Higgs Pseudo-Observables

Nuclear Physics B (Proc. Suppl.) 205–206 (2010) 16–19 www.elsevier.com/locate/npbps

Higgs Pseudo - Observables Giampiero PASSARINOa∗ a

Dipartimento di Fisica Teorica, Universit`a di Torino, Italy INFN, Sezione di Torino, Italy

All the questions in this talk are not really urgent, as nothing will be really measurable by LHC before 2013, 2014 or beyond but they will be if you consider as relevant to have results published in such a way that theorists can later enter their general model parameters, calculate resulting POs and see how well data constrains this model.

Experimenters (should) extract (or unfold) socalled realistic observables from raw data, e.g. M (γγ) in σ(pp → γγ + X) and need to present results in a form that can be useful for comparing them with theoretical predictions, i.e. the results should be transformed into POs. Theorists (should) compute pseudo-observables (PO) using the best available technology and satisfying a list of demands from the self-consistency of the underlying theory. The search for a mechanism explaining EWSB has been a major goal for many years, in particular the search for a SM Higgs boson. As a result of this an intense effort in the theoretical community has been made to produce the most accurate NLO and NNLO predictions [1–3]. There is, however, a point that has been ignored: the Higgs boson is an unstable particle and should be removed from the | in/out > bases in the H - space, without destroying unitarity of the theory. Therefore, concepts as the production or partial decay widths of an unstable particle should be replaced by a conventionalized definition which respects first principles of QFT. As an example, combine gg → H with H → γγ. The full process is pp → γγ + X where the signal is represented by pp → gg(→ H → γγ) + X and where we have a non-resonant background. The question is: how to extract from the data, without ambiguities, a PO H partial decay width into γγ which does not violate first principles? Once ∗ Work

supported by the European Community’s MarieCurie Research Training Network under contract MRTNCT-2006-035505 ‘Tools and Precision Calculations for Physics Discoveries at Colliders’

0920-5632/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2010.08.012

again, the Higgs boson doesn’t belong to the inbases and the matrix element < γγ out|H in > is not definable in QFT. Let us consider the following question [4]: how to combine gg → H with H → γγ? The full process is pp → γγ + X with a signal, pp → gg(→ H → γγ) + X plus a non-resonant background. The question is: how to extract from the data, without ambiguities, a PO H partial decay width into γγ which does not violate first principles? Once again, the Higgs boson ∈ | in >; < γγ out|H in > is not definable in QFT. 1. Complex poles Consider resummed propagators: given a skeleton expansion of the self-energy S = 16 π 4 i Σ with propagators that are resummed up to O (n) (0)

Δi (s) =

1 , s − m2i

 (n) (0) (0) Δi (s) = − Δi (s) 1 + Δi (s)  −1 (n) (n−1) (s) , × Σii s , Δi

(1)

a dressed propagator is the formal limit (n)

Δi (s) = lim Δi (s), n→∞  (0) (0) Δi (s) = − Δi (s) 1 + Δi (s)  −1 , × Σii s , Δi (s)

(2)

The Higgs boson complex pole (sH ) is the solution of the equation sH − MH2 + ΣHH (sH ) = 0,

G. Passarino / Nuclear Physics B (Proc. Suppl.) 205–206 (2010) 16–19

where MH2 is the renormalized mass; all local counter terms have been introduced to make the ∂ sH = 0, off-shell Σ UV finite. It follows that ∂ξ ∂ 2 Σ (s , M , ξ) = 0 [5]. As a consequence we H ∂ξ HH H (1)

∂ have ∂ξ ΣHH (sH , sH , ξ) = 0. Therefore, at oneloop, the Higgs complex pole is gauge parameter independent if the self-energy is computed at MH2 = sH .

17

μ5H σ(μH )gg→H ⊗ Γ(μH )H→γγ , s | s − s H |2 (7) (2 π)4 dΦf (PH , {pf }) 2

2 



×

S (Hc → f ) .

μH Γ (Hc → f ) =

(8)

spins

2. Extracting PO

We have four parameters, all PO,

At the parton level the S -matrix for the process i → f can be written as

sH = μ2H − i μH γH , Γ(μH )H→XX , σ(μH )ij→H ,

Sf i = Vi (s) ΔH (s) Vf (s) + Bif (s),

to use in a fit to the (box-detector) experimental distribution (of course, after folding with PDFs); these quantities are universal, uniquely defined, and in one-to-one correspondence with corrected experimental data; after that one could start comparing the results of the fit with a XM calculation. The proposed initial step is to unfold RO into something with idealized cuts and then use the PO approach to fit that:

(3)

where Vi is the production vertex i → H, Vf is the decay vertex H → f , ΔH is the re-summed propagator and Bif is the non-resonant background. 2.1. Modification of the LSZ reduction Consider the following matrix element < f out | | i in >= < f out | H > < H | i in >  + < f out | n > < n | i in >

(4)

n = H

where {n} ⊕ H is a complete set of states. We define ΣHH (s) − ΣHH (sH ) , ΠHH (s) = s − sH  −1 , ΔHH (s) = (s − sH )−1 1 + ΠHH (s) ZH = 1 + ΠHH ,

(5)

and write Sf i = Vi (s)

1 Vf (s) + Bif (s), s − sH

−1/2

Vi (s) = ZH

(s) Vi (s),

(6)

arriving at the following definition: −1/2

S (Hc → f ) = ZH (sH ) Vf (sH ), S (i → Hc ) S (Hc → f ) + non-res. Sf i = s − sH Our main result can now be summarized for gg → γγ:

S (s ) S (s ) 2 1

i H f H

dΦf (PH , {pf })

= s s − sH

(9)

• RD = real data • RO = from real data → distributions with cuts ≡ RO, e.g. diphoton pairs (E, p) → M (γγ) • PO = transform the universal intuition of a QFT-non-existing quantity into an archetype, e. g. σ(gg → H), Γ(H → γγ) • ROth (mH , Γ(H → γγ), . . .) fitted to ROexp (e.g. RO = M (γγ)) defines and extracts mH etc. Why PO language? POs are the the moneta franca of LHC, creating an awful lot of wealth. There are reasons why the chain MCT → detector simulation-selection cuts → realistic distributions → unfolded distributions should be replaced by MCT → box acceptance → unfolded (particle-level) distributions. Since detectorexperimental issues are a moving target only PO are the Frankish language to understand LHC. POs transform the universal intuition of a QFTnon-existing quantity into an archetype, an original model after which theoretical calculations

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G. Passarino / Nuclear Physics B (Proc. Suppl.) 205–206 (2010) 16–19

are patterned without worries on generating and detector-simulating events for signal and background The necessary steps are: go via idealized (model-independent?) RO distributions and from there then go to the POs. One has to use a (new) MCT – the PO code – to fit ROs, trying to understand differences with a standard event generator plus detector simulation plus calibrating the method/event generator used (which differ from the PO-code in its theoretical content).

only when the corresponding Feynman diagram, in the limit of all (internal) stable particles, develops an imaginary part (e.g. above some normal threshold); however, in all cases where the analytical expression for the diagram is known, one can easily see that the result does not change when replacing z0 with zR , the second variant. 3.2. The di-logarithm (0,0)

Li2

(z)

(n,m) (z) Li2

3. Complexity To introduce the general setup consider a φ σ 2 theory with Mφ > 2 mσ ; the φ propagator is −1  . (10) Δ = s − Mφ2 + Σφφ (s) The inverse function, Δ−1 (s), is analytic in the entire s -plane except for a cut [ 4 m2σ → ∞ ]; is defined above the cut, Δ−1 (s + i 0), and the analytical continuation downwards is to the 2nd Riemann sheet −1 (s + i 0) = Δ−1 2 (s − i 0) = Δ −1 Δ (s − i 0) + 2 i π ρ(s),

(11)

where 2 i π ρ(s) is the discontinuity across the cut. We need a few definitions which will help the understanding of the procedure for the analytical continuation of functions defined through a parametric integral representation: 3.1. Logarithm ln(k) z = ln(0) z + 2 i π k, k = 0 , ±1 , . . . ,

(12)

where ln(0) z denotes the principal branch (−π < arg(z) ≤ +π). Let z± = z0 ± i 0 and z = zR + i zI , define  ln z ± 2 i π θ (−z 0 ) θ (∓zI ) ln± (z ; z± ) = ln z ± 2 i π θ (−zR ) θ (∓zI ) . (13) ±

The first definition of the ln -functions is most natural in defining analytical continuation of Feynman integrals with a smooth limit into the theory of stable particles; the reason is simple, in case some of the particles are taken to be unstable we have to perform analytical continuation

(0 < arg(z − 1) < 2 π), (0,0)

= Li2 (z) + 2 n π i ln(0) z + 4 m π2

(14)

Consider the following question: given Li2 (M 2 + i 0) = 1  dx  ln 1 − M 2 x − i 0 , − x 0

(15)

    Im Li2 M 2 + i 0 = π ln M 2 θ M 2 − 1 , how do we understand analytical continuation in terms of an integral representation? Let us consider the analytical continuation from z + = M 2 + i 0 to z = M 2 − i M Γ and define 1  dx −  ln 1 − z x ; 1 − z + x , I=− x 0 χ(x) = 1 − z x = 1 − (M 2 − i M Γ) x. (16) If M 2 > 1, χ crosses the positive imaginary axis: (0,0)

I = Li2

(z) + 2 i π ln M 2 ,

(17)

which is not the expected result. The mismatch can be understood by observing that ln− χ has a cut [0 , +i ∞] and, in the process of continuation, with x ∈ [0, 1], we have been crossing the cut. The solution consists in deforming the integration contour, therefore defining a new integral,  dx −  ln 1 − z x ; 1 − z + x , (18) IC = x C where C = C0 + C  ; C0 = {0 ≤ x ≤ 1/M 2 − ⊕ 2 u 1/M 2 + ≤ x ≤ 1, C  (u) : {x = u + i 1−M MΓ } , 1 1  M 2 +Γ2 ≤ u ≤ M 2 The integral over C is downwards on the first quadrant and upwards on the

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G. Passarino / Nuclear Physics B (Proc. Suppl.) 205–206 (2010) 16–19

t → t

0.5 0.4

s →

σCMCP/ σCMRP-1

m2H → sH  

θ

0.3 0.2 0.1

Figure 1. Complex Mandelstam invariants −

−

second (along the cut of ln ); integration of ln over C  gives −2 i π (ln M 2 − ln z), showing that (1,0) Li2 (z)

-0.1 300

320

340

360

380

400 420 μ [GeV]

440

460

480

500

H

= IC ,

(19)

the correct analytical continuation. Therefore we can extend our integral, by modifying the contour of integration, to reproduce the right analytical continuation: Lin

0

Analyt.Cont.

−→

Li− n+1 (z)



=

z 0

Li− n, dx − Li (x). x n

(20)

In performing analytical continuation of Feynman diagrams there is a natural choice: from real kinematics to complex Mandelstam invariants, see Figs. 1 and 2:   PO = σ gg → Hj (sH , s, t)  sH s 1− − βH cos θ t=− 2 s  sH s 1− + βssH cos θ u=− 2 s  sH 1/2 . (21) βH = 1 − 4 s 4. Schemes We define the following schemes: RMRP, the usual on-shell scheme where all masses and all Mandelstam invariants are real; CMRP, the complex mass scheme with complex internal W and Z poles (extendable to top complex pole) but with real, external, on-shell Higgs, W, Z , etc. legs and with the standard LSZ wavefunction renormalization; CMCP, the (complete) complex mass scheme with complex, external Higgs (W, Z, etc.) where the LSZ procedure is carried out at the Higgs complex pole (on the second Riemann sheet). No theoretical uncertainty; only the CMCP scheme is fully consistent.

√ Figure 2. σCMCP √/σCMRP (pp → H) at s = 3 TeV s = 10 TeV (dashed-blue) and (dotted-red), √ s = 14 TeV (black) REFERENCES 1. Charalampos Anastasiou, Radja Boughezal, and Frank Petriello. Mixed QCD-electroweak corrections to Higgs boson production in gluon fusion. JHEP, 04:003, 2009. 2. Massimiliano Grazzini. NNLO predictions for the Higgs boson signal in the W W and ZZ decay modes. Nucl. Phys. Proc. Suppl., 183:25–29, 2008. 3. Daniel de Florian and Massimiliano Grazzini. Higgs production through gluon fusion: updated cross sections at the Tevatron and the LHC. Phys. Lett., B674:291–294, 2009. 4. Giampiero Passarino, Christian Sturm, and Sandro Uccirati. Higgs Pseudo-Observables, Second Riemann Sheet and All That. Nucl. Phys., B834:77–115, 2010. 5. Pietro A. Grassi, Bernd A. Kniehl, and Alberto Sirlin. Width and partial widths of unstable particles in the light of the Nielsen identities. Phys. Rev., D65:085001, 2002.