. Wealso found a new, additional contribution D U , not considered in [18], which is a finite one-loopcorrection to the potential. D U emerges from the correction (cid:181) e to the field dependent masses(eq.(2.3)) of the states that “run” in the loop; when this correction (cid:181) e multiplies the pole 1 / e of the momentum integral, one obtains a finite term (last term in (2.5)) that gives D U . Also D U vanishes on the tree level ground state when V and its first derivatives vanish.The problem with the result in eq.(2.7) is that it depends on the unknown function m ( f , s ) ,which generates the subtraction scale after spontaneous scale symmetry breaking. Obviously, phys-ical observables cannot depend on the regularization done with this function. We should thendetermine this function from some general principles; then, the potential must respect the Callan-Symanzik equation, to enable us to make physical predictions.To this purpose, consider first a particular example of m ( f , s ) used in previous models [18, 22] m ( f , s ) = z (cid:0) x f f + x s s (cid:1) / (2.8)With this, one computes the expression of D U which in the particular limit l m → D U (cid:12)(cid:12)(cid:12) l m = = − h x f x s (cid:2) l f ( l f + l s ) f s + l s ( l f + l s ) f s (cid:3) + (cid:0) l f x f f + l s x s s (cid:1) − ( x f + x s ) l f l s f s i ( x f f + x s s ) − (2.9)This simplifies further if l m = l f l s which ensures spontaneous breaking of scale symmetry; how-ever the term (cid:181) x f x s l f f s remains even in this case, unless x f x s = m depends onone field only. Now, when l m →
0, the “visible” sector of higgs-like f is classically decoupledfrom the “hidden” sector of the dilaton s . Nevertheless, we see that in this limit the two sectorsstill interact at the quantum level, which is unacceptable. This situation is more general and ap-plies when the subtraction function depends on both fields. The reason for this is related to how D U is generated, from “evanescent” interactions introduced by scale invariance of the action in d = − e , see ˜ V ( f , s ) = m e V ( f , s ) . Since m ( f , s ) contains both fields, it brings interactionswith any term in V ( f , s ) , not only with that proportional to l m . This explains the presence in (2.9)of non-decoupling interactions terms proportional to l f .5 ne-loop scale invariant potential and effective operators D. M. Ghilencea
Similar considerations apply for a general subtraction function, which, up to a relabeling of thefields, can be written as m ( f , s ) = z s exp ( g ( f / s )) , with g an arbitrary continuous, differentiablefunction. Then one can show that D U vanishes in the decoupling limit ( l m =
0) only if g is aconstant . We conclude that the subtraction function must depend on the dilaton only, with m ( s ) = z s . Here z is some arbitrary dimensionless parameter, whose role will be clarified shortly .With m ( s ) = z s uniquely identified and with V of eq.(1.2) one obtains D U = l f l m f s − (cid:0) l f l m + l m − l f l s (cid:1) f − (cid:0) l m + l s (cid:1) l m f s − l s s (2.10)This simplifies further for our case with l m = l f l s of spontaneous symmetry breaking that gen-erates a subtraction scale z h s i . In the decoupling limit ( l m →
0) there are no quantum interactionsleft between f and s , since D U →
0. With D U of eq.(2.10) the one-loop potential becomes U ( f , s ) = V ( f , s ) + p h (cid:229) s = f , s M s ( f , s ) (cid:16) ln M s ( f , s ) z s − (cid:17) + D U ( f , s ) i (2.11)This quantum expression is scale invariant. It has a form and properties similar to those discussedin section 1 (text around eq.(1.4)).
3. More about quantum corrections
One notes the presence in D U of eq.(2.10) of higher dimensional non-polynomial operatorssuch as f / s , in addition to other finite quantum results induced by manifest scale invariance.It is expected that more such operators be generated at higher loop orders. Needless to say, thecorrection D U is missed in calculations that are not scale invariant such as the usual DR scheme,since the result depends on derivatives of m wrt s which vanish if m =constant. Finally, after aTaylor expansion, the above operator can be re-written as a series of standard effective polynomialoperators in fluctuations ˜ s , where s = h s i + ˜ sf s = f h s i (cid:16) − s h s i + s h s i + · · · (cid:17) (3.1)The logarithm ln ( z s ) of the Coleman-Weinberg (CW) term can also be expanded about h s i to recover the usual CW term obtained for m =constant ( = z h s i ), plus additional corrections. Inconclusion, U contains new quantum corrections that can be re-written as series of polynomialterms in ˜ s , suppressed by h s i . We discard an extra solution for g and thus m ( f , s ) which is not continuous in f = z ), where the latter is taken care of by the Callan-Symanzik equation, (see later). To be exact, one actually has m ( s ) = z s / ( d − ) , since the fields have mass dimension ( d − ) / m has massdimension 1. In our one-loop approximation and at this stage it is safe to take the limit d → m ( s ) = z s . ne-loop scale invariant potential and effective operators D. M. Ghilencea
At higher loops, the new operator f / s can render the theory non-renormalizable. If theinitial theory were regularized with m =constant, the scale symmetry is broken explicitly, there is noGoldstone mode, but such operator is never generated dynamically and the theory is renormalizableto all orders. Nevertheless such operators should be added “by hand” to the classical Lagrangian,since they respect its symmetries. The presence here of this non-polynomial operator is not toosurprising; since it is not forbidden by the scale symmetry (preserved by the quantum calculation)such operator is expected to emerge at some loop-order. Its origin is due to loop corrections with“evanescent” interactions dictated by scale invariance in d = − e .The potential U still depends on the dimensionless subtraction parameter z , which apparentlyprevents one from making predictions for physical observables. However, its presence is under-stood by analogy to the subtraction scale dependence (in a given order) in the “ordinary” regu-larization with m =constant. The Callan-Symanzik equation should be respected by the potentialand this will ensure that the physical observables do not depend on z (or on the subtraction scale m ( h s i ) = z h s i ). In our case the Callan-Symanzik equation for U of eq.(2.11) is dU ( l j , z ) d ln z = (cid:16) b l j ¶¶l j + ¶¶ z (cid:17) U ( l j , z ) = , sum over j = f , m , s ; b l j ≡ d a j d ln z (3.2)This condition is easily verified, since the only explicit dependence on z is via the CW term andthe (non-vanishing [25, 26]) beta functions b l j are found from the condition ( d / d ln z )( l j Z l j ) = j = f , m , s , fixed). In conclusion the one-loop potential is independent of z and respects theCallan-Symanzik equation for theories with this symmetry [25].Since U is scale invariant at the one-loop level, the necessary minimum conditions of vanishingfirst derivatives U f = U s =
0, with h s i 6 =
0, ensure that the ground state has vanishing vacuumenergy U min = f receives quantum corrections. Its mass is then m f = ( U ff + U ss ) min . One can compute m f in some approximation, such as l s ≪ | l m | ≪ l f ,when minimising the potential is easier. It is possible to show that the quantum correction to themass of f due to the Coleman-Weinberg part of the potential does not require additional tuning ofthe couplings in order to keep it light [18], well below the scale h s i , where h s i ≫ h f i (see alsotext after eq.(1.3)). This means that there are no quantum corrections to its mass of the type l f h s i or similar, that would require tuning the higgs quartic self-coupling l f .Further, the contribution to the (mass) of f due to the new correction D U that we found (notconsidered in [18]) is also under control. This correction is d m f = p ( D U ff + D U ss ) min It can be shown [4] that d m f contains only terms proportional to l m h s i or l s h s i (here l s = These are b l f = / ( p )( l f + l m ) , b l m = / ( p )( l f + l m + l s ) l m and b l s = / ( p )( l m + l s ) . Such tuning of l f would be the sign of re-introducing the hierarchy problem in the context discussed here. ne-loop scale invariant potential and effective operators D. M. Ghilencea l m / l f ≪ | l m | ). Therefore no tuning of the higgs self-coupling l f is required to keep d m f muchsmaller than the UV scale h s i .
4. Final remarks and conclusions
Scale invariant theories can provide an alternative to supersymmetry to address the mass hi-erarchy problem. The Standard Model classical Lagrangian has a scale symmetry in the limit ofvanishing tree-level higgs mass. As emphasized long ago by Bardeen, the usual regularization ofquantum corrections breaks this symmetry explicitly by the presence of the subtraction scale (viadimensional regularization, Pauli-Villars, etc) and introduces regularization artefacts. Obviously,in the real world scale symmetry is broken, but to preserve its ultraviolet properties, while generat-ing all the mass scales of the theory (including the subtraction scale), it is recommended that thissymmetry be broken only spontaneously (softly). This means the spectrum of the theory will con-tain an additional, massless (Goldstone) mode of this symmetry (dilaton s ), whose vev generatesthe subtraction scale. All other scales, vacuum expectations of scalar fields, are then related to h s i .In this talk we presented the consequences of such an approach, with a regularization thatpreserves scale symmetry, to compute the one-loop corrections to the scalar potential of a classi-cally scale invariant theory of a higgs-like f and dilaton s . One consequence is that the one-loopscalar potential contains additional finite quantum corrections ( D U ), beyond the familiar Coleman-Weinberg term, itself modified into a scale-invariant form (with m ( s ) = z s , where z is an arbitrarydimensionless parameter). The origin of D U is due to evanescent corrections (i.e. proportional to e ) to the field dependent masses that “run” in the one-loop diagram when these multiply the poleof its momentum integral, to give a finite D U . Also D U contains new, non-polynomial effectiveoperators of the type f / s . These can be Taylor expanded into a series of polynomial operators,suppressed by h s i ≫ h f i ; note that no dangerous operators of the opposite type s / f can begenerated. The quantum correction to the mass of f ( m f ) due to D U remains thus under control,with no tuning needed of the higgs self-coupling to keep m f much smaller than the dilaton scale h s i . It would be interesting to check if this behaviour survives in higher loop orders.It was also shown that the subtraction function cannot also depend on the higgs-like scalar f . This is because in the classical decoupling limit l m → f ) from thehidden sector (of s ), there exists a non-decoupling quantum interaction between these sectors. Asa result the subtraction function depends on s only m ( s ) = z s , as considered above, and is unique .Since physical observables cannot depend on arbitrary parameters such as z (or the subtractionscale m ( h s i ) = z h s i ), we checked that the Callan-Symanzik equation is respected by the potential.The above results can now be applied to the scale invariant version of the (classical) StandardModel Lagrangian to explore their phenomenological implications. The presence of the higherdimensional operators of the type found above can have implications for the stability of the SMground state at the high scale. 8 ne-loop scale invariant potential and effective operators D. M. Ghilencea
Acknowledgements:
This work was supported by a grant of the Romanian National Authorityfor Scientific Research CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0607.
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