GDR Terascale Avoiding the Goldstone Boson Catastrophe in general renormalisable field theories at two loops

Transcription

1 GDR Terascale Avoiding the Goldstone Boson Catastrophe in general renormalisable field theories at two loops Johannes Braathen in collaboration with Dr. Mark Goodsell arxiv: , to appear in JHEP Laboratoire de Physique Théorique et Hautes Énergies November 4, 016

2 The context Going Beyond the Standard Model 01: discovery of a SM-Higgs-like particle by ATLAS and CMS No Physics beyond the SM found yet properties of the Higgs as a probe for new Physics Higgs mass mh A tool to compute the Higgs mass effective potential V eff State of the art SM: V eff (relates m h λ) is known to full -loop (Ford, Jack and Jones 9) + leading QCD 3-loop and 4-loop (Martin 13, Martin 15) Some results for m h in specific SUSY theories: MSSM (leading SQCD 3-loop order); NMSSM (-loop); Dirac Gaugino models (leading SQCD -loop: J.B., Goodsell, Slavich 16) Generic theories: V eff computed to -loop (Martin 01), tadpoles and scalar masses (in gaugeless limit) implemented in SARAH (Goodsell, Nickel, Staub 15)

3 The effective potential V eff = V (0) + quantum corrections Quantum corrections = 1PI vacuum graphs computed loop by loop 1-loop ; -loop + ; etc. Expressed as a function of running tree-level masses of particles, in some minimal substraction scheme (MS, DR, etc.) First derivative of V eff : tadpole equation ( minimum condition), relates vev and mass-squared parameters Second derivative: same as self-energy diagrams, but with zero external momentum approximate scalar masses

4 The Goldstone Boson Catastrophe Beyond one loop, V eff only computed in Landau gauge Goldstones are treated as actual massless bosons i.e. (m G )OS = 0 By choice (simplicity) V eff is computed with running masses: (m G) run. = (m G) OS Π G ((m G) OS ) = Π G (0), where Π G is the Goldstone self-energy Under RG flow, (m G )run. may become 0 infrared divergence in V eff change sign imaginary part in V eff Goldstone boson catastrophe

5 Illustration: the abelian Goldstone model 1 complex scalar φ = 1 (v + h + ig), no gauge group and only a potential V (0) = µ φ + λ φ 4 v: true vev, to all orders in perturbation theory (PT) SM: G +, G 0 Goldstones do not mix, and can be treated separetely this model captures the behaviour of the GBC in the SM V eff at -loop order: [ ] V eff =V (0) + 1 f (m 16π h ) + f (m G ) [ + 1 λ (16 ) }{{} 1-loop no Goldstone ( 3 4 A(m G ) + 1 ) {}}{ A(m G )A(m h ) λ v I(m h, m G, m G ) + }{{} -loop ] +O(3-loop) where f (x) = x 4 (log x/q 3/), A(x) = x(log x/q 1) and I Tree-level masses: m h = µ + 3λv, m G = µ + λv

6 Illustration: the abelian Goldstone model Tree-level tadpole V (0) h Loop-corrected tadpole = 0 = µ v + λv 3 = mgv h=0,g=0 V eff h = 0 = mgv + λv [ ] h=0,g=0 16π 3A(mh) + A(mG) }{{} 1-loop + log m G Q [3λ (16 ) v A(m G) + 4λ3 v 3 ] regular for m G mh A(mh) {}}{ O(3-loop) }{{} -loop

7 Illustration: the abelian Goldstone model Tree-level tadpole equation V (0) h Loop-corrected tadpole equation + = 0 = µ v + λv 3 = mgv h=0,g=0 V eff h = 0 = mgv + λv [ ] h=0,g=0 16π 3A(mh) + A(mG) }{{} 1-loop GBC! {}}{ log m G Q (16 ) [3λ v A(m G) + 4λ3 v 3 m h ] regular for m G A(mh) {}}{ O(3-loop) } {{ } -loop

8 First approaches to the GBC By hand if m G < 0, drop the imaginary part of V eff tune the renormalisation scale Q to ensure m G > 0 (and even m G not too small) may be impossible to achieve and is completely ad hoc In automated codes (SARAH) For SUSY theories only Rely on the gauge-coupling dependent part of V (0) minimize full V eff = V (0) π V (1) + 1 (16π ) V () gaugeless compute tree-level masses with V (0) gaugeless (= turn off the D-term potential) yields a fake Goldstone mass of order O(m EW ) no GBC

9 Resummation of the Goldstone contribution SM: Martin ; Ellias-Miro, Espinosa, Konstandin MSSM: Kumar, Martin [Adapted from arxiv: ] ˆV eff = V eff π [ Power counting most divergent contribution to V eff at l-loop = ring of l 1 Goldstone propagators and l 1 insertions of 1PI subdiagrams Π g involving only heavy particles Π g obtained from Π G, Goldstone self-energy, by removing "soft" Goldstone terms Resumming Goldstone rings shifting the Goldstone tree-level mass by Π g in the 1-loop Goldstone term l 1 f (mg (Π g ) n ( ) n ] d + Π g ) f (m n! G) n=0 dm G l-loop resummed V eff, free of leading Goldstone boson catastrophe

10 Extending the resummation to generic theories arxiv: Generic theories: J.B., Goodsell arxiv: Real scalar fields ϕ 0 i = v i + φ 0 i, where v i are the vevs to all order in PT V (0) ({ϕ 0 i }) = V (0) (v i ) + 1 m 0,ij φ0 i φ0 j + 1 ijk ˆλ 0 6 φ0 i φ0 j φ0 k + 1 ijkl ˆλ 0 4 φ0 i φ0 j φ0 k φ0 l m 0,ij solution of the tree-level tadpole equation To work in minimum of loop-corrected V eff new couplings m ij Diagonalise to work with mass eigenstates in both bases (φ 0 i, m 0,ij ) φ0 i = R ij φj (φ 0 i, m ij ) φ0 i =R ij φ j ( φ i, m i ) (no loop corrections) (φ i, m i ) (with loop corrections) Single out the Goldstone boson(s), index G, G,... and its/their mass(es) m G = i 1 ( R ig ) (V eff V (0) ) v i = O(1-loop) φ 0 i =0 φ 0 i

11 Our solution: setting the Goldstone boson on-shell arxiv: Issues with the resummation taking derivatives of ˆV eff can be very difficult (involves derivatives of the rotation matrices, etc.) in practice resummation was only used to find the tadpole equations. the choice of "soft" Goldstone terms to remove from Π G to find Π g may be ambiguous and it is difficult to justify which terms to keep Setting the Goldstone boson on-shell Adopt an on-shell scheme for the Goldstone(s): replace (m G )run. by (m G )OS (= 0) and Π G (0) (m G) run. = (m G) OS Π G ((m G) OS ) = Π G (0) This can be done directly in the tadpole equations or mass diagrams!

12 Canceling the IR divergences in the tadpole equations arxiv: loop tadpole diagrams involving scalars only: The GBC also appears in diagrams with scalars and fermions or gauge bosons, and is cured with the same procedure we present the purely scalar case.

13 Canceling the IR divergences in the tadpole equations arxiv: loop tadpole diagrams involving scalars only: Some diagrams of T SS and T SSSS topologies diverge for m G 0

14 Canceling the IR divergences in the tadpole equations arxiv: What happens when setting the Goldstone on-shell? Contribution of the Goldstone(s) to the 1-loop tadpole: T S G A(m G ) = m G( log m G Q 1 ) At 1-loop order the scalar-only diagrams in Π G (0) are (mg )run. = (mg) OS }{{} =0 p = 0 G G p = 0 G G+ Shifting m G by a 1-loop quantity, Π G(0), in the 1-loop tadpole -loop shift! A((mG) run. ) = A(0) }{{} =0 log m G Q }{{} 1-loop Π G (0) }{{} 1-loop

15 Canceling the IR divergences in the tadpole equations arxiv: loop divergent tadpole diagrams shifting the Goldstone term in the 1-loop tadpole T S the divergent parts from the diagrams and the shift will cancel out!

16 Canceling the IR divergences in the mass diagrams arxiv: Earlier literature: inclusion of momentum cures all the IR divergences We found true at 1-loop order at -loop, diagrams that still diverge for mg external momentum included 0 even with

17 Canceling the IR divergences in the mass diagrams arxiv: Earlier literature: inclusion of momentum cures all the IR divergences We found true at 1-loop order at -loop, diagrams that still diverge for mg external momentum included 0 even with

18 Canceling the IR divergences in the mass diagrams arxiv: Earlier literature: inclusion of momentum cures all the IR divergences We found true at 1-loop order at -loop, diagrams that still diverge for mg external momentum included 0 even with

19 Canceling the IR divergences in the mass diagrams arxiv: Earlier literature: inclusion of momentum cures all the IR divergences We found true at 1-loop order at -loop, diagrams that still diverge for mg external momentum included 0 even with

20 Canceling the IR divergences in the mass diagrams arxiv:

21 cancels the divergence in the V, X, Y, W diagrams! Canceling the IR divergences in the mass diagrams arxiv: Setting the Goldstone(s) on-shell in mass diagrams Goldstone contributions to the 1-loop scalar self-energy Π (1) ij (s = -p ) = s i G }{{} j + s i G }{{} k j + cure W and X diagrams cure V and Y diagrams Again, shifting the Goldstone mass to on-shell scheme gives (m G )run. = p = 0 G -loop shift to the mass diagrams G p = 0 G G+ δπ (1) ij (s) = s i Π G (0) G j s i G Π G (0) k j

22 Our results Results for generic theories (scalars, fermions, gauge bosons), avoiding the Goldstone boson catastrophe full two-loop tadpole equations two-loop mass diagrams for neutral scalars in gaugeless limit, in a generalised effective potential approach (i.e. neglect terms of order O(s) and higher) Numerical implementation (soon): SARAH and/or stand-alone code no more numerical instability associated with the GBC in particular useful for automated study of non-susy theories (for which there was previously no way of evading the GBC)

23 Outlook Further work on the GBC investigate further the link between resummation and on-shell method extend the solution of GBC to higher loop order on-shell method still working? how to formalise/prove the resummation prescription? (i.e. how to find Π g ) extend mass-diagram calculations to quartic order in the gauge couplings (go beyond the gaugeless limit) Apply similar techniques to address other IR divergences

24 Thank you for your attention!

25 Backup

26 More details on the resummation of Goldstone contributions R l so l ( Πg d d k i(π) d k mg ( ) l 1 d (Π g) l 1 (l 1)! dm G = 1 (Π g ) l 1 16π (l 1)! ( d R l = 1 16π f (m G + Π g ) dm G ) l 1 d d k i(π) d log(k mg) ) l 1 f (mg) where f (x) = x 4 (log x 3 )

27 More details about the calculations for the scalar-only tadpole Divergent terms From T SS : From T SSSS : V () S φ 0 r V () S φ 0 r ϕ=v ϕ=v 1 4 Rrp λ GGll λ GGp log m G A(m l ) l G 1 4 RrpλpGG λ Gkl λ Gkl log m G P SS(m k, m l ) Setting the Goldstone mass on-shell Π (1),S GG ( p ) = 1 λggjj A(m j ) 1 (λgjk ) B(p, m j, m k ) Hence a -loop shift: V () S ((m φ 0 G )OS ) = r V () S φ 0 r 1 m G (m G )OS 4 RrpλGGp log(m G )OS ( λ GGjj A(m j ) (λgjk ) B(0, m j, m k ) ).

28 The full -loop tadpole equation free of GBC ˆV () φ 0 r =R rp ϕ=v [ T p SS + T p SSS + T p SSSS + T p SSFF + T p FFFS + T p SSV + T p VS + T p VVS + T p FFV + T p FF V + T p gauge ]. Notations: see ,

29 The full -loop tadpole equation free of GBC The all-scalar diagrams are T p SS = 1 4 j,k,l G + 1 k,l G λ jkll λ jkp P SS (m j, m k)a(m l ) λ Gkll λ Gkp P SS (0, m k)a(m l ), T p SSS = 1 6 λpjkl λ jkl f SSS (mj, mk, ml ) m 0, G T p SSSS = 1 λ pjj λ jkl λ j kl U 0 (mj, mj 4, m k, ml ) (j,j ) (G,G ) (k,l) (G,G ) λ pgg λ Gkl λ G kl R SS (m k, m l ), where by (j, j ) (G, G ) we mean that j, j are not both Goldstone indices.

30 The full -loop tadpole equation free of GBC The fermion-scalar diagrams are T p { 1 SSFF = y IJk y IJl λ klp f (0,0,1) FFS (mi, mj; mk, ml ) (k,l) (G,G ) ] } Re [y IJk y I J k M II M JJ λ klp U 0 (mk, ml, mi, mj) T p FFFS =T p FFFS + 1 λgg p y IJG ( y IJG I(m I, mj, 0) (mi + mj)r SS (mi, mj) ) ] λ GG p Re [y IJG y I J G M II M JJ R SS (mi, mj), m G 0,

31 The full -loop tadpole equation free of GBC The gauge boson-scalar tadpoles are T p SSV =T p SSV m G 0, T p VS = 1 4 g abii g abp f (1,0) VS (ma, mb; mi ) m G g aaik λ ikp f (0,1) VS (ma; mi, mk), (i,k) (G,G ) T p VVS = 1 g abi g cbi g acp f (1,0,0) VVS (ma, mc; mb, mi ) m G g abi g abj λ ijp f (0,0,1) VVS (ma, mb; mi, mj ) (i,j) (G,G ) 1 4 g abg g abg λ GG p R VV (m a, m b).

32 The full -loop tadpole equation free of GBC The gauge boson-fermion and gauge diagrams are not affected by the Goldstone boson catastrophe T p FFV =g aj I g K bjre[m KI y I Ip ]f (1,0,0) FFV (m I, m K ; m J, m a) T p FF V + 1 g I aj g I bjg abp f (0,0,1) (mi, mj; ma, mb), FFV aj =gi gi aj Re[y II p M JJ ][ f FF V (mi, mj, ma) + MI f (1,0,0) (mi, mi ; m J, ma) ] FF V FF V + gi aj gi aj Re[MIK M KI MJJ y KK p]f (1,0,0) (mi, mi ; m J, ma) + 1 g I aj gi bj g abp M II MJJ (0,0,1) f (m FF V I, mj; ma, mb), T p gauge = 1 4 g abc g dbc g adp f gauge (1,0,0) (ma, md; mb, mc).