Combination of KLOE $\sigma (e^{+}e^{-} \rightarrow \pi ^{+}\pi ^{-}\gamma (\gamma))$ measurements and determination of $a_{\mu }^{\pi^{+}\pi^{-}}$ in the energy range 0.10 < s < 0.95 GeV$^{2}$

P u b l i s h e d f o r S I S S A b y S p r i n g e r R ec e iv e d : Novem ber 9, 2017

A c c e p te d : March, 3, 2018 P u b lish e d : M arch 28, 2018

Combination of KLOE a ( e + e - n + n - Y ( 7 )) measurements and determination of a^+n in the energy range 0.10 < s < 0.95 G eV2

T h e K L O E -2 collaboration

A . A nastasi,ec D . Babusci,c M . Berlowski,c,v C. Bloise,c F. Bossi,c P. B ranchini,s A . Budano,rs L. Caldeira B alkestahl,” B. C ao,u F. C eradini,rs P. C iam brone,c

F. C urciarello,c E. C zerw iński,6 G. D ’A gostini,” 0 E. D a n e ,c V . D e Leo,q E. D e Lucia,c A . D e Santis,c P. D e Sim one,c A. Di Cicco,rs A. Di D om enico,”-’0 D . D om en ici,c A . D ’ U ffizi,c A. Fantini,pq G. F an tin i,d P. Ferm ani,c S. Fiore,to A. Gajos,6

P. G auzzi,” 0 S. G iovannella,c E. G raziani,s V . L. Ivanov,gh T . Johansson,” X . K an g ,c D . K isielew ska-Kam ińska,6 E. A . K ozyrev,gh W . K rzem ien ,v A. Kupsc,u S. Loffredo,rs P. A. Lukin,gh G. M a n d a g lio ,f “ M . M a rtin i,'c,m R. Messi,pq S. M is c e tti,c G. M o rello ,c D . M o ricciani,q P. M o skal,6 A. Passeri,s V . P a te ra,1’0 E. Perez del R io,c N . R aha,q P. Santangelo,c M . Schioppa,jfc A. Selce,rs M . Silarski,6 F. Sirghi,c E. P. Solodov,gh L. T o rto ra ,s G. Venanzoni*1 W . W islickiv and M . W o lk e ,u

A . Keshavarziw1 S.E. Muller® and T . Teubnerw a I N F N S e z io n e d i C a ta n ia , C a ta n ia , Ita ly

bI n s titu te o f P h y sic s, J a g ie llo n ia n U n ive rsity, Cracow, P o la n d cL a b o ra to ri N a z io n a li di F rascati d e ll’IN F N , F rascati, Ita ly d G ra n S a sso S c ie n c e In s titu te , L ’A quila, Ita ly

e D ip a r tim e n to di S c ie n ze M a te m a tic h e e In fo r m a tic h e , S c ie n z e F isic h e e S c ie n z e della Terra, U n iv e rsita d i M e ssin a , M e ssin a , Ita ly

f D ip a r tim e n to di S c ie n ze C h im ich e , B iologiche, F a rm a c e u tic h e ed A m b ie n ta li, U n iv e rsita d i M e ssin a , M e ssin a , Ita ly

g B u d k e r I n s titu te o f N u c le a r P h y sic s, N o v o sib irsk , R u s s ia h N o v o s ib ir s k S ta te U n ive rsity, N o v o sib irsk , R u s s ia

1 I N F N S e z io n e d i P isa , P isa , Ita ly

j D ip a r tim e n to di F isica, U n iv e rsita della C alabria, R e n d e , Ita ly k I N F N G ruppo collegato di C o sen za , R e n d e , Ita ly

1 D ip a r tim e n to di S c ie n ze di B a s e ed A p p lic a te p e r l ’In g eg n e ria , U niversita, “S a p ie n z a ”, R o m a , Ita ly

1 Corresponding author.

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

m D ip a r tim e n to di S c ie n ze e T ecnologie applicate, U n iv e rsita “G uglielm o M a r c o n i”, R o m a , Ita ly n D ip a r tim e n to di F isica, U n iv e rsita “S a p ie n z a ”, R o m a , Ita ly

oI N F N S e z io n e d i R o m a , R o m a , Ita ly

p D ip a r tim e n to di F isica, U n iv e rsita “T o r V erg a ta ”, R o m a , Ita ly qI N F N S e z io n e d i R o m a T o r Vergata, R o m a , Ita ly

r D ip a r tim e n to di M a te m a tic a e F isica , U n iv e rsita “R o m a T r e ”, R o m a , Ita ly s I N F N S e z io n e d i R o m a Tre, R o m a , Ita ly

t E N E A , D e p a r tm e n t o f F u sio n a n d T echnology f o r N u c le a r S a fe ty a n d S ec u rity , F ra sc a ti ( R M ) , Ita ly

u D e p a r tm e n t o f P h y s ic s a n d A s tr o n o m y , U ppsala U n ive rsity, Uppsala, S w e d e n v N a tio n a l C e n tre f o r N u c le a r R esearch, W arsaw , P o la n d

w D e p a r tm e n t o f M a th e m a tic a l S cien c es, U n iv e rsity o f L iverpool, L iverp o o l L 6 9 3 B X , U .K . x D e p a r tm e n t o f I n fo r m a tio n S e rv ic e s a n d C o m p u tin g & I n s titu te o f R a d ia tio n P h y sic s,

H e lm h o ltz -Z e n tr u m D re sd e n -R o s se n d o rf, D resd en , G e rm a n y

E-m ail: g r a z i a n o . v e n a n z o n i @ l n f . i n f n . i t , a . i . K e s h a v a r z i @ l i v e r p o o l . a c . u k

A b s t r a c t : T h e th re e precisio n m e a s u re m e n ts o f th e cross sec tio n a ( e + e - ^ n + n - Y (y)) usin g in itia l s ta te ra d ia tio n by th e K L O E c o lla b o ra tio n p ro v id e an im p o rta n t in p u t for th e p re d ic tio n of th e h a d ro n ic c o n trib u tio n to th e an o m alo u s m a g n e tic m o m e n t o f th e m u o n . T h ese m e a su re m e n ts a re c o rre la te d for b o th s ta tis tic a l a n d sy ste m a tic u n c e rta in tie s an d , th e re fo re , th e sim u lta n e o u s use of th e s e m e a su re m e n ts re q u ires co varian ce m a tric e s t h a t fully d e sc rib e th e c o rre la tio n s. W e p re se n t th e c o n s tru c tio n of th e s e co varian ce m a tric e s a n d use th e m to d e te rm in e a co m b in ed K L O E m e a s u re m e n t for a ( e + e - ^ n + n - Y(Y)).

W e find, from th is co m b in a tio n , a tw o -p io n c o n trib u tio n to th e m u o n m a g n e tic a n o m a ly in th e en e rg y ra n g e 0.10 < s < 0.95 G eV2 o f a^ + n = (489.8 ± 1.7stat ± 4.8sys) x 10-1 0 .

D a ta v ec to rs a n d covarian ce m a tric e s a re av ailab le a t h t t p :/ /w w w .ln f .in f n .it/k lo e /p p g / pp g _ 2 0 1 7 /p p g _ 2 0 1 7 .h tm l.

K e y w o r d s : e + -e - E x p e rim e n ts

A r X i v e P r i n t : 1711.03085

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

C o n t e n t s

1 I n tr o d u c tio n 1

2 M e a s u r e m e n ts o f ct0( e + e - ^ n + n - 7(7)) b y t h e K L O E c o lla b o r a tio n 2

2.1 D e te rm in a tio n o f th e n + n - cross sec tio n 2

2.2 T h e K L O E m e a s u re m e n ts 3

3 C o n s t r u c tin g t h e K L O E c o m b in a tio n c o v a r ia n c e m a tr ic e s 5

3.1 S ta tis tic a l c o rre la tio n s 6

3.2 S y ste m a tic c o rre la tio n s 9

4 C o m b in a tio n a n d r e s u lts 11

4.1 T h e co m b in ed K L O E e + e - ^ n + n - y (y ) cross sec tio n 11 4.2 C o m p a riso n w ith re su lts from th e C M D -2, SN D , B a B a r a n d B E S III

e x p e rim e n ts 14

5 C o n c lu s io n s 18

A P r o p e r tie s o f a c o v a r ia n c e m a tr ix 19

1 I n t r o d u c t i o n

T h e K L O E c o lla b o ra tio n have m ad e th re e precise m e a su re m e n ts of th e cross sectio n a (e + e - ^ n + n - y (7)) in 2008 [1, 2], 2010 [3 , 4] a n d 2012 [5, 6].1 T h ese m e a su re m e n ts a re cru cial for e s tim a tin g th e h a d ro n ic v a c u u m p o la ris a tio n (H V P ) c o n trib u tio n to th e an o m alo u s m a g n e tic m o m e n t of th e m uon, ajjVP, w hich is p re se n tly th e lim itin g fa c to r in th e precisio n o f th e S ta n d a rd M odel (SM ) p re d ic tio n , a^M. T h is SM p re d ic tio n d is­

agrees w ith th e e x p e rim e n ta l value, a^xp [8- 11], by a p p ro x im a te ly 3.5 s ta n d a r d d ev ia tio n s o r h ig h er [12- 19], m a k in g it a n in te re stin g p ro b e of p o te n tia l physics b ey o n d th e SM. C u r­

rently, th e u n c e rta in tie s of a ^ M a n d a^xp are of c o m p a ra b le m a g n itu d e . H ow ever, w ith new e x p e rim e n ta l efforts a t F e rm ila b [20] a n d J-P A R C [2 1] set to im prov e th e e x p e rim e n ta l e rro r by a fa c to r of fo u r co m p a re d to th e B N L m e a su re m e n ts [8- 10], it is im p e ra tiv e t h a t th e SM p re d ic tio n is also im proved.

T h e H V P c o n trib u tio n to a^M c a n be d e te rm in e d usin g a d isp e rsio n in te g ra l an d th e cross sec tio n a °(e + e - ^ h a d ro n s ), w hich is b a re (u n d ressed of all v ac u u m p o la ris a tio n

1The KLOE collaboration also made a measurement of a(e+e- ^ n + n -7(7)) in 2005 [7]. However, this is now considered to be superseded by the 2008 measurement, as discussed in [1].

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

(V P ) effects) as in d ic a te d by th e su p e rsc rip t ‘0 ’, b u t in clu d es final s ta te ra d ia tio n (F S R ).

A t lead in g o rd e r (L O ), th e d isp e rsio n in te g ra l is 1 f ^

aL° ’HVP = 4^3 J d s " ° a d ( s ) K ( s ) , (1 .1)

w h ere s th = m^0 is th e h a d ro n ic p ro d u c tio n th re s h o ld , ^ ° ad(s) is th e b a re cross sec tio n of th e p rocess e + e - ^ h a d ro n s a n d K ( s ) is a w ell-know n kernel fu n c tio n [2 2 , 23]. T h e c o n tri­

b u tio n of th e n + n - final s ta te to th e an o m alo u s m a g n e tic m o m e n t o f th e m u on, a^+ n , is over 70% of th e to ta l e s tim a te o f aHVP a n d is also th e la rg e st c o n trib u to r to its u n c e rta in ty . C o nseq u en tly , th e th re e m e a su re m e n ts of th e cross sec tio n a °(e + e - ^ n + n - Y (y)) by th e K L O E c o lla b o ra tio n a re in v alu ab le to p recisely d e te rm in e a^ n .

T h e sim u lta n e o u s in p u t of th e K L O E m e a s u re m e n ts in to e q u a tio n ( 1.1) req u ires a d e ta ile d an a ly sis to a t t a i n th e c o rre c t c o m b in a tio n o f th e th re e , w h ich w ill h ave a n o n ­ triv ia l influence o n a^ + n a n d p ro v id e a n im p o rta n t c o m p a riso n w ith o th e r e x p e rim e n ta l m e a su re m e n ts of an n . T h e K L O E m e a su re m e n ts of " nn(7) are, in p a r t, h ig h ly c o rre la te d , n e c e s sita tin g th e c o n s tru c tio n of full s ta tis tic a l a n d s y s te m a tic cov arian ce m a tric e s to be used in an y c o m b in a tio n of th e s e d a ta . To co m b in e th e d a t a w ith o u t th e c o rre la tio n s w ould re su lt in an u n d e re s tim a te o f th e u n c e rta in ty of a HVP an d , p o te n tia lly , a b ias of its m ean value. T h e c o n s tru c tio n o f th e se covarian ce m a tric e s m u st b e s ta tis tic a lly ro b u s t in o rd e r to e n su re t h a t th e y c o rre c tly d esc rib e th e c o rre la te d re la tio n s h ip of th e th re e m e a su re m e n ts.

T h e m a in p u rp o se of th is w ork is to fo rm u la te th e cov arian ce m a tric e s re q u ired to d e te rm in e th e co rre c t co m b in a tio n . In sec tio n 2 , th e th re e K L O E m e a su re m e n ts of

" ( e + e - ^ n + n - Y (y)) [1-6] are review ed an d , in som e cases, u p d a te d in o rd e r to en su re a co n siste n t c o m b in a tio n . S ectio n 3 th e n focuses on th e c o n s tru c tio n o f th e s ta tis tic a l a n d sy s te m a tic covarian ce m a tric e s for th e c o m b in a tio n o f th e K L O E m e a su re m e n ts. In sec tio n 4 , th e s e m a tric e s are th e n used to co m bine th e th re e m e a su re m e n ts in to a single m e a su re m e n t of " ° (e + e - ^ n + n -^y(^y)) , w hich we use to p ro v id e a n e s tim a te o f a^ + n . W e th e n c o m p a re o u r re su lts w ith th e in d iv id u a l K L O E m e a su re m e n ts an d o th e r e x p e rim e n ta l m e a su re m e n ts o f " nn(7).

2 M e a s u r e m e n t s o f ( e + e - ^ n + n - Y (7 ) ) b y t h e K L O E c o l l a b o r a t i o n

2.1 D e t e r m in a tio n o f t h e n+n- c r o ss s e c t io n

D A $ N E [24] is a h ig h lu m in o sity e + e - co llid er t h a t o p e ra te s p re d o m in a n tly a t th e c e n tre of m ass en e rg y eq u a l to th e <fi m eson m ass, 7% = m ^ = 1.0194 G eV [11]. T h e K L O E d e te c to r has b ee n used to o b ta in m e a su re m e n ts o f th e p ro cess e + e - ^ n + n - Y(y) [1, 3 , 5, 2 5 , 26].

T h ese m e a su re m e n ts are achieved th ro u g h ra d ia tiv e re tu rn , w h ere th e d iffe ren tial cross sec­

tio n is m e a su re d as a fu n c tio n of th e in v a ria n t m ass o f th e p io n p air, a/s7 = . T h e cross sec tio n a nn = " ( e + e - ^ n + n - ) is th e n d e te rm in e d a c co rd in g to [27- 30] u sin g th e re la tio n

s d a dnMn ^ = " n n ( M - ) # ( M n , s) , (2.1)

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

w h ere H is th e r a d ia to r fu n c tio n d e sc rib in g th e em ission o f p h o to n s in th e in itia l s ta te [31- 34]. E q u a tio n (2.1) is valid n eg lec tin g th e c o n trib u tio n from F S R , a lth o u g h it is p ro p e rly a c c o u n te d for in th e K L O E an aly ses [1, 3 , 5 , 35].

T h e K L O E c o lla b o ra tio n h ave p erfo rm ed th re e m e a s u re m e n ts of th e cross sectio n a ( e + e - — n + n - y ( y )) [1- 6]. All th re e p u b lish e d cross sectio n s are b a re (u n d ressed o f all V P effects) a n d in clu d in g F S R . F or th e first tw o, w hich for th e p u rp o se s o f th is s tu d y we sh all d e n o te as K L O E 0 8 [1] a n d K L O E 1 0 [3], th e b a re cross sec tio n is o b ta in e d by [3 6 , 37]

^ ( ^y^ O = ff^u (7)(s ,)|1 - (2 .2)

w h ere th e s u p e rsc rip t ‘0 ’ in d ic a te s t h a t th e cross sec tio n is b are, th e s u b sc rip t (y ) in d ic a te s t h a t th e cross sec tio n in cludes F S R , a nn(7)( s ') is o b ta in e d u sin g e q u a tio n (2.1) a n d n ( s ') is th e v ac u u m p o la risa tio n c o n ta in in g b o th real a n d im a g in a ry p a r ts [38].2

F o r th e th ir d m e a s u re m e n t o f n(7)(s '), n a m e ly K L O E 1 2 [5], a re cip ro ca l re la tio n to e q u a tio n (2.1) w as u tilised , allow ing for a b in -b y -b in n o rm a lis a tio n o f th e n + n - cross sec tio n by th e ^ + ^ - cross section. F o r th e sam e in v a ria n t m ass sq u a re d , th e ra tio of th e n + n - y a n d ^ + ^ - y d iffe ren tial cross sectio ns allow s th e re la tio n

-A

V

A

w h ere s ' = = M 2^. T h is n o rm a lis a tio n h as m a n y ad v a n ta g e s co n c ern in g th e d e te r­

m in a tio n o f th e cross section. Im p o rta n tly , th e ra tio in e q u a tio n ( 2 .3 ) b en e fits fro m th e c a n c e lla tio n of th e r a d ia to r fu n c tio n for in itia l s ta te ra d ia tio n (IS R ) a n d of th e V P co r­

re c tio n , m a n ife stly re s u ltin g in a b a re cross sectio n. T h erefo re, th e u n d re ssin g p ro c e d u re d e sc rib e d by e q u a tio n ( 2.2) is n o t ap p lied to K L O E 1 2 , a lth o u g h th e F S R c o n trib u tio n to th e n + n - p ro d u c tio n m u st a g a in b e in clu d ed .

T h e p io n form fa c to r, |F n |2, is d e te rm in e d for all th re e m e a su re m e n ts to be

^ i 2 = n i f - w ( > - a * m ) , ^

w h ere a = a ( 0 ) , (s ') = y/1 — 4 m l / s ' an d is th e inclusive F S R c o rre c tio n assu m in g p o in t-lik e pions [39].

2 .2 T h e K L O E m e a s u r e m e n ts

T h e e x p e rim e n ta l an a ly sis of each K L O E m e a s u re m e n t o f a ( e + e - — n + n - y ( y ) ) h as b een review ed an d , in som e cases, u p d a te d in o rd e r to e n su re a m ore p recise a n d c o n siste n t c o m b in a tio n o f th e th re e m e a su re m e n ts. In th e follow ing, each m e a su re m e n t is d iscussed ind iv id u ally , w h ere an y chan g es to th e re sp ectiv e an aly sis a re e x p lic itly s ta te d .

2T h e c o r r e c tio n u s e d p r e v io u s ly fo r K L O E 0 8 [1 , 2] a n d K L O E 1 0 [3 , 4] c o n ta in e d o n ly t h e r e a l p a r t o f t h e V P . T h is h a s b e e n u p d a t e d in t h i s a n a ly s is t o in c o r p o r a te t h e fu ll V P w i t h b o t h t h e r e a l a n d im a g in a r y p a r t s , w h e r e t h e im a g in a r y p a r t is s m a ll a n d s u b - le a d in g c o m p a r e d t o t h e r e a l c o n tr i b u t io n (s e e s e c tio n 2.2 fo r m o r e d e ta i ls ) .

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

T h e K L O E 0 8 m e a s u re m e n t co n sists o f 60 d a t a p o in ts in th e ra n g e 0.35 < s ' < 0.95 G eV2, covering th e d o m in a n t p re so n an ce s tr u c tu r e a n d th e p — w in terfe ren c e reg ion in th e n + n - final s ta te . T h e u n c e rta in tie s o f th e cross sec tio n are d o m in a te d by th e sy ste m a tic s u n c e rta in tie s , esp ecially in th e region w h ere th e cross sec tio n is large. T h e K L O E 0 8 d a t a h av e b ee n u p d a te d w ith re sp e c t to [1] to in c o rp o ra te th e follow ing n ecessary changes:

• T h e d a t a have b ee n u n d re sse d of V P effects u sin g a n u p d a te d ro u tin e [3 6 , 37] co m ­ p a re d to th e one u sed p re v io u sly [40], w h ich now c o rre c ts th e d a t a u sin g a m ore a p p r o p ria te en e rg y g rid p a r a m e tris a tio n for th e d e te rm in a tio n o f th e V P.

• T h e V P c o rre c tio n c o n ta in s b o th re al a n d im a g in a ry p a rts , w h e reas p re v io u sly th e d a t a w ere o n ly c o rre c te d for th e real p a r t of th e V P.

• T h e d a t a a re n o t ro u n d e d as th e y w ere in [1] to en su re t h a t th e s ta tis tic a l an d sy ste m a tic u n c e rta in tie s co rre sp o n d to th e v aria n ces t h a t e n te r in to th e d iag o n al elem en ts of th e c o rre sp o n d in g co v arian ce m atrice s.

• T h e c a lc u la tio n o f th e cross sec tio n h as b een u p d a te d w ith re s p e c t to th e p recision of in p u t p a ra m e te rs a n d fu n d a m e n ta l c o n s ta n ts [1 1].

W e find, u sin g th e u p d a te d d a ta for K L O E 0 8 , a c o n trib u tio n to th e an o m alo u s m a g n e tic m o m e n t of th e m u o n of

(K L O E 0 8 , 0.35 < s ' < 0.95 G eV2) = (386.6 ± 0 . 4 ^ ± 3.3sys) x 10-10, (2.5) w hich e x h ib its a d ec rea se in th e m e a n value o f a^ + n w h en c o m p a re d to th e e s tim a te q u o te d in [1] t h a t is larg ely d u e to th e u p d a te d d e te r m in a tio n o f th e V P . T h e u p d a te d cross sec tio n a n d p io n form fa c to r v ec to rs w ith co rre sp o n d in g cov arian ce m a tric e s for th e s ta tis tic a l an d sy s te m a tic u n c e rta in tie s are av ailab le from [41].

T h e K L O E 1 0 m e a su re m e n t to ta ls 75 d a t a p o in ts in th e ra n g e 0.1 < s ' < 0.85 G eV2. T h is an aly sis [3] selected ev en ts t h a t in clu d ed a p h o to n d e te c te d in th e c a lo rim e te r a t large p o la r angle, allow ing th e m e a su re m e n t to b e ta k e n a t low er s' closer to th re sh o ld . T h e fifty en e rg y bins of th e d a ta in th e ra n g e 0.35 < s ' < 0.85 G eV2 are id e n tic a l to th e fifty K L O E 0 8 bins in th e sam e in terv a l. T h e K L O E 1 0 cross sec tio n h as b een u p d a te d in th e sam e w ay as K L O E 0 8 , w ith th e a p p lic a tio n o f th e im p ro v ed V P c o rre c tio n [3 6 , 37], th e n o n -ro u n d e d d a t a a n d im p ro v ed p a r a m e te r precisio n re su ltin g in

a £ + ( K L O E 1 0 , 0.10 < s ' < 0.85 G eV2) = (477.9 ± 2.0stat ± 6.7sys) x 10-10, (2.6) w hich, like o b served w ith K L O E 0 8 , re su lts in a d ec rea se in th e m ean v alue of a^ + n co m ­ p a re d to th e e s tim a te in [3]. T h e u p d a te d K L O E 1 0 d a t a v ec to rs a n d covarian ce m a tric e s a re available from [41].

T h e K L O E 1 2 m e a su re m e n t w as d e te rm in e d as a p + p - Y n o rm alised cross sectio n, as d e sc rib e d briefly in sec tio n 2 .1 . T h e p + p - cross sec tio n w as m e a su re d for th e analy sis, w h e reas th e K L O E 0 8 n + n - d a ta w ere u sed as th e in p u t in to e q u a tio n ( 2.3) , w ith th e K L O E 1 2 m e a su re m e n t h av in g a n id e n tic a l b in n in g a n d en e rg y ra n g e to K L O E 0 8 . As th e se

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

m e a su re m e n ts sh a re th e sam e tw o -p io n d a ta , K L O E 0 8 a n d K L O E 1 2 a re hig h ly c o rre la te d a n d it is im p e ra tiv e t h a t th e y b e tr e a te d as such in an y c o m b in a tio n of th e tw o m e a su re ­ m en ts. T h e K L O E 12 cross sec tio n h as b ee n u p d a te d w ith re sp e c t to th e use of n o n -ro u n d e d d a t a an d in p u t p a r a m e te r precision. T h e ra tio in e q u a tio n (2.3) b en e fits from th e ca n cella­

tio n o f th e V P c o rre c tio n an d , th e re fo re , d oes n o t re q u ire a n u p d a te d V P c o rre c tio n as w ith th e K L O E 0 8 a n d K L O E 1 0 cross sec tio n d a ta . F o r th e c o n trib u tio n to th e m u o n m a g n e tic anom aly, from th e K L O E 12 d a t a alone, we find

a £ + ( K L O E 1 2 , 0.35 < s ' < 0.95 G eV2) = (385.1 ± 1.2stat ± 2.3sys) x 10-10. (2.7) H ere, th e e rro r h as re d u ced since [5], w h ere a flaw in th e p re v io u s e rro r c a lc u la tio n re su lte d in a n o v e re stim a tio n of th e p u b lish e d s y s te m a tic u n c e rta in ty an d , as a re su lt, th e re have b ee n n ecessary chan g es to th e K L O E 1 2 sy s te m a tic m a trix c o n s tru c tio n .3 T h e u p d a te d K L O E 1 2 d a ta a re available from [41].

3 C o n s t r u c t i n g t h e K L O E c o m b i n a t i o n c o v a r i a n c e m a t r i c e s

T h e flow of th e e x p e rim e n ta l an a ly se s for th e K L O E 0 8 , K L O E 1 0 a n d K L O E 1 2 m e a su re ­ m e n ts is show n in figure 1. In th e case o f th e K L O E 1 2 m e a su re m e n t, th e b eg in n in g of th e flow refers to th e m e a su re m e n t of p + p -7(7). T h e p o in t w h ere th e K L O E 0 8 n + n - Y(7) d a t a e n te rs is clea rly m ark e d . T h is d ia g ra m e x h ib its th e e x te n t of th e c o rre la tio n b etw een K L O E 0 8 a n d K L O E 1 2 , w ith c o rre la tio n s e x istin g for all elem en ts of th e K L O E 0 8 an a ly sis from th e o b serv ed s p e c tru m o f n + n -7(7) ev e n ts u p to th e a c c e p ta n c e co rrec tio n . In a d d itio n , th e d eg ree of c o rre la tio n b etw e en K L O E 0 8 a n d K L O E 1 0 o r K L O E 1 0 an d K L O E 1 2 is clea rly show n, w ith m an y p a r ts of th e e x p e rim e n ta l an a ly se s b ein g co m m o n to a p a ir of m e a su re m e n ts o r h av in g b ee n o b ta in e d th ro u g h a sim ila r m e th o d .

T h e K L O E s ta tis tic a l a n d sy s te m a tic c o m b in a tio n co v arian ce m a tric e s are 195 x 195 m a tric e s a n d are d e p ic te d in figure 2 . T h e y hav e b ee n ca refu lly c o n s tru c te d to sa tisfy th e necessary m a th e m a tic a l p ro p e rtie s of a cov arian ce m a trix , d e ta ils of w h ich a re d esc rib ed in a p p e n d ix A . T h e K L O E 0 8 , K L O E 1 0 a n d K L O E 1 2 d ia g o n a l blocks are sim p ly th e co­

v a ria n c e m a tric e s of th e in d iv id u a l m e a su re m e n ts. T h e K L O E 0 8 1 0 b lock d esc rib es th e c o rre la tio n b etw e en K L O E 0 8 a n d K L O E 1 0 , w ith co rre sp o n d in g d efin itio n s for K L O E 0812 a n d K L O E 1012. S ta tis tic a l u n c e rta in tie s are, in g en e ral, u n c o rre la te d (Pajstat |i= j = 0, w here P j is th e c o rre la tio n coefficient defined in a p p e n d ix A ) a n d o n ly c o n trib u te to th e d iag o n al elem e n ts of th e co rre sp o n d in g c o rre la tio n b lock of th e c o m b in a tio n cov arian ce m a trix . T h e ex c ep tio n s to th is a re th e un fo ld in g a n d u n sh iftin g c o rre c tio n s4 (see sec tio n 3.1) , w h ich b o th c o n trib u te to th e n o n -d iag o n a l elem e n ts of th e s ta tis tic a l m a trix (p°jstat = — 1 < p < 1).

F o r s y ste m a tic (sys) u n c e rta in tie s, all d a t a p o in ts a re ta k e n to b e 100% c o rre la te d o r a n ti­

c o rre la te d (p " sys = ± 1 ) . T h e re su ltin g c o rre la tio n s tru c tu re s for b o th th e s ta tis tic a l an d 3The KLOE12 systematic uncertainty has reduced from 2.7 x 10-10 given in [5] to 2.3 x 10-10 in this analysis.

4While the unfolding correction accounts for the smearing due to the detector resolution, the unshifting is a redistribution correction that accounts for photons emitted through final state radiation, which results in shifting the observed value of s' away from the squared invariant mass of the virtual photon [35].

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

F ig u r e 1. T h e flow o f th e exp erim en tal an alyses o f all three <r0( e + e - ^ n + n - y ( y ) ) cross sec­

tio n m easu rem ents. T h e p oin t w here th e K L O E 08 n + n - y ( y ) d a ta enter th e K L O E 12 an alysis is in d icated b y th e b old black arrows.

sy s te m a tic m a tr ix are show n in figure 3 . In th e follow ing, we o u tlin e th e c o rre la tio n s t h a t ex ist for a n d b etw e en th e in d iv id u a l m e a su re m e n ts for th e s ta tis tic a l a n d sy ste m a tic u n c e rta in tie s sep a rately .

3.1 S t a tis tic a l c o r r e la tio n s

O th e r th a n th o se t h a t ex ist as p a r t o f th e in d iv id u a l an aly ses for th e K L O E 0 8 , K L O E 1 0 a n d K L O E 1 2 d ia g o n a l su b -m a tric e s in th e s ta tis tic a l 195 x 195 c o m b in a tio n cov arian ce m a tr ix d e p ic te d in figure 2, th e only s ta tis tic a l c o rre la tio n s t h a t a re p re se n t are th o se d u e to th e tw o -p io n d a t a t h a t a re sh a re d b etw e en K L O E 0 8 a n d K L O E 1 2 . T h ese o ccu p y th e K L O E 08 12 a n d K L O E 1 2 0 8 blocks of th e s ta tis tic a l c o m b in a tio n cov arian ce m a trix . A s no s ta tis tic a l c o rre la tio n s ex ist b etw een K L O E 0 8 a n d K L O E 1 0 o r K L O E 1 0 a n d K L O E 1 2 , all elem e n ts of th e K L O E 0 8 1 0 (K L O E 100 8) a n d K L O E 1 01 2 (K L O E 1 21 0) c o rre la tio n blocks o f th e s ta tis tic a l co v arian ce m a trix are zero. T h is c a n be seen d ia g ra m m a tic a lly in figure 3 .

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

Figure 2. The KLOE n + n 7(7) com bination m atrix stru cture for b o th the statistical and sys­

tem atic covariance matrices.

The individual KLOE08, KLOE10 and KLOE12 statistical covariance matrices (cor­

responding to the diagonal blocks of the statistical combination m atrix given by figure ²) describe all statistical uncertainties inherent in the respective experimental analysis. The contributions to the statistical covariance matrices from the unfolding and unshifting pro­

cedures are partially correlated, where the correlation coefficients are defined by the unfold­

ing [42- 44] and unshifting [42] procedures themselves. Details regarding these procedures and all other statistical uncertainties (which are considered to be fully uncorrelated) can be found in [2, 4, 6].

The KLOE0812 statistical correlation block receives contributions from all corrections to the KLOE08 n+n - Y(7) d ata up until the point where these d ata enter the KLOE12 analysis. Following the experimental analysis flow for KLOE08 in figure 1, these include the detector resolution correction (unfolding), the correction for border efficiency in the acceptance, the pion identification efficiency (n /e likelihood), the tracking efficiency, the trigger corrections, the unshifting of ^ ( M ^ )² and the acceptance for the cuts in Pn and Pnn [2]. All corrections prior to the unfolding in the analysis flow are included in the unfolded KLOE08 n+n - Y(Y) spectrum and, therefore, manifestly enter the KLOE0812 cor­

relations through the correlations of the unfolding. As the unfolding (unf) and unshifting (uns) corrections are identically correlated for the KLOE08 and KLOE12 statistical co­

variance matrices, these correlations must be reflected in the KLOE0812 correlation block exactly in the form

0 8 1 2,^{u n f}/^{un s 1 2 0 8},^{u n f}/^{u n s 0 8},^{u n f} /^{u n s 12},^{u n f}/^{un s / o 1 \}

P j ' = Pji 7 = P j 7 = P j 7 . (3.1)

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

F ig u r e 3 . T h e correlation stru ctu re o f th e 195x195 sta tistic a l and sy stem a tic com b in ation m atrices.

In each case, th e axis on th e right represents th e overall correlation coefficient ( p j = —1 < p < 1), w here th e corresp ond ing colour in d icates th e degree o f correlation at each p oin t in th e resp ective m atrix (colour on lin e).

Not doing so would result in the statistical covariance m atrix having negative eigenvalues, therefore violating the condition th a t the covariance m atrix is a positive semi-definite m a­

trix. All remaining correlated statistical uncertainties only enter into the diagonal elements of the KLOE0812 correlation block, as they are fully correlated only for the same energy bins between the two measurements.

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

(a) Statistical correlation matrix

(b ) Systematic correlation matrix

All c o rre la tio n blocks in figure 2 receive c o n trib u tio n s from sy ste m a tic u n c e rta in tie s, as ca n b e seen clea rly in figure 3 . U nless s ta te d o th erw ise , for an y tw o b in s i a n d j , s y s te m a tic u n c e rta in tie s w h ere c o rre la tio n s ex ist are fu lly c o rre la te d ( p j = +1) o r a n ti-c o rre la te d (Pij = —1 ).

F o r th e in d iv id u a l m e a su re m e n ts, a p a r t from tw o ex c ep tio n s, all sources o f sy s te m a tic u n c e rta in ty a re fully c o rre la te d b etw e en all en e rg y bins. T h e first e x c e p tio n is th e sy ste m ­ a tic u n c e rta in ty d u e to th e u nfold in g, w h ich o n ly c o n trib u te s a t th e s h a rp d esc en t of th e cross sec tio n in th e p — w in terfe ren c e region. H ere, a n id e n tic a l u n fo ld in g u n c e rta in ty e n ­ te rs for five bins of th e K L O E 0 8 a n d K L O E 1 2 an a ly se s a n d is a n ti-c o rre la te d o nly for p a irs o f bins t h a t a re on d ifferen t sides of th is s h a rp d esc en t o f th e cross sectio n . F or K L O E 1 0 , th e o n ly tw o affected b in s a re th o se d ire c tly before a n d d ire c tly a fte r th e sh a rp d esc en t in th e cross sectio n, w h ere th e u n c e rta in tie s are fully a n ti-c o rre la te d b etw e en th e s e tw o bins. T h e second e x c e p tio n is th e w eig h ted b ac k g ro u n d s u b tra c tio n for K L O E 1 2 , w h ere in th e e x p e rim e n ta l an aly sis th e w eig hts of th e fitte d e + e - Y, nn Y a n d n n n b ac k g ro u n d s to th e ^ + ^ - y (y ) s p e c tru m a re d is trib u te d over n e ig h b o u rin g tw o -b in in terv a ls fro m 0.32 to 0.96 G eV 2. F o r th e K L O E 1 2 sy s te m a tic co varian ce m a trix , th is re su lts in o n ly n eig h ­ b o u rin g bins from 0.36 to 0.94 G e V2 b ein g c o rre la te d w ith each o th e r for th is b a c k g ro u n d s u b tra c tio n u n c e rta in ty , w h ere th e first a n d la st bin re m a in e n tire ly u n c o rre la te d in th is case. A co m p re h en siv e discu ssio n co n c ern in g th is a n d all o th e r s y s te m a tic u n c e rta in tie s fo r each m e a su re m e n t c a n b e fo u n d in [2 , 4 , 6].

Im p o rta n tly , for th e K L O E 1 2 s y s te m a tic co varian ce m a trix th e trig g e r, L3 (softw are trig g e r), tra c k m a ss , tra c k in g efficiency, a c c e p ta n c e a n d b a c k g ro u n d s u b tra c tio n c o rrec tio n s a re a p p lie d to b o th th e n + n - Y a n d ^ + ^ - y d a t a t h a t e n te r in to th e ra tio in e q u a tio n (2.3) an d , th e re fo re , th e co rre sp o n d in g u n c e rta in tie s from a given so urce b etw e en th e n + n - Y an d

^ + ^ - Y d a t a a re c o r re la te d.5 F orm ally, th e ra tio of th e s e c o rre c tio n u n c e rta in tie s re su lts in a re d u c tio n of th e to ta l u n c e rta in ty of a^ n , w h ere th e c o n trib u tio n s o f th e p o sitiv e c o rre la tio n s b etw e en th e K L O E 0 8 a n d K L O E 1 2 u n c e rta in tie s c o n trib u te n eg a tiv e ly to th e ov erall u n c e rta in ty d u e to th e p a r tia l d e riv a tiv e o f th e ^ + ^ - y d a t a in th e d e n o m in a to r of th e ra tio . H ow ever, th e u n c e rta in tie s d u e to a given so u rce a re defined in te rm s of th e ra tio o f n + n - Y over ^ + ^ - y, such t h a t th e c o n trib u tio n s from b o th d a t a sources are a lre a d y fully in c o rp o ra te d . T h erefo re, we d o n o t s e p a ra te ly a d d th e u n c e rta in tie s of th e s e c o rrec tio n s for th e n + n - Y d a t a to th e K L O E 1 2 sy s te m a tic co varian ce m a trix .

In a d d itio n , th e K L O E 1 2 s y s te m a tic u n c e rta in ty v e c to r for th e n o n -w eig h ted b ac k ­ g ro u n d s u b tra c tio n w as c o n s tru c te d in [5 , 6] such t h a t it c o n ta in e d th e ra tio of th e co n ­ tr ib u tio n s from th e c o rre c tio n s of th e ee ^ e e n n a n d ee ^ e e ^ b a c k g ro u n d processes, alo n g w ith a tra c k m a s s (M trk) ta il co rre c tio n , su m m ed in q u a d r a tu re . F o r th is an a ly sis, in o rd e r to c o rre c tly c o rre la te th e se in d e p e n d e n t sources o f sy s te m a tic u n c e rta in ty ac co rd in g

5This only refers to the correlation of uncertainties from a specific source between the n + n - Y analysis and the y,+ analysis that enter into the KLOE12 ratio. The correlation between the KLOE08 n +n - Y data and the KLOE12 cross section ratio are described in detail in the discussion of the KLOE0812 block of the systematic covariance matrix.

3 .2 S y s t e m a t ic c o r r e la tio n s

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

to e q u a tio n (A .5) , th e s e c o n trib u tio n s a re s e p a ra te d a n d c o rre la te d ind iv id u ally . T h is has c o n trib u te d to th e re d u c tio n of th e K L O E 1 2 e rro r e s tim a te in e q u a tio n ( 2.7) , w h ere p re ­ v iously th e c o rre la tio n of th e co m b in ed v e c to r re su lte d in a n in c o rre c t o v e re stim a te of th e sy s te m a tic u n c e rta in ty .

F o r K L O E 0 8 a n d K L O E 1 0 , th e c o n trib u tio n s to th e s y ste m a tic u n c e rta in ty from th e tra c k m a ss, tra c k in g efficiency, L3 (softw are trig g e r) efficiency, ac c e p ta n c e , lum in osity , r a ­ d ia to r fu n c tio n , v a c u u m p o la ris a tio n c o rre c tio n a n d final s ta te ra d ia tio n c o rre c tio n are co n sid ered to be fully c o rre la te d in th e K L O E 0 8 1 0 (K L O E 100 8) co varian ce m a trix blocks.

F o r th e c o rre la tio n o f th e sy ste m a tic u n c e rta in ty d u e to th e a c c e p ta n c e , on ly h a lf of th e K L O E 1 0 u n c e rta in ty is c o rre la te d w ith th e K L O E 0 8 u n c e rta in ty in o rd e r to en su re t h a t th e p h o to n d e te c tio n a c c e p ta n c e t h a t e n te rs in to th e K L O E 1 0 u n c e rta in ty ( th a t is n o t p re se n t in th e K L O E 0 8 an aly ses) is n o t c o rre la te d a n d o nly th e c o rre la tio n of th e pion tra c k s is d u ly ac c o u n te d for. Im p o rta n tly , a lth o u g h th e K L O E 0 8 a n d K L O E 1 0 m e a s u re m e n ts o nly o v erla p for th e 50 d a ta p o in ts in th e en e rg y ra n g e 0.35 to 0.85 G e V 2, all en e rg y b in s in th e 60 x 75 K L O E 0 8 1 0 (75 x 60 K L O E 10 08 ) c o rre la tio n block m u st b e fully c o rre la te d . N o te t h a t a p p ly in g 100% c o rre la tio n to only th e o v e rla p p in g 50 x 50 regio n w ould re su lt in th e s y s te m a tic m a tr ix h av in g n eg a tiv e eigenvalues.

As w ith th e s ta tis tic a l u n c e rta in tie s for K L O E 0 81 2 (K L O E 1 2 0 8 ), th e s y s te m a tic u n ­ c e rta in tie s in h e re n t in th e n + n - Y(y) d a t a sh a re d b etw e en th e tw o an aly ses are c o rre ­ la te d b etw e en th e K L O E 0 8 a n d K L O E 1 2 m e a su re m e n ts. T h ese in clu d e th e u n c e rta in tie s from th e L3 efficiency, th e b a c k g ro u n d s u b tra c tio n , th e tra c k m a s s (M trk), th e u n fo ld ­ ing, th e tra c k in g efficiency, th e trig g e r efficiency a n d th e a c c e p ta n c e from th e K L O E 0 8 an aly sis. T h e d e te rm in e d u n c e rta in tie s for th e L3, M trk , tra c k in g , trig g e r a n d accep ­ ta n c e c o rre c tio n s for K L O E 1 2 are fully c o rre la te d for K L O E 0 8 1 2 such t h a t th e a n ti­

c o rre la tio n t h a t o ccu rs d u e to th e ra tio in K L O E 1 2 is p ro p a g a te d accordingly. T h is is also tr u e for th e n o n -w eig h ted b a c k g ro u n d s u b tra c tio n c o n trib u tio n , e n su rin g t h a t only th e c o rre c tio n s for th e ee ^ e e n n b ac k g ro u n d fro m th e K L O E 0 8 an a ly sis a re c o rre­

la te d w ith th e ra tio o f th e c o rre c tio n s of th e ee ^ e e n n a n d ee ^ e e ^ b a c k g ro u n d p rocesses as th e y e n te r in th e K L O E 12 an aly sis. T h e u n fo ld in g u n c e rta in tie s for th e K L O E 0812 c o rre la tio n block are, in p a r t, a n ti-c o rre la te d as th e y are for K L O E 0 8 an d K L O E 1 2 in dividually. All o th e r sy s te m a tic u n c e rta in tie s are 100% c o rre la te d b etw een K L O E 0 8 a n d K L O E 1 2 .

W ith th e sam e n + n - Y(y) d a t a b ein g sh a re d b etw e en th e K L O E 0 8 a n d K L O E 1 2 m e a ­ s u re m e n ts, th e K L O E 1012 (K L O E 1 21 0) c o rre la tio n blocks follow a sim ilar s tru c tu r e to th e K L O E 0 8 1 0 (K L O E 1008) c o rre la tio n blocks. T h e ca v eats to th is s ta te m e n t a re t h a t th e re a re no c o rre la te d u n c e rta in tie s here d u e to th e lu m in osity, r a d ia to r fu n c tio n or vacu u m p o la ris a tio n co rre c tio n , as th e s e effects can cel in th e ra tio of th e n + n - Y(Y) d a t a over th e

^ + ^ -y(y) d a t a for th e K L O E 1 2 m e a su re m e n t (see sec tio n 2.1) . T h erefo re, th e c o rre la te d s y s te m a tic u n c e rta in tie s for K L O E 10 12 a re th e tra c k m a ss, tra c k in g efficiency, L3 efficiency, a c c e p ta n c e a n d final s ta te ra d ia tio n c o rre c tio n u n c e rta in tie s , w h ere it is a g a in n ec essary to c o rre la te on ly h a lf o f th e K L O E 1 0 a c c e p ta n c e u n c e rta in ty w ith K L O E 1 2 in o rd e r to en su re t h a t o nly th e effect d u e to th e a c c e p ta n c e o f th e p io n tra c k s is in c o rp o ra te d .

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

4 C o m b i n a t i o n a n d r e s u l t s

4 .1 T h e c o m b in e d K L O E e + e ^ n + n 7(7) c r o ss s e c t io n

F ollow ing th e m e th o d o lo g y o f sec tio n 3 yields full K L O E n + n - y (y ) s ta tis tic a l a n d sys­

te m a tic co v arian ce m a tric e s t h a t d esc rib e th e c o rre la tio n s t h a t ex ist b etw e en K L O E 0 8 , K L O E 1 0 a n d K L O E 1 2 . T h ese d a t a are co m b in ed in c o rp o ra tin g th e en e rg y d e p e n d e n t s ta tis tic a l a n d sy s te m a tic u n c e rta in tie s a n d co rre sp o n d in g c o rre la tio n s, u sin g a n ite ra tiv e m in im isa tio n of th e follow ing lin e a r %2 fu n c tio n [12]

195 195

X2 = n(7) (i) - j (n) ) ( ^ n(7) (j ) - ^ n ( Y ) (n )) . (4 .1) i= 1j=1

H ere, 0"On (Y)(i) is th e cross sec tio n v alue o f th e d a t a p o in t i c o n trib u tin g to th e com ­ b in ed cross sec tio n value <r0n (Y)(m ) a n d th e c o m b in a tio n cross sec tio n v e c to r w ith th e ele­

m e n ts lab elled by m c o n ta in s 85 d a t a p o in ts over th e en e rg y ra n g e 0.1 < s ' < 0.95 G eV 2, w ith th e 85 bins co rre sp o n d in g to th e 85 d is tin c t en e rg y bins o f th e th re e m e a su re m e n ts.

C - 1( i (m), j (n)) is sim p ly th e inverse of th e co varian ce m a tr ix C ( i (m), j (n)) , w hich is defind as th e sum of th e s ta tis tic a l covarian ce m a trix C sta t( i (m), j (n)) a n d th e s y s te m a tic covari­

an ce m a trix C sys( i (m), j (n)) . A t each ite ra tiv e sta g e of th e m in im isa tio n , it is defined as csys(i(m ) j,n))

C( i (m), j (n)) = C stat ( i(m), j (n)) + ( , j j s;n(Y)(m.}<r;n(Y)(n.), (4.2) nn(Y)( ) nn(Y) ( j)

w h ere th e q u a n titie s <r0n (Y)(m ) a n d <r0n (Y)(n ) are th e re s u ltin g com b in ed cross sec tio n values from th e p re v io u s ite ra tio n . T h is m e th o d h as b ee n a d a p te d fro m [45] (see also [18]), has b ee n a d v o c a te d to b e free of sy s te m a tic b ias a n d e x h ib its a sw ift convergence, a fte r only a few ite ra tio n s . W e also o b ta in an o u tp u t co v arian ce m a trix for th e co m b in ed s ta tis tic a l a n d sy ste m a tic u n c e rta in tie s t h a t d esc rib es th e c o rre la tio n s b etw e en th e d a t a p o in ts of th e re su ltin g cross sec tio n vector.

T h e K L O E c o m b in a tio n cross sec tio n a n d p io n form fa c to r d a t a are liste d in ta b le 1.

T h e in p u t cross sec tio n v ec to rs a n d c o m b in a tio n cov arian ce m atrice s, alon g w ith th e co m ­ b in ed o u tp u t cross sec tio n v e c to r a n d t o ta l co varian ce m a tr ix a re av ailable from [41].6 F or th e c o n trib u tio n to th e an o m alo u s m a g n e tic m o m e n t of th e m u o n in th e full en e rg y ran ge, th e K L O E c o m b in a tio n re su lts in

(0.10 < s ' < 0.95 G eV2) = (489.8 ± 1.7stat ± 4.8sys) x 10- 10. (4.3) T h e re su ltin g cross sec tio n a n d th e in d iv id u a l m e a su re m e n ts a re show n in figure 4 . In a d d i­

tio n , th e n o rm alised differences o f th e in d iv id u a l K L O E m e a su re m e n ts a n d th e co m b in a tio n a re show n in figure 5 . W e observe goo d a g reem en t b etw e en th e d a t a a n d th e co m b in atio n , esp e cially w ith K L O E 0 8 w hich d o m in a te s th e fit d u e to its sm aller s ta tis tic a l u n c e rta in ty

6The total output matrix given contains the contributions from both the statistical and systematic uncertainties, where the choice to use both as input into the data combination results in a solution that entangles the statistical and systematic sources of uncertainty.

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

KLOE combination

s' (GeV2) ^?n(7)(nb) IF (n)|2 s' (GeV2) n(7)(nb) IF (n)|2

0.105 47.27 ± 8.41 1.74 ± 0.31 0.535 1154.56 ± 6.81 35.96 ± 0.21 0.115 70.65 ± 10.44 2.04 ± 0.30 0.545 1207.69 ± 6.83 38.20 ± 0.22 0.125 80.13 ± 10.97 2.00 ± 0.27 0.555 1243.32 ± 10.13 39.94 ± 0.33 0.135 80.42 ± 11.27 1.82 ± 0.26 0.565 1285.35 ± 7.14 41.92 ± 0.23 0.145 87.58 ± 11.70 1.86 ± 0.25 0.575 1277.36 ± 7.32 42.29 ± 0.24 0.155 102.88 ± 12.35 2.10 ± 0.25 0.585 1279.89 ± 7.31 42.98 ± 0.25 0.165 115.16 ± 13.85 2.29 ± 0.28 0.595 1274.03 ± 10.32 43.27 ± 0.35 0.175 122.58 ± 13.42 2.40 ± 0.26 0.605 1228.97 ± 12.29 42.18 ± 0.42 0.185 126.19 ± 12.61 2.45 ± 0.24 0.615 950.47 ± 20.95 34.85 ± 0.77 0.195 146.34 ± 14.10 2.84 ± 0.27 0.625 803.87 ± 4.65 29.94 ± 0.17 0.205 144.18 ± 13.35 2.80 ± 0.26 0.635 781.82 ± 4.39 29.24 ± 0.16 0.215 147.47 ± 12.68 2.88 ± 0.25 0.645 731.86 ± 5.74 27.61 ± 0.22 0.225 154.64 ± 11.98 3.04 ± 0.24 0.655 679.26 ± 3.93 25.90 ± 0.15 0.235 170.47 ± 12.40 3.39 ± 0.25 0.665 620.73 ± 3.46 23.93 ± 0.13 0.245 168.96 ± 11.53 3.40 ± 0.23 0.675 569.26 ± 4.63 22.20 ± 0.18 0.255 176.55 ± 10.84 3.60 ± 0.22 0.685 518.39 ± 5.62 20.45 ± 0.22 0.265 202.38 ± 11.63 4.18 ± 0.24 0.695 471.79 ± 2.69 18.82 ± 0.11 0.275 203.28 ± 10.70 4.26 ± 0.22 0.705 431.19 ± 2.44 17.39 ± 0.10 0.285 215.28 ± 10.60 4.58 ± 0.23 0.715 386.51 ± 3.21 15.76 ± 0.13 0.295 225.63 ± 10.46 4.87 ± 0.23 0.725 356.81 ± 2.03 14.70 ± 0.08 0.305 236.90 ± 10.49 5.19 ± 0.23 0.735 327.36 ± 1.91 13.63 ± 0.08 0.315 244.65 ± 10.11 5.45 ± 0.23 0.745 299.08 ± 1.96 12.59 ± 0.08 0.325 248.45 ± 9.83 5.62 ± 0.22 0.755 273.28 ± 1.80 11.62 ± 0.08 0.335 255.64 ± 9.62 5.88 ± 0.22 0.765 249.34 ± 1.45 10.71 ± 0.06 0.345 280.05 ± 9.46 6.54 ± 0.22 0.775 228.91 ± 1.94 9.93 ± 0.08 0.355 305.24 ± 4.55 7.24 ± 0.11 0.785 211.31 ± 1.27 9.26 ± 0.06 0.365 330.21 ± 7.67 7.96 ± 0.18 0.795 196.17 ± 1.36 8.68 ± 0.06 0.375 349.58 ± 4.60 8.56 ± 0.11 0.805 183.29 ± 1.08 8.19 ± 0.05 0.385 376.70 ± 4.63 9.37 ± 0.12 0.815 170.45 ± 1.00 7.69 ± 0.05 0.395 400.82 ± 4.57 10.12 ± 0.12 0.825 157.72 ± 1.09 7.19 ± 0.05 0.405 433.99 ± 6.28 11.13 ± 0.16 0.835 146.52 ± 0.95 6.74 ± 0.04 0.415 465.70 ± 4.79 12.13 ± 0.12 0.845 136.86 ± 0.79 6.36 ± 0.04 0.425 506.53 ± 4.87 13.39 ± 0.13 0.855 126.97 ± 0.78 5.95 ± 0.04 0.435 544.42 ± 4.84 14.61 ± 0.13 0.865 119.05 ± 0.89 5.63 ± 0.04 0.445 585.65 ± 5.04 15.95 ± 0.14 0.875 111.33 ± 0.83 5.31 ± 0.04 0.455 640.09 ± 7.95 17.69 ± 0.22 0.885 104.92 ± 1.81 5.05 ± 0.09 0.465 691.86 ± 7.66 19.41 ± 0.21 0.895 98.60 ± 0.59 4.79 ± 0.03 0.475 740.82 ± 8.20 21.09 ± 0.23 0.905 93.05 ± 0.56 4.56 ± 0.03 0.485 822.23 ± 5.82 23.75 ± 0.17 0.915 87.66 ± 0.74 4.33 ± 0.04 0.495 895.61 ± 17.85 26.26 ± 0.52 0.925 82.76 ± 0.49 4.13 ± 0.02 0.505 953.15 ± 13.08 28.36 ± 0.39 0.935 78.84 ± 0.65 3.96 ± 0.03 0.515 1032.72 ± 6.28 31.20 ± 0.19 0.945 74.74 ± 0.64 3.79 ± 0.03

0.525 1078.01 ± 8.23 33.06 ± 0.25 - - -

T a b le 1. T h e com bined K L O E m easu rem ent o f th e n + n - Y(Y) bare cross sectio n and p ion form factor in 0.01 G eV2 intervals from 0.10 < s' < 0 .9 5 G e V 2. Here, s' d en o tes th e b in centre. For b o th <7^n(Y) and |F ( n ) |2, th e error show n is th e to ta l (sta tistic a l and sy stem a tic) uncertainty. T he errors have b een in flated according to th e local xmnin/d.o.f. in each en ergy bin, w here in flation is o n ly applied if x ^ m / d ^ .f . > 1.

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

Figure 4. The KLOE combination plotted with the individual cross section m easurem ents, where the KLOE combination is represented by the yellow band and the KLOE08, KLOE10 and KLOE12 cross section m easurem ents are given by the blue, black and pink markers, respectively (colour online). In all cases, the error bars shown are the statistical and system atic uncertainties summed in quadrature.

(a) N o r m a lis e d d iffe re n c e in t h e fu ll d a t a r a n g e (b ) N o r m a lis e d d iffe re n c e in t h e o v e r la p p in g d a t a ra n g e

Figure 5. The normalised difference of the KLOE com bination and the individual KLOE mea­

surements, where the yellow band represents the statistical and system atic uncertainties of the KLOE combination summed in quadrature and the KLOE08, KLOE10 and KLOE12 cross section m easurem ents are given by the blue, black and pink markers, respectively (colour online). Here, the errors bars of the individual m easurem ents are not shown in order to be able to distinguish the d a ta points, bu t are in good agreement w ith the KLOE combination.

when comparing to KLOE10 and KLOE12. KLOE12 exhibits the largest fluctuations when comparing to the fitted combination, but is well within the errors of the data. In plot (a) of figure 5, we note how the KLOE0810 and KLOE1012 systematic uncertainties have a non-trivial effect in the lower energy region where only the KLOE10 d ata exist, with the correlations providing an expected upward pull (which is well within the errors of the combination) to the KLOE combination cross section away from the KLOE10 d ata points.

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

(a) Cross section in the full data range (b ) Cross section in the overlapping data range

K L O E n + n 7(7) d a t a set a£ + n (0.35 < s ' < 0.85 G e V2) K L O E 0 8

K L O E 1 0 K L O E 12

3 7 8 .9 ± ° . 4stat ± 3 .2sys 376.0 ± 0.9stat ± 3.3sys 377.4 ± 1.2stat ± 2.3sys K L O E c o m b in a tio n 377.5 ± 0.5stat ± 2.1sys

T a b le 2 . C om p arative resu lts o f th e values o b tain ed for a^ n (0.35 < s' < 0.85 G eV ) from th e in d ivid u al K L O E m easu rem ents and th e full com b ination . R esu lts for a^j+n are given in u nits o f 10- 1 0 .

F ig u r e 6. C om parison o f estim a tes o f a^+n from th e K L O E com b in ation and th e in d ivid u al K L O E m easu rem ents in th e range 0.35 < s' < 0 .8 5 G eV 2. T h e K L O E com b in ation is represented b y th e yellow b and (colour o n lin e). In all cases, th e u n certain ties show n are th e sta tistic a l and sy stem a tic u n certain ties su m m ed in quadrature.

F o r th e o v erla p p in g en e rg y reg ion o f all th re e m e a su re m e n ts, th e e s tim a te s for a ^ +n from th e K L O E c o m b in a tio n a n d th e in d iv id u a l m e a s u re m e n ts are g iven in ta b le 2 an d figure 6. In all cases, th e e rro rs in clu d e all c o rre la tio n c o n trib u tio n s. F o r th e c o m b in a ­ tio n , th e y h ave b ee n in flated a c co rd in g to a local %min/ d . o . f . in each en e rg y b in if th e X m in /d .o .f. > 1 [11, 4 6 , 47], as show n in figure 7. T h is h as re su lte d in a n in crease to th e overall u n c e rta in ty of th e e s tim a te of a^ n of ~ 13%. T h e c o m b in a tio n agrees well w ith th e e s tim a te s from th e in d iv id u a l m e a su re m e n ts, w ith a m ark e d im p ro v em en t in th e overall u n c e rta in ty . W h ile th e s ta tis tic a l u n c e rta in ty o f a^ + n from th e c o m b in a tio n is d o m in a te d by K L O E 0 8 (w hich h as th e sm allest s ta tis tic a l u n c e rta in ty o f th e th re e in d iv id u a l m e a su re ­ m e n ts), th e c o m b in a tio n m e a n value of a^ n is closest to t h a t o b ta in e d w ith th e K L O E 1 2 d a t a alone, w hich h as th e sm allest s y s te m a tic an d , th e re fo re , th e sm allest t o ta l e rro r of th e th re e . T h is in tu r n leads to th e im prov ed s y ste m a tic e rro r of th e co m b in ed re s u lt a n d its m a rk e d ly im p ro v ed to ta l erro r.

4 .2 C o m p a r iso n w it h r e s u lts fro m t h e C M D -2 , S N D , B a B a r a n d B E S I I I e x p e r im e n ts

T h e a ( e + e - ^ n + n - ) cross sec tio n h as b ee n m e a su re d below 1 G eV by th e C M D -2 [48- 50], SN D [51], B a B a r [52] a n d B E S III [53] c o lla b o ra tio n s. T h e B a B a r a n d B E S III m e a su re ­

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

F ig u r e 7 . T h e effect o f th e local x ^ m / d ^ . f . error in flation on th e K L O E com b ination , w hich is ap plied in each en ergy bin if th e local x ^ m / d ^ .f . > 1. T h e to ta l effect on th e K L O E com b in ation is represented b y th e yellow b locks (colour on lin e). T h e relative con trib u tion s to each local x ^ m / d ^ . f . from th e K L O E 08, K L O E 10 and K L O E 12 m easu rem ents in d ivid u ally are given b y th e blue, black and pink m arkers, respectively.

(a) C ro s s s e c tio n in t h e r a n g e 0.6 < ' / s ' < 0.9 G e V (b) C ro s s s e c tio n in t h e p — u in te r f e r e n c e re g io n

Figure 8. The n + n cross section from th e KLOE combination, CMD-2 [48- 50], SND [51], B aB ar [52] and B ESIII [53] d a ta points. The KLOE com bination is represented by the yellow band (colour online). W here uncertainties are displayed, th ey represent the statistical and system atic uncertainties summed in quadrature. The uncertainties of the separate experim ental m easurem ents in figure (b) have been suppressed in order to improve readability.

ments, like the KLOE measurements, are obtained through radiative return. The CMD-2 and SND measurements are taken by energy scan, allowing us to compare the two m eth­

ods. All the experimental measurements are undressed of VP effects and include FSR, such th a t there is a consistent comparison of 0^ (7). The cross section measurements from each experiment and the KLOE combination are shown in figure 8.

The normalised difference of the d ata from these experiments w ith respect to the KLOE combination are shown in figure 9. In particular, we note th a t the KLOE combination is lower th an all other d ata at the p peak where the cross section is largest, but higher th an

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

Figure 9. The n + n - cross section from the KLOE com bination com pared to the CMD-2, SND, B aB ar and BESIII d ata points in the range 0.6 < a/s' < 0.9 GeV. The KLOE com bination is rep­

resented by the yellow band (colour online). In all cases, the uncertainties shown are th e statistical and system atic uncertainties summed in quadrature.

the other experimental d ata where the cross section drops off in the p — w interference region. This effect is evident in figure 9, where we note th a t for all cases (except for (c) BESIII, where the effect is less prominent), there is a sharp rise and fall of the difference in the experimental cross section at the p — w interference region due to KLOE having fewer bins in this region compared to the other experiments (see plot (b) of figure 8) .

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

(a) KLOE combination vs. other experiments

(b ) KLOE combination vs. BaBar (c) KLOE combination vs. BESIII

(d ) KLOE combination vs. CMD-2 (e) KLOE combination vs. SND

Figure 10. Estim ates of a^+n from the KLOE combination, CMD-2, SND, B aB ar and BESIII in the range 0.6 < a/s' < 0.9 GeV. The available CMD-2 d a ta have been combined following the prescription of [12]. The KLOE combination is represented by the yellow band (colour online).

In all cases, the uncertainties shown are the statistical and system atic uncertainties summed in quadrature.

n + n y(y) d ata set a£+n- (0.6 < V S < 0.9 GeV) CMD-2 fit (03,06) 372.4 ± 3.0

SND (04) 371.7 ± 5.0

B aB ar (09) 376.7 ± 2.7

BESIII (15) 368.2 ± 4.2

KLOE combination 366.9 ± 2.1

Table 3. Com parative results of the values obtained for a[[+n (0.6 < %/S < 0.9 GeV) from the KLOE combination and the CMD-2, SND, B aB ar and BESIII data. The available CMD-2 d ata have been combined following the prescription of [12]. Results for a[[+n are given in units of 10-10. In all cases, the uncertainties shown are the statistical and system atic uncertainties summed in quadrature.

The BaBar d ata are, in majority, higher th an the KLOE combination, whereas we observe th a t the other d ata sit mainly lower th an KLOE below the p peak and higher above it. We also note th a t our comparison of the KLOE combination with the BESIII d ata looks markedly different from th a t presented in [53], especially at higher energies.

However, in [53], the comparison has been made using a fit of the d ata to the Gounaris- Sakurai param etrisation [54], which does not provide an adequate description of the BESIII measurements of the n + n - cross section in the tail of the resonance. We therefore opt to compare, in plot (c) of figure 9, the published BESIII d ata points directly with our combination of the KLOE data.

Estim ates of the contribution to the anomalous magnetic moment of the muon from these experiments in the range 0.6 < V S < 0.9 GeV are shown in figure 10 and table 3,

J H E P 0 3 ( 2 0 1 8 ) 1 7 3

w h ere we have com b in ed th e availab le C M D -2 d a t a in to a single e s tim a te by ap p ly in g th e sam e m e th o d used to fit th e K L O E co m b in a tio n . W e o bserv e g o o d a g re e m e n t (w ith in 1 .5 a) b etw e en th e K L O E c o m b in a tio n a n d th e m e a s u re m e n ts by C M D -2, SND a n d B E S III. T h e m e a s u re m e n t by B a B a r, as ev id e n t from p lo t (b) of figure 9 , re su lts in a h ig h er e s tim a te o f a „ + n - ._{f l}

5 C o n c l u s i o n s

T h e K L O E c o lla b o ra tio n have p erfo rm ed th re e m e a su re m e n ts of th e a0 (e + e - ^ n + n - Y (y)) cross sec tio n below 1 G eV2 u sin g th e m e th o d of ra d ia tiv e re tu r n . T h e se m e a su re m e n ts are, in p a r t, hig h ly c o rre la te d . T h is is esp ecially tr u e for K L O E 0 8 a n d K L O E 1 2 w here, for th e K L O E 1 2 m e a su re m e n t, th e K L O E 0 8 n + n - Y(y) d a t a is n o rm alised by th e m easu red

^ + ^ - y ( y ) cross section. T h is h as n e c e s sita te d th e c o n s tru c tio n of s ta tis tic a l a n d sy s te m a tic c o m b in a tio n co v arian ce m a tric e s, w hich h ave b een ca refu lly b u ilt to satisfy th e re q u ired p ro p e rtie s of a covarian ce m a trix .

U sin g th e se co varian ce m a tric e s, th e th re e m e a su re m e n ts have b een co m b in ed to p ro ­ d u ce single v ec to rs for b o th th e tw o -p io n cross sec tio n a nn(Y) a n d th e p io n form fa c to r |F n |2, a lo n g w ith a co rre sp o n d in g co v arian ce m a trix for each. T h is c o m b in a tio n o f th e K L O E cross sec tio n d a t a re su lts in a n e s tim a te of th e tw o -p io n c o n trib u tio n to th e an o m alo u s m a g n e tic m o m en t o f th e m u o n of

a / + n (K L O E co m b in a tio n , 0.10 < s ' < 0.95 G eV2) = (489.8 ± 5.1) x 10-10, (5.1) w hich is co n siste n t w ith th e in d iv id u a l K L O E m e a s u re m e n ts a n d w ith in 1 .5 a o f th e C M D -2, SN D a n d B E S III m e a su re m e n ts, w hile th e difference w ith th e B a B a r d a t a is below 3 a .

A c k n o w l e d g m e n t s

W e w ould like to th a n k F ed o r Ig n a to v for n u m ero u s useful d iscu ssio n s a n d D a isu k e N o m u ra for his c o lla b o ra tio n in p ro d u c in g th e c o m p ila tio n a n d d e te r m in a tio n of th e e s tim a te s o f a^.

W e give sp ecial th a n k s to M au ric e B en a y o u n for his stu d ie s a n d d iscussions re g a rd in g th e d e te r m in a tio n of th e p io n form fa c to r. W e w ould also like to acknow ledge th e d iscu ssio ns w ith in th e W orking Group on R adiative Corrections and M C Generators fo r Low Energies (Radio M o nteC arL O W ) [h ttp :/ /w w w .ln f .in f n .it/w g /s ig h a d /] a n d The M uon (g — 2)^ T he­

ory In itiative co n c ern in g th is w ork. T h e w ork o f A lex K e sh a v arz i a n d T h o m a s T e u b n e r is su p p o rte d by S T F C u n d e r th e c o n s o lid a te d g ra n ts S T /N 5 0 4 1 3 0 /1 a n d S T /L 0 0 0 4 3 1 /1 , respectively.

T h e K L O E -2 c o lla b o ra tio n w ould like to w a rm ly th a n k fo rm e r K L O E colleagues for th e access to th e d a t a co llected d u rin g th e K L O E d a t a ta k in g c a m p a ig n . W e t h a n k th e D A $ N E te a m for th e ir efforts in m a in ta in in g low b a c k g ro u n d ru n n in g c o n d itio n s a n d th e ir co llab o ­ ra tio n d u rin g all d a t a ta k in g . W e w a n t to t h a n k o u r te c h n ic a l staff: G .F . F o rtu g n o a n d F . S b o rzacch i for th e ir d e d ic a tio n in e n su rin g efficient o p e ra tio n of th e K L O E c o m p u tin g facil­

ities; M . A nelli for his co n tin u o u s a tte n tio n to th e gas sy stem a n d d e te c to r safety; A. B alla, M . G a tta , G. C o rra d i a n d G. P a p a lin o fo r elec tro n ics m a in te n a n c e ; C. P isc ite lli for his help

J H E P 0 3 ( 2 0 1 8 ) 1 7 3